Properties

Label 1875.4.a.f
Level $1875$
Weight $4$
Character orbit 1875.a
Self dual yes
Analytic conductor $110.629$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1875 = 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1875.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(110.628581261\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \( x^{14} - 81 x^{12} - 7 x^{11} + 2512 x^{10} + 517 x^{9} - 36970 x^{8} - 12987 x^{7} + 257291 x^{6} + 125779 x^{5} - 718713 x^{4} - 371750 x^{3} + 579848 x^{2} + \cdots + 42064 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 5^{3} \)
Twist minimal: no (minimal twist has level 75)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + 4) q^{4} + 3 \beta_1 q^{6} + ( - \beta_{8} - \beta_{5} - \beta_{3} + 2 \beta_1 + 1) q^{7} + ( - \beta_{6} - 3 \beta_{3} - 3 \beta_1 - 3) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + 4) q^{4} + 3 \beta_1 q^{6} + ( - \beta_{8} - \beta_{5} - \beta_{3} + 2 \beta_1 + 1) q^{7} + ( - \beta_{6} - 3 \beta_{3} - 3 \beta_1 - 3) q^{8} + 9 q^{9} + ( - \beta_{13} - 2 \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - 4) q^{11} + ( - 3 \beta_{2} - 12) q^{12} + ( - \beta_{13} + 2 \beta_{12} - 2 \beta_{7} - \beta_{6} + \beta_{4} + 5 \beta_{3} - \beta_1 + 15) q^{13} + ( - \beta_{13} + 3 \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} - 2 \beta_{7} - 3 \beta_{5} + \cdots - 26) q^{14}+ \cdots + ( - 9 \beta_{13} - 18 \beta_{9} - 9 \beta_{8} + 9 \beta_{7} - 9 \beta_{6} + \cdots - 36) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 42 q^{3} + 50 q^{4} + 27 q^{7} - 21 q^{8} + 126 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 42 q^{3} + 50 q^{4} + 27 q^{7} - 21 q^{8} + 126 q^{9} - 33 q^{11} - 150 q^{12} + 188 q^{13} - 287 q^{14} + 82 q^{16} + 146 q^{17} - 184 q^{19} - 81 q^{21} + 277 q^{22} + 164 q^{23} + 63 q^{24} + 103 q^{26} - 378 q^{27} - 224 q^{28} + 252 q^{29} - 889 q^{31} + 422 q^{32} + 99 q^{33} - 230 q^{34} + 450 q^{36} + 642 q^{37} + 1833 q^{38} - 564 q^{39} - 164 q^{41} + 861 q^{42} + 696 q^{43} - 6 q^{44} - 419 q^{46} - 92 q^{47} - 246 q^{48} + 81 q^{49} - 438 q^{51} + 1956 q^{52} + 949 q^{53} - 2380 q^{56} + 552 q^{57} + 1041 q^{58} - 81 q^{59} - 496 q^{61} + 1454 q^{62} + 243 q^{63} - 3903 q^{64} - 831 q^{66} + 1926 q^{67} + 685 q^{68} - 492 q^{69} + 2498 q^{71} - 189 q^{72} - 1026 q^{73} - 707 q^{74} - 5704 q^{76} + 5434 q^{77} - 309 q^{78} - 695 q^{79} + 1134 q^{81} - 886 q^{82} + 5315 q^{83} + 672 q^{84} + 3997 q^{86} - 756 q^{87} + 2969 q^{88} - 1424 q^{89} - 1194 q^{91} + 1607 q^{92} + 2667 q^{93} - 629 q^{94} - 1266 q^{96} + 291 q^{97} - 1018 q^{98} - 297 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 81 x^{12} - 7 x^{11} + 2512 x^{10} + 517 x^{9} - 36970 x^{8} - 12987 x^{7} + 257291 x^{6} + 125779 x^{5} - 718713 x^{4} - 371750 x^{3} + 579848 x^{2} + \cdots + 42064 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 154777191 \nu^{13} + 214678293 \nu^{12} + 12768694032 \nu^{11} - 14915595999 \nu^{10} - 404069408515 \nu^{9} + \cdots - 29392108806288 ) / 9573422384000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 39842017881 \nu^{13} + 18793858963 \nu^{12} + 3381917381312 \nu^{11} + 617153824991 \nu^{10} - 107039813741365 \nu^{9} + \cdots - 61\!\cdots\!08 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 214678293 \nu^{13} - 231741561 \nu^{12} + 15999036336 \nu^{11} + 15269104723 \nu^{10} - 450292207345 \nu^{9} + \cdots - 6510547762224 ) / 9573422384000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 464331573 \nu^{13} - 644034879 \nu^{12} - 38306082096 \nu^{11} + 44746787997 \nu^{10} + 1212208225545 \nu^{9} + \cdots + 59456059266864 ) / 9573422384000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 52774069869 \nu^{13} - 146787187287 \nu^{12} - 4038749648088 \nu^{11} + 10590731904141 \nu^{10} + 114153083031385 \nu^{9} + \cdots - 24\!\cdots\!08 ) / 679712989264000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 198268707541 \nu^{13} - 330612632143 \nu^{12} - 15446080755832 \nu^{11} + 24130842178949 \nu^{10} + 453486628821265 \nu^{9} + \cdots - 30\!\cdots\!12 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 24723383429 \nu^{13} - 19698303673 \nu^{12} + 1994905817228 \nu^{11} + 1654834981079 \nu^{10} - 61648333032485 \nu^{9} + \cdots - 91\!\cdots\!52 ) / 203913896779200 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 8146109293 \nu^{13} + 6363639039 \nu^{12} + 663666222536 \nu^{11} - 440450615677 \nu^{10} - 20765197179345 \nu^{9} + \cdots - 212680826375024 ) / 53661551784000 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2089067853 \nu^{13} + 1186234119 \nu^{12} + 166990685056 \nu^{11} - 90007487317 \nu^{10} - 5108559447745 \nu^{9} + \cdots - 228794009350704 ) / 9573422384000 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 268898217 \nu^{13} - 60162011 \nu^{12} - 21662421664 \nu^{11} + 3229526593 \nu^{10} + 662126520605 \nu^{9} - 31650082226 \nu^{8} + \cdots + 16216865165616 ) / 957342238400 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 629711814493 \nu^{13} + 211929067839 \nu^{12} + 51141259110536 \nu^{11} - 12148429797277 \nu^{10} + \cdots - 10\!\cdots\!24 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + 3\beta_{3} + 19\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{13} - \beta_{11} + \beta_{10} + 2 \beta_{9} - \beta_{8} - 2 \beta_{6} - \beta_{5} - 2 \beta_{4} + 23 \beta_{2} - 3 \beta _1 + 229 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3 \beta_{13} + \beta_{12} + 3 \beta_{11} - 5 \beta_{10} + 2 \beta_{9} + 3 \beta_{8} - 4 \beta_{7} + 34 \beta_{6} - 15 \beta_{5} - 4 \beta_{4} + 78 \beta_{3} - 2 \beta_{2} + 400 \beta _1 + 60 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 41 \beta_{13} - 38 \beta_{11} + 28 \beta_{10} + 76 \beta_{9} - 34 \beta_{8} + \beta_{7} - 74 \beta_{6} - 9 \beta_{5} - 72 \beta_{4} + 141 \beta_{3} + 515 \beta_{2} - 164 \beta _1 + 4850 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 122 \beta_{13} + 49 \beta_{12} + 146 \beta_{11} - 202 \beta_{10} + 106 \beta_{9} + 128 \beta_{8} - 180 \beta_{7} + 931 \beta_{6} - 724 \beta_{5} - 174 \beta_{4} + 1565 \beta_{3} - 104 \beta_{2} + 8783 \beta _1 + 636 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1257 \beta_{13} - 9 \beta_{12} - 1152 \beta_{11} + 672 \beta_{10} + 2204 \beta_{9} - 872 \beta_{8} + 23 \beta_{7} - 2125 \beta_{6} + 421 \beta_{5} - 2046 \beta_{4} + 6944 \beta_{3} + 11581 \beta_{2} + \cdots + 107016 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3623 \beta_{13} + 1534 \beta_{12} + 5048 \beta_{11} - 6120 \beta_{10} + 3642 \beta_{9} + 4046 \beta_{8} - 5813 \beta_{7} + 23756 \beta_{6} - 24641 \beta_{5} - 5368 \beta_{4} + 27815 \beta_{3} - 3748 \beta_{2} + \cdots - 5982 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 34186 \beta_{13} - 247 \beta_{12} - 32506 \beta_{11} + 15550 \beta_{10} + 58426 \beta_{9} - 20316 \beta_{8} - 212 \beta_{7} - 56078 \beta_{6} + 26260 \beta_{5} - 53402 \beta_{4} + 239954 \beta_{3} + \cdots + 2413393 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 95312 \beta_{13} + 39355 \beta_{12} + 152581 \beta_{11} - 166261 \beta_{10} + 104498 \beta_{9} + 114853 \beta_{8} - 163757 \beta_{7} + 587178 \beta_{6} - 732762 \beta_{5} - 144636 \beta_{4} + \cdots - 620070 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 873549 \beta_{13} + 1297 \beta_{12} - 886792 \beta_{11} + 355354 \beta_{10} + 1487098 \beta_{9} - 455066 \beta_{8} - 37205 \beta_{7} - 1425812 \beta_{6} + 998693 \beta_{5} + \cdots + 55193599 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 2359274 \beta_{13} + 891798 \beta_{12} + 4304259 \beta_{11} - 4273639 \beta_{10} + 2722752 \beta_{9} + 3092949 \beta_{8} - 4288261 \beta_{7} + 14271698 \beta_{6} - 20388722 \beta_{5} + \cdots - 25133913 \) 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
4.81720
4.65416
4.24928
3.90684
1.73740
1.33976
−0.134967
−0.574607
−0.955230
−2.33957
−2.93781
−4.01755
−4.79301
−4.95189
−4.81720 −3.00000 15.2054 0 14.4516 −1.13492 −34.7100 9.00000 0
1.2 −4.65416 −3.00000 13.6612 0 13.9625 26.0445 −26.3483 9.00000 0
1.3 −4.24928 −3.00000 10.0564 0 12.7478 28.2853 −8.73812 9.00000 0
1.4 −3.90684 −3.00000 7.26339 0 11.7205 −22.0918 2.87782 9.00000 0
1.5 −1.73740 −3.00000 −4.98145 0 5.21219 −7.66213 22.5539 9.00000 0
1.6 −1.33976 −3.00000 −6.20504 0 4.01928 12.2101 19.0313 9.00000 0
1.7 0.134967 −3.00000 −7.98178 0 −0.404902 17.3099 −2.15702 9.00000 0
1.8 0.574607 −3.00000 −7.66983 0 −1.72382 2.67744 −9.00399 9.00000 0
1.9 0.955230 −3.00000 −7.08754 0 −2.86569 −12.4836 −14.4121 9.00000 0
1.10 2.33957 −3.00000 −2.52642 0 −7.01870 32.9322 −24.6273 9.00000 0
1.11 2.93781 −3.00000 0.630743 0 −8.81344 −18.9115 −21.6495 9.00000 0
1.12 4.01755 −3.00000 8.14069 0 −12.0526 −1.75849 0.565245 9.00000 0
1.13 4.79301 −3.00000 14.9730 0 −14.3790 −0.140520 33.4216 9.00000 0
1.14 4.95189 −3.00000 16.5212 0 −14.8557 −28.2766 42.1962 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
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 1875.4.a.f 14
5.b even 2 1 1875.4.a.g 14
25.e even 10 2 75.4.g.b 28
75.h odd 10 2 225.4.h.a 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.4.g.b 28 25.e even 10 2
225.4.h.a 28 75.h odd 10 2
1875.4.a.f 14 1.a even 1 1 trivial
1875.4.a.g 14 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} - 81 T_{2}^{12} + 7 T_{2}^{11} + 2512 T_{2}^{10} - 517 T_{2}^{9} - 36970 T_{2}^{8} + 12987 T_{2}^{7} + 257291 T_{2}^{6} - 125779 T_{2}^{5} - 718713 T_{2}^{4} + 371750 T_{2}^{3} + 579848 T_{2}^{2} + \cdots + 42064 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1875))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} - 81 T^{12} + 7 T^{11} + \cdots + 42064 \) Copy content Toggle raw display
$3$ \( (T + 3)^{14} \) Copy content Toggle raw display
$5$ \( T^{14} \) Copy content Toggle raw display
$7$ \( T^{14} - 27 T^{13} + \cdots + 4350562367600 \) Copy content Toggle raw display
$11$ \( T^{14} + 33 T^{13} + \cdots + 57\!\cdots\!96 \) Copy content Toggle raw display
$13$ \( T^{14} - 188 T^{13} + \cdots - 21\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{14} - 146 T^{13} + \cdots + 29\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{14} + 184 T^{13} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{14} - 164 T^{13} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{14} - 252 T^{13} + \cdots + 69\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{14} + 889 T^{13} + \cdots - 98\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{14} - 642 T^{13} + \cdots - 50\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{14} + 164 T^{13} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{14} - 696 T^{13} + \cdots - 55\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{14} + 92 T^{13} + \cdots - 35\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{14} - 949 T^{13} + \cdots - 14\!\cdots\!75 \) Copy content Toggle raw display
$59$ \( T^{14} + 81 T^{13} + \cdots - 44\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{14} + 496 T^{13} + \cdots - 10\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{14} - 1926 T^{13} + \cdots - 22\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{14} - 2498 T^{13} + \cdots + 16\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( T^{14} + 1026 T^{13} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{14} + 695 T^{13} + \cdots + 54\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{14} - 5315 T^{13} + \cdots + 23\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{14} + 1424 T^{13} + \cdots - 11\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{14} - 291 T^{13} + \cdots + 17\!\cdots\!71 \) Copy content Toggle raw display
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