Properties

Label 4025.2.a.ba
Level $4025$
Weight $2$
Character orbit 4025.a
Self dual yes
Analytic conductor $32.140$
Analytic rank $1$
Dimension $14$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4025 = 5^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4025.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.1397868136\)
Analytic rank: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \( x^{14} - x^{13} - 22 x^{12} + 18 x^{11} + 187 x^{10} - 118 x^{9} - 772 x^{8} + 346 x^{7} + 1581 x^{6} - 443 x^{5} - 1429 x^{4} + 193 x^{3} + 386 x^{2} - 3 x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{3} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{4} q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{7} + \beta_1) q^{6} - q^{7} + ( - \beta_{3} - \beta_1 - 1) q^{8} + ( - \beta_{10} + \beta_{4} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - \beta_{4} q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{7} + \beta_1) q^{6} - q^{7} + ( - \beta_{3} - \beta_1 - 1) q^{8} + ( - \beta_{10} + \beta_{4} + 1) q^{9} - \beta_{13} q^{11} + ( - \beta_{13} + \beta_{12} - 2 \beta_{9} + \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \cdots + 2 \beta_1) q^{12}+ \cdots + ( - \beta_{13} + \beta_{12} + 3 \beta_{8} + 2 \beta_{7} + \beta_{6} + 2 \beta_{4} - 2 \beta_{3} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - q^{2} - 6 q^{3} + 17 q^{4} - 4 q^{6} - 14 q^{7} - 9 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - q^{2} - 6 q^{3} + 17 q^{4} - 4 q^{6} - 14 q^{7} - 9 q^{8} + 18 q^{9} - 3 q^{11} - 11 q^{12} - 15 q^{13} + q^{14} + 23 q^{16} - 9 q^{17} - 17 q^{18} - 4 q^{19} + 6 q^{21} - 9 q^{22} + 14 q^{23} + 10 q^{24} - 5 q^{26} - 33 q^{27} - 17 q^{28} + 11 q^{29} - q^{31} - 24 q^{32} - 26 q^{33} - 6 q^{34} + 13 q^{36} - 18 q^{37} + 6 q^{38} + 6 q^{39} - 7 q^{41} + 4 q^{42} - 18 q^{43} - 16 q^{44} - q^{46} - 10 q^{47} - 40 q^{48} + 14 q^{49} + 28 q^{51} - 46 q^{52} - 5 q^{53} - 24 q^{54} + 9 q^{56} + 26 q^{57} - 2 q^{58} - 24 q^{59} - 6 q^{61} + 16 q^{62} - 18 q^{63} + 29 q^{64} + 27 q^{66} - 61 q^{67} - 35 q^{68} - 6 q^{69} + 11 q^{71} - 12 q^{72} - 28 q^{73} - 49 q^{74} - 27 q^{76} + 3 q^{77} - 38 q^{78} + 6 q^{79} + 26 q^{81} + 14 q^{82} - 16 q^{83} + 11 q^{84} + 46 q^{86} - 61 q^{87} - 58 q^{88} - 39 q^{89} + 15 q^{91} + 17 q^{92} - 21 q^{93} - 74 q^{94} + 41 q^{96} - 19 q^{97} - q^{98} - 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - x^{13} - 22 x^{12} + 18 x^{11} + 187 x^{10} - 118 x^{9} - 772 x^{8} + 346 x^{7} + 1581 x^{6} - 443 x^{5} - 1429 x^{4} + 193 x^{3} + 386 x^{2} - 3 x - 5 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu - 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 134 \nu^{13} + 37911 \nu^{12} + 9431 \nu^{11} - 833971 \nu^{10} - 178425 \nu^{9} + 6807185 \nu^{8} + 1395439 \nu^{7} - 25174921 \nu^{6} - 5051683 \nu^{5} + \cdots + 1719962 ) / 1242873 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 8617 \nu^{13} - 22078 \nu^{12} + 273594 \nu^{11} + 349093 \nu^{10} - 3046784 \nu^{9} - 1722211 \nu^{8} + 15312112 \nu^{7} + 2054981 \nu^{6} - 35751429 \nu^{5} + \cdots + 208671 ) / 414291 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3427 \nu^{13} - 14217 \nu^{12} - 50538 \nu^{11} + 246032 \nu^{10} + 208975 \nu^{9} - 1485425 \nu^{8} + 41160 \nu^{7} + 3684384 \nu^{6} - 1876665 \nu^{5} + \cdots - 136436 ) / 138097 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 37777 \nu^{13} + 6483 \nu^{12} - 831559 \nu^{11} - 153367 \nu^{10} + 6791373 \nu^{9} + 1291991 \nu^{8} - 25128557 \nu^{7} - 4839829 \nu^{6} + 40647488 \nu^{5} + \cdots - 670 ) / 1242873 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 17266 \nu^{13} + 28793 \nu^{12} + 353631 \nu^{11} - 477956 \nu^{10} - 2836277 \nu^{9} + 2649746 \nu^{8} + 11524948 \nu^{7} - 5168881 \nu^{6} + \cdots - 1259865 ) / 414291 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 74561 \nu^{13} - 92562 \nu^{12} - 1571588 \nu^{11} + 1647493 \nu^{10} + 12628452 \nu^{9} - 10583948 \nu^{8} - 48372151 \nu^{7} + 29999800 \nu^{6} + \cdots + 3221839 ) / 1242873 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 92428 \nu^{13} - 123405 \nu^{12} - 1867552 \nu^{11} + 2127977 \nu^{10} + 14235777 \nu^{9} - 12965704 \nu^{8} - 51178406 \nu^{7} + 33005087 \nu^{6} + \cdots + 4568996 ) / 1242873 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 114541 \nu^{13} + 106461 \nu^{12} + 2641672 \nu^{11} - 2117591 \nu^{10} - 23419830 \nu^{9} + 15754684 \nu^{8} + 99159767 \nu^{7} - 53798933 \nu^{6} + \cdots + 1997443 ) / 1242873 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 64550 \nu^{13} + 87137 \nu^{12} + 1331797 \nu^{11} - 1563334 \nu^{10} - 10454477 \nu^{9} + 10248993 \nu^{8} + 39148617 \nu^{7} - 30423831 \nu^{6} + \cdots + 1835824 ) / 414291 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 203639 \nu^{13} + 277101 \nu^{12} + 4169738 \nu^{11} - 4742359 \nu^{10} - 32522883 \nu^{9} + 28618406 \nu^{8} + 121398712 \nu^{7} - 72669595 \nu^{6} + \cdots - 2790322 ) / 1242873 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{13} + \beta_{10} + 2\beta_{9} - \beta_{7} + \beta_{4} + \beta_{3} + 7\beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{10} + 2\beta_{9} - \beta_{6} + \beta_{5} + 10\beta_{3} + \beta_{2} + 27\beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12 \beta_{13} - \beta_{12} + 9 \beta_{10} + 23 \beta_{9} - \beta_{8} - 10 \beta_{7} + \beta_{6} + 12 \beta_{4} + 12 \beta_{3} + 48 \beta_{2} + \beta _1 + 76 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3 \beta_{13} - \beta_{12} - 2 \beta_{11} - 12 \beta_{10} + 29 \beta_{9} - 2 \beta_{7} - 13 \beta_{6} + 15 \beta_{5} + 5 \beta_{4} + 83 \beta_{3} + 13 \beta_{2} + 155 \beta _1 + 78 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 111 \beta_{13} - 14 \beta_{12} - 2 \beta_{11} + 67 \beta_{10} + 205 \beta_{9} - 17 \beta_{8} - 78 \beta_{7} + 12 \beta_{6} + 4 \beta_{5} + 109 \beta_{4} + 110 \beta_{3} + 333 \beta_{2} + 17 \beta _1 + 453 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 57 \beta_{13} - 21 \beta_{12} - 33 \beta_{11} - 107 \beta_{10} + 307 \beta_{9} - 6 \beta_{8} - 37 \beta_{7} - 129 \beta_{6} + 161 \beta_{5} + 88 \beta_{4} + 654 \beta_{3} + 130 \beta_{2} + 932 \beta _1 + 567 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 945 \beta_{13} - 149 \beta_{12} - 38 \beta_{11} + 476 \beta_{10} + 1692 \beta_{9} - 194 \beta_{8} - 571 \beta_{7} + 93 \beta_{6} + 83 \beta_{5} + 906 \beta_{4} + 931 \beta_{3} + 2344 \beta_{2} + 186 \beta _1 + 2876 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 720 \beta_{13} - 281 \beta_{12} - 370 \beta_{11} - 854 \beta_{10} + 2872 \beta_{9} - 121 \beta_{8} - 444 \beta_{7} - 1155 \beta_{6} + 1513 \beta_{5} + 1043 \beta_{4} + 5055 \beta_{3} + 1186 \beta_{2} + 5815 \beta _1 + 4049 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 7758 \beta_{13} - 1431 \beta_{12} - 479 \beta_{11} + 3338 \beta_{10} + 13556 \beta_{9} - 1883 \beta_{8} - 4128 \beta_{7} + 546 \beta_{6} + 1117 \beta_{5} + 7294 \beta_{4} + 7643 \beta_{3} + 16718 \beta_{2} + \cdots + 19068 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 7632 \beta_{13} - 3084 \beta_{12} - 3546 \beta_{11} - 6459 \beta_{10} + 25222 \beta_{9} - 1596 \beta_{8} - 4431 \beta_{7} - 9803 \beta_{6} + 13271 \beta_{5} + 10489 \beta_{4} + 38763 \beta_{3} + 10361 \beta_{2} + \cdots + 28967 \) 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
2.79074
2.52898
2.17410
1.82692
1.31493
0.704319
0.117364
−0.116065
−0.582976
−1.25067
−1.76659
−1.83943
−2.25329
−2.64833
−2.79074 −2.29281 5.78823 0 6.39865 −1.00000 −10.5720 2.25700 0
1.2 −2.52898 1.80521 4.39572 0 −4.56532 −1.00000 −6.05871 0.258773 0
1.3 −2.17410 −1.37249 2.72672 0 2.98394 −1.00000 −1.57997 −1.11626 0
1.4 −1.82692 2.52976 1.33763 0 −4.62167 −1.00000 1.21009 3.39970 0
1.5 −1.31493 −3.36299 −0.270967 0 4.42209 −1.00000 2.98616 8.30972 0
1.6 −0.704319 2.54639 −1.50393 0 −1.79347 −1.00000 2.46789 3.48411 0
1.7 −0.117364 −0.701283 −1.98623 0 0.0823052 −1.00000 0.467839 −2.50820 0
1.8 0.116065 −1.59146 −1.98653 0 −0.184713 −1.00000 −0.462695 −0.467244 0
1.9 0.582976 0.367598 −1.66014 0 0.214301 −1.00000 −2.13377 −2.86487 0
1.10 1.25067 −3.28603 −0.435826 0 −4.10974 −1.00000 −3.04641 7.79801 0
1.11 1.76659 0.991358 1.12083 0 1.75132 −1.00000 −1.55314 −2.01721 0
1.12 1.83943 1.64563 1.38349 0 3.02702 −1.00000 −1.13403 −0.291898 0
1.13 2.25329 −2.73097 3.07734 0 −6.15367 −1.00000 2.42755 4.45818 0
1.14 2.64833 −0.547903 5.01366 0 −1.45103 −1.00000 7.98118 −2.69980 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
\(5\) \(-1\)
\(7\) \(1\)
\(23\) \(-1\)

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 4025.2.a.ba 14
5.b even 2 1 4025.2.a.bb yes 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4025.2.a.ba 14 1.a even 1 1 trivial
4025.2.a.bb yes 14 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4025))\):

\( T_{2}^{14} + T_{2}^{13} - 22 T_{2}^{12} - 18 T_{2}^{11} + 187 T_{2}^{10} + 118 T_{2}^{9} - 772 T_{2}^{8} - 346 T_{2}^{7} + 1581 T_{2}^{6} + 443 T_{2}^{5} - 1429 T_{2}^{4} - 193 T_{2}^{3} + 386 T_{2}^{2} + 3 T_{2} - 5 \) Copy content Toggle raw display
\( T_{3}^{14} + 6 T_{3}^{13} - 12 T_{3}^{12} - 121 T_{3}^{11} - 8 T_{3}^{10} + 909 T_{3}^{9} + 632 T_{3}^{8} - 3128 T_{3}^{7} - 3000 T_{3}^{6} + 4860 T_{3}^{5} + 5298 T_{3}^{4} - 2694 T_{3}^{3} - 3258 T_{3}^{2} + 135 T_{3} + 405 \) Copy content Toggle raw display
\( T_{11}^{14} + 3 T_{11}^{13} - 98 T_{11}^{12} - 265 T_{11}^{11} + 3589 T_{11}^{10} + 8946 T_{11}^{9} - 61324 T_{11}^{8} - 150514 T_{11}^{7} + 483981 T_{11}^{6} + 1283456 T_{11}^{5} - 1262144 T_{11}^{4} - 4537424 T_{11}^{3} + \cdots - 30528 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + T^{13} - 22 T^{12} - 18 T^{11} + \cdots - 5 \) Copy content Toggle raw display
$3$ \( T^{14} + 6 T^{13} - 12 T^{12} - 121 T^{11} + \cdots + 405 \) Copy content Toggle raw display
$5$ \( T^{14} \) Copy content Toggle raw display
$7$ \( (T + 1)^{14} \) Copy content Toggle raw display
$11$ \( T^{14} + 3 T^{13} - 98 T^{12} + \cdots - 30528 \) Copy content Toggle raw display
$13$ \( T^{14} + 15 T^{13} - 4 T^{12} + \cdots + 2762487 \) Copy content Toggle raw display
$17$ \( T^{14} + 9 T^{13} - 89 T^{12} + \cdots + 3043008 \) Copy content Toggle raw display
$19$ \( T^{14} + 4 T^{13} + \cdots + 1202640448 \) Copy content Toggle raw display
$23$ \( (T - 1)^{14} \) Copy content Toggle raw display
$29$ \( T^{14} - 11 T^{13} - 109 T^{12} + \cdots + 3379103 \) Copy content Toggle raw display
$31$ \( T^{14} + T^{13} - 316 T^{12} + \cdots - 57840829 \) Copy content Toggle raw display
$37$ \( T^{14} + 18 T^{13} + \cdots - 6636147392 \) Copy content Toggle raw display
$41$ \( T^{14} + 7 T^{13} + \cdots - 1303165251 \) Copy content Toggle raw display
$43$ \( T^{14} + 18 T^{13} + \cdots - 4740109632 \) Copy content Toggle raw display
$47$ \( T^{14} + 10 T^{13} + \cdots - 1176896709 \) Copy content Toggle raw display
$53$ \( T^{14} + 5 T^{13} + \cdots + 2549303360 \) Copy content Toggle raw display
$59$ \( T^{14} + 24 T^{13} - 218 T^{12} + \cdots - 19782035 \) Copy content Toggle raw display
$61$ \( T^{14} + 6 T^{13} + \cdots + 14122038080 \) Copy content Toggle raw display
$67$ \( T^{14} + 61 T^{13} + 1483 T^{12} + \cdots - 17784256 \) Copy content Toggle raw display
$71$ \( T^{14} - 11 T^{13} + \cdots + 1585190508535 \) Copy content Toggle raw display
$73$ \( T^{14} + 28 T^{13} + \cdots - 20947706379 \) Copy content Toggle raw display
$79$ \( T^{14} - 6 T^{13} + \cdots + 65236518720 \) Copy content Toggle raw display
$83$ \( T^{14} + 16 T^{13} + \cdots + 6763300416 \) Copy content Toggle raw display
$89$ \( T^{14} + 39 T^{13} + \cdots - 138345595456 \) Copy content Toggle raw display
$97$ \( T^{14} + 19 T^{13} + \cdots + 1258422080 \) Copy content Toggle raw display
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