Properties

Label 4025.2.a.y
Level $4025$
Weight $2$
Character orbit 4025.a
Self dual yes
Analytic conductor $32.140$
Analytic rank $1$
Dimension $12$
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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - 2 x^{11} - 14 x^{10} + 26 x^{9} + 71 x^{8} - 120 x^{7} - 162 x^{6} + 244 x^{5} + 170 x^{4} - 210 x^{3} - 81 x^{2} + 58 x + 17 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 805)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{6} q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{10} - \beta_{9} - \beta_{6} - 1) q^{6} - q^{7} + (\beta_{10} + \beta_{9} + \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_1 + 1) q^{8} + ( - \beta_{8} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{6} q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{10} - \beta_{9} - \beta_{6} - 1) q^{6} - q^{7} + (\beta_{10} + \beta_{9} + \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_1 + 1) q^{8} + ( - \beta_{8} + \beta_1 + 1) q^{9} + (\beta_{9} + \beta_{8} - \beta_{2} - 1) q^{11} + ( - \beta_{10} - \beta_{9} - \beta_{4} - \beta_{2} - 2) q^{12} + ( - \beta_{11} + \beta_{10} + \beta_{6} + \beta_{4} - \beta_{3} - \beta_1) q^{13} - \beta_1 q^{14} + (\beta_{11} + \beta_{10} + \beta_{6} + \beta_{5} + \beta_{2}) q^{16} + (\beta_{11} - \beta_{10} - \beta_{7} - \beta_{5} - \beta_{4} + \beta_{3}) q^{17} + ( - \beta_{9} - \beta_{8} - \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} + 2) q^{18} + (\beta_{10} + \beta_{8} - \beta_{5} - \beta_{2} - 3) q^{19} + \beta_{6} q^{21} + (\beta_{11} - \beta_{10} + \beta_{8} - 2 \beta_{7} - \beta_{6} - 2 \beta_{4} + \beta_{3} + \beta_{2} + 1) q^{22} + q^{23} + (\beta_{10} + \beta_{9} + \beta_{8} - \beta_{4} + \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 1) q^{24} + ( - 2 \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} + 2 \beta_{6} + \beta_{4} - \beta_{3} - 2 \beta_{2} + \cdots - 2) q^{26}+ \cdots + (3 \beta_{11} - \beta_{10} + 2 \beta_{8} - 3 \beta_{7} - 2 \beta_{6} - \beta_{5} - 2 \beta_{4} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{2} + 8 q^{4} - 6 q^{6} - 12 q^{7} + 6 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{2} + 8 q^{4} - 6 q^{6} - 12 q^{7} + 6 q^{8} + 8 q^{9} - 8 q^{11} - 12 q^{12} + 2 q^{13} - 2 q^{14} - 4 q^{16} - 8 q^{17} + 20 q^{18} - 26 q^{19} + 4 q^{22} + 12 q^{23} - 12 q^{24} - 22 q^{26} + 12 q^{27} - 8 q^{28} - 12 q^{29} - 50 q^{31} + 14 q^{32} + 4 q^{33} - 28 q^{34} - 18 q^{36} + 8 q^{37} - 4 q^{38} - 26 q^{39} - 4 q^{41} + 6 q^{42} + 26 q^{43} - 10 q^{44} + 2 q^{46} + 16 q^{47} - 40 q^{48} + 12 q^{49} - 32 q^{51} + 10 q^{52} - 18 q^{53} - 10 q^{54} - 6 q^{56} - 10 q^{57} - 18 q^{58} - 18 q^{59} + 8 q^{61} - 54 q^{62} - 8 q^{63} + 12 q^{64} - 2 q^{66} + 38 q^{67} - 36 q^{68} - 24 q^{71} + 18 q^{72} - 14 q^{73} + 36 q^{74} - 56 q^{76} + 8 q^{77} - 26 q^{78} - 44 q^{79} - 16 q^{81} - 44 q^{82} - 14 q^{83} + 12 q^{84} - 32 q^{86} + 16 q^{87} + 32 q^{88} - 10 q^{89} - 2 q^{91} + 8 q^{92} + 26 q^{93} + 18 q^{94} - 38 q^{96} - 4 q^{97} + 2 q^{98} - 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} - 14 x^{10} + 26 x^{9} + 71 x^{8} - 120 x^{7} - 162 x^{6} + 244 x^{5} + 170 x^{4} - 210 x^{3} - 81 x^{2} + 58 x + 17 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 17 \nu^{11} - 24 \nu^{10} - 218 \nu^{9} + 237 \nu^{8} + 966 \nu^{7} - 670 \nu^{6} - 1780 \nu^{5} + 537 \nu^{4} + 1297 \nu^{3} + 81 \nu^{2} - 241 \nu - 94 ) / 29 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 12 \nu^{11} + 5 \nu^{10} + 188 \nu^{9} - 53 \nu^{8} - 1035 \nu^{7} + 171 \nu^{6} + 2367 \nu^{5} - 159 \nu^{4} - 2125 \nu^{3} - 151 \nu^{2} + 571 \nu + 138 ) / 29 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 18 \nu^{11} - 22 \nu^{10} - 253 \nu^{9} + 239 \nu^{8} + 1277 \nu^{7} - 822 \nu^{6} - 2840 \nu^{5} + 1036 \nu^{4} + 2767 \nu^{3} - 281 \nu^{2} - 958 \nu - 120 ) / 29 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 20 \nu^{11} - 18 \nu^{10} - 294 \nu^{9} + 185 \nu^{8} + 1551 \nu^{7} - 575 \nu^{6} - 3539 \nu^{5} + 613 \nu^{4} + 3329 \nu^{3} - 77 \nu^{2} - 913 \nu - 114 ) / 29 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 28 \nu^{11} + 31 \nu^{10} + 400 \nu^{9} - 317 \nu^{8} - 2067 \nu^{7} + 950 \nu^{6} + 4740 \nu^{5} - 806 \nu^{4} - 4736 \nu^{3} - 275 \nu^{2} + 1574 \nu + 322 ) / 29 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 24 \nu^{11} - 10 \nu^{10} - 376 \nu^{9} + 77 \nu^{8} + 2128 \nu^{7} - 52 \nu^{6} - 5256 \nu^{5} - 523 \nu^{4} + 5468 \nu^{3} + 1056 \nu^{2} - 1751 \nu - 508 ) / 29 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 33 \nu^{11} + 21 \nu^{10} + 488 \nu^{9} - 182 \nu^{8} - 2578 \nu^{7} + 347 \nu^{6} + 5864 \nu^{5} + 179 \nu^{4} - 5561 \nu^{3} - 785 \nu^{2} + 1592 \nu + 336 ) / 29 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 35 \nu^{11} - 17 \nu^{10} - 529 \nu^{9} + 128 \nu^{8} + 2852 \nu^{7} - 71 \nu^{6} - 6592 \nu^{5} - 863 \nu^{4} + 6355 \nu^{3} + 1569 \nu^{2} - 1953 \nu - 591 ) / 29 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 73 \nu^{11} + 57 \nu^{10} + 1076 \nu^{9} - 552 \nu^{8} - 5680 \nu^{7} + 1468 \nu^{6} + 12971 \nu^{5} - 757 \nu^{4} - 12451 \nu^{3} - 1414 \nu^{2} + 3824 \nu + 1028 ) / 29 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} + \beta_{9} + \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 3\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + \beta_{10} + \beta_{6} + \beta_{5} + 7\beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7\beta_{10} + 8\beta_{9} + \beta_{8} + 8\beta_{7} + 8\beta_{6} + 7\beta_{5} + 7\beta_{4} + \beta_{3} + 12\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9 \beta_{11} + 9 \beta_{10} + \beta_{9} + \beta_{8} + 8 \beta_{6} + 9 \beta_{5} - \beta_{4} + \beta_{3} + 43 \beta_{2} + 2 \beta _1 + 76 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{11} + 43 \beta_{10} + 52 \beta_{9} + 11 \beta_{8} + 53 \beta_{7} + 52 \beta_{6} + 45 \beta_{5} + 43 \beta_{4} + 10 \beta_{3} + 2 \beta_{2} + 57 \beta _1 + 67 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 63 \beta_{11} + 63 \beta_{10} + 12 \beta_{9} + 13 \beta_{8} + 4 \beta_{7} + 53 \beta_{6} + 67 \beta_{5} - 10 \beta_{4} + 12 \beta_{3} + 257 \beta_{2} + 23 \beta _1 + 438 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 16 \beta_{11} + 259 \beta_{10} + 319 \beta_{9} + 90 \beta_{8} + 334 \beta_{7} + 318 \beta_{6} + 288 \beta_{5} + 254 \beta_{4} + 78 \beta_{3} + 32 \beta_{2} + 299 \beta _1 + 465 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 410 \beta_{11} + 412 \beta_{10} + 106 \beta_{9} + 124 \beta_{8} + 68 \beta_{7} + 337 \beta_{6} + 474 \beta_{5} - 69 \beta_{4} + 108 \beta_{3} + 1526 \beta_{2} + 184 \beta _1 + 2594 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 172 \beta_{11} + 1561 \beta_{10} + 1919 \beta_{9} + 667 \beta_{8} + 2073 \beta_{7} + 1905 \beta_{6} + 1851 \beta_{5} + 1473 \beta_{4} + 563 \beta_{3} + 337 \beta_{2} + 1655 \beta _1 + 3122 \) 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
−2.37028
−1.76402
−1.69164
−1.11343
−0.649823
−0.271475
0.792462
1.23080
1.31562
1.53413
2.43455
2.55311
−2.37028 −0.134036 3.61823 0 0.317703 −1.00000 −3.83566 −2.98203 0
1.2 −1.76402 −1.09514 1.11176 0 1.93184 −1.00000 1.56687 −1.80068 0
1.3 −1.69164 0.285164 0.861643 0 −0.482394 −1.00000 1.92569 −2.91868 0
1.4 −1.11343 2.87976 −0.760266 0 −3.20642 −1.00000 3.07337 5.29300 0
1.5 −0.649823 1.89575 −1.57773 0 −1.23191 −1.00000 2.32489 0.593883 0
1.6 −0.271475 −2.40463 −1.92630 0 0.652799 −1.00000 1.06589 2.78227 0
1.7 0.792462 −2.09947 −1.37200 0 −1.66375 −1.00000 −2.67218 1.40777 0
1.8 1.23080 −0.767488 −0.485126 0 −0.944626 −1.00000 −3.05870 −2.41096 0
1.9 1.31562 2.16083 −0.269134 0 2.84283 −1.00000 −2.98533 1.66917 0
1.10 1.53413 2.66039 0.353561 0 4.08139 −1.00000 −2.52585 4.07768 0
1.11 2.43455 −2.82471 3.92701 0 −6.87689 −1.00000 4.69139 4.97900 0
1.12 2.55311 −0.556415 4.51835 0 −1.42059 −1.00000 6.42962 −2.69040 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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.y 12
5.b even 2 1 4025.2.a.x 12
5.c odd 4 2 805.2.c.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
805.2.c.b 24 5.c odd 4 2
4025.2.a.x 12 5.b even 2 1
4025.2.a.y 12 1.a even 1 1 trivial

Hecke kernels

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

\( T_{2}^{12} - 2 T_{2}^{11} - 14 T_{2}^{10} + 26 T_{2}^{9} + 71 T_{2}^{8} - 120 T_{2}^{7} - 162 T_{2}^{6} + 244 T_{2}^{5} + 170 T_{2}^{4} - 210 T_{2}^{3} - 81 T_{2}^{2} + 58 T_{2} + 17 \) Copy content Toggle raw display
\( T_{3}^{12} - 22 T_{3}^{10} - 4 T_{3}^{9} + 174 T_{3}^{8} + 68 T_{3}^{7} - 581 T_{3}^{6} - 372 T_{3}^{5} + 669 T_{3}^{4} + 672 T_{3}^{3} + 75 T_{3}^{2} - 60 T_{3} - 8 \) Copy content Toggle raw display
\( T_{11}^{12} + 8 T_{11}^{11} - 25 T_{11}^{10} - 274 T_{11}^{9} + 188 T_{11}^{8} + 3456 T_{11}^{7} + 3 T_{11}^{6} - 19422 T_{11}^{5} - 6026 T_{11}^{4} + 45752 T_{11}^{3} + 21040 T_{11}^{2} - 28480 T_{11} - 7392 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 2 T^{11} - 14 T^{10} + 26 T^{9} + \cdots + 17 \) Copy content Toggle raw display
$3$ \( T^{12} - 22 T^{10} - 4 T^{9} + 174 T^{8} + \cdots - 8 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T + 1)^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + 8 T^{11} - 25 T^{10} + \cdots - 7392 \) Copy content Toggle raw display
$13$ \( T^{12} - 2 T^{11} - 75 T^{10} + \cdots - 1234 \) Copy content Toggle raw display
$17$ \( T^{12} + 8 T^{11} - 61 T^{10} + \cdots + 84256 \) Copy content Toggle raw display
$19$ \( T^{12} + 26 T^{11} + 198 T^{10} + \cdots + 1987488 \) Copy content Toggle raw display
$23$ \( (T - 1)^{12} \) Copy content Toggle raw display
$29$ \( T^{12} + 12 T^{11} - 90 T^{10} + \cdots - 6811136 \) Copy content Toggle raw display
$31$ \( T^{12} + 50 T^{11} + \cdots + 714551774 \) Copy content Toggle raw display
$37$ \( T^{12} - 8 T^{11} - 173 T^{10} + \cdots - 1732696 \) Copy content Toggle raw display
$41$ \( T^{12} + 4 T^{11} - 241 T^{10} + \cdots - 220612 \) Copy content Toggle raw display
$43$ \( T^{12} - 26 T^{11} + 62 T^{10} + \cdots + 81498208 \) Copy content Toggle raw display
$47$ \( T^{12} - 16 T^{11} - 196 T^{10} + \cdots - 23799312 \) Copy content Toggle raw display
$53$ \( T^{12} + 18 T^{11} + \cdots + 136913064 \) Copy content Toggle raw display
$59$ \( T^{12} + 18 T^{11} + \cdots - 632701728 \) Copy content Toggle raw display
$61$ \( T^{12} - 8 T^{11} - 197 T^{10} + \cdots + 7050656 \) Copy content Toggle raw display
$67$ \( T^{12} - 38 T^{11} + \cdots + 2026241824 \) Copy content Toggle raw display
$71$ \( T^{12} + 24 T^{11} - 68 T^{10} + \cdots + 65141008 \) Copy content Toggle raw display
$73$ \( T^{12} + 14 T^{11} - 487 T^{10} + \cdots + 58991502 \) Copy content Toggle raw display
$79$ \( T^{12} + 44 T^{11} + \cdots - 1360803976 \) Copy content Toggle raw display
$83$ \( T^{12} + 14 T^{11} - 261 T^{10} + \cdots - 11862896 \) Copy content Toggle raw display
$89$ \( T^{12} + 10 T^{11} + \cdots + 85724172576 \) Copy content Toggle raw display
$97$ \( T^{12} + 4 T^{11} + \cdots - 15480701824 \) Copy content Toggle raw display
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