Properties

Label 1449.2.a.r
Level 1449
Weight 2
Character orbit 1449.a
Self dual yes
Analytic conductor 11.570
Analytic rank 0
Dimension 5
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1449 = 3^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1449.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.5703232529\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.2147108.1
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 161)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( 2 + \beta_{2} ) q^{4} + ( 1 + \beta_{4} ) q^{5} + q^{7} + ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( 2 + \beta_{2} ) q^{4} + ( 1 + \beta_{4} ) q^{5} + q^{7} + ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{8} + ( -1 - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{10} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{11} + ( -1 + \beta_{3} + \beta_{4} ) q^{13} -\beta_{1} q^{14} + ( 1 + 3 \beta_{2} + \beta_{3} + \beta_{4} ) q^{16} + ( 3 - 2 \beta_{2} - \beta_{4} ) q^{17} + ( 2 - 2 \beta_{1} ) q^{19} + ( -1 + 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{4} ) q^{20} + ( 3 - \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{22} + q^{23} + ( 4 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{25} + ( -1 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{26} + ( 2 + \beta_{2} ) q^{28} + ( -\beta_{1} + 3 \beta_{2} ) q^{29} + ( 6 - \beta_{3} ) q^{31} + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{32} + ( 1 - 2 \beta_{1} + 4 \beta_{2} + 3 \beta_{4} ) q^{34} + ( 1 + \beta_{4} ) q^{35} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{37} + ( 8 - 2 \beta_{1} + 2 \beta_{2} ) q^{38} + ( -9 + 2 \beta_{1} - 6 \beta_{2} - 3 \beta_{4} ) q^{40} + ( -1 - \beta_{3} + \beta_{4} ) q^{41} + ( -2 + 2 \beta_{4} ) q^{43} + ( 5 - 3 \beta_{1} + 5 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} ) q^{44} -\beta_{1} q^{46} + ( -2 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{47} + q^{49} + ( 3 - 4 \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} ) q^{50} + ( -4 + \beta_{2} + \beta_{3} + 3 \beta_{4} ) q^{52} + ( -4 + 2 \beta_{2} ) q^{53} + ( 2 + 4 \beta_{1} - 2 \beta_{3} ) q^{55} + ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{56} + ( 4 - 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} ) q^{58} + ( -5 + 2 \beta_{1} + \beta_{4} ) q^{59} + ( -3 - 2 \beta_{1} - 2 \beta_{3} - \beta_{4} ) q^{61} + ( -6 \beta_{1} - \beta_{2} + \beta_{4} ) q^{62} + ( 5 + 2 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} ) q^{64} + ( 4 + 2 \beta_{1} - 2 \beta_{4} ) q^{65} + ( -3 + 5 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{67} + ( -1 - 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 5 \beta_{4} ) q^{68} + ( -1 - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{70} + ( -1 - 3 \beta_{3} - \beta_{4} ) q^{71} + ( 1 - 4 \beta_{2} - \beta_{3} - \beta_{4} ) q^{73} + ( -8 - 2 \beta_{4} ) q^{74} + ( 4 - 6 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} ) q^{76} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{77} + ( 7 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{79} + ( -3 + 8 \beta_{1} + 6 \beta_{2} + 3 \beta_{4} ) q^{80} + ( -1 + 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{82} + ( -2 - 2 \beta_{4} ) q^{83} + ( 1 - 3 \beta_{1} - 5 \beta_{2} + \beta_{3} + \beta_{4} ) q^{85} + ( -2 + 4 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} ) q^{86} + ( 3 - 5 \beta_{1} - 3 \beta_{2} + \beta_{3} - 5 \beta_{4} ) q^{88} + ( 5 - 2 \beta_{1} + 2 \beta_{3} + \beta_{4} ) q^{89} + ( -1 + \beta_{3} + \beta_{4} ) q^{91} + ( 2 + \beta_{2} ) q^{92} + ( -8 + 4 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{94} + ( -4 \beta_{2} - 4 \beta_{3} ) q^{95} + ( -3 + 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{4} ) q^{97} -\beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 2q^{2} + 12q^{4} + 4q^{5} + 5q^{7} - 3q^{8} + O(q^{10}) \) \( 5q - 2q^{2} + 12q^{4} + 4q^{5} + 5q^{7} - 3q^{8} - 8q^{10} + 4q^{11} - 6q^{13} - 2q^{14} + 10q^{16} + 12q^{17} + 6q^{19} + 14q^{22} + 5q^{23} + 19q^{25} - q^{26} + 12q^{28} + 4q^{29} + 30q^{31} - 8q^{32} + 6q^{34} + 4q^{35} + 4q^{37} + 40q^{38} - 50q^{40} - 6q^{41} - 12q^{43} + 26q^{44} - 2q^{46} - 10q^{47} + 5q^{49} + 2q^{50} - 21q^{52} - 16q^{53} + 18q^{55} - 3q^{56} + 13q^{58} - 22q^{59} - 18q^{61} - 15q^{62} + 25q^{64} + 26q^{65} - 2q^{67} - 12q^{68} - 8q^{70} - 4q^{71} - 2q^{73} - 38q^{74} + 10q^{76} + 4q^{77} + 30q^{79} + 10q^{80} - 7q^{82} - 8q^{83} - 12q^{85} - 8q^{86} + 4q^{88} + 20q^{89} - 6q^{91} + 12q^{92} - 25q^{94} - 8q^{95} - 12q^{97} - 2q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 2 x^{4} - 9 x^{3} + 17 x^{2} + 16 x - 27\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{4} - \nu^{3} - 8 \nu^{2} + 5 \nu + 11 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{4} + \nu^{3} - 10 \nu^{2} - 5 \nu + 19 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{4} - \beta_{3} + \beta_{2} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4} + \beta_{3} + 9 \beta_{2} + 21\)

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
2.69017
2.11948
1.23828
−1.50216
−2.54577
−2.69017 0 5.23702 3.51109 0 1.00000 −8.70812 0 −9.44544
1.2 −2.11948 0 2.49221 −2.40920 0 1.00000 −1.04322 0 5.10626
1.3 −1.23828 0 −0.466664 1.86253 0 1.00000 3.05442 0 −2.30633
1.4 1.50216 0 0.256481 3.82405 0 1.00000 −2.61904 0 5.74433
1.5 2.54577 0 4.48096 −2.78847 0 1.00000 6.31597 0 −7.09882
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

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 1449.2.a.r 5
3.b odd 2 1 161.2.a.d 5
12.b even 2 1 2576.2.a.bd 5
15.d odd 2 1 4025.2.a.p 5
21.c even 2 1 1127.2.a.h 5
69.c even 2 1 3703.2.a.j 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
161.2.a.d 5 3.b odd 2 1
1127.2.a.h 5 21.c even 2 1
1449.2.a.r 5 1.a even 1 1 trivial
2576.2.a.bd 5 12.b even 2 1
3703.2.a.j 5 69.c even 2 1
4025.2.a.p 5 15.d odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)
\(23\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1449))\):

\( T_{2}^{5} + 2 T_{2}^{4} - 9 T_{2}^{3} - 17 T_{2}^{2} + 16 T_{2} + 27 \)
\( T_{5}^{5} - 4 T_{5}^{4} - 14 T_{5}^{3} + 54 T_{5}^{2} + 52 T_{5} - 168 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + T^{2} - T^{3} + 2 T^{4} + 7 T^{5} + 4 T^{6} - 4 T^{7} + 8 T^{8} + 32 T^{9} + 32 T^{10} \)
$3$ 1
$5$ \( 1 - 4 T + 11 T^{2} - 26 T^{3} + 92 T^{4} - 228 T^{5} + 460 T^{6} - 650 T^{7} + 1375 T^{8} - 2500 T^{9} + 3125 T^{10} \)
$7$ \( ( 1 - T )^{5} \)
$11$ \( 1 - 4 T + 27 T^{2} - 28 T^{3} + 126 T^{4} + 400 T^{5} + 1386 T^{6} - 3388 T^{7} + 35937 T^{8} - 58564 T^{9} + 161051 T^{10} \)
$13$ \( 1 + 6 T + 56 T^{2} + 266 T^{3} + 1351 T^{4} + 4944 T^{5} + 17563 T^{6} + 44954 T^{7} + 123032 T^{8} + 171366 T^{9} + 371293 T^{10} \)
$17$ \( 1 - 12 T + 91 T^{2} - 430 T^{3} + 1692 T^{4} - 6148 T^{5} + 28764 T^{6} - 124270 T^{7} + 447083 T^{8} - 1002252 T^{9} + 1419857 T^{10} \)
$19$ \( 1 - 6 T + 67 T^{2} - 360 T^{3} + 2334 T^{4} - 9220 T^{5} + 44346 T^{6} - 129960 T^{7} + 459553 T^{8} - 781926 T^{9} + 2476099 T^{10} \)
$23$ \( ( 1 - T )^{5} \)
$29$ \( 1 - 4 T + 34 T^{2} - 214 T^{3} + 1217 T^{4} - 4232 T^{5} + 35293 T^{6} - 179974 T^{7} + 829226 T^{8} - 2829124 T^{9} + 20511149 T^{10} \)
$31$ \( 1 - 30 T + 502 T^{2} - 5646 T^{3} + 46995 T^{4} - 297598 T^{5} + 1456845 T^{6} - 5425806 T^{7} + 14955082 T^{8} - 27705630 T^{9} + 28629151 T^{10} \)
$37$ \( 1 - 4 T + 109 T^{2} - 216 T^{3} + 5126 T^{4} - 5064 T^{5} + 189662 T^{6} - 295704 T^{7} + 5521177 T^{8} - 7496644 T^{9} + 69343957 T^{10} \)
$41$ \( 1 + 6 T + 176 T^{2} + 838 T^{3} + 13295 T^{4} + 49000 T^{5} + 545095 T^{6} + 1408678 T^{7} + 12130096 T^{8} + 16954566 T^{9} + 115856201 T^{10} \)
$43$ \( 1 + 12 T + 191 T^{2} + 1696 T^{3} + 16226 T^{4} + 101736 T^{5} + 697718 T^{6} + 3135904 T^{7} + 15185837 T^{8} + 41025612 T^{9} + 147008443 T^{10} \)
$47$ \( 1 + 10 T + 110 T^{2} - 38 T^{3} - 3609 T^{4} - 58894 T^{5} - 169623 T^{6} - 83942 T^{7} + 11420530 T^{8} + 48796810 T^{9} + 229345007 T^{10} \)
$53$ \( 1 + 16 T + 317 T^{2} + 3240 T^{3} + 35942 T^{4} + 254032 T^{5} + 1904926 T^{6} + 9101160 T^{7} + 47194009 T^{8} + 126247696 T^{9} + 418195493 T^{10} \)
$59$ \( 1 + 22 T + 413 T^{2} + 5194 T^{3} + 54600 T^{4} + 458288 T^{5} + 3221400 T^{6} + 18080314 T^{7} + 84821527 T^{8} + 266581942 T^{9} + 714924299 T^{10} \)
$61$ \( 1 + 18 T + 339 T^{2} + 3954 T^{3} + 43744 T^{4} + 348488 T^{5} + 2668384 T^{6} + 14712834 T^{7} + 76946559 T^{8} + 249225138 T^{9} + 844596301 T^{10} \)
$67$ \( 1 + 2 T + 35 T^{2} - 204 T^{3} + 6790 T^{4} + 16644 T^{5} + 454930 T^{6} - 915756 T^{7} + 10526705 T^{8} + 40302242 T^{9} + 1350125107 T^{10} \)
$71$ \( 1 + 4 T + 254 T^{2} + 860 T^{3} + 30833 T^{4} + 86976 T^{5} + 2189143 T^{6} + 4335260 T^{7} + 90909394 T^{8} + 101646724 T^{9} + 1804229351 T^{10} \)
$73$ \( 1 + 2 T + 168 T^{2} + 946 T^{3} + 18175 T^{4} + 89144 T^{5} + 1326775 T^{6} + 5041234 T^{7} + 65354856 T^{8} + 56796482 T^{9} + 2073071593 T^{10} \)
$79$ \( 1 - 30 T + 703 T^{2} - 10676 T^{3} + 136374 T^{4} - 1310412 T^{5} + 10773546 T^{6} - 66628916 T^{7} + 346606417 T^{8} - 1168502430 T^{9} + 3077056399 T^{10} \)
$83$ \( 1 + 8 T + 359 T^{2} + 2224 T^{3} + 55778 T^{4} + 264336 T^{5} + 4629574 T^{6} + 15321136 T^{7} + 205271533 T^{8} + 379666568 T^{9} + 3939040643 T^{10} \)
$89$ \( 1 - 20 T + 511 T^{2} - 6422 T^{3} + 92672 T^{4} - 821572 T^{5} + 8247808 T^{6} - 50868662 T^{7} + 360239159 T^{8} - 1254844820 T^{9} + 5584059449 T^{10} \)
$97$ \( 1 + 12 T + 371 T^{2} + 3294 T^{3} + 61432 T^{4} + 417340 T^{5} + 5958904 T^{6} + 30993246 T^{7} + 338601683 T^{8} + 1062351372 T^{9} + 8587340257 T^{10} \)
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