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Properties

Label 275.4.a.j.1.3
Level $275$
Weight $4$
Character 275.1
Self dual yes
Analytic conductor $16.226$
Analytic rank $1$
Dimension $6$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [275,4,Mod(1,275)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("275.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 275.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(16.2255252516\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.2301792529.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 22x^{4} + 101x^{2} - 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 55)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.405276\) of defining polynomial
Character \(\chi\) \(=\) 275.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.405276 q^{2} +8.56932 q^{3} -7.83575 q^{4} -3.47294 q^{6} -27.4984 q^{7} +6.41784 q^{8} +46.4333 q^{9} -11.0000 q^{11} -67.1471 q^{12} -27.6235 q^{13} +11.1444 q^{14} +60.0850 q^{16} +9.63773 q^{17} -18.8183 q^{18} -112.560 q^{19} -235.643 q^{21} +4.45803 q^{22} -154.630 q^{23} +54.9966 q^{24} +11.1951 q^{26} +166.530 q^{27} +215.471 q^{28} -236.372 q^{29} -45.9456 q^{31} -75.6937 q^{32} -94.2626 q^{33} -3.90594 q^{34} -363.840 q^{36} +235.321 q^{37} +45.6177 q^{38} -236.715 q^{39} -55.5664 q^{41} +95.5003 q^{42} +306.070 q^{43} +86.1933 q^{44} +62.6679 q^{46} +133.999 q^{47} +514.888 q^{48} +413.164 q^{49} +82.5889 q^{51} +216.451 q^{52} -315.357 q^{53} -67.4907 q^{54} -176.481 q^{56} -964.562 q^{57} +95.7959 q^{58} +509.192 q^{59} -656.843 q^{61} +18.6206 q^{62} -1276.84 q^{63} -450.003 q^{64} +38.2023 q^{66} -173.172 q^{67} -75.5189 q^{68} -1325.08 q^{69} -183.924 q^{71} +298.002 q^{72} +83.2810 q^{73} -95.3697 q^{74} +881.991 q^{76} +302.483 q^{77} +95.9348 q^{78} +798.912 q^{79} +173.354 q^{81} +22.5197 q^{82} +442.779 q^{83} +1846.44 q^{84} -124.043 q^{86} -2025.55 q^{87} -70.5963 q^{88} -448.553 q^{89} +759.604 q^{91} +1211.65 q^{92} -393.722 q^{93} -54.3067 q^{94} -648.644 q^{96} +307.267 q^{97} -167.445 q^{98} -510.767 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{4} - 22 q^{6} + 24 q^{9} - 66 q^{11} - 18 q^{14} - 108 q^{16} - 258 q^{19} - 478 q^{21} - 6 q^{24} - 356 q^{26} - 494 q^{29} - 514 q^{31} - 6 q^{34} - 874 q^{36} - 560 q^{39} - 824 q^{41} + 44 q^{44}+ \cdots - 264 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.405276 −0.143287 −0.0716433 − 0.997430i \(-0.522824\pi\)
−0.0716433 + 0.997430i \(0.522824\pi\)
\(3\) 8.56932 1.64917 0.824584 − 0.565740i \(-0.191409\pi\)
0.824584 + 0.565740i \(0.191409\pi\)
\(4\) −7.83575 −0.979469
\(5\) 0 0
\(6\) −3.47294 −0.236303
\(7\) −27.4984 −1.48478 −0.742388 − 0.669970i \(-0.766307\pi\)
−0.742388 + 0.669970i \(0.766307\pi\)
\(8\) 6.41784 0.283631
\(9\) 46.4333 1.71975
\(10\) 0 0
\(11\) −11.0000 −0.301511
\(12\) −67.1471 −1.61531
\(13\) −27.6235 −0.589338 −0.294669 − 0.955599i \(-0.595209\pi\)
−0.294669 + 0.955599i \(0.595209\pi\)
\(14\) 11.1444 0.212748
\(15\) 0 0
\(16\) 60.0850 0.938828
\(17\) 9.63773 0.137500 0.0687498 − 0.997634i \(-0.478099\pi\)
0.0687498 + 0.997634i \(0.478099\pi\)
\(18\) −18.8183 −0.246417
\(19\) −112.560 −1.35911 −0.679553 − 0.733627i \(-0.737826\pi\)
−0.679553 + 0.733627i \(0.737826\pi\)
\(20\) 0 0
\(21\) −235.643 −2.44864
\(22\) 4.45803 0.0432025
\(23\) −154.630 −1.40186 −0.700928 − 0.713232i \(-0.747230\pi\)
−0.700928 + 0.713232i \(0.747230\pi\)
\(24\) 54.9966 0.467755
\(25\) 0 0
\(26\) 11.1951 0.0844442
\(27\) 166.530 1.18699
\(28\) 215.471 1.45429
\(29\) −236.372 −1.51356 −0.756780 − 0.653670i \(-0.773229\pi\)
−0.756780 + 0.653670i \(0.773229\pi\)
\(30\) 0 0
\(31\) −45.9456 −0.266196 −0.133098 − 0.991103i \(-0.542492\pi\)
−0.133098 + 0.991103i \(0.542492\pi\)
\(32\) −75.6937 −0.418153
\(33\) −94.2626 −0.497243
\(34\) −3.90594 −0.0197018
\(35\) 0 0
\(36\) −363.840 −1.68444
\(37\) 235.321 1.04558 0.522790 − 0.852461i \(-0.324891\pi\)
0.522790 + 0.852461i \(0.324891\pi\)
\(38\) 45.6177 0.194741
\(39\) −236.715 −0.971916
\(40\) 0 0
\(41\) −55.5664 −0.211659 −0.105829 − 0.994384i \(-0.533750\pi\)
−0.105829 + 0.994384i \(0.533750\pi\)
\(42\) 95.5003 0.350858
\(43\) 306.070 1.08547 0.542736 − 0.839904i \(-0.317389\pi\)
0.542736 + 0.839904i \(0.317389\pi\)
\(44\) 86.1933 0.295321
\(45\) 0 0
\(46\) 62.6679 0.200867
\(47\) 133.999 0.415868 0.207934 − 0.978143i \(-0.433326\pi\)
0.207934 + 0.978143i \(0.433326\pi\)
\(48\) 514.888 1.54829
\(49\) 413.164 1.20456
\(50\) 0 0
\(51\) 82.5889 0.226760
\(52\) 216.451 0.577238
\(53\) −315.357 −0.817315 −0.408657 − 0.912688i \(-0.634003\pi\)
−0.408657 + 0.912688i \(0.634003\pi\)
\(54\) −67.4907 −0.170080
\(55\) 0 0
\(56\) −176.481 −0.421129
\(57\) −964.562 −2.24139
\(58\) 95.7959 0.216873
\(59\) 509.192 1.12358 0.561790 − 0.827280i \(-0.310113\pi\)
0.561790 + 0.827280i \(0.310113\pi\)
\(60\) 0 0
\(61\) −656.843 −1.37869 −0.689345 − 0.724433i \(-0.742101\pi\)
−0.689345 + 0.724433i \(0.742101\pi\)
\(62\) 18.6206 0.0381423
\(63\) −1276.84 −2.55345
\(64\) −450.003 −0.878913
\(65\) 0 0
\(66\) 38.2023 0.0712482
\(67\) −173.172 −0.315766 −0.157883 − 0.987458i \(-0.550467\pi\)
−0.157883 + 0.987458i \(0.550467\pi\)
\(68\) −75.5189 −0.134677
\(69\) −1325.08 −2.31189
\(70\) 0 0
\(71\) −183.924 −0.307433 −0.153716 − 0.988115i \(-0.549124\pi\)
−0.153716 + 0.988115i \(0.549124\pi\)
\(72\) 298.002 0.487776
\(73\) 83.2810 0.133525 0.0667624 − 0.997769i \(-0.478733\pi\)
0.0667624 + 0.997769i \(0.478733\pi\)
\(74\) −95.3697 −0.149818
\(75\) 0 0
\(76\) 881.991 1.33120
\(77\) 302.483 0.447677
\(78\) 95.9348 0.139263
\(79\) 798.912 1.13778 0.568890 − 0.822414i \(-0.307373\pi\)
0.568890 + 0.822414i \(0.307373\pi\)
\(80\) 0 0
\(81\) 173.354 0.237797
\(82\) 22.5197 0.0303278
\(83\) 442.779 0.585558 0.292779 − 0.956180i \(-0.405420\pi\)
0.292779 + 0.956180i \(0.405420\pi\)
\(84\) 1846.44 2.39837
\(85\) 0 0
\(86\) −124.043 −0.155533
\(87\) −2025.55 −2.49611
\(88\) −70.5963 −0.0855180
\(89\) −448.553 −0.534230 −0.267115 − 0.963665i \(-0.586070\pi\)
−0.267115 + 0.963665i \(0.586070\pi\)
\(90\) 0 0
\(91\) 759.604 0.875034
\(92\) 1211.65 1.37307
\(93\) −393.722 −0.439001
\(94\) −54.3067 −0.0595883
\(95\) 0 0
\(96\) −648.644 −0.689604
\(97\) 307.267 0.321632 0.160816 − 0.986984i \(-0.448587\pi\)
0.160816 + 0.986984i \(0.448587\pi\)
\(98\) −167.445 −0.172597
\(99\) −510.767 −0.518525
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 275.4.a.j.1.3 6
3.2 odd 2 2475.4.a.bn.1.4 6
5.2 odd 4 55.4.b.a.34.3 ✓ 6
5.3 odd 4 55.4.b.a.34.4 yes 6
5.4 even 2 inner 275.4.a.j.1.4 6
15.2 even 4 495.4.c.a.199.4 6
15.8 even 4 495.4.c.a.199.3 6
15.14 odd 2 2475.4.a.bn.1.3 6
20.3 even 4 880.4.b.f.529.1 6
20.7 even 4 880.4.b.f.529.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.4.b.a.34.3 ✓ 6 5.2 odd 4
55.4.b.a.34.4 yes 6 5.3 odd 4
275.4.a.j.1.3 6 1.1 even 1 trivial
275.4.a.j.1.4 6 5.4 even 2 inner
495.4.c.a.199.3 6 15.8 even 4
495.4.c.a.199.4 6 15.2 even 4
880.4.b.f.529.1 6 20.3 even 4
880.4.b.f.529.6 6 20.7 even 4
2475.4.a.bn.1.3 6 15.14 odd 2
2475.4.a.bn.1.4 6 3.2 odd 2