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Properties

Label 9025.2.a.t.1.2
Level $9025$
Weight $2$
Character 9025.1
Self dual yes
Analytic conductor $72.065$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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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] = [9025,2,Mod(1,9025)] 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("9025.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9025, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9025 = 5^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9025.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,3,0,3,0,-5,-4,6,4,0,2,-15,-4,-11] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(72.0649878242\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 9025.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.61803 q^{2} -2.23607 q^{3} +4.85410 q^{4} -5.85410 q^{6} -4.23607 q^{7} +7.47214 q^{8} +2.00000 q^{9} +5.47214 q^{11} -10.8541 q^{12} -2.00000 q^{13} -11.0902 q^{14} +9.85410 q^{16} -4.85410 q^{17} +5.23607 q^{18} +9.47214 q^{21} +14.3262 q^{22} -1.23607 q^{23} -16.7082 q^{24} -5.23607 q^{26} +2.23607 q^{27} -20.5623 q^{28} +6.00000 q^{29} -1.85410 q^{31} +10.8541 q^{32} -12.2361 q^{33} -12.7082 q^{34} +9.70820 q^{36} +4.14590 q^{37} +4.47214 q^{39} -6.38197 q^{41} +24.7984 q^{42} +10.5623 q^{43} +26.5623 q^{44} -3.23607 q^{46} -7.61803 q^{47} -22.0344 q^{48} +10.9443 q^{49} +10.8541 q^{51} -9.70820 q^{52} -6.38197 q^{53} +5.85410 q^{54} -31.6525 q^{56} +15.7082 q^{58} -2.76393 q^{59} -5.56231 q^{61} -4.85410 q^{62} -8.47214 q^{63} +8.70820 q^{64} -32.0344 q^{66} -8.70820 q^{67} -23.5623 q^{68} +2.76393 q^{69} -4.52786 q^{71} +14.9443 q^{72} -10.7082 q^{73} +10.8541 q^{74} -23.1803 q^{77} +11.7082 q^{78} +1.00000 q^{79} -11.0000 q^{81} -16.7082 q^{82} +1.09017 q^{83} +45.9787 q^{84} +27.6525 q^{86} -13.4164 q^{87} +40.8885 q^{88} +3.70820 q^{89} +8.47214 q^{91} -6.00000 q^{92} +4.14590 q^{93} -19.9443 q^{94} -24.2705 q^{96} +7.38197 q^{97} +28.6525 q^{98} +10.9443 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} + 3 q^{4} - 5 q^{6} - 4 q^{7} + 6 q^{8} + 4 q^{9} + 2 q^{11} - 15 q^{12} - 4 q^{13} - 11 q^{14} + 13 q^{16} - 3 q^{17} + 6 q^{18} + 10 q^{21} + 13 q^{22} + 2 q^{23} - 20 q^{24} - 6 q^{26}+ \cdots + 4 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.61803 1.85123 0.925615 − 0.378467i \(-0.123549\pi\)
0.925615 + 0.378467i \(0.123549\pi\)
\(3\) −2.23607 −1.29099 −0.645497 − 0.763763i \(-0.723350\pi\)
−0.645497 + 0.763763i \(0.723350\pi\)
\(4\) 4.85410 2.42705
\(5\) 0 0
\(6\) −5.85410 −2.38993
\(7\) −4.23607 −1.60108 −0.800542 − 0.599277i \(-0.795455\pi\)
−0.800542 + 0.599277i \(0.795455\pi\)
\(8\) 7.47214 2.64180
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) 5.47214 1.64991 0.824956 − 0.565198i \(-0.191200\pi\)
0.824956 + 0.565198i \(0.191200\pi\)
\(12\) −10.8541 −3.13331
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −11.0902 −2.96397
\(15\) 0 0
\(16\) 9.85410 2.46353
\(17\) −4.85410 −1.17729 −0.588646 − 0.808391i \(-0.700339\pi\)
−0.588646 + 0.808391i \(0.700339\pi\)
\(18\) 5.23607 1.23415
\(19\) 0 0
\(20\) 0 0
\(21\) 9.47214 2.06699
\(22\) 14.3262 3.05436
\(23\) −1.23607 −0.257738 −0.128869 − 0.991662i \(-0.541135\pi\)
−0.128869 + 0.991662i \(0.541135\pi\)
\(24\) −16.7082 −3.41055
\(25\) 0 0
\(26\) −5.23607 −1.02688
\(27\) 2.23607 0.430331
\(28\) −20.5623 −3.88591
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −1.85410 −0.333007 −0.166503 − 0.986041i \(-0.553248\pi\)
−0.166503 + 0.986041i \(0.553248\pi\)
\(32\) 10.8541 1.91875
\(33\) −12.2361 −2.13003
\(34\) −12.7082 −2.17944
\(35\) 0 0
\(36\) 9.70820 1.61803
\(37\) 4.14590 0.681581 0.340791 − 0.940139i \(-0.389305\pi\)
0.340791 + 0.940139i \(0.389305\pi\)
\(38\) 0 0
\(39\) 4.47214 0.716115
\(40\) 0 0
\(41\) −6.38197 −0.996696 −0.498348 − 0.866977i \(-0.666060\pi\)
−0.498348 + 0.866977i \(0.666060\pi\)
\(42\) 24.7984 3.82647
\(43\) 10.5623 1.61074 0.805368 − 0.592775i \(-0.201968\pi\)
0.805368 + 0.592775i \(0.201968\pi\)
\(44\) 26.5623 4.00442
\(45\) 0 0
\(46\) −3.23607 −0.477132
\(47\) −7.61803 −1.11120 −0.555602 − 0.831448i \(-0.687512\pi\)
−0.555602 + 0.831448i \(0.687512\pi\)
\(48\) −22.0344 −3.18040
\(49\) 10.9443 1.56347
\(50\) 0 0
\(51\) 10.8541 1.51988
\(52\) −9.70820 −1.34629
\(53\) −6.38197 −0.876630 −0.438315 − 0.898821i \(-0.644425\pi\)
−0.438315 + 0.898821i \(0.644425\pi\)
\(54\) 5.85410 0.796642
\(55\) 0 0
\(56\) −31.6525 −4.22974
\(57\) 0 0
\(58\) 15.7082 2.06259
\(59\) −2.76393 −0.359833 −0.179917 − 0.983682i \(-0.557583\pi\)
−0.179917 + 0.983682i \(0.557583\pi\)
\(60\) 0 0
\(61\) −5.56231 −0.712180 −0.356090 − 0.934452i \(-0.615890\pi\)
−0.356090 + 0.934452i \(0.615890\pi\)
\(62\) −4.85410 −0.616472
\(63\) −8.47214 −1.06739
\(64\) 8.70820 1.08853
\(65\) 0 0
\(66\) −32.0344 −3.94317
\(67\) −8.70820 −1.06388 −0.531938 − 0.846783i \(-0.678536\pi\)
−0.531938 + 0.846783i \(0.678536\pi\)
\(68\) −23.5623 −2.85735
\(69\) 2.76393 0.332738
\(70\) 0 0
\(71\) −4.52786 −0.537359 −0.268679 − 0.963230i \(-0.586587\pi\)
−0.268679 + 0.963230i \(0.586587\pi\)
\(72\) 14.9443 1.76120
\(73\) −10.7082 −1.25330 −0.626650 − 0.779301i \(-0.715575\pi\)
−0.626650 + 0.779301i \(0.715575\pi\)
\(74\) 10.8541 1.26176
\(75\) 0 0
\(76\) 0 0
\(77\) −23.1803 −2.64164
\(78\) 11.7082 1.32569
\(79\) 1.00000 0.112509 0.0562544 − 0.998416i \(-0.482084\pi\)
0.0562544 + 0.998416i \(0.482084\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) −16.7082 −1.84511
\(83\) 1.09017 0.119662 0.0598308 − 0.998209i \(-0.480944\pi\)
0.0598308 + 0.998209i \(0.480944\pi\)
\(84\) 45.9787 5.01669
\(85\) 0 0
\(86\) 27.6525 2.98184
\(87\) −13.4164 −1.43839
\(88\) 40.8885 4.35873
\(89\) 3.70820 0.393069 0.196534 − 0.980497i \(-0.437031\pi\)
0.196534 + 0.980497i \(0.437031\pi\)
\(90\) 0 0
\(91\) 8.47214 0.888121
\(92\) −6.00000 −0.625543
\(93\) 4.14590 0.429910
\(94\) −19.9443 −2.05709
\(95\) 0 0
\(96\) −24.2705 −2.47710
\(97\) 7.38197 0.749525 0.374763 − 0.927121i \(-0.377724\pi\)
0.374763 + 0.927121i \(0.377724\pi\)
\(98\) 28.6525 2.89434
\(99\) 10.9443 1.09994
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 9025.2.a.t.1.2 yes 2
5.4 even 2 9025.2.a.m.1.1 yes 2
19.18 odd 2 9025.2.a.l.1.1 ✓ 2
95.94 odd 2 9025.2.a.u.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9025.2.a.l.1.1 ✓ 2 19.18 odd 2
9025.2.a.m.1.1 yes 2 5.4 even 2
9025.2.a.t.1.2 yes 2 1.1 even 1 trivial
9025.2.a.u.1.2 yes 2 95.94 odd 2