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Properties

Label 9025.2.a.ba.1.1
Level $9025$
Weight $2$
Character 9025.1
Self dual yes
Analytic conductor $72.065$
Analytic rank $0$
Dimension $3$
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 = [3,1,1,7,0,-6,-2,-6,4,0,-5,-4,-15,-7] 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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.361.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x + 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.28514\) 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)
 
\(f(q)\) \(=\) \(q-2.50702 q^{2} +1.22188 q^{3} +4.28514 q^{4} -3.06327 q^{6} +0.221876 q^{7} -5.72889 q^{8} -1.50702 q^{9} -0.778124 q^{11} +5.23591 q^{12} -5.00000 q^{13} -0.556248 q^{14} +5.79216 q^{16} -7.07730 q^{17} +3.77812 q^{18} +0.271105 q^{21} +1.95077 q^{22} -8.07730 q^{23} -7.00000 q^{24} +12.5351 q^{26} -5.50702 q^{27} +0.950771 q^{28} -0.221876 q^{29} -2.50702 q^{31} -3.06327 q^{32} -0.950771 q^{33} +17.7429 q^{34} -6.45779 q^{36} -1.90466 q^{37} -6.10938 q^{39} +7.23591 q^{41} -0.679666 q^{42} -7.29918 q^{43} -3.33437 q^{44} +20.2500 q^{46} +2.79216 q^{47} +7.07730 q^{48} -6.95077 q^{49} -8.64759 q^{51} -21.4257 q^{52} +4.38049 q^{53} +13.8062 q^{54} -1.27111 q^{56} +0.556248 q^{58} +2.79216 q^{59} -12.5843 q^{61} +6.28514 q^{62} -0.334372 q^{63} -3.90466 q^{64} +2.38360 q^{66} -10.5703 q^{67} -30.3273 q^{68} -9.86946 q^{69} +9.84139 q^{71} +8.63355 q^{72} +14.0773 q^{73} +4.77501 q^{74} -0.172647 q^{77} +15.3163 q^{78} -1.58432 q^{79} -2.20784 q^{81} -18.1406 q^{82} +9.52106 q^{83} +1.16172 q^{84} +18.2992 q^{86} -0.271105 q^{87} +4.45779 q^{88} -3.14057 q^{89} -1.10938 q^{91} -34.6124 q^{92} -3.06327 q^{93} -7.00000 q^{94} -3.74293 q^{96} -6.36245 q^{97} +17.4257 q^{98} +1.17265 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + q^{3} + 7 q^{4} - 6 q^{6} - 2 q^{7} - 6 q^{8} + 4 q^{9} - 5 q^{11} - 4 q^{12} - 15 q^{13} - 7 q^{14} + 3 q^{16} - q^{17} + 14 q^{18} + 12 q^{21} - 8 q^{22} - 4 q^{23} - 21 q^{24} - 5 q^{26}+ \cdots - 13 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.50702 −1.77273 −0.886365 − 0.462987i \(-0.846778\pi\)
−0.886365 + 0.462987i \(0.846778\pi\)
\(3\) 1.22188 0.705451 0.352725 − 0.935727i \(-0.385255\pi\)
0.352725 + 0.935727i \(0.385255\pi\)
\(4\) 4.28514 2.14257
\(5\) 0 0
\(6\) −3.06327 −1.25057
\(7\) 0.221876 0.0838613 0.0419307 − 0.999121i \(-0.486649\pi\)
0.0419307 + 0.999121i \(0.486649\pi\)
\(8\) −5.72889 −2.02547
\(9\) −1.50702 −0.502340
\(10\) 0 0
\(11\) −0.778124 −0.234613 −0.117307 − 0.993096i \(-0.537426\pi\)
−0.117307 + 0.993096i \(0.537426\pi\)
\(12\) 5.23591 1.51148
\(13\) −5.00000 −1.38675 −0.693375 − 0.720577i \(-0.743877\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) −0.556248 −0.148663
\(15\) 0 0
\(16\) 5.79216 1.44804
\(17\) −7.07730 −1.71650 −0.858249 − 0.513233i \(-0.828448\pi\)
−0.858249 + 0.513233i \(0.828448\pi\)
\(18\) 3.77812 0.890512
\(19\) 0 0
\(20\) 0 0
\(21\) 0.271105 0.0591600
\(22\) 1.95077 0.415906
\(23\) −8.07730 −1.68423 −0.842117 − 0.539295i \(-0.818691\pi\)
−0.842117 + 0.539295i \(0.818691\pi\)
\(24\) −7.00000 −1.42887
\(25\) 0 0
\(26\) 12.5351 2.45833
\(27\) −5.50702 −1.05983
\(28\) 0.950771 0.179679
\(29\) −0.221876 −0.0412014 −0.0206007 − 0.999788i \(-0.506558\pi\)
−0.0206007 + 0.999788i \(0.506558\pi\)
\(30\) 0 0
\(31\) −2.50702 −0.450274 −0.225137 − 0.974327i \(-0.572283\pi\)
−0.225137 + 0.974327i \(0.572283\pi\)
\(32\) −3.06327 −0.541514
\(33\) −0.950771 −0.165508
\(34\) 17.7429 3.04289
\(35\) 0 0
\(36\) −6.45779 −1.07630
\(37\) −1.90466 −0.313124 −0.156562 − 0.987668i \(-0.550041\pi\)
−0.156562 + 0.987668i \(0.550041\pi\)
\(38\) 0 0
\(39\) −6.10938 −0.978284
\(40\) 0 0
\(41\) 7.23591 1.13006 0.565030 − 0.825070i \(-0.308865\pi\)
0.565030 + 0.825070i \(0.308865\pi\)
\(42\) −0.679666 −0.104875
\(43\) −7.29918 −1.11311 −0.556557 − 0.830809i \(-0.687878\pi\)
−0.556557 + 0.830809i \(0.687878\pi\)
\(44\) −3.33437 −0.502675
\(45\) 0 0
\(46\) 20.2500 2.98569
\(47\) 2.79216 0.407279 0.203639 − 0.979046i \(-0.434723\pi\)
0.203639 + 0.979046i \(0.434723\pi\)
\(48\) 7.07730 1.02152
\(49\) −6.95077 −0.992967
\(50\) 0 0
\(51\) −8.64759 −1.21090
\(52\) −21.4257 −2.97121
\(53\) 4.38049 0.601706 0.300853 − 0.953671i \(-0.402729\pi\)
0.300853 + 0.953671i \(0.402729\pi\)
\(54\) 13.8062 1.87879
\(55\) 0 0
\(56\) −1.27111 −0.169859
\(57\) 0 0
\(58\) 0.556248 0.0730389
\(59\) 2.79216 0.363508 0.181754 − 0.983344i \(-0.441822\pi\)
0.181754 + 0.983344i \(0.441822\pi\)
\(60\) 0 0
\(61\) −12.5843 −1.61126 −0.805629 − 0.592421i \(-0.798172\pi\)
−0.805629 + 0.592421i \(0.798172\pi\)
\(62\) 6.28514 0.798214
\(63\) −0.334372 −0.0421269
\(64\) −3.90466 −0.488082
\(65\) 0 0
\(66\) 2.38360 0.293401
\(67\) −10.5703 −1.29137 −0.645683 − 0.763606i \(-0.723427\pi\)
−0.645683 + 0.763606i \(0.723427\pi\)
\(68\) −30.3273 −3.67772
\(69\) −9.86946 −1.18814
\(70\) 0 0
\(71\) 9.84139 1.16796 0.583979 − 0.811769i \(-0.301495\pi\)
0.583979 + 0.811769i \(0.301495\pi\)
\(72\) 8.63355 1.01747
\(73\) 14.0773 1.64762 0.823812 − 0.566863i \(-0.191843\pi\)
0.823812 + 0.566863i \(0.191843\pi\)
\(74\) 4.77501 0.555084
\(75\) 0 0
\(76\) 0 0
\(77\) −0.172647 −0.0196750
\(78\) 15.3163 1.73423
\(79\) −1.58432 −0.178250 −0.0891251 − 0.996020i \(-0.528407\pi\)
−0.0891251 + 0.996020i \(0.528407\pi\)
\(80\) 0 0
\(81\) −2.20784 −0.245315
\(82\) −18.1406 −2.00329
\(83\) 9.52106 1.04507 0.522536 − 0.852617i \(-0.324986\pi\)
0.522536 + 0.852617i \(0.324986\pi\)
\(84\) 1.16172 0.126755
\(85\) 0 0
\(86\) 18.2992 1.97325
\(87\) −0.271105 −0.0290655
\(88\) 4.45779 0.475202
\(89\) −3.14057 −0.332900 −0.166450 − 0.986050i \(-0.553230\pi\)
−0.166450 + 0.986050i \(0.553230\pi\)
\(90\) 0 0
\(91\) −1.10938 −0.116295
\(92\) −34.6124 −3.60859
\(93\) −3.06327 −0.317646
\(94\) −7.00000 −0.721995
\(95\) 0 0
\(96\) −3.74293 −0.382011
\(97\) −6.36245 −0.646009 −0.323004 − 0.946398i \(-0.604693\pi\)
−0.323004 + 0.946398i \(0.604693\pi\)
\(98\) 17.4257 1.76026
\(99\) 1.17265 0.117855
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.ba.1.1 3
5.4 even 2 1805.2.a.g.1.3 3
19.8 odd 6 475.2.e.d.26.1 6
19.12 odd 6 475.2.e.d.201.1 6
19.18 odd 2 9025.2.a.z.1.3 3
95.8 even 12 475.2.j.b.349.1 12
95.12 even 12 475.2.j.b.49.1 12
95.27 even 12 475.2.j.b.349.6 12
95.69 odd 6 95.2.e.b.11.3 ✓ 6
95.84 odd 6 95.2.e.b.26.3 yes 6
95.88 even 12 475.2.j.b.49.6 12
95.94 odd 2 1805.2.a.h.1.1 3
285.164 even 6 855.2.k.g.676.1 6
285.179 even 6 855.2.k.g.406.1 6
380.179 even 6 1520.2.q.j.881.2 6
380.259 even 6 1520.2.q.j.961.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.e.b.11.3 ✓ 6 95.69 odd 6
95.2.e.b.26.3 yes 6 95.84 odd 6
475.2.e.d.26.1 6 19.8 odd 6
475.2.e.d.201.1 6 19.12 odd 6
475.2.j.b.49.1 12 95.12 even 12
475.2.j.b.49.6 12 95.88 even 12
475.2.j.b.349.1 12 95.8 even 12
475.2.j.b.349.6 12 95.27 even 12
855.2.k.g.406.1 6 285.179 even 6
855.2.k.g.676.1 6 285.164 even 6
1520.2.q.j.881.2 6 380.179 even 6
1520.2.q.j.961.2 6 380.259 even 6
1805.2.a.g.1.3 3 5.4 even 2
1805.2.a.h.1.1 3 95.94 odd 2
9025.2.a.z.1.3 3 19.18 odd 2
9025.2.a.ba.1.1 3 1.1 even 1 trivial