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Properties

Label 3025.2.a.g.1.1
Level $3025$
Weight $2$
Character 3025.1
Self dual yes
Analytic conductor $24.155$
Analytic rank $0$
Dimension $1$
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] = [3025,2,Mod(1,3025)] 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("3025.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3025, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3025 = 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3025.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,1,3,-1,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(24.1547466114\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 605)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3025.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +3.00000 q^{3} -1.00000 q^{4} +3.00000 q^{6} +3.00000 q^{7} -3.00000 q^{8} +6.00000 q^{9} -3.00000 q^{12} -4.00000 q^{13} +3.00000 q^{14} -1.00000 q^{16} +6.00000 q^{18} +4.00000 q^{19} +9.00000 q^{21} +8.00000 q^{23} -9.00000 q^{24} -4.00000 q^{26} +9.00000 q^{27} -3.00000 q^{28} +6.00000 q^{29} -2.00000 q^{31} +5.00000 q^{32} -6.00000 q^{36} +8.00000 q^{37} +4.00000 q^{38} -12.0000 q^{39} -5.00000 q^{41} +9.00000 q^{42} -5.00000 q^{43} +8.00000 q^{46} +3.00000 q^{47} -3.00000 q^{48} +2.00000 q^{49} +4.00000 q^{52} -4.00000 q^{53} +9.00000 q^{54} -9.00000 q^{56} +12.0000 q^{57} +6.00000 q^{58} -2.00000 q^{59} -11.0000 q^{61} -2.00000 q^{62} +18.0000 q^{63} +7.00000 q^{64} +13.0000 q^{67} +24.0000 q^{69} +2.00000 q^{71} -18.0000 q^{72} +8.00000 q^{73} +8.00000 q^{74} -4.00000 q^{76} -12.0000 q^{78} +10.0000 q^{79} +9.00000 q^{81} -5.00000 q^{82} -4.00000 q^{83} -9.00000 q^{84} -5.00000 q^{86} +18.0000 q^{87} +1.00000 q^{89} -12.0000 q^{91} -8.00000 q^{92} -6.00000 q^{93} +3.00000 q^{94} +15.0000 q^{96} +8.00000 q^{97} +2.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107 0.353553 − 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) 3.00000 1.73205 0.866025 − 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 3.00000 1.22474
\(7\) 3.00000 1.13389 0.566947 − 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) −3.00000 −1.06066
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) 0 0
\(12\) −3.00000 −0.866025
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 3.00000 0.801784
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 6.00000 1.41421
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 9.00000 1.96396
\(22\) 0 0
\(23\) 8.00000 1.66812 0.834058 − 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) −9.00000 −1.83712
\(25\) 0 0
\(26\) −4.00000 −0.784465
\(27\) 9.00000 1.73205
\(28\) −3.00000 −0.566947
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 − 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 5.00000 0.883883
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −6.00000 −1.00000
\(37\) 8.00000 1.31519 0.657596 − 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 4.00000 0.648886
\(39\) −12.0000 −1.92154
\(40\) 0 0
\(41\) −5.00000 −0.780869 −0.390434 − 0.920631i \(-0.627675\pi\)
−0.390434 + 0.920631i \(0.627675\pi\)
\(42\) 9.00000 1.38873
\(43\) −5.00000 −0.762493 −0.381246 − 0.924473i \(-0.624505\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) 3.00000 0.437595 0.218797 − 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) −3.00000 −0.433013
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) 0 0
\(52\) 4.00000 0.554700
\(53\) −4.00000 −0.549442 −0.274721 − 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 9.00000 1.22474
\(55\) 0 0
\(56\) −9.00000 −1.20268
\(57\) 12.0000 1.58944
\(58\) 6.00000 0.787839
\(59\) −2.00000 −0.260378 −0.130189 − 0.991489i \(-0.541558\pi\)
−0.130189 + 0.991489i \(0.541558\pi\)
\(60\) 0 0
\(61\) −11.0000 −1.40841 −0.704203 − 0.709999i \(-0.748695\pi\)
−0.704203 + 0.709999i \(0.748695\pi\)
\(62\) −2.00000 −0.254000
\(63\) 18.0000 2.26779
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) 0 0
\(67\) 13.0000 1.58820 0.794101 − 0.607785i \(-0.207942\pi\)
0.794101 + 0.607785i \(0.207942\pi\)
\(68\) 0 0
\(69\) 24.0000 2.88926
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 − 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) −18.0000 −2.12132
\(73\) 8.00000 0.936329 0.468165 − 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) 8.00000 0.929981
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) −12.0000 −1.35873
\(79\) 10.0000 1.12509 0.562544 − 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) −5.00000 −0.552158
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) −9.00000 −0.981981
\(85\) 0 0
\(86\) −5.00000 −0.539164
\(87\) 18.0000 1.92980
\(88\) 0 0
\(89\) 1.00000 0.106000 0.0529999 − 0.998595i \(-0.483122\pi\)
0.0529999 + 0.998595i \(0.483122\pi\)
\(90\) 0 0
\(91\) −12.0000 −1.25794
\(92\) −8.00000 −0.834058
\(93\) −6.00000 −0.622171
\(94\) 3.00000 0.309426
\(95\) 0 0
\(96\) 15.0000 1.53093
\(97\) 8.00000 0.812277 0.406138 − 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 2.00000 0.202031
\(99\) 0 0
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 3025.2.a.g.1.1 1
5.4 even 2 605.2.a.a.1.1 ✓ 1
11.10 odd 2 3025.2.a.c.1.1 1
15.14 odd 2 5445.2.a.h.1.1 1
20.19 odd 2 9680.2.a.bf.1.1 1
55.4 even 10 605.2.g.d.511.1 4
55.9 even 10 605.2.g.d.81.1 4
55.14 even 10 605.2.g.d.251.1 4
55.19 odd 10 605.2.g.b.251.1 4
55.24 odd 10 605.2.g.b.81.1 4
55.29 odd 10 605.2.g.b.511.1 4
55.39 odd 10 605.2.g.b.366.1 4
55.49 even 10 605.2.g.d.366.1 4
55.54 odd 2 605.2.a.c.1.1 yes 1
165.164 even 2 5445.2.a.d.1.1 1
220.219 even 2 9680.2.a.be.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
605.2.a.a.1.1 ✓ 1 5.4 even 2
605.2.a.c.1.1 yes 1 55.54 odd 2
605.2.g.b.81.1 4 55.24 odd 10
605.2.g.b.251.1 4 55.19 odd 10
605.2.g.b.366.1 4 55.39 odd 10
605.2.g.b.511.1 4 55.29 odd 10
605.2.g.d.81.1 4 55.9 even 10
605.2.g.d.251.1 4 55.14 even 10
605.2.g.d.366.1 4 55.49 even 10
605.2.g.d.511.1 4 55.4 even 10
3025.2.a.c.1.1 1 11.10 odd 2
3025.2.a.g.1.1 1 1.1 even 1 trivial
5445.2.a.d.1.1 1 165.164 even 2
5445.2.a.h.1.1 1 15.14 odd 2
9680.2.a.be.1.1 1 220.219 even 2
9680.2.a.bf.1.1 1 20.19 odd 2