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Properties

Label 2475.2.a.t.1.2
Level $2475$
Weight $2$
Character 2475.1
Self dual yes
Analytic conductor $19.763$
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] = [2475,2,Mod(1,2475)] 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("2475.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2475, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2475.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,1,0,3,0,0,-5,6,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(19.7629745003\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 275)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.30278\) of defining polynomial
Character \(\chi\) \(=\) 2475.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.30278 q^{2} +3.30278 q^{4} -4.30278 q^{7} +3.00000 q^{8} +1.00000 q^{11} -5.00000 q^{13} -9.90833 q^{14} +0.302776 q^{16} -3.90833 q^{17} -1.00000 q^{19} +2.30278 q^{22} -3.69722 q^{23} -11.5139 q^{26} -14.2111 q^{28} +9.90833 q^{29} -4.21110 q^{31} -5.30278 q^{32} -9.00000 q^{34} -9.60555 q^{37} -2.30278 q^{38} -1.60555 q^{41} +7.21110 q^{43} +3.30278 q^{44} -8.51388 q^{46} -3.00000 q^{47} +11.5139 q^{49} -16.5139 q^{52} +2.30278 q^{53} -12.9083 q^{56} +22.8167 q^{58} -0.211103 q^{59} +2.90833 q^{61} -9.69722 q^{62} -12.8167 q^{64} +4.00000 q^{67} -12.9083 q^{68} -4.60555 q^{71} -2.90833 q^{73} -22.1194 q^{74} -3.30278 q^{76} -4.30278 q^{77} -0.0916731 q^{79} -3.69722 q^{82} +14.5139 q^{83} +16.6056 q^{86} +3.00000 q^{88} -5.30278 q^{89} +21.5139 q^{91} -12.2111 q^{92} -6.90833 q^{94} -11.6972 q^{97} +26.5139 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 3 q^{4} - 5 q^{7} + 6 q^{8} + 2 q^{11} - 10 q^{13} - 9 q^{14} - 3 q^{16} + 3 q^{17} - 2 q^{19} + q^{22} - 11 q^{23} - 5 q^{26} - 14 q^{28} + 9 q^{29} + 6 q^{31} - 7 q^{32} - 18 q^{34} - 12 q^{37}+ \cdots + 35 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.30278 1.62831 0.814154 − 0.580649i \(-0.197201\pi\)
0.814154 + 0.580649i \(0.197201\pi\)
\(3\) 0 0
\(4\) 3.30278 1.65139
\(5\) 0 0
\(6\) 0 0
\(7\) −4.30278 −1.62630 −0.813148 − 0.582057i \(-0.802248\pi\)
−0.813148 + 0.582057i \(0.802248\pi\)
\(8\) 3.00000 1.06066
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −5.00000 −1.38675 −0.693375 − 0.720577i \(-0.743877\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) −9.90833 −2.64811
\(15\) 0 0
\(16\) 0.302776 0.0756939
\(17\) −3.90833 −0.947909 −0.473954 − 0.880549i \(-0.657174\pi\)
−0.473954 + 0.880549i \(0.657174\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 − 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.30278 0.490953
\(23\) −3.69722 −0.770925 −0.385462 − 0.922724i \(-0.625958\pi\)
−0.385462 + 0.922724i \(0.625958\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −11.5139 −2.25806
\(27\) 0 0
\(28\) −14.2111 −2.68565
\(29\) 9.90833 1.83993 0.919965 − 0.392000i \(-0.128217\pi\)
0.919965 + 0.392000i \(0.128217\pi\)
\(30\) 0 0
\(31\) −4.21110 −0.756336 −0.378168 − 0.925737i \(-0.623446\pi\)
−0.378168 + 0.925737i \(0.623446\pi\)
\(32\) −5.30278 −0.937407
\(33\) 0 0
\(34\) −9.00000 −1.54349
\(35\) 0 0
\(36\) 0 0
\(37\) −9.60555 −1.57914 −0.789571 − 0.613659i \(-0.789697\pi\)
−0.789571 + 0.613659i \(0.789697\pi\)
\(38\) −2.30278 −0.373560
\(39\) 0 0
\(40\) 0 0
\(41\) −1.60555 −0.250745 −0.125372 − 0.992110i \(-0.540013\pi\)
−0.125372 + 0.992110i \(0.540013\pi\)
\(42\) 0 0
\(43\) 7.21110 1.09968 0.549841 − 0.835269i \(-0.314688\pi\)
0.549841 + 0.835269i \(0.314688\pi\)
\(44\) 3.30278 0.497912
\(45\) 0 0
\(46\) −8.51388 −1.25530
\(47\) −3.00000 −0.437595 −0.218797 − 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) 0 0
\(49\) 11.5139 1.64484
\(50\) 0 0
\(51\) 0 0
\(52\) −16.5139 −2.29006
\(53\) 2.30278 0.316311 0.158155 − 0.987414i \(-0.449445\pi\)
0.158155 + 0.987414i \(0.449445\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −12.9083 −1.72495
\(57\) 0 0
\(58\) 22.8167 2.99597
\(59\) −0.211103 −0.0274832 −0.0137416 − 0.999906i \(-0.504374\pi\)
−0.0137416 + 0.999906i \(0.504374\pi\)
\(60\) 0 0
\(61\) 2.90833 0.372373 0.186187 − 0.982514i \(-0.440387\pi\)
0.186187 + 0.982514i \(0.440387\pi\)
\(62\) −9.69722 −1.23155
\(63\) 0 0
\(64\) −12.8167 −1.60208
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 − 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) −12.9083 −1.56536
\(69\) 0 0
\(70\) 0 0
\(71\) −4.60555 −0.546578 −0.273289 − 0.961932i \(-0.588112\pi\)
−0.273289 + 0.961932i \(0.588112\pi\)
\(72\) 0 0
\(73\) −2.90833 −0.340394 −0.170197 − 0.985410i \(-0.554440\pi\)
−0.170197 + 0.985410i \(0.554440\pi\)
\(74\) −22.1194 −2.57133
\(75\) 0 0
\(76\) −3.30278 −0.378854
\(77\) −4.30278 −0.490347
\(78\) 0 0
\(79\) −0.0916731 −0.0103140 −0.00515701 − 0.999987i \(-0.501642\pi\)
−0.00515701 + 0.999987i \(0.501642\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −3.69722 −0.408290
\(83\) 14.5139 1.59311 0.796553 − 0.604569i \(-0.206655\pi\)
0.796553 + 0.604569i \(0.206655\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 16.6056 1.79062
\(87\) 0 0
\(88\) 3.00000 0.319801
\(89\) −5.30278 −0.562093 −0.281047 − 0.959694i \(-0.590682\pi\)
−0.281047 + 0.959694i \(0.590682\pi\)
\(90\) 0 0
\(91\) 21.5139 2.25527
\(92\) −12.2111 −1.27310
\(93\) 0 0
\(94\) −6.90833 −0.712540
\(95\) 0 0
\(96\) 0 0
\(97\) −11.6972 −1.18767 −0.593837 − 0.804586i \(-0.702387\pi\)
−0.593837 + 0.804586i \(0.702387\pi\)
\(98\) 26.5139 2.67831
\(99\) 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 2475.2.a.t.1.2 2
3.2 odd 2 275.2.a.e.1.1 ✓ 2
5.2 odd 4 2475.2.c.k.199.4 4
5.3 odd 4 2475.2.c.k.199.1 4
5.4 even 2 2475.2.a.o.1.1 2
12.11 even 2 4400.2.a.bs.1.1 2
15.2 even 4 275.2.b.c.199.1 4
15.8 even 4 275.2.b.c.199.4 4
15.14 odd 2 275.2.a.f.1.2 yes 2
33.32 even 2 3025.2.a.n.1.2 2
60.23 odd 4 4400.2.b.y.4049.2 4
60.47 odd 4 4400.2.b.y.4049.3 4
60.59 even 2 4400.2.a.bh.1.2 2
165.164 even 2 3025.2.a.h.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
275.2.a.e.1.1 ✓ 2 3.2 odd 2
275.2.a.f.1.2 yes 2 15.14 odd 2
275.2.b.c.199.1 4 15.2 even 4
275.2.b.c.199.4 4 15.8 even 4
2475.2.a.o.1.1 2 5.4 even 2
2475.2.a.t.1.2 2 1.1 even 1 trivial
2475.2.c.k.199.1 4 5.3 odd 4
2475.2.c.k.199.4 4 5.2 odd 4
3025.2.a.h.1.1 2 165.164 even 2
3025.2.a.n.1.2 2 33.32 even 2
4400.2.a.bh.1.2 2 60.59 even 2
4400.2.a.bs.1.1 2 12.11 even 2
4400.2.b.y.4049.2 4 60.23 odd 4
4400.2.b.y.4049.3 4 60.47 odd 4