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Properties

Label 3025.2.a.h.1.1
Level $3025$
Weight $2$
Character 3025.1
Self dual yes
Analytic conductor $24.155$
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] = [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 = [2,-1,1,3,0,6] 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: \(1\)
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 1.1
Root \(2.30278\) of defining polynomial
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)
 
\(f(q)\) \(=\) \(q-2.30278 q^{2} -1.30278 q^{3} +3.30278 q^{4} +3.00000 q^{6} -4.30278 q^{7} -3.00000 q^{8} -1.30278 q^{9} -4.30278 q^{12} -5.00000 q^{13} +9.90833 q^{14} +0.302776 q^{16} +3.90833 q^{17} +3.00000 q^{18} +1.00000 q^{19} +5.60555 q^{21} -3.69722 q^{23} +3.90833 q^{24} +11.5139 q^{26} +5.60555 q^{27} -14.2111 q^{28} +9.90833 q^{29} -4.21110 q^{31} +5.30278 q^{32} -9.00000 q^{34} -4.30278 q^{36} +9.60555 q^{37} -2.30278 q^{38} +6.51388 q^{39} -1.60555 q^{41} -12.9083 q^{42} +7.21110 q^{43} +8.51388 q^{46} -3.00000 q^{47} -0.394449 q^{48} +11.5139 q^{49} -5.09167 q^{51} -16.5139 q^{52} +2.30278 q^{53} -12.9083 q^{54} +12.9083 q^{56} -1.30278 q^{57} -22.8167 q^{58} +0.211103 q^{59} -2.90833 q^{61} +9.69722 q^{62} +5.60555 q^{63} -12.8167 q^{64} -4.00000 q^{67} +12.9083 q^{68} +4.81665 q^{69} +4.60555 q^{71} +3.90833 q^{72} -2.90833 q^{73} -22.1194 q^{74} +3.30278 q^{76} -15.0000 q^{78} +0.0916731 q^{79} -3.39445 q^{81} +3.69722 q^{82} -14.5139 q^{83} +18.5139 q^{84} -16.6056 q^{86} -12.9083 q^{87} +5.30278 q^{89} +21.5139 q^{91} -12.2111 q^{92} +5.48612 q^{93} +6.90833 q^{94} -6.90833 q^{96} +11.6972 q^{97} -26.5139 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + q^{3} + 3 q^{4} + 6 q^{6} - 5 q^{7} - 6 q^{8} + q^{9} - 5 q^{12} - 10 q^{13} + 9 q^{14} - 3 q^{16} - 3 q^{17} + 6 q^{18} + 2 q^{19} + 4 q^{21} - 11 q^{23} - 3 q^{24} + 5 q^{26} + 4 q^{27}+ \cdots - 35 q^{98}+O(q^{100}) \) 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 \(-0.802799\pi\)
−0.814154 + 0.580649i \(0.802799\pi\)
\(3\) −1.30278 −0.752158 −0.376079 − 0.926588i \(-0.622728\pi\)
−0.376079 + 0.926588i \(0.622728\pi\)
\(4\) 3.30278 1.65139
\(5\) 0 0
\(6\) 3.00000 1.22474
\(7\) −4.30278 −1.62630 −0.813148 − 0.582057i \(-0.802248\pi\)
−0.813148 + 0.582057i \(0.802248\pi\)
\(8\) −3.00000 −1.06066
\(9\) −1.30278 −0.434259
\(10\) 0 0
\(11\) 0 0
\(12\) −4.30278 −1.24210
\(13\) −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 \(-0.342826\pi\)
0.473954 + 0.880549i \(0.342826\pi\)
\(18\) 3.00000 0.707107
\(19\) 1.00000 0.229416 0.114708 − 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) 5.60555 1.22323
\(22\) 0 0
\(23\) −3.69722 −0.770925 −0.385462 − 0.922724i \(-0.625958\pi\)
−0.385462 + 0.922724i \(0.625958\pi\)
\(24\) 3.90833 0.797784
\(25\) 0 0
\(26\) 11.5139 2.25806
\(27\) 5.60555 1.07879
\(28\) −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
\(36\) −4.30278 −0.717129
\(37\) 9.60555 1.57914 0.789571 − 0.613659i \(-0.210303\pi\)
0.789571 + 0.613659i \(0.210303\pi\)
\(38\) −2.30278 −0.373560
\(39\) 6.51388 1.04306
\(40\) 0 0
\(41\) −1.60555 −0.250745 −0.125372 − 0.992110i \(-0.540013\pi\)
−0.125372 + 0.992110i \(0.540013\pi\)
\(42\) −12.9083 −1.99180
\(43\) 7.21110 1.09968 0.549841 − 0.835269i \(-0.314688\pi\)
0.549841 + 0.835269i \(0.314688\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 8.51388 1.25530
\(47\) −3.00000 −0.437595 −0.218797 − 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) −0.394449 −0.0569338
\(49\) 11.5139 1.64484
\(50\) 0 0
\(51\) −5.09167 −0.712977
\(52\) −16.5139 −2.29006
\(53\) 2.30278 0.316311 0.158155 − 0.987414i \(-0.449445\pi\)
0.158155 + 0.987414i \(0.449445\pi\)
\(54\) −12.9083 −1.75660
\(55\) 0 0
\(56\) 12.9083 1.72495
\(57\) −1.30278 −0.172557
\(58\) −22.8167 −2.99597
\(59\) 0.211103 0.0274832 0.0137416 − 0.999906i \(-0.495626\pi\)
0.0137416 + 0.999906i \(0.495626\pi\)
\(60\) 0 0
\(61\) −2.90833 −0.372373 −0.186187 − 0.982514i \(-0.559613\pi\)
−0.186187 + 0.982514i \(0.559613\pi\)
\(62\) 9.69722 1.23155
\(63\) 5.60555 0.706233
\(64\) −12.8167 −1.60208
\(65\) 0 0
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 12.9083 1.56536
\(69\) 4.81665 0.579857
\(70\) 0 0
\(71\) 4.60555 0.546578 0.273289 − 0.961932i \(-0.411888\pi\)
0.273289 + 0.961932i \(0.411888\pi\)
\(72\) 3.90833 0.460601
\(73\) −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
\(77\) 0 0
\(78\) −15.0000 −1.69842
\(79\) 0.0916731 0.0103140 0.00515701 − 0.999987i \(-0.498358\pi\)
0.00515701 + 0.999987i \(0.498358\pi\)
\(80\) 0 0
\(81\) −3.39445 −0.377161
\(82\) 3.69722 0.408290
\(83\) −14.5139 −1.59311 −0.796553 − 0.604569i \(-0.793345\pi\)
−0.796553 + 0.604569i \(0.793345\pi\)
\(84\) 18.5139 2.02003
\(85\) 0 0
\(86\) −16.6056 −1.79062
\(87\) −12.9083 −1.38392
\(88\) 0 0
\(89\) 5.30278 0.562093 0.281047 − 0.959694i \(-0.409318\pi\)
0.281047 + 0.959694i \(0.409318\pi\)
\(90\) 0 0
\(91\) 21.5139 2.25527
\(92\) −12.2111 −1.27310
\(93\) 5.48612 0.568884
\(94\) 6.90833 0.712540
\(95\) 0 0
\(96\) −6.90833 −0.705078
\(97\) 11.6972 1.18767 0.593837 − 0.804586i \(-0.297613\pi\)
0.593837 + 0.804586i \(0.297613\pi\)
\(98\) −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 3025.2.a.h.1.1 2
5.4 even 2 3025.2.a.n.1.2 2
11.10 odd 2 275.2.a.f.1.2 yes 2
33.32 even 2 2475.2.a.o.1.1 2
44.43 even 2 4400.2.a.bh.1.2 2
55.32 even 4 275.2.b.c.199.4 4
55.43 even 4 275.2.b.c.199.1 4
55.54 odd 2 275.2.a.e.1.1 ✓ 2
165.32 odd 4 2475.2.c.k.199.1 4
165.98 odd 4 2475.2.c.k.199.4 4
165.164 even 2 2475.2.a.t.1.2 2
220.43 odd 4 4400.2.b.y.4049.3 4
220.87 odd 4 4400.2.b.y.4049.2 4
220.219 even 2 4400.2.a.bs.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
275.2.a.e.1.1 ✓ 2 55.54 odd 2
275.2.a.f.1.2 yes 2 11.10 odd 2
275.2.b.c.199.1 4 55.43 even 4
275.2.b.c.199.4 4 55.32 even 4
2475.2.a.o.1.1 2 33.32 even 2
2475.2.a.t.1.2 2 165.164 even 2
2475.2.c.k.199.1 4 165.32 odd 4
2475.2.c.k.199.4 4 165.98 odd 4
3025.2.a.h.1.1 2 1.1 even 1 trivial
3025.2.a.n.1.2 2 5.4 even 2
4400.2.a.bh.1.2 2 44.43 even 2
4400.2.a.bs.1.1 2 220.219 even 2
4400.2.b.y.4049.2 4 220.87 odd 4
4400.2.b.y.4049.3 4 220.43 odd 4