Properties

Label 145.2.a.d.1.1
Level $145$
Weight $2$
Character 145.1
Self dual yes
Analytic conductor $1.158$
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] = [145,2,Mod(1,145)] 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("145.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(145, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 145 = 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 145.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(1.15783082931\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) 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 1.1
Root \(0.311108\) of defining polynomial
Character \(\chi\) \(=\) 145.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.21432 q^{2} -2.90321 q^{3} -0.525428 q^{4} -1.00000 q^{5} +3.52543 q^{6} +1.52543 q^{7} +3.06668 q^{8} +5.42864 q^{9} +1.21432 q^{10} +4.90321 q^{11} +1.52543 q^{12} -6.42864 q^{13} -1.85236 q^{14} +2.90321 q^{15} -2.67307 q^{16} +2.14764 q^{17} -6.59210 q^{18} +2.28100 q^{19} +0.525428 q^{20} -4.42864 q^{21} -5.95407 q^{22} +6.90321 q^{23} -8.90321 q^{24} +1.00000 q^{25} +7.80642 q^{26} -7.05086 q^{27} -0.801502 q^{28} +1.00000 q^{29} -3.52543 q^{30} +1.71900 q^{31} -2.88739 q^{32} -14.2351 q^{33} -2.60793 q^{34} -1.52543 q^{35} -2.85236 q^{36} +7.95407 q^{37} -2.76986 q^{38} +18.6637 q^{39} -3.06668 q^{40} -3.37778 q^{41} +5.37778 q^{42} -1.09679 q^{43} -2.57628 q^{44} -5.42864 q^{45} -8.38271 q^{46} +12.7096 q^{47} +7.76049 q^{48} -4.67307 q^{49} -1.21432 q^{50} -6.23506 q^{51} +3.37778 q^{52} +3.37778 q^{53} +8.56199 q^{54} -4.90321 q^{55} +4.67799 q^{56} -6.62222 q^{57} -1.21432 q^{58} -3.18421 q^{59} -1.52543 q^{60} -2.42864 q^{61} -2.08742 q^{62} +8.28100 q^{63} +8.85236 q^{64} +6.42864 q^{65} +17.2859 q^{66} -1.09679 q^{67} -1.12843 q^{68} -20.0415 q^{69} +1.85236 q^{70} +3.57136 q^{71} +16.6479 q^{72} +14.1891 q^{73} -9.65878 q^{74} -2.90321 q^{75} -1.19850 q^{76} +7.47949 q^{77} -22.6637 q^{78} +0.341219 q^{79} +2.67307 q^{80} +4.18421 q^{81} +4.10171 q^{82} -7.33185 q^{83} +2.32693 q^{84} -2.14764 q^{85} +1.33185 q^{86} -2.90321 q^{87} +15.0366 q^{88} +2.94914 q^{89} +6.59210 q^{90} -9.80642 q^{91} -3.62714 q^{92} -4.99063 q^{93} -15.4336 q^{94} -2.28100 q^{95} +8.38271 q^{96} -18.5763 q^{97} +5.67460 q^{98} +26.6178 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - 2 q^{3} + 5 q^{4} - 3 q^{5} + 4 q^{6} - 2 q^{7} + 9 q^{8} + 3 q^{9} - 3 q^{10} + 8 q^{11} - 2 q^{12} - 6 q^{13} - 12 q^{14} + 2 q^{15} + 5 q^{16} - 13 q^{18} - 5 q^{20} + 2 q^{22} + 14 q^{23}+ \cdots + 20 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.21432 −0.858654 −0.429327 − 0.903149i \(-0.641249\pi\)
−0.429327 + 0.903149i \(0.641249\pi\)
\(3\) −2.90321 −1.67617 −0.838085 − 0.545540i \(-0.816325\pi\)
−0.838085 + 0.545540i \(0.816325\pi\)
\(4\) −0.525428 −0.262714
\(5\) −1.00000 −0.447214
\(6\) 3.52543 1.43925
\(7\) 1.52543 0.576557 0.288279 − 0.957547i \(-0.406917\pi\)
0.288279 + 0.957547i \(0.406917\pi\)
\(8\) 3.06668 1.08423
\(9\) 5.42864 1.80955
\(10\) 1.21432 0.384002
\(11\) 4.90321 1.47837 0.739187 − 0.673500i \(-0.235210\pi\)
0.739187 + 0.673500i \(0.235210\pi\)
\(12\) 1.52543 0.440353
\(13\) −6.42864 −1.78298 −0.891492 − 0.453037i \(-0.850341\pi\)
−0.891492 + 0.453037i \(0.850341\pi\)
\(14\) −1.85236 −0.495063
\(15\) 2.90321 0.749606
\(16\) −2.67307 −0.668268
\(17\) 2.14764 0.520880 0.260440 − 0.965490i \(-0.416132\pi\)
0.260440 + 0.965490i \(0.416132\pi\)
\(18\) −6.59210 −1.55377
\(19\) 2.28100 0.523296 0.261648 − 0.965163i \(-0.415734\pi\)
0.261648 + 0.965163i \(0.415734\pi\)
\(20\) 0.525428 0.117489
\(21\) −4.42864 −0.966408
\(22\) −5.95407 −1.26941
\(23\) 6.90321 1.43942 0.719710 − 0.694275i \(-0.244275\pi\)
0.719710 + 0.694275i \(0.244275\pi\)
\(24\) −8.90321 −1.81736
\(25\) 1.00000 0.200000
\(26\) 7.80642 1.53097
\(27\) −7.05086 −1.35694
\(28\) −0.801502 −0.151470
\(29\) 1.00000 0.185695
\(30\) −3.52543 −0.643652
\(31\) 1.71900 0.308742 0.154371 − 0.988013i \(-0.450665\pi\)
0.154371 + 0.988013i \(0.450665\pi\)
\(32\) −2.88739 −0.510423
\(33\) −14.2351 −2.47801
\(34\) −2.60793 −0.447256
\(35\) −1.52543 −0.257844
\(36\) −2.85236 −0.475393
\(37\) 7.95407 1.30764 0.653820 − 0.756650i \(-0.273165\pi\)
0.653820 + 0.756650i \(0.273165\pi\)
\(38\) −2.76986 −0.449330
\(39\) 18.6637 2.98858
\(40\) −3.06668 −0.484884
\(41\) −3.37778 −0.527521 −0.263761 − 0.964588i \(-0.584963\pi\)
−0.263761 + 0.964588i \(0.584963\pi\)
\(42\) 5.37778 0.829810
\(43\) −1.09679 −0.167259 −0.0836293 − 0.996497i \(-0.526651\pi\)
−0.0836293 + 0.996497i \(0.526651\pi\)
\(44\) −2.57628 −0.388389
\(45\) −5.42864 −0.809254
\(46\) −8.38271 −1.23596
\(47\) 12.7096 1.85389 0.926945 − 0.375196i \(-0.122425\pi\)
0.926945 + 0.375196i \(0.122425\pi\)
\(48\) 7.76049 1.12013
\(49\) −4.67307 −0.667582
\(50\) −1.21432 −0.171731
\(51\) −6.23506 −0.873084
\(52\) 3.37778 0.468414
\(53\) 3.37778 0.463974 0.231987 − 0.972719i \(-0.425477\pi\)
0.231987 + 0.972719i \(0.425477\pi\)
\(54\) 8.56199 1.16514
\(55\) −4.90321 −0.661149
\(56\) 4.67799 0.625123
\(57\) −6.62222 −0.877134
\(58\) −1.21432 −0.159448
\(59\) −3.18421 −0.414549 −0.207274 − 0.978283i \(-0.566459\pi\)
−0.207274 + 0.978283i \(0.566459\pi\)
\(60\) −1.52543 −0.196932
\(61\) −2.42864 −0.310955 −0.155478 − 0.987839i \(-0.549692\pi\)
−0.155478 + 0.987839i \(0.549692\pi\)
\(62\) −2.08742 −0.265103
\(63\) 8.28100 1.04331
\(64\) 8.85236 1.10654
\(65\) 6.42864 0.797375
\(66\) 17.2859 2.12775
\(67\) −1.09679 −0.133994 −0.0669970 − 0.997753i \(-0.521342\pi\)
−0.0669970 + 0.997753i \(0.521342\pi\)
\(68\) −1.12843 −0.136842
\(69\) −20.0415 −2.41271
\(70\) 1.85236 0.221399
\(71\) 3.57136 0.423843 0.211921 − 0.977287i \(-0.432028\pi\)
0.211921 + 0.977287i \(0.432028\pi\)
\(72\) 16.6479 1.96197
\(73\) 14.1891 1.66071 0.830356 − 0.557233i \(-0.188137\pi\)
0.830356 + 0.557233i \(0.188137\pi\)
\(74\) −9.65878 −1.12281
\(75\) −2.90321 −0.335234
\(76\) −1.19850 −0.137477
\(77\) 7.47949 0.852368
\(78\) −22.6637 −2.56616
\(79\) 0.341219 0.0383902 0.0191951 − 0.999816i \(-0.493890\pi\)
0.0191951 + 0.999816i \(0.493890\pi\)
\(80\) 2.67307 0.298858
\(81\) 4.18421 0.464912
\(82\) 4.10171 0.452958
\(83\) −7.33185 −0.804775 −0.402388 − 0.915469i \(-0.631820\pi\)
−0.402388 + 0.915469i \(0.631820\pi\)
\(84\) 2.32693 0.253889
\(85\) −2.14764 −0.232945
\(86\) 1.33185 0.143617
\(87\) −2.90321 −0.311257
\(88\) 15.0366 1.60290
\(89\) 2.94914 0.312609 0.156304 − 0.987709i \(-0.450042\pi\)
0.156304 + 0.987709i \(0.450042\pi\)
\(90\) 6.59210 0.694869
\(91\) −9.80642 −1.02799
\(92\) −3.62714 −0.378155
\(93\) −4.99063 −0.517504
\(94\) −15.4336 −1.59185
\(95\) −2.28100 −0.234025
\(96\) 8.38271 0.855556
\(97\) −18.5763 −1.88614 −0.943068 − 0.332600i \(-0.892074\pi\)
−0.943068 + 0.332600i \(0.892074\pi\)
\(98\) 5.67460 0.573221
\(99\) 26.6178 2.67519
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 145.2.a.d.1.1 ✓ 3
3.2 odd 2 1305.2.a.o.1.3 3
4.3 odd 2 2320.2.a.s.1.3 3
5.2 odd 4 725.2.b.d.349.3 6
5.3 odd 4 725.2.b.d.349.4 6
5.4 even 2 725.2.a.d.1.3 3
7.6 odd 2 7105.2.a.p.1.1 3
8.3 odd 2 9280.2.a.bm.1.1 3
8.5 even 2 9280.2.a.bu.1.3 3
15.14 odd 2 6525.2.a.bh.1.1 3
29.28 even 2 4205.2.a.e.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.d.1.1 ✓ 3 1.1 even 1 trivial
725.2.a.d.1.3 3 5.4 even 2
725.2.b.d.349.3 6 5.2 odd 4
725.2.b.d.349.4 6 5.3 odd 4
1305.2.a.o.1.3 3 3.2 odd 2
2320.2.a.s.1.3 3 4.3 odd 2
4205.2.a.e.1.3 3 29.28 even 2
6525.2.a.bh.1.1 3 15.14 odd 2
7105.2.a.p.1.1 3 7.6 odd 2
9280.2.a.bm.1.1 3 8.3 odd 2
9280.2.a.bu.1.3 3 8.5 even 2