↑

Properties

Label 1665.2.a.p.1.4
Level $1665$
Weight $2$
Character 1665.1
Self dual yes
Analytic conductor $13.295$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1665,2,Mod(1,1665)] 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("1665.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1665, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1665 = 3^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1665.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,-2,0,10,5,0,11] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(13.2950919365\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.973904.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 8x^{3} + 6x^{2} + 19x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 185)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.10563\) of defining polynomial
Character \(\chi\) \(=\) 1665.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.13359 q^{2} -0.714970 q^{4} +1.00000 q^{5} +2.46164 q^{7} -3.07767 q^{8} +1.13359 q^{10} -1.71497 q^{11} +6.49255 q^{13} +2.79049 q^{14} -2.05888 q^{16} -3.32980 q^{17} +0.734568 q^{19} -0.714970 q^{20} -1.94408 q^{22} +2.08603 q^{23} +1.00000 q^{25} +7.35990 q^{26} -1.76000 q^{28} +4.21126 q^{29} +7.46459 q^{31} +3.82141 q^{32} -3.77463 q^{34} +2.46164 q^{35} +1.00000 q^{37} +0.832700 q^{38} -3.07767 q^{40} -1.71497 q^{41} +1.81885 q^{43} +1.22615 q^{44} +2.36471 q^{46} +0.882270 q^{47} -0.940340 q^{49} +1.13359 q^{50} -4.64198 q^{52} +7.03066 q^{53} -1.71497 q^{55} -7.57610 q^{56} +4.77385 q^{58} -0.387867 q^{59} -11.8224 q^{61} +8.46180 q^{62} +8.44967 q^{64} +6.49255 q^{65} +12.1086 q^{67} +2.38070 q^{68} +2.79049 q^{70} -13.7486 q^{71} +16.6719 q^{73} +1.13359 q^{74} -0.525194 q^{76} -4.22163 q^{77} +8.23253 q^{79} -2.05888 q^{80} -1.94408 q^{82} +4.80275 q^{83} -3.32980 q^{85} +2.06183 q^{86} +5.27811 q^{88} +1.52506 q^{89} +15.9823 q^{91} -1.49145 q^{92} +1.00013 q^{94} +0.734568 q^{95} -18.1588 q^{97} -1.06596 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{2} + 10 q^{4} + 5 q^{5} + 11 q^{7} - 6 q^{8} - 2 q^{10} + 5 q^{11} + 4 q^{13} + 8 q^{14} + 16 q^{16} - 4 q^{19} + 10 q^{20} - 8 q^{22} - 4 q^{23} + 5 q^{25} + 4 q^{26} + 28 q^{28} + 4 q^{29}+ \cdots + 18 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.13359 0.801570 0.400785 − 0.916172i \(-0.368737\pi\)
0.400785 + 0.916172i \(0.368737\pi\)
\(3\) 0 0
\(4\) −0.714970 −0.357485
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.46164 0.930412 0.465206 − 0.885203i \(-0.345980\pi\)
0.465206 + 0.885203i \(0.345980\pi\)
\(8\) −3.07767 −1.08812
\(9\) 0 0
\(10\) 1.13359 0.358473
\(11\) −1.71497 −0.517083 −0.258541 − 0.966000i \(-0.583242\pi\)
−0.258541 + 0.966000i \(0.583242\pi\)
\(12\) 0 0
\(13\) 6.49255 1.80071 0.900355 − 0.435156i \(-0.143307\pi\)
0.900355 + 0.435156i \(0.143307\pi\)
\(14\) 2.79049 0.745790
\(15\) 0 0
\(16\) −2.05888 −0.514719
\(17\) −3.32980 −0.807594 −0.403797 − 0.914849i \(-0.632310\pi\)
−0.403797 + 0.914849i \(0.632310\pi\)
\(18\) 0 0
\(19\) 0.734568 0.168521 0.0842607 − 0.996444i \(-0.473147\pi\)
0.0842607 + 0.996444i \(0.473147\pi\)
\(20\) −0.714970 −0.159872
\(21\) 0 0
\(22\) −1.94408 −0.414478
\(23\) 2.08603 0.434968 0.217484 − 0.976064i \(-0.430215\pi\)
0.217484 + 0.976064i \(0.430215\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 7.35990 1.44340
\(27\) 0 0
\(28\) −1.76000 −0.332608
\(29\) 4.21126 0.782011 0.391006 − 0.920388i \(-0.372127\pi\)
0.391006 + 0.920388i \(0.372127\pi\)
\(30\) 0 0
\(31\) 7.46459 1.34068 0.670340 − 0.742054i \(-0.266148\pi\)
0.670340 + 0.742054i \(0.266148\pi\)
\(32\) 3.82141 0.675536
\(33\) 0 0
\(34\) −3.77463 −0.647344
\(35\) 2.46164 0.416093
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 0.832700 0.135082
\(39\) 0 0
\(40\) −3.07767 −0.486622
\(41\) −1.71497 −0.267833 −0.133917 − 0.990993i \(-0.542755\pi\)
−0.133917 + 0.990993i \(0.542755\pi\)
\(42\) 0 0
\(43\) 1.81885 0.277372 0.138686 − 0.990336i \(-0.455712\pi\)
0.138686 + 0.990336i \(0.455712\pi\)
\(44\) 1.22615 0.184849
\(45\) 0 0
\(46\) 2.36471 0.348657
\(47\) 0.882270 0.128692 0.0643462 − 0.997928i \(-0.479504\pi\)
0.0643462 + 0.997928i \(0.479504\pi\)
\(48\) 0 0
\(49\) −0.940340 −0.134334
\(50\) 1.13359 0.160314
\(51\) 0 0
\(52\) −4.64198 −0.643727
\(53\) 7.03066 0.965735 0.482867 − 0.875693i \(-0.339595\pi\)
0.482867 + 0.875693i \(0.339595\pi\)
\(54\) 0 0
\(55\) −1.71497 −0.231247
\(56\) −7.57610 −1.01240
\(57\) 0 0
\(58\) 4.77385 0.626837
\(59\) −0.387867 −0.0504959 −0.0252480 − 0.999681i \(-0.508038\pi\)
−0.0252480 + 0.999681i \(0.508038\pi\)
\(60\) 0 0
\(61\) −11.8224 −1.51370 −0.756848 − 0.653590i \(-0.773262\pi\)
−0.756848 + 0.653590i \(0.773262\pi\)
\(62\) 8.46180 1.07465
\(63\) 0 0
\(64\) 8.44967 1.05621
\(65\) 6.49255 0.805302
\(66\) 0 0
\(67\) 12.1086 1.47930 0.739649 − 0.672992i \(-0.234991\pi\)
0.739649 + 0.672992i \(0.234991\pi\)
\(68\) 2.38070 0.288703
\(69\) 0 0
\(70\) 2.79049 0.333528
\(71\) −13.7486 −1.63166 −0.815828 − 0.578295i \(-0.803718\pi\)
−0.815828 + 0.578295i \(0.803718\pi\)
\(72\) 0 0
\(73\) 16.6719 1.95129 0.975646 − 0.219349i \(-0.0703933\pi\)
0.975646 + 0.219349i \(0.0703933\pi\)
\(74\) 1.13359 0.131777
\(75\) 0 0
\(76\) −0.525194 −0.0602439
\(77\) −4.22163 −0.481100
\(78\) 0 0
\(79\) 8.23253 0.926232 0.463116 − 0.886298i \(-0.346731\pi\)
0.463116 + 0.886298i \(0.346731\pi\)
\(80\) −2.05888 −0.230190
\(81\) 0 0
\(82\) −1.94408 −0.214687
\(83\) 4.80275 0.527171 0.263585 − 0.964636i \(-0.415095\pi\)
0.263585 + 0.964636i \(0.415095\pi\)
\(84\) 0 0
\(85\) −3.32980 −0.361167
\(86\) 2.06183 0.222333
\(87\) 0 0
\(88\) 5.27811 0.562648
\(89\) 1.52506 0.161656 0.0808280 − 0.996728i \(-0.474244\pi\)
0.0808280 + 0.996728i \(0.474244\pi\)
\(90\) 0 0
\(91\) 15.9823 1.67540
\(92\) −1.49145 −0.155494
\(93\) 0 0
\(94\) 1.00013 0.103156
\(95\) 0.734568 0.0753650
\(96\) 0 0
\(97\) −18.1588 −1.84375 −0.921875 − 0.387487i \(-0.873343\pi\)
−0.921875 + 0.387487i \(0.873343\pi\)
\(98\) −1.06596 −0.107678
\(99\) 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 1665.2.a.p.1.4 5
3.2 odd 2 185.2.a.e.1.2 ✓ 5
5.4 even 2 8325.2.a.ch.1.2 5
12.11 even 2 2960.2.a.w.1.4 5
15.2 even 4 925.2.b.f.149.4 10
15.8 even 4 925.2.b.f.149.7 10
15.14 odd 2 925.2.a.f.1.4 5
21.20 even 2 9065.2.a.k.1.2 5
111.110 odd 2 6845.2.a.f.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.a.e.1.2 ✓ 5 3.2 odd 2
925.2.a.f.1.4 5 15.14 odd 2
925.2.b.f.149.4 10 15.2 even 4
925.2.b.f.149.7 10 15.8 even 4
1665.2.a.p.1.4 5 1.1 even 1 trivial
2960.2.a.w.1.4 5 12.11 even 2
6845.2.a.f.1.4 5 111.110 odd 2
8325.2.a.ch.1.2 5 5.4 even 2
9065.2.a.k.1.2 5 21.20 even 2