Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2325,2,Mod(1,2325)] 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("2325.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2325 = 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2325.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,-3,-3,5,0,3,-2,-9,3,0,2] 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: \(18.5652184699\)
Analytic rank: \(1\)
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: no (minimal twist has level 465)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.17009\) of defining polynomial
Character \(\chi\) \(=\) 2325.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.53919 q^{2} -1.00000 q^{3} +0.369102 q^{4} +1.53919 q^{6} -4.87936 q^{7} +2.51026 q^{8} +1.00000 q^{9} +4.34017 q^{11} -0.369102 q^{12} +2.53919 q^{13} +7.51026 q^{14} -4.60197 q^{16} -2.63090 q^{17} -1.53919 q^{18} -7.41855 q^{19} +4.87936 q^{21} -6.68035 q^{22} -2.29072 q^{23} -2.51026 q^{24} -3.90829 q^{26} -1.00000 q^{27} -1.80098 q^{28} +6.09171 q^{29} -1.00000 q^{31} +2.06278 q^{32} -4.34017 q^{33} +4.04945 q^{34} +0.369102 q^{36} +5.80098 q^{37} +11.4186 q^{38} -2.53919 q^{39} -0.183417 q^{41} -7.51026 q^{42} +6.49693 q^{43} +1.60197 q^{44} +3.52586 q^{46} +9.80817 q^{47} +4.60197 q^{48} +16.8082 q^{49} +2.63090 q^{51} +0.937221 q^{52} +1.86603 q^{53} +1.53919 q^{54} -12.2485 q^{56} +7.41855 q^{57} -9.37629 q^{58} -7.90829 q^{59} -8.15676 q^{61} +1.53919 q^{62} -4.87936 q^{63} +6.02893 q^{64} +6.68035 q^{66} +10.4813 q^{67} -0.971071 q^{68} +2.29072 q^{69} +3.17009 q^{71} +2.51026 q^{72} +15.5597 q^{73} -8.92881 q^{74} -2.73820 q^{76} -21.1773 q^{77} +3.90829 q^{78} -6.23287 q^{79} +1.00000 q^{81} +0.282314 q^{82} -6.38962 q^{83} +1.80098 q^{84} -10.0000 q^{86} -6.09171 q^{87} +10.8950 q^{88} -7.51026 q^{89} -12.3896 q^{91} -0.845512 q^{92} +1.00000 q^{93} -15.0966 q^{94} -2.06278 q^{96} -16.2823 q^{97} -25.8710 q^{98} +4.34017 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} - 3 q^{3} + 5 q^{4} + 3 q^{6} - 2 q^{7} - 9 q^{8} + 3 q^{9} + 2 q^{11} - 5 q^{12} + 6 q^{13} + 6 q^{14} + 5 q^{16} - 4 q^{17} - 3 q^{18} - 8 q^{19} + 2 q^{21} + 2 q^{22} - 14 q^{23} + 9 q^{24}+ \cdots + 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.53919 −1.08837 −0.544185 − 0.838965i \(-0.683161\pi\)
−0.544185 + 0.838965i \(0.683161\pi\)
\(3\) −1.00000 −0.577350
\(4\) 0.369102 0.184551
\(5\) 0 0
\(6\) 1.53919 0.628371
\(7\) −4.87936 −1.84423 −0.922113 − 0.386921i \(-0.873538\pi\)
−0.922113 + 0.386921i \(0.873538\pi\)
\(8\) 2.51026 0.887511
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.34017 1.30861 0.654306 − 0.756230i \(-0.272961\pi\)
0.654306 + 0.756230i \(0.272961\pi\)
\(12\) −0.369102 −0.106551
\(13\) 2.53919 0.704244 0.352122 − 0.935954i \(-0.385460\pi\)
0.352122 + 0.935954i \(0.385460\pi\)
\(14\) 7.51026 2.00720
\(15\) 0 0
\(16\) −4.60197 −1.15049
\(17\) −2.63090 −0.638086 −0.319043 − 0.947740i \(-0.603362\pi\)
−0.319043 + 0.947740i \(0.603362\pi\)
\(18\) −1.53919 −0.362790
\(19\) −7.41855 −1.70193 −0.850966 − 0.525221i \(-0.823983\pi\)
−0.850966 + 0.525221i \(0.823983\pi\)
\(20\) 0 0
\(21\) 4.87936 1.06476
\(22\) −6.68035 −1.42425
\(23\) −2.29072 −0.477649 −0.238825 − 0.971063i \(-0.576762\pi\)
−0.238825 + 0.971063i \(0.576762\pi\)
\(24\) −2.51026 −0.512405
\(25\) 0 0
\(26\) −3.90829 −0.766479
\(27\) −1.00000 −0.192450
\(28\) −1.80098 −0.340354
\(29\) 6.09171 1.13120 0.565601 − 0.824679i \(-0.308644\pi\)
0.565601 + 0.824679i \(0.308644\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 2.06278 0.364651
\(33\) −4.34017 −0.755527
\(34\) 4.04945 0.694475
\(35\) 0 0
\(36\) 0.369102 0.0615171
\(37\) 5.80098 0.953676 0.476838 − 0.878991i \(-0.341783\pi\)
0.476838 + 0.878991i \(0.341783\pi\)
\(38\) 11.4186 1.85233
\(39\) −2.53919 −0.406596
\(40\) 0 0
\(41\) −0.183417 −0.0286450 −0.0143225 − 0.999897i \(-0.504559\pi\)
−0.0143225 + 0.999897i \(0.504559\pi\)
\(42\) −7.51026 −1.15886
\(43\) 6.49693 0.990772 0.495386 − 0.868673i \(-0.335027\pi\)
0.495386 + 0.868673i \(0.335027\pi\)
\(44\) 1.60197 0.241506
\(45\) 0 0
\(46\) 3.52586 0.519859
\(47\) 9.80817 1.43067 0.715334 − 0.698782i \(-0.246274\pi\)
0.715334 + 0.698782i \(0.246274\pi\)
\(48\) 4.60197 0.664237
\(49\) 16.8082 2.40117
\(50\) 0 0
\(51\) 2.63090 0.368399
\(52\) 0.937221 0.129969
\(53\) 1.86603 0.256319 0.128160 − 0.991754i \(-0.459093\pi\)
0.128160 + 0.991754i \(0.459093\pi\)
\(54\) 1.53919 0.209457
\(55\) 0 0
\(56\) −12.2485 −1.63677
\(57\) 7.41855 0.982611
\(58\) −9.37629 −1.23117
\(59\) −7.90829 −1.02957 −0.514786 − 0.857319i \(-0.672129\pi\)
−0.514786 + 0.857319i \(0.672129\pi\)
\(60\) 0 0
\(61\) −8.15676 −1.04437 −0.522183 − 0.852834i \(-0.674882\pi\)
−0.522183 + 0.852834i \(0.674882\pi\)
\(62\) 1.53919 0.195477
\(63\) −4.87936 −0.614742
\(64\) 6.02893 0.753616
\(65\) 0 0
\(66\) 6.68035 0.822294
\(67\) 10.4813 1.28050 0.640249 − 0.768167i \(-0.278831\pi\)
0.640249 + 0.768167i \(0.278831\pi\)
\(68\) −0.971071 −0.117760
\(69\) 2.29072 0.275771
\(70\) 0 0
\(71\) 3.17009 0.376220 0.188110 − 0.982148i \(-0.439764\pi\)
0.188110 + 0.982148i \(0.439764\pi\)
\(72\) 2.51026 0.295837
\(73\) 15.5597 1.82113 0.910563 − 0.413370i \(-0.135648\pi\)
0.910563 + 0.413370i \(0.135648\pi\)
\(74\) −8.92881 −1.03795
\(75\) 0 0
\(76\) −2.73820 −0.314094
\(77\) −21.1773 −2.41337
\(78\) 3.90829 0.442527
\(79\) −6.23287 −0.701252 −0.350626 − 0.936516i \(-0.614031\pi\)
−0.350626 + 0.936516i \(0.614031\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0.282314 0.0311764
\(83\) −6.38962 −0.701352 −0.350676 − 0.936497i \(-0.614048\pi\)
−0.350676 + 0.936497i \(0.614048\pi\)
\(84\) 1.80098 0.196503
\(85\) 0 0
\(86\) −10.0000 −1.07833
\(87\) −6.09171 −0.653100
\(88\) 10.8950 1.16141
\(89\) −7.51026 −0.796086 −0.398043 − 0.917367i \(-0.630311\pi\)
−0.398043 + 0.917367i \(0.630311\pi\)
\(90\) 0 0
\(91\) −12.3896 −1.29879
\(92\) −0.845512 −0.0881507
\(93\) 1.00000 0.103695
\(94\) −15.0966 −1.55710
\(95\) 0 0
\(96\) −2.06278 −0.210532
\(97\) −16.2823 −1.65322 −0.826609 − 0.562776i \(-0.809733\pi\)
−0.826609 + 0.562776i \(0.809733\pi\)
\(98\) −25.8710 −2.61336
\(99\) 4.34017 0.436204
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 2325.2.a.p.1.2 3
3.2 odd 2 6975.2.a.bi.1.2 3
5.2 odd 4 2325.2.c.l.1024.2 6
5.3 odd 4 2325.2.c.l.1024.5 6
5.4 even 2 465.2.a.g.1.2 ✓ 3
15.14 odd 2 1395.2.a.h.1.2 3
20.19 odd 2 7440.2.a.bm.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.a.g.1.2 ✓ 3 5.4 even 2
1395.2.a.h.1.2 3 15.14 odd 2
2325.2.a.p.1.2 3 1.1 even 1 trivial
2325.2.c.l.1024.2 6 5.2 odd 4
2325.2.c.l.1024.5 6 5.3 odd 4
6975.2.a.bi.1.2 3 3.2 odd 2
7440.2.a.bm.1.1 3 20.19 odd 2