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Properties

Label 2325.2.a.n.1.2
Level $2325$
Weight $2$
Character 2325.1
Self dual yes
Analytic conductor $18.565$
Analytic rank $0$
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] = [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 = [2,2,-2,2,0,-2,4,6,2,0,0] 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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) 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 \(1.41421\) 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)
 
\(f(q)\) \(=\) \(q+2.41421 q^{2} -1.00000 q^{3} +3.82843 q^{4} -2.41421 q^{6} +0.585786 q^{7} +4.41421 q^{8} +1.00000 q^{9} -2.82843 q^{11} -3.82843 q^{12} +2.58579 q^{13} +1.41421 q^{14} +3.00000 q^{16} +4.00000 q^{17} +2.41421 q^{18} +2.82843 q^{19} -0.585786 q^{21} -6.82843 q^{22} +6.00000 q^{23} -4.41421 q^{24} +6.24264 q^{26} -1.00000 q^{27} +2.24264 q^{28} +2.24264 q^{29} +1.00000 q^{31} -1.58579 q^{32} +2.82843 q^{33} +9.65685 q^{34} +3.82843 q^{36} +1.41421 q^{37} +6.82843 q^{38} -2.58579 q^{39} -0.828427 q^{41} -1.41421 q^{42} +11.3137 q^{43} -10.8284 q^{44} +14.4853 q^{46} +4.82843 q^{47} -3.00000 q^{48} -6.65685 q^{49} -4.00000 q^{51} +9.89949 q^{52} +4.00000 q^{53} -2.41421 q^{54} +2.58579 q^{56} -2.82843 q^{57} +5.41421 q^{58} -0.242641 q^{59} -10.4853 q^{61} +2.41421 q^{62} +0.585786 q^{63} -9.82843 q^{64} +6.82843 q^{66} -3.89949 q^{67} +15.3137 q^{68} -6.00000 q^{69} +9.89949 q^{71} +4.41421 q^{72} +5.89949 q^{73} +3.41421 q^{74} +10.8284 q^{76} -1.65685 q^{77} -6.24264 q^{78} -14.4853 q^{79} +1.00000 q^{81} -2.00000 q^{82} +0.343146 q^{83} -2.24264 q^{84} +27.3137 q^{86} -2.24264 q^{87} -12.4853 q^{88} +5.07107 q^{89} +1.51472 q^{91} +22.9706 q^{92} -1.00000 q^{93} +11.6569 q^{94} +1.58579 q^{96} -15.6569 q^{97} -16.0711 q^{98} -2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} + 4 q^{7} + 6 q^{8} + 2 q^{9} - 2 q^{12} + 8 q^{13} + 6 q^{16} + 8 q^{17} + 2 q^{18} - 4 q^{21} - 8 q^{22} + 12 q^{23} - 6 q^{24} + 4 q^{26} - 2 q^{27} - 4 q^{28}+ \cdots - 18 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.41421 1.70711 0.853553 − 0.521005i \(-0.174443\pi\)
0.853553 + 0.521005i \(0.174443\pi\)
\(3\) −1.00000 −0.577350
\(4\) 3.82843 1.91421
\(5\) 0 0
\(6\) −2.41421 −0.985599
\(7\) 0.585786 0.221406 0.110703 − 0.993854i \(-0.464690\pi\)
0.110703 + 0.993854i \(0.464690\pi\)
\(8\) 4.41421 1.56066
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.82843 −0.852803 −0.426401 − 0.904534i \(-0.640219\pi\)
−0.426401 + 0.904534i \(0.640219\pi\)
\(12\) −3.82843 −1.10517
\(13\) 2.58579 0.717168 0.358584 − 0.933497i \(-0.383260\pi\)
0.358584 + 0.933497i \(0.383260\pi\)
\(14\) 1.41421 0.377964
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) 4.00000 0.970143 0.485071 − 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 2.41421 0.569036
\(19\) 2.82843 0.648886 0.324443 − 0.945905i \(-0.394823\pi\)
0.324443 + 0.945905i \(0.394823\pi\)
\(20\) 0 0
\(21\) −0.585786 −0.127829
\(22\) −6.82843 −1.45583
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) −4.41421 −0.901048
\(25\) 0 0
\(26\) 6.24264 1.22428
\(27\) −1.00000 −0.192450
\(28\) 2.24264 0.423819
\(29\) 2.24264 0.416448 0.208224 − 0.978081i \(-0.433232\pi\)
0.208224 + 0.978081i \(0.433232\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) −1.58579 −0.280330
\(33\) 2.82843 0.492366
\(34\) 9.65685 1.65614
\(35\) 0 0
\(36\) 3.82843 0.638071
\(37\) 1.41421 0.232495 0.116248 − 0.993220i \(-0.462913\pi\)
0.116248 + 0.993220i \(0.462913\pi\)
\(38\) 6.82843 1.10772
\(39\) −2.58579 −0.414057
\(40\) 0 0
\(41\) −0.828427 −0.129379 −0.0646893 − 0.997905i \(-0.520606\pi\)
−0.0646893 + 0.997905i \(0.520606\pi\)
\(42\) −1.41421 −0.218218
\(43\) 11.3137 1.72532 0.862662 − 0.505781i \(-0.168795\pi\)
0.862662 + 0.505781i \(0.168795\pi\)
\(44\) −10.8284 −1.63245
\(45\) 0 0
\(46\) 14.4853 2.13574
\(47\) 4.82843 0.704298 0.352149 − 0.935944i \(-0.385451\pi\)
0.352149 + 0.935944i \(0.385451\pi\)
\(48\) −3.00000 −0.433013
\(49\) −6.65685 −0.950979
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) 9.89949 1.37281
\(53\) 4.00000 0.549442 0.274721 − 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) −2.41421 −0.328533
\(55\) 0 0
\(56\) 2.58579 0.345540
\(57\) −2.82843 −0.374634
\(58\) 5.41421 0.710921
\(59\) −0.242641 −0.0315891 −0.0157946 − 0.999875i \(-0.505028\pi\)
−0.0157946 + 0.999875i \(0.505028\pi\)
\(60\) 0 0
\(61\) −10.4853 −1.34250 −0.671251 − 0.741230i \(-0.734243\pi\)
−0.671251 + 0.741230i \(0.734243\pi\)
\(62\) 2.41421 0.306605
\(63\) 0.585786 0.0738022
\(64\) −9.82843 −1.22855
\(65\) 0 0
\(66\) 6.82843 0.840521
\(67\) −3.89949 −0.476399 −0.238200 − 0.971216i \(-0.576557\pi\)
−0.238200 + 0.971216i \(0.576557\pi\)
\(68\) 15.3137 1.85706
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) 9.89949 1.17485 0.587427 − 0.809277i \(-0.300141\pi\)
0.587427 + 0.809277i \(0.300141\pi\)
\(72\) 4.41421 0.520220
\(73\) 5.89949 0.690484 0.345242 − 0.938514i \(-0.387797\pi\)
0.345242 + 0.938514i \(0.387797\pi\)
\(74\) 3.41421 0.396894
\(75\) 0 0
\(76\) 10.8284 1.24211
\(77\) −1.65685 −0.188816
\(78\) −6.24264 −0.706840
\(79\) −14.4853 −1.62972 −0.814861 − 0.579657i \(-0.803187\pi\)
−0.814861 + 0.579657i \(0.803187\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) 0.343146 0.0376651 0.0188326 − 0.999823i \(-0.494005\pi\)
0.0188326 + 0.999823i \(0.494005\pi\)
\(84\) −2.24264 −0.244692
\(85\) 0 0
\(86\) 27.3137 2.94531
\(87\) −2.24264 −0.240436
\(88\) −12.4853 −1.33094
\(89\) 5.07107 0.537532 0.268766 − 0.963205i \(-0.413384\pi\)
0.268766 + 0.963205i \(0.413384\pi\)
\(90\) 0 0
\(91\) 1.51472 0.158786
\(92\) 22.9706 2.39485
\(93\) −1.00000 −0.103695
\(94\) 11.6569 1.20231
\(95\) 0 0
\(96\) 1.58579 0.161849
\(97\) −15.6569 −1.58971 −0.794856 − 0.606798i \(-0.792454\pi\)
−0.794856 + 0.606798i \(0.792454\pi\)
\(98\) −16.0711 −1.62342
\(99\) −2.82843 −0.284268
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.n.1.2 2
3.2 odd 2 6975.2.a.u.1.1 2
5.2 odd 4 2325.2.c.i.1024.4 4
5.3 odd 4 2325.2.c.i.1024.1 4
5.4 even 2 465.2.a.c.1.1 ✓ 2
15.14 odd 2 1395.2.a.g.1.2 2
20.19 odd 2 7440.2.a.be.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.a.c.1.1 ✓ 2 5.4 even 2
1395.2.a.g.1.2 2 15.14 odd 2
2325.2.a.n.1.2 2 1.1 even 1 trivial
2325.2.c.i.1024.1 4 5.3 odd 4
2325.2.c.i.1024.4 4 5.2 odd 4
6975.2.a.u.1.1 2 3.2 odd 2
7440.2.a.be.1.1 2 20.19 odd 2