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Properties

Label 2325.2.c.i.1024.4
Level $2325$
Weight $2$
Character 2325.1024
Analytic conductor $18.565$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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(1024,2325)] 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("2325.1024"); 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, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2325 = 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2325.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-4,0,-4,0,0,-4,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.5652184699\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 465)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1024.4
Root \(0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 2325.1024
Dual form 2325.2.c.i.1024.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.41421i q^{2} +1.00000i q^{3} -3.82843 q^{4} -2.41421 q^{6} +0.585786i q^{7} -4.41421i q^{8} -1.00000 q^{9} -2.82843 q^{11} -3.82843i q^{12} -2.58579i q^{13} -1.41421 q^{14} +3.00000 q^{16} +4.00000i q^{17} -2.41421i q^{18} -2.82843 q^{19} -0.585786 q^{21} -6.82843i q^{22} -6.00000i q^{23} +4.41421 q^{24} +6.24264 q^{26} -1.00000i q^{27} -2.24264i q^{28} -2.24264 q^{29} +1.00000 q^{31} -1.58579i q^{32} -2.82843i q^{33} -9.65685 q^{34} +3.82843 q^{36} +1.41421i q^{37} -6.82843i q^{38} +2.58579 q^{39} -0.828427 q^{41} -1.41421i q^{42} -11.3137i q^{43} +10.8284 q^{44} +14.4853 q^{46} +4.82843i q^{47} +3.00000i q^{48} +6.65685 q^{49} -4.00000 q^{51} +9.89949i q^{52} -4.00000i q^{53} +2.41421 q^{54} +2.58579 q^{56} -2.82843i q^{57} -5.41421i q^{58} +0.242641 q^{59} -10.4853 q^{61} +2.41421i q^{62} -0.585786i q^{63} +9.82843 q^{64} +6.82843 q^{66} -3.89949i q^{67} -15.3137i q^{68} +6.00000 q^{69} +9.89949 q^{71} +4.41421i q^{72} -5.89949i q^{73} -3.41421 q^{74} +10.8284 q^{76} -1.65685i q^{77} +6.24264i q^{78} +14.4853 q^{79} +1.00000 q^{81} -2.00000i q^{82} -0.343146i q^{83} +2.24264 q^{84} +27.3137 q^{86} -2.24264i q^{87} +12.4853i q^{88} -5.07107 q^{89} +1.51472 q^{91} +22.9706i q^{92} +1.00000i q^{93} -11.6569 q^{94} +1.58579 q^{96} -15.6569i q^{97} +16.0711i q^{98} +2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9} + 12 q^{16} - 8 q^{21} + 12 q^{24} + 8 q^{26} + 8 q^{29} + 4 q^{31} - 16 q^{34} + 4 q^{36} + 16 q^{39} + 8 q^{41} + 32 q^{44} + 24 q^{46} + 4 q^{49} - 16 q^{51} + 4 q^{54}+ \cdots + 12 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2325\mathbb{Z}\right)^\times\).

\(n\) \(652\) \(776\) \(1801\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.41421i 1.70711i 0.521005 + 0.853553i \(0.325557\pi\)
−0.521005 + 0.853553i \(0.674443\pi\)
\(3\) 1.00000i 0.577350i
\(4\) −3.82843 −1.91421
\(5\) 0 0
\(6\) −2.41421 −0.985599
\(7\) 0.585786i 0.221406i 0.993854 + 0.110703i \(0.0353103\pi\)
−0.993854 + 0.110703i \(0.964690\pi\)
\(8\) − 4.41421i − 1.56066i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −2.82843 −0.852803 −0.426401 − 0.904534i \(-0.640219\pi\)
−0.426401 + 0.904534i \(0.640219\pi\)
\(12\) − 3.82843i − 1.10517i
\(13\) − 2.58579i − 0.717168i −0.933497 − 0.358584i \(-0.883260\pi\)
0.933497 − 0.358584i \(-0.116740\pi\)
\(14\) −1.41421 −0.377964
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) 4.00000i 0.970143i 0.874475 + 0.485071i \(0.161206\pi\)
−0.874475 + 0.485071i \(0.838794\pi\)
\(18\) − 2.41421i − 0.569036i
\(19\) −2.82843 −0.648886 −0.324443 − 0.945905i \(-0.605177\pi\)
−0.324443 + 0.945905i \(0.605177\pi\)
\(20\) 0 0
\(21\) −0.585786 −0.127829
\(22\) − 6.82843i − 1.45583i
\(23\) − 6.00000i − 1.25109i −0.780189 − 0.625543i \(-0.784877\pi\)
0.780189 − 0.625543i \(-0.215123\pi\)
\(24\) 4.41421 0.901048
\(25\) 0 0
\(26\) 6.24264 1.22428
\(27\) − 1.00000i − 0.192450i
\(28\) − 2.24264i − 0.423819i
\(29\) −2.24264 −0.416448 −0.208224 − 0.978081i \(-0.566768\pi\)
−0.208224 + 0.978081i \(0.566768\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) − 1.58579i − 0.280330i
\(33\) − 2.82843i − 0.492366i
\(34\) −9.65685 −1.65614
\(35\) 0 0
\(36\) 3.82843 0.638071
\(37\) 1.41421i 0.232495i 0.993220 + 0.116248i \(0.0370866\pi\)
−0.993220 + 0.116248i \(0.962913\pi\)
\(38\) − 6.82843i − 1.10772i
\(39\) 2.58579 0.414057
\(40\) 0 0
\(41\) −0.828427 −0.129379 −0.0646893 − 0.997905i \(-0.520606\pi\)
−0.0646893 + 0.997905i \(0.520606\pi\)
\(42\) − 1.41421i − 0.218218i
\(43\) − 11.3137i − 1.72532i −0.505781 − 0.862662i \(-0.668795\pi\)
0.505781 − 0.862662i \(-0.331205\pi\)
\(44\) 10.8284 1.63245
\(45\) 0 0
\(46\) 14.4853 2.13574
\(47\) 4.82843i 0.704298i 0.935944 + 0.352149i \(0.114549\pi\)
−0.935944 + 0.352149i \(0.885451\pi\)
\(48\) 3.00000i 0.433013i
\(49\) 6.65685 0.950979
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) 9.89949i 1.37281i
\(53\) − 4.00000i − 0.549442i −0.961524 − 0.274721i \(-0.911414\pi\)
0.961524 − 0.274721i \(-0.0885855\pi\)
\(54\) 2.41421 0.328533
\(55\) 0 0
\(56\) 2.58579 0.345540
\(57\) − 2.82843i − 0.374634i
\(58\) − 5.41421i − 0.710921i
\(59\) 0.242641 0.0315891 0.0157946 − 0.999875i \(-0.494972\pi\)
0.0157946 + 0.999875i \(0.494972\pi\)
\(60\) 0 0
\(61\) −10.4853 −1.34250 −0.671251 − 0.741230i \(-0.734243\pi\)
−0.671251 + 0.741230i \(0.734243\pi\)
\(62\) 2.41421i 0.306605i
\(63\) − 0.585786i − 0.0738022i
\(64\) 9.82843 1.22855
\(65\) 0 0
\(66\) 6.82843 0.840521
\(67\) − 3.89949i − 0.476399i −0.971216 − 0.238200i \(-0.923443\pi\)
0.971216 − 0.238200i \(-0.0765572\pi\)
\(68\) − 15.3137i − 1.85706i
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) 9.89949 1.17485 0.587427 − 0.809277i \(-0.300141\pi\)
0.587427 + 0.809277i \(0.300141\pi\)
\(72\) 4.41421i 0.520220i
\(73\) − 5.89949i − 0.690484i −0.938514 − 0.345242i \(-0.887797\pi\)
0.938514 − 0.345242i \(-0.112203\pi\)
\(74\) −3.41421 −0.396894
\(75\) 0 0
\(76\) 10.8284 1.24211
\(77\) − 1.65685i − 0.188816i
\(78\) 6.24264i 0.706840i
\(79\) 14.4853 1.62972 0.814861 − 0.579657i \(-0.196813\pi\)
0.814861 + 0.579657i \(0.196813\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 2.00000i − 0.220863i
\(83\) − 0.343146i − 0.0376651i −0.999823 − 0.0188326i \(-0.994005\pi\)
0.999823 − 0.0188326i \(-0.00599495\pi\)
\(84\) 2.24264 0.244692
\(85\) 0 0
\(86\) 27.3137 2.94531
\(87\) − 2.24264i − 0.240436i
\(88\) 12.4853i 1.33094i
\(89\) −5.07107 −0.537532 −0.268766 − 0.963205i \(-0.586616\pi\)
−0.268766 + 0.963205i \(0.586616\pi\)
\(90\) 0 0
\(91\) 1.51472 0.158786
\(92\) 22.9706i 2.39485i
\(93\) 1.00000i 0.103695i
\(94\) −11.6569 −1.20231
\(95\) 0 0
\(96\) 1.58579 0.161849
\(97\) − 15.6569i − 1.58971i −0.606798 − 0.794856i \(-0.707546\pi\)
0.606798 − 0.794856i \(-0.292454\pi\)
\(98\) 16.0711i 1.62342i
\(99\) 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.c.i.1024.4 4
5.2 odd 4 465.2.a.c.1.1 ✓ 2
5.3 odd 4 2325.2.a.n.1.2 2
5.4 even 2 inner 2325.2.c.i.1024.1 4
15.2 even 4 1395.2.a.g.1.2 2
15.8 even 4 6975.2.a.u.1.1 2
20.7 even 4 7440.2.a.be.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.a.c.1.1 ✓ 2 5.2 odd 4
1395.2.a.g.1.2 2 15.2 even 4
2325.2.a.n.1.2 2 5.3 odd 4
2325.2.c.i.1024.1 4 5.4 even 2 inner
2325.2.c.i.1024.4 4 1.1 even 1 trivial
6975.2.a.u.1.1 2 15.8 even 4
7440.2.a.be.1.1 2 20.7 even 4