Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1725,2,Mod(1174,1725)] 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("1725.1174"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1725, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1725 = 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1725.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-12,0,0,0,0,-4,0,16,0,0,20] 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: \(13.7741943487\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 69)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1174.4
Root \(-1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 1725.1174
Dual form 1725.2.b.o.1174.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.23607i q^{2} +1.00000i q^{3} -3.00000 q^{4} -2.23607 q^{6} -1.23607i q^{7} -2.23607i q^{8} -1.00000 q^{9} +4.00000 q^{11} -3.00000i q^{12} -4.47214i q^{13} +2.76393 q^{14} -1.00000 q^{16} -7.23607i q^{17} -2.23607i q^{18} -2.76393 q^{19} +1.23607 q^{21} +8.94427i q^{22} -1.00000i q^{23} +2.23607 q^{24} +10.0000 q^{26} -1.00000i q^{27} +3.70820i q^{28} +4.47214 q^{29} +2.47214 q^{31} -6.70820i q^{32} +4.00000i q^{33} +16.1803 q^{34} +3.00000 q^{36} -4.47214i q^{37} -6.18034i q^{38} +4.47214 q^{39} +6.94427 q^{41} +2.76393i q^{42} -7.70820i q^{43} -12.0000 q^{44} +2.23607 q^{46} -4.00000i q^{47} -1.00000i q^{48} +5.47214 q^{49} +7.23607 q^{51} +13.4164i q^{52} +0.763932i q^{53} +2.23607 q^{54} -2.76393 q^{56} -2.76393i q^{57} +10.0000i q^{58} -12.9443 q^{59} -4.47214 q^{61} +5.52786i q^{62} +1.23607i q^{63} +13.0000 q^{64} -8.94427 q^{66} +5.23607i q^{67} +21.7082i q^{68} +1.00000 q^{69} -8.00000 q^{71} +2.23607i q^{72} +10.9443i q^{73} +10.0000 q^{74} +8.29180 q^{76} -4.94427i q^{77} +10.0000i q^{78} +3.70820 q^{79} +1.00000 q^{81} +15.5279i q^{82} -4.00000i q^{83} -3.70820 q^{84} +17.2361 q^{86} +4.47214i q^{87} -8.94427i q^{88} -3.23607 q^{89} -5.52786 q^{91} +3.00000i q^{92} +2.47214i q^{93} +8.94427 q^{94} +6.70820 q^{96} -0.472136i q^{97} +12.2361i q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{4} - 4 q^{9} + 16 q^{11} + 20 q^{14} - 4 q^{16} - 20 q^{19} - 4 q^{21} + 40 q^{26} - 8 q^{31} + 20 q^{34} + 12 q^{36} - 8 q^{41} - 48 q^{44} + 4 q^{49} + 20 q^{51} - 20 q^{56} - 16 q^{59} + 52 q^{64}+ \cdots - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1725\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(1151\) \(1201\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.23607i 1.58114i 0.612372 + 0.790569i \(0.290215\pi\)
−0.612372 + 0.790569i \(0.709785\pi\)
\(3\) 1.00000i 0.577350i
\(4\) −3.00000 −1.50000
\(5\) 0 0
\(6\) −2.23607 −0.912871
\(7\) − 1.23607i − 0.467190i −0.972334 − 0.233595i \(-0.924951\pi\)
0.972334 − 0.233595i \(-0.0750489\pi\)
\(8\) − 2.23607i − 0.790569i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) − 3.00000i − 0.866025i
\(13\) − 4.47214i − 1.24035i −0.784465 − 0.620174i \(-0.787062\pi\)
0.784465 − 0.620174i \(-0.212938\pi\)
\(14\) 2.76393 0.738692
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) − 7.23607i − 1.75500i −0.479573 − 0.877502i \(-0.659208\pi\)
0.479573 − 0.877502i \(-0.340792\pi\)
\(18\) − 2.23607i − 0.527046i
\(19\) −2.76393 −0.634089 −0.317045 − 0.948411i \(-0.602691\pi\)
−0.317045 + 0.948411i \(0.602691\pi\)
\(20\) 0 0
\(21\) 1.23607 0.269732
\(22\) 8.94427i 1.90693i
\(23\) − 1.00000i − 0.208514i
\(24\) 2.23607 0.456435
\(25\) 0 0
\(26\) 10.0000 1.96116
\(27\) − 1.00000i − 0.192450i
\(28\) 3.70820i 0.700785i
\(29\) 4.47214 0.830455 0.415227 − 0.909718i \(-0.363702\pi\)
0.415227 + 0.909718i \(0.363702\pi\)
\(30\) 0 0
\(31\) 2.47214 0.444009 0.222004 − 0.975046i \(-0.428740\pi\)
0.222004 + 0.975046i \(0.428740\pi\)
\(32\) − 6.70820i − 1.18585i
\(33\) 4.00000i 0.696311i
\(34\) 16.1803 2.77491
\(35\) 0 0
\(36\) 3.00000 0.500000
\(37\) − 4.47214i − 0.735215i −0.929981 − 0.367607i \(-0.880177\pi\)
0.929981 − 0.367607i \(-0.119823\pi\)
\(38\) − 6.18034i − 1.00258i
\(39\) 4.47214 0.716115
\(40\) 0 0
\(41\) 6.94427 1.08451 0.542257 − 0.840213i \(-0.317570\pi\)
0.542257 + 0.840213i \(0.317570\pi\)
\(42\) 2.76393i 0.426484i
\(43\) − 7.70820i − 1.17549i −0.809046 − 0.587745i \(-0.800016\pi\)
0.809046 − 0.587745i \(-0.199984\pi\)
\(44\) −12.0000 −1.80907
\(45\) 0 0
\(46\) 2.23607 0.329690
\(47\) − 4.00000i − 0.583460i −0.956501 − 0.291730i \(-0.905769\pi\)
0.956501 − 0.291730i \(-0.0942309\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) 5.47214 0.781734
\(50\) 0 0
\(51\) 7.23607 1.01325
\(52\) 13.4164i 1.86052i
\(53\) 0.763932i 0.104934i 0.998623 + 0.0524671i \(0.0167085\pi\)
−0.998623 + 0.0524671i \(0.983292\pi\)
\(54\) 2.23607 0.304290
\(55\) 0 0
\(56\) −2.76393 −0.369346
\(57\) − 2.76393i − 0.366092i
\(58\) 10.0000i 1.31306i
\(59\) −12.9443 −1.68520 −0.842600 − 0.538539i \(-0.818976\pi\)
−0.842600 + 0.538539i \(0.818976\pi\)
\(60\) 0 0
\(61\) −4.47214 −0.572598 −0.286299 − 0.958140i \(-0.592425\pi\)
−0.286299 + 0.958140i \(0.592425\pi\)
\(62\) 5.52786i 0.702039i
\(63\) 1.23607i 0.155730i
\(64\) 13.0000 1.62500
\(65\) 0 0
\(66\) −8.94427 −1.10096
\(67\) 5.23607i 0.639688i 0.947470 + 0.319844i \(0.103630\pi\)
−0.947470 + 0.319844i \(0.896370\pi\)
\(68\) 21.7082i 2.63251i
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 2.23607i 0.263523i
\(73\) 10.9443i 1.28093i 0.767987 + 0.640465i \(0.221258\pi\)
−0.767987 + 0.640465i \(0.778742\pi\)
\(74\) 10.0000 1.16248
\(75\) 0 0
\(76\) 8.29180 0.951134
\(77\) − 4.94427i − 0.563452i
\(78\) 10.0000i 1.13228i
\(79\) 3.70820 0.417206 0.208603 − 0.978000i \(-0.433108\pi\)
0.208603 + 0.978000i \(0.433108\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 15.5279i 1.71477i
\(83\) − 4.00000i − 0.439057i −0.975606 − 0.219529i \(-0.929548\pi\)
0.975606 − 0.219529i \(-0.0704519\pi\)
\(84\) −3.70820 −0.404598
\(85\) 0 0
\(86\) 17.2361 1.85861
\(87\) 4.47214i 0.479463i
\(88\) − 8.94427i − 0.953463i
\(89\) −3.23607 −0.343023 −0.171511 − 0.985182i \(-0.554865\pi\)
−0.171511 + 0.985182i \(0.554865\pi\)
\(90\) 0 0
\(91\) −5.52786 −0.579478
\(92\) 3.00000i 0.312772i
\(93\) 2.47214i 0.256349i
\(94\) 8.94427 0.922531
\(95\) 0 0
\(96\) 6.70820 0.684653
\(97\) − 0.472136i − 0.0479381i −0.999713 − 0.0239691i \(-0.992370\pi\)
0.999713 − 0.0239691i \(-0.00763032\pi\)
\(98\) 12.2361i 1.23603i
\(99\) −4.00000 −0.402015
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 1725.2.b.o.1174.4 4
5.2 odd 4 1725.2.a.ba.1.1 2
5.3 odd 4 69.2.a.b.1.2 ✓ 2
5.4 even 2 inner 1725.2.b.o.1174.1 4
15.2 even 4 5175.2.a.bk.1.2 2
15.8 even 4 207.2.a.c.1.1 2
20.3 even 4 1104.2.a.m.1.1 2
35.13 even 4 3381.2.a.t.1.2 2
40.3 even 4 4416.2.a.bg.1.2 2
40.13 odd 4 4416.2.a.bm.1.2 2
55.43 even 4 8349.2.a.i.1.1 2
60.23 odd 4 3312.2.a.bb.1.2 2
115.68 even 4 1587.2.a.i.1.2 2
345.68 odd 4 4761.2.a.v.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
69.2.a.b.1.2 ✓ 2 5.3 odd 4
207.2.a.c.1.1 2 15.8 even 4
1104.2.a.m.1.1 2 20.3 even 4
1587.2.a.i.1.2 2 115.68 even 4
1725.2.a.ba.1.1 2 5.2 odd 4
1725.2.b.o.1174.1 4 5.4 even 2 inner
1725.2.b.o.1174.4 4 1.1 even 1 trivial
3312.2.a.bb.1.2 2 60.23 odd 4
3381.2.a.t.1.2 2 35.13 even 4
4416.2.a.bg.1.2 2 40.3 even 4
4416.2.a.bm.1.2 2 40.13 odd 4
4761.2.a.v.1.1 2 345.68 odd 4
5175.2.a.bk.1.2 2 15.2 even 4
8349.2.a.i.1.1 2 55.43 even 4