Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2175,2,Mod(1,2175)] 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("2175.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2175, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2175.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-1,-2,5,0,1,-2] 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: \(17.3674624396\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 435)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 2175.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.56155 q^{2} -1.00000 q^{3} +0.438447 q^{4} -1.56155 q^{6} -5.12311 q^{7} -2.43845 q^{8} +1.00000 q^{9} -1.43845 q^{11} -0.438447 q^{12} +2.00000 q^{13} -8.00000 q^{14} -4.68466 q^{16} +7.12311 q^{17} +1.56155 q^{18} +5.12311 q^{19} +5.12311 q^{21} -2.24621 q^{22} -6.56155 q^{23} +2.43845 q^{24} +3.12311 q^{26} -1.00000 q^{27} -2.24621 q^{28} +1.00000 q^{29} +4.00000 q^{31} -2.43845 q^{32} +1.43845 q^{33} +11.1231 q^{34} +0.438447 q^{36} +1.68466 q^{37} +8.00000 q^{38} -2.00000 q^{39} -1.68466 q^{41} +8.00000 q^{42} -7.68466 q^{43} -0.630683 q^{44} -10.2462 q^{46} +13.1231 q^{47} +4.68466 q^{48} +19.2462 q^{49} -7.12311 q^{51} +0.876894 q^{52} +3.43845 q^{53} -1.56155 q^{54} +12.4924 q^{56} -5.12311 q^{57} +1.56155 q^{58} +12.0000 q^{59} +0.876894 q^{61} +6.24621 q^{62} -5.12311 q^{63} +5.56155 q^{64} +2.24621 q^{66} +11.3693 q^{67} +3.12311 q^{68} +6.56155 q^{69} -2.87689 q^{71} -2.43845 q^{72} -1.68466 q^{73} +2.63068 q^{74} +2.24621 q^{76} +7.36932 q^{77} -3.12311 q^{78} -12.0000 q^{79} +1.00000 q^{81} -2.63068 q^{82} +2.56155 q^{83} +2.24621 q^{84} -12.0000 q^{86} -1.00000 q^{87} +3.50758 q^{88} +12.2462 q^{89} -10.2462 q^{91} -2.87689 q^{92} -4.00000 q^{93} +20.4924 q^{94} +2.43845 q^{96} +5.68466 q^{97} +30.0540 q^{98} -1.43845 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 2 q^{3} + 5 q^{4} + q^{6} - 2 q^{7} - 9 q^{8} + 2 q^{9} - 7 q^{11} - 5 q^{12} + 4 q^{13} - 16 q^{14} + 3 q^{16} + 6 q^{17} - q^{18} + 2 q^{19} + 2 q^{21} + 12 q^{22} - 9 q^{23} + 9 q^{24}+ \cdots - 7 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.56155 1.10418 0.552092 − 0.833783i \(-0.313830\pi\)
0.552092 + 0.833783i \(0.313830\pi\)
\(3\) −1.00000 −0.577350
\(4\) 0.438447 0.219224
\(5\) 0 0
\(6\) −1.56155 −0.637501
\(7\) −5.12311 −1.93635 −0.968176 − 0.250270i \(-0.919480\pi\)
−0.968176 + 0.250270i \(0.919480\pi\)
\(8\) −2.43845 −0.862121
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.43845 −0.433708 −0.216854 − 0.976204i \(-0.569580\pi\)
−0.216854 + 0.976204i \(0.569580\pi\)
\(12\) −0.438447 −0.126569
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −8.00000 −2.13809
\(15\) 0 0
\(16\) −4.68466 −1.17116
\(17\) 7.12311 1.72761 0.863803 − 0.503829i \(-0.168076\pi\)
0.863803 + 0.503829i \(0.168076\pi\)
\(18\) 1.56155 0.368062
\(19\) 5.12311 1.17532 0.587661 − 0.809108i \(-0.300049\pi\)
0.587661 + 0.809108i \(0.300049\pi\)
\(20\) 0 0
\(21\) 5.12311 1.11795
\(22\) −2.24621 −0.478894
\(23\) −6.56155 −1.36818 −0.684089 − 0.729398i \(-0.739800\pi\)
−0.684089 + 0.729398i \(0.739800\pi\)
\(24\) 2.43845 0.497746
\(25\) 0 0
\(26\) 3.12311 0.612491
\(27\) −1.00000 −0.192450
\(28\) −2.24621 −0.424494
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) −2.43845 −0.431061
\(33\) 1.43845 0.250402
\(34\) 11.1231 1.90760
\(35\) 0 0
\(36\) 0.438447 0.0730745
\(37\) 1.68466 0.276956 0.138478 − 0.990366i \(-0.455779\pi\)
0.138478 + 0.990366i \(0.455779\pi\)
\(38\) 8.00000 1.29777
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) −1.68466 −0.263099 −0.131550 − 0.991310i \(-0.541995\pi\)
−0.131550 + 0.991310i \(0.541995\pi\)
\(42\) 8.00000 1.23443
\(43\) −7.68466 −1.17190 −0.585950 − 0.810347i \(-0.699278\pi\)
−0.585950 + 0.810347i \(0.699278\pi\)
\(44\) −0.630683 −0.0950791
\(45\) 0 0
\(46\) −10.2462 −1.51072
\(47\) 13.1231 1.91420 0.957101 − 0.289755i \(-0.0935738\pi\)
0.957101 + 0.289755i \(0.0935738\pi\)
\(48\) 4.68466 0.676172
\(49\) 19.2462 2.74946
\(50\) 0 0
\(51\) −7.12311 −0.997434
\(52\) 0.876894 0.121603
\(53\) 3.43845 0.472307 0.236154 − 0.971716i \(-0.424113\pi\)
0.236154 + 0.971716i \(0.424113\pi\)
\(54\) −1.56155 −0.212500
\(55\) 0 0
\(56\) 12.4924 1.66937
\(57\) −5.12311 −0.678572
\(58\) 1.56155 0.205042
\(59\) 12.0000 1.56227 0.781133 − 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 0.876894 0.112275 0.0561374 − 0.998423i \(-0.482122\pi\)
0.0561374 + 0.998423i \(0.482122\pi\)
\(62\) 6.24621 0.793270
\(63\) −5.12311 −0.645451
\(64\) 5.56155 0.695194
\(65\) 0 0
\(66\) 2.24621 0.276489
\(67\) 11.3693 1.38898 0.694492 − 0.719501i \(-0.255629\pi\)
0.694492 + 0.719501i \(0.255629\pi\)
\(68\) 3.12311 0.378732
\(69\) 6.56155 0.789918
\(70\) 0 0
\(71\) −2.87689 −0.341425 −0.170712 − 0.985321i \(-0.554607\pi\)
−0.170712 + 0.985321i \(0.554607\pi\)
\(72\) −2.43845 −0.287374
\(73\) −1.68466 −0.197174 −0.0985872 − 0.995128i \(-0.531432\pi\)
−0.0985872 + 0.995128i \(0.531432\pi\)
\(74\) 2.63068 0.305811
\(75\) 0 0
\(76\) 2.24621 0.257658
\(77\) 7.36932 0.839812
\(78\) −3.12311 −0.353622
\(79\) −12.0000 −1.35011 −0.675053 − 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −2.63068 −0.290510
\(83\) 2.56155 0.281167 0.140583 − 0.990069i \(-0.455102\pi\)
0.140583 + 0.990069i \(0.455102\pi\)
\(84\) 2.24621 0.245082
\(85\) 0 0
\(86\) −12.0000 −1.29399
\(87\) −1.00000 −0.107211
\(88\) 3.50758 0.373909
\(89\) 12.2462 1.29810 0.649048 − 0.760748i \(-0.275167\pi\)
0.649048 + 0.760748i \(0.275167\pi\)
\(90\) 0 0
\(91\) −10.2462 −1.07409
\(92\) −2.87689 −0.299937
\(93\) −4.00000 −0.414781
\(94\) 20.4924 2.11363
\(95\) 0 0
\(96\) 2.43845 0.248873
\(97\) 5.68466 0.577190 0.288595 − 0.957451i \(-0.406812\pi\)
0.288595 + 0.957451i \(0.406812\pi\)
\(98\) 30.0540 3.03591
\(99\) −1.43845 −0.144569
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 2175.2.a.m.1.2 2
3.2 odd 2 6525.2.a.bc.1.1 2
5.2 odd 4 2175.2.c.h.349.3 4
5.3 odd 4 2175.2.c.h.349.2 4
5.4 even 2 435.2.a.h.1.1 ✓ 2
15.14 odd 2 1305.2.a.i.1.2 2
20.19 odd 2 6960.2.a.bx.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.h.1.1 ✓ 2 5.4 even 2
1305.2.a.i.1.2 2 15.14 odd 2
2175.2.a.m.1.2 2 1.1 even 1 trivial
2175.2.c.h.349.2 4 5.3 odd 4
2175.2.c.h.349.3 4 5.2 odd 4
6525.2.a.bc.1.1 2 3.2 odd 2
6960.2.a.bx.1.1 2 20.19 odd 2