Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1425,2,Mod(1,1425)] 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("1425.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1425, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1425.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,3,6,0,0,0,-3,3,0,-3] 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: \(11.3786822880\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.837.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 6x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.167449\) of defining polynomial
Character \(\chi\) \(=\) 1425.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.167449 q^{2} +1.00000 q^{3} -1.97196 q^{4} +0.167449 q^{6} +4.13941 q^{7} -0.665102 q^{8} +1.00000 q^{9} -4.80451 q^{11} -1.97196 q^{12} +4.00000 q^{13} +0.693141 q^{14} +3.83255 q^{16} +5.94392 q^{17} +0.167449 q^{18} -1.00000 q^{19} +4.13941 q^{21} -0.804512 q^{22} -7.13941 q^{23} -0.665102 q^{24} +0.669797 q^{26} +1.00000 q^{27} -8.16275 q^{28} -5.00000 q^{29} +8.80451 q^{31} +1.97196 q^{32} -4.80451 q^{33} +0.995305 q^{34} -1.97196 q^{36} +3.66510 q^{37} -0.167449 q^{38} +4.00000 q^{39} +0.195488 q^{41} +0.693141 q^{42} +4.00000 q^{43} +9.47431 q^{44} -1.19549 q^{46} +11.6090 q^{47} +3.83255 q^{48} +10.1347 q^{49} +5.94392 q^{51} -7.88784 q^{52} -2.33020 q^{53} +0.167449 q^{54} -2.75313 q^{56} -1.00000 q^{57} -0.837246 q^{58} +7.46961 q^{59} -6.60902 q^{61} +1.47431 q^{62} +4.13941 q^{63} -7.33490 q^{64} -0.804512 q^{66} +1.19549 q^{67} -11.7212 q^{68} -7.13941 q^{69} +9.46961 q^{71} -0.665102 q^{72} +0.330203 q^{73} +0.613718 q^{74} +1.97196 q^{76} -19.8878 q^{77} +0.669797 q^{78} +13.4182 q^{79} +1.00000 q^{81} +0.0327344 q^{82} -2.80451 q^{83} -8.16275 q^{84} +0.669797 q^{86} -5.00000 q^{87} +3.19549 q^{88} +7.27882 q^{89} +16.5576 q^{91} +14.0786 q^{92} +8.80451 q^{93} +1.94392 q^{94} +1.97196 q^{96} -11.2741 q^{97} +1.69705 q^{98} -4.80451 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 6 q^{4} - 3 q^{8} + 3 q^{9} - 3 q^{11} + 6 q^{12} + 12 q^{13} + 15 q^{14} + 12 q^{16} - 6 q^{17} - 3 q^{19} + 9 q^{22} - 9 q^{23} - 3 q^{24} + 3 q^{27} - 27 q^{28} - 15 q^{29} + 15 q^{31}+ \cdots - 3 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.167449 0.118404 0.0592022 − 0.998246i \(-0.481144\pi\)
0.0592022 + 0.998246i \(0.481144\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.97196 −0.985980
\(5\) 0 0
\(6\) 0.167449 0.0683608
\(7\) 4.13941 1.56455 0.782275 − 0.622933i \(-0.214059\pi\)
0.782275 + 0.622933i \(0.214059\pi\)
\(8\) −0.665102 −0.235149
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.80451 −1.44861 −0.724307 − 0.689477i \(-0.757840\pi\)
−0.724307 + 0.689477i \(0.757840\pi\)
\(12\) −1.97196 −0.569256
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0.693141 0.185250
\(15\) 0 0
\(16\) 3.83255 0.958138
\(17\) 5.94392 1.44161 0.720806 − 0.693136i \(-0.243772\pi\)
0.720806 + 0.693136i \(0.243772\pi\)
\(18\) 0.167449 0.0394682
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 4.13941 0.903293
\(22\) −0.804512 −0.171522
\(23\) −7.13941 −1.48867 −0.744335 − 0.667806i \(-0.767233\pi\)
−0.744335 + 0.667806i \(0.767233\pi\)
\(24\) −0.665102 −0.135763
\(25\) 0 0
\(26\) 0.669797 0.131358
\(27\) 1.00000 0.192450
\(28\) −8.16275 −1.54262
\(29\) −5.00000 −0.928477 −0.464238 − 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) 8.80451 1.58134 0.790668 − 0.612245i \(-0.209733\pi\)
0.790668 + 0.612245i \(0.209733\pi\)
\(32\) 1.97196 0.348597
\(33\) −4.80451 −0.836358
\(34\) 0.995305 0.170693
\(35\) 0 0
\(36\) −1.97196 −0.328660
\(37\) 3.66510 0.602539 0.301269 − 0.953539i \(-0.402590\pi\)
0.301269 + 0.953539i \(0.402590\pi\)
\(38\) −0.167449 −0.0271638
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) 0.195488 0.0305302 0.0152651 − 0.999883i \(-0.495141\pi\)
0.0152651 + 0.999883i \(0.495141\pi\)
\(42\) 0.693141 0.106954
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 9.47431 1.42831
\(45\) 0 0
\(46\) −1.19549 −0.176265
\(47\) 11.6090 1.69335 0.846675 − 0.532110i \(-0.178601\pi\)
0.846675 + 0.532110i \(0.178601\pi\)
\(48\) 3.83255 0.553181
\(49\) 10.1347 1.44782
\(50\) 0 0
\(51\) 5.94392 0.832315
\(52\) −7.88784 −1.09385
\(53\) −2.33020 −0.320078 −0.160039 − 0.987111i \(-0.551162\pi\)
−0.160039 + 0.987111i \(0.551162\pi\)
\(54\) 0.167449 0.0227869
\(55\) 0 0
\(56\) −2.75313 −0.367902
\(57\) −1.00000 −0.132453
\(58\) −0.837246 −0.109936
\(59\) 7.46961 0.972461 0.486230 − 0.873831i \(-0.338372\pi\)
0.486230 + 0.873831i \(0.338372\pi\)
\(60\) 0 0
\(61\) −6.60902 −0.846199 −0.423099 − 0.906083i \(-0.639058\pi\)
−0.423099 + 0.906083i \(0.639058\pi\)
\(62\) 1.47431 0.187237
\(63\) 4.13941 0.521517
\(64\) −7.33490 −0.916862
\(65\) 0 0
\(66\) −0.804512 −0.0990285
\(67\) 1.19549 0.146052 0.0730261 − 0.997330i \(-0.476734\pi\)
0.0730261 + 0.997330i \(0.476734\pi\)
\(68\) −11.7212 −1.42140
\(69\) −7.13941 −0.859484
\(70\) 0 0
\(71\) 9.46961 1.12384 0.561918 − 0.827193i \(-0.310064\pi\)
0.561918 + 0.827193i \(0.310064\pi\)
\(72\) −0.665102 −0.0783830
\(73\) 0.330203 0.0386474 0.0193237 − 0.999813i \(-0.493849\pi\)
0.0193237 + 0.999813i \(0.493849\pi\)
\(74\) 0.613718 0.0713433
\(75\) 0 0
\(76\) 1.97196 0.226199
\(77\) −19.8878 −2.26643
\(78\) 0.669797 0.0758395
\(79\) 13.4182 1.50967 0.754834 − 0.655915i \(-0.227717\pi\)
0.754834 + 0.655915i \(0.227717\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0.0327344 0.00361491
\(83\) −2.80451 −0.307835 −0.153918 − 0.988084i \(-0.549189\pi\)
−0.153918 + 0.988084i \(0.549189\pi\)
\(84\) −8.16275 −0.890629
\(85\) 0 0
\(86\) 0.669797 0.0722260
\(87\) −5.00000 −0.536056
\(88\) 3.19549 0.340640
\(89\) 7.27882 0.771553 0.385777 − 0.922592i \(-0.373934\pi\)
0.385777 + 0.922592i \(0.373934\pi\)
\(90\) 0 0
\(91\) 16.5576 1.73571
\(92\) 14.0786 1.46780
\(93\) 8.80451 0.912985
\(94\) 1.94392 0.200500
\(95\) 0 0
\(96\) 1.97196 0.201262
\(97\) −11.2741 −1.14471 −0.572357 − 0.820005i \(-0.693971\pi\)
−0.572357 + 0.820005i \(0.693971\pi\)
\(98\) 1.69705 0.171428
\(99\) −4.80451 −0.482872
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 1425.2.a.w.1.2 yes 3
3.2 odd 2 4275.2.a.bg.1.2 3
5.2 odd 4 1425.2.c.o.799.4 6
5.3 odd 4 1425.2.c.o.799.3 6
5.4 even 2 1425.2.a.t.1.2 ✓ 3
15.14 odd 2 4275.2.a.bf.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1425.2.a.t.1.2 ✓ 3 5.4 even 2
1425.2.a.w.1.2 yes 3 1.1 even 1 trivial
1425.2.c.o.799.3 6 5.3 odd 4
1425.2.c.o.799.4 6 5.2 odd 4
4275.2.a.bf.1.2 3 15.14 odd 2
4275.2.a.bg.1.2 3 3.2 odd 2