Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [275,6,Mod(1,275)] 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("275.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 275.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,7,36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(44.1055504486\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.21865.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 30x + 40 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.35507\) of defining polynomial
Character \(\chi\) \(=\) 275.1

$q$-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)
 
\(f(q)\) \(=\) \(q-6.40437 q^{2} +15.0652 q^{3} +9.01590 q^{4} -96.4830 q^{6} +122.399 q^{7} +147.199 q^{8} -16.0398 q^{9} +121.000 q^{11} +135.826 q^{12} +1042.06 q^{13} -783.886 q^{14} -1231.22 q^{16} -400.809 q^{17} +102.725 q^{18} +581.415 q^{19} +1843.96 q^{21} -774.928 q^{22} -66.9186 q^{23} +2217.58 q^{24} -6673.76 q^{26} -3902.49 q^{27} +1103.53 q^{28} +6780.57 q^{29} -3861.53 q^{31} +3174.84 q^{32} +1822.89 q^{33} +2566.93 q^{34} -144.613 q^{36} +14501.1 q^{37} -3723.60 q^{38} +15698.9 q^{39} -5669.29 q^{41} -11809.4 q^{42} -1853.94 q^{43} +1090.92 q^{44} +428.571 q^{46} -27384.9 q^{47} -18548.6 q^{48} -1825.55 q^{49} -6038.27 q^{51} +9395.15 q^{52} +16822.0 q^{53} +24992.9 q^{54} +18016.9 q^{56} +8759.14 q^{57} -43425.3 q^{58} -19863.8 q^{59} -24637.7 q^{61} +24730.6 q^{62} -1963.25 q^{63} +19066.3 q^{64} -11674.4 q^{66} +39950.3 q^{67} -3613.65 q^{68} -1008.14 q^{69} +24983.1 q^{71} -2361.04 q^{72} +81725.0 q^{73} -92870.5 q^{74} +5241.98 q^{76} +14810.2 q^{77} -100542. q^{78} -16805.9 q^{79} -54894.0 q^{81} +36308.2 q^{82} -20025.4 q^{83} +16625.0 q^{84} +11873.3 q^{86} +102151. q^{87} +17811.0 q^{88} +105988. q^{89} +127547. q^{91} -603.331 q^{92} -58174.7 q^{93} +175383. q^{94} +47829.6 q^{96} +138361. q^{97} +11691.5 q^{98} -1940.82 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 7 q^{2} + 36 q^{3} + 41 q^{4} + 101 q^{6} + 102 q^{7} + 15 q^{8} + 249 q^{9} + 363 q^{11} + 1237 q^{12} + 1646 q^{13} - 963 q^{14} - 2687 q^{16} + 1742 q^{17} + 3076 q^{18} - 10 q^{19} + 1236 q^{21}+ \cdots + 30129 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −6.40437 −1.13214 −0.566071 − 0.824356i \(-0.691537\pi\)
−0.566071 + 0.824356i \(0.691537\pi\)
\(3\) 15.0652 0.966433 0.483216 − 0.875501i \(-0.339468\pi\)
0.483216 + 0.875501i \(0.339468\pi\)
\(4\) 9.01590 0.281747
\(5\) 0 0
\(6\) −96.4830 −1.09414
\(7\) 122.399 0.944130 0.472065 − 0.881564i \(-0.343509\pi\)
0.472065 + 0.881564i \(0.343509\pi\)
\(8\) 147.199 0.813165
\(9\) −16.0398 −0.0660075
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) 135.826 0.272289
\(13\) 1042.06 1.71016 0.855079 − 0.518497i \(-0.173508\pi\)
0.855079 + 0.518497i \(0.173508\pi\)
\(14\) −783.886 −1.06889
\(15\) 0 0
\(16\) −1231.22 −1.20237
\(17\) −400.809 −0.336368 −0.168184 − 0.985756i \(-0.553790\pi\)
−0.168184 + 0.985756i \(0.553790\pi\)
\(18\) 102.725 0.0747299
\(19\) 581.415 0.369490 0.184745 − 0.982786i \(-0.440854\pi\)
0.184745 + 0.982786i \(0.440854\pi\)
\(20\) 0 0
\(21\) 1843.96 0.912438
\(22\) −774.928 −0.341354
\(23\) −66.9186 −0.0263771 −0.0131886 − 0.999913i \(-0.504198\pi\)
−0.0131886 + 0.999913i \(0.504198\pi\)
\(24\) 2217.58 0.785869
\(25\) 0 0
\(26\) −6673.76 −1.93614
\(27\) −3902.49 −1.03022
\(28\) 1103.53 0.266006
\(29\) 6780.57 1.49717 0.748585 − 0.663039i \(-0.230733\pi\)
0.748585 + 0.663039i \(0.230733\pi\)
\(30\) 0 0
\(31\) −3861.53 −0.721697 −0.360848 − 0.932624i \(-0.617513\pi\)
−0.360848 + 0.932624i \(0.617513\pi\)
\(32\) 3174.84 0.548084
\(33\) 1822.89 0.291390
\(34\) 2566.93 0.380817
\(35\) 0 0
\(36\) −144.613 −0.0185974
\(37\) 14501.1 1.74139 0.870697 − 0.491819i \(-0.163668\pi\)
0.870697 + 0.491819i \(0.163668\pi\)
\(38\) −3723.60 −0.418315
\(39\) 15698.9 1.65275
\(40\) 0 0
\(41\) −5669.29 −0.526707 −0.263353 − 0.964699i \(-0.584829\pi\)
−0.263353 + 0.964699i \(0.584829\pi\)
\(42\) −11809.4 −1.03301
\(43\) −1853.94 −0.152906 −0.0764531 − 0.997073i \(-0.524360\pi\)
−0.0764531 + 0.997073i \(0.524360\pi\)
\(44\) 1090.92 0.0849499
\(45\) 0 0
\(46\) 428.571 0.0298627
\(47\) −27384.9 −1.80828 −0.904142 − 0.427233i \(-0.859489\pi\)
−0.904142 + 0.427233i \(0.859489\pi\)
\(48\) −18548.6 −1.16201
\(49\) −1825.55 −0.108619
\(50\) 0 0
\(51\) −6038.27 −0.325077
\(52\) 9395.15 0.481832
\(53\) 16822.0 0.822601 0.411300 − 0.911500i \(-0.365075\pi\)
0.411300 + 0.911500i \(0.365075\pi\)
\(54\) 24992.9 1.16636
\(55\) 0 0
\(56\) 18016.9 0.767733
\(57\) 8759.14 0.357087
\(58\) −43425.3 −1.69501
\(59\) −19863.8 −0.742905 −0.371452 − 0.928452i \(-0.621140\pi\)
−0.371452 + 0.928452i \(0.621140\pi\)
\(60\) 0 0
\(61\) −24637.7 −0.847766 −0.423883 − 0.905717i \(-0.639333\pi\)
−0.423883 + 0.905717i \(0.639333\pi\)
\(62\) 24730.6 0.817064
\(63\) −1963.25 −0.0623197
\(64\) 19066.3 0.581856
\(65\) 0 0
\(66\) −11674.4 −0.329896
\(67\) 39950.3 1.08726 0.543630 − 0.839325i \(-0.317050\pi\)
0.543630 + 0.839325i \(0.317050\pi\)
\(68\) −3613.65 −0.0947707
\(69\) −1008.14 −0.0254917
\(70\) 0 0
\(71\) 24983.1 0.588167 0.294084 − 0.955780i \(-0.404986\pi\)
0.294084 + 0.955780i \(0.404986\pi\)
\(72\) −2361.04 −0.0536750
\(73\) 81725.0 1.79493 0.897466 − 0.441084i \(-0.145406\pi\)
0.897466 + 0.441084i \(0.145406\pi\)
\(74\) −92870.5 −1.97151
\(75\) 0 0
\(76\) 5241.98 0.104103
\(77\) 14810.2 0.284666
\(78\) −100542. −1.87115
\(79\) −16805.9 −0.302967 −0.151483 − 0.988460i \(-0.548405\pi\)
−0.151483 + 0.988460i \(0.548405\pi\)
\(80\) 0 0
\(81\) −54894.0 −0.929635
\(82\) 36308.2 0.596307
\(83\) −20025.4 −0.319070 −0.159535 − 0.987192i \(-0.550999\pi\)
−0.159535 + 0.987192i \(0.550999\pi\)
\(84\) 16625.0 0.257077
\(85\) 0 0
\(86\) 11873.3 0.173112
\(87\) 102151. 1.44691
\(88\) 17811.0 0.245178
\(89\) 105988. 1.41835 0.709173 − 0.705035i \(-0.249069\pi\)
0.709173 + 0.705035i \(0.249069\pi\)
\(90\) 0 0
\(91\) 127547. 1.61461
\(92\) −603.331 −0.00743167
\(93\) −58174.7 −0.697472
\(94\) 175383. 2.04723
\(95\) 0 0
\(96\) 47829.6 0.529687
\(97\) 138361. 1.49308 0.746542 − 0.665339i \(-0.231713\pi\)
0.746542 + 0.665339i \(0.231713\pi\)
\(98\) 11691.5 0.122972
\(99\) −1940.82 −0.0199020
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 275.6.a.c.1.1 3
5.2 odd 4 275.6.b.c.199.2 6
5.3 odd 4 275.6.b.c.199.5 6
5.4 even 2 55.6.a.a.1.3 ✓ 3
15.14 odd 2 495.6.a.f.1.1 3
20.19 odd 2 880.6.a.l.1.2 3
55.54 odd 2 605.6.a.b.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.6.a.a.1.3 ✓ 3 5.4 even 2
275.6.a.c.1.1 3 1.1 even 1 trivial
275.6.b.c.199.2 6 5.2 odd 4
275.6.b.c.199.5 6 5.3 odd 4
495.6.a.f.1.1 3 15.14 odd 2
605.6.a.b.1.1 3 55.54 odd 2
880.6.a.l.1.2 3 20.19 odd 2