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 = [4,5,0] 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: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 33x^{2} - 8x + 116 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 55)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(6.71109\) 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)
 
\(f(q)\) \(=\) \(q+8.50380 q^{2} +13.0311 q^{3} +40.3146 q^{4} +110.814 q^{6} -217.433 q^{7} +70.7058 q^{8} -73.1914 q^{9} -121.000 q^{11} +525.342 q^{12} -747.549 q^{13} -1849.01 q^{14} -688.799 q^{16} +677.308 q^{17} -622.405 q^{18} -1916.17 q^{19} -2833.39 q^{21} -1028.96 q^{22} +3515.96 q^{23} +921.372 q^{24} -6357.01 q^{26} -4120.31 q^{27} -8765.74 q^{28} +4658.87 q^{29} +371.680 q^{31} -8120.00 q^{32} -1576.76 q^{33} +5759.69 q^{34} -2950.68 q^{36} +1726.69 q^{37} -16294.7 q^{38} -9741.36 q^{39} -16374.9 q^{41} -24094.6 q^{42} +19279.9 q^{43} -4878.07 q^{44} +29899.1 q^{46} +5241.87 q^{47} -8975.79 q^{48} +30470.3 q^{49} +8826.04 q^{51} -30137.1 q^{52} +29695.5 q^{53} -35038.3 q^{54} -15373.8 q^{56} -24969.8 q^{57} +39618.1 q^{58} -13018.2 q^{59} -37124.4 q^{61} +3160.69 q^{62} +15914.3 q^{63} -47009.3 q^{64} -13408.4 q^{66} -34221.0 q^{67} +27305.4 q^{68} +45816.7 q^{69} -23751.7 q^{71} -5175.06 q^{72} +41963.4 q^{73} +14683.4 q^{74} -77249.8 q^{76} +26309.4 q^{77} -82838.5 q^{78} +35874.8 q^{79} -35906.5 q^{81} -139249. q^{82} -88476.3 q^{83} -114227. q^{84} +163952. q^{86} +60710.0 q^{87} -8555.40 q^{88} -103222. q^{89} +162542. q^{91} +141745. q^{92} +4843.38 q^{93} +44575.8 q^{94} -105812. q^{96} -13556.7 q^{97} +259113. q^{98} +8856.16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 5 q^{2} + 61 q^{4} - 157 q^{6} + 90 q^{7} + 135 q^{8} + 22 q^{9} - 484 q^{11} + 795 q^{12} - 820 q^{13} - 1687 q^{14} - 2671 q^{16} + 3800 q^{17} + 1610 q^{18} - 3394 q^{19} - 4708 q^{21} - 605 q^{22}+ \cdots - 2662 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.50380 1.50327 0.751637 − 0.659577i \(-0.229265\pi\)
0.751637 + 0.659577i \(0.229265\pi\)
\(3\) 13.0311 0.835943 0.417972 − 0.908460i \(-0.362741\pi\)
0.417972 + 0.908460i \(0.362741\pi\)
\(4\) 40.3146 1.25983
\(5\) 0 0
\(6\) 110.814 1.25665
\(7\) −217.433 −1.67719 −0.838593 − 0.544759i \(-0.816621\pi\)
−0.838593 + 0.544759i \(0.816621\pi\)
\(8\) 70.7058 0.390598
\(9\) −73.1914 −0.301199
\(10\) 0 0
\(11\) −121.000 −0.301511
\(12\) 525.342 1.05315
\(13\) −747.549 −1.22682 −0.613410 − 0.789764i \(-0.710203\pi\)
−0.613410 + 0.789764i \(0.710203\pi\)
\(14\) −1849.01 −2.52127
\(15\) 0 0
\(16\) −688.799 −0.672656
\(17\) 677.308 0.568412 0.284206 − 0.958763i \(-0.408270\pi\)
0.284206 + 0.958763i \(0.408270\pi\)
\(18\) −622.405 −0.452785
\(19\) −1916.17 −1.21773 −0.608864 − 0.793274i \(-0.708375\pi\)
−0.608864 + 0.793274i \(0.708375\pi\)
\(20\) 0 0
\(21\) −2833.39 −1.40203
\(22\) −1028.96 −0.453254
\(23\) 3515.96 1.38588 0.692939 − 0.720997i \(-0.256316\pi\)
0.692939 + 0.720997i \(0.256316\pi\)
\(24\) 921.372 0.326518
\(25\) 0 0
\(26\) −6357.01 −1.84425
\(27\) −4120.31 −1.08773
\(28\) −8765.74 −2.11297
\(29\) 4658.87 1.02869 0.514346 − 0.857583i \(-0.328035\pi\)
0.514346 + 0.857583i \(0.328035\pi\)
\(30\) 0 0
\(31\) 371.680 0.0694648 0.0347324 − 0.999397i \(-0.488942\pi\)
0.0347324 + 0.999397i \(0.488942\pi\)
\(32\) −8120.00 −1.40178
\(33\) −1576.76 −0.252046
\(34\) 5759.69 0.854480
\(35\) 0 0
\(36\) −2950.68 −0.379460
\(37\) 1726.69 0.207353 0.103677 − 0.994611i \(-0.466939\pi\)
0.103677 + 0.994611i \(0.466939\pi\)
\(38\) −16294.7 −1.83058
\(39\) −9741.36 −1.02555
\(40\) 0 0
\(41\) −16374.9 −1.52132 −0.760658 − 0.649153i \(-0.775124\pi\)
−0.760658 + 0.649153i \(0.775124\pi\)
\(42\) −24094.6 −2.10764
\(43\) 19279.9 1.59013 0.795065 − 0.606524i \(-0.207437\pi\)
0.795065 + 0.606524i \(0.207437\pi\)
\(44\) −4878.07 −0.379854
\(45\) 0 0
\(46\) 29899.1 2.08335
\(47\) 5241.87 0.346132 0.173066 − 0.984910i \(-0.444633\pi\)
0.173066 + 0.984910i \(0.444633\pi\)
\(48\) −8975.79 −0.562302
\(49\) 30470.3 1.81295
\(50\) 0 0
\(51\) 8826.04 0.475160
\(52\) −30137.1 −1.54559
\(53\) 29695.5 1.45212 0.726058 − 0.687633i \(-0.241350\pi\)
0.726058 + 0.687633i \(0.241350\pi\)
\(54\) −35038.3 −1.63515
\(55\) 0 0
\(56\) −15373.8 −0.655106
\(57\) −24969.8 −1.01795
\(58\) 39618.1 1.54640
\(59\) −13018.2 −0.486878 −0.243439 − 0.969916i \(-0.578275\pi\)
−0.243439 + 0.969916i \(0.578275\pi\)
\(60\) 0 0
\(61\) −37124.4 −1.27742 −0.638711 − 0.769447i \(-0.720532\pi\)
−0.638711 + 0.769447i \(0.720532\pi\)
\(62\) 3160.69 0.104425
\(63\) 15914.3 0.505167
\(64\) −47009.3 −1.43461
\(65\) 0 0
\(66\) −13408.4 −0.378895
\(67\) −34221.0 −0.931334 −0.465667 − 0.884960i \(-0.654185\pi\)
−0.465667 + 0.884960i \(0.654185\pi\)
\(68\) 27305.4 0.716104
\(69\) 45816.7 1.15851
\(70\) 0 0
\(71\) −23751.7 −0.559176 −0.279588 − 0.960120i \(-0.590198\pi\)
−0.279588 + 0.960120i \(0.590198\pi\)
\(72\) −5175.06 −0.117648
\(73\) 41963.4 0.921644 0.460822 − 0.887492i \(-0.347555\pi\)
0.460822 + 0.887492i \(0.347555\pi\)
\(74\) 14683.4 0.311708
\(75\) 0 0
\(76\) −77249.8 −1.53413
\(77\) 26309.4 0.505690
\(78\) −82838.5 −1.54169
\(79\) 35874.8 0.646728 0.323364 − 0.946275i \(-0.395186\pi\)
0.323364 + 0.946275i \(0.395186\pi\)
\(80\) 0 0
\(81\) −35906.5 −0.608080
\(82\) −139249. −2.28695
\(83\) −88476.3 −1.40972 −0.704859 − 0.709348i \(-0.748990\pi\)
−0.704859 + 0.709348i \(0.748990\pi\)
\(84\) −114227. −1.76632
\(85\) 0 0
\(86\) 163952. 2.39040
\(87\) 60710.0 0.859928
\(88\) −8555.40 −0.117770
\(89\) −103222. −1.38132 −0.690662 − 0.723178i \(-0.742681\pi\)
−0.690662 + 0.723178i \(0.742681\pi\)
\(90\) 0 0
\(91\) 162542. 2.05761
\(92\) 141745. 1.74597
\(93\) 4843.38 0.0580686
\(94\) 44575.8 0.520331
\(95\) 0 0
\(96\) −105812. −1.17181
\(97\) −13556.7 −0.146293 −0.0731467 − 0.997321i \(-0.523304\pi\)
−0.0731467 + 0.997321i \(0.523304\pi\)
\(98\) 259113. 2.72536
\(99\) 8856.16 0.0908150
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.d.1.4 4
5.2 odd 4 275.6.b.d.199.8 8
5.3 odd 4 275.6.b.d.199.1 8
5.4 even 2 55.6.a.b.1.1 ✓ 4
15.14 odd 2 495.6.a.g.1.4 4
20.19 odd 2 880.6.a.n.1.3 4
55.54 odd 2 605.6.a.c.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.6.a.b.1.1 ✓ 4 5.4 even 2
275.6.a.d.1.4 4 1.1 even 1 trivial
275.6.b.d.199.1 8 5.3 odd 4
275.6.b.d.199.8 8 5.2 odd 4
495.6.a.g.1.4 4 15.14 odd 2
605.6.a.c.1.4 4 55.54 odd 2
880.6.a.n.1.3 4 20.19 odd 2