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Properties

Label 4001.2.a.b.1.6
Level $4001$
Weight $2$
Character 4001.1
Self dual yes
Analytic conductor $31.948$
Analytic rank $0$
Dimension $184$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4001,2,Mod(1,4001)] 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("4001.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4001, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4001 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4001.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [184] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(31.9481458487\)
Analytic rank: \(0\)
Dimension: \(184\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Character \(\chi\) \(=\) 4001.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.71583 q^{2} +2.95270 q^{3} +5.37572 q^{4} +2.81232 q^{5} -8.01903 q^{6} -1.71576 q^{7} -9.16786 q^{8} +5.71846 q^{9} -7.63778 q^{10} +3.65762 q^{11} +15.8729 q^{12} -5.54091 q^{13} +4.65970 q^{14} +8.30396 q^{15} +14.1469 q^{16} -2.82348 q^{17} -15.5304 q^{18} +2.81693 q^{19} +15.1183 q^{20} -5.06613 q^{21} -9.93345 q^{22} +3.39196 q^{23} -27.0700 q^{24} +2.90916 q^{25} +15.0481 q^{26} +8.02681 q^{27} -9.22343 q^{28} +7.16279 q^{29} -22.5521 q^{30} +1.41203 q^{31} -20.0848 q^{32} +10.7999 q^{33} +7.66807 q^{34} -4.82527 q^{35} +30.7408 q^{36} +10.6344 q^{37} -7.65029 q^{38} -16.3607 q^{39} -25.7830 q^{40} +4.06503 q^{41} +13.7587 q^{42} +5.63493 q^{43} +19.6623 q^{44} +16.0822 q^{45} -9.21198 q^{46} -1.75490 q^{47} +41.7716 q^{48} -4.05617 q^{49} -7.90078 q^{50} -8.33689 q^{51} -29.7864 q^{52} +2.98389 q^{53} -21.7994 q^{54} +10.2864 q^{55} +15.7298 q^{56} +8.31756 q^{57} -19.4529 q^{58} -3.05256 q^{59} +44.6397 q^{60} -6.68573 q^{61} -3.83484 q^{62} -9.81150 q^{63} +26.2530 q^{64} -15.5828 q^{65} -29.3305 q^{66} -13.2281 q^{67} -15.1782 q^{68} +10.0155 q^{69} +13.1046 q^{70} -0.844264 q^{71} -52.4261 q^{72} +3.44574 q^{73} -28.8812 q^{74} +8.58989 q^{75} +15.1430 q^{76} -6.27558 q^{77} +44.4327 q^{78} +2.22690 q^{79} +39.7856 q^{80} +6.54541 q^{81} -11.0399 q^{82} -1.22741 q^{83} -27.2341 q^{84} -7.94053 q^{85} -15.3035 q^{86} +21.1496 q^{87} -33.5325 q^{88} -6.85737 q^{89} -43.6764 q^{90} +9.50686 q^{91} +18.2342 q^{92} +4.16932 q^{93} +4.76601 q^{94} +7.92211 q^{95} -59.3045 q^{96} +17.8276 q^{97} +11.0159 q^{98} +20.9159 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 184 q + 3 q^{2} + 28 q^{3} + 217 q^{4} + 15 q^{5} + 31 q^{6} + 49 q^{7} + 6 q^{8} + 210 q^{9} + 46 q^{10} + 25 q^{11} + 61 q^{12} + 52 q^{13} + 28 q^{14} + 59 q^{15} + 279 q^{16} + 16 q^{17} - 2 q^{18}+ \cdots + 53 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.71583 −1.92038 −0.960190 − 0.279348i \(-0.909882\pi\)
−0.960190 + 0.279348i \(0.909882\pi\)
\(3\) 2.95270 1.70474 0.852372 − 0.522936i \(-0.175163\pi\)
0.852372 + 0.522936i \(0.175163\pi\)
\(4\) 5.37572 2.68786
\(5\) 2.81232 1.25771 0.628855 − 0.777523i \(-0.283524\pi\)
0.628855 + 0.777523i \(0.283524\pi\)
\(6\) −8.01903 −3.27376
\(7\) −1.71576 −0.648496 −0.324248 − 0.945972i \(-0.605111\pi\)
−0.324248 + 0.945972i \(0.605111\pi\)
\(8\) −9.16786 −3.24133
\(9\) 5.71846 1.90615
\(10\) −7.63778 −2.41528
\(11\) 3.65762 1.10281 0.551406 − 0.834237i \(-0.314091\pi\)
0.551406 + 0.834237i \(0.314091\pi\)
\(12\) 15.8729 4.58211
\(13\) −5.54091 −1.53677 −0.768386 − 0.639987i \(-0.778940\pi\)
−0.768386 + 0.639987i \(0.778940\pi\)
\(14\) 4.65970 1.24536
\(15\) 8.30396 2.14407
\(16\) 14.1469 3.53672
\(17\) −2.82348 −0.684793 −0.342397 − 0.939555i \(-0.611239\pi\)
−0.342397 + 0.939555i \(0.611239\pi\)
\(18\) −15.5304 −3.66054
\(19\) 2.81693 0.646248 0.323124 − 0.946357i \(-0.395267\pi\)
0.323124 + 0.946357i \(0.395267\pi\)
\(20\) 15.1183 3.38054
\(21\) −5.06613 −1.10552
\(22\) −9.93345 −2.11782
\(23\) 3.39196 0.707273 0.353636 − 0.935383i \(-0.384945\pi\)
0.353636 + 0.935383i \(0.384945\pi\)
\(24\) −27.0700 −5.52564
\(25\) 2.90916 0.581832
\(26\) 15.0481 2.95118
\(27\) 8.02681 1.54476
\(28\) −9.22343 −1.74306
\(29\) 7.16279 1.33010 0.665048 − 0.746801i \(-0.268411\pi\)
0.665048 + 0.746801i \(0.268411\pi\)
\(30\) −22.5521 −4.11743
\(31\) 1.41203 0.253609 0.126804 − 0.991928i \(-0.459528\pi\)
0.126804 + 0.991928i \(0.459528\pi\)
\(32\) −20.0848 −3.55052
\(33\) 10.7999 1.88001
\(34\) 7.66807 1.31506
\(35\) −4.82527 −0.815619
\(36\) 30.7408 5.12347
\(37\) 10.6344 1.74828 0.874141 − 0.485672i \(-0.161425\pi\)
0.874141 + 0.485672i \(0.161425\pi\)
\(38\) −7.65029 −1.24104
\(39\) −16.3607 −2.61980
\(40\) −25.7830 −4.07665
\(41\) 4.06503 0.634850 0.317425 − 0.948283i \(-0.397182\pi\)
0.317425 + 0.948283i \(0.397182\pi\)
\(42\) 13.7587 2.12302
\(43\) 5.63493 0.859319 0.429659 − 0.902991i \(-0.358634\pi\)
0.429659 + 0.902991i \(0.358634\pi\)
\(44\) 19.6623 2.96420
\(45\) 16.0822 2.39739
\(46\) −9.21198 −1.35823
\(47\) −1.75490 −0.255979 −0.127989 − 0.991776i \(-0.540852\pi\)
−0.127989 + 0.991776i \(0.540852\pi\)
\(48\) 41.7716 6.02921
\(49\) −4.05617 −0.579454
\(50\) −7.90078 −1.11734
\(51\) −8.33689 −1.16740
\(52\) −29.7864 −4.13062
\(53\) 2.98389 0.409869 0.204934 − 0.978776i \(-0.434302\pi\)
0.204934 + 0.978776i \(0.434302\pi\)
\(54\) −21.7994 −2.96653
\(55\) 10.2864 1.38702
\(56\) 15.7298 2.10199
\(57\) 8.31756 1.10169
\(58\) −19.4529 −2.55429
\(59\) −3.05256 −0.397409 −0.198705 − 0.980059i \(-0.563673\pi\)
−0.198705 + 0.980059i \(0.563673\pi\)
\(60\) 44.6397 5.76296
\(61\) −6.68573 −0.856019 −0.428010 − 0.903774i \(-0.640785\pi\)
−0.428010 + 0.903774i \(0.640785\pi\)
\(62\) −3.83484 −0.487025
\(63\) −9.81150 −1.23613
\(64\) 26.2530 3.28163
\(65\) −15.5828 −1.93281
\(66\) −29.3305 −3.61034
\(67\) −13.2281 −1.61607 −0.808033 − 0.589138i \(-0.799468\pi\)
−0.808033 + 0.589138i \(0.799468\pi\)
\(68\) −15.1782 −1.84063
\(69\) 10.0155 1.20572
\(70\) 13.1046 1.56630
\(71\) −0.844264 −0.100196 −0.0500979 − 0.998744i \(-0.515953\pi\)
−0.0500979 + 0.998744i \(0.515953\pi\)
\(72\) −52.4261 −6.17847
\(73\) 3.44574 0.403293 0.201646 − 0.979458i \(-0.435371\pi\)
0.201646 + 0.979458i \(0.435371\pi\)
\(74\) −28.8812 −3.35737
\(75\) 8.58989 0.991875
\(76\) 15.1430 1.73702
\(77\) −6.27558 −0.715169
\(78\) 44.4327 5.03102
\(79\) 2.22690 0.250545 0.125273 − 0.992122i \(-0.460019\pi\)
0.125273 + 0.992122i \(0.460019\pi\)
\(80\) 39.7856 4.44817
\(81\) 6.54541 0.727268
\(82\) −11.0399 −1.21915
\(83\) −1.22741 −0.134726 −0.0673630 − 0.997729i \(-0.521459\pi\)
−0.0673630 + 0.997729i \(0.521459\pi\)
\(84\) −27.2341 −2.97148
\(85\) −7.94053 −0.861271
\(86\) −15.3035 −1.65022
\(87\) 21.1496 2.26747
\(88\) −33.5325 −3.57458
\(89\) −6.85737 −0.726880 −0.363440 − 0.931618i \(-0.618398\pi\)
−0.363440 + 0.931618i \(0.618398\pi\)
\(90\) −43.6764 −4.60389
\(91\) 9.50686 0.996589
\(92\) 18.2342 1.90105
\(93\) 4.16932 0.432338
\(94\) 4.76601 0.491576
\(95\) 7.92211 0.812792
\(96\) −59.3045 −6.05274
\(97\) 17.8276 1.81011 0.905057 − 0.425290i \(-0.139828\pi\)
0.905057 + 0.425290i \(0.139828\pi\)
\(98\) 11.0159 1.11277
\(99\) 20.9159 2.10213
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 4001.2.a.b.1.6 ✓ 184
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4001.2.a.b.1.6 ✓ 184 1.1 even 1 trivial