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Properties

Label 4001.2.a.b.1.9
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 1.9
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)
 
\(f(q)\) \(=\) \(q-2.66682 q^{2} +0.260886 q^{3} +5.11190 q^{4} -1.12952 q^{5} -0.695735 q^{6} -1.67133 q^{7} -8.29887 q^{8} -2.93194 q^{9} +3.01221 q^{10} +4.71305 q^{11} +1.33362 q^{12} -4.92080 q^{13} +4.45712 q^{14} -0.294675 q^{15} +11.9077 q^{16} -3.62116 q^{17} +7.81894 q^{18} -4.61861 q^{19} -5.77398 q^{20} -0.436026 q^{21} -12.5688 q^{22} -5.48210 q^{23} -2.16506 q^{24} -3.72419 q^{25} +13.1229 q^{26} -1.54756 q^{27} -8.54367 q^{28} +2.38742 q^{29} +0.785844 q^{30} +6.32942 q^{31} -15.1580 q^{32} +1.22957 q^{33} +9.65697 q^{34} +1.88779 q^{35} -14.9878 q^{36} +1.40769 q^{37} +12.3170 q^{38} -1.28377 q^{39} +9.37371 q^{40} -5.95370 q^{41} +1.16280 q^{42} +0.539714 q^{43} +24.0927 q^{44} +3.31167 q^{45} +14.6197 q^{46} -6.16466 q^{47} +3.10656 q^{48} -4.20666 q^{49} +9.93173 q^{50} -0.944710 q^{51} -25.1546 q^{52} -10.6448 q^{53} +4.12706 q^{54} -5.32347 q^{55} +13.8701 q^{56} -1.20493 q^{57} -6.36680 q^{58} -4.98420 q^{59} -1.50635 q^{60} +9.33444 q^{61} -16.8794 q^{62} +4.90023 q^{63} +16.6081 q^{64} +5.55812 q^{65} -3.27904 q^{66} +13.2191 q^{67} -18.5110 q^{68} -1.43020 q^{69} -5.03439 q^{70} -4.18398 q^{71} +24.3318 q^{72} +5.46343 q^{73} -3.75406 q^{74} -0.971590 q^{75} -23.6099 q^{76} -7.87706 q^{77} +3.42357 q^{78} -11.6527 q^{79} -13.4500 q^{80} +8.39208 q^{81} +15.8774 q^{82} -8.72403 q^{83} -2.22892 q^{84} +4.09016 q^{85} -1.43932 q^{86} +0.622844 q^{87} -39.1130 q^{88} -5.76793 q^{89} -8.83162 q^{90} +8.22427 q^{91} -28.0240 q^{92} +1.65126 q^{93} +16.4400 q^{94} +5.21680 q^{95} -3.95451 q^{96} -3.07817 q^{97} +11.2184 q^{98} -13.8184 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 184 q + 3 q^{2} + 28 q^{3} + 217 q^{4} + 15 q^{5} + 31 q^{6} + 49 q^{7} + 6 q^{8} + 210 q^{9} + 46 q^{10} + 25 q^{11} + 61 q^{12} + 52 q^{13} + 28 q^{14} + 59 q^{15} + 279 q^{16} + 16 q^{17} - 2 q^{18}+ \cdots + 53 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.66682 −1.88572 −0.942861 − 0.333185i \(-0.891877\pi\)
−0.942861 + 0.333185i \(0.891877\pi\)
\(3\) 0.260886 0.150623 0.0753113 − 0.997160i \(-0.476005\pi\)
0.0753113 + 0.997160i \(0.476005\pi\)
\(4\) 5.11190 2.55595
\(5\) −1.12952 −0.505135 −0.252568 − 0.967579i \(-0.581275\pi\)
−0.252568 + 0.967579i \(0.581275\pi\)
\(6\) −0.695735 −0.284033
\(7\) −1.67133 −0.631703 −0.315851 − 0.948809i \(-0.602290\pi\)
−0.315851 + 0.948809i \(0.602290\pi\)
\(8\) −8.29887 −2.93409
\(9\) −2.93194 −0.977313
\(10\) 3.01221 0.952545
\(11\) 4.71305 1.42104 0.710520 − 0.703677i \(-0.248460\pi\)
0.710520 + 0.703677i \(0.248460\pi\)
\(12\) 1.33362 0.384984
\(13\) −4.92080 −1.36478 −0.682392 − 0.730987i \(-0.739060\pi\)
−0.682392 + 0.730987i \(0.739060\pi\)
\(14\) 4.45712 1.19122
\(15\) −0.294675 −0.0760848
\(16\) 11.9077 2.97694
\(17\) −3.62116 −0.878261 −0.439130 − 0.898423i \(-0.644713\pi\)
−0.439130 + 0.898423i \(0.644713\pi\)
\(18\) 7.81894 1.84294
\(19\) −4.61861 −1.05958 −0.529791 − 0.848128i \(-0.677730\pi\)
−0.529791 + 0.848128i \(0.677730\pi\)
\(20\) −5.77398 −1.29110
\(21\) −0.436026 −0.0951487
\(22\) −12.5688 −2.67969
\(23\) −5.48210 −1.14310 −0.571548 − 0.820568i \(-0.693657\pi\)
−0.571548 + 0.820568i \(0.693657\pi\)
\(24\) −2.16506 −0.441941
\(25\) −3.72419 −0.744839
\(26\) 13.1229 2.57360
\(27\) −1.54756 −0.297828
\(28\) −8.54367 −1.61460
\(29\) 2.38742 0.443332 0.221666 − 0.975123i \(-0.428850\pi\)
0.221666 + 0.975123i \(0.428850\pi\)
\(30\) 0.785844 0.143475
\(31\) 6.32942 1.13680 0.568398 − 0.822753i \(-0.307563\pi\)
0.568398 + 0.822753i \(0.307563\pi\)
\(32\) −15.1580 −2.67958
\(33\) 1.22957 0.214041
\(34\) 9.65697 1.65616
\(35\) 1.88779 0.319095
\(36\) −14.9878 −2.49796
\(37\) 1.40769 0.231423 0.115712 − 0.993283i \(-0.463085\pi\)
0.115712 + 0.993283i \(0.463085\pi\)
\(38\) 12.3170 1.99808
\(39\) −1.28377 −0.205567
\(40\) 9.37371 1.48211
\(41\) −5.95370 −0.929812 −0.464906 − 0.885360i \(-0.653912\pi\)
−0.464906 + 0.885360i \(0.653912\pi\)
\(42\) 1.16280 0.179424
\(43\) 0.539714 0.0823056 0.0411528 − 0.999153i \(-0.486897\pi\)
0.0411528 + 0.999153i \(0.486897\pi\)
\(44\) 24.0927 3.63211
\(45\) 3.31167 0.493675
\(46\) 14.6197 2.15556
\(47\) −6.16466 −0.899208 −0.449604 − 0.893228i \(-0.648435\pi\)
−0.449604 + 0.893228i \(0.648435\pi\)
\(48\) 3.10656 0.448394
\(49\) −4.20666 −0.600952
\(50\) 9.93173 1.40456
\(51\) −0.944710 −0.132286
\(52\) −25.1546 −3.48832
\(53\) −10.6448 −1.46217 −0.731087 − 0.682284i \(-0.760987\pi\)
−0.731087 + 0.682284i \(0.760987\pi\)
\(54\) 4.12706 0.561621
\(55\) −5.32347 −0.717817
\(56\) 13.8701 1.85347
\(57\) −1.20493 −0.159597
\(58\) −6.36680 −0.836002
\(59\) −4.98420 −0.648887 −0.324444 − 0.945905i \(-0.605177\pi\)
−0.324444 + 0.945905i \(0.605177\pi\)
\(60\) −1.50635 −0.194469
\(61\) 9.33444 1.19515 0.597576 − 0.801812i \(-0.296131\pi\)
0.597576 + 0.801812i \(0.296131\pi\)
\(62\) −16.8794 −2.14368
\(63\) 4.90023 0.617371
\(64\) 16.6081 2.07601
\(65\) 5.55812 0.689400
\(66\) −3.27904 −0.403621
\(67\) 13.2191 1.61497 0.807483 − 0.589891i \(-0.200829\pi\)
0.807483 + 0.589891i \(0.200829\pi\)
\(68\) −18.5110 −2.24479
\(69\) −1.43020 −0.172176
\(70\) −5.03439 −0.601725
\(71\) −4.18398 −0.496548 −0.248274 − 0.968690i \(-0.579863\pi\)
−0.248274 + 0.968690i \(0.579863\pi\)
\(72\) 24.3318 2.86753
\(73\) 5.46343 0.639446 0.319723 − 0.947511i \(-0.396410\pi\)
0.319723 + 0.947511i \(0.396410\pi\)
\(74\) −3.75406 −0.436400
\(75\) −0.971590 −0.112190
\(76\) −23.6099 −2.70824
\(77\) −7.87706 −0.897674
\(78\) 3.42357 0.387643
\(79\) −11.6527 −1.31103 −0.655517 − 0.755181i \(-0.727549\pi\)
−0.655517 + 0.755181i \(0.727549\pi\)
\(80\) −13.4500 −1.50375
\(81\) 8.39208 0.932453
\(82\) 15.8774 1.75337
\(83\) −8.72403 −0.957587 −0.478793 − 0.877928i \(-0.658926\pi\)
−0.478793 + 0.877928i \(0.658926\pi\)
\(84\) −2.22892 −0.243195
\(85\) 4.09016 0.443640
\(86\) −1.43932 −0.155206
\(87\) 0.622844 0.0667759
\(88\) −39.1130 −4.16946
\(89\) −5.76793 −0.611399 −0.305700 − 0.952128i \(-0.598890\pi\)
−0.305700 + 0.952128i \(0.598890\pi\)
\(90\) −8.83162 −0.930934
\(91\) 8.22427 0.862137
\(92\) −28.0240 −2.92170
\(93\) 1.65126 0.171227
\(94\) 16.4400 1.69566
\(95\) 5.21680 0.535232
\(96\) −3.95451 −0.403606
\(97\) −3.07817 −0.312541 −0.156271 − 0.987714i \(-0.549947\pi\)
−0.156271 + 0.987714i \(0.549947\pi\)
\(98\) 11.2184 1.13323
\(99\) −13.8184 −1.38880
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.1.9 ✓ 184
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4001.2.a.b.1.9 ✓ 184 1.1 even 1 trivial