Properties

Label 1445.4.a.u.1.9
Level $1445$
Weight $4$
Character 1445.1
Self dual yes
Analytic conductor $85.258$
Analytic rank $1$
Dimension $21$
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] = [1445,4,Mod(1,1445)] 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("1445.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1445, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1445 = 5 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1445.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [21,-6,-18,60,105] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(85.2577599583\)
Analytic rank: \(1\)
Dimension: \(21\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Character \(\chi\) \(=\) 1445.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.98792 q^{2} -10.2526 q^{3} -4.04818 q^{4} +5.00000 q^{5} +20.3813 q^{6} +3.29700 q^{7} +23.9508 q^{8} +78.1153 q^{9} -9.93959 q^{10} +35.2225 q^{11} +41.5043 q^{12} -82.1744 q^{13} -6.55417 q^{14} -51.2629 q^{15} -15.2268 q^{16} -155.287 q^{18} +56.6670 q^{19} -20.2409 q^{20} -33.8027 q^{21} -70.0195 q^{22} -25.5431 q^{23} -245.557 q^{24} +25.0000 q^{25} +163.356 q^{26} -524.063 q^{27} -13.3469 q^{28} -198.600 q^{29} +101.906 q^{30} -11.4000 q^{31} -161.337 q^{32} -361.122 q^{33} +16.4850 q^{35} -316.225 q^{36} -56.6939 q^{37} -112.649 q^{38} +842.500 q^{39} +119.754 q^{40} +474.744 q^{41} +67.1971 q^{42} +69.2802 q^{43} -142.587 q^{44} +390.576 q^{45} +50.7775 q^{46} +69.1389 q^{47} +156.114 q^{48} -332.130 q^{49} -49.6980 q^{50} +332.657 q^{52} -104.993 q^{53} +1041.80 q^{54} +176.113 q^{55} +78.9658 q^{56} -580.983 q^{57} +394.801 q^{58} -304.454 q^{59} +207.521 q^{60} +468.232 q^{61} +22.6623 q^{62} +257.546 q^{63} +442.539 q^{64} -410.872 q^{65} +717.880 q^{66} -674.574 q^{67} +261.882 q^{69} -32.7708 q^{70} +907.834 q^{71} +1870.92 q^{72} +765.305 q^{73} +112.703 q^{74} -256.314 q^{75} -229.398 q^{76} +116.129 q^{77} -1674.82 q^{78} -1031.46 q^{79} -76.1339 q^{80} +3263.89 q^{81} -943.752 q^{82} -65.9065 q^{83} +136.840 q^{84} -137.723 q^{86} +2036.16 q^{87} +843.608 q^{88} -786.597 q^{89} -776.434 q^{90} -270.929 q^{91} +103.403 q^{92} +116.880 q^{93} -137.442 q^{94} +283.335 q^{95} +1654.12 q^{96} +847.611 q^{97} +660.247 q^{98} +2751.42 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 21 q - 6 q^{2} - 18 q^{3} + 60 q^{4} + 105 q^{5} + 45 q^{6} - 42 q^{7} - 21 q^{8} + 135 q^{9} - 30 q^{10} - 30 q^{11} - 216 q^{12} - 132 q^{13} + 162 q^{14} - 90 q^{15} + 48 q^{16} - 126 q^{18} - 405 q^{19}+ \cdots + 9066 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.98792 −0.702835 −0.351418 − 0.936219i \(-0.614300\pi\)
−0.351418 + 0.936219i \(0.614300\pi\)
\(3\) −10.2526 −1.97311 −0.986554 − 0.163433i \(-0.947743\pi\)
−0.986554 + 0.163433i \(0.947743\pi\)
\(4\) −4.04818 −0.506023
\(5\) 5.00000 0.447214
\(6\) 20.3813 1.38677
\(7\) 3.29700 0.178021 0.0890106 − 0.996031i \(-0.471629\pi\)
0.0890106 + 0.996031i \(0.471629\pi\)
\(8\) 23.9508 1.05849
\(9\) 78.1153 2.89316
\(10\) −9.93959 −0.314317
\(11\) 35.2225 0.965454 0.482727 − 0.875771i \(-0.339646\pi\)
0.482727 + 0.875771i \(0.339646\pi\)
\(12\) 41.5043 0.998438
\(13\) −82.1744 −1.75316 −0.876580 − 0.481255i \(-0.840181\pi\)
−0.876580 + 0.481255i \(0.840181\pi\)
\(14\) −6.55417 −0.125120
\(15\) −51.2629 −0.882401
\(16\) −15.2268 −0.237918
\(17\) 0 0
\(18\) −155.287 −2.03341
\(19\) 56.6670 0.684226 0.342113 − 0.939659i \(-0.388857\pi\)
0.342113 + 0.939659i \(0.388857\pi\)
\(20\) −20.2409 −0.226300
\(21\) −33.8027 −0.351255
\(22\) −70.0195 −0.678555
\(23\) −25.5431 −0.231569 −0.115785 − 0.993274i \(-0.536938\pi\)
−0.115785 + 0.993274i \(0.536938\pi\)
\(24\) −245.557 −2.08851
\(25\) 25.0000 0.200000
\(26\) 163.356 1.23218
\(27\) −524.063 −3.73541
\(28\) −13.3469 −0.0900828
\(29\) −198.600 −1.27169 −0.635847 − 0.771815i \(-0.719349\pi\)
−0.635847 + 0.771815i \(0.719349\pi\)
\(30\) 101.906 0.620183
\(31\) −11.4000 −0.0660486 −0.0330243 − 0.999455i \(-0.510514\pi\)
−0.0330243 + 0.999455i \(0.510514\pi\)
\(32\) −161.337 −0.891268
\(33\) −361.122 −1.90495
\(34\) 0 0
\(35\) 16.4850 0.0796135
\(36\) −316.225 −1.46400
\(37\) −56.6939 −0.251903 −0.125952 − 0.992036i \(-0.540198\pi\)
−0.125952 + 0.992036i \(0.540198\pi\)
\(38\) −112.649 −0.480898
\(39\) 842.500 3.45918
\(40\) 119.754 0.473369
\(41\) 474.744 1.80836 0.904178 − 0.427157i \(-0.140485\pi\)
0.904178 + 0.427157i \(0.140485\pi\)
\(42\) 67.1971 0.246875
\(43\) 69.2802 0.245701 0.122850 − 0.992425i \(-0.460796\pi\)
0.122850 + 0.992425i \(0.460796\pi\)
\(44\) −142.587 −0.488542
\(45\) 390.576 1.29386
\(46\) 50.7775 0.162755
\(47\) 69.1389 0.214573 0.107287 − 0.994228i \(-0.465784\pi\)
0.107287 + 0.994228i \(0.465784\pi\)
\(48\) 156.114 0.469439
\(49\) −332.130 −0.968308
\(50\) −49.6980 −0.140567
\(51\) 0 0
\(52\) 332.657 0.887139
\(53\) −104.993 −0.272113 −0.136056 − 0.990701i \(-0.543443\pi\)
−0.136056 + 0.990701i \(0.543443\pi\)
\(54\) 1041.80 2.62538
\(55\) 176.113 0.431764
\(56\) 78.9658 0.188433
\(57\) −580.983 −1.35005
\(58\) 394.801 0.893791
\(59\) −304.454 −0.671806 −0.335903 − 0.941897i \(-0.609041\pi\)
−0.335903 + 0.941897i \(0.609041\pi\)
\(60\) 207.521 0.446515
\(61\) 468.232 0.982803 0.491401 − 0.870933i \(-0.336485\pi\)
0.491401 + 0.870933i \(0.336485\pi\)
\(62\) 22.6623 0.0464213
\(63\) 257.546 0.515044
\(64\) 442.539 0.864333
\(65\) −410.872 −0.784037
\(66\) 717.880 1.33886
\(67\) −674.574 −1.23004 −0.615018 − 0.788513i \(-0.710851\pi\)
−0.615018 + 0.788513i \(0.710851\pi\)
\(68\) 0 0
\(69\) 261.882 0.456912
\(70\) −32.7708 −0.0559552
\(71\) 907.834 1.51747 0.758733 − 0.651402i \(-0.225819\pi\)
0.758733 + 0.651402i \(0.225819\pi\)
\(72\) 1870.92 3.06237
\(73\) 765.305 1.22702 0.613508 − 0.789688i \(-0.289758\pi\)
0.613508 + 0.789688i \(0.289758\pi\)
\(74\) 112.703 0.177047
\(75\) −256.314 −0.394622
\(76\) −229.398 −0.346234
\(77\) 116.129 0.171871
\(78\) −1674.82 −2.43123
\(79\) −1031.46 −1.46897 −0.734484 − 0.678626i \(-0.762576\pi\)
−0.734484 + 0.678626i \(0.762576\pi\)
\(80\) −76.1339 −0.106400
\(81\) 3263.89 4.47721
\(82\) −943.752 −1.27098
\(83\) −65.9065 −0.0871587 −0.0435794 − 0.999050i \(-0.513876\pi\)
−0.0435794 + 0.999050i \(0.513876\pi\)
\(84\) 136.840 0.177743
\(85\) 0 0
\(86\) −137.723 −0.172687
\(87\) 2036.16 2.50919
\(88\) 843.608 1.02192
\(89\) −786.597 −0.936844 −0.468422 − 0.883505i \(-0.655177\pi\)
−0.468422 + 0.883505i \(0.655177\pi\)
\(90\) −776.434 −0.909370
\(91\) −270.929 −0.312100
\(92\) 103.403 0.117179
\(93\) 116.880 0.130321
\(94\) −137.442 −0.150810
\(95\) 283.335 0.305995
\(96\) 1654.12 1.75857
\(97\) 847.611 0.887236 0.443618 − 0.896216i \(-0.353695\pi\)
0.443618 + 0.896216i \(0.353695\pi\)
\(98\) 660.247 0.680561
\(99\) 2751.42 2.79321
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 1445.4.a.u.1.9 ✓ 21
17.16 even 2 1445.4.a.v.1.9 yes 21
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1445.4.a.u.1.9 ✓ 21 1.1 even 1 trivial
1445.4.a.v.1.9 yes 21 17.16 even 2