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 = [8,4,2,48,-40] 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: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 48x^{6} + 112x^{5} + 767x^{4} - 496x^{3} - 3404x^{2} + 576x + 4128 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-3.13159\) of defining polynomial
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)
 
\(f(q)\) \(=\) \(q+4.13159 q^{2} +8.07129 q^{3} +9.07001 q^{4} -5.00000 q^{5} +33.3473 q^{6} +3.79835 q^{7} +4.42086 q^{8} +38.1458 q^{9} -20.6579 q^{10} +34.0439 q^{11} +73.2068 q^{12} +41.7701 q^{13} +15.6932 q^{14} -40.3565 q^{15} -54.2949 q^{16} +157.603 q^{18} +58.8011 q^{19} -45.3501 q^{20} +30.6576 q^{21} +140.655 q^{22} +98.6802 q^{23} +35.6821 q^{24} +25.0000 q^{25} +172.577 q^{26} +89.9609 q^{27} +34.4511 q^{28} +208.468 q^{29} -166.736 q^{30} -288.856 q^{31} -259.691 q^{32} +274.778 q^{33} -18.9918 q^{35} +345.983 q^{36} +317.685 q^{37} +242.942 q^{38} +337.138 q^{39} -22.1043 q^{40} -200.759 q^{41} +126.665 q^{42} -254.176 q^{43} +308.779 q^{44} -190.729 q^{45} +407.706 q^{46} +448.157 q^{47} -438.230 q^{48} -328.573 q^{49} +103.290 q^{50} +378.855 q^{52} +705.978 q^{53} +371.681 q^{54} -170.220 q^{55} +16.7920 q^{56} +474.601 q^{57} +861.305 q^{58} +78.2568 q^{59} -366.034 q^{60} +32.3885 q^{61} -1193.44 q^{62} +144.891 q^{63} -638.577 q^{64} -208.850 q^{65} +1135.27 q^{66} +571.193 q^{67} +796.477 q^{69} -78.4661 q^{70} -455.645 q^{71} +168.637 q^{72} -463.769 q^{73} +1312.54 q^{74} +201.782 q^{75} +533.327 q^{76} +129.311 q^{77} +1392.92 q^{78} -794.183 q^{79} +271.475 q^{80} -303.835 q^{81} -829.455 q^{82} +1459.49 q^{83} +278.065 q^{84} -1050.15 q^{86} +1682.61 q^{87} +150.503 q^{88} -1027.25 q^{89} -788.013 q^{90} +158.657 q^{91} +895.031 q^{92} -2331.44 q^{93} +1851.60 q^{94} -294.005 q^{95} -2096.04 q^{96} -606.193 q^{97} -1357.53 q^{98} +1298.63 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} + 2 q^{3} + 48 q^{4} - 40 q^{5} - 44 q^{6} - 32 q^{7} - 36 q^{8} + 162 q^{9} - 20 q^{10} + 44 q^{11} + 36 q^{12} + 86 q^{13} - 146 q^{14} - 10 q^{15} + 348 q^{16} - 8 q^{18} + 288 q^{19} - 240 q^{20}+ \cdots + 5258 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.13159 1.46074 0.730368 − 0.683053i \(-0.239348\pi\)
0.730368 + 0.683053i \(0.239348\pi\)
\(3\) 8.07129 1.55332 0.776661 − 0.629919i \(-0.216912\pi\)
0.776661 + 0.629919i \(0.216912\pi\)
\(4\) 9.07001 1.13375
\(5\) −5.00000 −0.447214
\(6\) 33.3473 2.26899
\(7\) 3.79835 0.205092 0.102546 − 0.994728i \(-0.467301\pi\)
0.102546 + 0.994728i \(0.467301\pi\)
\(8\) 4.42086 0.195376
\(9\) 38.1458 1.41281
\(10\) −20.6579 −0.653261
\(11\) 34.0439 0.933148 0.466574 − 0.884482i \(-0.345488\pi\)
0.466574 + 0.884482i \(0.345488\pi\)
\(12\) 73.2068 1.76108
\(13\) 41.7701 0.891148 0.445574 − 0.895245i \(-0.353000\pi\)
0.445574 + 0.895245i \(0.353000\pi\)
\(14\) 15.6932 0.299585
\(15\) −40.3565 −0.694666
\(16\) −54.2949 −0.848359
\(17\) 0 0
\(18\) 157.603 2.06374
\(19\) 58.8011 0.709994 0.354997 − 0.934867i \(-0.384482\pi\)
0.354997 + 0.934867i \(0.384482\pi\)
\(20\) −45.3501 −0.507029
\(21\) 30.6576 0.318573
\(22\) 140.655 1.36308
\(23\) 98.6802 0.894619 0.447310 − 0.894379i \(-0.352382\pi\)
0.447310 + 0.894379i \(0.352382\pi\)
\(24\) 35.6821 0.303482
\(25\) 25.0000 0.200000
\(26\) 172.577 1.30173
\(27\) 89.9609 0.641221
\(28\) 34.4511 0.232523
\(29\) 208.468 1.33488 0.667441 − 0.744663i \(-0.267390\pi\)
0.667441 + 0.744663i \(0.267390\pi\)
\(30\) −166.736 −1.01472
\(31\) −288.856 −1.67355 −0.836776 − 0.547545i \(-0.815563\pi\)
−0.836776 + 0.547545i \(0.815563\pi\)
\(32\) −259.691 −1.43460
\(33\) 274.778 1.44948
\(34\) 0 0
\(35\) −18.9918 −0.0917198
\(36\) 345.983 1.60177
\(37\) 317.685 1.41154 0.705771 − 0.708440i \(-0.250601\pi\)
0.705771 + 0.708440i \(0.250601\pi\)
\(38\) 242.942 1.03711
\(39\) 337.138 1.38424
\(40\) −22.1043 −0.0873749
\(41\) −200.759 −0.764716 −0.382358 − 0.924014i \(-0.624888\pi\)
−0.382358 + 0.924014i \(0.624888\pi\)
\(42\) 126.665 0.465352
\(43\) −254.176 −0.901431 −0.450715 − 0.892668i \(-0.648831\pi\)
−0.450715 + 0.892668i \(0.648831\pi\)
\(44\) 308.779 1.05796
\(45\) −190.729 −0.631826
\(46\) 407.706 1.30680
\(47\) 448.157 1.39086 0.695431 − 0.718593i \(-0.255214\pi\)
0.695431 + 0.718593i \(0.255214\pi\)
\(48\) −438.230 −1.31777
\(49\) −328.573 −0.957937
\(50\) 103.290 0.292147
\(51\) 0 0
\(52\) 378.855 1.01034
\(53\) 705.978 1.82969 0.914845 − 0.403806i \(-0.132313\pi\)
0.914845 + 0.403806i \(0.132313\pi\)
\(54\) 371.681 0.936656
\(55\) −170.220 −0.417316
\(56\) 16.7920 0.0400701
\(57\) 474.601 1.10285
\(58\) 861.305 1.94991
\(59\) 78.2568 0.172681 0.0863404 − 0.996266i \(-0.472483\pi\)
0.0863404 + 0.996266i \(0.472483\pi\)
\(60\) −366.034 −0.787579
\(61\) 32.3885 0.0679823 0.0339911 − 0.999422i \(-0.489178\pi\)
0.0339911 + 0.999422i \(0.489178\pi\)
\(62\) −1193.44 −2.44462
\(63\) 144.891 0.289755
\(64\) −638.577 −1.24722
\(65\) −208.850 −0.398534
\(66\) 1135.27 2.11731
\(67\) 571.193 1.04153 0.520763 − 0.853701i \(-0.325647\pi\)
0.520763 + 0.853701i \(0.325647\pi\)
\(68\) 0 0
\(69\) 796.477 1.38963
\(70\) −78.4661 −0.133979
\(71\) −455.645 −0.761621 −0.380811 − 0.924653i \(-0.624355\pi\)
−0.380811 + 0.924653i \(0.624355\pi\)
\(72\) 168.637 0.276029
\(73\) −463.769 −0.743562 −0.371781 − 0.928321i \(-0.621253\pi\)
−0.371781 + 0.928321i \(0.621253\pi\)
\(74\) 1312.54 2.06189
\(75\) 201.782 0.310664
\(76\) 533.327 0.804958
\(77\) 129.311 0.191381
\(78\) 1392.92 2.02201
\(79\) −794.183 −1.13105 −0.565523 − 0.824733i \(-0.691326\pi\)
−0.565523 + 0.824733i \(0.691326\pi\)
\(80\) 271.475 0.379397
\(81\) −303.835 −0.416784
\(82\) −829.455 −1.11705
\(83\) 1459.49 1.93012 0.965058 − 0.262035i \(-0.0843937\pi\)
0.965058 + 0.262035i \(0.0843937\pi\)
\(84\) 278.065 0.361183
\(85\) 0 0
\(86\) −1050.15 −1.31675
\(87\) 1682.61 2.07350
\(88\) 150.503 0.182315
\(89\) −1027.25 −1.22347 −0.611733 − 0.791064i \(-0.709528\pi\)
−0.611733 + 0.791064i \(0.709528\pi\)
\(90\) −788.013 −0.922932
\(91\) 158.657 0.182767
\(92\) 895.031 1.01428
\(93\) −2331.44 −2.59956
\(94\) 1851.60 2.03168
\(95\) −294.005 −0.317519
\(96\) −2096.04 −2.22840
\(97\) −606.193 −0.634531 −0.317266 − 0.948337i \(-0.602765\pi\)
−0.317266 + 0.948337i \(0.602765\pi\)
\(98\) −1357.53 −1.39929
\(99\) 1298.63 1.31836
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.r.1.7 yes 8
17.16 even 2 1445.4.a.q.1.7 ✓ 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1445.4.a.q.1.7 ✓ 8 17.16 even 2
1445.4.a.r.1.7 yes 8 1.1 even 1 trivial