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 = [5,2,1,34,-25] 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: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 20x^{3} + 38x^{2} + 69x - 126 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 85)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.90874\) 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)
 
\(f(q)\) \(=\) \(q-5.14668 q^{2} +1.13153 q^{3} +18.4883 q^{4} -5.00000 q^{5} -5.82364 q^{6} +26.5778 q^{7} -53.9797 q^{8} -25.7196 q^{9} +25.7334 q^{10} -65.6224 q^{11} +20.9201 q^{12} +44.3601 q^{13} -136.787 q^{14} -5.65767 q^{15} +129.910 q^{16} +132.371 q^{18} -37.6482 q^{19} -92.4413 q^{20} +30.0737 q^{21} +337.737 q^{22} -194.150 q^{23} -61.0799 q^{24} +25.0000 q^{25} -228.307 q^{26} -59.6541 q^{27} +491.377 q^{28} -157.743 q^{29} +29.1182 q^{30} -287.386 q^{31} -236.766 q^{32} -74.2540 q^{33} -132.889 q^{35} -475.511 q^{36} +96.6691 q^{37} +193.763 q^{38} +50.1950 q^{39} +269.898 q^{40} +106.333 q^{41} -154.779 q^{42} -142.879 q^{43} -1213.24 q^{44} +128.598 q^{45} +999.227 q^{46} +275.572 q^{47} +146.997 q^{48} +363.378 q^{49} -128.667 q^{50} +820.141 q^{52} -180.696 q^{53} +307.020 q^{54} +328.112 q^{55} -1434.66 q^{56} -42.6003 q^{57} +811.853 q^{58} +284.982 q^{59} -104.601 q^{60} +644.300 q^{61} +1479.08 q^{62} -683.571 q^{63} +179.280 q^{64} -221.800 q^{65} +382.161 q^{66} +396.688 q^{67} -219.687 q^{69} +683.936 q^{70} +573.342 q^{71} +1388.34 q^{72} -574.739 q^{73} -497.524 q^{74} +28.2884 q^{75} -696.050 q^{76} -1744.10 q^{77} -258.337 q^{78} +184.587 q^{79} -649.549 q^{80} +626.929 q^{81} -547.263 q^{82} +626.402 q^{83} +556.010 q^{84} +735.351 q^{86} -178.492 q^{87} +3542.28 q^{88} -454.081 q^{89} -661.853 q^{90} +1178.99 q^{91} -3589.50 q^{92} -325.187 q^{93} -1418.28 q^{94} +188.241 q^{95} -267.909 q^{96} -123.922 q^{97} -1870.19 q^{98} +1687.78 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{2} + q^{3} + 34 q^{4} - 25 q^{5} + 5 q^{6} - 10 q^{7} + 30 q^{8} - 30 q^{9} - 10 q^{10} - 126 q^{11} - 15 q^{12} + 83 q^{13} - 90 q^{14} - 5 q^{15} + 322 q^{16} - 97 q^{18} + 55 q^{19} - 170 q^{20}+ \cdots + 858 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.14668 −1.81962 −0.909812 − 0.415020i \(-0.863775\pi\)
−0.909812 + 0.415020i \(0.863775\pi\)
\(3\) 1.13153 0.217764 0.108882 − 0.994055i \(-0.465273\pi\)
0.108882 + 0.994055i \(0.465273\pi\)
\(4\) 18.4883 2.31103
\(5\) −5.00000 −0.447214
\(6\) −5.82364 −0.396248
\(7\) 26.5778 1.43506 0.717532 − 0.696525i \(-0.245271\pi\)
0.717532 + 0.696525i \(0.245271\pi\)
\(8\) −53.9797 −2.38559
\(9\) −25.7196 −0.952579
\(10\) 25.7334 0.813761
\(11\) −65.6224 −1.79872 −0.899359 − 0.437211i \(-0.855966\pi\)
−0.899359 + 0.437211i \(0.855966\pi\)
\(12\) 20.9201 0.503259
\(13\) 44.3601 0.946406 0.473203 − 0.880954i \(-0.343098\pi\)
0.473203 + 0.880954i \(0.343098\pi\)
\(14\) −136.787 −2.61128
\(15\) −5.65767 −0.0973870
\(16\) 129.910 2.02984
\(17\) 0 0
\(18\) 132.371 1.73334
\(19\) −37.6482 −0.454584 −0.227292 − 0.973827i \(-0.572987\pi\)
−0.227292 + 0.973827i \(0.572987\pi\)
\(20\) −92.4413 −1.03353
\(21\) 30.0737 0.312505
\(22\) 337.737 3.27299
\(23\) −194.150 −1.76013 −0.880067 − 0.474849i \(-0.842503\pi\)
−0.880067 + 0.474849i \(0.842503\pi\)
\(24\) −61.0799 −0.519495
\(25\) 25.0000 0.200000
\(26\) −228.307 −1.72210
\(27\) −59.6541 −0.425201
\(28\) 491.377 3.31648
\(29\) −157.743 −1.01007 −0.505037 − 0.863098i \(-0.668521\pi\)
−0.505037 + 0.863098i \(0.668521\pi\)
\(30\) 29.1182 0.177208
\(31\) −287.386 −1.66503 −0.832517 − 0.554000i \(-0.813101\pi\)
−0.832517 + 0.554000i \(0.813101\pi\)
\(32\) −236.766 −1.30796
\(33\) −74.2540 −0.391696
\(34\) 0 0
\(35\) −132.889 −0.641781
\(36\) −475.511 −2.20144
\(37\) 96.6691 0.429522 0.214761 − 0.976667i \(-0.431103\pi\)
0.214761 + 0.976667i \(0.431103\pi\)
\(38\) 193.763 0.827172
\(39\) 50.1950 0.206093
\(40\) 269.898 1.06687
\(41\) 106.333 0.405036 0.202518 − 0.979279i \(-0.435088\pi\)
0.202518 + 0.979279i \(0.435088\pi\)
\(42\) −154.779 −0.568642
\(43\) −142.879 −0.506716 −0.253358 − 0.967373i \(-0.581535\pi\)
−0.253358 + 0.967373i \(0.581535\pi\)
\(44\) −1213.24 −4.15690
\(45\) 128.598 0.426006
\(46\) 999.227 3.20278
\(47\) 275.572 0.855241 0.427621 − 0.903958i \(-0.359352\pi\)
0.427621 + 0.903958i \(0.359352\pi\)
\(48\) 146.997 0.442026
\(49\) 363.378 1.05941
\(50\) −128.667 −0.363925
\(51\) 0 0
\(52\) 820.141 2.18717
\(53\) −180.696 −0.468310 −0.234155 − 0.972199i \(-0.575232\pi\)
−0.234155 + 0.972199i \(0.575232\pi\)
\(54\) 307.020 0.773706
\(55\) 328.112 0.804411
\(56\) −1434.66 −3.42347
\(57\) −42.6003 −0.0989919
\(58\) 811.853 1.83796
\(59\) 284.982 0.628839 0.314420 − 0.949284i \(-0.398190\pi\)
0.314420 + 0.949284i \(0.398190\pi\)
\(60\) −104.601 −0.225064
\(61\) 644.300 1.35236 0.676182 − 0.736735i \(-0.263633\pi\)
0.676182 + 0.736735i \(0.263633\pi\)
\(62\) 1479.08 3.02974
\(63\) −683.571 −1.36701
\(64\) 179.280 0.350156
\(65\) −221.800 −0.423245
\(66\) 382.161 0.712739
\(67\) 396.688 0.723330 0.361665 − 0.932308i \(-0.382208\pi\)
0.361665 + 0.932308i \(0.382208\pi\)
\(68\) 0 0
\(69\) −219.687 −0.383294
\(70\) 683.936 1.16780
\(71\) 573.342 0.958354 0.479177 − 0.877718i \(-0.340935\pi\)
0.479177 + 0.877718i \(0.340935\pi\)
\(72\) 1388.34 2.27246
\(73\) −574.739 −0.921481 −0.460741 − 0.887535i \(-0.652416\pi\)
−0.460741 + 0.887535i \(0.652416\pi\)
\(74\) −497.524 −0.781568
\(75\) 28.2884 0.0435528
\(76\) −696.050 −1.05056
\(77\) −1744.10 −2.58128
\(78\) −258.337 −0.375012
\(79\) 184.587 0.262882 0.131441 − 0.991324i \(-0.458040\pi\)
0.131441 + 0.991324i \(0.458040\pi\)
\(80\) −649.549 −0.907772
\(81\) 626.929 0.859985
\(82\) −547.263 −0.737013
\(83\) 626.402 0.828392 0.414196 − 0.910188i \(-0.364063\pi\)
0.414196 + 0.910188i \(0.364063\pi\)
\(84\) 556.010 0.722210
\(85\) 0 0
\(86\) 735.351 0.922034
\(87\) −178.492 −0.219958
\(88\) 3542.28 4.29100
\(89\) −454.081 −0.540815 −0.270407 − 0.962746i \(-0.587158\pi\)
−0.270407 + 0.962746i \(0.587158\pi\)
\(90\) −661.853 −0.775171
\(91\) 1178.99 1.35815
\(92\) −3589.50 −4.06773
\(93\) −325.187 −0.362584
\(94\) −1418.28 −1.55622
\(95\) 188.241 0.203296
\(96\) −267.909 −0.284826
\(97\) −123.922 −0.129715 −0.0648577 − 0.997895i \(-0.520659\pi\)
−0.0648577 + 0.997895i \(0.520659\pi\)
\(98\) −1870.19 −1.92773
\(99\) 1687.78 1.71342
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.l.1.1 5
17.16 even 2 85.4.a.g.1.1 ✓ 5
51.50 odd 2 765.4.a.m.1.5 5
68.67 odd 2 1360.4.a.w.1.4 5
85.33 odd 4 425.4.b.i.324.9 10
85.67 odd 4 425.4.b.i.324.2 10
85.84 even 2 425.4.a.i.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.a.g.1.1 ✓ 5 17.16 even 2
425.4.a.i.1.5 5 85.84 even 2
425.4.b.i.324.2 10 85.67 odd 4
425.4.b.i.324.9 10 85.33 odd 4
765.4.a.m.1.5 5 51.50 odd 2
1360.4.a.w.1.4 5 68.67 odd 2
1445.4.a.l.1.1 5 1.1 even 1 trivial