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 1.8
Root \(-4.38110\) 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)
 
\(f(q)\) \(=\) \(q+5.38110 q^{2} -4.56319 q^{3} +20.9562 q^{4} -5.00000 q^{5} -24.5550 q^{6} -26.2712 q^{7} +69.7188 q^{8} -6.17728 q^{9} -26.9055 q^{10} +2.68424 q^{11} -95.6273 q^{12} +62.7100 q^{13} -141.368 q^{14} +22.8160 q^{15} +207.514 q^{16} -33.2405 q^{18} -119.645 q^{19} -104.781 q^{20} +119.881 q^{21} +14.4442 q^{22} +173.490 q^{23} -318.140 q^{24} +25.0000 q^{25} +337.449 q^{26} +151.394 q^{27} -550.546 q^{28} -117.425 q^{29} +122.775 q^{30} +145.290 q^{31} +558.902 q^{32} -12.2487 q^{33} +131.356 q^{35} -129.452 q^{36} +190.246 q^{37} -643.820 q^{38} -286.158 q^{39} -348.594 q^{40} +38.0668 q^{41} +645.090 q^{42} +450.875 q^{43} +56.2516 q^{44} +30.8864 q^{45} +933.568 q^{46} -353.578 q^{47} -946.925 q^{48} +347.177 q^{49} +134.527 q^{50} +1314.17 q^{52} +24.3584 q^{53} +814.668 q^{54} -13.4212 q^{55} -1831.60 q^{56} +545.962 q^{57} -631.874 q^{58} -496.458 q^{59} +478.137 q^{60} +825.205 q^{61} +781.820 q^{62} +162.285 q^{63} +1347.40 q^{64} -313.550 q^{65} -65.9115 q^{66} +864.475 q^{67} -791.669 q^{69} +706.840 q^{70} +521.227 q^{71} -430.672 q^{72} +192.589 q^{73} +1023.73 q^{74} -114.080 q^{75} -2507.30 q^{76} -70.5183 q^{77} -1539.84 q^{78} +847.542 q^{79} -1037.57 q^{80} -524.055 q^{81} +204.841 q^{82} +265.881 q^{83} +2512.25 q^{84} +2426.20 q^{86} +535.832 q^{87} +187.142 q^{88} +718.221 q^{89} +166.203 q^{90} -1647.47 q^{91} +3635.70 q^{92} -662.986 q^{93} -1902.64 q^{94} +598.224 q^{95} -2550.38 q^{96} -291.145 q^{97} +1868.19 q^{98} -16.5813 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} + 2 q^{3} + 48 q^{4} - 40 q^{5} - 44 q^{6} - 32 q^{7} - 36 q^{8} + 162 q^{9} - 20 q^{10} + 44 q^{11} + 36 q^{12} + 86 q^{13} - 146 q^{14} - 10 q^{15} + 348 q^{16} - 8 q^{18} + 288 q^{19} - 240 q^{20}+ \cdots + 5258 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.38110 1.90251 0.951253 − 0.308412i \(-0.0997974\pi\)
0.951253 + 0.308412i \(0.0997974\pi\)
\(3\) −4.56319 −0.878187 −0.439093 − 0.898441i \(-0.644700\pi\)
−0.439093 + 0.898441i \(0.644700\pi\)
\(4\) 20.9562 2.61953
\(5\) −5.00000 −0.447214
\(6\) −24.5550 −1.67076
\(7\) −26.2712 −1.41851 −0.709256 − 0.704951i \(-0.750969\pi\)
−0.709256 + 0.704951i \(0.750969\pi\)
\(8\) 69.7188 3.08116
\(9\) −6.17728 −0.228788
\(10\) −26.9055 −0.850827
\(11\) 2.68424 0.0735754 0.0367877 − 0.999323i \(-0.488287\pi\)
0.0367877 + 0.999323i \(0.488287\pi\)
\(12\) −95.6273 −2.30044
\(13\) 62.7100 1.33789 0.668947 − 0.743310i \(-0.266745\pi\)
0.668947 + 0.743310i \(0.266745\pi\)
\(14\) −141.368 −2.69873
\(15\) 22.8160 0.392737
\(16\) 207.514 3.24240
\(17\) 0 0
\(18\) −33.2405 −0.435271
\(19\) −119.645 −1.44465 −0.722326 − 0.691553i \(-0.756927\pi\)
−0.722326 + 0.691553i \(0.756927\pi\)
\(20\) −104.781 −1.17149
\(21\) 119.881 1.24572
\(22\) 14.4442 0.139978
\(23\) 173.490 1.57284 0.786418 − 0.617695i \(-0.211933\pi\)
0.786418 + 0.617695i \(0.211933\pi\)
\(24\) −318.140 −2.70584
\(25\) 25.0000 0.200000
\(26\) 337.449 2.54535
\(27\) 151.394 1.07911
\(28\) −550.546 −3.71583
\(29\) −117.425 −0.751905 −0.375952 − 0.926639i \(-0.622684\pi\)
−0.375952 + 0.926639i \(0.622684\pi\)
\(30\) 122.775 0.747185
\(31\) 145.290 0.841769 0.420885 − 0.907114i \(-0.361720\pi\)
0.420885 + 0.907114i \(0.361720\pi\)
\(32\) 558.902 3.08753
\(33\) −12.2487 −0.0646129
\(34\) 0 0
\(35\) 131.356 0.634378
\(36\) −129.452 −0.599317
\(37\) 190.246 0.845303 0.422652 − 0.906292i \(-0.361099\pi\)
0.422652 + 0.906292i \(0.361099\pi\)
\(38\) −643.820 −2.74846
\(39\) −286.158 −1.17492
\(40\) −348.594 −1.37794
\(41\) 38.0668 0.145001 0.0725004 − 0.997368i \(-0.476902\pi\)
0.0725004 + 0.997368i \(0.476902\pi\)
\(42\) 645.090 2.36999
\(43\) 450.875 1.59902 0.799509 − 0.600654i \(-0.205093\pi\)
0.799509 + 0.600654i \(0.205093\pi\)
\(44\) 56.2516 0.192733
\(45\) 30.8864 0.102317
\(46\) 933.568 2.99233
\(47\) −353.578 −1.09733 −0.548667 − 0.836041i \(-0.684865\pi\)
−0.548667 + 0.836041i \(0.684865\pi\)
\(48\) −946.925 −2.84744
\(49\) 347.177 1.01218
\(50\) 134.527 0.380501
\(51\) 0 0
\(52\) 1314.17 3.50465
\(53\) 24.3584 0.0631298 0.0315649 − 0.999502i \(-0.489951\pi\)
0.0315649 + 0.999502i \(0.489951\pi\)
\(54\) 814.668 2.05300
\(55\) −13.4212 −0.0329039
\(56\) −1831.60 −4.37067
\(57\) 545.962 1.26867
\(58\) −631.874 −1.43050
\(59\) −496.458 −1.09548 −0.547740 − 0.836648i \(-0.684512\pi\)
−0.547740 + 0.836648i \(0.684512\pi\)
\(60\) 478.137 1.02879
\(61\) 825.205 1.73208 0.866039 − 0.499977i \(-0.166658\pi\)
0.866039 + 0.499977i \(0.166658\pi\)
\(62\) 781.820 1.60147
\(63\) 162.285 0.324539
\(64\) 1347.40 2.63164
\(65\) −313.550 −0.598325
\(66\) −65.9115 −0.122927
\(67\) 864.475 1.57630 0.788152 − 0.615480i \(-0.211038\pi\)
0.788152 + 0.615480i \(0.211038\pi\)
\(68\) 0 0
\(69\) −791.669 −1.38124
\(70\) 706.840 1.20691
\(71\) 521.227 0.871243 0.435622 − 0.900130i \(-0.356529\pi\)
0.435622 + 0.900130i \(0.356529\pi\)
\(72\) −430.672 −0.704933
\(73\) 192.589 0.308778 0.154389 − 0.988010i \(-0.450659\pi\)
0.154389 + 0.988010i \(0.450659\pi\)
\(74\) 1023.73 1.60819
\(75\) −114.080 −0.175637
\(76\) −2507.30 −3.78431
\(77\) −70.5183 −0.104368
\(78\) −1539.84 −2.23529
\(79\) 847.542 1.20704 0.603518 − 0.797349i \(-0.293765\pi\)
0.603518 + 0.797349i \(0.293765\pi\)
\(80\) −1037.57 −1.45005
\(81\) −524.055 −0.718868
\(82\) 204.841 0.275865
\(83\) 265.881 0.351618 0.175809 − 0.984424i \(-0.443746\pi\)
0.175809 + 0.984424i \(0.443746\pi\)
\(84\) 2512.25 3.26320
\(85\) 0 0
\(86\) 2426.20 3.04214
\(87\) 535.832 0.660313
\(88\) 187.142 0.226698
\(89\) 718.221 0.855407 0.427704 − 0.903919i \(-0.359323\pi\)
0.427704 + 0.903919i \(0.359323\pi\)
\(90\) 166.203 0.194659
\(91\) −1647.47 −1.89782
\(92\) 3635.70 4.12009
\(93\) −662.986 −0.739231
\(94\) −1902.64 −2.08769
\(95\) 598.224 0.646068
\(96\) −2550.38 −2.71143
\(97\) −291.145 −0.304756 −0.152378 − 0.988322i \(-0.548693\pi\)
−0.152378 + 0.988322i \(0.548693\pi\)
\(98\) 1868.19 1.92567
\(99\) −16.5813 −0.0168332
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.1.8 yes 8
17.16 even 2 1445.4.a.q.1.8 ✓ 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1445.4.a.q.1.8 ✓ 8 17.16 even 2
1445.4.a.r.1.8 yes 8 1.1 even 1 trivial