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 1.2
Root \(5.30190\) 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)
 
\(f(q)\) \(=\) \(q-4.30190 q^{2} +9.17969 q^{3} +10.5063 q^{4} -5.00000 q^{5} -39.4901 q^{6} -13.6050 q^{7} -10.7820 q^{8} +57.2667 q^{9} +21.5095 q^{10} +51.5323 q^{11} +96.4449 q^{12} -67.8563 q^{13} +58.5274 q^{14} -45.8985 q^{15} -37.6677 q^{16} -246.356 q^{18} -81.6297 q^{19} -52.5317 q^{20} -124.890 q^{21} -221.687 q^{22} +49.9145 q^{23} -98.9753 q^{24} +25.0000 q^{25} +291.911 q^{26} +277.839 q^{27} -142.939 q^{28} +92.8555 q^{29} +197.450 q^{30} +154.486 q^{31} +248.298 q^{32} +473.051 q^{33} +68.0250 q^{35} +601.663 q^{36} -388.139 q^{37} +351.163 q^{38} -622.900 q^{39} +53.9099 q^{40} +504.273 q^{41} +537.263 q^{42} +79.9009 q^{43} +541.416 q^{44} -286.334 q^{45} -214.727 q^{46} -199.842 q^{47} -345.777 q^{48} -157.904 q^{49} -107.547 q^{50} -712.921 q^{52} -391.083 q^{53} -1195.24 q^{54} -257.662 q^{55} +146.689 q^{56} -749.335 q^{57} -399.455 q^{58} +821.717 q^{59} -482.224 q^{60} +541.688 q^{61} -664.582 q^{62} -779.114 q^{63} -766.813 q^{64} +339.282 q^{65} -2035.02 q^{66} +109.626 q^{67} +458.200 q^{69} -292.637 q^{70} +263.969 q^{71} -617.449 q^{72} +1149.07 q^{73} +1669.73 q^{74} +229.492 q^{75} -857.629 q^{76} -701.098 q^{77} +2679.65 q^{78} -18.5601 q^{79} +188.338 q^{80} +1004.28 q^{81} -2169.33 q^{82} +622.481 q^{83} -1312.13 q^{84} -343.726 q^{86} +852.385 q^{87} -555.621 q^{88} -675.544 q^{89} +1231.78 q^{90} +923.186 q^{91} +524.419 q^{92} +1418.13 q^{93} +859.698 q^{94} +408.148 q^{95} +2279.30 q^{96} +234.000 q^{97} +679.286 q^{98} +2951.09 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} + 2 q^{3} + 48 q^{4} - 40 q^{5} - 44 q^{6} - 32 q^{7} - 36 q^{8} + 162 q^{9} - 20 q^{10} + 44 q^{11} + 36 q^{12} + 86 q^{13} - 146 q^{14} - 10 q^{15} + 348 q^{16} - 8 q^{18} + 288 q^{19} - 240 q^{20}+ \cdots + 5258 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.30190 −1.52095 −0.760475 − 0.649367i \(-0.775034\pi\)
−0.760475 + 0.649367i \(0.775034\pi\)
\(3\) 9.17969 1.76663 0.883316 − 0.468778i \(-0.155305\pi\)
0.883316 + 0.468778i \(0.155305\pi\)
\(4\) 10.5063 1.31329
\(5\) −5.00000 −0.447214
\(6\) −39.4901 −2.68696
\(7\) −13.6050 −0.734601 −0.367301 − 0.930102i \(-0.619718\pi\)
−0.367301 + 0.930102i \(0.619718\pi\)
\(8\) −10.7820 −0.476501
\(9\) 57.2667 2.12099
\(10\) 21.5095 0.680190
\(11\) 51.5323 1.41251 0.706254 − 0.707959i \(-0.250384\pi\)
0.706254 + 0.707959i \(0.250384\pi\)
\(12\) 96.4449 2.32010
\(13\) −67.8563 −1.44769 −0.723844 − 0.689963i \(-0.757627\pi\)
−0.723844 + 0.689963i \(0.757627\pi\)
\(14\) 58.5274 1.11729
\(15\) −45.8985 −0.790062
\(16\) −37.6677 −0.588557
\(17\) 0 0
\(18\) −246.356 −3.22592
\(19\) −81.6297 −0.985639 −0.492819 − 0.870132i \(-0.664034\pi\)
−0.492819 + 0.870132i \(0.664034\pi\)
\(20\) −52.5317 −0.587322
\(21\) −124.890 −1.29777
\(22\) −221.687 −2.14835
\(23\) 49.9145 0.452517 0.226259 − 0.974067i \(-0.427351\pi\)
0.226259 + 0.974067i \(0.427351\pi\)
\(24\) −98.9753 −0.841802
\(25\) 25.0000 0.200000
\(26\) 291.911 2.20186
\(27\) 277.839 1.98038
\(28\) −142.939 −0.964746
\(29\) 92.8555 0.594580 0.297290 − 0.954787i \(-0.403917\pi\)
0.297290 + 0.954787i \(0.403917\pi\)
\(30\) 197.450 1.20165
\(31\) 154.486 0.895048 0.447524 − 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(32\) 248.298 1.37167
\(33\) 473.051 2.49538
\(34\) 0 0
\(35\) 68.0250 0.328524
\(36\) 601.663 2.78548
\(37\) −388.139 −1.72459 −0.862293 − 0.506410i \(-0.830972\pi\)
−0.862293 + 0.506410i \(0.830972\pi\)
\(38\) 351.163 1.49911
\(39\) −622.900 −2.55753
\(40\) 53.9099 0.213098
\(41\) 504.273 1.92083 0.960417 − 0.278566i \(-0.0898590\pi\)
0.960417 + 0.278566i \(0.0898590\pi\)
\(42\) 537.263 1.97385
\(43\) 79.9009 0.283367 0.141683 − 0.989912i \(-0.454748\pi\)
0.141683 + 0.989912i \(0.454748\pi\)
\(44\) 541.416 1.85503
\(45\) −286.334 −0.948535
\(46\) −214.727 −0.688257
\(47\) −199.842 −0.620210 −0.310105 − 0.950702i \(-0.600364\pi\)
−0.310105 + 0.950702i \(0.600364\pi\)
\(48\) −345.777 −1.03976
\(49\) −157.904 −0.460361
\(50\) −107.547 −0.304190
\(51\) 0 0
\(52\) −712.921 −1.90124
\(53\) −391.083 −1.01357 −0.506786 − 0.862072i \(-0.669167\pi\)
−0.506786 + 0.862072i \(0.669167\pi\)
\(54\) −1195.24 −3.01206
\(55\) −257.662 −0.631693
\(56\) 146.689 0.350038
\(57\) −749.335 −1.74126
\(58\) −399.455 −0.904328
\(59\) 821.717 1.81319 0.906597 − 0.421997i \(-0.138671\pi\)
0.906597 + 0.421997i \(0.138671\pi\)
\(60\) −482.224 −1.03758
\(61\) 541.688 1.13698 0.568492 − 0.822689i \(-0.307527\pi\)
0.568492 + 0.822689i \(0.307527\pi\)
\(62\) −664.582 −1.36132
\(63\) −779.114 −1.55808
\(64\) −766.813 −1.49768
\(65\) 339.282 0.647426
\(66\) −2035.02 −3.79535
\(67\) 109.626 0.199894 0.0999471 − 0.994993i \(-0.468133\pi\)
0.0999471 + 0.994993i \(0.468133\pi\)
\(68\) 0 0
\(69\) 458.200 0.799432
\(70\) −292.637 −0.499668
\(71\) 263.969 0.441230 0.220615 − 0.975361i \(-0.429194\pi\)
0.220615 + 0.975361i \(0.429194\pi\)
\(72\) −617.449 −1.01065
\(73\) 1149.07 1.84231 0.921153 − 0.389202i \(-0.127249\pi\)
0.921153 + 0.389202i \(0.127249\pi\)
\(74\) 1669.73 2.62301
\(75\) 229.492 0.353326
\(76\) −857.629 −1.29443
\(77\) −701.098 −1.03763
\(78\) 2679.65 3.88988
\(79\) −18.5601 −0.0264326 −0.0132163 − 0.999913i \(-0.504207\pi\)
−0.0132163 + 0.999913i \(0.504207\pi\)
\(80\) 188.338 0.263211
\(81\) 1004.28 1.37761
\(82\) −2169.33 −2.92149
\(83\) 622.481 0.823207 0.411603 − 0.911363i \(-0.364969\pi\)
0.411603 + 0.911363i \(0.364969\pi\)
\(84\) −1312.13 −1.70435
\(85\) 0 0
\(86\) −343.726 −0.430987
\(87\) 852.385 1.05040
\(88\) −555.621 −0.673061
\(89\) −675.544 −0.804579 −0.402289 − 0.915513i \(-0.631785\pi\)
−0.402289 + 0.915513i \(0.631785\pi\)
\(90\) 1231.78 1.44268
\(91\) 923.186 1.06347
\(92\) 524.419 0.594287
\(93\) 1418.13 1.58122
\(94\) 859.698 0.943310
\(95\) 408.148 0.440791
\(96\) 2279.30 2.42323
\(97\) 234.000 0.244940 0.122470 − 0.992472i \(-0.460919\pi\)
0.122470 + 0.992472i \(0.460919\pi\)
\(98\) 679.286 0.700186
\(99\) 2951.09 2.99591
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.1.2 yes 8
17.16 even 2 1445.4.a.q.1.2 ✓ 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1445.4.a.q.1.2 ✓ 8 17.16 even 2
1445.4.a.r.1.2 yes 8 1.1 even 1 trivial