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 = [3,3,-9,-1,15] 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: \(3\)
Coefficient field: 3.3.568.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 85)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.363328\) 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)
 
\(f(q)\) \(=\) \(q+4.50466 q^{2} -5.08998 q^{3} +12.2920 q^{4} +5.00000 q^{5} -22.9287 q^{6} +0.616696 q^{7} +19.3340 q^{8} -1.09206 q^{9} +22.5233 q^{10} +8.63535 q^{11} -62.5661 q^{12} +6.44012 q^{13} +2.77801 q^{14} -25.4499 q^{15} -11.2427 q^{16} -4.91934 q^{18} +7.96806 q^{19} +61.4600 q^{20} -3.13897 q^{21} +38.8994 q^{22} -66.1209 q^{23} -98.4099 q^{24} +25.0000 q^{25} +29.0106 q^{26} +142.988 q^{27} +7.58043 q^{28} -219.813 q^{29} -114.643 q^{30} +0.608686 q^{31} -205.317 q^{32} -43.9538 q^{33} +3.08348 q^{35} -13.4235 q^{36} +216.124 q^{37} +35.8934 q^{38} -32.7801 q^{39} +96.6701 q^{40} -355.508 q^{41} -14.1400 q^{42} +209.947 q^{43} +106.146 q^{44} -5.46028 q^{45} -297.852 q^{46} -324.626 q^{47} +57.2253 q^{48} -342.620 q^{49} +112.617 q^{50} +79.1619 q^{52} +189.591 q^{53} +644.114 q^{54} +43.1768 q^{55} +11.9232 q^{56} -40.5573 q^{57} -990.185 q^{58} -257.619 q^{59} -312.830 q^{60} -240.335 q^{61} +2.74192 q^{62} -0.673466 q^{63} -834.942 q^{64} +32.2006 q^{65} -197.997 q^{66} -66.9428 q^{67} +336.554 q^{69} +13.8900 q^{70} +131.664 q^{71} -21.1138 q^{72} -1172.76 q^{73} +973.565 q^{74} -127.250 q^{75} +97.9434 q^{76} +5.32539 q^{77} -147.663 q^{78} -707.825 q^{79} -56.2136 q^{80} -698.322 q^{81} -1601.44 q^{82} +1007.30 q^{83} -38.5843 q^{84} +945.740 q^{86} +1118.85 q^{87} +166.956 q^{88} +1013.01 q^{89} -24.5967 q^{90} +3.97159 q^{91} -812.758 q^{92} -3.09820 q^{93} -1462.33 q^{94} +39.8403 q^{95} +1045.06 q^{96} -973.147 q^{97} -1543.39 q^{98} -9.43029 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - 9 q^{3} - q^{4} + 15 q^{5} - 33 q^{6} - 34 q^{7} + 39 q^{8} + 60 q^{9} + 15 q^{10} - 52 q^{11} - 17 q^{12} + 19 q^{13} + 2 q^{14} - 45 q^{15} + 59 q^{16} - 153 q^{19} - 5 q^{20} + 286 q^{21}+ \cdots - 2524 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.50466 1.59264 0.796320 − 0.604876i \(-0.206777\pi\)
0.796320 + 0.604876i \(0.206777\pi\)
\(3\) −5.08998 −0.979568 −0.489784 − 0.871844i \(-0.662924\pi\)
−0.489784 + 0.871844i \(0.662924\pi\)
\(4\) 12.2920 1.53650
\(5\) 5.00000 0.447214
\(6\) −22.9287 −1.56010
\(7\) 0.616696 0.0332984 0.0166492 − 0.999861i \(-0.494700\pi\)
0.0166492 + 0.999861i \(0.494700\pi\)
\(8\) 19.3340 0.854451
\(9\) −1.09206 −0.0404465
\(10\) 22.5233 0.712250
\(11\) 8.63535 0.236696 0.118348 − 0.992972i \(-0.462240\pi\)
0.118348 + 0.992972i \(0.462240\pi\)
\(12\) −62.5661 −1.50511
\(13\) 6.44012 0.137397 0.0686987 − 0.997637i \(-0.478115\pi\)
0.0686987 + 0.997637i \(0.478115\pi\)
\(14\) 2.77801 0.0530324
\(15\) −25.4499 −0.438076
\(16\) −11.2427 −0.175668
\(17\) 0 0
\(18\) −4.91934 −0.0644167
\(19\) 7.96806 0.0962104 0.0481052 − 0.998842i \(-0.484682\pi\)
0.0481052 + 0.998842i \(0.484682\pi\)
\(20\) 61.4600 0.687144
\(21\) −3.13897 −0.0326181
\(22\) 38.8994 0.376972
\(23\) −66.1209 −0.599442 −0.299721 − 0.954027i \(-0.596894\pi\)
−0.299721 + 0.954027i \(0.596894\pi\)
\(24\) −98.4099 −0.836993
\(25\) 25.0000 0.200000
\(26\) 29.0106 0.218825
\(27\) 142.988 1.01919
\(28\) 7.58043 0.0511631
\(29\) −219.813 −1.40753 −0.703764 − 0.710434i \(-0.748499\pi\)
−0.703764 + 0.710434i \(0.748499\pi\)
\(30\) −114.643 −0.697697
\(31\) 0.608686 0.00352655 0.00176328 − 0.999998i \(-0.499439\pi\)
0.00176328 + 0.999998i \(0.499439\pi\)
\(32\) −205.317 −1.13423
\(33\) −43.9538 −0.231860
\(34\) 0 0
\(35\) 3.08348 0.0148915
\(36\) −13.4235 −0.0621461
\(37\) 216.124 0.960285 0.480142 − 0.877191i \(-0.340585\pi\)
0.480142 + 0.877191i \(0.340585\pi\)
\(38\) 35.8934 0.153229
\(39\) −32.7801 −0.134590
\(40\) 96.6701 0.382122
\(41\) −355.508 −1.35417 −0.677086 − 0.735904i \(-0.736757\pi\)
−0.677086 + 0.735904i \(0.736757\pi\)
\(42\) −14.1400 −0.0519489
\(43\) 209.947 0.744572 0.372286 − 0.928118i \(-0.378574\pi\)
0.372286 + 0.928118i \(0.378574\pi\)
\(44\) 106.146 0.363684
\(45\) −5.46028 −0.0180882
\(46\) −297.852 −0.954694
\(47\) −324.626 −1.00748 −0.503740 − 0.863855i \(-0.668043\pi\)
−0.503740 + 0.863855i \(0.668043\pi\)
\(48\) 57.2253 0.172078
\(49\) −342.620 −0.998891
\(50\) 112.617 0.318528
\(51\) 0 0
\(52\) 79.1619 0.211111
\(53\) 189.591 0.491364 0.245682 − 0.969351i \(-0.420988\pi\)
0.245682 + 0.969351i \(0.420988\pi\)
\(54\) 644.114 1.62320
\(55\) 43.1768 0.105854
\(56\) 11.9232 0.0284519
\(57\) −40.5573 −0.0942446
\(58\) −990.185 −2.24168
\(59\) −257.619 −0.568459 −0.284230 − 0.958756i \(-0.591738\pi\)
−0.284230 + 0.958756i \(0.591738\pi\)
\(60\) −312.830 −0.673104
\(61\) −240.335 −0.504455 −0.252227 − 0.967668i \(-0.581163\pi\)
−0.252227 + 0.967668i \(0.581163\pi\)
\(62\) 2.74192 0.00561653
\(63\) −0.673466 −0.00134681
\(64\) −834.942 −1.63075
\(65\) 32.2006 0.0614460
\(66\) −197.997 −0.369269
\(67\) −66.9428 −0.122065 −0.0610326 − 0.998136i \(-0.519439\pi\)
−0.0610326 + 0.998136i \(0.519439\pi\)
\(68\) 0 0
\(69\) 336.554 0.587194
\(70\) 13.8900 0.0237168
\(71\) 131.664 0.220080 0.110040 − 0.993927i \(-0.464902\pi\)
0.110040 + 0.993927i \(0.464902\pi\)
\(72\) −21.1138 −0.0345596
\(73\) −1172.76 −1.88029 −0.940145 − 0.340774i \(-0.889311\pi\)
−0.940145 + 0.340774i \(0.889311\pi\)
\(74\) 973.565 1.52939
\(75\) −127.250 −0.195914
\(76\) 97.9434 0.147827
\(77\) 5.32539 0.00788161
\(78\) −147.663 −0.214354
\(79\) −707.825 −1.00806 −0.504029 − 0.863687i \(-0.668149\pi\)
−0.504029 + 0.863687i \(0.668149\pi\)
\(80\) −56.2136 −0.0785609
\(81\) −698.322 −0.957918
\(82\) −1601.44 −2.15671
\(83\) 1007.30 1.33212 0.666058 − 0.745900i \(-0.267980\pi\)
0.666058 + 0.745900i \(0.267980\pi\)
\(84\) −38.5843 −0.0501177
\(85\) 0 0
\(86\) 945.740 1.18583
\(87\) 1118.85 1.37877
\(88\) 166.956 0.202245
\(89\) 1013.01 1.20650 0.603249 − 0.797553i \(-0.293872\pi\)
0.603249 + 0.797553i \(0.293872\pi\)
\(90\) −24.5967 −0.0288080
\(91\) 3.97159 0.00457512
\(92\) −812.758 −0.921042
\(93\) −3.09820 −0.00345450
\(94\) −1462.33 −1.60455
\(95\) 39.8403 0.0430266
\(96\) 1045.06 1.11105
\(97\) −973.147 −1.01864 −0.509320 − 0.860577i \(-0.670103\pi\)
−0.509320 + 0.860577i \(0.670103\pi\)
\(98\) −1543.39 −1.59087
\(99\) −9.43029 −0.00957353
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.k.1.3 3
17.16 even 2 85.4.a.f.1.3 ✓ 3
51.50 odd 2 765.4.a.k.1.1 3
68.67 odd 2 1360.4.a.p.1.2 3
85.33 odd 4 425.4.b.h.324.1 6
85.67 odd 4 425.4.b.h.324.6 6
85.84 even 2 425.4.a.f.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.a.f.1.3 ✓ 3 17.16 even 2
425.4.a.f.1.1 3 85.84 even 2
425.4.b.h.324.1 6 85.33 odd 4
425.4.b.h.324.6 6 85.67 odd 4
765.4.a.k.1.1 3 51.50 odd 2
1360.4.a.p.1.2 3 68.67 odd 2
1445.4.a.k.1.3 3 1.1 even 1 trivial