Properties

Label 1027.4.a.c.1.13
Level $1027$
Weight $4$
Character 1027.1
Self dual yes
Analytic conductor $60.595$
Analytic rank $1$
Dimension $56$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1027,4,Mod(1,1027)] 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("1027.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1027, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1027 = 13 \cdot 79 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1027.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [56,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(60.5949615759\)
Analytic rank: \(1\)
Dimension: \(56\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.13
Character \(\chi\) \(=\) 1027.1

$q$-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)
 
\(f(q)\) \(=\) \(q-3.75425 q^{2} +6.45205 q^{3} +6.09437 q^{4} -8.91793 q^{5} -24.2226 q^{6} -20.7363 q^{7} +7.15420 q^{8} +14.6289 q^{9} +33.4801 q^{10} +5.50070 q^{11} +39.3212 q^{12} -13.0000 q^{13} +77.8490 q^{14} -57.5389 q^{15} -75.6136 q^{16} +69.1805 q^{17} -54.9206 q^{18} +116.691 q^{19} -54.3492 q^{20} -133.791 q^{21} -20.6510 q^{22} -84.2973 q^{23} +46.1592 q^{24} -45.4704 q^{25} +48.8052 q^{26} -79.8188 q^{27} -126.374 q^{28} +165.428 q^{29} +216.015 q^{30} +202.107 q^{31} +226.639 q^{32} +35.4908 q^{33} -259.721 q^{34} +184.925 q^{35} +89.1541 q^{36} +73.2915 q^{37} -438.087 q^{38} -83.8766 q^{39} -63.8007 q^{40} +350.993 q^{41} +502.286 q^{42} -55.7931 q^{43} +33.5233 q^{44} -130.460 q^{45} +316.473 q^{46} -53.3527 q^{47} -487.863 q^{48} +86.9921 q^{49} +170.707 q^{50} +446.356 q^{51} -79.2268 q^{52} -621.381 q^{53} +299.659 q^{54} -49.0549 q^{55} -148.351 q^{56} +752.896 q^{57} -621.057 q^{58} +413.432 q^{59} -350.664 q^{60} -324.439 q^{61} -758.761 q^{62} -303.349 q^{63} -245.948 q^{64} +115.933 q^{65} -133.241 q^{66} -378.553 q^{67} +421.611 q^{68} -543.890 q^{69} -694.252 q^{70} -645.008 q^{71} +104.658 q^{72} -438.929 q^{73} -275.155 q^{74} -293.377 q^{75} +711.158 q^{76} -114.064 q^{77} +314.894 q^{78} +79.0000 q^{79} +674.317 q^{80} -909.976 q^{81} -1317.72 q^{82} -1085.09 q^{83} -815.374 q^{84} -616.947 q^{85} +209.461 q^{86} +1067.35 q^{87} +39.3531 q^{88} -270.342 q^{89} +489.778 q^{90} +269.571 q^{91} -513.739 q^{92} +1304.01 q^{93} +200.299 q^{94} -1040.64 q^{95} +1462.28 q^{96} -580.798 q^{97} -326.590 q^{98} +80.4694 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q - 16 q^{2} - 17 q^{3} + 208 q^{4} - 35 q^{5} + 29 q^{6} - 37 q^{7} - 156 q^{8} + 415 q^{9} - 67 q^{10} - 114 q^{11} - 196 q^{12} - 728 q^{13} - 183 q^{14} - 17 q^{15} + 680 q^{16} - 383 q^{17} - 342 q^{18}+ \cdots - 11354 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.75425 −1.32733 −0.663663 − 0.748031i \(-0.730999\pi\)
−0.663663 + 0.748031i \(0.730999\pi\)
\(3\) 6.45205 1.24170 0.620849 − 0.783930i \(-0.286788\pi\)
0.620849 + 0.783930i \(0.286788\pi\)
\(4\) 6.09437 0.761797
\(5\) −8.91793 −0.797644 −0.398822 − 0.917028i \(-0.630581\pi\)
−0.398822 + 0.917028i \(0.630581\pi\)
\(6\) −24.2226 −1.64814
\(7\) −20.7363 −1.11965 −0.559826 − 0.828610i \(-0.689132\pi\)
−0.559826 + 0.828610i \(0.689132\pi\)
\(8\) 7.15420 0.316174
\(9\) 14.6289 0.541812
\(10\) 33.4801 1.05873
\(11\) 5.50070 0.150775 0.0753874 − 0.997154i \(-0.475981\pi\)
0.0753874 + 0.997154i \(0.475981\pi\)
\(12\) 39.3212 0.945921
\(13\) −13.0000 −0.277350
\(14\) 77.8490 1.48614
\(15\) −57.5389 −0.990433
\(16\) −75.6136 −1.18146
\(17\) 69.1805 0.986984 0.493492 − 0.869750i \(-0.335720\pi\)
0.493492 + 0.869750i \(0.335720\pi\)
\(18\) −54.9206 −0.719162
\(19\) 116.691 1.40899 0.704493 − 0.709710i \(-0.251174\pi\)
0.704493 + 0.709710i \(0.251174\pi\)
\(20\) −54.3492 −0.607643
\(21\) −133.791 −1.39027
\(22\) −20.6510 −0.200128
\(23\) −84.2973 −0.764226 −0.382113 − 0.924116i \(-0.624803\pi\)
−0.382113 + 0.924116i \(0.624803\pi\)
\(24\) 46.1592 0.392592
\(25\) −45.4704 −0.363764
\(26\) 48.8052 0.368134
\(27\) −79.8188 −0.568931
\(28\) −126.374 −0.852947
\(29\) 165.428 1.05928 0.529641 − 0.848222i \(-0.322327\pi\)
0.529641 + 0.848222i \(0.322327\pi\)
\(30\) 216.015 1.31463
\(31\) 202.107 1.17095 0.585476 − 0.810689i \(-0.300907\pi\)
0.585476 + 0.810689i \(0.300907\pi\)
\(32\) 226.639 1.25201
\(33\) 35.4908 0.187217
\(34\) −259.721 −1.31005
\(35\) 184.925 0.893084
\(36\) 89.1541 0.412751
\(37\) 73.2915 0.325650 0.162825 − 0.986655i \(-0.447939\pi\)
0.162825 + 0.986655i \(0.447939\pi\)
\(38\) −438.087 −1.87019
\(39\) −83.8766 −0.344385
\(40\) −63.8007 −0.252194
\(41\) 350.993 1.33697 0.668487 − 0.743724i \(-0.266942\pi\)
0.668487 + 0.743724i \(0.266942\pi\)
\(42\) 502.286 1.84534
\(43\) −55.7931 −0.197869 −0.0989345 − 0.995094i \(-0.531543\pi\)
−0.0989345 + 0.995094i \(0.531543\pi\)
\(44\) 33.5233 0.114860
\(45\) −130.460 −0.432173
\(46\) 316.473 1.01438
\(47\) −53.3527 −0.165581 −0.0827903 − 0.996567i \(-0.526383\pi\)
−0.0827903 + 0.996567i \(0.526383\pi\)
\(48\) −487.863 −1.46702
\(49\) 86.9921 0.253621
\(50\) 170.707 0.482833
\(51\) 446.356 1.22554
\(52\) −79.2268 −0.211284
\(53\) −621.381 −1.61044 −0.805219 − 0.592977i \(-0.797952\pi\)
−0.805219 + 0.592977i \(0.797952\pi\)
\(54\) 299.659 0.755157
\(55\) −49.0549 −0.120265
\(56\) −148.351 −0.354005
\(57\) 752.896 1.74954
\(58\) −621.057 −1.40601
\(59\) 413.432 0.912277 0.456138 − 0.889909i \(-0.349232\pi\)
0.456138 + 0.889909i \(0.349232\pi\)
\(60\) −350.664 −0.754508
\(61\) −324.439 −0.680987 −0.340494 − 0.940247i \(-0.610594\pi\)
−0.340494 + 0.940247i \(0.610594\pi\)
\(62\) −758.761 −1.55424
\(63\) −303.349 −0.606641
\(64\) −245.948 −0.480368
\(65\) 115.933 0.221227
\(66\) −133.241 −0.248498
\(67\) −378.553 −0.690263 −0.345131 − 0.938554i \(-0.612166\pi\)
−0.345131 + 0.938554i \(0.612166\pi\)
\(68\) 421.611 0.751881
\(69\) −543.890 −0.948937
\(70\) −694.252 −1.18541
\(71\) −645.008 −1.07815 −0.539073 − 0.842259i \(-0.681225\pi\)
−0.539073 + 0.842259i \(0.681225\pi\)
\(72\) 104.658 0.171307
\(73\) −438.929 −0.703736 −0.351868 − 0.936050i \(-0.614453\pi\)
−0.351868 + 0.936050i \(0.614453\pi\)
\(74\) −275.155 −0.432244
\(75\) −293.377 −0.451684
\(76\) 711.158 1.07336
\(77\) −114.064 −0.168815
\(78\) 314.894 0.457111
\(79\) 79.0000 0.112509
\(80\) 674.317 0.942387
\(81\) −909.976 −1.24825
\(82\) −1317.72 −1.77460
\(83\) −1085.09 −1.43499 −0.717496 − 0.696562i \(-0.754712\pi\)
−0.717496 + 0.696562i \(0.754712\pi\)
\(84\) −815.374 −1.05910
\(85\) −616.947 −0.787262
\(86\) 209.461 0.262637
\(87\) 1067.35 1.31531
\(88\) 39.3531 0.0476711
\(89\) −270.342 −0.321980 −0.160990 − 0.986956i \(-0.551469\pi\)
−0.160990 + 0.986956i \(0.551469\pi\)
\(90\) 489.778 0.573635
\(91\) 269.571 0.310536
\(92\) −513.739 −0.582185
\(93\) 1304.01 1.45397
\(94\) 200.299 0.219779
\(95\) −1040.64 −1.12387
\(96\) 1462.28 1.55462
\(97\) −580.798 −0.607949 −0.303975 − 0.952680i \(-0.598314\pi\)
−0.303975 + 0.952680i \(0.598314\pi\)
\(98\) −326.590 −0.336638
\(99\) 80.4694 0.0816917
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 1027.4.a.c.1.13 ✓ 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1027.4.a.c.1.13 ✓ 56 1.1 even 1 trivial