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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [230,4,Mod(31,230)] 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("230.31"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(230, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([0, 6])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 230.g (of order \(11\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [70,-14,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.5704393013\)
Analytic rank: \(0\)
Dimension: \(70\)
Relative dimension: \(7\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label 41.5
Character \(\chi\) \(=\) 230.41
Dual form 230.4.g.d.101.5

$q$-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+(-1.91899 - 0.563465i) q^{2} +(1.16443 - 1.34382i) q^{3} +(3.36501 + 2.16256i) q^{4} +(-0.711574 + 4.94911i) q^{5} +(-2.99172 + 1.92266i) q^{6} +(-6.00332 - 13.1454i) q^{7} +(-5.23889 - 6.04600i) q^{8} +(3.39254 + 23.5956i) q^{9} +(4.15415 - 9.09632i) q^{10} +(-9.69089 + 2.84550i) q^{11} +(6.82441 - 2.00383i) q^{12} +(30.9160 - 67.6965i) q^{13} +(4.11329 + 28.6086i) q^{14} +(5.82214 + 6.71910i) q^{15} +(6.64664 + 14.5541i) q^{16} +(24.4958 - 15.7425i) q^{17} +(6.78507 - 47.1912i) q^{18} +(-33.5780 - 21.5793i) q^{19} +(-13.0972 + 15.1150i) q^{20} +(-24.6555 - 7.23952i) q^{21} +20.2000 q^{22} +(103.071 + 39.2852i) q^{23} -14.2250 q^{24} +(-23.9873 - 7.04331i) q^{25} +(-97.4719 + 112.489i) q^{26} +(76.0468 + 48.8723i) q^{27} +(8.22658 - 57.2171i) q^{28} +(173.571 - 111.547i) q^{29} +(-7.38662 - 16.1744i) q^{30} +(-89.0819 - 102.806i) q^{31} +(-4.55407 - 31.6743i) q^{32} +(-7.46050 + 16.3362i) q^{33} +(-55.8775 + 16.4071i) q^{34} +(69.3299 - 20.3571i) q^{35} +(-39.6111 + 86.7362i) q^{36} +(-38.6315 - 268.688i) q^{37} +(52.2766 + 60.3304i) q^{38} +(-54.9726 - 120.373i) q^{39} +(33.6501 - 21.6256i) q^{40} +(53.4518 - 371.765i) q^{41} +(43.2344 + 27.7851i) q^{42} +(-35.4675 + 40.9317i) q^{43} +(-38.7636 - 11.3820i) q^{44} -119.191 q^{45} +(-175.656 - 133.465i) q^{46} +55.9261 q^{47} +(27.2976 + 8.01531i) q^{48} +(87.8547 - 101.390i) q^{49} +(42.0627 + 27.0320i) q^{50} +(7.36850 - 51.2490i) q^{51} +(250.431 - 160.942i) q^{52} +(-260.758 - 570.980i) q^{53} +(-118.395 - 136.635i) q^{54} +(-7.18691 - 49.9861i) q^{55} +(-48.0265 + 105.163i) q^{56} +(-68.0978 + 19.9953i) q^{57} +(-395.932 + 116.256i) q^{58} +(73.6038 - 161.170i) q^{59} +(5.06109 + 35.2006i) q^{60} +(-212.542 - 245.286i) q^{61} +(113.019 + 247.478i) q^{62} +(289.808 - 186.248i) q^{63} +(-9.10815 + 63.3486i) q^{64} +(313.038 + 201.177i) q^{65} +(23.5215 - 27.1452i) q^{66} +(874.331 + 256.727i) q^{67} +116.473 q^{68} +(172.811 - 92.7644i) q^{69} -144.514 q^{70} +(300.044 + 88.1010i) q^{71} +(124.886 - 144.126i) q^{72} +(230.375 + 148.053i) q^{73} +(-77.2630 + 537.376i) q^{74} +(-37.3964 + 24.0332i) q^{75} +(-66.3239 - 145.229i) q^{76} +(95.5829 + 110.309i) q^{77} +(37.6655 + 261.970i) q^{78} +(-430.821 + 943.366i) q^{79} +(-76.7594 + 22.5386i) q^{80} +(-463.335 + 136.047i) q^{81} +(-312.050 + 683.294i) q^{82} +(-44.0419 - 306.318i) q^{83} +(-67.3103 - 77.6802i) q^{84} +(60.4808 + 132.434i) q^{85} +(91.1253 - 58.5627i) q^{86} +(52.2111 - 363.136i) q^{87} +(67.9734 + 43.6838i) q^{88} +(-317.587 + 366.514i) q^{89} +(228.726 + 67.1601i) q^{90} -1075.50 q^{91} +(261.879 + 355.093i) q^{92} -241.882 q^{93} +(-107.321 - 31.5124i) q^{94} +(130.691 - 150.826i) q^{95} +(-47.8675 - 30.7625i) q^{96} +(-134.691 + 936.798i) q^{97} +(-225.722 + 145.062i) q^{98} +(-100.018 - 219.009i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 70 q - 14 q^{2} + 3 q^{3} - 28 q^{4} - 35 q^{5} + 6 q^{6} - 78 q^{7} - 56 q^{8} - 24 q^{9} - 70 q^{10} - 15 q^{11} - 120 q^{12} - 270 q^{13} + 64 q^{14} + 15 q^{15} - 112 q^{16} + 114 q^{17} - 48 q^{18}+ \cdots + 11285 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(1\) \(e\left(\frac{6}{11}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.91899 − 0.563465i −0.678464 − 0.199215i
\(3\) 1.16443 − 1.34382i 0.224094 − 0.258618i −0.632558 − 0.774513i \(-0.717995\pi\)
0.856652 + 0.515895i \(0.172540\pi\)
\(4\) 3.36501 + 2.16256i 0.420627 + 0.270320i
\(5\) −0.711574 + 4.94911i −0.0636451 + 0.442662i
\(6\) −2.99172 + 1.92266i −0.203560 + 0.130820i
\(7\) −6.00332 − 13.1454i −0.324149 − 0.709786i 0.675470 − 0.737387i \(-0.263941\pi\)
−0.999619 + 0.0276006i \(0.991213\pi\)
\(8\) −5.23889 − 6.04600i −0.231528 − 0.267198i
\(9\) 3.39254 + 23.5956i 0.125650 + 0.873912i
\(10\) 4.15415 − 9.09632i 0.131366 − 0.287651i
\(11\) −9.69089 + 2.84550i −0.265629 + 0.0779956i −0.411835 − 0.911259i \(-0.635112\pi\)
0.146206 + 0.989254i \(0.453294\pi\)
\(12\) 6.82441 − 2.00383i 0.164170 − 0.0482046i
\(13\) 30.9160 − 67.6965i 0.659580 − 1.44428i −0.223332 − 0.974742i \(-0.571694\pi\)
0.882913 − 0.469537i \(-0.155579\pi\)
\(14\) 4.11329 + 28.6086i 0.0785231 + 0.546140i
\(15\) 5.82214 + 6.71910i 0.100218 + 0.115658i
\(16\) 6.64664 + 14.5541i 0.103854 + 0.227408i
\(17\) 24.4958 − 15.7425i 0.349477 − 0.224595i −0.354111 − 0.935203i \(-0.615216\pi\)
0.703588 + 0.710608i \(0.251580\pi\)
\(18\) 6.78507 − 47.1912i 0.0888476 − 0.617949i
\(19\) −33.5780 − 21.5793i −0.405438 − 0.260559i 0.321990 − 0.946743i \(-0.395648\pi\)
−0.727428 + 0.686184i \(0.759285\pi\)
\(20\) −13.0972 + 15.1150i −0.146431 + 0.168991i
\(21\) −24.6555 − 7.23952i −0.256204 − 0.0752282i
\(22\) 20.2000 0.195757
\(23\) 103.071 + 39.2852i 0.934427 + 0.356154i
\(24\) −14.2250 −0.120986
\(25\) −23.9873 − 7.04331i −0.191899 − 0.0563465i
\(26\) −97.4719 + 112.489i −0.735224 + 0.848493i
\(27\) 76.0468 + 48.8723i 0.542045 + 0.348351i
\(28\) 8.22658 − 57.2171i 0.0555242 − 0.386179i
\(29\) 173.571 − 111.547i 1.11142 − 0.714268i 0.149820 − 0.988713i \(-0.452131\pi\)
0.961603 + 0.274446i \(0.0884944\pi\)
\(30\) −7.38662 − 16.1744i −0.0449535 − 0.0984345i
\(31\) −89.0819 − 102.806i −0.516115 − 0.595629i 0.436539 − 0.899686i \(-0.356204\pi\)
−0.952654 + 0.304057i \(0.901659\pi\)
\(32\) −4.55407 − 31.6743i −0.0251579 − 0.174977i
\(33\) −7.46050 + 16.3362i −0.0393547 + 0.0861748i
\(34\) −55.8775 + 16.4071i −0.281850 + 0.0827587i
\(35\) 69.3299 − 20.3571i 0.334826 − 0.0983137i
\(36\) −39.6111 + 86.7362i −0.183385 + 0.401556i
\(37\) −38.6315 − 268.688i −0.171648 − 1.19384i −0.875402 − 0.483396i \(-0.839403\pi\)
0.703754 − 0.710444i \(-0.251506\pi\)
\(38\) 52.2766 + 60.3304i 0.223168 + 0.257549i
\(39\) −54.9726 − 120.373i −0.225709 − 0.494234i
\(40\) 33.6501 − 21.6256i 0.133014 − 0.0854828i
\(41\) 53.4518 − 371.765i 0.203604 − 1.41610i −0.589872 − 0.807497i \(-0.700822\pi\)
0.793476 − 0.608601i \(-0.208269\pi\)
\(42\) 43.2344 + 27.7851i 0.158838 + 0.102079i
\(43\) −35.4675 + 40.9317i −0.125785 + 0.145163i −0.815149 − 0.579252i \(-0.803345\pi\)
0.689364 + 0.724415i \(0.257890\pi\)
\(44\) −38.7636 − 11.3820i −0.132814 − 0.0389978i
\(45\) −119.191 −0.394844
\(46\) −175.656 − 133.465i −0.563024 − 0.427789i
\(47\) 55.9261 0.173567 0.0867836 − 0.996227i \(-0.472341\pi\)
0.0867836 + 0.996227i \(0.472341\pi\)
\(48\) 27.2976 + 8.01531i 0.0820849 + 0.0241023i
\(49\) 87.8547 − 101.390i 0.256136 − 0.295597i
\(50\) 42.0627 + 27.0320i 0.118971 + 0.0764582i
\(51\) 7.36850 − 51.2490i 0.0202313 − 0.140712i
\(52\) 250.431 − 160.942i 0.667855 − 0.429205i
\(53\) −260.758 − 570.980i −0.675808 − 1.47981i −0.867025 − 0.498265i \(-0.833971\pi\)
0.191217 − 0.981548i \(-0.438757\pi\)
\(54\) −118.395 − 136.635i −0.298361 − 0.344327i
\(55\) −7.18691 − 49.9861i −0.0176197 − 0.122548i
\(56\) −48.0265 + 105.163i −0.114604 + 0.250947i
\(57\) −68.0978 + 19.9953i −0.158242 + 0.0464640i
\(58\) −395.932 + 116.256i −0.896353 + 0.263193i
\(59\) 73.6038 − 161.170i 0.162414 − 0.355636i −0.810876 − 0.585218i \(-0.801009\pi\)
0.973289 + 0.229582i \(0.0737360\pi\)
\(60\) 5.06109 + 35.2006i 0.0108897 + 0.0757397i
\(61\) −212.542 − 245.286i −0.446118 − 0.514848i 0.487497 − 0.873124i \(-0.337910\pi\)
−0.933615 + 0.358277i \(0.883364\pi\)
\(62\) 113.019 + 247.478i 0.231507 + 0.506931i
\(63\) 289.808 − 186.248i 0.579562 − 0.372462i
\(64\) −9.10815 + 63.3486i −0.0177894 + 0.123728i
\(65\) 313.038 + 201.177i 0.597348 + 0.383892i
\(66\) 23.5215 − 27.1452i 0.0438681 − 0.0506265i
\(67\) 874.331 + 256.727i 1.59428 + 0.468122i 0.953946 − 0.299977i \(-0.0969791\pi\)
0.640331 + 0.768099i \(0.278797\pi\)
\(68\) 116.473 0.207712
\(69\) 172.811 − 92.7644i 0.301508 − 0.161848i
\(70\) −144.514 −0.246753
\(71\) 300.044 + 88.1010i 0.501531 + 0.147263i 0.522706 − 0.852513i \(-0.324922\pi\)
−0.0211749 + 0.999776i \(0.506741\pi\)
\(72\) 124.886 − 144.126i 0.204416 − 0.235909i
\(73\) 230.375 + 148.053i 0.369362 + 0.237374i 0.712133 − 0.702044i \(-0.247729\pi\)
−0.342771 + 0.939419i \(0.611366\pi\)
\(74\) −77.2630 + 537.376i −0.121374 + 0.844172i
\(75\) −37.3964 + 24.0332i −0.0575756 + 0.0370016i
\(76\) −66.3239 − 145.229i −0.100104 − 0.219196i
\(77\) 95.5829 + 110.309i 0.141463 + 0.163257i
\(78\) 37.6655 + 261.970i 0.0546767 + 0.380285i
\(79\) −430.821 + 943.366i −0.613559 + 1.34351i 0.306555 + 0.951853i \(0.400824\pi\)
−0.920113 + 0.391653i \(0.871903\pi\)
\(80\) −76.7594 + 22.5386i −0.107275 + 0.0314987i
\(81\) −463.335 + 136.047i −0.635576 + 0.186622i
\(82\) −312.050 + 683.294i −0.420246 + 0.920210i
\(83\) −44.0419 − 306.318i −0.0582437 − 0.405094i −0.997998 − 0.0632434i \(-0.979856\pi\)
0.939754 − 0.341850i \(-0.111054\pi\)
\(84\) −67.3103 − 77.6802i −0.0874304 − 0.100900i
\(85\) 60.4808 + 132.434i 0.0771772 + 0.168994i
\(86\) 91.1253 − 58.5627i 0.114259 − 0.0734299i
\(87\) 52.2111 − 363.136i 0.0643404 − 0.447497i
\(88\) 67.9734 + 43.6838i 0.0823408 + 0.0529172i
\(89\) −317.587 + 366.514i −0.378248 + 0.436522i −0.912671 − 0.408696i \(-0.865984\pi\)
0.534422 + 0.845218i \(0.320529\pi\)
\(90\) 228.726 + 67.1601i 0.267888 + 0.0786589i
\(91\) −1075.50 −1.23893
\(92\) 261.879 + 355.093i 0.296770 + 0.402403i
\(93\) −241.882 −0.269699
\(94\) −107.321 − 31.5124i −0.117759 − 0.0345772i
\(95\) 130.691 − 150.826i 0.141144 − 0.162889i
\(96\) −47.8675 − 30.7625i −0.0508901 − 0.0327051i
\(97\) −134.691 + 936.798i −0.140988 + 0.980593i 0.789364 + 0.613925i \(0.210410\pi\)
−0.930352 + 0.366667i \(0.880499\pi\)
\(98\) −225.722 + 145.062i −0.232667 + 0.149526i
\(99\) −100.018 − 219.009i −0.101537 − 0.222336i
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 230.4.g.d.41.5 ✓ 70
23.9 even 11 inner 230.4.g.d.101.5 yes 70
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.g.d.41.5 ✓ 70 1.1 even 1 trivial
230.4.g.d.101.5 yes 70 23.9 even 11 inner