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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,2,Mod(7,380)] 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("380.7"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 3, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.v (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [216,0] 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: no
Analytic conductor: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(216\)
Relative dimension: \(54\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 7.4
Character \(\chi\) \(=\) 380.7
Dual form 380.2.v.c.163.4

$q$-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+(-1.39454 - 0.235046i) q^{2} +(-0.550661 + 2.05509i) q^{3} +(1.88951 + 0.655563i) q^{4} +(-1.04297 - 1.97793i) q^{5} +(1.25096 - 2.73649i) q^{6} +(0.179304 - 0.179304i) q^{7} +(-2.48091 - 1.35833i) q^{8} +(-1.32211 - 0.763319i) q^{9} +(0.989568 + 3.00346i) q^{10} +5.59983i q^{11} +(-2.38772 + 3.52212i) q^{12} +(-0.144960 - 0.540998i) q^{13} +(-0.292192 + 0.207903i) q^{14} +(4.63916 - 1.05424i) q^{15} +(3.14047 + 2.47738i) q^{16} +(-1.48130 + 5.52829i) q^{17} +(1.66432 + 1.37524i) q^{18} +(-0.0837631 - 4.35809i) q^{19} +(-0.674047 - 4.42105i) q^{20} +(0.269751 + 0.467222i) q^{21} +(1.31621 - 7.80921i) q^{22} +(-7.59829 + 2.03596i) q^{23} +(4.15764 - 4.35053i) q^{24} +(-2.82442 + 4.12585i) q^{25} +(0.0749939 + 0.788518i) q^{26} +(-2.21657 + 2.21657i) q^{27} +(0.456341 - 0.221251i) q^{28} +(-4.45983 - 2.57488i) q^{29} +(-6.71730 + 0.379769i) q^{30} -2.50953i q^{31} +(-3.79723 - 4.19297i) q^{32} +(-11.5082 - 3.08360i) q^{33} +(3.36514 - 7.36127i) q^{34} +(-0.541660 - 0.167642i) q^{35} +(-1.99773 - 2.30902i) q^{36} +(-2.81317 - 2.81317i) q^{37} +(-0.907540 + 6.09724i) q^{38} +1.19163 q^{39} +(-0.0991599 + 6.32378i) q^{40} +(5.51038 + 9.54426i) q^{41} +(-0.266361 - 0.714966i) q^{42} +(0.103161 - 0.385001i) q^{43} +(-3.67104 + 10.5809i) q^{44} +(-0.130870 + 3.41116i) q^{45} +(11.0747 - 1.05329i) q^{46} +(0.811997 + 3.03041i) q^{47} +(-6.82059 + 5.08977i) q^{48} +6.93570i q^{49} +(4.90854 - 5.08982i) q^{50} +(-10.5455 - 6.08842i) q^{51} +(0.0807554 - 1.11725i) q^{52} +(-1.07892 - 4.02658i) q^{53} +(3.61211 - 2.57012i) q^{54} +(11.0761 - 5.84047i) q^{55} +(-0.688392 + 0.201283i) q^{56} +(9.00242 + 2.22769i) q^{57} +(5.61421 + 4.63905i) q^{58} +(-4.67284 - 8.09360i) q^{59} +(9.45684 + 1.04927i) q^{60} +(-5.16054 + 8.93832i) q^{61} +(-0.589854 + 3.49965i) q^{62} +(-0.373925 + 0.100193i) q^{63} +(4.30987 + 6.73981i) q^{64} +(-0.918867 + 0.850967i) q^{65} +(15.3239 + 7.00517i) q^{66} +(1.54566 + 5.76848i) q^{67} +(-6.42307 + 9.47465i) q^{68} -16.7363i q^{69} +(0.715966 + 0.361098i) q^{70} +(3.12358 - 1.80340i) q^{71} +(2.24319 + 3.68959i) q^{72} +(2.42726 + 0.650383i) q^{73} +(3.26186 + 4.58431i) q^{74} +(-6.92372 - 8.07639i) q^{75} +(2.69873 - 8.28956i) q^{76} +(1.00407 + 1.00407i) q^{77} +(-1.66177 - 0.280086i) q^{78} +(0.642802 + 1.11337i) q^{79} +(1.62466 - 8.79548i) q^{80} +(-5.62465 - 9.74217i) q^{81} +(-5.44113 - 14.6051i) q^{82} +(6.95875 + 6.95875i) q^{83} +(0.203403 + 1.05966i) q^{84} +(12.4795 - 2.83595i) q^{85} +(-0.234355 + 0.512653i) q^{86} +(7.74748 - 7.74748i) q^{87} +(7.60642 - 13.8927i) q^{88} +(-1.52733 - 0.881803i) q^{89} +(0.984281 - 4.72625i) q^{90} +(-0.122995 - 0.0710112i) q^{91} +(-15.6917 - 1.13421i) q^{92} +(5.15732 + 1.38190i) q^{93} +(-0.420080 - 4.41690i) q^{94} +(-8.53264 + 4.71105i) q^{95} +(10.7079 - 5.49476i) q^{96} +(-1.39017 + 5.18819i) q^{97} +(1.63021 - 9.67214i) q^{98} +(4.27445 - 7.40357i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 216 q + 12 q^{6} - 36 q^{8} + 4 q^{10} - 20 q^{12} - 8 q^{13} + 8 q^{16} - 16 q^{17} + 12 q^{18} - 48 q^{20} + 40 q^{21} + 16 q^{22} - 16 q^{25} + 24 q^{26} + 8 q^{28} - 12 q^{30} - 20 q^{32} + 20 q^{33}+ \cdots + 34 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{4}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.39454 − 0.235046i −0.986092 − 0.166202i
\(3\) −0.550661 + 2.05509i −0.317924 + 1.18651i 0.603312 + 0.797505i \(0.293847\pi\)
−0.921236 + 0.389004i \(0.872819\pi\)
\(4\) 1.88951 + 0.655563i 0.944754 + 0.327781i
\(5\) −1.04297 − 1.97793i −0.466432 − 0.884557i
\(6\) 1.25096 − 2.73649i 0.510703 − 1.11717i
\(7\) 0.179304 − 0.179304i 0.0677706 − 0.0677706i −0.672409 − 0.740180i \(-0.734740\pi\)
0.740180 + 0.672409i \(0.234740\pi\)
\(8\) −2.48091 − 1.35833i −0.877136 − 0.480243i
\(9\) −1.32211 − 0.763319i −0.440702 − 0.254440i
\(10\) 0.989568 + 3.00346i 0.312929 + 0.949777i
\(11\) 5.59983i 1.68841i 0.536020 + 0.844206i \(0.319927\pi\)
−0.536020 + 0.844206i \(0.680073\pi\)
\(12\) −2.38772 + 3.52212i −0.689276 + 1.01675i
\(13\) −0.144960 − 0.540998i −0.0402047 − 0.150046i 0.942906 − 0.333060i \(-0.108081\pi\)
−0.983110 + 0.183014i \(0.941415\pi\)
\(14\) −0.292192 + 0.207903i −0.0780916 + 0.0555644i
\(15\) 4.63916 − 1.05424i 1.19782 − 0.272203i
\(16\) 3.14047 + 2.47738i 0.785119 + 0.619345i
\(17\) −1.48130 + 5.52829i −0.359268 + 1.34081i 0.515759 + 0.856733i \(0.327510\pi\)
−0.875027 + 0.484073i \(0.839157\pi\)
\(18\) 1.66432 + 1.37524i 0.392284 + 0.324147i
\(19\) −0.0837631 − 4.35809i −0.0192166 − 0.999815i
\(20\) −0.674047 − 4.42105i −0.150721 − 0.988576i
\(21\) 0.269751 + 0.467222i 0.0588645 + 0.101956i
\(22\) 1.31621 − 7.80921i 0.280618 − 1.66493i
\(23\) −7.59829 + 2.03596i −1.58435 + 0.424526i −0.940270 − 0.340429i \(-0.889428\pi\)
−0.644083 + 0.764955i \(0.722761\pi\)
\(24\) 4.15764 − 4.35053i 0.848675 − 0.888048i
\(25\) −2.82442 + 4.12585i −0.564883 + 0.825171i
\(26\) 0.0749939 + 0.788518i 0.0147075 + 0.154641i
\(27\) −2.21657 + 2.21657i −0.426580 + 0.426580i
\(28\) 0.456341 − 0.221251i 0.0862404 − 0.0418125i
\(29\) −4.45983 − 2.57488i −0.828169 − 0.478144i 0.0250562 − 0.999686i \(-0.492024\pi\)
−0.853225 + 0.521542i \(0.825357\pi\)
\(30\) −6.71730 + 0.379769i −1.22641 + 0.0693360i
\(31\) − 2.50953i − 0.450725i −0.974275 − 0.225363i \(-0.927643\pi\)
0.974275 − 0.225363i \(-0.0723567\pi\)
\(32\) −3.79723 − 4.19297i −0.671262 − 0.741220i
\(33\) −11.5082 − 3.08360i −2.00331 − 0.536787i
\(34\) 3.36514 − 7.36127i 0.577117 − 1.26245i
\(35\) −0.541660 − 0.167642i −0.0915573 − 0.0283366i
\(36\) −1.99773 − 2.30902i −0.332955 − 0.384837i
\(37\) −2.81317 − 2.81317i −0.462482 − 0.462482i 0.436986 − 0.899468i \(-0.356046\pi\)
−0.899468 + 0.436986i \(0.856046\pi\)
\(38\) −0.907540 + 6.09724i −0.147222 + 0.989103i
\(39\) 1.19163 0.190813
\(40\) −0.0991599 + 6.32378i −0.0156786 + 0.999877i
\(41\) 5.51038 + 9.54426i 0.860577 + 1.49056i 0.871373 + 0.490621i \(0.163230\pi\)
−0.0107964 + 0.999942i \(0.503437\pi\)
\(42\) −0.266361 − 0.714966i −0.0411004 − 0.110322i
\(43\) 0.103161 − 0.385001i 0.0157319 − 0.0587121i −0.957614 − 0.288056i \(-0.906991\pi\)
0.973346 + 0.229343i \(0.0736579\pi\)
\(44\) −3.67104 + 10.5809i −0.553430 + 1.59513i
\(45\) −0.130870 + 3.41116i −0.0195089 + 0.508505i
\(46\) 11.0747 − 1.05329i 1.63287 − 0.155298i
\(47\) 0.811997 + 3.03041i 0.118442 + 0.442031i 0.999521 − 0.0309370i \(-0.00984914\pi\)
−0.881079 + 0.472968i \(0.843182\pi\)
\(48\) −6.82059 + 5.08977i −0.984467 + 0.734645i
\(49\) 6.93570i 0.990814i
\(50\) 4.90854 − 5.08982i 0.694172 − 0.719809i
\(51\) −10.5455 − 6.08842i −1.47666 − 0.852550i
\(52\) 0.0807554 − 1.11725i 0.0111988 − 0.154935i
\(53\) −1.07892 − 4.02658i −0.148201 − 0.553094i −0.999592 − 0.0285622i \(-0.990907\pi\)
0.851391 − 0.524532i \(-0.175760\pi\)
\(54\) 3.61211 − 2.57012i 0.491546 − 0.349748i
\(55\) 11.0761 − 5.84047i 1.49350 − 0.787528i
\(56\) −0.688392 + 0.201283i −0.0919903 + 0.0268976i
\(57\) 9.00242 + 2.22769i 1.19240 + 0.295065i
\(58\) 5.61421 + 4.63905i 0.737182 + 0.609137i
\(59\) −4.67284 − 8.09360i −0.608352 − 1.05370i −0.991512 − 0.130015i \(-0.958497\pi\)
0.383160 − 0.923682i \(-0.374836\pi\)
\(60\) 9.45684 + 1.04927i 1.22087 + 0.135460i
\(61\) −5.16054 + 8.93832i −0.660739 + 1.14443i 0.319682 + 0.947525i \(0.396424\pi\)
−0.980422 + 0.196909i \(0.936910\pi\)
\(62\) −0.589854 + 3.49965i −0.0749116 + 0.444456i
\(63\) −0.373925 + 0.100193i −0.0471102 + 0.0126231i
\(64\) 4.30987 + 6.73981i 0.538734 + 0.842476i
\(65\) −0.918867 + 0.850967i −0.113971 + 0.105549i
\(66\) 15.3239 + 7.00517i 1.88624 + 0.862276i
\(67\) 1.54566 + 5.76848i 0.188832 + 0.704732i 0.993778 + 0.111383i \(0.0355279\pi\)
−0.804945 + 0.593349i \(0.797805\pi\)
\(68\) −6.42307 + 9.47465i −0.778911 + 1.14897i
\(69\) − 16.7363i − 2.01482i
\(70\) 0.715966 + 0.361098i 0.0855742 + 0.0431595i
\(71\) 3.12358 − 1.80340i 0.370700 − 0.214024i −0.303064 − 0.952970i \(-0.598010\pi\)
0.673764 + 0.738946i \(0.264676\pi\)
\(72\) 2.24319 + 3.68959i 0.264363 + 0.434822i
\(73\) 2.42726 + 0.650383i 0.284090 + 0.0761216i 0.398050 − 0.917364i \(-0.369687\pi\)
−0.113960 + 0.993485i \(0.536354\pi\)
\(74\) 3.26186 + 4.58431i 0.379184 + 0.532915i
\(75\) −6.92372 − 8.07639i −0.799483 − 0.932581i
\(76\) 2.69873 − 8.28956i 0.309566 − 0.950878i
\(77\) 1.00407 + 1.00407i 0.114425 + 0.114425i
\(78\) −1.66177 − 0.280086i −0.188159 − 0.0317135i
\(79\) 0.642802 + 1.11337i 0.0723208 + 0.125263i 0.899918 − 0.436059i \(-0.143626\pi\)
−0.827597 + 0.561322i \(0.810293\pi\)
\(80\) 1.62466 − 8.79548i 0.181642 − 0.983365i
\(81\) −5.62465 − 9.74217i −0.624961 − 1.08246i
\(82\) −5.44113 − 14.6051i −0.600873 − 1.61286i
\(83\) 6.95875 + 6.95875i 0.763822 + 0.763822i 0.977011 − 0.213189i \(-0.0683849\pi\)
−0.213189 + 0.977011i \(0.568385\pi\)
\(84\) 0.203403 + 1.05966i 0.0221930 + 0.115618i
\(85\) 12.4795 − 2.83595i 1.35359 − 0.307601i
\(86\) −0.234355 + 0.512653i −0.0252711 + 0.0552808i
\(87\) 7.74748 − 7.74748i 0.830617 − 0.830617i
\(88\) 7.60642 − 13.8927i 0.810847 − 1.48097i
\(89\) −1.52733 − 0.881803i −0.161896 − 0.0934710i 0.416863 − 0.908969i \(-0.363130\pi\)
−0.578759 + 0.815498i \(0.696463\pi\)
\(90\) 0.984281 − 4.72625i 0.103752 − 0.498190i
\(91\) −0.122995 − 0.0710112i −0.0128934 − 0.00744400i
\(92\) −15.6917 − 1.13421i −1.63598 − 0.118249i
\(93\) 5.15732 + 1.38190i 0.534789 + 0.143296i
\(94\) −0.420080 − 4.41690i −0.0433280 − 0.455569i
\(95\) −8.53264 + 4.71105i −0.875431 + 0.483344i
\(96\) 10.7079 − 5.49476i 1.09287 − 0.560807i
\(97\) −1.39017 + 5.18819i −0.141150 + 0.526781i 0.858746 + 0.512401i \(0.171244\pi\)
−0.999897 + 0.0143794i \(0.995423\pi\)
\(98\) 1.63021 − 9.67214i 0.164676 − 0.977034i
\(99\) 4.27445 − 7.40357i 0.429599 − 0.744087i
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 380.2.v.c.7.4 ✓ 216
4.3 odd 2 inner 380.2.v.c.7.49 yes 216
5.3 odd 4 inner 380.2.v.c.83.24 yes 216
19.11 even 3 inner 380.2.v.c.87.33 yes 216
20.3 even 4 inner 380.2.v.c.83.33 yes 216
76.11 odd 6 inner 380.2.v.c.87.24 yes 216
95.68 odd 12 inner 380.2.v.c.163.49 yes 216
380.163 even 12 inner 380.2.v.c.163.4 yes 216
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.v.c.7.4 ✓ 216 1.1 even 1 trivial
380.2.v.c.7.49 yes 216 4.3 odd 2 inner
380.2.v.c.83.24 yes 216 5.3 odd 4 inner
380.2.v.c.83.33 yes 216 20.3 even 4 inner
380.2.v.c.87.24 yes 216 76.11 odd 6 inner
380.2.v.c.87.33 yes 216 19.11 even 3 inner
380.2.v.c.163.4 yes 216 380.163 even 12 inner
380.2.v.c.163.49 yes 216 95.68 odd 12 inner