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Properties

Label 1400.2.x.c.993.1
Level $1400$
Weight $2$
Character 1400.993
Analytic conductor $11.179$
Analytic rank $0$
Dimension $32$
Inner twists $8$

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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] = [1400,2,Mod(657,1400)] 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("1400.657"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1400, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 1, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1400.x (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.1790562830\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 993.1
Character \(\chi\) \(=\) 1400.993
Dual form 1400.2.x.c.657.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-2.09284 + 2.09284i) q^{3} +(0.510946 + 2.59595i) q^{7} -5.75996i q^{9} -3.94864 q^{11} +(1.69798 - 1.69798i) q^{13} +(2.66927 + 2.66927i) q^{17} -5.36102 q^{19} +(-6.50223 - 4.36357i) q^{21} +(-3.55527 - 3.55527i) q^{23} +(5.77616 + 5.77616i) q^{27} -7.24449i q^{29} +0.174160i q^{31} +(8.26386 - 8.26386i) q^{33} +(-4.85737 + 4.85737i) q^{37} +7.10719i q^{39} +0.732583i q^{41} +(-8.20187 - 8.20187i) q^{43} +(2.31716 + 2.31716i) q^{47} +(-6.47787 + 2.65278i) q^{49} -11.1727 q^{51} +(4.53449 + 4.53449i) q^{53} +(11.2198 - 11.2198i) q^{57} +13.1904 q^{59} -13.3556i q^{61} +(14.9525 - 2.94303i) q^{63} +(6.55658 - 6.55658i) q^{67} +14.8812 q^{69} +16.3312 q^{71} +(-7.02549 + 7.02549i) q^{73} +(-2.01754 - 10.2504i) q^{77} -6.63234i q^{79} -6.89727 q^{81} +(-10.4642 + 10.4642i) q^{83} +(15.1616 + 15.1616i) q^{87} -3.43554 q^{89} +(5.27543 + 3.54028i) q^{91} +(-0.364490 - 0.364490i) q^{93} +(-1.40892 - 1.40892i) q^{97} +22.7440i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q - 16 q^{11} - 40 q^{21} + 32 q^{51} + 128 q^{71}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(701\) \(801\) \(1177\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −2.09284 + 2.09284i −1.20830 + 1.20830i −0.236725 + 0.971577i \(0.576074\pi\)
−0.971577 + 0.236725i \(0.923926\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0.510946 + 2.59595i 0.193119 + 0.981175i
\(8\) 0 0
\(9\) − 5.75996i − 1.91999i
\(10\) 0 0
\(11\) −3.94864 −1.19056 −0.595279 − 0.803519i \(-0.702959\pi\)
−0.595279 + 0.803519i \(0.702959\pi\)
\(12\) 0 0
\(13\) 1.69798 − 1.69798i 0.470934 − 0.470934i −0.431283 − 0.902217i \(-0.641939\pi\)
0.902217 + 0.431283i \(0.141939\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.66927 + 2.66927i 0.647392 + 0.647392i 0.952362 − 0.304970i \(-0.0986464\pi\)
−0.304970 + 0.952362i \(0.598646\pi\)
\(18\) 0 0
\(19\) −5.36102 −1.22990 −0.614952 − 0.788565i \(-0.710824\pi\)
−0.614952 + 0.788565i \(0.710824\pi\)
\(20\) 0 0
\(21\) −6.50223 − 4.36357i −1.41890 − 0.952209i
\(22\) 0 0
\(23\) −3.55527 − 3.55527i −0.741325 − 0.741325i 0.231508 − 0.972833i \(-0.425634\pi\)
−0.972833 + 0.231508i \(0.925634\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.77616 + 5.77616i 1.11162 + 1.11162i
\(28\) 0 0
\(29\) − 7.24449i − 1.34527i −0.739975 − 0.672634i \(-0.765163\pi\)
0.739975 − 0.672634i \(-0.234837\pi\)
\(30\) 0 0
\(31\) 0.174160i 0.0312801i 0.999878 + 0.0156401i \(0.00497859\pi\)
−0.999878 + 0.0156401i \(0.995021\pi\)
\(32\) 0 0
\(33\) 8.26386 − 8.26386i 1.43855 − 1.43855i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −4.85737 + 4.85737i −0.798547 + 0.798547i −0.982866 − 0.184320i \(-0.940992\pi\)
0.184320 + 0.982866i \(0.440992\pi\)
\(38\) 0 0
\(39\) 7.10719i 1.13806i
\(40\) 0 0
\(41\) 0.732583i 0.114410i 0.998362 + 0.0572051i \(0.0182189\pi\)
−0.998362 + 0.0572051i \(0.981781\pi\)
\(42\) 0 0
\(43\) −8.20187 − 8.20187i −1.25077 − 1.25077i −0.955374 − 0.295400i \(-0.904547\pi\)
−0.295400 − 0.955374i \(-0.595453\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.31716 + 2.31716i 0.337993 + 0.337993i 0.855612 − 0.517619i \(-0.173181\pi\)
−0.517619 + 0.855612i \(0.673181\pi\)
\(48\) 0 0
\(49\) −6.47787 + 2.65278i −0.925410 + 0.378968i
\(50\) 0 0
\(51\) −11.1727 −1.56449
\(52\) 0 0
\(53\) 4.53449 + 4.53449i 0.622860 + 0.622860i 0.946262 − 0.323402i \(-0.104827\pi\)
−0.323402 + 0.946262i \(0.604827\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 11.2198 − 11.2198i 1.48609 − 1.48609i
\(58\) 0 0
\(59\) 13.1904 1.71724 0.858620 − 0.512613i \(-0.171322\pi\)
0.858620 + 0.512613i \(0.171322\pi\)
\(60\) 0 0
\(61\) − 13.3556i − 1.71001i −0.518622 − 0.855004i \(-0.673555\pi\)
0.518622 − 0.855004i \(-0.326445\pi\)
\(62\) 0 0
\(63\) 14.9525 − 2.94303i 1.88384 − 0.370787i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 6.55658 − 6.55658i 0.801014 − 0.801014i −0.182240 − 0.983254i \(-0.558335\pi\)
0.983254 + 0.182240i \(0.0583349\pi\)
\(68\) 0 0
\(69\) 14.8812 1.79149
\(70\) 0 0
\(71\) 16.3312 1.93816 0.969081 − 0.246742i \(-0.0793599\pi\)
0.969081 + 0.246742i \(0.0793599\pi\)
\(72\) 0 0
\(73\) −7.02549 + 7.02549i −0.822271 + 0.822271i −0.986433 − 0.164162i \(-0.947508\pi\)
0.164162 + 0.986433i \(0.447508\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.01754 − 10.2504i −0.229920 − 1.16815i
\(78\) 0 0
\(79\) − 6.63234i − 0.746197i −0.927792 − 0.373098i \(-0.878295\pi\)
0.927792 − 0.373098i \(-0.121705\pi\)
\(80\) 0 0
\(81\) −6.89727 −0.766363
\(82\) 0 0
\(83\) −10.4642 + 10.4642i −1.14860 + 1.14860i −0.161766 + 0.986829i \(0.551719\pi\)
−0.986829 + 0.161766i \(0.948281\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 15.1616 + 15.1616i 1.62549 + 1.62549i
\(88\) 0 0
\(89\) −3.43554 −0.364166 −0.182083 − 0.983283i \(-0.558284\pi\)
−0.182083 + 0.983283i \(0.558284\pi\)
\(90\) 0 0
\(91\) 5.27543 + 3.54028i 0.553015 + 0.371122i
\(92\) 0 0
\(93\) −0.364490 − 0.364490i −0.0377958 − 0.0377958i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.40892 − 1.40892i −0.143054 − 0.143054i 0.631953 − 0.775007i \(-0.282254\pi\)
−0.775007 + 0.631953i \(0.782254\pi\)
\(98\) 0 0
\(99\) 22.7440i 2.28586i
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 1400.2.x.c.993.1 yes 32
5.2 odd 4 inner 1400.2.x.c.657.15 yes 32
5.3 odd 4 inner 1400.2.x.c.657.2 yes 32
5.4 even 2 inner 1400.2.x.c.993.16 yes 32
7.6 odd 2 inner 1400.2.x.c.993.15 yes 32
35.13 even 4 inner 1400.2.x.c.657.16 yes 32
35.27 even 4 inner 1400.2.x.c.657.1 ✓ 32
35.34 odd 2 inner 1400.2.x.c.993.2 yes 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1400.2.x.c.657.1 ✓ 32 35.27 even 4 inner
1400.2.x.c.657.2 yes 32 5.3 odd 4 inner
1400.2.x.c.657.15 yes 32 5.2 odd 4 inner
1400.2.x.c.657.16 yes 32 35.13 even 4 inner
1400.2.x.c.993.1 yes 32 1.1 even 1 trivial
1400.2.x.c.993.2 yes 32 35.34 odd 2 inner
1400.2.x.c.993.15 yes 32 7.6 odd 2 inner
1400.2.x.c.993.16 yes 32 5.4 even 2 inner