↑

Properties

Label 575.2.p.a.524.1
Level $575$
Weight $2$
Character 575.524
Analytic conductor $4.591$
Analytic rank $0$
Dimension $20$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [575,2,Mod(49,575)] 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("575.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(575, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([11, 16])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 575.p (of order \(22\), degree \(10\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.59139811622\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{22})\)
Coefficient field: \(\Q(\zeta_{44})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) 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 115)
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label 524.1
Root \(-0.989821 + 0.142315i\) of defining polynomial
Character \(\chi\) \(=\) 575.524
Dual form 575.2.p.a.124.1

$q$-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+(-0.386758 + 1.31718i) q^{2} +(-0.755750 - 0.654861i) q^{3} +(0.0971309 + 0.0624222i) q^{4} +(1.15486 - 0.742184i) q^{6} +(-2.02730 + 0.925839i) q^{7} +(-2.19475 + 1.90176i) q^{8} +(-0.284630 - 1.97964i) q^{9} +(-1.98926 + 0.584100i) q^{11} +(-0.0325288 - 0.110783i) q^{12} +(0.932456 + 0.425839i) q^{13} +(-0.435418 - 3.02840i) q^{14} +(-1.56019 - 3.41635i) q^{16} +(-2.14059 - 3.33083i) q^{17} +(2.71763 + 0.390736i) q^{18} +(0.879900 + 0.565477i) q^{19} +(2.13843 + 0.627899i) q^{21} -2.84612i q^{22} +(-4.35110 - 2.01691i) q^{23} +2.90407 q^{24} +(-0.921540 + 1.06351i) q^{26} +(-2.70320 + 4.20627i) q^{27} +(-0.254707 - 0.0366213i) q^{28} +(-4.95274 + 3.18293i) q^{29} +(-2.78201 - 3.21061i) q^{31} +(-0.645667 + 0.0928329i) q^{32} +(1.88589 + 0.861256i) q^{33} +(5.21519 - 1.53132i) q^{34} +(0.0959274 - 0.210052i) q^{36} +(-5.92406 + 0.851751i) q^{37} +(-1.08514 + 0.940282i) q^{38} +(-0.425839 - 0.932456i) q^{39} +(1.36297 - 9.47964i) q^{41} +(-1.65411 + 2.57385i) q^{42} +(1.42270 + 1.23278i) q^{43} +(-0.229680 - 0.0674400i) q^{44} +(4.33945 - 4.95112i) q^{46} -7.61130i q^{47} +(-1.05812 + 3.60362i) q^{48} +(-1.33124 + 1.53634i) q^{49} +(-0.563476 + 3.91906i) q^{51} +(0.0639885 + 0.0995681i) q^{52} +(10.8563 - 4.95792i) q^{53} +(-4.49492 - 5.18741i) q^{54} +(2.68870 - 5.88744i) q^{56} +(-0.294675 - 1.00357i) q^{57} +(-2.27697 - 7.75466i) q^{58} +(-4.75662 + 10.4155i) q^{59} +(4.20330 + 4.85087i) q^{61} +(5.30490 - 2.42267i) q^{62} +(2.40986 + 3.74982i) q^{63} +(1.19644 - 8.32140i) q^{64} +(-1.86381 + 2.15095i) q^{66} +(-1.15453 + 3.93196i) q^{67} -0.457147i q^{68} +(1.96755 + 4.37364i) q^{69} +(-9.69553 - 2.84686i) q^{71} +(4.38950 + 3.80353i) q^{72} +(4.91945 - 7.65481i) q^{73} +(1.16927 - 8.13247i) q^{74} +(0.0501671 + 0.109851i) q^{76} +(3.49206 - 3.02588i) q^{77} +(1.39291 - 0.200270i) q^{78} +(-1.59168 + 3.48530i) q^{79} +(-0.959493 + 0.281733i) q^{81} +(11.9592 + 5.46160i) q^{82} +(-7.20866 + 1.03645i) q^{83} +(0.168513 + 0.194474i) q^{84} +(-2.17403 + 1.39717i) q^{86} +(5.82741 + 0.837855i) q^{87} +(3.25512 - 5.06506i) q^{88} +(-6.80606 + 7.85461i) q^{89} -2.28463 q^{91} +(-0.296726 - 0.467509i) q^{92} +4.24824i q^{93} +(10.0254 + 2.94373i) q^{94} +(0.548755 + 0.352663i) q^{96} +(-1.44652 - 0.207978i) q^{97} +(-1.50876 - 2.34767i) q^{98} +(1.72251 + 3.77178i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{4} + 12 q^{6} - 4 q^{9} + 16 q^{11} + 6 q^{14} + 10 q^{16} + 26 q^{19} + 12 q^{21} + 12 q^{24} + 22 q^{26} - 4 q^{29} - 40 q^{31} - 32 q^{34} + 48 q^{36} - 10 q^{41} - 38 q^{44} - 32 q^{46}+ \cdots - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/575\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(e\left(\frac{6}{11}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.386758 + 1.31718i −0.273479 + 0.931386i 0.702163 + 0.712017i \(0.252218\pi\)
−0.975642 + 0.219369i \(0.929600\pi\)
\(3\) −0.755750 − 0.654861i −0.436332 − 0.378084i 0.408815 − 0.912617i \(-0.365942\pi\)
−0.845148 + 0.534533i \(0.820487\pi\)
\(4\) 0.0971309 + 0.0624222i 0.0485654 + 0.0312111i
\(5\) 0 0
\(6\) 1.15486 − 0.742184i 0.471470 − 0.302995i
\(7\) −2.02730 + 0.925839i −0.766249 + 0.349934i −0.759895 − 0.650046i \(-0.774750\pi\)
−0.00635365 + 0.999980i \(0.502022\pi\)
\(8\) −2.19475 + 1.90176i −0.775962 + 0.672375i
\(9\) −0.284630 − 1.97964i −0.0948766 − 0.659881i
\(10\) 0 0
\(11\) −1.98926 + 0.584100i −0.599785 + 0.176113i −0.567510 − 0.823366i \(-0.692093\pi\)
−0.0322746 + 0.999479i \(0.510275\pi\)
\(12\) −0.0325288 − 0.110783i −0.00939024 − 0.0319802i
\(13\) 0.932456 + 0.425839i 0.258617 + 0.118106i 0.540505 − 0.841341i \(-0.318233\pi\)
−0.281888 + 0.959447i \(0.590961\pi\)
\(14\) −0.435418 − 3.02840i −0.116370 − 0.809373i
\(15\) 0 0
\(16\) −1.56019 − 3.41635i −0.390049 − 0.854088i
\(17\) −2.14059 − 3.33083i −0.519170 − 0.807845i 0.478354 − 0.878167i \(-0.341234\pi\)
−0.997524 + 0.0703222i \(0.977597\pi\)
\(18\) 2.71763 + 0.390736i 0.640550 + 0.0920972i
\(19\) 0.879900 + 0.565477i 0.201863 + 0.129729i 0.637667 − 0.770312i \(-0.279900\pi\)
−0.435804 + 0.900042i \(0.643536\pi\)
\(20\) 0 0
\(21\) 2.13843 + 0.627899i 0.466643 + 0.137019i
\(22\) − 2.84612i − 0.606794i
\(23\) −4.35110 − 2.01691i −0.907267 − 0.420554i
\(24\) 2.90407 0.592791
\(25\) 0 0
\(26\) −0.921540 + 1.06351i −0.180729 + 0.208572i
\(27\) −2.70320 + 4.20627i −0.520232 + 0.809497i
\(28\) −0.254707 − 0.0366213i −0.0481350 − 0.00692077i
\(29\) −4.95274 + 3.18293i −0.919700 + 0.591056i −0.912571 − 0.408919i \(-0.865906\pi\)
−0.00712969 + 0.999975i \(0.502269\pi\)
\(30\) 0 0
\(31\) −2.78201 − 3.21061i −0.499663 − 0.576642i 0.448759 − 0.893653i \(-0.351866\pi\)
−0.948422 + 0.317011i \(0.897321\pi\)
\(32\) −0.645667 + 0.0928329i −0.114139 + 0.0164107i
\(33\) 1.88589 + 0.861256i 0.328291 + 0.149925i
\(34\) 5.21519 − 1.53132i 0.894398 − 0.262619i
\(35\) 0 0
\(36\) 0.0959274 − 0.210052i 0.0159879 − 0.0350086i
\(37\) −5.92406 + 0.851751i −0.973910 + 0.140027i −0.610857 − 0.791741i \(-0.709175\pi\)
−0.363053 + 0.931768i \(0.618266\pi\)
\(38\) −1.08514 + 0.940282i −0.176033 + 0.152534i
\(39\) −0.425839 − 0.932456i −0.0681887 − 0.149313i
\(40\) 0 0
\(41\) 1.36297 − 9.47964i 0.212860 − 1.48047i −0.550684 − 0.834714i \(-0.685633\pi\)
0.763543 − 0.645757i \(-0.223458\pi\)
\(42\) −1.65411 + 2.57385i −0.255235 + 0.397153i
\(43\) 1.42270 + 1.23278i 0.216960 + 0.187997i 0.756560 − 0.653925i \(-0.226879\pi\)
−0.539599 + 0.841922i \(0.681424\pi\)
\(44\) −0.229680 − 0.0674400i −0.0346255 − 0.0101670i
\(45\) 0 0
\(46\) 4.33945 − 4.95112i 0.639817 − 0.730003i
\(47\) − 7.61130i − 1.11022i −0.831776 − 0.555111i \(-0.812676\pi\)
0.831776 − 0.555111i \(-0.187324\pi\)
\(48\) −1.05812 + 3.60362i −0.152726 + 0.520137i
\(49\) −1.33124 + 1.53634i −0.190177 + 0.219477i
\(50\) 0 0
\(51\) −0.563476 + 3.91906i −0.0789025 + 0.548779i
\(52\) 0.0639885 + 0.0995681i 0.00887361 + 0.0138076i
\(53\) 10.8563 − 4.95792i 1.49123 − 0.681022i 0.507662 − 0.861556i \(-0.330510\pi\)
0.983569 + 0.180534i \(0.0577826\pi\)
\(54\) −4.49492 − 5.18741i −0.611681 − 0.705917i
\(55\) 0 0
\(56\) 2.68870 − 5.88744i 0.359293 − 0.786742i
\(57\) −0.294675 − 1.00357i −0.0390307 − 0.132926i
\(58\) −2.27697 − 7.75466i −0.298981 − 1.01824i
\(59\) −4.75662 + 10.4155i −0.619259 + 1.35599i 0.296799 + 0.954940i \(0.404081\pi\)
−0.916057 + 0.401047i \(0.868646\pi\)
\(60\) 0 0
\(61\) 4.20330 + 4.85087i 0.538178 + 0.621090i 0.958087 − 0.286476i \(-0.0924838\pi\)
−0.419910 + 0.907566i \(0.637938\pi\)
\(62\) 5.30490 − 2.42267i 0.673723 − 0.307679i
\(63\) 2.40986 + 3.74982i 0.303614 + 0.472432i
\(64\) 1.19644 − 8.32140i 0.149555 − 1.04018i
\(65\) 0 0
\(66\) −1.86381 + 2.15095i −0.229419 + 0.264764i
\(67\) −1.15453 + 3.93196i −0.141048 + 0.480366i −0.999469 − 0.0325800i \(-0.989628\pi\)
0.858421 + 0.512946i \(0.171446\pi\)
\(68\) − 0.457147i − 0.0554372i
\(69\) 1.96755 + 4.37364i 0.236865 + 0.526525i
\(70\) 0 0
\(71\) −9.69553 − 2.84686i −1.15065 − 0.337861i −0.349856 − 0.936804i \(-0.613769\pi\)
−0.800792 + 0.598943i \(0.795588\pi\)
\(72\) 4.38950 + 3.80353i 0.517308 + 0.448250i
\(73\) 4.91945 − 7.65481i 0.575778 − 0.895927i −0.424176 − 0.905580i \(-0.639436\pi\)
0.999953 + 0.00965241i \(0.00307251\pi\)
\(74\) 1.16927 − 8.13247i 0.135925 − 0.945380i
\(75\) 0 0
\(76\) 0.0501671 + 0.109851i 0.00575456 + 0.0126007i
\(77\) 3.49206 − 3.02588i 0.397957 − 0.344831i
\(78\) 1.39291 − 0.200270i 0.157716 − 0.0226761i
\(79\) −1.59168 + 3.48530i −0.179078 + 0.392127i −0.977790 − 0.209587i \(-0.932788\pi\)
0.798712 + 0.601714i \(0.205515\pi\)
\(80\) 0 0
\(81\) −0.959493 + 0.281733i −0.106610 + 0.0313036i
\(82\) 11.9592 + 5.46160i 1.32068 + 0.603133i
\(83\) −7.20866 + 1.03645i −0.791254 + 0.113765i −0.526079 − 0.850435i \(-0.676339\pi\)
−0.265174 + 0.964201i \(0.585429\pi\)
\(84\) 0.168513 + 0.194474i 0.0183862 + 0.0212188i
\(85\) 0 0
\(86\) −2.17403 + 1.39717i −0.234432 + 0.150660i
\(87\) 5.82741 + 0.837855i 0.624764 + 0.0898274i
\(88\) 3.25512 − 5.06506i 0.346996 − 0.539937i
\(89\) −6.80606 + 7.85461i −0.721441 + 0.832587i −0.991480 − 0.130262i \(-0.958418\pi\)
0.270039 + 0.962850i \(0.412964\pi\)
\(90\) 0 0
\(91\) −2.28463 −0.239494
\(92\) −0.296726 − 0.467509i −0.0309359 − 0.0487412i
\(93\) 4.24824i 0.440522i
\(94\) 10.0254 + 2.94373i 1.03404 + 0.303623i
\(95\) 0 0
\(96\) 0.548755 + 0.352663i 0.0560071 + 0.0359936i
\(97\) −1.44652 − 0.207978i −0.146872 − 0.0211169i 0.0684868 − 0.997652i \(-0.478183\pi\)
−0.215358 + 0.976535i \(0.569092\pi\)
\(98\) −1.50876 − 2.34767i −0.152408 − 0.237151i
\(99\) 1.72251 + 3.77178i 0.173119 + 0.379078i
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 575.2.p.a.524.1 20
5.2 odd 4 575.2.k.a.501.1 10
5.3 odd 4 115.2.g.a.41.1 ✓ 10
5.4 even 2 inner 575.2.p.a.524.2 20
23.9 even 11 inner 575.2.p.a.124.2 20
115.3 odd 44 2645.2.a.n.1.3 5
115.9 even 22 inner 575.2.p.a.124.1 20
115.32 odd 44 575.2.k.a.101.1 10
115.43 even 44 2645.2.a.o.1.3 5
115.78 odd 44 115.2.g.a.101.1 yes 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.2.g.a.41.1 ✓ 10 5.3 odd 4
115.2.g.a.101.1 yes 10 115.78 odd 44
575.2.k.a.101.1 10 115.32 odd 44
575.2.k.a.501.1 10 5.2 odd 4
575.2.p.a.124.1 20 115.9 even 22 inner
575.2.p.a.124.2 20 23.9 even 11 inner
575.2.p.a.524.1 20 1.1 even 1 trivial
575.2.p.a.524.2 20 5.4 even 2 inner
2645.2.a.n.1.3 5 115.3 odd 44
2645.2.a.o.1.3 5 115.43 even 44