# Properties

 Label 23.4.c.a Level $23$ Weight $4$ Character orbit 23.c Analytic conductor $1.357$ Analytic rank $0$ Dimension $50$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [23,4,Mod(2,23)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(23, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([2]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("23.2");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$23$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 23.c (of order $$11$$, degree $$10$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$1.35704393013$$ Analytic rank: $$0$$ Dimension: $$50$$ Relative dimension: $$5$$ over $$\Q(\zeta_{11})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

## $q$-expansion

The algebraic $$q$$-expansion of this newform has not been computed, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$50 q - 11 q^{2} - 13 q^{3} - 27 q^{4} - 19 q^{5} - 4 q^{6} - 19 q^{7} + 28 q^{8} + 24 q^{9}+O(q^{10})$$ 50 * q - 11 * q^2 - 13 * q^3 - 27 * q^4 - 19 * q^5 - 4 * q^6 - 19 * q^7 + 28 * q^8 + 24 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$50 q - 11 q^{2} - 13 q^{3} - 27 q^{4} - 19 q^{5} - 4 q^{6} - 19 q^{7} + 28 q^{8} + 24 q^{9} + 47 q^{10} - 53 q^{11} + 36 q^{12} - 65 q^{13} + 117 q^{14} - 425 q^{15} - 499 q^{16} - 117 q^{17} + 24 q^{18} + 73 q^{19} + 529 q^{20} + 429 q^{21} + 310 q^{22} + 542 q^{23} + 1606 q^{24} + 246 q^{25} + 324 q^{26} + 65 q^{27} - 677 q^{28} - 497 q^{29} - 1041 q^{30} - 471 q^{31} - 915 q^{32} - 391 q^{33} - 2751 q^{34} - 737 q^{35} - 1865 q^{36} - 1071 q^{37} - 1504 q^{38} + 127 q^{39} + 1479 q^{40} + 569 q^{41} + 3059 q^{42} + 1615 q^{43} + 2518 q^{44} + 2768 q^{45} + 4041 q^{46} + 2904 q^{47} + 2702 q^{48} + 1226 q^{49} + 1322 q^{50} + 589 q^{51} - 2156 q^{52} + 391 q^{53} - 5862 q^{54} - 3323 q^{55} - 7028 q^{56} - 7623 q^{57} - 5639 q^{58} - 2445 q^{59} - 3157 q^{60} - 1059 q^{61} + 1468 q^{62} + 3155 q^{63} + 4570 q^{64} + 2641 q^{65} + 5206 q^{66} + 27 q^{67} + 8350 q^{68} + 4005 q^{69} + 9702 q^{70} + 3465 q^{71} + 5629 q^{72} + 435 q^{73} - 994 q^{74} - 7819 q^{75} - 3598 q^{76} - 5931 q^{77} - 8996 q^{78} - 2559 q^{79} - 14052 q^{80} - 4788 q^{81} - 3822 q^{82} - 3967 q^{83} - 8427 q^{84} + 299 q^{85} + 721 q^{86} + 8363 q^{87} + 5825 q^{88} + 3717 q^{89} + 16742 q^{90} + 7238 q^{91} + 9550 q^{92} + 12750 q^{93} + 6035 q^{94} + 4551 q^{95} + 2493 q^{96} - 2419 q^{97} - 5687 q^{98} - 755 q^{99}+O(q^{100})$$ 50 * q - 11 * q^2 - 13 * q^3 - 27 * q^4 - 19 * q^5 - 4 * q^6 - 19 * q^7 + 28 * q^8 + 24 * q^9 + 47 * q^10 - 53 * q^11 + 36 * q^12 - 65 * q^13 + 117 * q^14 - 425 * q^15 - 499 * q^16 - 117 * q^17 + 24 * q^18 + 73 * q^19 + 529 * q^20 + 429 * q^21 + 310 * q^22 + 542 * q^23 + 1606 * q^24 + 246 * q^25 + 324 * q^26 + 65 * q^27 - 677 * q^28 - 497 * q^29 - 1041 * q^30 - 471 * q^31 - 915 * q^32 - 391 * q^33 - 2751 * q^34 - 737 * q^35 - 1865 * q^36 - 1071 * q^37 - 1504 * q^38 + 127 * q^39 + 1479 * q^40 + 569 * q^41 + 3059 * q^42 + 1615 * q^43 + 2518 * q^44 + 2768 * q^45 + 4041 * q^46 + 2904 * q^47 + 2702 * q^48 + 1226 * q^49 + 1322 * q^50 + 589 * q^51 - 2156 * q^52 + 391 * q^53 - 5862 * q^54 - 3323 * q^55 - 7028 * q^56 - 7623 * q^57 - 5639 * q^58 - 2445 * q^59 - 3157 * q^60 - 1059 * q^61 + 1468 * q^62 + 3155 * q^63 + 4570 * q^64 + 2641 * q^65 + 5206 * q^66 + 27 * q^67 + 8350 * q^68 + 4005 * q^69 + 9702 * q^70 + 3465 * q^71 + 5629 * q^72 + 435 * q^73 - 994 * q^74 - 7819 * q^75 - 3598 * q^76 - 5931 * q^77 - 8996 * q^78 - 2559 * q^79 - 14052 * q^80 - 4788 * q^81 - 3822 * q^82 - 3967 * q^83 - 8427 * q^84 + 299 * q^85 + 721 * q^86 + 8363 * q^87 + 5825 * q^88 + 3717 * q^89 + 16742 * q^90 + 7238 * q^91 + 9550 * q^92 + 12750 * q^93 + 6035 * q^94 + 4551 * q^95 + 2493 * q^96 - 2419 * q^97 - 5687 * q^98 - 755 * q^99

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1 −3.24175 2.08335i 1.21610 + 8.45817i 2.84529 + 6.23031i 12.7969 + 3.75752i 13.6790 29.9528i −11.3462 + 13.0942i −0.631070 + 4.38919i −44.1554 + 12.9652i −33.6563 38.8414i
2.2 −2.68394 1.72487i −0.220662 1.53474i 0.905070 + 1.98183i −18.5174 5.43720i −2.05498 + 4.49977i 13.0047 15.0082i −2.64311 + 18.3833i 23.5996 6.92946i 40.3212 + 46.5331i
2.3 −0.442460 0.284352i −0.784628 5.45721i −3.20841 7.02543i 12.7367 + 3.73985i −1.20460 + 2.63770i −5.26652 + 6.07789i −1.17691 + 8.18558i −3.25915 + 0.956973i −4.57207 5.27645i
2.4 1.91736 + 1.23221i 0.776967 + 5.40392i −1.16540 2.55188i −0.907869 0.266574i −5.16905 + 11.3186i 8.82725 10.1872i 3.50483 24.3766i −2.69237 + 0.790552i −1.41223 1.62980i
2.5 3.73432 + 2.39990i −0.655499 4.55909i 4.86229 + 10.6469i −9.99480 2.93474i 8.49353 18.5982i −17.7583 + 20.4942i −2.34036 + 16.2775i 5.55065 1.62982i −30.2807 34.9458i
3.1 −0.589312 4.09876i 1.03962 2.27645i −8.77657 + 2.57703i −3.66646 + 4.23132i −9.94329 2.91961i 16.9812 10.9132i 1.97323 + 4.32077i 13.5798 + 15.6719i 19.5039 + 12.5344i
3.2 −0.132292 0.920110i 0.209861 0.459531i 6.84684 2.01041i 6.87249 7.93128i −0.450582 0.132303i −25.4560 + 16.3596i −5.84485 12.7984i 17.5141 + 20.2124i −8.20682 5.27420i
3.3 0.0282746 + 0.196654i −3.66027 + 8.01487i 7.63807 2.24274i −6.59261 + 7.60828i −1.67965 0.493191i 21.3237 13.7039i 1.31727 + 2.88443i −33.1594 38.2680i −1.68261 1.08135i
3.4 0.306014 + 2.12837i 3.96829 8.68933i 3.23961 0.951236i −11.4648 + 13.2311i 19.7085 + 5.78694i −2.65189 + 1.70427i 10.1620 + 22.2516i −42.0760 48.5583i −31.6691 20.3525i
3.5 0.619909 + 4.31156i −0.504759 + 1.10527i −10.5293 + 3.09169i 4.52322 5.22007i −5.07833 1.49113i 2.36872 1.52228i −5.38116 11.7831i 16.7144 + 19.2894i 25.3106 + 16.2662i
4.1 −2.10729 4.61431i 8.43369 2.47635i −11.6123 + 13.4013i −5.02341 3.22835i −29.1989 33.6973i 3.12135 + 21.7095i 47.3705 + 13.9092i 42.2809 27.1723i −4.31085 + 29.9826i
4.2 −1.58245 3.46509i −7.04544 + 2.06873i −4.26380 + 4.92069i −4.88780 3.14120i 18.3174 + 21.1394i −2.25200 15.6630i −5.44232 1.59801i 22.6448 14.5529i −3.14982 + 21.9075i
4.3 −0.308069 0.674576i 1.68484 0.494713i 4.87874 5.63037i 3.36936 + 2.16536i −0.852766 0.984145i 0.387311 + 2.69380i −10.9935 3.22799i −20.1199 + 12.9303i 0.422704 2.93997i
4.4 1.49850 + 3.28126i −7.64281 + 2.24413i −3.28227 + 3.78794i 15.0482 + 9.67090i −18.8163 21.7152i −2.56161 17.8164i 10.3412 + 3.03646i 30.6625 19.7056i −9.18297 + 63.8690i
4.5 1.61009 + 3.52560i 3.27467 0.961529i −4.59861 + 5.30707i −10.7624 6.91659i 8.66248 + 9.99703i −0.249078 1.73238i 3.63606 + 1.06764i −12.9149 + 8.29992i 7.05669 49.0804i
6.1 −2.10729 + 4.61431i 8.43369 + 2.47635i −11.6123 13.4013i −5.02341 + 3.22835i −29.1989 + 33.6973i 3.12135 21.7095i 47.3705 13.9092i 42.2809 + 27.1723i −4.31085 29.9826i
6.2 −1.58245 + 3.46509i −7.04544 2.06873i −4.26380 4.92069i −4.88780 + 3.14120i 18.3174 21.1394i −2.25200 + 15.6630i −5.44232 + 1.59801i 22.6448 + 14.5529i −3.14982 21.9075i
6.3 −0.308069 + 0.674576i 1.68484 + 0.494713i 4.87874 + 5.63037i 3.36936 2.16536i −0.852766 + 0.984145i 0.387311 2.69380i −10.9935 + 3.22799i −20.1199 12.9303i 0.422704 + 2.93997i
6.4 1.49850 3.28126i −7.64281 2.24413i −3.28227 3.78794i 15.0482 9.67090i −18.8163 + 21.7152i −2.56161 + 17.8164i 10.3412 3.03646i 30.6625 + 19.7056i −9.18297 63.8690i
6.5 1.61009 3.52560i 3.27467 + 0.961529i −4.59861 5.30707i −10.7624 + 6.91659i 8.66248 9.99703i −0.249078 + 1.73238i 3.63606 1.06764i −12.9149 8.29992i 7.05669 + 49.0804i
See all 50 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 23.4.c.a 50
3.b odd 2 1 207.4.i.a 50
23.c even 11 1 inner 23.4.c.a 50
23.c even 11 1 529.4.a.n 25
23.d odd 22 1 529.4.a.m 25
69.h odd 22 1 207.4.i.a 50

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.4.c.a 50 1.a even 1 1 trivial
23.4.c.a 50 23.c even 11 1 inner
207.4.i.a 50 3.b odd 2 1
207.4.i.a 50 69.h odd 22 1
529.4.a.m 25 23.d odd 22 1
529.4.a.n 25 23.c even 11 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(23, [\chi])$$.