# Properties

 Label 23.3.d.a Level 23 Weight 3 Character orbit 23.d Analytic conductor 0.627 Analytic rank 0 Dimension 30 CM No

# Related objects

## Newspace parameters

 Level: $$N$$ = $$23$$ Weight: $$k$$ = $$3$$ Character orbit: $$[\chi]$$ = 23.d (of order $$22$$ and degree $$10$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.626704608029$$ Analytic rank: $$0$$ Dimension: $$30$$ Relative dimension: $$3$$ over $$\Q(\zeta_{22})$$ Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$30q$$ $$\mathstrut -\mathstrut 11q^{2}$$ $$\mathstrut -\mathstrut 11q^{3}$$ $$\mathstrut -\mathstrut 23q^{4}$$ $$\mathstrut -\mathstrut 11q^{5}$$ $$\mathstrut +\mathstrut 22q^{6}$$ $$\mathstrut -\mathstrut 11q^{7}$$ $$\mathstrut +\mathstrut 10q^{8}$$ $$\mathstrut -\mathstrut 38q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$30q$$ $$\mathstrut -\mathstrut 11q^{2}$$ $$\mathstrut -\mathstrut 11q^{3}$$ $$\mathstrut -\mathstrut 23q^{4}$$ $$\mathstrut -\mathstrut 11q^{5}$$ $$\mathstrut +\mathstrut 22q^{6}$$ $$\mathstrut -\mathstrut 11q^{7}$$ $$\mathstrut +\mathstrut 10q^{8}$$ $$\mathstrut -\mathstrut 38q^{9}$$ $$\mathstrut -\mathstrut 11q^{10}$$ $$\mathstrut -\mathstrut 11q^{11}$$ $$\mathstrut -\mathstrut 14q^{12}$$ $$\mathstrut -\mathstrut 11q^{13}$$ $$\mathstrut -\mathstrut 11q^{14}$$ $$\mathstrut +\mathstrut 66q^{15}$$ $$\mathstrut +\mathstrut 73q^{16}$$ $$\mathstrut +\mathstrut 44q^{17}$$ $$\mathstrut +\mathstrut 126q^{18}$$ $$\mathstrut +\mathstrut 22q^{19}$$ $$\mathstrut +\mathstrut 77q^{20}$$ $$\mathstrut +\mathstrut 22q^{21}$$ $$\mathstrut +\mathstrut 36q^{23}$$ $$\mathstrut -\mathstrut 22q^{24}$$ $$\mathstrut -\mathstrut 152q^{25}$$ $$\mathstrut -\mathstrut 186q^{26}$$ $$\mathstrut -\mathstrut 62q^{27}$$ $$\mathstrut -\mathstrut 275q^{28}$$ $$\mathstrut -\mathstrut 88q^{29}$$ $$\mathstrut -\mathstrut 363q^{30}$$ $$\mathstrut -\mathstrut 110q^{31}$$ $$\mathstrut -\mathstrut 147q^{32}$$ $$\mathstrut -\mathstrut 132q^{33}$$ $$\mathstrut +\mathstrut 231q^{34}$$ $$\mathstrut +\mathstrut 209q^{35}$$ $$\mathstrut +\mathstrut 229q^{36}$$ $$\mathstrut +\mathstrut 341q^{37}$$ $$\mathstrut +\mathstrut 374q^{38}$$ $$\mathstrut +\mathstrut 295q^{39}$$ $$\mathstrut +\mathstrut 429q^{40}$$ $$\mathstrut +\mathstrut 77q^{41}$$ $$\mathstrut +\mathstrut 319q^{42}$$ $$\mathstrut +\mathstrut 77q^{43}$$ $$\mathstrut +\mathstrut 110q^{44}$$ $$\mathstrut -\mathstrut 99q^{46}$$ $$\mathstrut -\mathstrut 110q^{47}$$ $$\mathstrut -\mathstrut 550q^{48}$$ $$\mathstrut -\mathstrut 422q^{49}$$ $$\mathstrut -\mathstrut 396q^{50}$$ $$\mathstrut -\mathstrut 275q^{51}$$ $$\mathstrut -\mathstrut 472q^{52}$$ $$\mathstrut -\mathstrut 187q^{53}$$ $$\mathstrut -\mathstrut 198q^{54}$$ $$\mathstrut -\mathstrut 165q^{55}$$ $$\mathstrut +\mathstrut 176q^{56}$$ $$\mathstrut -\mathstrut 176q^{57}$$ $$\mathstrut -\mathstrut 13q^{58}$$ $$\mathstrut -\mathstrut q^{59}$$ $$\mathstrut +\mathstrut 539q^{60}$$ $$\mathstrut +\mathstrut 297q^{61}$$ $$\mathstrut +\mathstrut 82q^{62}$$ $$\mathstrut +\mathstrut 264q^{63}$$ $$\mathstrut +\mathstrut 386q^{64}$$ $$\mathstrut +\mathstrut 220q^{65}$$ $$\mathstrut +\mathstrut 264q^{66}$$ $$\mathstrut +\mathstrut 11q^{67}$$ $$\mathstrut -\mathstrut 66q^{69}$$ $$\mathstrut -\mathstrut 198q^{70}$$ $$\mathstrut -\mathstrut 176q^{71}$$ $$\mathstrut -\mathstrut 605q^{72}$$ $$\mathstrut -\mathstrut 121q^{73}$$ $$\mathstrut -\mathstrut 352q^{74}$$ $$\mathstrut +\mathstrut 154q^{75}$$ $$\mathstrut +\mathstrut 110q^{76}$$ $$\mathstrut +\mathstrut 110q^{77}$$ $$\mathstrut +\mathstrut 360q^{78}$$ $$\mathstrut +\mathstrut 33q^{79}$$ $$\mathstrut -\mathstrut 242q^{80}$$ $$\mathstrut +\mathstrut 494q^{81}$$ $$\mathstrut +\mathstrut 96q^{82}$$ $$\mathstrut -\mathstrut 154q^{83}$$ $$\mathstrut +\mathstrut 11q^{84}$$ $$\mathstrut +\mathstrut 275q^{85}$$ $$\mathstrut +\mathstrut 143q^{86}$$ $$\mathstrut +\mathstrut 271q^{87}$$ $$\mathstrut +\mathstrut 429q^{88}$$ $$\mathstrut +\mathstrut 121q^{89}$$ $$\mathstrut +\mathstrut 242q^{90}$$ $$\mathstrut +\mathstrut 166q^{92}$$ $$\mathstrut +\mathstrut 260q^{93}$$ $$\mathstrut -\mathstrut 295q^{94}$$ $$\mathstrut -\mathstrut 154q^{95}$$ $$\mathstrut -\mathstrut 419q^{96}$$ $$\mathstrut +\mathstrut 154q^{97}$$ $$\mathstrut +\mathstrut 77q^{98}$$ $$\mathstrut -\mathstrut 242q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

## 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.

Label $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 −3.10125 0.910608i −3.16934 + 3.65762i 5.42352 + 3.48548i −5.40865 0.777647i 13.1596 8.45714i −0.889564 + 0.406250i −5.17926 5.97719i −2.05259 14.2761i 16.0654 + 7.33684i
5.2 −1.80062 0.528710i 2.35762 2.72084i −0.402313 0.258551i 5.05070 + 0.726181i −5.68372 + 3.65270i −8.85488 + 4.04389i 5.50346 + 6.35133i −0.563759 3.92103i −8.71046 3.97793i
5.3 1.44626 + 0.424661i −0.590042 + 0.680945i −1.45368 0.934223i −1.77862 0.255727i −1.14252 + 0.734256i 2.68734 1.22727i −5.65401 6.52507i 1.16530 + 8.10482i −2.46375 1.12516i
7.1 −1.38322 + 1.59632i −3.80315 + 2.44414i −0.0656849 0.456848i 6.26010 2.85889i 1.35897 9.45184i 2.54176 + 8.65643i −6.28757 4.04078i 4.75142 10.4042i −4.09539 + 13.9476i
7.2 −0.881085 + 1.01683i 2.64748 1.70143i 0.311634 + 2.16747i −2.08252 + 0.951056i −0.602595 + 4.19114i −2.90589 9.89656i −7.00598 4.50247i 0.375553 0.822346i 0.867820 2.95552i
7.3 1.59301 1.83844i −1.87410 + 1.20441i −0.272894 1.89802i −2.69128 + 1.22907i −0.771235 + 5.36406i −1.33225 4.53722i 4.26162 + 2.73877i −1.67709 + 3.67232i −2.02769 + 6.90566i
10.1 −1.38322 1.59632i −3.80315 2.44414i −0.0656849 + 0.456848i 6.26010 + 2.85889i 1.35897 + 9.45184i 2.54176 8.65643i −6.28757 + 4.04078i 4.75142 + 10.4042i −4.09539 13.9476i
10.2 −0.881085 1.01683i 2.64748 + 1.70143i 0.311634 2.16747i −2.08252 0.951056i −0.602595 4.19114i −2.90589 + 9.89656i −7.00598 + 4.50247i 0.375553 + 0.822346i 0.867820 + 2.95552i
10.3 1.59301 + 1.83844i −1.87410 1.20441i −0.272894 + 1.89802i −2.69128 1.22907i −0.771235 5.36406i −1.33225 + 4.53722i 4.26162 2.73877i −1.67709 3.67232i −2.02769 6.90566i
11.1 −1.94274 + 1.24852i −0.365090 + 2.53926i 0.553766 1.21258i 0.682875 + 2.32566i −2.46105 5.38894i 6.72814 5.82996i −0.876504 6.09622i 2.32089 + 0.681474i −4.23029 3.66556i
11.2 0.0163142 0.0104845i 0.749667 5.21405i −1.66150 + 3.63819i 1.45779 + 4.96477i −0.0424364 0.0929228i −1.27914 + 1.10838i 0.0220779 + 0.153555i −17.9889 5.28201i 0.0758357 + 0.0657120i
11.3 1.20995 0.777587i −0.238691 + 1.66013i −0.802326 + 1.75685i −2.65558 9.04406i 1.00209 + 2.19428i −5.31379 + 4.60443i 1.21408 + 8.44409i 5.93637 + 1.74307i −10.2457 8.87791i
14.1 −3.10125 + 0.910608i −3.16934 3.65762i 5.42352 3.48548i −5.40865 + 0.777647i 13.1596 + 8.45714i −0.889564 0.406250i −5.17926 + 5.97719i −2.05259 + 14.2761i 16.0654 7.33684i
14.2 −1.80062 + 0.528710i 2.35762 + 2.72084i −0.402313 + 0.258551i 5.05070 0.726181i −5.68372 3.65270i −8.85488 4.04389i 5.50346 6.35133i −0.563759 + 3.92103i −8.71046 + 3.97793i
14.3 1.44626 0.424661i −0.590042 0.680945i −1.45368 + 0.934223i −1.77862 + 0.255727i −1.14252 0.734256i 2.68734 + 1.22727i −5.65401 + 6.52507i 1.16530 8.10482i −2.46375 + 1.12516i
15.1 −0.387684 + 2.69640i −0.141551 0.309954i −3.28232 0.963778i −0.353921 + 0.306674i 0.890639 0.261515i 4.93105 7.67286i −0.655341 + 1.43500i 5.81771 6.71400i −0.689708 1.07321i
15.2 0.123822 0.861198i −1.04228 2.28229i 3.11164 + 0.913660i −1.80972 + 1.56813i −2.09456 + 0.615017i −6.61670 + 10.2958i 2.61786 5.73232i 1.77128 2.04416i 1.12639 + 1.75270i
15.3 0.496456 3.45293i 1.53764 + 3.36695i −7.83827 2.30152i −4.84192 + 4.19555i 12.3892 3.63780i 3.55260 5.52795i −6.04175 + 13.2296i −3.07830 + 3.55255i 12.0831 + 18.8017i
17.1 −1.55977 + 3.41542i 0.201951 + 0.0592983i −6.61275 7.63152i 2.90325 + 4.51755i −0.517526 + 0.597257i 7.13192 + 1.02541i 21.9686 6.45058i −7.53401 4.84182i −19.9577 + 2.86949i
17.2 −0.282292 + 0.618133i 0.844537 + 0.247978i 2.31704 + 2.67401i −3.24760 5.05336i −0.391690 + 0.452034i −3.20136 0.460286i −4.91504 + 1.44319i −6.91953 4.44691i 4.04042 0.580925i
See all 30 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 21.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not have CM; other inner twists have not been computed.

## Hecke kernels

There are no other newforms in $$S_{3}^{\mathrm{new}}(23, [\chi])$$.