# Properties

 Label 5.10.b.a Level 5 Weight 10 Character orbit 5.b Analytic conductor 2.575 Analytic rank 0 Dimension 4 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$5$$ Weight: $$k$$ = $$10$$ Character orbit: $$[\chi]$$ = 5.b (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$2.57517918082$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} + \cdots)$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{5}\cdot 3^{2}\cdot 5^{2}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ \beta_{1} q^{2}$$ $$+ ( -\beta_{1} + \beta_{2} ) q^{3}$$ $$+ ( -342 - \beta_{3} ) q^{4}$$ $$+ ( 285 + 19 \beta_{1} + 7 \beta_{2} + \beta_{3} ) q^{5}$$ $$+ ( 702 + \beta_{3} ) q^{6}$$ $$+ ( 133 \beta_{1} - 49 \beta_{2} ) q^{7}$$ $$+ ( -684 \beta_{1} + 8 \beta_{2} ) q^{8}$$ $$+ ( 2907 + 18 \beta_{3} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ \beta_{1} q^{2}$$ $$+ ( -\beta_{1} + \beta_{2} ) q^{3}$$ $$+ ( -342 - \beta_{3} ) q^{4}$$ $$+ ( 285 + 19 \beta_{1} + 7 \beta_{2} + \beta_{3} ) q^{5}$$ $$+ ( 702 + \beta_{3} ) q^{6}$$ $$+ ( 133 \beta_{1} - 49 \beta_{2} ) q^{7}$$ $$+ ( -684 \beta_{1} + 8 \beta_{2} ) q^{8}$$ $$+ ( 2907 + 18 \beta_{3} ) q^{9}$$ $$+ ( -17290 + 1139 \beta_{1} - 8 \beta_{2} - 19 \beta_{3} ) q^{10}$$ $$+ ( 27492 - 20 \beta_{3} ) q^{11}$$ $$+ ( 1044 \beta_{1} + 504 \beta_{2} ) q^{12}$$ $$+ ( -1918 \beta_{1} - 110 \beta_{2} ) q^{13}$$ $$+ ( -106134 - 133 \beta_{3} ) q^{14}$$ $$+ ( -99180 - 987 \beta_{1} - 561 \beta_{2} + 152 \beta_{3} ) q^{15}$$ $$+ ( 407816 + 172 \beta_{3} ) q^{16}$$ $$+ ( -7448 \beta_{1} - 1632 \beta_{2} ) q^{17}$$ $$+ ( 18279 \beta_{1} - 144 \beta_{2} ) q^{18}$$ $$+ ( -159220 + 476 \beta_{3} ) q^{19}$$ $$+ ( -825570 - 23788 \beta_{1} + 3736 \beta_{2} - 627 \beta_{3} ) q^{20}$$ $$+ ( 880992 - 798 \beta_{3} ) q^{21}$$ $$+ ( 10412 \beta_{1} + 160 \beta_{2} ) q^{22}$$ $$+ ( 17005 \beta_{1} + 5199 \beta_{2} ) q^{23}$$ $$+ ( -608760 - 532 \beta_{3} ) q^{24}$$ $$+ ( -334475 + 45410 \beta_{1} - 8270 \beta_{2} + 1140 \beta_{3} ) q^{25}$$ $$+ ( 1654692 + 1918 \beta_{3} ) q^{26}$$ $$+ ( -35226 \beta_{1} + 7362 \beta_{2} ) q^{27}$$ $$+ ( -151620 \beta_{1} - 24024 \beta_{2} ) q^{28}$$ $$+ ( -882930 - 1976 \beta_{3} ) q^{29}$$ $$+ ( 928170 + 30628 \beta_{1} - 1216 \beta_{2} + 987 \beta_{3} ) q^{30}$$ $$+ ( -2646928 - 760 \beta_{3} ) q^{31}$$ $$+ ( 204496 \beta_{1} + 2720 \beta_{2} ) q^{32}$$ $$+ ( -13452 \beta_{1} + 44412 \beta_{2} ) q^{33}$$ $$+ ( 6608656 + 7448 \beta_{3} ) q^{34}$$ $$+ ( 3407460 + 144039 \beta_{1} + 26817 \beta_{2} - 9044 \beta_{3} ) q^{35}$$ $$+ ( -14099994 - 9063 \beta_{3} ) q^{36}$$ $$+ ( -335370 \beta_{1} - 51378 \beta_{2} ) q^{37}$$ $$+ ( 247284 \beta_{1} - 3808 \beta_{2} ) q^{38}$$ $$+ ( 421704 - 4008 \beta_{3} ) q^{39}$$ $$+ ( 10894600 - 777860 \beta_{1} + 920 \beta_{2} + 14060 \beta_{3} ) q^{40}$$ $$+ ( -4197138 + 24890 \beta_{3} ) q^{41}$$ $$+ ( 199500 \beta_{1} + 6384 \beta_{2} ) q^{42}$$ $$+ ( 719131 \beta_{1} - 132643 \beta_{2} ) q^{43}$$ $$+ ( 5159736 - 20652 \beta_{3} ) q^{44}$$ $$+ ( 13934295 + 366453 \beta_{1} - 89991 \beta_{2} + 8037 \beta_{3} ) q^{45}$$ $$+ ( -15312518 - 17005 \beta_{3} ) q^{46}$$ $$+ ( -1012111 \beta_{1} + 214259 \beta_{2} ) q^{47}$$ $$+ ( -528560 \beta_{1} + 262304 \beta_{2} ) q^{48}$$ $$+ ( -11730257 + 27930 \beta_{3} ) q^{49}$$ $$+ ( -37523100 + 639085 \beta_{1} - 9120 \beta_{2} - 45410 \beta_{3} ) q^{50}$$ $$+ ( 21004272 - 38456 \beta_{3} ) q^{51}$$ $$+ ( 2310648 \beta_{1} - 71664 \beta_{2} ) q^{52}$$ $$+ ( -1349666 \beta_{1} - 259794 \beta_{2} ) q^{53}$$ $$+ ( 28963980 + 35226 \beta_{3} ) q^{54}$$ $$+ ( -6726780 + 176548 \beta_{1} + 315044 \beta_{2} + 21792 \beta_{3} ) q^{55}$$ $$+ ( 78794520 + 83524 \beta_{3} ) q^{56}$$ $$+ ( -174932 \beta_{1} - 561916 \beta_{2} ) q^{57}$$ $$+ ( -2570434 \beta_{1} + 15808 \beta_{2} ) q^{58}$$ $$+ ( -115207260 - 52972 \beta_{3} ) q^{59}$$ $$+ ( -76751640 + 1265724 \beta_{1} - 295128 \beta_{2} + 47196 \beta_{3} ) q^{60}$$ $$+ ( 90122642 - 43150 \beta_{3} ) q^{61}$$ $$+ ( -3295968 \beta_{1} + 6080 \beta_{2} ) q^{62}$$ $$+ ( 2297043 \beta_{1} + 591633 \beta_{2} ) q^{63}$$ $$+ ( 33748768 - 116432 \beta_{3} ) q^{64}$$ $$+ ( 45973920 - 2201322 \beta_{1} + 77934 \beta_{2} + 21812 \beta_{3} ) q^{65}$$ $$+ ( 4737384 + 13452 \beta_{3} ) q^{66}$$ $$+ ( 6669647 \beta_{1} + 1448953 \beta_{2} ) q^{67}$$ $$+ ( 9155872 \beta_{1} - 895168 \beta_{2} ) q^{68}$$ $$+ ( -71631216 + 115786 \beta_{3} ) q^{69}$$ $$+ ( -127085490 - 4316116 \beta_{1} + 72352 \beta_{2} - 144039 \beta_{3} ) q^{70}$$ $$+ ( -11902968 - 91200 \beta_{3} ) q^{71}$$ $$+ ( -12480948 \beta_{1} - 1224 \beta_{2} ) q^{72}$$ $$+ ( -1847940 \beta_{1} + 90564 \beta_{2} ) q^{73}$$ $$+ ( 294215436 + 335370 \beta_{3} ) q^{74}$$ $$+ ( 164809800 - 465805 \beta_{1} - 1298915 \beta_{2} - 111720 \beta_{3} ) q^{75}$$ $$+ ( -292122360 - 3572 \beta_{3} ) q^{76}$$ $$+ ( 1533756 \beta_{1} - 2162748 \beta_{2} ) q^{77}$$ $$+ ( -3001128 \beta_{1} + 32064 \beta_{2} ) q^{78}$$ $$+ ( -182010880 - 267976 \beta_{3} ) q^{79}$$ $$+ ( 241460760 + 10722384 \beta_{1} + 1800352 \beta_{2} + 456836 \beta_{3} ) q^{80}$$ $$+ ( -85846959 + 458946 \beta_{3} ) q^{81}$$ $$+ ( 17058922 \beta_{1} - 199120 \beta_{2} ) q^{82}$$ $$+ ( -12737657 \beta_{1} + 1180353 \beta_{2} ) q^{83}$$ $$+ ( 279724536 - 608076 \beta_{3} ) q^{84}$$ $$+ ( 318854960 - 8731336 \beta_{1} + 988192 \beta_{2} - 75544 \beta_{3} ) q^{85}$$ $$+ ( -593976138 - 719131 \beta_{3} ) q^{86}$$ $$+ ( 2270082 \beta_{1} + 788766 \beta_{2} ) q^{87}$$ $$+ ( -7146128 \beta_{1} + 247136 \beta_{2} ) q^{88}$$ $$+ ( -395675190 + 185592 \beta_{3} ) q^{89}$$ $$+ ( -299272230 + 20797893 \beta_{1} - 64296 \beta_{2} - 366453 \beta_{3} ) q^{90}$$ $$+ ( 118330632 + 357504 \beta_{3} ) q^{91}$$ $$+ ( -21128228 \beta_{1} + 2797928 \beta_{2} ) q^{92}$$ $$+ ( 3180448 \beta_{1} - 2003968 \beta_{2} ) q^{93}$$ $$+ ( 831775426 + 1012111 \beta_{3} ) q^{94}$$ $$+ ( 301197900 + 5204860 \beta_{1} - 4032420 \beta_{2} - 23560 \beta_{3} ) q^{95}$$ $$+ ( 99834912 + 256176 \beta_{3} ) q^{96}$$ $$+ ( 14786464 \beta_{1} - 1954216 \beta_{2} ) q^{97}$$ $$+ ( 12121963 \beta_{1} - 223440 \beta_{2} ) q^{98}$$ $$+ ( -182196756 + 436716 \beta_{3} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q$$ $$\mathstrut -\mathstrut 1368q^{4}$$ $$\mathstrut +\mathstrut 1140q^{5}$$ $$\mathstrut +\mathstrut 2808q^{6}$$ $$\mathstrut +\mathstrut 11628q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$4q$$ $$\mathstrut -\mathstrut 1368q^{4}$$ $$\mathstrut +\mathstrut 1140q^{5}$$ $$\mathstrut +\mathstrut 2808q^{6}$$ $$\mathstrut +\mathstrut 11628q^{9}$$ $$\mathstrut -\mathstrut 69160q^{10}$$ $$\mathstrut +\mathstrut 109968q^{11}$$ $$\mathstrut -\mathstrut 424536q^{14}$$ $$\mathstrut -\mathstrut 396720q^{15}$$ $$\mathstrut +\mathstrut 1631264q^{16}$$ $$\mathstrut -\mathstrut 636880q^{19}$$ $$\mathstrut -\mathstrut 3302280q^{20}$$ $$\mathstrut +\mathstrut 3523968q^{21}$$ $$\mathstrut -\mathstrut 2435040q^{24}$$ $$\mathstrut -\mathstrut 1337900q^{25}$$ $$\mathstrut +\mathstrut 6618768q^{26}$$ $$\mathstrut -\mathstrut 3531720q^{29}$$ $$\mathstrut +\mathstrut 3712680q^{30}$$ $$\mathstrut -\mathstrut 10587712q^{31}$$ $$\mathstrut +\mathstrut 26434624q^{34}$$ $$\mathstrut +\mathstrut 13629840q^{35}$$ $$\mathstrut -\mathstrut 56399976q^{36}$$ $$\mathstrut +\mathstrut 1686816q^{39}$$ $$\mathstrut +\mathstrut 43578400q^{40}$$ $$\mathstrut -\mathstrut 16788552q^{41}$$ $$\mathstrut +\mathstrut 20638944q^{44}$$ $$\mathstrut +\mathstrut 55737180q^{45}$$ $$\mathstrut -\mathstrut 61250072q^{46}$$ $$\mathstrut -\mathstrut 46921028q^{49}$$ $$\mathstrut -\mathstrut 150092400q^{50}$$ $$\mathstrut +\mathstrut 84017088q^{51}$$ $$\mathstrut +\mathstrut 115855920q^{54}$$ $$\mathstrut -\mathstrut 26907120q^{55}$$ $$\mathstrut +\mathstrut 315178080q^{56}$$ $$\mathstrut -\mathstrut 460829040q^{59}$$ $$\mathstrut -\mathstrut 307006560q^{60}$$ $$\mathstrut +\mathstrut 360490568q^{61}$$ $$\mathstrut +\mathstrut 134995072q^{64}$$ $$\mathstrut +\mathstrut 183895680q^{65}$$ $$\mathstrut +\mathstrut 18949536q^{66}$$ $$\mathstrut -\mathstrut 286524864q^{69}$$ $$\mathstrut -\mathstrut 508341960q^{70}$$ $$\mathstrut -\mathstrut 47611872q^{71}$$ $$\mathstrut +\mathstrut 1176861744q^{74}$$ $$\mathstrut +\mathstrut 659239200q^{75}$$ $$\mathstrut -\mathstrut 1168489440q^{76}$$ $$\mathstrut -\mathstrut 728043520q^{79}$$ $$\mathstrut +\mathstrut 965843040q^{80}$$ $$\mathstrut -\mathstrut 343387836q^{81}$$ $$\mathstrut +\mathstrut 1118898144q^{84}$$ $$\mathstrut +\mathstrut 1275419840q^{85}$$ $$\mathstrut -\mathstrut 2375904552q^{86}$$ $$\mathstrut -\mathstrut 1582700760q^{89}$$ $$\mathstrut -\mathstrut 1197088920q^{90}$$ $$\mathstrut +\mathstrut 473322528q^{91}$$ $$\mathstrut +\mathstrut 3327101704q^{94}$$ $$\mathstrut +\mathstrut 1204791600q^{95}$$ $$\mathstrut +\mathstrut 399339648q^{96}$$ $$\mathstrut -\mathstrut 728787024q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4}\mathstrut +\mathstrut$$ $$45$$ $$x^{2}\mathstrut +\mathstrut$$ $$304$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 37 \nu$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$-\nu^{3} - 7 \nu$$ $$\beta_{3}$$ $$=$$ $$60 \nu^{2} + 1350$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{1}$$$$)/30$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3}\mathstrut -\mathstrut$$ $$1350$$$$)/60$$ $$\nu^{3}$$ $$=$$ $$($$$$-$$$$37$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$14$$ $$\beta_{1}$$$$)/30$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 − 2.87724i − 6.05982i 6.05982i 2.87724i
41.3193i 37.6407i −1195.29 1138.29 810.818i 1555.29 5315.22i 28233.0i 18266.2 −33502.5 47033.3i
4.2 0.843944i 179.263i 511.288 −568.288 1276.78i −151.288 8712.99i 863.597i −12452.2 −1077.53 + 479.603i
4.3 0.843944i 179.263i 511.288 −568.288 + 1276.78i −151.288 8712.99i 863.597i −12452.2 −1077.53 479.603i
4.4 41.3193i 37.6407i −1195.29 1138.29 + 810.818i 1555.29 5315.22i 28233.0i 18266.2 −33502.5 + 47033.3i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
5.b Even 1 yes

## Hecke kernels

There are no other newforms in $$S_{10}^{\mathrm{new}}(5, [\chi])$$.