# Properties

 Label 25.9.c.b Level $25$ Weight $9$ Character orbit 25.c Analytic conductor $10.184$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 25.c (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.1844652515$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ Defining polynomial: $$x^{6} - 2 x^{5} + 2 x^{4} - 30 x^{3} + 1089 x^{2} - 3168 x + 4608$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{6}\cdot 5^{2}$$ Twist minimal: no (minimal twist has level 5) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} + ( 12 + 12 \beta_{1} + \beta_{2} + \beta_{5} ) q^{3} + ( -10 \beta_{1} - \beta_{2} - 5 \beta_{3} + \beta_{4} + 5 \beta_{5} ) q^{4} + ( 270 + 7 \beta_{2} + 40 \beta_{3} + 7 \beta_{4} + 40 \beta_{5} ) q^{6} + ( 434 - 434 \beta_{1} - 119 \beta_{3} + 7 \beta_{4} ) q^{7} + ( 1326 + 1326 \beta_{1} - 2 \beta_{2} + 130 \beta_{5} ) q^{8} + ( 2823 \beta_{1} + 3 \beta_{2} - 285 \beta_{3} - 3 \beta_{4} + 285 \beta_{5} ) q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{2} + ( 12 + 12 \beta_{1} + \beta_{2} + \beta_{5} ) q^{3} + ( -10 \beta_{1} - \beta_{2} - 5 \beta_{3} + \beta_{4} + 5 \beta_{5} ) q^{4} + ( 270 + 7 \beta_{2} + 40 \beta_{3} + 7 \beta_{4} + 40 \beta_{5} ) q^{6} + ( 434 - 434 \beta_{1} - 119 \beta_{3} + 7 \beta_{4} ) q^{7} + ( 1326 + 1326 \beta_{1} - 2 \beta_{2} + 130 \beta_{5} ) q^{8} + ( 2823 \beta_{1} + 3 \beta_{2} - 285 \beta_{3} - 3 \beta_{4} + 285 \beta_{5} ) q^{9} + ( 3792 - 85 \beta_{2} + 25 \beta_{3} - 85 \beta_{4} + 25 \beta_{5} ) q^{11} + ( 7596 - 7596 \beta_{1} + 76 \beta_{3} - 92 \beta_{4} ) q^{12} + ( 20015 + 20015 \beta_{1} - 80 \beta_{2} - 554 \beta_{5} ) q^{13} + ( 31626 \beta_{1} + 77 \beta_{2} + 1260 \beta_{3} - 77 \beta_{4} - 1260 \beta_{5} ) q^{14} + ( 37132 + 374 \beta_{2} - 670 \beta_{3} + 374 \beta_{4} - 670 \beta_{5} ) q^{16} + ( 43617 - 43617 \beta_{1} + 2366 \beta_{3} + 466 \beta_{4} ) q^{17} + ( 75822 + 75822 \beta_{1} + 606 \beta_{2} + 171 \beta_{5} ) q^{18} + ( 54264 \beta_{1} - 684 \beta_{2} + 930 \beta_{3} + 684 \beta_{4} - 930 \beta_{5} ) q^{19} + ( 40068 - 511 \beta_{2} - 2695 \beta_{3} - 511 \beta_{4} - 2695 \beta_{5} ) q^{21} + ( 6310 - 6310 \beta_{1} - 2068 \beta_{3} - 970 \beta_{4} ) q^{22} + ( -4722 - 4722 \beta_{1} - 1581 \beta_{2} - 1859 \beta_{5} ) q^{23} + ( 49272 \beta_{1} + 2268 \beta_{2} - 6060 \beta_{3} - 2268 \beta_{4} + 6060 \beta_{5} ) q^{24} + ( -147684 - 1034 \beta_{2} + 20145 \beta_{3} - 1034 \beta_{4} + 20145 \beta_{5} ) q^{26} + ( -58572 + 58572 \beta_{1} - 20460 \beta_{3} - 156 \beta_{4} ) q^{27} + ( -223748 - 223748 \beta_{1} + 196 \beta_{2} + 18844 \beta_{5} ) q^{28} + ( -451044 \beta_{1} - 2386 \beta_{2} - 15980 \beta_{3} + 2386 \beta_{4} + 15980 \beta_{5} ) q^{29} + ( -99628 + 3045 \beta_{2} - 34425 \beta_{3} + 3045 \beta_{4} - 34425 \beta_{5} ) q^{31} + ( -516180 + 516180 \beta_{1} + 35236 \beta_{3} + 3660 \beta_{4} ) q^{32} + ( -697956 - 697956 \beta_{1} + 6862 \beta_{2} - 33818 \beta_{5} ) q^{33} + ( -631220 \beta_{1} - 5162 \beta_{2} + 47165 \beta_{3} + 5162 \beta_{4} - 47165 \beta_{5} ) q^{34} + ( -674778 + 3039 \beta_{2} + 22005 \beta_{3} + 3039 \beta_{4} + 22005 \beta_{5} ) q^{36} + ( 72653 - 72653 \beta_{1} + 7536 \beta_{3} - 1506 \beta_{4} ) q^{37} + ( -250116 - 250116 \beta_{1} - 10068 \beta_{2} + 18420 \beta_{5} ) q^{38} + ( -375300 \beta_{1} + 17417 \beta_{2} + 20785 \beta_{3} - 17417 \beta_{4} - 20785 \beta_{5} ) q^{39} + ( 422352 - 12505 \beta_{2} - 23675 \beta_{3} - 12505 \beta_{4} - 23675 \beta_{5} ) q^{41} + ( -718914 + 718914 \beta_{1} + 33292 \beta_{3} - 11522 \beta_{4} ) q^{42} + ( -105376 - 105376 \beta_{1} - 5323 \beta_{2} - 85519 \beta_{5} ) q^{43} + ( 1524720 \beta_{1} - 13872 \beta_{2} - 21760 \beta_{3} + 13872 \beta_{4} + 21760 \beta_{5} ) q^{44} + ( -500818 - 11345 \beta_{2} - 47600 \beta_{3} - 11345 \beta_{4} - 47600 \beta_{5} ) q^{46} + ( 2583906 - 2583906 \beta_{1} - 90779 \beta_{3} + 4263 \beta_{4} ) q^{47} + ( 3565608 + 3565608 \beta_{1} + 15784 \beta_{2} + 157816 \beta_{5} ) q^{48} + ( 1195649 \beta_{1} - 8575 \beta_{2} - 219275 \beta_{3} + 8575 \beta_{4} + 219275 \beta_{5} ) q^{49} + ( 5798544 + 52723 \beta_{2} + 246835 \beta_{3} + 52723 \beta_{4} + 246835 \beta_{5} ) q^{51} + ( 230594 - 230594 \beta_{1} - 275554 \beta_{3} + 48362 \beta_{4} ) q^{52} + ( 2162457 + 2162457 \beta_{1} + 4186 \beta_{2} + 271376 \beta_{5} ) q^{53} + ( 5442984 \beta_{1} + 21396 \beta_{2} + 38580 \beta_{3} - 21396 \beta_{4} - 38580 \beta_{5} ) q^{54} + ( -3082968 + 308 \beta_{2} + 11060 \beta_{3} + 308 \beta_{4} + 11060 \beta_{5} ) q^{56} + ( 5636916 - 5636916 \beta_{1} + 338880 \beta_{3} - 63132 \beta_{4} ) q^{57} + ( 4241136 + 4241136 \beta_{1} + 3328 \beta_{2} - 768320 \beta_{5} ) q^{58} + ( -1372608 \beta_{1} - 30152 \beta_{2} + 508490 \beta_{3} + 30152 \beta_{4} - 508490 \beta_{5} ) q^{59} + ( 4266032 - 95625 \beta_{2} - 466875 \beta_{3} - 95625 \beta_{4} - 466875 \beta_{5} ) q^{61} + ( -9144870 + 9144870 \beta_{1} + 445592 \beta_{3} - 32310 \beta_{4} ) q^{62} + ( -7604394 - 7604394 \beta_{1} - 27237 \beta_{2} + 367101 \beta_{5} ) q^{63} + ( 118376 \beta_{1} + 38548 \beta_{2} - 400060 \beta_{3} - 38548 \beta_{4} + 400060 \beta_{5} ) q^{64} + ( -8968140 + 7354 \beta_{2} - 302420 \beta_{3} + 7354 \beta_{4} - 302420 \beta_{5} ) q^{66} + ( 5539832 - 5539832 \beta_{1} - 101959 \beta_{3} + 103661 \beta_{4} ) q^{67} + ( -1400586 - 1400586 \beta_{1} - 36978 \beta_{2} + 105434 \beta_{5} ) q^{68} + ( -14569164 \beta_{1} + 7561 \beta_{2} + 447455 \beta_{3} - 7561 \beta_{4} - 447455 \beta_{5} ) q^{69} + ( -2912988 + 78125 \beta_{2} + 949375 \beta_{3} + 78125 \beta_{4} + 949375 \beta_{5} ) q^{71} + ( -13544946 + 13544946 \beta_{1} - 738030 \beta_{3} - 74658 \beta_{4} ) q^{72} + ( -18480997 - 18480997 \beta_{1} - 12756 \beta_{2} - 499584 \beta_{5} ) q^{73} + ( -1998552 \beta_{1} + 1500 \beta_{2} - 14725 \beta_{3} - 1500 \beta_{4} + 14725 \beta_{5} ) q^{74} + ( -9032136 + 133116 \beta_{2} - 436380 \beta_{3} + 133116 \beta_{4} - 436380 \beta_{5} ) q^{76} + ( -4353552 + 4353552 \beta_{1} + 41342 \beta_{3} + 110754 \beta_{4} ) q^{77} + ( -5459142 - 5459142 \beta_{1} + 167434 \beta_{2} + 982072 \beta_{5} ) q^{78} + ( -22879824 \beta_{1} - 122456 \beta_{2} - 1313680 \beta_{3} + 122456 \beta_{4} + 1313680 \beta_{5} ) q^{79} + ( 10214919 - 179613 \beta_{2} + 956565 \beta_{3} - 179613 \beta_{4} + 956565 \beta_{5} ) q^{81} + ( -6347570 + 6347570 \beta_{1} - 166228 \beta_{3} - 197410 \beta_{4} ) q^{82} + ( 2441256 + 2441256 \beta_{1} + 212813 \beta_{2} + 273261 \beta_{5} ) q^{83} + ( 1447824 \beta_{1} - 94976 \beta_{2} - 575680 \beta_{3} + 94976 \beta_{4} + 575680 \beta_{5} ) q^{84} + ( -22769346 - 117457 \beta_{2} + 146560 \beta_{3} - 117457 \beta_{4} + 146560 \beta_{5} ) q^{86} + ( 22156764 - 22156764 \beta_{1} + 284520 \beta_{3} + 150972 \beta_{4} ) q^{87} + ( 7348032 + 7348032 \beta_{1} - 371264 \beta_{2} - 137840 \beta_{5} ) q^{88} + ( 38523288 \beta_{1} + 442572 \beta_{2} + 666960 \beta_{3} - 442572 \beta_{4} - 666960 \beta_{5} ) q^{89} + ( 29991584 + 224203 \beta_{2} - 2877665 \beta_{3} + 224203 \beta_{4} - 2877665 \beta_{5} ) q^{91} + ( -11498148 + 11498148 \beta_{1} - 297684 \beta_{3} + 173396 \beta_{4} ) q^{92} + ( 16384884 + 16384884 \beta_{1} - 679018 \beta_{2} - 1434658 \beta_{5} ) q^{93} + ( 24130162 \beta_{1} + 65201 \beta_{2} + 3178480 \beta_{3} - 65201 \beta_{4} - 3178480 \beta_{5} ) q^{94} + ( 29428560 - 328088 \beta_{2} + 1746040 \beta_{3} - 328088 \beta_{4} + 1746040 \beta_{5} ) q^{96} + ( 31017911 - 31017911 \beta_{1} - 586904 \beta_{3} - 861272 \beta_{4} ) q^{97} + ( 58292850 + 58292850 \beta_{1} + 335650 \beta_{2} - 1563051 \beta_{5} ) q^{98} + ( 9802896 \beta_{1} - 486789 \beta_{2} + 615855 \beta_{3} + 486789 \beta_{4} - 615855 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 2q^{2} + 72q^{3} + 1752q^{6} + 2352q^{7} + 8220q^{8} + O(q^{10})$$ $$6q + 2q^{2} + 72q^{3} + 1752q^{6} + 2352q^{7} + 8220q^{8} + 23192q^{11} + 45912q^{12} + 119142q^{13} + 218616q^{16} + 265502q^{17} + 454062q^{18} + 231672q^{21} + 35664q^{22} - 28888q^{23} - 801388q^{26} - 392040q^{27} - 1305192q^{28} - 747648q^{31} - 3033928q^{32} - 4269096q^{33} - 3972804q^{36} + 454002q^{37} - 1443720q^{38} + 2489432q^{41} - 4223856q^{42} - 792648q^{43} - 3149928q^{46} + 15313352q^{47} + 21677712q^{48} + 35567712q^{51} + 735732q^{52} + 13509122q^{53} - 18454800q^{56} + 34625520q^{57} + 23903520q^{58} + 24111192q^{61} - 53913416q^{62} - 44837688q^{63} - 55047936q^{66} + 32827752q^{67} - 8118692q^{68} - 13992928q^{71} - 82596420q^{72} - 111859638q^{73} - 56470800q^{76} - 26260136q^{77} - 31125576q^{78} + 65834226q^{81} - 38023056q^{82} + 14768432q^{83} - 135560008q^{86} + 133207680q^{87} + 44555040q^{88} + 167542032q^{91} - 69931048q^{92} + 96798024q^{93} + 184867872q^{96} + 186656202q^{97} + 345959698q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2 x^{5} + 2 x^{4} - 30 x^{3} + 1089 x^{2} - 3168 x + 4608$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$39 \nu^{5} - 22 \nu^{4} - 10 \nu^{3} + 790 \nu^{2} + 41631 \nu - 62928$$$$)/66000$$ $$\beta_{2}$$ $$=$$ $$($$$$-273 \nu^{5} + 154 \nu^{4} + 70 \nu^{3} - 5530 \nu^{2} + 1028583 \nu - 21504$$$$)/66000$$ $$\beta_{3}$$ $$=$$ $$($$$$761 \nu^{5} - 2178 \nu^{4} - 4990 \nu^{3} - 45790 \nu^{2} + 838569 \nu - 2347872$$$$)/66000$$ $$\beta_{4}$$ $$=$$ $$($$$$-77 \nu^{5} + 146 \nu^{4} - 3570 \nu^{3} + 2030 \nu^{2} - 83733 \nu + 244704$$$$)/6000$$ $$\beta_{5}$$ $$=$$ $$($$$$-921 \nu^{5} - 1342 \nu^{4} - 610 \nu^{3} + 48190 \nu^{2} - 823209 \nu + 187392$$$$)/66000$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 7 \beta_{1} + 7$$$$)/20$$ $$\nu^{2}$$ $$=$$ $$($$$$10 \beta_{5} + \beta_{4} - 10 \beta_{3} - \beta_{2} + 446 \beta_{1}$$$$)/20$$ $$\nu^{3}$$ $$=$$ $$($$$$-31 \beta_{4} - 20 \beta_{3} - 283 \beta_{1} + 283$$$$)/20$$ $$\nu^{4}$$ $$=$$ $$($$$$-35 \beta_{5} + 5 \beta_{4} - 35 \beta_{3} + 5 \beta_{2} - 1348$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-400 \beta_{5} - 1019 \beta_{2} + 17267 \beta_{1} + 17267$$$$)/20$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/25\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-\beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 1.52966 − 1.52966i −4.23471 + 4.23471i 3.70505 − 3.70505i 1.52966 + 1.52966i −4.23471 − 4.23471i 3.70505 + 3.70505i
−15.2610 15.2610i 20.3321 20.3321i 209.796i 0 −620.576 2415.21 + 2415.21i −705.116 + 705.116i 5734.21i 0
7.2 4.39608 + 4.39608i −75.2981 + 75.2981i 217.349i 0 −662.032 −730.992 730.992i 2080.88 2080.88i 4778.60i 0
7.3 11.8649 + 11.8649i 90.9660 90.9660i 25.5528i 0 2158.61 −508.219 508.219i 2734.24 2734.24i 9988.61i 0
18.1 −15.2610 + 15.2610i 20.3321 + 20.3321i 209.796i 0 −620.576 2415.21 2415.21i −705.116 705.116i 5734.21i 0
18.2 4.39608 4.39608i −75.2981 75.2981i 217.349i 0 −662.032 −730.992 + 730.992i 2080.88 + 2080.88i 4778.60i 0
18.3 11.8649 11.8649i 90.9660 + 90.9660i 25.5528i 0 2158.61 −508.219 + 508.219i 2734.24 + 2734.24i 9988.61i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 18.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 25.9.c.b 6
5.b even 2 1 5.9.c.a 6
5.c odd 4 1 5.9.c.a 6
5.c odd 4 1 inner 25.9.c.b 6
15.d odd 2 1 45.9.g.a 6
15.e even 4 1 45.9.g.a 6
20.d odd 2 1 80.9.p.c 6
20.e even 4 1 80.9.p.c 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.9.c.a 6 5.b even 2 1
5.9.c.a 6 5.c odd 4 1
25.9.c.b 6 1.a even 1 1 trivial
25.9.c.b 6 5.c odd 4 1 inner
45.9.g.a 6 15.d odd 2 1
45.9.g.a 6 15.e even 4 1
80.9.p.c 6 20.d odd 2 1
80.9.p.c 6 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} - 2 T_{2}^{5} + 2 T_{2}^{4} - 2400 T_{2}^{3} + 153664 T_{2}^{2} - 1248128 T_{2} + 5068928$$ acting on $$S_{9}^{\mathrm{new}}(25, [\chi])$$.