# Properties

 Label 25.10.b.c Level 25 Weight 10 Character orbit 25.b Analytic conductor 12.876 Analytic rank 0 Dimension 6 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 25.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.8758959041$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} + \cdots)$$ Defining polynomial: $$x^{6} + 1305 x^{4} + 433104 x^{2} + 16000000$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 5^{8}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( -2 \beta_{2} - \beta_{4} ) q^{3} + ( -114 + \beta_{1} ) q^{4} + ( 1147 - 19 \beta_{1} - \beta_{5} ) q^{6} + ( 140 \beta_{2} + 4 \beta_{3} - 26 \beta_{4} ) q^{7} + ( 279 \beta_{2} - 27 \beta_{3} - 33 \beta_{4} ) q^{8} + ( -19438 + 88 \beta_{1} - 8 \beta_{5} ) q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{2} + ( -2 \beta_{2} - \beta_{4} ) q^{3} + ( -114 + \beta_{1} ) q^{4} + ( 1147 - 19 \beta_{1} - \beta_{5} ) q^{6} + ( 140 \beta_{2} + 4 \beta_{3} - 26 \beta_{4} ) q^{7} + ( 279 \beta_{2} - 27 \beta_{3} - 33 \beta_{4} ) q^{8} + ( -19438 + 88 \beta_{1} - 8 \beta_{5} ) q^{9} + ( -18264 + 76 \beta_{1} + 17 \beta_{5} ) q^{11} + ( 2431 \beta_{2} + 183 \beta_{3} + 221 \beta_{4} ) q^{12} + ( 928 \beta_{2} - 832 \beta_{3} + 460 \beta_{4} ) q^{13} + ( -90870 - 282 \beta_{1} - 30 \beta_{5} ) q^{14} + ( -233112 + 95 \beta_{1} - 6 \beta_{5} ) q^{16} + ( -3974 \beta_{2} + 1521 \beta_{3} - 168 \beta_{4} ) q^{17} + ( -29534 \beta_{2} - 5016 \beta_{3} - 2056 \beta_{4} ) q^{18} + ( -273096 + 92 \beta_{1} + 351 \beta_{5} ) q^{19} + ( -791154 - 1512 \beta_{1} - 420 \beta_{5} ) q^{21} + ( -28107 \beta_{2} + 3558 \beta_{3} - 4310 \beta_{4} ) q^{22} + ( -4764 \beta_{2} - 11160 \beta_{3} + 3426 \beta_{4} ) q^{23} + ( -934212 - 2625 \beta_{1} - 474 \beta_{5} ) q^{24} + ( -428628 + 4588 \beta_{1} + 1292 \beta_{5} ) q^{26} + ( 152846 \beta_{2} + 3792 \beta_{3} + 18163 \beta_{4} ) q^{27} + ( 15778 \beta_{2} - 238 \beta_{3} - 826 \beta_{4} ) q^{28} + ( -728268 + 9832 \beta_{1} - 508 \beta_{5} ) q^{29} + ( 1423984 + 2648 \beta_{1} - 534 \beta_{5} ) q^{31} + ( -101287 \beta_{2} - 18369 \beta_{3} - 19395 \beta_{4} ) q^{32} + ( 290078 \beta_{2} + 40071 \beta_{3} + 7288 \beta_{4} ) q^{33} + ( 2279959 + 775 \beta_{1} - 1689 \beta_{5} ) q^{34} + ( 8947148 - 44510 \beta_{1} - 1136 \beta_{5} ) q^{36} + ( -137524 \beta_{2} - 15866 \beta_{3} - 37572 \beta_{4} ) q^{37} + ( -300541 \beta_{2} + 113346 \beta_{3} - 40242 \beta_{4} ) q^{38} + ( 16805092 - 9016 \beta_{1} + 7832 \beta_{5} ) q^{39} + ( 1952709 + 73088 \beta_{1} - 4904 \beta_{5} ) q^{41} + ( -591486 \beta_{2} - 97776 \beta_{3} + 94416 \beta_{4} ) q^{42} + ( -34976 \beta_{2} + 83000 \beta_{3} - 2732 \beta_{4} ) q^{43} + ( 7346514 - 44675 \beta_{1} + 836 \beta_{5} ) q^{44} + ( 4736994 - 2322 \beta_{1} + 14586 \beta_{5} ) q^{46} + ( 894880 \beta_{2} + 121500 \beta_{3} - 9384 \beta_{4} ) q^{47} + ( 645113 \beta_{2} + 8151 \beta_{3} + 250021 \beta_{4} ) q^{48} + ( 2188595 - 93952 \beta_{1} - 17904 \beta_{5} ) q^{49} + ( -8551832 + 26108 \beta_{1} - 5311 \beta_{5} ) q^{51} + ( -560188 \beta_{2} - 123500 \beta_{3} - 52836 \beta_{4} ) q^{52} + ( 208484 \beta_{2} - 428022 \beta_{3} - 314856 \beta_{4} ) q^{53} + ( -94248481 + 480577 \beta_{1} + 14371 \beta_{5} ) q^{54} + ( -56459448 - 143838 \beta_{1} - 15948 \beta_{5} ) q^{56} + ( 2527034 \beta_{2} + 557025 \beta_{3} - 111584 \beta_{4} ) q^{57} + ( -1874400 \beta_{2} - 433104 \beta_{3} - 270608 \beta_{4} ) q^{58} + ( -1818504 - 193768 \beta_{1} - 21680 \beta_{5} ) q^{59} + ( 41881302 - 359960 \beta_{1} + 22980 \beta_{5} ) q^{61} + ( 1133970 \beta_{2} - 247716 \beta_{3} - 30780 \beta_{4} ) q^{62} + ( -1120728 \beta_{2} - 844344 \beta_{3} + 589692 \beta_{4} ) q^{63} + ( -55688032 - 474207 \beta_{1} - 4098 \beta_{5} ) q^{64} + ( -185832463 + 614329 \beta_{1} - 32783 \beta_{5} ) q^{66} + ( -6018994 \beta_{2} + 906256 \beta_{3} + 322747 \beta_{4} ) q^{67} + ( 232429 \beta_{2} + 200457 \beta_{3} + 67443 \beta_{4} ) q^{68} + ( 104612274 + 343296 \beta_{1} + 94824 \beta_{5} ) q^{69} + ( 98905212 + 786920 \beta_{1} + 48040 \beta_{5} ) q^{71} + ( -824178 \beta_{2} - 1741302 \beta_{3} + 536574 \beta_{4} ) q^{72} + ( -10112822 \beta_{2} + 497441 \beta_{3} - 56864 \beta_{4} ) q^{73} + ( 84128214 - 855578 \beta_{1} - 21706 \beta_{5} ) q^{74} + ( 29919854 - 370821 \beta_{1} + 26124 \beta_{5} ) q^{76} + ( 2388876 \beta_{2} + 1613286 \beta_{3} - 701052 \beta_{4} ) q^{77} + ( 17509892 \beta_{2} + 2827992 \beta_{3} - 532664 \beta_{4} ) q^{78} + ( 102948380 + 1255264 \beta_{1} - 75234 \beta_{5} ) q^{79} + ( 467073073 - 2224216 \beta_{1} - 147640 \beta_{5} ) q^{81} + ( -6514275 \beta_{2} - 3591696 \beta_{3} - 1892080 \beta_{4} ) q^{82} + ( 10762890 \beta_{2} + 1204200 \beta_{3} + 165897 \beta_{4} ) q^{83} + ( -12664932 - 249438 \beta_{1} - 22848 \beta_{5} ) q^{84} + ( 11233116 + 333580 \beta_{1} - 85732 \beta_{5} ) q^{86} + ( 20542904 \beta_{2} + 1017444 \beta_{3} + 2351284 \beta_{4} ) q^{87} + ( -1767237 \beta_{2} + 3303801 \beta_{3} - 821061 \beta_{4} ) q^{88} + ( 367582761 + 1107576 \beta_{1} + 4176 \beta_{5} ) q^{89} + ( 393981224 + 796432 \beta_{1} + 447552 \beta_{5} ) q^{91} + ( 1888602 \beta_{2} - 837846 \beta_{3} + 284622 \beta_{4} ) q^{92} + ( 280332 \beta_{2} - 337242 \beta_{3} - 540432 \beta_{4} ) q^{93} + ( -576367700 + 1342852 \beta_{1} - 130884 \beta_{5} ) q^{94} + ( -856923952 + 3592225 \beta_{1} - 818 \beta_{5} ) q^{96} + ( -29036476 \beta_{2} + 2021122 \beta_{3} + 62616 \beta_{4} ) q^{97} + ( 14210371 \beta_{2} - 3371616 \beta_{3} + 4998240 \beta_{4} ) q^{98} + ( 299406712 - 5902672 \beta_{1} - 82942 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 682q^{4} + 6842q^{6} - 116468q^{9} + O(q^{10})$$ $$6q - 682q^{4} + 6842q^{6} - 116468q^{9} - 109398q^{11} - 545844q^{14} - 1398494q^{16} - 1637690q^{19} - 4750788q^{21} - 5611470q^{24} - 2560008q^{26} - 4350960q^{29} + 8548132q^{31} + 13677926q^{34} + 53591596q^{36} + 100828184q^{39} + 11852622q^{41} + 43991406q^{44} + 28446492q^{46} + 12907858q^{49} - 51269398q^{51} - 564500990q^{54} - 339076260q^{56} - 11341920q^{59} + 250613852q^{61} - 335084802q^{64} - 1113831686q^{66} + 628549884q^{69} + 595101192q^{71} + 503014716q^{74} + 178829730q^{76} + 620050340q^{79} + 2797694726q^{81} - 76534164q^{84} + 67894392q^{86} + 2207720070q^{89} + 2366375312q^{91} - 3455782264q^{94} - 5134360898q^{96} + 1784469044q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 1305 x^{4} + 433104 x^{2} + 16000000$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-35 \nu^{4} - 17275 \nu^{2} + 2252704$$$$)/13392$$ $$\beta_{2}$$ $$=$$ $$($$$$77 \nu^{5} + 71485 \nu^{3} + 13296008 \nu$$$$)/3348000$$ $$\beta_{3}$$ $$=$$ $$($$$$-491 \nu^{5} - 908755 \nu^{3} - 378730064 \nu$$$$)/13392000$$ $$\beta_{4}$$ $$=$$ $$($$$$-1177 \nu^{5} - 1291985 \nu^{3} - 198655408 \nu$$$$)/13392000$$ $$\beta_{5}$$ $$=$$ $$($$$$-185 \nu^{4} - 171025 \nu^{2} - 22789928$$$$)/6696$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$25 \beta_{4} - 11 \beta_{3} + 78 \beta_{2}$$$$)/250$$ $$\nu^{2}$$ $$=$$ $$($$$$-21 \beta_{5} + 222 \beta_{1} - 108817$$$$)/250$$ $$\nu^{3}$$ $$=$$ $$($$$$-16225 \beta_{4} - 253 \beta_{3} - 62406 \beta_{2}$$$$)/250$$ $$\nu^{4}$$ $$=$$ $$($$$$2073 \beta_{5} - 41046 \beta_{1} + 13959941$$$$)/50$$ $$\nu^{5}$$ $$=$$ $$($$$$10746025 \beta_{4} + 2134309 \beta_{3} + 55337718 \beta_{2}$$$$)/250$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
24.1
 − 27.7229i 22.2334i − 6.48955i 6.48955i − 22.2334i 27.7229i
31.7828i 268.664i −498.143 0 8538.88 637.237i 440.406i −52497.3 0
24.2 21.4187i 210.171i 53.2406 0 −4501.59 9905.49i 12106.7i −24489.0 0
24.3 20.2014i 30.5073i 103.903 0 −616.291 4010.25i 12442.1i 18752.3 0
24.4 20.2014i 30.5073i 103.903 0 −616.291 4010.25i 12442.1i 18752.3 0
24.5 21.4187i 210.171i 53.2406 0 −4501.59 9905.49i 12106.7i −24489.0 0
24.6 31.7828i 268.664i −498.143 0 8538.88 637.237i 440.406i −52497.3 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 24.6 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.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 25.10.b.c 6
3.b odd 2 1 225.10.b.m 6
4.b odd 2 1 400.10.c.q 6
5.b even 2 1 inner 25.10.b.c 6
5.c odd 4 1 25.10.a.c 3
5.c odd 4 1 25.10.a.d yes 3
15.d odd 2 1 225.10.b.m 6
15.e even 4 1 225.10.a.m 3
15.e even 4 1 225.10.a.p 3
20.d odd 2 1 400.10.c.q 6
20.e even 4 1 400.10.a.u 3
20.e even 4 1 400.10.a.y 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.10.a.c 3 5.c odd 4 1
25.10.a.d yes 3 5.c odd 4 1
25.10.b.c 6 1.a even 1 1 trivial
25.10.b.c 6 5.b even 2 1 inner
225.10.a.m 3 15.e even 4 1
225.10.a.p 3 15.e even 4 1
225.10.b.m 6 3.b odd 2 1
225.10.b.m 6 15.d odd 2 1
400.10.a.u 3 20.e even 4 1
400.10.a.y 3 20.e even 4 1
400.10.c.q 6 4.b odd 2 1
400.10.c.q 6 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + 1877 T_{2}^{4} + 1062868 T_{2}^{2} + 189117504$$ acting on $$S_{10}^{\mathrm{new}}(25, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 1195 T^{2} + 1150932 T^{4} - 631348160 T^{6} + 301709918208 T^{8} - 82119774699520 T^{10} + 18014398509481984 T^{12}$$
$3$ $$1 - 815 T^{2} - 125981658 T^{4} - 6692923857195 T^{6} - 48807875547390762 T^{8} -$$$$12\!\cdots\!15$$$$T^{10} +$$$$58\!\cdots\!69$$$$T^{12}$$
$5$ 1
$7$ $$1 - 127514750 T^{2} + 7551328074598847 T^{4} -$$$$32\!\cdots\!00$$$$T^{6} +$$$$12\!\cdots\!03$$$$T^{8} -$$$$33\!\cdots\!50$$$$T^{10} +$$$$43\!\cdots\!49$$$$T^{12}$$
$11$ $$( 1 + 54699 T + 5816187440 T^{2} + 182671409832855 T^{3} + 13714265744571201040 T^{4} +$$$$30\!\cdots\!19$$$$T^{5} +$$$$13\!\cdots\!71$$$$T^{6} )^{2}$$
$13$ $$1 - 15844411230 T^{2} +$$$$30\!\cdots\!87$$$$T^{4} -$$$$30\!\cdots\!40$$$$T^{6} +$$$$33\!\cdots\!23$$$$T^{8} -$$$$20\!\cdots\!30$$$$T^{10} +$$$$14\!\cdots\!89$$$$T^{12}$$
$17$ $$1 - 597847713835 T^{2} +$$$$15\!\cdots\!02$$$$T^{4} -$$$$24\!\cdots\!55$$$$T^{6} +$$$$22\!\cdots\!18$$$$T^{8} -$$$$11\!\cdots\!35$$$$T^{10} +$$$$27\!\cdots\!29$$$$T^{12}$$
$19$ $$( 1 + 818845 T + 468173656712 T^{2} + 302339390932836385 T^{3} +$$$$15\!\cdots\!48$$$$T^{4} +$$$$85\!\cdots\!45$$$$T^{5} +$$$$33\!\cdots\!39$$$$T^{6} )^{2}$$
$23$ $$1 - 4962494312670 T^{2} +$$$$14\!\cdots\!07$$$$T^{4} -$$$$32\!\cdots\!60$$$$T^{6} +$$$$47\!\cdots\!83$$$$T^{8} -$$$$52\!\cdots\!70$$$$T^{10} +$$$$34\!\cdots\!09$$$$T^{12}$$
$29$ $$( 1 + 2175480 T + 32457706855607 T^{2} + 59155485309271560240 T^{3} +$$$$47\!\cdots\!83$$$$T^{4} +$$$$45\!\cdots\!80$$$$T^{5} +$$$$30\!\cdots\!09$$$$T^{6} )^{2}$$
$31$ $$( 1 - 4274066 T + 82851493809465 T^{2} -$$$$22\!\cdots\!20$$$$T^{3} +$$$$21\!\cdots\!15$$$$T^{4} -$$$$29\!\cdots\!06$$$$T^{5} +$$$$18\!\cdots\!11$$$$T^{6} )^{2}$$
$37$ $$1 - 575490501959730 T^{2} +$$$$15\!\cdots\!87$$$$T^{4} -$$$$24\!\cdots\!40$$$$T^{6} +$$$$25\!\cdots\!23$$$$T^{8} -$$$$16\!\cdots\!30$$$$T^{10} +$$$$48\!\cdots\!89$$$$T^{12}$$
$41$ $$( 1 - 5926311 T + 232043727124790 T^{2} -$$$$23\!\cdots\!95$$$$T^{3} +$$$$75\!\cdots\!90$$$$T^{4} -$$$$63\!\cdots\!31$$$$T^{5} +$$$$35\!\cdots\!81$$$$T^{6} )^{2}$$
$43$ $$1 - 2755277785986450 T^{2} +$$$$32\!\cdots\!47$$$$T^{4} -$$$$21\!\cdots\!00$$$$T^{6} +$$$$82\!\cdots\!03$$$$T^{8} -$$$$17\!\cdots\!50$$$$T^{10} +$$$$16\!\cdots\!49$$$$T^{12}$$
$47$ $$1 - 4564445424701290 T^{2} +$$$$10\!\cdots\!67$$$$T^{4} -$$$$14\!\cdots\!20$$$$T^{6} +$$$$12\!\cdots\!63$$$$T^{8} -$$$$71\!\cdots\!90$$$$T^{10} +$$$$19\!\cdots\!69$$$$T^{12}$$
$53$ $$1 - 830296241726290 T^{2} +$$$$19\!\cdots\!67$$$$T^{4} -$$$$24\!\cdots\!20$$$$T^{6} +$$$$21\!\cdots\!63$$$$T^{8} -$$$$98\!\cdots\!90$$$$T^{10} +$$$$12\!\cdots\!69$$$$T^{12}$$
$59$ $$( 1 + 5670960 T + 19337695182532817 T^{2} +$$$$14\!\cdots\!80$$$$T^{3} +$$$$16\!\cdots\!63$$$$T^{4} +$$$$42\!\cdots\!60$$$$T^{5} +$$$$65\!\cdots\!19$$$$T^{6} )^{2}$$
$61$ $$( 1 - 125306926 T + 22192130203358915 T^{2} -$$$$23\!\cdots\!20$$$$T^{3} +$$$$25\!\cdots\!15$$$$T^{4} -$$$$17\!\cdots\!06$$$$T^{5} +$$$$15\!\cdots\!21$$$$T^{6} )^{2}$$
$67$ $$1 - 53234092789742135 T^{2} +$$$$12\!\cdots\!02$$$$T^{4} -$$$$21\!\cdots\!55$$$$T^{6} +$$$$92\!\cdots\!18$$$$T^{8} -$$$$29\!\cdots\!35$$$$T^{10} +$$$$40\!\cdots\!29$$$$T^{12}$$
$71$ $$( 1 - 297550596 T + 87036096018332165 T^{2} -$$$$15\!\cdots\!20$$$$T^{3} +$$$$39\!\cdots\!15$$$$T^{4} -$$$$62\!\cdots\!56$$$$T^{5} +$$$$96\!\cdots\!91$$$$T^{6} )^{2}$$
$73$ $$1 - 155852935645686795 T^{2} +$$$$17\!\cdots\!82$$$$T^{4} -$$$$11\!\cdots\!35$$$$T^{6} +$$$$59\!\cdots\!58$$$$T^{8} -$$$$18\!\cdots\!95$$$$T^{10} +$$$$41\!\cdots\!09$$$$T^{12}$$
$79$ $$( 1 - 310025170 T + 176092553119892457 T^{2} -$$$$48\!\cdots\!60$$$$T^{3} +$$$$21\!\cdots\!83$$$$T^{4} -$$$$44\!\cdots\!70$$$$T^{5} +$$$$17\!\cdots\!59$$$$T^{6} )^{2}$$
$83$ $$1 - 836375861147057535 T^{2} +$$$$32\!\cdots\!02$$$$T^{4} -$$$$77\!\cdots\!55$$$$T^{6} +$$$$11\!\cdots\!18$$$$T^{8} -$$$$10\!\cdots\!35$$$$T^{10} +$$$$42\!\cdots\!29$$$$T^{12}$$
$89$ $$( 1 - 1103860035 T + 1320664213408319502 T^{2} -$$$$75\!\cdots\!55$$$$T^{3} +$$$$46\!\cdots\!18$$$$T^{4} -$$$$13\!\cdots\!35$$$$T^{5} +$$$$43\!\cdots\!29$$$$T^{6} )^{2}$$
$97$ $$1 - 2865108733311184890 T^{2} +$$$$43\!\cdots\!67$$$$T^{4} -$$$$40\!\cdots\!20$$$$T^{6} +$$$$25\!\cdots\!63$$$$T^{8} -$$$$95\!\cdots\!90$$$$T^{10} +$$$$19\!\cdots\!69$$$$T^{12}$$