# Properties

 Label 25.7.c.d Level $25$ Weight $7$ Character orbit 25.c Analytic conductor $5.751$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$25 = 5^{2}$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 25.c (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.75135209050$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.333061916000256.23 Defining polynomial: $$x^{8} + 90 x^{6} - 12 x^{5} + 3011 x^{4} + 528 x^{3} + 41202 x^{2} + 17580 x + 243850$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$3^{2}\cdot 5^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{2} + \beta_{3} ) q^{2} + ( 4 \beta_{4} + 3 \beta_{5} ) q^{3} + ( 22 \beta_{1} - \beta_{6} ) q^{4} + ( 306 - 3 \beta_{7} ) q^{6} + ( -24 \beta_{2} + 28 \beta_{3} ) q^{7} + ( 5 \beta_{4} - 43 \beta_{5} ) q^{8} + ( -418 \beta_{1} + 8 \beta_{6} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{2} + \beta_{3} ) q^{2} + ( 4 \beta_{4} + 3 \beta_{5} ) q^{3} + ( 22 \beta_{1} - \beta_{6} ) q^{4} + ( 306 - 3 \beta_{7} ) q^{6} + ( -24 \beta_{2} + 28 \beta_{3} ) q^{7} + ( 5 \beta_{4} - 43 \beta_{5} ) q^{8} + ( -418 \beta_{1} + 8 \beta_{6} ) q^{9} + ( 977 + 20 \beta_{7} ) q^{11} + ( 161 \beta_{2} + 369 \beta_{3} ) q^{12} + ( -216 \beta_{4} - 366 \beta_{5} ) q^{13} + ( -88 \beta_{1} - 28 \beta_{6} ) q^{14} + ( 14 - 21 \beta_{7} ) q^{16} + ( -424 \beta_{2} - 673 \beta_{3} ) q^{17} + ( 714 \beta_{4} + 1098 \beta_{5} ) q^{18} + ( 1617 \beta_{1} + 76 \beta_{6} ) q^{19} + ( -3340 - 136 \beta_{7} ) q^{21} + ( 237 \beta_{2} - 723 \beta_{3} ) q^{22} + ( -136 \beta_{4} + 230 \beta_{5} ) q^{23} + ( 2166 \beta_{1} - 177 \beta_{6} ) q^{24} + ( -24276 + 366 \beta_{7} ) q^{26} + ( 660 \beta_{2} - 1491 \beta_{3} ) q^{27} + ( -2484 \beta_{4} - 500 \beta_{5} ) q^{28} + ( 6218 \beta_{1} + 104 \beta_{6} ) q^{29} + ( 13602 - 40 \beta_{7} ) q^{31} + ( 471 \beta_{2} + 4551 \beta_{3} ) q^{32} + ( 2448 \beta_{4} - 3129 \beta_{5} ) q^{33} + ( -45926 \beta_{1} + 673 \beta_{6} ) q^{34} + ( 49244 - 586 \beta_{7} ) q^{36} + ( -4824 \beta_{2} - 2080 \beta_{3} ) q^{37} + ( 1195 \beta_{4} + 4843 \beta_{5} ) q^{38} + ( 77646 \beta_{1} - 1248 \beta_{6} ) q^{39} + ( 48937 + 160 \beta_{7} ) q^{41} + ( 1692 \beta_{2} + 8220 \beta_{3} ) q^{42} + ( 1224 \beta_{4} - 8668 \beta_{5} ) q^{43} + ( -78626 \beta_{1} - 557 \beta_{6} ) q^{44} + ( 2212 - 230 \beta_{7} ) q^{46} + ( 16176 \beta_{2} + 11214 \beta_{3} ) q^{47} + ( 1589 \beta_{4} + 6405 \beta_{5} ) q^{48} + ( 39839 \beta_{1} + 1920 \beta_{6} ) q^{49} + ( -148917 + 2268 \beta_{7} ) q^{51} + ( -23994 \beta_{2} - 31962 \beta_{3} ) q^{52} + ( -16496 \beta_{4} - 3362 \beta_{5} ) q^{53} + ( -24978 \beta_{1} + 1491 \beta_{6} ) q^{54} + ( -143864 - 1292 \beta_{7} ) q^{56} + ( 920 \beta_{2} - 18177 \beta_{3} ) q^{57} + ( -2370 \beta_{4} + 2622 \beta_{5} ) q^{58} + ( -2454 \beta_{1} - 2112 \beta_{6} ) q^{59} + ( 50412 - 2600 \beta_{7} ) q^{61} + ( 15082 \beta_{2} + 17002 \beta_{3} ) q^{62} + ( 14064 \beta_{4} + 10776 \beta_{5} ) q^{63} + ( 194650 \beta_{1} - 3207 \beta_{6} ) q^{64} + ( -1398 + 3129 \beta_{7} ) q^{66} + ( -924 \beta_{2} - 12603 \beta_{3} ) q^{67} + ( 43691 \beta_{4} + 60059 \beta_{5} ) q^{68} + ( 13434 \beta_{1} + 1056 \beta_{6} ) q^{69} + ( 103322 - 2800 \beta_{7} ) q^{71} + ( 25230 \beta_{2} + 28782 \beta_{3} ) q^{72} + ( 7464 \beta_{4} - 27055 \beta_{5} ) q^{73} + ( -310592 \beta_{1} + 2080 \beta_{6} ) q^{74} + ( 344882 + 21 \beta_{7} ) q^{76} + ( -83688 \beta_{2} + 29676 \beta_{3} ) q^{77} + ( -123822 \beta_{4} - 183726 \beta_{5} ) q^{78} + ( -218712 \beta_{1} + 3464 \beta_{6} ) q^{79} + ( 341055 + 792 \beta_{7} ) q^{81} + ( 43017 \beta_{2} + 35337 \beta_{3} ) q^{82} + ( -556 \beta_{4} + 88961 \beta_{5} ) q^{83} + ( 607336 \beta_{1} + 484 \beta_{6} ) q^{84} + ( -270632 + 8668 \beta_{7} ) q^{86} + ( 17280 \beta_{2} - 12858 \beta_{3} ) q^{87} + ( 73185 \beta_{4} - 14991 \beta_{5} ) q^{88} + ( -1158071 \beta_{1} - 488 \beta_{6} ) q^{89} + ( -429192 + 2448 \beta_{7} ) q^{91} + ( 19426 \beta_{2} + 7042 \beta_{3} ) q^{92} + ( 57328 \beta_{4} + 52926 \beta_{5} ) q^{93} + ( 1202580 \beta_{1} - 11214 \beta_{6} ) q^{94} + ( 458286 - 17733 \beta_{7} ) q^{96} + ( 113376 \beta_{2} + 87044 \beta_{3} ) q^{97} + ( 31201 \beta_{4} + 123361 \beta_{5} ) q^{98} + ( 392574 \beta_{1} - 384 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 2436q^{6} + O(q^{10})$$ $$8q + 2436q^{6} + 7896q^{11} + 28q^{16} - 27264q^{21} - 192744q^{26} + 108656q^{31} + 391608q^{36} + 392136q^{41} + 16776q^{46} - 1182264q^{51} - 1156080q^{56} + 392896q^{61} + 1332q^{66} + 815376q^{71} + 2759140q^{76} + 2731608q^{81} - 2130384q^{86} - 3423744q^{91} + 3595356q^{96} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 90 x^{6} - 12 x^{5} + 3011 x^{4} + 528 x^{3} + 41202 x^{2} + 17580 x + 243850$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$150 \nu^{7} - 462 \nu^{6} + 10174 \nu^{5} - 53319 \nu^{4} + 232394 \nu^{3} - 1499697 \nu^{2} + 1183910 \nu - 13362675$$$$)/5194175$$ $$\beta_{2}$$ $$=$$ $$($$$$282286 \nu^{7} - 15339522 \nu^{6} + 35907577 \nu^{5} - 1002858672 \nu^{4} + 1947856606 \nu^{3} - 21061149120 \nu^{2} + 15432103570 \nu - 99219003000$$$$)/ 8378204275$$ $$\beta_{3}$$ $$=$$ $$($$$$-121302 \nu^{7} - 349419 \nu^{6} - 14676598 \nu^{5} - 37861257 \nu^{4} - 662937606 \nu^{3} - 1239889533 \nu^{2} - 8625793435 \nu - 13779356625$$$$)/ 1675640855$$ $$\beta_{4}$$ $$=$$ $$($$$$963177 \nu^{7} - 1768660 \nu^{6} + 45747936 \nu^{5} - 88514646 \nu^{4} + 488036934 \nu^{3} + 4582556204 \nu^{2} - 7730209020 \nu + 98741527500$$$$)/ 8378204275$$ $$\beta_{5}$$ $$=$$ $$($$$$-679482 \nu^{7} + 81620 \nu^{6} - 44765601 \nu^{5} + 16276001 \nu^{4} - 929477859 \nu^{3} - 317810284 \nu^{2} - 3215199645 \nu - 1500016825$$$$)/ 1675640855$$ $$\beta_{6}$$ $$=$$ $$($$$$3755130 \nu^{7} - 720363 \nu^{6} + 351434266 \nu^{5} - 120109521 \nu^{4} + 12942876266 \nu^{3} - 753747093 \nu^{2} + 270337140290 \nu + 34258391175$$$$)/ 8378204275$$ $$\beta_{7}$$ $$=$$ $$($$$$144 \nu^{7} + 534 \nu^{6} + 12924 \nu^{5} + 34101 \nu^{4} + 351360 \nu^{3} + 846702 \nu^{2} + 2933460 \nu + 6566615$$$$)/56455$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{6} + 3 \beta_{3} - 8 \beta_{1}$$$$)/15$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} + 18 \beta_{5} + 30 \beta_{4} + 45 \beta_{1} - 338$$$$)/15$$ $$\nu^{3}$$ $$=$$ $$($$$$-9 \beta_{7} - 23 \beta_{6} - 9 \beta_{5} - 225 \beta_{3} - 45 \beta_{2} + 514 \beta_{1} + 72$$$$)/15$$ $$\nu^{4}$$ $$=$$ $$($$$$-15 \beta_{7} + 6 \beta_{6} - 364 \beta_{5} - 460 \beta_{4} + 36 \beta_{3} + 60 \beta_{2} - 2028 \beta_{1} + 2605$$$$)/5$$ $$\nu^{5}$$ $$=$$ $$($$$$690 \beta_{7} + 506 \beta_{6} + 2250 \beta_{5} + 450 \beta_{4} + 9588 \beta_{3} + 3375 \beta_{2} - 18898 \beta_{1} - 15420$$$$)/15$$ $$\nu^{6}$$ $$=$$ $$($$$$1406 \beta_{7} - 2025 \beta_{6} + 47088 \beta_{5} + 48780 \beta_{4} - 16380 \beta_{3} - 20700 \beta_{2} + 357345 \beta_{1} - 148528$$$$)/15$$ $$\nu^{7}$$ $$=$$ $$($$$$-34524 \beta_{7} - 6418 \beta_{6} - 201834 \beta_{5} - 70875 \beta_{4} - 337470 \beta_{3} - 158970 \beta_{2} + 455924 \beta_{1} + 1211742$$$$)/15$$

## 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
 −1.22474 − 6.44174i 1.22474 + 5.44174i −1.22474 + 2.99225i 1.22474 − 3.99225i −1.22474 + 6.44174i 1.22474 − 5.44174i −1.22474 − 2.99225i 1.22474 + 3.99225i
−8.83897 8.83897i −29.2322 + 29.2322i 92.2549i 0 516.765 −106.298 106.298i 249.744 249.744i 980.039i 0
7.2 −2.71525 2.71525i −16.9847 + 16.9847i 49.2549i 0 92.2354 383.600 + 383.600i −307.515 + 307.515i 152.039i 0
7.3 2.71525 + 2.71525i 16.9847 16.9847i 49.2549i 0 92.2354 −383.600 383.600i 307.515 307.515i 152.039i 0
7.4 8.83897 + 8.83897i 29.2322 29.2322i 92.2549i 0 516.765 106.298 + 106.298i −249.744 + 249.744i 980.039i 0
18.1 −8.83897 + 8.83897i −29.2322 29.2322i 92.2549i 0 516.765 −106.298 + 106.298i 249.744 + 249.744i 980.039i 0
18.2 −2.71525 + 2.71525i −16.9847 16.9847i 49.2549i 0 92.2354 383.600 383.600i −307.515 307.515i 152.039i 0
18.3 2.71525 2.71525i 16.9847 + 16.9847i 49.2549i 0 92.2354 −383.600 + 383.600i 307.515 + 307.515i 152.039i 0
18.4 8.83897 8.83897i 29.2322 + 29.2322i 92.2549i 0 516.765 106.298 106.298i −249.744 249.744i 980.039i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 18.4 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
5.c odd 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 25.7.c.d 8
3.b odd 2 1 225.7.g.f 8
5.b even 2 1 inner 25.7.c.d 8
5.c odd 4 2 inner 25.7.c.d 8
15.d odd 2 1 225.7.g.f 8
15.e even 4 2 225.7.g.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.7.c.d 8 1.a even 1 1 trivial
25.7.c.d 8 5.b even 2 1 inner
25.7.c.d 8 5.c odd 4 2 inner
225.7.g.f 8 3.b odd 2 1
225.7.g.f 8 15.d odd 2 1
225.7.g.f 8 15.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + 24633 T_{2}^{4} + 5308416$$ acting on $$S_{7}^{\mathrm{new}}(25, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 2759 T^{4} - 712704 T^{8} - 46288338944 T^{12} + 281474976710656 T^{16}$$
$3$ $$1 - 1286514 T^{4} + 851471881011 T^{8} - 363349552696317234 T^{12} +$$$$79\!\cdots\!61$$$$T^{16}$$
$5$ 1
$7$ $$1 - 6643223804 T^{4} - 41780575508959113594 T^{8} -$$$$12\!\cdots\!04$$$$T^{12} +$$$$36\!\cdots\!01$$$$T^{16}$$
$11$ $$( 1 - 1974 T + 2514791 T^{2} - 3497061414 T^{3} + 3138428376721 T^{4} )^{4}$$
$13$ $$1 + 35973943011076 T^{4} +$$$$81\!\cdots\!66$$$$T^{8} +$$$$19\!\cdots\!36$$$$T^{12} +$$$$29\!\cdots\!21$$$$T^{16}$$
$17$ $$1 + 26775383846206 T^{4} -$$$$58\!\cdots\!49$$$$T^{8} +$$$$90\!\cdots\!26$$$$T^{12} +$$$$11\!\cdots\!41$$$$T^{16}$$
$19$ $$( 1 - 124873274 T^{2} + 8008155794792091 T^{4} -$$$$27\!\cdots\!14$$$$T^{6} +$$$$48\!\cdots\!21$$$$T^{8} )^{2}$$
$23$ $$1 + 78499101955211716 T^{4} +$$$$24\!\cdots\!46$$$$T^{8} +$$$$37\!\cdots\!56$$$$T^{12} +$$$$23\!\cdots\!81$$$$T^{16}$$
$29$ $$( 1 - 2192372284 T^{2} + 1900738827587622246 T^{4} -$$$$77\!\cdots\!44$$$$T^{6} +$$$$12\!\cdots\!81$$$$T^{8} )^{2}$$
$31$ $$( 1 - 27164 T + 1951468086 T^{2} - 24108149990684 T^{3} + 787662783788549761 T^{4} )^{4}$$
$37$ $$1 - 741444200428936124 T^{4} +$$$$84\!\cdots\!66$$$$T^{8} -$$$$32\!\cdots\!64$$$$T^{12} +$$$$18\!\cdots\!21$$$$T^{16}$$
$41$ $$( 1 - 98034 T + 11774714771 T^{2} - 465671719162194 T^{3} + 22563490300366186081 T^{4} )^{4}$$
$43$ $$1 - 80644963714520140604 T^{4} +$$$$44\!\cdots\!06$$$$T^{8} -$$$$12\!\cdots\!04$$$$T^{12} +$$$$25\!\cdots\!01$$$$T^{16}$$
$47$ $$1 -$$$$33\!\cdots\!64$$$$T^{4} +$$$$53\!\cdots\!86$$$$T^{8} -$$$$45\!\cdots\!84$$$$T^{12} +$$$$18\!\cdots\!61$$$$T^{16}$$
$53$ $$1 -$$$$51\!\cdots\!64$$$$T^{4} +$$$$24\!\cdots\!86$$$$T^{8} -$$$$12\!\cdots\!84$$$$T^{12} +$$$$58\!\cdots\!61$$$$T^{16}$$
$59$ $$( 1 - 124036297564 T^{2} +$$$$74\!\cdots\!86$$$$T^{4} -$$$$22\!\cdots\!84$$$$T^{6} +$$$$31\!\cdots\!61$$$$T^{8} )^{2}$$
$61$ $$( 1 - 98224 T + 71610487266 T^{2} - 5060537251234864 T^{3} +$$$$26\!\cdots\!21$$$$T^{4} )^{4}$$
$67$ $$1 +$$$$24\!\cdots\!06$$$$T^{4} +$$$$28\!\cdots\!51$$$$T^{8} +$$$$16\!\cdots\!26$$$$T^{12} +$$$$44\!\cdots\!41$$$$T^{16}$$
$71$ $$( 1 - 203844 T + 227339661926 T^{2} - 26112474275592324 T^{3} +$$$$16\!\cdots\!41$$$$T^{4} )^{4}$$
$73$ $$1 +$$$$14\!\cdots\!66$$$$T^{4} +$$$$88\!\cdots\!71$$$$T^{8} +$$$$75\!\cdots\!06$$$$T^{12} +$$$$27\!\cdots\!81$$$$T^{16}$$
$79$ $$( 1 - 758046230084 T^{2} +$$$$25\!\cdots\!46$$$$T^{4} -$$$$44\!\cdots\!44$$$$T^{6} +$$$$34\!\cdots\!81$$$$T^{8} )^{2}$$
$83$ $$1 -$$$$42\!\cdots\!94$$$$T^{4} +$$$$67\!\cdots\!51$$$$T^{8} -$$$$48\!\cdots\!74$$$$T^{12} +$$$$13\!\cdots\!41$$$$T^{16}$$
$89$ $$( 1 + 697846531406 T^{2} +$$$$60\!\cdots\!51$$$$T^{4} +$$$$17\!\cdots\!26$$$$T^{6} +$$$$61\!\cdots\!41$$$$T^{8} )^{2}$$
$97$ $$1 -$$$$12\!\cdots\!64$$$$T^{4} +$$$$81\!\cdots\!86$$$$T^{8} -$$$$58\!\cdots\!84$$$$T^{12} +$$$$23\!\cdots\!61$$$$T^{16}$$