# Properties

 Label 7120.2.a.bj Level 7120 Weight 2 Character orbit 7120.a Self dual yes Analytic conductor 56.853 Analytic rank 0 Dimension 7 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$7120 = 2^{4} \cdot 5 \cdot 89$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 7120.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$56.8534862392$$ Analytic rank: $$0$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 445) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{6}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{5} ) q^{3} + q^{5} + ( 2 + \beta_{6} ) q^{7} + ( 1 + \beta_{2} + 2 \beta_{5} + \beta_{6} ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta_{5} ) q^{3} + q^{5} + ( 2 + \beta_{6} ) q^{7} + ( 1 + \beta_{2} + 2 \beta_{5} + \beta_{6} ) q^{9} + ( 1 - \beta_{4} - \beta_{5} + \beta_{6} ) q^{11} + ( -2 - \beta_{1} - \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{13} + ( 1 + \beta_{5} ) q^{15} + ( -2 - \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{17} + ( 3 \beta_{1} + \beta_{2} - \beta_{6} ) q^{19} + ( 2 - \beta_{2} + \beta_{3} + 3 \beta_{5} + \beta_{6} ) q^{21} + ( 1 - \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{23} + q^{25} + ( 3 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{27} + ( 1 + 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - 3 \beta_{6} ) q^{29} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} - 3 \beta_{6} ) q^{31} + ( -1 - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} ) q^{33} + ( 2 + \beta_{6} ) q^{35} + ( -1 - \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{37} + ( 1 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + 3 \beta_{6} ) q^{39} + ( -2 + 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{41} + ( 5 - \beta_{1} - \beta_{2} - \beta_{5} ) q^{43} + ( 1 + \beta_{2} + 2 \beta_{5} + \beta_{6} ) q^{45} + ( 2 + 3 \beta_{2} - \beta_{4} - \beta_{6} ) q^{47} + ( 2 + \beta_{2} + \beta_{3} + 4 \beta_{6} ) q^{49} + ( -5 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} + 4 \beta_{6} ) q^{51} + ( \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{53} + ( 1 - \beta_{4} - \beta_{5} + \beta_{6} ) q^{55} + ( 2 + 7 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + \beta_{4} - 3 \beta_{6} ) q^{57} + ( 2 - 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{59} + ( -1 - \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - \beta_{4} + \beta_{6} ) q^{61} + ( 7 + 2 \beta_{2} + 3 \beta_{3} - \beta_{4} + 4 \beta_{5} + 4 \beta_{6} ) q^{63} + ( -2 - \beta_{1} - \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{65} + ( -1 + \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{67} + ( 3 - 5 \beta_{1} - 5 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{69} + ( 2 - 2 \beta_{1} + 2 \beta_{3} - 3 \beta_{4} ) q^{71} + ( 3 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - 4 \beta_{5} - 3 \beta_{6} ) q^{73} + ( 1 + \beta_{5} ) q^{75} + ( 5 + 2 \beta_{2} - \beta_{4} - \beta_{5} + 3 \beta_{6} ) q^{77} + ( 1 + \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} - 2 \beta_{5} - 3 \beta_{6} ) q^{79} + ( 6 + 4 \beta_{1} - \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{81} + ( 8 - 4 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{83} + ( -2 - \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{85} + ( 6 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - 4 \beta_{6} ) q^{87} + q^{89} + ( -5 - 5 \beta_{1} - \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + 5 \beta_{5} + \beta_{6} ) q^{91} + ( 4 - 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{6} ) q^{93} + ( 3 \beta_{1} + \beta_{2} - \beta_{6} ) q^{95} + ( 1 - \beta_{1} + \beta_{3} + 2 \beta_{4} - 2 \beta_{6} ) q^{97} + ( -\beta_{1} - \beta_{2} + 3 \beta_{3} - \beta_{5} + 4 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7q + 8q^{3} + 7q^{5} + 16q^{7} + 11q^{9} + O(q^{10})$$ $$7q + 8q^{3} + 7q^{5} + 16q^{7} + 11q^{9} + 10q^{11} - 7q^{13} + 8q^{15} - 13q^{17} + 7q^{19} + 16q^{21} + 13q^{23} + 7q^{25} + 23q^{27} - 4q^{29} - q^{31} - 6q^{33} + 16q^{35} - 5q^{37} + 13q^{39} + 5q^{41} + 31q^{43} + 11q^{45} + 14q^{47} + 19q^{49} + q^{51} - 13q^{53} + 10q^{55} + 21q^{57} + 14q^{59} + 3q^{61} + 54q^{63} - 7q^{65} - q^{67} + 31q^{69} + 8q^{71} + 9q^{73} + 8q^{75} + 42q^{77} - 9q^{79} + 35q^{81} + 42q^{83} - 13q^{85} - 6q^{87} + 7q^{89} - 31q^{91} + 24q^{93} + 7q^{95} - 7q^{97} - 5q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - x^{6} - 8 x^{5} + 6 x^{4} + 19 x^{3} - 10 x^{2} - 12 x + 6$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} - 2$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} - \nu^{2} - 4 \nu + 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{4} - 5 \nu^{2} + 3$$ $$\beta_{4}$$ $$=$$ $$-\nu^{4} + \nu^{3} + 4 \nu^{2} - 2 \nu - 2$$ $$\beta_{5}$$ $$=$$ $$\nu^{5} - \nu^{4} - 6 \nu^{3} + 5 \nu^{2} + 8 \nu - 5$$ $$\beta_{6}$$ $$=$$ $$\nu^{6} - \nu^{5} - 7 \nu^{4} + 5 \nu^{3} + 13 \nu^{2} - 4 \nu - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} + \beta_{3} - \beta_{2} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{1} + 2$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_{1} + 2$$ $$\nu^{4}$$ $$=$$ $$\beta_{3} + 5 \beta_{1} + 7$$ $$\nu^{5}$$ $$=$$ $$\beta_{5} + 8 \beta_{4} + 9 \beta_{3} - 2 \beta_{2} + 6 \beta_{1} + 10$$ $$\nu^{6}$$ $$=$$ $$\beta_{6} + \beta_{5} + 8 \beta_{3} + \beta_{2} + 23 \beta_{1} + 29$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.07810 1.89340 0.498937 −1.96388 0.885013 −1.49803 2.26266
0 −2.10226 0 1.00000 0 4.72699 0 1.41950 0
1.2 0 −0.172659 0 1.00000 0 2.74591 0 −2.97019 0
1.3 0 0.459905 0 1.00000 0 −0.587818 0 −2.78849 0
1.4 0 0.931146 0 1.00000 0 0.580377 0 −2.13297 0
1.5 0 2.76669 0 1.00000 0 3.75132 0 4.65459 0
1.6 0 2.82660 0 1.00000 0 −0.0498231 0 4.98967 0
1.7 0 3.29058 0 1.00000 0 4.83304 0 7.82788 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7120.2.a.bj 7
4.b odd 2 1 445.2.a.f 7
12.b even 2 1 4005.2.a.o 7
20.d odd 2 1 2225.2.a.k 7

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
445.2.a.f 7 4.b odd 2 1
2225.2.a.k 7 20.d odd 2 1
4005.2.a.o 7 12.b even 2 1
7120.2.a.bj 7 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$
$$89$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7120))$$:

 $$T_{3}^{7} - 8 T_{3}^{6} + 16 T_{3}^{5} + 19 T_{3}^{4} - 89 T_{3}^{3} + 72 T_{3}^{2} - 8 T_{3} - 4$$ $$T_{7}^{7} - 16 T_{7}^{6} + 94 T_{7}^{5} - 236 T_{7}^{4} + 189 T_{7}^{3} + 96 T_{7}^{2} - 76 T_{7} - 4$$ $$T_{11}^{7} - 10 T_{11}^{6} + 14 T_{11}^{5} + 149 T_{11}^{4} - 639 T_{11}^{3} + 968 T_{11}^{2} - 608 T_{11} + 128$$ $$T_{13}^{7} + 7 T_{13}^{6} - 44 T_{13}^{5} - 378 T_{13}^{4} - 67 T_{13}^{3} + 4078 T_{13}^{2} + 9196 T_{13} + 5816$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$1 - 8 T + 37 T^{2} - 125 T^{3} + 340 T^{4} - 780 T^{5} + 1576 T^{6} - 2866 T^{7} + 4728 T^{8} - 7020 T^{9} + 9180 T^{10} - 10125 T^{11} + 8991 T^{12} - 5832 T^{13} + 2187 T^{14}$$
$5$ $$( 1 - T )^{7}$$
$7$ $$1 - 16 T + 143 T^{2} - 908 T^{3} + 4508 T^{4} - 18272 T^{5} + 61958 T^{6} - 177804 T^{7} + 433706 T^{8} - 895328 T^{9} + 1546244 T^{10} - 2180108 T^{11} + 2403401 T^{12} - 1882384 T^{13} + 823543 T^{14}$$
$11$ $$1 - 10 T + 91 T^{2} - 511 T^{3} + 2672 T^{4} - 10626 T^{5} + 41830 T^{6} - 136602 T^{7} + 460130 T^{8} - 1285746 T^{9} + 3556432 T^{10} - 7481551 T^{11} + 14655641 T^{12} - 17715610 T^{13} + 19487171 T^{14}$$
$13$ $$1 + 7 T + 47 T^{2} + 168 T^{3} + 622 T^{4} + 2167 T^{5} + 9118 T^{6} + 36132 T^{7} + 118534 T^{8} + 366223 T^{9} + 1366534 T^{10} + 4798248 T^{11} + 17450771 T^{12} + 33787663 T^{13} + 62748517 T^{14}$$
$17$ $$1 + 13 T + 108 T^{2} + 591 T^{3} + 3017 T^{4} + 14845 T^{5} + 78122 T^{6} + 344702 T^{7} + 1328074 T^{8} + 4290205 T^{9} + 14822521 T^{10} + 49360911 T^{11} + 153344556 T^{12} + 313788397 T^{13} + 410338673 T^{14}$$
$19$ $$1 - 7 T + 81 T^{2} - 366 T^{3} + 2784 T^{4} - 9257 T^{5} + 61132 T^{6} - 176376 T^{7} + 1161508 T^{8} - 3341777 T^{9} + 19095456 T^{10} - 47697486 T^{11} + 200564019 T^{12} - 329321167 T^{13} + 893871739 T^{14}$$
$23$ $$1 - 13 T + 144 T^{2} - 1105 T^{3} + 7943 T^{4} - 47855 T^{5} + 274740 T^{6} - 1359354 T^{7} + 6319020 T^{8} - 25315295 T^{9} + 96642481 T^{10} - 309224305 T^{11} + 926833392 T^{12} - 1924466557 T^{13} + 3404825447 T^{14}$$
$29$ $$1 + 4 T + 113 T^{2} + 625 T^{3} + 6740 T^{4} + 40902 T^{5} + 266814 T^{6} + 1538922 T^{7} + 7737606 T^{8} + 34398582 T^{9} + 164381860 T^{10} + 442050625 T^{11} + 2317759837 T^{12} + 2379293284 T^{13} + 17249876309 T^{14}$$
$31$ $$1 + T + 77 T^{2} + 34 T^{3} + 3570 T^{4} + 2367 T^{5} + 134894 T^{6} + 138904 T^{7} + 4181714 T^{8} + 2274687 T^{9} + 106353870 T^{10} + 31399714 T^{11} + 2204444627 T^{12} + 887503681 T^{13} + 27512614111 T^{14}$$
$37$ $$1 + 5 T + 168 T^{2} + 803 T^{3} + 14161 T^{4} + 59989 T^{5} + 771494 T^{6} + 2748646 T^{7} + 28545278 T^{8} + 82124941 T^{9} + 717297133 T^{10} + 1504951283 T^{11} + 11649784776 T^{12} + 12828632045 T^{13} + 94931877133 T^{14}$$
$41$ $$1 - 5 T + 92 T^{2} - 673 T^{3} + 7237 T^{4} - 46413 T^{5} + 387726 T^{6} - 2290138 T^{7} + 15896766 T^{8} - 78020253 T^{9} + 498781277 T^{10} - 1901737153 T^{11} + 10658770492 T^{12} - 23750521205 T^{13} + 194754273881 T^{14}$$
$43$ $$1 - 31 T + 687 T^{2} - 10436 T^{3} + 129854 T^{4} - 1291529 T^{5} + 10962460 T^{6} - 77409324 T^{7} + 471385780 T^{8} - 2388037121 T^{9} + 10324301978 T^{10} - 35678607236 T^{11} + 100994800341 T^{12} - 195962254519 T^{13} + 271818611107 T^{14}$$
$47$ $$1 - 14 T + 271 T^{2} - 2621 T^{3} + 31508 T^{4} - 240678 T^{5} + 2207362 T^{6} - 13822254 T^{7} + 103746014 T^{8} - 531657702 T^{9} + 3271255084 T^{10} - 12789643901 T^{11} + 62152496897 T^{12} - 150909014606 T^{13} + 506623120463 T^{14}$$
$53$ $$1 + 13 T + 269 T^{2} + 2568 T^{3} + 34060 T^{4} + 265121 T^{5} + 2673394 T^{6} + 17174660 T^{7} + 141689882 T^{8} + 744724889 T^{9} + 5070750620 T^{10} + 20262755208 T^{11} + 112494587617 T^{12} + 288136694677 T^{13} + 1174711139837 T^{14}$$
$59$ $$1 - 14 T + 253 T^{2} - 2446 T^{3} + 31472 T^{4} - 260310 T^{5} + 2563464 T^{6} - 17624256 T^{7} + 151244376 T^{8} - 906139110 T^{9} + 6463687888 T^{10} - 29639065006 T^{11} + 180875847647 T^{12} - 590527470974 T^{13} + 2488651484819 T^{14}$$
$61$ $$1 - 3 T + 222 T^{2} - 759 T^{3} + 29403 T^{4} - 83651 T^{5} + 2524030 T^{6} - 6527094 T^{7} + 153965830 T^{8} - 311265371 T^{9} + 6673922343 T^{10} - 10508993319 T^{11} + 187500378822 T^{12} - 154561123083 T^{13} + 3142742836021 T^{14}$$
$67$ $$1 + T + 328 T^{2} + 279 T^{3} + 53091 T^{4} + 39287 T^{5} + 5339384 T^{6} + 3265170 T^{7} + 357738728 T^{8} + 176359343 T^{9} + 15967808433 T^{10} + 5622162759 T^{11} + 442841035096 T^{12} + 90458382169 T^{13} + 6060711605323 T^{14}$$
$71$ $$1 - 8 T + 187 T^{2} - 1328 T^{3} + 22776 T^{4} - 168048 T^{5} + 2047314 T^{6} - 13028752 T^{7} + 145359294 T^{8} - 847129968 T^{9} + 8151780936 T^{10} - 33746712368 T^{11} + 337390888637 T^{12} - 1024802271368 T^{13} + 9095120158391 T^{14}$$
$73$ $$1 - 9 T + 286 T^{2} - 1613 T^{3} + 37567 T^{4} - 166989 T^{5} + 3689378 T^{6} - 14466938 T^{7} + 269324594 T^{8} - 889884381 T^{9} + 14614201639 T^{10} - 45806362733 T^{11} + 592898475598 T^{12} - 1362008036601 T^{13} + 11047398519097 T^{14}$$
$79$ $$1 + 9 T + 430 T^{2} + 2867 T^{3} + 81347 T^{4} + 413191 T^{5} + 9305818 T^{6} + 38366618 T^{7} + 735159622 T^{8} + 2578725031 T^{9} + 40107243533 T^{10} + 111669882227 T^{11} + 1323134251570 T^{12} + 2187787099689 T^{13} + 19203908986159 T^{14}$$
$83$ $$1 - 42 T + 1008 T^{2} - 16390 T^{3} + 206255 T^{4} - 2129142 T^{5} + 19880600 T^{6} - 179268272 T^{7} + 1650089800 T^{8} - 14667659238 T^{9} + 117933927685 T^{10} - 777841881190 T^{11} + 3970552968144 T^{12} - 13731495681498 T^{13} + 27136050989627 T^{14}$$
$89$ $$( 1 - T )^{7}$$
$97$ $$1 + 7 T + 609 T^{2} + 3724 T^{3} + 164856 T^{4} + 856227 T^{5} + 25710150 T^{6} + 108804964 T^{7} + 2493884550 T^{8} + 8056239843 T^{9} + 150459620088 T^{10} + 329683042444 T^{11} + 5229690216513 T^{12} + 5830804034503 T^{13} + 80798284478113 T^{14}$$