# Properties

 Label 688.6.a.e Level $688$ Weight $6$ Character orbit 688.a Self dual yes Analytic conductor $110.344$ Analytic rank $1$ Dimension $8$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$688 = 2^{4} \cdot 43$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 688.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$110.344068031$$ Analytic rank: $$1$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 4x^{7} - 173x^{6} + 462x^{5} + 9118x^{4} - 14192x^{3} - 167688x^{2} + 106368x + 681984$$ x^8 - 4*x^7 - 173*x^6 + 462*x^5 + 9118*x^4 - 14192*x^3 - 167688*x^2 + 106368*x + 681984 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 43) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + (\beta_{4} + 3) q^{3} + (\beta_{7} - \beta_{6} + \beta_{4} - 27) q^{5} + (\beta_{7} + 2 \beta_{6} + \beta_{5} - \beta_{4} + 4 \beta_{3} + 6 \beta_{2} + 4 \beta_1 + 17) q^{7} + ( - 4 \beta_{7} + \beta_{6} - 5 \beta_{5} + 9 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} + \cdots + 68) q^{9}+O(q^{10})$$ q + (b4 + 3) * q^3 + (b7 - b6 + b4 - 27) * q^5 + (b7 + 2*b6 + b5 - b4 + 4*b3 + 6*b2 + 4*b1 + 17) * q^7 + (-4*b7 + b6 - 5*b5 + 9*b4 + 2*b3 + 3*b2 + 9*b1 + 68) * q^9 $$q + (\beta_{4} + 3) q^{3} + (\beta_{7} - \beta_{6} + \beta_{4} - 27) q^{5} + (\beta_{7} + 2 \beta_{6} + \beta_{5} - \beta_{4} + 4 \beta_{3} + 6 \beta_{2} + 4 \beta_1 + 17) q^{7} + ( - 4 \beta_{7} + \beta_{6} - 5 \beta_{5} + 9 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} + \cdots + 68) q^{9}+ \cdots + (563 \beta_{7} + 304 \beta_{6} + 1398 \beta_{5} - 3541 \beta_{4} + \cdots - 30494) q^{99}+O(q^{100})$$ q + (b4 + 3) * q^3 + (b7 - b6 + b4 - 27) * q^5 + (b7 + 2*b6 + b5 - b4 + 4*b3 + 6*b2 + 4*b1 + 17) * q^7 + (-4*b7 + b6 - 5*b5 + 9*b4 + 2*b3 + 3*b2 + 9*b1 + 68) * q^9 + (-b7 - 11*b6 - 7*b4 - 10*b3 - 3*b2 - 9*b1 + 70) * q^11 + (-b7 + 4*b6 + 15*b5 - 11*b4 - 15*b3 - 13*b2 - 13*b1 - 316) * q^13 + (-2*b7 - 4*b6 + 8*b5 - 43*b4 - 7*b3 - 10*b2 + 9*b1 + 225) * q^15 + (4*b7 + 4*b6 - 5*b5 + 37*b4 + 13*b3 + 38*b2 + 12*b1 - 318) * q^17 + (32*b7 + 13*b6 - 54*b5 - 3*b4 + 4*b3 - 56*b2 - 62*b1 + 231) * q^19 + (-12*b7 + 7*b6 - 16*b5 + 58*b4 + 37*b3 - 33*b2 - 87*b1 - 272) * q^21 + (-35*b7 + 35*b6 + 77*b5 - 27*b4 + 61*b3 + 40*b2 + 112*b1 + 270) * q^23 + (-17*b7 + 68*b6 + 42*b5 - 72*b4 + 42*b3 - 65*b2 + 21*b1 + 527) * q^25 + (3*b7 + 28*b6 - 110*b5 + 107*b4 - 65*b3 - 132*b2 - 74*b1 + 1133) * q^27 + (61*b7 + 10*b6 - 76*b5 - 114*b4 + 22*b3 + 41*b2 - 54*b1 - 470) * q^29 + (-101*b7 - 44*b6 + 47*b5 - 46*b4 - 83*b3 - 73*b2 - 140*b1 - 683) * q^31 + (59*b7 - 16*b6 + 213*b5 - 155*b4 - 162*b3 + 42*b2 + 34*b1 - 1687) * q^33 + (28*b7 - 135*b6 - 84*b5 - 50*b4 - 207*b3 - 301*b2 - 323*b1 - 652) * q^35 + (-140*b7 - 110*b6 - 47*b5 - 321*b4 - 12*b3 + 45*b2 - 114*b1 - 293) * q^37 + (-b7 - 130*b6 + 63*b5 - 669*b4 - 100*b3 + 214*b2 + 242*b1 - 1261) * q^39 + (-69*b7 + 66*b6 + 80*b5 - 190*b4 - 148*b3 + 123*b2 + 173*b1 - 1325) * q^41 + 1849 * q^43 + (13*b7 + 60*b6 + 240*b5 - 330*b4 - 377*b3 - 353*b2 - 404*b1 - 5642) * q^45 + (-150*b7 + 57*b6 + 123*b5 + 183*b4 - 51*b3 - 136*b2 + 193*b1 + 9581) * q^47 + (-258*b7 - 143*b6 - 610*b5 - 16*b4 + 19*b3 - 321*b2 - 205*b1 + 1243) * q^49 + (-241*b7 + 161*b6 - 223*b5 + 206*b4 + 182*b3 + 22*b2 - 54*b1 + 10170) * q^51 + (-203*b7 + 103*b6 + 237*b5 - 279*b4 - 320*b3 - 372*b2 + 100*b1 - 7910) * q^53 + (161*b7 + 743*b6 + 444*b5 + 237*b4 + 499*b3 + 398*b2 + 516*b1 + 5843) * q^55 + (-175*b7 + 269*b6 - 399*b5 - 67*b4 + 886*b3 + 638*b2 + 1038*b1 + 59) * q^57 + (364*b7 - 427*b6 - 48*b5 - 922*b4 - 815*b3 + 285*b2 - 445*b1 + 3624) * q^59 + (323*b7 - b6 + 224*b5 - 387*b4 + 203*b3 - 542*b2 + 296*b1 - 10623) * q^61 + (-707*b7 + 233*b6 - 188*b5 - 511*b4 + 1309*b3 + 738*b2 + 484*b1 + 8183) * q^63 + (-577*b7 - 323*b6 - 984*b5 + 883*b4 + 811*b3 + 1158*b2 + 796*b1 - 119) * q^65 + (-437*b7 + 69*b6 + 448*b5 - 1225*b4 - 90*b3 + 635*b2 + 77*b1 - 3142) * q^67 + (289*b7 - 372*b6 - 94*b5 + 810*b4 + 77*b3 - 969*b2 - 1728*b1 - 11760) * q^69 + (822*b7 + 903*b6 + 133*b5 - 1250*b4 + 291*b3 + 334*b2 + 816*b1 + 1108) * q^71 + (357*b7 + 244*b6 - 409*b5 - 167*b4 + 212*b3 + 878*b2 + 280*b1 + 2089) * q^73 + (236*b7 - 287*b6 - 492*b5 + 212*b4 + 1008*b3 + 170*b2 - 276*b1 - 20682) * q^75 + (921*b7 + 149*b6 + 1342*b5 - 1179*b4 - 179*b3 + 1798*b2 + 1714*b1 - 27853) * q^77 + (838*b7 - 430*b6 - 1457*b5 - 86*b4 - 625*b3 - 157*b2 - 639*b1 - 19370) * q^79 + (605*b7 + 48*b6 - 591*b5 + 527*b4 - 653*b3 - 249*b2 + 467*b1 + 19968) * q^81 + (-1039*b7 + 32*b6 - 138*b5 + 1755*b4 + 1917*b3 + 408*b2 + 884*b1 + 9418) * q^83 + (-333*b7 + 633*b6 + 451*b5 - 3606*b4 - 1672*b3 - 1523*b2 - 1274*b1 + 18334) * q^85 + (-40*b7 + 464*b6 + 211*b5 - 1044*b4 + 791*b3 + 175*b2 - 639*b1 - 33148) * q^87 + (671*b7 - 343*b6 - 728*b5 + 1221*b4 + 1173*b3 - 346*b2 - 1528*b1 - 33025) * q^89 + (447*b7 - 531*b6 + 1296*b5 + 327*b4 - 2583*b3 - 2670*b2 - 2070*b1 - 51135) * q^91 + (319*b7 + 290*b6 + 1741*b5 - 2599*b4 + 315*b3 + 2872*b2 + 1312*b1 - 15393) * q^93 + (1046*b7 - 1008*b6 - 195*b5 - 1309*b4 + 142*b3 + 3631*b2 + 2330*b1 - 16497) * q^95 + (-1289*b7 + 923*b6 + 900*b5 - 986*b4 + 163*b3 - 2094*b2 - 1116*b1 + 17967) * q^97 + (563*b7 + 304*b6 + 1398*b5 - 3541*b4 - 1801*b3 - 732*b2 + 1668*b1 - 30494) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 26 q^{3} - 212 q^{5} + 136 q^{7} + 546 q^{9}+O(q^{10})$$ 8 * q + 26 * q^3 - 212 * q^5 + 136 * q^7 + 546 * q^9 $$8 q + 26 q^{3} - 212 q^{5} + 136 q^{7} + 546 q^{9} + 532 q^{11} - 2492 q^{13} + 1780 q^{15} - 2534 q^{17} + 1678 q^{19} - 2256 q^{21} + 2488 q^{23} + 4378 q^{25} + 8960 q^{27} - 4360 q^{29} - 5704 q^{31} - 12852 q^{33} - 5640 q^{35} - 3772 q^{37} - 11120 q^{39} - 10698 q^{41} + 14792 q^{43} - 44912 q^{45} + 77864 q^{47} + 7188 q^{49} + 80246 q^{51} - 62352 q^{53} + 49552 q^{55} - 808 q^{57} + 26224 q^{59} - 82540 q^{61} + 61768 q^{63} - 5000 q^{65} - 27784 q^{67} - 93776 q^{69} + 9504 q^{71} + 14260 q^{73} - 167420 q^{75} - 218140 q^{77} - 160248 q^{79} + 161076 q^{81} + 77176 q^{83} + 141096 q^{85} - 268136 q^{87} - 265692 q^{89} - 401148 q^{91} - 123860 q^{93} - 135884 q^{95} + 144742 q^{97} - 239516 q^{99}+O(q^{100})$$ 8 * q + 26 * q^3 - 212 * q^5 + 136 * q^7 + 546 * q^9 + 532 * q^11 - 2492 * q^13 + 1780 * q^15 - 2534 * q^17 + 1678 * q^19 - 2256 * q^21 + 2488 * q^23 + 4378 * q^25 + 8960 * q^27 - 4360 * q^29 - 5704 * q^31 - 12852 * q^33 - 5640 * q^35 - 3772 * q^37 - 11120 * q^39 - 10698 * q^41 + 14792 * q^43 - 44912 * q^45 + 77864 * q^47 + 7188 * q^49 + 80246 * q^51 - 62352 * q^53 + 49552 * q^55 - 808 * q^57 + 26224 * q^59 - 82540 * q^61 + 61768 * q^63 - 5000 * q^65 - 27784 * q^67 - 93776 * q^69 + 9504 * q^71 + 14260 * q^73 - 167420 * q^75 - 218140 * q^77 - 160248 * q^79 + 161076 * q^81 + 77176 * q^83 + 141096 * q^85 - 268136 * q^87 - 265692 * q^89 - 401148 * q^91 - 123860 * q^93 - 135884 * q^95 + 144742 * q^97 - 239516 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4x^{7} - 173x^{6} + 462x^{5} + 9118x^{4} - 14192x^{3} - 167688x^{2} + 106368x + 681984$$ :

 $$\beta_{1}$$ $$=$$ $$( - 12949 \nu^{7} + 681148 \nu^{6} - 2601727 \nu^{5} - 80176782 \nu^{4} + 378607946 \nu^{3} + 1837549088 \nu^{2} - 8768955576 \nu - 1162574592 ) / 547265856$$ (-12949*v^7 + 681148*v^6 - 2601727*v^5 - 80176782*v^4 + 378607946*v^3 + 1837549088*v^2 - 8768955576*v - 1162574592) / 547265856 $$\beta_{2}$$ $$=$$ $$( 36205 \nu^{7} - 1830508 \nu^{6} - 539057 \nu^{5} + 265985910 \nu^{4} - 188725898 \nu^{3} - 9341109104 \nu^{2} + 5760293592 \nu + 58885512576 ) / 1094531712$$ (36205*v^7 - 1830508*v^6 - 539057*v^5 + 265985910*v^4 - 188725898*v^3 - 9341109104*v^2 + 5760293592*v + 58885512576) / 1094531712 $$\beta_{3}$$ $$=$$ $$( - 67489 \nu^{7} + 623356 \nu^{6} + 8964549 \nu^{5} - 78907518 \nu^{4} - 250466958 \nu^{3} + 2271218384 \nu^{2} - 114021816 \nu - 8596825856 ) / 364843904$$ (-67489*v^7 + 623356*v^6 + 8964549*v^5 - 78907518*v^4 - 250466958*v^3 + 2271218384*v^2 - 114021816*v - 8596825856) / 364843904 $$\beta_{4}$$ $$=$$ $$( - 204587 \nu^{7} + 1315940 \nu^{6} + 28380103 \nu^{5} - 147318570 \nu^{4} - 993160922 \nu^{3} + 4310420080 \nu^{2} + 8605716312 \nu - 37440050688 ) / 1094531712$$ (-204587*v^7 + 1315940*v^6 + 28380103*v^5 - 147318570*v^4 - 993160922*v^3 + 4310420080*v^2 + 8605716312*v - 37440050688) / 1094531712 $$\beta_{5}$$ $$=$$ $$( - 309497 \nu^{7} + 2935676 \nu^{6} + 38047933 \nu^{5} - 372264846 \nu^{4} - 955734974 \nu^{3} + 11960213584 \nu^{2} + 2287797192 \nu - 76606929024 ) / 1094531712$$ (-309497*v^7 + 2935676*v^6 + 38047933*v^5 - 372264846*v^4 - 955734974*v^3 + 11960213584*v^2 + 2287797192*v - 76606929024) / 1094531712 $$\beta_{6}$$ $$=$$ $$( 73259 \nu^{7} - 190788 \nu^{6} - 10643911 \nu^{5} + 12835930 \nu^{4} + 347431274 \nu^{3} - 15275088 \nu^{2} - 785010872 \nu - 3291957568 ) / 182421952$$ (73259*v^7 - 190788*v^6 - 10643911*v^5 + 12835930*v^4 + 347431274*v^3 - 15275088*v^2 - 785010872*v - 3291957568) / 182421952 $$\beta_{7}$$ $$=$$ $$( - 34066 \nu^{7} + 259912 \nu^{6} + 4750721 \nu^{5} - 30187209 \nu^{4} - 176761426 \nu^{3} + 777094526 \nu^{2} + 1922056332 \nu - 2232499728 ) / 68408232$$ (-34066*v^7 + 259912*v^6 + 4750721*v^5 - 30187209*v^4 - 176761426*v^3 + 777094526*v^2 + 1922056332*v - 2232499728) / 68408232
 $$\nu$$ $$=$$ $$( \beta_{6} + \beta_{5} + \beta_{3} + 2\beta_{2} + 4 ) / 8$$ (b6 + b5 + b3 + 2*b2 + 4) / 8 $$\nu^{2}$$ $$=$$ $$( -2\beta_{7} + \beta_{6} - \beta_{5} + 8\beta_{4} + \beta_{3} + 180 ) / 4$$ (-2*b7 + b6 - b5 + 8*b4 + b3 + 180) / 4 $$\nu^{3}$$ $$=$$ $$( -14\beta_{7} + 33\beta_{6} + 15\beta_{5} + 32\beta_{4} + 65\beta_{3} + 64\beta_{2} + 4\beta _1 + 380 ) / 4$$ (-14*b7 + 33*b6 + 15*b5 + 32*b4 + 65*b3 + 64*b2 + 4*b1 + 380) / 4 $$\nu^{4}$$ $$=$$ $$( -99\beta_{7} + 84\beta_{6} - 117\beta_{5} + 488\beta_{4} + 140\beta_{3} + 63\beta_{2} + 46\beta _1 + 6796 ) / 2$$ (-99*b7 + 84*b6 - 117*b5 + 488*b4 + 140*b3 + 63*b2 + 46*b1 + 6796) / 2 $$\nu^{5}$$ $$=$$ $$( - 869 \beta_{7} + 1527 \beta_{6} - 276 \beta_{5} + 2768 \beta_{4} + 3711 \beta_{3} + 2515 \beta_{2} + 142 \beta _1 + 27000 ) / 2$$ (-869*b7 + 1527*b6 - 276*b5 + 2768*b4 + 3711*b3 + 2515*b2 + 142*b1 + 27000) / 2 $$\nu^{6}$$ $$=$$ $$( - 10339 \beta_{7} + 11590 \beta_{6} - 16231 \beta_{5} + 55680 \beta_{4} + 22478 \beta_{3} + 10127 \beta_{2} + 6850 \beta _1 + 639708 ) / 2$$ (-10339*b7 + 11590*b6 - 16231*b5 + 55680*b4 + 22478*b3 + 10127*b2 + 6850*b1 + 639708) / 2 $$\nu^{7}$$ $$=$$ $$( - 102849 \beta_{7} + 166838 \beta_{6} - 94865 \beta_{5} + 386624 \beta_{4} + 421838 \beta_{3} + 234341 \beta_{2} + 20926 \beta _1 + 3616388 ) / 2$$ (-102849*b7 + 166838*b6 - 94865*b5 + 386624*b4 + 421838*b3 + 234341*b2 + 20926*b1 + 3616388) / 2

## 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
 −2.08717 −6.09504 5.65705 2.58275 −5.06235 7.21373 10.9591 −9.16809
0 −25.6605 0 −61.4284 0 184.774 0 415.460 0
1.2 0 −11.1803 0 −63.3756 0 −223.489 0 −118.000 0
1.3 0 −7.84314 0 −107.102 0 25.5214 0 −181.485 0
1.4 0 −3.05838 0 27.7074 0 103.690 0 −233.646 0
1.5 0 9.19190 0 73.4416 0 4.24720 0 −158.509 0
1.6 0 11.2683 0 −9.80186 0 11.1041 0 −116.025 0
1.7 0 25.1057 0 −61.2927 0 166.001 0 387.294 0
1.8 0 28.1764 0 −10.1483 0 −135.849 0 550.912 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$43$$ $$-1$$

## 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 688.6.a.e 8
4.b odd 2 1 43.6.a.a 8
12.b even 2 1 387.6.a.c 8
20.d odd 2 1 1075.6.a.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.6.a.a 8 4.b odd 2 1
387.6.a.c 8 12.b even 2 1
688.6.a.e 8 1.a even 1 1 trivial
1075.6.a.a 8 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} - 26 T_{3}^{7} - 907 T_{3}^{6} + 22242 T_{3}^{5} + 184435 T_{3}^{4} - 3627880 T_{3}^{3} - 14906374 T_{3}^{2} + 156321960 T_{3} + 504223128$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(688))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8} - 26 T^{7} + \cdots + 504223128$$
$5$ $$T^{8} + 212 T^{7} + \cdots + 5172924974752$$
$7$ $$T^{8} + \cdots + 116222354316288$$
$11$ $$T^{8} - 532 T^{7} + \cdots + 16\!\cdots\!76$$
$13$ $$T^{8} + 2492 T^{7} + \cdots - 20\!\cdots\!44$$
$17$ $$T^{8} + 2534 T^{7} + \cdots + 37\!\cdots\!33$$
$19$ $$T^{8} - 1678 T^{7} + \cdots + 20\!\cdots\!68$$
$23$ $$T^{8} - 2488 T^{7} + \cdots - 18\!\cdots\!83$$
$29$ $$T^{8} + 4360 T^{7} + \cdots - 39\!\cdots\!36$$
$31$ $$T^{8} + 5704 T^{7} + \cdots + 10\!\cdots\!13$$
$37$ $$T^{8} + 3772 T^{7} + \cdots - 10\!\cdots\!04$$
$41$ $$T^{8} + 10698 T^{7} + \cdots + 17\!\cdots\!57$$
$43$ $$(T - 1849)^{8}$$
$47$ $$T^{8} - 77864 T^{7} + \cdots - 19\!\cdots\!04$$
$53$ $$T^{8} + 62352 T^{7} + \cdots + 55\!\cdots\!84$$
$59$ $$T^{8} - 26224 T^{7} + \cdots - 68\!\cdots\!68$$
$61$ $$T^{8} + 82540 T^{7} + \cdots + 23\!\cdots\!84$$
$67$ $$T^{8} + 27784 T^{7} + \cdots + 69\!\cdots\!48$$
$71$ $$T^{8} - 9504 T^{7} + \cdots - 15\!\cdots\!64$$
$73$ $$T^{8} - 14260 T^{7} + \cdots + 80\!\cdots\!92$$
$79$ $$T^{8} + 160248 T^{7} + \cdots + 19\!\cdots\!12$$
$83$ $$T^{8} - 77176 T^{7} + \cdots - 19\!\cdots\!08$$
$89$ $$T^{8} + 265692 T^{7} + \cdots - 13\!\cdots\!64$$
$97$ $$T^{8} - 144742 T^{7} + \cdots + 38\!\cdots\!17$$