# Properties

 Label 7.13.b.b Level $7$ Weight $13$ Character orbit 7.b Analytic conductor $6.398$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7$$ Weight: $$k$$ $$=$$ $$13$$ Character orbit: $$[\chi]$$ $$=$$ 7.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.39795672093$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ Defining polynomial: $$x^{6} - x^{5} + 238188x^{4} - 14589496x^{3} + 11212054600x^{2} - 101757597480x + 81251686776288$$ x^6 - x^5 + 238188*x^4 - 14589496*x^3 + 11212054600*x^2 - 101757597480*x + 81251686776288 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{11}\cdot 3\cdot 7^{3}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + \beta_{3} q^{3} + ( - 26 \beta_{2} + 2 \beta_1 + 2714) q^{4} + (\beta_{4} + 8 \beta_{3}) q^{5} + (\beta_{5} - \beta_{4} - 26 \beta_{3}) q^{6} + ( - \beta_{5} - \beta_{4} + 7 \beta_{3} + 674 \beta_{2} + 25 \beta_1 - 52450) q^{7} + (2024 \beta_{2} - 172800) q^{8} + (3126 \beta_{2} + 48 \beta_1 - 280143) q^{9}+O(q^{10})$$ q + b2 * q^2 + b3 * q^3 + (-26*b2 + 2*b1 + 2714) * q^4 + (b4 + 8*b3) * q^5 + (b5 - b4 - 26*b3) * q^6 + (-b5 - b4 + 7*b3 + 674*b2 + 25*b1 - 52450) * q^7 + (2024*b2 - 172800) * q^8 + (3126*b2 + 48*b1 - 280143) * q^9 $$q + \beta_{2} q^{2} + \beta_{3} q^{3} + ( - 26 \beta_{2} + 2 \beta_1 + 2714) q^{4} + (\beta_{4} + 8 \beta_{3}) q^{5} + (\beta_{5} - \beta_{4} - 26 \beta_{3}) q^{6} + ( - \beta_{5} - \beta_{4} + 7 \beta_{3} + 674 \beta_{2} + 25 \beta_1 - 52450) q^{7} + (2024 \beta_{2} - 172800) q^{8} + (3126 \beta_{2} + 48 \beta_1 - 280143) q^{9} + (25 \beta_{5} + \beta_{4} - 712 \beta_{3}) q^{10} + ( - 2222 \beta_{2} + 594 \beta_1 + 283932) q^{11} + ( - 26 \beta_{5} + 224 \beta_{4} + 2754 \beta_{3}) q^{12} + (2 \beta_{5} - 223 \beta_{4} + 2056 \beta_{3}) q^{13} + ( - 27 \beta_{5} - 223 \beta_{4} - 5348 \beta_{3} + \cdots + 4643190) q^{14}+ \cdots + (4615961790 \beta_{2} - 160053630 \beta_1 + 10982049804) q^{99}+O(q^{100})$$ q + b2 * q^2 + b3 * q^3 + (-26*b2 + 2*b1 + 2714) * q^4 + (b4 + 8*b3) * q^5 + (b5 - b4 - 26*b3) * q^6 + (-b5 - b4 + 7*b3 + 674*b2 + 25*b1 - 52450) * q^7 + (2024*b2 - 172800) * q^8 + (3126*b2 + 48*b1 - 280143) * q^9 + (25*b5 + b4 - 712*b3) * q^10 + (-2222*b2 + 594*b1 + 283932) * q^11 + (-26*b5 + 224*b4 + 2754*b3) * q^12 + (2*b5 - 223*b4 + 2056*b3) * q^13 + (-27*b5 - 223*b4 - 5348*b3 - 35849*b2 + 1998*b1 + 4643190) * q^14 + (87270*b2 - 7500*b1 - 6751980) * q^15 + (-118928*b2 - 4144*b1 + 2666896) * q^16 + (-378*b5 + 222*b4 - 15396*b3) * q^17 + (-295899*b2 + 7500*b1 + 21390300) * q^18 + (378*b5 - 1578*b4 - 903*b3) * q^19 + (-270*b5 + 1800*b4 + 126990*b3) * q^20 + (674*b5 + 1801*b4 - 77924*b3 + 643422*b2 + 7356*b1 - 5451588) * q^21 + (1152514*b2 + 11000*b1 - 13866600) * q^22 + (-1398254*b2 + 22950*b1 - 44463600) * q^23 + (2024*b5 - 2024*b4 - 225424*b3) * q^24 + (756750*b2 - 26040*b1 - 185989175) * q^25 + (-1701*b5 - 3649*b4 + 70276*b3) * q^26 + (3126*b5 + 1626*b4 + 154758*b3) * q^27 + (-5502*b5 + 1848*b4 + 69678*b3 + 5541830*b2 - 122150*b1 - 25040750) * q^28 + (4090324*b2 + 220752*b1 + 348290658) * q^29 + (-19258500*b2 - 20460*b1 + 578333700) * q^30 + (-2254*b5 - 46*b4 + 440104*b3) * q^31 + (-8187840*b2 - 345600*b1 - 110937600) * q^32 + (-2222*b5 + 61028*b4 + 152812*b3) * q^33 + (-18048*b5 - 60852*b4 - 1854852*b3) * q^34 + (21600*b5 - 62876*b4 + 185087*b3 + 17483190*b2 + 95040*b1 + 302181840) * q^35 + (26517078*b2 - 593406*b1 - 851631462) * q^36 + (-13168244*b2 - 45400*b1 - 804129350) * q^37 + (-21303*b5 + 64947*b4 + 2962050*b3) * q^38 + (-8824974*b2 + 1858548*b1 - 1610731404) * q^39 + (50600*b5 - 170776*b4 - 2823488*b3) * q^40 + (61452*b5 + 233698*b4 - 2800552*b3) * q^41 + (-35849*b5 + 233651*b4 + 4939900*b3 - 12139620*b2 + 1478100*b1 + 4397372100) * q^42 + (44513686*b2 - 841750*b1 - 4787084300) * q^43 + (-19715652*b2 + 158004*b1 + 6709064868) * q^44 + (87270*b5 - 298329*b4 - 2384472*b3) * q^45 + (23217754*b2 - 2199808*b1 - 9473226240) * q^46 + (-194670*b5 - 7270*b4 + 2913244*b3) * q^47 + (-118928*b5 - 291328*b4 + 7076816*b3) * q^48 + (-18746*b5 - 226604*b4 - 10152604*b3 - 49514052*b2 - 3380496*b1 + 3870525169) * q^49 + (-241209275*b2 + 836460*b1 + 5098002300) * q^50 + (224171424*b2 - 2815848*b1 + 12426313464) * q^51 + (-28866*b5 + 458184*b4 - 18054126*b3) * q^52 + (-34021832*b2 - 1004400*b1 - 17419974750) * q^53 + (235542*b5 + 506958*b4 + 12881208*b3) * q^54 + (57310*b5 + 17974*b4 + 29623022*b3) * q^55 + (118152*b5 - 278552*b4 - 12033952*b3 - 189025576*b2 - 276048*b1 + 18461176560) * q^56 + (-359549370*b2 + 12724200*b1 + 1153315800) * q^57 + (543268714*b2 + 13920200*b1 + 28325308200) * q^58 + (-535626*b5 + 116626*b4 + 20057351*b3) * q^59 + (693668880*b2 - 8328960*b1 - 103537854720) * q^60 + (456232*b5 + 727243*b4 - 21247432*b3) * q^61 + (401004*b5 - 907096*b4 - 24199700*b3) * q^62 + (111981*b5 - 446619*b4 - 20799681*b3 - 234147876*b2 - 4071075*b1 + 34920868050) * q^63 + (117331328*b2 - 8387456*b1 - 67418924416) * q^64 + (132499770*b2 - 23356080*b1 + 70044678720) * q^65 + (1152514*b5 - 63514*b4 - 47329964*b3) * q^66 + (-981960834*b2 + 35315850*b1 - 21636205900) * q^67 + (-1647864*b5 - 3338064*b4 + 39625416*b3) * q^68 + (-1398254*b5 + 3670304*b4 - 15407096*b3) * q^69 + (-516605*b5 + 3720229*b4 + 149349242*b3 - 22651500*b2 + 37437420*b1 + 119262959100) * q^70 + (501818548*b2 - 61389846*b1 - 2482470828) * q^71 + (-1139072376*b2 + 6885600*b1 + 91702677600) * q^72 + (274806*b5 - 4005144*b4 + 183946716*b3) * q^73 + (-523726006*b2 - 27516888*b1 - 89772443640) * q^74 + (756750*b5 - 3334710*b4 - 197383955*b3) * q^75 + (2155710*b5 - 323760*b4 - 226835910*b3) * q^76 + (-1260468*b5 - 155518*b4 - 12374516*b3 + 1096841086*b2 - 38342700*b1 + 45485950950) * q^77 + (1155635940*b2 + 30672300*b1 - 56139365700) * q^78 + (-1471919016*b2 - 7923318*b1 - 28751964172) * q^79 + (-3760560*b5 + 4387904*b4 - 73767248*b3) * q^80 + (108731034*b2 + 22819032*b1 - 274807047735) * q^81 + (2216998*b5 + 17624398*b4 + 303463400*b3) * q^82 + (-278532*b5 - 13530532*b4 + 30440461*b3) * q^83 + (5541830*b5 - 17634680*b4 - 130284630*b3 + 4095152208*b2 - 15978816*b1 - 57192754752) * q^84 + (5069274480*b2 + 191616120*b1 + 10375582680) * q^85 + (-7093428886*b2 + 67141872*b1 + 301345274160) * q^86 + (4090324*b5 + 17764124*b4 + 171743098*b3) * q^87 + (2716649936*b2 - 80379200*b1 - 77129448000) * q^88 + (840294*b5 - 4156644*b4 - 152674044*b3) * q^89 + (-5972475*b5 + 17764401*b4 + 707174988*b3) * q^90 + (-2232874*b5 + 21343574*b4 - 333057851*b3 - 1394160558*b2 + 10464216*b1 - 79117797528) * q^91 + (-7352377380*b2 - 104762700*b1 + 335550219300) * q^92 + (2914190760*b2 + 19540200*b1 - 357236623800) * q^93 + (-519736*b5 - 43275364*b4 - 1175859164*b3) * q^94 + (-6571136250*b2 - 115423380*b1 + 609685469100) * q^95 + (-8187840*b5 - 26026560*b4 + 211847040*b3) * q^96 + (11166106*b5 - 13771394*b4 + 212660924*b3) * q^97 + (-14323554*b5 + 4232746*b4 + 271886300*b3 + 543513481*b2 - 186921000*b1 - 344391150600) * q^98 + (4615961790*b2 - 160053630*b1 + 10982049804) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 16288 q^{4} - 314650 q^{7} - 1036800 q^{8} - 1680762 q^{9}+O(q^{10})$$ 6 * q + 16288 * q^4 - 314650 * q^7 - 1036800 * q^8 - 1680762 * q^9 $$6 q + 16288 q^{4} - 314650 q^{7} - 1036800 q^{8} - 1680762 q^{9} + 1704780 q^{11} + 27863136 q^{14} - 40526880 q^{15} + 15993088 q^{16} + 128356800 q^{18} - 32694816 q^{21} - 83177600 q^{22} - 266735700 q^{23} - 1115987130 q^{25} - 150488800 q^{28} + 2090185452 q^{29} + 3469961280 q^{30} - 666316800 q^{32} + 1813281120 q^{35} - 5110975584 q^{36} - 4824866900 q^{37} - 9660671328 q^{39} + 26387188800 q^{42} - 28724189300 q^{43} + 40254705216 q^{44} - 56843757056 q^{46} + 23216390022 q^{49} + 30589686720 q^{50} + 74552249088 q^{51} - 104521857300 q^{53} + 110766507264 q^{56} + 6945343200 q^{57} + 169979689600 q^{58} - 621243786240 q^{60} + 209517066150 q^{63} - 404530321408 q^{64} + 420221360160 q^{65} - 129746603700 q^{67} + 715652629440 q^{70} - 15017604660 q^{71} + 550229836800 q^{72} - 538689695616 q^{74} + 272839020300 q^{77} - 336774849600 q^{78} - 172527631668 q^{79} - 1648796648346 q^{81} - 343188486144 q^{84} + 62636728320 q^{85} + 1808205928704 q^{86} - 462937446400 q^{88} - 474685856736 q^{91} + 2013091790400 q^{92} - 2143380662400 q^{93} + 3657881967840 q^{95} - 2066720745600 q^{98} + 65572191564 q^{99}+O(q^{100})$$ 6 * q + 16288 * q^4 - 314650 * q^7 - 1036800 * q^8 - 1680762 * q^9 + 1704780 * q^11 + 27863136 * q^14 - 40526880 * q^15 + 15993088 * q^16 + 128356800 * q^18 - 32694816 * q^21 - 83177600 * q^22 - 266735700 * q^23 - 1115987130 * q^25 - 150488800 * q^28 + 2090185452 * q^29 + 3469961280 * q^30 - 666316800 * q^32 + 1813281120 * q^35 - 5110975584 * q^36 - 4824866900 * q^37 - 9660671328 * q^39 + 26387188800 * q^42 - 28724189300 * q^43 + 40254705216 * q^44 - 56843757056 * q^46 + 23216390022 * q^49 + 30589686720 * q^50 + 74552249088 * q^51 - 104521857300 * q^53 + 110766507264 * q^56 + 6945343200 * q^57 + 169979689600 * q^58 - 621243786240 * q^60 + 209517066150 * q^63 - 404530321408 * q^64 + 420221360160 * q^65 - 129746603700 * q^67 + 715652629440 * q^70 - 15017604660 * q^71 + 550229836800 * q^72 - 538689695616 * q^74 + 272839020300 * q^77 - 336774849600 * q^78 - 172527631668 * q^79 - 1648796648346 * q^81 - 343188486144 * q^84 + 62636728320 * q^85 + 1808205928704 * q^86 - 462937446400 * q^88 - 474685856736 * q^91 + 2013091790400 * q^92 - 2143380662400 * q^93 + 3657881967840 * q^95 - 2066720745600 * q^98 + 65572191564 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 238188x^{4} - 14589496x^{3} + 11212054600x^{2} - 101757597480x + 81251686776288$$ :

 $$\beta_{1}$$ $$=$$ $$( 316322973 \nu^{5} - 1880954567990 \nu^{4} - 229888629934646 \nu^{3} + \cdots + 67\!\cdots\!71 ) / 78\!\cdots\!45$$ (316322973*v^5 - 1880954567990*v^4 - 229888629934646*v^3 - 218209443632945729*v^2 - 2705491229165538204*v + 6734423159950735729371) / 7828093866043308345 $$\beta_{2}$$ $$=$$ $$( - 829712739 \nu^{5} - 436403341435 \nu^{4} - 197574368898172 \nu^{3} + \cdots - 23\!\cdots\!88 ) / 15\!\cdots\!90$$ (-829712739*v^5 - 436403341435*v^4 - 197574368898172*v^3 - 81768308016407698*v^2 - 1935016319862991968*v - 2379822241139324515488) / 15656187732086616690 $$\beta_{3}$$ $$=$$ $$( - 214467963392 \nu^{5} + 2208641196555 \nu^{4} + \cdots + 34\!\cdots\!96 ) / 10\!\cdots\!30$$ (-214467963392*v^5 + 2208641196555*v^4 - 45948827507107761*v^3 + 4092725646142317206*v^2 - 1598199332000021247324*v + 34898058588711375127896) / 109593314124606316830 $$\beta_{4}$$ $$=$$ $$( - 29099137708946 \nu^{5} + \cdots + 57\!\cdots\!28 ) / 98\!\cdots\!70$$ (-29099137708946*v^5 + 1226769311590575*v^4 - 6568958189589851583*v^3 + 670986264864802228418*v^2 - 209514530669342298139452*v + 5707822599244199925207528) / 986339827121456851470 $$\beta_{5}$$ $$=$$ $$( 36542487068476 \nu^{5} + \cdots - 31\!\cdots\!08 ) / 98\!\cdots\!70$$ (36542487068476*v^5 - 1253855067805785*v^4 + 11120643582720361623*v^3 - 490346455076467708858*v^2 + 848820538894816245552612*v - 3182286189481356997768008) / 986339827121456851470
 $$\nu$$ $$=$$ $$( \beta_{5} + 8\beta_{4} - 105\beta_{3} + 132\beta_{2} + 12\beta _1 + 108 ) / 672$$ (b5 + 8*b4 - 105*b3 + 132*b2 + 12*b1 + 108) / 672 $$\nu^{2}$$ $$=$$ $$( 43\beta_{5} - 160\beta_{4} + 5285\beta_{3} - 70276\beta_{2} + 7540\beta _1 - 17787180 ) / 224$$ (43*b5 - 160*b4 + 5285*b3 - 70276*b2 + 7540*b1 - 17787180) / 224 $$\nu^{3}$$ $$=$$ $$( 4319\beta_{5} - 435680\beta_{4} + 6648369\beta_{3} - 1152820\beta_{2} - 1586684\beta _1 + 1607875428 ) / 224$$ (4319*b5 - 435680*b4 + 6648369*b3 - 1152820*b2 - 1586684*b1 + 1607875428) / 224 $$\nu^{4}$$ $$=$$ $$( - 6442253 \beta_{5} + 79955072 \beta_{4} - 1576024835 \beta_{3} + 8005050140 \beta_{2} - 1545103820 \beta _1 + 2564427752724 ) / 224$$ (-6442253*b5 + 79955072*b4 - 1576024835*b3 + 8005050140*b2 - 1545103820*b1 + 2564427752724) / 224 $$\nu^{5}$$ $$=$$ $$( - 2655070297 \beta_{5} + 71240832448 \beta_{4} - 1193406700087 \beta_{3} - 1339548073876 \beta_{2} + 438107594404 \beta _1 - 621325934265084 ) / 224$$ (-2655070297*b5 + 71240832448*b4 - 1193406700087*b3 - 1339548073876*b2 + 438107594404*b1 - 621325934265084) / 224

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/7\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
6.1
 −1.79648 + 93.6923i −1.79648 − 93.6923i −50.4579 + 435.079i −50.4579 − 435.079i 52.7544 − 213.185i 52.7544 + 213.185i
−108.657 1048.40i 7710.33 23251.4i 113916.i −98544.2 64267.7i −392722. −567695. 2.52642e6i
6.2 −108.657 1048.40i 7710.33 23251.4i 113916.i −98544.2 + 64267.7i −392722. −567695. 2.52642e6i
6.3 17.4333 949.924i −3792.08 18388.3i 16560.3i −116360. 17366.5i −137515. −370914. 320568.i
6.4 17.4333 949.924i −3792.08 18388.3i 16560.3i −116360. + 17366.5i −137515. −370914. 320568.i
6.5 91.2237 658.189i 4225.75 20289.4i 60042.4i 57579.4 + 102596.i 11836.7 98227.8 1.85087e6i
6.6 91.2237 658.189i 4225.75 20289.4i 60042.4i 57579.4 102596.i 11836.7 98227.8 1.85087e6i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 6.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7.13.b.b 6
3.b odd 2 1 63.13.d.d 6
4.b odd 2 1 112.13.c.b 6
7.b odd 2 1 inner 7.13.b.b 6
7.c even 3 2 49.13.d.b 12
7.d odd 6 2 49.13.d.b 12
21.c even 2 1 63.13.d.d 6
28.d even 2 1 112.13.c.b 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.13.b.b 6 1.a even 1 1 trivial
7.13.b.b 6 7.b odd 2 1 inner
49.13.d.b 12 7.c even 3 2
49.13.d.b 12 7.d odd 6 2
63.13.d.d 6 3.b odd 2 1
63.13.d.d 6 21.c even 2 1
112.13.c.b 6 4.b odd 2 1
112.13.c.b 6 28.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - 10216T_{2} + 172800$$ acting on $$S_{13}^{\mathrm{new}}(7, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{3} - 10216 T + 172800)^{2}$$
$3$ $$T^{6} + 2434704 T^{4} + \cdots + 42\!\cdots\!00$$
$5$ $$T^{6} + 1290415440 T^{4} + \cdots + 75\!\cdots\!00$$
$7$ $$T^{6} + 314650 T^{5} + \cdots + 26\!\cdots\!01$$
$11$ $$(T^{3} - 852390 T^{2} + \cdots + 22\!\cdots\!48)^{2}$$
$13$ $$T^{6} + 65493616180944 T^{4} + \cdots + 33\!\cdots\!00$$
$17$ $$T^{6} + \cdots + 24\!\cdots\!00$$
$19$ $$T^{6} + \cdots + 62\!\cdots\!00$$
$23$ $$(T^{3} + 133367850 T^{2} + \cdots - 23\!\cdots\!00)^{2}$$
$29$ $$(T^{3} - 1045092726 T^{2} + \cdots + 41\!\cdots\!92)^{2}$$
$31$ $$T^{6} + \cdots + 37\!\cdots\!00$$
$37$ $$(T^{3} + 2412433450 T^{2} + \cdots - 10\!\cdots\!00)^{2}$$
$41$ $$T^{6} + \cdots + 49\!\cdots\!00$$
$43$ $$(T^{3} + 14362094650 T^{2} + \cdots + 36\!\cdots\!00)^{2}$$
$47$ $$T^{6} + \cdots + 20\!\cdots\!00$$
$53$ $$(T^{3} + 52260928650 T^{2} + \cdots + 49\!\cdots\!00)^{2}$$
$59$ $$T^{6} + \cdots + 27\!\cdots\!00$$
$61$ $$T^{6} + \cdots + 30\!\cdots\!00$$
$67$ $$(T^{3} + 64873301850 T^{2} + \cdots - 76\!\cdots\!00)^{2}$$
$71$ $$(T^{3} + 7508802330 T^{2} + \cdots - 18\!\cdots\!12)^{2}$$
$73$ $$T^{6} + \cdots + 76\!\cdots\!00$$
$79$ $$(T^{3} + 86263815834 T^{2} + \cdots - 66\!\cdots\!68)^{2}$$
$83$ $$T^{6} + \cdots + 13\!\cdots\!00$$
$89$ $$T^{6} + \cdots + 34\!\cdots\!00$$
$97$ $$T^{6} + \cdots + 30\!\cdots\!00$$