# Properties

 Label 13.12.a.b Level $13$ Weight $12$ Character orbit 13.a Self dual yes Analytic conductor $9.988$ Analytic rank $0$ Dimension $6$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [13,12,Mod(1,13)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(13, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 12, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("13.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$13$$ Weight: $$k$$ $$=$$ $$12$$ Character orbit: $$[\chi]$$ $$=$$ 13.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$9.98846134727$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{5} - 13546x^{4} + 130998x^{3} + 49403509x^{2} - 776207317x - 22123683244$$ x^6 - x^5 - 13546*x^4 + 130998*x^3 + 49403509*x^2 - 776207317*x - 22123683244 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q + (\beta_1 + 9) q^{2} + (\beta_{4} - 2 \beta_1 + 80) q^{3} + (3 \beta_{4} - 5 \beta_{3} + \cdots + 2550) q^{4}+ \cdots + ( - 6 \beta_{5} + 283 \beta_{4} + \cdots + 146078) q^{9}+O(q^{10})$$ q + (b1 + 9) * q^2 + (b4 - 2*b1 + 80) * q^3 + (3*b4 - 5*b3 - 2*b2 + 5*b1 + 2550) * q^4 + (-2*b5 + 11*b4 - 7*b3 + 11*b2 + 36*b1 + 547) * q^5 + (13*b5 - 25*b4 + 67*b3 - 42*b2 + 212*b1 - 9652) * q^6 + (-36*b5 + 3*b4 + 2*b3 + 86*b2 + 238*b1 - 740) * q^7 + (58*b5 - 85*b4 - 85*b3 - 106*b2 + 1983*b1 + 26038) * q^8 + (-6*b5 + 283*b4 - 303*b3 + 387*b2 - 296*b1 + 146078) * q^9 $$q + (\beta_1 + 9) q^{2} + (\beta_{4} - 2 \beta_1 + 80) q^{3} + (3 \beta_{4} - 5 \beta_{3} + \cdots + 2550) q^{4}+ \cdots + (3379140 \beta_{5} + \cdots - 39305917264) q^{99}+O(q^{100})$$ q + (b1 + 9) * q^2 + (b4 - 2*b1 + 80) * q^3 + (3*b4 - 5*b3 - 2*b2 + 5*b1 + 2550) * q^4 + (-2*b5 + 11*b4 - 7*b3 + 11*b2 + 36*b1 + 547) * q^5 + (13*b5 - 25*b4 + 67*b3 - 42*b2 + 212*b1 - 9652) * q^6 + (-36*b5 + 3*b4 + 2*b3 + 86*b2 + 238*b1 - 740) * q^7 + (58*b5 - 85*b4 - 85*b3 - 106*b2 + 1983*b1 + 26038) * q^8 + (-6*b5 + 283*b4 - 303*b3 + 387*b2 - 296*b1 + 146078) * q^9 + (-165*b5 - 717*b4 + 143*b3 - 682*b2 + 1324*b1 + 165810) * q^10 + (-176*b5 - 352*b4 + 374*b3 - 286*b2 - 1936*b1 + 46144) * q^11 + (1240*b5 + 2395*b4 + 547*b3 + 678*b2 - 14447*b1 + 661770) * q^12 - 371293 * q^13 + (-3217*b5 - 2639*b4 - 1067*b3 + 682*b2 - 15022*b1 + 1077830) * q^14 + (592*b5 + 3435*b4 + 1708*b3 + 2820*b2 - 32358*b1 + 999148) * q^15 + (3342*b5 + 3351*b4 - 8025*b3 + 942*b2 + 28599*b1 + 4167390) * q^16 + (1122*b5 - 1683*b4 + 8283*b3 - 4719*b2 - 29108*b1 + 3085713) * q^17 + (-4109*b5 - 29677*b4 + 5071*b3 - 22506*b2 + 178415*b1 + 218837) * q^18 + (5436*b5 + 3708*b4 - 3590*b3 - 1826*b2 + 96476*b1 + 381968) * q^19 + (-1452*b5 + 27429*b4 - 30579*b3 + 41994*b2 + 65275*b1 + 6291462) * q^20 + (-9982*b5 + 2331*b4 + 39365*b3 + 6495*b2 - 160272*b1 - 11234515) * q^21 + (-7392*b5 + 22836*b4 + 3652*b3 + 45496*b2 - 17986*b1 - 8764358) * q^22 + (-5308*b5 + 19924*b4 - 57656*b3 - 13400*b2 - 110396*b1 + 4171928) * q^23 + (56626*b5 - 57969*b4 + 111823*b3 - 133314*b2 + 433323*b1 - 40551434) * q^24 + (-15546*b5 - 27129*b4 - 21063*b3 - 55509*b2 - 465468*b1 - 2284944) * q^25 + (-371293*b1 - 3341637) * q^26 + (15396*b5 + 136123*b4 - 85260*b3 + 108396*b2 - 1448042*b1 + 13136012) * q^27 + (-136224*b5 - 39621*b4 - 160765*b3 + 270710*b2 + 303793*b1 - 56946286) * q^28 + (33836*b5 - 35648*b4 - 149216*b3 - 33392*b2 + 209764*b1 - 23903614) * q^29 + (31439*b5 - 238219*b4 + 449417*b3 - 221214*b2 + 884372*b1 - 139871244) * q^30 + (88872*b5 + 53226*b4 + 121118*b3 + 111866*b2 - 248156*b1 - 26285292) * q^31 + (51790*b5 - 59749*b4 - 99445*b3 - 362314*b2 + 1736739*b1 + 122963542) * q^32 + (-88176*b5 - 97406*b4 + 106854*b3 + 37026*b2 + 503692*b1 - 19271794) * q^33 + (139293*b5 + 339597*b4 + 420529*b3 + 164938*b2 + 2299628*b1 - 111573682) * q^34 + (-172640*b5 - 284257*b4 - 38572*b3 - 419812*b2 - 303646*b1 + 229414840) * q^35 + (-181148*b5 + 1589590*b4 - 1847642*b3 + 1213716*b2 - 1361198*b1 + 516667992) * q^36 + (-68022*b5 - 593697*b4 + 926681*b3 - 103525*b2 + 245056*b1 + 8057891) * q^37 + (340276*b5 + 176912*b4 - 378256*b3 - 855952*b2 + 1495810*b1 + 445243582) * q^38 + (-371293*b4 + 742586*b1 - 29703440) * q^39 + (-183738*b5 - 1335879*b4 - 321831*b3 - 1006254*b2 + 5878245*b1 + 38918138) * q^40 + (401112*b5 + 221010*b4 + 336438*b3 + 1254258*b2 - 5859452*b1 + 19902168) * q^41 + (-461117*b5 + 242635*b4 + 2615335*b3 + 1101414*b2 - 18198782*b1 - 876529992) * q^42 + (778212*b5 - 969141*b4 + 287624*b3 - 564472*b2 + 3420790*b1 + 232531052) * q^43 + (-474760*b5 - 1504022*b4 + 845146*b3 - 879932*b2 - 9209986*b1 - 260759596) * q^44 + (-541340*b5 + 1581600*b4 - 2619260*b3 + 265980*b2 - 12877620*b1 + 1291053790) * q^45 + (48444*b5 - 1575204*b4 - 1005460*b3 + 57560*b2 + 18560896*b1 - 454181032) * q^46 + (-862840*b5 + 926935*b4 - 54098*b3 + 989098*b2 + 4953834*b1 - 564611440) * q^47 + (2139990*b5 + 5474739*b4 - 1336125*b3 - 1448154*b2 - 18040893*b1 + 186540150) * q^48 + (-220206*b5 - 61425*b4 - 314043*b3 - 3402465*b2 + 24482808*b1 + 1194014170) * q^49 + (-249513*b5 + 946071*b4 - 504381*b3 + 5103486*b2 + 7157385*b1 - 2142522777) * q^50 + (-1299104*b5 + 2631437*b4 - 2901824*b3 + 165216*b2 + 21235142*b1 + 1489414640) * q^51 + (-1113879*b4 + 1856465*b3 + 742586*b2 - 1856465*b1 - 946797150) * q^52 + (1828456*b5 + 1792394*b4 + 3210302*b3 - 366454*b2 + 1213892*b1 + 92930260) * q^53 + (440887*b5 - 13583179*b4 + 11657065*b3 - 7863198*b2 + 35316812*b1 - 6401791612) * q^54 + (127932*b5 + 2256870*b4 + 1077028*b3 + 4139740*b2 + 25992392*b1 - 1303787152) * q^55 + (-5822222*b5 - 7368265*b4 - 9512905*b3 + 5808206*b2 - 47940109*b1 - 1133252810) * q^56 + (2735212*b5 - 723126*b4 + 1050502*b3 - 5070606*b2 + 27674736*b1 + 149042134) * q^57 + (1348392*b5 - 1223904*b4 - 9523456*b3 - 1525888*b2 + 1878898*b1 + 948777530) * q^58 + (1901160*b5 - 2637504*b4 + 2317086*b3 + 5483706*b2 + 2624824*b1 + 1677744960) * q^59 + (2371576*b5 + 20292401*b4 - 1592375*b3 + 6983346*b2 - 150211741*b1 + 440501054) * q^60 + (-5161776*b5 + 6345606*b4 - 1663798*b3 - 1283410*b2 - 34676732*b1 + 2628450024) * q^61 + (3597170*b5 - 3758678*b4 + 11038162*b3 - 13327820*b2 - 52202654*b1 - 1293186130) * q^62 + (-4504792*b5 - 9361248*b4 - 6093742*b3 - 3905130*b2 + 26282424*b1 + 4969079756) * q^63 + (829350*b5 + 11592399*b4 - 646465*b3 + 1491614*b2 + 119949031*b1 + 356676126) * q^64 + (742586*b5 - 4084223*b4 + 2599051*b3 - 4084223*b2 - 13366548*b1 - 203097271) * q^65 + (-5983958*b5 + 4955954*b4 - 4121030*b3 + 11365572*b2 - 59502556*b1 + 2014503584) * q^66 + (598596*b5 - 3056796*b4 - 2251246*b3 + 11492870*b2 + 33098212*b1 + 8417159624) * q^67 + (7539196*b5 + 6947627*b4 + 11846147*b3 - 27654250*b2 - 109082203*b1 + 2594502202) * q^68 + (10254036*b5 + 6854708*b4 + 1480188*b3 + 12675492*b2 - 187564684*b1 + 1848942532) * q^69 + (-5306253*b5 + 21601137*b4 - 19578635*b3 + 41439802*b2 + 255989048*b1 + 444533128) * q^70 + (-29692*b5 - 30103631*b4 + 7840990*b3 - 7765622*b2 - 94159670*b1 - 432755236) * q^71 + (-6153596*b5 - 68554974*b4 + 5531842*b3 - 41805852*b2 + 487935114*b1 - 1556813396) * q^72 + (-10208196*b5 - 12012804*b4 + 17868044*b3 - 31958476*b2 + 76542940*b1 + 687275734) * q^73 + (-6579193*b5 + 39767071*b4 + 5575819*b3 + 38031118*b2 - 210626004*b1 + 890864522) * q^74 + (1430400*b5 - 11090724*b4 - 2044824*b3 + 16456536*b2 - 119350392*b1 - 1266868440) * q^75 + (21597144*b5 + 17468370*b4 - 5889086*b3 - 19238732*b2 + 459973862*b1 + 9918921652) * q^76 + (5124584*b5 + 25381606*b4 + 9312730*b3 + 17065534*b2 + 204100428*b1 + 1481040278) * q^77 + (-4826809*b5 + 9282325*b4 - 24876631*b3 + 15594306*b2 - 78714116*b1 + 3583720036) * q^78 + (7883832*b5 - 18834912*b4 + 7081348*b3 - 26079764*b2 + 200836520*b1 - 15390231688) * q^79 + (-7438806*b5 + 20971173*b4 - 62432811*b3 + 7215498*b2 - 92529147*b1 + 14903928666) * q^80 + (-9387540*b5 + 70426144*b4 - 88062732*b3 + 29709612*b2 - 653802716*b1 + 11302986437) * q^81 + (2615658*b5 - 66796734*b4 + 65209546*b3 - 69694364*b2 - 151381858*b1 - 25108004518) * q^82 + (-21984700*b5 - 22134434*b4 + 5903266*b3 - 34055690*b2 - 213383824*b1 + 3433205480) * q^83 + (-10336492*b5 - 41449313*b4 + 140126615*b3 + 33717966*b2 - 1058559215*b1 - 69684056390) * q^84 + (3022074*b5 + 63677937*b4 + 17812883*b3 + 64114457*b2 + 173119180*b1 - 20683834449) * q^85 + (36770663*b5 + 55118941*b4 - 52534991*b3 - 23274350*b2 + 155123760*b1 + 19387212080) * q^86 + (36047172*b5 - 27135334*b4 + 4736640*b3 - 32960880*b2 - 221708104*b1 - 26830773320) * q^87 + (-10657188*b5 + 13757994*b4 - 16989206*b3 + 65524180*b2 - 406654702*b1 - 26371317292) * q^88 + (-16087552*b5 + 8418052*b4 + 71831428*b3 - 2847380*b2 + 327168696*b1 + 3392912798) * q^89 + (-18885640*b5 - 142363240*b4 + 31515800*b3 - 6101040*b2 + 1926245570*b1 - 46871086590) * q^90 + (13366548*b5 - 1113879*b4 - 742586*b3 - 31931198*b2 - 88367734*b1 + 274756820) * q^91 + (-11179568*b5 + 17068532*b4 - 109051756*b3 + 54525800*b2 - 334359476*b1 + 74398837912) * q^92 + (-26587108*b5 + 50630444*b4 - 138948616*b3 - 6459816*b2 + 394436828*b1 + 23599677120) * q^93 + (-48068805*b5 - 40341531*b4 + 19107769*b3 + 6186898*b2 - 628407070*b1 + 15521654022) * q^94 + (-4735528*b5 - 71894762*b4 + 3911956*b3 - 91540772*b2 - 221179364*b1 + 13766970128) * q^95 + (82811302*b5 - 24150969*b4 + 132320887*b3 - 117468402*b2 + 550048527*b1 - 483755858) * q^96 + (40239900*b5 + 116546820*b4 - 101502860*b3 + 90656812*b2 - 514515988*b1 - 33892937202) * q^97 + (45090951*b5 + 207366519*b4 - 152676861*b3 + 63276126*b2 + 1617642943*b1 + 118319290461) * q^98 + (3379140*b5 - 13992602*b4 + 56198214*b3 + 34729506*b2 + 514982368*b1 - 39305917264) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 55 q^{2} + 476 q^{3} + 15309 q^{4} + 3312 q^{5} - 57797 q^{6} - 4176 q^{7} + 158493 q^{8} + 876218 q^{9}+O(q^{10})$$ 6 * q + 55 * q^2 + 476 * q^3 + 15309 * q^4 + 3312 * q^5 - 57797 * q^6 - 4176 * q^7 + 158493 * q^8 + 876218 * q^9 $$6 q + 55 q^{2} + 476 q^{3} + 15309 q^{4} + 3312 q^{5} - 57797 q^{6} - 4176 q^{7} + 158493 q^{8} + 876218 q^{9} + 997497 q^{10} + 275060 q^{11} + 3949049 q^{12} - 2227758 q^{13} + 6462587 q^{14} + 5951652 q^{15} + 25038945 q^{16} + 18470848 q^{17} + 1544758 q^{18} + 2382612 q^{19} + 37821799 q^{20} - 67640772 q^{21} - 52649718 q^{22} + 25001944 q^{23} - 243039615 q^{24} - 14063202 q^{25} - 20421115 q^{26} + 77250908 q^{27} - 340836927 q^{28} - 142876028 q^{29} - 838796927 q^{30} - 158397468 q^{31} + 739784589 q^{32} - 115057792 q^{33} - 668802009 q^{34} + 1377003692 q^{35} + 3099344006 q^{36} + 47994456 q^{37} + 2673019714 q^{38} - 176735468 q^{39} + 242886231 q^{40} + 112037548 q^{41} - 5282633557 q^{42} + 1399191924 q^{43} - 1571975050 q^{44} + 7736061780 q^{45} - 2701412412 q^{46} - 3383597640 q^{47} + 1090782789 q^{48} + 7189538970 q^{49} - 12848613144 q^{50} + 8959562860 q^{51} - 5684124537 q^{52} + 546961604 q^{53} - 38372021519 q^{54} - 7803526248 q^{55} - 6807872407 q^{56} + 918537576 q^{57} + 5714690406 q^{58} + 10067834260 q^{59} + 2453022955 q^{60} + 15731821572 q^{61} - 7829475572 q^{62} + 29876175732 q^{63} + 2237284569 q^{64} - 1229722416 q^{65} + 12031833058 q^{66} + 50546073444 q^{67} + 15412804265 q^{68} + 10879166680 q^{69} + 2924449065 q^{70} - 2646136112 q^{71} - 8720745402 q^{72} + 4198695060 q^{73} + 5050454541 q^{74} - 7695720336 q^{75} + 59928748062 q^{76} + 9015828840 q^{77} + 21459621521 q^{78} - 92124930312 q^{79} + 89421404931 q^{80} + 67208776622 q^{81} - 150798850248 q^{82} + 20440296092 q^{83} - 419349915667 q^{84} - 124095891228 q^{85} + 116436457677 q^{86} - 161197597808 q^{87} - 158617438842 q^{88} + 20540234076 q^{89} - 279059693450 q^{90} + 1550519568 q^{91} + 446253814012 q^{92} + 142195723000 q^{93} + 92592053391 q^{94} + 82521342544 q^{95} - 2651637759 q^{96} - 203942467020 q^{97} + 711378915442 q^{98} - 235408311580 q^{99}+O(q^{100})$$ 6 * q + 55 * q^2 + 476 * q^3 + 15309 * q^4 + 3312 * q^5 - 57797 * q^6 - 4176 * q^7 + 158493 * q^8 + 876218 * q^9 + 997497 * q^10 + 275060 * q^11 + 3949049 * q^12 - 2227758 * q^13 + 6462587 * q^14 + 5951652 * q^15 + 25038945 * q^16 + 18470848 * q^17 + 1544758 * q^18 + 2382612 * q^19 + 37821799 * q^20 - 67640772 * q^21 - 52649718 * q^22 + 25001944 * q^23 - 243039615 * q^24 - 14063202 * q^25 - 20421115 * q^26 + 77250908 * q^27 - 340836927 * q^28 - 142876028 * q^29 - 838796927 * q^30 - 158397468 * q^31 + 739784589 * q^32 - 115057792 * q^33 - 668802009 * q^34 + 1377003692 * q^35 + 3099344006 * q^36 + 47994456 * q^37 + 2673019714 * q^38 - 176735468 * q^39 + 242886231 * q^40 + 112037548 * q^41 - 5282633557 * q^42 + 1399191924 * q^43 - 1571975050 * q^44 + 7736061780 * q^45 - 2701412412 * q^46 - 3383597640 * q^47 + 1090782789 * q^48 + 7189538970 * q^49 - 12848613144 * q^50 + 8959562860 * q^51 - 5684124537 * q^52 + 546961604 * q^53 - 38372021519 * q^54 - 7803526248 * q^55 - 6807872407 * q^56 + 918537576 * q^57 + 5714690406 * q^58 + 10067834260 * q^59 + 2453022955 * q^60 + 15731821572 * q^61 - 7829475572 * q^62 + 29876175732 * q^63 + 2237284569 * q^64 - 1229722416 * q^65 + 12031833058 * q^66 + 50546073444 * q^67 + 15412804265 * q^68 + 10879166680 * q^69 + 2924449065 * q^70 - 2646136112 * q^71 - 8720745402 * q^72 + 4198695060 * q^73 + 5050454541 * q^74 - 7695720336 * q^75 + 59928748062 * q^76 + 9015828840 * q^77 + 21459621521 * q^78 - 92124930312 * q^79 + 89421404931 * q^80 + 67208776622 * q^81 - 150798850248 * q^82 + 20440296092 * q^83 - 419349915667 * q^84 - 124095891228 * q^85 + 116436457677 * q^86 - 161197597808 * q^87 - 158617438842 * q^88 + 20540234076 * q^89 - 279059693450 * q^90 + 1550519568 * q^91 + 446253814012 * q^92 + 142195723000 * q^93 + 92592053391 * q^94 + 82521342544 * q^95 - 2651637759 * q^96 - 203942467020 * q^97 + 711378915442 * q^98 - 235408311580 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} - 13546x^{4} + 130998x^{3} + 49403509x^{2} - 776207317x - 22123683244$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -1925\nu^{5} - 86014\nu^{4} + 15916032\nu^{3} + 462431442\nu^{2} - 26906396267\nu - 114096339268 ) / 502193280$$ (-1925*v^5 - 86014*v^4 + 15916032*v^3 + 462431442*v^2 - 26906396267*v - 114096339268) / 502193280 $$\beta_{3}$$ $$=$$ $$( 4549 \nu^{5} + 297830 \nu^{4} - 39656088 \nu^{3} - 2035001754 \nu^{2} + 79116342019 \nu + 1739047527524 ) / 1004386560$$ (4549*v^5 + 297830*v^4 - 39656088*v^3 - 2035001754*v^2 + 79116342019*v + 1739047527524) / 1004386560 $$\beta_{4}$$ $$=$$ $$( 5015 \nu^{5} + 381698 \nu^{4} - 44872104 \nu^{3} - 2440298814 \nu^{2} + 100337716769 \nu + 1234012729676 ) / 1004386560$$ (5015*v^5 + 381698*v^4 - 44872104*v^3 - 2440298814*v^2 + 100337716769*v + 1234012729676) / 1004386560 $$\beta_{5}$$ $$=$$ $$( 1745\nu^{5} + 170366\nu^{4} - 12096168\nu^{3} - 1100201058\nu^{2} + 15895857863\nu + 715844708372 ) / 251096640$$ (1745*v^5 + 170366*v^4 - 12096168*v^3 - 1100201058*v^2 + 15895857863*v + 715844708372) / 251096640
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$3\beta_{4} - 5\beta_{3} - 2\beta_{2} - 13\beta _1 + 4517$$ 3*b4 - 5*b3 - 2*b2 - 13*b1 + 4517 $$\nu^{3}$$ $$=$$ $$58\beta_{5} - 166\beta_{4} + 50\beta_{3} - 52\beta_{2} + 6187\beta _1 - 59786$$ 58*b5 - 166*b4 + 50*b3 - 52*b2 + 6187*b1 - 59786 $$\nu^{4}$$ $$=$$ $$1254\beta_{5} + 26301\beta_{4} - 38115\beta_{3} - 8502\beta_{2} - 160011\beta _1 + 28173671$$ 1254*b5 + 26301*b4 - 38115*b3 - 8502*b2 - 160011*b1 + 28173671 $$\nu^{5}$$ $$=$$ $$423516\beta_{5} - 1827024\beta_{4} + 915360\beta_{3} - 791376\beta_{2} + 41203985\beta _1 - 727365996$$ 423516*b5 - 1827024*b4 + 915360*b3 - 791376*b2 + 41203985*b1 - 727365996

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −89.6176 −79.8045 −15.2405 36.5093 73.0961 76.0571
−80.6176 827.494 4451.20 7679.45 −66710.6 −6800.32 −193740. 507599. −619099.
1.2 −70.8045 −19.0788 2965.27 −11885.7 1350.86 −62220.3 −64947.1 −176783. 841561.
1.3 −6.24050 −573.204 −2009.06 −3613.82 3577.08 14401.4 25318.0 151416. 22552.1
1.4 45.5093 469.277 23.0963 4406.29 21356.5 63013.3 −92151.9 43074.3 200527.
1.5 82.0961 −696.954 4691.77 6795.70 −57217.2 67467.4 217044. 308598. 557901.
1.6 85.0571 468.466 5186.72 −69.9037 39846.4 −80037.3 266970. 42313.3 −5945.81
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$13$$ $$+1$$

## 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 13.12.a.b 6
3.b odd 2 1 117.12.a.d 6
4.b odd 2 1 208.12.a.h 6
13.b even 2 1 169.12.a.c 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.12.a.b 6 1.a even 1 1 trivial
117.12.a.d 6 3.b odd 2 1
169.12.a.c 6 13.b even 2 1
208.12.a.h 6 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} - 55T_{2}^{5} - 12286T_{2}^{4} + 603264T_{2}^{3} + 39388912T_{2}^{2} - 1594524928T_{2} - 11319915520$$ acting on $$S_{12}^{\mathrm{new}}(\Gamma_0(13))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + \cdots - 11319915520$$
$3$ $$T^{6} + \cdots - 13\!\cdots\!12$$
$5$ $$T^{6} + \cdots - 69\!\cdots\!00$$
$7$ $$T^{6} + \cdots - 20\!\cdots\!96$$
$11$ $$T^{6} + \cdots - 18\!\cdots\!92$$
$13$ $$(T + 371293)^{6}$$
$17$ $$T^{6} + \cdots - 14\!\cdots\!04$$
$19$ $$T^{6} + \cdots - 20\!\cdots\!84$$
$23$ $$T^{6} + \cdots + 21\!\cdots\!88$$
$29$ $$T^{6} + \cdots + 82\!\cdots\!12$$
$31$ $$T^{6} + \cdots - 54\!\cdots\!60$$
$37$ $$T^{6} + \cdots - 36\!\cdots\!32$$
$41$ $$T^{6} + \cdots + 89\!\cdots\!16$$
$43$ $$T^{6} + \cdots + 38\!\cdots\!60$$
$47$ $$T^{6} + \cdots + 46\!\cdots\!20$$
$53$ $$T^{6} + \cdots + 24\!\cdots\!72$$
$59$ $$T^{6} + \cdots - 28\!\cdots\!04$$
$61$ $$T^{6} + \cdots + 32\!\cdots\!92$$
$67$ $$T^{6} + \cdots - 12\!\cdots\!84$$
$71$ $$T^{6} + \cdots - 32\!\cdots\!16$$
$73$ $$T^{6} + \cdots - 73\!\cdots\!68$$
$79$ $$T^{6} + \cdots - 25\!\cdots\!96$$
$83$ $$T^{6} + \cdots - 50\!\cdots\!72$$
$89$ $$T^{6} + \cdots + 28\!\cdots\!00$$
$97$ $$T^{6} + \cdots - 13\!\cdots\!00$$