# Properties

 Label 13.10.b.a Level $13$ Weight $10$ Character orbit 13.b Analytic conductor $6.695$ Analytic rank $0$ Dimension $10$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [13,10,Mod(12,13)]

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([1]))

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("13.12");

S:= CuspForms(chi, 10);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$13$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 13.b (of order $$2$$, degree $$1$$, minimal)

## 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: no Analytic conductor: $$6.69546587013$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{10} + 3841x^{8} + 5134480x^{6} + 2823572208x^{4} + 614223235584x^{2} + 43308450164736$$ x^10 + 3841*x^8 + 5134480*x^6 + 2823572208*x^4 + 614223235584*x^2 + 43308450164736 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{7}\cdot 3^{4}\cdot 13^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{2} - 256) q^{4} + (\beta_{5} - 6 \beta_1) q^{5} + (\beta_{8} - 6 \beta_1) q^{6} + (\beta_{9} - \beta_{5} - 22 \beta_1) q^{7} + ( - \beta_{9} - \beta_{8} + \cdots - 144 \beta_1) q^{8}+ \cdots + ( - \beta_{4} - 33 \beta_{3} + \cdots + 6727) q^{9}+O(q^{10})$$ q + b1 * q^2 + b3 * q^3 + (b2 - 256) * q^4 + (b5 - 6*b1) * q^5 + (b8 - 6*b1) * q^6 + (b9 - b5 - 22*b1) * q^7 + (-b9 - b8 + b7 - 144*b1) * q^8 + (-b4 - 33*b3 + 16*b2 + 6727) * q^9 $$q + \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{2} - 256) q^{4} + (\beta_{5} - 6 \beta_1) q^{5} + (\beta_{8} - 6 \beta_1) q^{6} + (\beta_{9} - \beta_{5} - 22 \beta_1) q^{7} + ( - \beta_{9} - \beta_{8} + \cdots - 144 \beta_1) q^{8}+ \cdots + ( - 45438 \beta_{9} + \cdots + 7727926 \beta_1) q^{99}+O(q^{100})$$ q + b1 * q^2 + b3 * q^3 + (b2 - 256) * q^4 + (b5 - 6*b1) * q^5 + (b8 - 6*b1) * q^6 + (b9 - b5 - 22*b1) * q^7 + (-b9 - b8 + b7 - 144*b1) * q^8 + (-b4 - 33*b3 + 16*b2 + 6727) * q^9 + (-b6 + b4 - 26*b3 - 11*b2 + 4549) * q^10 + (-2*b9 + 2*b8 + 3*b7 - 8*b5 - 458*b1) * q^11 + (-b6 - 86*b3 + 31*b2 + 4619) * q^12 + (2*b9 + 8*b8 + 5*b7 + b6 + 7*b5 + b4 + 63*b3 - 48*b2 - 994*b1 - 11146) * q^13 + (2*b6 - 7*b4 - 24*b3 + 131*b2 + 16956) * q^14 + (27*b9 + 14*b8 - 13*b7 + 63*b5 + 988*b1) * q^15 + (-b6 + 6*b4 + 410*b3 - 121*b2 - 20889) * q^16 + (2*b6 + 7*b4 + 283*b3 + 72*b2 - 118898) * q^17 + (-78*b9 - 65*b8 + 14*b7 + 72*b5 + 691*b1) * q^18 + (-28*b9 + 166*b8 - 35*b7 - 62*b5 + 6582*b1) * q^19 + (89*b9 - 157*b8 - 17*b7 - 520*b5 + 6080*b1) * q^20 + (-36*b9 - 280*b8 - 14*b7 + 393*b5 - 118*b1) * q^21 + (b6 + 4*b4 - 1602*b3 - 1552*b2 + 351045) * q^22 + (16*b6 - 12*b4 + 2110*b3 - 1192*b2 + 221272) * q^23 + (-15*b9 + 237*b8 + 23*b7 - 960*b5 - 10124*b1) * q^24 + (16*b6 - 7*b4 - 6079*b3 + 3448*b2 - 1011461) * q^25 + (94*b9 + 285*b8 - 38*b7 - 18*b6 + 888*b5 - 5*b4 - 6256*b3 - 1957*b2 + 7297*b1 + 761084) * q^26 + (-36*b6 + 100*b4 + 10599*b3 + 488*b2 - 785776) * q^27 + (-85*b9 + 49*b8 + 133*b7 + 1912*b5 - 45828*b1) * q^28 + (18*b6 - 94*b4 + 18010*b3 + 616*b2 + 707644) * q^29 + (-37*b6 - 99*b4 - 8266*b3 + 9139*b2 - 757159) * q^30 + (454*b9 + 106*b8 - 119*b7 + 248*b5 + 73678*b1) * q^31 + (-3*b9 - 43*b8 + 395*b7 - 1392*b5 - 49460*b1) * q^32 + (-660*b9 - 8*b8 + 216*b7 - 2568*b5 + 64560*b1) * q^33 + (330*b9 + 639*b8 + 102*b7 + 1416*b5 - 150984*b1) * q^34 + (-20*b6 + 84*b4 - 33941*b3 - 16192*b2 + 3939328) * q^35 + (-99*b6 + 28*b4 + 20670*b3 - 9682*b2 + 2902937) * q^36 + (544*b9 - 496*b8 - 604*b7 - 409*b5 - 163554*b1) * q^37 + (-97*b6 + 106*b4 - 87926*b3 + 19530*b2 - 5035625) * q^38 + (-963*b9 - 1382*b8 + 17*b7 + 136*b6 - 231*b5 - 72*b4 - 49074*b3 - 11416*b2 + 226996*b1 + 1438108) * q^39 + (271*b6 - 542*b4 + 93690*b3 + 15579*b2 - 2304561) * q^40 + (-996*b9 - 1216*b8 - 448*b7 - 1048*b5 - 67464*b1) * q^41 + (-135*b6 + 609*b4 + 162354*b3 - 13515*b2 + 69879) * q^42 + (288*b6 + 536*b4 + 11041*b3 + 2632*b2 - 9230760) * q^43 + (760*b9 + 1196*b8 - 3424*b5 + 746084*b1) * q^44 + (1692*b9 - 6096*b8 - 1482*b7 + 3222*b5 + 345156*b1) * q^45 + (192*b9 + 5638*b8 - 1088*b7 + 16224*b5 + 683996*b1) * q^46 + (-655*b9 + 5784*b8 - 532*b7 - 11657*b5 - 808566*b1) * q^47 + (173*b6 - 870*b4 - 165522*b3 + 10105*b2 + 10190261) * q^48 + (242*b6 + 301*b4 + 122305*b3 - 8544*b2 - 10048741) * q^49 + (-4138*b9 - 7111*b8 + 3562*b7 + 15864*b5 - 2356433*b1) * q^50 + (204*b6 - 636*b4 - 270105*b3 - 17712*b2 + 7739328) * q^51 + (2959*b9 - 3127*b8 + 449*b7 - 529*b6 - 13336*b5 + 836*b4 - 156918*b3 + 14290*b2 + 1077156*b1 - 11344925) * q^52 + (-1040*b6 + 1088*b4 + 184100*b3 - 33528*b2 + 2981078) * q^53 + (6288*b9 + 6023*b8 + 400*b7 - 41760*b5 - 1053502*b1) * q^54 + (512*b6 - 2004*b4 + 248768*b3 + 52280*b2 + 13919168) * q^55 + (-1155*b6 - 1162*b4 - 118706*b3 - 39515*b2 + 43712221) * q^56 + (-10800*b9 - 728*b8 + 3536*b7 + 10398*b5 + 2906116*b1) * q^57 + (-6732*b9 + 18734*b8 + 572*b7 + 24048*b5 + 364954*b1) * q^58 + (9736*b9 - 13170*b8 + 2653*b7 + 11162*b5 + 1849718*b1) * q^59 + (-861*b9 - 17667*b8 + 1989*b7 + 3864*b5 - 3832992*b1) * q^60 + (-816*b6 + 3258*b4 + 17598*b3 + 127384*b2 - 23808422) * q^61 + (219*b6 - 2476*b4 - 64214*b3 + 179728*b2 - 56550473) * q^62 + (14520*b9 + 31654*b8 - 8197*b7 + 13914*b5 - 5052770*b1) * q^63 + (525*b6 + 1698*b4 + 177966*b3 - 226567*b2 + 27214949) * q^64 + (-2432*b9 + 23136*b8 + 238*b7 + 916*b6 - 10150*b5 - 3465*b4 + 37207*b3 + 131168*b2 + 3426764*b1 - 35103722) * q^65 + (1700*b6 + 1392*b4 + 53144*b3 - 86240*b2 - 49517068) * q^66 + (-9786*b9 - 16438*b8 - 553*b7 + 74100*b5 - 2597906*b1) * q^67 + (-803*b6 + 3020*b4 - 314818*b3 - 79725*b2 + 54963257) * q^68 + (2040*b6 - 378*b4 + 58866*b3 - 163608*b2 + 50486892) * q^69 + (21720*b9 - 19565*b8 - 16184*b7 - 25248*b5 + 10610090*b1) * q^70 + (253*b9 - 7920*b8 + 9988*b7 - 73845*b5 - 371726*b1) * q^71 + (-26934*b9 - 18122*b8 - 3250*b7 - 60192*b5 + 7021948*b1) * q^72 + (-780*b9 - 19176*b8 - 4692*b7 - 4746*b5 - 6589476*b1) * q^73 + (2053*b6 - 3673*b4 + 423794*b3 + 84267*b2 + 125868479) * q^74 + (-2500*b6 + 7476*b4 - 213560*b3 - 153392*b2 - 144411688) * q^75 + (-25742*b9 - 36094*b8 + 1046*b7 - 132496*b5 - 8947020*b1) * q^76 + (-3790*b6 - 616*b4 + 815928*b3 - 83416*b2 + 53237118) * q^77 + (4776*b9 - 17322*b8 - 10472*b7 + 633*b6 + 135744*b5 + 5547*b4 + 876546*b3 + 29325*b2 + 6281936*b1 - 174342213) * q^78 + (1720*b6 - 9864*b4 - 1727378*b3 + 33936*b2 - 44507008) * q^79 + (-7951*b9 + 31873*b8 + 7959*b7 + 32944*b5 - 5953092*b1) * q^80 + (-936*b6 + 296*b4 - 2091696*b3 + 192760*b2 + 148298881) * q^81 + (1716*b6 + 4928*b4 + 910840*b3 - 118432*b2 + 52037668) * q^82 + (15104*b9 + 51126*b8 + 24833*b7 + 82574*b5 + 3150398*b1) * q^83 + (35001*b9 + 20923*b8 - 20545*b7 + 27768*b5 + 4369000*b1) * q^84 + (8532*b9 + 37704*b8 + 3774*b7 - 165357*b5 + 6624582*b1) * q^85 + (25992*b9 + 62489*b8 + 6008*b7 + 237888*b5 - 10500126*b1) * q^86 + (-1272*b6 - 10668*b4 + 1931184*b3 + 168648*b2 + 479744424) * q^87 + (3500*b6 - 5936*b4 - 1484408*b3 + 125312*b2 - 393041540) * q^88 + (24440*b9 + 30168*b8 + 25256*b7 - 22334*b5 - 12686916*b1) * q^89 + (6048*b6 - 6930*b4 + 3829752*b3 + 804438*b2 - 264746916) * q^90 + (-47476*b9 - 36946*b8 + 29393*b7 - 6032*b6 - 65546*b5 + 5096*b4 + 1897025*b3 - 654264*b2 + 737438*b1 - 77484160) * q^91 + (-12390*b6 + 8928*b4 - 2463684*b3 + 560346*b2 - 412480734) * q^92 + (6480*b9 - 45664*b8 - 15188*b7 + 281688*b5 + 1072376*b1) * q^93 + (5750*b6 - 7727*b4 - 2996384*b3 - 471485*b2 + 621940624) * q^94 + (-15264*b6 + 1776*b4 - 1468830*b3 + 1238928*b2 + 104740800) * q^95 + (-74493*b9 - 40869*b8 + 21525*b7 - 262800*b5 + 2065476*b1) * q^96 + (-3980*b9 - 207960*b8 - 10452*b7 - 121588*b5 + 1489880*b1) * q^97 + (23334*b9 + 173901*b8 - 6006*b7 + 210648*b5 - 7466483*b1) * q^98 + (-45438*b9 + 149530*b8 - 5797*b7 - 399168*b5 + 7727926*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q - 2 q^{3} - 2562 q^{4} + 67304 q^{9}+O(q^{10})$$ 10 * q - 2 * q^3 - 2562 * q^4 + 67304 * q^9 $$10 q - 2 q^{3} - 2562 q^{4} + 67304 q^{9} + 45562 q^{10} + 46298 q^{12} - 111488 q^{13} + 169350 q^{14} - 209470 q^{16} - 1189686 q^{17} + 3516760 q^{22} + 2210916 q^{23} - 10109316 q^{25} + 7627230 q^{26} - 7880006 q^{27} + 7039224 q^{29} - 7573410 q^{30} + 39493506 q^{35} + 29007196 q^{36} - 50219652 q^{38} + 14502332 q^{39} - 23263606 q^{40} + 400842 q^{42} - 92334370 q^{43} + 102213790 q^{48} - 100714448 q^{49} + 77969322 q^{51} - 113165052 q^{52} + 29507556 q^{53} + 138590608 q^{55} + 437436342 q^{56} - 238375816 q^{61} - 565735320 q^{62} + 272247742 q^{64} - 351372138 q^{65} - 495101088 q^{66} + 550420050 q^{68} + 505082484 q^{69} + 1257672774 q^{74} - 1443387976 q^{75} + 530898576 q^{77} - 1745232606 q^{78} - 441679756 q^{79} + 1486784810 q^{81} + 518795296 q^{82} + 4793242032 q^{87} - 3927690208 q^{88} - 2656725444 q^{90} - 777339186 q^{91} - 4121025444 q^{92} + 6226353478 q^{94} + 1047837276 q^{95}+O(q^{100})$$ 10 * q - 2 * q^3 - 2562 * q^4 + 67304 * q^9 + 45562 * q^10 + 46298 * q^12 - 111488 * q^13 + 169350 * q^14 - 209470 * q^16 - 1189686 * q^17 + 3516760 * q^22 + 2210916 * q^23 - 10109316 * q^25 + 7627230 * q^26 - 7880006 * q^27 + 7039224 * q^29 - 7573410 * q^30 + 39493506 * q^35 + 29007196 * q^36 - 50219652 * q^38 + 14502332 * q^39 - 23263606 * q^40 + 400842 * q^42 - 92334370 * q^43 + 102213790 * q^48 - 100714448 * q^49 + 77969322 * q^51 - 113165052 * q^52 + 29507556 * q^53 + 138590608 * q^55 + 437436342 * q^56 - 238375816 * q^61 - 565735320 * q^62 + 272247742 * q^64 - 351372138 * q^65 - 495101088 * q^66 + 550420050 * q^68 + 505082484 * q^69 + 1257672774 * q^74 - 1443387976 * q^75 + 530898576 * q^77 - 1745232606 * q^78 - 441679756 * q^79 + 1486784810 * q^81 + 518795296 * q^82 + 4793242032 * q^87 - 3927690208 * q^88 - 2656725444 * q^90 - 777339186 * q^91 - 4121025444 * q^92 + 6226353478 * q^94 + 1047837276 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} + 3841x^{8} + 5134480x^{6} + 2823572208x^{4} + 614223235584x^{2} + 43308450164736$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + 768$$ v^2 + 768 $$\beta_{3}$$ $$=$$ $$( -101\nu^{8} - 335285\nu^{6} - 359803808\nu^{4} - 134078143536\nu^{2} - 13050018918144 ) / 12944968704$$ (-101*v^8 - 335285*v^6 - 359803808*v^4 - 134078143536*v^2 - 13050018918144) / 12944968704 $$\beta_{4}$$ $$=$$ $$( 128\nu^{8} + 466637\nu^{6} + 584699369\nu^{4} + 281289968700\nu^{2} + 36497540643552 ) / 202265136$$ (128*v^8 + 466637*v^6 + 584699369*v^4 + 281289968700*v^2 + 36497540643552) / 202265136 $$\beta_{5}$$ $$=$$ $$( 103037 \nu^{9} + 407466221 \nu^{7} + 517098453440 \nu^{5} + 225182669584944 \nu^{3} + 24\!\cdots\!24 \nu ) / 49242660950016$$ (103037*v^9 + 407466221*v^7 + 517098453440*v^5 + 225182669584944*v^3 + 24602442807930624*v) / 49242660950016 $$\beta_{6}$$ $$=$$ $$( 3871\nu^{8} + 20860879\nu^{6} + 32030013856\nu^{4} + 15796747994256\nu^{2} + 1898872235880192 ) / 6472484352$$ (3871*v^8 + 20860879*v^6 + 32030013856*v^4 + 15796747994256*v^2 + 1898872235880192) / 6472484352 $$\beta_{7}$$ $$=$$ $$( 18149 \nu^{9} + 73154197 \nu^{7} + 101232420736 \nu^{5} + 55804164878768 \nu^{3} + 10\!\cdots\!36 \nu ) / 2735703386112$$ (18149*v^9 + 73154197*v^7 + 101232420736*v^5 + 55804164878768*v^3 + 10382897932475136*v) / 2735703386112 $$\beta_{8}$$ $$=$$ $$( -101\nu^{9} - 335285\nu^{7} - 359803808\nu^{5} - 134078143536\nu^{3} - 12972349105920\nu ) / 12944968704$$ (-101*v^9 - 335285*v^7 - 359803808*v^5 - 134078143536*v^3 - 12972349105920*v) / 12944968704 $$\beta_{9}$$ $$=$$ $$( 118481 \nu^{9} + 432033281 \nu^{7} + 531812876480 \nu^{5} + 244210927479792 \nu^{3} + 29\!\cdots\!40 \nu ) / 8207110158336$$ (118481*v^9 + 432033281*v^7 + 531812876480*v^5 + 244210927479792*v^3 + 29787258465642240*v) / 8207110158336
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 768$$ b2 - 768 $$\nu^{3}$$ $$=$$ $$-\beta_{9} - \beta_{8} + \beta_{7} - 1168\beta_1$$ -b9 - b8 + b7 - 1168*b1 $$\nu^{4}$$ $$=$$ $$-\beta_{6} + 6\beta_{4} + 410\beta_{3} - 1657\beta_{2} + 896615$$ -b6 + 6*b4 + 410*b3 - 1657*b2 + 896615 $$\nu^{5}$$ $$=$$ $$2045\beta_{9} + 2005\beta_{8} - 1653\beta_{7} - 1392\beta_{5} + 1556172\beta_1$$ 2045*b9 + 2005*b8 - 1653*b7 - 1392*b5 + 1556172*b1 $$\nu^{6}$$ $$=$$ $$3085\beta_{6} - 13662\beta_{4} - 871634\beta_{3} + 2442489\beta_{2} - 1194377627$$ 3085*b6 - 13662*b4 - 871634*b3 + 2442489*b2 - 1194377627 $$\nu^{7}$$ $$=$$ $$-3338893\beta_{9} - 3045285\beta_{8} + 2439845\beta_{7} + 3945264\beta_{5} - 2164532636\beta_1$$ -3338893*b9 - 3045285*b8 + 2439845*b7 + 3945264*b5 - 2164532636*b1 $$\nu^{8}$$ $$=$$ $$-6678717\beta_{6} + 23978622\beta_{4} + 1304765106\beta_{3} - 3532803545\beta_{2} + 1661122838379$$ -6678717*b6 + 23978622*b4 + 1304765106*b3 - 3532803545*b2 + 1661122838379 $$\nu^{9}$$ $$=$$ $$5126337581\beta_{9} + 4165989317\beta_{8} - 3538276037\beta_{7} - 8138029104\beta_{5} + 3063857781212\beta_1$$ 5126337581*b9 + 4165989317*b8 - 3538276037*b7 - 8138029104*b5 + 3063857781212*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/13\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
12.1
 − 38.5911i − 36.6220i − 24.9226i − 15.7959i − 11.8282i 11.8282i 15.7959i 24.9226i 36.6220i 38.5911i
38.5911i −58.0612 −977.274 2510.84i 2240.65i 8007.46i 17955.4i −16311.9 96896.0
12.2 36.6220i 126.985 −829.173 2095.65i 4650.45i 3353.27i 11615.5i −3557.77 −76746.8
12.3 24.9226i −252.690 −109.136 825.595i 6297.70i 9476.84i 10040.4i 44169.4 −20576.0
12.4 15.7959i 217.907 262.489 1801.57i 3442.04i 6896.47i 12233.8i 27800.4 28457.4
12.5 11.8282i −35.1406 372.094 443.831i 415.650i 6276.11i 10457.2i −18448.1 −5249.72
12.6 11.8282i −35.1406 372.094 443.831i 415.650i 6276.11i 10457.2i −18448.1 −5249.72
12.7 15.7959i 217.907 262.489 1801.57i 3442.04i 6896.47i 12233.8i 27800.4 28457.4
12.8 24.9226i −252.690 −109.136 825.595i 6297.70i 9476.84i 10040.4i 44169.4 −20576.0
12.9 36.6220i 126.985 −829.173 2095.65i 4650.45i 3353.27i 11615.5i −3557.77 −76746.8
12.10 38.5911i −58.0612 −977.274 2510.84i 2240.65i 8007.46i 17955.4i −16311.9 96896.0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 12.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.10.b.a 10
3.b odd 2 1 117.10.b.c 10
4.b odd 2 1 208.10.f.b 10
13.b even 2 1 inner 13.10.b.a 10
13.d odd 4 2 169.10.a.e 10
39.d odd 2 1 117.10.b.c 10
52.b odd 2 1 208.10.f.b 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.10.b.a 10 1.a even 1 1 trivial
13.10.b.a 10 13.b even 2 1 inner
117.10.b.c 10 3.b odd 2 1
117.10.b.c 10 39.d odd 2 1
169.10.a.e 10 13.d odd 4 2
208.10.f.b 10 4.b odd 2 1
208.10.f.b 10 52.b odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{10}^{\mathrm{new}}(13, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} + \cdots + 43308450164736$$
$3$ $$(T^{5} + T^{4} + \cdots + 14266185264)^{2}$$
$5$ $$T^{10} + \cdots + 12\!\cdots\!00$$
$7$ $$T^{10} + \cdots + 12\!\cdots\!00$$
$11$ $$T^{10} + \cdots + 15\!\cdots\!00$$
$13$ $$T^{10} + \cdots + 13\!\cdots\!93$$
$17$ $$(T^{5} + \cdots + 94\!\cdots\!60)^{2}$$
$19$ $$T^{10} + \cdots + 35\!\cdots\!56$$
$23$ $$(T^{5} + \cdots - 20\!\cdots\!84)^{2}$$
$29$ $$(T^{5} + \cdots + 22\!\cdots\!60)^{2}$$
$31$ $$T^{10} + \cdots + 12\!\cdots\!00$$
$37$ $$T^{10} + \cdots + 58\!\cdots\!16$$
$41$ $$T^{10} + \cdots + 19\!\cdots\!00$$
$43$ $$(T^{5} + \cdots + 56\!\cdots\!88)^{2}$$
$47$ $$T^{10} + \cdots + 61\!\cdots\!44$$
$53$ $$(T^{5} + \cdots + 58\!\cdots\!16)^{2}$$
$59$ $$T^{10} + \cdots + 45\!\cdots\!76$$
$61$ $$(T^{5} + \cdots + 10\!\cdots\!08)^{2}$$
$67$ $$T^{10} + \cdots + 51\!\cdots\!04$$
$71$ $$T^{10} + \cdots + 62\!\cdots\!00$$
$73$ $$T^{10} + \cdots + 72\!\cdots\!96$$
$79$ $$(T^{5} + \cdots - 12\!\cdots\!00)^{2}$$
$83$ $$T^{10} + \cdots + 26\!\cdots\!36$$
$89$ $$T^{10} + \cdots + 62\!\cdots\!76$$
$97$ $$T^{10} + \cdots + 19\!\cdots\!44$$