# Properties

 Label 100.9.b.g Level $100$ Weight $9$ Character orbit 100.b Analytic conductor $40.738$ Analytic rank $0$ Dimension $20$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [100,9,Mod(51,100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(100, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("100.51");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$100 = 2^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 100.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: $$40.7378610061$$ Analytic rank: $$0$$ Dimension: $$20$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{20} - 94 x^{18} + 5343 x^{16} - 172772 x^{14} + 36131456 x^{12} - 3044563968 x^{10} + \cdots + 11\!\cdots\!76$$ x^20 - 94*x^18 + 5343*x^16 - 172772*x^14 + 36131456*x^12 - 3044563968*x^10 + 147994443776*x^8 - 2898633162752*x^6 + 367168164200448*x^4 - 26458647810801664*x^2 + 1152921504606846976 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{75}\cdot 3^{4}\cdot 5^{14}$$ Twist minimal: no (minimal twist has level 20) 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 $$1,\beta_1,\ldots,\beta_{19}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + ( - \beta_{8} - \beta_1) q^{3} + (\beta_{2} + 38) q^{4} + (\beta_{7} - \beta_{2} + 170) q^{6} + ( - \beta_{14} + 3 \beta_{8} + 3 \beta_1) q^{7} + (\beta_{3} + 38 \beta_1) q^{8} + (\beta_{10} - 2 \beta_{7} + \beta_{5} + \cdots - 130) q^{9}+O(q^{10})$$ q + b1 * q^2 + (-b8 - b1) * q^3 + (b2 + 38) * q^4 + (b7 - b2 + 170) * q^6 + (-b14 + 3*b8 + 3*b1) * q^7 + (b3 + 38*b1) * q^8 + (b10 - 2*b7 + b5 - 6*b2 - 130) * q^9 $$q + \beta_1 q^{2} + ( - \beta_{8} - \beta_1) q^{3} + (\beta_{2} + 38) q^{4} + (\beta_{7} - \beta_{2} + 170) q^{6} + ( - \beta_{14} + 3 \beta_{8} + 3 \beta_1) q^{7} + (\beta_{3} + 38 \beta_1) q^{8} + (\beta_{10} - 2 \beta_{7} + \beta_{5} + \cdots - 130) q^{9}+ \cdots + (342 \beta_{15} - 4299 \beta_{12} + \cdots - 25347) q^{99}+O(q^{100})$$ q + b1 * q^2 + (-b8 - b1) * q^3 + (b2 + 38) * q^4 + (b7 - b2 + 170) * q^6 + (-b14 + 3*b8 + 3*b1) * q^7 + (b3 + 38*b1) * q^8 + (b10 - 2*b7 + b5 - 6*b2 - 130) * q^9 + (-b12 - b7 + 2*b2 + 1) * q^11 + (-b18 - b14 - 96*b8 - b6 - b3 + 138*b1) * q^12 + (-b19 + b18 + b16 + 2*b8 - 2*b6 - 86*b1) * q^13 + (-b15 + 2*b10 - 3*b7 + b5 + 2*b4 + b2 - 442) * q^14 + (-b15 - 3*b12 + b11 + b9 + b7 + 3*b5 - 4*b4 + 38*b2 - 2942) * q^16 + (3*b19 + b18 + b16 - 4*b14 + b13 + 26*b8 - 14*b6 - 8*b3 - 846*b1) * q^17 + (b19 + 2*b18 - b17 + 2*b16 + 5*b14 + 4*b13 - 445*b8 - 30*b6 - 5*b3 - 288*b1) * q^18 + (2*b15 + b12 + 2*b11 - 2*b10 + 3*b9 - 21*b7 - b5 - 2*b4 + 50*b2 + 21) * q^19 + (-7*b10 + 28*b7 - 28*b5 + 28*b4 - 84*b2 + 20468) * q^21 + (-5*b19 + 4*b17 + 8*b16 + 17*b14 - 13*b13 + 248*b8 - 10*b6 + 6*b3 + 75*b1) * q^22 + (-b19 - 14*b18 - 2*b17 + 14*b16 + 2*b14 + b13 - 839*b8 - 3*b6 + 12*b3 - 387*b1) * q^23 + (-b15 - 7*b12 - 3*b11 - 52*b10 - 3*b9 + 105*b7 - 37*b5 + 24*b4 + 182*b2 + 7894) * q^24 + (19*b12 + 3*b11 + 3*b10 - 3*b9 + 20*b7 + 72*b5 - 25*b4 - 107*b2 - 22049) * q^26 + (6*b19 + 12*b18 + 12*b17 - 12*b16 - 48*b14 - 6*b13 - 1242*b8 - 138*b6 + 72*b3 + 6834*b1) * q^27 + (14*b19 + 7*b18 - 207*b14 - 42*b13 - 744*b8 - 133*b6 - 21*b3 - 688*b1) * q^28 + (8*b10 + 48*b9 + 56*b7 + 70*b5 - 100*b4 - 672*b2 - 33232) * q^29 + (2*b15 + 6*b12 - 14*b11 - 18*b10 + 43*b9 - 576*b7 - 57*b5 - 18*b4 + 776*b2 + 304) * q^31 + (-4*b18 + 24*b17 + 16*b16 - 132*b14 + 88*b13 + 280*b8 - 52*b6 + 24*b3 - 2832*b1) * q^32 + (-19*b19 - b18 + 16*b17 - b16 - 12*b14 - 19*b13 - 154*b8 + 54*b6 - 72*b3 + 6166*b1) * q^33 + (16*b15 + 6*b12 - 10*b11 + 266*b10 + 10*b9 - 44*b7 + 160*b5 + 82*b4 - 922*b2 - 217486) * q^34 + (18*b15 + 18*b12 - 6*b11 + 232*b10 - 54*b9 + 490*b7 + 178*b5 - 219*b2 - 359638) * q^36 + (34*b19 - 10*b18 + 32*b17 - 10*b16 - 88*b14 - 23*b13 + 188*b8 - 132*b6 - 272*b3 - 3804*b1) * q^37 + (23*b19 + 28*b17 - 40*b16 + 613*b14 + 247*b13 + 5440*b8 - 50*b6 + 86*b3 + 1703*b1) * q^38 + (-32*b15 + 22*b12 - 32*b10 + 192*b9 - 714*b7 + 64*b5 - 32*b4 + 5556*b2 + 2282) * q^39 + (-133*b10 + 240*b9 + 1022*b7 - 159*b5 - 224*b4 - 2790*b2 + 352371) * q^41 + (70*b19 - 42*b18 + 35*b17 - 70*b16 + 168*b14 - 455*b13 + 8827*b8 - 308*b6 - 91*b3 + 23240*b1) * q^42 + (24*b19 + 104*b18 + 48*b17 - 104*b16 + 58*b14 - 24*b13 + 11379*b8 + 508*b6 + 176*b3 - 14457*b1) * q^43 + (16*b15 + 56*b12 + 24*b11 - 432*b10 + 216*b9 - 40*b7 + 480*b5 - 16*b4 + 136*b2 - 133152) * q^44 + (-b15 - 176*b12 - 16*b11 - 582*b10 - 144*b9 + 1143*b7 + 289*b5 + 378*b4 - 377*b2 + 27890) * q^46 + (5*b19 - 66*b18 + 10*b17 + 66*b16 + 68*b14 - 5*b13 + 14363*b8 + 947*b6 + 212*b3 - 32533*b1) * q^47 + (-104*b19 - 36*b18 + 56*b17 - 48*b16 - 476*b14 - 848*b13 + 4376*b8 + 60*b6 + 140*b3 + 9408*b1) * q^48 + (-287*b10 + 448*b9 + 798*b7 - 259*b5 + 616*b4 - 14742*b2 - 591934) * q^49 + (-34*b15 - 46*b12 + 126*b11 - 254*b10 + 573*b9 - 4376*b7 - 191*b5 - 254*b4 + 12776*b2 + 5304) * q^51 + (30*b19 + 106*b18 - 56*b17 - 144*b16 - 316*b14 + 1070*b13 - 51872*b8 - 930*b6 - 174*b3 - 39446*b1) * q^52 + (265*b19 - 97*b18 - 64*b17 - 97*b16 - 40*b14 + 164*b13 - 2514*b8 + 1450*b6 + 112*b3 + 92918*b1) * q^53 + (-102*b15 - 96*b12 + 96*b11 + 540*b10 + 864*b9 + 1314*b7 + 678*b5 - 612*b4 + 7890*b2 - 1851876) * q^54 + (-137*b15 + 49*b12 + 21*b11 + 1548*b10 + 21*b9 + 737*b7 + 403*b5 + 1016*b4 - 570*b2 - 1752538) * q^56 + (-379*b19 + 31*b18 - 80*b17 + 31*b16 + 508*b14 + 101*b13 - 4298*b8 + 2238*b6 + 1256*b3 + 139878*b1) * q^57 + (-206*b19 - 128*b18 - 108*b17 + 216*b16 - 746*b14 + 1374*b13 + 19804*b8 - 2460*b6 - 760*b3 - 26944*b1) * q^58 + (206*b15 - 329*b12 - 114*b11 + 50*b10 + 981*b9 - 547*b7 + 953*b5 + 50*b4 + 34238*b2 + 14051) * q^59 + (577*b10 + 656*b9 - 1036*b7 + 1482*b5 - 616*b4 - 19092*b2 - 890108) * q^61 + (-208*b19 + 448*b18 - 248*b17 + 176*b16 - 304*b14 - 1224*b13 + 141544*b8 - 768*b6 + 696*b3 + 47344*b1) * q^62 + (-105*b19 - 126*b18 - 210*b17 + 126*b16 - 1080*b14 + 105*b13 - 69189*b8 + 3045*b6 - 1428*b3 - 246849*b1) * q^63 + (-104*b15 - 104*b12 - 72*b11 - 336*b10 + 1592*b9 + 168*b7 + 3032*b5 - 816*b4 - 3216*b2 - 1010272) * q^64 + (16*b15 + 460*b12 - 36*b11 - 1856*b10 + 804*b9 + 300*b7 - 80*b5 - 176*b4 + 6624*b2 + 1597856) * q^66 + (-204*b19 - 96*b18 - 408*b17 + 96*b16 + 1982*b14 + 204*b13 - 75531*b8 + 2232*b6 - 3072*b3 - 230655*b1) * q^67 + (284*b19 - 124*b18 - 368*b17 + 480*b16 + 4312*b14 - 836*b13 - 104512*b8 - 9108*b6 - 316*b3 - 255948*b1) * q^68 + (1275*b10 + 1776*b9 - 2248*b7 + 3198*b5 - 596*b4 - 56448*b2 - 5608826) * q^69 + (238*b15 + 448*b12 - 354*b11 + 610*b10 - 147*b9 + 10230*b7 + 1585*b5 + 610*b4 + 9708*b2 + 3770) * q^71 + (88*b19 - 520*b18 - 400*b17 + 416*b16 + 3584*b14 + 1000*b13 - 212272*b8 - 600*b6 + 481*b3 - 431298*b1) * q^72 + (-987*b19 + 727*b18 - 144*b17 + 727*b16 + 548*b14 - 687*b13 - 23882*b8 + 11430*b6 + 1528*b3 + 850678*b1) * q^73 + (288*b15 + 561*b12 - 303*b11 + 469*b10 + 1839*b9 - 584*b7 + 664*b5 + 1569*b4 - 3021*b2 - 957031) * q^74 + (544*b15 - 600*b12 - 184*b11 - 496*b10 + 2184*b9 - 5944*b7 + 1200*b5 - 4368*b4 + 3560*b2 - 2747008) * q^76 + (973*b19 - 21*b18 - 448*b17 - 21*b16 - 56*b14 - 483*b13 - 8666*b8 + 5026*b6 + 1232*b3 + 321958*b1) * q^77 + (334*b19 + 640*b18 - 88*b17 - 176*b16 - 7254*b14 - 386*b13 + 214576*b8 - 18276*b6 + 4284*b3 + 70382*b1) * q^78 + (-610*b15 + 1670*b12 + 462*b11 - 46*b10 + 2293*b9 + 6156*b7 + 3033*b5 - 46*b4 + 78112*b2 + 32292) * q^79 + (-1893*b10 - 1296*b9 - 1506*b7 - 2595*b5 + 4104*b4 + 9738*b2 - 7708326) * q^81 + (-315*b19 - 1778*b18 - 91*b17 + 182*b16 - 2723*b14 + 1582*b13 + 298873*b8 - 7830*b6 - 3403*b3 + 451045*b1) * q^82 + (36*b19 - 832*b18 + 72*b17 + 832*b16 + 5562*b14 - 36*b13 + 31579*b8 + 30536*b6 + 2240*b3 - 1550617*b1) * q^83 + (322*b15 - 182*b12 + 210*b11 - 1624*b10 + 1890*b9 - 9646*b7 - 6286*b5 + 4704*b4 + 29652*b2 - 5054364) * q^84 + (-102*b15 + 512*b12 + 384*b11 + 4012*b10 + 3456*b9 - 14443*b7 + 614*b5 - 4180*b4 - 19105*b2 + 4715562) * q^86 + (378*b19 + 1172*b18 + 756*b17 - 1172*b16 - 11382*b14 - 378*b13 + 131850*b8 + 11290*b6 + 3704*b3 - 433074*b1) * q^87 + (-560*b19 + 1312*b18 + 576*b17 - 1664*b16 + 2704*b14 + 3792*b13 - 167040*b8 - 36736*b6 - 568*b3 - 199424*b1) * q^88 + (-3716*b10 + 1488*b9 - 772*b7 + 4626*b5 - 5504*b4 - 13932*b2 - 5253568) * q^89 + (-814*b15 - 2016*b12 - 14*b11 + 1614*b10 - 1813*b9 + 34938*b7 + 4615*b5 + 1614*b4 - 6060*b2 - 1626) * q^91 + (-2002*b19 + 1157*b18 + 1408*b17 - 256*b16 + 15*b14 - 7946*b13 - 571048*b8 - 19911*b6 + 9*b3 - 169276*b1) * q^92 + (820*b19 - 2028*b18 + 896*b17 - 2028*b16 - 584*b14 + 2740*b13 - 98472*b8 + 49920*b6 - 3856*b3 + 3617224*b1) * q^93 + (-53*b15 - 1296*b12 + 80*b11 - 2958*b10 + 720*b9 - 14755*b7 + 2965*b5 + 946*b4 - 41223*b2 + 9622254) * q^94 + (-1004*b15 + 1228*b12 + 1020*b11 - 12624*b10 + 2172*b9 - 2964*b7 - 9756*b5 + 5056*b4 + 18056*b2 + 1447592) * q^96 + (311*b19 + 149*b18 + 1152*b17 + 149*b16 - 2764*b14 + 4041*b13 - 34446*b8 + 15346*b6 - 8984*b3 + 1309274*b1) * q^97 + (1365*b19 - 1022*b18 + 903*b17 - 1806*b16 + 3073*b14 - 10416*b13 + 346395*b8 - 61558*b6 - 15925*b3 - 495180*b1) * q^98 + (342*b15 - 4299*b12 - 138*b11 + 906*b10 - 2895*b9 + 14883*b7 + 453*b5 + 906*b4 - 63534*b2 - 25347) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20 q + 752 q^{4} + 3408 q^{6} - 2556 q^{9}+O(q^{10})$$ 20 * q + 752 * q^4 + 3408 * q^6 - 2556 * q^9 $$20 q + 752 q^{4} + 3408 q^{6} - 2556 q^{9} - 8848 q^{14} - 59200 q^{16} + 410256 q^{21} + 156672 q^{24} - 440448 q^{26} - 660136 q^{29} - 4342528 q^{34} - 7191312 q^{36} + 7068520 q^{41} - 2666880 q^{44} + 561168 q^{46} - 11719036 q^{49} - 37110816 q^{54} - 35044352 q^{56} - 17660440 q^{61} - 20201728 q^{64} + 31902720 q^{66} - 111747216 q^{69} - 19114368 q^{74} - 54998400 q^{76} - 154212444 q^{81} - 101289216 q^{84} + 94429648 q^{86} - 105006376 q^{89} + 192757872 q^{94} + 28850688 q^{96}+O(q^{100})$$ 20 * q + 752 * q^4 + 3408 * q^6 - 2556 * q^9 - 8848 * q^14 - 59200 * q^16 + 410256 * q^21 + 156672 * q^24 - 440448 * q^26 - 660136 * q^29 - 4342528 * q^34 - 7191312 * q^36 + 7068520 * q^41 - 2666880 * q^44 + 561168 * q^46 - 11719036 * q^49 - 37110816 * q^54 - 35044352 * q^56 - 17660440 * q^61 - 20201728 * q^64 + 31902720 * q^66 - 111747216 * q^69 - 19114368 * q^74 - 54998400 * q^76 - 154212444 * q^81 - 101289216 * q^84 + 94429648 * q^86 - 105006376 * q^89 + 192757872 * q^94 + 28850688 * q^96

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} - 94 x^{18} + 5343 x^{16} - 172772 x^{14} + 36131456 x^{12} - 3044563968 x^{10} + \cdots + 11\!\cdots\!76$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu$$ 2*v $$\beta_{2}$$ $$=$$ $$4\nu^{2} - 38$$ 4*v^2 - 38 $$\beta_{3}$$ $$=$$ $$8\nu^{3} - 76\nu$$ 8*v^3 - 76*v $$\beta_{4}$$ $$=$$ $$( 3302597 \nu^{18} - 1391343702 \nu^{16} + 29876009883 \nu^{14} + 1448679820940 \nu^{12} + \cdots - 37\!\cdots\!28 ) / 12\!\cdots\!32$$ (3302597*v^18 - 1391343702*v^16 + 29876009883*v^14 + 1448679820940*v^12 + 186292651580544*v^10 - 38687379151309824*v^8 + 677837057835925504*v^6 + 16327481853927751680*v^4 + 1614929506130542460928*v^2 - 379959887896387285680128) / 12647057731702751232 $$\beta_{5}$$ $$=$$ $$( - 4117787 \nu^{18} - 336684054 \nu^{16} + 5062818171 \nu^{14} + 677234017804 \nu^{12} + \cdots - 10\!\cdots\!56 ) / 31\!\cdots\!08$$ (-4117787*v^18 - 336684054*v^16 + 5062818171*v^14 + 677234017804*v^12 - 90559709717376*v^10 - 8689377821346816*v^8 + 175685887218679808*v^6 + 8959431859721207808*v^4 - 940647336463092940800*v^2 - 102697153346715837792256) / 3161764432925687808 $$\beta_{6}$$ $$=$$ $$( 10120793 \nu^{19} + 1538239314 \nu^{17} - 72882621561 \nu^{15} - 834264549700 \nu^{13} + \cdots + 47\!\cdots\!48 \nu ) / 16\!\cdots\!96$$ (10120793*v^19 + 1538239314*v^17 - 72882621561*v^15 - 834264549700*v^13 + 124572188012160*v^11 + 46269733902222336*v^9 - 2016433934423293952*v^7 - 342252525333774336*v^5 + 1227114972205899841536*v^3 + 476580632743716635803648*v) / 1618823389657952157696 $$\beta_{7}$$ $$=$$ $$( - 607811 \nu^{18} + 30995610 \nu^{16} - 223223901 \nu^{14} + 58863964588 \nu^{12} + \cdots + 60\!\cdots\!92 ) / 39\!\cdots\!76$$ (-607811*v^18 + 30995610*v^16 - 223223901*v^14 + 58863964588*v^12 - 18819124901760*v^10 + 857972427985920*v^8 - 7078665453568000*v^6 + 607645999476768768*v^4 - 188843249831686176768*v^2 + 6004190878028206702592) / 395220554115710976 $$\beta_{8}$$ $$=$$ $$( 21623225 \nu^{19} + 457010706 \nu^{17} - 11425127385 \nu^{15} - 2821562731204 \nu^{13} + \cdots + 20\!\cdots\!28 \nu ) / 32\!\cdots\!92$$ (21623225*v^19 + 457010706*v^17 - 11425127385*v^15 - 2821562731204*v^13 + 540171803713152*v^11 + 11249843890652160*v^9 - 314137908512030720*v^7 - 33683583372833587200*v^5 + 5450441813486387331072*v^3 + 201380656501864768995328*v) / 3237646779315904315392 $$\beta_{9}$$ $$=$$ $$( 5555591 \nu^{18} + 201530478 \nu^{16} + 2619368601 \nu^{14} - 925646290492 \nu^{12} + \cdots + 68\!\cdots\!24 ) / 31\!\cdots\!08$$ (5555591*v^18 + 201530478*v^16 + 2619368601*v^14 - 925646290492*v^12 + 142509661680000*v^10 + 4311891569900544*v^8 + 37101116020228096*v^6 - 13127098215658684416*v^4 + 1443269076159748374528*v^2 + 68695538635032991105024) / 3161764432925687808 $$\beta_{10}$$ $$=$$ $$( 3819187 \nu^{18} - 514637242 \nu^{16} + 8512799213 \nu^{14} - 487547106668 \nu^{12} + \cdots - 12\!\cdots\!60 ) / 14\!\cdots\!48$$ (3819187*v^18 - 514637242*v^16 + 8512799213*v^14 - 487547106668*v^12 + 163163551551360*v^10 - 14525785356653568*v^8 + 209626121754902528*v^6 - 5703158516135493632*v^4 + 1705651632619191795712*v^2 - 128455875919075669442560) / 1405228636855861248 $$\beta_{11}$$ $$=$$ $$( - 1687693 \nu^{18} - 208464954 \nu^{16} + 10100867373 \nu^{14} + 136486344596 \nu^{12} + \cdots - 44\!\cdots\!40 ) / 52\!\cdots\!68$$ (-1687693*v^18 - 208464954*v^16 + 10100867373*v^14 + 136486344596*v^12 - 49753044152448*v^10 - 4879682105594880*v^8 + 136004134929694720*v^6 + 5319564383326568448*v^4 - 705405618017198407680*v^2 - 44780080148469398896640) / 526960738820947968 $$\beta_{12}$$ $$=$$ $$( 11250533 \nu^{18} - 356503830 \nu^{16} + 6095187195 \nu^{14} - 1380868386292 \nu^{12} + \cdots - 16\!\cdots\!96 ) / 15\!\cdots\!04$$ (11250533*v^18 - 356503830*v^16 + 6095187195*v^14 - 1380868386292*v^12 + 320541156988032*v^10 - 7748665416698880*v^8 + 102852276112064512*v^6 - 17957218619410612224*v^4 + 3135517970758599966720*v^2 - 16797154910545217847296) / 1580882216462843904 $$\beta_{13}$$ $$=$$ $$( 3849039 \nu^{19} - 564836098 \nu^{17} + 12303037905 \nu^{15} - 1038892195932 \nu^{13} + \cdots - 10\!\cdots\!76 \nu ) / 59\!\cdots\!48$$ (3849039*v^19 - 564836098*v^17 + 12303037905*v^15 - 1038892195932*v^13 + 137976954729856*v^11 - 14016355039732736*v^9 + 271631947587387392*v^7 - 8211038787065610240*v^5 + 1292945949569799159808*v^3 - 104484713534621038411776*v) / 59956421839183413248 $$\beta_{14}$$ $$=$$ $$( - 32459431 \nu^{19} - 1527461550 \nu^{17} + 42771655815 \nu^{15} + \cdots - 61\!\cdots\!48 \nu ) / 26\!\cdots\!16$$ (-32459431*v^19 - 1527461550*v^17 + 42771655815*v^15 + 3554605976252*v^13 - 745081528209792*v^11 - 47291577343933440*v^9 + 930922400653508608*v^7 + 33103730770905661440*v^5 - 7655738336341515042816*v^3 - 616847663822970445365248*v) / 269803898276325359616 $$\beta_{15}$$ $$=$$ $$( - 92592211 \nu^{18} + 1719020154 \nu^{16} + 10180103091 \nu^{14} + 9232416044012 \nu^{12} + \cdots + 68\!\cdots\!68 ) / 31\!\cdots\!08$$ (-92592211*v^18 + 1719020154*v^16 + 10180103091*v^14 + 9232416044012*v^12 - 2687780325109632*v^10 + 41008816547681280*v^8 + 28603549117579264*v^6 + 91357350595003416576*v^4 - 27069301776863648022528*v^2 + 6832443285141264007168) / 3161764432925687808 $$\beta_{16}$$ $$=$$ $$( 383591717 \nu^{19} - 52518900630 \nu^{17} + 661582634811 \nu^{15} + \cdots - 13\!\cdots\!44 \nu ) / 26\!\cdots\!16$$ (383591717*v^19 - 52518900630*v^17 + 661582634811*v^15 - 39295815470836*v^13 + 16303953351421056*v^11 - 1502060657991306240*v^9 + 19111126812626255872*v^7 - 345444880039167393792*v^5 + 170347974610982082183168*v^3 - 13571022254537108822687744*v) / 269803898276325359616 $$\beta_{17}$$ $$=$$ $$( - 534030799 \nu^{19} + 38122392066 \nu^{17} - 484129093457 \nu^{15} + \cdots + 78\!\cdots\!20 \nu ) / 35\!\cdots\!88$$ (-534030799*v^19 + 38122392066*v^17 - 484129093457*v^15 + 73505585826396*v^13 - 18721554341361024*v^11 + 1032895461822194688*v^9 - 14126468624274685952*v^7 + 1121790072135629144064*v^5 - 196097183563737387237376*v^3 + 7854762096792945001758720*v) / 359738531035100479488 $$\beta_{18}$$ $$=$$ $$( 4125908705 \nu^{19} - 268226021022 \nu^{17} + 2193308998143 \nu^{15} + \cdots - 55\!\cdots\!12 \nu ) / 16\!\cdots\!96$$ (4125908705*v^19 - 268226021022*v^17 + 2193308998143*v^15 - 367271958114916*v^13 + 132583941598233216*v^11 - 7331022906650582016*v^9 + 69497946534708772864*v^7 - 3559304157917777756160*v^5 + 1330090010619669758607360*v^3 - 55731702531640581121114112*v) / 1618823389657952157696 $$\beta_{19}$$ $$=$$ $$( 1651996615 \nu^{19} - 148477643538 \nu^{17} + 2203492598361 \nu^{15} + \cdots - 33\!\cdots\!40 \nu ) / 53\!\cdots\!32$$ (1651996615*v^19 - 148477643538*v^17 + 2203492598361*v^15 - 168214241184572*v^13 + 59280743722180992*v^11 - 4087951125389214720*v^9 + 53102213053548068864*v^7 - 1740157820557571653632*v^5 + 600144468245718667100160*v^3 - 33435270712585421785661440*v) / 539607796552650719232
 $$\nu$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{2} + 38 ) / 4$$ (b2 + 38) / 4 $$\nu^{3}$$ $$=$$ $$( \beta_{3} + 38\beta_1 ) / 8$$ (b3 + 38*b1) / 8 $$\nu^{4}$$ $$=$$ $$( -\beta_{15} - 3\beta_{12} + \beta_{11} + \beta_{9} + \beta_{7} + 3\beta_{5} - 4\beta_{4} + 38\beta_{2} - 2942 ) / 16$$ (-b15 - 3*b12 + b11 + b9 + b7 + 3*b5 - 4*b4 + 38*b2 - 2942) / 16 $$\nu^{5}$$ $$=$$ $$( - \beta_{18} + 6 \beta_{17} + 4 \beta_{16} - 33 \beta_{14} + 22 \beta_{13} + 70 \beta_{8} + \cdots - 708 \beta_1 ) / 8$$ (-b18 + 6*b17 + 4*b16 - 33*b14 + 22*b13 + 70*b8 - 13*b6 + 6*b3 - 708*b1) / 8 $$\nu^{6}$$ $$=$$ $$( - 13 \beta_{15} - 13 \beta_{12} - 9 \beta_{11} - 42 \beta_{10} + 199 \beta_{9} + 21 \beta_{7} + \cdots - 126284 ) / 8$$ (-13*b15 - 13*b12 - 9*b11 - 42*b10 + 199*b9 + 21*b7 + 379*b5 - 102*b4 - 402*b2 - 126284) / 8 $$\nu^{7}$$ $$=$$ $$( - 302 \beta_{19} + 198 \beta_{18} - 76 \beta_{17} + 184 \beta_{16} - 2308 \beta_{14} + \cdots - 74600 \beta_1 ) / 8$$ (-302*b19 + 198*b18 - 76*b17 + 184*b16 - 2308*b14 + 286*b13 - 24356*b8 - 14054*b6 - 551*b3 - 74600*b1) / 8 $$\nu^{8}$$ $$=$$ $$( - 2173 \beta_{15} + 5761 \beta_{12} - 139 \beta_{11} + 2008 \beta_{10} - 6091 \beta_{9} + \cdots - 213160862 ) / 16$$ (-2173*b15 + 5761*b12 - 139*b11 + 2008*b10 - 6091*b9 + 56461*b7 + 9703*b5 - 1404*b4 + 42670*b2 - 213160862) / 16 $$\nu^{9}$$ $$=$$ $$( 6732 \beta_{19} - 3821 \beta_{18} - 7170 \beta_{17} - 9260 \beta_{16} - 169897 \beta_{14} + \cdots - 54757548 \beta_1 ) / 8$$ (6732*b19 - 3821*b18 - 7170*b17 - 9260*b16 - 169897*b14 + 21098*b13 - 4571346*b8 + 61823*b6 - 4584*b3 - 54757548*b1) / 8 $$\nu^{10}$$ $$=$$ $$( - 43657 \beta_{15} - 33706 \beta_{12} - 10014 \beta_{11} + 190175 \beta_{10} - 100502 \beta_{9} + \cdots - 75256117 ) / 4$$ (-43657*b15 - 33706*b12 - 10014*b11 + 190175*b10 - 100502*b9 + 1059909*b7 - 148659*b5 + 97511*b4 - 13877818*b2 - 75256117) / 4 $$\nu^{11}$$ $$=$$ $$( 580346 \beta_{19} - 1258084 \beta_{18} - 118064 \beta_{17} + 615200 \beta_{16} - 9935094 \beta_{14} + \cdots - 103249184 \beta_1 ) / 8$$ (580346*b19 - 1258084*b18 - 118064*b17 + 615200*b16 - 9935094*b14 - 3478174*b13 - 96666216*b8 + 8553832*b6 - 14399583*b3 - 103249184*b1) / 8 $$\nu^{12}$$ $$=$$ $$( 6059779 \beta_{15} + 28400713 \beta_{12} - 14017923 \beta_{11} - 10424432 \beta_{10} + \cdots + 213163934826 ) / 16$$ (6059779*b15 + 28400713*b12 - 14017923*b11 - 10424432*b10 - 24812163*b9 + 74008941*b7 - 58728329*b5 + 152069788*b4 - 190607826*b2 + 213163934826) / 16 $$\nu^{13}$$ $$=$$ $$( 41117344 \beta_{19} + 22827667 \beta_{18} - 43848850 \beta_{17} - 81976844 \beta_{16} + \cdots + 50040075636 \beta_1 ) / 8$$ (41117344*b19 + 22827667*b18 - 43848850*b17 - 81976844*b16 + 194656467*b14 - 713557154*b13 - 8922961490*b8 - 605195721*b6 - 29894022*b3 + 50040075636*b1) / 8 $$\nu^{14}$$ $$=$$ $$( 81463825 \beta_{15} + 243032197 \beta_{12} + 344100561 \beta_{11} + 922044958 \beta_{10} + \cdots - 2477399821968 ) / 8$$ (81463825*b15 + 243032197*b12 + 344100561*b11 + 922044958*b10 - 1605418911*b9 + 4093038615*b7 - 5786223599*b5 + 3186007546*b4 + 29290935290*b2 - 2477399821968) / 8 $$\nu^{15}$$ $$=$$ $$( 9617817002 \beta_{19} - 6285977854 \beta_{18} + 3398721388 \beta_{17} - 5988895864 \beta_{16} + \cdots - 1009722348984 \beta_1 ) / 8$$ (9617817002*b19 - 6285977854*b18 + 3398721388*b17 - 5988895864*b16 + 34611936576*b14 - 8599628914*b13 + 622089709940*b8 + 111624552630*b6 + 17203410541*b3 - 1009722348984*b1) / 8 $$\nu^{16}$$ $$=$$ $$( 31396928847 \beta_{15} - 178835848715 \beta_{12} + 3613266745 \beta_{11} - 19365091832 \beta_{10} + \cdots + 14\!\cdots\!66 ) / 16$$ (31396928847*b15 - 178835848715*b12 + 3613266745*b11 - 19365091832*b10 + 254841636985*b9 - 970768884671*b7 - 222541846109*b5 + 98802812580*b4 - 1788869543962*b2 + 1451823789245466) / 16 $$\nu^{17}$$ $$=$$ $$( - 103165004876 \beta_{19} - 41335429089 \beta_{18} + 222830891798 \beta_{17} + \cdots + 401708096627684 \beta_1 ) / 8$$ (-103165004876*b19 - 41335429089*b18 + 222830891798*b17 + 284134678244*b16 + 3118165926171*b14 - 1019072156278*b13 + 122827976301158*b8 - 5079131618293*b6 - 212262055644*b3 + 401708096627684*b1) / 8 $$\nu^{18}$$ $$=$$ $$( 942674446596 \beta_{15} + 765578160849 \beta_{12} + 206826061833 \beta_{11} - 5603630468293 \beta_{10} + \cdots - 21\!\cdots\!59 ) / 4$$ (942674446596*b15 + 765578160849*b12 + 206826061833*b11 - 5603630468293*b10 + 2996421299249*b9 - 26701510478280*b7 + 3373248208270*b5 + 436024400615*b4 + 113089484921976*b2 - 21060916700947659) / 4 $$\nu^{19}$$ $$=$$ $$( - 9760000253310 \beta_{19} + 35266025760656 \beta_{18} + 6286559214840 \beta_{17} + \cdots - 20\!\cdots\!60 \beta_1 ) / 8$$ (-9760000253310*b19 + 35266025760656*b18 + 6286559214840*b17 - 21513152013936*b16 + 229430616160790*b14 + 38221558085298*b13 + 2314559170464864*b8 - 382915783406468*b6 + 121054885930093*b3 - 20444027658224560*b1) / 8

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/100\mathbb{Z}\right)^\times$$.

 $$n$$ $$51$$ $$77$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
51.1
 −7.73824 − 2.02968i −7.73824 + 2.02968i −7.17826 − 3.53165i −7.17826 + 3.53165i −7.13604 − 3.61621i −7.13604 + 3.61621i −3.58697 − 7.15078i −3.58697 + 7.15078i −2.88145 − 7.46306i −2.88145 + 7.46306i 2.88145 − 7.46306i 2.88145 + 7.46306i 3.58697 − 7.15078i 3.58697 + 7.15078i 7.13604 − 3.61621i 7.13604 + 3.61621i 7.17826 − 3.53165i 7.17826 + 3.53165i 7.73824 − 2.02968i 7.73824 + 2.02968i
−15.4765 4.05935i 75.2390i 223.043 + 125.649i 0 −305.422 + 1164.44i 4411.35i −2941.87 2850.02i 900.091 0
51.2 −15.4765 + 4.05935i 75.2390i 223.043 125.649i 0 −305.422 1164.44i 4411.35i −2941.87 + 2850.02i 900.091 0
51.3 −14.3565 7.06329i 134.970i 156.220 + 202.809i 0 953.329 1937.69i 1863.96i −810.276 4015.06i −11655.8 0
51.4 −14.3565 + 7.06329i 134.970i 156.220 202.809i 0 953.329 + 1937.69i 1863.96i −810.276 + 4015.06i −11655.8 0
51.5 −14.2721 7.23243i 44.3270i 151.384 + 206.443i 0 −320.592 + 632.638i 2826.75i −667.476 4041.25i 4596.11 0
51.6 −14.2721 + 7.23243i 44.3270i 151.384 206.443i 0 −320.592 632.638i 2826.75i −667.476 + 4041.25i 4596.11 0
51.7 −7.17394 14.3016i 42.6663i −153.069 + 205.197i 0 −610.194 + 306.085i 869.685i 4032.75 + 717.054i 4740.59 0
51.8 −7.17394 + 14.3016i 42.6663i −153.069 205.197i 0 −610.194 306.085i 869.685i 4032.75 717.054i 4740.59 0
51.9 −5.76290 14.9261i 76.0331i −189.578 + 172.035i 0 1134.88 438.171i 269.408i 3660.34 + 1838.24i 779.974 0
51.10 −5.76290 + 14.9261i 76.0331i −189.578 172.035i 0 1134.88 + 438.171i 269.408i 3660.34 1838.24i 779.974 0
51.11 5.76290 14.9261i 76.0331i −189.578 172.035i 0 1134.88 + 438.171i 269.408i −3660.34 + 1838.24i 779.974 0
51.12 5.76290 + 14.9261i 76.0331i −189.578 + 172.035i 0 1134.88 438.171i 269.408i −3660.34 1838.24i 779.974 0
51.13 7.17394 14.3016i 42.6663i −153.069 205.197i 0 −610.194 306.085i 869.685i −4032.75 + 717.054i 4740.59 0
51.14 7.17394 + 14.3016i 42.6663i −153.069 + 205.197i 0 −610.194 + 306.085i 869.685i −4032.75 717.054i 4740.59 0
51.15 14.2721 7.23243i 44.3270i 151.384 206.443i 0 −320.592 632.638i 2826.75i 667.476 4041.25i 4596.11 0
51.16 14.2721 + 7.23243i 44.3270i 151.384 + 206.443i 0 −320.592 + 632.638i 2826.75i 667.476 + 4041.25i 4596.11 0
51.17 14.3565 7.06329i 134.970i 156.220 202.809i 0 953.329 + 1937.69i 1863.96i 810.276 4015.06i −11655.8 0
51.18 14.3565 + 7.06329i 134.970i 156.220 + 202.809i 0 953.329 1937.69i 1863.96i 810.276 + 4015.06i −11655.8 0
51.19 15.4765 4.05935i 75.2390i 223.043 125.649i 0 −305.422 1164.44i 4411.35i 2941.87 2850.02i 900.091 0
51.20 15.4765 + 4.05935i 75.2390i 223.043 + 125.649i 0 −305.422 + 1164.44i 4411.35i 2941.87 + 2850.02i 900.091 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 51.20 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.9.b.g 20
4.b odd 2 1 inner 100.9.b.g 20
5.b even 2 1 inner 100.9.b.g 20
5.c odd 4 2 20.9.d.c 20
20.d odd 2 1 inner 100.9.b.g 20
20.e even 4 2 20.9.d.c 20
40.i odd 4 2 320.9.h.g 20
40.k even 4 2 320.9.h.g 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.9.d.c 20 5.c odd 4 2
20.9.d.c 20 20.e even 4 2
100.9.b.g 20 1.a even 1 1 trivial
100.9.b.g 20 4.b odd 2 1 inner
100.9.b.g 20 5.b even 2 1 inner
100.9.b.g 20 20.d odd 2 1 inner
320.9.h.g 20 40.i odd 4 2
320.9.h.g 20 40.k even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{9}^{\mathrm{new}}(100, [\chi])$$:

 $$T_{3}^{10} + 33444 T_{3}^{8} + 357004800 T_{3}^{6} + 1615110935040 T_{3}^{4} + \cdots + 21\!\cdots\!00$$ T3^10 + 33444*T3^8 + 357004800*T3^6 + 1615110935040*T3^4 + 3119247097344000*T3^2 + 2132400444991488000 $$T_{13}^{10} - 4307527296 T_{13}^{8} + \cdots - 15\!\cdots\!00$$ T13^10 - 4307527296*T13^8 + 6777713303056951296*T13^6 - 4818723919967564715491721216*T13^4 + 1511377356832093209759389863212220416*T13^2 - 155657975855076136795350189556610720228966400

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{20} + \cdots + 12\!\cdots\!76$$
$3$ $$(T^{10} + \cdots + 21\!\cdots\!00)^{2}$$
$5$ $$T^{20}$$
$7$ $$(T^{10} + \cdots + 29\!\cdots\!00)^{2}$$
$11$ $$(T^{10} + \cdots + 15\!\cdots\!00)^{2}$$
$13$ $$(T^{10} + \cdots - 15\!\cdots\!00)^{2}$$
$17$ $$(T^{10} + \cdots - 57\!\cdots\!00)^{2}$$
$19$ $$(T^{10} + \cdots + 54\!\cdots\!00)^{2}$$
$23$ $$(T^{10} + \cdots + 72\!\cdots\!00)^{2}$$
$29$ $$(T^{5} + \cdots + 30\!\cdots\!52)^{4}$$
$31$ $$(T^{10} + \cdots + 51\!\cdots\!00)^{2}$$
$37$ $$(T^{10} + \cdots - 30\!\cdots\!00)^{2}$$
$41$ $$(T^{5} + \cdots + 51\!\cdots\!28)^{4}$$
$43$ $$(T^{10} + \cdots + 13\!\cdots\!00)^{2}$$
$47$ $$(T^{10} + \cdots + 25\!\cdots\!00)^{2}$$
$53$ $$(T^{10} + \cdots - 74\!\cdots\!00)^{2}$$
$59$ $$(T^{10} + \cdots + 53\!\cdots\!00)^{2}$$
$61$ $$(T^{5} + \cdots + 50\!\cdots\!68)^{4}$$
$67$ $$(T^{10} + \cdots + 19\!\cdots\!00)^{2}$$
$71$ $$(T^{10} + \cdots + 92\!\cdots\!00)^{2}$$
$73$ $$(T^{10} + \cdots - 86\!\cdots\!00)^{2}$$
$79$ $$(T^{10} + \cdots + 74\!\cdots\!00)^{2}$$
$83$ $$(T^{10} + \cdots + 78\!\cdots\!00)^{2}$$
$89$ $$(T^{5} + \cdots - 38\!\cdots\!28)^{4}$$
$97$ $$(T^{10} + \cdots - 34\!\cdots\!00)^{2}$$