# Properties

 Label 2025.2.a.j.1.1 Level $2025$ Weight $2$ Character 2025.1 Self dual yes Analytic conductor $16.170$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2025,2,Mod(1,2025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2025, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2025 = 3^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2025.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: $$16.1697064093$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 81) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 2025.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.73205 q^{2} +1.00000 q^{4} -2.00000 q^{7} +1.73205 q^{8} +O(q^{10})$$ $$q-1.73205 q^{2} +1.00000 q^{4} -2.00000 q^{7} +1.73205 q^{8} -3.46410 q^{11} +1.00000 q^{13} +3.46410 q^{14} -5.00000 q^{16} -5.19615 q^{17} +2.00000 q^{19} +6.00000 q^{22} -3.46410 q^{23} -1.73205 q^{26} -2.00000 q^{28} +1.73205 q^{29} +8.00000 q^{31} +5.19615 q^{32} +9.00000 q^{34} +7.00000 q^{37} -3.46410 q^{38} -6.92820 q^{41} -2.00000 q^{43} -3.46410 q^{44} +6.00000 q^{46} -6.92820 q^{47} -3.00000 q^{49} +1.00000 q^{52} -3.46410 q^{56} -3.00000 q^{58} +13.8564 q^{59} -7.00000 q^{61} -13.8564 q^{62} +1.00000 q^{64} +10.0000 q^{67} -5.19615 q^{68} -10.3923 q^{71} +7.00000 q^{73} -12.1244 q^{74} +2.00000 q^{76} +6.92820 q^{77} +2.00000 q^{79} +12.0000 q^{82} +13.8564 q^{83} +3.46410 q^{86} -6.00000 q^{88} -5.19615 q^{89} -2.00000 q^{91} -3.46410 q^{92} +12.0000 q^{94} -2.00000 q^{97} +5.19615 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} - 4 q^{7}+O(q^{10})$$ 2 * q + 2 * q^4 - 4 * q^7 $$2 q + 2 q^{4} - 4 q^{7} + 2 q^{13} - 10 q^{16} + 4 q^{19} + 12 q^{22} - 4 q^{28} + 16 q^{31} + 18 q^{34} + 14 q^{37} - 4 q^{43} + 12 q^{46} - 6 q^{49} + 2 q^{52} - 6 q^{58} - 14 q^{61} + 2 q^{64} + 20 q^{67} + 14 q^{73} + 4 q^{76} + 4 q^{79} + 24 q^{82} - 12 q^{88} - 4 q^{91} + 24 q^{94} - 4 q^{97}+O(q^{100})$$ 2 * q + 2 * q^4 - 4 * q^7 + 2 * q^13 - 10 * q^16 + 4 * q^19 + 12 * q^22 - 4 * q^28 + 16 * q^31 + 18 * q^34 + 14 * q^37 - 4 * q^43 + 12 * q^46 - 6 * q^49 + 2 * q^52 - 6 * q^58 - 14 * q^61 + 2 * q^64 + 20 * q^67 + 14 * q^73 + 4 * q^76 + 4 * q^79 + 24 * q^82 - 12 * q^88 - 4 * q^91 + 24 * q^94 - 4 * q^97

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.73205 −1.22474 −0.612372 0.790569i $$-0.709785\pi$$
−0.612372 + 0.790569i $$0.709785\pi$$
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −2.00000 −0.755929 −0.377964 0.925820i $$-0.623376\pi$$
−0.377964 + 0.925820i $$0.623376\pi$$
$$8$$ 1.73205 0.612372
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −3.46410 −1.04447 −0.522233 0.852803i $$-0.674901\pi$$
−0.522233 + 0.852803i $$0.674901\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350 0.138675 0.990338i $$-0.455716\pi$$
0.138675 + 0.990338i $$0.455716\pi$$
$$14$$ 3.46410 0.925820
$$15$$ 0 0
$$16$$ −5.00000 −1.25000
$$17$$ −5.19615 −1.26025 −0.630126 0.776493i $$-0.716997\pi$$
−0.630126 + 0.776493i $$0.716997\pi$$
$$18$$ 0 0
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 6.00000 1.27920
$$23$$ −3.46410 −0.722315 −0.361158 0.932505i $$-0.617618\pi$$
−0.361158 + 0.932505i $$0.617618\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −1.73205 −0.339683
$$27$$ 0 0
$$28$$ −2.00000 −0.377964
$$29$$ 1.73205 0.321634 0.160817 0.986984i $$-0.448587\pi$$
0.160817 + 0.986984i $$0.448587\pi$$
$$30$$ 0 0
$$31$$ 8.00000 1.43684 0.718421 0.695608i $$-0.244865\pi$$
0.718421 + 0.695608i $$0.244865\pi$$
$$32$$ 5.19615 0.918559
$$33$$ 0 0
$$34$$ 9.00000 1.54349
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 7.00000 1.15079 0.575396 0.817875i $$-0.304848\pi$$
0.575396 + 0.817875i $$0.304848\pi$$
$$38$$ −3.46410 −0.561951
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −6.92820 −1.08200 −0.541002 0.841021i $$-0.681955\pi$$
−0.541002 + 0.841021i $$0.681955\pi$$
$$42$$ 0 0
$$43$$ −2.00000 −0.304997 −0.152499 0.988304i $$-0.548732\pi$$
−0.152499 + 0.988304i $$0.548732\pi$$
$$44$$ −3.46410 −0.522233
$$45$$ 0 0
$$46$$ 6.00000 0.884652
$$47$$ −6.92820 −1.01058 −0.505291 0.862949i $$-0.668615\pi$$
−0.505291 + 0.862949i $$0.668615\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 1.00000 0.138675
$$53$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −3.46410 −0.462910
$$57$$ 0 0
$$58$$ −3.00000 −0.393919
$$59$$ 13.8564 1.80395 0.901975 0.431788i $$-0.142117\pi$$
0.901975 + 0.431788i $$0.142117\pi$$
$$60$$ 0 0
$$61$$ −7.00000 −0.896258 −0.448129 0.893969i $$-0.647910\pi$$
−0.448129 + 0.893969i $$0.647910\pi$$
$$62$$ −13.8564 −1.75977
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 10.0000 1.22169 0.610847 0.791748i $$-0.290829\pi$$
0.610847 + 0.791748i $$0.290829\pi$$
$$68$$ −5.19615 −0.630126
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −10.3923 −1.23334 −0.616670 0.787222i $$-0.711519\pi$$
−0.616670 + 0.787222i $$0.711519\pi$$
$$72$$ 0 0
$$73$$ 7.00000 0.819288 0.409644 0.912245i $$-0.365653\pi$$
0.409644 + 0.912245i $$0.365653\pi$$
$$74$$ −12.1244 −1.40943
$$75$$ 0 0
$$76$$ 2.00000 0.229416
$$77$$ 6.92820 0.789542
$$78$$ 0 0
$$79$$ 2.00000 0.225018 0.112509 0.993651i $$-0.464111\pi$$
0.112509 + 0.993651i $$0.464111\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 12.0000 1.32518
$$83$$ 13.8564 1.52094 0.760469 0.649374i $$-0.224969\pi$$
0.760469 + 0.649374i $$0.224969\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 3.46410 0.373544
$$87$$ 0 0
$$88$$ −6.00000 −0.639602
$$89$$ −5.19615 −0.550791 −0.275396 0.961331i $$-0.588809\pi$$
−0.275396 + 0.961331i $$0.588809\pi$$
$$90$$ 0 0
$$91$$ −2.00000 −0.209657
$$92$$ −3.46410 −0.361158
$$93$$ 0 0
$$94$$ 12.0000 1.23771
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −2.00000 −0.203069 −0.101535 0.994832i $$-0.532375\pi$$
−0.101535 + 0.994832i $$0.532375\pi$$
$$98$$ 5.19615 0.524891
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 6.92820 0.689382 0.344691 0.938716i $$-0.387984\pi$$
0.344691 + 0.938716i $$0.387984\pi$$
$$102$$ 0 0
$$103$$ −8.00000 −0.788263 −0.394132 0.919054i $$-0.628955\pi$$
−0.394132 + 0.919054i $$0.628955\pi$$
$$104$$ 1.73205 0.169842
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 0 0
$$109$$ 11.0000 1.05361 0.526804 0.849987i $$-0.323390\pi$$
0.526804 + 0.849987i $$0.323390\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 10.0000 0.944911
$$113$$ 1.73205 0.162938 0.0814688 0.996676i $$-0.474039\pi$$
0.0814688 + 0.996676i $$0.474039\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 1.73205 0.160817
$$117$$ 0 0
$$118$$ −24.0000 −2.20938
$$119$$ 10.3923 0.952661
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 12.1244 1.09769
$$123$$ 0 0
$$124$$ 8.00000 0.718421
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −2.00000 −0.177471 −0.0887357 0.996055i $$-0.528283\pi$$
−0.0887357 + 0.996055i $$0.528283\pi$$
$$128$$ −12.1244 −1.07165
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 3.46410 0.302660 0.151330 0.988483i $$-0.451644\pi$$
0.151330 + 0.988483i $$0.451644\pi$$
$$132$$ 0 0
$$133$$ −4.00000 −0.346844
$$134$$ −17.3205 −1.49626
$$135$$ 0 0
$$136$$ −9.00000 −0.771744
$$137$$ −1.73205 −0.147979 −0.0739895 0.997259i $$-0.523573\pi$$
−0.0739895 + 0.997259i $$0.523573\pi$$
$$138$$ 0 0
$$139$$ 8.00000 0.678551 0.339276 0.940687i $$-0.389818\pi$$
0.339276 + 0.940687i $$0.389818\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 18.0000 1.51053
$$143$$ −3.46410 −0.289683
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −12.1244 −1.00342
$$147$$ 0 0
$$148$$ 7.00000 0.575396
$$149$$ 8.66025 0.709476 0.354738 0.934966i $$-0.384570\pi$$
0.354738 + 0.934966i $$0.384570\pi$$
$$150$$ 0 0
$$151$$ 20.0000 1.62758 0.813788 0.581161i $$-0.197401\pi$$
0.813788 + 0.581161i $$0.197401\pi$$
$$152$$ 3.46410 0.280976
$$153$$ 0 0
$$154$$ −12.0000 −0.966988
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −17.0000 −1.35675 −0.678374 0.734717i $$-0.737315\pi$$
−0.678374 + 0.734717i $$0.737315\pi$$
$$158$$ −3.46410 −0.275589
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 6.92820 0.546019
$$162$$ 0 0
$$163$$ 16.0000 1.25322 0.626608 0.779334i $$-0.284443\pi$$
0.626608 + 0.779334i $$0.284443\pi$$
$$164$$ −6.92820 −0.541002
$$165$$ 0 0
$$166$$ −24.0000 −1.86276
$$167$$ 17.3205 1.34030 0.670151 0.742225i $$-0.266230\pi$$
0.670151 + 0.742225i $$0.266230\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −2.00000 −0.152499
$$173$$ 19.0526 1.44854 0.724270 0.689517i $$-0.242177\pi$$
0.724270 + 0.689517i $$0.242177\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 17.3205 1.30558
$$177$$ 0 0
$$178$$ 9.00000 0.674579
$$179$$ 20.7846 1.55351 0.776757 0.629800i $$-0.216863\pi$$
0.776757 + 0.629800i $$0.216863\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ 3.46410 0.256776
$$183$$ 0 0
$$184$$ −6.00000 −0.442326
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 18.0000 1.31629
$$188$$ −6.92820 −0.505291
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 17.3205 1.25327 0.626634 0.779314i $$-0.284432\pi$$
0.626634 + 0.779314i $$0.284432\pi$$
$$192$$ 0 0
$$193$$ 1.00000 0.0719816 0.0359908 0.999352i $$-0.488541\pi$$
0.0359908 + 0.999352i $$0.488541\pi$$
$$194$$ 3.46410 0.248708
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ 5.19615 0.370211 0.185105 0.982719i $$-0.440737\pi$$
0.185105 + 0.982719i $$0.440737\pi$$
$$198$$ 0 0
$$199$$ 20.0000 1.41776 0.708881 0.705328i $$-0.249200\pi$$
0.708881 + 0.705328i $$0.249200\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −12.0000 −0.844317
$$203$$ −3.46410 −0.243132
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 13.8564 0.965422
$$207$$ 0 0
$$208$$ −5.00000 −0.346688
$$209$$ −6.92820 −0.479234
$$210$$ 0 0
$$211$$ −10.0000 −0.688428 −0.344214 0.938891i $$-0.611855\pi$$
−0.344214 + 0.938891i $$0.611855\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −16.0000 −1.08615
$$218$$ −19.0526 −1.29040
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −5.19615 −0.349531
$$222$$ 0 0
$$223$$ −2.00000 −0.133930 −0.0669650 0.997755i $$-0.521332\pi$$
−0.0669650 + 0.997755i $$0.521332\pi$$
$$224$$ −10.3923 −0.694365
$$225$$ 0 0
$$226$$ −3.00000 −0.199557
$$227$$ 3.46410 0.229920 0.114960 0.993370i $$-0.463326\pi$$
0.114960 + 0.993370i $$0.463326\pi$$
$$228$$ 0 0
$$229$$ −1.00000 −0.0660819 −0.0330409 0.999454i $$-0.510519\pi$$
−0.0330409 + 0.999454i $$0.510519\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 3.00000 0.196960
$$233$$ 25.9808 1.70206 0.851028 0.525120i $$-0.175980\pi$$
0.851028 + 0.525120i $$0.175980\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 13.8564 0.901975
$$237$$ 0 0
$$238$$ −18.0000 −1.16677
$$239$$ −27.7128 −1.79259 −0.896296 0.443455i $$-0.853752\pi$$
−0.896296 + 0.443455i $$0.853752\pi$$
$$240$$ 0 0
$$241$$ 29.0000 1.86805 0.934027 0.357202i $$-0.116269\pi$$
0.934027 + 0.357202i $$0.116269\pi$$
$$242$$ −1.73205 −0.111340
$$243$$ 0 0
$$244$$ −7.00000 −0.448129
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 2.00000 0.127257
$$248$$ 13.8564 0.879883
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 10.3923 0.655956 0.327978 0.944685i $$-0.393633\pi$$
0.327978 + 0.944685i $$0.393633\pi$$
$$252$$ 0 0
$$253$$ 12.0000 0.754434
$$254$$ 3.46410 0.217357
$$255$$ 0 0
$$256$$ 19.0000 1.18750
$$257$$ −8.66025 −0.540212 −0.270106 0.962831i $$-0.587059\pi$$
−0.270106 + 0.962831i $$0.587059\pi$$
$$258$$ 0 0
$$259$$ −14.0000 −0.869918
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −6.00000 −0.370681
$$263$$ −6.92820 −0.427211 −0.213606 0.976920i $$-0.568521\pi$$
−0.213606 + 0.976920i $$0.568521\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 6.92820 0.424795
$$267$$ 0 0
$$268$$ 10.0000 0.610847
$$269$$ −15.5885 −0.950445 −0.475223 0.879866i $$-0.657632\pi$$
−0.475223 + 0.879866i $$0.657632\pi$$
$$270$$ 0 0
$$271$$ 2.00000 0.121491 0.0607457 0.998153i $$-0.480652\pi$$
0.0607457 + 0.998153i $$0.480652\pi$$
$$272$$ 25.9808 1.57532
$$273$$ 0 0
$$274$$ 3.00000 0.181237
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −2.00000 −0.120168 −0.0600842 0.998193i $$-0.519137\pi$$
−0.0600842 + 0.998193i $$0.519137\pi$$
$$278$$ −13.8564 −0.831052
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 12.1244 0.723278 0.361639 0.932318i $$-0.382217\pi$$
0.361639 + 0.932318i $$0.382217\pi$$
$$282$$ 0 0
$$283$$ 28.0000 1.66443 0.832214 0.554455i $$-0.187073\pi$$
0.832214 + 0.554455i $$0.187073\pi$$
$$284$$ −10.3923 −0.616670
$$285$$ 0 0
$$286$$ 6.00000 0.354787
$$287$$ 13.8564 0.817918
$$288$$ 0 0
$$289$$ 10.0000 0.588235
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 7.00000 0.409644
$$293$$ −19.0526 −1.11306 −0.556531 0.830827i $$-0.687868\pi$$
−0.556531 + 0.830827i $$0.687868\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 12.1244 0.704714
$$297$$ 0 0
$$298$$ −15.0000 −0.868927
$$299$$ −3.46410 −0.200334
$$300$$ 0 0
$$301$$ 4.00000 0.230556
$$302$$ −34.6410 −1.99337
$$303$$ 0 0
$$304$$ −10.0000 −0.573539
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 16.0000 0.913168 0.456584 0.889680i $$-0.349073\pi$$
0.456584 + 0.889680i $$0.349073\pi$$
$$308$$ 6.92820 0.394771
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −6.92820 −0.392862 −0.196431 0.980518i $$-0.562935\pi$$
−0.196431 + 0.980518i $$0.562935\pi$$
$$312$$ 0 0
$$313$$ 25.0000 1.41308 0.706542 0.707671i $$-0.250254\pi$$
0.706542 + 0.707671i $$0.250254\pi$$
$$314$$ 29.4449 1.66167
$$315$$ 0 0
$$316$$ 2.00000 0.112509
$$317$$ 8.66025 0.486408 0.243204 0.969975i $$-0.421801\pi$$
0.243204 + 0.969975i $$0.421801\pi$$
$$318$$ 0 0
$$319$$ −6.00000 −0.335936
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −12.0000 −0.668734
$$323$$ −10.3923 −0.578243
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −27.7128 −1.53487
$$327$$ 0 0
$$328$$ −12.0000 −0.662589
$$329$$ 13.8564 0.763928
$$330$$ 0 0
$$331$$ 2.00000 0.109930 0.0549650 0.998488i $$-0.482495\pi$$
0.0549650 + 0.998488i $$0.482495\pi$$
$$332$$ 13.8564 0.760469
$$333$$ 0 0
$$334$$ −30.0000 −1.64153
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −26.0000 −1.41631 −0.708155 0.706057i $$-0.750472\pi$$
−0.708155 + 0.706057i $$0.750472\pi$$
$$338$$ 20.7846 1.13053
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −27.7128 −1.50073
$$342$$ 0 0
$$343$$ 20.0000 1.07990
$$344$$ −3.46410 −0.186772
$$345$$ 0 0
$$346$$ −33.0000 −1.77409
$$347$$ −3.46410 −0.185963 −0.0929814 0.995668i $$-0.529640\pi$$
−0.0929814 + 0.995668i $$0.529640\pi$$
$$348$$ 0 0
$$349$$ 2.00000 0.107058 0.0535288 0.998566i $$-0.482953\pi$$
0.0535288 + 0.998566i $$0.482953\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −18.0000 −0.959403
$$353$$ 13.8564 0.737502 0.368751 0.929528i $$-0.379785\pi$$
0.368751 + 0.929528i $$0.379785\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −5.19615 −0.275396
$$357$$ 0 0
$$358$$ −36.0000 −1.90266
$$359$$ 10.3923 0.548485 0.274242 0.961661i $$-0.411573\pi$$
0.274242 + 0.961661i $$0.411573\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ −3.46410 −0.182069
$$363$$ 0 0
$$364$$ −2.00000 −0.104828
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −20.0000 −1.04399 −0.521996 0.852948i $$-0.674812\pi$$
−0.521996 + 0.852948i $$0.674812\pi$$
$$368$$ 17.3205 0.902894
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 10.0000 0.517780 0.258890 0.965907i $$-0.416643\pi$$
0.258890 + 0.965907i $$0.416643\pi$$
$$374$$ −31.1769 −1.61212
$$375$$ 0 0
$$376$$ −12.0000 −0.618853
$$377$$ 1.73205 0.0892052
$$378$$ 0 0
$$379$$ −16.0000 −0.821865 −0.410932 0.911666i $$-0.634797\pi$$
−0.410932 + 0.911666i $$0.634797\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −30.0000 −1.53493
$$383$$ 17.3205 0.885037 0.442518 0.896759i $$-0.354085\pi$$
0.442518 + 0.896759i $$0.354085\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −1.73205 −0.0881591
$$387$$ 0 0
$$388$$ −2.00000 −0.101535
$$389$$ 27.7128 1.40510 0.702548 0.711637i $$-0.252046\pi$$
0.702548 + 0.711637i $$0.252046\pi$$
$$390$$ 0 0
$$391$$ 18.0000 0.910299
$$392$$ −5.19615 −0.262445
$$393$$ 0 0
$$394$$ −9.00000 −0.453413
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −29.0000 −1.45547 −0.727734 0.685859i $$-0.759427\pi$$
−0.727734 + 0.685859i $$0.759427\pi$$
$$398$$ −34.6410 −1.73640
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −12.1244 −0.605461 −0.302731 0.953076i $$-0.597898\pi$$
−0.302731 + 0.953076i $$0.597898\pi$$
$$402$$ 0 0
$$403$$ 8.00000 0.398508
$$404$$ 6.92820 0.344691
$$405$$ 0 0
$$406$$ 6.00000 0.297775
$$407$$ −24.2487 −1.20196
$$408$$ 0 0
$$409$$ −19.0000 −0.939490 −0.469745 0.882802i $$-0.655654\pi$$
−0.469745 + 0.882802i $$0.655654\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −8.00000 −0.394132
$$413$$ −27.7128 −1.36366
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 5.19615 0.254762
$$417$$ 0 0
$$418$$ 12.0000 0.586939
$$419$$ −6.92820 −0.338465 −0.169232 0.985576i $$-0.554129\pi$$
−0.169232 + 0.985576i $$0.554129\pi$$
$$420$$ 0 0
$$421$$ −25.0000 −1.21843 −0.609213 0.793007i $$-0.708514\pi$$
−0.609213 + 0.793007i $$0.708514\pi$$
$$422$$ 17.3205 0.843149
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 14.0000 0.677507
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ −11.0000 −0.528626 −0.264313 0.964437i $$-0.585145\pi$$
−0.264313 + 0.964437i $$0.585145\pi$$
$$434$$ 27.7128 1.33026
$$435$$ 0 0
$$436$$ 11.0000 0.526804
$$437$$ −6.92820 −0.331421
$$438$$ 0 0
$$439$$ 20.0000 0.954548 0.477274 0.878755i $$-0.341625\pi$$
0.477274 + 0.878755i $$0.341625\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 9.00000 0.428086
$$443$$ 34.6410 1.64584 0.822922 0.568154i $$-0.192342\pi$$
0.822922 + 0.568154i $$0.192342\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 3.46410 0.164030
$$447$$ 0 0
$$448$$ −2.00000 −0.0944911
$$449$$ 20.7846 0.980886 0.490443 0.871473i $$-0.336835\pi$$
0.490443 + 0.871473i $$0.336835\pi$$
$$450$$ 0 0
$$451$$ 24.0000 1.13012
$$452$$ 1.73205 0.0814688
$$453$$ 0 0
$$454$$ −6.00000 −0.281594
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −29.0000 −1.35656 −0.678281 0.734802i $$-0.737275\pi$$
−0.678281 + 0.734802i $$0.737275\pi$$
$$458$$ 1.73205 0.0809334
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −13.8564 −0.645357 −0.322679 0.946509i $$-0.604583\pi$$
−0.322679 + 0.946509i $$0.604583\pi$$
$$462$$ 0 0
$$463$$ −8.00000 −0.371792 −0.185896 0.982569i $$-0.559519\pi$$
−0.185896 + 0.982569i $$0.559519\pi$$
$$464$$ −8.66025 −0.402042
$$465$$ 0 0
$$466$$ −45.0000 −2.08458
$$467$$ 20.7846 0.961797 0.480899 0.876776i $$-0.340311\pi$$
0.480899 + 0.876776i $$0.340311\pi$$
$$468$$ 0 0
$$469$$ −20.0000 −0.923514
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 24.0000 1.10469
$$473$$ 6.92820 0.318559
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 10.3923 0.476331
$$477$$ 0 0
$$478$$ 48.0000 2.19547
$$479$$ −24.2487 −1.10795 −0.553976 0.832533i $$-0.686890\pi$$
−0.553976 + 0.832533i $$0.686890\pi$$
$$480$$ 0 0
$$481$$ 7.00000 0.319173
$$482$$ −50.2295 −2.28789
$$483$$ 0 0
$$484$$ 1.00000 0.0454545
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 16.0000 0.725029 0.362515 0.931978i $$-0.381918\pi$$
0.362515 + 0.931978i $$0.381918\pi$$
$$488$$ −12.1244 −0.548844
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −17.3205 −0.781664 −0.390832 0.920462i $$-0.627813\pi$$
−0.390832 + 0.920462i $$0.627813\pi$$
$$492$$ 0 0
$$493$$ −9.00000 −0.405340
$$494$$ −3.46410 −0.155857
$$495$$ 0 0
$$496$$ −40.0000 −1.79605
$$497$$ 20.7846 0.932317
$$498$$ 0 0
$$499$$ −10.0000 −0.447661 −0.223831 0.974628i $$-0.571856\pi$$
−0.223831 + 0.974628i $$0.571856\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −18.0000 −0.803379
$$503$$ −20.7846 −0.926740 −0.463370 0.886165i $$-0.653360\pi$$
−0.463370 + 0.886165i $$0.653360\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −20.7846 −0.923989
$$507$$ 0 0
$$508$$ −2.00000 −0.0887357
$$509$$ −27.7128 −1.22835 −0.614174 0.789170i $$-0.710511\pi$$
−0.614174 + 0.789170i $$0.710511\pi$$
$$510$$ 0 0
$$511$$ −14.0000 −0.619324
$$512$$ −8.66025 −0.382733
$$513$$ 0 0
$$514$$ 15.0000 0.661622
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 24.0000 1.05552
$$518$$ 24.2487 1.06543
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −20.7846 −0.910590 −0.455295 0.890341i $$-0.650466\pi$$
−0.455295 + 0.890341i $$0.650466\pi$$
$$522$$ 0 0
$$523$$ −38.0000 −1.66162 −0.830812 0.556553i $$-0.812124\pi$$
−0.830812 + 0.556553i $$0.812124\pi$$
$$524$$ 3.46410 0.151330
$$525$$ 0 0
$$526$$ 12.0000 0.523225
$$527$$ −41.5692 −1.81078
$$528$$ 0 0
$$529$$ −11.0000 −0.478261
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −4.00000 −0.173422
$$533$$ −6.92820 −0.300094
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 17.3205 0.748132
$$537$$ 0 0
$$538$$ 27.0000 1.16405
$$539$$ 10.3923 0.447628
$$540$$ 0 0
$$541$$ 11.0000 0.472927 0.236463 0.971640i $$-0.424012\pi$$
0.236463 + 0.971640i $$0.424012\pi$$
$$542$$ −3.46410 −0.148796
$$543$$ 0 0
$$544$$ −27.0000 −1.15762
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −20.0000 −0.855138 −0.427569 0.903983i $$-0.640630\pi$$
−0.427569 + 0.903983i $$0.640630\pi$$
$$548$$ −1.73205 −0.0739895
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 3.46410 0.147576
$$552$$ 0 0
$$553$$ −4.00000 −0.170097
$$554$$ 3.46410 0.147176
$$555$$ 0 0
$$556$$ 8.00000 0.339276
$$557$$ −36.3731 −1.54118 −0.770588 0.637333i $$-0.780037\pi$$
−0.770588 + 0.637333i $$0.780037\pi$$
$$558$$ 0 0
$$559$$ −2.00000 −0.0845910
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −21.0000 −0.885832
$$563$$ −34.6410 −1.45994 −0.729972 0.683477i $$-0.760467\pi$$
−0.729972 + 0.683477i $$0.760467\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −48.4974 −2.03850
$$567$$ 0 0
$$568$$ −18.0000 −0.755263
$$569$$ 32.9090 1.37962 0.689808 0.723993i $$-0.257695\pi$$
0.689808 + 0.723993i $$0.257695\pi$$
$$570$$ 0 0
$$571$$ 8.00000 0.334790 0.167395 0.985890i $$-0.446465\pi$$
0.167395 + 0.985890i $$0.446465\pi$$
$$572$$ −3.46410 −0.144841
$$573$$ 0 0
$$574$$ −24.0000 −1.00174
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −11.0000 −0.457936 −0.228968 0.973434i $$-0.573535\pi$$
−0.228968 + 0.973434i $$0.573535\pi$$
$$578$$ −17.3205 −0.720438
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −27.7128 −1.14972
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 12.1244 0.501709
$$585$$ 0 0
$$586$$ 33.0000 1.36322
$$587$$ −38.1051 −1.57277 −0.786383 0.617739i $$-0.788049\pi$$
−0.786383 + 0.617739i $$0.788049\pi$$
$$588$$ 0 0
$$589$$ 16.0000 0.659269
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −35.0000 −1.43849
$$593$$ −15.5885 −0.640141 −0.320071 0.947394i $$-0.603707\pi$$
−0.320071 + 0.947394i $$0.603707\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 8.66025 0.354738
$$597$$ 0 0
$$598$$ 6.00000 0.245358
$$599$$ 13.8564 0.566157 0.283079 0.959097i $$-0.408644\pi$$
0.283079 + 0.959097i $$0.408644\pi$$
$$600$$ 0 0
$$601$$ −25.0000 −1.01977 −0.509886 0.860242i $$-0.670312\pi$$
−0.509886 + 0.860242i $$0.670312\pi$$
$$602$$ −6.92820 −0.282372
$$603$$ 0 0
$$604$$ 20.0000 0.813788
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −26.0000 −1.05531 −0.527654 0.849460i $$-0.676928\pi$$
−0.527654 + 0.849460i $$0.676928\pi$$
$$608$$ 10.3923 0.421464
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −6.92820 −0.280285
$$612$$ 0 0
$$613$$ 34.0000 1.37325 0.686624 0.727013i $$-0.259092\pi$$
0.686624 + 0.727013i $$0.259092\pi$$
$$614$$ −27.7128 −1.11840
$$615$$ 0 0
$$616$$ 12.0000 0.483494
$$617$$ 12.1244 0.488108 0.244054 0.969762i $$-0.421523\pi$$
0.244054 + 0.969762i $$0.421523\pi$$
$$618$$ 0 0
$$619$$ 20.0000 0.803868 0.401934 0.915669i $$-0.368338\pi$$
0.401934 + 0.915669i $$0.368338\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 12.0000 0.481156
$$623$$ 10.3923 0.416359
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −43.3013 −1.73067
$$627$$ 0 0
$$628$$ −17.0000 −0.678374
$$629$$ −36.3731 −1.45029
$$630$$ 0 0
$$631$$ 20.0000 0.796187 0.398094 0.917345i $$-0.369672\pi$$
0.398094 + 0.917345i $$0.369672\pi$$
$$632$$ 3.46410 0.137795
$$633$$ 0 0
$$634$$ −15.0000 −0.595726
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −3.00000 −0.118864
$$638$$ 10.3923 0.411435
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 22.5167 0.889355 0.444677 0.895691i $$-0.353318\pi$$
0.444677 + 0.895691i $$0.353318\pi$$
$$642$$ 0 0
$$643$$ −8.00000 −0.315489 −0.157745 0.987480i $$-0.550422\pi$$
−0.157745 + 0.987480i $$0.550422\pi$$
$$644$$ 6.92820 0.273009
$$645$$ 0 0
$$646$$ 18.0000 0.708201
$$647$$ 31.1769 1.22569 0.612845 0.790203i $$-0.290025\pi$$
0.612845 + 0.790203i $$0.290025\pi$$
$$648$$ 0 0
$$649$$ −48.0000 −1.88416
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 16.0000 0.626608
$$653$$ −13.8564 −0.542243 −0.271122 0.962545i $$-0.587395\pi$$
−0.271122 + 0.962545i $$0.587395\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 34.6410 1.35250
$$657$$ 0 0
$$658$$ −24.0000 −0.935617
$$659$$ −3.46410 −0.134942 −0.0674711 0.997721i $$-0.521493\pi$$
−0.0674711 + 0.997721i $$0.521493\pi$$
$$660$$ 0 0
$$661$$ 17.0000 0.661223 0.330612 0.943767i $$-0.392745\pi$$
0.330612 + 0.943767i $$0.392745\pi$$
$$662$$ −3.46410 −0.134636
$$663$$ 0 0
$$664$$ 24.0000 0.931381
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −6.00000 −0.232321
$$668$$ 17.3205 0.670151
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 24.2487 0.936111
$$672$$ 0 0
$$673$$ 25.0000 0.963679 0.481840 0.876259i $$-0.339969\pi$$
0.481840 + 0.876259i $$0.339969\pi$$
$$674$$ 45.0333 1.73462
$$675$$ 0 0
$$676$$ −12.0000 −0.461538
$$677$$ 13.8564 0.532545 0.266272 0.963898i $$-0.414208\pi$$
0.266272 + 0.963898i $$0.414208\pi$$
$$678$$ 0 0
$$679$$ 4.00000 0.153506
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 48.0000 1.83801
$$683$$ 20.7846 0.795301 0.397650 0.917537i $$-0.369826\pi$$
0.397650 + 0.917537i $$0.369826\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −34.6410 −1.32260
$$687$$ 0 0
$$688$$ 10.0000 0.381246
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 2.00000 0.0760836 0.0380418 0.999276i $$-0.487888\pi$$
0.0380418 + 0.999276i $$0.487888\pi$$
$$692$$ 19.0526 0.724270
$$693$$ 0 0
$$694$$ 6.00000 0.227757
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 36.0000 1.36360
$$698$$ −3.46410 −0.131118
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 46.7654 1.76630 0.883152 0.469087i $$-0.155417\pi$$
0.883152 + 0.469087i $$0.155417\pi$$
$$702$$ 0 0
$$703$$ 14.0000 0.528020
$$704$$ −3.46410 −0.130558
$$705$$ 0 0
$$706$$ −24.0000 −0.903252
$$707$$ −13.8564 −0.521124
$$708$$ 0 0
$$709$$ −25.0000 −0.938895 −0.469447 0.882960i $$-0.655547\pi$$
−0.469447 + 0.882960i $$0.655547\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −9.00000 −0.337289
$$713$$ −27.7128 −1.03785
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 20.7846 0.776757
$$717$$ 0 0
$$718$$ −18.0000 −0.671754
$$719$$ −10.3923 −0.387568 −0.193784 0.981044i $$-0.562076\pi$$
−0.193784 + 0.981044i $$0.562076\pi$$
$$720$$ 0 0
$$721$$ 16.0000 0.595871
$$722$$ 25.9808 0.966904
$$723$$ 0 0
$$724$$ 2.00000 0.0743294
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 34.0000 1.26099 0.630495 0.776193i $$-0.282852\pi$$
0.630495 + 0.776193i $$0.282852\pi$$
$$728$$ −3.46410 −0.128388
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 10.3923 0.384373
$$732$$ 0 0
$$733$$ 46.0000 1.69905 0.849524 0.527549i $$-0.176889\pi$$
0.849524 + 0.527549i $$0.176889\pi$$
$$734$$ 34.6410 1.27862
$$735$$ 0 0
$$736$$ −18.0000 −0.663489
$$737$$ −34.6410 −1.27602
$$738$$ 0 0
$$739$$ 20.0000 0.735712 0.367856 0.929883i $$-0.380092\pi$$
0.367856 + 0.929883i $$0.380092\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 6.92820 0.254171 0.127086 0.991892i $$-0.459438\pi$$
0.127086 + 0.991892i $$0.459438\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −17.3205 −0.634149
$$747$$ 0 0
$$748$$ 18.0000 0.658145
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −10.0000 −0.364905 −0.182453 0.983215i $$-0.558404\pi$$
−0.182453 + 0.983215i $$0.558404\pi$$
$$752$$ 34.6410 1.26323
$$753$$ 0 0
$$754$$ −3.00000 −0.109254
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −2.00000 −0.0726912 −0.0363456 0.999339i $$-0.511572\pi$$
−0.0363456 + 0.999339i $$0.511572\pi$$
$$758$$ 27.7128 1.00657
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 29.4449 1.06738 0.533688 0.845682i $$-0.320806\pi$$
0.533688 + 0.845682i $$0.320806\pi$$
$$762$$ 0 0
$$763$$ −22.0000 −0.796453
$$764$$ 17.3205 0.626634
$$765$$ 0 0
$$766$$ −30.0000 −1.08394
$$767$$ 13.8564 0.500326
$$768$$ 0 0
$$769$$ −1.00000 −0.0360609 −0.0180305 0.999837i $$-0.505740\pi$$
−0.0180305 + 0.999837i $$0.505740\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 1.00000 0.0359908
$$773$$ 25.9808 0.934463 0.467232 0.884135i $$-0.345251\pi$$
0.467232 + 0.884135i $$0.345251\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −3.46410 −0.124354
$$777$$ 0 0
$$778$$ −48.0000 −1.72088
$$779$$ −13.8564 −0.496457
$$780$$ 0 0
$$781$$ 36.0000 1.28818
$$782$$ −31.1769 −1.11488
$$783$$ 0 0
$$784$$ 15.0000 0.535714
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −26.0000 −0.926800 −0.463400 0.886149i $$-0.653371\pi$$
−0.463400 + 0.886149i $$0.653371\pi$$
$$788$$ 5.19615 0.185105
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −3.46410 −0.123169
$$792$$ 0 0
$$793$$ −7.00000 −0.248577
$$794$$ 50.2295 1.78258
$$795$$ 0 0
$$796$$ 20.0000 0.708881
$$797$$ 53.6936 1.90192 0.950962 0.309308i $$-0.100097\pi$$
0.950962 + 0.309308i $$0.100097\pi$$
$$798$$ 0 0
$$799$$ 36.0000 1.27359
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 21.0000 0.741536
$$803$$ −24.2487 −0.855718
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −13.8564 −0.488071
$$807$$ 0 0
$$808$$ 12.0000 0.422159
$$809$$ −46.7654 −1.64418 −0.822091 0.569355i $$-0.807193\pi$$
−0.822091 + 0.569355i $$0.807193\pi$$
$$810$$ 0 0
$$811$$ −16.0000 −0.561836 −0.280918 0.959732i $$-0.590639\pi$$
−0.280918 + 0.959732i $$0.590639\pi$$
$$812$$ −3.46410 −0.121566
$$813$$ 0 0
$$814$$ 42.0000 1.47210
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −4.00000 −0.139942
$$818$$ 32.9090 1.15063
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 12.1244 0.423143 0.211571 0.977363i $$-0.432142\pi$$
0.211571 + 0.977363i $$0.432142\pi$$
$$822$$ 0 0
$$823$$ 28.0000 0.976019 0.488009 0.872838i $$-0.337723\pi$$
0.488009 + 0.872838i $$0.337723\pi$$
$$824$$ −13.8564 −0.482711
$$825$$ 0 0
$$826$$ 48.0000 1.67013
$$827$$ 10.3923 0.361376 0.180688 0.983540i $$-0.442168\pi$$
0.180688 + 0.983540i $$0.442168\pi$$
$$828$$ 0 0
$$829$$ 2.00000 0.0694629 0.0347314 0.999397i $$-0.488942\pi$$
0.0347314 + 0.999397i $$0.488942\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 1.00000 0.0346688
$$833$$ 15.5885 0.540108
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −6.92820 −0.239617
$$837$$ 0 0
$$838$$ 12.0000 0.414533
$$839$$ −45.0333 −1.55472 −0.777361 0.629054i $$-0.783442\pi$$
−0.777361 + 0.629054i $$0.783442\pi$$
$$840$$ 0 0
$$841$$ −26.0000 −0.896552
$$842$$ 43.3013 1.49226
$$843$$ 0 0
$$844$$ −10.0000 −0.344214
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −2.00000 −0.0687208
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −24.2487 −0.831235
$$852$$ 0 0
$$853$$ 34.0000 1.16414 0.582069 0.813139i $$-0.302243\pi$$
0.582069 + 0.813139i $$0.302243\pi$$
$$854$$ −24.2487 −0.829774
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −22.5167 −0.769154 −0.384577 0.923093i $$-0.625653\pi$$
−0.384577 + 0.923093i $$0.625653\pi$$
$$858$$ 0 0
$$859$$ 44.0000 1.50126 0.750630 0.660722i $$-0.229750\pi$$
0.750630 + 0.660722i $$0.229750\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −31.1769 −1.06127 −0.530637 0.847599i $$-0.678047\pi$$
−0.530637 + 0.847599i $$0.678047\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 19.0526 0.647432
$$867$$ 0 0
$$868$$ −16.0000 −0.543075
$$869$$ −6.92820 −0.235023
$$870$$ 0 0
$$871$$ 10.0000 0.338837
$$872$$ 19.0526 0.645201
$$873$$ 0 0
$$874$$ 12.0000 0.405906
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −53.0000 −1.78968 −0.894841 0.446384i $$-0.852711\pi$$
−0.894841 + 0.446384i $$0.852711\pi$$
$$878$$ −34.6410 −1.16908
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 20.7846 0.700251 0.350126 0.936703i $$-0.386139\pi$$
0.350126 + 0.936703i $$0.386139\pi$$
$$882$$ 0 0
$$883$$ −56.0000 −1.88455 −0.942275 0.334840i $$-0.891318\pi$$
−0.942275 + 0.334840i $$0.891318\pi$$
$$884$$ −5.19615 −0.174766
$$885$$ 0 0
$$886$$ −60.0000 −2.01574
$$887$$ −3.46410 −0.116313 −0.0581566 0.998307i $$-0.518522\pi$$
−0.0581566 + 0.998307i $$0.518522\pi$$
$$888$$ 0 0
$$889$$ 4.00000 0.134156
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −2.00000 −0.0669650
$$893$$ −13.8564 −0.463687
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 24.2487 0.810093
$$897$$ 0 0
$$898$$ −36.0000 −1.20134
$$899$$ 13.8564 0.462137
$$900$$ 0 0
$$901$$ 0 0
$$902$$ −41.5692 −1.38410
$$903$$ 0 0
$$904$$ 3.00000 0.0997785
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 52.0000 1.72663 0.863316 0.504664i $$-0.168384\pi$$
0.863316 + 0.504664i $$0.168384\pi$$
$$908$$ 3.46410 0.114960
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −24.2487 −0.803396 −0.401698 0.915772i $$-0.631580\pi$$
−0.401698 + 0.915772i $$0.631580\pi$$
$$912$$ 0 0
$$913$$ −48.0000 −1.58857
$$914$$ 50.2295 1.66144
$$915$$ 0 0
$$916$$ −1.00000 −0.0330409
$$917$$ −6.92820 −0.228789
$$918$$ 0 0
$$919$$ 2.00000 0.0659739 0.0329870 0.999456i $$-0.489498\pi$$
0.0329870 + 0.999456i $$0.489498\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 24.0000 0.790398
$$923$$ −10.3923 −0.342067
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 13.8564 0.455350
$$927$$ 0 0
$$928$$ 9.00000 0.295439
$$929$$ −50.2295 −1.64798 −0.823988 0.566608i $$-0.808256\pi$$
−0.823988 + 0.566608i $$0.808256\pi$$
$$930$$ 0 0
$$931$$ −6.00000 −0.196642
$$932$$ 25.9808 0.851028
$$933$$ 0 0
$$934$$ −36.0000 −1.17796
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 25.0000 0.816714 0.408357 0.912822i $$-0.366102\pi$$
0.408357 + 0.912822i $$0.366102\pi$$
$$938$$ 34.6410 1.13107
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 50.2295 1.63743 0.818717 0.574197i $$-0.194686\pi$$
0.818717 + 0.574197i $$0.194686\pi$$
$$942$$ 0 0
$$943$$ 24.0000 0.781548
$$944$$ −69.2820 −2.25494
$$945$$ 0 0
$$946$$ −12.0000 −0.390154
$$947$$ −17.3205 −0.562841 −0.281420 0.959585i $$-0.590806\pi$$
−0.281420 + 0.959585i $$0.590806\pi$$
$$948$$ 0 0
$$949$$ 7.00000 0.227230
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 18.0000 0.583383
$$953$$ 5.19615 0.168320 0.0841599 0.996452i $$-0.473179\pi$$
0.0841599 + 0.996452i $$0.473179\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −27.7128 −0.896296
$$957$$ 0 0
$$958$$ 42.0000 1.35696
$$959$$ 3.46410 0.111862
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ −12.1244 −0.390905
$$963$$ 0 0
$$964$$ 29.0000 0.934027
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 46.0000 1.47926 0.739630 0.673014i $$-0.235000\pi$$
0.739630 + 0.673014i $$0.235000\pi$$
$$968$$ 1.73205 0.0556702
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 31.1769 1.00051 0.500257 0.865877i $$-0.333239\pi$$
0.500257 + 0.865877i $$0.333239\pi$$
$$972$$ 0 0
$$973$$ −16.0000 −0.512936
$$974$$ −27.7128 −0.887976
$$975$$ 0 0
$$976$$ 35.0000 1.12032
$$977$$ 48.4974 1.55157 0.775785 0.630997i $$-0.217354\pi$$
0.775785 + 0.630997i $$0.217354\pi$$
$$978$$ 0 0
$$979$$ 18.0000 0.575282
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 30.0000 0.957338
$$983$$ 34.6410 1.10488 0.552438 0.833554i $$-0.313697\pi$$
0.552438 + 0.833554i $$0.313697\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 15.5885 0.496438
$$987$$ 0 0
$$988$$ 2.00000 0.0636285
$$989$$ 6.92820 0.220304
$$990$$ 0 0
$$991$$ −34.0000 −1.08005 −0.540023 0.841650i $$-0.681584\pi$$
−0.540023 + 0.841650i $$0.681584\pi$$
$$992$$ 41.5692 1.31982
$$993$$ 0 0
$$994$$ −36.0000 −1.14185
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 7.00000 0.221692 0.110846 0.993838i $$-0.464644\pi$$
0.110846 + 0.993838i $$0.464644\pi$$
$$998$$ 17.3205 0.548271
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2025.2.a.j.1.1 2
3.2 odd 2 inner 2025.2.a.j.1.2 2
5.2 odd 4 2025.2.b.k.649.1 4
5.3 odd 4 2025.2.b.k.649.4 4
5.4 even 2 81.2.a.a.1.2 yes 2
15.2 even 4 2025.2.b.k.649.3 4
15.8 even 4 2025.2.b.k.649.2 4
15.14 odd 2 81.2.a.a.1.1 2
20.19 odd 2 1296.2.a.o.1.1 2
35.34 odd 2 3969.2.a.i.1.2 2
40.19 odd 2 5184.2.a.bq.1.2 2
40.29 even 2 5184.2.a.br.1.2 2
45.4 even 6 81.2.c.b.55.1 4
45.14 odd 6 81.2.c.b.55.2 4
45.29 odd 6 81.2.c.b.28.2 4
45.34 even 6 81.2.c.b.28.1 4
55.54 odd 2 9801.2.a.v.1.1 2
60.59 even 2 1296.2.a.o.1.2 2
105.104 even 2 3969.2.a.i.1.1 2
120.29 odd 2 5184.2.a.br.1.1 2
120.59 even 2 5184.2.a.bq.1.1 2
135.4 even 18 729.2.e.o.406.2 12
135.14 odd 18 729.2.e.o.163.2 12
135.29 odd 18 729.2.e.o.568.2 12
135.34 even 18 729.2.e.o.325.2 12
135.49 even 18 729.2.e.o.649.2 12
135.59 odd 18 729.2.e.o.649.1 12
135.74 odd 18 729.2.e.o.325.1 12
135.79 even 18 729.2.e.o.568.1 12
135.94 even 18 729.2.e.o.163.1 12
135.104 odd 18 729.2.e.o.406.1 12
135.119 odd 18 729.2.e.o.82.1 12
135.124 even 18 729.2.e.o.82.2 12
165.164 even 2 9801.2.a.v.1.2 2
180.59 even 6 1296.2.i.s.865.1 4
180.79 odd 6 1296.2.i.s.433.2 4
180.119 even 6 1296.2.i.s.433.1 4
180.139 odd 6 1296.2.i.s.865.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
81.2.a.a.1.1 2 15.14 odd 2
81.2.a.a.1.2 yes 2 5.4 even 2
81.2.c.b.28.1 4 45.34 even 6
81.2.c.b.28.2 4 45.29 odd 6
81.2.c.b.55.1 4 45.4 even 6
81.2.c.b.55.2 4 45.14 odd 6
729.2.e.o.82.1 12 135.119 odd 18
729.2.e.o.82.2 12 135.124 even 18
729.2.e.o.163.1 12 135.94 even 18
729.2.e.o.163.2 12 135.14 odd 18
729.2.e.o.325.1 12 135.74 odd 18
729.2.e.o.325.2 12 135.34 even 18
729.2.e.o.406.1 12 135.104 odd 18
729.2.e.o.406.2 12 135.4 even 18
729.2.e.o.568.1 12 135.79 even 18
729.2.e.o.568.2 12 135.29 odd 18
729.2.e.o.649.1 12 135.59 odd 18
729.2.e.o.649.2 12 135.49 even 18
1296.2.a.o.1.1 2 20.19 odd 2
1296.2.a.o.1.2 2 60.59 even 2
1296.2.i.s.433.1 4 180.119 even 6
1296.2.i.s.433.2 4 180.79 odd 6
1296.2.i.s.865.1 4 180.59 even 6
1296.2.i.s.865.2 4 180.139 odd 6
2025.2.a.j.1.1 2 1.1 even 1 trivial
2025.2.a.j.1.2 2 3.2 odd 2 inner
2025.2.b.k.649.1 4 5.2 odd 4
2025.2.b.k.649.2 4 15.8 even 4
2025.2.b.k.649.3 4 15.2 even 4
2025.2.b.k.649.4 4 5.3 odd 4
3969.2.a.i.1.1 2 105.104 even 2
3969.2.a.i.1.2 2 35.34 odd 2
5184.2.a.bq.1.1 2 120.59 even 2
5184.2.a.bq.1.2 2 40.19 odd 2
5184.2.a.br.1.1 2 120.29 odd 2
5184.2.a.br.1.2 2 40.29 even 2
9801.2.a.v.1.1 2 55.54 odd 2
9801.2.a.v.1.2 2 165.164 even 2