# Properties

 Label 2025.2.b.k.649.3 Level $2025$ Weight $2$ Character 2025.649 Analytic conductor $16.170$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2025,2,Mod(649,2025)]

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, 1]))

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

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

chi := DirichletCharacter("2025.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2025 = 3^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2025.b (of order $$2$$, degree $$1$$, not 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: $$16.1697064093$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 81) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 649.3 Root $$0.866025 + 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2025.649 Dual form 2025.2.b.k.649.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.73205i q^{2} -1.00000 q^{4} -2.00000i q^{7} +1.73205i q^{8} +O(q^{10})$$ $$q+1.73205i q^{2} -1.00000 q^{4} -2.00000i q^{7} +1.73205i q^{8} +3.46410 q^{11} -1.00000i q^{13} +3.46410 q^{14} -5.00000 q^{16} +5.19615i q^{17} -2.00000 q^{19} +6.00000i q^{22} -3.46410i q^{23} +1.73205 q^{26} +2.00000i q^{28} +1.73205 q^{29} +8.00000 q^{31} -5.19615i q^{32} -9.00000 q^{34} +7.00000i q^{37} -3.46410i q^{38} +6.92820 q^{41} +2.00000i q^{43} -3.46410 q^{44} +6.00000 q^{46} +6.92820i q^{47} +3.00000 q^{49} +1.00000i q^{52} +3.46410 q^{56} +3.00000i q^{58} +13.8564 q^{59} -7.00000 q^{61} +13.8564i q^{62} -1.00000 q^{64} +10.0000i q^{67} -5.19615i q^{68} +10.3923 q^{71} -7.00000i q^{73} -12.1244 q^{74} +2.00000 q^{76} -6.92820i q^{77} -2.00000 q^{79} +12.0000i q^{82} +13.8564i q^{83} -3.46410 q^{86} +6.00000i q^{88} -5.19615 q^{89} -2.00000 q^{91} +3.46410i q^{92} -12.0000 q^{94} -2.00000i q^{97} +5.19615i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4}+O(q^{10})$$ 4 * q - 4 * q^4 $$4 q - 4 q^{4} - 20 q^{16} - 8 q^{19} + 32 q^{31} - 36 q^{34} + 24 q^{46} + 12 q^{49} - 28 q^{61} - 4 q^{64} + 8 q^{76} - 8 q^{79} - 8 q^{91} - 48 q^{94}+O(q^{100})$$ 4 * q - 4 * q^4 - 20 * q^16 - 8 * q^19 + 32 * q^31 - 36 * q^34 + 24 * q^46 + 12 * q^49 - 28 * q^61 - 4 * q^64 + 8 * q^76 - 8 * q^79 - 8 * q^91 - 48 * q^94

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2025\mathbb{Z}\right)^\times$$.

 $$n$$ $$326$$ $$1702$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.73205i 1.22474i 0.790569 + 0.612372i $$0.209785\pi$$
−0.790569 + 0.612372i $$0.790215\pi$$
$$3$$ 0 0
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 2.00000i − 0.755929i −0.925820 0.377964i $$-0.876624\pi$$
0.925820 0.377964i $$-0.123376\pi$$
$$8$$ 1.73205i 0.612372i
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 3.46410 1.04447 0.522233 0.852803i $$-0.325099\pi$$
0.522233 + 0.852803i $$0.325099\pi$$
$$12$$ 0 0
$$13$$ − 1.00000i − 0.277350i −0.990338 0.138675i $$-0.955716\pi$$
0.990338 0.138675i $$-0.0442844\pi$$
$$14$$ 3.46410 0.925820
$$15$$ 0 0
$$16$$ −5.00000 −1.25000
$$17$$ 5.19615i 1.26025i 0.776493 + 0.630126i $$0.216997\pi$$
−0.776493 + 0.630126i $$0.783003\pi$$
$$18$$ 0 0
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 6.00000i 1.27920i
$$23$$ − 3.46410i − 0.722315i −0.932505 0.361158i $$-0.882382\pi$$
0.932505 0.361158i $$-0.117618\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 1.73205 0.339683
$$27$$ 0 0
$$28$$ 2.00000i 0.377964i
$$29$$ 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 $$0.244865\pi$$
$$32$$ − 5.19615i − 0.918559i
$$33$$ 0 0
$$34$$ −9.00000 −1.54349
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 7.00000i 1.15079i 0.817875 + 0.575396i $$0.195152\pi$$
−0.817875 + 0.575396i $$0.804848\pi$$
$$38$$ − 3.46410i − 0.561951i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 6.92820 1.08200 0.541002 0.841021i $$-0.318045\pi$$
0.541002 + 0.841021i $$0.318045\pi$$
$$42$$ 0 0
$$43$$ 2.00000i 0.304997i 0.988304 + 0.152499i $$0.0487319\pi$$
−0.988304 + 0.152499i $$0.951268\pi$$
$$44$$ −3.46410 −0.522233
$$45$$ 0 0
$$46$$ 6.00000 0.884652
$$47$$ 6.92820i 1.01058i 0.862949 + 0.505291i $$0.168615\pi$$
−0.862949 + 0.505291i $$0.831385\pi$$
$$48$$ 0 0
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 1.00000i 0.138675i
$$53$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 3.46410 0.462910
$$57$$ 0 0
$$58$$ 3.00000i 0.393919i
$$59$$ 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 $$0.647910\pi$$
$$62$$ 13.8564i 1.75977i
$$63$$ 0 0
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 10.0000i 1.22169i 0.791748 + 0.610847i $$0.209171\pi$$
−0.791748 + 0.610847i $$0.790829\pi$$
$$68$$ − 5.19615i − 0.630126i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 10.3923 1.23334 0.616670 0.787222i $$-0.288481\pi$$
0.616670 + 0.787222i $$0.288481\pi$$
$$72$$ 0 0
$$73$$ − 7.00000i − 0.819288i −0.912245 0.409644i $$-0.865653\pi$$
0.912245 0.409644i $$-0.134347\pi$$
$$74$$ −12.1244 −1.40943
$$75$$ 0 0
$$76$$ 2.00000 0.229416
$$77$$ − 6.92820i − 0.789542i
$$78$$ 0 0
$$79$$ −2.00000 −0.225018 −0.112509 0.993651i $$-0.535889\pi$$
−0.112509 + 0.993651i $$0.535889\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 12.0000i 1.32518i
$$83$$ 13.8564i 1.52094i 0.649374 + 0.760469i $$0.275031\pi$$
−0.649374 + 0.760469i $$0.724969\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −3.46410 −0.373544
$$87$$ 0 0
$$88$$ 6.00000i 0.639602i
$$89$$ −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
$$92$$ 3.46410i 0.361158i
$$93$$ 0 0
$$94$$ −12.0000 −1.23771
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 2.00000i − 0.203069i −0.994832 0.101535i $$-0.967625\pi$$
0.994832 0.101535i $$-0.0323753\pi$$
$$98$$ 5.19615i 0.524891i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −6.92820 −0.689382 −0.344691 0.938716i $$-0.612016\pi$$
−0.344691 + 0.938716i $$0.612016\pi$$
$$102$$ 0 0
$$103$$ 8.00000i 0.788263i 0.919054 + 0.394132i $$0.128955\pi$$
−0.919054 + 0.394132i $$0.871045\pi$$
$$104$$ 1.73205 0.169842
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$108$$ 0 0
$$109$$ −11.0000 −1.05361 −0.526804 0.849987i $$-0.676610\pi$$
−0.526804 + 0.849987i $$0.676610\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 10.0000i 0.944911i
$$113$$ 1.73205i 0.162938i 0.996676 + 0.0814688i $$0.0259611\pi$$
−0.996676 + 0.0814688i $$0.974039\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −1.73205 −0.160817
$$117$$ 0 0
$$118$$ 24.0000i 2.20938i
$$119$$ 10.3923 0.952661
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ − 12.1244i − 1.09769i
$$123$$ 0 0
$$124$$ −8.00000 −0.718421
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 2.00000i − 0.177471i −0.996055 0.0887357i $$-0.971717\pi$$
0.996055 0.0887357i $$-0.0282826\pi$$
$$128$$ − 12.1244i − 1.07165i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −3.46410 −0.302660 −0.151330 0.988483i $$-0.548356\pi$$
−0.151330 + 0.988483i $$0.548356\pi$$
$$132$$ 0 0
$$133$$ 4.00000i 0.346844i
$$134$$ −17.3205 −1.49626
$$135$$ 0 0
$$136$$ −9.00000 −0.771744
$$137$$ 1.73205i 0.147979i 0.997259 + 0.0739895i $$0.0235731\pi$$
−0.997259 + 0.0739895i $$0.976427\pi$$
$$138$$ 0 0
$$139$$ −8.00000 −0.678551 −0.339276 0.940687i $$-0.610182\pi$$
−0.339276 + 0.940687i $$0.610182\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 18.0000i 1.51053i
$$143$$ − 3.46410i − 0.289683i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 12.1244 1.00342
$$147$$ 0 0
$$148$$ − 7.00000i − 0.575396i
$$149$$ 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 $$0.197401\pi$$
$$152$$ − 3.46410i − 0.280976i
$$153$$ 0 0
$$154$$ 12.0000 0.966988
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 17.0000i − 1.35675i −0.734717 0.678374i $$-0.762685\pi$$
0.734717 0.678374i $$-0.237315\pi$$
$$158$$ − 3.46410i − 0.275589i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −6.92820 −0.546019
$$162$$ 0 0
$$163$$ − 16.0000i − 1.25322i −0.779334 0.626608i $$-0.784443\pi$$
0.779334 0.626608i $$-0.215557\pi$$
$$164$$ −6.92820 −0.541002
$$165$$ 0 0
$$166$$ −24.0000 −1.86276
$$167$$ − 17.3205i − 1.34030i −0.742225 0.670151i $$-0.766230\pi$$
0.742225 0.670151i $$-0.233770\pi$$
$$168$$ 0 0
$$169$$ 12.0000 0.923077
$$170$$ 0 0
$$171$$ 0 0
$$172$$ − 2.00000i − 0.152499i
$$173$$ 19.0526i 1.44854i 0.689517 + 0.724270i $$0.257823\pi$$
−0.689517 + 0.724270i $$0.742177\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −17.3205 −1.30558
$$177$$ 0 0
$$178$$ − 9.00000i − 0.674579i
$$179$$ 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 $$0.476318\pi$$
$$182$$ − 3.46410i − 0.256776i
$$183$$ 0 0
$$184$$ 6.00000 0.442326
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 18.0000i 1.31629i
$$188$$ − 6.92820i − 0.505291i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −17.3205 −1.25327 −0.626634 0.779314i $$-0.715568\pi$$
−0.626634 + 0.779314i $$0.715568\pi$$
$$192$$ 0 0
$$193$$ − 1.00000i − 0.0719816i −0.999352 0.0359908i $$-0.988541\pi$$
0.999352 0.0359908i $$-0.0114587\pi$$
$$194$$ 3.46410 0.248708
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ − 5.19615i − 0.370211i −0.982719 0.185105i $$-0.940737\pi$$
0.982719 0.185105i $$-0.0592626\pi$$
$$198$$ 0 0
$$199$$ −20.0000 −1.41776 −0.708881 0.705328i $$-0.750800\pi$$
−0.708881 + 0.705328i $$0.750800\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ − 12.0000i − 0.844317i
$$203$$ − 3.46410i − 0.243132i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −13.8564 −0.965422
$$207$$ 0 0
$$208$$ 5.00000i 0.346688i
$$209$$ −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
$$217$$ − 16.0000i − 1.08615i
$$218$$ − 19.0526i − 1.29040i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 5.19615 0.349531
$$222$$ 0 0
$$223$$ 2.00000i 0.133930i 0.997755 + 0.0669650i $$0.0213316\pi$$
−0.997755 + 0.0669650i $$0.978668\pi$$
$$224$$ −10.3923 −0.694365
$$225$$ 0 0
$$226$$ −3.00000 −0.199557
$$227$$ − 3.46410i − 0.229920i −0.993370 0.114960i $$-0.963326\pi$$
0.993370 0.114960i $$-0.0366741\pi$$
$$228$$ 0 0
$$229$$ 1.00000 0.0660819 0.0330409 0.999454i $$-0.489481\pi$$
0.0330409 + 0.999454i $$0.489481\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 3.00000i 0.196960i
$$233$$ 25.9808i 1.70206i 0.525120 + 0.851028i $$0.324020\pi$$
−0.525120 + 0.851028i $$0.675980\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −13.8564 −0.901975
$$237$$ 0 0
$$238$$ 18.0000i 1.16677i
$$239$$ −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 $$0.116269\pi$$
$$242$$ 1.73205i 0.111340i
$$243$$ 0 0
$$244$$ 7.00000 0.448129
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 2.00000i 0.127257i
$$248$$ 13.8564i 0.879883i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −10.3923 −0.655956 −0.327978 0.944685i $$-0.606367\pi$$
−0.327978 + 0.944685i $$0.606367\pi$$
$$252$$ 0 0
$$253$$ − 12.0000i − 0.754434i
$$254$$ 3.46410 0.217357
$$255$$ 0 0
$$256$$ 19.0000 1.18750
$$257$$ 8.66025i 0.540212i 0.962831 + 0.270106i $$0.0870587\pi$$
−0.962831 + 0.270106i $$0.912941\pi$$
$$258$$ 0 0
$$259$$ 14.0000 0.869918
$$260$$ 0 0
$$261$$ 0 0
$$262$$ − 6.00000i − 0.370681i
$$263$$ − 6.92820i − 0.427211i −0.976920 0.213606i $$-0.931479\pi$$
0.976920 0.213606i $$-0.0685208\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −6.92820 −0.424795
$$267$$ 0 0
$$268$$ − 10.0000i − 0.610847i
$$269$$ −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 $$0.480652\pi$$
$$272$$ − 25.9808i − 1.57532i
$$273$$ 0 0
$$274$$ −3.00000 −0.181237
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 2.00000i − 0.120168i −0.998193 0.0600842i $$-0.980863\pi$$
0.998193 0.0600842i $$-0.0191369\pi$$
$$278$$ − 13.8564i − 0.831052i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −12.1244 −0.723278 −0.361639 0.932318i $$-0.617783\pi$$
−0.361639 + 0.932318i $$0.617783\pi$$
$$282$$ 0 0
$$283$$ − 28.0000i − 1.66443i −0.554455 0.832214i $$-0.687073\pi$$
0.554455 0.832214i $$-0.312927\pi$$
$$284$$ −10.3923 −0.616670
$$285$$ 0 0
$$286$$ 6.00000 0.354787
$$287$$ − 13.8564i − 0.817918i
$$288$$ 0 0
$$289$$ −10.0000 −0.588235
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 7.00000i 0.409644i
$$293$$ − 19.0526i − 1.11306i −0.830827 0.556531i $$-0.812132\pi$$
0.830827 0.556531i $$-0.187868\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −12.1244 −0.704714
$$297$$ 0 0
$$298$$ 15.0000i 0.868927i
$$299$$ −3.46410 −0.200334
$$300$$ 0 0
$$301$$ 4.00000 0.230556
$$302$$ 34.6410i 1.99337i
$$303$$ 0 0
$$304$$ 10.0000 0.573539
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 16.0000i 0.913168i 0.889680 + 0.456584i $$0.150927\pi$$
−0.889680 + 0.456584i $$0.849073\pi$$
$$308$$ 6.92820i 0.394771i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 6.92820 0.392862 0.196431 0.980518i $$-0.437065\pi$$
0.196431 + 0.980518i $$0.437065\pi$$
$$312$$ 0 0
$$313$$ − 25.0000i − 1.41308i −0.707671 0.706542i $$-0.750254\pi$$
0.707671 0.706542i $$-0.249746\pi$$
$$314$$ 29.4449 1.66167
$$315$$ 0 0
$$316$$ 2.00000 0.112509
$$317$$ − 8.66025i − 0.486408i −0.969975 0.243204i $$-0.921801\pi$$
0.969975 0.243204i $$-0.0781985\pi$$
$$318$$ 0 0
$$319$$ 6.00000 0.335936
$$320$$ 0 0
$$321$$ 0 0
$$322$$ − 12.0000i − 0.668734i
$$323$$ − 10.3923i − 0.578243i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 27.7128 1.53487
$$327$$ 0 0
$$328$$ 12.0000i 0.662589i
$$329$$ 13.8564 0.763928
$$330$$ 0 0
$$331$$ 2.00000 0.109930 0.0549650 0.998488i $$-0.482495\pi$$
0.0549650 + 0.998488i $$0.482495\pi$$
$$332$$ − 13.8564i − 0.760469i
$$333$$ 0 0
$$334$$ 30.0000 1.64153
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 26.0000i − 1.41631i −0.706057 0.708155i $$-0.749528\pi$$
0.706057 0.708155i $$-0.250472\pi$$
$$338$$ 20.7846i 1.13053i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 27.7128 1.50073
$$342$$ 0 0
$$343$$ − 20.0000i − 1.07990i
$$344$$ −3.46410 −0.186772
$$345$$ 0 0
$$346$$ −33.0000 −1.77409
$$347$$ 3.46410i 0.185963i 0.995668 + 0.0929814i $$0.0296397\pi$$
−0.995668 + 0.0929814i $$0.970360\pi$$
$$348$$ 0 0
$$349$$ −2.00000 −0.107058 −0.0535288 0.998566i $$-0.517047\pi$$
−0.0535288 + 0.998566i $$0.517047\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ − 18.0000i − 0.959403i
$$353$$ 13.8564i 0.737502i 0.929528 + 0.368751i $$0.120215\pi$$
−0.929528 + 0.368751i $$0.879785\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 5.19615 0.275396
$$357$$ 0 0
$$358$$ 36.0000i 1.90266i
$$359$$ 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
$$362$$ 3.46410i 0.182069i
$$363$$ 0 0
$$364$$ 2.00000 0.104828
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 20.0000i − 1.04399i −0.852948 0.521996i $$-0.825188\pi$$
0.852948 0.521996i $$-0.174812\pi$$
$$368$$ 17.3205i 0.902894i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ − 10.0000i − 0.517780i −0.965907 0.258890i $$-0.916643\pi$$
0.965907 0.258890i $$-0.0833568\pi$$
$$374$$ −31.1769 −1.61212
$$375$$ 0 0
$$376$$ −12.0000 −0.618853
$$377$$ − 1.73205i − 0.0892052i
$$378$$ 0 0
$$379$$ 16.0000 0.821865 0.410932 0.911666i $$-0.365203\pi$$
0.410932 + 0.911666i $$0.365203\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ − 30.0000i − 1.53493i
$$383$$ 17.3205i 0.885037i 0.896759 + 0.442518i $$0.145915\pi$$
−0.896759 + 0.442518i $$0.854085\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 1.73205 0.0881591
$$387$$ 0 0
$$388$$ 2.00000i 0.101535i
$$389$$ 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
$$392$$ 5.19615i 0.262445i
$$393$$ 0 0
$$394$$ 9.00000 0.453413
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 29.0000i − 1.45547i −0.685859 0.727734i $$-0.740573\pi$$
0.685859 0.727734i $$-0.259427\pi$$
$$398$$ − 34.6410i − 1.73640i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 12.1244 0.605461 0.302731 0.953076i $$-0.402102\pi$$
0.302731 + 0.953076i $$0.402102\pi$$
$$402$$ 0 0
$$403$$ − 8.00000i − 0.398508i
$$404$$ 6.92820 0.344691
$$405$$ 0 0
$$406$$ 6.00000 0.297775
$$407$$ 24.2487i 1.20196i
$$408$$ 0 0
$$409$$ 19.0000 0.939490 0.469745 0.882802i $$-0.344346\pi$$
0.469745 + 0.882802i $$0.344346\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ − 8.00000i − 0.394132i
$$413$$ − 27.7128i − 1.36366i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −5.19615 −0.254762
$$417$$ 0 0
$$418$$ − 12.0000i − 0.586939i
$$419$$ −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 $$0.708514\pi$$
$$422$$ − 17.3205i − 0.843149i
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 14.0000i 0.677507i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ 11.0000i 0.528626i 0.964437 + 0.264313i $$0.0851452\pi$$
−0.964437 + 0.264313i $$0.914855\pi$$
$$434$$ 27.7128 1.33026
$$435$$ 0 0
$$436$$ 11.0000 0.526804
$$437$$ 6.92820i 0.331421i
$$438$$ 0 0
$$439$$ −20.0000 −0.954548 −0.477274 0.878755i $$-0.658375\pi$$
−0.477274 + 0.878755i $$0.658375\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 9.00000i 0.428086i
$$443$$ 34.6410i 1.64584i 0.568154 + 0.822922i $$0.307658\pi$$
−0.568154 + 0.822922i $$0.692342\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −3.46410 −0.164030
$$447$$ 0 0
$$448$$ 2.00000i 0.0944911i
$$449$$ 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
$$452$$ − 1.73205i − 0.0814688i
$$453$$ 0 0
$$454$$ 6.00000 0.281594
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 29.0000i − 1.35656i −0.734802 0.678281i $$-0.762725\pi$$
0.734802 0.678281i $$-0.237275\pi$$
$$458$$ 1.73205i 0.0809334i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 13.8564 0.645357 0.322679 0.946509i $$-0.395417\pi$$
0.322679 + 0.946509i $$0.395417\pi$$
$$462$$ 0 0
$$463$$ 8.00000i 0.371792i 0.982569 + 0.185896i $$0.0595187\pi$$
−0.982569 + 0.185896i $$0.940481\pi$$
$$464$$ −8.66025 −0.402042
$$465$$ 0 0
$$466$$ −45.0000 −2.08458
$$467$$ − 20.7846i − 0.961797i −0.876776 0.480899i $$-0.840311\pi$$
0.876776 0.480899i $$-0.159689\pi$$
$$468$$ 0 0
$$469$$ 20.0000 0.923514
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 24.0000i 1.10469i
$$473$$ 6.92820i 0.318559i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −10.3923 −0.476331
$$477$$ 0 0
$$478$$ − 48.0000i − 2.19547i
$$479$$ −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
$$482$$ 50.2295i 2.28789i
$$483$$ 0 0
$$484$$ −1.00000 −0.0454545
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 16.0000i 0.725029i 0.931978 + 0.362515i $$0.118082\pi$$
−0.931978 + 0.362515i $$0.881918\pi$$
$$488$$ − 12.1244i − 0.548844i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 17.3205 0.781664 0.390832 0.920462i $$-0.372187\pi$$
0.390832 + 0.920462i $$0.372187\pi$$
$$492$$ 0 0
$$493$$ 9.00000i 0.405340i
$$494$$ −3.46410 −0.155857
$$495$$ 0 0
$$496$$ −40.0000 −1.79605
$$497$$ − 20.7846i − 0.932317i
$$498$$ 0 0
$$499$$ 10.0000 0.447661 0.223831 0.974628i $$-0.428144\pi$$
0.223831 + 0.974628i $$0.428144\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ − 18.0000i − 0.803379i
$$503$$ − 20.7846i − 0.926740i −0.886165 0.463370i $$-0.846640\pi$$
0.886165 0.463370i $$-0.153360\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 20.7846 0.923989
$$507$$ 0 0
$$508$$ 2.00000i 0.0887357i
$$509$$ −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
$$512$$ 8.66025i 0.382733i
$$513$$ 0 0
$$514$$ −15.0000 −0.661622
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 24.0000i 1.05552i
$$518$$ 24.2487i 1.06543i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 20.7846 0.910590 0.455295 0.890341i $$-0.349534\pi$$
0.455295 + 0.890341i $$0.349534\pi$$
$$522$$ 0 0
$$523$$ 38.0000i 1.66162i 0.556553 + 0.830812i $$0.312124\pi$$
−0.556553 + 0.830812i $$0.687876\pi$$
$$524$$ 3.46410 0.151330
$$525$$ 0 0
$$526$$ 12.0000 0.523225
$$527$$ 41.5692i 1.81078i
$$528$$ 0 0
$$529$$ 11.0000 0.478261
$$530$$ 0 0
$$531$$ 0 0
$$532$$ − 4.00000i − 0.173422i
$$533$$ − 6.92820i − 0.300094i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −17.3205 −0.748132
$$537$$ 0 0
$$538$$ − 27.0000i − 1.16405i
$$539$$ 10.3923 0.447628
$$540$$ 0 0
$$541$$ 11.0000 0.472927 0.236463 0.971640i $$-0.424012\pi$$
0.236463 + 0.971640i $$0.424012\pi$$
$$542$$ 3.46410i 0.148796i
$$543$$ 0 0
$$544$$ 27.0000 1.15762
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 20.0000i − 0.855138i −0.903983 0.427569i $$-0.859370\pi$$
0.903983 0.427569i $$-0.140630\pi$$
$$548$$ − 1.73205i − 0.0739895i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −3.46410 −0.147576
$$552$$ 0 0
$$553$$ 4.00000i 0.170097i
$$554$$ 3.46410 0.147176
$$555$$ 0 0
$$556$$ 8.00000 0.339276
$$557$$ 36.3731i 1.54118i 0.637333 + 0.770588i $$0.280037\pi$$
−0.637333 + 0.770588i $$0.719963\pi$$
$$558$$ 0 0
$$559$$ 2.00000 0.0845910
$$560$$ 0 0
$$561$$ 0 0
$$562$$ − 21.0000i − 0.885832i
$$563$$ − 34.6410i − 1.45994i −0.683477 0.729972i $$-0.739533\pi$$
0.683477 0.729972i $$-0.260467\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 48.4974 2.03850
$$567$$ 0 0
$$568$$ 18.0000i 0.755263i
$$569$$ 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 $$0.446465\pi$$
$$572$$ 3.46410i 0.144841i
$$573$$ 0 0
$$574$$ 24.0000 1.00174
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 11.0000i − 0.457936i −0.973434 0.228968i $$-0.926465\pi$$
0.973434 0.228968i $$-0.0735351\pi$$
$$578$$ − 17.3205i − 0.720438i
$$579$$ 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
$$587$$ 38.1051i 1.57277i 0.617739 + 0.786383i $$0.288049\pi$$
−0.617739 + 0.786383i $$0.711951\pi$$
$$588$$ 0 0
$$589$$ −16.0000 −0.659269
$$590$$ 0 0
$$591$$ 0 0
$$592$$ − 35.0000i − 1.43849i
$$593$$ − 15.5885i − 0.640141i −0.947394 0.320071i $$-0.896293\pi$$
0.947394 0.320071i $$-0.103707\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −8.66025 −0.354738
$$597$$ 0 0
$$598$$ − 6.00000i − 0.245358i
$$599$$ 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 $$0.670312\pi$$
$$602$$ 6.92820i 0.282372i
$$603$$ 0 0
$$604$$ −20.0000 −0.813788
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 26.0000i − 1.05531i −0.849460 0.527654i $$-0.823072\pi$$
0.849460 0.527654i $$-0.176928\pi$$
$$608$$ 10.3923i 0.421464i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 6.92820 0.280285
$$612$$ 0 0
$$613$$ − 34.0000i − 1.37325i −0.727013 0.686624i $$-0.759092\pi$$
0.727013 0.686624i $$-0.240908\pi$$
$$614$$ −27.7128 −1.11840
$$615$$ 0 0
$$616$$ 12.0000 0.483494
$$617$$ − 12.1244i − 0.488108i −0.969762 0.244054i $$-0.921523\pi$$
0.969762 0.244054i $$-0.0784774\pi$$
$$618$$ 0 0
$$619$$ −20.0000 −0.803868 −0.401934 0.915669i $$-0.631662\pi$$
−0.401934 + 0.915669i $$0.631662\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 12.0000i 0.481156i
$$623$$ 10.3923i 0.416359i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 43.3013 1.73067
$$627$$ 0 0
$$628$$ 17.0000i 0.678374i
$$629$$ −36.3731 −1.45029
$$630$$ 0 0
$$631$$ 20.0000 0.796187 0.398094 0.917345i $$-0.369672\pi$$
0.398094 + 0.917345i $$0.369672\pi$$
$$632$$ − 3.46410i − 0.137795i
$$633$$ 0 0
$$634$$ 15.0000 0.595726
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 3.00000i − 0.118864i
$$638$$ 10.3923i 0.411435i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −22.5167 −0.889355 −0.444677 0.895691i $$-0.646682\pi$$
−0.444677 + 0.895691i $$0.646682\pi$$
$$642$$ 0 0
$$643$$ 8.00000i 0.315489i 0.987480 + 0.157745i $$0.0504223\pi$$
−0.987480 + 0.157745i $$0.949578\pi$$
$$644$$ 6.92820 0.273009
$$645$$ 0 0
$$646$$ 18.0000 0.708201
$$647$$ − 31.1769i − 1.22569i −0.790203 0.612845i $$-0.790025\pi$$
0.790203 0.612845i $$-0.209975\pi$$
$$648$$ 0 0
$$649$$ 48.0000 1.88416
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 16.0000i 0.626608i
$$653$$ − 13.8564i − 0.542243i −0.962545 0.271122i $$-0.912605\pi$$
0.962545 0.271122i $$-0.0873945\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −34.6410 −1.35250
$$657$$ 0 0
$$658$$ 24.0000i 0.935617i
$$659$$ −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 $$0.392745\pi$$
$$662$$ 3.46410i 0.134636i
$$663$$ 0 0
$$664$$ −24.0000 −0.931381
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 6.00000i − 0.232321i
$$668$$ 17.3205i 0.670151i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −24.2487 −0.936111
$$672$$ 0 0
$$673$$ − 25.0000i − 0.963679i −0.876259 0.481840i $$-0.839969\pi$$
0.876259 0.481840i $$-0.160031\pi$$
$$674$$ 45.0333 1.73462
$$675$$ 0 0
$$676$$ −12.0000 −0.461538
$$677$$ − 13.8564i − 0.532545i −0.963898 0.266272i $$-0.914208\pi$$
0.963898 0.266272i $$-0.0857921\pi$$
$$678$$ 0 0
$$679$$ −4.00000 −0.153506
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 48.0000i 1.83801i
$$683$$ 20.7846i 0.795301i 0.917537 + 0.397650i $$0.130174\pi$$
−0.917537 + 0.397650i $$0.869826\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 34.6410 1.32260
$$687$$ 0 0
$$688$$ − 10.0000i − 0.381246i
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 2.00000 0.0760836 0.0380418 0.999276i $$-0.487888\pi$$
0.0380418 + 0.999276i $$0.487888\pi$$
$$692$$ − 19.0526i − 0.724270i
$$693$$ 0 0
$$694$$ −6.00000 −0.227757
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 36.0000i 1.36360i
$$698$$ − 3.46410i − 0.131118i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −46.7654 −1.76630 −0.883152 0.469087i $$-0.844583\pi$$
−0.883152 + 0.469087i $$0.844583\pi$$
$$702$$ 0 0
$$703$$ − 14.0000i − 0.528020i
$$704$$ −3.46410 −0.130558
$$705$$ 0 0
$$706$$ −24.0000 −0.903252
$$707$$ 13.8564i 0.521124i
$$708$$ 0 0
$$709$$ 25.0000 0.938895 0.469447 0.882960i $$-0.344453\pi$$
0.469447 + 0.882960i $$0.344453\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ − 9.00000i − 0.337289i
$$713$$ − 27.7128i − 1.03785i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −20.7846 −0.776757
$$717$$ 0 0
$$718$$ 18.0000i 0.671754i
$$719$$ −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
$$722$$ − 25.9808i − 0.966904i
$$723$$ 0 0
$$724$$ −2.00000 −0.0743294
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 34.0000i 1.26099i 0.776193 + 0.630495i $$0.217148\pi$$
−0.776193 + 0.630495i $$0.782852\pi$$
$$728$$ − 3.46410i − 0.128388i
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −10.3923 −0.384373
$$732$$ 0 0
$$733$$ − 46.0000i − 1.69905i −0.527549 0.849524i $$-0.676889\pi$$
0.527549 0.849524i $$-0.323111\pi$$
$$734$$ 34.6410 1.27862
$$735$$ 0 0
$$736$$ −18.0000 −0.663489
$$737$$ 34.6410i 1.27602i
$$738$$ 0 0
$$739$$ −20.0000 −0.735712 −0.367856 0.929883i $$-0.619908\pi$$
−0.367856 + 0.929883i $$0.619908\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 6.92820i 0.254171i 0.991892 + 0.127086i $$0.0405623\pi$$
−0.991892 + 0.127086i $$0.959438\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 17.3205 0.634149
$$747$$ 0 0
$$748$$ − 18.0000i − 0.658145i
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −10.0000 −0.364905 −0.182453 0.983215i $$-0.558404\pi$$
−0.182453 + 0.983215i $$0.558404\pi$$
$$752$$ − 34.6410i − 1.26323i
$$753$$ 0 0
$$754$$ 3.00000 0.109254
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 2.00000i − 0.0726912i −0.999339 0.0363456i $$-0.988428\pi$$
0.999339 0.0363456i $$-0.0115717\pi$$
$$758$$ 27.7128i 1.00657i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −29.4449 −1.06738 −0.533688 0.845682i $$-0.679194\pi$$
−0.533688 + 0.845682i $$0.679194\pi$$
$$762$$ 0 0
$$763$$ 22.0000i 0.796453i
$$764$$ 17.3205 0.626634
$$765$$ 0 0
$$766$$ −30.0000 −1.08394
$$767$$ − 13.8564i − 0.500326i
$$768$$ 0 0
$$769$$ 1.00000 0.0360609 0.0180305 0.999837i $$-0.494260\pi$$
0.0180305 + 0.999837i $$0.494260\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 1.00000i 0.0359908i
$$773$$ 25.9808i 0.934463i 0.884135 + 0.467232i $$0.154749\pi$$
−0.884135 + 0.467232i $$0.845251\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 3.46410 0.124354
$$777$$ 0 0
$$778$$ 48.0000i 1.72088i
$$779$$ −13.8564 −0.496457
$$780$$ 0 0
$$781$$ 36.0000 1.28818
$$782$$ 31.1769i 1.11488i
$$783$$ 0 0
$$784$$ −15.0000 −0.535714
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 26.0000i − 0.926800i −0.886149 0.463400i $$-0.846629\pi$$
0.886149 0.463400i $$-0.153371\pi$$
$$788$$ 5.19615i 0.185105i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 3.46410 0.123169
$$792$$ 0 0
$$793$$ 7.00000i 0.248577i
$$794$$ 50.2295 1.78258
$$795$$ 0 0
$$796$$ 20.0000 0.708881
$$797$$ − 53.6936i − 1.90192i −0.309308 0.950962i $$-0.600097\pi$$
0.309308 0.950962i $$-0.399903\pi$$
$$798$$ 0 0
$$799$$ −36.0000 −1.27359
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 21.0000i 0.741536i
$$803$$ − 24.2487i − 0.855718i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 13.8564 0.488071
$$807$$ 0 0
$$808$$ − 12.0000i − 0.422159i
$$809$$ −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 $$0.590639\pi$$
$$812$$ 3.46410i 0.121566i
$$813$$ 0 0
$$814$$ −42.0000 −1.47210
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 4.00000i − 0.139942i
$$818$$ 32.9090i 1.15063i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −12.1244 −0.423143 −0.211571 0.977363i $$-0.567858\pi$$
−0.211571 + 0.977363i $$0.567858\pi$$
$$822$$ 0 0
$$823$$ − 28.0000i − 0.976019i −0.872838 0.488009i $$-0.837723\pi$$
0.872838 0.488009i $$-0.162277\pi$$
$$824$$ −13.8564 −0.482711
$$825$$ 0 0
$$826$$ 48.0000 1.67013
$$827$$ − 10.3923i − 0.361376i −0.983540 0.180688i $$-0.942168\pi$$
0.983540 0.180688i $$-0.0578324\pi$$
$$828$$ 0 0
$$829$$ −2.00000 −0.0694629 −0.0347314 0.999397i $$-0.511058\pi$$
−0.0347314 + 0.999397i $$0.511058\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 1.00000i 0.0346688i
$$833$$ 15.5885i 0.540108i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 6.92820 0.239617
$$837$$ 0 0
$$838$$ − 12.0000i − 0.414533i
$$839$$ −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
$$842$$ − 43.3013i − 1.49226i
$$843$$ 0 0
$$844$$ 10.0000 0.344214
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 2.00000i − 0.0687208i
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 24.2487 0.831235
$$852$$ 0 0
$$853$$ − 34.0000i − 1.16414i −0.813139 0.582069i $$-0.802243\pi$$
0.813139 0.582069i $$-0.197757\pi$$
$$854$$ −24.2487 −0.829774
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 22.5167i 0.769154i 0.923093 + 0.384577i $$0.125653\pi$$
−0.923093 + 0.384577i $$0.874347\pi$$
$$858$$ 0 0
$$859$$ −44.0000 −1.50126 −0.750630 0.660722i $$-0.770250\pi$$
−0.750630 + 0.660722i $$0.770250\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ − 31.1769i − 1.06127i −0.847599 0.530637i $$-0.821953\pi$$
0.847599 0.530637i $$-0.178047\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −19.0526 −0.647432
$$867$$ 0 0
$$868$$ 16.0000i 0.543075i
$$869$$ −6.92820 −0.235023
$$870$$ 0 0
$$871$$ 10.0000 0.338837
$$872$$ − 19.0526i − 0.645201i
$$873$$ 0 0
$$874$$ −12.0000 −0.405906
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 53.0000i − 1.78968i −0.446384 0.894841i $$-0.647289\pi$$
0.446384 0.894841i $$-0.352711\pi$$
$$878$$ − 34.6410i − 1.16908i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −20.7846 −0.700251 −0.350126 0.936703i $$-0.613861\pi$$
−0.350126 + 0.936703i $$0.613861\pi$$
$$882$$ 0 0
$$883$$ 56.0000i 1.88455i 0.334840 + 0.942275i $$0.391318\pi$$
−0.334840 + 0.942275i $$0.608682\pi$$
$$884$$ −5.19615 −0.174766
$$885$$ 0 0
$$886$$ −60.0000 −2.01574
$$887$$ 3.46410i 0.116313i 0.998307 + 0.0581566i $$0.0185223\pi$$
−0.998307 + 0.0581566i $$0.981478\pi$$
$$888$$ 0 0
$$889$$ −4.00000 −0.134156
$$890$$ 0 0
$$891$$ 0 0
$$892$$ − 2.00000i − 0.0669650i
$$893$$ − 13.8564i − 0.463687i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −24.2487 −0.810093
$$897$$ 0 0
$$898$$ 36.0000i 1.20134i
$$899$$ 13.8564 0.462137
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 41.5692i 1.38410i
$$903$$ 0 0
$$904$$ −3.00000 −0.0997785
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 52.0000i 1.72663i 0.504664 + 0.863316i $$0.331616\pi$$
−0.504664 + 0.863316i $$0.668384\pi$$
$$908$$ 3.46410i 0.114960i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 24.2487 0.803396 0.401698 0.915772i $$-0.368420\pi$$
0.401698 + 0.915772i $$0.368420\pi$$
$$912$$ 0 0
$$913$$ 48.0000i 1.58857i
$$914$$ 50.2295 1.66144
$$915$$ 0 0
$$916$$ −1.00000 −0.0330409
$$917$$ 6.92820i 0.228789i
$$918$$ 0 0
$$919$$ −2.00000 −0.0659739 −0.0329870 0.999456i $$-0.510502\pi$$
−0.0329870 + 0.999456i $$0.510502\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 24.0000i 0.790398i
$$923$$ − 10.3923i − 0.342067i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −13.8564 −0.455350
$$927$$ 0 0
$$928$$ − 9.00000i − 0.295439i
$$929$$ −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
$$932$$ − 25.9808i − 0.851028i
$$933$$ 0 0
$$934$$ 36.0000 1.17796
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 25.0000i 0.816714i 0.912822 + 0.408357i $$0.133898\pi$$
−0.912822 + 0.408357i $$0.866102\pi$$
$$938$$ 34.6410i 1.13107i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −50.2295 −1.63743 −0.818717 0.574197i $$-0.805314\pi$$
−0.818717 + 0.574197i $$0.805314\pi$$
$$942$$ 0 0
$$943$$ − 24.0000i − 0.781548i
$$944$$ −69.2820 −2.25494
$$945$$ 0 0
$$946$$ −12.0000 −0.390154
$$947$$ 17.3205i 0.562841i 0.959585 + 0.281420i $$0.0908056\pi$$
−0.959585 + 0.281420i $$0.909194\pi$$
$$948$$ 0 0
$$949$$ −7.00000 −0.227230
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 18.0000i 0.583383i
$$953$$ 5.19615i 0.168320i 0.996452 + 0.0841599i $$0.0268207\pi$$
−0.996452 + 0.0841599i $$0.973179\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 27.7128 0.896296
$$957$$ 0 0
$$958$$ − 42.0000i − 1.35696i
$$959$$ 3.46410 0.111862
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 12.1244i 0.390905i
$$963$$ 0 0
$$964$$ −29.0000 −0.934027
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 46.0000i 1.47926i 0.673014 + 0.739630i $$0.265000\pi$$
−0.673014 + 0.739630i $$0.735000\pi$$
$$968$$ 1.73205i 0.0556702i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −31.1769 −1.00051 −0.500257 0.865877i $$-0.666761\pi$$
−0.500257 + 0.865877i $$0.666761\pi$$
$$972$$ 0 0
$$973$$ 16.0000i 0.512936i
$$974$$ −27.7128 −0.887976
$$975$$ 0 0
$$976$$ 35.0000 1.12032
$$977$$ − 48.4974i − 1.55157i −0.630997 0.775785i $$-0.717354\pi$$
0.630997 0.775785i $$-0.282646\pi$$
$$978$$ 0 0
$$979$$ −18.0000 −0.575282
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 30.0000i 0.957338i
$$983$$ 34.6410i 1.10488i 0.833554 + 0.552438i $$0.186303\pi$$
−0.833554 + 0.552438i $$0.813697\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −15.5885 −0.496438
$$987$$ 0 0
$$988$$ − 2.00000i − 0.0636285i
$$989$$ 6.92820 0.220304
$$990$$ 0 0
$$991$$ −34.0000 −1.08005 −0.540023 0.841650i $$-0.681584\pi$$
−0.540023 + 0.841650i $$0.681584\pi$$
$$992$$ − 41.5692i − 1.31982i
$$993$$ 0 0
$$994$$ 36.0000 1.14185
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 7.00000i 0.221692i 0.993838 + 0.110846i $$0.0353561\pi$$
−0.993838 + 0.110846i $$0.964644\pi$$
$$998$$ 17.3205i 0.548271i
$$999$$ 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.b.k.649.3 4
3.2 odd 2 inner 2025.2.b.k.649.1 4
5.2 odd 4 81.2.a.a.1.1 2
5.3 odd 4 2025.2.a.j.1.2 2
5.4 even 2 inner 2025.2.b.k.649.2 4
15.2 even 4 81.2.a.a.1.2 yes 2
15.8 even 4 2025.2.a.j.1.1 2
15.14 odd 2 inner 2025.2.b.k.649.4 4
20.7 even 4 1296.2.a.o.1.2 2
35.27 even 4 3969.2.a.i.1.1 2
40.27 even 4 5184.2.a.bq.1.1 2
40.37 odd 4 5184.2.a.br.1.1 2
45.2 even 12 81.2.c.b.28.1 4
45.7 odd 12 81.2.c.b.28.2 4
45.22 odd 12 81.2.c.b.55.2 4
45.32 even 12 81.2.c.b.55.1 4
55.32 even 4 9801.2.a.v.1.2 2
60.47 odd 4 1296.2.a.o.1.1 2
105.62 odd 4 3969.2.a.i.1.2 2
120.77 even 4 5184.2.a.br.1.2 2
120.107 odd 4 5184.2.a.bq.1.2 2
135.2 even 36 729.2.e.o.568.1 12
135.7 odd 36 729.2.e.o.325.1 12
135.22 odd 36 729.2.e.o.649.1 12
135.32 even 36 729.2.e.o.649.2 12
135.47 even 36 729.2.e.o.325.2 12
135.52 odd 36 729.2.e.o.568.2 12
135.67 odd 36 729.2.e.o.163.2 12
135.77 even 36 729.2.e.o.406.2 12
135.92 even 36 729.2.e.o.82.2 12
135.97 odd 36 729.2.e.o.82.1 12
135.112 odd 36 729.2.e.o.406.1 12
135.122 even 36 729.2.e.o.163.1 12
165.32 odd 4 9801.2.a.v.1.1 2
180.7 even 12 1296.2.i.s.433.1 4
180.47 odd 12 1296.2.i.s.433.2 4
180.67 even 12 1296.2.i.s.865.1 4
180.167 odd 12 1296.2.i.s.865.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
81.2.a.a.1.1 2 5.2 odd 4
81.2.a.a.1.2 yes 2 15.2 even 4
81.2.c.b.28.1 4 45.2 even 12
81.2.c.b.28.2 4 45.7 odd 12
81.2.c.b.55.1 4 45.32 even 12
81.2.c.b.55.2 4 45.22 odd 12
729.2.e.o.82.1 12 135.97 odd 36
729.2.e.o.82.2 12 135.92 even 36
729.2.e.o.163.1 12 135.122 even 36
729.2.e.o.163.2 12 135.67 odd 36
729.2.e.o.325.1 12 135.7 odd 36
729.2.e.o.325.2 12 135.47 even 36
729.2.e.o.406.1 12 135.112 odd 36
729.2.e.o.406.2 12 135.77 even 36
729.2.e.o.568.1 12 135.2 even 36
729.2.e.o.568.2 12 135.52 odd 36
729.2.e.o.649.1 12 135.22 odd 36
729.2.e.o.649.2 12 135.32 even 36
1296.2.a.o.1.1 2 60.47 odd 4
1296.2.a.o.1.2 2 20.7 even 4
1296.2.i.s.433.1 4 180.7 even 12
1296.2.i.s.433.2 4 180.47 odd 12
1296.2.i.s.865.1 4 180.67 even 12
1296.2.i.s.865.2 4 180.167 odd 12
2025.2.a.j.1.1 2 15.8 even 4
2025.2.a.j.1.2 2 5.3 odd 4
2025.2.b.k.649.1 4 3.2 odd 2 inner
2025.2.b.k.649.2 4 5.4 even 2 inner
2025.2.b.k.649.3 4 1.1 even 1 trivial
2025.2.b.k.649.4 4 15.14 odd 2 inner
3969.2.a.i.1.1 2 35.27 even 4
3969.2.a.i.1.2 2 105.62 odd 4
5184.2.a.bq.1.1 2 40.27 even 4
5184.2.a.bq.1.2 2 120.107 odd 4
5184.2.a.br.1.1 2 40.37 odd 4
5184.2.a.br.1.2 2 120.77 even 4
9801.2.a.v.1.1 2 165.32 odd 4
9801.2.a.v.1.2 2 55.32 even 4