# Properties

 Label 2025.2.b.k.649.4 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 649.4 Root $$-0.866025 - 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2025.649 Dual form 2025.2.b.k.649.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.73205i q^{2} -1.00000 q^{4} +2.00000i q^{7} +1.73205i q^{8} +O(q^{10})$$ $$q+1.73205i q^{2} -1.00000 q^{4} +2.00000i q^{7} +1.73205i q^{8} -3.46410 q^{11} +1.00000i q^{13} -3.46410 q^{14} -5.00000 q^{16} +5.19615i q^{17} -2.00000 q^{19} -6.00000i q^{22} -3.46410i q^{23} -1.73205 q^{26} -2.00000i q^{28} -1.73205 q^{29} +8.00000 q^{31} -5.19615i q^{32} -9.00000 q^{34} -7.00000i q^{37} -3.46410i q^{38} -6.92820 q^{41} -2.00000i q^{43} +3.46410 q^{44} +6.00000 q^{46} +6.92820i q^{47} +3.00000 q^{49} -1.00000i q^{52} -3.46410 q^{56} -3.00000i q^{58} -13.8564 q^{59} -7.00000 q^{61} +13.8564i q^{62} -1.00000 q^{64} -10.0000i q^{67} -5.19615i q^{68} -10.3923 q^{71} +7.00000i q^{73} +12.1244 q^{74} +2.00000 q^{76} -6.92820i q^{77} -2.00000 q^{79} -12.0000i q^{82} +13.8564i q^{83} +3.46410 q^{86} -6.00000i q^{88} +5.19615 q^{89} -2.00000 q^{91} +3.46410i q^{92} -12.0000 q^{94} +2.00000i q^{97} +5.19615i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4}+O(q^{10})$$ 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 $$0.123376\pi$$
−0.925820 + 0.377964i $$0.876624\pi$$
$$8$$ 1.73205i 0.612372i
$$9$$ 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
$$13$$ 1.00000i 0.277350i 0.990338 + 0.138675i $$0.0442844\pi$$
−0.990338 + 0.138675i $$0.955716\pi$$
$$14$$ −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 $$-0.551413\pi$$
−0.160817 + 0.986984i $$0.551413\pi$$
$$30$$ 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 $$-0.804848\pi$$
0.817875 0.575396i $$-0.195152\pi$$
$$38$$ − 3.46410i − 0.561951i
$$39$$ 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
$$43$$ − 2.00000i − 0.304997i −0.988304 0.152499i $$-0.951268\pi$$
0.988304 0.152499i $$-0.0487319\pi$$
$$44$$ 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 $$-0.857883\pi$$
−0.901975 + 0.431788i $$0.857883\pi$$
$$60$$ 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 $$-0.790829\pi$$
0.791748 0.610847i $$-0.209171\pi$$
$$68$$ − 5.19615i − 0.630126i
$$69$$ 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
$$73$$ 7.00000i 0.819288i 0.912245 + 0.409644i $$0.134347\pi$$
−0.912245 + 0.409644i $$0.865653\pi$$
$$74$$ 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 $$-0.411191\pi$$
0.275396 + 0.961331i $$0.411191\pi$$
$$90$$ 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 $$0.0323753\pi$$
−0.994832 + 0.101535i $$0.967625\pi$$
$$98$$ 5.19615i 0.524891i
$$99$$ 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
$$103$$ − 8.00000i − 0.788263i −0.919054 0.394132i $$-0.871045\pi$$
0.919054 0.394132i $$-0.128955\pi$$
$$104$$ −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 $$0.0282826\pi$$
−0.996055 + 0.0887357i $$0.971717\pi$$
$$128$$ − 12.1244i − 1.07165i
$$129$$ 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
$$133$$ − 4.00000i − 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 $$-0.615430\pi$$
−0.354738 + 0.934966i $$0.615430\pi$$
$$150$$ 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 $$0.237315\pi$$
−0.734717 + 0.678374i $$0.762685\pi$$
$$158$$ − 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 $$0.215557\pi$$
−0.779334 + 0.626608i $$0.784443\pi$$
$$164$$ 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 $$-0.783137\pi$$
−0.776757 + 0.629800i $$0.783137\pi$$
$$180$$ 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 $$-0.284432\pi$$
0.626634 + 0.779314i $$0.284432\pi$$
$$192$$ 0 0
$$193$$ 1.00000i 0.0719816i 0.999352 + 0.0359908i $$0.0114587\pi$$
−0.999352 + 0.0359908i $$0.988541\pi$$
$$194$$ −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 $$-0.978668\pi$$
0.997755 0.0669650i $$-0.0213316\pi$$
$$224$$ 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 $$-0.146248\pi$$
0.896296 + 0.443455i $$0.146248\pi$$
$$240$$ 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 $$-0.393633\pi$$
0.327978 + 0.944685i $$0.393633\pi$$
$$252$$ 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 $$-0.342368\pi$$
0.475223 + 0.879866i $$0.342368\pi$$
$$270$$ 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 $$0.0191369\pi$$
−0.998193 + 0.0600842i $$0.980863\pi$$
$$278$$ − 13.8564i − 0.831052i
$$279$$ 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
$$283$$ 28.0000i 1.66443i 0.554455 + 0.832214i $$0.312927\pi$$
−0.554455 + 0.832214i $$0.687073\pi$$
$$284$$ 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 $$-0.849073\pi$$
0.889680 0.456584i $$-0.150927\pi$$
$$308$$ 6.92820i 0.394771i
$$309$$ 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
$$313$$ 25.0000i 1.41308i 0.707671 + 0.706542i $$0.249746\pi$$
−0.707671 + 0.706542i $$0.750254\pi$$
$$314$$ −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 $$0.250472\pi$$
−0.706057 + 0.708155i $$0.749528\pi$$
$$338$$ 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 $$-0.588427\pi$$
−0.274242 + 0.961661i $$0.588427\pi$$
$$360$$ 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 $$0.174812\pi$$
−0.852948 + 0.521996i $$0.825188\pi$$
$$368$$ 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 $$0.0833568\pi$$
−0.965907 + 0.258890i $$0.916643\pi$$
$$374$$ 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 $$-0.747954\pi$$
−0.702548 + 0.711637i $$0.747954\pi$$
$$390$$ 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 $$0.259427\pi$$
−0.685859 + 0.727734i $$0.740573\pi$$
$$398$$ − 34.6410i − 1.73640i
$$399$$ 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
$$403$$ 8.00000i 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 $$-0.445871\pi$$
0.169232 + 0.985576i $$0.445871\pi$$
$$420$$ 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 $$-0.914855\pi$$
0.964437 0.264313i $$-0.0851452\pi$$
$$434$$ −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 $$-0.663165\pi$$
−0.490443 + 0.871473i $$0.663165\pi$$
$$450$$ 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 $$0.237275\pi$$
−0.734802 + 0.678281i $$0.762725\pi$$
$$458$$ 1.73205i 0.0809334i
$$459$$ 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
$$463$$ − 8.00000i − 0.371792i −0.982569 0.185896i $$-0.940481\pi$$
0.982569 0.185896i $$-0.0595187\pi$$
$$464$$ 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 $$-0.313110\pi$$
0.553976 + 0.832533i $$0.313110\pi$$
$$480$$ 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 $$-0.881918\pi$$
0.931978 0.362515i $$-0.118082\pi$$
$$488$$ − 12.1244i − 0.548844i
$$489$$ 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
$$493$$ − 9.00000i − 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 $$-0.289489\pi$$
0.614174 + 0.789170i $$0.289489\pi$$
$$510$$ 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 $$-0.650466\pi$$
−0.455295 + 0.890341i $$0.650466\pi$$
$$522$$ 0 0
$$523$$ − 38.0000i − 1.66162i −0.556553 0.830812i $$-0.687876\pi$$
0.556553 0.830812i $$-0.312124\pi$$
$$524$$ −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 $$0.140630\pi$$
−0.903983 + 0.427569i $$0.859370\pi$$
$$548$$ − 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 $$-0.742305\pi$$
−0.689808 + 0.723993i $$0.742305\pi$$
$$570$$ 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 $$0.0735351\pi$$
−0.973434 + 0.228968i $$0.926465\pi$$
$$578$$ − 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 $$-0.591356\pi$$
−0.283079 + 0.959097i $$0.591356\pi$$
$$600$$ 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 $$0.176928\pi$$
−0.849460 + 0.527654i $$0.823072\pi$$
$$608$$ 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 $$0.240908\pi$$
−0.727013 + 0.686624i $$0.759092\pi$$
$$614$$ 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 $$-0.353318\pi$$
0.444677 + 0.895691i $$0.353318\pi$$
$$642$$ 0 0
$$643$$ − 8.00000i − 0.315489i −0.987480 0.157745i $$-0.949578\pi$$
0.987480 0.157745i $$-0.0504223\pi$$
$$644$$ −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 $$-0.478507\pi$$
0.0674711 + 0.997721i $$0.478507\pi$$
$$660$$ 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 $$0.160031\pi$$
−0.876259 + 0.481840i $$0.839969\pi$$
$$674$$ −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 $$-0.155417\pi$$
0.883152 + 0.469087i $$0.155417\pi$$
$$702$$ 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 $$-0.437924\pi$$
0.193784 + 0.981044i $$0.437924\pi$$
$$720$$ 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 $$-0.782852\pi$$
0.776193 0.630495i $$-0.217148\pi$$
$$728$$ − 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 $$0.323111\pi$$
−0.527549 + 0.849524i $$0.676889\pi$$
$$734$$ −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 $$0.0115717\pi$$
−0.999339 + 0.0363456i $$0.988428\pi$$
$$758$$ 27.7128i 1.00657i
$$759$$ 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
$$763$$ − 22.0000i − 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 $$0.153371\pi$$
−0.886149 + 0.463400i $$0.846629\pi$$
$$788$$ 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 $$-0.192807\pi$$
0.822091 + 0.569355i $$0.192807\pi$$
$$810$$ 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 $$-0.432142\pi$$
0.211571 + 0.977363i $$0.432142\pi$$
$$822$$ 0 0
$$823$$ 28.0000i 0.976019i 0.872838 + 0.488009i $$0.162277\pi$$
−0.872838 + 0.488009i $$0.837723\pi$$
$$824$$ 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 $$-0.216558\pi$$
0.777361 + 0.629054i $$0.216558\pi$$
$$840$$ 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 $$0.197757\pi$$
−0.813139 + 0.582069i $$0.802243\pi$$
$$854$$ 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 $$0.352711\pi$$
−0.446384 + 0.894841i $$0.647289\pi$$
$$878$$ − 34.6410i − 1.16908i
$$879$$ 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
$$883$$ − 56.0000i − 1.88455i −0.334840 0.942275i $$-0.608682\pi$$
0.334840 0.942275i $$-0.391318\pi$$
$$884$$ 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 $$-0.668384\pi$$
0.504664 0.863316i $$-0.331616\pi$$
$$908$$ 3.46410i 0.114960i
$$909$$ 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
$$913$$ − 48.0000i − 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 $$-0.191744\pi$$
0.823988 + 0.566608i $$0.191744\pi$$
$$930$$ 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 $$-0.866102\pi$$
0.912822 0.408357i $$-0.133898\pi$$
$$938$$ 34.6410i 1.13107i
$$939$$ 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
$$943$$ 24.0000i 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 $$-0.735000\pi$$
0.673014 0.739630i $$-0.265000\pi$$
$$968$$ 1.73205i 0.0556702i
$$969$$ 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
$$973$$ − 16.0000i − 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 $$-0.964644\pi$$
0.993838 0.110846i $$-0.0353561\pi$$
$$998$$ 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.649.4 4
3.2 odd 2 inner 2025.2.b.k.649.2 4
5.2 odd 4 2025.2.a.j.1.1 2
5.3 odd 4 81.2.a.a.1.2 yes 2
5.4 even 2 inner 2025.2.b.k.649.1 4
15.2 even 4 2025.2.a.j.1.2 2
15.8 even 4 81.2.a.a.1.1 2
15.14 odd 2 inner 2025.2.b.k.649.3 4
20.3 even 4 1296.2.a.o.1.1 2
35.13 even 4 3969.2.a.i.1.2 2
40.3 even 4 5184.2.a.bq.1.2 2
40.13 odd 4 5184.2.a.br.1.2 2
45.13 odd 12 81.2.c.b.55.1 4
45.23 even 12 81.2.c.b.55.2 4
45.38 even 12 81.2.c.b.28.2 4
45.43 odd 12 81.2.c.b.28.1 4
55.43 even 4 9801.2.a.v.1.1 2
60.23 odd 4 1296.2.a.o.1.2 2
105.83 odd 4 3969.2.a.i.1.1 2
120.53 even 4 5184.2.a.br.1.1 2
120.83 odd 4 5184.2.a.bq.1.1 2
135.13 odd 36 729.2.e.o.163.1 12
135.23 even 36 729.2.e.o.406.1 12
135.38 even 36 729.2.e.o.82.1 12
135.43 odd 36 729.2.e.o.82.2 12
135.58 odd 36 729.2.e.o.406.2 12
135.68 even 36 729.2.e.o.163.2 12
135.83 even 36 729.2.e.o.568.2 12
135.88 odd 36 729.2.e.o.325.2 12
135.103 odd 36 729.2.e.o.649.2 12
135.113 even 36 729.2.e.o.649.1 12
135.128 even 36 729.2.e.o.325.1 12
135.133 odd 36 729.2.e.o.568.1 12
165.98 odd 4 9801.2.a.v.1.2 2
180.23 odd 12 1296.2.i.s.865.1 4
180.43 even 12 1296.2.i.s.433.2 4
180.83 odd 12 1296.2.i.s.433.1 4
180.103 even 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 15.8 even 4
81.2.a.a.1.2 yes 2 5.3 odd 4
81.2.c.b.28.1 4 45.43 odd 12
81.2.c.b.28.2 4 45.38 even 12
81.2.c.b.55.1 4 45.13 odd 12
81.2.c.b.55.2 4 45.23 even 12
729.2.e.o.82.1 12 135.38 even 36
729.2.e.o.82.2 12 135.43 odd 36
729.2.e.o.163.1 12 135.13 odd 36
729.2.e.o.163.2 12 135.68 even 36
729.2.e.o.325.1 12 135.128 even 36
729.2.e.o.325.2 12 135.88 odd 36
729.2.e.o.406.1 12 135.23 even 36
729.2.e.o.406.2 12 135.58 odd 36
729.2.e.o.568.1 12 135.133 odd 36
729.2.e.o.568.2 12 135.83 even 36
729.2.e.o.649.1 12 135.113 even 36
729.2.e.o.649.2 12 135.103 odd 36
1296.2.a.o.1.1 2 20.3 even 4
1296.2.a.o.1.2 2 60.23 odd 4
1296.2.i.s.433.1 4 180.83 odd 12
1296.2.i.s.433.2 4 180.43 even 12
1296.2.i.s.865.1 4 180.23 odd 12
1296.2.i.s.865.2 4 180.103 even 12
2025.2.a.j.1.1 2 5.2 odd 4
2025.2.a.j.1.2 2 15.2 even 4
2025.2.b.k.649.1 4 5.4 even 2 inner
2025.2.b.k.649.2 4 3.2 odd 2 inner
2025.2.b.k.649.3 4 15.14 odd 2 inner
2025.2.b.k.649.4 4 1.1 even 1 trivial
3969.2.a.i.1.1 2 105.83 odd 4
3969.2.a.i.1.2 2 35.13 even 4
5184.2.a.bq.1.1 2 120.83 odd 4
5184.2.a.bq.1.2 2 40.3 even 4
5184.2.a.br.1.1 2 120.53 even 4
5184.2.a.br.1.2 2 40.13 odd 4
9801.2.a.v.1.1 2 55.43 even 4
9801.2.a.v.1.2 2 165.98 odd 4