Properties

 Label 825.2.c.a.199.2 Level $825$ Weight $2$ Character 825.199 Analytic conductor $6.588$ 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] = [825,2,Mod(199,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("825.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 825.c (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: $$6.58765816676$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 199.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 825.199 Dual form 825.2.c.a.199.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} +4.00000i q^{7} +3.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} +4.00000i q^{7} +3.00000i q^{8} -1.00000 q^{9} +1.00000 q^{11} +1.00000i q^{12} +2.00000i q^{13} -4.00000 q^{14} -1.00000 q^{16} -2.00000i q^{17} -1.00000i q^{18} -4.00000 q^{21} +1.00000i q^{22} -8.00000i q^{23} -3.00000 q^{24} -2.00000 q^{26} -1.00000i q^{27} +4.00000i q^{28} +6.00000 q^{29} -8.00000 q^{31} +5.00000i q^{32} +1.00000i q^{33} +2.00000 q^{34} -1.00000 q^{36} +6.00000i q^{37} -2.00000 q^{39} -2.00000 q^{41} -4.00000i q^{42} +1.00000 q^{44} +8.00000 q^{46} +8.00000i q^{47} -1.00000i q^{48} -9.00000 q^{49} +2.00000 q^{51} +2.00000i q^{52} -6.00000i q^{53} +1.00000 q^{54} -12.0000 q^{56} +6.00000i q^{58} +4.00000 q^{59} +6.00000 q^{61} -8.00000i q^{62} -4.00000i q^{63} -7.00000 q^{64} -1.00000 q^{66} -4.00000i q^{67} -2.00000i q^{68} +8.00000 q^{69} -3.00000i q^{72} +14.0000i q^{73} -6.00000 q^{74} +4.00000i q^{77} -2.00000i q^{78} +4.00000 q^{79} +1.00000 q^{81} -2.00000i q^{82} -12.0000i q^{83} -4.00000 q^{84} +6.00000i q^{87} +3.00000i q^{88} +6.00000 q^{89} -8.00000 q^{91} -8.00000i q^{92} -8.00000i q^{93} -8.00000 q^{94} -5.00000 q^{96} +2.00000i q^{97} -9.00000i q^{98} -1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^4 - 2 * q^6 - 2 * q^9 $$2 q + 2 q^{4} - 2 q^{6} - 2 q^{9} + 2 q^{11} - 8 q^{14} - 2 q^{16} - 8 q^{21} - 6 q^{24} - 4 q^{26} + 12 q^{29} - 16 q^{31} + 4 q^{34} - 2 q^{36} - 4 q^{39} - 4 q^{41} + 2 q^{44} + 16 q^{46} - 18 q^{49} + 4 q^{51} + 2 q^{54} - 24 q^{56} + 8 q^{59} + 12 q^{61} - 14 q^{64} - 2 q^{66} + 16 q^{69} - 12 q^{74} + 8 q^{79} + 2 q^{81} - 8 q^{84} + 12 q^{89} - 16 q^{91} - 16 q^{94} - 10 q^{96} - 2 q^{99}+O(q^{100})$$ 2 * q + 2 * q^4 - 2 * q^6 - 2 * q^9 + 2 * q^11 - 8 * q^14 - 2 * q^16 - 8 * q^21 - 6 * q^24 - 4 * q^26 + 12 * q^29 - 16 * q^31 + 4 * q^34 - 2 * q^36 - 4 * q^39 - 4 * q^41 + 2 * q^44 + 16 * q^46 - 18 * q^49 + 4 * q^51 + 2 * q^54 - 24 * q^56 + 8 * q^59 + 12 * q^61 - 14 * q^64 - 2 * q^66 + 16 * q^69 - 12 * q^74 + 8 * q^79 + 2 * q^81 - 8 * q^84 + 12 * q^89 - 16 * q^91 - 16 * q^94 - 10 * q^96 - 2 * q^99

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$$.

 $$n$$ $$376$$ $$551$$ $$727$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 0.707107i 0.935414 + 0.353553i $$0.115027\pi$$
−0.935414 + 0.353553i $$0.884973\pi$$
$$3$$ 1.00000i 0.577350i
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ −1.00000 −0.408248
$$7$$ 4.00000i 1.51186i 0.654654 + 0.755929i $$0.272814\pi$$
−0.654654 + 0.755929i $$0.727186\pi$$
$$8$$ 3.00000i 1.06066i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 1.00000i 0.288675i
$$13$$ 2.00000i 0.554700i 0.960769 + 0.277350i $$0.0894562\pi$$
−0.960769 + 0.277350i $$0.910544\pi$$
$$14$$ −4.00000 −1.06904
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ − 2.00000i − 0.485071i −0.970143 0.242536i $$-0.922021\pi$$
0.970143 0.242536i $$-0.0779791\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ −4.00000 −0.872872
$$22$$ 1.00000i 0.213201i
$$23$$ − 8.00000i − 1.66812i −0.551677 0.834058i $$-0.686012\pi$$
0.551677 0.834058i $$-0.313988\pi$$
$$24$$ −3.00000 −0.612372
$$25$$ 0 0
$$26$$ −2.00000 −0.392232
$$27$$ − 1.00000i − 0.192450i
$$28$$ 4.00000i 0.755929i
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ 5.00000i 0.883883i
$$33$$ 1.00000i 0.174078i
$$34$$ 2.00000 0.342997
$$35$$ 0 0
$$36$$ −1.00000 −0.166667
$$37$$ 6.00000i 0.986394i 0.869918 + 0.493197i $$0.164172\pi$$
−0.869918 + 0.493197i $$0.835828\pi$$
$$38$$ 0 0
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ − 4.00000i − 0.617213i
$$43$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$44$$ 1.00000 0.150756
$$45$$ 0 0
$$46$$ 8.00000 1.17954
$$47$$ 8.00000i 1.16692i 0.812142 + 0.583460i $$0.198301\pi$$
−0.812142 + 0.583460i $$0.801699\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ −9.00000 −1.28571
$$50$$ 0 0
$$51$$ 2.00000 0.280056
$$52$$ 2.00000i 0.277350i
$$53$$ − 6.00000i − 0.824163i −0.911147 0.412082i $$-0.864802\pi$$
0.911147 0.412082i $$-0.135198\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ −12.0000 −1.60357
$$57$$ 0 0
$$58$$ 6.00000i 0.787839i
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 0 0
$$61$$ 6.00000 0.768221 0.384111 0.923287i $$-0.374508\pi$$
0.384111 + 0.923287i $$0.374508\pi$$
$$62$$ − 8.00000i − 1.01600i
$$63$$ − 4.00000i − 0.503953i
$$64$$ −7.00000 −0.875000
$$65$$ 0 0
$$66$$ −1.00000 −0.123091
$$67$$ − 4.00000i − 0.488678i −0.969690 0.244339i $$-0.921429\pi$$
0.969690 0.244339i $$-0.0785709\pi$$
$$68$$ − 2.00000i − 0.242536i
$$69$$ 8.00000 0.963087
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ − 3.00000i − 0.353553i
$$73$$ 14.0000i 1.63858i 0.573382 + 0.819288i $$0.305631\pi$$
−0.573382 + 0.819288i $$0.694369\pi$$
$$74$$ −6.00000 −0.697486
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 4.00000i 0.455842i
$$78$$ − 2.00000i − 0.226455i
$$79$$ 4.00000 0.450035 0.225018 0.974355i $$-0.427756\pi$$
0.225018 + 0.974355i $$0.427756\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 2.00000i − 0.220863i
$$83$$ − 12.0000i − 1.31717i −0.752506 0.658586i $$-0.771155\pi$$
0.752506 0.658586i $$-0.228845\pi$$
$$84$$ −4.00000 −0.436436
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 6.00000i 0.643268i
$$88$$ 3.00000i 0.319801i
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ −8.00000 −0.838628
$$92$$ − 8.00000i − 0.834058i
$$93$$ − 8.00000i − 0.829561i
$$94$$ −8.00000 −0.825137
$$95$$ 0 0
$$96$$ −5.00000 −0.510310
$$97$$ 2.00000i 0.203069i 0.994832 + 0.101535i $$0.0323753\pi$$
−0.994832 + 0.101535i $$0.967625\pi$$
$$98$$ − 9.00000i − 0.909137i
$$99$$ −1.00000 −0.100504
$$100$$ 0 0
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ 2.00000i 0.198030i
$$103$$ − 8.00000i − 0.788263i −0.919054 0.394132i $$-0.871045\pi$$
0.919054 0.394132i $$-0.128955\pi$$
$$104$$ −6.00000 −0.588348
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ − 12.0000i − 1.16008i −0.814587 0.580042i $$-0.803036\pi$$
0.814587 0.580042i $$-0.196964\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 0 0
$$111$$ −6.00000 −0.569495
$$112$$ − 4.00000i − 0.377964i
$$113$$ 6.00000i 0.564433i 0.959351 + 0.282216i $$0.0910696\pi$$
−0.959351 + 0.282216i $$0.908930\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 6.00000 0.557086
$$117$$ − 2.00000i − 0.184900i
$$118$$ 4.00000i 0.368230i
$$119$$ 8.00000 0.733359
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 6.00000i 0.543214i
$$123$$ − 2.00000i − 0.180334i
$$124$$ −8.00000 −0.718421
$$125$$ 0 0
$$126$$ 4.00000 0.356348
$$127$$ − 4.00000i − 0.354943i −0.984126 0.177471i $$-0.943208\pi$$
0.984126 0.177471i $$-0.0567917\pi$$
$$128$$ 3.00000i 0.265165i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 1.00000i 0.0870388i
$$133$$ 0 0
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ 6.00000 0.514496
$$137$$ 2.00000i 0.170872i 0.996344 + 0.0854358i $$0.0272282\pi$$
−0.996344 + 0.0854358i $$0.972772\pi$$
$$138$$ 8.00000i 0.681005i
$$139$$ 8.00000 0.678551 0.339276 0.940687i $$-0.389818\pi$$
0.339276 + 0.940687i $$0.389818\pi$$
$$140$$ 0 0
$$141$$ −8.00000 −0.673722
$$142$$ 0 0
$$143$$ 2.00000i 0.167248i
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ −14.0000 −1.15865
$$147$$ − 9.00000i − 0.742307i
$$148$$ 6.00000i 0.493197i
$$149$$ 22.0000 1.80231 0.901155 0.433497i $$-0.142720\pi$$
0.901155 + 0.433497i $$0.142720\pi$$
$$150$$ 0 0
$$151$$ 20.0000 1.62758 0.813788 0.581161i $$-0.197401\pi$$
0.813788 + 0.581161i $$0.197401\pi$$
$$152$$ 0 0
$$153$$ 2.00000i 0.161690i
$$154$$ −4.00000 −0.322329
$$155$$ 0 0
$$156$$ −2.00000 −0.160128
$$157$$ 14.0000i 1.11732i 0.829396 + 0.558661i $$0.188685\pi$$
−0.829396 + 0.558661i $$0.811315\pi$$
$$158$$ 4.00000i 0.318223i
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ 32.0000 2.52195
$$162$$ 1.00000i 0.0785674i
$$163$$ − 4.00000i − 0.313304i −0.987654 0.156652i $$-0.949930\pi$$
0.987654 0.156652i $$-0.0500701\pi$$
$$164$$ −2.00000 −0.156174
$$165$$ 0 0
$$166$$ 12.0000 0.931381
$$167$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$168$$ − 12.0000i − 0.925820i
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 6.00000i 0.456172i 0.973641 + 0.228086i $$0.0732467\pi$$
−0.973641 + 0.228086i $$0.926753\pi$$
$$174$$ −6.00000 −0.454859
$$175$$ 0 0
$$176$$ −1.00000 −0.0753778
$$177$$ 4.00000i 0.300658i
$$178$$ 6.00000i 0.449719i
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ 22.0000 1.63525 0.817624 0.575753i $$-0.195291\pi$$
0.817624 + 0.575753i $$0.195291\pi$$
$$182$$ − 8.00000i − 0.592999i
$$183$$ 6.00000i 0.443533i
$$184$$ 24.0000 1.76930
$$185$$ 0 0
$$186$$ 8.00000 0.586588
$$187$$ − 2.00000i − 0.146254i
$$188$$ 8.00000i 0.583460i
$$189$$ 4.00000 0.290957
$$190$$ 0 0
$$191$$ 8.00000 0.578860 0.289430 0.957199i $$-0.406534\pi$$
0.289430 + 0.957199i $$0.406534\pi$$
$$192$$ − 7.00000i − 0.505181i
$$193$$ 14.0000i 1.00774i 0.863779 + 0.503871i $$0.168091\pi$$
−0.863779 + 0.503871i $$0.831909\pi$$
$$194$$ −2.00000 −0.143592
$$195$$ 0 0
$$196$$ −9.00000 −0.642857
$$197$$ − 14.0000i − 0.997459i −0.866758 0.498729i $$-0.833800\pi$$
0.866758 0.498729i $$-0.166200\pi$$
$$198$$ − 1.00000i − 0.0710669i
$$199$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$200$$ 0 0
$$201$$ 4.00000 0.282138
$$202$$ 2.00000i 0.140720i
$$203$$ 24.0000i 1.68447i
$$204$$ 2.00000 0.140028
$$205$$ 0 0
$$206$$ 8.00000 0.557386
$$207$$ 8.00000i 0.556038i
$$208$$ − 2.00000i − 0.138675i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$212$$ − 6.00000i − 0.412082i
$$213$$ 0 0
$$214$$ 12.0000 0.820303
$$215$$ 0 0
$$216$$ 3.00000 0.204124
$$217$$ − 32.0000i − 2.17230i
$$218$$ 2.00000i 0.135457i
$$219$$ −14.0000 −0.946032
$$220$$ 0 0
$$221$$ 4.00000 0.269069
$$222$$ − 6.00000i − 0.402694i
$$223$$ − 16.0000i − 1.07144i −0.844396 0.535720i $$-0.820040\pi$$
0.844396 0.535720i $$-0.179960\pi$$
$$224$$ −20.0000 −1.33631
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ 12.0000i 0.796468i 0.917284 + 0.398234i $$0.130377\pi$$
−0.917284 + 0.398234i $$0.869623\pi$$
$$228$$ 0 0
$$229$$ −6.00000 −0.396491 −0.198246 0.980152i $$-0.563524\pi$$
−0.198246 + 0.980152i $$0.563524\pi$$
$$230$$ 0 0
$$231$$ −4.00000 −0.263181
$$232$$ 18.0000i 1.18176i
$$233$$ − 30.0000i − 1.96537i −0.185296 0.982683i $$-0.559325\pi$$
0.185296 0.982683i $$-0.440675\pi$$
$$234$$ 2.00000 0.130744
$$235$$ 0 0
$$236$$ 4.00000 0.260378
$$237$$ 4.00000i 0.259828i
$$238$$ 8.00000i 0.518563i
$$239$$ −24.0000 −1.55243 −0.776215 0.630468i $$-0.782863\pi$$
−0.776215 + 0.630468i $$0.782863\pi$$
$$240$$ 0 0
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ 1.00000i 0.0642824i
$$243$$ 1.00000i 0.0641500i
$$244$$ 6.00000 0.384111
$$245$$ 0 0
$$246$$ 2.00000 0.127515
$$247$$ 0 0
$$248$$ − 24.0000i − 1.52400i
$$249$$ 12.0000 0.760469
$$250$$ 0 0
$$251$$ 4.00000 0.252478 0.126239 0.992000i $$-0.459709\pi$$
0.126239 + 0.992000i $$0.459709\pi$$
$$252$$ − 4.00000i − 0.251976i
$$253$$ − 8.00000i − 0.502956i
$$254$$ 4.00000 0.250982
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ − 14.0000i − 0.873296i −0.899632 0.436648i $$-0.856166\pi$$
0.899632 0.436648i $$-0.143834\pi$$
$$258$$ 0 0
$$259$$ −24.0000 −1.49129
$$260$$ 0 0
$$261$$ −6.00000 −0.371391
$$262$$ − 12.0000i − 0.741362i
$$263$$ 16.0000i 0.986602i 0.869859 + 0.493301i $$0.164210\pi$$
−0.869859 + 0.493301i $$0.835790\pi$$
$$264$$ −3.00000 −0.184637
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 6.00000i 0.367194i
$$268$$ − 4.00000i − 0.244339i
$$269$$ 2.00000 0.121942 0.0609711 0.998140i $$-0.480580\pi$$
0.0609711 + 0.998140i $$0.480580\pi$$
$$270$$ 0 0
$$271$$ 20.0000 1.21491 0.607457 0.794353i $$-0.292190\pi$$
0.607457 + 0.794353i $$0.292190\pi$$
$$272$$ 2.00000i 0.121268i
$$273$$ − 8.00000i − 0.484182i
$$274$$ −2.00000 −0.120824
$$275$$ 0 0
$$276$$ 8.00000 0.481543
$$277$$ − 26.0000i − 1.56219i −0.624413 0.781094i $$-0.714662\pi$$
0.624413 0.781094i $$-0.285338\pi$$
$$278$$ 8.00000i 0.479808i
$$279$$ 8.00000 0.478947
$$280$$ 0 0
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ − 8.00000i − 0.476393i
$$283$$ − 16.0000i − 0.951101i −0.879688 0.475551i $$-0.842249\pi$$
0.879688 0.475551i $$-0.157751\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ −2.00000 −0.118262
$$287$$ − 8.00000i − 0.472225i
$$288$$ − 5.00000i − 0.294628i
$$289$$ 13.0000 0.764706
$$290$$ 0 0
$$291$$ −2.00000 −0.117242
$$292$$ 14.0000i 0.819288i
$$293$$ 6.00000i 0.350524i 0.984522 + 0.175262i $$0.0560772\pi$$
−0.984522 + 0.175262i $$0.943923\pi$$
$$294$$ 9.00000 0.524891
$$295$$ 0 0
$$296$$ −18.0000 −1.04623
$$297$$ − 1.00000i − 0.0580259i
$$298$$ 22.0000i 1.27443i
$$299$$ 16.0000 0.925304
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 20.0000i 1.15087i
$$303$$ 2.00000i 0.114897i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ −2.00000 −0.114332
$$307$$ 32.0000i 1.82634i 0.407583 + 0.913168i $$0.366372\pi$$
−0.407583 + 0.913168i $$0.633628\pi$$
$$308$$ 4.00000i 0.227921i
$$309$$ 8.00000 0.455104
$$310$$ 0 0
$$311$$ −24.0000 −1.36092 −0.680458 0.732787i $$-0.738219\pi$$
−0.680458 + 0.732787i $$0.738219\pi$$
$$312$$ − 6.00000i − 0.339683i
$$313$$ 22.0000i 1.24351i 0.783210 + 0.621757i $$0.213581\pi$$
−0.783210 + 0.621757i $$0.786419\pi$$
$$314$$ −14.0000 −0.790066
$$315$$ 0 0
$$316$$ 4.00000 0.225018
$$317$$ 22.0000i 1.23564i 0.786318 + 0.617822i $$0.211985\pi$$
−0.786318 + 0.617822i $$0.788015\pi$$
$$318$$ 6.00000i 0.336463i
$$319$$ 6.00000 0.335936
$$320$$ 0 0
$$321$$ 12.0000 0.669775
$$322$$ 32.0000i 1.78329i
$$323$$ 0 0
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ 4.00000 0.221540
$$327$$ 2.00000i 0.110600i
$$328$$ − 6.00000i − 0.331295i
$$329$$ −32.0000 −1.76422
$$330$$ 0 0
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ − 12.0000i − 0.658586i
$$333$$ − 6.00000i − 0.328798i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 4.00000 0.218218
$$337$$ − 22.0000i − 1.19842i −0.800593 0.599208i $$-0.795482\pi$$
0.800593 0.599208i $$-0.204518\pi$$
$$338$$ 9.00000i 0.489535i
$$339$$ −6.00000 −0.325875
$$340$$ 0 0
$$341$$ −8.00000 −0.433224
$$342$$ 0 0
$$343$$ − 8.00000i − 0.431959i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ −6.00000 −0.322562
$$347$$ 4.00000i 0.214731i 0.994220 + 0.107366i $$0.0342415\pi$$
−0.994220 + 0.107366i $$0.965758\pi$$
$$348$$ 6.00000i 0.321634i
$$349$$ −6.00000 −0.321173 −0.160586 0.987022i $$-0.551338\pi$$
−0.160586 + 0.987022i $$0.551338\pi$$
$$350$$ 0 0
$$351$$ 2.00000 0.106752
$$352$$ 5.00000i 0.266501i
$$353$$ − 18.0000i − 0.958043i −0.877803 0.479022i $$-0.840992\pi$$
0.877803 0.479022i $$-0.159008\pi$$
$$354$$ −4.00000 −0.212598
$$355$$ 0 0
$$356$$ 6.00000 0.317999
$$357$$ 8.00000i 0.423405i
$$358$$ − 12.0000i − 0.634220i
$$359$$ 8.00000 0.422224 0.211112 0.977462i $$-0.432292\pi$$
0.211112 + 0.977462i $$0.432292\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ 22.0000i 1.15629i
$$363$$ 1.00000i 0.0524864i
$$364$$ −8.00000 −0.419314
$$365$$ 0 0
$$366$$ −6.00000 −0.313625
$$367$$ − 32.0000i − 1.67039i −0.549957 0.835193i $$-0.685356\pi$$
0.549957 0.835193i $$-0.314644\pi$$
$$368$$ 8.00000i 0.417029i
$$369$$ 2.00000 0.104116
$$370$$ 0 0
$$371$$ 24.0000 1.24602
$$372$$ − 8.00000i − 0.414781i
$$373$$ 2.00000i 0.103556i 0.998659 + 0.0517780i $$0.0164888\pi$$
−0.998659 + 0.0517780i $$0.983511\pi$$
$$374$$ 2.00000 0.103418
$$375$$ 0 0
$$376$$ −24.0000 −1.23771
$$377$$ 12.0000i 0.618031i
$$378$$ 4.00000i 0.205738i
$$379$$ −28.0000 −1.43826 −0.719132 0.694874i $$-0.755460\pi$$
−0.719132 + 0.694874i $$0.755460\pi$$
$$380$$ 0 0
$$381$$ 4.00000 0.204926
$$382$$ 8.00000i 0.409316i
$$383$$ 16.0000i 0.817562i 0.912633 + 0.408781i $$0.134046\pi$$
−0.912633 + 0.408781i $$0.865954\pi$$
$$384$$ −3.00000 −0.153093
$$385$$ 0 0
$$386$$ −14.0000 −0.712581
$$387$$ 0 0
$$388$$ 2.00000i 0.101535i
$$389$$ 18.0000 0.912636 0.456318 0.889817i $$-0.349168\pi$$
0.456318 + 0.889817i $$0.349168\pi$$
$$390$$ 0 0
$$391$$ −16.0000 −0.809155
$$392$$ − 27.0000i − 1.36371i
$$393$$ − 12.0000i − 0.605320i
$$394$$ 14.0000 0.705310
$$395$$ 0 0
$$396$$ −1.00000 −0.0502519
$$397$$ − 2.00000i − 0.100377i −0.998740 0.0501886i $$-0.984018\pi$$
0.998740 0.0501886i $$-0.0159822\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 26.0000 1.29838 0.649189 0.760627i $$-0.275108\pi$$
0.649189 + 0.760627i $$0.275108\pi$$
$$402$$ 4.00000i 0.199502i
$$403$$ − 16.0000i − 0.797017i
$$404$$ 2.00000 0.0995037
$$405$$ 0 0
$$406$$ −24.0000 −1.19110
$$407$$ 6.00000i 0.297409i
$$408$$ 6.00000i 0.297044i
$$409$$ −18.0000 −0.890043 −0.445021 0.895520i $$-0.646804\pi$$
−0.445021 + 0.895520i $$0.646804\pi$$
$$410$$ 0 0
$$411$$ −2.00000 −0.0986527
$$412$$ − 8.00000i − 0.394132i
$$413$$ 16.0000i 0.787309i
$$414$$ −8.00000 −0.393179
$$415$$ 0 0
$$416$$ −10.0000 −0.490290
$$417$$ 8.00000i 0.391762i
$$418$$ 0 0
$$419$$ 4.00000 0.195413 0.0977064 0.995215i $$-0.468849\pi$$
0.0977064 + 0.995215i $$0.468849\pi$$
$$420$$ 0 0
$$421$$ −26.0000 −1.26716 −0.633581 0.773676i $$-0.718416\pi$$
−0.633581 + 0.773676i $$0.718416\pi$$
$$422$$ 0 0
$$423$$ − 8.00000i − 0.388973i
$$424$$ 18.0000 0.874157
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 24.0000i 1.16144i
$$428$$ − 12.0000i − 0.580042i
$$429$$ −2.00000 −0.0965609
$$430$$ 0 0
$$431$$ −24.0000 −1.15604 −0.578020 0.816023i $$-0.696174\pi$$
−0.578020 + 0.816023i $$0.696174\pi$$
$$432$$ 1.00000i 0.0481125i
$$433$$ − 34.0000i − 1.63394i −0.576683 0.816968i $$-0.695653\pi$$
0.576683 0.816968i $$-0.304347\pi$$
$$434$$ 32.0000 1.53605
$$435$$ 0 0
$$436$$ 2.00000 0.0957826
$$437$$ 0 0
$$438$$ − 14.0000i − 0.668946i
$$439$$ 20.0000 0.954548 0.477274 0.878755i $$-0.341625\pi$$
0.477274 + 0.878755i $$0.341625\pi$$
$$440$$ 0 0
$$441$$ 9.00000 0.428571
$$442$$ 4.00000i 0.190261i
$$443$$ − 28.0000i − 1.33032i −0.746701 0.665160i $$-0.768363\pi$$
0.746701 0.665160i $$-0.231637\pi$$
$$444$$ −6.00000 −0.284747
$$445$$ 0 0
$$446$$ 16.0000 0.757622
$$447$$ 22.0000i 1.04056i
$$448$$ − 28.0000i − 1.32288i
$$449$$ −2.00000 −0.0943858 −0.0471929 0.998886i $$-0.515028\pi$$
−0.0471929 + 0.998886i $$0.515028\pi$$
$$450$$ 0 0
$$451$$ −2.00000 −0.0941763
$$452$$ 6.00000i 0.282216i
$$453$$ 20.0000i 0.939682i
$$454$$ −12.0000 −0.563188
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 18.0000i 0.842004i 0.907060 + 0.421002i $$0.138322\pi$$
−0.907060 + 0.421002i $$0.861678\pi$$
$$458$$ − 6.00000i − 0.280362i
$$459$$ −2.00000 −0.0933520
$$460$$ 0 0
$$461$$ −30.0000 −1.39724 −0.698620 0.715493i $$-0.746202\pi$$
−0.698620 + 0.715493i $$0.746202\pi$$
$$462$$ − 4.00000i − 0.186097i
$$463$$ − 16.0000i − 0.743583i −0.928316 0.371792i $$-0.878744\pi$$
0.928316 0.371792i $$-0.121256\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ 30.0000 1.38972
$$467$$ − 12.0000i − 0.555294i −0.960683 0.277647i $$-0.910445\pi$$
0.960683 0.277647i $$-0.0895545\pi$$
$$468$$ − 2.00000i − 0.0924500i
$$469$$ 16.0000 0.738811
$$470$$ 0 0
$$471$$ −14.0000 −0.645086
$$472$$ 12.0000i 0.552345i
$$473$$ 0 0
$$474$$ −4.00000 −0.183726
$$475$$ 0 0
$$476$$ 8.00000 0.366679
$$477$$ 6.00000i 0.274721i
$$478$$ − 24.0000i − 1.09773i
$$479$$ −8.00000 −0.365529 −0.182765 0.983157i $$-0.558505\pi$$
−0.182765 + 0.983157i $$0.558505\pi$$
$$480$$ 0 0
$$481$$ −12.0000 −0.547153
$$482$$ 10.0000i 0.455488i
$$483$$ 32.0000i 1.45605i
$$484$$ 1.00000 0.0454545
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ − 16.0000i − 0.725029i −0.931978 0.362515i $$-0.881918\pi$$
0.931978 0.362515i $$-0.118082\pi$$
$$488$$ 18.0000i 0.814822i
$$489$$ 4.00000 0.180886
$$490$$ 0 0
$$491$$ 4.00000 0.180517 0.0902587 0.995918i $$-0.471231\pi$$
0.0902587 + 0.995918i $$0.471231\pi$$
$$492$$ − 2.00000i − 0.0901670i
$$493$$ − 12.0000i − 0.540453i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 8.00000 0.359211
$$497$$ 0 0
$$498$$ 12.0000i 0.537733i
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 4.00000i 0.178529i
$$503$$ 32.0000i 1.42681i 0.700752 + 0.713405i $$0.252848\pi$$
−0.700752 + 0.713405i $$0.747152\pi$$
$$504$$ 12.0000 0.534522
$$505$$ 0 0
$$506$$ 8.00000 0.355643
$$507$$ 9.00000i 0.399704i
$$508$$ − 4.00000i − 0.177471i
$$509$$ −30.0000 −1.32973 −0.664863 0.746965i $$-0.731510\pi$$
−0.664863 + 0.746965i $$0.731510\pi$$
$$510$$ 0 0
$$511$$ −56.0000 −2.47729
$$512$$ − 11.0000i − 0.486136i
$$513$$ 0 0
$$514$$ 14.0000 0.617514
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 8.00000i 0.351840i
$$518$$ − 24.0000i − 1.05450i
$$519$$ −6.00000 −0.263371
$$520$$ 0 0
$$521$$ −30.0000 −1.31432 −0.657162 0.753749i $$-0.728243\pi$$
−0.657162 + 0.753749i $$0.728243\pi$$
$$522$$ − 6.00000i − 0.262613i
$$523$$ 16.0000i 0.699631i 0.936819 + 0.349816i $$0.113756\pi$$
−0.936819 + 0.349816i $$0.886244\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ −16.0000 −0.697633
$$527$$ 16.0000i 0.696971i
$$528$$ − 1.00000i − 0.0435194i
$$529$$ −41.0000 −1.78261
$$530$$ 0 0
$$531$$ −4.00000 −0.173585
$$532$$ 0 0
$$533$$ − 4.00000i − 0.173259i
$$534$$ −6.00000 −0.259645
$$535$$ 0 0
$$536$$ 12.0000 0.518321
$$537$$ − 12.0000i − 0.517838i
$$538$$ 2.00000i 0.0862261i
$$539$$ −9.00000 −0.387657
$$540$$ 0 0
$$541$$ 46.0000 1.97769 0.988847 0.148933i $$-0.0475840\pi$$
0.988847 + 0.148933i $$0.0475840\pi$$
$$542$$ 20.0000i 0.859074i
$$543$$ 22.0000i 0.944110i
$$544$$ 10.0000 0.428746
$$545$$ 0 0
$$546$$ 8.00000 0.342368
$$547$$ 8.00000i 0.342055i 0.985266 + 0.171028i $$0.0547087\pi$$
−0.985266 + 0.171028i $$0.945291\pi$$
$$548$$ 2.00000i 0.0854358i
$$549$$ −6.00000 −0.256074
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 24.0000i 1.02151i
$$553$$ 16.0000i 0.680389i
$$554$$ 26.0000 1.10463
$$555$$ 0 0
$$556$$ 8.00000 0.339276
$$557$$ − 14.0000i − 0.593199i −0.955002 0.296600i $$-0.904147\pi$$
0.955002 0.296600i $$-0.0958526\pi$$
$$558$$ 8.00000i 0.338667i
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 2.00000 0.0844401
$$562$$ − 18.0000i − 0.759284i
$$563$$ 44.0000i 1.85438i 0.374593 + 0.927189i $$0.377783\pi$$
−0.374593 + 0.927189i $$0.622217\pi$$
$$564$$ −8.00000 −0.336861
$$565$$ 0 0
$$566$$ 16.0000 0.672530
$$567$$ 4.00000i 0.167984i
$$568$$ 0 0
$$569$$ 42.0000 1.76073 0.880366 0.474295i $$-0.157297\pi$$
0.880366 + 0.474295i $$0.157297\pi$$
$$570$$ 0 0
$$571$$ −16.0000 −0.669579 −0.334790 0.942293i $$-0.608665\pi$$
−0.334790 + 0.942293i $$0.608665\pi$$
$$572$$ 2.00000i 0.0836242i
$$573$$ 8.00000i 0.334205i
$$574$$ 8.00000 0.333914
$$575$$ 0 0
$$576$$ 7.00000 0.291667
$$577$$ − 30.0000i − 1.24892i −0.781058 0.624458i $$-0.785320\pi$$
0.781058 0.624458i $$-0.214680\pi$$
$$578$$ 13.0000i 0.540729i
$$579$$ −14.0000 −0.581820
$$580$$ 0 0
$$581$$ 48.0000 1.99138
$$582$$ − 2.00000i − 0.0829027i
$$583$$ − 6.00000i − 0.248495i
$$584$$ −42.0000 −1.73797
$$585$$ 0 0
$$586$$ −6.00000 −0.247858
$$587$$ 28.0000i 1.15568i 0.816149 + 0.577842i $$0.196105\pi$$
−0.816149 + 0.577842i $$0.803895\pi$$
$$588$$ − 9.00000i − 0.371154i
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 14.0000 0.575883
$$592$$ − 6.00000i − 0.246598i
$$593$$ − 38.0000i − 1.56047i −0.625485 0.780236i $$-0.715099\pi$$
0.625485 0.780236i $$-0.284901\pi$$
$$594$$ 1.00000 0.0410305
$$595$$ 0 0
$$596$$ 22.0000 0.901155
$$597$$ 0 0
$$598$$ 16.0000i 0.654289i
$$599$$ 8.00000 0.326871 0.163436 0.986554i $$-0.447742\pi$$
0.163436 + 0.986554i $$0.447742\pi$$
$$600$$ 0 0
$$601$$ 26.0000 1.06056 0.530281 0.847822i $$-0.322086\pi$$
0.530281 + 0.847822i $$0.322086\pi$$
$$602$$ 0 0
$$603$$ 4.00000i 0.162893i
$$604$$ 20.0000 0.813788
$$605$$ 0 0
$$606$$ −2.00000 −0.0812444
$$607$$ − 4.00000i − 0.162355i −0.996700 0.0811775i $$-0.974132\pi$$
0.996700 0.0811775i $$-0.0258681\pi$$
$$608$$ 0 0
$$609$$ −24.0000 −0.972529
$$610$$ 0 0
$$611$$ −16.0000 −0.647291
$$612$$ 2.00000i 0.0808452i
$$613$$ − 14.0000i − 0.565455i −0.959200 0.282727i $$-0.908761\pi$$
0.959200 0.282727i $$-0.0912392\pi$$
$$614$$ −32.0000 −1.29141
$$615$$ 0 0
$$616$$ −12.0000 −0.483494
$$617$$ − 30.0000i − 1.20775i −0.797077 0.603877i $$-0.793622\pi$$
0.797077 0.603877i $$-0.206378\pi$$
$$618$$ 8.00000i 0.321807i
$$619$$ −44.0000 −1.76851 −0.884255 0.467005i $$-0.845333\pi$$
−0.884255 + 0.467005i $$0.845333\pi$$
$$620$$ 0 0
$$621$$ −8.00000 −0.321029
$$622$$ − 24.0000i − 0.962312i
$$623$$ 24.0000i 0.961540i
$$624$$ 2.00000 0.0800641
$$625$$ 0 0
$$626$$ −22.0000 −0.879297
$$627$$ 0 0
$$628$$ 14.0000i 0.558661i
$$629$$ 12.0000 0.478471
$$630$$ 0 0
$$631$$ 16.0000 0.636950 0.318475 0.947931i $$-0.396829\pi$$
0.318475 + 0.947931i $$0.396829\pi$$
$$632$$ 12.0000i 0.477334i
$$633$$ 0 0
$$634$$ −22.0000 −0.873732
$$635$$ 0 0
$$636$$ 6.00000 0.237915
$$637$$ − 18.0000i − 0.713186i
$$638$$ 6.00000i 0.237542i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ 12.0000i 0.473602i
$$643$$ − 20.0000i − 0.788723i −0.918955 0.394362i $$-0.870966\pi$$
0.918955 0.394362i $$-0.129034\pi$$
$$644$$ 32.0000 1.26098
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 8.00000i 0.314512i 0.987558 + 0.157256i $$0.0502649\pi$$
−0.987558 + 0.157256i $$0.949735\pi$$
$$648$$ 3.00000i 0.117851i
$$649$$ 4.00000 0.157014
$$650$$ 0 0
$$651$$ 32.0000 1.25418
$$652$$ − 4.00000i − 0.156652i
$$653$$ 2.00000i 0.0782660i 0.999234 + 0.0391330i $$0.0124596\pi$$
−0.999234 + 0.0391330i $$0.987540\pi$$
$$654$$ −2.00000 −0.0782062
$$655$$ 0 0
$$656$$ 2.00000 0.0780869
$$657$$ − 14.0000i − 0.546192i
$$658$$ − 32.0000i − 1.24749i
$$659$$ −4.00000 −0.155818 −0.0779089 0.996960i $$-0.524824\pi$$
−0.0779089 + 0.996960i $$0.524824\pi$$
$$660$$ 0 0
$$661$$ −26.0000 −1.01128 −0.505641 0.862744i $$-0.668744\pi$$
−0.505641 + 0.862744i $$0.668744\pi$$
$$662$$ − 20.0000i − 0.777322i
$$663$$ 4.00000i 0.155347i
$$664$$ 36.0000 1.39707
$$665$$ 0 0
$$666$$ 6.00000 0.232495
$$667$$ − 48.0000i − 1.85857i
$$668$$ 0 0
$$669$$ 16.0000 0.618596
$$670$$ 0 0
$$671$$ 6.00000 0.231627
$$672$$ − 20.0000i − 0.771517i
$$673$$ 46.0000i 1.77317i 0.462566 + 0.886585i $$0.346929\pi$$
−0.462566 + 0.886585i $$0.653071\pi$$
$$674$$ 22.0000 0.847408
$$675$$ 0 0
$$676$$ 9.00000 0.346154
$$677$$ 18.0000i 0.691796i 0.938272 + 0.345898i $$0.112426\pi$$
−0.938272 + 0.345898i $$0.887574\pi$$
$$678$$ − 6.00000i − 0.230429i
$$679$$ −8.00000 −0.307012
$$680$$ 0 0
$$681$$ −12.0000 −0.459841
$$682$$ − 8.00000i − 0.306336i
$$683$$ − 20.0000i − 0.765279i −0.923898 0.382639i $$-0.875015\pi$$
0.923898 0.382639i $$-0.124985\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 8.00000 0.305441
$$687$$ − 6.00000i − 0.228914i
$$688$$ 0 0
$$689$$ 12.0000 0.457164
$$690$$ 0 0
$$691$$ −28.0000 −1.06517 −0.532585 0.846376i $$-0.678779\pi$$
−0.532585 + 0.846376i $$0.678779\pi$$
$$692$$ 6.00000i 0.228086i
$$693$$ − 4.00000i − 0.151947i
$$694$$ −4.00000 −0.151838
$$695$$ 0 0
$$696$$ −18.0000 −0.682288
$$697$$ 4.00000i 0.151511i
$$698$$ − 6.00000i − 0.227103i
$$699$$ 30.0000 1.13470
$$700$$ 0 0
$$701$$ 50.0000 1.88847 0.944237 0.329267i $$-0.106802\pi$$
0.944237 + 0.329267i $$0.106802\pi$$
$$702$$ 2.00000i 0.0754851i
$$703$$ 0 0
$$704$$ −7.00000 −0.263822
$$705$$ 0 0
$$706$$ 18.0000 0.677439
$$707$$ 8.00000i 0.300871i
$$708$$ 4.00000i 0.150329i
$$709$$ −38.0000 −1.42712 −0.713560 0.700594i $$-0.752918\pi$$
−0.713560 + 0.700594i $$0.752918\pi$$
$$710$$ 0 0
$$711$$ −4.00000 −0.150012
$$712$$ 18.0000i 0.674579i
$$713$$ 64.0000i 2.39682i
$$714$$ −8.00000 −0.299392
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ − 24.0000i − 0.896296i
$$718$$ 8.00000i 0.298557i
$$719$$ −24.0000 −0.895049 −0.447524 0.894272i $$-0.647694\pi$$
−0.447524 + 0.894272i $$0.647694\pi$$
$$720$$ 0 0
$$721$$ 32.0000 1.19174
$$722$$ − 19.0000i − 0.707107i
$$723$$ 10.0000i 0.371904i
$$724$$ 22.0000 0.817624
$$725$$ 0 0
$$726$$ −1.00000 −0.0371135
$$727$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$728$$ − 24.0000i − 0.889499i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 6.00000i 0.221766i
$$733$$ − 30.0000i − 1.10808i −0.832492 0.554038i $$-0.813086\pi$$
0.832492 0.554038i $$-0.186914\pi$$
$$734$$ 32.0000 1.18114
$$735$$ 0 0
$$736$$ 40.0000 1.47442
$$737$$ − 4.00000i − 0.147342i
$$738$$ 2.00000i 0.0736210i
$$739$$ −8.00000 −0.294285 −0.147142 0.989115i $$-0.547008\pi$$
−0.147142 + 0.989115i $$0.547008\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 24.0000i 0.881068i
$$743$$ − 40.0000i − 1.46746i −0.679442 0.733729i $$-0.737778\pi$$
0.679442 0.733729i $$-0.262222\pi$$
$$744$$ 24.0000 0.879883
$$745$$ 0 0
$$746$$ −2.00000 −0.0732252
$$747$$ 12.0000i 0.439057i
$$748$$ − 2.00000i − 0.0731272i
$$749$$ 48.0000 1.75388
$$750$$ 0 0
$$751$$ −8.00000 −0.291924 −0.145962 0.989290i $$-0.546628\pi$$
−0.145962 + 0.989290i $$0.546628\pi$$
$$752$$ − 8.00000i − 0.291730i
$$753$$ 4.00000i 0.145768i
$$754$$ −12.0000 −0.437014
$$755$$ 0 0
$$756$$ 4.00000 0.145479
$$757$$ − 10.0000i − 0.363456i −0.983349 0.181728i $$-0.941831\pi$$
0.983349 0.181728i $$-0.0581691\pi$$
$$758$$ − 28.0000i − 1.01701i
$$759$$ 8.00000 0.290382
$$760$$ 0 0
$$761$$ 6.00000 0.217500 0.108750 0.994069i $$-0.465315\pi$$
0.108750 + 0.994069i $$0.465315\pi$$
$$762$$ 4.00000i 0.144905i
$$763$$ 8.00000i 0.289619i
$$764$$ 8.00000 0.289430
$$765$$ 0 0
$$766$$ −16.0000 −0.578103
$$767$$ 8.00000i 0.288863i
$$768$$ − 17.0000i − 0.613435i
$$769$$ −2.00000 −0.0721218 −0.0360609 0.999350i $$-0.511481\pi$$
−0.0360609 + 0.999350i $$0.511481\pi$$
$$770$$ 0 0
$$771$$ 14.0000 0.504198
$$772$$ 14.0000i 0.503871i
$$773$$ − 6.00000i − 0.215805i −0.994161 0.107903i $$-0.965587\pi$$
0.994161 0.107903i $$-0.0344134\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −6.00000 −0.215387
$$777$$ − 24.0000i − 0.860995i
$$778$$ 18.0000i 0.645331i
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ − 16.0000i − 0.572159i
$$783$$ − 6.00000i − 0.214423i
$$784$$ 9.00000 0.321429
$$785$$ 0 0
$$786$$ 12.0000 0.428026
$$787$$ − 8.00000i − 0.285169i −0.989783 0.142585i $$-0.954459\pi$$
0.989783 0.142585i $$-0.0455413\pi$$
$$788$$ − 14.0000i − 0.498729i
$$789$$ −16.0000 −0.569615
$$790$$ 0 0
$$791$$ −24.0000 −0.853342
$$792$$ − 3.00000i − 0.106600i
$$793$$ 12.0000i 0.426132i
$$794$$ 2.00000 0.0709773
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 10.0000i − 0.354218i −0.984191 0.177109i $$-0.943325\pi$$
0.984191 0.177109i $$-0.0566745\pi$$
$$798$$ 0 0
$$799$$ 16.0000 0.566039
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ 26.0000i 0.918092i
$$803$$ 14.0000i 0.494049i
$$804$$ 4.00000 0.141069
$$805$$ 0 0
$$806$$ 16.0000 0.563576
$$807$$ 2.00000i 0.0704033i
$$808$$ 6.00000i 0.211079i
$$809$$ −54.0000 −1.89854 −0.949269 0.314464i $$-0.898175\pi$$
−0.949269 + 0.314464i $$0.898175\pi$$
$$810$$ 0 0
$$811$$ −56.0000 −1.96643 −0.983213 0.182462i $$-0.941593\pi$$
−0.983213 + 0.182462i $$0.941593\pi$$
$$812$$ 24.0000i 0.842235i
$$813$$ 20.0000i 0.701431i
$$814$$ −6.00000 −0.210300
$$815$$ 0 0
$$816$$ −2.00000 −0.0700140
$$817$$ 0 0
$$818$$ − 18.0000i − 0.629355i
$$819$$ 8.00000 0.279543
$$820$$ 0 0
$$821$$ −14.0000 −0.488603 −0.244302 0.969699i $$-0.578559\pi$$
−0.244302 + 0.969699i $$0.578559\pi$$
$$822$$ − 2.00000i − 0.0697580i
$$823$$ − 24.0000i − 0.836587i −0.908312 0.418294i $$-0.862628\pi$$
0.908312 0.418294i $$-0.137372\pi$$
$$824$$ 24.0000 0.836080
$$825$$ 0 0
$$826$$ −16.0000 −0.556711
$$827$$ 20.0000i 0.695468i 0.937593 + 0.347734i $$0.113049\pi$$
−0.937593 + 0.347734i $$0.886951\pi$$
$$828$$ 8.00000i 0.278019i
$$829$$ 2.00000 0.0694629 0.0347314 0.999397i $$-0.488942\pi$$
0.0347314 + 0.999397i $$0.488942\pi$$
$$830$$ 0 0
$$831$$ 26.0000 0.901930
$$832$$ − 14.0000i − 0.485363i
$$833$$ 18.0000i 0.623663i
$$834$$ −8.00000 −0.277017
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 8.00000i 0.276520i
$$838$$ 4.00000i 0.138178i
$$839$$ 56.0000 1.93333 0.966667 0.256036i $$-0.0824164\pi$$
0.966667 + 0.256036i $$0.0824164\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ − 26.0000i − 0.896019i
$$843$$ − 18.0000i − 0.619953i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 8.00000 0.275046
$$847$$ 4.00000i 0.137442i
$$848$$ 6.00000i 0.206041i
$$849$$ 16.0000 0.549119
$$850$$ 0 0
$$851$$ 48.0000 1.64542
$$852$$ 0 0
$$853$$ 34.0000i 1.16414i 0.813139 + 0.582069i $$0.197757\pi$$
−0.813139 + 0.582069i $$0.802243\pi$$
$$854$$ −24.0000 −0.821263
$$855$$ 0 0
$$856$$ 36.0000 1.23045
$$857$$ − 10.0000i − 0.341593i −0.985306 0.170797i $$-0.945366\pi$$
0.985306 0.170797i $$-0.0546341\pi$$
$$858$$ − 2.00000i − 0.0682789i
$$859$$ 36.0000 1.22830 0.614152 0.789188i $$-0.289498\pi$$
0.614152 + 0.789188i $$0.289498\pi$$
$$860$$ 0 0
$$861$$ 8.00000 0.272639
$$862$$ − 24.0000i − 0.817443i
$$863$$ − 48.0000i − 1.63394i −0.576681 0.816970i $$-0.695652\pi$$
0.576681 0.816970i $$-0.304348\pi$$
$$864$$ 5.00000 0.170103
$$865$$ 0 0
$$866$$ 34.0000 1.15537
$$867$$ 13.0000i 0.441503i
$$868$$ − 32.0000i − 1.08615i
$$869$$ 4.00000 0.135691
$$870$$ 0 0
$$871$$ 8.00000 0.271070
$$872$$ 6.00000i 0.203186i
$$873$$ − 2.00000i − 0.0676897i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ −14.0000 −0.473016
$$877$$ 6.00000i 0.202606i 0.994856 + 0.101303i $$0.0323011\pi$$
−0.994856 + 0.101303i $$0.967699\pi$$
$$878$$ 20.0000i 0.674967i
$$879$$ −6.00000 −0.202375
$$880$$ 0 0
$$881$$ 26.0000 0.875962 0.437981 0.898984i $$-0.355694\pi$$
0.437981 + 0.898984i $$0.355694\pi$$
$$882$$ 9.00000i 0.303046i
$$883$$ 20.0000i 0.673054i 0.941674 + 0.336527i $$0.109252\pi$$
−0.941674 + 0.336527i $$0.890748\pi$$
$$884$$ 4.00000 0.134535
$$885$$ 0 0
$$886$$ 28.0000 0.940678
$$887$$ 8.00000i 0.268614i 0.990940 + 0.134307i $$0.0428808\pi$$
−0.990940 + 0.134307i $$0.957119\pi$$
$$888$$ − 18.0000i − 0.604040i
$$889$$ 16.0000 0.536623
$$890$$ 0 0
$$891$$ 1.00000 0.0335013
$$892$$ − 16.0000i − 0.535720i
$$893$$ 0 0
$$894$$ −22.0000 −0.735790
$$895$$ 0 0
$$896$$ −12.0000 −0.400892
$$897$$ 16.0000i 0.534224i
$$898$$ − 2.00000i − 0.0667409i
$$899$$ −48.0000 −1.60089
$$900$$ 0 0
$$901$$ −12.0000 −0.399778
$$902$$ − 2.00000i − 0.0665927i
$$903$$ 0 0
$$904$$ −18.0000 −0.598671
$$905$$ 0 0
$$906$$ −20.0000 −0.664455
$$907$$ 12.0000i 0.398453i 0.979953 + 0.199227i $$0.0638430\pi$$
−0.979953 + 0.199227i $$0.936157\pi$$
$$908$$ 12.0000i 0.398234i
$$909$$ −2.00000 −0.0663358
$$910$$ 0 0
$$911$$ 24.0000 0.795155 0.397578 0.917568i $$-0.369851\pi$$
0.397578 + 0.917568i $$0.369851\pi$$
$$912$$ 0 0
$$913$$ − 12.0000i − 0.397142i
$$914$$ −18.0000 −0.595387
$$915$$ 0 0
$$916$$ −6.00000 −0.198246
$$917$$ − 48.0000i − 1.58510i
$$918$$ − 2.00000i − 0.0660098i
$$919$$ 20.0000 0.659739 0.329870 0.944027i $$-0.392995\pi$$
0.329870 + 0.944027i $$0.392995\pi$$
$$920$$ 0 0
$$921$$ −32.0000 −1.05444
$$922$$ − 30.0000i − 0.987997i
$$923$$ 0 0
$$924$$ −4.00000 −0.131590
$$925$$ 0 0
$$926$$ 16.0000 0.525793
$$927$$ 8.00000i 0.262754i
$$928$$ 30.0000i 0.984798i
$$929$$ 6.00000 0.196854 0.0984268 0.995144i $$-0.468619\pi$$
0.0984268 + 0.995144i $$0.468619\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ − 30.0000i − 0.982683i
$$933$$ − 24.0000i − 0.785725i
$$934$$ 12.0000 0.392652
$$935$$ 0 0
$$936$$ 6.00000 0.196116
$$937$$ 26.0000i 0.849383i 0.905338 + 0.424691i $$0.139617\pi$$
−0.905338 + 0.424691i $$0.860383\pi$$
$$938$$ 16.0000i 0.522419i
$$939$$ −22.0000 −0.717943
$$940$$ 0 0
$$941$$ −54.0000 −1.76035 −0.880175 0.474650i $$-0.842575\pi$$
−0.880175 + 0.474650i $$0.842575\pi$$
$$942$$ − 14.0000i − 0.456145i
$$943$$ 16.0000i 0.521032i
$$944$$ −4.00000 −0.130189
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 12.0000i 0.389948i 0.980808 + 0.194974i $$0.0624622\pi$$
−0.980808 + 0.194974i $$0.937538\pi$$
$$948$$ 4.00000i 0.129914i
$$949$$ −28.0000 −0.908918
$$950$$ 0 0
$$951$$ −22.0000 −0.713399
$$952$$ 24.0000i 0.777844i
$$953$$ − 22.0000i − 0.712650i −0.934362 0.356325i $$-0.884030\pi$$
0.934362 0.356325i $$-0.115970\pi$$
$$954$$ −6.00000 −0.194257
$$955$$ 0 0
$$956$$ −24.0000 −0.776215
$$957$$ 6.00000i 0.193952i
$$958$$ − 8.00000i − 0.258468i
$$959$$ −8.00000 −0.258333
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ − 12.0000i − 0.386896i
$$963$$ 12.0000i 0.386695i
$$964$$ 10.0000 0.322078
$$965$$ 0 0
$$966$$ −32.0000 −1.02958
$$967$$ 4.00000i 0.128631i 0.997930 + 0.0643157i $$0.0204865\pi$$
−0.997930 + 0.0643157i $$0.979514\pi$$
$$968$$ 3.00000i 0.0964237i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −52.0000 −1.66876 −0.834380 0.551190i $$-0.814174\pi$$
−0.834380 + 0.551190i $$0.814174\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ 32.0000i 1.02587i
$$974$$ 16.0000 0.512673
$$975$$ 0 0
$$976$$ −6.00000 −0.192055
$$977$$ − 6.00000i − 0.191957i −0.995383 0.0959785i $$-0.969402\pi$$
0.995383 0.0959785i $$-0.0305980\pi$$
$$978$$ 4.00000i 0.127906i
$$979$$ 6.00000 0.191761
$$980$$ 0 0
$$981$$ −2.00000 −0.0638551
$$982$$ 4.00000i 0.127645i
$$983$$ − 24.0000i − 0.765481i −0.923856 0.382741i $$-0.874980\pi$$
0.923856 0.382741i $$-0.125020\pi$$
$$984$$ 6.00000 0.191273
$$985$$ 0 0
$$986$$ 12.0000 0.382158
$$987$$ − 32.0000i − 1.01857i
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 40.0000 1.27064 0.635321 0.772248i $$-0.280868\pi$$
0.635321 + 0.772248i $$0.280868\pi$$
$$992$$ − 40.0000i − 1.27000i
$$993$$ − 20.0000i − 0.634681i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 12.0000 0.380235
$$997$$ 14.0000i 0.443384i 0.975117 + 0.221692i $$0.0711580\pi$$
−0.975117 + 0.221692i $$0.928842\pi$$
$$998$$ 4.00000i 0.126618i
$$999$$ 6.00000 0.189832
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 825.2.c.a.199.2 2
3.2 odd 2 2475.2.c.d.199.1 2
5.2 odd 4 825.2.a.a.1.1 1
5.3 odd 4 33.2.a.a.1.1 1
5.4 even 2 inner 825.2.c.a.199.1 2
15.2 even 4 2475.2.a.g.1.1 1
15.8 even 4 99.2.a.b.1.1 1
15.14 odd 2 2475.2.c.d.199.2 2
20.3 even 4 528.2.a.g.1.1 1
35.13 even 4 1617.2.a.j.1.1 1
40.3 even 4 2112.2.a.j.1.1 1
40.13 odd 4 2112.2.a.bb.1.1 1
45.13 odd 12 891.2.e.e.298.1 2
45.23 even 12 891.2.e.g.298.1 2
45.38 even 12 891.2.e.g.595.1 2
45.43 odd 12 891.2.e.e.595.1 2
55.3 odd 20 363.2.e.e.130.1 4
55.8 even 20 363.2.e.g.130.1 4
55.13 even 20 363.2.e.g.202.1 4
55.18 even 20 363.2.e.g.148.1 4
55.28 even 20 363.2.e.g.124.1 4
55.32 even 4 9075.2.a.q.1.1 1
55.38 odd 20 363.2.e.e.124.1 4
55.43 even 4 363.2.a.b.1.1 1
55.48 odd 20 363.2.e.e.148.1 4
55.53 odd 20 363.2.e.e.202.1 4
60.23 odd 4 1584.2.a.o.1.1 1
65.38 odd 4 5577.2.a.a.1.1 1
85.33 odd 4 9537.2.a.m.1.1 1
105.83 odd 4 4851.2.a.b.1.1 1
120.53 even 4 6336.2.a.x.1.1 1
120.83 odd 4 6336.2.a.n.1.1 1
165.98 odd 4 1089.2.a.j.1.1 1
220.43 odd 4 5808.2.a.t.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
33.2.a.a.1.1 1 5.3 odd 4
99.2.a.b.1.1 1 15.8 even 4
363.2.a.b.1.1 1 55.43 even 4
363.2.e.e.124.1 4 55.38 odd 20
363.2.e.e.130.1 4 55.3 odd 20
363.2.e.e.148.1 4 55.48 odd 20
363.2.e.e.202.1 4 55.53 odd 20
363.2.e.g.124.1 4 55.28 even 20
363.2.e.g.130.1 4 55.8 even 20
363.2.e.g.148.1 4 55.18 even 20
363.2.e.g.202.1 4 55.13 even 20
528.2.a.g.1.1 1 20.3 even 4
825.2.a.a.1.1 1 5.2 odd 4
825.2.c.a.199.1 2 5.4 even 2 inner
825.2.c.a.199.2 2 1.1 even 1 trivial
891.2.e.e.298.1 2 45.13 odd 12
891.2.e.e.595.1 2 45.43 odd 12
891.2.e.g.298.1 2 45.23 even 12
891.2.e.g.595.1 2 45.38 even 12
1089.2.a.j.1.1 1 165.98 odd 4
1584.2.a.o.1.1 1 60.23 odd 4
1617.2.a.j.1.1 1 35.13 even 4
2112.2.a.j.1.1 1 40.3 even 4
2112.2.a.bb.1.1 1 40.13 odd 4
2475.2.a.g.1.1 1 15.2 even 4
2475.2.c.d.199.1 2 3.2 odd 2
2475.2.c.d.199.2 2 15.14 odd 2
4851.2.a.b.1.1 1 105.83 odd 4
5577.2.a.a.1.1 1 65.38 odd 4
5808.2.a.t.1.1 1 220.43 odd 4
6336.2.a.n.1.1 1 120.83 odd 4
6336.2.a.x.1.1 1 120.53 even 4
9075.2.a.q.1.1 1 55.32 even 4
9537.2.a.m.1.1 1 85.33 odd 4