Properties

 Label 1425.2.c.g.799.2 Level $1425$ Weight $2$ Character 1425.799 Analytic conductor $11.379$ Analytic rank $0$ Dimension $2$ Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1425,2,Mod(799,1425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1425, 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("1425.799");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1425 = 3 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1425.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: $$11.3786822880$$ 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 57) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 799.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1425.799 Dual form 1425.2.c.g.799.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.00000i q^{8} -1.00000 q^{9} -1.00000i q^{12} -6.00000i q^{13} -1.00000 q^{16} -6.00000i q^{17} -1.00000i q^{18} +1.00000 q^{19} -4.00000i q^{23} +3.00000 q^{24} +6.00000 q^{26} +1.00000i q^{27} -2.00000 q^{29} +8.00000 q^{31} +5.00000i q^{32} +6.00000 q^{34} -1.00000 q^{36} -10.0000i q^{37} +1.00000i q^{38} -6.00000 q^{39} -2.00000 q^{41} +4.00000i q^{43} +4.00000 q^{46} +12.0000i q^{47} +1.00000i q^{48} +7.00000 q^{49} -6.00000 q^{51} -6.00000i q^{52} +6.00000i q^{53} -1.00000 q^{54} -1.00000i q^{57} -2.00000i q^{58} +12.0000 q^{59} -2.00000 q^{61} +8.00000i q^{62} -7.00000 q^{64} -4.00000i q^{67} -6.00000i q^{68} -4.00000 q^{69} -3.00000i q^{72} -10.0000i q^{73} +10.0000 q^{74} +1.00000 q^{76} -6.00000i q^{78} +1.00000 q^{81} -2.00000i q^{82} -16.0000i q^{83} -4.00000 q^{86} +2.00000i q^{87} +2.00000 q^{89} -4.00000i q^{92} -8.00000i q^{93} -12.0000 q^{94} +5.00000 q^{96} +10.0000i q^{97} +7.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^4 + 2 * q^6 - 2 * q^9 $$2 q + 2 q^{4} + 2 q^{6} - 2 q^{9} - 2 q^{16} + 2 q^{19} + 6 q^{24} + 12 q^{26} - 4 q^{29} + 16 q^{31} + 12 q^{34} - 2 q^{36} - 12 q^{39} - 4 q^{41} + 8 q^{46} + 14 q^{49} - 12 q^{51} - 2 q^{54} + 24 q^{59} - 4 q^{61} - 14 q^{64} - 8 q^{69} + 20 q^{74} + 2 q^{76} + 2 q^{81} - 8 q^{86} + 4 q^{89} - 24 q^{94} + 10 q^{96}+O(q^{100})$$ 2 * q + 2 * q^4 + 2 * q^6 - 2 * q^9 - 2 * q^16 + 2 * q^19 + 6 * q^24 + 12 * q^26 - 4 * q^29 + 16 * q^31 + 12 * q^34 - 2 * q^36 - 12 * q^39 - 4 * q^41 + 8 * q^46 + 14 * q^49 - 12 * q^51 - 2 * q^54 + 24 * q^59 - 4 * q^61 - 14 * q^64 - 8 * q^69 + 20 * q^74 + 2 * q^76 + 2 * q^81 - 8 * q^86 + 4 * q^89 - 24 * q^94 + 10 * q^96

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times$$.

 $$n$$ $$476$$ $$1027$$ $$1351$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

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
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ 3.00000i 1.06066i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ − 1.00000i − 0.288675i
$$13$$ − 6.00000i − 1.66410i −0.554700 0.832050i $$-0.687167\pi$$
0.554700 0.832050i $$-0.312833\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ − 6.00000i − 1.45521i −0.685994 0.727607i $$-0.740633\pi$$
0.685994 0.727607i $$-0.259367\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ − 4.00000i − 0.834058i −0.908893 0.417029i $$-0.863071\pi$$
0.908893 0.417029i $$-0.136929\pi$$
$$24$$ 3.00000 0.612372
$$25$$ 0 0
$$26$$ 6.00000 1.17670
$$27$$ 1.00000i 0.192450i
$$28$$ 0 0
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ 8.00000 1.43684 0.718421 0.695608i $$-0.244865\pi$$
0.718421 + 0.695608i $$0.244865\pi$$
$$32$$ 5.00000i 0.883883i
$$33$$ 0 0
$$34$$ 6.00000 1.02899
$$35$$ 0 0
$$36$$ −1.00000 −0.166667
$$37$$ − 10.0000i − 1.64399i −0.569495 0.821995i $$-0.692861\pi$$
0.569495 0.821995i $$-0.307139\pi$$
$$38$$ 1.00000i 0.162221i
$$39$$ −6.00000 −0.960769
$$40$$ 0 0
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 0 0
$$43$$ 4.00000i 0.609994i 0.952353 + 0.304997i $$0.0986555\pi$$
−0.952353 + 0.304997i $$0.901344\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 4.00000 0.589768
$$47$$ 12.0000i 1.75038i 0.483779 + 0.875190i $$0.339264\pi$$
−0.483779 + 0.875190i $$0.660736\pi$$
$$48$$ 1.00000i 0.144338i
$$49$$ 7.00000 1.00000
$$50$$ 0 0
$$51$$ −6.00000 −0.840168
$$52$$ − 6.00000i − 0.832050i
$$53$$ 6.00000i 0.824163i 0.911147 + 0.412082i $$0.135198\pi$$
−0.911147 + 0.412082i $$0.864802\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 1.00000i − 0.132453i
$$58$$ − 2.00000i − 0.262613i
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ 8.00000i 1.01600i
$$63$$ 0 0
$$64$$ −7.00000 −0.875000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 4.00000i − 0.488678i −0.969690 0.244339i $$-0.921429\pi$$
0.969690 0.244339i $$-0.0785709\pi$$
$$68$$ − 6.00000i − 0.727607i
$$69$$ −4.00000 −0.481543
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ − 3.00000i − 0.353553i
$$73$$ − 10.0000i − 1.17041i −0.810885 0.585206i $$-0.801014\pi$$
0.810885 0.585206i $$-0.198986\pi$$
$$74$$ 10.0000 1.16248
$$75$$ 0 0
$$76$$ 1.00000 0.114708
$$77$$ 0 0
$$78$$ − 6.00000i − 0.679366i
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 2.00000i − 0.220863i
$$83$$ − 16.0000i − 1.75623i −0.478451 0.878114i $$-0.658802\pi$$
0.478451 0.878114i $$-0.341198\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −4.00000 −0.431331
$$87$$ 2.00000i 0.214423i
$$88$$ 0 0
$$89$$ 2.00000 0.212000 0.106000 0.994366i $$-0.466196\pi$$
0.106000 + 0.994366i $$0.466196\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ − 4.00000i − 0.417029i
$$93$$ − 8.00000i − 0.829561i
$$94$$ −12.0000 −1.23771
$$95$$ 0 0
$$96$$ 5.00000 0.510310
$$97$$ 10.0000i 1.01535i 0.861550 + 0.507673i $$0.169494\pi$$
−0.861550 + 0.507673i $$0.830506\pi$$
$$98$$ 7.00000i 0.707107i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ − 6.00000i − 0.594089i
$$103$$ − 8.00000i − 0.788263i −0.919054 0.394132i $$-0.871045\pi$$
0.919054 0.394132i $$-0.128955\pi$$
$$104$$ 18.0000 1.76505
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ 4.00000i 0.386695i 0.981130 + 0.193347i $$0.0619344\pi$$
−0.981130 + 0.193347i $$0.938066\pi$$
$$108$$ 1.00000i 0.0962250i
$$109$$ 10.0000 0.957826 0.478913 0.877862i $$-0.341031\pi$$
0.478913 + 0.877862i $$0.341031\pi$$
$$110$$ 0 0
$$111$$ −10.0000 −0.949158
$$112$$ 0 0
$$113$$ − 6.00000i − 0.564433i −0.959351 0.282216i $$-0.908930\pi$$
0.959351 0.282216i $$-0.0910696\pi$$
$$114$$ 1.00000 0.0936586
$$115$$ 0 0
$$116$$ −2.00000 −0.185695
$$117$$ 6.00000i 0.554700i
$$118$$ 12.0000i 1.10469i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ − 2.00000i − 0.181071i
$$123$$ 2.00000i 0.180334i
$$124$$ 8.00000 0.718421
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 8.00000i − 0.709885i −0.934888 0.354943i $$-0.884500\pi$$
0.934888 0.354943i $$-0.115500\pi$$
$$128$$ 3.00000i 0.265165i
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ 8.00000 0.698963 0.349482 0.936943i $$-0.386358\pi$$
0.349482 + 0.936943i $$0.386358\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ 18.0000 1.54349
$$137$$ 18.0000i 1.53784i 0.639343 + 0.768922i $$0.279207\pi$$
−0.639343 + 0.768922i $$0.720793\pi$$
$$138$$ − 4.00000i − 0.340503i
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ 12.0000 1.01058
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ 10.0000 0.827606
$$147$$ − 7.00000i − 0.577350i
$$148$$ − 10.0000i − 0.821995i
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 3.00000i 0.243332i
$$153$$ 6.00000i 0.485071i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −6.00000 −0.480384
$$157$$ − 2.00000i − 0.159617i −0.996810 0.0798087i $$-0.974569\pi$$
0.996810 0.0798087i $$-0.0254309\pi$$
$$158$$ 0 0
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 1.00000i 0.0785674i
$$163$$ 4.00000i 0.313304i 0.987654 + 0.156652i $$0.0500701\pi$$
−0.987654 + 0.156652i $$0.949930\pi$$
$$164$$ −2.00000 −0.156174
$$165$$ 0 0
$$166$$ 16.0000 1.24184
$$167$$ 24.0000i 1.85718i 0.371113 + 0.928588i $$0.378976\pi$$
−0.371113 + 0.928588i $$0.621024\pi$$
$$168$$ 0 0
$$169$$ −23.0000 −1.76923
$$170$$ 0 0
$$171$$ −1.00000 −0.0764719
$$172$$ 4.00000i 0.304997i
$$173$$ 22.0000i 1.67263i 0.548250 + 0.836315i $$0.315294\pi$$
−0.548250 + 0.836315i $$0.684706\pi$$
$$174$$ −2.00000 −0.151620
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 12.0000i − 0.901975i
$$178$$ 2.00000i 0.149906i
$$179$$ 4.00000 0.298974 0.149487 0.988764i $$-0.452238\pi$$
0.149487 + 0.988764i $$0.452238\pi$$
$$180$$ 0 0
$$181$$ 14.0000 1.04061 0.520306 0.853980i $$-0.325818\pi$$
0.520306 + 0.853980i $$0.325818\pi$$
$$182$$ 0 0
$$183$$ 2.00000i 0.147844i
$$184$$ 12.0000 0.884652
$$185$$ 0 0
$$186$$ 8.00000 0.586588
$$187$$ 0 0
$$188$$ 12.0000i 0.875190i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ 7.00000i 0.505181i
$$193$$ 14.0000i 1.00774i 0.863779 + 0.503871i $$0.168091\pi$$
−0.863779 + 0.503871i $$0.831909\pi$$
$$194$$ −10.0000 −0.717958
$$195$$ 0 0
$$196$$ 7.00000 0.500000
$$197$$ − 2.00000i − 0.142494i −0.997459 0.0712470i $$-0.977302\pi$$
0.997459 0.0712470i $$-0.0226979\pi$$
$$198$$ 0 0
$$199$$ 8.00000 0.567105 0.283552 0.958957i $$-0.408487\pi$$
0.283552 + 0.958957i $$0.408487\pi$$
$$200$$ 0 0
$$201$$ −4.00000 −0.282138
$$202$$ − 10.0000i − 0.703598i
$$203$$ 0 0
$$204$$ −6.00000 −0.420084
$$205$$ 0 0
$$206$$ 8.00000 0.557386
$$207$$ 4.00000i 0.278019i
$$208$$ 6.00000i 0.416025i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ 6.00000i 0.412082i
$$213$$ 0 0
$$214$$ −4.00000 −0.273434
$$215$$ 0 0
$$216$$ −3.00000 −0.204124
$$217$$ 0 0
$$218$$ 10.0000i 0.677285i
$$219$$ −10.0000 −0.675737
$$220$$ 0 0
$$221$$ −36.0000 −2.42162
$$222$$ − 10.0000i − 0.671156i
$$223$$ − 16.0000i − 1.07144i −0.844396 0.535720i $$-0.820040\pi$$
0.844396 0.535720i $$-0.179960\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 6.00000 0.399114
$$227$$ − 12.0000i − 0.796468i −0.917284 0.398234i $$-0.869623\pi$$
0.917284 0.398234i $$-0.130377\pi$$
$$228$$ − 1.00000i − 0.0662266i
$$229$$ −6.00000 −0.396491 −0.198246 0.980152i $$-0.563524\pi$$
−0.198246 + 0.980152i $$0.563524\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 6.00000i − 0.393919i
$$233$$ − 10.0000i − 0.655122i −0.944830 0.327561i $$-0.893773\pi$$
0.944830 0.327561i $$-0.106227\pi$$
$$234$$ −6.00000 −0.392232
$$235$$ 0 0
$$236$$ 12.0000 0.781133
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 12.0000 0.776215 0.388108 0.921614i $$-0.373129\pi$$
0.388108 + 0.921614i $$0.373129\pi$$
$$240$$ 0 0
$$241$$ −6.00000 −0.386494 −0.193247 0.981150i $$-0.561902\pi$$
−0.193247 + 0.981150i $$0.561902\pi$$
$$242$$ − 11.0000i − 0.707107i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ −2.00000 −0.128037
$$245$$ 0 0
$$246$$ −2.00000 −0.127515
$$247$$ − 6.00000i − 0.381771i
$$248$$ 24.0000i 1.52400i
$$249$$ −16.0000 −1.01396
$$250$$ 0 0
$$251$$ −24.0000 −1.51487 −0.757433 0.652913i $$-0.773547\pi$$
−0.757433 + 0.652913i $$0.773547\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 8.00000 0.501965
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ 14.0000i 0.873296i 0.899632 + 0.436648i $$0.143834\pi$$
−0.899632 + 0.436648i $$0.856166\pi$$
$$258$$ 4.00000i 0.249029i
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 2.00000 0.123797
$$262$$ 8.00000i 0.494242i
$$263$$ − 12.0000i − 0.739952i −0.929041 0.369976i $$-0.879366\pi$$
0.929041 0.369976i $$-0.120634\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 2.00000i − 0.122398i
$$268$$ − 4.00000i − 0.244339i
$$269$$ 6.00000 0.365826 0.182913 0.983129i $$-0.441447\pi$$
0.182913 + 0.983129i $$0.441447\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ 6.00000i 0.363803i
$$273$$ 0 0
$$274$$ −18.0000 −1.08742
$$275$$ 0 0
$$276$$ −4.00000 −0.240772
$$277$$ 22.0000i 1.32185i 0.750451 + 0.660926i $$0.229836\pi$$
−0.750451 + 0.660926i $$0.770164\pi$$
$$278$$ − 4.00000i − 0.239904i
$$279$$ −8.00000 −0.478947
$$280$$ 0 0
$$281$$ −10.0000 −0.596550 −0.298275 0.954480i $$-0.596411\pi$$
−0.298275 + 0.954480i $$0.596411\pi$$
$$282$$ 12.0000i 0.714590i
$$283$$ 20.0000i 1.18888i 0.804141 + 0.594438i $$0.202626\pi$$
−0.804141 + 0.594438i $$0.797374\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ − 5.00000i − 0.294628i
$$289$$ −19.0000 −1.11765
$$290$$ 0 0
$$291$$ 10.0000 0.586210
$$292$$ − 10.0000i − 0.585206i
$$293$$ 14.0000i 0.817889i 0.912559 + 0.408944i $$0.134103\pi$$
−0.912559 + 0.408944i $$0.865897\pi$$
$$294$$ 7.00000 0.408248
$$295$$ 0 0
$$296$$ 30.0000 1.74371
$$297$$ 0 0
$$298$$ − 6.00000i − 0.347571i
$$299$$ −24.0000 −1.38796
$$300$$ 0 0
$$301$$ 0 0
$$302$$ − 8.00000i − 0.460348i
$$303$$ 10.0000i 0.574485i
$$304$$ −1.00000 −0.0573539
$$305$$ 0 0
$$306$$ −6.00000 −0.342997
$$307$$ − 12.0000i − 0.684876i −0.939540 0.342438i $$-0.888747\pi$$
0.939540 0.342438i $$-0.111253\pi$$
$$308$$ 0 0
$$309$$ −8.00000 −0.455104
$$310$$ 0 0
$$311$$ 4.00000 0.226819 0.113410 0.993548i $$-0.463823\pi$$
0.113410 + 0.993548i $$0.463823\pi$$
$$312$$ − 18.0000i − 1.01905i
$$313$$ 22.0000i 1.24351i 0.783210 + 0.621757i $$0.213581\pi$$
−0.783210 + 0.621757i $$0.786419\pi$$
$$314$$ 2.00000 0.112867
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 6.00000i − 0.336994i −0.985702 0.168497i $$-0.946109\pi$$
0.985702 0.168497i $$-0.0538913\pi$$
$$318$$ 6.00000i 0.336463i
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 4.00000 0.223258
$$322$$ 0 0
$$323$$ − 6.00000i − 0.333849i
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −4.00000 −0.221540
$$327$$ − 10.0000i − 0.553001i
$$328$$ − 6.00000i − 0.331295i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 12.0000 0.659580 0.329790 0.944054i $$-0.393022\pi$$
0.329790 + 0.944054i $$0.393022\pi$$
$$332$$ − 16.0000i − 0.878114i
$$333$$ 10.0000i 0.547997i
$$334$$ −24.0000 −1.31322
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 22.0000i − 1.19842i −0.800593 0.599208i $$-0.795482\pi$$
0.800593 0.599208i $$-0.204518\pi$$
$$338$$ − 23.0000i − 1.25104i
$$339$$ −6.00000 −0.325875
$$340$$ 0 0
$$341$$ 0 0
$$342$$ − 1.00000i − 0.0540738i
$$343$$ 0 0
$$344$$ −12.0000 −0.646997
$$345$$ 0 0
$$346$$ −22.0000 −1.18273
$$347$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$348$$ 2.00000i 0.107211i
$$349$$ 2.00000 0.107058 0.0535288 0.998566i $$-0.482953\pi$$
0.0535288 + 0.998566i $$0.482953\pi$$
$$350$$ 0 0
$$351$$ 6.00000 0.320256
$$352$$ 0 0
$$353$$ 22.0000i 1.17094i 0.810693 + 0.585471i $$0.199090\pi$$
−0.810693 + 0.585471i $$0.800910\pi$$
$$354$$ 12.0000 0.637793
$$355$$ 0 0
$$356$$ 2.00000 0.106000
$$357$$ 0 0
$$358$$ 4.00000i 0.211407i
$$359$$ 20.0000 1.05556 0.527780 0.849381i $$-0.323025\pi$$
0.527780 + 0.849381i $$0.323025\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 14.0000i 0.735824i
$$363$$ 11.0000i 0.577350i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ −2.00000 −0.104542
$$367$$ 32.0000i 1.67039i 0.549957 + 0.835193i $$0.314644\pi$$
−0.549957 + 0.835193i $$0.685356\pi$$
$$368$$ 4.00000i 0.208514i
$$369$$ 2.00000 0.104116
$$370$$ 0 0
$$371$$ 0 0
$$372$$ − 8.00000i − 0.414781i
$$373$$ 10.0000i 0.517780i 0.965907 + 0.258890i $$0.0833568\pi$$
−0.965907 + 0.258890i $$0.916643\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −36.0000 −1.85656
$$377$$ 12.0000i 0.618031i
$$378$$ 0 0
$$379$$ −12.0000 −0.616399 −0.308199 0.951322i $$-0.599726\pi$$
−0.308199 + 0.951322i $$0.599726\pi$$
$$380$$ 0 0
$$381$$ −8.00000 −0.409852
$$382$$ − 12.0000i − 0.613973i
$$383$$ − 8.00000i − 0.408781i −0.978889 0.204390i $$-0.934479\pi$$
0.978889 0.204390i $$-0.0655212\pi$$
$$384$$ 3.00000 0.153093
$$385$$ 0 0
$$386$$ −14.0000 −0.712581
$$387$$ − 4.00000i − 0.203331i
$$388$$ 10.0000i 0.507673i
$$389$$ −30.0000 −1.52106 −0.760530 0.649303i $$-0.775061\pi$$
−0.760530 + 0.649303i $$0.775061\pi$$
$$390$$ 0 0
$$391$$ −24.0000 −1.21373
$$392$$ 21.0000i 1.06066i
$$393$$ − 8.00000i − 0.403547i
$$394$$ 2.00000 0.100759
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 14.0000i 0.702640i 0.936255 + 0.351320i $$0.114267\pi$$
−0.936255 + 0.351320i $$0.885733\pi$$
$$398$$ 8.00000i 0.401004i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 38.0000 1.89763 0.948815 0.315833i $$-0.102284\pi$$
0.948815 + 0.315833i $$0.102284\pi$$
$$402$$ − 4.00000i − 0.199502i
$$403$$ − 48.0000i − 2.39105i
$$404$$ −10.0000 −0.497519
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ − 18.0000i − 0.891133i
$$409$$ 14.0000 0.692255 0.346128 0.938187i $$-0.387496\pi$$
0.346128 + 0.938187i $$0.387496\pi$$
$$410$$ 0 0
$$411$$ 18.0000 0.887875
$$412$$ − 8.00000i − 0.394132i
$$413$$ 0 0
$$414$$ −4.00000 −0.196589
$$415$$ 0 0
$$416$$ 30.0000 1.47087
$$417$$ 4.00000i 0.195881i
$$418$$ 0 0
$$419$$ −8.00000 −0.390826 −0.195413 0.980721i $$-0.562605\pi$$
−0.195413 + 0.980721i $$0.562605\pi$$
$$420$$ 0 0
$$421$$ 14.0000 0.682318 0.341159 0.940006i $$-0.389181\pi$$
0.341159 + 0.940006i $$0.389181\pi$$
$$422$$ − 4.00000i − 0.194717i
$$423$$ − 12.0000i − 0.583460i
$$424$$ −18.0000 −0.874157
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 4.00000i 0.193347i
$$429$$ 0 0
$$430$$ 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
$$433$$ 14.0000i 0.672797i 0.941720 + 0.336399i $$0.109209\pi$$
−0.941720 + 0.336399i $$0.890791\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 10.0000 0.478913
$$437$$ − 4.00000i − 0.191346i
$$438$$ − 10.0000i − 0.477818i
$$439$$ 8.00000 0.381819 0.190910 0.981608i $$-0.438856\pi$$
0.190910 + 0.981608i $$0.438856\pi$$
$$440$$ 0 0
$$441$$ −7.00000 −0.333333
$$442$$ − 36.0000i − 1.71235i
$$443$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$444$$ −10.0000 −0.474579
$$445$$ 0 0
$$446$$ 16.0000 0.757622
$$447$$ 6.00000i 0.283790i
$$448$$ 0 0
$$449$$ 2.00000 0.0943858 0.0471929 0.998886i $$-0.484972\pi$$
0.0471929 + 0.998886i $$0.484972\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ − 6.00000i − 0.282216i
$$453$$ 8.00000i 0.375873i
$$454$$ 12.0000 0.563188
$$455$$ 0 0
$$456$$ 3.00000 0.140488
$$457$$ − 6.00000i − 0.280668i −0.990104 0.140334i $$-0.955182\pi$$
0.990104 0.140334i $$-0.0448177\pi$$
$$458$$ − 6.00000i − 0.280362i
$$459$$ 6.00000 0.280056
$$460$$ 0 0
$$461$$ −18.0000 −0.838344 −0.419172 0.907907i $$-0.637680\pi$$
−0.419172 + 0.907907i $$0.637680\pi$$
$$462$$ 0 0
$$463$$ − 32.0000i − 1.48717i −0.668644 0.743583i $$-0.733125\pi$$
0.668644 0.743583i $$-0.266875\pi$$
$$464$$ 2.00000 0.0928477
$$465$$ 0 0
$$466$$ 10.0000 0.463241
$$467$$ − 32.0000i − 1.48078i −0.672176 0.740392i $$-0.734640\pi$$
0.672176 0.740392i $$-0.265360\pi$$
$$468$$ 6.00000i 0.277350i
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −2.00000 −0.0921551
$$472$$ 36.0000i 1.65703i
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 6.00000i − 0.274721i
$$478$$ 12.0000i 0.548867i
$$479$$ −20.0000 −0.913823 −0.456912 0.889512i $$-0.651044\pi$$
−0.456912 + 0.889512i $$0.651044\pi$$
$$480$$ 0 0
$$481$$ −60.0000 −2.73576
$$482$$ − 6.00000i − 0.273293i
$$483$$ 0 0
$$484$$ −11.0000 −0.500000
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ 32.0000i 1.45006i 0.688718 + 0.725029i $$0.258174\pi$$
−0.688718 + 0.725029i $$0.741826\pi$$
$$488$$ − 6.00000i − 0.271607i
$$489$$ 4.00000 0.180886
$$490$$ 0 0
$$491$$ −32.0000 −1.44414 −0.722070 0.691820i $$-0.756809\pi$$
−0.722070 + 0.691820i $$0.756809\pi$$
$$492$$ 2.00000i 0.0901670i
$$493$$ 12.0000i 0.540453i
$$494$$ 6.00000 0.269953
$$495$$ 0 0
$$496$$ −8.00000 −0.359211
$$497$$ 0 0
$$498$$ − 16.0000i − 0.716977i
$$499$$ −28.0000 −1.25345 −0.626726 0.779240i $$-0.715605\pi$$
−0.626726 + 0.779240i $$0.715605\pi$$
$$500$$ 0 0
$$501$$ 24.0000 1.07224
$$502$$ − 24.0000i − 1.07117i
$$503$$ − 12.0000i − 0.535054i −0.963550 0.267527i $$-0.913794\pi$$
0.963550 0.267527i $$-0.0862064\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 23.0000i 1.02147i
$$508$$ − 8.00000i − 0.354943i
$$509$$ 22.0000 0.975133 0.487566 0.873086i $$-0.337885\pi$$
0.487566 + 0.873086i $$0.337885\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ − 11.0000i − 0.486136i
$$513$$ 1.00000i 0.0441511i
$$514$$ −14.0000 −0.617514
$$515$$ 0 0
$$516$$ 4.00000 0.176090
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 22.0000 0.965693
$$520$$ 0 0
$$521$$ 14.0000 0.613351 0.306676 0.951814i $$-0.400783\pi$$
0.306676 + 0.951814i $$0.400783\pi$$
$$522$$ 2.00000i 0.0875376i
$$523$$ 28.0000i 1.22435i 0.790721 + 0.612177i $$0.209706\pi$$
−0.790721 + 0.612177i $$0.790294\pi$$
$$524$$ 8.00000 0.349482
$$525$$ 0 0
$$526$$ 12.0000 0.523225
$$527$$ − 48.0000i − 2.09091i
$$528$$ 0 0
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ −12.0000 −0.520756
$$532$$ 0 0
$$533$$ 12.0000i 0.519778i
$$534$$ 2.00000 0.0865485
$$535$$ 0 0
$$536$$ 12.0000 0.518321
$$537$$ − 4.00000i − 0.172613i
$$538$$ 6.00000i 0.258678i
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 14.0000 0.601907 0.300954 0.953639i $$-0.402695\pi$$
0.300954 + 0.953639i $$0.402695\pi$$
$$542$$ 0 0
$$543$$ − 14.0000i − 0.600798i
$$544$$ 30.0000 1.28624
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 4.00000i 0.171028i 0.996337 + 0.0855138i $$0.0272532\pi$$
−0.996337 + 0.0855138i $$0.972747\pi$$
$$548$$ 18.0000i 0.768922i
$$549$$ 2.00000 0.0853579
$$550$$ 0 0
$$551$$ −2.00000 −0.0852029
$$552$$ − 12.0000i − 0.510754i
$$553$$ 0 0
$$554$$ −22.0000 −0.934690
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ 30.0000i 1.27114i 0.772043 + 0.635570i $$0.219235\pi$$
−0.772043 + 0.635570i $$0.780765\pi$$
$$558$$ − 8.00000i − 0.338667i
$$559$$ 24.0000 1.01509
$$560$$ 0 0
$$561$$ 0 0
$$562$$ − 10.0000i − 0.421825i
$$563$$ − 20.0000i − 0.842900i −0.906852 0.421450i $$-0.861521\pi$$
0.906852 0.421450i $$-0.138479\pi$$
$$564$$ 12.0000 0.505291
$$565$$ 0 0
$$566$$ −20.0000 −0.840663
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ 44.0000 1.84134 0.920671 0.390339i $$-0.127642\pi$$
0.920671 + 0.390339i $$0.127642\pi$$
$$572$$ 0 0
$$573$$ 12.0000i 0.501307i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 7.00000 0.291667
$$577$$ 18.0000i 0.749350i 0.927156 + 0.374675i $$0.122246\pi$$
−0.927156 + 0.374675i $$0.877754\pi$$
$$578$$ − 19.0000i − 0.790296i
$$579$$ 14.0000 0.581820
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 10.0000i 0.414513i
$$583$$ 0 0
$$584$$ 30.0000 1.24141
$$585$$ 0 0
$$586$$ −14.0000 −0.578335
$$587$$ 8.00000i 0.330195i 0.986277 + 0.165098i $$0.0527939\pi$$
−0.986277 + 0.165098i $$0.947206\pi$$
$$588$$ − 7.00000i − 0.288675i
$$589$$ 8.00000 0.329634
$$590$$ 0 0
$$591$$ −2.00000 −0.0822690
$$592$$ 10.0000i 0.410997i
$$593$$ 30.0000i 1.23195i 0.787765 + 0.615976i $$0.211238\pi$$
−0.787765 + 0.615976i $$0.788762\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −6.00000 −0.245770
$$597$$ − 8.00000i − 0.327418i
$$598$$ − 24.0000i − 0.981433i
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 0 0
$$601$$ 10.0000 0.407909 0.203954 0.978980i $$-0.434621\pi$$
0.203954 + 0.978980i $$0.434621\pi$$
$$602$$ 0 0
$$603$$ 4.00000i 0.162893i
$$604$$ −8.00000 −0.325515
$$605$$ 0 0
$$606$$ −10.0000 −0.406222
$$607$$ 24.0000i 0.974130i 0.873366 + 0.487065i $$0.161933\pi$$
−0.873366 + 0.487065i $$0.838067\pi$$
$$608$$ 5.00000i 0.202777i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 72.0000 2.91281
$$612$$ 6.00000i 0.242536i
$$613$$ − 6.00000i − 0.242338i −0.992632 0.121169i $$-0.961336\pi$$
0.992632 0.121169i $$-0.0386643\pi$$
$$614$$ 12.0000 0.484281
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 2.00000i 0.0805170i 0.999189 + 0.0402585i $$0.0128181\pi$$
−0.999189 + 0.0402585i $$0.987182\pi$$
$$618$$ − 8.00000i − 0.321807i
$$619$$ 28.0000 1.12542 0.562708 0.826656i $$-0.309760\pi$$
0.562708 + 0.826656i $$0.309760\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ 4.00000i 0.160385i
$$623$$ 0 0
$$624$$ 6.00000 0.240192
$$625$$ 0 0
$$626$$ −22.0000 −0.879297
$$627$$ 0 0
$$628$$ − 2.00000i − 0.0798087i
$$629$$ −60.0000 −2.39236
$$630$$ 0 0
$$631$$ −32.0000 −1.27390 −0.636950 0.770905i $$-0.719804\pi$$
−0.636950 + 0.770905i $$0.719804\pi$$
$$632$$ 0 0
$$633$$ 4.00000i 0.158986i
$$634$$ 6.00000 0.238290
$$635$$ 0 0
$$636$$ 6.00000 0.237915
$$637$$ − 42.0000i − 1.66410i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 38.0000 1.50091 0.750455 0.660922i $$-0.229834\pi$$
0.750455 + 0.660922i $$0.229834\pi$$
$$642$$ 4.00000i 0.157867i
$$643$$ − 20.0000i − 0.788723i −0.918955 0.394362i $$-0.870966\pi$$
0.918955 0.394362i $$-0.129034\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 6.00000 0.236067
$$647$$ 36.0000i 1.41531i 0.706560 + 0.707653i $$0.250246\pi$$
−0.706560 + 0.707653i $$0.749754\pi$$
$$648$$ 3.00000i 0.117851i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 4.00000i 0.156652i
$$653$$ − 14.0000i − 0.547862i −0.961749 0.273931i $$-0.911676\pi$$
0.961749 0.273931i $$-0.0883240\pi$$
$$654$$ 10.0000 0.391031
$$655$$ 0 0
$$656$$ 2.00000 0.0780869
$$657$$ 10.0000i 0.390137i
$$658$$ 0 0
$$659$$ 44.0000 1.71400 0.856998 0.515319i $$-0.172327\pi$$
0.856998 + 0.515319i $$0.172327\pi$$
$$660$$ 0 0
$$661$$ 6.00000 0.233373 0.116686 0.993169i $$-0.462773\pi$$
0.116686 + 0.993169i $$0.462773\pi$$
$$662$$ 12.0000i 0.466393i
$$663$$ 36.0000i 1.39812i
$$664$$ 48.0000 1.86276
$$665$$ 0 0
$$666$$ −10.0000 −0.387492
$$667$$ 8.00000i 0.309761i
$$668$$ 24.0000i 0.928588i
$$669$$ −16.0000 −0.618596
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 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
$$676$$ −23.0000 −0.884615
$$677$$ − 22.0000i − 0.845529i −0.906240 0.422764i $$-0.861060\pi$$
0.906240 0.422764i $$-0.138940\pi$$
$$678$$ − 6.00000i − 0.230429i
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −12.0000 −0.459841
$$682$$ 0 0
$$683$$ − 36.0000i − 1.37750i −0.724998 0.688751i $$-0.758159\pi$$
0.724998 0.688751i $$-0.241841\pi$$
$$684$$ −1.00000 −0.0382360
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 6.00000i 0.228914i
$$688$$ − 4.00000i − 0.152499i
$$689$$ 36.0000 1.37149
$$690$$ 0 0
$$691$$ 20.0000 0.760836 0.380418 0.924815i $$-0.375780\pi$$
0.380418 + 0.924815i $$0.375780\pi$$
$$692$$ 22.0000i 0.836315i
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ −6.00000 −0.227429
$$697$$ 12.0000i 0.454532i
$$698$$ 2.00000i 0.0757011i
$$699$$ −10.0000 −0.378235
$$700$$ 0 0
$$701$$ −18.0000 −0.679851 −0.339925 0.940452i $$-0.610402\pi$$
−0.339925 + 0.940452i $$0.610402\pi$$
$$702$$ 6.00000i 0.226455i
$$703$$ − 10.0000i − 0.377157i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ −22.0000 −0.827981
$$707$$ 0 0
$$708$$ − 12.0000i − 0.450988i
$$709$$ −38.0000 −1.42712 −0.713560 0.700594i $$-0.752918\pi$$
−0.713560 + 0.700594i $$0.752918\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 6.00000i 0.224860i
$$713$$ − 32.0000i − 1.19841i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 4.00000 0.149487
$$717$$ − 12.0000i − 0.448148i
$$718$$ 20.0000i 0.746393i
$$719$$ 20.0000 0.745874 0.372937 0.927857i $$-0.378351\pi$$
0.372937 + 0.927857i $$0.378351\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 1.00000i 0.0372161i
$$723$$ 6.00000i 0.223142i
$$724$$ 14.0000 0.520306
$$725$$ 0 0
$$726$$ −11.0000 −0.408248
$$727$$ 8.00000i 0.296704i 0.988935 + 0.148352i $$0.0473968\pi$$
−0.988935 + 0.148352i $$0.952603\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 24.0000 0.887672
$$732$$ 2.00000i 0.0739221i
$$733$$ − 46.0000i − 1.69905i −0.527549 0.849524i $$-0.676889\pi$$
0.527549 0.849524i $$-0.323111\pi$$
$$734$$ −32.0000 −1.18114
$$735$$ 0 0
$$736$$ 20.0000 0.737210
$$737$$ 0 0
$$738$$ 2.00000i 0.0736210i
$$739$$ 36.0000 1.32428 0.662141 0.749380i $$-0.269648\pi$$
0.662141 + 0.749380i $$0.269648\pi$$
$$740$$ 0 0
$$741$$ −6.00000 −0.220416
$$742$$ 0 0
$$743$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$744$$ 24.0000 0.879883
$$745$$ 0 0
$$746$$ −10.0000 −0.366126
$$747$$ 16.0000i 0.585409i
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 16.0000 0.583848 0.291924 0.956441i $$-0.405705\pi$$
0.291924 + 0.956441i $$0.405705\pi$$
$$752$$ − 12.0000i − 0.437595i
$$753$$ 24.0000i 0.874609i
$$754$$ −12.0000 −0.437014
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 38.0000i 1.38113i 0.723269 + 0.690567i $$0.242639\pi$$
−0.723269 + 0.690567i $$0.757361\pi$$
$$758$$ − 12.0000i − 0.435860i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 50.0000 1.81250 0.906249 0.422744i $$-0.138933\pi$$
0.906249 + 0.422744i $$0.138933\pi$$
$$762$$ − 8.00000i − 0.289809i
$$763$$ 0 0
$$764$$ −12.0000 −0.434145
$$765$$ 0 0
$$766$$ 8.00000 0.289052
$$767$$ − 72.0000i − 2.59977i
$$768$$ 17.0000i 0.613435i
$$769$$ −18.0000 −0.649097 −0.324548 0.945869i $$-0.605212\pi$$
−0.324548 + 0.945869i $$0.605212\pi$$
$$770$$ 0 0
$$771$$ 14.0000 0.504198
$$772$$ 14.0000i 0.503871i
$$773$$ − 18.0000i − 0.647415i −0.946157 0.323708i $$-0.895071\pi$$
0.946157 0.323708i $$-0.104929\pi$$
$$774$$ 4.00000 0.143777
$$775$$ 0 0
$$776$$ −30.0000 −1.07694
$$777$$ 0 0
$$778$$ − 30.0000i − 1.07555i
$$779$$ −2.00000 −0.0716574
$$780$$ 0 0
$$781$$ 0 0
$$782$$ − 24.0000i − 0.858238i
$$783$$ − 2.00000i − 0.0714742i
$$784$$ −7.00000 −0.250000
$$785$$ 0 0
$$786$$ 8.00000 0.285351
$$787$$ 44.0000i 1.56843i 0.620489 + 0.784215i $$0.286934\pi$$
−0.620489 + 0.784215i $$0.713066\pi$$
$$788$$ − 2.00000i − 0.0712470i
$$789$$ −12.0000 −0.427211
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 12.0000i 0.426132i
$$794$$ −14.0000 −0.496841
$$795$$ 0 0
$$796$$ 8.00000 0.283552
$$797$$ − 6.00000i − 0.212531i −0.994338 0.106265i $$-0.966111\pi$$
0.994338 0.106265i $$-0.0338893\pi$$
$$798$$ 0 0
$$799$$ 72.0000 2.54718
$$800$$ 0 0
$$801$$ −2.00000 −0.0706665
$$802$$ 38.0000i 1.34183i
$$803$$ 0 0
$$804$$ −4.00000 −0.141069
$$805$$ 0 0
$$806$$ 48.0000 1.69073
$$807$$ − 6.00000i − 0.211210i
$$808$$ − 30.0000i − 1.05540i
$$809$$ −26.0000 −0.914111 −0.457056 0.889438i $$-0.651096\pi$$
−0.457056 + 0.889438i $$0.651096\pi$$
$$810$$ 0 0
$$811$$ 44.0000 1.54505 0.772524 0.634985i $$-0.218994\pi$$
0.772524 + 0.634985i $$0.218994\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 6.00000 0.210042
$$817$$ 4.00000i 0.139942i
$$818$$ 14.0000i 0.489499i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −42.0000 −1.46581 −0.732905 0.680331i $$-0.761836\pi$$
−0.732905 + 0.680331i $$0.761836\pi$$
$$822$$ 18.0000i 0.627822i
$$823$$ 32.0000i 1.11545i 0.830026 + 0.557725i $$0.188326\pi$$
−0.830026 + 0.557725i $$0.811674\pi$$
$$824$$ 24.0000 0.836080
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 28.0000i − 0.973655i −0.873498 0.486828i $$-0.838154\pi$$
0.873498 0.486828i $$-0.161846\pi$$
$$828$$ 4.00000i 0.139010i
$$829$$ 10.0000 0.347314 0.173657 0.984806i $$-0.444442\pi$$
0.173657 + 0.984806i $$0.444442\pi$$
$$830$$ 0 0
$$831$$ 22.0000 0.763172
$$832$$ 42.0000i 1.45609i
$$833$$ − 42.0000i − 1.45521i
$$834$$ −4.00000 −0.138509
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 8.00000i 0.276520i
$$838$$ − 8.00000i − 0.276355i
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 14.0000i 0.482472i
$$843$$ 10.0000i 0.344418i
$$844$$ −4.00000 −0.137686
$$845$$ 0 0
$$846$$ 12.0000 0.412568
$$847$$ 0 0
$$848$$ − 6.00000i − 0.206041i
$$849$$ 20.0000 0.686398
$$850$$ 0 0
$$851$$ −40.0000 −1.37118
$$852$$ 0 0
$$853$$ − 22.0000i − 0.753266i −0.926363 0.376633i $$-0.877082\pi$$
0.926363 0.376633i $$-0.122918\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −12.0000 −0.410152
$$857$$ 30.0000i 1.02478i 0.858753 + 0.512390i $$0.171240\pi$$
−0.858753 + 0.512390i $$0.828760\pi$$
$$858$$ 0 0
$$859$$ 28.0000 0.955348 0.477674 0.878537i $$-0.341480\pi$$
0.477674 + 0.878537i $$0.341480\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ − 24.0000i − 0.817443i
$$863$$ 40.0000i 1.36162i 0.732462 + 0.680808i $$0.238371\pi$$
−0.732462 + 0.680808i $$0.761629\pi$$
$$864$$ −5.00000 −0.170103
$$865$$ 0 0
$$866$$ −14.0000 −0.475739
$$867$$ 19.0000i 0.645274i
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −24.0000 −0.813209
$$872$$ 30.0000i 1.01593i
$$873$$ − 10.0000i − 0.338449i
$$874$$ 4.00000 0.135302
$$875$$ 0 0
$$876$$ −10.0000 −0.337869
$$877$$ − 34.0000i − 1.14810i −0.818821 0.574049i $$-0.805372\pi$$
0.818821 0.574049i $$-0.194628\pi$$
$$878$$ 8.00000i 0.269987i
$$879$$ 14.0000 0.472208
$$880$$ 0 0
$$881$$ −38.0000 −1.28025 −0.640126 0.768270i $$-0.721118\pi$$
−0.640126 + 0.768270i $$0.721118\pi$$
$$882$$ − 7.00000i − 0.235702i
$$883$$ 36.0000i 1.21150i 0.795656 + 0.605748i $$0.207126\pi$$
−0.795656 + 0.605748i $$0.792874\pi$$
$$884$$ −36.0000 −1.21081
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 40.0000i − 1.34307i −0.740973 0.671534i $$-0.765636\pi$$
0.740973 0.671534i $$-0.234364\pi$$
$$888$$ − 30.0000i − 1.00673i
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ − 16.0000i − 0.535720i
$$893$$ 12.0000i 0.401565i
$$894$$ −6.00000 −0.200670
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 24.0000i 0.801337i
$$898$$ 2.00000i 0.0667409i
$$899$$ −16.0000 −0.533630
$$900$$ 0 0
$$901$$ 36.0000 1.19933
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 18.0000 0.598671
$$905$$ 0 0
$$906$$ −8.00000 −0.265782
$$907$$ 4.00000i 0.132818i 0.997792 + 0.0664089i $$0.0211542\pi$$
−0.997792 + 0.0664089i $$0.978846\pi$$
$$908$$ − 12.0000i − 0.398234i
$$909$$ 10.0000 0.331679
$$910$$ 0 0
$$911$$ −16.0000 −0.530104 −0.265052 0.964234i $$-0.585389\pi$$
−0.265052 + 0.964234i $$0.585389\pi$$
$$912$$ 1.00000i 0.0331133i
$$913$$ 0 0
$$914$$ 6.00000 0.198462
$$915$$ 0 0
$$916$$ −6.00000 −0.198246
$$917$$ 0 0
$$918$$ 6.00000i 0.198030i
$$919$$ 16.0000 0.527791 0.263896 0.964551i $$-0.414993\pi$$
0.263896 + 0.964551i $$0.414993\pi$$
$$920$$ 0 0
$$921$$ −12.0000 −0.395413
$$922$$ − 18.0000i − 0.592798i
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 32.0000 1.05159
$$927$$ 8.00000i 0.262754i
$$928$$ − 10.0000i − 0.328266i
$$929$$ −34.0000 −1.11550 −0.557752 0.830008i $$-0.688336\pi$$
−0.557752 + 0.830008i $$0.688336\pi$$
$$930$$ 0 0
$$931$$ 7.00000 0.229416
$$932$$ − 10.0000i − 0.327561i
$$933$$ − 4.00000i − 0.130954i
$$934$$ 32.0000 1.04707
$$935$$ 0 0
$$936$$ −18.0000 −0.588348
$$937$$ − 22.0000i − 0.718709i −0.933201 0.359354i $$-0.882997\pi$$
0.933201 0.359354i $$-0.117003\pi$$
$$938$$ 0 0
$$939$$ 22.0000 0.717943
$$940$$ 0 0
$$941$$ −22.0000 −0.717180 −0.358590 0.933495i $$-0.616742\pi$$
−0.358590 + 0.933495i $$0.616742\pi$$
$$942$$ − 2.00000i − 0.0651635i
$$943$$ 8.00000i 0.260516i
$$944$$ −12.0000 −0.390567
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 8.00000i − 0.259965i −0.991516 0.129983i $$-0.958508\pi$$
0.991516 0.129983i $$-0.0414921\pi$$
$$948$$ 0 0
$$949$$ −60.0000 −1.94768
$$950$$ 0 0
$$951$$ −6.00000 −0.194563
$$952$$ 0 0
$$953$$ − 38.0000i − 1.23094i −0.788160 0.615470i $$-0.788966\pi$$
0.788160 0.615470i $$-0.211034\pi$$
$$954$$ 6.00000 0.194257
$$955$$ 0 0
$$956$$ 12.0000 0.388108
$$957$$ 0 0
$$958$$ − 20.0000i − 0.646171i
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ − 60.0000i − 1.93448i
$$963$$ − 4.00000i − 0.128898i
$$964$$ −6.00000 −0.193247
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 32.0000i − 1.02905i −0.857475 0.514525i $$-0.827968\pi$$
0.857475 0.514525i $$-0.172032\pi$$
$$968$$ − 33.0000i − 1.06066i
$$969$$ −6.00000 −0.192748
$$970$$ 0 0
$$971$$ −36.0000 −1.15529 −0.577647 0.816286i $$-0.696029\pi$$
−0.577647 + 0.816286i $$0.696029\pi$$
$$972$$ − 1.00000i − 0.0320750i
$$973$$ 0 0
$$974$$ −32.0000 −1.02535
$$975$$ 0 0
$$976$$ 2.00000 0.0640184
$$977$$ − 42.0000i − 1.34370i −0.740688 0.671850i $$-0.765500\pi$$
0.740688 0.671850i $$-0.234500\pi$$
$$978$$ 4.00000i 0.127906i
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −10.0000 −0.319275
$$982$$ − 32.0000i − 1.02116i
$$983$$ 8.00000i 0.255160i 0.991828 + 0.127580i $$0.0407210\pi$$
−0.991828 + 0.127580i $$0.959279\pi$$
$$984$$ −6.00000 −0.191273
$$985$$ 0 0
$$986$$ −12.0000 −0.382158
$$987$$ 0 0
$$988$$ − 6.00000i − 0.190885i
$$989$$ 16.0000 0.508770
$$990$$ 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
$$993$$ − 12.0000i − 0.380808i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ −16.0000 −0.506979
$$997$$ − 58.0000i − 1.83688i −0.395562 0.918439i $$-0.629450\pi$$
0.395562 0.918439i $$-0.370550\pi$$
$$998$$ − 28.0000i − 0.886325i
$$999$$ 10.0000 0.316386
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 1425.2.c.g.799.2 2
5.2 odd 4 1425.2.a.a.1.1 1
5.3 odd 4 57.2.a.c.1.1 1
5.4 even 2 inner 1425.2.c.g.799.1 2
15.2 even 4 4275.2.a.m.1.1 1
15.8 even 4 171.2.a.a.1.1 1
20.3 even 4 912.2.a.b.1.1 1
35.13 even 4 2793.2.a.i.1.1 1
40.3 even 4 3648.2.a.bf.1.1 1
40.13 odd 4 3648.2.a.o.1.1 1
55.43 even 4 6897.2.a.a.1.1 1
60.23 odd 4 2736.2.a.s.1.1 1
65.38 odd 4 9633.2.a.h.1.1 1
95.18 even 4 1083.2.a.a.1.1 1
105.83 odd 4 8379.2.a.e.1.1 1
285.113 odd 4 3249.2.a.g.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.a.c.1.1 1 5.3 odd 4
171.2.a.a.1.1 1 15.8 even 4
912.2.a.b.1.1 1 20.3 even 4
1083.2.a.a.1.1 1 95.18 even 4
1425.2.a.a.1.1 1 5.2 odd 4
1425.2.c.g.799.1 2 5.4 even 2 inner
1425.2.c.g.799.2 2 1.1 even 1 trivial
2736.2.a.s.1.1 1 60.23 odd 4
2793.2.a.i.1.1 1 35.13 even 4
3249.2.a.g.1.1 1 285.113 odd 4
3648.2.a.o.1.1 1 40.13 odd 4
3648.2.a.bf.1.1 1 40.3 even 4
4275.2.a.m.1.1 1 15.2 even 4
6897.2.a.a.1.1 1 55.43 even 4
8379.2.a.e.1.1 1 105.83 odd 4
9633.2.a.h.1.1 1 65.38 odd 4