# Properties

 Label 1425.2.c.g.799.1 Level $1425$ Weight $2$ Character 1425.799 Analytic conductor $11.379$ 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] = [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 799.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1425.799 Dual form 1425.2.c.g.799.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} +1.00000 q^{6} -3.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} +1.00000 q^{6} -3.00000i q^{8} -1.00000 q^{9} +1.00000i q^{12} +6.00000i q^{13} -1.00000 q^{16} +6.00000i q^{17} +1.00000i q^{18} +1.00000 q^{19} +4.00000i q^{23} +3.00000 q^{24} +6.00000 q^{26} -1.00000i q^{27} -2.00000 q^{29} +8.00000 q^{31} -5.00000i q^{32} +6.00000 q^{34} -1.00000 q^{36} +10.0000i q^{37} -1.00000i q^{38} -6.00000 q^{39} -2.00000 q^{41} -4.00000i q^{43} +4.00000 q^{46} -12.0000i q^{47} -1.00000i q^{48} +7.00000 q^{49} -6.00000 q^{51} +6.00000i q^{52} -6.00000i q^{53} -1.00000 q^{54} +1.00000i q^{57} +2.00000i q^{58} +12.0000 q^{59} -2.00000 q^{61} -8.00000i q^{62} -7.00000 q^{64} +4.00000i q^{67} +6.00000i q^{68} -4.00000 q^{69} +3.00000i q^{72} +10.0000i q^{73} +10.0000 q^{74} +1.00000 q^{76} +6.00000i q^{78} +1.00000 q^{81} +2.00000i q^{82} +16.0000i q^{83} -4.00000 q^{86} -2.00000i q^{87} +2.00000 q^{89} +4.00000i q^{92} +8.00000i q^{93} -12.0000 q^{94} +5.00000 q^{96} -10.0000i q^{97} -7.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10})$$ 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 $$-0.884973\pi$$
0.935414 0.353553i $$-0.115027\pi$$
$$3$$ 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 $$0.312833\pi$$
−0.554700 + 0.832050i $$0.687167\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ 6.00000i 1.45521i 0.685994 + 0.727607i $$0.259367\pi$$
−0.685994 + 0.727607i $$0.740633\pi$$
$$18$$ 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 $$0.136929\pi$$
−0.908893 + 0.417029i $$0.863071\pi$$
$$24$$ 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 $$0.307139\pi$$
−0.569495 + 0.821995i $$0.692861\pi$$
$$38$$ − 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 $$-0.901344\pi$$
0.952353 0.304997i $$-0.0986555\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 4.00000 0.589768
$$47$$ − 12.0000i − 1.75038i −0.483779 0.875190i $$-0.660736\pi$$
0.483779 0.875190i $$-0.339264\pi$$
$$48$$ − 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 $$-0.864802\pi$$
0.911147 0.412082i $$-0.135198\pi$$
$$54$$ −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 $$0.0785709\pi$$
−0.969690 + 0.244339i $$0.921429\pi$$
$$68$$ 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 $$0.198986\pi$$
−0.810885 + 0.585206i $$0.801014\pi$$
$$74$$ 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 $$0.341198\pi$$
−0.478451 + 0.878114i $$0.658802\pi$$
$$84$$ 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 $$-0.830506\pi$$
0.861550 0.507673i $$-0.169494\pi$$
$$98$$ − 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 $$0.128955\pi$$
−0.919054 + 0.394132i $$0.871045\pi$$
$$104$$ 18.0000 1.76505
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ − 4.00000i − 0.386695i −0.981130 0.193347i $$-0.938066\pi$$
0.981130 0.193347i $$-0.0619344\pi$$
$$108$$ − 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 $$0.0910696\pi$$
−0.959351 + 0.282216i $$0.908930\pi$$
$$114$$ 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 $$0.115500\pi$$
−0.934888 + 0.354943i $$0.884500\pi$$
$$128$$ − 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 $$-0.720793\pi$$
0.639343 0.768922i $$-0.279207\pi$$
$$138$$ 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 $$0.0254309\pi$$
−0.996810 + 0.0798087i $$0.974569\pi$$
$$158$$ 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 $$-0.949930\pi$$
0.987654 0.156652i $$-0.0500701\pi$$
$$164$$ −2.00000 −0.156174
$$165$$ 0 0
$$166$$ 16.0000 1.24184
$$167$$ − 24.0000i − 1.85718i −0.371113 0.928588i $$-0.621024\pi$$
0.371113 0.928588i $$-0.378976\pi$$
$$168$$ 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 $$-0.684706\pi$$
0.548250 0.836315i $$-0.315294\pi$$
$$174$$ −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 $$-0.831909\pi$$
0.863779 0.503871i $$-0.168091\pi$$
$$194$$ −10.0000 −0.717958
$$195$$ 0 0
$$196$$ 7.00000 0.500000
$$197$$ 2.00000i 0.142494i 0.997459 + 0.0712470i $$0.0226979\pi$$
−0.997459 + 0.0712470i $$0.977302\pi$$
$$198$$ 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 $$0.179960\pi$$
−0.844396 + 0.535720i $$0.820040\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 6.00000 0.399114
$$227$$ 12.0000i 0.796468i 0.917284 + 0.398234i $$0.130377\pi$$
−0.917284 + 0.398234i $$0.869623\pi$$
$$228$$ 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 $$0.106227\pi$$
−0.944830 + 0.327561i $$0.893773\pi$$
$$234$$ −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 $$-0.856166\pi$$
0.899632 0.436648i $$-0.143834\pi$$
$$258$$ − 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 $$0.120634\pi$$
−0.929041 + 0.369976i $$0.879366\pi$$
$$264$$ 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 $$-0.770164\pi$$
0.750451 0.660926i $$-0.229836\pi$$
$$278$$ 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 $$-0.797374\pi$$
0.804141 0.594438i $$-0.202626\pi$$
$$284$$ 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 $$-0.865897\pi$$
0.912559 0.408944i $$-0.134103\pi$$
$$294$$ 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 $$0.111253\pi$$
−0.939540 + 0.342438i $$0.888747\pi$$
$$308$$ 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 $$-0.786419\pi$$
0.783210 0.621757i $$-0.213581\pi$$
$$314$$ 2.00000 0.112867
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 6.00000i 0.336994i 0.985702 + 0.168497i $$0.0538913\pi$$
−0.985702 + 0.168497i $$0.946109\pi$$
$$318$$ − 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 $$0.204518\pi$$
−0.800593 + 0.599208i $$0.795482\pi$$
$$338$$ 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 $$-0.800910\pi$$
0.810693 0.585471i $$-0.199090\pi$$
$$354$$ 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 $$-0.685356\pi$$
0.549957 0.835193i $$-0.314644\pi$$
$$368$$ − 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 $$-0.916643\pi$$
0.965907 0.258890i $$-0.0833568\pi$$
$$374$$ 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 $$0.0655212\pi$$
−0.978889 + 0.204390i $$0.934479\pi$$
$$384$$ 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 $$-0.885733\pi$$
0.936255 0.351320i $$-0.114267\pi$$
$$398$$ − 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 $$-0.890791\pi$$
0.941720 0.336399i $$-0.109209\pi$$
$$434$$ 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 $$0.0448177\pi$$
−0.990104 + 0.140334i $$0.955182\pi$$
$$458$$ 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 $$0.266875\pi$$
−0.668644 + 0.743583i $$0.733125\pi$$
$$464$$ 2.00000 0.0928477
$$465$$ 0 0
$$466$$ 10.0000 0.463241
$$467$$ 32.0000i 1.48078i 0.672176 + 0.740392i $$0.265360\pi$$
−0.672176 + 0.740392i $$0.734640\pi$$
$$468$$ − 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 $$-0.741826\pi$$
0.688718 0.725029i $$-0.258174\pi$$
$$488$$ 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 $$0.0862064\pi$$
−0.963550 + 0.267527i $$0.913794\pi$$
$$504$$ 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 $$-0.790294\pi$$
0.790721 0.612177i $$-0.209706\pi$$
$$524$$ 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 $$-0.972747\pi$$
0.996337 0.0855138i $$-0.0272532\pi$$
$$548$$ − 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 $$-0.780765\pi$$
0.772043 0.635570i $$-0.219235\pi$$
$$558$$ 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 $$0.138479\pi$$
−0.906852 + 0.421450i $$0.861521\pi$$
$$564$$ 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 $$-0.877754\pi$$
0.927156 0.374675i $$-0.122246\pi$$
$$578$$ 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 $$-0.947206\pi$$
0.986277 0.165098i $$-0.0527939\pi$$
$$588$$ 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 $$-0.788762\pi$$
0.787765 0.615976i $$-0.211238\pi$$
$$594$$ 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 $$-0.838067\pi$$
0.873366 0.487065i $$-0.161933\pi$$
$$608$$ − 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 $$0.0386643\pi$$
−0.992632 + 0.121169i $$0.961336\pi$$
$$614$$ 12.0000 0.484281
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 2.00000i − 0.0805170i −0.999189 0.0402585i $$-0.987182\pi$$
0.999189 0.0402585i $$-0.0128181\pi$$
$$618$$ 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 $$0.129034\pi$$
−0.918955 + 0.394362i $$0.870966\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 6.00000 0.236067
$$647$$ − 36.0000i − 1.41531i −0.706560 0.707653i $$-0.749754\pi$$
0.706560 0.707653i $$-0.250246\pi$$
$$648$$ − 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 $$0.0883240\pi$$
−0.961749 + 0.273931i $$0.911676\pi$$
$$654$$ 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 $$-0.653071\pi$$
0.462566 0.886585i $$-0.346929\pi$$
$$674$$ 22.0000 0.847408
$$675$$ 0 0
$$676$$ −23.0000 −0.884615
$$677$$ 22.0000i 0.845529i 0.906240 + 0.422764i $$0.138940\pi$$
−0.906240 + 0.422764i $$0.861060\pi$$
$$678$$ 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 $$0.241841\pi$$
−0.724998 + 0.688751i $$0.758159\pi$$
$$684$$ −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 $$-0.952603\pi$$
0.988935 0.148352i $$-0.0473968\pi$$
$$728$$ 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 $$0.323111\pi$$
−0.527549 + 0.849524i $$0.676889\pi$$
$$734$$ −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 $$-0.757361\pi$$
0.723269 0.690567i $$-0.242639\pi$$
$$758$$ 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 $$0.104929\pi$$
−0.946157 + 0.323708i $$0.895071\pi$$
$$774$$ 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 $$-0.713066\pi$$
0.620489 0.784215i $$-0.286934\pi$$
$$788$$ 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 $$0.0338893\pi$$
−0.994338 + 0.106265i $$0.966111\pi$$
$$798$$ 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 $$-0.811674\pi$$
0.830026 0.557725i $$-0.188326\pi$$
$$824$$ 24.0000 0.836080
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 28.0000i 0.973655i 0.873498 + 0.486828i $$0.161846\pi$$
−0.873498 + 0.486828i $$0.838154\pi$$
$$828$$ − 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 $$0.122918\pi$$
−0.926363 + 0.376633i $$0.877082\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −12.0000 −0.410152
$$857$$ − 30.0000i − 1.02478i −0.858753 0.512390i $$-0.828760\pi$$
0.858753 0.512390i $$-0.171240\pi$$
$$858$$ 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 $$-0.761629\pi$$
0.732462 0.680808i $$-0.238371\pi$$
$$864$$ −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 $$0.194628\pi$$
−0.818821 + 0.574049i $$0.805372\pi$$
$$878$$ − 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 $$-0.792874\pi$$
0.795656 0.605748i $$-0.207126\pi$$
$$884$$ −36.0000 −1.21081
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 40.0000i 1.34307i 0.740973 + 0.671534i $$0.234364\pi$$
−0.740973 + 0.671534i $$0.765636\pi$$
$$888$$ 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 $$-0.978846\pi$$
0.997792 0.0664089i $$-0.0211542\pi$$
$$908$$ 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 $$0.117003\pi$$
−0.933201 + 0.359354i $$0.882997\pi$$
$$938$$ 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 $$0.0414921\pi$$
−0.991516 + 0.129983i $$0.958508\pi$$
$$948$$ 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 $$0.211034\pi$$
−0.788160 + 0.615470i $$0.788966\pi$$
$$954$$ 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 $$0.172032\pi$$
−0.857475 + 0.514525i $$0.827968\pi$$
$$968$$ 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 $$0.234500\pi$$
−0.740688 + 0.671850i $$0.765500\pi$$
$$978$$ − 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 $$-0.959279\pi$$
0.991828 0.127580i $$-0.0407210\pi$$
$$984$$ −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 $$0.370550\pi$$
−0.395562 + 0.918439i $$0.629450\pi$$
$$998$$ 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.799.1 2
5.2 odd 4 57.2.a.c.1.1 1
5.3 odd 4 1425.2.a.a.1.1 1
5.4 even 2 inner 1425.2.c.g.799.2 2
15.2 even 4 171.2.a.a.1.1 1
15.8 even 4 4275.2.a.m.1.1 1
20.7 even 4 912.2.a.b.1.1 1
35.27 even 4 2793.2.a.i.1.1 1
40.27 even 4 3648.2.a.bf.1.1 1
40.37 odd 4 3648.2.a.o.1.1 1
55.32 even 4 6897.2.a.a.1.1 1
60.47 odd 4 2736.2.a.s.1.1 1
65.12 odd 4 9633.2.a.h.1.1 1
95.37 even 4 1083.2.a.a.1.1 1
105.62 odd 4 8379.2.a.e.1.1 1
285.227 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 5.2 odd 4
171.2.a.a.1.1 1 15.2 even 4
912.2.a.b.1.1 1 20.7 even 4
1083.2.a.a.1.1 1 95.37 even 4
1425.2.a.a.1.1 1 5.3 odd 4
1425.2.c.g.799.1 2 1.1 even 1 trivial
1425.2.c.g.799.2 2 5.4 even 2 inner
2736.2.a.s.1.1 1 60.47 odd 4
2793.2.a.i.1.1 1 35.27 even 4
3249.2.a.g.1.1 1 285.227 odd 4
3648.2.a.o.1.1 1 40.37 odd 4
3648.2.a.bf.1.1 1 40.27 even 4
4275.2.a.m.1.1 1 15.8 even 4
6897.2.a.a.1.1 1 55.32 even 4
8379.2.a.e.1.1 1 105.62 odd 4
9633.2.a.h.1.1 1 65.12 odd 4