# Properties

 Label 2450.2.c.c.99.2 Level $2450$ Weight $2$ Character 2450.99 Analytic conductor $19.563$ 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] = [2450,2,Mod(99,2450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2450, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2450.99");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2450 = 2 \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2450.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: $$19.5633484952$$ 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 14) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 99.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2450.99 Dual form 2450.2.c.c.99.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} -2.00000i q^{12} +4.00000i q^{13} +1.00000 q^{16} +6.00000i q^{17} -1.00000i q^{18} +2.00000 q^{19} +2.00000 q^{24} -4.00000 q^{26} +4.00000i q^{27} +6.00000 q^{29} +4.00000 q^{31} +1.00000i q^{32} -6.00000 q^{34} +1.00000 q^{36} -2.00000i q^{37} +2.00000i q^{38} -8.00000 q^{39} -6.00000 q^{41} +8.00000i q^{43} -12.0000i q^{47} +2.00000i q^{48} -12.0000 q^{51} -4.00000i q^{52} +6.00000i q^{53} -4.00000 q^{54} +4.00000i q^{57} +6.00000i q^{58} -6.00000 q^{59} -8.00000 q^{61} +4.00000i q^{62} -1.00000 q^{64} +4.00000i q^{67} -6.00000i q^{68} +1.00000i q^{72} -2.00000i q^{73} +2.00000 q^{74} -2.00000 q^{76} -8.00000i q^{78} -8.00000 q^{79} -11.0000 q^{81} -6.00000i q^{82} +6.00000i q^{83} -8.00000 q^{86} +12.0000i q^{87} -6.00000 q^{89} +8.00000i q^{93} +12.0000 q^{94} -2.00000 q^{96} -10.0000i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 4 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 - 4 * q^6 - 2 * q^9 $$2 q - 2 q^{4} - 4 q^{6} - 2 q^{9} + 2 q^{16} + 4 q^{19} + 4 q^{24} - 8 q^{26} + 12 q^{29} + 8 q^{31} - 12 q^{34} + 2 q^{36} - 16 q^{39} - 12 q^{41} - 24 q^{51} - 8 q^{54} - 12 q^{59} - 16 q^{61} - 2 q^{64} + 4 q^{74} - 4 q^{76} - 16 q^{79} - 22 q^{81} - 16 q^{86} - 12 q^{89} + 24 q^{94} - 4 q^{96}+O(q^{100})$$ 2 * q - 2 * q^4 - 4 * q^6 - 2 * q^9 + 2 * q^16 + 4 * q^19 + 4 * q^24 - 8 * q^26 + 12 * q^29 + 8 * q^31 - 12 * q^34 + 2 * q^36 - 16 * q^39 - 12 * q^41 - 24 * q^51 - 8 * q^54 - 12 * q^59 - 16 * q^61 - 2 * q^64 + 4 * q^74 - 4 * q^76 - 16 * q^79 - 22 * q^81 - 16 * q^86 - 12 * q^89 + 24 * q^94 - 4 * q^96

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2450\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$1177$$ $$\chi(n)$$ $$1$$ $$-1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 0.707107i
$$3$$ 2.00000i 1.15470i 0.816497 + 0.577350i $$0.195913\pi$$
−0.816497 + 0.577350i $$0.804087\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ −2.00000 −0.816497
$$7$$ 0 0
$$8$$ − 1.00000i − 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ − 2.00000i − 0.577350i
$$13$$ 4.00000i 1.10940i 0.832050 + 0.554700i $$0.187167\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ 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
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 2.00000 0.408248
$$25$$ 0 0
$$26$$ −4.00000 −0.784465
$$27$$ 4.00000i 0.769800i
$$28$$ 0 0
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 0 0
$$34$$ −6.00000 −1.02899
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ − 2.00000i − 0.328798i −0.986394 0.164399i $$-0.947432\pi$$
0.986394 0.164399i $$-0.0525685\pi$$
$$38$$ 2.00000i 0.324443i
$$39$$ −8.00000 −1.28103
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ 8.00000i 1.21999i 0.792406 + 0.609994i $$0.208828\pi$$
−0.792406 + 0.609994i $$0.791172\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 12.0000i − 1.75038i −0.483779 0.875190i $$-0.660736\pi$$
0.483779 0.875190i $$-0.339264\pi$$
$$48$$ 2.00000i 0.288675i
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −12.0000 −1.68034
$$52$$ − 4.00000i − 0.554700i
$$53$$ 6.00000i 0.824163i 0.911147 + 0.412082i $$0.135198\pi$$
−0.911147 + 0.412082i $$0.864802\pi$$
$$54$$ −4.00000 −0.544331
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 4.00000i 0.529813i
$$58$$ 6.00000i 0.787839i
$$59$$ −6.00000 −0.781133 −0.390567 0.920575i $$-0.627721\pi$$
−0.390567 + 0.920575i $$0.627721\pi$$
$$60$$ 0 0
$$61$$ −8.00000 −1.02430 −0.512148 0.858898i $$-0.671150\pi$$
−0.512148 + 0.858898i $$0.671150\pi$$
$$62$$ 4.00000i 0.508001i
$$63$$ 0 0
$$64$$ −1.00000 −0.125000
$$65$$ 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
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ − 2.00000i − 0.234082i −0.993127 0.117041i $$-0.962659\pi$$
0.993127 0.117041i $$-0.0373409\pi$$
$$74$$ 2.00000 0.232495
$$75$$ 0 0
$$76$$ −2.00000 −0.229416
$$77$$ 0 0
$$78$$ − 8.00000i − 0.905822i
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ − 6.00000i − 0.662589i
$$83$$ 6.00000i 0.658586i 0.944228 + 0.329293i $$0.106810\pi$$
−0.944228 + 0.329293i $$0.893190\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −8.00000 −0.862662
$$87$$ 12.0000i 1.28654i
$$88$$ 0 0
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 8.00000i 0.829561i
$$94$$ 12.0000 1.23771
$$95$$ 0 0
$$96$$ −2.00000 −0.204124
$$97$$ − 10.0000i − 1.01535i −0.861550 0.507673i $$-0.830506\pi$$
0.861550 0.507673i $$-0.169494\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ − 12.0000i − 1.18818i
$$103$$ 4.00000i 0.394132i 0.980390 + 0.197066i $$0.0631413\pi$$
−0.980390 + 0.197066i $$0.936859\pi$$
$$104$$ 4.00000 0.392232
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ − 12.0000i − 1.16008i −0.814587 0.580042i $$-0.803036\pi$$
0.814587 0.580042i $$-0.196964\pi$$
$$108$$ − 4.00000i − 0.384900i
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 0 0
$$111$$ 4.00000 0.379663
$$112$$ 0 0
$$113$$ 6.00000i 0.564433i 0.959351 + 0.282216i $$0.0910696\pi$$
−0.959351 + 0.282216i $$0.908930\pi$$
$$114$$ −4.00000 −0.374634
$$115$$ 0 0
$$116$$ −6.00000 −0.557086
$$117$$ − 4.00000i − 0.369800i
$$118$$ − 6.00000i − 0.552345i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ − 8.00000i − 0.724286i
$$123$$ − 12.0000i − 1.08200i
$$124$$ −4.00000 −0.359211
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 16.0000i 1.41977i 0.704317 + 0.709885i $$0.251253\pi$$
−0.704317 + 0.709885i $$0.748747\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ −16.0000 −1.40872
$$130$$ 0 0
$$131$$ −18.0000 −1.57267 −0.786334 0.617802i $$-0.788023\pi$$
−0.786334 + 0.617802i $$0.788023\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −4.00000 −0.345547
$$135$$ 0 0
$$136$$ 6.00000 0.514496
$$137$$ − 18.0000i − 1.53784i −0.639343 0.768922i $$-0.720793\pi$$
0.639343 0.768922i $$-0.279207\pi$$
$$138$$ 0 0
$$139$$ 14.0000 1.18746 0.593732 0.804663i $$-0.297654\pi$$
0.593732 + 0.804663i $$0.297654\pi$$
$$140$$ 0 0
$$141$$ 24.0000 2.02116
$$142$$ 0 0
$$143$$ 0 0
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 2.00000 0.165521
$$147$$ 0 0
$$148$$ 2.00000i 0.164399i
$$149$$ 18.0000 1.47462 0.737309 0.675556i $$-0.236096\pi$$
0.737309 + 0.675556i $$0.236096\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ − 2.00000i − 0.162221i
$$153$$ − 6.00000i − 0.485071i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 8.00000 0.640513
$$157$$ − 4.00000i − 0.319235i −0.987179 0.159617i $$-0.948974\pi$$
0.987179 0.159617i $$-0.0510260\pi$$
$$158$$ − 8.00000i − 0.636446i
$$159$$ −12.0000 −0.951662
$$160$$ 0 0
$$161$$ 0 0
$$162$$ − 11.0000i − 0.864242i
$$163$$ − 16.0000i − 1.25322i −0.779334 0.626608i $$-0.784443\pi$$
0.779334 0.626608i $$-0.215557\pi$$
$$164$$ 6.00000 0.468521
$$165$$ 0 0
$$166$$ −6.00000 −0.465690
$$167$$ − 12.0000i − 0.928588i −0.885681 0.464294i $$-0.846308\pi$$
0.885681 0.464294i $$-0.153692\pi$$
$$168$$ 0 0
$$169$$ −3.00000 −0.230769
$$170$$ 0 0
$$171$$ −2.00000 −0.152944
$$172$$ − 8.00000i − 0.609994i
$$173$$ 12.0000i 0.912343i 0.889892 + 0.456172i $$0.150780\pi$$
−0.889892 + 0.456172i $$0.849220\pi$$
$$174$$ −12.0000 −0.909718
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 12.0000i − 0.901975i
$$178$$ − 6.00000i − 0.449719i
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ −20.0000 −1.48659 −0.743294 0.668965i $$-0.766738\pi$$
−0.743294 + 0.668965i $$0.766738\pi$$
$$182$$ 0 0
$$183$$ − 16.0000i − 1.18275i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ −8.00000 −0.586588
$$187$$ 0 0
$$188$$ 12.0000i 0.875190i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 24.0000 1.73658 0.868290 0.496058i $$-0.165220\pi$$
0.868290 + 0.496058i $$0.165220\pi$$
$$192$$ − 2.00000i − 0.144338i
$$193$$ 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
$$196$$ 0 0
$$197$$ 18.0000i 1.28245i 0.767354 + 0.641223i $$0.221573\pi$$
−0.767354 + 0.641223i $$0.778427\pi$$
$$198$$ 0 0
$$199$$ 20.0000 1.41776 0.708881 0.705328i $$-0.249200\pi$$
0.708881 + 0.705328i $$0.249200\pi$$
$$200$$ 0 0
$$201$$ −8.00000 −0.564276
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 12.0000 0.840168
$$205$$ 0 0
$$206$$ −4.00000 −0.278693
$$207$$ 0 0
$$208$$ 4.00000i 0.277350i
$$209$$ 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
$$214$$ 12.0000 0.820303
$$215$$ 0 0
$$216$$ 4.00000 0.272166
$$217$$ 0 0
$$218$$ − 2.00000i − 0.135457i
$$219$$ 4.00000 0.270295
$$220$$ 0 0
$$221$$ −24.0000 −1.61441
$$222$$ 4.00000i 0.268462i
$$223$$ − 8.00000i − 0.535720i −0.963458 0.267860i $$-0.913684\pi$$
0.963458 0.267860i $$-0.0863164\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ 18.0000i 1.19470i 0.801980 + 0.597351i $$0.203780\pi$$
−0.801980 + 0.597351i $$0.796220\pi$$
$$228$$ − 4.00000i − 0.264906i
$$229$$ −4.00000 −0.264327 −0.132164 0.991228i $$-0.542192\pi$$
−0.132164 + 0.991228i $$0.542192\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 6.00000i − 0.393919i
$$233$$ − 6.00000i − 0.393073i −0.980497 0.196537i $$-0.937031\pi$$
0.980497 0.196537i $$-0.0629694\pi$$
$$234$$ 4.00000 0.261488
$$235$$ 0 0
$$236$$ 6.00000 0.390567
$$237$$ − 16.0000i − 1.03931i
$$238$$ 0 0
$$239$$ −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 $$0.395619\pi$$
$$242$$ − 11.0000i − 0.707107i
$$243$$ − 10.0000i − 0.641500i
$$244$$ 8.00000 0.512148
$$245$$ 0 0
$$246$$ 12.0000 0.765092
$$247$$ 8.00000i 0.509028i
$$248$$ − 4.00000i − 0.254000i
$$249$$ −12.0000 −0.760469
$$250$$ 0 0
$$251$$ 18.0000 1.13615 0.568075 0.822977i $$-0.307688\pi$$
0.568075 + 0.822977i $$0.307688\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ −16.0000 −1.00393
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 18.0000i 1.12281i 0.827541 + 0.561405i $$0.189739\pi$$
−0.827541 + 0.561405i $$0.810261\pi$$
$$258$$ − 16.0000i − 0.996116i
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −6.00000 −0.371391
$$262$$ − 18.0000i − 1.11204i
$$263$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 12.0000i − 0.734388i
$$268$$ − 4.00000i − 0.244339i
$$269$$ −12.0000 −0.731653 −0.365826 0.930683i $$-0.619214\pi$$
−0.365826 + 0.930683i $$0.619214\pi$$
$$270$$ 0 0
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ 6.00000i 0.363803i
$$273$$ 0 0
$$274$$ 18.0000 1.08742
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 10.0000i 0.600842i 0.953807 + 0.300421i $$0.0971271\pi$$
−0.953807 + 0.300421i $$0.902873\pi$$
$$278$$ 14.0000i 0.839664i
$$279$$ −4.00000 −0.239474
$$280$$ 0 0
$$281$$ −6.00000 −0.357930 −0.178965 0.983855i $$-0.557275\pi$$
−0.178965 + 0.983855i $$0.557275\pi$$
$$282$$ 24.0000i 1.42918i
$$283$$ 22.0000i 1.30776i 0.756596 + 0.653882i $$0.226861\pi$$
−0.756596 + 0.653882i $$0.773139\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ − 1.00000i − 0.0589256i
$$289$$ −19.0000 −1.11765
$$290$$ 0 0
$$291$$ 20.0000 1.17242
$$292$$ 2.00000i 0.117041i
$$293$$ − 24.0000i − 1.40209i −0.713115 0.701047i $$-0.752716\pi$$
0.713115 0.701047i $$-0.247284\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −2.00000 −0.116248
$$297$$ 0 0
$$298$$ 18.0000i 1.04271i
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 8.00000i 0.460348i
$$303$$ 0 0
$$304$$ 2.00000 0.114708
$$305$$ 0 0
$$306$$ 6.00000 0.342997
$$307$$ 2.00000i 0.114146i 0.998370 + 0.0570730i $$0.0181768\pi$$
−0.998370 + 0.0570730i $$0.981823\pi$$
$$308$$ 0 0
$$309$$ −8.00000 −0.455104
$$310$$ 0 0
$$311$$ 24.0000 1.36092 0.680458 0.732787i $$-0.261781\pi$$
0.680458 + 0.732787i $$0.261781\pi$$
$$312$$ 8.00000i 0.452911i
$$313$$ 10.0000i 0.565233i 0.959233 + 0.282617i $$0.0912024\pi$$
−0.959233 + 0.282617i $$0.908798\pi$$
$$314$$ 4.00000 0.225733
$$315$$ 0 0
$$316$$ 8.00000 0.450035
$$317$$ − 6.00000i − 0.336994i −0.985702 0.168497i $$-0.946109\pi$$
0.985702 0.168497i $$-0.0538913\pi$$
$$318$$ − 12.0000i − 0.672927i
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 24.0000 1.33955
$$322$$ 0 0
$$323$$ 12.0000i 0.667698i
$$324$$ 11.0000 0.611111
$$325$$ 0 0
$$326$$ 16.0000 0.886158
$$327$$ − 4.00000i − 0.221201i
$$328$$ 6.00000i 0.331295i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 8.00000 0.439720 0.219860 0.975531i $$-0.429440\pi$$
0.219860 + 0.975531i $$0.429440\pi$$
$$332$$ − 6.00000i − 0.329293i
$$333$$ 2.00000i 0.109599i
$$334$$ 12.0000 0.656611
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 14.0000i − 0.762629i −0.924445 0.381314i $$-0.875472\pi$$
0.924445 0.381314i $$-0.124528\pi$$
$$338$$ − 3.00000i − 0.163178i
$$339$$ −12.0000 −0.651751
$$340$$ 0 0
$$341$$ 0 0
$$342$$ − 2.00000i − 0.108148i
$$343$$ 0 0
$$344$$ 8.00000 0.431331
$$345$$ 0 0
$$346$$ −12.0000 −0.645124
$$347$$ 24.0000i 1.28839i 0.764862 + 0.644194i $$0.222807\pi$$
−0.764862 + 0.644194i $$0.777193\pi$$
$$348$$ − 12.0000i − 0.643268i
$$349$$ −28.0000 −1.49881 −0.749403 0.662114i $$-0.769659\pi$$
−0.749403 + 0.662114i $$0.769659\pi$$
$$350$$ 0 0
$$351$$ −16.0000 −0.854017
$$352$$ 0 0
$$353$$ − 18.0000i − 0.958043i −0.877803 0.479022i $$-0.840992\pi$$
0.877803 0.479022i $$-0.159008\pi$$
$$354$$ 12.0000 0.637793
$$355$$ 0 0
$$356$$ 6.00000 0.317999
$$357$$ 0 0
$$358$$ 12.0000i 0.634220i
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ − 20.0000i − 1.05118i
$$363$$ − 22.0000i − 1.15470i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 16.0000 0.836333
$$367$$ 8.00000i 0.417597i 0.977959 + 0.208798i $$0.0669552\pi$$
−0.977959 + 0.208798i $$0.933045\pi$$
$$368$$ 0 0
$$369$$ 6.00000 0.312348
$$370$$ 0 0
$$371$$ 0 0
$$372$$ − 8.00000i − 0.414781i
$$373$$ 14.0000i 0.724893i 0.932005 + 0.362446i $$0.118058\pi$$
−0.932005 + 0.362446i $$0.881942\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −12.0000 −0.618853
$$377$$ 24.0000i 1.23606i
$$378$$ 0 0
$$379$$ 16.0000 0.821865 0.410932 0.911666i $$-0.365203\pi$$
0.410932 + 0.911666i $$0.365203\pi$$
$$380$$ 0 0
$$381$$ −32.0000 −1.63941
$$382$$ 24.0000i 1.22795i
$$383$$ − 36.0000i − 1.83951i −0.392488 0.919757i $$-0.628386\pi$$
0.392488 0.919757i $$-0.371614\pi$$
$$384$$ 2.00000 0.102062
$$385$$ 0 0
$$386$$ −14.0000 −0.712581
$$387$$ − 8.00000i − 0.406663i
$$388$$ 10.0000i 0.507673i
$$389$$ −18.0000 −0.912636 −0.456318 0.889817i $$-0.650832\pi$$
−0.456318 + 0.889817i $$0.650832\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ − 36.0000i − 1.81596i
$$394$$ −18.0000 −0.906827
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 20.0000i 1.00377i 0.864934 + 0.501886i $$0.167360\pi$$
−0.864934 + 0.501886i $$0.832640\pi$$
$$398$$ 20.0000i 1.00251i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −18.0000 −0.898877 −0.449439 0.893311i $$-0.648376\pi$$
−0.449439 + 0.893311i $$0.648376\pi$$
$$402$$ − 8.00000i − 0.399004i
$$403$$ 16.0000i 0.797017i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 12.0000i 0.594089i
$$409$$ 14.0000 0.692255 0.346128 0.938187i $$-0.387496\pi$$
0.346128 + 0.938187i $$0.387496\pi$$
$$410$$ 0 0
$$411$$ 36.0000 1.77575
$$412$$ − 4.00000i − 0.197066i
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −4.00000 −0.196116
$$417$$ 28.0000i 1.37117i
$$418$$ 0 0
$$419$$ 6.00000 0.293119 0.146560 0.989202i $$-0.453180\pi$$
0.146560 + 0.989202i $$0.453180\pi$$
$$420$$ 0 0
$$421$$ −10.0000 −0.487370 −0.243685 0.969854i $$-0.578356\pi$$
−0.243685 + 0.969854i $$0.578356\pi$$
$$422$$ − 4.00000i − 0.194717i
$$423$$ 12.0000i 0.583460i
$$424$$ 6.00000 0.291386
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 12.0000i 0.580042i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 24.0000 1.15604 0.578020 0.816023i $$-0.303826\pi$$
0.578020 + 0.816023i $$0.303826\pi$$
$$432$$ 4.00000i 0.192450i
$$433$$ 34.0000i 1.63394i 0.576683 + 0.816968i $$0.304347\pi$$
−0.576683 + 0.816968i $$0.695653\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 2.00000 0.0957826
$$437$$ 0 0
$$438$$ 4.00000i 0.191127i
$$439$$ 8.00000 0.381819 0.190910 0.981608i $$-0.438856\pi$$
0.190910 + 0.981608i $$0.438856\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ − 24.0000i − 1.14156i
$$443$$ − 12.0000i − 0.570137i −0.958507 0.285069i $$-0.907984\pi$$
0.958507 0.285069i $$-0.0920164\pi$$
$$444$$ −4.00000 −0.189832
$$445$$ 0 0
$$446$$ 8.00000 0.378811
$$447$$ 36.0000i 1.70274i
$$448$$ 0 0
$$449$$ −18.0000 −0.849473 −0.424736 0.905317i $$-0.639633\pi$$
−0.424736 + 0.905317i $$0.639633\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ − 6.00000i − 0.282216i
$$453$$ 16.0000i 0.751746i
$$454$$ −18.0000 −0.844782
$$455$$ 0 0
$$456$$ 4.00000 0.187317
$$457$$ 10.0000i 0.467780i 0.972263 + 0.233890i $$0.0751456\pi$$
−0.972263 + 0.233890i $$0.924854\pi$$
$$458$$ − 4.00000i − 0.186908i
$$459$$ −24.0000 −1.12022
$$460$$ 0 0
$$461$$ −12.0000 −0.558896 −0.279448 0.960161i $$-0.590151\pi$$
−0.279448 + 0.960161i $$0.590151\pi$$
$$462$$ 0 0
$$463$$ 32.0000i 1.48717i 0.668644 + 0.743583i $$0.266875\pi$$
−0.668644 + 0.743583i $$0.733125\pi$$
$$464$$ 6.00000 0.278543
$$465$$ 0 0
$$466$$ 6.00000 0.277945
$$467$$ − 6.00000i − 0.277647i −0.990317 0.138823i $$-0.955668\pi$$
0.990317 0.138823i $$-0.0443321\pi$$
$$468$$ 4.00000i 0.184900i
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 8.00000 0.368621
$$472$$ 6.00000i 0.276172i
$$473$$ 0 0
$$474$$ 16.0000 0.734904
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 6.00000i − 0.274721i
$$478$$ − 24.0000i − 1.09773i
$$479$$ −36.0000 −1.64488 −0.822441 0.568850i $$-0.807388\pi$$
−0.822441 + 0.568850i $$0.807388\pi$$
$$480$$ 0 0
$$481$$ 8.00000 0.364769
$$482$$ 10.0000i 0.455488i
$$483$$ 0 0
$$484$$ 11.0000 0.500000
$$485$$ 0 0
$$486$$ 10.0000 0.453609
$$487$$ 16.0000i 0.725029i 0.931978 + 0.362515i $$0.118082\pi$$
−0.931978 + 0.362515i $$0.881918\pi$$
$$488$$ 8.00000i 0.362143i
$$489$$ 32.0000 1.44709
$$490$$ 0 0
$$491$$ −12.0000 −0.541552 −0.270776 0.962642i $$-0.587280\pi$$
−0.270776 + 0.962642i $$0.587280\pi$$
$$492$$ 12.0000i 0.541002i
$$493$$ 36.0000i 1.62136i
$$494$$ −8.00000 −0.359937
$$495$$ 0 0
$$496$$ 4.00000 0.179605
$$497$$ 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
$$501$$ 24.0000 1.07224
$$502$$ 18.0000i 0.803379i
$$503$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 6.00000i − 0.266469i
$$508$$ − 16.0000i − 0.709885i
$$509$$ 36.0000 1.59567 0.797836 0.602875i $$-0.205978\pi$$
0.797836 + 0.602875i $$0.205978\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 1.00000i 0.0441942i
$$513$$ 8.00000i 0.353209i
$$514$$ −18.0000 −0.793946
$$515$$ 0 0
$$516$$ 16.0000 0.704361
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −24.0000 −1.05348
$$520$$ 0 0
$$521$$ −6.00000 −0.262865 −0.131432 0.991325i $$-0.541958\pi$$
−0.131432 + 0.991325i $$0.541958\pi$$
$$522$$ − 6.00000i − 0.262613i
$$523$$ − 2.00000i − 0.0874539i −0.999044 0.0437269i $$-0.986077\pi$$
0.999044 0.0437269i $$-0.0139232\pi$$
$$524$$ 18.0000 0.786334
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 24.0000i 1.04546i
$$528$$ 0 0
$$529$$ 23.0000 1.00000
$$530$$ 0 0
$$531$$ 6.00000 0.260378
$$532$$ 0 0
$$533$$ − 24.0000i − 1.03956i
$$534$$ 12.0000 0.519291
$$535$$ 0 0
$$536$$ 4.00000 0.172774
$$537$$ 24.0000i 1.03568i
$$538$$ − 12.0000i − 0.517357i
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 38.0000 1.63375 0.816874 0.576816i $$-0.195705\pi$$
0.816874 + 0.576816i $$0.195705\pi$$
$$542$$ 16.0000i 0.687259i
$$543$$ − 40.0000i − 1.71656i
$$544$$ −6.00000 −0.257248
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 8.00000i − 0.342055i −0.985266 0.171028i $$-0.945291\pi$$
0.985266 0.171028i $$-0.0547087\pi$$
$$548$$ 18.0000i 0.768922i
$$549$$ 8.00000 0.341432
$$550$$ 0 0
$$551$$ 12.0000 0.511217
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −10.0000 −0.424859
$$555$$ 0 0
$$556$$ −14.0000 −0.593732
$$557$$ − 6.00000i − 0.254228i −0.991888 0.127114i $$-0.959429\pi$$
0.991888 0.127114i $$-0.0405714\pi$$
$$558$$ − 4.00000i − 0.169334i
$$559$$ −32.0000 −1.35346
$$560$$ 0 0
$$561$$ 0 0
$$562$$ − 6.00000i − 0.253095i
$$563$$ − 30.0000i − 1.26435i −0.774826 0.632175i $$-0.782163\pi$$
0.774826 0.632175i $$-0.217837\pi$$
$$564$$ −24.0000 −1.01058
$$565$$ 0 0
$$566$$ −22.0000 −0.924729
$$567$$ 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
$$571$$ 32.0000 1.33916 0.669579 0.742741i $$-0.266474\pi$$
0.669579 + 0.742741i $$0.266474\pi$$
$$572$$ 0 0
$$573$$ 48.0000i 2.00523i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 2.00000i 0.0832611i 0.999133 + 0.0416305i $$0.0132552\pi$$
−0.999133 + 0.0416305i $$0.986745\pi$$
$$578$$ − 19.0000i − 0.790296i
$$579$$ −28.0000 −1.16364
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 20.0000i 0.829027i
$$583$$ 0 0
$$584$$ −2.00000 −0.0827606
$$585$$ 0 0
$$586$$ 24.0000 0.991431
$$587$$ − 42.0000i − 1.73353i −0.498721 0.866763i $$-0.666197\pi$$
0.498721 0.866763i $$-0.333803\pi$$
$$588$$ 0 0
$$589$$ 8.00000 0.329634
$$590$$ 0 0
$$591$$ −36.0000 −1.48084
$$592$$ − 2.00000i − 0.0821995i
$$593$$ 6.00000i 0.246390i 0.992382 + 0.123195i $$0.0393141\pi$$
−0.992382 + 0.123195i $$0.960686\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −18.0000 −0.737309
$$597$$ 40.0000i 1.63709i
$$598$$ 0 0
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 0 0
$$601$$ −26.0000 −1.06056 −0.530281 0.847822i $$-0.677914\pi$$
−0.530281 + 0.847822i $$0.677914\pi$$
$$602$$ 0 0
$$603$$ − 4.00000i − 0.162893i
$$604$$ −8.00000 −0.325515
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 32.0000i 1.29884i 0.760430 + 0.649420i $$0.224988\pi$$
−0.760430 + 0.649420i $$0.775012\pi$$
$$608$$ 2.00000i 0.0811107i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 48.0000 1.94187
$$612$$ 6.00000i 0.242536i
$$613$$ 2.00000i 0.0807792i 0.999184 + 0.0403896i $$0.0128599\pi$$
−0.999184 + 0.0403896i $$0.987140\pi$$
$$614$$ −2.00000 −0.0807134
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 6.00000i − 0.241551i −0.992680 0.120775i $$-0.961462\pi$$
0.992680 0.120775i $$-0.0385381\pi$$
$$618$$ − 8.00000i − 0.321807i
$$619$$ 26.0000 1.04503 0.522514 0.852631i $$-0.324994\pi$$
0.522514 + 0.852631i $$0.324994\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 24.0000i 0.962312i
$$623$$ 0 0
$$624$$ −8.00000 −0.320256
$$625$$ 0 0
$$626$$ −10.0000 −0.399680
$$627$$ 0 0
$$628$$ 4.00000i 0.159617i
$$629$$ 12.0000 0.478471
$$630$$ 0 0
$$631$$ −16.0000 −0.636950 −0.318475 0.947931i $$-0.603171\pi$$
−0.318475 + 0.947931i $$0.603171\pi$$
$$632$$ 8.00000i 0.318223i
$$633$$ − 8.00000i − 0.317971i
$$634$$ 6.00000 0.238290
$$635$$ 0 0
$$636$$ 12.0000 0.475831
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −18.0000 −0.710957 −0.355479 0.934684i $$-0.615682\pi$$
−0.355479 + 0.934684i $$0.615682\pi$$
$$642$$ 24.0000i 0.947204i
$$643$$ − 14.0000i − 0.552106i −0.961142 0.276053i $$-0.910973\pi$$
0.961142 0.276053i $$-0.0890266\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −12.0000 −0.472134
$$647$$ − 12.0000i − 0.471769i −0.971781 0.235884i $$-0.924201\pi$$
0.971781 0.235884i $$-0.0757987\pi$$
$$648$$ 11.0000i 0.432121i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 16.0000i 0.626608i
$$653$$ 18.0000i 0.704394i 0.935926 + 0.352197i $$0.114565\pi$$
−0.935926 + 0.352197i $$0.885435\pi$$
$$654$$ 4.00000 0.156412
$$655$$ 0 0
$$656$$ −6.00000 −0.234261
$$657$$ 2.00000i 0.0780274i
$$658$$ 0 0
$$659$$ 24.0000 0.934907 0.467454 0.884018i $$-0.345171\pi$$
0.467454 + 0.884018i $$0.345171\pi$$
$$660$$ 0 0
$$661$$ 40.0000 1.55582 0.777910 0.628376i $$-0.216280\pi$$
0.777910 + 0.628376i $$0.216280\pi$$
$$662$$ 8.00000i 0.310929i
$$663$$ − 48.0000i − 1.86417i
$$664$$ 6.00000 0.232845
$$665$$ 0 0
$$666$$ −2.00000 −0.0774984
$$667$$ 0 0
$$668$$ 12.0000i 0.464294i
$$669$$ 16.0000 0.618596
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 26.0000i 1.00223i 0.865382 + 0.501113i $$0.167076\pi$$
−0.865382 + 0.501113i $$0.832924\pi$$
$$674$$ 14.0000 0.539260
$$675$$ 0 0
$$676$$ 3.00000 0.115385
$$677$$ − 12.0000i − 0.461197i −0.973049 0.230599i $$-0.925932\pi$$
0.973049 0.230599i $$-0.0740685\pi$$
$$678$$ − 12.0000i − 0.460857i
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −36.0000 −1.37952
$$682$$ 0 0
$$683$$ − 12.0000i − 0.459167i −0.973289 0.229584i $$-0.926264\pi$$
0.973289 0.229584i $$-0.0737364\pi$$
$$684$$ 2.00000 0.0764719
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 8.00000i − 0.305219i
$$688$$ 8.00000i 0.304997i
$$689$$ −24.0000 −0.914327
$$690$$ 0 0
$$691$$ 46.0000 1.74992 0.874961 0.484193i $$-0.160887\pi$$
0.874961 + 0.484193i $$0.160887\pi$$
$$692$$ − 12.0000i − 0.456172i
$$693$$ 0 0
$$694$$ −24.0000 −0.911028
$$695$$ 0 0
$$696$$ 12.0000 0.454859
$$697$$ − 36.0000i − 1.36360i
$$698$$ − 28.0000i − 1.05982i
$$699$$ 12.0000 0.453882
$$700$$ 0 0
$$701$$ 18.0000 0.679851 0.339925 0.940452i $$-0.389598\pi$$
0.339925 + 0.940452i $$0.389598\pi$$
$$702$$ − 16.0000i − 0.603881i
$$703$$ − 4.00000i − 0.150863i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 18.0000 0.677439
$$707$$ 0 0
$$708$$ 12.0000i 0.450988i
$$709$$ 46.0000 1.72757 0.863783 0.503864i $$-0.168089\pi$$
0.863783 + 0.503864i $$0.168089\pi$$
$$710$$ 0 0
$$711$$ 8.00000 0.300023
$$712$$ 6.00000i 0.224860i
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ − 48.0000i − 1.79259i
$$718$$ 24.0000i 0.895672i
$$719$$ 12.0000 0.447524 0.223762 0.974644i $$-0.428166\pi$$
0.223762 + 0.974644i $$0.428166\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ − 15.0000i − 0.558242i
$$723$$ 20.0000i 0.743808i
$$724$$ 20.0000 0.743294
$$725$$ 0 0
$$726$$ 22.0000 0.816497
$$727$$ 44.0000i 1.63187i 0.578144 + 0.815935i $$0.303777\pi$$
−0.578144 + 0.815935i $$0.696223\pi$$
$$728$$ 0 0
$$729$$ −13.0000 −0.481481
$$730$$ 0 0
$$731$$ −48.0000 −1.77534
$$732$$ 16.0000i 0.591377i
$$733$$ 40.0000i 1.47743i 0.674016 + 0.738717i $$0.264568\pi$$
−0.674016 + 0.738717i $$0.735432\pi$$
$$734$$ −8.00000 −0.295285
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 6.00000i 0.220863i
$$739$$ 16.0000 0.588570 0.294285 0.955718i $$-0.404919\pi$$
0.294285 + 0.955718i $$0.404919\pi$$
$$740$$ 0 0
$$741$$ −16.0000 −0.587775
$$742$$ 0 0
$$743$$ 24.0000i 0.880475i 0.897881 + 0.440237i $$0.145106\pi$$
−0.897881 + 0.440237i $$0.854894\pi$$
$$744$$ 8.00000 0.293294
$$745$$ 0 0
$$746$$ −14.0000 −0.512576
$$747$$ − 6.00000i − 0.219529i
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −40.0000 −1.45962 −0.729810 0.683650i $$-0.760392\pi$$
−0.729810 + 0.683650i $$0.760392\pi$$
$$752$$ − 12.0000i − 0.437595i
$$753$$ 36.0000i 1.31191i
$$754$$ −24.0000 −0.874028
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 2.00000i − 0.0726912i −0.999339 0.0363456i $$-0.988428\pi$$
0.999339 0.0363456i $$-0.0115717\pi$$
$$758$$ 16.0000i 0.581146i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 18.0000 0.652499 0.326250 0.945284i $$-0.394215\pi$$
0.326250 + 0.945284i $$0.394215\pi$$
$$762$$ − 32.0000i − 1.15924i
$$763$$ 0 0
$$764$$ −24.0000 −0.868290
$$765$$ 0 0
$$766$$ 36.0000 1.30073
$$767$$ − 24.0000i − 0.866590i
$$768$$ 2.00000i 0.0721688i
$$769$$ 14.0000 0.504853 0.252426 0.967616i $$-0.418771\pi$$
0.252426 + 0.967616i $$0.418771\pi$$
$$770$$ 0 0
$$771$$ −36.0000 −1.29651
$$772$$ − 14.0000i − 0.503871i
$$773$$ − 24.0000i − 0.863220i −0.902060 0.431610i $$-0.857946\pi$$
0.902060 0.431610i $$-0.142054\pi$$
$$774$$ 8.00000 0.287554
$$775$$ 0 0
$$776$$ −10.0000 −0.358979
$$777$$ 0 0
$$778$$ − 18.0000i − 0.645331i
$$779$$ −12.0000 −0.429945
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 24.0000i 0.857690i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 36.0000 1.28408
$$787$$ − 22.0000i − 0.784215i −0.919919 0.392108i $$-0.871746\pi$$
0.919919 0.392108i $$-0.128254\pi$$
$$788$$ − 18.0000i − 0.641223i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ − 32.0000i − 1.13635i
$$794$$ −20.0000 −0.709773
$$795$$ 0 0
$$796$$ −20.0000 −0.708881
$$797$$ − 12.0000i − 0.425062i −0.977154 0.212531i $$-0.931829\pi$$
0.977154 0.212531i $$-0.0681706\pi$$
$$798$$ 0 0
$$799$$ 72.0000 2.54718
$$800$$ 0 0
$$801$$ 6.00000 0.212000
$$802$$ − 18.0000i − 0.635602i
$$803$$ 0 0
$$804$$ 8.00000 0.282138
$$805$$ 0 0
$$806$$ −16.0000 −0.563576
$$807$$ − 24.0000i − 0.844840i
$$808$$ 0 0
$$809$$ −6.00000 −0.210949 −0.105474 0.994422i $$-0.533636\pi$$
−0.105474 + 0.994422i $$0.533636\pi$$
$$810$$ 0 0
$$811$$ −2.00000 −0.0702295 −0.0351147 0.999383i $$-0.511180\pi$$
−0.0351147 + 0.999383i $$0.511180\pi$$
$$812$$ 0 0
$$813$$ 32.0000i 1.12229i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ −12.0000 −0.420084
$$817$$ 16.0000i 0.559769i
$$818$$ 14.0000i 0.489499i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 6.00000 0.209401 0.104701 0.994504i $$-0.466612\pi$$
0.104701 + 0.994504i $$0.466612\pi$$
$$822$$ 36.0000i 1.25564i
$$823$$ − 40.0000i − 1.39431i −0.716919 0.697156i $$-0.754448\pi$$
0.716919 0.697156i $$-0.245552\pi$$
$$824$$ 4.00000 0.139347
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 36.0000i 1.25184i 0.779886 + 0.625921i $$0.215277\pi$$
−0.779886 + 0.625921i $$0.784723\pi$$
$$828$$ 0 0
$$829$$ 56.0000 1.94496 0.972480 0.232986i $$-0.0748495\pi$$
0.972480 + 0.232986i $$0.0748495\pi$$
$$830$$ 0 0
$$831$$ −20.0000 −0.693792
$$832$$ − 4.00000i − 0.138675i
$$833$$ 0 0
$$834$$ −28.0000 −0.969561
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 16.0000i 0.553041i
$$838$$ 6.00000i 0.207267i
$$839$$ 12.0000 0.414286 0.207143 0.978311i $$-0.433583\pi$$
0.207143 + 0.978311i $$0.433583\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ − 10.0000i − 0.344623i
$$843$$ − 12.0000i − 0.413302i
$$844$$ 4.00000 0.137686
$$845$$ 0 0
$$846$$ −12.0000 −0.412568
$$847$$ 0 0
$$848$$ 6.00000i 0.206041i
$$849$$ −44.0000 −1.51008
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ − 44.0000i − 1.50653i −0.657716 0.753266i $$-0.728477\pi$$
0.657716 0.753266i $$-0.271523\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −12.0000 −0.410152
$$857$$ − 18.0000i − 0.614868i −0.951569 0.307434i $$-0.900530\pi$$
0.951569 0.307434i $$-0.0994704\pi$$
$$858$$ 0 0
$$859$$ 14.0000 0.477674 0.238837 0.971060i $$-0.423234\pi$$
0.238837 + 0.971060i $$0.423234\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 24.0000i 0.817443i
$$863$$ − 24.0000i − 0.816970i −0.912765 0.408485i $$-0.866057\pi$$
0.912765 0.408485i $$-0.133943\pi$$
$$864$$ −4.00000 −0.136083
$$865$$ 0 0
$$866$$ −34.0000 −1.15537
$$867$$ − 38.0000i − 1.29055i
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −16.0000 −0.542139
$$872$$ 2.00000i 0.0677285i
$$873$$ 10.0000i 0.338449i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ −4.00000 −0.135147
$$877$$ 22.0000i 0.742887i 0.928456 + 0.371444i $$0.121137\pi$$
−0.928456 + 0.371444i $$0.878863\pi$$
$$878$$ 8.00000i 0.269987i
$$879$$ 48.0000 1.61900
$$880$$ 0 0
$$881$$ 54.0000 1.81931 0.909653 0.415369i $$-0.136347\pi$$
0.909653 + 0.415369i $$0.136347\pi$$
$$882$$ 0 0
$$883$$ 20.0000i 0.673054i 0.941674 + 0.336527i $$0.109252\pi$$
−0.941674 + 0.336527i $$0.890748\pi$$
$$884$$ 24.0000 0.807207
$$885$$ 0 0
$$886$$ 12.0000 0.403148
$$887$$ − 36.0000i − 1.20876i −0.796696 0.604381i $$-0.793421\pi$$
0.796696 0.604381i $$-0.206579\pi$$
$$888$$ − 4.00000i − 0.134231i
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 8.00000i 0.267860i
$$893$$ − 24.0000i − 0.803129i
$$894$$ −36.0000 −1.20402
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ − 18.0000i − 0.600668i
$$899$$ 24.0000 0.800445
$$900$$ 0 0
$$901$$ −36.0000 −1.19933
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 6.00000 0.199557
$$905$$ 0 0
$$906$$ −16.0000 −0.531564
$$907$$ − 44.0000i − 1.46100i −0.682915 0.730498i $$-0.739288\pi$$
0.682915 0.730498i $$-0.260712\pi$$
$$908$$ − 18.0000i − 0.597351i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 48.0000 1.59031 0.795155 0.606406i $$-0.207389\pi$$
0.795155 + 0.606406i $$0.207389\pi$$
$$912$$ 4.00000i 0.132453i
$$913$$ 0 0
$$914$$ −10.0000 −0.330771
$$915$$ 0 0
$$916$$ 4.00000 0.132164
$$917$$ 0 0
$$918$$ − 24.0000i − 0.792118i
$$919$$ −56.0000 −1.84727 −0.923635 0.383274i $$-0.874797\pi$$
−0.923635 + 0.383274i $$0.874797\pi$$
$$920$$ 0 0
$$921$$ −4.00000 −0.131804
$$922$$ − 12.0000i − 0.395199i
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −32.0000 −1.05159
$$927$$ − 4.00000i − 0.131377i
$$928$$ 6.00000i 0.196960i
$$929$$ 6.00000 0.196854 0.0984268 0.995144i $$-0.468619\pi$$
0.0984268 + 0.995144i $$0.468619\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 6.00000i 0.196537i
$$933$$ 48.0000i 1.57145i
$$934$$ 6.00000 0.196326
$$935$$ 0 0
$$936$$ −4.00000 −0.130744
$$937$$ 2.00000i 0.0653372i 0.999466 + 0.0326686i $$0.0104006\pi$$
−0.999466 + 0.0326686i $$0.989599\pi$$
$$938$$ 0 0
$$939$$ −20.0000 −0.652675
$$940$$ 0 0
$$941$$ 24.0000 0.782378 0.391189 0.920310i $$-0.372064\pi$$
0.391189 + 0.920310i $$0.372064\pi$$
$$942$$ 8.00000i 0.260654i
$$943$$ 0 0
$$944$$ −6.00000 −0.195283
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 24.0000i − 0.779895i −0.920837 0.389948i $$-0.872493\pi$$
0.920837 0.389948i $$-0.127507\pi$$
$$948$$ 16.0000i 0.519656i
$$949$$ 8.00000 0.259691
$$950$$ 0 0
$$951$$ 12.0000 0.389127
$$952$$ 0 0
$$953$$ − 54.0000i − 1.74923i −0.484817 0.874616i $$-0.661114\pi$$
0.484817 0.874616i $$-0.338886\pi$$
$$954$$ 6.00000 0.194257
$$955$$ 0 0
$$956$$ 24.0000 0.776215
$$957$$ 0 0
$$958$$ − 36.0000i − 1.16311i
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 8.00000i 0.257930i
$$963$$ 12.0000i 0.386695i
$$964$$ −10.0000 −0.322078
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 32.0000i − 1.02905i −0.857475 0.514525i $$-0.827968\pi$$
0.857475 0.514525i $$-0.172032\pi$$
$$968$$ 11.0000i 0.353553i
$$969$$ −24.0000 −0.770991
$$970$$ 0 0
$$971$$ 6.00000 0.192549 0.0962746 0.995355i $$-0.469307\pi$$
0.0962746 + 0.995355i $$0.469307\pi$$
$$972$$ 10.0000i 0.320750i
$$973$$ 0 0
$$974$$ −16.0000 −0.512673
$$975$$ 0 0
$$976$$ −8.00000 −0.256074
$$977$$ 6.00000i 0.191957i 0.995383 + 0.0959785i $$0.0305980\pi$$
−0.995383 + 0.0959785i $$0.969402\pi$$
$$978$$ 32.0000i 1.02325i
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 2.00000 0.0638551
$$982$$ − 12.0000i − 0.382935i
$$983$$ 36.0000i 1.14822i 0.818778 + 0.574111i $$0.194652\pi$$
−0.818778 + 0.574111i $$0.805348\pi$$
$$984$$ −12.0000 −0.382546
$$985$$ 0 0
$$986$$ −36.0000 −1.14647
$$987$$ 0 0
$$988$$ − 8.00000i − 0.254514i
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ 4.00000i 0.127000i
$$993$$ 16.0000i 0.507745i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 12.0000 0.380235
$$997$$ 8.00000i 0.253363i 0.991943 + 0.126681i $$0.0404325\pi$$
−0.991943 + 0.126681i $$0.959567\pi$$
$$998$$ 4.00000i 0.126618i
$$999$$ 8.00000 0.253109
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 2450.2.c.c.99.2 2
5.2 odd 4 98.2.a.a.1.1 1
5.3 odd 4 2450.2.a.t.1.1 1
5.4 even 2 inner 2450.2.c.c.99.1 2
7.6 odd 2 350.2.c.d.99.2 2
15.2 even 4 882.2.a.i.1.1 1
20.7 even 4 784.2.a.b.1.1 1
21.20 even 2 3150.2.g.j.2899.1 2
28.27 even 2 2800.2.g.h.449.2 2
35.2 odd 12 98.2.c.a.67.1 2
35.12 even 12 98.2.c.b.67.1 2
35.13 even 4 350.2.a.f.1.1 1
35.17 even 12 98.2.c.b.79.1 2
35.27 even 4 14.2.a.a.1.1 1
35.32 odd 12 98.2.c.a.79.1 2
35.34 odd 2 350.2.c.d.99.1 2
40.27 even 4 3136.2.a.z.1.1 1
40.37 odd 4 3136.2.a.e.1.1 1
60.47 odd 4 7056.2.a.bd.1.1 1
105.2 even 12 882.2.g.d.361.1 2
105.17 odd 12 882.2.g.c.667.1 2
105.32 even 12 882.2.g.d.667.1 2
105.47 odd 12 882.2.g.c.361.1 2
105.62 odd 4 126.2.a.b.1.1 1
105.83 odd 4 3150.2.a.i.1.1 1
105.104 even 2 3150.2.g.j.2899.2 2
140.27 odd 4 112.2.a.c.1.1 1
140.47 odd 12 784.2.i.c.753.1 2
140.67 even 12 784.2.i.i.177.1 2
140.83 odd 4 2800.2.a.g.1.1 1
140.87 odd 12 784.2.i.c.177.1 2
140.107 even 12 784.2.i.i.753.1 2
140.139 even 2 2800.2.g.h.449.1 2
280.27 odd 4 448.2.a.a.1.1 1
280.237 even 4 448.2.a.g.1.1 1
315.97 even 12 1134.2.f.l.757.1 2
315.167 odd 12 1134.2.f.f.379.1 2
315.202 even 12 1134.2.f.l.379.1 2
315.272 odd 12 1134.2.f.f.757.1 2
385.307 odd 4 1694.2.a.e.1.1 1
420.167 even 4 1008.2.a.h.1.1 1
455.272 even 4 2366.2.a.j.1.1 1
455.307 odd 4 2366.2.d.b.337.1 2
455.447 odd 4 2366.2.d.b.337.2 2
560.27 odd 4 1792.2.b.g.897.1 2
560.237 even 4 1792.2.b.c.897.1 2
560.307 odd 4 1792.2.b.g.897.2 2
560.517 even 4 1792.2.b.c.897.2 2
595.237 even 4 4046.2.a.f.1.1 1
665.132 odd 4 5054.2.a.c.1.1 1
805.482 odd 4 7406.2.a.a.1.1 1
840.587 even 4 4032.2.a.r.1.1 1
840.797 odd 4 4032.2.a.w.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
14.2.a.a.1.1 1 35.27 even 4
98.2.a.a.1.1 1 5.2 odd 4
98.2.c.a.67.1 2 35.2 odd 12
98.2.c.a.79.1 2 35.32 odd 12
98.2.c.b.67.1 2 35.12 even 12
98.2.c.b.79.1 2 35.17 even 12
112.2.a.c.1.1 1 140.27 odd 4
126.2.a.b.1.1 1 105.62 odd 4
350.2.a.f.1.1 1 35.13 even 4
350.2.c.d.99.1 2 35.34 odd 2
350.2.c.d.99.2 2 7.6 odd 2
448.2.a.a.1.1 1 280.27 odd 4
448.2.a.g.1.1 1 280.237 even 4
784.2.a.b.1.1 1 20.7 even 4
784.2.i.c.177.1 2 140.87 odd 12
784.2.i.c.753.1 2 140.47 odd 12
784.2.i.i.177.1 2 140.67 even 12
784.2.i.i.753.1 2 140.107 even 12
882.2.a.i.1.1 1 15.2 even 4
882.2.g.c.361.1 2 105.47 odd 12
882.2.g.c.667.1 2 105.17 odd 12
882.2.g.d.361.1 2 105.2 even 12
882.2.g.d.667.1 2 105.32 even 12
1008.2.a.h.1.1 1 420.167 even 4
1134.2.f.f.379.1 2 315.167 odd 12
1134.2.f.f.757.1 2 315.272 odd 12
1134.2.f.l.379.1 2 315.202 even 12
1134.2.f.l.757.1 2 315.97 even 12
1694.2.a.e.1.1 1 385.307 odd 4
1792.2.b.c.897.1 2 560.237 even 4
1792.2.b.c.897.2 2 560.517 even 4
1792.2.b.g.897.1 2 560.27 odd 4
1792.2.b.g.897.2 2 560.307 odd 4
2366.2.a.j.1.1 1 455.272 even 4
2366.2.d.b.337.1 2 455.307 odd 4
2366.2.d.b.337.2 2 455.447 odd 4
2450.2.a.t.1.1 1 5.3 odd 4
2450.2.c.c.99.1 2 5.4 even 2 inner
2450.2.c.c.99.2 2 1.1 even 1 trivial
2800.2.a.g.1.1 1 140.83 odd 4
2800.2.g.h.449.1 2 140.139 even 2
2800.2.g.h.449.2 2 28.27 even 2
3136.2.a.e.1.1 1 40.37 odd 4
3136.2.a.z.1.1 1 40.27 even 4
3150.2.a.i.1.1 1 105.83 odd 4
3150.2.g.j.2899.1 2 21.20 even 2
3150.2.g.j.2899.2 2 105.104 even 2
4032.2.a.r.1.1 1 840.587 even 4
4032.2.a.w.1.1 1 840.797 odd 4
4046.2.a.f.1.1 1 595.237 even 4
5054.2.a.c.1.1 1 665.132 odd 4
7056.2.a.bd.1.1 1 60.47 odd 4
7406.2.a.a.1.1 1 805.482 odd 4