# Properties

 Label 2175.2.c.a.349.1 Level $2175$ Weight $2$ Character 2175.349 Analytic conductor $17.367$ 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] = [2175,2,Mod(349,2175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2175, 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("2175.349");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2175 = 3 \cdot 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2175.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: $$17.3674624396$$ 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 435) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 349.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2175.349 Dual form 2175.2.c.a.349.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} -4.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} -4.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} -1.00000i q^{12} -6.00000i q^{13} -4.00000 q^{14} -1.00000 q^{16} +2.00000i q^{17} +1.00000i q^{18} -8.00000 q^{19} -4.00000 q^{21} +4.00000i q^{23} -3.00000 q^{24} -6.00000 q^{26} +1.00000i q^{27} -4.00000i q^{28} -1.00000 q^{29} +4.00000 q^{31} -5.00000i q^{32} +2.00000 q^{34} -1.00000 q^{36} +6.00000i q^{37} +8.00000i q^{38} -6.00000 q^{39} +2.00000 q^{41} +4.00000i q^{42} +4.00000i q^{43} +4.00000 q^{46} +1.00000i q^{48} -9.00000 q^{49} +2.00000 q^{51} -6.00000i q^{52} -6.00000i q^{53} +1.00000 q^{54} -12.0000 q^{56} +8.00000i q^{57} +1.00000i q^{58} +12.0000 q^{59} +6.00000 q^{61} -4.00000i q^{62} +4.00000i q^{63} -7.00000 q^{64} -8.00000i q^{67} +2.00000i q^{68} +4.00000 q^{69} +16.0000 q^{71} +3.00000i q^{72} +6.00000i q^{73} +6.00000 q^{74} -8.00000 q^{76} +6.00000i q^{78} -12.0000 q^{79} +1.00000 q^{81} -2.00000i q^{82} +16.0000i q^{83} -4.00000 q^{84} +4.00000 q^{86} +1.00000i q^{87} -2.00000 q^{89} -24.0000 q^{91} +4.00000i q^{92} -4.00000i q^{93} -5.00000 q^{96} -14.0000i q^{97} +9.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^4 - 2 * q^6 - 2 * q^9 $$2 q + 2 q^{4} - 2 q^{6} - 2 q^{9} - 8 q^{14} - 2 q^{16} - 16 q^{19} - 8 q^{21} - 6 q^{24} - 12 q^{26} - 2 q^{29} + 8 q^{31} + 4 q^{34} - 2 q^{36} - 12 q^{39} + 4 q^{41} + 8 q^{46} - 18 q^{49} + 4 q^{51} + 2 q^{54} - 24 q^{56} + 24 q^{59} + 12 q^{61} - 14 q^{64} + 8 q^{69} + 32 q^{71} + 12 q^{74} - 16 q^{76} - 24 q^{79} + 2 q^{81} - 8 q^{84} + 8 q^{86} - 4 q^{89} - 48 q^{91} - 10 q^{96}+O(q^{100})$$ 2 * q + 2 * q^4 - 2 * q^6 - 2 * q^9 - 8 * q^14 - 2 * q^16 - 16 * q^19 - 8 * q^21 - 6 * q^24 - 12 * q^26 - 2 * q^29 + 8 * q^31 + 4 * q^34 - 2 * q^36 - 12 * q^39 + 4 * q^41 + 8 * q^46 - 18 * q^49 + 4 * q^51 + 2 * q^54 - 24 * q^56 + 24 * q^59 + 12 * q^61 - 14 * q^64 + 8 * q^69 + 32 * q^71 + 12 * q^74 - 16 * q^76 - 24 * q^79 + 2 * q^81 - 8 * q^84 + 8 * q^86 - 4 * q^89 - 48 * q^91 - 10 * q^96

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2175\mathbb{Z}\right)^\times$$.

 $$n$$ $$901$$ $$1451$$ $$2002$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$7$$ − 4.00000i − 1.51186i −0.654654 0.755929i $$-0.727186\pi$$
0.654654 0.755929i $$-0.272814\pi$$
$$8$$ − 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 $$-0.312833\pi$$
$$14$$ −4.00000 −1.06904
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ 2.00000i 0.485071i 0.970143 + 0.242536i $$0.0779791\pi$$
−0.970143 + 0.242536i $$0.922021\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ −8.00000 −1.83533 −0.917663 0.397360i $$-0.869927\pi$$
−0.917663 + 0.397360i $$0.869927\pi$$
$$20$$ 0 0
$$21$$ −4.00000 −0.872872
$$22$$ 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
$$28$$ − 4.00000i − 0.755929i
$$29$$ −1.00000 −0.185695
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ − 5.00000i − 0.883883i
$$33$$ 0 0
$$34$$ 2.00000 0.342997
$$35$$ 0 0
$$36$$ −1.00000 −0.166667
$$37$$ 6.00000i 0.986394i 0.869918 + 0.493197i $$0.164172\pi$$
−0.869918 + 0.493197i $$0.835828\pi$$
$$38$$ 8.00000i 1.29777i
$$39$$ −6.00000 −0.960769
$$40$$ 0 0
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ 4.00000i 0.617213i
$$43$$ 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
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ 1.00000i 0.144338i
$$49$$ −9.00000 −1.28571
$$50$$ 0 0
$$51$$ 2.00000 0.280056
$$52$$ − 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
$$56$$ −12.0000 −1.60357
$$57$$ 8.00000i 1.05963i
$$58$$ 1.00000i 0.131306i
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 0 0
$$61$$ 6.00000 0.768221 0.384111 0.923287i $$-0.374508\pi$$
0.384111 + 0.923287i $$0.374508\pi$$
$$62$$ − 4.00000i − 0.508001i
$$63$$ 4.00000i 0.503953i
$$64$$ −7.00000 −0.875000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 8.00000i − 0.977356i −0.872464 0.488678i $$-0.837479\pi$$
0.872464 0.488678i $$-0.162521\pi$$
$$68$$ 2.00000i 0.242536i
$$69$$ 4.00000 0.481543
$$70$$ 0 0
$$71$$ 16.0000 1.89885 0.949425 0.313993i $$-0.101667\pi$$
0.949425 + 0.313993i $$0.101667\pi$$
$$72$$ 3.00000i 0.353553i
$$73$$ 6.00000i 0.702247i 0.936329 + 0.351123i $$0.114200\pi$$
−0.936329 + 0.351123i $$0.885800\pi$$
$$74$$ 6.00000 0.697486
$$75$$ 0 0
$$76$$ −8.00000 −0.917663
$$77$$ 0 0
$$78$$ 6.00000i 0.679366i
$$79$$ −12.0000 −1.35011 −0.675053 0.737769i $$-0.735879\pi$$
−0.675053 + 0.737769i $$0.735879\pi$$
$$80$$ 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 $$0.658802\pi$$
$$84$$ −4.00000 −0.436436
$$85$$ 0 0
$$86$$ 4.00000 0.431331
$$87$$ 1.00000i 0.107211i
$$88$$ 0 0
$$89$$ −2.00000 −0.212000 −0.106000 0.994366i $$-0.533804\pi$$
−0.106000 + 0.994366i $$0.533804\pi$$
$$90$$ 0 0
$$91$$ −24.0000 −2.51588
$$92$$ 4.00000i 0.417029i
$$93$$ − 4.00000i − 0.414781i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ −5.00000 −0.510310
$$97$$ − 14.0000i − 1.42148i −0.703452 0.710742i $$-0.748359\pi$$
0.703452 0.710742i $$-0.251641\pi$$
$$98$$ 9.00000i 0.909137i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ − 2.00000i − 0.198030i
$$103$$ − 4.00000i − 0.394132i −0.980390 0.197066i $$-0.936859\pi$$
0.980390 0.197066i $$-0.0631413\pi$$
$$104$$ −18.0000 −1.76505
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ − 16.0000i − 1.54678i −0.633932 0.773389i $$-0.718560\pi$$
0.633932 0.773389i $$-0.281440\pi$$
$$108$$ 1.00000i 0.0962250i
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 0 0
$$111$$ 6.00000 0.569495
$$112$$ 4.00000i 0.377964i
$$113$$ − 2.00000i − 0.188144i −0.995565 0.0940721i $$-0.970012\pi$$
0.995565 0.0940721i $$-0.0299884\pi$$
$$114$$ 8.00000 0.749269
$$115$$ 0 0
$$116$$ −1.00000 −0.0928477
$$117$$ 6.00000i 0.554700i
$$118$$ − 12.0000i − 1.10469i
$$119$$ 8.00000 0.733359
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ − 6.00000i − 0.543214i
$$123$$ − 2.00000i − 0.180334i
$$124$$ 4.00000 0.359211
$$125$$ 0 0
$$126$$ 4.00000 0.356348
$$127$$ − 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
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ 32.0000i 2.77475i
$$134$$ −8.00000 −0.691095
$$135$$ 0 0
$$136$$ 6.00000 0.514496
$$137$$ − 6.00000i − 0.512615i −0.966595 0.256307i $$-0.917494\pi$$
0.966595 0.256307i $$-0.0825059\pi$$
$$138$$ − 4.00000i − 0.340503i
$$139$$ −20.0000 −1.69638 −0.848189 0.529694i $$-0.822307\pi$$
−0.848189 + 0.529694i $$0.822307\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ − 16.0000i − 1.34269i
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ 6.00000 0.496564
$$147$$ 9.00000i 0.742307i
$$148$$ 6.00000i 0.493197i
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ 0 0
$$151$$ 16.0000 1.30206 0.651031 0.759051i $$-0.274337\pi$$
0.651031 + 0.759051i $$0.274337\pi$$
$$152$$ 24.0000i 1.94666i
$$153$$ − 2.00000i − 0.161690i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −6.00000 −0.480384
$$157$$ 14.0000i 1.11732i 0.829396 + 0.558661i $$0.188685\pi$$
−0.829396 + 0.558661i $$0.811315\pi$$
$$158$$ 12.0000i 0.954669i
$$159$$ −6.00000 −0.475831
$$160$$ 0 0
$$161$$ 16.0000 1.26098
$$162$$ − 1.00000i − 0.0785674i
$$163$$ − 12.0000i − 0.939913i −0.882690 0.469956i $$-0.844270\pi$$
0.882690 0.469956i $$-0.155730\pi$$
$$164$$ 2.00000 0.156174
$$165$$ 0 0
$$166$$ 16.0000 1.24184
$$167$$ − 20.0000i − 1.54765i −0.633402 0.773823i $$-0.718342\pi$$
0.633402 0.773823i $$-0.281658\pi$$
$$168$$ 12.0000i 0.925820i
$$169$$ −23.0000 −1.76923
$$170$$ 0 0
$$171$$ 8.00000 0.611775
$$172$$ 4.00000i 0.304997i
$$173$$ 2.00000i 0.152057i 0.997106 + 0.0760286i $$0.0242240\pi$$
−0.997106 + 0.0760286i $$0.975776\pi$$
$$174$$ 1.00000 0.0758098
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 12.0000i − 0.901975i
$$178$$ 2.00000i 0.149906i
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ −10.0000 −0.743294 −0.371647 0.928374i $$-0.621207\pi$$
−0.371647 + 0.928374i $$0.621207\pi$$
$$182$$ 24.0000i 1.77900i
$$183$$ − 6.00000i − 0.443533i
$$184$$ 12.0000 0.884652
$$185$$ 0 0
$$186$$ −4.00000 −0.293294
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 4.00000 0.290957
$$190$$ 0 0
$$191$$ −4.00000 −0.289430 −0.144715 0.989473i $$-0.546227\pi$$
−0.144715 + 0.989473i $$0.546227\pi$$
$$192$$ 7.00000i 0.505181i
$$193$$ − 2.00000i − 0.143963i −0.997406 0.0719816i $$-0.977068\pi$$
0.997406 0.0719816i $$-0.0229323\pi$$
$$194$$ −14.0000 −1.00514
$$195$$ 0 0
$$196$$ −9.00000 −0.642857
$$197$$ 6.00000i 0.427482i 0.976890 + 0.213741i $$0.0685649\pi$$
−0.976890 + 0.213741i $$0.931435\pi$$
$$198$$ 0 0
$$199$$ 16.0000 1.13421 0.567105 0.823646i $$-0.308063\pi$$
0.567105 + 0.823646i $$0.308063\pi$$
$$200$$ 0 0
$$201$$ −8.00000 −0.564276
$$202$$ 10.0000i 0.703598i
$$203$$ 4.00000i 0.280745i
$$204$$ 2.00000 0.140028
$$205$$ 0 0
$$206$$ −4.00000 −0.278693
$$207$$ − 4.00000i − 0.278019i
$$208$$ 6.00000i 0.416025i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 8.00000 0.550743 0.275371 0.961338i $$-0.411199\pi$$
0.275371 + 0.961338i $$0.411199\pi$$
$$212$$ − 6.00000i − 0.412082i
$$213$$ − 16.0000i − 1.09630i
$$214$$ −16.0000 −1.09374
$$215$$ 0 0
$$216$$ 3.00000 0.204124
$$217$$ − 16.0000i − 1.08615i
$$218$$ − 2.00000i − 0.135457i
$$219$$ 6.00000 0.405442
$$220$$ 0 0
$$221$$ 12.0000 0.807207
$$222$$ − 6.00000i − 0.402694i
$$223$$ − 12.0000i − 0.803579i −0.915732 0.401790i $$-0.868388\pi$$
0.915732 0.401790i $$-0.131612\pi$$
$$224$$ −20.0000 −1.33631
$$225$$ 0 0
$$226$$ −2.00000 −0.133038
$$227$$ 8.00000i 0.530979i 0.964114 + 0.265489i $$0.0855335\pi$$
−0.964114 + 0.265489i $$0.914466\pi$$
$$228$$ 8.00000i 0.529813i
$$229$$ −14.0000 −0.925146 −0.462573 0.886581i $$-0.653074\pi$$
−0.462573 + 0.886581i $$0.653074\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 3.00000i 0.196960i
$$233$$ − 18.0000i − 1.17922i −0.807688 0.589610i $$-0.799282\pi$$
0.807688 0.589610i $$-0.200718\pi$$
$$234$$ 6.00000 0.392232
$$235$$ 0 0
$$236$$ 12.0000 0.781133
$$237$$ 12.0000i 0.779484i
$$238$$ − 8.00000i − 0.518563i
$$239$$ −24.0000 −1.55243 −0.776215 0.630468i $$-0.782863\pi$$
−0.776215 + 0.630468i $$0.782863\pi$$
$$240$$ 0 0
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ 11.0000i 0.707107i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ 6.00000 0.384111
$$245$$ 0 0
$$246$$ −2.00000 −0.127515
$$247$$ 48.0000i 3.05417i
$$248$$ − 12.0000i − 0.762001i
$$249$$ 16.0000 1.01396
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 4.00000i 0.251976i
$$253$$ 0 0
$$254$$ −8.00000 −0.501965
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ 26.0000i 1.62184i 0.585160 + 0.810918i $$0.301032\pi$$
−0.585160 + 0.810918i $$0.698968\pi$$
$$258$$ − 4.00000i − 0.249029i
$$259$$ 24.0000 1.49129
$$260$$ 0 0
$$261$$ 1.00000 0.0618984
$$262$$ 0 0
$$263$$ − 8.00000i − 0.493301i −0.969104 0.246651i $$-0.920670\pi$$
0.969104 0.246651i $$-0.0793300\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 32.0000 1.96205
$$267$$ 2.00000i 0.122398i
$$268$$ − 8.00000i − 0.488678i
$$269$$ 18.0000 1.09748 0.548740 0.835993i $$-0.315108\pi$$
0.548740 + 0.835993i $$0.315108\pi$$
$$270$$ 0 0
$$271$$ −20.0000 −1.21491 −0.607457 0.794353i $$-0.707810\pi$$
−0.607457 + 0.794353i $$0.707810\pi$$
$$272$$ − 2.00000i − 0.121268i
$$273$$ 24.0000i 1.45255i
$$274$$ −6.00000 −0.362473
$$275$$ 0 0
$$276$$ 4.00000 0.240772
$$277$$ − 2.00000i − 0.120168i −0.998193 0.0600842i $$-0.980863\pi$$
0.998193 0.0600842i $$-0.0191369\pi$$
$$278$$ 20.0000i 1.19952i
$$279$$ −4.00000 −0.239474
$$280$$ 0 0
$$281$$ −6.00000 −0.357930 −0.178965 0.983855i $$-0.557275\pi$$
−0.178965 + 0.983855i $$0.557275\pi$$
$$282$$ 0 0
$$283$$ − 8.00000i − 0.475551i −0.971320 0.237775i $$-0.923582\pi$$
0.971320 0.237775i $$-0.0764182\pi$$
$$284$$ 16.0000 0.949425
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 8.00000i − 0.472225i
$$288$$ 5.00000i 0.294628i
$$289$$ 13.0000 0.764706
$$290$$ 0 0
$$291$$ −14.0000 −0.820695
$$292$$ 6.00000i 0.351123i
$$293$$ 26.0000i 1.51894i 0.650545 + 0.759468i $$0.274541\pi$$
−0.650545 + 0.759468i $$0.725459\pi$$
$$294$$ 9.00000 0.524891
$$295$$ 0 0
$$296$$ 18.0000 1.04623
$$297$$ 0 0
$$298$$ − 10.0000i − 0.579284i
$$299$$ 24.0000 1.38796
$$300$$ 0 0
$$301$$ 16.0000 0.922225
$$302$$ − 16.0000i − 0.920697i
$$303$$ 10.0000i 0.574485i
$$304$$ 8.00000 0.458831
$$305$$ 0 0
$$306$$ −2.00000 −0.114332
$$307$$ − 20.0000i − 1.14146i −0.821138 0.570730i $$-0.806660\pi$$
0.821138 0.570730i $$-0.193340\pi$$
$$308$$ 0 0
$$309$$ −4.00000 −0.227552
$$310$$ 0 0
$$311$$ −28.0000 −1.58773 −0.793867 0.608091i $$-0.791935\pi$$
−0.793867 + 0.608091i $$0.791935\pi$$
$$312$$ 18.0000i 1.01905i
$$313$$ − 10.0000i − 0.565233i −0.959233 0.282617i $$-0.908798\pi$$
0.959233 0.282617i $$-0.0912024\pi$$
$$314$$ 14.0000 0.790066
$$315$$ 0 0
$$316$$ −12.0000 −0.675053
$$317$$ 30.0000i 1.68497i 0.538721 + 0.842484i $$0.318908\pi$$
−0.538721 + 0.842484i $$0.681092\pi$$
$$318$$ 6.00000i 0.336463i
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −16.0000 −0.893033
$$322$$ − 16.0000i − 0.891645i
$$323$$ − 16.0000i − 0.890264i
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −12.0000 −0.664619
$$327$$ − 2.00000i − 0.110600i
$$328$$ − 6.00000i − 0.331295i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −16.0000 −0.879440 −0.439720 0.898135i $$-0.644922\pi$$
−0.439720 + 0.898135i $$0.644922\pi$$
$$332$$ 16.0000i 0.878114i
$$333$$ − 6.00000i − 0.328798i
$$334$$ −20.0000 −1.09435
$$335$$ 0 0
$$336$$ 4.00000 0.218218
$$337$$ − 30.0000i − 1.63420i −0.576493 0.817102i $$-0.695579\pi$$
0.576493 0.817102i $$-0.304421\pi$$
$$338$$ 23.0000i 1.25104i
$$339$$ −2.00000 −0.108625
$$340$$ 0 0
$$341$$ 0 0
$$342$$ − 8.00000i − 0.432590i
$$343$$ 8.00000i 0.431959i
$$344$$ 12.0000 0.646997
$$345$$ 0 0
$$346$$ 2.00000 0.107521
$$347$$ − 24.0000i − 1.28839i −0.764862 0.644194i $$-0.777193\pi$$
0.764862 0.644194i $$-0.222807\pi$$
$$348$$ 1.00000i 0.0536056i
$$349$$ 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
$$353$$ 6.00000i 0.319348i 0.987170 + 0.159674i $$0.0510443\pi$$
−0.987170 + 0.159674i $$0.948956\pi$$
$$354$$ −12.0000 −0.637793
$$355$$ 0 0
$$356$$ −2.00000 −0.106000
$$357$$ − 8.00000i − 0.423405i
$$358$$ − 12.0000i − 0.634220i
$$359$$ 36.0000 1.90001 0.950004 0.312239i $$-0.101079\pi$$
0.950004 + 0.312239i $$0.101079\pi$$
$$360$$ 0 0
$$361$$ 45.0000 2.36842
$$362$$ 10.0000i 0.525588i
$$363$$ 11.0000i 0.577350i
$$364$$ −24.0000 −1.25794
$$365$$ 0 0
$$366$$ −6.00000 −0.313625
$$367$$ − 24.0000i − 1.25279i −0.779506 0.626395i $$-0.784530\pi$$
0.779506 0.626395i $$-0.215470\pi$$
$$368$$ − 4.00000i − 0.208514i
$$369$$ −2.00000 −0.104116
$$370$$ 0 0
$$371$$ −24.0000 −1.24602
$$372$$ − 4.00000i − 0.207390i
$$373$$ 34.0000i 1.76045i 0.474554 + 0.880227i $$0.342610\pi$$
−0.474554 + 0.880227i $$0.657390\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 6.00000i 0.309016i
$$378$$ − 4.00000i − 0.205738i
$$379$$ 16.0000 0.821865 0.410932 0.911666i $$-0.365203\pi$$
0.410932 + 0.911666i $$0.365203\pi$$
$$380$$ 0 0
$$381$$ −8.00000 −0.409852
$$382$$ 4.00000i 0.204658i
$$383$$ − 28.0000i − 1.43073i −0.698749 0.715367i $$-0.746260\pi$$
0.698749 0.715367i $$-0.253740\pi$$
$$384$$ −3.00000 −0.153093
$$385$$ 0 0
$$386$$ −2.00000 −0.101797
$$387$$ − 4.00000i − 0.203331i
$$388$$ − 14.0000i − 0.710742i
$$389$$ 26.0000 1.31825 0.659126 0.752032i $$-0.270926\pi$$
0.659126 + 0.752032i $$0.270926\pi$$
$$390$$ 0 0
$$391$$ −8.00000 −0.404577
$$392$$ 27.0000i 1.36371i
$$393$$ 0 0
$$394$$ 6.00000 0.302276
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 22.0000i 1.10415i 0.833795 + 0.552074i $$0.186163\pi$$
−0.833795 + 0.552074i $$0.813837\pi$$
$$398$$ − 16.0000i − 0.802008i
$$399$$ 32.0000 1.60200
$$400$$ 0 0
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ 8.00000i 0.399004i
$$403$$ − 24.0000i − 1.19553i
$$404$$ −10.0000 −0.497519
$$405$$ 0 0
$$406$$ 4.00000 0.198517
$$407$$ 0 0
$$408$$ − 6.00000i − 0.297044i
$$409$$ 6.00000 0.296681 0.148340 0.988936i $$-0.452607\pi$$
0.148340 + 0.988936i $$0.452607\pi$$
$$410$$ 0 0
$$411$$ −6.00000 −0.295958
$$412$$ − 4.00000i − 0.197066i
$$413$$ − 48.0000i − 2.36193i
$$414$$ −4.00000 −0.196589
$$415$$ 0 0
$$416$$ −30.0000 −1.47087
$$417$$ 20.0000i 0.979404i
$$418$$ 0 0
$$419$$ −4.00000 −0.195413 −0.0977064 0.995215i $$-0.531151\pi$$
−0.0977064 + 0.995215i $$0.531151\pi$$
$$420$$ 0 0
$$421$$ 30.0000 1.46211 0.731055 0.682318i $$-0.239028\pi$$
0.731055 + 0.682318i $$0.239028\pi$$
$$422$$ − 8.00000i − 0.389434i
$$423$$ 0 0
$$424$$ −18.0000 −0.874157
$$425$$ 0 0
$$426$$ −16.0000 −0.775203
$$427$$ − 24.0000i − 1.16144i
$$428$$ − 16.0000i − 0.773389i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 32.0000 1.54139 0.770693 0.637207i $$-0.219910\pi$$
0.770693 + 0.637207i $$0.219910\pi$$
$$432$$ − 1.00000i − 0.0481125i
$$433$$ 14.0000i 0.672797i 0.941720 + 0.336399i $$0.109209\pi$$
−0.941720 + 0.336399i $$0.890791\pi$$
$$434$$ −16.0000 −0.768025
$$435$$ 0 0
$$436$$ 2.00000 0.0957826
$$437$$ − 32.0000i − 1.53077i
$$438$$ − 6.00000i − 0.286691i
$$439$$ 24.0000 1.14546 0.572729 0.819745i $$-0.305885\pi$$
0.572729 + 0.819745i $$0.305885\pi$$
$$440$$ 0 0
$$441$$ 9.00000 0.428571
$$442$$ − 12.0000i − 0.570782i
$$443$$ − 20.0000i − 0.950229i −0.879924 0.475114i $$-0.842407\pi$$
0.879924 0.475114i $$-0.157593\pi$$
$$444$$ 6.00000 0.284747
$$445$$ 0 0
$$446$$ −12.0000 −0.568216
$$447$$ − 10.0000i − 0.472984i
$$448$$ 28.0000i 1.32288i
$$449$$ 6.00000 0.283158 0.141579 0.989927i $$-0.454782\pi$$
0.141579 + 0.989927i $$0.454782\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ − 2.00000i − 0.0940721i
$$453$$ − 16.0000i − 0.751746i
$$454$$ 8.00000 0.375459
$$455$$ 0 0
$$456$$ 24.0000 1.12390
$$457$$ − 22.0000i − 1.02912i −0.857455 0.514558i $$-0.827956\pi$$
0.857455 0.514558i $$-0.172044\pi$$
$$458$$ 14.0000i 0.654177i
$$459$$ −2.00000 −0.0933520
$$460$$ 0 0
$$461$$ −18.0000 −0.838344 −0.419172 0.907907i $$-0.637680\pi$$
−0.419172 + 0.907907i $$0.637680\pi$$
$$462$$ 0 0
$$463$$ − 36.0000i − 1.67306i −0.547920 0.836531i $$-0.684580\pi$$
0.547920 0.836531i $$-0.315420\pi$$
$$464$$ 1.00000 0.0464238
$$465$$ 0 0
$$466$$ −18.0000 −0.833834
$$467$$ 12.0000i 0.555294i 0.960683 + 0.277647i $$0.0895545\pi$$
−0.960683 + 0.277647i $$0.910445\pi$$
$$468$$ 6.00000i 0.277350i
$$469$$ −32.0000 −1.47762
$$470$$ 0 0
$$471$$ 14.0000 0.645086
$$472$$ − 36.0000i − 1.65703i
$$473$$ 0 0
$$474$$ 12.0000 0.551178
$$475$$ 0 0
$$476$$ 8.00000 0.366679
$$477$$ 6.00000i 0.274721i
$$478$$ 24.0000i 1.09773i
$$479$$ −12.0000 −0.548294 −0.274147 0.961688i $$-0.588395\pi$$
−0.274147 + 0.961688i $$0.588395\pi$$
$$480$$ 0 0
$$481$$ 36.0000 1.64146
$$482$$ − 2.00000i − 0.0910975i
$$483$$ − 16.0000i − 0.728025i
$$484$$ −11.0000 −0.500000
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ − 20.0000i − 0.906287i −0.891438 0.453143i $$-0.850303\pi$$
0.891438 0.453143i $$-0.149697\pi$$
$$488$$ − 18.0000i − 0.814822i
$$489$$ −12.0000 −0.542659
$$490$$ 0 0
$$491$$ −24.0000 −1.08310 −0.541552 0.840667i $$-0.682163\pi$$
−0.541552 + 0.840667i $$0.682163\pi$$
$$492$$ − 2.00000i − 0.0901670i
$$493$$ − 2.00000i − 0.0900755i
$$494$$ 48.0000 2.15962
$$495$$ 0 0
$$496$$ −4.00000 −0.179605
$$497$$ − 64.0000i − 2.87079i
$$498$$ − 16.0000i − 0.716977i
$$499$$ −36.0000 −1.61158 −0.805791 0.592200i $$-0.798259\pi$$
−0.805791 + 0.592200i $$0.798259\pi$$
$$500$$ 0 0
$$501$$ −20.0000 −0.893534
$$502$$ 0 0
$$503$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$504$$ 12.0000 0.534522
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 23.0000i 1.02147i
$$508$$ − 8.00000i − 0.354943i
$$509$$ −30.0000 −1.32973 −0.664863 0.746965i $$-0.731510\pi$$
−0.664863 + 0.746965i $$0.731510\pi$$
$$510$$ 0 0
$$511$$ 24.0000 1.06170
$$512$$ 11.0000i 0.486136i
$$513$$ − 8.00000i − 0.353209i
$$514$$ 26.0000 1.14681
$$515$$ 0 0
$$516$$ 4.00000 0.176090
$$517$$ 0 0
$$518$$ − 24.0000i − 1.05450i
$$519$$ 2.00000 0.0877903
$$520$$ 0 0
$$521$$ −6.00000 −0.262865 −0.131432 0.991325i $$-0.541958\pi$$
−0.131432 + 0.991325i $$0.541958\pi$$
$$522$$ − 1.00000i − 0.0437688i
$$523$$ 16.0000i 0.699631i 0.936819 + 0.349816i $$0.113756\pi$$
−0.936819 + 0.349816i $$0.886244\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ −8.00000 −0.348817
$$527$$ 8.00000i 0.348485i
$$528$$ 0 0
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ −12.0000 −0.520756
$$532$$ 32.0000i 1.38738i
$$533$$ − 12.0000i − 0.519778i
$$534$$ 2.00000 0.0865485
$$535$$ 0 0
$$536$$ −24.0000 −1.03664
$$537$$ − 12.0000i − 0.517838i
$$538$$ − 18.0000i − 0.776035i
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 22.0000 0.945854 0.472927 0.881102i $$-0.343197\pi$$
0.472927 + 0.881102i $$0.343197\pi$$
$$542$$ 20.0000i 0.859074i
$$543$$ 10.0000i 0.429141i
$$544$$ 10.0000 0.428746
$$545$$ 0 0
$$546$$ 24.0000 1.02711
$$547$$ − 32.0000i − 1.36822i −0.729378 0.684111i $$-0.760191\pi$$
0.729378 0.684111i $$-0.239809\pi$$
$$548$$ − 6.00000i − 0.256307i
$$549$$ −6.00000 −0.256074
$$550$$ 0 0
$$551$$ 8.00000 0.340811
$$552$$ − 12.0000i − 0.510754i
$$553$$ 48.0000i 2.04117i
$$554$$ −2.00000 −0.0849719
$$555$$ 0 0
$$556$$ −20.0000 −0.848189
$$557$$ 14.0000i 0.593199i 0.955002 + 0.296600i $$0.0958526\pi$$
−0.955002 + 0.296600i $$0.904147\pi$$
$$558$$ 4.00000i 0.169334i
$$559$$ 24.0000 1.01509
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 6.00000i 0.253095i
$$563$$ 4.00000i 0.168580i 0.996441 + 0.0842900i $$0.0268622\pi$$
−0.996441 + 0.0842900i $$0.973138\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −8.00000 −0.336265
$$567$$ − 4.00000i − 0.167984i
$$568$$ − 48.0000i − 2.01404i
$$569$$ −18.0000 −0.754599 −0.377300 0.926091i $$-0.623147\pi$$
−0.377300 + 0.926091i $$0.623147\pi$$
$$570$$ 0 0
$$571$$ 12.0000 0.502184 0.251092 0.967963i $$-0.419210\pi$$
0.251092 + 0.967963i $$0.419210\pi$$
$$572$$ 0 0
$$573$$ 4.00000i 0.167102i
$$574$$ −8.00000 −0.333914
$$575$$ 0 0
$$576$$ 7.00000 0.291667
$$577$$ 2.00000i 0.0832611i 0.999133 + 0.0416305i $$0.0132552\pi$$
−0.999133 + 0.0416305i $$0.986745\pi$$
$$578$$ − 13.0000i − 0.540729i
$$579$$ −2.00000 −0.0831172
$$580$$ 0 0
$$581$$ 64.0000 2.65517
$$582$$ 14.0000i 0.580319i
$$583$$ 0 0
$$584$$ 18.0000 0.744845
$$585$$ 0 0
$$586$$ 26.0000 1.07405
$$587$$ − 16.0000i − 0.660391i −0.943913 0.330195i $$-0.892885\pi$$
0.943913 0.330195i $$-0.107115\pi$$
$$588$$ 9.00000i 0.371154i
$$589$$ −32.0000 −1.31854
$$590$$ 0 0
$$591$$ 6.00000 0.246807
$$592$$ − 6.00000i − 0.246598i
$$593$$ 6.00000i 0.246390i 0.992382 + 0.123195i $$0.0393141\pi$$
−0.992382 + 0.123195i $$0.960686\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 10.0000 0.409616
$$597$$ − 16.0000i − 0.654836i
$$598$$ − 24.0000i − 0.981433i
$$599$$ 36.0000 1.47092 0.735460 0.677568i $$-0.236966\pi$$
0.735460 + 0.677568i $$0.236966\pi$$
$$600$$ 0 0
$$601$$ 10.0000 0.407909 0.203954 0.978980i $$-0.434621\pi$$
0.203954 + 0.978980i $$0.434621\pi$$
$$602$$ − 16.0000i − 0.652111i
$$603$$ 8.00000i 0.325785i
$$604$$ 16.0000 0.651031
$$605$$ 0 0
$$606$$ 10.0000 0.406222
$$607$$ − 32.0000i − 1.29884i −0.760430 0.649420i $$-0.775012\pi$$
0.760430 0.649420i $$-0.224988\pi$$
$$608$$ 40.0000i 1.62221i
$$609$$ 4.00000 0.162088
$$610$$ 0 0
$$611$$ 0 0
$$612$$ − 2.00000i − 0.0808452i
$$613$$ 2.00000i 0.0807792i 0.999184 + 0.0403896i $$0.0128599\pi$$
−0.999184 + 0.0403896i $$0.987140\pi$$
$$614$$ −20.0000 −0.807134
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 42.0000i 1.69086i 0.534089 + 0.845428i $$0.320655\pi$$
−0.534089 + 0.845428i $$0.679345\pi$$
$$618$$ 4.00000i 0.160904i
$$619$$ 8.00000 0.321547 0.160774 0.986991i $$-0.448601\pi$$
0.160774 + 0.986991i $$0.448601\pi$$
$$620$$ 0 0
$$621$$ −4.00000 −0.160514
$$622$$ 28.0000i 1.12270i
$$623$$ 8.00000i 0.320513i
$$624$$ 6.00000 0.240192
$$625$$ 0 0
$$626$$ −10.0000 −0.399680
$$627$$ 0 0
$$628$$ 14.0000i 0.558661i
$$629$$ −12.0000 −0.478471
$$630$$ 0 0
$$631$$ 48.0000 1.91085 0.955425 0.295234i $$-0.0953977\pi$$
0.955425 + 0.295234i $$0.0953977\pi$$
$$632$$ 36.0000i 1.43200i
$$633$$ − 8.00000i − 0.317971i
$$634$$ 30.0000 1.19145
$$635$$ 0 0
$$636$$ −6.00000 −0.237915
$$637$$ 54.0000i 2.13956i
$$638$$ 0 0
$$639$$ −16.0000 −0.632950
$$640$$ 0 0
$$641$$ −6.00000 −0.236986 −0.118493 0.992955i $$-0.537806\pi$$
−0.118493 + 0.992955i $$0.537806\pi$$
$$642$$ 16.0000i 0.631470i
$$643$$ 16.0000i 0.630978i 0.948929 + 0.315489i $$0.102169\pi$$
−0.948929 + 0.315489i $$0.897831\pi$$
$$644$$ 16.0000 0.630488
$$645$$ 0 0
$$646$$ −16.0000 −0.629512
$$647$$ 28.0000i 1.10079i 0.834903 + 0.550397i $$0.185524\pi$$
−0.834903 + 0.550397i $$0.814476\pi$$
$$648$$ − 3.00000i − 0.117851i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −16.0000 −0.627089
$$652$$ − 12.0000i − 0.469956i
$$653$$ 50.0000i 1.95665i 0.207072 + 0.978326i $$0.433606\pi$$
−0.207072 + 0.978326i $$0.566394\pi$$
$$654$$ −2.00000 −0.0782062
$$655$$ 0 0
$$656$$ −2.00000 −0.0780869
$$657$$ − 6.00000i − 0.234082i
$$658$$ 0 0
$$659$$ 48.0000 1.86981 0.934907 0.354892i $$-0.115482\pi$$
0.934907 + 0.354892i $$0.115482\pi$$
$$660$$ 0 0
$$661$$ −10.0000 −0.388955 −0.194477 0.980907i $$-0.562301\pi$$
−0.194477 + 0.980907i $$0.562301\pi$$
$$662$$ 16.0000i 0.621858i
$$663$$ − 12.0000i − 0.466041i
$$664$$ 48.0000 1.86276
$$665$$ 0 0
$$666$$ −6.00000 −0.232495
$$667$$ − 4.00000i − 0.154881i
$$668$$ − 20.0000i − 0.773823i
$$669$$ −12.0000 −0.463947
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 20.0000i 0.771517i
$$673$$ 30.0000i 1.15642i 0.815890 + 0.578208i $$0.196248\pi$$
−0.815890 + 0.578208i $$0.803752\pi$$
$$674$$ −30.0000 −1.15556
$$675$$ 0 0
$$676$$ −23.0000 −0.884615
$$677$$ − 26.0000i − 0.999261i −0.866239 0.499631i $$-0.833469\pi$$
0.866239 0.499631i $$-0.166531\pi$$
$$678$$ 2.00000i 0.0768095i
$$679$$ −56.0000 −2.14908
$$680$$ 0 0
$$681$$ 8.00000 0.306561
$$682$$ 0 0
$$683$$ 24.0000i 0.918334i 0.888350 + 0.459167i $$0.151852\pi$$
−0.888350 + 0.459167i $$0.848148\pi$$
$$684$$ 8.00000 0.305888
$$685$$ 0 0
$$686$$ 8.00000 0.305441
$$687$$ 14.0000i 0.534133i
$$688$$ − 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 $$0.375780\pi$$
$$692$$ 2.00000i 0.0760286i
$$693$$ 0 0
$$694$$ −24.0000 −0.911028
$$695$$ 0 0
$$696$$ 3.00000 0.113715
$$697$$ 4.00000i 0.151511i
$$698$$ − 2.00000i − 0.0757011i
$$699$$ −18.0000 −0.680823
$$700$$ 0 0
$$701$$ −34.0000 −1.28416 −0.642081 0.766637i $$-0.721929\pi$$
−0.642081 + 0.766637i $$0.721929\pi$$
$$702$$ − 6.00000i − 0.226455i
$$703$$ − 48.0000i − 1.81035i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 6.00000 0.225813
$$707$$ 40.0000i 1.50435i
$$708$$ − 12.0000i − 0.450988i
$$709$$ 10.0000 0.375558 0.187779 0.982211i $$-0.439871\pi$$
0.187779 + 0.982211i $$0.439871\pi$$
$$710$$ 0 0
$$711$$ 12.0000 0.450035
$$712$$ 6.00000i 0.224860i
$$713$$ 16.0000i 0.599205i
$$714$$ −8.00000 −0.299392
$$715$$ 0 0
$$716$$ 12.0000 0.448461
$$717$$ 24.0000i 0.896296i
$$718$$ − 36.0000i − 1.34351i
$$719$$ 40.0000 1.49175 0.745874 0.666087i $$-0.232032\pi$$
0.745874 + 0.666087i $$0.232032\pi$$
$$720$$ 0 0
$$721$$ −16.0000 −0.595871
$$722$$ − 45.0000i − 1.67473i
$$723$$ − 2.00000i − 0.0743808i
$$724$$ −10.0000 −0.371647
$$725$$ 0 0
$$726$$ 11.0000 0.408248
$$727$$ 32.0000i 1.18681i 0.804902 + 0.593407i $$0.202218\pi$$
−0.804902 + 0.593407i $$0.797782\pi$$
$$728$$ 72.0000i 2.66850i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −8.00000 −0.295891
$$732$$ − 6.00000i − 0.221766i
$$733$$ − 46.0000i − 1.69905i −0.527549 0.849524i $$-0.676889\pi$$
0.527549 0.849524i $$-0.323111\pi$$
$$734$$ −24.0000 −0.885856
$$735$$ 0 0
$$736$$ 20.0000 0.737210
$$737$$ 0 0
$$738$$ 2.00000i 0.0736210i
$$739$$ 24.0000 0.882854 0.441427 0.897297i $$-0.354472\pi$$
0.441427 + 0.897297i $$0.354472\pi$$
$$740$$ 0 0
$$741$$ 48.0000 1.76332
$$742$$ 24.0000i 0.881068i
$$743$$ 32.0000i 1.17397i 0.809599 + 0.586983i $$0.199684\pi$$
−0.809599 + 0.586983i $$0.800316\pi$$
$$744$$ −12.0000 −0.439941
$$745$$ 0 0
$$746$$ 34.0000 1.24483
$$747$$ − 16.0000i − 0.585409i
$$748$$ 0 0
$$749$$ −64.0000 −2.33851
$$750$$ 0 0
$$751$$ −4.00000 −0.145962 −0.0729810 0.997333i $$-0.523251\pi$$
−0.0729810 + 0.997333i $$0.523251\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 6.00000 0.218507
$$755$$ 0 0
$$756$$ 4.00000 0.145479
$$757$$ − 10.0000i − 0.363456i −0.983349 0.181728i $$-0.941831\pi$$
0.983349 0.181728i $$-0.0581691\pi$$
$$758$$ − 16.0000i − 0.581146i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −38.0000 −1.37750 −0.688749 0.724999i $$-0.741840\pi$$
−0.688749 + 0.724999i $$0.741840\pi$$
$$762$$ 8.00000i 0.289809i
$$763$$ − 8.00000i − 0.289619i
$$764$$ −4.00000 −0.144715
$$765$$ 0 0
$$766$$ −28.0000 −1.01168
$$767$$ − 72.0000i − 2.59977i
$$768$$ 17.0000i 0.613435i
$$769$$ 30.0000 1.08183 0.540914 0.841078i $$-0.318079\pi$$
0.540914 + 0.841078i $$0.318079\pi$$
$$770$$ 0 0
$$771$$ 26.0000 0.936367
$$772$$ − 2.00000i − 0.0719816i
$$773$$ − 6.00000i − 0.215805i −0.994161 0.107903i $$-0.965587\pi$$
0.994161 0.107903i $$-0.0344134\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ 0 0
$$776$$ −42.0000 −1.50771
$$777$$ − 24.0000i − 0.860995i
$$778$$ − 26.0000i − 0.932145i
$$779$$ −16.0000 −0.573259
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 8.00000i 0.286079i
$$783$$ − 1.00000i − 0.0357371i
$$784$$ 9.00000 0.321429
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 24.0000i − 0.855508i −0.903895 0.427754i $$-0.859305\pi$$
0.903895 0.427754i $$-0.140695\pi$$
$$788$$ 6.00000i 0.213741i
$$789$$ −8.00000 −0.284808
$$790$$ 0 0
$$791$$ −8.00000 −0.284447
$$792$$ 0 0
$$793$$ − 36.0000i − 1.27840i
$$794$$ 22.0000 0.780751
$$795$$ 0 0
$$796$$ 16.0000 0.567105
$$797$$ 30.0000i 1.06265i 0.847167 + 0.531327i $$0.178307\pi$$
−0.847167 + 0.531327i $$0.821693\pi$$
$$798$$ − 32.0000i − 1.13279i
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 2.00000 0.0706665
$$802$$ − 18.0000i − 0.635602i
$$803$$ 0 0
$$804$$ −8.00000 −0.282138
$$805$$ 0 0
$$806$$ −24.0000 −0.845364
$$807$$ − 18.0000i − 0.633630i
$$808$$ 30.0000i 1.05540i
$$809$$ 30.0000 1.05474 0.527372 0.849635i $$-0.323177\pi$$
0.527372 + 0.849635i $$0.323177\pi$$
$$810$$ 0 0
$$811$$ −28.0000 −0.983213 −0.491606 0.870817i $$-0.663590\pi$$
−0.491606 + 0.870817i $$0.663590\pi$$
$$812$$ 4.00000i 0.140372i
$$813$$ 20.0000i 0.701431i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ −2.00000 −0.0700140
$$817$$ − 32.0000i − 1.11954i
$$818$$ − 6.00000i − 0.209785i
$$819$$ 24.0000 0.838628
$$820$$ 0 0
$$821$$ 22.0000 0.767805 0.383903 0.923374i $$-0.374580\pi$$
0.383903 + 0.923374i $$0.374580\pi$$
$$822$$ 6.00000i 0.209274i
$$823$$ 8.00000i 0.278862i 0.990232 + 0.139431i $$0.0445274\pi$$
−0.990232 + 0.139431i $$0.955473\pi$$
$$824$$ −12.0000 −0.418040
$$825$$ 0 0
$$826$$ −48.0000 −1.67013
$$827$$ 12.0000i 0.417281i 0.977992 + 0.208640i $$0.0669038\pi$$
−0.977992 + 0.208640i $$0.933096\pi$$
$$828$$ − 4.00000i − 0.139010i
$$829$$ 26.0000 0.903017 0.451509 0.892267i $$-0.350886\pi$$
0.451509 + 0.892267i $$0.350886\pi$$
$$830$$ 0 0
$$831$$ −2.00000 −0.0693792
$$832$$ 42.0000i 1.45609i
$$833$$ − 18.0000i − 0.623663i
$$834$$ 20.0000 0.692543
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 4.00000i 0.138260i
$$838$$ 4.00000i 0.138178i
$$839$$ −20.0000 −0.690477 −0.345238 0.938515i $$-0.612202\pi$$
−0.345238 + 0.938515i $$0.612202\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ − 30.0000i − 1.03387i
$$843$$ 6.00000i 0.206651i
$$844$$ 8.00000 0.275371
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 44.0000i 1.51186i
$$848$$ 6.00000i 0.206041i
$$849$$ −8.00000 −0.274559
$$850$$ 0 0
$$851$$ −24.0000 −0.822709
$$852$$ − 16.0000i − 0.548151i
$$853$$ 26.0000i 0.890223i 0.895475 + 0.445112i $$0.146836\pi$$
−0.895475 + 0.445112i $$0.853164\pi$$
$$854$$ −24.0000 −0.821263
$$855$$ 0 0
$$856$$ −48.0000 −1.64061
$$857$$ 18.0000i 0.614868i 0.951569 + 0.307434i $$0.0994704\pi$$
−0.951569 + 0.307434i $$0.900530\pi$$
$$858$$ 0 0
$$859$$ −8.00000 −0.272956 −0.136478 0.990643i $$-0.543578\pi$$
−0.136478 + 0.990643i $$0.543578\pi$$
$$860$$ 0 0
$$861$$ −8.00000 −0.272639
$$862$$ − 32.0000i − 1.08992i
$$863$$ 52.0000i 1.77010i 0.465495 + 0.885050i $$0.345876\pi$$
−0.465495 + 0.885050i $$0.654124\pi$$
$$864$$ 5.00000 0.170103
$$865$$ 0 0
$$866$$ 14.0000 0.475739
$$867$$ − 13.0000i − 0.441503i
$$868$$ − 16.0000i − 0.543075i
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −48.0000 −1.62642
$$872$$ − 6.00000i − 0.203186i
$$873$$ 14.0000i 0.473828i
$$874$$ −32.0000 −1.08242
$$875$$ 0 0
$$876$$ 6.00000 0.202721
$$877$$ 22.0000i 0.742887i 0.928456 + 0.371444i $$0.121137\pi$$
−0.928456 + 0.371444i $$0.878863\pi$$
$$878$$ − 24.0000i − 0.809961i
$$879$$ 26.0000 0.876958
$$880$$ 0 0
$$881$$ 42.0000 1.41502 0.707508 0.706705i $$-0.249819\pi$$
0.707508 + 0.706705i $$0.249819\pi$$
$$882$$ − 9.00000i − 0.303046i
$$883$$ − 16.0000i − 0.538443i −0.963078 0.269221i $$-0.913234\pi$$
0.963078 0.269221i $$-0.0867663\pi$$
$$884$$ 12.0000 0.403604
$$885$$ 0 0
$$886$$ −20.0000 −0.671913
$$887$$ − 32.0000i − 1.07445i −0.843437 0.537227i $$-0.819472\pi$$
0.843437 0.537227i $$-0.180528\pi$$
$$888$$ − 18.0000i − 0.604040i
$$889$$ −32.0000 −1.07325
$$890$$ 0 0
$$891$$ 0 0
$$892$$ − 12.0000i − 0.401790i
$$893$$ 0 0
$$894$$ −10.0000 −0.334450
$$895$$ 0 0
$$896$$ −12.0000 −0.400892
$$897$$ − 24.0000i − 0.801337i
$$898$$ − 6.00000i − 0.200223i
$$899$$ −4.00000 −0.133407
$$900$$ 0 0
$$901$$ 12.0000 0.399778
$$902$$ 0 0
$$903$$ − 16.0000i − 0.532447i
$$904$$ −6.00000 −0.199557
$$905$$ 0 0
$$906$$ −16.0000 −0.531564
$$907$$ 28.0000i 0.929725i 0.885383 + 0.464862i $$0.153896\pi$$
−0.885383 + 0.464862i $$0.846104\pi$$
$$908$$ 8.00000i 0.265489i
$$909$$ 10.0000 0.331679
$$910$$ 0 0
$$911$$ 20.0000 0.662630 0.331315 0.943520i $$-0.392508\pi$$
0.331315 + 0.943520i $$0.392508\pi$$
$$912$$ − 8.00000i − 0.264906i
$$913$$ 0 0
$$914$$ −22.0000 −0.727695
$$915$$ 0 0
$$916$$ −14.0000 −0.462573
$$917$$ 0 0
$$918$$ 2.00000i 0.0660098i
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 0 0
$$921$$ −20.0000 −0.659022
$$922$$ 18.0000i 0.592798i
$$923$$ − 96.0000i − 3.15988i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −36.0000 −1.18303
$$927$$ 4.00000i 0.131377i
$$928$$ 5.00000i 0.164133i
$$929$$ −18.0000 −0.590561 −0.295280 0.955411i $$-0.595413\pi$$
−0.295280 + 0.955411i $$0.595413\pi$$
$$930$$ 0 0
$$931$$ 72.0000 2.35970
$$932$$ − 18.0000i − 0.589610i
$$933$$ 28.0000i 0.916679i
$$934$$ 12.0000 0.392652
$$935$$ 0 0
$$936$$ 18.0000 0.588348
$$937$$ − 38.0000i − 1.24141i −0.784046 0.620703i $$-0.786847\pi$$
0.784046 0.620703i $$-0.213153\pi$$
$$938$$ 32.0000i 1.04484i
$$939$$ −10.0000 −0.326338
$$940$$ 0 0
$$941$$ −18.0000 −0.586783 −0.293392 0.955992i $$-0.594784\pi$$
−0.293392 + 0.955992i $$0.594784\pi$$
$$942$$ − 14.0000i − 0.456145i
$$943$$ 8.00000i 0.260516i
$$944$$ −12.0000 −0.390567
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 28.0000i 0.909878i 0.890523 + 0.454939i $$0.150339\pi$$
−0.890523 + 0.454939i $$0.849661\pi$$
$$948$$ 12.0000i 0.389742i
$$949$$ 36.0000 1.16861
$$950$$ 0 0
$$951$$ 30.0000 0.972817
$$952$$ − 24.0000i − 0.777844i
$$953$$ − 18.0000i − 0.583077i −0.956559 0.291539i $$-0.905833\pi$$
0.956559 0.291539i $$-0.0941672\pi$$
$$954$$ 6.00000 0.194257
$$955$$ 0 0
$$956$$ −24.0000 −0.776215
$$957$$ 0 0
$$958$$ 12.0000i 0.387702i
$$959$$ −24.0000 −0.775000
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ − 36.0000i − 1.16069i
$$963$$ 16.0000i 0.515593i
$$964$$ 2.00000 0.0644157
$$965$$ 0 0
$$966$$ −16.0000 −0.514792
$$967$$ 56.0000i 1.80084i 0.435023 + 0.900419i $$0.356740\pi$$
−0.435023 + 0.900419i $$0.643260\pi$$
$$968$$ 33.0000i 1.06066i
$$969$$ −16.0000 −0.513994
$$970$$ 0 0
$$971$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$972$$ − 1.00000i − 0.0320750i
$$973$$ 80.0000i 2.56468i
$$974$$ −20.0000 −0.640841
$$975$$ 0 0
$$976$$ −6.00000 −0.192055
$$977$$ − 6.00000i − 0.191957i −0.995383 0.0959785i $$-0.969402\pi$$
0.995383 0.0959785i $$-0.0305980\pi$$
$$978$$ 12.0000i 0.383718i
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −2.00000 −0.0638551
$$982$$ 24.0000i 0.765871i
$$983$$ 16.0000i 0.510321i 0.966899 + 0.255160i $$0.0821283\pi$$
−0.966899 + 0.255160i $$0.917872\pi$$
$$984$$ −6.00000 −0.191273
$$985$$ 0 0
$$986$$ −2.00000 −0.0636930
$$987$$ 0 0
$$988$$ 48.0000i 1.52708i
$$989$$ −16.0000 −0.508770
$$990$$ 0 0
$$991$$ −8.00000 −0.254128 −0.127064 0.991894i $$-0.540555\pi$$
−0.127064 + 0.991894i $$0.540555\pi$$
$$992$$ − 20.0000i − 0.635001i
$$993$$ 16.0000i 0.507745i
$$994$$ −64.0000 −2.02996
$$995$$ 0 0
$$996$$ 16.0000 0.506979
$$997$$ 38.0000i 1.20347i 0.798695 + 0.601736i $$0.205524\pi$$
−0.798695 + 0.601736i $$0.794476\pi$$
$$998$$ 36.0000i 1.13956i
$$999$$ −6.00000 −0.189832
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2175.2.c.a.349.1 2
5.2 odd 4 2175.2.a.h.1.1 1
5.3 odd 4 435.2.a.a.1.1 1
5.4 even 2 inner 2175.2.c.a.349.2 2
15.2 even 4 6525.2.a.e.1.1 1
15.8 even 4 1305.2.a.e.1.1 1
20.3 even 4 6960.2.a.w.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.a.1.1 1 5.3 odd 4
1305.2.a.e.1.1 1 15.8 even 4
2175.2.a.h.1.1 1 5.2 odd 4
2175.2.c.a.349.1 2 1.1 even 1 trivial
2175.2.c.a.349.2 2 5.4 even 2 inner
6525.2.a.e.1.1 1 15.2 even 4
6960.2.a.w.1.1 1 20.3 even 4