# Properties

 Label 975.2.c.b.274.1 Level $975$ Weight $2$ Character 975.274 Analytic conductor $7.785$ Analytic rank $1$ 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] = [975,2,Mod(274,975)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(975, 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("975.274");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$975 = 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 975.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: $$7.78541419707$$ Analytic rank: $$1$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 195) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 274.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 975.274 Dual form 975.2.c.b.274.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000i q^{2} +1.00000i q^{3} -2.00000 q^{4} +2.00000 q^{6} +3.00000i q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q-2.00000i q^{2} +1.00000i q^{3} -2.00000 q^{4} +2.00000 q^{6} +3.00000i q^{7} -1.00000 q^{9} -5.00000 q^{11} -2.00000i q^{12} +1.00000i q^{13} +6.00000 q^{14} -4.00000 q^{16} -5.00000i q^{17} +2.00000i q^{18} -2.00000 q^{19} -3.00000 q^{21} +10.0000i q^{22} -1.00000i q^{23} +2.00000 q^{26} -1.00000i q^{27} -6.00000i q^{28} -10.0000 q^{29} -2.00000 q^{31} +8.00000i q^{32} -5.00000i q^{33} -10.0000 q^{34} +2.00000 q^{36} +3.00000i q^{37} +4.00000i q^{38} -1.00000 q^{39} -9.00000 q^{41} +6.00000i q^{42} -4.00000i q^{43} +10.0000 q^{44} -2.00000 q^{46} -10.0000i q^{47} -4.00000i q^{48} -2.00000 q^{49} +5.00000 q^{51} -2.00000i q^{52} +9.00000i q^{53} -2.00000 q^{54} -2.00000i q^{57} +20.0000i q^{58} -11.0000 q^{61} +4.00000i q^{62} -3.00000i q^{63} +8.00000 q^{64} -10.0000 q^{66} +4.00000i q^{67} +10.0000i q^{68} +1.00000 q^{69} +15.0000 q^{71} +6.00000i q^{73} +6.00000 q^{74} +4.00000 q^{76} -15.0000i q^{77} +2.00000i q^{78} +11.0000 q^{79} +1.00000 q^{81} +18.0000i q^{82} +8.00000i q^{83} +6.00000 q^{84} -8.00000 q^{86} -10.0000i q^{87} +11.0000 q^{89} -3.00000 q^{91} +2.00000i q^{92} -2.00000i q^{93} -20.0000 q^{94} -8.00000 q^{96} +9.00000i q^{97} +4.00000i q^{98} +5.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{4} + 4 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 4 * q^4 + 4 * q^6 - 2 * q^9 $$2 q - 4 q^{4} + 4 q^{6} - 2 q^{9} - 10 q^{11} + 12 q^{14} - 8 q^{16} - 4 q^{19} - 6 q^{21} + 4 q^{26} - 20 q^{29} - 4 q^{31} - 20 q^{34} + 4 q^{36} - 2 q^{39} - 18 q^{41} + 20 q^{44} - 4 q^{46} - 4 q^{49} + 10 q^{51} - 4 q^{54} - 22 q^{61} + 16 q^{64} - 20 q^{66} + 2 q^{69} + 30 q^{71} + 12 q^{74} + 8 q^{76} + 22 q^{79} + 2 q^{81} + 12 q^{84} - 16 q^{86} + 22 q^{89} - 6 q^{91} - 40 q^{94} - 16 q^{96} + 10 q^{99}+O(q^{100})$$ 2 * q - 4 * q^4 + 4 * q^6 - 2 * q^9 - 10 * q^11 + 12 * q^14 - 8 * q^16 - 4 * q^19 - 6 * q^21 + 4 * q^26 - 20 * q^29 - 4 * q^31 - 20 * q^34 + 4 * q^36 - 2 * q^39 - 18 * q^41 + 20 * q^44 - 4 * q^46 - 4 * q^49 + 10 * q^51 - 4 * q^54 - 22 * q^61 + 16 * q^64 - 20 * q^66 + 2 * q^69 + 30 * q^71 + 12 * q^74 + 8 * q^76 + 22 * q^79 + 2 * q^81 + 12 * q^84 - 16 * q^86 + 22 * q^89 - 6 * q^91 - 40 * q^94 - 16 * q^96 + 10 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/975\mathbb{Z}\right)^\times$$.

 $$n$$ $$301$$ $$326$$ $$352$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 2.00000i − 1.41421i −0.707107 0.707107i $$-0.750000\pi$$
0.707107 0.707107i $$-0.250000\pi$$
$$3$$ 1.00000i 0.577350i
$$4$$ −2.00000 −1.00000
$$5$$ 0 0
$$6$$ 2.00000 0.816497
$$7$$ 3.00000i 1.13389i 0.823754 + 0.566947i $$0.191875\pi$$
−0.823754 + 0.566947i $$0.808125\pi$$
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −5.00000 −1.50756 −0.753778 0.657129i $$-0.771771\pi$$
−0.753778 + 0.657129i $$0.771771\pi$$
$$12$$ − 2.00000i − 0.577350i
$$13$$ 1.00000i 0.277350i
$$14$$ 6.00000 1.60357
$$15$$ 0 0
$$16$$ −4.00000 −1.00000
$$17$$ − 5.00000i − 1.21268i −0.795206 0.606339i $$-0.792637\pi$$
0.795206 0.606339i $$-0.207363\pi$$
$$18$$ 2.00000i 0.471405i
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ 0 0
$$21$$ −3.00000 −0.654654
$$22$$ 10.0000i 2.13201i
$$23$$ − 1.00000i − 0.208514i −0.994550 0.104257i $$-0.966753\pi$$
0.994550 0.104257i $$-0.0332465\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 2.00000 0.392232
$$27$$ − 1.00000i − 0.192450i
$$28$$ − 6.00000i − 1.13389i
$$29$$ −10.0000 −1.85695 −0.928477 0.371391i $$-0.878881\pi$$
−0.928477 + 0.371391i $$0.878881\pi$$
$$30$$ 0 0
$$31$$ −2.00000 −0.359211 −0.179605 0.983739i $$-0.557482\pi$$
−0.179605 + 0.983739i $$0.557482\pi$$
$$32$$ 8.00000i 1.41421i
$$33$$ − 5.00000i − 0.870388i
$$34$$ −10.0000 −1.71499
$$35$$ 0 0
$$36$$ 2.00000 0.333333
$$37$$ 3.00000i 0.493197i 0.969118 + 0.246598i $$0.0793129\pi$$
−0.969118 + 0.246598i $$0.920687\pi$$
$$38$$ 4.00000i 0.648886i
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ −9.00000 −1.40556 −0.702782 0.711405i $$-0.748059\pi$$
−0.702782 + 0.711405i $$0.748059\pi$$
$$42$$ 6.00000i 0.925820i
$$43$$ − 4.00000i − 0.609994i −0.952353 0.304997i $$-0.901344\pi$$
0.952353 0.304997i $$-0.0986555\pi$$
$$44$$ 10.0000 1.50756
$$45$$ 0 0
$$46$$ −2.00000 −0.294884
$$47$$ − 10.0000i − 1.45865i −0.684167 0.729325i $$-0.739834\pi$$
0.684167 0.729325i $$-0.260166\pi$$
$$48$$ − 4.00000i − 0.577350i
$$49$$ −2.00000 −0.285714
$$50$$ 0 0
$$51$$ 5.00000 0.700140
$$52$$ − 2.00000i − 0.277350i
$$53$$ 9.00000i 1.23625i 0.786082 + 0.618123i $$0.212106\pi$$
−0.786082 + 0.618123i $$0.787894\pi$$
$$54$$ −2.00000 −0.272166
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 2.00000i − 0.264906i
$$58$$ 20.0000i 2.62613i
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ −11.0000 −1.40841 −0.704203 0.709999i $$-0.748695\pi$$
−0.704203 + 0.709999i $$0.748695\pi$$
$$62$$ 4.00000i 0.508001i
$$63$$ − 3.00000i − 0.377964i
$$64$$ 8.00000 1.00000
$$65$$ 0 0
$$66$$ −10.0000 −1.23091
$$67$$ 4.00000i 0.488678i 0.969690 + 0.244339i $$0.0785709\pi$$
−0.969690 + 0.244339i $$0.921429\pi$$
$$68$$ 10.0000i 1.21268i
$$69$$ 1.00000 0.120386
$$70$$ 0 0
$$71$$ 15.0000 1.78017 0.890086 0.455792i $$-0.150644\pi$$
0.890086 + 0.455792i $$0.150644\pi$$
$$72$$ 0 0
$$73$$ 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
$$76$$ 4.00000 0.458831
$$77$$ − 15.0000i − 1.70941i
$$78$$ 2.00000i 0.226455i
$$79$$ 11.0000 1.23760 0.618798 0.785550i $$-0.287620\pi$$
0.618798 + 0.785550i $$0.287620\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 18.0000i 1.98777i
$$83$$ 8.00000i 0.878114i 0.898459 + 0.439057i $$0.144687\pi$$
−0.898459 + 0.439057i $$0.855313\pi$$
$$84$$ 6.00000 0.654654
$$85$$ 0 0
$$86$$ −8.00000 −0.862662
$$87$$ − 10.0000i − 1.07211i
$$88$$ 0 0
$$89$$ 11.0000 1.16600 0.582999 0.812473i $$-0.301879\pi$$
0.582999 + 0.812473i $$0.301879\pi$$
$$90$$ 0 0
$$91$$ −3.00000 −0.314485
$$92$$ 2.00000i 0.208514i
$$93$$ − 2.00000i − 0.207390i
$$94$$ −20.0000 −2.06284
$$95$$ 0 0
$$96$$ −8.00000 −0.816497
$$97$$ 9.00000i 0.913812i 0.889515 + 0.456906i $$0.151042\pi$$
−0.889515 + 0.456906i $$0.848958\pi$$
$$98$$ 4.00000i 0.404061i
$$99$$ 5.00000 0.502519
$$100$$ 0 0
$$101$$ −12.0000 −1.19404 −0.597022 0.802225i $$-0.703650\pi$$
−0.597022 + 0.802225i $$0.703650\pi$$
$$102$$ − 10.0000i − 0.990148i
$$103$$ 4.00000i 0.394132i 0.980390 + 0.197066i $$0.0631413\pi$$
−0.980390 + 0.197066i $$0.936859\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 18.0000 1.74831
$$107$$ − 3.00000i − 0.290021i −0.989430 0.145010i $$-0.953678\pi$$
0.989430 0.145010i $$-0.0463216\pi$$
$$108$$ 2.00000i 0.192450i
$$109$$ −16.0000 −1.53252 −0.766261 0.642529i $$-0.777885\pi$$
−0.766261 + 0.642529i $$0.777885\pi$$
$$110$$ 0 0
$$111$$ −3.00000 −0.284747
$$112$$ − 12.0000i − 1.13389i
$$113$$ 2.00000i 0.188144i 0.995565 + 0.0940721i $$0.0299884\pi$$
−0.995565 + 0.0940721i $$0.970012\pi$$
$$114$$ −4.00000 −0.374634
$$115$$ 0 0
$$116$$ 20.0000 1.85695
$$117$$ − 1.00000i − 0.0924500i
$$118$$ 0 0
$$119$$ 15.0000 1.37505
$$120$$ 0 0
$$121$$ 14.0000 1.27273
$$122$$ 22.0000i 1.99179i
$$123$$ − 9.00000i − 0.811503i
$$124$$ 4.00000 0.359211
$$125$$ 0 0
$$126$$ −6.00000 −0.534522
$$127$$ − 14.0000i − 1.24230i −0.783692 0.621150i $$-0.786666\pi$$
0.783692 0.621150i $$-0.213334\pi$$
$$128$$ 0 0
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ 6.00000 0.524222 0.262111 0.965038i $$-0.415581\pi$$
0.262111 + 0.965038i $$0.415581\pi$$
$$132$$ 10.0000i 0.870388i
$$133$$ − 6.00000i − 0.520266i
$$134$$ 8.00000 0.691095
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 6.00000i − 0.512615i −0.966595 0.256307i $$-0.917494\pi$$
0.966595 0.256307i $$-0.0825059\pi$$
$$138$$ − 2.00000i − 0.170251i
$$139$$ 17.0000 1.44192 0.720961 0.692976i $$-0.243701\pi$$
0.720961 + 0.692976i $$0.243701\pi$$
$$140$$ 0 0
$$141$$ 10.0000 0.842152
$$142$$ − 30.0000i − 2.51754i
$$143$$ − 5.00000i − 0.418121i
$$144$$ 4.00000 0.333333
$$145$$ 0 0
$$146$$ 12.0000 0.993127
$$147$$ − 2.00000i − 0.164957i
$$148$$ − 6.00000i − 0.493197i
$$149$$ 7.00000 0.573462 0.286731 0.958011i $$-0.407431\pi$$
0.286731 + 0.958011i $$0.407431\pi$$
$$150$$ 0 0
$$151$$ −12.0000 −0.976546 −0.488273 0.872691i $$-0.662373\pi$$
−0.488273 + 0.872691i $$0.662373\pi$$
$$152$$ 0 0
$$153$$ 5.00000i 0.404226i
$$154$$ −30.0000 −2.41747
$$155$$ 0 0
$$156$$ 2.00000 0.160128
$$157$$ − 22.0000i − 1.75579i −0.478852 0.877896i $$-0.658947\pi$$
0.478852 0.877896i $$-0.341053\pi$$
$$158$$ − 22.0000i − 1.75023i
$$159$$ −9.00000 −0.713746
$$160$$ 0 0
$$161$$ 3.00000 0.236433
$$162$$ − 2.00000i − 0.157135i
$$163$$ − 11.0000i − 0.861586i −0.902451 0.430793i $$-0.858234\pi$$
0.902451 0.430793i $$-0.141766\pi$$
$$164$$ 18.0000 1.40556
$$165$$ 0 0
$$166$$ 16.0000 1.24184
$$167$$ 8.00000i 0.619059i 0.950890 + 0.309529i $$0.100171\pi$$
−0.950890 + 0.309529i $$0.899829\pi$$
$$168$$ 0 0
$$169$$ −1.00000 −0.0769231
$$170$$ 0 0
$$171$$ 2.00000 0.152944
$$172$$ 8.00000i 0.609994i
$$173$$ − 2.00000i − 0.152057i −0.997106 0.0760286i $$-0.975776\pi$$
0.997106 0.0760286i $$-0.0242240\pi$$
$$174$$ −20.0000 −1.51620
$$175$$ 0 0
$$176$$ 20.0000 1.50756
$$177$$ 0 0
$$178$$ − 22.0000i − 1.64897i
$$179$$ 6.00000 0.448461 0.224231 0.974536i $$-0.428013\pi$$
0.224231 + 0.974536i $$0.428013\pi$$
$$180$$ 0 0
$$181$$ −23.0000 −1.70958 −0.854788 0.518977i $$-0.826313\pi$$
−0.854788 + 0.518977i $$0.826313\pi$$
$$182$$ 6.00000i 0.444750i
$$183$$ − 11.0000i − 0.813143i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ −4.00000 −0.293294
$$187$$ 25.0000i 1.82818i
$$188$$ 20.0000i 1.45865i
$$189$$ 3.00000 0.218218
$$190$$ 0 0
$$191$$ −20.0000 −1.44715 −0.723575 0.690246i $$-0.757502\pi$$
−0.723575 + 0.690246i $$0.757502\pi$$
$$192$$ 8.00000i 0.577350i
$$193$$ 13.0000i 0.935760i 0.883792 + 0.467880i $$0.154982\pi$$
−0.883792 + 0.467880i $$0.845018\pi$$
$$194$$ 18.0000 1.29232
$$195$$ 0 0
$$196$$ 4.00000 0.285714
$$197$$ 12.0000i 0.854965i 0.904024 + 0.427482i $$0.140599\pi$$
−0.904024 + 0.427482i $$0.859401\pi$$
$$198$$ − 10.0000i − 0.710669i
$$199$$ −4.00000 −0.283552 −0.141776 0.989899i $$-0.545281\pi$$
−0.141776 + 0.989899i $$0.545281\pi$$
$$200$$ 0 0
$$201$$ −4.00000 −0.282138
$$202$$ 24.0000i 1.68863i
$$203$$ − 30.0000i − 2.10559i
$$204$$ −10.0000 −0.700140
$$205$$ 0 0
$$206$$ 8.00000 0.557386
$$207$$ 1.00000i 0.0695048i
$$208$$ − 4.00000i − 0.277350i
$$209$$ 10.0000 0.691714
$$210$$ 0 0
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ − 18.0000i − 1.23625i
$$213$$ 15.0000i 1.02778i
$$214$$ −6.00000 −0.410152
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 6.00000i − 0.407307i
$$218$$ 32.0000i 2.16731i
$$219$$ −6.00000 −0.405442
$$220$$ 0 0
$$221$$ 5.00000 0.336336
$$222$$ 6.00000i 0.402694i
$$223$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$224$$ −24.0000 −1.60357
$$225$$ 0 0
$$226$$ 4.00000 0.266076
$$227$$ 18.0000i 1.19470i 0.801980 + 0.597351i $$0.203780\pi$$
−0.801980 + 0.597351i $$0.796220\pi$$
$$228$$ 4.00000i 0.264906i
$$229$$ 14.0000 0.925146 0.462573 0.886581i $$-0.346926\pi$$
0.462573 + 0.886581i $$0.346926\pi$$
$$230$$ 0 0
$$231$$ 15.0000 0.986928
$$232$$ 0 0
$$233$$ − 25.0000i − 1.63780i −0.573933 0.818902i $$-0.694583\pi$$
0.573933 0.818902i $$-0.305417\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 11.0000i 0.714527i
$$238$$ − 30.0000i − 1.94461i
$$239$$ −15.0000 −0.970269 −0.485135 0.874439i $$-0.661229\pi$$
−0.485135 + 0.874439i $$0.661229\pi$$
$$240$$ 0 0
$$241$$ −14.0000 −0.901819 −0.450910 0.892570i $$-0.648900\pi$$
−0.450910 + 0.892570i $$0.648900\pi$$
$$242$$ − 28.0000i − 1.79991i
$$243$$ 1.00000i 0.0641500i
$$244$$ 22.0000 1.40841
$$245$$ 0 0
$$246$$ −18.0000 −1.14764
$$247$$ − 2.00000i − 0.127257i
$$248$$ 0 0
$$249$$ −8.00000 −0.506979
$$250$$ 0 0
$$251$$ 20.0000 1.26239 0.631194 0.775625i $$-0.282565\pi$$
0.631194 + 0.775625i $$0.282565\pi$$
$$252$$ 6.00000i 0.377964i
$$253$$ 5.00000i 0.314347i
$$254$$ −28.0000 −1.75688
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ − 18.0000i − 1.12281i −0.827541 0.561405i $$-0.810261\pi$$
0.827541 0.561405i $$-0.189739\pi$$
$$258$$ − 8.00000i − 0.498058i
$$259$$ −9.00000 −0.559233
$$260$$ 0 0
$$261$$ 10.0000 0.618984
$$262$$ − 12.0000i − 0.741362i
$$263$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −12.0000 −0.735767
$$267$$ 11.0000i 0.673189i
$$268$$ − 8.00000i − 0.488678i
$$269$$ −32.0000 −1.95107 −0.975537 0.219834i $$-0.929448\pi$$
−0.975537 + 0.219834i $$0.929448\pi$$
$$270$$ 0 0
$$271$$ −2.00000 −0.121491 −0.0607457 0.998153i $$-0.519348\pi$$
−0.0607457 + 0.998153i $$0.519348\pi$$
$$272$$ 20.0000i 1.21268i
$$273$$ − 3.00000i − 0.181568i
$$274$$ −12.0000 −0.724947
$$275$$ 0 0
$$276$$ −2.00000 −0.120386
$$277$$ − 26.0000i − 1.56219i −0.624413 0.781094i $$-0.714662\pi$$
0.624413 0.781094i $$-0.285338\pi$$
$$278$$ − 34.0000i − 2.03918i
$$279$$ 2.00000 0.119737
$$280$$ 0 0
$$281$$ −10.0000 −0.596550 −0.298275 0.954480i $$-0.596411\pi$$
−0.298275 + 0.954480i $$0.596411\pi$$
$$282$$ − 20.0000i − 1.19098i
$$283$$ − 8.00000i − 0.475551i −0.971320 0.237775i $$-0.923582\pi$$
0.971320 0.237775i $$-0.0764182\pi$$
$$284$$ −30.0000 −1.78017
$$285$$ 0 0
$$286$$ −10.0000 −0.591312
$$287$$ − 27.0000i − 1.59376i
$$288$$ − 8.00000i − 0.471405i
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ −9.00000 −0.527589
$$292$$ − 12.0000i − 0.702247i
$$293$$ 24.0000i 1.40209i 0.713115 + 0.701047i $$0.247284\pi$$
−0.713115 + 0.701047i $$0.752716\pi$$
$$294$$ −4.00000 −0.233285
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 5.00000i 0.290129i
$$298$$ − 14.0000i − 0.810998i
$$299$$ 1.00000 0.0578315
$$300$$ 0 0
$$301$$ 12.0000 0.691669
$$302$$ 24.0000i 1.38104i
$$303$$ − 12.0000i − 0.689382i
$$304$$ 8.00000 0.458831
$$305$$ 0 0
$$306$$ 10.0000 0.571662
$$307$$ 19.0000i 1.08439i 0.840254 + 0.542194i $$0.182406\pi$$
−0.840254 + 0.542194i $$0.817594\pi$$
$$308$$ 30.0000i 1.70941i
$$309$$ −4.00000 −0.227552
$$310$$ 0 0
$$311$$ 24.0000 1.36092 0.680458 0.732787i $$-0.261781\pi$$
0.680458 + 0.732787i $$0.261781\pi$$
$$312$$ 0 0
$$313$$ − 10.0000i − 0.565233i −0.959233 0.282617i $$-0.908798\pi$$
0.959233 0.282617i $$-0.0912024\pi$$
$$314$$ −44.0000 −2.48306
$$315$$ 0 0
$$316$$ −22.0000 −1.23760
$$317$$ − 12.0000i − 0.673987i −0.941507 0.336994i $$-0.890590\pi$$
0.941507 0.336994i $$-0.109410\pi$$
$$318$$ 18.0000i 1.00939i
$$319$$ 50.0000 2.79946
$$320$$ 0 0
$$321$$ 3.00000 0.167444
$$322$$ − 6.00000i − 0.334367i
$$323$$ 10.0000i 0.556415i
$$324$$ −2.00000 −0.111111
$$325$$ 0 0
$$326$$ −22.0000 −1.21847
$$327$$ − 16.0000i − 0.884802i
$$328$$ 0 0
$$329$$ 30.0000 1.65395
$$330$$ 0 0
$$331$$ 32.0000 1.75888 0.879440 0.476011i $$-0.157918\pi$$
0.879440 + 0.476011i $$0.157918\pi$$
$$332$$ − 16.0000i − 0.878114i
$$333$$ − 3.00000i − 0.164399i
$$334$$ 16.0000 0.875481
$$335$$ 0 0
$$336$$ 12.0000 0.654654
$$337$$ − 4.00000i − 0.217894i −0.994048 0.108947i $$-0.965252\pi$$
0.994048 0.108947i $$-0.0347479\pi$$
$$338$$ 2.00000i 0.108786i
$$339$$ −2.00000 −0.108625
$$340$$ 0 0
$$341$$ 10.0000 0.541530
$$342$$ − 4.00000i − 0.216295i
$$343$$ 15.0000i 0.809924i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ −4.00000 −0.215041
$$347$$ 1.00000i 0.0536828i 0.999640 + 0.0268414i $$0.00854491\pi$$
−0.999640 + 0.0268414i $$0.991455\pi$$
$$348$$ 20.0000i 1.07211i
$$349$$ −20.0000 −1.07058 −0.535288 0.844670i $$-0.679797\pi$$
−0.535288 + 0.844670i $$0.679797\pi$$
$$350$$ 0 0
$$351$$ 1.00000 0.0533761
$$352$$ − 40.0000i − 2.13201i
$$353$$ − 14.0000i − 0.745145i −0.928003 0.372572i $$-0.878476\pi$$
0.928003 0.372572i $$-0.121524\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −22.0000 −1.16600
$$357$$ 15.0000i 0.793884i
$$358$$ − 12.0000i − 0.634220i
$$359$$ −16.0000 −0.844448 −0.422224 0.906492i $$-0.638750\pi$$
−0.422224 + 0.906492i $$0.638750\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 46.0000i 2.41771i
$$363$$ 14.0000i 0.734809i
$$364$$ 6.00000 0.314485
$$365$$ 0 0
$$366$$ −22.0000 −1.14996
$$367$$ − 8.00000i − 0.417597i −0.977959 0.208798i $$-0.933045\pi$$
0.977959 0.208798i $$-0.0669552\pi$$
$$368$$ 4.00000i 0.208514i
$$369$$ 9.00000 0.468521
$$370$$ 0 0
$$371$$ −27.0000 −1.40177
$$372$$ 4.00000i 0.207390i
$$373$$ 16.0000i 0.828449i 0.910175 + 0.414224i $$0.135947\pi$$
−0.910175 + 0.414224i $$0.864053\pi$$
$$374$$ 50.0000 2.58544
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 10.0000i − 0.515026i
$$378$$ − 6.00000i − 0.308607i
$$379$$ −6.00000 −0.308199 −0.154100 0.988055i $$-0.549248\pi$$
−0.154100 + 0.988055i $$0.549248\pi$$
$$380$$ 0 0
$$381$$ 14.0000 0.717242
$$382$$ 40.0000i 2.04658i
$$383$$ − 18.0000i − 0.919757i −0.887982 0.459879i $$-0.847893\pi$$
0.887982 0.459879i $$-0.152107\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 26.0000 1.32337
$$387$$ 4.00000i 0.203331i
$$388$$ − 18.0000i − 0.913812i
$$389$$ 24.0000 1.21685 0.608424 0.793612i $$-0.291802\pi$$
0.608424 + 0.793612i $$0.291802\pi$$
$$390$$ 0 0
$$391$$ −5.00000 −0.252861
$$392$$ 0 0
$$393$$ 6.00000i 0.302660i
$$394$$ 24.0000 1.20910
$$395$$ 0 0
$$396$$ −10.0000 −0.502519
$$397$$ 19.0000i 0.953583i 0.879017 + 0.476791i $$0.158200\pi$$
−0.879017 + 0.476791i $$0.841800\pi$$
$$398$$ 8.00000i 0.401004i
$$399$$ 6.00000 0.300376
$$400$$ 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
$$403$$ − 2.00000i − 0.0996271i
$$404$$ 24.0000 1.19404
$$405$$ 0 0
$$406$$ −60.0000 −2.97775
$$407$$ − 15.0000i − 0.743522i
$$408$$ 0 0
$$409$$ 26.0000 1.28562 0.642809 0.766027i $$-0.277769\pi$$
0.642809 + 0.766027i $$0.277769\pi$$
$$410$$ 0 0
$$411$$ 6.00000 0.295958
$$412$$ − 8.00000i − 0.394132i
$$413$$ 0 0
$$414$$ 2.00000 0.0982946
$$415$$ 0 0
$$416$$ −8.00000 −0.392232
$$417$$ 17.0000i 0.832494i
$$418$$ − 20.0000i − 0.978232i
$$419$$ −26.0000 −1.27018 −0.635092 0.772437i $$-0.719038\pi$$
−0.635092 + 0.772437i $$0.719038\pi$$
$$420$$ 0 0
$$421$$ 32.0000 1.55958 0.779792 0.626038i $$-0.215325\pi$$
0.779792 + 0.626038i $$0.215325\pi$$
$$422$$ 8.00000i 0.389434i
$$423$$ 10.0000i 0.486217i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 30.0000 1.45350
$$427$$ − 33.0000i − 1.59698i
$$428$$ 6.00000i 0.290021i
$$429$$ 5.00000 0.241402
$$430$$ 0 0
$$431$$ −16.0000 −0.770693 −0.385346 0.922772i $$-0.625918\pi$$
−0.385346 + 0.922772i $$0.625918\pi$$
$$432$$ 4.00000i 0.192450i
$$433$$ 24.0000i 1.15337i 0.816968 + 0.576683i $$0.195653\pi$$
−0.816968 + 0.576683i $$0.804347\pi$$
$$434$$ −12.0000 −0.576018
$$435$$ 0 0
$$436$$ 32.0000 1.53252
$$437$$ 2.00000i 0.0956730i
$$438$$ 12.0000i 0.573382i
$$439$$ 33.0000 1.57500 0.787502 0.616312i $$-0.211374\pi$$
0.787502 + 0.616312i $$0.211374\pi$$
$$440$$ 0 0
$$441$$ 2.00000 0.0952381
$$442$$ − 10.0000i − 0.475651i
$$443$$ 35.0000i 1.66290i 0.555599 + 0.831450i $$0.312489\pi$$
−0.555599 + 0.831450i $$0.687511\pi$$
$$444$$ 6.00000 0.284747
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 7.00000i 0.331089i
$$448$$ 24.0000i 1.13389i
$$449$$ −15.0000 −0.707894 −0.353947 0.935266i $$-0.615161\pi$$
−0.353947 + 0.935266i $$0.615161\pi$$
$$450$$ 0 0
$$451$$ 45.0000 2.11897
$$452$$ − 4.00000i − 0.188144i
$$453$$ − 12.0000i − 0.563809i
$$454$$ 36.0000 1.68956
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 13.0000i 0.608114i 0.952654 + 0.304057i $$0.0983414\pi$$
−0.952654 + 0.304057i $$0.901659\pi$$
$$458$$ − 28.0000i − 1.30835i
$$459$$ −5.00000 −0.233380
$$460$$ 0 0
$$461$$ 3.00000 0.139724 0.0698620 0.997557i $$-0.477744\pi$$
0.0698620 + 0.997557i $$0.477744\pi$$
$$462$$ − 30.0000i − 1.39573i
$$463$$ 5.00000i 0.232370i 0.993228 + 0.116185i $$0.0370665\pi$$
−0.993228 + 0.116185i $$0.962933\pi$$
$$464$$ 40.0000 1.85695
$$465$$ 0 0
$$466$$ −50.0000 −2.31621
$$467$$ 29.0000i 1.34196i 0.741475 + 0.670980i $$0.234126\pi$$
−0.741475 + 0.670980i $$0.765874\pi$$
$$468$$ 2.00000i 0.0924500i
$$469$$ −12.0000 −0.554109
$$470$$ 0 0
$$471$$ 22.0000 1.01371
$$472$$ 0 0
$$473$$ 20.0000i 0.919601i
$$474$$ 22.0000 1.01049
$$475$$ 0 0
$$476$$ −30.0000 −1.37505
$$477$$ − 9.00000i − 0.412082i
$$478$$ 30.0000i 1.37217i
$$479$$ −5.00000 −0.228456 −0.114228 0.993455i $$-0.536439\pi$$
−0.114228 + 0.993455i $$0.536439\pi$$
$$480$$ 0 0
$$481$$ −3.00000 −0.136788
$$482$$ 28.0000i 1.27537i
$$483$$ 3.00000i 0.136505i
$$484$$ −28.0000 −1.27273
$$485$$ 0 0
$$486$$ 2.00000 0.0907218
$$487$$ − 7.00000i − 0.317200i −0.987343 0.158600i $$-0.949302\pi$$
0.987343 0.158600i $$-0.0506981\pi$$
$$488$$ 0 0
$$489$$ 11.0000 0.497437
$$490$$ 0 0
$$491$$ 16.0000 0.722070 0.361035 0.932552i $$-0.382424\pi$$
0.361035 + 0.932552i $$0.382424\pi$$
$$492$$ 18.0000i 0.811503i
$$493$$ 50.0000i 2.25189i
$$494$$ −4.00000 −0.179969
$$495$$ 0 0
$$496$$ 8.00000 0.359211
$$497$$ 45.0000i 2.01853i
$$498$$ 16.0000i 0.716977i
$$499$$ −34.0000 −1.52205 −0.761025 0.648723i $$-0.775303\pi$$
−0.761025 + 0.648723i $$0.775303\pi$$
$$500$$ 0 0
$$501$$ −8.00000 −0.357414
$$502$$ − 40.0000i − 1.78529i
$$503$$ 12.0000i 0.535054i 0.963550 + 0.267527i $$0.0862064\pi$$
−0.963550 + 0.267527i $$0.913794\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 10.0000 0.444554
$$507$$ − 1.00000i − 0.0444116i
$$508$$ 28.0000i 1.24230i
$$509$$ −21.0000 −0.930809 −0.465404 0.885098i $$-0.654091\pi$$
−0.465404 + 0.885098i $$0.654091\pi$$
$$510$$ 0 0
$$511$$ −18.0000 −0.796273
$$512$$ − 32.0000i − 1.41421i
$$513$$ 2.00000i 0.0883022i
$$514$$ −36.0000 −1.58789
$$515$$ 0 0
$$516$$ −8.00000 −0.352180
$$517$$ 50.0000i 2.19900i
$$518$$ 18.0000i 0.790875i
$$519$$ 2.00000 0.0877903
$$520$$ 0 0
$$521$$ −22.0000 −0.963837 −0.481919 0.876216i $$-0.660060\pi$$
−0.481919 + 0.876216i $$0.660060\pi$$
$$522$$ − 20.0000i − 0.875376i
$$523$$ − 20.0000i − 0.874539i −0.899331 0.437269i $$-0.855946\pi$$
0.899331 0.437269i $$-0.144054\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 10.0000i 0.435607i
$$528$$ 20.0000i 0.870388i
$$529$$ 22.0000 0.956522
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 12.0000i 0.520266i
$$533$$ − 9.00000i − 0.389833i
$$534$$ 22.0000 0.952033
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 6.00000i 0.258919i
$$538$$ 64.0000i 2.75924i
$$539$$ 10.0000 0.430730
$$540$$ 0 0
$$541$$ 38.0000 1.63375 0.816874 0.576816i $$-0.195705\pi$$
0.816874 + 0.576816i $$0.195705\pi$$
$$542$$ 4.00000i 0.171815i
$$543$$ − 23.0000i − 0.987024i
$$544$$ 40.0000 1.71499
$$545$$ 0 0
$$546$$ −6.00000 −0.256776
$$547$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$548$$ 12.0000i 0.512615i
$$549$$ 11.0000 0.469469
$$550$$ 0 0
$$551$$ 20.0000 0.852029
$$552$$ 0 0
$$553$$ 33.0000i 1.40330i
$$554$$ −52.0000 −2.20927
$$555$$ 0 0
$$556$$ −34.0000 −1.44192
$$557$$ 30.0000i 1.27114i 0.772043 + 0.635570i $$0.219235\pi$$
−0.772043 + 0.635570i $$0.780765\pi$$
$$558$$ − 4.00000i − 0.169334i
$$559$$ 4.00000 0.169182
$$560$$ 0 0
$$561$$ −25.0000 −1.05550
$$562$$ 20.0000i 0.843649i
$$563$$ − 41.0000i − 1.72794i −0.503540 0.863972i $$-0.667969\pi$$
0.503540 0.863972i $$-0.332031\pi$$
$$564$$ −20.0000 −0.842152
$$565$$ 0 0
$$566$$ −16.0000 −0.672530
$$567$$ 3.00000i 0.125988i
$$568$$ 0 0
$$569$$ −16.0000 −0.670755 −0.335377 0.942084i $$-0.608864\pi$$
−0.335377 + 0.942084i $$0.608864\pi$$
$$570$$ 0 0
$$571$$ 17.0000 0.711428 0.355714 0.934595i $$-0.384238\pi$$
0.355714 + 0.934595i $$0.384238\pi$$
$$572$$ 10.0000i 0.418121i
$$573$$ − 20.0000i − 0.835512i
$$574$$ −54.0000 −2.25392
$$575$$ 0 0
$$576$$ −8.00000 −0.333333
$$577$$ 21.0000i 0.874241i 0.899403 + 0.437121i $$0.144002\pi$$
−0.899403 + 0.437121i $$0.855998\pi$$
$$578$$ 16.0000i 0.665512i
$$579$$ −13.0000 −0.540262
$$580$$ 0 0
$$581$$ −24.0000 −0.995688
$$582$$ 18.0000i 0.746124i
$$583$$ − 45.0000i − 1.86371i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 48.0000 1.98286
$$587$$ − 42.0000i − 1.73353i −0.498721 0.866763i $$-0.666197\pi$$
0.498721 0.866763i $$-0.333803\pi$$
$$588$$ 4.00000i 0.164957i
$$589$$ 4.00000 0.164817
$$590$$ 0 0
$$591$$ −12.0000 −0.493614
$$592$$ − 12.0000i − 0.493197i
$$593$$ − 36.0000i − 1.47834i −0.673517 0.739171i $$-0.735217\pi$$
0.673517 0.739171i $$-0.264783\pi$$
$$594$$ 10.0000 0.410305
$$595$$ 0 0
$$596$$ −14.0000 −0.573462
$$597$$ − 4.00000i − 0.163709i
$$598$$ − 2.00000i − 0.0817861i
$$599$$ −4.00000 −0.163436 −0.0817178 0.996656i $$-0.526041\pi$$
−0.0817178 + 0.996656i $$0.526041\pi$$
$$600$$ 0 0
$$601$$ −5.00000 −0.203954 −0.101977 0.994787i $$-0.532517\pi$$
−0.101977 + 0.994787i $$0.532517\pi$$
$$602$$ − 24.0000i − 0.978167i
$$603$$ − 4.00000i − 0.162893i
$$604$$ 24.0000 0.976546
$$605$$ 0 0
$$606$$ −24.0000 −0.974933
$$607$$ 40.0000i 1.62355i 0.583970 + 0.811775i $$0.301498\pi$$
−0.583970 + 0.811775i $$0.698502\pi$$
$$608$$ − 16.0000i − 0.648886i
$$609$$ 30.0000 1.21566
$$610$$ 0 0
$$611$$ 10.0000 0.404557
$$612$$ − 10.0000i − 0.404226i
$$613$$ 3.00000i 0.121169i 0.998163 + 0.0605844i $$0.0192964\pi$$
−0.998163 + 0.0605844i $$0.980704\pi$$
$$614$$ 38.0000 1.53356
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 22.0000i − 0.885687i −0.896599 0.442843i $$-0.853970\pi$$
0.896599 0.442843i $$-0.146030\pi$$
$$618$$ 8.00000i 0.321807i
$$619$$ 2.00000 0.0803868 0.0401934 0.999192i $$-0.487203\pi$$
0.0401934 + 0.999192i $$0.487203\pi$$
$$620$$ 0 0
$$621$$ −1.00000 −0.0401286
$$622$$ − 48.0000i − 1.92462i
$$623$$ 33.0000i 1.32212i
$$624$$ 4.00000 0.160128
$$625$$ 0 0
$$626$$ −20.0000 −0.799361
$$627$$ 10.0000i 0.399362i
$$628$$ 44.0000i 1.75579i
$$629$$ 15.0000 0.598089
$$630$$ 0 0
$$631$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$632$$ 0 0
$$633$$ − 4.00000i − 0.158986i
$$634$$ −24.0000 −0.953162
$$635$$ 0 0
$$636$$ 18.0000 0.713746
$$637$$ − 2.00000i − 0.0792429i
$$638$$ − 100.000i − 3.95904i
$$639$$ −15.0000 −0.593391
$$640$$ 0 0
$$641$$ −36.0000 −1.42191 −0.710957 0.703235i $$-0.751738\pi$$
−0.710957 + 0.703235i $$0.751738\pi$$
$$642$$ − 6.00000i − 0.236801i
$$643$$ 1.00000i 0.0394362i 0.999806 + 0.0197181i $$0.00627687\pi$$
−0.999806 + 0.0197181i $$0.993723\pi$$
$$644$$ −6.00000 −0.236433
$$645$$ 0 0
$$646$$ 20.0000 0.786889
$$647$$ − 21.0000i − 0.825595i −0.910823 0.412798i $$-0.864552\pi$$
0.910823 0.412798i $$-0.135448\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 6.00000 0.235159
$$652$$ 22.0000i 0.861586i
$$653$$ − 30.0000i − 1.17399i −0.809590 0.586995i $$-0.800311\pi$$
0.809590 0.586995i $$-0.199689\pi$$
$$654$$ −32.0000 −1.25130
$$655$$ 0 0
$$656$$ 36.0000 1.40556
$$657$$ − 6.00000i − 0.234082i
$$658$$ − 60.0000i − 2.33904i
$$659$$ −12.0000 −0.467454 −0.233727 0.972302i $$-0.575092\pi$$
−0.233727 + 0.972302i $$0.575092\pi$$
$$660$$ 0 0
$$661$$ −16.0000 −0.622328 −0.311164 0.950356i $$-0.600719\pi$$
−0.311164 + 0.950356i $$0.600719\pi$$
$$662$$ − 64.0000i − 2.48743i
$$663$$ 5.00000i 0.194184i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ −6.00000 −0.232495
$$667$$ 10.0000i 0.387202i
$$668$$ − 16.0000i − 0.619059i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 55.0000 2.12325
$$672$$ − 24.0000i − 0.925820i
$$673$$ 22.0000i 0.848038i 0.905653 + 0.424019i $$0.139381\pi$$
−0.905653 + 0.424019i $$0.860619\pi$$
$$674$$ −8.00000 −0.308148
$$675$$ 0 0
$$676$$ 2.00000 0.0769231
$$677$$ 7.00000i 0.269032i 0.990911 + 0.134516i $$0.0429479\pi$$
−0.990911 + 0.134516i $$0.957052\pi$$
$$678$$ 4.00000i 0.153619i
$$679$$ −27.0000 −1.03616
$$680$$ 0 0
$$681$$ −18.0000 −0.689761
$$682$$ − 20.0000i − 0.765840i
$$683$$ 4.00000i 0.153056i 0.997067 + 0.0765279i $$0.0243834\pi$$
−0.997067 + 0.0765279i $$0.975617\pi$$
$$684$$ −4.00000 −0.152944
$$685$$ 0 0
$$686$$ 30.0000 1.14541
$$687$$ 14.0000i 0.534133i
$$688$$ 16.0000i 0.609994i
$$689$$ −9.00000 −0.342873
$$690$$ 0 0
$$691$$ 6.00000 0.228251 0.114125 0.993466i $$-0.463593\pi$$
0.114125 + 0.993466i $$0.463593\pi$$
$$692$$ 4.00000i 0.152057i
$$693$$ 15.0000i 0.569803i
$$694$$ 2.00000 0.0759190
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 45.0000i 1.70450i
$$698$$ 40.0000i 1.51402i
$$699$$ 25.0000 0.945587
$$700$$ 0 0
$$701$$ −4.00000 −0.151078 −0.0755390 0.997143i $$-0.524068\pi$$
−0.0755390 + 0.997143i $$0.524068\pi$$
$$702$$ − 2.00000i − 0.0754851i
$$703$$ − 6.00000i − 0.226294i
$$704$$ −40.0000 −1.50756
$$705$$ 0 0
$$706$$ −28.0000 −1.05379
$$707$$ − 36.0000i − 1.35392i
$$708$$ 0 0
$$709$$ −20.0000 −0.751116 −0.375558 0.926799i $$-0.622549\pi$$
−0.375558 + 0.926799i $$0.622549\pi$$
$$710$$ 0 0
$$711$$ −11.0000 −0.412532
$$712$$ 0 0
$$713$$ 2.00000i 0.0749006i
$$714$$ 30.0000 1.12272
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ − 15.0000i − 0.560185i
$$718$$ 32.0000i 1.19423i
$$719$$ −12.0000 −0.447524 −0.223762 0.974644i $$-0.571834\pi$$
−0.223762 + 0.974644i $$0.571834\pi$$
$$720$$ 0 0
$$721$$ −12.0000 −0.446903
$$722$$ 30.0000i 1.11648i
$$723$$ − 14.0000i − 0.520666i
$$724$$ 46.0000 1.70958
$$725$$ 0 0
$$726$$ 28.0000 1.03918
$$727$$ − 6.00000i − 0.222528i −0.993791 0.111264i $$-0.964510\pi$$
0.993791 0.111264i $$-0.0354899\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −20.0000 −0.739727
$$732$$ 22.0000i 0.813143i
$$733$$ − 15.0000i − 0.554038i −0.960864 0.277019i $$-0.910654\pi$$
0.960864 0.277019i $$-0.0893464\pi$$
$$734$$ −16.0000 −0.590571
$$735$$ 0 0
$$736$$ 8.00000 0.294884
$$737$$ − 20.0000i − 0.736709i
$$738$$ − 18.0000i − 0.662589i
$$739$$ −38.0000 −1.39785 −0.698926 0.715194i $$-0.746338\pi$$
−0.698926 + 0.715194i $$0.746338\pi$$
$$740$$ 0 0
$$741$$ 2.00000 0.0734718
$$742$$ 54.0000i 1.98240i
$$743$$ 6.00000i 0.220119i 0.993925 + 0.110059i $$0.0351041\pi$$
−0.993925 + 0.110059i $$0.964896\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 32.0000 1.17160
$$747$$ − 8.00000i − 0.292705i
$$748$$ − 50.0000i − 1.82818i
$$749$$ 9.00000 0.328853
$$750$$ 0 0
$$751$$ −45.0000 −1.64207 −0.821037 0.570875i $$-0.806604\pi$$
−0.821037 + 0.570875i $$0.806604\pi$$
$$752$$ 40.0000i 1.45865i
$$753$$ 20.0000i 0.728841i
$$754$$ −20.0000 −0.728357
$$755$$ 0 0
$$756$$ −6.00000 −0.218218
$$757$$ − 36.0000i − 1.30844i −0.756303 0.654221i $$-0.772997\pi$$
0.756303 0.654221i $$-0.227003\pi$$
$$758$$ 12.0000i 0.435860i
$$759$$ −5.00000 −0.181489
$$760$$ 0 0
$$761$$ −14.0000 −0.507500 −0.253750 0.967270i $$-0.581664\pi$$
−0.253750 + 0.967270i $$0.581664\pi$$
$$762$$ − 28.0000i − 1.01433i
$$763$$ − 48.0000i − 1.73772i
$$764$$ 40.0000 1.44715
$$765$$ 0 0
$$766$$ −36.0000 −1.30073
$$767$$ 0 0
$$768$$ 16.0000i 0.577350i
$$769$$ 12.0000 0.432731 0.216366 0.976312i $$-0.430580\pi$$
0.216366 + 0.976312i $$0.430580\pi$$
$$770$$ 0 0
$$771$$ 18.0000 0.648254
$$772$$ − 26.0000i − 0.935760i
$$773$$ 24.0000i 0.863220i 0.902060 + 0.431610i $$0.142054\pi$$
−0.902060 + 0.431610i $$0.857946\pi$$
$$774$$ 8.00000 0.287554
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 9.00000i − 0.322873i
$$778$$ − 48.0000i − 1.72088i
$$779$$ 18.0000 0.644917
$$780$$ 0 0
$$781$$ −75.0000 −2.68371
$$782$$ 10.0000i 0.357599i
$$783$$ 10.0000i 0.357371i
$$784$$ 8.00000 0.285714
$$785$$ 0 0
$$786$$ 12.0000 0.428026
$$787$$ − 44.0000i − 1.56843i −0.620489 0.784215i $$-0.713066\pi$$
0.620489 0.784215i $$-0.286934\pi$$
$$788$$ − 24.0000i − 0.854965i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −6.00000 −0.213335
$$792$$ 0 0
$$793$$ − 11.0000i − 0.390621i
$$794$$ 38.0000 1.34857
$$795$$ 0 0
$$796$$ 8.00000 0.283552
$$797$$ 5.00000i 0.177109i 0.996071 + 0.0885545i $$0.0282248\pi$$
−0.996071 + 0.0885545i $$0.971775\pi$$
$$798$$ − 12.0000i − 0.424795i
$$799$$ −50.0000 −1.76887
$$800$$ 0 0
$$801$$ −11.0000 −0.388666
$$802$$ 36.0000i 1.27120i
$$803$$ − 30.0000i − 1.05868i
$$804$$ 8.00000 0.282138
$$805$$ 0 0
$$806$$ −4.00000 −0.140894
$$807$$ − 32.0000i − 1.12645i
$$808$$ 0 0
$$809$$ 6.00000 0.210949 0.105474 0.994422i $$-0.466364\pi$$
0.105474 + 0.994422i $$0.466364\pi$$
$$810$$ 0 0
$$811$$ −4.00000 −0.140459 −0.0702295 0.997531i $$-0.522373\pi$$
−0.0702295 + 0.997531i $$0.522373\pi$$
$$812$$ 60.0000i 2.10559i
$$813$$ − 2.00000i − 0.0701431i
$$814$$ −30.0000 −1.05150
$$815$$ 0 0
$$816$$ −20.0000 −0.700140
$$817$$ 8.00000i 0.279885i
$$818$$ − 52.0000i − 1.81814i
$$819$$ 3.00000 0.104828
$$820$$ 0 0
$$821$$ −41.0000 −1.43091 −0.715455 0.698659i $$-0.753781\pi$$
−0.715455 + 0.698659i $$0.753781\pi$$
$$822$$ − 12.0000i − 0.418548i
$$823$$ − 48.0000i − 1.67317i −0.547833 0.836587i $$-0.684547\pi$$
0.547833 0.836587i $$-0.315453\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 42.0000i 1.46048i 0.683189 + 0.730242i $$0.260592\pi$$
−0.683189 + 0.730242i $$0.739408\pi$$
$$828$$ − 2.00000i − 0.0695048i
$$829$$ −14.0000 −0.486240 −0.243120 0.969996i $$-0.578171\pi$$
−0.243120 + 0.969996i $$0.578171\pi$$
$$830$$ 0 0
$$831$$ 26.0000 0.901930
$$832$$ 8.00000i 0.277350i
$$833$$ 10.0000i 0.346479i
$$834$$ 34.0000 1.17732
$$835$$ 0 0
$$836$$ −20.0000 −0.691714
$$837$$ 2.00000i 0.0691301i
$$838$$ 52.0000i 1.79631i
$$839$$ −7.00000 −0.241667 −0.120833 0.992673i $$-0.538557\pi$$
−0.120833 + 0.992673i $$0.538557\pi$$
$$840$$ 0 0
$$841$$ 71.0000 2.44828
$$842$$ − 64.0000i − 2.20559i
$$843$$ − 10.0000i − 0.344418i
$$844$$ 8.00000 0.275371
$$845$$ 0 0
$$846$$ 20.0000 0.687614
$$847$$ 42.0000i 1.44314i
$$848$$ − 36.0000i − 1.23625i
$$849$$ 8.00000 0.274559
$$850$$ 0 0
$$851$$ 3.00000 0.102839
$$852$$ − 30.0000i − 1.02778i
$$853$$ 51.0000i 1.74621i 0.487535 + 0.873103i $$0.337896\pi$$
−0.487535 + 0.873103i $$0.662104\pi$$
$$854$$ −66.0000 −2.25847
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 17.0000i 0.580709i 0.956919 + 0.290354i $$0.0937732\pi$$
−0.956919 + 0.290354i $$0.906227\pi$$
$$858$$ − 10.0000i − 0.341394i
$$859$$ −35.0000 −1.19418 −0.597092 0.802173i $$-0.703677\pi$$
−0.597092 + 0.802173i $$0.703677\pi$$
$$860$$ 0 0
$$861$$ 27.0000 0.920158
$$862$$ 32.0000i 1.08992i
$$863$$ − 22.0000i − 0.748889i −0.927249 0.374444i $$-0.877833\pi$$
0.927249 0.374444i $$-0.122167\pi$$
$$864$$ 8.00000 0.272166
$$865$$ 0 0
$$866$$ 48.0000 1.63111
$$867$$ − 8.00000i − 0.271694i
$$868$$ 12.0000i 0.407307i
$$869$$ −55.0000 −1.86575
$$870$$ 0 0
$$871$$ −4.00000 −0.135535
$$872$$ 0 0
$$873$$ − 9.00000i − 0.304604i
$$874$$ 4.00000 0.135302
$$875$$ 0 0
$$876$$ 12.0000 0.405442
$$877$$ − 22.0000i − 0.742887i −0.928456 0.371444i $$-0.878863\pi$$
0.928456 0.371444i $$-0.121137\pi$$
$$878$$ − 66.0000i − 2.22739i
$$879$$ −24.0000 −0.809500
$$880$$ 0 0
$$881$$ −14.0000 −0.471672 −0.235836 0.971793i $$-0.575783\pi$$
−0.235836 + 0.971793i $$0.575783\pi$$
$$882$$ − 4.00000i − 0.134687i
$$883$$ 12.0000i 0.403832i 0.979403 + 0.201916i $$0.0647168\pi$$
−0.979403 + 0.201916i $$0.935283\pi$$
$$884$$ −10.0000 −0.336336
$$885$$ 0 0
$$886$$ 70.0000 2.35170
$$887$$ 15.0000i 0.503651i 0.967773 + 0.251825i $$0.0810309\pi$$
−0.967773 + 0.251825i $$0.918969\pi$$
$$888$$ 0 0
$$889$$ 42.0000 1.40863
$$890$$ 0 0
$$891$$ −5.00000 −0.167506
$$892$$ 0 0
$$893$$ 20.0000i 0.669274i
$$894$$ 14.0000 0.468230
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 1.00000i 0.0333890i
$$898$$ 30.0000i 1.00111i
$$899$$ 20.0000 0.667037
$$900$$ 0 0
$$901$$ 45.0000 1.49917
$$902$$ − 90.0000i − 2.99667i
$$903$$ 12.0000i 0.399335i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ −24.0000 −0.797347
$$907$$ − 2.00000i − 0.0664089i −0.999449 0.0332045i $$-0.989429\pi$$
0.999449 0.0332045i $$-0.0105712\pi$$
$$908$$ − 36.0000i − 1.19470i
$$909$$ 12.0000 0.398015
$$910$$ 0 0
$$911$$ 8.00000 0.265052 0.132526 0.991180i $$-0.457691\pi$$
0.132526 + 0.991180i $$0.457691\pi$$
$$912$$ 8.00000i 0.264906i
$$913$$ − 40.0000i − 1.32381i
$$914$$ 26.0000 0.860004
$$915$$ 0 0
$$916$$ −28.0000 −0.925146
$$917$$ 18.0000i 0.594412i
$$918$$ 10.0000i 0.330049i
$$919$$ −29.0000 −0.956622 −0.478311 0.878191i $$-0.658751\pi$$
−0.478311 + 0.878191i $$0.658751\pi$$
$$920$$ 0 0
$$921$$ −19.0000 −0.626071
$$922$$ − 6.00000i − 0.197599i
$$923$$ 15.0000i 0.493731i
$$924$$ −30.0000 −0.986928
$$925$$ 0 0
$$926$$ 10.0000 0.328620
$$927$$ − 4.00000i − 0.131377i
$$928$$ − 80.0000i − 2.62613i
$$929$$ −21.0000 −0.688988 −0.344494 0.938789i $$-0.611949\pi$$
−0.344494 + 0.938789i $$0.611949\pi$$
$$930$$ 0 0
$$931$$ 4.00000 0.131095
$$932$$ 50.0000i 1.63780i
$$933$$ 24.0000i 0.785725i
$$934$$ 58.0000 1.89782
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 2.00000i − 0.0653372i −0.999466 0.0326686i $$-0.989599\pi$$
0.999466 0.0326686i $$-0.0104006\pi$$
$$938$$ 24.0000i 0.783628i
$$939$$ 10.0000 0.326338
$$940$$ 0 0
$$941$$ −23.0000 −0.749779 −0.374889 0.927070i $$-0.622319\pi$$
−0.374889 + 0.927070i $$0.622319\pi$$
$$942$$ − 44.0000i − 1.43360i
$$943$$ 9.00000i 0.293080i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 40.0000 1.30051
$$947$$ − 36.0000i − 1.16984i −0.811090 0.584921i $$-0.801125\pi$$
0.811090 0.584921i $$-0.198875\pi$$
$$948$$ − 22.0000i − 0.714527i
$$949$$ −6.00000 −0.194768
$$950$$ 0 0
$$951$$ 12.0000 0.389127
$$952$$ 0 0
$$953$$ − 11.0000i − 0.356325i −0.984001 0.178162i $$-0.942985\pi$$
0.984001 0.178162i $$-0.0570153\pi$$
$$954$$ −18.0000 −0.582772
$$955$$ 0 0
$$956$$ 30.0000 0.970269
$$957$$ 50.0000i 1.61627i
$$958$$ 10.0000i 0.323085i
$$959$$ 18.0000 0.581250
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ 6.00000i 0.193448i
$$963$$ 3.00000i 0.0966736i
$$964$$ 28.0000 0.901819
$$965$$ 0 0
$$966$$ 6.00000 0.193047
$$967$$ 40.0000i 1.28631i 0.765735 + 0.643157i $$0.222376\pi$$
−0.765735 + 0.643157i $$0.777624\pi$$
$$968$$ 0 0
$$969$$ −10.0000 −0.321246
$$970$$ 0 0
$$971$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$972$$ − 2.00000i − 0.0641500i
$$973$$ 51.0000i 1.63498i
$$974$$ −14.0000 −0.448589
$$975$$ 0 0
$$976$$ 44.0000 1.40841
$$977$$ − 12.0000i − 0.383914i −0.981403 0.191957i $$-0.938517\pi$$
0.981403 0.191957i $$-0.0614834\pi$$
$$978$$ − 22.0000i − 0.703482i
$$979$$ −55.0000 −1.75781
$$980$$ 0 0
$$981$$ 16.0000 0.510841
$$982$$ − 32.0000i − 1.02116i
$$983$$ 36.0000i 1.14822i 0.818778 + 0.574111i $$0.194652\pi$$
−0.818778 + 0.574111i $$0.805348\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 100.000 3.18465
$$987$$ 30.0000i 0.954911i
$$988$$ 4.00000i 0.127257i
$$989$$ −4.00000 −0.127193
$$990$$ 0 0
$$991$$ 39.0000 1.23888 0.619438 0.785046i $$-0.287361\pi$$
0.619438 + 0.785046i $$0.287361\pi$$
$$992$$ − 16.0000i − 0.508001i
$$993$$ 32.0000i 1.01549i
$$994$$ 90.0000 2.85463
$$995$$ 0 0
$$996$$ 16.0000 0.506979
$$997$$ 16.0000i 0.506725i 0.967371 + 0.253363i $$0.0815366\pi$$
−0.967371 + 0.253363i $$0.918463\pi$$
$$998$$ 68.0000i 2.15250i
$$999$$ 3.00000 0.0949158
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 975.2.c.b.274.1 2
3.2 odd 2 2925.2.c.d.2224.2 2
5.2 odd 4 195.2.a.d.1.1 1
5.3 odd 4 975.2.a.b.1.1 1
5.4 even 2 inner 975.2.c.b.274.2 2
15.2 even 4 585.2.a.a.1.1 1
15.8 even 4 2925.2.a.t.1.1 1
15.14 odd 2 2925.2.c.d.2224.1 2
20.7 even 4 3120.2.a.n.1.1 1
35.27 even 4 9555.2.a.t.1.1 1
60.47 odd 4 9360.2.a.w.1.1 1
65.12 odd 4 2535.2.a.b.1.1 1
195.77 even 4 7605.2.a.v.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.a.d.1.1 1 5.2 odd 4
585.2.a.a.1.1 1 15.2 even 4
975.2.a.b.1.1 1 5.3 odd 4
975.2.c.b.274.1 2 1.1 even 1 trivial
975.2.c.b.274.2 2 5.4 even 2 inner
2535.2.a.b.1.1 1 65.12 odd 4
2925.2.a.t.1.1 1 15.8 even 4
2925.2.c.d.2224.1 2 15.14 odd 2
2925.2.c.d.2224.2 2 3.2 odd 2
3120.2.a.n.1.1 1 20.7 even 4
7605.2.a.v.1.1 1 195.77 even 4
9360.2.a.w.1.1 1 60.47 odd 4
9555.2.a.t.1.1 1 35.27 even 4