# Properties

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

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.00000i q^{2} -1.00000i q^{3} -2.00000 q^{4} +2.00000 q^{6} -3.00000i q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q+2.00000i q^{2} -1.00000i q^{3} -2.00000 q^{4} +2.00000 q^{6} -3.00000i q^{7} -1.00000 q^{9} -5.00000 q^{11} +2.00000i q^{12} -1.00000i q^{13} +6.00000 q^{14} -4.00000 q^{16} +5.00000i q^{17} -2.00000i q^{18} -2.00000 q^{19} -3.00000 q^{21} -10.0000i q^{22} +1.00000i q^{23} +2.00000 q^{26} +1.00000i q^{27} +6.00000i q^{28} -10.0000 q^{29} -2.00000 q^{31} -8.00000i q^{32} +5.00000i q^{33} -10.0000 q^{34} +2.00000 q^{36} -3.00000i q^{37} -4.00000i q^{38} -1.00000 q^{39} -9.00000 q^{41} -6.00000i q^{42} +4.00000i q^{43} +10.0000 q^{44} -2.00000 q^{46} +10.0000i q^{47} +4.00000i q^{48} -2.00000 q^{49} +5.00000 q^{51} +2.00000i q^{52} -9.00000i q^{53} -2.00000 q^{54} +2.00000i q^{57} -20.0000i q^{58} -11.0000 q^{61} -4.00000i q^{62} +3.00000i q^{63} +8.00000 q^{64} -10.0000 q^{66} -4.00000i q^{67} -10.0000i q^{68} +1.00000 q^{69} +15.0000 q^{71} -6.00000i q^{73} +6.00000 q^{74} +4.00000 q^{76} +15.0000i q^{77} -2.00000i q^{78} +11.0000 q^{79} +1.00000 q^{81} -18.0000i q^{82} -8.00000i q^{83} +6.00000 q^{84} -8.00000 q^{86} +10.0000i q^{87} +11.0000 q^{89} -3.00000 q^{91} -2.00000i q^{92} +2.00000i q^{93} -20.0000 q^{94} -8.00000 q^{96} -9.00000i q^{97} -4.00000i q^{98} +5.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{4} + 4 q^{6} - 2 q^{9}+O(q^{10})$$ 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 $$0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$3$$ − 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 $$-0.808125\pi$$
0.823754 0.566947i $$-0.191875\pi$$
$$8$$ 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 $$0.207363\pi$$
−0.795206 + 0.606339i $$0.792637\pi$$
$$18$$ − 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 $$0.0332465\pi$$
−0.994550 + 0.104257i $$0.966753\pi$$
$$24$$ 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 $$-0.920687\pi$$
0.969118 0.246598i $$-0.0793129\pi$$
$$38$$ − 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 $$0.0986555\pi$$
−0.952353 + 0.304997i $$0.901344\pi$$
$$44$$ 10.0000 1.50756
$$45$$ 0 0
$$46$$ −2.00000 −0.294884
$$47$$ 10.0000i 1.45865i 0.684167 + 0.729325i $$0.260166\pi$$
−0.684167 + 0.729325i $$0.739834\pi$$
$$48$$ 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 $$-0.787894\pi$$
0.786082 0.618123i $$-0.212106\pi$$
$$54$$ −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 $$-0.921429\pi$$
0.969690 0.244339i $$-0.0785709\pi$$
$$68$$ − 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 $$-0.885800\pi$$
0.936329 0.351123i $$-0.114200\pi$$
$$74$$ 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 $$-0.855313\pi$$
0.898459 0.439057i $$-0.144687\pi$$
$$84$$ 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 $$-0.848958\pi$$
0.889515 0.456906i $$-0.151042\pi$$
$$98$$ − 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 $$-0.936859\pi$$
0.980390 0.197066i $$-0.0631413\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 18.0000 1.74831
$$107$$ 3.00000i 0.290021i 0.989430 + 0.145010i $$0.0463216\pi$$
−0.989430 + 0.145010i $$0.953678\pi$$
$$108$$ − 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 $$-0.970012\pi$$
0.995565 0.0940721i $$-0.0299884\pi$$
$$114$$ −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 $$0.213334\pi$$
−0.783692 + 0.621150i $$0.786666\pi$$
$$128$$ 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 $$0.0825059\pi$$
−0.966595 + 0.256307i $$0.917494\pi$$
$$138$$ 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 $$0.341053\pi$$
−0.478852 + 0.877896i $$0.658947\pi$$
$$158$$ 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 $$0.141766\pi$$
−0.902451 + 0.430793i $$0.858234\pi$$
$$164$$ 18.0000 1.40556
$$165$$ 0 0
$$166$$ 16.0000 1.24184
$$167$$ − 8.00000i − 0.619059i −0.950890 0.309529i $$-0.899829\pi$$
0.950890 0.309529i $$-0.100171\pi$$
$$168$$ 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 $$0.0242240\pi$$
−0.997106 + 0.0760286i $$0.975776\pi$$
$$174$$ −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 $$-0.845018\pi$$
0.883792 0.467880i $$-0.154982\pi$$
$$194$$ 18.0000 1.29232
$$195$$ 0 0
$$196$$ 4.00000 0.285714
$$197$$ − 12.0000i − 0.854965i −0.904024 0.427482i $$-0.859401\pi$$
0.904024 0.427482i $$-0.140599\pi$$
$$198$$ 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 $$-0.796220\pi$$
0.801980 0.597351i $$-0.203780\pi$$
$$228$$ − 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 $$0.305417\pi$$
−0.573933 + 0.818902i $$0.694583\pi$$
$$234$$ −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 $$0.189739\pi$$
−0.827541 + 0.561405i $$0.810261\pi$$
$$258$$ 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 $$0.285338\pi$$
−0.624413 + 0.781094i $$0.714662\pi$$
$$278$$ 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 $$0.0764182\pi$$
−0.971320 + 0.237775i $$0.923582\pi$$
$$284$$ −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 $$-0.752716\pi$$
0.713115 0.701047i $$-0.247284\pi$$
$$294$$ −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 $$-0.817594\pi$$
0.840254 0.542194i $$-0.182406\pi$$
$$308$$ − 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 $$0.0912024\pi$$
−0.959233 + 0.282617i $$0.908798\pi$$
$$314$$ −44.0000 −2.48306
$$315$$ 0 0
$$316$$ −22.0000 −1.23760
$$317$$ 12.0000i 0.673987i 0.941507 + 0.336994i $$0.109410\pi$$
−0.941507 + 0.336994i $$0.890590\pi$$
$$318$$ − 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 $$0.0347479\pi$$
−0.994048 + 0.108947i $$0.965252\pi$$
$$338$$ − 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 $$-0.991455\pi$$
0.999640 0.0268414i $$-0.00854491\pi$$
$$348$$ − 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 $$0.121524\pi$$
−0.928003 + 0.372572i $$0.878476\pi$$
$$354$$ 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 $$0.0669552\pi$$
−0.977959 + 0.208798i $$0.933045\pi$$
$$368$$ − 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 $$-0.864053\pi$$
0.910175 0.414224i $$-0.135947\pi$$
$$374$$ 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 $$0.152107\pi$$
−0.887982 + 0.459879i $$0.847893\pi$$
$$384$$ 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 $$-0.841800\pi$$
0.879017 0.476791i $$-0.158200\pi$$
$$398$$ − 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 $$-0.804347\pi$$
0.816968 0.576683i $$-0.195653\pi$$
$$434$$ −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 $$-0.687511\pi$$
0.555599 0.831450i $$-0.312489\pi$$
$$444$$ 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 $$-0.901659\pi$$
0.952654 0.304057i $$-0.0983414\pi$$
$$458$$ 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 $$-0.962933\pi$$
0.993228 0.116185i $$-0.0370665\pi$$
$$464$$ 40.0000 1.85695
$$465$$ 0 0
$$466$$ −50.0000 −2.31621
$$467$$ − 29.0000i − 1.34196i −0.741475 0.670980i $$-0.765874\pi$$
0.741475 0.670980i $$-0.234126\pi$$
$$468$$ − 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 $$0.0506981\pi$$
−0.987343 + 0.158600i $$0.949302\pi$$
$$488$$ 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 $$-0.913794\pi$$
0.963550 0.267527i $$-0.0862064\pi$$
$$504$$ 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 $$0.144054\pi$$
−0.899331 + 0.437269i $$0.855946\pi$$
$$524$$ −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 $$-0.780765\pi$$
0.772043 0.635570i $$-0.219235\pi$$
$$558$$ 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 $$0.332031\pi$$
−0.503540 + 0.863972i $$0.667969\pi$$
$$564$$ −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 $$-0.855998\pi$$
0.899403 0.437121i $$-0.144002\pi$$
$$578$$ − 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 $$0.333803\pi$$
−0.498721 + 0.866763i $$0.666197\pi$$
$$588$$ − 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 $$0.264783\pi$$
−0.673517 + 0.739171i $$0.735217\pi$$
$$594$$ 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 $$-0.698502\pi$$
0.583970 0.811775i $$-0.301498\pi$$
$$608$$ 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 $$-0.980704\pi$$
0.998163 0.0605844i $$-0.0192964\pi$$
$$614$$ 38.0000 1.53356
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 22.0000i 0.885687i 0.896599 + 0.442843i $$0.146030\pi$$
−0.896599 + 0.442843i $$0.853970\pi$$
$$618$$ − 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 $$-0.993723\pi$$
0.999806 0.0197181i $$-0.00627687\pi$$
$$644$$ −6.00000 −0.236433
$$645$$ 0 0
$$646$$ 20.0000 0.786889
$$647$$ 21.0000i 0.825595i 0.910823 + 0.412798i $$0.135448\pi$$
−0.910823 + 0.412798i $$0.864552\pi$$
$$648$$ 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 $$0.199689\pi$$
−0.809590 + 0.586995i $$0.800311\pi$$
$$654$$ −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 $$-0.860619\pi$$
0.905653 0.424019i $$-0.139381\pi$$
$$674$$ −8.00000 −0.308148
$$675$$ 0 0
$$676$$ 2.00000 0.0769231
$$677$$ − 7.00000i − 0.269032i −0.990911 0.134516i $$-0.957052\pi$$
0.990911 0.134516i $$-0.0429479\pi$$
$$678$$ − 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 $$-0.975617\pi$$
0.997067 0.0765279i $$-0.0243834\pi$$
$$684$$ −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 $$0.0354899\pi$$
−0.993791 + 0.111264i $$0.964510\pi$$
$$728$$ 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 $$0.0893464\pi$$
−0.960864 + 0.277019i $$0.910654\pi$$
$$734$$ −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 $$-0.964896\pi$$
0.993925 0.110059i $$-0.0351041\pi$$
$$744$$ 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 $$0.227003\pi$$
−0.756303 + 0.654221i $$0.772997\pi$$
$$758$$ − 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 $$-0.857946\pi$$
0.902060 0.431610i $$-0.142054\pi$$
$$774$$ 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 $$0.286934\pi$$
−0.620489 + 0.784215i $$0.713066\pi$$
$$788$$ 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 $$-0.971775\pi$$
0.996071 0.0885545i $$-0.0282248\pi$$
$$798$$ 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 $$0.315453\pi$$
−0.547833 + 0.836587i $$0.684547\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 42.0000i − 1.46048i −0.683189 0.730242i $$-0.739408\pi$$
0.683189 0.730242i $$-0.260592\pi$$
$$828$$ 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 $$-0.662104\pi$$
0.487535 0.873103i $$-0.337896\pi$$
$$854$$ −66.0000 −2.25847
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 17.0000i − 0.580709i −0.956919 0.290354i $$-0.906227\pi$$
0.956919 0.290354i $$-0.0937732\pi$$
$$858$$ 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 $$0.122167\pi$$
−0.927249 + 0.374444i $$0.877833\pi$$
$$864$$ 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 $$0.121137\pi$$
−0.928456 + 0.371444i $$0.878863\pi$$
$$878$$ 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 $$-0.935283\pi$$
0.979403 0.201916i $$-0.0647168\pi$$
$$884$$ −10.0000 −0.336336
$$885$$ 0 0
$$886$$ 70.0000 2.35170
$$887$$ − 15.0000i − 0.503651i −0.967773 0.251825i $$-0.918969\pi$$
0.967773 0.251825i $$-0.0810309\pi$$
$$888$$ 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 $$0.0105712\pi$$
−0.999449 + 0.0332045i $$0.989429\pi$$
$$908$$ 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 $$0.0104006\pi$$
−0.999466 + 0.0326686i $$0.989599\pi$$
$$938$$ − 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 $$0.198875\pi$$
−0.811090 + 0.584921i $$0.801125\pi$$
$$948$$ 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 $$0.0570153\pi$$
−0.984001 + 0.178162i $$0.942985\pi$$
$$954$$ −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 $$-0.777624\pi$$
0.765735 0.643157i $$-0.222376\pi$$
$$968$$ 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 $$0.0614834\pi$$
−0.981403 + 0.191957i $$0.938517\pi$$
$$978$$ 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 $$-0.805348\pi$$
0.818778 0.574111i $$-0.194652\pi$$
$$984$$ 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 $$-0.918463\pi$$
0.967371 0.253363i $$-0.0815366\pi$$
$$998$$ − 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.274.2 2
3.2 odd 2 2925.2.c.d.2224.1 2
5.2 odd 4 975.2.a.b.1.1 1
5.3 odd 4 195.2.a.d.1.1 1
5.4 even 2 inner 975.2.c.b.274.1 2
15.2 even 4 2925.2.a.t.1.1 1
15.8 even 4 585.2.a.a.1.1 1
15.14 odd 2 2925.2.c.d.2224.2 2
20.3 even 4 3120.2.a.n.1.1 1
35.13 even 4 9555.2.a.t.1.1 1
60.23 odd 4 9360.2.a.w.1.1 1
65.38 odd 4 2535.2.a.b.1.1 1
195.38 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 5.3 odd 4
585.2.a.a.1.1 1 15.8 even 4
975.2.a.b.1.1 1 5.2 odd 4
975.2.c.b.274.1 2 5.4 even 2 inner
975.2.c.b.274.2 2 1.1 even 1 trivial
2535.2.a.b.1.1 1 65.38 odd 4
2925.2.a.t.1.1 1 15.2 even 4
2925.2.c.d.2224.1 2 3.2 odd 2
2925.2.c.d.2224.2 2 15.14 odd 2
3120.2.a.n.1.1 1 20.3 even 4
7605.2.a.v.1.1 1 195.38 even 4
9360.2.a.w.1.1 1 60.23 odd 4
9555.2.a.t.1.1 1 35.13 even 4