# Properties

 Label 2925.2.c.d.2224.2 Level $2925$ Weight $2$ Character 2925.2224 Analytic conductor $23.356$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2925,2,Mod(2224,2925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2925, 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("2925.2224");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2925 = 3^{2} \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2925.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: $$23.3562425912$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 195) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 2224.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2925.2224 Dual form 2925.2.c.d.2224.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.00000i q^{2} -2.00000 q^{4} +3.00000i q^{7} +O(q^{10})$$ $$q+2.00000i q^{2} -2.00000 q^{4} +3.00000i q^{7} +5.00000 q^{11} +1.00000i q^{13} -6.00000 q^{14} -4.00000 q^{16} +5.00000i q^{17} -2.00000 q^{19} +10.0000i q^{22} +1.00000i q^{23} -2.00000 q^{26} -6.00000i q^{28} +10.0000 q^{29} -2.00000 q^{31} -8.00000i q^{32} -10.0000 q^{34} +3.00000i q^{37} -4.00000i q^{38} +9.00000 q^{41} -4.00000i q^{43} -10.0000 q^{44} -2.00000 q^{46} +10.0000i q^{47} -2.00000 q^{49} -2.00000i q^{52} -9.00000i q^{53} +20.0000i q^{58} -11.0000 q^{61} -4.00000i q^{62} +8.00000 q^{64} +4.00000i q^{67} -10.0000i q^{68} -15.0000 q^{71} +6.00000i q^{73} -6.00000 q^{74} +4.00000 q^{76} +15.0000i q^{77} +11.0000 q^{79} +18.0000i q^{82} -8.00000i q^{83} +8.00000 q^{86} -11.0000 q^{89} -3.00000 q^{91} -2.00000i q^{92} -20.0000 q^{94} +9.00000i q^{97} -4.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{4}+O(q^{10})$$ 2 * q - 4 * q^4 $$2 q - 4 q^{4} + 10 q^{11} - 12 q^{14} - 8 q^{16} - 4 q^{19} - 4 q^{26} + 20 q^{29} - 4 q^{31} - 20 q^{34} + 18 q^{41} - 20 q^{44} - 4 q^{46} - 4 q^{49} - 22 q^{61} + 16 q^{64} - 30 q^{71} - 12 q^{74} + 8 q^{76} + 22 q^{79} + 16 q^{86} - 22 q^{89} - 6 q^{91} - 40 q^{94}+O(q^{100})$$ 2 * q - 4 * q^4 + 10 * q^11 - 12 * q^14 - 8 * q^16 - 4 * q^19 - 4 * q^26 + 20 * q^29 - 4 * q^31 - 20 * q^34 + 18 * q^41 - 20 * q^44 - 4 * q^46 - 4 * q^49 - 22 * q^61 + 16 * q^64 - 30 * q^71 - 12 * q^74 + 8 * q^76 + 22 * q^79 + 16 * q^86 - 22 * q^89 - 6 * q^91 - 40 * q^94

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2925\mathbb{Z}\right)^\times$$.

 $$n$$ $$326$$ $$352$$ $$2251$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$0.750000\pi$$
$$3$$ 0 0
$$4$$ −2.00000 −1.00000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 3.00000i 1.13389i 0.823754 + 0.566947i $$0.191875\pi$$
−0.823754 + 0.566947i $$0.808125\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 5.00000 1.50756 0.753778 0.657129i $$-0.228229\pi$$
0.753778 + 0.657129i $$0.228229\pi$$
$$12$$ 0 0
$$13$$ 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 $$0.792637\pi$$
$$18$$ 0 0
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 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
$$27$$ 0 0
$$28$$ − 6.00000i − 1.13389i
$$29$$ 10.0000 1.85695 0.928477 0.371391i $$-0.121119\pi$$
0.928477 + 0.371391i $$0.121119\pi$$
$$30$$ 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
$$33$$ 0 0
$$34$$ −10.0000 −1.71499
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 3.00000i 0.493197i 0.969118 + 0.246598i $$0.0793129\pi$$
−0.969118 + 0.246598i $$0.920687\pi$$
$$38$$ − 4.00000i − 0.648886i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 9.00000 1.40556 0.702782 0.711405i $$-0.251941\pi$$
0.702782 + 0.711405i $$0.251941\pi$$
$$42$$ 0 0
$$43$$ − 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 $$0.260166\pi$$
−0.684167 + 0.729325i $$0.739834\pi$$
$$48$$ 0 0
$$49$$ −2.00000 −0.285714
$$50$$ 0 0
$$51$$ 0 0
$$52$$ − 2.00000i − 0.277350i
$$53$$ − 9.00000i − 1.23625i −0.786082 0.618123i $$-0.787894\pi$$
0.786082 0.618123i $$-0.212106\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 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
$$63$$ 0 0
$$64$$ 8.00000 1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 4.00000i 0.488678i 0.969690 + 0.244339i $$0.0785709\pi$$
−0.969690 + 0.244339i $$0.921429\pi$$
$$68$$ − 10.0000i − 1.21268i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −15.0000 −1.78017 −0.890086 0.455792i $$-0.849356\pi$$
−0.890086 + 0.455792i $$0.849356\pi$$
$$72$$ 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
$$78$$ 0 0
$$79$$ 11.0000 1.23760 0.618798 0.785550i $$-0.287620\pi$$
0.618798 + 0.785550i $$0.287620\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 18.0000i 1.98777i
$$83$$ − 8.00000i − 0.878114i −0.898459 0.439057i $$-0.855313\pi$$
0.898459 0.439057i $$-0.144687\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 8.00000 0.862662
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −11.0000 −1.16600 −0.582999 0.812473i $$-0.698121\pi$$
−0.582999 + 0.812473i $$0.698121\pi$$
$$90$$ 0 0
$$91$$ −3.00000 −0.314485
$$92$$ − 2.00000i − 0.208514i
$$93$$ 0 0
$$94$$ −20.0000 −2.06284
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 9.00000i 0.913812i 0.889515 + 0.456906i $$0.151042\pi$$
−0.889515 + 0.456906i $$0.848958\pi$$
$$98$$ − 4.00000i − 0.404061i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 12.0000 1.19404 0.597022 0.802225i $$-0.296350\pi$$
0.597022 + 0.802225i $$0.296350\pi$$
$$102$$ 0 0
$$103$$ 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 $$0.0463216\pi$$
−0.989430 + 0.145010i $$0.953678\pi$$
$$108$$ 0 0
$$109$$ −16.0000 −1.53252 −0.766261 0.642529i $$-0.777885\pi$$
−0.766261 + 0.642529i $$0.777885\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ − 12.0000i − 1.13389i
$$113$$ − 2.00000i − 0.188144i −0.995565 0.0940721i $$-0.970012\pi$$
0.995565 0.0940721i $$-0.0299884\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −20.0000 −1.85695
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −15.0000 −1.37505
$$120$$ 0 0
$$121$$ 14.0000 1.27273
$$122$$ − 22.0000i − 1.99179i
$$123$$ 0 0
$$124$$ 4.00000 0.359211
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 14.0000i − 1.24230i −0.783692 0.621150i $$-0.786666\pi$$
0.783692 0.621150i $$-0.213334\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −6.00000 −0.524222 −0.262111 0.965038i $$-0.584419\pi$$
−0.262111 + 0.965038i $$0.584419\pi$$
$$132$$ 0 0
$$133$$ − 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 $$0.917494\pi$$
$$138$$ 0 0
$$139$$ 17.0000 1.44192 0.720961 0.692976i $$-0.243701\pi$$
0.720961 + 0.692976i $$0.243701\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ − 30.0000i − 2.51754i
$$143$$ 5.00000i 0.418121i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −12.0000 −0.993127
$$147$$ 0 0
$$148$$ − 6.00000i − 0.493197i
$$149$$ −7.00000 −0.573462 −0.286731 0.958011i $$-0.592569\pi$$
−0.286731 + 0.958011i $$0.592569\pi$$
$$150$$ 0 0
$$151$$ −12.0000 −0.976546 −0.488273 0.872691i $$-0.662373\pi$$
−0.488273 + 0.872691i $$0.662373\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ −30.0000 −2.41747
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 22.0000i − 1.75579i −0.478852 0.877896i $$-0.658947\pi$$
0.478852 0.877896i $$-0.341053\pi$$
$$158$$ 22.0000i 1.75023i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −3.00000 −0.236433
$$162$$ 0 0
$$163$$ − 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 $$-0.899829\pi$$
0.950890 0.309529i $$-0.100171\pi$$
$$168$$ 0 0
$$169$$ −1.00000 −0.0769231
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 8.00000i 0.609994i
$$173$$ 2.00000i 0.152057i 0.997106 + 0.0760286i $$0.0242240\pi$$
−0.997106 + 0.0760286i $$0.975776\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −20.0000 −1.50756
$$177$$ 0 0
$$178$$ − 22.0000i − 1.64897i
$$179$$ −6.00000 −0.448461 −0.224231 0.974536i $$-0.571987\pi$$
−0.224231 + 0.974536i $$0.571987\pi$$
$$180$$ 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
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 25.0000i 1.82818i
$$188$$ − 20.0000i − 1.45865i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 20.0000 1.44715 0.723575 0.690246i $$-0.242498\pi$$
0.723575 + 0.690246i $$0.242498\pi$$
$$192$$ 0 0
$$193$$ 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 $$-0.859401\pi$$
0.904024 0.427482i $$-0.140599\pi$$
$$198$$ 0 0
$$199$$ −4.00000 −0.283552 −0.141776 0.989899i $$-0.545281\pi$$
−0.141776 + 0.989899i $$0.545281\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 24.0000i 1.68863i
$$203$$ 30.0000i 2.10559i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −8.00000 −0.557386
$$207$$ 0 0
$$208$$ − 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
$$213$$ 0 0
$$214$$ −6.00000 −0.410152
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 6.00000i − 0.407307i
$$218$$ − 32.0000i − 2.16731i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −5.00000 −0.336336
$$222$$ 0 0
$$223$$ 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 $$-0.203780\pi$$
$$228$$ 0 0
$$229$$ 14.0000 0.925146 0.462573 0.886581i $$-0.346926\pi$$
0.462573 + 0.886581i $$0.346926\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 25.0000i 1.63780i 0.573933 + 0.818902i $$0.305417\pi$$
−0.573933 + 0.818902i $$0.694583\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ − 30.0000i − 1.94461i
$$239$$ 15.0000 0.970269 0.485135 0.874439i $$-0.338771\pi$$
0.485135 + 0.874439i $$0.338771\pi$$
$$240$$ 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
$$243$$ 0 0
$$244$$ 22.0000 1.40841
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 2.00000i − 0.127257i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −20.0000 −1.26239 −0.631194 0.775625i $$-0.717435\pi$$
−0.631194 + 0.775625i $$0.717435\pi$$
$$252$$ 0 0
$$253$$ 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 $$0.810261\pi$$
$$258$$ 0 0
$$259$$ −9.00000 −0.559233
$$260$$ 0 0
$$261$$ 0 0
$$262$$ − 12.0000i − 0.741362i
$$263$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 12.0000 0.735767
$$267$$ 0 0
$$268$$ − 8.00000i − 0.488678i
$$269$$ 32.0000 1.95107 0.975537 0.219834i $$-0.0705517\pi$$
0.975537 + 0.219834i $$0.0705517\pi$$
$$270$$ 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
$$273$$ 0 0
$$274$$ −12.0000 −0.724947
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 26.0000i − 1.56219i −0.624413 0.781094i $$-0.714662\pi$$
0.624413 0.781094i $$-0.285338\pi$$
$$278$$ 34.0000i 2.03918i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 10.0000 0.596550 0.298275 0.954480i $$-0.403589\pi$$
0.298275 + 0.954480i $$0.403589\pi$$
$$282$$ 0 0
$$283$$ − 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
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ 0 0
$$292$$ − 12.0000i − 0.702247i
$$293$$ − 24.0000i − 1.40209i −0.713115 0.701047i $$-0.752716\pi$$
0.713115 0.701047i $$-0.247284\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ − 14.0000i − 0.810998i
$$299$$ −1.00000 −0.0578315
$$300$$ 0 0
$$301$$ 12.0000 0.691669
$$302$$ − 24.0000i − 1.38104i
$$303$$ 0 0
$$304$$ 8.00000 0.458831
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 19.0000i 1.08439i 0.840254 + 0.542194i $$0.182406\pi$$
−0.840254 + 0.542194i $$0.817594\pi$$
$$308$$ − 30.0000i − 1.70941i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −24.0000 −1.36092 −0.680458 0.732787i $$-0.738219\pi$$
−0.680458 + 0.732787i $$0.738219\pi$$
$$312$$ 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 $$0.109410\pi$$
−0.941507 + 0.336994i $$0.890590\pi$$
$$318$$ 0 0
$$319$$ 50.0000 2.79946
$$320$$ 0 0
$$321$$ 0 0
$$322$$ − 6.00000i − 0.334367i
$$323$$ − 10.0000i − 0.556415i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 22.0000 1.21847
$$327$$ 0 0
$$328$$ 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
$$333$$ 0 0
$$334$$ 16.0000 0.875481
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 4.00000i − 0.217894i −0.994048 0.108947i $$-0.965252\pi$$
0.994048 0.108947i $$-0.0347479\pi$$
$$338$$ − 2.00000i − 0.108786i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −10.0000 −0.541530
$$342$$ 0 0
$$343$$ 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 $$-0.00854491\pi$$
$$348$$ 0 0
$$349$$ −20.0000 −1.07058 −0.535288 0.844670i $$-0.679797\pi$$
−0.535288 + 0.844670i $$0.679797\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ − 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
$$357$$ 0 0
$$358$$ − 12.0000i − 0.634220i
$$359$$ 16.0000 0.844448 0.422224 0.906492i $$-0.361250\pi$$
0.422224 + 0.906492i $$0.361250\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ − 46.0000i − 2.41771i
$$363$$ 0 0
$$364$$ 6.00000 0.314485
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 8.00000i − 0.417597i −0.977959 0.208798i $$-0.933045\pi$$
0.977959 0.208798i $$-0.0669552\pi$$
$$368$$ − 4.00000i − 0.208514i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 27.0000 1.40177
$$372$$ 0 0
$$373$$ 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
$$378$$ 0 0
$$379$$ −6.00000 −0.308199 −0.154100 0.988055i $$-0.549248\pi$$
−0.154100 + 0.988055i $$0.549248\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 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
$$387$$ 0 0
$$388$$ − 18.0000i − 0.913812i
$$389$$ −24.0000 −1.21685 −0.608424 0.793612i $$-0.708198\pi$$
−0.608424 + 0.793612i $$0.708198\pi$$
$$390$$ 0 0
$$391$$ −5.00000 −0.252861
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 24.0000 1.20910
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 19.0000i 0.953583i 0.879017 + 0.476791i $$0.158200\pi$$
−0.879017 + 0.476791i $$0.841800\pi$$
$$398$$ − 8.00000i − 0.401004i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ 0 0
$$403$$ − 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
$$411$$ 0 0
$$412$$ − 8.00000i − 0.394132i
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 8.00000 0.392232
$$417$$ 0 0
$$418$$ − 20.0000i − 0.978232i
$$419$$ 26.0000 1.27018 0.635092 0.772437i $$-0.280962\pi$$
0.635092 + 0.772437i $$0.280962\pi$$
$$420$$ 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
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 33.0000i − 1.59698i
$$428$$ − 6.00000i − 0.290021i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 16.0000 0.770693 0.385346 0.922772i $$-0.374082\pi$$
0.385346 + 0.922772i $$0.374082\pi$$
$$432$$ 0 0
$$433$$ 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
$$438$$ 0 0
$$439$$ 33.0000 1.57500 0.787502 0.616312i $$-0.211374\pi$$
0.787502 + 0.616312i $$0.211374\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ − 10.0000i − 0.475651i
$$443$$ − 35.0000i − 1.66290i −0.555599 0.831450i $$-0.687511\pi$$
0.555599 0.831450i $$-0.312489\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 24.0000i 1.13389i
$$449$$ 15.0000 0.707894 0.353947 0.935266i $$-0.384839\pi$$
0.353947 + 0.935266i $$0.384839\pi$$
$$450$$ 0 0
$$451$$ 45.0000 2.11897
$$452$$ 4.00000i 0.188144i
$$453$$ 0 0
$$454$$ 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
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −3.00000 −0.139724 −0.0698620 0.997557i $$-0.522256\pi$$
−0.0698620 + 0.997557i $$0.522256\pi$$
$$462$$ 0 0
$$463$$ 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 $$-0.765874\pi$$
0.741475 0.670980i $$-0.234126\pi$$
$$468$$ 0 0
$$469$$ −12.0000 −0.554109
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ − 20.0000i − 0.919601i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 30.0000 1.37505
$$477$$ 0 0
$$478$$ 30.0000i 1.37217i
$$479$$ 5.00000 0.228456 0.114228 0.993455i $$-0.463561\pi$$
0.114228 + 0.993455i $$0.463561\pi$$
$$480$$ 0 0
$$481$$ −3.00000 −0.136788
$$482$$ − 28.0000i − 1.27537i
$$483$$ 0 0
$$484$$ −28.0000 −1.27273
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 7.00000i − 0.317200i −0.987343 0.158600i $$-0.949302\pi$$
0.987343 0.158600i $$-0.0506981\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −16.0000 −0.722070 −0.361035 0.932552i $$-0.617576\pi$$
−0.361035 + 0.932552i $$0.617576\pi$$
$$492$$ 0 0
$$493$$ 50.0000i 2.25189i
$$494$$ 4.00000 0.179969
$$495$$ 0 0
$$496$$ 8.00000 0.359211
$$497$$ − 45.0000i − 2.01853i
$$498$$ 0 0
$$499$$ −34.0000 −1.52205 −0.761025 0.648723i $$-0.775303\pi$$
−0.761025 + 0.648723i $$0.775303\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ − 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
$$507$$ 0 0
$$508$$ 28.0000i 1.24230i
$$509$$ 21.0000 0.930809 0.465404 0.885098i $$-0.345909\pi$$
0.465404 + 0.885098i $$0.345909\pi$$
$$510$$ 0 0
$$511$$ −18.0000 −0.796273
$$512$$ 32.0000i 1.41421i
$$513$$ 0 0
$$514$$ −36.0000 −1.58789
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 50.0000i 2.19900i
$$518$$ − 18.0000i − 0.790875i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 22.0000 0.963837 0.481919 0.876216i $$-0.339940\pi$$
0.481919 + 0.876216i $$0.339940\pi$$
$$522$$ 0 0
$$523$$ − 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
$$528$$ 0 0
$$529$$ 22.0000 0.956522
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 12.0000i 0.520266i
$$533$$ 9.00000i 0.389833i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 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
$$543$$ 0 0
$$544$$ 40.0000 1.71499
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$548$$ − 12.0000i − 0.512615i
$$549$$ 0 0
$$550$$ 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 $$-0.219235\pi$$
$$558$$ 0 0
$$559$$ 4.00000 0.169182
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 20.0000i 0.843649i
$$563$$ 41.0000i 1.72794i 0.503540 + 0.863972i $$0.332031\pi$$
−0.503540 + 0.863972i $$0.667969\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 16.0000 0.672530
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 16.0000 0.670755 0.335377 0.942084i $$-0.391136\pi$$
0.335377 + 0.942084i $$0.391136\pi$$
$$570$$ 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
$$573$$ 0 0
$$574$$ −54.0000 −2.25392
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 21.0000i 0.874241i 0.899403 + 0.437121i $$0.144002\pi$$
−0.899403 + 0.437121i $$0.855998\pi$$
$$578$$ − 16.0000i − 0.665512i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 24.0000 0.995688
$$582$$ 0 0
$$583$$ − 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 $$0.666197\pi$$
$$588$$ 0 0
$$589$$ 4.00000 0.164817
$$590$$ 0 0
$$591$$ 0 0
$$592$$ − 12.0000i − 0.493197i
$$593$$ 36.0000i 1.47834i 0.673517 + 0.739171i $$0.264783\pi$$
−0.673517 + 0.739171i $$0.735217\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 14.0000 0.573462
$$597$$ 0 0
$$598$$ − 2.00000i − 0.0817861i
$$599$$ 4.00000 0.163436 0.0817178 0.996656i $$-0.473959\pi$$
0.0817178 + 0.996656i $$0.473959\pi$$
$$600$$ 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
$$603$$ 0 0
$$604$$ 24.0000 0.976546
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 40.0000i 1.62355i 0.583970 + 0.811775i $$0.301498\pi$$
−0.583970 + 0.811775i $$0.698502\pi$$
$$608$$ 16.0000i 0.648886i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −10.0000 −0.404557
$$612$$ 0 0
$$613$$ 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 $$0.146030\pi$$
−0.896599 + 0.442843i $$0.853970\pi$$
$$618$$ 0 0
$$619$$ 2.00000 0.0803868 0.0401934 0.999192i $$-0.487203\pi$$
0.0401934 + 0.999192i $$0.487203\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ − 48.0000i − 1.92462i
$$623$$ − 33.0000i − 1.32212i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 20.0000 0.799361
$$627$$ 0 0
$$628$$ 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
$$633$$ 0 0
$$634$$ −24.0000 −0.953162
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 2.00000i − 0.0792429i
$$638$$ 100.000i 3.95904i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 36.0000 1.42191 0.710957 0.703235i $$-0.248262\pi$$
0.710957 + 0.703235i $$0.248262\pi$$
$$642$$ 0 0
$$643$$ 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 $$0.135448\pi$$
−0.910823 + 0.412798i $$0.864552\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 22.0000i 0.861586i
$$653$$ 30.0000i 1.17399i 0.809590 + 0.586995i $$0.199689\pi$$
−0.809590 + 0.586995i $$0.800311\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −36.0000 −1.40556
$$657$$ 0 0
$$658$$ − 60.0000i − 2.33904i
$$659$$ 12.0000 0.467454 0.233727 0.972302i $$-0.424908\pi$$
0.233727 + 0.972302i $$0.424908\pi$$
$$660$$ 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
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 10.0000i 0.387202i
$$668$$ 16.0000i 0.619059i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −55.0000 −2.12325
$$672$$ 0 0
$$673$$ 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 $$-0.957052\pi$$
0.990911 0.134516i $$-0.0429479\pi$$
$$678$$ 0 0
$$679$$ −27.0000 −1.03616
$$680$$ 0 0
$$681$$ 0 0
$$682$$ − 20.0000i − 0.765840i
$$683$$ − 4.00000i − 0.153056i −0.997067 0.0765279i $$-0.975617\pi$$
0.997067 0.0765279i $$-0.0243834\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −30.0000 −1.14541
$$687$$ 0 0
$$688$$ 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
$$693$$ 0 0
$$694$$ 2.00000 0.0759190
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 45.0000i 1.70450i
$$698$$ − 40.0000i − 1.51402i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 4.00000 0.151078 0.0755390 0.997143i $$-0.475932\pi$$
0.0755390 + 0.997143i $$0.475932\pi$$
$$702$$ 0 0
$$703$$ − 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
$$711$$ 0 0
$$712$$ 0 0
$$713$$ − 2.00000i − 0.0749006i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 12.0000 0.448461
$$717$$ 0 0
$$718$$ 32.0000i 1.19423i
$$719$$ 12.0000 0.447524 0.223762 0.974644i $$-0.428166\pi$$
0.223762 + 0.974644i $$0.428166\pi$$
$$720$$ 0 0
$$721$$ −12.0000 −0.446903
$$722$$ − 30.0000i − 1.11648i
$$723$$ 0 0
$$724$$ 46.0000 1.70958
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 6.00000i − 0.222528i −0.993791 0.111264i $$-0.964510\pi$$
0.993791 0.111264i $$-0.0354899\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 20.0000 0.739727
$$732$$ 0 0
$$733$$ − 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
$$738$$ 0 0
$$739$$ −38.0000 −1.39785 −0.698926 0.715194i $$-0.746338\pi$$
−0.698926 + 0.715194i $$0.746338\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 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
$$747$$ 0 0
$$748$$ − 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
$$753$$ 0 0
$$754$$ −20.0000 −0.728357
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 36.0000i − 1.30844i −0.756303 0.654221i $$-0.772997\pi$$
0.756303 0.654221i $$-0.227003\pi$$
$$758$$ − 12.0000i − 0.435860i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 14.0000 0.507500 0.253750 0.967270i $$-0.418336\pi$$
0.253750 + 0.967270i $$0.418336\pi$$
$$762$$ 0 0
$$763$$ − 48.0000i − 1.73772i
$$764$$ −40.0000 −1.44715
$$765$$ 0 0
$$766$$ −36.0000 −1.30073
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 12.0000 0.432731 0.216366 0.976312i $$-0.430580\pi$$
0.216366 + 0.976312i $$0.430580\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ − 26.0000i − 0.935760i
$$773$$ − 24.0000i − 0.863220i −0.902060 0.431610i $$-0.857946\pi$$
0.902060 0.431610i $$-0.142054\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ − 48.0000i − 1.72088i
$$779$$ −18.0000 −0.644917
$$780$$ 0 0
$$781$$ −75.0000 −2.68371
$$782$$ − 10.0000i − 0.357599i
$$783$$ 0 0
$$784$$ 8.00000 0.285714
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 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 $$-0.971775\pi$$
0.996071 0.0885545i $$-0.0282248\pi$$
$$798$$ 0 0
$$799$$ −50.0000 −1.76887
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 36.0000i 1.27120i
$$803$$ 30.0000i 1.05868i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 4.00000 0.140894
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −6.00000 −0.210949 −0.105474 0.994422i $$-0.533636\pi$$
−0.105474 + 0.994422i $$0.533636\pi$$
$$810$$ 0 0
$$811$$ −4.00000 −0.140459 −0.0702295 0.997531i $$-0.522373\pi$$
−0.0702295 + 0.997531i $$0.522373\pi$$
$$812$$ − 60.0000i − 2.10559i
$$813$$ 0 0
$$814$$ −30.0000 −1.05150
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 8.00000i 0.279885i
$$818$$ 52.0000i 1.81814i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 41.0000 1.43091 0.715455 0.698659i $$-0.246219\pi$$
0.715455 + 0.698659i $$0.246219\pi$$
$$822$$ 0 0
$$823$$ − 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 $$-0.739408\pi$$
0.683189 0.730242i $$-0.260592\pi$$
$$828$$ 0 0
$$829$$ −14.0000 −0.486240 −0.243120 0.969996i $$-0.578171\pi$$
−0.243120 + 0.969996i $$0.578171\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 8.00000i 0.277350i
$$833$$ − 10.0000i − 0.346479i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 20.0000 0.691714
$$837$$ 0 0
$$838$$ 52.0000i 1.79631i
$$839$$ 7.00000 0.241667 0.120833 0.992673i $$-0.461443\pi$$
0.120833 + 0.992673i $$0.461443\pi$$
$$840$$ 0 0
$$841$$ 71.0000 2.44828
$$842$$ 64.0000i 2.20559i
$$843$$ 0 0
$$844$$ 8.00000 0.275371
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 42.0000i 1.44314i
$$848$$ 36.0000i 1.23625i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −3.00000 −0.102839
$$852$$ 0 0
$$853$$ 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 $$-0.906227\pi$$
0.956919 0.290354i $$-0.0937732\pi$$
$$858$$ 0 0
$$859$$ −35.0000 −1.19418 −0.597092 0.802173i $$-0.703677\pi$$
−0.597092 + 0.802173i $$0.703677\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 32.0000i 1.08992i
$$863$$ 22.0000i 0.748889i 0.927249 + 0.374444i $$0.122167\pi$$
−0.927249 + 0.374444i $$0.877833\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −48.0000 −1.63111
$$867$$ 0 0
$$868$$ 12.0000i 0.407307i
$$869$$ 55.0000 1.86575
$$870$$ 0 0
$$871$$ −4.00000 −0.135535
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 4.00000 0.135302
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 22.0000i − 0.742887i −0.928456 0.371444i $$-0.878863\pi$$
0.928456 0.371444i $$-0.121137\pi$$
$$878$$ 66.0000i 2.22739i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 14.0000 0.471672 0.235836 0.971793i $$-0.424217\pi$$
0.235836 + 0.971793i $$0.424217\pi$$
$$882$$ 0 0
$$883$$ 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 $$-0.918969\pi$$
0.967773 0.251825i $$-0.0810309\pi$$
$$888$$ 0 0
$$889$$ 42.0000 1.40863
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ − 20.0000i − 0.669274i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 30.0000i 1.00111i
$$899$$ −20.0000 −0.667037
$$900$$ 0 0
$$901$$ 45.0000 1.49917
$$902$$ 90.0000i 2.99667i
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 2.00000i − 0.0664089i −0.999449 0.0332045i $$-0.989429\pi$$
0.999449 0.0332045i $$-0.0105712\pi$$
$$908$$ 36.0000i 1.19470i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −8.00000 −0.265052 −0.132526 0.991180i $$-0.542309\pi$$
−0.132526 + 0.991180i $$0.542309\pi$$
$$912$$ 0 0
$$913$$ − 40.0000i − 1.32381i
$$914$$ −26.0000 −0.860004
$$915$$ 0 0
$$916$$ −28.0000 −0.925146
$$917$$ − 18.0000i − 0.594412i
$$918$$ 0 0
$$919$$ −29.0000 −0.956622 −0.478311 0.878191i $$-0.658751\pi$$
−0.478311 + 0.878191i $$0.658751\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ − 6.00000i − 0.197599i
$$923$$ − 15.0000i − 0.493731i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −10.0000 −0.328620
$$927$$ 0 0
$$928$$ − 80.0000i − 2.62613i
$$929$$ 21.0000 0.688988 0.344494 0.938789i $$-0.388051\pi$$
0.344494 + 0.938789i $$0.388051\pi$$
$$930$$ 0 0
$$931$$ 4.00000 0.131095
$$932$$ − 50.0000i − 1.63780i
$$933$$ 0 0
$$934$$ 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
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 23.0000 0.749779 0.374889 0.927070i $$-0.377681\pi$$
0.374889 + 0.927070i $$0.377681\pi$$
$$942$$ 0 0
$$943$$ 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 $$0.801125\pi$$
$$948$$ 0 0
$$949$$ −6.00000 −0.194768
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 11.0000i 0.356325i 0.984001 + 0.178162i $$0.0570153\pi$$
−0.984001 + 0.178162i $$0.942985\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −30.0000 −0.970269
$$957$$ 0 0
$$958$$ 10.0000i 0.323085i
$$959$$ −18.0000 −0.581250
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ − 6.00000i − 0.193448i
$$963$$ 0 0
$$964$$ 28.0000 0.901819
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 40.0000i 1.28631i 0.765735 + 0.643157i $$0.222376\pi$$
−0.765735 + 0.643157i $$0.777624\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$972$$ 0 0
$$973$$ 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 $$0.938517\pi$$
$$978$$ 0 0
$$979$$ −55.0000 −1.75781
$$980$$ 0 0
$$981$$ 0 0
$$982$$ − 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
$$987$$ 0 0
$$988$$ 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
$$993$$ 0 0
$$994$$ 90.0000 2.85463
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 16.0000i 0.506725i 0.967371 + 0.253363i $$0.0815366\pi$$
−0.967371 + 0.253363i $$0.918463\pi$$
$$998$$ − 68.0000i − 2.15250i
$$999$$ 0 0
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 2925.2.c.d.2224.2 2
3.2 odd 2 975.2.c.b.274.1 2
5.2 odd 4 585.2.a.a.1.1 1
5.3 odd 4 2925.2.a.t.1.1 1
5.4 even 2 inner 2925.2.c.d.2224.1 2
15.2 even 4 195.2.a.d.1.1 1
15.8 even 4 975.2.a.b.1.1 1
15.14 odd 2 975.2.c.b.274.2 2
20.7 even 4 9360.2.a.w.1.1 1
60.47 odd 4 3120.2.a.n.1.1 1
65.12 odd 4 7605.2.a.v.1.1 1
105.62 odd 4 9555.2.a.t.1.1 1
195.77 even 4 2535.2.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.a.d.1.1 1 15.2 even 4
585.2.a.a.1.1 1 5.2 odd 4
975.2.a.b.1.1 1 15.8 even 4
975.2.c.b.274.1 2 3.2 odd 2
975.2.c.b.274.2 2 15.14 odd 2
2535.2.a.b.1.1 1 195.77 even 4
2925.2.a.t.1.1 1 5.3 odd 4
2925.2.c.d.2224.1 2 5.4 even 2 inner
2925.2.c.d.2224.2 2 1.1 even 1 trivial
3120.2.a.n.1.1 1 60.47 odd 4
7605.2.a.v.1.1 1 65.12 odd 4
9360.2.a.w.1.1 1 20.7 even 4
9555.2.a.t.1.1 1 105.62 odd 4