# Properties

 Label 1850.2.b.d.149.2 Level $1850$ Weight $2$ Character 1850.149 Analytic conductor $14.772$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1850,2,Mod(149,1850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1850, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1850.149");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1850 = 2 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1850.b (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: $$14.7723243739$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 370) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 149.2 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1850.149 Dual form 1850.2.b.d.149.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{8} +3.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{8} +3.00000 q^{9} -4.00000 q^{11} +2.00000i q^{13} +1.00000 q^{16} +2.00000i q^{17} +3.00000i q^{18} +4.00000 q^{19} -4.00000i q^{22} -2.00000 q^{26} +6.00000 q^{29} -4.00000 q^{31} +1.00000i q^{32} -2.00000 q^{34} -3.00000 q^{36} +1.00000i q^{37} +4.00000i q^{38} -6.00000 q^{41} +4.00000i q^{43} +4.00000 q^{44} +8.00000i q^{47} +7.00000 q^{49} -2.00000i q^{52} +10.0000i q^{53} +6.00000i q^{58} -4.00000 q^{59} +10.0000 q^{61} -4.00000i q^{62} -1.00000 q^{64} +8.00000i q^{67} -2.00000i q^{68} -3.00000i q^{72} +10.0000i q^{73} -1.00000 q^{74} -4.00000 q^{76} +4.00000 q^{79} +9.00000 q^{81} -6.00000i q^{82} -4.00000 q^{86} +4.00000i q^{88} -2.00000 q^{89} -8.00000 q^{94} -6.00000i q^{97} +7.00000i q^{98} -12.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 6 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 + 6 * q^9 $$2 q - 2 q^{4} + 6 q^{9} - 8 q^{11} + 2 q^{16} + 8 q^{19} - 4 q^{26} + 12 q^{29} - 8 q^{31} - 4 q^{34} - 6 q^{36} - 12 q^{41} + 8 q^{44} + 14 q^{49} - 8 q^{59} + 20 q^{61} - 2 q^{64} - 2 q^{74} - 8 q^{76} + 8 q^{79} + 18 q^{81} - 8 q^{86} - 4 q^{89} - 16 q^{94} - 24 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 + 6 * q^9 - 8 * q^11 + 2 * q^16 + 8 * q^19 - 4 * q^26 + 12 * q^29 - 8 * q^31 - 4 * q^34 - 6 * q^36 - 12 * q^41 + 8 * q^44 + 14 * q^49 - 8 * q^59 + 20 * q^61 - 2 * q^64 - 2 * q^74 - 8 * q^76 + 8 * q^79 + 18 * q^81 - 8 * q^86 - 4 * q^89 - 16 * q^94 - 24 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times$$.

 $$n$$ $$1001$$ $$1777$$ $$\chi(n)$$ $$1$$ $$-1$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 0.707107i
$$3$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ 3.00000 1.00000
$$10$$ 0 0
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ 0 0
$$13$$ 2.00000i 0.554700i 0.960769 + 0.277350i $$0.0894562\pi$$
−0.960769 + 0.277350i $$0.910544\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 2.00000i 0.485071i 0.970143 + 0.242536i $$0.0779791\pi$$
−0.970143 + 0.242536i $$0.922021\pi$$
$$18$$ 3.00000i 0.707107i
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ − 4.00000i − 0.852803i
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −2.00000 −0.392232
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 0 0
$$34$$ −2.00000 −0.342997
$$35$$ 0 0
$$36$$ −3.00000 −0.500000
$$37$$ 1.00000i 0.164399i
$$38$$ 4.00000i 0.648886i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ 4.00000i 0.609994i 0.952353 + 0.304997i $$0.0986555\pi$$
−0.952353 + 0.304997i $$0.901344\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 8.00000i 1.16692i 0.812142 + 0.583460i $$0.198301\pi$$
−0.812142 + 0.583460i $$0.801699\pi$$
$$48$$ 0 0
$$49$$ 7.00000 1.00000
$$50$$ 0 0
$$51$$ 0 0
$$52$$ − 2.00000i − 0.277350i
$$53$$ 10.0000i 1.37361i 0.726844 + 0.686803i $$0.240986\pi$$
−0.726844 + 0.686803i $$0.759014\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 6.00000i 0.787839i
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ 10.0000 1.28037 0.640184 0.768221i $$-0.278858\pi$$
0.640184 + 0.768221i $$0.278858\pi$$
$$62$$ − 4.00000i − 0.508001i
$$63$$ 0 0
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 8.00000i 0.977356i 0.872464 + 0.488678i $$0.162521\pi$$
−0.872464 + 0.488678i $$0.837479\pi$$
$$68$$ − 2.00000i − 0.242536i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ − 3.00000i − 0.353553i
$$73$$ 10.0000i 1.17041i 0.810885 + 0.585206i $$0.198986\pi$$
−0.810885 + 0.585206i $$0.801014\pi$$
$$74$$ −1.00000 −0.116248
$$75$$ 0 0
$$76$$ −4.00000 −0.458831
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 4.00000 0.450035 0.225018 0.974355i $$-0.427756\pi$$
0.225018 + 0.974355i $$0.427756\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ − 6.00000i − 0.662589i
$$83$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −4.00000 −0.431331
$$87$$ 0 0
$$88$$ 4.00000i 0.426401i
$$89$$ −2.00000 −0.212000 −0.106000 0.994366i $$-0.533804\pi$$
−0.106000 + 0.994366i $$0.533804\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ −8.00000 −0.825137
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 6.00000i − 0.609208i −0.952479 0.304604i $$-0.901476\pi$$
0.952479 0.304604i $$-0.0985241\pi$$
$$98$$ 7.00000i 0.707107i
$$99$$ −12.0000 −1.20605
$$100$$ 0 0
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ −10.0000 −0.971286
$$107$$ 8.00000i 0.773389i 0.922208 + 0.386695i $$0.126383\pi$$
−0.922208 + 0.386695i $$0.873617\pi$$
$$108$$ 0 0
$$109$$ −18.0000 −1.72409 −0.862044 0.506834i $$-0.830816\pi$$
−0.862044 + 0.506834i $$0.830816\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 14.0000i 1.31701i 0.752577 + 0.658505i $$0.228811\pi$$
−0.752577 + 0.658505i $$0.771189\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −6.00000 −0.557086
$$117$$ 6.00000i 0.554700i
$$118$$ − 4.00000i − 0.368230i
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 10.0000i 0.905357i
$$123$$ 0 0
$$124$$ 4.00000 0.359211
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 8.00000i − 0.709885i −0.934888 0.354943i $$-0.884500\pi$$
0.934888 0.354943i $$-0.115500\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −8.00000 −0.691095
$$135$$ 0 0
$$136$$ 2.00000 0.171499
$$137$$ 6.00000i 0.512615i 0.966595 + 0.256307i $$0.0825059\pi$$
−0.966595 + 0.256307i $$0.917494\pi$$
$$138$$ 0 0
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ − 8.00000i − 0.668994i
$$144$$ 3.00000 0.250000
$$145$$ 0 0
$$146$$ −10.0000 −0.827606
$$147$$ 0 0
$$148$$ − 1.00000i − 0.0821995i
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ 0 0
$$151$$ −24.0000 −1.95309 −0.976546 0.215308i $$-0.930924\pi$$
−0.976546 + 0.215308i $$0.930924\pi$$
$$152$$ − 4.00000i − 0.324443i
$$153$$ 6.00000i 0.485071i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 6.00000i 0.478852i 0.970915 + 0.239426i $$0.0769593\pi$$
−0.970915 + 0.239426i $$0.923041\pi$$
$$158$$ 4.00000i 0.318223i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 9.00000i 0.707107i
$$163$$ − 4.00000i − 0.313304i −0.987654 0.156652i $$-0.949930\pi$$
0.987654 0.156652i $$-0.0500701\pi$$
$$164$$ 6.00000 0.468521
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 16.0000i 1.23812i 0.785345 + 0.619059i $$0.212486\pi$$
−0.785345 + 0.619059i $$0.787514\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ 12.0000 0.917663
$$172$$ − 4.00000i − 0.304997i
$$173$$ − 14.0000i − 1.06440i −0.846619 0.532200i $$-0.821365\pi$$
0.846619 0.532200i $$-0.178635\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −4.00000 −0.301511
$$177$$ 0 0
$$178$$ − 2.00000i − 0.149906i
$$179$$ −4.00000 −0.298974 −0.149487 0.988764i $$-0.547762\pi$$
−0.149487 + 0.988764i $$0.547762\pi$$
$$180$$ 0 0
$$181$$ −2.00000 −0.148659 −0.0743294 0.997234i $$-0.523682\pi$$
−0.0743294 + 0.997234i $$0.523682\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 8.00000i − 0.585018i
$$188$$ − 8.00000i − 0.583460i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ 0 0
$$193$$ 14.0000i 1.00774i 0.863779 + 0.503871i $$0.168091\pi$$
−0.863779 + 0.503871i $$0.831909\pi$$
$$194$$ 6.00000 0.430775
$$195$$ 0 0
$$196$$ −7.00000 −0.500000
$$197$$ − 2.00000i − 0.142494i −0.997459 0.0712470i $$-0.977302\pi$$
0.997459 0.0712470i $$-0.0226979\pi$$
$$198$$ − 12.0000i − 0.852803i
$$199$$ 20.0000 1.41776 0.708881 0.705328i $$-0.249200\pi$$
0.708881 + 0.705328i $$0.249200\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 6.00000i 0.422159i
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 2.00000i 0.138675i
$$209$$ −16.0000 −1.10674
$$210$$ 0 0
$$211$$ 28.0000 1.92760 0.963800 0.266627i $$-0.0859092\pi$$
0.963800 + 0.266627i $$0.0859092\pi$$
$$212$$ − 10.0000i − 0.686803i
$$213$$ 0 0
$$214$$ −8.00000 −0.546869
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ − 18.0000i − 1.21911i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −4.00000 −0.269069
$$222$$ 0 0
$$223$$ − 16.0000i − 1.07144i −0.844396 0.535720i $$-0.820040\pi$$
0.844396 0.535720i $$-0.179960\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −14.0000 −0.931266
$$227$$ − 20.0000i − 1.32745i −0.747978 0.663723i $$-0.768975\pi$$
0.747978 0.663723i $$-0.231025\pi$$
$$228$$ 0 0
$$229$$ 18.0000 1.18947 0.594737 0.803921i $$-0.297256\pi$$
0.594737 + 0.803921i $$0.297256\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 6.00000i − 0.393919i
$$233$$ − 22.0000i − 1.44127i −0.693316 0.720634i $$-0.743851\pi$$
0.693316 0.720634i $$-0.256149\pi$$
$$234$$ −6.00000 −0.392232
$$235$$ 0 0
$$236$$ 4.00000 0.260378
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −12.0000 −0.776215 −0.388108 0.921614i $$-0.626871\pi$$
−0.388108 + 0.921614i $$0.626871\pi$$
$$240$$ 0 0
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ 5.00000i 0.321412i
$$243$$ 0 0
$$244$$ −10.0000 −0.640184
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 8.00000i 0.509028i
$$248$$ 4.00000i 0.254000i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 20.0000 1.26239 0.631194 0.775625i $$-0.282565\pi$$
0.631194 + 0.775625i $$0.282565\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 8.00000 0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 14.0000i − 0.873296i −0.899632 0.436648i $$-0.856166\pi$$
0.899632 0.436648i $$-0.143834\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 18.0000 1.11417
$$262$$ − 12.0000i − 0.741362i
$$263$$ − 32.0000i − 1.97320i −0.163144 0.986602i $$-0.552164\pi$$
0.163144 0.986602i $$-0.447836\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ − 8.00000i − 0.488678i
$$269$$ −14.0000 −0.853595 −0.426798 0.904347i $$-0.640358\pi$$
−0.426798 + 0.904347i $$0.640358\pi$$
$$270$$ 0 0
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ 2.00000i 0.121268i
$$273$$ 0 0
$$274$$ −6.00000 −0.362473
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 22.0000i 1.32185i 0.750451 + 0.660926i $$0.229836\pi$$
−0.750451 + 0.660926i $$0.770164\pi$$
$$278$$ 4.00000i 0.239904i
$$279$$ −12.0000 −0.718421
$$280$$ 0 0
$$281$$ 2.00000 0.119310 0.0596550 0.998219i $$-0.481000\pi$$
0.0596550 + 0.998219i $$0.481000\pi$$
$$282$$ 0 0
$$283$$ 4.00000i 0.237775i 0.992908 + 0.118888i $$0.0379328\pi$$
−0.992908 + 0.118888i $$0.962067\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 8.00000 0.473050
$$287$$ 0 0
$$288$$ 3.00000i 0.176777i
$$289$$ 13.0000 0.764706
$$290$$ 0 0
$$291$$ 0 0
$$292$$ − 10.0000i − 0.585206i
$$293$$ 18.0000i 1.05157i 0.850617 + 0.525786i $$0.176229\pi$$
−0.850617 + 0.525786i $$0.823771\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 1.00000 0.0581238
$$297$$ 0 0
$$298$$ 10.0000i 0.579284i
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ − 24.0000i − 1.38104i
$$303$$ 0 0
$$304$$ 4.00000 0.229416
$$305$$ 0 0
$$306$$ −6.00000 −0.342997
$$307$$ 32.0000i 1.82634i 0.407583 + 0.913168i $$0.366372\pi$$
−0.407583 + 0.913168i $$0.633628\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 12.0000 0.680458 0.340229 0.940343i $$-0.389495\pi$$
0.340229 + 0.940343i $$0.389495\pi$$
$$312$$ 0 0
$$313$$ − 10.0000i − 0.565233i −0.959233 0.282617i $$-0.908798\pi$$
0.959233 0.282617i $$-0.0912024\pi$$
$$314$$ −6.00000 −0.338600
$$315$$ 0 0
$$316$$ −4.00000 −0.225018
$$317$$ − 2.00000i − 0.112331i −0.998421 0.0561656i $$-0.982113\pi$$
0.998421 0.0561656i $$-0.0178875\pi$$
$$318$$ 0 0
$$319$$ −24.0000 −1.34374
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 8.00000i 0.445132i
$$324$$ −9.00000 −0.500000
$$325$$ 0 0
$$326$$ 4.00000 0.221540
$$327$$ 0 0
$$328$$ 6.00000i 0.331295i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 20.0000 1.09930 0.549650 0.835395i $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ 0 0
$$333$$ 3.00000i 0.164399i
$$334$$ −16.0000 −0.875481
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 34.0000i − 1.85210i −0.377403 0.926049i $$-0.623183\pi$$
0.377403 0.926049i $$-0.376817\pi$$
$$338$$ 9.00000i 0.489535i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 16.0000 0.866449
$$342$$ 12.0000i 0.648886i
$$343$$ 0 0
$$344$$ 4.00000 0.215666
$$345$$ 0 0
$$346$$ 14.0000 0.752645
$$347$$ − 4.00000i − 0.214731i −0.994220 0.107366i $$-0.965758\pi$$
0.994220 0.107366i $$-0.0342415\pi$$
$$348$$ 0 0
$$349$$ 18.0000 0.963518 0.481759 0.876304i $$-0.339998\pi$$
0.481759 + 0.876304i $$0.339998\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ − 4.00000i − 0.213201i
$$353$$ 6.00000i 0.319348i 0.987170 + 0.159674i $$0.0510443\pi$$
−0.987170 + 0.159674i $$0.948956\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 2.00000 0.106000
$$357$$ 0 0
$$358$$ − 4.00000i − 0.211407i
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ − 2.00000i − 0.105118i
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 8.00000i − 0.417597i −0.977959 0.208798i $$-0.933045\pi$$
0.977959 0.208798i $$-0.0669552\pi$$
$$368$$ 0 0
$$369$$ −18.0000 −0.937043
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ − 14.0000i − 0.724893i −0.932005 0.362446i $$-0.881942\pi$$
0.932005 0.362446i $$-0.118058\pi$$
$$374$$ 8.00000 0.413670
$$375$$ 0 0
$$376$$ 8.00000 0.412568
$$377$$ 12.0000i 0.618031i
$$378$$ 0 0
$$379$$ 28.0000 1.43826 0.719132 0.694874i $$-0.244540\pi$$
0.719132 + 0.694874i $$0.244540\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ − 12.0000i − 0.613973i
$$383$$ − 24.0000i − 1.22634i −0.789950 0.613171i $$-0.789894\pi$$
0.789950 0.613171i $$-0.210106\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −14.0000 −0.712581
$$387$$ 12.0000i 0.609994i
$$388$$ 6.00000i 0.304604i
$$389$$ 14.0000 0.709828 0.354914 0.934899i $$-0.384510\pi$$
0.354914 + 0.934899i $$0.384510\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ − 7.00000i − 0.353553i
$$393$$ 0 0
$$394$$ 2.00000 0.100759
$$395$$ 0 0
$$396$$ 12.0000 0.603023
$$397$$ − 34.0000i − 1.70641i −0.521575 0.853206i $$-0.674655\pi$$
0.521575 0.853206i $$-0.325345\pi$$
$$398$$ 20.0000i 1.00251i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −22.0000 −1.09863 −0.549314 0.835616i $$-0.685111\pi$$
−0.549314 + 0.835616i $$0.685111\pi$$
$$402$$ 0 0
$$403$$ − 8.00000i − 0.398508i
$$404$$ −6.00000 −0.298511
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 4.00000i − 0.198273i
$$408$$ 0 0
$$409$$ −2.00000 −0.0988936 −0.0494468 0.998777i $$-0.515746\pi$$
−0.0494468 + 0.998777i $$0.515746\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −2.00000 −0.0980581
$$417$$ 0 0
$$418$$ − 16.0000i − 0.782586i
$$419$$ −28.0000 −1.36789 −0.683945 0.729534i $$-0.739737\pi$$
−0.683945 + 0.729534i $$0.739737\pi$$
$$420$$ 0 0
$$421$$ −6.00000 −0.292422 −0.146211 0.989253i $$-0.546708\pi$$
−0.146211 + 0.989253i $$0.546708\pi$$
$$422$$ 28.0000i 1.36302i
$$423$$ 24.0000i 1.16692i
$$424$$ 10.0000 0.485643
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ − 8.00000i − 0.386695i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −20.0000 −0.963366 −0.481683 0.876346i $$-0.659974\pi$$
−0.481683 + 0.876346i $$0.659974\pi$$
$$432$$ 0 0
$$433$$ − 14.0000i − 0.672797i −0.941720 0.336399i $$-0.890791\pi$$
0.941720 0.336399i $$-0.109209\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 18.0000 0.862044
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 20.0000 0.954548 0.477274 0.878755i $$-0.341625\pi$$
0.477274 + 0.878755i $$0.341625\pi$$
$$440$$ 0 0
$$441$$ 21.0000 1.00000
$$442$$ − 4.00000i − 0.190261i
$$443$$ − 24.0000i − 1.14027i −0.821549 0.570137i $$-0.806890\pi$$
0.821549 0.570137i $$-0.193110\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 16.0000 0.757622
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 6.00000 0.283158 0.141579 0.989927i $$-0.454782\pi$$
0.141579 + 0.989927i $$0.454782\pi$$
$$450$$ 0 0
$$451$$ 24.0000 1.13012
$$452$$ − 14.0000i − 0.658505i
$$453$$ 0 0
$$454$$ 20.0000 0.938647
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 18.0000i 0.842004i 0.907060 + 0.421002i $$0.138322\pi$$
−0.907060 + 0.421002i $$0.861678\pi$$
$$458$$ 18.0000i 0.841085i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 10.0000 0.465746 0.232873 0.972507i $$-0.425187\pi$$
0.232873 + 0.972507i $$0.425187\pi$$
$$462$$ 0 0
$$463$$ − 40.0000i − 1.85896i −0.368875 0.929479i $$-0.620257\pi$$
0.368875 0.929479i $$-0.379743\pi$$
$$464$$ 6.00000 0.278543
$$465$$ 0 0
$$466$$ 22.0000 1.01913
$$467$$ − 28.0000i − 1.29569i −0.761774 0.647843i $$-0.775671\pi$$
0.761774 0.647843i $$-0.224329\pi$$
$$468$$ − 6.00000i − 0.277350i
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 4.00000i 0.184115i
$$473$$ − 16.0000i − 0.735681i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 30.0000i 1.37361i
$$478$$ − 12.0000i − 0.548867i
$$479$$ −28.0000 −1.27935 −0.639676 0.768644i $$-0.720932\pi$$
−0.639676 + 0.768644i $$0.720932\pi$$
$$480$$ 0 0
$$481$$ −2.00000 −0.0911922
$$482$$ 2.00000i 0.0910975i
$$483$$ 0 0
$$484$$ −5.00000 −0.227273
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 16.0000i 0.725029i 0.931978 + 0.362515i $$0.118082\pi$$
−0.931978 + 0.362515i $$0.881918\pi$$
$$488$$ − 10.0000i − 0.452679i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 20.0000 0.902587 0.451294 0.892375i $$-0.350963\pi$$
0.451294 + 0.892375i $$0.350963\pi$$
$$492$$ 0 0
$$493$$ 12.0000i 0.540453i
$$494$$ −8.00000 −0.359937
$$495$$ 0 0
$$496$$ −4.00000 −0.179605
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 20.0000 0.895323 0.447661 0.894203i $$-0.352257\pi$$
0.447661 + 0.894203i $$0.352257\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 20.0000i 0.892644i
$$503$$ 24.0000i 1.07011i 0.844818 + 0.535054i $$0.179709\pi$$
−0.844818 + 0.535054i $$0.820291\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 8.00000i 0.354943i
$$509$$ 18.0000 0.797836 0.398918 0.916987i $$-0.369386\pi$$
0.398918 + 0.916987i $$0.369386\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 1.00000i 0.0441942i
$$513$$ 0 0
$$514$$ 14.0000 0.617514
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 32.0000i − 1.40736i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −38.0000 −1.66481 −0.832405 0.554168i $$-0.813037\pi$$
−0.832405 + 0.554168i $$0.813037\pi$$
$$522$$ 18.0000i 0.787839i
$$523$$ 12.0000i 0.524723i 0.964970 + 0.262362i $$0.0845013\pi$$
−0.964970 + 0.262362i $$0.915499\pi$$
$$524$$ 12.0000 0.524222
$$525$$ 0 0
$$526$$ 32.0000 1.39527
$$527$$ − 8.00000i − 0.348485i
$$528$$ 0 0
$$529$$ 23.0000 1.00000
$$530$$ 0 0
$$531$$ −12.0000 −0.520756
$$532$$ 0 0
$$533$$ − 12.0000i − 0.519778i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 8.00000 0.345547
$$537$$ 0 0
$$538$$ − 14.0000i − 0.603583i
$$539$$ −28.0000 −1.20605
$$540$$ 0 0
$$541$$ −6.00000 −0.257960 −0.128980 0.991647i $$-0.541170\pi$$
−0.128980 + 0.991647i $$0.541170\pi$$
$$542$$ − 8.00000i − 0.343629i
$$543$$ 0 0
$$544$$ −2.00000 −0.0857493
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 4.00000i − 0.171028i −0.996337 0.0855138i $$-0.972747\pi$$
0.996337 0.0855138i $$-0.0272532\pi$$
$$548$$ − 6.00000i − 0.256307i
$$549$$ 30.0000 1.28037
$$550$$ 0 0
$$551$$ 24.0000 1.02243
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −22.0000 −0.934690
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ − 2.00000i − 0.0847427i −0.999102 0.0423714i $$-0.986509\pi$$
0.999102 0.0423714i $$-0.0134913\pi$$
$$558$$ − 12.0000i − 0.508001i
$$559$$ −8.00000 −0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 2.00000i 0.0843649i
$$563$$ 36.0000i 1.51722i 0.651546 + 0.758610i $$0.274121\pi$$
−0.651546 + 0.758610i $$0.725879\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −4.00000 −0.168133
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 30.0000 1.25767 0.628833 0.777541i $$-0.283533\pi$$
0.628833 + 0.777541i $$0.283533\pi$$
$$570$$ 0 0
$$571$$ 12.0000 0.502184 0.251092 0.967963i $$-0.419210\pi$$
0.251092 + 0.967963i $$0.419210\pi$$
$$572$$ 8.00000i 0.334497i
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ −3.00000 −0.125000
$$577$$ − 46.0000i − 1.91501i −0.288425 0.957503i $$-0.593132\pi$$
0.288425 0.957503i $$-0.406868\pi$$
$$578$$ 13.0000i 0.540729i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ − 40.0000i − 1.65663i
$$584$$ 10.0000 0.413803
$$585$$ 0 0
$$586$$ −18.0000 −0.743573
$$587$$ 12.0000i 0.495293i 0.968850 + 0.247647i $$0.0796572\pi$$
−0.968850 + 0.247647i $$0.920343\pi$$
$$588$$ 0 0
$$589$$ −16.0000 −0.659269
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 1.00000i 0.0410997i
$$593$$ 34.0000i 1.39621i 0.715994 + 0.698106i $$0.245974\pi$$
−0.715994 + 0.698106i $$0.754026\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −10.0000 −0.409616
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 0 0
$$601$$ 10.0000 0.407909 0.203954 0.978980i $$-0.434621\pi$$
0.203954 + 0.978980i $$0.434621\pi$$
$$602$$ 0 0
$$603$$ 24.0000i 0.977356i
$$604$$ 24.0000 0.976546
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 24.0000i − 0.974130i −0.873366 0.487065i $$-0.838067\pi$$
0.873366 0.487065i $$-0.161933\pi$$
$$608$$ 4.00000i 0.162221i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −16.0000 −0.647291
$$612$$ − 6.00000i − 0.242536i
$$613$$ − 6.00000i − 0.242338i −0.992632 0.121169i $$-0.961336\pi$$
0.992632 0.121169i $$-0.0386643\pi$$
$$614$$ −32.0000 −1.29141
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 30.0000i 1.20775i 0.797077 + 0.603877i $$0.206378\pi$$
−0.797077 + 0.603877i $$0.793622\pi$$
$$618$$ 0 0
$$619$$ −4.00000 −0.160774 −0.0803868 0.996764i $$-0.525616\pi$$
−0.0803868 + 0.996764i $$0.525616\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 12.0000i 0.481156i
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 10.0000 0.399680
$$627$$ 0 0
$$628$$ − 6.00000i − 0.239426i
$$629$$ −2.00000 −0.0797452
$$630$$ 0 0
$$631$$ −36.0000 −1.43314 −0.716569 0.697517i $$-0.754288\pi$$
−0.716569 + 0.697517i $$0.754288\pi$$
$$632$$ − 4.00000i − 0.159111i
$$633$$ 0 0
$$634$$ 2.00000 0.0794301
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 14.0000i 0.554700i
$$638$$ − 24.0000i − 0.950169i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 2.00000 0.0789953 0.0394976 0.999220i $$-0.487424\pi$$
0.0394976 + 0.999220i $$0.487424\pi$$
$$642$$ 0 0
$$643$$ − 44.0000i − 1.73519i −0.497271 0.867595i $$-0.665665\pi$$
0.497271 0.867595i $$-0.334335\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −8.00000 −0.314756
$$647$$ − 8.00000i − 0.314512i −0.987558 0.157256i $$-0.949735\pi$$
0.987558 0.157256i $$-0.0502649\pi$$
$$648$$ − 9.00000i − 0.353553i
$$649$$ 16.0000 0.628055
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 4.00000i 0.156652i
$$653$$ 34.0000i 1.33052i 0.746611 + 0.665261i $$0.231680\pi$$
−0.746611 + 0.665261i $$0.768320\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −6.00000 −0.234261
$$657$$ 30.0000i 1.17041i
$$658$$ 0 0
$$659$$ −20.0000 −0.779089 −0.389545 0.921008i $$-0.627368\pi$$
−0.389545 + 0.921008i $$0.627368\pi$$
$$660$$ 0 0
$$661$$ −22.0000 −0.855701 −0.427850 0.903850i $$-0.640729\pi$$
−0.427850 + 0.903850i $$0.640729\pi$$
$$662$$ 20.0000i 0.777322i
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ −3.00000 −0.116248
$$667$$ 0 0
$$668$$ − 16.0000i − 0.619059i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −40.0000 −1.54418
$$672$$ 0 0
$$673$$ 10.0000i 0.385472i 0.981251 + 0.192736i $$0.0617360\pi$$
−0.981251 + 0.192736i $$0.938264\pi$$
$$674$$ 34.0000 1.30963
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ − 2.00000i − 0.0768662i −0.999261 0.0384331i $$-0.987763\pi$$
0.999261 0.0384331i $$-0.0122367\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 16.0000i 0.612672i
$$683$$ 36.0000i 1.37750i 0.724998 + 0.688751i $$0.241841\pi$$
−0.724998 + 0.688751i $$0.758159\pi$$
$$684$$ −12.0000 −0.458831
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 4.00000i 0.152499i
$$689$$ −20.0000 −0.761939
$$690$$ 0 0
$$691$$ −20.0000 −0.760836 −0.380418 0.924815i $$-0.624220\pi$$
−0.380418 + 0.924815i $$0.624220\pi$$
$$692$$ 14.0000i 0.532200i
$$693$$ 0 0
$$694$$ 4.00000 0.151838
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 12.0000i − 0.454532i
$$698$$ 18.0000i 0.681310i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 50.0000 1.88847 0.944237 0.329267i $$-0.106802\pi$$
0.944237 + 0.329267i $$0.106802\pi$$
$$702$$ 0 0
$$703$$ 4.00000i 0.150863i
$$704$$ 4.00000 0.150756
$$705$$ 0 0
$$706$$ −6.00000 −0.225813
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 30.0000 1.12667 0.563337 0.826227i $$-0.309517\pi$$
0.563337 + 0.826227i $$0.309517\pi$$
$$710$$ 0 0
$$711$$ 12.0000 0.450035
$$712$$ 2.00000i 0.0749532i
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 4.00000 0.149487
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −32.0000 −1.19340 −0.596699 0.802465i $$-0.703521\pi$$
−0.596699 + 0.802465i $$0.703521\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ − 3.00000i − 0.111648i
$$723$$ 0 0
$$724$$ 2.00000 0.0743294
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 16.0000i − 0.593407i −0.954970 0.296704i $$-0.904113\pi$$
0.954970 0.296704i $$-0.0958873\pi$$
$$728$$ 0 0
$$729$$ 27.0000 1.00000
$$730$$ 0 0
$$731$$ −8.00000 −0.295891
$$732$$ 0 0
$$733$$ 34.0000i 1.25582i 0.778287 + 0.627909i $$0.216089\pi$$
−0.778287 + 0.627909i $$0.783911\pi$$
$$734$$ 8.00000 0.295285
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 32.0000i − 1.17874i
$$738$$ − 18.0000i − 0.662589i
$$739$$ −36.0000 −1.32428 −0.662141 0.749380i $$-0.730352\pi$$
−0.662141 + 0.749380i $$0.730352\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ − 24.0000i − 0.880475i −0.897881 0.440237i $$-0.854894\pi$$
0.897881 0.440237i $$-0.145106\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 14.0000 0.512576
$$747$$ 0 0
$$748$$ 8.00000i 0.292509i
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −40.0000 −1.45962 −0.729810 0.683650i $$-0.760392\pi$$
−0.729810 + 0.683650i $$0.760392\pi$$
$$752$$ 8.00000i 0.291730i
$$753$$ 0 0
$$754$$ −12.0000 −0.437014
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 10.0000i − 0.363456i −0.983349 0.181728i $$-0.941831\pi$$
0.983349 0.181728i $$-0.0581691\pi$$
$$758$$ 28.0000i 1.01701i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 42.0000 1.52250 0.761249 0.648459i $$-0.224586\pi$$
0.761249 + 0.648459i $$0.224586\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 12.0000 0.434145
$$765$$ 0 0
$$766$$ 24.0000 0.867155
$$767$$ − 8.00000i − 0.288863i
$$768$$ 0 0
$$769$$ 30.0000 1.08183 0.540914 0.841078i $$-0.318079\pi$$
0.540914 + 0.841078i $$0.318079\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ − 14.0000i − 0.503871i
$$773$$ − 22.0000i − 0.791285i −0.918405 0.395643i $$-0.870522\pi$$
0.918405 0.395643i $$-0.129478\pi$$
$$774$$ −12.0000 −0.431331
$$775$$ 0 0
$$776$$ −6.00000 −0.215387
$$777$$ 0 0
$$778$$ 14.0000i 0.501924i
$$779$$ −24.0000 −0.859889
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 7.00000 0.250000
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 48.0000i − 1.71102i −0.517790 0.855508i $$-0.673245\pi$$
0.517790 0.855508i $$-0.326755\pi$$
$$788$$ 2.00000i 0.0712470i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 12.0000i 0.426401i
$$793$$ 20.0000i 0.710221i
$$794$$ 34.0000 1.20661
$$795$$ 0 0
$$796$$ −20.0000 −0.708881
$$797$$ 30.0000i 1.06265i 0.847167 + 0.531327i $$0.178307\pi$$
−0.847167 + 0.531327i $$0.821693\pi$$
$$798$$ 0 0
$$799$$ −16.0000 −0.566039
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ − 22.0000i − 0.776847i
$$803$$ − 40.0000i − 1.41157i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 8.00000 0.281788
$$807$$ 0 0
$$808$$ − 6.00000i − 0.211079i
$$809$$ −26.0000 −0.914111 −0.457056 0.889438i $$-0.651096\pi$$
−0.457056 + 0.889438i $$0.651096\pi$$
$$810$$ 0 0
$$811$$ −20.0000 −0.702295 −0.351147 0.936320i $$-0.614208\pi$$
−0.351147 + 0.936320i $$0.614208\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 4.00000 0.140200
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 16.0000i 0.559769i
$$818$$ − 2.00000i − 0.0699284i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 6.00000 0.209401 0.104701 0.994504i $$-0.466612\pi$$
0.104701 + 0.994504i $$0.466612\pi$$
$$822$$ 0 0
$$823$$ 8.00000i 0.278862i 0.990232 + 0.139431i $$0.0445274\pi$$
−0.990232 + 0.139431i $$0.955473\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 12.0000i − 0.417281i −0.977992 0.208640i $$-0.933096\pi$$
0.977992 0.208640i $$-0.0669038\pi$$
$$828$$ 0 0
$$829$$ 14.0000 0.486240 0.243120 0.969996i $$-0.421829\pi$$
0.243120 + 0.969996i $$0.421829\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ − 2.00000i − 0.0693375i
$$833$$ 14.0000i 0.485071i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 16.0000 0.553372
$$837$$ 0 0
$$838$$ − 28.0000i − 0.967244i
$$839$$ −24.0000 −0.828572 −0.414286 0.910147i $$-0.635969\pi$$
−0.414286 + 0.910147i $$0.635969\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ − 6.00000i − 0.206774i
$$843$$ 0 0
$$844$$ −28.0000 −0.963800
$$845$$ 0 0
$$846$$ −24.0000 −0.825137
$$847$$ 0 0
$$848$$ 10.0000i 0.343401i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 10.0000i 0.342393i 0.985237 + 0.171197i $$0.0547634\pi$$
−0.985237 + 0.171197i $$0.945237\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 8.00000 0.273434
$$857$$ 18.0000i 0.614868i 0.951569 + 0.307434i $$0.0994704\pi$$
−0.951569 + 0.307434i $$0.900530\pi$$
$$858$$ 0 0
$$859$$ −20.0000 −0.682391 −0.341196 0.939992i $$-0.610832\pi$$
−0.341196 + 0.939992i $$0.610832\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ − 20.0000i − 0.681203i
$$863$$ 24.0000i 0.816970i 0.912765 + 0.408485i $$0.133943\pi$$
−0.912765 + 0.408485i $$0.866057\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 14.0000 0.475739
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −16.0000 −0.542763
$$870$$ 0 0
$$871$$ −16.0000 −0.542139
$$872$$ 18.0000i 0.609557i
$$873$$ − 18.0000i − 0.609208i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 42.0000i − 1.41824i −0.705088 0.709120i $$-0.749093\pi$$
0.705088 0.709120i $$-0.250907\pi$$
$$878$$ 20.0000i 0.674967i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −30.0000 −1.01073 −0.505363 0.862907i $$-0.668641\pi$$
−0.505363 + 0.862907i $$0.668641\pi$$
$$882$$ 21.0000i 0.707107i
$$883$$ 20.0000i 0.673054i 0.941674 + 0.336527i $$0.109252\pi$$
−0.941674 + 0.336527i $$0.890748\pi$$
$$884$$ 4.00000 0.134535
$$885$$ 0 0
$$886$$ 24.0000 0.806296
$$887$$ − 56.0000i − 1.88030i −0.340766 0.940148i $$-0.610687\pi$$
0.340766 0.940148i $$-0.389313\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −36.0000 −1.20605
$$892$$ 16.0000i 0.535720i
$$893$$ 32.0000i 1.07084i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 6.00000i 0.200223i
$$899$$ −24.0000 −0.800445
$$900$$ 0 0
$$901$$ −20.0000 −0.666297
$$902$$ 24.0000i 0.799113i
$$903$$ 0 0
$$904$$ 14.0000 0.465633
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 20.0000i 0.664089i 0.943264 + 0.332045i $$0.107738\pi$$
−0.943264 + 0.332045i $$0.892262\pi$$
$$908$$ 20.0000i 0.663723i
$$909$$ 18.0000 0.597022
$$910$$ 0 0
$$911$$ −36.0000 −1.19273 −0.596367 0.802712i $$-0.703390\pi$$
−0.596367 + 0.802712i $$0.703390\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ −18.0000 −0.595387
$$915$$ 0 0
$$916$$ −18.0000 −0.594737
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −20.0000 −0.659739 −0.329870 0.944027i $$-0.607005\pi$$
−0.329870 + 0.944027i $$0.607005\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 10.0000i 0.329332i
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 40.0000 1.31448
$$927$$ 0 0
$$928$$ 6.00000i 0.196960i
$$929$$ −18.0000 −0.590561 −0.295280 0.955411i $$-0.595413\pi$$
−0.295280 + 0.955411i $$0.595413\pi$$
$$930$$ 0 0
$$931$$ 28.0000 0.917663
$$932$$ 22.0000i 0.720634i
$$933$$ 0 0
$$934$$ 28.0000 0.916188
$$935$$ 0 0
$$936$$ 6.00000 0.196116
$$937$$ 14.0000i 0.457360i 0.973502 + 0.228680i $$0.0734410\pi$$
−0.973502 + 0.228680i $$0.926559\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −42.0000 −1.36916 −0.684580 0.728937i $$-0.740015\pi$$
−0.684580 + 0.728937i $$0.740015\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ −4.00000 −0.130189
$$945$$ 0 0
$$946$$ 16.0000 0.520205
$$947$$ 4.00000i 0.129983i 0.997886 + 0.0649913i $$0.0207020\pi$$
−0.997886 + 0.0649913i $$0.979298\pi$$
$$948$$ 0 0
$$949$$ −20.0000 −0.649227
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ − 6.00000i − 0.194359i −0.995267 0.0971795i $$-0.969018\pi$$
0.995267 0.0971795i $$-0.0309821\pi$$
$$954$$ −30.0000 −0.971286
$$955$$ 0 0
$$956$$ 12.0000 0.388108
$$957$$ 0 0
$$958$$ − 28.0000i − 0.904639i
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ − 2.00000i − 0.0644826i
$$963$$ 24.0000i 0.773389i
$$964$$ −2.00000 −0.0644157
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 8.00000i 0.257263i 0.991692 + 0.128631i $$0.0410584\pi$$
−0.991692 + 0.128631i $$0.958942\pi$$
$$968$$ − 5.00000i − 0.160706i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −12.0000 −0.385098 −0.192549 0.981287i $$-0.561675\pi$$
−0.192549 + 0.981287i $$0.561675\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −16.0000 −0.512673
$$975$$ 0 0
$$976$$ 10.0000 0.320092
$$977$$ 50.0000i 1.59964i 0.600239 + 0.799821i $$0.295072\pi$$
−0.600239 + 0.799821i $$0.704928\pi$$
$$978$$ 0 0
$$979$$ 8.00000 0.255681
$$980$$ 0 0
$$981$$ −54.0000 −1.72409
$$982$$ 20.0000i 0.638226i
$$983$$ 24.0000i 0.765481i 0.923856 + 0.382741i $$0.125020\pi$$
−0.923856 + 0.382741i $$0.874980\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −12.0000 −0.382158
$$987$$ 0 0
$$988$$ − 8.00000i − 0.254514i
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 20.0000 0.635321 0.317660 0.948205i $$-0.397103\pi$$
0.317660 + 0.948205i $$0.397103\pi$$
$$992$$ − 4.00000i − 0.127000i
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 10.0000i − 0.316703i −0.987383 0.158352i $$-0.949382\pi$$
0.987383 0.158352i $$-0.0506179\pi$$
$$998$$ 20.0000i 0.633089i
$$999$$ 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 1850.2.b.d.149.2 2
5.2 odd 4 370.2.a.b.1.1 1
5.3 odd 4 1850.2.a.k.1.1 1
5.4 even 2 inner 1850.2.b.d.149.1 2
15.2 even 4 3330.2.a.w.1.1 1
20.7 even 4 2960.2.a.g.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.b.1.1 1 5.2 odd 4
1850.2.a.k.1.1 1 5.3 odd 4
1850.2.b.d.149.1 2 5.4 even 2 inner
1850.2.b.d.149.2 2 1.1 even 1 trivial
2960.2.a.g.1.1 1 20.7 even 4
3330.2.a.w.1.1 1 15.2 even 4