# Properties

 Label 400.4.c.f.49.1 Level $400$ Weight $4$ Character 400.49 Analytic conductor $23.601$ 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] = [400,4,Mod(49,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("400.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 400.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: $$23.6007640023$$ 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: $$2$$ Twist minimal: no (minimal twist has level 40) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 49.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 400.49 Dual form 400.4.c.f.49.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-6.00000i q^{3} +34.0000i q^{7} -9.00000 q^{9} +O(q^{10})$$ $$q-6.00000i q^{3} +34.0000i q^{7} -9.00000 q^{9} -16.0000 q^{11} -58.0000i q^{13} -70.0000i q^{17} +4.00000 q^{19} +204.000 q^{21} -134.000i q^{23} -108.000i q^{27} +242.000 q^{29} -100.000 q^{31} +96.0000i q^{33} -438.000i q^{37} -348.000 q^{39} -138.000 q^{41} +178.000i q^{43} -22.0000i q^{47} -813.000 q^{49} -420.000 q^{51} -162.000i q^{53} -24.0000i q^{57} -268.000 q^{59} +250.000 q^{61} -306.000i q^{63} -422.000i q^{67} -804.000 q^{69} +852.000 q^{71} -306.000i q^{73} -544.000i q^{77} -456.000 q^{79} -891.000 q^{81} +434.000i q^{83} -1452.00i q^{87} +726.000 q^{89} +1972.00 q^{91} +600.000i q^{93} +1378.00i q^{97} +144.000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 18 q^{9}+O(q^{10})$$ 2 * q - 18 * q^9 $$2 q - 18 q^{9} - 32 q^{11} + 8 q^{19} + 408 q^{21} + 484 q^{29} - 200 q^{31} - 696 q^{39} - 276 q^{41} - 1626 q^{49} - 840 q^{51} - 536 q^{59} + 500 q^{61} - 1608 q^{69} + 1704 q^{71} - 912 q^{79} - 1782 q^{81} + 1452 q^{89} + 3944 q^{91} + 288 q^{99}+O(q^{100})$$ 2 * q - 18 * q^9 - 32 * q^11 + 8 * q^19 + 408 * q^21 + 484 * q^29 - 200 * q^31 - 696 * q^39 - 276 * q^41 - 1626 * q^49 - 840 * q^51 - 536 * q^59 + 500 * q^61 - 1608 * q^69 + 1704 * q^71 - 912 * q^79 - 1782 * q^81 + 1452 * q^89 + 3944 * q^91 + 288 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$177$$ $$351$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ − 6.00000i − 1.15470i −0.816497 0.577350i $$-0.804087\pi$$
0.816497 0.577350i $$-0.195913\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 34.0000i 1.83583i 0.396780 + 0.917914i $$0.370128\pi$$
−0.396780 + 0.917914i $$0.629872\pi$$
$$8$$ 0 0
$$9$$ −9.00000 −0.333333
$$10$$ 0 0
$$11$$ −16.0000 −0.438562 −0.219281 0.975662i $$-0.570371\pi$$
−0.219281 + 0.975662i $$0.570371\pi$$
$$12$$ 0 0
$$13$$ − 58.0000i − 1.23741i −0.785624 0.618704i $$-0.787658\pi$$
0.785624 0.618704i $$-0.212342\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 70.0000i − 0.998676i −0.866407 0.499338i $$-0.833577\pi$$
0.866407 0.499338i $$-0.166423\pi$$
$$18$$ 0 0
$$19$$ 4.00000 0.0482980 0.0241490 0.999708i $$-0.492312\pi$$
0.0241490 + 0.999708i $$0.492312\pi$$
$$20$$ 0 0
$$21$$ 204.000 2.11983
$$22$$ 0 0
$$23$$ − 134.000i − 1.21482i −0.794387 0.607412i $$-0.792208\pi$$
0.794387 0.607412i $$-0.207792\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 108.000i − 0.769800i
$$28$$ 0 0
$$29$$ 242.000 1.54960 0.774798 0.632209i $$-0.217852\pi$$
0.774798 + 0.632209i $$0.217852\pi$$
$$30$$ 0 0
$$31$$ −100.000 −0.579372 −0.289686 0.957122i $$-0.593551\pi$$
−0.289686 + 0.957122i $$0.593551\pi$$
$$32$$ 0 0
$$33$$ 96.0000i 0.506408i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 438.000i − 1.94613i −0.230534 0.973064i $$-0.574047\pi$$
0.230534 0.973064i $$-0.425953\pi$$
$$38$$ 0 0
$$39$$ −348.000 −1.42884
$$40$$ 0 0
$$41$$ −138.000 −0.525658 −0.262829 0.964842i $$-0.584656\pi$$
−0.262829 + 0.964842i $$0.584656\pi$$
$$42$$ 0 0
$$43$$ 178.000i 0.631273i 0.948880 + 0.315637i $$0.102218\pi$$
−0.948880 + 0.315637i $$0.897782\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 22.0000i − 0.0682772i −0.999417 0.0341386i $$-0.989131\pi$$
0.999417 0.0341386i $$-0.0108688\pi$$
$$48$$ 0 0
$$49$$ −813.000 −2.37026
$$50$$ 0 0
$$51$$ −420.000 −1.15317
$$52$$ 0 0
$$53$$ − 162.000i − 0.419857i −0.977717 0.209928i $$-0.932677\pi$$
0.977717 0.209928i $$-0.0673231\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 24.0000i − 0.0557698i
$$58$$ 0 0
$$59$$ −268.000 −0.591367 −0.295683 0.955286i $$-0.595547\pi$$
−0.295683 + 0.955286i $$0.595547\pi$$
$$60$$ 0 0
$$61$$ 250.000 0.524741 0.262371 0.964967i $$-0.415496\pi$$
0.262371 + 0.964967i $$0.415496\pi$$
$$62$$ 0 0
$$63$$ − 306.000i − 0.611942i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 422.000i − 0.769485i −0.923024 0.384743i $$-0.874290\pi$$
0.923024 0.384743i $$-0.125710\pi$$
$$68$$ 0 0
$$69$$ −804.000 −1.40276
$$70$$ 0 0
$$71$$ 852.000 1.42414 0.712069 0.702109i $$-0.247758\pi$$
0.712069 + 0.702109i $$0.247758\pi$$
$$72$$ 0 0
$$73$$ − 306.000i − 0.490611i −0.969446 0.245305i $$-0.921112\pi$$
0.969446 0.245305i $$-0.0788882\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 544.000i − 0.805124i
$$78$$ 0 0
$$79$$ −456.000 −0.649418 −0.324709 0.945814i $$-0.605266\pi$$
−0.324709 + 0.945814i $$0.605266\pi$$
$$80$$ 0 0
$$81$$ −891.000 −1.22222
$$82$$ 0 0
$$83$$ 434.000i 0.573948i 0.957938 + 0.286974i $$0.0926493\pi$$
−0.957938 + 0.286974i $$0.907351\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ − 1452.00i − 1.78932i
$$88$$ 0 0
$$89$$ 726.000 0.864672 0.432336 0.901712i $$-0.357689\pi$$
0.432336 + 0.901712i $$0.357689\pi$$
$$90$$ 0 0
$$91$$ 1972.00 2.27167
$$92$$ 0 0
$$93$$ 600.000i 0.669001i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 1378.00i 1.44242i 0.692717 + 0.721210i $$0.256414\pi$$
−0.692717 + 0.721210i $$0.743586\pi$$
$$98$$ 0 0
$$99$$ 144.000 0.146187
$$100$$ 0 0
$$101$$ 126.000 0.124133 0.0620667 0.998072i $$-0.480231\pi$$
0.0620667 + 0.998072i $$0.480231\pi$$
$$102$$ 0 0
$$103$$ − 1262.00i − 1.20727i −0.797262 0.603634i $$-0.793719\pi$$
0.797262 0.603634i $$-0.206281\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 510.000i − 0.460781i −0.973098 0.230390i $$-0.926000\pi$$
0.973098 0.230390i $$-0.0740003\pi$$
$$108$$ 0 0
$$109$$ −26.0000 −0.0228472 −0.0114236 0.999935i $$-0.503636\pi$$
−0.0114236 + 0.999935i $$0.503636\pi$$
$$110$$ 0 0
$$111$$ −2628.00 −2.24720
$$112$$ 0 0
$$113$$ − 1242.00i − 1.03396i −0.855997 0.516980i $$-0.827056\pi$$
0.855997 0.516980i $$-0.172944\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 522.000i 0.412469i
$$118$$ 0 0
$$119$$ 2380.00 1.83340
$$120$$ 0 0
$$121$$ −1075.00 −0.807663
$$122$$ 0 0
$$123$$ 828.000i 0.606978i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 978.000i 0.683334i 0.939821 + 0.341667i $$0.110992\pi$$
−0.939821 + 0.341667i $$0.889008\pi$$
$$128$$ 0 0
$$129$$ 1068.00 0.728931
$$130$$ 0 0
$$131$$ 912.000 0.608258 0.304129 0.952631i $$-0.401635\pi$$
0.304129 + 0.952631i $$0.401635\pi$$
$$132$$ 0 0
$$133$$ 136.000i 0.0886669i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 926.000i − 0.577471i −0.957409 0.288735i $$-0.906765\pi$$
0.957409 0.288735i $$-0.0932348\pi$$
$$138$$ 0 0
$$139$$ 516.000 0.314867 0.157434 0.987530i $$-0.449678\pi$$
0.157434 + 0.987530i $$0.449678\pi$$
$$140$$ 0 0
$$141$$ −132.000 −0.0788398
$$142$$ 0 0
$$143$$ 928.000i 0.542680i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 4878.00i 2.73694i
$$148$$ 0 0
$$149$$ 958.000 0.526728 0.263364 0.964697i $$-0.415168\pi$$
0.263364 + 0.964697i $$0.415168\pi$$
$$150$$ 0 0
$$151$$ −332.000 −0.178926 −0.0894628 0.995990i $$-0.528515\pi$$
−0.0894628 + 0.995990i $$0.528515\pi$$
$$152$$ 0 0
$$153$$ 630.000i 0.332892i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 1022.00i − 0.519519i −0.965673 0.259759i $$-0.916357\pi$$
0.965673 0.259759i $$-0.0836433\pi$$
$$158$$ 0 0
$$159$$ −972.000 −0.484809
$$160$$ 0 0
$$161$$ 4556.00 2.23021
$$162$$ 0 0
$$163$$ − 926.000i − 0.444969i −0.974936 0.222484i $$-0.928583\pi$$
0.974936 0.222484i $$-0.0714166\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 654.000i − 0.303042i −0.988454 0.151521i $$-0.951583\pi$$
0.988454 0.151521i $$-0.0484171\pi$$
$$168$$ 0 0
$$169$$ −1167.00 −0.531179
$$170$$ 0 0
$$171$$ −36.0000 −0.0160993
$$172$$ 0 0
$$173$$ 1294.00i 0.568676i 0.958724 + 0.284338i $$0.0917738\pi$$
−0.958724 + 0.284338i $$0.908226\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 1608.00i 0.682851i
$$178$$ 0 0
$$179$$ −2836.00 −1.18420 −0.592102 0.805863i $$-0.701702\pi$$
−0.592102 + 0.805863i $$0.701702\pi$$
$$180$$ 0 0
$$181$$ 1742.00 0.715369 0.357685 0.933842i $$-0.383566\pi$$
0.357685 + 0.933842i $$0.383566\pi$$
$$182$$ 0 0
$$183$$ − 1500.00i − 0.605919i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 1120.00i 0.437981i
$$188$$ 0 0
$$189$$ 3672.00 1.41322
$$190$$ 0 0
$$191$$ −4460.00 −1.68960 −0.844802 0.535079i $$-0.820282\pi$$
−0.844802 + 0.535079i $$0.820282\pi$$
$$192$$ 0 0
$$193$$ 3782.00i 1.41054i 0.708939 + 0.705270i $$0.249174\pi$$
−0.708939 + 0.705270i $$0.750826\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 4474.00i 1.61807i 0.587762 + 0.809034i $$0.300009\pi$$
−0.587762 + 0.809034i $$0.699991\pi$$
$$198$$ 0 0
$$199$$ 3608.00 1.28525 0.642624 0.766182i $$-0.277846\pi$$
0.642624 + 0.766182i $$0.277846\pi$$
$$200$$ 0 0
$$201$$ −2532.00 −0.888525
$$202$$ 0 0
$$203$$ 8228.00i 2.84479i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 1206.00i 0.404941i
$$208$$ 0 0
$$209$$ −64.0000 −0.0211817
$$210$$ 0 0
$$211$$ 256.000 0.0835250 0.0417625 0.999128i $$-0.486703\pi$$
0.0417625 + 0.999128i $$0.486703\pi$$
$$212$$ 0 0
$$213$$ − 5112.00i − 1.64445i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 3400.00i − 1.06363i
$$218$$ 0 0
$$219$$ −1836.00 −0.566509
$$220$$ 0 0
$$221$$ −4060.00 −1.23577
$$222$$ 0 0
$$223$$ − 5158.00i − 1.54890i −0.632634 0.774451i $$-0.718026\pi$$
0.632634 0.774451i $$-0.281974\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 2226.00i 0.650858i 0.945566 + 0.325429i $$0.105509\pi$$
−0.945566 + 0.325429i $$0.894491\pi$$
$$228$$ 0 0
$$229$$ −2086.00 −0.601951 −0.300975 0.953632i $$-0.597312\pi$$
−0.300975 + 0.953632i $$0.597312\pi$$
$$230$$ 0 0
$$231$$ −3264.00 −0.929677
$$232$$ 0 0
$$233$$ 5718.00i 1.60772i 0.594819 + 0.803860i $$0.297224\pi$$
−0.594819 + 0.803860i $$0.702776\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 2736.00i 0.749883i
$$238$$ 0 0
$$239$$ −3624.00 −0.980825 −0.490412 0.871491i $$-0.663154\pi$$
−0.490412 + 0.871491i $$0.663154\pi$$
$$240$$ 0 0
$$241$$ −82.0000 −0.0219174 −0.0109587 0.999940i $$-0.503488\pi$$
−0.0109587 + 0.999940i $$0.503488\pi$$
$$242$$ 0 0
$$243$$ 2430.00i 0.641500i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 232.000i − 0.0597644i
$$248$$ 0 0
$$249$$ 2604.00 0.662738
$$250$$ 0 0
$$251$$ 5040.00 1.26742 0.633709 0.773571i $$-0.281532\pi$$
0.633709 + 0.773571i $$0.281532\pi$$
$$252$$ 0 0
$$253$$ 2144.00i 0.532775i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 2310.00i − 0.560676i −0.959901 0.280338i $$-0.909553\pi$$
0.959901 0.280338i $$-0.0904466\pi$$
$$258$$ 0 0
$$259$$ 14892.0 3.57276
$$260$$ 0 0
$$261$$ −2178.00 −0.516532
$$262$$ 0 0
$$263$$ − 4110.00i − 0.963625i −0.876274 0.481813i $$-0.839979\pi$$
0.876274 0.481813i $$-0.160021\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 4356.00i − 0.998438i
$$268$$ 0 0
$$269$$ −746.000 −0.169087 −0.0845435 0.996420i $$-0.526943\pi$$
−0.0845435 + 0.996420i $$0.526943\pi$$
$$270$$ 0 0
$$271$$ 4596.00 1.03021 0.515105 0.857127i $$-0.327753\pi$$
0.515105 + 0.857127i $$0.327753\pi$$
$$272$$ 0 0
$$273$$ − 11832.0i − 2.62310i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 2206.00i − 0.478504i −0.970957 0.239252i $$-0.923098\pi$$
0.970957 0.239252i $$-0.0769023\pi$$
$$278$$ 0 0
$$279$$ 900.000 0.193124
$$280$$ 0 0
$$281$$ 8278.00 1.75738 0.878691 0.477392i $$-0.158418\pi$$
0.878691 + 0.477392i $$0.158418\pi$$
$$282$$ 0 0
$$283$$ 1178.00i 0.247438i 0.992317 + 0.123719i $$0.0394821\pi$$
−0.992317 + 0.123719i $$0.960518\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 4692.00i − 0.965017i
$$288$$ 0 0
$$289$$ 13.0000 0.00264604
$$290$$ 0 0
$$291$$ 8268.00 1.66556
$$292$$ 0 0
$$293$$ − 106.000i − 0.0211351i −0.999944 0.0105676i $$-0.996636\pi$$
0.999944 0.0105676i $$-0.00336382\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 1728.00i 0.337605i
$$298$$ 0 0
$$299$$ −7772.00 −1.50323
$$300$$ 0 0
$$301$$ −6052.00 −1.15891
$$302$$ 0 0
$$303$$ − 756.000i − 0.143337i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 8134.00i − 1.51216i −0.654482 0.756078i $$-0.727113\pi$$
0.654482 0.756078i $$-0.272887\pi$$
$$308$$ 0 0
$$309$$ −7572.00 −1.39403
$$310$$ 0 0
$$311$$ 4396.00 0.801525 0.400763 0.916182i $$-0.368745\pi$$
0.400763 + 0.916182i $$0.368745\pi$$
$$312$$ 0 0
$$313$$ − 4826.00i − 0.871507i −0.900066 0.435753i $$-0.856482\pi$$
0.900066 0.435753i $$-0.143518\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 7026.00i 1.24486i 0.782677 + 0.622428i $$0.213854\pi$$
−0.782677 + 0.622428i $$0.786146\pi$$
$$318$$ 0 0
$$319$$ −3872.00 −0.679594
$$320$$ 0 0
$$321$$ −3060.00 −0.532064
$$322$$ 0 0
$$323$$ − 280.000i − 0.0482341i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 156.000i 0.0263817i
$$328$$ 0 0
$$329$$ 748.000 0.125345
$$330$$ 0 0
$$331$$ −8808.00 −1.46263 −0.731316 0.682038i $$-0.761094\pi$$
−0.731316 + 0.682038i $$0.761094\pi$$
$$332$$ 0 0
$$333$$ 3942.00i 0.648710i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 5602.00i 0.905520i 0.891632 + 0.452760i $$0.149561\pi$$
−0.891632 + 0.452760i $$0.850439\pi$$
$$338$$ 0 0
$$339$$ −7452.00 −1.19391
$$340$$ 0 0
$$341$$ 1600.00 0.254090
$$342$$ 0 0
$$343$$ − 15980.0i − 2.51557i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 6634.00i 1.02632i 0.858294 + 0.513158i $$0.171525\pi$$
−0.858294 + 0.513158i $$0.828475\pi$$
$$348$$ 0 0
$$349$$ −3198.00 −0.490501 −0.245251 0.969460i $$-0.578870\pi$$
−0.245251 + 0.969460i $$0.578870\pi$$
$$350$$ 0 0
$$351$$ −6264.00 −0.952557
$$352$$ 0 0
$$353$$ 5230.00i 0.788569i 0.918988 + 0.394284i $$0.129008\pi$$
−0.918988 + 0.394284i $$0.870992\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ − 14280.0i − 2.11702i
$$358$$ 0 0
$$359$$ −312.000 −0.0458683 −0.0229342 0.999737i $$-0.507301\pi$$
−0.0229342 + 0.999737i $$0.507301\pi$$
$$360$$ 0 0
$$361$$ −6843.00 −0.997667
$$362$$ 0 0
$$363$$ 6450.00i 0.932609i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 10790.0i − 1.53470i −0.641231 0.767348i $$-0.721576\pi$$
0.641231 0.767348i $$-0.278424\pi$$
$$368$$ 0 0
$$369$$ 1242.00 0.175219
$$370$$ 0 0
$$371$$ 5508.00 0.770785
$$372$$ 0 0
$$373$$ 4190.00i 0.581635i 0.956778 + 0.290818i $$0.0939273\pi$$
−0.956778 + 0.290818i $$0.906073\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 14036.0i − 1.91748i
$$378$$ 0 0
$$379$$ −6980.00 −0.946012 −0.473006 0.881059i $$-0.656831\pi$$
−0.473006 + 0.881059i $$0.656831\pi$$
$$380$$ 0 0
$$381$$ 5868.00 0.789047
$$382$$ 0 0
$$383$$ 13962.0i 1.86273i 0.364089 + 0.931364i $$0.381380\pi$$
−0.364089 + 0.931364i $$0.618620\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 1602.00i − 0.210424i
$$388$$ 0 0
$$389$$ −3810.00 −0.496593 −0.248296 0.968684i $$-0.579871\pi$$
−0.248296 + 0.968684i $$0.579871\pi$$
$$390$$ 0 0
$$391$$ −9380.00 −1.21321
$$392$$ 0 0
$$393$$ − 5472.00i − 0.702356i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 9158.00i − 1.15775i −0.815416 0.578875i $$-0.803492\pi$$
0.815416 0.578875i $$-0.196508\pi$$
$$398$$ 0 0
$$399$$ 816.000 0.102384
$$400$$ 0 0
$$401$$ 4866.00 0.605976 0.302988 0.952994i $$-0.402016\pi$$
0.302988 + 0.952994i $$0.402016\pi$$
$$402$$ 0 0
$$403$$ 5800.00i 0.716920i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 7008.00i 0.853498i
$$408$$ 0 0
$$409$$ −13486.0 −1.63042 −0.815208 0.579169i $$-0.803377\pi$$
−0.815208 + 0.579169i $$0.803377\pi$$
$$410$$ 0 0
$$411$$ −5556.00 −0.666806
$$412$$ 0 0
$$413$$ − 9112.00i − 1.08565i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ − 3096.00i − 0.363577i
$$418$$ 0 0
$$419$$ 5628.00 0.656195 0.328098 0.944644i $$-0.393593\pi$$
0.328098 + 0.944644i $$0.393593\pi$$
$$420$$ 0 0
$$421$$ 7938.00 0.918942 0.459471 0.888193i $$-0.348039\pi$$
0.459471 + 0.888193i $$0.348039\pi$$
$$422$$ 0 0
$$423$$ 198.000i 0.0227591i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 8500.00i 0.963334i
$$428$$ 0 0
$$429$$ 5568.00 0.626633
$$430$$ 0 0
$$431$$ −1916.00 −0.214131 −0.107066 0.994252i $$-0.534145\pi$$
−0.107066 + 0.994252i $$0.534145\pi$$
$$432$$ 0 0
$$433$$ 16510.0i 1.83238i 0.400746 + 0.916189i $$0.368751\pi$$
−0.400746 + 0.916189i $$0.631249\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 536.000i − 0.0586736i
$$438$$ 0 0
$$439$$ −1256.00 −0.136550 −0.0682752 0.997667i $$-0.521750\pi$$
−0.0682752 + 0.997667i $$0.521750\pi$$
$$440$$ 0 0
$$441$$ 7317.00 0.790087
$$442$$ 0 0
$$443$$ − 12222.0i − 1.31080i −0.755282 0.655400i $$-0.772500\pi$$
0.755282 0.655400i $$-0.227500\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ − 5748.00i − 0.608213i
$$448$$ 0 0
$$449$$ 5946.00 0.624965 0.312482 0.949924i $$-0.398840\pi$$
0.312482 + 0.949924i $$0.398840\pi$$
$$450$$ 0 0
$$451$$ 2208.00 0.230534
$$452$$ 0 0
$$453$$ 1992.00i 0.206606i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 1258.00i 0.128768i 0.997925 + 0.0643838i $$0.0205082\pi$$
−0.997925 + 0.0643838i $$0.979492\pi$$
$$458$$ 0 0
$$459$$ −7560.00 −0.768781
$$460$$ 0 0
$$461$$ 16422.0 1.65911 0.829554 0.558426i $$-0.188595\pi$$
0.829554 + 0.558426i $$0.188595\pi$$
$$462$$ 0 0
$$463$$ 2658.00i 0.266799i 0.991062 + 0.133399i $$0.0425893\pi$$
−0.991062 + 0.133399i $$0.957411\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 3686.00i − 0.365241i −0.983183 0.182621i $$-0.941542\pi$$
0.983183 0.182621i $$-0.0584580\pi$$
$$468$$ 0 0
$$469$$ 14348.0 1.41264
$$470$$ 0 0
$$471$$ −6132.00 −0.599889
$$472$$ 0 0
$$473$$ − 2848.00i − 0.276852i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 1458.00i 0.139952i
$$478$$ 0 0
$$479$$ 88.0000 0.00839420 0.00419710 0.999991i $$-0.498664\pi$$
0.00419710 + 0.999991i $$0.498664\pi$$
$$480$$ 0 0
$$481$$ −25404.0 −2.40816
$$482$$ 0 0
$$483$$ − 27336.0i − 2.57522i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 14714.0i 1.36911i 0.728963 + 0.684553i $$0.240003\pi$$
−0.728963 + 0.684553i $$0.759997\pi$$
$$488$$ 0 0
$$489$$ −5556.00 −0.513806
$$490$$ 0 0
$$491$$ 7344.00 0.675010 0.337505 0.941324i $$-0.390417\pi$$
0.337505 + 0.941324i $$0.390417\pi$$
$$492$$ 0 0
$$493$$ − 16940.0i − 1.54754i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 28968.0i 2.61447i
$$498$$ 0 0
$$499$$ 1604.00 0.143898 0.0719488 0.997408i $$-0.477078\pi$$
0.0719488 + 0.997408i $$0.477078\pi$$
$$500$$ 0 0
$$501$$ −3924.00 −0.349923
$$502$$ 0 0
$$503$$ 14802.0i 1.31210i 0.754715 + 0.656052i $$0.227775\pi$$
−0.754715 + 0.656052i $$0.772225\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 7002.00i 0.613353i
$$508$$ 0 0
$$509$$ 22514.0 1.96054 0.980271 0.197660i $$-0.0633342\pi$$
0.980271 + 0.197660i $$0.0633342\pi$$
$$510$$ 0 0
$$511$$ 10404.0 0.900677
$$512$$ 0 0
$$513$$ − 432.000i − 0.0371799i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 352.000i 0.0299438i
$$518$$ 0 0
$$519$$ 7764.00 0.656651
$$520$$ 0 0
$$521$$ −6710.00 −0.564243 −0.282121 0.959379i $$-0.591038\pi$$
−0.282121 + 0.959379i $$0.591038\pi$$
$$522$$ 0 0
$$523$$ 7930.00i 0.663011i 0.943453 + 0.331505i $$0.107557\pi$$
−0.943453 + 0.331505i $$0.892443\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 7000.00i 0.578605i
$$528$$ 0 0
$$529$$ −5789.00 −0.475795
$$530$$ 0 0
$$531$$ 2412.00 0.197122
$$532$$ 0 0
$$533$$ 8004.00i 0.650454i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 17016.0i 1.36740i
$$538$$ 0 0
$$539$$ 13008.0 1.03951
$$540$$ 0 0
$$541$$ 4918.00 0.390834 0.195417 0.980720i $$-0.437394\pi$$
0.195417 + 0.980720i $$0.437394\pi$$
$$542$$ 0 0
$$543$$ − 10452.0i − 0.826037i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 3922.00i 0.306568i 0.988182 + 0.153284i $$0.0489849\pi$$
−0.988182 + 0.153284i $$0.951015\pi$$
$$548$$ 0 0
$$549$$ −2250.00 −0.174914
$$550$$ 0 0
$$551$$ 968.000 0.0748424
$$552$$ 0 0
$$553$$ − 15504.0i − 1.19222i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 17786.0i 1.35299i 0.736446 + 0.676496i $$0.236503\pi$$
−0.736446 + 0.676496i $$0.763497\pi$$
$$558$$ 0 0
$$559$$ 10324.0 0.781143
$$560$$ 0 0
$$561$$ 6720.00 0.505737
$$562$$ 0 0
$$563$$ 20266.0i 1.51707i 0.651633 + 0.758535i $$0.274084\pi$$
−0.651633 + 0.758535i $$0.725916\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ − 30294.0i − 2.24379i
$$568$$ 0 0
$$569$$ −13358.0 −0.984177 −0.492088 0.870545i $$-0.663766\pi$$
−0.492088 + 0.870545i $$0.663766\pi$$
$$570$$ 0 0
$$571$$ −16360.0 −1.19903 −0.599514 0.800364i $$-0.704639\pi$$
−0.599514 + 0.800364i $$0.704639\pi$$
$$572$$ 0 0
$$573$$ 26760.0i 1.95099i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 15574.0i − 1.12366i −0.827251 0.561832i $$-0.810097\pi$$
0.827251 0.561832i $$-0.189903\pi$$
$$578$$ 0 0
$$579$$ 22692.0 1.62875
$$580$$ 0 0
$$581$$ −14756.0 −1.05367
$$582$$ 0 0
$$583$$ 2592.00i 0.184133i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 6654.00i − 0.467870i −0.972252 0.233935i $$-0.924840\pi$$
0.972252 0.233935i $$-0.0751604\pi$$
$$588$$ 0 0
$$589$$ −400.000 −0.0279825
$$590$$ 0 0
$$591$$ 26844.0 1.86838
$$592$$ 0 0
$$593$$ 17742.0i 1.22863i 0.789062 + 0.614314i $$0.210567\pi$$
−0.789062 + 0.614314i $$0.789433\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ − 21648.0i − 1.48408i
$$598$$ 0 0
$$599$$ 15840.0 1.08048 0.540238 0.841512i $$-0.318334\pi$$
0.540238 + 0.841512i $$0.318334\pi$$
$$600$$ 0 0
$$601$$ −3002.00 −0.203751 −0.101875 0.994797i $$-0.532484\pi$$
−0.101875 + 0.994797i $$0.532484\pi$$
$$602$$ 0 0
$$603$$ 3798.00i 0.256495i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 23610.0i 1.57875i 0.613912 + 0.789374i $$0.289595\pi$$
−0.613912 + 0.789374i $$0.710405\pi$$
$$608$$ 0 0
$$609$$ 49368.0 3.28488
$$610$$ 0 0
$$611$$ −1276.00 −0.0844868
$$612$$ 0 0
$$613$$ − 23850.0i − 1.57144i −0.618583 0.785720i $$-0.712293\pi$$
0.618583 0.785720i $$-0.287707\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 5334.00i − 0.348037i −0.984742 0.174018i $$-0.944325\pi$$
0.984742 0.174018i $$-0.0556752\pi$$
$$618$$ 0 0
$$619$$ −2164.00 −0.140515 −0.0702573 0.997529i $$-0.522382\pi$$
−0.0702573 + 0.997529i $$0.522382\pi$$
$$620$$ 0 0
$$621$$ −14472.0 −0.935171
$$622$$ 0 0
$$623$$ 24684.0i 1.58739i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 384.000i 0.0244585i
$$628$$ 0 0
$$629$$ −30660.0 −1.94355
$$630$$ 0 0
$$631$$ 25220.0 1.59111 0.795557 0.605879i $$-0.207179\pi$$
0.795557 + 0.605879i $$0.207179\pi$$
$$632$$ 0 0
$$633$$ − 1536.00i − 0.0964463i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 47154.0i 2.93298i
$$638$$ 0 0
$$639$$ −7668.00 −0.474713
$$640$$ 0 0
$$641$$ −12306.0 −0.758280 −0.379140 0.925339i $$-0.623780\pi$$
−0.379140 + 0.925339i $$0.623780\pi$$
$$642$$ 0 0
$$643$$ − 27414.0i − 1.68134i −0.541547 0.840671i $$-0.682161\pi$$
0.541547 0.840671i $$-0.317839\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 21834.0i 1.32671i 0.748304 + 0.663356i $$0.230869\pi$$
−0.748304 + 0.663356i $$0.769131\pi$$
$$648$$ 0 0
$$649$$ 4288.00 0.259351
$$650$$ 0 0
$$651$$ −20400.0 −1.22817
$$652$$ 0 0
$$653$$ 23998.0i 1.43815i 0.694931 + 0.719077i $$0.255435\pi$$
−0.694931 + 0.719077i $$0.744565\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 2754.00i 0.163537i
$$658$$ 0 0
$$659$$ −32004.0 −1.89180 −0.945902 0.324452i $$-0.894820\pi$$
−0.945902 + 0.324452i $$0.894820\pi$$
$$660$$ 0 0
$$661$$ −8526.00 −0.501699 −0.250849 0.968026i $$-0.580710\pi$$
−0.250849 + 0.968026i $$0.580710\pi$$
$$662$$ 0 0
$$663$$ 24360.0i 1.42694i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 32428.0i − 1.88248i
$$668$$ 0 0
$$669$$ −30948.0 −1.78852
$$670$$ 0 0
$$671$$ −4000.00 −0.230132
$$672$$ 0 0
$$673$$ − 8178.00i − 0.468408i −0.972187 0.234204i $$-0.924752\pi$$
0.972187 0.234204i $$-0.0752484\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 16646.0i − 0.944989i −0.881334 0.472495i $$-0.843354\pi$$
0.881334 0.472495i $$-0.156646\pi$$
$$678$$ 0 0
$$679$$ −46852.0 −2.64803
$$680$$ 0 0
$$681$$ 13356.0 0.751546
$$682$$ 0 0
$$683$$ − 22446.0i − 1.25750i −0.777608 0.628750i $$-0.783567\pi$$
0.777608 0.628750i $$-0.216433\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 12516.0i 0.695073i
$$688$$ 0 0
$$689$$ −9396.00 −0.519534
$$690$$ 0 0
$$691$$ −35336.0 −1.94536 −0.972681 0.232147i $$-0.925425\pi$$
−0.972681 + 0.232147i $$0.925425\pi$$
$$692$$ 0 0
$$693$$ 4896.00i 0.268375i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 9660.00i 0.524962i
$$698$$ 0 0
$$699$$ 34308.0 1.85643
$$700$$ 0 0
$$701$$ 3482.00 0.187608 0.0938041 0.995591i $$-0.470097\pi$$
0.0938041 + 0.995591i $$0.470097\pi$$
$$702$$ 0 0
$$703$$ − 1752.00i − 0.0939942i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 4284.00i 0.227887i
$$708$$ 0 0
$$709$$ 19402.0 1.02773 0.513863 0.857872i $$-0.328214\pi$$
0.513863 + 0.857872i $$0.328214\pi$$
$$710$$ 0 0
$$711$$ 4104.00 0.216473
$$712$$ 0 0
$$713$$ 13400.0i 0.703834i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 21744.0i 1.13256i
$$718$$ 0 0
$$719$$ −9896.00 −0.513294 −0.256647 0.966505i $$-0.582618\pi$$
−0.256647 + 0.966505i $$0.582618\pi$$
$$720$$ 0 0
$$721$$ 42908.0 2.21633
$$722$$ 0 0
$$723$$ 492.000i 0.0253080i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 494.000i − 0.0252014i −0.999921 0.0126007i $$-0.995989\pi$$
0.999921 0.0126007i $$-0.00401104\pi$$
$$728$$ 0 0
$$729$$ −9477.00 −0.481481
$$730$$ 0 0
$$731$$ 12460.0 0.630437
$$732$$ 0 0
$$733$$ − 9282.00i − 0.467720i −0.972270 0.233860i $$-0.924864\pi$$
0.972270 0.233860i $$-0.0751357\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 6752.00i 0.337467i
$$738$$ 0 0
$$739$$ −3252.00 −0.161877 −0.0809383 0.996719i $$-0.525792\pi$$
−0.0809383 + 0.996719i $$0.525792\pi$$
$$740$$ 0 0
$$741$$ −1392.00 −0.0690100
$$742$$ 0 0
$$743$$ − 4710.00i − 0.232561i −0.993216 0.116281i $$-0.962903\pi$$
0.993216 0.116281i $$-0.0370972\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ − 3906.00i − 0.191316i
$$748$$ 0 0
$$749$$ 17340.0 0.845914
$$750$$ 0 0
$$751$$ −25764.0 −1.25185 −0.625927 0.779882i $$-0.715279\pi$$
−0.625927 + 0.779882i $$0.715279\pi$$
$$752$$ 0 0
$$753$$ − 30240.0i − 1.46349i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 30094.0i − 1.44489i −0.691426 0.722447i $$-0.743017\pi$$
0.691426 0.722447i $$-0.256983\pi$$
$$758$$ 0 0
$$759$$ 12864.0 0.615196
$$760$$ 0 0
$$761$$ 22362.0 1.06521 0.532603 0.846365i $$-0.321214\pi$$
0.532603 + 0.846365i $$0.321214\pi$$
$$762$$ 0 0
$$763$$ − 884.000i − 0.0419436i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 15544.0i 0.731762i
$$768$$ 0 0
$$769$$ 30398.0 1.42546 0.712731 0.701438i $$-0.247458\pi$$
0.712731 + 0.701438i $$0.247458\pi$$
$$770$$ 0 0
$$771$$ −13860.0 −0.647413
$$772$$ 0 0
$$773$$ − 1290.00i − 0.0600234i −0.999550 0.0300117i $$-0.990446\pi$$
0.999550 0.0300117i $$-0.00955445\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 89352.0i − 4.12546i
$$778$$ 0 0
$$779$$ −552.000 −0.0253883
$$780$$ 0 0
$$781$$ −13632.0 −0.624573
$$782$$ 0 0
$$783$$ − 26136.0i − 1.19288i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 14.0000i 0 0.000634112i −1.00000 0.000317056i $$-0.999899\pi$$
1.00000 0.000317056i $$-0.000100922\pi$$
$$788$$ 0 0
$$789$$ −24660.0 −1.11270
$$790$$ 0 0
$$791$$ 42228.0 1.89817
$$792$$ 0 0
$$793$$ − 14500.0i − 0.649319i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 38814.0i − 1.72505i −0.506017 0.862523i $$-0.668883\pi$$
0.506017 0.862523i $$-0.331117\pi$$
$$798$$ 0 0
$$799$$ −1540.00 −0.0681868
$$800$$ 0 0
$$801$$ −6534.00 −0.288224
$$802$$ 0 0
$$803$$ 4896.00i 0.215163i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 4476.00i 0.195245i
$$808$$ 0 0
$$809$$ −27402.0 −1.19086 −0.595428 0.803408i $$-0.703018\pi$$
−0.595428 + 0.803408i $$0.703018\pi$$
$$810$$ 0 0
$$811$$ 28576.0 1.23729 0.618643 0.785672i $$-0.287683\pi$$
0.618643 + 0.785672i $$0.287683\pi$$
$$812$$ 0 0
$$813$$ − 27576.0i − 1.18958i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 712.000i 0.0304893i
$$818$$ 0 0
$$819$$ −17748.0 −0.757223
$$820$$ 0 0
$$821$$ 31762.0 1.35018 0.675092 0.737733i $$-0.264104\pi$$
0.675092 + 0.737733i $$0.264104\pi$$
$$822$$ 0 0
$$823$$ 20506.0i 0.868523i 0.900787 + 0.434261i $$0.142991\pi$$
−0.900787 + 0.434261i $$0.857009\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 13014.0i − 0.547208i −0.961842 0.273604i $$-0.911784\pi$$
0.961842 0.273604i $$-0.0882158\pi$$
$$828$$ 0 0
$$829$$ 22790.0 0.954800 0.477400 0.878686i $$-0.341579\pi$$
0.477400 + 0.878686i $$0.341579\pi$$
$$830$$ 0 0
$$831$$ −13236.0 −0.552529
$$832$$ 0 0
$$833$$ 56910.0i 2.36712i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 10800.0i 0.446001i
$$838$$ 0 0
$$839$$ 23696.0 0.975062 0.487531 0.873106i $$-0.337898\pi$$
0.487531 + 0.873106i $$0.337898\pi$$
$$840$$ 0 0
$$841$$ 34175.0 1.40125
$$842$$ 0 0
$$843$$ − 49668.0i − 2.02925i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 36550.0i − 1.48273i
$$848$$ 0 0
$$849$$ 7068.00 0.285716
$$850$$ 0 0
$$851$$ −58692.0 −2.36420
$$852$$ 0 0
$$853$$ − 5306.00i − 0.212982i −0.994314 0.106491i $$-0.966038\pi$$
0.994314 0.106491i $$-0.0339616\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 21054.0i − 0.839196i −0.907710 0.419598i $$-0.862171\pi$$
0.907710 0.419598i $$-0.137829\pi$$
$$858$$ 0 0
$$859$$ 7364.00 0.292499 0.146249 0.989248i $$-0.453280\pi$$
0.146249 + 0.989248i $$0.453280\pi$$
$$860$$ 0 0
$$861$$ −28152.0 −1.11431
$$862$$ 0 0
$$863$$ 17226.0i 0.679467i 0.940522 + 0.339733i $$0.110337\pi$$
−0.940522 + 0.339733i $$0.889663\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ − 78.0000i − 0.00305539i
$$868$$ 0 0
$$869$$ 7296.00 0.284810
$$870$$ 0 0
$$871$$ −24476.0 −0.952167
$$872$$ 0 0
$$873$$ − 12402.0i − 0.480807i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 21202.0i 0.816352i 0.912903 + 0.408176i $$0.133835\pi$$
−0.912903 + 0.408176i $$0.866165\pi$$
$$878$$ 0 0
$$879$$ −636.000 −0.0244047
$$880$$ 0 0
$$881$$ −29490.0 −1.12774 −0.563872 0.825862i $$-0.690689\pi$$
−0.563872 + 0.825862i $$0.690689\pi$$
$$882$$ 0 0
$$883$$ 2570.00i 0.0979472i 0.998800 + 0.0489736i $$0.0155950\pi$$
−0.998800 + 0.0489736i $$0.984405\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 36334.0i − 1.37540i −0.725997 0.687698i $$-0.758621\pi$$
0.725997 0.687698i $$-0.241379\pi$$
$$888$$ 0 0
$$889$$ −33252.0 −1.25448
$$890$$ 0 0
$$891$$ 14256.0 0.536020
$$892$$ 0 0
$$893$$ − 88.0000i − 0.00329766i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 46632.0i 1.73578i
$$898$$ 0 0
$$899$$ −24200.0 −0.897792
$$900$$ 0 0
$$901$$ −11340.0 −0.419301
$$902$$ 0 0
$$903$$ 36312.0i 1.33819i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 12474.0i 0.456662i 0.973584 + 0.228331i $$0.0733268\pi$$
−0.973584 + 0.228331i $$0.926673\pi$$
$$908$$ 0 0
$$909$$ −1134.00 −0.0413778
$$910$$ 0 0
$$911$$ 41132.0 1.49590 0.747949 0.663756i $$-0.231039\pi$$
0.747949 + 0.663756i $$0.231039\pi$$
$$912$$ 0 0
$$913$$ − 6944.00i − 0.251712i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 31008.0i 1.11666i
$$918$$ 0 0
$$919$$ −38416.0 −1.37892 −0.689460 0.724324i $$-0.742152\pi$$
−0.689460 + 0.724324i $$0.742152\pi$$
$$920$$ 0 0
$$921$$ −48804.0 −1.74609
$$922$$ 0 0
$$923$$ − 49416.0i − 1.76224i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 11358.0i 0.402423i
$$928$$ 0 0
$$929$$ −41302.0 −1.45864 −0.729319 0.684174i $$-0.760163\pi$$
−0.729319 + 0.684174i $$0.760163\pi$$
$$930$$ 0 0
$$931$$ −3252.00 −0.114479
$$932$$ 0 0
$$933$$ − 26376.0i − 0.925521i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 26150.0i − 0.911722i −0.890051 0.455861i $$-0.849331\pi$$
0.890051 0.455861i $$-0.150669\pi$$
$$938$$ 0 0
$$939$$ −28956.0 −1.00633
$$940$$ 0 0
$$941$$ 35254.0 1.22130 0.610652 0.791899i $$-0.290907\pi$$
0.610652 + 0.791899i $$0.290907\pi$$
$$942$$ 0 0
$$943$$ 18492.0i 0.638582i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 18550.0i − 0.636530i −0.948002 0.318265i $$-0.896900\pi$$
0.948002 0.318265i $$-0.103100\pi$$
$$948$$ 0 0
$$949$$ −17748.0 −0.607086
$$950$$ 0 0
$$951$$ 42156.0 1.43744
$$952$$ 0 0
$$953$$ − 17322.0i − 0.588788i −0.955684 0.294394i $$-0.904882\pi$$
0.955684 0.294394i $$-0.0951177\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 23232.0i 0.784727i
$$958$$ 0 0
$$959$$ 31484.0 1.06014
$$960$$ 0 0
$$961$$ −19791.0 −0.664328
$$962$$ 0 0
$$963$$ 4590.00i 0.153594i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 35190.0i − 1.17025i −0.810942 0.585126i $$-0.801045\pi$$
0.810942 0.585126i $$-0.198955\pi$$
$$968$$ 0 0
$$969$$ −1680.00 −0.0556960
$$970$$ 0 0
$$971$$ 40696.0 1.34500 0.672501 0.740096i $$-0.265220\pi$$
0.672501 + 0.740096i $$0.265220\pi$$
$$972$$ 0 0
$$973$$ 17544.0i 0.578042i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 44306.0i 1.45084i 0.688304 + 0.725422i $$0.258355\pi$$
−0.688304 + 0.725422i $$0.741645\pi$$
$$978$$ 0 0
$$979$$ −11616.0 −0.379212
$$980$$ 0 0
$$981$$ 234.000 0.00761574
$$982$$ 0 0
$$983$$ − 18798.0i − 0.609932i −0.952363 0.304966i $$-0.901355\pi$$
0.952363 0.304966i $$-0.0986451\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 4488.00i − 0.144736i
$$988$$ 0 0
$$989$$ 23852.0 0.766885
$$990$$ 0 0
$$991$$ −2468.00 −0.0791106 −0.0395553 0.999217i $$-0.512594\pi$$
−0.0395553 + 0.999217i $$0.512594\pi$$
$$992$$ 0 0
$$993$$ 52848.0i 1.68890i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 61086.0i − 1.94043i −0.242237 0.970217i $$-0.577881\pi$$
0.242237 0.970217i $$-0.422119\pi$$
$$998$$ 0 0
$$999$$ −47304.0 −1.49813
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 400.4.c.f.49.1 2
4.3 odd 2 200.4.c.c.49.2 2
5.2 odd 4 400.4.a.e.1.1 1
5.3 odd 4 80.4.a.e.1.1 1
5.4 even 2 inner 400.4.c.f.49.2 2
12.11 even 2 1800.4.f.j.649.1 2
15.8 even 4 720.4.a.bd.1.1 1
20.3 even 4 40.4.a.a.1.1 1
20.7 even 4 200.4.a.i.1.1 1
20.19 odd 2 200.4.c.c.49.1 2
40.3 even 4 320.4.a.l.1.1 1
40.13 odd 4 320.4.a.c.1.1 1
40.27 even 4 1600.4.a.j.1.1 1
40.37 odd 4 1600.4.a.br.1.1 1
60.23 odd 4 360.4.a.h.1.1 1
60.47 odd 4 1800.4.a.bi.1.1 1
60.59 even 2 1800.4.f.j.649.2 2
80.3 even 4 1280.4.d.p.641.1 2
80.13 odd 4 1280.4.d.a.641.2 2
80.43 even 4 1280.4.d.p.641.2 2
80.53 odd 4 1280.4.d.a.641.1 2
140.83 odd 4 1960.4.a.h.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
40.4.a.a.1.1 1 20.3 even 4
80.4.a.e.1.1 1 5.3 odd 4
200.4.a.i.1.1 1 20.7 even 4
200.4.c.c.49.1 2 20.19 odd 2
200.4.c.c.49.2 2 4.3 odd 2
320.4.a.c.1.1 1 40.13 odd 4
320.4.a.l.1.1 1 40.3 even 4
360.4.a.h.1.1 1 60.23 odd 4
400.4.a.e.1.1 1 5.2 odd 4
400.4.c.f.49.1 2 1.1 even 1 trivial
400.4.c.f.49.2 2 5.4 even 2 inner
720.4.a.bd.1.1 1 15.8 even 4
1280.4.d.a.641.1 2 80.53 odd 4
1280.4.d.a.641.2 2 80.13 odd 4
1280.4.d.p.641.1 2 80.3 even 4
1280.4.d.p.641.2 2 80.43 even 4
1600.4.a.j.1.1 1 40.27 even 4
1600.4.a.br.1.1 1 40.37 odd 4
1800.4.a.bi.1.1 1 60.47 odd 4
1800.4.f.j.649.1 2 12.11 even 2
1800.4.f.j.649.2 2 60.59 even 2
1960.4.a.h.1.1 1 140.83 odd 4