# Properties

 Label 6525.2.a.t.1.2 Level $6525$ Weight $2$ Character 6525.1 Self dual yes Analytic conductor $52.102$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6525,2,Mod(1,6525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6525, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6525.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6525 = 3^{2} \cdot 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6525.a (trivial)

## 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: yes Analytic conductor: $$52.1023873189$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{21})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 5$$ x^2 - x - 5 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 435) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.79129$$ of defining polynomial Character $$\chi$$ $$=$$ 6525.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.79129 q^{2} +1.20871 q^{4} -1.00000 q^{7} -1.41742 q^{8} +O(q^{10})$$ $$q+1.79129 q^{2} +1.20871 q^{4} -1.00000 q^{7} -1.41742 q^{8} -5.00000 q^{11} +4.58258 q^{13} -1.79129 q^{14} -4.95644 q^{16} -3.00000 q^{17} +3.58258 q^{19} -8.95644 q^{22} -4.00000 q^{23} +8.20871 q^{26} -1.20871 q^{28} -1.00000 q^{29} +4.00000 q^{31} -6.04356 q^{32} -5.37386 q^{34} +4.00000 q^{37} +6.41742 q^{38} +9.16515 q^{41} +9.58258 q^{43} -6.04356 q^{44} -7.16515 q^{46} +10.5826 q^{47} -6.00000 q^{49} +5.53901 q^{52} +0.417424 q^{53} +1.41742 q^{56} -1.79129 q^{58} +7.58258 q^{59} +12.7477 q^{61} +7.16515 q^{62} -0.912878 q^{64} +4.16515 q^{67} -3.62614 q^{68} +9.58258 q^{71} -4.00000 q^{73} +7.16515 q^{74} +4.33030 q^{76} +5.00000 q^{77} +7.58258 q^{79} +16.4174 q^{82} -11.5826 q^{83} +17.1652 q^{86} +7.08712 q^{88} -1.41742 q^{89} -4.58258 q^{91} -4.83485 q^{92} +18.9564 q^{94} -11.5826 q^{97} -10.7477 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 7 q^{4} - 2 q^{7} - 12 q^{8}+O(q^{10})$$ 2 * q - q^2 + 7 * q^4 - 2 * q^7 - 12 * q^8 $$2 q - q^{2} + 7 q^{4} - 2 q^{7} - 12 q^{8} - 10 q^{11} + q^{14} + 13 q^{16} - 6 q^{17} - 2 q^{19} + 5 q^{22} - 8 q^{23} + 21 q^{26} - 7 q^{28} - 2 q^{29} + 8 q^{31} - 35 q^{32} + 3 q^{34} + 8 q^{37} + 22 q^{38} + 10 q^{43} - 35 q^{44} + 4 q^{46} + 12 q^{47} - 12 q^{49} - 21 q^{52} + 10 q^{53} + 12 q^{56} + q^{58} + 6 q^{59} - 2 q^{61} - 4 q^{62} + 44 q^{64} - 10 q^{67} - 21 q^{68} + 10 q^{71} - 8 q^{73} - 4 q^{74} - 28 q^{76} + 10 q^{77} + 6 q^{79} + 42 q^{82} - 14 q^{83} + 16 q^{86} + 60 q^{88} - 12 q^{89} - 28 q^{92} + 15 q^{94} - 14 q^{97} + 6 q^{98}+O(q^{100})$$ 2 * q - q^2 + 7 * q^4 - 2 * q^7 - 12 * q^8 - 10 * q^11 + q^14 + 13 * q^16 - 6 * q^17 - 2 * q^19 + 5 * q^22 - 8 * q^23 + 21 * q^26 - 7 * q^28 - 2 * q^29 + 8 * q^31 - 35 * q^32 + 3 * q^34 + 8 * q^37 + 22 * q^38 + 10 * q^43 - 35 * q^44 + 4 * q^46 + 12 * q^47 - 12 * q^49 - 21 * q^52 + 10 * q^53 + 12 * q^56 + q^58 + 6 * q^59 - 2 * q^61 - 4 * q^62 + 44 * q^64 - 10 * q^67 - 21 * q^68 + 10 * q^71 - 8 * q^73 - 4 * q^74 - 28 * q^76 + 10 * q^77 + 6 * q^79 + 42 * q^82 - 14 * q^83 + 16 * q^86 + 60 * q^88 - 12 * q^89 - 28 * q^92 + 15 * q^94 - 14 * q^97 + 6 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.79129 1.26663 0.633316 0.773893i $$-0.281693\pi$$
0.633316 + 0.773893i $$0.281693\pi$$
$$3$$ 0 0
$$4$$ 1.20871 0.604356
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −1.00000 −0.377964 −0.188982 0.981981i $$-0.560519\pi$$
−0.188982 + 0.981981i $$0.560519\pi$$
$$8$$ −1.41742 −0.501135
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −5.00000 −1.50756 −0.753778 0.657129i $$-0.771771\pi$$
−0.753778 + 0.657129i $$0.771771\pi$$
$$12$$ 0 0
$$13$$ 4.58258 1.27098 0.635489 0.772110i $$-0.280799\pi$$
0.635489 + 0.772110i $$0.280799\pi$$
$$14$$ −1.79129 −0.478742
$$15$$ 0 0
$$16$$ −4.95644 −1.23911
$$17$$ −3.00000 −0.727607 −0.363803 0.931476i $$-0.618522\pi$$
−0.363803 + 0.931476i $$0.618522\pi$$
$$18$$ 0 0
$$19$$ 3.58258 0.821899 0.410950 0.911658i $$-0.365197\pi$$
0.410950 + 0.911658i $$0.365197\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −8.95644 −1.90952
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 8.20871 1.60986
$$27$$ 0 0
$$28$$ −1.20871 −0.228425
$$29$$ −1.00000 −0.185695
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ −6.04356 −1.06836
$$33$$ 0 0
$$34$$ −5.37386 −0.921610
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 4.00000 0.657596 0.328798 0.944400i $$-0.393356\pi$$
0.328798 + 0.944400i $$0.393356\pi$$
$$38$$ 6.41742 1.04104
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 9.16515 1.43136 0.715678 0.698430i $$-0.246118\pi$$
0.715678 + 0.698430i $$0.246118\pi$$
$$42$$ 0 0
$$43$$ 9.58258 1.46133 0.730665 0.682737i $$-0.239210\pi$$
0.730665 + 0.682737i $$0.239210\pi$$
$$44$$ −6.04356 −0.911101
$$45$$ 0 0
$$46$$ −7.16515 −1.05644
$$47$$ 10.5826 1.54363 0.771814 0.635849i $$-0.219350\pi$$
0.771814 + 0.635849i $$0.219350\pi$$
$$48$$ 0 0
$$49$$ −6.00000 −0.857143
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 5.53901 0.768123
$$53$$ 0.417424 0.0573376 0.0286688 0.999589i $$-0.490873\pi$$
0.0286688 + 0.999589i $$0.490873\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 1.41742 0.189411
$$57$$ 0 0
$$58$$ −1.79129 −0.235208
$$59$$ 7.58258 0.987167 0.493584 0.869698i $$-0.335687\pi$$
0.493584 + 0.869698i $$0.335687\pi$$
$$60$$ 0 0
$$61$$ 12.7477 1.63218 0.816090 0.577925i $$-0.196138\pi$$
0.816090 + 0.577925i $$0.196138\pi$$
$$62$$ 7.16515 0.909975
$$63$$ 0 0
$$64$$ −0.912878 −0.114110
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 4.16515 0.508854 0.254427 0.967092i $$-0.418113\pi$$
0.254427 + 0.967092i $$0.418113\pi$$
$$68$$ −3.62614 −0.439734
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 9.58258 1.13724 0.568621 0.822599i $$-0.307477\pi$$
0.568621 + 0.822599i $$0.307477\pi$$
$$72$$ 0 0
$$73$$ −4.00000 −0.468165 −0.234082 0.972217i $$-0.575209\pi$$
−0.234082 + 0.972217i $$0.575209\pi$$
$$74$$ 7.16515 0.832932
$$75$$ 0 0
$$76$$ 4.33030 0.496720
$$77$$ 5.00000 0.569803
$$78$$ 0 0
$$79$$ 7.58258 0.853106 0.426553 0.904462i $$-0.359728\pi$$
0.426553 + 0.904462i $$0.359728\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 16.4174 1.81300
$$83$$ −11.5826 −1.27135 −0.635676 0.771956i $$-0.719279\pi$$
−0.635676 + 0.771956i $$0.719279\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 17.1652 1.85097
$$87$$ 0 0
$$88$$ 7.08712 0.755490
$$89$$ −1.41742 −0.150247 −0.0751233 0.997174i $$-0.523935\pi$$
−0.0751233 + 0.997174i $$0.523935\pi$$
$$90$$ 0 0
$$91$$ −4.58258 −0.480384
$$92$$ −4.83485 −0.504068
$$93$$ 0 0
$$94$$ 18.9564 1.95521
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −11.5826 −1.17603 −0.588016 0.808849i $$-0.700091\pi$$
−0.588016 + 0.808849i $$0.700091\pi$$
$$98$$ −10.7477 −1.08568
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 0.582576 0.0579684 0.0289842 0.999580i $$-0.490773\pi$$
0.0289842 + 0.999580i $$0.490773\pi$$
$$102$$ 0 0
$$103$$ −15.1652 −1.49427 −0.747133 0.664674i $$-0.768570\pi$$
−0.747133 + 0.664674i $$0.768570\pi$$
$$104$$ −6.49545 −0.636932
$$105$$ 0 0
$$106$$ 0.747727 0.0726257
$$107$$ 5.16515 0.499334 0.249667 0.968332i $$-0.419679\pi$$
0.249667 + 0.968332i $$0.419679\pi$$
$$108$$ 0 0
$$109$$ 14.1652 1.35678 0.678388 0.734704i $$-0.262679\pi$$
0.678388 + 0.734704i $$0.262679\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 4.95644 0.468339
$$113$$ −14.1652 −1.33255 −0.666273 0.745708i $$-0.732111\pi$$
−0.666273 + 0.745708i $$0.732111\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −1.20871 −0.112226
$$117$$ 0 0
$$118$$ 13.5826 1.25038
$$119$$ 3.00000 0.275010
$$120$$ 0 0
$$121$$ 14.0000 1.27273
$$122$$ 22.8348 2.06737
$$123$$ 0 0
$$124$$ 4.83485 0.434182
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −2.00000 −0.177471 −0.0887357 0.996055i $$-0.528283\pi$$
−0.0887357 + 0.996055i $$0.528283\pi$$
$$128$$ 10.4519 0.923826
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 15.0000 1.31056 0.655278 0.755388i $$-0.272551\pi$$
0.655278 + 0.755388i $$0.272551\pi$$
$$132$$ 0 0
$$133$$ −3.58258 −0.310649
$$134$$ 7.46099 0.644531
$$135$$ 0 0
$$136$$ 4.25227 0.364629
$$137$$ 16.3303 1.39519 0.697596 0.716491i $$-0.254253\pi$$
0.697596 + 0.716491i $$0.254253\pi$$
$$138$$ 0 0
$$139$$ −9.41742 −0.798776 −0.399388 0.916782i $$-0.630777\pi$$
−0.399388 + 0.916782i $$0.630777\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 17.1652 1.44047
$$143$$ −22.9129 −1.91607
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −7.16515 −0.592992
$$147$$ 0 0
$$148$$ 4.83485 0.397422
$$149$$ −16.7477 −1.37203 −0.686014 0.727589i $$-0.740641\pi$$
−0.686014 + 0.727589i $$0.740641\pi$$
$$150$$ 0 0
$$151$$ 7.16515 0.583092 0.291546 0.956557i $$-0.405830\pi$$
0.291546 + 0.956557i $$0.405830\pi$$
$$152$$ −5.07803 −0.411883
$$153$$ 0 0
$$154$$ 8.95644 0.721730
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −10.7477 −0.857762 −0.428881 0.903361i $$-0.641092\pi$$
−0.428881 + 0.903361i $$0.641092\pi$$
$$158$$ 13.5826 1.08057
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 4.00000 0.315244
$$162$$ 0 0
$$163$$ −7.58258 −0.593913 −0.296957 0.954891i $$-0.595972\pi$$
−0.296957 + 0.954891i $$0.595972\pi$$
$$164$$ 11.0780 0.865049
$$165$$ 0 0
$$166$$ −20.7477 −1.61034
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ 8.00000 0.615385
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 11.5826 0.883163
$$173$$ −3.16515 −0.240642 −0.120321 0.992735i $$-0.538392\pi$$
−0.120321 + 0.992735i $$0.538392\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 24.7822 1.86803
$$177$$ 0 0
$$178$$ −2.53901 −0.190307
$$179$$ −4.74773 −0.354862 −0.177431 0.984133i $$-0.556779\pi$$
−0.177431 + 0.984133i $$0.556779\pi$$
$$180$$ 0 0
$$181$$ 16.1652 1.20155 0.600773 0.799420i $$-0.294860\pi$$
0.600773 + 0.799420i $$0.294860\pi$$
$$182$$ −8.20871 −0.608470
$$183$$ 0 0
$$184$$ 5.66970 0.417976
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 15.0000 1.09691
$$188$$ 12.7913 0.932901
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 4.00000 0.289430 0.144715 0.989473i $$-0.453773\pi$$
0.144715 + 0.989473i $$0.453773\pi$$
$$192$$ 0 0
$$193$$ 20.3303 1.46341 0.731704 0.681623i $$-0.238726\pi$$
0.731704 + 0.681623i $$0.238726\pi$$
$$194$$ −20.7477 −1.48960
$$195$$ 0 0
$$196$$ −7.25227 −0.518019
$$197$$ −16.3303 −1.16349 −0.581743 0.813373i $$-0.697629\pi$$
−0.581743 + 0.813373i $$0.697629\pi$$
$$198$$ 0 0
$$199$$ 13.4174 0.951136 0.475568 0.879679i $$-0.342243\pi$$
0.475568 + 0.879679i $$0.342243\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 1.04356 0.0734247
$$203$$ 1.00000 0.0701862
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −27.1652 −1.89269
$$207$$ 0 0
$$208$$ −22.7133 −1.57488
$$209$$ −17.9129 −1.23906
$$210$$ 0 0
$$211$$ 20.3303 1.39960 0.699798 0.714341i $$-0.253273\pi$$
0.699798 + 0.714341i $$0.253273\pi$$
$$212$$ 0.504546 0.0346523
$$213$$ 0 0
$$214$$ 9.25227 0.632472
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −4.00000 −0.271538
$$218$$ 25.3739 1.71853
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −13.7477 −0.924772
$$222$$ 0 0
$$223$$ −7.00000 −0.468755 −0.234377 0.972146i $$-0.575305\pi$$
−0.234377 + 0.972146i $$0.575305\pi$$
$$224$$ 6.04356 0.403802
$$225$$ 0 0
$$226$$ −25.3739 −1.68784
$$227$$ −7.58258 −0.503273 −0.251637 0.967822i $$-0.580969\pi$$
−0.251637 + 0.967822i $$0.580969\pi$$
$$228$$ 0 0
$$229$$ −17.1652 −1.13431 −0.567153 0.823613i $$-0.691955\pi$$
−0.567153 + 0.823613i $$0.691955\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 1.41742 0.0930585
$$233$$ −13.1652 −0.862478 −0.431239 0.902238i $$-0.641923\pi$$
−0.431239 + 0.902238i $$0.641923\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 9.16515 0.596601
$$237$$ 0 0
$$238$$ 5.37386 0.348336
$$239$$ 26.0000 1.68180 0.840900 0.541190i $$-0.182026\pi$$
0.840900 + 0.541190i $$0.182026\pi$$
$$240$$ 0 0
$$241$$ 29.3303 1.88933 0.944665 0.328035i $$-0.106387\pi$$
0.944665 + 0.328035i $$0.106387\pi$$
$$242$$ 25.0780 1.61208
$$243$$ 0 0
$$244$$ 15.4083 0.986417
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 16.4174 1.04462
$$248$$ −5.66970 −0.360026
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −16.1652 −1.02034 −0.510168 0.860075i $$-0.670417\pi$$
−0.510168 + 0.860075i $$0.670417\pi$$
$$252$$ 0 0
$$253$$ 20.0000 1.25739
$$254$$ −3.58258 −0.224791
$$255$$ 0 0
$$256$$ 20.5481 1.28426
$$257$$ 12.7477 0.795181 0.397591 0.917563i $$-0.369846\pi$$
0.397591 + 0.917563i $$0.369846\pi$$
$$258$$ 0 0
$$259$$ −4.00000 −0.248548
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 26.8693 1.65999
$$263$$ 30.3303 1.87025 0.935123 0.354322i $$-0.115288\pi$$
0.935123 + 0.354322i $$0.115288\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −6.41742 −0.393478
$$267$$ 0 0
$$268$$ 5.03447 0.307529
$$269$$ −22.5826 −1.37688 −0.688442 0.725291i $$-0.741705\pi$$
−0.688442 + 0.725291i $$0.741705\pi$$
$$270$$ 0 0
$$271$$ 1.16515 0.0707779 0.0353890 0.999374i $$-0.488733\pi$$
0.0353890 + 0.999374i $$0.488733\pi$$
$$272$$ 14.8693 0.901585
$$273$$ 0 0
$$274$$ 29.2523 1.76719
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 24.9129 1.49687 0.748435 0.663208i $$-0.230806\pi$$
0.748435 + 0.663208i $$0.230806\pi$$
$$278$$ −16.8693 −1.01175
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 0.417424 0.0249014 0.0124507 0.999922i $$-0.496037\pi$$
0.0124507 + 0.999922i $$0.496037\pi$$
$$282$$ 0 0
$$283$$ 0.834849 0.0496266 0.0248133 0.999692i $$-0.492101\pi$$
0.0248133 + 0.999692i $$0.492101\pi$$
$$284$$ 11.5826 0.687299
$$285$$ 0 0
$$286$$ −41.0436 −2.42696
$$287$$ −9.16515 −0.541002
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −4.83485 −0.282938
$$293$$ 30.1652 1.76227 0.881133 0.472868i $$-0.156781\pi$$
0.881133 + 0.472868i $$0.156781\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −5.66970 −0.329544
$$297$$ 0 0
$$298$$ −30.0000 −1.73785
$$299$$ −18.3303 −1.06007
$$300$$ 0 0
$$301$$ −9.58258 −0.552330
$$302$$ 12.8348 0.738563
$$303$$ 0 0
$$304$$ −17.7568 −1.01842
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$308$$ 6.04356 0.344364
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 3.00000 0.170114 0.0850572 0.996376i $$-0.472893\pi$$
0.0850572 + 0.996376i $$0.472893\pi$$
$$312$$ 0 0
$$313$$ −10.5826 −0.598163 −0.299081 0.954228i $$-0.596680\pi$$
−0.299081 + 0.954228i $$0.596680\pi$$
$$314$$ −19.2523 −1.08647
$$315$$ 0 0
$$316$$ 9.16515 0.515580
$$317$$ −25.0000 −1.40414 −0.702070 0.712108i $$-0.747741\pi$$
−0.702070 + 0.712108i $$0.747741\pi$$
$$318$$ 0 0
$$319$$ 5.00000 0.279946
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 7.16515 0.399298
$$323$$ −10.7477 −0.598020
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −13.5826 −0.752269
$$327$$ 0 0
$$328$$ −12.9909 −0.717303
$$329$$ −10.5826 −0.583436
$$330$$ 0 0
$$331$$ −8.33030 −0.457875 −0.228937 0.973441i $$-0.573525\pi$$
−0.228937 + 0.973441i $$0.573525\pi$$
$$332$$ −14.0000 −0.768350
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 21.1652 1.15294 0.576470 0.817119i $$-0.304430\pi$$
0.576470 + 0.817119i $$0.304430\pi$$
$$338$$ 14.3303 0.779466
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −20.0000 −1.08306
$$342$$ 0 0
$$343$$ 13.0000 0.701934
$$344$$ −13.5826 −0.732323
$$345$$ 0 0
$$346$$ −5.66970 −0.304805
$$347$$ 7.16515 0.384645 0.192323 0.981332i $$-0.438398\pi$$
0.192323 + 0.981332i $$0.438398\pi$$
$$348$$ 0 0
$$349$$ −4.33030 −0.231796 −0.115898 0.993261i $$-0.536975\pi$$
−0.115898 + 0.993261i $$0.536975\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 30.2178 1.61061
$$353$$ −14.8348 −0.789579 −0.394790 0.918772i $$-0.629183\pi$$
−0.394790 + 0.918772i $$0.629183\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −1.71326 −0.0908025
$$357$$ 0 0
$$358$$ −8.50455 −0.449479
$$359$$ 27.1652 1.43372 0.716861 0.697216i $$-0.245578\pi$$
0.716861 + 0.697216i $$0.245578\pi$$
$$360$$ 0 0
$$361$$ −6.16515 −0.324482
$$362$$ 28.9564 1.52192
$$363$$ 0 0
$$364$$ −5.53901 −0.290323
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 37.4955 1.95725 0.978623 0.205661i $$-0.0659343\pi$$
0.978623 + 0.205661i $$0.0659343\pi$$
$$368$$ 19.8258 1.03349
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −0.417424 −0.0216716
$$372$$ 0 0
$$373$$ −28.3303 −1.46689 −0.733444 0.679750i $$-0.762088\pi$$
−0.733444 + 0.679750i $$0.762088\pi$$
$$374$$ 26.8693 1.38938
$$375$$ 0 0
$$376$$ −15.0000 −0.773566
$$377$$ −4.58258 −0.236015
$$378$$ 0 0
$$379$$ −26.0000 −1.33553 −0.667765 0.744372i $$-0.732749\pi$$
−0.667765 + 0.744372i $$0.732749\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 7.16515 0.366601
$$383$$ 3.58258 0.183061 0.0915305 0.995802i $$-0.470824\pi$$
0.0915305 + 0.995802i $$0.470824\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 36.4174 1.85360
$$387$$ 0 0
$$388$$ −14.0000 −0.710742
$$389$$ 6.58258 0.333750 0.166875 0.985978i $$-0.446632\pi$$
0.166875 + 0.985978i $$0.446632\pi$$
$$390$$ 0 0
$$391$$ 12.0000 0.606866
$$392$$ 8.50455 0.429544
$$393$$ 0 0
$$394$$ −29.2523 −1.47371
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −10.8348 −0.543785 −0.271893 0.962328i $$-0.587650\pi$$
−0.271893 + 0.962328i $$0.587650\pi$$
$$398$$ 24.0345 1.20474
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −12.4174 −0.620097 −0.310048 0.950721i $$-0.600345\pi$$
−0.310048 + 0.950721i $$0.600345\pi$$
$$402$$ 0 0
$$403$$ 18.3303 0.913097
$$404$$ 0.704166 0.0350336
$$405$$ 0 0
$$406$$ 1.79129 0.0889001
$$407$$ −20.0000 −0.991363
$$408$$ 0 0
$$409$$ −2.74773 −0.135866 −0.0679332 0.997690i $$-0.521640\pi$$
−0.0679332 + 0.997690i $$0.521640\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −18.3303 −0.903069
$$413$$ −7.58258 −0.373114
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −27.6951 −1.35786
$$417$$ 0 0
$$418$$ −32.0871 −1.56943
$$419$$ 37.1652 1.81564 0.907818 0.419364i $$-0.137747\pi$$
0.907818 + 0.419364i $$0.137747\pi$$
$$420$$ 0 0
$$421$$ 4.41742 0.215292 0.107646 0.994189i $$-0.465669\pi$$
0.107646 + 0.994189i $$0.465669\pi$$
$$422$$ 36.4174 1.77277
$$423$$ 0 0
$$424$$ −0.591667 −0.0287339
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −12.7477 −0.616906
$$428$$ 6.24318 0.301776
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 39.1652 1.88652 0.943259 0.332057i $$-0.107742\pi$$
0.943259 + 0.332057i $$0.107742\pi$$
$$432$$ 0 0
$$433$$ −25.0780 −1.20517 −0.602587 0.798053i $$-0.705863\pi$$
−0.602587 + 0.798053i $$0.705863\pi$$
$$434$$ −7.16515 −0.343938
$$435$$ 0 0
$$436$$ 17.1216 0.819975
$$437$$ −14.3303 −0.685511
$$438$$ 0 0
$$439$$ −16.9129 −0.807208 −0.403604 0.914934i $$-0.632243\pi$$
−0.403604 + 0.914934i $$0.632243\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −24.6261 −1.17135
$$443$$ 0.582576 0.0276790 0.0138395 0.999904i $$-0.495595\pi$$
0.0138395 + 0.999904i $$0.495595\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −12.5390 −0.593740
$$447$$ 0 0
$$448$$ 0.912878 0.0431295
$$449$$ 30.0780 1.41947 0.709735 0.704469i $$-0.248815\pi$$
0.709735 + 0.704469i $$0.248815\pi$$
$$450$$ 0 0
$$451$$ −45.8258 −2.15785
$$452$$ −17.1216 −0.805332
$$453$$ 0 0
$$454$$ −13.5826 −0.637462
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 3.74773 0.175311 0.0876556 0.996151i $$-0.472062\pi$$
0.0876556 + 0.996151i $$0.472062\pi$$
$$458$$ −30.7477 −1.43675
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 9.16515 0.426864 0.213432 0.976958i $$-0.431536\pi$$
0.213432 + 0.976958i $$0.431536\pi$$
$$462$$ 0 0
$$463$$ 0.165151 0.00767524 0.00383762 0.999993i $$-0.498778\pi$$
0.00383762 + 0.999993i $$0.498778\pi$$
$$464$$ 4.95644 0.230097
$$465$$ 0 0
$$466$$ −23.5826 −1.09244
$$467$$ −8.00000 −0.370196 −0.185098 0.982720i $$-0.559260\pi$$
−0.185098 + 0.982720i $$0.559260\pi$$
$$468$$ 0 0
$$469$$ −4.16515 −0.192329
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −10.7477 −0.494704
$$473$$ −47.9129 −2.20304
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 3.62614 0.166204
$$477$$ 0 0
$$478$$ 46.5735 2.13022
$$479$$ 19.1652 0.875678 0.437839 0.899053i $$-0.355744\pi$$
0.437839 + 0.899053i $$0.355744\pi$$
$$480$$ 0 0
$$481$$ 18.3303 0.835790
$$482$$ 52.5390 2.39309
$$483$$ 0 0
$$484$$ 16.9220 0.769180
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −2.33030 −0.105596 −0.0527980 0.998605i $$-0.516814\pi$$
−0.0527980 + 0.998605i $$0.516814\pi$$
$$488$$ −18.0689 −0.817942
$$489$$ 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
$$493$$ 3.00000 0.135113
$$494$$ 29.4083 1.32314
$$495$$ 0 0
$$496$$ −19.8258 −0.890203
$$497$$ −9.58258 −0.429837
$$498$$ 0 0
$$499$$ −13.4174 −0.600646 −0.300323 0.953837i $$-0.597095\pi$$
−0.300323 + 0.953837i $$0.597095\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −28.9564 −1.29239
$$503$$ −22.9129 −1.02163 −0.510817 0.859689i $$-0.670657\pi$$
−0.510817 + 0.859689i $$0.670657\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 35.8258 1.59265
$$507$$ 0 0
$$508$$ −2.41742 −0.107256
$$509$$ −26.7477 −1.18557 −0.592786 0.805360i $$-0.701972\pi$$
−0.592786 + 0.805360i $$0.701972\pi$$
$$510$$ 0 0
$$511$$ 4.00000 0.176950
$$512$$ 15.9038 0.702855
$$513$$ 0 0
$$514$$ 22.8348 1.00720
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −52.9129 −2.32711
$$518$$ −7.16515 −0.314819
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −43.0780 −1.88728 −0.943641 0.330970i $$-0.892624\pi$$
−0.943641 + 0.330970i $$0.892624\pi$$
$$522$$ 0 0
$$523$$ 33.3303 1.45743 0.728716 0.684816i $$-0.240117\pi$$
0.728716 + 0.684816i $$0.240117\pi$$
$$524$$ 18.1307 0.792043
$$525$$ 0 0
$$526$$ 54.3303 2.36891
$$527$$ −12.0000 −0.522728
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −4.33030 −0.187742
$$533$$ 42.0000 1.81922
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −5.90379 −0.255005
$$537$$ 0 0
$$538$$ −40.4519 −1.74400
$$539$$ 30.0000 1.29219
$$540$$ 0 0
$$541$$ −31.4955 −1.35410 −0.677048 0.735939i $$-0.736741\pi$$
−0.677048 + 0.735939i $$0.736741\pi$$
$$542$$ 2.08712 0.0896495
$$543$$ 0 0
$$544$$ 18.1307 0.777347
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −4.16515 −0.178089 −0.0890445 0.996028i $$-0.528381\pi$$
−0.0890445 + 0.996028i $$0.528381\pi$$
$$548$$ 19.7386 0.843193
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −3.58258 −0.152623
$$552$$ 0 0
$$553$$ −7.58258 −0.322444
$$554$$ 44.6261 1.89598
$$555$$ 0 0
$$556$$ −11.3830 −0.482745
$$557$$ −22.7477 −0.963852 −0.481926 0.876212i $$-0.660063\pi$$
−0.481926 + 0.876212i $$0.660063\pi$$
$$558$$ 0 0
$$559$$ 43.9129 1.85732
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0.747727 0.0315410
$$563$$ −14.5826 −0.614582 −0.307291 0.951616i $$-0.599423\pi$$
−0.307291 + 0.951616i $$0.599423\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 1.49545 0.0628586
$$567$$ 0 0
$$568$$ −13.5826 −0.569912
$$569$$ −28.5826 −1.19824 −0.599122 0.800658i $$-0.704484\pi$$
−0.599122 + 0.800658i $$0.704484\pi$$
$$570$$ 0 0
$$571$$ 12.0000 0.502184 0.251092 0.967963i $$-0.419210\pi$$
0.251092 + 0.967963i $$0.419210\pi$$
$$572$$ −27.6951 −1.15799
$$573$$ 0 0
$$574$$ −16.4174 −0.685250
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 19.1652 0.797856 0.398928 0.916982i $$-0.369382\pi$$
0.398928 + 0.916982i $$0.369382\pi$$
$$578$$ −14.3303 −0.596062
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 11.5826 0.480526
$$582$$ 0 0
$$583$$ −2.08712 −0.0864397
$$584$$ 5.66970 0.234614
$$585$$ 0 0
$$586$$ 54.0345 2.23214
$$587$$ −11.5826 −0.478064 −0.239032 0.971012i $$-0.576830\pi$$
−0.239032 + 0.971012i $$0.576830\pi$$
$$588$$ 0 0
$$589$$ 14.3303 0.590470
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −19.8258 −0.814834
$$593$$ 8.41742 0.345662 0.172831 0.984951i $$-0.444709\pi$$
0.172831 + 0.984951i $$0.444709\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −20.2432 −0.829193
$$597$$ 0 0
$$598$$ −32.8348 −1.34272
$$599$$ −0.165151 −0.00674790 −0.00337395 0.999994i $$-0.501074\pi$$
−0.00337395 + 0.999994i $$0.501074\pi$$
$$600$$ 0 0
$$601$$ −32.3303 −1.31878 −0.659390 0.751801i $$-0.729185\pi$$
−0.659390 + 0.751801i $$0.729185\pi$$
$$602$$ −17.1652 −0.699599
$$603$$ 0 0
$$604$$ 8.66061 0.352395
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −3.58258 −0.145412 −0.0727061 0.997353i $$-0.523164\pi$$
−0.0727061 + 0.997353i $$0.523164\pi$$
$$608$$ −21.6515 −0.878085
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 48.4955 1.96192
$$612$$ 0 0
$$613$$ 43.7477 1.76695 0.883477 0.468474i $$-0.155196\pi$$
0.883477 + 0.468474i $$0.155196\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ −7.08712 −0.285548
$$617$$ 15.4955 0.623823 0.311912 0.950111i $$-0.399031\pi$$
0.311912 + 0.950111i $$0.399031\pi$$
$$618$$ 0 0
$$619$$ 13.1652 0.529152 0.264576 0.964365i $$-0.414768\pi$$
0.264576 + 0.964365i $$0.414768\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 5.37386 0.215472
$$623$$ 1.41742 0.0567879
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −18.9564 −0.757652
$$627$$ 0 0
$$628$$ −12.9909 −0.518394
$$629$$ −12.0000 −0.478471
$$630$$ 0 0
$$631$$ 10.9129 0.434435 0.217217 0.976123i $$-0.430302\pi$$
0.217217 + 0.976123i $$0.430302\pi$$
$$632$$ −10.7477 −0.427522
$$633$$ 0 0
$$634$$ −44.7822 −1.77853
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −27.4955 −1.08941
$$638$$ 8.95644 0.354589
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 6.58258 0.259996 0.129998 0.991514i $$-0.458503\pi$$
0.129998 + 0.991514i $$0.458503\pi$$
$$642$$ 0 0
$$643$$ −43.6606 −1.72181 −0.860903 0.508769i $$-0.830101\pi$$
−0.860903 + 0.508769i $$0.830101\pi$$
$$644$$ 4.83485 0.190520
$$645$$ 0 0
$$646$$ −19.2523 −0.757471
$$647$$ 24.7477 0.972934 0.486467 0.873699i $$-0.338285\pi$$
0.486467 + 0.873699i $$0.338285\pi$$
$$648$$ 0 0
$$649$$ −37.9129 −1.48821
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −9.16515 −0.358935
$$653$$ 26.1652 1.02392 0.511961 0.859009i $$-0.328919\pi$$
0.511961 + 0.859009i $$0.328919\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −45.4265 −1.77361
$$657$$ 0 0
$$658$$ −18.9564 −0.738999
$$659$$ −18.1652 −0.707614 −0.353807 0.935318i $$-0.615113\pi$$
−0.353807 + 0.935318i $$0.615113\pi$$
$$660$$ 0 0
$$661$$ 25.0000 0.972387 0.486194 0.873851i $$-0.338385\pi$$
0.486194 + 0.873851i $$0.338385\pi$$
$$662$$ −14.9220 −0.579959
$$663$$ 0 0
$$664$$ 16.4174 0.637120
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 4.00000 0.154881
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −63.7386 −2.46060
$$672$$ 0 0
$$673$$ −1.74773 −0.0673699 −0.0336850 0.999433i $$-0.510724\pi$$
−0.0336850 + 0.999433i $$0.510724\pi$$
$$674$$ 37.9129 1.46035
$$675$$ 0 0
$$676$$ 9.66970 0.371911
$$677$$ 44.8258 1.72279 0.861397 0.507932i $$-0.169590\pi$$
0.861397 + 0.507932i $$0.169590\pi$$
$$678$$ 0 0
$$679$$ 11.5826 0.444498
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −35.8258 −1.37184
$$683$$ 7.16515 0.274167 0.137083 0.990560i $$-0.456227\pi$$
0.137083 + 0.990560i $$0.456227\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 23.2867 0.889092
$$687$$ 0 0
$$688$$ −47.4955 −1.81075
$$689$$ 1.91288 0.0728749
$$690$$ 0 0
$$691$$ 42.9129 1.63248 0.816241 0.577711i $$-0.196054\pi$$
0.816241 + 0.577711i $$0.196054\pi$$
$$692$$ −3.82576 −0.145433
$$693$$ 0 0
$$694$$ 12.8348 0.487204
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −27.4955 −1.04146
$$698$$ −7.75682 −0.293600
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 23.0780 0.871645 0.435823 0.900033i $$-0.356458\pi$$
0.435823 + 0.900033i $$0.356458\pi$$
$$702$$ 0 0
$$703$$ 14.3303 0.540478
$$704$$ 4.56439 0.172027
$$705$$ 0 0
$$706$$ −26.5735 −1.00011
$$707$$ −0.582576 −0.0219100
$$708$$ 0 0
$$709$$ 10.0000 0.375558 0.187779 0.982211i $$-0.439871\pi$$
0.187779 + 0.982211i $$0.439871\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 2.00909 0.0752939
$$713$$ −16.0000 −0.599205
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −5.73864 −0.214463
$$717$$ 0 0
$$718$$ 48.6606 1.81600
$$719$$ −7.91288 −0.295101 −0.147550 0.989055i $$-0.547139\pi$$
−0.147550 + 0.989055i $$0.547139\pi$$
$$720$$ 0 0
$$721$$ 15.1652 0.564780
$$722$$ −11.0436 −0.410999
$$723$$ 0 0
$$724$$ 19.5390 0.726162
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −9.66970 −0.358629 −0.179315 0.983792i $$-0.557388\pi$$
−0.179315 + 0.983792i $$0.557388\pi$$
$$728$$ 6.49545 0.240738
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −28.7477 −1.06327
$$732$$ 0 0
$$733$$ 38.4174 1.41898 0.709490 0.704716i $$-0.248925\pi$$
0.709490 + 0.704716i $$0.248925\pi$$
$$734$$ 67.1652 2.47911
$$735$$ 0 0
$$736$$ 24.1742 0.891074
$$737$$ −20.8258 −0.767127
$$738$$ 0 0
$$739$$ 19.2523 0.708206 0.354103 0.935206i $$-0.384786\pi$$
0.354103 + 0.935206i $$0.384786\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −0.747727 −0.0274499
$$743$$ 29.7477 1.09134 0.545669 0.838001i $$-0.316276\pi$$
0.545669 + 0.838001i $$0.316276\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −50.7477 −1.85801
$$747$$ 0 0
$$748$$ 18.1307 0.662923
$$749$$ −5.16515 −0.188731
$$750$$ 0 0
$$751$$ −17.4955 −0.638418 −0.319209 0.947684i $$-0.603417\pi$$
−0.319209 + 0.947684i $$0.603417\pi$$
$$752$$ −52.4519 −1.91272
$$753$$ 0 0
$$754$$ −8.20871 −0.298944
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 2.33030 0.0846963 0.0423481 0.999103i $$-0.486516\pi$$
0.0423481 + 0.999103i $$0.486516\pi$$
$$758$$ −46.5735 −1.69163
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −36.4174 −1.32013 −0.660065 0.751208i $$-0.729471\pi$$
−0.660065 + 0.751208i $$0.729471\pi$$
$$762$$ 0 0
$$763$$ −14.1652 −0.512813
$$764$$ 4.83485 0.174919
$$765$$ 0 0
$$766$$ 6.41742 0.231871
$$767$$ 34.7477 1.25467
$$768$$ 0 0
$$769$$ 25.5826 0.922531 0.461266 0.887262i $$-0.347396\pi$$
0.461266 + 0.887262i $$0.347396\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 24.5735 0.884419
$$773$$ −5.16515 −0.185778 −0.0928888 0.995676i $$-0.529610\pi$$
−0.0928888 + 0.995676i $$0.529610\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 16.4174 0.589351
$$777$$ 0 0
$$778$$ 11.7913 0.422738
$$779$$ 32.8348 1.17643
$$780$$ 0 0
$$781$$ −47.9129 −1.71446
$$782$$ 21.4955 0.768676
$$783$$ 0 0
$$784$$ 29.7386 1.06209
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 24.0000 0.855508 0.427754 0.903895i $$-0.359305\pi$$
0.427754 + 0.903895i $$0.359305\pi$$
$$788$$ −19.7386 −0.703160
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 14.1652 0.503655
$$792$$ 0 0
$$793$$ 58.4174 2.07446
$$794$$ −19.4083 −0.688776
$$795$$ 0 0
$$796$$ 16.2178 0.574825
$$797$$ 32.3303 1.14520 0.572599 0.819836i $$-0.305935\pi$$
0.572599 + 0.819836i $$0.305935\pi$$
$$798$$ 0 0
$$799$$ −31.7477 −1.12315
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −22.2432 −0.785434
$$803$$ 20.0000 0.705785
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 32.8348 1.15656
$$807$$ 0 0
$$808$$ −0.825757 −0.0290500
$$809$$ −44.0780 −1.54970 −0.774851 0.632145i $$-0.782175\pi$$
−0.774851 + 0.632145i $$0.782175\pi$$
$$810$$ 0 0
$$811$$ 32.5826 1.14413 0.572064 0.820209i $$-0.306143\pi$$
0.572064 + 0.820209i $$0.306143\pi$$
$$812$$ 1.20871 0.0424175
$$813$$ 0 0
$$814$$ −35.8258 −1.25569
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 34.3303 1.20107
$$818$$ −4.92197 −0.172093
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 31.4955 1.09920 0.549599 0.835428i $$-0.314780\pi$$
0.549599 + 0.835428i $$0.314780\pi$$
$$822$$ 0 0
$$823$$ −51.0780 −1.78047 −0.890234 0.455503i $$-0.849459\pi$$
−0.890234 + 0.455503i $$0.849459\pi$$
$$824$$ 21.4955 0.748830
$$825$$ 0 0
$$826$$ −13.5826 −0.472598
$$827$$ −43.8258 −1.52397 −0.761985 0.647594i $$-0.775775\pi$$
−0.761985 + 0.647594i $$0.775775\pi$$
$$828$$ 0 0
$$829$$ 4.00000 0.138926 0.0694629 0.997585i $$-0.477871\pi$$
0.0694629 + 0.997585i $$0.477871\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −4.18333 −0.145031
$$833$$ 18.0000 0.623663
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −21.6515 −0.748833
$$837$$ 0 0
$$838$$ 66.5735 2.29974
$$839$$ 48.4955 1.67425 0.837125 0.547012i $$-0.184235\pi$$
0.837125 + 0.547012i $$0.184235\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ 7.91288 0.272696
$$843$$ 0 0
$$844$$ 24.5735 0.845854
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −14.0000 −0.481046
$$848$$ −2.06894 −0.0710476
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −16.0000 −0.548473
$$852$$ 0 0
$$853$$ −20.7477 −0.710389 −0.355194 0.934792i $$-0.615585\pi$$
−0.355194 + 0.934792i $$0.615585\pi$$
$$854$$ −22.8348 −0.781392
$$855$$ 0 0
$$856$$ −7.32121 −0.250234
$$857$$ −46.6606 −1.59390 −0.796948 0.604048i $$-0.793554\pi$$
−0.796948 + 0.604048i $$0.793554\pi$$
$$858$$ 0 0
$$859$$ −47.5826 −1.62350 −0.811748 0.584007i $$-0.801484\pi$$
−0.811748 + 0.584007i $$0.801484\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 70.1561 2.38952
$$863$$ −6.41742 −0.218452 −0.109226 0.994017i $$-0.534837\pi$$
−0.109226 + 0.994017i $$0.534837\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −44.9220 −1.52651
$$867$$ 0 0
$$868$$ −4.83485 −0.164105
$$869$$ −37.9129 −1.28611
$$870$$ 0 0
$$871$$ 19.0871 0.646742
$$872$$ −20.0780 −0.679928
$$873$$ 0 0
$$874$$ −25.6697 −0.868290
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −19.6697 −0.664198 −0.332099 0.943244i $$-0.607757\pi$$
−0.332099 + 0.943244i $$0.607757\pi$$
$$878$$ −30.2958 −1.02243
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 32.0780 1.08074 0.540368 0.841429i $$-0.318285\pi$$
0.540368 + 0.841429i $$0.318285\pi$$
$$882$$ 0 0
$$883$$ −16.0000 −0.538443 −0.269221 0.963078i $$-0.586766\pi$$
−0.269221 + 0.963078i $$0.586766\pi$$
$$884$$ −16.6170 −0.558892
$$885$$ 0 0
$$886$$ 1.04356 0.0350591
$$887$$ −44.9129 −1.50803 −0.754013 0.656859i $$-0.771885\pi$$
−0.754013 + 0.656859i $$0.771885\pi$$
$$888$$ 0 0
$$889$$ 2.00000 0.0670778
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −8.46099 −0.283295
$$893$$ 37.9129 1.26871
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −10.4519 −0.349173
$$897$$ 0 0
$$898$$ 53.8784 1.79795
$$899$$ −4.00000 −0.133407
$$900$$ 0 0
$$901$$ −1.25227 −0.0417193
$$902$$ −82.0871 −2.73320
$$903$$ 0 0
$$904$$ 20.0780 0.667785
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −23.5826 −0.783047 −0.391523 0.920168i $$-0.628052\pi$$
−0.391523 + 0.920168i $$0.628052\pi$$
$$908$$ −9.16515 −0.304156
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 50.8258 1.68393 0.841966 0.539530i $$-0.181398\pi$$
0.841966 + 0.539530i $$0.181398\pi$$
$$912$$ 0 0
$$913$$ 57.9129 1.91664
$$914$$ 6.71326 0.222055
$$915$$ 0 0
$$916$$ −20.7477 −0.685524
$$917$$ −15.0000 −0.495344
$$918$$ 0 0
$$919$$ 18.9129 0.623878 0.311939 0.950102i $$-0.399021\pi$$
0.311939 + 0.950102i $$0.399021\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 16.4174 0.540679
$$923$$ 43.9129 1.44541
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0.295834 0.00972170
$$927$$ 0 0
$$928$$ 6.04356 0.198390
$$929$$ −15.6697 −0.514106 −0.257053 0.966397i $$-0.582752\pi$$
−0.257053 + 0.966397i $$0.582752\pi$$
$$930$$ 0 0
$$931$$ −21.4955 −0.704485
$$932$$ −15.9129 −0.521244
$$933$$ 0 0
$$934$$ −14.3303 −0.468902
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 18.5826 0.607066 0.303533 0.952821i $$-0.401834\pi$$
0.303533 + 0.952821i $$0.401834\pi$$
$$938$$ −7.46099 −0.243610
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −19.9129 −0.649141 −0.324571 0.945861i $$-0.605220\pi$$
−0.324571 + 0.945861i $$0.605220\pi$$
$$942$$ 0 0
$$943$$ −36.6606 −1.19383
$$944$$ −37.5826 −1.22321
$$945$$ 0 0
$$946$$ −85.8258 −2.79044
$$947$$ −54.9129 −1.78443 −0.892214 0.451612i $$-0.850849\pi$$
−0.892214 + 0.451612i $$0.850849\pi$$
$$948$$ 0 0
$$949$$ −18.3303 −0.595027
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −4.25227 −0.137817
$$953$$ −37.5826 −1.21742 −0.608710 0.793393i $$-0.708313\pi$$
−0.608710 + 0.793393i $$0.708313\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 31.4265 1.01641
$$957$$ 0 0
$$958$$ 34.3303 1.10916
$$959$$ −16.3303 −0.527333
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 32.8348 1.05864
$$963$$ 0 0
$$964$$ 35.4519 1.14183
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −9.25227 −0.297533 −0.148767 0.988872i $$-0.547530\pi$$
−0.148767 + 0.988872i $$0.547530\pi$$
$$968$$ −19.8439 −0.637808
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −20.0000 −0.641831 −0.320915 0.947108i $$-0.603990\pi$$
−0.320915 + 0.947108i $$0.603990\pi$$
$$972$$ 0 0
$$973$$ 9.41742 0.301909
$$974$$ −4.17424 −0.133751
$$975$$ 0 0
$$976$$ −63.1833 −2.02245
$$977$$ −2.50455 −0.0801275 −0.0400638 0.999197i $$-0.512756\pi$$
−0.0400638 + 0.999197i $$0.512756\pi$$
$$978$$ 0 0
$$979$$ 7.08712 0.226505
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −28.6606 −0.914597
$$983$$ 55.1652 1.75950 0.879748 0.475441i $$-0.157712\pi$$
0.879748 + 0.475441i $$0.157712\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 5.37386 0.171139
$$987$$ 0 0
$$988$$ 19.8439 0.631320
$$989$$ −38.3303 −1.21883
$$990$$ 0 0
$$991$$ 16.0780 0.510735 0.255368 0.966844i $$-0.417803\pi$$
0.255368 + 0.966844i $$0.417803\pi$$
$$992$$ −24.1742 −0.767533
$$993$$ 0 0
$$994$$ −17.1652 −0.544446
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 0.834849 0.0264399 0.0132200 0.999913i $$-0.495792\pi$$
0.0132200 + 0.999913i $$0.495792\pi$$
$$998$$ −24.0345 −0.760798
$$999$$ 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 6525.2.a.t.1.2 2
3.2 odd 2 2175.2.a.r.1.1 2
5.4 even 2 1305.2.a.m.1.1 2
15.2 even 4 2175.2.c.f.349.2 4
15.8 even 4 2175.2.c.f.349.3 4
15.14 odd 2 435.2.a.f.1.2 2
60.59 even 2 6960.2.a.bw.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.f.1.2 2 15.14 odd 2
1305.2.a.m.1.1 2 5.4 even 2
2175.2.a.r.1.1 2 3.2 odd 2
2175.2.c.f.349.2 4 15.2 even 4
2175.2.c.f.349.3 4 15.8 even 4
6525.2.a.t.1.2 2 1.1 even 1 trivial
6960.2.a.bw.1.1 2 60.59 even 2