# Properties

 Label 6525.2.a.bi.1.1 Level $6525$ Weight $2$ Character 6525.1 Self dual yes Analytic conductor $52.102$ Analytic rank $1$ Dimension $4$ 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: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.2225.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 5x^{2} + 2x + 4$$ x^4 - x^3 - 5*x^2 + 2*x + 4 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 435) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.75660$$ 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)

 $$f(q)$$ $$=$$ $$q-2.75660 q^{2} +5.59883 q^{4} -0.393832 q^{7} -9.92054 q^{8} +O(q^{10})$$ $$q-2.75660 q^{2} +5.59883 q^{4} -0.393832 q^{7} -9.92054 q^{8} +0.393832 q^{11} +2.56511 q^{13} +1.08564 q^{14} +16.1493 q^{16} +2.07830 q^{17} -0.958939 q^{19} -1.08564 q^{22} +6.15661 q^{23} -7.07097 q^{26} -2.20500 q^{28} +1.00000 q^{29} -10.1566 q^{31} -24.6760 q^{32} -5.72905 q^{34} +7.34192 q^{37} +2.64341 q^{38} +1.65745 q^{41} -10.3279 q^{43} +2.20500 q^{44} -16.9713 q^{46} -11.5915 q^{47} -6.84490 q^{49} +14.3616 q^{52} -12.3279 q^{53} +3.90703 q^{56} -2.75660 q^{58} +9.54022 q^{59} -6.25340 q^{61} +27.9977 q^{62} +35.7232 q^{64} +7.42023 q^{67} +11.6361 q^{68} -5.98533 q^{71} -3.34192 q^{73} -20.2387 q^{74} -5.36894 q^{76} -0.155104 q^{77} -2.06745 q^{79} -4.56892 q^{82} +6.41000 q^{83} +28.4698 q^{86} -3.90703 q^{88} -15.8302 q^{89} -1.01022 q^{91} +34.4698 q^{92} +31.9531 q^{94} +18.4575 q^{97} +18.8686 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 3 q^{2} + 5 q^{4} - 2 q^{7} - 12 q^{8}+O(q^{10})$$ 4 * q - 3 * q^2 + 5 * q^4 - 2 * q^7 - 12 * q^8 $$4 q - 3 q^{2} + 5 q^{4} - 2 q^{7} - 12 q^{8} + 2 q^{11} + 8 q^{13} + 3 q^{14} + 11 q^{16} - 10 q^{17} - 2 q^{19} - 3 q^{22} - 12 q^{23} + 7 q^{26} + 9 q^{28} + 4 q^{29} - 4 q^{31} - 17 q^{32} - q^{34} + 16 q^{37} - 10 q^{38} + 12 q^{41} - 2 q^{43} - 9 q^{44} - 8 q^{46} - 12 q^{47} + 6 q^{49} + 3 q^{52} - 10 q^{53} - 3 q^{58} - 2 q^{59} - 26 q^{61} + 20 q^{62} + 34 q^{64} - 2 q^{67} + 9 q^{68} + 10 q^{71} - 48 q^{74} + 16 q^{76} - 34 q^{77} + 22 q^{79} - 38 q^{82} - 10 q^{83} + 4 q^{86} + 4 q^{89} - 8 q^{91} + 28 q^{92} + 39 q^{94} + 22 q^{97} + 34 q^{98}+O(q^{100})$$ 4 * q - 3 * q^2 + 5 * q^4 - 2 * q^7 - 12 * q^8 + 2 * q^11 + 8 * q^13 + 3 * q^14 + 11 * q^16 - 10 * q^17 - 2 * q^19 - 3 * q^22 - 12 * q^23 + 7 * q^26 + 9 * q^28 + 4 * q^29 - 4 * q^31 - 17 * q^32 - q^34 + 16 * q^37 - 10 * q^38 + 12 * q^41 - 2 * q^43 - 9 * q^44 - 8 * q^46 - 12 * q^47 + 6 * q^49 + 3 * q^52 - 10 * q^53 - 3 * q^58 - 2 * q^59 - 26 * q^61 + 20 * q^62 + 34 * q^64 - 2 * q^67 + 9 * q^68 + 10 * q^71 - 48 * q^74 + 16 * q^76 - 34 * q^77 + 22 * q^79 - 38 * q^82 - 10 * q^83 + 4 * q^86 + 4 * q^89 - 8 * q^91 + 28 * q^92 + 39 * q^94 + 22 * q^97 + 34 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.75660 −1.94921 −0.974605 0.223932i $$-0.928110\pi$$
−0.974605 + 0.223932i $$0.928110\pi$$
$$3$$ 0 0
$$4$$ 5.59883 2.79942
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −0.393832 −0.148855 −0.0744273 0.997226i $$-0.523713\pi$$
−0.0744273 + 0.997226i $$0.523713\pi$$
$$8$$ −9.92054 −3.50744
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 0.393832 0.118745 0.0593725 0.998236i $$-0.481090\pi$$
0.0593725 + 0.998236i $$0.481090\pi$$
$$12$$ 0 0
$$13$$ 2.56511 0.711433 0.355716 0.934594i $$-0.384237\pi$$
0.355716 + 0.934594i $$0.384237\pi$$
$$14$$ 1.08564 0.290149
$$15$$ 0 0
$$16$$ 16.1493 4.03732
$$17$$ 2.07830 0.504063 0.252031 0.967719i $$-0.418901\pi$$
0.252031 + 0.967719i $$0.418901\pi$$
$$18$$ 0 0
$$19$$ −0.958939 −0.219996 −0.109998 0.993932i $$-0.535084\pi$$
−0.109998 + 0.993932i $$0.535084\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −1.08564 −0.231459
$$23$$ 6.15661 1.28374 0.641871 0.766813i $$-0.278159\pi$$
0.641871 + 0.766813i $$0.278159\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −7.07097 −1.38673
$$27$$ 0 0
$$28$$ −2.20500 −0.416706
$$29$$ 1.00000 0.185695
$$30$$ 0 0
$$31$$ −10.1566 −1.82418 −0.912090 0.409989i $$-0.865532\pi$$
−0.912090 + 0.409989i $$0.865532\pi$$
$$32$$ −24.6760 −4.36214
$$33$$ 0 0
$$34$$ −5.72905 −0.982524
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 7.34192 1.20700 0.603502 0.797361i $$-0.293771\pi$$
0.603502 + 0.797361i $$0.293771\pi$$
$$38$$ 2.64341 0.428818
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 1.65745 0.258850 0.129425 0.991589i $$-0.458687\pi$$
0.129425 + 0.991589i $$0.458687\pi$$
$$42$$ 0 0
$$43$$ −10.3279 −1.57499 −0.787494 0.616323i $$-0.788622\pi$$
−0.787494 + 0.616323i $$0.788622\pi$$
$$44$$ 2.20500 0.332417
$$45$$ 0 0
$$46$$ −16.9713 −2.50228
$$47$$ −11.5915 −1.69079 −0.845397 0.534138i $$-0.820636\pi$$
−0.845397 + 0.534138i $$0.820636\pi$$
$$48$$ 0 0
$$49$$ −6.84490 −0.977842
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 14.3616 1.99160
$$53$$ −12.3279 −1.69336 −0.846682 0.532099i $$-0.821404\pi$$
−0.846682 + 0.532099i $$0.821404\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 3.90703 0.522099
$$57$$ 0 0
$$58$$ −2.75660 −0.361959
$$59$$ 9.54022 1.24203 0.621015 0.783798i $$-0.286720\pi$$
0.621015 + 0.783798i $$0.286720\pi$$
$$60$$ 0 0
$$61$$ −6.25340 −0.800665 −0.400333 0.916370i $$-0.631105\pi$$
−0.400333 + 0.916370i $$0.631105\pi$$
$$62$$ 27.9977 3.55571
$$63$$ 0 0
$$64$$ 35.7232 4.46540
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 7.42023 0.906525 0.453262 0.891377i $$-0.350260\pi$$
0.453262 + 0.891377i $$0.350260\pi$$
$$68$$ 11.6361 1.41108
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −5.98533 −0.710328 −0.355164 0.934804i $$-0.615575\pi$$
−0.355164 + 0.934804i $$0.615575\pi$$
$$72$$ 0 0
$$73$$ −3.34192 −0.391142 −0.195571 0.980690i $$-0.562656\pi$$
−0.195571 + 0.980690i $$0.562656\pi$$
$$74$$ −20.2387 −2.35270
$$75$$ 0 0
$$76$$ −5.36894 −0.615860
$$77$$ −0.155104 −0.0176757
$$78$$ 0 0
$$79$$ −2.06745 −0.232607 −0.116303 0.993214i $$-0.537104\pi$$
−0.116303 + 0.993214i $$0.537104\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −4.56892 −0.504553
$$83$$ 6.41000 0.703589 0.351795 0.936077i $$-0.385572\pi$$
0.351795 + 0.936077i $$0.385572\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 28.4698 3.06998
$$87$$ 0 0
$$88$$ −3.90703 −0.416491
$$89$$ −15.8302 −1.67800 −0.839000 0.544131i $$-0.816860\pi$$
−0.839000 + 0.544131i $$0.816860\pi$$
$$90$$ 0 0
$$91$$ −1.01022 −0.105900
$$92$$ 34.4698 3.59373
$$93$$ 0 0
$$94$$ 31.9531 3.29571
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 18.4575 1.87407 0.937036 0.349233i $$-0.113558\pi$$
0.937036 + 0.349233i $$0.113558\pi$$
$$98$$ 18.8686 1.90602
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 12.8038 1.27403 0.637015 0.770852i $$-0.280169\pi$$
0.637015 + 0.770852i $$0.280169\pi$$
$$102$$ 0 0
$$103$$ −4.86979 −0.479834 −0.239917 0.970793i $$-0.577120\pi$$
−0.239917 + 0.970793i $$0.577120\pi$$
$$104$$ −25.4472 −2.49531
$$105$$ 0 0
$$106$$ 33.9830 3.30072
$$107$$ −2.34255 −0.226463 −0.113231 0.993569i $$-0.536120\pi$$
−0.113231 + 0.993569i $$0.536120\pi$$
$$108$$ 0 0
$$109$$ 8.55044 0.818984 0.409492 0.912314i $$-0.365706\pi$$
0.409492 + 0.912314i $$0.365706\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −6.36011 −0.600973
$$113$$ −11.2085 −1.05441 −0.527204 0.849739i $$-0.676760\pi$$
−0.527204 + 0.849739i $$0.676760\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 5.59883 0.519839
$$117$$ 0 0
$$118$$ −26.2985 −2.42098
$$119$$ −0.818503 −0.0750321
$$120$$ 0 0
$$121$$ −10.8449 −0.985900
$$122$$ 17.2381 1.56066
$$123$$ 0 0
$$124$$ −56.8652 −5.10664
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −20.1566 −1.78861 −0.894305 0.447458i $$-0.852329\pi$$
−0.894305 + 0.447458i $$0.852329\pi$$
$$128$$ −49.1226 −4.34187
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 5.50235 0.480742 0.240371 0.970681i $$-0.422731\pi$$
0.240371 + 0.970681i $$0.422731\pi$$
$$132$$ 0 0
$$133$$ 0.377661 0.0327474
$$134$$ −20.4546 −1.76701
$$135$$ 0 0
$$136$$ −20.6179 −1.76797
$$137$$ 7.49853 0.640643 0.320321 0.947309i $$-0.396209\pi$$
0.320321 + 0.947309i $$0.396209\pi$$
$$138$$ 0 0
$$139$$ −9.35277 −0.793292 −0.396646 0.917972i $$-0.629826\pi$$
−0.396646 + 0.917972i $$0.629826\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 16.4992 1.38458
$$143$$ 1.01022 0.0844790
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 9.21234 0.762418
$$147$$ 0 0
$$148$$ 41.1062 3.37891
$$149$$ 11.5402 0.945411 0.472706 0.881220i $$-0.343277\pi$$
0.472706 + 0.881220i $$0.343277\pi$$
$$150$$ 0 0
$$151$$ 6.08212 0.494956 0.247478 0.968894i $$-0.420398\pi$$
0.247478 + 0.968894i $$0.420398\pi$$
$$152$$ 9.51320 0.771622
$$153$$ 0 0
$$154$$ 0.427559 0.0344537
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 11.5953 0.925407 0.462704 0.886513i $$-0.346879\pi$$
0.462704 + 0.886513i $$0.346879\pi$$
$$158$$ 5.69914 0.453399
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −2.42467 −0.191091
$$162$$ 0 0
$$163$$ 0.855118 0.0669780 0.0334890 0.999439i $$-0.489338\pi$$
0.0334890 + 0.999439i $$0.489338\pi$$
$$164$$ 9.27979 0.724630
$$165$$ 0 0
$$166$$ −17.6698 −1.37144
$$167$$ 7.73194 0.598315 0.299158 0.954204i $$-0.403294\pi$$
0.299158 + 0.954204i $$0.403294\pi$$
$$168$$ 0 0
$$169$$ −6.42023 −0.493863
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −57.8241 −4.40905
$$173$$ 8.07449 0.613892 0.306946 0.951727i $$-0.400693\pi$$
0.306946 + 0.951727i $$0.400693\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 6.36011 0.479411
$$177$$ 0 0
$$178$$ 43.6376 3.27077
$$179$$ −12.4100 −0.927567 −0.463784 0.885949i $$-0.653508\pi$$
−0.463784 + 0.885949i $$0.653508\pi$$
$$180$$ 0 0
$$181$$ −2.49765 −0.185649 −0.0928246 0.995682i $$-0.529590\pi$$
−0.0928246 + 0.995682i $$0.529590\pi$$
$$182$$ 2.78478 0.206421
$$183$$ 0 0
$$184$$ −61.0769 −4.50265
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0.818503 0.0598549
$$188$$ −64.8989 −4.73324
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 6.32085 0.457361 0.228680 0.973502i $$-0.426559\pi$$
0.228680 + 0.973502i $$0.426559\pi$$
$$192$$ 0 0
$$193$$ −25.2921 −1.82057 −0.910284 0.413984i $$-0.864137\pi$$
−0.910284 + 0.413984i $$0.864137\pi$$
$$194$$ −50.8798 −3.65296
$$195$$ 0 0
$$196$$ −38.3234 −2.73739
$$197$$ −21.2047 −1.51077 −0.755386 0.655280i $$-0.772551\pi$$
−0.755386 + 0.655280i $$0.772551\pi$$
$$198$$ 0 0
$$199$$ −2.64723 −0.187657 −0.0938285 0.995588i $$-0.529911\pi$$
−0.0938285 + 0.995588i $$0.529911\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −35.2950 −2.48335
$$203$$ −0.393832 −0.0276416
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 13.4240 0.935297
$$207$$ 0 0
$$208$$ 41.4246 2.87228
$$209$$ −0.377661 −0.0261234
$$210$$ 0 0
$$211$$ 2.00000 0.137686 0.0688428 0.997628i $$-0.478069\pi$$
0.0688428 + 0.997628i $$0.478069\pi$$
$$212$$ −69.0218 −4.74043
$$213$$ 0 0
$$214$$ 6.45747 0.441423
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 4.00000 0.271538
$$218$$ −23.5701 −1.59637
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 5.33107 0.358607
$$222$$ 0 0
$$223$$ −12.5504 −0.840440 −0.420220 0.907422i $$-0.638047\pi$$
−0.420220 + 0.907422i $$0.638047\pi$$
$$224$$ 9.71820 0.649324
$$225$$ 0 0
$$226$$ 30.8974 2.05526
$$227$$ 1.30149 0.0863829 0.0431914 0.999067i $$-0.486247\pi$$
0.0431914 + 0.999067i $$0.486247\pi$$
$$228$$ 0 0
$$229$$ 3.28682 0.217199 0.108600 0.994086i $$-0.465363\pi$$
0.108600 + 0.994086i $$0.465363\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −9.92054 −0.651315
$$233$$ 17.7115 1.16032 0.580159 0.814503i $$-0.302990\pi$$
0.580159 + 0.814503i $$0.302990\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 53.4141 3.47696
$$237$$ 0 0
$$238$$ 2.25628 0.146253
$$239$$ −21.2651 −1.37553 −0.687763 0.725935i $$-0.741407\pi$$
−0.687763 + 0.725935i $$0.741407\pi$$
$$240$$ 0 0
$$241$$ 15.3177 0.986697 0.493349 0.869832i $$-0.335773\pi$$
0.493349 + 0.869832i $$0.335773\pi$$
$$242$$ 29.8950 1.92172
$$243$$ 0 0
$$244$$ −35.0117 −2.24140
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −2.45978 −0.156512
$$248$$ 100.759 6.39820
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −26.8917 −1.69739 −0.848696 0.528882i $$-0.822612\pi$$
−0.848696 + 0.528882i $$0.822612\pi$$
$$252$$ 0 0
$$253$$ 2.42467 0.152438
$$254$$ 55.5637 3.48637
$$255$$ 0 0
$$256$$ 63.9648 3.99780
$$257$$ −6.85512 −0.427611 −0.213805 0.976876i $$-0.568586\pi$$
−0.213805 + 0.976876i $$0.568586\pi$$
$$258$$ 0 0
$$259$$ −2.89149 −0.179668
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −15.1678 −0.937067
$$263$$ −13.6294 −0.840423 −0.420212 0.907426i $$-0.638044\pi$$
−0.420212 + 0.907426i $$0.638044\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −1.04106 −0.0638315
$$267$$ 0 0
$$268$$ 41.5446 2.53774
$$269$$ 8.46129 0.515894 0.257947 0.966159i $$-0.416954\pi$$
0.257947 + 0.966159i $$0.416954\pi$$
$$270$$ 0 0
$$271$$ 6.34255 0.385282 0.192641 0.981269i $$-0.438295\pi$$
0.192641 + 0.981269i $$0.438295\pi$$
$$272$$ 33.5631 2.03506
$$273$$ 0 0
$$274$$ −20.6704 −1.24875
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 17.1464 1.03023 0.515113 0.857122i $$-0.327750\pi$$
0.515113 + 0.857122i $$0.327750\pi$$
$$278$$ 25.7818 1.54629
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 12.2985 0.733670 0.366835 0.930286i $$-0.380441\pi$$
0.366835 + 0.930286i $$0.380441\pi$$
$$282$$ 0 0
$$283$$ −25.1830 −1.49697 −0.748487 0.663149i $$-0.769219\pi$$
−0.748487 + 0.663149i $$0.769219\pi$$
$$284$$ −33.5109 −1.98851
$$285$$ 0 0
$$286$$ −2.78478 −0.164667
$$287$$ −0.652757 −0.0385311
$$288$$ 0 0
$$289$$ −12.6807 −0.745921
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −18.7109 −1.09497
$$293$$ 12.3170 0.719569 0.359784 0.933035i $$-0.382850\pi$$
0.359784 + 0.933035i $$0.382850\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −72.8358 −4.23350
$$297$$ 0 0
$$298$$ −31.8117 −1.84280
$$299$$ 15.7924 0.913296
$$300$$ 0 0
$$301$$ 4.06745 0.234444
$$302$$ −16.7660 −0.964773
$$303$$ 0 0
$$304$$ −15.4862 −0.888193
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 22.7379 1.29772 0.648860 0.760908i $$-0.275246\pi$$
0.648860 + 0.760908i $$0.275246\pi$$
$$308$$ −0.868401 −0.0494817
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 5.33810 0.302696 0.151348 0.988481i $$-0.451639\pi$$
0.151348 + 0.988481i $$0.451639\pi$$
$$312$$ 0 0
$$313$$ 1.96338 0.110977 0.0554885 0.998459i $$-0.482328\pi$$
0.0554885 + 0.998459i $$0.482328\pi$$
$$314$$ −31.9636 −1.80381
$$315$$ 0 0
$$316$$ −11.5753 −0.651163
$$317$$ 1.29064 0.0724895 0.0362448 0.999343i $$-0.488460\pi$$
0.0362448 + 0.999343i $$0.488460\pi$$
$$318$$ 0 0
$$319$$ 0.393832 0.0220504
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 6.68384 0.372476
$$323$$ −1.99297 −0.110892
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −2.35722 −0.130554
$$327$$ 0 0
$$328$$ −16.4428 −0.907902
$$329$$ 4.56511 0.251683
$$330$$ 0 0
$$331$$ 28.9971 1.59382 0.796911 0.604096i $$-0.206466\pi$$
0.796911 + 0.604096i $$0.206466\pi$$
$$332$$ 35.8885 1.96964
$$333$$ 0 0
$$334$$ −21.3138 −1.16624
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 16.7854 0.914356 0.457178 0.889375i $$-0.348860\pi$$
0.457178 + 0.889375i $$0.348860\pi$$
$$338$$ 17.6980 0.962643
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −4.00000 −0.216612
$$342$$ 0 0
$$343$$ 5.45257 0.294411
$$344$$ 102.458 5.52417
$$345$$ 0 0
$$346$$ −22.2581 −1.19660
$$347$$ −17.8681 −0.959210 −0.479605 0.877485i $$-0.659220\pi$$
−0.479605 + 0.877485i $$0.659220\pi$$
$$348$$ 0 0
$$349$$ 8.73789 0.467728 0.233864 0.972269i $$-0.424863\pi$$
0.233864 + 0.972269i $$0.424863\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −9.71820 −0.517982
$$353$$ −22.6017 −1.20297 −0.601484 0.798885i $$-0.705424\pi$$
−0.601484 + 0.798885i $$0.705424\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −88.6308 −4.69742
$$357$$ 0 0
$$358$$ 34.2094 1.80802
$$359$$ 13.1830 0.695772 0.347886 0.937537i $$-0.386900\pi$$
0.347886 + 0.937537i $$0.386900\pi$$
$$360$$ 0 0
$$361$$ −18.0804 −0.951602
$$362$$ 6.88503 0.361869
$$363$$ 0 0
$$364$$ −5.65607 −0.296458
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −17.4511 −0.910938 −0.455469 0.890252i $$-0.650528\pi$$
−0.455469 + 0.890252i $$0.650528\pi$$
$$368$$ 99.4247 5.18287
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 4.85512 0.252065
$$372$$ 0 0
$$373$$ 32.2018 1.66734 0.833672 0.552260i $$-0.186234\pi$$
0.833672 + 0.552260i $$0.186234\pi$$
$$374$$ −2.25628 −0.116670
$$375$$ 0 0
$$376$$ 114.994 5.93036
$$377$$ 2.56511 0.132110
$$378$$ 0 0
$$379$$ 32.3660 1.66253 0.831265 0.555876i $$-0.187617\pi$$
0.831265 + 0.555876i $$0.187617\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −17.4240 −0.891492
$$383$$ −25.8739 −1.32209 −0.661047 0.750345i $$-0.729887\pi$$
−0.661047 + 0.750345i $$0.729887\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 69.7203 3.54867
$$387$$ 0 0
$$388$$ 103.340 5.24631
$$389$$ −32.6103 −1.65341 −0.826703 0.562639i $$-0.809786\pi$$
−0.826703 + 0.562639i $$0.809786\pi$$
$$390$$ 0 0
$$391$$ 12.7953 0.647086
$$392$$ 67.9051 3.42972
$$393$$ 0 0
$$394$$ 58.4528 2.94481
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −4.39534 −0.220596 −0.110298 0.993899i $$-0.535180\pi$$
−0.110298 + 0.993899i $$0.535180\pi$$
$$398$$ 7.29734 0.365783
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 19.0658 0.952099 0.476049 0.879418i $$-0.342068\pi$$
0.476049 + 0.879418i $$0.342068\pi$$
$$402$$ 0 0
$$403$$ −26.0528 −1.29778
$$404$$ 71.6865 3.56654
$$405$$ 0 0
$$406$$ 1.08564 0.0538793
$$407$$ 2.89149 0.143326
$$408$$ 0 0
$$409$$ −1.38235 −0.0683530 −0.0341765 0.999416i $$-0.510881\pi$$
−0.0341765 + 0.999416i $$0.510881\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −27.2651 −1.34326
$$413$$ −3.75725 −0.184882
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −63.2965 −3.10337
$$417$$ 0 0
$$418$$ 1.04106 0.0509199
$$419$$ −2.86979 −0.140198 −0.0700991 0.997540i $$-0.522332\pi$$
−0.0700991 + 0.997540i $$0.522332\pi$$
$$420$$ 0 0
$$421$$ 13.6440 0.664970 0.332485 0.943109i $$-0.392113\pi$$
0.332485 + 0.943109i $$0.392113\pi$$
$$422$$ −5.51320 −0.268378
$$423$$ 0 0
$$424$$ 122.299 5.93938
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 2.46279 0.119183
$$428$$ −13.1155 −0.633964
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 11.9707 0.576607 0.288303 0.957539i $$-0.406909\pi$$
0.288303 + 0.957539i $$0.406909\pi$$
$$432$$ 0 0
$$433$$ −17.9907 −0.864576 −0.432288 0.901736i $$-0.642294\pi$$
−0.432288 + 0.901736i $$0.642294\pi$$
$$434$$ −11.0264 −0.529284
$$435$$ 0 0
$$436$$ 47.8725 2.29268
$$437$$ −5.90381 −0.282418
$$438$$ 0 0
$$439$$ −16.2226 −0.774260 −0.387130 0.922025i $$-0.626534\pi$$
−0.387130 + 0.922025i $$0.626534\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −14.6956 −0.698999
$$443$$ −35.3762 −1.68078 −0.840388 0.541986i $$-0.817673\pi$$
−0.840388 + 0.541986i $$0.817673\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 34.5965 1.63819
$$447$$ 0 0
$$448$$ −14.0690 −0.664696
$$449$$ −41.6971 −1.96781 −0.983903 0.178702i $$-0.942810\pi$$
−0.983903 + 0.178702i $$0.942810\pi$$
$$450$$ 0 0
$$451$$ 0.652757 0.0307371
$$452$$ −62.7546 −2.95173
$$453$$ 0 0
$$454$$ −3.58768 −0.168378
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 16.1111 0.753646 0.376823 0.926285i $$-0.377017\pi$$
0.376823 + 0.926285i $$0.377017\pi$$
$$458$$ −9.06045 −0.423367
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −17.3396 −0.807586 −0.403793 0.914850i $$-0.632308\pi$$
−0.403793 + 0.914850i $$0.632308\pi$$
$$462$$ 0 0
$$463$$ −19.3177 −0.897768 −0.448884 0.893590i $$-0.648178\pi$$
−0.448884 + 0.893590i $$0.648178\pi$$
$$464$$ 16.1493 0.749711
$$465$$ 0 0
$$466$$ −48.8235 −2.26170
$$467$$ −8.00000 −0.370196 −0.185098 0.982720i $$-0.559260\pi$$
−0.185098 + 0.982720i $$0.559260\pi$$
$$468$$ 0 0
$$469$$ −2.92232 −0.134940
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −94.6441 −4.35635
$$473$$ −4.06745 −0.187022
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −4.58266 −0.210046
$$477$$ 0 0
$$478$$ 58.6194 2.68119
$$479$$ 40.3877 1.84536 0.922681 0.385565i $$-0.125994\pi$$
0.922681 + 0.385565i $$0.125994\pi$$
$$480$$ 0 0
$$481$$ 18.8328 0.858702
$$482$$ −42.2246 −1.92328
$$483$$ 0 0
$$484$$ −60.7188 −2.75994
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −2.84171 −0.128770 −0.0643850 0.997925i $$-0.520509\pi$$
−0.0643850 + 0.997925i $$0.520509\pi$$
$$488$$ 62.0371 2.80829
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −0.157863 −0.00712428 −0.00356214 0.999994i $$-0.501134\pi$$
−0.00356214 + 0.999994i $$0.501134\pi$$
$$492$$ 0 0
$$493$$ 2.07830 0.0936021
$$494$$ 6.78063 0.305075
$$495$$ 0 0
$$496$$ −164.022 −7.36480
$$497$$ 2.35722 0.105736
$$498$$ 0 0
$$499$$ 2.64723 0.118506 0.0592531 0.998243i $$-0.481128\pi$$
0.0592531 + 0.998243i $$0.481128\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 74.1297 3.30857
$$503$$ 13.9545 0.622200 0.311100 0.950377i $$-0.399303\pi$$
0.311100 + 0.950377i $$0.399303\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −6.68384 −0.297133
$$507$$ 0 0
$$508$$ −112.853 −5.00706
$$509$$ 16.4921 0.731001 0.365500 0.930811i $$-0.380898\pi$$
0.365500 + 0.930811i $$0.380898\pi$$
$$510$$ 0 0
$$511$$ 1.31616 0.0582233
$$512$$ −78.0802 −3.45069
$$513$$ 0 0
$$514$$ 18.8968 0.833502
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −4.56511 −0.200773
$$518$$ 7.97066 0.350211
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 12.2253 0.535601 0.267800 0.963474i $$-0.413703\pi$$
0.267800 + 0.963474i $$0.413703\pi$$
$$522$$ 0 0
$$523$$ −1.66659 −0.0728748 −0.0364374 0.999336i $$-0.511601\pi$$
−0.0364374 + 0.999336i $$0.511601\pi$$
$$524$$ 30.8067 1.34580
$$525$$ 0 0
$$526$$ 37.5707 1.63816
$$527$$ −21.1085 −0.919501
$$528$$ 0 0
$$529$$ 14.9038 0.647992
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 2.11446 0.0916736
$$533$$ 4.25154 0.184155
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −73.6126 −3.17958
$$537$$ 0 0
$$538$$ −23.3244 −1.00558
$$539$$ −2.69574 −0.116114
$$540$$ 0 0
$$541$$ −34.6558 −1.48997 −0.744984 0.667083i $$-0.767543\pi$$
−0.744984 + 0.667083i $$0.767543\pi$$
$$542$$ −17.4839 −0.750996
$$543$$ 0 0
$$544$$ −51.2842 −2.19879
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −38.2530 −1.63558 −0.817791 0.575515i $$-0.804801\pi$$
−0.817791 + 0.575515i $$0.804801\pi$$
$$548$$ 41.9830 1.79343
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −0.958939 −0.0408522
$$552$$ 0 0
$$553$$ 0.814230 0.0346246
$$554$$ −47.2657 −2.00813
$$555$$ 0 0
$$556$$ −52.3646 −2.22075
$$557$$ −28.7760 −1.21928 −0.609639 0.792679i $$-0.708686\pi$$
−0.609639 + 0.792679i $$0.708686\pi$$
$$558$$ 0 0
$$559$$ −26.4921 −1.12050
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −33.9022 −1.43008
$$563$$ −32.9809 −1.38998 −0.694989 0.719020i $$-0.744591\pi$$
−0.694989 + 0.719020i $$0.744591\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 69.4194 2.91792
$$567$$ 0 0
$$568$$ 59.3777 2.49143
$$569$$ −16.8038 −0.704453 −0.352227 0.935915i $$-0.614575\pi$$
−0.352227 + 0.935915i $$0.614575\pi$$
$$570$$ 0 0
$$571$$ −6.21703 −0.260175 −0.130087 0.991503i $$-0.541526\pi$$
−0.130087 + 0.991503i $$0.541526\pi$$
$$572$$ 5.65607 0.236492
$$573$$ 0 0
$$574$$ 1.79939 0.0751051
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −11.9977 −0.499470 −0.249735 0.968314i $$-0.580344\pi$$
−0.249735 + 0.968314i $$0.580344\pi$$
$$578$$ 34.9555 1.45396
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −2.52447 −0.104733
$$582$$ 0 0
$$583$$ −4.85512 −0.201078
$$584$$ 33.1537 1.37191
$$585$$ 0 0
$$586$$ −33.9531 −1.40259
$$587$$ −11.2194 −0.463073 −0.231536 0.972826i $$-0.574375\pi$$
−0.231536 + 0.972826i $$0.574375\pi$$
$$588$$ 0 0
$$589$$ 9.73957 0.401312
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 118.567 4.87306
$$593$$ −7.69682 −0.316071 −0.158035 0.987433i $$-0.550516\pi$$
−0.158035 + 0.987433i $$0.550516\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 64.6118 2.64660
$$597$$ 0 0
$$598$$ −43.5332 −1.78020
$$599$$ −20.0543 −0.819396 −0.409698 0.912221i $$-0.634366\pi$$
−0.409698 + 0.912221i $$0.634366\pi$$
$$600$$ 0 0
$$601$$ 3.10851 0.126799 0.0633995 0.997988i $$-0.479806\pi$$
0.0633995 + 0.997988i $$0.479806\pi$$
$$602$$ −11.2123 −0.456981
$$603$$ 0 0
$$604$$ 34.0528 1.38559
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −37.0441 −1.50357 −0.751786 0.659407i $$-0.770807\pi$$
−0.751786 + 0.659407i $$0.770807\pi$$
$$608$$ 23.6628 0.959652
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −29.7334 −1.20289
$$612$$ 0 0
$$613$$ 13.3839 0.540569 0.270284 0.962781i $$-0.412882\pi$$
0.270284 + 0.962781i $$0.412882\pi$$
$$614$$ −62.6792 −2.52953
$$615$$ 0 0
$$616$$ 1.53871 0.0619966
$$617$$ −12.6364 −0.508721 −0.254361 0.967109i $$-0.581865\pi$$
−0.254361 + 0.967109i $$0.581865\pi$$
$$618$$ 0 0
$$619$$ −41.2447 −1.65776 −0.828882 0.559424i $$-0.811022\pi$$
−0.828882 + 0.559424i $$0.811022\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −14.7150 −0.590018
$$623$$ 6.23446 0.249778
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −5.41226 −0.216318
$$627$$ 0 0
$$628$$ 64.9203 2.59060
$$629$$ 15.2587 0.608406
$$630$$ 0 0
$$631$$ 29.8009 1.18635 0.593177 0.805072i $$-0.297873\pi$$
0.593177 + 0.805072i $$0.297873\pi$$
$$632$$ 20.5103 0.815854
$$633$$ 0 0
$$634$$ −3.55777 −0.141297
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −17.5579 −0.695669
$$638$$ −1.08564 −0.0429808
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −17.7774 −0.702167 −0.351083 0.936344i $$-0.614187\pi$$
−0.351083 + 0.936344i $$0.614187\pi$$
$$642$$ 0 0
$$643$$ 44.5534 1.75702 0.878508 0.477727i $$-0.158539\pi$$
0.878508 + 0.477727i $$0.158539\pi$$
$$644$$ −13.5753 −0.534943
$$645$$ 0 0
$$646$$ 5.49381 0.216151
$$647$$ 42.9339 1.68790 0.843952 0.536418i $$-0.180223\pi$$
0.843952 + 0.536418i $$0.180223\pi$$
$$648$$ 0 0
$$649$$ 3.75725 0.147485
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 4.78766 0.187499
$$653$$ −19.8426 −0.776500 −0.388250 0.921554i $$-0.626920\pi$$
−0.388250 + 0.921554i $$0.626920\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 26.7666 1.04506
$$657$$ 0 0
$$658$$ −12.5842 −0.490582
$$659$$ −23.0483 −0.897836 −0.448918 0.893573i $$-0.648190\pi$$
−0.448918 + 0.893573i $$0.648190\pi$$
$$660$$ 0 0
$$661$$ −36.2079 −1.40832 −0.704162 0.710040i $$-0.748677\pi$$
−0.704162 + 0.710040i $$0.748677\pi$$
$$662$$ −79.9332 −3.10669
$$663$$ 0 0
$$664$$ −63.5907 −2.46780
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 6.15661 0.238385
$$668$$ 43.2898 1.67493
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −2.46279 −0.0950749
$$672$$ 0 0
$$673$$ −22.9000 −0.882731 −0.441365 0.897327i $$-0.645506\pi$$
−0.441365 + 0.897327i $$0.645506\pi$$
$$674$$ −46.2705 −1.78227
$$675$$ 0 0
$$676$$ −35.9458 −1.38253
$$677$$ 36.0068 1.38385 0.691927 0.721967i $$-0.256762\pi$$
0.691927 + 0.721967i $$0.256762\pi$$
$$678$$ 0 0
$$679$$ −7.26915 −0.278964
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 11.0264 0.422222
$$683$$ 9.12896 0.349310 0.174655 0.984630i $$-0.444119\pi$$
0.174655 + 0.984630i $$0.444119\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −15.0305 −0.573869
$$687$$ 0 0
$$688$$ −166.788 −6.35873
$$689$$ −31.6223 −1.20472
$$690$$ 0 0
$$691$$ 5.85956 0.222908 0.111454 0.993770i $$-0.464449\pi$$
0.111454 + 0.993770i $$0.464449\pi$$
$$692$$ 45.2077 1.71854
$$693$$ 0 0
$$694$$ 49.2552 1.86970
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 3.44469 0.130477
$$698$$ −24.0868 −0.911700
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 7.56955 0.285898 0.142949 0.989730i $$-0.454342\pi$$
0.142949 + 0.989730i $$0.454342\pi$$
$$702$$ 0 0
$$703$$ −7.04046 −0.265536
$$704$$ 14.0690 0.530244
$$705$$ 0 0
$$706$$ 62.3039 2.34484
$$707$$ −5.04256 −0.189645
$$708$$ 0 0
$$709$$ 14.5209 0.545342 0.272671 0.962107i $$-0.412093\pi$$
0.272671 + 0.962107i $$0.412093\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 157.044 5.88549
$$713$$ −62.5302 −2.34178
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −69.4815 −2.59665
$$717$$ 0 0
$$718$$ −36.3402 −1.35621
$$719$$ −12.2164 −0.455596 −0.227798 0.973708i $$-0.573153\pi$$
−0.227798 + 0.973708i $$0.573153\pi$$
$$720$$ 0 0
$$721$$ 1.91788 0.0714255
$$722$$ 49.8405 1.85487
$$723$$ 0 0
$$724$$ −13.9839 −0.519709
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 40.8856 1.51636 0.758182 0.652044i $$-0.226088\pi$$
0.758182 + 0.652044i $$0.226088\pi$$
$$728$$ 10.0219 0.371438
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −21.4645 −0.793892
$$732$$ 0 0
$$733$$ −25.4915 −0.941550 −0.470775 0.882253i $$-0.656026\pi$$
−0.470775 + 0.882253i $$0.656026\pi$$
$$734$$ 48.1056 1.77561
$$735$$ 0 0
$$736$$ −151.920 −5.59986
$$737$$ 2.92232 0.107645
$$738$$ 0 0
$$739$$ 9.56830 0.351975 0.175988 0.984392i $$-0.443688\pi$$
0.175988 + 0.984392i $$0.443688\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −13.3836 −0.491328
$$743$$ −18.0455 −0.662025 −0.331013 0.943626i $$-0.607390\pi$$
−0.331013 + 0.943626i $$0.607390\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −88.7673 −3.25000
$$747$$ 0 0
$$748$$ 4.58266 0.167559
$$749$$ 0.922572 0.0337100
$$750$$ 0 0
$$751$$ 11.9255 0.435168 0.217584 0.976042i $$-0.430182\pi$$
0.217584 + 0.976042i $$0.430182\pi$$
$$752$$ −187.194 −6.82627
$$753$$ 0 0
$$754$$ −7.07097 −0.257510
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −5.86152 −0.213041 −0.106520 0.994311i $$-0.533971\pi$$
−0.106520 + 0.994311i $$0.533971\pi$$
$$758$$ −89.2201 −3.24062
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −49.8739 −1.80793 −0.903963 0.427610i $$-0.859356\pi$$
−0.903963 + 0.427610i $$0.859356\pi$$
$$762$$ 0 0
$$763$$ −3.36744 −0.121909
$$764$$ 35.3894 1.28034
$$765$$ 0 0
$$766$$ 71.3239 2.57704
$$767$$ 24.4717 0.883621
$$768$$ 0 0
$$769$$ 29.5121 1.06423 0.532117 0.846671i $$-0.321396\pi$$
0.532117 + 0.846671i $$0.321396\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −141.607 −5.09653
$$773$$ −2.41935 −0.0870180 −0.0435090 0.999053i $$-0.513854\pi$$
−0.0435090 + 0.999053i $$0.513854\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −183.108 −6.57320
$$777$$ 0 0
$$778$$ 89.8934 3.22283
$$779$$ −1.58939 −0.0569460
$$780$$ 0 0
$$781$$ −2.35722 −0.0843479
$$782$$ −35.2715 −1.26131
$$783$$ 0 0
$$784$$ −110.540 −3.94786
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 51.1549 1.82348 0.911738 0.410772i $$-0.134741\pi$$
0.911738 + 0.410772i $$0.134741\pi$$
$$788$$ −118.722 −4.22928
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 4.41428 0.156954
$$792$$ 0 0
$$793$$ −16.0406 −0.569619
$$794$$ 12.1162 0.429987
$$795$$ 0 0
$$796$$ −14.8214 −0.525330
$$797$$ 28.8662 1.02249 0.511247 0.859434i $$-0.329184\pi$$
0.511247 + 0.859434i $$0.329184\pi$$
$$798$$ 0 0
$$799$$ −24.0907 −0.852266
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −52.5567 −1.85584
$$803$$ −1.31616 −0.0464462
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 71.8171 2.52965
$$807$$ 0 0
$$808$$ −127.021 −4.46858
$$809$$ −19.0426 −0.669501 −0.334750 0.942307i $$-0.608652\pi$$
−0.334750 + 0.942307i $$0.608652\pi$$
$$810$$ 0 0
$$811$$ −20.8783 −0.733137 −0.366569 0.930391i $$-0.619467\pi$$
−0.366569 + 0.930391i $$0.619467\pi$$
$$812$$ −2.20500 −0.0773804
$$813$$ 0 0
$$814$$ −7.97066 −0.279372
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 9.90381 0.346491
$$818$$ 3.81060 0.133234
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −3.60466 −0.125804 −0.0629018 0.998020i $$-0.520035\pi$$
−0.0629018 + 0.998020i $$0.520035\pi$$
$$822$$ 0 0
$$823$$ −27.2288 −0.949135 −0.474567 0.880219i $$-0.657395\pi$$
−0.474567 + 0.880219i $$0.657395\pi$$
$$824$$ 48.3109 1.68299
$$825$$ 0 0
$$826$$ 10.3572 0.360374
$$827$$ 47.3936 1.64804 0.824019 0.566562i $$-0.191727\pi$$
0.824019 + 0.566562i $$0.191727\pi$$
$$828$$ 0 0
$$829$$ −41.8388 −1.45312 −0.726560 0.687103i $$-0.758882\pi$$
−0.726560 + 0.687103i $$0.758882\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 91.6339 3.17683
$$833$$ −14.2258 −0.492894
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −2.11446 −0.0731302
$$837$$ 0 0
$$838$$ 7.91085 0.273276
$$839$$ 31.6385 1.09228 0.546141 0.837693i $$-0.316096\pi$$
0.546141 + 0.837693i $$0.316096\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ −37.6111 −1.29617
$$843$$ 0 0
$$844$$ 11.1977 0.385440
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 4.27107 0.146756
$$848$$ −199.086 −6.83665
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 45.2013 1.54948
$$852$$ 0 0
$$853$$ −44.2000 −1.51338 −0.756690 0.653773i $$-0.773185\pi$$
−0.756690 + 0.653773i $$0.773185\pi$$
$$854$$ −6.78892 −0.232312
$$855$$ 0 0
$$856$$ 23.2394 0.794305
$$857$$ 14.4775 0.494541 0.247270 0.968947i $$-0.420466\pi$$
0.247270 + 0.968947i $$0.420466\pi$$
$$858$$ 0 0
$$859$$ 22.5883 0.770703 0.385352 0.922770i $$-0.374080\pi$$
0.385352 + 0.922770i $$0.374080\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −32.9983 −1.12393
$$863$$ −22.9900 −0.782590 −0.391295 0.920265i $$-0.627973\pi$$
−0.391295 + 0.920265i $$0.627973\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 49.5930 1.68524
$$867$$ 0 0
$$868$$ 22.3953 0.760147
$$869$$ −0.814230 −0.0276209
$$870$$ 0 0
$$871$$ 19.0337 0.644931
$$872$$ −84.8250 −2.87254
$$873$$ 0 0
$$874$$ 16.2744 0.550491
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −13.5741 −0.458364 −0.229182 0.973384i $$-0.573605\pi$$
−0.229182 + 0.973384i $$0.573605\pi$$
$$878$$ 44.7191 1.50920
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −9.91235 −0.333956 −0.166978 0.985961i $$-0.553401\pi$$
−0.166978 + 0.985961i $$0.553401\pi$$
$$882$$ 0 0
$$883$$ −39.3192 −1.32320 −0.661598 0.749859i $$-0.730121\pi$$
−0.661598 + 0.749859i $$0.730121\pi$$
$$884$$ 29.8478 1.00389
$$885$$ 0 0
$$886$$ 97.5180 3.27618
$$887$$ −59.2520 −1.98949 −0.994743 0.102403i $$-0.967347\pi$$
−0.994743 + 0.102403i $$0.967347\pi$$
$$888$$ 0 0
$$889$$ 7.93832 0.266243
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −70.2678 −2.35274
$$893$$ 11.1155 0.371968
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 19.3461 0.646307
$$897$$ 0 0
$$898$$ 114.942 3.83567
$$899$$ −10.1566 −0.338742
$$900$$ 0 0
$$901$$ −25.6211 −0.853562
$$902$$ −1.79939 −0.0599131
$$903$$ 0 0
$$904$$ 111.195 3.69828
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 5.91210 0.196308 0.0981541 0.995171i $$-0.468706\pi$$
0.0981541 + 0.995171i $$0.468706\pi$$
$$908$$ 7.28682 0.241822
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 12.4185 0.411445 0.205722 0.978610i $$-0.434046\pi$$
0.205722 + 0.978610i $$0.434046\pi$$
$$912$$ 0 0
$$913$$ 2.52447 0.0835476
$$914$$ −44.4118 −1.46901
$$915$$ 0 0
$$916$$ 18.4024 0.608031
$$917$$ −2.16700 −0.0715607
$$918$$ 0 0
$$919$$ 16.4332 0.542081 0.271041 0.962568i $$-0.412632\pi$$
0.271041 + 0.962568i $$0.412632\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 47.7983 1.57415
$$923$$ −15.3530 −0.505351
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 53.2510 1.74994
$$927$$ 0 0
$$928$$ −24.6760 −0.810029
$$929$$ −13.8358 −0.453936 −0.226968 0.973902i $$-0.572881\pi$$
−0.226968 + 0.973902i $$0.572881\pi$$
$$930$$ 0 0
$$931$$ 6.56384 0.215121
$$932$$ 99.1637 3.24822
$$933$$ 0 0
$$934$$ 22.0528 0.721589
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −20.7528 −0.677964 −0.338982 0.940793i $$-0.610083\pi$$
−0.338982 + 0.940793i $$0.610083\pi$$
$$938$$ 8.05567 0.263027
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −40.1666 −1.30940 −0.654698 0.755891i $$-0.727204\pi$$
−0.654698 + 0.755891i $$0.727204\pi$$
$$942$$ 0 0
$$943$$ 10.2043 0.332297
$$944$$ 154.068 5.01447
$$945$$ 0 0
$$946$$ 11.2123 0.364544
$$947$$ −17.4144 −0.565894 −0.282947 0.959136i $$-0.591312\pi$$
−0.282947 + 0.959136i $$0.591312\pi$$
$$948$$ 0 0
$$949$$ −8.57239 −0.278271
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 8.11999 0.263170
$$953$$ 13.5121 0.437701 0.218851 0.975758i $$-0.429769\pi$$
0.218851 + 0.975758i $$0.429769\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −119.060 −3.85067
$$957$$ 0 0
$$958$$ −111.333 −3.59700
$$959$$ −2.95316 −0.0953626
$$960$$ 0 0
$$961$$ 72.1567 2.32763
$$962$$ −51.9145 −1.67379
$$963$$ 0 0
$$964$$ 85.7610 2.76218
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 24.0598 0.773712 0.386856 0.922140i $$-0.373561\pi$$
0.386856 + 0.922140i $$0.373561\pi$$
$$968$$ 107.587 3.45798
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −34.7877 −1.11639 −0.558195 0.829710i $$-0.688506\pi$$
−0.558195 + 0.829710i $$0.688506\pi$$
$$972$$ 0 0
$$973$$ 3.68342 0.118085
$$974$$ 7.83344 0.251000
$$975$$ 0 0
$$976$$ −100.988 −3.23254
$$977$$ 35.7566 1.14396 0.571978 0.820269i $$-0.306176\pi$$
0.571978 + 0.820269i $$0.306176\pi$$
$$978$$ 0 0
$$979$$ −6.23446 −0.199254
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0.435166 0.0138867
$$983$$ −7.19064 −0.229346 −0.114673 0.993403i $$-0.536582\pi$$
−0.114673 + 0.993403i $$0.536582\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −5.72905 −0.182450
$$987$$ 0 0
$$988$$ −13.7719 −0.438143
$$989$$ −63.5847 −2.02188
$$990$$ 0 0
$$991$$ −19.9123 −0.632537 −0.316268 0.948670i $$-0.602430\pi$$
−0.316268 + 0.948670i $$0.602430\pi$$
$$992$$ 250.624 7.95733
$$993$$ 0 0
$$994$$ −6.49790 −0.206101
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 1.57302 0.0498179 0.0249089 0.999690i $$-0.492070\pi$$
0.0249089 + 0.999690i $$0.492070\pi$$
$$998$$ −7.29734 −0.230993
$$999$$ 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.bi.1.1 4
3.2 odd 2 2175.2.a.v.1.4 4
5.4 even 2 1305.2.a.r.1.4 4
15.2 even 4 2175.2.c.n.349.8 8
15.8 even 4 2175.2.c.n.349.1 8
15.14 odd 2 435.2.a.j.1.1 4
60.59 even 2 6960.2.a.co.1.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.j.1.1 4 15.14 odd 2
1305.2.a.r.1.4 4 5.4 even 2
2175.2.a.v.1.4 4 3.2 odd 2
2175.2.c.n.349.1 8 15.8 even 4
2175.2.c.n.349.8 8 15.2 even 4
6525.2.a.bi.1.1 4 1.1 even 1 trivial
6960.2.a.co.1.2 4 60.59 even 2