# Properties

 Label 9680.2.a.cg.1.1 Level $9680$ Weight $2$ Character 9680.1 Self dual yes Analytic conductor $77.295$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9680,2,Mod(1,9680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9680 = 2^{4} \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9680.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: $$77.2951891566$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.568.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 6x - 2$$ x^3 - x^2 - 6*x - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 4840) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.76156$$ of defining polynomial Character $$\chi$$ $$=$$ 9680.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.62620 q^{3} +1.00000 q^{5} +2.62620 q^{7} +3.89692 q^{9} +O(q^{10})$$ $$q-2.62620 q^{3} +1.00000 q^{5} +2.62620 q^{7} +3.89692 q^{9} -1.72928 q^{13} -2.62620 q^{15} +0.626198 q^{17} +8.14931 q^{19} -6.89692 q^{21} +5.52311 q^{23} +1.00000 q^{25} -2.35548 q^{27} -2.89692 q^{29} +6.89692 q^{31} +2.62620 q^{35} -0.896916 q^{37} +4.54144 q^{39} -5.25240 q^{41} +9.52311 q^{43} +3.89692 q^{45} +0.270718 q^{47} -0.103084 q^{49} -1.64452 q^{51} -0.896916 q^{53} -21.4017 q^{57} -7.04623 q^{59} +9.40171 q^{61} +10.2341 q^{63} -1.72928 q^{65} +12.9817 q^{67} -14.5048 q^{69} -13.4017 q^{71} +11.5231 q^{73} -2.62620 q^{75} +12.0000 q^{79} -5.50479 q^{81} -12.5693 q^{83} +0.626198 q^{85} +7.60788 q^{87} +9.60788 q^{89} -4.54144 q^{91} -18.1127 q^{93} +8.14931 q^{95} +6.00000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{3} + 3 q^{5} - q^{7} + 8 q^{9}+O(q^{10})$$ 3 * q + q^3 + 3 * q^5 - q^7 + 8 * q^9 $$3 q + q^{3} + 3 q^{5} - q^{7} + 8 q^{9} + q^{15} - 7 q^{17} + 3 q^{19} - 17 q^{21} + 4 q^{23} + 3 q^{25} + 7 q^{27} - 5 q^{29} + 17 q^{31} - q^{35} + q^{37} + 24 q^{39} + 2 q^{41} + 16 q^{43} + 8 q^{45} + 6 q^{47} - 4 q^{49} - 19 q^{51} + q^{53} - 25 q^{57} + 4 q^{59} - 11 q^{61} - 10 q^{63} + 16 q^{67} - 8 q^{69} - q^{71} + 22 q^{73} + q^{75} + 36 q^{79} + 19 q^{81} - 7 q^{85} - 9 q^{87} - 3 q^{89} - 24 q^{91} + 13 q^{93} + 3 q^{95} + 18 q^{97}+O(q^{100})$$ 3 * q + q^3 + 3 * q^5 - q^7 + 8 * q^9 + q^15 - 7 * q^17 + 3 * q^19 - 17 * q^21 + 4 * q^23 + 3 * q^25 + 7 * q^27 - 5 * q^29 + 17 * q^31 - q^35 + q^37 + 24 * q^39 + 2 * q^41 + 16 * q^43 + 8 * q^45 + 6 * q^47 - 4 * q^49 - 19 * q^51 + q^53 - 25 * q^57 + 4 * q^59 - 11 * q^61 - 10 * q^63 + 16 * q^67 - 8 * q^69 - q^71 + 22 * q^73 + q^75 + 36 * q^79 + 19 * q^81 - 7 * q^85 - 9 * q^87 - 3 * q^89 - 24 * q^91 + 13 * q^93 + 3 * q^95 + 18 * q^97

## 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
$$3$$ −2.62620 −1.51624 −0.758118 0.652117i $$-0.773881\pi$$
−0.758118 + 0.652117i $$0.773881\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 2.62620 0.992610 0.496305 0.868148i $$-0.334690\pi$$
0.496305 + 0.868148i $$0.334690\pi$$
$$8$$ 0 0
$$9$$ 3.89692 1.29897
$$10$$ 0 0
$$11$$ 0 0
$$12$$ 0 0
$$13$$ −1.72928 −0.479616 −0.239808 0.970820i $$-0.577085\pi$$
−0.239808 + 0.970820i $$0.577085\pi$$
$$14$$ 0 0
$$15$$ −2.62620 −0.678081
$$16$$ 0 0
$$17$$ 0.626198 0.151875 0.0759377 0.997113i $$-0.475805\pi$$
0.0759377 + 0.997113i $$0.475805\pi$$
$$18$$ 0 0
$$19$$ 8.14931 1.86958 0.934790 0.355200i $$-0.115587\pi$$
0.934790 + 0.355200i $$0.115587\pi$$
$$20$$ 0 0
$$21$$ −6.89692 −1.50503
$$22$$ 0 0
$$23$$ 5.52311 1.15165 0.575824 0.817573i $$-0.304681\pi$$
0.575824 + 0.817573i $$0.304681\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −2.35548 −0.453312
$$28$$ 0 0
$$29$$ −2.89692 −0.537944 −0.268972 0.963148i $$-0.586684\pi$$
−0.268972 + 0.963148i $$0.586684\pi$$
$$30$$ 0 0
$$31$$ 6.89692 1.23872 0.619361 0.785106i $$-0.287392\pi$$
0.619361 + 0.785106i $$0.287392\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 2.62620 0.443908
$$36$$ 0 0
$$37$$ −0.896916 −0.147452 −0.0737261 0.997279i $$-0.523489\pi$$
−0.0737261 + 0.997279i $$0.523489\pi$$
$$38$$ 0 0
$$39$$ 4.54144 0.727212
$$40$$ 0 0
$$41$$ −5.25240 −0.820286 −0.410143 0.912021i $$-0.634521\pi$$
−0.410143 + 0.912021i $$0.634521\pi$$
$$42$$ 0 0
$$43$$ 9.52311 1.45226 0.726131 0.687557i $$-0.241317\pi$$
0.726131 + 0.687557i $$0.241317\pi$$
$$44$$ 0 0
$$45$$ 3.89692 0.580918
$$46$$ 0 0
$$47$$ 0.270718 0.0394883 0.0197442 0.999805i $$-0.493715\pi$$
0.0197442 + 0.999805i $$0.493715\pi$$
$$48$$ 0 0
$$49$$ −0.103084 −0.0147262
$$50$$ 0 0
$$51$$ −1.64452 −0.230279
$$52$$ 0 0
$$53$$ −0.896916 −0.123201 −0.0616005 0.998101i $$-0.519620\pi$$
−0.0616005 + 0.998101i $$0.519620\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −21.4017 −2.83473
$$58$$ 0 0
$$59$$ −7.04623 −0.917341 −0.458670 0.888606i $$-0.651674\pi$$
−0.458670 + 0.888606i $$0.651674\pi$$
$$60$$ 0 0
$$61$$ 9.40171 1.20377 0.601883 0.798584i $$-0.294417\pi$$
0.601883 + 0.798584i $$0.294417\pi$$
$$62$$ 0 0
$$63$$ 10.2341 1.28937
$$64$$ 0 0
$$65$$ −1.72928 −0.214491
$$66$$ 0 0
$$67$$ 12.9817 1.58596 0.792982 0.609245i $$-0.208527\pi$$
0.792982 + 0.609245i $$0.208527\pi$$
$$68$$ 0 0
$$69$$ −14.5048 −1.74617
$$70$$ 0 0
$$71$$ −13.4017 −1.59049 −0.795245 0.606288i $$-0.792658\pi$$
−0.795245 + 0.606288i $$0.792658\pi$$
$$72$$ 0 0
$$73$$ 11.5231 1.34868 0.674339 0.738422i $$-0.264429\pi$$
0.674339 + 0.738422i $$0.264429\pi$$
$$74$$ 0 0
$$75$$ −2.62620 −0.303247
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 12.0000 1.35011 0.675053 0.737769i $$-0.264121\pi$$
0.675053 + 0.737769i $$0.264121\pi$$
$$80$$ 0 0
$$81$$ −5.50479 −0.611644
$$82$$ 0 0
$$83$$ −12.5693 −1.37966 −0.689832 0.723969i $$-0.742316\pi$$
−0.689832 + 0.723969i $$0.742316\pi$$
$$84$$ 0 0
$$85$$ 0.626198 0.0679207
$$86$$ 0 0
$$87$$ 7.60788 0.815650
$$88$$ 0 0
$$89$$ 9.60788 1.01843 0.509216 0.860639i $$-0.329935\pi$$
0.509216 + 0.860639i $$0.329935\pi$$
$$90$$ 0 0
$$91$$ −4.54144 −0.476072
$$92$$ 0 0
$$93$$ −18.1127 −1.87820
$$94$$ 0 0
$$95$$ 8.14931 0.836102
$$96$$ 0 0
$$97$$ 6.00000 0.609208 0.304604 0.952479i $$-0.401476\pi$$
0.304604 + 0.952479i $$0.401476\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −9.25240 −0.920648 −0.460324 0.887751i $$-0.652267\pi$$
−0.460324 + 0.887751i $$0.652267\pi$$
$$102$$ 0 0
$$103$$ 8.27072 0.814938 0.407469 0.913219i $$-0.366411\pi$$
0.407469 + 0.913219i $$0.366411\pi$$
$$104$$ 0 0
$$105$$ −6.89692 −0.673070
$$106$$ 0 0
$$107$$ −14.7755 −1.42840 −0.714201 0.699940i $$-0.753210\pi$$
−0.714201 + 0.699940i $$0.753210\pi$$
$$108$$ 0 0
$$109$$ 16.8401 1.61299 0.806493 0.591244i $$-0.201363\pi$$
0.806493 + 0.591244i $$0.201363\pi$$
$$110$$ 0 0
$$111$$ 2.35548 0.223572
$$112$$ 0 0
$$113$$ 14.2986 1.34510 0.672551 0.740051i $$-0.265199\pi$$
0.672551 + 0.740051i $$0.265199\pi$$
$$114$$ 0 0
$$115$$ 5.52311 0.515033
$$116$$ 0 0
$$117$$ −6.73887 −0.623008
$$118$$ 0 0
$$119$$ 1.64452 0.150753
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 0 0
$$123$$ 13.7938 1.24375
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −5.52311 −0.490097 −0.245049 0.969511i $$-0.578804\pi$$
−0.245049 + 0.969511i $$0.578804\pi$$
$$128$$ 0 0
$$129$$ −25.0096 −2.20197
$$130$$ 0 0
$$131$$ 9.10308 0.795340 0.397670 0.917528i $$-0.369819\pi$$
0.397670 + 0.917528i $$0.369819\pi$$
$$132$$ 0 0
$$133$$ 21.4017 1.85576
$$134$$ 0 0
$$135$$ −2.35548 −0.202727
$$136$$ 0 0
$$137$$ 0.206167 0.0176141 0.00880704 0.999961i $$-0.497197\pi$$
0.00880704 + 0.999961i $$0.497197\pi$$
$$138$$ 0 0
$$139$$ −21.9634 −1.86291 −0.931454 0.363860i $$-0.881459\pi$$
−0.931454 + 0.363860i $$0.881459\pi$$
$$140$$ 0 0
$$141$$ −0.710960 −0.0598736
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −2.89692 −0.240576
$$146$$ 0 0
$$147$$ 0.270718 0.0223285
$$148$$ 0 0
$$149$$ −5.64452 −0.462417 −0.231209 0.972904i $$-0.574268\pi$$
−0.231209 + 0.972904i $$0.574268\pi$$
$$150$$ 0 0
$$151$$ 15.4586 1.25800 0.629000 0.777405i $$-0.283465\pi$$
0.629000 + 0.777405i $$0.283465\pi$$
$$152$$ 0 0
$$153$$ 2.44024 0.197282
$$154$$ 0 0
$$155$$ 6.89692 0.553974
$$156$$ 0 0
$$157$$ −15.4017 −1.22919 −0.614595 0.788843i $$-0.710681\pi$$
−0.614595 + 0.788843i $$0.710681\pi$$
$$158$$ 0 0
$$159$$ 2.35548 0.186802
$$160$$ 0 0
$$161$$ 14.5048 1.14314
$$162$$ 0 0
$$163$$ −14.2139 −1.11332 −0.556658 0.830742i $$-0.687917\pi$$
−0.556658 + 0.830742i $$0.687917\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 18.0848 1.39944 0.699720 0.714417i $$-0.253308\pi$$
0.699720 + 0.714417i $$0.253308\pi$$
$$168$$ 0 0
$$169$$ −10.0096 −0.769968
$$170$$ 0 0
$$171$$ 31.7572 2.42853
$$172$$ 0 0
$$173$$ 7.52311 0.571972 0.285986 0.958234i $$-0.407679\pi$$
0.285986 + 0.958234i $$0.407679\pi$$
$$174$$ 0 0
$$175$$ 2.62620 0.198522
$$176$$ 0 0
$$177$$ 18.5048 1.39091
$$178$$ 0 0
$$179$$ −23.7572 −1.77570 −0.887848 0.460137i $$-0.847800\pi$$
−0.887848 + 0.460137i $$0.847800\pi$$
$$180$$ 0 0
$$181$$ −18.2986 −1.36013 −0.680063 0.733154i $$-0.738048\pi$$
−0.680063 + 0.733154i $$0.738048\pi$$
$$182$$ 0 0
$$183$$ −24.6907 −1.82519
$$184$$ 0 0
$$185$$ −0.896916 −0.0659426
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −6.18596 −0.449962
$$190$$ 0 0
$$191$$ 12.0000 0.868290 0.434145 0.900843i $$-0.357051\pi$$
0.434145 + 0.900843i $$0.357051\pi$$
$$192$$ 0 0
$$193$$ 0.626198 0.0450747 0.0225374 0.999746i $$-0.492826\pi$$
0.0225374 + 0.999746i $$0.492826\pi$$
$$194$$ 0 0
$$195$$ 4.54144 0.325219
$$196$$ 0 0
$$197$$ −25.0741 −1.78646 −0.893229 0.449602i $$-0.851566\pi$$
−0.893229 + 0.449602i $$0.851566\pi$$
$$198$$ 0 0
$$199$$ 8.14931 0.577689 0.288845 0.957376i $$-0.406729\pi$$
0.288845 + 0.957376i $$0.406729\pi$$
$$200$$ 0 0
$$201$$ −34.0925 −2.40470
$$202$$ 0 0
$$203$$ −7.60788 −0.533968
$$204$$ 0 0
$$205$$ −5.25240 −0.366843
$$206$$ 0 0
$$207$$ 21.5231 1.49596
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −6.89692 −0.474803 −0.237402 0.971412i $$-0.576296\pi$$
−0.237402 + 0.971412i $$0.576296\pi$$
$$212$$ 0 0
$$213$$ 35.1955 2.41156
$$214$$ 0 0
$$215$$ 9.52311 0.649471
$$216$$ 0 0
$$217$$ 18.1127 1.22957
$$218$$ 0 0
$$219$$ −30.2620 −2.04492
$$220$$ 0 0
$$221$$ −1.08287 −0.0728419
$$222$$ 0 0
$$223$$ 3.31695 0.222119 0.111060 0.993814i $$-0.464576\pi$$
0.111060 + 0.993814i $$0.464576\pi$$
$$224$$ 0 0
$$225$$ 3.89692 0.259794
$$226$$ 0 0
$$227$$ −1.22449 −0.0812722 −0.0406361 0.999174i $$-0.512938\pi$$
−0.0406361 + 0.999174i $$0.512938\pi$$
$$228$$ 0 0
$$229$$ −28.8034 −1.90338 −0.951692 0.307055i $$-0.900656\pi$$
−0.951692 + 0.307055i $$0.900656\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 10.1772 0.666732 0.333366 0.942798i $$-0.391816\pi$$
0.333366 + 0.942798i $$0.391816\pi$$
$$234$$ 0 0
$$235$$ 0.270718 0.0176597
$$236$$ 0 0
$$237$$ −31.5144 −2.04708
$$238$$ 0 0
$$239$$ 1.79383 0.116033 0.0580167 0.998316i $$-0.481522\pi$$
0.0580167 + 0.998316i $$0.481522\pi$$
$$240$$ 0 0
$$241$$ −21.2524 −1.36899 −0.684494 0.729019i $$-0.739977\pi$$
−0.684494 + 0.729019i $$0.739977\pi$$
$$242$$ 0 0
$$243$$ 21.5231 1.38071
$$244$$ 0 0
$$245$$ −0.103084 −0.00658578
$$246$$ 0 0
$$247$$ −14.0925 −0.896682
$$248$$ 0 0
$$249$$ 33.0096 2.09190
$$250$$ 0 0
$$251$$ 6.50479 0.410579 0.205289 0.978701i $$-0.434186\pi$$
0.205289 + 0.978701i $$0.434186\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ −1.64452 −0.102984
$$256$$ 0 0
$$257$$ −2.71096 −0.169105 −0.0845525 0.996419i $$-0.526946\pi$$
−0.0845525 + 0.996419i $$0.526946\pi$$
$$258$$ 0 0
$$259$$ −2.35548 −0.146362
$$260$$ 0 0
$$261$$ −11.2890 −0.698774
$$262$$ 0 0
$$263$$ −5.67243 −0.349777 −0.174888 0.984588i $$-0.555956\pi$$
−0.174888 + 0.984588i $$0.555956\pi$$
$$264$$ 0 0
$$265$$ −0.896916 −0.0550971
$$266$$ 0 0
$$267$$ −25.2322 −1.54418
$$268$$ 0 0
$$269$$ 6.71096 0.409174 0.204587 0.978848i $$-0.434415\pi$$
0.204587 + 0.978848i $$0.434415\pi$$
$$270$$ 0 0
$$271$$ −7.45856 −0.453075 −0.226538 0.974002i $$-0.572741\pi$$
−0.226538 + 0.974002i $$0.572741\pi$$
$$272$$ 0 0
$$273$$ 11.9267 0.721837
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −11.9354 −0.717132 −0.358566 0.933504i $$-0.616734\pi$$
−0.358566 + 0.933504i $$0.616734\pi$$
$$278$$ 0 0
$$279$$ 26.8767 1.60907
$$280$$ 0 0
$$281$$ −10.7476 −0.641148 −0.320574 0.947223i $$-0.603876\pi$$
−0.320574 + 0.947223i $$0.603876\pi$$
$$282$$ 0 0
$$283$$ 7.01832 0.417196 0.208598 0.978001i $$-0.433110\pi$$
0.208598 + 0.978001i $$0.433110\pi$$
$$284$$ 0 0
$$285$$ −21.4017 −1.26773
$$286$$ 0 0
$$287$$ −13.7938 −0.814224
$$288$$ 0 0
$$289$$ −16.6079 −0.976934
$$290$$ 0 0
$$291$$ −15.7572 −0.923703
$$292$$ 0 0
$$293$$ 21.4865 1.25525 0.627626 0.778515i $$-0.284027\pi$$
0.627626 + 0.778515i $$0.284027\pi$$
$$294$$ 0 0
$$295$$ −7.04623 −0.410247
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −9.55102 −0.552350
$$300$$ 0 0
$$301$$ 25.0096 1.44153
$$302$$ 0 0
$$303$$ 24.2986 1.39592
$$304$$ 0 0
$$305$$ 9.40171 0.538340
$$306$$ 0 0
$$307$$ 20.3265 1.16010 0.580048 0.814582i $$-0.303034\pi$$
0.580048 + 0.814582i $$0.303034\pi$$
$$308$$ 0 0
$$309$$ −21.7205 −1.23564
$$310$$ 0 0
$$311$$ 5.40171 0.306303 0.153151 0.988203i $$-0.451058\pi$$
0.153151 + 0.988203i $$0.451058\pi$$
$$312$$ 0 0
$$313$$ 2.00000 0.113047 0.0565233 0.998401i $$-0.481998\pi$$
0.0565233 + 0.998401i $$0.481998\pi$$
$$314$$ 0 0
$$315$$ 10.2341 0.576625
$$316$$ 0 0
$$317$$ 6.39212 0.359017 0.179509 0.983756i $$-0.442549\pi$$
0.179509 + 0.983756i $$0.442549\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 38.8034 2.16580
$$322$$ 0 0
$$323$$ 5.10308 0.283943
$$324$$ 0 0
$$325$$ −1.72928 −0.0959233
$$326$$ 0 0
$$327$$ −44.2253 −2.44567
$$328$$ 0 0
$$329$$ 0.710960 0.0391965
$$330$$ 0 0
$$331$$ 14.9171 0.819919 0.409960 0.912104i $$-0.365543\pi$$
0.409960 + 0.912104i $$0.365543\pi$$
$$332$$ 0 0
$$333$$ −3.49521 −0.191536
$$334$$ 0 0
$$335$$ 12.9817 0.709265
$$336$$ 0 0
$$337$$ −18.1772 −0.990176 −0.495088 0.868843i $$-0.664864\pi$$
−0.495088 + 0.868843i $$0.664864\pi$$
$$338$$ 0 0
$$339$$ −37.5510 −2.03949
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −18.6541 −1.00723
$$344$$ 0 0
$$345$$ −14.5048 −0.780912
$$346$$ 0 0
$$347$$ 15.3169 0.822257 0.411128 0.911577i $$-0.365135\pi$$
0.411128 + 0.911577i $$0.365135\pi$$
$$348$$ 0 0
$$349$$ 5.79383 0.310137 0.155068 0.987904i $$-0.450440\pi$$
0.155068 + 0.987904i $$0.450440\pi$$
$$350$$ 0 0
$$351$$ 4.07329 0.217416
$$352$$ 0 0
$$353$$ 29.8863 1.59069 0.795343 0.606159i $$-0.207291\pi$$
0.795343 + 0.606159i $$0.207291\pi$$
$$354$$ 0 0
$$355$$ −13.4017 −0.711289
$$356$$ 0 0
$$357$$ −4.31884 −0.228577
$$358$$ 0 0
$$359$$ −33.3082 −1.75794 −0.878970 0.476877i $$-0.841769\pi$$
−0.878970 + 0.476877i $$0.841769\pi$$
$$360$$ 0 0
$$361$$ 47.4113 2.49533
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 11.5231 0.603147
$$366$$ 0 0
$$367$$ −9.11078 −0.475579 −0.237789 0.971317i $$-0.576423\pi$$
−0.237789 + 0.971317i $$0.576423\pi$$
$$368$$ 0 0
$$369$$ −20.4681 −1.06553
$$370$$ 0 0
$$371$$ −2.35548 −0.122290
$$372$$ 0 0
$$373$$ 7.52311 0.389532 0.194766 0.980850i $$-0.437605\pi$$
0.194766 + 0.980850i $$0.437605\pi$$
$$374$$ 0 0
$$375$$ −2.62620 −0.135616
$$376$$ 0 0
$$377$$ 5.00958 0.258007
$$378$$ 0 0
$$379$$ 9.55102 0.490603 0.245301 0.969447i $$-0.421113\pi$$
0.245301 + 0.969447i $$0.421113\pi$$
$$380$$ 0 0
$$381$$ 14.5048 0.743103
$$382$$ 0 0
$$383$$ 10.7755 0.550603 0.275301 0.961358i $$-0.411222\pi$$
0.275301 + 0.961358i $$0.411222\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 37.1108 1.88645
$$388$$ 0 0
$$389$$ 26.7110 1.35430 0.677150 0.735845i $$-0.263215\pi$$
0.677150 + 0.735845i $$0.263215\pi$$
$$390$$ 0 0
$$391$$ 3.45856 0.174907
$$392$$ 0 0
$$393$$ −23.9065 −1.20592
$$394$$ 0 0
$$395$$ 12.0000 0.603786
$$396$$ 0 0
$$397$$ −18.5972 −0.933369 −0.466685 0.884424i $$-0.654552\pi$$
−0.466685 + 0.884424i $$0.654552\pi$$
$$398$$ 0 0
$$399$$ −56.2051 −2.81378
$$400$$ 0 0
$$401$$ 19.4017 0.968875 0.484438 0.874826i $$-0.339024\pi$$
0.484438 + 0.874826i $$0.339024\pi$$
$$402$$ 0 0
$$403$$ −11.9267 −0.594112
$$404$$ 0 0
$$405$$ −5.50479 −0.273535
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 6.80342 0.336407 0.168204 0.985752i $$-0.446203\pi$$
0.168204 + 0.985752i $$0.446203\pi$$
$$410$$ 0 0
$$411$$ −0.541436 −0.0267071
$$412$$ 0 0
$$413$$ −18.5048 −0.910561
$$414$$ 0 0
$$415$$ −12.5693 −0.617005
$$416$$ 0 0
$$417$$ 57.6801 2.82461
$$418$$ 0 0
$$419$$ −0.840061 −0.0410397 −0.0205198 0.999789i $$-0.506532\pi$$
−0.0205198 + 0.999789i $$0.506532\pi$$
$$420$$ 0 0
$$421$$ 22.5972 1.10132 0.550661 0.834729i $$-0.314376\pi$$
0.550661 + 0.834729i $$0.314376\pi$$
$$422$$ 0 0
$$423$$ 1.05497 0.0512942
$$424$$ 0 0
$$425$$ 0.626198 0.0303751
$$426$$ 0 0
$$427$$ 24.6907 1.19487
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0.129102 0.00621861 0.00310930 0.999995i $$-0.499010\pi$$
0.00310930 + 0.999995i $$0.499010\pi$$
$$432$$ 0 0
$$433$$ 2.41233 0.115929 0.0579647 0.998319i $$-0.481539\pi$$
0.0579647 + 0.998319i $$0.481539\pi$$
$$434$$ 0 0
$$435$$ 7.60788 0.364770
$$436$$ 0 0
$$437$$ 45.0096 2.15310
$$438$$ 0 0
$$439$$ −2.20617 −0.105295 −0.0526473 0.998613i $$-0.516766\pi$$
−0.0526473 + 0.998613i $$0.516766\pi$$
$$440$$ 0 0
$$441$$ −0.401709 −0.0191290
$$442$$ 0 0
$$443$$ 29.5789 1.40534 0.702669 0.711517i $$-0.251992\pi$$
0.702669 + 0.711517i $$0.251992\pi$$
$$444$$ 0 0
$$445$$ 9.60788 0.455457
$$446$$ 0 0
$$447$$ 14.8236 0.701134
$$448$$ 0 0
$$449$$ −20.5048 −0.967681 −0.483840 0.875156i $$-0.660759\pi$$
−0.483840 + 0.875156i $$0.660759\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ −40.5972 −1.90743
$$454$$ 0 0
$$455$$ −4.54144 −0.212906
$$456$$ 0 0
$$457$$ 7.37380 0.344932 0.172466 0.985015i $$-0.444827\pi$$
0.172466 + 0.985015i $$0.444827\pi$$
$$458$$ 0 0
$$459$$ −1.47500 −0.0688470
$$460$$ 0 0
$$461$$ −35.0664 −1.63321 −0.816603 0.577199i $$-0.804146\pi$$
−0.816603 + 0.577199i $$0.804146\pi$$
$$462$$ 0 0
$$463$$ 9.11078 0.423414 0.211707 0.977333i $$-0.432098\pi$$
0.211707 + 0.977333i $$0.432098\pi$$
$$464$$ 0 0
$$465$$ −18.1127 −0.839955
$$466$$ 0 0
$$467$$ 34.3834 1.59107 0.795537 0.605905i $$-0.207189\pi$$
0.795537 + 0.605905i $$0.207189\pi$$
$$468$$ 0 0
$$469$$ 34.0925 1.57424
$$470$$ 0 0
$$471$$ 40.4479 1.86374
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 8.14931 0.373916
$$476$$ 0 0
$$477$$ −3.49521 −0.160035
$$478$$ 0 0
$$479$$ −31.8863 −1.45692 −0.728461 0.685087i $$-0.759764\pi$$
−0.728461 + 0.685087i $$0.759764\pi$$
$$480$$ 0 0
$$481$$ 1.55102 0.0707205
$$482$$ 0 0
$$483$$ −38.0925 −1.73327
$$484$$ 0 0
$$485$$ 6.00000 0.272446
$$486$$ 0 0
$$487$$ 35.0741 1.58936 0.794680 0.607028i $$-0.207638\pi$$
0.794680 + 0.607028i $$0.207638\pi$$
$$488$$ 0 0
$$489$$ 37.3284 1.68805
$$490$$ 0 0
$$491$$ 20.0202 0.903499 0.451750 0.892145i $$-0.350800\pi$$
0.451750 + 0.892145i $$0.350800\pi$$
$$492$$ 0 0
$$493$$ −1.81404 −0.0817004
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −35.1955 −1.57874
$$498$$ 0 0
$$499$$ −12.8401 −0.574800 −0.287400 0.957811i $$-0.592791\pi$$
−0.287400 + 0.957811i $$0.592791\pi$$
$$500$$ 0 0
$$501$$ −47.4942 −2.12188
$$502$$ 0 0
$$503$$ −40.0279 −1.78476 −0.892378 0.451289i $$-0.850965\pi$$
−0.892378 + 0.451289i $$0.850965\pi$$
$$504$$ 0 0
$$505$$ −9.25240 −0.411726
$$506$$ 0 0
$$507$$ 26.2872 1.16745
$$508$$ 0 0
$$509$$ −42.5972 −1.88809 −0.944045 0.329817i $$-0.893013\pi$$
−0.944045 + 0.329817i $$0.893013\pi$$
$$510$$ 0 0
$$511$$ 30.2620 1.33871
$$512$$ 0 0
$$513$$ −19.1955 −0.847504
$$514$$ 0 0
$$515$$ 8.27072 0.364451
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −19.7572 −0.867244
$$520$$ 0 0
$$521$$ 2.41233 0.105686 0.0528432 0.998603i $$-0.483172\pi$$
0.0528432 + 0.998603i $$0.483172\pi$$
$$522$$ 0 0
$$523$$ 10.3632 0.453150 0.226575 0.973994i $$-0.427247\pi$$
0.226575 + 0.973994i $$0.427247\pi$$
$$524$$ 0 0
$$525$$ −6.89692 −0.301006
$$526$$ 0 0
$$527$$ 4.31884 0.188131
$$528$$ 0 0
$$529$$ 7.50479 0.326295
$$530$$ 0 0
$$531$$ −27.4586 −1.19160
$$532$$ 0 0
$$533$$ 9.08287 0.393423
$$534$$ 0 0
$$535$$ −14.7755 −0.638801
$$536$$ 0 0
$$537$$ 62.3911 2.69237
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 24.5616 1.05599 0.527994 0.849248i $$-0.322944\pi$$
0.527994 + 0.849248i $$0.322944\pi$$
$$542$$ 0 0
$$543$$ 48.0558 2.06227
$$544$$ 0 0
$$545$$ 16.8401 0.721349
$$546$$ 0 0
$$547$$ −25.8217 −1.10406 −0.552029 0.833825i $$-0.686146\pi$$
−0.552029 + 0.833825i $$0.686146\pi$$
$$548$$ 0 0
$$549$$ 36.6377 1.56366
$$550$$ 0 0
$$551$$ −23.6079 −1.00573
$$552$$ 0 0
$$553$$ 31.5144 1.34013
$$554$$ 0 0
$$555$$ 2.35548 0.0999846
$$556$$ 0 0
$$557$$ −32.7755 −1.38874 −0.694371 0.719617i $$-0.744318\pi$$
−0.694371 + 0.719617i $$0.744318\pi$$
$$558$$ 0 0
$$559$$ −16.4681 −0.696528
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −9.82174 −0.413937 −0.206968 0.978348i $$-0.566360\pi$$
−0.206968 + 0.978348i $$0.566360\pi$$
$$564$$ 0 0
$$565$$ 14.2986 0.601548
$$566$$ 0 0
$$567$$ −14.4567 −0.607123
$$568$$ 0 0
$$569$$ 1.19658 0.0501634 0.0250817 0.999685i $$-0.492015\pi$$
0.0250817 + 0.999685i $$0.492015\pi$$
$$570$$ 0 0
$$571$$ 6.22638 0.260566 0.130283 0.991477i $$-0.458411\pi$$
0.130283 + 0.991477i $$0.458411\pi$$
$$572$$ 0 0
$$573$$ −31.5144 −1.31653
$$574$$ 0 0
$$575$$ 5.52311 0.230330
$$576$$ 0 0
$$577$$ 43.3082 1.80294 0.901472 0.432837i $$-0.142487\pi$$
0.901472 + 0.432837i $$0.142487\pi$$
$$578$$ 0 0
$$579$$ −1.64452 −0.0683439
$$580$$ 0 0
$$581$$ −33.0096 −1.36947
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ −6.73887 −0.278618
$$586$$ 0 0
$$587$$ 19.7649 0.815784 0.407892 0.913030i $$-0.366264\pi$$
0.407892 + 0.913030i $$0.366264\pi$$
$$588$$ 0 0
$$589$$ 56.2051 2.31589
$$590$$ 0 0
$$591$$ 65.8496 2.70869
$$592$$ 0 0
$$593$$ 4.60599 0.189145 0.0945726 0.995518i $$-0.469852\pi$$
0.0945726 + 0.995518i $$0.469852\pi$$
$$594$$ 0 0
$$595$$ 1.64452 0.0674188
$$596$$ 0 0
$$597$$ −21.4017 −0.875914
$$598$$ 0 0
$$599$$ 46.9527 1.91844 0.959218 0.282666i $$-0.0912190\pi$$
0.959218 + 0.282666i $$0.0912190\pi$$
$$600$$ 0 0
$$601$$ −21.2524 −0.866903 −0.433452 0.901177i $$-0.642705\pi$$
−0.433452 + 0.901177i $$0.642705\pi$$
$$602$$ 0 0
$$603$$ 50.5885 2.06012
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 34.3834 1.39558 0.697789 0.716303i $$-0.254167\pi$$
0.697789 + 0.716303i $$0.254167\pi$$
$$608$$ 0 0
$$609$$ 19.9798 0.809622
$$610$$ 0 0
$$611$$ −0.468148 −0.0189392
$$612$$ 0 0
$$613$$ −28.3632 −1.14558 −0.572789 0.819703i $$-0.694139\pi$$
−0.572789 + 0.819703i $$0.694139\pi$$
$$614$$ 0 0
$$615$$ 13.7938 0.556221
$$616$$ 0 0
$$617$$ 45.5144 1.83234 0.916170 0.400790i $$-0.131264\pi$$
0.916170 + 0.400790i $$0.131264\pi$$
$$618$$ 0 0
$$619$$ 20.4277 0.821060 0.410530 0.911847i $$-0.365344\pi$$
0.410530 + 0.911847i $$0.365344\pi$$
$$620$$ 0 0
$$621$$ −13.0096 −0.522057
$$622$$ 0 0
$$623$$ 25.2322 1.01091
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −0.561647 −0.0223943
$$630$$ 0 0
$$631$$ 42.1127 1.67648 0.838239 0.545302i $$-0.183585\pi$$
0.838239 + 0.545302i $$0.183585\pi$$
$$632$$ 0 0
$$633$$ 18.1127 0.719914
$$634$$ 0 0
$$635$$ −5.52311 −0.219178
$$636$$ 0 0
$$637$$ 0.178261 0.00706295
$$638$$ 0 0
$$639$$ −52.2253 −2.06600
$$640$$ 0 0
$$641$$ 21.9065 0.865255 0.432627 0.901573i $$-0.357587\pi$$
0.432627 + 0.901573i $$0.357587\pi$$
$$642$$ 0 0
$$643$$ −42.6262 −1.68101 −0.840507 0.541801i $$-0.817743\pi$$
−0.840507 + 0.541801i $$0.817743\pi$$
$$644$$ 0 0
$$645$$ −25.0096 −0.984751
$$646$$ 0 0
$$647$$ −1.65222 −0.0649553 −0.0324777 0.999472i $$-0.510340\pi$$
−0.0324777 + 0.999472i $$0.510340\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −47.5675 −1.86432
$$652$$ 0 0
$$653$$ 14.3188 0.560339 0.280170 0.959950i $$-0.409609\pi$$
0.280170 + 0.959950i $$0.409609\pi$$
$$654$$ 0 0
$$655$$ 9.10308 0.355687
$$656$$ 0 0
$$657$$ 44.9046 1.75190
$$658$$ 0 0
$$659$$ 38.9527 1.51738 0.758691 0.651450i $$-0.225839\pi$$
0.758691 + 0.651450i $$0.225839\pi$$
$$660$$ 0 0
$$661$$ 39.6801 1.54338 0.771689 0.636000i $$-0.219412\pi$$
0.771689 + 0.636000i $$0.219412\pi$$
$$662$$ 0 0
$$663$$ 2.84384 0.110446
$$664$$ 0 0
$$665$$ 21.4017 0.829923
$$666$$ 0 0
$$667$$ −16.0000 −0.619522
$$668$$ 0 0
$$669$$ −8.71096 −0.336785
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 23.5433 0.907528 0.453764 0.891122i $$-0.350081\pi$$
0.453764 + 0.891122i $$0.350081\pi$$
$$674$$ 0 0
$$675$$ −2.35548 −0.0906625
$$676$$ 0 0
$$677$$ 37.4865 1.44072 0.720361 0.693599i $$-0.243976\pi$$
0.720361 + 0.693599i $$0.243976\pi$$
$$678$$ 0 0
$$679$$ 15.7572 0.604705
$$680$$ 0 0
$$681$$ 3.21575 0.123228
$$682$$ 0 0
$$683$$ 17.5433 0.671277 0.335638 0.941991i $$-0.391048\pi$$
0.335638 + 0.941991i $$0.391048\pi$$
$$684$$ 0 0
$$685$$ 0.206167 0.00787725
$$686$$ 0 0
$$687$$ 75.6435 2.88598
$$688$$ 0 0
$$689$$ 1.55102 0.0590892
$$690$$ 0 0
$$691$$ −19.2158 −0.731002 −0.365501 0.930811i $$-0.619102\pi$$
−0.365501 + 0.930811i $$0.619102\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −21.9634 −0.833118
$$696$$ 0 0
$$697$$ −3.28904 −0.124581
$$698$$ 0 0
$$699$$ −26.7274 −1.01092
$$700$$ 0 0
$$701$$ −8.02021 −0.302919 −0.151460 0.988463i $$-0.548397\pi$$
−0.151460 + 0.988463i $$0.548397\pi$$
$$702$$ 0 0
$$703$$ −7.30925 −0.275674
$$704$$ 0 0
$$705$$ −0.710960 −0.0267763
$$706$$ 0 0
$$707$$ −24.2986 −0.913844
$$708$$ 0 0
$$709$$ −3.90754 −0.146751 −0.0733754 0.997304i $$-0.523377\pi$$
−0.0733754 + 0.997304i $$0.523377\pi$$
$$710$$ 0 0
$$711$$ 46.7630 1.75375
$$712$$ 0 0
$$713$$ 38.0925 1.42657
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −4.71096 −0.175934
$$718$$ 0 0
$$719$$ 17.3459 0.646893 0.323446 0.946247i $$-0.395158\pi$$
0.323446 + 0.946247i $$0.395158\pi$$
$$720$$ 0 0
$$721$$ 21.7205 0.808915
$$722$$ 0 0
$$723$$ 55.8130 2.07571
$$724$$ 0 0
$$725$$ −2.89692 −0.107589
$$726$$ 0 0
$$727$$ 8.85258 0.328324 0.164162 0.986433i $$-0.447508\pi$$
0.164162 + 0.986433i $$0.447508\pi$$
$$728$$ 0 0
$$729$$ −40.0096 −1.48184
$$730$$ 0 0
$$731$$ 5.96336 0.220563
$$732$$ 0 0
$$733$$ 5.31695 0.196386 0.0981930 0.995167i $$-0.468694\pi$$
0.0981930 + 0.995167i $$0.468694\pi$$
$$734$$ 0 0
$$735$$ 0.270718 0.00998559
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 9.55102 0.351340 0.175670 0.984449i $$-0.443791\pi$$
0.175670 + 0.984449i $$0.443791\pi$$
$$740$$ 0 0
$$741$$ 37.0096 1.35958
$$742$$ 0 0
$$743$$ −6.79572 −0.249311 −0.124655 0.992200i $$-0.539783\pi$$
−0.124655 + 0.992200i $$0.539783\pi$$
$$744$$ 0 0
$$745$$ −5.64452 −0.206799
$$746$$ 0 0
$$747$$ −48.9817 −1.79215
$$748$$ 0 0
$$749$$ −38.8034 −1.41785
$$750$$ 0 0
$$751$$ 3.72159 0.135803 0.0679013 0.997692i $$-0.478370\pi$$
0.0679013 + 0.997692i $$0.478370\pi$$
$$752$$ 0 0
$$753$$ −17.0829 −0.622534
$$754$$ 0 0
$$755$$ 15.4586 0.562595
$$756$$ 0 0
$$757$$ 9.32946 0.339085 0.169543 0.985523i $$-0.445771\pi$$
0.169543 + 0.985523i $$0.445771\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −2.80342 −0.101624 −0.0508119 0.998708i $$-0.516181\pi$$
−0.0508119 + 0.998708i $$0.516181\pi$$
$$762$$ 0 0
$$763$$ 44.2253 1.60106
$$764$$ 0 0
$$765$$ 2.44024 0.0882271
$$766$$ 0 0
$$767$$ 12.1849 0.439972
$$768$$ 0 0
$$769$$ −29.4373 −1.06154 −0.530768 0.847517i $$-0.678097\pi$$
−0.530768 + 0.847517i $$0.678097\pi$$
$$770$$ 0 0
$$771$$ 7.11952 0.256403
$$772$$ 0 0
$$773$$ −5.30925 −0.190960 −0.0954802 0.995431i $$-0.530439\pi$$
−0.0954802 + 0.995431i $$0.530439\pi$$
$$774$$ 0 0
$$775$$ 6.89692 0.247745
$$776$$ 0 0
$$777$$ 6.18596 0.221920
$$778$$ 0 0
$$779$$ −42.8034 −1.53359
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 6.82363 0.243857
$$784$$ 0 0
$$785$$ −15.4017 −0.549711
$$786$$ 0 0
$$787$$ −48.1974 −1.71805 −0.859026 0.511931i $$-0.828930\pi$$
−0.859026 + 0.511931i $$0.828930\pi$$
$$788$$ 0 0
$$789$$ 14.8969 0.530344
$$790$$ 0 0
$$791$$ 37.5510 1.33516
$$792$$ 0 0
$$793$$ −16.2582 −0.577346
$$794$$ 0 0
$$795$$ 2.35548 0.0835403
$$796$$ 0 0
$$797$$ −2.59725 −0.0919993 −0.0459997 0.998941i $$-0.514647\pi$$
−0.0459997 + 0.998941i $$0.514647\pi$$
$$798$$ 0 0
$$799$$ 0.169523 0.00599730
$$800$$ 0 0
$$801$$ 37.4411 1.32292
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 14.5048 0.511227
$$806$$ 0 0
$$807$$ −17.6243 −0.620405
$$808$$ 0 0
$$809$$ 18.3757 0.646055 0.323027 0.946390i $$-0.395299\pi$$
0.323027 + 0.946390i $$0.395299\pi$$
$$810$$ 0 0
$$811$$ 51.4942 1.80820 0.904102 0.427316i $$-0.140541\pi$$
0.904102 + 0.427316i $$0.140541\pi$$
$$812$$ 0 0
$$813$$ 19.5877 0.686969
$$814$$ 0 0
$$815$$ −14.2139 −0.497890
$$816$$ 0 0
$$817$$ 77.6068 2.71512
$$818$$ 0 0
$$819$$ −17.6976 −0.618404
$$820$$ 0 0
$$821$$ −23.2890 −0.812793 −0.406397 0.913697i $$-0.633215\pi$$
−0.406397 + 0.913697i $$0.633215\pi$$
$$822$$ 0 0
$$823$$ 42.2899 1.47413 0.737066 0.675820i $$-0.236210\pi$$
0.737066 + 0.675820i $$0.236210\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 2.64641 0.0920247 0.0460123 0.998941i $$-0.485349\pi$$
0.0460123 + 0.998941i $$0.485349\pi$$
$$828$$ 0 0
$$829$$ 8.20617 0.285012 0.142506 0.989794i $$-0.454484\pi$$
0.142506 + 0.989794i $$0.454484\pi$$
$$830$$ 0 0
$$831$$ 31.3449 1.08734
$$832$$ 0 0
$$833$$ −0.0645508 −0.00223655
$$834$$ 0 0
$$835$$ 18.0848 0.625849
$$836$$ 0 0
$$837$$ −16.2455 −0.561528
$$838$$ 0 0
$$839$$ −22.6339 −0.781409 −0.390704 0.920516i $$-0.627769\pi$$
−0.390704 + 0.920516i $$0.627769\pi$$
$$840$$ 0 0
$$841$$ −20.6079 −0.710616
$$842$$ 0 0
$$843$$ 28.2253 0.972132
$$844$$ 0 0
$$845$$ −10.0096 −0.344340
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −18.4315 −0.632568
$$850$$ 0 0
$$851$$ −4.95377 −0.169813
$$852$$ 0 0
$$853$$ 28.2341 0.966716 0.483358 0.875423i $$-0.339417\pi$$
0.483358 + 0.875423i $$0.339417\pi$$
$$854$$ 0 0
$$855$$ 31.7572 1.08607
$$856$$ 0 0
$$857$$ −25.0539 −0.855826 −0.427913 0.903820i $$-0.640751\pi$$
−0.427913 + 0.903820i $$0.640751\pi$$
$$858$$ 0 0
$$859$$ −21.5510 −0.735311 −0.367656 0.929962i $$-0.619839\pi$$
−0.367656 + 0.929962i $$0.619839\pi$$
$$860$$ 0 0
$$861$$ 36.2253 1.23456
$$862$$ 0 0
$$863$$ 20.6831 0.704059 0.352030 0.935989i $$-0.385492\pi$$
0.352030 + 0.935989i $$0.385492\pi$$
$$864$$ 0 0
$$865$$ 7.52311 0.255794
$$866$$ 0 0
$$867$$ 43.6156 1.48126
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −22.4490 −0.760655
$$872$$ 0 0
$$873$$ 23.3815 0.791344
$$874$$ 0 0
$$875$$ 2.62620 0.0887817
$$876$$ 0 0
$$877$$ −57.6714 −1.94742 −0.973712 0.227782i $$-0.926853\pi$$
−0.973712 + 0.227782i $$0.926853\pi$$
$$878$$ 0 0
$$879$$ −56.4277 −1.90326
$$880$$ 0 0
$$881$$ −45.5144 −1.53342 −0.766709 0.641995i $$-0.778107\pi$$
−0.766709 + 0.641995i $$0.778107\pi$$
$$882$$ 0 0
$$883$$ 42.6820 1.43636 0.718182 0.695855i $$-0.244975\pi$$
0.718182 + 0.695855i $$0.244975\pi$$
$$884$$ 0 0
$$885$$ 18.5048 0.622032
$$886$$ 0 0
$$887$$ 40.5693 1.36219 0.681093 0.732197i $$-0.261505\pi$$
0.681093 + 0.732197i $$0.261505\pi$$
$$888$$ 0 0
$$889$$ −14.5048 −0.486475
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 2.20617 0.0738266
$$894$$ 0 0
$$895$$ −23.7572 −0.794115
$$896$$ 0 0
$$897$$ 25.0829 0.837493
$$898$$ 0 0
$$899$$ −19.9798 −0.666363
$$900$$ 0 0
$$901$$ −0.561647 −0.0187112
$$902$$ 0 0
$$903$$ −65.6801 −2.18570
$$904$$ 0 0
$$905$$ −18.2986 −0.608267
$$906$$ 0 0
$$907$$ −50.1406 −1.66489 −0.832445 0.554107i $$-0.813060\pi$$
−0.832445 + 0.554107i $$0.813060\pi$$
$$908$$ 0 0
$$909$$ −36.0558 −1.19590
$$910$$ 0 0
$$911$$ −32.5770 −1.07933 −0.539663 0.841881i $$-0.681448\pi$$
−0.539663 + 0.841881i $$0.681448\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ −24.6907 −0.816251
$$916$$ 0 0
$$917$$ 23.9065 0.789462
$$918$$ 0 0
$$919$$ −10.5048 −0.346521 −0.173261 0.984876i $$-0.555430\pi$$
−0.173261 + 0.984876i $$0.555430\pi$$
$$920$$ 0 0
$$921$$ −53.3815 −1.75898
$$922$$ 0 0
$$923$$ 23.1753 0.762825
$$924$$ 0 0
$$925$$ −0.896916 −0.0294904
$$926$$ 0 0
$$927$$ 32.2303 1.05858
$$928$$ 0 0
$$929$$ 34.5038 1.13203 0.566016 0.824394i $$-0.308484\pi$$
0.566016 + 0.824394i $$0.308484\pi$$
$$930$$ 0 0
$$931$$ −0.840061 −0.0275319
$$932$$ 0 0
$$933$$ −14.1860 −0.464427
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 8.06455 0.263457 0.131729 0.991286i $$-0.457947\pi$$
0.131729 + 0.991286i $$0.457947\pi$$
$$938$$ 0 0
$$939$$ −5.25240 −0.171405
$$940$$ 0 0
$$941$$ 8.81985 0.287519 0.143759 0.989613i $$-0.454081\pi$$
0.143759 + 0.989613i $$0.454081\pi$$
$$942$$ 0 0
$$943$$ −29.0096 −0.944682
$$944$$ 0 0
$$945$$ −6.18596 −0.201229
$$946$$ 0 0
$$947$$ 48.1772 1.56555 0.782775 0.622305i $$-0.213804\pi$$
0.782775 + 0.622305i $$0.213804\pi$$
$$948$$ 0 0
$$949$$ −19.9267 −0.646848
$$950$$ 0 0
$$951$$ −16.7870 −0.544355
$$952$$ 0 0
$$953$$ −53.6358 −1.73743 −0.868717 0.495309i $$-0.835055\pi$$
−0.868717 + 0.495309i $$0.835055\pi$$
$$954$$ 0 0
$$955$$ 12.0000 0.388311
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0.541436 0.0174839
$$960$$ 0 0
$$961$$ 16.5675 0.534434
$$962$$ 0 0
$$963$$ −57.5789 −1.85546
$$964$$ 0 0
$$965$$ 0.626198 0.0201580
$$966$$ 0 0
$$967$$ −32.1214 −1.03295 −0.516477 0.856301i $$-0.672757\pi$$
−0.516477 + 0.856301i $$0.672757\pi$$
$$968$$ 0 0
$$969$$ −13.4017 −0.430525
$$970$$ 0 0
$$971$$ 40.8959 1.31241 0.656206 0.754582i $$-0.272160\pi$$
0.656206 + 0.754582i $$0.272160\pi$$
$$972$$ 0 0
$$973$$ −57.6801 −1.84914
$$974$$ 0 0
$$975$$ 4.54144 0.145442
$$976$$ 0 0
$$977$$ −35.6801 −1.14151 −0.570754 0.821121i $$-0.693349\pi$$
−0.570754 + 0.821121i $$0.693349\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 65.6243 2.09522
$$982$$ 0 0
$$983$$ 28.9817 0.924372 0.462186 0.886783i $$-0.347065\pi$$
0.462186 + 0.886783i $$0.347065\pi$$
$$984$$ 0 0
$$985$$ −25.0741 −0.798928
$$986$$ 0 0
$$987$$ −1.86712 −0.0594311
$$988$$ 0 0
$$989$$ 52.5972 1.67250
$$990$$ 0 0
$$991$$ 15.5877 0.495159 0.247579 0.968868i $$-0.420365\pi$$
0.247579 + 0.968868i $$0.420365\pi$$
$$992$$ 0 0
$$993$$ −39.1753 −1.24319
$$994$$ 0 0
$$995$$ 8.14931 0.258351
$$996$$ 0 0
$$997$$ 19.1512 0.606525 0.303262 0.952907i $$-0.401924\pi$$
0.303262 + 0.952907i $$0.401924\pi$$
$$998$$ 0 0
$$999$$ 2.11267 0.0668419
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 9680.2.a.cg.1.1 3
4.3 odd 2 4840.2.a.r.1.3 yes 3
11.10 odd 2 9680.2.a.ch.1.1 3
44.43 even 2 4840.2.a.q.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
4840.2.a.q.1.3 3 44.43 even 2
4840.2.a.r.1.3 yes 3 4.3 odd 2
9680.2.a.cg.1.1 3 1.1 even 1 trivial
9680.2.a.ch.1.1 3 11.10 odd 2