# Properties

 Label 3332.2.a.n.1.2 Level $3332$ Weight $2$ Character 3332.1 Self dual yes Analytic conductor $26.606$ 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] = [3332,2,Mod(1,3332)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3332, 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("3332.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3332 = 2^{2} \cdot 7^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3332.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: $$26.6061539535$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 476) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.30278$$ of defining polynomial Character $$\chi$$ $$=$$ 3332.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.30278 q^{3} -0.302776 q^{5} +7.90833 q^{9} +O(q^{10})$$ $$q+3.30278 q^{3} -0.302776 q^{5} +7.90833 q^{9} -2.60555 q^{11} -0.605551 q^{13} -1.00000 q^{15} +1.00000 q^{17} +6.00000 q^{19} +4.60555 q^{23} -4.90833 q^{25} +16.2111 q^{27} -1.39445 q^{29} +10.3028 q^{31} -8.60555 q^{33} -7.21110 q^{37} -2.00000 q^{39} +8.51388 q^{41} -0.697224 q^{43} -2.39445 q^{45} -10.0000 q^{47} +3.30278 q^{51} +4.30278 q^{53} +0.788897 q^{55} +19.8167 q^{57} +9.21110 q^{59} +0.697224 q^{61} +0.183346 q^{65} +6.30278 q^{67} +15.2111 q^{69} -2.00000 q^{71} -4.51388 q^{73} -16.2111 q^{75} -6.00000 q^{79} +29.8167 q^{81} +5.21110 q^{83} -0.302776 q^{85} -4.60555 q^{87} +2.00000 q^{89} +34.0278 q^{93} -1.81665 q^{95} +12.9083 q^{97} -20.6056 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{3} + 3 q^{5} + 5 q^{9}+O(q^{10})$$ 2 * q + 3 * q^3 + 3 * q^5 + 5 * q^9 $$2 q + 3 q^{3} + 3 q^{5} + 5 q^{9} + 2 q^{11} + 6 q^{13} - 2 q^{15} + 2 q^{17} + 12 q^{19} + 2 q^{23} + q^{25} + 18 q^{27} - 10 q^{29} + 17 q^{31} - 10 q^{33} - 4 q^{39} - q^{41} - 5 q^{43} - 12 q^{45} - 20 q^{47} + 3 q^{51} + 5 q^{53} + 16 q^{55} + 18 q^{57} + 4 q^{59} + 5 q^{61} + 22 q^{65} + 9 q^{67} + 16 q^{69} - 4 q^{71} + 9 q^{73} - 18 q^{75} - 12 q^{79} + 38 q^{81} - 4 q^{83} + 3 q^{85} - 2 q^{87} + 4 q^{89} + 32 q^{93} + 18 q^{95} + 15 q^{97} - 34 q^{99}+O(q^{100})$$ 2 * q + 3 * q^3 + 3 * q^5 + 5 * q^9 + 2 * q^11 + 6 * q^13 - 2 * q^15 + 2 * q^17 + 12 * q^19 + 2 * q^23 + q^25 + 18 * q^27 - 10 * q^29 + 17 * q^31 - 10 * q^33 - 4 * q^39 - q^41 - 5 * q^43 - 12 * q^45 - 20 * q^47 + 3 * q^51 + 5 * q^53 + 16 * q^55 + 18 * q^57 + 4 * q^59 + 5 * q^61 + 22 * q^65 + 9 * q^67 + 16 * q^69 - 4 * q^71 + 9 * q^73 - 18 * q^75 - 12 * q^79 + 38 * q^81 - 4 * q^83 + 3 * q^85 - 2 * q^87 + 4 * q^89 + 32 * q^93 + 18 * q^95 + 15 * q^97 - 34 * q^99

## 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
$$3$$ 3.30278 1.90686 0.953429 0.301617i $$-0.0975264\pi$$
0.953429 + 0.301617i $$0.0975264\pi$$
$$4$$ 0 0
$$5$$ −0.302776 −0.135405 −0.0677027 0.997706i $$-0.521567\pi$$
−0.0677027 + 0.997706i $$0.521567\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 7.90833 2.63611
$$10$$ 0 0
$$11$$ −2.60555 −0.785603 −0.392802 0.919623i $$-0.628494\pi$$
−0.392802 + 0.919623i $$0.628494\pi$$
$$12$$ 0 0
$$13$$ −0.605551 −0.167950 −0.0839749 0.996468i $$-0.526762\pi$$
−0.0839749 + 0.996468i $$0.526762\pi$$
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ 0 0
$$17$$ 1.00000 0.242536
$$18$$ 0 0
$$19$$ 6.00000 1.37649 0.688247 0.725476i $$-0.258380\pi$$
0.688247 + 0.725476i $$0.258380\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 4.60555 0.960324 0.480162 0.877180i $$-0.340578\pi$$
0.480162 + 0.877180i $$0.340578\pi$$
$$24$$ 0 0
$$25$$ −4.90833 −0.981665
$$26$$ 0 0
$$27$$ 16.2111 3.11983
$$28$$ 0 0
$$29$$ −1.39445 −0.258943 −0.129471 0.991583i $$-0.541328\pi$$
−0.129471 + 0.991583i $$0.541328\pi$$
$$30$$ 0 0
$$31$$ 10.3028 1.85043 0.925217 0.379439i $$-0.123883\pi$$
0.925217 + 0.379439i $$0.123883\pi$$
$$32$$ 0 0
$$33$$ −8.60555 −1.49803
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −7.21110 −1.18550 −0.592749 0.805387i $$-0.701957\pi$$
−0.592749 + 0.805387i $$0.701957\pi$$
$$38$$ 0 0
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ 8.51388 1.32964 0.664822 0.747002i $$-0.268507\pi$$
0.664822 + 0.747002i $$0.268507\pi$$
$$42$$ 0 0
$$43$$ −0.697224 −0.106326 −0.0531629 0.998586i $$-0.516930\pi$$
−0.0531629 + 0.998586i $$0.516930\pi$$
$$44$$ 0 0
$$45$$ −2.39445 −0.356943
$$46$$ 0 0
$$47$$ −10.0000 −1.45865 −0.729325 0.684167i $$-0.760166\pi$$
−0.729325 + 0.684167i $$0.760166\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 3.30278 0.462481
$$52$$ 0 0
$$53$$ 4.30278 0.591032 0.295516 0.955338i $$-0.404508\pi$$
0.295516 + 0.955338i $$0.404508\pi$$
$$54$$ 0 0
$$55$$ 0.788897 0.106375
$$56$$ 0 0
$$57$$ 19.8167 2.62478
$$58$$ 0 0
$$59$$ 9.21110 1.19918 0.599592 0.800306i $$-0.295330\pi$$
0.599592 + 0.800306i $$0.295330\pi$$
$$60$$ 0 0
$$61$$ 0.697224 0.0892704 0.0446352 0.999003i $$-0.485787\pi$$
0.0446352 + 0.999003i $$0.485787\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0.183346 0.0227413
$$66$$ 0 0
$$67$$ 6.30278 0.770007 0.385003 0.922915i $$-0.374200\pi$$
0.385003 + 0.922915i $$0.374200\pi$$
$$68$$ 0 0
$$69$$ 15.2111 1.83120
$$70$$ 0 0
$$71$$ −2.00000 −0.237356 −0.118678 0.992933i $$-0.537866\pi$$
−0.118678 + 0.992933i $$0.537866\pi$$
$$72$$ 0 0
$$73$$ −4.51388 −0.528309 −0.264155 0.964480i $$-0.585093\pi$$
−0.264155 + 0.964480i $$0.585093\pi$$
$$74$$ 0 0
$$75$$ −16.2111 −1.87190
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −6.00000 −0.675053 −0.337526 0.941316i $$-0.609590\pi$$
−0.337526 + 0.941316i $$0.609590\pi$$
$$80$$ 0 0
$$81$$ 29.8167 3.31296
$$82$$ 0 0
$$83$$ 5.21110 0.571993 0.285996 0.958231i $$-0.407675\pi$$
0.285996 + 0.958231i $$0.407675\pi$$
$$84$$ 0 0
$$85$$ −0.302776 −0.0328406
$$86$$ 0 0
$$87$$ −4.60555 −0.493767
$$88$$ 0 0
$$89$$ 2.00000 0.212000 0.106000 0.994366i $$-0.466196\pi$$
0.106000 + 0.994366i $$0.466196\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 34.0278 3.52851
$$94$$ 0 0
$$95$$ −1.81665 −0.186385
$$96$$ 0 0
$$97$$ 12.9083 1.31064 0.655321 0.755350i $$-0.272533\pi$$
0.655321 + 0.755350i $$0.272533\pi$$
$$98$$ 0 0
$$99$$ −20.6056 −2.07094
$$100$$ 0 0
$$101$$ −16.4222 −1.63407 −0.817035 0.576588i $$-0.804384\pi$$
−0.817035 + 0.576588i $$0.804384\pi$$
$$102$$ 0 0
$$103$$ 1.21110 0.119333 0.0596667 0.998218i $$-0.480996\pi$$
0.0596667 + 0.998218i $$0.480996\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −16.6056 −1.60532 −0.802660 0.596437i $$-0.796582\pi$$
−0.802660 + 0.596437i $$0.796582\pi$$
$$108$$ 0 0
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 0 0
$$111$$ −23.8167 −2.26058
$$112$$ 0 0
$$113$$ 4.78890 0.450502 0.225251 0.974301i $$-0.427680\pi$$
0.225251 + 0.974301i $$0.427680\pi$$
$$114$$ 0 0
$$115$$ −1.39445 −0.130033
$$116$$ 0 0
$$117$$ −4.78890 −0.442734
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −4.21110 −0.382828
$$122$$ 0 0
$$123$$ 28.1194 2.53544
$$124$$ 0 0
$$125$$ 3.00000 0.268328
$$126$$ 0 0
$$127$$ 20.1194 1.78531 0.892655 0.450740i $$-0.148840\pi$$
0.892655 + 0.450740i $$0.148840\pi$$
$$128$$ 0 0
$$129$$ −2.30278 −0.202748
$$130$$ 0 0
$$131$$ −18.4222 −1.60956 −0.804778 0.593576i $$-0.797716\pi$$
−0.804778 + 0.593576i $$0.797716\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −4.90833 −0.422442
$$136$$ 0 0
$$137$$ −13.9083 −1.18827 −0.594134 0.804366i $$-0.702505\pi$$
−0.594134 + 0.804366i $$0.702505\pi$$
$$138$$ 0 0
$$139$$ 19.5139 1.65515 0.827573 0.561358i $$-0.189721\pi$$
0.827573 + 0.561358i $$0.189721\pi$$
$$140$$ 0 0
$$141$$ −33.0278 −2.78144
$$142$$ 0 0
$$143$$ 1.57779 0.131942
$$144$$ 0 0
$$145$$ 0.422205 0.0350622
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −5.90833 −0.484029 −0.242015 0.970273i $$-0.577808\pi$$
−0.242015 + 0.970273i $$0.577808\pi$$
$$150$$ 0 0
$$151$$ −11.1194 −0.904886 −0.452443 0.891793i $$-0.649447\pi$$
−0.452443 + 0.891793i $$0.649447\pi$$
$$152$$ 0 0
$$153$$ 7.90833 0.639350
$$154$$ 0 0
$$155$$ −3.11943 −0.250559
$$156$$ 0 0
$$157$$ 21.8167 1.74116 0.870579 0.492028i $$-0.163744\pi$$
0.870579 + 0.492028i $$0.163744\pi$$
$$158$$ 0 0
$$159$$ 14.2111 1.12701
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −0.183346 −0.0143608 −0.00718039 0.999974i $$-0.502286\pi$$
−0.00718039 + 0.999974i $$0.502286\pi$$
$$164$$ 0 0
$$165$$ 2.60555 0.202842
$$166$$ 0 0
$$167$$ −23.1194 −1.78904 −0.894518 0.447033i $$-0.852481\pi$$
−0.894518 + 0.447033i $$0.852481\pi$$
$$168$$ 0 0
$$169$$ −12.6333 −0.971793
$$170$$ 0 0
$$171$$ 47.4500 3.62859
$$172$$ 0 0
$$173$$ 5.90833 0.449202 0.224601 0.974451i $$-0.427892\pi$$
0.224601 + 0.974451i $$0.427892\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 30.4222 2.28667
$$178$$ 0 0
$$179$$ 21.9083 1.63751 0.818753 0.574146i $$-0.194666\pi$$
0.818753 + 0.574146i $$0.194666\pi$$
$$180$$ 0 0
$$181$$ −21.6333 −1.60799 −0.803996 0.594635i $$-0.797296\pi$$
−0.803996 + 0.594635i $$0.797296\pi$$
$$182$$ 0 0
$$183$$ 2.30278 0.170226
$$184$$ 0 0
$$185$$ 2.18335 0.160523
$$186$$ 0 0
$$187$$ −2.60555 −0.190537
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 8.72498 0.631317 0.315659 0.948873i $$-0.397775\pi$$
0.315659 + 0.948873i $$0.397775\pi$$
$$192$$ 0 0
$$193$$ −12.6056 −0.907367 −0.453684 0.891163i $$-0.649890\pi$$
−0.453684 + 0.891163i $$0.649890\pi$$
$$194$$ 0 0
$$195$$ 0.605551 0.0433644
$$196$$ 0 0
$$197$$ 13.0278 0.928189 0.464095 0.885786i $$-0.346380\pi$$
0.464095 + 0.885786i $$0.346380\pi$$
$$198$$ 0 0
$$199$$ 4.51388 0.319980 0.159990 0.987119i $$-0.448854\pi$$
0.159990 + 0.987119i $$0.448854\pi$$
$$200$$ 0 0
$$201$$ 20.8167 1.46829
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −2.57779 −0.180041
$$206$$ 0 0
$$207$$ 36.4222 2.53152
$$208$$ 0 0
$$209$$ −15.6333 −1.08138
$$210$$ 0 0
$$211$$ 9.02776 0.621496 0.310748 0.950492i $$-0.399420\pi$$
0.310748 + 0.950492i $$0.399420\pi$$
$$212$$ 0 0
$$213$$ −6.60555 −0.452605
$$214$$ 0 0
$$215$$ 0.211103 0.0143971
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −14.9083 −1.00741
$$220$$ 0 0
$$221$$ −0.605551 −0.0407338
$$222$$ 0 0
$$223$$ −21.6333 −1.44867 −0.724337 0.689446i $$-0.757854\pi$$
−0.724337 + 0.689446i $$0.757854\pi$$
$$224$$ 0 0
$$225$$ −38.8167 −2.58778
$$226$$ 0 0
$$227$$ −10.9083 −0.724011 −0.362006 0.932176i $$-0.617908\pi$$
−0.362006 + 0.932176i $$0.617908\pi$$
$$228$$ 0 0
$$229$$ −15.8167 −1.04519 −0.522597 0.852580i $$-0.675037\pi$$
−0.522597 + 0.852580i $$0.675037\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −1.21110 −0.0793420 −0.0396710 0.999213i $$-0.512631\pi$$
−0.0396710 + 0.999213i $$0.512631\pi$$
$$234$$ 0 0
$$235$$ 3.02776 0.197509
$$236$$ 0 0
$$237$$ −19.8167 −1.28723
$$238$$ 0 0
$$239$$ −8.69722 −0.562577 −0.281288 0.959623i $$-0.590762\pi$$
−0.281288 + 0.959623i $$0.590762\pi$$
$$240$$ 0 0
$$241$$ −3.09167 −0.199152 −0.0995761 0.995030i $$-0.531749\pi$$
−0.0995761 + 0.995030i $$0.531749\pi$$
$$242$$ 0 0
$$243$$ 49.8444 3.19752
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −3.63331 −0.231182
$$248$$ 0 0
$$249$$ 17.2111 1.09071
$$250$$ 0 0
$$251$$ −23.2111 −1.46507 −0.732536 0.680728i $$-0.761663\pi$$
−0.732536 + 0.680728i $$0.761663\pi$$
$$252$$ 0 0
$$253$$ −12.0000 −0.754434
$$254$$ 0 0
$$255$$ −1.00000 −0.0626224
$$256$$ 0 0
$$257$$ −7.81665 −0.487589 −0.243795 0.969827i $$-0.578392\pi$$
−0.243795 + 0.969827i $$0.578392\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −11.0278 −0.682601
$$262$$ 0 0
$$263$$ −6.42221 −0.396010 −0.198005 0.980201i $$-0.563446\pi$$
−0.198005 + 0.980201i $$0.563446\pi$$
$$264$$ 0 0
$$265$$ −1.30278 −0.0800289
$$266$$ 0 0
$$267$$ 6.60555 0.404253
$$268$$ 0 0
$$269$$ 6.00000 0.365826 0.182913 0.983129i $$-0.441447\pi$$
0.182913 + 0.983129i $$0.441447\pi$$
$$270$$ 0 0
$$271$$ 10.1833 0.618594 0.309297 0.950965i $$-0.399906\pi$$
0.309297 + 0.950965i $$0.399906\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 12.7889 0.771200
$$276$$ 0 0
$$277$$ −11.0278 −0.662594 −0.331297 0.943527i $$-0.607486\pi$$
−0.331297 + 0.943527i $$0.607486\pi$$
$$278$$ 0 0
$$279$$ 81.4777 4.87794
$$280$$ 0 0
$$281$$ −20.3028 −1.21116 −0.605581 0.795784i $$-0.707059\pi$$
−0.605581 + 0.795784i $$0.707059\pi$$
$$282$$ 0 0
$$283$$ −10.3028 −0.612436 −0.306218 0.951961i $$-0.599064\pi$$
−0.306218 + 0.951961i $$0.599064\pi$$
$$284$$ 0 0
$$285$$ −6.00000 −0.355409
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ 42.6333 2.49921
$$292$$ 0 0
$$293$$ 19.8167 1.15770 0.578851 0.815434i $$-0.303501\pi$$
0.578851 + 0.815434i $$0.303501\pi$$
$$294$$ 0 0
$$295$$ −2.78890 −0.162376
$$296$$ 0 0
$$297$$ −42.2389 −2.45095
$$298$$ 0 0
$$299$$ −2.78890 −0.161286
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −54.2389 −3.11594
$$304$$ 0 0
$$305$$ −0.211103 −0.0120877
$$306$$ 0 0
$$307$$ 26.0000 1.48390 0.741949 0.670456i $$-0.233902\pi$$
0.741949 + 0.670456i $$0.233902\pi$$
$$308$$ 0 0
$$309$$ 4.00000 0.227552
$$310$$ 0 0
$$311$$ −16.7250 −0.948387 −0.474193 0.880421i $$-0.657260\pi$$
−0.474193 + 0.880421i $$0.657260\pi$$
$$312$$ 0 0
$$313$$ −26.7250 −1.51059 −0.755293 0.655388i $$-0.772505\pi$$
−0.755293 + 0.655388i $$0.772505\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −5.39445 −0.302982 −0.151491 0.988459i $$-0.548408\pi$$
−0.151491 + 0.988459i $$0.548408\pi$$
$$318$$ 0 0
$$319$$ 3.63331 0.203426
$$320$$ 0 0
$$321$$ −54.8444 −3.06112
$$322$$ 0 0
$$323$$ 6.00000 0.333849
$$324$$ 0 0
$$325$$ 2.97224 0.164870
$$326$$ 0 0
$$327$$ −6.60555 −0.365288
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −32.7250 −1.79873 −0.899364 0.437201i $$-0.855970\pi$$
−0.899364 + 0.437201i $$0.855970\pi$$
$$332$$ 0 0
$$333$$ −57.0278 −3.12510
$$334$$ 0 0
$$335$$ −1.90833 −0.104263
$$336$$ 0 0
$$337$$ 25.2111 1.37334 0.686668 0.726971i $$-0.259073\pi$$
0.686668 + 0.726971i $$0.259073\pi$$
$$338$$ 0 0
$$339$$ 15.8167 0.859043
$$340$$ 0 0
$$341$$ −26.8444 −1.45371
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −4.60555 −0.247955
$$346$$ 0 0
$$347$$ 6.00000 0.322097 0.161048 0.986947i $$-0.448512\pi$$
0.161048 + 0.986947i $$0.448512\pi$$
$$348$$ 0 0
$$349$$ −4.78890 −0.256344 −0.128172 0.991752i $$-0.540911\pi$$
−0.128172 + 0.991752i $$0.540911\pi$$
$$350$$ 0 0
$$351$$ −9.81665 −0.523974
$$352$$ 0 0
$$353$$ 2.60555 0.138680 0.0693398 0.997593i $$-0.477911\pi$$
0.0693398 + 0.997593i $$0.477911\pi$$
$$354$$ 0 0
$$355$$ 0.605551 0.0321393
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −35.5139 −1.87435 −0.937175 0.348859i $$-0.886569\pi$$
−0.937175 + 0.348859i $$0.886569\pi$$
$$360$$ 0 0
$$361$$ 17.0000 0.894737
$$362$$ 0 0
$$363$$ −13.9083 −0.729998
$$364$$ 0 0
$$365$$ 1.36669 0.0715359
$$366$$ 0 0
$$367$$ 5.72498 0.298842 0.149421 0.988774i $$-0.452259\pi$$
0.149421 + 0.988774i $$0.452259\pi$$
$$368$$ 0 0
$$369$$ 67.3305 3.50509
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 8.69722 0.450325 0.225163 0.974321i $$-0.427709\pi$$
0.225163 + 0.974321i $$0.427709\pi$$
$$374$$ 0 0
$$375$$ 9.90833 0.511664
$$376$$ 0 0
$$377$$ 0.844410 0.0434893
$$378$$ 0 0
$$379$$ 22.2389 1.14233 0.571167 0.820834i $$-0.306491\pi$$
0.571167 + 0.820834i $$0.306491\pi$$
$$380$$ 0 0
$$381$$ 66.4500 3.40433
$$382$$ 0 0
$$383$$ −33.0278 −1.68764 −0.843820 0.536627i $$-0.819698\pi$$
−0.843820 + 0.536627i $$0.819698\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −5.51388 −0.280286
$$388$$ 0 0
$$389$$ −24.9083 −1.26290 −0.631451 0.775416i $$-0.717540\pi$$
−0.631451 + 0.775416i $$0.717540\pi$$
$$390$$ 0 0
$$391$$ 4.60555 0.232913
$$392$$ 0 0
$$393$$ −60.8444 −3.06919
$$394$$ 0 0
$$395$$ 1.81665 0.0914058
$$396$$ 0 0
$$397$$ 16.9083 0.848605 0.424302 0.905521i $$-0.360519\pi$$
0.424302 + 0.905521i $$0.360519\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 24.0000 1.19850 0.599251 0.800561i $$-0.295465\pi$$
0.599251 + 0.800561i $$0.295465\pi$$
$$402$$ 0 0
$$403$$ −6.23886 −0.310780
$$404$$ 0 0
$$405$$ −9.02776 −0.448593
$$406$$ 0 0
$$407$$ 18.7889 0.931331
$$408$$ 0 0
$$409$$ −33.0278 −1.63312 −0.816559 0.577262i $$-0.804121\pi$$
−0.816559 + 0.577262i $$0.804121\pi$$
$$410$$ 0 0
$$411$$ −45.9361 −2.26586
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −1.57779 −0.0774509
$$416$$ 0 0
$$417$$ 64.4500 3.15613
$$418$$ 0 0
$$419$$ −2.30278 −0.112498 −0.0562490 0.998417i $$-0.517914\pi$$
−0.0562490 + 0.998417i $$0.517914\pi$$
$$420$$ 0 0
$$421$$ 9.90833 0.482902 0.241451 0.970413i $$-0.422377\pi$$
0.241451 + 0.970413i $$0.422377\pi$$
$$422$$ 0 0
$$423$$ −79.0833 −3.84516
$$424$$ 0 0
$$425$$ −4.90833 −0.238089
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 5.21110 0.251594
$$430$$ 0 0
$$431$$ −38.8444 −1.87107 −0.935535 0.353235i $$-0.885082\pi$$
−0.935535 + 0.353235i $$0.885082\pi$$
$$432$$ 0 0
$$433$$ 31.6333 1.52020 0.760100 0.649806i $$-0.225150\pi$$
0.760100 + 0.649806i $$0.225150\pi$$
$$434$$ 0 0
$$435$$ 1.39445 0.0668587
$$436$$ 0 0
$$437$$ 27.6333 1.32188
$$438$$ 0 0
$$439$$ −3.30278 −0.157633 −0.0788164 0.996889i $$-0.525114\pi$$
−0.0788164 + 0.996889i $$0.525114\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 18.4222 0.875265 0.437633 0.899154i $$-0.355817\pi$$
0.437633 + 0.899154i $$0.355817\pi$$
$$444$$ 0 0
$$445$$ −0.605551 −0.0287059
$$446$$ 0 0
$$447$$ −19.5139 −0.922975
$$448$$ 0 0
$$449$$ 18.2389 0.860745 0.430372 0.902651i $$-0.358382\pi$$
0.430372 + 0.902651i $$0.358382\pi$$
$$450$$ 0 0
$$451$$ −22.1833 −1.04457
$$452$$ 0 0
$$453$$ −36.7250 −1.72549
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 16.9083 0.790938 0.395469 0.918479i $$-0.370582\pi$$
0.395469 + 0.918479i $$0.370582\pi$$
$$458$$ 0 0
$$459$$ 16.2111 0.756669
$$460$$ 0 0
$$461$$ 5.21110 0.242705 0.121353 0.992609i $$-0.461277\pi$$
0.121353 + 0.992609i $$0.461277\pi$$
$$462$$ 0 0
$$463$$ −17.5416 −0.815229 −0.407614 0.913154i $$-0.633639\pi$$
−0.407614 + 0.913154i $$0.633639\pi$$
$$464$$ 0 0
$$465$$ −10.3028 −0.477780
$$466$$ 0 0
$$467$$ −2.78890 −0.129055 −0.0645274 0.997916i $$-0.520554\pi$$
−0.0645274 + 0.997916i $$0.520554\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 72.0555 3.32014
$$472$$ 0 0
$$473$$ 1.81665 0.0835298
$$474$$ 0 0
$$475$$ −29.4500 −1.35126
$$476$$ 0 0
$$477$$ 34.0278 1.55802
$$478$$ 0 0
$$479$$ 20.6972 0.945680 0.472840 0.881148i $$-0.343229\pi$$
0.472840 + 0.881148i $$0.343229\pi$$
$$480$$ 0 0
$$481$$ 4.36669 0.199104
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −3.90833 −0.177468
$$486$$ 0 0
$$487$$ −29.6333 −1.34281 −0.671407 0.741089i $$-0.734310\pi$$
−0.671407 + 0.741089i $$0.734310\pi$$
$$488$$ 0 0
$$489$$ −0.605551 −0.0273840
$$490$$ 0 0
$$491$$ 34.3028 1.54806 0.774031 0.633147i $$-0.218237\pi$$
0.774031 + 0.633147i $$0.218237\pi$$
$$492$$ 0 0
$$493$$ −1.39445 −0.0628028
$$494$$ 0 0
$$495$$ 6.23886 0.280416
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −6.00000 −0.268597 −0.134298 0.990941i $$-0.542878\pi$$
−0.134298 + 0.990941i $$0.542878\pi$$
$$500$$ 0 0
$$501$$ −76.3583 −3.41144
$$502$$ 0 0
$$503$$ −36.1194 −1.61049 −0.805243 0.592945i $$-0.797965\pi$$
−0.805243 + 0.592945i $$0.797965\pi$$
$$504$$ 0 0
$$505$$ 4.97224 0.221262
$$506$$ 0 0
$$507$$ −41.7250 −1.85307
$$508$$ 0 0
$$509$$ −10.6056 −0.470083 −0.235041 0.971985i $$-0.575523\pi$$
−0.235041 + 0.971985i $$0.575523\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 97.2666 4.29443
$$514$$ 0 0
$$515$$ −0.366692 −0.0161584
$$516$$ 0 0
$$517$$ 26.0555 1.14592
$$518$$ 0 0
$$519$$ 19.5139 0.856564
$$520$$ 0 0
$$521$$ 32.3305 1.41643 0.708213 0.705999i $$-0.249502\pi$$
0.708213 + 0.705999i $$0.249502\pi$$
$$522$$ 0 0
$$523$$ 19.0278 0.832026 0.416013 0.909359i $$-0.363427\pi$$
0.416013 + 0.909359i $$0.363427\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 10.3028 0.448796
$$528$$ 0 0
$$529$$ −1.78890 −0.0777781
$$530$$ 0 0
$$531$$ 72.8444 3.16118
$$532$$ 0 0
$$533$$ −5.15559 −0.223313
$$534$$ 0 0
$$535$$ 5.02776 0.217369
$$536$$ 0 0
$$537$$ 72.3583 3.12249
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −17.8167 −0.765998 −0.382999 0.923749i $$-0.625109\pi$$
−0.382999 + 0.923749i $$0.625109\pi$$
$$542$$ 0 0
$$543$$ −71.4500 −3.06621
$$544$$ 0 0
$$545$$ 0.605551 0.0259390
$$546$$ 0 0
$$547$$ 25.4500 1.08816 0.544081 0.839033i $$-0.316878\pi$$
0.544081 + 0.839033i $$0.316878\pi$$
$$548$$ 0 0
$$549$$ 5.51388 0.235327
$$550$$ 0 0
$$551$$ −8.36669 −0.356433
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 7.21110 0.306094
$$556$$ 0 0
$$557$$ −3.21110 −0.136059 −0.0680294 0.997683i $$-0.521671\pi$$
−0.0680294 + 0.997683i $$0.521671\pi$$
$$558$$ 0 0
$$559$$ 0.422205 0.0178574
$$560$$ 0 0
$$561$$ −8.60555 −0.363327
$$562$$ 0 0
$$563$$ 18.2389 0.768676 0.384338 0.923192i $$-0.374430\pi$$
0.384338 + 0.923192i $$0.374430\pi$$
$$564$$ 0 0
$$565$$ −1.44996 −0.0610003
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −23.4861 −0.984589 −0.492295 0.870429i $$-0.663842\pi$$
−0.492295 + 0.870429i $$0.663842\pi$$
$$570$$ 0 0
$$571$$ 9.63331 0.403141 0.201571 0.979474i $$-0.435395\pi$$
0.201571 + 0.979474i $$0.435395\pi$$
$$572$$ 0 0
$$573$$ 28.8167 1.20383
$$574$$ 0 0
$$575$$ −22.6056 −0.942717
$$576$$ 0 0
$$577$$ −2.97224 −0.123736 −0.0618681 0.998084i $$-0.519706\pi$$
−0.0618681 + 0.998084i $$0.519706\pi$$
$$578$$ 0 0
$$579$$ −41.6333 −1.73022
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −11.2111 −0.464316
$$584$$ 0 0
$$585$$ 1.44996 0.0599485
$$586$$ 0 0
$$587$$ 2.78890 0.115110 0.0575551 0.998342i $$-0.481670\pi$$
0.0575551 + 0.998342i $$0.481670\pi$$
$$588$$ 0 0
$$589$$ 61.8167 2.54711
$$590$$ 0 0
$$591$$ 43.0278 1.76993
$$592$$ 0 0
$$593$$ 33.6333 1.38115 0.690577 0.723259i $$-0.257357\pi$$
0.690577 + 0.723259i $$0.257357\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 14.9083 0.610157
$$598$$ 0 0
$$599$$ 21.5416 0.880167 0.440084 0.897957i $$-0.354949\pi$$
0.440084 + 0.897957i $$0.354949\pi$$
$$600$$ 0 0
$$601$$ −4.78890 −0.195343 −0.0976716 0.995219i $$-0.531139\pi$$
−0.0976716 + 0.995219i $$0.531139\pi$$
$$602$$ 0 0
$$603$$ 49.8444 2.02982
$$604$$ 0 0
$$605$$ 1.27502 0.0518369
$$606$$ 0 0
$$607$$ 26.9083 1.09218 0.546088 0.837728i $$-0.316117\pi$$
0.546088 + 0.837728i $$0.316117\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 6.05551 0.244980
$$612$$ 0 0
$$613$$ 15.9361 0.643652 0.321826 0.946799i $$-0.395703\pi$$
0.321826 + 0.946799i $$0.395703\pi$$
$$614$$ 0 0
$$615$$ −8.51388 −0.343313
$$616$$ 0 0
$$617$$ −2.60555 −0.104896 −0.0524478 0.998624i $$-0.516702\pi$$
−0.0524478 + 0.998624i $$0.516702\pi$$
$$618$$ 0 0
$$619$$ 8.00000 0.321547 0.160774 0.986991i $$-0.448601\pi$$
0.160774 + 0.986991i $$0.448601\pi$$
$$620$$ 0 0
$$621$$ 74.6611 2.99605
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 23.6333 0.945332
$$626$$ 0 0
$$627$$ −51.6333 −2.06204
$$628$$ 0 0
$$629$$ −7.21110 −0.287525
$$630$$ 0 0
$$631$$ −12.1194 −0.482467 −0.241233 0.970467i $$-0.577552\pi$$
−0.241233 + 0.970467i $$0.577552\pi$$
$$632$$ 0 0
$$633$$ 29.8167 1.18511
$$634$$ 0 0
$$635$$ −6.09167 −0.241741
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −15.8167 −0.625697
$$640$$ 0 0
$$641$$ −36.4222 −1.43859 −0.719295 0.694704i $$-0.755535\pi$$
−0.719295 + 0.694704i $$0.755535\pi$$
$$642$$ 0 0
$$643$$ 12.3305 0.486269 0.243134 0.969993i $$-0.421824\pi$$
0.243134 + 0.969993i $$0.421824\pi$$
$$644$$ 0 0
$$645$$ 0.697224 0.0274532
$$646$$ 0 0
$$647$$ 34.6056 1.36048 0.680242 0.732987i $$-0.261875\pi$$
0.680242 + 0.732987i $$0.261875\pi$$
$$648$$ 0 0
$$649$$ −24.0000 −0.942082
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 2.00000 0.0782660 0.0391330 0.999234i $$-0.487540\pi$$
0.0391330 + 0.999234i $$0.487540\pi$$
$$654$$ 0 0
$$655$$ 5.57779 0.217942
$$656$$ 0 0
$$657$$ −35.6972 −1.39268
$$658$$ 0 0
$$659$$ −46.5416 −1.81300 −0.906502 0.422201i $$-0.861258\pi$$
−0.906502 + 0.422201i $$0.861258\pi$$
$$660$$ 0 0
$$661$$ 27.6333 1.07481 0.537406 0.843324i $$-0.319404\pi$$
0.537406 + 0.843324i $$0.319404\pi$$
$$662$$ 0 0
$$663$$ −2.00000 −0.0776736
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −6.42221 −0.248669
$$668$$ 0 0
$$669$$ −71.4500 −2.76242
$$670$$ 0 0
$$671$$ −1.81665 −0.0701311
$$672$$ 0 0
$$673$$ 36.8444 1.42025 0.710124 0.704077i $$-0.248639\pi$$
0.710124 + 0.704077i $$0.248639\pi$$
$$674$$ 0 0
$$675$$ −79.5694 −3.06263
$$676$$ 0 0
$$677$$ −22.0000 −0.845529 −0.422764 0.906240i $$-0.638940\pi$$
−0.422764 + 0.906240i $$0.638940\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −36.0278 −1.38059
$$682$$ 0 0
$$683$$ 25.0278 0.957660 0.478830 0.877908i $$-0.341061\pi$$
0.478830 + 0.877908i $$0.341061\pi$$
$$684$$ 0 0
$$685$$ 4.21110 0.160898
$$686$$ 0 0
$$687$$ −52.2389 −1.99304
$$688$$ 0 0
$$689$$ −2.60555 −0.0992636
$$690$$ 0 0
$$691$$ −0.697224 −0.0265237 −0.0132618 0.999912i $$-0.504221\pi$$
−0.0132618 + 0.999912i $$0.504221\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −5.90833 −0.224116
$$696$$ 0 0
$$697$$ 8.51388 0.322486
$$698$$ 0 0
$$699$$ −4.00000 −0.151294
$$700$$ 0 0
$$701$$ 3.21110 0.121282 0.0606408 0.998160i $$-0.480686\pi$$
0.0606408 + 0.998160i $$0.480686\pi$$
$$702$$ 0 0
$$703$$ −43.2666 −1.63183
$$704$$ 0 0
$$705$$ 10.0000 0.376622
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −45.8167 −1.72068 −0.860340 0.509720i $$-0.829749\pi$$
−0.860340 + 0.509720i $$0.829749\pi$$
$$710$$ 0 0
$$711$$ −47.4500 −1.77951
$$712$$ 0 0
$$713$$ 47.4500 1.77702
$$714$$ 0 0
$$715$$ −0.477718 −0.0178656
$$716$$ 0 0
$$717$$ −28.7250 −1.07275
$$718$$ 0 0
$$719$$ 20.5139 0.765039 0.382519 0.923948i $$-0.375057\pi$$
0.382519 + 0.923948i $$0.375057\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −10.2111 −0.379755
$$724$$ 0 0
$$725$$ 6.84441 0.254195
$$726$$ 0 0
$$727$$ 37.2666 1.38214 0.691071 0.722787i $$-0.257139\pi$$
0.691071 + 0.722787i $$0.257139\pi$$
$$728$$ 0 0
$$729$$ 75.1749 2.78426
$$730$$ 0 0
$$731$$ −0.697224 −0.0257878
$$732$$ 0 0
$$733$$ −27.6333 −1.02066 −0.510330 0.859979i $$-0.670477\pi$$
−0.510330 + 0.859979i $$0.670477\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −16.4222 −0.604920
$$738$$ 0 0
$$739$$ −14.0917 −0.518371 −0.259185 0.965828i $$-0.583454\pi$$
−0.259185 + 0.965828i $$0.583454\pi$$
$$740$$ 0 0
$$741$$ −12.0000 −0.440831
$$742$$ 0 0
$$743$$ −40.6056 −1.48967 −0.744837 0.667247i $$-0.767473\pi$$
−0.744837 + 0.667247i $$0.767473\pi$$
$$744$$ 0 0
$$745$$ 1.78890 0.0655401
$$746$$ 0 0
$$747$$ 41.2111 1.50784
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −36.7889 −1.34245 −0.671223 0.741256i $$-0.734231\pi$$
−0.671223 + 0.741256i $$0.734231\pi$$
$$752$$ 0 0
$$753$$ −76.6611 −2.79368
$$754$$ 0 0
$$755$$ 3.36669 0.122526
$$756$$ 0 0
$$757$$ 33.9083 1.23242 0.616210 0.787582i $$-0.288667\pi$$
0.616210 + 0.787582i $$0.288667\pi$$
$$758$$ 0 0
$$759$$ −39.6333 −1.43860
$$760$$ 0 0
$$761$$ −39.3944 −1.42805 −0.714024 0.700121i $$-0.753129\pi$$
−0.714024 + 0.700121i $$0.753129\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −2.39445 −0.0865715
$$766$$ 0 0
$$767$$ −5.57779 −0.201403
$$768$$ 0 0
$$769$$ 9.81665 0.353998 0.176999 0.984211i $$-0.443361\pi$$
0.176999 + 0.984211i $$0.443361\pi$$
$$770$$ 0 0
$$771$$ −25.8167 −0.929764
$$772$$ 0 0
$$773$$ −50.8444 −1.82875 −0.914373 0.404872i $$-0.867316\pi$$
−0.914373 + 0.404872i $$0.867316\pi$$
$$774$$ 0 0
$$775$$ −50.5694 −1.81651
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 51.0833 1.83025
$$780$$ 0 0
$$781$$ 5.21110 0.186468
$$782$$ 0 0
$$783$$ −22.6056 −0.807856
$$784$$ 0 0
$$785$$ −6.60555 −0.235762
$$786$$ 0 0
$$787$$ −14.4222 −0.514096 −0.257048 0.966399i $$-0.582750\pi$$
−0.257048 + 0.966399i $$0.582750\pi$$
$$788$$ 0 0
$$789$$ −21.2111 −0.755135
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −0.422205 −0.0149929
$$794$$ 0 0
$$795$$ −4.30278 −0.152604
$$796$$ 0 0
$$797$$ −29.2111 −1.03471 −0.517355 0.855771i $$-0.673083\pi$$
−0.517355 + 0.855771i $$0.673083\pi$$
$$798$$ 0 0
$$799$$ −10.0000 −0.353775
$$800$$ 0 0
$$801$$ 15.8167 0.558854
$$802$$ 0 0
$$803$$ 11.7611 0.415042
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 19.8167 0.697579
$$808$$ 0 0
$$809$$ −19.4500 −0.683824 −0.341912 0.939732i $$-0.611075\pi$$
−0.341912 + 0.939732i $$0.611075\pi$$
$$810$$ 0 0
$$811$$ −37.3583 −1.31183 −0.655913 0.754836i $$-0.727716\pi$$
−0.655913 + 0.754836i $$0.727716\pi$$
$$812$$ 0 0
$$813$$ 33.6333 1.17957
$$814$$ 0 0
$$815$$ 0.0555128 0.00194453
$$816$$ 0 0
$$817$$ −4.18335 −0.146357
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 11.5778 0.404068 0.202034 0.979379i $$-0.435245\pi$$
0.202034 + 0.979379i $$0.435245\pi$$
$$822$$ 0 0
$$823$$ 31.0278 1.08156 0.540780 0.841164i $$-0.318129\pi$$
0.540780 + 0.841164i $$0.318129\pi$$
$$824$$ 0 0
$$825$$ 42.2389 1.47057
$$826$$ 0 0
$$827$$ −38.8444 −1.35075 −0.675376 0.737473i $$-0.736019\pi$$
−0.675376 + 0.737473i $$0.736019\pi$$
$$828$$ 0 0
$$829$$ 36.6056 1.27136 0.635682 0.771951i $$-0.280719\pi$$
0.635682 + 0.771951i $$0.280719\pi$$
$$830$$ 0 0
$$831$$ −36.4222 −1.26347
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 7.00000 0.242245
$$836$$ 0 0
$$837$$ 167.019 5.77303
$$838$$ 0 0
$$839$$ −6.78890 −0.234379 −0.117189 0.993110i $$-0.537388\pi$$
−0.117189 + 0.993110i $$0.537388\pi$$
$$840$$ 0 0
$$841$$ −27.0555 −0.932949
$$842$$ 0 0
$$843$$ −67.0555 −2.30951
$$844$$ 0 0
$$845$$ 3.82506 0.131586
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −34.0278 −1.16783
$$850$$ 0 0
$$851$$ −33.2111 −1.13846
$$852$$ 0 0
$$853$$ 4.78890 0.163969 0.0819844 0.996634i $$-0.473874\pi$$
0.0819844 + 0.996634i $$0.473874\pi$$
$$854$$ 0 0
$$855$$ −14.3667 −0.491331
$$856$$ 0 0
$$857$$ 5.51388 0.188350 0.0941752 0.995556i $$-0.469979\pi$$
0.0941752 + 0.995556i $$0.469979\pi$$
$$858$$ 0 0
$$859$$ 2.36669 0.0807505 0.0403753 0.999185i $$-0.487145\pi$$
0.0403753 + 0.999185i $$0.487145\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −37.9361 −1.29136 −0.645680 0.763608i $$-0.723426\pi$$
−0.645680 + 0.763608i $$0.723426\pi$$
$$864$$ 0 0
$$865$$ −1.78890 −0.0608243
$$866$$ 0 0
$$867$$ 3.30278 0.112168
$$868$$ 0 0
$$869$$ 15.6333 0.530324
$$870$$ 0 0
$$871$$ −3.81665 −0.129322
$$872$$ 0 0
$$873$$ 102.083 3.45500
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −27.8167 −0.939302 −0.469651 0.882852i $$-0.655620\pi$$
−0.469651 + 0.882852i $$0.655620\pi$$
$$878$$ 0 0
$$879$$ 65.4500 2.20757
$$880$$ 0 0
$$881$$ −18.1194 −0.610459 −0.305230 0.952279i $$-0.598733\pi$$
−0.305230 + 0.952279i $$0.598733\pi$$
$$882$$ 0 0
$$883$$ 23.2750 0.783267 0.391633 0.920121i $$-0.371910\pi$$
0.391633 + 0.920121i $$0.371910\pi$$
$$884$$ 0 0
$$885$$ −9.21110 −0.309628
$$886$$ 0 0
$$887$$ 44.7250 1.50172 0.750859 0.660463i $$-0.229640\pi$$
0.750859 + 0.660463i $$0.229640\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −77.6888 −2.60267
$$892$$ 0 0
$$893$$ −60.0000 −2.00782
$$894$$ 0 0
$$895$$ −6.63331 −0.221727
$$896$$ 0 0
$$897$$ −9.21110 −0.307550
$$898$$ 0 0
$$899$$ −14.3667 −0.479156
$$900$$ 0 0
$$901$$ 4.30278 0.143346
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 6.55004 0.217731
$$906$$ 0 0
$$907$$ −32.4222 −1.07656 −0.538281 0.842766i $$-0.680926\pi$$
−0.538281 + 0.842766i $$0.680926\pi$$
$$908$$ 0 0
$$909$$ −129.872 −4.30759
$$910$$ 0 0
$$911$$ 18.4222 0.610355 0.305177 0.952296i $$-0.401284\pi$$
0.305177 + 0.952296i $$0.401284\pi$$
$$912$$ 0 0
$$913$$ −13.5778 −0.449359
$$914$$ 0 0
$$915$$ −0.697224 −0.0230495
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −28.5139 −0.940586 −0.470293 0.882510i $$-0.655852\pi$$
−0.470293 + 0.882510i $$0.655852\pi$$
$$920$$ 0 0
$$921$$ 85.8722 2.82958
$$922$$ 0 0
$$923$$ 1.21110 0.0398639
$$924$$ 0 0
$$925$$ 35.3944 1.16376
$$926$$ 0 0
$$927$$ 9.57779 0.314576
$$928$$ 0 0
$$929$$ 28.3028 0.928584 0.464292 0.885682i $$-0.346309\pi$$
0.464292 + 0.885682i $$0.346309\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −55.2389 −1.80844
$$934$$ 0 0
$$935$$ 0.788897 0.0257997
$$936$$ 0 0
$$937$$ 10.7889 0.352458 0.176229 0.984349i $$-0.443610\pi$$
0.176229 + 0.984349i $$0.443610\pi$$
$$938$$ 0 0
$$939$$ −88.2666 −2.88047
$$940$$ 0 0
$$941$$ 40.3305 1.31474 0.657369 0.753569i $$-0.271669\pi$$
0.657369 + 0.753569i $$0.271669\pi$$
$$942$$ 0 0
$$943$$ 39.2111 1.27689
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −30.2389 −0.982631 −0.491315 0.870982i $$-0.663484\pi$$
−0.491315 + 0.870982i $$0.663484\pi$$
$$948$$ 0 0
$$949$$ 2.73338 0.0887294
$$950$$ 0 0
$$951$$ −17.8167 −0.577745
$$952$$ 0 0
$$953$$ −41.7527 −1.35250 −0.676252 0.736670i $$-0.736397\pi$$
−0.676252 + 0.736670i $$0.736397\pi$$
$$954$$ 0 0
$$955$$ −2.64171 −0.0854838
$$956$$ 0 0
$$957$$ 12.0000 0.387905
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 75.1472 2.42410
$$962$$ 0 0
$$963$$ −131.322 −4.23180
$$964$$ 0 0
$$965$$ 3.81665 0.122862
$$966$$ 0 0
$$967$$ −16.6972 −0.536947 −0.268473 0.963287i $$-0.586519\pi$$
−0.268473 + 0.963287i $$0.586519\pi$$
$$968$$ 0 0
$$969$$ 19.8167 0.636603
$$970$$ 0 0
$$971$$ −30.2389 −0.970411 −0.485206 0.874400i $$-0.661255\pi$$
−0.485206 + 0.874400i $$0.661255\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 9.81665 0.314385
$$976$$ 0 0
$$977$$ −5.51388 −0.176405 −0.0882023 0.996103i $$-0.528112\pi$$
−0.0882023 + 0.996103i $$0.528112\pi$$
$$978$$ 0 0
$$979$$ −5.21110 −0.166548
$$980$$ 0 0
$$981$$ −15.8167 −0.504987
$$982$$ 0 0
$$983$$ 3.51388 0.112075 0.0560377 0.998429i $$-0.482153\pi$$
0.0560377 + 0.998429i $$0.482153\pi$$
$$984$$ 0 0
$$985$$ −3.94449 −0.125682
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −3.21110 −0.102107
$$990$$ 0 0
$$991$$ 6.00000 0.190596 0.0952981 0.995449i $$-0.469620\pi$$
0.0952981 + 0.995449i $$0.469620\pi$$
$$992$$ 0 0
$$993$$ −108.083 −3.42992
$$994$$ 0 0
$$995$$ −1.36669 −0.0433271
$$996$$ 0 0
$$997$$ −22.0917 −0.699650 −0.349825 0.936815i $$-0.613759\pi$$
−0.349825 + 0.936815i $$0.613759\pi$$
$$998$$ 0 0
$$999$$ −116.900 −3.69855
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 3332.2.a.n.1.2 2
7.6 odd 2 476.2.a.a.1.1 2
21.20 even 2 4284.2.a.p.1.1 2
28.27 even 2 1904.2.a.l.1.2 2
56.13 odd 2 7616.2.a.z.1.2 2
56.27 even 2 7616.2.a.m.1.1 2
119.118 odd 2 8092.2.a.n.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
476.2.a.a.1.1 2 7.6 odd 2
1904.2.a.l.1.2 2 28.27 even 2
3332.2.a.n.1.2 2 1.1 even 1 trivial
4284.2.a.p.1.1 2 21.20 even 2
7616.2.a.m.1.1 2 56.27 even 2
7616.2.a.z.1.2 2 56.13 odd 2
8092.2.a.n.1.2 2 119.118 odd 2