# Properties

 Label 476.2.a.a.1.1 Level $476$ Weight $2$ Character 476.1 Self dual yes Analytic conductor $3.801$ Analytic rank $1$ 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] = [476,2,Mod(1,476)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(476, 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("476.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$476 = 2^{2} \cdot 7 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 476.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: $$3.80087913621$$ Analytic rank: $$1$$ 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: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$2.30278$$ of defining polynomial Character $$\chi$$ $$=$$ 476.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.30278 q^{3} +0.302776 q^{5} +1.00000 q^{7} +7.90833 q^{9} +O(q^{10})$$ $$q-3.30278 q^{3} +0.302776 q^{5} +1.00000 q^{7} +7.90833 q^{9} -2.60555 q^{11} +0.605551 q^{13} -1.00000 q^{15} -1.00000 q^{17} -6.00000 q^{19} -3.30278 q^{21} +4.60555 q^{23} -4.90833 q^{25} -16.2111 q^{27} -1.39445 q^{29} -10.3028 q^{31} +8.60555 q^{33} +0.302776 q^{35} -7.21110 q^{37} -2.00000 q^{39} -8.51388 q^{41} -0.697224 q^{43} +2.39445 q^{45} +10.0000 q^{47} +1.00000 q^{49} +3.30278 q^{51} +4.30278 q^{53} -0.788897 q^{55} +19.8167 q^{57} -9.21110 q^{59} -0.697224 q^{61} +7.90833 q^{63} +0.183346 q^{65} +6.30278 q^{67} -15.2111 q^{69} -2.00000 q^{71} +4.51388 q^{73} +16.2111 q^{75} -2.60555 q^{77} -6.00000 q^{79} +29.8167 q^{81} -5.21110 q^{83} -0.302776 q^{85} +4.60555 q^{87} -2.00000 q^{89} +0.605551 q^{91} +34.0278 q^{93} -1.81665 q^{95} -12.9083 q^{97} -20.6056 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{3} - 3 q^{5} + 2 q^{7} + 5 q^{9}+O(q^{10})$$ 2 * q - 3 * q^3 - 3 * q^5 + 2 * q^7 + 5 * q^9 $$2 q - 3 q^{3} - 3 q^{5} + 2 q^{7} + 5 q^{9} + 2 q^{11} - 6 q^{13} - 2 q^{15} - 2 q^{17} - 12 q^{19} - 3 q^{21} + 2 q^{23} + q^{25} - 18 q^{27} - 10 q^{29} - 17 q^{31} + 10 q^{33} - 3 q^{35} - 4 q^{39} + q^{41} - 5 q^{43} + 12 q^{45} + 20 q^{47} + 2 q^{49} + 3 q^{51} + 5 q^{53} - 16 q^{55} + 18 q^{57} - 4 q^{59} - 5 q^{61} + 5 q^{63} + 22 q^{65} + 9 q^{67} - 16 q^{69} - 4 q^{71} - 9 q^{73} + 18 q^{75} + 2 q^{77} - 12 q^{79} + 38 q^{81} + 4 q^{83} + 3 q^{85} + 2 q^{87} - 4 q^{89} - 6 q^{91} + 32 q^{93} + 18 q^{95} - 15 q^{97} - 34 q^{99}+O(q^{100})$$ 2 * q - 3 * q^3 - 3 * q^5 + 2 * q^7 + 5 * q^9 + 2 * q^11 - 6 * q^13 - 2 * q^15 - 2 * q^17 - 12 * q^19 - 3 * q^21 + 2 * q^23 + q^25 - 18 * q^27 - 10 * q^29 - 17 * q^31 + 10 * q^33 - 3 * q^35 - 4 * q^39 + q^41 - 5 * q^43 + 12 * q^45 + 20 * q^47 + 2 * q^49 + 3 * q^51 + 5 * q^53 - 16 * q^55 + 18 * q^57 - 4 * q^59 - 5 * q^61 + 5 * q^63 + 22 * q^65 + 9 * q^67 - 16 * q^69 - 4 * q^71 - 9 * q^73 + 18 * q^75 + 2 * q^77 - 12 * q^79 + 38 * q^81 + 4 * q^83 + 3 * q^85 + 2 * q^87 - 4 * q^89 - 6 * q^91 + 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 $$-0.902474\pi$$
−0.953429 + 0.301617i $$0.902474\pi$$
$$4$$ 0 0
$$5$$ 0.302776 0.135405 0.0677027 0.997706i $$-0.478433\pi$$
0.0677027 + 0.997706i $$0.478433\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 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 $$-0.473238\pi$$
0.0839749 + 0.996468i $$0.473238\pi$$
$$14$$ 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 $$-0.741620\pi$$
−0.688247 + 0.725476i $$0.741620\pi$$
$$20$$ 0 0
$$21$$ −3.30278 −0.720725
$$22$$ 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 $$-0.876117\pi$$
−0.925217 + 0.379439i $$0.876117\pi$$
$$32$$ 0 0
$$33$$ 8.60555 1.49803
$$34$$ 0 0
$$35$$ 0.302776 0.0511784
$$36$$ 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 $$-0.731493\pi$$
−0.664822 + 0.747002i $$0.731493\pi$$
$$42$$ 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 $$-0.239834\pi$$
0.729325 + 0.684167i $$0.239834\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 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 $$-0.704670\pi$$
−0.599592 + 0.800306i $$0.704670\pi$$
$$60$$ 0 0
$$61$$ −0.697224 −0.0892704 −0.0446352 0.999003i $$-0.514213\pi$$
−0.0446352 + 0.999003i $$0.514213\pi$$
$$62$$ 0 0
$$63$$ 7.90833 0.996356
$$64$$ 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 $$-0.414907\pi$$
0.264155 + 0.964480i $$0.414907\pi$$
$$74$$ 0 0
$$75$$ 16.2111 1.87190
$$76$$ 0 0
$$77$$ −2.60555 −0.296930
$$78$$ 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 $$-0.592325\pi$$
−0.285996 + 0.958231i $$0.592325\pi$$
$$84$$ 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 $$-0.533804\pi$$
−0.106000 + 0.994366i $$0.533804\pi$$
$$90$$ 0 0
$$91$$ 0.605551 0.0634790
$$92$$ 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 $$-0.727467\pi$$
−0.655321 + 0.755350i $$0.727467\pi$$
$$98$$ 0 0
$$99$$ −20.6056 −2.07094
$$100$$ 0 0
$$101$$ 16.4222 1.63407 0.817035 0.576588i $$-0.195616\pi$$
0.817035 + 0.576588i $$0.195616\pi$$
$$102$$ 0 0
$$103$$ −1.21110 −0.119333 −0.0596667 0.998218i $$-0.519004\pi$$
−0.0596667 + 0.998218i $$0.519004\pi$$
$$104$$ 0 0
$$105$$ −1.00000 −0.0975900
$$106$$ 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
$$119$$ −1.00000 −0.0916698
$$120$$ 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 $$-0.202284\pi$$
0.804778 + 0.593576i $$0.202284\pi$$
$$132$$ 0 0
$$133$$ −6.00000 −0.520266
$$134$$ 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 $$-0.810279\pi$$
−0.827573 + 0.561358i $$0.810279\pi$$
$$140$$ 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
$$147$$ −3.30278 −0.272408
$$148$$ 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 $$-0.836256\pi$$
−0.870579 + 0.492028i $$0.836256\pi$$
$$158$$ 0 0
$$159$$ −14.2111 −1.12701
$$160$$ 0 0
$$161$$ 4.60555 0.362968
$$162$$ 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 $$-0.147519\pi$$
0.894518 + 0.447033i $$0.147519\pi$$
$$168$$ 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 $$-0.572108\pi$$
−0.224601 + 0.974451i $$0.572108\pi$$
$$174$$ 0 0
$$175$$ −4.90833 −0.371035
$$176$$ 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 $$-0.202704\pi$$
0.803996 + 0.594635i $$0.202704\pi$$
$$182$$ 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
$$189$$ −16.2111 −1.17918
$$190$$ 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 $$-0.551146\pi$$
−0.159990 + 0.987119i $$0.551146\pi$$
$$200$$ 0 0
$$201$$ −20.8167 −1.46829
$$202$$ 0 0
$$203$$ −1.39445 −0.0978711
$$204$$ 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
$$217$$ −10.3028 −0.699398
$$218$$ 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 $$-0.242146\pi$$
0.724337 + 0.689446i $$0.242146\pi$$
$$224$$ 0 0
$$225$$ −38.8167 −2.58778
$$226$$ 0 0
$$227$$ 10.9083 0.724011 0.362006 0.932176i $$-0.382092\pi$$
0.362006 + 0.932176i $$0.382092\pi$$
$$228$$ 0 0
$$229$$ 15.8167 1.04519 0.522597 0.852580i $$-0.324963\pi$$
0.522597 + 0.852580i $$0.324963\pi$$
$$230$$ 0 0
$$231$$ 8.60555 0.566204
$$232$$ 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 $$-0.468251\pi$$
0.0995761 + 0.995030i $$0.468251\pi$$
$$242$$ 0 0
$$243$$ −49.8444 −3.19752
$$244$$ 0 0
$$245$$ 0.302776 0.0193436
$$246$$ 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 $$-0.238337\pi$$
0.732536 + 0.680728i $$0.238337\pi$$
$$252$$ 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 $$-0.421608\pi$$
0.243795 + 0.969827i $$0.421608\pi$$
$$258$$ 0 0
$$259$$ −7.21110 −0.448076
$$260$$ 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 $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ 0 0
$$271$$ −10.1833 −0.618594 −0.309297 0.950965i $$-0.600094\pi$$
−0.309297 + 0.950965i $$0.600094\pi$$
$$272$$ 0 0
$$273$$ −2.00000 −0.121046
$$274$$ 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 $$-0.400936\pi$$
0.306218 + 0.951961i $$0.400936\pi$$
$$284$$ 0 0
$$285$$ 6.00000 0.355409
$$286$$ 0 0
$$287$$ −8.51388 −0.502558
$$288$$ 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 $$-0.696499\pi$$
−0.578851 + 0.815434i $$0.696499\pi$$
$$294$$ 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
$$301$$ −0.697224 −0.0401873
$$302$$ 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 $$-0.766098\pi$$
−0.741949 + 0.670456i $$0.766098\pi$$
$$308$$ 0 0
$$309$$ 4.00000 0.227552
$$310$$ 0 0
$$311$$ 16.7250 0.948387 0.474193 0.880421i $$-0.342740\pi$$
0.474193 + 0.880421i $$0.342740\pi$$
$$312$$ 0 0
$$313$$ 26.7250 1.51059 0.755293 0.655388i $$-0.227495\pi$$
0.755293 + 0.655388i $$0.227495\pi$$
$$314$$ 0 0
$$315$$ 2.39445 0.134912
$$316$$ 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
$$329$$ 10.0000 0.551318
$$330$$ 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
$$343$$ 1.00000 0.0539949
$$344$$ 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 $$-0.459089\pi$$
0.128172 + 0.991752i $$0.459089\pi$$
$$350$$ 0 0
$$351$$ −9.81665 −0.523974
$$352$$ 0 0
$$353$$ −2.60555 −0.138680 −0.0693398 0.997593i $$-0.522089\pi$$
−0.0693398 + 0.997593i $$0.522089\pi$$
$$354$$ 0 0
$$355$$ −0.605551 −0.0321393
$$356$$ 0 0
$$357$$ 3.30278 0.174801
$$358$$ 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 $$-0.547741\pi$$
−0.149421 + 0.988774i $$0.547741\pi$$
$$368$$ 0 0
$$369$$ −67.3305 −3.50509
$$370$$ 0 0
$$371$$ 4.30278 0.223389
$$372$$ 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 $$-0.180302\pi$$
0.843820 + 0.536627i $$0.180302\pi$$
$$384$$ 0 0
$$385$$ −0.788897 −0.0402059
$$386$$ 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 $$-0.639481\pi$$
−0.424302 + 0.905521i $$0.639481\pi$$
$$398$$ 0 0
$$399$$ 19.8167 0.992074
$$400$$ 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 $$-0.195879\pi$$
0.816559 + 0.577262i $$0.195879\pi$$
$$410$$ 0 0
$$411$$ 45.9361 2.26586
$$412$$ 0 0
$$413$$ −9.21110 −0.453249
$$414$$ 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 $$-0.482086\pi$$
0.0562490 + 0.998417i $$0.482086\pi$$
$$420$$ 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
$$427$$ −0.697224 −0.0337411
$$428$$ 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 $$-0.774850\pi$$
−0.760100 + 0.649806i $$0.774850\pi$$
$$434$$ 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 $$-0.474886\pi$$
0.0788164 + 0.996889i $$0.474886\pi$$
$$440$$ 0 0
$$441$$ 7.90833 0.376587
$$442$$ 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
$$455$$ 0.183346 0.00859540
$$456$$ 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 $$-0.538723\pi$$
−0.121353 + 0.992609i $$0.538723\pi$$
$$462$$ 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 $$-0.479446\pi$$
0.0645274 + 0.997916i $$0.479446\pi$$
$$468$$ 0 0
$$469$$ 6.30278 0.291035
$$470$$ 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 $$-0.656771\pi$$
−0.472840 + 0.881148i $$0.656771\pi$$
$$480$$ 0 0
$$481$$ −4.36669 −0.199104
$$482$$ 0 0
$$483$$ −15.2111 −0.692129
$$484$$ 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
$$497$$ −2.00000 −0.0897123
$$498$$ 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 $$-0.202035\pi$$
0.805243 + 0.592945i $$0.202035\pi$$
$$504$$ 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 $$-0.424477\pi$$
0.235041 + 0.971985i $$0.424477\pi$$
$$510$$ 0 0
$$511$$ 4.51388 0.199682
$$512$$ 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 $$-0.750498\pi$$
−0.708213 + 0.705999i $$0.750498\pi$$
$$522$$ 0 0
$$523$$ −19.0278 −0.832026 −0.416013 0.909359i $$-0.636573\pi$$
−0.416013 + 0.909359i $$0.636573\pi$$
$$524$$ 0 0
$$525$$ 16.2111 0.707511
$$526$$ 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
$$539$$ −2.60555 −0.112229
$$540$$ 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
$$553$$ −6.00000 −0.255146
$$554$$ 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 $$-0.625570\pi$$
−0.384338 + 0.923192i $$0.625570\pi$$
$$564$$ 0 0
$$565$$ 1.44996 0.0610003
$$566$$ 0 0
$$567$$ 29.8167 1.25218
$$568$$ 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 $$-0.480294\pi$$
0.0618681 + 0.998084i $$0.480294\pi$$
$$578$$ 0 0
$$579$$ 41.6333 1.73022
$$580$$ 0 0
$$581$$ −5.21110 −0.216193
$$582$$ 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 $$-0.518330\pi$$
−0.0575551 + 0.998342i $$0.518330\pi$$
$$588$$ 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 $$-0.742643\pi$$
−0.690577 + 0.723259i $$0.742643\pi$$
$$594$$ 0 0
$$595$$ −0.302776 −0.0124126
$$596$$ 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 $$-0.468861\pi$$
0.0976716 + 0.995219i $$0.468861\pi$$
$$602$$ 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 $$-0.683883\pi$$
−0.546088 + 0.837728i $$0.683883\pi$$
$$608$$ 0 0
$$609$$ 4.60555 0.186626
$$610$$ 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 $$-0.551399\pi$$
−0.160774 + 0.986991i $$0.551399\pi$$
$$620$$ 0 0
$$621$$ −74.6611 −2.99605
$$622$$ 0 0
$$623$$ −2.00000 −0.0801283
$$624$$ 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
$$637$$ 0.605551 0.0239928
$$638$$ 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 $$-0.578176\pi$$
−0.243134 + 0.969993i $$0.578176\pi$$
$$644$$ 0 0
$$645$$ 0.697224 0.0274532
$$646$$ 0 0
$$647$$ −34.6056 −1.36048 −0.680242 0.732987i $$-0.738125\pi$$
−0.680242 + 0.732987i $$0.738125\pi$$
$$648$$ 0 0
$$649$$ 24.0000 0.942082
$$650$$ 0 0
$$651$$ 34.0278 1.33365
$$652$$ 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 $$-0.680596\pi$$
−0.537406 + 0.843324i $$0.680596\pi$$
$$662$$ 0 0
$$663$$ 2.00000 0.0776736
$$664$$ 0 0
$$665$$ −1.81665 −0.0704468
$$666$$ 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 $$-0.361060\pi$$
0.422764 + 0.906240i $$0.361060\pi$$
$$678$$ 0 0
$$679$$ −12.9083 −0.495376
$$680$$ 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 $$-0.495779\pi$$
0.0132618 + 0.999912i $$0.495779\pi$$
$$692$$ 0 0
$$693$$ −20.6056 −0.782740
$$694$$ 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
$$707$$ 16.4222 0.617621
$$708$$ 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 $$-0.624943\pi$$
−0.382519 + 0.923948i $$0.624943\pi$$
$$720$$ 0 0
$$721$$ −1.21110 −0.0451038
$$722$$ 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 $$-0.742861\pi$$
−0.691071 + 0.722787i $$0.742861\pi$$
$$728$$ 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 $$-0.329523\pi$$
0.510330 + 0.859979i $$0.329523\pi$$
$$734$$ 0 0
$$735$$ −1.00000 −0.0368856
$$736$$ 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
$$749$$ −16.6056 −0.606754
$$750$$ 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 $$-0.246871\pi$$
0.714024 + 0.700121i $$0.246871\pi$$
$$762$$ 0 0
$$763$$ −2.00000 −0.0724049
$$764$$ 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 $$-0.556639\pi$$
−0.176999 + 0.984211i $$0.556639\pi$$
$$770$$ 0 0
$$771$$ −25.8167 −0.929764
$$772$$ 0 0
$$773$$ 50.8444 1.82875 0.914373 0.404872i $$-0.132684\pi$$
0.914373 + 0.404872i $$0.132684\pi$$
$$774$$ 0 0
$$775$$ 50.5694 1.81651
$$776$$ 0 0
$$777$$ 23.8167 0.854418
$$778$$ 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 $$-0.417250\pi$$
0.257048 + 0.966399i $$0.417250\pi$$
$$788$$ 0 0
$$789$$ 21.2111 0.755135
$$790$$ 0 0
$$791$$ 4.78890 0.170274
$$792$$ 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 $$-0.326917\pi$$
0.517355 + 0.855771i $$0.326917\pi$$
$$798$$ 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
$$805$$ 1.39445 0.0491479
$$806$$ 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 $$-0.272284\pi$$
0.655913 + 0.754836i $$0.272284\pi$$
$$812$$ 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
$$819$$ 4.78890 0.167338
$$820$$ 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 $$-0.719281\pi$$
−0.635682 + 0.771951i $$0.719281\pi$$
$$830$$ 0 0
$$831$$ 36.4222 1.26347
$$832$$ 0 0
$$833$$ −1.00000 −0.0346479
$$834$$ 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 $$-0.462612\pi$$
0.117189 + 0.993110i $$0.462612\pi$$
$$840$$ 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
$$847$$ −4.21110 −0.144695
$$848$$ 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 $$-0.526126\pi$$
−0.0819844 + 0.996634i $$0.526126\pi$$
$$854$$ 0 0
$$855$$ −14.3667 −0.491331
$$856$$ 0 0
$$857$$ −5.51388 −0.188350 −0.0941752 0.995556i $$-0.530021\pi$$
−0.0941752 + 0.995556i $$0.530021\pi$$
$$858$$ 0 0
$$859$$ −2.36669 −0.0807505 −0.0403753 0.999185i $$-0.512855\pi$$
−0.0403753 + 0.999185i $$0.512855\pi$$
$$860$$ 0 0
$$861$$ 28.1194 0.958308
$$862$$ 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
$$875$$ −3.00000 −0.101419
$$876$$ 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 $$-0.401267\pi$$
0.305230 + 0.952279i $$0.401267\pi$$
$$882$$ 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 $$-0.770360\pi$$
−0.750859 + 0.660463i $$0.770360\pi$$
$$888$$ 0 0
$$889$$ 20.1194 0.674784
$$890$$ 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
$$903$$ 2.30278 0.0766316
$$904$$ 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
$$917$$ 18.4222 0.608355
$$918$$ 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 $$-0.653691\pi$$
−0.464292 + 0.885682i $$0.653691\pi$$
$$930$$ 0 0
$$931$$ −6.00000 −0.196642
$$932$$ 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 $$-0.556390\pi$$
−0.176229 + 0.984349i $$0.556390\pi$$
$$938$$ 0 0
$$939$$ −88.2666 −2.88047
$$940$$ 0 0
$$941$$ −40.3305 −1.31474 −0.657369 0.753569i $$-0.728331\pi$$
−0.657369 + 0.753569i $$0.728331\pi$$
$$942$$ 0 0
$$943$$ −39.2111 −1.27689
$$944$$ 0 0
$$945$$ −4.90833 −0.159668
$$946$$ 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
$$959$$ −13.9083 −0.449123
$$960$$ 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 $$-0.338745\pi$$
0.485206 + 0.874400i $$0.338745\pi$$
$$972$$ 0 0
$$973$$ −19.5139 −0.625586
$$974$$ 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 $$-0.517847\pi$$
−0.0560377 + 0.998429i $$0.517847\pi$$
$$984$$ 0 0
$$985$$ 3.94449 0.125682
$$986$$ 0 0
$$987$$ −33.0278 −1.05129
$$988$$ 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 $$-0.386241\pi$$
0.349825 + 0.936815i $$0.386241\pi$$
$$998$$ 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 476.2.a.a.1.1 2
3.2 odd 2 4284.2.a.p.1.1 2
4.3 odd 2 1904.2.a.l.1.2 2
7.6 odd 2 3332.2.a.n.1.2 2
8.3 odd 2 7616.2.a.m.1.1 2
8.5 even 2 7616.2.a.z.1.2 2
17.16 even 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 1.1 even 1 trivial
1904.2.a.l.1.2 2 4.3 odd 2
3332.2.a.n.1.2 2 7.6 odd 2
4284.2.a.p.1.1 2 3.2 odd 2
7616.2.a.m.1.1 2 8.3 odd 2
7616.2.a.z.1.2 2 8.5 even 2
8092.2.a.n.1.2 2 17.16 even 2