# Properties

 Label 7800.2.a.bn.1.2 Level $7800$ Weight $2$ Character 7800.1 Self dual yes Analytic conductor $62.283$ 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] = [7800,2,Mod(1,7800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7800.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: $$62.2833135766$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-0.363328$$ of defining polynomial Character $$\chi$$ $$=$$ 7800.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -1.36333 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -1.36333 q^{7} +1.00000 q^{9} -5.64600 q^{11} -1.00000 q^{13} +2.14134 q^{17} +2.41468 q^{19} -1.36333 q^{21} -2.72666 q^{23} +1.00000 q^{27} -5.28267 q^{29} -1.64600 q^{31} -5.64600 q^{33} +3.86799 q^{37} -1.00000 q^{39} +5.86799 q^{41} -9.00933 q^{43} +3.77801 q^{47} -5.14134 q^{49} +2.14134 q^{51} +12.0093 q^{53} +2.41468 q^{57} +6.19269 q^{59} +11.5946 q^{61} -1.36333 q^{63} -0.324695 q^{67} -2.72666 q^{69} +11.8680 q^{71} +8.28267 q^{73} +7.69735 q^{77} +2.31198 q^{79} +1.00000 q^{81} +10.3727 q^{83} -5.28267 q^{87} +3.73599 q^{89} +1.36333 q^{91} -1.64600 q^{93} -16.8480 q^{97} -5.64600 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{3} - 2 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^3 - 2 * q^7 + 3 * q^9 $$3 q + 3 q^{3} - 2 q^{7} + 3 q^{9} + 2 q^{11} - 3 q^{13} - 2 q^{17} + 3 q^{19} - 2 q^{21} - 4 q^{23} + 3 q^{27} + q^{29} + 14 q^{31} + 2 q^{33} - q^{37} - 3 q^{39} + 5 q^{41} - 6 q^{43} + 5 q^{47} - 7 q^{49} - 2 q^{51} + 15 q^{53} + 3 q^{57} + 8 q^{59} + 18 q^{61} - 2 q^{63} - 3 q^{67} - 4 q^{69} + 23 q^{71} + 8 q^{73} + 2 q^{77} + 7 q^{79} + 3 q^{81} + 8 q^{83} + q^{87} - 14 q^{89} + 2 q^{91} + 14 q^{93} + 2 q^{99}+O(q^{100})$$ 3 * q + 3 * q^3 - 2 * q^7 + 3 * q^9 + 2 * q^11 - 3 * q^13 - 2 * q^17 + 3 * q^19 - 2 * q^21 - 4 * q^23 + 3 * q^27 + q^29 + 14 * q^31 + 2 * q^33 - q^37 - 3 * q^39 + 5 * q^41 - 6 * q^43 + 5 * q^47 - 7 * q^49 - 2 * q^51 + 15 * q^53 + 3 * q^57 + 8 * q^59 + 18 * q^61 - 2 * q^63 - 3 * q^67 - 4 * q^69 + 23 * q^71 + 8 * q^73 + 2 * q^77 + 7 * q^79 + 3 * q^81 + 8 * q^83 + q^87 - 14 * q^89 + 2 * q^91 + 14 * q^93 + 2 * 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
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −1.36333 −0.515290 −0.257645 0.966240i $$-0.582946\pi$$
−0.257645 + 0.966240i $$0.582946\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −5.64600 −1.70233 −0.851167 0.524896i $$-0.824104\pi$$
−0.851167 + 0.524896i $$0.824104\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 2.14134 0.519350 0.259675 0.965696i $$-0.416385\pi$$
0.259675 + 0.965696i $$0.416385\pi$$
$$18$$ 0 0
$$19$$ 2.41468 0.553966 0.276983 0.960875i $$-0.410666\pi$$
0.276983 + 0.960875i $$0.410666\pi$$
$$20$$ 0 0
$$21$$ −1.36333 −0.297503
$$22$$ 0 0
$$23$$ −2.72666 −0.568547 −0.284274 0.958743i $$-0.591752\pi$$
−0.284274 + 0.958743i $$0.591752\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −5.28267 −0.980968 −0.490484 0.871450i $$-0.663180\pi$$
−0.490484 + 0.871450i $$0.663180\pi$$
$$30$$ 0 0
$$31$$ −1.64600 −0.295630 −0.147815 0.989015i $$-0.547224\pi$$
−0.147815 + 0.989015i $$0.547224\pi$$
$$32$$ 0 0
$$33$$ −5.64600 −0.982843
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 3.86799 0.635894 0.317947 0.948108i $$-0.397007\pi$$
0.317947 + 0.948108i $$0.397007\pi$$
$$38$$ 0 0
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ 5.86799 0.916426 0.458213 0.888842i $$-0.348490\pi$$
0.458213 + 0.888842i $$0.348490\pi$$
$$42$$ 0 0
$$43$$ −9.00933 −1.37391 −0.686955 0.726700i $$-0.741053\pi$$
−0.686955 + 0.726700i $$0.741053\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 3.77801 0.551079 0.275540 0.961290i $$-0.411143\pi$$
0.275540 + 0.961290i $$0.411143\pi$$
$$48$$ 0 0
$$49$$ −5.14134 −0.734477
$$50$$ 0 0
$$51$$ 2.14134 0.299847
$$52$$ 0 0
$$53$$ 12.0093 1.64961 0.824804 0.565419i $$-0.191285\pi$$
0.824804 + 0.565419i $$0.191285\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 2.41468 0.319832
$$58$$ 0 0
$$59$$ 6.19269 0.806219 0.403110 0.915152i $$-0.367929\pi$$
0.403110 + 0.915152i $$0.367929\pi$$
$$60$$ 0 0
$$61$$ 11.5946 1.48454 0.742271 0.670099i $$-0.233749\pi$$
0.742271 + 0.670099i $$0.233749\pi$$
$$62$$ 0 0
$$63$$ −1.36333 −0.171763
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −0.324695 −0.0396678 −0.0198339 0.999803i $$-0.506314\pi$$
−0.0198339 + 0.999803i $$0.506314\pi$$
$$68$$ 0 0
$$69$$ −2.72666 −0.328251
$$70$$ 0 0
$$71$$ 11.8680 1.40847 0.704236 0.709966i $$-0.251290\pi$$
0.704236 + 0.709966i $$0.251290\pi$$
$$72$$ 0 0
$$73$$ 8.28267 0.969413 0.484707 0.874677i $$-0.338926\pi$$
0.484707 + 0.874677i $$0.338926\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 7.69735 0.877195
$$78$$ 0 0
$$79$$ 2.31198 0.260118 0.130059 0.991506i $$-0.458483\pi$$
0.130059 + 0.991506i $$0.458483\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 10.3727 1.13855 0.569274 0.822148i $$-0.307225\pi$$
0.569274 + 0.822148i $$0.307225\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −5.28267 −0.566362
$$88$$ 0 0
$$89$$ 3.73599 0.396014 0.198007 0.980201i $$-0.436553\pi$$
0.198007 + 0.980201i $$0.436553\pi$$
$$90$$ 0 0
$$91$$ 1.36333 0.142916
$$92$$ 0 0
$$93$$ −1.64600 −0.170682
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −16.8480 −1.71066 −0.855328 0.518086i $$-0.826645\pi$$
−0.855328 + 0.518086i $$0.826645\pi$$
$$98$$ 0 0
$$99$$ −5.64600 −0.567444
$$100$$ 0 0
$$101$$ −5.87732 −0.584815 −0.292408 0.956294i $$-0.594456\pi$$
−0.292408 + 0.956294i $$0.594456\pi$$
$$102$$ 0 0
$$103$$ −14.0187 −1.38130 −0.690650 0.723189i $$-0.742675\pi$$
−0.690650 + 0.723189i $$0.742675\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −10.8773 −1.05155 −0.525775 0.850624i $$-0.676225\pi$$
−0.525775 + 0.850624i $$0.676225\pi$$
$$108$$ 0 0
$$109$$ −3.58532 −0.343411 −0.171706 0.985148i $$-0.554928\pi$$
−0.171706 + 0.985148i $$0.554928\pi$$
$$110$$ 0 0
$$111$$ 3.86799 0.367134
$$112$$ 0 0
$$113$$ 18.5653 1.74648 0.873240 0.487290i $$-0.162014\pi$$
0.873240 + 0.487290i $$0.162014\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −1.00000 −0.0924500
$$118$$ 0 0
$$119$$ −2.91934 −0.267616
$$120$$ 0 0
$$121$$ 20.8773 1.89794
$$122$$ 0 0
$$123$$ 5.86799 0.529099
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −0.858664 −0.0761941 −0.0380970 0.999274i $$-0.512130\pi$$
−0.0380970 + 0.999274i $$0.512130\pi$$
$$128$$ 0 0
$$129$$ −9.00933 −0.793227
$$130$$ 0 0
$$131$$ −1.86799 −0.163207 −0.0816036 0.996665i $$-0.526004\pi$$
−0.0816036 + 0.996665i $$0.526004\pi$$
$$132$$ 0 0
$$133$$ −3.29200 −0.285453
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −5.24404 −0.448028 −0.224014 0.974586i $$-0.571916\pi$$
−0.224014 + 0.974586i $$0.571916\pi$$
$$138$$ 0 0
$$139$$ 8.17997 0.693816 0.346908 0.937899i $$-0.387232\pi$$
0.346908 + 0.937899i $$0.387232\pi$$
$$140$$ 0 0
$$141$$ 3.77801 0.318166
$$142$$ 0 0
$$143$$ 5.64600 0.472142
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −5.14134 −0.424050
$$148$$ 0 0
$$149$$ 9.73599 0.797603 0.398801 0.917037i $$-0.369426\pi$$
0.398801 + 0.917037i $$0.369426\pi$$
$$150$$ 0 0
$$151$$ 1.18336 0.0963004 0.0481502 0.998840i $$-0.484667\pi$$
0.0481502 + 0.998840i $$0.484667\pi$$
$$152$$ 0 0
$$153$$ 2.14134 0.173117
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 17.9800 1.43496 0.717481 0.696578i $$-0.245295\pi$$
0.717481 + 0.696578i $$0.245295\pi$$
$$158$$ 0 0
$$159$$ 12.0093 0.952402
$$160$$ 0 0
$$161$$ 3.71733 0.292966
$$162$$ 0 0
$$163$$ 4.28267 0.335445 0.167722 0.985834i $$-0.446359\pi$$
0.167722 + 0.985834i $$0.446359\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −6.41468 −0.496383 −0.248191 0.968711i $$-0.579836\pi$$
−0.248191 + 0.968711i $$0.579836\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 2.41468 0.184655
$$172$$ 0 0
$$173$$ 17.1800 1.30617 0.653084 0.757285i $$-0.273475\pi$$
0.653084 + 0.757285i $$0.273475\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 6.19269 0.465471
$$178$$ 0 0
$$179$$ 7.11203 0.531578 0.265789 0.964031i $$-0.414368\pi$$
0.265789 + 0.964031i $$0.414368\pi$$
$$180$$ 0 0
$$181$$ 25.4333 1.89045 0.945223 0.326427i $$-0.105845\pi$$
0.945223 + 0.326427i $$0.105845\pi$$
$$182$$ 0 0
$$183$$ 11.5946 0.857101
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −12.0900 −0.884107
$$188$$ 0 0
$$189$$ −1.36333 −0.0991675
$$190$$ 0 0
$$191$$ 16.1214 1.16650 0.583250 0.812292i $$-0.301781\pi$$
0.583250 + 0.812292i $$0.301781\pi$$
$$192$$ 0 0
$$193$$ 10.0187 0.721159 0.360579 0.932729i $$-0.382579\pi$$
0.360579 + 0.932729i $$0.382579\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −10.0187 −0.713800 −0.356900 0.934143i $$-0.616166\pi$$
−0.356900 + 0.934143i $$0.616166\pi$$
$$198$$ 0 0
$$199$$ −15.4427 −1.09470 −0.547351 0.836903i $$-0.684364\pi$$
−0.547351 + 0.836903i $$0.684364\pi$$
$$200$$ 0 0
$$201$$ −0.324695 −0.0229022
$$202$$ 0 0
$$203$$ 7.20202 0.505482
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −2.72666 −0.189516
$$208$$ 0 0
$$209$$ −13.6333 −0.943034
$$210$$ 0 0
$$211$$ 17.8387 1.22807 0.614033 0.789280i $$-0.289546\pi$$
0.614033 + 0.789280i $$0.289546\pi$$
$$212$$ 0 0
$$213$$ 11.8680 0.813181
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 2.24404 0.152335
$$218$$ 0 0
$$219$$ 8.28267 0.559691
$$220$$ 0 0
$$221$$ −2.14134 −0.144042
$$222$$ 0 0
$$223$$ −6.56534 −0.439648 −0.219824 0.975540i $$-0.570548\pi$$
−0.219824 + 0.975540i $$0.570548\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 13.3820 0.888194 0.444097 0.895979i $$-0.353525\pi$$
0.444097 + 0.895979i $$0.353525\pi$$
$$228$$ 0 0
$$229$$ 9.88665 0.653328 0.326664 0.945140i $$-0.394075\pi$$
0.326664 + 0.945140i $$0.394075\pi$$
$$230$$ 0 0
$$231$$ 7.69735 0.506449
$$232$$ 0 0
$$233$$ −26.8667 −1.76009 −0.880047 0.474886i $$-0.842489\pi$$
−0.880047 + 0.474886i $$0.842489\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 2.31198 0.150179
$$238$$ 0 0
$$239$$ 27.4847 1.77784 0.888918 0.458066i $$-0.151458\pi$$
0.888918 + 0.458066i $$0.151458\pi$$
$$240$$ 0 0
$$241$$ −10.0187 −0.645358 −0.322679 0.946508i $$-0.604584\pi$$
−0.322679 + 0.946508i $$0.604584\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −2.41468 −0.153642
$$248$$ 0 0
$$249$$ 10.3727 0.657340
$$250$$ 0 0
$$251$$ −24.1507 −1.52438 −0.762188 0.647355i $$-0.775875\pi$$
−0.762188 + 0.647355i $$0.775875\pi$$
$$252$$ 0 0
$$253$$ 15.3947 0.967857
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 1.15066 0.0717765 0.0358882 0.999356i $$-0.488574\pi$$
0.0358882 + 0.999356i $$0.488574\pi$$
$$258$$ 0 0
$$259$$ −5.27334 −0.327670
$$260$$ 0 0
$$261$$ −5.28267 −0.326989
$$262$$ 0 0
$$263$$ −2.28267 −0.140756 −0.0703778 0.997520i $$-0.522420\pi$$
−0.0703778 + 0.997520i $$0.522420\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 3.73599 0.228639
$$268$$ 0 0
$$269$$ 8.17064 0.498173 0.249086 0.968481i $$-0.419870\pi$$
0.249086 + 0.968481i $$0.419870\pi$$
$$270$$ 0 0
$$271$$ 21.5433 1.30866 0.654331 0.756208i $$-0.272950\pi$$
0.654331 + 0.756208i $$0.272950\pi$$
$$272$$ 0 0
$$273$$ 1.36333 0.0825124
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −23.1307 −1.38979 −0.694894 0.719112i $$-0.744549\pi$$
−0.694894 + 0.719112i $$0.744549\pi$$
$$278$$ 0 0
$$279$$ −1.64600 −0.0985435
$$280$$ 0 0
$$281$$ 4.96137 0.295970 0.147985 0.988990i $$-0.452721\pi$$
0.147985 + 0.988990i $$0.452721\pi$$
$$282$$ 0 0
$$283$$ 4.74531 0.282080 0.141040 0.990004i $$-0.454955\pi$$
0.141040 + 0.990004i $$0.454955\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −8.00000 −0.472225
$$288$$ 0 0
$$289$$ −12.4147 −0.730275
$$290$$ 0 0
$$291$$ −16.8480 −0.987648
$$292$$ 0 0
$$293$$ 29.7360 1.73719 0.868597 0.495518i $$-0.165022\pi$$
0.868597 + 0.495518i $$0.165022\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −5.64600 −0.327614
$$298$$ 0 0
$$299$$ 2.72666 0.157687
$$300$$ 0 0
$$301$$ 12.2827 0.707961
$$302$$ 0 0
$$303$$ −5.87732 −0.337643
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 18.4520 1.05311 0.526555 0.850141i $$-0.323483\pi$$
0.526555 + 0.850141i $$0.323483\pi$$
$$308$$ 0 0
$$309$$ −14.0187 −0.797494
$$310$$ 0 0
$$311$$ 15.9160 0.902511 0.451255 0.892395i $$-0.350976\pi$$
0.451255 + 0.892395i $$0.350976\pi$$
$$312$$ 0 0
$$313$$ −1.99067 −0.112519 −0.0562597 0.998416i $$-0.517917\pi$$
−0.0562597 + 0.998416i $$0.517917\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −2.26401 −0.127160 −0.0635799 0.997977i $$-0.520252\pi$$
−0.0635799 + 0.997977i $$0.520252\pi$$
$$318$$ 0 0
$$319$$ 29.8260 1.66993
$$320$$ 0 0
$$321$$ −10.8773 −0.607113
$$322$$ 0 0
$$323$$ 5.17064 0.287702
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −3.58532 −0.198269
$$328$$ 0 0
$$329$$ −5.15066 −0.283965
$$330$$ 0 0
$$331$$ 1.69867 0.0933674 0.0466837 0.998910i $$-0.485135\pi$$
0.0466837 + 0.998910i $$0.485135\pi$$
$$332$$ 0 0
$$333$$ 3.86799 0.211965
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −20.7067 −1.12796 −0.563982 0.825787i $$-0.690731\pi$$
−0.563982 + 0.825787i $$0.690731\pi$$
$$338$$ 0 0
$$339$$ 18.5653 1.00833
$$340$$ 0 0
$$341$$ 9.29332 0.503261
$$342$$ 0 0
$$343$$ 16.5526 0.893758
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 7.97070 0.427889 0.213945 0.976846i $$-0.431369\pi$$
0.213945 + 0.976846i $$0.431369\pi$$
$$348$$ 0 0
$$349$$ 6.56534 0.351435 0.175717 0.984441i $$-0.443776\pi$$
0.175717 + 0.984441i $$0.443776\pi$$
$$350$$ 0 0
$$351$$ −1.00000 −0.0533761
$$352$$ 0 0
$$353$$ −20.4520 −1.08855 −0.544275 0.838907i $$-0.683195\pi$$
−0.544275 + 0.838907i $$0.683195\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −2.91934 −0.154508
$$358$$ 0 0
$$359$$ −11.8994 −0.628025 −0.314012 0.949419i $$-0.601673\pi$$
−0.314012 + 0.949419i $$0.601673\pi$$
$$360$$ 0 0
$$361$$ −13.1693 −0.693122
$$362$$ 0 0
$$363$$ 20.8773 1.09578
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −23.5853 −1.23114 −0.615572 0.788081i $$-0.711075\pi$$
−0.615572 + 0.788081i $$0.711075\pi$$
$$368$$ 0 0
$$369$$ 5.86799 0.305475
$$370$$ 0 0
$$371$$ −16.3727 −0.850026
$$372$$ 0 0
$$373$$ −25.1693 −1.30322 −0.651609 0.758555i $$-0.725906\pi$$
−0.651609 + 0.758555i $$0.725906\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 5.28267 0.272071
$$378$$ 0 0
$$379$$ −8.83530 −0.453839 −0.226919 0.973914i $$-0.572865\pi$$
−0.226919 + 0.973914i $$0.572865\pi$$
$$380$$ 0 0
$$381$$ −0.858664 −0.0439907
$$382$$ 0 0
$$383$$ −27.9053 −1.42589 −0.712947 0.701218i $$-0.752640\pi$$
−0.712947 + 0.701218i $$0.752640\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −9.00933 −0.457970
$$388$$ 0 0
$$389$$ −0.697352 −0.0353571 −0.0176786 0.999844i $$-0.505628\pi$$
−0.0176786 + 0.999844i $$0.505628\pi$$
$$390$$ 0 0
$$391$$ −5.83869 −0.295275
$$392$$ 0 0
$$393$$ −1.86799 −0.0942278
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 17.8867 0.897705 0.448853 0.893606i $$-0.351833\pi$$
0.448853 + 0.893606i $$0.351833\pi$$
$$398$$ 0 0
$$399$$ −3.29200 −0.164806
$$400$$ 0 0
$$401$$ 25.1120 1.25404 0.627018 0.779005i $$-0.284275\pi$$
0.627018 + 0.779005i $$0.284275\pi$$
$$402$$ 0 0
$$403$$ 1.64600 0.0819931
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −21.8387 −1.08250
$$408$$ 0 0
$$409$$ −23.6774 −1.17077 −0.585385 0.810755i $$-0.699057\pi$$
−0.585385 + 0.810755i $$0.699057\pi$$
$$410$$ 0 0
$$411$$ −5.24404 −0.258669
$$412$$ 0 0
$$413$$ −8.44267 −0.415436
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 8.17997 0.400575
$$418$$ 0 0
$$419$$ −18.7347 −0.915248 −0.457624 0.889146i $$-0.651299\pi$$
−0.457624 + 0.889146i $$0.651299\pi$$
$$420$$ 0 0
$$421$$ 4.26401 0.207815 0.103908 0.994587i $$-0.466865\pi$$
0.103908 + 0.994587i $$0.466865\pi$$
$$422$$ 0 0
$$423$$ 3.77801 0.183693
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −15.8073 −0.764969
$$428$$ 0 0
$$429$$ 5.64600 0.272591
$$430$$ 0 0
$$431$$ 20.1693 0.971522 0.485761 0.874092i $$-0.338543\pi$$
0.485761 + 0.874092i $$0.338543\pi$$
$$432$$ 0 0
$$433$$ −20.6974 −0.994651 −0.497326 0.867564i $$-0.665685\pi$$
−0.497326 + 0.867564i $$0.665685\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −6.58400 −0.314956
$$438$$ 0 0
$$439$$ −2.31198 −0.110345 −0.0551723 0.998477i $$-0.517571\pi$$
−0.0551723 + 0.998477i $$0.517571\pi$$
$$440$$ 0 0
$$441$$ −5.14134 −0.244826
$$442$$ 0 0
$$443$$ 2.25337 0.107061 0.0535304 0.998566i $$-0.482953\pi$$
0.0535304 + 0.998566i $$0.482953\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 9.73599 0.460496
$$448$$ 0 0
$$449$$ 22.1693 1.04624 0.523118 0.852261i $$-0.324769\pi$$
0.523118 + 0.852261i $$0.324769\pi$$
$$450$$ 0 0
$$451$$ −33.1307 −1.56006
$$452$$ 0 0
$$453$$ 1.18336 0.0555990
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 31.4720 1.47220 0.736098 0.676875i $$-0.236666\pi$$
0.736098 + 0.676875i $$0.236666\pi$$
$$458$$ 0 0
$$459$$ 2.14134 0.0999490
$$460$$ 0 0
$$461$$ 17.7360 0.826047 0.413024 0.910720i $$-0.364473\pi$$
0.413024 + 0.910720i $$0.364473\pi$$
$$462$$ 0 0
$$463$$ −0.431266 −0.0200426 −0.0100213 0.999950i $$-0.503190\pi$$
−0.0100213 + 0.999950i $$0.503190\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −5.30265 −0.245377 −0.122689 0.992445i $$-0.539152\pi$$
−0.122689 + 0.992445i $$0.539152\pi$$
$$468$$ 0 0
$$469$$ 0.442666 0.0204404
$$470$$ 0 0
$$471$$ 17.9800 0.828476
$$472$$ 0 0
$$473$$ 50.8667 2.33885
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 12.0093 0.549869
$$478$$ 0 0
$$479$$ −2.07934 −0.0950074 −0.0475037 0.998871i $$-0.515127\pi$$
−0.0475037 + 0.998871i $$0.515127\pi$$
$$480$$ 0 0
$$481$$ −3.86799 −0.176365
$$482$$ 0 0
$$483$$ 3.71733 0.169144
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −15.3633 −0.696179 −0.348089 0.937461i $$-0.613169\pi$$
−0.348089 + 0.937461i $$0.613169\pi$$
$$488$$ 0 0
$$489$$ 4.28267 0.193669
$$490$$ 0 0
$$491$$ 28.2173 1.27343 0.636714 0.771100i $$-0.280293\pi$$
0.636714 + 0.771100i $$0.280293\pi$$
$$492$$ 0 0
$$493$$ −11.3120 −0.509466
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −16.1800 −0.725771
$$498$$ 0 0
$$499$$ 7.15405 0.320259 0.160130 0.987096i $$-0.448809\pi$$
0.160130 + 0.987096i $$0.448809\pi$$
$$500$$ 0 0
$$501$$ −6.41468 −0.286587
$$502$$ 0 0
$$503$$ 5.09337 0.227102 0.113551 0.993532i $$-0.463777\pi$$
0.113551 + 0.993532i $$0.463777\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 1.00000 0.0444116
$$508$$ 0 0
$$509$$ −21.4906 −0.952555 −0.476278 0.879295i $$-0.658014\pi$$
−0.476278 + 0.879295i $$0.658014\pi$$
$$510$$ 0 0
$$511$$ −11.2920 −0.499529
$$512$$ 0 0
$$513$$ 2.41468 0.106611
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −21.3306 −0.938120
$$518$$ 0 0
$$519$$ 17.1800 0.754117
$$520$$ 0 0
$$521$$ 25.7360 1.12751 0.563757 0.825941i $$-0.309355\pi$$
0.563757 + 0.825941i $$0.309355\pi$$
$$522$$ 0 0
$$523$$ −0.972014 −0.0425032 −0.0212516 0.999774i $$-0.506765\pi$$
−0.0212516 + 0.999774i $$0.506765\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −3.52464 −0.153536
$$528$$ 0 0
$$529$$ −15.5653 −0.676754
$$530$$ 0 0
$$531$$ 6.19269 0.268740
$$532$$ 0 0
$$533$$ −5.86799 −0.254171
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 7.11203 0.306907
$$538$$ 0 0
$$539$$ 29.0280 1.25032
$$540$$ 0 0
$$541$$ 13.6774 0.588036 0.294018 0.955800i $$-0.405007\pi$$
0.294018 + 0.955800i $$0.405007\pi$$
$$542$$ 0 0
$$543$$ 25.4333 1.09145
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −33.7546 −1.44324 −0.721622 0.692287i $$-0.756603\pi$$
−0.721622 + 0.692287i $$0.756603\pi$$
$$548$$ 0 0
$$549$$ 11.5946 0.494848
$$550$$ 0 0
$$551$$ −12.7560 −0.543422
$$552$$ 0 0
$$553$$ −3.15198 −0.134036
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 10.2827 0.435691 0.217845 0.975983i $$-0.430097\pi$$
0.217845 + 0.975983i $$0.430097\pi$$
$$558$$ 0 0
$$559$$ 9.00933 0.381054
$$560$$ 0 0
$$561$$ −12.0900 −0.510440
$$562$$ 0 0
$$563$$ −35.3841 −1.49126 −0.745630 0.666360i $$-0.767851\pi$$
−0.745630 + 0.666360i $$0.767851\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −1.36333 −0.0572544
$$568$$ 0 0
$$569$$ 21.7160 0.910382 0.455191 0.890394i $$-0.349571\pi$$
0.455191 + 0.890394i $$0.349571\pi$$
$$570$$ 0 0
$$571$$ 24.9507 1.04416 0.522078 0.852898i $$-0.325157\pi$$
0.522078 + 0.852898i $$0.325157\pi$$
$$572$$ 0 0
$$573$$ 16.1214 0.673479
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −27.1893 −1.13191 −0.565953 0.824438i $$-0.691492\pi$$
−0.565953 + 0.824438i $$0.691492\pi$$
$$578$$ 0 0
$$579$$ 10.0187 0.416361
$$580$$ 0 0
$$581$$ −14.1413 −0.586681
$$582$$ 0 0
$$583$$ −67.8047 −2.80818
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −47.5220 −1.96144 −0.980721 0.195411i $$-0.937396\pi$$
−0.980721 + 0.195411i $$0.937396\pi$$
$$588$$ 0 0
$$589$$ −3.97456 −0.163769
$$590$$ 0 0
$$591$$ −10.0187 −0.412112
$$592$$ 0 0
$$593$$ 17.9453 0.736923 0.368462 0.929643i $$-0.379885\pi$$
0.368462 + 0.929643i $$0.379885\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −15.4427 −0.632026
$$598$$ 0 0
$$599$$ 15.4720 0.632168 0.316084 0.948731i $$-0.397632\pi$$
0.316084 + 0.948731i $$0.397632\pi$$
$$600$$ 0 0
$$601$$ 45.7453 1.86599 0.932995 0.359889i $$-0.117185\pi$$
0.932995 + 0.359889i $$0.117185\pi$$
$$602$$ 0 0
$$603$$ −0.324695 −0.0132226
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −15.3467 −0.622905 −0.311453 0.950262i $$-0.600815\pi$$
−0.311453 + 0.950262i $$0.600815\pi$$
$$608$$ 0 0
$$609$$ 7.20202 0.291840
$$610$$ 0 0
$$611$$ −3.77801 −0.152842
$$612$$ 0 0
$$613$$ −5.67738 −0.229307 −0.114654 0.993406i $$-0.536576\pi$$
−0.114654 + 0.993406i $$0.536576\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 21.8867 0.881123 0.440562 0.897722i $$-0.354779\pi$$
0.440562 + 0.897722i $$0.354779\pi$$
$$618$$ 0 0
$$619$$ 21.4134 0.860676 0.430338 0.902668i $$-0.358394\pi$$
0.430338 + 0.902668i $$0.358394\pi$$
$$620$$ 0 0
$$621$$ −2.72666 −0.109417
$$622$$ 0 0
$$623$$ −5.09337 −0.204062
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −13.6333 −0.544461
$$628$$ 0 0
$$629$$ 8.28267 0.330252
$$630$$ 0 0
$$631$$ 35.1893 1.40086 0.700432 0.713719i $$-0.252991\pi$$
0.700432 + 0.713719i $$0.252991\pi$$
$$632$$ 0 0
$$633$$ 17.8387 0.709024
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 5.14134 0.203707
$$638$$ 0 0
$$639$$ 11.8680 0.469491
$$640$$ 0 0
$$641$$ 33.5547 1.32533 0.662665 0.748916i $$-0.269425\pi$$
0.662665 + 0.748916i $$0.269425\pi$$
$$642$$ 0 0
$$643$$ 18.7160 0.738087 0.369044 0.929412i $$-0.379685\pi$$
0.369044 + 0.929412i $$0.379685\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −16.5653 −0.651251 −0.325625 0.945499i $$-0.605575\pi$$
−0.325625 + 0.945499i $$0.605575\pi$$
$$648$$ 0 0
$$649$$ −34.9639 −1.37245
$$650$$ 0 0
$$651$$ 2.24404 0.0879508
$$652$$ 0 0
$$653$$ 26.5454 1.03880 0.519400 0.854531i $$-0.326155\pi$$
0.519400 + 0.854531i $$0.326155\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 8.28267 0.323138
$$658$$ 0 0
$$659$$ 36.3306 1.41524 0.707620 0.706593i $$-0.249769\pi$$
0.707620 + 0.706593i $$0.249769\pi$$
$$660$$ 0 0
$$661$$ 22.2279 0.864566 0.432283 0.901738i $$-0.357708\pi$$
0.432283 + 0.901738i $$0.357708\pi$$
$$662$$ 0 0
$$663$$ −2.14134 −0.0831626
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 14.4040 0.557726
$$668$$ 0 0
$$669$$ −6.56534 −0.253831
$$670$$ 0 0
$$671$$ −65.4634 −2.52719
$$672$$ 0 0
$$673$$ 1.46264 0.0563807 0.0281903 0.999603i $$-0.491026\pi$$
0.0281903 + 0.999603i $$0.491026\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 18.5653 0.713524 0.356762 0.934195i $$-0.383881\pi$$
0.356762 + 0.934195i $$0.383881\pi$$
$$678$$ 0 0
$$679$$ 22.9694 0.881484
$$680$$ 0 0
$$681$$ 13.3820 0.512799
$$682$$ 0 0
$$683$$ 6.03138 0.230784 0.115392 0.993320i $$-0.463188\pi$$
0.115392 + 0.993320i $$0.463188\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 9.88665 0.377199
$$688$$ 0 0
$$689$$ −12.0093 −0.457519
$$690$$ 0 0
$$691$$ 14.1566 0.538543 0.269271 0.963064i $$-0.413217\pi$$
0.269271 + 0.963064i $$0.413217\pi$$
$$692$$ 0 0
$$693$$ 7.69735 0.292398
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 12.5653 0.475946
$$698$$ 0 0
$$699$$ −26.8667 −1.01619
$$700$$ 0 0
$$701$$ 6.26270 0.236539 0.118269 0.992982i $$-0.462265\pi$$
0.118269 + 0.992982i $$0.462265\pi$$
$$702$$ 0 0
$$703$$ 9.33996 0.352263
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 8.01272 0.301349
$$708$$ 0 0
$$709$$ 6.28267 0.235951 0.117975 0.993017i $$-0.462360\pi$$
0.117975 + 0.993017i $$0.462360\pi$$
$$710$$ 0 0
$$711$$ 2.31198 0.0867059
$$712$$ 0 0
$$713$$ 4.48808 0.168080
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 27.4847 1.02643
$$718$$ 0 0
$$719$$ −28.9253 −1.07873 −0.539366 0.842072i $$-0.681336\pi$$
−0.539366 + 0.842072i $$0.681336\pi$$
$$720$$ 0 0
$$721$$ 19.1120 0.711769
$$722$$ 0 0
$$723$$ −10.0187 −0.372598
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −40.2427 −1.49252 −0.746260 0.665655i $$-0.768152\pi$$
−0.746260 + 0.665655i $$0.768152\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −19.2920 −0.713540
$$732$$ 0 0
$$733$$ 19.7907 0.730987 0.365494 0.930814i $$-0.380900\pi$$
0.365494 + 0.930814i $$0.380900\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 1.83323 0.0675278
$$738$$ 0 0
$$739$$ −18.1634 −0.668151 −0.334075 0.942546i $$-0.608424\pi$$
−0.334075 + 0.942546i $$0.608424\pi$$
$$740$$ 0 0
$$741$$ −2.41468 −0.0887055
$$742$$ 0 0
$$743$$ 24.2661 0.890236 0.445118 0.895472i $$-0.353162\pi$$
0.445118 + 0.895472i $$0.353162\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 10.3727 0.379516
$$748$$ 0 0
$$749$$ 14.8294 0.541853
$$750$$ 0 0
$$751$$ 28.2720 1.03166 0.515830 0.856691i $$-0.327483\pi$$
0.515830 + 0.856691i $$0.327483\pi$$
$$752$$ 0 0
$$753$$ −24.1507 −0.880099
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 37.0853 1.34789 0.673944 0.738783i $$-0.264599\pi$$
0.673944 + 0.738783i $$0.264599\pi$$
$$758$$ 0 0
$$759$$ 15.3947 0.558792
$$760$$ 0 0
$$761$$ 28.4919 1.03283 0.516416 0.856338i $$-0.327266\pi$$
0.516416 + 0.856338i $$0.327266\pi$$
$$762$$ 0 0
$$763$$ 4.88797 0.176956
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −6.19269 −0.223605
$$768$$ 0 0
$$769$$ −27.6774 −0.998072 −0.499036 0.866581i $$-0.666312\pi$$
−0.499036 + 0.866581i $$0.666312\pi$$
$$770$$ 0 0
$$771$$ 1.15066 0.0414402
$$772$$ 0 0
$$773$$ 32.6426 1.17407 0.587037 0.809560i $$-0.300294\pi$$
0.587037 + 0.809560i $$0.300294\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −5.27334 −0.189180
$$778$$ 0 0
$$779$$ 14.1693 0.507669
$$780$$ 0 0
$$781$$ −67.0067 −2.39769
$$782$$ 0 0
$$783$$ −5.28267 −0.188787
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −50.6740 −1.80633 −0.903166 0.429291i $$-0.858764\pi$$
−0.903166 + 0.429291i $$0.858764\pi$$
$$788$$ 0 0
$$789$$ −2.28267 −0.0812653
$$790$$ 0 0
$$791$$ −25.3107 −0.899943
$$792$$ 0 0
$$793$$ −11.5946 −0.411738
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −24.3586 −0.862827 −0.431413 0.902154i $$-0.641985\pi$$
−0.431413 + 0.902154i $$0.641985\pi$$
$$798$$ 0 0
$$799$$ 8.08998 0.286203
$$800$$ 0 0
$$801$$ 3.73599 0.132005
$$802$$ 0 0
$$803$$ −46.7640 −1.65026
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 8.17064 0.287620
$$808$$ 0 0
$$809$$ −50.6027 −1.77909 −0.889547 0.456843i $$-0.848980\pi$$
−0.889547 + 0.456843i $$0.848980\pi$$
$$810$$ 0 0
$$811$$ 18.4168 0.646700 0.323350 0.946280i $$-0.395191\pi$$
0.323350 + 0.946280i $$0.395191\pi$$
$$812$$ 0 0
$$813$$ 21.5433 0.755556
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −21.7546 −0.761099
$$818$$ 0 0
$$819$$ 1.36333 0.0476385
$$820$$ 0 0
$$821$$ −31.4906 −1.09903 −0.549515 0.835484i $$-0.685188\pi$$
−0.549515 + 0.835484i $$0.685188\pi$$
$$822$$ 0 0
$$823$$ 18.7347 0.653049 0.326525 0.945189i $$-0.394122\pi$$
0.326525 + 0.945189i $$0.394122\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −16.7580 −0.582734 −0.291367 0.956611i $$-0.594110\pi$$
−0.291367 + 0.956611i $$0.594110\pi$$
$$828$$ 0 0
$$829$$ −20.1014 −0.698150 −0.349075 0.937095i $$-0.613504\pi$$
−0.349075 + 0.937095i $$0.613504\pi$$
$$830$$ 0 0
$$831$$ −23.1307 −0.802395
$$832$$ 0 0
$$833$$ −11.0093 −0.381451
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −1.64600 −0.0568941
$$838$$ 0 0
$$839$$ −6.82936 −0.235776 −0.117888 0.993027i $$-0.537612\pi$$
−0.117888 + 0.993027i $$0.537612\pi$$
$$840$$ 0 0
$$841$$ −1.09337 −0.0377026
$$842$$ 0 0
$$843$$ 4.96137 0.170879
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −28.4626 −0.977988
$$848$$ 0 0
$$849$$ 4.74531 0.162859
$$850$$ 0 0
$$851$$ −10.5467 −0.361536
$$852$$ 0 0
$$853$$ 10.2279 0.350198 0.175099 0.984551i $$-0.443975\pi$$
0.175099 + 0.984551i $$0.443975\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 31.1307 1.06340 0.531702 0.846931i $$-0.321553\pi$$
0.531702 + 0.846931i $$0.321553\pi$$
$$858$$ 0 0
$$859$$ −33.0093 −1.12626 −0.563132 0.826367i $$-0.690404\pi$$
−0.563132 + 0.826367i $$0.690404\pi$$
$$860$$ 0 0
$$861$$ −8.00000 −0.272639
$$862$$ 0 0
$$863$$ 44.4206 1.51210 0.756048 0.654516i $$-0.227128\pi$$
0.756048 + 0.654516i $$0.227128\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −12.4147 −0.421625
$$868$$ 0 0
$$869$$ −13.0534 −0.442807
$$870$$ 0 0
$$871$$ 0.324695 0.0110019
$$872$$ 0 0
$$873$$ −16.8480 −0.570219
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 15.8867 0.536454 0.268227 0.963356i $$-0.413562\pi$$
0.268227 + 0.963356i $$0.413562\pi$$
$$878$$ 0 0
$$879$$ 29.7360 1.00297
$$880$$ 0 0
$$881$$ 3.13201 0.105520 0.0527600 0.998607i $$-0.483198\pi$$
0.0527600 + 0.998607i $$0.483198\pi$$
$$882$$ 0 0
$$883$$ −33.1307 −1.11494 −0.557468 0.830198i $$-0.688227\pi$$
−0.557468 + 0.830198i $$0.688227\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −19.8200 −0.665491 −0.332746 0.943017i $$-0.607975\pi$$
−0.332746 + 0.943017i $$0.607975\pi$$
$$888$$ 0 0
$$889$$ 1.17064 0.0392620
$$890$$ 0 0
$$891$$ −5.64600 −0.189148
$$892$$ 0 0
$$893$$ 9.12268 0.305279
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 2.72666 0.0910404
$$898$$ 0 0
$$899$$ 8.69528 0.290004
$$900$$ 0 0
$$901$$ 25.7160 0.856724
$$902$$ 0 0
$$903$$ 12.2827 0.408742
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −14.6426 −0.486200 −0.243100 0.970001i $$-0.578164\pi$$
−0.243100 + 0.970001i $$0.578164\pi$$
$$908$$ 0 0
$$909$$ −5.87732 −0.194938
$$910$$ 0 0
$$911$$ −37.8973 −1.25559 −0.627797 0.778377i $$-0.716043\pi$$
−0.627797 + 0.778377i $$0.716043\pi$$
$$912$$ 0 0
$$913$$ −58.5640 −1.93819
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 2.54669 0.0840990
$$918$$ 0 0
$$919$$ 26.2534 0.866019 0.433009 0.901389i $$-0.357452\pi$$
0.433009 + 0.901389i $$0.357452\pi$$
$$920$$ 0 0
$$921$$ 18.4520 0.608014
$$922$$ 0 0
$$923$$ −11.8680 −0.390640
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −14.0187 −0.460433
$$928$$ 0 0
$$929$$ 52.2066 1.71284 0.856422 0.516276i $$-0.172682\pi$$
0.856422 + 0.516276i $$0.172682\pi$$
$$930$$ 0 0
$$931$$ −12.4147 −0.406875
$$932$$ 0 0
$$933$$ 15.9160 0.521065
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −16.3213 −0.533194 −0.266597 0.963808i $$-0.585899\pi$$
−0.266597 + 0.963808i $$0.585899\pi$$
$$938$$ 0 0
$$939$$ −1.99067 −0.0649631
$$940$$ 0 0
$$941$$ 35.7546 1.16557 0.582784 0.812627i $$-0.301963\pi$$
0.582784 + 0.812627i $$0.301963\pi$$
$$942$$ 0 0
$$943$$ −16.0000 −0.521032
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 8.73260 0.283771 0.141886 0.989883i $$-0.454683\pi$$
0.141886 + 0.989883i $$0.454683\pi$$
$$948$$ 0 0
$$949$$ −8.28267 −0.268867
$$950$$ 0 0
$$951$$ −2.26401 −0.0734157
$$952$$ 0 0
$$953$$ 28.8280 0.933832 0.466916 0.884302i $$-0.345365\pi$$
0.466916 + 0.884302i $$0.345365\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 29.8260 0.964137
$$958$$ 0 0
$$959$$ 7.14935 0.230864
$$960$$ 0 0
$$961$$ −28.2907 −0.912603
$$962$$ 0 0
$$963$$ −10.8773 −0.350517
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 38.0687 1.22421 0.612103 0.790778i $$-0.290324\pi$$
0.612103 + 0.790778i $$0.290324\pi$$
$$968$$ 0 0
$$969$$ 5.17064 0.166105
$$970$$ 0 0
$$971$$ 1.72534 0.0553687 0.0276844 0.999617i $$-0.491187\pi$$
0.0276844 + 0.999617i $$0.491187\pi$$
$$972$$ 0 0
$$973$$ −11.1520 −0.357516
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 21.7173 0.694799 0.347399 0.937717i $$-0.387065\pi$$
0.347399 + 0.937717i $$0.387065\pi$$
$$978$$ 0 0
$$979$$ −21.0934 −0.674147
$$980$$ 0 0
$$981$$ −3.58532 −0.114470
$$982$$ 0 0
$$983$$ 13.0153 0.415123 0.207561 0.978222i $$-0.433447\pi$$
0.207561 + 0.978222i $$0.433447\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −5.15066 −0.163947
$$988$$ 0 0
$$989$$ 24.5653 0.781133
$$990$$ 0 0
$$991$$ 23.1413 0.735109 0.367554 0.930002i $$-0.380195\pi$$
0.367554 + 0.930002i $$0.380195\pi$$
$$992$$ 0 0
$$993$$ 1.69867 0.0539057
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 34.9707 1.10753 0.553767 0.832672i $$-0.313190\pi$$
0.553767 + 0.832672i $$0.313190\pi$$
$$998$$ 0 0
$$999$$ 3.86799 0.122378
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 7800.2.a.bn.1.2 yes 3
5.4 even 2 7800.2.a.bm.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
7800.2.a.bm.1.2 3 5.4 even 2
7800.2.a.bn.1.2 yes 3 1.1 even 1 trivial