# Properties

 Label 2175.2.a.m.1.2 Level $2175$ Weight $2$ Character 2175.1 Self dual yes Analytic conductor $17.367$ 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] = [2175,2,Mod(1,2175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2175, 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("2175.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2175 = 3 \cdot 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2175.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: $$17.3674624396$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 435) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 2175.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.56155 q^{2} -1.00000 q^{3} +0.438447 q^{4} -1.56155 q^{6} -5.12311 q^{7} -2.43845 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.56155 q^{2} -1.00000 q^{3} +0.438447 q^{4} -1.56155 q^{6} -5.12311 q^{7} -2.43845 q^{8} +1.00000 q^{9} -1.43845 q^{11} -0.438447 q^{12} +2.00000 q^{13} -8.00000 q^{14} -4.68466 q^{16} +7.12311 q^{17} +1.56155 q^{18} +5.12311 q^{19} +5.12311 q^{21} -2.24621 q^{22} -6.56155 q^{23} +2.43845 q^{24} +3.12311 q^{26} -1.00000 q^{27} -2.24621 q^{28} +1.00000 q^{29} +4.00000 q^{31} -2.43845 q^{32} +1.43845 q^{33} +11.1231 q^{34} +0.438447 q^{36} +1.68466 q^{37} +8.00000 q^{38} -2.00000 q^{39} -1.68466 q^{41} +8.00000 q^{42} -7.68466 q^{43} -0.630683 q^{44} -10.2462 q^{46} +13.1231 q^{47} +4.68466 q^{48} +19.2462 q^{49} -7.12311 q^{51} +0.876894 q^{52} +3.43845 q^{53} -1.56155 q^{54} +12.4924 q^{56} -5.12311 q^{57} +1.56155 q^{58} +12.0000 q^{59} +0.876894 q^{61} +6.24621 q^{62} -5.12311 q^{63} +5.56155 q^{64} +2.24621 q^{66} +11.3693 q^{67} +3.12311 q^{68} +6.56155 q^{69} -2.87689 q^{71} -2.43845 q^{72} -1.68466 q^{73} +2.63068 q^{74} +2.24621 q^{76} +7.36932 q^{77} -3.12311 q^{78} -12.0000 q^{79} +1.00000 q^{81} -2.63068 q^{82} +2.56155 q^{83} +2.24621 q^{84} -12.0000 q^{86} -1.00000 q^{87} +3.50758 q^{88} +12.2462 q^{89} -10.2462 q^{91} -2.87689 q^{92} -4.00000 q^{93} +20.4924 q^{94} +2.43845 q^{96} +5.68466 q^{97} +30.0540 q^{98} -1.43845 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - 2 q^{3} + 5 q^{4} + q^{6} - 2 q^{7} - 9 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - q^2 - 2 * q^3 + 5 * q^4 + q^6 - 2 * q^7 - 9 * q^8 + 2 * q^9 $$2 q - q^{2} - 2 q^{3} + 5 q^{4} + q^{6} - 2 q^{7} - 9 q^{8} + 2 q^{9} - 7 q^{11} - 5 q^{12} + 4 q^{13} - 16 q^{14} + 3 q^{16} + 6 q^{17} - q^{18} + 2 q^{19} + 2 q^{21} + 12 q^{22} - 9 q^{23} + 9 q^{24} - 2 q^{26} - 2 q^{27} + 12 q^{28} + 2 q^{29} + 8 q^{31} - 9 q^{32} + 7 q^{33} + 14 q^{34} + 5 q^{36} - 9 q^{37} + 16 q^{38} - 4 q^{39} + 9 q^{41} + 16 q^{42} - 3 q^{43} - 26 q^{44} - 4 q^{46} + 18 q^{47} - 3 q^{48} + 22 q^{49} - 6 q^{51} + 10 q^{52} + 11 q^{53} + q^{54} - 8 q^{56} - 2 q^{57} - q^{58} + 24 q^{59} + 10 q^{61} - 4 q^{62} - 2 q^{63} + 7 q^{64} - 12 q^{66} - 2 q^{67} - 2 q^{68} + 9 q^{69} - 14 q^{71} - 9 q^{72} + 9 q^{73} + 30 q^{74} - 12 q^{76} - 10 q^{77} + 2 q^{78} - 24 q^{79} + 2 q^{81} - 30 q^{82} + q^{83} - 12 q^{84} - 24 q^{86} - 2 q^{87} + 40 q^{88} + 8 q^{89} - 4 q^{91} - 14 q^{92} - 8 q^{93} + 8 q^{94} + 9 q^{96} - q^{97} + 23 q^{98} - 7 q^{99}+O(q^{100})$$ 2 * q - q^2 - 2 * q^3 + 5 * q^4 + q^6 - 2 * q^7 - 9 * q^8 + 2 * q^9 - 7 * q^11 - 5 * q^12 + 4 * q^13 - 16 * q^14 + 3 * q^16 + 6 * q^17 - q^18 + 2 * q^19 + 2 * q^21 + 12 * q^22 - 9 * q^23 + 9 * q^24 - 2 * q^26 - 2 * q^27 + 12 * q^28 + 2 * q^29 + 8 * q^31 - 9 * q^32 + 7 * q^33 + 14 * q^34 + 5 * q^36 - 9 * q^37 + 16 * q^38 - 4 * q^39 + 9 * q^41 + 16 * q^42 - 3 * q^43 - 26 * q^44 - 4 * q^46 + 18 * q^47 - 3 * q^48 + 22 * q^49 - 6 * q^51 + 10 * q^52 + 11 * q^53 + q^54 - 8 * q^56 - 2 * q^57 - q^58 + 24 * q^59 + 10 * q^61 - 4 * q^62 - 2 * q^63 + 7 * q^64 - 12 * q^66 - 2 * q^67 - 2 * q^68 + 9 * q^69 - 14 * q^71 - 9 * q^72 + 9 * q^73 + 30 * q^74 - 12 * q^76 - 10 * q^77 + 2 * q^78 - 24 * q^79 + 2 * q^81 - 30 * q^82 + q^83 - 12 * q^84 - 24 * q^86 - 2 * q^87 + 40 * q^88 + 8 * q^89 - 4 * q^91 - 14 * q^92 - 8 * q^93 + 8 * q^94 + 9 * q^96 - q^97 + 23 * q^98 - 7 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.56155 1.10418 0.552092 0.833783i $$-0.313830\pi$$
0.552092 + 0.833783i $$0.313830\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ 0.438447 0.219224
$$5$$ 0 0
$$6$$ −1.56155 −0.637501
$$7$$ −5.12311 −1.93635 −0.968176 0.250270i $$-0.919480\pi$$
−0.968176 + 0.250270i $$0.919480\pi$$
$$8$$ −2.43845 −0.862121
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −1.43845 −0.433708 −0.216854 0.976204i $$-0.569580\pi$$
−0.216854 + 0.976204i $$0.569580\pi$$
$$12$$ −0.438447 −0.126569
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ −8.00000 −2.13809
$$15$$ 0 0
$$16$$ −4.68466 −1.17116
$$17$$ 7.12311 1.72761 0.863803 0.503829i $$-0.168076\pi$$
0.863803 + 0.503829i $$0.168076\pi$$
$$18$$ 1.56155 0.368062
$$19$$ 5.12311 1.17532 0.587661 0.809108i $$-0.300049\pi$$
0.587661 + 0.809108i $$0.300049\pi$$
$$20$$ 0 0
$$21$$ 5.12311 1.11795
$$22$$ −2.24621 −0.478894
$$23$$ −6.56155 −1.36818 −0.684089 0.729398i $$-0.739800\pi$$
−0.684089 + 0.729398i $$0.739800\pi$$
$$24$$ 2.43845 0.497746
$$25$$ 0 0
$$26$$ 3.12311 0.612491
$$27$$ −1.00000 −0.192450
$$28$$ −2.24621 −0.424494
$$29$$ 1.00000 0.185695
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ −2.43845 −0.431061
$$33$$ 1.43845 0.250402
$$34$$ 11.1231 1.90760
$$35$$ 0 0
$$36$$ 0.438447 0.0730745
$$37$$ 1.68466 0.276956 0.138478 0.990366i $$-0.455779\pi$$
0.138478 + 0.990366i $$0.455779\pi$$
$$38$$ 8.00000 1.29777
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ −1.68466 −0.263099 −0.131550 0.991310i $$-0.541995\pi$$
−0.131550 + 0.991310i $$0.541995\pi$$
$$42$$ 8.00000 1.23443
$$43$$ −7.68466 −1.17190 −0.585950 0.810347i $$-0.699278\pi$$
−0.585950 + 0.810347i $$0.699278\pi$$
$$44$$ −0.630683 −0.0950791
$$45$$ 0 0
$$46$$ −10.2462 −1.51072
$$47$$ 13.1231 1.91420 0.957101 0.289755i $$-0.0935738\pi$$
0.957101 + 0.289755i $$0.0935738\pi$$
$$48$$ 4.68466 0.676172
$$49$$ 19.2462 2.74946
$$50$$ 0 0
$$51$$ −7.12311 −0.997434
$$52$$ 0.876894 0.121603
$$53$$ 3.43845 0.472307 0.236154 0.971716i $$-0.424113\pi$$
0.236154 + 0.971716i $$0.424113\pi$$
$$54$$ −1.56155 −0.212500
$$55$$ 0 0
$$56$$ 12.4924 1.66937
$$57$$ −5.12311 −0.678572
$$58$$ 1.56155 0.205042
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 0 0
$$61$$ 0.876894 0.112275 0.0561374 0.998423i $$-0.482122\pi$$
0.0561374 + 0.998423i $$0.482122\pi$$
$$62$$ 6.24621 0.793270
$$63$$ −5.12311 −0.645451
$$64$$ 5.56155 0.695194
$$65$$ 0 0
$$66$$ 2.24621 0.276489
$$67$$ 11.3693 1.38898 0.694492 0.719501i $$-0.255629\pi$$
0.694492 + 0.719501i $$0.255629\pi$$
$$68$$ 3.12311 0.378732
$$69$$ 6.56155 0.789918
$$70$$ 0 0
$$71$$ −2.87689 −0.341425 −0.170712 0.985321i $$-0.554607\pi$$
−0.170712 + 0.985321i $$0.554607\pi$$
$$72$$ −2.43845 −0.287374
$$73$$ −1.68466 −0.197174 −0.0985872 0.995128i $$-0.531432\pi$$
−0.0985872 + 0.995128i $$0.531432\pi$$
$$74$$ 2.63068 0.305811
$$75$$ 0 0
$$76$$ 2.24621 0.257658
$$77$$ 7.36932 0.839812
$$78$$ −3.12311 −0.353622
$$79$$ −12.0000 −1.35011 −0.675053 0.737769i $$-0.735879\pi$$
−0.675053 + 0.737769i $$0.735879\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −2.63068 −0.290510
$$83$$ 2.56155 0.281167 0.140583 0.990069i $$-0.455102\pi$$
0.140583 + 0.990069i $$0.455102\pi$$
$$84$$ 2.24621 0.245082
$$85$$ 0 0
$$86$$ −12.0000 −1.29399
$$87$$ −1.00000 −0.107211
$$88$$ 3.50758 0.373909
$$89$$ 12.2462 1.29810 0.649048 0.760748i $$-0.275167\pi$$
0.649048 + 0.760748i $$0.275167\pi$$
$$90$$ 0 0
$$91$$ −10.2462 −1.07409
$$92$$ −2.87689 −0.299937
$$93$$ −4.00000 −0.414781
$$94$$ 20.4924 2.11363
$$95$$ 0 0
$$96$$ 2.43845 0.248873
$$97$$ 5.68466 0.577190 0.288595 0.957451i $$-0.406812\pi$$
0.288595 + 0.957451i $$0.406812\pi$$
$$98$$ 30.0540 3.03591
$$99$$ −1.43845 −0.144569
$$100$$ 0 0
$$101$$ −8.56155 −0.851906 −0.425953 0.904745i $$-0.640061\pi$$
−0.425953 + 0.904745i $$0.640061\pi$$
$$102$$ −11.1231 −1.10135
$$103$$ 2.87689 0.283469 0.141734 0.989905i $$-0.454732\pi$$
0.141734 + 0.989905i $$0.454732\pi$$
$$104$$ −4.87689 −0.478219
$$105$$ 0 0
$$106$$ 5.36932 0.521514
$$107$$ −16.4924 −1.59438 −0.797191 0.603727i $$-0.793682\pi$$
−0.797191 + 0.603727i $$0.793682\pi$$
$$108$$ −0.438447 −0.0421896
$$109$$ −5.68466 −0.544492 −0.272246 0.962228i $$-0.587766\pi$$
−0.272246 + 0.962228i $$0.587766\pi$$
$$110$$ 0 0
$$111$$ −1.68466 −0.159901
$$112$$ 24.0000 2.26779
$$113$$ 4.87689 0.458780 0.229390 0.973335i $$-0.426327\pi$$
0.229390 + 0.973335i $$0.426327\pi$$
$$114$$ −8.00000 −0.749269
$$115$$ 0 0
$$116$$ 0.438447 0.0407088
$$117$$ 2.00000 0.184900
$$118$$ 18.7386 1.72503
$$119$$ −36.4924 −3.34525
$$120$$ 0 0
$$121$$ −8.93087 −0.811897
$$122$$ 1.36932 0.123972
$$123$$ 1.68466 0.151901
$$124$$ 1.75379 0.157495
$$125$$ 0 0
$$126$$ −8.00000 −0.712697
$$127$$ −4.31534 −0.382925 −0.191462 0.981500i $$-0.561323\pi$$
−0.191462 + 0.981500i $$0.561323\pi$$
$$128$$ 13.5616 1.19868
$$129$$ 7.68466 0.676596
$$130$$ 0 0
$$131$$ 10.2462 0.895216 0.447608 0.894230i $$-0.352276\pi$$
0.447608 + 0.894230i $$0.352276\pi$$
$$132$$ 0.630683 0.0548939
$$133$$ −26.2462 −2.27584
$$134$$ 17.7538 1.53369
$$135$$ 0 0
$$136$$ −17.3693 −1.48941
$$137$$ 15.1231 1.29205 0.646027 0.763315i $$-0.276429\pi$$
0.646027 + 0.763315i $$0.276429\pi$$
$$138$$ 10.2462 0.872215
$$139$$ −17.9309 −1.52088 −0.760438 0.649410i $$-0.775016\pi$$
−0.760438 + 0.649410i $$0.775016\pi$$
$$140$$ 0 0
$$141$$ −13.1231 −1.10516
$$142$$ −4.49242 −0.376996
$$143$$ −2.87689 −0.240578
$$144$$ −4.68466 −0.390388
$$145$$ 0 0
$$146$$ −2.63068 −0.217717
$$147$$ −19.2462 −1.58740
$$148$$ 0.738634 0.0607153
$$149$$ 0.246211 0.0201704 0.0100852 0.999949i $$-0.496790\pi$$
0.0100852 + 0.999949i $$0.496790\pi$$
$$150$$ 0 0
$$151$$ 4.31534 0.351178 0.175589 0.984464i $$-0.443817\pi$$
0.175589 + 0.984464i $$0.443817\pi$$
$$152$$ −12.4924 −1.01327
$$153$$ 7.12311 0.575869
$$154$$ 11.5076 0.927307
$$155$$ 0 0
$$156$$ −0.876894 −0.0702077
$$157$$ 14.0000 1.11732 0.558661 0.829396i $$-0.311315\pi$$
0.558661 + 0.829396i $$0.311315\pi$$
$$158$$ −18.7386 −1.49077
$$159$$ −3.43845 −0.272687
$$160$$ 0 0
$$161$$ 33.6155 2.64927
$$162$$ 1.56155 0.122687
$$163$$ 17.9309 1.40445 0.702227 0.711953i $$-0.252189\pi$$
0.702227 + 0.711953i $$0.252189\pi$$
$$164$$ −0.738634 −0.0576776
$$165$$ 0 0
$$166$$ 4.00000 0.310460
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ −12.4924 −0.963811
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 5.12311 0.391774
$$172$$ −3.36932 −0.256908
$$173$$ −12.5616 −0.955037 −0.477519 0.878622i $$-0.658464\pi$$
−0.477519 + 0.878622i $$0.658464\pi$$
$$174$$ −1.56155 −0.118381
$$175$$ 0 0
$$176$$ 6.73863 0.507944
$$177$$ −12.0000 −0.901975
$$178$$ 19.1231 1.43334
$$179$$ 1.12311 0.0839449 0.0419724 0.999119i $$-0.486636\pi$$
0.0419724 + 0.999119i $$0.486636\pi$$
$$180$$ 0 0
$$181$$ 25.6847 1.90913 0.954563 0.298010i $$-0.0963228\pi$$
0.954563 + 0.298010i $$0.0963228\pi$$
$$182$$ −16.0000 −1.18600
$$183$$ −0.876894 −0.0648219
$$184$$ 16.0000 1.17954
$$185$$ 0 0
$$186$$ −6.24621 −0.457994
$$187$$ −10.2462 −0.749277
$$188$$ 5.75379 0.419638
$$189$$ 5.12311 0.372651
$$190$$ 0 0
$$191$$ −9.93087 −0.718573 −0.359286 0.933227i $$-0.616980\pi$$
−0.359286 + 0.933227i $$0.616980\pi$$
$$192$$ −5.56155 −0.401371
$$193$$ 10.0000 0.719816 0.359908 0.932988i $$-0.382808\pi$$
0.359908 + 0.932988i $$0.382808\pi$$
$$194$$ 8.87689 0.637324
$$195$$ 0 0
$$196$$ 8.43845 0.602746
$$197$$ −4.56155 −0.324997 −0.162499 0.986709i $$-0.551955\pi$$
−0.162499 + 0.986709i $$0.551955\pi$$
$$198$$ −2.24621 −0.159631
$$199$$ −11.0540 −0.783596 −0.391798 0.920051i $$-0.628147\pi$$
−0.391798 + 0.920051i $$0.628147\pi$$
$$200$$ 0 0
$$201$$ −11.3693 −0.801930
$$202$$ −13.3693 −0.940662
$$203$$ −5.12311 −0.359572
$$204$$ −3.12311 −0.218661
$$205$$ 0 0
$$206$$ 4.49242 0.313002
$$207$$ −6.56155 −0.456059
$$208$$ −9.36932 −0.649645
$$209$$ −7.36932 −0.509746
$$210$$ 0 0
$$211$$ −2.87689 −0.198054 −0.0990268 0.995085i $$-0.531573\pi$$
−0.0990268 + 0.995085i $$0.531573\pi$$
$$212$$ 1.50758 0.103541
$$213$$ 2.87689 0.197122
$$214$$ −25.7538 −1.76049
$$215$$ 0 0
$$216$$ 2.43845 0.165915
$$217$$ −20.4924 −1.39112
$$218$$ −8.87689 −0.601219
$$219$$ 1.68466 0.113839
$$220$$ 0 0
$$221$$ 14.2462 0.958304
$$222$$ −2.63068 −0.176560
$$223$$ 18.2462 1.22186 0.610928 0.791686i $$-0.290796\pi$$
0.610928 + 0.791686i $$0.290796\pi$$
$$224$$ 12.4924 0.834685
$$225$$ 0 0
$$226$$ 7.61553 0.506577
$$227$$ 3.19224 0.211876 0.105938 0.994373i $$-0.466215\pi$$
0.105938 + 0.994373i $$0.466215\pi$$
$$228$$ −2.24621 −0.148759
$$229$$ 22.0000 1.45380 0.726900 0.686743i $$-0.240960\pi$$
0.726900 + 0.686743i $$0.240960\pi$$
$$230$$ 0 0
$$231$$ −7.36932 −0.484865
$$232$$ −2.43845 −0.160092
$$233$$ 10.3153 0.675780 0.337890 0.941186i $$-0.390287\pi$$
0.337890 + 0.941186i $$0.390287\pi$$
$$234$$ 3.12311 0.204164
$$235$$ 0 0
$$236$$ 5.26137 0.342486
$$237$$ 12.0000 0.779484
$$238$$ −56.9848 −3.69378
$$239$$ −13.1231 −0.848863 −0.424432 0.905460i $$-0.639526\pi$$
−0.424432 + 0.905460i $$0.639526\pi$$
$$240$$ 0 0
$$241$$ −25.0540 −1.61387 −0.806934 0.590641i $$-0.798875\pi$$
−0.806934 + 0.590641i $$0.798875\pi$$
$$242$$ −13.9460 −0.896484
$$243$$ −1.00000 −0.0641500
$$244$$ 0.384472 0.0246133
$$245$$ 0 0
$$246$$ 2.63068 0.167726
$$247$$ 10.2462 0.651951
$$248$$ −9.75379 −0.619366
$$249$$ −2.56155 −0.162332
$$250$$ 0 0
$$251$$ 2.24621 0.141780 0.0708898 0.997484i $$-0.477416\pi$$
0.0708898 + 0.997484i $$0.477416\pi$$
$$252$$ −2.24621 −0.141498
$$253$$ 9.43845 0.593390
$$254$$ −6.73863 −0.422819
$$255$$ 0 0
$$256$$ 10.0540 0.628373
$$257$$ −11.4384 −0.713511 −0.356755 0.934198i $$-0.616117\pi$$
−0.356755 + 0.934198i $$0.616117\pi$$
$$258$$ 12.0000 0.747087
$$259$$ −8.63068 −0.536285
$$260$$ 0 0
$$261$$ 1.00000 0.0618984
$$262$$ 16.0000 0.988483
$$263$$ 5.75379 0.354794 0.177397 0.984139i $$-0.443232\pi$$
0.177397 + 0.984139i $$0.443232\pi$$
$$264$$ −3.50758 −0.215876
$$265$$ 0 0
$$266$$ −40.9848 −2.51294
$$267$$ −12.2462 −0.749456
$$268$$ 4.98485 0.304498
$$269$$ 11.7538 0.716641 0.358321 0.933599i $$-0.383349\pi$$
0.358321 + 0.933599i $$0.383349\pi$$
$$270$$ 0 0
$$271$$ 17.1231 1.04015 0.520077 0.854119i $$-0.325903\pi$$
0.520077 + 0.854119i $$0.325903\pi$$
$$272$$ −33.3693 −2.02331
$$273$$ 10.2462 0.620129
$$274$$ 23.6155 1.42667
$$275$$ 0 0
$$276$$ 2.87689 0.173169
$$277$$ −0.876894 −0.0526875 −0.0263437 0.999653i $$-0.508386\pi$$
−0.0263437 + 0.999653i $$0.508386\pi$$
$$278$$ −28.0000 −1.67933
$$279$$ 4.00000 0.239474
$$280$$ 0 0
$$281$$ −23.6155 −1.40878 −0.704392 0.709811i $$-0.748780\pi$$
−0.704392 + 0.709811i $$0.748780\pi$$
$$282$$ −20.4924 −1.22031
$$283$$ −6.87689 −0.408789 −0.204394 0.978889i $$-0.565523\pi$$
−0.204394 + 0.978889i $$0.565523\pi$$
$$284$$ −1.26137 −0.0748483
$$285$$ 0 0
$$286$$ −4.49242 −0.265643
$$287$$ 8.63068 0.509453
$$288$$ −2.43845 −0.143687
$$289$$ 33.7386 1.98463
$$290$$ 0 0
$$291$$ −5.68466 −0.333241
$$292$$ −0.738634 −0.0432253
$$293$$ 21.3693 1.24841 0.624204 0.781261i $$-0.285423\pi$$
0.624204 + 0.781261i $$0.285423\pi$$
$$294$$ −30.0540 −1.75278
$$295$$ 0 0
$$296$$ −4.10795 −0.238770
$$297$$ 1.43845 0.0834672
$$298$$ 0.384472 0.0222719
$$299$$ −13.1231 −0.758929
$$300$$ 0 0
$$301$$ 39.3693 2.26921
$$302$$ 6.73863 0.387765
$$303$$ 8.56155 0.491848
$$304$$ −24.0000 −1.37649
$$305$$ 0 0
$$306$$ 11.1231 0.635866
$$307$$ 31.6847 1.80834 0.904169 0.427174i $$-0.140491\pi$$
0.904169 + 0.427174i $$0.140491\pi$$
$$308$$ 3.23106 0.184107
$$309$$ −2.87689 −0.163661
$$310$$ 0 0
$$311$$ 16.3153 0.925158 0.462579 0.886578i $$-0.346924\pi$$
0.462579 + 0.886578i $$0.346924\pi$$
$$312$$ 4.87689 0.276100
$$313$$ 21.3693 1.20787 0.603933 0.797035i $$-0.293600\pi$$
0.603933 + 0.797035i $$0.293600\pi$$
$$314$$ 21.8617 1.23373
$$315$$ 0 0
$$316$$ −5.26137 −0.295975
$$317$$ −4.87689 −0.273914 −0.136957 0.990577i $$-0.543732\pi$$
−0.136957 + 0.990577i $$0.543732\pi$$
$$318$$ −5.36932 −0.301096
$$319$$ −1.43845 −0.0805376
$$320$$ 0 0
$$321$$ 16.4924 0.920517
$$322$$ 52.4924 2.92529
$$323$$ 36.4924 2.03049
$$324$$ 0.438447 0.0243582
$$325$$ 0 0
$$326$$ 28.0000 1.55078
$$327$$ 5.68466 0.314362
$$328$$ 4.10795 0.226824
$$329$$ −67.2311 −3.70657
$$330$$ 0 0
$$331$$ 10.2462 0.563183 0.281591 0.959534i $$-0.409138\pi$$
0.281591 + 0.959534i $$0.409138\pi$$
$$332$$ 1.12311 0.0616384
$$333$$ 1.68466 0.0923187
$$334$$ 0 0
$$335$$ 0 0
$$336$$ −24.0000 −1.30931
$$337$$ 7.75379 0.422376 0.211188 0.977445i $$-0.432267\pi$$
0.211188 + 0.977445i $$0.432267\pi$$
$$338$$ −14.0540 −0.764435
$$339$$ −4.87689 −0.264877
$$340$$ 0 0
$$341$$ −5.75379 −0.311585
$$342$$ 8.00000 0.432590
$$343$$ −62.7386 −3.38757
$$344$$ 18.7386 1.01032
$$345$$ 0 0
$$346$$ −19.6155 −1.05454
$$347$$ 12.1771 0.653700 0.326850 0.945076i $$-0.394013\pi$$
0.326850 + 0.945076i $$0.394013\pi$$
$$348$$ −0.438447 −0.0235032
$$349$$ 0.0691303 0.00370046 0.00185023 0.999998i $$-0.499411\pi$$
0.00185023 + 0.999998i $$0.499411\pi$$
$$350$$ 0 0
$$351$$ −2.00000 −0.106752
$$352$$ 3.50758 0.186955
$$353$$ −30.4924 −1.62295 −0.811474 0.584389i $$-0.801334\pi$$
−0.811474 + 0.584389i $$0.801334\pi$$
$$354$$ −18.7386 −0.995947
$$355$$ 0 0
$$356$$ 5.36932 0.284573
$$357$$ 36.4924 1.93138
$$358$$ 1.75379 0.0926906
$$359$$ −3.19224 −0.168480 −0.0842399 0.996446i $$-0.526846\pi$$
−0.0842399 + 0.996446i $$0.526846\pi$$
$$360$$ 0 0
$$361$$ 7.24621 0.381380
$$362$$ 40.1080 2.10803
$$363$$ 8.93087 0.468749
$$364$$ −4.49242 −0.235467
$$365$$ 0 0
$$366$$ −1.36932 −0.0715753
$$367$$ 19.6847 1.02753 0.513765 0.857931i $$-0.328250\pi$$
0.513765 + 0.857931i $$0.328250\pi$$
$$368$$ 30.7386 1.60236
$$369$$ −1.68466 −0.0876998
$$370$$ 0 0
$$371$$ −17.6155 −0.914553
$$372$$ −1.75379 −0.0909297
$$373$$ −13.3693 −0.692237 −0.346118 0.938191i $$-0.612500\pi$$
−0.346118 + 0.938191i $$0.612500\pi$$
$$374$$ −16.0000 −0.827340
$$375$$ 0 0
$$376$$ −32.0000 −1.65027
$$377$$ 2.00000 0.103005
$$378$$ 8.00000 0.411476
$$379$$ 8.00000 0.410932 0.205466 0.978664i $$-0.434129\pi$$
0.205466 + 0.978664i $$0.434129\pi$$
$$380$$ 0 0
$$381$$ 4.31534 0.221082
$$382$$ −15.5076 −0.793437
$$383$$ −11.0540 −0.564832 −0.282416 0.959292i $$-0.591136\pi$$
−0.282416 + 0.959292i $$0.591136\pi$$
$$384$$ −13.5616 −0.692060
$$385$$ 0 0
$$386$$ 15.6155 0.794809
$$387$$ −7.68466 −0.390633
$$388$$ 2.49242 0.126534
$$389$$ 14.8078 0.750783 0.375392 0.926866i $$-0.377508\pi$$
0.375392 + 0.926866i $$0.377508\pi$$
$$390$$ 0 0
$$391$$ −46.7386 −2.36367
$$392$$ −46.9309 −2.37037
$$393$$ −10.2462 −0.516853
$$394$$ −7.12311 −0.358857
$$395$$ 0 0
$$396$$ −0.630683 −0.0316930
$$397$$ −24.2462 −1.21688 −0.608441 0.793599i $$-0.708205\pi$$
−0.608441 + 0.793599i $$0.708205\pi$$
$$398$$ −17.2614 −0.865234
$$399$$ 26.2462 1.31395
$$400$$ 0 0
$$401$$ 35.6155 1.77855 0.889277 0.457368i $$-0.151208\pi$$
0.889277 + 0.457368i $$0.151208\pi$$
$$402$$ −17.7538 −0.885479
$$403$$ 8.00000 0.398508
$$404$$ −3.75379 −0.186758
$$405$$ 0 0
$$406$$ −8.00000 −0.397033
$$407$$ −2.42329 −0.120118
$$408$$ 17.3693 0.859909
$$409$$ 17.3693 0.858857 0.429429 0.903101i $$-0.358715\pi$$
0.429429 + 0.903101i $$0.358715\pi$$
$$410$$ 0 0
$$411$$ −15.1231 −0.745968
$$412$$ 1.26137 0.0621431
$$413$$ −61.4773 −3.02510
$$414$$ −10.2462 −0.503574
$$415$$ 0 0
$$416$$ −4.87689 −0.239109
$$417$$ 17.9309 0.878078
$$418$$ −11.5076 −0.562854
$$419$$ −27.3693 −1.33708 −0.668539 0.743677i $$-0.733080\pi$$
−0.668539 + 0.743677i $$0.733080\pi$$
$$420$$ 0 0
$$421$$ −10.0000 −0.487370 −0.243685 0.969854i $$-0.578356\pi$$
−0.243685 + 0.969854i $$0.578356\pi$$
$$422$$ −4.49242 −0.218688
$$423$$ 13.1231 0.638067
$$424$$ −8.38447 −0.407186
$$425$$ 0 0
$$426$$ 4.49242 0.217659
$$427$$ −4.49242 −0.217404
$$428$$ −7.23106 −0.349526
$$429$$ 2.87689 0.138898
$$430$$ 0 0
$$431$$ −21.1231 −1.01746 −0.508732 0.860925i $$-0.669886\pi$$
−0.508732 + 0.860925i $$0.669886\pi$$
$$432$$ 4.68466 0.225391
$$433$$ 4.06913 0.195550 0.0977750 0.995209i $$-0.468827\pi$$
0.0977750 + 0.995209i $$0.468827\pi$$
$$434$$ −32.0000 −1.53605
$$435$$ 0 0
$$436$$ −2.49242 −0.119365
$$437$$ −33.6155 −1.60805
$$438$$ 2.63068 0.125699
$$439$$ 12.4924 0.596231 0.298115 0.954530i $$-0.403642\pi$$
0.298115 + 0.954530i $$0.403642\pi$$
$$440$$ 0 0
$$441$$ 19.2462 0.916486
$$442$$ 22.2462 1.05814
$$443$$ −6.24621 −0.296766 −0.148383 0.988930i $$-0.547407\pi$$
−0.148383 + 0.988930i $$0.547407\pi$$
$$444$$ −0.738634 −0.0350540
$$445$$ 0 0
$$446$$ 28.4924 1.34916
$$447$$ −0.246211 −0.0116454
$$448$$ −28.4924 −1.34614
$$449$$ 11.4384 0.539814 0.269907 0.962886i $$-0.413007\pi$$
0.269907 + 0.962886i $$0.413007\pi$$
$$450$$ 0 0
$$451$$ 2.42329 0.114108
$$452$$ 2.13826 0.100575
$$453$$ −4.31534 −0.202752
$$454$$ 4.98485 0.233950
$$455$$ 0 0
$$456$$ 12.4924 0.585011
$$457$$ 8.87689 0.415244 0.207622 0.978209i $$-0.433428\pi$$
0.207622 + 0.978209i $$0.433428\pi$$
$$458$$ 34.3542 1.60526
$$459$$ −7.12311 −0.332478
$$460$$ 0 0
$$461$$ −18.1771 −0.846591 −0.423296 0.905992i $$-0.639127\pi$$
−0.423296 + 0.905992i $$0.639127\pi$$
$$462$$ −11.5076 −0.535381
$$463$$ −25.6155 −1.19045 −0.595227 0.803557i $$-0.702938\pi$$
−0.595227 + 0.803557i $$0.702938\pi$$
$$464$$ −4.68466 −0.217480
$$465$$ 0 0
$$466$$ 16.1080 0.746186
$$467$$ 3.36932 0.155913 0.0779567 0.996957i $$-0.475160\pi$$
0.0779567 + 0.996957i $$0.475160\pi$$
$$468$$ 0.876894 0.0405345
$$469$$ −58.2462 −2.68956
$$470$$ 0 0
$$471$$ −14.0000 −0.645086
$$472$$ −29.2614 −1.34686
$$473$$ 11.0540 0.508262
$$474$$ 18.7386 0.860694
$$475$$ 0 0
$$476$$ −16.0000 −0.733359
$$477$$ 3.43845 0.157436
$$478$$ −20.4924 −0.937302
$$479$$ −22.2462 −1.01646 −0.508228 0.861223i $$-0.669699\pi$$
−0.508228 + 0.861223i $$0.669699\pi$$
$$480$$ 0 0
$$481$$ 3.36932 0.153628
$$482$$ −39.1231 −1.78201
$$483$$ −33.6155 −1.52956
$$484$$ −3.91571 −0.177987
$$485$$ 0 0
$$486$$ −1.56155 −0.0708335
$$487$$ −34.2462 −1.55184 −0.775922 0.630829i $$-0.782715\pi$$
−0.775922 + 0.630829i $$0.782715\pi$$
$$488$$ −2.13826 −0.0967945
$$489$$ −17.9309 −0.810862
$$490$$ 0 0
$$491$$ 6.73863 0.304110 0.152055 0.988372i $$-0.451411\pi$$
0.152055 + 0.988372i $$0.451411\pi$$
$$492$$ 0.738634 0.0333002
$$493$$ 7.12311 0.320809
$$494$$ 16.0000 0.719874
$$495$$ 0 0
$$496$$ −18.7386 −0.841389
$$497$$ 14.7386 0.661118
$$498$$ −4.00000 −0.179244
$$499$$ 42.7386 1.91324 0.956622 0.291332i $$-0.0940984\pi$$
0.956622 + 0.291332i $$0.0940984\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 3.50758 0.156551
$$503$$ 39.3693 1.75539 0.877696 0.479219i $$-0.159080\pi$$
0.877696 + 0.479219i $$0.159080\pi$$
$$504$$ 12.4924 0.556457
$$505$$ 0 0
$$506$$ 14.7386 0.655212
$$507$$ 9.00000 0.399704
$$508$$ −1.89205 −0.0839461
$$509$$ 18.4924 0.819662 0.409831 0.912161i $$-0.365588\pi$$
0.409831 + 0.912161i $$0.365588\pi$$
$$510$$ 0 0
$$511$$ 8.63068 0.381799
$$512$$ −11.4233 −0.504843
$$513$$ −5.12311 −0.226191
$$514$$ −17.8617 −0.787848
$$515$$ 0 0
$$516$$ 3.36932 0.148326
$$517$$ −18.8769 −0.830205
$$518$$ −13.4773 −0.592157
$$519$$ 12.5616 0.551391
$$520$$ 0 0
$$521$$ −0.246211 −0.0107867 −0.00539336 0.999985i $$-0.501717\pi$$
−0.00539336 + 0.999985i $$0.501717\pi$$
$$522$$ 1.56155 0.0683473
$$523$$ −18.7386 −0.819383 −0.409692 0.912224i $$-0.634364\pi$$
−0.409692 + 0.912224i $$0.634364\pi$$
$$524$$ 4.49242 0.196252
$$525$$ 0 0
$$526$$ 8.98485 0.391758
$$527$$ 28.4924 1.24115
$$528$$ −6.73863 −0.293261
$$529$$ 20.0540 0.871912
$$530$$ 0 0
$$531$$ 12.0000 0.520756
$$532$$ −11.5076 −0.498917
$$533$$ −3.36932 −0.145941
$$534$$ −19.1231 −0.827538
$$535$$ 0 0
$$536$$ −27.7235 −1.19747
$$537$$ −1.12311 −0.0484656
$$538$$ 18.3542 0.791304
$$539$$ −27.6847 −1.19246
$$540$$ 0 0
$$541$$ −40.7386 −1.75149 −0.875745 0.482773i $$-0.839629\pi$$
−0.875745 + 0.482773i $$0.839629\pi$$
$$542$$ 26.7386 1.14852
$$543$$ −25.6847 −1.10223
$$544$$ −17.3693 −0.744703
$$545$$ 0 0
$$546$$ 16.0000 0.684737
$$547$$ 42.7386 1.82737 0.913686 0.406421i $$-0.133223\pi$$
0.913686 + 0.406421i $$0.133223\pi$$
$$548$$ 6.63068 0.283249
$$549$$ 0.876894 0.0374249
$$550$$ 0 0
$$551$$ 5.12311 0.218252
$$552$$ −16.0000 −0.681005
$$553$$ 61.4773 2.61428
$$554$$ −1.36932 −0.0581767
$$555$$ 0 0
$$556$$ −7.86174 −0.333412
$$557$$ 20.4233 0.865363 0.432681 0.901547i $$-0.357568\pi$$
0.432681 + 0.901547i $$0.357568\pi$$
$$558$$ 6.24621 0.264423
$$559$$ −15.3693 −0.650053
$$560$$ 0 0
$$561$$ 10.2462 0.432595
$$562$$ −36.8769 −1.55556
$$563$$ −9.12311 −0.384493 −0.192247 0.981347i $$-0.561577\pi$$
−0.192247 + 0.981347i $$0.561577\pi$$
$$564$$ −5.75379 −0.242278
$$565$$ 0 0
$$566$$ −10.7386 −0.451378
$$567$$ −5.12311 −0.215150
$$568$$ 7.01515 0.294349
$$569$$ −16.2462 −0.681077 −0.340538 0.940231i $$-0.610609\pi$$
−0.340538 + 0.940231i $$0.610609\pi$$
$$570$$ 0 0
$$571$$ 39.0540 1.63436 0.817179 0.576384i $$-0.195537\pi$$
0.817179 + 0.576384i $$0.195537\pi$$
$$572$$ −1.26137 −0.0527404
$$573$$ 9.93087 0.414868
$$574$$ 13.4773 0.562530
$$575$$ 0 0
$$576$$ 5.56155 0.231731
$$577$$ 30.4924 1.26942 0.634708 0.772752i $$-0.281120\pi$$
0.634708 + 0.772752i $$0.281120\pi$$
$$578$$ 52.6847 2.19139
$$579$$ −10.0000 −0.415586
$$580$$ 0 0
$$581$$ −13.1231 −0.544438
$$582$$ −8.87689 −0.367959
$$583$$ −4.94602 −0.204843
$$584$$ 4.10795 0.169988
$$585$$ 0 0
$$586$$ 33.3693 1.37847
$$587$$ −24.4924 −1.01091 −0.505455 0.862853i $$-0.668675\pi$$
−0.505455 + 0.862853i $$0.668675\pi$$
$$588$$ −8.43845 −0.347996
$$589$$ 20.4924 0.844376
$$590$$ 0 0
$$591$$ 4.56155 0.187637
$$592$$ −7.89205 −0.324361
$$593$$ −36.2462 −1.48845 −0.744227 0.667927i $$-0.767182\pi$$
−0.744227 + 0.667927i $$0.767182\pi$$
$$594$$ 2.24621 0.0921632
$$595$$ 0 0
$$596$$ 0.107951 0.00442183
$$597$$ 11.0540 0.452409
$$598$$ −20.4924 −0.837997
$$599$$ −20.9848 −0.857418 −0.428709 0.903443i $$-0.641031\pi$$
−0.428709 + 0.903443i $$0.641031\pi$$
$$600$$ 0 0
$$601$$ 41.3693 1.68749 0.843745 0.536745i $$-0.180346\pi$$
0.843745 + 0.536745i $$0.180346\pi$$
$$602$$ 61.4773 2.50563
$$603$$ 11.3693 0.462994
$$604$$ 1.89205 0.0769864
$$605$$ 0 0
$$606$$ 13.3693 0.543091
$$607$$ 16.0000 0.649420 0.324710 0.945814i $$-0.394733\pi$$
0.324710 + 0.945814i $$0.394733\pi$$
$$608$$ −12.4924 −0.506635
$$609$$ 5.12311 0.207599
$$610$$ 0 0
$$611$$ 26.2462 1.06181
$$612$$ 3.12311 0.126244
$$613$$ 2.00000 0.0807792 0.0403896 0.999184i $$-0.487140\pi$$
0.0403896 + 0.999184i $$0.487140\pi$$
$$614$$ 49.4773 1.99674
$$615$$ 0 0
$$616$$ −17.9697 −0.724019
$$617$$ −8.24621 −0.331980 −0.165990 0.986127i $$-0.553082\pi$$
−0.165990 + 0.986127i $$0.553082\pi$$
$$618$$ −4.49242 −0.180712
$$619$$ 29.1231 1.17056 0.585278 0.810833i $$-0.300985\pi$$
0.585278 + 0.810833i $$0.300985\pi$$
$$620$$ 0 0
$$621$$ 6.56155 0.263306
$$622$$ 25.4773 1.02155
$$623$$ −62.7386 −2.51357
$$624$$ 9.36932 0.375073
$$625$$ 0 0
$$626$$ 33.3693 1.33371
$$627$$ 7.36932 0.294302
$$628$$ 6.13826 0.244943
$$629$$ 12.0000 0.478471
$$630$$ 0 0
$$631$$ 32.0000 1.27390 0.636950 0.770905i $$-0.280196\pi$$
0.636950 + 0.770905i $$0.280196\pi$$
$$632$$ 29.2614 1.16395
$$633$$ 2.87689 0.114346
$$634$$ −7.61553 −0.302451
$$635$$ 0 0
$$636$$ −1.50758 −0.0597793
$$637$$ 38.4924 1.52513
$$638$$ −2.24621 −0.0889284
$$639$$ −2.87689 −0.113808
$$640$$ 0 0
$$641$$ 19.4384 0.767773 0.383886 0.923380i $$-0.374585\pi$$
0.383886 + 0.923380i $$0.374585\pi$$
$$642$$ 25.7538 1.01642
$$643$$ 18.7386 0.738980 0.369490 0.929235i $$-0.379532\pi$$
0.369490 + 0.929235i $$0.379532\pi$$
$$644$$ 14.7386 0.580784
$$645$$ 0 0
$$646$$ 56.9848 2.24204
$$647$$ −5.93087 −0.233167 −0.116583 0.993181i $$-0.537194\pi$$
−0.116583 + 0.993181i $$0.537194\pi$$
$$648$$ −2.43845 −0.0957913
$$649$$ −17.2614 −0.677568
$$650$$ 0 0
$$651$$ 20.4924 0.803161
$$652$$ 7.86174 0.307889
$$653$$ −12.2462 −0.479231 −0.239616 0.970868i $$-0.577021\pi$$
−0.239616 + 0.970868i $$0.577021\pi$$
$$654$$ 8.87689 0.347114
$$655$$ 0 0
$$656$$ 7.89205 0.308133
$$657$$ −1.68466 −0.0657248
$$658$$ −104.985 −4.09274
$$659$$ 6.56155 0.255602 0.127801 0.991800i $$-0.459208\pi$$
0.127801 + 0.991800i $$0.459208\pi$$
$$660$$ 0 0
$$661$$ 1.05398 0.0409949 0.0204974 0.999790i $$-0.493475\pi$$
0.0204974 + 0.999790i $$0.493475\pi$$
$$662$$ 16.0000 0.621858
$$663$$ −14.2462 −0.553277
$$664$$ −6.24621 −0.242400
$$665$$ 0 0
$$666$$ 2.63068 0.101937
$$667$$ −6.56155 −0.254064
$$668$$ 0 0
$$669$$ −18.2462 −0.705439
$$670$$ 0 0
$$671$$ −1.26137 −0.0486945
$$672$$ −12.4924 −0.481906
$$673$$ 6.00000 0.231283 0.115642 0.993291i $$-0.463108\pi$$
0.115642 + 0.993291i $$0.463108\pi$$
$$674$$ 12.1080 0.466381
$$675$$ 0 0
$$676$$ −3.94602 −0.151770
$$677$$ −13.8617 −0.532750 −0.266375 0.963869i $$-0.585826\pi$$
−0.266375 + 0.963869i $$0.585826\pi$$
$$678$$ −7.61553 −0.292473
$$679$$ −29.1231 −1.11764
$$680$$ 0 0
$$681$$ −3.19224 −0.122327
$$682$$ −8.98485 −0.344047
$$683$$ 21.4384 0.820319 0.410160 0.912014i $$-0.365473\pi$$
0.410160 + 0.912014i $$0.365473\pi$$
$$684$$ 2.24621 0.0858860
$$685$$ 0 0
$$686$$ −97.9697 −3.74050
$$687$$ −22.0000 −0.839352
$$688$$ 36.0000 1.37249
$$689$$ 6.87689 0.261989
$$690$$ 0 0
$$691$$ −4.00000 −0.152167 −0.0760836 0.997101i $$-0.524242\pi$$
−0.0760836 + 0.997101i $$0.524242\pi$$
$$692$$ −5.50758 −0.209367
$$693$$ 7.36932 0.279937
$$694$$ 19.0152 0.721805
$$695$$ 0 0
$$696$$ 2.43845 0.0924291
$$697$$ −12.0000 −0.454532
$$698$$ 0.107951 0.00408599
$$699$$ −10.3153 −0.390162
$$700$$ 0 0
$$701$$ 3.75379 0.141779 0.0708893 0.997484i $$-0.477416\pi$$
0.0708893 + 0.997484i $$0.477416\pi$$
$$702$$ −3.12311 −0.117874
$$703$$ 8.63068 0.325512
$$704$$ −8.00000 −0.301511
$$705$$ 0 0
$$706$$ −47.6155 −1.79203
$$707$$ 43.8617 1.64959
$$708$$ −5.26137 −0.197734
$$709$$ −12.4233 −0.466567 −0.233283 0.972409i $$-0.574947\pi$$
−0.233283 + 0.972409i $$0.574947\pi$$
$$710$$ 0 0
$$711$$ −12.0000 −0.450035
$$712$$ −29.8617 −1.11912
$$713$$ −26.2462 −0.982928
$$714$$ 56.9848 2.13260
$$715$$ 0 0
$$716$$ 0.492423 0.0184027
$$717$$ 13.1231 0.490091
$$718$$ −4.98485 −0.186033
$$719$$ −4.49242 −0.167539 −0.0837695 0.996485i $$-0.526696\pi$$
−0.0837695 + 0.996485i $$0.526696\pi$$
$$720$$ 0 0
$$721$$ −14.7386 −0.548895
$$722$$ 11.3153 0.421113
$$723$$ 25.0540 0.931767
$$724$$ 11.2614 0.418525
$$725$$ 0 0
$$726$$ 13.9460 0.517586
$$727$$ 38.7386 1.43674 0.718368 0.695663i $$-0.244889\pi$$
0.718368 + 0.695663i $$0.244889\pi$$
$$728$$ 24.9848 0.926000
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −54.7386 −2.02458
$$732$$ −0.384472 −0.0142105
$$733$$ 22.9848 0.848965 0.424482 0.905436i $$-0.360456\pi$$
0.424482 + 0.905436i $$0.360456\pi$$
$$734$$ 30.7386 1.13458
$$735$$ 0 0
$$736$$ 16.0000 0.589768
$$737$$ −16.3542 −0.602413
$$738$$ −2.63068 −0.0968368
$$739$$ −24.0000 −0.882854 −0.441427 0.897297i $$-0.645528\pi$$
−0.441427 + 0.897297i $$0.645528\pi$$
$$740$$ 0 0
$$741$$ −10.2462 −0.376404
$$742$$ −27.5076 −1.00983
$$743$$ −34.8769 −1.27951 −0.639755 0.768579i $$-0.720964\pi$$
−0.639755 + 0.768579i $$0.720964\pi$$
$$744$$ 9.75379 0.357591
$$745$$ 0 0
$$746$$ −20.8769 −0.764357
$$747$$ 2.56155 0.0937223
$$748$$ −4.49242 −0.164259
$$749$$ 84.4924 3.08729
$$750$$ 0 0
$$751$$ 48.4924 1.76951 0.884757 0.466053i $$-0.154324\pi$$
0.884757 + 0.466053i $$0.154324\pi$$
$$752$$ −61.4773 −2.24185
$$753$$ −2.24621 −0.0818565
$$754$$ 3.12311 0.113737
$$755$$ 0 0
$$756$$ 2.24621 0.0816939
$$757$$ −5.68466 −0.206612 −0.103306 0.994650i $$-0.532942\pi$$
−0.103306 + 0.994650i $$0.532942\pi$$
$$758$$ 12.4924 0.453745
$$759$$ −9.43845 −0.342594
$$760$$ 0 0
$$761$$ −24.8769 −0.901787 −0.450893 0.892578i $$-0.648895\pi$$
−0.450893 + 0.892578i $$0.648895\pi$$
$$762$$ 6.73863 0.244115
$$763$$ 29.1231 1.05433
$$764$$ −4.35416 −0.157528
$$765$$ 0 0
$$766$$ −17.2614 −0.623679
$$767$$ 24.0000 0.866590
$$768$$ −10.0540 −0.362792
$$769$$ 43.6155 1.57282 0.786408 0.617707i $$-0.211938\pi$$
0.786408 + 0.617707i $$0.211938\pi$$
$$770$$ 0 0
$$771$$ 11.4384 0.411946
$$772$$ 4.38447 0.157801
$$773$$ −24.7386 −0.889787 −0.444893 0.895584i $$-0.646758\pi$$
−0.444893 + 0.895584i $$0.646758\pi$$
$$774$$ −12.0000 −0.431331
$$775$$ 0 0
$$776$$ −13.8617 −0.497607
$$777$$ 8.63068 0.309624
$$778$$ 23.1231 0.829004
$$779$$ −8.63068 −0.309226
$$780$$ 0 0
$$781$$ 4.13826 0.148079
$$782$$ −72.9848 −2.60993
$$783$$ −1.00000 −0.0357371
$$784$$ −90.1619 −3.22007
$$785$$ 0 0
$$786$$ −16.0000 −0.570701
$$787$$ 31.2311 1.11327 0.556633 0.830758i $$-0.312093\pi$$
0.556633 + 0.830758i $$0.312093\pi$$
$$788$$ −2.00000 −0.0712470
$$789$$ −5.75379 −0.204840
$$790$$ 0 0
$$791$$ −24.9848 −0.888359
$$792$$ 3.50758 0.124636
$$793$$ 1.75379 0.0622789
$$794$$ −37.8617 −1.34366
$$795$$ 0 0
$$796$$ −4.84658 −0.171783
$$797$$ −38.4924 −1.36347 −0.681736 0.731598i $$-0.738775\pi$$
−0.681736 + 0.731598i $$0.738775\pi$$
$$798$$ 40.9848 1.45085
$$799$$ 93.4773 3.30699
$$800$$ 0 0
$$801$$ 12.2462 0.432699
$$802$$ 55.6155 1.96385
$$803$$ 2.42329 0.0855161
$$804$$ −4.98485 −0.175802
$$805$$ 0 0
$$806$$ 12.4924 0.440027
$$807$$ −11.7538 −0.413753
$$808$$ 20.8769 0.734447
$$809$$ −22.1771 −0.779705 −0.389852 0.920877i $$-0.627474\pi$$
−0.389852 + 0.920877i $$0.627474\pi$$
$$810$$ 0 0
$$811$$ 44.1771 1.55127 0.775634 0.631183i $$-0.217431\pi$$
0.775634 + 0.631183i $$0.217431\pi$$
$$812$$ −2.24621 −0.0788266
$$813$$ −17.1231 −0.600534
$$814$$ −3.78410 −0.132633
$$815$$ 0 0
$$816$$ 33.3693 1.16816
$$817$$ −39.3693 −1.37736
$$818$$ 27.1231 0.948337
$$819$$ −10.2462 −0.358032
$$820$$ 0 0
$$821$$ 55.6155 1.94100 0.970498 0.241111i $$-0.0775116\pi$$
0.970498 + 0.241111i $$0.0775116\pi$$
$$822$$ −23.6155 −0.823686
$$823$$ −16.0000 −0.557725 −0.278862 0.960331i $$-0.589957\pi$$
−0.278862 + 0.960331i $$0.589957\pi$$
$$824$$ −7.01515 −0.244385
$$825$$ 0 0
$$826$$ −96.0000 −3.34027
$$827$$ −53.6155 −1.86439 −0.932197 0.361951i $$-0.882111\pi$$
−0.932197 + 0.361951i $$0.882111\pi$$
$$828$$ −2.87689 −0.0999790
$$829$$ −4.24621 −0.147477 −0.0737385 0.997278i $$-0.523493\pi$$
−0.0737385 + 0.997278i $$0.523493\pi$$
$$830$$ 0 0
$$831$$ 0.876894 0.0304191
$$832$$ 11.1231 0.385624
$$833$$ 137.093 4.74998
$$834$$ 28.0000 0.969561
$$835$$ 0 0
$$836$$ −3.23106 −0.111748
$$837$$ −4.00000 −0.138260
$$838$$ −42.7386 −1.47638
$$839$$ 42.7386 1.47550 0.737751 0.675073i $$-0.235888\pi$$
0.737751 + 0.675073i $$0.235888\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ −15.6155 −0.538147
$$843$$ 23.6155 0.813362
$$844$$ −1.26137 −0.0434180
$$845$$ 0 0
$$846$$ 20.4924 0.704544
$$847$$ 45.7538 1.57212
$$848$$ −16.1080 −0.553149
$$849$$ 6.87689 0.236014
$$850$$ 0 0
$$851$$ −11.0540 −0.378925
$$852$$ 1.26137 0.0432137
$$853$$ −19.4384 −0.665560 −0.332780 0.943005i $$-0.607987\pi$$
−0.332780 + 0.943005i $$0.607987\pi$$
$$854$$ −7.01515 −0.240054
$$855$$ 0 0
$$856$$ 40.2159 1.37455
$$857$$ 15.4384 0.527367 0.263684 0.964609i $$-0.415062\pi$$
0.263684 + 0.964609i $$0.415062\pi$$
$$858$$ 4.49242 0.153369
$$859$$ −23.3693 −0.797351 −0.398675 0.917092i $$-0.630530\pi$$
−0.398675 + 0.917092i $$0.630530\pi$$
$$860$$ 0 0
$$861$$ −8.63068 −0.294133
$$862$$ −32.9848 −1.12347
$$863$$ −40.9848 −1.39514 −0.697570 0.716516i $$-0.745735\pi$$
−0.697570 + 0.716516i $$0.745735\pi$$
$$864$$ 2.43845 0.0829577
$$865$$ 0 0
$$866$$ 6.35416 0.215923
$$867$$ −33.7386 −1.14582
$$868$$ −8.98485 −0.304966
$$869$$ 17.2614 0.585552
$$870$$ 0 0
$$871$$ 22.7386 0.770469
$$872$$ 13.8617 0.469418
$$873$$ 5.68466 0.192397
$$874$$ −52.4924 −1.77558
$$875$$ 0 0
$$876$$ 0.738634 0.0249561
$$877$$ −8.87689 −0.299751 −0.149876 0.988705i $$-0.547887\pi$$
−0.149876 + 0.988705i $$0.547887\pi$$
$$878$$ 19.5076 0.658349
$$879$$ −21.3693 −0.720769
$$880$$ 0 0
$$881$$ −8.06913 −0.271856 −0.135928 0.990719i $$-0.543402\pi$$
−0.135928 + 0.990719i $$0.543402\pi$$
$$882$$ 30.0540 1.01197
$$883$$ 42.7386 1.43827 0.719135 0.694871i $$-0.244538\pi$$
0.719135 + 0.694871i $$0.244538\pi$$
$$884$$ 6.24621 0.210083
$$885$$ 0 0
$$886$$ −9.75379 −0.327685
$$887$$ 43.8617 1.47273 0.736367 0.676583i $$-0.236540\pi$$
0.736367 + 0.676583i $$0.236540\pi$$
$$888$$ 4.10795 0.137854
$$889$$ 22.1080 0.741477
$$890$$ 0 0
$$891$$ −1.43845 −0.0481898
$$892$$ 8.00000 0.267860
$$893$$ 67.2311 2.24980
$$894$$ −0.384472 −0.0128587
$$895$$ 0 0
$$896$$ −69.4773 −2.32107
$$897$$ 13.1231 0.438168
$$898$$ 17.8617 0.596054
$$899$$ 4.00000 0.133407
$$900$$ 0 0
$$901$$ 24.4924 0.815961
$$902$$ 3.78410 0.125997
$$903$$ −39.3693 −1.31013
$$904$$ −11.8920 −0.395524
$$905$$ 0 0
$$906$$ −6.73863 −0.223876
$$907$$ −31.0540 −1.03113 −0.515565 0.856850i $$-0.672418\pi$$
−0.515565 + 0.856850i $$0.672418\pi$$
$$908$$ 1.39963 0.0464482
$$909$$ −8.56155 −0.283969
$$910$$ 0 0
$$911$$ −18.5616 −0.614972 −0.307486 0.951553i $$-0.599488\pi$$
−0.307486 + 0.951553i $$0.599488\pi$$
$$912$$ 24.0000 0.794719
$$913$$ −3.68466 −0.121944
$$914$$ 13.8617 0.458506
$$915$$ 0 0
$$916$$ 9.64584 0.318707
$$917$$ −52.4924 −1.73345
$$918$$ −11.1231 −0.367117
$$919$$ 30.7386 1.01397 0.506987 0.861954i $$-0.330759\pi$$
0.506987 + 0.861954i $$0.330759\pi$$
$$920$$ 0 0
$$921$$ −31.6847 −1.04404
$$922$$ −28.3845 −0.934793
$$923$$ −5.75379 −0.189388
$$924$$ −3.23106 −0.106294
$$925$$ 0 0
$$926$$ −40.0000 −1.31448
$$927$$ 2.87689 0.0944896
$$928$$ −2.43845 −0.0800460
$$929$$ 4.87689 0.160006 0.0800029 0.996795i $$-0.474507\pi$$
0.0800029 + 0.996795i $$0.474507\pi$$
$$930$$ 0 0
$$931$$ 98.6004 3.23150
$$932$$ 4.52273 0.148147
$$933$$ −16.3153 −0.534140
$$934$$ 5.26137 0.172157
$$935$$ 0 0
$$936$$ −4.87689 −0.159406
$$937$$ −39.1231 −1.27810 −0.639048 0.769167i $$-0.720672\pi$$
−0.639048 + 0.769167i $$0.720672\pi$$
$$938$$ −90.9545 −2.96977
$$939$$ −21.3693 −0.697361
$$940$$ 0 0
$$941$$ −0.738634 −0.0240788 −0.0120394 0.999928i $$-0.503832\pi$$
−0.0120394 + 0.999928i $$0.503832\pi$$
$$942$$ −21.8617 −0.712294
$$943$$ 11.0540 0.359967
$$944$$ −56.2159 −1.82967
$$945$$ 0 0
$$946$$ 17.2614 0.561215
$$947$$ 3.36932 0.109488 0.0547440 0.998500i $$-0.482566\pi$$
0.0547440 + 0.998500i $$0.482566\pi$$
$$948$$ 5.26137 0.170881
$$949$$ −3.36932 −0.109373
$$950$$ 0 0
$$951$$ 4.87689 0.158144
$$952$$ 88.9848 2.88402
$$953$$ 34.4924 1.11732 0.558660 0.829397i $$-0.311316\pi$$
0.558660 + 0.829397i $$0.311316\pi$$
$$954$$ 5.36932 0.173838
$$955$$ 0 0
$$956$$ −5.75379 −0.186091
$$957$$ 1.43845 0.0464984
$$958$$ −34.7386 −1.12235
$$959$$ −77.4773 −2.50187
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 5.26137 0.169633
$$963$$ −16.4924 −0.531461
$$964$$ −10.9848 −0.353798
$$965$$ 0 0
$$966$$ −52.4924 −1.68892
$$967$$ −27.0540 −0.869997 −0.434999 0.900431i $$-0.643251\pi$$
−0.434999 + 0.900431i $$0.643251\pi$$
$$968$$ 21.7775 0.699954
$$969$$ −36.4924 −1.17231
$$970$$ 0 0
$$971$$ −44.6695 −1.43351 −0.716756 0.697324i $$-0.754374\pi$$
−0.716756 + 0.697324i $$0.754374\pi$$
$$972$$ −0.438447 −0.0140632
$$973$$ 91.8617 2.94495
$$974$$ −53.4773 −1.71352
$$975$$ 0 0
$$976$$ −4.10795 −0.131492
$$977$$ 7.43845 0.237977 0.118989 0.992896i $$-0.462035\pi$$
0.118989 + 0.992896i $$0.462035\pi$$
$$978$$ −28.0000 −0.895341
$$979$$ −17.6155 −0.562995
$$980$$ 0 0
$$981$$ −5.68466 −0.181497
$$982$$ 10.5227 0.335794
$$983$$ −27.8617 −0.888651 −0.444326 0.895865i $$-0.646557\pi$$
−0.444326 + 0.895865i $$0.646557\pi$$
$$984$$ −4.10795 −0.130957
$$985$$ 0 0
$$986$$ 11.1231 0.354232
$$987$$ 67.2311 2.13999
$$988$$ 4.49242 0.142923
$$989$$ 50.4233 1.60337
$$990$$ 0 0
$$991$$ 4.94602 0.157116 0.0785578 0.996910i $$-0.474968\pi$$
0.0785578 + 0.996910i $$0.474968\pi$$
$$992$$ −9.75379 −0.309683
$$993$$ −10.2462 −0.325154
$$994$$ 23.0152 0.729996
$$995$$ 0 0
$$996$$ −1.12311 −0.0355870
$$997$$ −54.0388 −1.71143 −0.855713 0.517450i $$-0.826881\pi$$
−0.855713 + 0.517450i $$0.826881\pi$$
$$998$$ 66.7386 2.11257
$$999$$ −1.68466 −0.0533002
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 2175.2.a.m.1.2 2
3.2 odd 2 6525.2.a.bc.1.1 2
5.2 odd 4 2175.2.c.h.349.3 4
5.3 odd 4 2175.2.c.h.349.2 4
5.4 even 2 435.2.a.h.1.1 2
15.14 odd 2 1305.2.a.i.1.2 2
20.19 odd 2 6960.2.a.bx.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.h.1.1 2 5.4 even 2
1305.2.a.i.1.2 2 15.14 odd 2
2175.2.a.m.1.2 2 1.1 even 1 trivial
2175.2.c.h.349.2 4 5.3 odd 4
2175.2.c.h.349.3 4 5.2 odd 4
6525.2.a.bc.1.1 2 3.2 odd 2
6960.2.a.bx.1.1 2 20.19 odd 2