# Properties

 Label 4275.2.a.z.1.3 Level $4275$ Weight $2$ Character 4275.1 Self dual yes Analytic conductor $34.136$ Analytic rank $1$ 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] = [4275,2,Mod(1,4275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4275, 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("4275.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4275 = 3^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4275.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: $$34.1360468641$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 2x + 1$$ x^3 - x^2 - 2*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 475) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-1.24698$$ of defining polynomial Character $$\chi$$ $$=$$ 4275.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.246980 q^{2} -1.93900 q^{4} +1.69202 q^{7} -0.972853 q^{8} +O(q^{10})$$ $$q+0.246980 q^{2} -1.93900 q^{4} +1.69202 q^{7} -0.972853 q^{8} +0.911854 q^{11} +1.55496 q^{13} +0.417895 q^{14} +3.63773 q^{16} -5.29590 q^{17} -1.00000 q^{19} +0.225209 q^{22} -4.24698 q^{23} +0.384043 q^{26} -3.28083 q^{28} -5.00969 q^{29} +1.82908 q^{31} +2.84415 q^{32} -1.30798 q^{34} +6.29590 q^{37} -0.246980 q^{38} -4.18060 q^{41} +7.31767 q^{43} -1.76809 q^{44} -1.04892 q^{46} +2.04892 q^{47} -4.13706 q^{49} -3.01507 q^{52} +2.70171 q^{53} -1.64609 q^{56} -1.23729 q^{58} -9.87800 q^{59} +0.542877 q^{61} +0.451747 q^{62} -6.57301 q^{64} -13.9976 q^{67} +10.2687 q^{68} +12.8780 q^{71} -2.80731 q^{73} +1.55496 q^{74} +1.93900 q^{76} +1.54288 q^{77} +1.59419 q^{79} -1.03252 q^{82} -12.2349 q^{83} +1.80731 q^{86} -0.887100 q^{88} -2.91723 q^{89} +2.63102 q^{91} +8.23490 q^{92} +0.506041 q^{94} +1.55496 q^{97} -1.02177 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 4 q^{2} + 4 q^{4} - 9 q^{8}+O(q^{10})$$ 3 * q - 4 * q^2 + 4 * q^4 - 9 * q^8 $$3 q - 4 q^{2} + 4 q^{4} - 9 q^{8} - q^{11} + 5 q^{13} + 7 q^{14} + 18 q^{16} - 2 q^{17} - 3 q^{19} - q^{22} - 8 q^{23} - 9 q^{26} - 21 q^{28} + 7 q^{29} - 5 q^{31} - 27 q^{32} - 9 q^{34} + 5 q^{37} + 4 q^{38} - q^{41} + 5 q^{43} + 15 q^{44} + 6 q^{46} - 3 q^{47} - 7 q^{49} + 16 q^{52} - 19 q^{53} + 35 q^{56} - 21 q^{58} - 10 q^{59} - 17 q^{61} + 23 q^{62} + 49 q^{64} - q^{67} + 23 q^{68} + 19 q^{71} - q^{73} + 5 q^{74} - 4 q^{76} - 14 q^{77} + 18 q^{79} + 6 q^{82} - 13 q^{83} - 2 q^{86} - 46 q^{88} - 2 q^{89} - 7 q^{91} + q^{92} + 11 q^{94} + 5 q^{97}+O(q^{100})$$ 3 * q - 4 * q^2 + 4 * q^4 - 9 * q^8 - q^11 + 5 * q^13 + 7 * q^14 + 18 * q^16 - 2 * q^17 - 3 * q^19 - q^22 - 8 * q^23 - 9 * q^26 - 21 * q^28 + 7 * q^29 - 5 * q^31 - 27 * q^32 - 9 * q^34 + 5 * q^37 + 4 * q^38 - q^41 + 5 * q^43 + 15 * q^44 + 6 * q^46 - 3 * q^47 - 7 * q^49 + 16 * q^52 - 19 * q^53 + 35 * q^56 - 21 * q^58 - 10 * q^59 - 17 * q^61 + 23 * q^62 + 49 * q^64 - q^67 + 23 * q^68 + 19 * q^71 - q^73 + 5 * q^74 - 4 * q^76 - 14 * q^77 + 18 * q^79 + 6 * q^82 - 13 * q^83 - 2 * q^86 - 46 * q^88 - 2 * q^89 - 7 * q^91 + q^92 + 11 * q^94 + 5 * q^97

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.246980 0.174641 0.0873205 0.996180i $$-0.472170\pi$$
0.0873205 + 0.996180i $$0.472170\pi$$
$$3$$ 0 0
$$4$$ −1.93900 −0.969501
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 1.69202 0.639524 0.319762 0.947498i $$-0.396397\pi$$
0.319762 + 0.947498i $$0.396397\pi$$
$$8$$ −0.972853 −0.343955
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 0.911854 0.274934 0.137467 0.990506i $$-0.456104\pi$$
0.137467 + 0.990506i $$0.456104\pi$$
$$12$$ 0 0
$$13$$ 1.55496 0.431268 0.215634 0.976474i $$-0.430818\pi$$
0.215634 + 0.976474i $$0.430818\pi$$
$$14$$ 0.417895 0.111687
$$15$$ 0 0
$$16$$ 3.63773 0.909432
$$17$$ −5.29590 −1.28444 −0.642222 0.766519i $$-0.721987\pi$$
−0.642222 + 0.766519i $$0.721987\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0.225209 0.0480148
$$23$$ −4.24698 −0.885556 −0.442778 0.896631i $$-0.646007\pi$$
−0.442778 + 0.896631i $$0.646007\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0.384043 0.0753170
$$27$$ 0 0
$$28$$ −3.28083 −0.620019
$$29$$ −5.00969 −0.930276 −0.465138 0.885238i $$-0.653995\pi$$
−0.465138 + 0.885238i $$0.653995\pi$$
$$30$$ 0 0
$$31$$ 1.82908 0.328513 0.164257 0.986418i $$-0.447477\pi$$
0.164257 + 0.986418i $$0.447477\pi$$
$$32$$ 2.84415 0.502779
$$33$$ 0 0
$$34$$ −1.30798 −0.224316
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 6.29590 1.03504 0.517520 0.855671i $$-0.326855\pi$$
0.517520 + 0.855671i $$0.326855\pi$$
$$38$$ −0.246980 −0.0400654
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −4.18060 −0.652901 −0.326450 0.945214i $$-0.605853\pi$$
−0.326450 + 0.945214i $$0.605853\pi$$
$$42$$ 0 0
$$43$$ 7.31767 1.11593 0.557967 0.829863i $$-0.311582\pi$$
0.557967 + 0.829863i $$0.311582\pi$$
$$44$$ −1.76809 −0.266549
$$45$$ 0 0
$$46$$ −1.04892 −0.154654
$$47$$ 2.04892 0.298865 0.149433 0.988772i $$-0.452255\pi$$
0.149433 + 0.988772i $$0.452255\pi$$
$$48$$ 0 0
$$49$$ −4.13706 −0.591009
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −3.01507 −0.418114
$$53$$ 2.70171 0.371108 0.185554 0.982634i $$-0.440592\pi$$
0.185554 + 0.982634i $$0.440592\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −1.64609 −0.219968
$$57$$ 0 0
$$58$$ −1.23729 −0.162464
$$59$$ −9.87800 −1.28601 −0.643003 0.765864i $$-0.722312\pi$$
−0.643003 + 0.765864i $$0.722312\pi$$
$$60$$ 0 0
$$61$$ 0.542877 0.0695082 0.0347541 0.999396i $$-0.488935\pi$$
0.0347541 + 0.999396i $$0.488935\pi$$
$$62$$ 0.451747 0.0573719
$$63$$ 0 0
$$64$$ −6.57301 −0.821626
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −13.9976 −1.71008 −0.855040 0.518562i $$-0.826467\pi$$
−0.855040 + 0.518562i $$0.826467\pi$$
$$68$$ 10.2687 1.24527
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 12.8780 1.52834 0.764169 0.645016i $$-0.223149\pi$$
0.764169 + 0.645016i $$0.223149\pi$$
$$72$$ 0 0
$$73$$ −2.80731 −0.328571 −0.164286 0.986413i $$-0.552532\pi$$
−0.164286 + 0.986413i $$0.552532\pi$$
$$74$$ 1.55496 0.180760
$$75$$ 0 0
$$76$$ 1.93900 0.222419
$$77$$ 1.54288 0.175827
$$78$$ 0 0
$$79$$ 1.59419 0.179360 0.0896800 0.995971i $$-0.471416\pi$$
0.0896800 + 0.995971i $$0.471416\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −1.03252 −0.114023
$$83$$ −12.2349 −1.34295 −0.671477 0.741025i $$-0.734340\pi$$
−0.671477 + 0.741025i $$0.734340\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 1.80731 0.194888
$$87$$ 0 0
$$88$$ −0.887100 −0.0945652
$$89$$ −2.91723 −0.309226 −0.154613 0.987975i $$-0.549413\pi$$
−0.154613 + 0.987975i $$0.549413\pi$$
$$90$$ 0 0
$$91$$ 2.63102 0.275806
$$92$$ 8.23490 0.858547
$$93$$ 0 0
$$94$$ 0.506041 0.0521941
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 1.55496 0.157882 0.0789410 0.996879i $$-0.474846\pi$$
0.0789410 + 0.996879i $$0.474846\pi$$
$$98$$ −1.02177 −0.103214
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 16.6015 1.65191 0.825955 0.563737i $$-0.190637\pi$$
0.825955 + 0.563737i $$0.190637\pi$$
$$102$$ 0 0
$$103$$ −4.84548 −0.477439 −0.238720 0.971089i $$-0.576728\pi$$
−0.238720 + 0.971089i $$0.576728\pi$$
$$104$$ −1.51275 −0.148337
$$105$$ 0 0
$$106$$ 0.667267 0.0648107
$$107$$ 4.46681 0.431823 0.215912 0.976413i $$-0.430728\pi$$
0.215912 + 0.976413i $$0.430728\pi$$
$$108$$ 0 0
$$109$$ 18.8267 1.80327 0.901635 0.432498i $$-0.142368\pi$$
0.901635 + 0.432498i $$0.142368\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 6.15511 0.581603
$$113$$ −20.0368 −1.88491 −0.942453 0.334337i $$-0.891488\pi$$
−0.942453 + 0.334337i $$0.891488\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 9.71379 0.901903
$$117$$ 0 0
$$118$$ −2.43967 −0.224589
$$119$$ −8.96077 −0.821433
$$120$$ 0 0
$$121$$ −10.1685 −0.924411
$$122$$ 0.134079 0.0121390
$$123$$ 0 0
$$124$$ −3.54660 −0.318494
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −17.8702 −1.58573 −0.792863 0.609399i $$-0.791411\pi$$
−0.792863 + 0.609399i $$0.791411\pi$$
$$128$$ −7.31170 −0.646269
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −7.44265 −0.650267 −0.325134 0.945668i $$-0.605409\pi$$
−0.325134 + 0.945668i $$0.605409\pi$$
$$132$$ 0 0
$$133$$ −1.69202 −0.146717
$$134$$ −3.45712 −0.298650
$$135$$ 0 0
$$136$$ 5.15213 0.441791
$$137$$ −5.68664 −0.485843 −0.242921 0.970046i $$-0.578106\pi$$
−0.242921 + 0.970046i $$0.578106\pi$$
$$138$$ 0 0
$$139$$ 3.61596 0.306701 0.153351 0.988172i $$-0.450994\pi$$
0.153351 + 0.988172i $$0.450994\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 3.18060 0.266910
$$143$$ 1.41789 0.118570
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −0.693349 −0.0573820
$$147$$ 0 0
$$148$$ −12.2078 −1.00347
$$149$$ −3.29052 −0.269570 −0.134785 0.990875i $$-0.543034\pi$$
−0.134785 + 0.990875i $$0.543034\pi$$
$$150$$ 0 0
$$151$$ −10.2131 −0.831133 −0.415566 0.909563i $$-0.636417\pi$$
−0.415566 + 0.909563i $$0.636417\pi$$
$$152$$ 0.972853 0.0789088
$$153$$ 0 0
$$154$$ 0.381059 0.0307066
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 14.3448 1.14484 0.572420 0.819960i $$-0.306005\pi$$
0.572420 + 0.819960i $$0.306005\pi$$
$$158$$ 0.393732 0.0313236
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −7.18598 −0.566335
$$162$$ 0 0
$$163$$ −19.5308 −1.52977 −0.764885 0.644167i $$-0.777204\pi$$
−0.764885 + 0.644167i $$0.777204\pi$$
$$164$$ 8.10620 0.632988
$$165$$ 0 0
$$166$$ −3.02177 −0.234535
$$167$$ −11.8823 −0.919481 −0.459741 0.888053i $$-0.652058\pi$$
−0.459741 + 0.888053i $$0.652058\pi$$
$$168$$ 0 0
$$169$$ −10.5821 −0.814008
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −14.1890 −1.08190
$$173$$ −15.4722 −1.17633 −0.588164 0.808741i $$-0.700149\pi$$
−0.588164 + 0.808741i $$0.700149\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 3.31708 0.250034
$$177$$ 0 0
$$178$$ −0.720497 −0.0540035
$$179$$ −2.16421 −0.161761 −0.0808803 0.996724i $$-0.525773\pi$$
−0.0808803 + 0.996724i $$0.525773\pi$$
$$180$$ 0 0
$$181$$ 16.8974 1.25597 0.627986 0.778225i $$-0.283879\pi$$
0.627986 + 0.778225i $$0.283879\pi$$
$$182$$ 0.649809 0.0481670
$$183$$ 0 0
$$184$$ 4.13169 0.304592
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −4.82908 −0.353138
$$188$$ −3.97285 −0.289750
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −5.92394 −0.428641 −0.214320 0.976763i $$-0.568754\pi$$
−0.214320 + 0.976763i $$0.568754\pi$$
$$192$$ 0 0
$$193$$ −4.43535 −0.319264 −0.159632 0.987177i $$-0.551031\pi$$
−0.159632 + 0.987177i $$0.551031\pi$$
$$194$$ 0.384043 0.0275727
$$195$$ 0 0
$$196$$ 8.02177 0.572984
$$197$$ −16.4722 −1.17359 −0.586797 0.809734i $$-0.699612\pi$$
−0.586797 + 0.809734i $$0.699612\pi$$
$$198$$ 0 0
$$199$$ −24.4131 −1.73060 −0.865300 0.501255i $$-0.832872\pi$$
−0.865300 + 0.501255i $$0.832872\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 4.10023 0.288491
$$203$$ −8.47650 −0.594934
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −1.19673 −0.0833804
$$207$$ 0 0
$$208$$ 5.65651 0.392209
$$209$$ −0.911854 −0.0630743
$$210$$ 0 0
$$211$$ −4.34050 −0.298813 −0.149406 0.988776i $$-0.547736\pi$$
−0.149406 + 0.988776i $$0.547736\pi$$
$$212$$ −5.23862 −0.359790
$$213$$ 0 0
$$214$$ 1.10321 0.0754140
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 3.09485 0.210092
$$218$$ 4.64981 0.314925
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −8.23490 −0.553939
$$222$$ 0 0
$$223$$ 26.2379 1.75702 0.878509 0.477725i $$-0.158539\pi$$
0.878509 + 0.477725i $$0.158539\pi$$
$$224$$ 4.81236 0.321540
$$225$$ 0 0
$$226$$ −4.94869 −0.329182
$$227$$ −14.6853 −0.974699 −0.487349 0.873207i $$-0.662036\pi$$
−0.487349 + 0.873207i $$0.662036\pi$$
$$228$$ 0 0
$$229$$ −21.6407 −1.43006 −0.715029 0.699095i $$-0.753587\pi$$
−0.715029 + 0.699095i $$0.753587\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 4.87369 0.319973
$$233$$ −27.1183 −1.77658 −0.888289 0.459286i $$-0.848105\pi$$
−0.888289 + 0.459286i $$0.848105\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 19.1535 1.24678
$$237$$ 0 0
$$238$$ −2.21313 −0.143456
$$239$$ 11.5308 0.745865 0.372933 0.927858i $$-0.378352\pi$$
0.372933 + 0.927858i $$0.378352\pi$$
$$240$$ 0 0
$$241$$ 11.8194 0.761354 0.380677 0.924708i $$-0.375691\pi$$
0.380677 + 0.924708i $$0.375691\pi$$
$$242$$ −2.51142 −0.161440
$$243$$ 0 0
$$244$$ −1.05264 −0.0673883
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −1.55496 −0.0989396
$$248$$ −1.77943 −0.112994
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 9.66487 0.610041 0.305021 0.952346i $$-0.401337\pi$$
0.305021 + 0.952346i $$0.401337\pi$$
$$252$$ 0 0
$$253$$ −3.87263 −0.243470
$$254$$ −4.41358 −0.276933
$$255$$ 0 0
$$256$$ 11.3402 0.708761
$$257$$ −14.1860 −0.884897 −0.442449 0.896794i $$-0.645890\pi$$
−0.442449 + 0.896794i $$0.645890\pi$$
$$258$$ 0 0
$$259$$ 10.6528 0.661932
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −1.83818 −0.113563
$$263$$ 21.7942 1.34389 0.671943 0.740603i $$-0.265460\pi$$
0.671943 + 0.740603i $$0.265460\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −0.417895 −0.0256228
$$267$$ 0 0
$$268$$ 27.1414 1.65792
$$269$$ 24.7265 1.50760 0.753800 0.657104i $$-0.228219\pi$$
0.753800 + 0.657104i $$0.228219\pi$$
$$270$$ 0 0
$$271$$ −13.2295 −0.803636 −0.401818 0.915720i $$-0.631622\pi$$
−0.401818 + 0.915720i $$0.631622\pi$$
$$272$$ −19.2650 −1.16811
$$273$$ 0 0
$$274$$ −1.40449 −0.0848481
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −0.560335 −0.0336673 −0.0168336 0.999858i $$-0.505359\pi$$
−0.0168336 + 0.999858i $$0.505359\pi$$
$$278$$ 0.893068 0.0535626
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −27.6039 −1.64671 −0.823355 0.567527i $$-0.807900\pi$$
−0.823355 + 0.567527i $$0.807900\pi$$
$$282$$ 0 0
$$283$$ −15.9608 −0.948769 −0.474385 0.880318i $$-0.657329\pi$$
−0.474385 + 0.880318i $$0.657329\pi$$
$$284$$ −24.9705 −1.48172
$$285$$ 0 0
$$286$$ 0.350191 0.0207072
$$287$$ −7.07367 −0.417546
$$288$$ 0 0
$$289$$ 11.0465 0.649796
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 5.44339 0.318550
$$293$$ −25.3327 −1.47995 −0.739977 0.672632i $$-0.765164\pi$$
−0.739977 + 0.672632i $$0.765164\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −6.12498 −0.356007
$$297$$ 0 0
$$298$$ −0.812691 −0.0470779
$$299$$ −6.60388 −0.381912
$$300$$ 0 0
$$301$$ 12.3817 0.713666
$$302$$ −2.52243 −0.145150
$$303$$ 0 0
$$304$$ −3.63773 −0.208638
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −11.5574 −0.659613 −0.329806 0.944049i $$-0.606983\pi$$
−0.329806 + 0.944049i $$0.606983\pi$$
$$308$$ −2.99164 −0.170464
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 18.3424 1.04010 0.520052 0.854135i $$-0.325913\pi$$
0.520052 + 0.854135i $$0.325913\pi$$
$$312$$ 0 0
$$313$$ −18.9119 −1.06896 −0.534481 0.845181i $$-0.679493\pi$$
−0.534481 + 0.845181i $$0.679493\pi$$
$$314$$ 3.54288 0.199936
$$315$$ 0 0
$$316$$ −3.09113 −0.173890
$$317$$ −20.2784 −1.13895 −0.569475 0.822008i $$-0.692854\pi$$
−0.569475 + 0.822008i $$0.692854\pi$$
$$318$$ 0 0
$$319$$ −4.56810 −0.255765
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −1.77479 −0.0989052
$$323$$ 5.29590 0.294672
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −4.82371 −0.267160
$$327$$ 0 0
$$328$$ 4.06711 0.224569
$$329$$ 3.46681 0.191132
$$330$$ 0 0
$$331$$ −4.77479 −0.262446 −0.131223 0.991353i $$-0.541890\pi$$
−0.131223 + 0.991353i $$0.541890\pi$$
$$332$$ 23.7235 1.30200
$$333$$ 0 0
$$334$$ −2.93469 −0.160579
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 24.3967 1.32897 0.664487 0.747300i $$-0.268650\pi$$
0.664487 + 0.747300i $$0.268650\pi$$
$$338$$ −2.61356 −0.142159
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 1.66786 0.0903196
$$342$$ 0 0
$$343$$ −18.8442 −1.01749
$$344$$ −7.11901 −0.383832
$$345$$ 0 0
$$346$$ −3.82132 −0.205435
$$347$$ 9.99761 0.536700 0.268350 0.963322i $$-0.413522\pi$$
0.268350 + 0.963322i $$0.413522\pi$$
$$348$$ 0 0
$$349$$ 21.9584 1.17541 0.587703 0.809077i $$-0.300033\pi$$
0.587703 + 0.809077i $$0.300033\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 2.59345 0.138231
$$353$$ 36.8786 1.96285 0.981425 0.191848i $$-0.0614479\pi$$
0.981425 + 0.191848i $$0.0614479\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 5.65651 0.299795
$$357$$ 0 0
$$358$$ −0.534516 −0.0282500
$$359$$ −16.5187 −0.871824 −0.435912 0.899989i $$-0.643574\pi$$
−0.435912 + 0.899989i $$0.643574\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 4.17331 0.219344
$$363$$ 0 0
$$364$$ −5.10156 −0.267394
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −6.83148 −0.356600 −0.178300 0.983976i $$-0.557060\pi$$
−0.178300 + 0.983976i $$0.557060\pi$$
$$368$$ −15.4494 −0.805353
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 4.57135 0.237333
$$372$$ 0 0
$$373$$ −4.76271 −0.246604 −0.123302 0.992369i $$-0.539348\pi$$
−0.123302 + 0.992369i $$0.539348\pi$$
$$374$$ −1.19269 −0.0616723
$$375$$ 0 0
$$376$$ −1.99330 −0.102796
$$377$$ −7.78986 −0.401198
$$378$$ 0 0
$$379$$ 14.3773 0.738514 0.369257 0.929327i $$-0.379612\pi$$
0.369257 + 0.929327i $$0.379612\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −1.46309 −0.0748583
$$383$$ 30.6708 1.56721 0.783603 0.621261i $$-0.213379\pi$$
0.783603 + 0.621261i $$0.213379\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −1.09544 −0.0557565
$$387$$ 0 0
$$388$$ −3.01507 −0.153067
$$389$$ 12.9215 0.655148 0.327574 0.944825i $$-0.393769\pi$$
0.327574 + 0.944825i $$0.393769\pi$$
$$390$$ 0 0
$$391$$ 22.4916 1.13745
$$392$$ 4.02475 0.203281
$$393$$ 0 0
$$394$$ −4.06829 −0.204958
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −29.8471 −1.49798 −0.748992 0.662579i $$-0.769462\pi$$
−0.748992 + 0.662579i $$0.769462\pi$$
$$398$$ −6.02954 −0.302234
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 28.7101 1.43371 0.716856 0.697221i $$-0.245580\pi$$
0.716856 + 0.697221i $$0.245580\pi$$
$$402$$ 0 0
$$403$$ 2.84415 0.141677
$$404$$ −32.1903 −1.60153
$$405$$ 0 0
$$406$$ −2.09352 −0.103900
$$407$$ 5.74094 0.284568
$$408$$ 0 0
$$409$$ −13.6203 −0.673479 −0.336739 0.941598i $$-0.609324\pi$$
−0.336739 + 0.941598i $$0.609324\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 9.39539 0.462878
$$413$$ −16.7138 −0.822432
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 4.42253 0.216833
$$417$$ 0 0
$$418$$ −0.225209 −0.0110153
$$419$$ 2.13946 0.104519 0.0522596 0.998634i $$-0.483358\pi$$
0.0522596 + 0.998634i $$0.483358\pi$$
$$420$$ 0 0
$$421$$ −15.0562 −0.733795 −0.366897 0.930261i $$-0.619580\pi$$
−0.366897 + 0.930261i $$0.619580\pi$$
$$422$$ −1.07202 −0.0521849
$$423$$ 0 0
$$424$$ −2.62837 −0.127645
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0.918559 0.0444522
$$428$$ −8.66115 −0.418653
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −4.37435 −0.210705 −0.105353 0.994435i $$-0.533597\pi$$
−0.105353 + 0.994435i $$0.533597\pi$$
$$432$$ 0 0
$$433$$ −33.1564 −1.59340 −0.796698 0.604377i $$-0.793422\pi$$
−0.796698 + 0.604377i $$0.793422\pi$$
$$434$$ 0.764365 0.0366907
$$435$$ 0 0
$$436$$ −36.5050 −1.74827
$$437$$ 4.24698 0.203161
$$438$$ 0 0
$$439$$ 32.2368 1.53858 0.769290 0.638900i $$-0.220610\pi$$
0.769290 + 0.638900i $$0.220610\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −2.03385 −0.0967405
$$443$$ −21.7362 −1.03272 −0.516358 0.856373i $$-0.672713\pi$$
−0.516358 + 0.856373i $$0.672713\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 6.48022 0.306847
$$447$$ 0 0
$$448$$ −11.1217 −0.525450
$$449$$ 24.9584 1.17786 0.588929 0.808185i $$-0.299550\pi$$
0.588929 + 0.808185i $$0.299550\pi$$
$$450$$ 0 0
$$451$$ −3.81210 −0.179505
$$452$$ 38.8514 1.82742
$$453$$ 0 0
$$454$$ −3.62697 −0.170222
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 13.1347 0.614414 0.307207 0.951643i $$-0.400606\pi$$
0.307207 + 0.951643i $$0.400606\pi$$
$$458$$ −5.34481 −0.249747
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 35.3726 1.64746 0.823732 0.566979i $$-0.191888\pi$$
0.823732 + 0.566979i $$0.191888\pi$$
$$462$$ 0 0
$$463$$ 14.6963 0.682997 0.341498 0.939882i $$-0.389066\pi$$
0.341498 + 0.939882i $$0.389066\pi$$
$$464$$ −18.2239 −0.846022
$$465$$ 0 0
$$466$$ −6.69766 −0.310263
$$467$$ 33.2121 1.53687 0.768435 0.639927i $$-0.221036\pi$$
0.768435 + 0.639927i $$0.221036\pi$$
$$468$$ 0 0
$$469$$ −23.6843 −1.09364
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 9.60984 0.442329
$$473$$ 6.67264 0.306809
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 17.3749 0.796379
$$477$$ 0 0
$$478$$ 2.84787 0.130259
$$479$$ −24.6219 −1.12500 −0.562502 0.826796i $$-0.690161\pi$$
−0.562502 + 0.826796i $$0.690161\pi$$
$$480$$ 0 0
$$481$$ 9.78986 0.446379
$$482$$ 2.91915 0.132964
$$483$$ 0 0
$$484$$ 19.7168 0.896217
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −29.5646 −1.33970 −0.669851 0.742496i $$-0.733642\pi$$
−0.669851 + 0.742496i $$0.733642\pi$$
$$488$$ −0.528139 −0.0239077
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −36.2978 −1.63810 −0.819049 0.573724i $$-0.805498\pi$$
−0.819049 + 0.573724i $$0.805498\pi$$
$$492$$ 0 0
$$493$$ 26.5308 1.19489
$$494$$ −0.384043 −0.0172789
$$495$$ 0 0
$$496$$ 6.65371 0.298760
$$497$$ 21.7899 0.977409
$$498$$ 0 0
$$499$$ 7.84415 0.351152 0.175576 0.984466i $$-0.443821\pi$$
0.175576 + 0.984466i $$0.443821\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 2.38703 0.106538
$$503$$ 20.4166 0.910330 0.455165 0.890407i $$-0.349580\pi$$
0.455165 + 0.890407i $$0.349580\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −0.956459 −0.0425198
$$507$$ 0 0
$$508$$ 34.6504 1.53736
$$509$$ 14.7530 0.653916 0.326958 0.945039i $$-0.393976\pi$$
0.326958 + 0.945039i $$0.393976\pi$$
$$510$$ 0 0
$$511$$ −4.75004 −0.210129
$$512$$ 17.4242 0.770048
$$513$$ 0 0
$$514$$ −3.50365 −0.154539
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 1.86831 0.0821683
$$518$$ 2.63102 0.115600
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −37.1487 −1.62751 −0.813756 0.581206i $$-0.802581\pi$$
−0.813756 + 0.581206i $$0.802581\pi$$
$$522$$ 0 0
$$523$$ −16.3623 −0.715472 −0.357736 0.933823i $$-0.616451\pi$$
−0.357736 + 0.933823i $$0.616451\pi$$
$$524$$ 14.4313 0.630434
$$525$$ 0 0
$$526$$ 5.38271 0.234698
$$527$$ −9.68664 −0.421957
$$528$$ 0 0
$$529$$ −4.96316 −0.215790
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 3.28083 0.142242
$$533$$ −6.50066 −0.281575
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 13.6176 0.588191
$$537$$ 0 0
$$538$$ 6.10693 0.263289
$$539$$ −3.77240 −0.162489
$$540$$ 0 0
$$541$$ −2.08947 −0.0898335 −0.0449168 0.998991i $$-0.514302\pi$$
−0.0449168 + 0.998991i $$0.514302\pi$$
$$542$$ −3.26742 −0.140348
$$543$$ 0 0
$$544$$ −15.0623 −0.645792
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −1.86054 −0.0795511 −0.0397756 0.999209i $$-0.512664\pi$$
−0.0397756 + 0.999209i $$0.512664\pi$$
$$548$$ 11.0264 0.471025
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 5.00969 0.213420
$$552$$ 0 0
$$553$$ 2.69740 0.114705
$$554$$ −0.138391 −0.00587968
$$555$$ 0 0
$$556$$ −7.01134 −0.297347
$$557$$ −9.80061 −0.415265 −0.207633 0.978207i $$-0.566576\pi$$
−0.207633 + 0.978207i $$0.566576\pi$$
$$558$$ 0 0
$$559$$ 11.3787 0.481266
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −6.81759 −0.287583
$$563$$ 38.0170 1.60222 0.801112 0.598514i $$-0.204242\pi$$
0.801112 + 0.598514i $$0.204242\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −3.94198 −0.165694
$$567$$ 0 0
$$568$$ −12.5284 −0.525680
$$569$$ 7.32975 0.307279 0.153640 0.988127i $$-0.450901\pi$$
0.153640 + 0.988127i $$0.450901\pi$$
$$570$$ 0 0
$$571$$ 37.8775 1.58513 0.792563 0.609791i $$-0.208746\pi$$
0.792563 + 0.609791i $$0.208746\pi$$
$$572$$ −2.74930 −0.114954
$$573$$ 0 0
$$574$$ −1.74705 −0.0729206
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −28.6993 −1.19477 −0.597384 0.801955i $$-0.703793\pi$$
−0.597384 + 0.801955i $$0.703793\pi$$
$$578$$ 2.72827 0.113481
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −20.7017 −0.858852
$$582$$ 0 0
$$583$$ 2.46357 0.102030
$$584$$ 2.73110 0.113014
$$585$$ 0 0
$$586$$ −6.25667 −0.258461
$$587$$ 3.72348 0.153684 0.0768422 0.997043i $$-0.475516\pi$$
0.0768422 + 0.997043i $$0.475516\pi$$
$$588$$ 0 0
$$589$$ −1.82908 −0.0753661
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 22.9028 0.941297
$$593$$ 27.7399 1.13914 0.569570 0.821943i $$-0.307110\pi$$
0.569570 + 0.821943i $$0.307110\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 6.38032 0.261348
$$597$$ 0 0
$$598$$ −1.63102 −0.0666975
$$599$$ 23.5579 0.962551 0.481276 0.876569i $$-0.340174\pi$$
0.481276 + 0.876569i $$0.340174\pi$$
$$600$$ 0 0
$$601$$ −7.36898 −0.300587 −0.150293 0.988641i $$-0.548022\pi$$
−0.150293 + 0.988641i $$0.548022\pi$$
$$602$$ 3.05802 0.124635
$$603$$ 0 0
$$604$$ 19.8033 0.805783
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 28.4198 1.15352 0.576762 0.816912i $$-0.304316\pi$$
0.576762 + 0.816912i $$0.304316\pi$$
$$608$$ −2.84415 −0.115346
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 3.18598 0.128891
$$612$$ 0 0
$$613$$ 6.48129 0.261777 0.130888 0.991397i $$-0.458217\pi$$
0.130888 + 0.991397i $$0.458217\pi$$
$$614$$ −2.85443 −0.115195
$$615$$ 0 0
$$616$$ −1.50099 −0.0604767
$$617$$ 24.4650 0.984924 0.492462 0.870334i $$-0.336097\pi$$
0.492462 + 0.870334i $$0.336097\pi$$
$$618$$ 0 0
$$619$$ −22.2457 −0.894128 −0.447064 0.894502i $$-0.647530\pi$$
−0.447064 + 0.894502i $$0.647530\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 4.53020 0.181645
$$623$$ −4.93602 −0.197757
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −4.67084 −0.186684
$$627$$ 0 0
$$628$$ −27.8146 −1.10992
$$629$$ −33.3424 −1.32945
$$630$$ 0 0
$$631$$ −15.5888 −0.620581 −0.310290 0.950642i $$-0.600426\pi$$
−0.310290 + 0.950642i $$0.600426\pi$$
$$632$$ −1.55091 −0.0616919
$$633$$ 0 0
$$634$$ −5.00836 −0.198907
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −6.43296 −0.254883
$$638$$ −1.12823 −0.0446670
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −8.17496 −0.322892 −0.161446 0.986882i $$-0.551616\pi$$
−0.161446 + 0.986882i $$0.551616\pi$$
$$642$$ 0 0
$$643$$ 11.1836 0.441038 0.220519 0.975383i $$-0.429225\pi$$
0.220519 + 0.975383i $$0.429225\pi$$
$$644$$ 13.9336 0.549062
$$645$$ 0 0
$$646$$ 1.30798 0.0514617
$$647$$ −16.2121 −0.637362 −0.318681 0.947862i $$-0.603240\pi$$
−0.318681 + 0.947862i $$0.603240\pi$$
$$648$$ 0 0
$$649$$ −9.00730 −0.353567
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 37.8702 1.48311
$$653$$ −24.7952 −0.970312 −0.485156 0.874427i $$-0.661237\pi$$
−0.485156 + 0.874427i $$0.661237\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −15.2079 −0.593769
$$657$$ 0 0
$$658$$ 0.856232 0.0333794
$$659$$ 20.2868 0.790262 0.395131 0.918625i $$-0.370699\pi$$
0.395131 + 0.918625i $$0.370699\pi$$
$$660$$ 0 0
$$661$$ 20.4222 0.794332 0.397166 0.917747i $$-0.369994\pi$$
0.397166 + 0.917747i $$0.369994\pi$$
$$662$$ −1.17928 −0.0458339
$$663$$ 0 0
$$664$$ 11.9028 0.461917
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 21.2760 0.823812
$$668$$ 23.0398 0.891437
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0.495024 0.0191102
$$672$$ 0 0
$$673$$ 28.7254 1.10728 0.553641 0.832755i $$-0.313238\pi$$
0.553641 + 0.832755i $$0.313238\pi$$
$$674$$ 6.02549 0.232093
$$675$$ 0 0
$$676$$ 20.5187 0.789181
$$677$$ −44.0062 −1.69130 −0.845648 0.533740i $$-0.820786\pi$$
−0.845648 + 0.533740i $$0.820786\pi$$
$$678$$ 0 0
$$679$$ 2.63102 0.100969
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0.411927 0.0157735
$$683$$ −8.48427 −0.324642 −0.162321 0.986738i $$-0.551898\pi$$
−0.162321 + 0.986738i $$0.551898\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −4.65412 −0.177695
$$687$$ 0 0
$$688$$ 26.6197 1.01487
$$689$$ 4.20105 0.160047
$$690$$ 0 0
$$691$$ 20.2586 0.770673 0.385336 0.922776i $$-0.374085\pi$$
0.385336 + 0.922776i $$0.374085\pi$$
$$692$$ 30.0006 1.14045
$$693$$ 0 0
$$694$$ 2.46921 0.0937297
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 22.1400 0.838614
$$698$$ 5.42327 0.205274
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 23.4101 0.884188 0.442094 0.896969i $$-0.354236\pi$$
0.442094 + 0.896969i $$0.354236\pi$$
$$702$$ 0 0
$$703$$ −6.29590 −0.237454
$$704$$ −5.99362 −0.225893
$$705$$ 0 0
$$706$$ 9.10826 0.342794
$$707$$ 28.0901 1.05644
$$708$$ 0 0
$$709$$ 30.3472 1.13971 0.569857 0.821744i $$-0.306999\pi$$
0.569857 + 0.821744i $$0.306999\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 2.83804 0.106360
$$713$$ −7.76809 −0.290917
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 4.19641 0.156827
$$717$$ 0 0
$$718$$ −4.07979 −0.152256
$$719$$ 38.3230 1.42921 0.714604 0.699529i $$-0.246607\pi$$
0.714604 + 0.699529i $$0.246607\pi$$
$$720$$ 0 0
$$721$$ −8.19865 −0.305334
$$722$$ 0.246980 0.00919163
$$723$$ 0 0
$$724$$ −32.7640 −1.21767
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 2.69069 0.0997923 0.0498961 0.998754i $$-0.484111\pi$$
0.0498961 + 0.998754i $$0.484111\pi$$
$$728$$ −2.55960 −0.0948650
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −38.7536 −1.43335
$$732$$ 0 0
$$733$$ 18.9651 0.700491 0.350246 0.936658i $$-0.386098\pi$$
0.350246 + 0.936658i $$0.386098\pi$$
$$734$$ −1.68724 −0.0622770
$$735$$ 0 0
$$736$$ −12.0790 −0.445240
$$737$$ −12.7638 −0.470160
$$738$$ 0 0
$$739$$ 29.8278 1.09723 0.548616 0.836075i $$-0.315155\pi$$
0.548616 + 0.836075i $$0.315155\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 1.12903 0.0414480
$$743$$ −8.78448 −0.322271 −0.161136 0.986932i $$-0.551516\pi$$
−0.161136 + 0.986932i $$0.551516\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −1.17629 −0.0430671
$$747$$ 0 0
$$748$$ 9.36360 0.342367
$$749$$ 7.55794 0.276161
$$750$$ 0 0
$$751$$ −18.1142 −0.660998 −0.330499 0.943806i $$-0.607217\pi$$
−0.330499 + 0.943806i $$0.607217\pi$$
$$752$$ 7.45340 0.271798
$$753$$ 0 0
$$754$$ −1.92394 −0.0700656
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −0.222816 −0.00809840 −0.00404920 0.999992i $$-0.501289\pi$$
−0.00404920 + 0.999992i $$0.501289\pi$$
$$758$$ 3.55091 0.128975
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 6.07798 0.220327 0.110163 0.993913i $$-0.464863\pi$$
0.110163 + 0.993913i $$0.464863\pi$$
$$762$$ 0 0
$$763$$ 31.8552 1.15323
$$764$$ 11.4865 0.415568
$$765$$ 0 0
$$766$$ 7.57507 0.273698
$$767$$ −15.3599 −0.554613
$$768$$ 0 0
$$769$$ −14.6267 −0.527453 −0.263726 0.964598i $$-0.584952\pi$$
−0.263726 + 0.964598i $$0.584952\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 8.60015 0.309526
$$773$$ 4.05728 0.145930 0.0729651 0.997334i $$-0.476754\pi$$
0.0729651 + 0.997334i $$0.476754\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −1.51275 −0.0543044
$$777$$ 0 0
$$778$$ 3.19136 0.114416
$$779$$ 4.18060 0.149786
$$780$$ 0 0
$$781$$ 11.7429 0.420192
$$782$$ 5.55496 0.198645
$$783$$ 0 0
$$784$$ −15.0495 −0.537482
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 19.5657 0.697442 0.348721 0.937227i $$-0.386616\pi$$
0.348721 + 0.937227i $$0.386616\pi$$
$$788$$ 31.9396 1.13780
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −33.9028 −1.20544
$$792$$ 0 0
$$793$$ 0.844150 0.0299767
$$794$$ −7.37163 −0.261609
$$795$$ 0 0
$$796$$ 47.3370 1.67782
$$797$$ −38.9051 −1.37809 −0.689046 0.724718i $$-0.741970\pi$$
−0.689046 + 0.724718i $$0.741970\pi$$
$$798$$ 0 0
$$799$$ −10.8509 −0.383876
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 7.09080 0.250385
$$803$$ −2.55986 −0.0903355
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0.702447 0.0247426
$$807$$ 0 0
$$808$$ −16.1508 −0.568183
$$809$$ 22.5730 0.793625 0.396812 0.917900i $$-0.370116\pi$$
0.396812 + 0.917900i $$0.370116\pi$$
$$810$$ 0 0
$$811$$ −46.3605 −1.62794 −0.813968 0.580909i $$-0.802697\pi$$
−0.813968 + 0.580909i $$0.802697\pi$$
$$812$$ 16.4359 0.576789
$$813$$ 0 0
$$814$$ 1.41789 0.0496972
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −7.31767 −0.256013
$$818$$ −3.36393 −0.117617
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 17.8194 0.621901 0.310951 0.950426i $$-0.399353\pi$$
0.310951 + 0.950426i $$0.399353\pi$$
$$822$$ 0 0
$$823$$ −32.5394 −1.13425 −0.567126 0.823631i $$-0.691945\pi$$
−0.567126 + 0.823631i $$0.691945\pi$$
$$824$$ 4.71394 0.164218
$$825$$ 0 0
$$826$$ −4.12797 −0.143630
$$827$$ −39.8256 −1.38487 −0.692436 0.721479i $$-0.743463\pi$$
−0.692436 + 0.721479i $$0.743463\pi$$
$$828$$ 0 0
$$829$$ 38.7525 1.34593 0.672966 0.739674i $$-0.265020\pi$$
0.672966 + 0.739674i $$0.265020\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −10.2208 −0.354341
$$833$$ 21.9095 0.759118
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 1.76809 0.0611505
$$837$$ 0 0
$$838$$ 0.528402 0.0182533
$$839$$ −2.76749 −0.0955445 −0.0477723 0.998858i $$-0.515212\pi$$
−0.0477723 + 0.998858i $$0.515212\pi$$
$$840$$ 0 0
$$841$$ −3.90302 −0.134587
$$842$$ −3.71858 −0.128151
$$843$$ 0 0
$$844$$ 8.41624 0.289699
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −17.2054 −0.591183
$$848$$ 9.82808 0.337498
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −26.7385 −0.916586
$$852$$ 0 0
$$853$$ 24.1390 0.826503 0.413252 0.910617i $$-0.364393\pi$$
0.413252 + 0.910617i $$0.364393\pi$$
$$854$$ 0.226865 0.00776317
$$855$$ 0 0
$$856$$ −4.34555 −0.148528
$$857$$ 3.16229 0.108022 0.0540109 0.998540i $$-0.482799\pi$$
0.0540109 + 0.998540i $$0.482799\pi$$
$$858$$ 0 0
$$859$$ 4.86426 0.165967 0.0829833 0.996551i $$-0.473555\pi$$
0.0829833 + 0.996551i $$0.473555\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −1.08038 −0.0367978
$$863$$ −4.54958 −0.154870 −0.0774348 0.996997i $$-0.524673\pi$$
−0.0774348 + 0.996997i $$0.524673\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −8.18896 −0.278272
$$867$$ 0 0
$$868$$ −6.00092 −0.203684
$$869$$ 1.45367 0.0493122
$$870$$ 0 0
$$871$$ −21.7657 −0.737502
$$872$$ −18.3156 −0.620245
$$873$$ 0 0
$$874$$ 1.04892 0.0354802
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −18.6595 −0.630086 −0.315043 0.949077i $$-0.602019\pi$$
−0.315043 + 0.949077i $$0.602019\pi$$
$$878$$ 7.96184 0.268699
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −19.4306 −0.654632 −0.327316 0.944915i $$-0.606144\pi$$
−0.327316 + 0.944915i $$0.606144\pi$$
$$882$$ 0 0
$$883$$ 25.6437 0.862979 0.431490 0.902118i $$-0.357988\pi$$
0.431490 + 0.902118i $$0.357988\pi$$
$$884$$ 15.9675 0.537044
$$885$$ 0 0
$$886$$ −5.36839 −0.180354
$$887$$ 31.0062 1.04109 0.520544 0.853835i $$-0.325729\pi$$
0.520544 + 0.853835i $$0.325729\pi$$
$$888$$ 0 0
$$889$$ −30.2368 −1.01411
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −50.8753 −1.70343
$$893$$ −2.04892 −0.0685644
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −12.3716 −0.413305
$$897$$ 0 0
$$898$$ 6.16421 0.205702
$$899$$ −9.16315 −0.305608
$$900$$ 0 0
$$901$$ −14.3080 −0.476668
$$902$$ −0.941511 −0.0313489
$$903$$ 0 0
$$904$$ 19.4929 0.648324
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −17.7676 −0.589964 −0.294982 0.955503i $$-0.595314\pi$$
−0.294982 + 0.955503i $$0.595314\pi$$
$$908$$ 28.4748 0.944971
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −41.7313 −1.38262 −0.691309 0.722559i $$-0.742966\pi$$
−0.691309 + 0.722559i $$0.742966\pi$$
$$912$$ 0 0
$$913$$ −11.1564 −0.369224
$$914$$ 3.24400 0.107302
$$915$$ 0 0
$$916$$ 41.9614 1.38644
$$917$$ −12.5931 −0.415862
$$918$$ 0 0
$$919$$ 30.4088 1.00309 0.501547 0.865130i $$-0.332764\pi$$
0.501547 + 0.865130i $$0.332764\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 8.73630 0.287715
$$923$$ 20.0248 0.659123
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 3.62969 0.119279
$$927$$ 0 0
$$928$$ −14.2483 −0.467724
$$929$$ 28.8219 0.945616 0.472808 0.881165i $$-0.343240\pi$$
0.472808 + 0.881165i $$0.343240\pi$$
$$930$$ 0 0
$$931$$ 4.13706 0.135587
$$932$$ 52.5824 1.72239
$$933$$ 0 0
$$934$$ 8.20270 0.268401
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −50.4601 −1.64846 −0.824230 0.566255i $$-0.808392\pi$$
−0.824230 + 0.566255i $$0.808392\pi$$
$$938$$ −5.84953 −0.190994
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −1.10082 −0.0358857 −0.0179428 0.999839i $$-0.505712\pi$$
−0.0179428 + 0.999839i $$0.505712\pi$$
$$942$$ 0 0
$$943$$ 17.7549 0.578180
$$944$$ −35.9335 −1.16954
$$945$$ 0 0
$$946$$ 1.64801 0.0535813
$$947$$ 13.9353 0.452836 0.226418 0.974030i $$-0.427299\pi$$
0.226418 + 0.974030i $$0.427299\pi$$
$$948$$ 0 0
$$949$$ −4.36526 −0.141702
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 8.71751 0.282536
$$953$$ 41.3400 1.33913 0.669567 0.742751i $$-0.266480\pi$$
0.669567 + 0.742751i $$0.266480\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −22.3582 −0.723117
$$957$$ 0 0
$$958$$ −6.08111 −0.196472
$$959$$ −9.62192 −0.310708
$$960$$ 0 0
$$961$$ −27.6544 −0.892079
$$962$$ 2.41789 0.0779561
$$963$$ 0 0
$$964$$ −22.9178 −0.738133
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −5.26875 −0.169432 −0.0847158 0.996405i $$-0.526998\pi$$
−0.0847158 + 0.996405i $$0.526998\pi$$
$$968$$ 9.89248 0.317956
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 5.15346 0.165382 0.0826911 0.996575i $$-0.473649\pi$$
0.0826911 + 0.996575i $$0.473649\pi$$
$$972$$ 0 0
$$973$$ 6.11828 0.196143
$$974$$ −7.30186 −0.233967
$$975$$ 0 0
$$976$$ 1.97484 0.0632130
$$977$$ −4.77612 −0.152802 −0.0764008 0.997077i $$-0.524343\pi$$
−0.0764008 + 0.997077i $$0.524343\pi$$
$$978$$ 0 0
$$979$$ −2.66009 −0.0850168
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −8.96482 −0.286079
$$983$$ 28.9758 0.924186 0.462093 0.886832i $$-0.347099\pi$$
0.462093 + 0.886832i $$0.347099\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 6.55257 0.208676
$$987$$ 0 0
$$988$$ 3.01507 0.0959220
$$989$$ −31.0780 −0.988222
$$990$$ 0 0
$$991$$ −38.5042 −1.22313 −0.611564 0.791195i $$-0.709459\pi$$
−0.611564 + 0.791195i $$0.709459\pi$$
$$992$$ 5.20219 0.165170
$$993$$ 0 0
$$994$$ 5.38165 0.170696
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 10.1491 0.321427 0.160713 0.987001i $$-0.448621\pi$$
0.160713 + 0.987001i $$0.448621\pi$$
$$998$$ 1.93735 0.0613256
$$999$$ 0 0
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 4275.2.a.z.1.3 3
3.2 odd 2 475.2.a.h.1.1 yes 3
5.4 even 2 4275.2.a.bn.1.1 3
12.11 even 2 7600.2.a.bn.1.3 3
15.2 even 4 475.2.b.c.324.3 6
15.8 even 4 475.2.b.c.324.4 6
15.14 odd 2 475.2.a.d.1.3 3
57.56 even 2 9025.2.a.w.1.3 3
60.59 even 2 7600.2.a.bw.1.1 3
285.284 even 2 9025.2.a.be.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
475.2.a.d.1.3 3 15.14 odd 2
475.2.a.h.1.1 yes 3 3.2 odd 2
475.2.b.c.324.3 6 15.2 even 4
475.2.b.c.324.4 6 15.8 even 4
4275.2.a.z.1.3 3 1.1 even 1 trivial
4275.2.a.bn.1.1 3 5.4 even 2
7600.2.a.bn.1.3 3 12.11 even 2
7600.2.a.bw.1.1 3 60.59 even 2
9025.2.a.w.1.3 3 57.56 even 2
9025.2.a.be.1.1 3 285.284 even 2