# Properties

 Label 6525.2.a.bb.1.2 Level $6525$ Weight $2$ Character 6525.1 Self dual yes Analytic conductor $52.102$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6525,2,Mod(1,6525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6525, 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("6525.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6525 = 3^{2} \cdot 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6525.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: $$52.1023873189$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1305) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 6525.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.61803 q^{2} +0.618034 q^{4} +4.23607 q^{7} -2.23607 q^{8} +O(q^{10})$$ $$q+1.61803 q^{2} +0.618034 q^{4} +4.23607 q^{7} -2.23607 q^{8} -0.236068 q^{11} -1.00000 q^{13} +6.85410 q^{14} -4.85410 q^{16} -7.47214 q^{17} +2.47214 q^{19} -0.381966 q^{22} -4.47214 q^{23} -1.61803 q^{26} +2.61803 q^{28} -1.00000 q^{29} -8.00000 q^{31} -3.38197 q^{32} -12.0902 q^{34} +4.00000 q^{38} -6.00000 q^{41} +6.00000 q^{43} -0.145898 q^{44} -7.23607 q^{46} -3.76393 q^{47} +10.9443 q^{49} -0.618034 q^{52} -2.47214 q^{53} -9.47214 q^{56} -1.61803 q^{58} -6.00000 q^{59} -8.47214 q^{61} -12.9443 q^{62} +4.23607 q^{64} +1.29180 q^{67} -4.61803 q^{68} -6.47214 q^{71} +6.00000 q^{73} +1.52786 q^{76} -1.00000 q^{77} -6.00000 q^{79} -9.70820 q^{82} +2.47214 q^{83} +9.70820 q^{86} +0.527864 q^{88} -9.94427 q^{89} -4.23607 q^{91} -2.76393 q^{92} -6.09017 q^{94} -15.4164 q^{97} +17.7082 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - q^{4} + 4 q^{7}+O(q^{10})$$ 2 * q + q^2 - q^4 + 4 * q^7 $$2 q + q^{2} - q^{4} + 4 q^{7} + 4 q^{11} - 2 q^{13} + 7 q^{14} - 3 q^{16} - 6 q^{17} - 4 q^{19} - 3 q^{22} - q^{26} + 3 q^{28} - 2 q^{29} - 16 q^{31} - 9 q^{32} - 13 q^{34} + 8 q^{38} - 12 q^{41} + 12 q^{43} - 7 q^{44} - 10 q^{46} - 12 q^{47} + 4 q^{49} + q^{52} + 4 q^{53} - 10 q^{56} - q^{58} - 12 q^{59} - 8 q^{61} - 8 q^{62} + 4 q^{64} + 16 q^{67} - 7 q^{68} - 4 q^{71} + 12 q^{73} + 12 q^{76} - 2 q^{77} - 12 q^{79} - 6 q^{82} - 4 q^{83} + 6 q^{86} + 10 q^{88} - 2 q^{89} - 4 q^{91} - 10 q^{92} - q^{94} - 4 q^{97} + 22 q^{98}+O(q^{100})$$ 2 * q + q^2 - q^4 + 4 * q^7 + 4 * q^11 - 2 * q^13 + 7 * q^14 - 3 * q^16 - 6 * q^17 - 4 * q^19 - 3 * q^22 - q^26 + 3 * q^28 - 2 * q^29 - 16 * q^31 - 9 * q^32 - 13 * q^34 + 8 * q^38 - 12 * q^41 + 12 * q^43 - 7 * q^44 - 10 * q^46 - 12 * q^47 + 4 * q^49 + q^52 + 4 * q^53 - 10 * q^56 - q^58 - 12 * q^59 - 8 * q^61 - 8 * q^62 + 4 * q^64 + 16 * q^67 - 7 * q^68 - 4 * q^71 + 12 * q^73 + 12 * q^76 - 2 * q^77 - 12 * q^79 - 6 * q^82 - 4 * q^83 + 6 * q^86 + 10 * q^88 - 2 * q^89 - 4 * q^91 - 10 * q^92 - q^94 - 4 * q^97 + 22 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.61803 1.14412 0.572061 0.820211i $$-0.306144\pi$$
0.572061 + 0.820211i $$0.306144\pi$$
$$3$$ 0 0
$$4$$ 0.618034 0.309017
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 4.23607 1.60108 0.800542 0.599277i $$-0.204545\pi$$
0.800542 + 0.599277i $$0.204545\pi$$
$$8$$ −2.23607 −0.790569
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −0.236068 −0.0711772 −0.0355886 0.999367i $$-0.511331\pi$$
−0.0355886 + 0.999367i $$0.511331\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350 −0.138675 0.990338i $$-0.544284\pi$$
−0.138675 + 0.990338i $$0.544284\pi$$
$$14$$ 6.85410 1.83184
$$15$$ 0 0
$$16$$ −4.85410 −1.21353
$$17$$ −7.47214 −1.81226 −0.906130 0.423000i $$-0.860977\pi$$
−0.906130 + 0.423000i $$0.860977\pi$$
$$18$$ 0 0
$$19$$ 2.47214 0.567147 0.283573 0.958951i $$-0.408480\pi$$
0.283573 + 0.958951i $$0.408480\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −0.381966 −0.0814354
$$23$$ −4.47214 −0.932505 −0.466252 0.884652i $$-0.654396\pi$$
−0.466252 + 0.884652i $$0.654396\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −1.61803 −0.317323
$$27$$ 0 0
$$28$$ 2.61803 0.494762
$$29$$ −1.00000 −0.185695
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ −3.38197 −0.597853
$$33$$ 0 0
$$34$$ −12.0902 −2.07345
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$38$$ 4.00000 0.648886
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ 6.00000 0.914991 0.457496 0.889212i $$-0.348747\pi$$
0.457496 + 0.889212i $$0.348747\pi$$
$$44$$ −0.145898 −0.0219950
$$45$$ 0 0
$$46$$ −7.23607 −1.06690
$$47$$ −3.76393 −0.549026 −0.274513 0.961583i $$-0.588517\pi$$
−0.274513 + 0.961583i $$0.588517\pi$$
$$48$$ 0 0
$$49$$ 10.9443 1.56347
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −0.618034 −0.0857059
$$53$$ −2.47214 −0.339574 −0.169787 0.985481i $$-0.554308\pi$$
−0.169787 + 0.985481i $$0.554308\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −9.47214 −1.26577
$$57$$ 0 0
$$58$$ −1.61803 −0.212458
$$59$$ −6.00000 −0.781133 −0.390567 0.920575i $$-0.627721\pi$$
−0.390567 + 0.920575i $$0.627721\pi$$
$$60$$ 0 0
$$61$$ −8.47214 −1.08475 −0.542373 0.840138i $$-0.682474\pi$$
−0.542373 + 0.840138i $$0.682474\pi$$
$$62$$ −12.9443 −1.64392
$$63$$ 0 0
$$64$$ 4.23607 0.529508
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 1.29180 0.157818 0.0789090 0.996882i $$-0.474856\pi$$
0.0789090 + 0.996882i $$0.474856\pi$$
$$68$$ −4.61803 −0.560019
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −6.47214 −0.768101 −0.384051 0.923312i $$-0.625471\pi$$
−0.384051 + 0.923312i $$0.625471\pi$$
$$72$$ 0 0
$$73$$ 6.00000 0.702247 0.351123 0.936329i $$-0.385800\pi$$
0.351123 + 0.936329i $$0.385800\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 1.52786 0.175258
$$77$$ −1.00000 −0.113961
$$78$$ 0 0
$$79$$ −6.00000 −0.675053 −0.337526 0.941316i $$-0.609590\pi$$
−0.337526 + 0.941316i $$0.609590\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −9.70820 −1.07209
$$83$$ 2.47214 0.271352 0.135676 0.990753i $$-0.456679\pi$$
0.135676 + 0.990753i $$0.456679\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 9.70820 1.04686
$$87$$ 0 0
$$88$$ 0.527864 0.0562705
$$89$$ −9.94427 −1.05409 −0.527045 0.849837i $$-0.676700\pi$$
−0.527045 + 0.849837i $$0.676700\pi$$
$$90$$ 0 0
$$91$$ −4.23607 −0.444061
$$92$$ −2.76393 −0.288160
$$93$$ 0 0
$$94$$ −6.09017 −0.628153
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −15.4164 −1.56530 −0.782650 0.622463i $$-0.786132\pi$$
−0.782650 + 0.622463i $$0.786132\pi$$
$$98$$ 17.7082 1.78880
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 6.52786 0.649547 0.324773 0.945792i $$-0.394712\pi$$
0.324773 + 0.945792i $$0.394712\pi$$
$$102$$ 0 0
$$103$$ 0.944272 0.0930419 0.0465209 0.998917i $$-0.485187\pi$$
0.0465209 + 0.998917i $$0.485187\pi$$
$$104$$ 2.23607 0.219265
$$105$$ 0 0
$$106$$ −4.00000 −0.388514
$$107$$ 4.47214 0.432338 0.216169 0.976356i $$-0.430644\pi$$
0.216169 + 0.976356i $$0.430644\pi$$
$$108$$ 0 0
$$109$$ 18.4164 1.76397 0.881986 0.471276i $$-0.156206\pi$$
0.881986 + 0.471276i $$0.156206\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −20.5623 −1.94296
$$113$$ 1.47214 0.138487 0.0692435 0.997600i $$-0.477941\pi$$
0.0692435 + 0.997600i $$0.477941\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −0.618034 −0.0573830
$$117$$ 0 0
$$118$$ −9.70820 −0.893713
$$119$$ −31.6525 −2.90158
$$120$$ 0 0
$$121$$ −10.9443 −0.994934
$$122$$ −13.7082 −1.24108
$$123$$ 0 0
$$124$$ −4.94427 −0.444009
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 12.0000 1.06483 0.532414 0.846484i $$-0.321285\pi$$
0.532414 + 0.846484i $$0.321285\pi$$
$$128$$ 13.6180 1.20368
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −4.23607 −0.370107 −0.185053 0.982728i $$-0.559246\pi$$
−0.185053 + 0.982728i $$0.559246\pi$$
$$132$$ 0 0
$$133$$ 10.4721 0.908049
$$134$$ 2.09017 0.180563
$$135$$ 0 0
$$136$$ 16.7082 1.43272
$$137$$ 6.94427 0.593289 0.296645 0.954988i $$-0.404132\pi$$
0.296645 + 0.954988i $$0.404132\pi$$
$$138$$ 0 0
$$139$$ 7.76393 0.658528 0.329264 0.944238i $$-0.393199\pi$$
0.329264 + 0.944238i $$0.393199\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −10.4721 −0.878802
$$143$$ 0.236068 0.0197410
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 9.70820 0.803457
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −5.52786 −0.452860 −0.226430 0.974027i $$-0.572705\pi$$
−0.226430 + 0.974027i $$0.572705\pi$$
$$150$$ 0 0
$$151$$ −12.0000 −0.976546 −0.488273 0.872691i $$-0.662373\pi$$
−0.488273 + 0.872691i $$0.662373\pi$$
$$152$$ −5.52786 −0.448369
$$153$$ 0 0
$$154$$ −1.61803 −0.130385
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 15.4164 1.23036 0.615182 0.788385i $$-0.289083\pi$$
0.615182 + 0.788385i $$0.289083\pi$$
$$158$$ −9.70820 −0.772343
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −18.9443 −1.49302
$$162$$ 0 0
$$163$$ 22.9443 1.79713 0.898567 0.438836i $$-0.144609\pi$$
0.898567 + 0.438836i $$0.144609\pi$$
$$164$$ −3.70820 −0.289562
$$165$$ 0 0
$$166$$ 4.00000 0.310460
$$167$$ 13.4164 1.03819 0.519096 0.854716i $$-0.326269\pi$$
0.519096 + 0.854716i $$0.326269\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 3.70820 0.282748
$$173$$ −11.8885 −0.903869 −0.451935 0.892051i $$-0.649266\pi$$
−0.451935 + 0.892051i $$0.649266\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 1.14590 0.0863753
$$177$$ 0 0
$$178$$ −16.0902 −1.20601
$$179$$ 14.9443 1.11699 0.558494 0.829509i $$-0.311380\pi$$
0.558494 + 0.829509i $$0.311380\pi$$
$$180$$ 0 0
$$181$$ −22.4164 −1.66620 −0.833099 0.553124i $$-0.813436\pi$$
−0.833099 + 0.553124i $$0.813436\pi$$
$$182$$ −6.85410 −0.508060
$$183$$ 0 0
$$184$$ 10.0000 0.737210
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 1.76393 0.128991
$$188$$ −2.32624 −0.169658
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 8.94427 0.647185 0.323592 0.946197i $$-0.395109\pi$$
0.323592 + 0.946197i $$0.395109\pi$$
$$192$$ 0 0
$$193$$ −20.0000 −1.43963 −0.719816 0.694165i $$-0.755774\pi$$
−0.719816 + 0.694165i $$0.755774\pi$$
$$194$$ −24.9443 −1.79089
$$195$$ 0 0
$$196$$ 6.76393 0.483138
$$197$$ 24.9443 1.77721 0.888603 0.458677i $$-0.151677\pi$$
0.888603 + 0.458677i $$0.151677\pi$$
$$198$$ 0 0
$$199$$ −20.1246 −1.42660 −0.713298 0.700861i $$-0.752799\pi$$
−0.713298 + 0.700861i $$0.752799\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 10.5623 0.743161
$$203$$ −4.23607 −0.297314
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 1.52786 0.106451
$$207$$ 0 0
$$208$$ 4.85410 0.336571
$$209$$ −0.583592 −0.0403679
$$210$$ 0 0
$$211$$ −0.944272 −0.0650064 −0.0325032 0.999472i $$-0.510348\pi$$
−0.0325032 + 0.999472i $$0.510348\pi$$
$$212$$ −1.52786 −0.104934
$$213$$ 0 0
$$214$$ 7.23607 0.494647
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −33.8885 −2.30050
$$218$$ 29.7984 2.01820
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 7.47214 0.502630
$$222$$ 0 0
$$223$$ −13.1803 −0.882621 −0.441310 0.897355i $$-0.645486\pi$$
−0.441310 + 0.897355i $$0.645486\pi$$
$$224$$ −14.3262 −0.957212
$$225$$ 0 0
$$226$$ 2.38197 0.158446
$$227$$ −13.5279 −0.897876 −0.448938 0.893563i $$-0.648198\pi$$
−0.448938 + 0.893563i $$0.648198\pi$$
$$228$$ 0 0
$$229$$ 4.00000 0.264327 0.132164 0.991228i $$-0.457808\pi$$
0.132164 + 0.991228i $$0.457808\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 2.23607 0.146805
$$233$$ −18.0000 −1.17922 −0.589610 0.807688i $$-0.700718\pi$$
−0.589610 + 0.807688i $$0.700718\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −3.70820 −0.241384
$$237$$ 0 0
$$238$$ −51.2148 −3.31976
$$239$$ −20.9443 −1.35477 −0.677386 0.735628i $$-0.736887\pi$$
−0.677386 + 0.735628i $$0.736887\pi$$
$$240$$ 0 0
$$241$$ −5.00000 −0.322078 −0.161039 0.986948i $$-0.551485\pi$$
−0.161039 + 0.986948i $$0.551485\pi$$
$$242$$ −17.7082 −1.13833
$$243$$ 0 0
$$244$$ −5.23607 −0.335205
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −2.47214 −0.157298
$$248$$ 17.8885 1.13592
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 18.5967 1.17382 0.586908 0.809654i $$-0.300345\pi$$
0.586908 + 0.809654i $$0.300345\pi$$
$$252$$ 0 0
$$253$$ 1.05573 0.0663731
$$254$$ 19.4164 1.21829
$$255$$ 0 0
$$256$$ 13.5623 0.847644
$$257$$ −13.5279 −0.843845 −0.421922 0.906632i $$-0.638645\pi$$
−0.421922 + 0.906632i $$0.638645\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −6.85410 −0.423448
$$263$$ 24.9443 1.53813 0.769065 0.639171i $$-0.220722\pi$$
0.769065 + 0.639171i $$0.220722\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 16.9443 1.03892
$$267$$ 0 0
$$268$$ 0.798374 0.0487684
$$269$$ −9.47214 −0.577526 −0.288763 0.957401i $$-0.593244\pi$$
−0.288763 + 0.957401i $$0.593244\pi$$
$$270$$ 0 0
$$271$$ −3.52786 −0.214302 −0.107151 0.994243i $$-0.534173\pi$$
−0.107151 + 0.994243i $$0.534173\pi$$
$$272$$ 36.2705 2.19922
$$273$$ 0 0
$$274$$ 11.2361 0.678796
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −19.9443 −1.19834 −0.599168 0.800624i $$-0.704502\pi$$
−0.599168 + 0.800624i $$0.704502\pi$$
$$278$$ 12.5623 0.753437
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −22.4721 −1.34058 −0.670288 0.742101i $$-0.733829\pi$$
−0.670288 + 0.742101i $$0.733829\pi$$
$$282$$ 0 0
$$283$$ −4.94427 −0.293906 −0.146953 0.989143i $$-0.546947\pi$$
−0.146953 + 0.989143i $$0.546947\pi$$
$$284$$ −4.00000 −0.237356
$$285$$ 0 0
$$286$$ 0.381966 0.0225861
$$287$$ −25.4164 −1.50028
$$288$$ 0 0
$$289$$ 38.8328 2.28428
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 3.70820 0.217006
$$293$$ 9.00000 0.525786 0.262893 0.964825i $$-0.415323\pi$$
0.262893 + 0.964825i $$0.415323\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ −8.94427 −0.518128
$$299$$ 4.47214 0.258630
$$300$$ 0 0
$$301$$ 25.4164 1.46498
$$302$$ −19.4164 −1.11729
$$303$$ 0 0
$$304$$ −12.0000 −0.688247
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 26.8328 1.53143 0.765715 0.643180i $$-0.222385\pi$$
0.765715 + 0.643180i $$0.222385\pi$$
$$308$$ −0.618034 −0.0352158
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 20.1246 1.14116 0.570581 0.821241i $$-0.306718\pi$$
0.570581 + 0.821241i $$0.306718\pi$$
$$312$$ 0 0
$$313$$ −28.4164 −1.60619 −0.803095 0.595851i $$-0.796815\pi$$
−0.803095 + 0.595851i $$0.796815\pi$$
$$314$$ 24.9443 1.40769
$$315$$ 0 0
$$316$$ −3.70820 −0.208603
$$317$$ −20.8885 −1.17322 −0.586609 0.809870i $$-0.699537\pi$$
−0.586609 + 0.809870i $$0.699537\pi$$
$$318$$ 0 0
$$319$$ 0.236068 0.0132173
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −30.6525 −1.70820
$$323$$ −18.4721 −1.02782
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 37.1246 2.05614
$$327$$ 0 0
$$328$$ 13.4164 0.740797
$$329$$ −15.9443 −0.879036
$$330$$ 0 0
$$331$$ −9.88854 −0.543524 −0.271762 0.962365i $$-0.587606\pi$$
−0.271762 + 0.962365i $$0.587606\pi$$
$$332$$ 1.52786 0.0838524
$$333$$ 0 0
$$334$$ 21.7082 1.18782
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −22.9443 −1.24985 −0.624927 0.780683i $$-0.714871\pi$$
−0.624927 + 0.780683i $$0.714871\pi$$
$$338$$ −19.4164 −1.05611
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 1.88854 0.102270
$$342$$ 0 0
$$343$$ 16.7082 0.902158
$$344$$ −13.4164 −0.723364
$$345$$ 0 0
$$346$$ −19.2361 −1.03414
$$347$$ −29.8885 −1.60450 −0.802251 0.596987i $$-0.796364\pi$$
−0.802251 + 0.596987i $$0.796364\pi$$
$$348$$ 0 0
$$349$$ −22.9443 −1.22818 −0.614089 0.789237i $$-0.710477\pi$$
−0.614089 + 0.789237i $$0.710477\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0.798374 0.0425535
$$353$$ 1.88854 0.100517 0.0502585 0.998736i $$-0.483995\pi$$
0.0502585 + 0.998736i $$0.483995\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −6.14590 −0.325732
$$357$$ 0 0
$$358$$ 24.1803 1.27797
$$359$$ 35.7771 1.88824 0.944121 0.329598i $$-0.106913\pi$$
0.944121 + 0.329598i $$0.106913\pi$$
$$360$$ 0 0
$$361$$ −12.8885 −0.678344
$$362$$ −36.2705 −1.90634
$$363$$ 0 0
$$364$$ −2.61803 −0.137222
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −1.41641 −0.0739359 −0.0369679 0.999316i $$-0.511770\pi$$
−0.0369679 + 0.999316i $$0.511770\pi$$
$$368$$ 21.7082 1.13162
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −10.4721 −0.543686
$$372$$ 0 0
$$373$$ 28.8328 1.49291 0.746453 0.665438i $$-0.231755\pi$$
0.746453 + 0.665438i $$0.231755\pi$$
$$374$$ 2.85410 0.147582
$$375$$ 0 0
$$376$$ 8.41641 0.434043
$$377$$ 1.00000 0.0515026
$$378$$ 0 0
$$379$$ −2.58359 −0.132710 −0.0663551 0.997796i $$-0.521137\pi$$
−0.0663551 + 0.997796i $$0.521137\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 14.4721 0.740459
$$383$$ 18.0000 0.919757 0.459879 0.887982i $$-0.347893\pi$$
0.459879 + 0.887982i $$0.347893\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −32.3607 −1.64712
$$387$$ 0 0
$$388$$ −9.52786 −0.483704
$$389$$ −19.4721 −0.987276 −0.493638 0.869667i $$-0.664333\pi$$
−0.493638 + 0.869667i $$0.664333\pi$$
$$390$$ 0 0
$$391$$ 33.4164 1.68994
$$392$$ −24.4721 −1.23603
$$393$$ 0 0
$$394$$ 40.3607 2.03334
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 7.88854 0.395915 0.197957 0.980211i $$-0.436569\pi$$
0.197957 + 0.980211i $$0.436569\pi$$
$$398$$ −32.5623 −1.63220
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −10.3607 −0.517388 −0.258694 0.965959i $$-0.583292\pi$$
−0.258694 + 0.965959i $$0.583292\pi$$
$$402$$ 0 0
$$403$$ 8.00000 0.398508
$$404$$ 4.03444 0.200721
$$405$$ 0 0
$$406$$ −6.85410 −0.340163
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 27.4164 1.35565 0.677827 0.735221i $$-0.262922\pi$$
0.677827 + 0.735221i $$0.262922\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0.583592 0.0287515
$$413$$ −25.4164 −1.25066
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 3.38197 0.165815
$$417$$ 0 0
$$418$$ −0.944272 −0.0461858
$$419$$ −37.4164 −1.82791 −0.913956 0.405814i $$-0.866988\pi$$
−0.913956 + 0.405814i $$0.866988\pi$$
$$420$$ 0 0
$$421$$ −17.4164 −0.848824 −0.424412 0.905469i $$-0.639519\pi$$
−0.424412 + 0.905469i $$0.639519\pi$$
$$422$$ −1.52786 −0.0743753
$$423$$ 0 0
$$424$$ 5.52786 0.268457
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −35.8885 −1.73677
$$428$$ 2.76393 0.133600
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 12.4721 0.600762 0.300381 0.953819i $$-0.402886\pi$$
0.300381 + 0.953819i $$0.402886\pi$$
$$432$$ 0 0
$$433$$ 39.3050 1.88888 0.944438 0.328690i $$-0.106607\pi$$
0.944438 + 0.328690i $$0.106607\pi$$
$$434$$ −54.8328 −2.63206
$$435$$ 0 0
$$436$$ 11.3820 0.545097
$$437$$ −11.0557 −0.528867
$$438$$ 0 0
$$439$$ −6.23607 −0.297631 −0.148816 0.988865i $$-0.547546\pi$$
−0.148816 + 0.988865i $$0.547546\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 12.0902 0.575071
$$443$$ −5.76393 −0.273853 −0.136926 0.990581i $$-0.543722\pi$$
−0.136926 + 0.990581i $$0.543722\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −21.3262 −1.00983
$$447$$ 0 0
$$448$$ 17.9443 0.847787
$$449$$ 41.9443 1.97947 0.989736 0.142906i $$-0.0456446\pi$$
0.989736 + 0.142906i $$0.0456446\pi$$
$$450$$ 0 0
$$451$$ 1.41641 0.0666960
$$452$$ 0.909830 0.0427948
$$453$$ 0 0
$$454$$ −21.8885 −1.02728
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 23.3607 1.09277 0.546383 0.837535i $$-0.316004\pi$$
0.546383 + 0.837535i $$0.316004\pi$$
$$458$$ 6.47214 0.302423
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −41.7771 −1.94575 −0.972876 0.231325i $$-0.925694\pi$$
−0.972876 + 0.231325i $$0.925694\pi$$
$$462$$ 0 0
$$463$$ 7.76393 0.360821 0.180410 0.983591i $$-0.442257\pi$$
0.180410 + 0.983591i $$0.442257\pi$$
$$464$$ 4.85410 0.225346
$$465$$ 0 0
$$466$$ −29.1246 −1.34917
$$467$$ −24.0000 −1.11059 −0.555294 0.831654i $$-0.687394\pi$$
−0.555294 + 0.831654i $$0.687394\pi$$
$$468$$ 0 0
$$469$$ 5.47214 0.252680
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 13.4164 0.617540
$$473$$ −1.41641 −0.0651265
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −19.5623 −0.896637
$$477$$ 0 0
$$478$$ −33.8885 −1.55003
$$479$$ −33.8885 −1.54841 −0.774204 0.632937i $$-0.781849\pi$$
−0.774204 + 0.632937i $$0.781849\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ −8.09017 −0.368497
$$483$$ 0 0
$$484$$ −6.76393 −0.307451
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 7.05573 0.319726 0.159863 0.987139i $$-0.448895\pi$$
0.159863 + 0.987139i $$0.448895\pi$$
$$488$$ 18.9443 0.857567
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 14.8328 0.669396 0.334698 0.942326i $$-0.391366\pi$$
0.334698 + 0.942326i $$0.391366\pi$$
$$492$$ 0 0
$$493$$ 7.47214 0.336528
$$494$$ −4.00000 −0.179969
$$495$$ 0 0
$$496$$ 38.8328 1.74364
$$497$$ −27.4164 −1.22979
$$498$$ 0 0
$$499$$ −26.1246 −1.16950 −0.584749 0.811214i $$-0.698807\pi$$
−0.584749 + 0.811214i $$0.698807\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 30.0902 1.34299
$$503$$ 14.1246 0.629785 0.314893 0.949127i $$-0.398031\pi$$
0.314893 + 0.949127i $$0.398031\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 1.70820 0.0759389
$$507$$ 0 0
$$508$$ 7.41641 0.329050
$$509$$ 38.3607 1.70031 0.850154 0.526535i $$-0.176509\pi$$
0.850154 + 0.526535i $$0.176509\pi$$
$$510$$ 0 0
$$511$$ 25.4164 1.12436
$$512$$ −5.29180 −0.233867
$$513$$ 0 0
$$514$$ −21.8885 −0.965462
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0.888544 0.0390781
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −12.4721 −0.546414 −0.273207 0.961955i $$-0.588084\pi$$
−0.273207 + 0.961955i $$0.588084\pi$$
$$522$$ 0 0
$$523$$ 33.5410 1.46665 0.733323 0.679880i $$-0.237968\pi$$
0.733323 + 0.679880i $$0.237968\pi$$
$$524$$ −2.61803 −0.114369
$$525$$ 0 0
$$526$$ 40.3607 1.75981
$$527$$ 59.7771 2.60393
$$528$$ 0 0
$$529$$ −3.00000 −0.130435
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 6.47214 0.280603
$$533$$ 6.00000 0.259889
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −2.88854 −0.124766
$$537$$ 0 0
$$538$$ −15.3262 −0.660761
$$539$$ −2.58359 −0.111283
$$540$$ 0 0
$$541$$ 2.00000 0.0859867 0.0429934 0.999075i $$-0.486311\pi$$
0.0429934 + 0.999075i $$0.486311\pi$$
$$542$$ −5.70820 −0.245188
$$543$$ 0 0
$$544$$ 25.2705 1.08346
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 30.7082 1.31299 0.656494 0.754331i $$-0.272039\pi$$
0.656494 + 0.754331i $$0.272039\pi$$
$$548$$ 4.29180 0.183336
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −2.47214 −0.105317
$$552$$ 0 0
$$553$$ −25.4164 −1.08082
$$554$$ −32.2705 −1.37104
$$555$$ 0 0
$$556$$ 4.79837 0.203496
$$557$$ −31.4164 −1.33116 −0.665578 0.746328i $$-0.731815\pi$$
−0.665578 + 0.746328i $$0.731815\pi$$
$$558$$ 0 0
$$559$$ −6.00000 −0.253773
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −36.3607 −1.53378
$$563$$ 19.1803 0.808355 0.404177 0.914681i $$-0.367558\pi$$
0.404177 + 0.914681i $$0.367558\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −8.00000 −0.336265
$$567$$ 0 0
$$568$$ 14.4721 0.607237
$$569$$ 31.9443 1.33917 0.669587 0.742734i $$-0.266471\pi$$
0.669587 + 0.742734i $$0.266471\pi$$
$$570$$ 0 0
$$571$$ 30.8328 1.29031 0.645157 0.764050i $$-0.276792\pi$$
0.645157 + 0.764050i $$0.276792\pi$$
$$572$$ 0.145898 0.00610030
$$573$$ 0 0
$$574$$ −41.1246 −1.71651
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 29.0557 1.20961 0.604803 0.796375i $$-0.293252\pi$$
0.604803 + 0.796375i $$0.293252\pi$$
$$578$$ 62.8328 2.61350
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 10.4721 0.434457
$$582$$ 0 0
$$583$$ 0.583592 0.0241699
$$584$$ −13.4164 −0.555175
$$585$$ 0 0
$$586$$ 14.5623 0.601563
$$587$$ −6.94427 −0.286621 −0.143310 0.989678i $$-0.545775\pi$$
−0.143310 + 0.989678i $$0.545775\pi$$
$$588$$ 0 0
$$589$$ −19.7771 −0.814901
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −12.4721 −0.512169 −0.256085 0.966654i $$-0.582433\pi$$
−0.256085 + 0.966654i $$0.582433\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −3.41641 −0.139942
$$597$$ 0 0
$$598$$ 7.23607 0.295905
$$599$$ 29.0689 1.18772 0.593861 0.804568i $$-0.297603\pi$$
0.593861 + 0.804568i $$0.297603\pi$$
$$600$$ 0 0
$$601$$ 34.8328 1.42086 0.710430 0.703768i $$-0.248500\pi$$
0.710430 + 0.703768i $$0.248500\pi$$
$$602$$ 41.1246 1.67611
$$603$$ 0 0
$$604$$ −7.41641 −0.301769
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 23.4164 0.950443 0.475221 0.879866i $$-0.342368\pi$$
0.475221 + 0.879866i $$0.342368\pi$$
$$608$$ −8.36068 −0.339070
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 3.76393 0.152272
$$612$$ 0 0
$$613$$ 23.0000 0.928961 0.464481 0.885583i $$-0.346241\pi$$
0.464481 + 0.885583i $$0.346241\pi$$
$$614$$ 43.4164 1.75214
$$615$$ 0 0
$$616$$ 2.23607 0.0900937
$$617$$ 26.9443 1.08474 0.542368 0.840141i $$-0.317528\pi$$
0.542368 + 0.840141i $$0.317528\pi$$
$$618$$ 0 0
$$619$$ 35.3050 1.41903 0.709513 0.704692i $$-0.248915\pi$$
0.709513 + 0.704692i $$0.248915\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 32.5623 1.30563
$$623$$ −42.1246 −1.68769
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −45.9787 −1.83768
$$627$$ 0 0
$$628$$ 9.52786 0.380203
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −48.1246 −1.91581 −0.957905 0.287084i $$-0.907314\pi$$
−0.957905 + 0.287084i $$0.907314\pi$$
$$632$$ 13.4164 0.533676
$$633$$ 0 0
$$634$$ −33.7984 −1.34230
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −10.9443 −0.433628
$$638$$ 0.381966 0.0151222
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 3.94427 0.155789 0.0778947 0.996962i $$-0.475180\pi$$
0.0778947 + 0.996962i $$0.475180\pi$$
$$642$$ 0 0
$$643$$ −45.5410 −1.79596 −0.897981 0.440034i $$-0.854966\pi$$
−0.897981 + 0.440034i $$0.854966\pi$$
$$644$$ −11.7082 −0.461368
$$645$$ 0 0
$$646$$ −29.8885 −1.17595
$$647$$ 46.9443 1.84557 0.922785 0.385316i $$-0.125907\pi$$
0.922785 + 0.385316i $$0.125907\pi$$
$$648$$ 0 0
$$649$$ 1.41641 0.0555989
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 14.1803 0.555345
$$653$$ −16.8885 −0.660900 −0.330450 0.943824i $$-0.607200\pi$$
−0.330450 + 0.943824i $$0.607200\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 29.1246 1.13713
$$657$$ 0 0
$$658$$ −25.7984 −1.00573
$$659$$ −36.4853 −1.42127 −0.710633 0.703563i $$-0.751591\pi$$
−0.710633 + 0.703563i $$0.751591\pi$$
$$660$$ 0 0
$$661$$ 0.416408 0.0161964 0.00809819 0.999967i $$-0.497422\pi$$
0.00809819 + 0.999967i $$0.497422\pi$$
$$662$$ −16.0000 −0.621858
$$663$$ 0 0
$$664$$ −5.52786 −0.214523
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 4.47214 0.173162
$$668$$ 8.29180 0.320819
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 2.00000 0.0772091
$$672$$ 0 0
$$673$$ −28.4164 −1.09537 −0.547686 0.836684i $$-0.684491\pi$$
−0.547686 + 0.836684i $$0.684491\pi$$
$$674$$ −37.1246 −1.42999
$$675$$ 0 0
$$676$$ −7.41641 −0.285246
$$677$$ −17.9443 −0.689654 −0.344827 0.938666i $$-0.612062\pi$$
−0.344827 + 0.938666i $$0.612062\pi$$
$$678$$ 0 0
$$679$$ −65.3050 −2.50617
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 3.05573 0.117010
$$683$$ −19.0557 −0.729147 −0.364574 0.931175i $$-0.618785\pi$$
−0.364574 + 0.931175i $$0.618785\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 27.0344 1.03218
$$687$$ 0 0
$$688$$ −29.1246 −1.11037
$$689$$ 2.47214 0.0941809
$$690$$ 0 0
$$691$$ −28.7082 −1.09211 −0.546056 0.837749i $$-0.683871\pi$$
−0.546056 + 0.837749i $$0.683871\pi$$
$$692$$ −7.34752 −0.279311
$$693$$ 0 0
$$694$$ −48.3607 −1.83575
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 44.8328 1.69816
$$698$$ −37.1246 −1.40519
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −34.4721 −1.30199 −0.650997 0.759080i $$-0.725649\pi$$
−0.650997 + 0.759080i $$0.725649\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ −1.00000 −0.0376889
$$705$$ 0 0
$$706$$ 3.05573 0.115004
$$707$$ 27.6525 1.03998
$$708$$ 0 0
$$709$$ 14.0000 0.525781 0.262891 0.964826i $$-0.415324\pi$$
0.262891 + 0.964826i $$0.415324\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 22.2361 0.833332
$$713$$ 35.7771 1.33986
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 9.23607 0.345168
$$717$$ 0 0
$$718$$ 57.8885 2.16038
$$719$$ 23.8885 0.890892 0.445446 0.895309i $$-0.353045\pi$$
0.445446 + 0.895309i $$0.353045\pi$$
$$720$$ 0 0
$$721$$ 4.00000 0.148968
$$722$$ −20.8541 −0.776109
$$723$$ 0 0
$$724$$ −13.8541 −0.514884
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 48.2492 1.78946 0.894732 0.446603i $$-0.147366\pi$$
0.894732 + 0.446603i $$0.147366\pi$$
$$728$$ 9.47214 0.351061
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −44.8328 −1.65820
$$732$$ 0 0
$$733$$ 3.52786 0.130305 0.0651523 0.997875i $$-0.479247\pi$$
0.0651523 + 0.997875i $$0.479247\pi$$
$$734$$ −2.29180 −0.0845917
$$735$$ 0 0
$$736$$ 15.1246 0.557501
$$737$$ −0.304952 −0.0112330
$$738$$ 0 0
$$739$$ 43.4164 1.59710 0.798549 0.601930i $$-0.205601\pi$$
0.798549 + 0.601930i $$0.205601\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −16.9443 −0.622044
$$743$$ −39.7639 −1.45880 −0.729399 0.684089i $$-0.760200\pi$$
−0.729399 + 0.684089i $$0.760200\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 46.6525 1.70807
$$747$$ 0 0
$$748$$ 1.09017 0.0398606
$$749$$ 18.9443 0.692209
$$750$$ 0 0
$$751$$ −10.1115 −0.368972 −0.184486 0.982835i $$-0.559062\pi$$
−0.184486 + 0.982835i $$0.559062\pi$$
$$752$$ 18.2705 0.666257
$$753$$ 0 0
$$754$$ 1.61803 0.0589253
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 28.8328 1.04795 0.523973 0.851735i $$-0.324449\pi$$
0.523973 + 0.851735i $$0.324449\pi$$
$$758$$ −4.18034 −0.151837
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −44.4721 −1.61211 −0.806057 0.591838i $$-0.798402\pi$$
−0.806057 + 0.591838i $$0.798402\pi$$
$$762$$ 0 0
$$763$$ 78.0132 2.82427
$$764$$ 5.52786 0.199991
$$765$$ 0 0
$$766$$ 29.1246 1.05231
$$767$$ 6.00000 0.216647
$$768$$ 0 0
$$769$$ −17.3050 −0.624033 −0.312016 0.950077i $$-0.601004\pi$$
−0.312016 + 0.950077i $$0.601004\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −12.3607 −0.444871
$$773$$ 20.8328 0.749304 0.374652 0.927165i $$-0.377762\pi$$
0.374652 + 0.927165i $$0.377762\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 34.4721 1.23748
$$777$$ 0 0
$$778$$ −31.5066 −1.12957
$$779$$ −14.8328 −0.531441
$$780$$ 0 0
$$781$$ 1.52786 0.0546713
$$782$$ 54.0689 1.93350
$$783$$ 0 0
$$784$$ −53.1246 −1.89731
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −27.7771 −0.990146 −0.495073 0.868851i $$-0.664859\pi$$
−0.495073 + 0.868851i $$0.664859\pi$$
$$788$$ 15.4164 0.549187
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 6.23607 0.221729
$$792$$ 0 0
$$793$$ 8.47214 0.300854
$$794$$ 12.7639 0.452975
$$795$$ 0 0
$$796$$ −12.4377 −0.440842
$$797$$ 42.7214 1.51327 0.756634 0.653839i $$-0.226842\pi$$
0.756634 + 0.653839i $$0.226842\pi$$
$$798$$ 0 0
$$799$$ 28.1246 0.994977
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −16.7639 −0.591955
$$803$$ −1.41641 −0.0499839
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 12.9443 0.455943
$$807$$ 0 0
$$808$$ −14.5967 −0.513512
$$809$$ −19.9443 −0.701203 −0.350602 0.936525i $$-0.614023\pi$$
−0.350602 + 0.936525i $$0.614023\pi$$
$$810$$ 0 0
$$811$$ 25.1803 0.884201 0.442101 0.896965i $$-0.354233\pi$$
0.442101 + 0.896965i $$0.354233\pi$$
$$812$$ −2.61803 −0.0918750
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 14.8328 0.518935
$$818$$ 44.3607 1.55103
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 31.0557 1.08385 0.541926 0.840426i $$-0.317695\pi$$
0.541926 + 0.840426i $$0.317695\pi$$
$$822$$ 0 0
$$823$$ 2.00000 0.0697156 0.0348578 0.999392i $$-0.488902\pi$$
0.0348578 + 0.999392i $$0.488902\pi$$
$$824$$ −2.11146 −0.0735561
$$825$$ 0 0
$$826$$ −41.1246 −1.43091
$$827$$ 6.11146 0.212516 0.106258 0.994339i $$-0.466113\pi$$
0.106258 + 0.994339i $$0.466113\pi$$
$$828$$ 0 0
$$829$$ 15.8885 0.551832 0.275916 0.961182i $$-0.411019\pi$$
0.275916 + 0.961182i $$0.411019\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −4.23607 −0.146859
$$833$$ −81.7771 −2.83341
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −0.360680 −0.0124744
$$837$$ 0 0
$$838$$ −60.5410 −2.09135
$$839$$ 5.29180 0.182693 0.0913465 0.995819i $$-0.470883\pi$$
0.0913465 + 0.995819i $$0.470883\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ −28.1803 −0.971159
$$843$$ 0 0
$$844$$ −0.583592 −0.0200881
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −46.3607 −1.59297
$$848$$ 12.0000 0.412082
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ −51.3050 −1.75665 −0.878324 0.478066i $$-0.841338\pi$$
−0.878324 + 0.478066i $$0.841338\pi$$
$$854$$ −58.0689 −1.98708
$$855$$ 0 0
$$856$$ −10.0000 −0.341793
$$857$$ −48.7214 −1.66429 −0.832145 0.554558i $$-0.812887\pi$$
−0.832145 + 0.554558i $$0.812887\pi$$
$$858$$ 0 0
$$859$$ 36.8328 1.25672 0.628360 0.777923i $$-0.283727\pi$$
0.628360 + 0.777923i $$0.283727\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 20.1803 0.687345
$$863$$ 45.5279 1.54979 0.774893 0.632092i $$-0.217804\pi$$
0.774893 + 0.632092i $$0.217804\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 63.5967 2.16111
$$867$$ 0 0
$$868$$ −20.9443 −0.710895
$$869$$ 1.41641 0.0480483
$$870$$ 0 0
$$871$$ −1.29180 −0.0437708
$$872$$ −41.1803 −1.39454
$$873$$ 0 0
$$874$$ −17.8885 −0.605089
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 19.8885 0.671588 0.335794 0.941935i $$-0.390995\pi$$
0.335794 + 0.941935i $$0.390995\pi$$
$$878$$ −10.0902 −0.340527
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −15.1115 −0.509118 −0.254559 0.967057i $$-0.581930\pi$$
−0.254559 + 0.967057i $$0.581930\pi$$
$$882$$ 0 0
$$883$$ 14.8328 0.499164 0.249582 0.968354i $$-0.419707\pi$$
0.249582 + 0.968354i $$0.419707\pi$$
$$884$$ 4.61803 0.155321
$$885$$ 0 0
$$886$$ −9.32624 −0.313321
$$887$$ −5.18034 −0.173939 −0.0869694 0.996211i $$-0.527718\pi$$
−0.0869694 + 0.996211i $$0.527718\pi$$
$$888$$ 0 0
$$889$$ 50.8328 1.70488
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −8.14590 −0.272745
$$893$$ −9.30495 −0.311378
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 57.6869 1.92718
$$897$$ 0 0
$$898$$ 67.8673 2.26476
$$899$$ 8.00000 0.266815
$$900$$ 0 0
$$901$$ 18.4721 0.615396
$$902$$ 2.29180 0.0763085
$$903$$ 0 0
$$904$$ −3.29180 −0.109484
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 24.8328 0.824560 0.412280 0.911057i $$-0.364733\pi$$
0.412280 + 0.911057i $$0.364733\pi$$
$$908$$ −8.36068 −0.277459
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −21.7639 −0.721071 −0.360536 0.932745i $$-0.617406\pi$$
−0.360536 + 0.932745i $$0.617406\pi$$
$$912$$ 0 0
$$913$$ −0.583592 −0.0193141
$$914$$ 37.7984 1.25026
$$915$$ 0 0
$$916$$ 2.47214 0.0816817
$$917$$ −17.9443 −0.592572
$$918$$ 0 0
$$919$$ 21.5410 0.710573 0.355286 0.934758i $$-0.384383\pi$$
0.355286 + 0.934758i $$0.384383\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −67.5967 −2.22618
$$923$$ 6.47214 0.213033
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 12.5623 0.412823
$$927$$ 0 0
$$928$$ 3.38197 0.111018
$$929$$ 9.05573 0.297109 0.148554 0.988904i $$-0.452538\pi$$
0.148554 + 0.988904i $$0.452538\pi$$
$$930$$ 0 0
$$931$$ 27.0557 0.886716
$$932$$ −11.1246 −0.364399
$$933$$ 0 0
$$934$$ −38.8328 −1.27065
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −45.4721 −1.48551 −0.742755 0.669563i $$-0.766481\pi$$
−0.742755 + 0.669563i $$0.766481\pi$$
$$938$$ 8.85410 0.289097
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 13.4164 0.437362 0.218681 0.975796i $$-0.429825\pi$$
0.218681 + 0.975796i $$0.429825\pi$$
$$942$$ 0 0
$$943$$ 26.8328 0.873797
$$944$$ 29.1246 0.947925
$$945$$ 0 0
$$946$$ −2.29180 −0.0745127
$$947$$ 57.5410 1.86983 0.934916 0.354869i $$-0.115475\pi$$
0.934916 + 0.354869i $$0.115475\pi$$
$$948$$ 0 0
$$949$$ −6.00000 −0.194768
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 70.7771 2.29390
$$953$$ −1.63932 −0.0531028 −0.0265514 0.999647i $$-0.508453\pi$$
−0.0265514 + 0.999647i $$0.508453\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −12.9443 −0.418648
$$957$$ 0 0
$$958$$ −54.8328 −1.77157
$$959$$ 29.4164 0.949905
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 0 0
$$963$$ 0 0
$$964$$ −3.09017 −0.0995277
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −11.4164 −0.367127 −0.183563 0.983008i $$-0.558763\pi$$
−0.183563 + 0.983008i $$0.558763\pi$$
$$968$$ 24.4721 0.786564
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 10.1115 0.324492 0.162246 0.986750i $$-0.448126\pi$$
0.162246 + 0.986750i $$0.448126\pi$$
$$972$$ 0 0
$$973$$ 32.8885 1.05436
$$974$$ 11.4164 0.365805
$$975$$ 0 0
$$976$$ 41.1246 1.31637
$$977$$ 48.9443 1.56587 0.782933 0.622106i $$-0.213723\pi$$
0.782933 + 0.622106i $$0.213723\pi$$
$$978$$ 0 0
$$979$$ 2.34752 0.0750272
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 24.0000 0.765871
$$983$$ −0.944272 −0.0301176 −0.0150588 0.999887i $$-0.504794\pi$$
−0.0150588 + 0.999887i $$0.504794\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 12.0902 0.385029
$$987$$ 0 0
$$988$$ −1.52786 −0.0486078
$$989$$ −26.8328 −0.853234
$$990$$ 0 0
$$991$$ −2.70820 −0.0860289 −0.0430145 0.999074i $$-0.513696\pi$$
−0.0430145 + 0.999074i $$0.513696\pi$$
$$992$$ 27.0557 0.859020
$$993$$ 0 0
$$994$$ −44.3607 −1.40704
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −12.9443 −0.409949 −0.204975 0.978767i $$-0.565711\pi$$
−0.204975 + 0.978767i $$0.565711\pi$$
$$998$$ −42.2705 −1.33805
$$999$$ 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 6525.2.a.bb.1.2 2
3.2 odd 2 6525.2.a.r.1.1 2
5.4 even 2 1305.2.a.h.1.1 2
15.14 odd 2 1305.2.a.l.1.2 yes 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1305.2.a.h.1.1 2 5.4 even 2
1305.2.a.l.1.2 yes 2 15.14 odd 2
6525.2.a.r.1.1 2 3.2 odd 2
6525.2.a.bb.1.2 2 1.1 even 1 trivial