# Properties

 Label 6525.2.a.ce.1.6 Level $6525$ Weight $2$ Character 6525.1 Self dual yes Analytic conductor $52.102$ Analytic rank $1$ Dimension $12$ Inner twists $2$

# 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: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 20x^{10} + 148x^{8} - 502x^{6} + 792x^{4} - 496x^{2} + 45$$ x^12 - 20*x^10 + 148*x^8 - 502*x^6 + 792*x^4 - 496*x^2 + 45 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 1305) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.6 Root $$-0.328889$$ 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)

 $$f(q)$$ $$=$$ $$q-0.328889 q^{2} -1.89183 q^{4} -1.86588 q^{7} +1.27998 q^{8} +O(q^{10})$$ $$q-0.328889 q^{2} -1.89183 q^{4} -1.86588 q^{7} +1.27998 q^{8} -0.564087 q^{11} -4.95986 q^{13} +0.613668 q^{14} +3.36269 q^{16} +7.06879 q^{17} -3.32971 q^{19} +0.185522 q^{22} +2.36471 q^{23} +1.63124 q^{26} +3.52994 q^{28} +1.00000 q^{29} +4.87352 q^{31} -3.66591 q^{32} -2.32485 q^{34} -8.50474 q^{37} +1.09510 q^{38} -7.86030 q^{41} +4.39693 q^{43} +1.06716 q^{44} -0.777727 q^{46} +7.06879 q^{47} -3.51848 q^{49} +9.38323 q^{52} +4.18169 q^{53} -2.38829 q^{56} -0.328889 q^{58} +9.01198 q^{59} -1.26059 q^{61} -1.60285 q^{62} -5.51971 q^{64} -6.61254 q^{67} -13.3730 q^{68} +5.46480 q^{71} +5.77659 q^{73} +2.79711 q^{74} +6.29925 q^{76} +1.05252 q^{77} +12.1018 q^{79} +2.58516 q^{82} +9.41830 q^{83} -1.44610 q^{86} -0.722020 q^{88} +0.622547 q^{89} +9.25453 q^{91} -4.47364 q^{92} -2.32485 q^{94} +19.2638 q^{97} +1.15719 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 16 q^{4}+O(q^{10})$$ 12 * q + 16 * q^4 $$12 q + 16 q^{4} - 12 q^{11} - 16 q^{14} + 16 q^{16} - 20 q^{19} - 56 q^{26} + 12 q^{29} - 16 q^{31} + 4 q^{34} - 32 q^{41} - 68 q^{44} + 20 q^{46} - 4 q^{49} - 76 q^{56} - 44 q^{59} - 52 q^{61} + 36 q^{64} - 20 q^{71} - 20 q^{74} - 28 q^{76} - 4 q^{79} - 36 q^{86} - 68 q^{89} + 48 q^{91} + 4 q^{94}+O(q^{100})$$ 12 * q + 16 * q^4 - 12 * q^11 - 16 * q^14 + 16 * q^16 - 20 * q^19 - 56 * q^26 + 12 * q^29 - 16 * q^31 + 4 * q^34 - 32 * q^41 - 68 * q^44 + 20 * q^46 - 4 * q^49 - 76 * q^56 - 44 * q^59 - 52 * q^61 + 36 * q^64 - 20 * q^71 - 20 * q^74 - 28 * q^76 - 4 * q^79 - 36 * q^86 - 68 * q^89 + 48 * q^91 + 4 * q^94

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.328889 −0.232560 −0.116280 0.993216i $$-0.537097\pi$$
−0.116280 + 0.993216i $$0.537097\pi$$
$$3$$ 0 0
$$4$$ −1.89183 −0.945916
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −1.86588 −0.705237 −0.352619 0.935767i $$-0.614709\pi$$
−0.352619 + 0.935767i $$0.614709\pi$$
$$8$$ 1.27998 0.452541
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −0.564087 −0.170079 −0.0850393 0.996378i $$-0.527102\pi$$
−0.0850393 + 0.996378i $$0.527102\pi$$
$$12$$ 0 0
$$13$$ −4.95986 −1.37562 −0.687809 0.725891i $$-0.741428\pi$$
−0.687809 + 0.725891i $$0.741428\pi$$
$$14$$ 0.613668 0.164010
$$15$$ 0 0
$$16$$ 3.36269 0.840673
$$17$$ 7.06879 1.71443 0.857217 0.514956i $$-0.172192\pi$$
0.857217 + 0.514956i $$0.172192\pi$$
$$18$$ 0 0
$$19$$ −3.32971 −0.763887 −0.381944 0.924186i $$-0.624745\pi$$
−0.381944 + 0.924186i $$0.624745\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0.185522 0.0395534
$$23$$ 2.36471 0.493076 0.246538 0.969133i $$-0.420707\pi$$
0.246538 + 0.969133i $$0.420707\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 1.63124 0.319913
$$27$$ 0 0
$$28$$ 3.52994 0.667095
$$29$$ 1.00000 0.185695
$$30$$ 0 0
$$31$$ 4.87352 0.875310 0.437655 0.899143i $$-0.355809\pi$$
0.437655 + 0.899143i $$0.355809\pi$$
$$32$$ −3.66591 −0.648048
$$33$$ 0 0
$$34$$ −2.32485 −0.398708
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −8.50474 −1.39817 −0.699085 0.715038i $$-0.746409\pi$$
−0.699085 + 0.715038i $$0.746409\pi$$
$$38$$ 1.09510 0.177649
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −7.86030 −1.22757 −0.613786 0.789473i $$-0.710354\pi$$
−0.613786 + 0.789473i $$0.710354\pi$$
$$42$$ 0 0
$$43$$ 4.39693 0.670526 0.335263 0.942125i $$-0.391175\pi$$
0.335263 + 0.942125i $$0.391175\pi$$
$$44$$ 1.06716 0.160880
$$45$$ 0 0
$$46$$ −0.777727 −0.114670
$$47$$ 7.06879 1.03109 0.515544 0.856863i $$-0.327590\pi$$
0.515544 + 0.856863i $$0.327590\pi$$
$$48$$ 0 0
$$49$$ −3.51848 −0.502640
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 9.38323 1.30122
$$53$$ 4.18169 0.574400 0.287200 0.957871i $$-0.407276\pi$$
0.287200 + 0.957871i $$0.407276\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −2.38829 −0.319149
$$57$$ 0 0
$$58$$ −0.328889 −0.0431852
$$59$$ 9.01198 1.17326 0.586630 0.809855i $$-0.300454\pi$$
0.586630 + 0.809855i $$0.300454\pi$$
$$60$$ 0 0
$$61$$ −1.26059 −0.161402 −0.0807009 0.996738i $$-0.525716\pi$$
−0.0807009 + 0.996738i $$0.525716\pi$$
$$62$$ −1.60285 −0.203562
$$63$$ 0 0
$$64$$ −5.51971 −0.689964
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −6.61254 −0.807851 −0.403925 0.914792i $$-0.632354\pi$$
−0.403925 + 0.914792i $$0.632354\pi$$
$$68$$ −13.3730 −1.62171
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 5.46480 0.648552 0.324276 0.945962i $$-0.394879\pi$$
0.324276 + 0.945962i $$0.394879\pi$$
$$72$$ 0 0
$$73$$ 5.77659 0.676099 0.338049 0.941128i $$-0.390233\pi$$
0.338049 + 0.941128i $$0.390233\pi$$
$$74$$ 2.79711 0.325158
$$75$$ 0 0
$$76$$ 6.29925 0.722573
$$77$$ 1.05252 0.119946
$$78$$ 0 0
$$79$$ 12.1018 1.36156 0.680782 0.732486i $$-0.261640\pi$$
0.680782 + 0.732486i $$0.261640\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 2.58516 0.285484
$$83$$ 9.41830 1.03379 0.516896 0.856048i $$-0.327087\pi$$
0.516896 + 0.856048i $$0.327087\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −1.44610 −0.155937
$$87$$ 0 0
$$88$$ −0.722020 −0.0769676
$$89$$ 0.622547 0.0659899 0.0329949 0.999456i $$-0.489495\pi$$
0.0329949 + 0.999456i $$0.489495\pi$$
$$90$$ 0 0
$$91$$ 9.25453 0.970138
$$92$$ −4.47364 −0.466409
$$93$$ 0 0
$$94$$ −2.32485 −0.239790
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 19.2638 1.95594 0.977970 0.208743i $$-0.0669373\pi$$
0.977970 + 0.208743i $$0.0669373\pi$$
$$98$$ 1.15719 0.116894
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −9.33018 −0.928387 −0.464194 0.885734i $$-0.653656\pi$$
−0.464194 + 0.885734i $$0.653656\pi$$
$$102$$ 0 0
$$103$$ 7.79546 0.768110 0.384055 0.923310i $$-0.374527\pi$$
0.384055 + 0.923310i $$0.374527\pi$$
$$104$$ −6.34853 −0.622524
$$105$$ 0 0
$$106$$ −1.37531 −0.133582
$$107$$ −15.5446 −1.50276 −0.751378 0.659873i $$-0.770610\pi$$
−0.751378 + 0.659873i $$0.770610\pi$$
$$108$$ 0 0
$$109$$ −3.89814 −0.373374 −0.186687 0.982419i $$-0.559775\pi$$
−0.186687 + 0.982419i $$0.559775\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −6.27439 −0.592874
$$113$$ −6.89464 −0.648593 −0.324297 0.945955i $$-0.605128\pi$$
−0.324297 + 0.945955i $$0.605128\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −1.89183 −0.175652
$$117$$ 0 0
$$118$$ −2.96394 −0.272853
$$119$$ −13.1895 −1.20908
$$120$$ 0 0
$$121$$ −10.6818 −0.971073
$$122$$ 0.414593 0.0375355
$$123$$ 0 0
$$124$$ −9.21989 −0.827970
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −6.04498 −0.536405 −0.268203 0.963363i $$-0.586430\pi$$
−0.268203 + 0.963363i $$0.586430\pi$$
$$128$$ 9.14720 0.808506
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −7.99224 −0.698285 −0.349143 0.937070i $$-0.613527\pi$$
−0.349143 + 0.937070i $$0.613527\pi$$
$$132$$ 0 0
$$133$$ 6.21284 0.538722
$$134$$ 2.17479 0.187873
$$135$$ 0 0
$$136$$ 9.04791 0.775852
$$137$$ −14.9670 −1.27872 −0.639361 0.768907i $$-0.720801\pi$$
−0.639361 + 0.768907i $$0.720801\pi$$
$$138$$ 0 0
$$139$$ −5.25453 −0.445683 −0.222842 0.974855i $$-0.571533\pi$$
−0.222842 + 0.974855i $$0.571533\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −1.79731 −0.150827
$$143$$ 2.79779 0.233963
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −1.89986 −0.157233
$$147$$ 0 0
$$148$$ 16.0895 1.32255
$$149$$ 2.97589 0.243795 0.121897 0.992543i $$-0.461102\pi$$
0.121897 + 0.992543i $$0.461102\pi$$
$$150$$ 0 0
$$151$$ −12.8646 −1.04690 −0.523452 0.852055i $$-0.675356\pi$$
−0.523452 + 0.852055i $$0.675356\pi$$
$$152$$ −4.26196 −0.345690
$$153$$ 0 0
$$154$$ −0.346162 −0.0278945
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 8.72737 0.696520 0.348260 0.937398i $$-0.386773\pi$$
0.348260 + 0.937398i $$0.386773\pi$$
$$158$$ −3.98016 −0.316645
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −4.41227 −0.347736
$$162$$ 0 0
$$163$$ −13.4872 −1.05640 −0.528199 0.849120i $$-0.677133\pi$$
−0.528199 + 0.849120i $$0.677133\pi$$
$$164$$ 14.8704 1.16118
$$165$$ 0 0
$$166$$ −3.09757 −0.240418
$$167$$ 24.7351 1.91406 0.957029 0.289993i $$-0.0936528\pi$$
0.957029 + 0.289993i $$0.0936528\pi$$
$$168$$ 0 0
$$169$$ 11.6003 0.892327
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −8.31826 −0.634262
$$173$$ −15.4531 −1.17488 −0.587440 0.809268i $$-0.699864\pi$$
−0.587440 + 0.809268i $$0.699864\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −1.89685 −0.142980
$$177$$ 0 0
$$178$$ −0.204749 −0.0153466
$$179$$ −17.3358 −1.29573 −0.647867 0.761753i $$-0.724339\pi$$
−0.647867 + 0.761753i $$0.724339\pi$$
$$180$$ 0 0
$$181$$ −2.04972 −0.152355 −0.0761773 0.997094i $$-0.524271\pi$$
−0.0761773 + 0.997094i $$0.524271\pi$$
$$182$$ −3.04371 −0.225615
$$183$$ 0 0
$$184$$ 3.02678 0.223137
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −3.98741 −0.291588
$$188$$ −13.3730 −0.975324
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −1.76575 −0.127765 −0.0638824 0.997957i $$-0.520348\pi$$
−0.0638824 + 0.997957i $$0.520348\pi$$
$$192$$ 0 0
$$193$$ −8.81831 −0.634756 −0.317378 0.948299i $$-0.602802\pi$$
−0.317378 + 0.948299i $$0.602802\pi$$
$$194$$ −6.33564 −0.454873
$$195$$ 0 0
$$196$$ 6.65638 0.475455
$$197$$ −13.4231 −0.956356 −0.478178 0.878263i $$-0.658703\pi$$
−0.478178 + 0.878263i $$0.658703\pi$$
$$198$$ 0 0
$$199$$ −6.08459 −0.431325 −0.215663 0.976468i $$-0.569191\pi$$
−0.215663 + 0.976468i $$0.569191\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 3.06859 0.215905
$$203$$ −1.86588 −0.130959
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −2.56384 −0.178631
$$207$$ 0 0
$$208$$ −16.6785 −1.15645
$$209$$ 1.87824 0.129921
$$210$$ 0 0
$$211$$ 9.05366 0.623280 0.311640 0.950200i $$-0.399122\pi$$
0.311640 + 0.950200i $$0.399122\pi$$
$$212$$ −7.91106 −0.543334
$$213$$ 0 0
$$214$$ 5.11245 0.349480
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −9.09342 −0.617302
$$218$$ 1.28206 0.0868318
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −35.0602 −2.35841
$$222$$ 0 0
$$223$$ −18.5300 −1.24086 −0.620432 0.784260i $$-0.713043\pi$$
−0.620432 + 0.784260i $$0.713043\pi$$
$$224$$ 6.84016 0.457028
$$225$$ 0 0
$$226$$ 2.26757 0.150837
$$227$$ 23.6011 1.56646 0.783229 0.621734i $$-0.213571\pi$$
0.783229 + 0.621734i $$0.213571\pi$$
$$228$$ 0 0
$$229$$ 10.5987 0.700384 0.350192 0.936678i $$-0.386116\pi$$
0.350192 + 0.936678i $$0.386116\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 1.27998 0.0840348
$$233$$ 8.62938 0.565329 0.282665 0.959219i $$-0.408782\pi$$
0.282665 + 0.959219i $$0.408782\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −17.0492 −1.10981
$$237$$ 0 0
$$238$$ 4.33789 0.281184
$$239$$ −24.0914 −1.55834 −0.779171 0.626811i $$-0.784360\pi$$
−0.779171 + 0.626811i $$0.784360\pi$$
$$240$$ 0 0
$$241$$ −25.6988 −1.65541 −0.827703 0.561166i $$-0.810353\pi$$
−0.827703 + 0.561166i $$0.810353\pi$$
$$242$$ 3.51313 0.225832
$$243$$ 0 0
$$244$$ 2.38482 0.152673
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 16.5149 1.05082
$$248$$ 6.23801 0.396114
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −25.0072 −1.57844 −0.789219 0.614112i $$-0.789514\pi$$
−0.789219 + 0.614112i $$0.789514\pi$$
$$252$$ 0 0
$$253$$ −1.33390 −0.0838617
$$254$$ 1.98813 0.124746
$$255$$ 0 0
$$256$$ 8.03101 0.501938
$$257$$ 1.73788 0.108406 0.0542030 0.998530i $$-0.482738\pi$$
0.0542030 + 0.998530i $$0.482738\pi$$
$$258$$ 0 0
$$259$$ 15.8688 0.986042
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 2.62856 0.162393
$$263$$ 8.87689 0.547372 0.273686 0.961819i $$-0.411757\pi$$
0.273686 + 0.961819i $$0.411757\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −2.04333 −0.125285
$$267$$ 0 0
$$268$$ 12.5098 0.764159
$$269$$ −6.55657 −0.399761 −0.199881 0.979820i $$-0.564055\pi$$
−0.199881 + 0.979820i $$0.564055\pi$$
$$270$$ 0 0
$$271$$ −20.8413 −1.26602 −0.633008 0.774145i $$-0.718180\pi$$
−0.633008 + 0.774145i $$0.718180\pi$$
$$272$$ 23.7702 1.44128
$$273$$ 0 0
$$274$$ 4.92250 0.297379
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 17.7687 1.06762 0.533810 0.845605i $$-0.320760\pi$$
0.533810 + 0.845605i $$0.320760\pi$$
$$278$$ 1.72815 0.103648
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 14.8520 0.885995 0.442998 0.896523i $$-0.353915\pi$$
0.442998 + 0.896523i $$0.353915\pi$$
$$282$$ 0 0
$$283$$ 13.8856 0.825414 0.412707 0.910864i $$-0.364583\pi$$
0.412707 + 0.910864i $$0.364583\pi$$
$$284$$ −10.3385 −0.613476
$$285$$ 0 0
$$286$$ −0.920163 −0.0544104
$$287$$ 14.6664 0.865730
$$288$$ 0 0
$$289$$ 32.9678 1.93928
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −10.9283 −0.639533
$$293$$ 7.21254 0.421361 0.210681 0.977555i $$-0.432432\pi$$
0.210681 + 0.977555i $$0.432432\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −10.8859 −0.632730
$$297$$ 0 0
$$298$$ −0.978738 −0.0566968
$$299$$ −11.7286 −0.678285
$$300$$ 0 0
$$301$$ −8.20417 −0.472880
$$302$$ 4.23101 0.243467
$$303$$ 0 0
$$304$$ −11.1968 −0.642179
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 21.2856 1.21483 0.607417 0.794383i $$-0.292206\pi$$
0.607417 + 0.794383i $$0.292206\pi$$
$$308$$ −1.99119 −0.113459
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −29.1250 −1.65153 −0.825764 0.564015i $$-0.809256\pi$$
−0.825764 + 0.564015i $$0.809256\pi$$
$$312$$ 0 0
$$313$$ 26.1401 1.47752 0.738762 0.673966i $$-0.235411\pi$$
0.738762 + 0.673966i $$0.235411\pi$$
$$314$$ −2.87034 −0.161982
$$315$$ 0 0
$$316$$ −22.8946 −1.28792
$$317$$ 7.83033 0.439795 0.219898 0.975523i $$-0.429428\pi$$
0.219898 + 0.975523i $$0.429428\pi$$
$$318$$ 0 0
$$319$$ −0.564087 −0.0315828
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 1.45115 0.0808693
$$323$$ −23.5370 −1.30963
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 4.43579 0.245676
$$327$$ 0 0
$$328$$ −10.0610 −0.555527
$$329$$ −13.1895 −0.727163
$$330$$ 0 0
$$331$$ −18.8874 −1.03814 −0.519072 0.854731i $$-0.673722\pi$$
−0.519072 + 0.854731i $$0.673722\pi$$
$$332$$ −17.8178 −0.977881
$$333$$ 0 0
$$334$$ −8.13509 −0.445132
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −8.33059 −0.453796 −0.226898 0.973919i $$-0.572858\pi$$
−0.226898 + 0.973919i $$0.572858\pi$$
$$338$$ −3.81519 −0.207519
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −2.74909 −0.148872
$$342$$ 0 0
$$343$$ 19.6263 1.05972
$$344$$ 5.62799 0.303441
$$345$$ 0 0
$$346$$ 5.08236 0.273230
$$347$$ −32.5993 −1.75002 −0.875010 0.484104i $$-0.839146\pi$$
−0.875010 + 0.484104i $$0.839146\pi$$
$$348$$ 0 0
$$349$$ 33.6370 1.80055 0.900273 0.435326i $$-0.143367\pi$$
0.900273 + 0.435326i $$0.143367\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 2.06789 0.110219
$$353$$ 2.85951 0.152196 0.0760982 0.997100i $$-0.475754\pi$$
0.0760982 + 0.997100i $$0.475754\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −1.17775 −0.0624209
$$357$$ 0 0
$$358$$ 5.70154 0.301335
$$359$$ −33.6883 −1.77800 −0.889001 0.457905i $$-0.848600\pi$$
−0.889001 + 0.457905i $$0.848600\pi$$
$$360$$ 0 0
$$361$$ −7.91305 −0.416477
$$362$$ 0.674130 0.0354315
$$363$$ 0 0
$$364$$ −17.5080 −0.917669
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −9.63376 −0.502878 −0.251439 0.967873i $$-0.580904\pi$$
−0.251439 + 0.967873i $$0.580904\pi$$
$$368$$ 7.95180 0.414516
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −7.80255 −0.405088
$$372$$ 0 0
$$373$$ −5.28344 −0.273566 −0.136783 0.990601i $$-0.543676\pi$$
−0.136783 + 0.990601i $$0.543676\pi$$
$$374$$ 1.31141 0.0678116
$$375$$ 0 0
$$376$$ 9.04791 0.466610
$$377$$ −4.95986 −0.255446
$$378$$ 0 0
$$379$$ −2.62792 −0.134987 −0.0674936 0.997720i $$-0.521500\pi$$
−0.0674936 + 0.997720i $$0.521500\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0.580734 0.0297129
$$383$$ −0.626829 −0.0320295 −0.0160147 0.999872i $$-0.505098\pi$$
−0.0160147 + 0.999872i $$0.505098\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 2.90024 0.147618
$$387$$ 0 0
$$388$$ −36.4438 −1.85016
$$389$$ −26.3950 −1.33828 −0.669140 0.743136i $$-0.733338\pi$$
−0.669140 + 0.743136i $$0.733338\pi$$
$$390$$ 0 0
$$391$$ 16.7156 0.845347
$$392$$ −4.50359 −0.227465
$$393$$ 0 0
$$394$$ 4.41471 0.222410
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −27.8333 −1.39691 −0.698457 0.715652i $$-0.746130\pi$$
−0.698457 + 0.715652i $$0.746130\pi$$
$$398$$ 2.00115 0.100309
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −11.0561 −0.552117 −0.276058 0.961141i $$-0.589028\pi$$
−0.276058 + 0.961141i $$0.589028\pi$$
$$402$$ 0 0
$$403$$ −24.1720 −1.20409
$$404$$ 17.6511 0.878176
$$405$$ 0 0
$$406$$ 0.613668 0.0304558
$$407$$ 4.79741 0.237799
$$408$$ 0 0
$$409$$ 1.57317 0.0777882 0.0388941 0.999243i $$-0.487617\pi$$
0.0388941 + 0.999243i $$0.487617\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −14.7477 −0.726567
$$413$$ −16.8153 −0.827427
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 18.1824 0.891467
$$417$$ 0 0
$$418$$ −0.617733 −0.0302143
$$419$$ −35.9256 −1.75508 −0.877540 0.479504i $$-0.840817\pi$$
−0.877540 + 0.479504i $$0.840817\pi$$
$$420$$ 0 0
$$421$$ −15.5075 −0.755788 −0.377894 0.925849i $$-0.623352\pi$$
−0.377894 + 0.925849i $$0.623352\pi$$
$$422$$ −2.97765 −0.144950
$$423$$ 0 0
$$424$$ 5.35248 0.259940
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 2.35211 0.113827
$$428$$ 29.4078 1.42148
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −16.9390 −0.815924 −0.407962 0.912999i $$-0.633760\pi$$
−0.407962 + 0.912999i $$0.633760\pi$$
$$432$$ 0 0
$$433$$ 40.5136 1.94696 0.973480 0.228772i $$-0.0734710\pi$$
0.973480 + 0.228772i $$0.0734710\pi$$
$$434$$ 2.99072 0.143559
$$435$$ 0 0
$$436$$ 7.37463 0.353181
$$437$$ −7.87380 −0.376655
$$438$$ 0 0
$$439$$ −23.4655 −1.11995 −0.559973 0.828511i $$-0.689188\pi$$
−0.559973 + 0.828511i $$0.689188\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 11.5309 0.548470
$$443$$ −16.3994 −0.779161 −0.389580 0.920992i $$-0.627380\pi$$
−0.389580 + 0.920992i $$0.627380\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 6.09432 0.288575
$$447$$ 0 0
$$448$$ 10.2991 0.486588
$$449$$ 8.92984 0.421425 0.210713 0.977548i $$-0.432422\pi$$
0.210713 + 0.977548i $$0.432422\pi$$
$$450$$ 0 0
$$451$$ 4.43389 0.208784
$$452$$ 13.0435 0.613515
$$453$$ 0 0
$$454$$ −7.76213 −0.364295
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 3.16068 0.147850 0.0739252 0.997264i $$-0.476447\pi$$
0.0739252 + 0.997264i $$0.476447\pi$$
$$458$$ −3.48580 −0.162881
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −3.85936 −0.179748 −0.0898741 0.995953i $$-0.528646\pi$$
−0.0898741 + 0.995953i $$0.528646\pi$$
$$462$$ 0 0
$$463$$ −24.5493 −1.14090 −0.570451 0.821331i $$-0.693232\pi$$
−0.570451 + 0.821331i $$0.693232\pi$$
$$464$$ 3.36269 0.156109
$$465$$ 0 0
$$466$$ −2.83811 −0.131473
$$467$$ −16.7566 −0.775405 −0.387703 0.921785i $$-0.626731\pi$$
−0.387703 + 0.921785i $$0.626731\pi$$
$$468$$ 0 0
$$469$$ 12.3382 0.569727
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 11.5352 0.530949
$$473$$ −2.48025 −0.114042
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 24.9524 1.14369
$$477$$ 0 0
$$478$$ 7.92339 0.362407
$$479$$ 18.4125 0.841289 0.420644 0.907226i $$-0.361804\pi$$
0.420644 + 0.907226i $$0.361804\pi$$
$$480$$ 0 0
$$481$$ 42.1823 1.92335
$$482$$ 8.45205 0.384981
$$483$$ 0 0
$$484$$ 20.2082 0.918554
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −18.3658 −0.832232 −0.416116 0.909311i $$-0.636609\pi$$
−0.416116 + 0.909311i $$0.636609\pi$$
$$488$$ −1.61353 −0.0730410
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −17.2651 −0.779165 −0.389582 0.920992i $$-0.627381\pi$$
−0.389582 + 0.920992i $$0.627381\pi$$
$$492$$ 0 0
$$493$$ 7.06879 0.318362
$$494$$ −5.43156 −0.244378
$$495$$ 0 0
$$496$$ 16.3882 0.735850
$$497$$ −10.1967 −0.457383
$$498$$ 0 0
$$499$$ −22.2484 −0.995976 −0.497988 0.867184i $$-0.665928\pi$$
−0.497988 + 0.867184i $$0.665928\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 8.22458 0.367081
$$503$$ 25.4171 1.13329 0.566646 0.823961i $$-0.308241\pi$$
0.566646 + 0.823961i $$0.308241\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0.438706 0.0195028
$$507$$ 0 0
$$508$$ 11.4361 0.507394
$$509$$ −12.5897 −0.558028 −0.279014 0.960287i $$-0.590008\pi$$
−0.279014 + 0.960287i $$0.590008\pi$$
$$510$$ 0 0
$$511$$ −10.7784 −0.476810
$$512$$ −20.9357 −0.925236
$$513$$ 0 0
$$514$$ −0.571570 −0.0252109
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −3.98741 −0.175366
$$518$$ −5.21909 −0.229313
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −25.3294 −1.10970 −0.554851 0.831949i $$-0.687225\pi$$
−0.554851 + 0.831949i $$0.687225\pi$$
$$522$$ 0 0
$$523$$ −40.2732 −1.76102 −0.880512 0.474023i $$-0.842801\pi$$
−0.880512 + 0.474023i $$0.842801\pi$$
$$524$$ 15.1200 0.660519
$$525$$ 0 0
$$526$$ −2.91951 −0.127297
$$527$$ 34.4499 1.50066
$$528$$ 0 0
$$529$$ −17.4081 −0.756876
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −11.7537 −0.509586
$$533$$ 38.9860 1.68867
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −8.46393 −0.365586
$$537$$ 0 0
$$538$$ 2.15638 0.0929683
$$539$$ 1.98473 0.0854883
$$540$$ 0 0
$$541$$ 10.5720 0.454527 0.227263 0.973833i $$-0.427022\pi$$
0.227263 + 0.973833i $$0.427022\pi$$
$$542$$ 6.85446 0.294424
$$543$$ 0 0
$$544$$ −25.9136 −1.11103
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −28.0175 −1.19794 −0.598971 0.800770i $$-0.704424\pi$$
−0.598971 + 0.800770i $$0.704424\pi$$
$$548$$ 28.3151 1.20956
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −3.32971 −0.141850
$$552$$ 0 0
$$553$$ −22.5806 −0.960225
$$554$$ −5.84394 −0.248285
$$555$$ 0 0
$$556$$ 9.94068 0.421579
$$557$$ 38.8895 1.64780 0.823901 0.566734i $$-0.191793\pi$$
0.823901 + 0.566734i $$0.191793\pi$$
$$558$$ 0 0
$$559$$ −21.8082 −0.922389
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −4.88465 −0.206047
$$563$$ −30.3735 −1.28009 −0.640045 0.768337i $$-0.721084\pi$$
−0.640045 + 0.768337i $$0.721084\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −4.56683 −0.191958
$$567$$ 0 0
$$568$$ 6.99483 0.293497
$$569$$ 46.3254 1.94206 0.971031 0.238953i $$-0.0768043\pi$$
0.971031 + 0.238953i $$0.0768043\pi$$
$$570$$ 0 0
$$571$$ 38.9012 1.62796 0.813982 0.580890i $$-0.197295\pi$$
0.813982 + 0.580890i $$0.197295\pi$$
$$572$$ −5.29295 −0.221310
$$573$$ 0 0
$$574$$ −4.82361 −0.201334
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 24.5426 1.02172 0.510861 0.859663i $$-0.329327\pi$$
0.510861 + 0.859663i $$0.329327\pi$$
$$578$$ −10.8427 −0.450998
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −17.5734 −0.729069
$$582$$ 0 0
$$583$$ −2.35884 −0.0976931
$$584$$ 7.39392 0.305963
$$585$$ 0 0
$$586$$ −2.37212 −0.0979915
$$587$$ −42.3399 −1.74755 −0.873777 0.486328i $$-0.838336\pi$$
−0.873777 + 0.486328i $$0.838336\pi$$
$$588$$ 0 0
$$589$$ −16.2274 −0.668638
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −28.5988 −1.17540
$$593$$ −4.17650 −0.171508 −0.0857541 0.996316i $$-0.527330\pi$$
−0.0857541 + 0.996316i $$0.527330\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −5.62989 −0.230609
$$597$$ 0 0
$$598$$ 3.85742 0.157742
$$599$$ 11.9559 0.488505 0.244252 0.969712i $$-0.421457\pi$$
0.244252 + 0.969712i $$0.421457\pi$$
$$600$$ 0 0
$$601$$ −25.1790 −1.02707 −0.513537 0.858068i $$-0.671665\pi$$
−0.513537 + 0.858068i $$0.671665\pi$$
$$602$$ 2.69826 0.109973
$$603$$ 0 0
$$604$$ 24.3376 0.990282
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 19.4870 0.790953 0.395476 0.918476i $$-0.370580\pi$$
0.395476 + 0.918476i $$0.370580\pi$$
$$608$$ 12.2064 0.495035
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −35.0602 −1.41839
$$612$$ 0 0
$$613$$ −27.7763 −1.12187 −0.560937 0.827859i $$-0.689559\pi$$
−0.560937 + 0.827859i $$0.689559\pi$$
$$614$$ −7.00060 −0.282521
$$615$$ 0 0
$$616$$ 1.34720 0.0542804
$$617$$ −12.7828 −0.514616 −0.257308 0.966329i $$-0.582836\pi$$
−0.257308 + 0.966329i $$0.582836\pi$$
$$618$$ 0 0
$$619$$ −5.83385 −0.234482 −0.117241 0.993103i $$-0.537405\pi$$
−0.117241 + 0.993103i $$0.537405\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 9.57890 0.384079
$$623$$ −1.16160 −0.0465385
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −8.59718 −0.343612
$$627$$ 0 0
$$628$$ −16.5107 −0.658850
$$629$$ −60.1182 −2.39707
$$630$$ 0 0
$$631$$ −9.97085 −0.396933 −0.198467 0.980108i $$-0.563596\pi$$
−0.198467 + 0.980108i $$0.563596\pi$$
$$632$$ 15.4901 0.616164
$$633$$ 0 0
$$634$$ −2.57531 −0.102279
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 17.4512 0.691441
$$638$$ 0.185522 0.00734488
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 47.6315 1.88133 0.940667 0.339332i $$-0.110201\pi$$
0.940667 + 0.339332i $$0.110201\pi$$
$$642$$ 0 0
$$643$$ −7.26701 −0.286583 −0.143291 0.989681i $$-0.545769\pi$$
−0.143291 + 0.989681i $$0.545769\pi$$
$$644$$ 8.34728 0.328929
$$645$$ 0 0
$$646$$ 7.74106 0.304568
$$647$$ −17.7415 −0.697488 −0.348744 0.937218i $$-0.613392\pi$$
−0.348744 + 0.937218i $$0.613392\pi$$
$$648$$ 0 0
$$649$$ −5.08354 −0.199546
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 25.5155 0.999264
$$653$$ 23.4342 0.917050 0.458525 0.888682i $$-0.348378\pi$$
0.458525 + 0.888682i $$0.348378\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −26.4318 −1.03199
$$657$$ 0 0
$$658$$ 4.33789 0.169109
$$659$$ 30.4650 1.18675 0.593373 0.804928i $$-0.297796\pi$$
0.593373 + 0.804928i $$0.297796\pi$$
$$660$$ 0 0
$$661$$ 1.35516 0.0527095 0.0263547 0.999653i $$-0.491610\pi$$
0.0263547 + 0.999653i $$0.491610\pi$$
$$662$$ 6.21184 0.241430
$$663$$ 0 0
$$664$$ 12.0552 0.467834
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 2.36471 0.0915620
$$668$$ −46.7946 −1.81054
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0.711081 0.0274510
$$672$$ 0 0
$$673$$ −6.26746 −0.241593 −0.120796 0.992677i $$-0.538545\pi$$
−0.120796 + 0.992677i $$0.538545\pi$$
$$674$$ 2.73984 0.105535
$$675$$ 0 0
$$676$$ −21.9457 −0.844066
$$677$$ −32.3653 −1.24390 −0.621950 0.783057i $$-0.713659\pi$$
−0.621950 + 0.783057i $$0.713659\pi$$
$$678$$ 0 0
$$679$$ −35.9440 −1.37940
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0.904145 0.0346215
$$683$$ −29.0003 −1.10967 −0.554833 0.831962i $$-0.687218\pi$$
−0.554833 + 0.831962i $$0.687218\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −6.45486 −0.246448
$$687$$ 0 0
$$688$$ 14.7855 0.563694
$$689$$ −20.7406 −0.790155
$$690$$ 0 0
$$691$$ 39.6602 1.50875 0.754373 0.656446i $$-0.227941\pi$$
0.754373 + 0.656446i $$0.227941\pi$$
$$692$$ 29.2347 1.11134
$$693$$ 0 0
$$694$$ 10.7215 0.406984
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −55.5628 −2.10459
$$698$$ −11.0628 −0.418734
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 1.21005 0.0457029 0.0228514 0.999739i $$-0.492726\pi$$
0.0228514 + 0.999739i $$0.492726\pi$$
$$702$$ 0 0
$$703$$ 28.3183 1.06804
$$704$$ 3.11359 0.117348
$$705$$ 0 0
$$706$$ −0.940462 −0.0353947
$$707$$ 17.4090 0.654733
$$708$$ 0 0
$$709$$ −2.37420 −0.0891651 −0.0445826 0.999006i $$-0.514196\pi$$
−0.0445826 + 0.999006i $$0.514196\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0.796848 0.0298631
$$713$$ 11.5245 0.431595
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 32.7963 1.22566
$$717$$ 0 0
$$718$$ 11.0797 0.413491
$$719$$ −3.46098 −0.129073 −0.0645364 0.997915i $$-0.520557\pi$$
−0.0645364 + 0.997915i $$0.520557\pi$$
$$720$$ 0 0
$$721$$ −14.5454 −0.541700
$$722$$ 2.60252 0.0968556
$$723$$ 0 0
$$724$$ 3.87773 0.144115
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −0.967841 −0.0358952 −0.0179476 0.999839i $$-0.505713\pi$$
−0.0179476 + 0.999839i $$0.505713\pi$$
$$728$$ 11.8456 0.439027
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 31.0810 1.14957
$$732$$ 0 0
$$733$$ 4.61626 0.170506 0.0852528 0.996359i $$-0.472830\pi$$
0.0852528 + 0.996359i $$0.472830\pi$$
$$734$$ 3.16844 0.116949
$$735$$ 0 0
$$736$$ −8.66883 −0.319537
$$737$$ 3.73005 0.137398
$$738$$ 0 0
$$739$$ −5.94453 −0.218673 −0.109336 0.994005i $$-0.534873\pi$$
−0.109336 + 0.994005i $$0.534873\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 2.56617 0.0942071
$$743$$ 11.0789 0.406444 0.203222 0.979133i $$-0.434859\pi$$
0.203222 + 0.979133i $$0.434859\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 1.73766 0.0636204
$$747$$ 0 0
$$748$$ 7.54351 0.275818
$$749$$ 29.0044 1.05980
$$750$$ 0 0
$$751$$ 16.3810 0.597750 0.298875 0.954292i $$-0.403389\pi$$
0.298875 + 0.954292i $$0.403389\pi$$
$$752$$ 23.7702 0.866809
$$753$$ 0 0
$$754$$ 1.63124 0.0594064
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 27.9468 1.01575 0.507873 0.861432i $$-0.330432\pi$$
0.507873 + 0.861432i $$0.330432\pi$$
$$758$$ 0.864293 0.0313925
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 16.1657 0.586006 0.293003 0.956112i $$-0.405345\pi$$
0.293003 + 0.956112i $$0.405345\pi$$
$$762$$ 0 0
$$763$$ 7.27348 0.263318
$$764$$ 3.34049 0.120855
$$765$$ 0 0
$$766$$ 0.206157 0.00744876
$$767$$ −44.6982 −1.61396
$$768$$ 0 0
$$769$$ −17.8664 −0.644279 −0.322139 0.946692i $$-0.604402\pi$$
−0.322139 + 0.946692i $$0.604402\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 16.6828 0.600426
$$773$$ 16.7961 0.604114 0.302057 0.953290i $$-0.402327\pi$$
0.302057 + 0.953290i $$0.402327\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 24.6573 0.885144
$$777$$ 0 0
$$778$$ 8.68103 0.311230
$$779$$ 26.1725 0.937726
$$780$$ 0 0
$$781$$ −3.08262 −0.110305
$$782$$ −5.49759 −0.196593
$$783$$ 0 0
$$784$$ −11.8316 −0.422556
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −43.1607 −1.53851 −0.769256 0.638941i $$-0.779373\pi$$
−0.769256 + 0.638941i $$0.779373\pi$$
$$788$$ 25.3942 0.904632
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 12.8646 0.457412
$$792$$ 0 0
$$793$$ 6.25235 0.222027
$$794$$ 9.15407 0.324866
$$795$$ 0 0
$$796$$ 11.5110 0.407998
$$797$$ 19.7554 0.699772 0.349886 0.936792i $$-0.386220\pi$$
0.349886 + 0.936792i $$0.386220\pi$$
$$798$$ 0 0
$$799$$ 49.9678 1.76773
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 3.63624 0.128400
$$803$$ −3.25850 −0.114990
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 7.94990 0.280023
$$807$$ 0 0
$$808$$ −11.9424 −0.420134
$$809$$ −23.6721 −0.832268 −0.416134 0.909303i $$-0.636615\pi$$
−0.416134 + 0.909303i $$0.636615\pi$$
$$810$$ 0 0
$$811$$ 18.5443 0.651178 0.325589 0.945511i $$-0.394437\pi$$
0.325589 + 0.945511i $$0.394437\pi$$
$$812$$ 3.52994 0.123877
$$813$$ 0 0
$$814$$ −1.57781 −0.0553024
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −14.6405 −0.512206
$$818$$ −0.517398 −0.0180904
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 41.5778 1.45108 0.725538 0.688182i $$-0.241591\pi$$
0.725538 + 0.688182i $$0.241591\pi$$
$$822$$ 0 0
$$823$$ 31.1276 1.08504 0.542519 0.840043i $$-0.317471\pi$$
0.542519 + 0.840043i $$0.317471\pi$$
$$824$$ 9.97804 0.347601
$$825$$ 0 0
$$826$$ 5.53037 0.192426
$$827$$ 42.5406 1.47928 0.739641 0.673001i $$-0.234995\pi$$
0.739641 + 0.673001i $$0.234995\pi$$
$$828$$ 0 0
$$829$$ 2.61307 0.0907558 0.0453779 0.998970i $$-0.485551\pi$$
0.0453779 + 0.998970i $$0.485551\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 27.3770 0.949127
$$833$$ −24.8714 −0.861743
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −3.55332 −0.122894
$$837$$ 0 0
$$838$$ 11.8155 0.408161
$$839$$ −30.9618 −1.06892 −0.534461 0.845193i $$-0.679485\pi$$
−0.534461 + 0.845193i $$0.679485\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ 5.10024 0.175766
$$843$$ 0 0
$$844$$ −17.1280 −0.589571
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 19.9310 0.684837
$$848$$ 14.0617 0.482882
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −20.1113 −0.689405
$$852$$ 0 0
$$853$$ −26.3048 −0.900659 −0.450329 0.892863i $$-0.648693\pi$$
−0.450329 + 0.892863i $$0.648693\pi$$
$$854$$ −0.773583 −0.0264715
$$855$$ 0 0
$$856$$ −19.8968 −0.680059
$$857$$ 5.37241 0.183518 0.0917590 0.995781i $$-0.470751\pi$$
0.0917590 + 0.995781i $$0.470751\pi$$
$$858$$ 0 0
$$859$$ −37.9952 −1.29638 −0.648189 0.761479i $$-0.724473\pi$$
−0.648189 + 0.761479i $$0.724473\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 5.57105 0.189751
$$863$$ 13.5782 0.462206 0.231103 0.972929i $$-0.425767\pi$$
0.231103 + 0.972929i $$0.425767\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −13.3245 −0.452784
$$867$$ 0 0
$$868$$ 17.2032 0.583916
$$869$$ −6.82649 −0.231573
$$870$$ 0 0
$$871$$ 32.7973 1.11129
$$872$$ −4.98954 −0.168967
$$873$$ 0 0
$$874$$ 2.58960 0.0875946
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −32.1671 −1.08621 −0.543104 0.839666i $$-0.682751\pi$$
−0.543104 + 0.839666i $$0.682751\pi$$
$$878$$ 7.71753 0.260454
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 47.6183 1.60430 0.802151 0.597121i $$-0.203689\pi$$
0.802151 + 0.597121i $$0.203689\pi$$
$$882$$ 0 0
$$883$$ −17.9986 −0.605701 −0.302851 0.953038i $$-0.597938\pi$$
−0.302851 + 0.953038i $$0.597938\pi$$
$$884$$ 66.3281 2.23085
$$885$$ 0 0
$$886$$ 5.39359 0.181201
$$887$$ −28.7126 −0.964076 −0.482038 0.876150i $$-0.660103\pi$$
−0.482038 + 0.876150i $$0.660103\pi$$
$$888$$ 0 0
$$889$$ 11.2792 0.378293
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 35.0557 1.17375
$$893$$ −23.5370 −0.787636
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −17.0676 −0.570188
$$897$$ 0 0
$$898$$ −2.93693 −0.0980065
$$899$$ 4.87352 0.162541
$$900$$ 0 0
$$901$$ 29.5595 0.984770
$$902$$ −1.45826 −0.0485546
$$903$$ 0 0
$$904$$ −8.82500 −0.293515
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −29.4971 −0.979434 −0.489717 0.871881i $$-0.662900\pi$$
−0.489717 + 0.871881i $$0.662900\pi$$
$$908$$ −44.6492 −1.48174
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 33.9486 1.12477 0.562384 0.826876i $$-0.309884\pi$$
0.562384 + 0.826876i $$0.309884\pi$$
$$912$$ 0 0
$$913$$ −5.31274 −0.175826
$$914$$ −1.03951 −0.0343840
$$915$$ 0 0
$$916$$ −20.0510 −0.662504
$$917$$ 14.9126 0.492457
$$918$$ 0 0
$$919$$ −17.6898 −0.583531 −0.291766 0.956490i $$-0.594243\pi$$
−0.291766 + 0.956490i $$0.594243\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 1.26930 0.0418022
$$923$$ −27.1047 −0.892161
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 8.07399 0.265328
$$927$$ 0 0
$$928$$ −3.66591 −0.120339
$$929$$ 3.74299 0.122804 0.0614018 0.998113i $$-0.480443\pi$$
0.0614018 + 0.998113i $$0.480443\pi$$
$$930$$ 0 0
$$931$$ 11.7155 0.383960
$$932$$ −16.3253 −0.534754
$$933$$ 0 0
$$934$$ 5.51107 0.180328
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 2.12550 0.0694369 0.0347185 0.999397i $$-0.488947\pi$$
0.0347185 + 0.999397i $$0.488947\pi$$
$$938$$ −4.05791 −0.132495
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 54.9017 1.78974 0.894871 0.446324i $$-0.147267\pi$$
0.894871 + 0.446324i $$0.147267\pi$$
$$942$$ 0 0
$$943$$ −18.5873 −0.605287
$$944$$ 30.3045 0.986328
$$945$$ 0 0
$$946$$ 0.815727 0.0265216
$$947$$ 32.5730 1.05848 0.529240 0.848472i $$-0.322477\pi$$
0.529240 + 0.848472i $$0.322477\pi$$
$$948$$ 0 0
$$949$$ −28.6511 −0.930054
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −16.8823 −0.547160
$$953$$ −8.68090 −0.281202 −0.140601 0.990066i $$-0.544903\pi$$
−0.140601 + 0.990066i $$0.544903\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 45.5769 1.47406
$$957$$ 0 0
$$958$$ −6.05567 −0.195650
$$959$$ 27.9268 0.901802
$$960$$ 0 0
$$961$$ −7.24878 −0.233832
$$962$$ −13.8733 −0.447293
$$963$$ 0 0
$$964$$ 48.6179 1.56588
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 53.2460 1.71228 0.856138 0.516747i $$-0.172857\pi$$
0.856138 + 0.516747i $$0.172857\pi$$
$$968$$ −13.6725 −0.439451
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −53.7614 −1.72529 −0.862643 0.505814i $$-0.831192\pi$$
−0.862643 + 0.505814i $$0.831192\pi$$
$$972$$ 0 0
$$973$$ 9.80433 0.314312
$$974$$ 6.04030 0.193544
$$975$$ 0 0
$$976$$ −4.23897 −0.135686
$$977$$ 13.2134 0.422735 0.211367 0.977407i $$-0.432208\pi$$
0.211367 + 0.977407i $$0.432208\pi$$
$$978$$ 0 0
$$979$$ −0.351171 −0.0112235
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 5.67831 0.181202
$$983$$ −31.7530 −1.01276 −0.506381 0.862310i $$-0.669017\pi$$
−0.506381 + 0.862310i $$0.669017\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −2.32485 −0.0740382
$$987$$ 0 0
$$988$$ −31.2434 −0.993985
$$989$$ 10.3975 0.330621
$$990$$ 0 0
$$991$$ 12.5237 0.397830 0.198915 0.980017i $$-0.436258\pi$$
0.198915 + 0.980017i $$0.436258\pi$$
$$992$$ −17.8659 −0.567243
$$993$$ 0 0
$$994$$ 3.35357 0.106369
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 28.9494 0.916837 0.458418 0.888736i $$-0.348416\pi$$
0.458418 + 0.888736i $$0.348416\pi$$
$$998$$ 7.31725 0.231624
$$999$$ 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.ce.1.6 12
3.2 odd 2 6525.2.a.cf.1.7 12
5.2 odd 4 1305.2.c.k.784.6 12
5.3 odd 4 1305.2.c.k.784.7 yes 12
5.4 even 2 inner 6525.2.a.ce.1.7 12
15.2 even 4 1305.2.c.l.784.7 yes 12
15.8 even 4 1305.2.c.l.784.6 yes 12
15.14 odd 2 6525.2.a.cf.1.6 12

By twisted newform
Twist Min Dim Char Parity Ord Type
1305.2.c.k.784.6 12 5.2 odd 4
1305.2.c.k.784.7 yes 12 5.3 odd 4
1305.2.c.l.784.6 yes 12 15.8 even 4
1305.2.c.l.784.7 yes 12 15.2 even 4
6525.2.a.ce.1.6 12 1.1 even 1 trivial
6525.2.a.ce.1.7 12 5.4 even 2 inner
6525.2.a.cf.1.6 12 15.14 odd 2
6525.2.a.cf.1.7 12 3.2 odd 2