# Properties

 Label 4275.2.a.bn.1.2 Level $4275$ Weight $2$ Character 4275.1 Self dual yes Analytic conductor $34.136$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [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: $$0$$ 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 1.2 Root $$0.445042$$ 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)

 $$f(q)$$ $$=$$ $$q+1.44504 q^{2} +0.0881460 q^{4} -1.35690 q^{7} -2.76271 q^{8} +O(q^{10})$$ $$q+1.44504 q^{2} +0.0881460 q^{4} -1.35690 q^{7} -2.76271 q^{8} -4.85086 q^{11} -0.198062 q^{13} -1.96077 q^{14} -4.16852 q^{16} -1.13706 q^{17} -1.00000 q^{19} -7.00969 q^{22} +2.55496 q^{23} -0.286208 q^{26} -0.119605 q^{28} +10.2349 q^{29} +2.51573 q^{31} -0.498271 q^{32} -1.64310 q^{34} +0.137063 q^{37} -1.44504 q^{38} +11.7506 q^{41} +7.59179 q^{43} -0.427583 q^{44} +3.69202 q^{46} +2.69202 q^{47} -5.15883 q^{49} -0.0174584 q^{52} +12.8780 q^{53} +3.74871 q^{56} +14.7899 q^{58} -5.82371 q^{59} -7.58211 q^{61} +3.63533 q^{62} +7.61702 q^{64} -8.01507 q^{67} -0.100228 q^{68} +8.82371 q^{71} +11.9705 q^{73} +0.198062 q^{74} -0.0881460 q^{76} +6.58211 q^{77} +10.7409 q^{79} +16.9801 q^{82} +3.77479 q^{83} +10.9705 q^{86} +13.4015 q^{88} -9.36658 q^{89} +0.268750 q^{91} +0.225209 q^{92} +3.89008 q^{94} -0.198062 q^{97} -7.45473 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 4 q^{2} + 4 q^{4} + 9 q^{8}+O(q^{10})$$ 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.44504 1.02180 0.510899 0.859640i $$-0.329312\pi$$
0.510899 + 0.859640i $$0.329312\pi$$
$$3$$ 0 0
$$4$$ 0.0881460 0.0440730
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −1.35690 −0.512858 −0.256429 0.966563i $$-0.582546\pi$$
−0.256429 + 0.966563i $$0.582546\pi$$
$$8$$ −2.76271 −0.976765
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −4.85086 −1.46259 −0.731294 0.682062i $$-0.761083\pi$$
−0.731294 + 0.682062i $$0.761083\pi$$
$$12$$ 0 0
$$13$$ −0.198062 −0.0549326 −0.0274663 0.999623i $$-0.508744\pi$$
−0.0274663 + 0.999623i $$0.508744\pi$$
$$14$$ −1.96077 −0.524038
$$15$$ 0 0
$$16$$ −4.16852 −1.04213
$$17$$ −1.13706 −0.275778 −0.137889 0.990448i $$-0.544032\pi$$
−0.137889 + 0.990448i $$0.544032\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −7.00969 −1.49447
$$23$$ 2.55496 0.532746 0.266373 0.963870i $$-0.414175\pi$$
0.266373 + 0.963870i $$0.414175\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −0.286208 −0.0561301
$$27$$ 0 0
$$28$$ −0.119605 −0.0226032
$$29$$ 10.2349 1.90057 0.950286 0.311377i $$-0.100790\pi$$
0.950286 + 0.311377i $$0.100790\pi$$
$$30$$ 0 0
$$31$$ 2.51573 0.451838 0.225919 0.974146i $$-0.427461\pi$$
0.225919 + 0.974146i $$0.427461\pi$$
$$32$$ −0.498271 −0.0880827
$$33$$ 0 0
$$34$$ −1.64310 −0.281790
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 0.137063 0.0225331 0.0112665 0.999937i $$-0.496414\pi$$
0.0112665 + 0.999937i $$0.496414\pi$$
$$38$$ −1.44504 −0.234417
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 11.7506 1.83514 0.917570 0.397575i $$-0.130148\pi$$
0.917570 + 0.397575i $$0.130148\pi$$
$$42$$ 0 0
$$43$$ 7.59179 1.15774 0.578869 0.815421i $$-0.303494\pi$$
0.578869 + 0.815421i $$0.303494\pi$$
$$44$$ −0.427583 −0.0644606
$$45$$ 0 0
$$46$$ 3.69202 0.544359
$$47$$ 2.69202 0.392672 0.196336 0.980537i $$-0.437096\pi$$
0.196336 + 0.980537i $$0.437096\pi$$
$$48$$ 0 0
$$49$$ −5.15883 −0.736976
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −0.0174584 −0.00242104
$$53$$ 12.8780 1.76893 0.884465 0.466607i $$-0.154524\pi$$
0.884465 + 0.466607i $$0.154524\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 3.74871 0.500942
$$57$$ 0 0
$$58$$ 14.7899 1.94200
$$59$$ −5.82371 −0.758182 −0.379091 0.925359i $$-0.623763\pi$$
−0.379091 + 0.925359i $$0.623763\pi$$
$$60$$ 0 0
$$61$$ −7.58211 −0.970789 −0.485395 0.874295i $$-0.661324\pi$$
−0.485395 + 0.874295i $$0.661324\pi$$
$$62$$ 3.63533 0.461688
$$63$$ 0 0
$$64$$ 7.61702 0.952128
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −8.01507 −0.979196 −0.489598 0.871948i $$-0.662856\pi$$
−0.489598 + 0.871948i $$0.662856\pi$$
$$68$$ −0.100228 −0.0121544
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 8.82371 1.04718 0.523591 0.851970i $$-0.324592\pi$$
0.523591 + 0.851970i $$0.324592\pi$$
$$72$$ 0 0
$$73$$ 11.9705 1.40104 0.700518 0.713635i $$-0.252952\pi$$
0.700518 + 0.713635i $$0.252952\pi$$
$$74$$ 0.198062 0.0230243
$$75$$ 0 0
$$76$$ −0.0881460 −0.0101110
$$77$$ 6.58211 0.750101
$$78$$ 0 0
$$79$$ 10.7409 1.20845 0.604225 0.796814i $$-0.293483\pi$$
0.604225 + 0.796814i $$0.293483\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 16.9801 1.87514
$$83$$ 3.77479 0.414337 0.207169 0.978305i $$-0.433575\pi$$
0.207169 + 0.978305i $$0.433575\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 10.9705 1.18298
$$87$$ 0 0
$$88$$ 13.4015 1.42860
$$89$$ −9.36658 −0.992856 −0.496428 0.868078i $$-0.665355\pi$$
−0.496428 + 0.868078i $$0.665355\pi$$
$$90$$ 0 0
$$91$$ 0.268750 0.0281726
$$92$$ 0.225209 0.0234797
$$93$$ 0 0
$$94$$ 3.89008 0.401232
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −0.198062 −0.0201102 −0.0100551 0.999949i $$-0.503201\pi$$
−0.0100551 + 0.999949i $$0.503201\pi$$
$$98$$ −7.45473 −0.753042
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −11.5090 −1.14519 −0.572595 0.819838i $$-0.694063\pi$$
−0.572595 + 0.819838i $$0.694063\pi$$
$$102$$ 0 0
$$103$$ −15.1564 −1.49341 −0.746704 0.665156i $$-0.768365\pi$$
−0.746704 + 0.665156i $$0.768365\pi$$
$$104$$ 0.547188 0.0536562
$$105$$ 0 0
$$106$$ 18.6093 1.80749
$$107$$ 2.65279 0.256455 0.128228 0.991745i $$-0.459071\pi$$
0.128228 + 0.991745i $$0.459071\pi$$
$$108$$ 0 0
$$109$$ −2.49934 −0.239393 −0.119696 0.992811i $$-0.538192\pi$$
−0.119696 + 0.992811i $$0.538192\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 5.65625 0.534465
$$113$$ 8.52781 0.802229 0.401114 0.916028i $$-0.368623\pi$$
0.401114 + 0.916028i $$0.368623\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0.902165 0.0837639
$$117$$ 0 0
$$118$$ −8.41550 −0.774710
$$119$$ 1.54288 0.141435
$$120$$ 0 0
$$121$$ 12.5308 1.13916
$$122$$ −10.9565 −0.991951
$$123$$ 0 0
$$124$$ 0.221751 0.0199139
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −20.4088 −1.81099 −0.905494 0.424359i $$-0.860499\pi$$
−0.905494 + 0.424359i $$0.860499\pi$$
$$128$$ 12.0035 1.06097
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 13.2131 1.15444 0.577218 0.816590i $$-0.304138\pi$$
0.577218 + 0.816590i $$0.304138\pi$$
$$132$$ 0 0
$$133$$ 1.35690 0.117658
$$134$$ −11.5821 −1.00054
$$135$$ 0 0
$$136$$ 3.14138 0.269371
$$137$$ −6.86054 −0.586136 −0.293068 0.956092i $$-0.594676\pi$$
−0.293068 + 0.956092i $$0.594676\pi$$
$$138$$ 0 0
$$139$$ 4.28621 0.363551 0.181776 0.983340i $$-0.441816\pi$$
0.181776 + 0.983340i $$0.441816\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 12.7506 1.07001
$$143$$ 0.960771 0.0803437
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 17.2978 1.43158
$$147$$ 0 0
$$148$$ 0.0120816 0.000993100 0
$$149$$ 15.3545 1.25789 0.628945 0.777450i $$-0.283487\pi$$
0.628945 + 0.777450i $$0.283487\pi$$
$$150$$ 0 0
$$151$$ −10.2295 −0.832467 −0.416233 0.909258i $$-0.636650\pi$$
−0.416233 + 0.909258i $$0.636650\pi$$
$$152$$ 2.76271 0.224085
$$153$$ 0 0
$$154$$ 9.51142 0.766452
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −3.17092 −0.253067 −0.126533 0.991962i $$-0.540385\pi$$
−0.126533 + 0.991962i $$0.540385\pi$$
$$158$$ 15.5211 1.23479
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −3.46681 −0.273223
$$162$$ 0 0
$$163$$ 4.63773 0.363255 0.181627 0.983367i $$-0.441864\pi$$
0.181627 + 0.983367i $$0.441864\pi$$
$$164$$ 1.03577 0.0808801
$$165$$ 0 0
$$166$$ 5.45473 0.423369
$$167$$ −19.6286 −1.51891 −0.759454 0.650560i $$-0.774534\pi$$
−0.759454 + 0.650560i $$0.774534\pi$$
$$168$$ 0 0
$$169$$ −12.9608 −0.996982
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0.669186 0.0510250
$$173$$ 20.5646 1.56350 0.781751 0.623591i $$-0.214327\pi$$
0.781751 + 0.623591i $$0.214327\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 20.2209 1.52421
$$177$$ 0 0
$$178$$ −13.5351 −1.01450
$$179$$ −6.92154 −0.517340 −0.258670 0.965966i $$-0.583284\pi$$
−0.258670 + 0.965966i $$0.583284\pi$$
$$180$$ 0 0
$$181$$ −17.6461 −1.31162 −0.655812 0.754925i $$-0.727673\pi$$
−0.655812 + 0.754925i $$0.727673\pi$$
$$182$$ 0.388355 0.0287868
$$183$$ 0 0
$$184$$ −7.05861 −0.520367
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 5.51573 0.403350
$$188$$ 0.237291 0.0173062
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −6.92931 −0.501387 −0.250694 0.968066i $$-0.580659\pi$$
−0.250694 + 0.968066i $$0.580659\pi$$
$$192$$ 0 0
$$193$$ 21.0368 1.51426 0.757132 0.653262i $$-0.226600\pi$$
0.757132 + 0.653262i $$0.226600\pi$$
$$194$$ −0.286208 −0.0205486
$$195$$ 0 0
$$196$$ −0.454731 −0.0324808
$$197$$ 21.5646 1.53642 0.768209 0.640199i $$-0.221148\pi$$
0.768209 + 0.640199i $$0.221148\pi$$
$$198$$ 0 0
$$199$$ 21.9909 1.55889 0.779447 0.626468i $$-0.215500\pi$$
0.779447 + 0.626468i $$0.215500\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −16.6310 −1.17015
$$203$$ −13.8877 −0.974725
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −21.9017 −1.52596
$$207$$ 0 0
$$208$$ 0.825627 0.0572469
$$209$$ 4.85086 0.335541
$$210$$ 0 0
$$211$$ −20.6233 −1.41976 −0.709882 0.704321i $$-0.751252\pi$$
−0.709882 + 0.704321i $$0.751252\pi$$
$$212$$ 1.13514 0.0779620
$$213$$ 0 0
$$214$$ 3.83340 0.262046
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −3.41358 −0.231729
$$218$$ −3.61165 −0.244611
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0.225209 0.0151492
$$222$$ 0 0
$$223$$ −7.97716 −0.534190 −0.267095 0.963670i $$-0.586064\pi$$
−0.267095 + 0.963670i $$0.586064\pi$$
$$224$$ 0.676102 0.0451740
$$225$$ 0 0
$$226$$ 12.3230 0.819717
$$227$$ 19.7942 1.31379 0.656893 0.753984i $$-0.271871\pi$$
0.656893 + 0.753984i $$0.271871\pi$$
$$228$$ 0 0
$$229$$ −4.03385 −0.266564 −0.133282 0.991078i $$-0.542552\pi$$
−0.133282 + 0.991078i $$0.542552\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −28.2760 −1.85641
$$233$$ 26.8159 1.75677 0.878385 0.477953i $$-0.158621\pi$$
0.878385 + 0.477953i $$0.158621\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −0.513337 −0.0334154
$$237$$ 0 0
$$238$$ 2.22952 0.144518
$$239$$ −3.36227 −0.217487 −0.108744 0.994070i $$-0.534683\pi$$
−0.108744 + 0.994070i $$0.534683\pi$$
$$240$$ 0 0
$$241$$ 27.7506 1.78758 0.893788 0.448491i $$-0.148038\pi$$
0.893788 + 0.448491i $$0.148038\pi$$
$$242$$ 18.1075 1.16400
$$243$$ 0 0
$$244$$ −0.668332 −0.0427856
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0.198062 0.0126024
$$248$$ −6.95023 −0.441340
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 5.59419 0.353102 0.176551 0.984291i $$-0.443506\pi$$
0.176551 + 0.984291i $$0.443506\pi$$
$$252$$ 0 0
$$253$$ −12.3937 −0.779187
$$254$$ −29.4916 −1.85047
$$255$$ 0 0
$$256$$ 2.11146 0.131966
$$257$$ 10.4668 0.652902 0.326451 0.945214i $$-0.394147\pi$$
0.326451 + 0.945214i $$0.394147\pi$$
$$258$$ 0 0
$$259$$ −0.185981 −0.0115563
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 19.0935 1.17960
$$263$$ 15.4795 0.954506 0.477253 0.878766i $$-0.341633\pi$$
0.477253 + 0.878766i $$0.341633\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 1.96077 0.120223
$$267$$ 0 0
$$268$$ −0.706496 −0.0431561
$$269$$ −9.13036 −0.556688 −0.278344 0.960481i $$-0.589785\pi$$
−0.278344 + 0.960481i $$0.589785\pi$$
$$270$$ 0 0
$$271$$ 7.44265 0.452109 0.226054 0.974115i $$-0.427417\pi$$
0.226054 + 0.974115i $$0.427417\pi$$
$$272$$ 4.73987 0.287397
$$273$$ 0 0
$$274$$ −9.91377 −0.598913
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 11.4155 0.685891 0.342946 0.939355i $$-0.388575\pi$$
0.342946 + 0.939355i $$0.388575\pi$$
$$278$$ 6.19375 0.371476
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −21.5060 −1.28294 −0.641471 0.767147i $$-0.721676\pi$$
−0.641471 + 0.767147i $$0.721676\pi$$
$$282$$ 0 0
$$283$$ 5.45712 0.324392 0.162196 0.986759i $$-0.448142\pi$$
0.162196 + 0.986759i $$0.448142\pi$$
$$284$$ 0.777775 0.0461524
$$285$$ 0 0
$$286$$ 1.38835 0.0820951
$$287$$ −15.9444 −0.941167
$$288$$ 0 0
$$289$$ −15.7071 −0.923946
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 1.05515 0.0617479
$$293$$ 7.39075 0.431772 0.215886 0.976419i $$-0.430736\pi$$
0.215886 + 0.976419i $$0.430736\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −0.378666 −0.0220095
$$297$$ 0 0
$$298$$ 22.1879 1.28531
$$299$$ −0.506041 −0.0292651
$$300$$ 0 0
$$301$$ −10.3013 −0.593756
$$302$$ −14.7821 −0.850613
$$303$$ 0 0
$$304$$ 4.16852 0.239081
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 32.2131 1.83850 0.919250 0.393674i $$-0.128796\pi$$
0.919250 + 0.393674i $$0.128796\pi$$
$$308$$ 0.580186 0.0330592
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −14.8442 −0.841735 −0.420867 0.907122i $$-0.638274\pi$$
−0.420867 + 0.907122i $$0.638274\pi$$
$$312$$ 0 0
$$313$$ 13.1491 0.743234 0.371617 0.928386i $$-0.378804\pi$$
0.371617 + 0.928386i $$0.378804\pi$$
$$314$$ −4.58211 −0.258583
$$315$$ 0 0
$$316$$ 0.946771 0.0532600
$$317$$ −5.13467 −0.288392 −0.144196 0.989549i $$-0.546060\pi$$
−0.144196 + 0.989549i $$0.546060\pi$$
$$318$$ 0 0
$$319$$ −49.6480 −2.77975
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −5.00969 −0.279179
$$323$$ 1.13706 0.0632679
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 6.70171 0.371173
$$327$$ 0 0
$$328$$ −32.4636 −1.79250
$$329$$ −3.65279 −0.201385
$$330$$ 0 0
$$331$$ 2.00969 0.110462 0.0552312 0.998474i $$-0.482410\pi$$
0.0552312 + 0.998474i $$0.482410\pi$$
$$332$$ 0.332733 0.0182611
$$333$$ 0 0
$$334$$ −28.3642 −1.55202
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 1.31873 0.0718359 0.0359180 0.999355i $$-0.488564\pi$$
0.0359180 + 0.999355i $$0.488564\pi$$
$$338$$ −18.7289 −1.01872
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −12.2034 −0.660853
$$342$$ 0 0
$$343$$ 16.4983 0.890823
$$344$$ −20.9739 −1.13084
$$345$$ 0 0
$$346$$ 29.7168 1.59758
$$347$$ 12.0151 0.645003 0.322501 0.946569i $$-0.395476\pi$$
0.322501 + 0.946569i $$0.395476\pi$$
$$348$$ 0 0
$$349$$ −10.5579 −0.565154 −0.282577 0.959245i $$-0.591189\pi$$
−0.282577 + 0.959245i $$0.591189\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 2.41704 0.128829
$$353$$ −1.01102 −0.0538110 −0.0269055 0.999638i $$-0.508565\pi$$
−0.0269055 + 0.999638i $$0.508565\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −0.825627 −0.0437581
$$357$$ 0 0
$$358$$ −10.0019 −0.528618
$$359$$ 5.14244 0.271408 0.135704 0.990749i $$-0.456670\pi$$
0.135704 + 0.990749i $$0.456670\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −25.4993 −1.34022
$$363$$ 0 0
$$364$$ 0.0236892 0.00124165
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 29.5308 1.54149 0.770747 0.637141i $$-0.219883\pi$$
0.770747 + 0.637141i $$0.219883\pi$$
$$368$$ −10.6504 −0.555190
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −17.4741 −0.907210
$$372$$ 0 0
$$373$$ −8.78986 −0.455121 −0.227561 0.973764i $$-0.573075\pi$$
−0.227561 + 0.973764i $$0.573075\pi$$
$$374$$ 7.97046 0.412143
$$375$$ 0 0
$$376$$ −7.43727 −0.383548
$$377$$ −2.02715 −0.104403
$$378$$ 0 0
$$379$$ 19.1511 0.983724 0.491862 0.870673i $$-0.336316\pi$$
0.491862 + 0.870673i $$0.336316\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −10.0131 −0.512317
$$383$$ −6.99894 −0.357629 −0.178814 0.983883i $$-0.557226\pi$$
−0.178814 + 0.983883i $$0.557226\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 30.3991 1.54727
$$387$$ 0 0
$$388$$ −0.0174584 −0.000886316 0
$$389$$ −8.08575 −0.409964 −0.204982 0.978766i $$-0.565714\pi$$
−0.204982 + 0.978766i $$0.565714\pi$$
$$390$$ 0 0
$$391$$ −2.90515 −0.146920
$$392$$ 14.2524 0.719853
$$393$$ 0 0
$$394$$ 31.1618 1.56991
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 17.7006 0.888370 0.444185 0.895935i $$-0.353493\pi$$
0.444185 + 0.895935i $$0.353493\pi$$
$$398$$ 31.7778 1.59288
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 15.5418 0.776121 0.388061 0.921634i $$-0.373145\pi$$
0.388061 + 0.921634i $$0.373145\pi$$
$$402$$ 0 0
$$403$$ −0.498271 −0.0248207
$$404$$ −1.01447 −0.0504720
$$405$$ 0 0
$$406$$ −20.0683 −0.995973
$$407$$ −0.664874 −0.0329566
$$408$$ 0 0
$$409$$ 13.1661 0.651023 0.325512 0.945538i $$-0.394463\pi$$
0.325512 + 0.945538i $$0.394463\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −1.33598 −0.0658190
$$413$$ 7.90217 0.388840
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0.0986887 0.00483861
$$417$$ 0 0
$$418$$ 7.00969 0.342855
$$419$$ 25.1739 1.22983 0.614913 0.788595i $$-0.289191\pi$$
0.614913 + 0.788595i $$0.289191\pi$$
$$420$$ 0 0
$$421$$ 26.9420 1.31307 0.656536 0.754295i $$-0.272021\pi$$
0.656536 + 0.754295i $$0.272021\pi$$
$$422$$ −29.8015 −1.45071
$$423$$ 0 0
$$424$$ −35.5782 −1.72783
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 10.2881 0.497877
$$428$$ 0.233833 0.0113027
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −18.9487 −0.912726 −0.456363 0.889794i $$-0.650848\pi$$
−0.456363 + 0.889794i $$0.650848\pi$$
$$432$$ 0 0
$$433$$ 3.68904 0.177284 0.0886419 0.996064i $$-0.471747\pi$$
0.0886419 + 0.996064i $$0.471747\pi$$
$$434$$ −4.93277 −0.236781
$$435$$ 0 0
$$436$$ −0.220306 −0.0105508
$$437$$ −2.55496 −0.122220
$$438$$ 0 0
$$439$$ −25.6926 −1.22624 −0.613121 0.789989i $$-0.710086\pi$$
−0.613121 + 0.789989i $$0.710086\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0.325437 0.0154795
$$443$$ −27.3653 −1.30016 −0.650081 0.759865i $$-0.725265\pi$$
−0.650081 + 0.759865i $$0.725265\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −11.5273 −0.545835
$$447$$ 0 0
$$448$$ −10.3355 −0.488307
$$449$$ −7.55794 −0.356681 −0.178341 0.983969i $$-0.557073\pi$$
−0.178341 + 0.983969i $$0.557073\pi$$
$$450$$ 0 0
$$451$$ −57.0006 −2.68405
$$452$$ 0.751692 0.0353566
$$453$$ 0 0
$$454$$ 28.6034 1.34242
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 7.85623 0.367499 0.183750 0.982973i $$-0.441176\pi$$
0.183750 + 0.982973i $$0.441176\pi$$
$$458$$ −5.82908 −0.272375
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −3.87907 −0.180666 −0.0903331 0.995912i $$-0.528793\pi$$
−0.0903331 + 0.995912i $$0.528793\pi$$
$$462$$ 0 0
$$463$$ 13.0954 0.608597 0.304298 0.952577i $$-0.401578\pi$$
0.304298 + 0.952577i $$0.401578\pi$$
$$464$$ −42.6644 −1.98065
$$465$$ 0 0
$$466$$ 38.7502 1.79507
$$467$$ 6.44026 0.298020 0.149010 0.988836i $$-0.452391\pi$$
0.149010 + 0.988836i $$0.452391\pi$$
$$468$$ 0 0
$$469$$ 10.8756 0.502189
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 16.0892 0.740566
$$473$$ −36.8267 −1.69329
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0.135998 0.00623348
$$477$$ 0 0
$$478$$ −4.85862 −0.222228
$$479$$ −5.69096 −0.260026 −0.130013 0.991512i $$-0.541502\pi$$
−0.130013 + 0.991512i $$0.541502\pi$$
$$480$$ 0 0
$$481$$ −0.0271471 −0.00123780
$$482$$ 40.1008 1.82654
$$483$$ 0 0
$$484$$ 1.10454 0.0502063
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 12.9632 0.587417 0.293709 0.955895i $$-0.405110\pi$$
0.293709 + 0.955895i $$0.405110\pi$$
$$488$$ 20.9472 0.948233
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 19.6045 0.884737 0.442369 0.896833i $$-0.354138\pi$$
0.442369 + 0.896833i $$0.354138\pi$$
$$492$$ 0 0
$$493$$ −11.6377 −0.524137
$$494$$ 0.286208 0.0128771
$$495$$ 0 0
$$496$$ −10.4869 −0.470875
$$497$$ −11.9729 −0.537056
$$498$$ 0 0
$$499$$ 5.49827 0.246136 0.123068 0.992398i $$-0.460727\pi$$
0.123068 + 0.992398i $$0.460727\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 8.08383 0.360799
$$503$$ −35.6939 −1.59151 −0.795757 0.605616i $$-0.792927\pi$$
−0.795757 + 0.605616i $$0.792927\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −17.9095 −0.796173
$$507$$ 0 0
$$508$$ −1.79895 −0.0798157
$$509$$ 16.4450 0.728914 0.364457 0.931220i $$-0.381255\pi$$
0.364457 + 0.931220i $$0.381255\pi$$
$$510$$ 0 0
$$511$$ −16.2427 −0.718533
$$512$$ −20.9558 −0.926123
$$513$$ 0 0
$$514$$ 15.1250 0.667134
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −13.0586 −0.574317
$$518$$ −0.268750 −0.0118082
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 26.5435 1.16289 0.581445 0.813586i $$-0.302487\pi$$
0.581445 + 0.813586i $$0.302487\pi$$
$$522$$ 0 0
$$523$$ 24.1685 1.05682 0.528408 0.848991i $$-0.322789\pi$$
0.528408 + 0.848991i $$0.322789\pi$$
$$524$$ 1.16468 0.0508795
$$525$$ 0 0
$$526$$ 22.3685 0.975313
$$527$$ −2.86054 −0.124607
$$528$$ 0 0
$$529$$ −16.4722 −0.716182
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0.119605 0.00518553
$$533$$ −2.32736 −0.100809
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 22.1433 0.956445
$$537$$ 0 0
$$538$$ −13.1938 −0.568823
$$539$$ 25.0248 1.07789
$$540$$ 0 0
$$541$$ 9.80386 0.421501 0.210750 0.977540i $$-0.432409\pi$$
0.210750 + 0.977540i $$0.432409\pi$$
$$542$$ 10.7549 0.461964
$$543$$ 0 0
$$544$$ 0.566566 0.0242913
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −21.1739 −0.905331 −0.452665 0.891681i $$-0.649527\pi$$
−0.452665 + 0.891681i $$0.649527\pi$$
$$548$$ −0.604729 −0.0258328
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −10.2349 −0.436021
$$552$$ 0 0
$$553$$ −14.5743 −0.619764
$$554$$ 16.4959 0.700843
$$555$$ 0 0
$$556$$ 0.377812 0.0160228
$$557$$ 24.4077 1.03419 0.517094 0.855928i $$-0.327014\pi$$
0.517094 + 0.855928i $$0.327014\pi$$
$$558$$ 0 0
$$559$$ −1.50365 −0.0635975
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −31.0771 −1.31091
$$563$$ 14.4849 0.610464 0.305232 0.952278i $$-0.401266\pi$$
0.305232 + 0.952278i $$0.401266\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 7.88577 0.331464
$$567$$ 0 0
$$568$$ −24.3773 −1.02285
$$569$$ −0.811626 −0.0340251 −0.0170126 0.999855i $$-0.505416\pi$$
−0.0170126 + 0.999855i $$0.505416\pi$$
$$570$$ 0 0
$$571$$ −37.6588 −1.57597 −0.787985 0.615694i $$-0.788876\pi$$
−0.787985 + 0.615694i $$0.788876\pi$$
$$572$$ 0.0846882 0.00354099
$$573$$ 0 0
$$574$$ −23.0403 −0.961683
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −8.89307 −0.370223 −0.185112 0.982718i $$-0.559265\pi$$
−0.185112 + 0.982718i $$0.559265\pi$$
$$578$$ −22.6974 −0.944087
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −5.12200 −0.212496
$$582$$ 0 0
$$583$$ −62.4693 −2.58721
$$584$$ −33.0709 −1.36848
$$585$$ 0 0
$$586$$ 10.6799 0.441184
$$587$$ 20.3327 0.839222 0.419611 0.907704i $$-0.362167\pi$$
0.419611 + 0.907704i $$0.362167\pi$$
$$588$$ 0 0
$$589$$ −2.51573 −0.103659
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −0.571352 −0.0234824
$$593$$ 17.0049 0.698308 0.349154 0.937065i $$-0.386469\pi$$
0.349154 + 0.937065i $$0.386469\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 1.35344 0.0554390
$$597$$ 0 0
$$598$$ −0.731250 −0.0299030
$$599$$ 12.4004 0.506668 0.253334 0.967379i $$-0.418473\pi$$
0.253334 + 0.967379i $$0.418473\pi$$
$$600$$ 0 0
$$601$$ −9.73125 −0.396946 −0.198473 0.980106i $$-0.563598\pi$$
−0.198473 + 0.980106i $$0.563598\pi$$
$$602$$ −14.8858 −0.606699
$$603$$ 0 0
$$604$$ −0.901691 −0.0366893
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 23.4282 0.950920 0.475460 0.879737i $$-0.342282\pi$$
0.475460 + 0.879737i $$0.342282\pi$$
$$608$$ 0.498271 0.0202076
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −0.533188 −0.0215705
$$612$$ 0 0
$$613$$ −28.1424 −1.13666 −0.568331 0.822800i $$-0.692411\pi$$
−0.568331 + 0.822800i $$0.692411\pi$$
$$614$$ 46.5493 1.87858
$$615$$ 0 0
$$616$$ −18.1844 −0.732672
$$617$$ 36.4805 1.46865 0.734326 0.678797i $$-0.237498\pi$$
0.734326 + 0.678797i $$0.237498\pi$$
$$618$$ 0 0
$$619$$ −38.2097 −1.53578 −0.767888 0.640584i $$-0.778692\pi$$
−0.767888 + 0.640584i $$0.778692\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −21.4504 −0.860083
$$623$$ 12.7095 0.509195
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 19.0011 0.759435
$$627$$ 0 0
$$628$$ −0.279503 −0.0111534
$$629$$ −0.155850 −0.00621413
$$630$$ 0 0
$$631$$ −12.5235 −0.498553 −0.249276 0.968432i $$-0.580193\pi$$
−0.249276 + 0.968432i $$0.580193\pi$$
$$632$$ −29.6741 −1.18037
$$633$$ 0 0
$$634$$ −7.41981 −0.294678
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 1.02177 0.0404840
$$638$$ −71.7434 −2.84035
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −37.3564 −1.47549 −0.737745 0.675080i $$-0.764109\pi$$
−0.737745 + 0.675080i $$0.764109\pi$$
$$642$$ 0 0
$$643$$ 14.5483 0.573727 0.286864 0.957971i $$-0.407387\pi$$
0.286864 + 0.957971i $$0.407387\pi$$
$$644$$ −0.305586 −0.0120418
$$645$$ 0 0
$$646$$ 1.64310 0.0646471
$$647$$ −23.4403 −0.921532 −0.460766 0.887522i $$-0.652425\pi$$
−0.460766 + 0.887522i $$0.652425\pi$$
$$648$$ 0 0
$$649$$ 28.2500 1.10891
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0.408797 0.0160097
$$653$$ 27.1903 1.06404 0.532019 0.846732i $$-0.321433\pi$$
0.532019 + 0.846732i $$0.321433\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −48.9828 −1.91246
$$657$$ 0 0
$$658$$ −5.27844 −0.205775
$$659$$ −2.71486 −0.105756 −0.0528779 0.998601i $$-0.516839\pi$$
−0.0528779 + 0.998601i $$0.516839\pi$$
$$660$$ 0 0
$$661$$ −9.41311 −0.366128 −0.183064 0.983101i $$-0.558601\pi$$
−0.183064 + 0.983101i $$0.558601\pi$$
$$662$$ 2.90408 0.112870
$$663$$ 0 0
$$664$$ −10.4286 −0.404710
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 26.1497 1.01252
$$668$$ −1.73019 −0.0669429
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 36.7797 1.41986
$$672$$ 0 0
$$673$$ 44.8001 1.72692 0.863459 0.504419i $$-0.168293\pi$$
0.863459 + 0.504419i $$0.168293\pi$$
$$674$$ 1.90562 0.0734019
$$675$$ 0 0
$$676$$ −1.14244 −0.0439400
$$677$$ −32.9197 −1.26521 −0.632604 0.774475i $$-0.718014\pi$$
−0.632604 + 0.774475i $$0.718014\pi$$
$$678$$ 0 0
$$679$$ 0.268750 0.0103137
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −17.6345 −0.675259
$$683$$ 20.3448 0.778473 0.389236 0.921138i $$-0.372739\pi$$
0.389236 + 0.921138i $$0.372739\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 23.8407 0.910242
$$687$$ 0 0
$$688$$ −31.6466 −1.20651
$$689$$ −2.55065 −0.0971719
$$690$$ 0 0
$$691$$ −46.1473 −1.75553 −0.877764 0.479094i $$-0.840965\pi$$
−0.877764 + 0.479094i $$0.840965\pi$$
$$692$$ 1.81269 0.0689082
$$693$$ 0 0
$$694$$ 17.3623 0.659063
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −13.3612 −0.506092
$$698$$ −15.2567 −0.577473
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −13.1933 −0.498303 −0.249152 0.968464i $$-0.580152\pi$$
−0.249152 + 0.968464i $$0.580152\pi$$
$$702$$ 0 0
$$703$$ −0.137063 −0.00516944
$$704$$ −36.9491 −1.39257
$$705$$ 0 0
$$706$$ −1.46096 −0.0549840
$$707$$ 15.6165 0.587321
$$708$$ 0 0
$$709$$ 41.1860 1.54677 0.773386 0.633935i $$-0.218562\pi$$
0.773386 + 0.633935i $$0.218562\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 25.8771 0.969787
$$713$$ 6.42758 0.240715
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −0.610106 −0.0228007
$$717$$ 0 0
$$718$$ 7.43104 0.277324
$$719$$ 35.6256 1.32861 0.664306 0.747461i $$-0.268727\pi$$
0.664306 + 0.747461i $$0.268727\pi$$
$$720$$ 0 0
$$721$$ 20.5657 0.765907
$$722$$ 1.44504 0.0537789
$$723$$ 0 0
$$724$$ −1.55543 −0.0578072
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −20.0116 −0.742189 −0.371095 0.928595i $$-0.621017\pi$$
−0.371095 + 0.928595i $$0.621017\pi$$
$$728$$ −0.742478 −0.0275181
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −8.63235 −0.319279
$$732$$ 0 0
$$733$$ 18.9952 0.701604 0.350802 0.936450i $$-0.385909\pi$$
0.350802 + 0.936450i $$0.385909\pi$$
$$734$$ 42.6732 1.57510
$$735$$ 0 0
$$736$$ −1.27306 −0.0469257
$$737$$ 38.8799 1.43216
$$738$$ 0 0
$$739$$ 48.1704 1.77198 0.885989 0.463706i $$-0.153481\pi$$
0.885989 + 0.463706i $$0.153481\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −25.2508 −0.926987
$$743$$ −13.2446 −0.485897 −0.242948 0.970039i $$-0.578115\pi$$
−0.242948 + 0.970039i $$0.578115\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −12.7017 −0.465043
$$747$$ 0 0
$$748$$ 0.486189 0.0177768
$$749$$ −3.59956 −0.131525
$$750$$ 0 0
$$751$$ 12.0562 0.439937 0.219969 0.975507i $$-0.429404\pi$$
0.219969 + 0.975507i $$0.429404\pi$$
$$752$$ −11.2218 −0.409215
$$753$$ 0 0
$$754$$ −2.92931 −0.106679
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −15.0054 −0.545380 −0.272690 0.962102i $$-0.587913\pi$$
−0.272690 + 0.962102i $$0.587913\pi$$
$$758$$ 27.6741 1.00517
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −44.3967 −1.60938 −0.804690 0.593695i $$-0.797668\pi$$
−0.804690 + 0.593695i $$0.797668\pi$$
$$762$$ 0 0
$$763$$ 3.39134 0.122775
$$764$$ −0.610791 −0.0220976
$$765$$ 0 0
$$766$$ −10.1138 −0.365425
$$767$$ 1.15346 0.0416489
$$768$$ 0 0
$$769$$ −39.7211 −1.43238 −0.716190 0.697906i $$-0.754115\pi$$
−0.716190 + 0.697906i $$0.754115\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 1.85431 0.0667382
$$773$$ −1.72779 −0.0621444 −0.0310722 0.999517i $$-0.509892\pi$$
−0.0310722 + 0.999517i $$0.509892\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0.547188 0.0196429
$$777$$ 0 0
$$778$$ −11.6843 −0.418901
$$779$$ −11.7506 −0.421010
$$780$$ 0 0
$$781$$ −42.8025 −1.53159
$$782$$ −4.19806 −0.150122
$$783$$ 0 0
$$784$$ 21.5047 0.768025
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −42.6329 −1.51970 −0.759850 0.650098i $$-0.774728\pi$$
−0.759850 + 0.650098i $$0.774728\pi$$
$$788$$ 1.90084 0.0677145
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −11.5714 −0.411430
$$792$$ 0 0
$$793$$ 1.50173 0.0533280
$$794$$ 25.5782 0.907735
$$795$$ 0 0
$$796$$ 1.93841 0.0687051
$$797$$ 38.5864 1.36680 0.683401 0.730044i $$-0.260500\pi$$
0.683401 + 0.730044i $$0.260500\pi$$
$$798$$ 0 0
$$799$$ −3.06100 −0.108290
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 22.4586 0.793040
$$803$$ −58.0670 −2.04914
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −0.720023 −0.0253617
$$807$$ 0 0
$$808$$ 31.7961 1.11858
$$809$$ 8.38298 0.294730 0.147365 0.989082i $$-0.452921\pi$$
0.147365 + 0.989082i $$0.452921\pi$$
$$810$$ 0 0
$$811$$ −0.340765 −0.0119659 −0.00598295 0.999982i $$-0.501904\pi$$
−0.00598295 + 0.999982i $$0.501904\pi$$
$$812$$ −1.22414 −0.0429590
$$813$$ 0 0
$$814$$ −0.960771 −0.0336750
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −7.59179 −0.265603
$$818$$ 19.0256 0.665215
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 33.7506 1.17791 0.588953 0.808168i $$-0.299540\pi$$
0.588953 + 0.808168i $$0.299540\pi$$
$$822$$ 0 0
$$823$$ −37.2669 −1.29904 −0.649522 0.760343i $$-0.725031\pi$$
−0.649522 + 0.760343i $$0.725031\pi$$
$$824$$ 41.8728 1.45871
$$825$$ 0 0
$$826$$ 11.4190 0.397316
$$827$$ −21.1691 −0.736122 −0.368061 0.929802i $$-0.619978\pi$$
−0.368061 + 0.929802i $$0.619978\pi$$
$$828$$ 0 0
$$829$$ −31.0374 −1.07797 −0.538987 0.842314i $$-0.681193\pi$$
−0.538987 + 0.842314i $$0.681193\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −1.50864 −0.0523028
$$833$$ 5.86592 0.203242
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0.427583 0.0147883
$$837$$ 0 0
$$838$$ 36.3773 1.25663
$$839$$ −33.2403 −1.14758 −0.573791 0.819002i $$-0.694528\pi$$
−0.573791 + 0.819002i $$0.694528\pi$$
$$840$$ 0 0
$$841$$ 75.7531 2.61218
$$842$$ 38.9323 1.34170
$$843$$ 0 0
$$844$$ −1.81786 −0.0625732
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −17.0030 −0.584229
$$848$$ −53.6822 −1.84346
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0.350191 0.0120044
$$852$$ 0 0
$$853$$ 24.3086 0.832310 0.416155 0.909294i $$-0.363377\pi$$
0.416155 + 0.909294i $$0.363377\pi$$
$$854$$ 14.8668 0.508731
$$855$$ 0 0
$$856$$ −7.32889 −0.250496
$$857$$ −57.3889 −1.96037 −0.980185 0.198087i $$-0.936527\pi$$
−0.980185 + 0.198087i $$0.936527\pi$$
$$858$$ 0 0
$$859$$ −13.8135 −0.471312 −0.235656 0.971837i $$-0.575724\pi$$
−0.235656 + 0.971837i $$0.575724\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −27.3817 −0.932623
$$863$$ −9.01938 −0.307023 −0.153512 0.988147i $$-0.549058\pi$$
−0.153512 + 0.988147i $$0.549058\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 5.33081 0.181148
$$867$$ 0 0
$$868$$ −0.300894 −0.0102130
$$869$$ −52.1027 −1.76746
$$870$$ 0 0
$$871$$ 1.58748 0.0537898
$$872$$ 6.90494 0.233831
$$873$$ 0 0
$$874$$ −3.69202 −0.124884
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 2.37675 0.0802570 0.0401285 0.999195i $$-0.487223\pi$$
0.0401285 + 0.999195i $$0.487223\pi$$
$$878$$ −37.1269 −1.25297
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 7.99330 0.269301 0.134650 0.990893i $$-0.457009\pi$$
0.134650 + 0.990893i $$0.457009\pi$$
$$882$$ 0 0
$$883$$ 1.76377 0.0593557 0.0296779 0.999560i $$-0.490552\pi$$
0.0296779 + 0.999560i $$0.490552\pi$$
$$884$$ 0.0198513 0.000667672 0
$$885$$ 0 0
$$886$$ −39.5439 −1.32850
$$887$$ 45.9197 1.54183 0.770917 0.636936i $$-0.219798\pi$$
0.770917 + 0.636936i $$0.219798\pi$$
$$888$$ 0 0
$$889$$ 27.6926 0.928780
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −0.703155 −0.0235434
$$893$$ −2.69202 −0.0900851
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −16.2874 −0.544125
$$897$$ 0 0
$$898$$ −10.9215 −0.364457
$$899$$ 25.7482 0.858752
$$900$$ 0 0
$$901$$ −14.6431 −0.487833
$$902$$ −82.3682 −2.74256
$$903$$ 0 0
$$904$$ −23.5599 −0.783589
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −55.0549 −1.82807 −0.914034 0.405638i $$-0.867049\pi$$
−0.914034 + 0.405638i $$0.867049\pi$$
$$908$$ 1.74478 0.0579024
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −51.8998 −1.71952 −0.859758 0.510702i $$-0.829386\pi$$
−0.859758 + 0.510702i $$0.829386\pi$$
$$912$$ 0 0
$$913$$ −18.3110 −0.606004
$$914$$ 11.3526 0.375510
$$915$$ 0 0
$$916$$ −0.355568 −0.0117483
$$917$$ −17.9288 −0.592062
$$918$$ 0 0
$$919$$ 11.4614 0.378078 0.189039 0.981970i $$-0.439463\pi$$
0.189039 + 0.981970i $$0.439463\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −5.60541 −0.184604
$$923$$ −1.74764 −0.0575244
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 18.9235 0.621864
$$927$$ 0 0
$$928$$ −5.09975 −0.167408
$$929$$ −36.5295 −1.19849 −0.599246 0.800565i $$-0.704533\pi$$
−0.599246 + 0.800565i $$0.704533\pi$$
$$930$$ 0 0
$$931$$ 5.15883 0.169074
$$932$$ 2.36372 0.0774261
$$933$$ 0 0
$$934$$ 9.30644 0.304516
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 48.7845 1.59372 0.796860 0.604164i $$-0.206493\pi$$
0.796860 + 0.604164i $$0.206493\pi$$
$$938$$ 15.7157 0.513136
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 18.1817 0.592705 0.296353 0.955079i $$-0.404230\pi$$
0.296353 + 0.955079i $$0.404230\pi$$
$$942$$ 0 0
$$943$$ 30.0224 0.977663
$$944$$ 24.2763 0.790125
$$945$$ 0 0
$$946$$ −53.2161 −1.73021
$$947$$ −7.55150 −0.245391 −0.122695 0.992444i $$-0.539154\pi$$
−0.122695 + 0.992444i $$0.539154\pi$$
$$948$$ 0 0
$$949$$ −2.37090 −0.0769626
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −4.26252 −0.138149
$$953$$ 13.8592 0.448944 0.224472 0.974481i $$-0.427934\pi$$
0.224472 + 0.974481i $$0.427934\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −0.296371 −0.00958532
$$957$$ 0 0
$$958$$ −8.22367 −0.265695
$$959$$ 9.30904 0.300605
$$960$$ 0 0
$$961$$ −24.6711 −0.795842
$$962$$ −0.0392287 −0.00126478
$$963$$ 0 0
$$964$$ 2.44611 0.0787838
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −4.89977 −0.157566 −0.0787830 0.996892i $$-0.525103\pi$$
−0.0787830 + 0.996892i $$0.525103\pi$$
$$968$$ −34.6189 −1.11269
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −14.5133 −0.465755 −0.232878 0.972506i $$-0.574814\pi$$
−0.232878 + 0.972506i $$0.574814\pi$$
$$972$$ 0 0
$$973$$ −5.81594 −0.186450
$$974$$ 18.7323 0.600222
$$975$$ 0 0
$$976$$ 31.6062 1.01169
$$977$$ −19.6644 −0.629120 −0.314560 0.949238i $$-0.601857\pi$$
−0.314560 + 0.949238i $$0.601857\pi$$
$$978$$ 0 0
$$979$$ 45.4359 1.45214
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 28.3293 0.904023
$$983$$ −15.4397 −0.492449 −0.246224 0.969213i $$-0.579190\pi$$
−0.246224 + 0.969213i $$0.579190\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −16.8170 −0.535562
$$987$$ 0 0
$$988$$ 0.0174584 0.000555426 0
$$989$$ 19.3967 0.616780
$$990$$ 0 0
$$991$$ 11.9377 0.379213 0.189606 0.981860i $$-0.439279\pi$$
0.189606 + 0.981860i $$0.439279\pi$$
$$992$$ −1.25352 −0.0397991
$$993$$ 0 0
$$994$$ −17.3013 −0.548763
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −17.9390 −0.568134 −0.284067 0.958804i $$-0.591684\pi$$
−0.284067 + 0.958804i $$0.591684\pi$$
$$998$$ 7.94523 0.251502
$$999$$ 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.bn.1.2 3
3.2 odd 2 475.2.a.d.1.2 3
5.4 even 2 4275.2.a.z.1.2 3
12.11 even 2 7600.2.a.bw.1.3 3
15.2 even 4 475.2.b.c.324.2 6
15.8 even 4 475.2.b.c.324.5 6
15.14 odd 2 475.2.a.h.1.2 yes 3
57.56 even 2 9025.2.a.be.1.2 3
60.59 even 2 7600.2.a.bn.1.1 3
285.284 even 2 9025.2.a.w.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
475.2.a.d.1.2 3 3.2 odd 2
475.2.a.h.1.2 yes 3 15.14 odd 2
475.2.b.c.324.2 6 15.2 even 4
475.2.b.c.324.5 6 15.8 even 4
4275.2.a.z.1.2 3 5.4 even 2
4275.2.a.bn.1.2 3 1.1 even 1 trivial
7600.2.a.bn.1.1 3 60.59 even 2
7600.2.a.bw.1.3 3 12.11 even 2
9025.2.a.w.1.2 3 285.284 even 2
9025.2.a.be.1.2 3 57.56 even 2