# Properties

 Label 2025.2.b.d.649.2 Level $2025$ Weight $2$ Character 2025.649 Analytic conductor $16.170$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2025,2,Mod(649,2025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2025.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2025 = 3^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2025.b (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$16.1697064093$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 45) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 649.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 2025.649 Dual form 2025.2.b.d.649.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} +1.00000 q^{4} +3.00000i q^{7} +3.00000i q^{8} +O(q^{10})$$ $$q+1.00000i q^{2} +1.00000 q^{4} +3.00000i q^{7} +3.00000i q^{8} +2.00000 q^{11} -2.00000i q^{13} -3.00000 q^{14} -1.00000 q^{16} +4.00000i q^{17} +8.00000 q^{19} +2.00000i q^{22} -3.00000i q^{23} +2.00000 q^{26} +3.00000i q^{28} -1.00000 q^{29} +5.00000i q^{32} -4.00000 q^{34} +4.00000i q^{37} +8.00000i q^{38} -5.00000 q^{41} -8.00000i q^{43} +2.00000 q^{44} +3.00000 q^{46} +7.00000i q^{47} -2.00000 q^{49} -2.00000i q^{52} +2.00000i q^{53} -9.00000 q^{56} -1.00000i q^{58} -14.0000 q^{59} +7.00000 q^{61} -7.00000 q^{64} +3.00000i q^{67} +4.00000i q^{68} -2.00000 q^{71} +4.00000i q^{73} -4.00000 q^{74} +8.00000 q^{76} +6.00000i q^{77} +6.00000 q^{79} -5.00000i q^{82} -9.00000i q^{83} +8.00000 q^{86} +6.00000i q^{88} -15.0000 q^{89} +6.00000 q^{91} -3.00000i q^{92} -7.00000 q^{94} -2.00000i q^{97} -2.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4}+O(q^{10})$$ 2 * q + 2 * q^4 $$2 q + 2 q^{4} + 4 q^{11} - 6 q^{14} - 2 q^{16} + 16 q^{19} + 4 q^{26} - 2 q^{29} - 8 q^{34} - 10 q^{41} + 4 q^{44} + 6 q^{46} - 4 q^{49} - 18 q^{56} - 28 q^{59} + 14 q^{61} - 14 q^{64} - 4 q^{71} - 8 q^{74} + 16 q^{76} + 12 q^{79} + 16 q^{86} - 30 q^{89} + 12 q^{91} - 14 q^{94}+O(q^{100})$$ 2 * q + 2 * q^4 + 4 * q^11 - 6 * q^14 - 2 * q^16 + 16 * q^19 + 4 * q^26 - 2 * q^29 - 8 * q^34 - 10 * q^41 + 4 * q^44 + 6 * q^46 - 4 * q^49 - 18 * q^56 - 28 * q^59 + 14 * q^61 - 14 * q^64 - 4 * q^71 - 8 * q^74 + 16 * q^76 + 12 * q^79 + 16 * q^86 - 30 * q^89 + 12 * q^91 - 14 * q^94

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2025\mathbb{Z}\right)^\times$$.

 $$n$$ $$326$$ $$1702$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 0.707107i 0.935414 + 0.353553i $$0.115027\pi$$
−0.935414 + 0.353553i $$0.884973\pi$$
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 3.00000i 1.13389i 0.823754 + 0.566947i $$0.191875\pi$$
−0.823754 + 0.566947i $$0.808125\pi$$
$$8$$ 3.00000i 1.06066i
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ 0 0
$$13$$ − 2.00000i − 0.554700i −0.960769 0.277350i $$-0.910544\pi$$
0.960769 0.277350i $$-0.0894562\pi$$
$$14$$ −3.00000 −0.801784
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ 4.00000i 0.970143i 0.874475 + 0.485071i $$0.161206\pi$$
−0.874475 + 0.485071i $$0.838794\pi$$
$$18$$ 0 0
$$19$$ 8.00000 1.83533 0.917663 0.397360i $$-0.130073\pi$$
0.917663 + 0.397360i $$0.130073\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 2.00000i 0.426401i
$$23$$ − 3.00000i − 0.625543i −0.949828 0.312772i $$-0.898743\pi$$
0.949828 0.312772i $$-0.101257\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 2.00000 0.392232
$$27$$ 0 0
$$28$$ 3.00000i 0.566947i
$$29$$ −1.00000 −0.185695 −0.0928477 0.995680i $$-0.529597\pi$$
−0.0928477 + 0.995680i $$0.529597\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 5.00000i 0.883883i
$$33$$ 0 0
$$34$$ −4.00000 −0.685994
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 4.00000i 0.657596i 0.944400 + 0.328798i $$0.106644\pi$$
−0.944400 + 0.328798i $$0.893356\pi$$
$$38$$ 8.00000i 1.29777i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −5.00000 −0.780869 −0.390434 0.920631i $$-0.627675\pi$$
−0.390434 + 0.920631i $$0.627675\pi$$
$$42$$ 0 0
$$43$$ − 8.00000i − 1.21999i −0.792406 0.609994i $$-0.791172\pi$$
0.792406 0.609994i $$-0.208828\pi$$
$$44$$ 2.00000 0.301511
$$45$$ 0 0
$$46$$ 3.00000 0.442326
$$47$$ 7.00000i 1.02105i 0.859861 + 0.510527i $$0.170550\pi$$
−0.859861 + 0.510527i $$0.829450\pi$$
$$48$$ 0 0
$$49$$ −2.00000 −0.285714
$$50$$ 0 0
$$51$$ 0 0
$$52$$ − 2.00000i − 0.277350i
$$53$$ 2.00000i 0.274721i 0.990521 + 0.137361i $$0.0438619\pi$$
−0.990521 + 0.137361i $$0.956138\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −9.00000 −1.20268
$$57$$ 0 0
$$58$$ − 1.00000i − 0.131306i
$$59$$ −14.0000 −1.82264 −0.911322 0.411693i $$-0.864937\pi$$
−0.911322 + 0.411693i $$0.864937\pi$$
$$60$$ 0 0
$$61$$ 7.00000 0.896258 0.448129 0.893969i $$-0.352090\pi$$
0.448129 + 0.893969i $$0.352090\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ −7.00000 −0.875000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 3.00000i 0.366508i 0.983066 + 0.183254i $$0.0586631\pi$$
−0.983066 + 0.183254i $$0.941337\pi$$
$$68$$ 4.00000i 0.485071i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −2.00000 −0.237356 −0.118678 0.992933i $$-0.537866\pi$$
−0.118678 + 0.992933i $$0.537866\pi$$
$$72$$ 0 0
$$73$$ 4.00000i 0.468165i 0.972217 + 0.234082i $$0.0752085\pi$$
−0.972217 + 0.234082i $$0.924791\pi$$
$$74$$ −4.00000 −0.464991
$$75$$ 0 0
$$76$$ 8.00000 0.917663
$$77$$ 6.00000i 0.683763i
$$78$$ 0 0
$$79$$ 6.00000 0.675053 0.337526 0.941316i $$-0.390410\pi$$
0.337526 + 0.941316i $$0.390410\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ − 5.00000i − 0.552158i
$$83$$ − 9.00000i − 0.987878i −0.869496 0.493939i $$-0.835557\pi$$
0.869496 0.493939i $$-0.164443\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 8.00000 0.862662
$$87$$ 0 0
$$88$$ 6.00000i 0.639602i
$$89$$ −15.0000 −1.59000 −0.794998 0.606612i $$-0.792528\pi$$
−0.794998 + 0.606612i $$0.792528\pi$$
$$90$$ 0 0
$$91$$ 6.00000 0.628971
$$92$$ − 3.00000i − 0.312772i
$$93$$ 0 0
$$94$$ −7.00000 −0.721995
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 2.00000i − 0.203069i −0.994832 0.101535i $$-0.967625\pi$$
0.994832 0.101535i $$-0.0323753\pi$$
$$98$$ − 2.00000i − 0.202031i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 18.0000 1.79107 0.895533 0.444994i $$-0.146794\pi$$
0.895533 + 0.444994i $$0.146794\pi$$
$$102$$ 0 0
$$103$$ 8.00000i 0.788263i 0.919054 + 0.394132i $$0.128955\pi$$
−0.919054 + 0.394132i $$0.871045\pi$$
$$104$$ 6.00000 0.588348
$$105$$ 0 0
$$106$$ −2.00000 −0.194257
$$107$$ 3.00000i 0.290021i 0.989430 + 0.145010i $$0.0463216\pi$$
−0.989430 + 0.145010i $$0.953678\pi$$
$$108$$ 0 0
$$109$$ −5.00000 −0.478913 −0.239457 0.970907i $$-0.576969\pi$$
−0.239457 + 0.970907i $$0.576969\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ − 3.00000i − 0.283473i
$$113$$ 8.00000i 0.752577i 0.926503 + 0.376288i $$0.122800\pi$$
−0.926503 + 0.376288i $$0.877200\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −1.00000 −0.0928477
$$117$$ 0 0
$$118$$ − 14.0000i − 1.28880i
$$119$$ −12.0000 −1.10004
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 7.00000i 0.633750i
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 5.00000i 0.443678i 0.975083 + 0.221839i $$0.0712060\pi$$
−0.975083 + 0.221839i $$0.928794\pi$$
$$128$$ 3.00000i 0.265165i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 6.00000 0.524222 0.262111 0.965038i $$-0.415581\pi$$
0.262111 + 0.965038i $$0.415581\pi$$
$$132$$ 0 0
$$133$$ 24.0000i 2.08106i
$$134$$ −3.00000 −0.259161
$$135$$ 0 0
$$136$$ −12.0000 −1.02899
$$137$$ 12.0000i 1.02523i 0.858619 + 0.512615i $$0.171323\pi$$
−0.858619 + 0.512615i $$0.828677\pi$$
$$138$$ 0 0
$$139$$ 16.0000 1.35710 0.678551 0.734553i $$-0.262608\pi$$
0.678551 + 0.734553i $$0.262608\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ − 2.00000i − 0.167836i
$$143$$ − 4.00000i − 0.334497i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −4.00000 −0.331042
$$147$$ 0 0
$$148$$ 4.00000i 0.328798i
$$149$$ 17.0000 1.39269 0.696347 0.717705i $$-0.254807\pi$$
0.696347 + 0.717705i $$0.254807\pi$$
$$150$$ 0 0
$$151$$ −2.00000 −0.162758 −0.0813788 0.996683i $$-0.525932\pi$$
−0.0813788 + 0.996683i $$0.525932\pi$$
$$152$$ 24.0000i 1.94666i
$$153$$ 0 0
$$154$$ −6.00000 −0.483494
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 14.0000i − 1.11732i −0.829396 0.558661i $$-0.811315\pi$$
0.829396 0.558661i $$-0.188685\pi$$
$$158$$ 6.00000i 0.477334i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 9.00000 0.709299
$$162$$ 0 0
$$163$$ − 4.00000i − 0.313304i −0.987654 0.156652i $$-0.949930\pi$$
0.987654 0.156652i $$-0.0500701\pi$$
$$164$$ −5.00000 −0.390434
$$165$$ 0 0
$$166$$ 9.00000 0.698535
$$167$$ − 9.00000i − 0.696441i −0.937413 0.348220i $$-0.886786\pi$$
0.937413 0.348220i $$-0.113214\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ 0 0
$$172$$ − 8.00000i − 0.609994i
$$173$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −2.00000 −0.150756
$$177$$ 0 0
$$178$$ − 15.0000i − 1.12430i
$$179$$ −2.00000 −0.149487 −0.0747435 0.997203i $$-0.523814\pi$$
−0.0747435 + 0.997203i $$0.523814\pi$$
$$180$$ 0 0
$$181$$ −7.00000 −0.520306 −0.260153 0.965567i $$-0.583773\pi$$
−0.260153 + 0.965567i $$0.583773\pi$$
$$182$$ 6.00000i 0.444750i
$$183$$ 0 0
$$184$$ 9.00000 0.663489
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 8.00000i 0.585018i
$$188$$ 7.00000i 0.510527i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ 0 0
$$193$$ − 10.0000i − 0.719816i −0.932988 0.359908i $$-0.882808\pi$$
0.932988 0.359908i $$-0.117192\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 0 0
$$196$$ −2.00000 −0.142857
$$197$$ − 12.0000i − 0.854965i −0.904024 0.427482i $$-0.859401\pi$$
0.904024 0.427482i $$-0.140599\pi$$
$$198$$ 0 0
$$199$$ −4.00000 −0.283552 −0.141776 0.989899i $$-0.545281\pi$$
−0.141776 + 0.989899i $$0.545281\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 18.0000i 1.26648i
$$203$$ − 3.00000i − 0.210559i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −8.00000 −0.557386
$$207$$ 0 0
$$208$$ 2.00000i 0.138675i
$$209$$ 16.0000 1.10674
$$210$$ 0 0
$$211$$ −22.0000 −1.51454 −0.757271 0.653101i $$-0.773468\pi$$
−0.757271 + 0.653101i $$0.773468\pi$$
$$212$$ 2.00000i 0.137361i
$$213$$ 0 0
$$214$$ −3.00000 −0.205076
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ − 5.00000i − 0.338643i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 8.00000 0.538138
$$222$$ 0 0
$$223$$ − 19.0000i − 1.27233i −0.771551 0.636167i $$-0.780519\pi$$
0.771551 0.636167i $$-0.219481\pi$$
$$224$$ −15.0000 −1.00223
$$225$$ 0 0
$$226$$ −8.00000 −0.532152
$$227$$ − 4.00000i − 0.265489i −0.991150 0.132745i $$-0.957621\pi$$
0.991150 0.132745i $$-0.0423790\pi$$
$$228$$ 0 0
$$229$$ −15.0000 −0.991228 −0.495614 0.868543i $$-0.665057\pi$$
−0.495614 + 0.868543i $$0.665057\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 3.00000i − 0.196960i
$$233$$ − 24.0000i − 1.57229i −0.618041 0.786146i $$-0.712073\pi$$
0.618041 0.786146i $$-0.287927\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −14.0000 −0.911322
$$237$$ 0 0
$$238$$ − 12.0000i − 0.777844i
$$239$$ −8.00000 −0.517477 −0.258738 0.965947i $$-0.583307\pi$$
−0.258738 + 0.965947i $$0.583307\pi$$
$$240$$ 0 0
$$241$$ −11.0000 −0.708572 −0.354286 0.935137i $$-0.615276\pi$$
−0.354286 + 0.935137i $$0.615276\pi$$
$$242$$ − 7.00000i − 0.449977i
$$243$$ 0 0
$$244$$ 7.00000 0.448129
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 16.0000i − 1.01806i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ − 6.00000i − 0.377217i
$$254$$ −5.00000 −0.313728
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ − 6.00000i − 0.374270i −0.982334 0.187135i $$-0.940080\pi$$
0.982334 0.187135i $$-0.0599201\pi$$
$$258$$ 0 0
$$259$$ −12.0000 −0.745644
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 6.00000i 0.370681i
$$263$$ 16.0000i 0.986602i 0.869859 + 0.493301i $$0.164210\pi$$
−0.869859 + 0.493301i $$0.835790\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −24.0000 −1.47153
$$267$$ 0 0
$$268$$ 3.00000i 0.183254i
$$269$$ 25.0000 1.52428 0.762138 0.647414i $$-0.224150\pi$$
0.762138 + 0.647414i $$0.224150\pi$$
$$270$$ 0 0
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ − 4.00000i − 0.242536i
$$273$$ 0 0
$$274$$ −12.0000 −0.724947
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 12.0000i − 0.721010i −0.932757 0.360505i $$-0.882604\pi$$
0.932757 0.360505i $$-0.117396\pi$$
$$278$$ 16.0000i 0.959616i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 15.0000 0.894825 0.447412 0.894328i $$-0.352346\pi$$
0.447412 + 0.894328i $$0.352346\pi$$
$$282$$ 0 0
$$283$$ 21.0000i 1.24832i 0.781296 + 0.624160i $$0.214559\pi$$
−0.781296 + 0.624160i $$0.785441\pi$$
$$284$$ −2.00000 −0.118678
$$285$$ 0 0
$$286$$ 4.00000 0.236525
$$287$$ − 15.0000i − 0.885422i
$$288$$ 0 0
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 4.00000i 0.234082i
$$293$$ − 12.0000i − 0.701047i −0.936554 0.350524i $$-0.886004\pi$$
0.936554 0.350524i $$-0.113996\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −12.0000 −0.697486
$$297$$ 0 0
$$298$$ 17.0000i 0.984784i
$$299$$ −6.00000 −0.346989
$$300$$ 0 0
$$301$$ 24.0000 1.38334
$$302$$ − 2.00000i − 0.115087i
$$303$$ 0 0
$$304$$ −8.00000 −0.458831
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 7.00000i − 0.399511i −0.979846 0.199756i $$-0.935985\pi$$
0.979846 0.199756i $$-0.0640148\pi$$
$$308$$ 6.00000i 0.341882i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ 14.0000i 0.791327i 0.918396 + 0.395663i $$0.129485\pi$$
−0.918396 + 0.395663i $$0.870515\pi$$
$$314$$ 14.0000 0.790066
$$315$$ 0 0
$$316$$ 6.00000 0.337526
$$317$$ − 34.0000i − 1.90963i −0.297200 0.954815i $$-0.596053\pi$$
0.297200 0.954815i $$-0.403947\pi$$
$$318$$ 0 0
$$319$$ −2.00000 −0.111979
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 9.00000i 0.501550i
$$323$$ 32.0000i 1.78053i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 4.00000 0.221540
$$327$$ 0 0
$$328$$ − 15.0000i − 0.828236i
$$329$$ −21.0000 −1.15777
$$330$$ 0 0
$$331$$ −6.00000 −0.329790 −0.164895 0.986311i $$-0.552728\pi$$
−0.164895 + 0.986311i $$0.552728\pi$$
$$332$$ − 9.00000i − 0.493939i
$$333$$ 0 0
$$334$$ 9.00000 0.492458
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 8.00000i 0.435788i 0.975972 + 0.217894i $$0.0699187\pi$$
−0.975972 + 0.217894i $$0.930081\pi$$
$$338$$ 9.00000i 0.489535i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 15.0000i 0.809924i
$$344$$ 24.0000 1.29399
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 4.00000i 0.214731i 0.994220 + 0.107366i $$0.0342415\pi$$
−0.994220 + 0.107366i $$0.965758\pi$$
$$348$$ 0 0
$$349$$ 5.00000 0.267644 0.133822 0.991005i $$-0.457275\pi$$
0.133822 + 0.991005i $$0.457275\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 10.0000i 0.533002i
$$353$$ − 24.0000i − 1.27739i −0.769460 0.638696i $$-0.779474\pi$$
0.769460 0.638696i $$-0.220526\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −15.0000 −0.794998
$$357$$ 0 0
$$358$$ − 2.00000i − 0.105703i
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 0 0
$$361$$ 45.0000 2.36842
$$362$$ − 7.00000i − 0.367912i
$$363$$ 0 0
$$364$$ 6.00000 0.314485
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 24.0000i − 1.25279i −0.779506 0.626395i $$-0.784530\pi$$
0.779506 0.626395i $$-0.215470\pi$$
$$368$$ 3.00000i 0.156386i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −6.00000 −0.311504
$$372$$ 0 0
$$373$$ 10.0000i 0.517780i 0.965907 + 0.258890i $$0.0833568\pi$$
−0.965907 + 0.258890i $$0.916643\pi$$
$$374$$ −8.00000 −0.413670
$$375$$ 0 0
$$376$$ −21.0000 −1.08299
$$377$$ 2.00000i 0.103005i
$$378$$ 0 0
$$379$$ 26.0000 1.33553 0.667765 0.744372i $$-0.267251\pi$$
0.667765 + 0.744372i $$0.267251\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ − 8.00000i − 0.409316i
$$383$$ − 36.0000i − 1.83951i −0.392488 0.919757i $$-0.628386\pi$$
0.392488 0.919757i $$-0.371614\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 10.0000 0.508987
$$387$$ 0 0
$$388$$ − 2.00000i − 0.101535i
$$389$$ 33.0000 1.67317 0.836583 0.547840i $$-0.184550\pi$$
0.836583 + 0.547840i $$0.184550\pi$$
$$390$$ 0 0
$$391$$ 12.0000 0.606866
$$392$$ − 6.00000i − 0.303046i
$$393$$ 0 0
$$394$$ 12.0000 0.604551
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 34.0000i − 1.70641i −0.521575 0.853206i $$-0.674655\pi$$
0.521575 0.853206i $$-0.325345\pi$$
$$398$$ − 4.00000i − 0.200502i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 18.0000 0.895533
$$405$$ 0 0
$$406$$ 3.00000 0.148888
$$407$$ 8.00000i 0.396545i
$$408$$ 0 0
$$409$$ −14.0000 −0.692255 −0.346128 0.938187i $$-0.612504\pi$$
−0.346128 + 0.938187i $$0.612504\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 8.00000i 0.394132i
$$413$$ − 42.0000i − 2.06668i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 10.0000 0.490290
$$417$$ 0 0
$$418$$ 16.0000i 0.782586i
$$419$$ 26.0000 1.27018 0.635092 0.772437i $$-0.280962\pi$$
0.635092 + 0.772437i $$0.280962\pi$$
$$420$$ 0 0
$$421$$ 34.0000 1.65706 0.828529 0.559946i $$-0.189178\pi$$
0.828529 + 0.559946i $$0.189178\pi$$
$$422$$ − 22.0000i − 1.07094i
$$423$$ 0 0
$$424$$ −6.00000 −0.291386
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 21.0000i 1.01626i
$$428$$ 3.00000i 0.145010i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 30.0000 1.44505 0.722525 0.691345i $$-0.242982\pi$$
0.722525 + 0.691345i $$0.242982\pi$$
$$432$$ 0 0
$$433$$ 28.0000i 1.34559i 0.739827 + 0.672797i $$0.234907\pi$$
−0.739827 + 0.672797i $$0.765093\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −5.00000 −0.239457
$$437$$ − 24.0000i − 1.14808i
$$438$$ 0 0
$$439$$ −28.0000 −1.33637 −0.668184 0.743996i $$-0.732928\pi$$
−0.668184 + 0.743996i $$0.732928\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 8.00000i 0.380521i
$$443$$ − 15.0000i − 0.712672i −0.934358 0.356336i $$-0.884026\pi$$
0.934358 0.356336i $$-0.115974\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 19.0000 0.899676
$$447$$ 0 0
$$448$$ − 21.0000i − 0.992157i
$$449$$ −26.0000 −1.22702 −0.613508 0.789689i $$-0.710242\pi$$
−0.613508 + 0.789689i $$0.710242\pi$$
$$450$$ 0 0
$$451$$ −10.0000 −0.470882
$$452$$ 8.00000i 0.376288i
$$453$$ 0 0
$$454$$ 4.00000 0.187729
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 20.0000i 0.935561i 0.883845 + 0.467780i $$0.154946\pi$$
−0.883845 + 0.467780i $$0.845054\pi$$
$$458$$ − 15.0000i − 0.700904i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −9.00000 −0.419172 −0.209586 0.977790i $$-0.567212\pi$$
−0.209586 + 0.977790i $$0.567212\pi$$
$$462$$ 0 0
$$463$$ − 36.0000i − 1.67306i −0.547920 0.836531i $$-0.684580\pi$$
0.547920 0.836531i $$-0.315420\pi$$
$$464$$ 1.00000 0.0464238
$$465$$ 0 0
$$466$$ 24.0000 1.11178
$$467$$ 20.0000i 0.925490i 0.886492 + 0.462745i $$0.153135\pi$$
−0.886492 + 0.462745i $$0.846865\pi$$
$$468$$ 0 0
$$469$$ −9.00000 −0.415581
$$470$$ 0 0
$$471$$ 0 0
$$472$$ − 42.0000i − 1.93321i
$$473$$ − 16.0000i − 0.735681i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −12.0000 −0.550019
$$477$$ 0 0
$$478$$ − 8.00000i − 0.365911i
$$479$$ −18.0000 −0.822441 −0.411220 0.911536i $$-0.634897\pi$$
−0.411220 + 0.911536i $$0.634897\pi$$
$$480$$ 0 0
$$481$$ 8.00000 0.364769
$$482$$ − 11.0000i − 0.501036i
$$483$$ 0 0
$$484$$ −7.00000 −0.318182
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 16.0000i 0.725029i 0.931978 + 0.362515i $$0.118082\pi$$
−0.931978 + 0.362515i $$0.881918\pi$$
$$488$$ 21.0000i 0.950625i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −20.0000 −0.902587 −0.451294 0.892375i $$-0.649037\pi$$
−0.451294 + 0.892375i $$0.649037\pi$$
$$492$$ 0 0
$$493$$ − 4.00000i − 0.180151i
$$494$$ 16.0000 0.719874
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 6.00000i − 0.269137i
$$498$$ 0 0
$$499$$ 32.0000 1.43252 0.716258 0.697835i $$-0.245853\pi$$
0.716258 + 0.697835i $$0.245853\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 7.00000i 0.312115i 0.987748 + 0.156057i $$0.0498784\pi$$
−0.987748 + 0.156057i $$0.950122\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 6.00000 0.266733
$$507$$ 0 0
$$508$$ 5.00000i 0.221839i
$$509$$ 43.0000 1.90594 0.952971 0.303062i $$-0.0980090\pi$$
0.952971 + 0.303062i $$0.0980090\pi$$
$$510$$ 0 0
$$511$$ −12.0000 −0.530849
$$512$$ − 11.0000i − 0.486136i
$$513$$ 0 0
$$514$$ 6.00000 0.264649
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 14.0000i 0.615719i
$$518$$ − 12.0000i − 0.527250i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −11.0000 −0.481919 −0.240959 0.970535i $$-0.577462\pi$$
−0.240959 + 0.970535i $$0.577462\pi$$
$$522$$ 0 0
$$523$$ − 29.0000i − 1.26808i −0.773300 0.634041i $$-0.781395\pi$$
0.773300 0.634041i $$-0.218605\pi$$
$$524$$ 6.00000 0.262111
$$525$$ 0 0
$$526$$ −16.0000 −0.697633
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 14.0000 0.608696
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 24.0000i 1.04053i
$$533$$ 10.0000i 0.433148i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −9.00000 −0.388741
$$537$$ 0 0
$$538$$ 25.0000i 1.07783i
$$539$$ −4.00000 −0.172292
$$540$$ 0 0
$$541$$ −39.0000 −1.67674 −0.838370 0.545101i $$-0.816491\pi$$
−0.838370 + 0.545101i $$0.816491\pi$$
$$542$$ − 8.00000i − 0.343629i
$$543$$ 0 0
$$544$$ −20.0000 −0.857493
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 29.0000i 1.23995i 0.784621 + 0.619975i $$0.212857\pi$$
−0.784621 + 0.619975i $$0.787143\pi$$
$$548$$ 12.0000i 0.512615i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −8.00000 −0.340811
$$552$$ 0 0
$$553$$ 18.0000i 0.765438i
$$554$$ 12.0000 0.509831
$$555$$ 0 0
$$556$$ 16.0000 0.678551
$$557$$ − 30.0000i − 1.27114i −0.772043 0.635570i $$-0.780765\pi$$
0.772043 0.635570i $$-0.219235\pi$$
$$558$$ 0 0
$$559$$ −16.0000 −0.676728
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 15.0000i 0.632737i
$$563$$ 21.0000i 0.885044i 0.896758 + 0.442522i $$0.145916\pi$$
−0.896758 + 0.442522i $$0.854084\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −21.0000 −0.882696
$$567$$ 0 0
$$568$$ − 6.00000i − 0.251754i
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ 32.0000 1.33916 0.669579 0.742741i $$-0.266474\pi$$
0.669579 + 0.742741i $$0.266474\pi$$
$$572$$ − 4.00000i − 0.167248i
$$573$$ 0 0
$$574$$ 15.0000 0.626088
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 10.0000i 0.416305i 0.978096 + 0.208153i $$0.0667451\pi$$
−0.978096 + 0.208153i $$0.933255\pi$$
$$578$$ 1.00000i 0.0415945i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 27.0000 1.12015
$$582$$ 0 0
$$583$$ 4.00000i 0.165663i
$$584$$ −12.0000 −0.496564
$$585$$ 0 0
$$586$$ 12.0000 0.495715
$$587$$ − 33.0000i − 1.36206i −0.732257 0.681028i $$-0.761533\pi$$
0.732257 0.681028i $$-0.238467\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ − 4.00000i − 0.164399i
$$593$$ − 20.0000i − 0.821302i −0.911793 0.410651i $$-0.865302\pi$$
0.911793 0.410651i $$-0.134698\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 17.0000 0.696347
$$597$$ 0 0
$$598$$ − 6.00000i − 0.245358i
$$599$$ −10.0000 −0.408589 −0.204294 0.978909i $$-0.565490\pi$$
−0.204294 + 0.978909i $$0.565490\pi$$
$$600$$ 0 0
$$601$$ 2.00000 0.0815817 0.0407909 0.999168i $$-0.487012\pi$$
0.0407909 + 0.999168i $$0.487012\pi$$
$$602$$ 24.0000i 0.978167i
$$603$$ 0 0
$$604$$ −2.00000 −0.0813788
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 41.0000i 1.66414i 0.554672 + 0.832069i $$0.312844\pi$$
−0.554672 + 0.832069i $$0.687156\pi$$
$$608$$ 40.0000i 1.62221i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 14.0000 0.566379
$$612$$ 0 0
$$613$$ − 44.0000i − 1.77714i −0.458738 0.888572i $$-0.651698\pi$$
0.458738 0.888572i $$-0.348302\pi$$
$$614$$ 7.00000 0.282497
$$615$$ 0 0
$$616$$ −18.0000 −0.725241
$$617$$ − 36.0000i − 1.44931i −0.689114 0.724653i $$-0.742000\pi$$
0.689114 0.724653i $$-0.258000\pi$$
$$618$$ 0 0
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ − 45.0000i − 1.80289i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −14.0000 −0.559553
$$627$$ 0 0
$$628$$ − 14.0000i − 0.558661i
$$629$$ −16.0000 −0.637962
$$630$$ 0 0
$$631$$ 16.0000 0.636950 0.318475 0.947931i $$-0.396829\pi$$
0.318475 + 0.947931i $$0.396829\pi$$
$$632$$ 18.0000i 0.716002i
$$633$$ 0 0
$$634$$ 34.0000 1.35031
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 4.00000i 0.158486i
$$638$$ − 2.00000i − 0.0791808i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −33.0000 −1.30342 −0.651711 0.758468i $$-0.725948\pi$$
−0.651711 + 0.758468i $$0.725948\pi$$
$$642$$ 0 0
$$643$$ − 9.00000i − 0.354925i −0.984128 0.177463i $$-0.943211\pi$$
0.984128 0.177463i $$-0.0567889\pi$$
$$644$$ 9.00000 0.354650
$$645$$ 0 0
$$646$$ −32.0000 −1.25902
$$647$$ − 17.0000i − 0.668339i −0.942513 0.334169i $$-0.891544\pi$$
0.942513 0.334169i $$-0.108456\pi$$
$$648$$ 0 0
$$649$$ −28.0000 −1.09910
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 4.00000i − 0.156652i
$$653$$ 4.00000i 0.156532i 0.996933 + 0.0782660i $$0.0249384\pi$$
−0.996933 + 0.0782660i $$0.975062\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 5.00000 0.195217
$$657$$ 0 0
$$658$$ − 21.0000i − 0.818665i
$$659$$ −8.00000 −0.311636 −0.155818 0.987786i $$-0.549801\pi$$
−0.155818 + 0.987786i $$0.549801\pi$$
$$660$$ 0 0
$$661$$ −14.0000 −0.544537 −0.272268 0.962221i $$-0.587774\pi$$
−0.272268 + 0.962221i $$0.587774\pi$$
$$662$$ − 6.00000i − 0.233197i
$$663$$ 0 0
$$664$$ 27.0000 1.04780
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 3.00000i 0.116160i
$$668$$ − 9.00000i − 0.348220i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 14.0000 0.540464
$$672$$ 0 0
$$673$$ 6.00000i 0.231283i 0.993291 + 0.115642i $$0.0368924\pi$$
−0.993291 + 0.115642i $$0.963108\pi$$
$$674$$ −8.00000 −0.308148
$$675$$ 0 0
$$676$$ 9.00000 0.346154
$$677$$ − 42.0000i − 1.61419i −0.590421 0.807096i $$-0.701038\pi$$
0.590421 0.807096i $$-0.298962\pi$$
$$678$$ 0 0
$$679$$ 6.00000 0.230259
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ − 12.0000i − 0.459167i −0.973289 0.229584i $$-0.926264\pi$$
0.973289 0.229584i $$-0.0737364\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −15.0000 −0.572703
$$687$$ 0 0
$$688$$ 8.00000i 0.304997i
$$689$$ 4.00000 0.152388
$$690$$ 0 0
$$691$$ −14.0000 −0.532585 −0.266293 0.963892i $$-0.585799\pi$$
−0.266293 + 0.963892i $$0.585799\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ −4.00000 −0.151838
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 20.0000i − 0.757554i
$$698$$ 5.00000i 0.189253i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −23.0000 −0.868698 −0.434349 0.900745i $$-0.643022\pi$$
−0.434349 + 0.900745i $$0.643022\pi$$
$$702$$ 0 0
$$703$$ 32.0000i 1.20690i
$$704$$ −14.0000 −0.527645
$$705$$ 0 0
$$706$$ 24.0000 0.903252
$$707$$ 54.0000i 2.03088i
$$708$$ 0 0
$$709$$ 41.0000 1.53979 0.769894 0.638172i $$-0.220309\pi$$
0.769894 + 0.638172i $$0.220309\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ − 45.0000i − 1.68645i
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −2.00000 −0.0747435
$$717$$ 0 0
$$718$$ 24.0000i 0.895672i
$$719$$ 6.00000 0.223762 0.111881 0.993722i $$-0.464312\pi$$
0.111881 + 0.993722i $$0.464312\pi$$
$$720$$ 0 0
$$721$$ −24.0000 −0.893807
$$722$$ 45.0000i 1.67473i
$$723$$ 0 0
$$724$$ −7.00000 −0.260153
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 23.0000i − 0.853023i −0.904482 0.426511i $$-0.859742\pi$$
0.904482 0.426511i $$-0.140258\pi$$
$$728$$ 18.0000i 0.667124i
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 32.0000 1.18356
$$732$$ 0 0
$$733$$ − 34.0000i − 1.25582i −0.778287 0.627909i $$-0.783911\pi$$
0.778287 0.627909i $$-0.216089\pi$$
$$734$$ 24.0000 0.885856
$$735$$ 0 0
$$736$$ 15.0000 0.552907
$$737$$ 6.00000i 0.221013i
$$738$$ 0 0
$$739$$ 2.00000 0.0735712 0.0367856 0.999323i $$-0.488288\pi$$
0.0367856 + 0.999323i $$0.488288\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ − 6.00000i − 0.220267i
$$743$$ 29.0000i 1.06391i 0.846774 + 0.531953i $$0.178542\pi$$
−0.846774 + 0.531953i $$0.821458\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −10.0000 −0.366126
$$747$$ 0 0
$$748$$ 8.00000i 0.292509i
$$749$$ −9.00000 −0.328853
$$750$$ 0 0
$$751$$ 10.0000 0.364905 0.182453 0.983215i $$-0.441596\pi$$
0.182453 + 0.983215i $$0.441596\pi$$
$$752$$ − 7.00000i − 0.255264i
$$753$$ 0 0
$$754$$ −2.00000 −0.0728357
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 26.0000i 0.944986i 0.881334 + 0.472493i $$0.156646\pi$$
−0.881334 + 0.472493i $$0.843354\pi$$
$$758$$ 26.0000i 0.944363i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −15.0000 −0.543750 −0.271875 0.962333i $$-0.587644\pi$$
−0.271875 + 0.962333i $$0.587644\pi$$
$$762$$ 0 0
$$763$$ − 15.0000i − 0.543036i
$$764$$ −8.00000 −0.289430
$$765$$ 0 0
$$766$$ 36.0000 1.30073
$$767$$ 28.0000i 1.01102i
$$768$$ 0 0
$$769$$ −5.00000 −0.180305 −0.0901523 0.995928i $$-0.528735\pi$$
−0.0901523 + 0.995928i $$0.528735\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ − 10.0000i − 0.359908i
$$773$$ 24.0000i 0.863220i 0.902060 + 0.431610i $$0.142054\pi$$
−0.902060 + 0.431610i $$0.857946\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 6.00000 0.215387
$$777$$ 0 0
$$778$$ 33.0000i 1.18311i
$$779$$ −40.0000 −1.43315
$$780$$ 0 0
$$781$$ −4.00000 −0.143131
$$782$$ 12.0000i 0.429119i
$$783$$ 0 0
$$784$$ 2.00000 0.0714286
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 28.0000i − 0.998092i −0.866575 0.499046i $$-0.833684\pi$$
0.866575 0.499046i $$-0.166316\pi$$
$$788$$ − 12.0000i − 0.427482i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −24.0000 −0.853342
$$792$$ 0 0
$$793$$ − 14.0000i − 0.497155i
$$794$$ 34.0000 1.20661
$$795$$ 0 0
$$796$$ −4.00000 −0.141776
$$797$$ 26.0000i 0.920967i 0.887668 + 0.460484i $$0.152324\pi$$
−0.887668 + 0.460484i $$0.847676\pi$$
$$798$$ 0 0
$$799$$ −28.0000 −0.990569
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 18.0000i 0.635602i
$$803$$ 8.00000i 0.282314i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 54.0000i 1.89971i
$$809$$ 26.0000 0.914111 0.457056 0.889438i $$-0.348904\pi$$
0.457056 + 0.889438i $$0.348904\pi$$
$$810$$ 0 0
$$811$$ −42.0000 −1.47482 −0.737410 0.675446i $$-0.763951\pi$$
−0.737410 + 0.675446i $$0.763951\pi$$
$$812$$ − 3.00000i − 0.105279i
$$813$$ 0 0
$$814$$ −8.00000 −0.280400
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 64.0000i − 2.23908i
$$818$$ − 14.0000i − 0.489499i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −15.0000 −0.523504 −0.261752 0.965135i $$-0.584300\pi$$
−0.261752 + 0.965135i $$0.584300\pi$$
$$822$$ 0 0
$$823$$ 53.0000i 1.84746i 0.383040 + 0.923732i $$0.374877\pi$$
−0.383040 + 0.923732i $$0.625123\pi$$
$$824$$ −24.0000 −0.836080
$$825$$ 0 0
$$826$$ 42.0000 1.46137
$$827$$ 37.0000i 1.28662i 0.765607 + 0.643308i $$0.222439\pi$$
−0.765607 + 0.643308i $$0.777561\pi$$
$$828$$ 0 0
$$829$$ 3.00000 0.104194 0.0520972 0.998642i $$-0.483409\pi$$
0.0520972 + 0.998642i $$0.483409\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 14.0000i 0.485363i
$$833$$ − 8.00000i − 0.277184i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 16.0000 0.553372
$$837$$ 0 0
$$838$$ 26.0000i 0.898155i
$$839$$ −40.0000 −1.38095 −0.690477 0.723355i $$-0.742599\pi$$
−0.690477 + 0.723355i $$0.742599\pi$$
$$840$$ 0 0
$$841$$ −28.0000 −0.965517
$$842$$ 34.0000i 1.17172i
$$843$$ 0 0
$$844$$ −22.0000 −0.757271
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 21.0000i − 0.721569i
$$848$$ − 2.00000i − 0.0686803i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 12.0000 0.411355
$$852$$ 0 0
$$853$$ − 54.0000i − 1.84892i −0.381273 0.924462i $$-0.624514\pi$$
0.381273 0.924462i $$-0.375486\pi$$
$$854$$ −21.0000 −0.718605
$$855$$ 0 0
$$856$$ −9.00000 −0.307614
$$857$$ 10.0000i 0.341593i 0.985306 + 0.170797i $$0.0546341\pi$$
−0.985306 + 0.170797i $$0.945366\pi$$
$$858$$ 0 0
$$859$$ −22.0000 −0.750630 −0.375315 0.926897i $$-0.622466\pi$$
−0.375315 + 0.926897i $$0.622466\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 30.0000i 1.02180i
$$863$$ − 17.0000i − 0.578687i −0.957225 0.289343i $$-0.906563\pi$$
0.957225 0.289343i $$-0.0934369\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −28.0000 −0.951479
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 12.0000 0.407072
$$870$$ 0 0
$$871$$ 6.00000 0.203302
$$872$$ − 15.0000i − 0.507964i
$$873$$ 0 0
$$874$$ 24.0000 0.811812
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 18.0000i 0.607817i 0.952701 + 0.303908i $$0.0982917\pi$$
−0.952701 + 0.303908i $$0.901708\pi$$
$$878$$ − 28.0000i − 0.944954i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 35.0000 1.17918 0.589590 0.807703i $$-0.299289\pi$$
0.589590 + 0.807703i $$0.299289\pi$$
$$882$$ 0 0
$$883$$ 23.0000i 0.774012i 0.922077 + 0.387006i $$0.126491\pi$$
−0.922077 + 0.387006i $$0.873509\pi$$
$$884$$ 8.00000 0.269069
$$885$$ 0 0
$$886$$ 15.0000 0.503935
$$887$$ 36.0000i 1.20876i 0.796696 + 0.604381i $$0.206579\pi$$
−0.796696 + 0.604381i $$0.793421\pi$$
$$888$$ 0 0
$$889$$ −15.0000 −0.503084
$$890$$ 0 0
$$891$$ 0 0
$$892$$ − 19.0000i − 0.636167i
$$893$$ 56.0000i 1.87397i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −9.00000 −0.300669
$$897$$ 0 0
$$898$$ − 26.0000i − 0.867631i
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −8.00000 −0.266519
$$902$$ − 10.0000i − 0.332964i
$$903$$ 0 0
$$904$$ −24.0000 −0.798228
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 51.0000i 1.69343i 0.532049 + 0.846714i $$0.321422\pi$$
−0.532049 + 0.846714i $$0.678578\pi$$
$$908$$ − 4.00000i − 0.132745i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 50.0000 1.65657 0.828287 0.560304i $$-0.189316\pi$$
0.828287 + 0.560304i $$0.189316\pi$$
$$912$$ 0 0
$$913$$ − 18.0000i − 0.595713i
$$914$$ −20.0000 −0.661541
$$915$$ 0 0
$$916$$ −15.0000 −0.495614
$$917$$ 18.0000i 0.594412i
$$918$$ 0 0
$$919$$ −10.0000 −0.329870 −0.164935 0.986304i $$-0.552741\pi$$
−0.164935 + 0.986304i $$0.552741\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ − 9.00000i − 0.296399i
$$923$$ 4.00000i 0.131662i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 36.0000 1.18303
$$927$$ 0 0
$$928$$ − 5.00000i − 0.164133i
$$929$$ 14.0000 0.459325 0.229663 0.973270i $$-0.426238\pi$$
0.229663 + 0.973270i $$0.426238\pi$$
$$930$$ 0 0
$$931$$ −16.0000 −0.524379
$$932$$ − 24.0000i − 0.786146i
$$933$$ 0 0
$$934$$ −20.0000 −0.654420
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$938$$ − 9.00000i − 0.293860i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 7.00000 0.228193 0.114097 0.993470i $$-0.463603\pi$$
0.114097 + 0.993470i $$0.463603\pi$$
$$942$$ 0 0
$$943$$ 15.0000i 0.488467i
$$944$$ 14.0000 0.455661
$$945$$ 0 0
$$946$$ 16.0000 0.520205
$$947$$ 57.0000i 1.85225i 0.377215 + 0.926126i $$0.376882\pi$$
−0.377215 + 0.926126i $$0.623118\pi$$
$$948$$ 0 0
$$949$$ 8.00000 0.259691
$$950$$ 0 0
$$951$$ 0 0
$$952$$ − 36.0000i − 1.16677i
$$953$$ 26.0000i 0.842223i 0.907009 + 0.421111i $$0.138360\pi$$
−0.907009 + 0.421111i $$0.861640\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −8.00000 −0.258738
$$957$$ 0 0
$$958$$ − 18.0000i − 0.581554i
$$959$$ −36.0000 −1.16250
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 8.00000i 0.257930i
$$963$$ 0 0
$$964$$ −11.0000 −0.354286
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 41.0000i − 1.31847i −0.751936 0.659236i $$-0.770880\pi$$
0.751936 0.659236i $$-0.229120\pi$$
$$968$$ − 21.0000i − 0.674966i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 36.0000 1.15529 0.577647 0.816286i $$-0.303971\pi$$
0.577647 + 0.816286i $$0.303971\pi$$
$$972$$ 0 0
$$973$$ 48.0000i 1.53881i
$$974$$ −16.0000 −0.512673
$$975$$ 0 0
$$976$$ −7.00000 −0.224065
$$977$$ − 38.0000i − 1.21573i −0.794041 0.607864i $$-0.792027\pi$$
0.794041 0.607864i $$-0.207973\pi$$
$$978$$ 0 0
$$979$$ −30.0000 −0.958804
$$980$$ 0 0
$$981$$ 0 0
$$982$$ − 20.0000i − 0.638226i
$$983$$ − 3.00000i − 0.0956851i −0.998855 0.0478426i $$-0.984765\pi$$
0.998855 0.0478426i $$-0.0152346\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 4.00000 0.127386
$$987$$ 0 0
$$988$$ − 16.0000i − 0.509028i
$$989$$ −24.0000 −0.763156
$$990$$ 0 0
$$991$$ 26.0000 0.825917 0.412959 0.910750i $$-0.364495\pi$$
0.412959 + 0.910750i $$0.364495\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 6.00000 0.190308
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 18.0000i 0.570066i 0.958518 + 0.285033i $$0.0920045\pi$$
−0.958518 + 0.285033i $$0.907995\pi$$
$$998$$ 32.0000i 1.01294i
$$999$$ 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 2025.2.b.d.649.2 2
3.2 odd 2 2025.2.b.c.649.1 2
5.2 odd 4 405.2.a.b.1.1 1
5.3 odd 4 2025.2.a.e.1.1 1
5.4 even 2 inner 2025.2.b.d.649.1 2
9.2 odd 6 225.2.k.a.49.1 4
9.4 even 3 675.2.k.a.424.1 4
9.5 odd 6 225.2.k.a.124.2 4
9.7 even 3 675.2.k.a.199.2 4
15.2 even 4 405.2.a.e.1.1 1
15.8 even 4 2025.2.a.b.1.1 1
15.14 odd 2 2025.2.b.c.649.2 2
20.7 even 4 6480.2.a.x.1.1 1
45.2 even 12 45.2.e.a.31.1 yes 2
45.4 even 6 675.2.k.a.424.2 4
45.7 odd 12 135.2.e.a.91.1 2
45.13 odd 12 675.2.e.a.451.1 2
45.14 odd 6 225.2.k.a.124.1 4
45.22 odd 12 135.2.e.a.46.1 2
45.23 even 12 225.2.e.a.151.1 2
45.29 odd 6 225.2.k.a.49.2 4
45.32 even 12 45.2.e.a.16.1 2
45.34 even 6 675.2.k.a.199.1 4
45.38 even 12 225.2.e.a.76.1 2
45.43 odd 12 675.2.e.a.226.1 2
60.47 odd 4 6480.2.a.k.1.1 1
180.7 even 12 2160.2.q.a.1441.1 2
180.47 odd 12 720.2.q.d.481.1 2
180.67 even 12 2160.2.q.a.721.1 2
180.167 odd 12 720.2.q.d.241.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
45.2.e.a.16.1 2 45.32 even 12
45.2.e.a.31.1 yes 2 45.2 even 12
135.2.e.a.46.1 2 45.22 odd 12
135.2.e.a.91.1 2 45.7 odd 12
225.2.e.a.76.1 2 45.38 even 12
225.2.e.a.151.1 2 45.23 even 12
225.2.k.a.49.1 4 9.2 odd 6
225.2.k.a.49.2 4 45.29 odd 6
225.2.k.a.124.1 4 45.14 odd 6
225.2.k.a.124.2 4 9.5 odd 6
405.2.a.b.1.1 1 5.2 odd 4
405.2.a.e.1.1 1 15.2 even 4
675.2.e.a.226.1 2 45.43 odd 12
675.2.e.a.451.1 2 45.13 odd 12
675.2.k.a.199.1 4 45.34 even 6
675.2.k.a.199.2 4 9.7 even 3
675.2.k.a.424.1 4 9.4 even 3
675.2.k.a.424.2 4 45.4 even 6
720.2.q.d.241.1 2 180.167 odd 12
720.2.q.d.481.1 2 180.47 odd 12
2025.2.a.b.1.1 1 15.8 even 4
2025.2.a.e.1.1 1 5.3 odd 4
2025.2.b.c.649.1 2 3.2 odd 2
2025.2.b.c.649.2 2 15.14 odd 2
2025.2.b.d.649.1 2 5.4 even 2 inner
2025.2.b.d.649.2 2 1.1 even 1 trivial
2160.2.q.a.721.1 2 180.67 even 12
2160.2.q.a.1441.1 2 180.7 even 12
6480.2.a.k.1.1 1 60.47 odd 4
6480.2.a.x.1.1 1 20.7 even 4