# Properties

 Label 2025.2.a.x.1.4 Level $2025$ Weight $2$ Character 2025.1 Self dual yes Analytic conductor $16.170$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2025,2,Mod(1,2025)]

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, 0]))

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

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

chi := DirichletCharacter("2025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2025 = 3^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2025.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: $$16.1697064093$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{3}, \sqrt{11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 7x^{2} + 4$$ x^4 - 7*x^2 + 4 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 405) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$2.52434$$ of defining polynomial Character $$\chi$$ $$=$$ 2025.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.52434 q^{2} +4.37228 q^{4} +3.46410 q^{7} +5.98844 q^{8} +O(q^{10})$$ $$q+2.52434 q^{2} +4.37228 q^{4} +3.46410 q^{7} +5.98844 q^{8} -1.37228 q^{11} -4.10891 q^{13} +8.74456 q^{14} +6.37228 q^{16} +2.52434 q^{17} +5.37228 q^{19} -3.46410 q^{22} +5.04868 q^{23} -10.3723 q^{26} +15.1460 q^{28} -5.74456 q^{29} +0.627719 q^{31} +4.10891 q^{32} +6.37228 q^{34} +7.57301 q^{37} +13.5615 q^{38} +1.37228 q^{41} -3.46410 q^{43} -6.00000 q^{44} +12.7446 q^{46} -8.51278 q^{47} +5.00000 q^{49} -17.9653 q^{52} -5.34363 q^{53} +20.7446 q^{56} -14.5012 q^{58} -7.37228 q^{59} +3.62772 q^{61} +1.58457 q^{62} -2.37228 q^{64} -8.21782 q^{67} +11.0371 q^{68} +4.11684 q^{71} +7.57301 q^{73} +19.1168 q^{74} +23.4891 q^{76} -4.75372 q^{77} -4.74456 q^{79} +3.46410 q^{82} -5.34363 q^{83} -8.74456 q^{86} -8.21782 q^{88} +3.00000 q^{89} -14.2337 q^{91} +22.0742 q^{92} -21.4891 q^{94} +18.6101 q^{97} +12.6217 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 6 q^{4}+O(q^{10})$$ 4 * q + 6 * q^4 $$4 q + 6 q^{4} + 6 q^{11} + 12 q^{14} + 14 q^{16} + 10 q^{19} - 30 q^{26} + 14 q^{31} + 14 q^{34} - 6 q^{41} - 24 q^{44} + 28 q^{46} + 20 q^{49} + 60 q^{56} - 18 q^{59} + 26 q^{61} + 2 q^{64} - 18 q^{71} + 42 q^{74} + 48 q^{76} + 4 q^{79} - 12 q^{86} + 12 q^{89} + 12 q^{91} - 40 q^{94}+O(q^{100})$$ 4 * q + 6 * q^4 + 6 * q^11 + 12 * q^14 + 14 * q^16 + 10 * q^19 - 30 * q^26 + 14 * q^31 + 14 * q^34 - 6 * q^41 - 24 * q^44 + 28 * q^46 + 20 * q^49 + 60 * q^56 - 18 * q^59 + 26 * q^61 + 2 * q^64 - 18 * q^71 + 42 * q^74 + 48 * q^76 + 4 * q^79 - 12 * q^86 + 12 * q^89 + 12 * q^91 - 40 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.52434 1.78498 0.892488 0.451071i $$-0.148958\pi$$
0.892488 + 0.451071i $$0.148958\pi$$
$$3$$ 0 0
$$4$$ 4.37228 2.18614
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 3.46410 1.30931 0.654654 0.755929i $$-0.272814\pi$$
0.654654 + 0.755929i $$0.272814\pi$$
$$8$$ 5.98844 2.11723
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −1.37228 −0.413758 −0.206879 0.978366i $$-0.566331\pi$$
−0.206879 + 0.978366i $$0.566331\pi$$
$$12$$ 0 0
$$13$$ −4.10891 −1.13961 −0.569804 0.821781i $$-0.692981\pi$$
−0.569804 + 0.821781i $$0.692981\pi$$
$$14$$ 8.74456 2.33708
$$15$$ 0 0
$$16$$ 6.37228 1.59307
$$17$$ 2.52434 0.612242 0.306121 0.951993i $$-0.400969\pi$$
0.306121 + 0.951993i $$0.400969\pi$$
$$18$$ 0 0
$$19$$ 5.37228 1.23249 0.616243 0.787556i $$-0.288654\pi$$
0.616243 + 0.787556i $$0.288654\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −3.46410 −0.738549
$$23$$ 5.04868 1.05272 0.526361 0.850261i $$-0.323556\pi$$
0.526361 + 0.850261i $$0.323556\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −10.3723 −2.03417
$$27$$ 0 0
$$28$$ 15.1460 2.86233
$$29$$ −5.74456 −1.06674 −0.533369 0.845883i $$-0.679074\pi$$
−0.533369 + 0.845883i $$0.679074\pi$$
$$30$$ 0 0
$$31$$ 0.627719 0.112742 0.0563708 0.998410i $$-0.482047\pi$$
0.0563708 + 0.998410i $$0.482047\pi$$
$$32$$ 4.10891 0.726360
$$33$$ 0 0
$$34$$ 6.37228 1.09284
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 7.57301 1.24500 0.622498 0.782621i $$-0.286118\pi$$
0.622498 + 0.782621i $$0.286118\pi$$
$$38$$ 13.5615 2.19996
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 1.37228 0.214314 0.107157 0.994242i $$-0.465825\pi$$
0.107157 + 0.994242i $$0.465825\pi$$
$$42$$ 0 0
$$43$$ −3.46410 −0.528271 −0.264135 0.964486i $$-0.585087\pi$$
−0.264135 + 0.964486i $$0.585087\pi$$
$$44$$ −6.00000 −0.904534
$$45$$ 0 0
$$46$$ 12.7446 1.87908
$$47$$ −8.51278 −1.24172 −0.620858 0.783923i $$-0.713216\pi$$
−0.620858 + 0.783923i $$0.713216\pi$$
$$48$$ 0 0
$$49$$ 5.00000 0.714286
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −17.9653 −2.49134
$$53$$ −5.34363 −0.734004 −0.367002 0.930220i $$-0.619616\pi$$
−0.367002 + 0.930220i $$0.619616\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 20.7446 2.77211
$$57$$ 0 0
$$58$$ −14.5012 −1.90410
$$59$$ −7.37228 −0.959789 −0.479895 0.877326i $$-0.659325\pi$$
−0.479895 + 0.877326i $$0.659325\pi$$
$$60$$ 0 0
$$61$$ 3.62772 0.464482 0.232241 0.972658i $$-0.425394\pi$$
0.232241 + 0.972658i $$0.425394\pi$$
$$62$$ 1.58457 0.201241
$$63$$ 0 0
$$64$$ −2.37228 −0.296535
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −8.21782 −1.00397 −0.501983 0.864877i $$-0.667396\pi$$
−0.501983 + 0.864877i $$0.667396\pi$$
$$68$$ 11.0371 1.33845
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 4.11684 0.488579 0.244290 0.969702i $$-0.421445\pi$$
0.244290 + 0.969702i $$0.421445\pi$$
$$72$$ 0 0
$$73$$ 7.57301 0.886354 0.443177 0.896434i $$-0.353851\pi$$
0.443177 + 0.896434i $$0.353851\pi$$
$$74$$ 19.1168 2.22229
$$75$$ 0 0
$$76$$ 23.4891 2.69439
$$77$$ −4.75372 −0.541737
$$78$$ 0 0
$$79$$ −4.74456 −0.533805 −0.266903 0.963724i $$-0.586000\pi$$
−0.266903 + 0.963724i $$0.586000\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 3.46410 0.382546
$$83$$ −5.34363 −0.586540 −0.293270 0.956030i $$-0.594743\pi$$
−0.293270 + 0.956030i $$0.594743\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −8.74456 −0.942950
$$87$$ 0 0
$$88$$ −8.21782 −0.876023
$$89$$ 3.00000 0.317999 0.159000 0.987279i $$-0.449173\pi$$
0.159000 + 0.987279i $$0.449173\pi$$
$$90$$ 0 0
$$91$$ −14.2337 −1.49210
$$92$$ 22.0742 2.30140
$$93$$ 0 0
$$94$$ −21.4891 −2.21643
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 18.6101 1.88957 0.944786 0.327688i $$-0.106269\pi$$
0.944786 + 0.327688i $$0.106269\pi$$
$$98$$ 12.6217 1.27498
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 10.6277 1.05750 0.528749 0.848778i $$-0.322661\pi$$
0.528749 + 0.848778i $$0.322661\pi$$
$$102$$ 0 0
$$103$$ −6.92820 −0.682656 −0.341328 0.939944i $$-0.610877\pi$$
−0.341328 + 0.939944i $$0.610877\pi$$
$$104$$ −24.6060 −2.41281
$$105$$ 0 0
$$106$$ −13.4891 −1.31018
$$107$$ −13.2665 −1.28252 −0.641260 0.767323i $$-0.721588\pi$$
−0.641260 + 0.767323i $$0.721588\pi$$
$$108$$ 0 0
$$109$$ −1.74456 −0.167099 −0.0835494 0.996504i $$-0.526626\pi$$
−0.0835494 + 0.996504i $$0.526626\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 22.0742 2.08582
$$113$$ −9.45254 −0.889220 −0.444610 0.895724i $$-0.646658\pi$$
−0.444610 + 0.895724i $$0.646658\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −25.1168 −2.33204
$$117$$ 0 0
$$118$$ −18.6101 −1.71320
$$119$$ 8.74456 0.801613
$$120$$ 0 0
$$121$$ −9.11684 −0.828804
$$122$$ 9.15759 0.829089
$$123$$ 0 0
$$124$$ 2.74456 0.246469
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −10.3923 −0.922168 −0.461084 0.887357i $$-0.652539\pi$$
−0.461084 + 0.887357i $$0.652539\pi$$
$$128$$ −14.2063 −1.25567
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 1.37228 0.119897 0.0599484 0.998201i $$-0.480906\pi$$
0.0599484 + 0.998201i $$0.480906\pi$$
$$132$$ 0 0
$$133$$ 18.6101 1.61370
$$134$$ −20.7446 −1.79206
$$135$$ 0 0
$$136$$ 15.1168 1.29626
$$137$$ −2.22938 −0.190469 −0.0952346 0.995455i $$-0.530360\pi$$
−0.0952346 + 0.995455i $$0.530360\pi$$
$$138$$ 0 0
$$139$$ −6.11684 −0.518824 −0.259412 0.965767i $$-0.583529\pi$$
−0.259412 + 0.965767i $$0.583529\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 10.3923 0.872103
$$143$$ 5.63858 0.471522
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 19.1168 1.58212
$$147$$ 0 0
$$148$$ 33.1113 2.72174
$$149$$ −16.3723 −1.34127 −0.670635 0.741788i $$-0.733978\pi$$
−0.670635 + 0.741788i $$0.733978\pi$$
$$150$$ 0 0
$$151$$ 18.1168 1.47433 0.737164 0.675714i $$-0.236165\pi$$
0.737164 + 0.675714i $$0.236165\pi$$
$$152$$ 32.1716 2.60946
$$153$$ 0 0
$$154$$ −12.0000 −0.966988
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −21.4294 −1.71025 −0.855127 0.518419i $$-0.826521\pi$$
−0.855127 + 0.518419i $$0.826521\pi$$
$$158$$ −11.9769 −0.952829
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 17.4891 1.37834
$$162$$ 0 0
$$163$$ −4.75372 −0.372340 −0.186170 0.982518i $$-0.559608\pi$$
−0.186170 + 0.982518i $$0.559608\pi$$
$$164$$ 6.00000 0.468521
$$165$$ 0 0
$$166$$ −13.4891 −1.04696
$$167$$ −0.294954 −0.0228242 −0.0114121 0.999935i $$-0.503633\pi$$
−0.0114121 + 0.999935i $$0.503633\pi$$
$$168$$ 0 0
$$169$$ 3.88316 0.298704
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −15.1460 −1.15487
$$173$$ 7.86797 0.598190 0.299095 0.954223i $$-0.403315\pi$$
0.299095 + 0.954223i $$0.403315\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −8.74456 −0.659146
$$177$$ 0 0
$$178$$ 7.57301 0.567621
$$179$$ 22.1168 1.65309 0.826545 0.562870i $$-0.190303\pi$$
0.826545 + 0.562870i $$0.190303\pi$$
$$180$$ 0 0
$$181$$ 14.8614 1.10464 0.552320 0.833632i $$-0.313743\pi$$
0.552320 + 0.833632i $$0.313743\pi$$
$$182$$ −35.9306 −2.66336
$$183$$ 0 0
$$184$$ 30.2337 2.22886
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −3.46410 −0.253320
$$188$$ −37.2203 −2.71457
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −13.3723 −0.967584 −0.483792 0.875183i $$-0.660741\pi$$
−0.483792 + 0.875183i $$0.660741\pi$$
$$192$$ 0 0
$$193$$ 21.4294 1.54252 0.771262 0.636518i $$-0.219626\pi$$
0.771262 + 0.636518i $$0.219626\pi$$
$$194$$ 46.9783 3.37284
$$195$$ 0 0
$$196$$ 21.8614 1.56153
$$197$$ −23.0140 −1.63968 −0.819840 0.572593i $$-0.805937\pi$$
−0.819840 + 0.572593i $$0.805937\pi$$
$$198$$ 0 0
$$199$$ 16.0000 1.13421 0.567105 0.823646i $$-0.308063\pi$$
0.567105 + 0.823646i $$0.308063\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 26.8280 1.88761
$$203$$ −19.8997 −1.39669
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −17.4891 −1.21853
$$207$$ 0 0
$$208$$ −26.1831 −1.81547
$$209$$ −7.37228 −0.509951
$$210$$ 0 0
$$211$$ 26.8614 1.84922 0.924608 0.380921i $$-0.124393\pi$$
0.924608 + 0.380921i $$0.124393\pi$$
$$212$$ −23.3639 −1.60464
$$213$$ 0 0
$$214$$ −33.4891 −2.28927
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 2.17448 0.147613
$$218$$ −4.40387 −0.298267
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −10.3723 −0.697715
$$222$$ 0 0
$$223$$ 1.28962 0.0863594 0.0431797 0.999067i $$-0.486251\pi$$
0.0431797 + 0.999067i $$0.486251\pi$$
$$224$$ 14.2337 0.951028
$$225$$ 0 0
$$226$$ −23.8614 −1.58724
$$227$$ −0.294954 −0.0195768 −0.00978838 0.999952i $$-0.503116\pi$$
−0.00978838 + 0.999952i $$0.503116\pi$$
$$228$$ 0 0
$$229$$ 22.6060 1.49384 0.746922 0.664911i $$-0.231531\pi$$
0.746922 + 0.664911i $$0.231531\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −34.4010 −2.25853
$$233$$ −9.45254 −0.619257 −0.309628 0.950858i $$-0.600205\pi$$
−0.309628 + 0.950858i $$0.600205\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −32.2337 −2.09823
$$237$$ 0 0
$$238$$ 22.0742 1.43086
$$239$$ 22.9783 1.48634 0.743170 0.669103i $$-0.233321\pi$$
0.743170 + 0.669103i $$0.233321\pi$$
$$240$$ 0 0
$$241$$ −24.4891 −1.57748 −0.788742 0.614725i $$-0.789267\pi$$
−0.788742 + 0.614725i $$0.789267\pi$$
$$242$$ −23.0140 −1.47940
$$243$$ 0 0
$$244$$ 15.8614 1.01542
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −22.0742 −1.40455
$$248$$ 3.75906 0.238700
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −14.2337 −0.898422 −0.449211 0.893426i $$-0.648295\pi$$
−0.449211 + 0.893426i $$0.648295\pi$$
$$252$$ 0 0
$$253$$ −6.92820 −0.435572
$$254$$ −26.2337 −1.64605
$$255$$ 0 0
$$256$$ −31.1168 −1.94480
$$257$$ −23.0140 −1.43557 −0.717787 0.696263i $$-0.754845\pi$$
−0.717787 + 0.696263i $$0.754845\pi$$
$$258$$ 0 0
$$259$$ 26.2337 1.63008
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 3.46410 0.214013
$$263$$ −26.5330 −1.63609 −0.818047 0.575151i $$-0.804943\pi$$
−0.818047 + 0.575151i $$0.804943\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 46.9783 2.88042
$$267$$ 0 0
$$268$$ −35.9306 −2.19481
$$269$$ −5.23369 −0.319104 −0.159552 0.987190i $$-0.551005\pi$$
−0.159552 + 0.987190i $$0.551005\pi$$
$$270$$ 0 0
$$271$$ 16.7446 1.01716 0.508580 0.861015i $$-0.330171\pi$$
0.508580 + 0.861015i $$0.330171\pi$$
$$272$$ 16.0858 0.975344
$$273$$ 0 0
$$274$$ −5.62772 −0.339983
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −2.17448 −0.130652 −0.0653260 0.997864i $$-0.520809\pi$$
−0.0653260 + 0.997864i $$0.520809\pi$$
$$278$$ −15.4410 −0.926088
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −4.37228 −0.260828 −0.130414 0.991460i $$-0.541631\pi$$
−0.130414 + 0.991460i $$0.541631\pi$$
$$282$$ 0 0
$$283$$ −27.7128 −1.64736 −0.823678 0.567058i $$-0.808082\pi$$
−0.823678 + 0.567058i $$0.808082\pi$$
$$284$$ 18.0000 1.06810
$$285$$ 0 0
$$286$$ 14.2337 0.841656
$$287$$ 4.75372 0.280603
$$288$$ 0 0
$$289$$ −10.6277 −0.625160
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 33.1113 1.93769
$$293$$ −2.52434 −0.147473 −0.0737367 0.997278i $$-0.523492\pi$$
−0.0737367 + 0.997278i $$0.523492\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 45.3505 2.63595
$$297$$ 0 0
$$298$$ −41.3292 −2.39413
$$299$$ −20.7446 −1.19969
$$300$$ 0 0
$$301$$ −12.0000 −0.691669
$$302$$ 45.7330 2.63164
$$303$$ 0 0
$$304$$ 34.2337 1.96344
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 4.75372 0.271309 0.135655 0.990756i $$-0.456686\pi$$
0.135655 + 0.990756i $$0.456686\pi$$
$$308$$ −20.7846 −1.18431
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −12.8614 −0.729303 −0.364652 0.931144i $$-0.618812\pi$$
−0.364652 + 0.931144i $$0.618812\pi$$
$$312$$ 0 0
$$313$$ −5.39853 −0.305143 −0.152572 0.988292i $$-0.548755\pi$$
−0.152572 + 0.988292i $$0.548755\pi$$
$$314$$ −54.0951 −3.05276
$$315$$ 0 0
$$316$$ −20.7446 −1.16697
$$317$$ −6.98311 −0.392210 −0.196105 0.980583i $$-0.562829\pi$$
−0.196105 + 0.980583i $$0.562829\pi$$
$$318$$ 0 0
$$319$$ 7.88316 0.441372
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 44.1485 2.46030
$$323$$ 13.5615 0.754579
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −12.0000 −0.664619
$$327$$ 0 0
$$328$$ 8.21782 0.453753
$$329$$ −29.4891 −1.62579
$$330$$ 0 0
$$331$$ −8.11684 −0.446142 −0.223071 0.974802i $$-0.571608\pi$$
−0.223071 + 0.974802i $$0.571608\pi$$
$$332$$ −23.3639 −1.28226
$$333$$ 0 0
$$334$$ −0.744563 −0.0407407
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 6.92820 0.377403 0.188702 0.982034i $$-0.439572\pi$$
0.188702 + 0.982034i $$0.439572\pi$$
$$338$$ 9.80240 0.533180
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −0.861407 −0.0466478
$$342$$ 0 0
$$343$$ −6.92820 −0.374088
$$344$$ −20.7446 −1.11847
$$345$$ 0 0
$$346$$ 19.8614 1.06776
$$347$$ −23.6588 −1.27007 −0.635036 0.772483i $$-0.719015\pi$$
−0.635036 + 0.772483i $$0.719015\pi$$
$$348$$ 0 0
$$349$$ 13.6060 0.728311 0.364155 0.931338i $$-0.381358\pi$$
0.364155 + 0.931338i $$0.381358\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −5.63858 −0.300537
$$353$$ 8.51278 0.453089 0.226545 0.974001i $$-0.427257\pi$$
0.226545 + 0.974001i $$0.427257\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 13.1168 0.695191
$$357$$ 0 0
$$358$$ 55.8304 2.95073
$$359$$ 28.1168 1.48395 0.741975 0.670427i $$-0.233889\pi$$
0.741975 + 0.670427i $$0.233889\pi$$
$$360$$ 0 0
$$361$$ 9.86141 0.519021
$$362$$ 37.5152 1.97176
$$363$$ 0 0
$$364$$ −62.2337 −3.26193
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 23.3639 1.21958 0.609792 0.792562i $$-0.291253\pi$$
0.609792 + 0.792562i $$0.291253\pi$$
$$368$$ 32.1716 1.67706
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −18.5109 −0.961037
$$372$$ 0 0
$$373$$ −11.6819 −0.604867 −0.302434 0.953170i $$-0.597799\pi$$
−0.302434 + 0.953170i $$0.597799\pi$$
$$374$$ −8.74456 −0.452171
$$375$$ 0 0
$$376$$ −50.9783 −2.62900
$$377$$ 23.6039 1.21566
$$378$$ 0 0
$$379$$ 21.4891 1.10382 0.551911 0.833903i $$-0.313899\pi$$
0.551911 + 0.833903i $$0.313899\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −33.7562 −1.72712
$$383$$ 35.3407 1.80583 0.902913 0.429822i $$-0.141424\pi$$
0.902913 + 0.429822i $$0.141424\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 54.0951 2.75337
$$387$$ 0 0
$$388$$ 81.3687 4.13087
$$389$$ 23.4891 1.19095 0.595473 0.803375i $$-0.296965\pi$$
0.595473 + 0.803375i $$0.296965\pi$$
$$390$$ 0 0
$$391$$ 12.7446 0.644520
$$392$$ 29.9422 1.51231
$$393$$ 0 0
$$394$$ −58.0951 −2.92679
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −12.3267 −0.618661 −0.309331 0.950955i $$-0.600105\pi$$
−0.309331 + 0.950955i $$0.600105\pi$$
$$398$$ 40.3894 2.02454
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −19.6277 −0.980161 −0.490081 0.871677i $$-0.663033\pi$$
−0.490081 + 0.871677i $$0.663033\pi$$
$$402$$ 0 0
$$403$$ −2.57924 −0.128481
$$404$$ 46.4674 2.31184
$$405$$ 0 0
$$406$$ −50.2337 −2.49306
$$407$$ −10.3923 −0.515127
$$408$$ 0 0
$$409$$ −5.86141 −0.289828 −0.144914 0.989444i $$-0.546291\pi$$
−0.144914 + 0.989444i $$0.546291\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −30.2921 −1.49238
$$413$$ −25.5383 −1.25666
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −16.8832 −0.827765
$$417$$ 0 0
$$418$$ −18.6101 −0.910251
$$419$$ −5.48913 −0.268161 −0.134081 0.990970i $$-0.542808\pi$$
−0.134081 + 0.990970i $$0.542808\pi$$
$$420$$ 0 0
$$421$$ −10.2554 −0.499819 −0.249910 0.968269i $$-0.580401\pi$$
−0.249910 + 0.968269i $$0.580401\pi$$
$$422$$ 67.8073 3.30081
$$423$$ 0 0
$$424$$ −32.0000 −1.55406
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 12.5668 0.608149
$$428$$ −58.0049 −2.80377
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −18.3505 −0.883914 −0.441957 0.897036i $$-0.645716\pi$$
−0.441957 + 0.897036i $$0.645716\pi$$
$$432$$ 0 0
$$433$$ −2.81929 −0.135487 −0.0677433 0.997703i $$-0.521580\pi$$
−0.0677433 + 0.997703i $$0.521580\pi$$
$$434$$ 5.48913 0.263486
$$435$$ 0 0
$$436$$ −7.62772 −0.365301
$$437$$ 27.1229 1.29746
$$438$$ 0 0
$$439$$ 3.13859 0.149797 0.0748984 0.997191i $$-0.476137\pi$$
0.0748984 + 0.997191i $$0.476137\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −26.1831 −1.24541
$$443$$ 24.9484 1.18534 0.592668 0.805447i $$-0.298075\pi$$
0.592668 + 0.805447i $$0.298075\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 3.25544 0.154149
$$447$$ 0 0
$$448$$ −8.21782 −0.388256
$$449$$ 28.1168 1.32692 0.663458 0.748214i $$-0.269088\pi$$
0.663458 + 0.748214i $$0.269088\pi$$
$$450$$ 0 0
$$451$$ −1.88316 −0.0886744
$$452$$ −41.3292 −1.94396
$$453$$ 0 0
$$454$$ −0.744563 −0.0349441
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −8.86263 −0.414577 −0.207288 0.978280i $$-0.566464\pi$$
−0.207288 + 0.978280i $$0.566464\pi$$
$$458$$ 57.0651 2.66648
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 39.0951 1.82084 0.910420 0.413685i $$-0.135759\pi$$
0.910420 + 0.413685i $$0.135759\pi$$
$$462$$ 0 0
$$463$$ 13.8564 0.643962 0.321981 0.946746i $$-0.395651\pi$$
0.321981 + 0.946746i $$0.395651\pi$$
$$464$$ −36.6060 −1.69939
$$465$$ 0 0
$$466$$ −23.8614 −1.10536
$$467$$ 14.8511 0.687226 0.343613 0.939111i $$-0.388349\pi$$
0.343613 + 0.939111i $$0.388349\pi$$
$$468$$ 0 0
$$469$$ −28.4674 −1.31450
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −44.1485 −2.03210
$$473$$ 4.75372 0.218576
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 38.2337 1.75244
$$477$$ 0 0
$$478$$ 58.0049 2.65308
$$479$$ 21.6060 0.987202 0.493601 0.869688i $$-0.335680\pi$$
0.493601 + 0.869688i $$0.335680\pi$$
$$480$$ 0 0
$$481$$ −31.1168 −1.41881
$$482$$ −61.8188 −2.81577
$$483$$ 0 0
$$484$$ −39.8614 −1.81188
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 30.2921 1.37266 0.686332 0.727288i $$-0.259220\pi$$
0.686332 + 0.727288i $$0.259220\pi$$
$$488$$ 21.7244 0.983416
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 37.3723 1.68659 0.843294 0.537453i $$-0.180613\pi$$
0.843294 + 0.537453i $$0.180613\pi$$
$$492$$ 0 0
$$493$$ −14.5012 −0.653102
$$494$$ −55.7228 −2.50709
$$495$$ 0 0
$$496$$ 4.00000 0.179605
$$497$$ 14.2612 0.639701
$$498$$ 0 0
$$499$$ −18.6277 −0.833891 −0.416946 0.908931i $$-0.636899\pi$$
−0.416946 + 0.908931i $$0.636899\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −35.9306 −1.60366
$$503$$ −10.0974 −0.450219 −0.225109 0.974334i $$-0.572274\pi$$
−0.225109 + 0.974334i $$0.572274\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −17.4891 −0.777486
$$507$$ 0 0
$$508$$ −45.4381 −2.01599
$$509$$ −23.4891 −1.04114 −0.520569 0.853820i $$-0.674280\pi$$
−0.520569 + 0.853820i $$0.674280\pi$$
$$510$$ 0 0
$$511$$ 26.2337 1.16051
$$512$$ −50.1369 −2.21576
$$513$$ 0 0
$$514$$ −58.0951 −2.56246
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 11.6819 0.513770
$$518$$ 66.2227 2.90966
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 18.0000 0.788594 0.394297 0.918983i $$-0.370988\pi$$
0.394297 + 0.918983i $$0.370988\pi$$
$$522$$ 0 0
$$523$$ 15.1460 0.662290 0.331145 0.943580i $$-0.392565\pi$$
0.331145 + 0.943580i $$0.392565\pi$$
$$524$$ 6.00000 0.262111
$$525$$ 0 0
$$526$$ −66.9783 −2.92039
$$527$$ 1.58457 0.0690251
$$528$$ 0 0
$$529$$ 2.48913 0.108223
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 81.3687 3.52778
$$533$$ −5.63858 −0.244234
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −49.2119 −2.12563
$$537$$ 0 0
$$538$$ −13.2116 −0.569592
$$539$$ −6.86141 −0.295542
$$540$$ 0 0
$$541$$ −15.2337 −0.654947 −0.327474 0.944860i $$-0.606197\pi$$
−0.327474 + 0.944860i $$0.606197\pi$$
$$542$$ 42.2689 1.81561
$$543$$ 0 0
$$544$$ 10.3723 0.444708
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −6.92820 −0.296229 −0.148114 0.988970i $$-0.547320\pi$$
−0.148114 + 0.988970i $$0.547320\pi$$
$$548$$ −9.74749 −0.416392
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −30.8614 −1.31474
$$552$$ 0 0
$$553$$ −16.4356 −0.698915
$$554$$ −5.48913 −0.233211
$$555$$ 0 0
$$556$$ −26.7446 −1.13422
$$557$$ −21.7244 −0.920491 −0.460246 0.887792i $$-0.652239\pi$$
−0.460246 + 0.887792i $$0.652239\pi$$
$$558$$ 0 0
$$559$$ 14.2337 0.602021
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −11.0371 −0.465573
$$563$$ −0.994667 −0.0419202 −0.0209601 0.999780i $$-0.506672\pi$$
−0.0209601 + 0.999780i $$0.506672\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −69.9565 −2.94049
$$567$$ 0 0
$$568$$ 24.6535 1.03444
$$569$$ 2.48913 0.104350 0.0521748 0.998638i $$-0.483385\pi$$
0.0521748 + 0.998638i $$0.483385\pi$$
$$570$$ 0 0
$$571$$ 32.8614 1.37521 0.687604 0.726086i $$-0.258663\pi$$
0.687604 + 0.726086i $$0.258663\pi$$
$$572$$ 24.6535 1.03081
$$573$$ 0 0
$$574$$ 12.0000 0.500870
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 28.3576 1.18054 0.590272 0.807205i $$-0.299021\pi$$
0.590272 + 0.807205i $$0.299021\pi$$
$$578$$ −26.8280 −1.11590
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −18.5109 −0.767960
$$582$$ 0 0
$$583$$ 7.33296 0.303700
$$584$$ 45.3505 1.87662
$$585$$ 0 0
$$586$$ −6.37228 −0.263237
$$587$$ −9.80240 −0.404588 −0.202294 0.979325i $$-0.564840\pi$$
−0.202294 + 0.979325i $$0.564840\pi$$
$$588$$ 0 0
$$589$$ 3.37228 0.138952
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 48.2574 1.98337
$$593$$ 38.1600 1.56704 0.783522 0.621364i $$-0.213421\pi$$
0.783522 + 0.621364i $$0.213421\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −71.5842 −2.93220
$$597$$ 0 0
$$598$$ −52.3663 −2.14142
$$599$$ −7.37228 −0.301223 −0.150612 0.988593i $$-0.548124\pi$$
−0.150612 + 0.988593i $$0.548124\pi$$
$$600$$ 0 0
$$601$$ 27.9783 1.14126 0.570628 0.821208i $$-0.306700\pi$$
0.570628 + 0.821208i $$0.306700\pi$$
$$602$$ −30.2921 −1.23461
$$603$$ 0 0
$$604$$ 79.2119 3.22309
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 36.3354 1.47481 0.737404 0.675452i $$-0.236051\pi$$
0.737404 + 0.675452i $$0.236051\pi$$
$$608$$ 22.0742 0.895228
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 34.9783 1.41507
$$612$$ 0 0
$$613$$ 9.50744 0.384002 0.192001 0.981395i $$-0.438502\pi$$
0.192001 + 0.981395i $$0.438502\pi$$
$$614$$ 12.0000 0.484281
$$615$$ 0 0
$$616$$ −28.4674 −1.14698
$$617$$ 22.4241 0.902760 0.451380 0.892332i $$-0.350932\pi$$
0.451380 + 0.892332i $$0.350932\pi$$
$$618$$ 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
$$622$$ −32.4665 −1.30179
$$623$$ 10.3923 0.416359
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −13.6277 −0.544673
$$627$$ 0 0
$$628$$ −93.6955 −3.73886
$$629$$ 19.1168 0.762238
$$630$$ 0 0
$$631$$ 0.627719 0.0249891 0.0124945 0.999922i $$-0.496023\pi$$
0.0124945 + 0.999922i $$0.496023\pi$$
$$632$$ −28.4125 −1.13019
$$633$$ 0 0
$$634$$ −17.6277 −0.700086
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −20.5446 −0.814005
$$638$$ 19.8997 0.787839
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 37.9783 1.50005 0.750025 0.661409i $$-0.230041\pi$$
0.750025 + 0.661409i $$0.230041\pi$$
$$642$$ 0 0
$$643$$ 18.6101 0.733912 0.366956 0.930238i $$-0.380400\pi$$
0.366956 + 0.930238i $$0.380400\pi$$
$$644$$ 76.4674 3.01324
$$645$$ 0 0
$$646$$ 34.2337 1.34691
$$647$$ 4.45877 0.175292 0.0876461 0.996152i $$-0.472066\pi$$
0.0876461 + 0.996152i $$0.472066\pi$$
$$648$$ 0 0
$$649$$ 10.1168 0.397121
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −20.7846 −0.813988
$$653$$ 20.1947 0.790280 0.395140 0.918621i $$-0.370696\pi$$
0.395140 + 0.918621i $$0.370696\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 8.74456 0.341418
$$657$$ 0 0
$$658$$ −74.4405 −2.90199
$$659$$ −32.7446 −1.27555 −0.637774 0.770224i $$-0.720144\pi$$
−0.637774 + 0.770224i $$0.720144\pi$$
$$660$$ 0 0
$$661$$ 19.2337 0.748104 0.374052 0.927408i $$-0.377968\pi$$
0.374052 + 0.927408i $$0.377968\pi$$
$$662$$ −20.4897 −0.796353
$$663$$ 0 0
$$664$$ −32.0000 −1.24184
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −29.0024 −1.12298
$$668$$ −1.28962 −0.0498969
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −4.97825 −0.192183
$$672$$ 0 0
$$673$$ 19.2549 0.742223 0.371112 0.928588i $$-0.378977\pi$$
0.371112 + 0.928588i $$0.378977\pi$$
$$674$$ 17.4891 0.673656
$$675$$ 0 0
$$676$$ 16.9783 0.653010
$$677$$ 26.5330 1.01975 0.509873 0.860250i $$-0.329692\pi$$
0.509873 + 0.860250i $$0.329692\pi$$
$$678$$ 0 0
$$679$$ 64.4674 2.47403
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −2.17448 −0.0832652
$$683$$ −0.589907 −0.0225722 −0.0112861 0.999936i $$-0.503593\pi$$
−0.0112861 + 0.999936i $$0.503593\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −17.4891 −0.667738
$$687$$ 0 0
$$688$$ −22.0742 −0.841572
$$689$$ 21.9565 0.836476
$$690$$ 0 0
$$691$$ −11.7228 −0.445957 −0.222978 0.974823i $$-0.571578\pi$$
−0.222978 + 0.974823i $$0.571578\pi$$
$$692$$ 34.4010 1.30773
$$693$$ 0 0
$$694$$ −59.7228 −2.26705
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 3.46410 0.131212
$$698$$ 34.3461 1.30002
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 17.2337 0.650907 0.325454 0.945558i $$-0.394483\pi$$
0.325454 + 0.945558i $$0.394483\pi$$
$$702$$ 0 0
$$703$$ 40.6844 1.53444
$$704$$ 3.25544 0.122694
$$705$$ 0 0
$$706$$ 21.4891 0.808754
$$707$$ 36.8155 1.38459
$$708$$ 0 0
$$709$$ 0.649468 0.0243913 0.0121956 0.999926i $$-0.496118\pi$$
0.0121956 + 0.999926i $$0.496118\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 17.9653 0.673279
$$713$$ 3.16915 0.118686
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 96.7011 3.61389
$$717$$ 0 0
$$718$$ 70.9764 2.64882
$$719$$ −28.1168 −1.04858 −0.524291 0.851539i $$-0.675669\pi$$
−0.524291 + 0.851539i $$0.675669\pi$$
$$720$$ 0 0
$$721$$ −24.0000 −0.893807
$$722$$ 24.8935 0.926441
$$723$$ 0 0
$$724$$ 64.9783 2.41490
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −10.7971 −0.400441 −0.200220 0.979751i $$-0.564166\pi$$
−0.200220 + 0.979751i $$0.564166\pi$$
$$728$$ −85.2376 −3.15911
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −8.74456 −0.323429
$$732$$ 0 0
$$733$$ 23.3639 0.862964 0.431482 0.902122i $$-0.357991\pi$$
0.431482 + 0.902122i $$0.357991\pi$$
$$734$$ 58.9783 2.17693
$$735$$ 0 0
$$736$$ 20.7446 0.764655
$$737$$ 11.2772 0.415400
$$738$$ 0 0
$$739$$ 11.3723 0.418336 0.209168 0.977880i $$-0.432924\pi$$
0.209168 + 0.977880i $$0.432924\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −46.7277 −1.71543
$$743$$ −42.5639 −1.56152 −0.780759 0.624833i $$-0.785167\pi$$
−0.780759 + 0.624833i $$0.785167\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −29.4891 −1.07967
$$747$$ 0 0
$$748$$ −15.1460 −0.553794
$$749$$ −45.9565 −1.67921
$$750$$ 0 0
$$751$$ −35.7228 −1.30354 −0.651772 0.758415i $$-0.725974\pi$$
−0.651772 + 0.758415i $$0.725974\pi$$
$$752$$ −54.2458 −1.97814
$$753$$ 0 0
$$754$$ 59.5842 2.16993
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −41.5692 −1.51086 −0.755429 0.655230i $$-0.772572\pi$$
−0.755429 + 0.655230i $$0.772572\pi$$
$$758$$ 54.2458 1.97030
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −9.51087 −0.344769 −0.172384 0.985030i $$-0.555147\pi$$
−0.172384 + 0.985030i $$0.555147\pi$$
$$762$$ 0 0
$$763$$ −6.04334 −0.218784
$$764$$ −58.4674 −2.11528
$$765$$ 0 0
$$766$$ 89.2119 3.22336
$$767$$ 30.2921 1.09378
$$768$$ 0 0
$$769$$ 6.48913 0.234004 0.117002 0.993132i $$-0.462672\pi$$
0.117002 + 0.993132i $$0.462672\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 93.6955 3.37217
$$773$$ 18.2603 0.656776 0.328388 0.944543i $$-0.393495\pi$$
0.328388 + 0.944543i $$0.393495\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 111.446 4.00066
$$777$$ 0 0
$$778$$ 59.2945 2.12581
$$779$$ 7.37228 0.264139
$$780$$ 0 0
$$781$$ −5.64947 −0.202154
$$782$$ 32.1716 1.15045
$$783$$ 0 0
$$784$$ 31.8614 1.13791
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 17.3205 0.617409 0.308705 0.951158i $$-0.400105\pi$$
0.308705 + 0.951158i $$0.400105\pi$$
$$788$$ −100.624 −3.58457
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −32.7446 −1.16426
$$792$$ 0 0
$$793$$ −14.9060 −0.529327
$$794$$ −31.1168 −1.10430
$$795$$ 0 0
$$796$$ 69.9565 2.47954
$$797$$ 48.8473 1.73026 0.865130 0.501548i $$-0.167236\pi$$
0.865130 + 0.501548i $$0.167236\pi$$
$$798$$ 0 0
$$799$$ −21.4891 −0.760231
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −49.5470 −1.74957
$$803$$ −10.3923 −0.366736
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −6.51087 −0.229336
$$807$$ 0 0
$$808$$ 63.6434 2.23897
$$809$$ −21.0000 −0.738321 −0.369160 0.929366i $$-0.620355\pi$$
−0.369160 + 0.929366i $$0.620355\pi$$
$$810$$ 0 0
$$811$$ 42.1168 1.47892 0.739461 0.673199i $$-0.235080\pi$$
0.739461 + 0.673199i $$0.235080\pi$$
$$812$$ −87.0073 −3.05336
$$813$$ 0 0
$$814$$ −26.2337 −0.919490
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −18.6101 −0.651086
$$818$$ −14.7962 −0.517336
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −16.7228 −0.583630 −0.291815 0.956475i $$-0.594259\pi$$
−0.291815 + 0.956475i $$0.594259\pi$$
$$822$$ 0 0
$$823$$ 44.1485 1.53892 0.769459 0.638696i $$-0.220526\pi$$
0.769459 + 0.638696i $$0.220526\pi$$
$$824$$ −41.4891 −1.44534
$$825$$ 0 0
$$826$$ −64.4674 −2.24311
$$827$$ 15.7359 0.547192 0.273596 0.961845i $$-0.411787\pi$$
0.273596 + 0.961845i $$0.411787\pi$$
$$828$$ 0 0
$$829$$ 28.3505 0.984655 0.492327 0.870410i $$-0.336146\pi$$
0.492327 + 0.870410i $$0.336146\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 9.74749 0.337934
$$833$$ 12.6217 0.437316
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −32.2337 −1.11483
$$837$$ 0 0
$$838$$ −13.8564 −0.478662
$$839$$ 17.1386 0.591690 0.295845 0.955236i $$-0.404399\pi$$
0.295845 + 0.955236i $$0.404399\pi$$
$$840$$ 0 0
$$841$$ 4.00000 0.137931
$$842$$ −25.8882 −0.892166
$$843$$ 0 0
$$844$$ 117.446 4.04265
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −31.5817 −1.08516
$$848$$ −34.0511 −1.16932
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 38.2337 1.31063
$$852$$ 0 0
$$853$$ −48.9022 −1.67438 −0.837189 0.546913i $$-0.815803\pi$$
−0.837189 + 0.546913i $$0.815803\pi$$
$$854$$ 31.7228 1.08553
$$855$$ 0 0
$$856$$ −79.4456 −2.71540
$$857$$ −26.8829 −0.918301 −0.459150 0.888359i $$-0.651846\pi$$
−0.459150 + 0.888359i $$0.651846\pi$$
$$858$$ 0 0
$$859$$ −49.3288 −1.68308 −0.841538 0.540198i $$-0.818349\pi$$
−0.841538 + 0.540198i $$0.818349\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −46.3229 −1.57777
$$863$$ 9.80240 0.333677 0.166839 0.985984i $$-0.446644\pi$$
0.166839 + 0.985984i $$0.446644\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −7.11684 −0.241840
$$867$$ 0 0
$$868$$ 9.50744 0.322704
$$869$$ 6.51087 0.220866
$$870$$ 0 0
$$871$$ 33.7663 1.14413
$$872$$ −10.4472 −0.353787
$$873$$ 0 0
$$874$$ 68.4674 2.31594
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −11.0371 −0.372697 −0.186348 0.982484i $$-0.559665\pi$$
−0.186348 + 0.982484i $$0.559665\pi$$
$$878$$ 7.92287 0.267384
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −28.1168 −0.947281 −0.473640 0.880718i $$-0.657060\pi$$
−0.473640 + 0.880718i $$0.657060\pi$$
$$882$$ 0 0
$$883$$ −44.5532 −1.49934 −0.749668 0.661815i $$-0.769787\pi$$
−0.749668 + 0.661815i $$0.769787\pi$$
$$884$$ −45.3505 −1.52530
$$885$$ 0 0
$$886$$ 62.9783 2.11580
$$887$$ 9.21249 0.309325 0.154663 0.987967i $$-0.450571\pi$$
0.154663 + 0.987967i $$0.450571\pi$$
$$888$$ 0 0
$$889$$ −36.0000 −1.20740
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 5.63858 0.188794
$$893$$ −45.7330 −1.53040
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −49.2119 −1.64406
$$897$$ 0 0
$$898$$ 70.9764 2.36851
$$899$$ −3.60597 −0.120266
$$900$$ 0 0
$$901$$ −13.4891 −0.449388
$$902$$ −4.75372 −0.158282
$$903$$ 0 0
$$904$$ −56.6060 −1.88269
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −58.0049 −1.92602 −0.963010 0.269466i $$-0.913153\pi$$
−0.963010 + 0.269466i $$0.913153\pi$$
$$908$$ −1.28962 −0.0427976
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −21.6060 −0.715838 −0.357919 0.933753i $$-0.616514\pi$$
−0.357919 + 0.933753i $$0.616514\pi$$
$$912$$ 0 0
$$913$$ 7.33296 0.242686
$$914$$ −22.3723 −0.740009
$$915$$ 0 0
$$916$$ 98.8397 3.26575
$$917$$ 4.75372 0.156982
$$918$$ 0 0
$$919$$ 3.64947 0.120385 0.0601924 0.998187i $$-0.480829\pi$$
0.0601924 + 0.998187i $$0.480829\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 98.6892 3.25016
$$923$$ −16.9157 −0.556789
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 34.9783 1.14946
$$927$$ 0 0
$$928$$ −23.6039 −0.774836
$$929$$ 26.4891 0.869080 0.434540 0.900653i $$-0.356911\pi$$
0.434540 + 0.900653i $$0.356911\pi$$
$$930$$ 0 0
$$931$$ 26.8614 0.880347
$$932$$ −41.3292 −1.35378
$$933$$ 0 0
$$934$$ 37.4891 1.22668
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 27.4728 0.897496 0.448748 0.893658i $$-0.351870\pi$$
0.448748 + 0.893658i $$0.351870\pi$$
$$938$$ −71.8613 −2.34635
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −21.8614 −0.712661 −0.356331 0.934360i $$-0.615972\pi$$
−0.356331 + 0.934360i $$0.615972\pi$$
$$942$$ 0 0
$$943$$ 6.92820 0.225613
$$944$$ −46.9783 −1.52901
$$945$$ 0 0
$$946$$ 12.0000 0.390154
$$947$$ 36.5205 1.18676 0.593379 0.804923i $$-0.297794\pi$$
0.593379 + 0.804923i $$0.297794\pi$$
$$948$$ 0 0
$$949$$ −31.1168 −1.01010
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 52.3663 1.69720
$$953$$ 21.7244 0.703721 0.351861 0.936052i $$-0.385549\pi$$
0.351861 + 0.936052i $$0.385549\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 100.467 3.24935
$$957$$ 0 0
$$958$$ 54.5408 1.76213
$$959$$ −7.72281 −0.249383
$$960$$ 0 0
$$961$$ −30.6060 −0.987289
$$962$$ −78.5494 −2.53254
$$963$$ 0 0
$$964$$ −107.073 −3.44860
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 36.3354 1.16847 0.584234 0.811585i $$-0.301395\pi$$
0.584234 + 0.811585i $$0.301395\pi$$
$$968$$ −54.5957 −1.75477
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 44.5842 1.43078 0.715388 0.698728i $$-0.246250\pi$$
0.715388 + 0.698728i $$0.246250\pi$$
$$972$$ 0 0
$$973$$ −21.1894 −0.679300
$$974$$ 76.4674 2.45017
$$975$$ 0 0
$$976$$ 23.1168 0.739952
$$977$$ −54.8357 −1.75435 −0.877175 0.480171i $$-0.840575\pi$$
−0.877175 + 0.480171i $$0.840575\pi$$
$$978$$ 0 0
$$979$$ −4.11684 −0.131575
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 94.3403 3.01052
$$983$$ 57.0102 1.81834 0.909171 0.416422i $$-0.136716\pi$$
0.909171 + 0.416422i $$0.136716\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −36.6060 −1.16577
$$987$$ 0 0
$$988$$ −96.5147 −3.07054
$$989$$ −17.4891 −0.556122
$$990$$ 0 0
$$991$$ −1.60597 −0.0510153 −0.0255076 0.999675i $$-0.508120\pi$$
−0.0255076 + 0.999675i $$0.508120\pi$$
$$992$$ 2.57924 0.0818910
$$993$$ 0 0
$$994$$ 36.0000 1.14185
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −10.6324 −0.336730 −0.168365 0.985725i $$-0.553849\pi$$
−0.168365 + 0.985725i $$0.553849\pi$$
$$998$$ −47.0227 −1.48848
$$999$$ 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.a.x.1.4 4
3.2 odd 2 2025.2.a.w.1.1 4
5.2 odd 4 405.2.b.b.244.4 yes 4
5.3 odd 4 405.2.b.b.244.1 yes 4
5.4 even 2 inner 2025.2.a.x.1.1 4
15.2 even 4 405.2.b.a.244.1 4
15.8 even 4 405.2.b.a.244.4 yes 4
15.14 odd 2 2025.2.a.w.1.4 4
45.2 even 12 405.2.j.b.109.1 4
45.7 odd 12 405.2.j.d.109.2 4
45.13 odd 12 405.2.j.d.379.2 4
45.22 odd 12 405.2.j.a.379.1 4
45.23 even 12 405.2.j.b.379.1 4
45.32 even 12 405.2.j.e.379.2 4
45.38 even 12 405.2.j.e.109.2 4
45.43 odd 12 405.2.j.a.109.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
405.2.b.a.244.1 4 15.2 even 4
405.2.b.a.244.4 yes 4 15.8 even 4
405.2.b.b.244.1 yes 4 5.3 odd 4
405.2.b.b.244.4 yes 4 5.2 odd 4
405.2.j.a.109.1 4 45.43 odd 12
405.2.j.a.379.1 4 45.22 odd 12
405.2.j.b.109.1 4 45.2 even 12
405.2.j.b.379.1 4 45.23 even 12
405.2.j.d.109.2 4 45.7 odd 12
405.2.j.d.379.2 4 45.13 odd 12
405.2.j.e.109.2 4 45.38 even 12
405.2.j.e.379.2 4 45.32 even 12
2025.2.a.w.1.1 4 3.2 odd 2
2025.2.a.w.1.4 4 15.14 odd 2
2025.2.a.x.1.1 4 5.4 even 2 inner
2025.2.a.x.1.4 4 1.1 even 1 trivial