# Properties

 Label 2025.2.a.w.1.2 Level $2025$ Weight $2$ Character 2025.1 Self dual yes Analytic conductor $16.170$ Analytic rank $0$ Dimension $4$ CM no 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 1.2 Root $$-0.792287$$ 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)

 $$f(q)$$ $$=$$ $$q-0.792287 q^{2} -1.37228 q^{4} -3.46410 q^{7} +2.67181 q^{8} +O(q^{10})$$ $$q-0.792287 q^{2} -1.37228 q^{4} -3.46410 q^{7} +2.67181 q^{8} -4.37228 q^{11} -5.84096 q^{13} +2.74456 q^{14} +0.627719 q^{16} -0.792287 q^{17} -0.372281 q^{19} +3.46410 q^{22} -1.58457 q^{23} +4.62772 q^{26} +4.75372 q^{28} -5.74456 q^{29} +6.37228 q^{31} -5.84096 q^{32} +0.627719 q^{34} +2.37686 q^{37} +0.294954 q^{38} +4.37228 q^{41} +3.46410 q^{43} +6.00000 q^{44} +1.25544 q^{46} -1.87953 q^{47} +5.00000 q^{49} +8.01544 q^{52} -11.9769 q^{53} -9.25544 q^{56} +4.55134 q^{58} +1.62772 q^{59} +9.37228 q^{61} -5.04868 q^{62} +3.37228 q^{64} -11.6819 q^{67} +1.08724 q^{68} +13.1168 q^{71} +2.37686 q^{73} -1.88316 q^{74} +0.510875 q^{76} +15.1460 q^{77} +6.74456 q^{79} -3.46410 q^{82} -11.9769 q^{83} -2.74456 q^{86} -11.6819 q^{88} -3.00000 q^{89} +20.2337 q^{91} +2.17448 q^{92} +1.48913 q^{94} +1.28962 q^{97} -3.96143 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 6 q^{4}+O(q^{10})$$ 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.792287 −0.560232 −0.280116 0.959966i $$-0.590373\pi$$
−0.280116 + 0.959966i $$0.590373\pi$$
$$3$$ 0 0
$$4$$ −1.37228 −0.686141
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −3.46410 −1.30931 −0.654654 0.755929i $$-0.727186\pi$$
−0.654654 + 0.755929i $$0.727186\pi$$
$$8$$ 2.67181 0.944629
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −4.37228 −1.31829 −0.659146 0.752015i $$-0.729082\pi$$
−0.659146 + 0.752015i $$0.729082\pi$$
$$12$$ 0 0
$$13$$ −5.84096 −1.61999 −0.809996 0.586436i $$-0.800531\pi$$
−0.809996 + 0.586436i $$0.800531\pi$$
$$14$$ 2.74456 0.733515
$$15$$ 0 0
$$16$$ 0.627719 0.156930
$$17$$ −0.792287 −0.192158 −0.0960789 0.995374i $$-0.530630\pi$$
−0.0960789 + 0.995374i $$0.530630\pi$$
$$18$$ 0 0
$$19$$ −0.372281 −0.0854072 −0.0427036 0.999088i $$-0.513597\pi$$
−0.0427036 + 0.999088i $$0.513597\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 3.46410 0.738549
$$23$$ −1.58457 −0.330407 −0.165203 0.986260i $$-0.552828\pi$$
−0.165203 + 0.986260i $$0.552828\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 4.62772 0.907570
$$27$$ 0 0
$$28$$ 4.75372 0.898369
$$29$$ −5.74456 −1.06674 −0.533369 0.845883i $$-0.679074\pi$$
−0.533369 + 0.845883i $$0.679074\pi$$
$$30$$ 0 0
$$31$$ 6.37228 1.14450 0.572248 0.820081i $$-0.306072\pi$$
0.572248 + 0.820081i $$0.306072\pi$$
$$32$$ −5.84096 −1.03255
$$33$$ 0 0
$$34$$ 0.627719 0.107653
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 2.37686 0.390754 0.195377 0.980728i $$-0.437407\pi$$
0.195377 + 0.980728i $$0.437407\pi$$
$$38$$ 0.294954 0.0478478
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 4.37228 0.682836 0.341418 0.939912i $$-0.389093\pi$$
0.341418 + 0.939912i $$0.389093\pi$$
$$42$$ 0 0
$$43$$ 3.46410 0.528271 0.264135 0.964486i $$-0.414913\pi$$
0.264135 + 0.964486i $$0.414913\pi$$
$$44$$ 6.00000 0.904534
$$45$$ 0 0
$$46$$ 1.25544 0.185104
$$47$$ −1.87953 −0.274157 −0.137079 0.990560i $$-0.543771\pi$$
−0.137079 + 0.990560i $$0.543771\pi$$
$$48$$ 0 0
$$49$$ 5.00000 0.714286
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 8.01544 1.11154
$$53$$ −11.9769 −1.64515 −0.822575 0.568656i $$-0.807464\pi$$
−0.822575 + 0.568656i $$0.807464\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −9.25544 −1.23681
$$57$$ 0 0
$$58$$ 4.55134 0.597621
$$59$$ 1.62772 0.211911 0.105955 0.994371i $$-0.466210\pi$$
0.105955 + 0.994371i $$0.466210\pi$$
$$60$$ 0 0
$$61$$ 9.37228 1.20000 0.599999 0.800001i $$-0.295168\pi$$
0.599999 + 0.800001i $$0.295168\pi$$
$$62$$ −5.04868 −0.641182
$$63$$ 0 0
$$64$$ 3.37228 0.421535
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −11.6819 −1.42717 −0.713587 0.700566i $$-0.752931\pi$$
−0.713587 + 0.700566i $$0.752931\pi$$
$$68$$ 1.08724 0.131847
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 13.1168 1.55668 0.778341 0.627841i $$-0.216061\pi$$
0.778341 + 0.627841i $$0.216061\pi$$
$$72$$ 0 0
$$73$$ 2.37686 0.278191 0.139095 0.990279i $$-0.455581\pi$$
0.139095 + 0.990279i $$0.455581\pi$$
$$74$$ −1.88316 −0.218912
$$75$$ 0 0
$$76$$ 0.510875 0.0586013
$$77$$ 15.1460 1.72605
$$78$$ 0 0
$$79$$ 6.74456 0.758823 0.379411 0.925228i $$-0.376127\pi$$
0.379411 + 0.925228i $$0.376127\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −3.46410 −0.382546
$$83$$ −11.9769 −1.31463 −0.657317 0.753615i $$-0.728309\pi$$
−0.657317 + 0.753615i $$0.728309\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −2.74456 −0.295954
$$87$$ 0 0
$$88$$ −11.6819 −1.24530
$$89$$ −3.00000 −0.317999 −0.159000 0.987279i $$-0.550827\pi$$
−0.159000 + 0.987279i $$0.550827\pi$$
$$90$$ 0 0
$$91$$ 20.2337 2.12107
$$92$$ 2.17448 0.226705
$$93$$ 0 0
$$94$$ 1.48913 0.153592
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 1.28962 0.130941 0.0654706 0.997855i $$-0.479145\pi$$
0.0654706 + 0.997855i $$0.479145\pi$$
$$98$$ −3.96143 −0.400165
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −16.3723 −1.62910 −0.814551 0.580091i $$-0.803017\pi$$
−0.814551 + 0.580091i $$0.803017\pi$$
$$102$$ 0 0
$$103$$ 6.92820 0.682656 0.341328 0.939944i $$-0.389123\pi$$
0.341328 + 0.939944i $$0.389123\pi$$
$$104$$ −15.6060 −1.53029
$$105$$ 0 0
$$106$$ 9.48913 0.921665
$$107$$ 13.2665 1.28252 0.641260 0.767323i $$-0.278412\pi$$
0.641260 + 0.767323i $$0.278412\pi$$
$$108$$ 0 0
$$109$$ 9.74456 0.933360 0.466680 0.884426i $$-0.345450\pi$$
0.466680 + 0.884426i $$0.345450\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −2.17448 −0.205469
$$113$$ −6.13592 −0.577218 −0.288609 0.957447i $$-0.593193\pi$$
−0.288609 + 0.957447i $$0.593193\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 7.88316 0.731933
$$117$$ 0 0
$$118$$ −1.28962 −0.118719
$$119$$ 2.74456 0.251594
$$120$$ 0 0
$$121$$ 8.11684 0.737895
$$122$$ −7.42554 −0.672276
$$123$$ 0 0
$$124$$ −8.74456 −0.785285
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 10.3923 0.922168 0.461084 0.887357i $$-0.347461\pi$$
0.461084 + 0.887357i $$0.347461\pi$$
$$128$$ 9.01011 0.796389
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 4.37228 0.382008 0.191004 0.981589i $$-0.438826\pi$$
0.191004 + 0.981589i $$0.438826\pi$$
$$132$$ 0 0
$$133$$ 1.28962 0.111824
$$134$$ 9.25544 0.799548
$$135$$ 0 0
$$136$$ −2.11684 −0.181518
$$137$$ 14.3537 1.22632 0.613161 0.789958i $$-0.289898\pi$$
0.613161 + 0.789958i $$0.289898\pi$$
$$138$$ 0 0
$$139$$ 11.1168 0.942918 0.471459 0.881888i $$-0.343727\pi$$
0.471459 + 0.881888i $$0.343727\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −10.3923 −0.872103
$$143$$ 25.5383 2.13562
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −1.88316 −0.155851
$$147$$ 0 0
$$148$$ −3.26172 −0.268112
$$149$$ 10.6277 0.870657 0.435328 0.900272i $$-0.356632\pi$$
0.435328 + 0.900272i $$0.356632\pi$$
$$150$$ 0 0
$$151$$ 0.883156 0.0718702 0.0359351 0.999354i $$-0.488559\pi$$
0.0359351 + 0.999354i $$0.488559\pi$$
$$152$$ −0.994667 −0.0806781
$$153$$ 0 0
$$154$$ −12.0000 −0.966988
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 11.4795 0.916167 0.458084 0.888909i $$-0.348536\pi$$
0.458084 + 0.888909i $$0.348536\pi$$
$$158$$ −5.34363 −0.425116
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 5.48913 0.432604
$$162$$ 0 0
$$163$$ −15.1460 −1.18633 −0.593164 0.805082i $$-0.702122\pi$$
−0.593164 + 0.805082i $$0.702122\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ 9.48913 0.736499
$$167$$ −13.5615 −1.04942 −0.524708 0.851282i $$-0.675826\pi$$
−0.524708 + 0.851282i $$0.675826\pi$$
$$168$$ 0 0
$$169$$ 21.1168 1.62437
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −4.75372 −0.362468
$$173$$ 11.1846 0.850349 0.425174 0.905111i $$-0.360213\pi$$
0.425174 + 0.905111i $$0.360213\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −2.74456 −0.206879
$$177$$ 0 0
$$178$$ 2.37686 0.178153
$$179$$ −4.88316 −0.364984 −0.182492 0.983207i $$-0.558416\pi$$
−0.182492 + 0.983207i $$0.558416\pi$$
$$180$$ 0 0
$$181$$ −13.8614 −1.03031 −0.515155 0.857097i $$-0.672266\pi$$
−0.515155 + 0.857097i $$0.672266\pi$$
$$182$$ −16.0309 −1.18829
$$183$$ 0 0
$$184$$ −4.23369 −0.312112
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 3.46410 0.253320
$$188$$ 2.57924 0.188110
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 7.62772 0.551922 0.275961 0.961169i $$-0.411004\pi$$
0.275961 + 0.961169i $$0.411004\pi$$
$$192$$ 0 0
$$193$$ −11.4795 −0.826316 −0.413158 0.910659i $$-0.635574\pi$$
−0.413158 + 0.910659i $$0.635574\pi$$
$$194$$ −1.02175 −0.0733573
$$195$$ 0 0
$$196$$ −6.86141 −0.490100
$$197$$ −6.43087 −0.458181 −0.229090 0.973405i $$-0.573575\pi$$
−0.229090 + 0.973405i $$0.573575\pi$$
$$198$$ 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
$$202$$ 12.9715 0.912675
$$203$$ 19.8997 1.39669
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −5.48913 −0.382445
$$207$$ 0 0
$$208$$ −3.66648 −0.254225
$$209$$ 1.62772 0.112592
$$210$$ 0 0
$$211$$ −1.86141 −0.128145 −0.0640723 0.997945i $$-0.520409\pi$$
−0.0640723 + 0.997945i $$0.520409\pi$$
$$212$$ 16.4356 1.12880
$$213$$ 0 0
$$214$$ −10.5109 −0.718509
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −22.0742 −1.49850
$$218$$ −7.72049 −0.522898
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 4.62772 0.311294
$$222$$ 0 0
$$223$$ 18.6101 1.24623 0.623113 0.782132i $$-0.285868\pi$$
0.623113 + 0.782132i $$0.285868\pi$$
$$224$$ 20.2337 1.35192
$$225$$ 0 0
$$226$$ 4.86141 0.323376
$$227$$ −13.5615 −0.900105 −0.450053 0.893002i $$-0.648595\pi$$
−0.450053 + 0.893002i $$0.648595\pi$$
$$228$$ 0 0
$$229$$ −17.6060 −1.16344 −0.581718 0.813391i $$-0.697619\pi$$
−0.581718 + 0.813391i $$0.697619\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −15.3484 −1.00767
$$233$$ −6.13592 −0.401977 −0.200989 0.979594i $$-0.564415\pi$$
−0.200989 + 0.979594i $$0.564415\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −2.23369 −0.145401
$$237$$ 0 0
$$238$$ −2.17448 −0.140951
$$239$$ 22.9783 1.48634 0.743170 0.669103i $$-0.233321\pi$$
0.743170 + 0.669103i $$0.233321\pi$$
$$240$$ 0 0
$$241$$ −1.51087 −0.0973240 −0.0486620 0.998815i $$-0.515496\pi$$
−0.0486620 + 0.998815i $$0.515496\pi$$
$$242$$ −6.43087 −0.413392
$$243$$ 0 0
$$244$$ −12.8614 −0.823367
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 2.17448 0.138359
$$248$$ 17.0256 1.08112
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −20.2337 −1.27714 −0.638570 0.769564i $$-0.720474\pi$$
−0.638570 + 0.769564i $$0.720474\pi$$
$$252$$ 0 0
$$253$$ 6.92820 0.435572
$$254$$ −8.23369 −0.516628
$$255$$ 0 0
$$256$$ −13.8832 −0.867697
$$257$$ −6.43087 −0.401147 −0.200573 0.979679i $$-0.564280\pi$$
−0.200573 + 0.979679i $$0.564280\pi$$
$$258$$ 0 0
$$259$$ −8.23369 −0.511616
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −3.46410 −0.214013
$$263$$ 26.5330 1.63609 0.818047 0.575151i $$-0.195057\pi$$
0.818047 + 0.575151i $$0.195057\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −1.02175 −0.0626475
$$267$$ 0 0
$$268$$ 16.0309 0.979242
$$269$$ −29.2337 −1.78241 −0.891205 0.453601i $$-0.850139\pi$$
−0.891205 + 0.453601i $$0.850139\pi$$
$$270$$ 0 0
$$271$$ 5.25544 0.319245 0.159623 0.987178i $$-0.448972\pi$$
0.159623 + 0.987178i $$0.448972\pi$$
$$272$$ −0.497333 −0.0301553
$$273$$ 0 0
$$274$$ −11.3723 −0.687025
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 22.0742 1.32631 0.663156 0.748481i $$-0.269217\pi$$
0.663156 + 0.748481i $$0.269217\pi$$
$$278$$ −8.80773 −0.528253
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −1.37228 −0.0818634 −0.0409317 0.999162i $$-0.513033\pi$$
−0.0409317 + 0.999162i $$0.513033\pi$$
$$282$$ 0 0
$$283$$ 27.7128 1.64736 0.823678 0.567058i $$-0.191918\pi$$
0.823678 + 0.567058i $$0.191918\pi$$
$$284$$ −18.0000 −1.06810
$$285$$ 0 0
$$286$$ −20.2337 −1.19644
$$287$$ −15.1460 −0.894042
$$288$$ 0 0
$$289$$ −16.3723 −0.963075
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −3.26172 −0.190878
$$293$$ 0.792287 0.0462859 0.0231430 0.999732i $$-0.492633\pi$$
0.0231430 + 0.999732i $$0.492633\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 6.35053 0.369117
$$297$$ 0 0
$$298$$ −8.42020 −0.487769
$$299$$ 9.25544 0.535256
$$300$$ 0 0
$$301$$ −12.0000 −0.691669
$$302$$ −0.699713 −0.0402640
$$303$$ 0 0
$$304$$ −0.233688 −0.0134029
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 15.1460 0.864429 0.432215 0.901771i $$-0.357732\pi$$
0.432215 + 0.901771i $$0.357732\pi$$
$$308$$ −20.7846 −1.18431
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −15.8614 −0.899418 −0.449709 0.893175i $$-0.648472\pi$$
−0.449709 + 0.893175i $$0.648472\pi$$
$$312$$ 0 0
$$313$$ −24.4511 −1.38206 −0.691029 0.722827i $$-0.742842\pi$$
−0.691029 + 0.722827i $$0.742842\pi$$
$$314$$ −9.09509 −0.513266
$$315$$ 0 0
$$316$$ −9.25544 −0.520659
$$317$$ 29.4998 1.65687 0.828436 0.560084i $$-0.189231\pi$$
0.828436 + 0.560084i $$0.189231\pi$$
$$318$$ 0 0
$$319$$ 25.1168 1.40627
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −4.34896 −0.242358
$$323$$ 0.294954 0.0164117
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 12.0000 0.664619
$$327$$ 0 0
$$328$$ 11.6819 0.645026
$$329$$ 6.51087 0.358956
$$330$$ 0 0
$$331$$ 9.11684 0.501107 0.250554 0.968103i $$-0.419387\pi$$
0.250554 + 0.968103i $$0.419387\pi$$
$$332$$ 16.4356 0.902023
$$333$$ 0 0
$$334$$ 10.7446 0.587916
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −6.92820 −0.377403 −0.188702 0.982034i $$-0.560428\pi$$
−0.188702 + 0.982034i $$0.560428\pi$$
$$338$$ −16.7306 −0.910025
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −27.8614 −1.50878
$$342$$ 0 0
$$343$$ 6.92820 0.374088
$$344$$ 9.25544 0.499020
$$345$$ 0 0
$$346$$ −8.86141 −0.476392
$$347$$ 2.87419 0.154295 0.0771474 0.997020i $$-0.475419\pi$$
0.0771474 + 0.997020i $$0.475419\pi$$
$$348$$ 0 0
$$349$$ −26.6060 −1.42418 −0.712092 0.702086i $$-0.752252\pi$$
−0.712092 + 0.702086i $$0.752252\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 25.5383 1.36120
$$353$$ 1.87953 0.100037 0.0500186 0.998748i $$-0.484072\pi$$
0.0500186 + 0.998748i $$0.484072\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 4.11684 0.218192
$$357$$ 0 0
$$358$$ 3.86886 0.204476
$$359$$ −10.8832 −0.574391 −0.287196 0.957872i $$-0.592723\pi$$
−0.287196 + 0.957872i $$0.592723\pi$$
$$360$$ 0 0
$$361$$ −18.8614 −0.992706
$$362$$ 10.9822 0.577212
$$363$$ 0 0
$$364$$ −27.7663 −1.45535
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 16.4356 0.857934 0.428967 0.903320i $$-0.358878\pi$$
0.428967 + 0.903320i $$0.358878\pi$$
$$368$$ −0.994667 −0.0518506
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 41.4891 2.15401
$$372$$ 0 0
$$373$$ −8.21782 −0.425503 −0.212751 0.977106i $$-0.568242\pi$$
−0.212751 + 0.977106i $$0.568242\pi$$
$$374$$ −2.74456 −0.141918
$$375$$ 0 0
$$376$$ −5.02175 −0.258977
$$377$$ 33.5538 1.72811
$$378$$ 0 0
$$379$$ −1.48913 −0.0764912 −0.0382456 0.999268i $$-0.512177\pi$$
−0.0382456 + 0.999268i $$0.512177\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −6.04334 −0.309204
$$383$$ −11.0920 −0.566776 −0.283388 0.959005i $$-0.591458\pi$$
−0.283388 + 0.959005i $$0.591458\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 9.09509 0.462928
$$387$$ 0 0
$$388$$ −1.76972 −0.0898440
$$389$$ −0.510875 −0.0259024 −0.0129512 0.999916i $$-0.504123\pi$$
−0.0129512 + 0.999916i $$0.504123\pi$$
$$390$$ 0 0
$$391$$ 1.25544 0.0634902
$$392$$ 13.3591 0.674735
$$393$$ 0 0
$$394$$ 5.09509 0.256687
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −17.5229 −0.879449 −0.439724 0.898133i $$-0.644924\pi$$
−0.439724 + 0.898133i $$0.644924\pi$$
$$398$$ −12.6766 −0.635420
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 25.3723 1.26703 0.633516 0.773730i $$-0.281611\pi$$
0.633516 + 0.773730i $$0.281611\pi$$
$$402$$ 0 0
$$403$$ −37.2203 −1.85407
$$404$$ 22.4674 1.11779
$$405$$ 0 0
$$406$$ −15.7663 −0.782469
$$407$$ −10.3923 −0.515127
$$408$$ 0 0
$$409$$ 22.8614 1.13042 0.565212 0.824946i $$-0.308794\pi$$
0.565212 + 0.824946i $$0.308794\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −9.50744 −0.468398
$$413$$ −5.63858 −0.277457
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 34.1168 1.67272
$$417$$ 0 0
$$418$$ −1.28962 −0.0630774
$$419$$ −17.4891 −0.854400 −0.427200 0.904157i $$-0.640500\pi$$
−0.427200 + 0.904157i $$0.640500\pi$$
$$420$$ 0 0
$$421$$ −21.7446 −1.05977 −0.529883 0.848071i $$-0.677764\pi$$
−0.529883 + 0.848071i $$0.677764\pi$$
$$422$$ 1.47477 0.0717906
$$423$$ 0 0
$$424$$ −32.0000 −1.55406
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −32.4665 −1.57117
$$428$$ −18.2054 −0.879990
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −33.3505 −1.60644 −0.803219 0.595683i $$-0.796881\pi$$
−0.803219 + 0.595683i $$0.796881\pi$$
$$432$$ 0 0
$$433$$ 12.7692 0.613647 0.306823 0.951766i $$-0.400734\pi$$
0.306823 + 0.951766i $$0.400734\pi$$
$$434$$ 17.4891 0.839505
$$435$$ 0 0
$$436$$ −13.3723 −0.640416
$$437$$ 0.589907 0.0282191
$$438$$ 0 0
$$439$$ 31.8614 1.52066 0.760331 0.649536i $$-0.225037\pi$$
0.760331 + 0.649536i $$0.225037\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −3.66648 −0.174397
$$443$$ −21.4843 −1.02075 −0.510375 0.859952i $$-0.670494\pi$$
−0.510375 + 0.859952i $$0.670494\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −14.7446 −0.698175
$$447$$ 0 0
$$448$$ −11.6819 −0.551919
$$449$$ −10.8832 −0.513608 −0.256804 0.966464i $$-0.582669\pi$$
−0.256804 + 0.966464i $$0.582669\pi$$
$$450$$ 0 0
$$451$$ −19.1168 −0.900177
$$452$$ 8.42020 0.396053
$$453$$ 0 0
$$454$$ 10.7446 0.504267
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −20.9870 −0.981730 −0.490865 0.871236i $$-0.663319\pi$$
−0.490865 + 0.871236i $$0.663319\pi$$
$$458$$ 13.9490 0.651793
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 24.0951 1.12222 0.561110 0.827741i $$-0.310374\pi$$
0.561110 + 0.827741i $$0.310374\pi$$
$$462$$ 0 0
$$463$$ −13.8564 −0.643962 −0.321981 0.946746i $$-0.604349\pi$$
−0.321981 + 0.946746i $$0.604349\pi$$
$$464$$ −3.60597 −0.167403
$$465$$ 0 0
$$466$$ 4.86141 0.225200
$$467$$ −18.3152 −0.847525 −0.423763 0.905773i $$-0.639291\pi$$
−0.423763 + 0.905773i $$0.639291\pi$$
$$468$$ 0 0
$$469$$ 40.4674 1.86861
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 4.34896 0.200177
$$473$$ −15.1460 −0.696415
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −3.76631 −0.172629
$$477$$ 0 0
$$478$$ −18.2054 −0.832694
$$479$$ 18.6060 0.850128 0.425064 0.905163i $$-0.360252\pi$$
0.425064 + 0.905163i $$0.360252\pi$$
$$480$$ 0 0
$$481$$ −13.8832 −0.633017
$$482$$ 1.19705 0.0545240
$$483$$ 0 0
$$484$$ −11.1386 −0.506300
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 9.50744 0.430823 0.215412 0.976523i $$-0.430891\pi$$
0.215412 + 0.976523i $$0.430891\pi$$
$$488$$ 25.0410 1.13355
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −31.6277 −1.42734 −0.713669 0.700483i $$-0.752968\pi$$
−0.713669 + 0.700483i $$0.752968\pi$$
$$492$$ 0 0
$$493$$ 4.55134 0.204982
$$494$$ −1.72281 −0.0775130
$$495$$ 0 0
$$496$$ 4.00000 0.179605
$$497$$ −45.4381 −2.03818
$$498$$ 0 0
$$499$$ −24.3723 −1.09105 −0.545527 0.838094i $$-0.683670\pi$$
−0.545527 + 0.838094i $$0.683670\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 16.0309 0.715494
$$503$$ 3.16915 0.141305 0.0706527 0.997501i $$-0.477492\pi$$
0.0706527 + 0.997501i $$0.477492\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −5.48913 −0.244021
$$507$$ 0 0
$$508$$ −14.2612 −0.632737
$$509$$ 0.510875 0.0226441 0.0113221 0.999936i $$-0.496396\pi$$
0.0113221 + 0.999936i $$0.496396\pi$$
$$510$$ 0 0
$$511$$ −8.23369 −0.364237
$$512$$ −7.02078 −0.310277
$$513$$ 0 0
$$514$$ 5.09509 0.224735
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 8.21782 0.361419
$$518$$ 6.52344 0.286624
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −18.0000 −0.788594 −0.394297 0.918983i $$-0.629012\pi$$
−0.394297 + 0.918983i $$0.629012\pi$$
$$522$$ 0 0
$$523$$ 4.75372 0.207866 0.103933 0.994584i $$-0.466857\pi$$
0.103933 + 0.994584i $$0.466857\pi$$
$$524$$ −6.00000 −0.262111
$$525$$ 0 0
$$526$$ −21.0217 −0.916592
$$527$$ −5.04868 −0.219924
$$528$$ 0 0
$$529$$ −20.4891 −0.890832
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −1.76972 −0.0767272
$$533$$ −25.5383 −1.10619
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −31.2119 −1.34815
$$537$$ 0 0
$$538$$ 23.1615 0.998562
$$539$$ −21.8614 −0.941637
$$540$$ 0 0
$$541$$ 19.2337 0.826921 0.413460 0.910522i $$-0.364320\pi$$
0.413460 + 0.910522i $$0.364320\pi$$
$$542$$ −4.16381 −0.178851
$$543$$ 0 0
$$544$$ 4.62772 0.198412
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 6.92820 0.296229 0.148114 0.988970i $$-0.452680\pi$$
0.148114 + 0.988970i $$0.452680\pi$$
$$548$$ −19.6974 −0.841430
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 2.13859 0.0911071
$$552$$ 0 0
$$553$$ −23.3639 −0.993532
$$554$$ −17.4891 −0.743042
$$555$$ 0 0
$$556$$ −15.2554 −0.646975
$$557$$ −25.0410 −1.06102 −0.530511 0.847678i $$-0.678000\pi$$
−0.530511 + 0.847678i $$0.678000\pi$$
$$558$$ 0 0
$$559$$ −20.2337 −0.855794
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 1.08724 0.0458625
$$563$$ 32.1716 1.35587 0.677935 0.735122i $$-0.262875\pi$$
0.677935 + 0.735122i $$0.262875\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −21.9565 −0.922901
$$567$$ 0 0
$$568$$ 35.0458 1.47049
$$569$$ 20.4891 0.858949 0.429474 0.903079i $$-0.358699\pi$$
0.429474 + 0.903079i $$0.358699\pi$$
$$570$$ 0 0
$$571$$ 4.13859 0.173195 0.0865974 0.996243i $$-0.472401\pi$$
0.0865974 + 0.996243i $$0.472401\pi$$
$$572$$ −35.0458 −1.46534
$$573$$ 0 0
$$574$$ 12.0000 0.500870
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −18.4077 −0.766325 −0.383162 0.923681i $$-0.625165\pi$$
−0.383162 + 0.923681i $$0.625165\pi$$
$$578$$ 12.9715 0.539545
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 41.4891 1.72126
$$582$$ 0 0
$$583$$ 52.3663 2.16879
$$584$$ 6.35053 0.262787
$$585$$ 0 0
$$586$$ −0.627719 −0.0259308
$$587$$ 16.7306 0.690546 0.345273 0.938502i $$-0.387786\pi$$
0.345273 + 0.938502i $$0.387786\pi$$
$$588$$ 0 0
$$589$$ −2.37228 −0.0977481
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 1.49200 0.0613208
$$593$$ 1.67715 0.0688722 0.0344361 0.999407i $$-0.489036\pi$$
0.0344361 + 0.999407i $$0.489036\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −14.5842 −0.597393
$$597$$ 0 0
$$598$$ −7.33296 −0.299867
$$599$$ 1.62772 0.0665068 0.0332534 0.999447i $$-0.489413\pi$$
0.0332534 + 0.999447i $$0.489413\pi$$
$$600$$ 0 0
$$601$$ −17.9783 −0.733348 −0.366674 0.930349i $$-0.619504\pi$$
−0.366674 + 0.930349i $$0.619504\pi$$
$$602$$ 9.50744 0.387494
$$603$$ 0 0
$$604$$ −1.21194 −0.0493131
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 43.2636 1.75602 0.878008 0.478647i $$-0.158872\pi$$
0.878008 + 0.478647i $$0.158872\pi$$
$$608$$ 2.17448 0.0881869
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 10.9783 0.444132
$$612$$ 0 0
$$613$$ 30.2921 1.22348 0.611742 0.791057i $$-0.290469\pi$$
0.611742 + 0.791057i $$0.290469\pi$$
$$614$$ −12.0000 −0.484281
$$615$$ 0 0
$$616$$ 40.4674 1.63048
$$617$$ −20.6920 −0.833030 −0.416515 0.909129i $$-0.636749\pi$$
−0.416515 + 0.909129i $$0.636749\pi$$
$$618$$ 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
$$622$$ 12.5668 0.503882
$$623$$ 10.3923 0.416359
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 19.3723 0.774272
$$627$$ 0 0
$$628$$ −15.7532 −0.628620
$$629$$ −1.88316 −0.0750863
$$630$$ 0 0
$$631$$ 6.37228 0.253677 0.126838 0.991923i $$-0.459517\pi$$
0.126838 + 0.991923i $$0.459517\pi$$
$$632$$ 18.0202 0.716806
$$633$$ 0 0
$$634$$ −23.3723 −0.928232
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −29.2048 −1.15714
$$638$$ −19.8997 −0.787839
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 7.97825 0.315122 0.157561 0.987509i $$-0.449637\pi$$
0.157561 + 0.987509i $$0.449637\pi$$
$$642$$ 0 0
$$643$$ 1.28962 0.0508577 0.0254288 0.999677i $$-0.491905\pi$$
0.0254288 + 0.999677i $$0.491905\pi$$
$$644$$ −7.53262 −0.296827
$$645$$ 0 0
$$646$$ −0.233688 −0.00919433
$$647$$ −28.7075 −1.12861 −0.564304 0.825567i $$-0.690855\pi$$
−0.564304 + 0.825567i $$0.690855\pi$$
$$648$$ 0 0
$$649$$ −7.11684 −0.279361
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 20.7846 0.813988
$$653$$ −6.33830 −0.248037 −0.124018 0.992280i $$-0.539578\pi$$
−0.124018 + 0.992280i $$0.539578\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 2.74456 0.107157
$$657$$ 0 0
$$658$$ −5.15848 −0.201099
$$659$$ 21.2554 0.827994 0.413997 0.910278i $$-0.364132\pi$$
0.413997 + 0.910278i $$0.364132\pi$$
$$660$$ 0 0
$$661$$ −15.2337 −0.592522 −0.296261 0.955107i $$-0.595740\pi$$
−0.296261 + 0.955107i $$0.595740\pi$$
$$662$$ −7.22316 −0.280736
$$663$$ 0 0
$$664$$ −32.0000 −1.24184
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 9.10268 0.352457
$$668$$ 18.6101 0.720047
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −40.9783 −1.58195
$$672$$ 0 0
$$673$$ 10.5947 0.408395 0.204198 0.978930i $$-0.434542\pi$$
0.204198 + 0.978930i $$0.434542\pi$$
$$674$$ 5.48913 0.211433
$$675$$ 0 0
$$676$$ −28.9783 −1.11455
$$677$$ −26.5330 −1.01975 −0.509873 0.860250i $$-0.670308\pi$$
−0.509873 + 0.860250i $$0.670308\pi$$
$$678$$ 0 0
$$679$$ −4.46738 −0.171442
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 22.0742 0.845266
$$683$$ −27.1229 −1.03783 −0.518915 0.854826i $$-0.673664\pi$$
−0.518915 + 0.854826i $$0.673664\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −5.48913 −0.209576
$$687$$ 0 0
$$688$$ 2.17448 0.0829013
$$689$$ 69.9565 2.66513
$$690$$ 0 0
$$691$$ 45.7228 1.73938 0.869689 0.493600i $$-0.164319\pi$$
0.869689 + 0.493600i $$0.164319\pi$$
$$692$$ −15.3484 −0.583459
$$693$$ 0 0
$$694$$ −2.27719 −0.0864408
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −3.46410 −0.131212
$$698$$ 21.0796 0.797873
$$699$$ 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
$$703$$ −0.884861 −0.0333732
$$704$$ −14.7446 −0.555707
$$705$$ 0 0
$$706$$ −1.48913 −0.0560440
$$707$$ 56.7152 2.13300
$$708$$ 0 0
$$709$$ 52.3505 1.96607 0.983033 0.183430i $$-0.0587201\pi$$
0.983033 + 0.183430i $$0.0587201\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −8.01544 −0.300391
$$713$$ −10.0974 −0.378149
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 6.70106 0.250431
$$717$$ 0 0
$$718$$ 8.62258 0.321792
$$719$$ 10.8832 0.405873 0.202937 0.979192i $$-0.434951\pi$$
0.202937 + 0.979192i $$0.434951\pi$$
$$720$$ 0 0
$$721$$ −24.0000 −0.893807
$$722$$ 14.9436 0.556145
$$723$$ 0 0
$$724$$ 19.0217 0.706938
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −48.9022 −1.81368 −0.906841 0.421473i $$-0.861513\pi$$
−0.906841 + 0.421473i $$0.861513\pi$$
$$728$$ 54.0607 2.00362
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −2.74456 −0.101511
$$732$$ 0 0
$$733$$ 16.4356 0.607064 0.303532 0.952821i $$-0.401834\pi$$
0.303532 + 0.952821i $$0.401834\pi$$
$$734$$ −13.0217 −0.480642
$$735$$ 0 0
$$736$$ 9.25544 0.341160
$$737$$ 51.0767 1.88143
$$738$$ 0 0
$$739$$ 5.62772 0.207019 0.103509 0.994628i $$-0.466993\pi$$
0.103509 + 0.994628i $$0.466993\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −32.8713 −1.20674
$$743$$ −9.39764 −0.344766 −0.172383 0.985030i $$-0.555147\pi$$
−0.172383 + 0.985030i $$0.555147\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 6.51087 0.238380
$$747$$ 0 0
$$748$$ −4.75372 −0.173813
$$749$$ −45.9565 −1.67921
$$750$$ 0 0
$$751$$ 21.7228 0.792677 0.396338 0.918105i $$-0.370281\pi$$
0.396338 + 0.918105i $$0.370281\pi$$
$$752$$ −1.17981 −0.0430234
$$753$$ 0 0
$$754$$ −26.5842 −0.968140
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 41.5692 1.51086 0.755429 0.655230i $$-0.227428\pi$$
0.755429 + 0.655230i $$0.227428\pi$$
$$758$$ 1.17981 0.0428528
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 32.4891 1.17773 0.588865 0.808231i $$-0.299575\pi$$
0.588865 + 0.808231i $$0.299575\pi$$
$$762$$ 0 0
$$763$$ −33.7562 −1.22205
$$764$$ −10.4674 −0.378696
$$765$$ 0 0
$$766$$ 8.78806 0.317526
$$767$$ −9.50744 −0.343294
$$768$$ 0 0
$$769$$ −16.4891 −0.594613 −0.297307 0.954782i $$-0.596088\pi$$
−0.297307 + 0.954782i $$0.596088\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 15.7532 0.566969
$$773$$ 21.5769 0.776067 0.388034 0.921645i $$-0.373154\pi$$
0.388034 + 0.921645i $$0.373154\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 3.44563 0.123691
$$777$$ 0 0
$$778$$ 0.404759 0.0145113
$$779$$ −1.62772 −0.0583191
$$780$$ 0 0
$$781$$ −57.3505 −2.05216
$$782$$ −0.994667 −0.0355692
$$783$$ 0 0
$$784$$ 3.13859 0.112093
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −17.3205 −0.617409 −0.308705 0.951158i $$-0.599895\pi$$
−0.308705 + 0.951158i $$0.599895\pi$$
$$788$$ 8.82496 0.314376
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 21.2554 0.755756
$$792$$ 0 0
$$793$$ −54.7431 −1.94399
$$794$$ 13.8832 0.492695
$$795$$ 0 0
$$796$$ −21.9565 −0.778228
$$797$$ 25.6309 0.907893 0.453947 0.891029i $$-0.350016\pi$$
0.453947 + 0.891029i $$0.350016\pi$$
$$798$$ 0 0
$$799$$ 1.48913 0.0526815
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −20.1021 −0.709831
$$803$$ −10.3923 −0.366736
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 29.4891 1.03871
$$807$$ 0 0
$$808$$ −43.7437 −1.53890
$$809$$ 21.0000 0.738321 0.369160 0.929366i $$-0.379645\pi$$
0.369160 + 0.929366i $$0.379645\pi$$
$$810$$ 0 0
$$811$$ 24.8832 0.873766 0.436883 0.899518i $$-0.356082\pi$$
0.436883 + 0.899518i $$0.356082\pi$$
$$812$$ −27.3081 −0.958325
$$813$$ 0 0
$$814$$ 8.23369 0.288591
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −1.28962 −0.0451181
$$818$$ −18.1128 −0.633299
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −40.7228 −1.42124 −0.710618 0.703578i $$-0.751585\pi$$
−0.710618 + 0.703578i $$0.751585\pi$$
$$822$$ 0 0
$$823$$ −4.34896 −0.151595 −0.0757977 0.997123i $$-0.524150\pi$$
−0.0757977 + 0.997123i $$0.524150\pi$$
$$824$$ 18.5109 0.644857
$$825$$ 0 0
$$826$$ 4.46738 0.155440
$$827$$ 22.3692 0.777853 0.388926 0.921269i $$-0.372846\pi$$
0.388926 + 0.921269i $$0.372846\pi$$
$$828$$ 0 0
$$829$$ −23.3505 −0.810997 −0.405499 0.914096i $$-0.632902\pi$$
−0.405499 + 0.914096i $$0.632902\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −19.6974 −0.682883
$$833$$ −3.96143 −0.137256
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −2.23369 −0.0772537
$$837$$ 0 0
$$838$$ 13.8564 0.478662
$$839$$ −45.8614 −1.58331 −0.791656 0.610967i $$-0.790781\pi$$
−0.791656 + 0.610967i $$0.790781\pi$$
$$840$$ 0 0
$$841$$ 4.00000 0.137931
$$842$$ 17.2279 0.593714
$$843$$ 0 0
$$844$$ 2.55437 0.0879252
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −28.1176 −0.966131
$$848$$ −7.51811 −0.258173
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −3.76631 −0.129108
$$852$$ 0 0
$$853$$ −10.7971 −0.369684 −0.184842 0.982768i $$-0.559177\pi$$
−0.184842 + 0.982768i $$0.559177\pi$$
$$854$$ 25.7228 0.880217
$$855$$ 0 0
$$856$$ 35.4456 1.21151
$$857$$ 49.3995 1.68746 0.843728 0.536772i $$-0.180356\pi$$
0.843728 + 0.536772i $$0.180356\pi$$
$$858$$ 0 0
$$859$$ 48.3288 1.64896 0.824478 0.565893i $$-0.191469\pi$$
0.824478 + 0.565893i $$0.191469\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 26.4232 0.899978
$$863$$ −16.7306 −0.569516 −0.284758 0.958599i $$-0.591913\pi$$
−0.284758 + 0.958599i $$0.591913\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −10.1168 −0.343784
$$867$$ 0 0
$$868$$ 30.2921 1.02818
$$869$$ −29.4891 −1.00035
$$870$$ 0 0
$$871$$ 68.2337 2.31201
$$872$$ 26.0357 0.881679
$$873$$ 0 0
$$874$$ −0.467376 −0.0158092
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 1.08724 0.0367135 0.0183568 0.999832i $$-0.494157\pi$$
0.0183568 + 0.999832i $$0.494157\pi$$
$$878$$ −25.2434 −0.851923
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 10.8832 0.366663 0.183331 0.983051i $$-0.441312\pi$$
0.183331 + 0.983051i $$0.441312\pi$$
$$882$$ 0 0
$$883$$ −54.9455 −1.84906 −0.924532 0.381104i $$-0.875544\pi$$
−0.924532 + 0.381104i $$0.875544\pi$$
$$884$$ −6.35053 −0.213592
$$885$$ 0 0
$$886$$ 17.0217 0.571857
$$887$$ −43.8535 −1.47246 −0.736228 0.676733i $$-0.763395\pi$$
−0.736228 + 0.676733i $$0.763395\pi$$
$$888$$ 0 0
$$889$$ −36.0000 −1.20740
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −25.5383 −0.855087
$$893$$ 0.699713 0.0234150
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −31.2119 −1.04272
$$897$$ 0 0
$$898$$ 8.62258 0.287739
$$899$$ −36.6060 −1.22088
$$900$$ 0 0
$$901$$ 9.48913 0.316129
$$902$$ 15.1460 0.504308
$$903$$ 0 0
$$904$$ −16.3940 −0.545257
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 18.2054 0.604499 0.302250 0.953229i $$-0.402262\pi$$
0.302250 + 0.953229i $$0.402262\pi$$
$$908$$ 18.6101 0.617599
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −18.6060 −0.616443 −0.308222 0.951315i $$-0.599734\pi$$
−0.308222 + 0.951315i $$0.599734\pi$$
$$912$$ 0 0
$$913$$ 52.3663 1.73307
$$914$$ 16.6277 0.549996
$$915$$ 0 0
$$916$$ 24.1603 0.798280
$$917$$ −15.1460 −0.500166
$$918$$ 0 0
$$919$$ 55.3505 1.82585 0.912923 0.408132i $$-0.133820\pi$$
0.912923 + 0.408132i $$0.133820\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −19.0902 −0.628703
$$923$$ −76.6150 −2.52181
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 10.9783 0.360768
$$927$$ 0 0
$$928$$ 33.5538 1.10146
$$929$$ −3.51087 −0.115188 −0.0575940 0.998340i $$-0.518343\pi$$
−0.0575940 + 0.998340i $$0.518343\pi$$
$$930$$ 0 0
$$931$$ −1.86141 −0.0610051
$$932$$ 8.42020 0.275813
$$933$$ 0 0
$$934$$ 14.5109 0.474810
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 22.2766 0.727745 0.363873 0.931449i $$-0.381454\pi$$
0.363873 + 0.931449i $$0.381454\pi$$
$$938$$ −32.0618 −1.04685
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −6.86141 −0.223675 −0.111838 0.993726i $$-0.535674\pi$$
−0.111838 + 0.993726i $$0.535674\pi$$
$$942$$ 0 0
$$943$$ −6.92820 −0.225613
$$944$$ 1.02175 0.0332551
$$945$$ 0 0
$$946$$ 12.0000 0.390154
$$947$$ 43.1538 1.40231 0.701155 0.713009i $$-0.252668\pi$$
0.701155 + 0.713009i $$0.252668\pi$$
$$948$$ 0 0
$$949$$ −13.8832 −0.450666
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 7.33296 0.237663
$$953$$ 25.0410 0.811158 0.405579 0.914060i $$-0.367070\pi$$
0.405579 + 0.914060i $$0.367070\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −31.5326 −1.01984
$$957$$ 0 0
$$958$$ −14.7413 −0.476269
$$959$$ −49.7228 −1.60563
$$960$$ 0 0
$$961$$ 9.60597 0.309870
$$962$$ 10.9994 0.354636
$$963$$ 0 0
$$964$$ 2.07335 0.0667780
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 43.2636 1.39126 0.695632 0.718399i $$-0.255125\pi$$
0.695632 + 0.718399i $$0.255125\pi$$
$$968$$ 21.6867 0.697037
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 41.5842 1.33450 0.667251 0.744833i $$-0.267471\pi$$
0.667251 + 0.744833i $$0.267471\pi$$
$$972$$ 0 0
$$973$$ −38.5099 −1.23457
$$974$$ −7.53262 −0.241361
$$975$$ 0 0
$$976$$ 5.88316 0.188315
$$977$$ −28.3027 −0.905484 −0.452742 0.891642i $$-0.649554\pi$$
−0.452742 + 0.891642i $$0.649554\pi$$
$$978$$ 0 0
$$979$$ 13.1168 0.419216
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 25.0582 0.799640
$$983$$ 50.3770 1.60678 0.803388 0.595456i $$-0.203029\pi$$
0.803388 + 0.595456i $$0.203029\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −3.60597 −0.114837
$$987$$ 0 0
$$988$$ −2.98400 −0.0949337
$$989$$ −5.48913 −0.174544
$$990$$ 0 0
$$991$$ 38.6060 1.22636 0.613180 0.789944i $$-0.289890\pi$$
0.613180 + 0.789944i $$0.289890\pi$$
$$992$$ −37.2203 −1.18174
$$993$$ 0 0
$$994$$ 36.0000 1.14185
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 60.3817 1.91231 0.956154 0.292864i $$-0.0946082\pi$$
0.956154 + 0.292864i $$0.0946082\pi$$
$$998$$ 19.3098 0.611242
$$999$$ 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.w.1.2 4
3.2 odd 2 2025.2.a.x.1.3 4
5.2 odd 4 405.2.b.a.244.2 4
5.3 odd 4 405.2.b.a.244.3 yes 4
5.4 even 2 inner 2025.2.a.w.1.3 4
15.2 even 4 405.2.b.b.244.3 yes 4
15.8 even 4 405.2.b.b.244.2 yes 4
15.14 odd 2 2025.2.a.x.1.2 4
45.2 even 12 405.2.j.a.109.2 4
45.7 odd 12 405.2.j.e.109.1 4
45.13 odd 12 405.2.j.e.379.1 4
45.22 odd 12 405.2.j.b.379.2 4
45.23 even 12 405.2.j.a.379.2 4
45.32 even 12 405.2.j.d.379.1 4
45.38 even 12 405.2.j.d.109.1 4
45.43 odd 12 405.2.j.b.109.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
405.2.b.a.244.2 4 5.2 odd 4
405.2.b.a.244.3 yes 4 5.3 odd 4
405.2.b.b.244.2 yes 4 15.8 even 4
405.2.b.b.244.3 yes 4 15.2 even 4
405.2.j.a.109.2 4 45.2 even 12
405.2.j.a.379.2 4 45.23 even 12
405.2.j.b.109.2 4 45.43 odd 12
405.2.j.b.379.2 4 45.22 odd 12
405.2.j.d.109.1 4 45.38 even 12
405.2.j.d.379.1 4 45.32 even 12
405.2.j.e.109.1 4 45.7 odd 12
405.2.j.e.379.1 4 45.13 odd 12
2025.2.a.w.1.2 4 1.1 even 1 trivial
2025.2.a.w.1.3 4 5.4 even 2 inner
2025.2.a.x.1.2 4 15.14 odd 2
2025.2.a.x.1.3 4 3.2 odd 2