# Properties

 Label 2025.2.a.x.1.2 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 1.2 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 $$-0.272814\pi$$
0.654654 + 0.755929i $$0.272814\pi$$
$$8$$ 2.67181 0.944629
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 4.37228 1.31829 0.659146 0.752015i $$-0.270918\pi$$
0.659146 + 0.752015i $$0.270918\pi$$
$$12$$ 0 0
$$13$$ 5.84096 1.61999 0.809996 0.586436i $$-0.199469\pi$$
0.809996 + 0.586436i $$0.199469\pi$$
$$14$$ −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 $$-0.320926\pi$$
0.533369 + 0.845883i $$0.320926\pi$$
$$30$$ 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 $$-0.562593\pi$$
−0.195377 + 0.980728i $$0.562593\pi$$
$$38$$ 0.294954 0.0478478
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −4.37228 −0.682836 −0.341418 0.939912i $$-0.610907\pi$$
−0.341418 + 0.939912i $$0.610907\pi$$
$$42$$ 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
$$46$$ 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 $$-0.533790\pi$$
−0.105955 + 0.994371i $$0.533790\pi$$
$$60$$ 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 $$-0.247069\pi$$
0.713587 + 0.700566i $$0.247069\pi$$
$$68$$ 1.08724 0.131847
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −13.1168 −1.55668 −0.778341 0.627841i $$-0.783939\pi$$
−0.778341 + 0.627841i $$0.783939\pi$$
$$72$$ 0 0
$$73$$ −2.37686 −0.278191 −0.139095 0.990279i $$-0.544419\pi$$
−0.139095 + 0.990279i $$0.544419\pi$$
$$74$$ 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 $$-0.449173\pi$$
0.159000 + 0.987279i $$0.449173\pi$$
$$90$$ 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 $$-0.520855\pi$$
−0.0654706 + 0.997855i $$0.520855\pi$$
$$98$$ −3.96143 −0.400165
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 16.3723 1.62910 0.814551 0.580091i $$-0.196983\pi$$
0.814551 + 0.580091i $$0.196983\pi$$
$$102$$ 0 0
$$103$$ −6.92820 −0.682656 −0.341328 0.939944i $$-0.610877\pi$$
−0.341328 + 0.939944i $$0.610877\pi$$
$$104$$ 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 $$-0.652539\pi$$
−0.461084 + 0.887357i $$0.652539\pi$$
$$128$$ 9.01011 0.796389
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −4.37228 −0.382008 −0.191004 0.981589i $$-0.561174\pi$$
−0.191004 + 0.981589i $$0.561174\pi$$
$$132$$ 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 $$-0.643368\pi$$
−0.435328 + 0.900272i $$0.643368\pi$$
$$150$$ 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 $$-0.651464\pi$$
−0.458084 + 0.888909i $$0.651464\pi$$
$$158$$ −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 $$-0.297878\pi$$
0.593164 + 0.805082i $$0.297878\pi$$
$$164$$ 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 $$-0.441584\pi$$
0.182492 + 0.983207i $$0.441584\pi$$
$$180$$ 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 $$-0.588996\pi$$
−0.275961 + 0.961169i $$0.588996\pi$$
$$192$$ 0 0
$$193$$ 11.4795 0.826316 0.413158 0.910659i $$-0.364426\pi$$
0.413158 + 0.910659i $$0.364426\pi$$
$$194$$ 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 $$-0.714132\pi$$
−0.623113 + 0.782132i $$0.714132\pi$$
$$224$$ −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 $$-0.766679\pi$$
−0.743170 + 0.669103i $$0.766679\pi$$
$$240$$ 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 $$-0.279526\pi$$
0.638570 + 0.769564i $$0.279526\pi$$
$$252$$ 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 $$-0.149861\pi$$
0.891205 + 0.453601i $$0.149861\pi$$
$$270$$ 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 $$-0.730783\pi$$
−0.663156 + 0.748481i $$0.730783\pi$$
$$278$$ −8.80773 −0.528253
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 1.37228 0.0818634 0.0409317 0.999162i $$-0.486967\pi$$
0.0409317 + 0.999162i $$0.486967\pi$$
$$282$$ 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
$$286$$ −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 $$-0.642268\pi$$
−0.432215 + 0.901771i $$0.642268\pi$$
$$308$$ −20.7846 −1.18431
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 15.8614 0.899418 0.449709 0.893175i $$-0.351528\pi$$
0.449709 + 0.893175i $$0.351528\pi$$
$$312$$ 0 0
$$313$$ 24.4511 1.38206 0.691029 0.722827i $$-0.257158\pi$$
0.691029 + 0.722827i $$0.257158\pi$$
$$314$$ 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 $$-0.439572\pi$$
0.188702 + 0.982034i $$0.439572\pi$$
$$338$$ −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 $$-0.407277\pi$$
0.287196 + 0.957872i $$0.407277\pi$$
$$360$$ 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 $$-0.641122\pi$$
−0.428967 + 0.903320i $$0.641122\pi$$
$$368$$ −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 $$-0.431758\pi$$
0.212751 + 0.977106i $$0.431758\pi$$
$$374$$ 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 $$-0.495877\pi$$
0.0129512 + 0.999916i $$0.495877\pi$$
$$390$$ 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 $$-0.355076\pi$$
0.439724 + 0.898133i $$0.355076\pi$$
$$398$$ −12.6766 −0.635420
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −25.3723 −1.26703 −0.633516 0.773730i $$-0.718389\pi$$
−0.633516 + 0.773730i $$0.718389\pi$$
$$402$$ 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 $$-0.359500\pi$$
0.427200 + 0.904157i $$0.359500\pi$$
$$420$$ 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 $$-0.203119\pi$$
0.803219 + 0.595683i $$0.203119\pi$$
$$432$$ 0 0
$$433$$ −12.7692 −0.613647 −0.306823 0.951766i $$-0.599266\pi$$
−0.306823 + 0.951766i $$0.599266\pi$$
$$434$$ −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 $$-0.417331\pi$$
0.256804 + 0.966464i $$0.417331\pi$$
$$450$$ 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 $$-0.336681\pi$$
0.490865 + 0.871236i $$0.336681\pi$$
$$458$$ 13.9490 0.651793
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −24.0951 −1.12222 −0.561110 0.827741i $$-0.689626\pi$$
−0.561110 + 0.827741i $$0.689626\pi$$
$$462$$ 0 0
$$463$$ 13.8564 0.643962 0.321981 0.946746i $$-0.395651\pi$$
0.321981 + 0.946746i $$0.395651\pi$$
$$464$$ 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 $$-0.639748\pi$$
−0.425064 + 0.905163i $$0.639748\pi$$
$$480$$ 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 $$-0.569109\pi$$
−0.215412 + 0.976523i $$0.569109\pi$$
$$488$$ 25.0410 1.13355
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 31.6277 1.42734 0.713669 0.700483i $$-0.247032\pi$$
0.713669 + 0.700483i $$0.247032\pi$$
$$492$$ 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 $$-0.503604\pi$$
−0.0113221 + 0.999936i $$0.503604\pi$$
$$510$$ 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 $$-0.370988\pi$$
0.394297 + 0.918983i $$0.370988\pi$$
$$522$$ 0 0
$$523$$ −4.75372 −0.207866 −0.103933 0.994584i $$-0.533143\pi$$
−0.103933 + 0.994584i $$0.533143\pi$$
$$524$$ 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 $$-0.547320\pi$$
−0.148114 + 0.988970i $$0.547320\pi$$
$$548$$ −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 $$-0.641301\pi$$
−0.429474 + 0.903079i $$0.641301\pi$$
$$570$$ 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 $$-0.374835\pi$$
0.383162 + 0.923681i $$0.374835\pi$$
$$578$$ 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 $$-0.510587\pi$$
−0.0332534 + 0.999447i $$0.510587\pi$$
$$600$$ 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 $$-0.841128\pi$$
−0.878008 + 0.478647i $$0.841128\pi$$
$$608$$ 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 $$-0.709531\pi$$
−0.611742 + 0.791057i $$0.709531\pi$$
$$614$$ 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 $$-0.550363\pi$$
−0.157561 + 0.987509i $$0.550363\pi$$
$$642$$ 0 0
$$643$$ −1.28962 −0.0508577 −0.0254288 0.999677i $$-0.508095\pi$$
−0.0254288 + 0.999677i $$0.508095\pi$$
$$644$$ 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 $$-0.635868\pi$$
−0.413997 + 0.910278i $$0.635868\pi$$
$$660$$ 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 $$-0.565458\pi$$
−0.204198 + 0.978930i $$0.565458\pi$$
$$674$$ −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 $$-0.605517\pi$$
−0.325454 + 0.945558i $$0.605517\pi$$
$$702$$ 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 $$-0.565049\pi$$
−0.202937 + 0.979192i $$0.565049\pi$$
$$720$$ 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 $$-0.138487\pi$$
0.906841 + 0.421473i $$0.138487\pi$$
$$728$$ 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 $$-0.598166\pi$$
−0.303532 + 0.952821i $$0.598166\pi$$
$$734$$ 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 $$-0.772572\pi$$
−0.755429 + 0.655230i $$0.772572\pi$$
$$758$$ 1.17981 0.0428528
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −32.4891 −1.17773 −0.588865 0.808231i $$-0.700425\pi$$
−0.588865 + 0.808231i $$0.700425\pi$$
$$762$$ 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 $$-0.400105\pi$$
0.308705 + 0.951158i $$0.400105\pi$$
$$788$$ 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 $$-0.620355\pi$$
−0.369160 + 0.929366i $$0.620355\pi$$
$$810$$ 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 $$-0.248415\pi$$
0.710618 + 0.703578i $$0.248415\pi$$
$$822$$ 0 0
$$823$$ 4.34896 0.151595 0.0757977 0.997123i $$-0.475850\pi$$
0.0757977 + 0.997123i $$0.475850\pi$$
$$824$$ −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 $$-0.209219\pi$$
0.791656 + 0.610967i $$0.209219\pi$$
$$840$$ 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 $$-0.440823\pi$$
0.184842 + 0.982768i $$0.440823\pi$$
$$854$$ −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 $$-0.505843\pi$$
−0.0183568 + 0.999832i $$0.505843\pi$$
$$878$$ −25.2434 −0.851923
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −10.8832 −0.366663 −0.183331 0.983051i $$-0.558688\pi$$
−0.183331 + 0.983051i $$0.558688\pi$$
$$882$$ 0 0
$$883$$ 54.9455 1.84906 0.924532 0.381104i $$-0.124456\pi$$
0.924532 + 0.381104i $$0.124456\pi$$
$$884$$ 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 $$-0.597738\pi$$
−0.302250 + 0.953229i $$0.597738\pi$$
$$908$$ 18.6101 0.617599
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 18.6060 0.616443 0.308222 0.951315i $$-0.400266\pi$$
0.308222 + 0.951315i $$0.400266\pi$$
$$912$$ 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 $$-0.481657\pi$$
0.0575940 + 0.998340i $$0.481657\pi$$
$$930$$ 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 $$-0.618546\pi$$
−0.363873 + 0.931449i $$0.618546\pi$$
$$938$$ −32.0618 −1.04685
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 6.86141 0.223675 0.111838 0.993726i $$-0.464326\pi$$
0.111838 + 0.993726i $$0.464326\pi$$
$$942$$ 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 $$-0.744875\pi$$
−0.695632 + 0.718399i $$0.744875\pi$$
$$968$$ 21.6867 0.697037
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −41.5842 −1.33450 −0.667251 0.744833i $$-0.732529\pi$$
−0.667251 + 0.744833i $$0.732529\pi$$
$$972$$ 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 $$-0.905392\pi$$
−0.956154 + 0.292864i $$0.905392\pi$$
$$998$$ 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.x.1.2 4
3.2 odd 2 2025.2.a.w.1.3 4
5.2 odd 4 405.2.b.b.244.2 yes 4
5.3 odd 4 405.2.b.b.244.3 yes 4
5.4 even 2 inner 2025.2.a.x.1.3 4
15.2 even 4 405.2.b.a.244.3 yes 4
15.8 even 4 405.2.b.a.244.2 4
15.14 odd 2 2025.2.a.w.1.2 4
45.2 even 12 405.2.j.b.109.2 4
45.7 odd 12 405.2.j.d.109.1 4
45.13 odd 12 405.2.j.d.379.1 4
45.22 odd 12 405.2.j.a.379.2 4
45.23 even 12 405.2.j.b.379.2 4
45.32 even 12 405.2.j.e.379.1 4
45.38 even 12 405.2.j.e.109.1 4
45.43 odd 12 405.2.j.a.109.2 4

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