# Properties

 Label 6525.2.a.bw.1.7 Level $6525$ Weight $2$ Character 6525.1 Self dual yes Analytic conductor $52.102$ Analytic rank $0$ Dimension $7$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6525,2,Mod(1,6525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6525, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6525.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6525 = 3^{2} \cdot 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6525.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: $$52.1023873189$$ Analytic rank: $$0$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{7} - x^{6} - 13x^{5} + 12x^{4} + 47x^{3} - 37x^{2} - 35x + 18$$ x^7 - x^6 - 13*x^5 + 12*x^4 + 47*x^3 - 37*x^2 - 35*x + 18 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1305) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.7 Root $$2.57501$$ of defining polynomial Character $$\chi$$ $$=$$ 6525.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.57501 q^{2} +4.63069 q^{4} +2.24826 q^{7} +6.77405 q^{8} +O(q^{10})$$ $$q+2.57501 q^{2} +4.63069 q^{4} +2.24826 q^{7} +6.77405 q^{8} -3.75744 q^{11} +1.55119 q^{13} +5.78930 q^{14} +8.18188 q^{16} +3.55119 q^{17} -1.18909 q^{19} -9.67546 q^{22} +3.67991 q^{23} +3.99434 q^{26} +10.4110 q^{28} +1.00000 q^{29} +5.18909 q^{31} +7.52034 q^{32} +9.14436 q^{34} -0.969518 q^{37} -3.06193 q^{38} +11.3809 q^{41} +0.884393 q^{43} -17.3995 q^{44} +9.47581 q^{46} -11.2251 q^{47} -1.94533 q^{49} +7.18308 q^{52} +10.6567 q^{53} +15.2298 q^{56} +2.57501 q^{58} +8.49652 q^{59} -10.8743 q^{61} +13.3620 q^{62} +3.00121 q^{64} +4.05179 q^{67} +16.4445 q^{68} +6.37818 q^{71} +3.26017 q^{73} -2.49652 q^{74} -5.50631 q^{76} -8.44770 q^{77} +6.32579 q^{79} +29.3060 q^{82} +9.03930 q^{83} +2.27732 q^{86} -25.4531 q^{88} +1.46717 q^{89} +3.48748 q^{91} +17.0405 q^{92} -28.9047 q^{94} -12.3993 q^{97} -5.00924 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7 q + q^{2} + 13 q^{4} - 10 q^{7}+O(q^{10})$$ 7 * q + q^2 + 13 * q^4 - 10 * q^7 $$7 q + q^{2} + 13 q^{4} - 10 q^{7} + 3 q^{11} - 6 q^{13} + 9 q^{14} + 21 q^{16} + 8 q^{17} + 10 q^{19} - 9 q^{22} + 11 q^{23} - 3 q^{26} - 25 q^{28} + 7 q^{29} + 18 q^{31} + q^{32} - q^{34} - 13 q^{37} + 12 q^{38} + 13 q^{41} - 9 q^{43} + 37 q^{44} - 8 q^{46} + 2 q^{47} + 21 q^{49} + q^{52} + 5 q^{53} + 30 q^{56} + q^{58} + 8 q^{59} + 14 q^{61} - 8 q^{62} + 8 q^{64} - 14 q^{67} + 27 q^{68} + 8 q^{71} - 3 q^{73} + 34 q^{74} + 4 q^{76} - 28 q^{77} + 4 q^{79} + 20 q^{82} + 17 q^{83} - 4 q^{86} - 26 q^{88} + 20 q^{89} + 12 q^{91} + 60 q^{92} - 21 q^{94} - 13 q^{97} - 20 q^{98}+O(q^{100})$$ 7 * q + q^2 + 13 * q^4 - 10 * q^7 + 3 * q^11 - 6 * q^13 + 9 * q^14 + 21 * q^16 + 8 * q^17 + 10 * q^19 - 9 * q^22 + 11 * q^23 - 3 * q^26 - 25 * q^28 + 7 * q^29 + 18 * q^31 + q^32 - q^34 - 13 * q^37 + 12 * q^38 + 13 * q^41 - 9 * q^43 + 37 * q^44 - 8 * q^46 + 2 * q^47 + 21 * q^49 + q^52 + 5 * q^53 + 30 * q^56 + q^58 + 8 * q^59 + 14 * q^61 - 8 * q^62 + 8 * q^64 - 14 * q^67 + 27 * q^68 + 8 * q^71 - 3 * q^73 + 34 * q^74 + 4 * q^76 - 28 * q^77 + 4 * q^79 + 20 * q^82 + 17 * q^83 - 4 * q^86 - 26 * q^88 + 20 * q^89 + 12 * q^91 + 60 * q^92 - 21 * q^94 - 13 * q^97 - 20 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.57501 1.82081 0.910404 0.413720i $$-0.135771\pi$$
0.910404 + 0.413720i $$0.135771\pi$$
$$3$$ 0 0
$$4$$ 4.63069 2.31534
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 2.24826 0.849762 0.424881 0.905249i $$-0.360316\pi$$
0.424881 + 0.905249i $$0.360316\pi$$
$$8$$ 6.77405 2.39499
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −3.75744 −1.13291 −0.566456 0.824092i $$-0.691686\pi$$
−0.566456 + 0.824092i $$0.691686\pi$$
$$12$$ 0 0
$$13$$ 1.55119 0.430223 0.215112 0.976589i $$-0.430988\pi$$
0.215112 + 0.976589i $$0.430988\pi$$
$$14$$ 5.78930 1.54725
$$15$$ 0 0
$$16$$ 8.18188 2.04547
$$17$$ 3.55119 0.861291 0.430645 0.902521i $$-0.358286\pi$$
0.430645 + 0.902521i $$0.358286\pi$$
$$18$$ 0 0
$$19$$ −1.18909 −0.272796 −0.136398 0.990654i $$-0.543553\pi$$
−0.136398 + 0.990654i $$0.543553\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −9.67546 −2.06281
$$23$$ 3.67991 0.767314 0.383657 0.923476i $$-0.374664\pi$$
0.383657 + 0.923476i $$0.374664\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 3.99434 0.783354
$$27$$ 0 0
$$28$$ 10.4110 1.96749
$$29$$ 1.00000 0.185695
$$30$$ 0 0
$$31$$ 5.18909 0.931988 0.465994 0.884788i $$-0.345697\pi$$
0.465994 + 0.884788i $$0.345697\pi$$
$$32$$ 7.52034 1.32942
$$33$$ 0 0
$$34$$ 9.14436 1.56825
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −0.969518 −0.159388 −0.0796939 0.996819i $$-0.525394\pi$$
−0.0796939 + 0.996819i $$0.525394\pi$$
$$38$$ −3.06193 −0.496710
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 11.3809 1.77740 0.888700 0.458489i $$-0.151609\pi$$
0.888700 + 0.458489i $$0.151609\pi$$
$$42$$ 0 0
$$43$$ 0.884393 0.134869 0.0674343 0.997724i $$-0.478519\pi$$
0.0674343 + 0.997724i $$0.478519\pi$$
$$44$$ −17.3995 −2.62308
$$45$$ 0 0
$$46$$ 9.47581 1.39713
$$47$$ −11.2251 −1.63734 −0.818672 0.574262i $$-0.805289\pi$$
−0.818672 + 0.574262i $$0.805289\pi$$
$$48$$ 0 0
$$49$$ −1.94533 −0.277904
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 7.18308 0.996115
$$53$$ 10.6567 1.46381 0.731906 0.681406i $$-0.238631\pi$$
0.731906 + 0.681406i $$0.238631\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 15.2298 2.03517
$$57$$ 0 0
$$58$$ 2.57501 0.338116
$$59$$ 8.49652 1.10615 0.553076 0.833131i $$-0.313454\pi$$
0.553076 + 0.833131i $$0.313454\pi$$
$$60$$ 0 0
$$61$$ −10.8743 −1.39231 −0.696154 0.717892i $$-0.745107\pi$$
−0.696154 + 0.717892i $$0.745107\pi$$
$$62$$ 13.3620 1.69697
$$63$$ 0 0
$$64$$ 3.00121 0.375151
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 4.05179 0.495005 0.247502 0.968887i $$-0.420390\pi$$
0.247502 + 0.968887i $$0.420390\pi$$
$$68$$ 16.4445 1.99418
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 6.37818 0.756951 0.378476 0.925611i $$-0.376448\pi$$
0.378476 + 0.925611i $$0.376448\pi$$
$$72$$ 0 0
$$73$$ 3.26017 0.381573 0.190787 0.981632i $$-0.438896\pi$$
0.190787 + 0.981632i $$0.438896\pi$$
$$74$$ −2.49652 −0.290215
$$75$$ 0 0
$$76$$ −5.50631 −0.631617
$$77$$ −8.44770 −0.962705
$$78$$ 0 0
$$79$$ 6.32579 0.711707 0.355854 0.934542i $$-0.384190\pi$$
0.355854 + 0.934542i $$0.384190\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 29.3060 3.23630
$$83$$ 9.03930 0.992193 0.496096 0.868268i $$-0.334766\pi$$
0.496096 + 0.868268i $$0.334766\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 2.27732 0.245570
$$87$$ 0 0
$$88$$ −25.4531 −2.71331
$$89$$ 1.46717 0.155520 0.0777599 0.996972i $$-0.475223\pi$$
0.0777599 + 0.996972i $$0.475223\pi$$
$$90$$ 0 0
$$91$$ 3.48748 0.365588
$$92$$ 17.0405 1.77660
$$93$$ 0 0
$$94$$ −28.9047 −2.98129
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −12.3993 −1.25896 −0.629478 0.777018i $$-0.716731\pi$$
−0.629478 + 0.777018i $$0.716731\pi$$
$$98$$ −5.00924 −0.506010
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 3.45196 0.343483 0.171742 0.985142i $$-0.445061\pi$$
0.171742 + 0.985142i $$0.445061\pi$$
$$102$$ 0 0
$$103$$ −18.1975 −1.79306 −0.896528 0.442988i $$-0.853918\pi$$
−0.896528 + 0.442988i $$0.853918\pi$$
$$104$$ 10.5078 1.03038
$$105$$ 0 0
$$106$$ 27.4412 2.66532
$$107$$ −8.07919 −0.781044 −0.390522 0.920594i $$-0.627706\pi$$
−0.390522 + 0.920594i $$0.627706\pi$$
$$108$$ 0 0
$$109$$ −14.0114 −1.34205 −0.671026 0.741434i $$-0.734146\pi$$
−0.671026 + 0.741434i $$0.734146\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 18.3950 1.73816
$$113$$ 0.678491 0.0638270 0.0319135 0.999491i $$-0.489840\pi$$
0.0319135 + 0.999491i $$0.489840\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 4.63069 0.429948
$$117$$ 0 0
$$118$$ 21.8786 2.01409
$$119$$ 7.98400 0.731892
$$120$$ 0 0
$$121$$ 3.11836 0.283488
$$122$$ −28.0014 −2.53513
$$123$$ 0 0
$$124$$ 24.0291 2.15787
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 8.39928 0.745315 0.372658 0.927969i $$-0.378447\pi$$
0.372658 + 0.927969i $$0.378447\pi$$
$$128$$ −7.31254 −0.646343
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 7.34298 0.641559 0.320779 0.947154i $$-0.396055\pi$$
0.320779 + 0.947154i $$0.396055\pi$$
$$132$$ 0 0
$$133$$ −2.67339 −0.231812
$$134$$ 10.4334 0.901308
$$135$$ 0 0
$$136$$ 24.0559 2.06278
$$137$$ −1.80353 −0.154086 −0.0770429 0.997028i $$-0.524548\pi$$
−0.0770429 + 0.997028i $$0.524548\pi$$
$$138$$ 0 0
$$139$$ 0.707451 0.0600052 0.0300026 0.999550i $$-0.490448\pi$$
0.0300026 + 0.999550i $$0.490448\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 16.4239 1.37826
$$143$$ −5.82851 −0.487405
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 8.39496 0.694772
$$147$$ 0 0
$$148$$ −4.48953 −0.369037
$$149$$ 16.9930 1.39212 0.696062 0.717982i $$-0.254934\pi$$
0.696062 + 0.717982i $$0.254934\pi$$
$$150$$ 0 0
$$151$$ 7.86646 0.640163 0.320082 0.947390i $$-0.396290\pi$$
0.320082 + 0.947390i $$0.396290\pi$$
$$152$$ −8.05497 −0.653344
$$153$$ 0 0
$$154$$ −21.7529 −1.75290
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −9.85591 −0.786588 −0.393294 0.919413i $$-0.628664\pi$$
−0.393294 + 0.919413i $$0.628664\pi$$
$$158$$ 16.2890 1.29588
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 8.27340 0.652035
$$162$$ 0 0
$$163$$ −16.5786 −1.29853 −0.649267 0.760560i $$-0.724924\pi$$
−0.649267 + 0.760560i $$0.724924\pi$$
$$164$$ 52.7014 4.11529
$$165$$ 0 0
$$166$$ 23.2763 1.80659
$$167$$ −4.74360 −0.367071 −0.183536 0.983013i $$-0.558754\pi$$
−0.183536 + 0.983013i $$0.558754\pi$$
$$168$$ 0 0
$$169$$ −10.5938 −0.814908
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 4.09534 0.312267
$$173$$ −19.8201 −1.50689 −0.753446 0.657510i $$-0.771610\pi$$
−0.753446 + 0.657510i $$0.771610\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −30.7429 −2.31734
$$177$$ 0 0
$$178$$ 3.77798 0.283172
$$179$$ −7.42036 −0.554624 −0.277312 0.960780i $$-0.589443\pi$$
−0.277312 + 0.960780i $$0.589443\pi$$
$$180$$ 0 0
$$181$$ 19.8411 1.47478 0.737390 0.675468i $$-0.236058\pi$$
0.737390 + 0.675468i $$0.236058\pi$$
$$182$$ 8.98031 0.665665
$$183$$ 0 0
$$184$$ 24.9279 1.83771
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −13.3434 −0.975766
$$188$$ −51.9797 −3.79101
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −5.66409 −0.409839 −0.204920 0.978779i $$-0.565693\pi$$
−0.204920 + 0.978779i $$0.565693\pi$$
$$192$$ 0 0
$$193$$ −14.7896 −1.06458 −0.532289 0.846563i $$-0.678668\pi$$
−0.532289 + 0.846563i $$0.678668\pi$$
$$194$$ −31.9283 −2.29232
$$195$$ 0 0
$$196$$ −9.00820 −0.643443
$$197$$ 27.1349 1.93328 0.966640 0.256139i $$-0.0824504\pi$$
0.966640 + 0.256139i $$0.0824504\pi$$
$$198$$ 0 0
$$199$$ −9.53523 −0.675934 −0.337967 0.941158i $$-0.609739\pi$$
−0.337967 + 0.941158i $$0.609739\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 8.88885 0.625417
$$203$$ 2.24826 0.157797
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −46.8588 −3.26481
$$207$$ 0 0
$$208$$ 12.6917 0.880009
$$209$$ 4.46794 0.309054
$$210$$ 0 0
$$211$$ 21.5727 1.48513 0.742564 0.669775i $$-0.233609\pi$$
0.742564 + 0.669775i $$0.233609\pi$$
$$212$$ 49.3479 3.38923
$$213$$ 0 0
$$214$$ −20.8040 −1.42213
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 11.6664 0.791969
$$218$$ −36.0796 −2.44362
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 5.50858 0.370547
$$222$$ 0 0
$$223$$ −22.5140 −1.50765 −0.753824 0.657076i $$-0.771793\pi$$
−0.753824 + 0.657076i $$0.771793\pi$$
$$224$$ 16.9077 1.12969
$$225$$ 0 0
$$226$$ 1.74712 0.116217
$$227$$ 8.38063 0.556242 0.278121 0.960546i $$-0.410288\pi$$
0.278121 + 0.960546i $$0.410288\pi$$
$$228$$ 0 0
$$229$$ −11.5492 −0.763192 −0.381596 0.924329i $$-0.624625\pi$$
−0.381596 + 0.924329i $$0.624625\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 6.77405 0.444738
$$233$$ −8.85319 −0.579991 −0.289996 0.957028i $$-0.593654\pi$$
−0.289996 + 0.957028i $$0.593654\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 39.3447 2.56112
$$237$$ 0 0
$$238$$ 20.5589 1.33264
$$239$$ −13.9536 −0.902584 −0.451292 0.892376i $$-0.649037\pi$$
−0.451292 + 0.892376i $$0.649037\pi$$
$$240$$ 0 0
$$241$$ −24.2138 −1.55975 −0.779874 0.625937i $$-0.784717\pi$$
−0.779874 + 0.625937i $$0.784717\pi$$
$$242$$ 8.02983 0.516177
$$243$$ 0 0
$$244$$ −50.3554 −3.22367
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −1.84451 −0.117363
$$248$$ 35.1512 2.23210
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 10.1141 0.638397 0.319198 0.947688i $$-0.396586\pi$$
0.319198 + 0.947688i $$0.396586\pi$$
$$252$$ 0 0
$$253$$ −13.8270 −0.869299
$$254$$ 21.6282 1.35708
$$255$$ 0 0
$$256$$ −24.8323 −1.55202
$$257$$ −18.2290 −1.13709 −0.568546 0.822652i $$-0.692494\pi$$
−0.568546 + 0.822652i $$0.692494\pi$$
$$258$$ 0 0
$$259$$ −2.17973 −0.135442
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 18.9083 1.16816
$$263$$ 4.26573 0.263036 0.131518 0.991314i $$-0.458015\pi$$
0.131518 + 0.991314i $$0.458015\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −6.88401 −0.422086
$$267$$ 0 0
$$268$$ 18.7626 1.14611
$$269$$ −8.10868 −0.494395 −0.247197 0.968965i $$-0.579510\pi$$
−0.247197 + 0.968965i $$0.579510\pi$$
$$270$$ 0 0
$$271$$ 26.9259 1.63563 0.817815 0.575481i $$-0.195185\pi$$
0.817815 + 0.575481i $$0.195185\pi$$
$$272$$ 29.0554 1.76174
$$273$$ 0 0
$$274$$ −4.64410 −0.280561
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −29.1380 −1.75073 −0.875367 0.483459i $$-0.839380\pi$$
−0.875367 + 0.483459i $$0.839380\pi$$
$$278$$ 1.82169 0.109258
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −21.6594 −1.29209 −0.646047 0.763298i $$-0.723579\pi$$
−0.646047 + 0.763298i $$0.723579\pi$$
$$282$$ 0 0
$$283$$ 16.1608 0.960660 0.480330 0.877088i $$-0.340517\pi$$
0.480330 + 0.877088i $$0.340517\pi$$
$$284$$ 29.5354 1.75260
$$285$$ 0 0
$$286$$ −15.0085 −0.887471
$$287$$ 25.5872 1.51037
$$288$$ 0 0
$$289$$ −4.38903 −0.258178
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 15.0968 0.883473
$$293$$ 16.5348 0.965975 0.482988 0.875627i $$-0.339552\pi$$
0.482988 + 0.875627i $$0.339552\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −6.56756 −0.381732
$$297$$ 0 0
$$298$$ 43.7573 2.53479
$$299$$ 5.70825 0.330117
$$300$$ 0 0
$$301$$ 1.98834 0.114606
$$302$$ 20.2562 1.16561
$$303$$ 0 0
$$304$$ −9.72901 −0.557997
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 25.3923 1.44922 0.724608 0.689161i $$-0.242021\pi$$
0.724608 + 0.689161i $$0.242021\pi$$
$$308$$ −39.1187 −2.22899
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −16.8950 −0.958030 −0.479015 0.877807i $$-0.659006\pi$$
−0.479015 + 0.877807i $$0.659006\pi$$
$$312$$ 0 0
$$313$$ 22.1978 1.25469 0.627347 0.778740i $$-0.284141\pi$$
0.627347 + 0.778740i $$0.284141\pi$$
$$314$$ −25.3791 −1.43223
$$315$$ 0 0
$$316$$ 29.2927 1.64785
$$317$$ −26.1161 −1.46683 −0.733414 0.679783i $$-0.762074\pi$$
−0.733414 + 0.679783i $$0.762074\pi$$
$$318$$ 0 0
$$319$$ −3.75744 −0.210376
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 21.3041 1.18723
$$323$$ −4.22270 −0.234957
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −42.6900 −2.36438
$$327$$ 0 0
$$328$$ 77.0948 4.25685
$$329$$ −25.2369 −1.39135
$$330$$ 0 0
$$331$$ 24.3439 1.33806 0.669030 0.743236i $$-0.266710\pi$$
0.669030 + 0.743236i $$0.266710\pi$$
$$332$$ 41.8582 2.29727
$$333$$ 0 0
$$334$$ −12.2148 −0.668366
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 23.5099 1.28067 0.640333 0.768098i $$-0.278796\pi$$
0.640333 + 0.768098i $$0.278796\pi$$
$$338$$ −27.2792 −1.48379
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −19.4977 −1.05586
$$342$$ 0 0
$$343$$ −20.1114 −1.08591
$$344$$ 5.99092 0.323009
$$345$$ 0 0
$$346$$ −51.0369 −2.74376
$$347$$ −29.9492 −1.60776 −0.803879 0.594793i $$-0.797234\pi$$
−0.803879 + 0.594793i $$0.797234\pi$$
$$348$$ 0 0
$$349$$ −17.0927 −0.914951 −0.457476 0.889222i $$-0.651246\pi$$
−0.457476 + 0.889222i $$0.651246\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −28.2572 −1.50612
$$353$$ −6.40124 −0.340704 −0.170352 0.985383i $$-0.554490\pi$$
−0.170352 + 0.985383i $$0.554490\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 6.79400 0.360082
$$357$$ 0 0
$$358$$ −19.1075 −1.00986
$$359$$ 29.2509 1.54380 0.771902 0.635741i $$-0.219306\pi$$
0.771902 + 0.635741i $$0.219306\pi$$
$$360$$ 0 0
$$361$$ −17.5861 −0.925582
$$362$$ 51.0911 2.68529
$$363$$ 0 0
$$364$$ 16.1494 0.846461
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 13.2919 0.693832 0.346916 0.937896i $$-0.387229\pi$$
0.346916 + 0.937896i $$0.387229\pi$$
$$368$$ 30.1086 1.56952
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 23.9591 1.24389
$$372$$ 0 0
$$373$$ 19.7309 1.02163 0.510814 0.859691i $$-0.329344\pi$$
0.510814 + 0.859691i $$0.329344\pi$$
$$374$$ −34.3594 −1.77668
$$375$$ 0 0
$$376$$ −76.0391 −3.92142
$$377$$ 1.55119 0.0798905
$$378$$ 0 0
$$379$$ 25.4806 1.30885 0.654424 0.756128i $$-0.272911\pi$$
0.654424 + 0.756128i $$0.272911\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −14.5851 −0.746239
$$383$$ −0.542784 −0.0277350 −0.0138675 0.999904i $$-0.504414\pi$$
−0.0138675 + 0.999904i $$0.504414\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −38.0833 −1.93839
$$387$$ 0 0
$$388$$ −57.4171 −2.91491
$$389$$ −19.9301 −1.01050 −0.505249 0.862974i $$-0.668599\pi$$
−0.505249 + 0.862974i $$0.668599\pi$$
$$390$$ 0 0
$$391$$ 13.0681 0.660881
$$392$$ −13.1777 −0.665576
$$393$$ 0 0
$$394$$ 69.8726 3.52013
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −9.40286 −0.471916 −0.235958 0.971763i $$-0.575823\pi$$
−0.235958 + 0.971763i $$0.575823\pi$$
$$398$$ −24.5533 −1.23075
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 9.37578 0.468204 0.234102 0.972212i $$-0.424785\pi$$
0.234102 + 0.972212i $$0.424785\pi$$
$$402$$ 0 0
$$403$$ 8.04928 0.400963
$$404$$ 15.9850 0.795282
$$405$$ 0 0
$$406$$ 5.78930 0.287318
$$407$$ 3.64291 0.180572
$$408$$ 0 0
$$409$$ 19.7204 0.975111 0.487556 0.873092i $$-0.337889\pi$$
0.487556 + 0.873092i $$0.337889\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −84.2670 −4.15154
$$413$$ 19.1024 0.939967
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 11.6655 0.571948
$$417$$ 0 0
$$418$$ 11.5050 0.562728
$$419$$ −6.24026 −0.304857 −0.152428 0.988315i $$-0.548709\pi$$
−0.152428 + 0.988315i $$0.548709\pi$$
$$420$$ 0 0
$$421$$ 6.31722 0.307882 0.153941 0.988080i $$-0.450803\pi$$
0.153941 + 0.988080i $$0.450803\pi$$
$$422$$ 55.5500 2.70413
$$423$$ 0 0
$$424$$ 72.1891 3.50581
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −24.4482 −1.18313
$$428$$ −37.4122 −1.80839
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −37.8006 −1.82079 −0.910396 0.413737i $$-0.864223\pi$$
−0.910396 + 0.413737i $$0.864223\pi$$
$$432$$ 0 0
$$433$$ −37.8447 −1.81870 −0.909350 0.416032i $$-0.863420\pi$$
−0.909350 + 0.416032i $$0.863420\pi$$
$$434$$ 30.0412 1.44202
$$435$$ 0 0
$$436$$ −64.8825 −3.10731
$$437$$ −4.37575 −0.209321
$$438$$ 0 0
$$439$$ −15.5318 −0.741291 −0.370646 0.928774i $$-0.620864\pi$$
−0.370646 + 0.928774i $$0.620864\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 14.1847 0.674696
$$443$$ 5.24223 0.249066 0.124533 0.992215i $$-0.460257\pi$$
0.124533 + 0.992215i $$0.460257\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −57.9738 −2.74514
$$447$$ 0 0
$$448$$ 6.74749 0.318789
$$449$$ −28.0123 −1.32198 −0.660992 0.750393i $$-0.729864\pi$$
−0.660992 + 0.750393i $$0.729864\pi$$
$$450$$ 0 0
$$451$$ −42.7631 −2.01364
$$452$$ 3.14188 0.147781
$$453$$ 0 0
$$454$$ 21.5802 1.01281
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −1.90391 −0.0890610 −0.0445305 0.999008i $$-0.514179\pi$$
−0.0445305 + 0.999008i $$0.514179\pi$$
$$458$$ −29.7393 −1.38963
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 30.5294 1.42190 0.710949 0.703244i $$-0.248266\pi$$
0.710949 + 0.703244i $$0.248266\pi$$
$$462$$ 0 0
$$463$$ −32.9230 −1.53006 −0.765030 0.643995i $$-0.777276\pi$$
−0.765030 + 0.643995i $$0.777276\pi$$
$$464$$ 8.18188 0.379834
$$465$$ 0 0
$$466$$ −22.7971 −1.05605
$$467$$ 4.42775 0.204892 0.102446 0.994739i $$-0.467333\pi$$
0.102446 + 0.994739i $$0.467333\pi$$
$$468$$ 0 0
$$469$$ 9.10947 0.420636
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 57.5558 2.64922
$$473$$ −3.32305 −0.152794
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 36.9714 1.69458
$$477$$ 0 0
$$478$$ −35.9307 −1.64343
$$479$$ −8.27071 −0.377898 −0.188949 0.981987i $$-0.560508\pi$$
−0.188949 + 0.981987i $$0.560508\pi$$
$$480$$ 0 0
$$481$$ −1.50391 −0.0685723
$$482$$ −62.3508 −2.84000
$$483$$ 0 0
$$484$$ 14.4402 0.656371
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −39.9284 −1.80933 −0.904665 0.426124i $$-0.859879\pi$$
−0.904665 + 0.426124i $$0.859879\pi$$
$$488$$ −73.6628 −3.33456
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −33.3896 −1.50685 −0.753427 0.657532i $$-0.771601\pi$$
−0.753427 + 0.657532i $$0.771601\pi$$
$$492$$ 0 0
$$493$$ 3.55119 0.159938
$$494$$ −4.74964 −0.213696
$$495$$ 0 0
$$496$$ 42.4565 1.90635
$$497$$ 14.3398 0.643229
$$498$$ 0 0
$$499$$ −3.12737 −0.140000 −0.0700002 0.997547i $$-0.522300\pi$$
−0.0700002 + 0.997547i $$0.522300\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 26.0439 1.16240
$$503$$ 17.2563 0.769419 0.384710 0.923038i $$-0.374302\pi$$
0.384710 + 0.923038i $$0.374302\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −35.6048 −1.58283
$$507$$ 0 0
$$508$$ 38.8944 1.72566
$$509$$ −4.71879 −0.209157 −0.104578 0.994517i $$-0.533349\pi$$
−0.104578 + 0.994517i $$0.533349\pi$$
$$510$$ 0 0
$$511$$ 7.32970 0.324247
$$512$$ −49.3183 −2.17958
$$513$$ 0 0
$$514$$ −46.9398 −2.07043
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 42.1775 1.85496
$$518$$ −5.61283 −0.246613
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 40.0385 1.75412 0.877060 0.480381i $$-0.159502\pi$$
0.877060 + 0.480381i $$0.159502\pi$$
$$522$$ 0 0
$$523$$ 5.38694 0.235555 0.117777 0.993040i $$-0.462423\pi$$
0.117777 + 0.993040i $$0.462423\pi$$
$$524$$ 34.0030 1.48543
$$525$$ 0 0
$$526$$ 10.9843 0.478939
$$527$$ 18.4275 0.802713
$$528$$ 0 0
$$529$$ −9.45826 −0.411229
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −12.3796 −0.536725
$$533$$ 17.6540 0.764679
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 27.4470 1.18553
$$537$$ 0 0
$$538$$ −20.8799 −0.900198
$$539$$ 7.30945 0.314840
$$540$$ 0 0
$$541$$ 35.9892 1.54730 0.773649 0.633614i $$-0.218429\pi$$
0.773649 + 0.633614i $$0.218429\pi$$
$$542$$ 69.3345 2.97817
$$543$$ 0 0
$$544$$ 26.7062 1.14502
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −24.2745 −1.03790 −0.518951 0.854804i $$-0.673677\pi$$
−0.518951 + 0.854804i $$0.673677\pi$$
$$548$$ −8.35157 −0.356761
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −1.18909 −0.0506570
$$552$$ 0 0
$$553$$ 14.2220 0.604782
$$554$$ −75.0307 −3.18775
$$555$$ 0 0
$$556$$ 3.27598 0.138933
$$557$$ 4.93097 0.208932 0.104466 0.994528i $$-0.466687\pi$$
0.104466 + 0.994528i $$0.466687\pi$$
$$558$$ 0 0
$$559$$ 1.37186 0.0580236
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −55.7733 −2.35266
$$563$$ 34.7020 1.46252 0.731258 0.682101i $$-0.238933\pi$$
0.731258 + 0.682101i $$0.238933\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 41.6142 1.74918
$$567$$ 0 0
$$568$$ 43.2061 1.81289
$$569$$ 19.9633 0.836903 0.418452 0.908239i $$-0.362573\pi$$
0.418452 + 0.908239i $$0.362573\pi$$
$$570$$ 0 0
$$571$$ −7.22542 −0.302375 −0.151187 0.988505i $$-0.548310\pi$$
−0.151187 + 0.988505i $$0.548310\pi$$
$$572$$ −26.9900 −1.12851
$$573$$ 0 0
$$574$$ 65.8875 2.75009
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −17.6693 −0.735581 −0.367791 0.929909i $$-0.619886\pi$$
−0.367791 + 0.929909i $$0.619886\pi$$
$$578$$ −11.3018 −0.470093
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 20.3227 0.843128
$$582$$ 0 0
$$583$$ −40.0420 −1.65837
$$584$$ 22.0845 0.913864
$$585$$ 0 0
$$586$$ 42.5774 1.75886
$$587$$ 13.1365 0.542203 0.271102 0.962551i $$-0.412612\pi$$
0.271102 + 0.962551i $$0.412612\pi$$
$$588$$ 0 0
$$589$$ −6.17031 −0.254243
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −7.93248 −0.326023
$$593$$ 11.3675 0.466807 0.233403 0.972380i $$-0.425014\pi$$
0.233403 + 0.972380i $$0.425014\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 78.6894 3.22324
$$597$$ 0 0
$$598$$ 14.6988 0.601079
$$599$$ 8.98828 0.367251 0.183626 0.982996i $$-0.441217\pi$$
0.183626 + 0.982996i $$0.441217\pi$$
$$600$$ 0 0
$$601$$ −5.15704 −0.210360 −0.105180 0.994453i $$-0.533542\pi$$
−0.105180 + 0.994453i $$0.533542\pi$$
$$602$$ 5.12001 0.208676
$$603$$ 0 0
$$604$$ 36.4271 1.48220
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 36.3436 1.47514 0.737570 0.675271i $$-0.235973\pi$$
0.737570 + 0.675271i $$0.235973\pi$$
$$608$$ −8.94238 −0.362661
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −17.4122 −0.704423
$$612$$ 0 0
$$613$$ −18.9943 −0.767171 −0.383585 0.923505i $$-0.625311\pi$$
−0.383585 + 0.923505i $$0.625311\pi$$
$$614$$ 65.3855 2.63874
$$615$$ 0 0
$$616$$ −57.2251 −2.30567
$$617$$ −24.5060 −0.986574 −0.493287 0.869867i $$-0.664205\pi$$
−0.493287 + 0.869867i $$0.664205\pi$$
$$618$$ 0 0
$$619$$ 12.5314 0.503680 0.251840 0.967769i $$-0.418964\pi$$
0.251840 + 0.967769i $$0.418964\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −43.5049 −1.74439
$$623$$ 3.29858 0.132155
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 57.1596 2.28456
$$627$$ 0 0
$$628$$ −45.6396 −1.82122
$$629$$ −3.44294 −0.137279
$$630$$ 0 0
$$631$$ 17.8019 0.708681 0.354340 0.935116i $$-0.384705\pi$$
0.354340 + 0.935116i $$0.384705\pi$$
$$632$$ 42.8512 1.70453
$$633$$ 0 0
$$634$$ −67.2493 −2.67081
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −3.01758 −0.119561
$$638$$ −9.67546 −0.383055
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −20.3973 −0.805644 −0.402822 0.915278i $$-0.631971\pi$$
−0.402822 + 0.915278i $$0.631971\pi$$
$$642$$ 0 0
$$643$$ 23.6531 0.932787 0.466394 0.884577i $$-0.345553\pi$$
0.466394 + 0.884577i $$0.345553\pi$$
$$644$$ 38.3115 1.50968
$$645$$ 0 0
$$646$$ −10.8735 −0.427812
$$647$$ −17.5789 −0.691096 −0.345548 0.938401i $$-0.612307\pi$$
−0.345548 + 0.938401i $$0.612307\pi$$
$$648$$ 0 0
$$649$$ −31.9252 −1.25317
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −76.7702 −3.00655
$$653$$ 10.4035 0.407121 0.203560 0.979062i $$-0.434749\pi$$
0.203560 + 0.979062i $$0.434749\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 93.1172 3.63562
$$657$$ 0 0
$$658$$ −64.9852 −2.53339
$$659$$ 43.5638 1.69700 0.848502 0.529192i $$-0.177505\pi$$
0.848502 + 0.529192i $$0.177505\pi$$
$$660$$ 0 0
$$661$$ 18.2404 0.709469 0.354734 0.934967i $$-0.384571\pi$$
0.354734 + 0.934967i $$0.384571\pi$$
$$662$$ 62.6857 2.43635
$$663$$ 0 0
$$664$$ 61.2327 2.37629
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 3.67991 0.142487
$$668$$ −21.9661 −0.849895
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 40.8595 1.57736
$$672$$ 0 0
$$673$$ 25.8626 0.996929 0.498464 0.866910i $$-0.333897\pi$$
0.498464 + 0.866910i $$0.333897\pi$$
$$674$$ 60.5383 2.33185
$$675$$ 0 0
$$676$$ −49.0566 −1.88679
$$677$$ 30.3287 1.16563 0.582813 0.812606i $$-0.301952\pi$$
0.582813 + 0.812606i $$0.301952\pi$$
$$678$$ 0 0
$$679$$ −27.8768 −1.06981
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −50.2068 −1.92252
$$683$$ 39.4637 1.51004 0.755018 0.655704i $$-0.227628\pi$$
0.755018 + 0.655704i $$0.227628\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −51.7871 −1.97724
$$687$$ 0 0
$$688$$ 7.23599 0.275870
$$689$$ 16.5306 0.629766
$$690$$ 0 0
$$691$$ −24.5853 −0.935269 −0.467635 0.883922i $$-0.654894\pi$$
−0.467635 + 0.883922i $$0.654894\pi$$
$$692$$ −91.7805 −3.48897
$$693$$ 0 0
$$694$$ −77.1196 −2.92742
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 40.4158 1.53086
$$698$$ −44.0139 −1.66595
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −34.8635 −1.31678 −0.658388 0.752678i $$-0.728762\pi$$
−0.658388 + 0.752678i $$0.728762\pi$$
$$702$$ 0 0
$$703$$ 1.15285 0.0434804
$$704$$ −11.2769 −0.425012
$$705$$ 0 0
$$706$$ −16.4833 −0.620356
$$707$$ 7.76091 0.291879
$$708$$ 0 0
$$709$$ −24.4622 −0.918698 −0.459349 0.888256i $$-0.651917\pi$$
−0.459349 + 0.888256i $$0.651917\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 9.93868 0.372468
$$713$$ 19.0954 0.715128
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −34.3613 −1.28414
$$717$$ 0 0
$$718$$ 75.3214 2.81097
$$719$$ −27.1333 −1.01190 −0.505951 0.862562i $$-0.668858\pi$$
−0.505951 + 0.862562i $$0.668858\pi$$
$$720$$ 0 0
$$721$$ −40.9128 −1.52367
$$722$$ −45.2843 −1.68531
$$723$$ 0 0
$$724$$ 91.8780 3.41462
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −33.9854 −1.26045 −0.630224 0.776413i $$-0.717037\pi$$
−0.630224 + 0.776413i $$0.717037\pi$$
$$728$$ 23.6244 0.875578
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 3.14065 0.116161
$$732$$ 0 0
$$733$$ −17.2131 −0.635779 −0.317889 0.948128i $$-0.602974\pi$$
−0.317889 + 0.948128i $$0.602974\pi$$
$$734$$ 34.2268 1.26333
$$735$$ 0 0
$$736$$ 27.6742 1.02008
$$737$$ −15.2244 −0.560796
$$738$$ 0 0
$$739$$ −12.5603 −0.462039 −0.231019 0.972949i $$-0.574206\pi$$
−0.231019 + 0.972949i $$0.574206\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 61.6949 2.26489
$$743$$ −43.0323 −1.57870 −0.789350 0.613943i $$-0.789582\pi$$
−0.789350 + 0.613943i $$0.789582\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 50.8073 1.86019
$$747$$ 0 0
$$748$$ −61.7891 −2.25923
$$749$$ −18.1641 −0.663702
$$750$$ 0 0
$$751$$ −12.4133 −0.452969 −0.226485 0.974015i $$-0.572723\pi$$
−0.226485 + 0.974015i $$0.572723\pi$$
$$752$$ −91.8421 −3.34914
$$753$$ 0 0
$$754$$ 3.99434 0.145465
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −18.0439 −0.655815 −0.327907 0.944710i $$-0.606343\pi$$
−0.327907 + 0.944710i $$0.606343\pi$$
$$758$$ 65.6127 2.38316
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −8.20477 −0.297423 −0.148711 0.988881i $$-0.547513\pi$$
−0.148711 + 0.988881i $$0.547513\pi$$
$$762$$ 0 0
$$763$$ −31.5013 −1.14043
$$764$$ −26.2286 −0.948919
$$765$$ 0 0
$$766$$ −1.39768 −0.0505001
$$767$$ 13.1797 0.475893
$$768$$ 0 0
$$769$$ −23.5524 −0.849320 −0.424660 0.905353i $$-0.639606\pi$$
−0.424660 + 0.905353i $$0.639606\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −68.4859 −2.46486
$$773$$ 27.2326 0.979487 0.489743 0.871867i $$-0.337090\pi$$
0.489743 + 0.871867i $$0.337090\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −83.9933 −3.01518
$$777$$ 0 0
$$778$$ −51.3203 −1.83992
$$779$$ −13.5330 −0.484868
$$780$$ 0 0
$$781$$ −23.9657 −0.857559
$$782$$ 33.6504 1.20334
$$783$$ 0 0
$$784$$ −15.9164 −0.568444
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −20.2585 −0.722137 −0.361069 0.932539i $$-0.617588\pi$$
−0.361069 + 0.932539i $$0.617588\pi$$
$$788$$ 125.653 4.47621
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 1.52542 0.0542378
$$792$$ 0 0
$$793$$ −16.8681 −0.599004
$$794$$ −24.2125 −0.859269
$$795$$ 0 0
$$796$$ −44.1546 −1.56502
$$797$$ −27.8574 −0.986761 −0.493381 0.869814i $$-0.664239\pi$$
−0.493381 + 0.869814i $$0.664239\pi$$
$$798$$ 0 0
$$799$$ −39.8624 −1.41023
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 24.1427 0.852510
$$803$$ −12.2499 −0.432289
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 20.7270 0.730077
$$807$$ 0 0
$$808$$ 23.3838 0.822638
$$809$$ −17.5060 −0.615477 −0.307739 0.951471i $$-0.599572\pi$$
−0.307739 + 0.951471i $$0.599572\pi$$
$$810$$ 0 0
$$811$$ 39.1690 1.37541 0.687704 0.725991i $$-0.258619\pi$$
0.687704 + 0.725991i $$0.258619\pi$$
$$812$$ 10.4110 0.365354
$$813$$ 0 0
$$814$$ 9.38053 0.328787
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −1.05162 −0.0367917
$$818$$ 50.7803 1.77549
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 19.8338 0.692203 0.346101 0.938197i $$-0.387505\pi$$
0.346101 + 0.938197i $$0.387505\pi$$
$$822$$ 0 0
$$823$$ −10.3727 −0.361568 −0.180784 0.983523i $$-0.557864\pi$$
−0.180784 + 0.983523i $$0.557864\pi$$
$$824$$ −123.271 −4.29434
$$825$$ 0 0
$$826$$ 49.1889 1.71150
$$827$$ 3.35272 0.116586 0.0582928 0.998300i $$-0.481434\pi$$
0.0582928 + 0.998300i $$0.481434\pi$$
$$828$$ 0 0
$$829$$ −43.0572 −1.49544 −0.747719 0.664015i $$-0.768851\pi$$
−0.747719 + 0.664015i $$0.768851\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 4.65545 0.161399
$$833$$ −6.90823 −0.239356
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 20.6896 0.715566
$$837$$ 0 0
$$838$$ −16.0688 −0.555086
$$839$$ 39.3736 1.35933 0.679664 0.733523i $$-0.262125\pi$$
0.679664 + 0.733523i $$0.262125\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ 16.2669 0.560595
$$843$$ 0 0
$$844$$ 99.8965 3.43858
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 7.01089 0.240897
$$848$$ 87.1919 2.99418
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −3.56774 −0.122301
$$852$$ 0 0
$$853$$ 45.2743 1.55016 0.775081 0.631862i $$-0.217709\pi$$
0.775081 + 0.631862i $$0.217709\pi$$
$$854$$ −62.9544 −2.15426
$$855$$ 0 0
$$856$$ −54.7288 −1.87059
$$857$$ 24.1619 0.825353 0.412677 0.910878i $$-0.364594\pi$$
0.412677 + 0.910878i $$0.364594\pi$$
$$858$$ 0 0
$$859$$ −22.3680 −0.763185 −0.381593 0.924331i $$-0.624624\pi$$
−0.381593 + 0.924331i $$0.624624\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −97.3371 −3.31531
$$863$$ −2.98467 −0.101599 −0.0507997 0.998709i $$-0.516177\pi$$
−0.0507997 + 0.998709i $$0.516177\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −97.4505 −3.31150
$$867$$ 0 0
$$868$$ 54.0236 1.83368
$$869$$ −23.7688 −0.806301
$$870$$ 0 0
$$871$$ 6.28510 0.212963
$$872$$ −94.9141 −3.21420
$$873$$ 0 0
$$874$$ −11.2676 −0.381133
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −29.5037 −0.996269 −0.498135 0.867100i $$-0.665981\pi$$
−0.498135 + 0.867100i $$0.665981\pi$$
$$878$$ −39.9945 −1.34975
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 52.0937 1.75508 0.877541 0.479502i $$-0.159183\pi$$
0.877541 + 0.479502i $$0.159183\pi$$
$$882$$ 0 0
$$883$$ −22.2221 −0.747832 −0.373916 0.927463i $$-0.621985\pi$$
−0.373916 + 0.927463i $$0.621985\pi$$
$$884$$ 25.5085 0.857944
$$885$$ 0 0
$$886$$ 13.4988 0.453502
$$887$$ −38.6220 −1.29680 −0.648400 0.761300i $$-0.724562\pi$$
−0.648400 + 0.761300i $$0.724562\pi$$
$$888$$ 0 0
$$889$$ 18.8838 0.633341
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −104.255 −3.49072
$$893$$ 13.3476 0.446662
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −16.4405 −0.549238
$$897$$ 0 0
$$898$$ −72.1321 −2.40708
$$899$$ 5.18909 0.173066
$$900$$ 0 0
$$901$$ 37.8440 1.26077
$$902$$ −110.116 −3.66645
$$903$$ 0 0
$$904$$ 4.59613 0.152865
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 29.7861 0.989030 0.494515 0.869169i $$-0.335346\pi$$
0.494515 + 0.869169i $$0.335346\pi$$
$$908$$ 38.8080 1.28789
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 53.7163 1.77970 0.889850 0.456254i $$-0.150809\pi$$
0.889850 + 0.456254i $$0.150809\pi$$
$$912$$ 0 0
$$913$$ −33.9647 −1.12407
$$914$$ −4.90258 −0.162163
$$915$$ 0 0
$$916$$ −53.4807 −1.76705
$$917$$ 16.5089 0.545173
$$918$$ 0 0
$$919$$ 52.0358 1.71650 0.858251 0.513231i $$-0.171551\pi$$
0.858251 + 0.513231i $$0.171551\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 78.6137 2.58900
$$923$$ 9.89379 0.325658
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −84.7770 −2.78595
$$927$$ 0 0
$$928$$ 7.52034 0.246867
$$929$$ −52.5081 −1.72273 −0.861367 0.507982i $$-0.830391\pi$$
−0.861367 + 0.507982i $$0.830391\pi$$
$$930$$ 0 0
$$931$$ 2.31317 0.0758112
$$932$$ −40.9963 −1.34288
$$933$$ 0 0
$$934$$ 11.4015 0.373068
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −46.1947 −1.50912 −0.754558 0.656233i $$-0.772149\pi$$
−0.754558 + 0.656233i $$0.772149\pi$$
$$938$$ 23.4570 0.765898
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 29.8642 0.973545 0.486773 0.873529i $$-0.338174\pi$$
0.486773 + 0.873529i $$0.338174\pi$$
$$942$$ 0 0
$$943$$ 41.8807 1.36382
$$944$$ 69.5175 2.26260
$$945$$ 0 0
$$946$$ −8.55690 −0.278209
$$947$$ −11.7523 −0.381898 −0.190949 0.981600i $$-0.561157\pi$$
−0.190949 + 0.981600i $$0.561157\pi$$
$$948$$ 0 0
$$949$$ 5.05714 0.164162
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 54.0840 1.75287
$$953$$ 17.1653 0.556040 0.278020 0.960575i $$-0.410322\pi$$
0.278020 + 0.960575i $$0.410322\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −64.6147 −2.08979
$$957$$ 0 0
$$958$$ −21.2972 −0.688081
$$959$$ −4.05480 −0.130936
$$960$$ 0 0
$$961$$ −4.07332 −0.131397
$$962$$ −3.87258 −0.124857
$$963$$ 0 0
$$964$$ −112.126 −3.61135
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −20.2076 −0.649831 −0.324916 0.945743i $$-0.605336\pi$$
−0.324916 + 0.945743i $$0.605336\pi$$
$$968$$ 21.1239 0.678949
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 24.7827 0.795315 0.397658 0.917534i $$-0.369823\pi$$
0.397658 + 0.917534i $$0.369823\pi$$
$$972$$ 0 0
$$973$$ 1.59053 0.0509902
$$974$$ −102.816 −3.29444
$$975$$ 0 0
$$976$$ −88.9720 −2.84792
$$977$$ −38.5423 −1.23308 −0.616539 0.787324i $$-0.711466\pi$$
−0.616539 + 0.787324i $$0.711466\pi$$
$$978$$ 0 0
$$979$$ −5.51281 −0.176190
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −85.9787 −2.74369
$$983$$ −13.0762 −0.417065 −0.208532 0.978015i $$-0.566869\pi$$
−0.208532 + 0.978015i $$0.566869\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 9.14436 0.291216
$$987$$ 0 0
$$988$$ −8.54135 −0.271737
$$989$$ 3.25449 0.103487
$$990$$ 0 0
$$991$$ −36.8186 −1.16958 −0.584791 0.811184i $$-0.698823\pi$$
−0.584791 + 0.811184i $$0.698823\pi$$
$$992$$ 39.0237 1.23900
$$993$$ 0 0
$$994$$ 36.9252 1.17120
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 18.5570 0.587707 0.293854 0.955850i $$-0.405062\pi$$
0.293854 + 0.955850i $$0.405062\pi$$
$$998$$ −8.05302 −0.254914
$$999$$ 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 6525.2.a.bw.1.7 7
3.2 odd 2 6525.2.a.bv.1.1 7
5.4 even 2 1305.2.a.s.1.1 7
15.14 odd 2 1305.2.a.t.1.7 yes 7

By twisted newform
Twist Min Dim Char Parity Ord Type
1305.2.a.s.1.1 7 5.4 even 2
1305.2.a.t.1.7 yes 7 15.14 odd 2
6525.2.a.bv.1.1 7 3.2 odd 2
6525.2.a.bw.1.7 7 1.1 even 1 trivial