LMFDB - Embedded newform 1792.3.g.f.127.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1792,3,Mod(127,1792)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1792, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1792.127");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1792 = 2^{8} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1792.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$48.8284633734$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1997017344.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 14x^{6} + 53x^{4} + 56x^{2} + 4 $ x^8 + 14x^6 + 53x^4 + 56*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.8
Root			$2.92812i$ of defining polynomial
Character	$\chi$	$=$	1792.127
Dual form			1792.3.g.f.127.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.85623 q^{3} +5.78167i q^{5} -2.64575i q^{7} +25.2955 q^{9} +O(q^{10})$	$q+5.85623 q^{3} +5.78167i q^{5} -2.64575i q^{7} +25.2955 q^{9} +3.01966 q^{11} +9.78167i q^{13} +33.8588i q^{15} -11.6027 q^{17} +25.5687 q^{19} -15.4941i q^{21} -26.1571i q^{23} -8.42771 q^{25} +95.4301 q^{27} +1.56334i q^{29} -12.0107i q^{31} +17.6839 q^{33} +15.2969 q^{35} +70.6721i q^{37} +57.2837i q^{39} +49.8775 q^{41} -73.2730 q^{43} +146.250i q^{45} -44.2248i q^{47} -7.00000 q^{49} -67.9479 q^{51} +54.2355i q^{53} +17.4587i q^{55} +149.736 q^{57} +12.4706 q^{59} +35.6770i q^{61} -66.9255i q^{63} -56.5544 q^{65} +24.4891 q^{67} -153.182i q^{69} -11.0480i q^{71} -74.3713 q^{73} -49.3547 q^{75} -7.98928i q^{77} -22.8912i q^{79} +331.202 q^{81} +48.1934 q^{83} -67.0828i q^{85} +9.15529i q^{87} -67.4801 q^{89} +25.8799 q^{91} -70.3376i q^{93} +147.830i q^{95} -7.75949 q^{97} +76.3838 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} + 40 q^{9}+O(q^{10}) $ 8 * q + 8 * q^3 + 40 * q^9	$ 8 q + 8 q^{3} + 40 q^{9} + 32 q^{11} - 16 q^{17} + 88 q^{19} - 104 q^{25} + 176 q^{27} + 56 q^{35} - 144 q^{41} - 224 q^{43} - 56 q^{49} + 16 q^{51} + 400 q^{57} + 232 q^{59} - 304 q^{65} - 368 q^{67} - 272 q^{73} - 664 q^{75} + 504 q^{81} - 424 q^{83} + 80 q^{89} + 56 q^{91} + 528 q^{97} + 544 q^{99}+O(q^{100}) $ 8 * q + 8 * q^3 + 40 * q^9 + 32 * q^11 - 16 * q^17 + 88 * q^19 - 104 * q^25 + 176 * q^27 + 56 * q^35 - 144 * q^41 - 224 * q^43 - 56 * q^49 + 16 * q^51 + 400 * q^57 + 232 * q^59 - 304 * q^65 - 368 * q^67 - 272 * q^73 - 664 * q^75 + 504 * q^81 - 424 * q^83 + 80 * q^89 + 56 * q^91 + 528 * q^97 + 544 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times$.

$n$	$1023$	$1025$	$1541$
$\chi(n)$	$-1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	5.85623	1.95208	0.976039	−	0.217596i	$-0.0698215\pi$
$3$	5.85623	1.95208	0.976039	+	0.217596i	$0.0698215\pi$
$4$	0	0
$5$	5.78167i	1.15633i	0.815918	+	0.578167i	$0.196232\pi$
$5$	5.78167i	1.15633i	−0.815918	+	0.578167i	$0.803768\pi$
$6$	0	0
$7$	− 2.64575i	− 0.377964i
$7$	− 2.64575i	− 0.377964i
$8$	0	0
$9$	25.2955	2.81061
$10$	0	0
$11$	3.01966	0.274515	0.137257	−	0.990535i	$-0.456171\pi$
$11$	3.01966	0.274515	0.137257	+	0.990535i	$0.456171\pi$
$12$	0	0
$13$	9.78167i	0.752436i	0.926531	+	0.376218i	$0.122776\pi$
$13$	9.78167i	0.752436i	−0.926531	+	0.376218i	$0.877224\pi$
$14$	0	0
$15$	33.8588i	2.25725i
$16$	0	0
$17$	−11.6027	−0.682510	−0.341255	−	0.939971i	$-0.610852\pi$
$17$	−11.6027	−0.682510	−0.341255	+	0.939971i	$0.610852\pi$
$18$	0	0
$19$	25.5687	1.34572	0.672861	−	0.739769i	$-0.265065\pi$
$19$	25.5687	1.34572	0.672861	+	0.739769i	$0.265065\pi$
$20$	0	0
$21$	− 15.4941i	− 0.737816i
$22$	0	0
$23$	− 26.1571i	− 1.13726i	−0.822592	−	0.568632i	$-0.807473\pi$
$23$	− 26.1571i	− 1.13726i	0.822592	−	0.568632i	$-0.192527\pi$
$24$	0	0
$25$	−8.42771	−0.337109
$26$	0	0
$27$	95.4301	3.53445
$28$	0	0
$29$	1.56334i	0.0539083i	0.999637	+	0.0269542i	$0.00858081\pi$
$29$	1.56334i	0.0539083i	−0.999637	+	0.0269542i	$0.991419\pi$
$30$	0	0
$31$	− 12.0107i	− 0.387443i	−0.981057	−	0.193721i	$-0.937944\pi$
$31$	− 12.0107i	− 0.387443i	0.981057	−	0.193721i	$-0.0620557\pi$
$32$	0	0
$33$	17.6839	0.535875
$34$	0	0
$35$	15.2969	0.437053
$36$	0	0
$37$	70.6721i	1.91006i	0.296513	+	0.955029i	$0.404176\pi$
$37$	70.6721i	1.91006i	−0.296513	+	0.955029i	$0.595824\pi$
$38$	0	0
$39$	57.2837i	1.46881i
$40$	0	0
$41$	49.8775	1.21652	0.608262	−	0.793736i	$-0.291867\pi$
$41$	49.8775	1.21652	0.608262	+	0.793736i	$0.291867\pi$
$42$	0	0
$43$	−73.2730	−1.70402	−0.852012	−	0.523522i	$-0.824618\pi$
$43$	−73.2730	−1.70402	−0.852012	+	0.523522i	$0.824618\pi$
$44$	0	0
$45$	146.250i	3.25000i
$46$	0	0
$47$	− 44.2248i	− 0.940952i	−0.882413	−	0.470476i	$-0.844082\pi$
$47$	− 44.2248i	− 0.940952i	0.882413	−	0.470476i	$-0.155918\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	−67.9479	−1.33231
$52$	0	0
$53$	54.2355i	1.02331i	0.859191	+	0.511655i	$0.170968\pi$
$53$	54.2355i	1.02331i	−0.859191	+	0.511655i	$0.829032\pi$
$54$	0	0
$55$	17.4587i	0.317431i
$56$	0	0
$57$	149.736	2.62695
$58$	0	0
$59$	12.4706	0.211367	0.105683	−	0.994400i	$-0.466297\pi$
$59$	12.4706	0.211367	0.105683	+	0.994400i	$0.466297\pi$
$60$	0	0
$61$	35.6770i	0.584869i	0.956285	+	0.292435i	$0.0944654\pi$
$61$	35.6770i	0.584869i	−0.956285	+	0.292435i	$0.905535\pi$
$62$	0	0
$63$	− 66.9255i	− 1.06231i
$64$	0	0
$65$	−56.5544	−0.870068
$66$	0	0
$67$	24.4891	0.365509	0.182754	−	0.983159i	$-0.441499\pi$
$67$	24.4891	0.365509	0.182754	+	0.983159i	$0.441499\pi$
$68$	0	0
$69$	− 153.182i	− 2.22003i
$70$	0	0
$71$	− 11.0480i	− 0.155606i	−0.996969	−	0.0778030i	$-0.975209\pi$
$71$	− 11.0480i	− 0.155606i	0.996969	−	0.0778030i	$-0.0247905\pi$
$72$	0	0
$73$	−74.3713	−1.01879	−0.509393	−	0.860534i	$-0.670130\pi$
$73$	−74.3713	−1.01879	−0.509393	+	0.860534i	$0.670130\pi$
$74$	0	0
$75$	−49.3547	−0.658062
$76$	0	0
$77$	− 7.98928i	− 0.103757i
$78$	0	0
$79$	− 22.8912i	− 0.289762i	−0.989449	−	0.144881i	$-0.953720\pi$
$79$	− 22.8912i	− 0.289762i	0.989449	−	0.144881i	$-0.0462800\pi$
$80$	0	0
$81$	331.202	4.08891
$82$	0	0
$83$	48.1934	0.580644	0.290322	−	0.956929i	$-0.406238\pi$
$83$	48.1934	0.580644	0.290322	+	0.956929i	$0.406238\pi$
$84$	0	0
$85$	− 67.0828i	− 0.789210i
$86$	0	0
$87$	9.15529i	0.105233i
$88$	0	0
$89$	−67.4801	−0.758204	−0.379102	−	0.925355i	$-0.623767\pi$
$89$	−67.4801	−0.758204	−0.379102	+	0.925355i	$0.623767\pi$
$90$	0	0
$91$	25.8799	0.284394
$92$	0	0
$93$	− 70.3376i	− 0.756318i
$94$	0	0
$95$	147.830i	1.55610i
$96$	0	0
$97$	−7.75949	−0.0799947	−0.0399974	−	0.999200i	$-0.512735\pi$
$97$	−7.75949	−0.0799947	−0.0399974	+	0.999200i	$0.512735\pi$
$98$	0	0
$99$	76.3838	0.771554
$100$	0	0
$101$	102.524i	1.01509i	0.861624	+	0.507546i	$0.169447\pi$
$101$	102.524i	1.01509i	−0.861624	+	0.507546i	$0.830553\pi$
$102$	0	0
$103$	− 14.6874i	− 0.142596i	−0.997455	−	0.0712980i	$-0.977286\pi$
$103$	− 14.6874i	− 0.142596i	0.997455	−	0.0712980i	$-0.0227141\pi$
$104$	0	0
$105$	89.5820	0.853162
$106$	0	0
$107$	−70.0212	−0.654404	−0.327202	−	0.944954i	$-0.606106\pi$
$107$	−70.0212	−0.654404	−0.327202	+	0.944954i	$0.606106\pi$
$108$	0	0
$109$	− 65.9204i	− 0.604774i	−0.953185	−	0.302387i	$-0.902217\pi$
$109$	− 65.9204i	− 0.604774i	0.953185	−	0.302387i	$-0.0977835\pi$
$110$	0	0
$111$	413.873i	3.72858i
$112$	0	0
$113$	83.2184	0.736446	0.368223	−	0.929737i	$-0.379966\pi$
$113$	83.2184	0.736446	0.368223	+	0.929737i	$0.379966\pi$
$114$	0	0
$115$	151.232	1.31506
$116$	0	0
$117$	247.432i	2.11480i
$118$	0	0
$119$	30.6978i	0.257965i
$120$	0	0
$121$	−111.882	−0.924642
$122$	0	0
$123$	292.094	2.37475
$124$	0	0
$125$	95.8155i	0.766524i
$126$	0	0
$127$	− 102.292i	− 0.805452i	−0.915321	−	0.402726i	$-0.868063\pi$
$127$	− 102.292i	− 0.805452i	0.915321	−	0.402726i	$-0.131937\pi$
$128$	0	0
$129$	−429.104	−3.32639
$130$	0	0
$131$	89.1759	0.680732	0.340366	−	0.940293i	$-0.389449\pi$
$131$	89.1759	0.680732	0.340366	+	0.940293i	$0.389449\pi$
$132$	0	0
$133$	− 67.6484i	− 0.508635i
$134$	0	0
$135$	551.745i	4.08700i
$136$	0	0
$137$	11.3800	0.0830660	0.0415330	−	0.999137i	$-0.486776\pi$
$137$	11.3800	0.0830660	0.0415330	+	0.999137i	$0.486776\pi$
$138$	0	0
$139$	−121.256	−0.872343	−0.436172	−	0.899863i	$-0.643666\pi$
$139$	−121.256	−0.872343	−0.436172	+	0.899863i	$0.643666\pi$
$140$	0	0
$141$	− 258.991i	− 1.83681i
$142$	0	0
$143$	29.5374i	0.206555i
$144$	0	0
$145$	−9.03872	−0.0623360
$146$	0	0
$147$	−40.9936	−0.278868
$148$	0	0
$149$	63.5731i	0.426665i	0.976980	+	0.213333i	$0.0684318\pi$
$149$	63.5731i	0.426665i	−0.976980	+	0.213333i	$0.931568\pi$
$150$	0	0
$151$	− 122.414i	− 0.810689i	−0.914164	−	0.405344i	$-0.867152\pi$
$151$	− 122.414i	− 0.810689i	0.914164	−	0.405344i	$-0.132848\pi$
$152$	0	0
$153$	−293.495	−1.91827
$154$	0	0
$155$	69.4420	0.448013
$156$	0	0
$157$	39.6070i	0.252274i	0.992013	+	0.126137i	$0.0402578\pi$
$157$	39.6070i	0.252274i	−0.992013	+	0.126137i	$0.959742\pi$
$158$	0	0
$159$	317.616i	1.99758i
$160$	0	0
$161$	−69.2051	−0.429845
$162$	0	0
$163$	−32.9017	−0.201851	−0.100925	−	0.994894i	$-0.532180\pi$
$163$	−32.9017	−0.201851	−0.100925	+	0.994894i	$0.532180\pi$
$164$	0	0
$165$	102.242i	0.619650i
$166$	0	0
$167$	98.0405i	0.587069i	0.955949	+	0.293534i	$0.0948315\pi$
$167$	98.0405i	0.587069i	−0.955949	+	0.293534i	$0.905168\pi$
$168$	0	0
$169$	73.3189	0.433840
$170$	0	0
$171$	646.772	3.78229
$172$	0	0
$173$	− 336.891i	− 1.94734i	−0.227951	−	0.973672i	$-0.573203\pi$
$173$	− 336.891i	− 1.94734i	0.227951	−	0.973672i	$-0.426797\pi$
$174$	0	0
$175$	22.2976i	0.127415i
$176$	0	0
$177$	73.0309	0.412604
$178$	0	0
$179$	−51.4537	−0.287451	−0.143725	−	0.989618i	$-0.545908\pi$
$179$	−51.4537	−0.287451	−0.143725	+	0.989618i	$0.545908\pi$
$180$	0	0
$181$	− 281.807i	− 1.55695i	−0.627678	−	0.778473i	$-0.715994\pi$
$181$	− 281.807i	− 1.55695i	0.627678	−	0.778473i	$-0.284006\pi$
$182$	0	0
$183$	208.933i	1.14171i
$184$	0	0
$185$	−408.603	−2.20866
$186$	0	0
$187$	−35.0362	−0.187359
$188$	0	0
$189$	− 252.484i	− 1.33590i
$190$	0	0
$191$	165.004i	0.863897i	0.901898	+	0.431949i	$0.142174\pi$
$191$	165.004i	0.863897i	−0.901898	+	0.431949i	$0.857826\pi$
$192$	0	0
$193$	−64.6933	−0.335199	−0.167599	−	0.985855i	$-0.553601\pi$
$193$	−64.6933	−0.335199	−0.167599	+	0.985855i	$0.553601\pi$
$194$	0	0
$195$	−331.196	−1.69844
$196$	0	0
$197$	− 189.712i	− 0.963003i	−0.876445	−	0.481501i	$-0.840092\pi$
$197$	− 189.712i	− 0.963003i	0.876445	−	0.481501i	$-0.159908\pi$
$198$	0	0
$199$	− 205.590i	− 1.03312i	−0.856252	−	0.516559i	$-0.827213\pi$
$199$	− 205.590i	− 1.03312i	0.856252	−	0.516559i	$-0.172787\pi$
$200$	0	0
$201$	143.414	0.713502
$202$	0	0
$203$	4.13621	0.0203754
$204$	0	0
$205$	288.375i	1.40671i
$206$	0	0
$207$	− 661.655i	− 3.19640i
$208$	0	0
$209$	77.2089	0.369421
$210$	0	0
$211$	220.725	1.04609	0.523045	−	0.852305i	$-0.324796\pi$
$211$	220.725	1.04609	0.523045	+	0.852305i	$0.324796\pi$
$212$	0	0
$213$	− 64.6998i	− 0.303755i
$214$	0	0
$215$	− 423.641i	− 1.97042i
$216$	0	0
$217$	−31.7774	−0.146440
$218$	0	0
$219$	−435.536	−1.98875
$220$	0	0
$221$	− 113.493i	− 0.513545i
$222$	0	0
$223$	− 163.489i	− 0.733134i	−0.930392	−	0.366567i	$-0.880533\pi$
$223$	− 163.489i	− 0.733134i	0.930392	−	0.366567i	$-0.119467\pi$
$224$	0	0
$225$	−213.183	−0.947480
$226$	0	0
$227$	−221.227	−0.974569	−0.487285	−	0.873243i	$-0.662013\pi$
$227$	−221.227	−0.974569	−0.487285	+	0.873243i	$0.662013\pi$
$228$	0	0
$229$	− 233.563i	− 1.01993i	−0.860196	−	0.509964i	$-0.829659\pi$
$229$	− 233.563i	− 1.01993i	0.860196	−	0.509964i	$-0.170341\pi$
$230$	0	0
$231$	− 46.7871i	− 0.202542i
$232$	0	0
$233$	13.2573	0.0568983	0.0284491	−	0.999595i	$-0.490943\pi$
$233$	13.2573	0.0568983	0.0284491	+	0.999595i	$0.490943\pi$
$234$	0	0
$235$	255.693	1.08806
$236$	0	0
$237$	− 134.056i	− 0.565638i
$238$	0	0
$239$	353.231i	1.47796i	0.673730	+	0.738978i	$0.264691\pi$
$239$	353.231i	1.47796i	−0.673730	+	0.738978i	$0.735309\pi$
$240$	0	0
$241$	181.261	0.752121	0.376060	−	0.926595i	$-0.377279\pi$
$241$	181.261	0.752121	0.376060	+	0.926595i	$0.377279\pi$
$242$	0	0
$243$	1080.72	4.44742
$244$	0	0
$245$	− 40.4717i	− 0.165191i
$246$	0	0
$247$	250.105i	1.01257i
$248$	0	0
$249$	282.232	1.13346
$250$	0	0
$251$	−98.4522	−0.392240	−0.196120	−	0.980580i	$-0.562834\pi$
$251$	−98.4522	−0.392240	−0.196120	+	0.980580i	$0.562834\pi$
$252$	0	0
$253$	− 78.9856i	− 0.312196i
$254$	0	0
$255$	− 392.853i	− 1.54060i
$256$	0	0
$257$	−279.527	−1.08765	−0.543827	−	0.839197i	$-0.683025\pi$
$257$	−279.527	−1.08765	−0.543827	+	0.839197i	$0.683025\pi$
$258$	0	0
$259$	186.981	0.721934
$260$	0	0
$261$	39.5455i	0.151515i
$262$	0	0
$263$	− 45.3085i	− 0.172276i	−0.996283	−	0.0861378i	$-0.972547\pi$
$263$	− 45.3085i	− 0.172276i	0.996283	−	0.0861378i	$-0.0274525\pi$
$264$	0	0
$265$	−313.572	−1.18329
$266$	0	0
$267$	−395.179	−1.48007
$268$	0	0
$269$	− 220.227i	− 0.818686i	−0.912380	−	0.409343i	$-0.865758\pi$
$269$	− 220.227i	− 0.818686i	0.912380	−	0.409343i	$-0.134242\pi$
$270$	0	0
$271$	− 54.8463i	− 0.202385i	−0.994867	−	0.101193i	$-0.967734\pi$
$271$	− 54.8463i	− 0.202385i	0.994867	−	0.101193i	$-0.0322658\pi$
$272$	0	0
$273$	151.559	0.555160
$274$	0	0
$275$	−25.4489	−0.0925413
$276$	0	0
$277$	− 320.078i	− 1.15552i	−0.816208	−	0.577758i	$-0.803928\pi$
$277$	− 320.078i	− 1.15552i	0.816208	−	0.577758i	$-0.196072\pi$
$278$	0	0
$279$	− 303.817i	− 1.08895i
$280$	0	0
$281$	−526.445	−1.87347	−0.936734	−	0.350041i	$-0.886168\pi$
$281$	−526.445	−1.87347	−0.936734	+	0.350041i	$0.886168\pi$
$282$	0	0
$283$	−331.932	−1.17290	−0.586452	−	0.809984i	$-0.699476\pi$
$283$	−331.932	−1.17290	−0.586452	+	0.809984i	$0.699476\pi$
$284$	0	0
$285$	865.726i	3.03763i
$286$	0	0
$287$	− 131.963i	− 0.459803i
$288$	0	0
$289$	−154.378	−0.534180
$290$	0	0
$291$	−45.4414	−0.156156
$292$	0	0
$293$	240.715i	0.821554i	0.911736	+	0.410777i	$0.134742\pi$
$293$	240.715i	0.821554i	−0.911736	+	0.410777i	$0.865258\pi$
$294$	0	0
$295$	72.1011i	0.244410i
$296$	0	0
$297$	288.167	0.970259
$298$	0	0
$299$	255.860	0.855718
$300$	0	0
$301$	193.862i	0.644061i
$302$	0	0
$303$	600.407i	1.98154i
$304$	0	0
$305$	−206.273	−0.676304
$306$	0	0
$307$	−221.862	−0.722678	−0.361339	−	0.932435i	$-0.617680\pi$
$307$	−221.862	−0.722678	−0.361339	+	0.932435i	$0.617680\pi$
$308$	0	0
$309$	− 86.0128i	− 0.278359i
$310$	0	0
$311$	476.048i	1.53070i	0.643614	+	0.765350i	$0.277434\pi$
$311$	476.048i	1.53070i	−0.643614	+	0.765350i	$0.722566\pi$
$312$	0	0
$313$	495.170	1.58201	0.791006	−	0.611808i	$-0.209558\pi$
$313$	495.170	1.58201	0.791006	+	0.611808i	$0.209558\pi$
$314$	0	0
$315$	386.941	1.22839
$316$	0	0
$317$	− 493.029i	− 1.55530i	−0.628699	−	0.777649i	$-0.716412\pi$
$317$	− 493.029i	− 1.55530i	0.628699	−	0.777649i	$-0.283588\pi$
$318$	0	0
$319$	4.72077i	0.0147986i
$320$	0	0
$321$	−410.060	−1.27745
$322$	0	0
$323$	−296.665	−0.918468
$324$	0	0
$325$	− 82.4371i	− 0.253653i
$326$	0	0
$327$	− 386.045i	− 1.18057i
$328$	0	0
$329$	−117.008	−0.355647
$330$	0	0
$331$	586.929	1.77320	0.886599	−	0.462539i	$-0.153061\pi$
$331$	586.929	1.77320	0.886599	+	0.462539i	$0.153061\pi$
$332$	0	0
$333$	1787.69i	5.36842i
$334$	0	0
$335$	141.588i	0.422650i
$336$	0	0
$337$	22.2636	0.0660639	0.0330320	−	0.999454i	$-0.489484\pi$
$337$	22.2636	0.0660639	0.0330320	+	0.999454i	$0.489484\pi$
$338$	0	0
$339$	487.347	1.43760
$340$	0	0
$341$	− 36.2683i	− 0.106359i
$342$	0	0
$343$	18.5203i	0.0539949i
$344$	0	0
$345$	885.647	2.56709
$346$	0	0
$347$	264.797	0.763103	0.381552	−	0.924348i	$-0.375390\pi$
$347$	264.797	0.763103	0.381552	+	0.924348i	$0.375390\pi$
$348$	0	0
$349$	101.730i	0.291489i	0.989322	+	0.145745i	$0.0465578\pi$
$349$	101.730i	0.291489i	−0.989322	+	0.145745i	$0.953442\pi$
$350$	0	0
$351$	933.466i	2.65945i
$352$	0	0
$353$	472.399	1.33824	0.669121	−	0.743154i	$-0.266671\pi$
$353$	472.399	1.33824	0.669121	+	0.743154i	$0.266671\pi$
$354$	0	0
$355$	63.8760	0.179933
$356$	0	0
$357$	179.773i	0.503567i
$358$	0	0
$359$	473.453i	1.31881i	0.751788	+	0.659405i	$0.229192\pi$
$359$	473.453i	1.31881i	−0.751788	+	0.659405i	$0.770808\pi$
$360$	0	0
$361$	292.758	0.810965
$362$	0	0
$363$	−655.205	−1.80497
$364$	0	0
$365$	− 429.991i	− 1.17806i
$366$	0	0
$367$	− 467.459i	− 1.27373i	−0.770975	−	0.636866i	$-0.780231\pi$
$367$	− 467.459i	− 1.27373i	0.770975	−	0.636866i	$-0.219769\pi$
$368$	0	0
$369$	1261.67	3.41917
$370$	0	0
$371$	143.494	0.386775
$372$	0	0
$373$	− 534.491i	− 1.43295i	−0.697613	−	0.716475i	$-0.745754\pi$
$373$	− 534.491i	− 1.43295i	0.697613	−	0.716475i	$-0.254246\pi$
$374$	0	0
$375$	561.118i	1.49631i
$376$	0	0
$377$	−15.2921	−0.0405626
$378$	0	0
$379$	204.355	0.539196	0.269598	−	0.962973i	$-0.413109\pi$
$379$	204.355	0.539196	0.269598	+	0.962973i	$0.413109\pi$
$380$	0	0
$381$	− 599.048i	− 1.57231i
$382$	0	0
$383$	− 622.590i	− 1.62556i	−0.582570	−	0.812781i	$-0.697953\pi$
$383$	− 622.590i	− 1.62556i	0.582570	−	0.812781i	$-0.302047\pi$
$384$	0	0
$385$	46.1914	0.119978
$386$	0	0
$387$	−1853.48	−4.78934
$388$	0	0
$389$	510.568i	1.31251i	0.754537	+	0.656257i	$0.227861\pi$
$389$	510.568i	1.31251i	−0.754537	+	0.656257i	$0.772139\pi$
$390$	0	0
$391$	303.492i	0.776194i
$392$	0	0
$393$	522.235	1.32884
$394$	0	0
$395$	132.349	0.335062
$396$	0	0
$397$	− 591.459i	− 1.48982i	−0.667165	−	0.744910i	$-0.732492\pi$
$397$	− 591.459i	− 1.48982i	0.667165	−	0.744910i	$-0.267508\pi$
$398$	0	0
$399$	− 396.165i	− 0.992895i
$400$	0	0
$401$	29.0538	0.0724534	0.0362267	−	0.999344i	$-0.488466\pi$
$401$	29.0538	0.0724534	0.0362267	+	0.999344i	$0.488466\pi$
$402$	0	0
$403$	117.485	0.291526
$404$	0	0
$405$	1914.90i	4.72815i
$406$	0	0
$407$	213.406i	0.524339i
$408$	0	0
$409$	−120.749	−0.295230	−0.147615	−	0.989045i	$-0.547160\pi$
$409$	−120.749	−0.295230	−0.147615	+	0.989045i	$0.547160\pi$
$410$	0	0
$411$	66.6442	0.162151
$412$	0	0
$413$	− 32.9942i	− 0.0798891i
$414$	0	0
$415$	278.638i	0.671418i
$416$	0	0
$417$	−710.102	−1.70288
$418$	0	0
$419$	379.985	0.906885	0.453443	−	0.891286i	$-0.350196\pi$
$419$	379.985	0.906885	0.453443	+	0.891286i	$0.350196\pi$
$420$	0	0
$421$	536.480i	1.27430i	0.770740	+	0.637149i	$0.219887\pi$
$421$	536.480i	1.27430i	−0.770740	+	0.637149i	$0.780113\pi$
$422$	0	0
$423$	− 1118.69i	− 2.64465i
$424$	0	0
$425$	97.7840	0.230080
$426$	0	0
$427$	94.3926	0.221060
$428$	0	0
$429$	172.978i	0.403211i
$430$	0	0
$431$	384.935i	0.893120i	0.894754	+	0.446560i	$0.147351\pi$
$431$	384.935i	0.893120i	−0.894754	+	0.446560i	$0.852649\pi$
$432$	0	0
$433$	−667.115	−1.54068	−0.770340	−	0.637633i	$-0.779914\pi$
$433$	−667.115	−1.54068	−0.770340	+	0.637633i	$0.779914\pi$
$434$	0	0
$435$	−52.9329	−0.121685
$436$	0	0
$437$	− 668.802i	− 1.53044i
$438$	0	0
$439$	− 521.733i	− 1.18846i	−0.804296	−	0.594229i	$-0.797457\pi$
$439$	− 521.733i	− 1.18846i	0.804296	−	0.594229i	$-0.202543\pi$
$440$	0	0
$441$	−177.068	−0.401515
$442$	0	0
$443$	−204.156	−0.460848	−0.230424	−	0.973090i	$-0.574011\pi$
$443$	−204.156	−0.460848	−0.230424	+	0.973090i	$0.574011\pi$
$444$	0	0
$445$	− 390.148i	− 0.876737i
$446$	0	0
$447$	372.299i	0.832884i
$448$	0	0
$449$	513.424	1.14348	0.571742	−	0.820434i	$-0.306268\pi$
$449$	513.424	1.14348	0.571742	+	0.820434i	$0.306268\pi$
$450$	0	0
$451$	150.613	0.333954
$452$	0	0
$453$	− 716.885i	− 1.58253i
$454$	0	0
$455$	149.629i	0.328855i
$456$	0	0
$457$	134.029	0.293281	0.146641	−	0.989190i	$-0.453154\pi$
$457$	134.029	0.293281	0.146641	+	0.989190i	$0.453154\pi$
$458$	0	0
$459$	−1107.24	−2.41230
$460$	0	0
$461$	8.06740i	0.0174998i	0.999962	+	0.00874990i	$0.00278521\pi$
$461$	8.06740i	0.0174998i	−0.999962	+	0.00874990i	$0.997215\pi$
$462$	0	0
$463$	− 837.707i	− 1.80930i	−0.426152	−	0.904652i	$-0.640131\pi$
$463$	− 837.707i	− 1.80930i	0.426152	−	0.904652i	$-0.359869\pi$
$464$	0	0
$465$	406.669	0.874556
$466$	0	0
$467$	−495.780	−1.06163	−0.530814	−	0.847489i	$-0.678114\pi$
$467$	−495.780	−1.06163	−0.530814	+	0.847489i	$0.678114\pi$
$468$	0	0
$469$	− 64.7920i	− 0.138149i
$470$	0	0
$471$	231.948i	0.492458i
$472$	0	0
$473$	−221.260	−0.467780
$474$	0	0
$475$	−215.486	−0.453654
$476$	0	0
$477$	1371.91i	2.87613i
$478$	0	0
$479$	− 225.993i	− 0.471802i	−0.971777	−	0.235901i	$-0.924196\pi$
$479$	− 225.993i	− 0.471802i	0.971777	−	0.235901i	$-0.0758042\pi$
$480$	0	0
$481$	−691.292	−1.43720
$482$	0	0
$483$	−405.281	−0.839091
$484$	0	0
$485$	− 44.8628i	− 0.0925006i
$486$	0	0
$487$	− 728.173i	− 1.49522i	−0.664137	−	0.747611i	$-0.731201\pi$
$487$	− 728.173i	− 1.49522i	0.664137	−	0.747611i	$-0.268799\pi$
$488$	0	0
$489$	−192.680	−0.394028
$490$	0	0
$491$	−724.908	−1.47639	−0.738196	−	0.674587i	$-0.764322\pi$
$491$	−724.908	−1.47639	−0.738196	+	0.674587i	$0.764322\pi$
$492$	0	0
$493$	− 18.1389i	− 0.0367930i
$494$	0	0
$495$	441.626i	0.892174i
$496$	0	0
$497$	−29.2303	−0.0588135
$498$	0	0
$499$	−691.520	−1.38581	−0.692906	−	0.721028i	$-0.743670\pi$
$499$	−691.520	−1.38581	−0.692906	+	0.721028i	$0.743670\pi$
$500$	0	0
$501$	574.148i	1.14600i
$502$	0	0
$503$	518.951i	1.03171i	0.856675	+	0.515856i	$0.172526\pi$
$503$	518.951i	1.03171i	−0.856675	+	0.515856i	$0.827474\pi$
$504$	0	0
$505$	−592.762	−1.17379
$506$	0	0
$507$	429.373	0.846889
$508$	0	0
$509$	823.772i	1.61841i	0.587525	+	0.809206i	$0.300102\pi$
$509$	823.772i	1.61841i	−0.587525	+	0.809206i	$0.699898\pi$
$510$	0	0
$511$	196.768i	0.385065i
$512$	0	0
$513$	2440.02	4.75638
$514$	0	0
$515$	84.9177	0.164889
$516$	0	0
$517$	− 133.544i	− 0.258305i
$518$	0	0
$519$	− 1972.91i	− 3.80137i
$520$	0	0
$521$	533.963	1.02488	0.512440	−	0.858723i	$-0.328742\pi$
$521$	533.963	1.02488	0.512440	+	0.858723i	$0.328742\pi$
$522$	0	0
$523$	−709.660	−1.35690	−0.678451	−	0.734646i	$-0.737348\pi$
$523$	−709.660	−1.35690	−0.678451	+	0.734646i	$0.737348\pi$
$524$	0	0
$525$	130.580i	0.248724i
$526$	0	0
$527$	139.356i	0.264433i
$528$	0	0
$529$	−155.192	−0.293369
$530$	0	0
$531$	315.450	0.594069
$532$	0	0
$533$	487.885i	0.915356i
$534$	0	0
$535$	− 404.839i	− 0.756709i
$536$	0	0
$537$	−301.325	−0.561126
$538$	0	0
$539$	−21.1376	−0.0392164
$540$	0	0
$541$	− 666.282i	− 1.23157i	−0.787913	−	0.615787i	$-0.788838\pi$
$541$	− 666.282i	− 1.23157i	0.787913	−	0.615787i	$-0.211162\pi$
$542$	0	0
$543$	− 1650.33i	− 3.03928i
$544$	0	0
$545$	381.130	0.699321
$546$	0	0
$547$	−65.3904	−0.119544	−0.0597718	−	0.998212i	$-0.519037\pi$
$547$	−65.3904	−0.119544	−0.0597718	+	0.998212i	$0.519037\pi$
$548$	0	0
$549$	902.467i	1.64384i
$550$	0	0
$551$	39.9726i	0.0725456i
$552$	0	0
$553$	−60.5644	−0.109520
$554$	0	0
$555$	−2392.87	−4.31149
$556$	0	0
$557$	− 671.474i	− 1.20552i	−0.797923	−	0.602759i	$-0.794068\pi$
$557$	− 671.474i	− 1.20552i	0.797923	−	0.602759i	$-0.205932\pi$
$558$	0	0
$559$	− 716.733i	− 1.28217i
$560$	0	0
$561$	−205.180	−0.365740
$562$	0	0
$563$	629.362	1.11787	0.558936	−	0.829211i	$-0.311210\pi$
$563$	629.362	1.11787	0.558936	+	0.829211i	$0.311210\pi$
$564$	0	0
$565$	481.142i	0.851578i
$566$	0	0
$567$	− 876.277i	− 1.54546i
$568$	0	0
$569$	49.5333	0.0870533	0.0435267	−	0.999052i	$-0.486141\pi$
$569$	49.5333	0.0870533	0.0435267	+	0.999052i	$0.486141\pi$
$570$	0	0
$571$	988.807	1.73171	0.865856	−	0.500294i	$-0.166775\pi$
$571$	988.807	1.73171	0.865856	+	0.500294i	$0.166775\pi$
$572$	0	0
$573$	966.304i	1.68639i
$574$	0	0
$575$	220.444i	0.383381i
$576$	0	0
$577$	−63.1355	−0.109420	−0.0547101	−	0.998502i	$-0.517423\pi$
$577$	−63.1355	−0.109420	−0.0547101	+	0.998502i	$0.517423\pi$
$578$	0	0
$579$	−378.859	−0.654334
$580$	0	0
$581$	− 127.508i	− 0.219463i
$582$	0	0
$583$	163.773i	0.280914i
$584$	0	0
$585$	−1430.57	−2.44542
$586$	0	0
$587$	−9.88169	−0.0168342	−0.00841711	−	0.999965i	$-0.502679\pi$
$587$	−9.88169	−0.0168342	−0.00841711	+	0.999965i	$0.502679\pi$
$588$	0	0
$589$	− 307.098i	− 0.521390i
$590$	0	0
$591$	− 1111.00i	− 1.87986i
$592$	0	0
$593$	−147.865	−0.249351	−0.124676	−	0.992198i	$-0.539789\pi$
$593$	−147.865	−0.249351	−0.124676	+	0.992198i	$0.539789\pi$
$594$	0	0
$595$	−177.484	−0.298293
$596$	0	0
$597$	− 1203.99i	− 2.01673i
$598$	0	0
$599$	464.007i	0.774636i	0.921946	+	0.387318i	$0.126599\pi$
$599$	464.007i	0.774636i	−0.921946	+	0.387318i	$0.873401\pi$
$600$	0	0
$601$	−215.359	−0.358334	−0.179167	−	0.983819i	$-0.557340\pi$
$601$	−215.359	−0.358334	−0.179167	+	0.983819i	$0.557340\pi$
$602$	0	0
$603$	619.463	1.02730
$604$	0	0
$605$	− 646.863i	− 1.06919i
$606$	0	0
$607$	− 310.053i	− 0.510796i	−0.966836	−	0.255398i	$-0.917794\pi$
$607$	− 310.053i	− 0.510796i	0.966836	−	0.255398i	$-0.0822065\pi$
$608$	0	0
$609$	24.2226	0.0397744
$610$	0	0
$611$	432.592	0.708007
$612$	0	0
$613$	791.288i	1.29084i	0.763826	+	0.645422i	$0.223319\pi$
$613$	791.288i	1.29084i	−0.763826	+	0.645422i	$0.776681\pi$
$614$	0	0
$615$	1688.79i	2.74600i
$616$	0	0
$617$	−1139.17	−1.84631	−0.923153	−	0.384433i	$-0.874397\pi$
$617$	−1139.17	−1.84631	−0.923153	+	0.384433i	$0.874397\pi$
$618$	0	0
$619$	419.673	0.677985	0.338992	−	0.940789i	$-0.389914\pi$
$619$	419.673	0.677985	0.338992	+	0.940789i	$0.389914\pi$
$620$	0	0
$621$	− 2496.17i	− 4.01960i
$622$	0	0
$623$	178.536i	0.286574i
$624$	0	0
$625$	−764.666	−1.22347
$626$	0	0
$627$	452.153	0.721138
$628$	0	0
$629$	− 819.985i	− 1.30363i
$630$	0	0
$631$	506.397i	0.802530i	0.915962	+	0.401265i	$0.131429\pi$
$631$	506.397i	0.802530i	−0.915962	+	0.401265i	$0.868571\pi$
$632$	0	0
$633$	1292.62	2.04205
$634$	0	0
$635$	591.421	0.931372
$636$	0	0
$637$	− 68.4717i	− 0.107491i
$638$	0	0
$639$	− 279.465i	− 0.437347i
$640$	0	0
$641$	−501.692	−0.782671	−0.391335	−	0.920248i	$-0.627987\pi$
$641$	−501.692	−0.782671	−0.391335	+	0.920248i	$0.627987\pi$
$642$	0	0
$643$	−691.217	−1.07499	−0.537493	−	0.843268i	$-0.680629\pi$
$643$	−691.217	−1.07499	−0.537493	+	0.843268i	$0.680629\pi$
$644$	0	0
$645$	− 2480.94i	− 3.84642i
$646$	0	0
$647$	711.553i	1.09977i	0.835239	+	0.549886i	$0.185329\pi$
$647$	711.553i	1.09977i	−0.835239	+	0.549886i	$0.814671\pi$
$648$	0	0
$649$	37.6571	0.0580233
$650$	0	0
$651$	−186.096	−0.285861
$652$	0	0
$653$	455.205i	0.697099i	0.937290	+	0.348549i	$0.113326\pi$
$653$	455.205i	0.697099i	−0.937290	+	0.348549i	$0.886674\pi$
$654$	0	0
$655$	515.586i	0.787154i
$656$	0	0
$657$	−1881.26	−2.86341
$658$	0	0
$659$	−156.616	−0.237656	−0.118828	−	0.992915i	$-0.537914\pi$
$659$	−156.616	−0.237656	−0.118828	+	0.992915i	$0.537914\pi$
$660$	0	0
$661$	− 241.002i	− 0.364602i	−0.983243	−	0.182301i	$-0.941646\pi$
$661$	− 241.002i	− 0.364602i	0.983243	−	0.182301i	$-0.0583545\pi$
$662$	0	0
$663$	− 664.644i	− 1.00248i
$664$	0	0
$665$	391.121	0.588152
$666$	0	0
$667$	40.8924	0.0613080
$668$	0	0
$669$	− 957.429i	− 1.43114i
$670$	0	0
$671$	107.733i	0.160555i
$672$	0	0
$673$	623.074	0.925816	0.462908	−	0.886406i	$-0.346806\pi$
$673$	623.074	0.925816	0.462908	+	0.886406i	$0.346806\pi$
$674$	0	0
$675$	−804.258	−1.19149
$676$	0	0
$677$	− 343.461i	− 0.507328i	−0.967292	−	0.253664i	$-0.918364\pi$
$677$	− 343.461i	− 0.507328i	0.967292	−	0.253664i	$-0.0816358\pi$
$678$	0	0
$679$	20.5297i	0.0302352i
$680$	0	0
$681$	−1295.56	−1.90243
$682$	0	0
$683$	909.578	1.33174	0.665870	−	0.746068i	$-0.268061\pi$
$683$	909.578	1.33174	0.665870	+	0.746068i	$0.268061\pi$
$684$	0	0
$685$	65.7957i	0.0960521i
$686$	0	0
$687$	− 1367.80i	− 1.99098i
$688$	0	0
$689$	−530.514	−0.769976
$690$	0	0
$691$	325.753	0.471423	0.235711	−	0.971823i	$-0.424258\pi$
$691$	325.753	0.471423	0.235711	+	0.971823i	$0.424258\pi$
$692$	0	0
$693$	− 202.093i	− 0.291620i
$694$	0	0
$695$	− 701.061i	− 1.00872i
$696$	0	0
$697$	−578.712	−0.830290
$698$	0	0
$699$	77.6379	0.111070
$700$	0	0
$701$	− 414.315i	− 0.591034i	−0.955338	−	0.295517i	$-0.904508\pi$
$701$	− 414.315i	− 0.591034i	0.955338	−	0.295517i	$-0.0954918\pi$
$702$	0	0
$703$	1806.99i	2.57040i
$704$	0	0
$705$	1497.40	2.12397
$706$	0	0
$707$	271.254	0.383669
$708$	0	0
$709$	− 1146.48i	− 1.61703i	−0.588472	−	0.808517i	$-0.700270\pi$
$709$	− 1146.48i	− 1.61703i	0.588472	−	0.808517i	$-0.299730\pi$
$710$	0	0
$711$	− 579.044i	− 0.814408i
$712$	0	0
$713$	−314.165	−0.440624
$714$	0	0
$715$	−170.775	−0.238847
$716$	0	0
$717$	2068.60i	2.88508i
$718$	0	0
$719$	387.316i	0.538687i	0.963044	+	0.269344i	$0.0868067\pi$
$719$	387.316i	0.538687i	−0.963044	+	0.269344i	$0.913193\pi$
$720$	0	0
$721$	−38.8592	−0.0538963
$722$	0	0
$723$	1061.51	1.46820
$724$	0	0
$725$	− 13.1754i	− 0.0181730i
$726$	0	0
$727$	− 620.563i	− 0.853594i	−0.904347	−	0.426797i	$-0.859642\pi$
$727$	− 620.563i	− 0.853594i	0.904347	−	0.426797i	$-0.140358\pi$
$728$	0	0
$729$	3348.15	4.59280
$730$	0	0
$731$	850.163	1.16301
$732$	0	0
$733$	96.8796i	0.132169i	0.997814	+	0.0660843i	$0.0210506\pi$
$733$	96.8796i	0.132169i	−0.997814	+	0.0660843i	$0.978949\pi$
$734$	0	0
$735$	− 237.012i	− 0.322465i
$736$	0	0
$737$	73.9488	0.100338
$738$	0	0
$739$	−782.913	−1.05942	−0.529711	−	0.848178i	$-0.677700\pi$
$739$	−782.913	−1.05942	−0.529711	+	0.848178i	$0.677700\pi$
$740$	0	0
$741$	1464.67i	1.97661i
$742$	0	0
$743$	48.0641i	0.0646892i	0.999477	+	0.0323446i	$0.0102974\pi$
$743$	48.0641i	0.0646892i	−0.999477	+	0.0323446i	$0.989703\pi$
$744$	0	0
$745$	−367.559	−0.493368
$746$	0	0
$747$	1219.08	1.63196
$748$	0	0
$749$	185.259i	0.247341i
$750$	0	0
$751$	284.489i	0.378814i	0.981899	+	0.189407i	$0.0606565\pi$
$751$	284.489i	0.378814i	−0.981899	+	0.189407i	$0.939343\pi$
$752$	0	0
$753$	−576.559	−0.765683
$754$	0	0
$755$	707.757	0.937427
$756$	0	0
$757$	797.514i	1.05352i	0.850014	+	0.526759i	$0.176593\pi$
$757$	797.514i	1.05352i	−0.850014	+	0.526759i	$0.823407\pi$
$758$	0	0
$759$	− 462.558i	− 0.609431i
$760$	0	0
$761$	−1004.15	−1.31951	−0.659756	−	0.751480i	$-0.729340\pi$
$761$	−1004.15	−1.31951	−0.659756	+	0.751480i	$0.729340\pi$
$762$	0	0
$763$	−174.409	−0.228583
$764$	0	0
$765$	− 1696.89i	− 2.21816i
$766$	0	0
$767$	121.984i	0.159040i
$768$	0	0
$769$	−959.769	−1.24807	−0.624037	−	0.781395i	$-0.714509\pi$
$769$	−959.769	−1.24807	−0.624037	+	0.781395i	$0.714509\pi$
$770$	0	0
$771$	−1636.98	−2.12319
$772$	0	0
$773$	94.4740i	0.122217i	0.998131	+	0.0611086i	$0.0194636\pi$
$773$	94.4740i	0.122217i	−0.998131	+	0.0611086i	$0.980536\pi$
$774$	0	0
$775$	101.223i	0.130610i
$776$	0	0
$777$	1095.00	1.40927
$778$	0	0
$779$	1275.30	1.63710
$780$	0	0
$781$	− 33.3613i	− 0.0427162i
$782$	0	0
$783$	149.190i	0.190536i
$784$	0	0
$785$	−228.994	−0.291713
$786$	0	0
$787$	−108.864	−0.138328	−0.0691642	−	0.997605i	$-0.522033\pi$
$787$	−108.864	−0.138328	−0.0691642	+	0.997605i	$0.522033\pi$
$788$	0	0
$789$	− 265.337i	− 0.336295i
$790$	0	0
$791$	− 220.175i	− 0.278351i
$792$	0	0
$793$	−348.981	−0.440077
$794$	0	0
$795$	−1836.35	−2.30987
$796$	0	0
$797$	887.200i	1.11317i	0.830789	+	0.556587i	$0.187889\pi$
$797$	887.200i	1.11317i	−0.830789	+	0.556587i	$0.812111\pi$
$798$	0	0
$799$	513.125i	0.642209i
$800$	0	0
$801$	−1706.94	−2.13101
$802$	0	0
$803$	−224.576	−0.279672
$804$	0	0
$805$	− 400.121i	− 0.497045i
$806$	0	0
$807$	− 1289.70i	− 1.59814i
$808$	0	0
$809$	−485.529	−0.600160	−0.300080	−	0.953914i	$-0.597013\pi$
$809$	−485.529	−0.600160	−0.300080	+	0.953914i	$0.597013\pi$
$810$	0	0
$811$	−1118.90	−1.37966	−0.689830	−	0.723972i	$-0.742315\pi$
$811$	−1118.90	−1.37966	−0.689830	+	0.723972i	$0.742315\pi$
$812$	0	0
$813$	− 321.193i	− 0.395071i
$814$	0	0
$815$	− 190.227i	− 0.233407i
$816$	0	0
$817$	−1873.50	−2.29314
$818$	0	0
$819$	654.643	0.799320
$820$	0	0
$821$	486.539i	0.592617i	0.955092	+	0.296308i	$0.0957556\pi$
$821$	486.539i	0.592617i	−0.955092	+	0.296308i	$0.904244\pi$
$822$	0	0
$823$	− 1049.45i	− 1.27515i	−0.770389	−	0.637575i	$-0.779938\pi$
$823$	− 1049.45i	− 1.27515i	0.770389	−	0.637575i	$-0.220062\pi$
$824$	0	0
$825$	−149.035	−0.180648
$826$	0	0
$827$	−1472.00	−1.77993	−0.889964	−	0.456031i	$-0.849271\pi$
$827$	−1472.00	−1.77993	−0.889964	+	0.456031i	$0.849271\pi$
$828$	0	0
$829$	1049.47i	1.26595i	0.774174	+	0.632973i	$0.218166\pi$
$829$	1049.47i	1.26595i	−0.774174	+	0.632973i	$0.781834\pi$
$830$	0	0
$831$	− 1874.45i	− 2.25566i
$832$	0	0
$833$	81.2187	0.0975014
$834$	0	0
$835$	−566.838	−0.678848
$836$	0	0
$837$	− 1146.18i	− 1.36940i
$838$	0	0
$839$	− 1456.31i	− 1.73576i	−0.496771	−	0.867882i	$-0.665481\pi$
$839$	− 1456.31i	− 1.73576i	0.496771	−	0.867882i	$-0.334519\pi$
$840$	0	0
$841$	838.556	0.997094
$842$	0	0
$843$	−3082.98	−3.65716
$844$	0	0
$845$	423.906i	0.501664i
$846$	0	0
$847$	296.011i	0.349482i
$848$	0	0
$849$	−1943.87	−2.28960
$850$	0	0
$851$	1848.58	2.17224
$852$	0	0
$853$	− 121.003i	− 0.141855i	−0.997481	−	0.0709277i	$-0.977404\pi$
$853$	− 121.003i	− 0.141855i	0.997481	−	0.0709277i	$-0.0225960\pi$
$854$	0	0
$855$	3739.42i	4.37360i
$856$	0	0
$857$	−1405.84	−1.64041	−0.820207	−	0.572066i	$-0.806142\pi$
$857$	−1405.84	−1.64041	−0.820207	+	0.572066i	$0.806142\pi$
$858$	0	0
$859$	−991.383	−1.15411	−0.577057	−	0.816704i	$-0.695799\pi$
$859$	−991.383	−1.15411	−0.577057	+	0.816704i	$0.695799\pi$
$860$	0	0
$861$	− 772.808i	− 0.897571i
$862$	0	0
$863$	− 284.139i	− 0.329245i	−0.986357	−	0.164623i	$-0.947359\pi$
$863$	− 284.139i	− 0.329245i	0.986357	−	0.164623i	$-0.0526406\pi$
$864$	0	0
$865$	1947.79	2.25178
$866$	0	0
$867$	−904.074	−1.04276
$868$	0	0
$869$	− 69.1238i	− 0.0795440i
$870$	0	0
$871$	239.544i	0.275022i
$872$	0	0
$873$	−196.280	−0.224834
$874$	0	0
$875$	253.504	0.289719
$876$	0	0
$877$	− 255.174i	− 0.290963i	−0.989361	−	0.145481i	$-0.953527\pi$
$877$	− 255.174i	− 0.290963i	0.989361	−	0.145481i	$-0.0464731\pi$
$878$	0	0
$879$	1409.68i	1.60374i
$880$	0	0
$881$	−421.803	−0.478778	−0.239389	−	0.970924i	$-0.576947\pi$
$881$	−421.803	−0.478778	−0.239389	+	0.970924i	$0.576947\pi$
$882$	0	0
$883$	−1069.35	−1.21105	−0.605524	−	0.795827i	$-0.707036\pi$
$883$	−1069.35	−1.21105	−0.605524	+	0.795827i	$0.707036\pi$
$884$	0	0
$885$	422.241i	0.477108i
$886$	0	0
$887$	− 619.581i	− 0.698513i	−0.937027	−	0.349256i	$-0.886434\pi$
$887$	− 619.581i	− 0.698513i	0.937027	−	0.349256i	$-0.113566\pi$
$888$	0	0
$889$	−270.640	−0.304432
$890$	0	0
$891$	1000.12	1.12247
$892$	0	0
$893$	− 1130.77i	− 1.26626i
$894$	0	0
$895$	− 297.488i	− 0.332389i
$896$	0	0
$897$	1498.37	1.67043
$898$	0	0
$899$	18.7769	0.0208864
$900$	0	0
$901$	− 629.276i	− 0.698420i
$902$	0	0
$903$	1135.30i	1.25726i
$904$	0	0
$905$	1629.32	1.80035
$906$	0	0
$907$	−302.666	−0.333700	−0.166850	−	0.985982i	$-0.553360\pi$
$907$	−302.666	−0.333700	−0.166850	+	0.985982i	$0.553360\pi$
$908$	0	0
$909$	2593.40i	2.85303i
$910$	0	0
$911$	1004.97i	1.10315i	0.834127	+	0.551573i	$0.185972\pi$
$911$	1004.97i	1.10315i	−0.834127	+	0.551573i	$0.814028\pi$
$912$	0	0
$913$	145.528	0.159395
$914$	0	0
$915$	−1207.98	−1.32020
$916$	0	0
$917$	− 235.937i	− 0.257293i
$918$	0	0
$919$	− 73.2026i	− 0.0796546i	−0.999207	−	0.0398273i	$-0.987319\pi$
$919$	− 73.2026i	− 0.0796546i	0.999207	−	0.0398273i	$-0.0126808\pi$
$920$	0	0
$921$	−1299.28	−1.41072
$922$	0	0
$923$	108.068	0.117084
$924$	0	0
$925$	− 595.605i	− 0.643897i
$926$	0	0
$927$	− 371.525i	− 0.400782i
$928$	0	0
$929$	225.883	0.243146	0.121573	−	0.992582i	$-0.461206\pi$
$929$	225.883	0.243146	0.121573	+	0.992582i	$0.461206\pi$
$930$	0	0
$931$	−178.981	−0.192246
$932$	0	0
$933$	2787.85i	2.98805i
$934$	0	0
$935$	− 202.568i	− 0.216650i
$936$	0	0
$937$	1168.51	1.24708	0.623540	−	0.781792i	$-0.285694\pi$
$937$	1168.51	1.24708	0.623540	+	0.781792i	$0.285694\pi$
$938$	0	0
$939$	2899.83	3.08821
$940$	0	0
$941$	− 119.216i	− 0.126690i	−0.997992	−	0.0633452i	$-0.979823\pi$
$941$	− 119.216i	− 0.126690i	0.997992	−	0.0633452i	$-0.0201769\pi$
$942$	0	0
$943$	− 1304.65i	− 1.38351i
$944$	0	0
$945$	1459.78	1.54474
$946$	0	0
$947$	875.062	0.924036	0.462018	−	0.886871i	$-0.347126\pi$
$947$	875.062	0.924036	0.462018	+	0.886871i	$0.347126\pi$
$948$	0	0
$949$	− 727.476i	− 0.766571i
$950$	0	0
$951$	− 2887.30i	− 3.03606i
$952$	0	0
$953$	468.167	0.491256	0.245628	−	0.969364i	$-0.421006\pi$
$953$	468.167	0.491256	0.245628	+	0.969364i	$0.421006\pi$
$954$	0	0
$955$	−954.001	−0.998954
$956$	0	0
$957$	27.6459i	0.0288881i
$958$	0	0
$959$	− 30.1088i	− 0.0313960i
$960$	0	0
$961$	816.743	0.849888
$962$	0	0
$963$	−1771.22	−1.83927
$964$	0	0
$965$	− 374.036i	− 0.387602i
$966$	0	0
$967$	− 1139.67i	− 1.17856i	−0.807930	−	0.589279i	$-0.799412\pi$
$967$	− 1139.67i	− 1.17856i	0.807930	−	0.589279i	$-0.200588\pi$
$968$	0	0
$969$	−1737.34	−1.79292
$970$	0	0
$971$	237.061	0.244141	0.122071	−	0.992521i	$-0.461047\pi$
$971$	237.061	0.244141	0.122071	+	0.992521i	$0.461047\pi$
$972$	0	0
$973$	320.812i	0.329715i
$974$	0	0
$975$	− 482.771i	− 0.495150i
$976$	0	0
$977$	422.059	0.431994	0.215997	−	0.976394i	$-0.430700\pi$
$977$	422.059	0.431994	0.215997	+	0.976394i	$0.430700\pi$
$978$	0	0
$979$	−203.767	−0.208138
$980$	0	0
$981$	− 1667.49i	− 1.69978i
$982$	0	0
$983$	1614.06i	1.64197i	0.570946	+	0.820987i	$0.306576\pi$
$983$	1614.06i	1.64197i	−0.570946	+	0.820987i	$0.693424\pi$
$984$	0	0
$985$	1096.85	1.11355
$986$	0	0
$987$	−685.224	−0.694250
$988$	0	0
$989$	1916.61i	1.93792i
$990$	0	0
$991$	226.772i	0.228832i	0.993433	+	0.114416i	$0.0364996\pi$
$991$	226.772i	0.228832i	−0.993433	+	0.114416i	$0.963500\pi$
$992$	0	0
$993$	3437.19	3.46142
$994$	0	0
$995$	1188.66	1.19463
$996$	0	0
$997$	1555.56i	1.56024i	0.625629	+	0.780120i	$0.284842\pi$
$997$	1555.56i	1.56024i	−0.625629	+	0.780120i	$0.715158\pi$
$998$	0	0
$999$	6744.25i	6.75100i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1792.3.g.f.127.8		8
4.3	odd	2		1792.3.g.d.127.2		8
8.3	odd	2	inner	1792.3.g.f.127.7		8
8.5	even	2		1792.3.g.d.127.1		8
16.3	odd	4		448.3.d.e.127.1		8
16.5	even	4		224.3.d.b.127.1	✓	8
16.11	odd	4		224.3.d.b.127.8	yes	8
16.13	even	4		448.3.d.e.127.8		8
48.5	odd	4		2016.3.m.c.127.8		8
48.11	even	4		2016.3.m.c.127.7		8
112.27	even	4		1568.3.d.n.1471.1		8
112.69	odd	4		1568.3.d.n.1471.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.d.b.127.1	✓	8	16.5	even	4
224.3.d.b.127.8	yes	8	16.11	odd	4
448.3.d.e.127.1		8	16.3	odd	4
448.3.d.e.127.8		8	16.13	even	4
1568.3.d.n.1471.1		8	112.27	even	4
1568.3.d.n.1471.8		8	112.69	odd	4
1792.3.g.d.127.1		8	8.5	even	2
1792.3.g.d.127.2		8	4.3	odd	2
1792.3.g.f.127.7		8	8.3	odd	2	inner
1792.3.g.f.127.8		8	1.1	even	1	trivial
2016.3.m.c.127.7		8	48.11	even	4
2016.3.m.c.127.8		8	48.5	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1792 = 2^{8} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1792.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.85623 q^{3} +5.78167i q^{5} -2.64575i q^{7} +25.2955 q^{9} +O(q^{10})\)	\(q+5.85623 q^{3} +5.78167i q^{5} -2.64575i q^{7} +25.2955 q^{9} +3.01966 q^{11} +9.78167i q^{13} +33.8588i q^{15} -11.6027 q^{17} +25.5687 q^{19} -15.4941i q^{21} -26.1571i q^{23} -8.42771 q^{25} +95.4301 q^{27} +1.56334i q^{29} -12.0107i q^{31} +17.6839 q^{33} +15.2969 q^{35} +70.6721i q^{37} +57.2837i q^{39} +49.8775 q^{41} -73.2730 q^{43} +146.250i q^{45} -44.2248i q^{47} -7.00000 q^{49} -67.9479 q^{51} +54.2355i q^{53} +17.4587i q^{55} +149.736 q^{57} +12.4706 q^{59} +35.6770i q^{61} -66.9255i q^{63} -56.5544 q^{65} +24.4891 q^{67} -153.182i q^{69} -11.0480i q^{71} -74.3713 q^{73} -49.3547 q^{75} -7.98928i q^{77} -22.8912i q^{79} +331.202 q^{81} +48.1934 q^{83} -67.0828i q^{85} +9.15529i q^{87} -67.4801 q^{89} +25.8799 q^{91} -70.3376i q^{93} +147.830i q^{95} -7.75949 q^{97} +76.3838 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} + 40 q^{9}+O(q^{10}) \) 8 * q + 8 * q^3 + 40 * q^9	\( 8 q + 8 q^{3} + 40 q^{9} + 32 q^{11} - 16 q^{17} + 88 q^{19} - 104 q^{25} + 176 q^{27} + 56 q^{35} - 144 q^{41} - 224 q^{43} - 56 q^{49} + 16 q^{51} + 400 q^{57} + 232 q^{59} - 304 q^{65} - 368 q^{67} - 272 q^{73} - 664 q^{75} + 504 q^{81} - 424 q^{83} + 80 q^{89} + 56 q^{91} + 528 q^{97} + 544 q^{99}+O(q^{100}) \) 8 * q + 8 * q^3 + 40 * q^9 + 32 * q^11 - 16 * q^17 + 88 * q^19 - 104 * q^25 + 176 * q^27 + 56 * q^35 - 144 * q^41 - 224 * q^43 - 56 * q^49 + 16 * q^51 + 400 * q^57 + 232 * q^59 - 304 * q^65 - 368 * q^67 - 272 * q^73 - 664 * q^75 + 504 * q^81 - 424 * q^83 + 80 * q^89 + 56 * q^91 + 528 * q^97 + 544 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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