LMFDB - Embedded newform 1336.2.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1336,2,Mod(1,1336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1336, 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("1336.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1336 = 2^{3} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1336.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:	$10.6680137100$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 13x^{7} + 8x^{6} + 56x^{5} - 15x^{4} - 81x^{3} + 2x^{2} + 13x + 1 $ x^9 - x^8 - 13x^7 + 8x^6 + 56x^5 - 15x^4 - 81x^3 + 2x^2 + 13*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.37109$ of defining polynomial
Character	$\chi$	$=$	1336.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.37109 q^{3} +0.438298 q^{5} +3.46684 q^{7} +2.62208 q^{9} +O(q^{10})$	$q-2.37109 q^{3} +0.438298 q^{5} +3.46684 q^{7} +2.62208 q^{9} -3.15524 q^{11} -5.09258 q^{13} -1.03924 q^{15} -2.92014 q^{17} +3.68615 q^{19} -8.22019 q^{21} +7.97085 q^{23} -4.80790 q^{25} +0.896077 q^{27} +1.62351 q^{29} -2.09649 q^{31} +7.48136 q^{33} +1.51951 q^{35} -5.18350 q^{37} +12.0750 q^{39} -5.08699 q^{41} -1.41951 q^{43} +1.14925 q^{45} +1.50331 q^{47} +5.01896 q^{49} +6.92392 q^{51} -9.99742 q^{53} -1.38293 q^{55} -8.74021 q^{57} -14.5950 q^{59} +14.0140 q^{61} +9.09033 q^{63} -2.23207 q^{65} -14.9345 q^{67} -18.8996 q^{69} +0.264067 q^{71} +14.9868 q^{73} +11.4000 q^{75} -10.9387 q^{77} -12.5056 q^{79} -9.99093 q^{81} -6.13306 q^{83} -1.27989 q^{85} -3.84949 q^{87} -14.1925 q^{89} -17.6552 q^{91} +4.97097 q^{93} +1.61563 q^{95} -5.25254 q^{97} -8.27329 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - q^{3} - 8 q^{5} + 2 q^{7}+O(q^{10}) $ 9 * q - q^3 - 8 * q^5 + 2 * q^7	$ 9 q - q^{3} - 8 q^{5} + 2 q^{7} - 10 q^{11} - 13 q^{13} + 2 q^{15} - 8 q^{17} - q^{19} - 19 q^{21} - 3 q^{23} + 3 q^{25} - 10 q^{27} - 25 q^{29} - q^{31} - 12 q^{33} - 17 q^{35} - 35 q^{37} - 4 q^{39} - 16 q^{41} + 9 q^{43} - 24 q^{45} - q^{47} - q^{49} - 10 q^{51} - 29 q^{53} + 9 q^{55} - 17 q^{57} - 14 q^{59} - 28 q^{61} + 4 q^{63} - 31 q^{65} + 19 q^{67} - 19 q^{69} - 9 q^{71} - 7 q^{75} - 33 q^{77} - 18 q^{79} - 27 q^{81} - 13 q^{83} - 36 q^{85} + 18 q^{87} - 21 q^{89} + 20 q^{91} - 35 q^{93} - 12 q^{95} + 2 q^{97} - 3 q^{99}+O(q^{100}) $ 9 * q - q^3 - 8 * q^5 + 2 * q^7 - 10 * q^11 - 13 * q^13 + 2 * q^15 - 8 * q^17 - q^19 - 19 * q^21 - 3 * q^23 + 3 * q^25 - 10 * q^27 - 25 * q^29 - q^31 - 12 * q^33 - 17 * q^35 - 35 * q^37 - 4 * q^39 - 16 * q^41 + 9 * q^43 - 24 * q^45 - q^47 - q^49 - 10 * q^51 - 29 * q^53 + 9 * q^55 - 17 * q^57 - 14 * q^59 - 28 * q^61 + 4 * q^63 - 31 * q^65 + 19 * q^67 - 19 * q^69 - 9 * q^71 - 7 * q^75 - 33 * q^77 - 18 * q^79 - 27 * q^81 - 13 * q^83 - 36 * q^85 + 18 * q^87 - 21 * q^89 + 20 * q^91 - 35 * q^93 - 12 * q^95 + 2 * q^97 - 3 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−2.37109	−1.36895	−0.684476	−	0.729036i	$-0.739969\pi$
$3$	−2.37109	−1.36895	−0.684476	+	0.729036i	$0.739969\pi$
$4$	0	0
$5$	0.438298	0.196013	0.0980063	−	0.995186i	$-0.468753\pi$
$5$	0.438298	0.196013	0.0980063	+	0.995186i	$0.468753\pi$
$6$	0	0
$7$	3.46684	1.31034	0.655171	−	0.755481i	$-0.272597\pi$
$7$	3.46684	1.31034	0.655171	+	0.755481i	$0.272597\pi$
$8$	0	0
$9$	2.62208	0.874028
$10$	0	0
$11$	−3.15524	−0.951340	−0.475670	−	0.879624i	$-0.657794\pi$
$11$	−3.15524	−0.951340	−0.475670	+	0.879624i	$0.657794\pi$
$12$	0	0
$13$	−5.09258	−1.41243	−0.706214	−	0.707998i	$-0.749598\pi$
$13$	−5.09258	−1.41243	−0.706214	+	0.707998i	$0.749598\pi$
$14$	0	0
$15$	−1.03924	−0.268332
$16$	0	0
$17$	−2.92014	−0.708238	−0.354119	−	0.935200i	$-0.615219\pi$
$17$	−2.92014	−0.708238	−0.354119	+	0.935200i	$0.615219\pi$
$18$	0	0
$19$	3.68615	0.845661	0.422831	−	0.906209i	$-0.361036\pi$
$19$	3.68615	0.845661	0.422831	+	0.906209i	$0.361036\pi$
$20$	0	0
$21$	−8.22019	−1.79379
$22$	0	0
$23$	7.97085	1.66204	0.831019	−	0.556245i	$-0.187758\pi$
$23$	7.97085	1.66204	0.831019	+	0.556245i	$0.187758\pi$
$24$	0	0
$25$	−4.80790	−0.961579
$26$	0	0
$27$	0.896077	0.172450
$28$	0	0
$29$	1.62351	0.301478	0.150739	−	0.988574i	$-0.451835\pi$
$29$	1.62351	0.301478	0.150739	+	0.988574i	$0.451835\pi$
$30$	0	0
$31$	−2.09649	−0.376541	−0.188270	−	0.982117i	$-0.560288\pi$
$31$	−2.09649	−0.376541	−0.188270	+	0.982117i	$0.560288\pi$
$32$	0	0
$33$	7.48136	1.30234
$34$	0	0
$35$	1.51951	0.256844
$36$	0	0
$37$	−5.18350	−0.852163	−0.426081	−	0.904685i	$-0.640106\pi$
$37$	−5.18350	−0.852163	−0.426081	+	0.904685i	$0.640106\pi$
$38$	0	0
$39$	12.0750	1.93354
$40$	0	0
$41$	−5.08699	−0.794455	−0.397227	−	0.917720i	$-0.630028\pi$
$41$	−5.08699	−0.794455	−0.397227	+	0.917720i	$0.630028\pi$
$42$	0	0
$43$	−1.41951	−0.216473	−0.108236	−	0.994125i	$-0.534520\pi$
$43$	−1.41951	−0.216473	−0.108236	+	0.994125i	$0.534520\pi$
$44$	0	0
$45$	1.14925	0.171320
$46$	0	0
$47$	1.50331	0.219280	0.109640	−	0.993971i	$-0.465030\pi$
$47$	1.50331	0.219280	0.109640	+	0.993971i	$0.465030\pi$
$48$	0	0
$49$	5.01896	0.716995
$50$	0	0
$51$	6.92392	0.969543
$52$	0	0
$53$	−9.99742	−1.37325	−0.686625	−	0.727011i	$-0.740909\pi$
$53$	−9.99742	−1.37325	−0.686625	+	0.727011i	$0.740909\pi$
$54$	0	0
$55$	−1.38293	−0.186475
$56$	0	0
$57$	−8.74021	−1.15767
$58$	0	0
$59$	−14.5950	−1.90011	−0.950054	−	0.312085i	$-0.898973\pi$
$59$	−14.5950	−1.90011	−0.950054	+	0.312085i	$0.898973\pi$
$60$	0	0
$61$	14.0140	1.79431	0.897155	−	0.441715i	$-0.145630\pi$
$61$	14.0140	1.79431	0.897155	+	0.441715i	$0.145630\pi$
$62$	0	0
$63$	9.09033	1.14527
$64$	0	0
$65$	−2.23207	−0.276854
$66$	0	0
$67$	−14.9345	−1.82454	−0.912271	−	0.409587i	$-0.865673\pi$
$67$	−14.9345	−1.82454	−0.912271	+	0.409587i	$0.865673\pi$
$68$	0	0
$69$	−18.8996	−2.27525
$70$	0	0
$71$	0.264067	0.0313390	0.0156695	−	0.999877i	$-0.495012\pi$
$71$	0.264067	0.0313390	0.0156695	+	0.999877i	$0.495012\pi$
$72$	0	0
$73$	14.9868	1.75407	0.877036	−	0.480425i	$-0.159518\pi$
$73$	14.9868	1.75407	0.877036	+	0.480425i	$0.159518\pi$
$74$	0	0
$75$	11.4000	1.31635
$76$	0	0
$77$	−10.9387	−1.24658
$78$	0	0
$79$	−12.5056	−1.40699	−0.703493	−	0.710702i	$-0.748377\pi$
$79$	−12.5056	−1.40699	−0.703493	+	0.710702i	$0.748377\pi$
$80$	0	0
$81$	−9.99093	−1.11010
$82$	0	0
$83$	−6.13306	−0.673191	−0.336596	−	0.941649i	$-0.609276\pi$
$83$	−6.13306	−0.673191	−0.336596	+	0.941649i	$0.609276\pi$
$84$	0	0
$85$	−1.27989	−0.138824
$86$	0	0
$87$	−3.84949	−0.412709
$88$	0	0
$89$	−14.1925	−1.50440	−0.752202	−	0.658932i	$-0.771008\pi$
$89$	−14.1925	−1.50440	−0.752202	+	0.658932i	$0.771008\pi$
$90$	0	0
$91$	−17.6552	−1.85076
$92$	0	0
$93$	4.97097	0.515466
$94$	0	0
$95$	1.61563	0.165760
$96$	0	0
$97$	−5.25254	−0.533315	−0.266658	−	0.963791i	$-0.585919\pi$
$97$	−5.25254	−0.533315	−0.266658	+	0.963791i	$0.585919\pi$
$98$	0	0
$99$	−8.27329	−0.831497
$100$	0	0
$101$	−16.5047	−1.64227	−0.821137	−	0.570731i	$-0.806660\pi$
$101$	−16.5047	−1.64227	−0.821137	+	0.570731i	$0.806660\pi$
$102$	0	0
$103$	13.2586	1.30641	0.653206	−	0.757180i	$-0.273423\pi$
$103$	13.2586	1.30641	0.653206	+	0.757180i	$0.273423\pi$
$104$	0	0
$105$	−3.60289	−0.351606
$106$	0	0
$107$	−4.76847	−0.460985	−0.230493	−	0.973074i	$-0.574034\pi$
$107$	−4.76847	−0.460985	−0.230493	+	0.973074i	$0.574034\pi$
$108$	0	0
$109$	−5.15042	−0.493320	−0.246660	−	0.969102i	$-0.579333\pi$
$109$	−5.15042	−0.493320	−0.246660	+	0.969102i	$0.579333\pi$
$110$	0	0
$111$	12.2906	1.16657
$112$	0	0
$113$	−4.58963	−0.431756	−0.215878	−	0.976420i	$-0.569261\pi$
$113$	−4.58963	−0.431756	−0.215878	+	0.976420i	$0.569261\pi$
$114$	0	0
$115$	3.49361	0.325780
$116$	0	0
$117$	−13.3532	−1.23450
$118$	0	0
$119$	−10.1236	−0.928033
$120$	0	0
$121$	−1.04448	−0.0949527
$122$	0	0
$123$	12.0617	1.08757
$124$	0	0
$125$	−4.29878	−0.384494
$126$	0	0
$127$	5.67456	0.503535	0.251768	−	0.967788i	$-0.418988\pi$
$127$	5.67456	0.503535	0.251768	+	0.967788i	$0.418988\pi$
$128$	0	0
$129$	3.36578	0.296340
$130$	0	0
$131$	−7.97389	−0.696682	−0.348341	−	0.937368i	$-0.613255\pi$
$131$	−7.97389	−0.696682	−0.348341	+	0.937368i	$0.613255\pi$
$132$	0	0
$133$	12.7793	1.10811
$134$	0	0
$135$	0.392749	0.0338024
$136$	0	0
$137$	−20.8982	−1.78545	−0.892727	−	0.450599i	$-0.851211\pi$
$137$	−20.8982	−1.78545	−0.892727	+	0.450599i	$0.851211\pi$
$138$	0	0
$139$	14.4620	1.22665	0.613325	−	0.789830i	$-0.289831\pi$
$139$	14.4620	1.22665	0.613325	+	0.789830i	$0.289831\pi$
$140$	0	0
$141$	−3.56448	−0.300184
$142$	0	0
$143$	16.0683	1.34370
$144$	0	0
$145$	0.711581	0.0590936
$146$	0	0
$147$	−11.9004	−0.981531
$148$	0	0
$149$	20.6394	1.69085	0.845423	−	0.534098i	$-0.179348\pi$
$149$	20.6394	1.69085	0.845423	+	0.534098i	$0.179348\pi$
$150$	0	0
$151$	6.49476	0.528536	0.264268	−	0.964449i	$-0.414870\pi$
$151$	6.49476	0.528536	0.264268	+	0.964449i	$0.414870\pi$
$152$	0	0
$153$	−7.65684	−0.619019
$154$	0	0
$155$	−0.918887	−0.0738067
$156$	0	0
$157$	−8.62952	−0.688711	−0.344355	−	0.938839i	$-0.611903\pi$
$157$	−8.62952	−0.688711	−0.344355	+	0.938839i	$0.611903\pi$
$158$	0	0
$159$	23.7048	1.87991
$160$	0	0
$161$	27.6336	2.17784
$162$	0	0
$163$	2.65154	0.207685	0.103842	−	0.994594i	$-0.466886\pi$
$163$	2.65154	0.207685	0.103842	+	0.994594i	$0.466886\pi$
$164$	0	0
$165$	3.27906	0.255275
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	12.9344	0.994953
$170$	0	0
$171$	9.66540	0.739131
$172$	0	0
$173$	12.3814	0.941344	0.470672	−	0.882308i	$-0.344011\pi$
$173$	12.3814	0.941344	0.470672	+	0.882308i	$0.344011\pi$
$174$	0	0
$175$	−16.6682	−1.26000
$176$	0	0
$177$	34.6061	2.60116
$178$	0	0
$179$	−17.0972	−1.27790	−0.638951	−	0.769247i	$-0.720631\pi$
$179$	−17.0972	−1.27790	−0.638951	+	0.769247i	$0.720631\pi$
$180$	0	0
$181$	5.05933	0.376057	0.188028	−	0.982164i	$-0.439790\pi$
$181$	5.05933	0.376057	0.188028	+	0.982164i	$0.439790\pi$
$182$	0	0
$183$	−33.2285	−2.45632
$184$	0	0
$185$	−2.27192	−0.167035
$186$	0	0
$187$	9.21373	0.673775
$188$	0	0
$189$	3.10655	0.225969
$190$	0	0
$191$	7.09875	0.513648	0.256824	−	0.966458i	$-0.417324\pi$
$191$	7.09875	0.513648	0.256824	+	0.966458i	$0.417324\pi$
$192$	0	0
$193$	−8.88684	−0.639688	−0.319844	−	0.947470i	$-0.603631\pi$
$193$	−8.88684	−0.639688	−0.319844	+	0.947470i	$0.603631\pi$
$194$	0	0
$195$	5.29244	0.378999
$196$	0	0
$197$	12.3982	0.883333	0.441666	−	0.897179i	$-0.354388\pi$
$197$	12.3982	0.883333	0.441666	+	0.897179i	$0.354388\pi$
$198$	0	0
$199$	−4.99936	−0.354395	−0.177198	−	0.984175i	$-0.556703\pi$
$199$	−4.99936	−0.354395	−0.177198	+	0.984175i	$0.556703\pi$
$200$	0	0
$201$	35.4111	2.49771
$202$	0	0
$203$	5.62845	0.395040
$204$	0	0
$205$	−2.22962	−0.155723
$206$	0	0
$207$	20.9002	1.45267
$208$	0	0
$209$	−11.6307	−0.804511
$210$	0	0
$211$	−0.866793	−0.0596725	−0.0298363	−	0.999555i	$-0.509499\pi$
$211$	−0.866793	−0.0596725	−0.0298363	+	0.999555i	$0.509499\pi$
$212$	0	0
$213$	−0.626128	−0.0429016
$214$	0	0
$215$	−0.622166	−0.0424314
$216$	0	0
$217$	−7.26819	−0.493397
$218$	0	0
$219$	−35.5351	−2.40124
$220$	0	0
$221$	14.8710	1.00033
$222$	0	0
$223$	−6.29874	−0.421795	−0.210897	−	0.977508i	$-0.567639\pi$
$223$	−6.29874	−0.421795	−0.210897	+	0.977508i	$0.567639\pi$
$224$	0	0
$225$	−12.6067	−0.840447
$226$	0	0
$227$	21.9722	1.45835	0.729173	−	0.684329i	$-0.239905\pi$
$227$	21.9722	1.45835	0.729173	+	0.684329i	$0.239905\pi$
$228$	0	0
$229$	−15.6952	−1.03717	−0.518585	−	0.855026i	$-0.673541\pi$
$229$	−15.6952	−1.03717	−0.518585	+	0.855026i	$0.673541\pi$
$230$	0	0
$231$	25.9367	1.70651
$232$	0	0
$233$	−2.46260	−0.161331	−0.0806653	−	0.996741i	$-0.525704\pi$
$233$	−2.46260	−0.161331	−0.0806653	+	0.996741i	$0.525704\pi$
$234$	0	0
$235$	0.658897	0.0429817
$236$	0	0
$237$	29.6519	1.92610
$238$	0	0
$239$	−5.85890	−0.378981	−0.189490	−	0.981883i	$-0.560684\pi$
$239$	−5.85890	−0.378981	−0.189490	+	0.981883i	$0.560684\pi$
$240$	0	0
$241$	24.2813	1.56410	0.782048	−	0.623218i	$-0.214175\pi$
$241$	24.2813	1.56410	0.782048	+	0.623218i	$0.214175\pi$
$242$	0	0
$243$	21.0012	1.34723
$244$	0	0
$245$	2.19980	0.140540
$246$	0	0
$247$	−18.7720	−1.19444
$248$	0	0
$249$	14.5421	0.921566
$250$	0	0
$251$	−7.31852	−0.461941	−0.230971	−	0.972961i	$-0.574190\pi$
$251$	−7.31852	−0.461941	−0.230971	+	0.972961i	$0.574190\pi$
$252$	0	0
$253$	−25.1499	−1.58116
$254$	0	0
$255$	3.03474	0.190043
$256$	0	0
$257$	−17.0003	−1.06045	−0.530224	−	0.847858i	$-0.677892\pi$
$257$	−17.0003	−1.06045	−0.530224	+	0.847858i	$0.677892\pi$
$258$	0	0
$259$	−17.9704	−1.11662
$260$	0	0
$261$	4.25698	0.263500
$262$	0	0
$263$	−6.39185	−0.394138	−0.197069	−	0.980390i	$-0.563142\pi$
$263$	−6.39185	−0.394138	−0.197069	+	0.980390i	$0.563142\pi$
$264$	0	0
$265$	−4.38184	−0.269175
$266$	0	0
$267$	33.6518	2.05946
$268$	0	0
$269$	−12.3245	−0.751439	−0.375720	−	0.926733i	$-0.622604\pi$
$269$	−12.3245	−0.751439	−0.375720	+	0.926733i	$0.622604\pi$
$270$	0	0
$271$	23.7725	1.44408	0.722040	−	0.691852i	$-0.243205\pi$
$271$	23.7725	1.44408	0.722040	+	0.691852i	$0.243205\pi$
$272$	0	0
$273$	41.8620	2.53360
$274$	0	0
$275$	15.1700	0.914788
$276$	0	0
$277$	14.1514	0.850275	0.425138	−	0.905129i	$-0.360226\pi$
$277$	14.1514	0.850275	0.425138	+	0.905129i	$0.360226\pi$
$278$	0	0
$279$	−5.49717	−0.329107
$280$	0	0
$281$	−21.9816	−1.31131	−0.655655	−	0.755061i	$-0.727607\pi$
$281$	−21.9816	−1.31131	−0.655655	+	0.755061i	$0.727607\pi$
$282$	0	0
$283$	−4.55613	−0.270834	−0.135417	−	0.990789i	$-0.543237\pi$
$283$	−4.55613	−0.270834	−0.135417	+	0.990789i	$0.543237\pi$
$284$	0	0
$285$	−3.83081	−0.226918
$286$	0	0
$287$	−17.6358	−1.04101
$288$	0	0
$289$	−8.47279	−0.498399
$290$	0	0
$291$	12.4543	0.730082
$292$	0	0
$293$	4.16001	0.243030	0.121515	−	0.992590i	$-0.461225\pi$
$293$	4.16001	0.243030	0.121515	+	0.992590i	$0.461225\pi$
$294$	0	0
$295$	−6.39696	−0.372445
$296$	0	0
$297$	−2.82734	−0.164059
$298$	0	0
$299$	−40.5922	−2.34751
$300$	0	0
$301$	−4.92120	−0.283653
$302$	0	0
$303$	39.1341	2.24819
$304$	0	0
$305$	6.14231	0.351708
$306$	0	0
$307$	34.7903	1.98559	0.992794	−	0.119833i	$-0.0382359\pi$
$307$	34.7903	1.98559	0.992794	+	0.119833i	$0.0382359\pi$
$308$	0	0
$309$	−31.4375	−1.78842
$310$	0	0
$311$	−8.22293	−0.466280	−0.233140	−	0.972443i	$-0.574900\pi$
$311$	−8.22293	−0.466280	−0.233140	+	0.972443i	$0.574900\pi$
$312$	0	0
$313$	23.0423	1.30243	0.651215	−	0.758893i	$-0.274260\pi$
$313$	23.0423	1.30243	0.651215	+	0.758893i	$0.274260\pi$
$314$	0	0
$315$	3.98427	0.224488
$316$	0	0
$317$	10.1935	0.572524	0.286262	−	0.958151i	$-0.407587\pi$
$317$	10.1935	0.572524	0.286262	+	0.958151i	$0.407587\pi$
$318$	0	0
$319$	−5.12256	−0.286808
$320$	0	0
$321$	11.3065	0.631066
$322$	0	0
$323$	−10.7641	−0.598929
$324$	0	0
$325$	24.4846	1.35816
$326$	0	0
$327$	12.2121	0.675332
$328$	0	0
$329$	5.21173	0.287332
$330$	0	0
$331$	−4.94314	−0.271699	−0.135850	−	0.990729i	$-0.543376\pi$
$331$	−4.94314	−0.271699	−0.135850	+	0.990729i	$0.543376\pi$
$332$	0	0
$333$	−13.5916	−0.744814
$334$	0	0
$335$	−6.54577	−0.357633
$336$	0	0
$337$	6.70674	0.365339	0.182670	−	0.983174i	$-0.441526\pi$
$337$	6.70674	0.365339	0.182670	+	0.983174i	$0.441526\pi$
$338$	0	0
$339$	10.8824	0.591053
$340$	0	0
$341$	6.61492	0.358218
$342$	0	0
$343$	−6.86793	−0.370834
$344$	0	0
$345$	−8.28366	−0.445978
$346$	0	0
$347$	21.7240	1.16621	0.583103	−	0.812399i	$-0.301839\pi$
$347$	21.7240	1.16621	0.583103	+	0.812399i	$0.301839\pi$
$348$	0	0
$349$	6.56610	0.351476	0.175738	−	0.984437i	$-0.443769\pi$
$349$	6.56610	0.351476	0.175738	+	0.984437i	$0.443769\pi$
$350$	0	0
$351$	−4.56335	−0.243573
$352$	0	0
$353$	22.5234	1.19880	0.599400	−	0.800450i	$-0.295406\pi$
$353$	22.5234	1.19880	0.599400	+	0.800450i	$0.295406\pi$
$354$	0	0
$355$	0.115740	0.00614284
$356$	0	0
$357$	24.0041	1.27043
$358$	0	0
$359$	−4.06888	−0.214747	−0.107374	−	0.994219i	$-0.534244\pi$
$359$	−4.06888	−0.214747	−0.107374	+	0.994219i	$0.534244\pi$
$360$	0	0
$361$	−5.41228	−0.284857
$362$	0	0
$363$	2.47656	0.129986
$364$	0	0
$365$	6.56868	0.343820
$366$	0	0
$367$	−0.971416	−0.0507075	−0.0253538	−	0.999679i	$-0.508071\pi$
$367$	−0.971416	−0.0507075	−0.0253538	+	0.999679i	$0.508071\pi$
$368$	0	0
$369$	−13.3385	−0.694376
$370$	0	0
$371$	−34.6594	−1.79943
$372$	0	0
$373$	−9.58029	−0.496049	−0.248024	−	0.968754i	$-0.579781\pi$
$373$	−9.58029	−0.496049	−0.248024	+	0.968754i	$0.579781\pi$
$374$	0	0
$375$	10.1928	0.526354
$376$	0	0
$377$	−8.26786	−0.425816
$378$	0	0
$379$	1.40346	0.0720909	0.0360455	−	0.999350i	$-0.488524\pi$
$379$	1.40346	0.0720909	0.0360455	+	0.999350i	$0.488524\pi$
$380$	0	0
$381$	−13.4549	−0.689315
$382$	0	0
$383$	8.40929	0.429694	0.214847	−	0.976648i	$-0.431075\pi$
$383$	8.40929	0.429694	0.214847	+	0.976648i	$0.431075\pi$
$384$	0	0
$385$	−4.79440	−0.244345
$386$	0	0
$387$	−3.72206	−0.189203
$388$	0	0
$389$	−8.62780	−0.437447	−0.218723	−	0.975787i	$-0.570189\pi$
$389$	−8.62780	−0.437447	−0.218723	+	0.975787i	$0.570189\pi$
$390$	0	0
$391$	−23.2760	−1.17712
$392$	0	0
$393$	18.9068	0.953724
$394$	0	0
$395$	−5.48116	−0.275787
$396$	0	0
$397$	−28.8989	−1.45039	−0.725196	−	0.688542i	$-0.758251\pi$
$397$	−28.8989	−1.45039	−0.725196	+	0.688542i	$0.758251\pi$
$398$	0	0
$399$	−30.3009	−1.51694
$400$	0	0
$401$	−10.6157	−0.530124	−0.265062	−	0.964231i	$-0.585392\pi$
$401$	−10.6157	−0.530124	−0.265062	+	0.964231i	$0.585392\pi$
$402$	0	0
$403$	10.6765	0.531837
$404$	0	0
$405$	−4.37900	−0.217594
$406$	0	0
$407$	16.3552	0.810696
$408$	0	0
$409$	1.72469	0.0852805	0.0426402	−	0.999090i	$-0.486423\pi$
$409$	1.72469	0.0852805	0.0426402	+	0.999090i	$0.486423\pi$
$410$	0	0
$411$	49.5516	2.44420
$412$	0	0
$413$	−50.5985	−2.48979
$414$	0	0
$415$	−2.68811	−0.131954
$416$	0	0
$417$	−34.2907	−1.67922
$418$	0	0
$419$	5.72856	0.279859	0.139929	−	0.990162i	$-0.455312\pi$
$419$	5.72856	0.279859	0.139929	+	0.990162i	$0.455312\pi$
$420$	0	0
$421$	−14.3225	−0.698036	−0.349018	−	0.937116i	$-0.613485\pi$
$421$	−14.3225	−0.698036	−0.349018	+	0.937116i	$0.613485\pi$
$422$	0	0
$423$	3.94180	0.191657
$424$	0	0
$425$	14.0397	0.681026
$426$	0	0
$427$	48.5843	2.35116
$428$	0	0
$429$	−38.0994	−1.83946
$430$	0	0
$431$	−19.2576	−0.927604	−0.463802	−	0.885939i	$-0.653515\pi$
$431$	−19.2576	−0.927604	−0.463802	+	0.885939i	$0.653515\pi$
$432$	0	0
$433$	25.9630	1.24770	0.623851	−	0.781543i	$-0.285567\pi$
$433$	25.9630	1.24770	0.623851	+	0.781543i	$0.285567\pi$
$434$	0	0
$435$	−1.68722	−0.0808962
$436$	0	0
$437$	29.3818	1.40552
$438$	0	0
$439$	18.1276	0.865181	0.432591	−	0.901590i	$-0.357600\pi$
$439$	18.1276	0.865181	0.432591	+	0.901590i	$0.357600\pi$
$440$	0	0
$441$	13.1601	0.626673
$442$	0	0
$443$	24.0203	1.14124	0.570621	−	0.821214i	$-0.306703\pi$
$443$	24.0203	1.14124	0.570621	+	0.821214i	$0.306703\pi$
$444$	0	0
$445$	−6.22055	−0.294882
$446$	0	0
$447$	−48.9379	−2.31469
$448$	0	0
$449$	36.5880	1.72669	0.863347	−	0.504611i	$-0.168364\pi$
$449$	36.5880	1.72669	0.863347	+	0.504611i	$0.168364\pi$
$450$	0	0
$451$	16.0507	0.755797
$452$	0	0
$453$	−15.3997	−0.723540
$454$	0	0
$455$	−7.73821	−0.362773
$456$	0	0
$457$	−23.5714	−1.10263	−0.551313	−	0.834299i	$-0.685873\pi$
$457$	−23.5714	−1.10263	−0.551313	+	0.834299i	$0.685873\pi$
$458$	0	0
$459$	−2.61667	−0.122136
$460$	0	0
$461$	−24.0703	−1.12106	−0.560532	−	0.828133i	$-0.689403\pi$
$461$	−24.0703	−1.12106	−0.560532	+	0.828133i	$0.689403\pi$
$462$	0	0
$463$	29.2121	1.35760	0.678800	−	0.734323i	$-0.262500\pi$
$463$	29.2121	1.35760	0.678800	+	0.734323i	$0.262500\pi$
$464$	0	0
$465$	2.17877	0.101038
$466$	0	0
$467$	−28.0414	−1.29760	−0.648800	−	0.760959i	$-0.724729\pi$
$467$	−28.0414	−1.29760	−0.648800	+	0.760959i	$0.724729\pi$
$468$	0	0
$469$	−51.7756	−2.39077
$470$	0	0
$471$	20.4614	0.942812
$472$	0	0
$473$	4.47888	0.205939
$474$	0	0
$475$	−17.7226	−0.813170
$476$	0	0
$477$	−26.2141	−1.20026
$478$	0	0
$479$	−40.0593	−1.83036	−0.915178	−	0.403050i	$-0.867950\pi$
$479$	−40.0593	−1.83036	−0.915178	+	0.403050i	$0.867950\pi$
$480$	0	0
$481$	26.3974	1.20362
$482$	0	0
$483$	−65.5219	−2.98135
$484$	0	0
$485$	−2.30218	−0.104537
$486$	0	0
$487$	41.2697	1.87011	0.935054	−	0.354505i	$-0.115351\pi$
$487$	41.2697	1.87011	0.935054	+	0.354505i	$0.115351\pi$
$488$	0	0
$489$	−6.28705	−0.284310
$490$	0	0
$491$	20.2983	0.916051	0.458025	−	0.888939i	$-0.348557\pi$
$491$	20.2983	0.916051	0.458025	+	0.888939i	$0.348557\pi$
$492$	0	0
$493$	−4.74088	−0.213518
$494$	0	0
$495$	−3.62616	−0.162984
$496$	0	0
$497$	0.915478	0.0410648
$498$	0	0
$499$	−25.2231	−1.12914	−0.564571	−	0.825385i	$-0.690958\pi$
$499$	−25.2231	−1.12914	−0.564571	+	0.825385i	$0.690958\pi$
$500$	0	0
$501$	−2.37109	−0.105933
$502$	0	0
$503$	−6.63112	−0.295667	−0.147834	−	0.989012i	$-0.547230\pi$
$503$	−6.63112	−0.295667	−0.147834	+	0.989012i	$0.547230\pi$
$504$	0	0
$505$	−7.23395	−0.321907
$506$	0	0
$507$	−30.6686	−1.36204
$508$	0	0
$509$	36.7255	1.62783	0.813916	−	0.580983i	$-0.197332\pi$
$509$	36.7255	1.62783	0.813916	+	0.580983i	$0.197332\pi$
$510$	0	0
$511$	51.9568	2.29843
$512$	0	0
$513$	3.30308	0.145834
$514$	0	0
$515$	5.81123	0.256073
$516$	0	0
$517$	−4.74329	−0.208610
$518$	0	0
$519$	−29.3576	−1.28865
$520$	0	0
$521$	−1.84892	−0.0810026	−0.0405013	−	0.999179i	$-0.512895\pi$
$521$	−1.84892	−0.0810026	−0.0405013	+	0.999179i	$0.512895\pi$
$522$	0	0
$523$	−14.8138	−0.647761	−0.323881	−	0.946098i	$-0.604988\pi$
$523$	−14.8138	−0.647761	−0.323881	+	0.946098i	$0.604988\pi$
$524$	0	0
$525$	39.5218	1.72487
$526$	0	0
$527$	6.12204	0.266680
$528$	0	0
$529$	40.5345	1.76237
$530$	0	0
$531$	−38.2693	−1.66075
$532$	0	0
$533$	25.9059	1.12211
$534$	0	0
$535$	−2.09001	−0.0903590
$536$	0	0
$537$	40.5390	1.74939
$538$	0	0
$539$	−15.8360	−0.682106
$540$	0	0
$541$	39.0615	1.67938	0.839692	−	0.543063i	$-0.182736\pi$
$541$	39.0615	1.67938	0.839692	+	0.543063i	$0.182736\pi$
$542$	0	0
$543$	−11.9961	−0.514803
$544$	0	0
$545$	−2.25742	−0.0966971
$546$	0	0
$547$	−0.281521	−0.0120370	−0.00601849	−	0.999982i	$-0.501916\pi$
$547$	−0.281521	−0.0120370	−0.00601849	+	0.999982i	$0.501916\pi$
$548$	0	0
$549$	36.7459	1.56828
$550$	0	0
$551$	5.98451	0.254949
$552$	0	0
$553$	−43.3548	−1.84363
$554$	0	0
$555$	5.38693	0.228662
$556$	0	0
$557$	−11.4612	−0.485628	−0.242814	−	0.970073i	$-0.578070\pi$
$557$	−11.4612	−0.485628	−0.242814	+	0.970073i	$0.578070\pi$
$558$	0	0
$559$	7.22895	0.305752
$560$	0	0
$561$	−21.8466	−0.922365
$562$	0	0
$563$	13.0043	0.548065	0.274032	−	0.961720i	$-0.411642\pi$
$563$	13.0043	0.548065	0.274032	+	0.961720i	$0.411642\pi$
$564$	0	0
$565$	−2.01163	−0.0846297
$566$	0	0
$567$	−34.6369	−1.45461
$568$	0	0
$569$	2.81961	0.118204	0.0591021	−	0.998252i	$-0.481176\pi$
$569$	2.81961	0.118204	0.0591021	+	0.998252i	$0.481176\pi$
$570$	0	0
$571$	−26.4489	−1.10685	−0.553425	−	0.832899i	$-0.686679\pi$
$571$	−26.4489	−1.10685	−0.553425	+	0.832899i	$0.686679\pi$
$572$	0	0
$573$	−16.8318	−0.703159
$574$	0	0
$575$	−38.3230	−1.59818
$576$	0	0
$577$	−21.2121	−0.883069	−0.441535	−	0.897244i	$-0.645566\pi$
$577$	−21.2121	−0.883069	−0.441535	+	0.897244i	$0.645566\pi$
$578$	0	0
$579$	21.0715	0.875702
$580$	0	0
$581$	−21.2623	−0.882110
$582$	0	0
$583$	31.5442	1.30643
$584$	0	0
$585$	−5.85266	−0.241978
$586$	0	0
$587$	24.5753	1.01433	0.507166	−	0.861848i	$-0.330693\pi$
$587$	24.5753	1.01433	0.507166	+	0.861848i	$0.330693\pi$
$588$	0	0
$589$	−7.72798	−0.318426
$590$	0	0
$591$	−29.3972	−1.20924
$592$	0	0
$593$	35.1509	1.44347	0.721736	−	0.692168i	$-0.243344\pi$
$593$	35.1509	1.44347	0.721736	+	0.692168i	$0.243344\pi$
$594$	0	0
$595$	−4.43717	−0.181906
$596$	0	0
$597$	11.8539	0.485150
$598$	0	0
$599$	−16.4638	−0.672692	−0.336346	−	0.941738i	$-0.609191\pi$
$599$	−16.4638	−0.672692	−0.336346	+	0.941738i	$0.609191\pi$
$600$	0	0
$601$	6.37385	0.259995	0.129997	−	0.991514i	$-0.458503\pi$
$601$	6.37385	0.259995	0.129997	+	0.991514i	$0.458503\pi$
$602$	0	0
$603$	−39.1595	−1.59470
$604$	0	0
$605$	−0.457793	−0.0186119
$606$	0	0
$607$	−34.1234	−1.38503	−0.692513	−	0.721406i	$-0.743496\pi$
$607$	−34.1234	−1.38503	−0.692513	+	0.721406i	$0.743496\pi$
$608$	0	0
$609$	−13.3456	−0.540790
$610$	0	0
$611$	−7.65572	−0.309717
$612$	0	0
$613$	−16.1928	−0.654019	−0.327010	−	0.945021i	$-0.606041\pi$
$613$	−16.1928	−0.654019	−0.327010	+	0.945021i	$0.606041\pi$
$614$	0	0
$615$	5.28663	0.213178
$616$	0	0
$617$	3.48303	0.140221	0.0701107	−	0.997539i	$-0.477665\pi$
$617$	3.48303	0.140221	0.0701107	+	0.997539i	$0.477665\pi$
$618$	0	0
$619$	19.8980	0.799768	0.399884	−	0.916566i	$-0.369050\pi$
$619$	19.8980	0.799768	0.399884	+	0.916566i	$0.369050\pi$
$620$	0	0
$621$	7.14250	0.286619
$622$	0	0
$623$	−49.2032	−1.97128
$624$	0	0
$625$	22.1553	0.886213
$626$	0	0
$627$	27.5774	1.10134
$628$	0	0
$629$	15.1366	0.603534
$630$	0	0
$631$	27.3447	1.08858	0.544288	−	0.838898i	$-0.316800\pi$
$631$	27.3447	1.08858	0.544288	+	0.838898i	$0.316800\pi$
$632$	0	0
$633$	2.05525	0.0816887
$634$	0	0
$635$	2.48715	0.0986993
$636$	0	0
$637$	−25.5595	−1.01270
$638$	0	0
$639$	0.692406	0.0273911
$640$	0	0
$641$	−33.3065	−1.31553	−0.657764	−	0.753224i	$-0.728497\pi$
$641$	−33.3065	−1.31553	−0.657764	+	0.753224i	$0.728497\pi$
$642$	0	0
$643$	31.5983	1.24612	0.623058	−	0.782176i	$-0.285890\pi$
$643$	31.5983	1.24612	0.623058	+	0.782176i	$0.285890\pi$
$644$	0	0
$645$	1.47521	0.0580865
$646$	0	0
$647$	42.9779	1.68964	0.844819	−	0.535053i	$-0.179708\pi$
$647$	42.9779	1.68964	0.844819	+	0.535053i	$0.179708\pi$
$648$	0	0
$649$	46.0507	1.80765
$650$	0	0
$651$	17.2336	0.675436
$652$	0	0
$653$	28.5585	1.11758	0.558790	−	0.829309i	$-0.311266\pi$
$653$	28.5585	1.11758	0.558790	+	0.829309i	$0.311266\pi$
$654$	0	0
$655$	−3.49494	−0.136558
$656$	0	0
$657$	39.2966	1.53311
$658$	0	0
$659$	−7.75541	−0.302108	−0.151054	−	0.988526i	$-0.548267\pi$
$659$	−7.75541	−0.302108	−0.151054	+	0.988526i	$0.548267\pi$
$660$	0	0
$661$	23.4534	0.912230	0.456115	−	0.889921i	$-0.349241\pi$
$661$	23.4534	0.912230	0.456115	+	0.889921i	$0.349241\pi$
$662$	0	0
$663$	−35.2606	−1.36941
$664$	0	0
$665$	5.60113	0.217203
$666$	0	0
$667$	12.9408	0.501068
$668$	0	0
$669$	14.9349	0.577417
$670$	0	0
$671$	−44.2175	−1.70700
$672$	0	0
$673$	34.8832	1.34465	0.672323	−	0.740258i	$-0.265296\pi$
$673$	34.8832	1.34465	0.672323	+	0.740258i	$0.265296\pi$
$674$	0	0
$675$	−4.30825	−0.165824
$676$	0	0
$677$	2.51744	0.0967532	0.0483766	−	0.998829i	$-0.484595\pi$
$677$	2.51744	0.0967532	0.0483766	+	0.998829i	$0.484595\pi$
$678$	0	0
$679$	−18.2097	−0.698825
$680$	0	0
$681$	−52.0982	−1.99641
$682$	0	0
$683$	11.8663	0.454051	0.227026	−	0.973889i	$-0.427100\pi$
$683$	11.8663	0.454051	0.227026	+	0.973889i	$0.427100\pi$
$684$	0	0
$685$	−9.15963	−0.349972
$686$	0	0
$687$	37.2149	1.41984
$688$	0	0
$689$	50.9127	1.93962
$690$	0	0
$691$	−12.7794	−0.486150	−0.243075	−	0.970007i	$-0.578156\pi$
$691$	−12.7794	−0.486150	−0.243075	+	0.970007i	$0.578156\pi$
$692$	0	0
$693$	−28.6822	−1.08955
$694$	0	0
$695$	6.33866	0.240439
$696$	0	0
$697$	14.8547	0.562663
$698$	0	0
$699$	5.83906	0.220854
$700$	0	0
$701$	−48.1903	−1.82012	−0.910062	−	0.414472i	$-0.863966\pi$
$701$	−48.1903	−1.82012	−0.910062	+	0.414472i	$0.863966\pi$
$702$	0	0
$703$	−19.1072	−0.720641
$704$	0	0
$705$	−1.56231	−0.0588398
$706$	0	0
$707$	−57.2190	−2.15194
$708$	0	0
$709$	−36.1221	−1.35659	−0.678297	−	0.734788i	$-0.737282\pi$
$709$	−36.1221	−1.35659	−0.678297	+	0.734788i	$0.737282\pi$
$710$	0	0
$711$	−32.7906	−1.22974
$712$	0	0
$713$	−16.7108	−0.625825
$714$	0	0
$715$	7.04270	0.263382
$716$	0	0
$717$	13.8920	0.518806
$718$	0	0
$719$	−14.9868	−0.558912	−0.279456	−	0.960159i	$-0.590154\pi$
$719$	−14.9868	−0.558912	−0.279456	+	0.960159i	$0.590154\pi$
$720$	0	0
$721$	45.9656	1.71185
$722$	0	0
$723$	−57.5732	−2.14117
$724$	0	0
$725$	−7.80567	−0.289895
$726$	0	0
$727$	17.5115	0.649464	0.324732	−	0.945806i	$-0.394726\pi$
$727$	17.5115	0.649464	0.324732	+	0.945806i	$0.394726\pi$
$728$	0	0
$729$	−19.8230	−0.734185
$730$	0	0
$731$	4.14515	0.153314
$732$	0	0
$733$	−5.76049	−0.212768	−0.106384	−	0.994325i	$-0.533927\pi$
$733$	−5.76049	−0.212768	−0.106384	+	0.994325i	$0.533927\pi$
$734$	0	0
$735$	−5.21593	−0.192392
$736$	0	0
$737$	47.1220	1.73576
$738$	0	0
$739$	−8.11359	−0.298463	−0.149232	−	0.988802i	$-0.547680\pi$
$739$	−8.11359	−0.298463	−0.149232	+	0.988802i	$0.547680\pi$
$740$	0	0
$741$	44.5102	1.63512
$742$	0	0
$743$	−20.8906	−0.766401	−0.383201	−	0.923665i	$-0.625178\pi$
$743$	−20.8906	−0.766401	−0.383201	+	0.923665i	$0.625178\pi$
$744$	0	0
$745$	9.04620	0.331427
$746$	0	0
$747$	−16.0814	−0.588388
$748$	0	0
$749$	−16.5315	−0.604048
$750$	0	0
$751$	−0.164561	−0.00600491	−0.00300245	−	0.999995i	$-0.500956\pi$
$751$	−0.164561	−0.00600491	−0.00300245	+	0.999995i	$0.500956\pi$
$752$	0	0
$753$	17.3529	0.632375
$754$	0	0
$755$	2.84664	0.103600
$756$	0	0
$757$	−22.1836	−0.806278	−0.403139	−	0.915139i	$-0.632081\pi$
$757$	−22.1836	−0.806278	−0.403139	+	0.915139i	$0.632081\pi$
$758$	0	0
$759$	59.6328	2.16453
$760$	0	0
$761$	−15.0619	−0.545993	−0.272996	−	0.962015i	$-0.588015\pi$
$761$	−15.0619	−0.545993	−0.272996	+	0.962015i	$0.588015\pi$
$762$	0	0
$763$	−17.8557	−0.646418
$764$	0	0
$765$	−3.35598	−0.121336
$766$	0	0
$767$	74.3263	2.68377
$768$	0	0
$769$	−42.3357	−1.52667	−0.763333	−	0.646006i	$-0.776438\pi$
$769$	−42.3357	−1.52667	−0.763333	+	0.646006i	$0.776438\pi$
$770$	0	0
$771$	40.3092	1.45170
$772$	0	0
$773$	4.12075	0.148213	0.0741065	−	0.997250i	$-0.476390\pi$
$773$	4.12075	0.148213	0.0741065	+	0.997250i	$0.476390\pi$
$774$	0	0
$775$	10.0797	0.362074
$776$	0	0
$777$	42.6094	1.52860
$778$	0	0
$779$	−18.7514	−0.671840
$780$	0	0
$781$	−0.833194	−0.0298140
$782$	0	0
$783$	1.45479	0.0519900
$784$	0	0
$785$	−3.78230	−0.134996
$786$	0	0
$787$	32.2255	1.14871	0.574357	−	0.818605i	$-0.305252\pi$
$787$	32.2255	1.14871	0.574357	+	0.818605i	$0.305252\pi$
$788$	0	0
$789$	15.1557	0.539556
$790$	0	0
$791$	−15.9115	−0.565748
$792$	0	0
$793$	−71.3675	−2.53433
$794$	0	0
$795$	10.3898	0.368487
$796$	0	0
$797$	−39.5520	−1.40100	−0.700502	−	0.713650i	$-0.747041\pi$
$797$	−39.5520	−1.40100	−0.700502	+	0.713650i	$0.747041\pi$
$798$	0	0
$799$	−4.38987	−0.155302
$800$	0	0
$801$	−37.2140	−1.31489
$802$	0	0
$803$	−47.2869	−1.66872
$804$	0	0
$805$	12.1118	0.426884
$806$	0	0
$807$	29.2226	1.02868
$808$	0	0
$809$	−6.18632	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$809$	−6.18632	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$810$	0	0
$811$	24.9679	0.876742	0.438371	−	0.898794i	$-0.355556\pi$
$811$	24.9679	0.876742	0.438371	+	0.898794i	$0.355556\pi$
$812$	0	0
$813$	−56.3669	−1.97687
$814$	0	0
$815$	1.16216	0.0407088
$816$	0	0
$817$	−5.23251	−0.183062
$818$	0	0
$819$	−46.2933	−1.61762
$820$	0	0
$821$	17.1430	0.598294	0.299147	−	0.954207i	$-0.403298\pi$
$821$	17.1430	0.598294	0.299147	+	0.954207i	$0.403298\pi$
$822$	0	0
$823$	−10.4876	−0.365576	−0.182788	−	0.983152i	$-0.558512\pi$
$823$	−10.4876	−0.365576	−0.182788	+	0.983152i	$0.558512\pi$
$824$	0	0
$825$	−35.9696	−1.25230
$826$	0	0
$827$	38.9865	1.35569	0.677847	−	0.735203i	$-0.262913\pi$
$827$	38.9865	1.35569	0.677847	+	0.735203i	$0.262913\pi$
$828$	0	0
$829$	10.5160	0.365237	0.182619	−	0.983184i	$-0.441543\pi$
$829$	10.5160	0.365237	0.182619	+	0.983184i	$0.441543\pi$
$830$	0	0
$831$	−33.5543	−1.16399
$832$	0	0
$833$	−14.6561	−0.507803
$834$	0	0
$835$	0.438298	0.0151679
$836$	0	0
$837$	−1.87862	−0.0649345
$838$	0	0
$839$	16.3582	0.564749	0.282375	−	0.959304i	$-0.408878\pi$
$839$	16.3582	0.564749	0.282375	+	0.959304i	$0.408878\pi$
$840$	0	0
$841$	−26.3642	−0.909111
$842$	0	0
$843$	52.1203	1.79512
$844$	0	0
$845$	5.66911	0.195023
$846$	0	0
$847$	−3.62104	−0.124420
$848$	0	0
$849$	10.8030	0.370758
$850$	0	0
$851$	−41.3169	−1.41633
$852$	0	0
$853$	41.4298	1.41853	0.709264	−	0.704943i	$-0.249027\pi$
$853$	41.4298	1.41853	0.709264	+	0.704943i	$0.249027\pi$
$854$	0	0
$855$	4.23632	0.144879
$856$	0	0
$857$	−9.04873	−0.309099	−0.154549	−	0.987985i	$-0.549393\pi$
$857$	−9.04873	−0.309099	−0.154549	+	0.987985i	$0.549393\pi$
$858$	0	0
$859$	−57.6110	−1.96566	−0.982831	−	0.184509i	$-0.940930\pi$
$859$	−57.6110	−1.96566	−0.982831	+	0.184509i	$0.940930\pi$
$860$	0	0
$861$	41.8161	1.42509
$862$	0	0
$863$	−55.6614	−1.89474	−0.947368	−	0.320147i	$-0.896268\pi$
$863$	−55.6614	−1.89474	−0.947368	+	0.320147i	$0.896268\pi$
$864$	0	0
$865$	5.42676	0.184515
$866$	0	0
$867$	20.0898	0.682285
$868$	0	0
$869$	39.4580	1.33852
$870$	0	0
$871$	76.0553	2.57703
$872$	0	0
$873$	−13.7726	−0.466132
$874$	0	0
$875$	−14.9032	−0.503819
$876$	0	0
$877$	−48.7523	−1.64625	−0.823124	−	0.567861i	$-0.807771\pi$
$877$	−48.7523	−1.64625	−0.823124	+	0.567861i	$0.807771\pi$
$878$	0	0
$879$	−9.86377	−0.332697
$880$	0	0
$881$	−1.91879	−0.0646458	−0.0323229	−	0.999477i	$-0.510290\pi$
$881$	−1.91879	−0.0646458	−0.0323229	+	0.999477i	$0.510290\pi$
$882$	0	0
$883$	31.8094	1.07047	0.535236	−	0.844702i	$-0.320223\pi$
$883$	31.8094	1.07047	0.535236	+	0.844702i	$0.320223\pi$
$884$	0	0
$885$	15.1678	0.509860
$886$	0	0
$887$	−37.8817	−1.27194	−0.635971	−	0.771713i	$-0.719400\pi$
$887$	−37.8817	−1.27194	−0.635971	+	0.771713i	$0.719400\pi$
$888$	0	0
$889$	19.6728	0.659803
$890$	0	0
$891$	31.5238	1.05609
$892$	0	0
$893$	5.54142	0.185437
$894$	0	0
$895$	−7.49365	−0.250485
$896$	0	0
$897$	96.2479	3.21362
$898$	0	0
$899$	−3.40367	−0.113519
$900$	0	0
$901$	29.1938	0.972588
$902$	0	0
$903$	11.6686	0.388307
$904$	0	0
$905$	2.21749	0.0737119
$906$	0	0
$907$	−19.2242	−0.638330	−0.319165	−	0.947699i	$-0.603402\pi$
$907$	−19.2242	−0.638330	−0.319165	+	0.947699i	$0.603402\pi$
$908$	0	0
$909$	−43.2766	−1.43539
$910$	0	0
$911$	−14.5213	−0.481112	−0.240556	−	0.970635i	$-0.577330\pi$
$911$	−14.5213	−0.481112	−0.240556	+	0.970635i	$0.577330\pi$
$912$	0	0
$913$	19.3513	0.640433
$914$	0	0
$915$	−14.5640	−0.481471
$916$	0	0
$917$	−27.6442	−0.912891
$918$	0	0
$919$	−14.9936	−0.494593	−0.247297	−	0.968940i	$-0.579542\pi$
$919$	−14.9936	−0.494593	−0.247297	+	0.968940i	$0.579542\pi$
$920$	0	0
$921$	−82.4911	−2.71817
$922$	0	0
$923$	−1.34478	−0.0442641
$924$	0	0
$925$	24.9217	0.819422
$926$	0	0
$927$	34.7653	1.14184
$928$	0	0
$929$	−46.6038	−1.52902	−0.764511	−	0.644611i	$-0.777019\pi$
$929$	−46.6038	−1.52902	−0.764511	+	0.644611i	$0.777019\pi$
$930$	0	0
$931$	18.5007	0.606335
$932$	0	0
$933$	19.4973	0.638314
$934$	0	0
$935$	4.03836	0.132068
$936$	0	0
$937$	17.4913	0.571415	0.285708	−	0.958317i	$-0.407771\pi$
$937$	17.4913	0.571415	0.285708	+	0.958317i	$0.407771\pi$
$938$	0	0
$939$	−54.6355	−1.78296
$940$	0	0
$941$	−17.0531	−0.555914	−0.277957	−	0.960594i	$-0.589657\pi$
$941$	−17.0531	−0.555914	−0.277957	+	0.960594i	$0.589657\pi$
$942$	0	0
$943$	−40.5477	−1.32041
$944$	0	0
$945$	1.36160	0.0442927
$946$	0	0
$947$	50.9810	1.65666	0.828330	−	0.560241i	$-0.189291\pi$
$947$	50.9810	1.65666	0.828330	+	0.560241i	$0.189291\pi$
$948$	0	0
$949$	−76.3214	−2.47750
$950$	0	0
$951$	−24.1697	−0.783758
$952$	0	0
$953$	−5.23412	−0.169550	−0.0847749	−	0.996400i	$-0.527017\pi$
$953$	−5.23412	−0.169550	−0.0847749	+	0.996400i	$0.527017\pi$
$954$	0	0
$955$	3.11137	0.100681
$956$	0	0
$957$	12.1461	0.392627
$958$	0	0
$959$	−72.4507	−2.33955
$960$	0	0
$961$	−26.6047	−0.858217
$962$	0	0
$963$	−12.5033	−0.402914
$964$	0	0
$965$	−3.89508	−0.125387
$966$	0	0
$967$	−52.2934	−1.68164	−0.840822	−	0.541312i	$-0.817928\pi$
$967$	−52.2934	−1.68164	−0.840822	+	0.541312i	$0.817928\pi$
$968$	0	0
$969$	25.5226	0.819905
$970$	0	0
$971$	−11.4376	−0.367051	−0.183526	−	0.983015i	$-0.558751\pi$
$971$	−11.4376	−0.367051	−0.183526	+	0.983015i	$0.558751\pi$
$972$	0	0
$973$	50.1374	1.60733
$974$	0	0
$975$	−58.0553	−1.85926
$976$	0	0
$977$	−16.2518	−0.519942	−0.259971	−	0.965616i	$-0.583713\pi$
$977$	−16.2518	−0.519942	−0.259971	+	0.965616i	$0.583713\pi$
$978$	0	0
$979$	44.7808	1.43120
$980$	0	0
$981$	−13.5048	−0.431176
$982$	0	0
$983$	−31.7381	−1.01229	−0.506144	−	0.862449i	$-0.668930\pi$
$983$	−31.7381	−1.01229	−0.506144	+	0.862449i	$0.668930\pi$
$984$	0	0
$985$	5.43409	0.173144
$986$	0	0
$987$	−12.3575	−0.393343
$988$	0	0
$989$	−11.3147	−0.359785
$990$	0	0
$991$	57.0028	1.81075	0.905376	−	0.424610i	$-0.139589\pi$
$991$	57.0028	1.81075	0.905376	+	0.424610i	$0.139589\pi$
$992$	0	0
$993$	11.7206	0.371943
$994$	0	0
$995$	−2.19121	−0.0694660
$996$	0	0
$997$	35.2136	1.11522	0.557612	−	0.830102i	$-0.311718\pi$
$997$	35.2136	1.11522	0.557612	+	0.830102i	$0.311718\pi$
$998$	0	0
$999$	−4.64482	−0.146956

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1336.2.a.c.1.2	✓	9
4.3	odd	2		2672.2.a.m.1.8		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1336.2.a.c.1.2	✓	9	1.1	even	1	trivial
2672.2.a.m.1.8		9	4.3	odd	2

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:	\( N \)	\(=\)	\( 1336 = 2^{3} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1336.a (trivial)

\(f(q)\)	\(=\)	\(q-2.37109 q^{3} +0.438298 q^{5} +3.46684 q^{7} +2.62208 q^{9} +O(q^{10})\)	\(q-2.37109 q^{3} +0.438298 q^{5} +3.46684 q^{7} +2.62208 q^{9} -3.15524 q^{11} -5.09258 q^{13} -1.03924 q^{15} -2.92014 q^{17} +3.68615 q^{19} -8.22019 q^{21} +7.97085 q^{23} -4.80790 q^{25} +0.896077 q^{27} +1.62351 q^{29} -2.09649 q^{31} +7.48136 q^{33} +1.51951 q^{35} -5.18350 q^{37} +12.0750 q^{39} -5.08699 q^{41} -1.41951 q^{43} +1.14925 q^{45} +1.50331 q^{47} +5.01896 q^{49} +6.92392 q^{51} -9.99742 q^{53} -1.38293 q^{55} -8.74021 q^{57} -14.5950 q^{59} +14.0140 q^{61} +9.09033 q^{63} -2.23207 q^{65} -14.9345 q^{67} -18.8996 q^{69} +0.264067 q^{71} +14.9868 q^{73} +11.4000 q^{75} -10.9387 q^{77} -12.5056 q^{79} -9.99093 q^{81} -6.13306 q^{83} -1.27989 q^{85} -3.84949 q^{87} -14.1925 q^{89} -17.6552 q^{91} +4.97097 q^{93} +1.61563 q^{95} -5.25254 q^{97} -8.27329 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - q^{3} - 8 q^{5} + 2 q^{7}+O(q^{10}) \) 9 * q - q^3 - 8 * q^5 + 2 * q^7	\( 9 q - q^{3} - 8 q^{5} + 2 q^{7} - 10 q^{11} - 13 q^{13} + 2 q^{15} - 8 q^{17} - q^{19} - 19 q^{21} - 3 q^{23} + 3 q^{25} - 10 q^{27} - 25 q^{29} - q^{31} - 12 q^{33} - 17 q^{35} - 35 q^{37} - 4 q^{39} - 16 q^{41} + 9 q^{43} - 24 q^{45} - q^{47} - q^{49} - 10 q^{51} - 29 q^{53} + 9 q^{55} - 17 q^{57} - 14 q^{59} - 28 q^{61} + 4 q^{63} - 31 q^{65} + 19 q^{67} - 19 q^{69} - 9 q^{71} - 7 q^{75} - 33 q^{77} - 18 q^{79} - 27 q^{81} - 13 q^{83} - 36 q^{85} + 18 q^{87} - 21 q^{89} + 20 q^{91} - 35 q^{93} - 12 q^{95} + 2 q^{97} - 3 q^{99}+O(q^{100}) \) 9 * q - q^3 - 8 * q^5 + 2 * q^7 - 10 * q^11 - 13 * q^13 + 2 * q^15 - 8 * q^17 - q^19 - 19 * q^21 - 3 * q^23 + 3 * q^25 - 10 * q^27 - 25 * q^29 - q^31 - 12 * q^33 - 17 * q^35 - 35 * q^37 - 4 * q^39 - 16 * q^41 + 9 * q^43 - 24 * q^45 - q^47 - q^49 - 10 * q^51 - 29 * q^53 + 9 * q^55 - 17 * q^57 - 14 * q^59 - 28 * q^61 + 4 * q^63 - 31 * q^65 + 19 * q^67 - 19 * q^69 - 9 * q^71 - 7 * q^75 - 33 * q^77 - 18 * q^79 - 27 * q^81 - 13 * q^83 - 36 * q^85 + 18 * q^87 - 21 * q^89 + 20 * q^91 - 35 * q^93 - 12 * q^95 + 2 * q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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