LMFDB - Embedded newform 1336.2.a.d.1.4

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:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} - 25 x^{10} + 20 x^{9} + 224 x^{8} - 135 x^{7} - 865 x^{6} + 330 x^{5} + 1341 x^{4} + \cdots + 64 $ x^12 - x^11 - 25x^10 + 20x^9 + 224x^8 - 135x^7 - 865x^6 + 330x^5 + 1341x^4 - 127x^3 - 652x^2 - 48x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.79442$ 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)

$f(q)$	$=$	$q-1.79442 q^{3} +3.50243 q^{5} -0.885520 q^{7} +0.219933 q^{9} +O(q^{10})$	$q-1.79442 q^{3} +3.50243 q^{5} -0.885520 q^{7} +0.219933 q^{9} -2.51830 q^{11} +3.89440 q^{13} -6.28483 q^{15} +2.93944 q^{17} -1.92980 q^{19} +1.58899 q^{21} +3.51146 q^{23} +7.26705 q^{25} +4.98860 q^{27} +1.32572 q^{29} +4.81178 q^{31} +4.51888 q^{33} -3.10147 q^{35} -7.25055 q^{37} -6.98819 q^{39} -1.70623 q^{41} +2.90464 q^{43} +0.770302 q^{45} -1.51461 q^{47} -6.21585 q^{49} -5.27459 q^{51} -2.91792 q^{53} -8.82018 q^{55} +3.46287 q^{57} +10.6020 q^{59} +9.21641 q^{61} -0.194755 q^{63} +13.6399 q^{65} +5.89286 q^{67} -6.30102 q^{69} -1.68411 q^{71} +13.6907 q^{73} -13.0401 q^{75} +2.23000 q^{77} +10.7622 q^{79} -9.61143 q^{81} +10.7502 q^{83} +10.2952 q^{85} -2.37890 q^{87} +11.6229 q^{89} -3.44857 q^{91} -8.63433 q^{93} -6.75901 q^{95} +8.31541 q^{97} -0.553859 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - q^{3} + 8 q^{5} - 4 q^{7} + 15 q^{9}+O(q^{10}) $ 12 * q - q^3 + 8 * q^5 - 4 * q^7 + 15 * q^9	$ 12 q - q^{3} + 8 q^{5} - 4 q^{7} + 15 q^{9} + 6 q^{11} + 13 q^{13} + 4 q^{15} + 10 q^{17} - q^{19} + 13 q^{21} + 3 q^{23} + 18 q^{25} - 10 q^{27} + 29 q^{29} - 3 q^{31} + 3 q^{35} + 41 q^{37} + 10 q^{39} + 20 q^{41} - q^{43} + 42 q^{45} + 5 q^{47} + 20 q^{49} + 14 q^{51} + 39 q^{53} + 3 q^{55} + 3 q^{57} + 8 q^{59} + 30 q^{61} - 2 q^{63} + 21 q^{65} - 9 q^{67} + 33 q^{69} + 29 q^{71} + 12 q^{73} + q^{75} + 19 q^{77} - 2 q^{79} + 24 q^{81} - 5 q^{83} + 44 q^{85} - 2 q^{87} + 7 q^{89} - 4 q^{91} + 37 q^{93} + 18 q^{95} - 14 q^{97} + 5 q^{99}+O(q^{100}) $ 12 * q - q^3 + 8 * q^5 - 4 * q^7 + 15 * q^9 + 6 * q^11 + 13 * q^13 + 4 * q^15 + 10 * q^17 - q^19 + 13 * q^21 + 3 * q^23 + 18 * q^25 - 10 * q^27 + 29 * q^29 - 3 * q^31 + 3 * q^35 + 41 * q^37 + 10 * q^39 + 20 * q^41 - q^43 + 42 * q^45 + 5 * q^47 + 20 * q^49 + 14 * q^51 + 39 * q^53 + 3 * q^55 + 3 * q^57 + 8 * q^59 + 30 * q^61 - 2 * q^63 + 21 * q^65 - 9 * q^67 + 33 * q^69 + 29 * q^71 + 12 * q^73 + q^75 + 19 * q^77 - 2 * q^79 + 24 * q^81 - 5 * q^83 + 44 * q^85 - 2 * q^87 + 7 * q^89 - 4 * q^91 + 37 * q^93 + 18 * q^95 - 14 * q^97 + 5 * 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
$3$	−1.79442	−1.03601	−0.518004	−	0.855378i	$-0.673325\pi$
$3$	−1.79442	−1.03601	−0.518004	+	0.855378i	$0.673325\pi$
$4$	0	0
$5$	3.50243	1.56634	0.783168	−	0.621810i	$-0.213602\pi$
$5$	3.50243	1.56634	0.783168	+	0.621810i	$0.213602\pi$
$6$	0	0
$7$	−0.885520	−0.334695	−0.167347	−	0.985898i	$-0.553520\pi$
$7$	−0.885520	−0.334695	−0.167347	+	0.985898i	$0.553520\pi$
$8$	0	0
$9$	0.219933	0.0733112
$10$	0	0
$11$	−2.51830	−0.759296	−0.379648	−	0.925131i	$-0.623955\pi$
$11$	−2.51830	−0.759296	−0.379648	+	0.925131i	$0.623955\pi$
$12$	0	0
$13$	3.89440	1.08011	0.540057	−	0.841629i	$-0.318403\pi$
$13$	3.89440	1.08011	0.540057	+	0.841629i	$0.318403\pi$
$14$	0	0
$15$	−6.28483	−1.62274
$16$	0	0
$17$	2.93944	0.712920	0.356460	−	0.934311i	$-0.383984\pi$
$17$	2.93944	0.712920	0.356460	+	0.934311i	$0.383984\pi$
$18$	0	0
$19$	−1.92980	−0.442727	−0.221364	−	0.975191i	$-0.571051\pi$
$19$	−1.92980	−0.442727	−0.221364	+	0.975191i	$0.571051\pi$
$20$	0	0
$21$	1.58899	0.346746
$22$	0	0
$23$	3.51146	0.732190	0.366095	−	0.930578i	$-0.380695\pi$
$23$	3.51146	0.732190	0.366095	+	0.930578i	$0.380695\pi$
$24$	0	0
$25$	7.26705	1.45341
$26$	0	0
$27$	4.98860	0.960056
$28$	0	0
$29$	1.32572	0.246180	0.123090	−	0.992395i	$-0.460720\pi$
$29$	1.32572	0.246180	0.123090	+	0.992395i	$0.460720\pi$
$30$	0	0
$31$	4.81178	0.864220	0.432110	−	0.901821i	$-0.357769\pi$
$31$	4.81178	0.864220	0.432110	+	0.901821i	$0.357769\pi$
$32$	0	0
$33$	4.51888	0.786636
$34$	0	0
$35$	−3.10147	−0.524245
$36$	0	0
$37$	−7.25055	−1.19198	−0.595991	−	0.802991i	$-0.703241\pi$
$37$	−7.25055	−1.19198	−0.595991	+	0.802991i	$0.703241\pi$
$38$	0	0
$39$	−6.98819	−1.11901
$40$	0	0
$41$	−1.70623	−0.266469	−0.133234	−	0.991085i	$-0.542536\pi$
$41$	−1.70623	−0.266469	−0.133234	+	0.991085i	$0.542536\pi$
$42$	0	0
$43$	2.90464	0.442954	0.221477	−	0.975166i	$-0.428912\pi$
$43$	2.90464	0.442954	0.221477	+	0.975166i	$0.428912\pi$
$44$	0	0
$45$	0.770302	0.114830
$46$	0	0
$47$	−1.51461	−0.220929	−0.110465	−	0.993880i	$-0.535234\pi$
$47$	−1.51461	−0.220929	−0.110465	+	0.993880i	$0.535234\pi$
$48$	0	0
$49$	−6.21585	−0.887979
$50$	0	0
$51$	−5.27459	−0.738590
$52$	0	0
$53$	−2.91792	−0.400807	−0.200403	−	0.979713i	$-0.564225\pi$
$53$	−2.91792	−0.400807	−0.200403	+	0.979713i	$0.564225\pi$
$54$	0	0
$55$	−8.82018	−1.18931
$56$	0	0
$57$	3.46287	0.458668
$58$	0	0
$59$	10.6020	1.38026	0.690129	−	0.723687i	$-0.257554\pi$
$59$	10.6020	1.38026	0.690129	+	0.723687i	$0.257554\pi$
$60$	0	0
$61$	9.21641	1.18004	0.590020	−	0.807388i	$-0.299120\pi$
$61$	9.21641	1.18004	0.590020	+	0.807388i	$0.299120\pi$
$62$	0	0
$63$	−0.194755	−0.0245369
$64$	0	0
$65$	13.6399	1.69182
$66$	0	0
$67$	5.89286	0.719927	0.359964	−	0.932966i	$-0.382789\pi$
$67$	5.89286	0.719927	0.359964	+	0.932966i	$0.382789\pi$
$68$	0	0
$69$	−6.30102	−0.758554
$70$	0	0
$71$	−1.68411	−0.199867	−0.0999337	−	0.994994i	$-0.531863\pi$
$71$	−1.68411	−0.199867	−0.0999337	+	0.994994i	$0.531863\pi$
$72$	0	0
$73$	13.6907	1.60237	0.801187	−	0.598414i	$-0.204202\pi$
$73$	13.6907	1.60237	0.801187	+	0.598414i	$0.204202\pi$
$74$	0	0
$75$	−13.0401	−1.50574
$76$	0	0
$77$	2.23000	0.254133
$78$	0	0
$79$	10.7622	1.21085	0.605423	−	0.795904i	$-0.293004\pi$
$79$	10.7622	1.21085	0.605423	+	0.795904i	$0.293004\pi$
$80$	0	0
$81$	−9.61143	−1.06794
$82$	0	0
$83$	10.7502	1.17999	0.589994	−	0.807408i	$-0.299130\pi$
$83$	10.7502	1.17999	0.589994	+	0.807408i	$0.299130\pi$
$84$	0	0
$85$	10.2952	1.11667
$86$	0	0
$87$	−2.37890	−0.255045
$88$	0	0
$89$	11.6229	1.23203	0.616013	−	0.787736i	$-0.288747\pi$
$89$	11.6229	1.23203	0.616013	+	0.787736i	$0.288747\pi$
$90$	0	0
$91$	−3.44857	−0.361509
$92$	0	0
$93$	−8.63433	−0.895339
$94$	0	0
$95$	−6.75901	−0.693459
$96$	0	0
$97$	8.31541	0.844302	0.422151	−	0.906526i	$-0.361275\pi$
$97$	8.31541	0.844302	0.422151	+	0.906526i	$0.361275\pi$
$98$	0	0
$99$	−0.553859	−0.0556649
$100$	0	0
$101$	5.50930	0.548196	0.274098	−	0.961702i	$-0.411621\pi$
$101$	5.50930	0.548196	0.274098	+	0.961702i	$0.411621\pi$
$102$	0	0
$103$	5.52105	0.544006	0.272003	−	0.962296i	$-0.412314\pi$
$103$	5.52105	0.544006	0.272003	+	0.962296i	$0.412314\pi$
$104$	0	0
$105$	5.56534	0.543122
$106$	0	0
$107$	−10.5161	−1.01663	−0.508317	−	0.861170i	$-0.669732\pi$
$107$	−10.5161	−1.01663	−0.508317	+	0.861170i	$0.669732\pi$
$108$	0	0
$109$	2.31086	0.221341	0.110670	−	0.993857i	$-0.464700\pi$
$109$	2.31086	0.221341	0.110670	+	0.993857i	$0.464700\pi$
$110$	0	0
$111$	13.0105	1.23490
$112$	0	0
$113$	−10.3560	−0.974210	−0.487105	−	0.873343i	$-0.661947\pi$
$113$	−10.3560	−0.974210	−0.487105	+	0.873343i	$0.661947\pi$
$114$	0	0
$115$	12.2987	1.14686
$116$	0	0
$117$	0.856510	0.0791844
$118$	0	0
$119$	−2.60294	−0.238611
$120$	0	0
$121$	−4.65816	−0.423469
$122$	0	0
$123$	3.06170	0.276064
$124$	0	0
$125$	7.94018	0.710191
$126$	0	0
$127$	−16.0562	−1.42476	−0.712378	−	0.701796i	$-0.752382\pi$
$127$	−16.0562	−1.42476	−0.712378	+	0.701796i	$0.752382\pi$
$128$	0	0
$129$	−5.21214	−0.458904
$130$	0	0
$131$	−2.53648	−0.221614	−0.110807	−	0.993842i	$-0.535343\pi$
$131$	−2.53648	−0.221614	−0.110807	+	0.993842i	$0.535343\pi$
$132$	0	0
$133$	1.70888	0.148179
$134$	0	0
$135$	17.4722	1.50377
$136$	0	0
$137$	16.9088	1.44461	0.722307	−	0.691572i	$-0.243082\pi$
$137$	16.9088	1.44461	0.722307	+	0.691572i	$0.243082\pi$
$138$	0	0
$139$	6.54065	0.554771	0.277385	−	0.960759i	$-0.410532\pi$
$139$	6.54065	0.554771	0.277385	+	0.960759i	$0.410532\pi$
$140$	0	0
$141$	2.71785	0.228884
$142$	0	0
$143$	−9.80728	−0.820126
$144$	0	0
$145$	4.64325	0.385601
$146$	0	0
$147$	11.1538	0.919953
$148$	0	0
$149$	−13.4889	−1.10506	−0.552528	−	0.833494i	$-0.686337\pi$
$149$	−13.4889	−1.10506	−0.552528	+	0.833494i	$0.686337\pi$
$150$	0	0
$151$	0.776936	0.0632262	0.0316131	−	0.999500i	$-0.489936\pi$
$151$	0.776936	0.0632262	0.0316131	+	0.999500i	$0.489936\pi$
$152$	0	0
$153$	0.646482	0.0522650
$154$	0	0
$155$	16.8529	1.35366
$156$	0	0
$157$	−8.30687	−0.662961	−0.331480	−	0.943462i	$-0.607548\pi$
$157$	−8.30687	−0.662961	−0.331480	+	0.943462i	$0.607548\pi$
$158$	0	0
$159$	5.23596	0.415239
$160$	0	0
$161$	−3.10947	−0.245060
$162$	0	0
$163$	8.79344	0.688755	0.344378	−	0.938831i	$-0.388090\pi$
$163$	8.79344	0.688755	0.344378	+	0.938831i	$0.388090\pi$
$164$	0	0
$165$	15.8271	1.23214
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	2.16639	0.166645
$170$	0	0
$171$	−0.424428	−0.0324568
$172$	0	0
$173$	−11.7301	−0.891826	−0.445913	−	0.895076i	$-0.647121\pi$
$173$	−11.7301	−0.891826	−0.445913	+	0.895076i	$0.647121\pi$
$174$	0	0
$175$	−6.43511	−0.486449
$176$	0	0
$177$	−19.0243	−1.42996
$178$	0	0
$179$	18.4565	1.37950	0.689752	−	0.724046i	$-0.257720\pi$
$179$	18.4565	1.37950	0.689752	+	0.724046i	$0.257720\pi$
$180$	0	0
$181$	5.91101	0.439362	0.219681	−	0.975572i	$-0.429498\pi$
$181$	5.91101	0.439362	0.219681	+	0.975572i	$0.429498\pi$
$182$	0	0
$183$	−16.5381	−1.22253
$184$	0	0
$185$	−25.3946	−1.86705
$186$	0	0
$187$	−7.40240	−0.541317
$188$	0	0
$189$	−4.41750	−0.321326
$190$	0	0
$191$	−10.6577	−0.771163	−0.385581	−	0.922674i	$-0.625999\pi$
$191$	−10.6577	−0.771163	−0.385581	+	0.922674i	$0.625999\pi$
$192$	0	0
$193$	−20.4476	−1.47185	−0.735924	−	0.677064i	$-0.763252\pi$
$193$	−20.4476	−1.47185	−0.735924	+	0.677064i	$0.763252\pi$
$194$	0	0
$195$	−24.4757	−1.75274
$196$	0	0
$197$	23.8510	1.69932	0.849658	−	0.527334i	$-0.176808\pi$
$197$	23.8510	1.69932	0.849658	+	0.527334i	$0.176808\pi$
$198$	0	0
$199$	−11.9623	−0.847983	−0.423992	−	0.905666i	$-0.639371\pi$
$199$	−11.9623	−0.847983	−0.423992	+	0.905666i	$0.639371\pi$
$200$	0	0
$201$	−10.5742	−0.745850
$202$	0	0
$203$	−1.17395	−0.0823953
$204$	0	0
$205$	−5.97597	−0.417380
$206$	0	0
$207$	0.772287	0.0536777
$208$	0	0
$209$	4.85982	0.336161
$210$	0	0
$211$	3.45990	0.238189	0.119095	−	0.992883i	$-0.462001\pi$
$211$	3.45990	0.238189	0.119095	+	0.992883i	$0.462001\pi$
$212$	0	0
$213$	3.02200	0.207064
$214$	0	0
$215$	10.1733	0.693815
$216$	0	0
$217$	−4.26092	−0.289250
$218$	0	0
$219$	−24.5668	−1.66007
$220$	0	0
$221$	11.4474	0.770034
$222$	0	0
$223$	−4.55775	−0.305210	−0.152605	−	0.988287i	$-0.548766\pi$
$223$	−4.55775	−0.305210	−0.152605	+	0.988287i	$0.548766\pi$
$224$	0	0
$225$	1.59827	0.106551
$226$	0	0
$227$	6.38378	0.423706	0.211853	−	0.977301i	$-0.432050\pi$
$227$	6.38378	0.423706	0.211853	+	0.977301i	$0.432050\pi$
$228$	0	0
$229$	24.5461	1.62205	0.811026	−	0.585010i	$-0.198909\pi$
$229$	24.5461	1.62205	0.811026	+	0.585010i	$0.198909\pi$
$230$	0	0
$231$	−4.00156	−0.263283
$232$	0	0
$233$	−6.84008	−0.448109	−0.224054	−	0.974577i	$-0.571929\pi$
$233$	−6.84008	−0.448109	−0.224054	+	0.974577i	$0.571929\pi$
$234$	0	0
$235$	−5.30483	−0.346049
$236$	0	0
$237$	−19.3120	−1.25445
$238$	0	0
$239$	−19.4596	−1.25874	−0.629370	−	0.777106i	$-0.716687\pi$
$239$	−19.4596	−1.25874	−0.629370	+	0.777106i	$0.716687\pi$
$240$	0	0
$241$	−20.8779	−1.34487	−0.672433	−	0.740158i	$-0.734751\pi$
$241$	−20.8779	−1.34487	−0.672433	+	0.740158i	$0.734751\pi$
$242$	0	0
$243$	2.28112	0.146334
$244$	0	0
$245$	−21.7706	−1.39087
$246$	0	0
$247$	−7.51543	−0.478195
$248$	0	0
$249$	−19.2903	−1.22248
$250$	0	0
$251$	1.69442	0.106951	0.0534754	−	0.998569i	$-0.482970\pi$
$251$	1.69442	0.106951	0.0534754	+	0.998569i	$0.482970\pi$
$252$	0	0
$253$	−8.84291	−0.555949
$254$	0	0
$255$	−18.4739	−1.15688
$256$	0	0
$257$	4.44004	0.276962	0.138481	−	0.990365i	$-0.455778\pi$
$257$	4.44004	0.276962	0.138481	+	0.990365i	$0.455778\pi$
$258$	0	0
$259$	6.42050	0.398951
$260$	0	0
$261$	0.291571	0.0180478
$262$	0	0
$263$	−23.1390	−1.42681	−0.713405	−	0.700752i	$-0.752848\pi$
$263$	−23.1390	−1.42681	−0.713405	+	0.700752i	$0.752848\pi$
$264$	0	0
$265$	−10.2198	−0.627798
$266$	0	0
$267$	−20.8564	−1.27639
$268$	0	0
$269$	18.7732	1.14462	0.572312	−	0.820036i	$-0.306047\pi$
$269$	18.7732	1.14462	0.572312	+	0.820036i	$0.306047\pi$
$270$	0	0
$271$	−6.11219	−0.371289	−0.185645	−	0.982617i	$-0.559437\pi$
$271$	−6.11219	−0.371289	−0.185645	+	0.982617i	$0.559437\pi$
$272$	0	0
$273$	6.18818	0.374526
$274$	0	0
$275$	−18.3006	−1.10357
$276$	0	0
$277$	−0.267321	−0.0160618	−0.00803089	−	0.999968i	$-0.502556\pi$
$277$	−0.267321	−0.0160618	−0.00803089	+	0.999968i	$0.502556\pi$
$278$	0	0
$279$	1.05827	0.0633570
$280$	0	0
$281$	7.00733	0.418022	0.209011	−	0.977913i	$-0.432976\pi$
$281$	7.00733	0.418022	0.209011	+	0.977913i	$0.432976\pi$
$282$	0	0
$283$	−25.5972	−1.52159	−0.760797	−	0.648990i	$-0.775192\pi$
$283$	−25.5972	−1.52159	−0.760797	+	0.648990i	$0.775192\pi$
$284$	0	0
$285$	12.1285	0.718429
$286$	0	0
$287$	1.51090	0.0891858
$288$	0	0
$289$	−8.35967	−0.491745
$290$	0	0
$291$	−14.9213	−0.874703
$292$	0	0
$293$	−7.97950	−0.466168	−0.233084	−	0.972457i	$-0.574882\pi$
$293$	−7.97950	−0.466168	−0.233084	+	0.972457i	$0.574882\pi$
$294$	0	0
$295$	37.1327	2.16195
$296$	0	0
$297$	−12.5628	−0.728967
$298$	0	0
$299$	13.6750	0.790848
$300$	0	0
$301$	−2.57212	−0.148255
$302$	0	0
$303$	−9.88598	−0.567935
$304$	0	0
$305$	32.2799	1.84834
$306$	0	0
$307$	23.1702	1.32239	0.661197	−	0.750212i	$-0.270049\pi$
$307$	23.1702	1.32239	0.661197	+	0.750212i	$0.270049\pi$
$308$	0	0
$309$	−9.90708	−0.563594
$310$	0	0
$311$	−9.59826	−0.544267	−0.272134	−	0.962259i	$-0.587729\pi$
$311$	−9.59826	−0.544267	−0.272134	+	0.962259i	$0.587729\pi$
$312$	0	0
$313$	−1.48733	−0.0840688	−0.0420344	−	0.999116i	$-0.513384\pi$
$313$	−1.48733	−0.0840688	−0.0420344	+	0.999116i	$0.513384\pi$
$314$	0	0
$315$	−0.682118	−0.0384330
$316$	0	0
$317$	−10.8905	−0.611670	−0.305835	−	0.952085i	$-0.598936\pi$
$317$	−10.8905	−0.611670	−0.305835	+	0.952085i	$0.598936\pi$
$318$	0	0
$319$	−3.33857	−0.186924
$320$	0	0
$321$	18.8703	1.05324
$322$	0	0
$323$	−5.67255	−0.315629
$324$	0	0
$325$	28.3008	1.56985
$326$	0	0
$327$	−4.14665	−0.229311
$328$	0	0
$329$	1.34122	0.0739439
$330$	0	0
$331$	−26.9295	−1.48018	−0.740090	−	0.672507i	$-0.765217\pi$
$331$	−26.9295	−1.48018	−0.740090	+	0.672507i	$0.765217\pi$
$332$	0	0
$333$	−1.59464	−0.0873856
$334$	0	0
$335$	20.6394	1.12765
$336$	0	0
$337$	−35.5174	−1.93476	−0.967378	−	0.253337i	$-0.918472\pi$
$337$	−35.5174	−1.93476	−0.967378	+	0.253337i	$0.918472\pi$
$338$	0	0
$339$	18.5830	1.00929
$340$	0	0
$341$	−12.1175	−0.656199
$342$	0	0
$343$	11.7029	0.631897
$344$	0	0
$345$	−22.0689	−1.18815
$346$	0	0
$347$	−11.7934	−0.633104	−0.316552	−	0.948575i	$-0.602525\pi$
$347$	−11.7934	−0.633104	−0.316552	+	0.948575i	$0.602525\pi$
$348$	0	0
$349$	14.0834	0.753866	0.376933	−	0.926241i	$-0.376979\pi$
$349$	14.0834	0.753866	0.376933	+	0.926241i	$0.376979\pi$
$350$	0	0
$351$	19.4276	1.03697
$352$	0	0
$353$	−10.0668	−0.535800	−0.267900	−	0.963447i	$-0.586330\pi$
$353$	−10.0668	−0.535800	−0.267900	+	0.963447i	$0.586330\pi$
$354$	0	0
$355$	−5.89849	−0.313060
$356$	0	0
$357$	4.67075	0.247202
$358$	0	0
$359$	−2.68076	−0.141485	−0.0707425	−	0.997495i	$-0.522537\pi$
$359$	−2.68076	−0.141485	−0.0707425	+	0.997495i	$0.522537\pi$
$360$	0	0
$361$	−15.2759	−0.803993
$362$	0	0
$363$	8.35869	0.438717
$364$	0	0
$365$	47.9508	2.50986
$366$	0	0
$367$	−28.8873	−1.50790	−0.753952	−	0.656929i	$-0.771855\pi$
$367$	−28.8873	−1.50790	−0.753952	+	0.656929i	$0.771855\pi$
$368$	0	0
$369$	−0.375258	−0.0195351
$370$	0	0
$371$	2.58387	0.134148
$372$	0	0
$373$	−7.16234	−0.370852	−0.185426	−	0.982658i	$-0.559366\pi$
$373$	−7.16234	−0.370852	−0.185426	+	0.982658i	$0.559366\pi$
$374$	0	0
$375$	−14.2480	−0.735763
$376$	0	0
$377$	5.16290	0.265903
$378$	0	0
$379$	2.57457	0.132247	0.0661233	−	0.997811i	$-0.478937\pi$
$379$	2.57457	0.132247	0.0661233	+	0.997811i	$0.478937\pi$
$380$	0	0
$381$	28.8115	1.47606
$382$	0	0
$383$	1.61188	0.0823630	0.0411815	−	0.999152i	$-0.486888\pi$
$383$	1.61188	0.0823630	0.0411815	+	0.999152i	$0.486888\pi$
$384$	0	0
$385$	7.81045	0.398057
$386$	0	0
$387$	0.638828	0.0324735
$388$	0	0
$389$	9.63429	0.488478	0.244239	−	0.969715i	$-0.421462\pi$
$389$	9.63429	0.488478	0.244239	+	0.969715i	$0.421462\pi$
$390$	0	0
$391$	10.3217	0.521993
$392$	0	0
$393$	4.55151	0.229593
$394$	0	0
$395$	37.6940	1.89659
$396$	0	0
$397$	−24.2626	−1.21771	−0.608854	−	0.793283i	$-0.708370\pi$
$397$	−24.2626	−1.21771	−0.608854	+	0.793283i	$0.708370\pi$
$398$	0	0
$399$	−3.06644	−0.153514
$400$	0	0
$401$	−8.59869	−0.429398	−0.214699	−	0.976680i	$-0.568877\pi$
$401$	−8.59869	−0.429398	−0.214699	+	0.976680i	$0.568877\pi$
$402$	0	0
$403$	18.7390	0.933456
$404$	0	0
$405$	−33.6634	−1.67275
$406$	0	0
$407$	18.2591	0.905068
$408$	0	0
$409$	20.9278	1.03481	0.517407	−	0.855740i	$-0.326897\pi$
$409$	20.9278	1.03481	0.517407	+	0.855740i	$0.326897\pi$
$410$	0	0
$411$	−30.3414	−1.49663
$412$	0	0
$413$	−9.38824	−0.461965
$414$	0	0
$415$	37.6519	1.84826
$416$	0	0
$417$	−11.7367	−0.574746
$418$	0	0
$419$	34.3738	1.67927	0.839636	−	0.543149i	$-0.182768\pi$
$419$	34.3738	1.67927	0.839636	+	0.543149i	$0.182768\pi$
$420$	0	0
$421$	−5.11019	−0.249056	−0.124528	−	0.992216i	$-0.539742\pi$
$421$	−5.11019	−0.249056	−0.124528	+	0.992216i	$0.539742\pi$
$422$	0	0
$423$	−0.333114	−0.0161966
$424$	0	0
$425$	21.3611	1.03616
$426$	0	0
$427$	−8.16131	−0.394954
$428$	0	0
$429$	17.5984	0.849657
$430$	0	0
$431$	20.7222	0.998151	0.499076	−	0.866558i	$-0.333673\pi$
$431$	20.7222	0.998151	0.499076	+	0.866558i	$0.333673\pi$
$432$	0	0
$433$	8.62079	0.414289	0.207144	−	0.978310i	$-0.433583\pi$
$433$	8.62079	0.414289	0.207144	+	0.978310i	$0.433583\pi$
$434$	0	0
$435$	−8.33193	−0.399486
$436$	0	0
$437$	−6.77642	−0.324160
$438$	0	0
$439$	−5.12401	−0.244556	−0.122278	−	0.992496i	$-0.539020\pi$
$439$	−5.12401	−0.244556	−0.122278	+	0.992496i	$0.539020\pi$
$440$	0	0
$441$	−1.36707	−0.0650988
$442$	0	0
$443$	18.0372	0.856974	0.428487	−	0.903548i	$-0.359047\pi$
$443$	18.0372	0.856974	0.428487	+	0.903548i	$0.359047\pi$
$444$	0	0
$445$	40.7085	1.92977
$446$	0	0
$447$	24.2048	1.14485
$448$	0	0
$449$	−16.1882	−0.763970	−0.381985	−	0.924168i	$-0.624759\pi$
$449$	−16.1882	−0.763970	−0.381985	+	0.924168i	$0.624759\pi$
$450$	0	0
$451$	4.29681	0.202329
$452$	0	0
$453$	−1.39415	−0.0655028
$454$	0	0
$455$	−12.0784	−0.566244
$456$	0	0
$457$	−35.6829	−1.66918	−0.834589	−	0.550874i	$-0.814295\pi$
$457$	−35.6829	−1.66918	−0.834589	+	0.550874i	$0.814295\pi$
$458$	0	0
$459$	14.6637	0.684443
$460$	0	0
$461$	10.5436	0.491064	0.245532	−	0.969389i	$-0.421037\pi$
$461$	10.5436	0.491064	0.245532	+	0.969389i	$0.421037\pi$
$462$	0	0
$463$	−18.6661	−0.867488	−0.433744	−	0.901036i	$-0.642808\pi$
$463$	−18.6661	−0.867488	−0.433744	+	0.901036i	$0.642808\pi$
$464$	0	0
$465$	−30.2412	−1.40240
$466$	0	0
$467$	−22.7142	−1.05109	−0.525545	−	0.850766i	$-0.676138\pi$
$467$	−22.7142	−1.05109	−0.525545	+	0.850766i	$0.676138\pi$
$468$	0	0
$469$	−5.21824	−0.240956
$470$	0	0
$471$	14.9060	0.686832
$472$	0	0
$473$	−7.31477	−0.336333
$474$	0	0
$475$	−14.0240	−0.643464
$476$	0	0
$477$	−0.641748	−0.0293836
$478$	0	0
$479$	27.4933	1.25620	0.628101	−	0.778132i	$-0.283832\pi$
$479$	27.4933	1.25620	0.628101	+	0.778132i	$0.283832\pi$
$480$	0	0
$481$	−28.2366	−1.28748
$482$	0	0
$483$	5.57968	0.253884
$484$	0	0
$485$	29.1242	1.32246
$486$	0	0
$487$	−28.3989	−1.28688	−0.643438	−	0.765498i	$-0.722493\pi$
$487$	−28.3989	−1.28688	−0.643438	+	0.765498i	$0.722493\pi$
$488$	0	0
$489$	−15.7791	−0.713555
$490$	0	0
$491$	19.4766	0.878967	0.439484	−	0.898251i	$-0.355161\pi$
$491$	19.4766	0.878967	0.439484	+	0.898251i	$0.355161\pi$
$492$	0	0
$493$	3.89688	0.175507
$494$	0	0
$495$	−1.93985	−0.0871899
$496$	0	0
$497$	1.49131	0.0668946
$498$	0	0
$499$	2.93188	0.131249	0.0656246	−	0.997844i	$-0.479096\pi$
$499$	2.93188	0.131249	0.0656246	+	0.997844i	$0.479096\pi$
$500$	0	0
$501$	1.79442	0.0801687
$502$	0	0
$503$	24.8869	1.10965	0.554825	−	0.831967i	$-0.312785\pi$
$503$	24.8869	1.10965	0.554825	+	0.831967i	$0.312785\pi$
$504$	0	0
$505$	19.2960	0.858659
$506$	0	0
$507$	−3.88740	−0.172646
$508$	0	0
$509$	18.7185	0.829685	0.414842	−	0.909893i	$-0.363837\pi$
$509$	18.7185	0.829685	0.414842	+	0.909893i	$0.363837\pi$
$510$	0	0
$511$	−12.1234	−0.536307
$512$	0	0
$513$	−9.62701	−0.425043
$514$	0	0
$515$	19.3371	0.852096
$516$	0	0
$517$	3.81425	0.167751
$518$	0	0
$519$	21.0488	0.923938
$520$	0	0
$521$	17.8140	0.780444	0.390222	−	0.920721i	$-0.372398\pi$
$521$	17.8140	0.780444	0.390222	+	0.920721i	$0.372398\pi$
$522$	0	0
$523$	−36.7363	−1.60637	−0.803183	−	0.595732i	$-0.796862\pi$
$523$	−36.7363	−1.60637	−0.803183	+	0.595732i	$0.796862\pi$
$524$	0	0
$525$	11.5473	0.503965
$526$	0	0
$527$	14.1439	0.616120
$528$	0	0
$529$	−10.6697	−0.463898
$530$	0	0
$531$	2.33173	0.101188
$532$	0	0
$533$	−6.64477	−0.287817
$534$	0	0
$535$	−36.8321	−1.59239
$536$	0	0
$537$	−33.1187	−1.42918
$538$	0	0
$539$	15.6534	0.674239
$540$	0	0
$541$	17.8489	0.767385	0.383693	−	0.923461i	$-0.374652\pi$
$541$	17.8489	0.767385	0.383693	+	0.923461i	$0.374652\pi$
$542$	0	0
$543$	−10.6068	−0.455182
$544$	0	0
$545$	8.09365	0.346694
$546$	0	0
$547$	−5.45439	−0.233213	−0.116606	−	0.993178i	$-0.537202\pi$
$547$	−5.45439	−0.233213	−0.116606	+	0.993178i	$0.537202\pi$
$548$	0	0
$549$	2.02700	0.0865101
$550$	0	0
$551$	−2.55838	−0.108991
$552$	0	0
$553$	−9.53018	−0.405264
$554$	0	0
$555$	45.5684	1.93427
$556$	0	0
$557$	19.0782	0.808370	0.404185	−	0.914677i	$-0.367555\pi$
$557$	19.0782	0.808370	0.404185	+	0.914677i	$0.367555\pi$
$558$	0	0
$559$	11.3119	0.478441
$560$	0	0
$561$	13.2830	0.560809
$562$	0	0
$563$	−1.06146	−0.0447350	−0.0223675	−	0.999750i	$-0.507120\pi$
$563$	−1.06146	−0.0447350	−0.0223675	+	0.999750i	$0.507120\pi$
$564$	0	0
$565$	−36.2712	−1.52594
$566$	0	0
$567$	8.51111	0.357433
$568$	0	0
$569$	−22.4628	−0.941691	−0.470845	−	0.882216i	$-0.656051\pi$
$569$	−22.4628	−0.941691	−0.470845	+	0.882216i	$0.656051\pi$
$570$	0	0
$571$	−38.0205	−1.59111	−0.795554	−	0.605882i	$-0.792820\pi$
$571$	−38.0205	−1.59111	−0.795554	+	0.605882i	$0.792820\pi$
$572$	0	0
$573$	19.1243	0.798930
$574$	0	0
$575$	25.5179	1.06417
$576$	0	0
$577$	−23.6441	−0.984315	−0.492158	−	0.870506i	$-0.663792\pi$
$577$	−23.6441	−0.984315	−0.492158	+	0.870506i	$0.663792\pi$
$578$	0	0
$579$	36.6915	1.52485
$580$	0	0
$581$	−9.51952	−0.394936
$582$	0	0
$583$	7.34819	0.304331
$584$	0	0
$585$	2.99987	0.124029
$586$	0	0
$587$	−16.8300	−0.694648	−0.347324	−	0.937745i	$-0.612910\pi$
$587$	−16.8300	−0.694648	−0.347324	+	0.937745i	$0.612910\pi$
$588$	0	0
$589$	−9.28578	−0.382614
$590$	0	0
$591$	−42.7987	−1.76050
$592$	0	0
$593$	−29.5694	−1.21427	−0.607134	−	0.794600i	$-0.707681\pi$
$593$	−29.5694	−1.21427	−0.607134	+	0.794600i	$0.707681\pi$
$594$	0	0
$595$	−9.11661	−0.373745
$596$	0	0
$597$	21.4653	0.878517
$598$	0	0
$599$	43.4609	1.77576	0.887882	−	0.460072i	$-0.152176\pi$
$599$	43.4609	1.77576	0.887882	+	0.460072i	$0.152176\pi$
$600$	0	0
$601$	34.9856	1.42709	0.713547	−	0.700608i	$-0.247088\pi$
$601$	34.9856	1.42709	0.713547	+	0.700608i	$0.247088\pi$
$602$	0	0
$603$	1.29604	0.0527787
$604$	0	0
$605$	−16.3149	−0.663295
$606$	0	0
$607$	22.5279	0.914378	0.457189	−	0.889370i	$-0.348856\pi$
$607$	22.5279	0.914378	0.457189	+	0.889370i	$0.348856\pi$
$608$	0	0
$609$	2.10656	0.0853622
$610$	0	0
$611$	−5.89852	−0.238629
$612$	0	0
$613$	−8.85992	−0.357849	−0.178924	−	0.983863i	$-0.557262\pi$
$613$	−8.85992	−0.357849	−0.178924	+	0.983863i	$0.557262\pi$
$614$	0	0
$615$	10.7234	0.432409
$616$	0	0
$617$	−30.0697	−1.21056	−0.605280	−	0.796013i	$-0.706939\pi$
$617$	−30.0697	−1.21056	−0.605280	+	0.796013i	$0.706939\pi$
$618$	0	0
$619$	−43.1458	−1.73418	−0.867088	−	0.498155i	$-0.834011\pi$
$619$	−43.1458	−1.73418	−0.867088	+	0.498155i	$0.834011\pi$
$620$	0	0
$621$	17.5173	0.702943
$622$	0	0
$623$	−10.2923	−0.412353
$624$	0	0
$625$	−8.52527	−0.341011
$626$	0	0
$627$	−8.72055	−0.348265
$628$	0	0
$629$	−21.3126	−0.849788
$630$	0	0
$631$	−23.0561	−0.917849	−0.458925	−	0.888475i	$-0.651765\pi$
$631$	−23.0561	−0.917849	−0.458925	+	0.888475i	$0.651765\pi$
$632$	0	0
$633$	−6.20850	−0.246766
$634$	0	0
$635$	−56.2357	−2.23165
$636$	0	0
$637$	−24.2071	−0.959118
$638$	0	0
$639$	−0.370393	−0.0146525
$640$	0	0
$641$	36.0738	1.42483	0.712415	−	0.701758i	$-0.247601\pi$
$641$	36.0738	1.42483	0.712415	+	0.701758i	$0.247601\pi$
$642$	0	0
$643$	8.28221	0.326618	0.163309	−	0.986575i	$-0.447783\pi$
$643$	8.28221	0.326618	0.163309	+	0.986575i	$0.447783\pi$
$644$	0	0
$645$	−18.2552	−0.718798
$646$	0	0
$647$	25.1191	0.987534	0.493767	−	0.869594i	$-0.335620\pi$
$647$	25.1191	0.987534	0.493767	+	0.869594i	$0.335620\pi$
$648$	0	0
$649$	−26.6989	−1.04802
$650$	0	0
$651$	7.64587	0.299665
$652$	0	0
$653$	−36.0981	−1.41263	−0.706314	−	0.707899i	$-0.749643\pi$
$653$	−36.0981	−1.41263	−0.706314	+	0.707899i	$0.749643\pi$
$654$	0	0
$655$	−8.88386	−0.347121
$656$	0	0
$657$	3.01104	0.117472
$658$	0	0
$659$	46.8891	1.82654	0.913270	−	0.407354i	$-0.133548\pi$
$659$	46.8891	1.82654	0.913270	+	0.407354i	$0.133548\pi$
$660$	0	0
$661$	−25.8778	−1.00653	−0.503266	−	0.864132i	$-0.667868\pi$
$661$	−25.8778	−1.00653	−0.503266	+	0.864132i	$0.667868\pi$
$662$	0	0
$663$	−20.5414	−0.797761
$664$	0	0
$665$	5.98523	0.232097
$666$	0	0
$667$	4.65522	0.180251
$668$	0	0
$669$	8.17851	0.316199
$670$	0	0
$671$	−23.2097	−0.896000
$672$	0	0
$673$	16.5570	0.638227	0.319114	−	0.947716i	$-0.396615\pi$
$673$	16.5570	0.638227	0.319114	+	0.947716i	$0.396615\pi$
$674$	0	0
$675$	36.2524	1.39535
$676$	0	0
$677$	−22.4990	−0.864707	−0.432354	−	0.901704i	$-0.642317\pi$
$677$	−22.4990	−0.864707	−0.432354	+	0.901704i	$0.642317\pi$
$678$	0	0
$679$	−7.36346	−0.282584
$680$	0	0
$681$	−11.4552	−0.438963
$682$	0	0
$683$	35.0116	1.33968	0.669840	−	0.742505i	$-0.266363\pi$
$683$	35.0116	1.33968	0.669840	+	0.742505i	$0.266363\pi$
$684$	0	0
$685$	59.2219	2.26275
$686$	0	0
$687$	−44.0459	−1.68046
$688$	0	0
$689$	−11.3636	−0.432917
$690$	0	0
$691$	1.11136	0.0422783	0.0211391	−	0.999777i	$-0.493271\pi$
$691$	1.11136	0.0422783	0.0211391	+	0.999777i	$0.493271\pi$
$692$	0	0
$693$	0.490453	0.0186308
$694$	0	0
$695$	22.9082	0.868957
$696$	0	0
$697$	−5.01538	−0.189971
$698$	0	0
$699$	12.2740	0.464244
$700$	0	0
$701$	−1.58157	−0.0597350	−0.0298675	−	0.999554i	$-0.509509\pi$
$701$	−1.58157	−0.0597350	−0.0298675	+	0.999554i	$0.509509\pi$
$702$	0	0
$703$	13.9921	0.527723
$704$	0	0
$705$	9.51909	0.358510
$706$	0	0
$707$	−4.87859	−0.183478
$708$	0	0
$709$	14.2311	0.534461	0.267230	−	0.963633i	$-0.413892\pi$
$709$	14.2311	0.534461	0.267230	+	0.963633i	$0.413892\pi$
$710$	0	0
$711$	2.36698	0.0887686
$712$	0	0
$713$	16.8963	0.632773
$714$	0	0
$715$	−34.3494	−1.28459
$716$	0	0
$717$	34.9187	1.30406
$718$	0	0
$719$	−0.495144	−0.0184657	−0.00923287	−	0.999957i	$-0.502939\pi$
$719$	−0.495144	−0.0184657	−0.00923287	+	0.999957i	$0.502939\pi$
$720$	0	0
$721$	−4.88900	−0.182076
$722$	0	0
$723$	37.4637	1.39329
$724$	0	0
$725$	9.63408	0.357801
$726$	0	0
$727$	−1.63007	−0.0604558	−0.0302279	−	0.999543i	$-0.509623\pi$
$727$	−1.63007	−0.0604558	−0.0302279	+	0.999543i	$0.509623\pi$
$728$	0	0
$729$	24.7410	0.916334
$730$	0	0
$731$	8.53804	0.315791
$732$	0	0
$733$	8.24748	0.304628	0.152314	−	0.988332i	$-0.451328\pi$
$733$	8.24748	0.304628	0.152314	+	0.988332i	$0.451328\pi$
$734$	0	0
$735$	39.0656	1.44096
$736$	0	0
$737$	−14.8400	−0.546638
$738$	0	0
$739$	24.0304	0.883971	0.441985	−	0.897022i	$-0.354274\pi$
$739$	24.0304	0.883971	0.441985	+	0.897022i	$0.354274\pi$
$740$	0	0
$741$	13.4858	0.495414
$742$	0	0
$743$	47.1259	1.72888	0.864442	−	0.502733i	$-0.167672\pi$
$743$	47.1259	1.72888	0.864442	+	0.502733i	$0.167672\pi$
$744$	0	0
$745$	−47.2441	−1.73089
$746$	0	0
$747$	2.36433	0.0865063
$748$	0	0
$749$	9.31225	0.340262
$750$	0	0
$751$	−18.7706	−0.684950	−0.342475	−	0.939527i	$-0.611265\pi$
$751$	−18.7706	−0.684950	−0.342475	+	0.939527i	$0.611265\pi$
$752$	0	0
$753$	−3.04049	−0.110802
$754$	0	0
$755$	2.72117	0.0990335
$756$	0	0
$757$	22.5474	0.819500	0.409750	−	0.912198i	$-0.365616\pi$
$757$	22.5474	0.819500	0.409750	+	0.912198i	$0.365616\pi$
$758$	0	0
$759$	15.8679	0.575967
$760$	0	0
$761$	−10.6268	−0.385221	−0.192611	−	0.981275i	$-0.561695\pi$
$761$	−10.6268	−0.385221	−0.192611	+	0.981275i	$0.561695\pi$
$762$	0	0
$763$	−2.04632	−0.0740816
$764$	0	0
$765$	2.26426	0.0818645
$766$	0	0
$767$	41.2883	1.49083
$768$	0	0
$769$	33.5903	1.21130	0.605649	−	0.795732i	$-0.292914\pi$
$769$	33.5903	1.21130	0.605649	+	0.795732i	$0.292914\pi$
$770$	0	0
$771$	−7.96729	−0.286935
$772$	0	0
$773$	−4.31618	−0.155242	−0.0776210	−	0.996983i	$-0.524732\pi$
$773$	−4.31618	−0.155242	−0.0776210	+	0.996983i	$0.524732\pi$
$774$	0	0
$775$	34.9674	1.25607
$776$	0	0
$777$	−11.5211	−0.413316
$778$	0	0
$779$	3.29269	0.117973
$780$	0	0
$781$	4.24110	0.151759
$782$	0	0
$783$	6.61349	0.236347
$784$	0	0
$785$	−29.0943	−1.03842
$786$	0	0
$787$	−5.97812	−0.213097	−0.106548	−	0.994308i	$-0.533980\pi$
$787$	−5.97812	−0.213097	−0.106548	+	0.994308i	$0.533980\pi$
$788$	0	0
$789$	41.5209	1.47818
$790$	0	0
$791$	9.17044	0.326063
$792$	0	0
$793$	35.8924	1.27458
$794$	0	0
$795$	18.3386	0.650404
$796$	0	0
$797$	42.5108	1.50581	0.752904	−	0.658130i	$-0.228652\pi$
$797$	42.5108	1.50581	0.752904	+	0.658130i	$0.228652\pi$
$798$	0	0
$799$	−4.45212	−0.157505
$800$	0	0
$801$	2.55627	0.0903213
$802$	0	0
$803$	−34.4773	−1.21668
$804$	0	0
$805$	−10.8907	−0.383847
$806$	0	0
$807$	−33.6870	−1.18584
$808$	0	0
$809$	30.7160	1.07992	0.539959	−	0.841691i	$-0.318440\pi$
$809$	30.7160	1.07992	0.539959	+	0.841691i	$0.318440\pi$
$810$	0	0
$811$	−2.07707	−0.0729357	−0.0364678	−	0.999335i	$-0.511611\pi$
$811$	−2.07707	−0.0729357	−0.0364678	+	0.999335i	$0.511611\pi$
$812$	0	0
$813$	10.9678	0.384658
$814$	0	0
$815$	30.7984	1.07882
$816$	0	0
$817$	−5.60539	−0.196108
$818$	0	0
$819$	−0.758456	−0.0265026
$820$	0	0
$821$	1.26392	0.0441110	0.0220555	−	0.999757i	$-0.492979\pi$
$821$	1.26392	0.0441110	0.0220555	+	0.999757i	$0.492979\pi$
$822$	0	0
$823$	−11.7077	−0.408106	−0.204053	−	0.978960i	$-0.565411\pi$
$823$	−11.7077	−0.408106	−0.204053	+	0.978960i	$0.565411\pi$
$824$	0	0
$825$	32.8389	1.14330
$826$	0	0
$827$	7.96678	0.277032	0.138516	−	0.990360i	$-0.455767\pi$
$827$	7.96678	0.277032	0.138516	+	0.990360i	$0.455767\pi$
$828$	0	0
$829$	−3.58840	−0.124630	−0.0623151	−	0.998057i	$-0.519848\pi$
$829$	−3.58840	−0.124630	−0.0623151	+	0.998057i	$0.519848\pi$
$830$	0	0
$831$	0.479686	0.0166401
$832$	0	0
$833$	−18.2712	−0.633058
$834$	0	0
$835$	−3.50243	−0.121207
$836$	0	0
$837$	24.0040	0.829700
$838$	0	0
$839$	31.1160	1.07424	0.537122	−	0.843505i	$-0.319511\pi$
$839$	31.1160	1.07424	0.537122	+	0.843505i	$0.319511\pi$
$840$	0	0
$841$	−27.2425	−0.939395
$842$	0	0
$843$	−12.5741	−0.433074
$844$	0	0
$845$	7.58763	0.261022
$846$	0	0
$847$	4.12489	0.141733
$848$	0	0
$849$	45.9320	1.57638
$850$	0	0
$851$	−25.4600	−0.872757
$852$	0	0
$853$	38.7097	1.32539	0.662697	−	0.748888i	$-0.269412\pi$
$853$	38.7097	1.32539	0.662697	+	0.748888i	$0.269412\pi$
$854$	0	0
$855$	−1.48653	−0.0508383
$856$	0	0
$857$	−4.71633	−0.161107	−0.0805534	−	0.996750i	$-0.525669\pi$
$857$	−4.71633	−0.161107	−0.0805534	+	0.996750i	$0.525669\pi$
$858$	0	0
$859$	−4.44140	−0.151539	−0.0757694	−	0.997125i	$-0.524141\pi$
$859$	−4.44140	−0.151539	−0.0757694	+	0.997125i	$0.524141\pi$
$860$	0	0
$861$	−2.71119	−0.0923972
$862$	0	0
$863$	−27.6818	−0.942298	−0.471149	−	0.882054i	$-0.656161\pi$
$863$	−27.6818	−0.942298	−0.471149	+	0.882054i	$0.656161\pi$
$864$	0	0
$865$	−41.0840	−1.39690
$866$	0	0
$867$	15.0007	0.509452
$868$	0	0
$869$	−27.1026	−0.919391
$870$	0	0
$871$	22.9492	0.777603
$872$	0	0
$873$	1.82884	0.0618967
$874$	0	0
$875$	−7.03119	−0.237697
$876$	0	0
$877$	−10.6327	−0.359042	−0.179521	−	0.983754i	$-0.557455\pi$
$877$	−10.6327	−0.359042	−0.179521	+	0.983754i	$0.557455\pi$
$878$	0	0
$879$	14.3186	0.482953
$880$	0	0
$881$	−6.58181	−0.221747	−0.110873	−	0.993835i	$-0.535365\pi$
$881$	−6.58181	−0.221747	−0.110873	+	0.993835i	$0.535365\pi$
$882$	0	0
$883$	24.6555	0.829724	0.414862	−	0.909884i	$-0.363830\pi$
$883$	24.6555	0.829724	0.414862	+	0.909884i	$0.363830\pi$
$884$	0	0
$885$	−66.6315	−2.23979
$886$	0	0
$887$	−49.5484	−1.66367	−0.831836	−	0.555022i	$-0.812710\pi$
$887$	−49.5484	−1.66367	−0.831836	+	0.555022i	$0.812710\pi$
$888$	0	0
$889$	14.2181	0.476859
$890$	0	0
$891$	24.2045	0.810880
$892$	0	0
$893$	2.92291	0.0978113
$894$	0	0
$895$	64.6427	2.16077
$896$	0	0
$897$	−24.5387	−0.819324
$898$	0	0
$899$	6.37907	0.212754
$900$	0	0
$901$	−8.57706	−0.285743
$902$	0	0
$903$	4.61546	0.153593
$904$	0	0
$905$	20.7029	0.688188
$906$	0	0
$907$	−59.9459	−1.99047	−0.995236	−	0.0974963i	$-0.968917\pi$
$907$	−59.9459	−1.99047	−0.995236	+	0.0974963i	$0.968917\pi$
$908$	0	0
$909$	1.21168	0.0401889
$910$	0	0
$911$	44.1354	1.46227	0.731136	−	0.682232i	$-0.238991\pi$
$911$	44.1354	1.46227	0.731136	+	0.682232i	$0.238991\pi$
$912$	0	0
$913$	−27.0722	−0.895960
$914$	0	0
$915$	−57.9236	−1.91489
$916$	0	0
$917$	2.24611	0.0741729
$918$	0	0
$919$	48.9112	1.61343	0.806716	−	0.590940i	$-0.201243\pi$
$919$	48.9112	1.61343	0.806716	+	0.590940i	$0.201243\pi$
$920$	0	0
$921$	−41.5770	−1.37001
$922$	0	0
$923$	−6.55861	−0.215879
$924$	0	0
$925$	−52.6901	−1.73244
$926$	0	0
$927$	1.21426	0.0398817
$928$	0	0
$929$	54.4181	1.78540	0.892701	−	0.450650i	$-0.148808\pi$
$929$	54.4181	1.78540	0.892701	+	0.450650i	$0.148808\pi$
$930$	0	0
$931$	11.9954	0.393132
$932$	0	0
$933$	17.2233	0.563865
$934$	0	0
$935$	−25.9264	−0.847885
$936$	0	0
$937$	56.3450	1.84071	0.920355	−	0.391083i	$-0.127900\pi$
$937$	56.3450	1.84071	0.920355	+	0.391083i	$0.127900\pi$
$938$	0	0
$939$	2.66889	0.0870959
$940$	0	0
$941$	−40.2631	−1.31254	−0.656270	−	0.754526i	$-0.727867\pi$
$941$	−40.2631	−1.31254	−0.656270	+	0.754526i	$0.727867\pi$
$942$	0	0
$943$	−5.99137	−0.195106
$944$	0	0
$945$	−15.4720	−0.503305
$946$	0	0
$947$	−22.8685	−0.743128	−0.371564	−	0.928407i	$-0.621178\pi$
$947$	−22.8685	−0.743128	−0.371564	+	0.928407i	$0.621178\pi$
$948$	0	0
$949$	53.3171	1.73075
$950$	0	0
$951$	19.5420	0.633694
$952$	0	0
$953$	−38.9214	−1.26079	−0.630394	−	0.776276i	$-0.717107\pi$
$953$	−38.9214	−1.26079	−0.630394	+	0.776276i	$0.717107\pi$
$954$	0	0
$955$	−37.3278	−1.20790
$956$	0	0
$957$	5.99078	0.193654
$958$	0	0
$959$	−14.9731	−0.483505
$960$	0	0
$961$	−7.84682	−0.253123
$962$	0	0
$963$	−2.31285	−0.0745306
$964$	0	0
$965$	−71.6163	−2.30541
$966$	0	0
$967$	14.9687	0.481363	0.240681	−	0.970604i	$-0.422629\pi$
$967$	14.9687	0.481363	0.240681	+	0.970604i	$0.422629\pi$
$968$	0	0
$969$	10.1789	0.326994
$970$	0	0
$971$	−19.8940	−0.638428	−0.319214	−	0.947683i	$-0.603419\pi$
$971$	−19.8940	−0.638428	−0.319214	+	0.947683i	$0.603419\pi$
$972$	0	0
$973$	−5.79187	−0.185679
$974$	0	0
$975$	−50.7835	−1.62637
$976$	0	0
$977$	−59.0802	−1.89014	−0.945071	−	0.326865i	$-0.894008\pi$
$977$	−59.0802	−1.89014	−0.945071	+	0.326865i	$0.894008\pi$
$978$	0	0
$979$	−29.2700	−0.935473
$980$	0	0
$981$	0.508236	0.0162267
$982$	0	0
$983$	−38.1136	−1.21564	−0.607818	−	0.794077i	$-0.707955\pi$
$983$	−38.1136	−1.21564	−0.607818	+	0.794077i	$0.707955\pi$
$984$	0	0
$985$	83.5367	2.66170
$986$	0	0
$987$	−2.40671	−0.0766064
$988$	0	0
$989$	10.1995	0.324326
$990$	0	0
$991$	36.2501	1.15152	0.575761	−	0.817618i	$-0.304706\pi$
$991$	36.2501	1.15152	0.575761	+	0.817618i	$0.304706\pi$
$992$	0	0
$993$	48.3228	1.53348
$994$	0	0
$995$	−41.8971	−1.32823
$996$	0	0
$997$	−20.9256	−0.662721	−0.331360	−	0.943504i	$-0.607508\pi$
$997$	−20.9256	−0.662721	−0.331360	+	0.943504i	$0.607508\pi$
$998$	0	0
$999$	−36.1701	−1.14437

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1336.2.a.d.1.4	✓	12
4.3	odd	2		2672.2.a.p.1.9		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1336.2.a.d.1.4	✓	12	1.1	even	1	trivial
2672.2.a.p.1.9		12	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)

\(f(q)\)	\(=\)	\(q-1.79442 q^{3} +3.50243 q^{5} -0.885520 q^{7} +0.219933 q^{9} +O(q^{10})\)	\(q-1.79442 q^{3} +3.50243 q^{5} -0.885520 q^{7} +0.219933 q^{9} -2.51830 q^{11} +3.89440 q^{13} -6.28483 q^{15} +2.93944 q^{17} -1.92980 q^{19} +1.58899 q^{21} +3.51146 q^{23} +7.26705 q^{25} +4.98860 q^{27} +1.32572 q^{29} +4.81178 q^{31} +4.51888 q^{33} -3.10147 q^{35} -7.25055 q^{37} -6.98819 q^{39} -1.70623 q^{41} +2.90464 q^{43} +0.770302 q^{45} -1.51461 q^{47} -6.21585 q^{49} -5.27459 q^{51} -2.91792 q^{53} -8.82018 q^{55} +3.46287 q^{57} +10.6020 q^{59} +9.21641 q^{61} -0.194755 q^{63} +13.6399 q^{65} +5.89286 q^{67} -6.30102 q^{69} -1.68411 q^{71} +13.6907 q^{73} -13.0401 q^{75} +2.23000 q^{77} +10.7622 q^{79} -9.61143 q^{81} +10.7502 q^{83} +10.2952 q^{85} -2.37890 q^{87} +11.6229 q^{89} -3.44857 q^{91} -8.63433 q^{93} -6.75901 q^{95} +8.31541 q^{97} -0.553859 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - q^{3} + 8 q^{5} - 4 q^{7} + 15 q^{9}+O(q^{10}) \) 12 * q - q^3 + 8 * q^5 - 4 * q^7 + 15 * q^9	\( 12 q - q^{3} + 8 q^{5} - 4 q^{7} + 15 q^{9} + 6 q^{11} + 13 q^{13} + 4 q^{15} + 10 q^{17} - q^{19} + 13 q^{21} + 3 q^{23} + 18 q^{25} - 10 q^{27} + 29 q^{29} - 3 q^{31} + 3 q^{35} + 41 q^{37} + 10 q^{39} + 20 q^{41} - q^{43} + 42 q^{45} + 5 q^{47} + 20 q^{49} + 14 q^{51} + 39 q^{53} + 3 q^{55} + 3 q^{57} + 8 q^{59} + 30 q^{61} - 2 q^{63} + 21 q^{65} - 9 q^{67} + 33 q^{69} + 29 q^{71} + 12 q^{73} + q^{75} + 19 q^{77} - 2 q^{79} + 24 q^{81} - 5 q^{83} + 44 q^{85} - 2 q^{87} + 7 q^{89} - 4 q^{91} + 37 q^{93} + 18 q^{95} - 14 q^{97} + 5 q^{99}+O(q^{100}) \) 12 * q - q^3 + 8 * q^5 - 4 * q^7 + 15 * q^9 + 6 * q^11 + 13 * q^13 + 4 * q^15 + 10 * q^17 - q^19 + 13 * q^21 + 3 * q^23 + 18 * q^25 - 10 * q^27 + 29 * q^29 - 3 * q^31 + 3 * q^35 + 41 * q^37 + 10 * q^39 + 20 * q^41 - q^43 + 42 * q^45 + 5 * q^47 + 20 * q^49 + 14 * q^51 + 39 * q^53 + 3 * q^55 + 3 * q^57 + 8 * q^59 + 30 * q^61 - 2 * q^63 + 21 * q^65 - 9 * q^67 + 33 * q^69 + 29 * q^71 + 12 * q^73 + q^75 + 19 * q^77 - 2 * q^79 + 24 * q^81 - 5 * q^83 + 44 * q^85 - 2 * q^87 + 7 * q^89 - 4 * q^91 + 37 * q^93 + 18 * q^95 - 14 * q^97 + 5 * q^99

Introduction

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