LMFDB - Embedded newform 4304.2.a.n.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4304,2,Mod(1,4304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4304, 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("4304.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4304 = 2^{4} \cdot 269 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4304.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:	$34.3676130300$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 4 x^{17} - 28 x^{16} + 121 x^{15} + 293 x^{14} - 1427 x^{13} - 1440 x^{12} + 8461 x^{11} + \cdots + 1006 $ x^18 - 4x^17 - 28x^16 + 121x^15 + 293x^14 - 1427x^13 - 1440x^12 + 8461x^11 + 3463x^10 - 27446x^9 - 3364x^8 + 49073x^7 - 1604x^6 - 45902x^5 + 7013x^4 + 20409x^3 - 5085x^2 - 3241*x + 1006
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2152)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$0.801198$ of defining polynomial
Character	$\chi$	$=$	4304.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.801198 q^{3} -3.55277 q^{5} +0.142030 q^{7} -2.35808 q^{9} -5.09556 q^{11} -3.44639 q^{13} -2.84647 q^{15} +5.28924 q^{17} -4.97197 q^{19} +0.113794 q^{21} -1.58205 q^{23} +7.62217 q^{25} -4.29289 q^{27} +5.53031 q^{29} +0.382134 q^{31} -4.08255 q^{33} -0.504599 q^{35} -6.21978 q^{37} -2.76125 q^{39} -11.4440 q^{41} +9.71230 q^{43} +8.37772 q^{45} +9.64910 q^{47} -6.97983 q^{49} +4.23773 q^{51} -10.7665 q^{53} +18.1033 q^{55} -3.98354 q^{57} -2.10144 q^{59} +8.92802 q^{61} -0.334918 q^{63} +12.2442 q^{65} -12.4585 q^{67} -1.26754 q^{69} +7.78439 q^{71} +8.61715 q^{73} +6.10687 q^{75} -0.723721 q^{77} +11.8091 q^{79} +3.63479 q^{81} -5.61859 q^{83} -18.7914 q^{85} +4.43087 q^{87} -12.7423 q^{89} -0.489491 q^{91} +0.306165 q^{93} +17.6643 q^{95} +8.41309 q^{97} +12.0157 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 4 q^{3} - 6 q^{5} + 17 q^{7} + 18 q^{9} + 10 q^{11} - 3 q^{13} + 19 q^{15} - 6 q^{17} + 3 q^{19} + q^{21} + 37 q^{23} + 18 q^{25} + 13 q^{27} - 3 q^{29} + 35 q^{31} - 9 q^{33} + 12 q^{35} - 8 q^{37}+ \cdots + 41 q^{99}+O(q^{100}) $ 18 * q + 4 * q^3 - 6 * q^5 + 17 * q^7 + 18 * q^9 + 10 * q^11 - 3 * q^13 + 19 * q^15 - 6 * q^17 + 3 * q^19 + q^21 + 37 * q^23 + 18 * q^25 + 13 * q^27 - 3 * q^29 + 35 * q^31 - 9 * q^33 + 12 * q^35 - 8 * q^37 + 44 * q^39 - 12 * q^41 + 15 * q^43 - 25 * q^45 + 28 * q^47 + 19 * q^49 + 9 * q^51 - 33 * q^53 + 35 * q^55 - 26 * q^57 + 22 * q^59 + 8 * q^61 + 75 * q^63 - 20 * q^65 + 25 * q^67 - 3 * q^69 + 55 * q^71 - 8 * q^73 + 28 * q^75 - 8 * q^77 + 80 * q^79 + 22 * q^81 + 22 * q^83 + 13 * q^85 + 55 * q^87 - 26 * q^89 + 19 * q^91 - 21 * q^93 + 56 * q^95 - 28 * q^97 + 41 * q^99

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
$3$	0.801198	0.462572	0.231286	−	0.972886i	$-0.425707\pi$
$3$	0.801198	0.462572	0.231286	+	0.972886i	$0.425707\pi$
$4$	0	0
$5$	−3.55277	−1.58885	−0.794423	−	0.607364i	$-0.792227\pi$
$5$	−3.55277	−1.58885	−0.794423	+	0.607364i	$0.792227\pi$
$6$	0	0
$7$	0.142030	0.0536822	0.0268411	−	0.999640i	$-0.491455\pi$
$7$	0.142030	0.0536822	0.0268411	+	0.999640i	$0.491455\pi$
$8$	0	0
$9$	−2.35808	−0.786027
$10$	0	0
$11$	−5.09556	−1.53637	−0.768184	−	0.640229i	$-0.778839\pi$
$11$	−5.09556	−1.53637	−0.768184	+	0.640229i	$0.778839\pi$
$12$	0	0
$13$	−3.44639	−0.955858	−0.477929	−	0.878398i	$-0.658612\pi$
$13$	−3.44639	−0.955858	−0.477929	+	0.878398i	$0.658612\pi$
$14$	0	0
$15$	−2.84647	−0.734956
$16$	0	0
$17$	5.28924	1.28283	0.641414	−	0.767195i	$-0.278348\pi$
$17$	5.28924	1.28283	0.641414	+	0.767195i	$0.278348\pi$
$18$	0	0
$19$	−4.97197	−1.14065	−0.570324	−	0.821420i	$-0.693183\pi$
$19$	−4.97197	−1.14065	−0.570324	+	0.821420i	$0.693183\pi$
$20$	0	0
$21$	0.113794	0.0248319
$22$	0	0
$23$	−1.58205	−0.329881	−0.164940	−	0.986304i	$-0.552743\pi$
$23$	−1.58205	−0.329881	−0.164940	+	0.986304i	$0.552743\pi$
$24$	0	0
$25$	7.62217	1.52443
$26$	0	0
$27$	−4.29289	−0.826166
$28$	0	0
$29$	5.53031	1.02695	0.513476	−	0.858104i	$-0.328358\pi$
$29$	5.53031	1.02695	0.513476	+	0.858104i	$0.328358\pi$
$30$	0	0
$31$	0.382134	0.0686334	0.0343167	−	0.999411i	$-0.489075\pi$
$31$	0.382134	0.0686334	0.0343167	+	0.999411i	$0.489075\pi$
$32$	0	0
$33$	−4.08255	−0.710681
$34$	0	0
$35$	−0.504599	−0.0852928
$36$	0	0
$37$	−6.21978	−1.02252	−0.511262	−	0.859425i	$-0.670822\pi$
$37$	−6.21978	−1.02252	−0.511262	+	0.859425i	$0.670822\pi$
$38$	0	0
$39$	−2.76125	−0.442153
$40$	0	0
$41$	−11.4440	−1.78725	−0.893623	−	0.448818i	$-0.851845\pi$
$41$	−11.4440	−1.78725	−0.893623	+	0.448818i	$0.851845\pi$
$42$	0	0
$43$	9.71230	1.48111	0.740556	−	0.671995i	$-0.234562\pi$
$43$	9.71230	1.48111	0.740556	+	0.671995i	$0.234562\pi$
$44$	0	0
$45$	8.37772	1.24888
$46$	0	0
$47$	9.64910	1.40747	0.703733	−	0.710464i	$-0.251515\pi$
$47$	9.64910	1.40747	0.703733	+	0.710464i	$0.251515\pi$
$48$	0	0
$49$	−6.97983	−0.997118
$50$	0	0
$51$	4.23773	0.593400
$52$	0	0
$53$	−10.7665	−1.47889	−0.739444	−	0.673218i	$-0.764911\pi$
$53$	−10.7665	−1.47889	−0.739444	+	0.673218i	$0.764911\pi$
$54$	0	0
$55$	18.1033	2.44105
$56$	0	0
$57$	−3.98354	−0.527632
$58$	0	0
$59$	−2.10144	−0.273584	−0.136792	−	0.990600i	$-0.543679\pi$
$59$	−2.10144	−0.273584	−0.136792	+	0.990600i	$0.543679\pi$
$60$	0	0
$61$	8.92802	1.14312	0.571558	−	0.820562i	$-0.306339\pi$
$61$	8.92802	1.14312	0.571558	+	0.820562i	$0.306339\pi$
$62$	0	0
$63$	−0.334918	−0.0421957
$64$	0	0
$65$	12.2442	1.51871
$66$	0	0
$67$	−12.4585	−1.52205	−0.761024	−	0.648724i	$-0.775303\pi$
$67$	−12.4585	−1.52205	−0.761024	+	0.648724i	$0.775303\pi$
$68$	0	0
$69$	−1.26754	−0.152594
$70$	0	0
$71$	7.78439	0.923837	0.461919	−	0.886922i	$-0.347161\pi$
$71$	7.78439	0.923837	0.461919	+	0.886922i	$0.347161\pi$
$72$	0	0
$73$	8.61715	1.00856	0.504281	−	0.863540i	$-0.331758\pi$
$73$	8.61715	1.00856	0.504281	+	0.863540i	$0.331758\pi$
$74$	0	0
$75$	6.10687	0.705160
$76$	0	0
$77$	−0.723721	−0.0824756
$78$	0	0
$79$	11.8091	1.32863	0.664314	−	0.747453i	$-0.268724\pi$
$79$	11.8091	1.32863	0.664314	+	0.747453i	$0.268724\pi$
$80$	0	0
$81$	3.63479	0.403866
$82$	0	0
$83$	−5.61859	−0.616720	−0.308360	−	0.951270i	$-0.599780\pi$
$83$	−5.61859	−0.616720	−0.308360	+	0.951270i	$0.599780\pi$
$84$	0	0
$85$	−18.7914	−2.03822
$86$	0	0
$87$	4.43087	0.475039
$88$	0	0
$89$	−12.7423	−1.35069	−0.675343	−	0.737504i	$-0.736004\pi$
$89$	−12.7423	−1.35069	−0.675343	+	0.737504i	$0.736004\pi$
$90$	0	0
$91$	−0.489491	−0.0513126
$92$	0	0
$93$	0.306165	0.0317479
$94$	0	0
$95$	17.6643	1.81232
$96$	0	0
$97$	8.41309	0.854220	0.427110	−	0.904200i	$-0.359532\pi$
$97$	8.41309	0.854220	0.427110	+	0.904200i	$0.359532\pi$
$98$	0	0
$99$	12.0157	1.20763
$100$	0	0
$101$	13.0360	1.29713	0.648565	−	0.761160i	$-0.275370\pi$
$101$	13.0360	1.29713	0.648565	+	0.761160i	$0.275370\pi$
$102$	0	0
$103$	14.0245	1.38188	0.690939	−	0.722913i	$-0.257197\pi$
$103$	14.0245	1.38188	0.690939	+	0.722913i	$0.257197\pi$
$104$	0	0
$105$	−0.404284	−0.0394541
$106$	0	0
$107$	18.8252	1.81990	0.909950	−	0.414717i	$-0.136119\pi$
$107$	18.8252	1.81990	0.909950	+	0.414717i	$0.136119\pi$
$108$	0	0
$109$	2.99688	0.287049	0.143525	−	0.989647i	$-0.454156\pi$
$109$	2.99688	0.287049	0.143525	+	0.989647i	$0.454156\pi$
$110$	0	0
$111$	−4.98327	−0.472991
$112$	0	0
$113$	−3.40804	−0.320602	−0.160301	−	0.987068i	$-0.551246\pi$
$113$	−3.40804	−0.320602	−0.160301	+	0.987068i	$0.551246\pi$
$114$	0	0
$115$	5.62067	0.524130
$116$	0	0
$117$	8.12688	0.751330
$118$	0	0
$119$	0.751229	0.0688650
$120$	0	0
$121$	14.9647	1.36043
$122$	0	0
$123$	−9.16888	−0.826730
$124$	0	0
$125$	−9.31596	−0.833245
$126$	0	0
$127$	13.9165	1.23489	0.617446	−	0.786613i	$-0.288167\pi$
$127$	13.9165	1.23489	0.617446	+	0.786613i	$0.288167\pi$
$128$	0	0
$129$	7.78147	0.685121
$130$	0	0
$131$	−5.62320	−0.491301	−0.245651	−	0.969358i	$-0.579002\pi$
$131$	−5.62320	−0.491301	−0.245651	+	0.969358i	$0.579002\pi$
$132$	0	0
$133$	−0.706168	−0.0612325
$134$	0	0
$135$	15.2516	1.31265
$136$	0	0
$137$	2.70088	0.230752	0.115376	−	0.993322i	$-0.463193\pi$
$137$	2.70088	0.230752	0.115376	+	0.993322i	$0.463193\pi$
$138$	0	0
$139$	−12.6711	−1.07475	−0.537376	−	0.843343i	$-0.680584\pi$
$139$	−12.6711	−1.07475	−0.537376	+	0.843343i	$0.680584\pi$
$140$	0	0
$141$	7.73085	0.651055
$142$	0	0
$143$	17.5613	1.46855
$144$	0	0
$145$	−19.6479	−1.63167
$146$	0	0
$147$	−5.59223	−0.461239
$148$	0	0
$149$	−13.8940	−1.13824	−0.569120	−	0.822254i	$-0.692716\pi$
$149$	−13.8940	−1.13824	−0.569120	+	0.822254i	$0.692716\pi$
$150$	0	0
$151$	−8.32426	−0.677419	−0.338710	−	0.940891i	$-0.609990\pi$
$151$	−8.32426	−0.677419	−0.338710	+	0.940891i	$0.609990\pi$
$152$	0	0
$153$	−12.4724	−1.00834
$154$	0	0
$155$	−1.35764	−0.109048
$156$	0	0
$157$	2.32917	0.185888	0.0929442	−	0.995671i	$-0.470372\pi$
$157$	2.32917	0.185888	0.0929442	+	0.995671i	$0.470372\pi$
$158$	0	0
$159$	−8.62607	−0.684092
$160$	0	0
$161$	−0.224699	−0.0177087
$162$	0	0
$163$	12.8628	1.00749	0.503745	−	0.863852i	$-0.331955\pi$
$163$	12.8628	1.00749	0.503745	+	0.863852i	$0.331955\pi$
$164$	0	0
$165$	14.5044	1.12916
$166$	0	0
$167$	−17.0141	−1.31659	−0.658295	−	0.752760i	$-0.728722\pi$
$167$	−17.0141	−1.31659	−0.658295	+	0.752760i	$0.728722\pi$
$168$	0	0
$169$	−1.12236	−0.0863357
$170$	0	0
$171$	11.7243	0.896581
$172$	0	0
$173$	−9.70735	−0.738036	−0.369018	−	0.929422i	$-0.620306\pi$
$173$	−9.70735	−0.738036	−0.369018	+	0.929422i	$0.620306\pi$
$174$	0	0
$175$	1.08257	0.0818349
$176$	0	0
$177$	−1.68367	−0.126552
$178$	0	0
$179$	11.6628	0.871716	0.435858	−	0.900015i	$-0.356445\pi$
$179$	11.6628	0.871716	0.435858	+	0.900015i	$0.356445\pi$
$180$	0	0
$181$	−0.149308	−0.0110980	−0.00554899	−	0.999985i	$-0.501766\pi$
$181$	−0.149308	−0.0110980	−0.00554899	+	0.999985i	$0.501766\pi$
$182$	0	0
$183$	7.15312	0.528774
$184$	0	0
$185$	22.0974	1.62464
$186$	0	0
$187$	−26.9516	−1.97090
$188$	0	0
$189$	−0.609717	−0.0443504
$190$	0	0
$191$	−2.65407	−0.192042	−0.0960208	−	0.995379i	$-0.530612\pi$
$191$	−2.65407	−0.192042	−0.0960208	+	0.995379i	$0.530612\pi$
$192$	0	0
$193$	5.83060	0.419696	0.209848	−	0.977734i	$-0.432703\pi$
$193$	5.83060	0.419696	0.209848	+	0.977734i	$0.432703\pi$
$194$	0	0
$195$	9.81007	0.702514
$196$	0	0
$197$	−20.5474	−1.46394	−0.731972	−	0.681335i	$-0.761400\pi$
$197$	−20.5474	−1.46394	−0.731972	+	0.681335i	$0.761400\pi$
$198$	0	0
$199$	−20.1328	−1.42718	−0.713588	−	0.700566i	$-0.752931\pi$
$199$	−20.1328	−1.42718	−0.713588	+	0.700566i	$0.752931\pi$
$200$	0	0
$201$	−9.98173	−0.704057
$202$	0	0
$203$	0.785468	0.0551290
$204$	0	0
$205$	40.6577	2.83966
$206$	0	0
$207$	3.73061	0.259295
$208$	0	0
$209$	25.3350	1.75246
$210$	0	0
$211$	7.56873	0.521053	0.260526	−	0.965467i	$-0.416104\pi$
$211$	7.56873	0.521053	0.260526	+	0.965467i	$0.416104\pi$
$212$	0	0
$213$	6.23684	0.427341
$214$	0	0
$215$	−34.5055	−2.35326
$216$	0	0
$217$	0.0542744	0.00368439
$218$	0	0
$219$	6.90405	0.466532
$220$	0	0
$221$	−18.2288	−1.22620
$222$	0	0
$223$	26.2466	1.75760	0.878801	−	0.477188i	$-0.158344\pi$
$223$	26.2466	1.75760	0.878801	+	0.477188i	$0.158344\pi$
$224$	0	0
$225$	−17.9737	−1.19825
$226$	0	0
$227$	26.7791	1.77739	0.888697	−	0.458495i	$-0.151611\pi$
$227$	26.7791	1.77739	0.888697	+	0.458495i	$0.151611\pi$
$228$	0	0
$229$	21.4657	1.41849	0.709247	−	0.704961i	$-0.249035\pi$
$229$	21.4657	1.41849	0.709247	+	0.704961i	$0.249035\pi$
$230$	0	0
$231$	−0.579844	−0.0381509
$232$	0	0
$233$	−20.6557	−1.35320	−0.676600	−	0.736351i	$-0.736547\pi$
$233$	−20.6557	−1.35320	−0.676600	+	0.736351i	$0.736547\pi$
$234$	0	0
$235$	−34.2810	−2.23625
$236$	0	0
$237$	9.46144	0.614586
$238$	0	0
$239$	−3.68982	−0.238674	−0.119337	−	0.992854i	$-0.538077\pi$
$239$	−3.68982	−0.238674	−0.119337	+	0.992854i	$0.538077\pi$
$240$	0	0
$241$	−30.5306	−1.96665	−0.983324	−	0.181864i	$-0.941787\pi$
$241$	−30.5306	−1.96665	−0.983324	+	0.181864i	$0.941787\pi$
$242$	0	0
$243$	15.7908	1.01298
$244$	0	0
$245$	24.7977	1.58427
$246$	0	0
$247$	17.1354	1.09030
$248$	0	0
$249$	−4.50160	−0.285277
$250$	0	0
$251$	10.9051	0.688326	0.344163	−	0.938910i	$-0.388163\pi$
$251$	10.9051	0.688326	0.344163	+	0.938910i	$0.388163\pi$
$252$	0	0
$253$	8.06144	0.506819
$254$	0	0
$255$	−15.0557	−0.942822
$256$	0	0
$257$	−1.09648	−0.0683965	−0.0341983	−	0.999415i	$-0.510888\pi$
$257$	−1.09648	−0.0683965	−0.0341983	+	0.999415i	$0.510888\pi$
$258$	0	0
$259$	−0.883393	−0.0548914
$260$	0	0
$261$	−13.0409	−0.807212
$262$	0	0
$263$	26.4167	1.62892	0.814462	−	0.580216i	$-0.197032\pi$
$263$	26.4167	1.62892	0.814462	+	0.580216i	$0.197032\pi$
$264$	0	0
$265$	38.2508	2.34973
$266$	0	0
$267$	−10.2091	−0.624790
$268$	0	0
$269$	1.00000	0.0609711
$269$	1.00000	0.0609711
$270$	0	0
$271$	−1.64796	−0.100106	−0.0500532	−	0.998747i	$-0.515939\pi$
$271$	−1.64796	−0.100106	−0.0500532	+	0.998747i	$0.515939\pi$
$272$	0	0
$273$	−0.392179	−0.0237358
$274$	0	0
$275$	−38.8392	−2.34209
$276$	0	0
$277$	28.2663	1.69836	0.849178	−	0.528106i	$-0.177098\pi$
$277$	28.2663	1.69836	0.849178	+	0.528106i	$0.177098\pi$
$278$	0	0
$279$	−0.901104	−0.0539477
$280$	0	0
$281$	−25.2079	−1.50378	−0.751888	−	0.659290i	$-0.770857\pi$
$281$	−25.2079	−1.50378	−0.751888	+	0.659290i	$0.770857\pi$
$282$	0	0
$283$	−5.84396	−0.347387	−0.173694	−	0.984800i	$-0.555570\pi$
$283$	−5.84396	−0.347387	−0.173694	+	0.984800i	$0.555570\pi$
$284$	0	0
$285$	14.1526	0.838327
$286$	0	0
$287$	−1.62538	−0.0959433
$288$	0	0
$289$	10.9760	0.645648
$290$	0	0
$291$	6.74056	0.395138
$292$	0	0
$293$	18.7907	1.09777	0.548884	−	0.835899i	$-0.315053\pi$
$293$	18.7907	1.09777	0.548884	+	0.835899i	$0.315053\pi$
$294$	0	0
$295$	7.46593	0.434683
$296$	0	0
$297$	21.8746	1.26930
$298$	0	0
$299$	5.45238	0.315319
$300$	0	0
$301$	1.37943	0.0795093
$302$	0	0
$303$	10.4444	0.600016
$304$	0	0
$305$	−31.7192	−1.81624
$306$	0	0
$307$	−8.38000	−0.478272	−0.239136	−	0.970986i	$-0.576864\pi$
$307$	−8.38000	−0.478272	−0.239136	+	0.970986i	$0.576864\pi$
$308$	0	0
$309$	11.2364	0.639219
$310$	0	0
$311$	−1.71579	−0.0972936	−0.0486468	−	0.998816i	$-0.515491\pi$
$311$	−1.71579	−0.0972936	−0.0486468	+	0.998816i	$0.515491\pi$
$312$	0	0
$313$	25.4846	1.44047	0.720237	−	0.693728i	$-0.244033\pi$
$313$	25.4846	1.44047	0.720237	+	0.693728i	$0.244033\pi$
$314$	0	0
$315$	1.18989	0.0670424
$316$	0	0
$317$	−9.20822	−0.517185	−0.258593	−	0.965986i	$-0.583259\pi$
$317$	−9.20822	−0.517185	−0.258593	+	0.965986i	$0.583259\pi$
$318$	0	0
$319$	−28.1800	−1.57778
$320$	0	0
$321$	15.0827	0.841835
$322$	0	0
$323$	−26.2979	−1.46326
$324$	0	0
$325$	−26.2690	−1.45714
$326$	0	0
$327$	2.40110	0.132781
$328$	0	0
$329$	1.37046	0.0755559
$330$	0	0
$331$	26.1658	1.43820	0.719101	−	0.694905i	$-0.244554\pi$
$331$	26.1658	1.43820	0.719101	+	0.694905i	$0.244554\pi$
$332$	0	0
$333$	14.6667	0.803732
$334$	0	0
$335$	44.2622	2.41830
$336$	0	0
$337$	−6.82467	−0.371763	−0.185882	−	0.982572i	$-0.559514\pi$
$337$	−6.82467	−0.371763	−0.185882	+	0.982572i	$0.559514\pi$
$338$	0	0
$339$	−2.73052	−0.148301
$340$	0	0
$341$	−1.94719	−0.105446
$342$	0	0
$343$	−1.98555	−0.107210
$344$	0	0
$345$	4.50327	0.242448
$346$	0	0
$347$	−17.2455	−0.925788	−0.462894	−	0.886414i	$-0.653189\pi$
$347$	−17.2455	−0.925788	−0.462894	+	0.886414i	$0.653189\pi$
$348$	0	0
$349$	−31.9670	−1.71115	−0.855576	−	0.517677i	$-0.826797\pi$
$349$	−31.9670	−1.71115	−0.855576	+	0.517677i	$0.826797\pi$
$350$	0	0
$351$	14.7950	0.789698
$352$	0	0
$353$	−26.5074	−1.41085	−0.705424	−	0.708785i	$-0.749243\pi$
$353$	−26.5074	−1.41085	−0.705424	+	0.708785i	$0.749243\pi$
$354$	0	0
$355$	−27.6561	−1.46784
$356$	0	0
$357$	0.601883	0.0318550
$358$	0	0
$359$	−8.56430	−0.452006	−0.226003	−	0.974127i	$-0.572566\pi$
$359$	−8.56430	−0.452006	−0.226003	+	0.974127i	$0.572566\pi$
$360$	0	0
$361$	5.72051	0.301080
$362$	0	0
$363$	11.9897	0.629296
$364$	0	0
$365$	−30.6147	−1.60245
$366$	0	0
$367$	4.49087	0.234421	0.117211	−	0.993107i	$-0.462605\pi$
$367$	4.49087	0.234421	0.117211	+	0.993107i	$0.462605\pi$
$368$	0	0
$369$	26.9858	1.40482
$370$	0	0
$371$	−1.52916	−0.0793899
$372$	0	0
$373$	−5.91255	−0.306140	−0.153070	−	0.988215i	$-0.548916\pi$
$373$	−5.91255	−0.306140	−0.153070	+	0.988215i	$0.548916\pi$
$374$	0	0
$375$	−7.46393	−0.385436
$376$	0	0
$377$	−19.0596	−0.981620
$378$	0	0
$379$	−0.145933	−0.00749606	−0.00374803	−	0.999993i	$-0.501193\pi$
$379$	−0.145933	−0.00749606	−0.00374803	+	0.999993i	$0.501193\pi$
$380$	0	0
$381$	11.1499	0.571227
$382$	0	0
$383$	10.0575	0.513915	0.256957	−	0.966423i	$-0.417280\pi$
$383$	10.0575	0.513915	0.256957	+	0.966423i	$0.417280\pi$
$384$	0	0
$385$	2.57121	0.131041
$386$	0	0
$387$	−22.9024	−1.16419
$388$	0	0
$389$	−28.1159	−1.42553	−0.712765	−	0.701403i	$-0.752557\pi$
$389$	−28.1159	−1.42553	−0.712765	+	0.701403i	$0.752557\pi$
$390$	0	0
$391$	−8.36785	−0.423180
$392$	0	0
$393$	−4.50530	−0.227262
$394$	0	0
$395$	−41.9550	−2.11099
$396$	0	0
$397$	26.2520	1.31755	0.658776	−	0.752339i	$-0.271074\pi$
$397$	26.2520	1.31755	0.658776	+	0.752339i	$0.271074\pi$
$398$	0	0
$399$	−0.565781	−0.0283245
$400$	0	0
$401$	−6.87368	−0.343255	−0.171628	−	0.985162i	$-0.554903\pi$
$401$	−6.87368	−0.343255	−0.171628	+	0.985162i	$0.554903\pi$
$402$	0	0
$403$	−1.31699	−0.0656037
$404$	0	0
$405$	−12.9136	−0.641681
$406$	0	0
$407$	31.6932	1.57097
$408$	0	0
$409$	15.7448	0.778531	0.389265	−	0.921126i	$-0.372729\pi$
$409$	15.7448	0.778531	0.389265	+	0.921126i	$0.372729\pi$
$410$	0	0
$411$	2.16394	0.106739
$412$	0	0
$413$	−0.298467	−0.0146866
$414$	0	0
$415$	19.9615	0.979874
$416$	0	0
$417$	−10.1521	−0.497150
$418$	0	0
$419$	−24.3510	−1.18962	−0.594811	−	0.803865i	$-0.702773\pi$
$419$	−24.3510	−1.18962	−0.594811	+	0.803865i	$0.702773\pi$
$420$	0	0
$421$	15.4946	0.755160	0.377580	−	0.925977i	$-0.376756\pi$
$421$	15.4946	0.755160	0.377580	+	0.925977i	$0.376756\pi$
$422$	0	0
$423$	−22.7534	−1.10631
$424$	0	0
$425$	40.3154	1.95559
$426$	0	0
$427$	1.26804	0.0613650
$428$	0	0
$429$	14.0701	0.679310
$430$	0	0
$431$	24.5516	1.18261	0.591305	−	0.806448i	$-0.298613\pi$
$431$	24.5516	1.18261	0.591305	+	0.806448i	$0.298613\pi$
$432$	0	0
$433$	−4.92431	−0.236647	−0.118324	−	0.992975i	$-0.537752\pi$
$433$	−4.92431	−0.236647	−0.118324	+	0.992975i	$0.537752\pi$
$434$	0	0
$435$	−15.7419	−0.754765
$436$	0	0
$437$	7.86593	0.376278
$438$	0	0
$439$	5.49766	0.262389	0.131195	−	0.991357i	$-0.458119\pi$
$439$	5.49766	0.262389	0.131195	+	0.991357i	$0.458119\pi$
$440$	0	0
$441$	16.4590	0.783762
$442$	0	0
$443$	−10.5997	−0.503609	−0.251804	−	0.967778i	$-0.581024\pi$
$443$	−10.5997	−0.503609	−0.251804	+	0.967778i	$0.581024\pi$
$444$	0	0
$445$	45.2706	2.14603
$446$	0	0
$447$	−11.1318	−0.526518
$448$	0	0
$449$	−17.0603	−0.805124	−0.402562	−	0.915393i	$-0.631880\pi$
$449$	−17.0603	−0.805124	−0.402562	+	0.915393i	$0.631880\pi$
$450$	0	0
$451$	58.3133	2.74587
$452$	0	0
$453$	−6.66939	−0.313355
$454$	0	0
$455$	1.73905	0.0815278
$456$	0	0
$457$	36.3474	1.70026	0.850131	−	0.526571i	$-0.176523\pi$
$457$	36.3474	1.70026	0.850131	+	0.526571i	$0.176523\pi$
$458$	0	0
$459$	−22.7061	−1.05983
$460$	0	0
$461$	−10.8274	−0.504280	−0.252140	−	0.967691i	$-0.581134\pi$
$461$	−10.8274	−0.504280	−0.252140	+	0.967691i	$0.581134\pi$
$462$	0	0
$463$	33.7594	1.56893	0.784466	−	0.620172i	$-0.212937\pi$
$463$	33.7594	1.56893	0.784466	+	0.620172i	$0.212937\pi$
$464$	0	0
$465$	−1.08773	−0.0504425
$466$	0	0
$467$	29.6160	1.37047	0.685233	−	0.728324i	$-0.259700\pi$
$467$	29.6160	1.37047	0.685233	+	0.728324i	$0.259700\pi$
$468$	0	0
$469$	−1.76948	−0.0817069
$470$	0	0
$471$	1.86613	0.0859868
$472$	0	0
$473$	−49.4896	−2.27553
$474$	0	0
$475$	−37.8972	−1.73884
$476$	0	0
$477$	25.3882	1.16245
$478$	0	0
$479$	−8.28091	−0.378364	−0.189182	−	0.981942i	$-0.560584\pi$
$479$	−8.28091	−0.378364	−0.189182	+	0.981942i	$0.560584\pi$
$480$	0	0
$481$	21.4358	0.977388
$482$	0	0
$483$	−0.180028	−0.00819156
$484$	0	0
$485$	−29.8898	−1.35722
$486$	0	0
$487$	−12.9498	−0.586814	−0.293407	−	0.955988i	$-0.594789\pi$
$487$	−12.9498	−0.586814	−0.293407	+	0.955988i	$0.594789\pi$
$488$	0	0
$489$	10.3056	0.466037
$490$	0	0
$491$	−34.8081	−1.57087	−0.785434	−	0.618946i	$-0.787560\pi$
$491$	−34.8081	−1.57087	−0.785434	+	0.618946i	$0.787560\pi$
$492$	0	0
$493$	29.2511	1.31740
$494$	0	0
$495$	−42.6891	−1.91873
$496$	0	0
$497$	1.10562	0.0495936
$498$	0	0
$499$	−1.98014	−0.0886434	−0.0443217	−	0.999017i	$-0.514113\pi$
$499$	−1.98014	−0.0886434	−0.0443217	+	0.999017i	$0.514113\pi$
$500$	0	0
$501$	−13.6317	−0.609018
$502$	0	0
$503$	12.2052	0.544203	0.272101	−	0.962269i	$-0.412281\pi$
$503$	12.2052	0.544203	0.272101	+	0.962269i	$0.412281\pi$
$504$	0	0
$505$	−46.3139	−2.06094
$506$	0	0
$507$	−0.899236	−0.0399365
$508$	0	0
$509$	−5.62094	−0.249144	−0.124572	−	0.992211i	$-0.539756\pi$
$509$	−5.62094	−0.249144	−0.124572	+	0.992211i	$0.539756\pi$
$510$	0	0
$511$	1.22389	0.0541418
$512$	0	0
$513$	21.3441	0.942366
$514$	0	0
$515$	−49.8259	−2.19559
$516$	0	0
$517$	−49.1676	−2.16239
$518$	0	0
$519$	−7.77751	−0.341395
$520$	0	0
$521$	−33.6681	−1.47503	−0.737513	−	0.675333i	$-0.764000\pi$
$521$	−33.6681	−1.47503	−0.737513	+	0.675333i	$0.764000\pi$
$522$	0	0
$523$	−13.3997	−0.585927	−0.292964	−	0.956124i	$-0.594641\pi$
$523$	−13.3997	−0.585927	−0.292964	+	0.956124i	$0.594641\pi$
$524$	0	0
$525$	0.867357	0.0378546
$526$	0	0
$527$	2.02120	0.0880448
$528$	0	0
$529$	−20.4971	−0.891179
$530$	0	0
$531$	4.95537	0.215045
$532$	0	0
$533$	39.4404	1.70835
$534$	0	0
$535$	−66.8816	−2.89154
$536$	0	0
$537$	9.34419	0.403231
$538$	0	0
$539$	35.5661	1.53194
$540$	0	0
$541$	37.8128	1.62570	0.812850	−	0.582474i	$-0.197915\pi$
$541$	37.8128	1.62570	0.812850	+	0.582474i	$0.197915\pi$
$542$	0	0
$543$	−0.119625	−0.00513361
$544$	0	0
$545$	−10.6472	−0.456077
$546$	0	0
$547$	−12.0734	−0.516222	−0.258111	−	0.966115i	$-0.583100\pi$
$547$	−12.0734	−0.516222	−0.258111	+	0.966115i	$0.583100\pi$
$548$	0	0
$549$	−21.0530	−0.898520
$550$	0	0
$551$	−27.4965	−1.17139
$552$	0	0
$553$	1.67724	0.0713237
$554$	0	0
$555$	17.7044	0.751511
$556$	0	0
$557$	13.2024	0.559404	0.279702	−	0.960087i	$-0.409764\pi$
$557$	13.2024	0.559404	0.279702	+	0.960087i	$0.409764\pi$
$558$	0	0
$559$	−33.4724	−1.41573
$560$	0	0
$561$	−21.5936	−0.911682
$562$	0	0
$563$	6.12913	0.258312	0.129156	−	0.991624i	$-0.458773\pi$
$563$	6.12913	0.258312	0.129156	+	0.991624i	$0.458773\pi$
$564$	0	0
$565$	12.1080	0.509387
$566$	0	0
$567$	0.516248	0.0216804
$568$	0	0
$569$	39.1804	1.64253	0.821264	−	0.570548i	$-0.193269\pi$
$569$	39.1804	1.64253	0.821264	+	0.570548i	$0.193269\pi$
$570$	0	0
$571$	13.9089	0.582069	0.291034	−	0.956713i	$-0.406001\pi$
$571$	13.9089	0.582069	0.291034	+	0.956713i	$0.406001\pi$
$572$	0	0
$573$	−2.12644	−0.0888331
$574$	0	0
$575$	−12.0587	−0.502882
$576$	0	0
$577$	3.80449	0.158383	0.0791915	−	0.996859i	$-0.474766\pi$
$577$	3.80449	0.158383	0.0791915	+	0.996859i	$0.474766\pi$
$578$	0	0
$579$	4.67147	0.194139
$580$	0	0
$581$	−0.798007	−0.0331069
$582$	0	0
$583$	54.8611	2.27212
$584$	0	0
$585$	−28.8729	−1.19375
$586$	0	0
$587$	3.45013	0.142402	0.0712011	−	0.997462i	$-0.477317\pi$
$587$	3.45013	0.142402	0.0712011	+	0.997462i	$0.477317\pi$
$588$	0	0
$589$	−1.89996	−0.0782866
$590$	0	0
$591$	−16.4626	−0.677179
$592$	0	0
$593$	−8.38305	−0.344251	−0.172125	−	0.985075i	$-0.555063\pi$
$593$	−8.38305	−0.344251	−0.172125	+	0.985075i	$0.555063\pi$
$594$	0	0
$595$	−2.66894	−0.109416
$596$	0	0
$597$	−16.1304	−0.660171
$598$	0	0
$599$	−5.07230	−0.207248	−0.103624	−	0.994617i	$-0.533044\pi$
$599$	−5.07230	−0.207248	−0.103624	+	0.994617i	$0.533044\pi$
$600$	0	0
$601$	−23.0191	−0.938967	−0.469483	−	0.882941i	$-0.655560\pi$
$601$	−23.0191	−0.938967	−0.469483	+	0.882941i	$0.655560\pi$
$602$	0	0
$603$	29.3781	1.19637
$604$	0	0
$605$	−53.1661	−2.16151
$606$	0	0
$607$	38.6041	1.56689	0.783445	−	0.621461i	$-0.213460\pi$
$607$	38.6041	1.56689	0.783445	+	0.621461i	$0.213460\pi$
$608$	0	0
$609$	0.629316	0.0255012
$610$	0	0
$611$	−33.2546	−1.34534
$612$	0	0
$613$	−42.6445	−1.72240	−0.861198	−	0.508269i	$-0.830285\pi$
$613$	−42.6445	−1.72240	−0.861198	+	0.508269i	$0.830285\pi$
$614$	0	0
$615$	32.5749	1.31355
$616$	0	0
$617$	1.54860	0.0623444	0.0311722	−	0.999514i	$-0.490076\pi$
$617$	1.54860	0.0623444	0.0311722	+	0.999514i	$0.490076\pi$
$618$	0	0
$619$	−24.3055	−0.976922	−0.488461	−	0.872586i	$-0.662442\pi$
$619$	−24.3055	−0.976922	−0.488461	+	0.872586i	$0.662442\pi$
$620$	0	0
$621$	6.79157	0.272536
$622$	0	0
$623$	−1.80979	−0.0725078
$624$	0	0
$625$	−5.01339	−0.200536
$626$	0	0
$627$	20.2983	0.810638
$628$	0	0
$629$	−32.8979	−1.31172
$630$	0	0
$631$	6.16144	0.245283	0.122641	−	0.992451i	$-0.460863\pi$
$631$	6.16144	0.245283	0.122641	+	0.992451i	$0.460863\pi$
$632$	0	0
$633$	6.06405	0.241024
$634$	0	0
$635$	−49.4422	−1.96206
$636$	0	0
$637$	24.0552	0.953103
$638$	0	0
$639$	−18.3562	−0.726161
$640$	0	0
$641$	41.4384	1.63672	0.818358	−	0.574708i	$-0.194884\pi$
$641$	41.4384	1.63672	0.818358	+	0.574708i	$0.194884\pi$
$642$	0	0
$643$	−5.51179	−0.217364	−0.108682	−	0.994077i	$-0.534663\pi$
$643$	−5.51179	−0.217364	−0.108682	+	0.994077i	$0.534663\pi$
$644$	0	0
$645$	−27.6458	−1.08855
$646$	0	0
$647$	2.54222	0.0999451	0.0499726	−	0.998751i	$-0.484087\pi$
$647$	2.54222	0.0999451	0.0499726	+	0.998751i	$0.484087\pi$
$648$	0	0
$649$	10.7080	0.420326
$650$	0	0
$651$	0.0434846	0.00170430
$652$	0	0
$653$	17.0401	0.666832	0.333416	−	0.942780i	$-0.391799\pi$
$653$	17.0401	0.666832	0.333416	+	0.942780i	$0.391799\pi$
$654$	0	0
$655$	19.9779	0.780602
$656$	0	0
$657$	−20.3199	−0.792756
$658$	0	0
$659$	−15.3350	−0.597366	−0.298683	−	0.954352i	$-0.596547\pi$
$659$	−15.3350	−0.597366	−0.298683	+	0.954352i	$0.596547\pi$
$660$	0	0
$661$	37.9168	1.47479	0.737396	−	0.675461i	$-0.236055\pi$
$661$	37.9168	1.47479	0.737396	+	0.675461i	$0.236055\pi$
$662$	0	0
$663$	−14.6049	−0.567206
$664$	0	0
$665$	2.50885	0.0972891
$666$	0	0
$667$	−8.74924	−0.338772
$668$	0	0
$669$	21.0287	0.813018
$670$	0	0
$671$	−45.4933	−1.75625
$672$	0	0
$673$	21.4804	0.828008	0.414004	−	0.910275i	$-0.364130\pi$
$673$	21.4804	0.828008	0.414004	+	0.910275i	$0.364130\pi$
$674$	0	0
$675$	−32.7211	−1.25944
$676$	0	0
$677$	−35.6115	−1.36866	−0.684330	−	0.729172i	$-0.739905\pi$
$677$	−35.6115	−1.36866	−0.684330	+	0.729172i	$0.739905\pi$
$678$	0	0
$679$	1.19491	0.0458564
$680$	0	0
$681$	21.4554	0.822173
$682$	0	0
$683$	29.8857	1.14354	0.571772	−	0.820412i	$-0.306256\pi$
$683$	29.8857	1.14354	0.571772	+	0.820412i	$0.306256\pi$
$684$	0	0
$685$	−9.59561	−0.366629
$686$	0	0
$687$	17.1983	0.656155
$688$	0	0
$689$	37.1055	1.41361
$690$	0	0
$691$	17.8935	0.680700	0.340350	−	0.940299i	$-0.389454\pi$
$691$	17.8935	0.680700	0.340350	+	0.940299i	$0.389454\pi$
$692$	0	0
$693$	1.70659	0.0648281
$694$	0	0
$695$	45.0176	1.70761
$696$	0	0
$697$	−60.5298	−2.29273
$698$	0	0
$699$	−16.5493	−0.625952
$700$	0	0
$701$	31.7463	1.19904	0.599520	−	0.800359i	$-0.295358\pi$
$701$	31.7463	1.19904	0.599520	+	0.800359i	$0.295358\pi$
$702$	0	0
$703$	30.9246	1.16634
$704$	0	0
$705$	−27.4659	−1.03443
$706$	0	0
$707$	1.85150	0.0696328
$708$	0	0
$709$	18.4531	0.693019	0.346510	−	0.938046i	$-0.387367\pi$
$709$	18.4531	0.693019	0.346510	+	0.938046i	$0.387367\pi$
$710$	0	0
$711$	−27.8468	−1.04434
$712$	0	0
$713$	−0.604557	−0.0226408
$714$	0	0
$715$	−62.3913	−2.33330
$716$	0	0
$717$	−2.95627	−0.110404
$718$	0	0
$719$	−6.75645	−0.251973	−0.125987	−	0.992032i	$-0.540210\pi$
$719$	−6.75645	−0.251973	−0.125987	+	0.992032i	$0.540210\pi$
$720$	0	0
$721$	1.99190	0.0741823
$722$	0	0
$723$	−24.4610	−0.909716
$724$	0	0
$725$	42.1529	1.56552
$726$	0	0
$727$	−29.9437	−1.11055	−0.555276	−	0.831666i	$-0.687387\pi$
$727$	−29.9437	−1.11055	−0.555276	+	0.831666i	$0.687387\pi$
$728$	0	0
$729$	1.74722	0.0647120
$730$	0	0
$731$	51.3706	1.90001
$732$	0	0
$733$	9.61089	0.354986	0.177493	−	0.984122i	$-0.443201\pi$
$733$	9.61089	0.354986	0.177493	+	0.984122i	$0.443201\pi$
$734$	0	0
$735$	19.8679	0.732838
$736$	0	0
$737$	63.4830	2.33843
$738$	0	0
$739$	−9.80839	−0.360808	−0.180404	−	0.983593i	$-0.557740\pi$
$739$	−9.80839	−0.360808	−0.180404	+	0.983593i	$0.557740\pi$
$740$	0	0
$741$	13.7288	0.504341
$742$	0	0
$743$	−39.8812	−1.46310	−0.731550	−	0.681787i	$-0.761203\pi$
$743$	−39.8812	−1.46310	−0.731550	+	0.681787i	$0.761203\pi$
$744$	0	0
$745$	49.3622	1.80849
$746$	0	0
$747$	13.2491	0.484759
$748$	0	0
$749$	2.67374	0.0976963
$750$	0	0
$751$	30.9473	1.12928	0.564641	−	0.825337i	$-0.309015\pi$
$751$	30.9473	1.12928	0.564641	+	0.825337i	$0.309015\pi$
$752$	0	0
$753$	8.73717	0.318400
$754$	0	0
$755$	29.5742	1.07631
$756$	0	0
$757$	−10.8965	−0.396041	−0.198021	−	0.980198i	$-0.563451\pi$
$757$	−10.8965	−0.396041	−0.198021	+	0.980198i	$0.563451\pi$
$758$	0	0
$759$	6.45881	0.234440
$760$	0	0
$761$	−38.0921	−1.38084	−0.690419	−	0.723410i	$-0.742574\pi$
$761$	−38.0921	−1.38084	−0.690419	+	0.723410i	$0.742574\pi$
$762$	0	0
$763$	0.425647	0.0154094
$764$	0	0
$765$	44.3117	1.60209
$766$	0	0
$767$	7.24239	0.261508
$768$	0	0
$769$	18.5966	0.670609	0.335304	−	0.942110i	$-0.391161\pi$
$769$	18.5966	0.670609	0.335304	+	0.942110i	$0.391161\pi$
$770$	0	0
$771$	−0.878498	−0.0316383
$772$	0	0
$773$	17.8334	0.641421	0.320711	−	0.947177i	$-0.396078\pi$
$773$	17.8334	0.641421	0.320711	+	0.947177i	$0.396078\pi$
$774$	0	0
$775$	2.91269	0.104627
$776$	0	0
$777$	−0.707773	−0.0253912
$778$	0	0
$779$	56.8991	2.03862
$780$	0	0
$781$	−39.6658	−1.41935
$782$	0	0
$783$	−23.7410	−0.848433
$784$	0	0
$785$	−8.27502	−0.295348
$786$	0	0
$787$	55.7464	1.98715	0.993573	−	0.113198i	$-0.0361093\pi$
$787$	55.7464	1.98715	0.993573	+	0.113198i	$0.0361093\pi$
$788$	0	0
$789$	21.1650	0.753495
$790$	0	0
$791$	−0.484043	−0.0172106
$792$	0	0
$793$	−30.7695	−1.09266
$794$	0	0
$795$	30.6464	1.08692
$796$	0	0
$797$	17.6090	0.623744	0.311872	−	0.950124i	$-0.399044\pi$
$797$	17.6090	0.623744	0.311872	+	0.950124i	$0.399044\pi$
$798$	0	0
$799$	51.0364	1.80554
$800$	0	0
$801$	30.0475	1.06168
$802$	0	0
$803$	−43.9092	−1.54952
$804$	0	0
$805$	0.798302	0.0281365
$806$	0	0
$807$	0.801198	0.0282035
$808$	0	0
$809$	32.0038	1.12519	0.562597	−	0.826731i	$-0.309802\pi$
$809$	32.0038	1.12519	0.562597	+	0.826731i	$0.309802\pi$
$810$	0	0
$811$	−36.6445	−1.28676	−0.643380	−	0.765547i	$-0.722469\pi$
$811$	−36.6445	−1.28676	−0.643380	+	0.765547i	$0.722469\pi$
$812$	0	0
$813$	−1.32034	−0.0463064
$814$	0	0
$815$	−45.6985	−1.60075
$816$	0	0
$817$	−48.2893	−1.68943
$818$	0	0
$819$	1.15426	0.0403331
$820$	0	0
$821$	−2.44510	−0.0853347	−0.0426673	−	0.999089i	$-0.513586\pi$
$821$	−2.44510	−0.0853347	−0.0426673	+	0.999089i	$0.513586\pi$
$822$	0	0
$823$	−43.4422	−1.51430	−0.757150	−	0.653241i	$-0.773409\pi$
$823$	−43.4422	−1.51430	−0.757150	+	0.653241i	$0.773409\pi$
$824$	0	0
$825$	−31.1179	−1.08339
$826$	0	0
$827$	23.5761	0.819820	0.409910	−	0.912126i	$-0.365560\pi$
$827$	23.5761	0.819820	0.409910	+	0.912126i	$0.365560\pi$
$828$	0	0
$829$	−5.41698	−0.188139	−0.0940697	−	0.995566i	$-0.529988\pi$
$829$	−5.41698	−0.188139	−0.0940697	+	0.995566i	$0.529988\pi$
$830$	0	0
$831$	22.6469	0.785612
$832$	0	0
$833$	−36.9179	−1.27913
$834$	0	0
$835$	60.4472	2.09186
$836$	0	0
$837$	−1.64046	−0.0567026
$838$	0	0
$839$	−35.1327	−1.21291	−0.606457	−	0.795116i	$-0.707410\pi$
$839$	−35.1327	−1.21291	−0.606457	+	0.795116i	$0.707410\pi$
$840$	0	0
$841$	1.58429	0.0546307
$842$	0	0
$843$	−20.1965	−0.695605
$844$	0	0
$845$	3.98750	0.137174
$846$	0	0
$847$	2.12543	0.0730308
$848$	0	0
$849$	−4.68217	−0.160692
$850$	0	0
$851$	9.84002	0.337311
$852$	0	0
$853$	−50.2052	−1.71899	−0.859497	−	0.511141i	$-0.829223\pi$
$853$	−50.2052	−1.71899	−0.859497	+	0.511141i	$0.829223\pi$
$854$	0	0
$855$	−41.6538	−1.42453
$856$	0	0
$857$	11.6711	0.398677	0.199339	−	0.979931i	$-0.436121\pi$
$857$	11.6711	0.398677	0.199339	+	0.979931i	$0.436121\pi$
$858$	0	0
$859$	4.54823	0.155184	0.0775918	−	0.996985i	$-0.475277\pi$
$859$	4.54823	0.155184	0.0775918	+	0.996985i	$0.475277\pi$
$860$	0	0
$861$	−1.30225	−0.0443807
$862$	0	0
$863$	28.6408	0.974945	0.487473	−	0.873138i	$-0.337919\pi$
$863$	28.6408	0.974945	0.487473	+	0.873138i	$0.337919\pi$
$864$	0	0
$865$	34.4880	1.17263
$866$	0	0
$867$	8.79396	0.298659
$868$	0	0
$869$	−60.1740	−2.04126
$870$	0	0
$871$	42.9369	1.45486
$872$	0	0
$873$	−19.8388	−0.671440
$874$	0	0
$875$	−1.32314	−0.0447304
$876$	0	0
$877$	19.0303	0.642606	0.321303	−	0.946976i	$-0.395879\pi$
$877$	19.0303	0.642606	0.321303	+	0.946976i	$0.395879\pi$
$878$	0	0
$879$	15.0551	0.507796
$880$	0	0
$881$	−2.53478	−0.0853990	−0.0426995	−	0.999088i	$-0.513596\pi$
$881$	−2.53478	−0.0853990	−0.0426995	+	0.999088i	$0.513596\pi$
$882$	0	0
$883$	40.7580	1.37162	0.685809	−	0.727782i	$-0.259449\pi$
$883$	40.7580	1.37162	0.685809	+	0.727782i	$0.259449\pi$
$884$	0	0
$885$	5.98169	0.201072
$886$	0	0
$887$	42.7811	1.43645	0.718224	−	0.695812i	$-0.244955\pi$
$887$	42.7811	1.43645	0.718224	+	0.695812i	$0.244955\pi$
$888$	0	0
$889$	1.97656	0.0662918
$890$	0	0
$891$	−18.5213	−0.620486
$892$	0	0
$893$	−47.9751	−1.60543
$894$	0	0
$895$	−41.4351	−1.38502
$896$	0	0
$897$	4.36844	0.145858
$898$	0	0
$899$	2.11332	0.0704832
$900$	0	0
$901$	−56.9464	−1.89716
$902$	0	0
$903$	1.10520	0.0367788
$904$	0	0
$905$	0.530457	0.0176330
$906$	0	0
$907$	−39.5376	−1.31282	−0.656412	−	0.754402i	$-0.727927\pi$
$907$	−39.5376	−1.31282	−0.656412	+	0.754402i	$0.727927\pi$
$908$	0	0
$909$	−30.7399	−1.01958
$910$	0	0
$911$	−54.0502	−1.79076	−0.895382	−	0.445299i	$-0.853097\pi$
$911$	−54.0502	−1.79076	−0.895382	+	0.445299i	$0.853097\pi$
$912$	0	0
$913$	28.6298	0.947509
$914$	0	0
$915$	−25.4134	−0.840140
$916$	0	0
$917$	−0.798661	−0.0263741
$918$	0	0
$919$	51.9582	1.71394	0.856971	−	0.515365i	$-0.172344\pi$
$919$	51.9582	1.71394	0.856971	+	0.515365i	$0.172344\pi$
$920$	0	0
$921$	−6.71404	−0.221235
$922$	0	0
$923$	−26.8281	−0.883057
$924$	0	0
$925$	−47.4082	−1.55877
$926$	0	0
$927$	−33.0710	−1.08619
$928$	0	0
$929$	52.5137	1.72292	0.861460	−	0.507826i	$-0.169551\pi$
$929$	52.5137	1.72292	0.861460	+	0.507826i	$0.169551\pi$
$930$	0	0
$931$	34.7035	1.13736
$932$	0	0
$933$	−1.37469	−0.0450053
$934$	0	0
$935$	95.7528	3.13145
$936$	0	0
$937$	37.4481	1.22338	0.611688	−	0.791099i	$-0.290491\pi$
$937$	37.4481	1.22338	0.611688	+	0.791099i	$0.290491\pi$
$938$	0	0
$939$	20.4182	0.666323
$940$	0	0
$941$	−39.0492	−1.27297	−0.636484	−	0.771290i	$-0.719612\pi$
$941$	−39.0492	−1.27297	−0.636484	+	0.771290i	$0.719612\pi$
$942$	0	0
$943$	18.1050	0.589578
$944$	0	0
$945$	2.16619	0.0704660
$946$	0	0
$947$	−7.07707	−0.229974	−0.114987	−	0.993367i	$-0.536683\pi$
$947$	−7.07707	−0.229974	−0.114987	+	0.993367i	$0.536683\pi$
$948$	0	0
$949$	−29.6981	−0.964041
$950$	0	0
$951$	−7.37761	−0.239235
$952$	0	0
$953$	−25.8353	−0.836888	−0.418444	−	0.908243i	$-0.637424\pi$
$953$	−25.8353	−0.836888	−0.418444	+	0.908243i	$0.637424\pi$
$954$	0	0
$955$	9.42929	0.305125
$956$	0	0
$957$	−22.5778	−0.729835
$958$	0	0
$959$	0.383606	0.0123873
$960$	0	0
$961$	−30.8540	−0.995289
$962$	0	0
$963$	−44.3914	−1.43049
$964$	0	0
$965$	−20.7148	−0.666832
$966$	0	0
$967$	30.4711	0.979885	0.489942	−	0.871755i	$-0.337018\pi$
$967$	30.4711	0.979885	0.489942	+	0.871755i	$0.337018\pi$
$968$	0	0
$969$	−21.0699	−0.676861
$970$	0	0
$971$	46.7058	1.49886	0.749429	−	0.662084i	$-0.230328\pi$
$971$	46.7058	1.49886	0.749429	+	0.662084i	$0.230328\pi$
$972$	0	0
$973$	−1.79968	−0.0576950
$974$	0	0
$975$	−21.0467	−0.674033
$976$	0	0
$977$	32.5862	1.04253	0.521263	−	0.853396i	$-0.325461\pi$
$977$	32.5862	1.04253	0.521263	+	0.853396i	$0.325461\pi$
$978$	0	0
$979$	64.9294	2.07515
$980$	0	0
$981$	−7.06689	−0.225629
$982$	0	0
$983$	−11.1617	−0.356002	−0.178001	−	0.984030i	$-0.556963\pi$
$983$	−11.1617	−0.356002	−0.178001	+	0.984030i	$0.556963\pi$
$984$	0	0
$985$	73.0002	2.32598
$986$	0	0
$987$	1.09801	0.0349500
$988$	0	0
$989$	−15.3654	−0.488590
$990$	0	0
$991$	11.2149	0.356253	0.178127	−	0.984008i	$-0.442996\pi$
$991$	11.2149	0.356253	0.178127	+	0.984008i	$0.442996\pi$
$992$	0	0
$993$	20.9640	0.665272
$994$	0	0
$995$	71.5271	2.26756
$996$	0	0
$997$	−27.1338	−0.859336	−0.429668	−	0.902987i	$-0.641369\pi$
$997$	−27.1338	−0.859336	−0.429668	+	0.902987i	$0.641369\pi$
$998$	0	0
$999$	26.7008	0.844775

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4304.2.a.n.1.12		18
4.3	odd	2		2152.2.a.d.1.7	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2152.2.a.d.1.7	✓	18	4.3	odd	2
4304.2.a.n.1.12		18	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4304 = 2^{4} \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4304.a (trivial)

\(f(q)\)	\(=\)	\(q+0.801198 q^{3} -3.55277 q^{5} +0.142030 q^{7} -2.35808 q^{9} -5.09556 q^{11} -3.44639 q^{13} -2.84647 q^{15} +5.28924 q^{17} -4.97197 q^{19} +0.113794 q^{21} -1.58205 q^{23} +7.62217 q^{25} -4.29289 q^{27} +5.53031 q^{29} +0.382134 q^{31} -4.08255 q^{33} -0.504599 q^{35} -6.21978 q^{37} -2.76125 q^{39} -11.4440 q^{41} +9.71230 q^{43} +8.37772 q^{45} +9.64910 q^{47} -6.97983 q^{49} +4.23773 q^{51} -10.7665 q^{53} +18.1033 q^{55} -3.98354 q^{57} -2.10144 q^{59} +8.92802 q^{61} -0.334918 q^{63} +12.2442 q^{65} -12.4585 q^{67} -1.26754 q^{69} +7.78439 q^{71} +8.61715 q^{73} +6.10687 q^{75} -0.723721 q^{77} +11.8091 q^{79} +3.63479 q^{81} -5.61859 q^{83} -18.7914 q^{85} +4.43087 q^{87} -12.7423 q^{89} -0.489491 q^{91} +0.306165 q^{93} +17.6643 q^{95} +8.41309 q^{97} +12.0157 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 4 q^{3} - 6 q^{5} + 17 q^{7} + 18 q^{9} + 10 q^{11} - 3 q^{13} + 19 q^{15} - 6 q^{17} + 3 q^{19} + q^{21} + 37 q^{23} + 18 q^{25} + 13 q^{27} - 3 q^{29} + 35 q^{31} - 9 q^{33} + 12 q^{35} - 8 q^{37}+ \cdots + 41 q^{99}+O(q^{100}) \) 18 * q + 4 * q^3 - 6 * q^5 + 17 * q^7 + 18 * q^9 + 10 * q^11 - 3 * q^13 + 19 * q^15 - 6 * q^17 + 3 * q^19 + q^21 + 37 * q^23 + 18 * q^25 + 13 * q^27 - 3 * q^29 + 35 * q^31 - 9 * q^33 + 12 * q^35 - 8 * q^37 + 44 * q^39 - 12 * q^41 + 15 * q^43 - 25 * q^45 + 28 * q^47 + 19 * q^49 + 9 * q^51 - 33 * q^53 + 35 * q^55 - 26 * q^57 + 22 * q^59 + 8 * q^61 + 75 * q^63 - 20 * q^65 + 25 * q^67 - 3 * q^69 + 55 * q^71 - 8 * q^73 + 28 * q^75 - 8 * q^77 + 80 * q^79 + 22 * q^81 + 22 * q^83 + 13 * q^85 + 55 * q^87 - 26 * q^89 + 19 * q^91 - 21 * q^93 + 56 * q^95 - 28 * q^97 + 41 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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