LMFDB - Embedded newform 3680.2.a.x.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3680,2,Mod(1,3680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3680 = 2^{5} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3680.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:	$29.3849479438$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.876604.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 9x^{3} + 8x^{2} + 18x - 16 $ x^5 - x^4 - 9x^3 + 8x^2 + 18*x - 16
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			1.1
Root			$0.895130$ of defining polynomial
Character	$\chi$	$=$	3680.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.94825 q^{3} +1.00000 q^{5} -2.13809 q^{7} +5.69219 q^{9} +O(q^{10})$	$q-2.94825 q^{3} +1.00000 q^{5} -2.13809 q^{7} +5.69219 q^{9} -5.67228 q^{11} -5.04212 q^{13} -2.94825 q^{15} +0.545195 q^{17} -6.74531 q^{19} +6.30361 q^{21} +1.00000 q^{23} +1.00000 q^{25} -7.93725 q^{27} -5.11378 q^{29} -4.64464 q^{31} +16.7233 q^{33} -2.13809 q^{35} -2.21748 q^{37} +14.8655 q^{39} -10.2264 q^{41} +10.5377 q^{43} +5.69219 q^{45} +2.61280 q^{47} -2.42859 q^{49} -1.60737 q^{51} +8.67366 q^{53} -5.67228 q^{55} +19.8869 q^{57} +0.777152 q^{59} -7.00137 q^{61} -12.1704 q^{63} -5.04212 q^{65} -7.27040 q^{67} -2.94825 q^{69} +3.40168 q^{71} -6.34436 q^{73} -2.94825 q^{75} +12.1278 q^{77} -4.07651 q^{79} +6.32446 q^{81} -1.51409 q^{83} +0.545195 q^{85} +15.0767 q^{87} -9.79799 q^{89} +10.7805 q^{91} +13.6936 q^{93} -6.74531 q^{95} +4.60389 q^{97} -32.2877 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{3} + 5 q^{5} - q^{7} + 6 q^{9}+O(q^{10}) $ 5 * q - q^3 + 5 * q^5 - q^7 + 6 * q^9	$ 5 q - q^{3} + 5 q^{5} - q^{7} + 6 q^{9} - 3 q^{11} + 7 q^{13} - q^{15} + 9 q^{17} - q^{19} + 20 q^{21} + 5 q^{23} + 5 q^{25} - 4 q^{27} - 10 q^{29} - 21 q^{31} + 7 q^{33} - q^{35} + 8 q^{37} + 24 q^{39} - 13 q^{41} + 6 q^{43} + 6 q^{45} + 24 q^{49} + 17 q^{51} - 6 q^{53} - 3 q^{55} + 26 q^{57} - 18 q^{59} - 11 q^{61} - 4 q^{63} + 7 q^{65} - 38 q^{67} - q^{69} + 21 q^{71} - 12 q^{73} - q^{75} + 46 q^{77} + 18 q^{79} + 9 q^{81} - 20 q^{83} + 9 q^{85} + 6 q^{87} - 16 q^{89} + 3 q^{91} + 22 q^{93} - q^{95} + 29 q^{97} - 29 q^{99}+O(q^{100}) $ 5 * q - q^3 + 5 * q^5 - q^7 + 6 * q^9 - 3 * q^11 + 7 * q^13 - q^15 + 9 * q^17 - q^19 + 20 * q^21 + 5 * q^23 + 5 * q^25 - 4 * q^27 - 10 * q^29 - 21 * q^31 + 7 * q^33 - q^35 + 8 * q^37 + 24 * q^39 - 13 * q^41 + 6 * q^43 + 6 * q^45 + 24 * q^49 + 17 * q^51 - 6 * q^53 - 3 * q^55 + 26 * q^57 - 18 * q^59 - 11 * q^61 - 4 * q^63 + 7 * q^65 - 38 * q^67 - q^69 + 21 * q^71 - 12 * q^73 - q^75 + 46 * q^77 + 18 * q^79 + 9 * q^81 - 20 * q^83 + 9 * q^85 + 6 * q^87 - 16 * q^89 + 3 * q^91 + 22 * q^93 - q^95 + 29 * q^97 - 29 * 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
$3$	−2.94825	−1.70217	−0.851087	−	0.525025i	$-0.824056\pi$
$3$	−2.94825	−1.70217	−0.851087	+	0.525025i	$0.824056\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.13809	−0.808120	−0.404060	−	0.914732i	$-0.632401\pi$
$7$	−2.13809	−0.808120	−0.404060	+	0.914732i	$0.632401\pi$
$8$	0	0
$9$	5.69219	1.89740
$10$	0	0
$11$	−5.67228	−1.71026	−0.855129	−	0.518416i	$-0.826522\pi$
$11$	−5.67228	−1.71026	−0.855129	+	0.518416i	$0.826522\pi$
$12$	0	0
$13$	−5.04212	−1.39843	−0.699217	−	0.714910i	$-0.746468\pi$
$13$	−5.04212	−1.39843	−0.699217	+	0.714910i	$0.746468\pi$
$14$	0	0
$15$	−2.94825	−0.761235
$16$	0	0
$17$	0.545195	0.132229	0.0661146	−	0.997812i	$-0.478940\pi$
$17$	0.545195	0.132229	0.0661146	+	0.997812i	$0.478940\pi$
$18$	0	0
$19$	−6.74531	−1.54748	−0.773740	−	0.633503i	$-0.781617\pi$
$19$	−6.74531	−1.54748	−0.773740	+	0.633503i	$0.781617\pi$
$20$	0	0
$21$	6.30361	1.37556
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−7.93725	−1.52753
$28$	0	0
$29$	−5.11378	−0.949605	−0.474803	−	0.880092i	$-0.657481\pi$
$29$	−5.11378	−0.949605	−0.474803	+	0.880092i	$0.657481\pi$
$30$	0	0
$31$	−4.64464	−0.834202	−0.417101	−	0.908860i	$-0.636954\pi$
$31$	−4.64464	−0.834202	−0.417101	+	0.908860i	$0.636954\pi$
$32$	0	0
$33$	16.7233	2.91116
$34$	0	0
$35$	−2.13809	−0.361402
$36$	0	0
$37$	−2.21748	−0.364551	−0.182275	−	0.983248i	$-0.558346\pi$
$37$	−2.21748	−0.364551	−0.182275	+	0.983248i	$0.558346\pi$
$38$	0	0
$39$	14.8655	2.38038
$40$	0	0
$41$	−10.2264	−1.59709	−0.798547	−	0.601933i	$-0.794397\pi$
$41$	−10.2264	−1.59709	−0.798547	+	0.601933i	$0.794397\pi$
$42$	0	0
$43$	10.5377	1.60698	0.803491	−	0.595317i	$-0.202974\pi$
$43$	10.5377	1.60698	0.803491	+	0.595317i	$0.202974\pi$
$44$	0	0
$45$	5.69219	0.848542
$46$	0	0
$47$	2.61280	0.381116	0.190558	−	0.981676i	$-0.438970\pi$
$47$	2.61280	0.381116	0.190558	+	0.981676i	$0.438970\pi$
$48$	0	0
$49$	−2.42859	−0.346942
$50$	0	0
$51$	−1.60737	−0.225077
$52$	0	0
$53$	8.67366	1.19142	0.595709	−	0.803200i	$-0.296871\pi$
$53$	8.67366	1.19142	0.595709	+	0.803200i	$0.296871\pi$
$54$	0	0
$55$	−5.67228	−0.764850
$56$	0	0
$57$	19.8869	2.63408
$58$	0	0
$59$	0.777152	0.101177	0.0505883	−	0.998720i	$-0.483890\pi$
$59$	0.777152	0.101177	0.0505883	+	0.998720i	$0.483890\pi$
$60$	0	0
$61$	−7.00137	−0.896434	−0.448217	−	0.893925i	$-0.647941\pi$
$61$	−7.00137	−0.896434	−0.448217	+	0.893925i	$0.647941\pi$
$62$	0	0
$63$	−12.1704	−1.53332
$64$	0	0
$65$	−5.04212	−0.625399
$66$	0	0
$67$	−7.27040	−0.888221	−0.444110	−	0.895972i	$-0.646480\pi$
$67$	−7.27040	−0.888221	−0.444110	+	0.895972i	$0.646480\pi$
$68$	0	0
$69$	−2.94825	−0.354928
$70$	0	0
$71$	3.40168	0.403705	0.201853	−	0.979416i	$-0.435304\pi$
$71$	3.40168	0.403705	0.201853	+	0.979416i	$0.435304\pi$
$72$	0	0
$73$	−6.34436	−0.742552	−0.371276	−	0.928523i	$-0.621080\pi$
$73$	−6.34436	−0.742552	−0.371276	+	0.928523i	$0.621080\pi$
$74$	0	0
$75$	−2.94825	−0.340435
$76$	0	0
$77$	12.1278	1.38209
$78$	0	0
$79$	−4.07651	−0.458644	−0.229322	−	0.973351i	$-0.573651\pi$
$79$	−4.07651	−0.458644	−0.229322	+	0.973351i	$0.573651\pi$
$80$	0	0
$81$	6.32446	0.702717
$82$	0	0
$83$	−1.51409	−0.166193	−0.0830964	−	0.996542i	$-0.526481\pi$
$83$	−1.51409	−0.166193	−0.0830964	+	0.996542i	$0.526481\pi$
$84$	0	0
$85$	0.545195	0.0591347
$86$	0	0
$87$	15.0767	1.61639
$88$	0	0
$89$	−9.79799	−1.03859	−0.519293	−	0.854596i	$-0.673805\pi$
$89$	−9.79799	−1.03859	−0.519293	+	0.854596i	$0.673805\pi$
$90$	0	0
$91$	10.7805	1.13010
$92$	0	0
$93$	13.6936	1.41996
$94$	0	0
$95$	−6.74531	−0.692054
$96$	0	0
$97$	4.60389	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$97$	4.60389	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$98$	0	0
$99$	−32.2877	−3.24504
$100$	0	0
$101$	−12.6915	−1.26285	−0.631425	−	0.775437i	$-0.717529\pi$
$101$	−12.6915	−1.26285	−0.631425	+	0.775437i	$0.717529\pi$
$102$	0	0
$103$	−10.3271	−1.01756	−0.508778	−	0.860898i	$-0.669903\pi$
$103$	−10.3271	−1.01756	−0.508778	+	0.860898i	$0.669903\pi$
$104$	0	0
$105$	6.30361	0.615170
$106$	0	0
$107$	−11.9221	−1.15255	−0.576275	−	0.817256i	$-0.695494\pi$
$107$	−11.9221	−1.15255	−0.576275	+	0.817256i	$0.695494\pi$
$108$	0	0
$109$	−7.06552	−0.676754	−0.338377	−	0.941011i	$-0.609878\pi$
$109$	−7.06552	−0.676754	−0.338377	+	0.941011i	$0.609878\pi$
$110$	0	0
$111$	6.53768	0.620529
$112$	0	0
$113$	19.5134	1.83567	0.917835	−	0.396963i	$-0.129936\pi$
$113$	19.5134	1.83567	0.917835	+	0.396963i	$0.129936\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	−28.7007	−2.65338
$118$	0	0
$119$	−1.16567	−0.106857
$120$	0	0
$121$	21.1748	1.92498
$122$	0	0
$123$	30.1500	2.71853
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	14.1007	1.25123	0.625616	−	0.780131i	$-0.284848\pi$
$127$	14.1007	1.25123	0.625616	+	0.780131i	$0.284848\pi$
$128$	0	0
$129$	−31.0677	−2.73536
$130$	0	0
$131$	19.0470	1.66414	0.832071	−	0.554669i	$-0.187155\pi$
$131$	19.0470	1.66414	0.832071	+	0.554669i	$0.187155\pi$
$132$	0	0
$133$	14.4221	1.25055
$134$	0	0
$135$	−7.93725	−0.683130
$136$	0	0
$137$	10.5943	0.905129	0.452565	−	0.891732i	$-0.350509\pi$
$137$	10.5943	0.905129	0.452565	+	0.891732i	$0.350509\pi$
$138$	0	0
$139$	14.3997	1.22136	0.610681	−	0.791877i	$-0.290896\pi$
$139$	14.3997	1.22136	0.610681	+	0.791877i	$0.290896\pi$
$140$	0	0
$141$	−7.70319	−0.648725
$142$	0	0
$143$	28.6004	2.39168
$144$	0	0
$145$	−5.11378	−0.424676
$146$	0	0
$147$	7.16010	0.590555
$148$	0	0
$149$	−2.81908	−0.230948	−0.115474	−	0.993311i	$-0.536839\pi$
$149$	−2.81908	−0.230948	−0.115474	+	0.993311i	$0.536839\pi$
$150$	0	0
$151$	−0.704562	−0.0573365	−0.0286682	−	0.999589i	$-0.509127\pi$
$151$	−0.704562	−0.0573365	−0.0286682	+	0.999589i	$0.509127\pi$
$152$	0	0
$153$	3.10335	0.250891
$154$	0	0
$155$	−4.64464	−0.373066
$156$	0	0
$157$	−16.0793	−1.28326	−0.641632	−	0.767012i	$-0.721743\pi$
$157$	−16.0793	−1.28326	−0.641632	+	0.767012i	$0.721743\pi$
$158$	0	0
$159$	−25.5721	−2.02800
$160$	0	0
$161$	−2.13809	−0.168505
$162$	0	0
$163$	−12.1665	−0.952951	−0.476475	−	0.879188i	$-0.658086\pi$
$163$	−12.1665	−0.952951	−0.476475	+	0.879188i	$0.658086\pi$
$164$	0	0
$165$	16.7233	1.30191
$166$	0	0
$167$	−22.6031	−1.74908	−0.874538	−	0.484956i	$-0.838835\pi$
$167$	−22.6031	−1.74908	−0.874538	+	0.484956i	$0.838835\pi$
$168$	0	0
$169$	12.4230	0.955617
$170$	0	0
$171$	−38.3956	−2.93618
$172$	0	0
$173$	6.50538	0.494595	0.247297	−	0.968940i	$-0.420458\pi$
$173$	6.50538	0.494595	0.247297	+	0.968940i	$0.420458\pi$
$174$	0	0
$175$	−2.13809	−0.161624
$176$	0	0
$177$	−2.29124	−0.172220
$178$	0	0
$179$	−9.90981	−0.740694	−0.370347	−	0.928893i	$-0.620761\pi$
$179$	−9.90981	−0.740694	−0.370347	+	0.928893i	$0.620761\pi$
$180$	0	0
$181$	2.95505	0.219647	0.109824	−	0.993951i	$-0.464971\pi$
$181$	2.95505	0.219647	0.109824	+	0.993951i	$0.464971\pi$
$182$	0	0
$183$	20.6418	1.52589
$184$	0	0
$185$	−2.21748	−0.163032
$186$	0	0
$187$	−3.09250	−0.226146
$188$	0	0
$189$	16.9705	1.23442
$190$	0	0
$191$	5.46155	0.395184	0.197592	−	0.980284i	$-0.436688\pi$
$191$	5.46155	0.395184	0.197592	+	0.980284i	$0.436688\pi$
$192$	0	0
$193$	−0.778917	−0.0560677	−0.0280338	−	0.999607i	$-0.508925\pi$
$193$	−0.778917	−0.0560677	−0.0280338	+	0.999607i	$0.508925\pi$
$194$	0	0
$195$	14.8655	1.06454
$196$	0	0
$197$	16.9363	1.20666	0.603332	−	0.797490i	$-0.293839\pi$
$197$	16.9363	1.20666	0.603332	+	0.797490i	$0.293839\pi$
$198$	0	0
$199$	−6.78489	−0.480968	−0.240484	−	0.970653i	$-0.577306\pi$
$199$	−6.78489	−0.480968	−0.240484	+	0.970653i	$0.577306\pi$
$200$	0	0
$201$	21.4350	1.51191
$202$	0	0
$203$	10.9337	0.767395
$204$	0	0
$205$	−10.2264	−0.714242
$206$	0	0
$207$	5.69219	0.395635
$208$	0	0
$209$	38.2613	2.64659
$210$	0	0
$211$	16.8197	1.15792	0.578959	−	0.815357i	$-0.303459\pi$
$211$	16.8197	1.15792	0.578959	+	0.815357i	$0.303459\pi$
$212$	0	0
$213$	−10.0290	−0.687177
$214$	0	0
$215$	10.5377	0.718664
$216$	0	0
$217$	9.93063	0.674135
$218$	0	0
$219$	18.7048	1.26395
$220$	0	0
$221$	−2.74894	−0.184914
$222$	0	0
$223$	20.2013	1.35278	0.676391	−	0.736543i	$-0.263543\pi$
$223$	20.2013	1.35278	0.676391	+	0.736543i	$0.263543\pi$
$224$	0	0
$225$	5.69219	0.379479
$226$	0	0
$227$	−20.2353	−1.34306	−0.671532	−	0.740976i	$-0.734363\pi$
$227$	−20.2353	−1.34306	−0.671532	+	0.740976i	$0.734363\pi$
$228$	0	0
$229$	21.2264	1.40268	0.701340	−	0.712827i	$-0.252585\pi$
$229$	21.2264	1.40268	0.701340	+	0.712827i	$0.252585\pi$
$230$	0	0
$231$	−35.7559	−2.35256
$232$	0	0
$233$	−13.1243	−0.859799	−0.429900	−	0.902877i	$-0.641451\pi$
$233$	−13.1243	−0.859799	−0.429900	+	0.902877i	$0.641451\pi$
$234$	0	0
$235$	2.61280	0.170440
$236$	0	0
$237$	12.0186	0.780691
$238$	0	0
$239$	−8.02691	−0.519218	−0.259609	−	0.965714i	$-0.583594\pi$
$239$	−8.02691	−0.519218	−0.259609	+	0.965714i	$0.583594\pi$
$240$	0	0
$241$	0.788006	0.0507599	0.0253800	−	0.999678i	$-0.491920\pi$
$241$	0.788006	0.0507599	0.0253800	+	0.999678i	$0.491920\pi$
$242$	0	0
$243$	5.16567	0.331378
$244$	0	0
$245$	−2.42859	−0.155157
$246$	0	0
$247$	34.0107	2.16405
$248$	0	0
$249$	4.46392	0.282889
$250$	0	0
$251$	30.8373	1.94643	0.973216	−	0.229891i	$-0.0738368\pi$
$251$	30.8373	1.94643	0.973216	+	0.229891i	$0.0738368\pi$
$252$	0	0
$253$	−5.67228	−0.356613
$254$	0	0
$255$	−1.60737	−0.100658
$256$	0	0
$257$	13.2362	0.825649	0.412824	−	0.910811i	$-0.364542\pi$
$257$	13.2362	0.825649	0.412824	+	0.910811i	$0.364542\pi$
$258$	0	0
$259$	4.74115	0.294601
$260$	0	0
$261$	−29.1086	−1.80178
$262$	0	0
$263$	26.7546	1.64976	0.824879	−	0.565310i	$-0.191243\pi$
$263$	26.7546	1.64976	0.824879	+	0.565310i	$0.191243\pi$
$264$	0	0
$265$	8.67366	0.532818
$266$	0	0
$267$	28.8870	1.76785
$268$	0	0
$269$	23.2672	1.41863	0.709313	−	0.704893i	$-0.249005\pi$
$269$	23.2672	1.41863	0.709313	+	0.704893i	$0.249005\pi$
$270$	0	0
$271$	−0.119207	−0.00724133	−0.00362067	−	0.999993i	$-0.501152\pi$
$271$	−0.119207	−0.00724133	−0.00362067	+	0.999993i	$0.501152\pi$
$272$	0	0
$273$	−31.7836	−1.92363
$274$	0	0
$275$	−5.67228	−0.342051
$276$	0	0
$277$	6.49882	0.390476	0.195238	−	0.980756i	$-0.437452\pi$
$277$	6.49882	0.390476	0.195238	+	0.980756i	$0.437452\pi$
$278$	0	0
$279$	−26.4382	−1.58281
$280$	0	0
$281$	3.59753	0.214610	0.107305	−	0.994226i	$-0.465778\pi$
$281$	3.59753	0.214610	0.107305	+	0.994226i	$0.465778\pi$
$282$	0	0
$283$	−18.7552	−1.11488	−0.557439	−	0.830218i	$-0.688216\pi$
$283$	−18.7552	−1.11488	−0.557439	+	0.830218i	$0.688216\pi$
$284$	0	0
$285$	19.8869	1.17800
$286$	0	0
$287$	21.8649	1.29064
$288$	0	0
$289$	−16.7028	−0.982515
$290$	0	0
$291$	−13.5734	−0.795688
$292$	0	0
$293$	1.19938	0.0700687	0.0350344	−	0.999386i	$-0.488846\pi$
$293$	1.19938	0.0700687	0.0350344	+	0.999386i	$0.488846\pi$
$294$	0	0
$295$	0.777152	0.0452476
$296$	0	0
$297$	45.0223	2.61246
$298$	0	0
$299$	−5.04212	−0.291594
$300$	0	0
$301$	−22.5305	−1.29863
$302$	0	0
$303$	37.4177	2.14959
$304$	0	0
$305$	−7.00137	−0.400898
$306$	0	0
$307$	−8.14771	−0.465014	−0.232507	−	0.972595i	$-0.574693\pi$
$307$	−8.14771	−0.465014	−0.232507	+	0.972595i	$0.574693\pi$
$308$	0	0
$309$	30.4468	1.73206
$310$	0	0
$311$	33.7342	1.91289	0.956447	−	0.291907i	$-0.0942898\pi$
$311$	33.7342	1.91289	0.956447	+	0.291907i	$0.0942898\pi$
$312$	0	0
$313$	3.24043	0.183160	0.0915799	−	0.995798i	$-0.470808\pi$
$313$	3.24043	0.183160	0.0915799	+	0.995798i	$0.470808\pi$
$314$	0	0
$315$	−12.1704	−0.685724
$316$	0	0
$317$	−5.89626	−0.331167	−0.165584	−	0.986196i	$-0.552951\pi$
$317$	−5.89626	−0.331167	−0.165584	+	0.986196i	$0.552951\pi$
$318$	0	0
$319$	29.0068	1.62407
$320$	0	0
$321$	35.1492	1.96184
$322$	0	0
$323$	−3.67751	−0.204622
$324$	0	0
$325$	−5.04212	−0.279687
$326$	0	0
$327$	20.8309	1.15195
$328$	0	0
$329$	−5.58638	−0.307987
$330$	0	0
$331$	−1.57807	−0.0867383	−0.0433691	−	0.999059i	$-0.513809\pi$
$331$	−1.57807	−0.0867383	−0.0433691	+	0.999059i	$0.513809\pi$
$332$	0	0
$333$	−12.6223	−0.691698
$334$	0	0
$335$	−7.27040	−0.397224
$336$	0	0
$337$	−1.55835	−0.0848888	−0.0424444	−	0.999099i	$-0.513515\pi$
$337$	−1.55835	−0.0848888	−0.0424444	+	0.999099i	$0.513515\pi$
$338$	0	0
$339$	−57.5305	−3.12463
$340$	0	0
$341$	26.3457	1.42670
$342$	0	0
$343$	20.1591	1.08849
$344$	0	0
$345$	−2.94825	−0.158729
$346$	0	0
$347$	−32.4997	−1.74468	−0.872339	−	0.488902i	$-0.837397\pi$
$347$	−32.4997	−1.74468	−0.872339	+	0.488902i	$0.837397\pi$
$348$	0	0
$349$	20.9841	1.12325	0.561626	−	0.827391i	$-0.310176\pi$
$349$	20.9841	1.12325	0.561626	+	0.827391i	$0.310176\pi$
$350$	0	0
$351$	40.0206	2.13614
$352$	0	0
$353$	−30.2614	−1.61065	−0.805327	−	0.592831i	$-0.798010\pi$
$353$	−30.2614	−1.61065	−0.805327	+	0.592831i	$0.798010\pi$
$354$	0	0
$355$	3.40168	0.180543
$356$	0	0
$357$	3.43670	0.181889
$358$	0	0
$359$	12.8907	0.680347	0.340173	−	0.940363i	$-0.389514\pi$
$359$	12.8907	0.680347	0.340173	+	0.940363i	$0.389514\pi$
$360$	0	0
$361$	26.4992	1.39470
$362$	0	0
$363$	−62.4286	−3.27665
$364$	0	0
$365$	−6.34436	−0.332079
$366$	0	0
$367$	−9.49484	−0.495627	−0.247813	−	0.968808i	$-0.579712\pi$
$367$	−9.49484	−0.495627	−0.247813	+	0.968808i	$0.579712\pi$
$368$	0	0
$369$	−58.2105	−3.03032
$370$	0	0
$371$	−18.5450	−0.962809
$372$	0	0
$373$	−15.3732	−0.795996	−0.397998	−	0.917386i	$-0.630295\pi$
$373$	−15.3732	−0.795996	−0.397998	+	0.917386i	$0.630295\pi$
$374$	0	0
$375$	−2.94825	−0.152247
$376$	0	0
$377$	25.7843	1.32796
$378$	0	0
$379$	−34.7952	−1.78731	−0.893655	−	0.448756i	$-0.851867\pi$
$379$	−34.7952	−1.78731	−0.893655	+	0.448756i	$0.851867\pi$
$380$	0	0
$381$	−41.5723	−2.12982
$382$	0	0
$383$	26.6026	1.35933	0.679666	−	0.733522i	$-0.262125\pi$
$383$	26.6026	1.35933	0.679666	+	0.733522i	$0.262125\pi$
$384$	0	0
$385$	12.1278	0.618091
$386$	0	0
$387$	59.9825	3.04908
$388$	0	0
$389$	−33.0795	−1.67720	−0.838598	−	0.544751i	$-0.816624\pi$
$389$	−33.0795	−1.67720	−0.838598	+	0.544751i	$0.816624\pi$
$390$	0	0
$391$	0.545195	0.0275717
$392$	0	0
$393$	−56.1553	−2.83266
$394$	0	0
$395$	−4.07651	−0.205112
$396$	0	0
$397$	33.8019	1.69647	0.848235	−	0.529620i	$-0.177666\pi$
$397$	33.8019	1.69647	0.848235	+	0.529620i	$0.177666\pi$
$398$	0	0
$399$	−42.5198	−2.12865
$400$	0	0
$401$	−23.1390	−1.15551	−0.577754	−	0.816211i	$-0.696071\pi$
$401$	−23.1390	−1.15551	−0.577754	+	0.816211i	$0.696071\pi$
$402$	0	0
$403$	23.4188	1.16658
$404$	0	0
$405$	6.32446	0.314265
$406$	0	0
$407$	12.5782	0.623476
$408$	0	0
$409$	−22.9838	−1.13648	−0.568239	−	0.822864i	$-0.692375\pi$
$409$	−22.9838	−1.13648	−0.568239	+	0.822864i	$0.692375\pi$
$410$	0	0
$411$	−31.2346	−1.54069
$412$	0	0
$413$	−1.66162	−0.0817629
$414$	0	0
$415$	−1.51409	−0.0743237
$416$	0	0
$417$	−42.4538	−2.07897
$418$	0	0
$419$	23.5935	1.15262	0.576308	−	0.817233i	$-0.304493\pi$
$419$	23.5935	1.15262	0.576308	+	0.817233i	$0.304493\pi$
$420$	0	0
$421$	35.0768	1.70954	0.854769	−	0.519009i	$-0.173699\pi$
$421$	35.0768	1.70954	0.854769	+	0.519009i	$0.173699\pi$
$422$	0	0
$423$	14.8725	0.723128
$424$	0	0
$425$	0.545195	0.0264458
$426$	0	0
$427$	14.9695	0.724427
$428$	0	0
$429$	−84.3210	−4.07106
$430$	0	0
$431$	−18.0434	−0.869119	−0.434559	−	0.900643i	$-0.643096\pi$
$431$	−18.0434	−0.869119	−0.434559	+	0.900643i	$0.643096\pi$
$432$	0	0
$433$	−22.0698	−1.06061	−0.530304	−	0.847808i	$-0.677922\pi$
$433$	−22.0698	−1.06061	−0.530304	+	0.847808i	$0.677922\pi$
$434$	0	0
$435$	15.0767	0.722873
$436$	0	0
$437$	−6.74531	−0.322672
$438$	0	0
$439$	21.0314	1.00377	0.501887	−	0.864933i	$-0.332639\pi$
$439$	21.0314	1.00377	0.501887	+	0.864933i	$0.332639\pi$
$440$	0	0
$441$	−13.8240	−0.658286
$442$	0	0
$443$	−13.4291	−0.638037	−0.319019	−	0.947748i	$-0.603353\pi$
$443$	−13.4291	−0.638037	−0.319019	+	0.947748i	$0.603353\pi$
$444$	0	0
$445$	−9.79799	−0.464469
$446$	0	0
$447$	8.31135	0.393113
$448$	0	0
$449$	−6.57879	−0.310472	−0.155236	−	0.987877i	$-0.549614\pi$
$449$	−6.57879	−0.310472	−0.155236	+	0.987877i	$0.549614\pi$
$450$	0	0
$451$	58.0069	2.73144
$452$	0	0
$453$	2.07723	0.0975966
$454$	0	0
$455$	10.7805	0.505397
$456$	0	0
$457$	40.3604	1.88798	0.943990	−	0.329973i	$-0.107040\pi$
$457$	40.3604	1.88798	0.943990	+	0.329973i	$0.107040\pi$
$458$	0	0
$459$	−4.32735	−0.201983
$460$	0	0
$461$	−3.83330	−0.178534	−0.0892672	−	0.996008i	$-0.528453\pi$
$461$	−3.83330	−0.178534	−0.0892672	+	0.996008i	$0.528453\pi$
$462$	0	0
$463$	−31.6251	−1.46974	−0.734871	−	0.678207i	$-0.762757\pi$
$463$	−31.6251	−1.46974	−0.734871	+	0.678207i	$0.762757\pi$
$464$	0	0
$465$	13.6936	0.635024
$466$	0	0
$467$	10.3994	0.481225	0.240612	−	0.970621i	$-0.422652\pi$
$467$	10.3994	0.481225	0.240612	+	0.970621i	$0.422652\pi$
$468$	0	0
$469$	15.5447	0.717789
$470$	0	0
$471$	47.4057	2.18434
$472$	0	0
$473$	−59.7727	−2.74835
$474$	0	0
$475$	−6.74531	−0.309496
$476$	0	0
$477$	49.3721	2.26059
$478$	0	0
$479$	−37.7475	−1.72473	−0.862365	−	0.506288i	$-0.831017\pi$
$479$	−37.7475	−1.72473	−0.862365	+	0.506288i	$0.831017\pi$
$480$	0	0
$481$	11.1808	0.509800
$482$	0	0
$483$	6.30361	0.286824
$484$	0	0
$485$	4.60389	0.209052
$486$	0	0
$487$	−8.15350	−0.369470	−0.184735	−	0.982788i	$-0.559143\pi$
$487$	−8.15350	−0.369470	−0.184735	+	0.982788i	$0.559143\pi$
$488$	0	0
$489$	35.8698	1.62209
$490$	0	0
$491$	−24.9268	−1.12493	−0.562466	−	0.826820i	$-0.690147\pi$
$491$	−24.9268	−1.12493	−0.562466	+	0.826820i	$0.690147\pi$
$492$	0	0
$493$	−2.78801	−0.125565
$494$	0	0
$495$	−32.2877	−1.45122
$496$	0	0
$497$	−7.27309	−0.326243
$498$	0	0
$499$	−8.59825	−0.384911	−0.192455	−	0.981306i	$-0.561645\pi$
$499$	−8.59825	−0.384911	−0.192455	+	0.981306i	$0.561645\pi$
$500$	0	0
$501$	66.6395	2.97723
$502$	0	0
$503$	−39.6459	−1.76772	−0.883861	−	0.467750i	$-0.845065\pi$
$503$	−39.6459	−1.76772	−0.883861	+	0.467750i	$0.845065\pi$
$504$	0	0
$505$	−12.6915	−0.564763
$506$	0	0
$507$	−36.6262	−1.62663
$508$	0	0
$509$	−29.7612	−1.31914	−0.659572	−	0.751642i	$-0.729262\pi$
$509$	−29.7612	−1.31914	−0.659572	+	0.751642i	$0.729262\pi$
$510$	0	0
$511$	13.5648	0.600071
$512$	0	0
$513$	53.5393	2.36382
$514$	0	0
$515$	−10.3271	−0.455065
$516$	0	0
$517$	−14.8205	−0.651806
$518$	0	0
$519$	−19.1795	−0.841886
$520$	0	0
$521$	−30.9544	−1.35614	−0.678069	−	0.734998i	$-0.737183\pi$
$521$	−30.9544	−1.35614	−0.678069	+	0.734998i	$0.737183\pi$
$522$	0	0
$523$	−28.3392	−1.23919	−0.619593	−	0.784923i	$-0.712702\pi$
$523$	−28.3392	−1.23919	−0.619593	+	0.784923i	$0.712702\pi$
$524$	0	0
$525$	6.30361	0.275112
$526$	0	0
$527$	−2.53223	−0.110306
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	4.42370	0.191972
$532$	0	0
$533$	51.5627	2.23343
$534$	0	0
$535$	−11.9221	−0.515436
$536$	0	0
$537$	29.2166	1.26079
$538$	0	0
$539$	13.7757	0.593360
$540$	0	0
$541$	2.72062	0.116969	0.0584844	−	0.998288i	$-0.481373\pi$
$541$	2.72062	0.116969	0.0584844	+	0.998288i	$0.481373\pi$
$542$	0	0
$543$	−8.71224	−0.373878
$544$	0	0
$545$	−7.06552	−0.302653
$546$	0	0
$547$	34.8646	1.49070	0.745352	−	0.666671i	$-0.232282\pi$
$547$	34.8646	1.49070	0.745352	+	0.666671i	$0.232282\pi$
$548$	0	0
$549$	−39.8532	−1.70089
$550$	0	0
$551$	34.4940	1.46950
$552$	0	0
$553$	8.71593	0.370639
$554$	0	0
$555$	6.53768	0.277509
$556$	0	0
$557$	−0.424068	−0.0179684	−0.00898418	−	0.999960i	$-0.502860\pi$
$557$	−0.424068	−0.0179684	−0.00898418	+	0.999960i	$0.502860\pi$
$558$	0	0
$559$	−53.1323	−2.24726
$560$	0	0
$561$	9.11746	0.384940
$562$	0	0
$563$	37.2454	1.56971	0.784853	−	0.619682i	$-0.212738\pi$
$563$	37.2454	1.56971	0.784853	+	0.619682i	$0.212738\pi$
$564$	0	0
$565$	19.5134	0.820936
$566$	0	0
$567$	−13.5222	−0.567880
$568$	0	0
$569$	2.31757	0.0971574	0.0485787	−	0.998819i	$-0.484531\pi$
$569$	2.31757	0.0971574	0.0485787	+	0.998819i	$0.484531\pi$
$570$	0	0
$571$	33.6079	1.40645	0.703224	−	0.710968i	$-0.251743\pi$
$571$	33.6079	1.40645	0.703224	+	0.710968i	$0.251743\pi$
$572$	0	0
$573$	−16.1020	−0.672672
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	7.59589	0.316221	0.158110	−	0.987421i	$-0.449460\pi$
$577$	7.59589	0.316221	0.158110	+	0.987421i	$0.449460\pi$
$578$	0	0
$579$	2.29644	0.0954369
$580$	0	0
$581$	3.23725	0.134304
$582$	0	0
$583$	−49.1994	−2.03763
$584$	0	0
$585$	−28.7007	−1.18663
$586$	0	0
$587$	−31.6838	−1.30773	−0.653864	−	0.756612i	$-0.726853\pi$
$587$	−31.6838	−1.30773	−0.653864	+	0.756612i	$0.726853\pi$
$588$	0	0
$589$	31.3295	1.29091
$590$	0	0
$591$	−49.9326	−2.05395
$592$	0	0
$593$	−33.4910	−1.37531	−0.687656	−	0.726037i	$-0.741360\pi$
$593$	−33.4910	−1.37531	−0.687656	+	0.726037i	$0.741360\pi$
$594$	0	0
$595$	−1.16567	−0.0477879
$596$	0	0
$597$	20.0036	0.818691
$598$	0	0
$599$	8.59861	0.351330	0.175665	−	0.984450i	$-0.443792\pi$
$599$	8.59861	0.351330	0.175665	+	0.984450i	$0.443792\pi$
$600$	0	0
$601$	2.84676	0.116122	0.0580608	−	0.998313i	$-0.481508\pi$
$601$	2.84676	0.116122	0.0580608	+	0.998313i	$0.481508\pi$
$602$	0	0
$603$	−41.3845	−1.68531
$604$	0	0
$605$	21.1748	0.860877
$606$	0	0
$607$	−33.1716	−1.34639	−0.673197	−	0.739463i	$-0.735080\pi$
$607$	−33.1716	−1.34639	−0.673197	+	0.739463i	$0.735080\pi$
$608$	0	0
$609$	−32.2353	−1.30624
$610$	0	0
$611$	−13.1741	−0.532965
$612$	0	0
$613$	43.2327	1.74615	0.873076	−	0.487583i	$-0.162121\pi$
$613$	43.2327	1.74615	0.873076	+	0.487583i	$0.162121\pi$
$614$	0	0
$615$	30.1500	1.21576
$616$	0	0
$617$	−5.55753	−0.223738	−0.111869	−	0.993723i	$-0.535684\pi$
$617$	−5.55753	−0.223738	−0.111869	+	0.993723i	$0.535684\pi$
$618$	0	0
$619$	2.89126	0.116210	0.0581048	−	0.998310i	$-0.481494\pi$
$619$	2.89126	0.116210	0.0581048	+	0.998310i	$0.481494\pi$
$620$	0	0
$621$	−7.93725	−0.318511
$622$	0	0
$623$	20.9489	0.839302
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−112.804	−4.50496
$628$	0	0
$629$	−1.20896	−0.0482043
$630$	0	0
$631$	10.2178	0.406763	0.203381	−	0.979100i	$-0.434807\pi$
$631$	10.2178	0.406763	0.203381	+	0.979100i	$0.434807\pi$
$632$	0	0
$633$	−49.5888	−1.97098
$634$	0	0
$635$	14.1007	0.559568
$636$	0	0
$637$	12.2453	0.485175
$638$	0	0
$639$	19.3630	0.765989
$640$	0	0
$641$	−28.0639	−1.10846	−0.554229	−	0.832364i	$-0.686987\pi$
$641$	−28.0639	−1.10846	−0.554229	+	0.832364i	$0.686987\pi$
$642$	0	0
$643$	−17.4578	−0.688467	−0.344234	−	0.938884i	$-0.611861\pi$
$643$	−17.4578	−0.688467	−0.344234	+	0.938884i	$0.611861\pi$
$644$	0	0
$645$	−31.0677	−1.22329
$646$	0	0
$647$	46.1560	1.81458	0.907290	−	0.420506i	$-0.138147\pi$
$647$	46.1560	1.81458	0.907290	+	0.420506i	$0.138147\pi$
$648$	0	0
$649$	−4.40823	−0.173038
$650$	0	0
$651$	−29.2780	−1.14750
$652$	0	0
$653$	33.6636	1.31736	0.658680	−	0.752423i	$-0.271115\pi$
$653$	33.6636	1.31736	0.658680	+	0.752423i	$0.271115\pi$
$654$	0	0
$655$	19.0470	0.744227
$656$	0	0
$657$	−36.1133	−1.40891
$658$	0	0
$659$	−4.63197	−0.180436	−0.0902180	−	0.995922i	$-0.528756\pi$
$659$	−4.63197	−0.180436	−0.0902180	+	0.995922i	$0.528756\pi$
$660$	0	0
$661$	11.7286	0.456191	0.228095	−	0.973639i	$-0.426750\pi$
$661$	11.7286	0.456191	0.228095	+	0.973639i	$0.426750\pi$
$662$	0	0
$663$	8.10457	0.314755
$664$	0	0
$665$	14.4221	0.559263
$666$	0	0
$667$	−5.11378	−0.198006
$668$	0	0
$669$	−59.5587	−2.30267
$670$	0	0
$671$	39.7138	1.53313
$672$	0	0
$673$	−42.4703	−1.63711	−0.818555	−	0.574428i	$-0.805224\pi$
$673$	−42.4703	−1.63711	−0.818555	+	0.574428i	$0.805224\pi$
$674$	0	0
$675$	−7.93725	−0.305505
$676$	0	0
$677$	−20.4373	−0.785471	−0.392735	−	0.919651i	$-0.628471\pi$
$677$	−20.4373	−0.785471	−0.392735	+	0.919651i	$0.628471\pi$
$678$	0	0
$679$	−9.84350	−0.377759
$680$	0	0
$681$	59.6588	2.28613
$682$	0	0
$683$	0.289606	0.0110815	0.00554073	−	0.999985i	$-0.498236\pi$
$683$	0.289606	0.0110815	0.00554073	+	0.999985i	$0.498236\pi$
$684$	0	0
$685$	10.5943	0.404786
$686$	0	0
$687$	−62.5808	−2.38761
$688$	0	0
$689$	−43.7337	−1.66612
$690$	0	0
$691$	−1.96253	−0.0746583	−0.0373291	−	0.999303i	$-0.511885\pi$
$691$	−1.96253	−0.0746583	−0.0373291	+	0.999303i	$0.511885\pi$
$692$	0	0
$693$	69.0339	2.62238
$694$	0	0
$695$	14.3997	0.546210
$696$	0	0
$697$	−5.57537	−0.211182
$698$	0	0
$699$	38.6936	1.46353
$700$	0	0
$701$	−37.2259	−1.40600	−0.703001	−	0.711189i	$-0.748157\pi$
$701$	−37.2259	−1.40600	−0.703001	+	0.711189i	$0.748157\pi$
$702$	0	0
$703$	14.9576	0.564136
$704$	0	0
$705$	−7.70319	−0.290119
$706$	0	0
$707$	27.1355	1.02053
$708$	0	0
$709$	−18.1006	−0.679782	−0.339891	−	0.940465i	$-0.610390\pi$
$709$	−18.1006	−0.679782	−0.339891	+	0.940465i	$0.610390\pi$
$710$	0	0
$711$	−23.2043	−0.870229
$712$	0	0
$713$	−4.64464	−0.173943
$714$	0	0
$715$	28.6004	1.06959
$716$	0	0
$717$	23.6654	0.883799
$718$	0	0
$719$	−29.5147	−1.10071	−0.550357	−	0.834930i	$-0.685508\pi$
$719$	−29.5147	−1.10071	−0.550357	+	0.834930i	$0.685508\pi$
$720$	0	0
$721$	22.0801	0.822307
$722$	0	0
$723$	−2.32324	−0.0864023
$724$	0	0
$725$	−5.11378	−0.189921
$726$	0	0
$727$	−2.92069	−0.108322	−0.0541611	−	0.998532i	$-0.517248\pi$
$727$	−2.92069	−0.108322	−0.0541611	+	0.998532i	$0.517248\pi$
$728$	0	0
$729$	−34.2031	−1.26678
$730$	0	0
$731$	5.74509	0.212490
$732$	0	0
$733$	−22.8857	−0.845304	−0.422652	−	0.906292i	$-0.638901\pi$
$733$	−22.8857	−0.845304	−0.422652	+	0.906292i	$0.638901\pi$
$734$	0	0
$735$	7.16010	0.264104
$736$	0	0
$737$	41.2398	1.51909
$738$	0	0
$739$	−3.48850	−0.128327	−0.0641633	−	0.997939i	$-0.520438\pi$
$739$	−3.48850	−0.128327	−0.0641633	+	0.997939i	$0.520438\pi$
$740$	0	0
$741$	−100.272	−3.68359
$742$	0	0
$743$	9.54065	0.350013	0.175006	−	0.984567i	$-0.444005\pi$
$743$	9.54065	0.350013	0.175006	+	0.984567i	$0.444005\pi$
$744$	0	0
$745$	−2.81908	−0.103283
$746$	0	0
$747$	−8.61848	−0.315334
$748$	0	0
$749$	25.4904	0.931398
$750$	0	0
$751$	−6.84212	−0.249673	−0.124836	−	0.992177i	$-0.539841\pi$
$751$	−6.84212	−0.249673	−0.124836	+	0.992177i	$0.539841\pi$
$752$	0	0
$753$	−90.9161	−3.31317
$754$	0	0
$755$	−0.704562	−0.0256416
$756$	0	0
$757$	40.2647	1.46344	0.731722	−	0.681603i	$-0.238717\pi$
$757$	40.2647	1.46344	0.731722	+	0.681603i	$0.238717\pi$
$758$	0	0
$759$	16.7233	0.607018
$760$	0	0
$761$	−46.1615	−1.67335	−0.836677	−	0.547697i	$-0.815505\pi$
$761$	−46.1615	−1.67335	−0.836677	+	0.547697i	$0.815505\pi$
$762$	0	0
$763$	15.1067	0.546898
$764$	0	0
$765$	3.10335	0.112202
$766$	0	0
$767$	−3.91850	−0.141489
$768$	0	0
$769$	50.4559	1.81949	0.909744	−	0.415171i	$-0.136278\pi$
$769$	50.4559	1.81949	0.909744	+	0.415171i	$0.136278\pi$
$770$	0	0
$771$	−39.0235	−1.40540
$772$	0	0
$773$	7.52270	0.270573	0.135286	−	0.990807i	$-0.456805\pi$
$773$	7.52270	0.270573	0.135286	+	0.990807i	$0.456805\pi$
$774$	0	0
$775$	−4.64464	−0.166840
$776$	0	0
$777$	−13.9781	−0.501462
$778$	0	0
$779$	68.9802	2.47147
$780$	0	0
$781$	−19.2953	−0.690440
$782$	0	0
$783$	40.5894	1.45055
$784$	0	0
$785$	−16.0793	−0.573893
$786$	0	0
$787$	−16.7815	−0.598195	−0.299098	−	0.954223i	$-0.596686\pi$
$787$	−16.7815	−0.598195	−0.299098	+	0.954223i	$0.596686\pi$
$788$	0	0
$789$	−78.8792	−2.80817
$790$	0	0
$791$	−41.7214	−1.48344
$792$	0	0
$793$	35.3018	1.25360
$794$	0	0
$795$	−25.5721	−0.906950
$796$	0	0
$797$	8.89231	0.314982	0.157491	−	0.987520i	$-0.449659\pi$
$797$	8.89231	0.314982	0.157491	+	0.987520i	$0.449659\pi$
$798$	0	0
$799$	1.42448	0.0503946
$800$	0	0
$801$	−55.7720	−1.97061
$802$	0	0
$803$	35.9870	1.26995
$804$	0	0
$805$	−2.13809	−0.0753576
$806$	0	0
$807$	−68.5976	−2.41475
$808$	0	0
$809$	31.6388	1.11236	0.556180	−	0.831062i	$-0.312266\pi$
$809$	31.6388	1.11236	0.556180	+	0.831062i	$0.312266\pi$
$810$	0	0
$811$	−12.2436	−0.429932	−0.214966	−	0.976622i	$-0.568964\pi$
$811$	−12.2436	−0.429932	−0.214966	+	0.976622i	$0.568964\pi$
$812$	0	0
$813$	0.351453	0.0123260
$814$	0	0
$815$	−12.1665	−0.426173
$816$	0	0
$817$	−71.0799	−2.48677
$818$	0	0
$819$	61.3646	2.14425
$820$	0	0
$821$	53.9559	1.88307	0.941536	−	0.336912i	$-0.109382\pi$
$821$	53.9559	1.88307	0.941536	+	0.336912i	$0.109382\pi$
$822$	0	0
$823$	−26.8887	−0.937282	−0.468641	−	0.883389i	$-0.655256\pi$
$823$	−26.8887	−0.937282	−0.468641	+	0.883389i	$0.655256\pi$
$824$	0	0
$825$	16.7233	0.582231
$826$	0	0
$827$	10.6209	0.369324	0.184662	−	0.982802i	$-0.440881\pi$
$827$	10.6209	0.369324	0.184662	+	0.982802i	$0.440881\pi$
$828$	0	0
$829$	−23.6878	−0.822710	−0.411355	−	0.911475i	$-0.634944\pi$
$829$	−23.6878	−0.822710	−0.411355	+	0.911475i	$0.634944\pi$
$830$	0	0
$831$	−19.1602	−0.664658
$832$	0	0
$833$	−1.32406	−0.0458758
$834$	0	0
$835$	−22.6031	−0.782211
$836$	0	0
$837$	36.8657	1.27426
$838$	0	0
$839$	−11.7151	−0.404449	−0.202225	−	0.979339i	$-0.564817\pi$
$839$	−11.7151	−0.404449	−0.202225	+	0.979339i	$0.564817\pi$
$840$	0	0
$841$	−2.84925	−0.0982499
$842$	0	0
$843$	−10.6064	−0.365304
$844$	0	0
$845$	12.4230	0.427365
$846$	0	0
$847$	−45.2735	−1.55562
$848$	0	0
$849$	55.2949	1.89772
$850$	0	0
$851$	−2.21748	−0.0760141
$852$	0	0
$853$	36.8775	1.26266	0.631331	−	0.775513i	$-0.282509\pi$
$853$	36.8775	1.26266	0.631331	+	0.775513i	$0.282509\pi$
$854$	0	0
$855$	−38.3956	−1.31310
$856$	0	0
$857$	−45.6431	−1.55914	−0.779569	−	0.626317i	$-0.784562\pi$
$857$	−45.6431	−1.55914	−0.779569	+	0.626317i	$0.784562\pi$
$858$	0	0
$859$	−32.3689	−1.10441	−0.552206	−	0.833708i	$-0.686214\pi$
$859$	−32.3689	−1.10441	−0.552206	+	0.833708i	$0.686214\pi$
$860$	0	0
$861$	−64.4632	−2.19690
$862$	0	0
$863$	24.9540	0.849443	0.424722	−	0.905324i	$-0.360372\pi$
$863$	24.9540	0.849443	0.424722	+	0.905324i	$0.360372\pi$
$864$	0	0
$865$	6.50538	0.221190
$866$	0	0
$867$	49.2440	1.67241
$868$	0	0
$869$	23.1231	0.784399
$870$	0	0
$871$	36.6583	1.24212
$872$	0	0
$873$	26.2062	0.886946
$874$	0	0
$875$	−2.13809	−0.0722805
$876$	0	0
$877$	−37.1405	−1.25415	−0.627073	−	0.778960i	$-0.715747\pi$
$877$	−37.1405	−1.25415	−0.627073	+	0.778960i	$0.715747\pi$
$878$	0	0
$879$	−3.53608	−0.119269
$880$	0	0
$881$	21.1005	0.710894	0.355447	−	0.934696i	$-0.384329\pi$
$881$	21.1005	0.710894	0.355447	+	0.934696i	$0.384329\pi$
$882$	0	0
$883$	−3.57577	−0.120334	−0.0601672	−	0.998188i	$-0.519163\pi$
$883$	−3.57577	−0.120334	−0.0601672	+	0.998188i	$0.519163\pi$
$884$	0	0
$885$	−2.29124	−0.0770192
$886$	0	0
$887$	54.6213	1.83401	0.917003	−	0.398881i	$-0.130601\pi$
$887$	54.6213	1.83401	0.917003	+	0.398881i	$0.130601\pi$
$888$	0	0
$889$	−30.1484	−1.01115
$890$	0	0
$891$	−35.8741	−1.20183
$892$	0	0
$893$	−17.6241	−0.589769
$894$	0	0
$895$	−9.90981	−0.331248
$896$	0	0
$897$	14.8655	0.496343
$898$	0	0
$899$	23.7517	0.792162
$900$	0	0
$901$	4.72883	0.157540
$902$	0	0
$903$	66.4255	2.21050
$904$	0	0
$905$	2.95505	0.0982293
$906$	0	0
$907$	38.5957	1.28155	0.640775	−	0.767729i	$-0.278613\pi$
$907$	38.5957	1.28155	0.640775	+	0.767729i	$0.278613\pi$
$908$	0	0
$909$	−72.2423	−2.39613
$910$	0	0
$911$	9.81812	0.325289	0.162644	−	0.986685i	$-0.447998\pi$
$911$	9.81812	0.325289	0.162644	+	0.986685i	$0.447998\pi$
$912$	0	0
$913$	8.58834	0.284233
$914$	0	0
$915$	20.6418	0.682397
$916$	0	0
$917$	−40.7241	−1.34483
$918$	0	0
$919$	8.64000	0.285007	0.142504	−	0.989794i	$-0.454485\pi$
$919$	8.64000	0.285007	0.142504	+	0.989794i	$0.454485\pi$
$920$	0	0
$921$	24.0215	0.791535
$922$	0	0
$923$	−17.1517	−0.564555
$924$	0	0
$925$	−2.21748	−0.0729102
$926$	0	0
$927$	−58.7836	−1.93071
$928$	0	0
$929$	−21.4687	−0.704367	−0.352183	−	0.935931i	$-0.614561\pi$
$929$	−21.4687	−0.704367	−0.352183	+	0.935931i	$0.614561\pi$
$930$	0	0
$931$	16.3816	0.536886
$932$	0	0
$933$	−99.4570	−3.25608
$934$	0	0
$935$	−3.09250	−0.101136
$936$	0	0
$937$	20.9844	0.685531	0.342765	−	0.939421i	$-0.388636\pi$
$937$	20.9844	0.685531	0.342765	+	0.939421i	$0.388636\pi$
$938$	0	0
$939$	−9.55360	−0.311770
$940$	0	0
$941$	−26.4942	−0.863687	−0.431844	−	0.901948i	$-0.642137\pi$
$941$	−26.4942	−0.863687	−0.431844	+	0.901948i	$0.642137\pi$
$942$	0	0
$943$	−10.2264	−0.333017
$944$	0	0
$945$	16.9705	0.552051
$946$	0	0
$947$	−12.9572	−0.421053	−0.210527	−	0.977588i	$-0.567518\pi$
$947$	−12.9572	−0.421053	−0.210527	+	0.977588i	$0.567518\pi$
$948$	0	0
$949$	31.9891	1.03841
$950$	0	0
$951$	17.3837	0.563704
$952$	0	0
$953$	3.87158	0.125413	0.0627064	−	0.998032i	$-0.480027\pi$
$953$	3.87158	0.125413	0.0627064	+	0.998032i	$0.480027\pi$
$954$	0	0
$955$	5.46155	0.176732
$956$	0	0
$957$	−85.5194	−2.76445
$958$	0	0
$959$	−22.6514	−0.731453
$960$	0	0
$961$	−9.42733	−0.304108
$962$	0	0
$963$	−67.8626	−2.18684
$964$	0	0
$965$	−0.778917	−0.0250742
$966$	0	0
$967$	34.7264	1.11673	0.558363	−	0.829597i	$-0.311430\pi$
$967$	34.7264	1.11673	0.558363	+	0.829597i	$0.311430\pi$
$968$	0	0
$969$	10.8422	0.348302
$970$	0	0
$971$	−10.4154	−0.334245	−0.167123	−	0.985936i	$-0.553448\pi$
$971$	−10.4154	−0.334245	−0.167123	+	0.985936i	$0.553448\pi$
$972$	0	0
$973$	−30.7877	−0.987008
$974$	0	0
$975$	14.8655	0.476076
$976$	0	0
$977$	−40.6802	−1.30148	−0.650738	−	0.759303i	$-0.725540\pi$
$977$	−40.6802	−1.30148	−0.650738	+	0.759303i	$0.725540\pi$
$978$	0	0
$979$	55.5770	1.77625
$980$	0	0
$981$	−40.2183	−1.28407
$982$	0	0
$983$	22.5016	0.717688	0.358844	−	0.933397i	$-0.383171\pi$
$983$	22.5016	0.717688	0.358844	+	0.933397i	$0.383171\pi$
$984$	0	0
$985$	16.9363	0.539637
$986$	0	0
$987$	16.4701	0.524248
$988$	0	0
$989$	10.5377	0.335079
$990$	0	0
$991$	−8.71659	−0.276892	−0.138446	−	0.990370i	$-0.544211\pi$
$991$	−8.71659	−0.276892	−0.138446	+	0.990370i	$0.544211\pi$
$992$	0	0
$993$	4.65253	0.147644
$994$	0	0
$995$	−6.78489	−0.215095
$996$	0	0
$997$	21.1186	0.668832	0.334416	−	0.942426i	$-0.391461\pi$
$997$	21.1186	0.668832	0.334416	+	0.942426i	$0.391461\pi$
$998$	0	0
$999$	17.6007	0.556861

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3680.2.a.x.1.1	✓	5
4.3	odd	2		3680.2.a.ba.1.5	yes	5
8.3	odd	2		7360.2.a.cl.1.1		5
8.5	even	2		7360.2.a.cq.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3680.2.a.x.1.1	✓	5	1.1	even	1	trivial
3680.2.a.ba.1.5	yes	5	4.3	odd	2
7360.2.a.cl.1.1		5	8.3	odd	2
7360.2.a.cq.1.5		5	8.5	even	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:	\( N \)	\(=\)	\( 3680 = 2^{5} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3680.a (trivial)

\(f(q)\)	\(=\)	\(q-2.94825 q^{3} +1.00000 q^{5} -2.13809 q^{7} +5.69219 q^{9} +O(q^{10})\)	\(q-2.94825 q^{3} +1.00000 q^{5} -2.13809 q^{7} +5.69219 q^{9} -5.67228 q^{11} -5.04212 q^{13} -2.94825 q^{15} +0.545195 q^{17} -6.74531 q^{19} +6.30361 q^{21} +1.00000 q^{23} +1.00000 q^{25} -7.93725 q^{27} -5.11378 q^{29} -4.64464 q^{31} +16.7233 q^{33} -2.13809 q^{35} -2.21748 q^{37} +14.8655 q^{39} -10.2264 q^{41} +10.5377 q^{43} +5.69219 q^{45} +2.61280 q^{47} -2.42859 q^{49} -1.60737 q^{51} +8.67366 q^{53} -5.67228 q^{55} +19.8869 q^{57} +0.777152 q^{59} -7.00137 q^{61} -12.1704 q^{63} -5.04212 q^{65} -7.27040 q^{67} -2.94825 q^{69} +3.40168 q^{71} -6.34436 q^{73} -2.94825 q^{75} +12.1278 q^{77} -4.07651 q^{79} +6.32446 q^{81} -1.51409 q^{83} +0.545195 q^{85} +15.0767 q^{87} -9.79799 q^{89} +10.7805 q^{91} +13.6936 q^{93} -6.74531 q^{95} +4.60389 q^{97} -32.2877 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{3} + 5 q^{5} - q^{7} + 6 q^{9}+O(q^{10}) \) 5 * q - q^3 + 5 * q^5 - q^7 + 6 * q^9	\( 5 q - q^{3} + 5 q^{5} - q^{7} + 6 q^{9} - 3 q^{11} + 7 q^{13} - q^{15} + 9 q^{17} - q^{19} + 20 q^{21} + 5 q^{23} + 5 q^{25} - 4 q^{27} - 10 q^{29} - 21 q^{31} + 7 q^{33} - q^{35} + 8 q^{37} + 24 q^{39} - 13 q^{41} + 6 q^{43} + 6 q^{45} + 24 q^{49} + 17 q^{51} - 6 q^{53} - 3 q^{55} + 26 q^{57} - 18 q^{59} - 11 q^{61} - 4 q^{63} + 7 q^{65} - 38 q^{67} - q^{69} + 21 q^{71} - 12 q^{73} - q^{75} + 46 q^{77} + 18 q^{79} + 9 q^{81} - 20 q^{83} + 9 q^{85} + 6 q^{87} - 16 q^{89} + 3 q^{91} + 22 q^{93} - q^{95} + 29 q^{97} - 29 q^{99}+O(q^{100}) \) 5 * q - q^3 + 5 * q^5 - q^7 + 6 * q^9 - 3 * q^11 + 7 * q^13 - q^15 + 9 * q^17 - q^19 + 20 * q^21 + 5 * q^23 + 5 * q^25 - 4 * q^27 - 10 * q^29 - 21 * q^31 + 7 * q^33 - q^35 + 8 * q^37 + 24 * q^39 - 13 * q^41 + 6 * q^43 + 6 * q^45 + 24 * q^49 + 17 * q^51 - 6 * q^53 - 3 * q^55 + 26 * q^57 - 18 * q^59 - 11 * q^61 - 4 * q^63 + 7 * q^65 - 38 * q^67 - q^69 + 21 * q^71 - 12 * q^73 - q^75 + 46 * q^77 + 18 * q^79 + 9 * q^81 - 20 * q^83 + 9 * q^85 + 6 * q^87 - 16 * q^89 + 3 * q^91 + 22 * q^93 - q^95 + 29 * q^97 - 29 * q^99

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