LMFDB - Embedded newform 2300.2.a.o.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2300,2,Mod(1,2300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2300, 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("2300.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2300 = 2^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2300.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:	$18.3655924649$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.143376304.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 12x^{4} + 22x^{2} - 6x - 1 $ x^6 - 12x^4 + 22x^2 - 6*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 460)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.65047$ of defining polynomial
Character	$\chi$	$=$	2300.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.80150 q^{3} +4.50896 q^{7} +4.84843 q^{9} +O(q^{10})$	$q+2.80150 q^{3} +4.50896 q^{7} +4.84843 q^{9} -4.10479 q^{11} -4.10245 q^{13} -2.26588 q^{17} +6.77484 q^{19} +12.6319 q^{21} +1.00000 q^{23} +5.17837 q^{27} +4.13863 q^{29} +1.84124 q^{31} -11.4996 q^{33} +11.1155 q^{37} -11.4930 q^{39} +8.36833 q^{41} +5.43473 q^{43} -0.593285 q^{47} +13.3307 q^{49} -6.34787 q^{51} +1.70512 q^{53} +18.9797 q^{57} +6.19772 q^{59} -11.3814 q^{61} +21.8614 q^{63} -5.78978 q^{67} +2.80150 q^{69} -11.9915 q^{71} -0.363592 q^{73} -18.5083 q^{77} +1.75692 q^{79} -0.0380417 q^{81} -9.72171 q^{83} +11.5944 q^{87} -17.2208 q^{89} -18.4978 q^{91} +5.15825 q^{93} +4.38314 q^{97} -19.9018 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{3} + 9 q^{7} + 10 q^{9}+O(q^{10}) $ 6 * q + 4 * q^3 + 9 * q^7 + 10 * q^9	$ 6 q + 4 q^{3} + 9 q^{7} + 10 q^{9} + 2 q^{11} + 8 q^{13} + 5 q^{17} + 4 q^{19} + 6 q^{23} + 22 q^{27} + 5 q^{29} + 9 q^{31} - 10 q^{33} + 21 q^{37} - 8 q^{39} - q^{41} + 16 q^{43} + 16 q^{47} + 19 q^{49} - 12 q^{51} - q^{53} + 12 q^{57} - 11 q^{59} - 4 q^{61} + 19 q^{63} + 25 q^{67} + 4 q^{69} - 17 q^{71} - 14 q^{73} + 20 q^{77} + 10 q^{79} + 14 q^{81} + 21 q^{83} + 64 q^{87} - 24 q^{89} - 4 q^{91} + 4 q^{97} - 16 q^{99}+O(q^{100}) $ 6 * q + 4 * q^3 + 9 * q^7 + 10 * q^9 + 2 * q^11 + 8 * q^13 + 5 * q^17 + 4 * q^19 + 6 * q^23 + 22 * q^27 + 5 * q^29 + 9 * q^31 - 10 * q^33 + 21 * q^37 - 8 * q^39 - q^41 + 16 * q^43 + 16 * q^47 + 19 * q^49 - 12 * q^51 - q^53 + 12 * q^57 - 11 * q^59 - 4 * q^61 + 19 * q^63 + 25 * q^67 + 4 * q^69 - 17 * q^71 - 14 * q^73 + 20 * q^77 + 10 * q^79 + 14 * q^81 + 21 * q^83 + 64 * q^87 - 24 * q^89 - 4 * q^91 + 4 * q^97 - 16 * 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
$3$	2.80150	1.61745	0.808725	−	0.588187i	$-0.200158\pi$
$3$	2.80150	1.61745	0.808725	+	0.588187i	$0.200158\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.50896	1.70423	0.852113	−	0.523357i	$-0.175321\pi$
$7$	4.50896	1.70423	0.852113	+	0.523357i	$0.175321\pi$
$8$	0	0
$9$	4.84843	1.61614
$10$	0	0
$11$	−4.10479	−1.23764	−0.618820	−	0.785533i	$-0.712389\pi$
$11$	−4.10479	−1.23764	−0.618820	+	0.785533i	$0.712389\pi$
$12$	0	0
$13$	−4.10245	−1.13781	−0.568907	−	0.822402i	$-0.692634\pi$
$13$	−4.10245	−1.13781	−0.568907	+	0.822402i	$0.692634\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.26588	−0.549556	−0.274778	−	0.961508i	$-0.588604\pi$
$17$	−2.26588	−0.549556	−0.274778	+	0.961508i	$0.588604\pi$
$18$	0	0
$19$	6.77484	1.55425	0.777127	−	0.629343i	$-0.216676\pi$
$19$	6.77484	1.55425	0.777127	+	0.629343i	$0.216676\pi$
$20$	0	0
$21$	12.6319	2.75650
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.17837	0.996579
$28$	0	0
$29$	4.13863	0.768525	0.384263	−	0.923224i	$-0.374456\pi$
$29$	4.13863	0.768525	0.384263	+	0.923224i	$0.374456\pi$
$30$	0	0
$31$	1.84124	0.330697	0.165349	−	0.986235i	$-0.447125\pi$
$31$	1.84124	0.330697	0.165349	+	0.986235i	$0.447125\pi$
$32$	0	0
$33$	−11.4996	−2.00182
$34$	0	0
$35$	0	0
$36$	0	0
$37$	11.1155	1.82738	0.913690	−	0.406411i	$-0.133220\pi$
$37$	11.1155	1.82738	0.913690	+	0.406411i	$0.133220\pi$
$38$	0	0
$39$	−11.4930	−1.84036
$40$	0	0
$41$	8.36833	1.30691	0.653457	−	0.756964i	$-0.273318\pi$
$41$	8.36833	1.30691	0.653457	+	0.756964i	$0.273318\pi$
$42$	0	0
$43$	5.43473	0.828789	0.414395	−	0.910097i	$-0.363993\pi$
$43$	5.43473	0.828789	0.414395	+	0.910097i	$0.363993\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.593285	−0.0865396	−0.0432698	−	0.999063i	$-0.513778\pi$
$47$	−0.593285	−0.0865396	−0.0432698	+	0.999063i	$0.513778\pi$
$48$	0	0
$49$	13.3307	1.90439
$50$	0	0
$51$	−6.34787	−0.888879
$52$	0	0
$53$	1.70512	0.234216	0.117108	−	0.993119i	$-0.462638\pi$
$53$	1.70512	0.234216	0.117108	+	0.993119i	$0.462638\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	18.9797	2.51393
$58$	0	0
$59$	6.19772	0.806874	0.403437	−	0.915007i	$-0.367815\pi$
$59$	6.19772	0.806874	0.403437	+	0.915007i	$0.367815\pi$
$60$	0	0
$61$	−11.3814	−1.45724	−0.728620	−	0.684919i	$-0.759838\pi$
$61$	−11.3814	−1.45724	−0.728620	+	0.684919i	$0.759838\pi$
$62$	0	0
$63$	21.8614	2.75427
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−5.78978	−0.707335	−0.353667	−	0.935371i	$-0.615066\pi$
$67$	−5.78978	−0.707335	−0.353667	+	0.935371i	$0.615066\pi$
$68$	0	0
$69$	2.80150	0.337261
$70$	0	0
$71$	−11.9915	−1.42312	−0.711562	−	0.702623i	$-0.752012\pi$
$71$	−11.9915	−1.42312	−0.711562	+	0.702623i	$0.752012\pi$
$72$	0	0
$73$	−0.363592	−0.0425552	−0.0212776	−	0.999774i	$-0.506773\pi$
$73$	−0.363592	−0.0425552	−0.0212776	+	0.999774i	$0.506773\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−18.5083	−2.10922
$78$	0	0
$79$	1.75692	0.197669	0.0988344	−	0.995104i	$-0.468489\pi$
$79$	1.75692	0.197669	0.0988344	+	0.995104i	$0.468489\pi$
$80$	0	0
$81$	−0.0380417	−0.00422685
$82$	0	0
$83$	−9.72171	−1.06710	−0.533548	−	0.845770i	$-0.679142\pi$
$83$	−9.72171	−1.06710	−0.533548	+	0.845770i	$0.679142\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	11.5944	1.24305
$88$	0	0
$89$	−17.2208	−1.82540	−0.912698	−	0.408634i	$-0.866005\pi$
$89$	−17.2208	−1.82540	−0.912698	+	0.408634i	$0.866005\pi$
$90$	0	0
$91$	−18.4978	−1.93909
$92$	0	0
$93$	5.15825	0.534886
$94$	0	0
$95$	0	0
$96$	0	0
$97$	4.38314	0.445040	0.222520	−	0.974928i	$-0.428572\pi$
$97$	4.38314	0.445040	0.222520	+	0.974928i	$0.428572\pi$
$98$	0	0
$99$	−19.9018	−2.00020
$100$	0	0
$101$	−3.21428	−0.319833	−0.159917	−	0.987131i	$-0.551123\pi$
$101$	−3.21428	−0.319833	−0.159917	+	0.987131i	$0.551123\pi$
$102$	0	0
$103$	−2.09140	−0.206072	−0.103036	−	0.994678i	$-0.532856\pi$
$103$	−2.09140	−0.206072	−0.103036	+	0.994678i	$0.532856\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.81411	−0.175377	−0.0876885	−	0.996148i	$-0.527948\pi$
$107$	−1.81411	−0.175377	−0.0876885	+	0.996148i	$0.527948\pi$
$108$	0	0
$109$	3.16716	0.303359	0.151679	−	0.988430i	$-0.451532\pi$
$109$	3.16716	0.303359	0.151679	+	0.988430i	$0.451532\pi$
$110$	0	0
$111$	31.1402	2.95570
$112$	0	0
$113$	−13.0407	−1.22677	−0.613384	−	0.789785i	$-0.710192\pi$
$113$	−13.0407	−1.22677	−0.613384	+	0.789785i	$0.710192\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−19.8904	−1.83887
$118$	0	0
$119$	−10.2168	−0.936568
$120$	0	0
$121$	5.84927	0.531752
$122$	0	0
$123$	23.4439	2.11387
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−12.0290	−1.06740	−0.533700	−	0.845674i	$-0.679199\pi$
$127$	−12.0290	−1.06740	−0.533700	+	0.845674i	$0.679199\pi$
$128$	0	0
$129$	15.2254	1.34052
$130$	0	0
$131$	1.33814	0.116914	0.0584571	−	0.998290i	$-0.481382\pi$
$131$	1.33814	0.116914	0.0584571	+	0.998290i	$0.481382\pi$
$132$	0	0
$133$	30.5475	2.64880
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.53176	0.728917	0.364459	−	0.931220i	$-0.381254\pi$
$137$	8.53176	0.728917	0.364459	+	0.931220i	$0.381254\pi$
$138$	0	0
$139$	−19.2021	−1.62870	−0.814352	−	0.580371i	$-0.802908\pi$
$139$	−19.2021	−1.62870	−0.814352	+	0.580371i	$0.802908\pi$
$140$	0	0
$141$	−1.66209	−0.139973
$142$	0	0
$143$	16.8397	1.40820
$144$	0	0
$145$	0	0
$146$	0	0
$147$	37.3461	3.08025
$148$	0	0
$149$	−13.9657	−1.14412	−0.572058	−	0.820213i	$-0.693855\pi$
$149$	−13.9657	−1.14412	−0.572058	+	0.820213i	$0.693855\pi$
$150$	0	0
$151$	−9.38572	−0.763799	−0.381900	−	0.924204i	$-0.624730\pi$
$151$	−9.38572	−0.763799	−0.381900	+	0.924204i	$0.624730\pi$
$152$	0	0
$153$	−10.9859	−0.888161
$154$	0	0
$155$	0	0
$156$	0	0
$157$	22.3649	1.78491	0.892457	−	0.451133i	$-0.148980\pi$
$157$	22.3649	1.78491	0.892457	+	0.451133i	$0.148980\pi$
$158$	0	0
$159$	4.77690	0.378833
$160$	0	0
$161$	4.50896	0.355356
$162$	0	0
$163$	−5.91327	−0.463163	−0.231581	−	0.972816i	$-0.574390\pi$
$163$	−5.91327	−0.463163	−0.231581	+	0.972816i	$0.574390\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.73957	−0.676288	−0.338144	−	0.941094i	$-0.609799\pi$
$167$	−8.73957	−0.676288	−0.338144	+	0.941094i	$0.609799\pi$
$168$	0	0
$169$	3.83010	0.294623
$170$	0	0
$171$	32.8473	2.51190
$172$	0	0
$173$	10.5284	0.800462	0.400231	−	0.916414i	$-0.368930\pi$
$173$	10.5284	0.800462	0.400231	+	0.916414i	$0.368930\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	17.3629	1.30508
$178$	0	0
$179$	−3.29655	−0.246396	−0.123198	−	0.992382i	$-0.539315\pi$
$179$	−3.29655	−0.246396	−0.123198	+	0.992382i	$0.539315\pi$
$180$	0	0
$181$	15.7994	1.17436	0.587180	−	0.809456i	$-0.300238\pi$
$181$	15.7994	1.17436	0.587180	+	0.809456i	$0.300238\pi$
$182$	0	0
$183$	−31.8850	−2.35701
$184$	0	0
$185$	0	0
$186$	0	0
$187$	9.30095	0.680153
$188$	0	0
$189$	23.3491	1.69840
$190$	0	0
$191$	−19.9965	−1.44689	−0.723447	−	0.690380i	$-0.757443\pi$
$191$	−19.9965	−1.44689	−0.723447	+	0.690380i	$0.757443\pi$
$192$	0	0
$193$	−0.613407	−0.0441540	−0.0220770	−	0.999756i	$-0.507028\pi$
$193$	−0.613407	−0.0441540	−0.0220770	+	0.999756i	$0.507028\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.63996	0.330584	0.165292	−	0.986245i	$-0.447143\pi$
$197$	4.63996	0.330584	0.165292	+	0.986245i	$0.447143\pi$
$198$	0	0
$199$	−13.1446	−0.931795	−0.465898	−	0.884839i	$-0.654269\pi$
$199$	−13.1446	−0.931795	−0.465898	+	0.884839i	$0.654269\pi$
$200$	0	0
$201$	−16.2201	−1.14408
$202$	0	0
$203$	18.6609	1.30974
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.84843	0.336989
$208$	0	0
$209$	−27.8093	−1.92361
$210$	0	0
$211$	−6.76863	−0.465972	−0.232986	−	0.972480i	$-0.574850\pi$
$211$	−6.76863	−0.465972	−0.232986	+	0.972480i	$0.574850\pi$
$212$	0	0
$213$	−33.5941	−2.30183
$214$	0	0
$215$	0	0
$216$	0	0
$217$	8.30209	0.563583
$218$	0	0
$219$	−1.01860	−0.0688308
$220$	0	0
$221$	9.29565	0.625293
$222$	0	0
$223$	−5.46851	−0.366199	−0.183099	−	0.983094i	$-0.558613\pi$
$223$	−5.46851	−0.366199	−0.183099	+	0.983094i	$0.558613\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−5.55927	−0.368981	−0.184491	−	0.982834i	$-0.559064\pi$
$227$	−5.55927	−0.368981	−0.184491	+	0.982834i	$0.559064\pi$
$228$	0	0
$229$	−6.33693	−0.418756	−0.209378	−	0.977835i	$-0.567144\pi$
$229$	−6.33693	−0.418756	−0.209378	+	0.977835i	$0.567144\pi$
$230$	0	0
$231$	−51.8511	−3.41155
$232$	0	0
$233$	9.84557	0.645005	0.322502	−	0.946569i	$-0.395476\pi$
$233$	9.84557	0.645005	0.322502	+	0.946569i	$0.395476\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	4.92201	0.319719
$238$	0	0
$239$	25.9526	1.67873	0.839366	−	0.543567i	$-0.182926\pi$
$239$	25.9526	1.67873	0.839366	+	0.543567i	$0.182926\pi$
$240$	0	0
$241$	−12.3220	−0.793729	−0.396865	−	0.917877i	$-0.629902\pi$
$241$	−12.3220	−0.793729	−0.396865	+	0.917877i	$0.629902\pi$
$242$	0	0
$243$	−15.6417	−1.00342
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−27.7934	−1.76845
$248$	0	0
$249$	−27.2354	−1.72597
$250$	0	0
$251$	−7.68797	−0.485261	−0.242630	−	0.970119i	$-0.578010\pi$
$251$	−7.68797	−0.485261	−0.242630	+	0.970119i	$0.578010\pi$
$252$	0	0
$253$	−4.10479	−0.258066
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−19.7126	−1.22964	−0.614818	−	0.788669i	$-0.710771\pi$
$257$	−19.7126	−1.22964	−0.614818	+	0.788669i	$0.710771\pi$
$258$	0	0
$259$	50.1195	3.11427
$260$	0	0
$261$	20.0659	1.24205
$262$	0	0
$263$	−2.96836	−0.183037	−0.0915183	−	0.995803i	$-0.529172\pi$
$263$	−2.96836	−0.183037	−0.0915183	+	0.995803i	$0.529172\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−48.2440	−2.95249
$268$	0	0
$269$	0.417916	0.0254808	0.0127404	−	0.999919i	$-0.495944\pi$
$269$	0.417916	0.0254808	0.0127404	+	0.999919i	$0.495944\pi$
$270$	0	0
$271$	10.1776	0.618245	0.309122	−	0.951022i	$-0.399965\pi$
$271$	10.1776	0.618245	0.309122	+	0.951022i	$0.399965\pi$
$272$	0	0
$273$	−51.8216	−3.13639
$274$	0	0
$275$	0	0
$276$	0	0
$277$	18.8176	1.13064	0.565321	−	0.824871i	$-0.308752\pi$
$277$	18.8176	1.13064	0.565321	+	0.824871i	$0.308752\pi$
$278$	0	0
$279$	8.92713	0.534454
$280$	0	0
$281$	−25.9609	−1.54870	−0.774350	−	0.632757i	$-0.781923\pi$
$281$	−25.9609	−1.54870	−0.774350	+	0.632757i	$0.781923\pi$
$282$	0	0
$283$	22.2612	1.32329	0.661646	−	0.749817i	$-0.269858\pi$
$283$	22.2612	1.32329	0.661646	+	0.749817i	$0.269858\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	37.7325	2.22728
$288$	0	0
$289$	−11.8658	−0.697988
$290$	0	0
$291$	12.2794	0.719830
$292$	0	0
$293$	2.21032	0.129128	0.0645641	−	0.997914i	$-0.479434\pi$
$293$	2.21032	0.129128	0.0645641	+	0.997914i	$0.479434\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−21.2561	−1.23341
$298$	0	0
$299$	−4.10245	−0.237251
$300$	0	0
$301$	24.5050	1.41244
$302$	0	0
$303$	−9.00483	−0.517314
$304$	0	0
$305$	0	0
$306$	0	0
$307$	19.7730	1.12851	0.564253	−	0.825602i	$-0.309164\pi$
$307$	19.7730	1.12851	0.564253	+	0.825602i	$0.309164\pi$
$308$	0	0
$309$	−5.85906	−0.333310
$310$	0	0
$311$	12.1593	0.689490	0.344745	−	0.938696i	$-0.387965\pi$
$311$	12.1593	0.689490	0.344745	+	0.938696i	$0.387965\pi$
$312$	0	0
$313$	−34.0793	−1.92628	−0.963139	−	0.269003i	$-0.913306\pi$
$313$	−34.0793	−1.92628	−0.963139	+	0.269003i	$0.913306\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.66161	0.205657	0.102828	−	0.994699i	$-0.467211\pi$
$317$	3.66161	0.205657	0.102828	+	0.994699i	$0.467211\pi$
$318$	0	0
$319$	−16.9882	−0.951157
$320$	0	0
$321$	−5.08225	−0.283663
$322$	0	0
$323$	−15.3510	−0.854150
$324$	0	0
$325$	0	0
$326$	0	0
$327$	8.87281	0.490667
$328$	0	0
$329$	−2.67510	−0.147483
$330$	0	0
$331$	3.50062	0.192412	0.0962058	−	0.995361i	$-0.469329\pi$
$331$	3.50062	0.192412	0.0962058	+	0.995361i	$0.469329\pi$
$332$	0	0
$333$	53.8928	2.95331
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−33.3643	−1.81747	−0.908735	−	0.417373i	$-0.862951\pi$
$337$	−33.3643	−1.81747	−0.908735	+	0.417373i	$0.862951\pi$
$338$	0	0
$339$	−36.5336	−1.98423
$340$	0	0
$341$	−7.55791	−0.409284
$342$	0	0
$343$	28.5450	1.54128
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5.80416	0.311584	0.155792	−	0.987790i	$-0.450207\pi$
$347$	5.80416	0.311584	0.155792	+	0.987790i	$0.450207\pi$
$348$	0	0
$349$	−0.496058	−0.0265534	−0.0132767	−	0.999912i	$-0.504226\pi$
$349$	−0.496058	−0.0265534	−0.0132767	+	0.999912i	$0.504226\pi$
$350$	0	0
$351$	−21.2440	−1.13392
$352$	0	0
$353$	−5.80547	−0.308994	−0.154497	−	0.987993i	$-0.549376\pi$
$353$	−5.80547	−0.308994	−0.154497	+	0.987993i	$0.549376\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−28.6223	−1.51485
$358$	0	0
$359$	28.3473	1.49611	0.748057	−	0.663635i	$-0.230987\pi$
$359$	28.3473	1.49611	0.748057	+	0.663635i	$0.230987\pi$
$360$	0	0
$361$	26.8984	1.41571
$362$	0	0
$363$	16.3868	0.860081
$364$	0	0
$365$	0	0
$366$	0	0
$367$	24.2615	1.26644	0.633219	−	0.773972i	$-0.281733\pi$
$367$	24.2615	1.26644	0.633219	+	0.773972i	$0.281733\pi$
$368$	0	0
$369$	40.5732	2.11216
$370$	0	0
$371$	7.68832	0.399158
$372$	0	0
$373$	22.6261	1.17153	0.585767	−	0.810479i	$-0.300793\pi$
$373$	22.6261	1.17153	0.585767	+	0.810479i	$0.300793\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−16.9785	−0.874439
$378$	0	0
$379$	6.43791	0.330693	0.165347	−	0.986236i	$-0.447126\pi$
$379$	6.43791	0.330693	0.165347	+	0.986236i	$0.447126\pi$
$380$	0	0
$381$	−33.6993	−1.72647
$382$	0	0
$383$	14.5145	0.741657	0.370829	−	0.928701i	$-0.379074\pi$
$383$	14.5145	0.741657	0.370829	+	0.928701i	$0.379074\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	26.3499	1.33944
$388$	0	0
$389$	−26.1850	−1.32763	−0.663815	−	0.747897i	$-0.731064\pi$
$389$	−26.1850	−1.32763	−0.663815	+	0.747897i	$0.731064\pi$
$390$	0	0
$391$	−2.26588	−0.114590
$392$	0	0
$393$	3.74882	0.189103
$394$	0	0
$395$	0	0
$396$	0	0
$397$	20.6074	1.03426	0.517129	−	0.855908i	$-0.327001\pi$
$397$	20.6074	1.03426	0.517129	+	0.855908i	$0.327001\pi$
$398$	0	0
$399$	85.5789	4.28430
$400$	0	0
$401$	12.4900	0.623719	0.311859	−	0.950128i	$-0.399048\pi$
$401$	12.4900	0.623719	0.311859	+	0.950128i	$0.399048\pi$
$402$	0	0
$403$	−7.55361	−0.376272
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−45.6268	−2.26164
$408$	0	0
$409$	9.11740	0.450826	0.225413	−	0.974263i	$-0.427627\pi$
$409$	9.11740	0.450826	0.225413	+	0.974263i	$0.427627\pi$
$410$	0	0
$411$	23.9018	1.17899
$412$	0	0
$413$	27.9453	1.37510
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−53.7949	−2.63435
$418$	0	0
$419$	−7.12358	−0.348010	−0.174005	−	0.984745i	$-0.555671\pi$
$419$	−7.12358	−0.348010	−0.174005	+	0.984745i	$0.555671\pi$
$420$	0	0
$421$	7.13707	0.347840	0.173920	−	0.984760i	$-0.444357\pi$
$421$	7.13707	0.347840	0.173920	+	0.984760i	$0.444357\pi$
$422$	0	0
$423$	−2.87650	−0.139860
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−51.3183	−2.48347
$428$	0	0
$429$	47.1764	2.27770
$430$	0	0
$431$	−14.2840	−0.688036	−0.344018	−	0.938963i	$-0.611788\pi$
$431$	−14.2840	−0.688036	−0.344018	+	0.938963i	$0.611788\pi$
$432$	0	0
$433$	−17.5102	−0.841485	−0.420743	−	0.907180i	$-0.638230\pi$
$433$	−17.5102	−0.841485	−0.420743	+	0.907180i	$0.638230\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.77484	0.324084
$438$	0	0
$439$	0.991828	0.0473374	0.0236687	−	0.999720i	$-0.492465\pi$
$439$	0.991828	0.0473374	0.0236687	+	0.999720i	$0.492465\pi$
$440$	0	0
$441$	64.6330	3.07776
$442$	0	0
$443$	2.99095	0.142105	0.0710523	−	0.997473i	$-0.477364\pi$
$443$	2.99095	0.142105	0.0710523	+	0.997473i	$0.477364\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−39.1250	−1.85055
$448$	0	0
$449$	20.8981	0.986244	0.493122	−	0.869960i	$-0.335856\pi$
$449$	20.8981	0.986244	0.493122	+	0.869960i	$0.335856\pi$
$450$	0	0
$451$	−34.3502	−1.61749
$452$	0	0
$453$	−26.2941	−1.23541
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−33.4482	−1.56464	−0.782320	−	0.622877i	$-0.785964\pi$
$457$	−33.4482	−1.56464	−0.782320	+	0.622877i	$0.785964\pi$
$458$	0	0
$459$	−11.7336	−0.547676
$460$	0	0
$461$	8.53331	0.397436	0.198718	−	0.980057i	$-0.436322\pi$
$461$	8.53331	0.397436	0.198718	+	0.980057i	$0.436322\pi$
$462$	0	0
$463$	13.5937	0.631751	0.315876	−	0.948801i	$-0.397702\pi$
$463$	13.5937	0.631751	0.315876	+	0.948801i	$0.397702\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	31.3442	1.45044	0.725219	−	0.688518i	$-0.241738\pi$
$467$	31.3442	1.45044	0.725219	+	0.688518i	$0.241738\pi$
$468$	0	0
$469$	−26.1059	−1.20546
$470$	0	0
$471$	62.6554	2.88701
$472$	0	0
$473$	−22.3084	−1.02574
$474$	0	0
$475$	0	0
$476$	0	0
$477$	8.26715	0.378527
$478$	0	0
$479$	20.0701	0.917027	0.458513	−	0.888687i	$-0.348382\pi$
$479$	20.0701	0.917027	0.458513	+	0.888687i	$0.348382\pi$
$480$	0	0
$481$	−45.6009	−2.07922
$482$	0	0
$483$	12.6319	0.574770
$484$	0	0
$485$	0	0
$486$	0	0
$487$	26.8679	1.21750	0.608750	−	0.793362i	$-0.291671\pi$
$487$	26.8679	1.21750	0.608750	+	0.793362i	$0.291671\pi$
$488$	0	0
$489$	−16.5660	−0.749143
$490$	0	0
$491$	33.2592	1.50097	0.750483	−	0.660890i	$-0.229821\pi$
$491$	33.2592	1.50097	0.750483	+	0.660890i	$0.229821\pi$
$492$	0	0
$493$	−9.37764	−0.422348
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−54.0690	−2.42533
$498$	0	0
$499$	−24.4726	−1.09554	−0.547772	−	0.836627i	$-0.684524\pi$
$499$	−24.4726	−1.09554	−0.547772	+	0.836627i	$0.684524\pi$
$500$	0	0
$501$	−24.4839	−1.09386
$502$	0	0
$503$	−39.8607	−1.77730	−0.888650	−	0.458585i	$-0.848356\pi$
$503$	−39.8607	−1.77730	−0.888650	+	0.458585i	$0.848356\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	10.7300	0.476537
$508$	0	0
$509$	20.4456	0.906237	0.453119	−	0.891450i	$-0.350311\pi$
$509$	20.4456	0.906237	0.453119	+	0.891450i	$0.350311\pi$
$510$	0	0
$511$	−1.63942	−0.0725237
$512$	0	0
$513$	35.0826	1.54894
$514$	0	0
$515$	0	0
$516$	0	0
$517$	2.43531	0.107105
$518$	0	0
$519$	29.4955	1.29471
$520$	0	0
$521$	−30.3985	−1.33178	−0.665892	−	0.746048i	$-0.731949\pi$
$521$	−30.3985	−1.33178	−0.665892	+	0.746048i	$0.731949\pi$
$522$	0	0
$523$	−4.27912	−0.187113	−0.0935564	−	0.995614i	$-0.529824\pi$
$523$	−4.27912	−0.187113	−0.0935564	+	0.995614i	$0.529824\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.17203	−0.181737
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	30.0492	1.30402
$532$	0	0
$533$	−34.3306	−1.48703
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−9.23529	−0.398532
$538$	0	0
$539$	−54.7198	−2.35695
$540$	0	0
$541$	−3.60876	−0.155153	−0.0775764	−	0.996986i	$-0.524718\pi$
$541$	−3.60876	−0.155153	−0.0775764	+	0.996986i	$0.524718\pi$
$542$	0	0
$543$	44.2621	1.89947
$544$	0	0
$545$	0	0
$546$	0	0
$547$	30.6519	1.31058	0.655290	−	0.755377i	$-0.272546\pi$
$547$	30.6519	1.31058	0.655290	+	0.755377i	$0.272546\pi$
$548$	0	0
$549$	−55.1819	−2.35511
$550$	0	0
$551$	28.0386	1.19448
$552$	0	0
$553$	7.92187	0.336872
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−45.9504	−1.94698	−0.973491	−	0.228725i	$-0.926544\pi$
$557$	−45.9504	−1.94698	−0.973491	+	0.228725i	$0.926544\pi$
$558$	0	0
$559$	−22.2957	−0.943009
$560$	0	0
$561$	26.0566	1.10011
$562$	0	0
$563$	4.30591	0.181472	0.0907362	−	0.995875i	$-0.471078\pi$
$563$	4.30591	0.181472	0.0907362	+	0.995875i	$0.471078\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−0.171528	−0.00720351
$568$	0	0
$569$	29.4627	1.23514	0.617571	−	0.786515i	$-0.288117\pi$
$569$	29.4627	1.23514	0.617571	+	0.786515i	$0.288117\pi$
$570$	0	0
$571$	−31.7317	−1.32793	−0.663966	−	0.747763i	$-0.731128\pi$
$571$	−31.7317	−1.32793	−0.663966	+	0.747763i	$0.731128\pi$
$572$	0	0
$573$	−56.0202	−2.34028
$574$	0	0
$575$	0	0
$576$	0	0
$577$	24.1386	1.00490	0.502451	−	0.864606i	$-0.332432\pi$
$577$	24.1386	1.00490	0.502451	+	0.864606i	$0.332432\pi$
$578$	0	0
$579$	−1.71846	−0.0714169
$580$	0	0
$581$	−43.8348	−1.81857
$582$	0	0
$583$	−6.99915	−0.289875
$584$	0	0
$585$	0	0
$586$	0	0
$587$	33.9551	1.40148	0.700738	−	0.713419i	$-0.252854\pi$
$587$	33.9551	1.40148	0.700738	+	0.713419i	$0.252854\pi$
$588$	0	0
$589$	12.4741	0.513988
$590$	0	0
$591$	12.9989	0.534702
$592$	0	0
$593$	−6.84201	−0.280968	−0.140484	−	0.990083i	$-0.544866\pi$
$593$	−6.84201	−0.280968	−0.140484	+	0.990083i	$0.544866\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−36.8246	−1.50713
$598$	0	0
$599$	26.6172	1.08755	0.543775	−	0.839231i	$-0.316995\pi$
$599$	26.6172	1.08755	0.543775	+	0.839231i	$0.316995\pi$
$600$	0	0
$601$	15.0702	0.614728	0.307364	−	0.951592i	$-0.400553\pi$
$601$	15.0702	0.614728	0.307364	+	0.951592i	$0.400553\pi$
$602$	0	0
$603$	−28.0713	−1.14315
$604$	0	0
$605$	0	0
$606$	0	0
$607$	43.4205	1.76238	0.881191	−	0.472761i	$-0.156742\pi$
$607$	43.4205	1.76238	0.881191	+	0.472761i	$0.156742\pi$
$608$	0	0
$609$	52.2787	2.11844
$610$	0	0
$611$	2.43392	0.0984660
$612$	0	0
$613$	4.59633	0.185644	0.0928220	−	0.995683i	$-0.470411\pi$
$613$	4.59633	0.185644	0.0928220	+	0.995683i	$0.470411\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	30.4809	1.22712	0.613558	−	0.789650i	$-0.289738\pi$
$617$	30.4809	1.22712	0.613558	+	0.789650i	$0.289738\pi$
$618$	0	0
$619$	−37.0511	−1.48921	−0.744605	−	0.667506i	$-0.767362\pi$
$619$	−37.0511	−1.48921	−0.744605	+	0.667506i	$0.767362\pi$
$620$	0	0
$621$	5.17837	0.207801
$622$	0	0
$623$	−77.6477	−3.11089
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−77.9078	−3.11134
$628$	0	0
$629$	−25.1864	−1.00425
$630$	0	0
$631$	−29.3035	−1.16655	−0.583276	−	0.812274i	$-0.698230\pi$
$631$	−29.3035	−1.16655	−0.583276	+	0.812274i	$0.698230\pi$
$632$	0	0
$633$	−18.9624	−0.753686
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−54.6886	−2.16684
$638$	0	0
$639$	−58.1397	−2.29997
$640$	0	0
$641$	−11.3307	−0.447536	−0.223768	−	0.974642i	$-0.571836\pi$
$641$	−11.3307	−0.447536	−0.223768	+	0.974642i	$0.571836\pi$
$642$	0	0
$643$	28.2340	1.11344	0.556721	−	0.830700i	$-0.312059\pi$
$643$	28.2340	1.11344	0.556721	+	0.830700i	$0.312059\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	30.0192	1.18018	0.590089	−	0.807338i	$-0.299093\pi$
$647$	30.0192	1.18018	0.590089	+	0.807338i	$0.299093\pi$
$648$	0	0
$649$	−25.4403	−0.998619
$650$	0	0
$651$	23.2584	0.911567
$652$	0	0
$653$	23.7192	0.928204	0.464102	−	0.885782i	$-0.346377\pi$
$653$	23.7192	0.928204	0.464102	+	0.885782i	$0.346377\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−1.76285	−0.0687752
$658$	0	0
$659$	−16.8488	−0.656336	−0.328168	−	0.944619i	$-0.606431\pi$
$659$	−16.8488	−0.656336	−0.328168	+	0.944619i	$0.606431\pi$
$660$	0	0
$661$	−23.4941	−0.913816	−0.456908	−	0.889514i	$-0.651043\pi$
$661$	−23.4941	−0.913816	−0.456908	+	0.889514i	$0.651043\pi$
$662$	0	0
$663$	26.0418	1.01138
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4.13863	0.160249
$668$	0	0
$669$	−15.3201	−0.592308
$670$	0	0
$671$	46.7182	1.80354
$672$	0	0
$673$	−19.9329	−0.768358	−0.384179	−	0.923259i	$-0.625515\pi$
$673$	−19.9329	−0.768358	−0.384179	+	0.923259i	$0.625515\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−33.9965	−1.30659	−0.653296	−	0.757102i	$-0.726614\pi$
$677$	−33.9965	−1.30659	−0.653296	+	0.757102i	$0.726614\pi$
$678$	0	0
$679$	19.7634	0.758450
$680$	0	0
$681$	−15.5743	−0.596809
$682$	0	0
$683$	11.8223	0.452367	0.226183	−	0.974085i	$-0.427375\pi$
$683$	11.8223	0.452367	0.226183	+	0.974085i	$0.427375\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−17.7529	−0.677316
$688$	0	0
$689$	−6.99517	−0.266495
$690$	0	0
$691$	−39.0765	−1.48654	−0.743271	−	0.668991i	$-0.766727\pi$
$691$	−39.0765	−1.48654	−0.743271	+	0.668991i	$0.766727\pi$
$692$	0	0
$693$	−89.7362	−3.40880
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−18.9616	−0.718222
$698$	0	0
$699$	27.5824	1.04326
$700$	0	0
$701$	−36.3454	−1.37275	−0.686373	−	0.727250i	$-0.740798\pi$
$701$	−36.3454	−1.37275	−0.686373	+	0.727250i	$0.740798\pi$
$702$	0	0
$703$	75.3059	2.84021
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−14.4931	−0.545068
$708$	0	0
$709$	−16.6781	−0.626359	−0.313179	−	0.949694i	$-0.601394\pi$
$709$	−16.6781	−0.626359	−0.313179	+	0.949694i	$0.601394\pi$
$710$	0	0
$711$	8.51829	0.319461
$712$	0	0
$713$	1.84124	0.0689551
$714$	0	0
$715$	0	0
$716$	0	0
$717$	72.7062	2.71526
$718$	0	0
$719$	−24.1134	−0.899278	−0.449639	−	0.893210i	$-0.648447\pi$
$719$	−24.1134	−0.899278	−0.449639	+	0.893210i	$0.648447\pi$
$720$	0	0
$721$	−9.43003	−0.351193
$722$	0	0
$723$	−34.5201	−1.28382
$724$	0	0
$725$	0	0
$726$	0	0
$727$	3.05594	0.113339	0.0566693	−	0.998393i	$-0.481952\pi$
$727$	3.05594	0.113339	0.0566693	+	0.998393i	$0.481952\pi$
$728$	0	0
$729$	−43.7062	−1.61875
$730$	0	0
$731$	−12.3144	−0.455466
$732$	0	0
$733$	49.2230	1.81809	0.909046	−	0.416696i	$-0.136812\pi$
$733$	49.2230	1.81809	0.909046	+	0.416696i	$0.136812\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	23.7658	0.875425
$738$	0	0
$739$	13.1179	0.482548	0.241274	−	0.970457i	$-0.422435\pi$
$739$	13.1179	0.482548	0.241274	+	0.970457i	$0.422435\pi$
$740$	0	0
$741$	−77.8634	−2.86038
$742$	0	0
$743$	−14.9149	−0.547173	−0.273587	−	0.961847i	$-0.588210\pi$
$743$	−14.9149	−0.547173	−0.273587	+	0.961847i	$0.588210\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−47.1350	−1.72458
$748$	0	0
$749$	−8.17977	−0.298882
$750$	0	0
$751$	7.40642	0.270264	0.135132	−	0.990828i	$-0.456854\pi$
$751$	7.40642	0.270264	0.135132	+	0.990828i	$0.456854\pi$
$752$	0	0
$753$	−21.5379	−0.784884
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−1.37246	−0.0498831	−0.0249415	−	0.999689i	$-0.507940\pi$
$757$	−1.37246	−0.0498831	−0.0249415	+	0.999689i	$0.507940\pi$
$758$	0	0
$759$	−11.4996	−0.417408
$760$	0	0
$761$	−33.9083	−1.22918	−0.614588	−	0.788849i	$-0.710678\pi$
$761$	−33.9083	−1.22918	−0.614588	+	0.788849i	$0.710678\pi$
$762$	0	0
$763$	14.2806	0.516992
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−25.4258	−0.918073
$768$	0	0
$769$	7.17267	0.258653	0.129327	−	0.991602i	$-0.458718\pi$
$769$	7.17267	0.258653	0.129327	+	0.991602i	$0.458718\pi$
$770$	0	0
$771$	−55.2248	−1.98887
$772$	0	0
$773$	−3.13634	−0.112806	−0.0564031	−	0.998408i	$-0.517963\pi$
$773$	−3.13634	−0.112806	−0.0564031	+	0.998408i	$0.517963\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	140.410	5.03718
$778$	0	0
$779$	56.6941	2.03128
$780$	0	0
$781$	49.2224	1.76131
$782$	0	0
$783$	21.4314	0.765896
$784$	0	0
$785$	0	0
$786$	0	0
$787$	0.546252	0.0194718	0.00973589	−	0.999953i	$-0.496901\pi$
$787$	0.546252	0.0194718	0.00973589	+	0.999953i	$0.496901\pi$
$788$	0	0
$789$	−8.31586	−0.296052
$790$	0	0
$791$	−58.8001	−2.09069
$792$	0	0
$793$	46.6916	1.65807
$794$	0	0
$795$	0	0
$796$	0	0
$797$	45.0453	1.59559	0.797794	−	0.602931i	$-0.206000\pi$
$797$	45.0453	1.59559	0.797794	+	0.602931i	$0.206000\pi$
$798$	0	0
$799$	1.34431	0.0475584
$800$	0	0
$801$	−83.4936	−2.95010
$802$	0	0
$803$	1.49247	0.0526680
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.17079	0.0412139
$808$	0	0
$809$	42.0227	1.47744	0.738719	−	0.674013i	$-0.235431\pi$
$809$	42.0227	1.47744	0.738719	+	0.674013i	$0.235431\pi$
$810$	0	0
$811$	14.9218	0.523974	0.261987	−	0.965071i	$-0.415622\pi$
$811$	14.9218	0.523974	0.261987	+	0.965071i	$0.415622\pi$
$812$	0	0
$813$	28.5126	0.999980
$814$	0	0
$815$	0	0
$816$	0	0
$817$	36.8194	1.28815
$818$	0	0
$819$	−89.6851	−3.13385
$820$	0	0
$821$	32.2317	1.12489	0.562447	−	0.826833i	$-0.309860\pi$
$821$	32.2317	1.12489	0.562447	+	0.826833i	$0.309860\pi$
$822$	0	0
$823$	31.3609	1.09317	0.546586	−	0.837403i	$-0.315927\pi$
$823$	31.3609	1.09317	0.546586	+	0.837403i	$0.315927\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	29.6787	1.03203	0.516015	−	0.856580i	$-0.327415\pi$
$827$	29.6787	1.03203	0.516015	+	0.856580i	$0.327415\pi$
$828$	0	0
$829$	17.4868	0.607342	0.303671	−	0.952777i	$-0.401788\pi$
$829$	17.4868	0.607342	0.303671	+	0.952777i	$0.401788\pi$
$830$	0	0
$831$	52.7177	1.82876
$832$	0	0
$833$	−30.2058	−1.04657
$834$	0	0
$835$	0	0
$836$	0	0
$837$	9.53465	0.329566
$838$	0	0
$839$	22.2177	0.767039	0.383520	−	0.923533i	$-0.374712\pi$
$839$	22.2177	0.767039	0.383520	+	0.923533i	$0.374712\pi$
$840$	0	0
$841$	−11.8717	−0.409369
$842$	0	0
$843$	−72.7297	−2.50494
$844$	0	0
$845$	0	0
$846$	0	0
$847$	26.3741	0.906225
$848$	0	0
$849$	62.3649	2.14036
$850$	0	0
$851$	11.1155	0.381035
$852$	0	0
$853$	26.5885	0.910374	0.455187	−	0.890396i	$-0.349572\pi$
$853$	26.5885	0.910374	0.455187	+	0.890396i	$0.349572\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	21.0394	0.718692	0.359346	−	0.933204i	$-0.383000\pi$
$857$	21.0394	0.718692	0.359346	+	0.933204i	$0.383000\pi$
$858$	0	0
$859$	−42.7924	−1.46006	−0.730028	−	0.683417i	$-0.760493\pi$
$859$	−42.7924	−1.46006	−0.730028	+	0.683417i	$0.760493\pi$
$860$	0	0
$861$	105.708	3.60251
$862$	0	0
$863$	−44.5471	−1.51640	−0.758200	−	0.652022i	$-0.773921\pi$
$863$	−44.5471	−1.51640	−0.758200	+	0.652022i	$0.773921\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−33.2421	−1.12896
$868$	0	0
$869$	−7.21177	−0.244643
$870$	0	0
$871$	23.7523	0.804816
$872$	0	0
$873$	21.2513	0.719249
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−12.1814	−0.411336	−0.205668	−	0.978622i	$-0.565937\pi$
$877$	−12.1814	−0.411336	−0.205668	+	0.978622i	$0.565937\pi$
$878$	0	0
$879$	6.19222	0.208858
$880$	0	0
$881$	−15.3075	−0.515723	−0.257862	−	0.966182i	$-0.583018\pi$
$881$	−15.3075	−0.515723	−0.257862	+	0.966182i	$0.583018\pi$
$882$	0	0
$883$	24.1308	0.812065	0.406033	−	0.913859i	$-0.366912\pi$
$883$	24.1308	0.812065	0.406033	+	0.913859i	$0.366912\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−17.9353	−0.602210	−0.301105	−	0.953591i	$-0.597355\pi$
$887$	−17.9353	−0.602210	−0.301105	+	0.953591i	$0.597355\pi$
$888$	0	0
$889$	−54.2383	−1.81909
$890$	0	0
$891$	0.156153	0.00523132
$892$	0	0
$893$	−4.01941	−0.134505
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−11.4930	−0.383741
$898$	0	0
$899$	7.62024	0.254149
$900$	0	0
$901$	−3.86359	−0.128715
$902$	0	0
$903$	68.6509	2.28456
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−52.7750	−1.75236	−0.876182	−	0.481980i	$-0.839918\pi$
$907$	−52.7750	−1.75236	−0.876182	+	0.481980i	$0.839918\pi$
$908$	0	0
$909$	−15.5842	−0.516896
$910$	0	0
$911$	23.2848	0.771461	0.385731	−	0.922611i	$-0.373949\pi$
$911$	23.2848	0.771461	0.385731	+	0.922611i	$0.373949\pi$
$912$	0	0
$913$	39.9055	1.32068
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6.03364	0.199248
$918$	0	0
$919$	−9.79940	−0.323253	−0.161626	−	0.986852i	$-0.551674\pi$
$919$	−9.79940	−0.323253	−0.161626	+	0.986852i	$0.551674\pi$
$920$	0	0
$921$	55.3942	1.82530
$922$	0	0
$923$	49.1944	1.61925
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−10.1400	−0.333041
$928$	0	0
$929$	−33.0707	−1.08501	−0.542507	−	0.840051i	$-0.682525\pi$
$929$	−33.0707	−1.08501	−0.542507	+	0.840051i	$0.682525\pi$
$930$	0	0
$931$	90.3135	2.95990
$932$	0	0
$933$	34.0643	1.11521
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−31.0078	−1.01298	−0.506491	−	0.862245i	$-0.669058\pi$
$937$	−31.0078	−1.01298	−0.506491	+	0.862245i	$0.669058\pi$
$938$	0	0
$939$	−95.4734	−3.11566
$940$	0	0
$941$	16.6560	0.542971	0.271486	−	0.962442i	$-0.412485\pi$
$941$	16.6560	0.542971	0.271486	+	0.962442i	$0.412485\pi$
$942$	0	0
$943$	8.36833	0.272510
$944$	0	0
$945$	0	0
$946$	0	0
$947$	16.3503	0.531312	0.265656	−	0.964068i	$-0.414411\pi$
$947$	16.3503	0.531312	0.265656	+	0.964068i	$0.414411\pi$
$948$	0	0
$949$	1.49162	0.0484199
$950$	0	0
$951$	10.2580	0.332639
$952$	0	0
$953$	−14.2223	−0.460704	−0.230352	−	0.973107i	$-0.573988\pi$
$953$	−14.2223	−0.460704	−0.230352	+	0.973107i	$0.573988\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−47.5925	−1.53845
$958$	0	0
$959$	38.4693	1.24224
$960$	0	0
$961$	−27.6098	−0.890639
$962$	0	0
$963$	−8.79560	−0.283434
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−31.2933	−1.00632	−0.503162	−	0.864192i	$-0.667830\pi$
$967$	−31.2933	−1.00632	−0.503162	+	0.864192i	$0.667830\pi$
$968$	0	0
$969$	−43.0058	−1.38154
$970$	0	0
$971$	−31.4671	−1.00983	−0.504914	−	0.863170i	$-0.668476\pi$
$971$	−31.4671	−1.00983	−0.504914	+	0.863170i	$0.668476\pi$
$972$	0	0
$973$	−86.5817	−2.77568
$974$	0	0
$975$	0	0
$976$	0	0
$977$	27.8139	0.889846	0.444923	−	0.895569i	$-0.353231\pi$
$977$	27.8139	0.889846	0.444923	+	0.895569i	$0.353231\pi$
$978$	0	0
$979$	70.6875	2.25918
$980$	0	0
$981$	15.3557	0.490271
$982$	0	0
$983$	−57.5702	−1.83621	−0.918103	−	0.396343i	$-0.870279\pi$
$983$	−57.5702	−1.83621	−0.918103	+	0.396343i	$0.870279\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−7.49431	−0.238546
$988$	0	0
$989$	5.43473	0.172814
$990$	0	0
$991$	−48.4772	−1.53993	−0.769964	−	0.638088i	$-0.779726\pi$
$991$	−48.4772	−1.53993	−0.769964	+	0.638088i	$0.779726\pi$
$992$	0	0
$993$	9.80701	0.311216
$994$	0	0
$995$	0	0
$996$	0	0
$997$	20.6703	0.654634	0.327317	−	0.944915i	$-0.393856\pi$
$997$	20.6703	0.654634	0.327317	+	0.944915i	$0.393856\pi$
$998$	0	0
$999$	57.5603	1.82113

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2300.2.a.o.1.5		6
4.3	odd	2		9200.2.a.cx.1.2		6
5.2	odd	4		460.2.c.a.369.2	✓	12
5.3	odd	4		460.2.c.a.369.11	yes	12
5.4	even	2		2300.2.a.n.1.2		6
15.2	even	4		4140.2.f.b.829.11		12
15.8	even	4		4140.2.f.b.829.12		12
20.3	even	4		1840.2.e.f.369.2		12
20.7	even	4		1840.2.e.f.369.11		12
20.19	odd	2		9200.2.a.cy.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.2.c.a.369.2	✓	12	5.2	odd	4
460.2.c.a.369.11	yes	12	5.3	odd	4
1840.2.e.f.369.2		12	20.3	even	4
1840.2.e.f.369.11		12	20.7	even	4
2300.2.a.n.1.2		6	5.4	even	2
2300.2.a.o.1.5		6	1.1	even	1	trivial
4140.2.f.b.829.11		12	15.2	even	4
4140.2.f.b.829.12		12	15.8	even	4
9200.2.a.cx.1.2		6	4.3	odd	2
9200.2.a.cy.1.5		6	20.19	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:	\( N \)	\(=\)	\( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2300.a (trivial)

\(f(q)\)	\(=\)	\(q+2.80150 q^{3} +4.50896 q^{7} +4.84843 q^{9} +O(q^{10})\)	\(q+2.80150 q^{3} +4.50896 q^{7} +4.84843 q^{9} -4.10479 q^{11} -4.10245 q^{13} -2.26588 q^{17} +6.77484 q^{19} +12.6319 q^{21} +1.00000 q^{23} +5.17837 q^{27} +4.13863 q^{29} +1.84124 q^{31} -11.4996 q^{33} +11.1155 q^{37} -11.4930 q^{39} +8.36833 q^{41} +5.43473 q^{43} -0.593285 q^{47} +13.3307 q^{49} -6.34787 q^{51} +1.70512 q^{53} +18.9797 q^{57} +6.19772 q^{59} -11.3814 q^{61} +21.8614 q^{63} -5.78978 q^{67} +2.80150 q^{69} -11.9915 q^{71} -0.363592 q^{73} -18.5083 q^{77} +1.75692 q^{79} -0.0380417 q^{81} -9.72171 q^{83} +11.5944 q^{87} -17.2208 q^{89} -18.4978 q^{91} +5.15825 q^{93} +4.38314 q^{97} -19.9018 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{3} + 9 q^{7} + 10 q^{9}+O(q^{10}) \) 6 * q + 4 * q^3 + 9 * q^7 + 10 * q^9	\( 6 q + 4 q^{3} + 9 q^{7} + 10 q^{9} + 2 q^{11} + 8 q^{13} + 5 q^{17} + 4 q^{19} + 6 q^{23} + 22 q^{27} + 5 q^{29} + 9 q^{31} - 10 q^{33} + 21 q^{37} - 8 q^{39} - q^{41} + 16 q^{43} + 16 q^{47} + 19 q^{49} - 12 q^{51} - q^{53} + 12 q^{57} - 11 q^{59} - 4 q^{61} + 19 q^{63} + 25 q^{67} + 4 q^{69} - 17 q^{71} - 14 q^{73} + 20 q^{77} + 10 q^{79} + 14 q^{81} + 21 q^{83} + 64 q^{87} - 24 q^{89} - 4 q^{91} + 4 q^{97} - 16 q^{99}+O(q^{100}) \) 6 * q + 4 * q^3 + 9 * q^7 + 10 * q^9 + 2 * q^11 + 8 * q^13 + 5 * q^17 + 4 * q^19 + 6 * q^23 + 22 * q^27 + 5 * q^29 + 9 * q^31 - 10 * q^33 + 21 * q^37 - 8 * q^39 - q^41 + 16 * q^43 + 16 * q^47 + 19 * q^49 - 12 * q^51 - q^53 + 12 * q^57 - 11 * q^59 - 4 * q^61 + 19 * q^63 + 25 * q^67 + 4 * q^69 - 17 * q^71 - 14 * q^73 + 20 * q^77 + 10 * q^79 + 14 * q^81 + 21 * q^83 + 64 * q^87 - 24 * q^89 - 4 * q^91 + 4 * q^97 - 16 * q^99

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