LMFDB - Embedded newform 4544.2.a.bf.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4544,2,Mod(1,4544)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4544, 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("4544.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4544 = 2^{6} \cdot 71 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4544.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:	$36.2840226785$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.2373841.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 9x^{3} + 15x^{2} + 19x - 28 $ x^5 - 2x^4 - 9x^3 + 15x^2 + 19x - 28
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 568)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.84621$ of defining polynomial
Character	$\chi$	$=$	4544.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.84621 q^{3} +2.12124 q^{5} +2.89595 q^{7} +0.408501 q^{9} +O(q^{10})$	$q-1.84621 q^{3} +2.12124 q^{5} +2.89595 q^{7} +0.408501 q^{9} -4.89595 q^{13} -3.91627 q^{15} +3.18300 q^{17} +0.591499 q^{19} -5.34654 q^{21} -1.71295 q^{23} -0.500325 q^{25} +4.78446 q^{27} -9.19275 q^{29} +5.34654 q^{31} +6.14301 q^{35} -2.20498 q^{37} +9.03896 q^{39} -1.28247 q^{41} -3.32477 q^{43} +0.866530 q^{45} -6.52954 q^{47} +1.38652 q^{49} -5.87649 q^{51} -7.73306 q^{53} -1.09203 q^{57} +0.0588384 q^{59} -12.3420 q^{61} +1.18300 q^{63} -10.3855 q^{65} -7.83253 q^{67} +3.16247 q^{69} +1.00000 q^{71} +1.73472 q^{73} +0.923707 q^{75} -9.54673 q^{79} -10.0586 q^{81} +5.99688 q^{83} +6.75191 q^{85} +16.9718 q^{87} -3.90505 q^{89} -14.1784 q^{91} -9.87085 q^{93} +1.25471 q^{95} +11.4014 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{3} - 7 q^{5} + q^{7} + 7 q^{9}+O(q^{10}) $ 5 * q + 2 * q^3 - 7 * q^5 + q^7 + 7 * q^9	$ 5 q + 2 q^{3} - 7 q^{5} + q^{7} + 7 q^{9} - 11 q^{13} - 6 q^{15} + 6 q^{17} - 2 q^{19} - 5 q^{21} - 5 q^{23} + 8 q^{25} + 5 q^{27} - 13 q^{29} + 5 q^{31} - 7 q^{37} + q^{39} + 8 q^{41} - 8 q^{43} - 7 q^{45} - q^{47} + 6 q^{49} - 14 q^{51} - 16 q^{53} - 9 q^{57} - 4 q^{59} - 22 q^{61} - 4 q^{63} + 14 q^{65} - 12 q^{67} - 13 q^{69} + 5 q^{71} - 8 q^{73} - 17 q^{75} + 15 q^{79} - 27 q^{81} - q^{83} - 14 q^{85} + 17 q^{87} - 4 q^{89} - 43 q^{91} + q^{93} - 8 q^{97}+O(q^{100}) $ 5 * q + 2 * q^3 - 7 * q^5 + q^7 + 7 * q^9 - 11 * q^13 - 6 * q^15 + 6 * q^17 - 2 * q^19 - 5 * q^21 - 5 * q^23 + 8 * q^25 + 5 * q^27 - 13 * q^29 + 5 * q^31 - 7 * q^37 + q^39 + 8 * q^41 - 8 * q^43 - 7 * q^45 - q^47 + 6 * q^49 - 14 * q^51 - 16 * q^53 - 9 * q^57 - 4 * q^59 - 22 * q^61 - 4 * q^63 + 14 * q^65 - 12 * q^67 - 13 * q^69 + 5 * q^71 - 8 * q^73 - 17 * q^75 + 15 * q^79 - 27 * q^81 - q^83 - 14 * q^85 + 17 * q^87 - 4 * q^89 - 43 * q^91 + q^93 - 8 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.84621	−1.06591	−0.532956	−	0.846143i	$-0.678919\pi$
$3$	−1.84621	−1.06591	−0.532956	+	0.846143i	$0.678919\pi$
$4$	0	0
$5$	2.12124	0.948649	0.474324	−	0.880350i	$-0.342692\pi$
$5$	2.12124	0.948649	0.474324	+	0.880350i	$0.342692\pi$
$6$	0	0
$7$	2.89595	1.09457	0.547283	−	0.836948i	$-0.315662\pi$
$7$	2.89595	1.09457	0.547283	+	0.836948i	$0.315662\pi$
$8$	0	0
$9$	0.408501	0.136167
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−4.89595	−1.35789	−0.678946	−	0.734188i	$-0.737563\pi$
$13$	−4.89595	−1.35789	−0.678946	+	0.734188i	$0.737563\pi$
$14$	0	0
$15$	−3.91627	−1.01118
$16$	0	0
$17$	3.18300	0.771990	0.385995	−	0.922501i	$-0.373858\pi$
$17$	3.18300	0.771990	0.385995	+	0.922501i	$0.373858\pi$
$18$	0	0
$19$	0.591499	0.135699	0.0678496	−	0.997696i	$-0.478386\pi$
$19$	0.591499	0.135699	0.0678496	+	0.997696i	$0.478386\pi$
$20$	0	0
$21$	−5.34654	−1.16671
$22$	0	0
$23$	−1.71295	−0.357175	−0.178588	−	0.983924i	$-0.557153\pi$
$23$	−1.71295	−0.357175	−0.178588	+	0.983924i	$0.557153\pi$
$24$	0	0
$25$	−0.500325	−0.100065
$26$	0	0
$27$	4.78446	0.920769
$28$	0	0
$29$	−9.19275	−1.70705	−0.853525	−	0.521051i	$-0.825540\pi$
$29$	−9.19275	−1.70705	−0.853525	+	0.521051i	$0.825540\pi$
$30$	0	0
$31$	5.34654	0.960267	0.480133	−	0.877196i	$-0.340588\pi$
$31$	5.34654	0.960267	0.480133	+	0.877196i	$0.340588\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	6.14301	1.03836
$36$	0	0
$37$	−2.20498	−0.362496	−0.181248	−	0.983437i	$-0.558014\pi$
$37$	−2.20498	−0.362496	−0.181248	+	0.983437i	$0.558014\pi$
$38$	0	0
$39$	9.03896	1.44739
$40$	0	0
$41$	−1.28247	−0.200288	−0.100144	−	0.994973i	$-0.531930\pi$
$41$	−1.28247	−0.200288	−0.100144	+	0.994973i	$0.531930\pi$
$42$	0	0
$43$	−3.32477	−0.507022	−0.253511	−	0.967332i	$-0.581585\pi$
$43$	−3.32477	−0.507022	−0.253511	+	0.967332i	$0.581585\pi$
$44$	0	0
$45$	0.866530	0.129175
$46$	0	0
$47$	−6.52954	−0.952431	−0.476215	−	0.879329i	$-0.657992\pi$
$47$	−6.52954	−0.952431	−0.476215	+	0.879329i	$0.657992\pi$
$48$	0	0
$49$	1.38652	0.198075
$50$	0	0
$51$	−5.87649	−0.822873
$52$	0	0
$53$	−7.73306	−1.06222	−0.531109	−	0.847304i	$-0.678224\pi$
$53$	−7.73306	−1.06222	−0.531109	+	0.847304i	$0.678224\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.09203	−0.144643
$58$	0	0
$59$	0.0588384	0.00766010	0.00383005	−	0.999993i	$-0.498781\pi$
$59$	0.0588384	0.00766010	0.00383005	+	0.999993i	$0.498781\pi$
$60$	0	0
$61$	−12.3420	−1.58023	−0.790113	−	0.612961i	$-0.789978\pi$
$61$	−12.3420	−1.58023	−0.790113	+	0.612961i	$0.789978\pi$
$62$	0	0
$63$	1.18300	0.149044
$64$	0	0
$65$	−10.3855	−1.28816
$66$	0	0
$67$	−7.83253	−0.956896	−0.478448	−	0.878116i	$-0.658801\pi$
$67$	−7.83253	−0.956896	−0.478448	+	0.878116i	$0.658801\pi$
$68$	0	0
$69$	3.16247	0.380717
$70$	0	0
$71$	1.00000	0.118678
$71$	1.00000	0.118678
$72$	0	0
$73$	1.73472	0.203034	0.101517	−	0.994834i	$-0.467630\pi$
$73$	1.73472	0.203034	0.101517	+	0.994834i	$0.467630\pi$
$74$	0	0
$75$	0.923707	0.106661
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−9.54673	−1.07409	−0.537045	−	0.843553i	$-0.680460\pi$
$79$	−9.54673	−1.07409	−0.537045	+	0.843553i	$0.680460\pi$
$80$	0	0
$81$	−10.0586	−1.11763
$82$	0	0
$83$	5.99688	0.658243	0.329121	−	0.944288i	$-0.393247\pi$
$83$	5.99688	0.658243	0.329121	+	0.944288i	$0.393247\pi$
$84$	0	0
$85$	6.75191	0.732348
$86$	0	0
$87$	16.9718	1.81956
$88$	0	0
$89$	−3.90505	−0.413935	−0.206967	−	0.978348i	$-0.566359\pi$
$89$	−3.90505	−0.413935	−0.206967	+	0.978348i	$0.566359\pi$
$90$	0	0
$91$	−14.1784	−1.48630
$92$	0	0
$93$	−9.87085	−1.02356
$94$	0	0
$95$	1.25471	0.128731
$96$	0	0
$97$	11.4014	1.15764	0.578821	−	0.815455i	$-0.303513\pi$
$97$	11.4014	1.15764	0.578821	+	0.815455i	$0.303513\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	14.2800	1.42091	0.710456	−	0.703741i	$-0.248489\pi$
$101$	14.2800	1.42091	0.710456	+	0.703741i	$0.248489\pi$
$102$	0	0
$103$	2.03379	0.200395	0.100198	−	0.994968i	$-0.468053\pi$
$103$	2.03379	0.200395	0.100198	+	0.994968i	$0.468053\pi$
$104$	0	0
$105$	−11.3413	−1.10680
$106$	0	0
$107$	20.2563	1.95825	0.979127	−	0.203247i	$-0.0651494\pi$
$107$	20.2563	1.95825	0.979127	+	0.203247i	$0.0651494\pi$
$108$	0	0
$109$	12.3111	1.17919	0.589594	−	0.807700i	$-0.299288\pi$
$109$	12.3111	1.17919	0.589594	+	0.807700i	$0.299288\pi$
$110$	0	0
$111$	4.07086	0.386389
$112$	0	0
$113$	−6.50901	−0.612316	−0.306158	−	0.951981i	$-0.599044\pi$
$113$	−6.50901	−0.612316	−0.306158	+	0.951981i	$0.599044\pi$
$114$	0	0
$115$	−3.63359	−0.338834
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	9.21780	0.844994
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	2.36771	0.213490
$124$	0	0
$125$	−11.6675	−1.04358
$126$	0	0
$127$	7.00065	0.621207	0.310604	−	0.950540i	$-0.399469\pi$
$127$	7.00065	0.621207	0.310604	+	0.950540i	$0.399469\pi$
$128$	0	0
$129$	6.13823	0.540441
$130$	0	0
$131$	11.5673	1.01064	0.505318	−	0.862933i	$-0.331375\pi$
$131$	11.5673	1.01064	0.505318	+	0.862933i	$0.331375\pi$
$132$	0	0
$133$	1.71295	0.148532
$134$	0	0
$135$	10.1490	0.873487
$136$	0	0
$137$	−6.18365	−0.528305	−0.264152	−	0.964481i	$-0.585092\pi$
$137$	−6.18365	−0.528305	−0.264152	+	0.964481i	$0.585092\pi$
$138$	0	0
$139$	10.9944	0.932533	0.466266	−	0.884644i	$-0.345599\pi$
$139$	10.9944	0.932533	0.466266	+	0.884644i	$0.345599\pi$
$140$	0	0
$141$	12.0549	1.01521
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−19.5001	−1.61939
$146$	0	0
$147$	−2.55981	−0.211130
$148$	0	0
$149$	4.64721	0.380715	0.190357	−	0.981715i	$-0.439035\pi$
$149$	4.64721	0.380715	0.190357	+	0.981715i	$0.439035\pi$
$150$	0	0
$151$	1.76727	0.143818	0.0719090	−	0.997411i	$-0.477091\pi$
$151$	1.76727	0.143818	0.0719090	+	0.997411i	$0.477091\pi$
$152$	0	0
$153$	1.30026	0.105120
$154$	0	0
$155$	11.3413	0.910956
$156$	0	0
$157$	−17.9040	−1.42890	−0.714449	−	0.699688i	$-0.753322\pi$
$157$	−17.9040	−1.42890	−0.714449	+	0.699688i	$0.753322\pi$
$158$	0	0
$159$	14.2769	1.13223
$160$	0	0
$161$	−4.96062	−0.390952
$162$	0	0
$163$	−13.2613	−1.03871	−0.519354	−	0.854559i	$-0.673827\pi$
$163$	−13.2613	−1.03871	−0.519354	+	0.854559i	$0.673827\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−22.2385	−1.72087	−0.860434	−	0.509563i	$-0.829807\pi$
$167$	−22.2385	−1.72087	−0.860434	+	0.509563i	$0.829807\pi$
$168$	0	0
$169$	10.9703	0.843871
$170$	0	0
$171$	0.241628	0.0184777
$172$	0	0
$173$	−22.5639	−1.71550	−0.857752	−	0.514064i	$-0.828139\pi$
$173$	−22.5639	−1.71550	−0.857752	+	0.514064i	$0.828139\pi$
$174$	0	0
$175$	−1.44892	−0.109528
$176$	0	0
$177$	−0.108628	−0.00816499
$178$	0	0
$179$	−5.93325	−0.443472	−0.221736	−	0.975107i	$-0.571172\pi$
$179$	−5.93325	−0.443472	−0.221736	+	0.975107i	$0.571172\pi$
$180$	0	0
$181$	19.8058	1.47215	0.736075	−	0.676900i	$-0.236677\pi$
$181$	19.8058	1.47215	0.736075	+	0.676900i	$0.236677\pi$
$182$	0	0
$183$	22.7859	1.68438
$184$	0	0
$185$	−4.67729	−0.343881
$186$	0	0
$187$	0	0
$188$	0	0
$189$	13.8555	1.00784
$190$	0	0
$191$	−26.1436	−1.89169	−0.945843	−	0.324625i	$-0.894762\pi$
$191$	−26.1436	−1.89169	−0.945843	+	0.324625i	$0.894762\pi$
$192$	0	0
$193$	3.42549	0.246572	0.123286	−	0.992371i	$-0.460657\pi$
$193$	3.42549	0.246572	0.123286	+	0.992371i	$0.460657\pi$
$194$	0	0
$195$	19.1738	1.37307
$196$	0	0
$197$	−19.1969	−1.36772	−0.683860	−	0.729613i	$-0.739700\pi$
$197$	−19.1969	−1.36772	−0.683860	+	0.729613i	$0.739700\pi$
$198$	0	0
$199$	−13.4630	−0.954366	−0.477183	−	0.878804i	$-0.658342\pi$
$199$	−13.4630	−0.954366	−0.477183	+	0.878804i	$0.658342\pi$
$200$	0	0
$201$	14.4605	1.01997
$202$	0	0
$203$	−26.6217	−1.86848
$204$	0	0
$205$	−2.72043	−0.190003
$206$	0	0
$207$	−0.699742	−0.0486354
$208$	0	0
$209$	0	0
$210$	0	0
$211$	2.04063	0.140483	0.0702416	−	0.997530i	$-0.477623\pi$
$211$	2.04063	0.140483	0.0702416	+	0.997530i	$0.477623\pi$
$212$	0	0
$213$	−1.84621	−0.126500
$214$	0	0
$215$	−7.05264	−0.480986
$216$	0	0
$217$	15.4833	1.05108
$218$	0	0
$219$	−3.20266	−0.216416
$220$	0	0
$221$	−15.5838	−1.04828
$222$	0	0
$223$	−7.75397	−0.519244	−0.259622	−	0.965710i	$-0.583598\pi$
$223$	−7.75397	−0.519244	−0.259622	+	0.965710i	$0.583598\pi$
$224$	0	0
$225$	−0.204383	−0.0136256
$226$	0	0
$227$	9.02510	0.599017	0.299509	−	0.954094i	$-0.403177\pi$
$227$	9.02510	0.599017	0.299509	+	0.954094i	$0.403177\pi$
$228$	0	0
$229$	14.3227	0.946469	0.473235	−	0.880936i	$-0.343086\pi$
$229$	14.3227	0.946469	0.473235	+	0.880936i	$0.343086\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−0.225503	−0.0147732	−0.00738659	−	0.999973i	$-0.502351\pi$
$233$	−0.225503	−0.0147732	−0.00738659	+	0.999973i	$0.502351\pi$
$234$	0	0
$235$	−13.8507	−0.903522
$236$	0	0
$237$	17.6253	1.14489
$238$	0	0
$239$	−3.92105	−0.253632	−0.126816	−	0.991926i	$-0.540476\pi$
$239$	−3.92105	−0.253632	−0.126816	+	0.991926i	$0.540476\pi$
$240$	0	0
$241$	−10.7764	−0.694167	−0.347083	−	0.937834i	$-0.612828\pi$
$241$	−10.7764	−0.694167	−0.347083	+	0.937834i	$0.612828\pi$
$242$	0	0
$243$	4.21699	0.270520
$244$	0	0
$245$	2.94115	0.187903
$246$	0	0
$247$	−2.89595	−0.184265
$248$	0	0
$249$	−11.0715	−0.701628
$250$	0	0
$251$	−12.5852	−0.794374	−0.397187	−	0.917738i	$-0.630014\pi$
$251$	−12.5852	−0.794374	−0.397187	+	0.917738i	$0.630014\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−12.4655	−0.780618
$256$	0	0
$257$	7.46654	0.465750	0.232875	−	0.972507i	$-0.425187\pi$
$257$	7.46654	0.465750	0.232875	+	0.972507i	$0.425187\pi$
$258$	0	0
$259$	−6.38550	−0.396776
$260$	0	0
$261$	−3.75525	−0.232444
$262$	0	0
$263$	5.46546	0.337015	0.168507	−	0.985700i	$-0.446105\pi$
$263$	5.46546	0.337015	0.168507	+	0.985700i	$0.446105\pi$
$264$	0	0
$265$	−16.4037	−1.00767
$266$	0	0
$267$	7.20955	0.441218
$268$	0	0
$269$	−4.98054	−0.303669	−0.151834	−	0.988406i	$-0.548518\pi$
$269$	−4.98054	−0.303669	−0.151834	+	0.988406i	$0.548518\pi$
$270$	0	0
$271$	−5.67814	−0.344923	−0.172461	−	0.985016i	$-0.555172\pi$
$271$	−5.67814	−0.344923	−0.172461	+	0.985016i	$0.555172\pi$
$272$	0	0
$273$	26.1764	1.58427
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−31.9890	−1.92203	−0.961017	−	0.276489i	$-0.910829\pi$
$277$	−31.9890	−1.92203	−0.961017	+	0.276489i	$0.910829\pi$
$278$	0	0
$279$	2.18407	0.130757
$280$	0	0
$281$	8.40954	0.501671	0.250835	−	0.968030i	$-0.419295\pi$
$281$	8.40954	0.501671	0.250835	+	0.968030i	$0.419295\pi$
$282$	0	0
$283$	7.18430	0.427062	0.213531	−	0.976936i	$-0.431504\pi$
$283$	7.18430	0.427062	0.213531	+	0.976936i	$0.431504\pi$
$284$	0	0
$285$	−2.31647	−0.137216
$286$	0	0
$287$	−3.71397	−0.219229
$288$	0	0
$289$	−6.86852	−0.404031
$290$	0	0
$291$	−21.0495	−1.23394
$292$	0	0
$293$	12.9686	0.757636	0.378818	−	0.925471i	$-0.376331\pi$
$293$	12.9686	0.757636	0.378818	+	0.925471i	$0.376331\pi$
$294$	0	0
$295$	0.124811	0.00726675
$296$	0	0
$297$	0	0
$298$	0	0
$299$	8.38652	0.485005
$300$	0	0
$301$	−9.62836	−0.554969
$302$	0	0
$303$	−26.3639	−1.51457
$304$	0	0
$305$	−26.1803	−1.49908
$306$	0	0
$307$	28.4199	1.62201	0.811004	−	0.585040i	$-0.198921\pi$
$307$	28.4199	1.62201	0.811004	+	0.585040i	$0.198921\pi$
$308$	0	0
$309$	−3.75481	−0.213603
$310$	0	0
$311$	0.613327	0.0347786	0.0173893	−	0.999849i	$-0.494465\pi$
$311$	0.613327	0.0347786	0.0173893	+	0.999849i	$0.494465\pi$
$312$	0	0
$313$	2.75023	0.155452	0.0777260	−	0.996975i	$-0.475234\pi$
$313$	2.75023	0.155452	0.0777260	+	0.996975i	$0.475234\pi$
$314$	0	0
$315$	2.50943	0.141390
$316$	0	0
$317$	10.0195	0.562751	0.281376	−	0.959598i	$-0.409209\pi$
$317$	10.0195	0.562751	0.281376	+	0.959598i	$0.409209\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−37.3975	−2.08733
$322$	0	0
$323$	1.88274	0.104758
$324$	0	0
$325$	2.44957	0.135878
$326$	0	0
$327$	−22.7289	−1.25691
$328$	0	0
$329$	−18.9092	−1.04250
$330$	0	0
$331$	−33.1615	−1.82272	−0.911359	−	0.411612i	$-0.864966\pi$
$331$	−33.1615	−1.82272	−0.911359	+	0.411612i	$0.864966\pi$
$332$	0	0
$333$	−0.900735	−0.0493600
$334$	0	0
$335$	−16.6147	−0.907759
$336$	0	0
$337$	−16.7681	−0.913416	−0.456708	−	0.889617i	$-0.650972\pi$
$337$	−16.7681	−0.913416	−0.456708	+	0.889617i	$0.650972\pi$
$338$	0	0
$339$	12.0170	0.652675
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−16.2563	−0.877760
$344$	0	0
$345$	6.70837	0.361167
$346$	0	0
$347$	26.5827	1.42704	0.713518	−	0.700637i	$-0.247101\pi$
$347$	26.5827	1.42704	0.713518	+	0.700637i	$0.247101\pi$
$348$	0	0
$349$	−6.80273	−0.364142	−0.182071	−	0.983285i	$-0.558280\pi$
$349$	−6.80273	−0.364142	−0.182071	+	0.983285i	$0.558280\pi$
$350$	0	0
$351$	−23.4245	−1.25031
$352$	0	0
$353$	−22.4352	−1.19410	−0.597052	−	0.802202i	$-0.703661\pi$
$353$	−22.4352	−1.19410	−0.597052	+	0.802202i	$0.703661\pi$
$354$	0	0
$355$	2.12124	0.112584
$356$	0	0
$357$	−17.0180	−0.900689
$358$	0	0
$359$	−24.7658	−1.30709	−0.653545	−	0.756887i	$-0.726719\pi$
$359$	−24.7658	−1.30709	−0.653545	+	0.756887i	$0.726719\pi$
$360$	0	0
$361$	−18.6501	−0.981586
$362$	0	0
$363$	20.3083	1.06591
$364$	0	0
$365$	3.67977	0.192608
$366$	0	0
$367$	−30.7235	−1.60375	−0.801876	−	0.597490i	$-0.796165\pi$
$367$	−30.7235	−1.60375	−0.801876	+	0.597490i	$0.796165\pi$
$368$	0	0
$369$	−0.523891	−0.0272727
$370$	0	0
$371$	−22.3945	−1.16267
$372$	0	0
$373$	35.1243	1.81867	0.909333	−	0.416069i	$-0.136593\pi$
$373$	35.1243	1.81867	0.909333	+	0.416069i	$0.136593\pi$
$374$	0	0
$375$	21.5407	1.11236
$376$	0	0
$377$	45.0072	2.31799
$378$	0	0
$379$	−14.2522	−0.732088	−0.366044	−	0.930597i	$-0.619288\pi$
$379$	−14.2522	−0.732088	−0.366044	+	0.930597i	$0.619288\pi$
$380$	0	0
$381$	−12.9247	−0.662152
$382$	0	0
$383$	27.5148	1.40594	0.702970	−	0.711220i	$-0.251857\pi$
$383$	27.5148	1.40594	0.702970	+	0.711220i	$0.251857\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.35817	−0.0690397
$388$	0	0
$389$	−6.32315	−0.320597	−0.160298	−	0.987069i	$-0.551246\pi$
$389$	−6.32315	−0.320597	−0.160298	+	0.987069i	$0.551246\pi$
$390$	0	0
$391$	−5.45232	−0.275736
$392$	0	0
$393$	−21.3556	−1.07725
$394$	0	0
$395$	−20.2509	−1.01894
$396$	0	0
$397$	20.1305	1.01032	0.505161	−	0.863025i	$-0.331433\pi$
$397$	20.1305	1.01032	0.505161	+	0.863025i	$0.331433\pi$
$398$	0	0
$399$	−3.16247	−0.158322
$400$	0	0
$401$	−4.58777	−0.229102	−0.114551	−	0.993417i	$-0.536543\pi$
$401$	−4.58777	−0.229102	−0.114551	+	0.993417i	$0.536543\pi$
$402$	0	0
$403$	−26.1764	−1.30394
$404$	0	0
$405$	−21.3368	−1.06023
$406$	0	0
$407$	0	0
$408$	0	0
$409$	35.7338	1.76692	0.883462	−	0.468503i	$-0.155206\pi$
$409$	35.7338	1.76692	0.883462	+	0.468503i	$0.155206\pi$
$410$	0	0
$411$	11.4163	0.563126
$412$	0	0
$413$	0.170393	0.00838449
$414$	0	0
$415$	12.7208	0.624441
$416$	0	0
$417$	−20.2980	−0.993997
$418$	0	0
$419$	14.5188	0.709288	0.354644	−	0.935001i	$-0.384602\pi$
$419$	14.5188	0.709288	0.354644	+	0.935001i	$0.384602\pi$
$420$	0	0
$421$	−32.5411	−1.58595	−0.792977	−	0.609251i	$-0.791470\pi$
$421$	−32.5411	−1.58595	−0.792977	+	0.609251i	$0.791470\pi$
$422$	0	0
$423$	−2.66732	−0.129690
$424$	0	0
$425$	−1.59253	−0.0772493
$426$	0	0
$427$	−35.7417	−1.72966
$428$	0	0
$429$	0	0
$430$	0	0
$431$	6.89805	0.332268	0.166134	−	0.986103i	$-0.446872\pi$
$431$	6.89805	0.332268	0.166134	+	0.986103i	$0.446872\pi$
$432$	0	0
$433$	14.4414	0.694011	0.347006	−	0.937863i	$-0.387199\pi$
$433$	14.4414	0.694011	0.347006	+	0.937863i	$0.387199\pi$
$434$	0	0
$435$	36.0013	1.72613
$436$	0	0
$437$	−1.01321	−0.0484684
$438$	0	0
$439$	−0.819908	−0.0391321	−0.0195660	−	0.999809i	$-0.506228\pi$
$439$	−0.819908	−0.0391321	−0.0195660	+	0.999809i	$0.506228\pi$
$440$	0	0
$441$	0.566396	0.0269712
$442$	0	0
$443$	−28.8418	−1.37032	−0.685158	−	0.728394i	$-0.740267\pi$
$443$	−28.8418	−1.37032	−0.685158	+	0.728394i	$0.740267\pi$
$444$	0	0
$445$	−8.28356	−0.392679
$446$	0	0
$447$	−8.57974	−0.405808
$448$	0	0
$449$	−7.83544	−0.369777	−0.184889	−	0.982759i	$-0.559192\pi$
$449$	−7.83544	−0.369777	−0.184889	+	0.982759i	$0.559192\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−3.26275	−0.153297
$454$	0	0
$455$	−30.0759	−1.40998
$456$	0	0
$457$	3.38527	0.158356	0.0791781	−	0.996860i	$-0.474770\pi$
$457$	3.38527	0.158356	0.0791781	+	0.996860i	$0.474770\pi$
$458$	0	0
$459$	15.2289	0.710825
$460$	0	0
$461$	−29.8419	−1.38987	−0.694937	−	0.719071i	$-0.744568\pi$
$461$	−29.8419	−1.38987	−0.694937	+	0.719071i	$0.744568\pi$
$462$	0	0
$463$	9.22383	0.428668	0.214334	−	0.976760i	$-0.431242\pi$
$463$	9.22383	0.428668	0.214334	+	0.976760i	$0.431242\pi$
$464$	0	0
$465$	−20.9385	−0.970998
$466$	0	0
$467$	26.0406	1.20502	0.602508	−	0.798113i	$-0.294168\pi$
$467$	26.0406	1.20502	0.602508	+	0.798113i	$0.294168\pi$
$468$	0	0
$469$	−22.6826	−1.04739
$470$	0	0
$471$	33.0546	1.52308
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−0.295942	−0.0135788
$476$	0	0
$477$	−3.15896	−0.144639
$478$	0	0
$479$	26.7472	1.22211	0.611054	−	0.791589i	$-0.290746\pi$
$479$	26.7472	1.22211	0.611054	+	0.791589i	$0.290746\pi$
$480$	0	0
$481$	10.7955	0.492230
$482$	0	0
$483$	9.15836	0.416720
$484$	0	0
$485$	24.1853	1.09820
$486$	0	0
$487$	43.4279	1.96790	0.983952	−	0.178432i	$-0.0571024\pi$
$487$	43.4279	1.96790	0.983952	+	0.178432i	$0.0571024\pi$
$488$	0	0
$489$	24.4833	1.10717
$490$	0	0
$491$	−37.2986	−1.68326	−0.841632	−	0.540052i	$-0.818405\pi$
$491$	−37.2986	−1.68326	−0.841632	+	0.540052i	$0.818405\pi$
$492$	0	0
$493$	−29.2605	−1.31783
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.89595	0.129901
$498$	0	0
$499$	−0.0953661	−0.00426917	−0.00213459	−	0.999998i	$-0.500679\pi$
$499$	−0.0953661	−0.00426917	−0.00213459	+	0.999998i	$0.500679\pi$
$500$	0	0
$501$	41.0570	1.83429
$502$	0	0
$503$	18.3609	0.818671	0.409335	−	0.912384i	$-0.365761\pi$
$503$	18.3609	0.818671	0.409335	+	0.912384i	$0.365761\pi$
$504$	0	0
$505$	30.2914	1.34795
$506$	0	0
$507$	−20.2535	−0.899491
$508$	0	0
$509$	−4.00915	−0.177703	−0.0888513	−	0.996045i	$-0.528320\pi$
$509$	−4.00915	−0.177703	−0.0888513	+	0.996045i	$0.528320\pi$
$510$	0	0
$511$	5.02367	0.222234
$512$	0	0
$513$	2.83000	0.124948
$514$	0	0
$515$	4.31416	0.190105
$516$	0	0
$517$	0	0
$518$	0	0
$519$	41.6578	1.82857
$520$	0	0
$521$	4.93200	0.216075	0.108037	−	0.994147i	$-0.465543\pi$
$521$	4.93200	0.216075	0.108037	+	0.994147i	$0.465543\pi$
$522$	0	0
$523$	−31.9287	−1.39614	−0.698071	−	0.716028i	$-0.745958\pi$
$523$	−31.9287	−1.39614	−0.698071	+	0.716028i	$0.745958\pi$
$524$	0	0
$525$	2.67501	0.116747
$526$	0	0
$527$	17.0180	0.741317
$528$	0	0
$529$	−20.0658	−0.872426
$530$	0	0
$531$	0.0240355	0.00104305
$532$	0	0
$533$	6.27891	0.271970
$534$	0	0
$535$	42.9686	1.85770
$536$	0	0
$537$	10.9540	0.472702
$538$	0	0
$539$	0	0
$540$	0	0
$541$	21.0516	0.905081	0.452540	−	0.891744i	$-0.350518\pi$
$541$	21.0516	0.905081	0.452540	+	0.891744i	$0.350518\pi$
$542$	0	0
$543$	−36.5656	−1.56918
$544$	0	0
$545$	26.1148	1.11863
$546$	0	0
$547$	11.0798	0.473739	0.236869	−	0.971542i	$-0.423879\pi$
$547$	11.0798	0.473739	0.236869	+	0.971542i	$0.423879\pi$
$548$	0	0
$549$	−5.04170	−0.215175
$550$	0	0
$551$	−5.43750	−0.231645
$552$	0	0
$553$	−27.6468	−1.17566
$554$	0	0
$555$	8.63528	0.366547
$556$	0	0
$557$	27.4912	1.16484	0.582419	−	0.812889i	$-0.302106\pi$
$557$	27.4912	1.16484	0.582419	+	0.812889i	$0.302106\pi$
$558$	0	0
$559$	16.2779	0.688482
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−34.1770	−1.44039	−0.720194	−	0.693773i	$-0.755947\pi$
$563$	−34.1770	−1.44039	−0.720194	+	0.693773i	$0.755947\pi$
$564$	0	0
$565$	−13.8072	−0.580873
$566$	0	0
$567$	−29.1293	−1.22331
$568$	0	0
$569$	11.3608	0.476268	0.238134	−	0.971232i	$-0.423464\pi$
$569$	11.3608	0.476268	0.238134	+	0.971232i	$0.423464\pi$
$570$	0	0
$571$	−44.0337	−1.84276	−0.921378	−	0.388669i	$-0.872935\pi$
$571$	−44.0337	−1.84276	−0.921378	+	0.388669i	$0.872935\pi$
$572$	0	0
$573$	48.2667	2.01637
$574$	0	0
$575$	0.857033	0.0357407
$576$	0	0
$577$	−29.8584	−1.24302	−0.621511	−	0.783406i	$-0.713481\pi$
$577$	−29.8584	−1.24302	−0.621511	+	0.783406i	$0.713481\pi$
$578$	0	0
$579$	−6.32417	−0.262824
$580$	0	0
$581$	17.3666	0.720490
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−4.24249	−0.175405
$586$	0	0
$587$	−16.5006	−0.681051	−0.340525	−	0.940235i	$-0.610605\pi$
$587$	−16.5006	−0.681051	−0.340525	+	0.940235i	$0.610605\pi$
$588$	0	0
$589$	3.16247	0.130307
$590$	0	0
$591$	35.4415	1.45787
$592$	0	0
$593$	35.2319	1.44680	0.723401	−	0.690428i	$-0.242578\pi$
$593$	35.2319	1.44680	0.723401	+	0.690428i	$0.242578\pi$
$594$	0	0
$595$	19.5532	0.801603
$596$	0	0
$597$	24.8556	1.01727
$598$	0	0
$599$	42.7628	1.74724	0.873621	−	0.486606i	$-0.161765\pi$
$599$	42.7628	1.74724	0.873621	+	0.486606i	$0.161765\pi$
$600$	0	0
$601$	−45.1838	−1.84308	−0.921542	−	0.388278i	$-0.873070\pi$
$601$	−45.1838	−1.84308	−0.921542	+	0.388278i	$0.873070\pi$
$602$	0	0
$603$	−3.19960	−0.130298
$604$	0	0
$605$	−23.3337	−0.948649
$606$	0	0
$607$	9.62775	0.390779	0.195389	−	0.980726i	$-0.437403\pi$
$607$	9.62775	0.390779	0.195389	+	0.980726i	$0.437403\pi$
$608$	0	0
$609$	49.1494	1.99163
$610$	0	0
$611$	31.9683	1.29330
$612$	0	0
$613$	−20.7363	−0.837532	−0.418766	−	0.908094i	$-0.637537\pi$
$613$	−20.7363	−0.837532	−0.418766	+	0.908094i	$0.637537\pi$
$614$	0	0
$615$	5.02250	0.202527
$616$	0	0
$617$	−15.2352	−0.613347	−0.306674	−	0.951815i	$-0.599216\pi$
$617$	−15.2352	−0.613347	−0.306674	+	0.951815i	$0.599216\pi$
$618$	0	0
$619$	18.6644	0.750185	0.375093	−	0.926987i	$-0.377611\pi$
$619$	18.6644	0.750185	0.375093	+	0.926987i	$0.377611\pi$
$620$	0	0
$621$	−8.19554	−0.328876
$622$	0	0
$623$	−11.3088	−0.453079
$624$	0	0
$625$	−22.2480	−0.889922
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−7.01844	−0.279843
$630$	0	0
$631$	−25.1632	−1.00173	−0.500865	−	0.865525i	$-0.666985\pi$
$631$	−25.1632	−1.00173	−0.500865	+	0.865525i	$0.666985\pi$
$632$	0	0
$633$	−3.76745	−0.149743
$634$	0	0
$635$	14.8501	0.589308
$636$	0	0
$637$	−6.78834	−0.268964
$638$	0	0
$639$	0.408501	0.0161600
$640$	0	0
$641$	35.1879	1.38984	0.694919	−	0.719088i	$-0.255440\pi$
$641$	35.1879	1.38984	0.694919	+	0.719088i	$0.255440\pi$
$642$	0	0
$643$	−28.6315	−1.12912	−0.564558	−	0.825393i	$-0.690953\pi$
$643$	−28.6315	−1.12912	−0.564558	+	0.825393i	$0.690953\pi$
$644$	0	0
$645$	13.0207	0.512689
$646$	0	0
$647$	−26.5629	−1.04430	−0.522148	−	0.852855i	$-0.674869\pi$
$647$	−26.5629	−1.04430	−0.522148	+	0.852855i	$0.674869\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−28.5855	−1.12035
$652$	0	0
$653$	7.87024	0.307986	0.153993	−	0.988072i	$-0.450787\pi$
$653$	7.87024	0.307986	0.153993	+	0.988072i	$0.450787\pi$
$654$	0	0
$655$	24.5370	0.958739
$656$	0	0
$657$	0.708635	0.0276465
$658$	0	0
$659$	30.8459	1.20159	0.600793	−	0.799404i	$-0.294851\pi$
$659$	30.8459	1.20159	0.600793	+	0.799404i	$0.294851\pi$
$660$	0	0
$661$	−21.4020	−0.832443	−0.416221	−	0.909263i	$-0.636646\pi$
$661$	−21.4020	−0.832443	−0.416221	+	0.909263i	$0.636646\pi$
$662$	0	0
$663$	28.7710	1.11737
$664$	0	0
$665$	3.63359	0.140904
$666$	0	0
$667$	15.7467	0.609716
$668$	0	0
$669$	14.3155	0.553468
$670$	0	0
$671$	0	0
$672$	0	0
$673$	30.8041	1.18741	0.593705	−	0.804683i	$-0.297665\pi$
$673$	30.8041	1.18741	0.593705	+	0.804683i	$0.297665\pi$
$674$	0	0
$675$	−2.39379	−0.0921369
$676$	0	0
$677$	11.8448	0.455232	0.227616	−	0.973751i	$-0.426907\pi$
$677$	11.8448	0.455232	0.227616	+	0.973751i	$0.426907\pi$
$678$	0	0
$679$	33.0180	1.26712
$680$	0	0
$681$	−16.6623	−0.638499
$682$	0	0
$683$	8.19442	0.313551	0.156775	−	0.987634i	$-0.449890\pi$
$683$	8.19442	0.313551	0.156775	+	0.987634i	$0.449890\pi$
$684$	0	0
$685$	−13.1170	−0.501176
$686$	0	0
$687$	−26.4427	−1.00885
$688$	0	0
$689$	37.8607	1.44238
$690$	0	0
$691$	20.2620	0.770803	0.385401	−	0.922749i	$-0.374063\pi$
$691$	20.2620	0.770803	0.385401	+	0.922749i	$0.374063\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	23.3218	0.884646
$696$	0	0
$697$	−4.08210	−0.154621
$698$	0	0
$699$	0.416326	0.0157469
$700$	0	0
$701$	−24.8561	−0.938802	−0.469401	−	0.882985i	$-0.655530\pi$
$701$	−24.8561	−0.938802	−0.469401	+	0.882985i	$0.655530\pi$
$702$	0	0
$703$	−1.30424	−0.0491904
$704$	0	0
$705$	25.5714	0.963075
$706$	0	0
$707$	41.3541	1.55528
$708$	0	0
$709$	−13.9828	−0.525136	−0.262568	−	0.964914i	$-0.584569\pi$
$709$	−13.9828	−0.525136	−0.262568	+	0.964914i	$0.584569\pi$
$710$	0	0
$711$	−3.89985	−0.146256
$712$	0	0
$713$	−9.15836	−0.342983
$714$	0	0
$715$	0	0
$716$	0	0
$717$	7.23910	0.270349
$718$	0	0
$719$	−18.9747	−0.707636	−0.353818	−	0.935314i	$-0.615117\pi$
$719$	−18.9747	−0.707636	−0.353818	+	0.935314i	$0.615117\pi$
$720$	0	0
$721$	5.88975	0.219346
$722$	0	0
$723$	19.8955	0.739920
$724$	0	0
$725$	4.59937	0.170816
$726$	0	0
$727$	−15.7809	−0.585283	−0.292641	−	0.956222i	$-0.594534\pi$
$727$	−15.7809	−0.585283	−0.292641	+	0.956222i	$0.594534\pi$
$728$	0	0
$729$	22.3904	0.829275
$730$	0	0
$731$	−10.5827	−0.391416
$732$	0	0
$733$	−43.6230	−1.61125	−0.805626	−	0.592424i	$-0.798171\pi$
$733$	−43.6230	−1.61125	−0.805626	+	0.592424i	$0.798171\pi$
$734$	0	0
$735$	−5.42999	−0.200288
$736$	0	0
$737$	0	0
$738$	0	0
$739$	39.7178	1.46104	0.730522	−	0.682889i	$-0.239277\pi$
$739$	39.7178	1.46104	0.730522	+	0.682889i	$0.239277\pi$
$740$	0	0
$741$	5.34654	0.196410
$742$	0	0
$743$	−26.1266	−0.958492	−0.479246	−	0.877681i	$-0.659090\pi$
$743$	−26.1266	−0.958492	−0.479246	+	0.877681i	$0.659090\pi$
$744$	0	0
$745$	9.85787	0.361164
$746$	0	0
$747$	2.44973	0.0896309
$748$	0	0
$749$	58.6614	2.14344
$750$	0	0
$751$	1.97638	0.0721192	0.0360596	−	0.999350i	$-0.488519\pi$
$751$	1.97638	0.0721192	0.0360596	+	0.999350i	$0.488519\pi$
$752$	0	0
$753$	23.2350	0.846732
$754$	0	0
$755$	3.74880	0.136433
$756$	0	0
$757$	23.6416	0.859269	0.429635	−	0.903003i	$-0.358642\pi$
$757$	23.6416	0.859269	0.429635	+	0.903003i	$0.358642\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−11.5673	−0.419315	−0.209657	−	0.977775i	$-0.567235\pi$
$761$	−11.5673	−0.419315	−0.209657	+	0.977775i	$0.567235\pi$
$762$	0	0
$763$	35.6523	1.29070
$764$	0	0
$765$	2.75816	0.0997216
$766$	0	0
$767$	−0.288070	−0.0104016
$768$	0	0
$769$	−12.4941	−0.450550	−0.225275	−	0.974295i	$-0.572328\pi$
$769$	−12.4941	−0.450550	−0.225275	+	0.974295i	$0.572328\pi$
$770$	0	0
$771$	−13.7848	−0.496448
$772$	0	0
$773$	−40.9253	−1.47198	−0.735990	−	0.676993i	$-0.763283\pi$
$773$	−40.9253	−1.47198	−0.735990	+	0.676993i	$0.763283\pi$
$774$	0	0
$775$	−2.67501	−0.0960892
$776$	0	0
$777$	11.7890	0.422928
$778$	0	0
$779$	−0.758581	−0.0271790
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−43.9823	−1.57180
$784$	0	0
$785$	−37.9788	−1.35552
$786$	0	0
$787$	−35.1648	−1.25349	−0.626745	−	0.779225i	$-0.715613\pi$
$787$	−35.1648	−1.25349	−0.626745	+	0.779225i	$0.715613\pi$
$788$	0	0
$789$	−10.0904	−0.359228
$790$	0	0
$791$	−18.8498	−0.670220
$792$	0	0
$793$	60.4256	2.14578
$794$	0	0
$795$	30.2847	1.07409
$796$	0	0
$797$	47.1347	1.66960	0.834799	−	0.550556i	$-0.185584\pi$
$797$	47.1347	1.66960	0.834799	+	0.550556i	$0.185584\pi$
$798$	0	0
$799$	−20.7835	−0.735267
$800$	0	0
$801$	−1.59522	−0.0563642
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−10.5227	−0.370876
$806$	0	0
$807$	9.19514	0.323684
$808$	0	0
$809$	−44.1948	−1.55381	−0.776903	−	0.629621i	$-0.783210\pi$
$809$	−44.1948	−1.55381	−0.776903	+	0.629621i	$0.783210\pi$
$810$	0	0
$811$	30.7202	1.07873	0.539365	−	0.842072i	$-0.318664\pi$
$811$	30.7202	1.07873	0.539365	+	0.842072i	$0.318664\pi$
$812$	0	0
$813$	10.4831	0.367657
$814$	0	0
$815$	−28.1305	−0.985370
$816$	0	0
$817$	−1.96660	−0.0688025
$818$	0	0
$819$	−5.79190	−0.202385
$820$	0	0
$821$	20.8876	0.728982	0.364491	−	0.931207i	$-0.381243\pi$
$821$	20.8876	0.728982	0.364491	+	0.931207i	$0.381243\pi$
$822$	0	0
$823$	19.0556	0.664235	0.332118	−	0.943238i	$-0.392237\pi$
$823$	19.0556	0.664235	0.332118	+	0.943238i	$0.392237\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	5.11423	0.177839	0.0889195	−	0.996039i	$-0.471659\pi$
$827$	5.11423	0.177839	0.0889195	+	0.996039i	$0.471659\pi$
$828$	0	0
$829$	−43.0999	−1.49692	−0.748461	−	0.663179i	$-0.769207\pi$
$829$	−43.0999	−1.49692	−0.748461	+	0.663179i	$0.769207\pi$
$830$	0	0
$831$	59.0585	2.04872
$832$	0	0
$833$	4.41330	0.152912
$834$	0	0
$835$	−47.1733	−1.63250
$836$	0	0
$837$	25.5803	0.884184
$838$	0	0
$839$	50.1109	1.73002	0.865010	−	0.501754i	$-0.167312\pi$
$839$	50.1109	1.73002	0.865010	+	0.501754i	$0.167312\pi$
$840$	0	0
$841$	55.5067	1.91402
$842$	0	0
$843$	−15.5258	−0.534737
$844$	0	0
$845$	23.2707	0.800537
$846$	0	0
$847$	−31.8554	−1.09457
$848$	0	0
$849$	−13.2637	−0.455211
$850$	0	0
$851$	3.77702	0.129475
$852$	0	0
$853$	−10.7455	−0.367918	−0.183959	−	0.982934i	$-0.558891\pi$
$853$	−10.7455	−0.367918	−0.183959	+	0.982934i	$0.558891\pi$
$854$	0	0
$855$	0.512552	0.0175289
$856$	0	0
$857$	29.2443	0.998968	0.499484	−	0.866323i	$-0.333523\pi$
$857$	29.2443	0.998968	0.499484	+	0.866323i	$0.333523\pi$
$858$	0	0
$859$	54.7763	1.86894	0.934472	−	0.356037i	$-0.115872\pi$
$859$	54.7763	1.86894	0.934472	+	0.356037i	$0.115872\pi$
$860$	0	0
$861$	6.85678	0.233679
$862$	0	0
$863$	20.1059	0.684413	0.342206	−	0.939625i	$-0.388826\pi$
$863$	20.1059	0.684413	0.342206	+	0.939625i	$0.388826\pi$
$864$	0	0
$865$	−47.8636	−1.62741
$866$	0	0
$867$	12.6808	0.430661
$868$	0	0
$869$	0	0
$870$	0	0
$871$	38.3477	1.29936
$872$	0	0
$873$	4.65750	0.157633
$874$	0	0
$875$	−33.7886	−1.14226
$876$	0	0
$877$	31.4559	1.06219	0.531095	−	0.847312i	$-0.321781\pi$
$877$	31.4559	1.06219	0.531095	+	0.847312i	$0.321781\pi$
$878$	0	0
$879$	−23.9429	−0.807573
$880$	0	0
$881$	3.07854	0.103719	0.0518593	−	0.998654i	$-0.483485\pi$
$881$	3.07854	0.103719	0.0518593	+	0.998654i	$0.483485\pi$
$882$	0	0
$883$	21.0968	0.709964	0.354982	−	0.934873i	$-0.384487\pi$
$883$	21.0968	0.709964	0.354982	+	0.934873i	$0.384487\pi$
$884$	0	0
$885$	−0.230427	−0.00774571
$886$	0	0
$887$	−20.9132	−0.702195	−0.351097	−	0.936339i	$-0.614191\pi$
$887$	−20.9132	−0.702195	−0.351097	+	0.936339i	$0.614191\pi$
$888$	0	0
$889$	20.2735	0.679952
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−3.86221	−0.129244
$894$	0	0
$895$	−12.5859	−0.420699
$896$	0	0
$897$	−15.4833	−0.516972
$898$	0	0
$899$	−49.1494	−1.63922
$900$	0	0
$901$	−24.6143	−0.820022
$902$	0	0
$903$	17.7760	0.591548
$904$	0	0
$905$	42.0128	1.39655
$906$	0	0
$907$	11.7638	0.390609	0.195305	−	0.980743i	$-0.437430\pi$
$907$	11.7638	0.390609	0.195305	+	0.980743i	$0.437430\pi$
$908$	0	0
$909$	5.83339	0.193481
$910$	0	0
$911$	9.13553	0.302674	0.151337	−	0.988482i	$-0.451642\pi$
$911$	9.13553	0.302674	0.151337	+	0.988482i	$0.451642\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	48.3344	1.59789
$916$	0	0
$917$	33.4982	1.10621
$918$	0	0
$919$	−41.1603	−1.35775	−0.678877	−	0.734252i	$-0.737533\pi$
$919$	−41.1603	−1.35775	−0.678877	+	0.734252i	$0.737533\pi$
$920$	0	0
$921$	−52.4692	−1.72892
$922$	0	0
$923$	−4.89595	−0.161152
$924$	0	0
$925$	1.10321	0.0362732
$926$	0	0
$927$	0.830804	0.0272872
$928$	0	0
$929$	7.46300	0.244853	0.122426	−	0.992478i	$-0.460932\pi$
$929$	7.46300	0.244853	0.122426	+	0.992478i	$0.460932\pi$
$930$	0	0
$931$	0.820127	0.0268786
$932$	0	0
$933$	−1.13233	−0.0370709
$934$	0	0
$935$	0	0
$936$	0	0
$937$	29.7488	0.971852	0.485926	−	0.874000i	$-0.338482\pi$
$937$	29.7488	0.971852	0.485926	+	0.874000i	$0.338482\pi$
$938$	0	0
$939$	−5.07751	−0.165698
$940$	0	0
$941$	22.8130	0.743684	0.371842	−	0.928296i	$-0.378726\pi$
$941$	22.8130	0.743684	0.371842	+	0.928296i	$0.378726\pi$
$942$	0	0
$943$	2.19681	0.0715380
$944$	0	0
$945$	29.3910	0.956089
$946$	0	0
$947$	30.8035	1.00098	0.500490	−	0.865742i	$-0.333153\pi$
$947$	30.8035	1.00098	0.500490	+	0.865742i	$0.333153\pi$
$948$	0	0
$949$	−8.49311	−0.275698
$950$	0	0
$951$	−18.4981	−0.599843
$952$	0	0
$953$	−3.99894	−0.129538	−0.0647692	−	0.997900i	$-0.520631\pi$
$953$	−3.99894	−0.129538	−0.0647692	+	0.997900i	$0.520631\pi$
$954$	0	0
$955$	−55.4570	−1.79455
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−17.9075	−0.578264
$960$	0	0
$961$	−2.41453	−0.0778881
$962$	0	0
$963$	8.27474	0.266650
$964$	0	0
$965$	7.26629	0.233910
$966$	0	0
$967$	19.9529	0.641643	0.320821	−	0.947140i	$-0.396041\pi$
$967$	19.9529	0.641643	0.320821	+	0.947140i	$0.396041\pi$
$968$	0	0
$969$	−3.47594	−0.111663
$970$	0	0
$971$	−26.0885	−0.837219	−0.418609	−	0.908166i	$-0.637482\pi$
$971$	−26.0885	−0.837219	−0.418609	+	0.908166i	$0.637482\pi$
$972$	0	0
$973$	31.8392	1.02072
$974$	0	0
$975$	−4.52242	−0.144833
$976$	0	0
$977$	37.7739	1.20849	0.604246	−	0.796798i	$-0.293474\pi$
$977$	37.7739	1.20849	0.604246	+	0.796798i	$0.293474\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	5.02909	0.160566
$982$	0	0
$983$	45.3328	1.44589	0.722946	−	0.690904i	$-0.242787\pi$
$983$	45.3328	1.44589	0.722946	+	0.690904i	$0.242787\pi$
$984$	0	0
$985$	−40.7212	−1.29749
$986$	0	0
$987$	34.9104	1.11121
$988$	0	0
$989$	5.69516	0.181096
$990$	0	0
$991$	−51.6163	−1.63965	−0.819823	−	0.572617i	$-0.805928\pi$
$991$	−51.6163	−1.63965	−0.819823	+	0.572617i	$0.805928\pi$
$992$	0	0
$993$	61.2231	1.94286
$994$	0	0
$995$	−28.5583	−0.905359
$996$	0	0
$997$	−60.2601	−1.90846	−0.954228	−	0.299080i	$-0.903320\pi$
$997$	−60.2601	−1.90846	−0.954228	+	0.299080i	$0.903320\pi$
$998$	0	0
$999$	−10.5496	−0.333775

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4544.2.a.bf.1.2		5
4.3	odd	2		4544.2.a.be.1.4		5
8.3	odd	2		568.2.a.e.1.2	✓	5
8.5	even	2		1136.2.a.n.1.4		5
24.11	even	2		5112.2.a.n.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
568.2.a.e.1.2	✓	5	8.3	odd	2
1136.2.a.n.1.4		5	8.5	even	2
4544.2.a.be.1.4		5	4.3	odd	2
4544.2.a.bf.1.2		5	1.1	even	1	trivial
5112.2.a.n.1.5		5	24.11	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:	\( N \)	\(=\)	\( 4544 = 2^{6} \cdot 71 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4544.a (trivial)

\(f(q)\)	\(=\)	\(q-1.84621 q^{3} +2.12124 q^{5} +2.89595 q^{7} +0.408501 q^{9} +O(q^{10})\)	\(q-1.84621 q^{3} +2.12124 q^{5} +2.89595 q^{7} +0.408501 q^{9} -4.89595 q^{13} -3.91627 q^{15} +3.18300 q^{17} +0.591499 q^{19} -5.34654 q^{21} -1.71295 q^{23} -0.500325 q^{25} +4.78446 q^{27} -9.19275 q^{29} +5.34654 q^{31} +6.14301 q^{35} -2.20498 q^{37} +9.03896 q^{39} -1.28247 q^{41} -3.32477 q^{43} +0.866530 q^{45} -6.52954 q^{47} +1.38652 q^{49} -5.87649 q^{51} -7.73306 q^{53} -1.09203 q^{57} +0.0588384 q^{59} -12.3420 q^{61} +1.18300 q^{63} -10.3855 q^{65} -7.83253 q^{67} +3.16247 q^{69} +1.00000 q^{71} +1.73472 q^{73} +0.923707 q^{75} -9.54673 q^{79} -10.0586 q^{81} +5.99688 q^{83} +6.75191 q^{85} +16.9718 q^{87} -3.90505 q^{89} -14.1784 q^{91} -9.87085 q^{93} +1.25471 q^{95} +11.4014 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{3} - 7 q^{5} + q^{7} + 7 q^{9}+O(q^{10}) \) 5 * q + 2 * q^3 - 7 * q^5 + q^7 + 7 * q^9	\( 5 q + 2 q^{3} - 7 q^{5} + q^{7} + 7 q^{9} - 11 q^{13} - 6 q^{15} + 6 q^{17} - 2 q^{19} - 5 q^{21} - 5 q^{23} + 8 q^{25} + 5 q^{27} - 13 q^{29} + 5 q^{31} - 7 q^{37} + q^{39} + 8 q^{41} - 8 q^{43} - 7 q^{45} - q^{47} + 6 q^{49} - 14 q^{51} - 16 q^{53} - 9 q^{57} - 4 q^{59} - 22 q^{61} - 4 q^{63} + 14 q^{65} - 12 q^{67} - 13 q^{69} + 5 q^{71} - 8 q^{73} - 17 q^{75} + 15 q^{79} - 27 q^{81} - q^{83} - 14 q^{85} + 17 q^{87} - 4 q^{89} - 43 q^{91} + q^{93} - 8 q^{97}+O(q^{100}) \) 5 * q + 2 * q^3 - 7 * q^5 + q^7 + 7 * q^9 - 11 * q^13 - 6 * q^15 + 6 * q^17 - 2 * q^19 - 5 * q^21 - 5 * q^23 + 8 * q^25 + 5 * q^27 - 13 * q^29 + 5 * q^31 - 7 * q^37 + q^39 + 8 * q^41 - 8 * q^43 - 7 * q^45 - q^47 + 6 * q^49 - 14 * q^51 - 16 * q^53 - 9 * q^57 - 4 * q^59 - 22 * q^61 - 4 * q^63 + 14 * q^65 - 12 * q^67 - 13 * q^69 + 5 * q^71 - 8 * q^73 - 17 * q^75 + 15 * q^79 - 27 * q^81 - q^83 - 14 * q^85 + 17 * q^87 - 4 * q^89 - 43 * q^91 + q^93 - 8 * q^97

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