LMFDB - Embedded newform 1444.2.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1444,2,Mod(1,1444)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1444, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1444.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1444 = 2^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1444.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:	$11.5303980519$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.20319417.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 9x^{4} + 19x^{3} + 27x^{2} - 27x - 27 $ x^6 - 3x^5 - 9x^4 + 19x^3 + 27x^2 - 27*x - 27
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 76)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.812576$ of defining polynomial
Character	$\chi$	$=$	1444.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.69196 q^{3} -1.28220 q^{5} -3.34467 q^{7} +4.24666 q^{9} +O(q^{10})$	$q-2.69196 q^{3} -1.28220 q^{5} -3.34467 q^{7} +4.24666 q^{9} -5.65641 q^{11} -0.442077 q^{13} +3.45165 q^{15} -4.75211 q^{17} +9.00371 q^{21} -3.67370 q^{23} -3.35595 q^{25} -3.35595 q^{27} +3.01826 q^{29} -10.7825 q^{31} +15.2268 q^{33} +4.28855 q^{35} -1.81198 q^{37} +1.19006 q^{39} +4.03555 q^{41} -2.69288 q^{43} -5.44508 q^{45} +6.06741 q^{47} +4.18678 q^{49} +12.7925 q^{51} +1.63213 q^{53} +7.25268 q^{55} +8.55249 q^{59} +2.59561 q^{61} -14.2036 q^{63} +0.566834 q^{65} -4.03325 q^{67} +9.88947 q^{69} -11.0941 q^{71} -7.36464 q^{73} +9.03409 q^{75} +18.9188 q^{77} +10.1592 q^{79} -3.70588 q^{81} +4.55683 q^{83} +6.09318 q^{85} -8.12503 q^{87} +12.9569 q^{89} +1.47860 q^{91} +29.0260 q^{93} -11.9018 q^{97} -24.0208 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{3} - 3 q^{5} - 3 q^{7} + 9 q^{9}+O(q^{10}) $ 6 * q + 3 * q^3 - 3 * q^5 - 3 * q^7 + 9 * q^9	$ 6 q + 3 q^{3} - 3 q^{5} - 3 q^{7} + 9 q^{9} - 3 q^{11} + 12 q^{13} - 6 q^{17} + 21 q^{21} + 9 q^{25} + 9 q^{27} + 21 q^{29} - 6 q^{31} + 9 q^{33} + 3 q^{35} - 6 q^{37} + 30 q^{39} + 36 q^{41} - 18 q^{43} - 24 q^{45} + 30 q^{47} - 9 q^{49} + 24 q^{51} + 18 q^{53} - 15 q^{55} + 21 q^{59} + 9 q^{61} + 6 q^{63} + 33 q^{65} + 18 q^{67} + 33 q^{69} - 12 q^{71} - 24 q^{73} + 21 q^{75} + 12 q^{77} + 9 q^{79} - 6 q^{81} - 3 q^{83} - 12 q^{85} + 18 q^{87} + 45 q^{89} + 9 q^{91} + 15 q^{97} - 33 q^{99}+O(q^{100}) $ 6 * q + 3 * q^3 - 3 * q^5 - 3 * q^7 + 9 * q^9 - 3 * q^11 + 12 * q^13 - 6 * q^17 + 21 * q^21 + 9 * q^25 + 9 * q^27 + 21 * q^29 - 6 * q^31 + 9 * q^33 + 3 * q^35 - 6 * q^37 + 30 * q^39 + 36 * q^41 - 18 * q^43 - 24 * q^45 + 30 * q^47 - 9 * q^49 + 24 * q^51 + 18 * q^53 - 15 * q^55 + 21 * q^59 + 9 * q^61 + 6 * q^63 + 33 * q^65 + 18 * q^67 + 33 * q^69 - 12 * q^71 - 24 * q^73 + 21 * q^75 + 12 * q^77 + 9 * q^79 - 6 * q^81 - 3 * q^83 - 12 * q^85 + 18 * q^87 + 45 * q^89 + 9 * q^91 + 15 * q^97 - 33 * 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
$3$	−2.69196	−1.55420	−0.777102	−	0.629374i	$-0.783311\pi$
$3$	−2.69196	−1.55420	−0.777102	+	0.629374i	$0.783311\pi$
$4$	0	0
$5$	−1.28220	−0.573419	−0.286710	−	0.958018i	$-0.592562\pi$
$5$	−1.28220	−0.573419	−0.286710	+	0.958018i	$0.592562\pi$
$6$	0	0
$7$	−3.34467	−1.26416	−0.632082	−	0.774901i	$-0.717800\pi$
$7$	−3.34467	−1.26416	−0.632082	+	0.774901i	$0.717800\pi$
$8$	0	0
$9$	4.24666	1.41555
$10$	0	0
$11$	−5.65641	−1.70547	−0.852736	−	0.522342i	$-0.825059\pi$
$11$	−5.65641	−1.70547	−0.852736	+	0.522342i	$0.825059\pi$
$12$	0	0
$13$	−0.442077	−0.122610	−0.0613051	−	0.998119i	$-0.519526\pi$
$13$	−0.442077	−0.122610	−0.0613051	+	0.998119i	$0.519526\pi$
$14$	0	0
$15$	3.45165	0.891211
$16$	0	0
$17$	−4.75211	−1.15256	−0.576278	−	0.817254i	$-0.695495\pi$
$17$	−4.75211	−1.15256	−0.576278	+	0.817254i	$0.695495\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	9.00371	1.96477
$22$	0	0
$23$	−3.67370	−0.766020	−0.383010	−	0.923744i	$-0.625113\pi$
$23$	−3.67370	−0.766020	−0.383010	+	0.923744i	$0.625113\pi$
$24$	0	0
$25$	−3.35595	−0.671190
$26$	0	0
$27$	−3.35595	−0.645853
$28$	0	0
$29$	3.01826	0.560476	0.280238	−	0.959931i	$-0.409587\pi$
$29$	3.01826	0.560476	0.280238	+	0.959931i	$0.409587\pi$
$30$	0	0
$31$	−10.7825	−1.93659	−0.968293	−	0.249817i	$-0.919629\pi$
$31$	−10.7825	−1.93659	−0.968293	+	0.249817i	$0.919629\pi$
$32$	0	0
$33$	15.2268	2.65065
$34$	0	0
$35$	4.28855	0.724896
$36$	0	0
$37$	−1.81198	−0.297888	−0.148944	−	0.988846i	$-0.547587\pi$
$37$	−1.81198	−0.297888	−0.148944	+	0.988846i	$0.547587\pi$
$38$	0	0
$39$	1.19006	0.190561
$40$	0	0
$41$	4.03555	0.630247	0.315123	−	0.949051i	$-0.397954\pi$
$41$	4.03555	0.630247	0.315123	+	0.949051i	$0.397954\pi$
$42$	0	0
$43$	−2.69288	−0.410660	−0.205330	−	0.978693i	$-0.565827\pi$
$43$	−2.69288	−0.410660	−0.205330	+	0.978693i	$0.565827\pi$
$44$	0	0
$45$	−5.44508	−0.811705
$46$	0	0
$47$	6.06741	0.885022	0.442511	−	0.896763i	$-0.354088\pi$
$47$	6.06741	0.885022	0.442511	+	0.896763i	$0.354088\pi$
$48$	0	0
$49$	4.18678	0.598112
$50$	0	0
$51$	12.7925	1.79131
$52$	0	0
$53$	1.63213	0.224191	0.112095	−	0.993697i	$-0.464244\pi$
$53$	1.63213	0.224191	0.112095	+	0.993697i	$0.464244\pi$
$54$	0	0
$55$	7.25268	0.977951
$56$	0	0
$57$	0	0
$58$	0	0
$59$	8.55249	1.11344	0.556720	−	0.830700i	$-0.312060\pi$
$59$	8.55249	1.11344	0.556720	+	0.830700i	$0.312060\pi$
$60$	0	0
$61$	2.59561	0.332334	0.166167	−	0.986098i	$-0.446861\pi$
$61$	2.59561	0.332334	0.166167	+	0.986098i	$0.446861\pi$
$62$	0	0
$63$	−14.2036	−1.78949
$64$	0	0
$65$	0.566834	0.0703071
$66$	0	0
$67$	−4.03325	−0.492740	−0.246370	−	0.969176i	$-0.579238\pi$
$67$	−4.03325	−0.492740	−0.246370	+	0.969176i	$0.579238\pi$
$68$	0	0
$69$	9.88947	1.19055
$70$	0	0
$71$	−11.0941	−1.31663	−0.658316	−	0.752742i	$-0.728731\pi$
$71$	−11.0941	−1.31663	−0.658316	+	0.752742i	$0.728731\pi$
$72$	0	0
$73$	−7.36464	−0.861966	−0.430983	−	0.902360i	$-0.641833\pi$
$73$	−7.36464	−0.861966	−0.430983	+	0.902360i	$0.641833\pi$
$74$	0	0
$75$	9.03409	1.04317
$76$	0	0
$77$	18.9188	2.15600
$78$	0	0
$79$	10.1592	1.14300	0.571502	−	0.820601i	$-0.306361\pi$
$79$	10.1592	1.14300	0.571502	+	0.820601i	$0.306361\pi$
$80$	0	0
$81$	−3.70588	−0.411764
$82$	0	0
$83$	4.55683	0.500177	0.250089	−	0.968223i	$-0.419540\pi$
$83$	4.55683	0.500177	0.250089	+	0.968223i	$0.419540\pi$
$84$	0	0
$85$	6.09318	0.660898
$86$	0	0
$87$	−8.12503	−0.871095
$88$	0	0
$89$	12.9569	1.37343	0.686713	−	0.726929i	$-0.259053\pi$
$89$	12.9569	1.37343	0.686713	+	0.726929i	$0.259053\pi$
$90$	0	0
$91$	1.47860	0.154999
$92$	0	0
$93$	29.0260	3.00985
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−11.9018	−1.20845	−0.604224	−	0.796814i	$-0.706517\pi$
$97$	−11.9018	−1.20845	−0.604224	+	0.796814i	$0.706517\pi$
$98$	0	0
$99$	−24.0208	−2.41419
$100$	0	0
$101$	10.1962	1.01456	0.507279	−	0.861782i	$-0.330651\pi$
$101$	10.1962	1.01456	0.507279	+	0.861782i	$0.330651\pi$
$102$	0	0
$103$	−4.52677	−0.446036	−0.223018	−	0.974814i	$-0.571591\pi$
$103$	−4.52677	−0.446036	−0.223018	+	0.974814i	$0.571591\pi$
$104$	0	0
$105$	−11.5446	−1.12664
$106$	0	0
$107$	1.27883	0.123629	0.0618146	−	0.998088i	$-0.480311\pi$
$107$	1.27883	0.123629	0.0618146	+	0.998088i	$0.480311\pi$
$108$	0	0
$109$	6.42336	0.615246	0.307623	−	0.951508i	$-0.400466\pi$
$109$	6.42336	0.615246	0.307623	+	0.951508i	$0.400466\pi$
$110$	0	0
$111$	4.87778	0.462978
$112$	0	0
$113$	−1.77037	−0.166542	−0.0832710	−	0.996527i	$-0.526537\pi$
$113$	−1.77037	−0.166542	−0.0832710	+	0.996527i	$0.526537\pi$
$114$	0	0
$115$	4.71044	0.439251
$116$	0	0
$117$	−1.87735	−0.173561
$118$	0	0
$119$	15.8942	1.45702
$120$	0	0
$121$	20.9950	1.90864
$122$	0	0
$123$	−10.8635	−0.979532
$124$	0	0
$125$	10.7140	0.958293
$126$	0	0
$127$	13.4819	1.19633	0.598165	−	0.801373i	$-0.295897\pi$
$127$	13.4819	1.19633	0.598165	+	0.801373i	$0.295897\pi$
$128$	0	0
$129$	7.24912	0.638249
$130$	0	0
$131$	−10.1327	−0.885298	−0.442649	−	0.896695i	$-0.645961\pi$
$131$	−10.1327	−0.885298	−0.442649	+	0.896695i	$0.645961\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	4.30302	0.370345
$136$	0	0
$137$	−3.49669	−0.298742	−0.149371	−	0.988781i	$-0.547725\pi$
$137$	−3.49669	−0.298742	−0.149371	+	0.988781i	$0.547725\pi$
$138$	0	0
$139$	10.6572	0.903933	0.451967	−	0.892035i	$-0.350723\pi$
$139$	10.6572	0.903933	0.451967	+	0.892035i	$0.350723\pi$
$140$	0	0
$141$	−16.3332	−1.37551
$142$	0	0
$143$	2.50057	0.209108
$144$	0	0
$145$	−3.87002	−0.321388
$146$	0	0
$147$	−11.2707	−0.929589
$148$	0	0
$149$	−8.42125	−0.689896	−0.344948	−	0.938622i	$-0.612103\pi$
$149$	−8.42125	−0.689896	−0.344948	+	0.938622i	$0.612103\pi$
$150$	0	0
$151$	−11.3794	−0.926039	−0.463020	−	0.886348i	$-0.653234\pi$
$151$	−11.3794	−0.926039	−0.463020	+	0.886348i	$0.653234\pi$
$152$	0	0
$153$	−20.1806	−1.63150
$154$	0	0
$155$	13.8253	1.11048
$156$	0	0
$157$	−4.67602	−0.373187	−0.186593	−	0.982437i	$-0.559745\pi$
$157$	−4.67602	−0.373187	−0.186593	+	0.982437i	$0.559745\pi$
$158$	0	0
$159$	−4.39364	−0.348438
$160$	0	0
$161$	12.2873	0.968376
$162$	0	0
$163$	−1.82998	−0.143335	−0.0716676	−	0.997429i	$-0.522832\pi$
$163$	−1.82998	−0.143335	−0.0716676	+	0.997429i	$0.522832\pi$
$164$	0	0
$165$	−19.5239	−1.51994
$166$	0	0
$167$	−19.9744	−1.54566	−0.772832	−	0.634610i	$-0.781161\pi$
$167$	−19.9744	−1.54566	−0.772832	+	0.634610i	$0.781161\pi$
$168$	0	0
$169$	−12.8046	−0.984967
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−4.67339	−0.355311	−0.177656	−	0.984093i	$-0.556851\pi$
$173$	−4.67339	−0.355311	−0.177656	+	0.984093i	$0.556851\pi$
$174$	0	0
$175$	11.2245	0.848495
$176$	0	0
$177$	−23.0230	−1.73051
$178$	0	0
$179$	7.64250	0.571227	0.285613	−	0.958345i	$-0.407803\pi$
$179$	7.64250	0.571227	0.285613	+	0.958345i	$0.407803\pi$
$180$	0	0
$181$	6.09446	0.452998	0.226499	−	0.974011i	$-0.427272\pi$
$181$	6.09446	0.452998	0.226499	+	0.974011i	$0.427272\pi$
$182$	0	0
$183$	−6.98728	−0.516515
$184$	0	0
$185$	2.32333	0.170815
$186$	0	0
$187$	26.8799	1.96565
$188$	0	0
$189$	11.2245	0.816465
$190$	0	0
$191$	7.93272	0.573992	0.286996	−	0.957932i	$-0.407343\pi$
$191$	7.93272	0.573992	0.286996	+	0.957932i	$0.407343\pi$
$192$	0	0
$193$	−6.84941	−0.493032	−0.246516	−	0.969139i	$-0.579286\pi$
$193$	−6.84941	−0.493032	−0.246516	+	0.969139i	$0.579286\pi$
$194$	0	0
$195$	−1.52589	−0.109272
$196$	0	0
$197$	−25.5509	−1.82043	−0.910213	−	0.414141i	$-0.864082\pi$
$197$	−25.5509	−1.82043	−0.910213	+	0.414141i	$0.864082\pi$
$198$	0	0
$199$	−7.62715	−0.540675	−0.270337	−	0.962766i	$-0.587135\pi$
$199$	−7.62715	−0.540675	−0.270337	+	0.962766i	$0.587135\pi$
$200$	0	0
$201$	10.8574	0.765819
$202$	0	0
$203$	−10.0951	−0.708534
$204$	0	0
$205$	−5.17440	−0.361396
$206$	0	0
$207$	−15.6010	−1.08434
$208$	0	0
$209$	0	0
$210$	0	0
$211$	11.4709	0.789690	0.394845	−	0.918748i	$-0.370798\pi$
$211$	11.4709	0.789690	0.394845	+	0.918748i	$0.370798\pi$
$212$	0	0
$213$	29.8650	2.04632
$214$	0	0
$215$	3.45282	0.235480
$216$	0	0
$217$	36.0637	2.44816
$218$	0	0
$219$	19.8253	1.33967
$220$	0	0
$221$	2.10080	0.141315
$222$	0	0
$223$	−2.79878	−0.187420	−0.0937102	−	0.995600i	$-0.529873\pi$
$223$	−2.79878	−0.187420	−0.0937102	+	0.995600i	$0.529873\pi$
$224$	0	0
$225$	−14.2516	−0.950105
$226$	0	0
$227$	−6.05837	−0.402108	−0.201054	−	0.979580i	$-0.564437\pi$
$227$	−6.05837	−0.402108	−0.201054	+	0.979580i	$0.564437\pi$
$228$	0	0
$229$	−8.97380	−0.593005	−0.296503	−	0.955032i	$-0.595820\pi$
$229$	−8.97380	−0.593005	−0.296503	+	0.955032i	$0.595820\pi$
$230$	0	0
$231$	−50.9287	−3.35086
$232$	0	0
$233$	−7.34058	−0.480897	−0.240449	−	0.970662i	$-0.577295\pi$
$233$	−7.34058	−0.480897	−0.240449	+	0.970662i	$0.577295\pi$
$234$	0	0
$235$	−7.77966	−0.507489
$236$	0	0
$237$	−27.3483	−1.77646
$238$	0	0
$239$	−19.1710	−1.24007	−0.620036	−	0.784574i	$-0.712882\pi$
$239$	−19.1710	−1.24007	−0.620036	+	0.784574i	$0.712882\pi$
$240$	0	0
$241$	−12.1273	−0.781191	−0.390596	−	0.920562i	$-0.627731\pi$
$241$	−12.1273	−0.781191	−0.390596	+	0.920562i	$0.627731\pi$
$242$	0	0
$243$	20.0439	1.28582
$244$	0	0
$245$	−5.36832	−0.342969
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−12.2668	−0.777378
$250$	0	0
$251$	14.6132	0.922379	0.461189	−	0.887302i	$-0.347423\pi$
$251$	14.6132	0.922379	0.461189	+	0.887302i	$0.347423\pi$
$252$	0	0
$253$	20.7800	1.30643
$254$	0	0
$255$	−16.4026	−1.02717
$256$	0	0
$257$	2.59119	0.161634	0.0808171	−	0.996729i	$-0.474247\pi$
$257$	2.59119	0.161634	0.0808171	+	0.996729i	$0.474247\pi$
$258$	0	0
$259$	6.06047	0.376579
$260$	0	0
$261$	12.8175	0.793383
$262$	0	0
$263$	−23.0162	−1.41924	−0.709619	−	0.704585i	$-0.751133\pi$
$263$	−23.0162	−1.41924	−0.709619	+	0.704585i	$0.751133\pi$
$264$	0	0
$265$	−2.09273	−0.128555
$266$	0	0
$267$	−34.8794	−2.13459
$268$	0	0
$269$	26.1828	1.59639	0.798195	−	0.602399i	$-0.205788\pi$
$269$	26.1828	1.59639	0.798195	+	0.602399i	$0.205788\pi$
$270$	0	0
$271$	−14.7762	−0.897588	−0.448794	−	0.893635i	$-0.648146\pi$
$271$	−14.7762	−0.897588	−0.448794	+	0.893635i	$0.648146\pi$
$272$	0	0
$273$	−3.98034	−0.240901
$274$	0	0
$275$	18.9826	1.14470
$276$	0	0
$277$	−14.7470	−0.886061	−0.443030	−	0.896507i	$-0.646097\pi$
$277$	−14.7470	−0.886061	−0.443030	+	0.896507i	$0.646097\pi$
$278$	0	0
$279$	−45.7894	−2.74134
$280$	0	0
$281$	13.9406	0.831624	0.415812	−	0.909451i	$-0.363498\pi$
$281$	13.9406	0.831624	0.415812	+	0.909451i	$0.363498\pi$
$282$	0	0
$283$	−18.2889	−1.08716	−0.543580	−	0.839357i	$-0.682932\pi$
$283$	−18.2889	−1.08716	−0.543580	+	0.839357i	$0.682932\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−13.4976	−0.796736
$288$	0	0
$289$	5.58253	0.328384
$290$	0	0
$291$	32.0393	1.87818
$292$	0	0
$293$	24.7475	1.44576	0.722882	−	0.690972i	$-0.242817\pi$
$293$	24.7475	1.44576	0.722882	+	0.690972i	$0.242817\pi$
$294$	0	0
$295$	−10.9660	−0.638468
$296$	0	0
$297$	18.9826	1.10148
$298$	0	0
$299$	1.62406	0.0939219
$300$	0	0
$301$	9.00677	0.519141
$302$	0	0
$303$	−27.4477	−1.57683
$304$	0	0
$305$	−3.32810	−0.190567
$306$	0	0
$307$	−22.9330	−1.30886	−0.654429	−	0.756124i	$-0.727091\pi$
$307$	−22.9330	−1.30886	−0.654429	+	0.756124i	$0.727091\pi$
$308$	0	0
$309$	12.1859	0.693232
$310$	0	0
$311$	29.6666	1.68224	0.841119	−	0.540850i	$-0.181897\pi$
$311$	29.6666	1.68224	0.841119	+	0.540850i	$0.181897\pi$
$312$	0	0
$313$	−20.7248	−1.17143	−0.585717	−	0.810516i	$-0.699187\pi$
$313$	−20.7248	−1.17143	−0.585717	+	0.810516i	$0.699187\pi$
$314$	0	0
$315$	18.2120	1.02613
$316$	0	0
$317$	−32.4136	−1.82053	−0.910264	−	0.414028i	$-0.864121\pi$
$317$	−32.4136	−1.82053	−0.910264	+	0.414028i	$0.864121\pi$
$318$	0	0
$319$	−17.0725	−0.955877
$320$	0	0
$321$	−3.44256	−0.192145
$322$	0	0
$323$	0	0
$324$	0	0
$325$	1.48359	0.0822948
$326$	0	0
$327$	−17.2914	−0.956218
$328$	0	0
$329$	−20.2934	−1.11881
$330$	0	0
$331$	−21.2388	−1.16739	−0.583694	−	0.811974i	$-0.698393\pi$
$331$	−21.2388	−1.16739	−0.583694	+	0.811974i	$0.698393\pi$
$332$	0	0
$333$	−7.69486	−0.421675
$334$	0	0
$335$	5.17146	0.282547
$336$	0	0
$337$	−11.7668	−0.640980	−0.320490	−	0.947252i	$-0.603847\pi$
$337$	−11.7668	−0.640980	−0.320490	+	0.947252i	$0.603847\pi$
$338$	0	0
$339$	4.76575	0.258840
$340$	0	0
$341$	60.9900	3.30279
$342$	0	0
$343$	9.40926	0.508052
$344$	0	0
$345$	−12.6803	−0.682686
$346$	0	0
$347$	25.7460	1.38212	0.691059	−	0.722799i	$-0.257145\pi$
$347$	25.7460	1.38212	0.691059	+	0.722799i	$0.257145\pi$
$348$	0	0
$349$	10.1537	0.543517	0.271759	−	0.962365i	$-0.412395\pi$
$349$	10.1537	0.543517	0.271759	+	0.962365i	$0.412395\pi$
$350$	0	0
$351$	1.48359	0.0791882
$352$	0	0
$353$	15.2411	0.811201	0.405600	−	0.914050i	$-0.367062\pi$
$353$	15.2411	0.811201	0.405600	+	0.914050i	$0.367062\pi$
$354$	0	0
$355$	14.2250	0.754983
$356$	0	0
$357$	−42.7866	−2.26451
$358$	0	0
$359$	−18.7653	−0.990394	−0.495197	−	0.868781i	$-0.664904\pi$
$359$	−18.7653	−0.990394	−0.495197	+	0.868781i	$0.664904\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	−56.5178	−2.96641
$364$	0	0
$365$	9.44298	0.494268
$366$	0	0
$367$	12.1097	0.632119	0.316060	−	0.948739i	$-0.397640\pi$
$367$	12.1097	0.632119	0.316060	+	0.948739i	$0.397640\pi$
$368$	0	0
$369$	17.1376	0.892147
$370$	0	0
$371$	−5.45894	−0.283414
$372$	0	0
$373$	−33.1556	−1.71673	−0.858366	−	0.513038i	$-0.828520\pi$
$373$	−33.1556	−1.71673	−0.858366	+	0.513038i	$0.828520\pi$
$374$	0	0
$375$	−28.8418	−1.48938
$376$	0	0
$377$	−1.33430	−0.0687201
$378$	0	0
$379$	−19.5956	−1.00656	−0.503280	−	0.864123i	$-0.667874\pi$
$379$	−19.5956	−1.00656	−0.503280	+	0.864123i	$0.667874\pi$
$380$	0	0
$381$	−36.2929	−1.85934
$382$	0	0
$383$	25.0491	1.27995	0.639974	−	0.768396i	$-0.278945\pi$
$383$	25.0491	1.27995	0.639974	+	0.768396i	$0.278945\pi$
$384$	0	0
$385$	−24.2578	−1.23629
$386$	0	0
$387$	−11.4357	−0.581310
$388$	0	0
$389$	−16.5821	−0.840745	−0.420372	−	0.907352i	$-0.638101\pi$
$389$	−16.5821	−0.840745	−0.420372	+	0.907352i	$0.638101\pi$
$390$	0	0
$391$	17.4578	0.882881
$392$	0	0
$393$	27.2768	1.37594
$394$	0	0
$395$	−13.0262	−0.655420
$396$	0	0
$397$	29.8911	1.50019	0.750095	−	0.661330i	$-0.230008\pi$
$397$	29.8911	1.50019	0.750095	+	0.661330i	$0.230008\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	20.1885	1.00816	0.504082	−	0.863656i	$-0.331831\pi$
$401$	20.1885	1.00816	0.504082	+	0.863656i	$0.331831\pi$
$402$	0	0
$403$	4.76668	0.237445
$404$	0	0
$405$	4.75170	0.236114
$406$	0	0
$407$	10.2493	0.508039
$408$	0	0
$409$	−11.9018	−0.588508	−0.294254	−	0.955727i	$-0.595071\pi$
$409$	−11.9018	−0.588508	−0.294254	+	0.955727i	$0.595071\pi$
$410$	0	0
$411$	9.41295	0.464306
$412$	0	0
$413$	−28.6052	−1.40757
$414$	0	0
$415$	−5.84279	−0.286811
$416$	0	0
$417$	−28.6888	−1.40490
$418$	0	0
$419$	17.0337	0.832151	0.416076	−	0.909330i	$-0.363405\pi$
$419$	17.0337	0.832151	0.416076	+	0.909330i	$0.363405\pi$
$420$	0	0
$421$	5.47848	0.267005	0.133502	−	0.991049i	$-0.457378\pi$
$421$	5.47848	0.267005	0.133502	+	0.991049i	$0.457378\pi$
$422$	0	0
$423$	25.7662	1.25279
$424$	0	0
$425$	15.9478	0.773584
$426$	0	0
$427$	−8.68144	−0.420124
$428$	0	0
$429$	−6.73144	−0.324997
$430$	0	0
$431$	0.293535	0.0141391	0.00706953	−	0.999975i	$-0.497750\pi$
$431$	0.293535	0.0141391	0.00706953	+	0.999975i	$0.497750\pi$
$432$	0	0
$433$	21.7700	1.04620	0.523101	−	0.852271i	$-0.324775\pi$
$433$	21.7700	1.04620	0.523101	+	0.852271i	$0.324775\pi$
$434$	0	0
$435$	10.4180	0.499503
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−13.6860	−0.653199	−0.326599	−	0.945163i	$-0.605903\pi$
$439$	−13.6860	−0.653199	−0.326599	+	0.945163i	$0.605903\pi$
$440$	0	0
$441$	17.7798	0.846659
$442$	0	0
$443$	−9.50161	−0.451435	−0.225718	−	0.974193i	$-0.572473\pi$
$443$	−9.50161	−0.451435	−0.225718	+	0.974193i	$0.572473\pi$
$444$	0	0
$445$	−16.6134	−0.787549
$446$	0	0
$447$	22.6697	1.07224
$448$	0	0
$449$	38.4822	1.81609	0.908043	−	0.418877i	$-0.137576\pi$
$449$	38.4822	1.81609	0.908043	+	0.418877i	$0.137576\pi$
$450$	0	0
$451$	−22.8267	−1.07487
$452$	0	0
$453$	30.6328	1.43925
$454$	0	0
$455$	−1.89587	−0.0888797
$456$	0	0
$457$	12.3537	0.577880	0.288940	−	0.957347i	$-0.406697\pi$
$457$	12.3537	0.577880	0.288940	+	0.957347i	$0.406697\pi$
$458$	0	0
$459$	15.9478	0.744381
$460$	0	0
$461$	3.31110	0.154213	0.0771065	−	0.997023i	$-0.475432\pi$
$461$	3.31110	0.154213	0.0771065	+	0.997023i	$0.475432\pi$
$462$	0	0
$463$	−4.16441	−0.193536	−0.0967682	−	0.995307i	$-0.530851\pi$
$463$	−4.16441	−0.193536	−0.0967682	+	0.995307i	$0.530851\pi$
$464$	0	0
$465$	−37.2172	−1.72591
$466$	0	0
$467$	39.2187	1.81483	0.907413	−	0.420241i	$-0.138054\pi$
$467$	39.2187	1.81483	0.907413	+	0.420241i	$0.138054\pi$
$468$	0	0
$469$	13.4899	0.622905
$470$	0	0
$471$	12.5877	0.580009
$472$	0	0
$473$	15.2320	0.700369
$474$	0	0
$475$	0	0
$476$	0	0
$477$	6.93111	0.317354
$478$	0	0
$479$	−7.42103	−0.339075	−0.169538	−	0.985524i	$-0.554227\pi$
$479$	−7.42103	−0.339075	−0.169538	+	0.985524i	$0.554227\pi$
$480$	0	0
$481$	0.801035	0.0365241
$482$	0	0
$483$	−33.0770	−1.50505
$484$	0	0
$485$	15.2606	0.692948
$486$	0	0
$487$	32.6140	1.47788	0.738940	−	0.673771i	$-0.235327\pi$
$487$	32.6140	1.47788	0.738940	+	0.673771i	$0.235327\pi$
$488$	0	0
$489$	4.92624	0.222772
$490$	0	0
$491$	−24.1272	−1.08885	−0.544423	−	0.838811i	$-0.683251\pi$
$491$	−24.1272	−1.08885	−0.544423	+	0.838811i	$0.683251\pi$
$492$	0	0
$493$	−14.3431	−0.645980
$494$	0	0
$495$	30.7996	1.38434
$496$	0	0
$497$	37.1062	1.66444
$498$	0	0
$499$	17.1228	0.766521	0.383260	−	0.923640i	$-0.374801\pi$
$499$	17.1228	0.766521	0.383260	+	0.923640i	$0.374801\pi$
$500$	0	0
$501$	53.7703	2.40228
$502$	0	0
$503$	25.0649	1.11759	0.558794	−	0.829306i	$-0.311264\pi$
$503$	25.0649	1.11759	0.558794	+	0.829306i	$0.311264\pi$
$504$	0	0
$505$	−13.0736	−0.581767
$506$	0	0
$507$	34.4694	1.53084
$508$	0	0
$509$	−13.0588	−0.578822	−0.289411	−	0.957205i	$-0.593459\pi$
$509$	−13.0588	−0.578822	−0.289411	+	0.957205i	$0.593459\pi$
$510$	0	0
$511$	24.6323	1.08967
$512$	0	0
$513$	0	0
$514$	0	0
$515$	5.80425	0.255766
$516$	0	0
$517$	−34.3198	−1.50938
$518$	0	0
$519$	12.5806	0.552226
$520$	0	0
$521$	19.1979	0.841076	0.420538	−	0.907275i	$-0.361841\pi$
$521$	19.1979	0.841076	0.420538	+	0.907275i	$0.361841\pi$
$522$	0	0
$523$	42.2288	1.84654	0.923269	−	0.384154i	$-0.125507\pi$
$523$	42.2288	1.84654	0.923269	+	0.384154i	$0.125507\pi$
$524$	0	0
$525$	−30.2160	−1.31873
$526$	0	0
$527$	51.2394	2.23202
$528$	0	0
$529$	−9.50390	−0.413213
$530$	0	0
$531$	36.3195	1.57613
$532$	0	0
$533$	−1.78402	−0.0772747
$534$	0	0
$535$	−1.63972	−0.0708914
$536$	0	0
$537$	−20.5733	−0.887804
$538$	0	0
$539$	−23.6822	−1.02006
$540$	0	0
$541$	0.908031	0.0390393	0.0195197	−	0.999809i	$-0.493786\pi$
$541$	0.908031	0.0390393	0.0195197	+	0.999809i	$0.493786\pi$
$542$	0	0
$543$	−16.4060	−0.704051
$544$	0	0
$545$	−8.23606	−0.352794
$546$	0	0
$547$	34.6089	1.47977	0.739885	−	0.672734i	$-0.234880\pi$
$547$	34.6089	1.47977	0.739885	+	0.672734i	$0.234880\pi$
$548$	0	0
$549$	11.0227	0.470436
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−33.9792	−1.44494
$554$	0	0
$555$	−6.25431	−0.265481
$556$	0	0
$557$	7.26969	0.308027	0.154013	−	0.988069i	$-0.450780\pi$
$557$	7.26969	0.308027	0.154013	+	0.988069i	$0.450780\pi$
$558$	0	0
$559$	1.19046	0.0503511
$560$	0	0
$561$	−72.3596	−3.05503
$562$	0	0
$563$	−2.45317	−0.103389	−0.0516944	−	0.998663i	$-0.516462\pi$
$563$	−2.45317	−0.103389	−0.0516944	+	0.998663i	$0.516462\pi$
$564$	0	0
$565$	2.26997	0.0954984
$566$	0	0
$567$	12.3949	0.520538
$568$	0	0
$569$	18.1326	0.760159	0.380080	−	0.924954i	$-0.375897\pi$
$569$	18.1326	0.760159	0.380080	+	0.924954i	$0.375897\pi$
$570$	0	0
$571$	−25.1171	−1.05112	−0.525560	−	0.850757i	$-0.676144\pi$
$571$	−25.1171	−1.05112	−0.525560	+	0.850757i	$0.676144\pi$
$572$	0	0
$573$	−21.3546	−0.892101
$574$	0	0
$575$	12.3288	0.514145
$576$	0	0
$577$	−20.6753	−0.860722	−0.430361	−	0.902657i	$-0.641614\pi$
$577$	−20.6753	−0.860722	−0.430361	+	0.902657i	$0.641614\pi$
$578$	0	0
$579$	18.4384	0.766272
$580$	0	0
$581$	−15.2411	−0.632307
$582$	0	0
$583$	−9.23201	−0.382351
$584$	0	0
$585$	2.40715	0.0995233
$586$	0	0
$587$	−25.0153	−1.03249	−0.516247	−	0.856440i	$-0.672671\pi$
$587$	−25.0153	−1.03249	−0.516247	+	0.856440i	$0.672671\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	68.7820	2.82931
$592$	0	0
$593$	−37.4535	−1.53803	−0.769016	−	0.639229i	$-0.779253\pi$
$593$	−37.4535	−1.53803	−0.769016	+	0.639229i	$0.779253\pi$
$594$	0	0
$595$	−20.3796	−0.835483
$596$	0	0
$597$	20.5320	0.840319
$598$	0	0
$599$	−41.9168	−1.71267	−0.856336	−	0.516419i	$-0.827265\pi$
$599$	−41.9168	−1.71267	−0.856336	+	0.516419i	$0.827265\pi$
$600$	0	0
$601$	−24.3133	−0.991760	−0.495880	−	0.868391i	$-0.665154\pi$
$601$	−24.3133	−0.991760	−0.495880	+	0.868391i	$0.665154\pi$
$602$	0	0
$603$	−17.1278	−0.697500
$604$	0	0
$605$	−26.9199	−1.09445
$606$	0	0
$607$	−19.2612	−0.781790	−0.390895	−	0.920435i	$-0.627834\pi$
$607$	−19.2612	−0.781790	−0.390895	+	0.920435i	$0.627834\pi$
$608$	0	0
$609$	27.1755	1.10121
$610$	0	0
$611$	−2.68226	−0.108513
$612$	0	0
$613$	16.0959	0.650108	0.325054	−	0.945695i	$-0.394618\pi$
$613$	16.0959	0.650108	0.325054	+	0.945695i	$0.394618\pi$
$614$	0	0
$615$	13.9293	0.561683
$616$	0	0
$617$	23.0188	0.926700	0.463350	−	0.886175i	$-0.346647\pi$
$617$	23.0188	0.926700	0.463350	+	0.886175i	$0.346647\pi$
$618$	0	0
$619$	3.28899	0.132196	0.0660979	−	0.997813i	$-0.478945\pi$
$619$	3.28899	0.132196	0.0660979	+	0.997813i	$0.478945\pi$
$620$	0	0
$621$	12.3288	0.494737
$622$	0	0
$623$	−43.3364	−1.73624
$624$	0	0
$625$	3.04216	0.121686
$626$	0	0
$627$	0	0
$628$	0	0
$629$	8.61072	0.343332
$630$	0	0
$631$	42.0411	1.67363	0.836816	−	0.547485i	$-0.184415\pi$
$631$	42.0411	1.67363	0.836816	+	0.547485i	$0.184415\pi$
$632$	0	0
$633$	−30.8792	−1.22734
$634$	0	0
$635$	−17.2866	−0.685999
$636$	0	0
$637$	−1.85088	−0.0733346
$638$	0	0
$639$	−47.1130	−1.86376
$640$	0	0
$641$	18.8123	0.743042	0.371521	−	0.928424i	$-0.378836\pi$
$641$	18.8123	0.743042	0.371521	+	0.928424i	$0.378836\pi$
$642$	0	0
$643$	15.8236	0.624021	0.312010	−	0.950079i	$-0.398998\pi$
$643$	15.8236	0.624021	0.312010	+	0.950079i	$0.398998\pi$
$644$	0	0
$645$	−9.29485	−0.365984
$646$	0	0
$647$	−8.04659	−0.316344	−0.158172	−	0.987412i	$-0.550560\pi$
$647$	−8.04659	−0.316344	−0.158172	+	0.987412i	$0.550560\pi$
$648$	0	0
$649$	−48.3764	−1.89894
$650$	0	0
$651$	−97.0821	−3.80495
$652$	0	0
$653$	−41.3583	−1.61848	−0.809238	−	0.587481i	$-0.800120\pi$
$653$	−41.3583	−1.61848	−0.809238	+	0.587481i	$0.800120\pi$
$654$	0	0
$655$	12.9922	0.507647
$656$	0	0
$657$	−31.2751	−1.22016
$658$	0	0
$659$	1.67610	0.0652915	0.0326457	−	0.999467i	$-0.489607\pi$
$659$	1.67610	0.0652915	0.0326457	+	0.999467i	$0.489607\pi$
$660$	0	0
$661$	11.1120	0.432207	0.216104	−	0.976370i	$-0.430665\pi$
$661$	11.1120	0.432207	0.216104	+	0.976370i	$0.430665\pi$
$662$	0	0
$663$	−5.65527	−0.219632
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−11.0882	−0.429336
$668$	0	0
$669$	7.53421	0.291290
$670$	0	0
$671$	−14.6818	−0.566786
$672$	0	0
$673$	−50.1601	−1.93353	−0.966764	−	0.255669i	$-0.917704\pi$
$673$	−50.1601	−1.93353	−0.966764	+	0.255669i	$0.917704\pi$
$674$	0	0
$675$	11.2624	0.433490
$676$	0	0
$677$	24.5938	0.945218	0.472609	−	0.881272i	$-0.343312\pi$
$677$	24.5938	0.945218	0.472609	+	0.881272i	$0.343312\pi$
$678$	0	0
$679$	39.8076	1.52768
$680$	0	0
$681$	16.3089	0.624958
$682$	0	0
$683$	40.1865	1.53769	0.768846	−	0.639434i	$-0.220831\pi$
$683$	40.1865	1.53769	0.768846	+	0.639434i	$0.220831\pi$
$684$	0	0
$685$	4.48347	0.171305
$686$	0	0
$687$	24.1571	0.921651
$688$	0	0
$689$	−0.721529	−0.0274881
$690$	0	0
$691$	−9.38100	−0.356870	−0.178435	−	0.983952i	$-0.557103\pi$
$691$	−9.38100	−0.356870	−0.178435	+	0.983952i	$0.557103\pi$
$692$	0	0
$693$	80.3417	3.05193
$694$	0	0
$695$	−13.6647	−0.518333
$696$	0	0
$697$	−19.1774	−0.726394
$698$	0	0
$699$	19.7605	0.747413
$700$	0	0
$701$	−37.1748	−1.40407	−0.702036	−	0.712141i	$-0.747726\pi$
$701$	−37.1748	−1.40407	−0.702036	+	0.712141i	$0.747726\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	20.9425	0.788741
$706$	0	0
$707$	−34.1028	−1.28257
$708$	0	0
$709$	−31.2732	−1.17449	−0.587246	−	0.809409i	$-0.699788\pi$
$709$	−31.2732	−1.17449	−0.587246	+	0.809409i	$0.699788\pi$
$710$	0	0
$711$	43.1428	1.61798
$712$	0	0
$713$	39.6115	1.48346
$714$	0	0
$715$	−3.20625	−0.119907
$716$	0	0
$717$	51.6077	1.92732
$718$	0	0
$719$	26.0486	0.971450	0.485725	−	0.874112i	$-0.338556\pi$
$719$	26.0486	0.971450	0.485725	+	0.874112i	$0.338556\pi$
$720$	0	0
$721$	15.1405	0.563863
$722$	0	0
$723$	32.6463	1.21413
$724$	0	0
$725$	−10.1291	−0.376186
$726$	0	0
$727$	−2.07973	−0.0771328	−0.0385664	−	0.999256i	$-0.512279\pi$
$727$	−2.07973	−0.0771328	−0.0385664	+	0.999256i	$0.512279\pi$
$728$	0	0
$729$	−42.8399	−1.58666
$730$	0	0
$731$	12.7968	0.473308
$732$	0	0
$733$	−6.92827	−0.255901	−0.127951	−	0.991781i	$-0.540840\pi$
$733$	−6.92827	−0.255901	−0.127951	+	0.991781i	$0.540840\pi$
$734$	0	0
$735$	14.4513	0.533044
$736$	0	0
$737$	22.8137	0.840355
$738$	0	0
$739$	17.3210	0.637165	0.318582	−	0.947895i	$-0.396793\pi$
$739$	17.3210	0.637165	0.318582	+	0.947895i	$0.396793\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	52.1550	1.91338	0.956690	−	0.291108i	$-0.0940239\pi$
$743$	52.1550	1.91338	0.956690	+	0.291108i	$0.0940239\pi$
$744$	0	0
$745$	10.7978	0.395600
$746$	0	0
$747$	19.3513	0.708027
$748$	0	0
$749$	−4.27726	−0.156288
$750$	0	0
$751$	11.6198	0.424013	0.212006	−	0.977268i	$-0.432000\pi$
$751$	11.6198	0.424013	0.212006	+	0.977268i	$0.432000\pi$
$752$	0	0
$753$	−39.3382	−1.43357
$754$	0	0
$755$	14.5907	0.531009
$756$	0	0
$757$	−27.2620	−0.990856	−0.495428	−	0.868649i	$-0.664989\pi$
$757$	−27.2620	−0.990856	−0.495428	+	0.868649i	$0.664989\pi$
$758$	0	0
$759$	−55.9389	−2.03045
$760$	0	0
$761$	32.8211	1.18976	0.594882	−	0.803813i	$-0.297199\pi$
$761$	32.8211	1.18976	0.594882	+	0.803813i	$0.297199\pi$
$762$	0	0
$763$	−21.4840	−0.777772
$764$	0	0
$765$	25.8756	0.935535
$766$	0	0
$767$	−3.78086	−0.136519
$768$	0	0
$769$	2.45229	0.0884319	0.0442159	−	0.999022i	$-0.485921\pi$
$769$	2.45229	0.0884319	0.0442159	+	0.999022i	$0.485921\pi$
$770$	0	0
$771$	−6.97539	−0.251213
$772$	0	0
$773$	9.70810	0.349176	0.174588	−	0.984642i	$-0.444141\pi$
$773$	9.70810	0.349176	0.174588	+	0.984642i	$0.444141\pi$
$774$	0	0
$775$	36.1854	1.29982
$776$	0	0
$777$	−16.3145	−0.585281
$778$	0	0
$779$	0	0
$780$	0	0
$781$	62.7530	2.24548
$782$	0	0
$783$	−10.1291	−0.361985
$784$	0	0
$785$	5.99561	0.213993
$786$	0	0
$787$	32.4028	1.15503	0.577517	−	0.816379i	$-0.304022\pi$
$787$	32.4028	1.15503	0.577517	+	0.816379i	$0.304022\pi$
$788$	0	0
$789$	61.9587	2.20579
$790$	0	0
$791$	5.92128	0.210536
$792$	0	0
$793$	−1.14746	−0.0407475
$794$	0	0
$795$	5.63354	0.199801
$796$	0	0
$797$	−14.1556	−0.501417	−0.250708	−	0.968063i	$-0.580664\pi$
$797$	−14.1556	−0.501417	−0.250708	+	0.968063i	$0.580664\pi$
$798$	0	0
$799$	−28.8330	−1.02004
$800$	0	0
$801$	55.0234	1.94416
$802$	0	0
$803$	41.6574	1.47006
$804$	0	0
$805$	−15.7548	−0.555285
$806$	0	0
$807$	−70.4830	−2.48112
$808$	0	0
$809$	−4.19788	−0.147589	−0.0737947	−	0.997273i	$-0.523511\pi$
$809$	−4.19788	−0.147589	−0.0737947	+	0.997273i	$0.523511\pi$
$810$	0	0
$811$	45.6564	1.60321	0.801607	−	0.597852i	$-0.203979\pi$
$811$	45.6564	1.60321	0.801607	+	0.597852i	$0.203979\pi$
$812$	0	0
$813$	39.7769	1.39504
$814$	0	0
$815$	2.34641	0.0821912
$816$	0	0
$817$	0	0
$818$	0	0
$819$	6.27911	0.219410
$820$	0	0
$821$	4.15692	0.145078	0.0725388	−	0.997366i	$-0.476890\pi$
$821$	4.15692	0.145078	0.0725388	+	0.997366i	$0.476890\pi$
$822$	0	0
$823$	−46.3619	−1.61607	−0.808037	−	0.589131i	$-0.799470\pi$
$823$	−46.3619	−1.61607	−0.808037	+	0.589131i	$0.799470\pi$
$824$	0	0
$825$	−51.1005	−1.77909
$826$	0	0
$827$	−30.5080	−1.06087	−0.530434	−	0.847726i	$-0.677971\pi$
$827$	−30.5080	−1.06087	−0.530434	+	0.847726i	$0.677971\pi$
$828$	0	0
$829$	7.28342	0.252964	0.126482	−	0.991969i	$-0.459631\pi$
$829$	7.28342	0.252964	0.126482	+	0.991969i	$0.459631\pi$
$830$	0	0
$831$	39.6983	1.37712
$832$	0	0
$833$	−19.8961	−0.689357
$834$	0	0
$835$	25.6113	0.886314
$836$	0	0
$837$	36.1854	1.25075
$838$	0	0
$839$	−13.2993	−0.459144	−0.229572	−	0.973292i	$-0.573733\pi$
$839$	−13.2993	−0.459144	−0.229572	+	0.973292i	$0.573733\pi$
$840$	0	0
$841$	−19.8901	−0.685866
$842$	0	0
$843$	−37.5274	−1.29251
$844$	0	0
$845$	16.4181	0.564799
$846$	0	0
$847$	−70.2213	−2.41283
$848$	0	0
$849$	49.2330	1.68967
$850$	0	0
$851$	6.65668	0.228188
$852$	0	0
$853$	16.3649	0.560323	0.280161	−	0.959953i	$-0.409612\pi$
$853$	16.3649	0.560323	0.280161	+	0.959953i	$0.409612\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	28.6087	0.977255	0.488628	−	0.872492i	$-0.337498\pi$
$857$	28.6087	0.977255	0.488628	+	0.872492i	$0.337498\pi$
$858$	0	0
$859$	−5.40768	−0.184508	−0.0922538	−	0.995736i	$-0.529407\pi$
$859$	−5.40768	−0.184508	−0.0922538	+	0.995736i	$0.529407\pi$
$860$	0	0
$861$	36.3349	1.23829
$862$	0	0
$863$	−42.0613	−1.43178	−0.715892	−	0.698211i	$-0.753980\pi$
$863$	−42.0613	−1.43178	−0.715892	+	0.698211i	$0.753980\pi$
$864$	0	0
$865$	5.99224	0.203742
$866$	0	0
$867$	−15.0280	−0.510376
$868$	0	0
$869$	−57.4648	−1.94936
$870$	0	0
$871$	1.78301	0.0604150
$872$	0	0
$873$	−50.5430	−1.71062
$874$	0	0
$875$	−35.8349	−1.21144
$876$	0	0
$877$	11.7349	0.396259	0.198129	−	0.980176i	$-0.436513\pi$
$877$	11.7349	0.396259	0.198129	+	0.980176i	$0.436513\pi$
$878$	0	0
$879$	−66.6193	−2.24701
$880$	0	0
$881$	23.5318	0.792806	0.396403	−	0.918077i	$-0.370258\pi$
$881$	23.5318	0.792806	0.396403	+	0.918077i	$0.370258\pi$
$882$	0	0
$883$	−3.56204	−0.119872	−0.0599361	−	0.998202i	$-0.519090\pi$
$883$	−3.56204	−0.119872	−0.0599361	+	0.998202i	$0.519090\pi$
$884$	0	0
$885$	29.5202	0.992310
$886$	0	0
$887$	−24.5918	−0.825711	−0.412856	−	0.910796i	$-0.635469\pi$
$887$	−24.5918	−0.825711	−0.412856	+	0.910796i	$0.635469\pi$
$888$	0	0
$889$	−45.0926	−1.51236
$890$	0	0
$891$	20.9620	0.702253
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−9.79924	−0.327553
$896$	0	0
$897$	−4.37191	−0.145974
$898$	0	0
$899$	−32.5442	−1.08541
$900$	0	0
$901$	−7.75607	−0.258392
$902$	0	0
$903$	−24.2459	−0.806852
$904$	0	0
$905$	−7.81434	−0.259758
$906$	0	0
$907$	2.96372	0.0984088	0.0492044	−	0.998789i	$-0.484331\pi$
$907$	2.96372	0.0984088	0.0492044	+	0.998789i	$0.484331\pi$
$908$	0	0
$909$	43.2997	1.43616
$910$	0	0
$911$	31.6411	1.04832	0.524158	−	0.851621i	$-0.324380\pi$
$911$	31.6411	1.04832	0.524158	+	0.851621i	$0.324380\pi$
$912$	0	0
$913$	−25.7753	−0.853039
$914$	0	0
$915$	8.95912	0.296179
$916$	0	0
$917$	33.8905	1.11916
$918$	0	0
$919$	39.8066	1.31310	0.656549	−	0.754284i	$-0.272016\pi$
$919$	39.8066	1.31310	0.656549	+	0.754284i	$0.272016\pi$
$920$	0	0
$921$	61.7349	2.03423
$922$	0	0
$923$	4.90447	0.161433
$924$	0	0
$925$	6.08092	0.199939
$926$	0	0
$927$	−19.2237	−0.631388
$928$	0	0
$929$	−2.42005	−0.0793992	−0.0396996	−	0.999212i	$-0.512640\pi$
$929$	−2.42005	−0.0793992	−0.0396996	+	0.999212i	$0.512640\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−79.8613	−2.61454
$934$	0	0
$935$	−34.4655	−1.12714
$936$	0	0
$937$	14.8154	0.483997	0.241999	−	0.970277i	$-0.422197\pi$
$937$	14.8154	0.483997	0.241999	+	0.970277i	$0.422197\pi$
$938$	0	0
$939$	55.7903	1.82065
$940$	0	0
$941$	14.7737	0.481610	0.240805	−	0.970573i	$-0.422589\pi$
$941$	14.7737	0.481610	0.240805	+	0.970573i	$0.422589\pi$
$942$	0	0
$943$	−14.8254	−0.482782
$944$	0	0
$945$	−14.3921	−0.468177
$946$	0	0
$947$	50.0448	1.62624	0.813119	−	0.582097i	$-0.197768\pi$
$947$	50.0448	1.62624	0.813119	+	0.582097i	$0.197768\pi$
$948$	0	0
$949$	3.25574	0.105686
$950$	0	0
$951$	87.2561	2.82947
$952$	0	0
$953$	5.78937	0.187536	0.0937680	−	0.995594i	$-0.470109\pi$
$953$	5.78937	0.187536	0.0937680	+	0.995594i	$0.470109\pi$
$954$	0	0
$955$	−10.1714	−0.329138
$956$	0	0
$957$	45.9585	1.48563
$958$	0	0
$959$	11.6952	0.377659
$960$	0	0
$961$	85.2613	2.75037
$962$	0	0
$963$	5.43076	0.175004
$964$	0	0
$965$	8.78235	0.282714
$966$	0	0
$967$	31.7279	1.02030	0.510151	−	0.860085i	$-0.329590\pi$
$967$	31.7279	1.02030	0.510151	+	0.860085i	$0.329590\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−9.01674	−0.289361	−0.144680	−	0.989478i	$-0.546215\pi$
$971$	−9.01674	−0.289361	−0.144680	+	0.989478i	$0.546215\pi$
$972$	0	0
$973$	−35.6448	−1.14272
$974$	0	0
$975$	−3.99377	−0.127903
$976$	0	0
$977$	−42.4099	−1.35681	−0.678407	−	0.734686i	$-0.737329\pi$
$977$	−42.4099	−1.35681	−0.678407	+	0.734686i	$0.737329\pi$
$978$	0	0
$979$	−73.2894	−2.34234
$980$	0	0
$981$	27.2778	0.870913
$982$	0	0
$983$	−50.7523	−1.61875	−0.809374	−	0.587294i	$-0.800193\pi$
$983$	−50.7523	−1.61875	−0.809374	+	0.587294i	$0.800193\pi$
$984$	0	0
$985$	32.7615	1.04387
$986$	0	0
$987$	54.6292	1.73887
$988$	0	0
$989$	9.89283	0.314574
$990$	0	0
$991$	−10.0696	−0.319871	−0.159936	−	0.987127i	$-0.551129\pi$
$991$	−10.0696	−0.319871	−0.159936	+	0.987127i	$0.551129\pi$
$992$	0	0
$993$	57.1739	1.81436
$994$	0	0
$995$	9.77957	0.310033
$996$	0	0
$997$	−32.7409	−1.03692	−0.518458	−	0.855103i	$-0.673494\pi$
$997$	−32.7409	−1.03692	−0.518458	+	0.855103i	$0.673494\pi$
$998$	0	0
$999$	6.08092	0.192392

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1444.2.a.h.1.1		6
4.3	odd	2		5776.2.a.bw.1.6		6
19.7	even	3		1444.2.e.g.429.6		12
19.8	odd	6		1444.2.e.h.653.1		12
19.9	even	9		76.2.i.a.5.1	✓	12
19.11	even	3		1444.2.e.g.653.6		12
19.12	odd	6		1444.2.e.h.429.1		12
19.17	even	9		76.2.i.a.61.1	yes	12
19.18	odd	2		1444.2.a.g.1.6		6
57.17	odd	18		684.2.bo.c.289.2		12
57.47	odd	18		684.2.bo.c.613.2		12
76.47	odd	18		304.2.u.e.81.2		12
76.55	odd	18		304.2.u.e.289.2		12
76.75	even	2		5776.2.a.by.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.2.i.a.5.1	✓	12	19.9	even	9
76.2.i.a.61.1	yes	12	19.17	even	9
304.2.u.e.81.2		12	76.47	odd	18
304.2.u.e.289.2		12	76.55	odd	18
684.2.bo.c.289.2		12	57.17	odd	18
684.2.bo.c.613.2		12	57.47	odd	18
1444.2.a.g.1.6		6	19.18	odd	2
1444.2.a.h.1.1		6	1.1	even	1	trivial
1444.2.e.g.429.6		12	19.7	even	3
1444.2.e.g.653.6		12	19.11	even	3
1444.2.e.h.429.1		12	19.12	odd	6
1444.2.e.h.653.1		12	19.8	odd	6
5776.2.a.bw.1.6		6	4.3	odd	2
5776.2.a.by.1.1		6	76.75	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:	\( N \)	\(=\)	\( 1444 = 2^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1444.a (trivial)

\(f(q)\)	\(=\)	\(q-2.69196 q^{3} -1.28220 q^{5} -3.34467 q^{7} +4.24666 q^{9} +O(q^{10})\)	\(q-2.69196 q^{3} -1.28220 q^{5} -3.34467 q^{7} +4.24666 q^{9} -5.65641 q^{11} -0.442077 q^{13} +3.45165 q^{15} -4.75211 q^{17} +9.00371 q^{21} -3.67370 q^{23} -3.35595 q^{25} -3.35595 q^{27} +3.01826 q^{29} -10.7825 q^{31} +15.2268 q^{33} +4.28855 q^{35} -1.81198 q^{37} +1.19006 q^{39} +4.03555 q^{41} -2.69288 q^{43} -5.44508 q^{45} +6.06741 q^{47} +4.18678 q^{49} +12.7925 q^{51} +1.63213 q^{53} +7.25268 q^{55} +8.55249 q^{59} +2.59561 q^{61} -14.2036 q^{63} +0.566834 q^{65} -4.03325 q^{67} +9.88947 q^{69} -11.0941 q^{71} -7.36464 q^{73} +9.03409 q^{75} +18.9188 q^{77} +10.1592 q^{79} -3.70588 q^{81} +4.55683 q^{83} +6.09318 q^{85} -8.12503 q^{87} +12.9569 q^{89} +1.47860 q^{91} +29.0260 q^{93} -11.9018 q^{97} -24.0208 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{3} - 3 q^{5} - 3 q^{7} + 9 q^{9}+O(q^{10}) \) 6 * q + 3 * q^3 - 3 * q^5 - 3 * q^7 + 9 * q^9	\( 6 q + 3 q^{3} - 3 q^{5} - 3 q^{7} + 9 q^{9} - 3 q^{11} + 12 q^{13} - 6 q^{17} + 21 q^{21} + 9 q^{25} + 9 q^{27} + 21 q^{29} - 6 q^{31} + 9 q^{33} + 3 q^{35} - 6 q^{37} + 30 q^{39} + 36 q^{41} - 18 q^{43} - 24 q^{45} + 30 q^{47} - 9 q^{49} + 24 q^{51} + 18 q^{53} - 15 q^{55} + 21 q^{59} + 9 q^{61} + 6 q^{63} + 33 q^{65} + 18 q^{67} + 33 q^{69} - 12 q^{71} - 24 q^{73} + 21 q^{75} + 12 q^{77} + 9 q^{79} - 6 q^{81} - 3 q^{83} - 12 q^{85} + 18 q^{87} + 45 q^{89} + 9 q^{91} + 15 q^{97} - 33 q^{99}+O(q^{100}) \) 6 * q + 3 * q^3 - 3 * q^5 - 3 * q^7 + 9 * q^9 - 3 * q^11 + 12 * q^13 - 6 * q^17 + 21 * q^21 + 9 * q^25 + 9 * q^27 + 21 * q^29 - 6 * q^31 + 9 * q^33 + 3 * q^35 - 6 * q^37 + 30 * q^39 + 36 * q^41 - 18 * q^43 - 24 * q^45 + 30 * q^47 - 9 * q^49 + 24 * q^51 + 18 * q^53 - 15 * q^55 + 21 * q^59 + 9 * q^61 + 6 * q^63 + 33 * q^65 + 18 * q^67 + 33 * q^69 - 12 * q^71 - 24 * q^73 + 21 * q^75 + 12 * q^77 + 9 * q^79 - 6 * q^81 - 3 * q^83 - 12 * q^85 + 18 * q^87 + 45 * q^89 + 9 * q^91 + 15 * q^97 - 33 * q^99

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