LMFDB - Embedded newform 3344.2.a.x.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3344,2,Mod(1,3344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3344, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3344.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3344 = 2^{4} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3344.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:	$26.7019744359$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 12x^{4} + 28x^{3} + 16x^{2} - 60x + 27 $ x^6 - 2x^5 - 12x^4 + 28x^3 + 16x^2 - 60*x + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 836)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.25580$ of defining polynomial
Character	$\chi$	$=$	3344.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.25580 q^{3} -2.84405 q^{5} +1.37411 q^{7} +7.60023 q^{9} +O(q^{10})$	$q-3.25580 q^{3} -2.84405 q^{5} +1.37411 q^{7} +7.60023 q^{9} -1.00000 q^{11} +4.50348 q^{13} +9.25967 q^{15} +8.04165 q^{17} -1.00000 q^{19} -4.47381 q^{21} +1.93269 q^{23} +3.08863 q^{25} -14.9774 q^{27} -6.01198 q^{29} +2.25270 q^{31} +3.25580 q^{33} -3.90803 q^{35} +6.40263 q^{37} -14.6624 q^{39} -4.47806 q^{41} -2.91137 q^{43} -21.6155 q^{45} -11.0070 q^{47} -5.11183 q^{49} -26.1820 q^{51} -2.68887 q^{53} +2.84405 q^{55} +3.25580 q^{57} +13.7377 q^{59} +0.975342 q^{61} +10.4435 q^{63} -12.8081 q^{65} +6.49551 q^{67} -6.29244 q^{69} +15.7674 q^{71} +4.99924 q^{73} -10.0560 q^{75} -1.37411 q^{77} +4.33532 q^{79} +25.9629 q^{81} -4.62489 q^{83} -22.8709 q^{85} +19.5738 q^{87} -15.6624 q^{89} +6.18826 q^{91} -7.33433 q^{93} +2.84405 q^{95} -0.0256605 q^{97} -7.60023 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{3} - 2 q^{7} + 10 q^{9}+O(q^{10}) $ 6 * q + 2 * q^3 - 2 * q^7 + 10 * q^9	$ 6 q + 2 q^{3} - 2 q^{7} + 10 q^{9} - 6 q^{11} + 2 q^{13} + 12 q^{15} + 12 q^{17} - 6 q^{19} + 2 q^{21} + 2 q^{23} + 26 q^{25} - 16 q^{27} + 4 q^{29} + 20 q^{31} - 2 q^{33} - 20 q^{35} + 22 q^{37} - 8 q^{39} - 2 q^{41} - 10 q^{43} - 6 q^{45} - 16 q^{47} + 48 q^{49} - 36 q^{51} + 12 q^{53} - 2 q^{57} + 14 q^{59} + 12 q^{61} + 4 q^{63} + 10 q^{65} + 12 q^{67} + 30 q^{69} + 30 q^{71} + 24 q^{73} + 50 q^{75} + 2 q^{77} + 10 q^{81} + 14 q^{83} + 12 q^{85} + 30 q^{87} - 14 q^{89} - 20 q^{91} + 14 q^{93} - 46 q^{97} - 10 q^{99}+O(q^{100}) $ 6 * q + 2 * q^3 - 2 * q^7 + 10 * q^9 - 6 * q^11 + 2 * q^13 + 12 * q^15 + 12 * q^17 - 6 * q^19 + 2 * q^21 + 2 * q^23 + 26 * q^25 - 16 * q^27 + 4 * q^29 + 20 * q^31 - 2 * q^33 - 20 * q^35 + 22 * q^37 - 8 * q^39 - 2 * q^41 - 10 * q^43 - 6 * q^45 - 16 * q^47 + 48 * q^49 - 36 * q^51 + 12 * q^53 - 2 * q^57 + 14 * q^59 + 12 * q^61 + 4 * q^63 + 10 * q^65 + 12 * q^67 + 30 * q^69 + 30 * q^71 + 24 * q^73 + 50 * q^75 + 2 * q^77 + 10 * q^81 + 14 * q^83 + 12 * q^85 + 30 * q^87 - 14 * q^89 - 20 * q^91 + 14 * q^93 - 46 * q^97 - 10 * 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
$3$	−3.25580	−1.87974	−0.939869	−	0.341536i	$-0.889053\pi$
$3$	−3.25580	−1.87974	−0.939869	+	0.341536i	$0.889053\pi$
$4$	0	0
$5$	−2.84405	−1.27190	−0.635949	−	0.771731i	$-0.719391\pi$
$5$	−2.84405	−1.27190	−0.635949	+	0.771731i	$0.719391\pi$
$6$	0	0
$7$	1.37411	0.519363	0.259682	−	0.965694i	$-0.416382\pi$
$7$	1.37411	0.519363	0.259682	+	0.965694i	$0.416382\pi$
$8$	0	0
$9$	7.60023	2.53341
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	4.50348	1.24904	0.624521	−	0.781008i	$-0.285294\pi$
$13$	4.50348	1.24904	0.624521	+	0.781008i	$0.285294\pi$
$14$	0	0
$15$	9.25967	2.39084
$16$	0	0
$17$	8.04165	1.95039	0.975194	−	0.221353i	$-0.0710474\pi$
$17$	8.04165	1.95039	0.975194	+	0.221353i	$0.0710474\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−4.47381	−0.976266
$22$	0	0
$23$	1.93269	0.402993	0.201496	−	0.979489i	$-0.435420\pi$
$23$	1.93269	0.402993	0.201496	+	0.979489i	$0.435420\pi$
$24$	0	0
$25$	3.08863	0.617727
$26$	0	0
$27$	−14.9774	−2.88241
$28$	0	0
$29$	−6.01198	−1.11640	−0.558198	−	0.829707i	$-0.688507\pi$
$29$	−6.01198	−1.11640	−0.558198	+	0.829707i	$0.688507\pi$
$30$	0	0
$31$	2.25270	0.404596	0.202298	−	0.979324i	$-0.435159\pi$
$31$	2.25270	0.404596	0.202298	+	0.979324i	$0.435159\pi$
$32$	0	0
$33$	3.25580	0.566762
$34$	0	0
$35$	−3.90803	−0.660577
$36$	0	0
$37$	6.40263	1.05259	0.526293	−	0.850303i	$-0.323581\pi$
$37$	6.40263	1.05259	0.526293	+	0.850303i	$0.323581\pi$
$38$	0	0
$39$	−14.6624	−2.34787
$40$	0	0
$41$	−4.47806	−0.699356	−0.349678	−	0.936870i	$-0.613709\pi$
$41$	−4.47806	−0.699356	−0.349678	+	0.936870i	$0.613709\pi$
$42$	0	0
$43$	−2.91137	−0.443979	−0.221990	−	0.975049i	$-0.571255\pi$
$43$	−2.91137	−0.443979	−0.221990	+	0.975049i	$0.571255\pi$
$44$	0	0
$45$	−21.6155	−3.22224
$46$	0	0
$47$	−11.0070	−1.60553	−0.802766	−	0.596295i	$-0.796639\pi$
$47$	−11.0070	−1.60553	−0.802766	+	0.596295i	$0.796639\pi$
$48$	0	0
$49$	−5.11183	−0.730262
$50$	0	0
$51$	−26.1820	−3.66622
$52$	0	0
$53$	−2.68887	−0.369344	−0.184672	−	0.982800i	$-0.559122\pi$
$53$	−2.68887	−0.369344	−0.184672	+	0.982800i	$0.559122\pi$
$54$	0	0
$55$	2.84405	0.383492
$56$	0	0
$57$	3.25580	0.431241
$58$	0	0
$59$	13.7377	1.78850	0.894250	−	0.447568i	$-0.147710\pi$
$59$	13.7377	1.78850	0.894250	+	0.447568i	$0.147710\pi$
$60$	0	0
$61$	0.975342	0.124880	0.0624399	−	0.998049i	$-0.480112\pi$
$61$	0.975342	0.124880	0.0624399	+	0.998049i	$0.480112\pi$
$62$	0	0
$63$	10.4435	1.31576
$64$	0	0
$65$	−12.8081	−1.58865
$66$	0	0
$67$	6.49551	0.793553	0.396777	−	0.917915i	$-0.370129\pi$
$67$	6.49551	0.793553	0.396777	+	0.917915i	$0.370129\pi$
$68$	0	0
$69$	−6.29244	−0.757521
$70$	0	0
$71$	15.7674	1.87125	0.935623	−	0.353001i	$-0.114839\pi$
$71$	15.7674	1.87125	0.935623	+	0.353001i	$0.114839\pi$
$72$	0	0
$73$	4.99924	0.585116	0.292558	−	0.956248i	$-0.405493\pi$
$73$	4.99924	0.585116	0.292558	+	0.956248i	$0.405493\pi$
$74$	0	0
$75$	−10.0560	−1.16116
$76$	0	0
$77$	−1.37411	−0.156594
$78$	0	0
$79$	4.33532	0.487762	0.243881	−	0.969805i	$-0.421579\pi$
$79$	4.33532	0.487762	0.243881	+	0.969805i	$0.421579\pi$
$80$	0	0
$81$	25.9629	2.88476
$82$	0	0
$83$	−4.62489	−0.507648	−0.253824	−	0.967250i	$-0.581688\pi$
$83$	−4.62489	−0.507648	−0.253824	+	0.967250i	$0.581688\pi$
$84$	0	0
$85$	−22.8709	−2.48070
$86$	0	0
$87$	19.5738	2.09853
$88$	0	0
$89$	−15.6624	−1.66022	−0.830108	−	0.557603i	$-0.811721\pi$
$89$	−15.6624	−1.66022	−0.830108	+	0.557603i	$0.811721\pi$
$90$	0	0
$91$	6.18826	0.648706
$92$	0	0
$93$	−7.33433	−0.760535
$94$	0	0
$95$	2.84405	0.291794
$96$	0	0
$97$	−0.0256605	−0.00260543	−0.00130271	−	0.999999i	$-0.500415\pi$
$97$	−0.0256605	−0.00260543	−0.00130271	+	0.999999i	$0.500415\pi$
$98$	0	0
$99$	−7.60023	−0.763852
$100$	0	0
$101$	−8.15805	−0.811756	−0.405878	−	0.913927i	$-0.633034\pi$
$101$	−8.15805	−0.811756	−0.405878	+	0.913927i	$0.633034\pi$
$102$	0	0
$103$	−0.718539	−0.0707998	−0.0353999	−	0.999373i	$-0.511270\pi$
$103$	−0.718539	−0.0707998	−0.0353999	+	0.999373i	$0.511270\pi$
$104$	0	0
$105$	12.7238	1.24171
$106$	0	0
$107$	8.00621	0.773989	0.386995	−	0.922082i	$-0.373513\pi$
$107$	8.00621	0.773989	0.386995	+	0.922082i	$0.373513\pi$
$108$	0	0
$109$	14.1835	1.35853	0.679265	−	0.733893i	$-0.262299\pi$
$109$	14.1835	1.35853	0.679265	+	0.733893i	$0.262299\pi$
$110$	0	0
$111$	−20.8457	−1.97859
$112$	0	0
$113$	−2.09274	−0.196868	−0.0984340	−	0.995144i	$-0.531383\pi$
$113$	−2.09274	−0.196868	−0.0984340	+	0.995144i	$0.531383\pi$
$114$	0	0
$115$	−5.49666	−0.512566
$116$	0	0
$117$	34.2275	3.16434
$118$	0	0
$119$	11.0501	1.01296
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	14.5797	1.31461
$124$	0	0
$125$	5.43603	0.486213
$126$	0	0
$127$	−11.1750	−0.991625	−0.495813	−	0.868430i	$-0.665130\pi$
$127$	−11.1750	−0.991625	−0.495813	+	0.868430i	$0.665130\pi$
$128$	0	0
$129$	9.47883	0.834564
$130$	0	0
$131$	−16.4227	−1.43486	−0.717430	−	0.696630i	$-0.754682\pi$
$131$	−16.4227	−1.43486	−0.717430	+	0.696630i	$0.754682\pi$
$132$	0	0
$133$	−1.37411	−0.119150
$134$	0	0
$135$	42.5966	3.66613
$136$	0	0
$137$	−15.2891	−1.30624	−0.653118	−	0.757256i	$-0.726539\pi$
$137$	−15.2891	−1.30624	−0.653118	+	0.757256i	$0.726539\pi$
$138$	0	0
$139$	8.58178	0.727897	0.363949	−	0.931419i	$-0.381428\pi$
$139$	8.58178	0.727897	0.363949	+	0.931419i	$0.381428\pi$
$140$	0	0
$141$	35.8365	3.01798
$142$	0	0
$143$	−4.50348	−0.376600
$144$	0	0
$145$	17.0984	1.41994
$146$	0	0
$147$	16.6431	1.37270
$148$	0	0
$149$	3.94686	0.323340	0.161670	−	0.986845i	$-0.448312\pi$
$149$	3.94686	0.323340	0.161670	+	0.986845i	$0.448312\pi$
$150$	0	0
$151$	−11.8893	−0.967540	−0.483770	−	0.875195i	$-0.660733\pi$
$151$	−11.8893	−0.967540	−0.483770	+	0.875195i	$0.660733\pi$
$152$	0	0
$153$	61.1184	4.94113
$154$	0	0
$155$	−6.40679	−0.514606
$156$	0	0
$157$	−8.17294	−0.652272	−0.326136	−	0.945323i	$-0.605747\pi$
$157$	−8.17294	−0.652272	−0.326136	+	0.945323i	$0.605747\pi$
$158$	0	0
$159$	8.75442	0.694270
$160$	0	0
$161$	2.65571	0.209300
$162$	0	0
$163$	14.2668	1.11746	0.558730	−	0.829349i	$-0.311289\pi$
$163$	14.2668	1.11746	0.558730	+	0.829349i	$0.311289\pi$
$164$	0	0
$165$	−9.25967	−0.720864
$166$	0	0
$167$	10.5470	0.816155	0.408078	−	0.912947i	$-0.366199\pi$
$167$	10.5470	0.816155	0.408078	+	0.912947i	$0.366199\pi$
$168$	0	0
$169$	7.28137	0.560105
$170$	0	0
$171$	−7.60023	−0.581204
$172$	0	0
$173$	21.0352	1.59928	0.799638	−	0.600483i	$-0.205025\pi$
$173$	21.0352	1.59928	0.799638	+	0.600483i	$0.205025\pi$
$174$	0	0
$175$	4.24411	0.320824
$176$	0	0
$177$	−44.7273	−3.36191
$178$	0	0
$179$	20.0241	1.49667	0.748336	−	0.663320i	$-0.230853\pi$
$179$	20.0241	1.49667	0.748336	+	0.663320i	$0.230853\pi$
$180$	0	0
$181$	21.9906	1.63455	0.817273	−	0.576250i	$-0.195485\pi$
$181$	21.9906	1.63455	0.817273	+	0.576250i	$0.195485\pi$
$182$	0	0
$183$	−3.17552	−0.234741
$184$	0	0
$185$	−18.2094	−1.33878
$186$	0	0
$187$	−8.04165	−0.588064
$188$	0	0
$189$	−20.5806	−1.49702
$190$	0	0
$191$	11.2007	0.810455	0.405227	−	0.914216i	$-0.367192\pi$
$191$	11.2007	0.810455	0.405227	+	0.914216i	$0.367192\pi$
$192$	0	0
$193$	−6.21314	−0.447232	−0.223616	−	0.974677i	$-0.571786\pi$
$193$	−6.21314	−0.447232	−0.223616	+	0.974677i	$0.571786\pi$
$194$	0	0
$195$	41.7008	2.98625
$196$	0	0
$197$	−27.4256	−1.95399	−0.976996	−	0.213256i	$-0.931593\pi$
$197$	−27.4256	−1.95399	−0.976996	+	0.213256i	$0.931593\pi$
$198$	0	0
$199$	−6.61358	−0.468825	−0.234412	−	0.972137i	$-0.575317\pi$
$199$	−6.61358	−0.468825	−0.234412	+	0.972137i	$0.575317\pi$
$200$	0	0
$201$	−21.1481	−1.49167
$202$	0	0
$203$	−8.26110	−0.579815
$204$	0	0
$205$	12.7358	0.889510
$206$	0	0
$207$	14.6889	1.02095
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	1.89929	0.130753	0.0653764	−	0.997861i	$-0.479175\pi$
$211$	1.89929	0.130753	0.0653764	+	0.997861i	$0.479175\pi$
$212$	0	0
$213$	−51.3355	−3.51745
$214$	0	0
$215$	8.28008	0.564697
$216$	0	0
$217$	3.09544	0.210132
$218$	0	0
$219$	−16.2765	−1.09986
$220$	0	0
$221$	36.2155	2.43612
$222$	0	0
$223$	−2.22689	−0.149124	−0.0745618	−	0.997216i	$-0.523756\pi$
$223$	−2.22689	−0.149124	−0.0745618	+	0.997216i	$0.523756\pi$
$224$	0	0
$225$	23.4743	1.56496
$226$	0	0
$227$	−23.5602	−1.56375	−0.781873	−	0.623437i	$-0.785736\pi$
$227$	−23.5602	−1.56375	−0.781873	+	0.623437i	$0.785736\pi$
$228$	0	0
$229$	3.66034	0.241882	0.120941	−	0.992660i	$-0.461409\pi$
$229$	3.66034	0.241882	0.120941	+	0.992660i	$0.461409\pi$
$230$	0	0
$231$	4.47381	0.294355
$232$	0	0
$233$	20.4349	1.33873	0.669366	−	0.742933i	$-0.266566\pi$
$233$	20.4349	1.33873	0.669366	+	0.742933i	$0.266566\pi$
$234$	0	0
$235$	31.3044	2.04207
$236$	0	0
$237$	−14.1149	−0.916863
$238$	0	0
$239$	10.9516	0.708398	0.354199	−	0.935170i	$-0.384754\pi$
$239$	10.9516	0.708398	0.354199	+	0.935170i	$0.384754\pi$
$240$	0	0
$241$	−14.6061	−0.940859	−0.470429	−	0.882438i	$-0.655901\pi$
$241$	−14.6061	−0.940859	−0.470429	+	0.882438i	$0.655901\pi$
$242$	0	0
$243$	−39.5975	−2.54018
$244$	0	0
$245$	14.5383	0.928819
$246$	0	0
$247$	−4.50348	−0.286550
$248$	0	0
$249$	15.0577	0.954245
$250$	0	0
$251$	−19.0800	−1.20432	−0.602160	−	0.798375i	$-0.705693\pi$
$251$	−19.0800	−1.20432	−0.602160	+	0.798375i	$0.705693\pi$
$252$	0	0
$253$	−1.93269	−0.121507
$254$	0	0
$255$	74.4630	4.66306
$256$	0	0
$257$	5.21670	0.325409	0.162704	−	0.986675i	$-0.447978\pi$
$257$	5.21670	0.325409	0.162704	+	0.986675i	$0.447978\pi$
$258$	0	0
$259$	8.79789	0.546674
$260$	0	0
$261$	−45.6925	−2.82829
$262$	0	0
$263$	7.85899	0.484606	0.242303	−	0.970201i	$-0.422097\pi$
$263$	7.85899	0.484606	0.242303	+	0.970201i	$0.422097\pi$
$264$	0	0
$265$	7.64728	0.469769
$266$	0	0
$267$	50.9938	3.12077
$268$	0	0
$269$	−20.0890	−1.22485	−0.612425	−	0.790529i	$-0.709806\pi$
$269$	−20.0890	−1.22485	−0.612425	+	0.790529i	$0.709806\pi$
$270$	0	0
$271$	14.8510	0.902135	0.451068	−	0.892490i	$-0.351043\pi$
$271$	14.8510	0.902135	0.451068	+	0.892490i	$0.351043\pi$
$272$	0	0
$273$	−20.1477	−1.21940
$274$	0	0
$275$	−3.08863	−0.186252
$276$	0	0
$277$	26.1259	1.56976	0.784878	−	0.619650i	$-0.212726\pi$
$277$	26.1259	1.56976	0.784878	+	0.619650i	$0.212726\pi$
$278$	0	0
$279$	17.1210	1.02501
$280$	0	0
$281$	29.6886	1.77107	0.885537	−	0.464569i	$-0.153791\pi$
$281$	29.6886	1.77107	0.885537	+	0.464569i	$0.153791\pi$
$282$	0	0
$283$	16.9597	1.00815	0.504076	−	0.863659i	$-0.331833\pi$
$283$	16.9597	1.00815	0.504076	+	0.863659i	$0.331833\pi$
$284$	0	0
$285$	−9.25967	−0.548495
$286$	0	0
$287$	−6.15333	−0.363220
$288$	0	0
$289$	47.6682	2.80401
$290$	0	0
$291$	0.0835455	0.00489752
$292$	0	0
$293$	−20.6438	−1.20603	−0.603013	−	0.797731i	$-0.706033\pi$
$293$	−20.6438	−1.20603	−0.603013	+	0.797731i	$0.706033\pi$
$294$	0	0
$295$	−39.0708	−2.27479
$296$	0	0
$297$	14.9774	0.869079
$298$	0	0
$299$	8.70382	0.503355
$300$	0	0
$301$	−4.00052	−0.230586
$302$	0	0
$303$	26.5610	1.52589
$304$	0	0
$305$	−2.77392	−0.158834
$306$	0	0
$307$	19.6211	1.11983	0.559917	−	0.828548i	$-0.310833\pi$
$307$	19.6211	1.11983	0.559917	+	0.828548i	$0.310833\pi$
$308$	0	0
$309$	2.33942	0.133085
$310$	0	0
$311$	−1.11898	−0.0634514	−0.0317257	−	0.999497i	$-0.510100\pi$
$311$	−1.11898	−0.0634514	−0.0317257	+	0.999497i	$0.510100\pi$
$312$	0	0
$313$	27.1699	1.53573	0.767866	−	0.640610i	$-0.221318\pi$
$313$	27.1699	1.53573	0.767866	+	0.640610i	$0.221318\pi$
$314$	0	0
$315$	−29.7019	−1.67351
$316$	0	0
$317$	−1.34452	−0.0755160	−0.0377580	−	0.999287i	$-0.512022\pi$
$317$	−1.34452	−0.0755160	−0.0377580	+	0.999287i	$0.512022\pi$
$318$	0	0
$319$	6.01198	0.336606
$320$	0	0
$321$	−26.0666	−1.45490
$322$	0	0
$323$	−8.04165	−0.447450
$324$	0	0
$325$	13.9096	0.771567
$326$	0	0
$327$	−46.1786	−2.55368
$328$	0	0
$329$	−15.1247	−0.833854
$330$	0	0
$331$	27.4981	1.51143	0.755717	−	0.654898i	$-0.227288\pi$
$331$	27.4981	1.51143	0.755717	+	0.654898i	$0.227288\pi$
$332$	0	0
$333$	48.6615	2.66663
$334$	0	0
$335$	−18.4736	−1.00932
$336$	0	0
$337$	−6.19159	−0.337277	−0.168639	−	0.985678i	$-0.553937\pi$
$337$	−6.19159	−0.337277	−0.168639	+	0.985678i	$0.553937\pi$
$338$	0	0
$339$	6.81353	0.370060
$340$	0	0
$341$	−2.25270	−0.121990
$342$	0	0
$343$	−16.6429	−0.898634
$344$	0	0
$345$	17.8960	0.963490
$346$	0	0
$347$	−17.7460	−0.952655	−0.476327	−	0.879268i	$-0.658032\pi$
$347$	−17.7460	−0.952655	−0.476327	+	0.879268i	$0.658032\pi$
$348$	0	0
$349$	−4.41289	−0.236217	−0.118108	−	0.993001i	$-0.537683\pi$
$349$	−4.41289	−0.236217	−0.118108	+	0.993001i	$0.537683\pi$
$350$	0	0
$351$	−67.4507	−3.60025
$352$	0	0
$353$	−17.1553	−0.913085	−0.456542	−	0.889702i	$-0.650912\pi$
$353$	−17.1553	−0.913085	−0.456542	+	0.889702i	$0.650912\pi$
$354$	0	0
$355$	−44.8433	−2.38004
$356$	0	0
$357$	−35.9769	−1.90410
$358$	0	0
$359$	25.5639	1.34921	0.674604	−	0.738180i	$-0.264314\pi$
$359$	25.5639	1.34921	0.674604	+	0.738180i	$0.264314\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−3.25580	−0.170885
$364$	0	0
$365$	−14.2181	−0.744209
$366$	0	0
$367$	−10.8295	−0.565293	−0.282647	−	0.959224i	$-0.591212\pi$
$367$	−10.8295	−0.565293	−0.282647	+	0.959224i	$0.591212\pi$
$368$	0	0
$369$	−34.0343	−1.77176
$370$	0	0
$371$	−3.69479	−0.191824
$372$	0	0
$373$	14.1638	0.733375	0.366688	−	0.930344i	$-0.380492\pi$
$373$	14.1638	0.733375	0.366688	+	0.930344i	$0.380492\pi$
$374$	0	0
$375$	−17.6986	−0.913952
$376$	0	0
$377$	−27.0749	−1.39443
$378$	0	0
$379$	−20.2669	−1.04104	−0.520521	−	0.853849i	$-0.674262\pi$
$379$	−20.2669	−1.04104	−0.520521	+	0.853849i	$0.674262\pi$
$380$	0	0
$381$	36.3837	1.86399
$382$	0	0
$383$	7.05175	0.360328	0.180164	−	0.983637i	$-0.442337\pi$
$383$	7.05175	0.360328	0.180164	+	0.983637i	$0.442337\pi$
$384$	0	0
$385$	3.90803	0.199172
$386$	0	0
$387$	−22.1271	−1.12478
$388$	0	0
$389$	−16.5312	−0.838167	−0.419084	−	0.907948i	$-0.637649\pi$
$389$	−16.5312	−0.838167	−0.419084	+	0.907948i	$0.637649\pi$
$390$	0	0
$391$	15.5420	0.785992
$392$	0	0
$393$	53.4691	2.69716
$394$	0	0
$395$	−12.3299	−0.620383
$396$	0	0
$397$	−5.46890	−0.274476	−0.137238	−	0.990538i	$-0.543823\pi$
$397$	−5.46890	−0.274476	−0.137238	+	0.990538i	$0.543823\pi$
$398$	0	0
$399$	4.47381	0.223971
$400$	0	0
$401$	24.9701	1.24695	0.623473	−	0.781845i	$-0.285721\pi$
$401$	24.9701	1.24695	0.623473	+	0.781845i	$0.285721\pi$
$402$	0	0
$403$	10.1450	0.505358
$404$	0	0
$405$	−73.8397	−3.66913
$406$	0	0
$407$	−6.40263	−0.317367
$408$	0	0
$409$	−10.5729	−0.522796	−0.261398	−	0.965231i	$-0.584184\pi$
$409$	−10.5729	−0.522796	−0.261398	+	0.965231i	$0.584184\pi$
$410$	0	0
$411$	49.7783	2.45538
$412$	0	0
$413$	18.8771	0.928881
$414$	0	0
$415$	13.1534	0.645677
$416$	0	0
$417$	−27.9406	−1.36826
$418$	0	0
$419$	−14.1571	−0.691618	−0.345809	−	0.938305i	$-0.612395\pi$
$419$	−14.1571	−0.691618	−0.345809	+	0.938305i	$0.612395\pi$
$420$	0	0
$421$	8.26726	0.402921	0.201461	−	0.979497i	$-0.435431\pi$
$421$	8.26726	0.402921	0.201461	+	0.979497i	$0.435431\pi$
$422$	0	0
$423$	−83.6555	−4.06747
$424$	0	0
$425$	24.8377	1.20481
$426$	0	0
$427$	1.34022	0.0648579
$428$	0	0
$429$	14.6624	0.707909
$430$	0	0
$431$	21.7714	1.04869	0.524346	−	0.851505i	$-0.324310\pi$
$431$	21.7714	1.04869	0.524346	+	0.851505i	$0.324310\pi$
$432$	0	0
$433$	24.8734	1.19534	0.597670	−	0.801742i	$-0.296093\pi$
$433$	24.8734	1.19534	0.597670	+	0.801742i	$0.296093\pi$
$434$	0	0
$435$	−55.6689	−2.66912
$436$	0	0
$437$	−1.93269	−0.0924529
$438$	0	0
$439$	39.4333	1.88205	0.941025	−	0.338337i	$-0.109864\pi$
$439$	39.4333	1.88205	0.941025	+	0.338337i	$0.109864\pi$
$440$	0	0
$441$	−38.8511	−1.85005
$442$	0	0
$443$	−1.69836	−0.0806917	−0.0403458	−	0.999186i	$-0.512846\pi$
$443$	−1.69836	−0.0806917	−0.0403458	+	0.999186i	$0.512846\pi$
$444$	0	0
$445$	44.5448	2.11163
$446$	0	0
$447$	−12.8502	−0.607793
$448$	0	0
$449$	−9.77645	−0.461379	−0.230690	−	0.973027i	$-0.574098\pi$
$449$	−9.77645	−0.461379	−0.230690	+	0.973027i	$0.574098\pi$
$450$	0	0
$451$	4.47806	0.210864
$452$	0	0
$453$	38.7093	1.81872
$454$	0	0
$455$	−17.5997	−0.825089
$456$	0	0
$457$	−8.68741	−0.406380	−0.203190	−	0.979139i	$-0.565131\pi$
$457$	−8.68741	−0.406380	−0.203190	+	0.979139i	$0.565131\pi$
$458$	0	0
$459$	−120.443	−5.62182
$460$	0	0
$461$	11.0150	0.513019	0.256510	−	0.966542i	$-0.417427\pi$
$461$	11.0150	0.513019	0.256510	+	0.966542i	$0.417427\pi$
$462$	0	0
$463$	13.1676	0.611951	0.305975	−	0.952039i	$-0.401017\pi$
$463$	13.1676	0.611951	0.305975	+	0.952039i	$0.401017\pi$
$464$	0	0
$465$	20.8592	0.967323
$466$	0	0
$467$	−27.7203	−1.28274	−0.641372	−	0.767230i	$-0.721634\pi$
$467$	−27.7203	−1.28274	−0.641372	+	0.767230i	$0.721634\pi$
$468$	0	0
$469$	8.92552	0.412142
$470$	0	0
$471$	26.6095	1.22610
$472$	0	0
$473$	2.91137	0.133865
$474$	0	0
$475$	−3.08863	−0.141716
$476$	0	0
$477$	−20.4360	−0.935701
$478$	0	0
$479$	−15.0478	−0.687551	−0.343776	−	0.939052i	$-0.611706\pi$
$479$	−15.0478	−0.687551	−0.343776	+	0.939052i	$0.611706\pi$
$480$	0	0
$481$	28.8342	1.31472
$482$	0	0
$483$	−8.64648	−0.393428
$484$	0	0
$485$	0.0729798	0.00331384
$486$	0	0
$487$	−27.6712	−1.25390	−0.626950	−	0.779060i	$-0.715697\pi$
$487$	−27.6712	−1.25390	−0.626950	+	0.779060i	$0.715697\pi$
$488$	0	0
$489$	−46.4498	−2.10053
$490$	0	0
$491$	16.6990	0.753615	0.376808	−	0.926292i	$-0.377022\pi$
$491$	16.6990	0.753615	0.376808	+	0.926292i	$0.377022\pi$
$492$	0	0
$493$	−48.3463	−2.17741
$494$	0	0
$495$	21.6155	0.971543
$496$	0	0
$497$	21.6661	0.971856
$498$	0	0
$499$	0.367424	0.0164482	0.00822408	−	0.999966i	$-0.497382\pi$
$499$	0.367424	0.0164482	0.00822408	+	0.999966i	$0.497382\pi$
$500$	0	0
$501$	−34.3391	−1.53416
$502$	0	0
$503$	−9.17194	−0.408957	−0.204478	−	0.978871i	$-0.565550\pi$
$503$	−9.17194	−0.408957	−0.204478	+	0.978871i	$0.565550\pi$
$504$	0	0
$505$	23.2019	1.03247
$506$	0	0
$507$	−23.7067	−1.05285
$508$	0	0
$509$	7.17681	0.318107	0.159053	−	0.987270i	$-0.449156\pi$
$509$	7.17681	0.318107	0.159053	+	0.987270i	$0.449156\pi$
$510$	0	0
$511$	6.86948	0.303888
$512$	0	0
$513$	14.9774	0.661270
$514$	0	0
$515$	2.04356	0.0900502
$516$	0	0
$517$	11.0070	0.484086
$518$	0	0
$519$	−68.4863	−3.00622
$520$	0	0
$521$	−31.7113	−1.38930	−0.694648	−	0.719349i	$-0.744440\pi$
$521$	−31.7113	−1.38930	−0.694648	+	0.719349i	$0.744440\pi$
$522$	0	0
$523$	36.1546	1.58093	0.790465	−	0.612507i	$-0.209839\pi$
$523$	36.1546	1.58093	0.790465	+	0.612507i	$0.209839\pi$
$524$	0	0
$525$	−13.8180	−0.603066
$526$	0	0
$527$	18.1154	0.789120
$528$	0	0
$529$	−19.2647	−0.837597
$530$	0	0
$531$	104.410	4.53101
$532$	0	0
$533$	−20.1669	−0.873525
$534$	0	0
$535$	−22.7701	−0.984436
$536$	0	0
$537$	−65.1945	−2.81335
$538$	0	0
$539$	5.11183	0.220182
$540$	0	0
$541$	42.3309	1.81995	0.909973	−	0.414667i	$-0.136102\pi$
$541$	42.3309	1.81995	0.909973	+	0.414667i	$0.136102\pi$
$542$	0	0
$543$	−71.5969	−3.07252
$544$	0	0
$545$	−40.3385	−1.72791
$546$	0	0
$547$	−30.3262	−1.29665	−0.648327	−	0.761362i	$-0.724531\pi$
$547$	−30.3262	−1.29665	−0.648327	+	0.761362i	$0.724531\pi$
$548$	0	0
$549$	7.41283	0.316372
$550$	0	0
$551$	6.01198	0.256119
$552$	0	0
$553$	5.95719	0.253325
$554$	0	0
$555$	59.2862	2.51656
$556$	0	0
$557$	−19.1989	−0.813485	−0.406743	−	0.913543i	$-0.633335\pi$
$557$	−19.1989	−0.813485	−0.406743	+	0.913543i	$0.633335\pi$
$558$	0	0
$559$	−13.1113	−0.554549
$560$	0	0
$561$	26.1820	1.10541
$562$	0	0
$563$	−15.1018	−0.636463	−0.318232	−	0.948013i	$-0.603089\pi$
$563$	−15.1018	−0.636463	−0.318232	+	0.948013i	$0.603089\pi$
$564$	0	0
$565$	5.95185	0.250396
$566$	0	0
$567$	35.6757	1.49824
$568$	0	0
$569$	14.1342	0.592538	0.296269	−	0.955104i	$-0.404257\pi$
$569$	14.1342	0.592538	0.296269	+	0.955104i	$0.404257\pi$
$570$	0	0
$571$	29.9321	1.25262	0.626310	−	0.779574i	$-0.284565\pi$
$571$	29.9321	1.25262	0.626310	+	0.779574i	$0.284565\pi$
$572$	0	0
$573$	−36.4673	−1.52344
$574$	0	0
$575$	5.96936	0.248940
$576$	0	0
$577$	−32.6995	−1.36130	−0.680649	−	0.732610i	$-0.738302\pi$
$577$	−32.6995	−1.36130	−0.680649	+	0.732610i	$0.738302\pi$
$578$	0	0
$579$	20.2288	0.840678
$580$	0	0
$581$	−6.35509	−0.263654
$582$	0	0
$583$	2.68887	0.111362
$584$	0	0
$585$	−97.3449	−4.02472
$586$	0	0
$587$	30.9045	1.27557	0.637783	−	0.770216i	$-0.279852\pi$
$587$	30.9045	1.27557	0.637783	+	0.770216i	$0.279852\pi$
$588$	0	0
$589$	−2.25270	−0.0928208
$590$	0	0
$591$	89.2923	3.67299
$592$	0	0
$593$	17.0980	0.702131	0.351065	−	0.936351i	$-0.385819\pi$
$593$	17.0980	0.702131	0.351065	+	0.936351i	$0.385819\pi$
$594$	0	0
$595$	−31.4270	−1.28838
$596$	0	0
$597$	21.5325	0.881267
$598$	0	0
$599$	−1.28146	−0.0523591	−0.0261795	−	0.999657i	$-0.508334\pi$
$599$	−1.28146	−0.0523591	−0.0261795	+	0.999657i	$0.508334\pi$
$600$	0	0
$601$	−6.62399	−0.270198	−0.135099	−	0.990832i	$-0.543135\pi$
$601$	−6.62399	−0.270198	−0.135099	+	0.990832i	$0.543135\pi$
$602$	0	0
$603$	49.3674	2.01040
$604$	0	0
$605$	−2.84405	−0.115627
$606$	0	0
$607$	29.1755	1.18420	0.592099	−	0.805865i	$-0.298299\pi$
$607$	29.1755	1.18420	0.592099	+	0.805865i	$0.298299\pi$
$608$	0	0
$609$	26.8965	1.08990
$610$	0	0
$611$	−49.5697	−2.00538
$612$	0	0
$613$	14.9815	0.605099	0.302549	−	0.953134i	$-0.402162\pi$
$613$	14.9815	0.605099	0.302549	+	0.953134i	$0.402162\pi$
$614$	0	0
$615$	−41.4654	−1.67204
$616$	0	0
$617$	13.8278	0.556687	0.278344	−	0.960482i	$-0.410215\pi$
$617$	13.8278	0.556687	0.278344	+	0.960482i	$0.410215\pi$
$618$	0	0
$619$	21.0469	0.845947	0.422974	−	0.906142i	$-0.360986\pi$
$619$	21.0469	0.845947	0.422974	+	0.906142i	$0.360986\pi$
$620$	0	0
$621$	−28.9467	−1.16159
$622$	0	0
$623$	−21.5219	−0.862255
$624$	0	0
$625$	−30.9035	−1.23614
$626$	0	0
$627$	−3.25580	−0.130024
$628$	0	0
$629$	51.4878	2.05295
$630$	0	0
$631$	−38.2872	−1.52419	−0.762094	−	0.647466i	$-0.775829\pi$
$631$	−38.2872	−1.52419	−0.762094	+	0.647466i	$0.775829\pi$
$632$	0	0
$633$	−6.18372	−0.245781
$634$	0	0
$635$	31.7824	1.26125
$636$	0	0
$637$	−23.0211	−0.912128
$638$	0	0
$639$	119.836	4.74064
$640$	0	0
$641$	14.3840	0.568132	0.284066	−	0.958805i	$-0.408316\pi$
$641$	14.3840	0.568132	0.284066	+	0.958805i	$0.408316\pi$
$642$	0	0
$643$	−38.6271	−1.52331	−0.761653	−	0.647985i	$-0.775612\pi$
$643$	−38.6271	−1.52331	−0.761653	+	0.647985i	$0.775612\pi$
$644$	0	0
$645$	−26.9583	−1.06148
$646$	0	0
$647$	17.6026	0.692028	0.346014	−	0.938229i	$-0.387535\pi$
$647$	17.6026	0.692028	0.346014	+	0.938229i	$0.387535\pi$
$648$	0	0
$649$	−13.7377	−0.539253
$650$	0	0
$651$	−10.0781	−0.394994
$652$	0	0
$653$	5.96203	0.233312	0.116656	−	0.993172i	$-0.462782\pi$
$653$	5.96203	0.233312	0.116656	+	0.993172i	$0.462782\pi$
$654$	0	0
$655$	46.7071	1.82500
$656$	0	0
$657$	37.9954	1.48234
$658$	0	0
$659$	−6.15239	−0.239663	−0.119831	−	0.992794i	$-0.538235\pi$
$659$	−6.15239	−0.239663	−0.119831	+	0.992794i	$0.538235\pi$
$660$	0	0
$661$	38.0127	1.47852	0.739262	−	0.673418i	$-0.235175\pi$
$661$	38.0127	1.47852	0.739262	+	0.673418i	$0.235175\pi$
$662$	0	0
$663$	−117.910	−4.57926
$664$	0	0
$665$	3.90803	0.151547
$666$	0	0
$667$	−11.6193	−0.449900
$668$	0	0
$669$	7.25031	0.280313
$670$	0	0
$671$	−0.975342	−0.0376526
$672$	0	0
$673$	15.2077	0.586216	0.293108	−	0.956079i	$-0.405311\pi$
$673$	15.2077	0.586216	0.293108	+	0.956079i	$0.405311\pi$
$674$	0	0
$675$	−46.2598	−1.78054
$676$	0	0
$677$	−0.280407	−0.0107769	−0.00538846	−	0.999985i	$-0.501715\pi$
$677$	−0.280407	−0.0107769	−0.00538846	+	0.999985i	$0.501715\pi$
$678$	0	0
$679$	−0.0352602	−0.00135316
$680$	0	0
$681$	76.7074	2.93943
$682$	0	0
$683$	7.60480	0.290990	0.145495	−	0.989359i	$-0.453523\pi$
$683$	7.60480	0.290990	0.145495	+	0.989359i	$0.453523\pi$
$684$	0	0
$685$	43.4830	1.66140
$686$	0	0
$687$	−11.9173	−0.454675
$688$	0	0
$689$	−12.1093	−0.461327
$690$	0	0
$691$	−19.5795	−0.744839	−0.372420	−	0.928064i	$-0.621472\pi$
$691$	−19.5795	−0.744839	−0.372420	+	0.928064i	$0.621472\pi$
$692$	0	0
$693$	−10.4435	−0.396717
$694$	0	0
$695$	−24.4070	−0.925812
$696$	0	0
$697$	−36.0110	−1.36401
$698$	0	0
$699$	−66.5318	−2.51646
$700$	0	0
$701$	40.5251	1.53061	0.765305	−	0.643667i	$-0.222588\pi$
$701$	40.5251	1.53061	0.765305	+	0.643667i	$0.222588\pi$
$702$	0	0
$703$	−6.40263	−0.241480
$704$	0	0
$705$	−101.921	−3.83856
$706$	0	0
$707$	−11.2100	−0.421596
$708$	0	0
$709$	22.9931	0.863526	0.431763	−	0.901987i	$-0.357892\pi$
$709$	22.9931	0.863526	0.431763	+	0.901987i	$0.357892\pi$
$710$	0	0
$711$	32.9494	1.23570
$712$	0	0
$713$	4.35376	0.163049
$714$	0	0
$715$	12.8081	0.478997
$716$	0	0
$717$	−35.6561	−1.33160
$718$	0	0
$719$	23.9307	0.892464	0.446232	−	0.894917i	$-0.352766\pi$
$719$	23.9307	0.892464	0.446232	+	0.894917i	$0.352766\pi$
$720$	0	0
$721$	−0.987349	−0.0367708
$722$	0	0
$723$	47.5544	1.76857
$724$	0	0
$725$	−18.5688	−0.689628
$726$	0	0
$727$	−10.0644	−0.373268	−0.186634	−	0.982430i	$-0.559758\pi$
$727$	−10.0644	−0.373268	−0.186634	+	0.982430i	$0.559758\pi$
$728$	0	0
$729$	51.0331	1.89011
$730$	0	0
$731$	−23.4122	−0.865931
$732$	0	0
$733$	−25.8983	−0.956577	−0.478289	−	0.878203i	$-0.658743\pi$
$733$	−25.8983	−0.956577	−0.478289	+	0.878203i	$0.658743\pi$
$734$	0	0
$735$	−47.3339	−1.74594
$736$	0	0
$737$	−6.49551	−0.239265
$738$	0	0
$739$	30.5526	1.12390	0.561948	−	0.827173i	$-0.310052\pi$
$739$	30.5526	1.12390	0.561948	+	0.827173i	$0.310052\pi$
$740$	0	0
$741$	14.6624	0.538638
$742$	0	0
$743$	16.0916	0.590343	0.295171	−	0.955444i	$-0.404623\pi$
$743$	16.0916	0.590343	0.295171	+	0.955444i	$0.404623\pi$
$744$	0	0
$745$	−11.2251	−0.411255
$746$	0	0
$747$	−35.1503	−1.28608
$748$	0	0
$749$	11.0014	0.401981
$750$	0	0
$751$	31.4829	1.14883	0.574413	−	0.818565i	$-0.305230\pi$
$751$	31.4829	1.14883	0.574413	+	0.818565i	$0.305230\pi$
$752$	0	0
$753$	62.1207	2.26380
$754$	0	0
$755$	33.8139	1.23061
$756$	0	0
$757$	7.80967	0.283847	0.141924	−	0.989878i	$-0.454671\pi$
$757$	7.80967	0.283847	0.141924	+	0.989878i	$0.454671\pi$
$758$	0	0
$759$	6.29244	0.228401
$760$	0	0
$761$	−24.1930	−0.876996	−0.438498	−	0.898732i	$-0.644489\pi$
$761$	−24.1930	−0.876996	−0.438498	+	0.898732i	$0.644489\pi$
$762$	0	0
$763$	19.4896	0.705570
$764$	0	0
$765$	−173.824	−6.28462
$766$	0	0
$767$	61.8676	2.23391
$768$	0	0
$769$	43.9975	1.58659	0.793296	−	0.608837i	$-0.208363\pi$
$769$	43.9975	1.58659	0.793296	+	0.608837i	$0.208363\pi$
$770$	0	0
$771$	−16.9845	−0.611683
$772$	0	0
$773$	3.76339	0.135360	0.0676798	−	0.997707i	$-0.478440\pi$
$773$	3.76339	0.135360	0.0676798	+	0.997707i	$0.478440\pi$
$774$	0	0
$775$	6.95776	0.249930
$776$	0	0
$777$	−28.6442	−1.02760
$778$	0	0
$779$	4.47806	0.160443
$780$	0	0
$781$	−15.7674	−0.564202
$782$	0	0
$783$	90.0441	3.21791
$784$	0	0
$785$	23.2443	0.829624
$786$	0	0
$787$	6.12260	0.218247	0.109124	−	0.994028i	$-0.465196\pi$
$787$	6.12260	0.218247	0.109124	+	0.994028i	$0.465196\pi$
$788$	0	0
$789$	−25.5873	−0.910932
$790$	0	0
$791$	−2.87564	−0.102246
$792$	0	0
$793$	4.39244	0.155980
$794$	0	0
$795$	−24.8980	−0.883042
$796$	0	0
$797$	26.2100	0.928407	0.464203	−	0.885729i	$-0.346341\pi$
$797$	26.2100	0.928407	0.464203	+	0.885729i	$0.346341\pi$
$798$	0	0
$799$	−88.5142	−3.13141
$800$	0	0
$801$	−119.038	−4.20601
$802$	0	0
$803$	−4.99924	−0.176419
$804$	0	0
$805$	−7.55299	−0.266208
$806$	0	0
$807$	65.4059	2.30240
$808$	0	0
$809$	−0.682663	−0.0240011	−0.0120006	−	0.999928i	$-0.503820\pi$
$809$	−0.682663	−0.0240011	−0.0120006	+	0.999928i	$0.503820\pi$
$810$	0	0
$811$	−41.0850	−1.44269	−0.721345	−	0.692576i	$-0.756476\pi$
$811$	−41.0850	−1.44269	−0.721345	+	0.692576i	$0.756476\pi$
$812$	0	0
$813$	−48.3520	−1.69578
$814$	0	0
$815$	−40.5755	−1.42130
$816$	0	0
$817$	2.91137	0.101856
$818$	0	0
$819$	47.0322	1.64344
$820$	0	0
$821$	41.8409	1.46026	0.730130	−	0.683309i	$-0.239460\pi$
$821$	41.8409	1.46026	0.730130	+	0.683309i	$0.239460\pi$
$822$	0	0
$823$	27.5677	0.960948	0.480474	−	0.877009i	$-0.340465\pi$
$823$	27.5677	0.960948	0.480474	+	0.877009i	$0.340465\pi$
$824$	0	0
$825$	10.0560	0.350104
$826$	0	0
$827$	22.3995	0.778906	0.389453	−	0.921046i	$-0.372664\pi$
$827$	22.3995	0.778906	0.389453	+	0.921046i	$0.372664\pi$
$828$	0	0
$829$	18.8022	0.653028	0.326514	−	0.945192i	$-0.394126\pi$
$829$	18.8022	0.653028	0.326514	+	0.945192i	$0.394126\pi$
$830$	0	0
$831$	−85.0609	−2.95073
$832$	0	0
$833$	−41.1076	−1.42429
$834$	0	0
$835$	−29.9964	−1.03807
$836$	0	0
$837$	−33.7396	−1.16621
$838$	0	0
$839$	30.3399	1.04745	0.523724	−	0.851888i	$-0.324542\pi$
$839$	30.3399	1.04745	0.523724	+	0.851888i	$0.324542\pi$
$840$	0	0
$841$	7.14392	0.246342
$842$	0	0
$843$	−96.6602	−3.32915
$844$	0	0
$845$	−20.7086	−0.712397
$846$	0	0
$847$	1.37411	0.0472148
$848$	0	0
$849$	−55.2176	−1.89506
$850$	0	0
$851$	12.3743	0.424185
$852$	0	0
$853$	22.5600	0.772440	0.386220	−	0.922407i	$-0.373781\pi$
$853$	22.5600	0.772440	0.386220	+	0.922407i	$0.373781\pi$
$854$	0	0
$855$	21.6155	0.739233
$856$	0	0
$857$	9.70243	0.331429	0.165714	−	0.986174i	$-0.447007\pi$
$857$	9.70243	0.331429	0.165714	+	0.986174i	$0.447007\pi$
$858$	0	0
$859$	−27.7185	−0.945743	−0.472872	−	0.881131i	$-0.656783\pi$
$859$	−27.7185	−0.945743	−0.472872	+	0.881131i	$0.656783\pi$
$860$	0	0
$861$	20.0340	0.682757
$862$	0	0
$863$	−11.2399	−0.382611	−0.191306	−	0.981531i	$-0.561272\pi$
$863$	−11.2399	−0.382611	−0.191306	+	0.981531i	$0.561272\pi$
$864$	0	0
$865$	−59.8252	−2.03412
$866$	0	0
$867$	−155.198	−5.27080
$868$	0	0
$869$	−4.33532	−0.147066
$870$	0	0
$871$	29.2524	0.991181
$872$	0	0
$873$	−0.195026	−0.00660063
$874$	0	0
$875$	7.46967	0.252521
$876$	0	0
$877$	33.2559	1.12297	0.561486	−	0.827486i	$-0.310230\pi$
$877$	33.2559	1.12297	0.561486	+	0.827486i	$0.310230\pi$
$878$	0	0
$879$	67.2122	2.26701
$880$	0	0
$881$	30.9264	1.04194	0.520968	−	0.853576i	$-0.325571\pi$
$881$	30.9264	1.04194	0.520968	+	0.853576i	$0.325571\pi$
$882$	0	0
$883$	−39.6834	−1.33545	−0.667726	−	0.744407i	$-0.732732\pi$
$883$	−39.6834	−1.33545	−0.667726	+	0.744407i	$0.732732\pi$
$884$	0	0
$885$	127.207	4.27601
$886$	0	0
$887$	−10.3257	−0.346701	−0.173351	−	0.984860i	$-0.555459\pi$
$887$	−10.3257	−0.346701	−0.173351	+	0.984860i	$0.555459\pi$
$888$	0	0
$889$	−15.3557	−0.515013
$890$	0	0
$891$	−25.9629	−0.869788
$892$	0	0
$893$	11.0070	0.368334
$894$	0	0
$895$	−56.9496	−1.90362
$896$	0	0
$897$	−28.3379	−0.946175
$898$	0	0
$899$	−13.5432	−0.451690
$900$	0	0
$901$	−21.6229	−0.720365
$902$	0	0
$903$	13.0249	0.433442
$904$	0	0
$905$	−62.5423	−2.07898
$906$	0	0
$907$	−32.0397	−1.06386	−0.531931	−	0.846788i	$-0.678533\pi$
$907$	−32.0397	−1.06386	−0.531931	+	0.846788i	$0.678533\pi$
$908$	0	0
$909$	−62.0031	−2.05651
$910$	0	0
$911$	37.1616	1.23122	0.615609	−	0.788052i	$-0.288910\pi$
$911$	37.1616	1.23122	0.615609	+	0.788052i	$0.288910\pi$
$912$	0	0
$913$	4.62489	0.153062
$914$	0	0
$915$	9.03134	0.298567
$916$	0	0
$917$	−22.5666	−0.745213
$918$	0	0
$919$	35.1368	1.15906	0.579528	−	0.814952i	$-0.303237\pi$
$919$	35.1368	1.15906	0.579528	+	0.814952i	$0.303237\pi$
$920$	0	0
$921$	−63.8823	−2.10499
$922$	0	0
$923$	71.0082	2.33726
$924$	0	0
$925$	19.7754	0.650211
$926$	0	0
$927$	−5.46107	−0.179365
$928$	0	0
$929$	−7.85026	−0.257559	−0.128779	−	0.991673i	$-0.541106\pi$
$929$	−7.85026	−0.257559	−0.128779	+	0.991673i	$0.541106\pi$
$930$	0	0
$931$	5.11183	0.167534
$932$	0	0
$933$	3.64317	0.119272
$934$	0	0
$935$	22.8709	0.747958
$936$	0	0
$937$	5.47492	0.178858	0.0894289	−	0.995993i	$-0.471496\pi$
$937$	5.47492	0.178858	0.0894289	+	0.995993i	$0.471496\pi$
$938$	0	0
$939$	−88.4597	−2.88677
$940$	0	0
$941$	20.8056	0.678244	0.339122	−	0.940742i	$-0.389870\pi$
$941$	20.8056	0.678244	0.339122	+	0.940742i	$0.389870\pi$
$942$	0	0
$943$	−8.65469	−0.281835
$944$	0	0
$945$	58.5323	1.90405
$946$	0	0
$947$	−25.2333	−0.819973	−0.409986	−	0.912092i	$-0.634467\pi$
$947$	−25.2333	−0.819973	−0.409986	+	0.912092i	$0.634467\pi$
$948$	0	0
$949$	22.5140	0.730835
$950$	0	0
$951$	4.37750	0.141950
$952$	0	0
$953$	−29.1735	−0.945023	−0.472511	−	0.881325i	$-0.656652\pi$
$953$	−29.1735	−0.945023	−0.472511	+	0.881325i	$0.656652\pi$
$954$	0	0
$955$	−31.8554	−1.03082
$956$	0	0
$957$	−19.5738	−0.632731
$958$	0	0
$959$	−21.0088	−0.678411
$960$	0	0
$961$	−25.9254	−0.836302
$962$	0	0
$963$	60.8490	1.96083
$964$	0	0
$965$	17.6705	0.568834
$966$	0	0
$967$	5.49314	0.176647	0.0883237	−	0.996092i	$-0.471849\pi$
$967$	5.49314	0.176647	0.0883237	+	0.996092i	$0.471849\pi$
$968$	0	0
$969$	26.1820	0.841088
$970$	0	0
$971$	−15.5655	−0.499521	−0.249760	−	0.968308i	$-0.580352\pi$
$971$	−15.5655	−0.499521	−0.249760	+	0.968308i	$0.580352\pi$
$972$	0	0
$973$	11.7923	0.378043
$974$	0	0
$975$	−45.2869	−1.45034
$976$	0	0
$977$	−39.3827	−1.25996	−0.629982	−	0.776609i	$-0.716938\pi$
$977$	−39.3827	−1.25996	−0.629982	+	0.776609i	$0.716938\pi$
$978$	0	0
$979$	15.6624	0.500574
$980$	0	0
$981$	107.798	3.44172
$982$	0	0
$983$	14.6297	0.466614	0.233307	−	0.972403i	$-0.425045\pi$
$983$	14.6297	0.466614	0.233307	+	0.972403i	$0.425045\pi$
$984$	0	0
$985$	77.9998	2.48528
$986$	0	0
$987$	49.2431	1.56743
$988$	0	0
$989$	−5.62676	−0.178920
$990$	0	0
$991$	24.1290	0.766482	0.383241	−	0.923648i	$-0.374808\pi$
$991$	24.1290	0.766482	0.383241	+	0.923648i	$0.374808\pi$
$992$	0	0
$993$	−89.5285	−2.84110
$994$	0	0
$995$	18.8094	0.596297
$996$	0	0
$997$	−60.4987	−1.91601	−0.958007	−	0.286745i	$-0.907427\pi$
$997$	−60.4987	−1.91601	−0.958007	+	0.286745i	$0.907427\pi$
$998$	0	0
$999$	−95.8951	−3.03399

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.x.1.1		6
4.3	odd	2		836.2.a.d.1.6	✓	6
12.11	even	2		7524.2.a.r.1.5		6
44.43	even	2		9196.2.a.k.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
836.2.a.d.1.6	✓	6	4.3	odd	2
3344.2.a.x.1.1		6	1.1	even	1	trivial
7524.2.a.r.1.5		6	12.11	even	2
9196.2.a.k.1.6		6	44.43	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:	\( N \)	\(=\)	\( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3344.a (trivial)

\(f(q)\)	\(=\)	\(q-3.25580 q^{3} -2.84405 q^{5} +1.37411 q^{7} +7.60023 q^{9} +O(q^{10})\)	\(q-3.25580 q^{3} -2.84405 q^{5} +1.37411 q^{7} +7.60023 q^{9} -1.00000 q^{11} +4.50348 q^{13} +9.25967 q^{15} +8.04165 q^{17} -1.00000 q^{19} -4.47381 q^{21} +1.93269 q^{23} +3.08863 q^{25} -14.9774 q^{27} -6.01198 q^{29} +2.25270 q^{31} +3.25580 q^{33} -3.90803 q^{35} +6.40263 q^{37} -14.6624 q^{39} -4.47806 q^{41} -2.91137 q^{43} -21.6155 q^{45} -11.0070 q^{47} -5.11183 q^{49} -26.1820 q^{51} -2.68887 q^{53} +2.84405 q^{55} +3.25580 q^{57} +13.7377 q^{59} +0.975342 q^{61} +10.4435 q^{63} -12.8081 q^{65} +6.49551 q^{67} -6.29244 q^{69} +15.7674 q^{71} +4.99924 q^{73} -10.0560 q^{75} -1.37411 q^{77} +4.33532 q^{79} +25.9629 q^{81} -4.62489 q^{83} -22.8709 q^{85} +19.5738 q^{87} -15.6624 q^{89} +6.18826 q^{91} -7.33433 q^{93} +2.84405 q^{95} -0.0256605 q^{97} -7.60023 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{3} - 2 q^{7} + 10 q^{9}+O(q^{10}) \) 6 * q + 2 * q^3 - 2 * q^7 + 10 * q^9	\( 6 q + 2 q^{3} - 2 q^{7} + 10 q^{9} - 6 q^{11} + 2 q^{13} + 12 q^{15} + 12 q^{17} - 6 q^{19} + 2 q^{21} + 2 q^{23} + 26 q^{25} - 16 q^{27} + 4 q^{29} + 20 q^{31} - 2 q^{33} - 20 q^{35} + 22 q^{37} - 8 q^{39} - 2 q^{41} - 10 q^{43} - 6 q^{45} - 16 q^{47} + 48 q^{49} - 36 q^{51} + 12 q^{53} - 2 q^{57} + 14 q^{59} + 12 q^{61} + 4 q^{63} + 10 q^{65} + 12 q^{67} + 30 q^{69} + 30 q^{71} + 24 q^{73} + 50 q^{75} + 2 q^{77} + 10 q^{81} + 14 q^{83} + 12 q^{85} + 30 q^{87} - 14 q^{89} - 20 q^{91} + 14 q^{93} - 46 q^{97} - 10 q^{99}+O(q^{100}) \) 6 * q + 2 * q^3 - 2 * q^7 + 10 * q^9 - 6 * q^11 + 2 * q^13 + 12 * q^15 + 12 * q^17 - 6 * q^19 + 2 * q^21 + 2 * q^23 + 26 * q^25 - 16 * q^27 + 4 * q^29 + 20 * q^31 - 2 * q^33 - 20 * q^35 + 22 * q^37 - 8 * q^39 - 2 * q^41 - 10 * q^43 - 6 * q^45 - 16 * q^47 + 48 * q^49 - 36 * q^51 + 12 * q^53 - 2 * q^57 + 14 * q^59 + 12 * q^61 + 4 * q^63 + 10 * q^65 + 12 * q^67 + 30 * q^69 + 30 * q^71 + 24 * q^73 + 50 * q^75 + 2 * q^77 + 10 * q^81 + 14 * q^83 + 12 * q^85 + 30 * q^87 - 14 * q^89 - 20 * q^91 + 14 * q^93 - 46 * q^97 - 10 * q^99

Introduction

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