LMFDB - Embedded newform 6032.2.a.ba.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6032,2,Mod(1,6032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6032, 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("6032.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6032 = 2^{4} \cdot 13 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6032.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:	$48.1657624992$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 21x^{8} + 40x^{7} + 138x^{6} - 243x^{5} - 318x^{4} + 448x^{3} + 312x^{2} - 240x - 128 $ x^10 - 2x^9 - 21x^8 + 40x^7 + 138x^6 - 243x^5 - 318x^4 + 448x^3 + 312x^2 - 240*x - 128
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 3016)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.29348$ of defining polynomial
Character	$\chi$	$=$	6032.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.29348 q^{3} -3.79162 q^{5} +2.99662 q^{7} -1.32691 q^{9} +O(q^{10})$	$q-1.29348 q^{3} -3.79162 q^{5} +2.99662 q^{7} -1.32691 q^{9} +2.34268 q^{11} +1.00000 q^{13} +4.90439 q^{15} +1.52459 q^{17} +5.79068 q^{19} -3.87607 q^{21} -4.28115 q^{23} +9.37642 q^{25} +5.59677 q^{27} -1.00000 q^{29} -0.366222 q^{31} -3.03021 q^{33} -11.3621 q^{35} +3.94749 q^{37} -1.29348 q^{39} -4.70380 q^{41} -1.35389 q^{43} +5.03115 q^{45} -11.2725 q^{47} +1.97976 q^{49} -1.97202 q^{51} +6.22764 q^{53} -8.88257 q^{55} -7.49012 q^{57} +9.59312 q^{59} -0.912713 q^{61} -3.97626 q^{63} -3.79162 q^{65} -2.96039 q^{67} +5.53758 q^{69} -12.6079 q^{71} +1.27258 q^{73} -12.1282 q^{75} +7.02013 q^{77} +4.00116 q^{79} -3.25857 q^{81} +0.0710861 q^{83} -5.78067 q^{85} +1.29348 q^{87} -2.95173 q^{89} +2.99662 q^{91} +0.473700 q^{93} -21.9561 q^{95} -0.0872232 q^{97} -3.10853 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 2 q^{3} + 5 q^{5} + 2 q^{7} + 16 q^{9}+O(q^{10}) $ 10 * q - 2 * q^3 + 5 * q^5 + 2 * q^7 + 16 * q^9	$ 10 q - 2 q^{3} + 5 q^{5} + 2 q^{7} + 16 q^{9} - 4 q^{11} + 10 q^{13} - 8 q^{15} + 10 q^{17} + q^{19} + q^{21} - 23 q^{23} + 25 q^{25} - 2 q^{27} - 10 q^{29} + 13 q^{31} + 15 q^{33} + 12 q^{35} - 7 q^{37} - 2 q^{39} + 16 q^{41} + 12 q^{43} + 55 q^{45} - 11 q^{47} + 25 q^{51} + 11 q^{53} - 22 q^{55} - 6 q^{57} + 11 q^{59} + 34 q^{61} - 37 q^{63} + 5 q^{65} + 23 q^{67} + 2 q^{69} + 4 q^{71} + 39 q^{73} - 11 q^{75} + 32 q^{77} - 5 q^{79} + 38 q^{81} - 6 q^{83} + 45 q^{85} + 2 q^{87} - 24 q^{89} + 2 q^{91} + 13 q^{93} - 33 q^{95} + 19 q^{97}+O(q^{100}) $ 10 * q - 2 * q^3 + 5 * q^5 + 2 * q^7 + 16 * q^9 - 4 * q^11 + 10 * q^13 - 8 * q^15 + 10 * q^17 + q^19 + q^21 - 23 * q^23 + 25 * q^25 - 2 * q^27 - 10 * q^29 + 13 * q^31 + 15 * q^33 + 12 * q^35 - 7 * q^37 - 2 * q^39 + 16 * q^41 + 12 * q^43 + 55 * q^45 - 11 * q^47 + 25 * q^51 + 11 * q^53 - 22 * q^55 - 6 * q^57 + 11 * q^59 + 34 * q^61 - 37 * q^63 + 5 * q^65 + 23 * q^67 + 2 * q^69 + 4 * q^71 + 39 * q^73 - 11 * q^75 + 32 * q^77 - 5 * q^79 + 38 * q^81 - 6 * q^83 + 45 * q^85 + 2 * q^87 - 24 * q^89 + 2 * q^91 + 13 * q^93 - 33 * q^95 + 19 * 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
$3$	−1.29348	−0.746790	−0.373395	−	0.927672i	$-0.621806\pi$
$3$	−1.29348	−0.746790	−0.373395	+	0.927672i	$0.621806\pi$
$4$	0	0
$5$	−3.79162	−1.69567	−0.847833	−	0.530263i	$-0.822093\pi$
$5$	−3.79162	−1.69567	−0.847833	+	0.530263i	$0.822093\pi$
$6$	0	0
$7$	2.99662	1.13262	0.566309	−	0.824193i	$-0.308371\pi$
$7$	2.99662	1.13262	0.566309	+	0.824193i	$0.308371\pi$
$8$	0	0
$9$	−1.32691	−0.442304
$10$	0	0
$11$	2.34268	0.706345	0.353172	−	0.935558i	$-0.385103\pi$
$11$	2.34268	0.706345	0.353172	+	0.935558i	$0.385103\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	4.90439	1.26631
$16$	0	0
$17$	1.52459	0.369767	0.184884	−	0.982760i	$-0.440809\pi$
$17$	1.52459	0.369767	0.184884	+	0.982760i	$0.440809\pi$
$18$	0	0
$19$	5.79068	1.32847	0.664237	−	0.747522i	$-0.268757\pi$
$19$	5.79068	1.32847	0.664237	+	0.747522i	$0.268757\pi$
$20$	0	0
$21$	−3.87607	−0.845828
$22$	0	0
$23$	−4.28115	−0.892681	−0.446341	−	0.894863i	$-0.647273\pi$
$23$	−4.28115	−0.892681	−0.446341	+	0.894863i	$0.647273\pi$
$24$	0	0
$25$	9.37642	1.87528
$26$	0	0
$27$	5.59677	1.07710
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−0.366222	−0.0657754	−0.0328877	−	0.999459i	$-0.510470\pi$
$31$	−0.366222	−0.0657754	−0.0328877	+	0.999459i	$0.510470\pi$
$32$	0	0
$33$	−3.03021	−0.527492
$34$	0	0
$35$	−11.3621	−1.92054
$36$	0	0
$37$	3.94749	0.648963	0.324481	−	0.945892i	$-0.394810\pi$
$37$	3.94749	0.648963	0.324481	+	0.945892i	$0.394810\pi$
$38$	0	0
$39$	−1.29348	−0.207122
$40$	0	0
$41$	−4.70380	−0.734610	−0.367305	−	0.930100i	$-0.619720\pi$
$41$	−4.70380	−0.734610	−0.367305	+	0.930100i	$0.619720\pi$
$42$	0	0
$43$	−1.35389	−0.206467	−0.103233	−	0.994657i	$-0.532919\pi$
$43$	−1.35389	−0.206467	−0.103233	+	0.994657i	$0.532919\pi$
$44$	0	0
$45$	5.03115	0.750000
$46$	0	0
$47$	−11.2725	−1.64426	−0.822132	−	0.569298i	$-0.807215\pi$
$47$	−11.2725	−1.64426	−0.822132	+	0.569298i	$0.807215\pi$
$48$	0	0
$49$	1.97976	0.282823
$50$	0	0
$51$	−1.97202	−0.276138
$52$	0	0
$53$	6.22764	0.855432	0.427716	−	0.903913i	$-0.359318\pi$
$53$	6.22764	0.855432	0.427716	+	0.903913i	$0.359318\pi$
$54$	0	0
$55$	−8.88257	−1.19773
$56$	0	0
$57$	−7.49012	−0.992091
$58$	0	0
$59$	9.59312	1.24892	0.624459	−	0.781058i	$-0.285319\pi$
$59$	9.59312	1.24892	0.624459	+	0.781058i	$0.285319\pi$
$60$	0	0
$61$	−0.912713	−0.116861	−0.0584305	−	0.998291i	$-0.518610\pi$
$61$	−0.912713	−0.116861	−0.0584305	+	0.998291i	$0.518610\pi$
$62$	0	0
$63$	−3.97626	−0.500961
$64$	0	0
$65$	−3.79162	−0.470293
$66$	0	0
$67$	−2.96039	−0.361669	−0.180834	−	0.983514i	$-0.557880\pi$
$67$	−2.96039	−0.361669	−0.180834	+	0.983514i	$0.557880\pi$
$68$	0	0
$69$	5.53758	0.666646
$70$	0	0
$71$	−12.6079	−1.49628	−0.748142	−	0.663538i	$-0.769054\pi$
$71$	−12.6079	−1.49628	−0.748142	+	0.663538i	$0.769054\pi$
$72$	0	0
$73$	1.27258	0.148944	0.0744720	−	0.997223i	$-0.476273\pi$
$73$	1.27258	0.148944	0.0744720	+	0.997223i	$0.476273\pi$
$74$	0	0
$75$	−12.1282	−1.40044
$76$	0	0
$77$	7.02013	0.800019
$78$	0	0
$79$	4.00116	0.450166	0.225083	−	0.974340i	$-0.427735\pi$
$79$	4.00116	0.450166	0.225083	+	0.974340i	$0.427735\pi$
$80$	0	0
$81$	−3.25857	−0.362063
$82$	0	0
$83$	0.0710861	0.00780271	0.00390136	−	0.999992i	$-0.498758\pi$
$83$	0.0710861	0.00780271	0.00390136	+	0.999992i	$0.498758\pi$
$84$	0	0
$85$	−5.78067	−0.627002
$86$	0	0
$87$	1.29348	0.138675
$88$	0	0
$89$	−2.95173	−0.312883	−0.156441	−	0.987687i	$-0.550002\pi$
$89$	−2.95173	−0.312883	−0.156441	+	0.987687i	$0.550002\pi$
$90$	0	0
$91$	2.99662	0.314132
$92$	0	0
$93$	0.473700	0.0491204
$94$	0	0
$95$	−21.9561	−2.25265
$96$	0	0
$97$	−0.0872232	−0.00885617	−0.00442809	−	0.999990i	$-0.501410\pi$
$97$	−0.0872232	−0.00885617	−0.00442809	+	0.999990i	$0.501410\pi$
$98$	0	0
$99$	−3.10853	−0.312419
$100$	0	0
$101$	2.73728	0.272370	0.136185	−	0.990683i	$-0.456516\pi$
$101$	2.73728	0.272370	0.136185	+	0.990683i	$0.456516\pi$
$102$	0	0
$103$	−16.4827	−1.62409	−0.812044	−	0.583596i	$-0.801645\pi$
$103$	−16.4827	−1.62409	−0.812044	+	0.583596i	$0.801645\pi$
$104$	0	0
$105$	14.6966	1.43424
$106$	0	0
$107$	−1.54447	−0.149310	−0.0746548	−	0.997209i	$-0.523785\pi$
$107$	−1.54447	−0.149310	−0.0746548	+	0.997209i	$0.523785\pi$
$108$	0	0
$109$	5.94929	0.569839	0.284919	−	0.958551i	$-0.408033\pi$
$109$	5.94929	0.569839	0.284919	+	0.958551i	$0.408033\pi$
$110$	0	0
$111$	−5.10599	−0.484639
$112$	0	0
$113$	7.86363	0.739748	0.369874	−	0.929082i	$-0.379401\pi$
$113$	7.86363	0.739748	0.369874	+	0.929082i	$0.379401\pi$
$114$	0	0
$115$	16.2325	1.51369
$116$	0	0
$117$	−1.32691	−0.122673
$118$	0	0
$119$	4.56862	0.418805
$120$	0	0
$121$	−5.51185	−0.501077
$122$	0	0
$123$	6.08427	0.548600
$124$	0	0
$125$	−16.5937	−1.48419
$126$	0	0
$127$	0.189180	0.0167870	0.00839352	−	0.999965i	$-0.497328\pi$
$127$	0.189180	0.0167870	0.00839352	+	0.999965i	$0.497328\pi$
$128$	0	0
$129$	1.75123	0.154187
$130$	0	0
$131$	−5.25565	−0.459189	−0.229594	−	0.973286i	$-0.573740\pi$
$131$	−5.25565	−0.459189	−0.229594	+	0.973286i	$0.573740\pi$
$132$	0	0
$133$	17.3525	1.50465
$134$	0	0
$135$	−21.2209	−1.82640
$136$	0	0
$137$	15.2818	1.30561	0.652805	−	0.757526i	$-0.273592\pi$
$137$	15.2818	1.30561	0.652805	+	0.757526i	$0.273592\pi$
$138$	0	0
$139$	16.4693	1.39691	0.698453	−	0.715656i	$-0.253872\pi$
$139$	16.4693	1.39691	0.698453	+	0.715656i	$0.253872\pi$
$140$	0	0
$141$	14.5807	1.22792
$142$	0	0
$143$	2.34268	0.195905
$144$	0	0
$145$	3.79162	0.314877
$146$	0	0
$147$	−2.56078	−0.211209
$148$	0	0
$149$	−5.10867	−0.418518	−0.209259	−	0.977860i	$-0.567105\pi$
$149$	−5.10867	−0.418518	−0.209259	+	0.977860i	$0.567105\pi$
$150$	0	0
$151$	6.15334	0.500752	0.250376	−	0.968149i	$-0.419446\pi$
$151$	6.15334	0.500752	0.250376	+	0.968149i	$0.419446\pi$
$152$	0	0
$153$	−2.02300	−0.163550
$154$	0	0
$155$	1.38858	0.111533
$156$	0	0
$157$	22.3276	1.78194	0.890968	−	0.454065i	$-0.150027\pi$
$157$	22.3276	1.78194	0.890968	+	0.454065i	$0.150027\pi$
$158$	0	0
$159$	−8.05532	−0.638829
$160$	0	0
$161$	−12.8290	−1.01107
$162$	0	0
$163$	9.15840	0.717341	0.358670	−	0.933464i	$-0.383230\pi$
$163$	9.15840	0.717341	0.358670	+	0.933464i	$0.383230\pi$
$164$	0	0
$165$	11.4894	0.894450
$166$	0	0
$167$	5.75042	0.444981	0.222490	−	0.974935i	$-0.428581\pi$
$167$	5.75042	0.444981	0.222490	+	0.974935i	$0.428581\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−7.68373	−0.587589
$172$	0	0
$173$	−0.262266	−0.0199397	−0.00996985	−	0.999950i	$-0.503174\pi$
$173$	−0.262266	−0.0199397	−0.00996985	+	0.999950i	$0.503174\pi$
$174$	0	0
$175$	28.0976	2.12398
$176$	0	0
$177$	−12.4085	−0.932680
$178$	0	0
$179$	−9.77423	−0.730560	−0.365280	−	0.930898i	$-0.619027\pi$
$179$	−9.77423	−0.730560	−0.365280	+	0.930898i	$0.619027\pi$
$180$	0	0
$181$	12.6498	0.940251	0.470125	−	0.882600i	$-0.344209\pi$
$181$	12.6498	0.940251	0.470125	+	0.882600i	$0.344209\pi$
$182$	0	0
$183$	1.18058	0.0872706
$184$	0	0
$185$	−14.9674	−1.10042
$186$	0	0
$187$	3.57162	0.261183
$188$	0	0
$189$	16.7714	1.21994
$190$	0	0
$191$	−12.2352	−0.885309	−0.442654	−	0.896692i	$-0.645963\pi$
$191$	−12.2352	−0.885309	−0.442654	+	0.896692i	$0.645963\pi$
$192$	0	0
$193$	4.28508	0.308447	0.154224	−	0.988036i	$-0.450712\pi$
$193$	4.28508	0.308447	0.154224	+	0.988036i	$0.450712\pi$
$194$	0	0
$195$	4.90439	0.351210
$196$	0	0
$197$	−14.5082	−1.03366	−0.516832	−	0.856087i	$-0.672889\pi$
$197$	−14.5082	−1.03366	−0.516832	+	0.856087i	$0.672889\pi$
$198$	0	0
$199$	12.5410	0.889007	0.444504	−	0.895777i	$-0.353380\pi$
$199$	12.5410	0.889007	0.444504	+	0.895777i	$0.353380\pi$
$200$	0	0
$201$	3.82920	0.270091
$202$	0	0
$203$	−2.99662	−0.210322
$204$	0	0
$205$	17.8350	1.24565
$206$	0	0
$207$	5.68071	0.394837
$208$	0	0
$209$	13.5657	0.938360
$210$	0	0
$211$	−6.04892	−0.416425	−0.208213	−	0.978084i	$-0.566765\pi$
$211$	−6.04892	−0.416425	−0.208213	+	0.978084i	$0.566765\pi$
$212$	0	0
$213$	16.3081	1.11741
$214$	0	0
$215$	5.13345	0.350099
$216$	0	0
$217$	−1.09743	−0.0744983
$218$	0	0
$219$	−1.64605	−0.111230
$220$	0	0
$221$	1.52459	0.102555
$222$	0	0
$223$	11.9985	0.803480	0.401740	−	0.915754i	$-0.368406\pi$
$223$	11.9985	0.803480	0.401740	+	0.915754i	$0.368406\pi$
$224$	0	0
$225$	−12.4417	−0.829446
$226$	0	0
$227$	−24.1460	−1.60263	−0.801314	−	0.598243i	$-0.795866\pi$
$227$	−24.1460	−1.60263	−0.801314	+	0.598243i	$0.795866\pi$
$228$	0	0
$229$	−20.6916	−1.36734	−0.683669	−	0.729792i	$-0.739617\pi$
$229$	−20.6916	−1.36734	−0.683669	+	0.729792i	$0.739617\pi$
$230$	0	0
$231$	−9.08040	−0.597446
$232$	0	0
$233$	27.4915	1.80103	0.900513	−	0.434830i	$-0.143191\pi$
$233$	27.4915	1.80103	0.900513	+	0.434830i	$0.143191\pi$
$234$	0	0
$235$	42.7411	2.78812
$236$	0	0
$237$	−5.17542	−0.336180
$238$	0	0
$239$	20.6257	1.33416	0.667081	−	0.744985i	$-0.267543\pi$
$239$	20.6257	1.33416	0.667081	+	0.744985i	$0.267543\pi$
$240$	0	0
$241$	−9.56479	−0.616122	−0.308061	−	0.951367i	$-0.599680\pi$
$241$	−9.56479	−0.616122	−0.308061	+	0.951367i	$0.599680\pi$
$242$	0	0
$243$	−12.5754	−0.806714
$244$	0	0
$245$	−7.50650	−0.479573
$246$	0	0
$247$	5.79068	0.368452
$248$	0	0
$249$	−0.0919484	−0.00582699
$250$	0	0
$251$	17.3508	1.09518	0.547588	−	0.836748i	$-0.315546\pi$
$251$	17.3508	1.09518	0.547588	+	0.836748i	$0.315546\pi$
$252$	0	0
$253$	−10.0294	−0.630541
$254$	0	0
$255$	7.47717	0.468239
$256$	0	0
$257$	−18.3043	−1.14179	−0.570897	−	0.821022i	$-0.693404\pi$
$257$	−18.3043	−1.14179	−0.570897	+	0.821022i	$0.693404\pi$
$258$	0	0
$259$	11.8291	0.735027
$260$	0	0
$261$	1.32691	0.0821338
$262$	0	0
$263$	27.8893	1.71973	0.859864	−	0.510523i	$-0.170548\pi$
$263$	27.8893	1.71973	0.859864	+	0.510523i	$0.170548\pi$
$264$	0	0
$265$	−23.6129	−1.45053
$266$	0	0
$267$	3.81800	0.233658
$268$	0	0
$269$	16.1940	0.987368	0.493684	−	0.869641i	$-0.335650\pi$
$269$	16.1940	0.987368	0.493684	+	0.869641i	$0.335650\pi$
$270$	0	0
$271$	11.0162	0.669185	0.334592	−	0.942363i	$-0.391401\pi$
$271$	11.0162	0.669185	0.334592	+	0.942363i	$0.391401\pi$
$272$	0	0
$273$	−3.87607	−0.234590
$274$	0	0
$275$	21.9660	1.32460
$276$	0	0
$277$	−24.0942	−1.44768	−0.723840	−	0.689968i	$-0.757624\pi$
$277$	−24.0942	−1.44768	−0.723840	+	0.689968i	$0.757624\pi$
$278$	0	0
$279$	0.485944	0.0290927
$280$	0	0
$281$	29.2919	1.74741	0.873705	−	0.486455i	$-0.161710\pi$
$281$	29.2919	1.74741	0.873705	+	0.486455i	$0.161710\pi$
$282$	0	0
$283$	2.32786	0.138377	0.0691884	−	0.997604i	$-0.477959\pi$
$283$	2.32786	0.138377	0.0691884	+	0.997604i	$0.477959\pi$
$284$	0	0
$285$	28.3997	1.68226
$286$	0	0
$287$	−14.0955	−0.832033
$288$	0	0
$289$	−14.6756	−0.863272
$290$	0	0
$291$	0.112821	0.00661371
$292$	0	0
$293$	2.54173	0.148490	0.0742448	−	0.997240i	$-0.476345\pi$
$293$	2.54173	0.148490	0.0742448	+	0.997240i	$0.476345\pi$
$294$	0	0
$295$	−36.3735	−2.11775
$296$	0	0
$297$	13.1114	0.760803
$298$	0	0
$299$	−4.28115	−0.247585
$300$	0	0
$301$	−4.05711	−0.233848
$302$	0	0
$303$	−3.54062	−0.203403
$304$	0	0
$305$	3.46067	0.198157
$306$	0	0
$307$	25.6883	1.46611	0.733055	−	0.680170i	$-0.238094\pi$
$307$	25.6883	1.46611	0.733055	+	0.680170i	$0.238094\pi$
$308$	0	0
$309$	21.3200	1.21285
$310$	0	0
$311$	5.50725	0.312287	0.156144	−	0.987734i	$-0.450094\pi$
$311$	5.50725	0.312287	0.156144	+	0.987734i	$0.450094\pi$
$312$	0	0
$313$	14.2447	0.805158	0.402579	−	0.915385i	$-0.368114\pi$
$313$	14.2447	0.805158	0.402579	+	0.915385i	$0.368114\pi$
$314$	0	0
$315$	15.0765	0.849463
$316$	0	0
$317$	31.7855	1.78525	0.892625	−	0.450800i	$-0.148861\pi$
$317$	31.7855	1.78525	0.892625	+	0.450800i	$0.148861\pi$
$318$	0	0
$319$	−2.34268	−0.131165
$320$	0	0
$321$	1.99774	0.111503
$322$	0	0
$323$	8.82841	0.491226
$324$	0	0
$325$	9.37642	0.520110
$326$	0	0
$327$	−7.69528	−0.425550
$328$	0	0
$329$	−33.7795	−1.86232
$330$	0	0
$331$	−4.04734	−0.222462	−0.111231	−	0.993795i	$-0.535479\pi$
$331$	−4.04734	−0.222462	−0.111231	+	0.993795i	$0.535479\pi$
$332$	0	0
$333$	−5.23797	−0.287039
$334$	0	0
$335$	11.2247	0.613270
$336$	0	0
$337$	21.4761	1.16988	0.584940	−	0.811077i	$-0.301118\pi$
$337$	21.4761	1.16988	0.584940	+	0.811077i	$0.301118\pi$
$338$	0	0
$339$	−10.1714	−0.552437
$340$	0	0
$341$	−0.857941	−0.0464601
$342$	0	0
$343$	−15.0438	−0.812288
$344$	0	0
$345$	−20.9964	−1.13041
$346$	0	0
$347$	7.43025	0.398877	0.199438	−	0.979910i	$-0.436088\pi$
$347$	7.43025	0.398877	0.199438	+	0.979910i	$0.436088\pi$
$348$	0	0
$349$	−33.1553	−1.77476	−0.887381	−	0.461037i	$-0.847478\pi$
$349$	−33.1553	−1.77476	−0.887381	+	0.461037i	$0.847478\pi$
$350$	0	0
$351$	5.59677	0.298733
$352$	0	0
$353$	17.8840	0.951869	0.475934	−	0.879481i	$-0.342110\pi$
$353$	17.8840	0.951869	0.475934	+	0.879481i	$0.342110\pi$
$354$	0	0
$355$	47.8045	2.53720
$356$	0	0
$357$	−5.90941	−0.312759
$358$	0	0
$359$	7.84389	0.413985	0.206992	−	0.978343i	$-0.433632\pi$
$359$	7.84389	0.413985	0.206992	+	0.978343i	$0.433632\pi$
$360$	0	0
$361$	14.5320	0.764842
$362$	0	0
$363$	7.12946	0.374199
$364$	0	0
$365$	−4.82514	−0.252559
$366$	0	0
$367$	4.97521	0.259704	0.129852	−	0.991533i	$-0.458550\pi$
$367$	4.97521	0.259704	0.129852	+	0.991533i	$0.458550\pi$
$368$	0	0
$369$	6.24153	0.324921
$370$	0	0
$371$	18.6619	0.968878
$372$	0	0
$373$	11.1867	0.579224	0.289612	−	0.957144i	$-0.406474\pi$
$373$	11.1867	0.579224	0.289612	+	0.957144i	$0.406474\pi$
$374$	0	0
$375$	21.4637	1.10838
$376$	0	0
$377$	−1.00000	−0.0515026
$378$	0	0
$379$	10.2887	0.528497	0.264248	−	0.964455i	$-0.414876\pi$
$379$	10.2887	0.528497	0.264248	+	0.964455i	$0.414876\pi$
$380$	0	0
$381$	−0.244701	−0.0125364
$382$	0	0
$383$	−35.4730	−1.81258	−0.906292	−	0.422652i	$-0.861099\pi$
$383$	−35.4730	−1.81258	−0.906292	+	0.422652i	$0.861099\pi$
$384$	0	0
$385$	−26.6177	−1.35656
$386$	0	0
$387$	1.79650	0.0913211
$388$	0	0
$389$	23.9731	1.21548	0.607741	−	0.794135i	$-0.292076\pi$
$389$	23.9731	1.21548	0.607741	+	0.794135i	$0.292076\pi$
$390$	0	0
$391$	−6.52699	−0.330084
$392$	0	0
$393$	6.79808	0.342918
$394$	0	0
$395$	−15.1709	−0.763331
$396$	0	0
$397$	24.9857	1.25400	0.626998	−	0.779021i	$-0.284283\pi$
$397$	24.9857	1.25400	0.626998	+	0.779021i	$0.284283\pi$
$398$	0	0
$399$	−22.4451	−1.12366
$400$	0	0
$401$	−8.95635	−0.447259	−0.223629	−	0.974674i	$-0.571791\pi$
$401$	−8.95635	−0.447259	−0.223629	+	0.974674i	$0.571791\pi$
$402$	0	0
$403$	−0.366222	−0.0182428
$404$	0	0
$405$	12.3553	0.613938
$406$	0	0
$407$	9.24770	0.458391
$408$	0	0
$409$	26.6613	1.31831	0.659157	−	0.752005i	$-0.270913\pi$
$409$	26.6613	1.31831	0.659157	+	0.752005i	$0.270913\pi$
$410$	0	0
$411$	−19.7666	−0.975017
$412$	0	0
$413$	28.7470	1.41455
$414$	0	0
$415$	−0.269532	−0.0132308
$416$	0	0
$417$	−21.3027	−1.04320
$418$	0	0
$419$	8.86876	0.433267	0.216634	−	0.976253i	$-0.430492\pi$
$419$	8.86876	0.433267	0.216634	+	0.976253i	$0.430492\pi$
$420$	0	0
$421$	29.2969	1.42784	0.713922	−	0.700226i	$-0.246917\pi$
$421$	29.2969	1.42784	0.713922	+	0.700226i	$0.246917\pi$
$422$	0	0
$423$	14.9576	0.727264
$424$	0	0
$425$	14.2952	0.693418
$426$	0	0
$427$	−2.73506	−0.132359
$428$	0	0
$429$	−3.03021	−0.146300
$430$	0	0
$431$	−15.4987	−0.746548	−0.373274	−	0.927721i	$-0.621765\pi$
$431$	−15.4987	−0.746548	−0.373274	+	0.927721i	$0.621765\pi$
$432$	0	0
$433$	−18.1970	−0.874493	−0.437246	−	0.899342i	$-0.644046\pi$
$433$	−18.1970	−0.874493	−0.437246	+	0.899342i	$0.644046\pi$
$434$	0	0
$435$	−4.90439	−0.235147
$436$	0	0
$437$	−24.7908	−1.18590
$438$	0	0
$439$	25.9041	1.23634	0.618168	−	0.786046i	$-0.287875\pi$
$439$	25.9041	1.23634	0.618168	+	0.786046i	$0.287875\pi$
$440$	0	0
$441$	−2.62697	−0.125094
$442$	0	0
$443$	−21.5215	−1.02252	−0.511259	−	0.859426i	$-0.670821\pi$
$443$	−21.5215	−1.02252	−0.511259	+	0.859426i	$0.670821\pi$
$444$	0	0
$445$	11.1919	0.530545
$446$	0	0
$447$	6.60795	0.312545
$448$	0	0
$449$	9.17460	0.432976	0.216488	−	0.976285i	$-0.430540\pi$
$449$	9.17460	0.432976	0.216488	+	0.976285i	$0.430540\pi$
$450$	0	0
$451$	−11.0195	−0.518888
$452$	0	0
$453$	−7.95922	−0.373957
$454$	0	0
$455$	−11.3621	−0.532662
$456$	0	0
$457$	−4.66304	−0.218128	−0.109064	−	0.994035i	$-0.534785\pi$
$457$	−4.66304	−0.218128	−0.109064	+	0.994035i	$0.534785\pi$
$458$	0	0
$459$	8.53277	0.398276
$460$	0	0
$461$	−19.8008	−0.922216	−0.461108	−	0.887344i	$-0.652548\pi$
$461$	−19.8008	−0.922216	−0.461108	+	0.887344i	$0.652548\pi$
$462$	0	0
$463$	1.04775	0.0486932	0.0243466	−	0.999704i	$-0.492249\pi$
$463$	1.04775	0.0486932	0.0243466	+	0.999704i	$0.492249\pi$
$464$	0	0
$465$	−1.79609	−0.0832918
$466$	0	0
$467$	−1.41705	−0.0655734	−0.0327867	−	0.999462i	$-0.510438\pi$
$467$	−1.41705	−0.0655734	−0.0327867	+	0.999462i	$0.510438\pi$
$468$	0	0
$469$	−8.87117	−0.409632
$470$	0	0
$471$	−28.8803	−1.33073
$472$	0	0
$473$	−3.17174	−0.145837
$474$	0	0
$475$	54.2959	2.49126
$476$	0	0
$477$	−8.26354	−0.378361
$478$	0	0
$479$	−17.5371	−0.801291	−0.400646	−	0.916233i	$-0.631214\pi$
$479$	−17.5371	−0.801291	−0.400646	+	0.916233i	$0.631214\pi$
$480$	0	0
$481$	3.94749	0.179990
$482$	0	0
$483$	16.5940	0.755055
$484$	0	0
$485$	0.330718	0.0150171
$486$	0	0
$487$	−41.9410	−1.90053	−0.950263	−	0.311447i	$-0.899186\pi$
$487$	−41.9410	−1.90053	−0.950263	+	0.311447i	$0.899186\pi$
$488$	0	0
$489$	−11.8462	−0.535703
$490$	0	0
$491$	8.47639	0.382534	0.191267	−	0.981538i	$-0.438740\pi$
$491$	8.47639	0.382534	0.191267	+	0.981538i	$0.438740\pi$
$492$	0	0
$493$	−1.52459	−0.0686640
$494$	0	0
$495$	11.7864	0.529759
$496$	0	0
$497$	−37.7812	−1.69472
$498$	0	0
$499$	−16.0338	−0.717774	−0.358887	−	0.933381i	$-0.616844\pi$
$499$	−16.0338	−0.717774	−0.358887	+	0.933381i	$0.616844\pi$
$500$	0	0
$501$	−7.43805	−0.332307
$502$	0	0
$503$	−13.3358	−0.594613	−0.297306	−	0.954782i	$-0.596088\pi$
$503$	−13.3358	−0.594613	−0.297306	+	0.954782i	$0.596088\pi$
$504$	0	0
$505$	−10.3787	−0.461848
$506$	0	0
$507$	−1.29348	−0.0574454
$508$	0	0
$509$	7.36408	0.326407	0.163203	−	0.986592i	$-0.447817\pi$
$509$	7.36408	0.326407	0.163203	+	0.986592i	$0.447817\pi$
$510$	0	0
$511$	3.81344	0.168697
$512$	0	0
$513$	32.4091	1.43090
$514$	0	0
$515$	62.4962	2.75391
$516$	0	0
$517$	−26.4079	−1.16142
$518$	0	0
$519$	0.339235	0.0148908
$520$	0	0
$521$	42.4778	1.86099	0.930493	−	0.366310i	$-0.119379\pi$
$521$	42.4778	1.86099	0.930493	+	0.366310i	$0.119379\pi$
$522$	0	0
$523$	14.1576	0.619068	0.309534	−	0.950888i	$-0.399827\pi$
$523$	14.1576	0.619068	0.309534	+	0.950888i	$0.399827\pi$
$524$	0	0
$525$	−36.3437	−1.58617
$526$	0	0
$527$	−0.558337	−0.0243216
$528$	0	0
$529$	−4.67176	−0.203120
$530$	0	0
$531$	−12.7292	−0.552402
$532$	0	0
$533$	−4.70380	−0.203744
$534$	0	0
$535$	5.85605	0.253179
$536$	0	0
$537$	12.6428	0.545575
$538$	0	0
$539$	4.63794	0.199770
$540$	0	0
$541$	41.5794	1.78764	0.893820	−	0.448427i	$-0.148015\pi$
$541$	41.5794	1.78764	0.893820	+	0.448427i	$0.148015\pi$
$542$	0	0
$543$	−16.3622	−0.702170
$544$	0	0
$545$	−22.5575	−0.966257
$546$	0	0
$547$	24.7246	1.05715	0.528574	−	0.848887i	$-0.322727\pi$
$547$	24.7246	1.05715	0.528574	+	0.848887i	$0.322727\pi$
$548$	0	0
$549$	1.21109	0.0516881
$550$	0	0
$551$	−5.79068	−0.246691
$552$	0	0
$553$	11.9900	0.509866
$554$	0	0
$555$	19.3600	0.821786
$556$	0	0
$557$	−9.46818	−0.401180	−0.200590	−	0.979675i	$-0.564286\pi$
$557$	−9.46818	−0.401180	−0.200590	+	0.979675i	$0.564286\pi$
$558$	0	0
$559$	−1.35389	−0.0572635
$560$	0	0
$561$	−4.61982	−0.195049
$562$	0	0
$563$	23.4481	0.988221	0.494110	−	0.869399i	$-0.335494\pi$
$563$	23.4481	0.988221	0.494110	+	0.869399i	$0.335494\pi$
$564$	0	0
$565$	−29.8159	−1.25437
$566$	0	0
$567$	−9.76470	−0.410079
$568$	0	0
$569$	24.8602	1.04219	0.521096	−	0.853498i	$-0.325523\pi$
$569$	24.8602	1.04219	0.521096	+	0.853498i	$0.325523\pi$
$570$	0	0
$571$	−10.3237	−0.432032	−0.216016	−	0.976390i	$-0.569306\pi$
$571$	−10.3237	−0.432032	−0.216016	+	0.976390i	$0.569306\pi$
$572$	0	0
$573$	15.8260	0.661140
$574$	0	0
$575$	−40.1419	−1.67403
$576$	0	0
$577$	−22.0564	−0.918221	−0.459110	−	0.888379i	$-0.651832\pi$
$577$	−22.0564	−0.918221	−0.459110	+	0.888379i	$0.651832\pi$
$578$	0	0
$579$	−5.54267	−0.230345
$580$	0	0
$581$	0.213018	0.00883749
$582$	0	0
$583$	14.5894	0.604230
$584$	0	0
$585$	5.03115	0.208013
$586$	0	0
$587$	−15.0269	−0.620226	−0.310113	−	0.950700i	$-0.600367\pi$
$587$	−15.0269	−0.620226	−0.310113	+	0.950700i	$0.600367\pi$
$588$	0	0
$589$	−2.12067	−0.0873808
$590$	0	0
$591$	18.7660	0.771930
$592$	0	0
$593$	22.7819	0.935540	0.467770	−	0.883850i	$-0.345057\pi$
$593$	22.7819	0.935540	0.467770	+	0.883850i	$0.345057\pi$
$594$	0	0
$595$	−17.3225	−0.710153
$596$	0	0
$597$	−16.2215	−0.663902
$598$	0	0
$599$	−1.02845	−0.0420213	−0.0210107	−	0.999779i	$-0.506688\pi$
$599$	−1.02845	−0.0420213	−0.0210107	+	0.999779i	$0.506688\pi$
$600$	0	0
$601$	24.0412	0.980660	0.490330	−	0.871537i	$-0.336876\pi$
$601$	24.0412	0.980660	0.490330	+	0.871537i	$0.336876\pi$
$602$	0	0
$603$	3.92817	0.159968
$604$	0	0
$605$	20.8989	0.849659
$606$	0	0
$607$	−25.1353	−1.02021	−0.510104	−	0.860113i	$-0.670393\pi$
$607$	−25.1353	−1.02021	−0.510104	+	0.860113i	$0.670393\pi$
$608$	0	0
$609$	3.87607	0.157066
$610$	0	0
$611$	−11.2725	−0.456037
$612$	0	0
$613$	21.5519	0.870472	0.435236	−	0.900316i	$-0.356665\pi$
$613$	21.5519	0.870472	0.435236	+	0.900316i	$0.356665\pi$
$614$	0	0
$615$	−23.0693	−0.930242
$616$	0	0
$617$	17.0979	0.688337	0.344169	−	0.938908i	$-0.388161\pi$
$617$	17.0979	0.688337	0.344169	+	0.938908i	$0.388161\pi$
$618$	0	0
$619$	2.39676	0.0963341	0.0481670	−	0.998839i	$-0.484662\pi$
$619$	2.39676	0.0963341	0.0481670	+	0.998839i	$0.484662\pi$
$620$	0	0
$621$	−23.9606	−0.961506
$622$	0	0
$623$	−8.84523	−0.354377
$624$	0	0
$625$	16.0351	0.641406
$626$	0	0
$627$	−17.5470	−0.700759
$628$	0	0
$629$	6.01829	0.239965
$630$	0	0
$631$	26.7335	1.06424	0.532121	−	0.846668i	$-0.321395\pi$
$631$	26.7335	1.06424	0.532121	+	0.846668i	$0.321395\pi$
$632$	0	0
$633$	7.82416	0.310982
$634$	0	0
$635$	−0.717301	−0.0284652
$636$	0	0
$637$	1.97976	0.0784409
$638$	0	0
$639$	16.7296	0.661813
$640$	0	0
$641$	29.8251	1.17802	0.589010	−	0.808126i	$-0.299518\pi$
$641$	29.8251	1.17802	0.589010	+	0.808126i	$0.299518\pi$
$642$	0	0
$643$	−24.5854	−0.969552	−0.484776	−	0.874638i	$-0.661099\pi$
$643$	−24.5854	−0.969552	−0.484776	+	0.874638i	$0.661099\pi$
$644$	0	0
$645$	−6.64001	−0.261450
$646$	0	0
$647$	−44.9924	−1.76883	−0.884416	−	0.466698i	$-0.845443\pi$
$647$	−44.9924	−1.76883	−0.884416	+	0.466698i	$0.845443\pi$
$648$	0	0
$649$	22.4736	0.882167
$650$	0	0
$651$	1.41950	0.0556346
$652$	0	0
$653$	−10.5553	−0.413059	−0.206530	−	0.978440i	$-0.566217\pi$
$653$	−10.5553	−0.413059	−0.206530	+	0.978440i	$0.566217\pi$
$654$	0	0
$655$	19.9275	0.778630
$656$	0	0
$657$	−1.68860	−0.0658786
$658$	0	0
$659$	7.95177	0.309757	0.154878	−	0.987934i	$-0.450501\pi$
$659$	7.95177	0.309757	0.154878	+	0.987934i	$0.450501\pi$
$660$	0	0
$661$	−43.7422	−1.70137	−0.850687	−	0.525672i	$-0.823814\pi$
$661$	−43.7422	−1.70137	−0.850687	+	0.525672i	$0.823814\pi$
$662$	0	0
$663$	−1.97202	−0.0765870
$664$	0	0
$665$	−65.7942	−2.55139
$666$	0	0
$667$	4.28115	0.165767
$668$	0	0
$669$	−15.5198	−0.600031
$670$	0	0
$671$	−2.13820	−0.0825441
$672$	0	0
$673$	28.3972	1.09463	0.547315	−	0.836927i	$-0.315650\pi$
$673$	28.3972	1.09463	0.547315	+	0.836927i	$0.315650\pi$
$674$	0	0
$675$	52.4777	2.01987
$676$	0	0
$677$	−23.2724	−0.894430	−0.447215	−	0.894426i	$-0.647584\pi$
$677$	−23.2724	−0.894430	−0.447215	+	0.894426i	$0.647584\pi$
$678$	0	0
$679$	−0.261375	−0.0100307
$680$	0	0
$681$	31.2324	1.19683
$682$	0	0
$683$	44.8105	1.71463	0.857314	−	0.514794i	$-0.172132\pi$
$683$	44.8105	1.71463	0.857314	+	0.514794i	$0.172132\pi$
$684$	0	0
$685$	−57.9427	−2.21388
$686$	0	0
$687$	26.7641	1.02112
$688$	0	0
$689$	6.22764	0.237254
$690$	0	0
$691$	−15.9330	−0.606121	−0.303060	−	0.952971i	$-0.598008\pi$
$691$	−15.9330	−0.606121	−0.303060	+	0.952971i	$0.598008\pi$
$692$	0	0
$693$	−9.31510	−0.353852
$694$	0	0
$695$	−62.4454	−2.36869
$696$	0	0
$697$	−7.17136	−0.271635
$698$	0	0
$699$	−35.5596	−1.34499
$700$	0	0
$701$	20.2473	0.764731	0.382366	−	0.924011i	$-0.375109\pi$
$701$	20.2473	0.764731	0.382366	+	0.924011i	$0.375109\pi$
$702$	0	0
$703$	22.8586	0.862130
$704$	0	0
$705$	−55.2847	−2.08214
$706$	0	0
$707$	8.20261	0.308491
$708$	0	0
$709$	30.1884	1.13375	0.566875	−	0.823804i	$-0.308152\pi$
$709$	30.1884	1.13375	0.566875	+	0.823804i	$0.308152\pi$
$710$	0	0
$711$	−5.30919	−0.199110
$712$	0	0
$713$	1.56785	0.0587164
$714$	0	0
$715$	−8.88257	−0.332189
$716$	0	0
$717$	−26.6789	−0.996340
$718$	0	0
$719$	−12.5066	−0.466416	−0.233208	−	0.972427i	$-0.574922\pi$
$719$	−12.5066	−0.466416	−0.233208	+	0.972427i	$0.574922\pi$
$720$	0	0
$721$	−49.3924	−1.83947
$722$	0	0
$723$	12.3719	0.460114
$724$	0	0
$725$	−9.37642	−0.348231
$726$	0	0
$727$	23.9071	0.886667	0.443333	−	0.896357i	$-0.353796\pi$
$727$	23.9071	0.886667	0.443333	+	0.896357i	$0.353796\pi$
$728$	0	0
$729$	26.0417	0.964509
$730$	0	0
$731$	−2.06413	−0.0763446
$732$	0	0
$733$	16.9508	0.626091	0.313045	−	0.949738i	$-0.398651\pi$
$733$	16.9508	0.626091	0.313045	+	0.949738i	$0.398651\pi$
$734$	0	0
$735$	9.70950	0.358140
$736$	0	0
$737$	−6.93524	−0.255463
$738$	0	0
$739$	20.7187	0.762150	0.381075	−	0.924544i	$-0.375554\pi$
$739$	20.7187	0.762150	0.381075	+	0.924544i	$0.375554\pi$
$740$	0	0
$741$	−7.49012	−0.275157
$742$	0	0
$743$	19.7341	0.723974	0.361987	−	0.932183i	$-0.382098\pi$
$743$	19.7341	0.723974	0.361987	+	0.932183i	$0.382098\pi$
$744$	0	0
$745$	19.3701	0.709667
$746$	0	0
$747$	−0.0943250	−0.00345117
$748$	0	0
$749$	−4.62820	−0.169111
$750$	0	0
$751$	50.8653	1.85610	0.928051	−	0.372454i	$-0.121483\pi$
$751$	50.8653	1.85610	0.928051	+	0.372454i	$0.121483\pi$
$752$	0	0
$753$	−22.4429	−0.817867
$754$	0	0
$755$	−23.3312	−0.849108
$756$	0	0
$757$	17.9576	0.652679	0.326339	−	0.945253i	$-0.394185\pi$
$757$	17.9576	0.652679	0.326339	+	0.945253i	$0.394185\pi$
$758$	0	0
$759$	12.9728	0.470882
$760$	0	0
$761$	42.4218	1.53779	0.768894	−	0.639376i	$-0.220807\pi$
$761$	42.4218	1.53779	0.768894	+	0.639376i	$0.220807\pi$
$762$	0	0
$763$	17.8278	0.645410
$764$	0	0
$765$	7.67044	0.277325
$766$	0	0
$767$	9.59312	0.346388
$768$	0	0
$769$	−17.0406	−0.614501	−0.307251	−	0.951629i	$-0.599409\pi$
$769$	−17.0406	−0.614501	−0.307251	+	0.951629i	$0.599409\pi$
$770$	0	0
$771$	23.6763	0.852681
$772$	0	0
$773$	−22.5707	−0.811811	−0.405906	−	0.913915i	$-0.633044\pi$
$773$	−22.5707	−0.811811	−0.405906	+	0.913915i	$0.633044\pi$
$774$	0	0
$775$	−3.43385	−0.123347
$776$	0	0
$777$	−15.3007	−0.548911
$778$	0	0
$779$	−27.2382	−0.975910
$780$	0	0
$781$	−29.5363	−1.05689
$782$	0	0
$783$	−5.59677	−0.200012
$784$	0	0
$785$	−84.6579	−3.02157
$786$	0	0
$787$	11.0849	0.395134	0.197567	−	0.980289i	$-0.436696\pi$
$787$	11.0849	0.395134	0.197567	+	0.980289i	$0.436696\pi$
$788$	0	0
$789$	−36.0742	−1.28428
$790$	0	0
$791$	23.5644	0.837852
$792$	0	0
$793$	−0.912713	−0.0324114
$794$	0	0
$795$	30.5428	1.08324
$796$	0	0
$797$	−37.8694	−1.34140	−0.670702	−	0.741727i	$-0.734007\pi$
$797$	−37.8694	−1.34140	−0.670702	+	0.741727i	$0.734007\pi$
$798$	0	0
$799$	−17.1859	−0.607994
$800$	0	0
$801$	3.91669	0.138389
$802$	0	0
$803$	2.98125	0.105206
$804$	0	0
$805$	48.6427	1.71443
$806$	0	0
$807$	−20.9466	−0.737357
$808$	0	0
$809$	−34.7938	−1.22328	−0.611642	−	0.791135i	$-0.709491\pi$
$809$	−34.7938	−1.22328	−0.611642	+	0.791135i	$0.709491\pi$
$810$	0	0
$811$	−34.0969	−1.19730	−0.598652	−	0.801009i	$-0.704297\pi$
$811$	−34.0969	−1.19730	−0.598652	+	0.801009i	$0.704297\pi$
$812$	0	0
$813$	−14.2492	−0.499741
$814$	0	0
$815$	−34.7252	−1.21637
$816$	0	0
$817$	−7.83996	−0.274285
$818$	0	0
$819$	−3.97626	−0.138942
$820$	0	0
$821$	−19.8240	−0.691863	−0.345931	−	0.938260i	$-0.612437\pi$
$821$	−19.8240	−0.691863	−0.345931	+	0.938260i	$0.612437\pi$
$822$	0	0
$823$	36.0880	1.25795	0.628974	−	0.777427i	$-0.283475\pi$
$823$	36.0880	1.25795	0.628974	+	0.777427i	$0.283475\pi$
$824$	0	0
$825$	−28.4125	−0.989196
$826$	0	0
$827$	1.46777	0.0510395	0.0255198	−	0.999674i	$-0.491876\pi$
$827$	1.46777	0.0510395	0.0255198	+	0.999674i	$0.491876\pi$
$828$	0	0
$829$	−56.4859	−1.96184	−0.980919	−	0.194418i	$-0.937718\pi$
$829$	−56.4859	−1.96184	−0.980919	+	0.194418i	$0.937718\pi$
$830$	0	0
$831$	31.1653	1.08111
$832$	0	0
$833$	3.01832	0.104578
$834$	0	0
$835$	−21.8034	−0.754539
$836$	0	0
$837$	−2.04966	−0.0708466
$838$	0	0
$839$	−50.7456	−1.75193	−0.875966	−	0.482372i	$-0.839775\pi$
$839$	−50.7456	−1.75193	−0.875966	+	0.482372i	$0.839775\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−37.8885	−1.30495
$844$	0	0
$845$	−3.79162	−0.130436
$846$	0	0
$847$	−16.5169	−0.567529
$848$	0	0
$849$	−3.01104	−0.103338
$850$	0	0
$851$	−16.8998	−0.579317
$852$	0	0
$853$	16.2540	0.556527	0.278263	−	0.960505i	$-0.410241\pi$
$853$	16.2540	0.556527	0.278263	+	0.960505i	$0.410241\pi$
$854$	0	0
$855$	29.1338	0.996355
$856$	0	0
$857$	12.8205	0.437938	0.218969	−	0.975732i	$-0.429731\pi$
$857$	12.8205	0.437938	0.218969	+	0.975732i	$0.429731\pi$
$858$	0	0
$859$	2.75272	0.0939215	0.0469608	−	0.998897i	$-0.485046\pi$
$859$	2.75272	0.0939215	0.0469608	+	0.998897i	$0.485046\pi$
$860$	0	0
$861$	18.2323	0.621354
$862$	0	0
$863$	−14.4049	−0.490348	−0.245174	−	0.969479i	$-0.578845\pi$
$863$	−14.4049	−0.490348	−0.245174	+	0.969479i	$0.578845\pi$
$864$	0	0
$865$	0.994414	0.0338111
$866$	0	0
$867$	18.9826	0.644683
$868$	0	0
$869$	9.37344	0.317972
$870$	0	0
$871$	−2.96039	−0.100309
$872$	0	0
$873$	0.115738	0.00391712
$874$	0	0
$875$	−49.7252	−1.68102
$876$	0	0
$877$	−22.2721	−0.752074	−0.376037	−	0.926605i	$-0.622713\pi$
$877$	−22.2721	−0.752074	−0.376037	+	0.926605i	$0.622713\pi$
$878$	0	0
$879$	−3.28768	−0.110891
$880$	0	0
$881$	−46.2371	−1.55777	−0.778883	−	0.627169i	$-0.784214\pi$
$881$	−46.2371	−1.55777	−0.778883	+	0.627169i	$0.784214\pi$
$882$	0	0
$883$	−57.9505	−1.95019	−0.975095	−	0.221786i	$-0.928811\pi$
$883$	−57.9505	−1.95019	−0.975095	+	0.221786i	$0.928811\pi$
$884$	0	0
$885$	47.0484	1.58151
$886$	0	0
$887$	−21.2352	−0.713008	−0.356504	−	0.934294i	$-0.616031\pi$
$887$	−21.2352	−0.713008	−0.356504	+	0.934294i	$0.616031\pi$
$888$	0	0
$889$	0.566902	0.0190133
$890$	0	0
$891$	−7.63378	−0.255741
$892$	0	0
$893$	−65.2755	−2.18436
$894$	0	0
$895$	37.0602	1.23879
$896$	0	0
$897$	5.53758	0.184894
$898$	0	0
$899$	0.366222	0.0122142
$900$	0	0
$901$	9.49459	0.316311
$902$	0	0
$903$	5.24778	0.174635
$904$	0	0
$905$	−47.9632	−1.59435
$906$	0	0
$907$	−26.7115	−0.886941	−0.443470	−	0.896289i	$-0.646253\pi$
$907$	−26.7115	−0.886941	−0.443470	+	0.896289i	$0.646253\pi$
$908$	0	0
$909$	−3.63213	−0.120470
$910$	0	0
$911$	−43.0362	−1.42585	−0.712927	−	0.701238i	$-0.752631\pi$
$911$	−43.0362	−1.42585	−0.712927	+	0.701238i	$0.752631\pi$
$912$	0	0
$913$	0.166532	0.00551141
$914$	0	0
$915$	−4.47630	−0.147982
$916$	0	0
$917$	−15.7492	−0.520085
$918$	0	0
$919$	−19.4591	−0.641897	−0.320949	−	0.947097i	$-0.604002\pi$
$919$	−19.4591	−0.641897	−0.320949	+	0.947097i	$0.604002\pi$
$920$	0	0
$921$	−33.2273	−1.09488
$922$	0	0
$923$	−12.6079	−0.414995
$924$	0	0
$925$	37.0133	1.21699
$926$	0	0
$927$	21.8711	0.718341
$928$	0	0
$929$	−7.23165	−0.237263	−0.118631	−	0.992938i	$-0.537851\pi$
$929$	−7.23165	−0.237263	−0.118631	+	0.992938i	$0.537851\pi$
$930$	0	0
$931$	11.4641	0.375722
$932$	0	0
$933$	−7.12351	−0.233213
$934$	0	0
$935$	−13.5423	−0.442879
$936$	0	0
$937$	20.2243	0.660699	0.330349	−	0.943859i	$-0.392833\pi$
$937$	20.2243	0.660699	0.330349	+	0.943859i	$0.392833\pi$
$938$	0	0
$939$	−18.4252	−0.601284
$940$	0	0
$941$	35.8164	1.16758	0.583790	−	0.811904i	$-0.301569\pi$
$941$	35.8164	1.16758	0.583790	+	0.811904i	$0.301569\pi$
$942$	0	0
$943$	20.1377	0.655773
$944$	0	0
$945$	−63.5909	−2.06861
$946$	0	0
$947$	25.4580	0.827273	0.413636	−	0.910442i	$-0.364258\pi$
$947$	25.4580	0.827273	0.413636	+	0.910442i	$0.364258\pi$
$948$	0	0
$949$	1.27258	0.0413097
$950$	0	0
$951$	−41.1138	−1.33321
$952$	0	0
$953$	13.0679	0.423311	0.211656	−	0.977344i	$-0.432114\pi$
$953$	13.0679	0.423311	0.211656	+	0.977344i	$0.432114\pi$
$954$	0	0
$955$	46.3913	1.50119
$956$	0	0
$957$	3.03021	0.0979527
$958$	0	0
$959$	45.7937	1.47876
$960$	0	0
$961$	−30.8659	−0.995674
$962$	0	0
$963$	2.04938	0.0660403
$964$	0	0
$965$	−16.2474	−0.523023
$966$	0	0
$967$	−3.38001	−0.108694	−0.0543468	−	0.998522i	$-0.517308\pi$
$967$	−3.38001	−0.108694	−0.0543468	+	0.998522i	$0.517308\pi$
$968$	0	0
$969$	−11.4194	−0.366843
$970$	0	0
$971$	−3.17096	−0.101761	−0.0508804	−	0.998705i	$-0.516203\pi$
$971$	−3.17096	−0.101761	−0.0508804	+	0.998705i	$0.516203\pi$
$972$	0	0
$973$	49.3523	1.58216
$974$	0	0
$975$	−12.1282	−0.388413
$976$	0	0
$977$	43.3505	1.38690	0.693452	−	0.720503i	$-0.256089\pi$
$977$	43.3505	1.38690	0.693452	+	0.720503i	$0.256089\pi$
$978$	0	0
$979$	−6.91496	−0.221003
$980$	0	0
$981$	−7.89419	−0.252042
$982$	0	0
$983$	37.9058	1.20901	0.604503	−	0.796603i	$-0.293372\pi$
$983$	37.9058	1.20901	0.604503	+	0.796603i	$0.293372\pi$
$984$	0	0
$985$	55.0095	1.75275
$986$	0	0
$987$	43.6930	1.39076
$988$	0	0
$989$	5.79622	0.184309
$990$	0	0
$991$	30.1686	0.958336	0.479168	−	0.877723i	$-0.340938\pi$
$991$	30.1686	0.958336	0.479168	+	0.877723i	$0.340938\pi$
$992$	0	0
$993$	5.23515	0.166133
$994$	0	0
$995$	−47.5507	−1.50746
$996$	0	0
$997$	21.4031	0.677841	0.338921	−	0.940815i	$-0.389938\pi$
$997$	21.4031	0.677841	0.338921	+	0.940815i	$0.389938\pi$
$998$	0	0
$999$	22.0932	0.698997

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6032.2.a.ba.1.4		10
4.3	odd	2		3016.2.a.g.1.7	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3016.2.a.g.1.7	✓	10	4.3	odd	2
6032.2.a.ba.1.4		10	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6032 = 2^{4} \cdot 13 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6032.a (trivial)

\(f(q)\)	\(=\)	\(q-1.29348 q^{3} -3.79162 q^{5} +2.99662 q^{7} -1.32691 q^{9} +O(q^{10})\)	\(q-1.29348 q^{3} -3.79162 q^{5} +2.99662 q^{7} -1.32691 q^{9} +2.34268 q^{11} +1.00000 q^{13} +4.90439 q^{15} +1.52459 q^{17} +5.79068 q^{19} -3.87607 q^{21} -4.28115 q^{23} +9.37642 q^{25} +5.59677 q^{27} -1.00000 q^{29} -0.366222 q^{31} -3.03021 q^{33} -11.3621 q^{35} +3.94749 q^{37} -1.29348 q^{39} -4.70380 q^{41} -1.35389 q^{43} +5.03115 q^{45} -11.2725 q^{47} +1.97976 q^{49} -1.97202 q^{51} +6.22764 q^{53} -8.88257 q^{55} -7.49012 q^{57} +9.59312 q^{59} -0.912713 q^{61} -3.97626 q^{63} -3.79162 q^{65} -2.96039 q^{67} +5.53758 q^{69} -12.6079 q^{71} +1.27258 q^{73} -12.1282 q^{75} +7.02013 q^{77} +4.00116 q^{79} -3.25857 q^{81} +0.0710861 q^{83} -5.78067 q^{85} +1.29348 q^{87} -2.95173 q^{89} +2.99662 q^{91} +0.473700 q^{93} -21.9561 q^{95} -0.0872232 q^{97} -3.10853 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{3} + 5 q^{5} + 2 q^{7} + 16 q^{9}+O(q^{10}) \) 10 * q - 2 * q^3 + 5 * q^5 + 2 * q^7 + 16 * q^9	\( 10 q - 2 q^{3} + 5 q^{5} + 2 q^{7} + 16 q^{9} - 4 q^{11} + 10 q^{13} - 8 q^{15} + 10 q^{17} + q^{19} + q^{21} - 23 q^{23} + 25 q^{25} - 2 q^{27} - 10 q^{29} + 13 q^{31} + 15 q^{33} + 12 q^{35} - 7 q^{37} - 2 q^{39} + 16 q^{41} + 12 q^{43} + 55 q^{45} - 11 q^{47} + 25 q^{51} + 11 q^{53} - 22 q^{55} - 6 q^{57} + 11 q^{59} + 34 q^{61} - 37 q^{63} + 5 q^{65} + 23 q^{67} + 2 q^{69} + 4 q^{71} + 39 q^{73} - 11 q^{75} + 32 q^{77} - 5 q^{79} + 38 q^{81} - 6 q^{83} + 45 q^{85} + 2 q^{87} - 24 q^{89} + 2 q^{91} + 13 q^{93} - 33 q^{95} + 19 q^{97}+O(q^{100}) \) 10 * q - 2 * q^3 + 5 * q^5 + 2 * q^7 + 16 * q^9 - 4 * q^11 + 10 * q^13 - 8 * q^15 + 10 * q^17 + q^19 + q^21 - 23 * q^23 + 25 * q^25 - 2 * q^27 - 10 * q^29 + 13 * q^31 + 15 * q^33 + 12 * q^35 - 7 * q^37 - 2 * q^39 + 16 * q^41 + 12 * q^43 + 55 * q^45 - 11 * q^47 + 25 * q^51 + 11 * q^53 - 22 * q^55 - 6 * q^57 + 11 * q^59 + 34 * q^61 - 37 * q^63 + 5 * q^65 + 23 * q^67 + 2 * q^69 + 4 * q^71 + 39 * q^73 - 11 * q^75 + 32 * q^77 - 5 * q^79 + 38 * q^81 - 6 * q^83 + 45 * q^85 + 2 * q^87 - 24 * q^89 + 2 * q^91 + 13 * q^93 - 33 * q^95 + 19 * q^97

Introduction

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