LMFDB - Embedded newform 3360.2.a.bi.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3360,2,Mod(1,3360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3360, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3360.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3360.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:	$26.8297350792$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	3360.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -5.80642 q^{11} +5.05086 q^{13} -1.00000 q^{15} +0.755569 q^{17} +1.80642 q^{19} +1.00000 q^{21} -8.85728 q^{23} +1.00000 q^{25} -1.00000 q^{27} +0.755569 q^{29} +10.6637 q^{31} +5.80642 q^{33} -1.00000 q^{35} -0.755569 q^{37} -5.05086 q^{39} -9.61285 q^{41} -1.24443 q^{43} +1.00000 q^{45} +8.85728 q^{47} +1.00000 q^{49} -0.755569 q^{51} -7.80642 q^{53} -5.80642 q^{55} -1.80642 q^{57} -8.00000 q^{59} -10.8573 q^{61} -1.00000 q^{63} +5.05086 q^{65} -10.3684 q^{67} +8.85728 q^{69} +3.05086 q^{71} -1.05086 q^{73} -1.00000 q^{75} +5.80642 q^{77} -4.00000 q^{79} +1.00000 q^{81} +7.61285 q^{83} +0.755569 q^{85} -0.755569 q^{87} +8.10171 q^{89} -5.05086 q^{91} -10.6637 q^{93} +1.80642 q^{95} +5.05086 q^{97} -5.80642 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 3 * q^5 - 3 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} + 3 q^{5} - 3 q^{7} + 3 q^{9} - 4 q^{11} + 2 q^{13} - 3 q^{15} + 2 q^{17} - 8 q^{19} + 3 q^{21} + 3 q^{25} - 3 q^{27} + 2 q^{29} - 8 q^{31} + 4 q^{33} - 3 q^{35} - 2 q^{37} - 2 q^{39} - 2 q^{41} - 4 q^{43} + 3 q^{45} + 3 q^{49} - 2 q^{51} - 10 q^{53} - 4 q^{55} + 8 q^{57} - 24 q^{59} - 6 q^{61} - 3 q^{63} + 2 q^{65} - 4 q^{67} - 4 q^{71} + 10 q^{73} - 3 q^{75} + 4 q^{77} - 12 q^{79} + 3 q^{81} - 4 q^{83} + 2 q^{85} - 2 q^{87} - 2 q^{89} - 2 q^{91} + 8 q^{93} - 8 q^{95} + 2 q^{97} - 4 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 3 * q^5 - 3 * q^7 + 3 * q^9 - 4 * q^11 + 2 * q^13 - 3 * q^15 + 2 * q^17 - 8 * q^19 + 3 * q^21 + 3 * q^25 - 3 * q^27 + 2 * q^29 - 8 * q^31 + 4 * q^33 - 3 * q^35 - 2 * q^37 - 2 * q^39 - 2 * q^41 - 4 * q^43 + 3 * q^45 + 3 * q^49 - 2 * q^51 - 10 * q^53 - 4 * q^55 + 8 * q^57 - 24 * q^59 - 6 * q^61 - 3 * q^63 + 2 * q^65 - 4 * q^67 - 4 * q^71 + 10 * q^73 - 3 * q^75 + 4 * q^77 - 12 * q^79 + 3 * q^81 - 4 * q^83 + 2 * q^85 - 2 * q^87 - 2 * q^89 - 2 * q^91 + 8 * q^93 - 8 * q^95 + 2 * q^97 - 4 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.80642	−1.75070	−0.875351	−	0.483487i	$-0.839370\pi$
$11$	−5.80642	−1.75070	−0.875351	+	0.483487i	$0.839370\pi$
$12$	0	0
$13$	5.05086	1.40086	0.700428	−	0.713723i	$-0.252993\pi$
$13$	5.05086	1.40086	0.700428	+	0.713723i	$0.252993\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	0.755569	0.183252	0.0916262	−	0.995793i	$-0.470794\pi$
$17$	0.755569	0.183252	0.0916262	+	0.995793i	$0.470794\pi$
$18$	0	0
$19$	1.80642	0.414422	0.207211	−	0.978296i	$-0.433561\pi$
$19$	1.80642	0.414422	0.207211	+	0.978296i	$0.433561\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−8.85728	−1.84687	−0.923435	−	0.383754i	$-0.874631\pi$
$23$	−8.85728	−1.84687	−0.923435	+	0.383754i	$0.874631\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	0.755569	0.140306	0.0701528	−	0.997536i	$-0.477651\pi$
$29$	0.755569	0.140306	0.0701528	+	0.997536i	$0.477651\pi$
$30$	0	0
$31$	10.6637	1.91526	0.957629	−	0.288005i	$-0.0929921\pi$
$31$	10.6637	1.91526	0.957629	+	0.288005i	$0.0929921\pi$
$32$	0	0
$33$	5.80642	1.01077
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−0.755569	−0.124215	−0.0621074	−	0.998069i	$-0.519782\pi$
$37$	−0.755569	−0.124215	−0.0621074	+	0.998069i	$0.519782\pi$
$38$	0	0
$39$	−5.05086	−0.808784
$40$	0	0
$41$	−9.61285	−1.50127	−0.750637	−	0.660715i	$-0.770253\pi$
$41$	−9.61285	−1.50127	−0.750637	+	0.660715i	$0.770253\pi$
$42$	0	0
$43$	−1.24443	−0.189774	−0.0948870	−	0.995488i	$-0.530249\pi$
$43$	−1.24443	−0.189774	−0.0948870	+	0.995488i	$0.530249\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	8.85728	1.29197	0.645983	−	0.763351i	$-0.276447\pi$
$47$	8.85728	1.29197	0.645983	+	0.763351i	$0.276447\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.755569	−0.105801
$52$	0	0
$53$	−7.80642	−1.07229	−0.536147	−	0.844124i	$-0.680121\pi$
$53$	−7.80642	−1.07229	−0.536147	+	0.844124i	$0.680121\pi$
$54$	0	0
$55$	−5.80642	−0.782938
$56$	0	0
$57$	−1.80642	−0.239267
$58$	0	0
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	−10.8573	−1.39013	−0.695066	−	0.718946i	$-0.744625\pi$
$61$	−10.8573	−1.39013	−0.695066	+	0.718946i	$0.744625\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	5.05086	0.626481
$66$	0	0
$67$	−10.3684	−1.26670	−0.633352	−	0.773864i	$-0.718321\pi$
$67$	−10.3684	−1.26670	−0.633352	+	0.773864i	$0.718321\pi$
$68$	0	0
$69$	8.85728	1.06629
$70$	0	0
$71$	3.05086	0.362070	0.181035	−	0.983477i	$-0.442055\pi$
$71$	3.05086	0.362070	0.181035	+	0.983477i	$0.442055\pi$
$72$	0	0
$73$	−1.05086	−0.122993	−0.0614966	−	0.998107i	$-0.519587\pi$
$73$	−1.05086	−0.122993	−0.0614966	+	0.998107i	$0.519587\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	5.80642	0.661703
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	7.61285	0.835619	0.417809	−	0.908535i	$-0.362798\pi$
$83$	7.61285	0.835619	0.417809	+	0.908535i	$0.362798\pi$
$84$	0	0
$85$	0.755569	0.0819529
$86$	0	0
$87$	−0.755569	−0.0810055
$88$	0	0
$89$	8.10171	0.858780	0.429390	−	0.903119i	$-0.358729\pi$
$89$	8.10171	0.858780	0.429390	+	0.903119i	$0.358729\pi$
$90$	0	0
$91$	−5.05086	−0.529473
$92$	0	0
$93$	−10.6637	−1.10577
$94$	0	0
$95$	1.80642	0.185335
$96$	0	0
$97$	5.05086	0.512837	0.256418	−	0.966566i	$-0.417458\pi$
$97$	5.05086	0.512837	0.256418	+	0.966566i	$0.417458\pi$
$98$	0	0
$99$	−5.80642	−0.583568
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	0	0
$107$	4.85728	0.469571	0.234785	−	0.972047i	$-0.424561\pi$
$107$	4.85728	0.469571	0.234785	+	0.972047i	$0.424561\pi$
$108$	0	0
$109$	−4.48886	−0.429955	−0.214978	−	0.976619i	$-0.568968\pi$
$109$	−4.48886	−0.429955	−0.214978	+	0.976619i	$0.568968\pi$
$110$	0	0
$111$	0.755569	0.0717154
$112$	0	0
$113$	−11.4193	−1.07423	−0.537117	−	0.843508i	$-0.680487\pi$
$113$	−11.4193	−1.07423	−0.537117	+	0.843508i	$0.680487\pi$
$114$	0	0
$115$	−8.85728	−0.825946
$116$	0	0
$117$	5.05086	0.466952
$118$	0	0
$119$	−0.755569	−0.0692629
$120$	0	0
$121$	22.7146	2.06496
$122$	0	0
$123$	9.61285	0.866761
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−14.1017	−1.25132	−0.625662	−	0.780094i	$-0.715171\pi$
$127$	−14.1017	−1.25132	−0.625662	+	0.780094i	$0.715171\pi$
$128$	0	0
$129$	1.24443	0.109566
$130$	0	0
$131$	−14.1017	−1.23207	−0.616036	−	0.787718i	$-0.711262\pi$
$131$	−14.1017	−1.23207	−0.616036	+	0.787718i	$0.711262\pi$
$132$	0	0
$133$	−1.80642	−0.156637
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−10.2953	−0.879586	−0.439793	−	0.898099i	$-0.644948\pi$
$137$	−10.2953	−0.879586	−0.439793	+	0.898099i	$0.644948\pi$
$138$	0	0
$139$	−6.19358	−0.525332	−0.262666	−	0.964887i	$-0.584602\pi$
$139$	−6.19358	−0.525332	−0.262666	+	0.964887i	$0.584602\pi$
$140$	0	0
$141$	−8.85728	−0.745917
$142$	0	0
$143$	−29.3274	−2.45248
$144$	0	0
$145$	0.755569	0.0627466
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−1.14272	−0.0936154	−0.0468077	−	0.998904i	$-0.514905\pi$
$149$	−1.14272	−0.0936154	−0.0468077	+	0.998904i	$0.514905\pi$
$150$	0	0
$151$	−15.6128	−1.27056	−0.635278	−	0.772284i	$-0.719114\pi$
$151$	−15.6128	−1.27056	−0.635278	+	0.772284i	$0.719114\pi$
$152$	0	0
$153$	0.755569	0.0610841
$154$	0	0
$155$	10.6637	0.856529
$156$	0	0
$157$	−9.05086	−0.722337	−0.361168	−	0.932501i	$-0.617622\pi$
$157$	−9.05086	−0.722337	−0.361168	+	0.932501i	$0.617622\pi$
$158$	0	0
$159$	7.80642	0.619090
$160$	0	0
$161$	8.85728	0.698051
$162$	0	0
$163$	9.24443	0.724080	0.362040	−	0.932163i	$-0.382080\pi$
$163$	9.24443	0.724080	0.362040	+	0.932163i	$0.382080\pi$
$164$	0	0
$165$	5.80642	0.452029
$166$	0	0
$167$	10.7556	0.832291	0.416145	−	0.909298i	$-0.363381\pi$
$167$	10.7556	0.832291	0.416145	+	0.909298i	$0.363381\pi$
$168$	0	0
$169$	12.5111	0.962395
$170$	0	0
$171$	1.80642	0.138141
$172$	0	0
$173$	9.61285	0.730851	0.365426	−	0.930841i	$-0.380923\pi$
$173$	9.61285	0.730851	0.365426	+	0.930841i	$0.380923\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	8.00000	0.601317
$178$	0	0
$179$	−21.8064	−1.62989	−0.814944	−	0.579539i	$-0.803233\pi$
$179$	−21.8064	−1.62989	−0.814944	+	0.579539i	$0.803233\pi$
$180$	0	0
$181$	−15.2444	−1.13311	−0.566555	−	0.824024i	$-0.691724\pi$
$181$	−15.2444	−1.13311	−0.566555	+	0.824024i	$0.691724\pi$
$182$	0	0
$183$	10.8573	0.802593
$184$	0	0
$185$	−0.755569	−0.0555505
$186$	0	0
$187$	−4.38715	−0.320820
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−3.05086	−0.220752	−0.110376	−	0.993890i	$-0.535206\pi$
$191$	−3.05086	−0.220752	−0.110376	+	0.993890i	$0.535206\pi$
$192$	0	0
$193$	4.48886	0.323115	0.161558	−	0.986863i	$-0.448348\pi$
$193$	4.48886	0.323115	0.161558	+	0.986863i	$0.448348\pi$
$194$	0	0
$195$	−5.05086	−0.361699
$196$	0	0
$197$	−1.70471	−0.121456	−0.0607279	−	0.998154i	$-0.519342\pi$
$197$	−1.70471	−0.121456	−0.0607279	+	0.998154i	$0.519342\pi$
$198$	0	0
$199$	−20.5620	−1.45760	−0.728801	−	0.684726i	$-0.759922\pi$
$199$	−20.5620	−1.45760	−0.728801	+	0.684726i	$0.759922\pi$
$200$	0	0
$201$	10.3684	0.731332
$202$	0	0
$203$	−0.755569	−0.0530305
$204$	0	0
$205$	−9.61285	−0.671390
$206$	0	0
$207$	−8.85728	−0.615623
$208$	0	0
$209$	−10.4889	−0.725530
$210$	0	0
$211$	21.7146	1.49489	0.747446	−	0.664323i	$-0.231280\pi$
$211$	21.7146	1.49489	0.747446	+	0.664323i	$0.231280\pi$
$212$	0	0
$213$	−3.05086	−0.209041
$214$	0	0
$215$	−1.24443	−0.0848695
$216$	0	0
$217$	−10.6637	−0.723899
$218$	0	0
$219$	1.05086	0.0710102
$220$	0	0
$221$	3.81627	0.256710
$222$	0	0
$223$	8.59057	0.575267	0.287634	−	0.957741i	$-0.407131\pi$
$223$	8.59057	0.575267	0.287634	+	0.957741i	$0.407131\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−12.5906	−0.835666	−0.417833	−	0.908524i	$-0.637210\pi$
$227$	−12.5906	−0.835666	−0.417833	+	0.908524i	$0.637210\pi$
$228$	0	0
$229$	−13.8796	−0.917187	−0.458593	−	0.888646i	$-0.651647\pi$
$229$	−13.8796	−0.917187	−0.458593	+	0.888646i	$0.651647\pi$
$230$	0	0
$231$	−5.80642	−0.382035
$232$	0	0
$233$	1.31756	0.0863163	0.0431582	−	0.999068i	$-0.486258\pi$
$233$	1.31756	0.0863163	0.0431582	+	0.999068i	$0.486258\pi$
$234$	0	0
$235$	8.85728	0.577785
$236$	0	0
$237$	4.00000	0.259828
$238$	0	0
$239$	3.05086	0.197343	0.0986717	−	0.995120i	$-0.468541\pi$
$239$	3.05086	0.197343	0.0986717	+	0.995120i	$0.468541\pi$
$240$	0	0
$241$	13.6128	0.876881	0.438440	−	0.898760i	$-0.355531\pi$
$241$	13.6128	0.876881	0.438440	+	0.898760i	$0.355531\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	9.12399	0.580545
$248$	0	0
$249$	−7.61285	−0.482445
$250$	0	0
$251$	2.48886	0.157096	0.0785478	−	0.996910i	$-0.474972\pi$
$251$	2.48886	0.157096	0.0785478	+	0.996910i	$0.474972\pi$
$252$	0	0
$253$	51.4291	3.23332
$254$	0	0
$255$	−0.755569	−0.0473155
$256$	0	0
$257$	−0.368416	−0.0229812	−0.0114906	−	0.999934i	$-0.503658\pi$
$257$	−0.368416	−0.0229812	−0.0114906	+	0.999934i	$0.503658\pi$
$258$	0	0
$259$	0.755569	0.0469488
$260$	0	0
$261$	0.755569	0.0467685
$262$	0	0
$263$	30.1847	1.86127	0.930634	−	0.365952i	$-0.119257\pi$
$263$	30.1847	1.86127	0.930634	+	0.365952i	$0.119257\pi$
$264$	0	0
$265$	−7.80642	−0.479545
$266$	0	0
$267$	−8.10171	−0.495817
$268$	0	0
$269$	−30.7368	−1.87406	−0.937029	−	0.349252i	$-0.886436\pi$
$269$	−30.7368	−1.87406	−0.937029	+	0.349252i	$0.886436\pi$
$270$	0	0
$271$	−18.6637	−1.13374	−0.566870	−	0.823808i	$-0.691846\pi$
$271$	−18.6637	−1.13374	−0.566870	+	0.823808i	$0.691846\pi$
$272$	0	0
$273$	5.05086	0.305692
$274$	0	0
$275$	−5.80642	−0.350141
$276$	0	0
$277$	−19.2444	−1.15629	−0.578143	−	0.815936i	$-0.696222\pi$
$277$	−19.2444	−1.15629	−0.578143	+	0.815936i	$0.696222\pi$
$278$	0	0
$279$	10.6637	0.638419
$280$	0	0
$281$	−9.61285	−0.573454	−0.286727	−	0.958012i	$-0.592567\pi$
$281$	−9.61285	−0.573454	−0.286727	+	0.958012i	$0.592567\pi$
$282$	0	0
$283$	−11.2257	−0.667298	−0.333649	−	0.942697i	$-0.608280\pi$
$283$	−11.2257	−0.667298	−0.333649	+	0.942697i	$0.608280\pi$
$284$	0	0
$285$	−1.80642	−0.107003
$286$	0	0
$287$	9.61285	0.567428
$288$	0	0
$289$	−16.4291	−0.966419
$290$	0	0
$291$	−5.05086	−0.296086
$292$	0	0
$293$	−24.1017	−1.40804	−0.704018	−	0.710182i	$-0.748613\pi$
$293$	−24.1017	−1.40804	−0.704018	+	0.710182i	$0.748613\pi$
$294$	0	0
$295$	−8.00000	−0.465778
$296$	0	0
$297$	5.80642	0.336923
$298$	0	0
$299$	−44.7368	−2.58720
$300$	0	0
$301$	1.24443	0.0717278
$302$	0	0
$303$	10.0000	0.574485
$304$	0	0
$305$	−10.8573	−0.621686
$306$	0	0
$307$	−5.12399	−0.292441	−0.146221	−	0.989252i	$-0.546711\pi$
$307$	−5.12399	−0.292441	−0.146221	+	0.989252i	$0.546711\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	23.2257	1.31701	0.658504	−	0.752577i	$-0.271189\pi$
$311$	23.2257	1.31701	0.658504	+	0.752577i	$0.271189\pi$
$312$	0	0
$313$	3.33630	0.188579	0.0942893	−	0.995545i	$-0.469942\pi$
$313$	3.33630	0.188579	0.0942893	+	0.995545i	$0.469942\pi$
$314$	0	0
$315$	−1.00000	−0.0563436
$316$	0	0
$317$	−28.0098	−1.57319	−0.786595	−	0.617470i	$-0.788158\pi$
$317$	−28.0098	−1.57319	−0.786595	+	0.617470i	$0.788158\pi$
$318$	0	0
$319$	−4.38715	−0.245633
$320$	0	0
$321$	−4.85728	−0.271107
$322$	0	0
$323$	1.36488	0.0759438
$324$	0	0
$325$	5.05086	0.280171
$326$	0	0
$327$	4.48886	0.248235
$328$	0	0
$329$	−8.85728	−0.488318
$330$	0	0
$331$	2.28544	0.125619	0.0628096	−	0.998026i	$-0.479994\pi$
$331$	2.28544	0.125619	0.0628096	+	0.998026i	$0.479994\pi$
$332$	0	0
$333$	−0.755569	−0.0414049
$334$	0	0
$335$	−10.3684	−0.566487
$336$	0	0
$337$	27.7146	1.50971	0.754854	−	0.655893i	$-0.227708\pi$
$337$	27.7146	1.50971	0.754854	+	0.655893i	$0.227708\pi$
$338$	0	0
$339$	11.4193	0.620210
$340$	0	0
$341$	−61.9180	−3.35305
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	8.85728	0.476860
$346$	0	0
$347$	−1.24443	−0.0668046	−0.0334023	−	0.999442i	$-0.510634\pi$
$347$	−1.24443	−0.0668046	−0.0334023	+	0.999442i	$0.510634\pi$
$348$	0	0
$349$	−28.5718	−1.52942	−0.764708	−	0.644377i	$-0.777117\pi$
$349$	−28.5718	−1.52942	−0.764708	+	0.644377i	$0.777117\pi$
$350$	0	0
$351$	−5.05086	−0.269595
$352$	0	0
$353$	33.6958	1.79345	0.896724	−	0.442591i	$-0.145940\pi$
$353$	33.6958	1.79345	0.896724	+	0.442591i	$0.145940\pi$
$354$	0	0
$355$	3.05086	0.161923
$356$	0	0
$357$	0.755569	0.0399889
$358$	0	0
$359$	−28.7654	−1.51818	−0.759090	−	0.650985i	$-0.774356\pi$
$359$	−28.7654	−1.51818	−0.759090	+	0.650985i	$0.774356\pi$
$360$	0	0
$361$	−15.7368	−0.828254
$362$	0	0
$363$	−22.7146	−1.19221
$364$	0	0
$365$	−1.05086	−0.0550043
$366$	0	0
$367$	−9.89829	−0.516687	−0.258343	−	0.966053i	$-0.583177\pi$
$367$	−9.89829	−0.516687	−0.258343	+	0.966053i	$0.583177\pi$
$368$	0	0
$369$	−9.61285	−0.500425
$370$	0	0
$371$	7.80642	0.405289
$372$	0	0
$373$	31.5941	1.63588	0.817941	−	0.575303i	$-0.195116\pi$
$373$	31.5941	1.63588	0.817941	+	0.575303i	$0.195116\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	3.81627	0.196548
$378$	0	0
$379$	−35.8163	−1.83976	−0.919879	−	0.392202i	$-0.871713\pi$
$379$	−35.8163	−1.83976	−0.919879	+	0.392202i	$0.871713\pi$
$380$	0	0
$381$	14.1017	0.722453
$382$	0	0
$383$	33.9813	1.73636	0.868181	−	0.496248i	$-0.165289\pi$
$383$	33.9813	1.73636	0.868181	+	0.496248i	$0.165289\pi$
$384$	0	0
$385$	5.80642	0.295923
$386$	0	0
$387$	−1.24443	−0.0632580
$388$	0	0
$389$	−34.8573	−1.76733	−0.883667	−	0.468116i	$-0.844933\pi$
$389$	−34.8573	−1.76733	−0.883667	+	0.468116i	$0.844933\pi$
$390$	0	0
$391$	−6.69228	−0.338443
$392$	0	0
$393$	14.1017	0.711337
$394$	0	0
$395$	−4.00000	−0.201262
$396$	0	0
$397$	29.9911	1.50521	0.752605	−	0.658472i	$-0.228797\pi$
$397$	29.9911	1.50521	0.752605	+	0.658472i	$0.228797\pi$
$398$	0	0
$399$	1.80642	0.0904343
$400$	0	0
$401$	23.5111	1.17409	0.587045	−	0.809554i	$-0.300291\pi$
$401$	23.5111	1.17409	0.587045	+	0.809554i	$0.300291\pi$
$402$	0	0
$403$	53.8608	2.68300
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	4.38715	0.217463
$408$	0	0
$409$	39.9180	1.97382	0.986909	−	0.161281i	$-0.0515626\pi$
$409$	39.9180	1.97382	0.986909	+	0.161281i	$0.0515626\pi$
$410$	0	0
$411$	10.2953	0.507829
$412$	0	0
$413$	8.00000	0.393654
$414$	0	0
$415$	7.61285	0.373700
$416$	0	0
$417$	6.19358	0.303301
$418$	0	0
$419$	1.12399	0.0549103	0.0274551	−	0.999623i	$-0.491260\pi$
$419$	1.12399	0.0549103	0.0274551	+	0.999623i	$0.491260\pi$
$420$	0	0
$421$	29.2257	1.42437	0.712187	−	0.701990i	$-0.247705\pi$
$421$	29.2257	1.42437	0.712187	+	0.701990i	$0.247705\pi$
$422$	0	0
$423$	8.85728	0.430656
$424$	0	0
$425$	0.755569	0.0366505
$426$	0	0
$427$	10.8573	0.525421
$428$	0	0
$429$	29.3274	1.41594
$430$	0	0
$431$	−36.1748	−1.74248	−0.871240	−	0.490857i	$-0.836684\pi$
$431$	−36.1748	−1.74248	−0.871240	+	0.490857i	$0.836684\pi$
$432$	0	0
$433$	−8.46028	−0.406575	−0.203288	−	0.979119i	$-0.565163\pi$
$433$	−8.46028	−0.406575	−0.203288	+	0.979119i	$0.565163\pi$
$434$	0	0
$435$	−0.755569	−0.0362267
$436$	0	0
$437$	−16.0000	−0.765384
$438$	0	0
$439$	−15.0509	−0.718338	−0.359169	−	0.933273i	$-0.616940\pi$
$439$	−15.0509	−0.718338	−0.359169	+	0.933273i	$0.616940\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	37.9813	1.80454	0.902272	−	0.431167i	$-0.141898\pi$
$443$	37.9813	1.80454	0.902272	+	0.431167i	$0.141898\pi$
$444$	0	0
$445$	8.10171	0.384058
$446$	0	0
$447$	1.14272	0.0540488
$448$	0	0
$449$	19.7146	0.930388	0.465194	−	0.885209i	$-0.345985\pi$
$449$	19.7146	0.930388	0.465194	+	0.885209i	$0.345985\pi$
$450$	0	0
$451$	55.8163	2.62829
$452$	0	0
$453$	15.6128	0.733556
$454$	0	0
$455$	−5.05086	−0.236788
$456$	0	0
$457$	3.89829	0.182354	0.0911772	−	0.995835i	$-0.470937\pi$
$457$	3.89829	0.182354	0.0911772	+	0.995835i	$0.470937\pi$
$458$	0	0
$459$	−0.755569	−0.0352669
$460$	0	0
$461$	−0.285442	−0.0132944	−0.00664718	−	0.999978i	$-0.502116\pi$
$461$	−0.285442	−0.0132944	−0.00664718	+	0.999978i	$0.502116\pi$
$462$	0	0
$463$	−23.8163	−1.10684	−0.553418	−	0.832904i	$-0.686677\pi$
$463$	−23.8163	−1.10684	−0.553418	+	0.832904i	$0.686677\pi$
$464$	0	0
$465$	−10.6637	−0.494517
$466$	0	0
$467$	−26.1017	−1.20784	−0.603922	−	0.797044i	$-0.706396\pi$
$467$	−26.1017	−1.20784	−0.603922	+	0.797044i	$0.706396\pi$
$468$	0	0
$469$	10.3684	0.478769
$470$	0	0
$471$	9.05086	0.417041
$472$	0	0
$473$	7.22570	0.332238
$474$	0	0
$475$	1.80642	0.0828844
$476$	0	0
$477$	−7.80642	−0.357432
$478$	0	0
$479$	−4.97773	−0.227438	−0.113719	−	0.993513i	$-0.536276\pi$
$479$	−4.97773	−0.227438	−0.113719	+	0.993513i	$0.536276\pi$
$480$	0	0
$481$	−3.81627	−0.174007
$482$	0	0
$483$	−8.85728	−0.403020
$484$	0	0
$485$	5.05086	0.229348
$486$	0	0
$487$	4.38715	0.198801	0.0994004	−	0.995048i	$-0.468308\pi$
$487$	4.38715	0.198801	0.0994004	+	0.995048i	$0.468308\pi$
$488$	0	0
$489$	−9.24443	−0.418048
$490$	0	0
$491$	5.21585	0.235388	0.117694	−	0.993050i	$-0.462450\pi$
$491$	5.21585	0.235388	0.117694	+	0.993050i	$0.462450\pi$
$492$	0	0
$493$	0.570884	0.0257113
$494$	0	0
$495$	−5.80642	−0.260979
$496$	0	0
$497$	−3.05086	−0.136850
$498$	0	0
$499$	28.5906	1.27989	0.639945	−	0.768421i	$-0.278957\pi$
$499$	28.5906	1.27989	0.639945	+	0.768421i	$0.278957\pi$
$500$	0	0
$501$	−10.7556	−0.480523
$502$	0	0
$503$	−14.1847	−0.632464	−0.316232	−	0.948682i	$-0.602418\pi$
$503$	−14.1847	−0.632464	−0.316232	+	0.948682i	$0.602418\pi$
$504$	0	0
$505$	−10.0000	−0.444994
$506$	0	0
$507$	−12.5111	−0.555639
$508$	0	0
$509$	26.7368	1.18509	0.592545	−	0.805538i	$-0.298123\pi$
$509$	26.7368	1.18509	0.592545	+	0.805538i	$0.298123\pi$
$510$	0	0
$511$	1.05086	0.0464871
$512$	0	0
$513$	−1.80642	−0.0797556
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−51.4291	−2.26185
$518$	0	0
$519$	−9.61285	−0.421957
$520$	0	0
$521$	−35.3274	−1.54772	−0.773861	−	0.633356i	$-0.781677\pi$
$521$	−35.3274	−1.54772	−0.773861	+	0.633356i	$0.781677\pi$
$522$	0	0
$523$	10.1017	0.441717	0.220858	−	0.975306i	$-0.429114\pi$
$523$	10.1017	0.441717	0.220858	+	0.975306i	$0.429114\pi$
$524$	0	0
$525$	1.00000	0.0436436
$526$	0	0
$527$	8.05716	0.350975
$528$	0	0
$529$	55.4514	2.41093
$530$	0	0
$531$	−8.00000	−0.347170
$532$	0	0
$533$	−48.5531	−2.10307
$534$	0	0
$535$	4.85728	0.209998
$536$	0	0
$537$	21.8064	0.941017
$538$	0	0
$539$	−5.80642	−0.250100
$540$	0	0
$541$	−6.73683	−0.289639	−0.144820	−	0.989458i	$-0.546260\pi$
$541$	−6.73683	−0.289639	−0.144820	+	0.989458i	$0.546260\pi$
$542$	0	0
$543$	15.2444	0.654201
$544$	0	0
$545$	−4.48886	−0.192282
$546$	0	0
$547$	41.0607	1.75563	0.877814	−	0.479001i	$-0.159001\pi$
$547$	41.0607	1.75563	0.877814	+	0.479001i	$0.159001\pi$
$548$	0	0
$549$	−10.8573	−0.463377
$550$	0	0
$551$	1.36488	0.0581457
$552$	0	0
$553$	4.00000	0.170097
$554$	0	0
$555$	0.755569	0.0320721
$556$	0	0
$557$	−31.0321	−1.31487	−0.657437	−	0.753510i	$-0.728359\pi$
$557$	−31.0321	−1.31487	−0.657437	+	0.753510i	$0.728359\pi$
$558$	0	0
$559$	−6.28544	−0.265846
$560$	0	0
$561$	4.38715	0.185226
$562$	0	0
$563$	32.2034	1.35721	0.678606	−	0.734502i	$-0.262584\pi$
$563$	32.2034	1.35721	0.678606	+	0.734502i	$0.262584\pi$
$564$	0	0
$565$	−11.4193	−0.480412
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−18.7368	−0.785489	−0.392744	−	0.919648i	$-0.628474\pi$
$569$	−18.7368	−0.785489	−0.392744	+	0.919648i	$0.628474\pi$
$570$	0	0
$571$	24.3872	1.02057	0.510285	−	0.860005i	$-0.329540\pi$
$571$	24.3872	1.02057	0.510285	+	0.860005i	$0.329540\pi$
$572$	0	0
$573$	3.05086	0.127451
$574$	0	0
$575$	−8.85728	−0.369374
$576$	0	0
$577$	−28.6637	−1.19329	−0.596643	−	0.802507i	$-0.703499\pi$
$577$	−28.6637	−1.19329	−0.596643	+	0.802507i	$0.703499\pi$
$578$	0	0
$579$	−4.48886	−0.186551
$580$	0	0
$581$	−7.61285	−0.315834
$582$	0	0
$583$	45.3274	1.87727
$584$	0	0
$585$	5.05086	0.208827
$586$	0	0
$587$	40.2034	1.65937	0.829686	−	0.558230i	$-0.188519\pi$
$587$	40.2034	1.65937	0.829686	+	0.558230i	$0.188519\pi$
$588$	0	0
$589$	19.2632	0.793725
$590$	0	0
$591$	1.70471	0.0701225
$592$	0	0
$593$	−17.4924	−0.718327	−0.359163	−	0.933275i	$-0.616938\pi$
$593$	−17.4924	−0.718327	−0.359163	+	0.933275i	$0.616938\pi$
$594$	0	0
$595$	−0.755569	−0.0309753
$596$	0	0
$597$	20.5620	0.841546
$598$	0	0
$599$	1.15257	0.0470925	0.0235463	−	0.999723i	$-0.492504\pi$
$599$	1.15257	0.0470925	0.0235463	+	0.999723i	$0.492504\pi$
$600$	0	0
$601$	11.1240	0.453757	0.226878	−	0.973923i	$-0.427148\pi$
$601$	11.1240	0.453757	0.226878	+	0.973923i	$0.427148\pi$
$602$	0	0
$603$	−10.3684	−0.422235
$604$	0	0
$605$	22.7146	0.923478
$606$	0	0
$607$	−18.8385	−0.764633	−0.382316	−	0.924031i	$-0.624874\pi$
$607$	−18.8385	−0.764633	−0.382316	+	0.924031i	$0.624874\pi$
$608$	0	0
$609$	0.755569	0.0306172
$610$	0	0
$611$	44.7368	1.80986
$612$	0	0
$613$	−12.9590	−0.523409	−0.261704	−	0.965148i	$-0.584285\pi$
$613$	−12.9590	−0.523409	−0.261704	+	0.965148i	$0.584285\pi$
$614$	0	0
$615$	9.61285	0.387627
$616$	0	0
$617$	29.5210	1.18847	0.594235	−	0.804291i	$-0.297455\pi$
$617$	29.5210	1.18847	0.594235	+	0.804291i	$0.297455\pi$
$618$	0	0
$619$	11.5210	0.463067	0.231534	−	0.972827i	$-0.425626\pi$
$619$	11.5210	0.463067	0.231534	+	0.972827i	$0.425626\pi$
$620$	0	0
$621$	8.85728	0.355430
$622$	0	0
$623$	−8.10171	−0.324588
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	10.4889	0.418885
$628$	0	0
$629$	−0.570884	−0.0227626
$630$	0	0
$631$	33.9180	1.35025	0.675127	−	0.737702i	$-0.264089\pi$
$631$	33.9180	1.35025	0.675127	+	0.737702i	$0.264089\pi$
$632$	0	0
$633$	−21.7146	−0.863076
$634$	0	0
$635$	−14.1017	−0.559609
$636$	0	0
$637$	5.05086	0.200122
$638$	0	0
$639$	3.05086	0.120690
$640$	0	0
$641$	23.3274	0.921377	0.460689	−	0.887562i	$-0.347603\pi$
$641$	23.3274	0.921377	0.460689	+	0.887562i	$0.347603\pi$
$642$	0	0
$643$	−11.8163	−0.465988	−0.232994	−	0.972478i	$-0.574852\pi$
$643$	−11.8163	−0.465988	−0.232994	+	0.972478i	$0.574852\pi$
$644$	0	0
$645$	1.24443	0.0489994
$646$	0	0
$647$	28.4701	1.11928	0.559638	−	0.828737i	$-0.310940\pi$
$647$	28.4701	1.11928	0.559638	+	0.828737i	$0.310940\pi$
$648$	0	0
$649$	46.4514	1.82338
$650$	0	0
$651$	10.6637	0.417943
$652$	0	0
$653$	8.78415	0.343750	0.171875	−	0.985119i	$-0.445017\pi$
$653$	8.78415	0.343750	0.171875	+	0.985119i	$0.445017\pi$
$654$	0	0
$655$	−14.1017	−0.550999
$656$	0	0
$657$	−1.05086	−0.0409978
$658$	0	0
$659$	3.31756	0.129234	0.0646169	−	0.997910i	$-0.479417\pi$
$659$	3.31756	0.129234	0.0646169	+	0.997910i	$0.479417\pi$
$660$	0	0
$661$	23.7975	0.925617	0.462808	−	0.886458i	$-0.346842\pi$
$661$	23.7975	0.925617	0.462808	+	0.886458i	$0.346842\pi$
$662$	0	0
$663$	−3.81627	−0.148212
$664$	0	0
$665$	−1.80642	−0.0700501
$666$	0	0
$667$	−6.69228	−0.259126
$668$	0	0
$669$	−8.59057	−0.332131
$670$	0	0
$671$	63.0420	2.43371
$672$	0	0
$673$	−6.59057	−0.254048	−0.127024	−	0.991900i	$-0.540543\pi$
$673$	−6.59057	−0.254048	−0.127024	+	0.991900i	$0.540543\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	29.8163	1.14593	0.572966	−	0.819579i	$-0.305793\pi$
$677$	29.8163	1.14593	0.572966	+	0.819579i	$0.305793\pi$
$678$	0	0
$679$	−5.05086	−0.193834
$680$	0	0
$681$	12.5906	0.482472
$682$	0	0
$683$	−18.1847	−0.695818	−0.347909	−	0.937528i	$-0.613108\pi$
$683$	−18.1847	−0.695818	−0.347909	+	0.937528i	$0.613108\pi$
$684$	0	0
$685$	−10.2953	−0.393363
$686$	0	0
$687$	13.8796	0.529538
$688$	0	0
$689$	−39.4291	−1.50213
$690$	0	0
$691$	−29.2355	−1.11217	−0.556086	−	0.831125i	$-0.687698\pi$
$691$	−29.2355	−1.11217	−0.556086	+	0.831125i	$0.687698\pi$
$692$	0	0
$693$	5.80642	0.220568
$694$	0	0
$695$	−6.19358	−0.234936
$696$	0	0
$697$	−7.26317	−0.275112
$698$	0	0
$699$	−1.31756	−0.0498347
$700$	0	0
$701$	23.9813	0.905760	0.452880	−	0.891572i	$-0.350397\pi$
$701$	23.9813	0.905760	0.452880	+	0.891572i	$0.350397\pi$
$702$	0	0
$703$	−1.36488	−0.0514773
$704$	0	0
$705$	−8.85728	−0.333584
$706$	0	0
$707$	10.0000	0.376089
$708$	0	0
$709$	32.4889	1.22014	0.610072	−	0.792346i	$-0.291140\pi$
$709$	32.4889	1.22014	0.610072	+	0.792346i	$0.291140\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	0	0
$713$	−94.4514	−3.53723
$714$	0	0
$715$	−29.3274	−1.09678
$716$	0	0
$717$	−3.05086	−0.113936
$718$	0	0
$719$	12.3872	0.461963	0.230981	−	0.972958i	$-0.425806\pi$
$719$	12.3872	0.461963	0.230981	+	0.972958i	$0.425806\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−13.6128	−0.506267
$724$	0	0
$725$	0.755569	0.0280611
$726$	0	0
$727$	17.1240	0.635093	0.317547	−	0.948243i	$-0.397141\pi$
$727$	17.1240	0.635093	0.317547	+	0.948243i	$0.397141\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−0.940253	−0.0347765
$732$	0	0
$733$	16.8474	0.622274	0.311137	−	0.950365i	$-0.399290\pi$
$733$	16.8474	0.622274	0.311137	+	0.950365i	$0.399290\pi$
$734$	0	0
$735$	−1.00000	−0.0368856
$736$	0	0
$737$	60.2034	2.21762
$738$	0	0
$739$	5.89829	0.216972	0.108486	−	0.994098i	$-0.465400\pi$
$739$	5.89829	0.216972	0.108486	+	0.994098i	$0.465400\pi$
$740$	0	0
$741$	−9.12399	−0.335178
$742$	0	0
$743$	34.7556	1.27506	0.637529	−	0.770426i	$-0.279957\pi$
$743$	34.7556	1.27506	0.637529	+	0.770426i	$0.279957\pi$
$744$	0	0
$745$	−1.14272	−0.0418661
$746$	0	0
$747$	7.61285	0.278540
$748$	0	0
$749$	−4.85728	−0.177481
$750$	0	0
$751$	0.977725	0.0356777	0.0178388	−	0.999841i	$-0.494321\pi$
$751$	0.977725	0.0356777	0.0178388	+	0.999841i	$0.494321\pi$
$752$	0	0
$753$	−2.48886	−0.0906992
$754$	0	0
$755$	−15.6128	−0.568210
$756$	0	0
$757$	−38.0830	−1.38415	−0.692075	−	0.721826i	$-0.743303\pi$
$757$	−38.0830	−1.38415	−0.692075	+	0.721826i	$0.743303\pi$
$758$	0	0
$759$	−51.4291	−1.86676
$760$	0	0
$761$	28.8385	1.04540	0.522698	−	0.852518i	$-0.324925\pi$
$761$	28.8385	1.04540	0.522698	+	0.852518i	$0.324925\pi$
$762$	0	0
$763$	4.48886	0.162508
$764$	0	0
$765$	0.755569	0.0273176
$766$	0	0
$767$	−40.4068	−1.45901
$768$	0	0
$769$	−4.63512	−0.167147	−0.0835734	−	0.996502i	$-0.526633\pi$
$769$	−4.63512	−0.167147	−0.0835734	+	0.996502i	$0.526633\pi$
$770$	0	0
$771$	0.368416	0.0132682
$772$	0	0
$773$	50.5531	1.81827	0.909134	−	0.416503i	$-0.136744\pi$
$773$	50.5531	1.81827	0.909134	+	0.416503i	$0.136744\pi$
$774$	0	0
$775$	10.6637	0.383052
$776$	0	0
$777$	−0.755569	−0.0271059
$778$	0	0
$779$	−17.3649	−0.622161
$780$	0	0
$781$	−17.7146	−0.633877
$782$	0	0
$783$	−0.755569	−0.0270018
$784$	0	0
$785$	−9.05086	−0.323039
$786$	0	0
$787$	11.4094	0.406702	0.203351	−	0.979106i	$-0.434817\pi$
$787$	11.4094	0.406702	0.203351	+	0.979106i	$0.434817\pi$
$788$	0	0
$789$	−30.1847	−1.07460
$790$	0	0
$791$	11.4193	0.406023
$792$	0	0
$793$	−54.8385	−1.94737
$794$	0	0
$795$	7.80642	0.276865
$796$	0	0
$797$	9.61285	0.340505	0.170252	−	0.985401i	$-0.445542\pi$
$797$	9.61285	0.340505	0.170252	+	0.985401i	$0.445542\pi$
$798$	0	0
$799$	6.69228	0.236756
$800$	0	0
$801$	8.10171	0.286260
$802$	0	0
$803$	6.10171	0.215325
$804$	0	0
$805$	8.85728	0.312178
$806$	0	0
$807$	30.7368	1.08199
$808$	0	0
$809$	−50.5531	−1.77735	−0.888676	−	0.458535i	$-0.848374\pi$
$809$	−50.5531	−1.77735	−0.888676	+	0.458535i	$0.848374\pi$
$810$	0	0
$811$	37.2355	1.30752	0.653758	−	0.756703i	$-0.273191\pi$
$811$	37.2355	1.30752	0.653758	+	0.756703i	$0.273191\pi$
$812$	0	0
$813$	18.6637	0.654565
$814$	0	0
$815$	9.24443	0.323818
$816$	0	0
$817$	−2.24797	−0.0786465
$818$	0	0
$819$	−5.05086	−0.176491
$820$	0	0
$821$	8.22216	0.286955	0.143478	−	0.989654i	$-0.454171\pi$
$821$	8.22216	0.286955	0.143478	+	0.989654i	$0.454171\pi$
$822$	0	0
$823$	−2.83854	−0.0989454	−0.0494727	−	0.998775i	$-0.515754\pi$
$823$	−2.83854	−0.0989454	−0.0494727	+	0.998775i	$0.515754\pi$
$824$	0	0
$825$	5.80642	0.202154
$826$	0	0
$827$	30.5718	1.06309	0.531543	−	0.847031i	$-0.321612\pi$
$827$	30.5718	1.06309	0.531543	+	0.847031i	$0.321612\pi$
$828$	0	0
$829$	37.4924	1.30216	0.651082	−	0.759007i	$-0.274315\pi$
$829$	37.4924	1.30216	0.651082	+	0.759007i	$0.274315\pi$
$830$	0	0
$831$	19.2444	0.667582
$832$	0	0
$833$	0.755569	0.0261789
$834$	0	0
$835$	10.7556	0.372212
$836$	0	0
$837$	−10.6637	−0.368591
$838$	0	0
$839$	20.3872	0.703843	0.351921	−	0.936030i	$-0.385528\pi$
$839$	20.3872	0.703843	0.351921	+	0.936030i	$0.385528\pi$
$840$	0	0
$841$	−28.4291	−0.980314
$842$	0	0
$843$	9.61285	0.331084
$844$	0	0
$845$	12.5111	0.430396
$846$	0	0
$847$	−22.7146	−0.780481
$848$	0	0
$849$	11.2257	0.385265
$850$	0	0
$851$	6.69228	0.229409
$852$	0	0
$853$	−1.23459	−0.0422715	−0.0211357	−	0.999777i	$-0.506728\pi$
$853$	−1.23459	−0.0422715	−0.0211357	+	0.999777i	$0.506728\pi$
$854$	0	0
$855$	1.80642	0.0617784
$856$	0	0
$857$	−36.5718	−1.24927	−0.624635	−	0.780917i	$-0.714752\pi$
$857$	−36.5718	−1.24927	−0.624635	+	0.780917i	$0.714752\pi$
$858$	0	0
$859$	−8.09187	−0.276091	−0.138045	−	0.990426i	$-0.544082\pi$
$859$	−8.09187	−0.276091	−0.138045	+	0.990426i	$0.544082\pi$
$860$	0	0
$861$	−9.61285	−0.327605
$862$	0	0
$863$	19.8796	0.676708	0.338354	−	0.941019i	$-0.390130\pi$
$863$	19.8796	0.676708	0.338354	+	0.941019i	$0.390130\pi$
$864$	0	0
$865$	9.61285	0.326847
$866$	0	0
$867$	16.4291	0.557962
$868$	0	0
$869$	23.2257	0.787878
$870$	0	0
$871$	−52.3694	−1.77447
$872$	0	0
$873$	5.05086	0.170946
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	11.9813	0.404579	0.202289	−	0.979326i	$-0.435162\pi$
$877$	11.9813	0.404579	0.202289	+	0.979326i	$0.435162\pi$
$878$	0	0
$879$	24.1017	0.812931
$880$	0	0
$881$	−4.87601	−0.164277	−0.0821386	−	0.996621i	$-0.526175\pi$
$881$	−4.87601	−0.164277	−0.0821386	+	0.996621i	$0.526175\pi$
$882$	0	0
$883$	−25.0607	−0.843360	−0.421680	−	0.906745i	$-0.638559\pi$
$883$	−25.0607	−0.843360	−0.421680	+	0.906745i	$0.638559\pi$
$884$	0	0
$885$	8.00000	0.268917
$886$	0	0
$887$	10.1650	0.341307	0.170654	−	0.985331i	$-0.445412\pi$
$887$	10.1650	0.341307	0.170654	+	0.985331i	$0.445412\pi$
$888$	0	0
$889$	14.1017	0.472956
$890$	0	0
$891$	−5.80642	−0.194523
$892$	0	0
$893$	16.0000	0.535420
$894$	0	0
$895$	−21.8064	−0.728908
$896$	0	0
$897$	44.7368	1.49372
$898$	0	0
$899$	8.05716	0.268721
$900$	0	0
$901$	−5.89829	−0.196501
$902$	0	0
$903$	−1.24443	−0.0414121
$904$	0	0
$905$	−15.2444	−0.506742
$906$	0	0
$907$	31.3461	1.04083	0.520416	−	0.853913i	$-0.325777\pi$
$907$	31.3461	1.04083	0.520416	+	0.853913i	$0.325777\pi$
$908$	0	0
$909$	−10.0000	−0.331679
$910$	0	0
$911$	−7.25428	−0.240345	−0.120172	−	0.992753i	$-0.538345\pi$
$911$	−7.25428	−0.240345	−0.120172	+	0.992753i	$0.538345\pi$
$912$	0	0
$913$	−44.2034	−1.46292
$914$	0	0
$915$	10.8573	0.358931
$916$	0	0
$917$	14.1017	0.465679
$918$	0	0
$919$	−51.2257	−1.68978	−0.844890	−	0.534940i	$-0.820334\pi$
$919$	−51.2257	−1.68978	−0.844890	+	0.534940i	$0.820334\pi$
$920$	0	0
$921$	5.12399	0.168841
$922$	0	0
$923$	15.4094	0.507207
$924$	0	0
$925$	−0.755569	−0.0248429
$926$	0	0
$927$	0	0
$928$	0	0
$929$	20.8385	0.683690	0.341845	−	0.939756i	$-0.388948\pi$
$929$	20.8385	0.683690	0.341845	+	0.939756i	$0.388948\pi$
$930$	0	0
$931$	1.80642	0.0592032
$932$	0	0
$933$	−23.2257	−0.760375
$934$	0	0
$935$	−4.38715	−0.143475
$936$	0	0
$937$	−46.5620	−1.52111	−0.760557	−	0.649271i	$-0.775074\pi$
$937$	−46.5620	−1.52111	−0.760557	+	0.649271i	$0.775074\pi$
$938$	0	0
$939$	−3.33630	−0.108876
$940$	0	0
$941$	−42.4068	−1.38242	−0.691212	−	0.722652i	$-0.742923\pi$
$941$	−42.4068	−1.38242	−0.691212	+	0.722652i	$0.742923\pi$
$942$	0	0
$943$	85.1437	2.77266
$944$	0	0
$945$	1.00000	0.0325300
$946$	0	0
$947$	−54.9215	−1.78471	−0.892355	−	0.451335i	$-0.850948\pi$
$947$	−54.9215	−1.78471	−0.892355	+	0.451335i	$0.850948\pi$
$948$	0	0
$949$	−5.30772	−0.172296
$950$	0	0
$951$	28.0098	0.908281
$952$	0	0
$953$	3.03212	0.0982200	0.0491100	−	0.998793i	$-0.484362\pi$
$953$	3.03212	0.0982200	0.0491100	+	0.998793i	$0.484362\pi$
$954$	0	0
$955$	−3.05086	−0.0987234
$956$	0	0
$957$	4.38715	0.141816
$958$	0	0
$959$	10.2953	0.332452
$960$	0	0
$961$	82.7146	2.66821
$962$	0	0
$963$	4.85728	0.156524
$964$	0	0
$965$	4.48886	0.144502
$966$	0	0
$967$	45.9180	1.47662	0.738311	−	0.674460i	$-0.235624\pi$
$967$	45.9180	1.47662	0.738311	+	0.674460i	$0.235624\pi$
$968$	0	0
$969$	−1.36488	−0.0438462
$970$	0	0
$971$	50.6548	1.62559	0.812795	−	0.582550i	$-0.197945\pi$
$971$	50.6548	1.62559	0.812795	+	0.582550i	$0.197945\pi$
$972$	0	0
$973$	6.19358	0.198557
$974$	0	0
$975$	−5.05086	−0.161757
$976$	0	0
$977$	−0.930409	−0.0297664	−0.0148832	−	0.999889i	$-0.504738\pi$
$977$	−0.930409	−0.0297664	−0.0148832	+	0.999889i	$0.504738\pi$
$978$	0	0
$979$	−47.0420	−1.50347
$980$	0	0
$981$	−4.48886	−0.143318
$982$	0	0
$983$	31.5496	1.00627	0.503137	−	0.864206i	$-0.332179\pi$
$983$	31.5496	1.00627	0.503137	+	0.864206i	$0.332179\pi$
$984$	0	0
$985$	−1.70471	−0.0543167
$986$	0	0
$987$	8.85728	0.281930
$988$	0	0
$989$	11.0223	0.350488
$990$	0	0
$991$	−4.77430	−0.151661	−0.0758304	−	0.997121i	$-0.524161\pi$
$991$	−4.77430	−0.151661	−0.0758304	+	0.997121i	$0.524161\pi$
$992$	0	0
$993$	−2.28544	−0.0725263
$994$	0	0
$995$	−20.5620	−0.651859
$996$	0	0
$997$	−0.460282	−0.0145773	−0.00728864	−	0.999973i	$-0.502320\pi$
$997$	−0.460282	−0.0145773	−0.00728864	+	0.999973i	$0.502320\pi$
$998$	0	0
$999$	0.755569	0.0239051

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3360.2.a.bi.1.1	✓	3
4.3	odd	2		3360.2.a.bj.1.3	yes	3
8.3	odd	2		6720.2.a.da.1.1		3
8.5	even	2		6720.2.a.db.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3360.2.a.bi.1.1	✓	3	1.1	even	1	trivial
3360.2.a.bj.1.3	yes	3	4.3	odd	2
6720.2.a.da.1.1		3	8.3	odd	2
6720.2.a.db.1.3		3	8.5	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:	\( N \)	\(=\)	\( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3360.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -5.80642 q^{11} +5.05086 q^{13} -1.00000 q^{15} +0.755569 q^{17} +1.80642 q^{19} +1.00000 q^{21} -8.85728 q^{23} +1.00000 q^{25} -1.00000 q^{27} +0.755569 q^{29} +10.6637 q^{31} +5.80642 q^{33} -1.00000 q^{35} -0.755569 q^{37} -5.05086 q^{39} -9.61285 q^{41} -1.24443 q^{43} +1.00000 q^{45} +8.85728 q^{47} +1.00000 q^{49} -0.755569 q^{51} -7.80642 q^{53} -5.80642 q^{55} -1.80642 q^{57} -8.00000 q^{59} -10.8573 q^{61} -1.00000 q^{63} +5.05086 q^{65} -10.3684 q^{67} +8.85728 q^{69} +3.05086 q^{71} -1.05086 q^{73} -1.00000 q^{75} +5.80642 q^{77} -4.00000 q^{79} +1.00000 q^{81} +7.61285 q^{83} +0.755569 q^{85} -0.755569 q^{87} +8.10171 q^{89} -5.05086 q^{91} -10.6637 q^{93} +1.80642 q^{95} +5.05086 q^{97} -5.80642 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 3 * q^5 - 3 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} + 3 q^{5} - 3 q^{7} + 3 q^{9} - 4 q^{11} + 2 q^{13} - 3 q^{15} + 2 q^{17} - 8 q^{19} + 3 q^{21} + 3 q^{25} - 3 q^{27} + 2 q^{29} - 8 q^{31} + 4 q^{33} - 3 q^{35} - 2 q^{37} - 2 q^{39} - 2 q^{41} - 4 q^{43} + 3 q^{45} + 3 q^{49} - 2 q^{51} - 10 q^{53} - 4 q^{55} + 8 q^{57} - 24 q^{59} - 6 q^{61} - 3 q^{63} + 2 q^{65} - 4 q^{67} - 4 q^{71} + 10 q^{73} - 3 q^{75} + 4 q^{77} - 12 q^{79} + 3 q^{81} - 4 q^{83} + 2 q^{85} - 2 q^{87} - 2 q^{89} - 2 q^{91} + 8 q^{93} - 8 q^{95} + 2 q^{97} - 4 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 3 * q^5 - 3 * q^7 + 3 * q^9 - 4 * q^11 + 2 * q^13 - 3 * q^15 + 2 * q^17 - 8 * q^19 + 3 * q^21 + 3 * q^25 - 3 * q^27 + 2 * q^29 - 8 * q^31 + 4 * q^33 - 3 * q^35 - 2 * q^37 - 2 * q^39 - 2 * q^41 - 4 * q^43 + 3 * q^45 + 3 * q^49 - 2 * q^51 - 10 * q^53 - 4 * q^55 + 8 * q^57 - 24 * q^59 - 6 * q^61 - 3 * q^63 + 2 * q^65 - 4 * q^67 - 4 * q^71 + 10 * q^73 - 3 * q^75 + 4 * q^77 - 12 * q^79 + 3 * q^81 - 4 * q^83 + 2 * q^85 - 2 * q^87 - 2 * q^89 - 2 * q^91 + 8 * q^93 - 8 * q^95 + 2 * q^97 - 4 * q^99

Introduction

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