LMFDB - Embedded newform 7728.2.a.bs.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7728,2,Mod(1,7728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7728, 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("7728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7728.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:	$61.7083906820$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3864)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.86081$ of defining polynomial
Character	$\chi$	$=$	7728.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -0.462598 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -0.462598 q^{5} +1.00000 q^{7} +1.00000 q^{9} -1.32340 q^{11} +1.78600 q^{13} +0.462598 q^{15} -2.39821 q^{17} +7.04502 q^{19} -1.00000 q^{21} +1.00000 q^{23} -4.78600 q^{25} -1.00000 q^{27} -3.32340 q^{29} -7.04502 q^{31} +1.32340 q^{33} -0.462598 q^{35} -7.32340 q^{37} -1.78600 q^{39} +1.04502 q^{41} +1.13919 q^{43} -0.462598 q^{45} +11.0152 q^{47} +1.00000 q^{49} +2.39821 q^{51} +0.591380 q^{53} +0.612205 q^{55} -7.04502 q^{57} +6.46260 q^{59} +0.860806 q^{61} +1.00000 q^{63} -0.826202 q^{65} -14.7022 q^{67} -1.00000 q^{69} +2.73202 q^{71} -15.3234 q^{73} +4.78600 q^{75} -1.32340 q^{77} +4.39821 q^{79} +1.00000 q^{81} -12.7666 q^{83} +1.10941 q^{85} +3.32340 q^{87} +7.37883 q^{89} +1.78600 q^{91} +7.04502 q^{93} -3.25901 q^{95} -8.11982 q^{97} -1.32340 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + q^5 + 3 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} + q^{5} + 3 q^{7} + 3 q^{9} + 4 q^{11} - 5 q^{13} - q^{15} - 4 q^{17} + 2 q^{19} - 3 q^{21} + 3 q^{23} - 4 q^{25} - 3 q^{27} - 2 q^{29} - 2 q^{31} - 4 q^{33} + q^{35} - 14 q^{37} + 5 q^{39} - 16 q^{41} + 9 q^{43} + q^{45} - 10 q^{47} + 3 q^{49} + 4 q^{51} + q^{53} + 9 q^{55} - 2 q^{57} + 17 q^{59} - 3 q^{61} + 3 q^{63} - 20 q^{65} - 13 q^{67} - 3 q^{69} + q^{71} - 38 q^{73} + 4 q^{75} + 4 q^{77} + 10 q^{79} + 3 q^{81} - 8 q^{83} - 15 q^{85} + 2 q^{87} - q^{89} - 5 q^{91} + 2 q^{93} - q^{95} - 10 q^{97} + 4 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + q^5 + 3 * q^7 + 3 * q^9 + 4 * q^11 - 5 * q^13 - q^15 - 4 * q^17 + 2 * q^19 - 3 * q^21 + 3 * q^23 - 4 * q^25 - 3 * q^27 - 2 * q^29 - 2 * q^31 - 4 * q^33 + q^35 - 14 * q^37 + 5 * q^39 - 16 * q^41 + 9 * q^43 + q^45 - 10 * q^47 + 3 * q^49 + 4 * q^51 + q^53 + 9 * q^55 - 2 * q^57 + 17 * q^59 - 3 * q^61 + 3 * q^63 - 20 * q^65 - 13 * q^67 - 3 * q^69 + q^71 - 38 * q^73 + 4 * q^75 + 4 * q^77 + 10 * q^79 + 3 * q^81 - 8 * q^83 - 15 * q^85 + 2 * q^87 - q^89 - 5 * q^91 + 2 * q^93 - q^95 - 10 * 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
$5$	−0.462598	−0.206880	−0.103440	−	0.994636i	$-0.532985\pi$
$5$	−0.462598	−0.206880	−0.103440	+	0.994636i	$0.532985\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.32340	−0.399021	−0.199511	−	0.979896i	$-0.563935\pi$
$11$	−1.32340	−0.399021	−0.199511	+	0.979896i	$0.563935\pi$
$12$	0	0
$13$	1.78600	0.495348	0.247674	−	0.968843i	$-0.420334\pi$
$13$	1.78600	0.495348	0.247674	+	0.968843i	$0.420334\pi$
$14$	0	0
$15$	0.462598	0.119442
$16$	0	0
$17$	−2.39821	−0.581651	−0.290825	−	0.956776i	$-0.593930\pi$
$17$	−2.39821	−0.581651	−0.290825	+	0.956776i	$0.593930\pi$
$18$	0	0
$19$	7.04502	1.61624	0.808119	−	0.589020i	$-0.200486\pi$
$19$	7.04502	1.61624	0.808119	+	0.589020i	$0.200486\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.78600	−0.957201
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−3.32340	−0.617141	−0.308570	−	0.951202i	$-0.599851\pi$
$29$	−3.32340	−0.617141	−0.308570	+	0.951202i	$0.599851\pi$
$30$	0	0
$31$	−7.04502	−1.26532	−0.632661	−	0.774429i	$-0.718037\pi$
$31$	−7.04502	−1.26532	−0.632661	+	0.774429i	$0.718037\pi$
$32$	0	0
$33$	1.32340	0.230375
$34$	0	0
$35$	−0.462598	−0.0781934
$36$	0	0
$37$	−7.32340	−1.20396	−0.601980	−	0.798511i	$-0.705621\pi$
$37$	−7.32340	−1.20396	−0.601980	+	0.798511i	$0.705621\pi$
$38$	0	0
$39$	−1.78600	−0.285989
$40$	0	0
$41$	1.04502	0.163204	0.0816020	−	0.996665i	$-0.473996\pi$
$41$	1.04502	0.163204	0.0816020	+	0.996665i	$0.473996\pi$
$42$	0	0
$43$	1.13919	0.173725	0.0868627	−	0.996220i	$-0.472316\pi$
$43$	1.13919	0.173725	0.0868627	+	0.996220i	$0.472316\pi$
$44$	0	0
$45$	−0.462598	−0.0689601
$46$	0	0
$47$	11.0152	1.60674	0.803368	−	0.595483i	$-0.203039\pi$
$47$	11.0152	1.60674	0.803368	+	0.595483i	$0.203039\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.39821	0.335816
$52$	0	0
$53$	0.591380	0.0812323	0.0406162	−	0.999175i	$-0.487068\pi$
$53$	0.591380	0.0812323	0.0406162	+	0.999175i	$0.487068\pi$
$54$	0	0
$55$	0.612205	0.0825497
$56$	0	0
$57$	−7.04502	−0.933135
$58$	0	0
$59$	6.46260	0.841359	0.420679	−	0.907209i	$-0.361792\pi$
$59$	6.46260	0.841359	0.420679	+	0.907209i	$0.361792\pi$
$60$	0	0
$61$	0.860806	0.110215	0.0551074	−	0.998480i	$-0.482450\pi$
$61$	0.860806	0.110215	0.0551074	+	0.998480i	$0.482450\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−0.826202	−0.102478
$66$	0	0
$67$	−14.7022	−1.79616	−0.898082	−	0.439828i	$-0.855039\pi$
$67$	−14.7022	−1.79616	−0.898082	+	0.439828i	$0.855039\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	2.73202	0.324232	0.162116	−	0.986772i	$-0.448168\pi$
$71$	2.73202	0.324232	0.162116	+	0.986772i	$0.448168\pi$
$72$	0	0
$73$	−15.3234	−1.79347	−0.896734	−	0.442569i	$-0.854067\pi$
$73$	−15.3234	−1.79347	−0.896734	+	0.442569i	$0.854067\pi$
$74$	0	0
$75$	4.78600	0.552640
$76$	0	0
$77$	−1.32340	−0.150816
$78$	0	0
$79$	4.39821	0.494837	0.247418	−	0.968909i	$-0.420418\pi$
$79$	4.39821	0.494837	0.247418	+	0.968909i	$0.420418\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−12.7666	−1.40132	−0.700660	−	0.713496i	$-0.747111\pi$
$83$	−12.7666	−1.40132	−0.700660	+	0.713496i	$0.747111\pi$
$84$	0	0
$85$	1.10941	0.120332
$86$	0	0
$87$	3.32340	0.356306
$88$	0	0
$89$	7.37883	0.782155	0.391077	−	0.920358i	$-0.372102\pi$
$89$	7.37883	0.782155	0.391077	+	0.920358i	$0.372102\pi$
$90$	0	0
$91$	1.78600	0.187224
$92$	0	0
$93$	7.04502	0.730534
$94$	0	0
$95$	−3.25901	−0.334368
$96$	0	0
$97$	−8.11982	−0.824443	−0.412221	−	0.911084i	$-0.635247\pi$
$97$	−8.11982	−0.824443	−0.412221	+	0.911084i	$0.635247\pi$
$98$	0	0
$99$	−1.32340	−0.133007
$100$	0	0
$101$	5.75622	0.572765	0.286382	−	0.958115i	$-0.407547\pi$
$101$	5.75622	0.572765	0.286382	+	0.958115i	$0.407547\pi$
$102$	0	0
$103$	−11.4432	−1.12753	−0.563767	−	0.825934i	$-0.690648\pi$
$103$	−11.4432	−1.12753	−0.563767	+	0.825934i	$0.690648\pi$
$104$	0	0
$105$	0.462598	0.0451450
$106$	0	0
$107$	8.71120	0.842143	0.421072	−	0.907027i	$-0.361654\pi$
$107$	8.71120	0.842143	0.421072	+	0.907027i	$0.361654\pi$
$108$	0	0
$109$	−11.4778	−1.09938	−0.549688	−	0.835370i	$-0.685253\pi$
$109$	−11.4778	−1.09938	−0.549688	+	0.835370i	$0.685253\pi$
$110$	0	0
$111$	7.32340	0.695107
$112$	0	0
$113$	11.3490	1.06763	0.533814	−	0.845602i	$-0.320758\pi$
$113$	11.3490	1.06763	0.533814	+	0.845602i	$0.320758\pi$
$114$	0	0
$115$	−0.462598	−0.0431375
$116$	0	0
$117$	1.78600	0.165116
$118$	0	0
$119$	−2.39821	−0.219843
$120$	0	0
$121$	−9.24860	−0.840782
$122$	0	0
$123$	−1.04502	−0.0942259
$124$	0	0
$125$	4.52699	0.404906
$126$	0	0
$127$	16.7112	1.48288	0.741440	−	0.671020i	$-0.234143\pi$
$127$	16.7112	1.48288	0.741440	+	0.671020i	$0.234143\pi$
$128$	0	0
$129$	−1.13919	−0.100300
$130$	0	0
$131$	12.0990	1.05709	0.528547	−	0.848904i	$-0.322737\pi$
$131$	12.0990	1.05709	0.528547	+	0.848904i	$0.322737\pi$
$132$	0	0
$133$	7.04502	0.610880
$134$	0	0
$135$	0.462598	0.0398141
$136$	0	0
$137$	4.37738	0.373985	0.186993	−	0.982361i	$-0.440126\pi$
$137$	4.37738	0.373985	0.186993	+	0.982361i	$0.440126\pi$
$138$	0	0
$139$	9.10941	0.772650	0.386325	−	0.922363i	$-0.373744\pi$
$139$	9.10941	0.772650	0.386325	+	0.922363i	$0.373744\pi$
$140$	0	0
$141$	−11.0152	−0.927650
$142$	0	0
$143$	−2.36360	−0.197654
$144$	0	0
$145$	1.53740	0.127674
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−6.36842	−0.521721	−0.260861	−	0.965376i	$-0.584006\pi$
$149$	−6.36842	−0.521721	−0.260861	+	0.965376i	$0.584006\pi$
$150$	0	0
$151$	7.31444	0.595241	0.297620	−	0.954684i	$-0.403807\pi$
$151$	7.31444	0.595241	0.297620	+	0.954684i	$0.403807\pi$
$152$	0	0
$153$	−2.39821	−0.193884
$154$	0	0
$155$	3.25901	0.261770
$156$	0	0
$157$	9.31444	0.743373	0.371687	−	0.928358i	$-0.378780\pi$
$157$	9.31444	0.743373	0.371687	+	0.928358i	$0.378780\pi$
$158$	0	0
$159$	−0.591380	−0.0468995
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	2.59138	0.202973	0.101486	−	0.994837i	$-0.467640\pi$
$163$	2.59138	0.202973	0.101486	+	0.994837i	$0.467640\pi$
$164$	0	0
$165$	−0.612205	−0.0476601
$166$	0	0
$167$	0.895410	0.0692889	0.0346444	−	0.999400i	$-0.488970\pi$
$167$	0.895410	0.0692889	0.0346444	+	0.999400i	$0.488970\pi$
$168$	0	0
$169$	−9.81019	−0.754630
$170$	0	0
$171$	7.04502	0.538746
$172$	0	0
$173$	−12.4882	−0.949463	−0.474732	−	0.880131i	$-0.657455\pi$
$173$	−12.4882	−0.949463	−0.474732	+	0.880131i	$0.657455\pi$
$174$	0	0
$175$	−4.78600	−0.361788
$176$	0	0
$177$	−6.46260	−0.485759
$178$	0	0
$179$	22.1455	1.65523	0.827615	−	0.561297i	$-0.189697\pi$
$179$	22.1455	1.65523	0.827615	+	0.561297i	$0.189697\pi$
$180$	0	0
$181$	−17.9702	−1.33572	−0.667858	−	0.744289i	$-0.732789\pi$
$181$	−17.9702	−1.33572	−0.667858	+	0.744289i	$0.732789\pi$
$182$	0	0
$183$	−0.860806	−0.0636326
$184$	0	0
$185$	3.38780	0.249076
$186$	0	0
$187$	3.17380	0.232091
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−7.31444	−0.529254	−0.264627	−	0.964351i	$-0.585249\pi$
$191$	−7.31444	−0.529254	−0.264627	+	0.964351i	$0.585249\pi$
$192$	0	0
$193$	−21.5720	−1.55279	−0.776393	−	0.630249i	$-0.782953\pi$
$193$	−21.5720	−1.55279	−0.776393	+	0.630249i	$0.782953\pi$
$194$	0	0
$195$	0.826202	0.0591656
$196$	0	0
$197$	−14.4120	−1.02681	−0.513406	−	0.858146i	$-0.671616\pi$
$197$	−14.4120	−1.02681	−0.513406	+	0.858146i	$0.671616\pi$
$198$	0	0
$199$	−6.33382	−0.448992	−0.224496	−	0.974475i	$-0.572074\pi$
$199$	−6.33382	−0.448992	−0.224496	+	0.974475i	$0.572074\pi$
$200$	0	0
$201$	14.7022	1.03702
$202$	0	0
$203$	−3.32340	−0.233257
$204$	0	0
$205$	−0.483423	−0.0337637
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−9.32340	−0.644913
$210$	0	0
$211$	5.06584	0.348747	0.174373	−	0.984680i	$-0.444210\pi$
$211$	5.06584	0.348747	0.174373	+	0.984680i	$0.444210\pi$
$212$	0	0
$213$	−2.73202	−0.187195
$214$	0	0
$215$	−0.526989	−0.0359404
$216$	0	0
$217$	−7.04502	−0.478247
$218$	0	0
$219$	15.3234	1.03546
$220$	0	0
$221$	−4.28320	−0.288120
$222$	0	0
$223$	−23.3282	−1.56217	−0.781087	−	0.624423i	$-0.785334\pi$
$223$	−23.3282	−1.56217	−0.781087	+	0.624423i	$0.785334\pi$
$224$	0	0
$225$	−4.78600	−0.319067
$226$	0	0
$227$	−14.5318	−0.964510	−0.482255	−	0.876031i	$-0.660182\pi$
$227$	−14.5318	−0.964510	−0.482255	+	0.876031i	$0.660182\pi$
$228$	0	0
$229$	4.83102	0.319243	0.159621	−	0.987178i	$-0.448973\pi$
$229$	4.83102	0.319243	0.159621	+	0.987178i	$0.448973\pi$
$230$	0	0
$231$	1.32340	0.0870736
$232$	0	0
$233$	15.2203	0.997113	0.498556	−	0.866857i	$-0.333864\pi$
$233$	15.2203	0.997113	0.498556	+	0.866857i	$0.333864\pi$
$234$	0	0
$235$	−5.09563	−0.332402
$236$	0	0
$237$	−4.39821	−0.285694
$238$	0	0
$239$	−22.6427	−1.46463	−0.732316	−	0.680965i	$-0.761561\pi$
$239$	−22.6427	−1.46463	−0.732316	+	0.680965i	$0.761561\pi$
$240$	0	0
$241$	−16.9162	−1.08967	−0.544835	−	0.838543i	$-0.683408\pi$
$241$	−16.9162	−1.08967	−0.544835	+	0.838543i	$0.683408\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−0.462598	−0.0295543
$246$	0	0
$247$	12.5824	0.800600
$248$	0	0
$249$	12.7666	0.809052
$250$	0	0
$251$	−20.0692	−1.26676	−0.633379	−	0.773842i	$-0.718332\pi$
$251$	−20.0692	−1.26676	−0.633379	+	0.773842i	$0.718332\pi$
$252$	0	0
$253$	−1.32340	−0.0832017
$254$	0	0
$255$	−1.10941	−0.0694738
$256$	0	0
$257$	−26.5485	−1.65605	−0.828024	−	0.560692i	$-0.810535\pi$
$257$	−26.5485	−1.65605	−0.828024	+	0.560692i	$0.810535\pi$
$258$	0	0
$259$	−7.32340	−0.455054
$260$	0	0
$261$	−3.32340	−0.205714
$262$	0	0
$263$	−11.3442	−0.699515	−0.349758	−	0.936840i	$-0.613736\pi$
$263$	−11.3442	−0.699515	−0.349758	+	0.936840i	$0.613736\pi$
$264$	0	0
$265$	−0.273572	−0.0168054
$266$	0	0
$267$	−7.37883	−0.451577
$268$	0	0
$269$	3.63640	0.221715	0.110858	−	0.993836i	$-0.464640\pi$
$269$	3.63640	0.221715	0.110858	+	0.993836i	$0.464640\pi$
$270$	0	0
$271$	−27.9106	−1.69545	−0.847725	−	0.530436i	$-0.822028\pi$
$271$	−27.9106	−1.69545	−0.847725	+	0.530436i	$0.822028\pi$
$272$	0	0
$273$	−1.78600	−0.108094
$274$	0	0
$275$	6.33382	0.381944
$276$	0	0
$277$	10.9508	0.657972	0.328986	−	0.944335i	$-0.393293\pi$
$277$	10.9508	0.657972	0.328986	+	0.944335i	$0.393293\pi$
$278$	0	0
$279$	−7.04502	−0.421774
$280$	0	0
$281$	−1.01523	−0.0605635	−0.0302817	−	0.999541i	$-0.509640\pi$
$281$	−1.01523	−0.0605635	−0.0302817	+	0.999541i	$0.509640\pi$
$282$	0	0
$283$	7.49865	0.445749	0.222874	−	0.974847i	$-0.428456\pi$
$283$	7.49865	0.445749	0.222874	+	0.974847i	$0.428456\pi$
$284$	0	0
$285$	3.25901	0.193047
$286$	0	0
$287$	1.04502	0.0616853
$288$	0	0
$289$	−11.2486	−0.661682
$290$	0	0
$291$	8.11982	0.475992
$292$	0	0
$293$	18.7964	1.09810	0.549049	−	0.835790i	$-0.314990\pi$
$293$	18.7964	1.09810	0.549049	+	0.835790i	$0.314990\pi$
$294$	0	0
$295$	−2.98959	−0.174061
$296$	0	0
$297$	1.32340	0.0767917
$298$	0	0
$299$	1.78600	0.103287
$300$	0	0
$301$	1.13919	0.0656621
$302$	0	0
$303$	−5.75622	−0.330686
$304$	0	0
$305$	−0.398207	−0.0228013
$306$	0	0
$307$	16.0811	0.917795	0.458898	−	0.888489i	$-0.348244\pi$
$307$	16.0811	0.917795	0.458898	+	0.888489i	$0.348244\pi$
$308$	0	0
$309$	11.4432	0.650982
$310$	0	0
$311$	−22.5437	−1.27833	−0.639167	−	0.769068i	$-0.720721\pi$
$311$	−22.5437	−1.27833	−0.639167	+	0.769068i	$0.720721\pi$
$312$	0	0
$313$	−8.90437	−0.503305	−0.251652	−	0.967818i	$-0.580974\pi$
$313$	−8.90437	−0.503305	−0.251652	+	0.967818i	$0.580974\pi$
$314$	0	0
$315$	−0.462598	−0.0260645
$316$	0	0
$317$	7.66618	0.430576	0.215288	−	0.976551i	$-0.430931\pi$
$317$	7.66618	0.430576	0.215288	+	0.976551i	$0.430931\pi$
$318$	0	0
$319$	4.39821	0.246252
$320$	0	0
$321$	−8.71120	−0.486212
$322$	0	0
$323$	−16.8954	−0.940086
$324$	0	0
$325$	−8.54781	−0.474147
$326$	0	0
$327$	11.4778	0.634725
$328$	0	0
$329$	11.0152	0.607289
$330$	0	0
$331$	25.2340	1.38699	0.693494	−	0.720462i	$-0.256070\pi$
$331$	25.2340	1.38699	0.693494	+	0.720462i	$0.256070\pi$
$332$	0	0
$333$	−7.32340	−0.401320
$334$	0	0
$335$	6.80123	0.371591
$336$	0	0
$337$	19.8760	1.08272	0.541358	−	0.840792i	$-0.317910\pi$
$337$	19.8760	1.08272	0.541358	+	0.840792i	$0.317910\pi$
$338$	0	0
$339$	−11.3490	−0.616396
$340$	0	0
$341$	9.32340	0.504891
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0.462598	0.0249055
$346$	0	0
$347$	−4.95498	−0.265997	−0.132999	−	0.991116i	$-0.542461\pi$
$347$	−4.95498	−0.265997	−0.132999	+	0.991116i	$0.542461\pi$
$348$	0	0
$349$	0.831019	0.0444834	0.0222417	−	0.999753i	$-0.492920\pi$
$349$	0.831019	0.0444834	0.0222417	+	0.999753i	$0.492920\pi$
$350$	0	0
$351$	−1.78600	−0.0953298
$352$	0	0
$353$	−22.8358	−1.21543	−0.607714	−	0.794156i	$-0.707913\pi$
$353$	−22.8358	−1.21543	−0.607714	+	0.794156i	$0.707913\pi$
$354$	0	0
$355$	−1.26383	−0.0670771
$356$	0	0
$357$	2.39821	0.126927
$358$	0	0
$359$	15.3878	0.812137	0.406068	−	0.913843i	$-0.366899\pi$
$359$	15.3878	0.812137	0.406068	+	0.913843i	$0.366899\pi$
$360$	0	0
$361$	30.6323	1.61222
$362$	0	0
$363$	9.24860	0.485426
$364$	0	0
$365$	7.08858	0.371033
$366$	0	0
$367$	10.6724	0.557097	0.278549	−	0.960422i	$-0.410147\pi$
$367$	10.6724	0.557097	0.278549	+	0.960422i	$0.410147\pi$
$368$	0	0
$369$	1.04502	0.0544014
$370$	0	0
$371$	0.591380	0.0307029
$372$	0	0
$373$	4.85666	0.251468	0.125734	−	0.992064i	$-0.459871\pi$
$373$	4.85666	0.251468	0.125734	+	0.992064i	$0.459871\pi$
$374$	0	0
$375$	−4.52699	−0.233773
$376$	0	0
$377$	−5.93561	−0.305699
$378$	0	0
$379$	−14.1496	−0.726816	−0.363408	−	0.931630i	$-0.618387\pi$
$379$	−14.1496	−0.726816	−0.363408	+	0.931630i	$0.618387\pi$
$380$	0	0
$381$	−16.7112	−0.856141
$382$	0	0
$383$	−31.6710	−1.61831	−0.809156	−	0.587594i	$-0.800075\pi$
$383$	−31.6710	−1.61831	−0.809156	+	0.587594i	$0.800075\pi$
$384$	0	0
$385$	0.612205	0.0312008
$386$	0	0
$387$	1.13919	0.0579085
$388$	0	0
$389$	26.8358	1.36063	0.680315	−	0.732919i	$-0.261843\pi$
$389$	26.8358	1.36063	0.680315	+	0.732919i	$0.261843\pi$
$390$	0	0
$391$	−2.39821	−0.121283
$392$	0	0
$393$	−12.0990	−0.610314
$394$	0	0
$395$	−2.03460	−0.102372
$396$	0	0
$397$	6.11086	0.306695	0.153348	−	0.988172i	$-0.450995\pi$
$397$	6.11086	0.306695	0.153348	+	0.988172i	$0.450995\pi$
$398$	0	0
$399$	−7.04502	−0.352692
$400$	0	0
$401$	−23.6918	−1.18311	−0.591557	−	0.806263i	$-0.701486\pi$
$401$	−23.6918	−1.18311	−0.591557	+	0.806263i	$0.701486\pi$
$402$	0	0
$403$	−12.5824	−0.626775
$404$	0	0
$405$	−0.462598	−0.0229867
$406$	0	0
$407$	9.69182	0.480406
$408$	0	0
$409$	−33.2847	−1.64582	−0.822910	−	0.568172i	$-0.807651\pi$
$409$	−33.2847	−1.64582	−0.822910	+	0.568172i	$0.807651\pi$
$410$	0	0
$411$	−4.37738	−0.215920
$412$	0	0
$413$	6.46260	0.318004
$414$	0	0
$415$	5.90582	0.289905
$416$	0	0
$417$	−9.10941	−0.446090
$418$	0	0
$419$	27.3103	1.33420	0.667098	−	0.744970i	$-0.267536\pi$
$419$	27.3103	1.33420	0.667098	+	0.744970i	$0.267536\pi$
$420$	0	0
$421$	24.8012	1.20874	0.604369	−	0.796705i	$-0.293425\pi$
$421$	24.8012	1.20874	0.604369	+	0.796705i	$0.293425\pi$
$422$	0	0
$423$	11.0152	0.535579
$424$	0	0
$425$	11.4778	0.556756
$426$	0	0
$427$	0.860806	0.0416573
$428$	0	0
$429$	2.36360	0.114116
$430$	0	0
$431$	−31.8552	−1.53441	−0.767206	−	0.641401i	$-0.778353\pi$
$431$	−31.8552	−1.53441	−0.767206	+	0.641401i	$0.778353\pi$
$432$	0	0
$433$	−38.0513	−1.82863	−0.914314	−	0.405006i	$-0.867269\pi$
$433$	−38.0513	−1.82863	−0.914314	+	0.405006i	$0.867269\pi$
$434$	0	0
$435$	−1.53740	−0.0737128
$436$	0	0
$437$	7.04502	0.337009
$438$	0	0
$439$	−9.15297	−0.436848	−0.218424	−	0.975854i	$-0.570092\pi$
$439$	−9.15297	−0.436848	−0.218424	+	0.975854i	$0.570092\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	14.5872	0.693060	0.346530	−	0.938039i	$-0.387360\pi$
$443$	14.5872	0.693060	0.346530	+	0.938039i	$0.387360\pi$
$444$	0	0
$445$	−3.41344	−0.161812
$446$	0	0
$447$	6.36842	0.301216
$448$	0	0
$449$	−18.4834	−0.872287	−0.436143	−	0.899877i	$-0.643656\pi$
$449$	−18.4834	−0.872287	−0.436143	+	0.899877i	$0.643656\pi$
$450$	0	0
$451$	−1.38298	−0.0651219
$452$	0	0
$453$	−7.31444	−0.343662
$454$	0	0
$455$	−0.826202	−0.0387329
$456$	0	0
$457$	−30.2951	−1.41714	−0.708572	−	0.705639i	$-0.750660\pi$
$457$	−30.2951	−1.41714	−0.708572	+	0.705639i	$0.750660\pi$
$458$	0	0
$459$	2.39821	0.111939
$460$	0	0
$461$	−9.38780	−0.437233	−0.218617	−	0.975811i	$-0.570154\pi$
$461$	−9.38780	−0.437233	−0.218617	+	0.975811i	$0.570154\pi$
$462$	0	0
$463$	10.4882	0.487430	0.243715	−	0.969847i	$-0.421634\pi$
$463$	10.4882	0.487430	0.243715	+	0.969847i	$0.421634\pi$
$464$	0	0
$465$	−3.25901	−0.151133
$466$	0	0
$467$	−10.1198	−0.468289	−0.234145	−	0.972202i	$-0.575229\pi$
$467$	−10.1198	−0.468289	−0.234145	+	0.972202i	$0.575229\pi$
$468$	0	0
$469$	−14.7022	−0.678886
$470$	0	0
$471$	−9.31444	−0.429187
$472$	0	0
$473$	−1.50761	−0.0693202
$474$	0	0
$475$	−33.7175	−1.54706
$476$	0	0
$477$	0.591380	0.0270774
$478$	0	0
$479$	19.2251	0.878416	0.439208	−	0.898385i	$-0.355259\pi$
$479$	19.2251	0.878416	0.439208	+	0.898385i	$0.355259\pi$
$480$	0	0
$481$	−13.0796	−0.596379
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	3.75622	0.170561
$486$	0	0
$487$	−7.90101	−0.358029	−0.179014	−	0.983846i	$-0.557291\pi$
$487$	−7.90101	−0.358029	−0.179014	+	0.983846i	$0.557291\pi$
$488$	0	0
$489$	−2.59138	−0.117186
$490$	0	0
$491$	−39.5797	−1.78621	−0.893104	−	0.449850i	$-0.851477\pi$
$491$	−39.5797	−1.78621	−0.893104	+	0.449850i	$0.851477\pi$
$492$	0	0
$493$	7.97021	0.358960
$494$	0	0
$495$	0.612205	0.0275166
$496$	0	0
$497$	2.73202	0.122548
$498$	0	0
$499$	−37.2299	−1.66664	−0.833320	−	0.552792i	$-0.813563\pi$
$499$	−37.2299	−1.66664	−0.833320	+	0.552792i	$0.813563\pi$
$500$	0	0
$501$	−0.895410	−0.0400040
$502$	0	0
$503$	−13.2798	−0.592119	−0.296059	−	0.955170i	$-0.595673\pi$
$503$	−13.2798	−0.592119	−0.296059	+	0.955170i	$0.595673\pi$
$504$	0	0
$505$	−2.66282	−0.118494
$506$	0	0
$507$	9.81019	0.435686
$508$	0	0
$509$	10.9252	0.484251	0.242125	−	0.970245i	$-0.422155\pi$
$509$	10.9252	0.484251	0.242125	+	0.970245i	$0.422155\pi$
$510$	0	0
$511$	−15.3234	−0.677867
$512$	0	0
$513$	−7.04502	−0.311045
$514$	0	0
$515$	5.29362	0.233265
$516$	0	0
$517$	−14.5776	−0.641122
$518$	0	0
$519$	12.4882	0.548173
$520$	0	0
$521$	37.4432	1.64042	0.820209	−	0.572064i	$-0.193857\pi$
$521$	37.4432	1.64042	0.820209	+	0.572064i	$0.193857\pi$
$522$	0	0
$523$	16.7964	0.734456	0.367228	−	0.930131i	$-0.380307\pi$
$523$	16.7964	0.734456	0.367228	+	0.930131i	$0.380307\pi$
$524$	0	0
$525$	4.78600	0.208878
$526$	0	0
$527$	16.8954	0.735976
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	6.46260	0.280453
$532$	0	0
$533$	1.86640	0.0808428
$534$	0	0
$535$	−4.02979	−0.174223
$536$	0	0
$537$	−22.1455	−0.955647
$538$	0	0
$539$	−1.32340	−0.0570031
$540$	0	0
$541$	−14.9252	−0.641684	−0.320842	−	0.947133i	$-0.603966\pi$
$541$	−14.9252	−0.641684	−0.320842	+	0.947133i	$0.603966\pi$
$542$	0	0
$543$	17.9702	0.771176
$544$	0	0
$545$	5.30962	0.227439
$546$	0	0
$547$	7.08858	0.303086	0.151543	−	0.988451i	$-0.451576\pi$
$547$	7.08858	0.303086	0.151543	+	0.988451i	$0.451576\pi$
$548$	0	0
$549$	0.860806	0.0367383
$550$	0	0
$551$	−23.4134	−0.997446
$552$	0	0
$553$	4.39821	0.187031
$554$	0	0
$555$	−3.38780	−0.143804
$556$	0	0
$557$	27.4224	1.16192	0.580962	−	0.813931i	$-0.302676\pi$
$557$	27.4224	1.16192	0.580962	+	0.813931i	$0.302676\pi$
$558$	0	0
$559$	2.03460	0.0860546
$560$	0	0
$561$	−3.17380	−0.133998
$562$	0	0
$563$	−47.4272	−1.99882	−0.999409	−	0.0343631i	$-0.989060\pi$
$563$	−47.4272	−1.99882	−0.999409	+	0.0343631i	$0.989060\pi$
$564$	0	0
$565$	−5.25005	−0.220871
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−16.9044	−0.708668	−0.354334	−	0.935119i	$-0.615292\pi$
$569$	−16.9044	−0.708668	−0.354334	+	0.935119i	$0.615292\pi$
$570$	0	0
$571$	25.7514	1.07766	0.538831	−	0.842414i	$-0.318866\pi$
$571$	25.7514	1.07766	0.538831	+	0.842414i	$0.318866\pi$
$572$	0	0
$573$	7.31444	0.305565
$574$	0	0
$575$	−4.78600	−0.199590
$576$	0	0
$577$	21.9225	0.912646	0.456323	−	0.889814i	$-0.349166\pi$
$577$	21.9225	0.912646	0.456323	+	0.889814i	$0.349166\pi$
$578$	0	0
$579$	21.5720	0.896502
$580$	0	0
$581$	−12.7666	−0.529649
$582$	0	0
$583$	−0.782635	−0.0324134
$584$	0	0
$585$	−0.826202	−0.0341592
$586$	0	0
$587$	−34.0048	−1.40353	−0.701764	−	0.712409i	$-0.747604\pi$
$587$	−34.0048	−1.40353	−0.701764	+	0.712409i	$0.747604\pi$
$588$	0	0
$589$	−49.6323	−2.04506
$590$	0	0
$591$	14.4120	0.592830
$592$	0	0
$593$	10.3088	0.423334	0.211667	−	0.977342i	$-0.432111\pi$
$593$	10.3088	0.423334	0.211667	+	0.977342i	$0.432111\pi$
$594$	0	0
$595$	1.10941	0.0454813
$596$	0	0
$597$	6.33382	0.259226
$598$	0	0
$599$	−32.3941	−1.32359	−0.661793	−	0.749687i	$-0.730204\pi$
$599$	−32.3941	−1.32359	−0.661793	+	0.749687i	$0.730204\pi$
$600$	0	0
$601$	−33.8850	−1.38220	−0.691099	−	0.722760i	$-0.742873\pi$
$601$	−33.8850	−1.38220	−0.691099	+	0.722760i	$0.742873\pi$
$602$	0	0
$603$	−14.7022	−0.598721
$604$	0	0
$605$	4.27839	0.173941
$606$	0	0
$607$	−12.0948	−0.490915	−0.245457	−	0.969407i	$-0.578938\pi$
$607$	−12.0948	−0.490915	−0.245457	+	0.969407i	$0.578938\pi$
$608$	0	0
$609$	3.32340	0.134671
$610$	0	0
$611$	19.6732	0.795894
$612$	0	0
$613$	13.8594	0.559774	0.279887	−	0.960033i	$-0.409703\pi$
$613$	13.8594	0.559774	0.279887	+	0.960033i	$0.409703\pi$
$614$	0	0
$615$	0.483423	0.0194935
$616$	0	0
$617$	−22.3428	−0.899486	−0.449743	−	0.893158i	$-0.648484\pi$
$617$	−22.3428	−0.899486	−0.449743	+	0.893158i	$0.648484\pi$
$618$	0	0
$619$	22.1157	0.888904	0.444452	−	0.895803i	$-0.353398\pi$
$619$	22.1157	0.888904	0.444452	+	0.895803i	$0.353398\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	7.37883	0.295627
$624$	0	0
$625$	21.8358	0.873433
$626$	0	0
$627$	9.32340	0.372341
$628$	0	0
$629$	17.5630	0.700284
$630$	0	0
$631$	−25.0063	−0.995484	−0.497742	−	0.867325i	$-0.665837\pi$
$631$	−25.0063	−0.995484	−0.497742	+	0.867325i	$0.665837\pi$
$632$	0	0
$633$	−5.06584	−0.201349
$634$	0	0
$635$	−7.73057	−0.306778
$636$	0	0
$637$	1.78600	0.0707640
$638$	0	0
$639$	2.73202	0.108077
$640$	0	0
$641$	−11.3193	−0.447084	−0.223542	−	0.974694i	$-0.571762\pi$
$641$	−11.3193	−0.447084	−0.223542	+	0.974694i	$0.571762\pi$
$642$	0	0
$643$	−17.7652	−0.700590	−0.350295	−	0.936639i	$-0.613919\pi$
$643$	−17.7652	−0.700590	−0.350295	+	0.936639i	$0.613919\pi$
$644$	0	0
$645$	0.526989	0.0207502
$646$	0	0
$647$	−1.04020	−0.0408945	−0.0204472	−	0.999791i	$-0.506509\pi$
$647$	−1.04020	−0.0408945	−0.0204472	+	0.999791i	$0.506509\pi$
$648$	0	0
$649$	−8.55263	−0.335720
$650$	0	0
$651$	7.04502	0.276116
$652$	0	0
$653$	−37.1399	−1.45340	−0.726698	−	0.686957i	$-0.758946\pi$
$653$	−37.1399	−1.45340	−0.726698	+	0.686957i	$0.758946\pi$
$654$	0	0
$655$	−5.59698	−0.218692
$656$	0	0
$657$	−15.3234	−0.597823
$658$	0	0
$659$	21.5630	0.839977	0.419988	−	0.907529i	$-0.362034\pi$
$659$	21.5630	0.839977	0.419988	+	0.907529i	$0.362034\pi$
$660$	0	0
$661$	−19.5630	−0.760914	−0.380457	−	0.924799i	$-0.624233\pi$
$661$	−19.5630	−0.760914	−0.380457	+	0.924799i	$0.624233\pi$
$662$	0	0
$663$	4.28320	0.166346
$664$	0	0
$665$	−3.25901	−0.126379
$666$	0	0
$667$	−3.32340	−0.128683
$668$	0	0
$669$	23.3282	0.901921
$670$	0	0
$671$	−1.13919	−0.0439781
$672$	0	0
$673$	31.3926	1.21010	0.605048	−	0.796189i	$-0.293154\pi$
$673$	31.3926	1.21010	0.605048	+	0.796189i	$0.293154\pi$
$674$	0	0
$675$	4.78600	0.184213
$676$	0	0
$677$	27.1905	1.04501	0.522507	−	0.852635i	$-0.324997\pi$
$677$	27.1905	1.04501	0.522507	+	0.852635i	$0.324997\pi$
$678$	0	0
$679$	−8.11982	−0.311610
$680$	0	0
$681$	14.5318	0.556860
$682$	0	0
$683$	−0.338633	−0.0129574	−0.00647872	−	0.999979i	$-0.502062\pi$
$683$	−0.338633	−0.0129574	−0.00647872	+	0.999979i	$0.502062\pi$
$684$	0	0
$685$	−2.02497	−0.0773701
$686$	0	0
$687$	−4.83102	−0.184315
$688$	0	0
$689$	1.05621	0.0402383
$690$	0	0
$691$	−33.3788	−1.26979	−0.634895	−	0.772598i	$-0.718957\pi$
$691$	−33.3788	−1.26979	−0.634895	+	0.772598i	$0.718957\pi$
$692$	0	0
$693$	−1.32340	−0.0502720
$694$	0	0
$695$	−4.21400	−0.159846
$696$	0	0
$697$	−2.50617	−0.0949278
$698$	0	0
$699$	−15.2203	−0.575683
$700$	0	0
$701$	28.4030	1.07277	0.536384	−	0.843974i	$-0.319790\pi$
$701$	28.4030	1.07277	0.536384	+	0.843974i	$0.319790\pi$
$702$	0	0
$703$	−51.5935	−1.94589
$704$	0	0
$705$	5.09563	0.191912
$706$	0	0
$707$	5.75622	0.216485
$708$	0	0
$709$	−9.67515	−0.363358	−0.181679	−	0.983358i	$-0.558153\pi$
$709$	−9.67515	−0.363358	−0.181679	+	0.983358i	$0.558153\pi$
$710$	0	0
$711$	4.39821	0.164946
$712$	0	0
$713$	−7.04502	−0.263838
$714$	0	0
$715$	1.09340	0.0408908
$716$	0	0
$717$	22.6427	0.845606
$718$	0	0
$719$	−21.3055	−0.794560	−0.397280	−	0.917697i	$-0.630046\pi$
$719$	−21.3055	−0.794560	−0.397280	+	0.917697i	$0.630046\pi$
$720$	0	0
$721$	−11.4432	−0.426168
$722$	0	0
$723$	16.9162	0.629122
$724$	0	0
$725$	15.9058	0.590727
$726$	0	0
$727$	1.69182	0.0627463	0.0313731	−	0.999508i	$-0.490012\pi$
$727$	1.69182	0.0627463	0.0313731	+	0.999508i	$0.490012\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−2.73202	−0.101048
$732$	0	0
$733$	5.39194	0.199156	0.0995780	−	0.995030i	$-0.468251\pi$
$733$	5.39194	0.199156	0.0995780	+	0.995030i	$0.468251\pi$
$734$	0	0
$735$	0.462598	0.0170632
$736$	0	0
$737$	19.4570	0.716708
$738$	0	0
$739$	−37.2638	−1.37077	−0.685386	−	0.728180i	$-0.740367\pi$
$739$	−37.2638	−1.37077	−0.685386	+	0.728180i	$0.740367\pi$
$740$	0	0
$741$	−12.5824	−0.462227
$742$	0	0
$743$	35.0527	1.28596	0.642980	−	0.765883i	$-0.277698\pi$
$743$	35.0527	1.28596	0.642980	+	0.765883i	$0.277698\pi$
$744$	0	0
$745$	2.94602	0.107934
$746$	0	0
$747$	−12.7666	−0.467106
$748$	0	0
$749$	8.71120	0.318300
$750$	0	0
$751$	38.3851	1.40069	0.700346	−	0.713803i	$-0.253029\pi$
$751$	38.3851	1.40069	0.700346	+	0.713803i	$0.253029\pi$
$752$	0	0
$753$	20.0692	0.731363
$754$	0	0
$755$	−3.38365	−0.123144
$756$	0	0
$757$	−51.4439	−1.86976	−0.934880	−	0.354964i	$-0.884493\pi$
$757$	−51.4439	−1.86976	−0.934880	+	0.354964i	$0.884493\pi$
$758$	0	0
$759$	1.32340	0.0480365
$760$	0	0
$761$	33.0844	1.19931	0.599655	−	0.800259i	$-0.295305\pi$
$761$	33.0844	1.19931	0.599655	+	0.800259i	$0.295305\pi$
$762$	0	0
$763$	−11.4778	−0.415525
$764$	0	0
$765$	1.10941	0.0401107
$766$	0	0
$767$	11.5422	0.416765
$768$	0	0
$769$	−3.72161	−0.134205	−0.0671024	−	0.997746i	$-0.521375\pi$
$769$	−3.72161	−0.134205	−0.0671024	+	0.997746i	$0.521375\pi$
$770$	0	0
$771$	26.5485	0.956120
$772$	0	0
$773$	−13.6710	−0.491712	−0.245856	−	0.969306i	$-0.579069\pi$
$773$	−13.6710	−0.491712	−0.245856	+	0.969306i	$0.579069\pi$
$774$	0	0
$775$	33.7175	1.21117
$776$	0	0
$777$	7.32340	0.262726
$778$	0	0
$779$	7.36215	0.263777
$780$	0	0
$781$	−3.61557	−0.129375
$782$	0	0
$783$	3.32340	0.118769
$784$	0	0
$785$	−4.30885	−0.153789
$786$	0	0
$787$	−53.3587	−1.90203	−0.951016	−	0.309142i	$-0.899958\pi$
$787$	−53.3587	−1.90203	−0.951016	+	0.309142i	$0.899958\pi$
$788$	0	0
$789$	11.3442	0.403865
$790$	0	0
$791$	11.3490	0.403526
$792$	0	0
$793$	1.53740	0.0545947
$794$	0	0
$795$	0.273572	0.00970259
$796$	0	0
$797$	7.52362	0.266500	0.133250	−	0.991082i	$-0.457459\pi$
$797$	7.52362	0.266500	0.133250	+	0.991082i	$0.457459\pi$
$798$	0	0
$799$	−26.4168	−0.934559
$800$	0	0
$801$	7.37883	0.260718
$802$	0	0
$803$	20.2791	0.715632
$804$	0	0
$805$	−0.462598	−0.0163045
$806$	0	0
$807$	−3.63640	−0.128007
$808$	0	0
$809$	24.5437	0.862909	0.431455	−	0.902135i	$-0.358000\pi$
$809$	24.5437	0.862909	0.431455	+	0.902135i	$0.358000\pi$
$810$	0	0
$811$	−30.5991	−1.07448	−0.537240	−	0.843430i	$-0.680533\pi$
$811$	−30.5991	−1.07448	−0.537240	+	0.843430i	$0.680533\pi$
$812$	0	0
$813$	27.9106	0.978869
$814$	0	0
$815$	−1.19877	−0.0419910
$816$	0	0
$817$	8.02564	0.280782
$818$	0	0
$819$	1.78600	0.0624080
$820$	0	0
$821$	−40.1413	−1.40094	−0.700471	−	0.713681i	$-0.747027\pi$
$821$	−40.1413	−1.40094	−0.700471	+	0.713681i	$0.747027\pi$
$822$	0	0
$823$	49.6066	1.72918	0.864589	−	0.502480i	$-0.167579\pi$
$823$	49.6066	1.72918	0.864589	+	0.502480i	$0.167579\pi$
$824$	0	0
$825$	−6.33382	−0.220515
$826$	0	0
$827$	−0.0733538	−0.00255076	−0.00127538	−	0.999999i	$-0.500406\pi$
$827$	−0.0733538	−0.00255076	−0.00127538	+	0.999999i	$0.500406\pi$
$828$	0	0
$829$	17.0629	0.592620	0.296310	−	0.955092i	$-0.404244\pi$
$829$	17.0629	0.592620	0.296310	+	0.955092i	$0.404244\pi$
$830$	0	0
$831$	−10.9508	−0.379880
$832$	0	0
$833$	−2.39821	−0.0830930
$834$	0	0
$835$	−0.414215	−0.0143345
$836$	0	0
$837$	7.04502	0.243511
$838$	0	0
$839$	13.9452	0.481443	0.240722	−	0.970594i	$-0.422616\pi$
$839$	13.9452	0.481443	0.240722	+	0.970594i	$0.422616\pi$
$840$	0	0
$841$	−17.9550	−0.619137
$842$	0	0
$843$	1.01523	0.0349663
$844$	0	0
$845$	4.53818	0.156118
$846$	0	0
$847$	−9.24860	−0.317786
$848$	0	0
$849$	−7.49865	−0.257353
$850$	0	0
$851$	−7.32340	−0.251043
$852$	0	0
$853$	7.42240	0.254138	0.127069	−	0.991894i	$-0.459443\pi$
$853$	7.42240	0.254138	0.127069	+	0.991894i	$0.459443\pi$
$854$	0	0
$855$	−3.25901	−0.111456
$856$	0	0
$857$	−12.8352	−0.438441	−0.219220	−	0.975675i	$-0.570351\pi$
$857$	−12.8352	−0.438441	−0.219220	+	0.975675i	$0.570351\pi$
$858$	0	0
$859$	2.09003	0.0713110	0.0356555	−	0.999364i	$-0.488648\pi$
$859$	2.09003	0.0713110	0.0356555	+	0.999364i	$0.488648\pi$
$860$	0	0
$861$	−1.04502	−0.0356140
$862$	0	0
$863$	38.5180	1.31117	0.655584	−	0.755122i	$-0.272422\pi$
$863$	38.5180	1.31117	0.655584	+	0.755122i	$0.272422\pi$
$864$	0	0
$865$	5.77704	0.196425
$866$	0	0
$867$	11.2486	0.382023
$868$	0	0
$869$	−5.82061	−0.197451
$870$	0	0
$871$	−26.2582	−0.889726
$872$	0	0
$873$	−8.11982	−0.274814
$874$	0	0
$875$	4.52699	0.153040
$876$	0	0
$877$	−34.6773	−1.17097	−0.585484	−	0.810684i	$-0.699096\pi$
$877$	−34.6773	−1.17097	−0.585484	+	0.810684i	$0.699096\pi$
$878$	0	0
$879$	−18.7964	−0.633987
$880$	0	0
$881$	−3.20648	−0.108029	−0.0540146	−	0.998540i	$-0.517202\pi$
$881$	−3.20648	−0.108029	−0.0540146	+	0.998540i	$0.517202\pi$
$882$	0	0
$883$	−1.20840	−0.0406660	−0.0203330	−	0.999793i	$-0.506473\pi$
$883$	−1.20840	−0.0406660	−0.0203330	+	0.999793i	$0.506473\pi$
$884$	0	0
$885$	2.98959	0.100494
$886$	0	0
$887$	−34.2445	−1.14982	−0.574908	−	0.818218i	$-0.694962\pi$
$887$	−34.2445	−1.14982	−0.574908	+	0.818218i	$0.694962\pi$
$888$	0	0
$889$	16.7112	0.560476
$890$	0	0
$891$	−1.32340	−0.0443357
$892$	0	0
$893$	77.6025	2.59687
$894$	0	0
$895$	−10.2445	−0.342434
$896$	0	0
$897$	−1.78600	−0.0596329
$898$	0	0
$899$	23.4134	0.780882
$900$	0	0
$901$	−1.41825	−0.0472489
$902$	0	0
$903$	−1.13919	−0.0379100
$904$	0	0
$905$	8.31299	0.276333
$906$	0	0
$907$	−2.35174	−0.0780883	−0.0390442	−	0.999237i	$-0.512431\pi$
$907$	−2.35174	−0.0780883	−0.0390442	+	0.999237i	$0.512431\pi$
$908$	0	0
$909$	5.75622	0.190922
$910$	0	0
$911$	21.7216	0.719669	0.359835	−	0.933016i	$-0.382833\pi$
$911$	21.7216	0.719669	0.359835	+	0.933016i	$0.382833\pi$
$912$	0	0
$913$	16.8954	0.559156
$914$	0	0
$915$	0.398207	0.0131643
$916$	0	0
$917$	12.0990	0.399544
$918$	0	0
$919$	21.0242	0.693524	0.346762	−	0.937953i	$-0.387281\pi$
$919$	21.0242	0.693524	0.346762	+	0.937953i	$0.387281\pi$
$920$	0	0
$921$	−16.0811	−0.529889
$922$	0	0
$923$	4.87940	0.160607
$924$	0	0
$925$	35.0498	1.15243
$926$	0	0
$927$	−11.4432	−0.375845
$928$	0	0
$929$	2.44177	0.0801120	0.0400560	−	0.999197i	$-0.487246\pi$
$929$	2.44177	0.0801120	0.0400560	+	0.999197i	$0.487246\pi$
$930$	0	0
$931$	7.04502	0.230891
$932$	0	0
$933$	22.5437	0.738047
$934$	0	0
$935$	−1.46819	−0.0480151
$936$	0	0
$937$	3.69182	0.120607	0.0603033	−	0.998180i	$-0.480793\pi$
$937$	3.69182	0.120607	0.0603033	+	0.998180i	$0.480793\pi$
$938$	0	0
$939$	8.90437	0.290583
$940$	0	0
$941$	16.9460	0.552425	0.276212	−	0.961097i	$-0.410921\pi$
$941$	16.9460	0.552425	0.276212	+	0.961097i	$0.410921\pi$
$942$	0	0
$943$	1.04502	0.0340304
$944$	0	0
$945$	0.462598	0.0150483
$946$	0	0
$947$	−51.5637	−1.67560	−0.837798	−	0.545981i	$-0.816157\pi$
$947$	−51.5637	−1.67560	−0.837798	+	0.545981i	$0.816157\pi$
$948$	0	0
$949$	−27.3676	−0.888391
$950$	0	0
$951$	−7.66618	−0.248593
$952$	0	0
$953$	−21.1101	−0.683822	−0.341911	−	0.939732i	$-0.611074\pi$
$953$	−21.1101	−0.683822	−0.341911	+	0.939732i	$0.611074\pi$
$954$	0	0
$955$	3.38365	0.109492
$956$	0	0
$957$	−4.39821	−0.142174
$958$	0	0
$959$	4.37738	0.141353
$960$	0	0
$961$	18.6323	0.601040
$962$	0	0
$963$	8.71120	0.280714
$964$	0	0
$965$	9.97918	0.321241
$966$	0	0
$967$	12.1205	0.389769	0.194884	−	0.980826i	$-0.437567\pi$
$967$	12.1205	0.389769	0.194884	+	0.980826i	$0.437567\pi$
$968$	0	0
$969$	16.8954	0.542759
$970$	0	0
$971$	31.1697	1.00028	0.500141	−	0.865944i	$-0.333282\pi$
$971$	31.1697	1.00028	0.500141	+	0.865944i	$0.333282\pi$
$972$	0	0
$973$	9.10941	0.292034
$974$	0	0
$975$	8.54781	0.273749
$976$	0	0
$977$	13.7777	0.440788	0.220394	−	0.975411i	$-0.429266\pi$
$977$	13.7777	0.440788	0.220394	+	0.975411i	$0.429266\pi$
$978$	0	0
$979$	−9.76518	−0.312096
$980$	0	0
$981$	−11.4778	−0.366459
$982$	0	0
$983$	−50.8567	−1.62208	−0.811038	−	0.584994i	$-0.801097\pi$
$983$	−50.8567	−1.62208	−0.811038	+	0.584994i	$0.801097\pi$
$984$	0	0
$985$	6.66696	0.212427
$986$	0	0
$987$	−11.0152	−0.350619
$988$	0	0
$989$	1.13919	0.0362243
$990$	0	0
$991$	36.7208	1.16648	0.583238	−	0.812301i	$-0.301786\pi$
$991$	36.7208	1.16648	0.583238	+	0.812301i	$0.301786\pi$
$992$	0	0
$993$	−25.2340	−0.800778
$994$	0	0
$995$	2.93001	0.0928877
$996$	0	0
$997$	25.1655	0.797000	0.398500	−	0.917168i	$-0.369531\pi$
$997$	25.1655	0.797000	0.398500	+	0.917168i	$0.369531\pi$
$998$	0	0
$999$	7.32340	0.231702

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.bs.1.2		3
4.3	odd	2		3864.2.a.q.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3864.2.a.q.1.2	✓	3	4.3	odd	2
7728.2.a.bs.1.2		3	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:	\( N \)	\(=\)	\( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7728.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.462598 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -0.462598 q^{5} +1.00000 q^{7} +1.00000 q^{9} -1.32340 q^{11} +1.78600 q^{13} +0.462598 q^{15} -2.39821 q^{17} +7.04502 q^{19} -1.00000 q^{21} +1.00000 q^{23} -4.78600 q^{25} -1.00000 q^{27} -3.32340 q^{29} -7.04502 q^{31} +1.32340 q^{33} -0.462598 q^{35} -7.32340 q^{37} -1.78600 q^{39} +1.04502 q^{41} +1.13919 q^{43} -0.462598 q^{45} +11.0152 q^{47} +1.00000 q^{49} +2.39821 q^{51} +0.591380 q^{53} +0.612205 q^{55} -7.04502 q^{57} +6.46260 q^{59} +0.860806 q^{61} +1.00000 q^{63} -0.826202 q^{65} -14.7022 q^{67} -1.00000 q^{69} +2.73202 q^{71} -15.3234 q^{73} +4.78600 q^{75} -1.32340 q^{77} +4.39821 q^{79} +1.00000 q^{81} -12.7666 q^{83} +1.10941 q^{85} +3.32340 q^{87} +7.37883 q^{89} +1.78600 q^{91} +7.04502 q^{93} -3.25901 q^{95} -8.11982 q^{97} -1.32340 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + q^5 + 3 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} + q^{5} + 3 q^{7} + 3 q^{9} + 4 q^{11} - 5 q^{13} - q^{15} - 4 q^{17} + 2 q^{19} - 3 q^{21} + 3 q^{23} - 4 q^{25} - 3 q^{27} - 2 q^{29} - 2 q^{31} - 4 q^{33} + q^{35} - 14 q^{37} + 5 q^{39} - 16 q^{41} + 9 q^{43} + q^{45} - 10 q^{47} + 3 q^{49} + 4 q^{51} + q^{53} + 9 q^{55} - 2 q^{57} + 17 q^{59} - 3 q^{61} + 3 q^{63} - 20 q^{65} - 13 q^{67} - 3 q^{69} + q^{71} - 38 q^{73} + 4 q^{75} + 4 q^{77} + 10 q^{79} + 3 q^{81} - 8 q^{83} - 15 q^{85} + 2 q^{87} - q^{89} - 5 q^{91} + 2 q^{93} - q^{95} - 10 q^{97} + 4 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + q^5 + 3 * q^7 + 3 * q^9 + 4 * q^11 - 5 * q^13 - q^15 - 4 * q^17 + 2 * q^19 - 3 * q^21 + 3 * q^23 - 4 * q^25 - 3 * q^27 - 2 * q^29 - 2 * q^31 - 4 * q^33 + q^35 - 14 * q^37 + 5 * q^39 - 16 * q^41 + 9 * q^43 + q^45 - 10 * q^47 + 3 * q^49 + 4 * q^51 + q^53 + 9 * q^55 - 2 * q^57 + 17 * q^59 - 3 * q^61 + 3 * q^63 - 20 * q^65 - 13 * q^67 - 3 * q^69 + q^71 - 38 * q^73 + 4 * q^75 + 4 * q^77 + 10 * q^79 + 3 * q^81 - 8 * q^83 - 15 * q^85 + 2 * q^87 - q^89 - 5 * q^91 + 2 * q^93 - q^95 - 10 * q^97 + 4 * q^99

Introduction

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