LMFDB - Embedded newform 7440.2.a.bd.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7440,2,Mod(1,7440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7440, 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("7440.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7440.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:	$59.4086991038$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{65}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 16 $ x^2 - x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 930)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.53113$ of defining polynomial
Character	$\chi$	$=$	7440.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} +4.53113 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{5} +4.53113 q^{7} +1.00000 q^{9} -6.53113 q^{11} +6.00000 q^{13} +1.00000 q^{15} -4.00000 q^{17} -4.53113 q^{19} -4.53113 q^{21} -6.53113 q^{23} +1.00000 q^{25} -1.00000 q^{27} -1.00000 q^{31} +6.53113 q^{33} -4.53113 q^{35} -7.06226 q^{37} -6.00000 q^{39} +7.06226 q^{41} +8.53113 q^{43} -1.00000 q^{45} +5.06226 q^{47} +13.5311 q^{49} +4.00000 q^{51} -2.53113 q^{53} +6.53113 q^{55} +4.53113 q^{57} +9.06226 q^{59} +5.06226 q^{61} +4.53113 q^{63} -6.00000 q^{65} -5.06226 q^{67} +6.53113 q^{69} +12.5311 q^{71} -8.53113 q^{73} -1.00000 q^{75} -29.5934 q^{77} +8.53113 q^{79} +1.00000 q^{81} +8.00000 q^{83} +4.00000 q^{85} -6.53113 q^{89} +27.1868 q^{91} +1.00000 q^{93} +4.53113 q^{95} +16.1245 q^{97} -6.53113 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^5 + q^7 + 2 * q^9	$ 2 q - 2 q^{3} - 2 q^{5} + q^{7} + 2 q^{9} - 5 q^{11} + 12 q^{13} + 2 q^{15} - 8 q^{17} - q^{19} - q^{21} - 5 q^{23} + 2 q^{25} - 2 q^{27} - 2 q^{31} + 5 q^{33} - q^{35} + 2 q^{37} - 12 q^{39} - 2 q^{41} + 9 q^{43} - 2 q^{45} - 6 q^{47} + 19 q^{49} + 8 q^{51} + 3 q^{53} + 5 q^{55} + q^{57} + 2 q^{59} - 6 q^{61} + q^{63} - 12 q^{65} + 6 q^{67} + 5 q^{69} + 17 q^{71} - 9 q^{73} - 2 q^{75} - 35 q^{77} + 9 q^{79} + 2 q^{81} + 16 q^{83} + 8 q^{85} - 5 q^{89} + 6 q^{91} + 2 q^{93} + q^{95} - 5 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 + q^7 + 2 * q^9 - 5 * q^11 + 12 * q^13 + 2 * q^15 - 8 * q^17 - q^19 - q^21 - 5 * q^23 + 2 * q^25 - 2 * q^27 - 2 * q^31 + 5 * q^33 - q^35 + 2 * q^37 - 12 * q^39 - 2 * q^41 + 9 * q^43 - 2 * q^45 - 6 * q^47 + 19 * q^49 + 8 * q^51 + 3 * q^53 + 5 * q^55 + q^57 + 2 * q^59 - 6 * q^61 + q^63 - 12 * q^65 + 6 * q^67 + 5 * q^69 + 17 * q^71 - 9 * q^73 - 2 * q^75 - 35 * q^77 + 9 * q^79 + 2 * q^81 + 16 * q^83 + 8 * q^85 - 5 * q^89 + 6 * q^91 + 2 * q^93 + q^95 - 5 * 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
$7$	4.53113	1.71261	0.856303	−	0.516474i	$-0.172756\pi$
$7$	4.53113	1.71261	0.856303	+	0.516474i	$0.172756\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−6.53113	−1.96921	−0.984605	−	0.174796i	$-0.944074\pi$
$11$	−6.53113	−1.96921	−0.984605	+	0.174796i	$0.944074\pi$
$12$	0	0
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	−4.53113	−1.03951	−0.519756	−	0.854315i	$-0.673977\pi$
$19$	−4.53113	−1.03951	−0.519756	+	0.854315i	$0.673977\pi$
$20$	0	0
$21$	−4.53113	−0.988773
$22$	0	0
$23$	−6.53113	−1.36183	−0.680917	−	0.732360i	$-0.738419\pi$
$23$	−6.53113	−1.36183	−0.680917	+	0.732360i	$0.738419\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	6.53113	1.13692
$34$	0	0
$35$	−4.53113	−0.765901
$36$	0	0
$37$	−7.06226	−1.16103	−0.580514	−	0.814250i	$-0.697148\pi$
$37$	−7.06226	−1.16103	−0.580514	+	0.814250i	$0.697148\pi$
$38$	0	0
$39$	−6.00000	−0.960769
$40$	0	0
$41$	7.06226	1.10294	0.551470	−	0.834195i	$-0.314067\pi$
$41$	7.06226	1.10294	0.551470	+	0.834195i	$0.314067\pi$
$42$	0	0
$43$	8.53113	1.30098	0.650492	−	0.759513i	$-0.274563\pi$
$43$	8.53113	1.30098	0.650492	+	0.759513i	$0.274563\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	5.06226	0.738406	0.369203	−	0.929349i	$-0.379631\pi$
$47$	5.06226	0.738406	0.369203	+	0.929349i	$0.379631\pi$
$48$	0	0
$49$	13.5311	1.93302
$50$	0	0
$51$	4.00000	0.560112
$52$	0	0
$53$	−2.53113	−0.347677	−0.173839	−	0.984774i	$-0.555617\pi$
$53$	−2.53113	−0.347677	−0.173839	+	0.984774i	$0.555617\pi$
$54$	0	0
$55$	6.53113	0.880657
$56$	0	0
$57$	4.53113	0.600163
$58$	0	0
$59$	9.06226	1.17981	0.589903	−	0.807474i	$-0.299166\pi$
$59$	9.06226	1.17981	0.589903	+	0.807474i	$0.299166\pi$
$60$	0	0
$61$	5.06226	0.648156	0.324078	−	0.946030i	$-0.394946\pi$
$61$	5.06226	0.648156	0.324078	+	0.946030i	$0.394946\pi$
$62$	0	0
$63$	4.53113	0.570869
$64$	0	0
$65$	−6.00000	−0.744208
$66$	0	0
$67$	−5.06226	−0.618453	−0.309227	−	0.950988i	$-0.600070\pi$
$67$	−5.06226	−0.618453	−0.309227	+	0.950988i	$0.600070\pi$
$68$	0	0
$69$	6.53113	0.786256
$70$	0	0
$71$	12.5311	1.48717	0.743586	−	0.668641i	$-0.233124\pi$
$71$	12.5311	1.48717	0.743586	+	0.668641i	$0.233124\pi$
$72$	0	0
$73$	−8.53113	−0.998493	−0.499247	−	0.866460i	$-0.666390\pi$
$73$	−8.53113	−0.998493	−0.499247	+	0.866460i	$0.666390\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−29.5934	−3.37248
$78$	0	0
$79$	8.53113	0.959827	0.479913	−	0.877316i	$-0.340668\pi$
$79$	8.53113	0.959827	0.479913	+	0.877316i	$0.340668\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−6.53113	−0.692298	−0.346149	−	0.938180i	$-0.612511\pi$
$89$	−6.53113	−0.692298	−0.346149	+	0.938180i	$0.612511\pi$
$90$	0	0
$91$	27.1868	2.84995
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	4.53113	0.464884
$96$	0	0
$97$	16.1245	1.63720	0.818598	−	0.574367i	$-0.194752\pi$
$97$	16.1245	1.63720	0.818598	+	0.574367i	$0.194752\pi$
$98$	0	0
$99$	−6.53113	−0.656403
$100$	0	0
$101$	−9.46887	−0.942188	−0.471094	−	0.882083i	$-0.656141\pi$
$101$	−9.46887	−0.942188	−0.471094	+	0.882083i	$0.656141\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	4.53113	0.442193
$106$	0	0
$107$	−9.59339	−0.927428	−0.463714	−	0.885985i	$-0.653483\pi$
$107$	−9.59339	−0.927428	−0.463714	+	0.885985i	$0.653483\pi$
$108$	0	0
$109$	15.0623	1.44270	0.721351	−	0.692569i	$-0.243521\pi$
$109$	15.0623	1.44270	0.721351	+	0.692569i	$0.243521\pi$
$110$	0	0
$111$	7.06226	0.670320
$112$	0	0
$113$	−7.59339	−0.714326	−0.357163	−	0.934042i	$-0.616256\pi$
$113$	−7.59339	−0.714326	−0.357163	+	0.934042i	$0.616256\pi$
$114$	0	0
$115$	6.53113	0.609031
$116$	0	0
$117$	6.00000	0.554700
$118$	0	0
$119$	−18.1245	−1.66147
$120$	0	0
$121$	31.6556	2.87779
$122$	0	0
$123$	−7.06226	−0.636782
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−7.06226	−0.626674	−0.313337	−	0.949642i	$-0.601447\pi$
$127$	−7.06226	−0.626674	−0.313337	+	0.949642i	$0.601447\pi$
$128$	0	0
$129$	−8.53113	−0.751124
$130$	0	0
$131$	9.06226	0.791773	0.395887	−	0.918299i	$-0.370437\pi$
$131$	9.06226	0.791773	0.395887	+	0.918299i	$0.370437\pi$
$132$	0	0
$133$	−20.5311	−1.78027
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	17.0623	1.45773	0.728864	−	0.684659i	$-0.240049\pi$
$137$	17.0623	1.45773	0.728864	+	0.684659i	$0.240049\pi$
$138$	0	0
$139$	−6.00000	−0.508913	−0.254457	−	0.967084i	$-0.581897\pi$
$139$	−6.00000	−0.508913	−0.254457	+	0.967084i	$0.581897\pi$
$140$	0	0
$141$	−5.06226	−0.426319
$142$	0	0
$143$	−39.1868	−3.27696
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−13.5311	−1.11603
$148$	0	0
$149$	−18.5311	−1.51813	−0.759065	−	0.651015i	$-0.774343\pi$
$149$	−18.5311	−1.51813	−0.759065	+	0.651015i	$0.774343\pi$
$150$	0	0
$151$	−14.1245	−1.14944	−0.574718	−	0.818351i	$-0.694888\pi$
$151$	−14.1245	−1.14944	−0.574718	+	0.818351i	$0.694888\pi$
$152$	0	0
$153$	−4.00000	−0.323381
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	−22.5311	−1.79818	−0.899090	−	0.437764i	$-0.855771\pi$
$157$	−22.5311	−1.79818	−0.899090	+	0.437764i	$0.855771\pi$
$158$	0	0
$159$	2.53113	0.200732
$160$	0	0
$161$	−29.5934	−2.33229
$162$	0	0
$163$	−5.06226	−0.396507	−0.198253	−	0.980151i	$-0.563527\pi$
$163$	−5.06226	−0.396507	−0.198253	+	0.980151i	$0.563527\pi$
$164$	0	0
$165$	−6.53113	−0.508448
$166$	0	0
$167$	23.5934	1.82571	0.912856	−	0.408283i	$-0.133872\pi$
$167$	23.5934	1.82571	0.912856	+	0.408283i	$0.133872\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	0	0
$171$	−4.53113	−0.346504
$172$	0	0
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	0	0
$175$	4.53113	0.342521
$176$	0	0
$177$	−9.06226	−0.681161
$178$	0	0
$179$	3.06226	0.228884	0.114442	−	0.993430i	$-0.463492\pi$
$179$	3.06226	0.228884	0.114442	+	0.993430i	$0.463492\pi$
$180$	0	0
$181$	2.40661	0.178882	0.0894411	−	0.995992i	$-0.471492\pi$
$181$	2.40661	0.178882	0.0894411	+	0.995992i	$0.471492\pi$
$182$	0	0
$183$	−5.06226	−0.374213
$184$	0	0
$185$	7.06226	0.519228
$186$	0	0
$187$	26.1245	1.91041
$188$	0	0
$189$	−4.53113	−0.329591
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	0	0
$195$	6.00000	0.429669
$196$	0	0
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	8.53113	0.604756	0.302378	−	0.953188i	$-0.402220\pi$
$199$	8.53113	0.604756	0.302378	+	0.953188i	$0.402220\pi$
$200$	0	0
$201$	5.06226	0.357064
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−7.06226	−0.493249
$206$	0	0
$207$	−6.53113	−0.453945
$208$	0	0
$209$	29.5934	2.04702
$210$	0	0
$211$	1.59339	0.109693	0.0548466	−	0.998495i	$-0.482533\pi$
$211$	1.59339	0.109693	0.0548466	+	0.998495i	$0.482533\pi$
$212$	0	0
$213$	−12.5311	−0.858619
$214$	0	0
$215$	−8.53113	−0.581818
$216$	0	0
$217$	−4.53113	−0.307593
$218$	0	0
$219$	8.53113	0.576480
$220$	0	0
$221$	−24.0000	−1.61441
$222$	0	0
$223$	−2.00000	−0.133930	−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000	−0.133930	−0.0669650	+	0.997755i	$0.521332\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	24.5311	1.62819	0.814094	−	0.580733i	$-0.197234\pi$
$227$	24.5311	1.62819	0.814094	+	0.580733i	$0.197234\pi$
$228$	0	0
$229$	−5.59339	−0.369621	−0.184811	−	0.982774i	$-0.559167\pi$
$229$	−5.59339	−0.369621	−0.184811	+	0.982774i	$0.559167\pi$
$230$	0	0
$231$	29.5934	1.94710
$232$	0	0
$233$	1.46887	0.0962289	0.0481145	−	0.998842i	$-0.484679\pi$
$233$	1.46887	0.0962289	0.0481145	+	0.998842i	$0.484679\pi$
$234$	0	0
$235$	−5.06226	−0.330225
$236$	0	0
$237$	−8.53113	−0.554156
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−13.5311	−0.864472
$246$	0	0
$247$	−27.1868	−1.72985
$248$	0	0
$249$	−8.00000	−0.506979
$250$	0	0
$251$	−3.06226	−0.193288	−0.0966440	−	0.995319i	$-0.530811\pi$
$251$	−3.06226	−0.193288	−0.0966440	+	0.995319i	$0.530811\pi$
$252$	0	0
$253$	42.6556	2.68174
$254$	0	0
$255$	−4.00000	−0.250490
$256$	0	0
$257$	12.6556	0.789437	0.394719	−	0.918802i	$-0.370842\pi$
$257$	12.6556	0.789437	0.394719	+	0.918802i	$0.370842\pi$
$258$	0	0
$259$	−32.0000	−1.98838
$260$	0	0
$261$	0	0
$262$	0	0
$263$	23.0623	1.42208	0.711040	−	0.703152i	$-0.248225\pi$
$263$	23.0623	1.42208	0.711040	+	0.703152i	$0.248225\pi$
$264$	0	0
$265$	2.53113	0.155486
$266$	0	0
$267$	6.53113	0.399699
$268$	0	0
$269$	−19.1868	−1.16984	−0.584919	−	0.811092i	$-0.698874\pi$
$269$	−19.1868	−1.16984	−0.584919	+	0.811092i	$0.698874\pi$
$270$	0	0
$271$	11.4689	0.696684	0.348342	−	0.937367i	$-0.386745\pi$
$271$	11.4689	0.696684	0.348342	+	0.937367i	$0.386745\pi$
$272$	0	0
$273$	−27.1868	−1.64542
$274$	0	0
$275$	−6.53113	−0.393842
$276$	0	0
$277$	−15.0623	−0.905003	−0.452502	−	0.891764i	$-0.649468\pi$
$277$	−15.0623	−0.905003	−0.452502	+	0.891764i	$0.649468\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	15.0623	0.898539	0.449269	−	0.893396i	$-0.351684\pi$
$281$	15.0623	0.898539	0.449269	+	0.893396i	$0.351684\pi$
$282$	0	0
$283$	12.0000	0.713326	0.356663	−	0.934233i	$-0.383914\pi$
$283$	12.0000	0.713326	0.356663	+	0.934233i	$0.383914\pi$
$284$	0	0
$285$	−4.53113	−0.268401
$286$	0	0
$287$	32.0000	1.88890
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−16.1245	−0.945236
$292$	0	0
$293$	−14.0000	−0.817889	−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	0	0
$295$	−9.06226	−0.527625
$296$	0	0
$297$	6.53113	0.378975
$298$	0	0
$299$	−39.1868	−2.26623
$300$	0	0
$301$	38.6556	2.22807
$302$	0	0
$303$	9.46887	0.543972
$304$	0	0
$305$	−5.06226	−0.289864
$306$	0	0
$307$	30.1245	1.71930	0.859648	−	0.510886i	$-0.170683\pi$
$307$	30.1245	1.71930	0.859648	+	0.510886i	$0.170683\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	0	0
$313$	10.9377	0.618238	0.309119	−	0.951023i	$-0.399966\pi$
$313$	10.9377	0.618238	0.309119	+	0.951023i	$0.399966\pi$
$314$	0	0
$315$	−4.53113	−0.255300
$316$	0	0
$317$	12.9377	0.726656	0.363328	−	0.931661i	$-0.381640\pi$
$317$	12.9377	0.726656	0.363328	+	0.931661i	$0.381640\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	9.59339	0.535451
$322$	0	0
$323$	18.1245	1.00848
$324$	0	0
$325$	6.00000	0.332820
$326$	0	0
$327$	−15.0623	−0.832945
$328$	0	0
$329$	22.9377	1.26460
$330$	0	0
$331$	−12.9377	−0.711123	−0.355561	−	0.934653i	$-0.615710\pi$
$331$	−12.9377	−0.711123	−0.355561	+	0.934653i	$0.615710\pi$
$332$	0	0
$333$	−7.06226	−0.387009
$334$	0	0
$335$	5.06226	0.276581
$336$	0	0
$337$	−19.1868	−1.04517	−0.522585	−	0.852587i	$-0.675032\pi$
$337$	−19.1868	−1.04517	−0.522585	+	0.852587i	$0.675032\pi$
$338$	0	0
$339$	7.59339	0.412416
$340$	0	0
$341$	6.53113	0.353680
$342$	0	0
$343$	29.5934	1.59789
$344$	0	0
$345$	−6.53113	−0.351624
$346$	0	0
$347$	−10.1245	−0.543512	−0.271756	−	0.962366i	$-0.587604\pi$
$347$	−10.1245	−0.543512	−0.271756	+	0.962366i	$0.587604\pi$
$348$	0	0
$349$	8.93774	0.478426	0.239213	−	0.970967i	$-0.423111\pi$
$349$	8.93774	0.478426	0.239213	+	0.970967i	$0.423111\pi$
$350$	0	0
$351$	−6.00000	−0.320256
$352$	0	0
$353$	−6.93774	−0.369259	−0.184629	−	0.982808i	$-0.559108\pi$
$353$	−6.93774	−0.369259	−0.184629	+	0.982808i	$0.559108\pi$
$354$	0	0
$355$	−12.5311	−0.665083
$356$	0	0
$357$	18.1245	0.959251
$358$	0	0
$359$	−3.46887	−0.183080	−0.0915400	−	0.995801i	$-0.529179\pi$
$359$	−3.46887	−0.183080	−0.0915400	+	0.995801i	$0.529179\pi$
$360$	0	0
$361$	1.53113	0.0805857
$362$	0	0
$363$	−31.6556	−1.66149
$364$	0	0
$365$	8.53113	0.446540
$366$	0	0
$367$	10.0000	0.521996	0.260998	−	0.965339i	$-0.415948\pi$
$367$	10.0000	0.521996	0.260998	+	0.965339i	$0.415948\pi$
$368$	0	0
$369$	7.06226	0.367646
$370$	0	0
$371$	−11.4689	−0.595434
$372$	0	0
$373$	−18.5311	−0.959505	−0.479753	−	0.877404i	$-0.659274\pi$
$373$	−18.5311	−0.959505	−0.479753	+	0.877404i	$0.659274\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−21.5934	−1.10918	−0.554589	−	0.832124i	$-0.687124\pi$
$379$	−21.5934	−1.10918	−0.554589	+	0.832124i	$0.687124\pi$
$380$	0	0
$381$	7.06226	0.361810
$382$	0	0
$383$	3.06226	0.156474	0.0782370	−	0.996935i	$-0.475071\pi$
$383$	3.06226	0.156474	0.0782370	+	0.996935i	$0.475071\pi$
$384$	0	0
$385$	29.5934	1.50822
$386$	0	0
$387$	8.53113	0.433662
$388$	0	0
$389$	10.9377	0.554566	0.277283	−	0.960788i	$-0.410566\pi$
$389$	10.9377	0.554566	0.277283	+	0.960788i	$0.410566\pi$
$390$	0	0
$391$	26.1245	1.32117
$392$	0	0
$393$	−9.06226	−0.457130
$394$	0	0
$395$	−8.53113	−0.429248
$396$	0	0
$397$	37.7179	1.89301	0.946504	−	0.322693i	$-0.104588\pi$
$397$	37.7179	1.89301	0.946504	+	0.322693i	$0.104588\pi$
$398$	0	0
$399$	20.5311	1.02784
$400$	0	0
$401$	21.7179	1.08454	0.542270	−	0.840204i	$-0.317565\pi$
$401$	21.7179	1.08454	0.542270	+	0.840204i	$0.317565\pi$
$402$	0	0
$403$	−6.00000	−0.298881
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	46.1245	2.28631
$408$	0	0
$409$	7.06226	0.349206	0.174603	−	0.984639i	$-0.444136\pi$
$409$	7.06226	0.349206	0.174603	+	0.984639i	$0.444136\pi$
$410$	0	0
$411$	−17.0623	−0.841619
$412$	0	0
$413$	41.0623	2.02054
$414$	0	0
$415$	−8.00000	−0.392705
$416$	0	0
$417$	6.00000	0.293821
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−26.2490	−1.27930	−0.639650	−	0.768667i	$-0.720921\pi$
$421$	−26.2490	−1.27930	−0.639650	+	0.768667i	$0.720921\pi$
$422$	0	0
$423$	5.06226	0.246135
$424$	0	0
$425$	−4.00000	−0.194029
$426$	0	0
$427$	22.9377	1.11004
$428$	0	0
$429$	39.1868	1.89196
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	0	0
$433$	−25.5934	−1.22994	−0.614970	−	0.788551i	$-0.710832\pi$
$433$	−25.5934	−1.22994	−0.614970	+	0.788551i	$0.710832\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	29.5934	1.41564
$438$	0	0
$439$	25.0623	1.19616	0.598078	−	0.801438i	$-0.295931\pi$
$439$	25.0623	1.19616	0.598078	+	0.801438i	$0.295931\pi$
$440$	0	0
$441$	13.5311	0.644339
$442$	0	0
$443$	−14.4066	−0.684479	−0.342239	−	0.939613i	$-0.611185\pi$
$443$	−14.4066	−0.684479	−0.342239	+	0.939613i	$0.611185\pi$
$444$	0	0
$445$	6.53113	0.309605
$446$	0	0
$447$	18.5311	0.876492
$448$	0	0
$449$	−4.12452	−0.194648	−0.0973240	−	0.995253i	$-0.531028\pi$
$449$	−4.12452	−0.194648	−0.0973240	+	0.995253i	$0.531028\pi$
$450$	0	0
$451$	−46.1245	−2.17192
$452$	0	0
$453$	14.1245	0.663628
$454$	0	0
$455$	−27.1868	−1.27454
$456$	0	0
$457$	14.9377	0.698758	0.349379	−	0.936981i	$-0.386393\pi$
$457$	14.9377	0.698758	0.349379	+	0.936981i	$0.386393\pi$
$458$	0	0
$459$	4.00000	0.186704
$460$	0	0
$461$	24.0000	1.11779	0.558896	−	0.829238i	$-0.311225\pi$
$461$	24.0000	1.11779	0.558896	+	0.829238i	$0.311225\pi$
$462$	0	0
$463$	−13.1868	−0.612841	−0.306421	−	0.951896i	$-0.599131\pi$
$463$	−13.1868	−0.612841	−0.306421	+	0.951896i	$0.599131\pi$
$464$	0	0
$465$	−1.00000	−0.0463739
$466$	0	0
$467$	22.1245	1.02380	0.511900	−	0.859045i	$-0.328942\pi$
$467$	22.1245	1.02380	0.511900	+	0.859045i	$0.328942\pi$
$468$	0	0
$469$	−22.9377	−1.05917
$470$	0	0
$471$	22.5311	1.03818
$472$	0	0
$473$	−55.7179	−2.56191
$474$	0	0
$475$	−4.53113	−0.207902
$476$	0	0
$477$	−2.53113	−0.115892
$478$	0	0
$479$	−1.34436	−0.0614252	−0.0307126	−	0.999528i	$-0.509778\pi$
$479$	−1.34436	−0.0614252	−0.0307126	+	0.999528i	$0.509778\pi$
$480$	0	0
$481$	−42.3735	−1.93207
$482$	0	0
$483$	29.5934	1.34655
$484$	0	0
$485$	−16.1245	−0.732177
$486$	0	0
$487$	−23.0623	−1.04505	−0.522525	−	0.852624i	$-0.675010\pi$
$487$	−23.0623	−1.04505	−0.522525	+	0.852624i	$0.675010\pi$
$488$	0	0
$489$	5.06226	0.228923
$490$	0	0
$491$	7.59339	0.342685	0.171342	−	0.985212i	$-0.445190\pi$
$491$	7.59339	0.342685	0.171342	+	0.985212i	$0.445190\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	6.53113	0.293552
$496$	0	0
$497$	56.7802	2.54694
$498$	0	0
$499$	7.06226	0.316150	0.158075	−	0.987427i	$-0.449471\pi$
$499$	7.06226	0.316150	0.158075	+	0.987427i	$0.449471\pi$
$500$	0	0
$501$	−23.5934	−1.05407
$502$	0	0
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	0	0
$505$	9.46887	0.421359
$506$	0	0
$507$	−23.0000	−1.02147
$508$	0	0
$509$	38.1245	1.68984	0.844920	−	0.534893i	$-0.179648\pi$
$509$	38.1245	1.68984	0.844920	+	0.534893i	$0.179648\pi$
$510$	0	0
$511$	−38.6556	−1.71003
$512$	0	0
$513$	4.53113	0.200054
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−33.0623	−1.45408
$518$	0	0
$519$	−14.0000	−0.614532
$520$	0	0
$521$	−12.1245	−0.531185	−0.265592	−	0.964085i	$-0.585568\pi$
$521$	−12.1245	−0.531185	−0.265592	+	0.964085i	$0.585568\pi$
$522$	0	0
$523$	−9.59339	−0.419490	−0.209745	−	0.977756i	$-0.567263\pi$
$523$	−9.59339	−0.419490	−0.209745	+	0.977756i	$0.567263\pi$
$524$	0	0
$525$	−4.53113	−0.197755
$526$	0	0
$527$	4.00000	0.174243
$528$	0	0
$529$	19.6556	0.854593
$530$	0	0
$531$	9.06226	0.393268
$532$	0	0
$533$	42.3735	1.83540
$534$	0	0
$535$	9.59339	0.414758
$536$	0	0
$537$	−3.06226	−0.132146
$538$	0	0
$539$	−88.3735	−3.80652
$540$	0	0
$541$	−4.12452	−0.177327	−0.0886634	−	0.996062i	$-0.528260\pi$
$541$	−4.12452	−0.177327	−0.0886634	+	0.996062i	$0.528260\pi$
$542$	0	0
$543$	−2.40661	−0.103278
$544$	0	0
$545$	−15.0623	−0.645196
$546$	0	0
$547$	7.18677	0.307284	0.153642	−	0.988127i	$-0.450900\pi$
$547$	7.18677	0.307284	0.153642	+	0.988127i	$0.450900\pi$
$548$	0	0
$549$	5.06226	0.216052
$550$	0	0
$551$	0	0
$552$	0	0
$553$	38.6556	1.64381
$554$	0	0
$555$	−7.06226	−0.299776
$556$	0	0
$557$	−26.7802	−1.13471	−0.567356	−	0.823473i	$-0.692034\pi$
$557$	−26.7802	−1.13471	−0.567356	+	0.823473i	$0.692034\pi$
$558$	0	0
$559$	51.1868	2.16497
$560$	0	0
$561$	−26.1245	−1.10298
$562$	0	0
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	0	0
$565$	7.59339	0.319456
$566$	0	0
$567$	4.53113	0.190290
$568$	0	0
$569$	9.46887	0.396956	0.198478	−	0.980105i	$-0.436400\pi$
$569$	9.46887	0.396956	0.198478	+	0.980105i	$0.436400\pi$
$570$	0	0
$571$	−36.1245	−1.51176	−0.755882	−	0.654708i	$-0.772792\pi$
$571$	−36.1245	−1.51176	−0.755882	+	0.654708i	$0.772792\pi$
$572$	0	0
$573$	−8.00000	−0.334205
$574$	0	0
$575$	−6.53113	−0.272367
$576$	0	0
$577$	37.1868	1.54811	0.774053	−	0.633121i	$-0.218226\pi$
$577$	37.1868	1.54811	0.774053	+	0.633121i	$0.218226\pi$
$578$	0	0
$579$	−14.0000	−0.581820
$580$	0	0
$581$	36.2490	1.50386
$582$	0	0
$583$	16.5311	0.684649
$584$	0	0
$585$	−6.00000	−0.248069
$586$	0	0
$587$	44.2490	1.82635	0.913176	−	0.407564i	$-0.133622\pi$
$587$	44.2490	1.82635	0.913176	+	0.407564i	$0.133622\pi$
$588$	0	0
$589$	4.53113	0.186702
$590$	0	0
$591$	−6.00000	−0.246807
$592$	0	0
$593$	−14.0000	−0.574911	−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$594$	0	0
$595$	18.1245	0.743033
$596$	0	0
$597$	−8.53113	−0.349156
$598$	0	0
$599$	13.5934	0.555411	0.277705	−	0.960666i	$-0.410426\pi$
$599$	13.5934	0.555411	0.277705	+	0.960666i	$0.410426\pi$
$600$	0	0
$601$	30.0000	1.22373	0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000	1.22373	0.611863	+	0.790964i	$0.290420\pi$
$602$	0	0
$603$	−5.06226	−0.206151
$604$	0	0
$605$	−31.6556	−1.28698
$606$	0	0
$607$	14.4066	0.584746	0.292373	−	0.956304i	$-0.405555\pi$
$607$	14.4066	0.584746	0.292373	+	0.956304i	$0.405555\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	30.3735	1.22878
$612$	0	0
$613$	−30.0000	−1.21169	−0.605844	−	0.795583i	$-0.707165\pi$
$613$	−30.0000	−1.21169	−0.605844	+	0.795583i	$0.707165\pi$
$614$	0	0
$615$	7.06226	0.284778
$616$	0	0
$617$	−3.59339	−0.144664	−0.0723321	−	0.997381i	$-0.523044\pi$
$617$	−3.59339	−0.144664	−0.0723321	+	0.997381i	$0.523044\pi$
$618$	0	0
$619$	34.0000	1.36658	0.683288	−	0.730149i	$-0.260549\pi$
$619$	34.0000	1.36658	0.683288	+	0.730149i	$0.260549\pi$
$620$	0	0
$621$	6.53113	0.262085
$622$	0	0
$623$	−29.5934	−1.18563
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−29.5934	−1.18185
$628$	0	0
$629$	28.2490	1.12636
$630$	0	0
$631$	−30.6556	−1.22038	−0.610191	−	0.792254i	$-0.708907\pi$
$631$	−30.6556	−1.22038	−0.610191	+	0.792254i	$0.708907\pi$
$632$	0	0
$633$	−1.59339	−0.0633314
$634$	0	0
$635$	7.06226	0.280257
$636$	0	0
$637$	81.1868	3.21674
$638$	0	0
$639$	12.5311	0.495724
$640$	0	0
$641$	−22.0000	−0.868948	−0.434474	−	0.900684i	$-0.643066\pi$
$641$	−22.0000	−0.868948	−0.434474	+	0.900684i	$0.643066\pi$
$642$	0	0
$643$	17.5934	0.693815	0.346908	−	0.937899i	$-0.387232\pi$
$643$	17.5934	0.693815	0.346908	+	0.937899i	$0.387232\pi$
$644$	0	0
$645$	8.53113	0.335913
$646$	0	0
$647$	6.53113	0.256765	0.128383	−	0.991725i	$-0.459021\pi$
$647$	6.53113	0.256765	0.128383	+	0.991725i	$0.459021\pi$
$648$	0	0
$649$	−59.1868	−2.32328
$650$	0	0
$651$	4.53113	0.177589
$652$	0	0
$653$	38.2490	1.49680	0.748400	−	0.663248i	$-0.230822\pi$
$653$	38.2490	1.49680	0.748400	+	0.663248i	$0.230822\pi$
$654$	0	0
$655$	−9.06226	−0.354092
$656$	0	0
$657$	−8.53113	−0.332831
$658$	0	0
$659$	−16.0000	−0.623272	−0.311636	−	0.950202i	$-0.600877\pi$
$659$	−16.0000	−0.623272	−0.311636	+	0.950202i	$0.600877\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	0	0
$663$	24.0000	0.932083
$664$	0	0
$665$	20.5311	0.796163
$666$	0	0
$667$	0	0
$668$	0	0
$669$	2.00000	0.0773245
$670$	0	0
$671$	−33.0623	−1.27635
$672$	0	0
$673$	−9.06226	−0.349324	−0.174662	−	0.984628i	$-0.555883\pi$
$673$	−9.06226	−0.349324	−0.174662	+	0.984628i	$0.555883\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	27.5934	1.06050	0.530250	−	0.847841i	$-0.322098\pi$
$677$	27.5934	1.06050	0.530250	+	0.847841i	$0.322098\pi$
$678$	0	0
$679$	73.0623	2.80387
$680$	0	0
$681$	−24.5311	−0.940035
$682$	0	0
$683$	−7.46887	−0.285788	−0.142894	−	0.989738i	$-0.545641\pi$
$683$	−7.46887	−0.285788	−0.142894	+	0.989738i	$0.545641\pi$
$684$	0	0
$685$	−17.0623	−0.651915
$686$	0	0
$687$	5.59339	0.213401
$688$	0	0
$689$	−15.1868	−0.578570
$690$	0	0
$691$	−0.531129	−0.0202051	−0.0101025	−	0.999949i	$-0.503216\pi$
$691$	−0.531129	−0.0202051	−0.0101025	+	0.999949i	$0.503216\pi$
$692$	0	0
$693$	−29.5934	−1.12416
$694$	0	0
$695$	6.00000	0.227593
$696$	0	0
$697$	−28.2490	−1.07001
$698$	0	0
$699$	−1.46887	−0.0555578
$700$	0	0
$701$	−13.4689	−0.508712	−0.254356	−	0.967111i	$-0.581864\pi$
$701$	−13.4689	−0.508712	−0.254356	+	0.967111i	$0.581864\pi$
$702$	0	0
$703$	32.0000	1.20690
$704$	0	0
$705$	5.06226	0.190656
$706$	0	0
$707$	−42.9047	−1.61360
$708$	0	0
$709$	49.5934	1.86252	0.931259	−	0.364357i	$-0.118711\pi$
$709$	49.5934	1.86252	0.931259	+	0.364357i	$0.118711\pi$
$710$	0	0
$711$	8.53113	0.319942
$712$	0	0
$713$	6.53113	0.244593
$714$	0	0
$715$	39.1868	1.46550
$716$	0	0
$717$	−8.00000	−0.298765
$718$	0	0
$719$	−15.1868	−0.566371	−0.283186	−	0.959065i	$-0.591391\pi$
$719$	−15.1868	−0.566371	−0.283186	+	0.959065i	$0.591391\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−2.00000	−0.0743808
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−37.8424	−1.40350	−0.701749	−	0.712424i	$-0.747597\pi$
$727$	−37.8424	−1.40350	−0.701749	+	0.712424i	$0.747597\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−34.1245	−1.26214
$732$	0	0
$733$	−28.1245	−1.03880	−0.519401	−	0.854530i	$-0.673845\pi$
$733$	−28.1245	−1.03880	−0.519401	+	0.854530i	$0.673845\pi$
$734$	0	0
$735$	13.5311	0.499103
$736$	0	0
$737$	33.0623	1.21786
$738$	0	0
$739$	21.1868	0.779368	0.389684	−	0.920949i	$-0.372584\pi$
$739$	21.1868	0.779368	0.389684	+	0.920949i	$0.372584\pi$
$740$	0	0
$741$	27.1868	0.998731
$742$	0	0
$743$	28.6556	1.05127	0.525637	−	0.850709i	$-0.323827\pi$
$743$	28.6556	1.05127	0.525637	+	0.850709i	$0.323827\pi$
$744$	0	0
$745$	18.5311	0.678928
$746$	0	0
$747$	8.00000	0.292705
$748$	0	0
$749$	−43.4689	−1.58832
$750$	0	0
$751$	−30.9377	−1.12893	−0.564467	−	0.825456i	$-0.690918\pi$
$751$	−30.9377	−1.12893	−0.564467	+	0.825456i	$0.690918\pi$
$752$	0	0
$753$	3.06226	0.111595
$754$	0	0
$755$	14.1245	0.514044
$756$	0	0
$757$	26.0000	0.944986	0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000	0.944986	0.472493	+	0.881334i	$0.343354\pi$
$758$	0	0
$759$	−42.6556	−1.54830
$760$	0	0
$761$	11.5934	0.420260	0.210130	−	0.977673i	$-0.432611\pi$
$761$	11.5934	0.420260	0.210130	+	0.977673i	$0.432611\pi$
$762$	0	0
$763$	68.2490	2.47078
$764$	0	0
$765$	4.00000	0.144620
$766$	0	0
$767$	54.3735	1.96331
$768$	0	0
$769$	−4.40661	−0.158907	−0.0794533	−	0.996839i	$-0.525317\pi$
$769$	−4.40661	−0.158907	−0.0794533	+	0.996839i	$0.525317\pi$
$770$	0	0
$771$	−12.6556	−0.455782
$772$	0	0
$773$	22.5311	0.810388	0.405194	−	0.914231i	$-0.367204\pi$
$773$	22.5311	0.810388	0.405194	+	0.914231i	$0.367204\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	0	0
$777$	32.0000	1.14799
$778$	0	0
$779$	−32.0000	−1.14652
$780$	0	0
$781$	−81.8424	−2.92855
$782$	0	0
$783$	0	0
$784$	0	0
$785$	22.5311	0.804170
$786$	0	0
$787$	36.5311	1.30219	0.651097	−	0.758994i	$-0.274309\pi$
$787$	36.5311	1.30219	0.651097	+	0.758994i	$0.274309\pi$
$788$	0	0
$789$	−23.0623	−0.821038
$790$	0	0
$791$	−34.4066	−1.22336
$792$	0	0
$793$	30.3735	1.07860
$794$	0	0
$795$	−2.53113	−0.0897699
$796$	0	0
$797$	12.1245	0.429472	0.214736	−	0.976672i	$-0.431111\pi$
$797$	12.1245	0.429472	0.214736	+	0.976672i	$0.431111\pi$
$798$	0	0
$799$	−20.2490	−0.716359
$800$	0	0
$801$	−6.53113	−0.230766
$802$	0	0
$803$	55.7179	1.96624
$804$	0	0
$805$	29.5934	1.04303
$806$	0	0
$807$	19.1868	0.675406
$808$	0	0
$809$	15.5934	0.548234	0.274117	−	0.961696i	$-0.411614\pi$
$809$	15.5934	0.548234	0.274117	+	0.961696i	$0.411614\pi$
$810$	0	0
$811$	−32.7802	−1.15107	−0.575534	−	0.817778i	$-0.695206\pi$
$811$	−32.7802	−1.15107	−0.575534	+	0.817778i	$0.695206\pi$
$812$	0	0
$813$	−11.4689	−0.402231
$814$	0	0
$815$	5.06226	0.177323
$816$	0	0
$817$	−38.6556	−1.35239
$818$	0	0
$819$	27.1868	0.949983
$820$	0	0
$821$	42.1245	1.47016	0.735078	−	0.677983i	$-0.237146\pi$
$821$	42.1245	1.47016	0.735078	+	0.677983i	$0.237146\pi$
$822$	0	0
$823$	−40.1245	−1.39865	−0.699326	−	0.714803i	$-0.746517\pi$
$823$	−40.1245	−1.39865	−0.699326	+	0.714803i	$0.746517\pi$
$824$	0	0
$825$	6.53113	0.227385
$826$	0	0
$827$	−8.00000	−0.278187	−0.139094	−	0.990279i	$-0.544419\pi$
$827$	−8.00000	−0.278187	−0.139094	+	0.990279i	$0.544419\pi$
$828$	0	0
$829$	30.6556	1.06471	0.532357	−	0.846520i	$-0.321306\pi$
$829$	30.6556	1.06471	0.532357	+	0.846520i	$0.321306\pi$
$830$	0	0
$831$	15.0623	0.522504
$832$	0	0
$833$	−54.1245	−1.87530
$834$	0	0
$835$	−23.5934	−0.816483
$836$	0	0
$837$	1.00000	0.0345651
$838$	0	0
$839$	−33.8424	−1.16837	−0.584185	−	0.811621i	$-0.698586\pi$
$839$	−33.8424	−1.16837	−0.584185	+	0.811621i	$0.698586\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	−15.0623	−0.518772
$844$	0	0
$845$	−23.0000	−0.791224
$846$	0	0
$847$	143.436	4.92851
$848$	0	0
$849$	−12.0000	−0.411839
$850$	0	0
$851$	46.1245	1.58113
$852$	0	0
$853$	57.7179	1.97622	0.988112	−	0.153738i	$-0.0491311\pi$
$853$	57.7179	1.97622	0.988112	+	0.153738i	$0.0491311\pi$
$854$	0	0
$855$	4.53113	0.154961
$856$	0	0
$857$	14.0000	0.478231	0.239115	−	0.970991i	$-0.423143\pi$
$857$	14.0000	0.478231	0.239115	+	0.970991i	$0.423143\pi$
$858$	0	0
$859$	23.0623	0.786874	0.393437	−	0.919352i	$-0.371286\pi$
$859$	23.0623	0.786874	0.393437	+	0.919352i	$0.371286\pi$
$860$	0	0
$861$	−32.0000	−1.09056
$862$	0	0
$863$	−33.4689	−1.13929	−0.569647	−	0.821890i	$-0.692920\pi$
$863$	−33.4689	−1.13929	−0.569647	+	0.821890i	$0.692920\pi$
$864$	0	0
$865$	−14.0000	−0.476014
$866$	0	0
$867$	1.00000	0.0339618
$868$	0	0
$869$	−55.7179	−1.89010
$870$	0	0
$871$	−30.3735	−1.02917
$872$	0	0
$873$	16.1245	0.545732
$874$	0	0
$875$	−4.53113	−0.153180
$876$	0	0
$877$	11.8755	0.401007	0.200503	−	0.979693i	$-0.435742\pi$
$877$	11.8755	0.401007	0.200503	+	0.979693i	$0.435742\pi$
$878$	0	0
$879$	14.0000	0.472208
$880$	0	0
$881$	3.87548	0.130568	0.0652842	−	0.997867i	$-0.479205\pi$
$881$	3.87548	0.130568	0.0652842	+	0.997867i	$0.479205\pi$
$882$	0	0
$883$	13.3444	0.449073	0.224537	−	0.974466i	$-0.427913\pi$
$883$	13.3444	0.449073	0.224537	+	0.974466i	$0.427913\pi$
$884$	0	0
$885$	9.06226	0.304624
$886$	0	0
$887$	−23.1868	−0.778536	−0.389268	−	0.921125i	$-0.627272\pi$
$887$	−23.1868	−0.778536	−0.389268	+	0.921125i	$0.627272\pi$
$888$	0	0
$889$	−32.0000	−1.07325
$890$	0	0
$891$	−6.53113	−0.218801
$892$	0	0
$893$	−22.9377	−0.767582
$894$	0	0
$895$	−3.06226	−0.102360
$896$	0	0
$897$	39.1868	1.30841
$898$	0	0
$899$	0	0
$900$	0	0
$901$	10.1245	0.337297
$902$	0	0
$903$	−38.6556	−1.28638
$904$	0	0
$905$	−2.40661	−0.0799985
$906$	0	0
$907$	−32.2490	−1.07081	−0.535406	−	0.844595i	$-0.679841\pi$
$907$	−32.2490	−1.07081	−0.535406	+	0.844595i	$0.679841\pi$
$908$	0	0
$909$	−9.46887	−0.314063
$910$	0	0
$911$	16.0000	0.530104	0.265052	−	0.964234i	$-0.414611\pi$
$911$	16.0000	0.530104	0.265052	+	0.964234i	$0.414611\pi$
$912$	0	0
$913$	−52.2490	−1.72919
$914$	0	0
$915$	5.06226	0.167353
$916$	0	0
$917$	41.0623	1.35600
$918$	0	0
$919$	25.0623	0.826728	0.413364	−	0.910566i	$-0.364354\pi$
$919$	25.0623	0.826728	0.413364	+	0.910566i	$0.364354\pi$
$920$	0	0
$921$	−30.1245	−0.992637
$922$	0	0
$923$	75.1868	2.47480
$924$	0	0
$925$	−7.06226	−0.232206
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−51.8424	−1.70089	−0.850447	−	0.526060i	$-0.823669\pi$
$929$	−51.8424	−1.70089	−0.850447	+	0.526060i	$0.823669\pi$
$930$	0	0
$931$	−61.3113	−2.00940
$932$	0	0
$933$	−8.00000	−0.261908
$934$	0	0
$935$	−26.1245	−0.854363
$936$	0	0
$937$	−8.93774	−0.291983	−0.145992	−	0.989286i	$-0.546637\pi$
$937$	−8.93774	−0.291983	−0.145992	+	0.989286i	$0.546637\pi$
$938$	0	0
$939$	−10.9377	−0.356940
$940$	0	0
$941$	−54.1245	−1.76441	−0.882204	−	0.470867i	$-0.843941\pi$
$941$	−54.1245	−1.76441	−0.882204	+	0.470867i	$0.843941\pi$
$942$	0	0
$943$	−46.1245	−1.50202
$944$	0	0
$945$	4.53113	0.147398
$946$	0	0
$947$	−18.9377	−0.615394	−0.307697	−	0.951484i	$-0.599558\pi$
$947$	−18.9377	−0.615394	−0.307697	+	0.951484i	$0.599558\pi$
$948$	0	0
$949$	−51.1868	−1.66159
$950$	0	0
$951$	−12.9377	−0.419535
$952$	0	0
$953$	21.8755	0.708616	0.354308	−	0.935129i	$-0.384716\pi$
$953$	21.8755	0.708616	0.354308	+	0.935129i	$0.384716\pi$
$954$	0	0
$955$	−8.00000	−0.258874
$956$	0	0
$957$	0	0
$958$	0	0
$959$	77.3113	2.49651
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−9.59339	−0.309143
$964$	0	0
$965$	−14.0000	−0.450676
$966$	0	0
$967$	0.937742	0.0301558	0.0150779	−	0.999886i	$-0.495200\pi$
$967$	0.937742	0.0301558	0.0150779	+	0.999886i	$0.495200\pi$
$968$	0	0
$969$	−18.1245	−0.582243
$970$	0	0
$971$	−15.1868	−0.487367	−0.243683	−	0.969855i	$-0.578356\pi$
$971$	−15.1868	−0.487367	−0.243683	+	0.969855i	$0.578356\pi$
$972$	0	0
$973$	−27.1868	−0.871568
$974$	0	0
$975$	−6.00000	−0.192154
$976$	0	0
$977$	−38.0000	−1.21573	−0.607864	−	0.794041i	$-0.707973\pi$
$977$	−38.0000	−1.21573	−0.607864	+	0.794041i	$0.707973\pi$
$978$	0	0
$979$	42.6556	1.36328
$980$	0	0
$981$	15.0623	0.480901
$982$	0	0
$983$	−0.937742	−0.0299093	−0.0149547	−	0.999888i	$-0.504760\pi$
$983$	−0.937742	−0.0299093	−0.0149547	+	0.999888i	$0.504760\pi$
$984$	0	0
$985$	−6.00000	−0.191176
$986$	0	0
$987$	−22.9377	−0.730116
$988$	0	0
$989$	−55.7179	−1.77173
$990$	0	0
$991$	32.5311	1.03339	0.516693	−	0.856171i	$-0.327163\pi$
$991$	32.5311	1.03339	0.516693	+	0.856171i	$0.327163\pi$
$992$	0	0
$993$	12.9377	0.410567
$994$	0	0
$995$	−8.53113	−0.270455
$996$	0	0
$997$	−2.24903	−0.0712275	−0.0356138	−	0.999366i	$-0.511339\pi$
$997$	−2.24903	−0.0712275	−0.0356138	+	0.999366i	$0.511339\pi$
$998$	0	0
$999$	7.06226	0.223440

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7440.2.a.bd.1.2		2
4.3	odd	2		930.2.a.q.1.1	✓	2
12.11	even	2		2790.2.a.bf.1.1		2
20.3	even	4		4650.2.d.bg.3349.2		4
20.7	even	4		4650.2.d.bg.3349.3		4
20.19	odd	2		4650.2.a.bz.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.q.1.1	✓	2	4.3	odd	2
2790.2.a.bf.1.1		2	12.11	even	2
4650.2.a.bz.1.2		2	20.19	odd	2
4650.2.d.bg.3349.2		4	20.3	even	4
4650.2.d.bg.3349.3		4	20.7	even	4
7440.2.a.bd.1.2		2	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:	\( N \)	\(=\)	\( 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7440.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} +4.53113 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{5} +4.53113 q^{7} +1.00000 q^{9} -6.53113 q^{11} +6.00000 q^{13} +1.00000 q^{15} -4.00000 q^{17} -4.53113 q^{19} -4.53113 q^{21} -6.53113 q^{23} +1.00000 q^{25} -1.00000 q^{27} -1.00000 q^{31} +6.53113 q^{33} -4.53113 q^{35} -7.06226 q^{37} -6.00000 q^{39} +7.06226 q^{41} +8.53113 q^{43} -1.00000 q^{45} +5.06226 q^{47} +13.5311 q^{49} +4.00000 q^{51} -2.53113 q^{53} +6.53113 q^{55} +4.53113 q^{57} +9.06226 q^{59} +5.06226 q^{61} +4.53113 q^{63} -6.00000 q^{65} -5.06226 q^{67} +6.53113 q^{69} +12.5311 q^{71} -8.53113 q^{73} -1.00000 q^{75} -29.5934 q^{77} +8.53113 q^{79} +1.00000 q^{81} +8.00000 q^{83} +4.00000 q^{85} -6.53113 q^{89} +27.1868 q^{91} +1.00000 q^{93} +4.53113 q^{95} +16.1245 q^{97} -6.53113 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^5 + q^7 + 2 * q^9	\( 2 q - 2 q^{3} - 2 q^{5} + q^{7} + 2 q^{9} - 5 q^{11} + 12 q^{13} + 2 q^{15} - 8 q^{17} - q^{19} - q^{21} - 5 q^{23} + 2 q^{25} - 2 q^{27} - 2 q^{31} + 5 q^{33} - q^{35} + 2 q^{37} - 12 q^{39} - 2 q^{41} + 9 q^{43} - 2 q^{45} - 6 q^{47} + 19 q^{49} + 8 q^{51} + 3 q^{53} + 5 q^{55} + q^{57} + 2 q^{59} - 6 q^{61} + q^{63} - 12 q^{65} + 6 q^{67} + 5 q^{69} + 17 q^{71} - 9 q^{73} - 2 q^{75} - 35 q^{77} + 9 q^{79} + 2 q^{81} + 16 q^{83} + 8 q^{85} - 5 q^{89} + 6 q^{91} + 2 q^{93} + q^{95} - 5 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 + q^7 + 2 * q^9 - 5 * q^11 + 12 * q^13 + 2 * q^15 - 8 * q^17 - q^19 - q^21 - 5 * q^23 + 2 * q^25 - 2 * q^27 - 2 * q^31 + 5 * q^33 - q^35 + 2 * q^37 - 12 * q^39 - 2 * q^41 + 9 * q^43 - 2 * q^45 - 6 * q^47 + 19 * q^49 + 8 * q^51 + 3 * q^53 + 5 * q^55 + q^57 + 2 * q^59 - 6 * q^61 + q^63 - 12 * q^65 + 6 * q^67 + 5 * q^69 + 17 * q^71 - 9 * q^73 - 2 * q^75 - 35 * q^77 + 9 * q^79 + 2 * q^81 + 16 * q^83 + 8 * q^85 - 5 * q^89 + 6 * q^91 + 2 * q^93 + q^95 - 5 * q^99

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