LMFDB - Embedded newform 1380.2.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1380,2,Mod(1,1380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1380, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1380.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1380.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:	$11.0193554789$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.44949$ of defining polynomial
Character	$\chi$	$=$	1380.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +0.449490 q^{11} -2.44949 q^{13} +1.00000 q^{15} +1.44949 q^{17} -7.34847 q^{19} -1.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -4.55051 q^{29} +7.89898 q^{31} -0.449490 q^{33} -1.00000 q^{35} -0.101021 q^{37} +2.44949 q^{39} -5.44949 q^{41} +11.7980 q^{43} -1.00000 q^{45} -7.34847 q^{47} -6.00000 q^{49} -1.44949 q^{51} -10.3485 q^{53} -0.449490 q^{55} +7.34847 q^{57} -11.4495 q^{59} -4.44949 q^{61} +1.00000 q^{63} +2.44949 q^{65} -10.7980 q^{67} -1.00000 q^{69} -12.3485 q^{71} +7.34847 q^{73} -1.00000 q^{75} +0.449490 q^{77} +5.79796 q^{79} +1.00000 q^{81} -10.5505 q^{83} -1.44949 q^{85} +4.55051 q^{87} -12.8990 q^{89} -2.44949 q^{91} -7.89898 q^{93} +7.34847 q^{95} -4.89898 q^{97} +0.449490 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} - 4 q^{11} + 2 q^{15} - 2 q^{17} - 2 q^{21} + 2 q^{23} + 2 q^{25} - 2 q^{27} - 14 q^{29} + 6 q^{31} + 4 q^{33} - 2 q^{35} - 10 q^{37} - 6 q^{41} + 4 q^{43} - 2 q^{45} - 12 q^{49} + 2 q^{51} - 6 q^{53} + 4 q^{55} - 18 q^{59} - 4 q^{61} + 2 q^{63} - 2 q^{67} - 2 q^{69} - 10 q^{71} - 2 q^{75} - 4 q^{77} - 8 q^{79} + 2 q^{81} - 26 q^{83} + 2 q^{85} + 14 q^{87} - 16 q^{89} - 6 q^{93} - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 - 4 * q^11 + 2 * q^15 - 2 * q^17 - 2 * q^21 + 2 * q^23 + 2 * q^25 - 2 * q^27 - 14 * q^29 + 6 * q^31 + 4 * q^33 - 2 * q^35 - 10 * q^37 - 6 * q^41 + 4 * q^43 - 2 * q^45 - 12 * q^49 + 2 * q^51 - 6 * q^53 + 4 * q^55 - 18 * q^59 - 4 * q^61 + 2 * q^63 - 2 * q^67 - 2 * q^69 - 10 * q^71 - 2 * q^75 - 4 * q^77 - 8 * q^79 + 2 * q^81 - 26 * q^83 + 2 * q^85 + 14 * q^87 - 16 * q^89 - 6 * q^93 - 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	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.449490	0.135526	0.0677631	−	0.997701i	$-0.478414\pi$
$11$	0.449490	0.135526	0.0677631	+	0.997701i	$0.478414\pi$
$12$	0	0
$13$	−2.44949	−0.679366	−0.339683	−	0.940540i	$-0.610320\pi$
$13$	−2.44949	−0.679366	−0.339683	+	0.940540i	$0.610320\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	1.44949	0.351553	0.175776	−	0.984430i	$-0.443756\pi$
$17$	1.44949	0.351553	0.175776	+	0.984430i	$0.443756\pi$
$18$	0	0
$19$	−7.34847	−1.68585	−0.842927	−	0.538028i	$-0.819170\pi$
$19$	−7.34847	−1.68585	−0.842927	+	0.538028i	$0.819170\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−4.55051	−0.845009	−0.422504	−	0.906361i	$-0.638849\pi$
$29$	−4.55051	−0.845009	−0.422504	+	0.906361i	$0.638849\pi$
$30$	0	0
$31$	7.89898	1.41870	0.709349	−	0.704857i	$-0.248989\pi$
$31$	7.89898	1.41870	0.709349	+	0.704857i	$0.248989\pi$
$32$	0	0
$33$	−0.449490	−0.0782461
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−0.101021	−0.0166077	−0.00830384	−	0.999966i	$-0.502643\pi$
$37$	−0.101021	−0.0166077	−0.00830384	+	0.999966i	$0.502643\pi$
$38$	0	0
$39$	2.44949	0.392232
$40$	0	0
$41$	−5.44949	−0.851067	−0.425534	−	0.904943i	$-0.639914\pi$
$41$	−5.44949	−0.851067	−0.425534	+	0.904943i	$0.639914\pi$
$42$	0	0
$43$	11.7980	1.79917	0.899586	−	0.436744i	$-0.143868\pi$
$43$	11.7980	1.79917	0.899586	+	0.436744i	$0.143868\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−7.34847	−1.07188	−0.535942	−	0.844255i	$-0.680044\pi$
$47$	−7.34847	−1.07188	−0.535942	+	0.844255i	$0.680044\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	−1.44949	−0.202969
$52$	0	0
$53$	−10.3485	−1.42147	−0.710736	−	0.703459i	$-0.751638\pi$
$53$	−10.3485	−1.42147	−0.710736	+	0.703459i	$0.751638\pi$
$54$	0	0
$55$	−0.449490	−0.0606092
$56$	0	0
$57$	7.34847	0.973329
$58$	0	0
$59$	−11.4495	−1.49060	−0.745298	−	0.666731i	$-0.767693\pi$
$59$	−11.4495	−1.49060	−0.745298	+	0.666731i	$0.767693\pi$
$60$	0	0
$61$	−4.44949	−0.569699	−0.284849	−	0.958572i	$-0.591944\pi$
$61$	−4.44949	−0.569699	−0.284849	+	0.958572i	$0.591944\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	2.44949	0.303822
$66$	0	0
$67$	−10.7980	−1.31918	−0.659590	−	0.751625i	$-0.729270\pi$
$67$	−10.7980	−1.31918	−0.659590	+	0.751625i	$0.729270\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−12.3485	−1.46549	−0.732747	−	0.680501i	$-0.761762\pi$
$71$	−12.3485	−1.46549	−0.732747	+	0.680501i	$0.761762\pi$
$72$	0	0
$73$	7.34847	0.860073	0.430037	−	0.902811i	$-0.358501\pi$
$73$	7.34847	0.860073	0.430037	+	0.902811i	$0.358501\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	0.449490	0.0512241
$78$	0	0
$79$	5.79796	0.652321	0.326161	−	0.945314i	$-0.394245\pi$
$79$	5.79796	0.652321	0.326161	+	0.945314i	$0.394245\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−10.5505	−1.15807	−0.579034	−	0.815303i	$-0.696570\pi$
$83$	−10.5505	−1.15807	−0.579034	+	0.815303i	$0.696570\pi$
$84$	0	0
$85$	−1.44949	−0.157219
$86$	0	0
$87$	4.55051	0.487866
$88$	0	0
$89$	−12.8990	−1.36729	−0.683645	−	0.729815i	$-0.739606\pi$
$89$	−12.8990	−1.36729	−0.683645	+	0.729815i	$0.739606\pi$
$90$	0	0
$91$	−2.44949	−0.256776
$92$	0	0
$93$	−7.89898	−0.819086
$94$	0	0
$95$	7.34847	0.753937
$96$	0	0
$97$	−4.89898	−0.497416	−0.248708	−	0.968579i	$-0.580006\pi$
$97$	−4.89898	−0.497416	−0.248708	+	0.968579i	$0.580006\pi$
$98$	0	0
$99$	0.449490	0.0451754
$100$	0	0
$101$	−1.65153	−0.164333	−0.0821667	−	0.996619i	$-0.526184\pi$
$101$	−1.65153	−0.164333	−0.0821667	+	0.996619i	$0.526184\pi$
$102$	0	0
$103$	−6.89898	−0.679777	−0.339888	−	0.940466i	$-0.610389\pi$
$103$	−6.89898	−0.679777	−0.339888	+	0.940466i	$0.610389\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	0	0
$107$	16.3485	1.58047	0.790233	−	0.612806i	$-0.209959\pi$
$107$	16.3485	1.58047	0.790233	+	0.612806i	$0.209959\pi$
$108$	0	0
$109$	−13.5505	−1.29790	−0.648952	−	0.760830i	$-0.724792\pi$
$109$	−13.5505	−1.29790	−0.648952	+	0.760830i	$0.724792\pi$
$110$	0	0
$111$	0.101021	0.00958844
$112$	0	0
$113$	9.24745	0.869927	0.434963	−	0.900448i	$-0.356761\pi$
$113$	9.24745	0.869927	0.434963	+	0.900448i	$0.356761\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	−2.44949	−0.226455
$118$	0	0
$119$	1.44949	0.132875
$120$	0	0
$121$	−10.7980	−0.981633
$122$	0	0
$123$	5.44949	0.491364
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	18.2474	1.61920	0.809600	−	0.586983i	$-0.199684\pi$
$127$	18.2474	1.61920	0.809600	+	0.586983i	$0.199684\pi$
$128$	0	0
$129$	−11.7980	−1.03875
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−7.34847	−0.637193
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−10.6969	−0.913901	−0.456951	−	0.889492i	$-0.651058\pi$
$137$	−10.6969	−0.913901	−0.456951	+	0.889492i	$0.651058\pi$
$138$	0	0
$139$	8.79796	0.746233	0.373117	−	0.927784i	$-0.378289\pi$
$139$	8.79796	0.746233	0.373117	+	0.927784i	$0.378289\pi$
$140$	0	0
$141$	7.34847	0.618853
$142$	0	0
$143$	−1.10102	−0.0920720
$144$	0	0
$145$	4.55051	0.377899
$146$	0	0
$147$	6.00000	0.494872
$148$	0	0
$149$	−10.4495	−0.856056	−0.428028	−	0.903766i	$-0.640791\pi$
$149$	−10.4495	−0.856056	−0.428028	+	0.903766i	$0.640791\pi$
$150$	0	0
$151$	15.7980	1.28562	0.642810	−	0.766026i	$-0.277769\pi$
$151$	15.7980	1.28562	0.642810	+	0.766026i	$0.277769\pi$
$152$	0	0
$153$	1.44949	0.117184
$154$	0	0
$155$	−7.89898	−0.634461
$156$	0	0
$157$	11.0000	0.877896	0.438948	−	0.898513i	$-0.355351\pi$
$157$	11.0000	0.877896	0.438948	+	0.898513i	$0.355351\pi$
$158$	0	0
$159$	10.3485	0.820687
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	−17.5959	−1.37822	−0.689109	−	0.724657i	$-0.741998\pi$
$163$	−17.5959	−1.37822	−0.689109	+	0.724657i	$0.741998\pi$
$164$	0	0
$165$	0.449490	0.0349927
$166$	0	0
$167$	9.55051	0.739041	0.369520	−	0.929223i	$-0.379522\pi$
$167$	9.55051	0.739041	0.369520	+	0.929223i	$0.379522\pi$
$168$	0	0
$169$	−7.00000	−0.538462
$170$	0	0
$171$	−7.34847	−0.561951
$172$	0	0
$173$	−9.79796	−0.744925	−0.372463	−	0.928047i	$-0.621486\pi$
$173$	−9.79796	−0.744925	−0.372463	+	0.928047i	$0.621486\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	11.4495	0.860596
$178$	0	0
$179$	10.0000	0.747435	0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000	0.747435	0.373718	+	0.927543i	$0.378083\pi$
$180$	0	0
$181$	4.89898	0.364138	0.182069	−	0.983286i	$-0.441721\pi$
$181$	4.89898	0.364138	0.182069	+	0.983286i	$0.441721\pi$
$182$	0	0
$183$	4.44949	0.328916
$184$	0	0
$185$	0.101021	0.00742718
$186$	0	0
$187$	0.651531	0.0476446
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	17.1464	1.24067	0.620336	−	0.784336i	$-0.286996\pi$
$191$	17.1464	1.24067	0.620336	+	0.784336i	$0.286996\pi$
$192$	0	0
$193$	8.89898	0.640563	0.320281	−	0.947322i	$-0.396223\pi$
$193$	8.89898	0.640563	0.320281	+	0.947322i	$0.396223\pi$
$194$	0	0
$195$	−2.44949	−0.175412
$196$	0	0
$197$	−1.10102	−0.0784445	−0.0392222	−	0.999231i	$-0.512488\pi$
$197$	−1.10102	−0.0784445	−0.0392222	+	0.999231i	$0.512488\pi$
$198$	0	0
$199$	−4.69694	−0.332957	−0.166479	−	0.986045i	$-0.553240\pi$
$199$	−4.69694	−0.332957	−0.166479	+	0.986045i	$0.553240\pi$
$200$	0	0
$201$	10.7980	0.761629
$202$	0	0
$203$	−4.55051	−0.319383
$204$	0	0
$205$	5.44949	0.380609
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−3.30306	−0.228478
$210$	0	0
$211$	25.0000	1.72107	0.860535	−	0.509390i	$-0.170129\pi$
$211$	25.0000	1.72107	0.860535	+	0.509390i	$0.170129\pi$
$212$	0	0
$213$	12.3485	0.846103
$214$	0	0
$215$	−11.7980	−0.804614
$216$	0	0
$217$	7.89898	0.536218
$218$	0	0
$219$	−7.34847	−0.496564
$220$	0	0
$221$	−3.55051	−0.238833
$222$	0	0
$223$	14.0000	0.937509	0.468755	−	0.883328i	$-0.344703\pi$
$223$	14.0000	0.937509	0.468755	+	0.883328i	$0.344703\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	21.7980	1.44678	0.723391	−	0.690439i	$-0.242583\pi$
$227$	21.7980	1.44678	0.723391	+	0.690439i	$0.242583\pi$
$228$	0	0
$229$	6.69694	0.442546	0.221273	−	0.975212i	$-0.428979\pi$
$229$	6.69694	0.442546	0.221273	+	0.975212i	$0.428979\pi$
$230$	0	0
$231$	−0.449490	−0.0295743
$232$	0	0
$233$	−15.5959	−1.02172	−0.510861	−	0.859663i	$-0.670673\pi$
$233$	−15.5959	−1.02172	−0.510861	+	0.859663i	$0.670673\pi$
$234$	0	0
$235$	7.34847	0.479361
$236$	0	0
$237$	−5.79796	−0.376618
$238$	0	0
$239$	−21.0454	−1.36131	−0.680657	−	0.732602i	$-0.738306\pi$
$239$	−21.0454	−1.36131	−0.680657	+	0.732602i	$0.738306\pi$
$240$	0	0
$241$	−30.0454	−1.93539	−0.967697	−	0.252114i	$-0.918874\pi$
$241$	−30.0454	−1.93539	−0.967697	+	0.252114i	$0.918874\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	6.00000	0.383326
$246$	0	0
$247$	18.0000	1.14531
$248$	0	0
$249$	10.5505	0.668611
$250$	0	0
$251$	−18.0000	−1.13615	−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000	−1.13615	−0.568075	+	0.822977i	$0.692312\pi$
$252$	0	0
$253$	0.449490	0.0282592
$254$	0	0
$255$	1.44949	0.0907706
$256$	0	0
$257$	−4.24745	−0.264949	−0.132474	−	0.991186i	$-0.542292\pi$
$257$	−4.24745	−0.264949	−0.132474	+	0.991186i	$0.542292\pi$
$258$	0	0
$259$	−0.101021	−0.00627711
$260$	0	0
$261$	−4.55051	−0.281670
$262$	0	0
$263$	−0.348469	−0.0214875	−0.0107438	−	0.999942i	$-0.503420\pi$
$263$	−0.348469	−0.0214875	−0.0107438	+	0.999942i	$0.503420\pi$
$264$	0	0
$265$	10.3485	0.635701
$266$	0	0
$267$	12.8990	0.789405
$268$	0	0
$269$	15.2474	0.929653	0.464827	−	0.885402i	$-0.346117\pi$
$269$	15.2474	0.929653	0.464827	+	0.885402i	$0.346117\pi$
$270$	0	0
$271$	6.10102	0.370611	0.185305	−	0.982681i	$-0.440673\pi$
$271$	6.10102	0.370611	0.185305	+	0.982681i	$0.440673\pi$
$272$	0	0
$273$	2.44949	0.148250
$274$	0	0
$275$	0.449490	0.0271053
$276$	0	0
$277$	−0.202041	−0.0121395	−0.00606973	−	0.999982i	$-0.501932\pi$
$277$	−0.202041	−0.0121395	−0.00606973	+	0.999982i	$0.501932\pi$
$278$	0	0
$279$	7.89898	0.472900
$280$	0	0
$281$	−2.65153	−0.158177	−0.0790885	−	0.996868i	$-0.525201\pi$
$281$	−2.65153	−0.158177	−0.0790885	+	0.996868i	$0.525201\pi$
$282$	0	0
$283$	3.89898	0.231770	0.115885	−	0.993263i	$-0.463030\pi$
$283$	3.89898	0.231770	0.115885	+	0.993263i	$0.463030\pi$
$284$	0	0
$285$	−7.34847	−0.435286
$286$	0	0
$287$	−5.44949	−0.321673
$288$	0	0
$289$	−14.8990	−0.876411
$290$	0	0
$291$	4.89898	0.287183
$292$	0	0
$293$	27.0454	1.58001	0.790005	−	0.613101i	$-0.210078\pi$
$293$	27.0454	1.58001	0.790005	+	0.613101i	$0.210078\pi$
$294$	0	0
$295$	11.4495	0.666615
$296$	0	0
$297$	−0.449490	−0.0260820
$298$	0	0
$299$	−2.44949	−0.141658
$300$	0	0
$301$	11.7980	0.680023
$302$	0	0
$303$	1.65153	0.0948780
$304$	0	0
$305$	4.44949	0.254777
$306$	0	0
$307$	−20.2474	−1.15558	−0.577791	−	0.816184i	$-0.696085\pi$
$307$	−20.2474	−1.15558	−0.577791	+	0.816184i	$0.696085\pi$
$308$	0	0
$309$	6.89898	0.392469
$310$	0	0
$311$	−15.5959	−0.884363	−0.442182	−	0.896926i	$-0.645795\pi$
$311$	−15.5959	−0.884363	−0.442182	+	0.896926i	$0.645795\pi$
$312$	0	0
$313$	−4.30306	−0.243223	−0.121612	−	0.992578i	$-0.538806\pi$
$313$	−4.30306	−0.243223	−0.121612	+	0.992578i	$0.538806\pi$
$314$	0	0
$315$	−1.00000	−0.0563436
$316$	0	0
$317$	27.3485	1.53604	0.768022	−	0.640424i	$-0.221241\pi$
$317$	27.3485	1.53604	0.768022	+	0.640424i	$0.221241\pi$
$318$	0	0
$319$	−2.04541	−0.114521
$320$	0	0
$321$	−16.3485	−0.912483
$322$	0	0
$323$	−10.6515	−0.592667
$324$	0	0
$325$	−2.44949	−0.135873
$326$	0	0
$327$	13.5505	0.749345
$328$	0	0
$329$	−7.34847	−0.405134
$330$	0	0
$331$	20.7980	1.14316	0.571580	−	0.820547i	$-0.306331\pi$
$331$	20.7980	1.14316	0.571580	+	0.820547i	$0.306331\pi$
$332$	0	0
$333$	−0.101021	−0.00553589
$334$	0	0
$335$	10.7980	0.589956
$336$	0	0
$337$	23.7980	1.29636	0.648179	−	0.761488i	$-0.275531\pi$
$337$	23.7980	1.29636	0.648179	+	0.761488i	$0.275531\pi$
$338$	0	0
$339$	−9.24745	−0.502252
$340$	0	0
$341$	3.55051	0.192271
$342$	0	0
$343$	−13.0000	−0.701934
$344$	0	0
$345$	1.00000	0.0538382
$346$	0	0
$347$	28.4949	1.52969	0.764843	−	0.644217i	$-0.222816\pi$
$347$	28.4949	1.52969	0.764843	+	0.644217i	$0.222816\pi$
$348$	0	0
$349$	−33.4949	−1.79294	−0.896470	−	0.443104i	$-0.853877\pi$
$349$	−33.4949	−1.79294	−0.896470	+	0.443104i	$0.853877\pi$
$350$	0	0
$351$	2.44949	0.130744
$352$	0	0
$353$	5.55051	0.295424	0.147712	−	0.989030i	$-0.452809\pi$
$353$	5.55051	0.295424	0.147712	+	0.989030i	$0.452809\pi$
$354$	0	0
$355$	12.3485	0.655389
$356$	0	0
$357$	−1.44949	−0.0767151
$358$	0	0
$359$	−9.55051	−0.504057	−0.252028	−	0.967720i	$-0.581098\pi$
$359$	−9.55051	−0.504057	−0.252028	+	0.967720i	$0.581098\pi$
$360$	0	0
$361$	35.0000	1.84211
$362$	0	0
$363$	10.7980	0.566746
$364$	0	0
$365$	−7.34847	−0.384636
$366$	0	0
$367$	29.8990	1.56071	0.780357	−	0.625334i	$-0.215037\pi$
$367$	29.8990	1.56071	0.780357	+	0.625334i	$0.215037\pi$
$368$	0	0
$369$	−5.44949	−0.283689
$370$	0	0
$371$	−10.3485	−0.537266
$372$	0	0
$373$	10.2020	0.528242	0.264121	−	0.964490i	$-0.414918\pi$
$373$	10.2020	0.528242	0.264121	+	0.964490i	$0.414918\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	11.1464	0.574070
$378$	0	0
$379$	13.7980	0.708754	0.354377	−	0.935103i	$-0.384693\pi$
$379$	13.7980	0.708754	0.354377	+	0.935103i	$0.384693\pi$
$380$	0	0
$381$	−18.2474	−0.934845
$382$	0	0
$383$	21.2474	1.08569	0.542847	−	0.839832i	$-0.317346\pi$
$383$	21.2474	1.08569	0.542847	+	0.839832i	$0.317346\pi$
$384$	0	0
$385$	−0.449490	−0.0229081
$386$	0	0
$387$	11.7980	0.599724
$388$	0	0
$389$	−14.4949	−0.734920	−0.367460	−	0.930039i	$-0.619773\pi$
$389$	−14.4949	−0.734920	−0.367460	+	0.930039i	$0.619773\pi$
$390$	0	0
$391$	1.44949	0.0733038
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−5.79796	−0.291727
$396$	0	0
$397$	−37.7980	−1.89703	−0.948513	−	0.316739i	$-0.897412\pi$
$397$	−37.7980	−1.89703	−0.948513	+	0.316739i	$0.897412\pi$
$398$	0	0
$399$	7.34847	0.367884
$400$	0	0
$401$	13.1010	0.654234	0.327117	−	0.944984i	$-0.393923\pi$
$401$	13.1010	0.654234	0.327117	+	0.944984i	$0.393923\pi$
$402$	0	0
$403$	−19.3485	−0.963816
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−0.0454077	−0.00225078
$408$	0	0
$409$	19.6969	0.973951	0.486975	−	0.873416i	$-0.338100\pi$
$409$	19.6969	0.973951	0.486975	+	0.873416i	$0.338100\pi$
$410$	0	0
$411$	10.6969	0.527641
$412$	0	0
$413$	−11.4495	−0.563393
$414$	0	0
$415$	10.5505	0.517904
$416$	0	0
$417$	−8.79796	−0.430838
$418$	0	0
$419$	−18.6515	−0.911187	−0.455593	−	0.890188i	$-0.650573\pi$
$419$	−18.6515	−0.911187	−0.455593	+	0.890188i	$0.650573\pi$
$420$	0	0
$421$	−7.34847	−0.358142	−0.179071	−	0.983836i	$-0.557309\pi$
$421$	−7.34847	−0.358142	−0.179071	+	0.983836i	$0.557309\pi$
$422$	0	0
$423$	−7.34847	−0.357295
$424$	0	0
$425$	1.44949	0.0703106
$426$	0	0
$427$	−4.44949	−0.215326
$428$	0	0
$429$	1.10102	0.0531578
$430$	0	0
$431$	−36.4949	−1.75790	−0.878949	−	0.476916i	$-0.841754\pi$
$431$	−36.4949	−1.75790	−0.878949	+	0.476916i	$0.841754\pi$
$432$	0	0
$433$	−5.20204	−0.249994	−0.124997	−	0.992157i	$-0.539892\pi$
$433$	−5.20204	−0.249994	−0.124997	+	0.992157i	$0.539892\pi$
$434$	0	0
$435$	−4.55051	−0.218180
$436$	0	0
$437$	−7.34847	−0.351525
$438$	0	0
$439$	4.69694	0.224173	0.112086	−	0.993698i	$-0.464247\pi$
$439$	4.69694	0.224173	0.112086	+	0.993698i	$0.464247\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	0	0
$443$	−22.4495	−1.06661	−0.533304	−	0.845924i	$-0.679050\pi$
$443$	−22.4495	−1.06661	−0.533304	+	0.845924i	$0.679050\pi$
$444$	0	0
$445$	12.8990	0.611470
$446$	0	0
$447$	10.4495	0.494244
$448$	0	0
$449$	24.8434	1.17243	0.586215	−	0.810155i	$-0.300617\pi$
$449$	24.8434	1.17243	0.586215	+	0.810155i	$0.300617\pi$
$450$	0	0
$451$	−2.44949	−0.115342
$452$	0	0
$453$	−15.7980	−0.742253
$454$	0	0
$455$	2.44949	0.114834
$456$	0	0
$457$	−27.4949	−1.28616	−0.643079	−	0.765800i	$-0.722343\pi$
$457$	−27.4949	−1.28616	−0.643079	+	0.765800i	$0.722343\pi$
$458$	0	0
$459$	−1.44949	−0.0676564
$460$	0	0
$461$	−12.6969	−0.591355	−0.295678	−	0.955288i	$-0.595545\pi$
$461$	−12.6969	−0.591355	−0.295678	+	0.955288i	$0.595545\pi$
$462$	0	0
$463$	−7.34847	−0.341512	−0.170756	−	0.985313i	$-0.554621\pi$
$463$	−7.34847	−0.341512	−0.170756	+	0.985313i	$0.554621\pi$
$464$	0	0
$465$	7.89898	0.366306
$466$	0	0
$467$	2.14643	0.0993249	0.0496624	−	0.998766i	$-0.484185\pi$
$467$	2.14643	0.0993249	0.0496624	+	0.998766i	$0.484185\pi$
$468$	0	0
$469$	−10.7980	−0.498603
$470$	0	0
$471$	−11.0000	−0.506853
$472$	0	0
$473$	5.30306	0.243835
$474$	0	0
$475$	−7.34847	−0.337171
$476$	0	0
$477$	−10.3485	−0.473824
$478$	0	0
$479$	22.9444	1.04836	0.524178	−	0.851609i	$-0.324373\pi$
$479$	22.9444	1.04836	0.524178	+	0.851609i	$0.324373\pi$
$480$	0	0
$481$	0.247449	0.0112827
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	4.89898	0.222451
$486$	0	0
$487$	25.8434	1.17107	0.585537	−	0.810645i	$-0.300884\pi$
$487$	25.8434	1.17107	0.585537	+	0.810645i	$0.300884\pi$
$488$	0	0
$489$	17.5959	0.795715
$490$	0	0
$491$	−21.2474	−0.958884	−0.479442	−	0.877574i	$-0.659161\pi$
$491$	−21.2474	−0.958884	−0.479442	+	0.877574i	$0.659161\pi$
$492$	0	0
$493$	−6.59592	−0.297065
$494$	0	0
$495$	−0.449490	−0.0202031
$496$	0	0
$497$	−12.3485	−0.553905
$498$	0	0
$499$	26.7980	1.19964	0.599821	−	0.800134i	$-0.295239\pi$
$499$	26.7980	1.19964	0.599821	+	0.800134i	$0.295239\pi$
$500$	0	0
$501$	−9.55051	−0.426685
$502$	0	0
$503$	−16.1464	−0.719934	−0.359967	−	0.932965i	$-0.617212\pi$
$503$	−16.1464	−0.719934	−0.359967	+	0.932965i	$0.617212\pi$
$504$	0	0
$505$	1.65153	0.0734922
$506$	0	0
$507$	7.00000	0.310881
$508$	0	0
$509$	30.6969	1.36062	0.680309	−	0.732925i	$-0.261846\pi$
$509$	30.6969	1.36062	0.680309	+	0.732925i	$0.261846\pi$
$510$	0	0
$511$	7.34847	0.325077
$512$	0	0
$513$	7.34847	0.324443
$514$	0	0
$515$	6.89898	0.304005
$516$	0	0
$517$	−3.30306	−0.145268
$518$	0	0
$519$	9.79796	0.430083
$520$	0	0
$521$	14.2474	0.624192	0.312096	−	0.950051i	$-0.398969\pi$
$521$	14.2474	0.624192	0.312096	+	0.950051i	$0.398969\pi$
$522$	0	0
$523$	−10.2020	−0.446104	−0.223052	−	0.974807i	$-0.571602\pi$
$523$	−10.2020	−0.446104	−0.223052	+	0.974807i	$0.571602\pi$
$524$	0	0
$525$	−1.00000	−0.0436436
$526$	0	0
$527$	11.4495	0.498748
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−11.4495	−0.496866
$532$	0	0
$533$	13.3485	0.578186
$534$	0	0
$535$	−16.3485	−0.706806
$536$	0	0
$537$	−10.0000	−0.431532
$538$	0	0
$539$	−2.69694	−0.116165
$540$	0	0
$541$	32.4949	1.39706	0.698532	−	0.715578i	$-0.253837\pi$
$541$	32.4949	1.39706	0.698532	+	0.715578i	$0.253837\pi$
$542$	0	0
$543$	−4.89898	−0.210235
$544$	0	0
$545$	13.5505	0.580440
$546$	0	0
$547$	−21.7980	−0.932013	−0.466007	−	0.884781i	$-0.654308\pi$
$547$	−21.7980	−0.932013	−0.466007	+	0.884781i	$0.654308\pi$
$548$	0	0
$549$	−4.44949	−0.189900
$550$	0	0
$551$	33.4393	1.42456
$552$	0	0
$553$	5.79796	0.246554
$554$	0	0
$555$	−0.101021	−0.00428808
$556$	0	0
$557$	21.6515	0.917405	0.458702	−	0.888590i	$-0.348314\pi$
$557$	21.6515	0.917405	0.458702	+	0.888590i	$0.348314\pi$
$558$	0	0
$559$	−28.8990	−1.22230
$560$	0	0
$561$	−0.651531	−0.0275077
$562$	0	0
$563$	−32.3485	−1.36333	−0.681663	−	0.731667i	$-0.738743\pi$
$563$	−32.3485	−1.36333	−0.681663	+	0.731667i	$0.738743\pi$
$564$	0	0
$565$	−9.24745	−0.389043
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−20.6969	−0.867661	−0.433830	−	0.900995i	$-0.642838\pi$
$569$	−20.6969	−0.867661	−0.433830	+	0.900995i	$0.642838\pi$
$570$	0	0
$571$	−8.44949	−0.353600	−0.176800	−	0.984247i	$-0.556575\pi$
$571$	−8.44949	−0.353600	−0.176800	+	0.984247i	$0.556575\pi$
$572$	0	0
$573$	−17.1464	−0.716302
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	−13.7980	−0.574417	−0.287208	−	0.957868i	$-0.592727\pi$
$577$	−13.7980	−0.574417	−0.287208	+	0.957868i	$0.592727\pi$
$578$	0	0
$579$	−8.89898	−0.369829
$580$	0	0
$581$	−10.5505	−0.437709
$582$	0	0
$583$	−4.65153	−0.192647
$584$	0	0
$585$	2.44949	0.101274
$586$	0	0
$587$	−39.3939	−1.62596	−0.812980	−	0.582292i	$-0.802156\pi$
$587$	−39.3939	−1.62596	−0.812980	+	0.582292i	$0.802156\pi$
$588$	0	0
$589$	−58.0454	−2.39172
$590$	0	0
$591$	1.10102	0.0452899
$592$	0	0
$593$	0.247449	0.0101615	0.00508075	−	0.999987i	$-0.498383\pi$
$593$	0.247449	0.0101615	0.00508075	+	0.999987i	$0.498383\pi$
$594$	0	0
$595$	−1.44949	−0.0594233
$596$	0	0
$597$	4.69694	0.192233
$598$	0	0
$599$	24.4949	1.00083	0.500417	−	0.865784i	$-0.333180\pi$
$599$	24.4949	1.00083	0.500417	+	0.865784i	$0.333180\pi$
$600$	0	0
$601$	22.3939	0.913465	0.456733	−	0.889604i	$-0.349020\pi$
$601$	22.3939	0.913465	0.456733	+	0.889604i	$0.349020\pi$
$602$	0	0
$603$	−10.7980	−0.439727
$604$	0	0
$605$	10.7980	0.438999
$606$	0	0
$607$	−12.0454	−0.488908	−0.244454	−	0.969661i	$-0.578609\pi$
$607$	−12.0454	−0.488908	−0.244454	+	0.969661i	$0.578609\pi$
$608$	0	0
$609$	4.55051	0.184396
$610$	0	0
$611$	18.0000	0.728202
$612$	0	0
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	0	0
$615$	−5.44949	−0.219745
$616$	0	0
$617$	7.85357	0.316173	0.158086	−	0.987425i	$-0.449468\pi$
$617$	7.85357	0.316173	0.158086	+	0.987425i	$0.449468\pi$
$618$	0	0
$619$	−45.7980	−1.84078	−0.920388	−	0.391007i	$-0.872127\pi$
$619$	−45.7980	−1.84078	−0.920388	+	0.391007i	$0.872127\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−12.8990	−0.516787
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	3.30306	0.131912
$628$	0	0
$629$	−0.146428	−0.00583847
$630$	0	0
$631$	−8.04541	−0.320283	−0.160141	−	0.987094i	$-0.551195\pi$
$631$	−8.04541	−0.320283	−0.160141	+	0.987094i	$0.551195\pi$
$632$	0	0
$633$	−25.0000	−0.993661
$634$	0	0
$635$	−18.2474	−0.724128
$636$	0	0
$637$	14.6969	0.582314
$638$	0	0
$639$	−12.3485	−0.488498
$640$	0	0
$641$	0.449490	0.0177538	0.00887689	−	0.999961i	$-0.497174\pi$
$641$	0.449490	0.0177538	0.00887689	+	0.999961i	$0.497174\pi$
$642$	0	0
$643$	−25.4949	−1.00542	−0.502710	−	0.864455i	$-0.667664\pi$
$643$	−25.4949	−1.00542	−0.502710	+	0.864455i	$0.667664\pi$
$644$	0	0
$645$	11.7980	0.464544
$646$	0	0
$647$	25.3485	0.996551	0.498276	−	0.867019i	$-0.333967\pi$
$647$	25.3485	0.996551	0.498276	+	0.867019i	$0.333967\pi$
$648$	0	0
$649$	−5.14643	−0.202015
$650$	0	0
$651$	−7.89898	−0.309585
$652$	0	0
$653$	5.34847	0.209302	0.104651	−	0.994509i	$-0.466627\pi$
$653$	5.34847	0.209302	0.104651	+	0.994509i	$0.466627\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	7.34847	0.286691
$658$	0	0
$659$	−32.9444	−1.28333	−0.641666	−	0.766985i	$-0.721756\pi$
$659$	−32.9444	−1.28333	−0.641666	+	0.766985i	$0.721756\pi$
$660$	0	0
$661$	15.3031	0.595220	0.297610	−	0.954688i	$-0.403810\pi$
$661$	15.3031	0.595220	0.297610	+	0.954688i	$0.403810\pi$
$662$	0	0
$663$	3.55051	0.137890
$664$	0	0
$665$	7.34847	0.284961
$666$	0	0
$667$	−4.55051	−0.176196
$668$	0	0
$669$	−14.0000	−0.541271
$670$	0	0
$671$	−2.00000	−0.0772091
$672$	0	0
$673$	−32.4495	−1.25084	−0.625418	−	0.780290i	$-0.715072\pi$
$673$	−32.4495	−1.25084	−0.625418	+	0.780290i	$0.715072\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	6.34847	0.243991	0.121996	−	0.992531i	$-0.461071\pi$
$677$	6.34847	0.243991	0.121996	+	0.992531i	$0.461071\pi$
$678$	0	0
$679$	−4.89898	−0.188006
$680$	0	0
$681$	−21.7980	−0.835300
$682$	0	0
$683$	46.0454	1.76188	0.880939	−	0.473229i	$-0.156912\pi$
$683$	46.0454	1.76188	0.880939	+	0.473229i	$0.156912\pi$
$684$	0	0
$685$	10.6969	0.408709
$686$	0	0
$687$	−6.69694	−0.255504
$688$	0	0
$689$	25.3485	0.965700
$690$	0	0
$691$	−9.59592	−0.365046	−0.182523	−	0.983202i	$-0.558426\pi$
$691$	−9.59592	−0.365046	−0.182523	+	0.983202i	$0.558426\pi$
$692$	0	0
$693$	0.449490	0.0170747
$694$	0	0
$695$	−8.79796	−0.333726
$696$	0	0
$697$	−7.89898	−0.299195
$698$	0	0
$699$	15.5959	0.589892
$700$	0	0
$701$	−31.3485	−1.18402	−0.592008	−	0.805932i	$-0.701664\pi$
$701$	−31.3485	−1.18402	−0.592008	+	0.805932i	$0.701664\pi$
$702$	0	0
$703$	0.742346	0.0279981
$704$	0	0
$705$	−7.34847	−0.276759
$706$	0	0
$707$	−1.65153	−0.0621122
$708$	0	0
$709$	−18.6515	−0.700473	−0.350236	−	0.936661i	$-0.613899\pi$
$709$	−18.6515	−0.700473	−0.350236	+	0.936661i	$0.613899\pi$
$710$	0	0
$711$	5.79796	0.217440
$712$	0	0
$713$	7.89898	0.295819
$714$	0	0
$715$	1.10102	0.0411758
$716$	0	0
$717$	21.0454	0.785955
$718$	0	0
$719$	−4.34847	−0.162171	−0.0810853	−	0.996707i	$-0.525839\pi$
$719$	−4.34847	−0.162171	−0.0810853	+	0.996707i	$0.525839\pi$
$720$	0	0
$721$	−6.89898	−0.256931
$722$	0	0
$723$	30.0454	1.11740
$724$	0	0
$725$	−4.55051	−0.169002
$726$	0	0
$727$	−17.8990	−0.663836	−0.331918	−	0.943308i	$-0.607696\pi$
$727$	−17.8990	−0.663836	−0.331918	+	0.943308i	$0.607696\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	17.1010	0.632504
$732$	0	0
$733$	32.3939	1.19650	0.598248	−	0.801311i	$-0.295864\pi$
$733$	32.3939	1.19650	0.598248	+	0.801311i	$0.295864\pi$
$734$	0	0
$735$	−6.00000	−0.221313
$736$	0	0
$737$	−4.85357	−0.178784
$738$	0	0
$739$	46.5959	1.71406	0.857029	−	0.515268i	$-0.172308\pi$
$739$	46.5959	1.71406	0.857029	+	0.515268i	$0.172308\pi$
$740$	0	0
$741$	−18.0000	−0.661247
$742$	0	0
$743$	−53.7980	−1.97366	−0.986828	−	0.161774i	$-0.948278\pi$
$743$	−53.7980	−1.97366	−0.986828	+	0.161774i	$0.948278\pi$
$744$	0	0
$745$	10.4495	0.382840
$746$	0	0
$747$	−10.5505	−0.386023
$748$	0	0
$749$	16.3485	0.597360
$750$	0	0
$751$	10.8536	0.396052	0.198026	−	0.980197i	$-0.436547\pi$
$751$	10.8536	0.396052	0.198026	+	0.980197i	$0.436547\pi$
$752$	0	0
$753$	18.0000	0.655956
$754$	0	0
$755$	−15.7980	−0.574947
$756$	0	0
$757$	9.69694	0.352441	0.176221	−	0.984351i	$-0.443613\pi$
$757$	9.69694	0.352441	0.176221	+	0.984351i	$0.443613\pi$
$758$	0	0
$759$	−0.449490	−0.0163154
$760$	0	0
$761$	−42.1464	−1.52781	−0.763903	−	0.645331i	$-0.776720\pi$
$761$	−42.1464	−1.52781	−0.763903	+	0.645331i	$0.776720\pi$
$762$	0	0
$763$	−13.5505	−0.490561
$764$	0	0
$765$	−1.44949	−0.0524064
$766$	0	0
$767$	28.0454	1.01266
$768$	0	0
$769$	−0.853572	−0.0307806	−0.0153903	−	0.999882i	$-0.504899\pi$
$769$	−0.853572	−0.0307806	−0.0153903	+	0.999882i	$0.504899\pi$
$770$	0	0
$771$	4.24745	0.152968
$772$	0	0
$773$	1.10102	0.0396010	0.0198005	−	0.999804i	$-0.493697\pi$
$773$	1.10102	0.0396010	0.0198005	+	0.999804i	$0.493697\pi$
$774$	0	0
$775$	7.89898	0.283740
$776$	0	0
$777$	0.101021	0.00362409
$778$	0	0
$779$	40.0454	1.43478
$780$	0	0
$781$	−5.55051	−0.198613
$782$	0	0
$783$	4.55051	0.162622
$784$	0	0
$785$	−11.0000	−0.392607
$786$	0	0
$787$	21.2929	0.759008	0.379504	−	0.925190i	$-0.376095\pi$
$787$	21.2929	0.759008	0.379504	+	0.925190i	$0.376095\pi$
$788$	0	0
$789$	0.348469	0.0124058
$790$	0	0
$791$	9.24745	0.328801
$792$	0	0
$793$	10.8990	0.387034
$794$	0	0
$795$	−10.3485	−0.367022
$796$	0	0
$797$	32.5505	1.15300	0.576499	−	0.817098i	$-0.304418\pi$
$797$	32.5505	1.15300	0.576499	+	0.817098i	$0.304418\pi$
$798$	0	0
$799$	−10.6515	−0.376824
$800$	0	0
$801$	−12.8990	−0.455763
$802$	0	0
$803$	3.30306	0.116563
$804$	0	0
$805$	−1.00000	−0.0352454
$806$	0	0
$807$	−15.2474	−0.536736
$808$	0	0
$809$	41.4495	1.45729	0.728643	−	0.684893i	$-0.240151\pi$
$809$	41.4495	1.45729	0.728643	+	0.684893i	$0.240151\pi$
$810$	0	0
$811$	−13.2020	−0.463586	−0.231793	−	0.972765i	$-0.574459\pi$
$811$	−13.2020	−0.463586	−0.231793	+	0.972765i	$0.574459\pi$
$812$	0	0
$813$	−6.10102	−0.213972
$814$	0	0
$815$	17.5959	0.616358
$816$	0	0
$817$	−86.6969	−3.03314
$818$	0	0
$819$	−2.44949	−0.0855921
$820$	0	0
$821$	−4.49490	−0.156873	−0.0784365	−	0.996919i	$-0.524993\pi$
$821$	−4.49490	−0.156873	−0.0784365	+	0.996919i	$0.524993\pi$
$822$	0	0
$823$	−1.79796	−0.0626729	−0.0313365	−	0.999509i	$-0.509976\pi$
$823$	−1.79796	−0.0626729	−0.0313365	+	0.999509i	$0.509976\pi$
$824$	0	0
$825$	−0.449490	−0.0156492
$826$	0	0
$827$	−52.4393	−1.82349	−0.911746	−	0.410754i	$-0.865266\pi$
$827$	−52.4393	−1.82349	−0.911746	+	0.410754i	$0.865266\pi$
$828$	0	0
$829$	−31.6969	−1.10088	−0.550440	−	0.834875i	$-0.685540\pi$
$829$	−31.6969	−1.10088	−0.550440	+	0.834875i	$0.685540\pi$
$830$	0	0
$831$	0.202041	0.00700873
$832$	0	0
$833$	−8.69694	−0.301331
$834$	0	0
$835$	−9.55051	−0.330509
$836$	0	0
$837$	−7.89898	−0.273029
$838$	0	0
$839$	−2.00000	−0.0690477	−0.0345238	−	0.999404i	$-0.510991\pi$
$839$	−2.00000	−0.0690477	−0.0345238	+	0.999404i	$0.510991\pi$
$840$	0	0
$841$	−8.29286	−0.285961
$842$	0	0
$843$	2.65153	0.0913236
$844$	0	0
$845$	7.00000	0.240807
$846$	0	0
$847$	−10.7980	−0.371022
$848$	0	0
$849$	−3.89898	−0.133813
$850$	0	0
$851$	−0.101021	−0.00346294
$852$	0	0
$853$	−14.8990	−0.510131	−0.255066	−	0.966924i	$-0.582097\pi$
$853$	−14.8990	−0.510131	−0.255066	+	0.966924i	$0.582097\pi$
$854$	0	0
$855$	7.34847	0.251312
$856$	0	0
$857$	25.5959	0.874340	0.437170	−	0.899379i	$-0.355981\pi$
$857$	25.5959	0.874340	0.437170	+	0.899379i	$0.355981\pi$
$858$	0	0
$859$	16.5959	0.566245	0.283123	−	0.959084i	$-0.408630\pi$
$859$	16.5959	0.566245	0.283123	+	0.959084i	$0.408630\pi$
$860$	0	0
$861$	5.44949	0.185718
$862$	0	0
$863$	40.2929	1.37158	0.685792	−	0.727797i	$-0.259456\pi$
$863$	40.2929	1.37158	0.685792	+	0.727797i	$0.259456\pi$
$864$	0	0
$865$	9.79796	0.333141
$866$	0	0
$867$	14.8990	0.505996
$868$	0	0
$869$	2.60612	0.0884067
$870$	0	0
$871$	26.4495	0.896207
$872$	0	0
$873$	−4.89898	−0.165805
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−4.69694	−0.158604	−0.0793022	−	0.996851i	$-0.525269\pi$
$877$	−4.69694	−0.158604	−0.0793022	+	0.996851i	$0.525269\pi$
$878$	0	0
$879$	−27.0454	−0.912219
$880$	0	0
$881$	−44.2474	−1.49073	−0.745367	−	0.666654i	$-0.767726\pi$
$881$	−44.2474	−1.49073	−0.745367	+	0.666654i	$0.767726\pi$
$882$	0	0
$883$	−17.7526	−0.597421	−0.298710	−	0.954344i	$-0.596556\pi$
$883$	−17.7526	−0.597421	−0.298710	+	0.954344i	$0.596556\pi$
$884$	0	0
$885$	−11.4495	−0.384870
$886$	0	0
$887$	3.30306	0.110906	0.0554530	−	0.998461i	$-0.482340\pi$
$887$	3.30306	0.110906	0.0554530	+	0.998461i	$0.482340\pi$
$888$	0	0
$889$	18.2474	0.612000
$890$	0	0
$891$	0.449490	0.0150585
$892$	0	0
$893$	54.0000	1.80704
$894$	0	0
$895$	−10.0000	−0.334263
$896$	0	0
$897$	2.44949	0.0817861
$898$	0	0
$899$	−35.9444	−1.19881
$900$	0	0
$901$	−15.0000	−0.499722
$902$	0	0
$903$	−11.7980	−0.392611
$904$	0	0
$905$	−4.89898	−0.162848
$906$	0	0
$907$	−45.0000	−1.49420	−0.747100	−	0.664711i	$-0.768555\pi$
$907$	−45.0000	−1.49420	−0.747100	+	0.664711i	$0.768555\pi$
$908$	0	0
$909$	−1.65153	−0.0547778
$910$	0	0
$911$	43.5959	1.44440	0.722199	−	0.691686i	$-0.243132\pi$
$911$	43.5959	1.44440	0.722199	+	0.691686i	$0.243132\pi$
$912$	0	0
$913$	−4.74235	−0.156949
$914$	0	0
$915$	−4.44949	−0.147096
$916$	0	0
$917$	0	0
$918$	0	0
$919$	6.20204	0.204586	0.102293	−	0.994754i	$-0.467382\pi$
$919$	6.20204	0.204586	0.102293	+	0.994754i	$0.467382\pi$
$920$	0	0
$921$	20.2474	0.667176
$922$	0	0
$923$	30.2474	0.995607
$924$	0	0
$925$	−0.101021	−0.00332153
$926$	0	0
$927$	−6.89898	−0.226592
$928$	0	0
$929$	−34.8434	−1.14317	−0.571587	−	0.820542i	$-0.693672\pi$
$929$	−34.8434	−1.14317	−0.571587	+	0.820542i	$0.693672\pi$
$930$	0	0
$931$	44.0908	1.44502
$932$	0	0
$933$	15.5959	0.510587
$934$	0	0
$935$	−0.651531	−0.0213073
$936$	0	0
$937$	−36.4949	−1.19224	−0.596118	−	0.802897i	$-0.703291\pi$
$937$	−36.4949	−1.19224	−0.596118	+	0.802897i	$0.703291\pi$
$938$	0	0
$939$	4.30306	0.140425
$940$	0	0
$941$	−25.5505	−0.832923	−0.416461	−	0.909153i	$-0.636730\pi$
$941$	−25.5505	−0.832923	−0.416461	+	0.909153i	$0.636730\pi$
$942$	0	0
$943$	−5.44949	−0.177460
$944$	0	0
$945$	1.00000	0.0325300
$946$	0	0
$947$	−13.3031	−0.432291	−0.216146	−	0.976361i	$-0.569349\pi$
$947$	−13.3031	−0.432291	−0.216146	+	0.976361i	$0.569349\pi$
$948$	0	0
$949$	−18.0000	−0.584305
$950$	0	0
$951$	−27.3485	−0.886835
$952$	0	0
$953$	33.5959	1.08828	0.544139	−	0.838995i	$-0.316856\pi$
$953$	33.5959	1.08828	0.544139	+	0.838995i	$0.316856\pi$
$954$	0	0
$955$	−17.1464	−0.554845
$956$	0	0
$957$	2.04541	0.0661186
$958$	0	0
$959$	−10.6969	−0.345422
$960$	0	0
$961$	31.3939	1.01271
$962$	0	0
$963$	16.3485	0.526822
$964$	0	0
$965$	−8.89898	−0.286468
$966$	0	0
$967$	−21.3485	−0.686520	−0.343260	−	0.939240i	$-0.611531\pi$
$967$	−21.3485	−0.686520	−0.343260	+	0.939240i	$0.611531\pi$
$968$	0	0
$969$	10.6515	0.342176
$970$	0	0
$971$	−35.1010	−1.12645	−0.563223	−	0.826305i	$-0.690439\pi$
$971$	−35.1010	−1.12645	−0.563223	+	0.826305i	$0.690439\pi$
$972$	0	0
$973$	8.79796	0.282050
$974$	0	0
$975$	2.44949	0.0784465
$976$	0	0
$977$	−7.65153	−0.244794	−0.122397	−	0.992481i	$-0.539058\pi$
$977$	−7.65153	−0.244794	−0.122397	+	0.992481i	$0.539058\pi$
$978$	0	0
$979$	−5.79796	−0.185304
$980$	0	0
$981$	−13.5505	−0.432634
$982$	0	0
$983$	−26.1464	−0.833942	−0.416971	−	0.908920i	$-0.636908\pi$
$983$	−26.1464	−0.833942	−0.416971	+	0.908920i	$0.636908\pi$
$984$	0	0
$985$	1.10102	0.0350814
$986$	0	0
$987$	7.34847	0.233904
$988$	0	0
$989$	11.7980	0.375153
$990$	0	0
$991$	20.5959	0.654251	0.327125	−	0.944981i	$-0.393920\pi$
$991$	20.5959	0.654251	0.327125	+	0.944981i	$0.393920\pi$
$992$	0	0
$993$	−20.7980	−0.660003
$994$	0	0
$995$	4.69694	0.148903
$996$	0	0
$997$	−27.1010	−0.858298	−0.429149	−	0.903234i	$-0.641186\pi$
$997$	−27.1010	−0.858298	−0.429149	+	0.903234i	$0.641186\pi$
$998$	0	0
$999$	0.101021	0.00319615

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1380.2.a.f.1.2	✓	2
3.2	odd	2		4140.2.a.r.1.1		2
4.3	odd	2		5520.2.a.bl.1.1		2
5.2	odd	4		6900.2.f.h.6349.4		4
5.3	odd	4		6900.2.f.h.6349.2		4
5.4	even	2		6900.2.a.r.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.a.f.1.2	✓	2	1.1	even	1	trivial
4140.2.a.r.1.1		2	3.2	odd	2
5520.2.a.bl.1.1		2	4.3	odd	2
6900.2.a.r.1.2		2	5.4	even	2
6900.2.f.h.6349.2		4	5.3	odd	4
6900.2.f.h.6349.4		4	5.2	odd	4

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:	\( N \)	\(=\)	\( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1380.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +0.449490 q^{11} -2.44949 q^{13} +1.00000 q^{15} +1.44949 q^{17} -7.34847 q^{19} -1.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -4.55051 q^{29} +7.89898 q^{31} -0.449490 q^{33} -1.00000 q^{35} -0.101021 q^{37} +2.44949 q^{39} -5.44949 q^{41} +11.7980 q^{43} -1.00000 q^{45} -7.34847 q^{47} -6.00000 q^{49} -1.44949 q^{51} -10.3485 q^{53} -0.449490 q^{55} +7.34847 q^{57} -11.4495 q^{59} -4.44949 q^{61} +1.00000 q^{63} +2.44949 q^{65} -10.7980 q^{67} -1.00000 q^{69} -12.3485 q^{71} +7.34847 q^{73} -1.00000 q^{75} +0.449490 q^{77} +5.79796 q^{79} +1.00000 q^{81} -10.5505 q^{83} -1.44949 q^{85} +4.55051 q^{87} -12.8990 q^{89} -2.44949 q^{91} -7.89898 q^{93} +7.34847 q^{95} -4.89898 q^{97} +0.449490 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} - 4 q^{11} + 2 q^{15} - 2 q^{17} - 2 q^{21} + 2 q^{23} + 2 q^{25} - 2 q^{27} - 14 q^{29} + 6 q^{31} + 4 q^{33} - 2 q^{35} - 10 q^{37} - 6 q^{41} + 4 q^{43} - 2 q^{45} - 12 q^{49} + 2 q^{51} - 6 q^{53} + 4 q^{55} - 18 q^{59} - 4 q^{61} + 2 q^{63} - 2 q^{67} - 2 q^{69} - 10 q^{71} - 2 q^{75} - 4 q^{77} - 8 q^{79} + 2 q^{81} - 26 q^{83} + 2 q^{85} + 14 q^{87} - 16 q^{89} - 6 q^{93} - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 - 4 * q^11 + 2 * q^15 - 2 * q^17 - 2 * q^21 + 2 * q^23 + 2 * q^25 - 2 * q^27 - 14 * q^29 + 6 * q^31 + 4 * q^33 - 2 * q^35 - 10 * q^37 - 6 * q^41 + 4 * q^43 - 2 * q^45 - 12 * q^49 + 2 * q^51 - 6 * q^53 + 4 * q^55 - 18 * q^59 - 4 * q^61 + 2 * q^63 - 2 * q^67 - 2 * q^69 - 10 * q^71 - 2 * q^75 - 4 * q^77 - 8 * q^79 + 2 * q^81 - 26 * q^83 + 2 * q^85 + 14 * q^87 - 16 * q^89 - 6 * q^93 - 4 * q^99

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