LMFDB - Embedded newform 9408.2.a.dv.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9408,2,Mod(1,9408)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9408, 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("9408.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9408 = 2^{6} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9408.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:	$75.1232582216$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1176)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	9408.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -3.41421 q^{5} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.41421 q^{5} +1.00000 q^{9} -0.828427 q^{11} +4.24264 q^{13} -3.41421 q^{15} +7.41421 q^{17} -6.82843 q^{19} -4.82843 q^{23} +6.65685 q^{25} +1.00000 q^{27} -2.82843 q^{29} -2.82843 q^{31} -0.828427 q^{33} +1.65685 q^{37} +4.24264 q^{39} +10.2426 q^{41} +11.3137 q^{43} -3.41421 q^{45} -4.48528 q^{47} +7.41421 q^{51} +2.00000 q^{53} +2.82843 q^{55} -6.82843 q^{57} -8.48528 q^{59} -11.0711 q^{61} -14.4853 q^{65} -11.3137 q^{67} -4.82843 q^{69} -10.4853 q^{71} +7.75736 q^{73} +6.65685 q^{75} +13.6569 q^{79} +1.00000 q^{81} +4.00000 q^{83} -25.3137 q^{85} -2.82843 q^{87} +5.75736 q^{89} -2.82843 q^{93} +23.3137 q^{95} -0.242641 q^{97} -0.828427 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 4 q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 4 * q^5 + 2 * q^9	$ 2 q + 2 q^{3} - 4 q^{5} + 2 q^{9} + 4 q^{11} - 4 q^{15} + 12 q^{17} - 8 q^{19} - 4 q^{23} + 2 q^{25} + 2 q^{27} + 4 q^{33} - 8 q^{37} + 12 q^{41} - 4 q^{45} + 8 q^{47} + 12 q^{51} + 4 q^{53} - 8 q^{57} - 8 q^{61} - 12 q^{65} - 4 q^{69} - 4 q^{71} + 24 q^{73} + 2 q^{75} + 16 q^{79} + 2 q^{81} + 8 q^{83} - 28 q^{85} + 20 q^{89} + 24 q^{95} + 8 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 4 * q^5 + 2 * q^9 + 4 * q^11 - 4 * q^15 + 12 * q^17 - 8 * q^19 - 4 * q^23 + 2 * q^25 + 2 * q^27 + 4 * q^33 - 8 * q^37 + 12 * q^41 - 4 * q^45 + 8 * q^47 + 12 * q^51 + 4 * q^53 - 8 * q^57 - 8 * q^61 - 12 * q^65 - 4 * q^69 - 4 * q^71 + 24 * q^73 + 2 * q^75 + 16 * q^79 + 2 * q^81 + 8 * q^83 - 28 * q^85 + 20 * q^89 + 24 * q^95 + 8 * 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
$5$	−3.41421	−1.52688	−0.763441	−	0.645877i	$-0.776492\pi$
$5$	−3.41421	−1.52688	−0.763441	+	0.645877i	$0.776492\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.828427	−0.249780	−0.124890	−	0.992171i	$-0.539858\pi$
$11$	−0.828427	−0.249780	−0.124890	+	0.992171i	$0.539858\pi$
$12$	0	0
$13$	4.24264	1.17670	0.588348	−	0.808608i	$-0.299778\pi$
$13$	4.24264	1.17670	0.588348	+	0.808608i	$0.299778\pi$
$14$	0	0
$15$	−3.41421	−0.881546
$16$	0	0
$17$	7.41421	1.79821	0.899105	−	0.437732i	$-0.144218\pi$
$17$	7.41421	1.79821	0.899105	+	0.437732i	$0.144218\pi$
$18$	0	0
$19$	−6.82843	−1.56655	−0.783274	−	0.621676i	$-0.786452\pi$
$19$	−6.82843	−1.56655	−0.783274	+	0.621676i	$0.786452\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.82843	−1.00680	−0.503398	−	0.864054i	$-0.667917\pi$
$23$	−4.82843	−1.00680	−0.503398	+	0.864054i	$0.667917\pi$
$24$	0	0
$25$	6.65685	1.33137
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.82843	−0.525226	−0.262613	−	0.964901i	$-0.584584\pi$
$29$	−2.82843	−0.525226	−0.262613	+	0.964901i	$0.584584\pi$
$30$	0	0
$31$	−2.82843	−0.508001	−0.254000	−	0.967204i	$-0.581746\pi$
$31$	−2.82843	−0.508001	−0.254000	+	0.967204i	$0.581746\pi$
$32$	0	0
$33$	−0.828427	−0.144211
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.65685	0.272385	0.136193	−	0.990682i	$-0.456513\pi$
$37$	1.65685	0.272385	0.136193	+	0.990682i	$0.456513\pi$
$38$	0	0
$39$	4.24264	0.679366
$40$	0	0
$41$	10.2426	1.59963	0.799816	−	0.600245i	$-0.204930\pi$
$41$	10.2426	1.59963	0.799816	+	0.600245i	$0.204930\pi$
$42$	0	0
$43$	11.3137	1.72532	0.862662	−	0.505781i	$-0.168795\pi$
$43$	11.3137	1.72532	0.862662	+	0.505781i	$0.168795\pi$
$44$	0	0
$45$	−3.41421	−0.508961
$46$	0	0
$47$	−4.48528	−0.654246	−0.327123	−	0.944982i	$-0.606079\pi$
$47$	−4.48528	−0.654246	−0.327123	+	0.944982i	$0.606079\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	7.41421	1.03820
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	2.82843	0.381385
$56$	0	0
$57$	−6.82843	−0.904447
$58$	0	0
$59$	−8.48528	−1.10469	−0.552345	−	0.833616i	$-0.686267\pi$
$59$	−8.48528	−1.10469	−0.552345	+	0.833616i	$0.686267\pi$
$60$	0	0
$61$	−11.0711	−1.41750	−0.708752	−	0.705457i	$-0.750742\pi$
$61$	−11.0711	−1.41750	−0.708752	+	0.705457i	$0.750742\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−14.4853	−1.79668
$66$	0	0
$67$	−11.3137	−1.38219	−0.691095	−	0.722764i	$-0.742871\pi$
$67$	−11.3137	−1.38219	−0.691095	+	0.722764i	$0.742871\pi$
$68$	0	0
$69$	−4.82843	−0.581274
$70$	0	0
$71$	−10.4853	−1.24437	−0.622187	−	0.782869i	$-0.713756\pi$
$71$	−10.4853	−1.24437	−0.622187	+	0.782869i	$0.713756\pi$
$72$	0	0
$73$	7.75736	0.907930	0.453965	−	0.891019i	$-0.350009\pi$
$73$	7.75736	0.907930	0.453965	+	0.891019i	$0.350009\pi$
$74$	0	0
$75$	6.65685	0.768667
$76$	0	0
$77$	0	0
$78$	0	0
$79$	13.6569	1.53652	0.768258	−	0.640140i	$-0.221124\pi$
$79$	13.6569	1.53652	0.768258	+	0.640140i	$0.221124\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	−25.3137	−2.74566
$86$	0	0
$87$	−2.82843	−0.303239
$88$	0	0
$89$	5.75736	0.610279	0.305139	−	0.952308i	$-0.401297\pi$
$89$	5.75736	0.610279	0.305139	+	0.952308i	$0.401297\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−2.82843	−0.293294
$94$	0	0
$95$	23.3137	2.39194
$96$	0	0
$97$	−0.242641	−0.0246364	−0.0123182	−	0.999924i	$-0.503921\pi$
$97$	−0.242641	−0.0246364	−0.0123182	+	0.999924i	$0.503921\pi$
$98$	0	0
$99$	−0.828427	−0.0832601
$100$	0	0
$101$	10.7279	1.06747	0.533734	−	0.845652i	$-0.320788\pi$
$101$	10.7279	1.06747	0.533734	+	0.845652i	$0.320788\pi$
$102$	0	0
$103$	14.1421	1.39347	0.696733	−	0.717331i	$-0.254636\pi$
$103$	14.1421	1.39347	0.696733	+	0.717331i	$0.254636\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.8284	−1.24017	−0.620085	−	0.784534i	$-0.712902\pi$
$107$	−12.8284	−1.24017	−0.620085	+	0.784534i	$0.712902\pi$
$108$	0	0
$109$	3.31371	0.317396	0.158698	−	0.987327i	$-0.449270\pi$
$109$	3.31371	0.317396	0.158698	+	0.987327i	$0.449270\pi$
$110$	0	0
$111$	1.65685	0.157262
$112$	0	0
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	0	0
$115$	16.4853	1.53726
$116$	0	0
$117$	4.24264	0.392232
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.3137	−0.937610
$122$	0	0
$123$	10.2426	0.923548
$124$	0	0
$125$	−5.65685	−0.505964
$126$	0	0
$127$	20.0000	1.77471	0.887357	−	0.461084i	$-0.152539\pi$
$127$	20.0000	1.77471	0.887357	+	0.461084i	$0.152539\pi$
$128$	0	0
$129$	11.3137	0.996116
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−3.41421	−0.293849
$136$	0	0
$137$	−9.17157	−0.783580	−0.391790	−	0.920055i	$-0.628144\pi$
$137$	−9.17157	−0.783580	−0.391790	+	0.920055i	$0.628144\pi$
$138$	0	0
$139$	20.9706	1.77870	0.889350	−	0.457227i	$-0.151157\pi$
$139$	20.9706	1.77870	0.889350	+	0.457227i	$0.151157\pi$
$140$	0	0
$141$	−4.48528	−0.377729
$142$	0	0
$143$	−3.51472	−0.293916
$144$	0	0
$145$	9.65685	0.801958
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.31371	−0.107623	−0.0538116	−	0.998551i	$-0.517137\pi$
$149$	−1.31371	−0.107623	−0.0538116	+	0.998551i	$0.517137\pi$
$150$	0	0
$151$	−9.65685	−0.785864	−0.392932	−	0.919568i	$-0.628539\pi$
$151$	−9.65685	−0.785864	−0.392932	+	0.919568i	$0.628539\pi$
$152$	0	0
$153$	7.41421	0.599404
$154$	0	0
$155$	9.65685	0.775657
$156$	0	0
$157$	−0.242641	−0.0193648	−0.00968242	−	0.999953i	$-0.503082\pi$
$157$	−0.242641	−0.0193648	−0.00968242	+	0.999953i	$0.503082\pi$
$158$	0	0
$159$	2.00000	0.158610
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−5.65685	−0.443079	−0.221540	−	0.975151i	$-0.571108\pi$
$163$	−5.65685	−0.443079	−0.221540	+	0.975151i	$0.571108\pi$
$164$	0	0
$165$	2.82843	0.220193
$166$	0	0
$167$	−14.8284	−1.14746	−0.573729	−	0.819045i	$-0.694504\pi$
$167$	−14.8284	−1.14746	−0.573729	+	0.819045i	$0.694504\pi$
$168$	0	0
$169$	5.00000	0.384615
$170$	0	0
$171$	−6.82843	−0.522183
$172$	0	0
$173$	16.5858	1.26099	0.630497	−	0.776192i	$-0.282851\pi$
$173$	16.5858	1.26099	0.630497	+	0.776192i	$0.282851\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−8.48528	−0.637793
$178$	0	0
$179$	6.48528	0.484733	0.242366	−	0.970185i	$-0.422076\pi$
$179$	6.48528	0.484733	0.242366	+	0.970185i	$0.422076\pi$
$180$	0	0
$181$	7.07107	0.525588	0.262794	−	0.964852i	$-0.415356\pi$
$181$	7.07107	0.525588	0.262794	+	0.964852i	$0.415356\pi$
$182$	0	0
$183$	−11.0711	−0.818397
$184$	0	0
$185$	−5.65685	−0.415900
$186$	0	0
$187$	−6.14214	−0.449157
$188$	0	0
$189$	0	0
$190$	0	0
$191$	12.1421	0.878574	0.439287	−	0.898347i	$-0.355231\pi$
$191$	12.1421	0.878574	0.439287	+	0.898347i	$0.355231\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	0	0
$195$	−14.4853	−1.03731
$196$	0	0
$197$	2.68629	0.191390	0.0956952	−	0.995411i	$-0.469493\pi$
$197$	2.68629	0.191390	0.0956952	+	0.995411i	$0.469493\pi$
$198$	0	0
$199$	−5.65685	−0.401004	−0.200502	−	0.979693i	$-0.564257\pi$
$199$	−5.65685	−0.401004	−0.200502	+	0.979693i	$0.564257\pi$
$200$	0	0
$201$	−11.3137	−0.798007
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−34.9706	−2.44245
$206$	0	0
$207$	−4.82843	−0.335599
$208$	0	0
$209$	5.65685	0.391293
$210$	0	0
$211$	−1.65685	−0.114063	−0.0570313	−	0.998372i	$-0.518163\pi$
$211$	−1.65685	−0.114063	−0.0570313	+	0.998372i	$0.518163\pi$
$212$	0	0
$213$	−10.4853	−0.718440
$214$	0	0
$215$	−38.6274	−2.63437
$216$	0	0
$217$	0	0
$218$	0	0
$219$	7.75736	0.524194
$220$	0	0
$221$	31.4558	2.11595
$222$	0	0
$223$	13.6569	0.914531	0.457265	−	0.889330i	$-0.348829\pi$
$223$	13.6569	0.914531	0.457265	+	0.889330i	$0.348829\pi$
$224$	0	0
$225$	6.65685	0.443790
$226$	0	0
$227$	5.17157	0.343249	0.171625	−	0.985162i	$-0.445098\pi$
$227$	5.17157	0.343249	0.171625	+	0.985162i	$0.445098\pi$
$228$	0	0
$229$	25.4142	1.67942	0.839709	−	0.543036i	$-0.182725\pi$
$229$	25.4142	1.67942	0.839709	+	0.543036i	$0.182725\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−14.8284	−0.971443	−0.485721	−	0.874114i	$-0.661443\pi$
$233$	−14.8284	−0.971443	−0.485721	+	0.874114i	$0.661443\pi$
$234$	0	0
$235$	15.3137	0.998956
$236$	0	0
$237$	13.6569	0.887108
$238$	0	0
$239$	−12.8284	−0.829802	−0.414901	−	0.909867i	$-0.636184\pi$
$239$	−12.8284	−0.829802	−0.414901	+	0.909867i	$0.636184\pi$
$240$	0	0
$241$	13.8995	0.895345	0.447673	−	0.894198i	$-0.352253\pi$
$241$	13.8995	0.895345	0.447673	+	0.894198i	$0.352253\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−28.9706	−1.84335
$248$	0	0
$249$	4.00000	0.253490
$250$	0	0
$251$	6.14214	0.387688	0.193844	−	0.981032i	$-0.437904\pi$
$251$	6.14214	0.387688	0.193844	+	0.981032i	$0.437904\pi$
$252$	0	0
$253$	4.00000	0.251478
$254$	0	0
$255$	−25.3137	−1.58521
$256$	0	0
$257$	−3.41421	−0.212973	−0.106486	−	0.994314i	$-0.533960\pi$
$257$	−3.41421	−0.212973	−0.106486	+	0.994314i	$0.533960\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−2.82843	−0.175075
$262$	0	0
$263$	−20.1421	−1.24202	−0.621009	−	0.783804i	$-0.713277\pi$
$263$	−20.1421	−1.24202	−0.621009	+	0.783804i	$0.713277\pi$
$264$	0	0
$265$	−6.82843	−0.419467
$266$	0	0
$267$	5.75736	0.352345
$268$	0	0
$269$	0.100505	0.00612790	0.00306395	−	0.999995i	$-0.499025\pi$
$269$	0.100505	0.00612790	0.00306395	+	0.999995i	$0.499025\pi$
$270$	0	0
$271$	6.14214	0.373108	0.186554	−	0.982445i	$-0.440268\pi$
$271$	6.14214	0.373108	0.186554	+	0.982445i	$0.440268\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.51472	−0.332550
$276$	0	0
$277$	28.6274	1.72005	0.860027	−	0.510248i	$-0.170446\pi$
$277$	28.6274	1.72005	0.860027	+	0.510248i	$0.170446\pi$
$278$	0	0
$279$	−2.82843	−0.169334
$280$	0	0
$281$	1.17157	0.0698902	0.0349451	−	0.999389i	$-0.488874\pi$
$281$	1.17157	0.0698902	0.0349451	+	0.999389i	$0.488874\pi$
$282$	0	0
$283$	26.1421	1.55399	0.776994	−	0.629508i	$-0.216743\pi$
$283$	26.1421	1.55399	0.776994	+	0.629508i	$0.216743\pi$
$284$	0	0
$285$	23.3137	1.38098
$286$	0	0
$287$	0	0
$288$	0	0
$289$	37.9706	2.23356
$290$	0	0
$291$	−0.242641	−0.0142238
$292$	0	0
$293$	1.75736	0.102666	0.0513330	−	0.998682i	$-0.483653\pi$
$293$	1.75736	0.102666	0.0513330	+	0.998682i	$0.483653\pi$
$294$	0	0
$295$	28.9706	1.68673
$296$	0	0
$297$	−0.828427	−0.0480702
$298$	0	0
$299$	−20.4853	−1.18469
$300$	0	0
$301$	0	0
$302$	0	0
$303$	10.7279	0.616303
$304$	0	0
$305$	37.7990	2.16436
$306$	0	0
$307$	11.5147	0.657180	0.328590	−	0.944473i	$-0.393427\pi$
$307$	11.5147	0.657180	0.328590	+	0.944473i	$0.393427\pi$
$308$	0	0
$309$	14.1421	0.804518
$310$	0	0
$311$	23.7990	1.34952	0.674758	−	0.738039i	$-0.264248\pi$
$311$	23.7990	1.34952	0.674758	+	0.738039i	$0.264248\pi$
$312$	0	0
$313$	28.7279	1.62380	0.811899	−	0.583798i	$-0.198434\pi$
$313$	28.7279	1.62380	0.811899	+	0.583798i	$0.198434\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	22.0000	1.23564	0.617822	−	0.786318i	$-0.288015\pi$
$317$	22.0000	1.23564	0.617822	+	0.786318i	$0.288015\pi$
$318$	0	0
$319$	2.34315	0.131191
$320$	0	0
$321$	−12.8284	−0.716013
$322$	0	0
$323$	−50.6274	−2.81698
$324$	0	0
$325$	28.2426	1.56662
$326$	0	0
$327$	3.31371	0.183248
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−7.31371	−0.401998	−0.200999	−	0.979591i	$-0.564419\pi$
$331$	−7.31371	−0.401998	−0.200999	+	0.979591i	$0.564419\pi$
$332$	0	0
$333$	1.65685	0.0907951
$334$	0	0
$335$	38.6274	2.11044
$336$	0	0
$337$	10.3431	0.563427	0.281714	−	0.959499i	$-0.409097\pi$
$337$	10.3431	0.563427	0.281714	+	0.959499i	$0.409097\pi$
$338$	0	0
$339$	−10.0000	−0.543125
$340$	0	0
$341$	2.34315	0.126888
$342$	0	0
$343$	0	0
$344$	0	0
$345$	16.4853	0.887538
$346$	0	0
$347$	2.48528	0.133417	0.0667084	−	0.997773i	$-0.478750\pi$
$347$	2.48528	0.133417	0.0667084	+	0.997773i	$0.478750\pi$
$348$	0	0
$349$	−10.5858	−0.566644	−0.283322	−	0.959025i	$-0.591437\pi$
$349$	−10.5858	−0.566644	−0.283322	+	0.959025i	$0.591437\pi$
$350$	0	0
$351$	4.24264	0.226455
$352$	0	0
$353$	−27.6985	−1.47424	−0.737121	−	0.675761i	$-0.763815\pi$
$353$	−27.6985	−1.47424	−0.737121	+	0.675761i	$0.763815\pi$
$354$	0	0
$355$	35.7990	1.90001
$356$	0	0
$357$	0	0
$358$	0	0
$359$	28.8284	1.52151	0.760753	−	0.649041i	$-0.224830\pi$
$359$	28.8284	1.52151	0.760753	+	0.649041i	$0.224830\pi$
$360$	0	0
$361$	27.6274	1.45407
$362$	0	0
$363$	−10.3137	−0.541329
$364$	0	0
$365$	−26.4853	−1.38630
$366$	0	0
$367$	4.68629	0.244622	0.122311	−	0.992492i	$-0.460969\pi$
$367$	4.68629	0.244622	0.122311	+	0.992492i	$0.460969\pi$
$368$	0	0
$369$	10.2426	0.533211
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−5.31371	−0.275133	−0.137567	−	0.990493i	$-0.543928\pi$
$373$	−5.31371	−0.275133	−0.137567	+	0.990493i	$0.543928\pi$
$374$	0	0
$375$	−5.65685	−0.292119
$376$	0	0
$377$	−12.0000	−0.618031
$378$	0	0
$379$	23.3137	1.19754	0.598772	−	0.800919i	$-0.295655\pi$
$379$	23.3137	1.19754	0.598772	+	0.800919i	$0.295655\pi$
$380$	0	0
$381$	20.0000	1.02463
$382$	0	0
$383$	−8.97056	−0.458374	−0.229187	−	0.973382i	$-0.573607\pi$
$383$	−8.97056	−0.458374	−0.229187	+	0.973382i	$0.573607\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	11.3137	0.575108
$388$	0	0
$389$	35.7990	1.81508	0.907540	−	0.419965i	$-0.137958\pi$
$389$	35.7990	1.81508	0.907540	+	0.419965i	$0.137958\pi$
$390$	0	0
$391$	−35.7990	−1.81043
$392$	0	0
$393$	4.00000	0.201773
$394$	0	0
$395$	−46.6274	−2.34608
$396$	0	0
$397$	16.2426	0.815195	0.407597	−	0.913162i	$-0.366367\pi$
$397$	16.2426	0.815195	0.407597	+	0.913162i	$0.366367\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.17157	0.0585056	0.0292528	−	0.999572i	$-0.490687\pi$
$401$	1.17157	0.0585056	0.0292528	+	0.999572i	$0.490687\pi$
$402$	0	0
$403$	−12.0000	−0.597763
$404$	0	0
$405$	−3.41421	−0.169654
$406$	0	0
$407$	−1.37258	−0.0680364
$408$	0	0
$409$	−8.24264	−0.407572	−0.203786	−	0.979015i	$-0.565325\pi$
$409$	−8.24264	−0.407572	−0.203786	+	0.979015i	$0.565325\pi$
$410$	0	0
$411$	−9.17157	−0.452400
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−13.6569	−0.670389
$416$	0	0
$417$	20.9706	1.02693
$418$	0	0
$419$	−7.51472	−0.367118	−0.183559	−	0.983009i	$-0.558762\pi$
$419$	−7.51472	−0.367118	−0.183559	+	0.983009i	$0.558762\pi$
$420$	0	0
$421$	25.3137	1.23371	0.616857	−	0.787075i	$-0.288406\pi$
$421$	25.3137	1.23371	0.616857	+	0.787075i	$0.288406\pi$
$422$	0	0
$423$	−4.48528	−0.218082
$424$	0	0
$425$	49.3553	2.39409
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−3.51472	−0.169692
$430$	0	0
$431$	−8.14214	−0.392193	−0.196096	−	0.980585i	$-0.562827\pi$
$431$	−8.14214	−0.392193	−0.196096	+	0.980585i	$0.562827\pi$
$432$	0	0
$433$	0.928932	0.0446416	0.0223208	−	0.999751i	$-0.492894\pi$
$433$	0.928932	0.0446416	0.0223208	+	0.999751i	$0.492894\pi$
$434$	0	0
$435$	9.65685	0.463011
$436$	0	0
$437$	32.9706	1.57720
$438$	0	0
$439$	12.6863	0.605484	0.302742	−	0.953073i	$-0.402098\pi$
$439$	12.6863	0.605484	0.302742	+	0.953073i	$0.402098\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	11.1716	0.530777	0.265389	−	0.964141i	$-0.414500\pi$
$443$	11.1716	0.530777	0.265389	+	0.964141i	$0.414500\pi$
$444$	0	0
$445$	−19.6569	−0.931824
$446$	0	0
$447$	−1.31371	−0.0621363
$448$	0	0
$449$	−16.6274	−0.784696	−0.392348	−	0.919817i	$-0.628337\pi$
$449$	−16.6274	−0.784696	−0.392348	+	0.919817i	$0.628337\pi$
$450$	0	0
$451$	−8.48528	−0.399556
$452$	0	0
$453$	−9.65685	−0.453719
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−24.6274	−1.15202	−0.576011	−	0.817442i	$-0.695391\pi$
$457$	−24.6274	−1.15202	−0.576011	+	0.817442i	$0.695391\pi$
$458$	0	0
$459$	7.41421	0.346066
$460$	0	0
$461$	−16.3848	−0.763115	−0.381558	−	0.924345i	$-0.624612\pi$
$461$	−16.3848	−0.763115	−0.381558	+	0.924345i	$0.624612\pi$
$462$	0	0
$463$	1.65685	0.0770005	0.0385003	−	0.999259i	$-0.487742\pi$
$463$	1.65685	0.0770005	0.0385003	+	0.999259i	$0.487742\pi$
$464$	0	0
$465$	9.65685	0.447826
$466$	0	0
$467$	18.8284	0.871276	0.435638	−	0.900122i	$-0.356523\pi$
$467$	18.8284	0.871276	0.435638	+	0.900122i	$0.356523\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−0.242641	−0.0111803
$472$	0	0
$473$	−9.37258	−0.430952
$474$	0	0
$475$	−45.4558	−2.08566
$476$	0	0
$477$	2.00000	0.0915737
$478$	0	0
$479$	22.8284	1.04306	0.521529	−	0.853234i	$-0.325362\pi$
$479$	22.8284	1.04306	0.521529	+	0.853234i	$0.325362\pi$
$480$	0	0
$481$	7.02944	0.320515
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0.828427	0.0376169
$486$	0	0
$487$	4.97056	0.225238	0.112619	−	0.993638i	$-0.464076\pi$
$487$	4.97056	0.225238	0.112619	+	0.993638i	$0.464076\pi$
$488$	0	0
$489$	−5.65685	−0.255812
$490$	0	0
$491$	−19.1716	−0.865201	−0.432600	−	0.901586i	$-0.642404\pi$
$491$	−19.1716	−0.865201	−0.432600	+	0.901586i	$0.642404\pi$
$492$	0	0
$493$	−20.9706	−0.944467
$494$	0	0
$495$	2.82843	0.127128
$496$	0	0
$497$	0	0
$498$	0	0
$499$	24.9706	1.11784	0.558918	−	0.829223i	$-0.311217\pi$
$499$	24.9706	1.11784	0.558918	+	0.829223i	$0.311217\pi$
$500$	0	0
$501$	−14.8284	−0.662485
$502$	0	0
$503$	−19.3137	−0.861156	−0.430578	−	0.902553i	$-0.641690\pi$
$503$	−19.3137	−0.861156	−0.430578	+	0.902553i	$0.641690\pi$
$504$	0	0
$505$	−36.6274	−1.62990
$506$	0	0
$507$	5.00000	0.222058
$508$	0	0
$509$	36.3848	1.61273	0.806363	−	0.591420i	$-0.201433\pi$
$509$	36.3848	1.61273	0.806363	+	0.591420i	$0.201433\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−6.82843	−0.301482
$514$	0	0
$515$	−48.2843	−2.12766
$516$	0	0
$517$	3.71573	0.163418
$518$	0	0
$519$	16.5858	0.728035
$520$	0	0
$521$	14.2426	0.623981	0.311991	−	0.950085i	$-0.399004\pi$
$521$	14.2426	0.623981	0.311991	+	0.950085i	$0.399004\pi$
$522$	0	0
$523$	4.97056	0.217348	0.108674	−	0.994077i	$-0.465340\pi$
$523$	4.97056	0.217348	0.108674	+	0.994077i	$0.465340\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−20.9706	−0.913492
$528$	0	0
$529$	0.313708	0.0136395
$530$	0	0
$531$	−8.48528	−0.368230
$532$	0	0
$533$	43.4558	1.88228
$534$	0	0
$535$	43.7990	1.89360
$536$	0	0
$537$	6.48528	0.279861
$538$	0	0
$539$	0	0
$540$	0	0
$541$	18.0000	0.773880	0.386940	−	0.922105i	$-0.373532\pi$
$541$	18.0000	0.773880	0.386940	+	0.922105i	$0.373532\pi$
$542$	0	0
$543$	7.07107	0.303449
$544$	0	0
$545$	−11.3137	−0.484626
$546$	0	0
$547$	3.02944	0.129529	0.0647647	−	0.997901i	$-0.479370\pi$
$547$	3.02944	0.129529	0.0647647	+	0.997901i	$0.479370\pi$
$548$	0	0
$549$	−11.0711	−0.472502
$550$	0	0
$551$	19.3137	0.822792
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−5.65685	−0.240120
$556$	0	0
$557$	−32.6274	−1.38247	−0.691234	−	0.722631i	$-0.742933\pi$
$557$	−32.6274	−1.38247	−0.691234	+	0.722631i	$0.742933\pi$
$558$	0	0
$559$	48.0000	2.03018
$560$	0	0
$561$	−6.14214	−0.259321
$562$	0	0
$563$	−1.85786	−0.0782996	−0.0391498	−	0.999233i	$-0.512465\pi$
$563$	−1.85786	−0.0782996	−0.0391498	+	0.999233i	$0.512465\pi$
$564$	0	0
$565$	34.1421	1.43637
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−5.85786	−0.245574	−0.122787	−	0.992433i	$-0.539183\pi$
$569$	−5.85786	−0.245574	−0.122787	+	0.992433i	$0.539183\pi$
$570$	0	0
$571$	41.6569	1.74329	0.871643	−	0.490142i	$-0.163055\pi$
$571$	41.6569	1.74329	0.871643	+	0.490142i	$0.163055\pi$
$572$	0	0
$573$	12.1421	0.507245
$574$	0	0
$575$	−32.1421	−1.34042
$576$	0	0
$577$	13.2132	0.550073	0.275036	−	0.961434i	$-0.411310\pi$
$577$	13.2132	0.550073	0.275036	+	0.961434i	$0.411310\pi$
$578$	0	0
$579$	2.00000	0.0831172
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−1.65685	−0.0686199
$584$	0	0
$585$	−14.4853	−0.598893
$586$	0	0
$587$	35.7990	1.47758	0.738791	−	0.673934i	$-0.235397\pi$
$587$	35.7990	1.47758	0.738791	+	0.673934i	$0.235397\pi$
$588$	0	0
$589$	19.3137	0.795807
$590$	0	0
$591$	2.68629	0.110499
$592$	0	0
$593$	47.4142	1.94707	0.973534	−	0.228541i	$-0.0733956\pi$
$593$	47.4142	1.94707	0.973534	+	0.228541i	$0.0733956\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−5.65685	−0.231520
$598$	0	0
$599$	23.4558	0.958380	0.479190	−	0.877711i	$-0.340931\pi$
$599$	23.4558	0.958380	0.479190	+	0.877711i	$0.340931\pi$
$600$	0	0
$601$	−12.2426	−0.499388	−0.249694	−	0.968325i	$-0.580330\pi$
$601$	−12.2426	−0.499388	−0.249694	+	0.968325i	$0.580330\pi$
$602$	0	0
$603$	−11.3137	−0.460730
$604$	0	0
$605$	35.2132	1.43162
$606$	0	0
$607$	24.9706	1.01352	0.506762	−	0.862086i	$-0.330842\pi$
$607$	24.9706	1.01352	0.506762	+	0.862086i	$0.330842\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−19.0294	−0.769849
$612$	0	0
$613$	10.3431	0.417756	0.208878	−	0.977942i	$-0.433019\pi$
$613$	10.3431	0.417756	0.208878	+	0.977942i	$0.433019\pi$
$614$	0	0
$615$	−34.9706	−1.41015
$616$	0	0
$617$	−4.48528	−0.180571	−0.0902853	−	0.995916i	$-0.528778\pi$
$617$	−4.48528	−0.180571	−0.0902853	+	0.995916i	$0.528778\pi$
$618$	0	0
$619$	−33.6569	−1.35278	−0.676392	−	0.736542i	$-0.736457\pi$
$619$	−33.6569	−1.35278	−0.676392	+	0.736542i	$0.736457\pi$
$620$	0	0
$621$	−4.82843	−0.193758
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−13.9706	−0.558823
$626$	0	0
$627$	5.65685	0.225913
$628$	0	0
$629$	12.2843	0.489806
$630$	0	0
$631$	−3.02944	−0.120600	−0.0603000	−	0.998180i	$-0.519206\pi$
$631$	−3.02944	−0.120600	−0.0603000	+	0.998180i	$0.519206\pi$
$632$	0	0
$633$	−1.65685	−0.0658540
$634$	0	0
$635$	−68.2843	−2.70978
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−10.4853	−0.414791
$640$	0	0
$641$	35.1127	1.38687	0.693434	−	0.720520i	$-0.256097\pi$
$641$	35.1127	1.38687	0.693434	+	0.720520i	$0.256097\pi$
$642$	0	0
$643$	−31.7990	−1.25403	−0.627015	−	0.779007i	$-0.715723\pi$
$643$	−31.7990	−1.25403	−0.627015	+	0.779007i	$0.715723\pi$
$644$	0	0
$645$	−38.6274	−1.52095
$646$	0	0
$647$	−0.201010	−0.00790252	−0.00395126	−	0.999992i	$-0.501258\pi$
$647$	−0.201010	−0.00790252	−0.00395126	+	0.999992i	$0.501258\pi$
$648$	0	0
$649$	7.02944	0.275930
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−46.1421	−1.80568	−0.902841	−	0.429975i	$-0.858522\pi$
$653$	−46.1421	−1.80568	−0.902841	+	0.429975i	$0.858522\pi$
$654$	0	0
$655$	−13.6569	−0.533617
$656$	0	0
$657$	7.75736	0.302643
$658$	0	0
$659$	−21.5147	−0.838094	−0.419047	−	0.907964i	$-0.637636\pi$
$659$	−21.5147	−0.838094	−0.419047	+	0.907964i	$0.637636\pi$
$660$	0	0
$661$	−4.92893	−0.191713	−0.0958566	−	0.995395i	$-0.530559\pi$
$661$	−4.92893	−0.191713	−0.0958566	+	0.995395i	$0.530559\pi$
$662$	0	0
$663$	31.4558	1.22164
$664$	0	0
$665$	0	0
$666$	0	0
$667$	13.6569	0.528796
$668$	0	0
$669$	13.6569	0.528004
$670$	0	0
$671$	9.17157	0.354065
$672$	0	0
$673$	−23.3137	−0.898677	−0.449339	−	0.893361i	$-0.648340\pi$
$673$	−23.3137	−0.898677	−0.449339	+	0.893361i	$0.648340\pi$
$674$	0	0
$675$	6.65685	0.256222
$676$	0	0
$677$	−27.6985	−1.06454	−0.532270	−	0.846575i	$-0.678661\pi$
$677$	−27.6985	−1.06454	−0.532270	+	0.846575i	$0.678661\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	5.17157	0.198175
$682$	0	0
$683$	1.79899	0.0688364	0.0344182	−	0.999408i	$-0.489042\pi$
$683$	1.79899	0.0688364	0.0344182	+	0.999408i	$0.489042\pi$
$684$	0	0
$685$	31.3137	1.19644
$686$	0	0
$687$	25.4142	0.969613
$688$	0	0
$689$	8.48528	0.323263
$690$	0	0
$691$	−8.68629	−0.330442	−0.165221	−	0.986257i	$-0.552834\pi$
$691$	−8.68629	−0.330442	−0.165221	+	0.986257i	$0.552834\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−71.5980	−2.71587
$696$	0	0
$697$	75.9411	2.87648
$698$	0	0
$699$	−14.8284	−0.560863
$700$	0	0
$701$	6.14214	0.231985	0.115993	−	0.993250i	$-0.462995\pi$
$701$	6.14214	0.231985	0.115993	+	0.993250i	$0.462995\pi$
$702$	0	0
$703$	−11.3137	−0.426705
$704$	0	0
$705$	15.3137	0.576748
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−6.62742	−0.248898	−0.124449	−	0.992226i	$-0.539716\pi$
$709$	−6.62742	−0.248898	−0.124449	+	0.992226i	$0.539716\pi$
$710$	0	0
$711$	13.6569	0.512172
$712$	0	0
$713$	13.6569	0.511453
$714$	0	0
$715$	12.0000	0.448775
$716$	0	0
$717$	−12.8284	−0.479086
$718$	0	0
$719$	−6.62742	−0.247161	−0.123580	−	0.992335i	$-0.539438\pi$
$719$	−6.62742	−0.247161	−0.123580	+	0.992335i	$0.539438\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	13.8995	0.516928
$724$	0	0
$725$	−18.8284	−0.699270
$726$	0	0
$727$	−9.85786	−0.365608	−0.182804	−	0.983149i	$-0.558517\pi$
$727$	−9.85786	−0.365608	−0.182804	+	0.983149i	$0.558517\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	83.8823	3.10250
$732$	0	0
$733$	8.04163	0.297024	0.148512	−	0.988911i	$-0.452552\pi$
$733$	8.04163	0.297024	0.148512	+	0.988911i	$0.452552\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9.37258	0.345244
$738$	0	0
$739$	−32.9706	−1.21284	−0.606421	−	0.795144i	$-0.707395\pi$
$739$	−32.9706	−1.21284	−0.606421	+	0.795144i	$0.707395\pi$
$740$	0	0
$741$	−28.9706	−1.06426
$742$	0	0
$743$	−4.82843	−0.177138	−0.0885689	−	0.996070i	$-0.528229\pi$
$743$	−4.82843	−0.177138	−0.0885689	+	0.996070i	$0.528229\pi$
$744$	0	0
$745$	4.48528	0.164328
$746$	0	0
$747$	4.00000	0.146352
$748$	0	0
$749$	0	0
$750$	0	0
$751$	20.2843	0.740184	0.370092	−	0.928995i	$-0.379326\pi$
$751$	20.2843	0.740184	0.370092	+	0.928995i	$0.379326\pi$
$752$	0	0
$753$	6.14214	0.223832
$754$	0	0
$755$	32.9706	1.19992
$756$	0	0
$757$	−41.9411	−1.52438	−0.762188	−	0.647356i	$-0.775875\pi$
$757$	−41.9411	−1.52438	−0.762188	+	0.647356i	$0.775875\pi$
$758$	0	0
$759$	4.00000	0.145191
$760$	0	0
$761$	−31.8995	−1.15636	−0.578178	−	0.815911i	$-0.696236\pi$
$761$	−31.8995	−1.15636	−0.578178	+	0.815911i	$0.696236\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−25.3137	−0.915219
$766$	0	0
$767$	−36.0000	−1.29988
$768$	0	0
$769$	34.8701	1.25745	0.628723	−	0.777629i	$-0.283578\pi$
$769$	34.8701	1.25745	0.628723	+	0.777629i	$0.283578\pi$
$770$	0	0
$771$	−3.41421	−0.122960
$772$	0	0
$773$	−33.3553	−1.19971	−0.599854	−	0.800109i	$-0.704775\pi$
$773$	−33.3553	−1.19971	−0.599854	+	0.800109i	$0.704775\pi$
$774$	0	0
$775$	−18.8284	−0.676337
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−69.9411	−2.50590
$780$	0	0
$781$	8.68629	0.310820
$782$	0	0
$783$	−2.82843	−0.101080
$784$	0	0
$785$	0.828427	0.0295678
$786$	0	0
$787$	−15.3137	−0.545875	−0.272937	−	0.962032i	$-0.587995\pi$
$787$	−15.3137	−0.545875	−0.272937	+	0.962032i	$0.587995\pi$
$788$	0	0
$789$	−20.1421	−0.717079
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−46.9706	−1.66797
$794$	0	0
$795$	−6.82843	−0.242179
$796$	0	0
$797$	39.4142	1.39612	0.698062	−	0.716038i	$-0.254046\pi$
$797$	39.4142	1.39612	0.698062	+	0.716038i	$0.254046\pi$
$798$	0	0
$799$	−33.2548	−1.17647
$800$	0	0
$801$	5.75736	0.203426
$802$	0	0
$803$	−6.42641	−0.226783
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0.100505	0.00353795
$808$	0	0
$809$	−4.62742	−0.162691	−0.0813457	−	0.996686i	$-0.525922\pi$
$809$	−4.62742	−0.162691	−0.0813457	+	0.996686i	$0.525922\pi$
$810$	0	0
$811$	−25.6569	−0.900934	−0.450467	−	0.892793i	$-0.648743\pi$
$811$	−25.6569	−0.900934	−0.450467	+	0.892793i	$0.648743\pi$
$812$	0	0
$813$	6.14214	0.215414
$814$	0	0
$815$	19.3137	0.676530
$816$	0	0
$817$	−77.2548	−2.70280
$818$	0	0
$819$	0	0
$820$	0	0
$821$	10.6863	0.372954	0.186477	−	0.982459i	$-0.440293\pi$
$821$	10.6863	0.372954	0.186477	+	0.982459i	$0.440293\pi$
$822$	0	0
$823$	−24.9706	−0.870419	−0.435210	−	0.900329i	$-0.643326\pi$
$823$	−24.9706	−0.870419	−0.435210	+	0.900329i	$0.643326\pi$
$824$	0	0
$825$	−5.51472	−0.191998
$826$	0	0
$827$	−40.1421	−1.39588	−0.697939	−	0.716157i	$-0.745900\pi$
$827$	−40.1421	−1.39588	−0.697939	+	0.716157i	$0.745900\pi$
$828$	0	0
$829$	−34.3848	−1.19423	−0.597116	−	0.802155i	$-0.703687\pi$
$829$	−34.3848	−1.19423	−0.597116	+	0.802155i	$0.703687\pi$
$830$	0	0
$831$	28.6274	0.993074
$832$	0	0
$833$	0	0
$834$	0	0
$835$	50.6274	1.75203
$836$	0	0
$837$	−2.82843	−0.0977647
$838$	0	0
$839$	23.7990	0.821632	0.410816	−	0.911718i	$-0.365244\pi$
$839$	23.7990	0.821632	0.410816	+	0.911718i	$0.365244\pi$
$840$	0	0
$841$	−21.0000	−0.724138
$842$	0	0
$843$	1.17157	0.0403511
$844$	0	0
$845$	−17.0711	−0.587263
$846$	0	0
$847$	0	0
$848$	0	0
$849$	26.1421	0.897196
$850$	0	0
$851$	−8.00000	−0.274236
$852$	0	0
$853$	−4.92893	−0.168763	−0.0843817	−	0.996434i	$-0.526892\pi$
$853$	−4.92893	−0.168763	−0.0843817	+	0.996434i	$0.526892\pi$
$854$	0	0
$855$	23.3137	0.797312
$856$	0	0
$857$	−11.2132	−0.383036	−0.191518	−	0.981489i	$-0.561341\pi$
$857$	−11.2132	−0.383036	−0.191518	+	0.981489i	$0.561341\pi$
$858$	0	0
$859$	−2.54416	−0.0868055	−0.0434027	−	0.999058i	$-0.513820\pi$
$859$	−2.54416	−0.0868055	−0.0434027	+	0.999058i	$0.513820\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	54.0833	1.84102	0.920508	−	0.390724i	$-0.127775\pi$
$863$	54.0833	1.84102	0.920508	+	0.390724i	$0.127775\pi$
$864$	0	0
$865$	−56.6274	−1.92539
$866$	0	0
$867$	37.9706	1.28955
$868$	0	0
$869$	−11.3137	−0.383791
$870$	0	0
$871$	−48.0000	−1.62642
$872$	0	0
$873$	−0.242641	−0.00821214
$874$	0	0
$875$	0	0
$876$	0	0
$877$	28.2843	0.955092	0.477546	−	0.878607i	$-0.341526\pi$
$877$	28.2843	0.955092	0.477546	+	0.878607i	$0.341526\pi$
$878$	0	0
$879$	1.75736	0.0592743
$880$	0	0
$881$	30.0416	1.01213	0.506064	−	0.862496i	$-0.331100\pi$
$881$	30.0416	1.01213	0.506064	+	0.862496i	$0.331100\pi$
$882$	0	0
$883$	10.3431	0.348075	0.174037	−	0.984739i	$-0.444319\pi$
$883$	10.3431	0.348075	0.174037	+	0.984739i	$0.444319\pi$
$884$	0	0
$885$	28.9706	0.973835
$886$	0	0
$887$	−39.7990	−1.33632	−0.668160	−	0.744018i	$-0.732918\pi$
$887$	−39.7990	−1.33632	−0.668160	+	0.744018i	$0.732918\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−0.828427	−0.0277534
$892$	0	0
$893$	30.6274	1.02491
$894$	0	0
$895$	−22.1421	−0.740130
$896$	0	0
$897$	−20.4853	−0.683984
$898$	0	0
$899$	8.00000	0.266815
$900$	0	0
$901$	14.8284	0.494007
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−24.1421	−0.802512
$906$	0	0
$907$	−53.6569	−1.78165	−0.890823	−	0.454350i	$-0.849872\pi$
$907$	−53.6569	−1.78165	−0.890823	+	0.454350i	$0.849872\pi$
$908$	0	0
$909$	10.7279	0.355823
$910$	0	0
$911$	6.48528	0.214867	0.107433	−	0.994212i	$-0.465737\pi$
$911$	6.48528	0.214867	0.107433	+	0.994212i	$0.465737\pi$
$912$	0	0
$913$	−3.31371	−0.109668
$914$	0	0
$915$	37.7990	1.24960
$916$	0	0
$917$	0	0
$918$	0	0
$919$	31.3137	1.03294	0.516472	−	0.856304i	$-0.327245\pi$
$919$	31.3137	1.03294	0.516472	+	0.856304i	$0.327245\pi$
$920$	0	0
$921$	11.5147	0.379423
$922$	0	0
$923$	−44.4853	−1.46425
$924$	0	0
$925$	11.0294	0.362646
$926$	0	0
$927$	14.1421	0.464489
$928$	0	0
$929$	−15.8995	−0.521646	−0.260823	−	0.965387i	$-0.583994\pi$
$929$	−15.8995	−0.521646	−0.260823	+	0.965387i	$0.583994\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	23.7990	0.779144
$934$	0	0
$935$	20.9706	0.685811
$936$	0	0
$937$	51.3553	1.67771	0.838853	−	0.544358i	$-0.183227\pi$
$937$	51.3553	1.67771	0.838853	+	0.544358i	$0.183227\pi$
$938$	0	0
$939$	28.7279	0.937500
$940$	0	0
$941$	14.7279	0.480117	0.240058	−	0.970758i	$-0.422833\pi$
$941$	14.7279	0.480117	0.240058	+	0.970758i	$0.422833\pi$
$942$	0	0
$943$	−49.4558	−1.61050
$944$	0	0
$945$	0	0
$946$	0	0
$947$	58.7696	1.90975	0.954877	−	0.297002i	$-0.0959867\pi$
$947$	58.7696	1.90975	0.954877	+	0.297002i	$0.0959867\pi$
$948$	0	0
$949$	32.9117	1.06836
$950$	0	0
$951$	22.0000	0.713399
$952$	0	0
$953$	2.00000	0.0647864	0.0323932	−	0.999475i	$-0.489687\pi$
$953$	2.00000	0.0647864	0.0323932	+	0.999475i	$0.489687\pi$
$954$	0	0
$955$	−41.4558	−1.34148
$956$	0	0
$957$	2.34315	0.0757431
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−23.0000	−0.741935
$962$	0	0
$963$	−12.8284	−0.413390
$964$	0	0
$965$	−6.82843	−0.219815
$966$	0	0
$967$	−31.3137	−1.00698	−0.503490	−	0.864001i	$-0.667951\pi$
$967$	−31.3137	−1.00698	−0.503490	+	0.864001i	$0.667951\pi$
$968$	0	0
$969$	−50.6274	−1.62639
$970$	0	0
$971$	20.0000	0.641831	0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000	0.641831	0.320915	+	0.947108i	$0.396010\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	28.2426	0.904488
$976$	0	0
$977$	−7.79899	−0.249512	−0.124756	−	0.992187i	$-0.539815\pi$
$977$	−7.79899	−0.249512	−0.124756	+	0.992187i	$0.539815\pi$
$978$	0	0
$979$	−4.76955	−0.152436
$980$	0	0
$981$	3.31371	0.105799
$982$	0	0
$983$	1.37258	0.0437786	0.0218893	−	0.999760i	$-0.493032\pi$
$983$	1.37258	0.0437786	0.0218893	+	0.999760i	$0.493032\pi$
$984$	0	0
$985$	−9.17157	−0.292231
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−54.6274	−1.73705
$990$	0	0
$991$	−18.6274	−0.591719	−0.295860	−	0.955231i	$-0.595606\pi$
$991$	−18.6274	−0.591719	−0.295860	+	0.955231i	$0.595606\pi$
$992$	0	0
$993$	−7.31371	−0.232094
$994$	0	0
$995$	19.3137	0.612286
$996$	0	0
$997$	54.3848	1.72238	0.861192	−	0.508281i	$-0.169719\pi$
$997$	54.3848	1.72238	0.861192	+	0.508281i	$0.169719\pi$
$998$	0	0
$999$	1.65685	0.0524205

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9408.2.a.dv.1.1		2
4.3	odd	2		9408.2.a.dh.1.1		2
7.6	odd	2		9408.2.a.dr.1.2		2
8.3	odd	2		2352.2.a.bg.1.2		2
8.5	even	2		1176.2.a.l.1.2	✓	2
24.5	odd	2		3528.2.a.bc.1.1		2
24.11	even	2		7056.2.a.ce.1.1		2
28.27	even	2		9408.2.a.ed.1.2		2
56.3	even	6		2352.2.q.bg.961.2		4
56.5	odd	6		1176.2.q.m.361.2		4
56.11	odd	6		2352.2.q.ba.961.1		4
56.13	odd	2		1176.2.a.m.1.1	yes	2
56.19	even	6		2352.2.q.bg.1537.2		4
56.27	even	2		2352.2.a.z.1.1		2
56.37	even	6		1176.2.q.n.361.1		4
56.45	odd	6		1176.2.q.m.961.2		4
56.51	odd	6		2352.2.q.ba.1537.1		4
56.53	even	6		1176.2.q.n.961.1		4
168.5	even	6		3528.2.s.bc.361.1		4
168.53	odd	6		3528.2.s.bl.3313.2		4
168.83	odd	2		7056.2.a.cw.1.2		2
168.101	even	6		3528.2.s.bc.3313.1		4
168.125	even	2		3528.2.a.bm.1.2		2
168.149	odd	6		3528.2.s.bl.361.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1176.2.a.l.1.2	✓	2	8.5	even	2
1176.2.a.m.1.1	yes	2	56.13	odd	2
1176.2.q.m.361.2		4	56.5	odd	6
1176.2.q.m.961.2		4	56.45	odd	6
1176.2.q.n.361.1		4	56.37	even	6
1176.2.q.n.961.1		4	56.53	even	6
2352.2.a.z.1.1		2	56.27	even	2
2352.2.a.bg.1.2		2	8.3	odd	2
2352.2.q.ba.961.1		4	56.11	odd	6
2352.2.q.ba.1537.1		4	56.51	odd	6
2352.2.q.bg.961.2		4	56.3	even	6
2352.2.q.bg.1537.2		4	56.19	even	6
3528.2.a.bc.1.1		2	24.5	odd	2
3528.2.a.bm.1.2		2	168.125	even	2
3528.2.s.bc.361.1		4	168.5	even	6
3528.2.s.bc.3313.1		4	168.101	even	6
3528.2.s.bl.361.2		4	168.149	odd	6
3528.2.s.bl.3313.2		4	168.53	odd	6
7056.2.a.ce.1.1		2	24.11	even	2
7056.2.a.cw.1.2		2	168.83	odd	2
9408.2.a.dh.1.1		2	4.3	odd	2
9408.2.a.dr.1.2		2	7.6	odd	2
9408.2.a.dv.1.1		2	1.1	even	1	trivial
9408.2.a.ed.1.2		2	28.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 9408 = 2^{6} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9408.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.41421 q^{5} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.41421 q^{5} +1.00000 q^{9} -0.828427 q^{11} +4.24264 q^{13} -3.41421 q^{15} +7.41421 q^{17} -6.82843 q^{19} -4.82843 q^{23} +6.65685 q^{25} +1.00000 q^{27} -2.82843 q^{29} -2.82843 q^{31} -0.828427 q^{33} +1.65685 q^{37} +4.24264 q^{39} +10.2426 q^{41} +11.3137 q^{43} -3.41421 q^{45} -4.48528 q^{47} +7.41421 q^{51} +2.00000 q^{53} +2.82843 q^{55} -6.82843 q^{57} -8.48528 q^{59} -11.0711 q^{61} -14.4853 q^{65} -11.3137 q^{67} -4.82843 q^{69} -10.4853 q^{71} +7.75736 q^{73} +6.65685 q^{75} +13.6569 q^{79} +1.00000 q^{81} +4.00000 q^{83} -25.3137 q^{85} -2.82843 q^{87} +5.75736 q^{89} -2.82843 q^{93} +23.3137 q^{95} -0.242641 q^{97} -0.828427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 4 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 4 * q^5 + 2 * q^9	\( 2 q + 2 q^{3} - 4 q^{5} + 2 q^{9} + 4 q^{11} - 4 q^{15} + 12 q^{17} - 8 q^{19} - 4 q^{23} + 2 q^{25} + 2 q^{27} + 4 q^{33} - 8 q^{37} + 12 q^{41} - 4 q^{45} + 8 q^{47} + 12 q^{51} + 4 q^{53} - 8 q^{57} - 8 q^{61} - 12 q^{65} - 4 q^{69} - 4 q^{71} + 24 q^{73} + 2 q^{75} + 16 q^{79} + 2 q^{81} + 8 q^{83} - 28 q^{85} + 20 q^{89} + 24 q^{95} + 8 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 4 * q^5 + 2 * q^9 + 4 * q^11 - 4 * q^15 + 12 * q^17 - 8 * q^19 - 4 * q^23 + 2 * q^25 + 2 * q^27 + 4 * q^33 - 8 * q^37 + 12 * q^41 - 4 * q^45 + 8 * q^47 + 12 * q^51 + 4 * q^53 - 8 * q^57 - 8 * q^61 - 12 * q^65 - 4 * q^69 - 4 * q^71 + 24 * q^73 + 2 * q^75 + 16 * q^79 + 2 * q^81 + 8 * q^83 - 28 * q^85 + 20 * q^89 + 24 * q^95 + 8 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more