LMFDB - Embedded newform 448.2.f.d.447.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,2,Mod(447,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

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

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

chi := DirichletCharacter("448.447");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	448.f (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$3.57729801055$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	no (minimal twist has level 224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			447.5
Root			$0.382683 + 0.923880i$ of defining polynomial
Character	$\chi$	$=$	448.447
Dual form			448.2.f.d.447.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.08239 q^{3} -2.61313i q^{5} +(-2.61313 - 0.414214i) q^{7} -1.82843 q^{9} +O(q^{10})$	$q+1.08239 q^{3} -2.61313i q^{5} +(-2.61313 - 0.414214i) q^{7} -1.82843 q^{9} -2.00000i q^{11} -4.77791i q^{13} -2.82843i q^{15} -3.06147i q^{17} +4.14386 q^{19} +(-2.82843 - 0.448342i) q^{21} +7.65685i q^{23} -1.82843 q^{25} -5.22625 q^{27} -3.65685 q^{29} +3.06147 q^{31} -2.16478i q^{33} +(-1.08239 + 6.82843i) q^{35} +7.65685 q^{37} -5.17157i q^{39} -9.55582i q^{41} +3.65685i q^{43} +4.77791i q^{45} +7.39104 q^{47} +(6.65685 + 2.16478i) q^{49} -3.31371i q^{51} +2.00000 q^{53} -5.22625 q^{55} +4.48528 q^{57} -8.47343 q^{59} +2.61313i q^{61} +(4.77791 + 0.757359i) q^{63} -12.4853 q^{65} -15.6569i q^{67} +8.28772i q^{69} -8.82843i q^{71} +12.6173i q^{73} -1.97908 q^{75} +(-0.828427 + 5.22625i) q^{77} +12.8284i q^{79} -0.171573 q^{81} +11.5349 q^{83} -8.00000 q^{85} -3.95815 q^{87} +2.16478i q^{89} +(-1.97908 + 12.4853i) q^{91} +3.31371 q^{93} -10.8284i q^{95} +13.5140i q^{97} +3.65685i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^9	$ 8 q + 8 q^{9} + 8 q^{25} + 16 q^{29} + 16 q^{37} + 8 q^{49} + 16 q^{53} - 32 q^{57} - 32 q^{65} + 16 q^{77} - 24 q^{81} - 64 q^{85} - 64 q^{93}+O(q^{100}) $ 8 * q + 8 * q^9 + 8 * q^25 + 16 * q^29 + 16 * q^37 + 8 * q^49 + 16 * q^53 - 32 * q^57 - 32 * q^65 + 16 * q^77 - 24 * q^81 - 64 * q^85 - 64 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$-1$	$-1$	$1$

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
$3$	1.08239			0.624919			0.312460	−	0.949931i	$-0.398847\pi$
$3$	1.08239			0.624919			0.312460	+	0.949931i	$0.398847\pi$
$4$		0			0
$5$		−	2.61313i		−	1.16863i	−0.811529	−	0.584313i	$-0.801364\pi$
$5$		−	2.61313i		−	1.16863i	0.811529	−	0.584313i	$-0.198636\pi$
$6$		0			0
$7$	−2.61313	−	0.414214i	−0.987669	−	0.156558i
$7$	−2.61313	−	0.414214i	−0.987669	−	0.156558i
$8$		0			0
$9$	−1.82843			−0.609476
$10$		0			0
$11$		−	2.00000i		−	0.603023i	−0.953463	−	0.301511i	$-0.902509\pi$
$11$		−	2.00000i		−	0.603023i	0.953463	−	0.301511i	$-0.0974911\pi$
$12$		0			0
$13$		−	4.77791i		−	1.32515i	−0.748994	−	0.662577i	$-0.769463\pi$
$13$		−	4.77791i		−	1.32515i	0.748994	−	0.662577i	$-0.230537\pi$
$14$		0			0
$15$		−	2.82843i		−	0.730297i
$16$		0			0
$17$		−	3.06147i		−	0.742515i	−0.928530	−	0.371257i	$-0.878927\pi$
$17$		−	3.06147i		−	0.742515i	0.928530	−	0.371257i	$-0.121073\pi$
$18$		0			0
$19$	4.14386			0.950667			0.475333	−	0.879806i	$-0.342327\pi$
$19$	4.14386			0.950667			0.475333	+	0.879806i	$0.342327\pi$
$20$		0			0
$21$	−2.82843	−	0.448342i	−0.617213	−	0.0978361i
$22$		0			0
$23$			7.65685i			1.59656i	0.602284	+	0.798282i	$0.294258\pi$
$23$			7.65685i			1.59656i	−0.602284	+	0.798282i	$0.705742\pi$
$24$		0			0
$25$	−1.82843			−0.365685
$26$		0			0
$27$	−5.22625			−1.00579
$28$		0			0
$29$	−3.65685			−0.679061			−0.339530	−	0.940595i	$-0.610268\pi$
$29$	−3.65685			−0.679061			−0.339530	+	0.940595i	$0.610268\pi$
$30$		0			0
$31$	3.06147			0.549856			0.274928	−	0.961465i	$-0.411346\pi$
$31$	3.06147			0.549856			0.274928	+	0.961465i	$0.411346\pi$
$32$		0			0
$33$		−	2.16478i		−	0.376841i
$34$		0			0
$35$	−1.08239	+	6.82843i	−0.182958	+	1.15421i
$36$		0			0
$37$	7.65685			1.25878			0.629390	−	0.777090i	$-0.283305\pi$
$37$	7.65685			1.25878			0.629390	+	0.777090i	$0.283305\pi$
$38$		0			0
$39$		−	5.17157i		−	0.828114i
$40$		0			0
$41$		−	9.55582i		−	1.49237i	−0.665740	−	0.746184i	$-0.731884\pi$
$41$		−	9.55582i		−	1.49237i	0.665740	−	0.746184i	$-0.268116\pi$
$42$		0			0
$43$			3.65685i			0.557665i	0.960340	+	0.278833i	$0.0899474\pi$
$43$			3.65685i			0.557665i	−0.960340	+	0.278833i	$0.910053\pi$
$44$		0			0
$45$			4.77791i			0.712249i
$46$		0			0
$47$	7.39104			1.07809			0.539047	−	0.842276i	$-0.318785\pi$
$47$	7.39104			1.07809			0.539047	+	0.842276i	$0.318785\pi$
$48$		0			0
$49$	6.65685	+	2.16478i	0.950979	+	0.309255i
$50$		0			0
$51$		−	3.31371i		−	0.464012i
$52$		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
$55$	−5.22625			−0.704708
$56$		0			0
$57$	4.48528			0.594090
$58$		0			0
$59$	−8.47343			−1.10315			−0.551573	−	0.834126i	$-0.685972\pi$
$59$	−8.47343			−1.10315			−0.551573	+	0.834126i	$0.685972\pi$
$60$		0			0
$61$			2.61313i			0.334576i	0.985908	+	0.167288i	$0.0535011\pi$
$61$			2.61313i			0.334576i	−0.985908	+	0.167288i	$0.946499\pi$
$62$		0			0
$63$	4.77791	+	0.757359i	0.601960	+	0.0954183i
$64$		0			0
$65$	−12.4853			−1.54861
$66$		0			0
$67$		−	15.6569i		−	1.91279i	−0.292078	−	0.956395i	$-0.594347\pi$
$67$		−	15.6569i		−	1.91279i	0.292078	−	0.956395i	$-0.405653\pi$
$68$		0			0
$69$			8.28772i			0.997724i
$70$		0			0
$71$		−	8.82843i		−	1.04774i	−0.851798	−	0.523871i	$-0.824487\pi$
$71$		−	8.82843i		−	1.04774i	0.851798	−	0.523871i	$-0.175513\pi$
$72$		0			0
$73$			12.6173i			1.47674i	0.674395	+	0.738371i	$0.264405\pi$
$73$			12.6173i			1.47674i	−0.674395	+	0.738371i	$0.735595\pi$
$74$		0			0
$75$	−1.97908			−0.228524
$76$		0			0
$77$	−0.828427	+	5.22625i	−0.0944080	+	0.595587i
$78$		0			0
$79$			12.8284i			1.44331i	0.692252	+	0.721655i	$0.256618\pi$
$79$			12.8284i			1.44331i	−0.692252	+	0.721655i	$0.743382\pi$
$80$		0			0
$81$	−0.171573			−0.0190637
$82$		0			0
$83$	11.5349			1.26612			0.633060	−	0.774103i	$-0.281799\pi$
$83$	11.5349			1.26612			0.633060	+	0.774103i	$0.281799\pi$
$84$		0			0
$85$	−8.00000			−0.867722
$86$		0			0
$87$	−3.95815			−0.424358
$88$		0			0
$89$			2.16478i			0.229467i	0.993396	+	0.114733i	$0.0366014\pi$
$89$			2.16478i			0.229467i	−0.993396	+	0.114733i	$0.963399\pi$
$90$		0			0
$91$	−1.97908	+	12.4853i	−0.207463	+	1.30881i
$92$		0			0
$93$	3.31371			0.343616
$94$		0			0
$95$		−	10.8284i		−	1.11097i
$96$		0			0
$97$			13.5140i			1.37214i	0.727537	+	0.686068i	$0.240665\pi$
$97$			13.5140i			1.37214i	−0.727537	+	0.686068i	$0.759335\pi$
$98$		0			0
$99$			3.65685i			0.367528i
$100$		0			0
$101$		−	5.67459i		−	0.564643i	−0.959320	−	0.282322i	$-0.908895\pi$
$101$		−	5.67459i		−	0.564643i	0.959320	−	0.282322i	$-0.0911045\pi$
$102$		0			0
$103$	−16.9469			−1.66982			−0.834912	−	0.550384i	$-0.814481\pi$
$103$	−16.9469			−1.66982			−0.834912	+	0.550384i	$0.814481\pi$
$104$		0			0
$105$	−1.17157	+	7.39104i	−0.114334	+	0.721291i
$106$		0			0
$107$			0.343146i			0.0331732i	0.999862	+	0.0165866i	$0.00527991\pi$
$107$			0.343146i			0.0331732i	−0.999862	+	0.0165866i	$0.994720\pi$
$108$		0			0
$109$	5.31371			0.508961			0.254480	−	0.967078i	$-0.418096\pi$
$109$	5.31371			0.508961			0.254480	+	0.967078i	$0.418096\pi$
$110$		0			0
$111$	8.28772			0.786636
$112$		0			0
$113$	8.82843			0.830509			0.415254	−	0.909705i	$-0.363693\pi$
$113$	8.82843			0.830509			0.415254	+	0.909705i	$0.363693\pi$
$114$		0			0
$115$	20.0083			1.86579
$116$		0			0
$117$			8.73606i			0.807649i
$118$		0			0
$119$	−1.26810	+	8.00000i	−0.116247	+	0.733359i
$120$		0			0
$121$	7.00000			0.636364
$122$		0			0
$123$		−	10.3431i		−	0.932610i
$124$		0			0
$125$		−	8.28772i		−	0.741276i
$126$		0			0
$127$		−	3.65685i		−	0.324493i	−0.986750	−	0.162247i	$-0.948126\pi$
$127$		−	3.65685i		−	0.324493i	0.986750	−	0.162247i	$-0.0518740\pi$
$128$		0			0
$129$			3.95815i			0.348496i
$130$		0			0
$131$	−13.6997			−1.19695			−0.598473	−	0.801143i	$-0.704226\pi$
$131$	−13.6997			−1.19695			−0.598473	+	0.801143i	$0.704226\pi$
$132$		0			0
$133$	−10.8284	−	1.71644i	−0.938944	−	0.148834i
$134$		0			0
$135$			13.6569i			1.17539i
$136$		0			0
$137$	2.00000			0.170872			0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000			0.170872			0.0854358	+	0.996344i	$0.472772\pi$
$138$		0			0
$139$	1.97908			0.167863			0.0839315	−	0.996472i	$-0.473252\pi$
$139$	1.97908			0.167863			0.0839315	+	0.996472i	$0.473252\pi$
$140$		0			0
$141$	8.00000			0.673722
$142$		0			0
$143$	−9.55582			−0.799098
$144$		0			0
$145$			9.55582i			0.793568i
$146$		0			0
$147$	7.20533	+	2.34315i	0.594285	+	0.193259i
$148$		0			0
$149$	10.0000			0.819232			0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000			0.819232			0.409616	+	0.912258i	$0.365663\pi$
$150$		0			0
$151$		−	8.34315i		−	0.678956i	−0.940614	−	0.339478i	$-0.889750\pi$
$151$		−	8.34315i		−	0.678956i	0.940614	−	0.339478i	$-0.110250\pi$
$152$		0			0
$153$			5.59767i			0.452545i
$154$		0			0
$155$		−	8.00000i		−	0.642575i
$156$		0			0
$157$			5.67459i			0.452882i	0.974025	+	0.226441i	$0.0727090\pi$
$157$			5.67459i			0.452882i	−0.974025	+	0.226441i	$0.927291\pi$
$158$		0			0
$159$	2.16478			0.171679
$160$		0			0
$161$	3.17157	−	20.0083i	0.249955	−	1.57688i
$162$		0			0
$163$			17.3137i			1.35611i	0.735009	+	0.678057i	$0.237178\pi$
$163$			17.3137i			1.35611i	−0.735009	+	0.678057i	$0.762822\pi$
$164$		0			0
$165$	−5.65685			−0.440386
$166$		0			0
$167$	6.49435			0.502548			0.251274	−	0.967916i	$-0.419150\pi$
$167$	6.49435			0.502548			0.251274	+	0.967916i	$0.419150\pi$
$168$		0			0
$169$	−9.82843			−0.756033
$170$		0			0
$171$	−7.57675			−0.579408
$172$		0			0
$173$		−	10.9008i		−	0.828776i	−0.910100	−	0.414388i	$-0.863996\pi$
$173$		−	10.9008i		−	0.828776i	0.910100	−	0.414388i	$-0.136004\pi$
$174$		0			0
$175$	4.77791	+	0.757359i	0.361176	+	0.0572510i
$176$		0			0
$177$	−9.17157			−0.689378
$178$		0			0
$179$		−	2.00000i		−	0.149487i	−0.997203	−	0.0747435i	$-0.976186\pi$
$179$		−	2.00000i		−	0.149487i	0.997203	−	0.0747435i	$-0.0238138\pi$
$180$		0			0
$181$			1.71644i			0.127582i	0.997963	+	0.0637911i	$0.0203191\pi$
$181$			1.71644i			0.127582i	−0.997963	+	0.0637911i	$0.979681\pi$
$182$		0			0
$183$			2.82843i			0.209083i
$184$		0			0
$185$		−	20.0083i		−	1.47104i
$186$		0			0
$187$	−6.12293			−0.447753
$188$		0			0
$189$	13.6569	+	2.16478i	0.993390	+	0.157465i
$190$		0			0
$191$			4.82843i			0.349373i	0.984624	+	0.174686i	$0.0558911\pi$
$191$			4.82843i			0.349373i	−0.984624	+	0.174686i	$0.944109\pi$
$192$		0			0
$193$	−10.4853			−0.754747			−0.377374	−	0.926061i	$-0.623173\pi$
$193$	−10.4853			−0.754747			−0.377374	+	0.926061i	$0.623173\pi$
$194$		0			0
$195$	−13.5140			−0.967756
$196$		0			0
$197$	−3.65685			−0.260540			−0.130270	−	0.991479i	$-0.541584\pi$
$197$	−3.65685			−0.260540			−0.130270	+	0.991479i	$0.541584\pi$
$198$		0			0
$199$	12.6173			0.894416			0.447208	−	0.894430i	$-0.352418\pi$
$199$	12.6173			0.894416			0.447208	+	0.894430i	$0.352418\pi$
$200$		0			0
$201$		−	16.9469i		−	1.19534i
$202$		0			0
$203$	9.55582	+	1.51472i	0.670687	+	0.106312i
$204$		0			0
$205$	−24.9706			−1.74402
$206$		0			0
$207$		−	14.0000i		−	0.973067i
$208$		0			0
$209$		−	8.28772i		−	0.573274i
$210$		0			0
$211$		−	10.9706i		−	0.755245i	−0.925960	−	0.377622i	$-0.876742\pi$
$211$		−	10.9706i		−	0.755245i	0.925960	−	0.377622i	$-0.123258\pi$
$212$		0			0
$213$		−	9.55582i		−	0.654754i
$214$		0			0
$215$	9.55582			0.651702
$216$		0			0
$217$	−8.00000	−	1.26810i	−0.543075	−	0.0860843i
$218$		0			0
$219$			13.6569i			0.922845i
$220$		0			0
$221$	−14.6274			−0.983947
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$	3.34315			0.222876
$226$		0			0
$227$	−18.5545			−1.23151			−0.615753	−	0.787939i	$-0.711148\pi$
$227$	−18.5545			−1.23151			−0.615753	+	0.787939i	$0.711148\pi$
$228$		0			0
$229$			2.98454i			0.197224i	0.995126	+	0.0986121i	$0.0314403\pi$
$229$			2.98454i			0.197224i	−0.995126	+	0.0986121i	$0.968560\pi$
$230$		0			0
$231$	−0.896683	+	5.65685i	−0.0589974	+	0.372194i
$232$		0			0
$233$	18.0000			1.17922			0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000			1.17922			0.589610	+	0.807688i	$0.299282\pi$
$234$		0			0
$235$		−	19.3137i		−	1.25989i
$236$		0			0
$237$			13.8854i			0.901953i
$238$		0			0
$239$		−	8.34315i		−	0.539673i	−0.962906	−	0.269837i	$-0.913030\pi$
$239$		−	8.34315i		−	0.539673i	0.962906	−	0.269837i	$-0.0869697\pi$
$240$		0			0
$241$		−	23.9665i		−	1.54382i	−0.635734	−	0.771908i	$-0.719302\pi$
$241$		−	23.9665i		−	1.54382i	0.635734	−	0.771908i	$-0.280698\pi$
$242$		0			0
$243$	15.4930			0.993879
$244$		0			0
$245$	5.65685	−	17.3952i	0.361403	−	1.11134i
$246$		0			0
$247$		−	19.7990i		−	1.25978i
$248$		0			0
$249$	12.4853			0.791223
$250$		0			0
$251$	−1.08239			−0.0683200			−0.0341600	−	0.999416i	$-0.510876\pi$
$251$	−1.08239			−0.0683200			−0.0341600	+	0.999416i	$0.510876\pi$
$252$		0			0
$253$	15.3137			0.962765
$254$		0			0
$255$	−8.65914			−0.542256
$256$		0			0
$257$		−	6.12293i		−	0.381938i	−0.981596	−	0.190969i	$-0.938837\pi$
$257$		−	6.12293i		−	0.381938i	0.981596	−	0.190969i	$-0.0611630\pi$
$258$		0			0
$259$	−20.0083	−	3.17157i	−1.24326	−	0.197072i
$260$		0			0
$261$	6.68629			0.413871
$262$		0			0
$263$		−	4.14214i		−	0.255415i	−0.991812	−	0.127708i	$-0.959238\pi$
$263$		−	4.14214i		−	0.255415i	0.991812	−	0.127708i	$-0.0407619\pi$
$264$		0			0
$265$		−	5.22625i		−	0.321046i
$266$		0			0
$267$			2.34315i			0.143398i
$268$		0			0
$269$		−	13.4370i		−	0.819271i	−0.912249	−	0.409636i	$-0.865656\pi$
$269$		−	13.4370i		−	0.819271i	0.912249	−	0.409636i	$-0.134344\pi$
$270$		0			0
$271$	−1.79337			−0.108939			−0.0544696	−	0.998515i	$-0.517347\pi$
$271$	−1.79337			−0.108939			−0.0544696	+	0.998515i	$0.517347\pi$
$272$		0			0
$273$	−2.14214	+	13.5140i	−0.129648	+	0.817903i
$274$		0			0
$275$			3.65685i			0.220517i
$276$		0			0
$277$	26.9706			1.62050			0.810252	−	0.586082i	$-0.199330\pi$
$277$	26.9706			1.62050			0.810252	+	0.586082i	$0.199330\pi$
$278$		0			0
$279$	−5.59767			−0.335124
$280$		0			0
$281$	−28.6274			−1.70777			−0.853884	−	0.520463i	$-0.825759\pi$
$281$	−28.6274			−1.70777			−0.853884	+	0.520463i	$0.825759\pi$
$282$		0			0
$283$	−1.08239			−0.0643415			−0.0321708	−	0.999482i	$-0.510242\pi$
$283$	−1.08239			−0.0643415			−0.0321708	+	0.999482i	$0.510242\pi$
$284$		0			0
$285$		−	11.7206i		−	0.694269i
$286$		0			0
$287$	−3.95815	+	24.9706i	−0.233642	+	1.47397i
$288$		0			0
$289$	7.62742			0.448672
$290$		0			0
$291$			14.6274i			0.857474i
$292$		0			0
$293$		−	5.67459i		−	0.331513i	−0.986167	−	0.165757i	$-0.946993\pi$
$293$		−	5.67459i		−	0.331513i	0.986167	−	0.165757i	$-0.0530066\pi$
$294$		0			0
$295$			22.1421i			1.28916i
$296$		0			0
$297$			10.4525i			0.606516i
$298$		0			0
$299$	36.5838			2.11569
$300$		0			0
$301$	1.51472	−	9.55582i	0.0873069	−	0.550788i
$302$		0			0
$303$		−	6.14214i		−	0.352856i
$304$		0			0
$305$	6.82843			0.390995
$306$		0			0
$307$	11.5349			0.658331			0.329166	−	0.944272i	$-0.393233\pi$
$307$	11.5349			0.658331			0.329166	+	0.944272i	$0.393233\pi$
$308$		0			0
$309$	−18.3431			−1.04351
$310$		0			0
$311$	20.0083			1.13457			0.567284	−	0.823522i	$-0.307994\pi$
$311$	20.0083			1.13457			0.567284	+	0.823522i	$0.307994\pi$
$312$		0			0
$313$			15.6788i			0.886216i	0.896468	+	0.443108i	$0.146124\pi$
$313$			15.6788i			0.886216i	−0.896468	+	0.443108i	$0.853876\pi$
$314$		0			0
$315$	1.97908	−	12.4853i	0.111508	−	0.703466i
$316$		0			0
$317$	−27.6569			−1.55336			−0.776682	−	0.629893i	$-0.783099\pi$
$317$	−27.6569			−1.55336			−0.776682	+	0.629893i	$0.783099\pi$
$318$		0			0
$319$			7.31371i			0.409489i
$320$		0			0
$321$			0.371418i			0.0207305i
$322$		0			0
$323$		−	12.6863i		−	0.705884i
$324$		0			0
$325$			8.73606i			0.484589i
$326$		0			0
$327$	5.75152			0.318060
$328$		0			0
$329$	−19.3137	−	3.06147i	−1.06480	−	0.168784i
$330$		0			0
$331$			17.3137i			0.951647i	0.879541	+	0.475824i	$0.157850\pi$
$331$			17.3137i			0.951647i	−0.879541	+	0.475824i	$0.842150\pi$
$332$		0			0
$333$	−14.0000			−0.767195
$334$		0			0
$335$	−40.9133			−2.23533
$336$		0			0
$337$	8.82843			0.480915			0.240458	−	0.970660i	$-0.422703\pi$
$337$	8.82843			0.480915			0.240458	+	0.970660i	$0.422703\pi$
$338$		0			0
$339$	9.55582			0.519001
$340$		0			0
$341$		−	6.12293i		−	0.331576i
$342$		0			0
$343$	−16.4985	−	8.41421i	−0.890836	−	0.454325i
$344$		0			0
$345$	21.6569			1.16597
$346$		0			0
$347$			19.6569i			1.05524i	0.849482	+	0.527618i	$0.176915\pi$
$347$			19.6569i			1.05524i	−0.849482	+	0.527618i	$0.823085\pi$
$348$		0			0
$349$			8.73606i			0.467631i	0.972281	+	0.233815i	$0.0751211\pi$
$349$			8.73606i			0.467631i	−0.972281	+	0.233815i	$0.924879\pi$
$350$		0			0
$351$			24.9706i			1.33283i
$352$		0			0
$353$			29.5641i			1.57354i	0.617246	+	0.786770i	$0.288248\pi$
$353$			29.5641i			1.57354i	−0.617246	+	0.786770i	$0.711752\pi$
$354$		0			0
$355$	−23.0698			−1.22442
$356$		0			0
$357$	−1.37258	+	8.65914i	−0.0726448	+	0.458290i
$358$		0			0
$359$		−	14.9706i		−	0.790116i	−0.918656	−	0.395058i	$-0.870724\pi$
$359$		−	14.9706i		−	0.790116i	0.918656	−	0.395058i	$-0.129276\pi$
$360$		0			0
$361$	−1.82843			−0.0962330
$362$		0			0
$363$	7.57675			0.397676
$364$		0			0
$365$	32.9706			1.72576
$366$		0			0
$367$	10.4525			0.545616			0.272808	−	0.962068i	$-0.412048\pi$
$367$	10.4525			0.545616			0.272808	+	0.962068i	$0.412048\pi$
$368$		0			0
$369$			17.4721i			0.909562i
$370$		0			0
$371$	−5.22625	−	0.828427i	−0.271333	−	0.0430098i
$372$		0			0
$373$	−6.97056			−0.360922			−0.180461	−	0.983582i	$-0.557759\pi$
$373$	−6.97056			−0.360922			−0.180461	+	0.983582i	$0.557759\pi$
$374$		0			0
$375$		−	8.97056i		−	0.463238i
$376$		0			0
$377$			17.4721i			0.899860i
$378$		0			0
$379$		−	5.31371i		−	0.272947i	−0.990644	−	0.136473i	$-0.956423\pi$
$379$		−	5.31371i		−	0.272947i	0.990644	−	0.136473i	$-0.0435768\pi$
$380$		0			0
$381$		−	3.95815i		−	0.202782i
$382$		0			0
$383$	−11.7206			−0.598895			−0.299447	−	0.954113i	$-0.596802\pi$
$383$	−11.7206			−0.598895			−0.299447	+	0.954113i	$0.596802\pi$
$384$		0			0
$385$	13.6569	+	2.16478i	0.696018	+	0.110328i
$386$		0			0
$387$		−	6.68629i		−	0.339883i
$388$		0			0
$389$	−19.6569			−0.996642			−0.498321	−	0.866993i	$-0.666050\pi$
$389$	−19.6569			−0.996642			−0.498321	+	0.866993i	$0.666050\pi$
$390$		0			0
$391$	23.4412			1.18547
$392$		0			0
$393$	−14.8284			−0.747995
$394$		0			0
$395$	33.5223			1.68669
$396$		0			0
$397$			33.9706i			1.70494i	0.522778	+	0.852469i	$0.324896\pi$
$397$			33.9706i			1.70494i	−0.522778	+	0.852469i	$0.675104\pi$
$398$		0			0
$399$	−11.7206	−	1.85786i	−0.586764	−	0.0930096i
$400$		0			0
$401$	−15.1716			−0.757632			−0.378816	−	0.925472i	$-0.623669\pi$
$401$	−15.1716			−0.757632			−0.378816	+	0.925472i	$0.623669\pi$
$402$		0			0
$403$		−	14.6274i		−	0.728644i
$404$		0			0
$405$			0.448342i			0.0222783i
$406$		0			0
$407$		−	15.3137i		−	0.759072i
$408$		0			0
$409$			0.896683i			0.0443381i	0.999754	+	0.0221691i	$0.00705721\pi$
$409$			0.896683i			0.0443381i	−0.999754	+	0.0221691i	$0.992943\pi$
$410$		0			0
$411$	2.16478			0.106781
$412$		0			0
$413$	22.1421	+	3.50981i	1.08954	+	0.172706i
$414$		0			0
$415$		−	30.1421i		−	1.47962i
$416$		0			0
$417$	2.14214			0.104901
$418$		0			0
$419$	−34.6047			−1.69055			−0.845275	−	0.534332i	$-0.820563\pi$
$419$	−34.6047			−1.69055			−0.845275	+	0.534332i	$0.820563\pi$
$420$		0			0
$421$	21.3137			1.03877			0.519383	−	0.854541i	$-0.326162\pi$
$421$	21.3137			1.03877			0.519383	+	0.854541i	$0.326162\pi$
$422$		0			0
$423$	−13.5140			−0.657072
$424$		0			0
$425$			5.59767i			0.271527i
$426$		0			0
$427$	1.08239	−	6.82843i	0.0523806	−	0.330451i
$428$		0			0
$429$	−10.3431			−0.499372
$430$		0			0
$431$			12.3431i			0.594548i	0.954792	+	0.297274i	$0.0960775\pi$
$431$			12.3431i			0.594548i	−0.954792	+	0.297274i	$0.903922\pi$
$432$		0			0
$433$		−	5.59767i		−	0.269007i	−0.990913	−	0.134503i	$-0.957056\pi$
$433$		−	5.59767i		−	0.269007i	0.990913	−	0.134503i	$-0.0429439\pi$
$434$		0			0
$435$			10.3431i			0.495916i
$436$		0			0
$437$			31.7289i			1.51780i
$438$		0			0
$439$	22.5445			1.07599			0.537996	−	0.842948i	$-0.319182\pi$
$439$	22.5445			1.07599			0.537996	+	0.842948i	$0.319182\pi$
$440$		0			0
$441$	−12.1716	−	3.95815i	−0.579599	−	0.188483i
$442$		0			0
$443$		−	10.0000i		−	0.475114i	−0.971374	−	0.237557i	$-0.923653\pi$
$443$		−	10.0000i		−	0.475114i	0.971374	−	0.237557i	$-0.0763467\pi$
$444$		0			0
$445$	5.65685			0.268161
$446$		0			0
$447$	10.8239			0.511954
$448$		0			0
$449$	21.3137			1.00586			0.502928	−	0.864328i	$-0.332256\pi$
$449$	21.3137			1.00586			0.502928	+	0.864328i	$0.332256\pi$
$450$		0			0
$451$	−19.1116			−0.899932
$452$		0			0
$453$		−	9.03056i		−	0.424293i
$454$		0			0
$455$	32.6256	+	5.17157i	1.52951	+	0.242447i
$456$		0			0
$457$	37.1127			1.73606			0.868029	−	0.496513i	$-0.165386\pi$
$457$	37.1127			1.73606			0.868029	+	0.496513i	$0.165386\pi$
$458$		0			0
$459$			16.0000i			0.746816i
$460$		0			0
$461$		−	10.3756i		−	0.483239i	−0.970371	−	0.241619i	$-0.922321\pi$
$461$		−	10.3756i		−	0.483239i	0.970371	−	0.241619i	$-0.0776786\pi$
$462$		0			0
$463$			16.1421i			0.750189i	0.926987	+	0.375094i	$0.122390\pi$
$463$			16.1421i			0.750189i	−0.926987	+	0.375094i	$0.877610\pi$
$464$		0			0
$465$		−	8.65914i		−	0.401558i
$466$		0			0
$467$	−8.10201			−0.374916			−0.187458	−	0.982273i	$-0.560025\pi$
$467$	−8.10201			−0.374916			−0.187458	+	0.982273i	$0.560025\pi$
$468$		0			0
$469$	−6.48528	+	40.9133i	−0.299462	+	1.88920i
$470$		0			0
$471$			6.14214i			0.283015i
$472$		0			0
$473$	7.31371			0.336285
$474$		0			0
$475$	−7.57675			−0.347645
$476$		0			0
$477$	−3.65685			−0.167436
$478$		0			0
$479$	−30.0894			−1.37482			−0.687410	−	0.726269i	$-0.741252\pi$
$479$	−30.0894			−1.37482			−0.687410	+	0.726269i	$0.741252\pi$
$480$		0			0
$481$		−	36.5838i		−	1.66808i
$482$		0			0
$483$	3.43289	−	21.6569i	0.156202	−	0.985421i
$484$		0			0
$485$	35.3137			1.60351
$486$		0			0
$487$			30.2843i			1.37231i	0.727455	+	0.686156i	$0.240703\pi$
$487$			30.2843i			1.37231i	−0.727455	+	0.686156i	$0.759297\pi$
$488$		0			0
$489$			18.7402i			0.847462i
$490$		0			0
$491$		−	30.2843i		−	1.36671i	−0.730086	−	0.683355i	$-0.760520\pi$
$491$		−	30.2843i		−	1.36671i	0.730086	−	0.683355i	$-0.239480\pi$
$492$		0			0
$493$			11.1953i			0.504213i
$494$		0			0
$495$	9.55582			0.429502
$496$		0			0
$497$	−3.65685	+	23.0698i	−0.164032	+	1.03482i
$498$		0			0
$499$			17.3137i			0.775068i	0.921855	+	0.387534i	$0.126673\pi$
$499$			17.3137i			0.775068i	−0.921855	+	0.387534i	$0.873327\pi$
$500$		0			0
$501$	7.02944			0.314052
$502$		0			0
$503$	17.4721			0.779043			0.389522	−	0.921017i	$-0.372640\pi$
$503$	17.4721			0.779043			0.389522	+	0.921017i	$0.372640\pi$
$504$		0			0
$505$	−14.8284			−0.659856
$506$		0			0
$507$	−10.6382			−0.472460
$508$		0			0
$509$			26.0543i			1.15484i	0.816448	+	0.577419i	$0.195940\pi$
$509$			26.0543i			1.15484i	−0.816448	+	0.577419i	$0.804060\pi$
$510$		0			0
$511$	5.22625	−	32.9706i	0.231196	−	1.45853i
$512$		0			0
$513$	−21.6569			−0.956173
$514$		0			0
$515$			44.2843i			1.95140i
$516$		0			0
$517$		−	14.7821i		−	0.650115i
$518$		0			0
$519$		−	11.7990i		−	0.517918i
$520$		0			0
$521$		−	0.896683i		−	0.0392844i	−0.999807	−	0.0196422i	$-0.993747\pi$
$521$		−	0.896683i		−	0.0392844i	0.999807	−	0.0196422i	$-0.00625271\pi$
$522$		0			0
$523$	−18.9259			−0.827573			−0.413787	−	0.910374i	$-0.635794\pi$
$523$	−18.9259			−0.827573			−0.413787	+	0.910374i	$0.635794\pi$
$524$		0			0
$525$	5.17157	+	0.819760i	0.225706	+	0.0357773i
$526$		0			0
$527$		−	9.37258i		−	0.408276i
$528$		0			0
$529$	−35.6274			−1.54902
$530$		0			0
$531$	15.4930			0.672341
$532$		0			0
$533$	−45.6569			−1.97762
$534$		0			0
$535$	0.896683			0.0387670
$536$		0			0
$537$		−	2.16478i		−	0.0934173i
$538$		0			0
$539$	4.32957	−	13.3137i	0.186488	−	0.573462i
$540$		0			0
$541$	−25.3137			−1.08832			−0.544161	−	0.838981i	$-0.683152\pi$
$541$	−25.3137			−1.08832			−0.544161	+	0.838981i	$0.683152\pi$
$542$		0			0
$543$			1.85786i			0.0797286i
$544$		0			0
$545$		−	13.8854i		−	0.594785i
$546$		0			0
$547$			33.3137i			1.42439i	0.701981	+	0.712196i	$0.252299\pi$
$547$			33.3137i			1.42439i	−0.701981	+	0.712196i	$0.747701\pi$
$548$		0			0
$549$		−	4.77791i		−	0.203916i
$550$		0			0
$551$	−15.1535			−0.645560
$552$		0			0
$553$	5.31371	−	33.5223i	0.225962	−	1.42551i
$554$		0			0
$555$		−	21.6569i		−	0.919282i
$556$		0			0
$557$	5.31371			0.225149			0.112575	−	0.993643i	$-0.464090\pi$
$557$	5.31371			0.225149			0.112575	+	0.993643i	$0.464090\pi$
$558$		0			0
$559$	17.4721			0.738992
$560$		0			0
$561$	−6.62742			−0.279810
$562$		0			0
$563$	−4.51528			−0.190296			−0.0951481	−	0.995463i	$-0.530332\pi$
$563$	−4.51528			−0.190296			−0.0951481	+	0.995463i	$0.530332\pi$
$564$		0			0
$565$		−	23.0698i		−	0.970553i
$566$		0			0
$567$	0.448342	+	0.0710678i	0.0188286	+	0.00298457i
$568$		0			0
$569$	7.85786			0.329419			0.164709	−	0.986342i	$-0.447331\pi$
$569$	7.85786			0.329419			0.164709	+	0.986342i	$0.447331\pi$
$570$		0			0
$571$			8.34315i			0.349150i	0.984644	+	0.174575i	$0.0558551\pi$
$571$			8.34315i			0.349150i	−0.984644	+	0.174575i	$0.944145\pi$
$572$		0			0
$573$			5.22625i			0.218330i
$574$		0			0
$575$		−	14.0000i		−	0.583840i
$576$		0			0
$577$			35.6871i			1.48567i	0.669473	+	0.742836i	$0.266520\pi$
$577$			35.6871i			1.48567i	−0.669473	+	0.742836i	$0.733480\pi$
$578$		0			0
$579$	−11.3492			−0.471656
$580$		0			0
$581$	−30.1421	−	4.77791i	−1.25051	−	0.198221i
$582$		0			0
$583$		−	4.00000i		−	0.165663i
$584$		0			0
$585$	22.8284			0.943839
$586$		0			0
$587$	29.0070			1.19725			0.598624	−	0.801030i	$-0.295714\pi$
$587$	29.0070			1.19725			0.598624	+	0.801030i	$0.295714\pi$
$588$		0			0
$589$	12.6863			0.522730
$590$		0			0
$591$	−3.95815			−0.162817
$592$		0			0
$593$			1.79337i			0.0736447i	0.999322	+	0.0368224i	$0.0117236\pi$
$593$			1.79337i			0.0736447i	−0.999322	+	0.0368224i	$0.988276\pi$
$594$		0			0
$595$	20.9050	+	3.31371i	0.857022	+	0.135849i
$596$		0			0
$597$	13.6569			0.558938
$598$		0			0
$599$			31.1716i			1.27364i	0.771014	+	0.636818i	$0.219750\pi$
$599$			31.1716i			1.27364i	−0.771014	+	0.636818i	$0.780250\pi$
$600$		0			0
$601$		−	42.1814i		−	1.72062i	−0.509774	−	0.860308i	$-0.670271\pi$
$601$		−	42.1814i		−	1.72062i	0.509774	−	0.860308i	$-0.329729\pi$
$602$		0			0
$603$			28.6274i			1.16580i
$604$		0			0
$605$		−	18.2919i		−	0.743671i
$606$		0			0
$607$	−17.3183			−0.702927			−0.351464	−	0.936202i	$-0.614316\pi$
$607$	−17.3183			−0.702927			−0.351464	+	0.936202i	$0.614316\pi$
$608$		0			0
$609$	10.3431	+	1.63952i	0.419125	+	0.0664367i
$610$		0			0
$611$		−	35.3137i		−	1.42864i
$612$		0			0
$613$	5.31371			0.214619			0.107309	−	0.994226i	$-0.465776\pi$
$613$	5.31371			0.214619			0.107309	+	0.994226i	$0.465776\pi$
$614$		0			0
$615$	−27.0279			−1.08987
$616$		0			0
$617$	13.1127			0.527897			0.263949	−	0.964537i	$-0.414975\pi$
$617$	13.1127			0.527897			0.263949	+	0.964537i	$0.414975\pi$
$618$		0			0
$619$	−2.87576			−0.115586			−0.0577932	−	0.998329i	$-0.518406\pi$
$619$	−2.87576			−0.115586			−0.0577932	+	0.998329i	$0.518406\pi$
$620$		0			0
$621$		−	40.0166i		−	1.60581i
$622$		0			0
$623$	0.896683	−	5.65685i	0.0359248	−	0.226637i
$624$		0			0
$625$	−30.7990			−1.23196
$626$		0			0
$627$		−	8.97056i		−	0.358250i
$628$		0			0
$629$		−	23.4412i		−	0.934662i
$630$		0			0
$631$			26.4853i			1.05436i	0.849753	+	0.527181i	$0.176751\pi$
$631$			26.4853i			1.05436i	−0.849753	+	0.527181i	$0.823249\pi$
$632$		0			0
$633$		−	11.8745i		−	0.471967i
$634$		0			0
$635$	−9.55582			−0.379211
$636$		0			0
$637$	10.3431	−	31.8059i	0.409810	−	1.26019i
$638$		0			0
$639$			16.1421i			0.638573i
$640$		0			0
$641$	28.1421			1.11155			0.555774	−	0.831334i	$-0.312422\pi$
$641$	28.1421			1.11155			0.555774	+	0.831334i	$0.312422\pi$
$642$		0			0
$643$	17.1326			0.675642			0.337821	−	0.941210i	$-0.390310\pi$
$643$	17.1326			0.675642			0.337821	+	0.941210i	$0.390310\pi$
$644$		0			0
$645$	10.3431			0.407261
$646$		0			0
$647$	−4.70099			−0.184815			−0.0924074	−	0.995721i	$-0.529456\pi$
$647$	−4.70099			−0.184815			−0.0924074	+	0.995721i	$0.529456\pi$
$648$		0			0
$649$			16.9469i			0.665222i
$650$		0			0
$651$	−8.65914	−	1.37258i	−0.339378	−	0.0537958i
$652$		0			0
$653$	−2.68629			−0.105123			−0.0525614	−	0.998618i	$-0.516739\pi$
$653$	−2.68629			−0.105123			−0.0525614	+	0.998618i	$0.516739\pi$
$654$		0			0
$655$			35.7990i			1.39878i
$656$		0			0
$657$		−	23.0698i		−	0.900038i
$658$		0			0
$659$		−	17.0294i		−	0.663373i	−0.943390	−	0.331686i	$-0.892382\pi$
$659$		−	17.0294i		−	0.663373i	0.943390	−	0.331686i	$-0.107618\pi$
$660$		0			0
$661$		−	30.9092i		−	1.20223i	−0.799164	−	0.601114i	$-0.794724\pi$
$661$		−	30.9092i		−	1.20223i	0.799164	−	0.601114i	$-0.205276\pi$
$662$		0			0
$663$	−15.8326			−0.614887
$664$		0			0
$665$	−4.48528	+	28.2960i	−0.173932	+	1.09727i
$666$		0			0
$667$		−	28.0000i		−	1.08416i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	5.22625			0.201757
$672$		0			0
$673$	6.68629			0.257738			0.128869	−	0.991662i	$-0.458865\pi$
$673$	6.68629			0.257738			0.128869	+	0.991662i	$0.458865\pi$
$674$		0			0
$675$	9.55582			0.367804
$676$		0			0
$677$			24.4148i			0.938338i	0.883109	+	0.469169i	$0.155446\pi$
$677$			24.4148i			0.938338i	−0.883109	+	0.469169i	$0.844554\pi$
$678$		0			0
$679$	5.59767	−	35.3137i	0.214819	−	1.35522i
$680$		0			0
$681$	−20.0833			−0.769592
$682$		0			0
$683$			43.6569i			1.67048i	0.549883	+	0.835242i	$0.314672\pi$
$683$			43.6569i			1.67048i	−0.549883	+	0.835242i	$0.685328\pi$
$684$		0			0
$685$		−	5.22625i		−	0.199685i
$686$		0			0
$687$			3.23045i			0.123249i
$688$		0			0
$689$		−	9.55582i		−	0.364048i
$690$		0			0
$691$	32.4399			1.23407			0.617036	−	0.786935i	$-0.288333\pi$
$691$	32.4399			1.23407			0.617036	+	0.786935i	$0.288333\pi$
$692$		0			0
$693$	1.51472	−	9.55582i	0.0575394	−	0.362996i
$694$		0			0
$695$		−	5.17157i		−	0.196169i
$696$		0			0
$697$	−29.2548			−1.10811
$698$		0			0
$699$	19.4831			0.736917
$700$		0			0
$701$	−9.31371			−0.351774			−0.175887	−	0.984410i	$-0.556279\pi$
$701$	−9.31371			−0.351774			−0.175887	+	0.984410i	$0.556279\pi$
$702$		0			0
$703$	31.7289			1.19668
$704$		0			0
$705$		−	20.9050i		−	0.787328i
$706$		0			0
$707$	−2.35049	+	14.8284i	−0.0883994	+	0.557680i
$708$		0			0
$709$	18.0000			0.676004			0.338002	−	0.941145i	$-0.390249\pi$
$709$	18.0000			0.676004			0.338002	+	0.941145i	$0.390249\pi$
$710$		0			0
$711$		−	23.4558i		−	0.879663i
$712$		0			0
$713$			23.4412i			0.877880i
$714$		0			0
$715$			24.9706i			0.933846i
$716$		0			0
$717$		−	9.03056i		−	0.337252i
$718$		0			0
$719$	−34.4190			−1.28361			−0.641806	−	0.766867i	$-0.721814\pi$
$719$	−34.4190			−1.28361			−0.641806	+	0.766867i	$0.721814\pi$
$720$		0			0
$721$	44.2843	+	7.01962i	1.64923	+	0.261424i
$722$		0			0
$723$		−	25.9411i		−	0.964761i
$724$		0			0
$725$	6.68629			0.248323
$726$		0			0
$727$	−42.1814			−1.56442			−0.782211	−	0.623013i	$-0.785908\pi$
$727$	−42.1814			−1.56442			−0.782211	+	0.623013i	$0.785908\pi$
$728$		0			0
$729$	17.2843			0.640158
$730$		0			0
$731$	11.1953			0.414075
$732$		0			0
$733$		−	24.4148i		−	0.901782i	−0.892579	−	0.450891i	$-0.851106\pi$
$733$		−	24.4148i		−	0.901782i	0.892579	−	0.450891i	$-0.148894\pi$
$734$		0			0
$735$	6.12293	−	18.8284i	0.225848	−	0.694497i
$736$		0			0
$737$	−31.3137			−1.15346
$738$		0			0
$739$		−	16.6274i		−	0.611649i	−0.952088	−	0.305825i	$-0.901068\pi$
$739$		−	16.6274i		−	0.611649i	0.952088	−	0.305825i	$-0.0989322\pi$
$740$		0			0
$741$		−	21.4303i		−	0.787261i
$742$		0			0
$743$		−	24.3431i		−	0.893063i	−0.894768	−	0.446532i	$-0.852659\pi$
$743$		−	24.3431i		−	0.893063i	0.894768	−	0.446532i	$-0.147341\pi$
$744$		0			0
$745$		−	26.1313i		−	0.957375i
$746$		0			0
$747$	−21.0907			−0.771669
$748$		0			0
$749$	0.142136	−	0.896683i	0.00519352	−	0.0327641i
$750$		0			0
$751$		−	14.9706i		−	0.546284i	−0.961974	−	0.273142i	$-0.911937\pi$
$751$		−	14.9706i		−	0.546284i	0.961974	−	0.273142i	$-0.0880628\pi$
$752$		0			0
$753$	−1.17157			−0.0426945
$754$		0			0
$755$	−21.8017			−0.793445
$756$		0			0
$757$	−49.3137			−1.79234			−0.896169	−	0.443714i	$-0.853661\pi$
$757$	−49.3137			−1.79234			−0.896169	+	0.443714i	$0.853661\pi$
$758$		0			0
$759$	16.5754			0.601650
$760$		0			0
$761$		−	7.76245i		−	0.281389i	−0.990053	−	0.140694i	$-0.955067\pi$
$761$		−	7.76245i		−	0.281389i	0.990053	−	0.140694i	$-0.0449335\pi$
$762$		0			0
$763$	−13.8854	−	2.20101i	−0.502685	−	0.0796819i
$764$		0			0
$765$	14.6274			0.528855
$766$		0			0
$767$			40.4853i			1.46184i
$768$		0			0
$769$		−	13.5140i		−	0.487326i	−0.969860	−	0.243663i	$-0.921651\pi$
$769$		−	13.5140i		−	0.487326i	0.969860	−	0.243663i	$-0.0783491\pi$
$770$		0			0
$771$		−	6.62742i		−	0.238681i
$772$		0			0
$773$			26.9510i			0.969361i	0.874691	+	0.484680i	$0.161064\pi$
$773$			26.9510i			0.969361i	−0.874691	+	0.484680i	$0.838936\pi$
$774$		0			0
$775$	−5.59767			−0.201074
$776$		0			0
$777$	−21.6569	−	3.43289i	−0.776935	−	0.123154i
$778$		0			0
$779$		−	39.5980i		−	1.41874i
$780$		0			0
$781$	−17.6569			−0.631812
$782$		0			0
$783$	19.1116			0.682994
$784$		0			0
$785$	14.8284			0.529249
$786$		0			0
$787$	−4.51528			−0.160952			−0.0804761	−	0.996757i	$-0.525644\pi$
$787$	−4.51528			−0.160952			−0.0804761	+	0.996757i	$0.525644\pi$
$788$		0			0
$789$		−	4.48342i		−	0.159614i
$790$		0			0
$791$	−23.0698	−	3.65685i	−0.820267	−	0.130023i
$792$		0			0
$793$	12.4853			0.443365
$794$		0			0
$795$		−	5.65685i		−	0.200628i
$796$		0			0
$797$			41.3617i			1.46511i	0.680710	+	0.732553i	$0.261671\pi$
$797$			41.3617i			1.46511i	−0.680710	+	0.732553i	$0.738329\pi$
$798$		0			0
$799$		−	22.6274i		−	0.800500i
$800$		0			0
$801$		−	3.95815i		−	0.139854i
$802$		0			0
$803$	25.2346			0.890509
$804$		0			0
$805$	−52.2843	−	8.28772i	−1.84278	−	0.292104i
$806$		0			0
$807$		−	14.5442i		−	0.511979i
$808$		0			0
$809$	−17.5147			−0.615785			−0.307892	−	0.951421i	$-0.599624\pi$
$809$	−17.5147			−0.615785			−0.307892	+	0.951421i	$0.599624\pi$
$810$		0			0
$811$	−38.5628			−1.35412			−0.677062	−	0.735926i	$-0.736747\pi$
$811$	−38.5628			−1.35412			−0.677062	+	0.735926i	$0.736747\pi$
$812$		0			0
$813$	−1.94113			−0.0680782
$814$		0			0
$815$	45.2429			1.58479
$816$		0			0
$817$			15.1535i			0.530154i
$818$		0			0
$819$	3.61859	−	22.8284i	0.126444	−	0.797690i
$820$		0			0
$821$	49.5980			1.73098			0.865491	−	0.500925i	$-0.167007\pi$
$821$	49.5980			1.73098			0.865491	+	0.500925i	$0.167007\pi$
$822$		0			0
$823$			25.1127i			0.875374i	0.899127	+	0.437687i	$0.144202\pi$
$823$			25.1127i			0.875374i	−0.899127	+	0.437687i	$0.855798\pi$
$824$		0			0
$825$			3.95815i			0.137805i
$826$		0			0
$827$			43.2548i			1.50412i	0.659096	+	0.752059i	$0.270939\pi$
$827$			43.2548i			1.50412i	−0.659096	+	0.752059i	$0.729061\pi$
$828$		0			0
$829$			6.19986i			0.215330i	0.994187	+	0.107665i	$0.0343374\pi$
$829$			6.19986i			0.215330i	−0.994187	+	0.107665i	$0.965663\pi$
$830$		0			0
$831$	29.1927			1.01268
$832$		0			0
$833$	6.62742	−	20.3797i	0.229626	−	0.706116i
$834$		0			0
$835$		−	16.9706i		−	0.587291i
$836$		0			0
$837$	−16.0000			−0.553041
$838$		0			0
$839$	42.1814			1.45626			0.728132	−	0.685437i	$-0.240389\pi$
$839$	42.1814			1.45626			0.728132	+	0.685437i	$0.240389\pi$
$840$		0			0
$841$	−15.6274			−0.538876
$842$		0			0
$843$	−30.9861			−1.06722
$844$		0			0
$845$			25.6829i			0.883519i
$846$		0			0
$847$	−18.2919	−	2.89949i	−0.628516	−	0.0996278i
$848$		0			0
$849$	−1.17157			−0.0402083
$850$		0			0
$851$			58.6274i			2.00972i
$852$		0			0
$853$			31.8059i			1.08901i	0.838757	+	0.544506i	$0.183283\pi$
$853$			31.8059i			1.08901i	−0.838757	+	0.544506i	$0.816717\pi$
$854$		0			0
$855$			19.7990i			0.677111i
$856$		0			0
$857$			7.01962i			0.239786i	0.992787	+	0.119893i	$0.0382551\pi$
$857$			7.01962i			0.239786i	−0.992787	+	0.119893i	$0.961745\pi$
$858$		0			0
$859$	2.50434			0.0854470			0.0427235	−	0.999087i	$-0.486397\pi$
$859$	2.50434			0.0854470			0.0427235	+	0.999087i	$0.486397\pi$
$860$		0			0
$861$	−4.28427	+	27.0279i	−0.146008	+	0.921110i
$862$		0			0
$863$			27.4558i			0.934608i	0.884097	+	0.467304i	$0.154775\pi$
$863$			27.4558i			0.934608i	−0.884097	+	0.467304i	$0.845225\pi$
$864$		0			0
$865$	−28.4853			−0.968529
$866$		0			0
$867$	8.25586			0.280384
$868$		0			0
$869$	25.6569			0.870349
$870$		0			0
$871$	−74.8070			−2.53474
$872$		0			0
$873$		−	24.7093i		−	0.836283i
$874$		0			0
$875$	−3.43289	+	21.6569i	−0.116053	+	0.732135i
$876$		0			0
$877$	−0.343146			−0.0115872			−0.00579360	−	0.999983i	$-0.501844\pi$
$877$	−0.343146			−0.0115872			−0.00579360	+	0.999983i	$0.501844\pi$
$878$		0			0
$879$		−	6.14214i		−	0.207169i
$880$		0			0
$881$			27.7708i			0.935621i	0.883829	+	0.467811i	$0.154957\pi$
$881$			27.7708i			0.935621i	−0.883829	+	0.467811i	$0.845043\pi$
$882$		0			0
$883$			24.3431i			0.819212i	0.912263	+	0.409606i	$0.134334\pi$
$883$			24.3431i			0.819212i	−0.912263	+	0.409606i	$0.865666\pi$
$884$		0			0
$885$			23.9665i			0.805624i
$886$		0			0
$887$	55.1701			1.85243			0.926216	−	0.376993i	$-0.123042\pi$
$887$	55.1701			1.85243			0.926216	+	0.376993i	$0.123042\pi$
$888$		0			0
$889$	−1.51472	+	9.55582i	−0.0508020	+	0.320492i
$890$		0			0
$891$			0.343146i			0.0114958i
$892$		0			0
$893$	30.6274			1.02491
$894$		0			0
$895$	−5.22625			−0.174694
$896$		0			0
$897$	39.5980			1.32214
$898$		0			0
$899$	−11.1953			−0.373386
$900$		0			0
$901$		−	6.12293i		−	0.203985i
$902$		0			0
$903$	1.63952	−	10.3431i	0.0545598	−	0.344198i
$904$		0			0
$905$	4.48528			0.149096
$906$		0			0
$907$		−	26.0000i		−	0.863316i	−0.902037	−	0.431658i	$-0.857929\pi$
$907$		−	26.0000i		−	0.863316i	0.902037	−	0.431658i	$-0.142071\pi$
$908$		0			0
$909$			10.3756i			0.344136i
$910$		0			0
$911$		−	24.3431i		−	0.806524i	−0.915084	−	0.403262i	$-0.867876\pi$
$911$		−	24.3431i		−	0.806524i	0.915084	−	0.403262i	$-0.132124\pi$
$912$		0			0
$913$		−	23.0698i		−	0.763499i
$914$		0			0
$915$	7.39104			0.244340
$916$		0			0
$917$	35.7990	+	5.67459i	1.18219	+	0.187392i
$918$		0			0
$919$			29.7990i			0.982978i	0.870884	+	0.491489i	$0.163547\pi$
$919$			29.7990i			0.982978i	−0.870884	+	0.491489i	$0.836453\pi$
$920$		0			0
$921$	12.4853			0.411404
$922$		0			0
$923$	−42.1814			−1.38842
$924$		0			0
$925$	−14.0000			−0.460317
$926$		0			0
$927$	30.9861			1.01772
$928$		0			0
$929$			57.8602i			1.89833i	0.314777	+	0.949166i	$0.398070\pi$
$929$			57.8602i			1.89833i	−0.314777	+	0.949166i	$0.601930\pi$
$930$		0			0
$931$	27.5851	+	8.97056i	0.904064	+	0.293998i
$932$		0			0
$933$	21.6569			0.709014
$934$		0			0
$935$			16.0000i			0.523256i
$936$		0			0
$937$			44.7176i			1.46086i	0.682987	+	0.730431i	$0.260681\pi$
$937$			44.7176i			1.46086i	−0.682987	+	0.730431i	$0.739319\pi$
$938$		0			0
$939$			16.9706i			0.553813i
$940$		0			0
$941$			24.7862i			0.808008i	0.914757	+	0.404004i	$0.132382\pi$
$941$			24.7862i			0.808008i	−0.914757	+	0.404004i	$0.867618\pi$
$942$		0			0
$943$	73.1675			2.38266
$944$		0			0
$945$	5.65685	−	35.6871i	0.184017	−	1.16090i
$946$		0			0
$947$		−	28.3431i		−	0.921028i	−0.887652	−	0.460514i	$-0.847665\pi$
$947$		−	28.3431i		−	0.921028i	0.887652	−	0.460514i	$-0.152335\pi$
$948$		0			0
$949$	60.2843			1.95691
$950$		0			0
$951$	−29.9356			−0.970727
$952$		0			0
$953$	0.627417			0.0203240			0.0101620	−	0.999948i	$-0.496765\pi$
$953$	0.627417			0.0203240			0.0101620	+	0.999948i	$0.496765\pi$
$954$		0			0
$955$	12.6173			0.408286
$956$		0			0
$957$			7.91630i			0.255898i
$958$		0			0
$959$	−5.22625	−	0.828427i	−0.168764	−	0.0267513i
$960$		0			0
$961$	−21.6274			−0.697659
$962$		0			0
$963$		−	0.627417i		−	0.0202182i
$964$		0			0
$965$			27.3994i			0.882017i
$966$		0			0
$967$			17.0294i			0.547630i	0.961782	+	0.273815i	$0.0882856\pi$
$967$			17.0294i			0.547630i	−0.961782	+	0.273815i	$0.911714\pi$
$968$		0			0
$969$		−	13.7315i		−	0.441121i
$970$		0			0
$971$	−4.14386			−0.132983			−0.0664914	−	0.997787i	$-0.521180\pi$
$971$	−4.14386			−0.132983			−0.0664914	+	0.997787i	$0.521180\pi$
$972$		0			0
$973$	−5.17157	−	0.819760i	−0.165793	−	0.0262803i
$974$		0			0
$975$			9.45584i			0.302829i
$976$		0			0
$977$	16.6274			0.531958			0.265979	−	0.963979i	$-0.414305\pi$
$977$	16.6274			0.531958			0.265979	+	0.963979i	$0.414305\pi$
$978$		0			0
$979$	4.32957			0.138374
$980$		0			0
$981$	−9.71573			−0.310199
$982$		0			0
$983$	−33.5223			−1.06920			−0.534598	−	0.845107i	$-0.679537\pi$
$983$	−33.5223			−1.06920			−0.534598	+	0.845107i	$0.679537\pi$
$984$		0			0
$985$			9.55582i			0.304474i
$986$		0			0
$987$	−20.9050	−	3.31371i	−0.665414	−	0.105477i
$988$		0			0
$989$	−28.0000			−0.890348
$990$		0			0
$991$		−	14.4853i		−	0.460140i	−0.973174	−	0.230070i	$-0.926104\pi$
$991$		−	14.4853i		−	0.460140i	0.973174	−	0.230070i	$-0.0738955\pi$
$992$		0			0
$993$			18.7402i			0.594703i
$994$		0			0
$995$		−	32.9706i		−	1.04524i
$996$		0			0
$997$		−	17.3952i		−	0.550911i	−0.961314	−	0.275456i	$-0.911171\pi$
$997$		−	17.3952i		−	0.550911i	0.961314	−	0.275456i	$-0.0888287\pi$
$998$		0			0
$999$	−40.0166			−1.26607

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.2.f.d.447.5		8
3.2	odd	2		4032.2.b.p.3583.7		8
4.3	odd	2	inner	448.2.f.d.447.3		8
7.6	odd	2	inner	448.2.f.d.447.4		8
8.3	odd	2		224.2.f.a.223.6	yes	8
8.5	even	2		224.2.f.a.223.4	yes	8
12.11	even	2		4032.2.b.p.3583.8		8
16.3	odd	4		1792.2.e.g.895.3		8
16.5	even	4		1792.2.e.g.895.4		8
16.11	odd	4		1792.2.e.f.895.6		8
16.13	even	4		1792.2.e.f.895.5		8
21.20	even	2		4032.2.b.p.3583.2		8
24.5	odd	2		2016.2.b.b.1567.1		8
24.11	even	2		2016.2.b.b.1567.2		8
28.27	even	2	inner	448.2.f.d.447.6		8
56.3	even	6		1568.2.p.a.607.6		16
56.5	odd	6		1568.2.p.a.31.3		16
56.11	odd	6		1568.2.p.a.607.3		16
56.13	odd	2		224.2.f.a.223.5	yes	8
56.19	even	6		1568.2.p.a.31.5		16
56.27	even	2		224.2.f.a.223.3	✓	8
56.37	even	6		1568.2.p.a.31.6		16
56.45	odd	6		1568.2.p.a.607.4		16
56.51	odd	6		1568.2.p.a.31.4		16
56.53	even	6		1568.2.p.a.607.5		16
84.83	odd	2		4032.2.b.p.3583.1		8
112.13	odd	4		1792.2.e.f.895.4		8
112.27	even	4		1792.2.e.f.895.3		8
112.69	odd	4		1792.2.e.g.895.5		8
112.83	even	4		1792.2.e.g.895.6		8
168.83	odd	2		2016.2.b.b.1567.7		8
168.125	even	2		2016.2.b.b.1567.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.2.f.a.223.3	✓	8	56.27	even	2
224.2.f.a.223.4	yes	8	8.5	even	2
224.2.f.a.223.5	yes	8	56.13	odd	2
224.2.f.a.223.6	yes	8	8.3	odd	2
448.2.f.d.447.3		8	4.3	odd	2	inner
448.2.f.d.447.4		8	7.6	odd	2	inner
448.2.f.d.447.5		8	1.1	even	1	trivial
448.2.f.d.447.6		8	28.27	even	2	inner
1568.2.p.a.31.3		16	56.5	odd	6
1568.2.p.a.31.4		16	56.51	odd	6
1568.2.p.a.31.5		16	56.19	even	6
1568.2.p.a.31.6		16	56.37	even	6
1568.2.p.a.607.3		16	56.11	odd	6
1568.2.p.a.607.4		16	56.45	odd	6
1568.2.p.a.607.5		16	56.53	even	6
1568.2.p.a.607.6		16	56.3	even	6
1792.2.e.f.895.3		8	112.27	even	4
1792.2.e.f.895.4		8	112.13	odd	4
1792.2.e.f.895.5		8	16.13	even	4
1792.2.e.f.895.6		8	16.11	odd	4
1792.2.e.g.895.3		8	16.3	odd	4
1792.2.e.g.895.4		8	16.5	even	4
1792.2.e.g.895.5		8	112.69	odd	4
1792.2.e.g.895.6		8	112.83	even	4
2016.2.b.b.1567.1		8	24.5	odd	2
2016.2.b.b.1567.2		8	24.11	even	2
2016.2.b.b.1567.7		8	168.83	odd	2
2016.2.b.b.1567.8		8	168.125	even	2
4032.2.b.p.3583.1		8	84.83	odd	2
4032.2.b.p.3583.2		8	21.20	even	2
4032.2.b.p.3583.7		8	3.2	odd	2
4032.2.b.p.3583.8		8	12.11	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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	448.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.08239 q^{3} -2.61313i q^{5} +(-2.61313 - 0.414214i) q^{7} -1.82843 q^{9} +O(q^{10})\)	\(q+1.08239 q^{3} -2.61313i q^{5} +(-2.61313 - 0.414214i) q^{7} -1.82843 q^{9} -2.00000i q^{11} -4.77791i q^{13} -2.82843i q^{15} -3.06147i q^{17} +4.14386 q^{19} +(-2.82843 - 0.448342i) q^{21} +7.65685i q^{23} -1.82843 q^{25} -5.22625 q^{27} -3.65685 q^{29} +3.06147 q^{31} -2.16478i q^{33} +(-1.08239 + 6.82843i) q^{35} +7.65685 q^{37} -5.17157i q^{39} -9.55582i q^{41} +3.65685i q^{43} +4.77791i q^{45} +7.39104 q^{47} +(6.65685 + 2.16478i) q^{49} -3.31371i q^{51} +2.00000 q^{53} -5.22625 q^{55} +4.48528 q^{57} -8.47343 q^{59} +2.61313i q^{61} +(4.77791 + 0.757359i) q^{63} -12.4853 q^{65} -15.6569i q^{67} +8.28772i q^{69} -8.82843i q^{71} +12.6173i q^{73} -1.97908 q^{75} +(-0.828427 + 5.22625i) q^{77} +12.8284i q^{79} -0.171573 q^{81} +11.5349 q^{83} -8.00000 q^{85} -3.95815 q^{87} +2.16478i q^{89} +(-1.97908 + 12.4853i) q^{91} +3.31371 q^{93} -10.8284i q^{95} +13.5140i q^{97} +3.65685i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^9	\( 8 q + 8 q^{9} + 8 q^{25} + 16 q^{29} + 16 q^{37} + 8 q^{49} + 16 q^{53} - 32 q^{57} - 32 q^{65} + 16 q^{77} - 24 q^{81} - 64 q^{85} - 64 q^{93}+O(q^{100}) \) 8 * q + 8 * q^9 + 8 * q^25 + 16 * q^29 + 16 * q^37 + 8 * q^49 + 16 * q^53 - 32 * q^57 - 32 * q^65 + 16 * q^77 - 24 * q^81 - 64 * q^85 - 64 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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