LMFDB - Embedded newform 1792.2.e.f.895.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1792,2,Mod(895,1792)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1792, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1792.895");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1792 = 2^{8} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1792.e (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:	$14.3091920422$
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			895.6
Root			$0.382683 + 0.923880i$ of defining polynomial
Character	$\chi$	$=$	1792.895
Dual form			1792.2.e.f.895.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.08239i q^{3} +2.61313 q^{5} +(-2.61313 - 0.414214i) q^{7} +1.82843 q^{9} +O(q^{10})$	$q+1.08239i q^{3} +2.61313 q^{5} +(-2.61313 - 0.414214i) q^{7} +1.82843 q^{9} -2.00000 q^{11} -4.77791 q^{13} +2.82843i q^{15} -3.06147i q^{17} +4.14386i q^{19} +(0.448342 - 2.82843i) q^{21} +7.65685i q^{23} +1.82843 q^{25} +5.22625i q^{27} +3.65685i q^{29} -3.06147 q^{31} -2.16478i q^{33} +(-6.82843 - 1.08239i) q^{35} +7.65685i q^{37} -5.17157i q^{39} +9.55582i q^{41} +3.65685 q^{43} +4.77791 q^{45} -7.39104 q^{47} +(6.65685 + 2.16478i) q^{49} +3.31371 q^{51} +2.00000i q^{53} -5.22625 q^{55} -4.48528 q^{57} +8.47343i q^{59} +2.61313 q^{61} +(-4.77791 - 0.757359i) q^{63} -12.4853 q^{65} +15.6569 q^{67} -8.28772 q^{69} -8.82843i q^{71} -12.6173i q^{73} +1.97908i q^{75} +(5.22625 + 0.828427i) q^{77} -12.8284i q^{79} -0.171573 q^{81} +11.5349i q^{83} -8.00000i q^{85} -3.95815 q^{87} -2.16478i q^{89} +(12.4853 + 1.97908i) q^{91} -3.31371i q^{93} +10.8284i q^{95} +13.5140i q^{97} -3.65685 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^9	$ 8 q - 8 q^{9} - 16 q^{11} - 8 q^{25} - 32 q^{35} - 16 q^{43} + 8 q^{49} - 64 q^{51} + 32 q^{57} - 32 q^{65} + 80 q^{67} - 24 q^{81} + 32 q^{91} + 16 q^{99}+O(q^{100}) $ 8 * q - 8 * q^9 - 16 * q^11 - 8 * q^25 - 32 * q^35 - 16 * q^43 + 8 * q^49 - 64 * q^51 + 32 * q^57 - 32 * q^65 + 80 * q^67 - 24 * q^81 + 32 * q^91 + 16 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times$.

$n$	$1023$	$1025$	$1541$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$			1.08239i			0.624919i	0.949931	+	0.312460i	$0.101153\pi$
$3$			1.08239i			0.624919i	−0.949931	+	0.312460i	$0.898847\pi$
$4$		0			0
$5$	2.61313			1.16863			0.584313	−	0.811529i	$-0.301364\pi$
$5$	2.61313			1.16863			0.584313	+	0.811529i	$0.301364\pi$
$6$		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
$11$	−2.00000			−0.603023			−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000			−0.603023			−0.301511	+	0.953463i	$0.597491\pi$
$12$		0			0
$13$	−4.77791			−1.32515			−0.662577	−	0.748994i	$-0.730537\pi$
$13$	−4.77791			−1.32515			−0.662577	+	0.748994i	$0.730537\pi$
$14$		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
$19$			4.14386i			0.950667i	0.879806	+	0.475333i	$0.157673\pi$
$19$			4.14386i			0.950667i	−0.879806	+	0.475333i	$0.842327\pi$
$20$		0			0
$21$	0.448342	−	2.82843i	0.0978361	−	0.617213i
$22$		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
$27$			5.22625i			1.00579i
$28$		0			0
$29$			3.65685i			0.679061i	0.940595	+	0.339530i	$0.110268\pi$
$29$			3.65685i			0.679061i	−0.940595	+	0.339530i	$0.889732\pi$
$30$		0			0
$31$	−3.06147			−0.549856			−0.274928	−	0.961465i	$-0.588654\pi$
$31$	−3.06147			−0.549856			−0.274928	+	0.961465i	$0.588654\pi$
$32$		0			0
$33$		−	2.16478i		−	0.376841i
$34$		0			0
$35$	−6.82843	−	1.08239i	−1.15421	−	0.182958i
$36$		0			0
$37$			7.65685i			1.25878i	0.777090	+	0.629390i	$0.216695\pi$
$37$			7.65685i			1.25878i	−0.777090	+	0.629390i	$0.783305\pi$
$38$		0			0
$39$		−	5.17157i		−	0.828114i
$40$		0			0
$41$			9.55582i			1.49237i	0.665740	+	0.746184i	$0.268116\pi$
$41$			9.55582i			1.49237i	−0.665740	+	0.746184i	$0.731884\pi$
$42$		0			0
$43$	3.65685			0.557665			0.278833	−	0.960340i	$-0.410053\pi$
$43$	3.65685			0.557665			0.278833	+	0.960340i	$0.410053\pi$
$44$		0			0
$45$	4.77791			0.712249
$46$		0			0
$47$	−7.39104			−1.07809			−0.539047	−	0.842276i	$-0.681215\pi$
$47$	−7.39104			−1.07809			−0.539047	+	0.842276i	$0.681215\pi$
$48$		0			0
$49$	6.65685	+	2.16478i	0.950979	+	0.309255i
$50$		0			0
$51$	3.31371			0.464012
$52$		0			0
$53$			2.00000i			0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$			2.00000i			0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$		0			0
$55$	−5.22625			−0.704708
$56$		0			0
$57$	−4.48528			−0.594090
$58$		0			0
$59$			8.47343i			1.10315i	0.834126	+	0.551573i	$0.185972\pi$
$59$			8.47343i			1.10315i	−0.834126	+	0.551573i	$0.814028\pi$
$60$		0			0
$61$	2.61313			0.334576			0.167288	−	0.985908i	$-0.446499\pi$
$61$	2.61313			0.334576			0.167288	+	0.985908i	$0.446499\pi$
$62$		0			0
$63$	−4.77791	−	0.757359i	−0.601960	−	0.0954183i
$64$		0			0
$65$	−12.4853			−1.54861
$66$		0			0
$67$	15.6569			1.91279			0.956395	−	0.292078i	$-0.0943466\pi$
$67$	15.6569			1.91279			0.956395	+	0.292078i	$0.0943466\pi$
$68$		0			0
$69$	−8.28772			−0.997724
$70$		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	$-0.735595\pi$
$73$		−	12.6173i		−	1.47674i	0.674395	−	0.738371i	$-0.264405\pi$
$74$		0			0
$75$			1.97908i			0.228524i
$76$		0			0
$77$	5.22625	+	0.828427i	0.595587	+	0.0944080i
$78$		0			0
$79$		−	12.8284i		−	1.44331i	−0.692252	−	0.721655i	$-0.743382\pi$
$79$		−	12.8284i		−	1.44331i	0.692252	−	0.721655i	$-0.256618\pi$
$80$		0			0
$81$	−0.171573			−0.0190637
$82$		0			0
$83$			11.5349i			1.26612i	0.774103	+	0.633060i	$0.218201\pi$
$83$			11.5349i			1.26612i	−0.774103	+	0.633060i	$0.781799\pi$
$84$		0			0
$85$		−	8.00000i		−	0.867722i
$86$		0			0
$87$	−3.95815			−0.424358
$88$		0			0
$89$		−	2.16478i		−	0.229467i	−0.993396	−	0.114733i	$-0.963399\pi$
$89$		−	2.16478i		−	0.229467i	0.993396	−	0.114733i	$-0.0366014\pi$
$90$		0			0
$91$	12.4853	+	1.97908i	1.30881	+	0.207463i
$92$		0			0
$93$		−	3.31371i		−	0.343616i
$94$		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
$99$	−3.65685			−0.367528
$100$		0			0
$101$	5.67459			0.564643			0.282322	−	0.959320i	$-0.408895\pi$
$101$	5.67459			0.564643			0.282322	+	0.959320i	$0.408895\pi$
$102$		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
$107$	0.343146			0.0331732			0.0165866	−	0.999862i	$-0.494720\pi$
$107$	0.343146			0.0331732			0.0165866	+	0.999862i	$0.494720\pi$
$108$		0			0
$109$		−	5.31371i		−	0.508961i	−0.967078	−	0.254480i	$-0.918096\pi$
$109$		−	5.31371i		−	0.508961i	0.967078	−	0.254480i	$-0.0819045\pi$
$110$		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
$115$			20.0083i			1.86579i
$116$		0			0
$117$	−8.73606			−0.807649
$118$		0			0
$119$	−1.26810	+	8.00000i	−0.116247	+	0.733359i
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$	−10.3431			−0.932610
$124$		0			0
$125$	−8.28772			−0.741276
$126$		0			0
$127$			3.65685i			0.324493i	0.986750	+	0.162247i	$0.0518740\pi$
$127$			3.65685i			0.324493i	−0.986750	+	0.162247i	$0.948126\pi$
$128$		0			0
$129$			3.95815i			0.348496i
$130$		0			0
$131$		−	13.6997i		−	1.19695i	−0.801143	−	0.598473i	$-0.795774\pi$
$131$		−	13.6997i		−	1.19695i	0.801143	−	0.598473i	$-0.204226\pi$
$132$		0			0
$133$	1.71644	−	10.8284i	0.148834	−	0.938944i
$134$		0			0
$135$			13.6569i			1.17539i
$136$		0			0
$137$	−2.00000			−0.170872			−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000			−0.170872			−0.0854358	+	0.996344i	$0.527228\pi$
$138$		0			0
$139$		−	1.97908i		−	0.167863i	−0.996472	−	0.0839315i	$-0.973252\pi$
$139$		−	1.97908i		−	0.167863i	0.996472	−	0.0839315i	$-0.0267477\pi$
$140$		0			0
$141$		−	8.00000i		−	0.673722i
$142$		0			0
$143$	9.55582			0.799098
$144$		0			0
$145$			9.55582i			0.793568i
$146$		0			0
$147$	−2.34315	+	7.20533i	−0.193259	+	0.594285i
$148$		0			0
$149$			10.0000i			0.819232i	0.912258	+	0.409616i	$0.134337\pi$
$149$			10.0000i			0.819232i	−0.912258	+	0.409616i	$0.865663\pi$
$150$		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
$155$	−8.00000			−0.642575
$156$		0			0
$157$	5.67459			0.452882			0.226441	−	0.974025i	$-0.427291\pi$
$157$	5.67459			0.452882			0.226441	+	0.974025i	$0.427291\pi$
$158$		0			0
$159$	−2.16478			−0.171679
$160$		0			0
$161$	3.17157	−	20.0083i	0.249955	−	1.57688i
$162$		0			0
$163$	−17.3137			−1.35611			−0.678057	−	0.735009i	$-0.737178\pi$
$163$	−17.3137			−1.35611			−0.678057	+	0.735009i	$0.737178\pi$
$164$		0			0
$165$		−	5.65685i		−	0.440386i
$166$		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
$171$			7.57675i			0.579408i
$172$		0			0
$173$	−10.9008			−0.828776			−0.414388	−	0.910100i	$-0.636004\pi$
$173$	−10.9008			−0.828776			−0.414388	+	0.910100i	$0.636004\pi$
$174$		0			0
$175$	−4.77791	−	0.757359i	−0.361176	−	0.0572510i
$176$		0			0
$177$	−9.17157			−0.689378
$178$		0			0
$179$	2.00000			0.149487			0.0747435	−	0.997203i	$-0.476186\pi$
$179$	2.00000			0.149487			0.0747435	+	0.997203i	$0.476186\pi$
$180$		0			0
$181$	−1.71644			−0.127582			−0.0637911	−	0.997963i	$-0.520319\pi$
$181$	−1.71644			−0.127582			−0.0637911	+	0.997963i	$0.520319\pi$
$182$		0			0
$183$			2.82843i			0.209083i
$184$		0			0
$185$			20.0083i			1.47104i
$186$		0			0
$187$			6.12293i			0.447753i
$188$		0			0
$189$	2.16478	−	13.6569i	0.157465	−	0.993390i
$190$		0			0
$191$		−	4.82843i		−	0.349373i	−0.984624	−	0.174686i	$-0.944109\pi$
$191$		−	4.82843i		−	0.349373i	0.984624	−	0.174686i	$-0.0558911\pi$
$192$		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
$195$		−	13.5140i		−	0.967756i
$196$		0			0
$197$		−	3.65685i		−	0.260540i	−0.991479	−	0.130270i	$-0.958416\pi$
$197$		−	3.65685i		−	0.260540i	0.991479	−	0.130270i	$-0.0415844\pi$
$198$		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
$203$	1.51472	−	9.55582i	0.106312	−	0.670687i
$204$		0			0
$205$			24.9706i			1.74402i
$206$		0			0
$207$			14.0000i			0.973067i
$208$		0			0
$209$		−	8.28772i		−	0.573274i
$210$		0			0
$211$	10.9706			0.755245			0.377622	−	0.925960i	$-0.376742\pi$
$211$	10.9706			0.755245			0.377622	+	0.925960i	$0.376742\pi$
$212$		0			0
$213$	9.55582			0.654754
$214$		0			0
$215$	9.55582			0.651702
$216$		0			0
$217$	8.00000	+	1.26810i	0.543075	+	0.0860843i
$218$		0			0
$219$	13.6569			0.922845
$220$		0			0
$221$			14.6274i			0.983947i
$222$		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
$227$		−	18.5545i		−	1.23151i	−0.787939	−	0.615753i	$-0.788852\pi$
$227$		−	18.5545i		−	1.23151i	0.787939	−	0.615753i	$-0.211148\pi$
$228$		0			0
$229$	−2.98454			−0.197224			−0.0986121	−	0.995126i	$-0.531440\pi$
$229$	−2.98454			−0.197224			−0.0986121	+	0.995126i	$0.531440\pi$
$230$		0			0
$231$	−0.896683	+	5.65685i	−0.0589974	+	0.372194i
$232$		0			0
$233$	−18.0000			−1.17922			−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000			−1.17922			−0.589610	+	0.807688i	$0.700718\pi$
$234$		0			0
$235$	−19.3137			−1.25989
$236$		0			0
$237$	13.8854			0.901953
$238$		0			0
$239$			8.34315i			0.539673i	0.962906	+	0.269837i	$0.0869697\pi$
$239$			8.34315i			0.539673i	−0.962906	+	0.269837i	$0.913030\pi$
$240$		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
$243$			15.4930i			0.993879i
$244$		0			0
$245$	17.3952	+	5.65685i	1.11134	+	0.361403i
$246$		0			0
$247$		−	19.7990i		−	1.25978i
$248$		0			0
$249$	−12.4853			−0.791223
$250$		0			0
$251$			1.08239i			0.0683200i	0.999416	+	0.0341600i	$0.0108756\pi$
$251$			1.08239i			0.0683200i	−0.999416	+	0.0341600i	$0.989124\pi$
$252$		0			0
$253$		−	15.3137i		−	0.962765i
$254$		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
$259$	3.17157	−	20.0083i	0.197072	−	1.24326i
$260$		0			0
$261$			6.68629i			0.413871i
$262$		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
$267$	2.34315			0.143398
$268$		0			0
$269$	−13.4370			−0.819271			−0.409636	−	0.912249i	$-0.634344\pi$
$269$	−13.4370			−0.819271			−0.409636	+	0.912249i	$0.634344\pi$
$270$		0			0
$271$	1.79337			0.108939			0.0544696	−	0.998515i	$-0.482653\pi$
$271$	1.79337			0.108939			0.0544696	+	0.998515i	$0.482653\pi$
$272$		0			0
$273$	−2.14214	+	13.5140i	−0.129648	+	0.817903i
$274$		0			0
$275$	−3.65685			−0.220517
$276$		0			0
$277$			26.9706i			1.62050i	0.586082	+	0.810252i	$0.300670\pi$
$277$			26.9706i			1.62050i	−0.586082	+	0.810252i	$0.699330\pi$
$278$		0			0
$279$	−5.59767			−0.335124
$280$		0			0
$281$	28.6274			1.70777			0.853884	−	0.520463i	$-0.174241\pi$
$281$	28.6274			1.70777			0.853884	+	0.520463i	$0.174241\pi$
$282$		0			0
$283$			1.08239i			0.0643415i	0.999482	+	0.0321708i	$0.0102420\pi$
$283$			1.08239i			0.0643415i	−0.999482	+	0.0321708i	$0.989758\pi$
$284$		0			0
$285$	−11.7206			−0.694269
$286$		0			0
$287$	3.95815	−	24.9706i	0.233642	−	1.47397i
$288$		0			0
$289$	7.62742			0.448672
$290$		0			0
$291$	−14.6274			−0.857474
$292$		0			0
$293$	5.67459			0.331513			0.165757	−	0.986167i	$-0.446993\pi$
$293$	5.67459			0.331513			0.165757	+	0.986167i	$0.446993\pi$
$294$		0			0
$295$			22.1421i			1.28916i
$296$		0			0
$297$		−	10.4525i		−	0.606516i
$298$		0			0
$299$		−	36.5838i		−	2.11569i
$300$		0			0
$301$	−9.55582	−	1.51472i	−0.550788	−	0.0873069i
$302$		0			0
$303$			6.14214i			0.352856i
$304$		0			0
$305$	6.82843			0.390995
$306$		0			0
$307$			11.5349i			0.658331i	0.944272	+	0.329166i	$0.106767\pi$
$307$			11.5349i			0.658331i	−0.944272	+	0.329166i	$0.893233\pi$
$308$		0			0
$309$		−	18.3431i		−	1.04351i
$310$		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	$-0.853876\pi$
$313$		−	15.6788i		−	0.886216i	0.896468	−	0.443108i	$-0.146124\pi$
$314$		0			0
$315$	−12.4853	−	1.97908i	−0.703466	−	0.111508i
$316$		0			0
$317$			27.6569i			1.55336i	0.629893	+	0.776682i	$0.283099\pi$
$317$			27.6569i			1.55336i	−0.629893	+	0.776682i	$0.716901\pi$
$318$		0			0
$319$		−	7.31371i		−	0.409489i
$320$		0			0
$321$			0.371418i			0.0207305i
$322$		0			0
$323$	12.6863			0.705884
$324$		0			0
$325$	−8.73606			−0.484589
$326$		0			0
$327$	5.75152			0.318060
$328$		0			0
$329$	19.3137	+	3.06147i	1.06480	+	0.168784i
$330$		0			0
$331$	17.3137			0.951647			0.475824	−	0.879541i	$-0.342150\pi$
$331$	17.3137			0.951647			0.475824	+	0.879541i	$0.342150\pi$
$332$		0			0
$333$			14.0000i			0.767195i
$334$		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
$339$			9.55582i			0.519001i
$340$		0			0
$341$	6.12293			0.331576
$342$		0			0
$343$	−16.4985	−	8.41421i	−0.890836	−	0.454325i
$344$		0			0
$345$	−21.6569			−1.16597
$346$		0			0
$347$	19.6569			1.05524			0.527618	−	0.849482i	$-0.323085\pi$
$347$	19.6569			1.05524			0.527618	+	0.849482i	$0.323085\pi$
$348$		0			0
$349$	8.73606			0.467631			0.233815	−	0.972281i	$-0.424879\pi$
$349$	8.73606			0.467631			0.233815	+	0.972281i	$0.424879\pi$
$350$		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
$355$		−	23.0698i		−	1.22442i
$356$		0			0
$357$	−8.65914	−	1.37258i	−0.458290	−	0.0726448i
$358$		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
$363$		−	7.57675i		−	0.397676i
$364$		0			0
$365$		−	32.9706i		−	1.72576i
$366$		0			0
$367$	−10.4525			−0.545616			−0.272808	−	0.962068i	$-0.587952\pi$
$367$	−10.4525			−0.545616			−0.272808	+	0.962068i	$0.587952\pi$
$368$		0			0
$369$			17.4721i			0.909562i
$370$		0			0
$371$	0.828427	−	5.22625i	0.0430098	−	0.271333i
$372$		0			0
$373$		−	6.97056i		−	0.360922i	−0.983582	−	0.180461i	$-0.942241\pi$
$373$		−	6.97056i		−	0.360922i	0.983582	−	0.180461i	$-0.0577590\pi$
$374$		0			0
$375$		−	8.97056i		−	0.463238i
$376$		0			0
$377$		−	17.4721i		−	0.899860i
$378$		0			0
$379$	−5.31371			−0.272947			−0.136473	−	0.990644i	$-0.543577\pi$
$379$	−5.31371			−0.272947			−0.136473	+	0.990644i	$0.543577\pi$
$380$		0			0
$381$	−3.95815			−0.202782
$382$		0			0
$383$	11.7206			0.598895			0.299447	−	0.954113i	$-0.403198\pi$
$383$	11.7206			0.598895			0.299447	+	0.954113i	$0.403198\pi$
$384$		0			0
$385$	13.6569	+	2.16478i	0.696018	+	0.110328i
$386$		0			0
$387$	6.68629			0.339883
$388$		0			0
$389$		−	19.6569i		−	0.996642i	−0.866993	−	0.498321i	$-0.833950\pi$
$389$		−	19.6569i		−	0.996642i	0.866993	−	0.498321i	$-0.166050\pi$
$390$		0			0
$391$	23.4412			1.18547
$392$		0			0
$393$	14.8284			0.747995
$394$		0			0
$395$		−	33.5223i		−	1.68669i
$396$		0			0
$397$	33.9706			1.70494			0.852469	−	0.522778i	$-0.175104\pi$
$397$	33.9706			1.70494			0.852469	+	0.522778i	$0.175104\pi$
$398$		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
$403$	14.6274			0.728644
$404$		0			0
$405$	−0.448342			−0.0222783
$406$		0			0
$407$		−	15.3137i		−	0.759072i
$408$		0			0
$409$		−	0.896683i		−	0.0443381i	−0.999754	−	0.0221691i	$-0.992943\pi$
$409$		−	0.896683i		−	0.0443381i	0.999754	−	0.0221691i	$-0.00705721\pi$
$410$		0			0
$411$		−	2.16478i		−	0.106781i
$412$		0			0
$413$	3.50981	−	22.1421i	0.172706	−	1.08954i
$414$		0			0
$415$			30.1421i			1.47962i
$416$		0			0
$417$	2.14214			0.104901
$418$		0			0
$419$		−	34.6047i		−	1.69055i	−0.534332	−	0.845275i	$-0.679437\pi$
$419$		−	34.6047i		−	1.69055i	0.534332	−	0.845275i	$-0.320563\pi$
$420$		0			0
$421$			21.3137i			1.03877i	0.854541	+	0.519383i	$0.173838\pi$
$421$			21.3137i			1.03877i	−0.854541	+	0.519383i	$0.826162\pi$
$422$		0			0
$423$	−13.5140			−0.657072
$424$		0			0
$425$		−	5.59767i		−	0.271527i
$426$		0			0
$427$	−6.82843	−	1.08239i	−0.330451	−	0.0523806i
$428$		0			0
$429$			10.3431i			0.499372i
$430$		0			0
$431$		−	12.3431i		−	0.594548i	−0.954792	−	0.297274i	$-0.903922\pi$
$431$		−	12.3431i		−	0.594548i	0.954792	−	0.297274i	$-0.0960775\pi$
$432$		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
$435$	−10.3431			−0.495916
$436$		0			0
$437$	−31.7289			−1.51780
$438$		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
$443$	−10.0000			−0.475114			−0.237557	−	0.971374i	$-0.576347\pi$
$443$	−10.0000			−0.475114			−0.237557	+	0.971374i	$0.576347\pi$
$444$		0			0
$445$		−	5.65685i		−	0.268161i
$446$		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
$451$		−	19.1116i		−	0.899932i
$452$		0			0
$453$	9.03056			0.424293
$454$		0			0
$455$	32.6256	+	5.17157i	1.52951	+	0.242447i
$456$		0			0
$457$	−37.1127			−1.73606			−0.868029	−	0.496513i	$-0.834614\pi$
$457$	−37.1127			−1.73606			−0.868029	+	0.496513i	$0.834614\pi$
$458$		0			0
$459$	16.0000			0.746816
$460$		0			0
$461$	−10.3756			−0.483239			−0.241619	−	0.970371i	$-0.577679\pi$
$461$	−10.3756			−0.483239			−0.241619	+	0.970371i	$0.577679\pi$
$462$		0			0
$463$		−	16.1421i		−	0.750189i	−0.926987	−	0.375094i	$-0.877610\pi$
$463$		−	16.1421i		−	0.750189i	0.926987	−	0.375094i	$-0.122390\pi$
$464$		0			0
$465$		−	8.65914i		−	0.401558i
$466$		0			0
$467$		−	8.10201i		−	0.374916i	−0.982273	−	0.187458i	$-0.939975\pi$
$467$		−	8.10201i		−	0.374916i	0.982273	−	0.187458i	$-0.0600249\pi$
$468$		0			0
$469$	−40.9133	−	6.48528i	−1.88920	−	0.299462i
$470$		0			0
$471$			6.14214i			0.283015i
$472$		0			0
$473$	−7.31371			−0.336285
$474$		0			0
$475$			7.57675i			0.347645i
$476$		0			0
$477$			3.65685i			0.167436i
$478$		0			0
$479$	30.0894			1.37482			0.687410	−	0.726269i	$-0.258748\pi$
$479$	30.0894			1.37482			0.687410	+	0.726269i	$0.258748\pi$
$480$		0			0
$481$		−	36.5838i		−	1.66808i
$482$		0			0
$483$	21.6569	+	3.43289i	0.985421	+	0.156202i
$484$		0			0
$485$			35.3137i			1.60351i
$486$		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
$491$	−30.2843			−1.36671			−0.683355	−	0.730086i	$-0.739480\pi$
$491$	−30.2843			−1.36671			−0.683355	+	0.730086i	$0.739480\pi$
$492$		0			0
$493$	11.1953			0.504213
$494$		0			0
$495$	−9.55582			−0.429502
$496$		0			0
$497$	−3.65685	+	23.0698i	−0.164032	+	1.03482i
$498$		0			0
$499$	−17.3137			−0.775068			−0.387534	−	0.921855i	$-0.626673\pi$
$499$	−17.3137			−0.775068			−0.387534	+	0.921855i	$0.626673\pi$
$500$		0			0
$501$			7.02944i			0.314052i
$502$		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
$507$			10.6382i			0.472460i
$508$		0			0
$509$	26.0543			1.15484			0.577419	−	0.816448i	$-0.304060\pi$
$509$	26.0543			1.15484			0.577419	+	0.816448i	$0.304060\pi$
$510$		0			0
$511$	−5.22625	+	32.9706i	−0.231196	+	1.45853i
$512$		0			0
$513$	−21.6569			−0.956173
$514$		0			0
$515$	−44.2843			−1.95140
$516$		0			0
$517$	14.7821			0.650115
$518$		0			0
$519$		−	11.7990i		−	0.517918i
$520$		0			0
$521$			0.896683i			0.0392844i	0.999807	+	0.0196422i	$0.00625271\pi$
$521$			0.896683i			0.0392844i	−0.999807	+	0.0196422i	$0.993747\pi$
$522$		0			0
$523$			18.9259i			0.827573i	0.910374	+	0.413787i	$0.135794\pi$
$523$			18.9259i			0.827573i	−0.910374	+	0.413787i	$0.864206\pi$
$524$		0			0
$525$	0.819760	−	5.17157i	0.0357773	−	0.225706i
$526$		0			0
$527$			9.37258i			0.408276i
$528$		0			0
$529$	−35.6274			−1.54902
$530$		0			0
$531$			15.4930i			0.672341i
$532$		0			0
$533$		−	45.6569i		−	1.97762i
$534$		0			0
$535$	0.896683			0.0387670
$536$		0			0
$537$			2.16478i			0.0934173i
$538$		0			0
$539$	−13.3137	−	4.32957i	−0.573462	−	0.186488i
$540$		0			0
$541$			25.3137i			1.08832i	0.838981	+	0.544161i	$0.183152\pi$
$541$			25.3137i			1.08832i	−0.838981	+	0.544161i	$0.816848\pi$
$542$		0			0
$543$		−	1.85786i		−	0.0797286i
$544$		0			0
$545$		−	13.8854i		−	0.594785i
$546$		0			0
$547$	−33.3137			−1.42439			−0.712196	−	0.701981i	$-0.752299\pi$
$547$	−33.3137			−1.42439			−0.712196	+	0.701981i	$0.752299\pi$
$548$		0			0
$549$	4.77791			0.203916
$550$		0			0
$551$	−15.1535			−0.645560
$552$		0			0
$553$	−5.31371	+	33.5223i	−0.225962	+	1.42551i
$554$		0			0
$555$	−21.6569			−0.919282
$556$		0			0
$557$		−	5.31371i		−	0.225149i	−0.993643	−	0.112575i	$-0.964090\pi$
$557$		−	5.31371i		−	0.225149i	0.993643	−	0.112575i	$-0.0359097\pi$
$558$		0			0
$559$	−17.4721			−0.738992
$560$		0			0
$561$	−6.62742			−0.279810
$562$		0			0
$563$		−	4.51528i		−	0.190296i	−0.995463	−	0.0951481i	$-0.969668\pi$
$563$		−	4.51528i		−	0.190296i	0.995463	−	0.0951481i	$-0.0303325\pi$
$564$		0			0
$565$	23.0698			0.970553
$566$		0			0
$567$	0.448342	+	0.0710678i	0.0188286	+	0.00298457i
$568$		0			0
$569$	−7.85786			−0.329419			−0.164709	−	0.986342i	$-0.552669\pi$
$569$	−7.85786			−0.329419			−0.164709	+	0.986342i	$0.552669\pi$
$570$		0			0
$571$	8.34315			0.349150			0.174575	−	0.984644i	$-0.444145\pi$
$571$	8.34315			0.349150			0.174575	+	0.984644i	$0.444145\pi$
$572$		0			0
$573$	5.22625			0.218330
$574$		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
$579$		−	11.3492i		−	0.471656i
$580$		0			0
$581$	4.77791	−	30.1421i	0.198221	−	1.25051i
$582$		0			0
$583$		−	4.00000i		−	0.165663i
$584$		0			0
$585$	−22.8284			−0.943839
$586$		0			0
$587$		−	29.0070i		−	1.19725i	−0.801030	−	0.598624i	$-0.795714\pi$
$587$		−	29.0070i		−	1.19725i	0.801030	−	0.598624i	$-0.204286\pi$
$588$		0			0
$589$		−	12.6863i		−	0.522730i
$590$		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
$595$	−3.31371	+	20.9050i	−0.135849	+	0.857022i
$596$		0			0
$597$			13.6569i			0.558938i
$598$		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	$0.329729\pi$
$601$			42.1814i			1.72062i	−0.509774	+	0.860308i	$0.670271\pi$
$602$		0			0
$603$	28.6274			1.16580
$604$		0			0
$605$	−18.2919			−0.743671
$606$		0			0
$607$	17.3183			0.702927			0.351464	−	0.936202i	$-0.385684\pi$
$607$	17.3183			0.702927			0.351464	+	0.936202i	$0.385684\pi$
$608$		0			0
$609$	10.3431	+	1.63952i	0.419125	+	0.0664367i
$610$		0			0
$611$	35.3137			1.42864
$612$		0			0
$613$			5.31371i			0.214619i	0.994226	+	0.107309i	$0.0342235\pi$
$613$			5.31371i			0.214619i	−0.994226	+	0.107309i	$0.965776\pi$
$614$		0			0
$615$	−27.0279			−1.08987
$616$		0			0
$617$	−13.1127			−0.527897			−0.263949	−	0.964537i	$-0.585025\pi$
$617$	−13.1127			−0.527897			−0.263949	+	0.964537i	$0.585025\pi$
$618$		0			0
$619$			2.87576i			0.115586i	0.998329	+	0.0577932i	$0.0184064\pi$
$619$			2.87576i			0.115586i	−0.998329	+	0.0577932i	$0.981594\pi$
$620$		0			0
$621$	−40.0166			−1.60581
$622$		0			0
$623$	−0.896683	+	5.65685i	−0.0359248	+	0.226637i
$624$		0			0
$625$	−30.7990			−1.23196
$626$		0			0
$627$	8.97056			0.358250
$628$		0			0
$629$	23.4412			0.934662
$630$		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
$635$			9.55582i			0.379211i
$636$		0			0
$637$	−31.8059	−	10.3431i	−1.26019	−	0.409810i
$638$		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
$643$			17.1326i			0.675642i	0.941210	+	0.337821i	$0.109690\pi$
$643$			17.1326i			0.675642i	−0.941210	+	0.337821i	$0.890310\pi$
$644$		0			0
$645$			10.3431i			0.407261i
$646$		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
$651$	−1.37258	+	8.65914i	−0.0537958	+	0.339378i
$652$		0			0
$653$			2.68629i			0.105123i	0.998618	+	0.0525614i	$0.0167385\pi$
$653$			2.68629i			0.105123i	−0.998618	+	0.0525614i	$0.983261\pi$
$654$		0			0
$655$		−	35.7990i		−	1.39878i
$656$		0			0
$657$		−	23.0698i		−	0.900038i
$658$		0			0
$659$	17.0294			0.663373			0.331686	−	0.943390i	$-0.392382\pi$
$659$	17.0294			0.663373			0.331686	+	0.943390i	$0.392382\pi$
$660$		0			0
$661$	30.9092			1.20223			0.601114	−	0.799164i	$-0.294724\pi$
$661$	30.9092			1.20223			0.601114	+	0.799164i	$0.294724\pi$
$662$		0			0
$663$	−15.8326			−0.614887
$664$		0			0
$665$	4.48528	−	28.2960i	0.173932	−	1.09727i
$666$		0			0
$667$	−28.0000			−1.08416
$668$		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
$675$			9.55582i			0.367804i
$676$		0			0
$677$	−24.4148			−0.938338			−0.469169	−	0.883109i	$-0.655446\pi$
$677$	−24.4148			−0.938338			−0.469169	+	0.883109i	$0.655446\pi$
$678$		0			0
$679$	5.59767	−	35.3137i	0.214819	−	1.35522i
$680$		0			0
$681$	20.0833			0.769592
$682$		0			0
$683$	43.6569			1.67048			0.835242	−	0.549883i	$-0.185328\pi$
$683$	43.6569			1.67048			0.835242	+	0.549883i	$0.185328\pi$
$684$		0			0
$685$	−5.22625			−0.199685
$686$		0			0
$687$		−	3.23045i		−	0.123249i
$688$		0			0
$689$		−	9.55582i		−	0.364048i
$690$		0			0
$691$			32.4399i			1.23407i	0.786935	+	0.617036i	$0.211667\pi$
$691$			32.4399i			1.23407i	−0.786935	+	0.617036i	$0.788333\pi$
$692$		0			0
$693$	9.55582	+	1.51472i	0.362996	+	0.0575394i
$694$		0			0
$695$		−	5.17157i		−	0.196169i
$696$		0			0
$697$	29.2548			1.10811
$698$		0			0
$699$		−	19.4831i		−	0.736917i
$700$		0			0
$701$			9.31371i			0.351774i	0.984410	+	0.175887i	$0.0562793\pi$
$701$			9.31371i			0.351774i	−0.984410	+	0.175887i	$0.943721\pi$
$702$		0			0
$703$	−31.7289			−1.19668
$704$		0			0
$705$		−	20.9050i		−	0.787328i
$706$		0			0
$707$	−14.8284	−	2.35049i	−0.557680	−	0.0883994i
$708$		0			0
$709$			18.0000i			0.676004i	0.941145	+	0.338002i	$0.109751\pi$
$709$			18.0000i			0.676004i	−0.941145	+	0.338002i	$0.890249\pi$
$710$		0			0
$711$		−	23.4558i		−	0.879663i
$712$		0			0
$713$		−	23.4412i		−	0.877880i
$714$		0			0
$715$	24.9706			0.933846
$716$		0			0
$717$	−9.03056			−0.337252
$718$		0			0
$719$	34.4190			1.28361			0.641806	−	0.766867i	$-0.278186\pi$
$719$	34.4190			1.28361			0.641806	+	0.766867i	$0.278186\pi$
$720$		0			0
$721$	44.2843	+	7.01962i	1.64923	+	0.261424i
$722$		0			0
$723$	25.9411			0.964761
$724$		0			0
$725$			6.68629i			0.248323i
$726$		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
$731$		−	11.1953i		−	0.414075i
$732$		0			0
$733$	−24.4148			−0.901782			−0.450891	−	0.892579i	$-0.648894\pi$
$733$	−24.4148			−0.901782			−0.450891	+	0.892579i	$0.648894\pi$
$734$		0			0
$735$	−6.12293	+	18.8284i	−0.225848	+	0.694497i
$736$		0			0
$737$	−31.3137			−1.15346
$738$		0			0
$739$	16.6274			0.611649			0.305825	−	0.952088i	$-0.401068\pi$
$739$	16.6274			0.611649			0.305825	+	0.952088i	$0.401068\pi$
$740$		0			0
$741$	21.4303			0.787261
$742$		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
$747$			21.0907i			0.771669i
$748$		0			0
$749$	−0.896683	−	0.142136i	−0.0327641	−	0.00519352i
$750$		0			0
$751$			14.9706i			0.546284i	0.961974	+	0.273142i	$0.0880628\pi$
$751$			14.9706i			0.546284i	−0.961974	+	0.273142i	$0.911937\pi$
$752$		0			0
$753$	−1.17157			−0.0426945
$754$		0			0
$755$		−	21.8017i		−	0.793445i
$756$		0			0
$757$		−	49.3137i		−	1.79234i	−0.443714	−	0.896169i	$-0.646339\pi$
$757$		−	49.3137i		−	1.79234i	0.443714	−	0.896169i	$-0.353661\pi$
$758$		0			0
$759$	16.5754			0.601650
$760$		0			0
$761$			7.76245i			0.281389i	0.990053	+	0.140694i	$0.0449335\pi$
$761$			7.76245i			0.281389i	−0.990053	+	0.140694i	$0.955067\pi$
$762$		0			0
$763$	−2.20101	+	13.8854i	−0.0796819	+	0.502685i
$764$		0			0
$765$		−	14.6274i		−	0.528855i
$766$		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
$771$	6.62742			0.238681
$772$		0			0
$773$	−26.9510			−0.969361			−0.484680	−	0.874691i	$-0.661064\pi$
$773$	−26.9510			−0.969361			−0.484680	+	0.874691i	$0.661064\pi$
$774$		0			0
$775$	−5.59767			−0.201074
$776$		0			0
$777$	21.6569	+	3.43289i	0.776935	+	0.123154i
$778$		0			0
$779$	−39.5980			−1.41874
$780$		0			0
$781$			17.6569i			0.631812i
$782$		0			0
$783$	−19.1116			−0.682994
$784$		0			0
$785$	14.8284			0.529249
$786$		0			0
$787$		−	4.51528i		−	0.160952i	−0.996757	−	0.0804761i	$-0.974356\pi$
$787$		−	4.51528i		−	0.160952i	0.996757	−	0.0804761i	$-0.0256441\pi$
$788$		0			0
$789$	4.48342			0.159614
$790$		0			0
$791$	−23.0698	−	3.65685i	−0.820267	−	0.130023i
$792$		0			0
$793$	−12.4853			−0.443365
$794$		0			0
$795$	−5.65685			−0.200628
$796$		0			0
$797$	41.3617			1.46511			0.732553	−	0.680710i	$-0.238329\pi$
$797$	41.3617			1.46511			0.732553	+	0.680710i	$0.238329\pi$
$798$		0			0
$799$			22.6274i			0.800500i
$800$		0			0
$801$		−	3.95815i		−	0.139854i
$802$		0			0
$803$			25.2346i			0.890509i
$804$		0			0
$805$	8.28772	−	52.2843i	0.292104	−	1.84278i
$806$		0			0
$807$		−	14.5442i		−	0.511979i
$808$		0			0
$809$	17.5147			0.615785			0.307892	−	0.951421i	$-0.400376\pi$
$809$	17.5147			0.615785			0.307892	+	0.951421i	$0.400376\pi$
$810$		0			0
$811$			38.5628i			1.35412i	0.735926	+	0.677062i	$0.236747\pi$
$811$			38.5628i			1.35412i	−0.735926	+	0.677062i	$0.763253\pi$
$812$		0			0
$813$			1.94113i			0.0680782i
$814$		0			0
$815$	−45.2429			−1.58479
$816$		0			0
$817$			15.1535i			0.530154i
$818$		0			0
$819$	22.8284	+	3.61859i	0.797690	+	0.126444i
$820$		0			0
$821$			49.5980i			1.73098i	0.500925	+	0.865491i	$0.332993\pi$
$821$			49.5980i			1.73098i	−0.500925	+	0.865491i	$0.667007\pi$
$822$		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
$827$	43.2548			1.50412			0.752059	−	0.659096i	$-0.229061\pi$
$827$	43.2548			1.50412			0.752059	+	0.659096i	$0.229061\pi$
$828$		0			0
$829$	6.19986			0.215330			0.107665	−	0.994187i	$-0.465663\pi$
$829$	6.19986			0.215330			0.107665	+	0.994187i	$0.465663\pi$
$830$		0			0
$831$	−29.1927			−1.01268
$832$		0			0
$833$	6.62742	−	20.3797i	0.229626	−	0.706116i
$834$		0			0
$835$	16.9706			0.587291
$836$		0			0
$837$		−	16.0000i		−	0.553041i
$838$		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
$843$			30.9861i			1.06722i
$844$		0			0
$845$	25.6829			0.883519
$846$		0			0
$847$	18.2919	+	2.89949i	0.628516	+	0.0996278i
$848$		0			0
$849$	−1.17157			−0.0402083
$850$		0			0
$851$	−58.6274			−2.00972
$852$		0			0
$853$	−31.8059			−1.08901			−0.544506	−	0.838757i	$-0.683283\pi$
$853$	−31.8059			−1.08901			−0.544506	+	0.838757i	$0.683283\pi$
$854$		0			0
$855$			19.7990i			0.677111i
$856$		0			0
$857$		−	7.01962i		−	0.239786i	−0.992787	−	0.119893i	$-0.961745\pi$
$857$		−	7.01962i		−	0.239786i	0.992787	−	0.119893i	$-0.0382551\pi$
$858$		0			0
$859$		−	2.50434i		−	0.0854470i	−0.999087	−	0.0427235i	$-0.986397\pi$
$859$		−	2.50434i		−	0.0854470i	0.999087	−	0.0427235i	$-0.0136034\pi$
$860$		0			0
$861$	27.0279	+	4.28427i	0.921110	+	0.146008i
$862$		0			0
$863$		−	27.4558i		−	0.934608i	−0.884097	−	0.467304i	$-0.845225\pi$
$863$		−	27.4558i		−	0.934608i	0.884097	−	0.467304i	$-0.154775\pi$
$864$		0			0
$865$	−28.4853			−0.968529
$866$		0			0
$867$			8.25586i			0.280384i
$868$		0			0
$869$			25.6569i			0.870349i
$870$		0			0
$871$	−74.8070			−2.53474
$872$		0			0
$873$			24.7093i			0.836283i
$874$		0			0
$875$	21.6569	+	3.43289i	0.732135	+	0.116053i
$876$		0			0
$877$			0.343146i			0.0115872i	0.999983	+	0.00579360i	$0.00184417\pi$
$877$			0.343146i			0.0115872i	−0.999983	+	0.00579360i	$0.998156\pi$
$878$		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
$883$	−24.3431			−0.819212			−0.409606	−	0.912263i	$-0.634334\pi$
$883$	−24.3431			−0.819212			−0.409606	+	0.912263i	$0.634334\pi$
$884$		0			0
$885$	−23.9665			−0.805624
$886$		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
$891$	0.343146			0.0114958
$892$		0			0
$893$		−	30.6274i		−	1.02491i
$894$		0			0
$895$	5.22625			0.174694
$896$		0			0
$897$	39.5980			1.32214
$898$		0			0
$899$		−	11.1953i		−	0.373386i
$900$		0			0
$901$	6.12293			0.203985
$902$		0			0
$903$	1.63952	−	10.3431i	0.0545598	−	0.344198i
$904$		0			0
$905$	−4.48528			−0.149096
$906$		0			0
$907$	−26.0000			−0.863316			−0.431658	−	0.902037i	$-0.642071\pi$
$907$	−26.0000			−0.863316			−0.431658	+	0.902037i	$0.642071\pi$
$908$		0			0
$909$	10.3756			0.344136
$910$		0			0
$911$			24.3431i			0.806524i	0.915084	+	0.403262i	$0.132124\pi$
$911$			24.3431i			0.806524i	−0.915084	+	0.403262i	$0.867876\pi$
$912$		0			0
$913$		−	23.0698i		−	0.763499i
$914$		0			0
$915$			7.39104i			0.244340i
$916$		0			0
$917$	−5.67459	+	35.7990i	−0.187392	+	1.18219i
$918$		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
$923$			42.1814i			1.38842i
$924$		0			0
$925$			14.0000i			0.460317i
$926$		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
$931$	−8.97056	+	27.5851i	−0.293998	+	0.904064i
$932$		0			0
$933$			21.6569i			0.709014i
$934$		0			0
$935$			16.0000i			0.523256i
$936$		0			0
$937$		−	44.7176i		−	1.46086i	−0.682987	−	0.730431i	$-0.739319\pi$
$937$		−	44.7176i		−	1.46086i	0.682987	−	0.730431i	$-0.260681\pi$
$938$		0			0
$939$	16.9706			0.553813
$940$		0			0
$941$	24.7862			0.808008			0.404004	−	0.914757i	$-0.367618\pi$
$941$	24.7862			0.808008			0.404004	+	0.914757i	$0.367618\pi$
$942$		0			0
$943$	−73.1675			−2.38266
$944$		0			0
$945$	5.65685	−	35.6871i	0.184017	−	1.16090i
$946$		0			0
$947$	28.3431			0.921028			0.460514	−	0.887652i	$-0.347665\pi$
$947$	28.3431			0.921028			0.460514	+	0.887652i	$0.347665\pi$
$948$		0			0
$949$			60.2843i			1.95691i
$950$		0			0
$951$	−29.9356			−0.970727
$952$		0			0
$953$	−0.627417			−0.0203240			−0.0101620	−	0.999948i	$-0.503235\pi$
$953$	−0.627417			−0.0203240			−0.0101620	+	0.999948i	$0.503235\pi$
$954$		0			0
$955$		−	12.6173i		−	0.408286i
$956$		0			0
$957$	7.91630			0.255898
$958$		0			0
$959$	5.22625	+	0.828427i	0.168764	+	0.0267513i
$960$		0			0
$961$	−21.6274			−0.697659
$962$		0			0
$963$	0.627417			0.0202182
$964$		0			0
$965$	−27.3994			−0.882017
$966$		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
$971$			4.14386i			0.132983i	0.997787	+	0.0664914i	$0.0211805\pi$
$971$			4.14386i			0.132983i	−0.997787	+	0.0664914i	$0.978820\pi$
$972$		0			0
$973$	−0.819760	+	5.17157i	−0.0262803	+	0.165793i
$974$		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
$979$			4.32957i			0.138374i
$980$		0			0
$981$		−	9.71573i		−	0.310199i
$982$		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
$987$	−3.31371	+	20.9050i	−0.105477	+	0.665414i
$988$		0			0
$989$			28.0000i			0.890348i
$990$		0			0
$991$			14.4853i			0.460140i	0.973174	+	0.230070i	$0.0738955\pi$
$991$			14.4853i			0.460140i	−0.973174	+	0.230070i	$0.926104\pi$
$992$		0			0
$993$			18.7402i			0.594703i
$994$		0			0
$995$	32.9706			1.04524
$996$		0			0
$997$	17.3952			0.550911			0.275456	−	0.961314i	$-0.411171\pi$
$997$	17.3952			0.550911			0.275456	+	0.961314i	$0.411171\pi$
$998$		0			0
$999$	−40.0166			−1.26607

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1792.2.e.f.895.6		8
4.3	odd	2		1792.2.e.g.895.4		8
7.6	odd	2	inner	1792.2.e.f.895.3		8
8.3	odd	2	inner	1792.2.e.f.895.5		8
8.5	even	2		1792.2.e.g.895.3		8
16.3	odd	4		448.2.f.d.447.5		8
16.5	even	4		224.2.f.a.223.6	yes	8
16.11	odd	4		224.2.f.a.223.4	yes	8
16.13	even	4		448.2.f.d.447.3		8
28.27	even	2		1792.2.e.g.895.5		8
48.5	odd	4		2016.2.b.b.1567.2		8
48.11	even	4		2016.2.b.b.1567.1		8
48.29	odd	4		4032.2.b.p.3583.8		8
48.35	even	4		4032.2.b.p.3583.7		8
56.13	odd	2		1792.2.e.g.895.6		8
56.27	even	2	inner	1792.2.e.f.895.4		8
112.5	odd	12		1568.2.p.a.31.5		16
112.11	odd	12		1568.2.p.a.607.5		16
112.13	odd	4		448.2.f.d.447.6		8
112.27	even	4		224.2.f.a.223.5	yes	8
112.37	even	12		1568.2.p.a.31.4		16
112.53	even	12		1568.2.p.a.607.3		16
112.59	even	12		1568.2.p.a.607.4		16
112.69	odd	4		224.2.f.a.223.3	✓	8
112.75	even	12		1568.2.p.a.31.3		16
112.83	even	4		448.2.f.d.447.4		8
112.101	odd	12		1568.2.p.a.607.6		16
112.107	odd	12		1568.2.p.a.31.6		16
336.83	odd	4		4032.2.b.p.3583.2		8
336.125	even	4		4032.2.b.p.3583.1		8
336.251	odd	4		2016.2.b.b.1567.8		8
336.293	even	4		2016.2.b.b.1567.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.2.f.a.223.3	✓	8	112.69	odd	4
224.2.f.a.223.4	yes	8	16.11	odd	4
224.2.f.a.223.5	yes	8	112.27	even	4
224.2.f.a.223.6	yes	8	16.5	even	4
448.2.f.d.447.3		8	16.13	even	4
448.2.f.d.447.4		8	112.83	even	4
448.2.f.d.447.5		8	16.3	odd	4
448.2.f.d.447.6		8	112.13	odd	4
1568.2.p.a.31.3		16	112.75	even	12
1568.2.p.a.31.4		16	112.37	even	12
1568.2.p.a.31.5		16	112.5	odd	12
1568.2.p.a.31.6		16	112.107	odd	12
1568.2.p.a.607.3		16	112.53	even	12
1568.2.p.a.607.4		16	112.59	even	12
1568.2.p.a.607.5		16	112.11	odd	12
1568.2.p.a.607.6		16	112.101	odd	12
1792.2.e.f.895.3		8	7.6	odd	2	inner
1792.2.e.f.895.4		8	56.27	even	2	inner
1792.2.e.f.895.5		8	8.3	odd	2	inner
1792.2.e.f.895.6		8	1.1	even	1	trivial
1792.2.e.g.895.3		8	8.5	even	2
1792.2.e.g.895.4		8	4.3	odd	2
1792.2.e.g.895.5		8	28.27	even	2
1792.2.e.g.895.6		8	56.13	odd	2
2016.2.b.b.1567.1		8	48.11	even	4
2016.2.b.b.1567.2		8	48.5	odd	4
2016.2.b.b.1567.7		8	336.293	even	4
2016.2.b.b.1567.8		8	336.251	odd	4
4032.2.b.p.3583.1		8	336.125	even	4
4032.2.b.p.3583.2		8	336.83	odd	4
4032.2.b.p.3583.7		8	48.35	even	4
4032.2.b.p.3583.8		8	48.29	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:	\( N \)	\(=\)	\( 1792 = 2^{8} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1792.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.08239i q^{3} +2.61313 q^{5} +(-2.61313 - 0.414214i) q^{7} +1.82843 q^{9} +O(q^{10})\)	\(q+1.08239i q^{3} +2.61313 q^{5} +(-2.61313 - 0.414214i) q^{7} +1.82843 q^{9} -2.00000 q^{11} -4.77791 q^{13} +2.82843i q^{15} -3.06147i q^{17} +4.14386i q^{19} +(0.448342 - 2.82843i) q^{21} +7.65685i q^{23} +1.82843 q^{25} +5.22625i q^{27} +3.65685i q^{29} -3.06147 q^{31} -2.16478i q^{33} +(-6.82843 - 1.08239i) q^{35} +7.65685i q^{37} -5.17157i q^{39} +9.55582i q^{41} +3.65685 q^{43} +4.77791 q^{45} -7.39104 q^{47} +(6.65685 + 2.16478i) q^{49} +3.31371 q^{51} +2.00000i q^{53} -5.22625 q^{55} -4.48528 q^{57} +8.47343i q^{59} +2.61313 q^{61} +(-4.77791 - 0.757359i) q^{63} -12.4853 q^{65} +15.6569 q^{67} -8.28772 q^{69} -8.82843i q^{71} -12.6173i q^{73} +1.97908i q^{75} +(5.22625 + 0.828427i) q^{77} -12.8284i q^{79} -0.171573 q^{81} +11.5349i q^{83} -8.00000i q^{85} -3.95815 q^{87} -2.16478i q^{89} +(12.4853 + 1.97908i) q^{91} -3.31371i q^{93} +10.8284i q^{95} +13.5140i q^{97} -3.65685 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^9	\( 8 q - 8 q^{9} - 16 q^{11} - 8 q^{25} - 32 q^{35} - 16 q^{43} + 8 q^{49} - 64 q^{51} + 32 q^{57} - 32 q^{65} + 80 q^{67} - 24 q^{81} + 32 q^{91} + 16 q^{99}+O(q^{100}) \) 8 * q - 8 * q^9 - 16 * q^11 - 8 * q^25 - 32 * q^35 - 16 * q^43 + 8 * q^49 - 64 * q^51 + 32 * q^57 - 32 * q^65 + 80 * q^67 - 24 * q^81 + 32 * q^91 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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