LMFDB - Embedded newform 8550.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8550,2,Mod(1,8550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8550, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8550.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$68.2720937282$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2850)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8550.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{4} -4.00000 q^{7} -1.00000 q^{8} +O(q^{10})$

$q-1.00000 q^{2} +1.00000 q^{4} -4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{11} +4.00000 q^{14} +1.00000 q^{16} +8.00000 q^{17} +1.00000 q^{19} -1.00000 q^{22} +3.00000 q^{23} -4.00000 q^{28} +1.00000 q^{29} +1.00000 q^{31} -1.00000 q^{32} -8.00000 q^{34} -2.00000 q^{37} -1.00000 q^{38} +10.0000 q^{41} -8.00000 q^{43} +1.00000 q^{44} -3.00000 q^{46} +9.00000 q^{49} -3.00000 q^{53} +4.00000 q^{56} -1.00000 q^{58} -4.00000 q^{59} +5.00000 q^{61} -1.00000 q^{62} +1.00000 q^{64} -5.00000 q^{67} +8.00000 q^{68} +6.00000 q^{71} -13.0000 q^{73} +2.00000 q^{74} +1.00000 q^{76} -4.00000 q^{77} -5.00000 q^{79} -10.0000 q^{82} -11.0000 q^{83} +8.00000 q^{86} -1.00000 q^{88} -3.00000 q^{89} +3.00000 q^{92} +10.0000 q^{97} -9.00000 q^{98} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	4.00000	1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	8.00000	1.94029	0.970143	−	0.242536i	$-0.0779791\pi$
$17$	8.00000	1.94029	0.970143	+	0.242536i	$0.0779791\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	−1.00000	−0.213201
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	−4.00000	−0.755929
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	0	0
$31$	1.00000	0.179605	0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000	0.179605	0.0898027	+	0.995960i	$0.471376\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−8.00000	−1.37199
$35$	0	0
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−3.00000	−0.442326
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−3.00000	−0.412082	−0.206041	−	0.978543i	$-0.566058\pi$
$53$	−3.00000	−0.412082	−0.206041	+	0.978543i	$0.566058\pi$
$54$	0	0
$55$	0	0
$56$	4.00000	0.534522
$57$	0	0
$58$	−1.00000	−0.131306
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	−1.00000	−0.127000
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−5.00000	−0.610847	−0.305424	−	0.952217i	$-0.598798\pi$
$67$	−5.00000	−0.610847	−0.305424	+	0.952217i	$0.598798\pi$
$68$	8.00000	0.970143
$69$	0	0
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	−13.0000	−1.52153	−0.760767	−	0.649025i	$-0.775177\pi$
$73$	−13.0000	−1.52153	−0.760767	+	0.649025i	$0.775177\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	1.00000	0.114708
$77$	−4.00000	−0.455842
$78$	0	0
$79$	−5.00000	−0.562544	−0.281272	−	0.959628i	$-0.590756\pi$
$79$	−5.00000	−0.562544	−0.281272	+	0.959628i	$0.590756\pi$
$80$	0	0
$81$	0	0
$82$	−10.0000	−1.10432
$83$	−11.0000	−1.20741	−0.603703	−	0.797209i	$-0.706309\pi$
$83$	−11.0000	−1.20741	−0.603703	+	0.797209i	$0.706309\pi$
$84$	0	0
$85$	0	0
$86$	8.00000	0.862662
$87$	0	0
$88$	−1.00000	−0.106600
$89$	−3.00000	−0.317999	−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000	−0.317999	−0.159000	+	0.987279i	$0.550827\pi$
$90$	0	0
$91$	0	0
$92$	3.00000	0.312772
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	−9.00000	−0.909137
$99$	0	0
$100$	0	0
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	0	0
$103$	1.00000	0.0985329	0.0492665	−	0.998786i	$-0.484312\pi$
$103$	1.00000	0.0985329	0.0492665	+	0.998786i	$0.484312\pi$
$104$	0	0
$105$	0	0
$106$	3.00000	0.291386
$107$	−14.0000	−1.35343	−0.676716	−	0.736245i	$-0.736597\pi$
$107$	−14.0000	−1.35343	−0.676716	+	0.736245i	$0.736597\pi$
$108$	0	0
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	0	0
$111$	0	0
$112$	−4.00000	−0.377964
$113$	1.00000	0.0940721	0.0470360	−	0.998893i	$-0.485022\pi$
$113$	1.00000	0.0940721	0.0470360	+	0.998893i	$0.485022\pi$
$114$	0	0
$115$	0	0
$116$	1.00000	0.0928477
$117$	0	0
$118$	4.00000	0.368230
$119$	−32.0000	−2.93344
$120$	0	0
$121$	−10.0000	−0.909091
$122$	−5.00000	−0.452679
$123$	0	0
$124$	1.00000	0.0898027
$125$	0	0
$126$	0	0
$127$	−17.0000	−1.50851	−0.754253	−	0.656584i	$-0.772001\pi$
$127$	−17.0000	−1.50851	−0.754253	+	0.656584i	$0.772001\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	15.0000	1.31056	0.655278	−	0.755388i	$-0.272551\pi$
$131$	15.0000	1.31056	0.655278	+	0.755388i	$0.272551\pi$
$132$	0	0
$133$	−4.00000	−0.346844
$134$	5.00000	0.431934
$135$	0	0
$136$	−8.00000	−0.685994
$137$	−4.00000	−0.341743	−0.170872	−	0.985293i	$-0.554658\pi$
$137$	−4.00000	−0.341743	−0.170872	+	0.985293i	$0.554658\pi$
$138$	0	0
$139$	6.00000	0.508913	0.254457	−	0.967084i	$-0.418103\pi$
$139$	6.00000	0.508913	0.254457	+	0.967084i	$0.418103\pi$
$140$	0	0
$141$	0	0
$142$	−6.00000	−0.503509
$143$	0	0
$144$	0	0
$145$	0	0
$146$	13.0000	1.07589
$147$	0	0
$148$	−2.00000	−0.164399
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	−1.00000	−0.0811107
$153$	0	0
$154$	4.00000	0.322329
$155$	0	0
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	5.00000	0.397779
$159$	0	0
$160$	0	0
$161$	−12.0000	−0.945732
$162$	0	0
$163$	10.0000	0.783260	0.391630	−	0.920123i	$-0.371911\pi$
$163$	10.0000	0.783260	0.391630	+	0.920123i	$0.371911\pi$
$164$	10.0000	0.780869
$165$	0	0
$166$	11.0000	0.853766
$167$	22.0000	1.70241	0.851206	−	0.524832i	$-0.175872\pi$
$167$	22.0000	1.70241	0.851206	+	0.524832i	$0.175872\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	−8.00000	−0.609994
$173$	11.0000	0.836315	0.418157	−	0.908375i	$-0.362676\pi$
$173$	11.0000	0.836315	0.418157	+	0.908375i	$0.362676\pi$
$174$	0	0
$175$	0	0
$176$	1.00000	0.0753778
$177$	0	0
$178$	3.00000	0.224860
$179$	−20.0000	−1.49487	−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000	−1.49487	−0.747435	+	0.664335i	$0.768715\pi$
$180$	0	0
$181$	12.0000	0.891953	0.445976	−	0.895045i	$-0.352856\pi$
$181$	12.0000	0.891953	0.445976	+	0.895045i	$0.352856\pi$
$182$	0	0
$183$	0	0
$184$	−3.00000	−0.221163
$185$	0	0
$186$	0	0
$187$	8.00000	0.585018
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−13.0000	−0.940647	−0.470323	−	0.882494i	$-0.655863\pi$
$191$	−13.0000	−0.940647	−0.470323	+	0.882494i	$0.655863\pi$
$192$	0	0
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	9.00000	0.642857
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	26.0000	1.84309	0.921546	−	0.388270i	$-0.126927\pi$
$199$	26.0000	1.84309	0.921546	+	0.388270i	$0.126927\pi$
$200$	0	0
$201$	0	0
$202$	−12.0000	−0.844317
$203$	−4.00000	−0.280745
$204$	0	0
$205$	0	0
$206$	−1.00000	−0.0696733
$207$	0	0
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	25.0000	1.72107	0.860535	−	0.509390i	$-0.170129\pi$
$211$	25.0000	1.72107	0.860535	+	0.509390i	$0.170129\pi$
$212$	−3.00000	−0.206041
$213$	0	0
$214$	14.0000	0.957020
$215$	0	0
$216$	0	0
$217$	−4.00000	−0.271538
$218$	−6.00000	−0.406371
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	29.0000	1.94198	0.970992	−	0.239113i	$-0.0768565\pi$
$223$	29.0000	1.94198	0.970992	+	0.239113i	$0.0768565\pi$
$224$	4.00000	0.267261
$225$	0	0
$226$	−1.00000	−0.0665190
$227$	30.0000	1.99117	0.995585	−	0.0938647i	$-0.0299221\pi$
$227$	30.0000	1.99117	0.995585	+	0.0938647i	$0.0299221\pi$
$228$	0	0
$229$	13.0000	0.859064	0.429532	−	0.903052i	$-0.358679\pi$
$229$	13.0000	0.859064	0.429532	+	0.903052i	$0.358679\pi$
$230$	0	0
$231$	0	0
$232$	−1.00000	−0.0656532
$233$	4.00000	0.262049	0.131024	−	0.991379i	$-0.458173\pi$
$233$	4.00000	0.262049	0.131024	+	0.991379i	$0.458173\pi$
$234$	0	0
$235$	0	0
$236$	−4.00000	−0.260378
$237$	0	0
$238$	32.0000	2.07425
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	10.0000	0.642824
$243$	0	0
$244$	5.00000	0.320092
$245$	0	0
$246$	0	0
$247$	0	0
$248$	−1.00000	−0.0635001
$249$	0	0
$250$	0	0
$251$	−16.0000	−1.00991	−0.504956	−	0.863145i	$-0.668491\pi$
$251$	−16.0000	−1.00991	−0.504956	+	0.863145i	$0.668491\pi$
$252$	0	0
$253$	3.00000	0.188608
$254$	17.0000	1.06667
$255$	0	0
$256$	1.00000	0.0625000
$257$	27.0000	1.68421	0.842107	−	0.539311i	$-0.181315\pi$
$257$	27.0000	1.68421	0.842107	+	0.539311i	$0.181315\pi$
$258$	0	0
$259$	8.00000	0.497096
$260$	0	0
$261$	0	0
$262$	−15.0000	−0.926703
$263$	19.0000	1.17159	0.585795	−	0.810459i	$-0.300782\pi$
$263$	19.0000	1.17159	0.585795	+	0.810459i	$0.300782\pi$
$264$	0	0
$265$	0	0
$266$	4.00000	0.245256
$267$	0	0
$268$	−5.00000	−0.305424
$269$	26.0000	1.58525	0.792624	−	0.609711i	$-0.208714\pi$
$269$	26.0000	1.58525	0.792624	+	0.609711i	$0.208714\pi$
$270$	0	0
$271$	−14.0000	−0.850439	−0.425220	−	0.905090i	$-0.639803\pi$
$271$	−14.0000	−0.850439	−0.425220	+	0.905090i	$0.639803\pi$
$272$	8.00000	0.485071
$273$	0	0
$274$	4.00000	0.241649
$275$	0	0
$276$	0	0
$277$	17.0000	1.02143	0.510716	−	0.859750i	$-0.329381\pi$
$277$	17.0000	1.02143	0.510716	+	0.859750i	$0.329381\pi$
$278$	−6.00000	−0.359856
$279$	0	0
$280$	0	0
$281$	15.0000	0.894825	0.447412	−	0.894328i	$-0.352346\pi$
$281$	15.0000	0.894825	0.447412	+	0.894328i	$0.352346\pi$
$282$	0	0
$283$	32.0000	1.90220	0.951101	−	0.308879i	$-0.0999539\pi$
$283$	32.0000	1.90220	0.951101	+	0.308879i	$0.0999539\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	0	0
$287$	−40.0000	−2.36113
$288$	0	0
$289$	47.0000	2.76471
$290$	0	0
$291$	0	0
$292$	−13.0000	−0.760767
$293$	−1.00000	−0.0584206	−0.0292103	−	0.999573i	$-0.509299\pi$
$293$	−1.00000	−0.0584206	−0.0292103	+	0.999573i	$0.509299\pi$
$294$	0	0
$295$	0	0
$296$	2.00000	0.116248
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	32.0000	1.84445
$302$	0	0
$303$	0	0
$304$	1.00000	0.0573539
$305$	0	0
$306$	0	0
$307$	−5.00000	−0.285365	−0.142683	−	0.989769i	$-0.545573\pi$
$307$	−5.00000	−0.285365	−0.142683	+	0.989769i	$0.545573\pi$
$308$	−4.00000	−0.227921
$309$	0	0
$310$	0	0
$311$	16.0000	0.907277	0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000	0.907277	0.453638	+	0.891186i	$0.350126\pi$
$312$	0	0
$313$	−21.0000	−1.18699	−0.593495	−	0.804838i	$-0.702252\pi$
$313$	−21.0000	−1.18699	−0.593495	+	0.804838i	$0.702252\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	−5.00000	−0.281272
$317$	−9.00000	−0.505490	−0.252745	−	0.967533i	$-0.581333\pi$
$317$	−9.00000	−0.505490	−0.252745	+	0.967533i	$0.581333\pi$
$318$	0	0
$319$	1.00000	0.0559893
$320$	0	0
$321$	0	0
$322$	12.0000	0.668734
$323$	8.00000	0.445132
$324$	0	0
$325$	0	0
$326$	−10.0000	−0.553849
$327$	0	0
$328$	−10.0000	−0.552158
$329$	0	0
$330$	0	0
$331$	−1.00000	−0.0549650	−0.0274825	−	0.999622i	$-0.508749\pi$
$331$	−1.00000	−0.0549650	−0.0274825	+	0.999622i	$0.508749\pi$
$332$	−11.0000	−0.603703
$333$	0	0
$334$	−22.0000	−1.20379
$335$	0	0
$336$	0	0
$337$	−16.0000	−0.871576	−0.435788	−	0.900049i	$-0.643530\pi$
$337$	−16.0000	−0.871576	−0.435788	+	0.900049i	$0.643530\pi$
$338$	13.0000	0.707107
$339$	0	0
$340$	0	0
$341$	1.00000	0.0541530
$342$	0	0
$343$	−8.00000	−0.431959
$344$	8.00000	0.431331
$345$	0	0
$346$	−11.0000	−0.591364
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	−25.0000	−1.33822	−0.669110	−	0.743164i	$-0.733324\pi$
$349$	−25.0000	−1.33822	−0.669110	+	0.743164i	$0.733324\pi$
$350$	0	0
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	−3.00000	−0.159000
$357$	0	0
$358$	20.0000	1.05703
$359$	−16.0000	−0.844448	−0.422224	−	0.906492i	$-0.638750\pi$
$359$	−16.0000	−0.844448	−0.422224	+	0.906492i	$0.638750\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−12.0000	−0.630706
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−22.0000	−1.14839	−0.574195	−	0.818718i	$-0.694685\pi$
$367$	−22.0000	−1.14839	−0.574195	+	0.818718i	$0.694685\pi$
$368$	3.00000	0.156386
$369$	0	0
$370$	0	0
$371$	12.0000	0.623009
$372$	0	0
$373$	32.0000	1.65690	0.828449	−	0.560065i	$-0.189224\pi$
$373$	32.0000	1.65690	0.828449	+	0.560065i	$0.189224\pi$
$374$	−8.00000	−0.413670
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−32.0000	−1.64373	−0.821865	−	0.569683i	$-0.807066\pi$
$379$	−32.0000	−1.64373	−0.821865	+	0.569683i	$0.807066\pi$
$380$	0	0
$381$	0	0
$382$	13.0000	0.665138
$383$	18.0000	0.919757	0.459879	−	0.887982i	$-0.347893\pi$
$383$	18.0000	0.919757	0.459879	+	0.887982i	$0.347893\pi$
$384$	0	0
$385$	0	0
$386$	14.0000	0.712581
$387$	0	0
$388$	10.0000	0.507673
$389$	22.0000	1.11544	0.557722	−	0.830028i	$-0.311675\pi$
$389$	22.0000	1.11544	0.557722	+	0.830028i	$0.311675\pi$
$390$	0	0
$391$	24.0000	1.21373
$392$	−9.00000	−0.454569
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	35.0000	1.75660	0.878300	−	0.478110i	$-0.158678\pi$
$397$	35.0000	1.75660	0.878300	+	0.478110i	$0.158678\pi$
$398$	−26.0000	−1.30326
$399$	0	0
$400$	0	0
$401$	5.00000	0.249688	0.124844	−	0.992176i	$-0.460157\pi$
$401$	5.00000	0.249688	0.124844	+	0.992176i	$0.460157\pi$
$402$	0	0
$403$	0	0
$404$	12.0000	0.597022
$405$	0	0
$406$	4.00000	0.198517
$407$	−2.00000	−0.0991363
$408$	0	0
$409$	−18.0000	−0.890043	−0.445021	−	0.895520i	$-0.646804\pi$
$409$	−18.0000	−0.890043	−0.445021	+	0.895520i	$0.646804\pi$
$410$	0	0
$411$	0	0
$412$	1.00000	0.0492665
$413$	16.0000	0.787309
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	−1.00000	−0.0489116
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	−25.0000	−1.21698
$423$	0	0
$424$	3.00000	0.145693
$425$	0	0
$426$	0	0
$427$	−20.0000	−0.967868
$428$	−14.0000	−0.676716
$429$	0	0
$430$	0	0
$431$	14.0000	0.674356	0.337178	−	0.941441i	$-0.390528\pi$
$431$	14.0000	0.674356	0.337178	+	0.941441i	$0.390528\pi$
$432$	0	0
$433$	−22.0000	−1.05725	−0.528626	−	0.848855i	$-0.677293\pi$
$433$	−22.0000	−1.05725	−0.528626	+	0.848855i	$0.677293\pi$
$434$	4.00000	0.192006
$435$	0	0
$436$	6.00000	0.287348
$437$	3.00000	0.143509
$438$	0	0
$439$	7.00000	0.334092	0.167046	−	0.985949i	$-0.446577\pi$
$439$	7.00000	0.334092	0.167046	+	0.985949i	$0.446577\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−3.00000	−0.142534	−0.0712672	−	0.997457i	$-0.522704\pi$
$443$	−3.00000	−0.142534	−0.0712672	+	0.997457i	$0.522704\pi$
$444$	0	0
$445$	0	0
$446$	−29.0000	−1.37319
$447$	0	0
$448$	−4.00000	−0.188982
$449$	23.0000	1.08544	0.542719	−	0.839915i	$-0.317395\pi$
$449$	23.0000	1.08544	0.542719	+	0.839915i	$0.317395\pi$
$450$	0	0
$451$	10.0000	0.470882
$452$	1.00000	0.0470360
$453$	0	0
$454$	−30.0000	−1.40797
$455$	0	0
$456$	0	0
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	−13.0000	−0.607450
$459$	0	0
$460$	0	0
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	0	0
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	1.00000	0.0464238
$465$	0	0
$466$	−4.00000	−0.185296
$467$	3.00000	0.138823	0.0694117	−	0.997588i	$-0.477888\pi$
$467$	3.00000	0.138823	0.0694117	+	0.997588i	$0.477888\pi$
$468$	0	0
$469$	20.0000	0.923514
$470$	0	0
$471$	0	0
$472$	4.00000	0.184115
$473$	−8.00000	−0.367840
$474$	0	0
$475$	0	0
$476$	−32.0000	−1.46672
$477$	0	0
$478$	24.0000	1.09773
$479$	−21.0000	−0.959514	−0.479757	−	0.877401i	$-0.659275\pi$
$479$	−21.0000	−0.959514	−0.479757	+	0.877401i	$0.659275\pi$
$480$	0	0
$481$	0	0
$482$	−8.00000	−0.364390
$483$	0	0
$484$	−10.0000	−0.454545
$485$	0	0
$486$	0	0
$487$	−32.0000	−1.45006	−0.725029	−	0.688718i	$-0.758174\pi$
$487$	−32.0000	−1.45006	−0.725029	+	0.688718i	$0.758174\pi$
$488$	−5.00000	−0.226339
$489$	0	0
$490$	0	0
$491$	−4.00000	−0.180517	−0.0902587	−	0.995918i	$-0.528769\pi$
$491$	−4.00000	−0.180517	−0.0902587	+	0.995918i	$0.528769\pi$
$492$	0	0
$493$	8.00000	0.360302
$494$	0	0
$495$	0	0
$496$	1.00000	0.0449013
$497$	−24.0000	−1.07655
$498$	0	0
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	0	0
$501$	0	0
$502$	16.0000	0.714115
$503$	4.00000	0.178351	0.0891756	−	0.996016i	$-0.471577\pi$
$503$	4.00000	0.178351	0.0891756	+	0.996016i	$0.471577\pi$
$504$	0	0
$505$	0	0
$506$	−3.00000	−0.133366
$507$	0	0
$508$	−17.0000	−0.754253
$509$	−25.0000	−1.10811	−0.554053	−	0.832482i	$-0.686919\pi$
$509$	−25.0000	−1.10811	−0.554053	+	0.832482i	$0.686919\pi$
$510$	0	0
$511$	52.0000	2.30034
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−27.0000	−1.19092
$515$	0	0
$516$	0	0
$517$	0	0
$518$	−8.00000	−0.351500
$519$	0	0
$520$	0	0
$521$	1.00000	0.0438108	0.0219054	−	0.999760i	$-0.493027\pi$
$521$	1.00000	0.0438108	0.0219054	+	0.999760i	$0.493027\pi$
$522$	0	0
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	15.0000	0.655278
$525$	0	0
$526$	−19.0000	−0.828439
$527$	8.00000	0.348485
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	0	0
$532$	−4.00000	−0.173422
$533$	0	0
$534$	0	0
$535$	0	0
$536$	5.00000	0.215967
$537$	0	0
$538$	−26.0000	−1.12094
$539$	9.00000	0.387657
$540$	0	0
$541$	27.0000	1.16082	0.580410	−	0.814324i	$-0.302892\pi$
$541$	27.0000	1.16082	0.580410	+	0.814324i	$0.302892\pi$
$542$	14.0000	0.601351
$543$	0	0
$544$	−8.00000	−0.342997
$545$	0	0
$546$	0	0
$547$	21.0000	0.897895	0.448948	−	0.893558i	$-0.351799\pi$
$547$	21.0000	0.897895	0.448948	+	0.893558i	$0.351799\pi$
$548$	−4.00000	−0.170872
$549$	0	0
$550$	0	0
$551$	1.00000	0.0426014
$552$	0	0
$553$	20.0000	0.850487
$554$	−17.0000	−0.722261
$555$	0	0
$556$	6.00000	0.254457
$557$	−36.0000	−1.52537	−0.762684	−	0.646771i	$-0.776119\pi$
$557$	−36.0000	−1.52537	−0.762684	+	0.646771i	$0.776119\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−15.0000	−0.632737
$563$	−22.0000	−0.927189	−0.463595	−	0.886047i	$-0.653441\pi$
$563$	−22.0000	−0.927189	−0.463595	+	0.886047i	$0.653441\pi$
$564$	0	0
$565$	0	0
$566$	−32.0000	−1.34506
$567$	0	0
$568$	−6.00000	−0.251754
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	6.00000	0.251092	0.125546	−	0.992088i	$-0.459932\pi$
$571$	6.00000	0.251092	0.125546	+	0.992088i	$0.459932\pi$
$572$	0	0
$573$	0	0
$574$	40.0000	1.66957
$575$	0	0
$576$	0	0
$577$	−21.0000	−0.874241	−0.437121	−	0.899403i	$-0.644002\pi$
$577$	−21.0000	−0.874241	−0.437121	+	0.899403i	$0.644002\pi$
$578$	−47.0000	−1.95494
$579$	0	0
$580$	0	0
$581$	44.0000	1.82543
$582$	0	0
$583$	−3.00000	−0.124247
$584$	13.0000	0.537944
$585$	0	0
$586$	1.00000	0.0413096
$587$	33.0000	1.36206	0.681028	−	0.732257i	$-0.261533\pi$
$587$	33.0000	1.36206	0.681028	+	0.732257i	$0.261533\pi$
$588$	0	0
$589$	1.00000	0.0412043
$590$	0	0
$591$	0	0
$592$	−2.00000	−0.0821995
$593$	20.0000	0.821302	0.410651	−	0.911793i	$-0.365302\pi$
$593$	20.0000	0.821302	0.410651	+	0.911793i	$0.365302\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	−38.0000	−1.55005	−0.775026	−	0.631929i	$-0.782263\pi$
$601$	−38.0000	−1.55005	−0.775026	+	0.631929i	$0.782263\pi$
$602$	−32.0000	−1.30422
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	27.0000	1.09590	0.547948	−	0.836512i	$-0.315409\pi$
$607$	27.0000	1.09590	0.547948	+	0.836512i	$0.315409\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	38.0000	1.53481	0.767403	−	0.641165i	$-0.221549\pi$
$613$	38.0000	1.53481	0.767403	+	0.641165i	$0.221549\pi$
$614$	5.00000	0.201784
$615$	0	0
$616$	4.00000	0.161165
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	0	0
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	0	0
$622$	−16.0000	−0.641542
$623$	12.0000	0.480770
$624$	0	0
$625$	0	0
$626$	21.0000	0.839329
$627$	0	0
$628$	2.00000	0.0798087
$629$	−16.0000	−0.637962
$630$	0	0
$631$	44.0000	1.75161	0.875806	−	0.482663i	$-0.160330\pi$
$631$	44.0000	1.75161	0.875806	+	0.482663i	$0.160330\pi$
$632$	5.00000	0.198889
$633$	0	0
$634$	9.00000	0.357436
$635$	0	0
$636$	0	0
$637$	0	0
$638$	−1.00000	−0.0395904
$639$	0	0
$640$	0	0
$641$	−6.00000	−0.236986	−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000	−0.236986	−0.118493	+	0.992955i	$0.537806\pi$
$642$	0	0
$643$	2.00000	0.0788723	0.0394362	−	0.999222i	$-0.487444\pi$
$643$	2.00000	0.0788723	0.0394362	+	0.999222i	$0.487444\pi$
$644$	−12.0000	−0.472866
$645$	0	0
$646$	−8.00000	−0.314756
$647$	−43.0000	−1.69050	−0.845252	−	0.534368i	$-0.820550\pi$
$647$	−43.0000	−1.69050	−0.845252	+	0.534368i	$0.820550\pi$
$648$	0	0
$649$	−4.00000	−0.157014
$650$	0	0
$651$	0	0
$652$	10.0000	0.391630
$653$	24.0000	0.939193	0.469596	−	0.882881i	$-0.344399\pi$
$653$	24.0000	0.939193	0.469596	+	0.882881i	$0.344399\pi$
$654$	0	0
$655$	0	0
$656$	10.0000	0.390434
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−8.00000	−0.311164	−0.155582	−	0.987823i	$-0.549725\pi$
$661$	−8.00000	−0.311164	−0.155582	+	0.987823i	$0.549725\pi$
$662$	1.00000	0.0388661
$663$	0	0
$664$	11.0000	0.426883
$665$	0	0
$666$	0	0
$667$	3.00000	0.116160
$668$	22.0000	0.851206
$669$	0	0
$670$	0	0
$671$	5.00000	0.193023
$672$	0	0
$673$	16.0000	0.616755	0.308377	−	0.951264i	$-0.400214\pi$
$673$	16.0000	0.616755	0.308377	+	0.951264i	$0.400214\pi$
$674$	16.0000	0.616297
$675$	0	0
$676$	−13.0000	−0.500000
$677$	−3.00000	−0.115299	−0.0576497	−	0.998337i	$-0.518361\pi$
$677$	−3.00000	−0.115299	−0.0576497	+	0.998337i	$0.518361\pi$
$678$	0	0
$679$	−40.0000	−1.53506
$680$	0	0
$681$	0	0
$682$	−1.00000	−0.0382920
$683$	6.00000	0.229584	0.114792	−	0.993390i	$-0.463380\pi$
$683$	6.00000	0.229584	0.114792	+	0.993390i	$0.463380\pi$
$684$	0	0
$685$	0	0
$686$	8.00000	0.305441
$687$	0	0
$688$	−8.00000	−0.304997
$689$	0	0
$690$	0	0
$691$	10.0000	0.380418	0.190209	−	0.981744i	$-0.439083\pi$
$691$	10.0000	0.380418	0.190209	+	0.981744i	$0.439083\pi$
$692$	11.0000	0.418157
$693$	0	0
$694$	12.0000	0.455514
$695$	0	0
$696$	0	0
$697$	80.0000	3.03022
$698$	25.0000	0.946264
$699$	0	0
$700$	0	0
$701$	8.00000	0.302156	0.151078	−	0.988522i	$-0.451726\pi$
$701$	8.00000	0.302156	0.151078	+	0.988522i	$0.451726\pi$
$702$	0	0
$703$	−2.00000	−0.0754314
$704$	1.00000	0.0376889
$705$	0	0
$706$	0	0
$707$	−48.0000	−1.80523
$708$	0	0
$709$	−35.0000	−1.31445	−0.657226	−	0.753693i	$-0.728270\pi$
$709$	−35.0000	−1.31445	−0.657226	+	0.753693i	$0.728270\pi$
$710$	0	0
$711$	0	0
$712$	3.00000	0.112430
$713$	3.00000	0.112351
$714$	0	0
$715$	0	0
$716$	−20.0000	−0.747435
$717$	0	0
$718$	16.0000	0.597115
$719$	−29.0000	−1.08152	−0.540759	−	0.841178i	$-0.681863\pi$
$719$	−29.0000	−1.08152	−0.540759	+	0.841178i	$0.681863\pi$
$720$	0	0
$721$	−4.00000	−0.148968
$722$	−1.00000	−0.0372161
$723$	0	0
$724$	12.0000	0.445976
$725$	0	0
$726$	0	0
$727$	−18.0000	−0.667583	−0.333792	−	0.942647i	$-0.608328\pi$
$727$	−18.0000	−0.667583	−0.333792	+	0.942647i	$0.608328\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−64.0000	−2.36713
$732$	0	0
$733$	23.0000	0.849524	0.424762	−	0.905305i	$-0.360358\pi$
$733$	23.0000	0.849524	0.424762	+	0.905305i	$0.360358\pi$
$734$	22.0000	0.812035
$735$	0	0
$736$	−3.00000	−0.110581
$737$	−5.00000	−0.184177
$738$	0	0
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	0	0
$741$	0	0
$742$	−12.0000	−0.440534
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	0	0
$745$	0	0
$746$	−32.0000	−1.17160
$747$	0	0
$748$	8.00000	0.292509
$749$	56.0000	2.04620
$750$	0	0
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	23.0000	0.835949	0.417975	−	0.908459i	$-0.362740\pi$
$757$	23.0000	0.835949	0.417975	+	0.908459i	$0.362740\pi$
$758$	32.0000	1.16229
$759$	0	0
$760$	0	0
$761$	26.0000	0.942499	0.471250	−	0.882000i	$-0.343803\pi$
$761$	26.0000	0.942499	0.471250	+	0.882000i	$0.343803\pi$
$762$	0	0
$763$	−24.0000	−0.868858
$764$	−13.0000	−0.470323
$765$	0	0
$766$	−18.0000	−0.650366
$767$	0	0
$768$	0	0
$769$	−45.0000	−1.62274	−0.811371	−	0.584532i	$-0.801278\pi$
$769$	−45.0000	−1.62274	−0.811371	+	0.584532i	$0.801278\pi$
$770$	0	0
$771$	0	0
$772$	−14.0000	−0.503871
$773$	−54.0000	−1.94225	−0.971123	−	0.238581i	$-0.923318\pi$
$773$	−54.0000	−1.94225	−0.971123	+	0.238581i	$0.923318\pi$
$774$	0	0
$775$	0	0
$776$	−10.0000	−0.358979
$777$	0	0
$778$	−22.0000	−0.788738
$779$	10.0000	0.358287
$780$	0	0
$781$	6.00000	0.214697
$782$	−24.0000	−0.858238
$783$	0	0
$784$	9.00000	0.321429
$785$	0	0
$786$	0	0
$787$	1.00000	0.0356462	0.0178231	−	0.999841i	$-0.494326\pi$
$787$	1.00000	0.0356462	0.0178231	+	0.999841i	$0.494326\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−4.00000	−0.142224
$792$	0	0
$793$	0	0
$794$	−35.0000	−1.24210
$795$	0	0
$796$	26.0000	0.921546
$797$	46.0000	1.62940	0.814702	−	0.579880i	$-0.196901\pi$
$797$	46.0000	1.62940	0.814702	+	0.579880i	$0.196901\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	−5.00000	−0.176556
$803$	−13.0000	−0.458760
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	−12.0000	−0.422159
$809$	16.0000	0.562530	0.281265	−	0.959630i	$-0.409246\pi$
$809$	16.0000	0.562530	0.281265	+	0.959630i	$0.409246\pi$
$810$	0	0
$811$	−49.0000	−1.72062	−0.860311	−	0.509769i	$-0.829731\pi$
$811$	−49.0000	−1.72062	−0.860311	+	0.509769i	$0.829731\pi$
$812$	−4.00000	−0.140372
$813$	0	0
$814$	2.00000	0.0701000
$815$	0	0
$816$	0	0
$817$	−8.00000	−0.279885
$818$	18.0000	0.629355
$819$	0	0
$820$	0	0
$821$	10.0000	0.349002	0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000	0.349002	0.174501	+	0.984657i	$0.444169\pi$
$822$	0	0
$823$	−26.0000	−0.906303	−0.453152	−	0.891434i	$-0.649700\pi$
$823$	−26.0000	−0.906303	−0.453152	+	0.891434i	$0.649700\pi$
$824$	−1.00000	−0.0348367
$825$	0	0
$826$	−16.0000	−0.556711
$827$	28.0000	0.973655	0.486828	−	0.873498i	$-0.338154\pi$
$827$	28.0000	0.973655	0.486828	+	0.873498i	$0.338154\pi$
$828$	0	0
$829$	26.0000	0.903017	0.451509	−	0.892267i	$-0.350886\pi$
$829$	26.0000	0.903017	0.451509	+	0.892267i	$0.350886\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	72.0000	2.49465
$834$	0	0
$835$	0	0
$836$	1.00000	0.0345857
$837$	0	0
$838$	−12.0000	−0.414533
$839$	48.0000	1.65714	0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000	1.65714	0.828572	+	0.559883i	$0.189154\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	−22.0000	−0.758170
$843$	0	0
$844$	25.0000	0.860535
$845$	0	0
$846$	0	0
$847$	40.0000	1.37442
$848$	−3.00000	−0.103020
$849$	0	0
$850$	0	0
$851$	−6.00000	−0.205677
$852$	0	0
$853$	46.0000	1.57501	0.787505	−	0.616308i	$-0.211372\pi$
$853$	46.0000	1.57501	0.787505	+	0.616308i	$0.211372\pi$
$854$	20.0000	0.684386
$855$	0	0
$856$	14.0000	0.478510
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	0	0
$859$	34.0000	1.16007	0.580033	−	0.814593i	$-0.303040\pi$
$859$	34.0000	1.16007	0.580033	+	0.814593i	$0.303040\pi$
$860$	0	0
$861$	0	0
$862$	−14.0000	−0.476842
$863$	−28.0000	−0.953131	−0.476566	−	0.879139i	$-0.658119\pi$
$863$	−28.0000	−0.953131	−0.476566	+	0.879139i	$0.658119\pi$
$864$	0	0
$865$	0	0
$866$	22.0000	0.747590
$867$	0	0
$868$	−4.00000	−0.135769
$869$	−5.00000	−0.169613
$870$	0	0
$871$	0	0
$872$	−6.00000	−0.203186
$873$	0	0
$874$	−3.00000	−0.101477
$875$	0	0
$876$	0	0
$877$	42.0000	1.41824	0.709120	−	0.705088i	$-0.249093\pi$
$877$	42.0000	1.41824	0.709120	+	0.705088i	$0.249093\pi$
$878$	−7.00000	−0.236239
$879$	0	0
$880$	0	0
$881$	8.00000	0.269527	0.134763	−	0.990878i	$-0.456973\pi$
$881$	8.00000	0.269527	0.134763	+	0.990878i	$0.456973\pi$
$882$	0	0
$883$	−26.0000	−0.874970	−0.437485	−	0.899226i	$-0.644131\pi$
$883$	−26.0000	−0.874970	−0.437485	+	0.899226i	$0.644131\pi$
$884$	0	0
$885$	0	0
$886$	3.00000	0.100787
$887$	48.0000	1.61168	0.805841	−	0.592132i	$-0.201714\pi$
$887$	48.0000	1.61168	0.805841	+	0.592132i	$0.201714\pi$
$888$	0	0
$889$	68.0000	2.28065
$890$	0	0
$891$	0	0
$892$	29.0000	0.970992
$893$	0	0
$894$	0	0
$895$	0	0
$896$	4.00000	0.133631
$897$	0	0
$898$	−23.0000	−0.767520
$899$	1.00000	0.0333519
$900$	0	0
$901$	−24.0000	−0.799556
$902$	−10.0000	−0.332964
$903$	0	0
$904$	−1.00000	−0.0332595
$905$	0	0
$906$	0	0
$907$	−48.0000	−1.59381	−0.796907	−	0.604102i	$-0.793532\pi$
$907$	−48.0000	−1.59381	−0.796907	+	0.604102i	$0.793532\pi$
$908$	30.0000	0.995585
$909$	0	0
$910$	0	0
$911$	−18.0000	−0.596367	−0.298183	−	0.954509i	$-0.596381\pi$
$911$	−18.0000	−0.596367	−0.298183	+	0.954509i	$0.596381\pi$
$912$	0	0
$913$	−11.0000	−0.364047
$914$	22.0000	0.727695
$915$	0	0
$916$	13.0000	0.429532
$917$	−60.0000	−1.98137
$918$	0	0
$919$	−2.00000	−0.0659739	−0.0329870	−	0.999456i	$-0.510502\pi$
$919$	−2.00000	−0.0659739	−0.0329870	+	0.999456i	$0.510502\pi$
$920$	0	0
$921$	0	0
$922$	−12.0000	−0.395199
$923$	0	0
$924$	0	0
$925$	0	0
$926$	−4.00000	−0.131448
$927$	0	0
$928$	−1.00000	−0.0328266
$929$	12.0000	0.393707	0.196854	−	0.980433i	$-0.436928\pi$
$929$	12.0000	0.393707	0.196854	+	0.980433i	$0.436928\pi$
$930$	0	0
$931$	9.00000	0.294963
$932$	4.00000	0.131024
$933$	0	0
$934$	−3.00000	−0.0981630
$935$	0	0
$936$	0	0
$937$	42.0000	1.37208	0.686040	−	0.727564i	$-0.259347\pi$
$937$	42.0000	1.37208	0.686040	+	0.727564i	$0.259347\pi$
$938$	−20.0000	−0.653023
$939$	0	0
$940$	0	0
$941$	19.0000	0.619382	0.309691	−	0.950837i	$-0.399774\pi$
$941$	19.0000	0.619382	0.309691	+	0.950837i	$0.399774\pi$
$942$	0	0
$943$	30.0000	0.976934
$944$	−4.00000	−0.130189
$945$	0	0
$946$	8.00000	0.260102
$947$	52.0000	1.68977	0.844886	−	0.534946i	$-0.179668\pi$
$947$	52.0000	1.68977	0.844886	+	0.534946i	$0.179668\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	32.0000	1.03713
$953$	15.0000	0.485898	0.242949	−	0.970039i	$-0.421885\pi$
$953$	15.0000	0.485898	0.242949	+	0.970039i	$0.421885\pi$
$954$	0	0
$955$	0	0
$956$	−24.0000	−0.776215
$957$	0	0
$958$	21.0000	0.678479
$959$	16.0000	0.516667
$960$	0	0
$961$	−30.0000	−0.967742
$962$	0	0
$963$	0	0
$964$	8.00000	0.257663
$965$	0	0
$966$	0	0
$967$	38.0000	1.22200	0.610999	−	0.791632i	$-0.290768\pi$
$967$	38.0000	1.22200	0.610999	+	0.791632i	$0.290768\pi$
$968$	10.0000	0.321412
$969$	0	0
$970$	0	0
$971$	8.00000	0.256732	0.128366	−	0.991727i	$-0.459027\pi$
$971$	8.00000	0.256732	0.128366	+	0.991727i	$0.459027\pi$
$972$	0	0
$973$	−24.0000	−0.769405
$974$	32.0000	1.02535
$975$	0	0
$976$	5.00000	0.160046
$977$	−2.00000	−0.0639857	−0.0319928	−	0.999488i	$-0.510185\pi$
$977$	−2.00000	−0.0639857	−0.0319928	+	0.999488i	$0.510185\pi$
$978$	0	0
$979$	−3.00000	−0.0958804
$980$	0	0
$981$	0	0
$982$	4.00000	0.127645
$983$	−30.0000	−0.956851	−0.478426	−	0.878128i	$-0.658792\pi$
$983$	−30.0000	−0.956851	−0.478426	+	0.878128i	$0.658792\pi$
$984$	0	0
$985$	0	0
$986$	−8.00000	−0.254772
$987$	0	0
$988$	0	0
$989$	−24.0000	−0.763156
$990$	0	0
$991$	−25.0000	−0.794151	−0.397076	−	0.917786i	$-0.629975\pi$
$991$	−25.0000	−0.794151	−0.397076	+	0.917786i	$0.629975\pi$
$992$	−1.00000	−0.0317500
$993$	0	0
$994$	24.0000	0.761234
$995$	0	0
$996$	0	0
$997$	−13.0000	−0.411714	−0.205857	−	0.978582i	$-0.565998\pi$
$997$	−13.0000	−0.411714	−0.205857	+	0.978582i	$0.565998\pi$
$998$	4.00000	0.126618
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8550.2.a.c.1.1		1
3.2	odd	2		2850.2.a.w.1.1	yes	1
5.4	even	2		8550.2.a.bk.1.1		1
15.2	even	4		2850.2.d.o.799.2		2
15.8	even	4		2850.2.d.o.799.1		2
15.14	odd	2		2850.2.a.f.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2850.2.a.f.1.1	✓	1	15.14	odd	2
2850.2.a.w.1.1	yes	1	3.2	odd	2
2850.2.d.o.799.1		2	15.8	even	4
2850.2.d.o.799.2		2	15.2	even	4
8550.2.a.c.1.1		1	1.1	even	1	trivial
8550.2.a.bk.1.1		1	5.4	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:	\( N \)	\(=\)	\( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8550.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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