LMFDB - Embedded newform 3120.2.l.k.1249.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3120,2,Mod(1249,3120)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3120.1249"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3120, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3120.l (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,-4,0,0,0,0,0,0,0,0,0,-20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$24.9133254306$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.1
Root			$1.61803i$ of defining polynomial
Character	$\chi$	$=$	3120.1249
Dual form			3120.2.l.k.1249.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -2.23607i q^{5} +3.23607i q^{7} -1.00000 q^{9} -4.47214 q^{11} +1.00000i q^{13} -2.23607 q^{15} -7.23607i q^{17} -2.76393 q^{19} +3.23607 q^{21} -2.76393i q^{23} -5.00000 q^{25} +1.00000i q^{27} +3.70820 q^{29} +4.00000 q^{31} +4.47214i q^{33} +7.23607 q^{35} +10.9443i q^{37} +1.00000 q^{39} +3.52786 q^{41} +2.47214i q^{43} +2.23607i q^{45} +12.9443i q^{47} -3.47214 q^{49} -7.23607 q^{51} -0.472136i q^{53} +10.0000i q^{55} +2.76393i q^{57} -8.47214 q^{59} -10.9443 q^{61} -3.23607i q^{63} +2.23607 q^{65} -2.76393 q^{69} +2.47214 q^{71} +13.2361i q^{73} +5.00000i q^{75} -14.4721i q^{77} -4.00000 q^{79} +1.00000 q^{81} +4.94427i q^{83} -16.1803 q^{85} -3.70820i q^{87} -0.472136 q^{89} -3.23607 q^{91} -4.00000i q^{93} +6.18034i q^{95} +3.70820i q^{97} +4.47214 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9} - 20 q^{19} + 4 q^{21} - 20 q^{25} - 12 q^{29} + 16 q^{31} + 20 q^{35} + 4 q^{39} + 32 q^{41} + 4 q^{49} - 20 q^{51} - 16 q^{59} - 8 q^{61} - 20 q^{69} - 8 q^{71} - 16 q^{79} + 4 q^{81} - 20 q^{85}+ \cdots - 4 q^{91}+O(q^{100}) $ 4 * q - 4 * q^9 - 20 * q^19 + 4 * q^21 - 20 * q^25 - 12 * q^29 + 16 * q^31 + 20 * q^35 + 4 * q^39 + 32 * q^41 + 4 * q^49 - 20 * q^51 - 16 * q^59 - 8 * q^61 - 20 * q^69 - 8 * q^71 - 16 * q^79 + 4 * q^81 - 20 * q^85 + 16 * q^89 - 4 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3120\mathbb{Z}\right)^\times$.

$n$	$1951$	$2081$	$2341$	$2497$	$2641$
$\chi(n)$	$1$	$1$	$1$	$-1$	$1$

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
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	− 2.23607i	− 1.00000i
$5$	− 2.23607i	− 1.00000i
$6$	0	0
$7$	3.23607i	1.22312i	0.791199	+	0.611559i	$0.209457\pi$
$7$	3.23607i	1.22312i	−0.791199	+	0.611559i	$0.790543\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−4.47214	−1.34840	−0.674200	−	0.738549i	$-0.735511\pi$
$11$	−4.47214	−1.34840	−0.674200	+	0.738549i	$0.735511\pi$
$12$	0	0
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	0	0
$15$	−2.23607	−0.577350
$16$	0	0
$17$	− 7.23607i	− 1.75500i	−0.479573	−	0.877502i	$-0.659208\pi$
$17$	− 7.23607i	− 1.75500i	0.479573	−	0.877502i	$-0.340792\pi$
$18$	0	0
$19$	−2.76393	−0.634089	−0.317045	−	0.948411i	$-0.602691\pi$
$19$	−2.76393	−0.634089	−0.317045	+	0.948411i	$0.602691\pi$
$20$	0	0
$21$	3.23607	0.706168
$22$	0	0
$23$	− 2.76393i	− 0.576320i	−0.957582	−	0.288160i	$-0.906957\pi$
$23$	− 2.76393i	− 0.576320i	0.957582	−	0.288160i	$-0.0930434\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	3.70820	0.688596	0.344298	−	0.938860i	$-0.388117\pi$
$29$	3.70820	0.688596	0.344298	+	0.938860i	$0.388117\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	4.47214i	0.778499i
$34$	0	0
$35$	7.23607	1.22312
$36$	0	0
$37$	10.9443i	1.79923i	0.436687	+	0.899614i	$0.356152\pi$
$37$	10.9443i	1.79923i	−0.436687	+	0.899614i	$0.643848\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	3.52786	0.550960	0.275480	−	0.961307i	$-0.411163\pi$
$41$	3.52786	0.550960	0.275480	+	0.961307i	$0.411163\pi$
$42$	0	0
$43$	2.47214i	0.376997i	0.982073	+	0.188499i	$0.0603621\pi$
$43$	2.47214i	0.376997i	−0.982073	+	0.188499i	$0.939638\pi$
$44$	0	0
$45$	2.23607i	0.333333i
$46$	0	0
$47$	12.9443i	1.88812i	0.329779	+	0.944058i	$0.393026\pi$
$47$	12.9443i	1.88812i	−0.329779	+	0.944058i	$0.606974\pi$
$48$	0	0
$49$	−3.47214	−0.496019
$50$	0	0
$51$	−7.23607	−1.01325
$52$	0	0
$53$	− 0.472136i	− 0.0648529i	−0.999474	−	0.0324264i	$-0.989677\pi$
$53$	− 0.472136i	− 0.0648529i	0.999474	−	0.0324264i	$-0.0103235\pi$
$54$	0	0
$55$	10.0000i	1.34840i
$56$	0	0
$57$	2.76393i	0.366092i
$58$	0	0
$59$	−8.47214	−1.10298	−0.551489	−	0.834182i	$-0.685940\pi$
$59$	−8.47214	−1.10298	−0.551489	+	0.834182i	$0.685940\pi$
$60$	0	0
$61$	−10.9443	−1.40127	−0.700635	−	0.713520i	$-0.747100\pi$
$61$	−10.9443	−1.40127	−0.700635	+	0.713520i	$0.747100\pi$
$62$	0	0
$63$	− 3.23607i	− 0.407706i
$64$	0	0
$65$	2.23607	0.277350
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	−2.76393	−0.332738
$70$	0	0
$71$	2.47214	0.293389	0.146694	−	0.989182i	$-0.453137\pi$
$71$	2.47214	0.293389	0.146694	+	0.989182i	$0.453137\pi$
$72$	0	0
$73$	13.2361i	1.54916i	0.632473	+	0.774582i	$0.282040\pi$
$73$	13.2361i	1.54916i	−0.632473	+	0.774582i	$0.717960\pi$
$74$	0	0
$75$	5.00000i	0.577350i
$76$	0	0
$77$	− 14.4721i	− 1.64925i
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.94427i	0.542704i	0.962480	+	0.271352i	$0.0874708\pi$
$83$	4.94427i	0.542704i	−0.962480	+	0.271352i	$0.912529\pi$
$84$	0	0
$85$	−16.1803	−1.75500
$86$	0	0
$87$	− 3.70820i	− 0.397561i
$88$	0	0
$89$	−0.472136	−0.0500463	−0.0250232	−	0.999687i	$-0.507966\pi$
$89$	−0.472136	−0.0500463	−0.0250232	+	0.999687i	$0.507966\pi$
$90$	0	0
$91$	−3.23607	−0.339232
$92$	0	0
$93$	− 4.00000i	− 0.414781i
$94$	0	0
$95$	6.18034i	0.634089i
$96$	0	0
$97$	3.70820i	0.376511i	0.982120	+	0.188256i	$0.0602833\pi$
$97$	3.70820i	0.376511i	−0.982120	+	0.188256i	$0.939717\pi$
$98$	0	0
$99$	4.47214	0.449467
$100$	0	0
$101$	−9.23607	−0.919023	−0.459512	−	0.888172i	$-0.651976\pi$
$101$	−9.23607	−0.919023	−0.459512	+	0.888172i	$0.651976\pi$
$102$	0	0
$103$	− 8.47214i	− 0.834784i	−0.908726	−	0.417392i	$-0.862944\pi$
$103$	− 8.47214i	− 0.834784i	0.908726	−	0.417392i	$-0.137056\pi$
$104$	0	0
$105$	− 7.23607i	− 0.706168i
$106$	0	0
$107$	− 1.52786i	− 0.147704i	−0.997269	−	0.0738521i	$-0.976471\pi$
$107$	− 1.52786i	− 0.147704i	0.997269	−	0.0738521i	$-0.0235293\pi$
$108$	0	0
$109$	2.29180	0.219514	0.109757	−	0.993958i	$-0.464993\pi$
$109$	2.29180	0.219514	0.109757	+	0.993958i	$0.464993\pi$
$110$	0	0
$111$	10.9443	1.03878
$112$	0	0
$113$	18.6525i	1.75468i	0.479872	+	0.877339i	$0.340683\pi$
$113$	18.6525i	1.75468i	−0.479872	+	0.877339i	$0.659317\pi$
$114$	0	0
$115$	−6.18034	−0.576320
$116$	0	0
$117$	− 1.00000i	− 0.0924500i
$118$	0	0
$119$	23.4164	2.14658
$120$	0	0
$121$	9.00000	0.818182
$122$	0	0
$123$	− 3.52786i	− 0.318097i
$124$	0	0
$125$	11.1803i	1.00000i
$126$	0	0
$127$	21.4164i	1.90040i	0.311642	+	0.950199i	$0.399121\pi$
$127$	21.4164i	1.90040i	−0.311642	+	0.950199i	$0.600879\pi$
$128$	0	0
$129$	2.47214	0.217659
$130$	0	0
$131$	−9.70820	−0.848210	−0.424105	−	0.905613i	$-0.639411\pi$
$131$	−9.70820	−0.848210	−0.424105	+	0.905613i	$0.639411\pi$
$132$	0	0
$133$	− 8.94427i	− 0.775567i
$134$	0	0
$135$	2.23607	0.192450
$136$	0	0
$137$	19.8885i	1.69919i	0.527433	+	0.849596i	$0.323154\pi$
$137$	19.8885i	1.69919i	−0.527433	+	0.849596i	$0.676846\pi$
$138$	0	0
$139$	−6.47214	−0.548959	−0.274480	−	0.961593i	$-0.588506\pi$
$139$	−6.47214	−0.548959	−0.274480	+	0.961593i	$0.588506\pi$
$140$	0	0
$141$	12.9443	1.09010
$142$	0	0
$143$	− 4.47214i	− 0.373979i
$144$	0	0
$145$	− 8.29180i	− 0.688596i
$146$	0	0
$147$	3.47214i	0.286377i
$148$	0	0
$149$	−13.5279	−1.10825	−0.554123	−	0.832435i	$-0.686946\pi$
$149$	−13.5279	−1.10825	−0.554123	+	0.832435i	$0.686946\pi$
$150$	0	0
$151$	19.4164	1.58008	0.790042	−	0.613052i	$-0.210058\pi$
$151$	19.4164	1.58008	0.790042	+	0.613052i	$0.210058\pi$
$152$	0	0
$153$	7.23607i	0.585001i
$154$	0	0
$155$	− 8.94427i	− 0.718421i
$156$	0	0
$157$	18.9443i	1.51192i	0.654619	+	0.755959i	$0.272829\pi$
$157$	18.9443i	1.51192i	−0.654619	+	0.755959i	$0.727171\pi$
$158$	0	0
$159$	−0.472136	−0.0374428
$160$	0	0
$161$	8.94427	0.704907
$162$	0	0
$163$	− 12.9443i	− 1.01387i	−0.861983	−	0.506937i	$-0.830778\pi$
$163$	− 12.9443i	− 1.01387i	0.861983	−	0.506937i	$-0.169222\pi$
$164$	0	0
$165$	10.0000	0.778499
$166$	0	0
$167$	− 19.4164i	− 1.50249i	−0.660025	−	0.751243i	$-0.729454\pi$
$167$	− 19.4164i	− 1.50249i	0.660025	−	0.751243i	$-0.270546\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	2.76393	0.211363
$172$	0	0
$173$	− 22.9443i	− 1.74442i	−0.489131	−	0.872210i	$-0.662686\pi$
$173$	− 22.9443i	− 1.74442i	0.489131	−	0.872210i	$-0.337314\pi$
$174$	0	0
$175$	− 16.1803i	− 1.22312i
$176$	0	0
$177$	8.47214i	0.636805i
$178$	0	0
$179$	8.18034	0.611427	0.305714	−	0.952124i	$-0.401105\pi$
$179$	8.18034	0.611427	0.305714	+	0.952124i	$0.401105\pi$
$180$	0	0
$181$	17.4164	1.29455	0.647276	−	0.762256i	$-0.275908\pi$
$181$	17.4164	1.29455	0.647276	+	0.762256i	$0.275908\pi$
$182$	0	0
$183$	10.9443i	0.809024i
$184$	0	0
$185$	24.4721	1.79923
$186$	0	0
$187$	32.3607i	2.36645i
$188$	0	0
$189$	−3.23607	−0.235389
$190$	0	0
$191$	−22.4721	−1.62603	−0.813013	−	0.582245i	$-0.802174\pi$
$191$	−22.4721	−1.62603	−0.813013	+	0.582245i	$0.802174\pi$
$192$	0	0
$193$	− 8.29180i	− 0.596857i	−0.954432	−	0.298428i	$-0.903538\pi$
$193$	− 8.29180i	− 0.596857i	0.954432	−	0.298428i	$-0.0964624\pi$
$194$	0	0
$195$	− 2.23607i	− 0.160128i
$196$	0	0
$197$	14.9443i	1.06474i	0.846513	+	0.532368i	$0.178698\pi$
$197$	14.9443i	1.06474i	−0.846513	+	0.532368i	$0.821302\pi$
$198$	0	0
$199$	−0.944272	−0.0669377	−0.0334688	−	0.999440i	$-0.510655\pi$
$199$	−0.944272	−0.0669377	−0.0334688	+	0.999440i	$0.510655\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	12.0000i	0.842235i
$204$	0	0
$205$	− 7.88854i	− 0.550960i
$206$	0	0
$207$	2.76393i	0.192107i
$208$	0	0
$209$	12.3607	0.855006
$210$	0	0
$211$	−13.8885	−0.956127	−0.478063	−	0.878325i	$-0.658661\pi$
$211$	−13.8885	−0.956127	−0.478063	+	0.878325i	$0.658661\pi$
$212$	0	0
$213$	− 2.47214i	− 0.169388i
$214$	0	0
$215$	5.52786	0.376997
$216$	0	0
$217$	12.9443i	0.878714i
$218$	0	0
$219$	13.2361	0.894411
$220$	0	0
$221$	7.23607	0.486751
$222$	0	0
$223$	− 16.7639i	− 1.12260i	−0.827614	−	0.561298i	$-0.810302\pi$
$223$	− 16.7639i	− 1.12260i	0.827614	−	0.561298i	$-0.189698\pi$
$224$	0	0
$225$	5.00000	0.333333
$226$	0	0
$227$	7.05573i	0.468305i	0.972200	+	0.234153i	$0.0752315\pi$
$227$	7.05573i	0.468305i	−0.972200	+	0.234153i	$0.924768\pi$
$228$	0	0
$229$	−9.70820	−0.641536	−0.320768	−	0.947158i	$-0.603941\pi$
$229$	−9.70820	−0.641536	−0.320768	+	0.947158i	$0.603941\pi$
$230$	0	0
$231$	−14.4721	−0.952197
$232$	0	0
$233$	20.1803i	1.32206i	0.750360	+	0.661029i	$0.229880\pi$
$233$	20.1803i	1.32206i	−0.750360	+	0.661029i	$0.770120\pi$
$234$	0	0
$235$	28.9443	1.88812
$236$	0	0
$237$	4.00000i	0.259828i
$238$	0	0
$239$	−21.8885	−1.41585	−0.707926	−	0.706287i	$-0.750369\pi$
$239$	−21.8885	−1.41585	−0.707926	+	0.706287i	$0.750369\pi$
$240$	0	0
$241$	3.52786	0.227250	0.113625	−	0.993524i	$-0.463754\pi$
$241$	3.52786	0.227250	0.113625	+	0.993524i	$0.463754\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	7.76393i	0.496019i
$246$	0	0
$247$	− 2.76393i	− 0.175865i
$248$	0	0
$249$	4.94427	0.313331
$250$	0	0
$251$	−2.29180	−0.144657	−0.0723284	−	0.997381i	$-0.523043\pi$
$251$	−2.29180	−0.144657	−0.0723284	+	0.997381i	$0.523043\pi$
$252$	0	0
$253$	12.3607i	0.777109i
$254$	0	0
$255$	16.1803i	1.01325i
$256$	0	0
$257$	− 0.763932i	− 0.0476528i	−0.999716	−	0.0238264i	$-0.992415\pi$
$257$	− 0.763932i	− 0.0476528i	0.999716	−	0.0238264i	$-0.00758489\pi$
$258$	0	0
$259$	−35.4164	−2.20067
$260$	0	0
$261$	−3.70820	−0.229532
$262$	0	0
$263$	− 10.1803i	− 0.627747i	−0.949465	−	0.313873i	$-0.898373\pi$
$263$	− 10.1803i	− 0.627747i	0.949465	−	0.313873i	$-0.101627\pi$
$264$	0	0
$265$	−1.05573	−0.0648529
$266$	0	0
$267$	0.472136i	0.0288943i
$268$	0	0
$269$	−6.76393	−0.412404	−0.206202	−	0.978509i	$-0.566110\pi$
$269$	−6.76393	−0.412404	−0.206202	+	0.978509i	$0.566110\pi$
$270$	0	0
$271$	11.4164	0.693497	0.346749	−	0.937958i	$-0.387286\pi$
$271$	11.4164	0.693497	0.346749	+	0.937958i	$0.387286\pi$
$272$	0	0
$273$	3.23607i	0.195856i
$274$	0	0
$275$	22.3607	1.34840
$276$	0	0
$277$	− 20.8328i	− 1.25172i	−0.779934	−	0.625861i	$-0.784748\pi$
$277$	− 20.8328i	− 1.25172i	0.779934	−	0.625861i	$-0.215252\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	0	0
$281$	24.4721	1.45989	0.729943	−	0.683508i	$-0.239547\pi$
$281$	24.4721	1.45989	0.729943	+	0.683508i	$0.239547\pi$
$282$	0	0
$283$	11.4164i	0.678635i	0.940672	+	0.339318i	$0.110196\pi$
$283$	11.4164i	0.678635i	−0.940672	+	0.339318i	$0.889804\pi$
$284$	0	0
$285$	6.18034	0.366092
$286$	0	0
$287$	11.4164i	0.673889i
$288$	0	0
$289$	−35.3607	−2.08004
$290$	0	0
$291$	3.70820	0.217379
$292$	0	0
$293$	2.94427i	0.172006i	0.996295	+	0.0860031i	$0.0274095\pi$
$293$	2.94427i	0.172006i	−0.996295	+	0.0860031i	$0.972591\pi$
$294$	0	0
$295$	18.9443i	1.10298i
$296$	0	0
$297$	− 4.47214i	− 0.259500i
$298$	0	0
$299$	2.76393	0.159842
$300$	0	0
$301$	−8.00000	−0.461112
$302$	0	0
$303$	9.23607i	0.530598i
$304$	0	0
$305$	24.4721i	1.40127i
$306$	0	0
$307$	− 24.3607i	− 1.39034i	−0.718847	−	0.695169i	$-0.755330\pi$
$307$	− 24.3607i	− 1.39034i	0.718847	−	0.695169i	$-0.244670\pi$
$308$	0	0
$309$	−8.47214	−0.481963
$310$	0	0
$311$	22.4721	1.27428	0.637139	−	0.770749i	$-0.280118\pi$
$311$	22.4721	1.27428	0.637139	+	0.770749i	$0.280118\pi$
$312$	0	0
$313$	− 19.4164i	− 1.09748i	−0.835993	−	0.548740i	$-0.815108\pi$
$313$	− 19.4164i	− 1.09748i	0.835993	−	0.548740i	$-0.184892\pi$
$314$	0	0
$315$	−7.23607	−0.407706
$316$	0	0
$317$	− 13.4164i	− 0.753541i	−0.926307	−	0.376770i	$-0.877035\pi$
$317$	− 13.4164i	− 0.753541i	0.926307	−	0.376770i	$-0.122965\pi$
$318$	0	0
$319$	−16.5836	−0.928503
$320$	0	0
$321$	−1.52786	−0.0852771
$322$	0	0
$323$	20.0000i	1.11283i
$324$	0	0
$325$	− 5.00000i	− 0.277350i
$326$	0	0
$327$	− 2.29180i	− 0.126737i
$328$	0	0
$329$	−41.8885	−2.30939
$330$	0	0
$331$	2.76393	0.151919	0.0759597	−	0.997111i	$-0.475798\pi$
$331$	2.76393	0.151919	0.0759597	+	0.997111i	$0.475798\pi$
$332$	0	0
$333$	− 10.9443i	− 0.599742i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	6.47214i	0.352560i	0.984340	+	0.176280i	$0.0564064\pi$
$337$	6.47214i	0.352560i	−0.984340	+	0.176280i	$0.943594\pi$
$338$	0	0
$339$	18.6525	1.01306
$340$	0	0
$341$	−17.8885	−0.968719
$342$	0	0
$343$	11.4164i	0.616428i
$344$	0	0
$345$	6.18034i	0.332738i
$346$	0	0
$347$	1.52786i	0.0820200i	0.999159	+	0.0410100i	$0.0130576\pi$
$347$	1.52786i	0.0820200i	−0.999159	+	0.0410100i	$0.986942\pi$
$348$	0	0
$349$	15.2361	0.815568	0.407784	−	0.913078i	$-0.366302\pi$
$349$	15.2361	0.815568	0.407784	+	0.913078i	$0.366302\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	− 26.9443i	− 1.43410i	−0.697022	−	0.717049i	$-0.745492\pi$
$353$	− 26.9443i	− 1.43410i	0.697022	−	0.717049i	$-0.254508\pi$
$354$	0	0
$355$	− 5.52786i	− 0.293389i
$356$	0	0
$357$	− 23.4164i	− 1.23933i
$358$	0	0
$359$	6.47214	0.341586	0.170793	−	0.985307i	$-0.445367\pi$
$359$	6.47214	0.341586	0.170793	+	0.985307i	$0.445367\pi$
$360$	0	0
$361$	−11.3607	−0.597931
$362$	0	0
$363$	− 9.00000i	− 0.472377i
$364$	0	0
$365$	29.5967	1.54916
$366$	0	0
$367$	0.472136i	0.0246453i	0.999924	+	0.0123226i	$0.00392252\pi$
$367$	0.472136i	0.0246453i	−0.999924	+	0.0123226i	$0.996077\pi$
$368$	0	0
$369$	−3.52786	−0.183653
$370$	0	0
$371$	1.52786	0.0793227
$372$	0	0
$373$	2.00000i	0.103556i	0.998659	+	0.0517780i	$0.0164888\pi$
$373$	2.00000i	0.103556i	−0.998659	+	0.0517780i	$0.983511\pi$
$374$	0	0
$375$	11.1803	0.577350
$376$	0	0
$377$	3.70820i	0.190982i
$378$	0	0
$379$	−35.1246	−1.80423	−0.902115	−	0.431496i	$-0.857986\pi$
$379$	−35.1246	−1.80423	−0.902115	+	0.431496i	$0.857986\pi$
$380$	0	0
$381$	21.4164	1.09720
$382$	0	0
$383$	− 5.52786i	− 0.282461i	−0.989977	−	0.141230i	$-0.954894\pi$
$383$	− 5.52786i	− 0.282461i	0.989977	−	0.141230i	$-0.0451058\pi$
$384$	0	0
$385$	−32.3607	−1.64925
$386$	0	0
$387$	− 2.47214i	− 0.125666i
$388$	0	0
$389$	−12.6525	−0.641506	−0.320753	−	0.947163i	$-0.603936\pi$
$389$	−12.6525	−0.641506	−0.320753	+	0.947163i	$0.603936\pi$
$390$	0	0
$391$	−20.0000	−1.01144
$392$	0	0
$393$	9.70820i	0.489714i
$394$	0	0
$395$	8.94427i	0.450035i
$396$	0	0
$397$	− 17.4164i	− 0.874104i	−0.899436	−	0.437052i	$-0.856022\pi$
$397$	− 17.4164i	− 0.874104i	0.899436	−	0.437052i	$-0.143978\pi$
$398$	0	0
$399$	−8.94427	−0.447774
$400$	0	0
$401$	−24.4721	−1.22208	−0.611040	−	0.791600i	$-0.709249\pi$
$401$	−24.4721	−1.22208	−0.611040	+	0.791600i	$0.709249\pi$
$402$	0	0
$403$	4.00000i	0.199254i
$404$	0	0
$405$	− 2.23607i	− 0.111111i
$406$	0	0
$407$	− 48.9443i	− 2.42608i
$408$	0	0
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	0	0
$411$	19.8885	0.981030
$412$	0	0
$413$	− 27.4164i	− 1.34907i
$414$	0	0
$415$	11.0557	0.542704
$416$	0	0
$417$	6.47214i	0.316942i
$418$	0	0
$419$	27.5967	1.34819	0.674095	−	0.738645i	$-0.264534\pi$
$419$	27.5967	1.34819	0.674095	+	0.738645i	$0.264534\pi$
$420$	0	0
$421$	−13.7082	−0.668097	−0.334048	−	0.942556i	$-0.608415\pi$
$421$	−13.7082	−0.668097	−0.334048	+	0.942556i	$0.608415\pi$
$422$	0	0
$423$	− 12.9443i	− 0.629372i
$424$	0	0
$425$	36.1803i	1.75500i
$426$	0	0
$427$	− 35.4164i	− 1.71392i
$428$	0	0
$429$	−4.47214	−0.215917
$430$	0	0
$431$	14.8328	0.714472	0.357236	−	0.934014i	$-0.383719\pi$
$431$	14.8328	0.714472	0.357236	+	0.934014i	$0.383719\pi$
$432$	0	0
$433$	− 9.52786i	− 0.457880i	−0.973441	−	0.228940i	$-0.926474\pi$
$433$	− 9.52786i	− 0.457880i	0.973441	−	0.228940i	$-0.0735259\pi$
$434$	0	0
$435$	−8.29180	−0.397561
$436$	0	0
$437$	7.63932i	0.365438i
$438$	0	0
$439$	−4.94427	−0.235977	−0.117989	−	0.993015i	$-0.537645\pi$
$439$	−4.94427	−0.235977	−0.117989	+	0.993015i	$0.537645\pi$
$440$	0	0
$441$	3.47214	0.165340
$442$	0	0
$443$	22.4721i	1.06768i	0.845584	+	0.533842i	$0.179252\pi$
$443$	22.4721i	1.06768i	−0.845584	+	0.533842i	$0.820748\pi$
$444$	0	0
$445$	1.05573i	0.0500463i
$446$	0	0
$447$	13.5279i	0.639846i
$448$	0	0
$449$	−0.472136	−0.0222815	−0.0111407	−	0.999938i	$-0.503546\pi$
$449$	−0.472136	−0.0222815	−0.0111407	+	0.999938i	$0.503546\pi$
$450$	0	0
$451$	−15.7771	−0.742914
$452$	0	0
$453$	− 19.4164i	− 0.912262i
$454$	0	0
$455$	7.23607i	0.339232i
$456$	0	0
$457$	3.70820i	0.173462i	0.996232	+	0.0867312i	$0.0276421\pi$
$457$	3.70820i	0.173462i	−0.996232	+	0.0867312i	$0.972358\pi$
$458$	0	0
$459$	7.23607	0.337751
$460$	0	0
$461$	41.8885	1.95094	0.975472	−	0.220124i	$-0.0706461\pi$
$461$	41.8885	1.95094	0.975472	+	0.220124i	$0.0706461\pi$
$462$	0	0
$463$	22.2918i	1.03599i	0.855384	+	0.517994i	$0.173321\pi$
$463$	22.2918i	1.03599i	−0.855384	+	0.517994i	$0.826679\pi$
$464$	0	0
$465$	−8.94427	−0.414781
$466$	0	0
$467$	6.47214i	0.299495i	0.988724	+	0.149747i	$0.0478460\pi$
$467$	6.47214i	0.299495i	−0.988724	+	0.149747i	$0.952154\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	18.9443	0.872906
$472$	0	0
$473$	− 11.0557i	− 0.508343i
$474$	0	0
$475$	13.8197	0.634089
$476$	0	0
$477$	0.472136i	0.0216176i
$478$	0	0
$479$	−36.0000	−1.64488	−0.822441	−	0.568850i	$-0.807388\pi$
$479$	−36.0000	−1.64488	−0.822441	+	0.568850i	$0.807388\pi$
$480$	0	0
$481$	−10.9443	−0.499016
$482$	0	0
$483$	− 8.94427i	− 0.406978i
$484$	0	0
$485$	8.29180	0.376511
$486$	0	0
$487$	− 23.5967i	− 1.06927i	−0.845083	−	0.534635i	$-0.820449\pi$
$487$	− 23.5967i	− 1.06927i	0.845083	−	0.534635i	$-0.179551\pi$
$488$	0	0
$489$	−12.9443	−0.585360
$490$	0	0
$491$	16.1803	0.730209	0.365104	−	0.930967i	$-0.381033\pi$
$491$	16.1803	0.730209	0.365104	+	0.930967i	$0.381033\pi$
$492$	0	0
$493$	− 26.8328i	− 1.20849i
$494$	0	0
$495$	− 10.0000i	− 0.449467i
$496$	0	0
$497$	8.00000i	0.358849i
$498$	0	0
$499$	−39.1246	−1.75146	−0.875729	−	0.482803i	$-0.839619\pi$
$499$	−39.1246	−1.75146	−0.875729	+	0.482803i	$0.839619\pi$
$500$	0	0
$501$	−19.4164	−0.867461
$502$	0	0
$503$	30.7639i	1.37170i	0.727745	+	0.685848i	$0.240569\pi$
$503$	30.7639i	1.37170i	−0.727745	+	0.685848i	$0.759431\pi$
$504$	0	0
$505$	20.6525i	0.919023i
$506$	0	0
$507$	1.00000i	0.0444116i
$508$	0	0
$509$	15.0557	0.667333	0.333667	−	0.942691i	$-0.391714\pi$
$509$	15.0557	0.667333	0.333667	+	0.942691i	$0.391714\pi$
$510$	0	0
$511$	−42.8328	−1.89481
$512$	0	0
$513$	− 2.76393i	− 0.122031i
$514$	0	0
$515$	−18.9443	−0.834784
$516$	0	0
$517$	− 57.8885i	− 2.54594i
$518$	0	0
$519$	−22.9443	−1.00714
$520$	0	0
$521$	20.8328	0.912702	0.456351	−	0.889800i	$-0.349156\pi$
$521$	20.8328	0.912702	0.456351	+	0.889800i	$0.349156\pi$
$522$	0	0
$523$	− 8.00000i	− 0.349816i	−0.984585	−	0.174908i	$-0.944037\pi$
$523$	− 8.00000i	− 0.349816i	0.984585	−	0.174908i	$-0.0559627\pi$
$524$	0	0
$525$	−16.1803	−0.706168
$526$	0	0
$527$	− 28.9443i	− 1.26083i
$528$	0	0
$529$	15.3607	0.667856
$530$	0	0
$531$	8.47214	0.367659
$532$	0	0
$533$	3.52786i	0.152809i
$534$	0	0
$535$	−3.41641	−0.147704
$536$	0	0
$537$	− 8.18034i	− 0.353008i
$538$	0	0
$539$	15.5279	0.668832
$540$	0	0
$541$	−4.76393	−0.204817	−0.102409	−	0.994742i	$-0.532655\pi$
$541$	−4.76393	−0.204817	−0.102409	+	0.994742i	$0.532655\pi$
$542$	0	0
$543$	− 17.4164i	− 0.747410i
$544$	0	0
$545$	− 5.12461i	− 0.219514i
$546$	0	0
$547$	12.0000i	0.513083i	0.966533	+	0.256541i	$0.0825830\pi$
$547$	12.0000i	0.513083i	−0.966533	+	0.256541i	$0.917417\pi$
$548$	0	0
$549$	10.9443	0.467090
$550$	0	0
$551$	−10.2492	−0.436632
$552$	0	0
$553$	− 12.9443i	− 0.550446i
$554$	0	0
$555$	− 24.4721i	− 1.03878i
$556$	0	0
$557$	− 19.3050i	− 0.817977i	−0.912540	−	0.408989i	$-0.865882\pi$
$557$	− 19.3050i	− 0.817977i	0.912540	−	0.408989i	$-0.134118\pi$
$558$	0	0
$559$	−2.47214	−0.104560
$560$	0	0
$561$	32.3607	1.36627
$562$	0	0
$563$	19.4164i	0.818304i	0.912466	+	0.409152i	$0.134175\pi$
$563$	19.4164i	0.818304i	−0.912466	+	0.409152i	$0.865825\pi$
$564$	0	0
$565$	41.7082	1.75468
$566$	0	0
$567$	3.23607i	0.135902i
$568$	0	0
$569$	−34.9443	−1.46494	−0.732470	−	0.680799i	$-0.761633\pi$
$569$	−34.9443	−1.46494	−0.732470	+	0.680799i	$0.761633\pi$
$570$	0	0
$571$	8.58359	0.359212	0.179606	−	0.983739i	$-0.442518\pi$
$571$	8.58359	0.359212	0.179606	+	0.983739i	$0.442518\pi$
$572$	0	0
$573$	22.4721i	0.938787i
$574$	0	0
$575$	13.8197i	0.576320i
$576$	0	0
$577$	43.1246i	1.79530i	0.440708	+	0.897651i	$0.354727\pi$
$577$	43.1246i	1.79530i	−0.440708	+	0.897651i	$0.645273\pi$
$578$	0	0
$579$	−8.29180	−0.344595
$580$	0	0
$581$	−16.0000	−0.663792
$582$	0	0
$583$	2.11146i	0.0874476i
$584$	0	0
$585$	−2.23607	−0.0924500
$586$	0	0
$587$	− 2.11146i	− 0.0871491i	−0.999050	−	0.0435746i	$-0.986125\pi$
$587$	− 2.11146i	− 0.0871491i	0.999050	−	0.0435746i	$-0.0138746\pi$
$588$	0	0
$589$	−11.0557	−0.455543
$590$	0	0
$591$	14.9443	0.614725
$592$	0	0
$593$	3.52786i	0.144872i	0.997373	+	0.0724360i	$0.0230773\pi$
$593$	3.52786i	0.144872i	−0.997373	+	0.0724360i	$0.976923\pi$
$594$	0	0
$595$	− 52.3607i	− 2.14658i
$596$	0	0
$597$	0.944272i	0.0386465i
$598$	0	0
$599$	−6.11146	−0.249707	−0.124854	−	0.992175i	$-0.539846\pi$
$599$	−6.11146	−0.249707	−0.124854	+	0.992175i	$0.539846\pi$
$600$	0	0
$601$	13.4164	0.547267	0.273633	−	0.961834i	$-0.411775\pi$
$601$	13.4164	0.547267	0.273633	+	0.961834i	$0.411775\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 20.1246i	− 0.818182i
$606$	0	0
$607$	1.41641i	0.0574902i	0.999587	+	0.0287451i	$0.00915111\pi$
$607$	1.41641i	0.0574902i	−0.999587	+	0.0287451i	$0.990849\pi$
$608$	0	0
$609$	12.0000	0.486265
$610$	0	0
$611$	−12.9443	−0.523669
$612$	0	0
$613$	18.0000i	0.727013i	0.931592	+	0.363507i	$0.118421\pi$
$613$	18.0000i	0.727013i	−0.931592	+	0.363507i	$0.881579\pi$
$614$	0	0
$615$	−7.88854	−0.318097
$616$	0	0
$617$	− 35.8885i	− 1.44482i	−0.691466	−	0.722409i	$-0.743035\pi$
$617$	− 35.8885i	− 1.44482i	0.691466	−	0.722409i	$-0.256965\pi$
$618$	0	0
$619$	−25.2361	−1.01432	−0.507162	−	0.861851i	$-0.669305\pi$
$619$	−25.2361	−1.01432	−0.507162	+	0.861851i	$0.669305\pi$
$620$	0	0
$621$	2.76393	0.110913
$622$	0	0
$623$	− 1.52786i	− 0.0612126i
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	− 12.3607i	− 0.493638i
$628$	0	0
$629$	79.1935	3.15765
$630$	0	0
$631$	12.5836	0.500945	0.250472	−	0.968124i	$-0.419414\pi$
$631$	12.5836	0.500945	0.250472	+	0.968124i	$0.419414\pi$
$632$	0	0
$633$	13.8885i	0.552020i
$634$	0	0
$635$	47.8885	1.90040
$636$	0	0
$637$	− 3.47214i	− 0.137571i
$638$	0	0
$639$	−2.47214	−0.0977962
$640$	0	0
$641$	−23.3050	−0.920490	−0.460245	−	0.887792i	$-0.652238\pi$
$641$	−23.3050	−0.920490	−0.460245	+	0.887792i	$0.652238\pi$
$642$	0	0
$643$	− 19.0557i	− 0.751485i	−0.926724	−	0.375742i	$-0.877388\pi$
$643$	− 19.0557i	− 0.751485i	0.926724	−	0.375742i	$-0.122612\pi$
$644$	0	0
$645$	− 5.52786i	− 0.217659i
$646$	0	0
$647$	− 3.70820i	− 0.145785i	−0.997340	−	0.0728923i	$-0.976777\pi$
$647$	− 3.70820i	− 0.145785i	0.997340	−	0.0728923i	$-0.0232229\pi$
$648$	0	0
$649$	37.8885	1.48726
$650$	0	0
$651$	12.9443	0.507326
$652$	0	0
$653$	− 30.3607i	− 1.18811i	−0.804426	−	0.594053i	$-0.797527\pi$
$653$	− 30.3607i	− 1.18811i	0.804426	−	0.594053i	$-0.202473\pi$
$654$	0	0
$655$	21.7082i	0.848210i
$656$	0	0
$657$	− 13.2361i	− 0.516388i
$658$	0	0
$659$	−10.0689	−0.392228	−0.196114	−	0.980581i	$-0.562832\pi$
$659$	−10.0689	−0.392228	−0.196114	+	0.980581i	$0.562832\pi$
$660$	0	0
$661$	−41.7082	−1.62226	−0.811131	−	0.584865i	$-0.801147\pi$
$661$	−41.7082	−1.62226	−0.811131	+	0.584865i	$0.801147\pi$
$662$	0	0
$663$	− 7.23607i	− 0.281026i
$664$	0	0
$665$	−20.0000	−0.775567
$666$	0	0
$667$	− 10.2492i	− 0.396852i
$668$	0	0
$669$	−16.7639	−0.648131
$670$	0	0
$671$	48.9443	1.88947
$672$	0	0
$673$	44.0000i	1.69608i	0.529936	+	0.848038i	$0.322216\pi$
$673$	44.0000i	1.69608i	−0.529936	+	0.848038i	$0.677784\pi$
$674$	0	0
$675$	− 5.00000i	− 0.192450i
$676$	0	0
$677$	− 27.5279i	− 1.05798i	−0.848628	−	0.528991i	$-0.822571\pi$
$677$	− 27.5279i	− 1.05798i	0.848628	−	0.528991i	$-0.177429\pi$
$678$	0	0
$679$	−12.0000	−0.460518
$680$	0	0
$681$	7.05573	0.270376
$682$	0	0
$683$	41.8885i	1.60282i	0.598115	+	0.801410i	$0.295917\pi$
$683$	41.8885i	1.60282i	−0.598115	+	0.801410i	$0.704083\pi$
$684$	0	0
$685$	44.4721	1.69919
$686$	0	0
$687$	9.70820i	0.370391i
$688$	0	0
$689$	0.472136	0.0179869
$690$	0	0
$691$	−0.291796	−0.0111004	−0.00555022	−	0.999985i	$-0.501767\pi$
$691$	−0.291796	−0.0111004	−0.00555022	+	0.999985i	$0.501767\pi$
$692$	0	0
$693$	14.4721i	0.549751i
$694$	0	0
$695$	14.4721i	0.548959i
$696$	0	0
$697$	− 25.5279i	− 0.966937i
$698$	0	0
$699$	20.1803	0.763291
$700$	0	0
$701$	−24.0689	−0.909069	−0.454535	−	0.890729i	$-0.650194\pi$
$701$	−24.0689	−0.909069	−0.454535	+	0.890729i	$0.650194\pi$
$702$	0	0
$703$	− 30.2492i	− 1.14087i
$704$	0	0
$705$	− 28.9443i	− 1.09010i
$706$	0	0
$707$	− 29.8885i	− 1.12407i
$708$	0	0
$709$	−32.5410	−1.22210	−0.611052	−	0.791591i	$-0.709253\pi$
$709$	−32.5410	−1.22210	−0.611052	+	0.791591i	$0.709253\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	0	0
$713$	− 11.0557i	− 0.414040i
$714$	0	0
$715$	−10.0000	−0.373979
$716$	0	0
$717$	21.8885i	0.817443i
$718$	0	0
$719$	20.9443	0.781090	0.390545	−	0.920584i	$-0.372287\pi$
$719$	20.9443	0.781090	0.390545	+	0.920584i	$0.372287\pi$
$720$	0	0
$721$	27.4164	1.02104
$722$	0	0
$723$	− 3.52786i	− 0.131203i
$724$	0	0
$725$	−18.5410	−0.688596
$726$	0	0
$727$	18.0000i	0.667583i	0.942647	+	0.333792i	$0.108328\pi$
$727$	18.0000i	0.667583i	−0.942647	+	0.333792i	$0.891672\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	17.8885	0.661632
$732$	0	0
$733$	33.4164i	1.23426i	0.786860	+	0.617132i	$0.211705\pi$
$733$	33.4164i	1.23426i	−0.786860	+	0.617132i	$0.788295\pi$
$734$	0	0
$735$	7.76393	0.286377
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−0.291796	−0.0107339	−0.00536695	−	0.999986i	$-0.501708\pi$
$739$	−0.291796	−0.0107339	−0.00536695	+	0.999986i	$0.501708\pi$
$740$	0	0
$741$	−2.76393	−0.101536
$742$	0	0
$743$	− 28.3607i	− 1.04045i	−0.854029	−	0.520226i	$-0.825848\pi$
$743$	− 28.3607i	− 1.04045i	0.854029	−	0.520226i	$-0.174152\pi$
$744$	0	0
$745$	30.2492i	1.10825i
$746$	0	0
$747$	− 4.94427i	− 0.180901i
$748$	0	0
$749$	4.94427	0.180660
$750$	0	0
$751$	46.8328	1.70895	0.854477	−	0.519489i	$-0.173878\pi$
$751$	46.8328	1.70895	0.854477	+	0.519489i	$0.173878\pi$
$752$	0	0
$753$	2.29180i	0.0835177i
$754$	0	0
$755$	− 43.4164i	− 1.58008i
$756$	0	0
$757$	38.0000i	1.38113i	0.723269	+	0.690567i	$0.242639\pi$
$757$	38.0000i	1.38113i	−0.723269	+	0.690567i	$0.757361\pi$
$758$	0	0
$759$	12.3607	0.448664
$760$	0	0
$761$	7.52786	0.272885	0.136442	−	0.990648i	$-0.456433\pi$
$761$	7.52786	0.272885	0.136442	+	0.990648i	$0.456433\pi$
$762$	0	0
$763$	7.41641i	0.268492i
$764$	0	0
$765$	16.1803	0.585001
$766$	0	0
$767$	− 8.47214i	− 0.305911i
$768$	0	0
$769$	4.83282	0.174276	0.0871379	−	0.996196i	$-0.472228\pi$
$769$	4.83282	0.174276	0.0871379	+	0.996196i	$0.472228\pi$
$770$	0	0
$771$	−0.763932	−0.0275123
$772$	0	0
$773$	− 2.94427i	− 0.105898i	−0.998597	−	0.0529491i	$-0.983138\pi$
$773$	− 2.94427i	− 0.105898i	0.998597	−	0.0529491i	$-0.0168621\pi$
$774$	0	0
$775$	−20.0000	−0.718421
$776$	0	0
$777$	35.4164i	1.27056i
$778$	0	0
$779$	−9.75078	−0.349358
$780$	0	0
$781$	−11.0557	−0.395605
$782$	0	0
$783$	3.70820i	0.132520i
$784$	0	0
$785$	42.3607	1.51192
$786$	0	0
$787$	28.9443i	1.03175i	0.856663	+	0.515876i	$0.172533\pi$
$787$	28.9443i	1.03175i	−0.856663	+	0.515876i	$0.827467\pi$
$788$	0	0
$789$	−10.1803	−0.362430
$790$	0	0
$791$	−60.3607	−2.14618
$792$	0	0
$793$	− 10.9443i	− 0.388642i
$794$	0	0
$795$	1.05573i	0.0374428i
$796$	0	0
$797$	− 9.05573i	− 0.320770i	−0.987055	−	0.160385i	$-0.948726\pi$
$797$	− 9.05573i	− 0.320770i	0.987055	−	0.160385i	$-0.0512736\pi$
$798$	0	0
$799$	93.6656	3.31365
$800$	0	0
$801$	0.472136	0.0166821
$802$	0	0
$803$	− 59.1935i	− 2.08889i
$804$	0	0
$805$	− 20.0000i	− 0.704907i
$806$	0	0
$807$	6.76393i	0.238102i
$808$	0	0
$809$	−31.5279	−1.10846	−0.554230	−	0.832363i	$-0.686987\pi$
$809$	−31.5279	−1.10846	−0.554230	+	0.832363i	$0.686987\pi$
$810$	0	0
$811$	−4.29180	−0.150705	−0.0753527	−	0.997157i	$-0.524008\pi$
$811$	−4.29180	−0.150705	−0.0753527	+	0.997157i	$0.524008\pi$
$812$	0	0
$813$	− 11.4164i	− 0.400391i
$814$	0	0
$815$	−28.9443	−1.01387
$816$	0	0
$817$	− 6.83282i	− 0.239050i
$818$	0	0
$819$	3.23607	0.113077
$820$	0	0
$821$	−30.4721	−1.06348	−0.531742	−	0.846906i	$-0.678463\pi$
$821$	−30.4721	−1.06348	−0.531742	+	0.846906i	$0.678463\pi$
$822$	0	0
$823$	− 48.2492i	− 1.68186i	−0.541142	−	0.840931i	$-0.682008\pi$
$823$	− 48.2492i	− 1.68186i	0.541142	−	0.840931i	$-0.317992\pi$
$824$	0	0
$825$	− 22.3607i	− 0.778499i
$826$	0	0
$827$	− 41.8885i	− 1.45661i	−0.685254	−	0.728304i	$-0.740309\pi$
$827$	− 41.8885i	− 1.45661i	0.685254	−	0.728304i	$-0.259691\pi$
$828$	0	0
$829$	−42.7214	−1.48377	−0.741887	−	0.670525i	$-0.766069\pi$
$829$	−42.7214	−1.48377	−0.741887	+	0.670525i	$0.766069\pi$
$830$	0	0
$831$	−20.8328	−0.722682
$832$	0	0
$833$	25.1246i	0.870516i
$834$	0	0
$835$	−43.4164	−1.50249
$836$	0	0
$837$	4.00000i	0.138260i
$838$	0	0
$839$	−51.7771	−1.78754	−0.893772	−	0.448522i	$-0.851951\pi$
$839$	−51.7771	−1.78754	−0.893772	+	0.448522i	$0.851951\pi$
$840$	0	0
$841$	−15.2492	−0.525835
$842$	0	0
$843$	− 24.4721i	− 0.842865i
$844$	0	0
$845$	2.23607i	0.0769231i
$846$	0	0
$847$	29.1246i	1.00073i
$848$	0	0
$849$	11.4164	0.391810
$850$	0	0
$851$	30.2492	1.03693
$852$	0	0
$853$	41.4164i	1.41807i	0.705173	+	0.709035i	$0.250869\pi$
$853$	41.4164i	1.41807i	−0.705173	+	0.709035i	$0.749131\pi$
$854$	0	0
$855$	− 6.18034i	− 0.211363i
$856$	0	0
$857$	0.180340i	0.00616029i	0.999995	+	0.00308015i	$0.000980443\pi$
$857$	0.180340i	0.00616029i	−0.999995	+	0.00308015i	$0.999020\pi$
$858$	0	0
$859$	36.0000	1.22830	0.614152	−	0.789188i	$-0.289498\pi$
$859$	36.0000	1.22830	0.614152	+	0.789188i	$0.289498\pi$
$860$	0	0
$861$	11.4164	0.389070
$862$	0	0
$863$	− 22.4721i	− 0.764960i	−0.923964	−	0.382480i	$-0.875070\pi$
$863$	− 22.4721i	− 0.764960i	0.923964	−	0.382480i	$-0.124930\pi$
$864$	0	0
$865$	−51.3050	−1.74442
$866$	0	0
$867$	35.3607i	1.20091i
$868$	0	0
$869$	17.8885	0.606827
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 3.70820i	− 0.125504i
$874$	0	0
$875$	−36.1803	−1.22312
$876$	0	0
$877$	− 24.4721i	− 0.826365i	−0.910648	−	0.413183i	$-0.864417\pi$
$877$	− 24.4721i	− 0.826365i	0.910648	−	0.413183i	$-0.135583\pi$
$878$	0	0
$879$	2.94427	0.0993078
$880$	0	0
$881$	34.3607	1.15764	0.578820	−	0.815455i	$-0.303513\pi$
$881$	34.3607	1.15764	0.578820	+	0.815455i	$0.303513\pi$
$882$	0	0
$883$	− 25.8885i	− 0.871219i	−0.900136	−	0.435609i	$-0.856533\pi$
$883$	− 25.8885i	− 0.871219i	0.900136	−	0.435609i	$-0.143467\pi$
$884$	0	0
$885$	18.9443	0.636805
$886$	0	0
$887$	− 10.1803i	− 0.341822i	−0.985286	−	0.170911i	$-0.945329\pi$
$887$	− 10.1803i	− 0.341822i	0.985286	−	0.170911i	$-0.0546711\pi$
$888$	0	0
$889$	−69.3050	−2.32441
$890$	0	0
$891$	−4.47214	−0.149822
$892$	0	0
$893$	− 35.7771i	− 1.19723i
$894$	0	0
$895$	− 18.2918i	− 0.611427i
$896$	0	0
$897$	− 2.76393i	− 0.0922850i
$898$	0	0
$899$	14.8328	0.494702
$900$	0	0
$901$	−3.41641	−0.113817
$902$	0	0
$903$	8.00000i	0.266223i
$904$	0	0
$905$	− 38.9443i	− 1.29455i
$906$	0	0
$907$	45.5279i	1.51173i	0.654729	+	0.755864i	$0.272783\pi$
$907$	45.5279i	1.51173i	−0.654729	+	0.755864i	$0.727217\pi$
$908$	0	0
$909$	9.23607	0.306341
$910$	0	0
$911$	11.0557	0.366293	0.183146	−	0.983086i	$-0.441372\pi$
$911$	11.0557	0.366293	0.183146	+	0.983086i	$0.441372\pi$
$912$	0	0
$913$	− 22.1115i	− 0.731782i
$914$	0	0
$915$	24.4721	0.809024
$916$	0	0
$917$	− 31.4164i	− 1.03746i
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	−24.3607	−0.802712
$922$	0	0
$923$	2.47214i	0.0813713i
$924$	0	0
$925$	− 54.7214i	− 1.79923i
$926$	0	0
$927$	8.47214i	0.278261i
$928$	0	0
$929$	−32.4721	−1.06538	−0.532688	−	0.846312i	$-0.678818\pi$
$929$	−32.4721	−1.06538	−0.532688	+	0.846312i	$0.678818\pi$
$930$	0	0
$931$	9.59675	0.314521
$932$	0	0
$933$	− 22.4721i	− 0.735705i
$934$	0	0
$935$	72.3607	2.36645
$936$	0	0
$937$	− 19.7771i	− 0.646089i	−0.946384	−	0.323045i	$-0.895294\pi$
$937$	− 19.7771i	− 0.646089i	0.946384	−	0.323045i	$-0.104706\pi$
$938$	0	0
$939$	−19.4164	−0.633631
$940$	0	0
$941$	−37.5279	−1.22337	−0.611687	−	0.791100i	$-0.709509\pi$
$941$	−37.5279	−1.22337	−0.611687	+	0.791100i	$0.709509\pi$
$942$	0	0
$943$	− 9.75078i	− 0.317529i
$944$	0	0
$945$	7.23607i	0.235389i
$946$	0	0
$947$	4.94427i	0.160667i	0.996768	+	0.0803336i	$0.0255986\pi$
$947$	4.94427i	0.160667i	−0.996768	+	0.0803336i	$0.974401\pi$
$948$	0	0
$949$	−13.2361	−0.429661
$950$	0	0
$951$	−13.4164	−0.435057
$952$	0	0
$953$	29.1246i	0.943439i	0.881749	+	0.471719i	$0.156366\pi$
$953$	29.1246i	0.943439i	−0.881749	+	0.471719i	$0.843634\pi$
$954$	0	0
$955$	50.2492i	1.62603i
$956$	0	0
$957$	16.5836i	0.536071i
$958$	0	0
$959$	−64.3607	−2.07831
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	1.52786i	0.0492347i
$964$	0	0
$965$	−18.5410	−0.596857
$966$	0	0
$967$	− 56.5410i	− 1.81824i	−0.416538	−	0.909118i	$-0.636757\pi$
$967$	− 56.5410i	− 1.81824i	0.416538	−	0.909118i	$-0.363243\pi$
$968$	0	0
$969$	20.0000	0.642493
$970$	0	0
$971$	−23.2361	−0.745681	−0.372840	−	0.927895i	$-0.621616\pi$
$971$	−23.2361	−0.745681	−0.372840	+	0.927895i	$0.621616\pi$
$972$	0	0
$973$	− 20.9443i	− 0.671443i
$974$	0	0
$975$	−5.00000	−0.160128
$976$	0	0
$977$	10.3607i	0.331468i	0.986171	+	0.165734i	$0.0529992\pi$
$977$	10.3607i	0.331468i	−0.986171	+	0.165734i	$0.947001\pi$
$978$	0	0
$979$	2.11146	0.0674824
$980$	0	0
$981$	−2.29180	−0.0731714
$982$	0	0
$983$	7.41641i	0.236547i	0.992981	+	0.118273i	$0.0377359\pi$
$983$	7.41641i	0.236547i	−0.992981	+	0.118273i	$0.962264\pi$
$984$	0	0
$985$	33.4164	1.06474
$986$	0	0
$987$	41.8885i	1.33333i
$988$	0	0
$989$	6.83282	0.217271
$990$	0	0
$991$	−24.9443	−0.792381	−0.396190	−	0.918168i	$-0.629668\pi$
$991$	−24.9443	−0.792381	−0.396190	+	0.918168i	$0.629668\pi$
$992$	0	0
$993$	− 2.76393i	− 0.0877107i
$994$	0	0
$995$	2.11146i	0.0669377i
$996$	0	0
$997$	− 43.8885i	− 1.38996i	−0.719027	−	0.694982i	$-0.755412\pi$
$997$	− 43.8885i	− 1.38996i	0.719027	−	0.694982i	$-0.244588\pi$
$998$	0	0
$999$	−10.9443	−0.346261

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3120.2.l.k.1249.1		4
4.3	odd	2		390.2.e.e.79.3	yes	4
5.4	even	2	inner	3120.2.l.k.1249.4		4
12.11	even	2		1170.2.e.e.469.2		4
20.3	even	4		1950.2.a.bf.1.1		2
20.7	even	4		1950.2.a.be.1.2		2
20.19	odd	2		390.2.e.e.79.2	✓	4
60.23	odd	4		5850.2.a.cf.1.1		2
60.47	odd	4		5850.2.a.cm.1.2		2
60.59	even	2		1170.2.e.e.469.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.e.e.79.2	✓	4	20.19	odd	2
390.2.e.e.79.3	yes	4	4.3	odd	2
1170.2.e.e.469.2		4	12.11	even	2
1170.2.e.e.469.3		4	60.59	even	2
1950.2.a.be.1.2		2	20.7	even	4
1950.2.a.bf.1.1		2	20.3	even	4
3120.2.l.k.1249.1		4	1.1	even	1	trivial
3120.2.l.k.1249.4		4	5.4	even	2	inner
5850.2.a.cf.1.1		2	60.23	odd	4
5850.2.a.cm.1.2		2	60.47	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3120.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -2.23607i q^{5} +3.23607i q^{7} -1.00000 q^{9} -4.47214 q^{11} +1.00000i q^{13} -2.23607 q^{15} -7.23607i q^{17} -2.76393 q^{19} +3.23607 q^{21} -2.76393i q^{23} -5.00000 q^{25} +1.00000i q^{27} +3.70820 q^{29} +4.00000 q^{31} +4.47214i q^{33} +7.23607 q^{35} +10.9443i q^{37} +1.00000 q^{39} +3.52786 q^{41} +2.47214i q^{43} +2.23607i q^{45} +12.9443i q^{47} -3.47214 q^{49} -7.23607 q^{51} -0.472136i q^{53} +10.0000i q^{55} +2.76393i q^{57} -8.47214 q^{59} -10.9443 q^{61} -3.23607i q^{63} +2.23607 q^{65} -2.76393 q^{69} +2.47214 q^{71} +13.2361i q^{73} +5.00000i q^{75} -14.4721i q^{77} -4.00000 q^{79} +1.00000 q^{81} +4.94427i q^{83} -16.1803 q^{85} -3.70820i q^{87} -0.472136 q^{89} -3.23607 q^{91} -4.00000i q^{93} +6.18034i q^{95} +3.70820i q^{97} +4.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9} - 20 q^{19} + 4 q^{21} - 20 q^{25} - 12 q^{29} + 16 q^{31} + 20 q^{35} + 4 q^{39} + 32 q^{41} + 4 q^{49} - 20 q^{51} - 16 q^{59} - 8 q^{61} - 20 q^{69} - 8 q^{71} - 16 q^{79} + 4 q^{81} - 20 q^{85}+ \cdots - 4 q^{91}+O(q^{100}) \) 4 * q - 4 * q^9 - 20 * q^19 + 4 * q^21 - 20 * q^25 - 12 * q^29 + 16 * q^31 + 20 * q^35 + 4 * q^39 + 32 * q^41 + 4 * q^49 - 20 * q^51 - 16 * q^59 - 8 * q^61 - 20 * q^69 - 8 * q^71 - 16 * q^79 + 4 * q^81 - 20 * q^85 + 16 * q^89 - 4 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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