LMFDB - Embedded newform 4704.2.c.f.2353.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4704,2,Mod(2353,4704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4704, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("4704.2353");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4704 = 2^{5} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4704.c (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:	$37.5616291108$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + x^{14} - 2 x^{13} - 2 x^{12} - 4 x^{11} - 2 x^{10} + 16 x^{9} + 8 x^{8} + 32 x^{7} - 8 x^{6} + \cdots + 256 $ x^16 + x^14 - 2x^13 - 2x^12 - 4x^11 - 2x^10 + 16x^9 + 8x^8 + 32x^7 - 8x^6 - 32x^5 - 32x^4 - 64x^3 + 64x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{13} $
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2353.1
Root			$-0.236856 + 1.39424i$ of defining polynomial
Character	$\chi$	$=$	4704.2353
Dual form			4704.2.c.f.2353.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -3.56550i q^{5} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -3.56550i q^{5} -1.00000 q^{9} -4.07092i q^{11} +1.44065i q^{13} -3.56550 q^{15} -6.99110 q^{17} -0.301495i q^{19} +2.42499 q^{23} -7.71279 q^{25} +1.00000i q^{27} +0.151350i q^{29} -4.75438 q^{31} -4.07092 q^{33} +11.3431i q^{37} +1.44065 q^{39} +0.239424 q^{41} -1.32831i q^{43} +3.56550i q^{45} -6.35872 q^{47} +6.99110i q^{51} +5.98569i q^{53} -14.5149 q^{55} -0.301495 q^{57} -11.2428i q^{59} +4.20933i q^{61} +5.13663 q^{65} -4.38075i q^{67} -2.42499i q^{69} +8.46387 q^{71} -0.569448 q^{73} +7.71279i q^{75} +1.49383 q^{79} +1.00000 q^{81} +10.0352i q^{83} +24.9268i q^{85} +0.151350 q^{87} +3.66869 q^{89} +4.75438i q^{93} -1.07498 q^{95} -10.4657 q^{97} +4.07092i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{9}+O(q^{10}) $ 16 * q - 16 * q^9	$ 16 q - 16 q^{9} - 8 q^{23} - 16 q^{25} + 24 q^{31} + 24 q^{47} - 32 q^{55} - 8 q^{57} + 40 q^{71} + 8 q^{73} + 8 q^{79} + 16 q^{81} - 24 q^{87} + 24 q^{95} + 24 q^{97}+O(q^{100}) $ 16 * q - 16 * q^9 - 8 * q^23 - 16 * q^25 + 24 * q^31 + 24 * q^47 - 32 * q^55 - 8 * q^57 + 40 * q^71 + 8 * q^73 + 8 * q^79 + 16 * q^81 - 24 * q^87 + 24 * q^95 + 24 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4704\mathbb{Z}\right)^\times$.

$n$	$1471$	$1765$	$3137$	$4609$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$5$	− 3.56550i	− 1.59454i	−0.603623	−	0.797270i	$-0.706277\pi$
$5$	− 3.56550i	− 1.59454i	0.603623	−	0.797270i	$-0.293723\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 4.07092i	− 1.22743i	−0.789528	−	0.613714i	$-0.789675\pi$
$11$	− 4.07092i	− 1.22743i	0.789528	−	0.613714i	$-0.210325\pi$
$12$	0	0
$13$	1.44065i	0.399564i	0.979840	+	0.199782i	$0.0640233\pi$
$13$	1.44065i	0.399564i	−0.979840	+	0.199782i	$0.935977\pi$
$14$	0	0
$15$	−3.56550	−0.920608
$16$	0	0
$17$	−6.99110	−1.69559	−0.847796	−	0.530323i	$-0.822071\pi$
$17$	−6.99110	−1.69559	−0.847796	+	0.530323i	$0.822071\pi$
$18$	0	0
$19$	− 0.301495i	− 0.0691677i	−0.999402	−	0.0345838i	$-0.988989\pi$
$19$	− 0.301495i	− 0.0691677i	0.999402	−	0.0345838i	$-0.0110106\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.42499	0.505646	0.252823	−	0.967513i	$-0.418641\pi$
$23$	2.42499	0.505646	0.252823	+	0.967513i	$0.418641\pi$
$24$	0	0
$25$	−7.71279	−1.54256
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	0.151350i	0.0281050i	0.999901	+	0.0140525i	$0.00447319\pi$
$29$	0.151350i	0.0281050i	−0.999901	+	0.0140525i	$0.995527\pi$
$30$	0	0
$31$	−4.75438	−0.853912	−0.426956	−	0.904272i	$-0.640414\pi$
$31$	−4.75438	−0.853912	−0.426956	+	0.904272i	$0.640414\pi$
$32$	0	0
$33$	−4.07092	−0.708656
$34$	0	0
$35$	0	0
$36$	0	0
$37$	11.3431i	1.86479i	0.361443	+	0.932394i	$0.382284\pi$
$37$	11.3431i	1.86479i	−0.361443	+	0.932394i	$0.617716\pi$
$38$	0	0
$39$	1.44065	0.230688
$40$	0	0
$41$	0.239424	0.0373917	0.0186959	−	0.999825i	$-0.494049\pi$
$41$	0.239424	0.0373917	0.0186959	+	0.999825i	$0.494049\pi$
$42$	0	0
$43$	− 1.32831i	− 0.202566i	−0.994858	−	0.101283i	$-0.967705\pi$
$43$	− 1.32831i	− 0.202566i	0.994858	−	0.101283i	$-0.0322947\pi$
$44$	0	0
$45$	3.56550i	0.531513i
$46$	0	0
$47$	−6.35872	−0.927515	−0.463757	−	0.885962i	$-0.653499\pi$
$47$	−6.35872	−0.927515	−0.463757	+	0.885962i	$0.653499\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	6.99110i	0.978950i
$52$	0	0
$53$	5.98569i	0.822198i	0.911591	+	0.411099i	$0.134855\pi$
$53$	5.98569i	0.822198i	−0.911591	+	0.411099i	$0.865145\pi$
$54$	0	0
$55$	−14.5149	−1.95718
$56$	0	0
$57$	−0.301495	−0.0399340
$58$	0	0
$59$	− 11.2428i	− 1.46369i	−0.681473	−	0.731843i	$-0.738660\pi$
$59$	− 11.2428i	− 1.46369i	0.681473	−	0.731843i	$-0.261340\pi$
$60$	0	0
$61$	4.20933i	0.538950i	0.963007	+	0.269475i	$0.0868501\pi$
$61$	4.20933i	0.538950i	−0.963007	+	0.269475i	$0.913150\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.13663	0.637120
$66$	0	0
$67$	− 4.38075i	− 0.535194i	−0.963531	−	0.267597i	$-0.913771\pi$
$67$	− 4.38075i	− 0.535194i	0.963531	−	0.267597i	$-0.0862295\pi$
$68$	0	0
$69$	− 2.42499i	− 0.291935i
$70$	0	0
$71$	8.46387	1.00448	0.502238	−	0.864729i	$-0.332510\pi$
$71$	8.46387	1.00448	0.502238	+	0.864729i	$0.332510\pi$
$72$	0	0
$73$	−0.569448	−0.0666488	−0.0333244	−	0.999445i	$-0.510609\pi$
$73$	−0.569448	−0.0666488	−0.0333244	+	0.999445i	$0.510609\pi$
$74$	0	0
$75$	7.71279i	0.890596i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	1.49383	0.168069	0.0840347	−	0.996463i	$-0.473219\pi$
$79$	1.49383	0.168069	0.0840347	+	0.996463i	$0.473219\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.0352i	1.10151i	0.834668	+	0.550753i	$0.185660\pi$
$83$	10.0352i	1.10151i	−0.834668	+	0.550753i	$0.814340\pi$
$84$	0	0
$85$	24.9268i	2.70369i
$86$	0	0
$87$	0.151350	0.0162264
$88$	0	0
$89$	3.66869	0.388880	0.194440	−	0.980914i	$-0.437711\pi$
$89$	3.66869	0.388880	0.194440	+	0.980914i	$0.437711\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	4.75438i	0.493006i
$94$	0	0
$95$	−1.07498	−0.110291
$96$	0	0
$97$	−10.4657	−1.06264	−0.531318	−	0.847173i	$-0.678303\pi$
$97$	−10.4657	−1.06264	−0.531318	+	0.847173i	$0.678303\pi$
$98$	0	0
$99$	4.07092i	0.409143i
$100$	0	0
$101$	− 12.7337i	− 1.26705i	−0.773722	−	0.633526i	$-0.781607\pi$
$101$	− 12.7337i	− 1.26705i	0.773722	−	0.633526i	$-0.218393\pi$
$102$	0	0
$103$	−6.00875	−0.592060	−0.296030	−	0.955179i	$-0.595663\pi$
$103$	−6.00875	−0.592060	−0.296030	+	0.955179i	$0.595663\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 15.9659i	− 1.54349i	−0.635935	−	0.771743i	$-0.719385\pi$
$107$	− 15.9659i	− 1.54349i	0.635935	−	0.771743i	$-0.280615\pi$
$108$	0	0
$109$	13.8711i	1.32861i	0.747463	+	0.664304i	$0.231272\pi$
$109$	13.8711i	1.32861i	−0.747463	+	0.664304i	$0.768728\pi$
$110$	0	0
$111$	11.3431	1.07664
$112$	0	0
$113$	−16.4495	−1.54744	−0.773719	−	0.633529i	$-0.781606\pi$
$113$	−16.4495	−1.54744	−0.773719	+	0.633529i	$0.781606\pi$
$114$	0	0
$115$	− 8.64631i	− 0.806272i
$116$	0	0
$117$	− 1.44065i	− 0.133188i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−5.57239	−0.506581
$122$	0	0
$123$	− 0.239424i	− 0.0215881i
$124$	0	0
$125$	9.67245i	0.865131i
$126$	0	0
$127$	3.33297	0.295754	0.147877	−	0.989006i	$-0.452756\pi$
$127$	3.33297	0.295754	0.147877	+	0.989006i	$0.452756\pi$
$128$	0	0
$129$	−1.32831	−0.116951
$130$	0	0
$131$	4.08867i	0.357229i	0.983919	+	0.178614i	$0.0571614\pi$
$131$	4.08867i	0.357229i	−0.983919	+	0.178614i	$0.942839\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	3.56550	0.306869
$136$	0	0
$137$	8.28737	0.708037	0.354019	−	0.935238i	$-0.384815\pi$
$137$	8.28737	0.708037	0.354019	+	0.935238i	$0.384815\pi$
$138$	0	0
$139$	− 18.4180i	− 1.56219i	−0.624411	−	0.781096i	$-0.714661\pi$
$139$	− 18.4180i	− 1.56219i	0.624411	−	0.781096i	$-0.285339\pi$
$140$	0	0
$141$	6.35872i	0.535501i
$142$	0	0
$143$	5.86476	0.490436
$144$	0	0
$145$	0.539638	0.0448145
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 5.11203i	− 0.418794i	−0.977831	−	0.209397i	$-0.932850\pi$
$149$	− 5.11203i	− 0.418794i	0.977831	−	0.209397i	$-0.0671501\pi$
$150$	0	0
$151$	4.17126	0.339452	0.169726	−	0.985491i	$-0.445712\pi$
$151$	4.17126	0.339452	0.169726	+	0.985491i	$0.445712\pi$
$152$	0	0
$153$	6.99110	0.565197
$154$	0	0
$155$	16.9517i	1.36160i
$156$	0	0
$157$	− 9.04875i	− 0.722169i	−0.932533	−	0.361084i	$-0.882407\pi$
$157$	− 9.04875i	− 0.722169i	0.932533	−	0.361084i	$-0.117593\pi$
$158$	0	0
$159$	5.98569	0.474696
$160$	0	0
$161$	0	0
$162$	0	0
$163$	13.4512i	1.05358i	0.849996	+	0.526790i	$0.176604\pi$
$163$	13.4512i	1.05358i	−0.849996	+	0.526790i	$0.823396\pi$
$164$	0	0
$165$	14.5149i	1.12998i
$166$	0	0
$167$	17.2099	1.33174	0.665872	−	0.746066i	$-0.268060\pi$
$167$	17.2099	1.33174	0.665872	+	0.746066i	$0.268060\pi$
$168$	0	0
$169$	10.9245	0.840349
$170$	0	0
$171$	0.301495i	0.0230559i
$172$	0	0
$173$	3.44940i	0.262253i	0.991366	+	0.131126i	$0.0418594\pi$
$173$	3.44940i	0.262253i	−0.991366	+	0.131126i	$0.958141\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−11.2428	−0.845060
$178$	0	0
$179$	− 11.3017i	− 0.844726i	−0.906427	−	0.422363i	$-0.861201\pi$
$179$	− 11.3017i	− 0.844726i	0.906427	−	0.422363i	$-0.138799\pi$
$180$	0	0
$181$	10.0566i	0.747502i	0.927529	+	0.373751i	$0.121929\pi$
$181$	10.0566i	0.747502i	−0.927529	+	0.373751i	$0.878071\pi$
$182$	0	0
$183$	4.20933	0.311163
$184$	0	0
$185$	40.4437	2.97348
$186$	0	0
$187$	28.4602i	2.08122i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−21.4860	−1.55468	−0.777338	−	0.629084i	$-0.783430\pi$
$191$	−21.4860	−1.55468	−0.777338	+	0.629084i	$0.783430\pi$
$192$	0	0
$193$	−16.4160	−1.18165	−0.590823	−	0.806801i	$-0.701197\pi$
$193$	−16.4160	−1.18165	−0.590823	+	0.806801i	$0.701197\pi$
$194$	0	0
$195$	− 5.13663i	− 0.367841i
$196$	0	0
$197$	15.7045i	1.11890i	0.828864	+	0.559450i	$0.188988\pi$
$197$	15.7045i	1.11890i	−0.828864	+	0.559450i	$0.811012\pi$
$198$	0	0
$199$	6.27434	0.444776	0.222388	−	0.974958i	$-0.428615\pi$
$199$	6.27434	0.444776	0.222388	+	0.974958i	$0.428615\pi$
$200$	0	0
$201$	−4.38075	−0.308994
$202$	0	0
$203$	0	0
$204$	0	0
$205$	− 0.853666i	− 0.0596226i
$206$	0	0
$207$	−2.42499	−0.168549
$208$	0	0
$209$	−1.22736	−0.0848984
$210$	0	0
$211$	9.40458i	0.647438i	0.946153	+	0.323719i	$0.104933\pi$
$211$	9.40458i	0.647438i	−0.946153	+	0.323719i	$0.895067\pi$
$212$	0	0
$213$	− 8.46387i	− 0.579935i
$214$	0	0
$215$	−4.73610	−0.322999
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0.569448i	0.0384797i
$220$	0	0
$221$	− 10.0717i	− 0.677496i
$222$	0	0
$223$	4.34741	0.291124	0.145562	−	0.989349i	$-0.453501\pi$
$223$	4.34741	0.291124	0.145562	+	0.989349i	$0.453501\pi$
$224$	0	0
$225$	7.71279	0.514186
$226$	0	0
$227$	− 7.28203i	− 0.483326i	−0.970360	−	0.241663i	$-0.922307\pi$
$227$	− 7.28203i	− 0.483326i	0.970360	−	0.241663i	$-0.0776928\pi$
$228$	0	0
$229$	1.02222i	0.0675505i	0.999429	+	0.0337752i	$0.0107530\pi$
$229$	1.02222i	0.0675505i	−0.999429	+	0.0337752i	$0.989247\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−22.0661	−1.44560	−0.722800	−	0.691058i	$-0.757145\pi$
$233$	−22.0661	−1.44560	−0.722800	+	0.691058i	$0.757145\pi$
$234$	0	0
$235$	22.6720i	1.47896i
$236$	0	0
$237$	− 1.49383i	− 0.0970349i
$238$	0	0
$239$	−4.80475	−0.310794	−0.155397	−	0.987852i	$-0.549666\pi$
$239$	−4.80475	−0.310794	−0.155397	+	0.987852i	$0.549666\pi$
$240$	0	0
$241$	−23.7688	−1.53108	−0.765542	−	0.643386i	$-0.777529\pi$
$241$	−23.7688	−1.53108	−0.765542	+	0.643386i	$0.777529\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0.434348	0.0276369
$248$	0	0
$249$	10.0352	0.635955
$250$	0	0
$251$	14.0888i	0.889280i	0.895709	+	0.444640i	$0.146668\pi$
$251$	14.0888i	0.889280i	−0.895709	+	0.444640i	$0.853332\pi$
$252$	0	0
$253$	− 9.87195i	− 0.620644i
$254$	0	0
$255$	24.9268	1.56098
$256$	0	0
$257$	5.79924	0.361746	0.180873	−	0.983506i	$-0.442108\pi$
$257$	5.79924	0.361746	0.180873	+	0.983506i	$0.442108\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	− 0.151350i	− 0.00936832i
$262$	0	0
$263$	−2.43254	−0.149997	−0.0749983	−	0.997184i	$-0.523895\pi$
$263$	−2.43254	−0.149997	−0.0749983	+	0.997184i	$0.523895\pi$
$264$	0	0
$265$	21.3420	1.31103
$266$	0	0
$267$	− 3.66869i	− 0.224520i
$268$	0	0
$269$	5.75584i	0.350940i	0.984485	+	0.175470i	$0.0561445\pi$
$269$	5.75584i	0.350940i	−0.984485	+	0.175470i	$0.943856\pi$
$270$	0	0
$271$	17.0185	1.03380	0.516901	−	0.856045i	$-0.327086\pi$
$271$	17.0185	1.03380	0.516901	+	0.856045i	$0.327086\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	31.3982i	1.89338i
$276$	0	0
$277$	− 22.4704i	− 1.35011i	−0.737765	−	0.675057i	$-0.764119\pi$
$277$	− 22.4704i	− 1.35011i	0.737765	−	0.675057i	$-0.235881\pi$
$278$	0	0
$279$	4.75438	0.284637
$280$	0	0
$281$	7.29261	0.435041	0.217520	−	0.976056i	$-0.430203\pi$
$281$	7.29261	0.435041	0.217520	+	0.976056i	$0.430203\pi$
$282$	0	0
$283$	− 10.6377i	− 0.632344i	−0.948702	−	0.316172i	$-0.897602\pi$
$283$	− 10.6377i	− 0.632344i	0.948702	−	0.316172i	$-0.102398\pi$
$284$	0	0
$285$	1.07498i	0.0636763i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	31.8755	1.87503
$290$	0	0
$291$	10.4657i	0.613513i
$292$	0	0
$293$	26.4432i	1.54483i	0.635120	+	0.772414i	$0.280951\pi$
$293$	26.4432i	1.54483i	−0.635120	+	0.772414i	$0.719049\pi$
$294$	0	0
$295$	−40.0862	−2.33391
$296$	0	0
$297$	4.07092	0.236219
$298$	0	0
$299$	3.49356i	0.202038i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−12.7337	−0.731532
$304$	0	0
$305$	15.0084	0.859377
$306$	0	0
$307$	23.6553i	1.35008i	0.737780	+	0.675041i	$0.235874\pi$
$307$	23.6553i	1.35008i	−0.737780	+	0.675041i	$0.764126\pi$
$308$	0	0
$309$	6.00875i	0.341826i
$310$	0	0
$311$	−3.25213	−0.184411	−0.0922056	−	0.995740i	$-0.529392\pi$
$311$	−3.25213	−0.184411	−0.0922056	+	0.995740i	$0.529392\pi$
$312$	0	0
$313$	−6.95242	−0.392974	−0.196487	−	0.980506i	$-0.562953\pi$
$313$	−6.95242	−0.392974	−0.196487	+	0.980506i	$0.562953\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.28507i	0.353005i	0.984300	+	0.176502i	$0.0564783\pi$
$317$	6.28507i	0.353005i	−0.984300	+	0.176502i	$0.943522\pi$
$318$	0	0
$319$	0.616133	0.0344968
$320$	0	0
$321$	−15.9659	−0.891132
$322$	0	0
$323$	2.10778i	0.117280i
$324$	0	0
$325$	− 11.1114i	− 0.616350i
$326$	0	0
$327$	13.8711	0.767072
$328$	0	0
$329$	0	0
$330$	0	0
$331$	25.3553i	1.39365i	0.717239	+	0.696827i	$0.245406\pi$
$331$	25.3553i	1.39365i	−0.717239	+	0.696827i	$0.754594\pi$
$332$	0	0
$333$	− 11.3431i	− 0.621596i
$334$	0	0
$335$	−15.6196	−0.853388
$336$	0	0
$337$	−18.0761	−0.984668	−0.492334	−	0.870406i	$-0.663856\pi$
$337$	−18.0761	−0.984668	−0.492334	+	0.870406i	$0.663856\pi$
$338$	0	0
$339$	16.4495i	0.893413i
$340$	0	0
$341$	19.3547i	1.04812i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−8.64631	−0.465501
$346$	0	0
$347$	7.46130i	0.400543i	0.979740	+	0.200272i	$0.0641825\pi$
$347$	7.46130i	0.400543i	−0.979740	+	0.200272i	$0.935818\pi$
$348$	0	0
$349$	− 23.2053i	− 1.24215i	−0.783750	−	0.621077i	$-0.786695\pi$
$349$	− 23.2053i	− 1.24215i	0.783750	−	0.621077i	$-0.213305\pi$
$350$	0	0
$351$	−1.44065	−0.0768960
$352$	0	0
$353$	4.29061	0.228366	0.114183	−	0.993460i	$-0.463575\pi$
$353$	4.29061	0.228366	0.114183	+	0.993460i	$0.463575\pi$
$354$	0	0
$355$	− 30.1779i	− 1.60168i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−6.01411	−0.317413	−0.158706	−	0.987326i	$-0.550732\pi$
$359$	−6.01411	−0.317413	−0.158706	+	0.987326i	$0.550732\pi$
$360$	0	0
$361$	18.9091	0.995216
$362$	0	0
$363$	5.57239i	0.292475i
$364$	0	0
$365$	2.03037i	0.106274i
$366$	0	0
$367$	−15.5739	−0.812949	−0.406475	−	0.913662i	$-0.633242\pi$
$367$	−15.5739	−0.812949	−0.406475	+	0.913662i	$0.633242\pi$
$368$	0	0
$369$	−0.239424	−0.0124639
$370$	0	0
$371$	0	0
$372$	0	0
$373$	11.3685i	0.588641i	0.955707	+	0.294320i	$0.0950933\pi$
$373$	11.3685i	0.588641i	−0.955707	+	0.294320i	$0.904907\pi$
$374$	0	0
$375$	9.67245	0.499483
$376$	0	0
$377$	−0.218042	−0.0112297
$378$	0	0
$379$	31.9644i	1.64190i	0.571000	+	0.820950i	$0.306556\pi$
$379$	31.9644i	1.64190i	−0.571000	+	0.820950i	$0.693444\pi$
$380$	0	0
$381$	− 3.33297i	− 0.170753i
$382$	0	0
$383$	−9.50448	−0.485656	−0.242828	−	0.970069i	$-0.578075\pi$
$383$	−9.50448	−0.485656	−0.242828	+	0.970069i	$0.578075\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.32831i	0.0675219i
$388$	0	0
$389$	11.3605i	0.575998i	0.957631	+	0.287999i	$0.0929901\pi$
$389$	11.3605i	0.575998i	−0.957631	+	0.287999i	$0.907010\pi$
$390$	0	0
$391$	−16.9534	−0.857368
$392$	0	0
$393$	4.08867	0.206246
$394$	0	0
$395$	− 5.32626i	− 0.267993i
$396$	0	0
$397$	− 21.5933i	− 1.08374i	−0.840463	−	0.541868i	$-0.817717\pi$
$397$	− 21.5933i	− 1.08374i	0.840463	−	0.541868i	$-0.182283\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	28.2459	1.41053	0.705267	−	0.708942i	$-0.250827\pi$
$401$	28.2459	1.41053	0.705267	+	0.708942i	$0.250827\pi$
$402$	0	0
$403$	− 6.84938i	− 0.341192i
$404$	0	0
$405$	− 3.56550i	− 0.177171i
$406$	0	0
$407$	46.1767	2.28889
$408$	0	0
$409$	−13.4892	−0.666996	−0.333498	−	0.942751i	$-0.608229\pi$
$409$	−13.4892	−0.666996	−0.333498	+	0.942751i	$0.608229\pi$
$410$	0	0
$411$	− 8.28737i	− 0.408786i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	35.7805	1.75640
$416$	0	0
$417$	−18.4180	−0.901932
$418$	0	0
$419$	− 25.0575i	− 1.22414i	−0.790804	−	0.612069i	$-0.790337\pi$
$419$	− 25.0575i	− 1.22414i	0.790804	−	0.612069i	$-0.209663\pi$
$420$	0	0
$421$	− 23.0346i	− 1.12264i	−0.827600	−	0.561318i	$-0.810294\pi$
$421$	− 23.0346i	− 1.12264i	0.827600	−	0.561318i	$-0.189706\pi$
$422$	0	0
$423$	6.35872	0.309172
$424$	0	0
$425$	53.9209	2.61555
$426$	0	0
$427$	0	0
$428$	0	0
$429$	− 5.86476i	− 0.283153i
$430$	0	0
$431$	−0.185856	−0.00895238	−0.00447619	−	0.999990i	$-0.501425\pi$
$431$	−0.185856	−0.00895238	−0.00447619	+	0.999990i	$0.501425\pi$
$432$	0	0
$433$	−34.7454	−1.66976	−0.834878	−	0.550435i	$-0.814462\pi$
$433$	−34.7454	−1.66976	−0.834878	+	0.550435i	$0.814462\pi$
$434$	0	0
$435$	− 0.539638i	− 0.0258737i
$436$	0	0
$437$	− 0.731123i	− 0.0349743i
$438$	0	0
$439$	−27.1781	−1.29714	−0.648570	−	0.761155i	$-0.724632\pi$
$439$	−27.1781	−1.29714	−0.648570	+	0.761155i	$0.724632\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	3.96868i	0.188558i	0.995546	+	0.0942789i	$0.0300545\pi$
$443$	3.96868i	0.188558i	−0.995546	+	0.0942789i	$0.969945\pi$
$444$	0	0
$445$	− 13.0807i	− 0.620085i
$446$	0	0
$447$	−5.11203	−0.241791
$448$	0	0
$449$	−12.1637	−0.574040	−0.287020	−	0.957925i	$-0.592665\pi$
$449$	−12.1637	−0.574040	−0.287020	+	0.957925i	$0.592665\pi$
$450$	0	0
$451$	− 0.974675i	− 0.0458957i
$452$	0	0
$453$	− 4.17126i	− 0.195983i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	25.7582	1.20492	0.602459	−	0.798150i	$-0.294188\pi$
$457$	25.7582	1.20492	0.602459	+	0.798150i	$0.294188\pi$
$458$	0	0
$459$	− 6.99110i	− 0.326317i
$460$	0	0
$461$	− 17.0423i	− 0.793737i	−0.917875	−	0.396868i	$-0.870097\pi$
$461$	− 17.0423i	− 0.793737i	0.917875	−	0.396868i	$-0.129903\pi$
$462$	0	0
$463$	−34.1343	−1.58635	−0.793177	−	0.608991i	$-0.791574\pi$
$463$	−34.1343	−1.58635	−0.793177	+	0.608991i	$0.791574\pi$
$464$	0	0
$465$	16.9517	0.786118
$466$	0	0
$467$	− 16.2743i	− 0.753085i	−0.926399	−	0.376543i	$-0.877113\pi$
$467$	− 16.2743i	− 0.753085i	0.926399	−	0.376543i	$-0.122887\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−9.04875	−0.416944
$472$	0	0
$473$	−5.40746	−0.248635
$474$	0	0
$475$	2.32537i	0.106695i
$476$	0	0
$477$	− 5.98569i	− 0.274066i
$478$	0	0
$479$	−20.0091	−0.914239	−0.457119	−	0.889405i	$-0.651119\pi$
$479$	−20.0091	−0.914239	−0.457119	+	0.889405i	$0.651119\pi$
$480$	0	0
$481$	−16.3413	−0.745101
$482$	0	0
$483$	0	0
$484$	0	0
$485$	37.3156i	1.69442i
$486$	0	0
$487$	−16.5578	−0.750304	−0.375152	−	0.926963i	$-0.622410\pi$
$487$	−16.5578	−0.750304	−0.375152	+	0.926963i	$0.622410\pi$
$488$	0	0
$489$	13.4512	0.608284
$490$	0	0
$491$	9.49257i	0.428394i	0.976791	+	0.214197i	$0.0687134\pi$
$491$	9.49257i	0.428394i	−0.976791	+	0.214197i	$0.931287\pi$
$492$	0	0
$493$	− 1.05810i	− 0.0476545i
$494$	0	0
$495$	14.5149	0.652395
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 34.2301i	− 1.53235i	−0.642633	−	0.766174i	$-0.722158\pi$
$499$	− 34.2301i	− 1.53235i	0.642633	−	0.766174i	$-0.277842\pi$
$500$	0	0
$501$	− 17.2099i	− 0.768883i
$502$	0	0
$503$	−21.0469	−0.938436	−0.469218	−	0.883082i	$-0.655464\pi$
$503$	−21.0469	−0.938436	−0.469218	+	0.883082i	$0.655464\pi$
$504$	0	0
$505$	−45.4020	−2.02036
$506$	0	0
$507$	− 10.9245i	− 0.485176i
$508$	0	0
$509$	15.0455i	0.666881i	0.942771	+	0.333440i	$0.108210\pi$
$509$	15.0455i	0.666881i	−0.942771	+	0.333440i	$0.891790\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0.301495	0.0133113
$514$	0	0
$515$	21.4242i	0.944063i
$516$	0	0
$517$	25.8858i	1.13846i
$518$	0	0
$519$	3.44940	0.151412
$520$	0	0
$521$	8.97913	0.393383	0.196691	−	0.980465i	$-0.436980\pi$
$521$	8.97913	0.393383	0.196691	+	0.980465i	$0.436980\pi$
$522$	0	0
$523$	− 9.87353i	− 0.431739i	−0.976422	−	0.215870i	$-0.930741\pi$
$523$	− 9.87353i	− 0.431739i	0.976422	−	0.215870i	$-0.0692586\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	33.2384	1.44789
$528$	0	0
$529$	−17.1194	−0.744323
$530$	0	0
$531$	11.2428i	0.487896i
$532$	0	0
$533$	0.344925i	0.0149404i
$534$	0	0
$535$	−56.9266	−2.46115
$536$	0	0
$537$	−11.3017	−0.487703
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 19.7815i	− 0.850475i	−0.905082	−	0.425237i	$-0.860191\pi$
$541$	− 19.7815i	− 0.850475i	0.905082	−	0.425237i	$-0.139809\pi$
$542$	0	0
$543$	10.0566	0.431570
$544$	0	0
$545$	49.4573	2.11852
$546$	0	0
$547$	− 39.3961i	− 1.68446i	−0.539122	−	0.842228i	$-0.681244\pi$
$547$	− 39.3961i	− 1.68446i	0.539122	−	0.842228i	$-0.318756\pi$
$548$	0	0
$549$	− 4.20933i	− 0.179650i
$550$	0	0
$551$	0.0456312	0.00194395
$552$	0	0
$553$	0	0
$554$	0	0
$555$	− 40.4437i	− 1.71674i
$556$	0	0
$557$	6.75051i	0.286028i	0.989721	+	0.143014i	$0.0456794\pi$
$557$	6.75051i	0.286028i	−0.989721	+	0.143014i	$0.954321\pi$
$558$	0	0
$559$	1.91363	0.0809379
$560$	0	0
$561$	28.4602	1.20159
$562$	0	0
$563$	28.9861i	1.22162i	0.791778	+	0.610809i	$0.209156\pi$
$563$	28.9861i	1.22162i	−0.791778	+	0.610809i	$0.790844\pi$
$564$	0	0
$565$	58.6506i	2.46745i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−39.9668	−1.67550	−0.837748	−	0.546057i	$-0.816128\pi$
$569$	−39.9668	−1.67550	−0.837748	+	0.546057i	$0.816128\pi$
$570$	0	0
$571$	8.09846i	0.338910i	0.985538	+	0.169455i	$0.0542007\pi$
$571$	8.09846i	0.338910i	−0.985538	+	0.169455i	$0.945799\pi$
$572$	0	0
$573$	21.4860i	0.897592i
$574$	0	0
$575$	−18.7034	−0.779988
$576$	0	0
$577$	−47.5222	−1.97838	−0.989188	−	0.146653i	$-0.953150\pi$
$577$	−47.5222	−1.97838	−0.989188	+	0.146653i	$0.953150\pi$
$578$	0	0
$579$	16.4160i	0.682224i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	24.3673	1.00919
$584$	0	0
$585$	−5.13663	−0.212373
$586$	0	0
$587$	− 11.8785i	− 0.490279i	−0.969488	−	0.245140i	$-0.921166\pi$
$587$	− 11.8785i	− 0.490279i	0.969488	−	0.245140i	$-0.0788338\pi$
$588$	0	0
$589$	1.43342i	0.0590631i
$590$	0	0
$591$	15.7045	0.645997
$592$	0	0
$593$	−32.2320	−1.32361	−0.661806	−	0.749675i	$-0.730210\pi$
$593$	−32.2320	−1.32361	−0.661806	+	0.749675i	$0.730210\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 6.27434i	− 0.256792i
$598$	0	0
$599$	−39.6855	−1.62150	−0.810752	−	0.585390i	$-0.800942\pi$
$599$	−39.6855	−1.62150	−0.810752	+	0.585390i	$0.800942\pi$
$600$	0	0
$601$	−22.3599	−0.912081	−0.456040	−	0.889959i	$-0.650733\pi$
$601$	−22.3599	−0.912081	−0.456040	+	0.889959i	$0.650733\pi$
$602$	0	0
$603$	4.38075i	0.178398i
$604$	0	0
$605$	19.8684i	0.807764i
$606$	0	0
$607$	14.7751	0.599704	0.299852	−	0.953986i	$-0.403063\pi$
$607$	14.7751	0.599704	0.299852	+	0.953986i	$0.403063\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 9.16067i	− 0.370601i
$612$	0	0
$613$	26.2641i	1.06080i	0.847749	+	0.530398i	$0.177957\pi$
$613$	26.2641i	1.06080i	−0.847749	+	0.530398i	$0.822043\pi$
$614$	0	0
$615$	−0.853666	−0.0344231
$616$	0	0
$617$	33.8882	1.36429	0.682144	−	0.731218i	$-0.261048\pi$
$617$	33.8882	1.36429	0.682144	+	0.731218i	$0.261048\pi$
$618$	0	0
$619$	6.05216i	0.243257i	0.992576	+	0.121628i	$0.0388116\pi$
$619$	6.05216i	0.243257i	−0.992576	+	0.121628i	$0.961188\pi$
$620$	0	0
$621$	2.42499i	0.0973115i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−4.07682	−0.163073
$626$	0	0
$627$	1.22736i	0.0490161i
$628$	0	0
$629$	− 79.3005i	− 3.16192i
$630$	0	0
$631$	14.6555	0.583426	0.291713	−	0.956506i	$-0.405775\pi$
$631$	14.6555	0.583426	0.291713	+	0.956506i	$0.405775\pi$
$632$	0	0
$633$	9.40458	0.373798
$634$	0	0
$635$	− 11.8837i	− 0.471591i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−8.46387	−0.334826
$640$	0	0
$641$	15.5498	0.614178	0.307089	−	0.951681i	$-0.400645\pi$
$641$	15.5498	0.614178	0.307089	+	0.951681i	$0.400645\pi$
$642$	0	0
$643$	28.3180i	1.11675i	0.829587	+	0.558377i	$0.188576\pi$
$643$	28.3180i	1.11675i	−0.829587	+	0.558377i	$0.811424\pi$
$644$	0	0
$645$	4.73610i	0.186484i
$646$	0	0
$647$	25.8683	1.01699	0.508495	−	0.861065i	$-0.330202\pi$
$647$	25.8683	1.01699	0.508495	+	0.861065i	$0.330202\pi$
$648$	0	0
$649$	−45.7685	−1.79657
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 28.2649i	− 1.10609i	−0.833151	−	0.553045i	$-0.813466\pi$
$653$	− 28.2649i	− 1.10609i	0.833151	−	0.553045i	$-0.186534\pi$
$654$	0	0
$655$	14.5781	0.569615
$656$	0	0
$657$	0.569448	0.0222163
$658$	0	0
$659$	21.7217i	0.846159i	0.906093	+	0.423079i	$0.139051\pi$
$659$	21.7217i	0.846159i	−0.906093	+	0.423079i	$0.860949\pi$
$660$	0	0
$661$	− 31.7834i	− 1.23623i	−0.786087	−	0.618115i	$-0.787896\pi$
$661$	− 31.7834i	− 1.23623i	0.786087	−	0.618115i	$-0.212104\pi$
$662$	0	0
$663$	−10.0717	−0.391153
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.367022i	0.0142111i
$668$	0	0
$669$	− 4.34741i	− 0.168081i
$670$	0	0
$671$	17.1359	0.661523
$672$	0	0
$673$	16.1882	0.624009	0.312004	−	0.950081i	$-0.399000\pi$
$673$	16.1882	0.624009	0.312004	+	0.950081i	$0.399000\pi$
$674$	0	0
$675$	− 7.71279i	− 0.296865i
$676$	0	0
$677$	23.8876i	0.918077i	0.888416	+	0.459038i	$0.151806\pi$
$677$	23.8876i	0.918077i	−0.888416	+	0.459038i	$0.848194\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−7.28203	−0.279048
$682$	0	0
$683$	25.2315i	0.965456i	0.875770	+	0.482728i	$0.160354\pi$
$683$	25.2315i	0.965456i	−0.875770	+	0.482728i	$0.839646\pi$
$684$	0	0
$685$	− 29.5486i	− 1.12899i
$686$	0	0
$687$	1.02222	0.0390003
$688$	0	0
$689$	−8.62327	−0.328520
$690$	0	0
$691$	15.9452i	0.606584i	0.952898	+	0.303292i	$0.0980857\pi$
$691$	15.9452i	0.606584i	−0.952898	+	0.303292i	$0.901914\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−65.6693	−2.49098
$696$	0	0
$697$	−1.67384	−0.0634011
$698$	0	0
$699$	22.0661i	0.834617i
$700$	0	0
$701$	− 46.5568i	− 1.75842i	−0.476430	−	0.879212i	$-0.658069\pi$
$701$	− 46.5568i	− 1.75842i	0.476430	−	0.879212i	$-0.341931\pi$
$702$	0	0
$703$	3.41988	0.128983
$704$	0	0
$705$	22.6720	0.853877
$706$	0	0
$707$	0	0
$708$	0	0
$709$	5.40547i	0.203007i	0.994835	+	0.101503i	$0.0323653\pi$
$709$	5.40547i	0.203007i	−0.994835	+	0.101503i	$0.967635\pi$
$710$	0	0
$711$	−1.49383	−0.0560231
$712$	0	0
$713$	−11.5293	−0.431777
$714$	0	0
$715$	− 20.9108i	− 0.782019i
$716$	0	0
$717$	4.80475i	0.179437i
$718$	0	0
$719$	19.4807	0.726508	0.363254	−	0.931690i	$-0.381666\pi$
$719$	19.4807	0.726508	0.363254	+	0.931690i	$0.381666\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	23.7688i	0.883972i
$724$	0	0
$725$	− 1.16733i	− 0.0433535i
$726$	0	0
$727$	22.4025	0.830864	0.415432	−	0.909624i	$-0.363630\pi$
$727$	22.4025	0.830864	0.415432	+	0.909624i	$0.363630\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	9.28637i	0.343469i
$732$	0	0
$733$	− 46.5412i	− 1.71904i	−0.511103	−	0.859520i	$-0.670763\pi$
$733$	− 46.5412i	− 1.71904i	0.511103	−	0.859520i	$-0.329237\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−17.8337	−0.656912
$738$	0	0
$739$	− 31.9417i	− 1.17499i	−0.809226	−	0.587497i	$-0.800113\pi$
$739$	− 31.9417i	− 1.17499i	0.809226	−	0.587497i	$-0.199887\pi$
$740$	0	0
$741$	− 0.434348i	− 0.0159562i
$742$	0	0
$743$	−14.3888	−0.527874	−0.263937	−	0.964540i	$-0.585021\pi$
$743$	−14.3888	−0.527874	−0.263937	+	0.964540i	$0.585021\pi$
$744$	0	0
$745$	−18.2269	−0.667784
$746$	0	0
$747$	− 10.0352i	− 0.367169i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	30.3137	1.10616	0.553080	−	0.833128i	$-0.313452\pi$
$751$	30.3137	1.10616	0.553080	+	0.833128i	$0.313452\pi$
$752$	0	0
$753$	14.0888	0.513426
$754$	0	0
$755$	− 14.8726i	− 0.541270i
$756$	0	0
$757$	51.0353i	1.85491i	0.373937	+	0.927454i	$0.378007\pi$
$757$	51.0353i	1.85491i	−0.373937	+	0.927454i	$0.621993\pi$
$758$	0	0
$759$	−9.87195	−0.358329
$760$	0	0
$761$	−30.2111	−1.09515	−0.547576	−	0.836756i	$-0.684449\pi$
$761$	−30.2111	−1.09515	−0.547576	+	0.836756i	$0.684449\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	− 24.9268i	− 0.901229i
$766$	0	0
$767$	16.1969	0.584836
$768$	0	0
$769$	−4.26448	−0.153781	−0.0768906	−	0.997040i	$-0.524499\pi$
$769$	−4.26448	−0.153781	−0.0768906	+	0.997040i	$0.524499\pi$
$770$	0	0
$771$	− 5.79924i	− 0.208854i
$772$	0	0
$773$	5.39018i	0.193871i	0.995291	+	0.0969357i	$0.0309041\pi$
$773$	5.39018i	0.193871i	−0.995291	+	0.0969357i	$0.969096\pi$
$774$	0	0
$775$	36.6695	1.31721
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 0.0721851i	− 0.00258630i
$780$	0	0
$781$	− 34.4557i	− 1.23292i
$782$	0	0
$783$	−0.151350	−0.00540880
$784$	0	0
$785$	−32.2633	−1.15153
$786$	0	0
$787$	− 6.19873i	− 0.220961i	−0.993878	−	0.110480i	$-0.964761\pi$
$787$	− 6.19873i	− 0.220961i	0.993878	−	0.110480i	$-0.0352390\pi$
$788$	0	0
$789$	2.43254i	0.0866006i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−6.06416	−0.215345
$794$	0	0
$795$	− 21.3420i	− 0.756922i
$796$	0	0
$797$	− 2.24353i	− 0.0794699i	−0.999210	−	0.0397349i	$-0.987349\pi$
$797$	− 2.24353i	− 0.0794699i	0.999210	−	0.0397349i	$-0.0126514\pi$
$798$	0	0
$799$	44.4545	1.57269
$800$	0	0
$801$	−3.66869	−0.129627
$802$	0	0
$803$	2.31818i	0.0818067i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	5.75584	0.202615
$808$	0	0
$809$	−15.3369	−0.539216	−0.269608	−	0.962970i	$-0.586894\pi$
$809$	−15.3369	−0.539216	−0.269608	+	0.962970i	$0.586894\pi$
$810$	0	0
$811$	− 15.1509i	− 0.532019i	−0.963970	−	0.266010i	$-0.914295\pi$
$811$	− 15.1509i	− 0.532019i	0.963970	−	0.266010i	$-0.0857053\pi$
$812$	0	0
$813$	− 17.0185i	− 0.596866i
$814$	0	0
$815$	47.9602	1.67997
$816$	0	0
$817$	−0.400480	−0.0140110
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 1.63526i	− 0.0570710i	−0.999593	−	0.0285355i	$-0.990916\pi$
$821$	− 1.63526i	− 0.0570710i	0.999593	−	0.0285355i	$-0.00908437\pi$
$822$	0	0
$823$	−45.0135	−1.56907	−0.784535	−	0.620084i	$-0.787098\pi$
$823$	−45.0135	−1.56907	−0.784535	+	0.620084i	$0.787098\pi$
$824$	0	0
$825$	31.3982	1.09314
$826$	0	0
$827$	− 42.4945i	− 1.47768i	−0.673882	−	0.738839i	$-0.735374\pi$
$827$	− 42.4945i	− 1.47768i	0.673882	−	0.738839i	$-0.264626\pi$
$828$	0	0
$829$	45.7456i	1.58881i	0.607388	+	0.794405i	$0.292217\pi$
$829$	45.7456i	1.58881i	−0.607388	+	0.794405i	$0.707783\pi$
$830$	0	0
$831$	−22.4704	−0.779489
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 61.3620i	− 2.12352i
$836$	0	0
$837$	− 4.75438i	− 0.164335i
$838$	0	0
$839$	30.4207	1.05024	0.525120	−	0.851028i	$-0.324020\pi$
$839$	30.4207	1.05024	0.525120	+	0.851028i	$0.324020\pi$
$840$	0	0
$841$	28.9771	0.999210
$842$	0	0
$843$	− 7.29261i	− 0.251171i
$844$	0	0
$845$	− 38.9514i	− 1.33997i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−10.6377	−0.365084
$850$	0	0
$851$	27.5068i	0.942922i
$852$	0	0
$853$	0.386158i	0.0132218i	0.999978	+	0.00661090i	$0.00210433\pi$
$853$	0.386158i	0.0132218i	−0.999978	+	0.00661090i	$0.997896\pi$
$854$	0	0
$855$	1.07498	0.0367635
$856$	0	0
$857$	26.3234	0.899190	0.449595	−	0.893233i	$-0.351568\pi$
$857$	26.3234	0.899190	0.449595	+	0.893233i	$0.351568\pi$
$858$	0	0
$859$	− 23.2753i	− 0.794144i	−0.917787	−	0.397072i	$-0.870026\pi$
$859$	− 23.2753i	− 0.794144i	0.917787	−	0.397072i	$-0.129974\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−16.5123	−0.562084	−0.281042	−	0.959695i	$-0.590680\pi$
$863$	−16.5123	−0.562084	−0.281042	+	0.959695i	$0.590680\pi$
$864$	0	0
$865$	12.2988	0.418172
$866$	0	0
$867$	− 31.8755i	− 1.08255i
$868$	0	0
$869$	− 6.08128i	− 0.206293i
$870$	0	0
$871$	6.31111	0.213844
$872$	0	0
$873$	10.4657	0.354212
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 28.6736i	− 0.968237i	−0.875002	−	0.484119i	$-0.839140\pi$
$877$	− 28.6736i	− 0.968237i	0.875002	−	0.484119i	$-0.160860\pi$
$878$	0	0
$879$	26.4432	0.891907
$880$	0	0
$881$	−5.88181	−0.198163	−0.0990816	−	0.995079i	$-0.531590\pi$
$881$	−5.88181	−0.198163	−0.0990816	+	0.995079i	$0.531590\pi$
$882$	0	0
$883$	1.69703i	0.0571095i	0.999592	+	0.0285548i	$0.00909050\pi$
$883$	1.69703i	0.0571095i	−0.999592	+	0.0285548i	$0.990910\pi$
$884$	0	0
$885$	40.0862i	1.34748i
$886$	0	0
$887$	−57.7245	−1.93820	−0.969099	−	0.246672i	$-0.920663\pi$
$887$	−57.7245	−1.93820	−0.969099	+	0.246672i	$0.920663\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	− 4.07092i	− 0.136381i
$892$	0	0
$893$	1.91712i	0.0641540i
$894$	0	0
$895$	−40.2961	−1.34695
$896$	0	0
$897$	3.49356	0.116646
$898$	0	0
$899$	− 0.719575i	− 0.0239992i
$900$	0	0
$901$	− 41.8466i	− 1.39411i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	35.8568	1.19192
$906$	0	0
$907$	− 45.3277i	− 1.50508i	−0.658546	−	0.752540i	$-0.728828\pi$
$907$	− 45.3277i	− 1.50508i	0.658546	−	0.752540i	$-0.271172\pi$
$908$	0	0
$909$	12.7337i	0.422350i
$910$	0	0
$911$	26.4254	0.875511	0.437755	−	0.899094i	$-0.355774\pi$
$911$	26.4254	0.875511	0.437755	+	0.899094i	$0.355774\pi$
$912$	0	0
$913$	40.8525	1.35202
$914$	0	0
$915$	− 15.0084i	− 0.496162i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0.300665	0.00991803	0.00495901	−	0.999988i	$-0.498421\pi$
$919$	0.300665	0.00991803	0.00495901	+	0.999988i	$0.498421\pi$
$920$	0	0
$921$	23.6553	0.779470
$922$	0	0
$923$	12.1934i	0.401352i
$924$	0	0
$925$	− 87.4867i	− 2.87654i
$926$	0	0
$927$	6.00875	0.197353
$928$	0	0
$929$	38.6666	1.26861	0.634304	−	0.773084i	$-0.281287\pi$
$929$	38.6666	1.26861	0.634304	+	0.773084i	$0.281287\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	3.25213i	0.106470i
$934$	0	0
$935$	101.475	3.31858
$936$	0	0
$937$	−23.9292	−0.781734	−0.390867	−	0.920447i	$-0.627825\pi$
$937$	−23.9292	−0.781734	−0.390867	+	0.920447i	$0.627825\pi$
$938$	0	0
$939$	6.95242i	0.226884i
$940$	0	0
$941$	42.4067i	1.38242i	0.722654	+	0.691210i	$0.242922\pi$
$941$	42.4067i	1.38242i	−0.722654	+	0.691210i	$0.757078\pi$
$942$	0	0
$943$	0.580601	0.0189070
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 13.7810i	− 0.447823i	−0.974609	−	0.223911i	$-0.928117\pi$
$947$	− 13.7810i	− 0.447823i	0.974609	−	0.223911i	$-0.0718826\pi$
$948$	0	0
$949$	− 0.820373i	− 0.0266304i
$950$	0	0
$951$	6.28507	0.203807
$952$	0	0
$953$	57.0092	1.84671	0.923354	−	0.383950i	$-0.125436\pi$
$953$	57.0092	1.84671	0.923354	+	0.383950i	$0.125436\pi$
$954$	0	0
$955$	76.6085i	2.47899i
$956$	0	0
$957$	− 0.616133i	− 0.0199168i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−8.39587	−0.270834
$962$	0	0
$963$	15.9659i	0.514495i
$964$	0	0
$965$	58.5311i	1.88418i
$966$	0	0
$967$	20.0555	0.644941	0.322470	−	0.946580i	$-0.395487\pi$
$967$	20.0555	0.644941	0.322470	+	0.946580i	$0.395487\pi$
$968$	0	0
$969$	2.10778	0.0677117
$970$	0	0
$971$	1.13107i	0.0362977i	0.999835	+	0.0181488i	$0.00577727\pi$
$971$	1.13107i	0.0362977i	−0.999835	+	0.0181488i	$0.994223\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−11.1114	−0.355850
$976$	0	0
$977$	−31.0067	−0.991991	−0.495996	−	0.868325i	$-0.665197\pi$
$977$	−31.0067	−0.991991	−0.495996	+	0.868325i	$0.665197\pi$
$978$	0	0
$979$	− 14.9349i	− 0.477323i
$980$	0	0
$981$	− 13.8711i	− 0.442869i
$982$	0	0
$983$	−9.18804	−0.293053	−0.146527	−	0.989207i	$-0.546809\pi$
$983$	−9.18804	−0.293053	−0.146527	+	0.989207i	$0.546809\pi$
$984$	0	0
$985$	55.9944	1.78413
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 3.22115i	− 0.102427i
$990$	0	0
$991$	−40.0550	−1.27239	−0.636194	−	0.771529i	$-0.719492\pi$
$991$	−40.0550	−1.27239	−0.636194	+	0.771529i	$0.719492\pi$
$992$	0	0
$993$	25.3553	0.804627
$994$	0	0
$995$	− 22.3712i	− 0.709214i
$996$	0	0
$997$	− 26.0861i	− 0.826154i	−0.910696	−	0.413077i	$-0.864454\pi$
$997$	− 26.0861i	− 0.826154i	0.910696	−	0.413077i	$-0.135546\pi$
$998$	0	0
$999$	−11.3431	−0.358879

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4704.2.c.f.2353.1		16
4.3	odd	2		1176.2.c.f.589.2		16
7.3	odd	6		672.2.bk.a.625.9		32
7.5	odd	6		672.2.bk.a.529.8		32
7.6	odd	2		4704.2.c.e.2353.16		16
8.3	odd	2		1176.2.c.f.589.1		16
8.5	even	2	inner	4704.2.c.f.2353.16		16
21.5	even	6		2016.2.cr.e.1873.2		32
21.17	even	6		2016.2.cr.e.1297.15		32
28.3	even	6		168.2.bc.a.37.10	✓	32
28.19	even	6		168.2.bc.a.109.12	yes	32
28.27	even	2		1176.2.c.e.589.2		16
56.3	even	6		168.2.bc.a.37.12	yes	32
56.5	odd	6		672.2.bk.a.529.9		32
56.13	odd	2		4704.2.c.e.2353.1		16
56.19	even	6		168.2.bc.a.109.10	yes	32
56.27	even	2		1176.2.c.e.589.1		16
56.45	odd	6		672.2.bk.a.625.8		32
84.47	odd	6		504.2.cj.e.109.5		32
84.59	odd	6		504.2.cj.e.37.7		32
168.5	even	6		2016.2.cr.e.1873.15		32
168.59	odd	6		504.2.cj.e.37.5		32
168.101	even	6		2016.2.cr.e.1297.2		32
168.131	odd	6		504.2.cj.e.109.7		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.bc.a.37.10	✓	32	28.3	even	6
168.2.bc.a.37.12	yes	32	56.3	even	6
168.2.bc.a.109.10	yes	32	56.19	even	6
168.2.bc.a.109.12	yes	32	28.19	even	6
504.2.cj.e.37.5		32	168.59	odd	6
504.2.cj.e.37.7		32	84.59	odd	6
504.2.cj.e.109.5		32	84.47	odd	6
504.2.cj.e.109.7		32	168.131	odd	6
672.2.bk.a.529.8		32	7.5	odd	6
672.2.bk.a.529.9		32	56.5	odd	6
672.2.bk.a.625.8		32	56.45	odd	6
672.2.bk.a.625.9		32	7.3	odd	6
1176.2.c.e.589.1		16	56.27	even	2
1176.2.c.e.589.2		16	28.27	even	2
1176.2.c.f.589.1		16	8.3	odd	2
1176.2.c.f.589.2		16	4.3	odd	2
2016.2.cr.e.1297.2		32	168.101	even	6
2016.2.cr.e.1297.15		32	21.17	even	6
2016.2.cr.e.1873.2		32	21.5	even	6
2016.2.cr.e.1873.15		32	168.5	even	6
4704.2.c.e.2353.1		16	56.13	odd	2
4704.2.c.e.2353.16		16	7.6	odd	2
4704.2.c.f.2353.1		16	1.1	even	1	trivial
4704.2.c.f.2353.16		16	8.5	even	2	inner

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:	\( N \)	\(=\)	\( 4704 = 2^{5} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4704.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -3.56550i q^{5} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -3.56550i q^{5} -1.00000 q^{9} -4.07092i q^{11} +1.44065i q^{13} -3.56550 q^{15} -6.99110 q^{17} -0.301495i q^{19} +2.42499 q^{23} -7.71279 q^{25} +1.00000i q^{27} +0.151350i q^{29} -4.75438 q^{31} -4.07092 q^{33} +11.3431i q^{37} +1.44065 q^{39} +0.239424 q^{41} -1.32831i q^{43} +3.56550i q^{45} -6.35872 q^{47} +6.99110i q^{51} +5.98569i q^{53} -14.5149 q^{55} -0.301495 q^{57} -11.2428i q^{59} +4.20933i q^{61} +5.13663 q^{65} -4.38075i q^{67} -2.42499i q^{69} +8.46387 q^{71} -0.569448 q^{73} +7.71279i q^{75} +1.49383 q^{79} +1.00000 q^{81} +10.0352i q^{83} +24.9268i q^{85} +0.151350 q^{87} +3.66869 q^{89} +4.75438i q^{93} -1.07498 q^{95} -10.4657 q^{97} +4.07092i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{9}+O(q^{10}) \) 16 * q - 16 * q^9	\( 16 q - 16 q^{9} - 8 q^{23} - 16 q^{25} + 24 q^{31} + 24 q^{47} - 32 q^{55} - 8 q^{57} + 40 q^{71} + 8 q^{73} + 8 q^{79} + 16 q^{81} - 24 q^{87} + 24 q^{95} + 24 q^{97}+O(q^{100}) \) 16 * q - 16 * q^9 - 8 * q^23 - 16 * q^25 + 24 * q^31 + 24 * q^47 - 32 * q^55 - 8 * q^57 + 40 * q^71 + 8 * q^73 + 8 * q^79 + 16 * q^81 - 24 * q^87 + 24 * q^95 + 24 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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