LMFDB - Embedded newform 4704.2.c.e.2353.15

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			2353.15
Root			$0.284419 + 1.38532i$ of defining polynomial
Character	$\chi$	$=$	4704.2353
Dual form			4704.2.c.e.2353.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +2.29465i q^{5} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +2.29465i q^{5} -1.00000 q^{9} +3.88667i q^{11} -3.33413i q^{13} -2.29465 q^{15} +0.287121 q^{17} -2.78525i q^{19} -6.53425 q^{23} -0.265415 q^{25} -1.00000i q^{27} -5.53374i q^{29} -7.44308 q^{31} -3.88667 q^{33} -5.95539i q^{37} +3.33413 q^{39} -3.51617 q^{41} -11.2465i q^{43} -2.29465i q^{45} -0.0870075 q^{47} +0.287121i q^{51} +7.05820i q^{53} -8.91855 q^{55} +2.78525 q^{57} -4.35217i q^{59} +7.16714i q^{61} +7.65066 q^{65} -13.0255i q^{67} -6.53425i q^{69} +6.18835 q^{71} -13.8796 q^{73} -0.265415i q^{75} +8.99853 q^{79} +1.00000 q^{81} +17.6313i q^{83} +0.658842i q^{85} +5.53374 q^{87} -17.1839 q^{89} -7.44308i q^{93} +6.39118 q^{95} +6.46528 q^{97} -3.88667i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	2.29465i	1.02620i	0.858329	+	0.513099i	$0.171503\pi$
$5$	2.29465i	1.02620i	−0.858329	+	0.513099i	$0.828497\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	3.88667i	1.17188i	0.810356	+	0.585938i	$0.199274\pi$
$11$	3.88667i	1.17188i	−0.810356	+	0.585938i	$0.800726\pi$
$12$	0	0
$13$	− 3.33413i	− 0.924721i	−0.886692	−	0.462361i	$-0.847003\pi$
$13$	− 3.33413i	− 0.924721i	0.886692	−	0.462361i	$-0.152997\pi$
$14$	0	0
$15$	−2.29465	−0.592476
$16$	0	0
$17$	0.287121	0.0696370	0.0348185	−	0.999394i	$-0.488915\pi$
$17$	0.287121	0.0696370	0.0348185	+	0.999394i	$0.488915\pi$
$18$	0	0
$19$	− 2.78525i	− 0.638981i	−0.947589	−	0.319491i	$-0.896488\pi$
$19$	− 2.78525i	− 0.638981i	0.947589	−	0.319491i	$-0.103512\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.53425	−1.36249	−0.681243	−	0.732058i	$-0.738560\pi$
$23$	−6.53425	−1.36249	−0.681243	+	0.732058i	$0.738560\pi$
$24$	0	0
$25$	−0.265415	−0.0530829
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	− 5.53374i	− 1.02759i	−0.857913	−	0.513795i	$-0.828239\pi$
$29$	− 5.53374i	− 1.02759i	0.857913	−	0.513795i	$-0.171761\pi$
$30$	0	0
$31$	−7.44308	−1.33682	−0.668408	−	0.743795i	$-0.733024\pi$
$31$	−7.44308	−1.33682	−0.668408	+	0.743795i	$0.733024\pi$
$32$	0	0
$33$	−3.88667	−0.676583
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 5.95539i	− 0.979060i	−0.871987	−	0.489530i	$-0.837168\pi$
$37$	− 5.95539i	− 0.979060i	0.871987	−	0.489530i	$-0.162832\pi$
$38$	0	0
$39$	3.33413	0.533888
$40$	0	0
$41$	−3.51617	−0.549134	−0.274567	−	0.961568i	$-0.588534\pi$
$41$	−3.51617	−0.549134	−0.274567	+	0.961568i	$0.588534\pi$
$42$	0	0
$43$	− 11.2465i	− 1.71508i	−0.514416	−	0.857541i	$-0.671991\pi$
$43$	− 11.2465i	− 1.71508i	0.514416	−	0.857541i	$-0.328009\pi$
$44$	0	0
$45$	− 2.29465i	− 0.342066i
$46$	0	0
$47$	−0.0870075	−0.0126913	−0.00634567	−	0.999980i	$-0.502020\pi$
$47$	−0.0870075	−0.0126913	−0.00634567	+	0.999980i	$0.502020\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0.287121i	0.0402050i
$52$	0	0
$53$	7.05820i	0.969518i	0.874648	+	0.484759i	$0.161093\pi$
$53$	7.05820i	0.969518i	−0.874648	+	0.484759i	$0.838907\pi$
$54$	0	0
$55$	−8.91855	−1.20258
$56$	0	0
$57$	2.78525	0.368916
$58$	0	0
$59$	− 4.35217i	− 0.566605i	−0.959031	−	0.283302i	$-0.908570\pi$
$59$	− 4.35217i	− 0.566605i	0.959031	−	0.283302i	$-0.0914300\pi$
$60$	0	0
$61$	7.16714i	0.917659i	0.888524	+	0.458829i	$0.151731\pi$
$61$	7.16714i	0.917659i	−0.888524	+	0.458829i	$0.848269\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	7.65066	0.948947
$66$	0	0
$67$	− 13.0255i	− 1.59132i	−0.605743	−	0.795661i	$-0.707124\pi$
$67$	− 13.0255i	− 1.59132i	0.605743	−	0.795661i	$-0.292876\pi$
$68$	0	0
$69$	− 6.53425i	− 0.786631i
$70$	0	0
$71$	6.18835	0.734421	0.367211	−	0.930138i	$-0.380313\pi$
$71$	6.18835	0.734421	0.367211	+	0.930138i	$0.380313\pi$
$72$	0	0
$73$	−13.8796	−1.62448	−0.812240	−	0.583323i	$-0.801752\pi$
$73$	−13.8796	−1.62448	−0.812240	+	0.583323i	$0.801752\pi$
$74$	0	0
$75$	− 0.265415i	− 0.0306474i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.99853	1.01241	0.506207	−	0.862412i	$-0.331047\pi$
$79$	8.99853	1.01241	0.506207	+	0.862412i	$0.331047\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	17.6313i	1.93529i	0.252312	+	0.967646i	$0.418809\pi$
$83$	17.6313i	1.93529i	−0.252312	+	0.967646i	$0.581191\pi$
$84$	0	0
$85$	0.658842i	0.0714614i
$86$	0	0
$87$	5.53374	0.593280
$88$	0	0
$89$	−17.1839	−1.82149	−0.910744	−	0.412972i	$-0.864491\pi$
$89$	−17.1839	−1.82149	−0.910744	+	0.412972i	$0.864491\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 7.44308i	− 0.771811i
$94$	0	0
$95$	6.39118	0.655721
$96$	0	0
$97$	6.46528	0.656450	0.328225	−	0.944600i	$-0.393550\pi$
$97$	6.46528	0.656450	0.328225	+	0.944600i	$0.393550\pi$
$98$	0	0
$99$	− 3.88667i	− 0.390625i
$100$	0	0
$101$	− 11.8461i	− 1.17873i	−0.807867	−	0.589365i	$-0.799378\pi$
$101$	− 11.8461i	− 1.17873i	0.807867	−	0.589365i	$-0.200622\pi$
$102$	0	0
$103$	−1.62190	−0.159810	−0.0799051	−	0.996802i	$-0.525462\pi$
$103$	−1.62190	−0.159810	−0.0799051	+	0.996802i	$0.525462\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 5.02926i	− 0.486197i	−0.970002	−	0.243098i	$-0.921836\pi$
$107$	− 5.02926i	− 0.486197i	0.970002	−	0.243098i	$-0.0781638\pi$
$108$	0	0
$109$	0.204420i	0.0195799i	0.999952	+	0.00978994i	$0.00311628\pi$
$109$	0.204420i	0.0195799i	−0.999952	+	0.00978994i	$0.996884\pi$
$110$	0	0
$111$	5.95539	0.565260
$112$	0	0
$113$	−14.9203	−1.40358	−0.701792	−	0.712382i	$-0.747616\pi$
$113$	−14.9203	−1.40358	−0.701792	+	0.712382i	$0.747616\pi$
$114$	0	0
$115$	− 14.9938i	− 1.39818i
$116$	0	0
$117$	3.33413i	0.308240i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−4.10622	−0.373293
$122$	0	0
$123$	− 3.51617i	− 0.317043i
$124$	0	0
$125$	10.8642i	0.971725i
$126$	0	0
$127$	6.29267	0.558384	0.279192	−	0.960235i	$-0.409933\pi$
$127$	6.29267	0.558384	0.279192	+	0.960235i	$0.409933\pi$
$128$	0	0
$129$	11.2465	0.990203
$130$	0	0
$131$	− 1.24096i	− 0.108423i	−0.998529	−	0.0542116i	$-0.982735\pi$
$131$	− 1.24096i	− 0.108423i	0.998529	−	0.0542116i	$-0.0172646\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	2.29465	0.197492
$136$	0	0
$137$	1.59019	0.135859	0.0679296	−	0.997690i	$-0.478361\pi$
$137$	1.59019	0.135859	0.0679296	+	0.997690i	$0.478361\pi$
$138$	0	0
$139$	1.02981i	0.0873470i	0.999046	+	0.0436735i	$0.0139061\pi$
$139$	1.02981i	0.0873470i	−0.999046	+	0.0436735i	$0.986094\pi$
$140$	0	0
$141$	− 0.0870075i	− 0.00732735i
$142$	0	0
$143$	12.9587	1.08366
$144$	0	0
$145$	12.6980	1.05451
$146$	0	0
$147$	0	0
$148$	0	0
$149$	9.59178i	0.785789i	0.919584	+	0.392895i	$0.128526\pi$
$149$	9.59178i	0.785789i	−0.919584	+	0.392895i	$0.871474\pi$
$150$	0	0
$151$	−6.99222	−0.569019	−0.284510	−	0.958673i	$-0.591831\pi$
$151$	−6.99222	−0.569019	−0.284510	+	0.958673i	$0.591831\pi$
$152$	0	0
$153$	−0.287121	−0.0232123
$154$	0	0
$155$	− 17.0792i	− 1.37184i
$156$	0	0
$157$	− 17.3579i	− 1.38531i	−0.721269	−	0.692655i	$-0.756441\pi$
$157$	− 17.3579i	− 1.38531i	0.721269	−	0.692655i	$-0.243559\pi$
$158$	0	0
$159$	−7.05820	−0.559751
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 15.7220i	− 1.23144i	−0.787965	−	0.615720i	$-0.788865\pi$
$163$	− 15.7220i	− 1.23144i	0.787965	−	0.615720i	$-0.211135\pi$
$164$	0	0
$165$	− 8.91855i	− 0.694308i
$166$	0	0
$167$	0.102064	0.00789795	0.00394897	−	0.999992i	$-0.498743\pi$
$167$	0.102064	0.00789795	0.00394897	+	0.999992i	$0.498743\pi$
$168$	0	0
$169$	1.88358	0.144891
$170$	0	0
$171$	2.78525i	0.212994i
$172$	0	0
$173$	2.28777i	0.173936i	0.996211	+	0.0869678i	$0.0277177\pi$
$173$	2.28777i	0.173936i	−0.996211	+	0.0869678i	$0.972282\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.35217	0.327130
$178$	0	0
$179$	− 18.6231i	− 1.39196i	−0.718064	−	0.695978i	$-0.754971\pi$
$179$	− 18.6231i	− 1.39196i	0.718064	−	0.695978i	$-0.245029\pi$
$180$	0	0
$181$	13.6786i	1.01672i	0.861144	+	0.508361i	$0.169748\pi$
$181$	13.6786i	1.01672i	−0.861144	+	0.508361i	$0.830252\pi$
$182$	0	0
$183$	−7.16714	−0.529810
$184$	0	0
$185$	13.6655	1.00471
$186$	0	0
$187$	1.11594i	0.0816060i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	12.6346	0.914207	0.457103	−	0.889414i	$-0.348887\pi$
$191$	12.6346	0.914207	0.457103	+	0.889414i	$0.348887\pi$
$192$	0	0
$193$	2.60582	0.187571	0.0937856	−	0.995592i	$-0.470103\pi$
$193$	2.60582	0.187571	0.0937856	+	0.995592i	$0.470103\pi$
$194$	0	0
$195$	7.65066i	0.547875i
$196$	0	0
$197$	− 0.794777i	− 0.0566255i	−0.999599	−	0.0283127i	$-0.990987\pi$
$197$	− 0.794777i	− 0.0566255i	0.999599	−	0.0283127i	$-0.00901343\pi$
$198$	0	0
$199$	−10.7343	−0.760933	−0.380466	−	0.924795i	$-0.624237\pi$
$199$	−10.7343	−0.760933	−0.380466	+	0.924795i	$0.624237\pi$
$200$	0	0
$201$	13.0255	0.918750
$202$	0	0
$203$	0	0
$204$	0	0
$205$	− 8.06838i	− 0.563520i
$206$	0	0
$207$	6.53425	0.454162
$208$	0	0
$209$	10.8254	0.748807
$210$	0	0
$211$	0.556345i	0.0383004i	0.999817	+	0.0191502i	$0.00609607\pi$
$211$	0.556345i	0.0383004i	−0.999817	+	0.0191502i	$0.993904\pi$
$212$	0	0
$213$	6.18835i	0.424018i
$214$	0	0
$215$	25.8069	1.76001
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 13.8796i	− 0.937894i
$220$	0	0
$221$	− 0.957298i	− 0.0643948i
$222$	0	0
$223$	−11.1076	−0.743820	−0.371910	−	0.928269i	$-0.621297\pi$
$223$	−11.1076	−0.743820	−0.371910	+	0.928269i	$0.621297\pi$
$224$	0	0
$225$	0.265415	0.0176943
$226$	0	0
$227$	− 8.76788i	− 0.581945i	−0.956731	−	0.290972i	$-0.906021\pi$
$227$	− 8.76788i	− 0.581945i	0.956731	−	0.290972i	$-0.0939788\pi$
$228$	0	0
$229$	20.6962i	1.36764i	0.729648	+	0.683822i	$0.239684\pi$
$229$	20.6962i	1.36764i	−0.729648	+	0.683822i	$0.760316\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−5.72853	−0.375288	−0.187644	−	0.982237i	$-0.560085\pi$
$233$	−5.72853	−0.375288	−0.187644	+	0.982237i	$0.560085\pi$
$234$	0	0
$235$	− 0.199652i	− 0.0130238i
$236$	0	0
$237$	8.99853i	0.584517i
$238$	0	0
$239$	−2.27651	−0.147255	−0.0736276	−	0.997286i	$-0.523458\pi$
$239$	−2.27651	−0.147255	−0.0736276	+	0.997286i	$0.523458\pi$
$240$	0	0
$241$	23.5850	1.51925	0.759623	−	0.650364i	$-0.225384\pi$
$241$	23.5850	1.51925	0.759623	+	0.650364i	$0.225384\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−9.28640	−0.590879
$248$	0	0
$249$	−17.6313	−1.11734
$250$	0	0
$251$	− 12.4322i	− 0.784716i	−0.919813	−	0.392358i	$-0.871659\pi$
$251$	− 12.4322i	− 0.784716i	0.919813	−	0.392358i	$-0.128341\pi$
$252$	0	0
$253$	− 25.3965i	− 1.59666i
$254$	0	0
$255$	−0.658842	−0.0412583
$256$	0	0
$257$	0.282984	0.0176521	0.00882604	−	0.999961i	$-0.497191\pi$
$257$	0.282984	0.0176521	0.00882604	+	0.999961i	$0.497191\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	5.53374i	0.342530i
$262$	0	0
$263$	−14.3192	−0.882962	−0.441481	−	0.897271i	$-0.645547\pi$
$263$	−14.3192	−0.882962	−0.441481	+	0.897271i	$0.645547\pi$
$264$	0	0
$265$	−16.1961	−0.994918
$266$	0	0
$267$	− 17.1839i	− 1.05164i
$268$	0	0
$269$	− 16.8141i	− 1.02517i	−0.858636	−	0.512586i	$-0.828688\pi$
$269$	− 16.8141i	− 1.02517i	0.858636	−	0.512586i	$-0.171312\pi$
$270$	0	0
$271$	−26.8456	−1.63076	−0.815378	−	0.578929i	$-0.803471\pi$
$271$	−26.8456	−1.63076	−0.815378	+	0.578929i	$0.803471\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 1.03158i	− 0.0622066i
$276$	0	0
$277$	22.5030i	1.35207i	0.736868	+	0.676036i	$0.236304\pi$
$277$	22.5030i	1.35207i	−0.736868	+	0.676036i	$0.763696\pi$
$278$	0	0
$279$	7.44308	0.445605
$280$	0	0
$281$	12.8375	0.765824	0.382912	−	0.923785i	$-0.374921\pi$
$281$	12.8375	0.765824	0.382912	+	0.923785i	$0.374921\pi$
$282$	0	0
$283$	7.29790i	0.433815i	0.976192	+	0.216908i	$0.0695971\pi$
$283$	7.29790i	0.433815i	−0.976192	+	0.216908i	$0.930403\pi$
$284$	0	0
$285$	6.39118i	0.378581i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−16.9176	−0.995151
$290$	0	0
$291$	6.46528i	0.379001i
$292$	0	0
$293$	− 9.95674i	− 0.581679i	−0.956772	−	0.290840i	$-0.906065\pi$
$293$	− 9.95674i	− 0.581679i	0.956772	−	0.290840i	$-0.0939346\pi$
$294$	0	0
$295$	9.98671	0.581449
$296$	0	0
$297$	3.88667	0.225528
$298$	0	0
$299$	21.7860i	1.25992i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	11.8461	0.680540
$304$	0	0
$305$	−16.4461	−0.941700
$306$	0	0
$307$	− 14.9479i	− 0.853122i	−0.904459	−	0.426561i	$-0.859725\pi$
$307$	− 14.9479i	− 0.853122i	0.904459	−	0.426561i	$-0.140275\pi$
$308$	0	0
$309$	− 1.62190i	− 0.0922664i
$310$	0	0
$311$	31.2591	1.77254	0.886270	−	0.463169i	$-0.153288\pi$
$311$	31.2591	1.77254	0.886270	+	0.463169i	$0.153288\pi$
$312$	0	0
$313$	−3.06468	−0.173226	−0.0866130	−	0.996242i	$-0.527604\pi$
$313$	−3.06468	−0.173226	−0.0866130	+	0.996242i	$0.527604\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 16.2981i	− 0.915391i	−0.889109	−	0.457695i	$-0.848675\pi$
$317$	− 16.2981i	− 0.915391i	0.889109	−	0.457695i	$-0.151325\pi$
$318$	0	0
$319$	21.5078	1.20421
$320$	0	0
$321$	5.02926	0.280706
$322$	0	0
$323$	− 0.799705i	− 0.0444968i
$324$	0	0
$325$	0.884927i	0.0490869i
$326$	0	0
$327$	−0.204420	−0.0113044
$328$	0	0
$329$	0	0
$330$	0	0
$331$	2.76307i	0.151872i	0.997113	+	0.0759359i	$0.0241945\pi$
$331$	2.76307i	0.151872i	−0.997113	+	0.0759359i	$0.975806\pi$
$332$	0	0
$333$	5.95539i	0.326353i
$334$	0	0
$335$	29.8890	1.63301
$336$	0	0
$337$	25.8259	1.40682	0.703412	−	0.710782i	$-0.251659\pi$
$337$	25.8259	1.40682	0.703412	+	0.710782i	$0.251659\pi$
$338$	0	0
$339$	− 14.9203i	− 0.810359i
$340$	0	0
$341$	− 28.9288i	− 1.56658i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	14.9938	0.807240
$346$	0	0
$347$	12.2755i	0.658984i	0.944158	+	0.329492i	$0.106877\pi$
$347$	12.2755i	0.658984i	−0.944158	+	0.329492i	$0.893123\pi$
$348$	0	0
$349$	0.209753i	0.0112278i	0.999984	+	0.00561390i	$0.00178697\pi$
$349$	0.209753i	0.0112278i	−0.999984	+	0.00561390i	$0.998213\pi$
$350$	0	0
$351$	−3.33413	−0.177963
$352$	0	0
$353$	−19.5985	−1.04312	−0.521561	−	0.853214i	$-0.674650\pi$
$353$	−19.5985	−1.04312	−0.521561	+	0.853214i	$0.674650\pi$
$354$	0	0
$355$	14.2001i	0.753662i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	5.71112	0.301422	0.150711	−	0.988578i	$-0.451844\pi$
$359$	5.71112	0.301422	0.150711	+	0.988578i	$0.451844\pi$
$360$	0	0
$361$	11.2424	0.591703
$362$	0	0
$363$	− 4.10622i	− 0.215521i
$364$	0	0
$365$	− 31.8487i	− 1.66704i
$366$	0	0
$367$	−28.1275	−1.46824	−0.734122	−	0.679017i	$-0.762406\pi$
$367$	−28.1275	−1.46824	−0.734122	+	0.679017i	$0.762406\pi$
$368$	0	0
$369$	3.51617	0.183045
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 17.2035i	− 0.890764i	−0.895341	−	0.445382i	$-0.853068\pi$
$373$	− 17.2035i	− 0.890764i	0.895341	−	0.445382i	$-0.146932\pi$
$374$	0	0
$375$	−10.8642	−0.561026
$376$	0	0
$377$	−18.4502	−0.950234
$378$	0	0
$379$	− 10.1872i	− 0.523281i	−0.965165	−	0.261640i	$-0.915737\pi$
$379$	− 10.1872i	− 0.523281i	0.965165	−	0.261640i	$-0.0842635\pi$
$380$	0	0
$381$	6.29267i	0.322383i
$382$	0	0
$383$	19.8433	1.01394	0.506972	−	0.861962i	$-0.330765\pi$
$383$	19.8433	1.01394	0.506972	+	0.861962i	$0.330765\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	11.2465i	0.571694i
$388$	0	0
$389$	− 18.5333i	− 0.939677i	−0.882752	−	0.469838i	$-0.844312\pi$
$389$	− 18.5333i	− 0.939677i	0.882752	−	0.469838i	$-0.155688\pi$
$390$	0	0
$391$	−1.87612	−0.0948794
$392$	0	0
$393$	1.24096	0.0625982
$394$	0	0
$395$	20.6485i	1.03894i
$396$	0	0
$397$	− 7.98137i	− 0.400573i	−0.979737	−	0.200287i	$-0.935813\pi$
$397$	− 7.98137i	− 0.400573i	0.979737	−	0.200287i	$-0.0641874\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	36.0420	1.79985	0.899925	−	0.436045i	$-0.143621\pi$
$401$	36.0420	1.79985	0.899925	+	0.436045i	$0.143621\pi$
$402$	0	0
$403$	24.8162i	1.23618i
$404$	0	0
$405$	2.29465i	0.114022i
$406$	0	0
$407$	23.1466	1.14734
$408$	0	0
$409$	−29.5666	−1.46197	−0.730987	−	0.682391i	$-0.760940\pi$
$409$	−29.5666	−1.46197	−0.730987	+	0.682391i	$0.760940\pi$
$410$	0	0
$411$	1.59019i	0.0784384i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−40.4578	−1.98599
$416$	0	0
$417$	−1.02981	−0.0504298
$418$	0	0
$419$	8.73304i	0.426637i	0.976983	+	0.213318i	$0.0684271\pi$
$419$	8.73304i	0.426637i	−0.976983	+	0.213318i	$0.931573\pi$
$420$	0	0
$421$	1.77349i	0.0864344i	0.999066	+	0.0432172i	$0.0137607\pi$
$421$	1.77349i	0.0864344i	−0.999066	+	0.0432172i	$0.986239\pi$
$422$	0	0
$423$	0.0870075	0.00423045
$424$	0	0
$425$	−0.0762061	−0.00369654
$426$	0	0
$427$	0	0
$428$	0	0
$429$	12.9587i	0.625650i
$430$	0	0
$431$	−28.3672	−1.36640	−0.683200	−	0.730231i	$-0.739412\pi$
$431$	−28.3672	−1.36640	−0.683200	+	0.730231i	$0.739412\pi$
$432$	0	0
$433$	−6.53217	−0.313916	−0.156958	−	0.987605i	$-0.550169\pi$
$433$	−6.53217	−0.313916	−0.156958	+	0.987605i	$0.550169\pi$
$434$	0	0
$435$	12.6980i	0.608822i
$436$	0	0
$437$	18.1995i	0.870602i
$438$	0	0
$439$	−0.532322	−0.0254063	−0.0127032	−	0.999919i	$-0.504044\pi$
$439$	−0.532322	−0.0254063	−0.0127032	+	0.999919i	$0.504044\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 8.51970i	− 0.404783i	−0.979305	−	0.202392i	$-0.935129\pi$
$443$	− 8.51970i	− 0.404783i	0.979305	−	0.202392i	$-0.0648714\pi$
$444$	0	0
$445$	− 39.4310i	− 1.86921i
$446$	0	0
$447$	−9.59178	−0.453676
$448$	0	0
$449$	−15.0180	−0.708745	−0.354373	−	0.935104i	$-0.615306\pi$
$449$	−15.0180	−0.708745	−0.354373	+	0.935104i	$0.615306\pi$
$450$	0	0
$451$	− 13.6662i	− 0.643517i
$452$	0	0
$453$	− 6.99222i	− 0.328523i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−31.4552	−1.47141	−0.735705	−	0.677302i	$-0.763149\pi$
$457$	−31.4552	−1.47141	−0.735705	+	0.677302i	$0.763149\pi$
$458$	0	0
$459$	− 0.287121i	− 0.0134017i
$460$	0	0
$461$	− 4.36281i	− 0.203196i	−0.994826	−	0.101598i	$-0.967604\pi$
$461$	− 4.36281i	− 0.203196i	0.994826	−	0.101598i	$-0.0323956\pi$
$462$	0	0
$463$	−13.9787	−0.649646	−0.324823	−	0.945775i	$-0.605305\pi$
$463$	−13.9787	−0.649646	−0.324823	+	0.945775i	$0.605305\pi$
$464$	0	0
$465$	17.0792	0.792031
$466$	0	0
$467$	16.5383i	0.765302i	0.923893	+	0.382651i	$0.124989\pi$
$467$	16.5383i	0.765302i	−0.923893	+	0.382651i	$0.875011\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	17.3579	0.799810
$472$	0	0
$473$	43.7116	2.00986
$474$	0	0
$475$	0.739247i	0.0339190i
$476$	0	0
$477$	− 7.05820i	− 0.323173i
$478$	0	0
$479$	1.18401	0.0540990	0.0270495	−	0.999634i	$-0.491389\pi$
$479$	1.18401	0.0540990	0.0270495	+	0.999634i	$0.491389\pi$
$480$	0	0
$481$	−19.8560	−0.905357
$482$	0	0
$483$	0	0
$484$	0	0
$485$	14.8356i	0.673648i
$486$	0	0
$487$	24.0896	1.09160	0.545801	−	0.837915i	$-0.316225\pi$
$487$	24.0896	1.09160	0.545801	+	0.837915i	$0.316225\pi$
$488$	0	0
$489$	15.7220	0.710973
$490$	0	0
$491$	18.4971i	0.834761i	0.908732	+	0.417381i	$0.137052\pi$
$491$	18.4971i	0.834761i	−0.908732	+	0.417381i	$0.862948\pi$
$492$	0	0
$493$	− 1.58885i	− 0.0715584i
$494$	0	0
$495$	8.91855	0.400859
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 12.1283i	− 0.542937i	−0.962447	−	0.271468i	$-0.912491\pi$
$499$	− 12.1283i	− 0.542937i	0.962447	−	0.271468i	$-0.0875092\pi$
$500$	0	0
$501$	0.102064i	0.00455988i
$502$	0	0
$503$	7.61078	0.339348	0.169674	−	0.985500i	$-0.445729\pi$
$503$	7.61078	0.339348	0.169674	+	0.985500i	$0.445729\pi$
$504$	0	0
$505$	27.1826	1.20961
$506$	0	0
$507$	1.88358i	0.0836528i
$508$	0	0
$509$	16.0956i	0.713424i	0.934214	+	0.356712i	$0.116102\pi$
$509$	16.0956i	0.713424i	−0.934214	+	0.356712i	$0.883898\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−2.78525	−0.122972
$514$	0	0
$515$	− 3.72168i	− 0.163997i
$516$	0	0
$517$	− 0.338169i	− 0.0148727i
$518$	0	0
$519$	−2.28777	−0.100422
$520$	0	0
$521$	−19.2934	−0.845258	−0.422629	−	0.906303i	$-0.638893\pi$
$521$	−19.2934	−0.845258	−0.422629	+	0.906303i	$0.638893\pi$
$522$	0	0
$523$	− 32.2551i	− 1.41042i	−0.709001	−	0.705208i	$-0.750854\pi$
$523$	− 32.2551i	− 1.41042i	0.709001	−	0.705208i	$-0.249146\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.13706	−0.0930919
$528$	0	0
$529$	19.6964	0.856366
$530$	0	0
$531$	4.35217i	0.188868i
$532$	0	0
$533$	11.7234i	0.507796i
$534$	0	0
$535$	11.5404	0.498934
$536$	0	0
$537$	18.6231	0.803646
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 3.98877i	− 0.171491i	−0.996317	−	0.0857454i	$-0.972673\pi$
$541$	− 3.98877i	− 0.171491i	0.996317	−	0.0857454i	$-0.0273272\pi$
$542$	0	0
$543$	−13.6786	−0.587005
$544$	0	0
$545$	−0.469072	−0.0200928
$546$	0	0
$547$	12.1824i	0.520882i	0.965490	+	0.260441i	$0.0838679\pi$
$547$	12.1824i	0.520882i	−0.965490	+	0.260441i	$0.916132\pi$
$548$	0	0
$549$	− 7.16714i	− 0.305886i
$550$	0	0
$551$	−15.4129	−0.656611
$552$	0	0
$553$	0	0
$554$	0	0
$555$	13.6655i	0.580069i
$556$	0	0
$557$	− 2.98826i	− 0.126617i	−0.997994	−	0.0633084i	$-0.979835\pi$
$557$	− 2.98826i	− 0.126617i	0.997994	−	0.0633084i	$-0.0201652\pi$
$558$	0	0
$559$	−37.4974	−1.58597
$560$	0	0
$561$	−1.11594	−0.0471152
$562$	0	0
$563$	− 4.41292i	− 0.185982i	−0.995667	−	0.0929911i	$-0.970357\pi$
$563$	− 4.41292i	− 0.185982i	0.995667	−	0.0929911i	$-0.0296428\pi$
$564$	0	0
$565$	− 34.2368i	− 1.44035i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−12.3086	−0.516005	−0.258002	−	0.966144i	$-0.583064\pi$
$569$	−12.3086	−0.516005	−0.258002	+	0.966144i	$0.583064\pi$
$570$	0	0
$571$	30.3416i	1.26975i	0.772613	+	0.634877i	$0.218949\pi$
$571$	30.3416i	1.26975i	−0.772613	+	0.634877i	$0.781051\pi$
$572$	0	0
$573$	12.6346i	0.527818i
$574$	0	0
$575$	1.73429	0.0723247
$576$	0	0
$577$	−12.0512	−0.501697	−0.250848	−	0.968026i	$-0.580710\pi$
$577$	−12.0512	−0.501697	−0.250848	+	0.968026i	$0.580710\pi$
$578$	0	0
$579$	2.60582i	0.108294i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−27.4329	−1.13615
$584$	0	0
$585$	−7.65066	−0.316316
$586$	0	0
$587$	− 5.34331i	− 0.220542i	−0.993902	−	0.110271i	$-0.964828\pi$
$587$	− 5.34331i	− 0.220542i	0.993902	−	0.110271i	$-0.0351719\pi$
$588$	0	0
$589$	20.7309i	0.854200i
$590$	0	0
$591$	0.794777	0.0326927
$592$	0	0
$593$	9.49060	0.389732	0.194866	−	0.980830i	$-0.437573\pi$
$593$	9.49060	0.389732	0.194866	+	0.980830i	$0.437573\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 10.7343i	− 0.439325i
$598$	0	0
$599$	7.73951	0.316228	0.158114	−	0.987421i	$-0.449459\pi$
$599$	7.73951	0.316228	0.158114	+	0.987421i	$0.449459\pi$
$600$	0	0
$601$	−19.0198	−0.775835	−0.387917	−	0.921694i	$-0.626805\pi$
$601$	−19.0198	−0.775835	−0.387917	+	0.921694i	$0.626805\pi$
$602$	0	0
$603$	13.0255i	0.530440i
$604$	0	0
$605$	− 9.42234i	− 0.383072i
$606$	0	0
$607$	−9.19581	−0.373247	−0.186623	−	0.982432i	$-0.559754\pi$
$607$	−9.19581	−0.373247	−0.186623	+	0.982432i	$0.559754\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0.290094i	0.0117360i
$612$	0	0
$613$	− 26.4067i	− 1.06655i	−0.845940	−	0.533277i	$-0.820960\pi$
$613$	− 26.4067i	− 1.06655i	0.845940	−	0.533277i	$-0.179040\pi$
$614$	0	0
$615$	8.06838	0.325348
$616$	0	0
$617$	−45.5837	−1.83513	−0.917565	−	0.397585i	$-0.869848\pi$
$617$	−45.5837	−1.83513	−0.917565	+	0.397585i	$0.869848\pi$
$618$	0	0
$619$	− 49.0337i	− 1.97083i	−0.170169	−	0.985415i	$-0.554431\pi$
$619$	− 49.0337i	− 1.97083i	0.170169	−	0.985415i	$-0.445569\pi$
$620$	0	0
$621$	6.53425i	0.262210i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−26.2566	−1.05027
$626$	0	0
$627$	10.8254i	0.432324i
$628$	0	0
$629$	− 1.70992i	− 0.0681788i
$630$	0	0
$631$	12.5801	0.500804	0.250402	−	0.968142i	$-0.419437\pi$
$631$	12.5801	0.500804	0.250402	+	0.968142i	$0.419437\pi$
$632$	0	0
$633$	−0.556345	−0.0221127
$634$	0	0
$635$	14.4395i	0.573013i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−6.18835	−0.244807
$640$	0	0
$641$	−34.2258	−1.35184	−0.675920	−	0.736975i	$-0.736253\pi$
$641$	−34.2258	−1.35184	−0.675920	+	0.736975i	$0.736253\pi$
$642$	0	0
$643$	23.0442i	0.908776i	0.890804	+	0.454388i	$0.150142\pi$
$643$	23.0442i	0.908776i	−0.890804	+	0.454388i	$0.849858\pi$
$644$	0	0
$645$	25.8069i	1.01614i
$646$	0	0
$647$	−15.5820	−0.612590	−0.306295	−	0.951937i	$-0.599089\pi$
$647$	−15.5820	−0.612590	−0.306295	+	0.951937i	$0.599089\pi$
$648$	0	0
$649$	16.9155	0.663991
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 49.2269i	− 1.92640i	−0.268789	−	0.963199i	$-0.586623\pi$
$653$	− 49.2269i	− 1.92640i	0.268789	−	0.963199i	$-0.413377\pi$
$654$	0	0
$655$	2.84757	0.111264
$656$	0	0
$657$	13.8796	0.541493
$658$	0	0
$659$	5.33204i	0.207707i	0.994593	+	0.103853i	$0.0331173\pi$
$659$	5.33204i	0.207707i	−0.994593	+	0.103853i	$0.966883\pi$
$660$	0	0
$661$	38.6891i	1.50483i	0.658689	+	0.752416i	$0.271111\pi$
$661$	38.6891i	1.50483i	−0.658689	+	0.752416i	$0.728889\pi$
$662$	0	0
$663$	0.957298	0.0371784
$664$	0	0
$665$	0	0
$666$	0	0
$667$	36.1589i	1.40008i
$668$	0	0
$669$	− 11.1076i	− 0.429445i
$670$	0	0
$671$	−27.8563	−1.07538
$672$	0	0
$673$	−9.56678	−0.368772	−0.184386	−	0.982854i	$-0.559030\pi$
$673$	−9.56678	−0.368772	−0.184386	+	0.982854i	$0.559030\pi$
$674$	0	0
$675$	0.265415i	0.0102158i
$676$	0	0
$677$	17.7288i	0.681372i	0.940177	+	0.340686i	$0.110659\pi$
$677$	17.7288i	0.681372i	−0.940177	+	0.340686i	$0.889341\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	8.76788	0.335986
$682$	0	0
$683$	− 8.77302i	− 0.335690i	−0.985813	−	0.167845i	$-0.946319\pi$
$683$	− 8.77302i	− 0.335690i	0.985813	−	0.167845i	$-0.0536808\pi$
$684$	0	0
$685$	3.64893i	0.139418i
$686$	0	0
$687$	−20.6962	−0.789610
$688$	0	0
$689$	23.5329	0.896534
$690$	0	0
$691$	21.3823i	0.813421i	0.913557	+	0.406710i	$0.133324\pi$
$691$	21.3823i	0.813421i	−0.913557	+	0.406710i	$0.866676\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2.36304	−0.0896354
$696$	0	0
$697$	−1.00957	−0.0382400
$698$	0	0
$699$	− 5.72853i	− 0.216673i
$700$	0	0
$701$	− 13.4233i	− 0.506990i	−0.967337	−	0.253495i	$-0.918420\pi$
$701$	− 13.4233i	− 0.506990i	0.967337	−	0.253495i	$-0.0815802\pi$
$702$	0	0
$703$	−16.5873	−0.625601
$704$	0	0
$705$	0.199652	0.00751931
$706$	0	0
$707$	0	0
$708$	0	0
$709$	50.6521i	1.90228i	0.308762	+	0.951139i	$0.400085\pi$
$709$	50.6521i	1.90228i	−0.308762	+	0.951139i	$0.599915\pi$
$710$	0	0
$711$	−8.99853	−0.337471
$712$	0	0
$713$	48.6349	1.82139
$714$	0	0
$715$	29.7356i	1.11205i
$716$	0	0
$717$	− 2.27651i	− 0.0850179i
$718$	0	0
$719$	−28.9600	−1.08003	−0.540013	−	0.841657i	$-0.681581\pi$
$719$	−28.9600	−1.08003	−0.540013	+	0.841657i	$0.681581\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	23.5850i	0.877137i
$724$	0	0
$725$	1.46874i	0.0545475i
$726$	0	0
$727$	10.5092	0.389766	0.194883	−	0.980827i	$-0.437567\pi$
$727$	10.5092	0.389766	0.194883	+	0.980827i	$0.437567\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 3.22912i	− 0.119433i
$732$	0	0
$733$	− 28.1008i	− 1.03793i	−0.854797	−	0.518963i	$-0.826318\pi$
$733$	− 28.1008i	− 1.03793i	0.854797	−	0.518963i	$-0.173682\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	50.6259	1.86483
$738$	0	0
$739$	− 34.8553i	− 1.28217i	−0.767469	−	0.641086i	$-0.778484\pi$
$739$	− 34.8553i	− 1.28217i	0.767469	−	0.641086i	$-0.221516\pi$
$740$	0	0
$741$	− 9.28640i	− 0.341144i
$742$	0	0
$743$	−38.4805	−1.41171	−0.705857	−	0.708354i	$-0.749438\pi$
$743$	−38.4805	−1.41171	−0.705857	+	0.708354i	$0.749438\pi$
$744$	0	0
$745$	−22.0098	−0.806375
$746$	0	0
$747$	− 17.6313i	− 0.645097i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−0.0225322	−0.000822212 0	−0.000411106	−	1.00000i	$-0.500131\pi$
$751$	−0.0225322	−0.000822212 0	−0.000411106		1.00000i	$0.500131\pi$
$752$	0	0
$753$	12.4322	0.453056
$754$	0	0
$755$	− 16.0447i	− 0.583926i
$756$	0	0
$757$	30.0479i	1.09211i	0.837749	+	0.546055i	$0.183871\pi$
$757$	30.0479i	1.09211i	−0.837749	+	0.546055i	$0.816129\pi$
$758$	0	0
$759$	25.3965	0.921834
$760$	0	0
$761$	−17.1613	−0.622096	−0.311048	−	0.950394i	$-0.600680\pi$
$761$	−17.1613	−0.622096	−0.311048	+	0.950394i	$0.600680\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	− 0.658842i	− 0.0238205i
$766$	0	0
$767$	−14.5107	−0.523952
$768$	0	0
$769$	−47.2945	−1.70548	−0.852742	−	0.522332i	$-0.825062\pi$
$769$	−47.2945	−1.70548	−0.852742	+	0.522332i	$0.825062\pi$
$770$	0	0
$771$	0.282984i	0.0101914i
$772$	0	0
$773$	42.7753i	1.53852i	0.638935	+	0.769260i	$0.279375\pi$
$773$	42.7753i	1.53852i	−0.638935	+	0.769260i	$0.720625\pi$
$774$	0	0
$775$	1.97550	0.0709621
$776$	0	0
$777$	0	0
$778$	0	0
$779$	9.79343i	0.350886i
$780$	0	0
$781$	24.0521i	0.860651i
$782$	0	0
$783$	−5.53374	−0.197760
$784$	0	0
$785$	39.8303	1.42160
$786$	0	0
$787$	49.4625i	1.76315i	0.472046	+	0.881574i	$0.343515\pi$
$787$	49.4625i	1.76315i	−0.472046	+	0.881574i	$0.656485\pi$
$788$	0	0
$789$	− 14.3192i	− 0.509778i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	23.8962	0.848578
$794$	0	0
$795$	− 16.1961i	− 0.574416i
$796$	0	0
$797$	13.3755i	0.473783i	0.971536	+	0.236892i	$0.0761286\pi$
$797$	13.3755i	0.473783i	−0.971536	+	0.236892i	$0.923871\pi$
$798$	0	0
$799$	−0.0249817	−0.000883788 0
$800$	0	0
$801$	17.1839	0.607163
$802$	0	0
$803$	− 53.9453i	− 1.90369i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	16.8141	0.591883
$808$	0	0
$809$	14.2253	0.500136	0.250068	−	0.968228i	$-0.419547\pi$
$809$	14.2253	0.500136	0.250068	+	0.968228i	$0.419547\pi$
$810$	0	0
$811$	20.7849i	0.729855i	0.931036	+	0.364928i	$0.118906\pi$
$811$	20.7849i	0.729855i	−0.931036	+	0.364928i	$0.881094\pi$
$812$	0	0
$813$	− 26.8456i	− 0.941518i
$814$	0	0
$815$	36.0764	1.26370
$816$	0	0
$817$	−31.3245	−1.09590
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 4.96410i	− 0.173248i	−0.996241	−	0.0866241i	$-0.972392\pi$
$821$	− 4.96410i	− 0.173248i	0.996241	−	0.0866241i	$-0.0276079\pi$
$822$	0	0
$823$	−9.83498	−0.342826	−0.171413	−	0.985199i	$-0.554833\pi$
$823$	−9.83498	−0.342826	−0.171413	+	0.985199i	$0.554833\pi$
$824$	0	0
$825$	1.03158	0.0359150
$826$	0	0
$827$	41.4331i	1.44077i	0.693574	+	0.720385i	$0.256035\pi$
$827$	41.4331i	1.44077i	−0.693574	+	0.720385i	$0.743965\pi$
$828$	0	0
$829$	− 25.0029i	− 0.868386i	−0.900820	−	0.434193i	$-0.857034\pi$
$829$	− 25.0029i	− 0.868386i	0.900820	−	0.434193i	$-0.142966\pi$
$830$	0	0
$831$	−22.5030	−0.780619
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0.234201i	0.00810486i
$836$	0	0
$837$	7.44308i	0.257270i
$838$	0	0
$839$	−2.43551	−0.0840833	−0.0420416	−	0.999116i	$-0.513386\pi$
$839$	−2.43551	−0.0840833	−0.0420416	+	0.999116i	$0.513386\pi$
$840$	0	0
$841$	−1.62232	−0.0559420
$842$	0	0
$843$	12.8375i	0.442148i
$844$	0	0
$845$	4.32216i	0.148687i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−7.29790	−0.250463
$850$	0	0
$851$	38.9140i	1.33395i
$852$	0	0
$853$	− 51.1404i	− 1.75101i	−0.483206	−	0.875507i	$-0.660528\pi$
$853$	− 51.1404i	− 1.75101i	0.483206	−	0.875507i	$-0.339472\pi$
$854$	0	0
$855$	−6.39118	−0.218574
$856$	0	0
$857$	53.4628	1.82625	0.913127	−	0.407675i	$-0.133660\pi$
$857$	53.4628	1.82625	0.913127	+	0.407675i	$0.133660\pi$
$858$	0	0
$859$	16.5958i	0.566242i	0.959084	+	0.283121i	$0.0913697\pi$
$859$	16.5958i	0.566242i	−0.959084	+	0.283121i	$0.908630\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	40.8512	1.39059	0.695296	−	0.718724i	$-0.255273\pi$
$863$	40.8512	1.39059	0.695296	+	0.718724i	$0.255273\pi$
$864$	0	0
$865$	−5.24962	−0.178493
$866$	0	0
$867$	− 16.9176i	− 0.574551i
$868$	0	0
$869$	34.9743i	1.18642i
$870$	0	0
$871$	−43.4288	−1.47153
$872$	0	0
$873$	−6.46528	−0.218817
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 1.51953i	− 0.0513107i	−0.999671	−	0.0256554i	$-0.991833\pi$
$877$	− 1.51953i	− 0.0513107i	0.999671	−	0.0256554i	$-0.00816725\pi$
$878$	0	0
$879$	9.95674	0.335833
$880$	0	0
$881$	12.3299	0.415406	0.207703	−	0.978192i	$-0.433401\pi$
$881$	12.3299	0.415406	0.207703	+	0.978192i	$0.433401\pi$
$882$	0	0
$883$	− 12.0145i	− 0.404319i	−0.979353	−	0.202159i	$-0.935204\pi$
$883$	− 12.0145i	− 0.404319i	0.979353	−	0.202159i	$-0.0647959\pi$
$884$	0	0
$885$	9.98671i	0.335700i
$886$	0	0
$887$	23.0480	0.773875	0.386938	−	0.922106i	$-0.373533\pi$
$887$	23.0480	0.773875	0.386938	+	0.922106i	$0.373533\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	3.88667i	0.130208i
$892$	0	0
$893$	0.242338i	0.00810953i
$894$	0	0
$895$	42.7335	1.42842
$896$	0	0
$897$	−21.7860	−0.727415
$898$	0	0
$899$	41.1881i	1.37370i
$900$	0	0
$901$	2.02656i	0.0675144i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−31.3876	−1.04336
$906$	0	0
$907$	32.7639i	1.08791i	0.839115	+	0.543954i	$0.183073\pi$
$907$	32.7639i	1.08791i	−0.839115	+	0.543954i	$0.816927\pi$
$908$	0	0
$909$	11.8461i	0.392910i
$910$	0	0
$911$	−35.5607	−1.17818	−0.589090	−	0.808067i	$-0.700514\pi$
$911$	−35.5607	−1.17818	−0.589090	+	0.808067i	$0.700514\pi$
$912$	0	0
$913$	−68.5273	−2.26792
$914$	0	0
$915$	− 16.4461i	− 0.543691i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	27.5241	0.907937	0.453969	−	0.891018i	$-0.350008\pi$
$919$	27.5241	0.907937	0.453969	+	0.891018i	$0.350008\pi$
$920$	0	0
$921$	14.9479	0.492550
$922$	0	0
$923$	− 20.6327i	− 0.679135i
$924$	0	0
$925$	1.58065i	0.0519713i
$926$	0	0
$927$	1.62190	0.0532701
$928$	0	0
$929$	34.9480	1.14661	0.573303	−	0.819343i	$-0.305662\pi$
$929$	34.9480	1.14661	0.573303	+	0.819343i	$0.305662\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	31.2591i	1.02338i
$934$	0	0
$935$	−2.56070	−0.0837439
$936$	0	0
$937$	17.7423	0.579615	0.289807	−	0.957085i	$-0.406409\pi$
$937$	17.7423	0.579615	0.289807	+	0.957085i	$0.406409\pi$
$938$	0	0
$939$	− 3.06468i	− 0.100012i
$940$	0	0
$941$	22.5728i	0.735852i	0.929855	+	0.367926i	$0.119932\pi$
$941$	22.5728i	0.735852i	−0.929855	+	0.367926i	$0.880068\pi$
$942$	0	0
$943$	22.9755	0.748187
$944$	0	0
$945$	0	0
$946$	0	0
$947$	27.2714i	0.886203i	0.896471	+	0.443102i	$0.146122\pi$
$947$	27.2714i	0.886203i	−0.896471	+	0.443102i	$0.853878\pi$
$948$	0	0
$949$	46.2763i	1.50219i
$950$	0	0
$951$	16.2981	0.528501
$952$	0	0
$953$	−5.90986	−0.191439	−0.0957196	−	0.995408i	$-0.530515\pi$
$953$	−5.90986	−0.191439	−0.0957196	+	0.995408i	$0.530515\pi$
$954$	0	0
$955$	28.9920i	0.938157i
$956$	0	0
$957$	21.5078i	0.695250i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	24.3994	0.787077
$962$	0	0
$963$	5.02926i	0.162066i
$964$	0	0
$965$	5.97945i	0.192485i
$966$	0	0
$967$	−14.0032	−0.450312	−0.225156	−	0.974323i	$-0.572289\pi$
$967$	−14.0032	−0.450312	−0.225156	+	0.974323i	$0.572289\pi$
$968$	0	0
$969$	0.799705	0.0256902
$970$	0	0
$971$	− 9.13650i	− 0.293204i	−0.989196	−	0.146602i	$-0.953166\pi$
$971$	− 9.13650i	− 0.293204i	0.989196	−	0.146602i	$-0.0468337\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−0.884927	−0.0283403
$976$	0	0
$977$	21.1534	0.676758	0.338379	−	0.941010i	$-0.390121\pi$
$977$	21.1534	0.676758	0.338379	+	0.941010i	$0.390121\pi$
$978$	0	0
$979$	− 66.7881i	− 2.13456i
$980$	0	0
$981$	− 0.204420i	− 0.00652662i
$982$	0	0
$983$	−46.4009	−1.47996	−0.739979	−	0.672630i	$-0.765164\pi$
$983$	−46.4009	−1.47996	−0.739979	+	0.672630i	$0.765164\pi$
$984$	0	0
$985$	1.82373	0.0581090
$986$	0	0
$987$	0	0
$988$	0	0
$989$	73.4877i	2.33677i
$990$	0	0
$991$	−2.58561	−0.0821345	−0.0410672	−	0.999156i	$-0.513076\pi$
$991$	−2.58561	−0.0821345	−0.0410672	+	0.999156i	$0.513076\pi$
$992$	0	0
$993$	−2.76307	−0.0876833
$994$	0	0
$995$	− 24.6314i	− 0.780868i
$996$	0	0
$997$	− 20.7334i	− 0.656635i	−0.944567	−	0.328317i	$-0.893518\pi$
$997$	− 20.7334i	− 0.656635i	0.944567	−	0.328317i	$-0.106482\pi$
$998$	0	0
$999$	−5.95539	−0.188420

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4704.2.c.e.2353.15		16
4.3	odd	2		1176.2.c.e.589.15		16
7.2	even	3		672.2.bk.a.529.7		32
7.4	even	3		672.2.bk.a.625.10		32
7.6	odd	2		4704.2.c.f.2353.2		16
8.3	odd	2		1176.2.c.e.589.16		16
8.5	even	2	inner	4704.2.c.e.2353.2		16
21.2	odd	6		2016.2.cr.e.1873.4		32
21.11	odd	6		2016.2.cr.e.1297.13		32
28.11	odd	6		168.2.bc.a.37.6	yes	32
28.23	odd	6		168.2.bc.a.109.5	yes	32
28.27	even	2		1176.2.c.f.589.15		16
56.11	odd	6		168.2.bc.a.37.5	✓	32
56.13	odd	2		4704.2.c.f.2353.15		16
56.27	even	2		1176.2.c.f.589.16		16
56.37	even	6		672.2.bk.a.529.10		32
56.51	odd	6		168.2.bc.a.109.6	yes	32
56.53	even	6		672.2.bk.a.625.7		32
84.11	even	6		504.2.cj.e.37.11		32
84.23	even	6		504.2.cj.e.109.12		32
168.11	even	6		504.2.cj.e.37.12		32
168.53	odd	6		2016.2.cr.e.1297.4		32
168.107	even	6		504.2.cj.e.109.11		32
168.149	odd	6		2016.2.cr.e.1873.13		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.bc.a.37.5	✓	32	56.11	odd	6
168.2.bc.a.37.6	yes	32	28.11	odd	6
168.2.bc.a.109.5	yes	32	28.23	odd	6
168.2.bc.a.109.6	yes	32	56.51	odd	6
504.2.cj.e.37.11		32	84.11	even	6
504.2.cj.e.37.12		32	168.11	even	6
504.2.cj.e.109.11		32	168.107	even	6
504.2.cj.e.109.12		32	84.23	even	6
672.2.bk.a.529.7		32	7.2	even	3
672.2.bk.a.529.10		32	56.37	even	6
672.2.bk.a.625.7		32	56.53	even	6
672.2.bk.a.625.10		32	7.4	even	3
1176.2.c.e.589.15		16	4.3	odd	2
1176.2.c.e.589.16		16	8.3	odd	2
1176.2.c.f.589.15		16	28.27	even	2
1176.2.c.f.589.16		16	56.27	even	2
2016.2.cr.e.1297.4		32	168.53	odd	6
2016.2.cr.e.1297.13		32	21.11	odd	6
2016.2.cr.e.1873.4		32	21.2	odd	6
2016.2.cr.e.1873.13		32	168.149	odd	6
4704.2.c.e.2353.2		16	8.5	even	2	inner
4704.2.c.e.2353.15		16	1.1	even	1	trivial
4704.2.c.f.2353.2		16	7.6	odd	2
4704.2.c.f.2353.15		16	56.13	odd	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:	\( N \)	\(=\)	\( 4704 = 2^{5} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4704.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +2.29465i q^{5} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +2.29465i q^{5} -1.00000 q^{9} +3.88667i q^{11} -3.33413i q^{13} -2.29465 q^{15} +0.287121 q^{17} -2.78525i q^{19} -6.53425 q^{23} -0.265415 q^{25} -1.00000i q^{27} -5.53374i q^{29} -7.44308 q^{31} -3.88667 q^{33} -5.95539i q^{37} +3.33413 q^{39} -3.51617 q^{41} -11.2465i q^{43} -2.29465i q^{45} -0.0870075 q^{47} +0.287121i q^{51} +7.05820i q^{53} -8.91855 q^{55} +2.78525 q^{57} -4.35217i q^{59} +7.16714i q^{61} +7.65066 q^{65} -13.0255i q^{67} -6.53425i q^{69} +6.18835 q^{71} -13.8796 q^{73} -0.265415i q^{75} +8.99853 q^{79} +1.00000 q^{81} +17.6313i q^{83} +0.658842i q^{85} +5.53374 q^{87} -17.1839 q^{89} -7.44308i q^{93} +6.39118 q^{95} +6.46528 q^{97} -3.88667i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{9}+O(q^{10}) \) 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

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Database

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