LMFDB - Embedded newform 4620.2.a.x.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4620,2,Mod(1,4620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4620, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4620.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4620.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$36.8908857338$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.138892.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 10x^{2} + 2x + 12 $ x^4 - x^3 - 10x^2 + 2x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.44214$ of defining polynomial
Character	$\chi$	$=$	4620.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} +4.40616 q^{13} +1.00000 q^{15} -0.699032 q^{17} -5.58330 q^{19} +1.00000 q^{21} +6.18524 q^{23} +1.00000 q^{25} +1.00000 q^{27} -1.70713 q^{29} +4.40616 q^{31} +1.00000 q^{33} +1.00000 q^{35} -2.47811 q^{37} +4.40616 q^{39} +4.47811 q^{41} +8.57520 q^{43} +1.00000 q^{45} -7.89237 q^{47} +1.00000 q^{49} -0.699032 q^{51} -4.11329 q^{53} +1.00000 q^{55} -5.58330 q^{57} +3.10519 q^{59} +10.9814 q^{61} +1.00000 q^{63} +4.40616 q^{65} -9.69659 q^{67} +6.18524 q^{69} +9.29043 q^{71} -6.28233 q^{73} +1.00000 q^{75} +1.00000 q^{77} +7.36238 q^{79} +1.00000 q^{81} +4.62708 q^{83} -0.699032 q^{85} -1.70713 q^{87} +7.19334 q^{89} +4.40616 q^{91} +4.40616 q^{93} -5.58330 q^{95} -3.63518 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 4 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 4 * q^5 + 4 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} + 4 q^{5} + 4 q^{7} + 4 q^{9} + 4 q^{11} + 4 q^{13} + 4 q^{15} + 2 q^{17} + 8 q^{19} + 4 q^{21} + 4 q^{23} + 4 q^{25} + 4 q^{27} + 2 q^{29} + 4 q^{31} + 4 q^{33} + 4 q^{35} + 2 q^{37} + 4 q^{39} + 6 q^{41} + 8 q^{43} + 4 q^{45} - 2 q^{47} + 4 q^{49} + 2 q^{51} + 6 q^{53} + 4 q^{55} + 8 q^{57} - 6 q^{59} + 4 q^{61} + 4 q^{63} + 4 q^{65} + 14 q^{67} + 4 q^{69} - 2 q^{71} + 10 q^{73} + 4 q^{75} + 4 q^{77} - 8 q^{79} + 4 q^{81} + 12 q^{83} + 2 q^{85} + 2 q^{87} + 4 q^{89} + 4 q^{91} + 4 q^{93} + 8 q^{95} - 4 q^{97} + 4 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^5 + 4 * q^7 + 4 * q^9 + 4 * q^11 + 4 * q^13 + 4 * q^15 + 2 * q^17 + 8 * q^19 + 4 * q^21 + 4 * q^23 + 4 * q^25 + 4 * q^27 + 2 * q^29 + 4 * q^31 + 4 * q^33 + 4 * q^35 + 2 * q^37 + 4 * q^39 + 6 * q^41 + 8 * q^43 + 4 * q^45 - 2 * q^47 + 4 * q^49 + 2 * q^51 + 6 * q^53 + 4 * q^55 + 8 * q^57 - 6 * q^59 + 4 * q^61 + 4 * q^63 + 4 * q^65 + 14 * q^67 + 4 * q^69 - 2 * q^71 + 10 * q^73 + 4 * q^75 + 4 * q^77 - 8 * q^79 + 4 * q^81 + 12 * q^83 + 2 * q^85 + 2 * q^87 + 4 * q^89 + 4 * q^91 + 4 * q^93 + 8 * q^95 - 4 * q^97 + 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	4.40616	1.22205	0.611025	−	0.791612i	$-0.290758\pi$
$13$	4.40616	1.22205	0.611025	+	0.791612i	$0.290758\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−0.699032	−0.169540	−0.0847700	−	0.996401i	$-0.527016\pi$
$17$	−0.699032	−0.169540	−0.0847700	+	0.996401i	$0.527016\pi$
$18$	0	0
$19$	−5.58330	−1.28090	−0.640449	−	0.768001i	$-0.721252\pi$
$19$	−5.58330	−1.28090	−0.640449	+	0.768001i	$0.721252\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	6.18524	1.28971	0.644856	−	0.764304i	$-0.276917\pi$
$23$	6.18524	1.28971	0.644856	+	0.764304i	$0.276917\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.70713	−0.317006	−0.158503	−	0.987358i	$-0.550667\pi$
$29$	−1.70713	−0.317006	−0.158503	+	0.987358i	$0.550667\pi$
$30$	0	0
$31$	4.40616	0.791370	0.395685	−	0.918386i	$-0.370507\pi$
$31$	4.40616	0.791370	0.395685	+	0.918386i	$0.370507\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−2.47811	−0.407399	−0.203699	−	0.979033i	$-0.565297\pi$
$37$	−2.47811	−0.407399	−0.203699	+	0.979033i	$0.565297\pi$
$38$	0	0
$39$	4.40616	0.705551
$40$	0	0
$41$	4.47811	0.699363	0.349682	−	0.936869i	$-0.386290\pi$
$41$	4.47811	0.699363	0.349682	+	0.936869i	$0.386290\pi$
$42$	0	0
$43$	8.57520	1.30771	0.653853	−	0.756621i	$-0.273151\pi$
$43$	8.57520	1.30771	0.653853	+	0.756621i	$0.273151\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−7.89237	−1.15122	−0.575610	−	0.817724i	$-0.695235\pi$
$47$	−7.89237	−1.15122	−0.575610	+	0.817724i	$0.695235\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.699032	−0.0978840
$52$	0	0
$53$	−4.11329	−0.565004	−0.282502	−	0.959267i	$-0.591164\pi$
$53$	−4.11329	−0.565004	−0.282502	+	0.959267i	$0.591164\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	−5.58330	−0.739527
$58$	0	0
$59$	3.10519	0.404262	0.202131	−	0.979359i	$-0.435213\pi$
$59$	3.10519	0.404262	0.202131	+	0.979359i	$0.435213\pi$
$60$	0	0
$61$	10.9814	1.40602	0.703010	−	0.711180i	$-0.251839\pi$
$61$	10.9814	1.40602	0.703010	+	0.711180i	$0.251839\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	4.40616	0.546517
$66$	0	0
$67$	−9.69659	−1.18463	−0.592314	−	0.805707i	$-0.701785\pi$
$67$	−9.69659	−1.18463	−0.592314	+	0.805707i	$0.701785\pi$
$68$	0	0
$69$	6.18524	0.744615
$70$	0	0
$71$	9.29043	1.10257	0.551286	−	0.834316i	$-0.314137\pi$
$71$	9.29043	1.10257	0.551286	+	0.834316i	$0.314137\pi$
$72$	0	0
$73$	−6.28233	−0.735292	−0.367646	−	0.929966i	$-0.619836\pi$
$73$	−6.28233	−0.735292	−0.367646	+	0.929966i	$0.619836\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	7.36238	0.828333	0.414166	−	0.910201i	$-0.364073\pi$
$79$	7.36238	0.828333	0.414166	+	0.910201i	$0.364073\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.62708	0.507888	0.253944	−	0.967219i	$-0.418272\pi$
$83$	4.62708	0.507888	0.253944	+	0.967219i	$0.418272\pi$
$84$	0	0
$85$	−0.699032	−0.0758206
$86$	0	0
$87$	−1.70713	−0.183024
$88$	0	0
$89$	7.19334	0.762492	0.381246	−	0.924474i	$-0.375495\pi$
$89$	7.19334	0.762492	0.381246	+	0.924474i	$0.375495\pi$
$90$	0	0
$91$	4.40616	0.461891
$92$	0	0
$93$	4.40616	0.456898
$94$	0	0
$95$	−5.58330	−0.572835
$96$	0	0
$97$	−3.63518	−0.369097	−0.184548	−	0.982823i	$-0.559082\pi$
$97$	−3.63518	−0.369097	−0.184548	+	0.982823i	$0.559082\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−15.1666	−1.50913	−0.754567	−	0.656223i	$-0.772153\pi$
$101$	−15.1666	−1.50913	−0.754567	+	0.656223i	$0.772153\pi$
$102$	0	0
$103$	−9.38753	−0.924981	−0.462490	−	0.886624i	$-0.653044\pi$
$103$	−9.38753	−0.924981	−0.462490	+	0.886624i	$0.653044\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	0	0
$107$	12.7604	1.23360	0.616799	−	0.787120i	$-0.288429\pi$
$107$	12.7604	1.23360	0.616799	+	0.787120i	$0.288429\pi$
$108$	0	0
$109$	6.47811	0.620490	0.310245	−	0.950657i	$-0.399589\pi$
$109$	6.47811	0.620490	0.310245	+	0.950657i	$0.399589\pi$
$110$	0	0
$111$	−2.47811	−0.235212
$112$	0	0
$113$	−11.6714	−1.09796	−0.548979	−	0.835836i	$-0.684983\pi$
$113$	−11.6714	−1.09796	−0.548979	+	0.835836i	$0.684983\pi$
$114$	0	0
$115$	6.18524	0.576777
$116$	0	0
$117$	4.40616	0.407350
$118$	0	0
$119$	−0.699032	−0.0640801
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	4.47811	0.403778
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	14.0057	1.24280	0.621401	−	0.783493i	$-0.286564\pi$
$127$	14.0057	1.24280	0.621401	+	0.783493i	$0.286564\pi$
$128$	0	0
$129$	8.57520	0.755005
$130$	0	0
$131$	−8.67230	−0.757702	−0.378851	−	0.925458i	$-0.623681\pi$
$131$	−8.67230	−0.757702	−0.378851	+	0.925458i	$0.623681\pi$
$132$	0	0
$133$	−5.58330	−0.484134
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−12.2104	−1.04320	−0.521602	−	0.853189i	$-0.674665\pi$
$137$	−12.2104	−1.04320	−0.521602	+	0.853189i	$0.674665\pi$
$138$	0	0
$139$	−11.2548	−0.954615	−0.477308	−	0.878736i	$-0.658387\pi$
$139$	−11.2548	−0.954615	−0.477308	+	0.878736i	$0.658387\pi$
$140$	0	0
$141$	−7.89237	−0.664657
$142$	0	0
$143$	4.40616	0.368462
$144$	0	0
$145$	−1.70713	−0.141769
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−19.7128	−1.61493	−0.807467	−	0.589912i	$-0.799162\pi$
$149$	−19.7128	−1.61493	−0.807467	+	0.589912i	$0.799162\pi$
$150$	0	0
$151$	21.0947	1.71666	0.858329	−	0.513099i	$-0.171503\pi$
$151$	21.0947	1.71666	0.858329	+	0.513099i	$0.171503\pi$
$152$	0	0
$153$	−0.699032	−0.0565134
$154$	0	0
$155$	4.40616	0.353911
$156$	0	0
$157$	4.59140	0.366434	0.183217	−	0.983073i	$-0.441349\pi$
$157$	4.59140	0.366434	0.183217	+	0.983073i	$0.441349\pi$
$158$	0	0
$159$	−4.11329	−0.326205
$160$	0	0
$161$	6.18524	0.487465
$162$	0	0
$163$	−0.406162	−0.0318130	−0.0159065	−	0.999873i	$-0.505063\pi$
$163$	−0.406162	−0.0318130	−0.0159065	+	0.999873i	$0.505063\pi$
$164$	0	0
$165$	1.00000	0.0778499
$166$	0	0
$167$	5.48621	0.424536	0.212268	−	0.977212i	$-0.431915\pi$
$167$	5.48621	0.424536	0.212268	+	0.977212i	$0.431915\pi$
$168$	0	0
$169$	6.41426	0.493405
$170$	0	0
$171$	−5.58330	−0.426966
$172$	0	0
$173$	12.2104	0.928338	0.464169	−	0.885747i	$-0.346353\pi$
$173$	12.2104	0.928338	0.464169	+	0.885747i	$0.346353\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	3.10519	0.233401
$178$	0	0
$179$	−7.27424	−0.543702	−0.271851	−	0.962339i	$-0.587636\pi$
$179$	−7.27424	−0.543702	−0.271851	+	0.962339i	$0.587636\pi$
$180$	0	0
$181$	11.6246	0.864053	0.432027	−	0.901861i	$-0.357799\pi$
$181$	11.6246	0.864053	0.432027	+	0.901861i	$0.357799\pi$
$182$	0	0
$183$	10.9814	0.811766
$184$	0	0
$185$	−2.47811	−0.182194
$186$	0	0
$187$	−0.699032	−0.0511183
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	2.79613	0.202321	0.101160	−	0.994870i	$-0.467745\pi$
$191$	2.79613	0.202321	0.101160	+	0.994870i	$0.467745\pi$
$192$	0	0
$193$	25.1828	1.81270	0.906349	−	0.422530i	$-0.138858\pi$
$193$	25.1828	1.81270	0.906349	+	0.422530i	$0.138858\pi$
$194$	0	0
$195$	4.40616	0.315532
$196$	0	0
$197$	2.74038	0.195244	0.0976218	−	0.995224i	$-0.468876\pi$
$197$	2.74038	0.195244	0.0976218	+	0.995224i	$0.468876\pi$
$198$	0	0
$199$	15.0947	1.07003	0.535016	−	0.844842i	$-0.320306\pi$
$199$	15.0947	1.07003	0.535016	+	0.844842i	$0.320306\pi$
$200$	0	0
$201$	−9.69659	−0.683945
$202$	0	0
$203$	−1.70713	−0.119817
$204$	0	0
$205$	4.47811	0.312765
$206$	0	0
$207$	6.18524	0.429904
$208$	0	0
$209$	−5.58330	−0.386205
$210$	0	0
$211$	−21.9070	−1.50814	−0.754069	−	0.656795i	$-0.771912\pi$
$211$	−21.9070	−1.50814	−0.754069	+	0.656795i	$0.771912\pi$
$212$	0	0
$213$	9.29043	0.636570
$214$	0	0
$215$	8.57520	0.584824
$216$	0	0
$217$	4.40616	0.299110
$218$	0	0
$219$	−6.28233	−0.424521
$220$	0	0
$221$	−3.08005	−0.207186
$222$	0	0
$223$	8.57520	0.574238	0.287119	−	0.957895i	$-0.407302\pi$
$223$	8.57520	0.574238	0.287119	+	0.957895i	$0.407302\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	0.965169	0.0640605	0.0320303	−	0.999487i	$-0.489803\pi$
$227$	0.965169	0.0640605	0.0320303	+	0.999487i	$0.489803\pi$
$228$	0	0
$229$	−7.29043	−0.481765	−0.240883	−	0.970554i	$-0.577437\pi$
$229$	−7.29043	−0.481765	−0.240883	+	0.970554i	$0.577437\pi$
$230$	0	0
$231$	1.00000	0.0657952
$232$	0	0
$233$	−5.20229	−0.340813	−0.170407	−	0.985374i	$-0.554508\pi$
$233$	−5.20229	−0.340813	−0.170407	+	0.985374i	$0.554508\pi$
$234$	0	0
$235$	−7.89237	−0.514841
$236$	0	0
$237$	7.36238	0.478238
$238$	0	0
$239$	−0.575204	−0.0372069	−0.0186034	−	0.999827i	$-0.505922\pi$
$239$	−0.575204	−0.0372069	−0.0186034	+	0.999827i	$0.505922\pi$
$240$	0	0
$241$	0.868074	0.0559176	0.0279588	−	0.999609i	$-0.491099\pi$
$241$	0.868074	0.0559176	0.0279588	+	0.999609i	$0.491099\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−24.6009	−1.56532
$248$	0	0
$249$	4.62708	0.293230
$250$	0	0
$251$	2.52999	0.159691	0.0798457	−	0.996807i	$-0.474557\pi$
$251$	2.52999	0.159691	0.0798457	+	0.996807i	$0.474557\pi$
$252$	0	0
$253$	6.18524	0.388863
$254$	0	0
$255$	−0.699032	−0.0437751
$256$	0	0
$257$	9.50082	0.592645	0.296322	−	0.955088i	$-0.404240\pi$
$257$	9.50082	0.592645	0.296322	+	0.955088i	$0.404240\pi$
$258$	0	0
$259$	−2.47811	−0.153982
$260$	0	0
$261$	−1.70713	−0.105669
$262$	0	0
$263$	11.0282	0.680026	0.340013	−	0.940421i	$-0.389569\pi$
$263$	11.0282	0.680026	0.340013	+	0.940421i	$0.389569\pi$
$264$	0	0
$265$	−4.11329	−0.252677
$266$	0	0
$267$	7.19334	0.440225
$268$	0	0
$269$	10.3599	0.631657	0.315828	−	0.948816i	$-0.397718\pi$
$269$	10.3599	0.631657	0.315828	+	0.948816i	$0.397718\pi$
$270$	0	0
$271$	3.37292	0.204890	0.102445	−	0.994739i	$-0.467333\pi$
$271$	3.37292	0.204890	0.102445	+	0.994739i	$0.467333\pi$
$272$	0	0
$273$	4.40616	0.266673
$274$	0	0
$275$	1.00000	0.0603023
$276$	0	0
$277$	−2.97242	−0.178595	−0.0892976	−	0.996005i	$-0.528462\pi$
$277$	−2.97242	−0.178595	−0.0892976	+	0.996005i	$0.528462\pi$
$278$	0	0
$279$	4.40616	0.263790
$280$	0	0
$281$	11.2386	0.670436	0.335218	−	0.942141i	$-0.391190\pi$
$281$	11.2386	0.670436	0.335218	+	0.942141i	$0.391190\pi$
$282$	0	0
$283$	−7.50082	−0.445877	−0.222939	−	0.974832i	$-0.571565\pi$
$283$	−7.50082	−0.445877	−0.222939	+	0.974832i	$0.571565\pi$
$284$	0	0
$285$	−5.58330	−0.330726
$286$	0	0
$287$	4.47811	0.264334
$288$	0	0
$289$	−16.5114	−0.971256
$290$	0	0
$291$	−3.63518	−0.213098
$292$	0	0
$293$	−13.8494	−0.809093	−0.404546	−	0.914517i	$-0.632571\pi$
$293$	−13.8494	−0.809093	−0.404546	+	0.914517i	$0.632571\pi$
$294$	0	0
$295$	3.10519	0.180791
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	27.2532	1.57609
$300$	0	0
$301$	8.57520	0.494267
$302$	0	0
$303$	−15.1666	−0.871299
$304$	0	0
$305$	10.9814	0.628791
$306$	0	0
$307$	25.1828	1.43726	0.718629	−	0.695393i	$-0.244770\pi$
$307$	25.1828	1.43726	0.718629	+	0.695393i	$0.244770\pi$
$308$	0	0
$309$	−9.38753	−0.534038
$310$	0	0
$311$	−33.1109	−1.87754	−0.938772	−	0.344539i	$-0.888035\pi$
$311$	−33.1109	−1.87754	−0.938772	+	0.344539i	$0.888035\pi$
$312$	0	0
$313$	8.36482	0.472807	0.236404	−	0.971655i	$-0.424031\pi$
$313$	8.36482	0.472807	0.236404	+	0.971655i	$0.424031\pi$
$314$	0	0
$315$	1.00000	0.0563436
$316$	0	0
$317$	6.37048	0.357802	0.178901	−	0.983867i	$-0.442746\pi$
$317$	6.37048	0.357802	0.178901	+	0.983867i	$0.442746\pi$
$318$	0	0
$319$	−1.70713	−0.0955809
$320$	0	0
$321$	12.7604	0.712219
$322$	0	0
$323$	3.90291	0.217163
$324$	0	0
$325$	4.40616	0.244410
$326$	0	0
$327$	6.47811	0.358240
$328$	0	0
$329$	−7.89237	−0.435120
$330$	0	0
$331$	−13.8656	−0.762124	−0.381062	−	0.924549i	$-0.624442\pi$
$331$	−13.8656	−0.762124	−0.381062	+	0.924549i	$0.624442\pi$
$332$	0	0
$333$	−2.47811	−0.135800
$334$	0	0
$335$	−9.69659	−0.529782
$336$	0	0
$337$	−14.4475	−0.787006	−0.393503	−	0.919323i	$-0.628737\pi$
$337$	−14.4475	−0.787006	−0.393503	+	0.919323i	$0.628737\pi$
$338$	0	0
$339$	−11.6714	−0.633906
$340$	0	0
$341$	4.40616	0.238607
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	6.18524	0.333002
$346$	0	0
$347$	−11.0282	−0.592023	−0.296012	−	0.955184i	$-0.595657\pi$
$347$	−11.0282	−0.592023	−0.296012	+	0.955184i	$0.595657\pi$
$348$	0	0
$349$	9.83096	0.526239	0.263120	−	0.964763i	$-0.415249\pi$
$349$	9.83096	0.526239	0.263120	+	0.964763i	$0.415249\pi$
$350$	0	0
$351$	4.40616	0.235184
$352$	0	0
$353$	−28.7248	−1.52886	−0.764432	−	0.644704i	$-0.776981\pi$
$353$	−28.7248	−1.52886	−0.764432	+	0.644704i	$0.776981\pi$
$354$	0	0
$355$	9.29043	0.493085
$356$	0	0
$357$	−0.699032	−0.0369967
$358$	0	0
$359$	2.59140	0.136769	0.0683845	−	0.997659i	$-0.478216\pi$
$359$	2.59140	0.136769	0.0683845	+	0.997659i	$0.478216\pi$
$360$	0	0
$361$	12.1733	0.640698
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	−6.28233	−0.328832
$366$	0	0
$367$	−25.5817	−1.33535	−0.667677	−	0.744451i	$-0.732711\pi$
$367$	−25.5817	−1.33535	−0.667677	+	0.744451i	$0.732711\pi$
$368$	0	0
$369$	4.47811	0.233121
$370$	0	0
$371$	−4.11329	−0.213551
$372$	0	0
$373$	−6.96188	−0.360472	−0.180236	−	0.983623i	$-0.557686\pi$
$373$	−6.96188	−0.360472	−0.180236	+	0.983623i	$0.557686\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−7.52189	−0.387397
$378$	0	0
$379$	2.08090	0.106889	0.0534443	−	0.998571i	$-0.482980\pi$
$379$	2.08090	0.106889	0.0534443	+	0.998571i	$0.482980\pi$
$380$	0	0
$381$	14.0057	0.717532
$382$	0	0
$383$	23.0590	1.17826	0.589129	−	0.808039i	$-0.299471\pi$
$383$	23.0590	1.17826	0.589129	+	0.808039i	$0.299471\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	0	0
$387$	8.57520	0.435902
$388$	0	0
$389$	0.494306	0.0250623	0.0125312	−	0.999921i	$-0.496011\pi$
$389$	0.494306	0.0250623	0.0125312	+	0.999921i	$0.496011\pi$
$390$	0	0
$391$	−4.32368	−0.218658
$392$	0	0
$393$	−8.67230	−0.437460
$394$	0	0
$395$	7.36238	0.370442
$396$	0	0
$397$	14.7913	0.742352	0.371176	−	0.928563i	$-0.378955\pi$
$397$	14.7913	0.742352	0.371176	+	0.928563i	$0.378955\pi$
$398$	0	0
$399$	−5.58330	−0.279515
$400$	0	0
$401$	−7.06385	−0.352752	−0.176376	−	0.984323i	$-0.556437\pi$
$401$	−7.06385	−0.352752	−0.176376	+	0.984323i	$0.556437\pi$
$402$	0	0
$403$	19.4143	0.967093
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−2.47811	−0.122835
$408$	0	0
$409$	−6.07195	−0.300239	−0.150119	−	0.988668i	$-0.547966\pi$
$409$	−6.07195	−0.300239	−0.150119	+	0.988668i	$0.547966\pi$
$410$	0	0
$411$	−12.2104	−0.602294
$412$	0	0
$413$	3.10519	0.152797
$414$	0	0
$415$	4.62708	0.227135
$416$	0	0
$417$	−11.2548	−0.551147
$418$	0	0
$419$	4.13278	0.201899	0.100950	−	0.994892i	$-0.467812\pi$
$419$	4.13278	0.201899	0.100950	+	0.994892i	$0.467812\pi$
$420$	0	0
$421$	−13.2848	−0.647460	−0.323730	−	0.946149i	$-0.604937\pi$
$421$	−13.2848	−0.647460	−0.323730	+	0.946149i	$0.604937\pi$
$422$	0	0
$423$	−7.89237	−0.383740
$424$	0	0
$425$	−0.699032	−0.0339080
$426$	0	0
$427$	10.9814	0.531426
$428$	0	0
$429$	4.40616	0.212731
$430$	0	0
$431$	−29.5533	−1.42353	−0.711766	−	0.702417i	$-0.752104\pi$
$431$	−29.5533	−1.42353	−0.711766	+	0.702417i	$0.752104\pi$
$432$	0	0
$433$	16.5485	0.795269	0.397634	−	0.917544i	$-0.369831\pi$
$433$	16.5485	0.795269	0.397634	+	0.917544i	$0.369831\pi$
$434$	0	0
$435$	−1.70713	−0.0818506
$436$	0	0
$437$	−34.5341	−1.65199
$438$	0	0
$439$	−0.770980	−0.0367968	−0.0183984	−	0.999831i	$-0.505857\pi$
$439$	−0.770980	−0.0367968	−0.0183984	+	0.999831i	$0.505857\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	15.5689	0.739701	0.369850	−	0.929091i	$-0.379409\pi$
$443$	15.5689	0.739701	0.369850	+	0.929091i	$0.379409\pi$
$444$	0	0
$445$	7.19334	0.340997
$446$	0	0
$447$	−19.7128	−0.932383
$448$	0	0
$449$	−21.5008	−1.01469	−0.507343	−	0.861744i	$-0.669372\pi$
$449$	−21.5008	−1.01469	−0.507343	+	0.861744i	$0.669372\pi$
$450$	0	0
$451$	4.47811	0.210866
$452$	0	0
$453$	21.0947	0.991113
$454$	0	0
$455$	4.40616	0.206564
$456$	0	0
$457$	−18.5914	−0.869669	−0.434835	−	0.900510i	$-0.643193\pi$
$457$	−18.5914	−0.869669	−0.434835	+	0.900510i	$0.643193\pi$
$458$	0	0
$459$	−0.699032	−0.0326280
$460$	0	0
$461$	−27.8350	−1.29641	−0.648203	−	0.761467i	$-0.724479\pi$
$461$	−27.8350	−1.29641	−0.648203	+	0.761467i	$0.724479\pi$
$462$	0	0
$463$	14.0558	0.653226	0.326613	−	0.945158i	$-0.394093\pi$
$463$	14.0558	0.653226	0.326613	+	0.945158i	$0.394093\pi$
$464$	0	0
$465$	4.40616	0.204331
$466$	0	0
$467$	−32.3494	−1.49695	−0.748476	−	0.663162i	$-0.769214\pi$
$467$	−32.3494	−1.49695	−0.748476	+	0.663162i	$0.769214\pi$
$468$	0	0
$469$	−9.69659	−0.447747
$470$	0	0
$471$	4.59140	0.211561
$472$	0	0
$473$	8.57520	0.394288
$474$	0	0
$475$	−5.58330	−0.256179
$476$	0	0
$477$	−4.11329	−0.188335
$478$	0	0
$479$	−6.90376	−0.315441	−0.157720	−	0.987484i	$-0.550414\pi$
$479$	−6.90376	−0.315441	−0.157720	+	0.987484i	$0.550414\pi$
$480$	0	0
$481$	−10.9190	−0.497861
$482$	0	0
$483$	6.18524	0.281438
$484$	0	0
$485$	−3.63518	−0.165065
$486$	0	0
$487$	28.2451	1.27991	0.639953	−	0.768414i	$-0.278954\pi$
$487$	28.2451	1.27991	0.639953	+	0.768414i	$0.278954\pi$
$488$	0	0
$489$	−0.406162	−0.0183673
$490$	0	0
$491$	18.1285	0.818127	0.409064	−	0.912506i	$-0.365855\pi$
$491$	18.1285	0.818127	0.409064	+	0.912506i	$0.365855\pi$
$492$	0	0
$493$	1.19334	0.0537452
$494$	0	0
$495$	1.00000	0.0449467
$496$	0	0
$497$	9.29043	0.416733
$498$	0	0
$499$	27.8608	1.24722	0.623610	−	0.781736i	$-0.285666\pi$
$499$	27.8608	1.24722	0.623610	+	0.781736i	$0.285666\pi$
$500$	0	0
$501$	5.48621	0.245106
$502$	0	0
$503$	11.8148	0.526794	0.263397	−	0.964688i	$-0.415157\pi$
$503$	11.8148	0.526794	0.263397	+	0.964688i	$0.415157\pi$
$504$	0	0
$505$	−15.1666	−0.674905
$506$	0	0
$507$	6.41426	0.284867
$508$	0	0
$509$	10.3599	0.459196	0.229598	−	0.973285i	$-0.426259\pi$
$509$	10.3599	0.459196	0.229598	+	0.973285i	$0.426259\pi$
$510$	0	0
$511$	−6.28233	−0.277914
$512$	0	0
$513$	−5.58330	−0.246509
$514$	0	0
$515$	−9.38753	−0.413664
$516$	0	0
$517$	−7.89237	−0.347106
$518$	0	0
$519$	12.2104	0.535976
$520$	0	0
$521$	0.785591	0.0344174	0.0172087	−	0.999852i	$-0.494522\pi$
$521$	0.785591	0.0344174	0.0172087	+	0.999852i	$0.494522\pi$
$522$	0	0
$523$	13.8399	0.605177	0.302588	−	0.953121i	$-0.402149\pi$
$523$	13.8399	0.605177	0.302588	+	0.953121i	$0.402149\pi$
$524$	0	0
$525$	1.00000	0.0436436
$526$	0	0
$527$	−3.08005	−0.134169
$528$	0	0
$529$	15.2572	0.663356
$530$	0	0
$531$	3.10519	0.134754
$532$	0	0
$533$	19.7313	0.854656
$534$	0	0
$535$	12.7604	0.551682
$536$	0	0
$537$	−7.27424	−0.313906
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	11.0428	0.474766	0.237383	−	0.971416i	$-0.423710\pi$
$541$	11.0428	0.474766	0.237383	+	0.971416i	$0.423710\pi$
$542$	0	0
$543$	11.6246	0.498861
$544$	0	0
$545$	6.47811	0.277492
$546$	0	0
$547$	32.9960	1.41081	0.705403	−	0.708806i	$-0.250766\pi$
$547$	32.9960	1.41081	0.705403	+	0.708806i	$0.250766\pi$
$548$	0	0
$549$	10.9814	0.468673
$550$	0	0
$551$	9.53142	0.406052
$552$	0	0
$553$	7.36238	0.313080
$554$	0	0
$555$	−2.47811	−0.105190
$556$	0	0
$557$	−0.921539	−0.0390469	−0.0195234	−	0.999809i	$-0.506215\pi$
$557$	−0.921539	−0.0390469	−0.0195234	+	0.999809i	$0.506215\pi$
$558$	0	0
$559$	37.7837	1.59808
$560$	0	0
$561$	−0.699032	−0.0295131
$562$	0	0
$563$	−8.52999	−0.359496	−0.179748	−	0.983713i	$-0.557528\pi$
$563$	−8.52999	−0.359496	−0.179748	+	0.983713i	$0.557528\pi$
$564$	0	0
$565$	−11.6714	−0.491021
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−24.8024	−1.03977	−0.519885	−	0.854236i	$-0.674025\pi$
$569$	−24.8024	−1.03977	−0.519885	+	0.854236i	$0.674025\pi$
$570$	0	0
$571$	37.9432	1.58788	0.793938	−	0.607999i	$-0.208028\pi$
$571$	37.9432	1.58788	0.793938	+	0.607999i	$0.208028\pi$
$572$	0	0
$573$	2.79613	0.116810
$574$	0	0
$575$	6.18524	0.257942
$576$	0	0
$577$	−14.1439	−0.588818	−0.294409	−	0.955680i	$-0.595123\pi$
$577$	−14.1439	−0.588818	−0.294409	+	0.955680i	$0.595123\pi$
$578$	0	0
$579$	25.1828	1.04656
$580$	0	0
$581$	4.62708	0.191964
$582$	0	0
$583$	−4.11329	−0.170355
$584$	0	0
$585$	4.40616	0.182172
$586$	0	0
$587$	−25.8551	−1.06715	−0.533577	−	0.845751i	$-0.679153\pi$
$587$	−25.8551	−1.06715	−0.533577	+	0.845751i	$0.679153\pi$
$588$	0	0
$589$	−24.6009	−1.01366
$590$	0	0
$591$	2.74038	0.112724
$592$	0	0
$593$	−6.37048	−0.261604	−0.130802	−	0.991408i	$-0.541755\pi$
$593$	−6.37048	−0.261604	−0.130802	+	0.991408i	$0.541755\pi$
$594$	0	0
$595$	−0.699032	−0.0286575
$596$	0	0
$597$	15.0947	0.617783
$598$	0	0
$599$	5.18280	0.211764	0.105882	−	0.994379i	$-0.466233\pi$
$599$	5.18280	0.211764	0.105882	+	0.994379i	$0.466233\pi$
$600$	0	0
$601$	−15.9376	−0.650108	−0.325054	−	0.945696i	$-0.605382\pi$
$601$	−15.9376	−0.650108	−0.325054	+	0.945696i	$0.605382\pi$
$602$	0	0
$603$	−9.69659	−0.394876
$604$	0	0
$605$	1.00000	0.0406558
$606$	0	0
$607$	−6.17799	−0.250757	−0.125379	−	0.992109i	$-0.540015\pi$
$607$	−6.17799	−0.250757	−0.125379	+	0.992109i	$0.540015\pi$
$608$	0	0
$609$	−1.70713	−0.0691764
$610$	0	0
$611$	−34.7751	−1.40685
$612$	0	0
$613$	7.62952	0.308153	0.154077	−	0.988059i	$-0.450760\pi$
$613$	7.62952	0.308153	0.154077	+	0.988059i	$0.450760\pi$
$614$	0	0
$615$	4.47811	0.180575
$616$	0	0
$617$	20.1390	0.810766	0.405383	−	0.914147i	$-0.367138\pi$
$617$	20.1390	0.810766	0.405383	+	0.914147i	$0.367138\pi$
$618$	0	0
$619$	−32.3488	−1.30021	−0.650105	−	0.759845i	$-0.725275\pi$
$619$	−32.3488	−1.30021	−0.650105	+	0.759845i	$0.725275\pi$
$620$	0	0
$621$	6.18524	0.248205
$622$	0	0
$623$	7.19334	0.288195
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−5.58330	−0.222976
$628$	0	0
$629$	1.73228	0.0690704
$630$	0	0
$631$	−32.0760	−1.27693	−0.638463	−	0.769652i	$-0.720430\pi$
$631$	−32.0760	−1.27693	−0.638463	+	0.769652i	$0.720430\pi$
$632$	0	0
$633$	−21.9070	−0.870724
$634$	0	0
$635$	14.0057	0.555798
$636$	0	0
$637$	4.40616	0.174578
$638$	0	0
$639$	9.29043	0.367524
$640$	0	0
$641$	−13.8561	−0.547283	−0.273642	−	0.961832i	$-0.588228\pi$
$641$	−13.8561	−0.547283	−0.273642	+	0.961832i	$0.588228\pi$
$642$	0	0
$643$	31.7953	1.25388	0.626942	−	0.779066i	$-0.284306\pi$
$643$	31.7953	1.25388	0.626942	+	0.779066i	$0.284306\pi$
$644$	0	0
$645$	8.57520	0.337648
$646$	0	0
$647$	−20.9562	−0.823874	−0.411937	−	0.911212i	$-0.635148\pi$
$647$	−20.9562	−0.823874	−0.411937	+	0.911212i	$0.635148\pi$
$648$	0	0
$649$	3.10519	0.121889
$650$	0	0
$651$	4.40616	0.172691
$652$	0	0
$653$	−13.3675	−0.523109	−0.261555	−	0.965189i	$-0.584235\pi$
$653$	−13.3675	−0.523109	−0.261555	+	0.965189i	$0.584235\pi$
$654$	0	0
$655$	−8.67230	−0.338855
$656$	0	0
$657$	−6.28233	−0.245097
$658$	0	0
$659$	−27.3827	−1.06668	−0.533338	−	0.845902i	$-0.679063\pi$
$659$	−27.3827	−1.06668	−0.533338	+	0.845902i	$0.679063\pi$
$660$	0	0
$661$	1.41813	0.0551589	0.0275795	−	0.999620i	$-0.491220\pi$
$661$	1.41813	0.0551589	0.0275795	+	0.999620i	$0.491220\pi$
$662$	0	0
$663$	−3.08005	−0.119619
$664$	0	0
$665$	−5.58330	−0.216511
$666$	0	0
$667$	−10.5590	−0.408846
$668$	0	0
$669$	8.57520	0.331537
$670$	0	0
$671$	10.9814	0.423931
$672$	0	0
$673$	21.8618	0.842709	0.421355	−	0.906896i	$-0.361555\pi$
$673$	21.8618	0.842709	0.421355	+	0.906896i	$0.361555\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−45.4076	−1.74516	−0.872578	−	0.488475i	$-0.837553\pi$
$677$	−45.4076	−1.74516	−0.872578	+	0.488475i	$0.837553\pi$
$678$	0	0
$679$	−3.63518	−0.139505
$680$	0	0
$681$	0.965169	0.0369854
$682$	0	0
$683$	6.56238	0.251103	0.125551	−	0.992087i	$-0.459930\pi$
$683$	6.56238	0.251103	0.125551	+	0.992087i	$0.459930\pi$
$684$	0	0
$685$	−12.2104	−0.466535
$686$	0	0
$687$	−7.29043	−0.278147
$688$	0	0
$689$	−18.1238	−0.690463
$690$	0	0
$691$	34.0850	1.29665	0.648327	−	0.761362i	$-0.275469\pi$
$691$	34.0850	1.29665	0.648327	+	0.761362i	$0.275469\pi$
$692$	0	0
$693$	1.00000	0.0379869
$694$	0	0
$695$	−11.2548	−0.426917
$696$	0	0
$697$	−3.13034	−0.118570
$698$	0	0
$699$	−5.20229	−0.196769
$700$	0	0
$701$	−37.8300	−1.42882	−0.714409	−	0.699729i	$-0.753304\pi$
$701$	−37.8300	−1.42882	−0.714409	+	0.699729i	$0.753304\pi$
$702$	0	0
$703$	13.8360	0.521836
$704$	0	0
$705$	−7.89237	−0.297244
$706$	0	0
$707$	−15.1666	−0.570399
$708$	0	0
$709$	−1.16195	−0.0436378	−0.0218189	−	0.999762i	$-0.506946\pi$
$709$	−1.16195	−0.0436378	−0.0218189	+	0.999762i	$0.506946\pi$
$710$	0	0
$711$	7.36238	0.276111
$712$	0	0
$713$	27.2532	1.02064
$714$	0	0
$715$	4.40616	0.164781
$716$	0	0
$717$	−0.575204	−0.0214814
$718$	0	0
$719$	−50.0889	−1.86800	−0.934001	−	0.357271i	$-0.883707\pi$
$719$	−50.0889	−1.86800	−0.934001	+	0.357271i	$0.883707\pi$
$720$	0	0
$721$	−9.38753	−0.349610
$722$	0	0
$723$	0.868074	0.0322840
$724$	0	0
$725$	−1.70713	−0.0634012
$726$	0	0
$727$	−25.1723	−0.933588	−0.466794	−	0.884366i	$-0.654591\pi$
$727$	−25.1723	−0.933588	−0.466794	+	0.884366i	$0.654591\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−5.99434	−0.221709
$732$	0	0
$733$	−7.56890	−0.279564	−0.139782	−	0.990182i	$-0.544640\pi$
$733$	−7.56890	−0.279564	−0.139782	+	0.990182i	$0.544640\pi$
$734$	0	0
$735$	1.00000	0.0368856
$736$	0	0
$737$	−9.69659	−0.357179
$738$	0	0
$739$	−12.2823	−0.451813	−0.225906	−	0.974149i	$-0.572534\pi$
$739$	−12.2823	−0.451813	−0.225906	+	0.974149i	$0.572534\pi$
$740$	0	0
$741$	−24.6009	−0.903738
$742$	0	0
$743$	−26.5040	−0.972339	−0.486170	−	0.873865i	$-0.661606\pi$
$743$	−26.5040	−0.972339	−0.486170	+	0.873865i	$0.661606\pi$
$744$	0	0
$745$	−19.7128	−0.722221
$746$	0	0
$747$	4.62708	0.169296
$748$	0	0
$749$	12.7604	0.466256
$750$	0	0
$751$	16.8883	0.616265	0.308132	−	0.951344i	$-0.400296\pi$
$751$	16.8883	0.616265	0.308132	+	0.951344i	$0.400296\pi$
$752$	0	0
$753$	2.52999	0.0921979
$754$	0	0
$755$	21.0947	0.767713
$756$	0	0
$757$	−51.9741	−1.88903	−0.944515	−	0.328470i	$-0.893467\pi$
$757$	−51.9741	−1.88903	−0.944515	+	0.328470i	$0.893467\pi$
$758$	0	0
$759$	6.18524	0.224510
$760$	0	0
$761$	3.66191	0.132744	0.0663721	−	0.997795i	$-0.478858\pi$
$761$	3.66191	0.132744	0.0663721	+	0.997795i	$0.478858\pi$
$762$	0	0
$763$	6.47811	0.234523
$764$	0	0
$765$	−0.699032	−0.0252735
$766$	0	0
$767$	13.6820	0.494028
$768$	0	0
$769$	22.2517	0.802418	0.401209	−	0.915987i	$-0.368590\pi$
$769$	22.2517	0.802418	0.401209	+	0.915987i	$0.368590\pi$
$770$	0	0
$771$	9.50082	0.342164
$772$	0	0
$773$	−44.4370	−1.59829	−0.799143	−	0.601141i	$-0.794713\pi$
$773$	−44.4370	−1.59829	−0.799143	+	0.601141i	$0.794713\pi$
$774$	0	0
$775$	4.40616	0.158274
$776$	0	0
$777$	−2.47811	−0.0889017
$778$	0	0
$779$	−25.0026	−0.895813
$780$	0	0
$781$	9.29043	0.332438
$782$	0	0
$783$	−1.70713	−0.0610078
$784$	0	0
$785$	4.59140	0.163874
$786$	0	0
$787$	45.6940	1.62882	0.814408	−	0.580293i	$-0.197062\pi$
$787$	45.6940	1.62882	0.814408	+	0.580293i	$0.197062\pi$
$788$	0	0
$789$	11.0282	0.392613
$790$	0	0
$791$	−11.6714	−0.414989
$792$	0	0
$793$	48.3857	1.71823
$794$	0	0
$795$	−4.11329	−0.145883
$796$	0	0
$797$	−38.9562	−1.37990	−0.689950	−	0.723857i	$-0.742368\pi$
$797$	−38.9562	−1.37990	−0.689950	+	0.723857i	$0.742368\pi$
$798$	0	0
$799$	5.51702	0.195178
$800$	0	0
$801$	7.19334	0.254164
$802$	0	0
$803$	−6.28233	−0.221699
$804$	0	0
$805$	6.18524	0.218001
$806$	0	0
$807$	10.3599	0.364687
$808$	0	0
$809$	6.47652	0.227702	0.113851	−	0.993498i	$-0.463681\pi$
$809$	6.47652	0.227702	0.113851	+	0.993498i	$0.463681\pi$
$810$	0	0
$811$	45.4651	1.59650	0.798248	−	0.602328i	$-0.205760\pi$
$811$	45.4651	1.59650	0.798248	+	0.602328i	$0.205760\pi$
$812$	0	0
$813$	3.37292	0.118293
$814$	0	0
$815$	−0.406162	−0.0142272
$816$	0	0
$817$	−47.8780	−1.67504
$818$	0	0
$819$	4.40616	0.153964
$820$	0	0
$821$	−27.4757	−0.958908	−0.479454	−	0.877567i	$-0.659165\pi$
$821$	−27.4757	−0.958908	−0.479454	+	0.877567i	$0.659165\pi$
$822$	0	0
$823$	−24.8995	−0.867941	−0.433970	−	0.900927i	$-0.642888\pi$
$823$	−24.8995	−0.867941	−0.433970	+	0.900927i	$0.642888\pi$
$824$	0	0
$825$	1.00000	0.0348155
$826$	0	0
$827$	30.8843	1.07395	0.536976	−	0.843598i	$-0.319567\pi$
$827$	30.8843	1.07395	0.536976	+	0.843598i	$0.319567\pi$
$828$	0	0
$829$	−56.1179	−1.94906	−0.974528	−	0.224264i	$-0.928002\pi$
$829$	−56.1179	−1.94906	−0.974528	+	0.224264i	$0.928002\pi$
$830$	0	0
$831$	−2.97242	−0.103112
$832$	0	0
$833$	−0.699032	−0.0242200
$834$	0	0
$835$	5.48621	0.189858
$836$	0	0
$837$	4.40616	0.152299
$838$	0	0
$839$	0.607015	0.0209565	0.0104782	−	0.999945i	$-0.496665\pi$
$839$	0.607015	0.0209565	0.0104782	+	0.999945i	$0.496665\pi$
$840$	0	0
$841$	−26.0857	−0.899507
$842$	0	0
$843$	11.2386	0.387076
$844$	0	0
$845$	6.41426	0.220657
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−7.50082	−0.257427
$850$	0	0
$851$	−15.3277	−0.525427
$852$	0	0
$853$	11.1844	0.382946	0.191473	−	0.981498i	$-0.438674\pi$
$853$	11.1844	0.382946	0.191473	+	0.981498i	$0.438674\pi$
$854$	0	0
$855$	−5.58330	−0.190945
$856$	0	0
$857$	−5.25904	−0.179645	−0.0898227	−	0.995958i	$-0.528630\pi$
$857$	−5.25904	−0.179645	−0.0898227	+	0.995958i	$0.528630\pi$
$858$	0	0
$859$	−19.7670	−0.674440	−0.337220	−	0.941426i	$-0.609487\pi$
$859$	−19.7670	−0.674440	−0.337220	+	0.941426i	$0.609487\pi$
$860$	0	0
$861$	4.47811	0.152614
$862$	0	0
$863$	−3.80989	−0.129690	−0.0648450	−	0.997895i	$-0.520655\pi$
$863$	−3.80989	−0.129690	−0.0648450	+	0.997895i	$0.520655\pi$
$864$	0	0
$865$	12.2104	0.415166
$866$	0	0
$867$	−16.5114	−0.560755
$868$	0	0
$869$	7.36238	0.249752
$870$	0	0
$871$	−42.7248	−1.44767
$872$	0	0
$873$	−3.63518	−0.123032
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	32.5362	1.09867	0.549335	−	0.835602i	$-0.314881\pi$
$877$	32.5362	1.09867	0.549335	+	0.835602i	$0.314881\pi$
$878$	0	0
$879$	−13.8494	−0.467130
$880$	0	0
$881$	12.5379	0.422414	0.211207	−	0.977441i	$-0.432261\pi$
$881$	12.5379	0.422414	0.211207	+	0.977441i	$0.432261\pi$
$882$	0	0
$883$	−15.9070	−0.535313	−0.267656	−	0.963514i	$-0.586249\pi$
$883$	−15.9070	−0.535313	−0.267656	+	0.963514i	$0.586249\pi$
$884$	0	0
$885$	3.10519	0.104380
$886$	0	0
$887$	−44.2757	−1.48663	−0.743316	−	0.668941i	$-0.766748\pi$
$887$	−44.2757	−1.48663	−0.743316	+	0.668941i	$0.766748\pi$
$888$	0	0
$889$	14.0057	0.469735
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	44.0655	1.47460
$894$	0	0
$895$	−7.27424	−0.243151
$896$	0	0
$897$	27.2532	0.909957
$898$	0	0
$899$	−7.52189	−0.250869
$900$	0	0
$901$	2.87532	0.0957908
$902$	0	0
$903$	8.57520	0.285365
$904$	0	0
$905$	11.6246	0.386416
$906$	0	0
$907$	13.4739	0.447393	0.223696	−	0.974659i	$-0.428188\pi$
$907$	13.4739	0.447393	0.223696	+	0.974659i	$0.428188\pi$
$908$	0	0
$909$	−15.1666	−0.503045
$910$	0	0
$911$	1.69601	0.0561914	0.0280957	−	0.999605i	$-0.491056\pi$
$911$	1.69601	0.0561914	0.0280957	+	0.999605i	$0.491056\pi$
$912$	0	0
$913$	4.62708	0.153134
$914$	0	0
$915$	10.9814	0.363033
$916$	0	0
$917$	−8.67230	−0.286385
$918$	0	0
$919$	−36.5300	−1.20501	−0.602507	−	0.798114i	$-0.705831\pi$
$919$	−36.5300	−1.20501	−0.602507	+	0.798114i	$0.705831\pi$
$920$	0	0
$921$	25.1828	0.829802
$922$	0	0
$923$	40.9351	1.34740
$924$	0	0
$925$	−2.47811	−0.0814797
$926$	0	0
$927$	−9.38753	−0.308327
$928$	0	0
$929$	−45.9076	−1.50618	−0.753089	−	0.657918i	$-0.771437\pi$
$929$	−45.9076	−1.50618	−0.753089	+	0.657918i	$0.771437\pi$
$930$	0	0
$931$	−5.58330	−0.182985
$932$	0	0
$933$	−33.1109	−1.08400
$934$	0	0
$935$	−0.699032	−0.0228608
$936$	0	0
$937$	−44.1374	−1.44191	−0.720954	−	0.692983i	$-0.756296\pi$
$937$	−44.1374	−1.44191	−0.720954	+	0.692983i	$0.756296\pi$
$938$	0	0
$939$	8.36482	0.272976
$940$	0	0
$941$	−37.0638	−1.20825	−0.604123	−	0.796891i	$-0.706477\pi$
$941$	−37.0638	−1.20825	−0.604123	+	0.796891i	$0.706477\pi$
$942$	0	0
$943$	27.6982	0.901977
$944$	0	0
$945$	1.00000	0.0325300
$946$	0	0
$947$	17.9376	0.582893	0.291447	−	0.956587i	$-0.405863\pi$
$947$	17.9376	0.582893	0.291447	+	0.956587i	$0.405863\pi$
$948$	0	0
$949$	−27.6810	−0.898563
$950$	0	0
$951$	6.37048	0.206577
$952$	0	0
$953$	33.6917	1.09138	0.545691	−	0.837987i	$-0.316267\pi$
$953$	33.6917	1.09138	0.545691	+	0.837987i	$0.316267\pi$
$954$	0	0
$955$	2.79613	0.0904805
$956$	0	0
$957$	−1.70713	−0.0551837
$958$	0	0
$959$	−12.2104	−0.394294
$960$	0	0
$961$	−11.5857	−0.373734
$962$	0	0
$963$	12.7604	0.411200
$964$	0	0
$965$	25.1828	0.810663
$966$	0	0
$967$	55.9684	1.79982	0.899911	−	0.436073i	$-0.143631\pi$
$967$	55.9684	1.79982	0.899911	+	0.436073i	$0.143631\pi$
$968$	0	0
$969$	3.90291	0.125379
$970$	0	0
$971$	−5.72820	−0.183827	−0.0919134	−	0.995767i	$-0.529298\pi$
$971$	−5.72820	−0.183827	−0.0919134	+	0.995767i	$0.529298\pi$
$972$	0	0
$973$	−11.2548	−0.360811
$974$	0	0
$975$	4.40616	0.141110
$976$	0	0
$977$	−1.10848	−0.0354635	−0.0177317	−	0.999843i	$-0.505644\pi$
$977$	−1.10848	−0.0354635	−0.0177317	+	0.999843i	$0.505644\pi$
$978$	0	0
$979$	7.19334	0.229900
$980$	0	0
$981$	6.47811	0.206830
$982$	0	0
$983$	−6.81720	−0.217435	−0.108717	−	0.994073i	$-0.534674\pi$
$983$	−6.81720	−0.217435	−0.108717	+	0.994073i	$0.534674\pi$
$984$	0	0
$985$	2.74038	0.0873156
$986$	0	0
$987$	−7.89237	−0.251217
$988$	0	0
$989$	53.0397	1.68656
$990$	0	0
$991$	−59.0063	−1.87440	−0.937198	−	0.348797i	$-0.886590\pi$
$991$	−59.0063	−1.87440	−0.937198	+	0.348797i	$0.886590\pi$
$992$	0	0
$993$	−13.8656	−0.440013
$994$	0	0
$995$	15.0947	0.478533
$996$	0	0
$997$	−2.44730	−0.0775068	−0.0387534	−	0.999249i	$-0.512339\pi$
$997$	−2.44730	−0.0775068	−0.0387534	+	0.999249i	$0.512339\pi$
$998$	0	0
$999$	−2.47811	−0.0784039

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4620.2.a.x.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4620.2.a.x.1.3	✓	4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4620.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} +4.40616 q^{13} +1.00000 q^{15} -0.699032 q^{17} -5.58330 q^{19} +1.00000 q^{21} +6.18524 q^{23} +1.00000 q^{25} +1.00000 q^{27} -1.70713 q^{29} +4.40616 q^{31} +1.00000 q^{33} +1.00000 q^{35} -2.47811 q^{37} +4.40616 q^{39} +4.47811 q^{41} +8.57520 q^{43} +1.00000 q^{45} -7.89237 q^{47} +1.00000 q^{49} -0.699032 q^{51} -4.11329 q^{53} +1.00000 q^{55} -5.58330 q^{57} +3.10519 q^{59} +10.9814 q^{61} +1.00000 q^{63} +4.40616 q^{65} -9.69659 q^{67} +6.18524 q^{69} +9.29043 q^{71} -6.28233 q^{73} +1.00000 q^{75} +1.00000 q^{77} +7.36238 q^{79} +1.00000 q^{81} +4.62708 q^{83} -0.699032 q^{85} -1.70713 q^{87} +7.19334 q^{89} +4.40616 q^{91} +4.40616 q^{93} -5.58330 q^{95} -3.63518 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 4 * q^5 + 4 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} + 4 q^{5} + 4 q^{7} + 4 q^{9} + 4 q^{11} + 4 q^{13} + 4 q^{15} + 2 q^{17} + 8 q^{19} + 4 q^{21} + 4 q^{23} + 4 q^{25} + 4 q^{27} + 2 q^{29} + 4 q^{31} + 4 q^{33} + 4 q^{35} + 2 q^{37} + 4 q^{39} + 6 q^{41} + 8 q^{43} + 4 q^{45} - 2 q^{47} + 4 q^{49} + 2 q^{51} + 6 q^{53} + 4 q^{55} + 8 q^{57} - 6 q^{59} + 4 q^{61} + 4 q^{63} + 4 q^{65} + 14 q^{67} + 4 q^{69} - 2 q^{71} + 10 q^{73} + 4 q^{75} + 4 q^{77} - 8 q^{79} + 4 q^{81} + 12 q^{83} + 2 q^{85} + 2 q^{87} + 4 q^{89} + 4 q^{91} + 4 q^{93} + 8 q^{95} - 4 q^{97} + 4 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^5 + 4 * q^7 + 4 * q^9 + 4 * q^11 + 4 * q^13 + 4 * q^15 + 2 * q^17 + 8 * q^19 + 4 * q^21 + 4 * q^23 + 4 * q^25 + 4 * q^27 + 2 * q^29 + 4 * q^31 + 4 * q^33 + 4 * q^35 + 2 * q^37 + 4 * q^39 + 6 * q^41 + 8 * q^43 + 4 * q^45 - 2 * q^47 + 4 * q^49 + 2 * q^51 + 6 * q^53 + 4 * q^55 + 8 * q^57 - 6 * q^59 + 4 * q^61 + 4 * q^63 + 4 * q^65 + 14 * q^67 + 4 * q^69 - 2 * q^71 + 10 * q^73 + 4 * q^75 + 4 * q^77 - 8 * q^79 + 4 * q^81 + 12 * q^83 + 2 * q^85 + 2 * q^87 + 4 * q^89 + 4 * q^91 + 4 * q^93 + 8 * q^95 - 4 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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