LMFDB - Embedded newform 8624.2.a.cp.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8624,2,Mod(1,8624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8624, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8624.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8624 = 2^{4} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8624.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$68.8629867032$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.257.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 3 $ x^3 - x^2 - 4*x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 308)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.19869$ of defining polynomial
Character	$\chi$	$=$	8624.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.83424 q^{3} +1.19869 q^{5} +5.03293 q^{9} +O(q^{10})$	$q+2.83424 q^{3} +1.19869 q^{5} +5.03293 q^{9} -1.00000 q^{11} +0.834243 q^{13} +3.39738 q^{15} -4.03293 q^{17} +6.76183 q^{19} +0.0933442 q^{23} -3.56314 q^{25} +5.76183 q^{27} +3.46980 q^{29} -2.59607 q^{31} -2.83424 q^{33} +10.3974 q^{37} +2.36445 q^{39} +7.62901 q^{41} -0.894653 q^{43} +6.03293 q^{45} +0.906656 q^{47} -11.4303 q^{51} -3.16576 q^{53} -1.19869 q^{55} +19.1647 q^{57} +12.6290 q^{59} -10.0659 q^{61} +1.00000 q^{65} -8.46325 q^{67} +0.264560 q^{69} +13.7673 q^{71} +13.1712 q^{73} -10.0988 q^{75} +14.3304 q^{79} +1.23163 q^{81} +4.69596 q^{83} -4.83424 q^{85} +9.83424 q^{87} -14.5961 q^{89} -7.35790 q^{93} +8.10535 q^{95} -12.9001 q^{97} -5.03293 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 2 q^{5} + 4 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 - 2 * q^5 + 4 * q^9	$ 3 q + 3 q^{3} - 2 q^{5} + 4 q^{9} - 3 q^{11} - 3 q^{13} - q^{15} - q^{17} + 9 q^{19} - 5 q^{25} + 6 q^{27} + 5 q^{29} + 9 q^{31} - 3 q^{33} + 20 q^{37} + 7 q^{39} - 5 q^{41} - 8 q^{43} + 7 q^{45} + 3 q^{47} - 12 q^{51} - 15 q^{53} + 2 q^{55} + 2 q^{57} + 10 q^{59} - 8 q^{61} + 3 q^{65} + 8 q^{67} - 27 q^{69} - 3 q^{71} + 12 q^{73} + 3 q^{75} - 7 q^{79} - 13 q^{81} + 25 q^{83} - 9 q^{85} + 24 q^{87} - 27 q^{89} + 6 q^{93} + 19 q^{95} - 11 q^{97} - 4 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 2 * q^5 + 4 * q^9 - 3 * q^11 - 3 * q^13 - q^15 - q^17 + 9 * q^19 - 5 * q^25 + 6 * q^27 + 5 * q^29 + 9 * q^31 - 3 * q^33 + 20 * q^37 + 7 * q^39 - 5 * q^41 - 8 * q^43 + 7 * q^45 + 3 * q^47 - 12 * q^51 - 15 * q^53 + 2 * q^55 + 2 * q^57 + 10 * q^59 - 8 * q^61 + 3 * q^65 + 8 * q^67 - 27 * q^69 - 3 * q^71 + 12 * q^73 + 3 * q^75 - 7 * q^79 - 13 * q^81 + 25 * q^83 - 9 * q^85 + 24 * q^87 - 27 * q^89 + 6 * q^93 + 19 * q^95 - 11 * 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
$3$	2.83424	1.63635	0.818176	−	0.574969i	$-0.194986\pi$
$3$	2.83424	1.63635	0.818176	+	0.574969i	$0.194986\pi$
$4$	0	0
$5$	1.19869	0.536071	0.268036	−	0.963409i	$-0.413626\pi$
$5$	1.19869	0.536071	0.268036	+	0.963409i	$0.413626\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	5.03293	1.67764
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	0.834243	0.231377	0.115689	−	0.993286i	$-0.463092\pi$
$13$	0.834243	0.231377	0.115689	+	0.993286i	$0.463092\pi$
$14$	0	0
$15$	3.39738	0.877200
$16$	0	0
$17$	−4.03293	−0.978130	−0.489065	−	0.872247i	$-0.662662\pi$
$17$	−4.03293	−0.978130	−0.489065	+	0.872247i	$0.662662\pi$
$18$	0	0
$19$	6.76183	1.55127	0.775635	−	0.631182i	$-0.217430\pi$
$19$	6.76183	1.55127	0.775635	+	0.631182i	$0.217430\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.0933442	0.0194636	0.00973180	−	0.999953i	$-0.496902\pi$
$23$	0.0933442	0.0194636	0.00973180	+	0.999953i	$0.496902\pi$
$24$	0	0
$25$	−3.56314	−0.712628
$26$	0	0
$27$	5.76183	1.10886
$28$	0	0
$29$	3.46980	0.644325	0.322162	−	0.946684i	$-0.395590\pi$
$29$	3.46980	0.644325	0.322162	+	0.946684i	$0.395590\pi$
$30$	0	0
$31$	−2.59607	−0.466269	−0.233134	−	0.972445i	$-0.574898\pi$
$31$	−2.59607	−0.466269	−0.233134	+	0.972445i	$0.574898\pi$
$32$	0	0
$33$	−2.83424	−0.493378
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10.3974	1.70932	0.854660	−	0.519189i	$-0.173766\pi$
$37$	10.3974	1.70932	0.854660	+	0.519189i	$0.173766\pi$
$38$	0	0
$39$	2.36445	0.378615
$40$	0	0
$41$	7.62901	1.19145	0.595725	−	0.803188i	$-0.296865\pi$
$41$	7.62901	1.19145	0.595725	+	0.803188i	$0.296865\pi$
$42$	0	0
$43$	−0.894653	−0.136433	−0.0682166	−	0.997671i	$-0.521731\pi$
$43$	−0.894653	−0.136433	−0.0682166	+	0.997671i	$0.521731\pi$
$44$	0	0
$45$	6.03293	0.899337
$46$	0	0
$47$	0.906656	0.132249	0.0661247	−	0.997811i	$-0.478936\pi$
$47$	0.906656	0.132249	0.0661247	+	0.997811i	$0.478936\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−11.4303	−1.60056
$52$	0	0
$53$	−3.16576	−0.434850	−0.217425	−	0.976077i	$-0.569766\pi$
$53$	−3.16576	−0.434850	−0.217425	+	0.976077i	$0.569766\pi$
$54$	0	0
$55$	−1.19869	−0.161631
$56$	0	0
$57$	19.1647	2.53842
$58$	0	0
$59$	12.6290	1.64416	0.822078	−	0.569374i	$-0.192814\pi$
$59$	12.6290	1.64416	0.822078	+	0.569374i	$0.192814\pi$
$60$	0	0
$61$	−10.0659	−1.28880	−0.644401	−	0.764688i	$-0.722893\pi$
$61$	−10.0659	−1.28880	−0.644401	+	0.764688i	$0.722893\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	−8.46325	−1.03395	−0.516975	−	0.856000i	$-0.672942\pi$
$67$	−8.46325	−1.03395	−0.516975	+	0.856000i	$0.672942\pi$
$68$	0	0
$69$	0.264560	0.0318493
$70$	0	0
$71$	13.7673	1.63388	0.816938	−	0.576725i	$-0.195670\pi$
$71$	13.7673	1.63388	0.816938	+	0.576725i	$0.195670\pi$
$72$	0	0
$73$	13.1712	1.54157	0.770787	−	0.637093i	$-0.219863\pi$
$73$	13.1712	1.54157	0.770787	+	0.637093i	$0.219863\pi$
$74$	0	0
$75$	−10.0988	−1.16611
$76$	0	0
$77$	0	0
$78$	0	0
$79$	14.3304	1.61230	0.806150	−	0.591712i	$-0.201548\pi$
$79$	14.3304	1.61230	0.806150	+	0.591712i	$0.201548\pi$
$80$	0	0
$81$	1.23163	0.136847
$82$	0	0
$83$	4.69596	0.515449	0.257724	−	0.966218i	$-0.417027\pi$
$83$	4.69596	0.515449	0.257724	+	0.966218i	$0.417027\pi$
$84$	0	0
$85$	−4.83424	−0.524347
$86$	0	0
$87$	9.83424	1.05434
$88$	0	0
$89$	−14.5961	−1.54718	−0.773590	−	0.633686i	$-0.781541\pi$
$89$	−14.5961	−1.54718	−0.773590	+	0.633686i	$0.781541\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−7.35790	−0.762979
$94$	0	0
$95$	8.10535	0.831591
$96$	0	0
$97$	−12.9001	−1.30981	−0.654904	−	0.755712i	$-0.727291\pi$
$97$	−12.9001	−1.30981	−0.654904	+	0.755712i	$0.727291\pi$
$98$	0	0
$99$	−5.03293	−0.505829
$100$	0	0
$101$	4.23817	0.421714	0.210857	−	0.977517i	$-0.432375\pi$
$101$	4.23817	0.421714	0.210857	+	0.977517i	$0.432375\pi$
$102$	0	0
$103$	11.0329	1.08711	0.543554	−	0.839374i	$-0.317078\pi$
$103$	11.0329	1.08711	0.543554	+	0.839374i	$0.317078\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−0.656483	−0.0634647	−0.0317323	−	0.999496i	$-0.510102\pi$
$107$	−0.656483	−0.0634647	−0.0317323	+	0.999496i	$0.510102\pi$
$108$	0	0
$109$	2.32497	0.222692	0.111346	−	0.993782i	$-0.464484\pi$
$109$	2.32497	0.222692	0.111346	+	0.993782i	$0.464484\pi$
$110$	0	0
$111$	29.4687	2.79705
$112$	0	0
$113$	19.7673	1.85955	0.929775	−	0.368128i	$-0.120001\pi$
$113$	19.7673	1.85955	0.929775	+	0.368128i	$0.120001\pi$
$114$	0	0
$115$	0.111891	0.0104339
$116$	0	0
$117$	4.19869	0.388169
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	21.6225	1.94963
$124$	0	0
$125$	−10.2646	−0.918090
$126$	0	0
$127$	14.7738	1.31097	0.655483	−	0.755210i	$-0.272465\pi$
$127$	14.7738	1.31097	0.655483	+	0.755210i	$0.272465\pi$
$128$	0	0
$129$	−2.53566	−0.223253
$130$	0	0
$131$	−9.20415	−0.804170	−0.402085	−	0.915602i	$-0.631714\pi$
$131$	−9.20415	−0.804170	−0.402085	+	0.915602i	$0.631714\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	6.90666	0.594430
$136$	0	0
$137$	−20.2975	−1.73413	−0.867066	−	0.498193i	$-0.833997\pi$
$137$	−20.2975	−1.73413	−0.867066	+	0.498193i	$0.833997\pi$
$138$	0	0
$139$	6.27110	0.531908	0.265954	−	0.963986i	$-0.414313\pi$
$139$	6.27110	0.531908	0.265954	+	0.963986i	$0.414313\pi$
$140$	0	0
$141$	2.56968	0.216406
$142$	0	0
$143$	−0.834243	−0.0697629
$144$	0	0
$145$	4.15921	0.345404
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−7.21069	−0.590723	−0.295362	−	0.955386i	$-0.595440\pi$
$149$	−7.21069	−0.590723	−0.295362	+	0.955386i	$0.595440\pi$
$150$	0	0
$151$	4.03948	0.328728	0.164364	−	0.986400i	$-0.447443\pi$
$151$	4.03948	0.328728	0.164364	+	0.986400i	$0.447443\pi$
$152$	0	0
$153$	−20.2975	−1.64096
$154$	0	0
$155$	−3.11189	−0.249953
$156$	0	0
$157$	−5.58407	−0.445657	−0.222829	−	0.974858i	$-0.571529\pi$
$157$	−5.58407	−0.445657	−0.222829	+	0.974858i	$0.571529\pi$
$158$	0	0
$159$	−8.97252	−0.711567
$160$	0	0
$161$	0	0
$162$	0	0
$163$	9.00000	0.704934	0.352467	−	0.935824i	$-0.385343\pi$
$163$	9.00000	0.704934	0.352467	+	0.935824i	$0.385343\pi$
$164$	0	0
$165$	−3.39738	−0.264486
$166$	0	0
$167$	−15.6015	−1.20728	−0.603641	−	0.797256i	$-0.706284\pi$
$167$	−15.6015	−1.20728	−0.603641	+	0.797256i	$0.706284\pi$
$168$	0	0
$169$	−12.3040	−0.946464
$170$	0	0
$171$	34.0318	2.60248
$172$	0	0
$173$	22.7553	1.73005	0.865026	−	0.501727i	$-0.167302\pi$
$173$	22.7553	1.73005	0.865026	+	0.501727i	$0.167302\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	35.7937	2.69042
$178$	0	0
$179$	3.17122	0.237028	0.118514	−	0.992952i	$-0.462187\pi$
$179$	3.17122	0.237028	0.118514	+	0.992952i	$0.462187\pi$
$180$	0	0
$181$	−5.56860	−0.413911	−0.206955	−	0.978350i	$-0.566355\pi$
$181$	−5.56860	−0.413911	−0.206955	+	0.978350i	$0.566355\pi$
$182$	0	0
$183$	−28.5291	−2.10893
$184$	0	0
$185$	12.4633	0.916316
$186$	0	0
$187$	4.03293	0.294917
$188$	0	0
$189$	0	0
$190$	0	0
$191$	10.5237	0.761465	0.380733	−	0.924685i	$-0.375672\pi$
$191$	10.5237	0.761465	0.380733	+	0.924685i	$0.375672\pi$
$192$	0	0
$193$	1.02093	0.0734883	0.0367441	−	0.999325i	$-0.488301\pi$
$193$	1.02093	0.0734883	0.0367441	+	0.999325i	$0.488301\pi$
$194$	0	0
$195$	2.83424	0.202964
$196$	0	0
$197$	−22.2910	−1.58816	−0.794082	−	0.607810i	$-0.792048\pi$
$197$	−22.2910	−1.58816	−0.794082	+	0.607810i	$0.792048\pi$
$198$	0	0
$199$	−13.2700	−0.940687	−0.470343	−	0.882483i	$-0.655870\pi$
$199$	−13.2700	−0.940687	−0.470343	+	0.882483i	$0.655870\pi$
$200$	0	0
$201$	−23.9869	−1.69191
$202$	0	0
$203$	0	0
$204$	0	0
$205$	9.14483	0.638702
$206$	0	0
$207$	0.469795	0.0326530
$208$	0	0
$209$	−6.76183	−0.467726
$210$	0	0
$211$	9.12628	0.628279	0.314139	−	0.949377i	$-0.398284\pi$
$211$	9.12628	0.628279	0.314139	+	0.949377i	$0.398284\pi$
$212$	0	0
$213$	39.0198	2.67360
$214$	0	0
$215$	−1.07241	−0.0731379
$216$	0	0
$217$	0	0
$218$	0	0
$219$	37.3304	2.52256
$220$	0	0
$221$	−3.36445	−0.226317
$222$	0	0
$223$	0.139366	0.00933265	0.00466632	−	0.999989i	$-0.498515\pi$
$223$	0.139366	0.00933265	0.00466632	+	0.999989i	$0.498515\pi$
$224$	0	0
$225$	−17.9330	−1.19554
$226$	0	0
$227$	−25.1987	−1.67250	−0.836248	−	0.548352i	$-0.815256\pi$
$227$	−25.1987	−1.67250	−0.836248	+	0.548352i	$0.815256\pi$
$228$	0	0
$229$	1.21069	0.0800049	0.0400025	−	0.999200i	$-0.487263\pi$
$229$	1.21069	0.0800049	0.0400025	+	0.999200i	$0.487263\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	11.3315	0.742352	0.371176	−	0.928563i	$-0.378955\pi$
$233$	11.3315	0.742352	0.371176	+	0.928563i	$0.378955\pi$
$234$	0	0
$235$	1.08680	0.0708950
$236$	0	0
$237$	40.6159	2.63829
$238$	0	0
$239$	11.6805	0.755548	0.377774	−	0.925898i	$-0.376690\pi$
$239$	11.6805	0.755548	0.377774	+	0.925898i	$0.376690\pi$
$240$	0	0
$241$	−8.31058	−0.535332	−0.267666	−	0.963512i	$-0.586252\pi$
$241$	−8.31058	−0.535332	−0.267666	+	0.963512i	$0.586252\pi$
$242$	0	0
$243$	−13.7948	−0.884935
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.64101	0.358929
$248$	0	0
$249$	13.3095	0.843455
$250$	0	0
$251$	−2.03293	−0.128318	−0.0641588	−	0.997940i	$-0.520436\pi$
$251$	−2.03293	−0.128318	−0.0641588	+	0.997940i	$0.520436\pi$
$252$	0	0
$253$	−0.0933442	−0.00586850
$254$	0	0
$255$	−13.7014	−0.858016
$256$	0	0
$257$	−23.6674	−1.47633	−0.738166	−	0.674619i	$-0.764308\pi$
$257$	−23.6674	−1.47633	−0.738166	+	0.674619i	$0.764308\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	17.4633	1.08095
$262$	0	0
$263$	14.2251	0.877156	0.438578	−	0.898693i	$-0.355482\pi$
$263$	14.2251	0.877156	0.438578	+	0.898693i	$0.355482\pi$
$264$	0	0
$265$	−3.79476	−0.233111
$266$	0	0
$267$	−41.3688	−2.53173
$268$	0	0
$269$	−11.9330	−0.727571	−0.363785	−	0.931483i	$-0.618516\pi$
$269$	−11.9330	−0.727571	−0.363785	+	0.931483i	$0.618516\pi$
$270$	0	0
$271$	26.0713	1.58372	0.791860	−	0.610702i	$-0.209113\pi$
$271$	26.0713	1.58372	0.791860	+	0.610702i	$0.209113\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.56314	0.214865
$276$	0	0
$277$	−9.75529	−0.586138	−0.293069	−	0.956091i	$-0.594677\pi$
$277$	−9.75529	−0.586138	−0.293069	+	0.956091i	$0.594677\pi$
$278$	0	0
$279$	−13.0659	−0.782233
$280$	0	0
$281$	−25.1437	−1.49995	−0.749975	−	0.661466i	$-0.769934\pi$
$281$	−25.1437	−1.49995	−0.749975	+	0.661466i	$0.769934\pi$
$282$	0	0
$283$	18.1592	1.07945	0.539727	−	0.841840i	$-0.318528\pi$
$283$	18.1592	1.07945	0.539727	+	0.841840i	$0.318528\pi$
$284$	0	0
$285$	22.9725	1.36077
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−0.735440	−0.0432612
$290$	0	0
$291$	−36.5621	−2.14331
$292$	0	0
$293$	12.0604	0.704577	0.352288	−	0.935892i	$-0.385404\pi$
$293$	12.0604	0.704577	0.352288	+	0.935892i	$0.385404\pi$
$294$	0	0
$295$	15.1383	0.881385
$296$	0	0
$297$	−5.76183	−0.334335
$298$	0	0
$299$	0.0778717	0.00450344
$300$	0	0
$301$	0	0
$302$	0	0
$303$	12.0120	0.690072
$304$	0	0
$305$	−12.0659	−0.690890
$306$	0	0
$307$	−8.20415	−0.468236	−0.234118	−	0.972208i	$-0.575220\pi$
$307$	−8.20415	−0.468236	−0.234118	+	0.972208i	$0.575220\pi$
$308$	0	0
$309$	31.2700	1.77889
$310$	0	0
$311$	9.86172	0.559207	0.279603	−	0.960116i	$-0.409797\pi$
$311$	9.86172	0.559207	0.279603	+	0.960116i	$0.409797\pi$
$312$	0	0
$313$	−1.23817	−0.0699855	−0.0349927	−	0.999388i	$-0.511141\pi$
$313$	−1.23817	−0.0699855	−0.0349927	+	0.999388i	$0.511141\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−30.2844	−1.70094	−0.850471	−	0.526022i	$-0.823683\pi$
$317$	−30.2844	−1.70094	−0.850471	+	0.526022i	$0.823683\pi$
$318$	0	0
$319$	−3.46980	−0.194271
$320$	0	0
$321$	−1.86063	−0.103850
$322$	0	0
$323$	−27.2700	−1.51734
$324$	0	0
$325$	−2.97252	−0.164886
$326$	0	0
$327$	6.58953	0.364402
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−25.3424	−1.39295	−0.696473	−	0.717583i	$-0.745248\pi$
$331$	−25.3424	−1.39295	−0.696473	+	0.717583i	$0.745248\pi$
$332$	0	0
$333$	52.3293	2.86763
$334$	0	0
$335$	−10.1448	−0.554271
$336$	0	0
$337$	−5.35899	−0.291923	−0.145961	−	0.989290i	$-0.546628\pi$
$337$	−5.35899	−0.291923	−0.145961	+	0.989290i	$0.546628\pi$
$338$	0	0
$339$	56.0253	3.04288
$340$	0	0
$341$	2.59607	0.140585
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0.317126	0.0170735
$346$	0	0
$347$	−13.4698	−0.723096	−0.361548	−	0.932353i	$-0.617752\pi$
$347$	−13.4698	−0.723096	−0.361548	+	0.932353i	$0.617752\pi$
$348$	0	0
$349$	26.5477	1.42106	0.710532	−	0.703665i	$-0.248454\pi$
$349$	26.5477	1.42106	0.710532	+	0.703665i	$0.248454\pi$
$350$	0	0
$351$	4.80677	0.256566
$352$	0	0
$353$	19.4423	1.03481	0.517405	−	0.855741i	$-0.326898\pi$
$353$	19.4423	1.03481	0.517405	+	0.855741i	$0.326898\pi$
$354$	0	0
$355$	16.5027	0.875874
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−33.2964	−1.75732	−0.878659	−	0.477450i	$-0.841561\pi$
$359$	−33.2964	−1.75732	−0.878659	+	0.477450i	$0.841561\pi$
$360$	0	0
$361$	26.7224	1.40644
$362$	0	0
$363$	2.83424	0.148759
$364$	0	0
$365$	15.7882	0.826393
$366$	0	0
$367$	6.33043	0.330446	0.165223	−	0.986256i	$-0.447166\pi$
$367$	6.33043	0.330446	0.165223	+	0.986256i	$0.447166\pi$
$368$	0	0
$369$	38.3963	1.99883
$370$	0	0
$371$	0	0
$372$	0	0
$373$	5.92650	0.306863	0.153431	−	0.988159i	$-0.450968\pi$
$373$	5.92650	0.306863	0.153431	+	0.988159i	$0.450968\pi$
$374$	0	0
$375$	−29.0923	−1.50232
$376$	0	0
$377$	2.89465	0.149082
$378$	0	0
$379$	6.11189	0.313947	0.156973	−	0.987603i	$-0.449826\pi$
$379$	6.11189	0.313947	0.156973	+	0.987603i	$0.449826\pi$
$380$	0	0
$381$	41.8726	2.14520
$382$	0	0
$383$	20.3030	1.03743	0.518716	−	0.854946i	$-0.326410\pi$
$383$	20.3030	1.03743	0.518716	+	0.854946i	$0.326410\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4.50273	−0.228887
$388$	0	0
$389$	−17.1383	−0.868945	−0.434473	−	0.900685i	$-0.643065\pi$
$389$	−17.1383	−0.868945	−0.434473	+	0.900685i	$0.643065\pi$
$390$	0	0
$391$	−0.376451	−0.0190379
$392$	0	0
$393$	−26.0868	−1.31591
$394$	0	0
$395$	17.1778	0.864307
$396$	0	0
$397$	−16.2766	−0.816897	−0.408449	−	0.912781i	$-0.633930\pi$
$397$	−16.2766	−0.816897	−0.408449	+	0.912781i	$0.633930\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	7.17122	0.358113	0.179057	−	0.983839i	$-0.442695\pi$
$401$	7.17122	0.358113	0.179057	+	0.983839i	$0.442695\pi$
$402$	0	0
$403$	−2.16576	−0.107884
$404$	0	0
$405$	1.47634	0.0733599
$406$	0	0
$407$	−10.3974	−0.515379
$408$	0	0
$409$	−6.22270	−0.307693	−0.153846	−	0.988095i	$-0.549166\pi$
$409$	−6.22270	−0.307693	−0.153846	+	0.988095i	$0.549166\pi$
$410$	0	0
$411$	−57.5280	−2.83765
$412$	0	0
$413$	0	0
$414$	0	0
$415$	5.62901	0.276317
$416$	0	0
$417$	17.7738	0.870388
$418$	0	0
$419$	−16.0473	−0.783963	−0.391981	−	0.919973i	$-0.628210\pi$
$419$	−16.0473	−0.783963	−0.391981	+	0.919973i	$0.628210\pi$
$420$	0	0
$421$	−9.49619	−0.462816	−0.231408	−	0.972857i	$-0.574333\pi$
$421$	−9.49619	−0.462816	−0.231408	+	0.972857i	$0.574333\pi$
$422$	0	0
$423$	4.56314	0.221867
$424$	0	0
$425$	14.3699	0.697043
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−2.36445	−0.114157
$430$	0	0
$431$	−34.9080	−1.68146	−0.840729	−	0.541457i	$-0.817873\pi$
$431$	−34.9080	−1.68146	−0.840729	+	0.541457i	$0.817873\pi$
$432$	0	0
$433$	14.8737	0.714785	0.357393	−	0.933954i	$-0.383666\pi$
$433$	14.8737	0.714785	0.357393	+	0.933954i	$0.383666\pi$
$434$	0	0
$435$	11.7882	0.565202
$436$	0	0
$437$	0.631178	0.0301933
$438$	0	0
$439$	24.5147	1.17002	0.585012	−	0.811025i	$-0.301090\pi$
$439$	24.5147	1.17002	0.585012	+	0.811025i	$0.301090\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−2.66740	−0.126732	−0.0633660	−	0.997990i	$-0.520184\pi$
$443$	−2.66740	−0.126732	−0.0633660	+	0.997990i	$0.520184\pi$
$444$	0	0
$445$	−17.4962	−0.829399
$446$	0	0
$447$	−20.4369	−0.966630
$448$	0	0
$449$	−9.60808	−0.453433	−0.226717	−	0.973961i	$-0.572799\pi$
$449$	−9.60808	−0.453433	−0.226717	+	0.973961i	$0.572799\pi$
$450$	0	0
$451$	−7.62901	−0.359236
$452$	0	0
$453$	11.4489	0.537915
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−22.1647	−1.03682	−0.518410	−	0.855132i	$-0.673476\pi$
$457$	−22.1647	−1.03682	−0.518410	+	0.855132i	$0.673476\pi$
$458$	0	0
$459$	−23.2371	−1.08461
$460$	0	0
$461$	14.9725	0.697340	0.348670	−	0.937246i	$-0.386633\pi$
$461$	14.9725	0.697340	0.348670	+	0.937246i	$0.386633\pi$
$462$	0	0
$463$	−31.8122	−1.47844	−0.739220	−	0.673464i	$-0.764806\pi$
$463$	−31.8122	−1.47844	−0.739220	+	0.673464i	$0.764806\pi$
$464$	0	0
$465$	−8.81986	−0.409011
$466$	0	0
$467$	−28.1043	−1.30051	−0.650255	−	0.759716i	$-0.725338\pi$
$467$	−28.1043	−1.30051	−0.650255	+	0.759716i	$0.725338\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−15.8266	−0.729252
$472$	0	0
$473$	0.894653	0.0411362
$474$	0	0
$475$	−24.0933	−1.10548
$476$	0	0
$477$	−15.9330	−0.729524
$478$	0	0
$479$	11.2831	0.515538	0.257769	−	0.966207i	$-0.417013\pi$
$479$	11.2831	0.515538	0.257769	+	0.966207i	$0.417013\pi$
$480$	0	0
$481$	8.67395	0.395498
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−15.4633	−0.702150
$486$	0	0
$487$	27.9529	1.26667	0.633333	−	0.773879i	$-0.281686\pi$
$487$	27.9529	1.26667	0.633333	+	0.773879i	$0.281686\pi$
$488$	0	0
$489$	25.5082	1.15352
$490$	0	0
$491$	9.77076	0.440948	0.220474	−	0.975393i	$-0.429240\pi$
$491$	9.77076	0.440948	0.220474	+	0.975393i	$0.429240\pi$
$492$	0	0
$493$	−13.9935	−0.630234
$494$	0	0
$495$	−6.03293	−0.271160
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−31.3119	−1.40171	−0.700856	−	0.713303i	$-0.747199\pi$
$499$	−31.3119	−1.40171	−0.700856	+	0.713303i	$0.747199\pi$
$500$	0	0
$501$	−44.2185	−1.97554
$502$	0	0
$503$	0.779293	0.0347469	0.0173735	−	0.999849i	$-0.494470\pi$
$503$	0.779293	0.0347469	0.0173735	+	0.999849i	$0.494470\pi$
$504$	0	0
$505$	5.08026	0.226068
$506$	0	0
$507$	−34.8726	−1.54875
$508$	0	0
$509$	30.4118	1.34798	0.673989	−	0.738741i	$-0.264579\pi$
$509$	30.4118	1.34798	0.673989	+	0.738741i	$0.264579\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	38.9605	1.72015
$514$	0	0
$515$	13.2251	0.582767
$516$	0	0
$517$	−0.906656	−0.0398747
$518$	0	0
$519$	64.4940	2.83097
$520$	0	0
$521$	37.2545	1.63215	0.816076	−	0.577945i	$-0.196145\pi$
$521$	37.2545	1.63215	0.816076	+	0.577945i	$0.196145\pi$
$522$	0	0
$523$	−14.3754	−0.628591	−0.314295	−	0.949325i	$-0.601768\pi$
$523$	−14.3754	−0.628591	−0.314295	+	0.949325i	$0.601768\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	10.4698	0.456071
$528$	0	0
$529$	−22.9913	−0.999621
$530$	0	0
$531$	63.5610	2.75831
$532$	0	0
$533$	6.36445	0.275675
$534$	0	0
$535$	−0.786921	−0.0340216
$536$	0	0
$537$	8.98800	0.387861
$538$	0	0
$539$	0	0
$540$	0	0
$541$	30.3963	1.30684	0.653419	−	0.756996i	$-0.273334\pi$
$541$	30.3963	1.30684	0.653419	+	0.756996i	$0.273334\pi$
$542$	0	0
$543$	−15.7828	−0.677303
$544$	0	0
$545$	2.78692	0.119379
$546$	0	0
$547$	6.74744	0.288500	0.144250	−	0.989541i	$-0.453923\pi$
$547$	6.74744	0.288500	0.144250	+	0.989541i	$0.453923\pi$
$548$	0	0
$549$	−50.6609	−2.16215
$550$	0	0
$551$	23.4622	0.999522
$552$	0	0
$553$	0	0
$554$	0	0
$555$	35.3239	1.49942
$556$	0	0
$557$	22.5226	0.954312	0.477156	−	0.878819i	$-0.341668\pi$
$557$	22.5226	0.954312	0.477156	+	0.878819i	$0.341668\pi$
$558$	0	0
$559$	−0.746358	−0.0315676
$560$	0	0
$561$	11.4303	0.482588
$562$	0	0
$563$	−16.3490	−0.689027	−0.344514	−	0.938781i	$-0.611956\pi$
$563$	−16.3490	−0.689027	−0.344514	+	0.938781i	$0.611956\pi$
$564$	0	0
$565$	23.6949	0.996851
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−41.1097	−1.72341	−0.861705	−	0.507410i	$-0.830603\pi$
$569$	−41.1097	−1.72341	−0.861705	+	0.507410i	$0.830603\pi$
$570$	0	0
$571$	−41.6225	−1.74185	−0.870923	−	0.491420i	$-0.836478\pi$
$571$	−41.6225	−1.74185	−0.870923	+	0.491420i	$0.836478\pi$
$572$	0	0
$573$	29.8266	1.24602
$574$	0	0
$575$	−0.332598	−0.0138703
$576$	0	0
$577$	−7.43578	−0.309555	−0.154778	−	0.987949i	$-0.549466\pi$
$577$	−7.43578	−0.309555	−0.154778	+	0.987949i	$0.549466\pi$
$578$	0	0
$579$	2.89357	0.120253
$580$	0	0
$581$	0	0
$582$	0	0
$583$	3.16576	0.131112
$584$	0	0
$585$	5.03293	0.208086
$586$	0	0
$587$	−14.5631	−0.601085	−0.300543	−	0.953768i	$-0.597168\pi$
$587$	−14.5631	−0.601085	−0.300543	+	0.953768i	$0.597168\pi$
$588$	0	0
$589$	−17.5542	−0.723309
$590$	0	0
$591$	−63.1780	−2.59879
$592$	0	0
$593$	−19.7882	−0.812605	−0.406302	−	0.913739i	$-0.633182\pi$
$593$	−19.7882	−0.812605	−0.406302	+	0.913739i	$0.633182\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−37.6105	−1.53929
$598$	0	0
$599$	29.0923	1.18868	0.594339	−	0.804215i	$-0.297414\pi$
$599$	29.0923	1.18868	0.594339	+	0.804215i	$0.297414\pi$
$600$	0	0
$601$	−0.680489	−0.0277577	−0.0138789	−	0.999904i	$-0.504418\pi$
$601$	−0.680489	−0.0277577	−0.0138789	+	0.999904i	$0.504418\pi$
$602$	0	0
$603$	−42.5950	−1.73460
$604$	0	0
$605$	1.19869	0.0487337
$606$	0	0
$607$	41.1527	1.67034	0.835168	−	0.549995i	$-0.185371\pi$
$607$	41.1527	1.67034	0.835168	+	0.549995i	$0.185371\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0.756371	0.0305995
$612$	0	0
$613$	−11.9450	−0.482456	−0.241228	−	0.970468i	$-0.577550\pi$
$613$	−11.9450	−0.482456	−0.241228	+	0.970468i	$0.577550\pi$
$614$	0	0
$615$	25.9187	1.04514
$616$	0	0
$617$	20.3952	0.821080	0.410540	−	0.911843i	$-0.365340\pi$
$617$	20.3952	0.821080	0.410540	+	0.911843i	$0.365340\pi$
$618$	0	0
$619$	25.5367	1.02641	0.513204	−	0.858266i	$-0.328458\pi$
$619$	25.5367	1.02641	0.513204	+	0.858266i	$0.328458\pi$
$620$	0	0
$621$	0.537833	0.0215825
$622$	0	0
$623$	0	0
$624$	0	0
$625$	5.51166	0.220466
$626$	0	0
$627$	−19.1647	−0.765363
$628$	0	0
$629$	−41.9320	−1.67194
$630$	0	0
$631$	−11.0504	−0.439909	−0.219955	−	0.975510i	$-0.570591\pi$
$631$	−11.0504	−0.439909	−0.219955	+	0.975510i	$0.570591\pi$
$632$	0	0
$633$	25.8661	1.02808
$634$	0	0
$635$	17.7093	0.702771
$636$	0	0
$637$	0	0
$638$	0	0
$639$	69.2899	2.74106
$640$	0	0
$641$	22.7278	0.897695	0.448847	−	0.893608i	$-0.351835\pi$
$641$	22.7278	0.897695	0.448847	+	0.893608i	$0.351835\pi$
$642$	0	0
$643$	−4.31843	−0.170302	−0.0851510	−	0.996368i	$-0.527137\pi$
$643$	−4.31843	−0.170302	−0.0851510	+	0.996368i	$0.527137\pi$
$644$	0	0
$645$	−3.03948	−0.119679
$646$	0	0
$647$	−49.1241	−1.93127	−0.965634	−	0.259906i	$-0.916309\pi$
$647$	−49.1241	−1.93127	−0.965634	+	0.259906i	$0.916309\pi$
$648$	0	0
$649$	−12.6290	−0.495732
$650$	0	0
$651$	0	0
$652$	0	0
$653$	8.81024	0.344771	0.172386	−	0.985030i	$-0.444852\pi$
$653$	8.81024	0.344771	0.172386	+	0.985030i	$0.444852\pi$
$654$	0	0
$655$	−11.0329	−0.431092
$656$	0	0
$657$	66.2899	2.58621
$658$	0	0
$659$	19.5621	0.762029	0.381015	−	0.924569i	$-0.375575\pi$
$659$	19.5621	0.762029	0.381015	+	0.924569i	$0.375575\pi$
$660$	0	0
$661$	−4.86718	−0.189311	−0.0946556	−	0.995510i	$-0.530175\pi$
$661$	−4.86718	−0.189311	−0.0946556	+	0.995510i	$0.530175\pi$
$662$	0	0
$663$	−9.53566	−0.370335
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.323885	0.0125409
$668$	0	0
$669$	0.394998	0.0152715
$670$	0	0
$671$	10.0659	0.388589
$672$	0	0
$673$	−45.9080	−1.76962	−0.884811	−	0.465950i	$-0.845713\pi$
$673$	−45.9080	−1.76962	−0.884811	+	0.465950i	$0.845713\pi$
$674$	0	0
$675$	−20.5302	−0.790208
$676$	0	0
$677$	12.0460	0.462966	0.231483	−	0.972839i	$-0.425642\pi$
$677$	12.0460	0.462966	0.231483	+	0.972839i	$0.425642\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−71.4192	−2.73679
$682$	0	0
$683$	−7.79784	−0.298376	−0.149188	−	0.988809i	$-0.547666\pi$
$683$	−7.79784	−0.298376	−0.149188	+	0.988809i	$0.547666\pi$
$684$	0	0
$685$	−24.3304	−0.929618
$686$	0	0
$687$	3.43140	0.130916
$688$	0	0
$689$	−2.64101	−0.100615
$690$	0	0
$691$	4.95268	0.188409	0.0942044	−	0.995553i	$-0.469969\pi$
$691$	4.95268	0.188409	0.0942044	+	0.995553i	$0.469969\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	7.51712	0.285141
$696$	0	0
$697$	−30.7673	−1.16539
$698$	0	0
$699$	32.1163	1.21475
$700$	0	0
$701$	11.0275	0.416502	0.208251	−	0.978075i	$-0.433223\pi$
$701$	11.0275	0.416502	0.208251	+	0.978075i	$0.433223\pi$
$702$	0	0
$703$	70.3053	2.65162
$704$	0	0
$705$	3.08026	0.116009
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−39.3599	−1.47819	−0.739096	−	0.673600i	$-0.764747\pi$
$709$	−39.3599	−1.47819	−0.739096	+	0.673600i	$0.764747\pi$
$710$	0	0
$711$	72.1241	2.70487
$712$	0	0
$713$	−0.242328	−0.00907527
$714$	0	0
$715$	−1.00000	−0.0373979
$716$	0	0
$717$	33.1053	1.23634
$718$	0	0
$719$	−37.8726	−1.41241	−0.706206	−	0.708007i	$-0.749595\pi$
$719$	−37.8726	−1.41241	−0.706206	+	0.708007i	$0.749595\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−23.5542	−0.875991
$724$	0	0
$725$	−12.3634	−0.459164
$726$	0	0
$727$	21.3250	0.790899	0.395450	−	0.918488i	$-0.370589\pi$
$727$	21.3250	0.790899	0.395450	+	0.918488i	$0.370589\pi$
$728$	0	0
$729$	−42.7926	−1.58491
$730$	0	0
$731$	3.60808	0.133450
$732$	0	0
$733$	−26.5621	−0.981092	−0.490546	−	0.871415i	$-0.663203\pi$
$733$	−26.5621	−0.981092	−0.490546	+	0.871415i	$0.663203\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.46325	0.311748
$738$	0	0
$739$	−15.8870	−0.584414	−0.292207	−	0.956355i	$-0.594390\pi$
$739$	−15.8870	−0.584414	−0.292207	+	0.956355i	$0.594390\pi$
$740$	0	0
$741$	15.9880	0.587334
$742$	0	0
$743$	−34.9385	−1.28177	−0.640885	−	0.767637i	$-0.721432\pi$
$743$	−34.9385	−1.28177	−0.640885	+	0.767637i	$0.721432\pi$
$744$	0	0
$745$	−8.64340	−0.316670
$746$	0	0
$747$	23.6345	0.864740
$748$	0	0
$749$	0	0
$750$	0	0
$751$	12.9880	0.473939	0.236969	−	0.971517i	$-0.423846\pi$
$751$	12.9880	0.473939	0.236969	+	0.971517i	$0.423846\pi$
$752$	0	0
$753$	−5.76183	−0.209973
$754$	0	0
$755$	4.84209	0.176222
$756$	0	0
$757$	−27.0408	−0.982814	−0.491407	−	0.870930i	$-0.663517\pi$
$757$	−27.0408	−0.982814	−0.491407	+	0.870930i	$0.663517\pi$
$758$	0	0
$759$	−0.264560	−0.00960292
$760$	0	0
$761$	1.17776	0.0426938	0.0213469	−	0.999772i	$-0.493205\pi$
$761$	1.17776	0.0426938	0.0213469	+	0.999772i	$0.493205\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−24.3304	−0.879669
$766$	0	0
$767$	10.5357	0.380421
$768$	0	0
$769$	32.8530	1.18471	0.592355	−	0.805677i	$-0.298198\pi$
$769$	32.8530	1.18471	0.592355	+	0.805677i	$0.298198\pi$
$770$	0	0
$771$	−67.0792	−2.41580
$772$	0	0
$773$	−10.1668	−0.365676	−0.182838	−	0.983143i	$-0.558528\pi$
$773$	−10.1668	−0.365676	−0.182838	+	0.983143i	$0.558528\pi$
$774$	0	0
$775$	9.25017	0.332276
$776$	0	0
$777$	0	0
$778$	0	0
$779$	51.5861	1.84826
$780$	0	0
$781$	−13.7673	−0.492632
$782$	0	0
$783$	19.9924	0.714469
$784$	0	0
$785$	−6.69358	−0.238904
$786$	0	0
$787$	−8.63447	−0.307786	−0.153893	−	0.988088i	$-0.549181\pi$
$787$	−8.63447	−0.307786	−0.153893	+	0.988088i	$0.549181\pi$
$788$	0	0
$789$	40.3173	1.43534
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−8.39738	−0.298200
$794$	0	0
$795$	−10.7553	−0.381451
$796$	0	0
$797$	−20.9671	−0.742692	−0.371346	−	0.928495i	$-0.621104\pi$
$797$	−20.9671	−0.742692	−0.371346	+	0.928495i	$0.621104\pi$
$798$	0	0
$799$	−3.65648	−0.129357
$800$	0	0
$801$	−73.4611	−2.59562
$802$	0	0
$803$	−13.1712	−0.464802
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−33.8212	−1.19056
$808$	0	0
$809$	14.9629	0.526068	0.263034	−	0.964787i	$-0.415277\pi$
$809$	14.9629	0.526068	0.263034	+	0.964787i	$0.415277\pi$
$810$	0	0
$811$	53.1965	1.86798	0.933991	−	0.357296i	$-0.116301\pi$
$811$	53.1965	1.86798	0.933991	+	0.357296i	$0.116301\pi$
$812$	0	0
$813$	73.8925	2.59152
$814$	0	0
$815$	10.7882	0.377895
$816$	0	0
$817$	−6.04949	−0.211645
$818$	0	0
$819$	0	0
$820$	0	0
$821$	35.8726	1.25196	0.625982	−	0.779838i	$-0.284698\pi$
$821$	35.8726	1.25196	0.625982	+	0.779838i	$0.284698\pi$
$822$	0	0
$823$	12.6859	0.442204	0.221102	−	0.975251i	$-0.429035\pi$
$823$	12.6859	0.442204	0.221102	+	0.975251i	$0.429035\pi$
$824$	0	0
$825$	10.0988	0.351595
$826$	0	0
$827$	2.92412	0.101682	0.0508408	−	0.998707i	$-0.483810\pi$
$827$	2.92412	0.101682	0.0508408	+	0.998707i	$0.483810\pi$
$828$	0	0
$829$	−11.7793	−0.409112	−0.204556	−	0.978855i	$-0.565575\pi$
$829$	−11.7793	−0.409112	−0.204556	+	0.978855i	$0.565575\pi$
$830$	0	0
$831$	−27.6489	−0.959128
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−18.7014	−0.647189
$836$	0	0
$837$	−14.9581	−0.517029
$838$	0	0
$839$	18.5117	0.639093	0.319547	−	0.947571i	$-0.396469\pi$
$839$	18.5117	0.639093	0.319547	+	0.947571i	$0.396469\pi$
$840$	0	0
$841$	−16.9605	−0.584846
$842$	0	0
$843$	−71.2635	−2.45444
$844$	0	0
$845$	−14.7487	−0.507372
$846$	0	0
$847$	0	0
$848$	0	0
$849$	51.4676	1.76636
$850$	0	0
$851$	0.970535	0.0332695
$852$	0	0
$853$	−37.0342	−1.26803	−0.634014	−	0.773322i	$-0.718594\pi$
$853$	−37.0342	−1.26803	−0.634014	+	0.773322i	$0.718594\pi$
$854$	0	0
$855$	40.7937	1.39511
$856$	0	0
$857$	−4.23817	−0.144773	−0.0723866	−	0.997377i	$-0.523062\pi$
$857$	−4.23817	−0.144773	−0.0723866	+	0.997377i	$0.523062\pi$
$858$	0	0
$859$	3.41831	0.116631	0.0583157	−	0.998298i	$-0.481427\pi$
$859$	3.41831	0.116631	0.0583157	+	0.998298i	$0.481427\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−38.7202	−1.31805	−0.659025	−	0.752121i	$-0.729031\pi$
$863$	−38.7202	−1.31805	−0.659025	+	0.752121i	$0.729031\pi$
$864$	0	0
$865$	27.2766	0.927431
$866$	0	0
$867$	−2.08442	−0.0707905
$868$	0	0
$869$	−14.3304	−0.486127
$870$	0	0
$871$	−7.06041	−0.239233
$872$	0	0
$873$	−64.9254	−2.19739
$874$	0	0
$875$	0	0
$876$	0	0
$877$	46.7607	1.57900	0.789499	−	0.613752i	$-0.210340\pi$
$877$	46.7607	1.57900	0.789499	+	0.613752i	$0.210340\pi$
$878$	0	0
$879$	34.1821	1.15293
$880$	0	0
$881$	−19.6050	−0.660509	−0.330255	−	0.943892i	$-0.607135\pi$
$881$	−19.6050	−0.660509	−0.330255	+	0.943892i	$0.607135\pi$
$882$	0	0
$883$	−10.0395	−0.337855	−0.168928	−	0.985628i	$-0.554030\pi$
$883$	−10.0395	−0.337855	−0.168928	+	0.985628i	$0.554030\pi$
$884$	0	0
$885$	42.9056	1.44225
$886$	0	0
$887$	−46.1934	−1.55102	−0.775512	−	0.631333i	$-0.782508\pi$
$887$	−46.1934	−1.55102	−0.775512	+	0.631333i	$0.782508\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−1.23163	−0.0412610
$892$	0	0
$893$	6.13065	0.205154
$894$	0	0
$895$	3.80131	0.127064
$896$	0	0
$897$	0.220707	0.00736921
$898$	0	0
$899$	−9.00784	−0.300428
$900$	0	0
$901$	12.7673	0.425340
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−6.67503	−0.221886
$906$	0	0
$907$	−24.2580	−0.805474	−0.402737	−	0.915316i	$-0.631941\pi$
$907$	−24.2580	−0.805474	−0.402737	+	0.915316i	$0.631941\pi$
$908$	0	0
$909$	21.3304	0.707486
$910$	0	0
$911$	−22.7991	−0.755369	−0.377685	−	0.925934i	$-0.623280\pi$
$911$	−22.7991	−0.755369	−0.377685	+	0.925934i	$0.623280\pi$
$912$	0	0
$913$	−4.69596	−0.155414
$914$	0	0
$915$	−34.1976	−1.13054
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−17.6870	−0.583441	−0.291721	−	0.956504i	$-0.594228\pi$
$919$	−17.6870	−0.583441	−0.291721	+	0.956504i	$0.594228\pi$
$920$	0	0
$921$	−23.2526	−0.766198
$922$	0	0
$923$	11.4853	0.378042
$924$	0	0
$925$	−37.0473	−1.21811
$926$	0	0
$927$	55.5280	1.82378
$928$	0	0
$929$	−8.85626	−0.290564	−0.145282	−	0.989390i	$-0.546409\pi$
$929$	−8.85626	−0.290564	−0.145282	+	0.989390i	$0.546409\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	27.9505	0.915059
$934$	0	0
$935$	4.83424	0.158097
$936$	0	0
$937$	13.9880	0.456968	0.228484	−	0.973548i	$-0.426623\pi$
$937$	13.9880	0.456968	0.228484	+	0.973548i	$0.426623\pi$
$938$	0	0
$939$	−3.50927	−0.114521
$940$	0	0
$941$	33.7553	1.10039	0.550195	−	0.835036i	$-0.314553\pi$
$941$	33.7553	1.10039	0.550195	+	0.835036i	$0.314553\pi$
$942$	0	0
$943$	0.712123	0.0231899
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−40.9090	−1.32937	−0.664683	−	0.747126i	$-0.731433\pi$
$947$	−40.9090	−1.32937	−0.664683	+	0.747126i	$0.731433\pi$
$948$	0	0
$949$	10.9880	0.356685
$950$	0	0
$951$	−85.8334	−2.78334
$952$	0	0
$953$	−16.5147	−0.534965	−0.267482	−	0.963563i	$-0.586192\pi$
$953$	−16.5147	−0.534965	−0.267482	+	0.963563i	$0.586192\pi$
$954$	0	0
$955$	12.6146	0.408200
$956$	0	0
$957$	−9.83424	−0.317896
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−24.2604	−0.782594
$962$	0	0
$963$	−3.30404	−0.106471
$964$	0	0
$965$	1.22378	0.0393949
$966$	0	0
$967$	15.0024	0.482444	0.241222	−	0.970470i	$-0.422452\pi$
$967$	15.0024	0.482444	0.241222	+	0.970470i	$0.422452\pi$
$968$	0	0
$969$	−77.2899	−2.48291
$970$	0	0
$971$	−12.1832	−0.390978	−0.195489	−	0.980706i	$-0.562629\pi$
$971$	−12.1832	−0.390978	−0.195489	+	0.980706i	$0.562629\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−8.42486	−0.269811
$976$	0	0
$977$	39.5730	1.26605	0.633026	−	0.774131i	$-0.281813\pi$
$977$	39.5730	1.26605	0.633026	+	0.774131i	$0.281813\pi$
$978$	0	0
$979$	14.5961	0.466493
$980$	0	0
$981$	11.7014	0.373598
$982$	0	0
$983$	−22.6840	−0.723506	−0.361753	−	0.932274i	$-0.617822\pi$
$983$	−22.6840	−0.723506	−0.361753	+	0.932274i	$0.617822\pi$
$984$	0	0
$985$	−26.7200	−0.851369
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−0.0835106	−0.00265548
$990$	0	0
$991$	18.8696	0.599411	0.299706	−	0.954032i	$-0.403112\pi$
$991$	18.8696	0.599411	0.299706	+	0.954032i	$0.403112\pi$
$992$	0	0
$993$	−71.8266	−2.27935
$994$	0	0
$995$	−15.9067	−0.504275
$996$	0	0
$997$	26.9420	0.853261	0.426630	−	0.904426i	$-0.359701\pi$
$997$	26.9420	0.853261	0.426630	+	0.904426i	$0.359701\pi$
$998$	0	0
$999$	59.9080	1.89540

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8624.2.a.cp.1.3		3
4.3	odd	2		2156.2.a.g.1.1		3
7.2	even	3		1232.2.q.j.529.1		6
7.4	even	3		1232.2.q.j.177.1		6
7.6	odd	2		8624.2.a.cg.1.1		3
28.3	even	6		2156.2.i.j.177.1		6
28.11	odd	6		308.2.i.b.177.3	✓	6
28.19	even	6		2156.2.i.j.1145.1		6
28.23	odd	6		308.2.i.b.221.3	yes	6
28.27	even	2		2156.2.a.k.1.3		3
84.11	even	6		2772.2.s.e.793.3		6
84.23	even	6		2772.2.s.e.2377.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
308.2.i.b.177.3	✓	6	28.11	odd	6
308.2.i.b.221.3	yes	6	28.23	odd	6
1232.2.q.j.177.1		6	7.4	even	3
1232.2.q.j.529.1		6	7.2	even	3
2156.2.a.g.1.1		3	4.3	odd	2
2156.2.a.k.1.3		3	28.27	even	2
2156.2.i.j.177.1		6	28.3	even	6
2156.2.i.j.1145.1		6	28.19	even	6
2772.2.s.e.793.3		6	84.11	even	6
2772.2.s.e.2377.3		6	84.23	even	6
8624.2.a.cg.1.1		3	7.6	odd	2
8624.2.a.cp.1.3		3	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:	\( N \)	\(=\)	\( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8624.a (trivial)

\(f(q)\)	\(=\)	\(q+2.83424 q^{3} +1.19869 q^{5} +5.03293 q^{9} +O(q^{10})\)	\(q+2.83424 q^{3} +1.19869 q^{5} +5.03293 q^{9} -1.00000 q^{11} +0.834243 q^{13} +3.39738 q^{15} -4.03293 q^{17} +6.76183 q^{19} +0.0933442 q^{23} -3.56314 q^{25} +5.76183 q^{27} +3.46980 q^{29} -2.59607 q^{31} -2.83424 q^{33} +10.3974 q^{37} +2.36445 q^{39} +7.62901 q^{41} -0.894653 q^{43} +6.03293 q^{45} +0.906656 q^{47} -11.4303 q^{51} -3.16576 q^{53} -1.19869 q^{55} +19.1647 q^{57} +12.6290 q^{59} -10.0659 q^{61} +1.00000 q^{65} -8.46325 q^{67} +0.264560 q^{69} +13.7673 q^{71} +13.1712 q^{73} -10.0988 q^{75} +14.3304 q^{79} +1.23163 q^{81} +4.69596 q^{83} -4.83424 q^{85} +9.83424 q^{87} -14.5961 q^{89} -7.35790 q^{93} +8.10535 q^{95} -12.9001 q^{97} -5.03293 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 2 q^{5} + 4 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 - 2 * q^5 + 4 * q^9	\( 3 q + 3 q^{3} - 2 q^{5} + 4 q^{9} - 3 q^{11} - 3 q^{13} - q^{15} - q^{17} + 9 q^{19} - 5 q^{25} + 6 q^{27} + 5 q^{29} + 9 q^{31} - 3 q^{33} + 20 q^{37} + 7 q^{39} - 5 q^{41} - 8 q^{43} + 7 q^{45} + 3 q^{47} - 12 q^{51} - 15 q^{53} + 2 q^{55} + 2 q^{57} + 10 q^{59} - 8 q^{61} + 3 q^{65} + 8 q^{67} - 27 q^{69} - 3 q^{71} + 12 q^{73} + 3 q^{75} - 7 q^{79} - 13 q^{81} + 25 q^{83} - 9 q^{85} + 24 q^{87} - 27 q^{89} + 6 q^{93} + 19 q^{95} - 11 q^{97} - 4 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 2 * q^5 + 4 * q^9 - 3 * q^11 - 3 * q^13 - q^15 - q^17 + 9 * q^19 - 5 * q^25 + 6 * q^27 + 5 * q^29 + 9 * q^31 - 3 * q^33 + 20 * q^37 + 7 * q^39 - 5 * q^41 - 8 * q^43 + 7 * q^45 + 3 * q^47 - 12 * q^51 - 15 * q^53 + 2 * q^55 + 2 * q^57 + 10 * q^59 - 8 * q^61 + 3 * q^65 + 8 * q^67 - 27 * q^69 - 3 * q^71 + 12 * q^73 + 3 * q^75 - 7 * q^79 - 13 * q^81 + 25 * q^83 - 9 * q^85 + 24 * q^87 - 27 * q^89 + 6 * q^93 + 19 * q^95 - 11 * q^97 - 4 * q^99

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