LMFDB - Embedded newform 3640.2.a.z.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3640,2,Mod(1,3640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3640, 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("3640.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3640.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:	$29.0655463357$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.2112217.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 10x^{3} - x^{2} + 10x + 4 $ x^5 - 10x^3 - x^2 + 10x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.19075$ of defining polynomial
Character	$\chi$	$=$	3640.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.19075 q^{3} +1.00000 q^{5} +1.00000 q^{7} -1.58211 q^{9} +O(q^{10})$	$q-1.19075 q^{3} +1.00000 q^{5} +1.00000 q^{7} -1.58211 q^{9} -2.27215 q^{11} -1.00000 q^{13} -1.19075 q^{15} -3.66918 q^{17} -6.64691 q^{19} -1.19075 q^{21} +1.89065 q^{23} +1.00000 q^{25} +5.45615 q^{27} +1.30010 q^{29} +0.809249 q^{31} +2.70557 q^{33} +1.00000 q^{35} +6.16847 q^{37} +1.19075 q^{39} +4.96361 q^{41} +6.36064 q^{43} -1.58211 q^{45} -9.63138 q^{47} +1.00000 q^{49} +4.36908 q^{51} +10.1628 q^{53} -2.27215 q^{55} +7.91481 q^{57} +8.44063 q^{59} -1.46857 q^{61} -1.58211 q^{63} -1.00000 q^{65} -3.75767 q^{67} -2.25130 q^{69} +8.15328 q^{71} +12.8069 q^{73} -1.19075 q^{75} -2.27215 q^{77} -5.96219 q^{79} -1.75059 q^{81} -0.0113373 q^{83} -3.66918 q^{85} -1.54809 q^{87} +10.6271 q^{89} -1.00000 q^{91} -0.963614 q^{93} -6.64691 q^{95} +2.85704 q^{97} +3.59480 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{5} + 5 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q + 5 * q^5 + 5 * q^7 + 5 * q^9	$ 5 q + 5 q^{5} + 5 q^{7} + 5 q^{9} + 7 q^{11} - 5 q^{13} + 3 q^{17} + 3 q^{19} + 3 q^{23} + 5 q^{25} - 3 q^{27} + 7 q^{29} + 10 q^{31} + 17 q^{33} + 5 q^{35} + 10 q^{37} + 4 q^{43} + 5 q^{45} - 3 q^{47} + 5 q^{49} + 26 q^{53} + 7 q^{55} - 20 q^{57} + 3 q^{59} + 13 q^{61} + 5 q^{63} - 5 q^{65} + 12 q^{67} + 23 q^{69} + 8 q^{71} + q^{73} + 7 q^{77} - 6 q^{79} + 25 q^{81} - 8 q^{83} + 3 q^{85} - 43 q^{87} + 3 q^{89} - 5 q^{91} + 20 q^{93} + 3 q^{95} - 15 q^{97} + 16 q^{99}+O(q^{100}) $ 5 * q + 5 * q^5 + 5 * q^7 + 5 * q^9 + 7 * q^11 - 5 * q^13 + 3 * q^17 + 3 * q^19 + 3 * q^23 + 5 * q^25 - 3 * q^27 + 7 * q^29 + 10 * q^31 + 17 * q^33 + 5 * q^35 + 10 * q^37 + 4 * q^43 + 5 * q^45 - 3 * q^47 + 5 * q^49 + 26 * q^53 + 7 * q^55 - 20 * q^57 + 3 * q^59 + 13 * q^61 + 5 * q^63 - 5 * q^65 + 12 * q^67 + 23 * q^69 + 8 * q^71 + q^73 + 7 * q^77 - 6 * q^79 + 25 * q^81 - 8 * q^83 + 3 * q^85 - 43 * q^87 + 3 * q^89 - 5 * q^91 + 20 * q^93 + 3 * q^95 - 15 * q^97 + 16 * 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
$3$	−1.19075	−0.687480	−0.343740	−	0.939065i	$-0.611694\pi$
$3$	−1.19075	−0.687480	−0.343740	+	0.939065i	$0.611694\pi$
$4$	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
$9$	−1.58211	−0.527371
$10$	0	0
$11$	−2.27215	−0.685080	−0.342540	−	0.939503i	$-0.611287\pi$
$11$	−2.27215	−0.685080	−0.342540	+	0.939503i	$0.611287\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−1.19075	−0.307451
$16$	0	0
$17$	−3.66918	−0.889908	−0.444954	−	0.895554i	$-0.646780\pi$
$17$	−3.66918	−0.889908	−0.444954	+	0.895554i	$0.646780\pi$
$18$	0	0
$19$	−6.64691	−1.52490	−0.762452	−	0.647044i	$-0.776005\pi$
$19$	−6.64691	−1.52490	−0.762452	+	0.647044i	$0.776005\pi$
$20$	0	0
$21$	−1.19075	−0.259843
$22$	0	0
$23$	1.89065	0.394228	0.197114	−	0.980381i	$-0.436843\pi$
$23$	1.89065	0.394228	0.197114	+	0.980381i	$0.436843\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.45615	1.05004
$28$	0	0
$29$	1.30010	0.241422	0.120711	−	0.992688i	$-0.461482\pi$
$29$	1.30010	0.241422	0.120711	+	0.992688i	$0.461482\pi$
$30$	0	0
$31$	0.809249	0.145345	0.0726727	−	0.997356i	$-0.476847\pi$
$31$	0.809249	0.145345	0.0726727	+	0.997356i	$0.476847\pi$
$32$	0	0
$33$	2.70557	0.470979
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	6.16847	1.01409	0.507045	−	0.861919i	$-0.330738\pi$
$37$	6.16847	1.01409	0.507045	+	0.861919i	$0.330738\pi$
$38$	0	0
$39$	1.19075	0.190673
$40$	0	0
$41$	4.96361	0.775186	0.387593	−	0.921831i	$-0.373307\pi$
$41$	4.96361	0.775186	0.387593	+	0.921831i	$0.373307\pi$
$42$	0	0
$43$	6.36064	0.969989	0.484995	−	0.874517i	$-0.338822\pi$
$43$	6.36064	0.969989	0.484995	+	0.874517i	$0.338822\pi$
$44$	0	0
$45$	−1.58211	−0.235847
$46$	0	0
$47$	−9.63138	−1.40488	−0.702440	−	0.711743i	$-0.747906\pi$
$47$	−9.63138	−1.40488	−0.702440	+	0.711743i	$0.747906\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	4.36908	0.611794
$52$	0	0
$53$	10.1628	1.39597	0.697984	−	0.716113i	$-0.254081\pi$
$53$	10.1628	1.39597	0.697984	+	0.716113i	$0.254081\pi$
$54$	0	0
$55$	−2.27215	−0.306377
$56$	0	0
$57$	7.91481	1.04834
$58$	0	0
$59$	8.44063	1.09888	0.549438	−	0.835534i	$-0.314842\pi$
$59$	8.44063	1.09888	0.549438	+	0.835534i	$0.314842\pi$
$60$	0	0
$61$	−1.46857	−0.188032	−0.0940158	−	0.995571i	$-0.529970\pi$
$61$	−1.46857	−0.188032	−0.0940158	+	0.995571i	$0.529970\pi$
$62$	0	0
$63$	−1.58211	−0.199327
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−3.75767	−0.459073	−0.229536	−	0.973300i	$-0.573721\pi$
$67$	−3.75767	−0.459073	−0.229536	+	0.973300i	$0.573721\pi$
$68$	0	0
$69$	−2.25130	−0.271024
$70$	0	0
$71$	8.15328	0.967617	0.483808	−	0.875174i	$-0.339253\pi$
$71$	8.15328	0.967617	0.483808	+	0.875174i	$0.339253\pi$
$72$	0	0
$73$	12.8069	1.49894	0.749469	−	0.662039i	$-0.230309\pi$
$73$	12.8069	1.49894	0.749469	+	0.662039i	$0.230309\pi$
$74$	0	0
$75$	−1.19075	−0.137496
$76$	0	0
$77$	−2.27215	−0.258936
$78$	0	0
$79$	−5.96219	−0.670799	−0.335400	−	0.942076i	$-0.608871\pi$
$79$	−5.96219	−0.670799	−0.335400	+	0.942076i	$0.608871\pi$
$80$	0	0
$81$	−1.75059	−0.194510
$82$	0	0
$83$	−0.0113373	−0.00124443	−0.000622213	−	1.00000i	$-0.500198\pi$
$83$	−0.0113373	−0.00124443	−0.000622213		1.00000i	$0.500198\pi$
$84$	0	0
$85$	−3.66918	−0.397979
$86$	0	0
$87$	−1.54809	−0.165973
$88$	0	0
$89$	10.6271	1.12647	0.563237	−	0.826296i	$-0.309556\pi$
$89$	10.6271	1.12647	0.563237	+	0.826296i	$0.309556\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−0.963614	−0.0999221
$94$	0	0
$95$	−6.64691	−0.681958
$96$	0	0
$97$	2.85704	0.290088	0.145044	−	0.989425i	$-0.453668\pi$
$97$	2.85704	0.290088	0.145044	+	0.989425i	$0.453668\pi$
$98$	0	0
$99$	3.59480	0.361291
$100$	0	0
$101$	−1.66243	−0.165418	−0.0827091	−	0.996574i	$-0.526357\pi$
$101$	−1.66243	−0.165418	−0.0827091	+	0.996574i	$0.526357\pi$
$102$	0	0
$103$	−3.48268	−0.343159	−0.171579	−	0.985170i	$-0.554887\pi$
$103$	−3.48268	−0.343159	−0.171579	+	0.985170i	$0.554887\pi$
$104$	0	0
$105$	−1.19075	−0.116205
$106$	0	0
$107$	−16.3161	−1.57734	−0.788668	−	0.614819i	$-0.789229\pi$
$107$	−16.3161	−1.57734	−0.788668	+	0.614819i	$0.789229\pi$
$108$	0	0
$109$	10.1741	0.974506	0.487253	−	0.873261i	$-0.337999\pi$
$109$	10.1741	0.974506	0.487253	+	0.873261i	$0.337999\pi$
$110$	0	0
$111$	−7.34512	−0.697168
$112$	0	0
$113$	−5.04252	−0.474360	−0.237180	−	0.971466i	$-0.576223\pi$
$113$	−5.04252	−0.474360	−0.237180	+	0.971466i	$0.576223\pi$
$114$	0	0
$115$	1.89065	0.176304
$116$	0	0
$117$	1.58211	0.146266
$118$	0	0
$119$	−3.66918	−0.336353
$120$	0	0
$121$	−5.83732	−0.530665
$122$	0	0
$123$	−5.91043	−0.532925
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−7.05488	−0.626019	−0.313009	−	0.949750i	$-0.601337\pi$
$127$	−7.05488	−0.626019	−0.313009	+	0.949750i	$0.601337\pi$
$128$	0	0
$129$	−7.57394	−0.666848
$130$	0	0
$131$	18.6767	1.63179	0.815897	−	0.578197i	$-0.196244\pi$
$131$	18.6767	1.63179	0.815897	+	0.578197i	$0.196244\pi$
$132$	0	0
$133$	−6.64691	−0.576360
$134$	0	0
$135$	5.45615	0.469591
$136$	0	0
$137$	16.4311	1.40380	0.701902	−	0.712273i	$-0.252334\pi$
$137$	16.4311	1.40380	0.701902	+	0.712273i	$0.252334\pi$
$138$	0	0
$139$	9.50117	0.805879	0.402939	−	0.915227i	$-0.367989\pi$
$139$	9.50117	0.805879	0.402939	+	0.915227i	$0.367989\pi$
$140$	0	0
$141$	11.4686	0.965828
$142$	0	0
$143$	2.27215	0.190007
$144$	0	0
$145$	1.30010	0.107967
$146$	0	0
$147$	−1.19075	−0.0982115
$148$	0	0
$149$	−10.0326	−0.821902	−0.410951	−	0.911657i	$-0.634803\pi$
$149$	−10.0326	−0.821902	−0.410951	+	0.911657i	$0.634803\pi$
$150$	0	0
$151$	−9.31751	−0.758248	−0.379124	−	0.925346i	$-0.623775\pi$
$151$	−9.31751	−0.758248	−0.379124	+	0.925346i	$0.623775\pi$
$152$	0	0
$153$	5.80506	0.469311
$154$	0	0
$155$	0.809249	0.0650004
$156$	0	0
$157$	1.84286	0.147077	0.0735383	−	0.997292i	$-0.476571\pi$
$157$	1.84286	0.147077	0.0735383	+	0.997292i	$0.476571\pi$
$158$	0	0
$159$	−12.1014	−0.959701
$160$	0	0
$161$	1.89065	0.149004
$162$	0	0
$163$	0.929935	0.0728381	0.0364191	−	0.999337i	$-0.488405\pi$
$163$	0.929935	0.0728381	0.0364191	+	0.999337i	$0.488405\pi$
$164$	0	0
$165$	2.70557	0.210628
$166$	0	0
$167$	−16.3087	−1.26201	−0.631004	−	0.775780i	$-0.717357\pi$
$167$	−16.3087	−1.26201	−0.631004	+	0.775780i	$0.717357\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	10.5161	0.804190
$172$	0	0
$173$	10.0383	0.763195	0.381598	−	0.924329i	$-0.375374\pi$
$173$	10.0383	0.763195	0.381598	+	0.924329i	$0.375374\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−10.0507	−0.755456
$178$	0	0
$179$	26.5172	1.98199	0.990995	−	0.133899i	$-0.0427497\pi$
$179$	26.5172	1.98199	0.990995	+	0.133899i	$0.0427497\pi$
$180$	0	0
$181$	12.4587	0.926045	0.463022	−	0.886347i	$-0.346765\pi$
$181$	12.4587	0.926045	0.463022	+	0.886347i	$0.346765\pi$
$182$	0	0
$183$	1.74870	0.129268
$184$	0	0
$185$	6.16847	0.453515
$186$	0	0
$187$	8.33695	0.609658
$188$	0	0
$189$	5.45615	0.396877
$190$	0	0
$191$	4.07998	0.295217	0.147609	−	0.989046i	$-0.452842\pi$
$191$	4.07998	0.295217	0.147609	+	0.989046i	$0.452842\pi$
$192$	0	0
$193$	11.0408	0.794732	0.397366	−	0.917660i	$-0.369924\pi$
$193$	11.0408	0.794732	0.397366	+	0.917660i	$0.369924\pi$
$194$	0	0
$195$	1.19075	0.0852715
$196$	0	0
$197$	8.74600	0.623127	0.311563	−	0.950225i	$-0.399147\pi$
$197$	8.74600	0.623127	0.311563	+	0.950225i	$0.399147\pi$
$198$	0	0
$199$	−10.1324	−0.718269	−0.359134	−	0.933286i	$-0.616928\pi$
$199$	−10.1324	−0.718269	−0.359134	+	0.933286i	$0.616928\pi$
$200$	0	0
$201$	4.47445	0.315604
$202$	0	0
$203$	1.30010	0.0912491
$204$	0	0
$205$	4.96361	0.346674
$206$	0	0
$207$	−2.99122	−0.207904
$208$	0	0
$209$	15.1028	1.04468
$210$	0	0
$211$	13.0521	0.898544	0.449272	−	0.893395i	$-0.351683\pi$
$211$	13.0521	0.898544	0.449272	+	0.893395i	$0.351683\pi$
$212$	0	0
$213$	−9.70853	−0.665218
$214$	0	0
$215$	6.36064	0.433792
$216$	0	0
$217$	0.809249	0.0549354
$218$	0	0
$219$	−15.2499	−1.03049
$220$	0	0
$221$	3.66918	0.246816
$222$	0	0
$223$	−5.31131	−0.355672	−0.177836	−	0.984060i	$-0.556910\pi$
$223$	−5.31131	−0.355672	−0.177836	+	0.984060i	$0.556910\pi$
$224$	0	0
$225$	−1.58211	−0.105474
$226$	0	0
$227$	−24.6626	−1.63691	−0.818456	−	0.574570i	$-0.805169\pi$
$227$	−24.6626	−1.63691	−0.818456	+	0.574570i	$0.805169\pi$
$228$	0	0
$229$	29.2802	1.93489	0.967446	−	0.253078i	$-0.0814430\pi$
$229$	29.2802	1.93489	0.967446	+	0.253078i	$0.0814430\pi$
$230$	0	0
$231$	2.70557	0.178013
$232$	0	0
$233$	16.6300	1.08946	0.544732	−	0.838610i	$-0.316631\pi$
$233$	16.6300	1.08946	0.544732	+	0.838610i	$0.316631\pi$
$234$	0	0
$235$	−9.63138	−0.628282
$236$	0	0
$237$	7.09949	0.461161
$238$	0	0
$239$	1.44833	0.0936850	0.0468425	−	0.998902i	$-0.485084\pi$
$239$	1.44833	0.0936850	0.0468425	+	0.998902i	$0.485084\pi$
$240$	0	0
$241$	−17.9334	−1.15519	−0.577595	−	0.816324i	$-0.696009\pi$
$241$	−17.9334	−1.15519	−0.577595	+	0.816324i	$0.696009\pi$
$242$	0	0
$243$	−14.2840	−0.916316
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	6.64691	0.422932
$248$	0	0
$249$	0.0134998	0.000855518 0
$250$	0	0
$251$	−4.42544	−0.279331	−0.139666	−	0.990199i	$-0.544603\pi$
$251$	−4.42544	−0.279331	−0.139666	+	0.990199i	$0.544603\pi$
$252$	0	0
$253$	−4.29585	−0.270078
$254$	0	0
$255$	4.36908	0.273603
$256$	0	0
$257$	−24.9409	−1.55577	−0.777886	−	0.628406i	$-0.783708\pi$
$257$	−24.9409	−1.55577	−0.777886	+	0.628406i	$0.783708\pi$
$258$	0	0
$259$	6.16847	0.383290
$260$	0	0
$261$	−2.05690	−0.127319
$262$	0	0
$263$	28.1677	1.73690	0.868448	−	0.495781i	$-0.165118\pi$
$263$	28.1677	1.73690	0.868448	+	0.495781i	$0.165118\pi$
$264$	0	0
$265$	10.1628	0.624296
$266$	0	0
$267$	−12.6543	−0.774428
$268$	0	0
$269$	17.3512	1.05792	0.528962	−	0.848645i	$-0.322581\pi$
$269$	17.3512	1.05792	0.528962	+	0.848645i	$0.322581\pi$
$270$	0	0
$271$	3.47809	0.211279	0.105640	−	0.994404i	$-0.466311\pi$
$271$	3.47809	0.211279	0.105640	+	0.994404i	$0.466311\pi$
$272$	0	0
$273$	1.19075	0.0720675
$274$	0	0
$275$	−2.27215	−0.137016
$276$	0	0
$277$	13.1507	0.790150	0.395075	−	0.918649i	$-0.370719\pi$
$277$	13.1507	0.790150	0.395075	+	0.918649i	$0.370719\pi$
$278$	0	0
$279$	−1.28032	−0.0766509
$280$	0	0
$281$	29.0272	1.73162	0.865808	−	0.500376i	$-0.166805\pi$
$281$	29.0272	1.73162	0.865808	+	0.500376i	$0.166805\pi$
$282$	0	0
$283$	14.2586	0.847584	0.423792	−	0.905760i	$-0.360699\pi$
$283$	14.2586	0.847584	0.423792	+	0.905760i	$0.360699\pi$
$284$	0	0
$285$	7.91481	0.468833
$286$	0	0
$287$	4.96361	0.292993
$288$	0	0
$289$	−3.53710	−0.208064
$290$	0	0
$291$	−3.40202	−0.199430
$292$	0	0
$293$	−30.7222	−1.79481	−0.897405	−	0.441207i	$-0.854550\pi$
$293$	−30.7222	−1.79481	−0.897405	+	0.441207i	$0.854550\pi$
$294$	0	0
$295$	8.44063	0.491432
$296$	0	0
$297$	−12.3972	−0.719360
$298$	0	0
$299$	−1.89065	−0.109339
$300$	0	0
$301$	6.36064	0.366621
$302$	0	0
$303$	1.97954	0.113722
$304$	0	0
$305$	−1.46857	−0.0840903
$306$	0	0
$307$	−12.1431	−0.693043	−0.346521	−	0.938042i	$-0.612637\pi$
$307$	−12.1431	−0.693043	−0.346521	+	0.938042i	$0.612637\pi$
$308$	0	0
$309$	4.14701	0.235915
$310$	0	0
$311$	3.42748	0.194354	0.0971772	−	0.995267i	$-0.469019\pi$
$311$	3.42748	0.194354	0.0971772	+	0.995267i	$0.469019\pi$
$312$	0	0
$313$	−9.47424	−0.535516	−0.267758	−	0.963486i	$-0.586283\pi$
$313$	−9.47424	−0.535516	−0.267758	+	0.963486i	$0.586283\pi$
$314$	0	0
$315$	−1.58211	−0.0891419
$316$	0	0
$317$	5.85846	0.329044	0.164522	−	0.986373i	$-0.447392\pi$
$317$	5.85846	0.329044	0.164522	+	0.986373i	$0.447392\pi$
$318$	0	0
$319$	−2.95403	−0.165394
$320$	0	0
$321$	19.4284	1.08439
$322$	0	0
$323$	24.3887	1.35702
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	−12.1149	−0.669954
$328$	0	0
$329$	−9.63138	−0.530995
$330$	0	0
$331$	−18.7444	−1.03028	−0.515142	−	0.857105i	$-0.672261\pi$
$331$	−18.7444	−1.03028	−0.515142	+	0.857105i	$0.672261\pi$
$332$	0	0
$333$	−9.75922	−0.534802
$334$	0	0
$335$	−3.75767	−0.205304
$336$	0	0
$337$	10.6441	0.579823	0.289911	−	0.957053i	$-0.406374\pi$
$337$	10.6441	0.579823	0.289911	+	0.957053i	$0.406374\pi$
$338$	0	0
$339$	6.00438	0.326113
$340$	0	0
$341$	−1.83874	−0.0995732
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−2.25130	−0.121206
$346$	0	0
$347$	0.369143	0.0198166	0.00990831	−	0.999951i	$-0.496846\pi$
$347$	0.369143	0.0198166	0.00990831	+	0.999951i	$0.496846\pi$
$348$	0	0
$349$	0.188811	0.0101068	0.00505342	−	0.999987i	$-0.498391\pi$
$349$	0.188811	0.0101068	0.00505342	+	0.999987i	$0.498391\pi$
$350$	0	0
$351$	−5.45615	−0.291228
$352$	0	0
$353$	26.8806	1.43071	0.715356	−	0.698760i	$-0.246265\pi$
$353$	26.8806	1.43071	0.715356	+	0.698760i	$0.246265\pi$
$354$	0	0
$355$	8.15328	0.432731
$356$	0	0
$357$	4.36908	0.231236
$358$	0	0
$359$	−23.8725	−1.25994	−0.629971	−	0.776619i	$-0.716933\pi$
$359$	−23.8725	−1.25994	−0.629971	+	0.776619i	$0.716933\pi$
$360$	0	0
$361$	25.1814	1.32533
$362$	0	0
$363$	6.95079	0.364822
$364$	0	0
$365$	12.8069	0.670346
$366$	0	0
$367$	6.03596	0.315074	0.157537	−	0.987513i	$-0.449645\pi$
$367$	6.03596	0.315074	0.157537	+	0.987513i	$0.449645\pi$
$368$	0	0
$369$	−7.85299	−0.408811
$370$	0	0
$371$	10.1628	0.527627
$372$	0	0
$373$	−12.5612	−0.650394	−0.325197	−	0.945646i	$-0.605431\pi$
$373$	−12.5612	−0.650394	−0.325197	+	0.945646i	$0.605431\pi$
$374$	0	0
$375$	−1.19075	−0.0614901
$376$	0	0
$377$	−1.30010	−0.0669585
$378$	0	0
$379$	22.0608	1.13319	0.566593	−	0.823998i	$-0.308261\pi$
$379$	22.0608	1.13319	0.566593	+	0.823998i	$0.308261\pi$
$380$	0	0
$381$	8.40060	0.430376
$382$	0	0
$383$	33.8488	1.72959	0.864797	−	0.502121i	$-0.167447\pi$
$383$	33.8488	1.72959	0.864797	+	0.502121i	$0.167447\pi$
$384$	0	0
$385$	−2.27215	−0.115800
$386$	0	0
$387$	−10.0633	−0.511544
$388$	0	0
$389$	−4.60633	−0.233550	−0.116775	−	0.993158i	$-0.537256\pi$
$389$	−4.60633	−0.233550	−0.116775	+	0.993158i	$0.537256\pi$
$390$	0	0
$391$	−6.93715	−0.350827
$392$	0	0
$393$	−22.2393	−1.12183
$394$	0	0
$395$	−5.96219	−0.299991
$396$	0	0
$397$	−20.7950	−1.04367	−0.521835	−	0.853046i	$-0.674752\pi$
$397$	−20.7950	−1.04367	−0.521835	+	0.853046i	$0.674752\pi$
$398$	0	0
$399$	7.91481	0.396236
$400$	0	0
$401$	3.82370	0.190946	0.0954731	−	0.995432i	$-0.469564\pi$
$401$	3.82370	0.190946	0.0954731	+	0.995432i	$0.469564\pi$
$402$	0	0
$403$	−0.809249	−0.0403116
$404$	0	0
$405$	−1.75059	−0.0869873
$406$	0	0
$407$	−14.0157	−0.694733
$408$	0	0
$409$	−31.6753	−1.56624	−0.783122	−	0.621868i	$-0.786374\pi$
$409$	−31.6753	−1.56624	−0.783122	+	0.621868i	$0.786374\pi$
$410$	0	0
$411$	−19.5654	−0.965088
$412$	0	0
$413$	8.44063	0.415336
$414$	0	0
$415$	−0.0113373	−0.000556524 0
$416$	0	0
$417$	−11.3135	−0.554026
$418$	0	0
$419$	1.05885	0.0517284	0.0258642	−	0.999665i	$-0.491766\pi$
$419$	1.05885	0.0517284	0.0258642	+	0.999665i	$0.491766\pi$
$420$	0	0
$421$	23.5338	1.14697	0.573483	−	0.819218i	$-0.305592\pi$
$421$	23.5338	1.14697	0.573483	+	0.819218i	$0.305592\pi$
$422$	0	0
$423$	15.2379	0.740893
$424$	0	0
$425$	−3.66918	−0.177982
$426$	0	0
$427$	−1.46857	−0.0710692
$428$	0	0
$429$	−2.70557	−0.130626
$430$	0	0
$431$	37.0177	1.78308	0.891539	−	0.452944i	$-0.149626\pi$
$431$	37.0177	1.78308	0.891539	+	0.452944i	$0.149626\pi$
$432$	0	0
$433$	−4.52130	−0.217280	−0.108640	−	0.994081i	$-0.534650\pi$
$433$	−4.52130	−0.217280	−0.108640	+	0.994081i	$0.534650\pi$
$434$	0	0
$435$	−1.54809	−0.0742255
$436$	0	0
$437$	−12.5670	−0.601160
$438$	0	0
$439$	2.92027	0.139377	0.0696884	−	0.997569i	$-0.477800\pi$
$439$	2.92027	0.139377	0.0696884	+	0.997569i	$0.477800\pi$
$440$	0	0
$441$	−1.58211	−0.0753387
$442$	0	0
$443$	8.50657	0.404159	0.202080	−	0.979369i	$-0.435230\pi$
$443$	8.50657	0.404159	0.202080	+	0.979369i	$0.435230\pi$
$444$	0	0
$445$	10.6271	0.503774
$446$	0	0
$447$	11.9463	0.565042
$448$	0	0
$449$	30.2727	1.42866	0.714328	−	0.699811i	$-0.246733\pi$
$449$	30.2727	1.42866	0.714328	+	0.699811i	$0.246733\pi$
$450$	0	0
$451$	−11.2781	−0.531065
$452$	0	0
$453$	11.0948	0.521281
$454$	0	0
$455$	−1.00000	−0.0468807
$456$	0	0
$457$	1.19411	0.0558581	0.0279291	−	0.999610i	$-0.491109\pi$
$457$	1.19411	0.0558581	0.0279291	+	0.999610i	$0.491109\pi$
$458$	0	0
$459$	−20.0196	−0.934436
$460$	0	0
$461$	22.7878	1.06133	0.530666	−	0.847581i	$-0.321942\pi$
$461$	22.7878	1.06133	0.530666	+	0.847581i	$0.321942\pi$
$462$	0	0
$463$	−32.2710	−1.49976	−0.749880	−	0.661574i	$-0.769889\pi$
$463$	−32.2710	−1.49976	−0.749880	+	0.661574i	$0.769889\pi$
$464$	0	0
$465$	−0.963614	−0.0446865
$466$	0	0
$467$	3.05029	0.141151	0.0705753	−	0.997506i	$-0.477517\pi$
$467$	3.05029	0.141151	0.0705753	+	0.997506i	$0.477517\pi$
$468$	0	0
$469$	−3.75767	−0.173513
$470$	0	0
$471$	−2.19439	−0.101112
$472$	0	0
$473$	−14.4524	−0.664520
$474$	0	0
$475$	−6.64691	−0.304981
$476$	0	0
$477$	−16.0787	−0.736193
$478$	0	0
$479$	6.84185	0.312612	0.156306	−	0.987709i	$-0.450041\pi$
$479$	6.84185	0.312612	0.156306	+	0.987709i	$0.450041\pi$
$480$	0	0
$481$	−6.16847	−0.281258
$482$	0	0
$483$	−2.25130	−0.102437
$484$	0	0
$485$	2.85704	0.129731
$486$	0	0
$487$	−16.6999	−0.756745	−0.378372	−	0.925653i	$-0.623516\pi$
$487$	−16.6999	−0.756745	−0.378372	+	0.925653i	$0.623516\pi$
$488$	0	0
$489$	−1.10732	−0.0500748
$490$	0	0
$491$	21.0886	0.951716	0.475858	−	0.879522i	$-0.342138\pi$
$491$	21.0886	0.951716	0.475858	+	0.879522i	$0.342138\pi$
$492$	0	0
$493$	−4.77030	−0.214844
$494$	0	0
$495$	3.59480	0.161574
$496$	0	0
$497$	8.15328	0.365725
$498$	0	0
$499$	23.9558	1.07241	0.536206	−	0.844087i	$-0.319857\pi$
$499$	23.9558	1.07241	0.536206	+	0.844087i	$0.319857\pi$
$500$	0	0
$501$	19.4196	0.867605
$502$	0	0
$503$	−26.0133	−1.15987	−0.579937	−	0.814661i	$-0.696923\pi$
$503$	−26.0133	−1.15987	−0.579937	+	0.814661i	$0.696923\pi$
$504$	0	0
$505$	−1.66243	−0.0739773
$506$	0	0
$507$	−1.19075	−0.0528831
$508$	0	0
$509$	−17.4223	−0.772228	−0.386114	−	0.922451i	$-0.626183\pi$
$509$	−17.4223	−0.772228	−0.386114	+	0.922451i	$0.626183\pi$
$510$	0	0
$511$	12.8069	0.566546
$512$	0	0
$513$	−36.2665	−1.60121
$514$	0	0
$515$	−3.48268	−0.153465
$516$	0	0
$517$	21.8840	0.962456
$518$	0	0
$519$	−11.9531	−0.524682
$520$	0	0
$521$	−2.18832	−0.0958719	−0.0479360	−	0.998850i	$-0.515264\pi$
$521$	−2.18832	−0.0958719	−0.0479360	+	0.998850i	$0.515264\pi$
$522$	0	0
$523$	−5.56504	−0.243342	−0.121671	−	0.992570i	$-0.538825\pi$
$523$	−5.56504	−0.243342	−0.121671	+	0.992570i	$0.538825\pi$
$524$	0	0
$525$	−1.19075	−0.0519686
$526$	0	0
$527$	−2.96928	−0.129344
$528$	0	0
$529$	−19.4254	−0.844584
$530$	0	0
$531$	−13.3540	−0.579515
$532$	0	0
$533$	−4.96361	−0.214998
$534$	0	0
$535$	−16.3161	−0.705406
$536$	0	0
$537$	−31.5754	−1.36258
$538$	0	0
$539$	−2.27215	−0.0978686
$540$	0	0
$541$	−34.7646	−1.49465	−0.747324	−	0.664460i	$-0.768661\pi$
$541$	−34.7646	−1.49465	−0.747324	+	0.664460i	$0.768661\pi$
$542$	0	0
$543$	−14.8352	−0.636637
$544$	0	0
$545$	10.1741	0.435812
$546$	0	0
$547$	−1.03066	−0.0440678	−0.0220339	−	0.999757i	$-0.507014\pi$
$547$	−1.03066	−0.0440678	−0.0220339	+	0.999757i	$0.507014\pi$
$548$	0	0
$549$	2.32345	0.0991623
$550$	0	0
$551$	−8.64164	−0.368146
$552$	0	0
$553$	−5.96219	−0.253538
$554$	0	0
$555$	−7.34512	−0.311783
$556$	0	0
$557$	13.0002	0.550837	0.275418	−	0.961324i	$-0.411184\pi$
$557$	13.0002	0.550837	0.275418	+	0.961324i	$0.411184\pi$
$558$	0	0
$559$	−6.36064	−0.269027
$560$	0	0
$561$	−9.92723	−0.419128
$562$	0	0
$563$	−22.4940	−0.948010	−0.474005	−	0.880522i	$-0.657192\pi$
$563$	−22.4940	−0.948010	−0.474005	+	0.880522i	$0.657192\pi$
$564$	0	0
$565$	−5.04252	−0.212140
$566$	0	0
$567$	−1.75059	−0.0735177
$568$	0	0
$569$	−17.8742	−0.749324	−0.374662	−	0.927161i	$-0.622241\pi$
$569$	−17.8742	−0.749324	−0.374662	+	0.927161i	$0.622241\pi$
$570$	0	0
$571$	27.3124	1.14299	0.571493	−	0.820607i	$-0.306364\pi$
$571$	27.3124	1.14299	0.571493	+	0.820607i	$0.306364\pi$
$572$	0	0
$573$	−4.85824	−0.202956
$574$	0	0
$575$	1.89065	0.0788456
$576$	0	0
$577$	−29.6888	−1.23596	−0.617981	−	0.786193i	$-0.712049\pi$
$577$	−29.6888	−1.23596	−0.617981	+	0.786193i	$0.712049\pi$
$578$	0	0
$579$	−13.1468	−0.546363
$580$	0	0
$581$	−0.0113373	−0.000470349 0
$582$	0	0
$583$	−23.0915	−0.956350
$584$	0	0
$585$	1.58211	0.0654123
$586$	0	0
$587$	−8.81378	−0.363784	−0.181892	−	0.983319i	$-0.558222\pi$
$587$	−8.81378	−0.363784	−0.181892	+	0.983319i	$0.558222\pi$
$588$	0	0
$589$	−5.37900	−0.221638
$590$	0	0
$591$	−10.4143	−0.428387
$592$	0	0
$593$	−29.5411	−1.21311	−0.606554	−	0.795042i	$-0.707449\pi$
$593$	−29.5411	−1.21311	−0.606554	+	0.795042i	$0.707449\pi$
$594$	0	0
$595$	−3.66918	−0.150422
$596$	0	0
$597$	12.0652	0.493796
$598$	0	0
$599$	−19.1439	−0.782198	−0.391099	−	0.920349i	$-0.627905\pi$
$599$	−19.1439	−0.782198	−0.391099	+	0.920349i	$0.627905\pi$
$600$	0	0
$601$	−1.74175	−0.0710474	−0.0355237	−	0.999369i	$-0.511310\pi$
$601$	−1.74175	−0.0710474	−0.0355237	+	0.999369i	$0.511310\pi$
$602$	0	0
$603$	5.94506	0.242102
$604$	0	0
$605$	−5.83732	−0.237321
$606$	0	0
$607$	−34.6176	−1.40509	−0.702543	−	0.711642i	$-0.747952\pi$
$607$	−34.6176	−1.40509	−0.702543	+	0.711642i	$0.747952\pi$
$608$	0	0
$609$	−1.54809	−0.0627320
$610$	0	0
$611$	9.63138	0.389644
$612$	0	0
$613$	−4.64130	−0.187460	−0.0937301	−	0.995598i	$-0.529879\pi$
$613$	−4.64130	−0.187460	−0.0937301	+	0.995598i	$0.529879\pi$
$614$	0	0
$615$	−5.91043	−0.238331
$616$	0	0
$617$	12.9016	0.519397	0.259699	−	0.965690i	$-0.416377\pi$
$617$	12.9016	0.519397	0.259699	+	0.965690i	$0.416377\pi$
$618$	0	0
$619$	−7.21109	−0.289838	−0.144919	−	0.989444i	$-0.546292\pi$
$619$	−7.21109	−0.289838	−0.144919	+	0.989444i	$0.546292\pi$
$620$	0	0
$621$	10.3157	0.413954
$622$	0	0
$623$	10.6271	0.425767
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−17.9837	−0.718198
$628$	0	0
$629$	−22.6333	−0.902447
$630$	0	0
$631$	40.1147	1.59694	0.798470	−	0.602035i	$-0.205643\pi$
$631$	40.1147	1.59694	0.798470	+	0.602035i	$0.205643\pi$
$632$	0	0
$633$	−15.5418	−0.617731
$634$	0	0
$635$	−7.05488	−0.279964
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−12.8994	−0.510293
$640$	0	0
$641$	−23.2098	−0.916733	−0.458367	−	0.888763i	$-0.651565\pi$
$641$	−23.2098	−0.916733	−0.458367	+	0.888763i	$0.651565\pi$
$642$	0	0
$643$	−22.4185	−0.884099	−0.442049	−	0.896991i	$-0.645748\pi$
$643$	−22.4185	−0.884099	−0.442049	+	0.896991i	$0.645748\pi$
$644$	0	0
$645$	−7.57394	−0.298224
$646$	0	0
$647$	−47.8752	−1.88217	−0.941085	−	0.338170i	$-0.890192\pi$
$647$	−47.8752	−1.88217	−0.941085	+	0.338170i	$0.890192\pi$
$648$	0	0
$649$	−19.1784	−0.752818
$650$	0	0
$651$	−0.963614	−0.0377670
$652$	0	0
$653$	20.7888	0.813527	0.406764	−	0.913533i	$-0.366657\pi$
$653$	20.7888	0.813527	0.406764	+	0.913533i	$0.366657\pi$
$654$	0	0
$655$	18.6767	0.729760
$656$	0	0
$657$	−20.2620	−0.790496
$658$	0	0
$659$	1.35195	0.0526647	0.0263323	−	0.999653i	$-0.491617\pi$
$659$	1.35195	0.0526647	0.0263323	+	0.999653i	$0.491617\pi$
$660$	0	0
$661$	13.0697	0.508351	0.254175	−	0.967158i	$-0.418196\pi$
$661$	13.0697	0.508351	0.254175	+	0.967158i	$0.418196\pi$
$662$	0	0
$663$	−4.36908	−0.169681
$664$	0	0
$665$	−6.64691	−0.257756
$666$	0	0
$667$	2.45804	0.0951755
$668$	0	0
$669$	6.32445	0.244517
$670$	0	0
$671$	3.33682	0.128817
$672$	0	0
$673$	41.5460	1.60148	0.800741	−	0.599011i	$-0.204439\pi$
$673$	41.5460	1.60148	0.800741	+	0.599011i	$0.204439\pi$
$674$	0	0
$675$	5.45615	0.210007
$676$	0	0
$677$	41.4922	1.59467	0.797337	−	0.603535i	$-0.206241\pi$
$677$	41.4922	1.59467	0.797337	+	0.603535i	$0.206241\pi$
$678$	0	0
$679$	2.85704	0.109643
$680$	0	0
$681$	29.3670	1.12534
$682$	0	0
$683$	37.0546	1.41785	0.708927	−	0.705282i	$-0.249180\pi$
$683$	37.0546	1.41785	0.708927	+	0.705282i	$0.249180\pi$
$684$	0	0
$685$	16.4311	0.627800
$686$	0	0
$687$	−34.8655	−1.33020
$688$	0	0
$689$	−10.1628	−0.387172
$690$	0	0
$691$	43.1318	1.64081	0.820405	−	0.571783i	$-0.193748\pi$
$691$	43.1318	1.64081	0.820405	+	0.571783i	$0.193748\pi$
$692$	0	0
$693$	3.59480	0.136555
$694$	0	0
$695$	9.50117	0.360400
$696$	0	0
$697$	−18.2124	−0.689844
$698$	0	0
$699$	−19.8021	−0.748986
$700$	0	0
$701$	−2.09616	−0.0791710	−0.0395855	−	0.999216i	$-0.512604\pi$
$701$	−2.09616	−0.0791710	−0.0395855	+	0.999216i	$0.512604\pi$
$702$	0	0
$703$	−41.0013	−1.54639
$704$	0	0
$705$	11.4686	0.431931
$706$	0	0
$707$	−1.66243	−0.0625222
$708$	0	0
$709$	45.7315	1.71748	0.858741	−	0.512409i	$-0.171247\pi$
$709$	45.7315	1.71748	0.858741	+	0.512409i	$0.171247\pi$
$710$	0	0
$711$	9.43286	0.353760
$712$	0	0
$713$	1.53001	0.0572992
$714$	0	0
$715$	2.27215	0.0849737
$716$	0	0
$717$	−1.72461	−0.0644066
$718$	0	0
$719$	−10.3511	−0.386032	−0.193016	−	0.981196i	$-0.561827\pi$
$719$	−10.3511	−0.386032	−0.193016	+	0.981196i	$0.561827\pi$
$720$	0	0
$721$	−3.48268	−0.129702
$722$	0	0
$723$	21.3542	0.794170
$724$	0	0
$725$	1.30010	0.0482845
$726$	0	0
$727$	52.4453	1.94509	0.972544	−	0.232717i	$-0.0747616\pi$
$727$	52.4453	1.94509	0.972544	+	0.232717i	$0.0747616\pi$
$728$	0	0
$729$	22.2604	0.824459
$730$	0	0
$731$	−23.3384	−0.863201
$732$	0	0
$733$	−46.3549	−1.71216	−0.856078	−	0.516846i	$-0.827106\pi$
$733$	−46.3549	−1.71216	−0.856078	+	0.516846i	$0.827106\pi$
$734$	0	0
$735$	−1.19075	−0.0439215
$736$	0	0
$737$	8.53801	0.314502
$738$	0	0
$739$	−23.5393	−0.865907	−0.432953	−	0.901416i	$-0.642529\pi$
$739$	−23.5393	−0.865907	−0.432953	+	0.901416i	$0.642529\pi$
$740$	0	0
$741$	−7.91481	−0.290758
$742$	0	0
$743$	4.14613	0.152107	0.0760534	−	0.997104i	$-0.475768\pi$
$743$	4.14613	0.152107	0.0760534	+	0.997104i	$0.475768\pi$
$744$	0	0
$745$	−10.0326	−0.367566
$746$	0	0
$747$	0.0179368	0.000656274 0
$748$	0	0
$749$	−16.3161	−0.596177
$750$	0	0
$751$	4.78285	0.174529	0.0872643	−	0.996185i	$-0.472188\pi$
$751$	4.78285	0.174529	0.0872643	+	0.996185i	$0.472188\pi$
$752$	0	0
$753$	5.26959	0.192035
$754$	0	0
$755$	−9.31751	−0.339099
$756$	0	0
$757$	−11.0354	−0.401090	−0.200545	−	0.979685i	$-0.564271\pi$
$757$	−11.0354	−0.401090	−0.200545	+	0.979685i	$0.564271\pi$
$758$	0	0
$759$	5.11529	0.185673
$760$	0	0
$761$	−21.8658	−0.792634	−0.396317	−	0.918114i	$-0.629712\pi$
$761$	−21.8658	−0.792634	−0.396317	+	0.918114i	$0.629712\pi$
$762$	0	0
$763$	10.1741	0.368329
$764$	0	0
$765$	5.80506	0.209882
$766$	0	0
$767$	−8.44063	−0.304773
$768$	0	0
$769$	−4.44799	−0.160399	−0.0801994	−	0.996779i	$-0.525556\pi$
$769$	−4.44799	−0.160399	−0.0801994	+	0.996779i	$0.525556\pi$
$770$	0	0
$771$	29.6984	1.06956
$772$	0	0
$773$	−37.2302	−1.33908	−0.669539	−	0.742777i	$-0.733508\pi$
$773$	−37.2302	−1.33908	−0.669539	+	0.742777i	$0.733508\pi$
$774$	0	0
$775$	0.809249	0.0290691
$776$	0	0
$777$	−7.34512	−0.263505
$778$	0	0
$779$	−32.9927	−1.18209
$780$	0	0
$781$	−18.5255	−0.662895
$782$	0	0
$783$	7.09354	0.253503
$784$	0	0
$785$	1.84286	0.0657746
$786$	0	0
$787$	15.1816	0.541166	0.270583	−	0.962697i	$-0.412784\pi$
$787$	15.1816	0.541166	0.270583	+	0.962697i	$0.412784\pi$
$788$	0	0
$789$	−33.5407	−1.19408
$790$	0	0
$791$	−5.04252	−0.179291
$792$	0	0
$793$	1.46857	0.0521506
$794$	0	0
$795$	−12.1014	−0.429191
$796$	0	0
$797$	−20.2422	−0.717014	−0.358507	−	0.933527i	$-0.616714\pi$
$797$	−20.2422	−0.717014	−0.358507	+	0.933527i	$0.616714\pi$
$798$	0	0
$799$	35.3393	1.25021
$800$	0	0
$801$	−16.8133	−0.594069
$802$	0	0
$803$	−29.0993	−1.02689
$804$	0	0
$805$	1.89065	0.0666367
$806$	0	0
$807$	−20.6610	−0.727302
$808$	0	0
$809$	30.8874	1.08594	0.542971	−	0.839751i	$-0.317299\pi$
$809$	30.8874	1.08594	0.542971	+	0.839751i	$0.317299\pi$
$810$	0	0
$811$	−7.55820	−0.265404	−0.132702	−	0.991156i	$-0.542365\pi$
$811$	−7.55820	−0.265404	−0.132702	+	0.991156i	$0.542365\pi$
$812$	0	0
$813$	−4.14154	−0.145250
$814$	0	0
$815$	0.929935	0.0325742
$816$	0	0
$817$	−42.2786	−1.47914
$818$	0	0
$819$	1.58211	0.0552835
$820$	0	0
$821$	21.6555	0.755781	0.377891	−	0.925850i	$-0.376650\pi$
$821$	21.6555	0.755781	0.377891	+	0.925850i	$0.376650\pi$
$822$	0	0
$823$	−1.92454	−0.0670854	−0.0335427	−	0.999437i	$-0.510679\pi$
$823$	−1.92454	−0.0670854	−0.0335427	+	0.999437i	$0.510679\pi$
$824$	0	0
$825$	2.70557	0.0941958
$826$	0	0
$827$	0.325610	0.0113226	0.00566128	−	0.999984i	$-0.498198\pi$
$827$	0.325610	0.0113226	0.00566128	+	0.999984i	$0.498198\pi$
$828$	0	0
$829$	28.5898	0.992964	0.496482	−	0.868047i	$-0.334625\pi$
$829$	28.5898	0.992964	0.496482	+	0.868047i	$0.334625\pi$
$830$	0	0
$831$	−15.6592	−0.543213
$832$	0	0
$833$	−3.66918	−0.127130
$834$	0	0
$835$	−16.3087	−0.564387
$836$	0	0
$837$	4.41539	0.152618
$838$	0	0
$839$	−8.41808	−0.290624	−0.145312	−	0.989386i	$-0.546419\pi$
$839$	−8.41808	−0.290624	−0.145312	+	0.989386i	$0.546419\pi$
$840$	0	0
$841$	−27.3097	−0.941715
$842$	0	0
$843$	−34.5641	−1.19045
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	−5.83732	−0.200573
$848$	0	0
$849$	−16.9784	−0.582697
$850$	0	0
$851$	11.6624	0.399783
$852$	0	0
$853$	20.2111	0.692016	0.346008	−	0.938232i	$-0.387537\pi$
$853$	20.2111	0.692016	0.346008	+	0.938232i	$0.387537\pi$
$854$	0	0
$855$	10.5161	0.359645
$856$	0	0
$857$	5.01443	0.171290	0.0856448	−	0.996326i	$-0.472705\pi$
$857$	5.01443	0.171290	0.0856448	+	0.996326i	$0.472705\pi$
$858$	0	0
$859$	−0.0513930	−0.00175351	−0.000876753	−	1.00000i	$-0.500279\pi$
$859$	−0.0513930	−0.00175351	−0.000876753		1.00000i	$0.500279\pi$
$860$	0	0
$861$	−5.91043	−0.201427
$862$	0	0
$863$	−8.75563	−0.298045	−0.149023	−	0.988834i	$-0.547613\pi$
$863$	−8.75563	−0.298045	−0.149023	+	0.988834i	$0.547613\pi$
$864$	0	0
$865$	10.0383	0.341311
$866$	0	0
$867$	4.21180	0.143040
$868$	0	0
$869$	13.5470	0.459551
$870$	0	0
$871$	3.75767	0.127324
$872$	0	0
$873$	−4.52015	−0.152984
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−16.4399	−0.555138	−0.277569	−	0.960706i	$-0.589529\pi$
$877$	−16.4399	−0.555138	−0.277569	+	0.960706i	$0.589529\pi$
$878$	0	0
$879$	36.5825	1.23390
$880$	0	0
$881$	12.4941	0.420937	0.210468	−	0.977601i	$-0.432501\pi$
$881$	12.4941	0.420937	0.210468	+	0.977601i	$0.432501\pi$
$882$	0	0
$883$	52.7656	1.77570	0.887852	−	0.460129i	$-0.152197\pi$
$883$	52.7656	1.77570	0.887852	+	0.460129i	$0.152197\pi$
$884$	0	0
$885$	−10.0507	−0.337850
$886$	0	0
$887$	15.4337	0.518212	0.259106	−	0.965849i	$-0.416572\pi$
$887$	15.4337	0.518212	0.259106	+	0.965849i	$0.416572\pi$
$888$	0	0
$889$	−7.05488	−0.236613
$890$	0	0
$891$	3.97760	0.133255
$892$	0	0
$893$	64.0189	2.14231
$894$	0	0
$895$	26.5172	0.886373
$896$	0	0
$897$	2.25130	0.0751686
$898$	0	0
$899$	1.05210	0.0350896
$900$	0	0
$901$	−37.2892	−1.24228
$902$	0	0
$903$	−7.57394	−0.252045
$904$	0	0
$905$	12.4587	0.414140
$906$	0	0
$907$	−41.5220	−1.37872	−0.689358	−	0.724421i	$-0.742107\pi$
$907$	−41.5220	−1.37872	−0.689358	+	0.724421i	$0.742107\pi$
$908$	0	0
$909$	2.63016	0.0872368
$910$	0	0
$911$	−6.74146	−0.223354	−0.111677	−	0.993745i	$-0.535622\pi$
$911$	−6.74146	−0.223354	−0.111677	+	0.993745i	$0.535622\pi$
$912$	0	0
$913$	0.0257600	0.000852531 0
$914$	0	0
$915$	1.74870	0.0578104
$916$	0	0
$917$	18.6767	0.616760
$918$	0	0
$919$	−42.5644	−1.40407	−0.702035	−	0.712142i	$-0.747725\pi$
$919$	−42.5644	−1.40407	−0.702035	+	0.712142i	$0.747725\pi$
$920$	0	0
$921$	14.4594	0.476453
$922$	0	0
$923$	−8.15328	−0.268369
$924$	0	0
$925$	6.16847	0.202818
$926$	0	0
$927$	5.50999	0.180972
$928$	0	0
$929$	−8.55743	−0.280760	−0.140380	−	0.990098i	$-0.544832\pi$
$929$	−8.55743	−0.280760	−0.140380	+	0.990098i	$0.544832\pi$
$930$	0	0
$931$	−6.64691	−0.217844
$932$	0	0
$933$	−4.08127	−0.133615
$934$	0	0
$935$	8.33695	0.272647
$936$	0	0
$937$	6.91400	0.225871	0.112935	−	0.993602i	$-0.463975\pi$
$937$	6.91400	0.225871	0.112935	+	0.993602i	$0.463975\pi$
$938$	0	0
$939$	11.2815	0.368157
$940$	0	0
$941$	−8.42153	−0.274534	−0.137267	−	0.990534i	$-0.543832\pi$
$941$	−8.42153	−0.274534	−0.137267	+	0.990534i	$0.543832\pi$
$942$	0	0
$943$	9.38446	0.305600
$944$	0	0
$945$	5.45615	0.177489
$946$	0	0
$947$	1.35207	0.0439364	0.0219682	−	0.999759i	$-0.493007\pi$
$947$	1.35207	0.0439364	0.0219682	+	0.999759i	$0.493007\pi$
$948$	0	0
$949$	−12.8069	−0.415731
$950$	0	0
$951$	−6.97596	−0.226211
$952$	0	0
$953$	−37.9005	−1.22772	−0.613858	−	0.789416i	$-0.710383\pi$
$953$	−37.9005	−1.22772	−0.613858	+	0.789416i	$0.710383\pi$
$954$	0	0
$955$	4.07998	0.132025
$956$	0	0
$957$	3.51751	0.113705
$958$	0	0
$959$	16.4311	0.530588
$960$	0	0
$961$	−30.3451	−0.978875
$962$	0	0
$963$	25.8139	0.831841
$964$	0	0
$965$	11.0408	0.355415
$966$	0	0
$967$	−26.7625	−0.860624	−0.430312	−	0.902680i	$-0.641597\pi$
$967$	−26.7625	−0.860624	−0.430312	+	0.902680i	$0.641597\pi$
$968$	0	0
$969$	−29.0409	−0.932928
$970$	0	0
$971$	−8.93844	−0.286848	−0.143424	−	0.989661i	$-0.545811\pi$
$971$	−8.93844	−0.286848	−0.143424	+	0.989661i	$0.545811\pi$
$972$	0	0
$973$	9.50117	0.304594
$974$	0	0
$975$	1.19075	0.0381346
$976$	0	0
$977$	−15.8972	−0.508596	−0.254298	−	0.967126i	$-0.581844\pi$
$977$	−15.8972	−0.508596	−0.254298	+	0.967126i	$0.581844\pi$
$978$	0	0
$979$	−24.1465	−0.771724
$980$	0	0
$981$	−16.0966	−0.513926
$982$	0	0
$983$	7.46703	0.238161	0.119081	−	0.992885i	$-0.462005\pi$
$983$	7.46703	0.238161	0.119081	+	0.992885i	$0.462005\pi$
$984$	0	0
$985$	8.74600	0.278671
$986$	0	0
$987$	11.4686	0.365049
$988$	0	0
$989$	12.0258	0.382397
$990$	0	0
$991$	−55.2039	−1.75361	−0.876804	−	0.480847i	$-0.840329\pi$
$991$	−55.2039	−1.75361	−0.876804	+	0.480847i	$0.840329\pi$
$992$	0	0
$993$	22.3199	0.708300
$994$	0	0
$995$	−10.1324	−0.321219
$996$	0	0
$997$	9.94802	0.315057	0.157528	−	0.987514i	$-0.449647\pi$
$997$	9.94802	0.315057	0.157528	+	0.987514i	$0.449647\pi$
$998$	0	0
$999$	33.6561	1.06483

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3640.2.a.z.1.2	✓	5
4.3	odd	2		7280.2.a.cd.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3640.2.a.z.1.2	✓	5	1.1	even	1	trivial
7280.2.a.cd.1.4		5	4.3	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:	\( N \)	\(=\)	\( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3640.a (trivial)

\(f(q)\)	\(=\)	\(q-1.19075 q^{3} +1.00000 q^{5} +1.00000 q^{7} -1.58211 q^{9} +O(q^{10})\)	\(q-1.19075 q^{3} +1.00000 q^{5} +1.00000 q^{7} -1.58211 q^{9} -2.27215 q^{11} -1.00000 q^{13} -1.19075 q^{15} -3.66918 q^{17} -6.64691 q^{19} -1.19075 q^{21} +1.89065 q^{23} +1.00000 q^{25} +5.45615 q^{27} +1.30010 q^{29} +0.809249 q^{31} +2.70557 q^{33} +1.00000 q^{35} +6.16847 q^{37} +1.19075 q^{39} +4.96361 q^{41} +6.36064 q^{43} -1.58211 q^{45} -9.63138 q^{47} +1.00000 q^{49} +4.36908 q^{51} +10.1628 q^{53} -2.27215 q^{55} +7.91481 q^{57} +8.44063 q^{59} -1.46857 q^{61} -1.58211 q^{63} -1.00000 q^{65} -3.75767 q^{67} -2.25130 q^{69} +8.15328 q^{71} +12.8069 q^{73} -1.19075 q^{75} -2.27215 q^{77} -5.96219 q^{79} -1.75059 q^{81} -0.0113373 q^{83} -3.66918 q^{85} -1.54809 q^{87} +10.6271 q^{89} -1.00000 q^{91} -0.963614 q^{93} -6.64691 q^{95} +2.85704 q^{97} +3.59480 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{5} + 5 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q + 5 * q^5 + 5 * q^7 + 5 * q^9	\( 5 q + 5 q^{5} + 5 q^{7} + 5 q^{9} + 7 q^{11} - 5 q^{13} + 3 q^{17} + 3 q^{19} + 3 q^{23} + 5 q^{25} - 3 q^{27} + 7 q^{29} + 10 q^{31} + 17 q^{33} + 5 q^{35} + 10 q^{37} + 4 q^{43} + 5 q^{45} - 3 q^{47} + 5 q^{49} + 26 q^{53} + 7 q^{55} - 20 q^{57} + 3 q^{59} + 13 q^{61} + 5 q^{63} - 5 q^{65} + 12 q^{67} + 23 q^{69} + 8 q^{71} + q^{73} + 7 q^{77} - 6 q^{79} + 25 q^{81} - 8 q^{83} + 3 q^{85} - 43 q^{87} + 3 q^{89} - 5 q^{91} + 20 q^{93} + 3 q^{95} - 15 q^{97} + 16 q^{99}+O(q^{100}) \) 5 * q + 5 * q^5 + 5 * q^7 + 5 * q^9 + 7 * q^11 - 5 * q^13 + 3 * q^17 + 3 * q^19 + 3 * q^23 + 5 * q^25 - 3 * q^27 + 7 * q^29 + 10 * q^31 + 17 * q^33 + 5 * q^35 + 10 * q^37 + 4 * q^43 + 5 * q^45 - 3 * q^47 + 5 * q^49 + 26 * q^53 + 7 * q^55 - 20 * q^57 + 3 * q^59 + 13 * q^61 + 5 * q^63 - 5 * q^65 + 12 * q^67 + 23 * q^69 + 8 * q^71 + q^73 + 7 * q^77 - 6 * q^79 + 25 * q^81 - 8 * q^83 + 3 * q^85 - 43 * q^87 + 3 * q^89 - 5 * q^91 + 20 * q^93 + 3 * q^95 - 15 * q^97 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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