LMFDB - Embedded newform 3680.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3680,2,Mod(1,3680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3680, 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("3680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3680 = 2^{5} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3680.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:	$29.3849479438$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.30278$ of defining polynomial
Character	$\chi$	$=$	3680.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.30278 q^{3} -1.00000 q^{5} -0.302776 q^{7} +7.90833 q^{9} +O(q^{10})$	$q-3.30278 q^{3} -1.00000 q^{5} -0.302776 q^{7} +7.90833 q^{9} +3.30278 q^{11} -2.30278 q^{13} +3.30278 q^{15} +7.30278 q^{17} -2.90833 q^{19} +1.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} -16.2111 q^{27} +8.60555 q^{29} +4.30278 q^{31} -10.9083 q^{33} +0.302776 q^{35} +5.21110 q^{37} +7.60555 q^{39} -4.69722 q^{41} -4.00000 q^{43} -7.90833 q^{45} +3.39445 q^{47} -6.90833 q^{49} -24.1194 q^{51} -7.21110 q^{53} -3.30278 q^{55} +9.60555 q^{57} -1.39445 q^{59} +1.30278 q^{61} -2.39445 q^{63} +2.30278 q^{65} -9.21110 q^{67} +3.30278 q^{69} +5.30278 q^{71} +5.39445 q^{73} -3.30278 q^{75} -1.00000 q^{77} -4.00000 q^{79} +29.8167 q^{81} -11.2111 q^{83} -7.30278 q^{85} -28.4222 q^{87} +2.78890 q^{89} +0.697224 q^{91} -14.2111 q^{93} +2.90833 q^{95} -13.7250 q^{97} +26.1194 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{3} - 2 q^{5} + 3 q^{7} + 5 q^{9}+O(q^{10}) $ 2 * q - 3 * q^3 - 2 * q^5 + 3 * q^7 + 5 * q^9	$ 2 q - 3 q^{3} - 2 q^{5} + 3 q^{7} + 5 q^{9} + 3 q^{11} - q^{13} + 3 q^{15} + 11 q^{17} + 5 q^{19} + 2 q^{21} - 2 q^{23} + 2 q^{25} - 18 q^{27} + 10 q^{29} + 5 q^{31} - 11 q^{33} - 3 q^{35} - 4 q^{37} + 8 q^{39} - 13 q^{41} - 8 q^{43} - 5 q^{45} + 14 q^{47} - 3 q^{49} - 23 q^{51} - 3 q^{55} + 12 q^{57} - 10 q^{59} - q^{61} - 12 q^{63} + q^{65} - 4 q^{67} + 3 q^{69} + 7 q^{71} + 18 q^{73} - 3 q^{75} - 2 q^{77} - 8 q^{79} + 38 q^{81} - 8 q^{83} - 11 q^{85} - 28 q^{87} + 20 q^{89} + 5 q^{91} - 14 q^{93} - 5 q^{95} + 5 q^{97} + 27 q^{99}+O(q^{100}) $ 2 * q - 3 * q^3 - 2 * q^5 + 3 * q^7 + 5 * q^9 + 3 * q^11 - q^13 + 3 * q^15 + 11 * q^17 + 5 * q^19 + 2 * q^21 - 2 * q^23 + 2 * q^25 - 18 * q^27 + 10 * q^29 + 5 * q^31 - 11 * q^33 - 3 * q^35 - 4 * q^37 + 8 * q^39 - 13 * q^41 - 8 * q^43 - 5 * q^45 + 14 * q^47 - 3 * q^49 - 23 * q^51 - 3 * q^55 + 12 * q^57 - 10 * q^59 - q^61 - 12 * q^63 + q^65 - 4 * q^67 + 3 * q^69 + 7 * q^71 + 18 * q^73 - 3 * q^75 - 2 * q^77 - 8 * q^79 + 38 * q^81 - 8 * q^83 - 11 * q^85 - 28 * q^87 + 20 * q^89 + 5 * q^91 - 14 * q^93 - 5 * q^95 + 5 * q^97 + 27 * 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
$3$	−3.30278	−1.90686	−0.953429	−	0.301617i	$-0.902474\pi$
$3$	−3.30278	−1.90686	−0.953429	+	0.301617i	$0.902474\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−0.302776	−0.114438	−0.0572192	−	0.998362i	$-0.518223\pi$
$7$	−0.302776	−0.114438	−0.0572192	+	0.998362i	$0.518223\pi$
$8$	0	0
$9$	7.90833	2.63611
$10$	0	0
$11$	3.30278	0.995824	0.497912	−	0.867227i	$-0.334100\pi$
$11$	3.30278	0.995824	0.497912	+	0.867227i	$0.334100\pi$
$12$	0	0
$13$	−2.30278	−0.638675	−0.319338	−	0.947641i	$-0.603460\pi$
$13$	−2.30278	−0.638675	−0.319338	+	0.947641i	$0.603460\pi$
$14$	0	0
$15$	3.30278	0.852773
$16$	0	0
$17$	7.30278	1.77118	0.885592	−	0.464465i	$-0.153753\pi$
$17$	7.30278	1.77118	0.885592	+	0.464465i	$0.153753\pi$
$18$	0	0
$19$	−2.90833	−0.667216	−0.333608	−	0.942712i	$-0.608266\pi$
$19$	−2.90833	−0.667216	−0.333608	+	0.942712i	$0.608266\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−16.2111	−3.11983
$28$	0	0
$29$	8.60555	1.59801	0.799005	−	0.601324i	$-0.205360\pi$
$29$	8.60555	1.59801	0.799005	+	0.601324i	$0.205360\pi$
$30$	0	0
$31$	4.30278	0.772801	0.386401	−	0.922331i	$-0.373718\pi$
$31$	4.30278	0.772801	0.386401	+	0.922331i	$0.373718\pi$
$32$	0	0
$33$	−10.9083	−1.89890
$34$	0	0
$35$	0.302776	0.0511784
$36$	0	0
$37$	5.21110	0.856700	0.428350	−	0.903613i	$-0.359095\pi$
$37$	5.21110	0.856700	0.428350	+	0.903613i	$0.359095\pi$
$38$	0	0
$39$	7.60555	1.21786
$40$	0	0
$41$	−4.69722	−0.733583	−0.366792	−	0.930303i	$-0.619544\pi$
$41$	−4.69722	−0.733583	−0.366792	+	0.930303i	$0.619544\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	−7.90833	−1.17890
$46$	0	0
$47$	3.39445	0.495131	0.247566	−	0.968871i	$-0.420369\pi$
$47$	3.39445	0.495131	0.247566	+	0.968871i	$0.420369\pi$
$48$	0	0
$49$	−6.90833	−0.986904
$50$	0	0
$51$	−24.1194	−3.37740
$52$	0	0
$53$	−7.21110	−0.990521	−0.495261	−	0.868744i	$-0.664927\pi$
$53$	−7.21110	−0.990521	−0.495261	+	0.868744i	$0.664927\pi$
$54$	0	0
$55$	−3.30278	−0.445346
$56$	0	0
$57$	9.60555	1.27229
$58$	0	0
$59$	−1.39445	−0.181542	−0.0907709	−	0.995872i	$-0.528933\pi$
$59$	−1.39445	−0.181542	−0.0907709	+	0.995872i	$0.528933\pi$
$60$	0	0
$61$	1.30278	0.166803	0.0834017	−	0.996516i	$-0.473422\pi$
$61$	1.30278	0.166803	0.0834017	+	0.996516i	$0.473422\pi$
$62$	0	0
$63$	−2.39445	−0.301672
$64$	0	0
$65$	2.30278	0.285624
$66$	0	0
$67$	−9.21110	−1.12532	−0.562658	−	0.826690i	$-0.690221\pi$
$67$	−9.21110	−1.12532	−0.562658	+	0.826690i	$0.690221\pi$
$68$	0	0
$69$	3.30278	0.397607
$70$	0	0
$71$	5.30278	0.629324	0.314662	−	0.949204i	$-0.398109\pi$
$71$	5.30278	0.629324	0.314662	+	0.949204i	$0.398109\pi$
$72$	0	0
$73$	5.39445	0.631372	0.315686	−	0.948864i	$-0.397765\pi$
$73$	5.39445	0.631372	0.315686	+	0.948864i	$0.397765\pi$
$74$	0	0
$75$	−3.30278	−0.381372
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	29.8167	3.31296
$82$	0	0
$83$	−11.2111	−1.23058	−0.615289	−	0.788301i	$-0.710961\pi$
$83$	−11.2111	−1.23058	−0.615289	+	0.788301i	$0.710961\pi$
$84$	0	0
$85$	−7.30278	−0.792097
$86$	0	0
$87$	−28.4222	−3.04718
$88$	0	0
$89$	2.78890	0.295623	0.147811	−	0.989016i	$-0.452777\pi$
$89$	2.78890	0.295623	0.147811	+	0.989016i	$0.452777\pi$
$90$	0	0
$91$	0.697224	0.0730890
$92$	0	0
$93$	−14.2111	−1.47362
$94$	0	0
$95$	2.90833	0.298388
$96$	0	0
$97$	−13.7250	−1.39356	−0.696780	−	0.717285i	$-0.745385\pi$
$97$	−13.7250	−1.39356	−0.696780	+	0.717285i	$0.745385\pi$
$98$	0	0
$99$	26.1194	2.62510
$100$	0	0
$101$	12.6056	1.25430	0.627150	−	0.778899i	$-0.284221\pi$
$101$	12.6056	1.25430	0.627150	+	0.778899i	$0.284221\pi$
$102$	0	0
$103$	−9.11943	−0.898564	−0.449282	−	0.893390i	$-0.648320\pi$
$103$	−9.11943	−0.898564	−0.449282	+	0.893390i	$0.648320\pi$
$104$	0	0
$105$	−1.00000	−0.0975900
$106$	0	0
$107$	1.81665	0.175623	0.0878113	−	0.996137i	$-0.472013\pi$
$107$	1.81665	0.175623	0.0878113	+	0.996137i	$0.472013\pi$
$108$	0	0
$109$	13.6972	1.31196	0.655978	−	0.754780i	$-0.272256\pi$
$109$	13.6972	1.31196	0.655978	+	0.754780i	$0.272256\pi$
$110$	0	0
$111$	−17.2111	−1.63361
$112$	0	0
$113$	−3.21110	−0.302075	−0.151038	−	0.988528i	$-0.548261\pi$
$113$	−3.21110	−0.302075	−0.151038	+	0.988528i	$0.548261\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	−18.2111	−1.68362
$118$	0	0
$119$	−2.21110	−0.202691
$120$	0	0
$121$	−0.0916731	−0.00833392
$122$	0	0
$123$	15.5139	1.39884
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	13.3944	1.18857	0.594283	−	0.804256i	$-0.297436\pi$
$127$	13.3944	1.18857	0.594283	+	0.804256i	$0.297436\pi$
$128$	0	0
$129$	13.2111	1.16317
$130$	0	0
$131$	−0.788897	−0.0689263	−0.0344631	−	0.999406i	$-0.510972\pi$
$131$	−0.788897	−0.0689263	−0.0344631	+	0.999406i	$0.510972\pi$
$132$	0	0
$133$	0.880571	0.0763551
$134$	0	0
$135$	16.2111	1.39523
$136$	0	0
$137$	18.1194	1.54805	0.774024	−	0.633157i	$-0.218241\pi$
$137$	18.1194	1.54805	0.774024	+	0.633157i	$0.218241\pi$
$138$	0	0
$139$	11.8167	1.00228	0.501138	−	0.865368i	$-0.332915\pi$
$139$	11.8167	1.00228	0.501138	+	0.865368i	$0.332915\pi$
$140$	0	0
$141$	−11.2111	−0.944145
$142$	0	0
$143$	−7.60555	−0.636008
$144$	0	0
$145$	−8.60555	−0.714652
$146$	0	0
$147$	22.8167	1.88189
$148$	0	0
$149$	−2.90833	−0.238259	−0.119130	−	0.992879i	$-0.538010\pi$
$149$	−2.90833	−0.238259	−0.119130	+	0.992879i	$0.538010\pi$
$150$	0	0
$151$	15.7250	1.27968	0.639840	−	0.768508i	$-0.279000\pi$
$151$	15.7250	1.27968	0.639840	+	0.768508i	$0.279000\pi$
$152$	0	0
$153$	57.7527	4.66903
$154$	0	0
$155$	−4.30278	−0.345607
$156$	0	0
$157$	−3.39445	−0.270907	−0.135453	−	0.990784i	$-0.543249\pi$
$157$	−3.39445	−0.270907	−0.135453	+	0.990784i	$0.543249\pi$
$158$	0	0
$159$	23.8167	1.88878
$160$	0	0
$161$	0.302776	0.0238621
$162$	0	0
$163$	−6.90833	−0.541102	−0.270551	−	0.962706i	$-0.587206\pi$
$163$	−6.90833	−0.541102	−0.270551	+	0.962706i	$0.587206\pi$
$164$	0	0
$165$	10.9083	0.849212
$166$	0	0
$167$	13.2111	1.02231	0.511153	−	0.859490i	$-0.329219\pi$
$167$	13.2111	1.02231	0.511153	+	0.859490i	$0.329219\pi$
$168$	0	0
$169$	−7.69722	−0.592094
$170$	0	0
$171$	−23.0000	−1.75885
$172$	0	0
$173$	−14.5139	−1.10347	−0.551735	−	0.834020i	$-0.686034\pi$
$173$	−14.5139	−1.10347	−0.551735	+	0.834020i	$0.686034\pi$
$174$	0	0
$175$	−0.302776	−0.0228877
$176$	0	0
$177$	4.60555	0.346174
$178$	0	0
$179$	23.0278	1.72118	0.860588	−	0.509302i	$-0.170097\pi$
$179$	23.0278	1.72118	0.860588	+	0.509302i	$0.170097\pi$
$180$	0	0
$181$	21.6972	1.61274	0.806371	−	0.591410i	$-0.201429\pi$
$181$	21.6972	1.61274	0.806371	+	0.591410i	$0.201429\pi$
$182$	0	0
$183$	−4.30278	−0.318070
$184$	0	0
$185$	−5.21110	−0.383128
$186$	0	0
$187$	24.1194	1.76379
$188$	0	0
$189$	4.90833	0.357028
$190$	0	0
$191$	−0.183346	−0.0132665	−0.00663323	−	0.999978i	$-0.502111\pi$
$191$	−0.183346	−0.0132665	−0.00663323	+	0.999978i	$0.502111\pi$
$192$	0	0
$193$	9.39445	0.676227	0.338114	−	0.941105i	$-0.390211\pi$
$193$	9.39445	0.676227	0.338114	+	0.941105i	$0.390211\pi$
$194$	0	0
$195$	−7.60555	−0.544645
$196$	0	0
$197$	22.5139	1.60405	0.802024	−	0.597292i	$-0.203757\pi$
$197$	22.5139	1.60405	0.802024	+	0.597292i	$0.203757\pi$
$198$	0	0
$199$	−8.42221	−0.597034	−0.298517	−	0.954404i	$-0.596492\pi$
$199$	−8.42221	−0.597034	−0.298517	+	0.954404i	$0.596492\pi$
$200$	0	0
$201$	30.4222	2.14582
$202$	0	0
$203$	−2.60555	−0.182874
$204$	0	0
$205$	4.69722	0.328068
$206$	0	0
$207$	−7.90833	−0.549667
$208$	0	0
$209$	−9.60555	−0.664430
$210$	0	0
$211$	−28.4222	−1.95667	−0.978333	−	0.207039i	$-0.933617\pi$
$211$	−28.4222	−1.95667	−0.978333	+	0.207039i	$0.933617\pi$
$212$	0	0
$213$	−17.5139	−1.20003
$214$	0	0
$215$	4.00000	0.272798
$216$	0	0
$217$	−1.30278	−0.0884382
$218$	0	0
$219$	−17.8167	−1.20394
$220$	0	0
$221$	−16.8167	−1.13121
$222$	0	0
$223$	−14.4222	−0.965782	−0.482891	−	0.875680i	$-0.660413\pi$
$223$	−14.4222	−0.965782	−0.482891	+	0.875680i	$0.660413\pi$
$224$	0	0
$225$	7.90833	0.527222
$226$	0	0
$227$	25.8167	1.71351	0.856756	−	0.515722i	$-0.172476\pi$
$227$	25.8167	1.71351	0.856756	+	0.515722i	$0.172476\pi$
$228$	0	0
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	3.30278	0.217307
$232$	0	0
$233$	22.2389	1.45692	0.728458	−	0.685090i	$-0.240237\pi$
$233$	22.2389	1.45692	0.728458	+	0.685090i	$0.240237\pi$
$234$	0	0
$235$	−3.39445	−0.221429
$236$	0	0
$237$	13.2111	0.858153
$238$	0	0
$239$	29.2111	1.88951	0.944755	−	0.327779i	$-0.106300\pi$
$239$	29.2111	1.88951	0.944755	+	0.327779i	$0.106300\pi$
$240$	0	0
$241$	−27.6333	−1.78002	−0.890009	−	0.455943i	$-0.849302\pi$
$241$	−27.6333	−1.78002	−0.890009	+	0.455943i	$0.849302\pi$
$242$	0	0
$243$	−49.8444	−3.19752
$244$	0	0
$245$	6.90833	0.441357
$246$	0	0
$247$	6.69722	0.426134
$248$	0	0
$249$	37.0278	2.34654
$250$	0	0
$251$	8.48612	0.535639	0.267820	−	0.963469i	$-0.413697\pi$
$251$	8.48612	0.535639	0.267820	+	0.963469i	$0.413697\pi$
$252$	0	0
$253$	−3.30278	−0.207644
$254$	0	0
$255$	24.1194	1.51042
$256$	0	0
$257$	−26.6056	−1.65961	−0.829804	−	0.558054i	$-0.811548\pi$
$257$	−26.6056	−1.65961	−0.829804	+	0.558054i	$0.811548\pi$
$258$	0	0
$259$	−1.57779	−0.0980394
$260$	0	0
$261$	68.0555	4.21253
$262$	0	0
$263$	19.9083	1.22760	0.613800	−	0.789462i	$-0.289640\pi$
$263$	19.9083	1.22760	0.613800	+	0.789462i	$0.289640\pi$
$264$	0	0
$265$	7.21110	0.442975
$266$	0	0
$267$	−9.21110	−0.563710
$268$	0	0
$269$	−23.0278	−1.40403	−0.702014	−	0.712164i	$-0.747715\pi$
$269$	−23.0278	−1.40403	−0.702014	+	0.712164i	$0.747715\pi$
$270$	0	0
$271$	28.3028	1.71927	0.859636	−	0.510908i	$-0.170691\pi$
$271$	28.3028	1.71927	0.859636	+	0.510908i	$0.170691\pi$
$272$	0	0
$273$	−2.30278	−0.139370
$274$	0	0
$275$	3.30278	0.199165
$276$	0	0
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	0	0
$279$	34.0278	2.03719
$280$	0	0
$281$	−15.0278	−0.896481	−0.448240	−	0.893913i	$-0.647949\pi$
$281$	−15.0278	−0.896481	−0.448240	+	0.893913i	$0.647949\pi$
$282$	0	0
$283$	29.6333	1.76152	0.880759	−	0.473565i	$-0.157033\pi$
$283$	29.6333	1.76152	0.880759	+	0.473565i	$0.157033\pi$
$284$	0	0
$285$	−9.60555	−0.568984
$286$	0	0
$287$	1.42221	0.0839501
$288$	0	0
$289$	36.3305	2.13709
$290$	0	0
$291$	45.3305	2.65732
$292$	0	0
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	0	0
$295$	1.39445	0.0811879
$296$	0	0
$297$	−53.5416	−3.10680
$298$	0	0
$299$	2.30278	0.133173
$300$	0	0
$301$	1.21110	0.0698068
$302$	0	0
$303$	−41.6333	−2.39177
$304$	0	0
$305$	−1.30278	−0.0745967
$306$	0	0
$307$	−5.90833	−0.337206	−0.168603	−	0.985684i	$-0.553926\pi$
$307$	−5.90833	−0.337206	−0.168603	+	0.985684i	$0.553926\pi$
$308$	0	0
$309$	30.1194	1.71343
$310$	0	0
$311$	−10.4222	−0.590989	−0.295495	−	0.955344i	$-0.595484\pi$
$311$	−10.4222	−0.590989	−0.295495	+	0.955344i	$0.595484\pi$
$312$	0	0
$313$	26.7250	1.51059	0.755293	−	0.655388i	$-0.227495\pi$
$313$	26.7250	1.51059	0.755293	+	0.655388i	$0.227495\pi$
$314$	0	0
$315$	2.39445	0.134912
$316$	0	0
$317$	−30.3028	−1.70197	−0.850987	−	0.525187i	$-0.823995\pi$
$317$	−30.3028	−1.70197	−0.850987	+	0.525187i	$0.823995\pi$
$318$	0	0
$319$	28.4222	1.59134
$320$	0	0
$321$	−6.00000	−0.334887
$322$	0	0
$323$	−21.2389	−1.18176
$324$	0	0
$325$	−2.30278	−0.127735
$326$	0	0
$327$	−45.2389	−2.50171
$328$	0	0
$329$	−1.02776	−0.0566620
$330$	0	0
$331$	3.81665	0.209782	0.104891	−	0.994484i	$-0.466551\pi$
$331$	3.81665	0.209782	0.104891	+	0.994484i	$0.466551\pi$
$332$	0	0
$333$	41.2111	2.25835
$334$	0	0
$335$	9.21110	0.503256
$336$	0	0
$337$	−7.09167	−0.386308	−0.193154	−	0.981168i	$-0.561872\pi$
$337$	−7.09167	−0.386308	−0.193154	+	0.981168i	$0.561872\pi$
$338$	0	0
$339$	10.6056	0.576014
$340$	0	0
$341$	14.2111	0.769574
$342$	0	0
$343$	4.21110	0.227378
$344$	0	0
$345$	−3.30278	−0.177815
$346$	0	0
$347$	−14.0917	−0.756481	−0.378240	−	0.925707i	$-0.623471\pi$
$347$	−14.0917	−0.756481	−0.378240	+	0.925707i	$0.623471\pi$
$348$	0	0
$349$	−17.6333	−0.943889	−0.471945	−	0.881628i	$-0.656448\pi$
$349$	−17.6333	−0.943889	−0.471945	+	0.881628i	$0.656448\pi$
$350$	0	0
$351$	37.3305	1.99256
$352$	0	0
$353$	−21.2111	−1.12895	−0.564477	−	0.825449i	$-0.690922\pi$
$353$	−21.2111	−1.12895	−0.564477	+	0.825449i	$0.690922\pi$
$354$	0	0
$355$	−5.30278	−0.281442
$356$	0	0
$357$	7.30278	0.386504
$358$	0	0
$359$	−16.4222	−0.866731	−0.433365	−	0.901218i	$-0.642674\pi$
$359$	−16.4222	−0.866731	−0.433365	+	0.901218i	$0.642674\pi$
$360$	0	0
$361$	−10.5416	−0.554823
$362$	0	0
$363$	0.302776	0.0158916
$364$	0	0
$365$	−5.39445	−0.282358
$366$	0	0
$367$	−21.2111	−1.10721	−0.553605	−	0.832779i	$-0.686748\pi$
$367$	−21.2111	−1.10721	−0.553605	+	0.832779i	$0.686748\pi$
$368$	0	0
$369$	−37.1472	−1.93381
$370$	0	0
$371$	2.18335	0.113354
$372$	0	0
$373$	26.6056	1.37758	0.688792	−	0.724959i	$-0.258141\pi$
$373$	26.6056	1.37758	0.688792	+	0.724959i	$0.258141\pi$
$374$	0	0
$375$	3.30278	0.170555
$376$	0	0
$377$	−19.8167	−1.02061
$378$	0	0
$379$	37.5416	1.92838	0.964192	−	0.265205i	$-0.0854396\pi$
$379$	37.5416	1.92838	0.964192	+	0.265205i	$0.0854396\pi$
$380$	0	0
$381$	−44.2389	−2.26643
$382$	0	0
$383$	26.4222	1.35011	0.675056	−	0.737767i	$-0.264120\pi$
$383$	26.4222	1.35011	0.675056	+	0.737767i	$0.264120\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	0	0
$387$	−31.6333	−1.60801
$388$	0	0
$389$	−18.5139	−0.938691	−0.469345	−	0.883015i	$-0.655510\pi$
$389$	−18.5139	−0.938691	−0.469345	+	0.883015i	$0.655510\pi$
$390$	0	0
$391$	−7.30278	−0.369317
$392$	0	0
$393$	2.60555	0.131433
$394$	0	0
$395$	4.00000	0.201262
$396$	0	0
$397$	10.5139	0.527676	0.263838	−	0.964567i	$-0.415011\pi$
$397$	10.5139	0.527676	0.263838	+	0.964567i	$0.415011\pi$
$398$	0	0
$399$	−2.90833	−0.145598
$400$	0	0
$401$	−19.8167	−0.989596	−0.494798	−	0.869008i	$-0.664758\pi$
$401$	−19.8167	−0.989596	−0.494798	+	0.869008i	$0.664758\pi$
$402$	0	0
$403$	−9.90833	−0.493569
$404$	0	0
$405$	−29.8167	−1.48160
$406$	0	0
$407$	17.2111	0.853123
$408$	0	0
$409$	−15.0917	−0.746235	−0.373118	−	0.927784i	$-0.621711\pi$
$409$	−15.0917	−0.746235	−0.373118	+	0.927784i	$0.621711\pi$
$410$	0	0
$411$	−59.8444	−2.95191
$412$	0	0
$413$	0.422205	0.0207754
$414$	0	0
$415$	11.2111	0.550331
$416$	0	0
$417$	−39.0278	−1.91120
$418$	0	0
$419$	9.21110	0.449992	0.224996	−	0.974360i	$-0.427763\pi$
$419$	9.21110	0.449992	0.224996	+	0.974360i	$0.427763\pi$
$420$	0	0
$421$	−1.09167	−0.0532049	−0.0266024	−	0.999646i	$-0.508469\pi$
$421$	−1.09167	−0.0532049	−0.0266024	+	0.999646i	$0.508469\pi$
$422$	0	0
$423$	26.8444	1.30522
$424$	0	0
$425$	7.30278	0.354237
$426$	0	0
$427$	−0.394449	−0.0190887
$428$	0	0
$429$	25.1194	1.21278
$430$	0	0
$431$	−8.60555	−0.414515	−0.207257	−	0.978286i	$-0.566454\pi$
$431$	−8.60555	−0.414515	−0.207257	+	0.978286i	$0.566454\pi$
$432$	0	0
$433$	35.6972	1.71550	0.857750	−	0.514068i	$-0.171862\pi$
$433$	35.6972	1.71550	0.857750	+	0.514068i	$0.171862\pi$
$434$	0	0
$435$	28.4222	1.36274
$436$	0	0
$437$	2.90833	0.139124
$438$	0	0
$439$	18.7250	0.893695	0.446847	−	0.894610i	$-0.352547\pi$
$439$	18.7250	0.893695	0.446847	+	0.894610i	$0.352547\pi$
$440$	0	0
$441$	−54.6333	−2.60159
$442$	0	0
$443$	1.33053	0.0632155	0.0316077	−	0.999500i	$-0.489937\pi$
$443$	1.33053	0.0632155	0.0316077	+	0.999500i	$0.489937\pi$
$444$	0	0
$445$	−2.78890	−0.132206
$446$	0	0
$447$	9.60555	0.454327
$448$	0	0
$449$	−2.48612	−0.117327	−0.0586637	−	0.998278i	$-0.518684\pi$
$449$	−2.48612	−0.117327	−0.0586637	+	0.998278i	$0.518684\pi$
$450$	0	0
$451$	−15.5139	−0.730520
$452$	0	0
$453$	−51.9361	−2.44017
$454$	0	0
$455$	−0.697224	−0.0326864
$456$	0	0
$457$	15.2111	0.711545	0.355773	−	0.934573i	$-0.384218\pi$
$457$	15.2111	0.711545	0.355773	+	0.934573i	$0.384218\pi$
$458$	0	0
$459$	−118.386	−5.52579
$460$	0	0
$461$	35.4500	1.65107	0.825535	−	0.564351i	$-0.190874\pi$
$461$	35.4500	1.65107	0.825535	+	0.564351i	$0.190874\pi$
$462$	0	0
$463$	−12.7889	−0.594350	−0.297175	−	0.954823i	$-0.596045\pi$
$463$	−12.7889	−0.594350	−0.297175	+	0.954823i	$0.596045\pi$
$464$	0	0
$465$	14.2111	0.659024
$466$	0	0
$467$	17.0278	0.787951	0.393975	−	0.919121i	$-0.371100\pi$
$467$	17.0278	0.787951	0.393975	+	0.919121i	$0.371100\pi$
$468$	0	0
$469$	2.78890	0.128779
$470$	0	0
$471$	11.2111	0.516580
$472$	0	0
$473$	−13.2111	−0.607447
$474$	0	0
$475$	−2.90833	−0.133443
$476$	0	0
$477$	−57.0278	−2.61112
$478$	0	0
$479$	−6.00000	−0.274147	−0.137073	−	0.990561i	$-0.543770\pi$
$479$	−6.00000	−0.274147	−0.137073	+	0.990561i	$0.543770\pi$
$480$	0	0
$481$	−12.0000	−0.547153
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	13.7250	0.623219
$486$	0	0
$487$	39.4500	1.78765	0.893824	−	0.448418i	$-0.148012\pi$
$487$	39.4500	1.78765	0.893824	+	0.448418i	$0.148012\pi$
$488$	0	0
$489$	22.8167	1.03180
$490$	0	0
$491$	−8.60555	−0.388363	−0.194182	−	0.980966i	$-0.562205\pi$
$491$	−8.60555	−0.388363	−0.194182	+	0.980966i	$0.562205\pi$
$492$	0	0
$493$	62.8444	2.83037
$494$	0	0
$495$	−26.1194	−1.17398
$496$	0	0
$497$	−1.60555	−0.0720188
$498$	0	0
$499$	13.2111	0.591410	0.295705	−	0.955279i	$-0.404445\pi$
$499$	13.2111	0.591410	0.295705	+	0.955279i	$0.404445\pi$
$500$	0	0
$501$	−43.6333	−1.94939
$502$	0	0
$503$	−9.51388	−0.424203	−0.212101	−	0.977248i	$-0.568031\pi$
$503$	−9.51388	−0.424203	−0.212101	+	0.977248i	$0.568031\pi$
$504$	0	0
$505$	−12.6056	−0.560940
$506$	0	0
$507$	25.4222	1.12904
$508$	0	0
$509$	19.3944	0.859644	0.429822	−	0.902914i	$-0.358576\pi$
$509$	19.3944	0.859644	0.429822	+	0.902914i	$0.358576\pi$
$510$	0	0
$511$	−1.63331	−0.0722533
$512$	0	0
$513$	47.1472	2.08160
$514$	0	0
$515$	9.11943	0.401850
$516$	0	0
$517$	11.2111	0.493064
$518$	0	0
$519$	47.9361	2.10416
$520$	0	0
$521$	−36.4222	−1.59569	−0.797843	−	0.602865i	$-0.794026\pi$
$521$	−36.4222	−1.59569	−0.797843	+	0.602865i	$0.794026\pi$
$522$	0	0
$523$	26.0000	1.13690	0.568450	−	0.822718i	$-0.307543\pi$
$523$	26.0000	1.13690	0.568450	+	0.822718i	$0.307543\pi$
$524$	0	0
$525$	1.00000	0.0436436
$526$	0	0
$527$	31.4222	1.36877
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−11.0278	−0.478564
$532$	0	0
$533$	10.8167	0.468521
$534$	0	0
$535$	−1.81665	−0.0785408
$536$	0	0
$537$	−76.0555	−3.28204
$538$	0	0
$539$	−22.8167	−0.982783
$540$	0	0
$541$	26.4222	1.13598	0.567990	−	0.823036i	$-0.307721\pi$
$541$	26.4222	1.13598	0.567990	+	0.823036i	$0.307721\pi$
$542$	0	0
$543$	−71.6611	−3.07527
$544$	0	0
$545$	−13.6972	−0.586725
$546$	0	0
$547$	2.90833	0.124351	0.0621755	−	0.998065i	$-0.480196\pi$
$547$	2.90833	0.124351	0.0621755	+	0.998065i	$0.480196\pi$
$548$	0	0
$549$	10.3028	0.439712
$550$	0	0
$551$	−25.0278	−1.06622
$552$	0	0
$553$	1.21110	0.0515013
$554$	0	0
$555$	17.2111	0.730571
$556$	0	0
$557$	5.21110	0.220802	0.110401	−	0.993887i	$-0.464787\pi$
$557$	5.21110	0.220802	0.110401	+	0.993887i	$0.464787\pi$
$558$	0	0
$559$	9.21110	0.389588
$560$	0	0
$561$	−79.6611	−3.36329
$562$	0	0
$563$	−2.78890	−0.117538	−0.0587690	−	0.998272i	$-0.518718\pi$
$563$	−2.78890	−0.117538	−0.0587690	+	0.998272i	$0.518718\pi$
$564$	0	0
$565$	3.21110	0.135092
$566$	0	0
$567$	−9.02776	−0.379130
$568$	0	0
$569$	−18.8444	−0.789999	−0.394999	−	0.918681i	$-0.629255\pi$
$569$	−18.8444	−0.789999	−0.394999	+	0.918681i	$0.629255\pi$
$570$	0	0
$571$	7.30278	0.305612	0.152806	−	0.988256i	$-0.451169\pi$
$571$	7.30278	0.305612	0.152806	+	0.988256i	$0.451169\pi$
$572$	0	0
$573$	0.605551	0.0252973
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	7.21110	0.300202	0.150101	−	0.988671i	$-0.452040\pi$
$577$	7.21110	0.300202	0.150101	+	0.988671i	$0.452040\pi$
$578$	0	0
$579$	−31.0278	−1.28947
$580$	0	0
$581$	3.39445	0.140825
$582$	0	0
$583$	−23.8167	−0.986385
$584$	0	0
$585$	18.2111	0.752936
$586$	0	0
$587$	−30.9361	−1.27687	−0.638434	−	0.769676i	$-0.720418\pi$
$587$	−30.9361	−1.27687	−0.638434	+	0.769676i	$0.720418\pi$
$588$	0	0
$589$	−12.5139	−0.515625
$590$	0	0
$591$	−74.3583	−3.05869
$592$	0	0
$593$	7.81665	0.320991	0.160496	−	0.987037i	$-0.448691\pi$
$593$	7.81665	0.320991	0.160496	+	0.987037i	$0.448691\pi$
$594$	0	0
$595$	2.21110	0.0906464
$596$	0	0
$597$	27.8167	1.13846
$598$	0	0
$599$	15.9083	0.649997	0.324998	−	0.945715i	$-0.394636\pi$
$599$	15.9083	0.649997	0.324998	+	0.945715i	$0.394636\pi$
$600$	0	0
$601$	44.5416	1.81689	0.908446	−	0.418003i	$-0.137270\pi$
$601$	44.5416	1.81689	0.908446	+	0.418003i	$0.137270\pi$
$602$	0	0
$603$	−72.8444	−2.96645
$604$	0	0
$605$	0.0916731	0.00372704
$606$	0	0
$607$	35.6333	1.44631	0.723156	−	0.690685i	$-0.242691\pi$
$607$	35.6333	1.44631	0.723156	+	0.690685i	$0.242691\pi$
$608$	0	0
$609$	8.60555	0.348715
$610$	0	0
$611$	−7.81665	−0.316228
$612$	0	0
$613$	36.0555	1.45627	0.728134	−	0.685435i	$-0.240388\pi$
$613$	36.0555	1.45627	0.728134	+	0.685435i	$0.240388\pi$
$614$	0	0
$615$	−15.5139	−0.625580
$616$	0	0
$617$	8.51388	0.342756	0.171378	−	0.985205i	$-0.445178\pi$
$617$	8.51388	0.342756	0.171378	+	0.985205i	$0.445178\pi$
$618$	0	0
$619$	−28.7250	−1.15455	−0.577277	−	0.816548i	$-0.695885\pi$
$619$	−28.7250	−1.15455	−0.577277	+	0.816548i	$0.695885\pi$
$620$	0	0
$621$	16.2111	0.650529
$622$	0	0
$623$	−0.844410	−0.0338306
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	31.7250	1.26697
$628$	0	0
$629$	38.0555	1.51737
$630$	0	0
$631$	−1.39445	−0.0555121	−0.0277561	−	0.999615i	$-0.508836\pi$
$631$	−1.39445	−0.0555121	−0.0277561	+	0.999615i	$0.508836\pi$
$632$	0	0
$633$	93.8722	3.73108
$634$	0	0
$635$	−13.3944	−0.531542
$636$	0	0
$637$	15.9083	0.630311
$638$	0	0
$639$	41.9361	1.65897
$640$	0	0
$641$	15.6333	0.617479	0.308739	−	0.951147i	$-0.400093\pi$
$641$	15.6333	0.617479	0.308739	+	0.951147i	$0.400093\pi$
$642$	0	0
$643$	8.23886	0.324909	0.162454	−	0.986716i	$-0.448059\pi$
$643$	8.23886	0.324909	0.162454	+	0.986716i	$0.448059\pi$
$644$	0	0
$645$	−13.2111	−0.520187
$646$	0	0
$647$	−45.6333	−1.79403	−0.897015	−	0.442000i	$-0.854269\pi$
$647$	−45.6333	−1.79403	−0.897015	+	0.442000i	$0.854269\pi$
$648$	0	0
$649$	−4.60555	−0.180784
$650$	0	0
$651$	4.30278	0.168639
$652$	0	0
$653$	0.486122	0.0190234	0.00951171	−	0.999955i	$-0.496972\pi$
$653$	0.486122	0.0190234	0.00951171	+	0.999955i	$0.496972\pi$
$654$	0	0
$655$	0.788897	0.0308248
$656$	0	0
$657$	42.6611	1.66437
$658$	0	0
$659$	26.0555	1.01498	0.507489	−	0.861658i	$-0.330574\pi$
$659$	26.0555	1.01498	0.507489	+	0.861658i	$0.330574\pi$
$660$	0	0
$661$	0.0916731	0.00356567	0.00178283	−	0.999998i	$-0.499433\pi$
$661$	0.0916731	0.00356567	0.00178283	+	0.999998i	$0.499433\pi$
$662$	0	0
$663$	55.5416	2.15706
$664$	0	0
$665$	−0.880571	−0.0341471
$666$	0	0
$667$	−8.60555	−0.333208
$668$	0	0
$669$	47.6333	1.84161
$670$	0	0
$671$	4.30278	0.166107
$672$	0	0
$673$	−26.8444	−1.03478	−0.517388	−	0.855751i	$-0.673096\pi$
$673$	−26.8444	−1.03478	−0.517388	+	0.855751i	$0.673096\pi$
$674$	0	0
$675$	−16.2111	−0.623966
$676$	0	0
$677$	−12.7889	−0.491517	−0.245759	−	0.969331i	$-0.579037\pi$
$677$	−12.7889	−0.491517	−0.245759	+	0.969331i	$0.579037\pi$
$678$	0	0
$679$	4.15559	0.159477
$680$	0	0
$681$	−85.2666	−3.26742
$682$	0	0
$683$	11.8806	0.454597	0.227299	−	0.973825i	$-0.427011\pi$
$683$	11.8806	0.454597	0.227299	+	0.973825i	$0.427011\pi$
$684$	0	0
$685$	−18.1194	−0.692308
$686$	0	0
$687$	−46.2389	−1.76412
$688$	0	0
$689$	16.6056	0.632621
$690$	0	0
$691$	−7.39445	−0.281298	−0.140649	−	0.990060i	$-0.544919\pi$
$691$	−7.39445	−0.281298	−0.140649	+	0.990060i	$0.544919\pi$
$692$	0	0
$693$	−7.90833	−0.300412
$694$	0	0
$695$	−11.8167	−0.448231
$696$	0	0
$697$	−34.3028	−1.29931
$698$	0	0
$699$	−73.4500	−2.77813
$700$	0	0
$701$	17.9361	0.677437	0.338718	−	0.940888i	$-0.390007\pi$
$701$	17.9361	0.677437	0.338718	+	0.940888i	$0.390007\pi$
$702$	0	0
$703$	−15.1556	−0.571604
$704$	0	0
$705$	11.2111	0.422235
$706$	0	0
$707$	−3.81665	−0.143540
$708$	0	0
$709$	27.5416	1.03435	0.517174	−	0.855880i	$-0.326984\pi$
$709$	27.5416	1.03435	0.517174	+	0.855880i	$0.326984\pi$
$710$	0	0
$711$	−31.6333	−1.18634
$712$	0	0
$713$	−4.30278	−0.161140
$714$	0	0
$715$	7.60555	0.284431
$716$	0	0
$717$	−96.4777	−3.60303
$718$	0	0
$719$	−31.9361	−1.19101	−0.595507	−	0.803350i	$-0.703049\pi$
$719$	−31.9361	−1.19101	−0.595507	+	0.803350i	$0.703049\pi$
$720$	0	0
$721$	2.76114	0.102830
$722$	0	0
$723$	91.2666	3.39424
$724$	0	0
$725$	8.60555	0.319602
$726$	0	0
$727$	−10.6972	−0.396738	−0.198369	−	0.980127i	$-0.563564\pi$
$727$	−10.6972	−0.396738	−0.198369	+	0.980127i	$0.563564\pi$
$728$	0	0
$729$	75.1749	2.78426
$730$	0	0
$731$	−29.2111	−1.08041
$732$	0	0
$733$	−40.0555	−1.47948	−0.739742	−	0.672891i	$-0.765052\pi$
$733$	−40.0555	−1.47948	−0.739742	+	0.672891i	$0.765052\pi$
$734$	0	0
$735$	−22.8167	−0.841605
$736$	0	0
$737$	−30.4222	−1.12062
$738$	0	0
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	−22.1194	−0.812578
$742$	0	0
$743$	12.9083	0.473561	0.236780	−	0.971563i	$-0.423908\pi$
$743$	12.9083	0.473561	0.236780	+	0.971563i	$0.423908\pi$
$744$	0	0
$745$	2.90833	0.106553
$746$	0	0
$747$	−88.6611	−3.24394
$748$	0	0
$749$	−0.550039	−0.0200980
$750$	0	0
$751$	−47.8167	−1.74485	−0.872427	−	0.488744i	$-0.837455\pi$
$751$	−47.8167	−1.74485	−0.872427	+	0.488744i	$0.837455\pi$
$752$	0	0
$753$	−28.0278	−1.02139
$754$	0	0
$755$	−15.7250	−0.572291
$756$	0	0
$757$	47.2666	1.71793	0.858967	−	0.512031i	$-0.171107\pi$
$757$	47.2666	1.71793	0.858967	+	0.512031i	$0.171107\pi$
$758$	0	0
$759$	10.9083	0.395947
$760$	0	0
$761$	22.4861	0.815121	0.407561	−	0.913178i	$-0.366380\pi$
$761$	22.4861	0.815121	0.407561	+	0.913178i	$0.366380\pi$
$762$	0	0
$763$	−4.14719	−0.150138
$764$	0	0
$765$	−57.7527	−2.08805
$766$	0	0
$767$	3.21110	0.115946
$768$	0	0
$769$	−40.0555	−1.44444	−0.722219	−	0.691664i	$-0.756878\pi$
$769$	−40.0555	−1.44444	−0.722219	+	0.691664i	$0.756878\pi$
$770$	0	0
$771$	87.8722	3.16464
$772$	0	0
$773$	−26.4222	−0.950341	−0.475170	−	0.879894i	$-0.657614\pi$
$773$	−26.4222	−0.950341	−0.475170	+	0.879894i	$0.657614\pi$
$774$	0	0
$775$	4.30278	0.154560
$776$	0	0
$777$	5.21110	0.186947
$778$	0	0
$779$	13.6611	0.489458
$780$	0	0
$781$	17.5139	0.626696
$782$	0	0
$783$	−139.505	−4.98552
$784$	0	0
$785$	3.39445	0.121153
$786$	0	0
$787$	12.1833	0.434289	0.217145	−	0.976139i	$-0.430326\pi$
$787$	12.1833	0.434289	0.217145	+	0.976139i	$0.430326\pi$
$788$	0	0
$789$	−65.7527	−2.34086
$790$	0	0
$791$	0.972244	0.0345690
$792$	0	0
$793$	−3.00000	−0.106533
$794$	0	0
$795$	−23.8167	−0.844690
$796$	0	0
$797$	−25.3944	−0.899518	−0.449759	−	0.893150i	$-0.648490\pi$
$797$	−25.3944	−0.899518	−0.449759	+	0.893150i	$0.648490\pi$
$798$	0	0
$799$	24.7889	0.876968
$800$	0	0
$801$	22.0555	0.779293
$802$	0	0
$803$	17.8167	0.628736
$804$	0	0
$805$	−0.302776	−0.0106714
$806$	0	0
$807$	76.0555	2.67728
$808$	0	0
$809$	9.90833	0.348358	0.174179	−	0.984714i	$-0.444273\pi$
$809$	9.90833	0.348358	0.174179	+	0.984714i	$0.444273\pi$
$810$	0	0
$811$	45.8167	1.60884	0.804420	−	0.594061i	$-0.202476\pi$
$811$	45.8167	1.60884	0.804420	+	0.594061i	$0.202476\pi$
$812$	0	0
$813$	−93.4777	−3.27841
$814$	0	0
$815$	6.90833	0.241988
$816$	0	0
$817$	11.6333	0.406998
$818$	0	0
$819$	5.51388	0.192670
$820$	0	0
$821$	7.21110	0.251669	0.125835	−	0.992051i	$-0.459839\pi$
$821$	7.21110	0.251669	0.125835	+	0.992051i	$0.459839\pi$
$822$	0	0
$823$	27.2111	0.948519	0.474260	−	0.880385i	$-0.342716\pi$
$823$	27.2111	0.948519	0.474260	+	0.880385i	$0.342716\pi$
$824$	0	0
$825$	−10.9083	−0.379779
$826$	0	0
$827$	39.4500	1.37181	0.685905	−	0.727691i	$-0.259407\pi$
$827$	39.4500	1.37181	0.685905	+	0.727691i	$0.259407\pi$
$828$	0	0
$829$	−10.8444	−0.376642	−0.188321	−	0.982108i	$-0.560305\pi$
$829$	−10.8444	−0.376642	−0.188321	+	0.982108i	$0.560305\pi$
$830$	0	0
$831$	−6.60555	−0.229144
$832$	0	0
$833$	−50.4500	−1.74799
$834$	0	0
$835$	−13.2111	−0.457189
$836$	0	0
$837$	−69.7527	−2.41101
$838$	0	0
$839$	−19.3944	−0.669571	−0.334785	−	0.942294i	$-0.608664\pi$
$839$	−19.3944	−0.669571	−0.334785	+	0.942294i	$0.608664\pi$
$840$	0	0
$841$	45.0555	1.55364
$842$	0	0
$843$	49.6333	1.70946
$844$	0	0
$845$	7.69722	0.264793
$846$	0	0
$847$	0.0277564	0.000953720 0
$848$	0	0
$849$	−97.8722	−3.35896
$850$	0	0
$851$	−5.21110	−0.178634
$852$	0	0
$853$	−2.06392	−0.0706672	−0.0353336	−	0.999376i	$-0.511249\pi$
$853$	−2.06392	−0.0706672	−0.0353336	+	0.999376i	$0.511249\pi$
$854$	0	0
$855$	23.0000	0.786583
$856$	0	0
$857$	−45.6333	−1.55880	−0.779402	−	0.626524i	$-0.784477\pi$
$857$	−45.6333	−1.55880	−0.779402	+	0.626524i	$0.784477\pi$
$858$	0	0
$859$	−9.81665	−0.334940	−0.167470	−	0.985877i	$-0.553560\pi$
$859$	−9.81665	−0.334940	−0.167470	+	0.985877i	$0.553560\pi$
$860$	0	0
$861$	−4.69722	−0.160081
$862$	0	0
$863$	54.2389	1.84631	0.923156	−	0.384425i	$-0.125600\pi$
$863$	54.2389	1.84631	0.923156	+	0.384425i	$0.125600\pi$
$864$	0	0
$865$	14.5139	0.493487
$866$	0	0
$867$	−119.992	−4.07513
$868$	0	0
$869$	−13.2111	−0.448156
$870$	0	0
$871$	21.2111	0.718711
$872$	0	0
$873$	−108.542	−3.67358
$874$	0	0
$875$	0.302776	0.0102357
$876$	0	0
$877$	24.0917	0.813518	0.406759	−	0.913536i	$-0.366659\pi$
$877$	24.0917	0.813518	0.406759	+	0.913536i	$0.366659\pi$
$878$	0	0
$879$	59.4500	2.00520
$880$	0	0
$881$	−40.8444	−1.37608	−0.688042	−	0.725671i	$-0.741529\pi$
$881$	−40.8444	−1.37608	−0.688042	+	0.725671i	$0.741529\pi$
$882$	0	0
$883$	26.9083	0.905537	0.452769	−	0.891628i	$-0.350436\pi$
$883$	26.9083	0.905537	0.452769	+	0.891628i	$0.350436\pi$
$884$	0	0
$885$	−4.60555	−0.154814
$886$	0	0
$887$	47.6333	1.59937	0.799685	−	0.600420i	$-0.205000\pi$
$887$	47.6333	1.59937	0.799685	+	0.600420i	$0.205000\pi$
$888$	0	0
$889$	−4.05551	−0.136018
$890$	0	0
$891$	98.4777	3.29913
$892$	0	0
$893$	−9.87217	−0.330359
$894$	0	0
$895$	−23.0278	−0.769733
$896$	0	0
$897$	−7.60555	−0.253942
$898$	0	0
$899$	37.0278	1.23494
$900$	0	0
$901$	−52.6611	−1.75439
$902$	0	0
$903$	−4.00000	−0.133112
$904$	0	0
$905$	−21.6972	−0.721240
$906$	0	0
$907$	−15.3944	−0.511164	−0.255582	−	0.966787i	$-0.582267\pi$
$907$	−15.3944	−0.511164	−0.255582	+	0.966787i	$0.582267\pi$
$908$	0	0
$909$	99.6888	3.30647
$910$	0	0
$911$	−45.8167	−1.51797	−0.758987	−	0.651106i	$-0.774305\pi$
$911$	−45.8167	−1.51797	−0.758987	+	0.651106i	$0.774305\pi$
$912$	0	0
$913$	−37.0278	−1.22544
$914$	0	0
$915$	4.30278	0.142245
$916$	0	0
$917$	0.238859	0.00788782
$918$	0	0
$919$	−51.2666	−1.69113	−0.845565	−	0.533873i	$-0.820736\pi$
$919$	−51.2666	−1.69113	−0.845565	+	0.533873i	$0.820736\pi$
$920$	0	0
$921$	19.5139	0.643004
$922$	0	0
$923$	−12.2111	−0.401933
$924$	0	0
$925$	5.21110	0.171340
$926$	0	0
$927$	−72.1194	−2.36871
$928$	0	0
$929$	3.21110	0.105353	0.0526764	−	0.998612i	$-0.483225\pi$
$929$	3.21110	0.105353	0.0526764	+	0.998612i	$0.483225\pi$
$930$	0	0
$931$	20.0917	0.658478
$932$	0	0
$933$	34.4222	1.12693
$934$	0	0
$935$	−24.1194	−0.788790
$936$	0	0
$937$	44.5416	1.45511	0.727556	−	0.686048i	$-0.240656\pi$
$937$	44.5416	1.45511	0.727556	+	0.686048i	$0.240656\pi$
$938$	0	0
$939$	−88.2666	−2.88047
$940$	0	0
$941$	28.9361	0.943289	0.471645	−	0.881789i	$-0.343661\pi$
$941$	28.9361	0.943289	0.471645	+	0.881789i	$0.343661\pi$
$942$	0	0
$943$	4.69722	0.152963
$944$	0	0
$945$	−4.90833	−0.159668
$946$	0	0
$947$	−19.1472	−0.622200	−0.311100	−	0.950377i	$-0.600697\pi$
$947$	−19.1472	−0.622200	−0.311100	+	0.950377i	$0.600697\pi$
$948$	0	0
$949$	−12.4222	−0.403242
$950$	0	0
$951$	100.083	3.24542
$952$	0	0
$953$	19.4861	0.631217	0.315609	−	0.948889i	$-0.397791\pi$
$953$	19.4861	0.631217	0.315609	+	0.948889i	$0.397791\pi$
$954$	0	0
$955$	0.183346	0.00593294
$956$	0	0
$957$	−93.8722	−3.03446
$958$	0	0
$959$	−5.48612	−0.177156
$960$	0	0
$961$	−12.4861	−0.402778
$962$	0	0
$963$	14.3667	0.462960
$964$	0	0
$965$	−9.39445	−0.302418
$966$	0	0
$967$	3.81665	0.122735	0.0613677	−	0.998115i	$-0.480454\pi$
$967$	3.81665	0.122735	0.0613677	+	0.998115i	$0.480454\pi$
$968$	0	0
$969$	70.1472	2.25345
$970$	0	0
$971$	19.1194	0.613572	0.306786	−	0.951779i	$-0.400746\pi$
$971$	19.1194	0.613572	0.306786	+	0.951779i	$0.400746\pi$
$972$	0	0
$973$	−3.57779	−0.114699
$974$	0	0
$975$	7.60555	0.243573
$976$	0	0
$977$	2.72498	0.0871799	0.0435899	−	0.999050i	$-0.486120\pi$
$977$	2.72498	0.0871799	0.0435899	+	0.999050i	$0.486120\pi$
$978$	0	0
$979$	9.21110	0.294388
$980$	0	0
$981$	108.322	3.45846
$982$	0	0
$983$	36.0917	1.15115	0.575573	−	0.817751i	$-0.304779\pi$
$983$	36.0917	1.15115	0.575573	+	0.817751i	$0.304779\pi$
$984$	0	0
$985$	−22.5139	−0.717352
$986$	0	0
$987$	3.39445	0.108046
$988$	0	0
$989$	4.00000	0.127193
$990$	0	0
$991$	−57.3028	−1.82028	−0.910141	−	0.414298i	$-0.864027\pi$
$991$	−57.3028	−1.82028	−0.910141	+	0.414298i	$0.864027\pi$
$992$	0	0
$993$	−12.6056	−0.400025
$994$	0	0
$995$	8.42221	0.267002
$996$	0	0
$997$	−24.4222	−0.773459	−0.386729	−	0.922193i	$-0.626395\pi$
$997$	−24.4222	−0.773459	−0.386729	+	0.922193i	$0.626395\pi$
$998$	0	0
$999$	−84.4777	−2.67276

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3680.2.a.k.1.1	✓	2
4.3	odd	2		3680.2.a.p.1.2	yes	2
8.3	odd	2		7360.2.a.bd.1.1		2
8.5	even	2		7360.2.a.bv.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3680.2.a.k.1.1	✓	2	1.1	even	1	trivial
3680.2.a.p.1.2	yes	2	4.3	odd	2
7360.2.a.bd.1.1		2	8.3	odd	2
7360.2.a.bv.1.2		2	8.5	even	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:	\( N \)	\(=\)	\( 3680 = 2^{5} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3680.a (trivial)

\(f(q)\)	\(=\)	\(q-3.30278 q^{3} -1.00000 q^{5} -0.302776 q^{7} +7.90833 q^{9} +O(q^{10})\)	\(q-3.30278 q^{3} -1.00000 q^{5} -0.302776 q^{7} +7.90833 q^{9} +3.30278 q^{11} -2.30278 q^{13} +3.30278 q^{15} +7.30278 q^{17} -2.90833 q^{19} +1.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} -16.2111 q^{27} +8.60555 q^{29} +4.30278 q^{31} -10.9083 q^{33} +0.302776 q^{35} +5.21110 q^{37} +7.60555 q^{39} -4.69722 q^{41} -4.00000 q^{43} -7.90833 q^{45} +3.39445 q^{47} -6.90833 q^{49} -24.1194 q^{51} -7.21110 q^{53} -3.30278 q^{55} +9.60555 q^{57} -1.39445 q^{59} +1.30278 q^{61} -2.39445 q^{63} +2.30278 q^{65} -9.21110 q^{67} +3.30278 q^{69} +5.30278 q^{71} +5.39445 q^{73} -3.30278 q^{75} -1.00000 q^{77} -4.00000 q^{79} +29.8167 q^{81} -11.2111 q^{83} -7.30278 q^{85} -28.4222 q^{87} +2.78890 q^{89} +0.697224 q^{91} -14.2111 q^{93} +2.90833 q^{95} -13.7250 q^{97} +26.1194 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{3} - 2 q^{5} + 3 q^{7} + 5 q^{9}+O(q^{10}) \) 2 * q - 3 * q^3 - 2 * q^5 + 3 * q^7 + 5 * q^9	\( 2 q - 3 q^{3} - 2 q^{5} + 3 q^{7} + 5 q^{9} + 3 q^{11} - q^{13} + 3 q^{15} + 11 q^{17} + 5 q^{19} + 2 q^{21} - 2 q^{23} + 2 q^{25} - 18 q^{27} + 10 q^{29} + 5 q^{31} - 11 q^{33} - 3 q^{35} - 4 q^{37} + 8 q^{39} - 13 q^{41} - 8 q^{43} - 5 q^{45} + 14 q^{47} - 3 q^{49} - 23 q^{51} - 3 q^{55} + 12 q^{57} - 10 q^{59} - q^{61} - 12 q^{63} + q^{65} - 4 q^{67} + 3 q^{69} + 7 q^{71} + 18 q^{73} - 3 q^{75} - 2 q^{77} - 8 q^{79} + 38 q^{81} - 8 q^{83} - 11 q^{85} - 28 q^{87} + 20 q^{89} + 5 q^{91} - 14 q^{93} - 5 q^{95} + 5 q^{97} + 27 q^{99}+O(q^{100}) \) 2 * q - 3 * q^3 - 2 * q^5 + 3 * q^7 + 5 * q^9 + 3 * q^11 - q^13 + 3 * q^15 + 11 * q^17 + 5 * q^19 + 2 * q^21 - 2 * q^23 + 2 * q^25 - 18 * q^27 + 10 * q^29 + 5 * q^31 - 11 * q^33 - 3 * q^35 - 4 * q^37 + 8 * q^39 - 13 * q^41 - 8 * q^43 - 5 * q^45 + 14 * q^47 - 3 * q^49 - 23 * q^51 - 3 * q^55 + 12 * q^57 - 10 * q^59 - q^61 - 12 * q^63 + q^65 - 4 * q^67 + 3 * q^69 + 7 * q^71 + 18 * q^73 - 3 * q^75 - 2 * q^77 - 8 * q^79 + 38 * q^81 - 8 * q^83 - 11 * q^85 - 28 * q^87 + 20 * q^89 + 5 * q^91 - 14 * q^93 - 5 * q^95 + 5 * q^97 + 27 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more