LMFDB - Embedded newform 1680.2.a.v.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1680,2,Mod(1,1680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1680, 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("1680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1680.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:	$13.4148675396$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	1680.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +2.47214 q^{11} -4.47214 q^{13} -1.00000 q^{15} -2.00000 q^{17} -6.47214 q^{19} -1.00000 q^{21} -4.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -2.00000 q^{29} -10.4721 q^{31} +2.47214 q^{33} +1.00000 q^{35} +10.9443 q^{37} -4.47214 q^{39} -2.00000 q^{41} +8.94427 q^{43} -1.00000 q^{45} +4.94427 q^{47} +1.00000 q^{49} -2.00000 q^{51} -12.4721 q^{53} -2.47214 q^{55} -6.47214 q^{57} -8.94427 q^{59} -2.00000 q^{61} -1.00000 q^{63} +4.47214 q^{65} +4.00000 q^{67} -4.00000 q^{69} -14.4721 q^{71} -3.52786 q^{73} +1.00000 q^{75} -2.47214 q^{77} +4.94427 q^{79} +1.00000 q^{81} -0.944272 q^{83} +2.00000 q^{85} -2.00000 q^{87} -2.00000 q^{89} +4.47214 q^{91} -10.4721 q^{93} +6.47214 q^{95} -0.472136 q^{97} +2.47214 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} - 4 q^{11} - 2 q^{15} - 4 q^{17} - 4 q^{19} - 2 q^{21} - 8 q^{23} + 2 q^{25} + 2 q^{27} - 4 q^{29} - 12 q^{31} - 4 q^{33} + 2 q^{35} + 4 q^{37} - 4 q^{41} - 2 q^{45} - 8 q^{47} + 2 q^{49} - 4 q^{51} - 16 q^{53} + 4 q^{55} - 4 q^{57} - 4 q^{61} - 2 q^{63} + 8 q^{67} - 8 q^{69} - 20 q^{71} - 16 q^{73} + 2 q^{75} + 4 q^{77} - 8 q^{79} + 2 q^{81} + 16 q^{83} + 4 q^{85} - 4 q^{87} - 4 q^{89} - 12 q^{93} + 4 q^{95} + 8 q^{97} - 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 - 4 * q^11 - 2 * q^15 - 4 * q^17 - 4 * q^19 - 2 * q^21 - 8 * q^23 + 2 * q^25 + 2 * q^27 - 4 * q^29 - 12 * q^31 - 4 * q^33 + 2 * q^35 + 4 * q^37 - 4 * q^41 - 2 * q^45 - 8 * q^47 + 2 * q^49 - 4 * q^51 - 16 * q^53 + 4 * q^55 - 4 * q^57 - 4 * q^61 - 2 * q^63 + 8 * q^67 - 8 * q^69 - 20 * q^71 - 16 * q^73 + 2 * q^75 + 4 * q^77 - 8 * q^79 + 2 * q^81 + 16 * q^83 + 4 * q^85 - 4 * q^87 - 4 * q^89 - 12 * q^93 + 4 * q^95 + 8 * q^97 - 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.47214	0.745377	0.372689	−	0.927957i	$-0.378436\pi$
$11$	2.47214	0.745377	0.372689	+	0.927957i	$0.378436\pi$
$12$	0	0
$13$	−4.47214	−1.24035	−0.620174	−	0.784465i	$-0.712938\pi$
$13$	−4.47214	−1.24035	−0.620174	+	0.784465i	$0.712938\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	−6.47214	−1.48481	−0.742405	−	0.669951i	$-0.766315\pi$
$19$	−6.47214	−1.48481	−0.742405	+	0.669951i	$0.766315\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−10.4721	−1.88085	−0.940426	−	0.340000i	$-0.889573\pi$
$31$	−10.4721	−1.88085	−0.940426	+	0.340000i	$0.889573\pi$
$32$	0	0
$33$	2.47214	0.430344
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	10.9443	1.79923	0.899614	−	0.436687i	$-0.143848\pi$
$37$	10.9443	1.79923	0.899614	+	0.436687i	$0.143848\pi$
$38$	0	0
$39$	−4.47214	−0.716115
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	8.94427	1.36399	0.681994	−	0.731357i	$-0.261113\pi$
$43$	8.94427	1.36399	0.681994	+	0.731357i	$0.261113\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	4.94427	0.721196	0.360598	−	0.932721i	$-0.382573\pi$
$47$	4.94427	0.721196	0.360598	+	0.932721i	$0.382573\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.00000	−0.280056
$52$	0	0
$53$	−12.4721	−1.71318	−0.856590	−	0.515998i	$-0.827421\pi$
$53$	−12.4721	−1.71318	−0.856590	+	0.515998i	$0.827421\pi$
$54$	0	0
$55$	−2.47214	−0.333343
$56$	0	0
$57$	−6.47214	−0.857255
$58$	0	0
$59$	−8.94427	−1.16445	−0.582223	−	0.813029i	$-0.697817\pi$
$59$	−8.94427	−1.16445	−0.582223	+	0.813029i	$0.697817\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	4.47214	0.554700
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	0	0
$71$	−14.4721	−1.71753	−0.858763	−	0.512373i	$-0.828767\pi$
$71$	−14.4721	−1.71753	−0.858763	+	0.512373i	$0.828767\pi$
$72$	0	0
$73$	−3.52786	−0.412905	−0.206453	−	0.978457i	$-0.566192\pi$
$73$	−3.52786	−0.412905	−0.206453	+	0.978457i	$0.566192\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−2.47214	−0.281726
$78$	0	0
$79$	4.94427	0.556274	0.278137	−	0.960541i	$-0.410283\pi$
$79$	4.94427	0.556274	0.278137	+	0.960541i	$0.410283\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−0.944272	−0.103647	−0.0518237	−	0.998656i	$-0.516503\pi$
$83$	−0.944272	−0.103647	−0.0518237	+	0.998656i	$0.516503\pi$
$84$	0	0
$85$	2.00000	0.216930
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	4.47214	0.468807
$92$	0	0
$93$	−10.4721	−1.08591
$94$	0	0
$95$	6.47214	0.664027
$96$	0	0
$97$	−0.472136	−0.0479381	−0.0239691	−	0.999713i	$-0.507630\pi$
$97$	−0.472136	−0.0479381	−0.0239691	+	0.999713i	$0.507630\pi$
$98$	0	0
$99$	2.47214	0.248459
$100$	0	0
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	0	0
$107$	−4.94427	−0.477981	−0.238990	−	0.971022i	$-0.576816\pi$
$107$	−4.94427	−0.477981	−0.238990	+	0.971022i	$0.576816\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	10.9443	1.03878
$112$	0	0
$113$	−8.47214	−0.796992	−0.398496	−	0.917170i	$-0.630468\pi$
$113$	−8.47214	−0.796992	−0.398496	+	0.917170i	$0.630468\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	0	0
$117$	−4.47214	−0.413449
$118$	0	0
$119$	2.00000	0.183340
$120$	0	0
$121$	−4.88854	−0.444413
$122$	0	0
$123$	−2.00000	−0.180334
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−12.9443	−1.14862	−0.574309	−	0.818638i	$-0.694729\pi$
$127$	−12.9443	−1.14862	−0.574309	+	0.818638i	$0.694729\pi$
$128$	0	0
$129$	8.94427	0.787499
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	6.47214	0.561205
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	12.4721	1.06557	0.532783	−	0.846252i	$-0.321146\pi$
$137$	12.4721	1.06557	0.532783	+	0.846252i	$0.321146\pi$
$138$	0	0
$139$	19.4164	1.64688	0.823439	−	0.567405i	$-0.192052\pi$
$139$	19.4164	1.64688	0.823439	+	0.567405i	$0.192052\pi$
$140$	0	0
$141$	4.94427	0.416383
$142$	0	0
$143$	−11.0557	−0.924526
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	2.94427	0.241204	0.120602	−	0.992701i	$-0.461517\pi$
$149$	2.94427	0.241204	0.120602	+	0.992701i	$0.461517\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	0	0
$155$	10.4721	0.841142
$156$	0	0
$157$	8.47214	0.676150	0.338075	−	0.941119i	$-0.390224\pi$
$157$	8.47214	0.676150	0.338075	+	0.941119i	$0.390224\pi$
$158$	0	0
$159$	−12.4721	−0.989105
$160$	0	0
$161$	4.00000	0.315244
$162$	0	0
$163$	−0.944272	−0.0739611	−0.0369805	−	0.999316i	$-0.511774\pi$
$163$	−0.944272	−0.0739611	−0.0369805	+	0.999316i	$0.511774\pi$
$164$	0	0
$165$	−2.47214	−0.192456
$166$	0	0
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	7.00000	0.538462
$170$	0	0
$171$	−6.47214	−0.494937
$172$	0	0
$173$	14.9443	1.13619	0.568096	−	0.822962i	$-0.307680\pi$
$173$	14.9443	1.13619	0.568096	+	0.822962i	$0.307680\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−8.94427	−0.672293
$178$	0	0
$179$	2.47214	0.184776	0.0923881	−	0.995723i	$-0.470550\pi$
$179$	2.47214	0.184776	0.0923881	+	0.995723i	$0.470550\pi$
$180$	0	0
$181$	18.9443	1.40812	0.704058	−	0.710142i	$-0.251369\pi$
$181$	18.9443	1.40812	0.704058	+	0.710142i	$0.251369\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	0	0
$185$	−10.9443	−0.804639
$186$	0	0
$187$	−4.94427	−0.361561
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−27.4164	−1.98378	−0.991891	−	0.127093i	$-0.959435\pi$
$191$	−27.4164	−1.98378	−0.991891	+	0.127093i	$0.959435\pi$
$192$	0	0
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	0	0
$195$	4.47214	0.320256
$196$	0	0
$197$	24.4721	1.74357	0.871784	−	0.489891i	$-0.162963\pi$
$197$	24.4721	1.74357	0.871784	+	0.489891i	$0.162963\pi$
$198$	0	0
$199$	−0.583592	−0.0413697	−0.0206849	−	0.999786i	$-0.506585\pi$
$199$	−0.583592	−0.0413697	−0.0206849	+	0.999786i	$0.506585\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	2.00000	0.140372
$204$	0	0
$205$	2.00000	0.139686
$206$	0	0
$207$	−4.00000	−0.278019
$208$	0	0
$209$	−16.0000	−1.10674
$210$	0	0
$211$	−0.944272	−0.0650064	−0.0325032	−	0.999472i	$-0.510348\pi$
$211$	−0.944272	−0.0650064	−0.0325032	+	0.999472i	$0.510348\pi$
$212$	0	0
$213$	−14.4721	−0.991614
$214$	0	0
$215$	−8.94427	−0.609994
$216$	0	0
$217$	10.4721	0.710895
$218$	0	0
$219$	−3.52786	−0.238391
$220$	0	0
$221$	8.94427	0.601657
$222$	0	0
$223$	−4.94427	−0.331093	−0.165546	−	0.986202i	$-0.552939\pi$
$223$	−4.94427	−0.331093	−0.165546	+	0.986202i	$0.552939\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	16.9443	1.12463	0.562315	−	0.826923i	$-0.309911\pi$
$227$	16.9443	1.12463	0.562315	+	0.826923i	$0.309911\pi$
$228$	0	0
$229$	−11.8885	−0.785617	−0.392809	−	0.919620i	$-0.628496\pi$
$229$	−11.8885	−0.785617	−0.392809	+	0.919620i	$0.628496\pi$
$230$	0	0
$231$	−2.47214	−0.162655
$232$	0	0
$233$	17.4164	1.14099	0.570493	−	0.821302i	$-0.306752\pi$
$233$	17.4164	1.14099	0.570493	+	0.821302i	$0.306752\pi$
$234$	0	0
$235$	−4.94427	−0.322529
$236$	0	0
$237$	4.94427	0.321165
$238$	0	0
$239$	1.52786	0.0988293	0.0494147	−	0.998778i	$-0.484264\pi$
$239$	1.52786	0.0988293	0.0494147	+	0.998778i	$0.484264\pi$
$240$	0	0
$241$	−1.05573	−0.0680054	−0.0340027	−	0.999422i	$-0.510825\pi$
$241$	−1.05573	−0.0680054	−0.0340027	+	0.999422i	$0.510825\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	28.9443	1.84168
$248$	0	0
$249$	−0.944272	−0.0598408
$250$	0	0
$251$	0.944272	0.0596019	0.0298010	−	0.999556i	$-0.490513\pi$
$251$	0.944272	0.0596019	0.0298010	+	0.999556i	$0.490513\pi$
$252$	0	0
$253$	−9.88854	−0.621687
$254$	0	0
$255$	2.00000	0.125245
$256$	0	0
$257$	1.05573	0.0658545	0.0329273	−	0.999458i	$-0.489517\pi$
$257$	1.05573	0.0658545	0.0329273	+	0.999458i	$0.489517\pi$
$258$	0	0
$259$	−10.9443	−0.680044
$260$	0	0
$261$	−2.00000	−0.123797
$262$	0	0
$263$	−24.9443	−1.53813	−0.769065	−	0.639171i	$-0.779278\pi$
$263$	−24.9443	−1.53813	−0.769065	+	0.639171i	$0.779278\pi$
$264$	0	0
$265$	12.4721	0.766157
$266$	0	0
$267$	−2.00000	−0.122398
$268$	0	0
$269$	−23.8885	−1.45651	−0.728255	−	0.685306i	$-0.759668\pi$
$269$	−23.8885	−1.45651	−0.728255	+	0.685306i	$0.759668\pi$
$270$	0	0
$271$	10.4721	0.636137	0.318068	−	0.948068i	$-0.396966\pi$
$271$	10.4721	0.636137	0.318068	+	0.948068i	$0.396966\pi$
$272$	0	0
$273$	4.47214	0.270666
$274$	0	0
$275$	2.47214	0.149075
$276$	0	0
$277$	1.05573	0.0634326	0.0317163	−	0.999497i	$-0.489903\pi$
$277$	1.05573	0.0634326	0.0317163	+	0.999497i	$0.489903\pi$
$278$	0	0
$279$	−10.4721	−0.626950
$280$	0	0
$281$	6.94427	0.414261	0.207130	−	0.978313i	$-0.433588\pi$
$281$	6.94427	0.414261	0.207130	+	0.978313i	$0.433588\pi$
$282$	0	0
$283$	−12.0000	−0.713326	−0.356663	−	0.934233i	$-0.616086\pi$
$283$	−12.0000	−0.713326	−0.356663	+	0.934233i	$0.616086\pi$
$284$	0	0
$285$	6.47214	0.383376
$286$	0	0
$287$	2.00000	0.118056
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−0.472136	−0.0276771
$292$	0	0
$293$	22.9443	1.34042	0.670209	−	0.742172i	$-0.266204\pi$
$293$	22.9443	1.34042	0.670209	+	0.742172i	$0.266204\pi$
$294$	0	0
$295$	8.94427	0.520756
$296$	0	0
$297$	2.47214	0.143448
$298$	0	0
$299$	17.8885	1.03452
$300$	0	0
$301$	−8.94427	−0.515539
$302$	0	0
$303$	−14.0000	−0.804279
$304$	0	0
$305$	2.00000	0.114520
$306$	0	0
$307$	32.9443	1.88023	0.940114	−	0.340859i	$-0.110718\pi$
$307$	32.9443	1.88023	0.940114	+	0.340859i	$0.110718\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9.88854	0.560728	0.280364	−	0.959894i	$-0.409545\pi$
$311$	9.88854	0.560728	0.280364	+	0.959894i	$0.409545\pi$
$312$	0	0
$313$	9.41641	0.532247	0.266123	−	0.963939i	$-0.414257\pi$
$313$	9.41641	0.532247	0.266123	+	0.963939i	$0.414257\pi$
$314$	0	0
$315$	1.00000	0.0563436
$316$	0	0
$317$	−30.3607	−1.70523	−0.852613	−	0.522543i	$-0.824983\pi$
$317$	−30.3607	−1.70523	−0.852613	+	0.522543i	$0.824983\pi$
$318$	0	0
$319$	−4.94427	−0.276826
$320$	0	0
$321$	−4.94427	−0.275962
$322$	0	0
$323$	12.9443	0.720239
$324$	0	0
$325$	−4.47214	−0.248069
$326$	0	0
$327$	−2.00000	−0.110600
$328$	0	0
$329$	−4.94427	−0.272587
$330$	0	0
$331$	16.9443	0.931341	0.465671	−	0.884958i	$-0.345813\pi$
$331$	16.9443	0.931341	0.465671	+	0.884958i	$0.345813\pi$
$332$	0	0
$333$	10.9443	0.599742
$334$	0	0
$335$	−4.00000	−0.218543
$336$	0	0
$337$	11.8885	0.647610	0.323805	−	0.946124i	$-0.395038\pi$
$337$	11.8885	0.647610	0.323805	+	0.946124i	$0.395038\pi$
$338$	0	0
$339$	−8.47214	−0.460143
$340$	0	0
$341$	−25.8885	−1.40194
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	4.00000	0.215353
$346$	0	0
$347$	8.00000	0.429463	0.214731	−	0.976673i	$-0.431112\pi$
$347$	8.00000	0.429463	0.214731	+	0.976673i	$0.431112\pi$
$348$	0	0
$349$	23.8885	1.27872	0.639362	−	0.768906i	$-0.279198\pi$
$349$	23.8885	1.27872	0.639362	+	0.768906i	$0.279198\pi$
$350$	0	0
$351$	−4.47214	−0.238705
$352$	0	0
$353$	−27.8885	−1.48436	−0.742179	−	0.670202i	$-0.766207\pi$
$353$	−27.8885	−1.48436	−0.742179	+	0.670202i	$0.766207\pi$
$354$	0	0
$355$	14.4721	0.768101
$356$	0	0
$357$	2.00000	0.105851
$358$	0	0
$359$	−9.52786	−0.502861	−0.251431	−	0.967875i	$-0.580901\pi$
$359$	−9.52786	−0.502861	−0.251431	+	0.967875i	$0.580901\pi$
$360$	0	0
$361$	22.8885	1.20466
$362$	0	0
$363$	−4.88854	−0.256582
$364$	0	0
$365$	3.52786	0.184657
$366$	0	0
$367$	−20.9443	−1.09328	−0.546641	−	0.837367i	$-0.684094\pi$
$367$	−20.9443	−1.09328	−0.546641	+	0.837367i	$0.684094\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	0	0
$371$	12.4721	0.647521
$372$	0	0
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	8.94427	0.460653
$378$	0	0
$379$	2.11146	0.108458	0.0542291	−	0.998529i	$-0.482730\pi$
$379$	2.11146	0.108458	0.0542291	+	0.998529i	$0.482730\pi$
$380$	0	0
$381$	−12.9443	−0.663155
$382$	0	0
$383$	8.00000	0.408781	0.204390	−	0.978889i	$-0.434479\pi$
$383$	8.00000	0.408781	0.204390	+	0.978889i	$0.434479\pi$
$384$	0	0
$385$	2.47214	0.125992
$386$	0	0
$387$	8.94427	0.454663
$388$	0	0
$389$	10.9443	0.554897	0.277448	−	0.960741i	$-0.410511\pi$
$389$	10.9443	0.554897	0.277448	+	0.960741i	$0.410511\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	0	0
$393$	−4.00000	−0.201773
$394$	0	0
$395$	−4.94427	−0.248773
$396$	0	0
$397$	13.4164	0.673350	0.336675	−	0.941621i	$-0.390698\pi$
$397$	13.4164	0.673350	0.336675	+	0.941621i	$0.390698\pi$
$398$	0	0
$399$	6.47214	0.324012
$400$	0	0
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	0	0
$403$	46.8328	2.33291
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	27.0557	1.34110
$408$	0	0
$409$	−23.8885	−1.18121	−0.590606	−	0.806960i	$-0.701111\pi$
$409$	−23.8885	−1.18121	−0.590606	+	0.806960i	$0.701111\pi$
$410$	0	0
$411$	12.4721	0.615205
$412$	0	0
$413$	8.94427	0.440119
$414$	0	0
$415$	0.944272	0.0463525
$416$	0	0
$417$	19.4164	0.950826
$418$	0	0
$419$	−5.88854	−0.287674	−0.143837	−	0.989601i	$-0.545944\pi$
$419$	−5.88854	−0.287674	−0.143837	+	0.989601i	$0.545944\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	0	0
$423$	4.94427	0.240399
$424$	0	0
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	2.00000	0.0967868
$428$	0	0
$429$	−11.0557	−0.533776
$430$	0	0
$431$	9.52786	0.458941	0.229471	−	0.973316i	$-0.426301\pi$
$431$	9.52786	0.458941	0.229471	+	0.973316i	$0.426301\pi$
$432$	0	0
$433$	7.52786	0.361766	0.180883	−	0.983505i	$-0.442104\pi$
$433$	7.52786	0.361766	0.180883	+	0.983505i	$0.442104\pi$
$434$	0	0
$435$	2.00000	0.0958927
$436$	0	0
$437$	25.8885	1.23842
$438$	0	0
$439$	−10.4721	−0.499808	−0.249904	−	0.968271i	$-0.580399\pi$
$439$	−10.4721	−0.499808	−0.249904	+	0.968271i	$0.580399\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	8.00000	0.380091	0.190046	−	0.981775i	$-0.439136\pi$
$443$	8.00000	0.380091	0.190046	+	0.981775i	$0.439136\pi$
$444$	0	0
$445$	2.00000	0.0948091
$446$	0	0
$447$	2.94427	0.139259
$448$	0	0
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	0	0
$451$	−4.94427	−0.232817
$452$	0	0
$453$	16.0000	0.751746
$454$	0	0
$455$	−4.47214	−0.209657
$456$	0	0
$457$	−10.9443	−0.511951	−0.255976	−	0.966683i	$-0.582397\pi$
$457$	−10.9443	−0.511951	−0.255976	+	0.966683i	$0.582397\pi$
$458$	0	0
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	−31.8885	−1.48520	−0.742599	−	0.669737i	$-0.766407\pi$
$461$	−31.8885	−1.48520	−0.742599	+	0.669737i	$0.766407\pi$
$462$	0	0
$463$	−3.05573	−0.142012	−0.0710059	−	0.997476i	$-0.522621\pi$
$463$	−3.05573	−0.142012	−0.0710059	+	0.997476i	$0.522621\pi$
$464$	0	0
$465$	10.4721	0.485634
$466$	0	0
$467$	−8.94427	−0.413892	−0.206946	−	0.978352i	$-0.566352\pi$
$467$	−8.94427	−0.413892	−0.206946	+	0.978352i	$0.566352\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	0	0
$471$	8.47214	0.390375
$472$	0	0
$473$	22.1115	1.01669
$474$	0	0
$475$	−6.47214	−0.296962
$476$	0	0
$477$	−12.4721	−0.571060
$478$	0	0
$479$	−17.8885	−0.817348	−0.408674	−	0.912680i	$-0.634009\pi$
$479$	−17.8885	−0.817348	−0.408674	+	0.912680i	$0.634009\pi$
$480$	0	0
$481$	−48.9443	−2.23167
$482$	0	0
$483$	4.00000	0.182006
$484$	0	0
$485$	0.472136	0.0214386
$486$	0	0
$487$	3.05573	0.138468	0.0692341	−	0.997600i	$-0.477944\pi$
$487$	3.05573	0.138468	0.0692341	+	0.997600i	$0.477944\pi$
$488$	0	0
$489$	−0.944272	−0.0427015
$490$	0	0
$491$	−41.3050	−1.86407	−0.932033	−	0.362373i	$-0.881967\pi$
$491$	−41.3050	−1.86407	−0.932033	+	0.362373i	$0.881967\pi$
$492$	0	0
$493$	4.00000	0.180151
$494$	0	0
$495$	−2.47214	−0.111114
$496$	0	0
$497$	14.4721	0.649164
$498$	0	0
$499$	−21.8885	−0.979866	−0.489933	−	0.871760i	$-0.662979\pi$
$499$	−21.8885	−0.979866	−0.489933	+	0.871760i	$0.662979\pi$
$500$	0	0
$501$	8.00000	0.357414
$502$	0	0
$503$	−32.0000	−1.42681	−0.713405	−	0.700752i	$-0.752848\pi$
$503$	−32.0000	−1.42681	−0.713405	+	0.700752i	$0.752848\pi$
$504$	0	0
$505$	14.0000	0.622992
$506$	0	0
$507$	7.00000	0.310881
$508$	0	0
$509$	11.8885	0.526950	0.263475	−	0.964666i	$-0.415131\pi$
$509$	11.8885	0.526950	0.263475	+	0.964666i	$0.415131\pi$
$510$	0	0
$511$	3.52786	0.156064
$512$	0	0
$513$	−6.47214	−0.285752
$514$	0	0
$515$	0	0
$516$	0	0
$517$	12.2229	0.537563
$518$	0	0
$519$	14.9443	0.655981
$520$	0	0
$521$	15.8885	0.696090	0.348045	−	0.937478i	$-0.386846\pi$
$521$	15.8885	0.696090	0.348045	+	0.937478i	$0.386846\pi$
$522$	0	0
$523$	−8.94427	−0.391106	−0.195553	−	0.980693i	$-0.562650\pi$
$523$	−8.94427	−0.391106	−0.195553	+	0.980693i	$0.562650\pi$
$524$	0	0
$525$	−1.00000	−0.0436436
$526$	0	0
$527$	20.9443	0.912347
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−8.94427	−0.388148
$532$	0	0
$533$	8.94427	0.387419
$534$	0	0
$535$	4.94427	0.213760
$536$	0	0
$537$	2.47214	0.106681
$538$	0	0
$539$	2.47214	0.106482
$540$	0	0
$541$	23.8885	1.02705	0.513524	−	0.858075i	$-0.328340\pi$
$541$	23.8885	1.02705	0.513524	+	0.858075i	$0.328340\pi$
$542$	0	0
$543$	18.9443	0.812977
$544$	0	0
$545$	2.00000	0.0856706
$546$	0	0
$547$	29.8885	1.27794	0.638971	−	0.769231i	$-0.279360\pi$
$547$	29.8885	1.27794	0.638971	+	0.769231i	$0.279360\pi$
$548$	0	0
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	12.9443	0.551445
$552$	0	0
$553$	−4.94427	−0.210252
$554$	0	0
$555$	−10.9443	−0.464558
$556$	0	0
$557$	11.5279	0.488451	0.244226	−	0.969718i	$-0.421466\pi$
$557$	11.5279	0.488451	0.244226	+	0.969718i	$0.421466\pi$
$558$	0	0
$559$	−40.0000	−1.69182
$560$	0	0
$561$	−4.94427	−0.208747
$562$	0	0
$563$	21.8885	0.922492	0.461246	−	0.887272i	$-0.347403\pi$
$563$	21.8885	0.922492	0.461246	+	0.887272i	$0.347403\pi$
$564$	0	0
$565$	8.47214	0.356425
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−4.11146	−0.172361	−0.0861806	−	0.996280i	$-0.527466\pi$
$569$	−4.11146	−0.172361	−0.0861806	+	0.996280i	$0.527466\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	0	0
$573$	−27.4164	−1.14534
$574$	0	0
$575$	−4.00000	−0.166812
$576$	0	0
$577$	−34.3607	−1.43045	−0.715227	−	0.698892i	$-0.753677\pi$
$577$	−34.3607	−1.43045	−0.715227	+	0.698892i	$0.753677\pi$
$578$	0	0
$579$	−14.0000	−0.581820
$580$	0	0
$581$	0.944272	0.0391750
$582$	0	0
$583$	−30.8328	−1.27696
$584$	0	0
$585$	4.47214	0.184900
$586$	0	0
$587$	4.00000	0.165098	0.0825488	−	0.996587i	$-0.473694\pi$
$587$	4.00000	0.165098	0.0825488	+	0.996587i	$0.473694\pi$
$588$	0	0
$589$	67.7771	2.79271
$590$	0	0
$591$	24.4721	1.00665
$592$	0	0
$593$	−11.8885	−0.488204	−0.244102	−	0.969750i	$-0.578493\pi$
$593$	−11.8885	−0.488204	−0.244102	+	0.969750i	$0.578493\pi$
$594$	0	0
$595$	−2.00000	−0.0819920
$596$	0	0
$597$	−0.583592	−0.0238848
$598$	0	0
$599$	32.3607	1.32222	0.661111	−	0.750288i	$-0.270085\pi$
$599$	32.3607	1.32222	0.661111	+	0.750288i	$0.270085\pi$
$600$	0	0
$601$	21.0557	0.858881	0.429441	−	0.903095i	$-0.358711\pi$
$601$	21.0557	0.858881	0.429441	+	0.903095i	$0.358711\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	0	0
$605$	4.88854	0.198748
$606$	0	0
$607$	14.8328	0.602045	0.301023	−	0.953617i	$-0.402672\pi$
$607$	14.8328	0.602045	0.301023	+	0.953617i	$0.402672\pi$
$608$	0	0
$609$	2.00000	0.0810441
$610$	0	0
$611$	−22.1115	−0.894534
$612$	0	0
$613$	10.9443	0.442035	0.221017	−	0.975270i	$-0.429062\pi$
$613$	10.9443	0.442035	0.221017	+	0.975270i	$0.429062\pi$
$614$	0	0
$615$	2.00000	0.0806478
$616$	0	0
$617$	7.52786	0.303060	0.151530	−	0.988453i	$-0.451580\pi$
$617$	7.52786	0.303060	0.151530	+	0.988453i	$0.451580\pi$
$618$	0	0
$619$	−12.5836	−0.505777	−0.252889	−	0.967495i	$-0.581381\pi$
$619$	−12.5836	−0.505777	−0.252889	+	0.967495i	$0.581381\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	0	0
$623$	2.00000	0.0801283
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−16.0000	−0.638978
$628$	0	0
$629$	−21.8885	−0.872753
$630$	0	0
$631$	22.8328	0.908960	0.454480	−	0.890757i	$-0.349825\pi$
$631$	22.8328	0.908960	0.454480	+	0.890757i	$0.349825\pi$
$632$	0	0
$633$	−0.944272	−0.0375314
$634$	0	0
$635$	12.9443	0.513678
$636$	0	0
$637$	−4.47214	−0.177192
$638$	0	0
$639$	−14.4721	−0.572509
$640$	0	0
$641$	−36.8328	−1.45481	−0.727404	−	0.686209i	$-0.759274\pi$
$641$	−36.8328	−1.45481	−0.727404	+	0.686209i	$0.759274\pi$
$642$	0	0
$643$	32.9443	1.29920	0.649598	−	0.760278i	$-0.274937\pi$
$643$	32.9443	1.29920	0.649598	+	0.760278i	$0.274937\pi$
$644$	0	0
$645$	−8.94427	−0.352180
$646$	0	0
$647$	−33.8885	−1.33230	−0.666148	−	0.745820i	$-0.732058\pi$
$647$	−33.8885	−1.33230	−0.666148	+	0.745820i	$0.732058\pi$
$648$	0	0
$649$	−22.1115	−0.867951
$650$	0	0
$651$	10.4721	0.410435
$652$	0	0
$653$	−49.4164	−1.93381	−0.966907	−	0.255130i	$-0.917882\pi$
$653$	−49.4164	−1.93381	−0.966907	+	0.255130i	$0.917882\pi$
$654$	0	0
$655$	4.00000	0.156293
$656$	0	0
$657$	−3.52786	−0.137635
$658$	0	0
$659$	41.3050	1.60901	0.804506	−	0.593944i	$-0.202430\pi$
$659$	41.3050	1.60901	0.804506	+	0.593944i	$0.202430\pi$
$660$	0	0
$661$	−0.111456	−0.00433514	−0.00216757	−	0.999998i	$-0.500690\pi$
$661$	−0.111456	−0.00433514	−0.00216757	+	0.999998i	$0.500690\pi$
$662$	0	0
$663$	8.94427	0.347367
$664$	0	0
$665$	−6.47214	−0.250979
$666$	0	0
$667$	8.00000	0.309761
$668$	0	0
$669$	−4.94427	−0.191157
$670$	0	0
$671$	−4.94427	−0.190872
$672$	0	0
$673$	−44.8328	−1.72818	−0.864089	−	0.503339i	$-0.832105\pi$
$673$	−44.8328	−1.72818	−0.864089	+	0.503339i	$0.832105\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	38.9443	1.49675	0.748375	−	0.663276i	$-0.230834\pi$
$677$	38.9443	1.49675	0.748375	+	0.663276i	$0.230834\pi$
$678$	0	0
$679$	0.472136	0.0181189
$680$	0	0
$681$	16.9443	0.649306
$682$	0	0
$683$	−33.8885	−1.29671	−0.648355	−	0.761339i	$-0.724543\pi$
$683$	−33.8885	−1.29671	−0.648355	+	0.761339i	$0.724543\pi$
$684$	0	0
$685$	−12.4721	−0.476536
$686$	0	0
$687$	−11.8885	−0.453576
$688$	0	0
$689$	55.7771	2.12494
$690$	0	0
$691$	0.360680	0.0137209	0.00686045	−	0.999976i	$-0.497816\pi$
$691$	0.360680	0.0137209	0.00686045	+	0.999976i	$0.497816\pi$
$692$	0	0
$693$	−2.47214	−0.0939087
$694$	0	0
$695$	−19.4164	−0.736506
$696$	0	0
$697$	4.00000	0.151511
$698$	0	0
$699$	17.4164	0.658749
$700$	0	0
$701$	−34.0000	−1.28416	−0.642081	−	0.766637i	$-0.721929\pi$
$701$	−34.0000	−1.28416	−0.642081	+	0.766637i	$0.721929\pi$
$702$	0	0
$703$	−70.8328	−2.67151
$704$	0	0
$705$	−4.94427	−0.186212
$706$	0	0
$707$	14.0000	0.526524
$708$	0	0
$709$	−45.7771	−1.71919	−0.859597	−	0.510972i	$-0.829286\pi$
$709$	−45.7771	−1.71919	−0.859597	+	0.510972i	$0.829286\pi$
$710$	0	0
$711$	4.94427	0.185425
$712$	0	0
$713$	41.8885	1.56874
$714$	0	0
$715$	11.0557	0.413461
$716$	0	0
$717$	1.52786	0.0570591
$718$	0	0
$719$	−46.8328	−1.74657	−0.873285	−	0.487210i	$-0.838015\pi$
$719$	−46.8328	−1.74657	−0.873285	+	0.487210i	$0.838015\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−1.05573	−0.0392630
$724$	0	0
$725$	−2.00000	−0.0742781
$726$	0	0
$727$	−14.8328	−0.550119	−0.275059	−	0.961427i	$-0.588698\pi$
$727$	−14.8328	−0.550119	−0.275059	+	0.961427i	$0.588698\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−17.8885	−0.661632
$732$	0	0
$733$	37.4164	1.38201	0.691003	−	0.722852i	$-0.257169\pi$
$733$	37.4164	1.38201	0.691003	+	0.722852i	$0.257169\pi$
$734$	0	0
$735$	−1.00000	−0.0368856
$736$	0	0
$737$	9.88854	0.364249
$738$	0	0
$739$	−29.8885	−1.09947	−0.549734	−	0.835340i	$-0.685271\pi$
$739$	−29.8885	−1.09947	−0.549734	+	0.835340i	$0.685271\pi$
$740$	0	0
$741$	28.9443	1.06329
$742$	0	0
$743$	18.8328	0.690909	0.345455	−	0.938436i	$-0.387725\pi$
$743$	18.8328	0.690909	0.345455	+	0.938436i	$0.387725\pi$
$744$	0	0
$745$	−2.94427	−0.107870
$746$	0	0
$747$	−0.944272	−0.0345491
$748$	0	0
$749$	4.94427	0.180660
$750$	0	0
$751$	3.05573	0.111505	0.0557526	−	0.998445i	$-0.482244\pi$
$751$	3.05573	0.111505	0.0557526	+	0.998445i	$0.482244\pi$
$752$	0	0
$753$	0.944272	0.0344112
$754$	0	0
$755$	−16.0000	−0.582300
$756$	0	0
$757$	−3.88854	−0.141332	−0.0706658	−	0.997500i	$-0.522512\pi$
$757$	−3.88854	−0.141332	−0.0706658	+	0.997500i	$0.522512\pi$
$758$	0	0
$759$	−9.88854	−0.358931
$760$	0	0
$761$	7.88854	0.285959	0.142980	−	0.989726i	$-0.454332\pi$
$761$	7.88854	0.285959	0.142980	+	0.989726i	$0.454332\pi$
$762$	0	0
$763$	2.00000	0.0724049
$764$	0	0
$765$	2.00000	0.0723102
$766$	0	0
$767$	40.0000	1.44432
$768$	0	0
$769$	0.832816	0.0300321	0.0150161	−	0.999887i	$-0.495220\pi$
$769$	0.832816	0.0300321	0.0150161	+	0.999887i	$0.495220\pi$
$770$	0	0
$771$	1.05573	0.0380211
$772$	0	0
$773$	−25.0557	−0.901192	−0.450596	−	0.892728i	$-0.648788\pi$
$773$	−25.0557	−0.901192	−0.450596	+	0.892728i	$0.648788\pi$
$774$	0	0
$775$	−10.4721	−0.376170
$776$	0	0
$777$	−10.9443	−0.392624
$778$	0	0
$779$	12.9443	0.463777
$780$	0	0
$781$	−35.7771	−1.28020
$782$	0	0
$783$	−2.00000	−0.0714742
$784$	0	0
$785$	−8.47214	−0.302383
$786$	0	0
$787$	−48.9443	−1.74467	−0.872337	−	0.488904i	$-0.837397\pi$
$787$	−48.9443	−1.74467	−0.872337	+	0.488904i	$0.837397\pi$
$788$	0	0
$789$	−24.9443	−0.888040
$790$	0	0
$791$	8.47214	0.301234
$792$	0	0
$793$	8.94427	0.317620
$794$	0	0
$795$	12.4721	0.442341
$796$	0	0
$797$	−1.05573	−0.0373958	−0.0186979	−	0.999825i	$-0.505952\pi$
$797$	−1.05573	−0.0373958	−0.0186979	+	0.999825i	$0.505952\pi$
$798$	0	0
$799$	−9.88854	−0.349832
$800$	0	0
$801$	−2.00000	−0.0706665
$802$	0	0
$803$	−8.72136	−0.307770
$804$	0	0
$805$	−4.00000	−0.140981
$806$	0	0
$807$	−23.8885	−0.840917
$808$	0	0
$809$	21.0557	0.740280	0.370140	−	0.928976i	$-0.379310\pi$
$809$	21.0557	0.740280	0.370140	+	0.928976i	$0.379310\pi$
$810$	0	0
$811$	−28.5836	−1.00371	−0.501853	−	0.864953i	$-0.667348\pi$
$811$	−28.5836	−1.00371	−0.501853	+	0.864953i	$0.667348\pi$
$812$	0	0
$813$	10.4721	0.367274
$814$	0	0
$815$	0.944272	0.0330764
$816$	0	0
$817$	−57.8885	−2.02526
$818$	0	0
$819$	4.47214	0.156269
$820$	0	0
$821$	−37.7771	−1.31843	−0.659215	−	0.751955i	$-0.729111\pi$
$821$	−37.7771	−1.31843	−0.659215	+	0.751955i	$0.729111\pi$
$822$	0	0
$823$	−27.0557	−0.943103	−0.471552	−	0.881838i	$-0.656306\pi$
$823$	−27.0557	−0.943103	−0.471552	+	0.881838i	$0.656306\pi$
$824$	0	0
$825$	2.47214	0.0860687
$826$	0	0
$827$	4.94427	0.171929	0.0859646	−	0.996298i	$-0.472603\pi$
$827$	4.94427	0.171929	0.0859646	+	0.996298i	$0.472603\pi$
$828$	0	0
$829$	−30.9443	−1.07474	−0.537369	−	0.843347i	$-0.680582\pi$
$829$	−30.9443	−1.07474	−0.537369	+	0.843347i	$0.680582\pi$
$830$	0	0
$831$	1.05573	0.0366228
$832$	0	0
$833$	−2.00000	−0.0692959
$834$	0	0
$835$	−8.00000	−0.276851
$836$	0	0
$837$	−10.4721	−0.361970
$838$	0	0
$839$	−1.16718	−0.0402957	−0.0201478	−	0.999797i	$-0.506414\pi$
$839$	−1.16718	−0.0402957	−0.0201478	+	0.999797i	$0.506414\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	6.94427	0.239173
$844$	0	0
$845$	−7.00000	−0.240807
$846$	0	0
$847$	4.88854	0.167972
$848$	0	0
$849$	−12.0000	−0.411839
$850$	0	0
$851$	−43.7771	−1.50066
$852$	0	0
$853$	31.3050	1.07186	0.535931	−	0.844262i	$-0.319961\pi$
$853$	31.3050	1.07186	0.535931	+	0.844262i	$0.319961\pi$
$854$	0	0
$855$	6.47214	0.221342
$856$	0	0
$857$	−16.8328	−0.574998	−0.287499	−	0.957781i	$-0.592824\pi$
$857$	−16.8328	−0.574998	−0.287499	+	0.957781i	$0.592824\pi$
$858$	0	0
$859$	−41.5279	−1.41691	−0.708456	−	0.705755i	$-0.750608\pi$
$859$	−41.5279	−1.41691	−0.708456	+	0.705755i	$0.750608\pi$
$860$	0	0
$861$	2.00000	0.0681598
$862$	0	0
$863$	13.8885	0.472772	0.236386	−	0.971659i	$-0.424037\pi$
$863$	13.8885	0.472772	0.236386	+	0.971659i	$0.424037\pi$
$864$	0	0
$865$	−14.9443	−0.508120
$866$	0	0
$867$	−13.0000	−0.441503
$868$	0	0
$869$	12.2229	0.414634
$870$	0	0
$871$	−17.8885	−0.606130
$872$	0	0
$873$	−0.472136	−0.0159794
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−3.16718	−0.106948	−0.0534741	−	0.998569i	$-0.517029\pi$
$877$	−3.16718	−0.106948	−0.0534741	+	0.998569i	$0.517029\pi$
$878$	0	0
$879$	22.9443	0.773891
$880$	0	0
$881$	7.88854	0.265772	0.132886	−	0.991131i	$-0.457576\pi$
$881$	7.88854	0.265772	0.132886	+	0.991131i	$0.457576\pi$
$882$	0	0
$883$	2.11146	0.0710562	0.0355281	−	0.999369i	$-0.488689\pi$
$883$	2.11146	0.0710562	0.0355281	+	0.999369i	$0.488689\pi$
$884$	0	0
$885$	8.94427	0.300658
$886$	0	0
$887$	−22.8328	−0.766651	−0.383325	−	0.923613i	$-0.625221\pi$
$887$	−22.8328	−0.766651	−0.383325	+	0.923613i	$0.625221\pi$
$888$	0	0
$889$	12.9443	0.434137
$890$	0	0
$891$	2.47214	0.0828197
$892$	0	0
$893$	−32.0000	−1.07084
$894$	0	0
$895$	−2.47214	−0.0826344
$896$	0	0
$897$	17.8885	0.597281
$898$	0	0
$899$	20.9443	0.698531
$900$	0	0
$901$	24.9443	0.831014
$902$	0	0
$903$	−8.94427	−0.297647
$904$	0	0
$905$	−18.9443	−0.629729
$906$	0	0
$907$	−18.1115	−0.601381	−0.300691	−	0.953722i	$-0.597217\pi$
$907$	−18.1115	−0.601381	−0.300691	+	0.953722i	$0.597217\pi$
$908$	0	0
$909$	−14.0000	−0.464351
$910$	0	0
$911$	34.2492	1.13473	0.567364	−	0.823467i	$-0.307963\pi$
$911$	34.2492	1.13473	0.567364	+	0.823467i	$0.307963\pi$
$912$	0	0
$913$	−2.33437	−0.0772563
$914$	0	0
$915$	2.00000	0.0661180
$916$	0	0
$917$	4.00000	0.132092
$918$	0	0
$919$	52.9443	1.74647	0.873235	−	0.487299i	$-0.162018\pi$
$919$	52.9443	1.74647	0.873235	+	0.487299i	$0.162018\pi$
$920$	0	0
$921$	32.9443	1.08555
$922$	0	0
$923$	64.7214	2.13033
$924$	0	0
$925$	10.9443	0.359845
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−51.8885	−1.70241	−0.851204	−	0.524835i	$-0.824127\pi$
$929$	−51.8885	−1.70241	−0.851204	+	0.524835i	$0.824127\pi$
$930$	0	0
$931$	−6.47214	−0.212116
$932$	0	0
$933$	9.88854	0.323736
$934$	0	0
$935$	4.94427	0.161695
$936$	0	0
$937$	−43.5279	−1.42199	−0.710997	−	0.703195i	$-0.751756\pi$
$937$	−43.5279	−1.42199	−0.710997	+	0.703195i	$0.751756\pi$
$938$	0	0
$939$	9.41641	0.307293
$940$	0	0
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	0	0
$943$	8.00000	0.260516
$944$	0	0
$945$	1.00000	0.0325300
$946$	0	0
$947$	−17.8885	−0.581300	−0.290650	−	0.956830i	$-0.593871\pi$
$947$	−17.8885	−0.581300	−0.290650	+	0.956830i	$0.593871\pi$
$948$	0	0
$949$	15.7771	0.512146
$950$	0	0
$951$	−30.3607	−0.984512
$952$	0	0
$953$	−6.58359	−0.213263	−0.106632	−	0.994299i	$-0.534007\pi$
$953$	−6.58359	−0.213263	−0.106632	+	0.994299i	$0.534007\pi$
$954$	0	0
$955$	27.4164	0.887174
$956$	0	0
$957$	−4.94427	−0.159826
$958$	0	0
$959$	−12.4721	−0.402746
$960$	0	0
$961$	78.6656	2.53760
$962$	0	0
$963$	−4.94427	−0.159327
$964$	0	0
$965$	14.0000	0.450676
$966$	0	0
$967$	9.88854	0.317994	0.158997	−	0.987279i	$-0.449174\pi$
$967$	9.88854	0.317994	0.158997	+	0.987279i	$0.449174\pi$
$968$	0	0
$969$	12.9443	0.415830
$970$	0	0
$971$	−23.0557	−0.739894	−0.369947	−	0.929053i	$-0.620624\pi$
$971$	−23.0557	−0.739894	−0.369947	+	0.929053i	$0.620624\pi$
$972$	0	0
$973$	−19.4164	−0.622461
$974$	0	0
$975$	−4.47214	−0.143223
$976$	0	0
$977$	57.4164	1.83691	0.918457	−	0.395521i	$-0.129436\pi$
$977$	57.4164	1.83691	0.918457	+	0.395521i	$0.129436\pi$
$978$	0	0
$979$	−4.94427	−0.158020
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	0	0
$983$	−30.8328	−0.983414	−0.491707	−	0.870761i	$-0.663627\pi$
$983$	−30.8328	−0.983414	−0.491707	+	0.870761i	$0.663627\pi$
$984$	0	0
$985$	−24.4721	−0.779747
$986$	0	0
$987$	−4.94427	−0.157378
$988$	0	0
$989$	−35.7771	−1.13765
$990$	0	0
$991$	12.9443	0.411188	0.205594	−	0.978637i	$-0.434087\pi$
$991$	12.9443	0.411188	0.205594	+	0.978637i	$0.434087\pi$
$992$	0	0
$993$	16.9443	0.537710
$994$	0	0
$995$	0.583592	0.0185011
$996$	0	0
$997$	21.4164	0.678264	0.339132	−	0.940739i	$-0.389867\pi$
$997$	21.4164	0.678264	0.339132	+	0.940739i	$0.389867\pi$
$998$	0	0
$999$	10.9443	0.346261

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.2.a.v.1.2		2
3.2	odd	2		5040.2.a.bw.1.1		2
4.3	odd	2		105.2.a.b.1.2	✓	2
5.4	even	2		8400.2.a.cx.1.2		2
8.3	odd	2		6720.2.a.cx.1.2		2
8.5	even	2		6720.2.a.cs.1.1		2
12.11	even	2		315.2.a.d.1.1		2
20.3	even	4		525.2.d.c.274.1		4
20.7	even	4		525.2.d.c.274.4		4
20.19	odd	2		525.2.a.g.1.1		2
28.3	even	6		735.2.i.i.226.1		4
28.11	odd	6		735.2.i.k.226.1		4
28.19	even	6		735.2.i.i.361.1		4
28.23	odd	6		735.2.i.k.361.1		4
28.27	even	2		735.2.a.k.1.2		2
60.23	odd	4		1575.2.d.d.1324.3		4
60.47	odd	4		1575.2.d.d.1324.2		4
60.59	even	2		1575.2.a.r.1.2		2
84.83	odd	2		2205.2.a.w.1.1		2
140.139	even	2		3675.2.a.y.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.a.b.1.2	✓	2	4.3	odd	2
315.2.a.d.1.1		2	12.11	even	2
525.2.a.g.1.1		2	20.19	odd	2
525.2.d.c.274.1		4	20.3	even	4
525.2.d.c.274.4		4	20.7	even	4
735.2.a.k.1.2		2	28.27	even	2
735.2.i.i.226.1		4	28.3	even	6
735.2.i.i.361.1		4	28.19	even	6
735.2.i.k.226.1		4	28.11	odd	6
735.2.i.k.361.1		4	28.23	odd	6
1575.2.a.r.1.2		2	60.59	even	2
1575.2.d.d.1324.2		4	60.47	odd	4
1575.2.d.d.1324.3		4	60.23	odd	4
1680.2.a.v.1.2		2	1.1	even	1	trivial
2205.2.a.w.1.1		2	84.83	odd	2
3675.2.a.y.1.1		2	140.139	even	2
5040.2.a.bw.1.1		2	3.2	odd	2
6720.2.a.cs.1.1		2	8.5	even	2
6720.2.a.cx.1.2		2	8.3	odd	2
8400.2.a.cx.1.2		2	5.4	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:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1680.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +2.47214 q^{11} -4.47214 q^{13} -1.00000 q^{15} -2.00000 q^{17} -6.47214 q^{19} -1.00000 q^{21} -4.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -2.00000 q^{29} -10.4721 q^{31} +2.47214 q^{33} +1.00000 q^{35} +10.9443 q^{37} -4.47214 q^{39} -2.00000 q^{41} +8.94427 q^{43} -1.00000 q^{45} +4.94427 q^{47} +1.00000 q^{49} -2.00000 q^{51} -12.4721 q^{53} -2.47214 q^{55} -6.47214 q^{57} -8.94427 q^{59} -2.00000 q^{61} -1.00000 q^{63} +4.47214 q^{65} +4.00000 q^{67} -4.00000 q^{69} -14.4721 q^{71} -3.52786 q^{73} +1.00000 q^{75} -2.47214 q^{77} +4.94427 q^{79} +1.00000 q^{81} -0.944272 q^{83} +2.00000 q^{85} -2.00000 q^{87} -2.00000 q^{89} +4.47214 q^{91} -10.4721 q^{93} +6.47214 q^{95} -0.472136 q^{97} +2.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} - 4 q^{11} - 2 q^{15} - 4 q^{17} - 4 q^{19} - 2 q^{21} - 8 q^{23} + 2 q^{25} + 2 q^{27} - 4 q^{29} - 12 q^{31} - 4 q^{33} + 2 q^{35} + 4 q^{37} - 4 q^{41} - 2 q^{45} - 8 q^{47} + 2 q^{49} - 4 q^{51} - 16 q^{53} + 4 q^{55} - 4 q^{57} - 4 q^{61} - 2 q^{63} + 8 q^{67} - 8 q^{69} - 20 q^{71} - 16 q^{73} + 2 q^{75} + 4 q^{77} - 8 q^{79} + 2 q^{81} + 16 q^{83} + 4 q^{85} - 4 q^{87} - 4 q^{89} - 12 q^{93} + 4 q^{95} + 8 q^{97} - 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 - 4 * q^11 - 2 * q^15 - 4 * q^17 - 4 * q^19 - 2 * q^21 - 8 * q^23 + 2 * q^25 + 2 * q^27 - 4 * q^29 - 12 * q^31 - 4 * q^33 + 2 * q^35 + 4 * q^37 - 4 * q^41 - 2 * q^45 - 8 * q^47 + 2 * q^49 - 4 * q^51 - 16 * q^53 + 4 * q^55 - 4 * q^57 - 4 * q^61 - 2 * q^63 + 8 * q^67 - 8 * q^69 - 20 * q^71 - 16 * q^73 + 2 * q^75 + 4 * q^77 - 8 * q^79 + 2 * q^81 + 16 * q^83 + 4 * q^85 - 4 * q^87 - 4 * q^89 - 12 * q^93 + 4 * q^95 + 8 * q^97 - 4 * q^99

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