LMFDB - Embedded newform 1575.2.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,2,Mod(1,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1575.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1575.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:	$12.5764383184$
Analytic rank:	$0$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 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	$\chi$	$=$	1575.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607 q^{2} +3.00000 q^{4} -1.00000 q^{7} +2.23607 q^{8} +O(q^{10})$	$q+2.23607 q^{2} +3.00000 q^{4} -1.00000 q^{7} +2.23607 q^{8} +2.47214 q^{11} +4.47214 q^{13} -2.23607 q^{14} -1.00000 q^{16} -2.00000 q^{17} +6.47214 q^{19} +5.52786 q^{22} +4.00000 q^{23} +10.0000 q^{26} -3.00000 q^{28} +2.00000 q^{29} +10.4721 q^{31} -6.70820 q^{32} -4.47214 q^{34} -10.9443 q^{37} +14.4721 q^{38} +2.00000 q^{41} +8.94427 q^{43} +7.41641 q^{44} +8.94427 q^{46} -4.94427 q^{47} +1.00000 q^{49} +13.4164 q^{52} -12.4721 q^{53} -2.23607 q^{56} +4.47214 q^{58} -8.94427 q^{59} -2.00000 q^{61} +23.4164 q^{62} -13.0000 q^{64} +4.00000 q^{67} -6.00000 q^{68} -14.4721 q^{71} +3.52786 q^{73} -24.4721 q^{74} +19.4164 q^{76} -2.47214 q^{77} -4.94427 q^{79} +4.47214 q^{82} +0.944272 q^{83} +20.0000 q^{86} +5.52786 q^{88} +2.00000 q^{89} -4.47214 q^{91} +12.0000 q^{92} -11.0557 q^{94} +0.472136 q^{97} +2.23607 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{4} - 2 q^{7}+O(q^{10}) $ 2 * q + 6 * q^4 - 2 * q^7	$ 2 q + 6 q^{4} - 2 q^{7} - 4 q^{11} - 2 q^{16} - 4 q^{17} + 4 q^{19} + 20 q^{22} + 8 q^{23} + 20 q^{26} - 6 q^{28} + 4 q^{29} + 12 q^{31} - 4 q^{37} + 20 q^{38} + 4 q^{41} - 12 q^{44} + 8 q^{47} + 2 q^{49} - 16 q^{53} - 4 q^{61} + 20 q^{62} - 26 q^{64} + 8 q^{67} - 12 q^{68} - 20 q^{71} + 16 q^{73} - 40 q^{74} + 12 q^{76} + 4 q^{77} + 8 q^{79} - 16 q^{83} + 40 q^{86} + 20 q^{88} + 4 q^{89} + 24 q^{92} - 40 q^{94} - 8 q^{97}+O(q^{100}) $ 2 * q + 6 * q^4 - 2 * q^7 - 4 * q^11 - 2 * q^16 - 4 * q^17 + 4 * q^19 + 20 * q^22 + 8 * q^23 + 20 * q^26 - 6 * q^28 + 4 * q^29 + 12 * q^31 - 4 * q^37 + 20 * q^38 + 4 * q^41 - 12 * q^44 + 8 * q^47 + 2 * q^49 - 16 * q^53 - 4 * q^61 + 20 * q^62 - 26 * q^64 + 8 * q^67 - 12 * q^68 - 20 * q^71 + 16 * q^73 - 40 * q^74 + 12 * q^76 + 4 * q^77 + 8 * q^79 - 16 * q^83 + 40 * q^86 + 20 * q^88 + 4 * q^89 + 24 * q^92 - 40 * q^94 - 8 * q^97

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 $

$2$	2.23607	1.58114	0.790569	−	0.612372i	$-0.209785\pi$
$2$	2.23607	1.58114	0.790569	+	0.612372i	$0.209785\pi$
$3$	0	0
$3$	0	0
$4$	3.00000	1.50000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.23607	0.790569
$9$	0	0
$10$	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	$-0.287062\pi$
$13$	4.47214	1.24035	0.620174	+	0.784465i	$0.287062\pi$
$14$	−2.23607	−0.597614
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−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	$-0.233685\pi$
$19$	6.47214	1.48481	0.742405	+	0.669951i	$0.233685\pi$
$20$	0	0
$21$	0	0
$22$	5.52786	1.17854
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	0	0
$26$	10.0000	1.96116
$27$	0	0
$28$	−3.00000	−0.566947
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	10.4721	1.88085	0.940426	−	0.340000i	$-0.110427\pi$
$31$	10.4721	1.88085	0.940426	+	0.340000i	$0.110427\pi$
$32$	−6.70820	−1.18585
$33$	0	0
$34$	−4.47214	−0.766965
$35$	0	0
$36$	0	0
$37$	−10.9443	−1.79923	−0.899614	−	0.436687i	$-0.856152\pi$
$37$	−10.9443	−1.79923	−0.899614	+	0.436687i	$0.856152\pi$
$38$	14.4721	2.34769
$39$	0	0
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	8.94427	1.36399	0.681994	−	0.731357i	$-0.261113\pi$
$43$	8.94427	1.36399	0.681994	+	0.731357i	$0.261113\pi$
$44$	7.41641	1.11807
$45$	0	0
$46$	8.94427	1.31876
$47$	−4.94427	−0.721196	−0.360598	−	0.932721i	$-0.617427\pi$
$47$	−4.94427	−0.721196	−0.360598	+	0.932721i	$0.617427\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	13.4164	1.86052
$53$	−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
$55$	0	0
$56$	−2.23607	−0.298807
$57$	0	0
$58$	4.47214	0.587220
$59$	−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	$0.540868\pi$
$62$	23.4164	2.97389
$63$	0	0
$64$	−13.0000	−1.62500
$65$	0	0
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	−6.00000	−0.727607
$69$	0	0
$70$	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	$-0.433808\pi$
$73$	3.52786	0.412905	0.206453	+	0.978457i	$0.433808\pi$
$74$	−24.4721	−2.84483
$75$	0	0
$76$	19.4164	2.22721
$77$	−2.47214	−0.281726
$78$	0	0
$79$	−4.94427	−0.556274	−0.278137	−	0.960541i	$-0.589717\pi$
$79$	−4.94427	−0.556274	−0.278137	+	0.960541i	$0.589717\pi$
$80$	0	0
$81$	0	0
$82$	4.47214	0.493865
$83$	0.944272	0.103647	0.0518237	−	0.998656i	$-0.483497\pi$
$83$	0.944272	0.103647	0.0518237	+	0.998656i	$0.483497\pi$
$84$	0	0
$85$	0	0
$86$	20.0000	2.15666
$87$	0	0
$88$	5.52786	0.589272
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	−4.47214	−0.468807
$92$	12.0000	1.25109
$93$	0	0
$94$	−11.0557	−1.14031
$95$	0	0
$96$	0	0
$97$	0.472136	0.0479381	0.0239691	−	0.999713i	$-0.492370\pi$
$97$	0.472136	0.0479381	0.0239691	+	0.999713i	$0.492370\pi$
$98$	2.23607	0.225877
$99$	0	0
$100$	0	0
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	10.0000	0.980581
$105$	0	0
$106$	−27.8885	−2.70877
$107$	4.94427	0.477981	0.238990	−	0.971022i	$-0.423184\pi$
$107$	4.94427	0.477981	0.238990	+	0.971022i	$0.423184\pi$
$108$	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
$111$	0	0
$112$	1.00000	0.0944911
$113$	−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
$115$	0	0
$116$	6.00000	0.557086
$117$	0	0
$118$	−20.0000	−1.84115
$119$	2.00000	0.183340
$120$	0	0
$121$	−4.88854	−0.444413
$122$	−4.47214	−0.404888
$123$	0	0
$124$	31.4164	2.82128
$125$	0	0
$126$	0	0
$127$	−12.9443	−1.14862	−0.574309	−	0.818638i	$-0.694729\pi$
$127$	−12.9443	−1.14862	−0.574309	+	0.818638i	$0.694729\pi$
$128$	−15.6525	−1.38350
$129$	0	0
$130$	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
$134$	8.94427	0.772667
$135$	0	0
$136$	−4.47214	−0.383482
$137$	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	$-0.807948\pi$
$139$	−19.4164	−1.64688	−0.823439	+	0.567405i	$0.807948\pi$
$140$	0	0
$141$	0	0
$142$	−32.3607	−2.71565
$143$	11.0557	0.924526
$144$	0	0
$145$	0	0
$146$	7.88854	0.652861
$147$	0	0
$148$	−32.8328	−2.69884
$149$	−2.94427	−0.241204	−0.120602	−	0.992701i	$-0.538483\pi$
$149$	−2.94427	−0.241204	−0.120602	+	0.992701i	$0.538483\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	14.4721	1.17385
$153$	0	0
$154$	−5.52786	−0.445448
$155$	0	0
$156$	0	0
$157$	−8.47214	−0.676150	−0.338075	−	0.941119i	$-0.609776\pi$
$157$	−8.47214	−0.676150	−0.338075	+	0.941119i	$0.609776\pi$
$158$	−11.0557	−0.879547
$159$	0	0
$160$	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	$0.511774\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	2.11146	0.163881
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	7.00000	0.538462
$170$	0	0
$171$	0	0
$172$	26.8328	2.04598
$173$	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
$175$	0	0
$176$	−2.47214	−0.186344
$177$	0	0
$178$	4.47214	0.335201
$179$	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	$0.251369\pi$
$182$	−10.0000	−0.741249
$183$	0	0
$184$	8.94427	0.659380
$185$	0	0
$186$	0	0
$187$	−4.94427	−0.361561
$188$	−14.8328	−1.08179
$189$	0	0
$190$	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	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	1.05573	0.0757969
$195$	0	0
$196$	3.00000	0.214286
$197$	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	$-0.493415\pi$
$199$	0.583592	0.0413697	0.0206849	+	0.999786i	$0.493415\pi$
$200$	0	0
$201$	0	0
$202$	31.3050	2.20261
$203$	−2.00000	−0.140372
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	−4.47214	−0.310087
$209$	16.0000	1.10674
$210$	0	0
$211$	0.944272	0.0650064	0.0325032	−	0.999472i	$-0.489652\pi$
$211$	0.944272	0.0650064	0.0325032	+	0.999472i	$0.489652\pi$
$212$	−37.4164	−2.56977
$213$	0	0
$214$	11.0557	0.755754
$215$	0	0
$216$	0	0
$217$	−10.4721	−0.710895
$218$	−4.47214	−0.302891
$219$	0	0
$220$	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	$0.552939\pi$
$224$	6.70820	0.448211
$225$	0	0
$226$	−18.9443	−1.26015
$227$	−16.9443	−1.12463	−0.562315	−	0.826923i	$-0.690089\pi$
$227$	−16.9443	−1.12463	−0.562315	+	0.826923i	$0.690089\pi$
$228$	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
$231$	0	0
$232$	4.47214	0.293610
$233$	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
$235$	0	0
$236$	−26.8328	−1.74667
$237$	0	0
$238$	4.47214	0.289886
$239$	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	$0.510825\pi$
$242$	−10.9311	−0.702679
$243$	0	0
$244$	−6.00000	−0.384111
$245$	0	0
$246$	0	0
$247$	28.9443	1.84168
$248$	23.4164	1.48694
$249$	0	0
$250$	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
$254$	−28.9443	−1.81613
$255$	0	0
$256$	−9.00000	−0.562500
$257$	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
$261$	0	0
$262$	−8.94427	−0.552579
$263$	24.9443	1.53813	0.769065	−	0.639171i	$-0.220722\pi$
$263$	24.9443	1.53813	0.769065	+	0.639171i	$0.220722\pi$
$264$	0	0
$265$	0	0
$266$	−14.4721	−0.887344
$267$	0	0
$268$	12.0000	0.733017
$269$	23.8885	1.45651	0.728255	−	0.685306i	$-0.240332\pi$
$269$	23.8885	1.45651	0.728255	+	0.685306i	$0.240332\pi$
$270$	0	0
$271$	−10.4721	−0.636137	−0.318068	−	0.948068i	$-0.603034\pi$
$271$	−10.4721	−0.636137	−0.318068	+	0.948068i	$0.603034\pi$
$272$	2.00000	0.121268
$273$	0	0
$274$	27.8885	1.68481
$275$	0	0
$276$	0	0
$277$	−1.05573	−0.0634326	−0.0317163	−	0.999497i	$-0.510097\pi$
$277$	−1.05573	−0.0634326	−0.0317163	+	0.999497i	$0.510097\pi$
$278$	−43.4164	−2.60394
$279$	0	0
$280$	0	0
$281$	−6.94427	−0.414261	−0.207130	−	0.978313i	$-0.566412\pi$
$281$	−6.94427	−0.414261	−0.207130	+	0.978313i	$0.566412\pi$
$282$	0	0
$283$	−12.0000	−0.713326	−0.356663	−	0.934233i	$-0.616086\pi$
$283$	−12.0000	−0.713326	−0.356663	+	0.934233i	$0.616086\pi$
$284$	−43.4164	−2.57629
$285$	0	0
$286$	24.7214	1.46180
$287$	−2.00000	−0.118056
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	0	0
$292$	10.5836	0.619358
$293$	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
$295$	0	0
$296$	−24.4721	−1.42241
$297$	0	0
$298$	−6.58359	−0.381377
$299$	17.8885	1.03452
$300$	0	0
$301$	−8.94427	−0.515539
$302$	−35.7771	−2.05874
$303$	0	0
$304$	−6.47214	−0.371202
$305$	0	0
$306$	0	0
$307$	32.9443	1.88023	0.940114	−	0.340859i	$-0.110718\pi$
$307$	32.9443	1.88023	0.940114	+	0.340859i	$0.110718\pi$
$308$	−7.41641	−0.422589
$309$	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	$-0.585743\pi$
$313$	−9.41641	−0.532247	−0.266123	+	0.963939i	$0.585743\pi$
$314$	−18.9443	−1.06909
$315$	0	0
$316$	−14.8328	−0.834411
$317$	−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
$321$	0	0
$322$	−8.94427	−0.498445
$323$	−12.9443	−0.720239
$324$	0	0
$325$	0	0
$326$	−2.11146	−0.116943
$327$	0	0
$328$	4.47214	0.246932
$329$	4.94427	0.272587
$330$	0	0
$331$	−16.9443	−0.931341	−0.465671	−	0.884958i	$-0.654187\pi$
$331$	−16.9443	−0.931341	−0.465671	+	0.884958i	$0.654187\pi$
$332$	2.83282	0.155471
$333$	0	0
$334$	−17.8885	−0.978818
$335$	0	0
$336$	0	0
$337$	−11.8885	−0.647610	−0.323805	−	0.946124i	$-0.604962\pi$
$337$	−11.8885	−0.647610	−0.323805	+	0.946124i	$0.604962\pi$
$338$	15.6525	0.851382
$339$	0	0
$340$	0	0
$341$	25.8885	1.40194
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	20.0000	1.07833
$345$	0	0
$346$	33.4164	1.79648
$347$	−8.00000	−0.429463	−0.214731	−	0.976673i	$-0.568888\pi$
$347$	−8.00000	−0.429463	−0.214731	+	0.976673i	$0.568888\pi$
$348$	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
$351$	0	0
$352$	−16.5836	−0.883908
$353$	−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
$355$	0	0
$356$	6.00000	0.317999
$357$	0	0
$358$	5.52786	0.292157
$359$	−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
$362$	42.3607	2.22643
$363$	0	0
$364$	−13.4164	−0.703211
$365$	0	0
$366$	0	0
$367$	−20.9443	−1.09328	−0.546641	−	0.837367i	$-0.684094\pi$
$367$	−20.9443	−1.09328	−0.546641	+	0.837367i	$0.684094\pi$
$368$	−4.00000	−0.208514
$369$	0	0
$370$	0	0
$371$	12.4721	0.647521
$372$	0	0
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	−11.0557	−0.571678
$375$	0	0
$376$	−11.0557	−0.570156
$377$	8.94427	0.460653
$378$	0	0
$379$	−2.11146	−0.108458	−0.0542291	−	0.998529i	$-0.517270\pi$
$379$	−2.11146	−0.108458	−0.0542291	+	0.998529i	$0.517270\pi$
$380$	0	0
$381$	0	0
$382$	−61.3050	−3.13663
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	0	0
$385$	0	0
$386$	31.3050	1.59338
$387$	0	0
$388$	1.41641	0.0719072
$389$	−10.9443	−0.554897	−0.277448	−	0.960741i	$-0.589489\pi$
$389$	−10.9443	−0.554897	−0.277448	+	0.960741i	$0.589489\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	2.23607	0.112938
$393$	0	0
$394$	54.7214	2.75682
$395$	0	0
$396$	0	0
$397$	−13.4164	−0.673350	−0.336675	−	0.941621i	$-0.609302\pi$
$397$	−13.4164	−0.673350	−0.336675	+	0.941621i	$0.609302\pi$
$398$	1.30495	0.0654113
$399$	0	0
$400$	0	0
$401$	−10.0000	−0.499376	−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000	−0.499376	−0.249688	+	0.968326i	$0.580328\pi$
$402$	0	0
$403$	46.8328	2.33291
$404$	42.0000	2.08958
$405$	0	0
$406$	−4.47214	−0.221948
$407$	−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
$411$	0	0
$412$	0	0
$413$	8.94427	0.440119
$414$	0	0
$415$	0	0
$416$	−30.0000	−1.47087
$417$	0	0
$418$	35.7771	1.74991
$419$	−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	$0.319894\pi$
$422$	2.11146	0.102784
$423$	0	0
$424$	−27.8885	−1.35439
$425$	0	0
$426$	0	0
$427$	2.00000	0.0967868
$428$	14.8328	0.716971
$429$	0	0
$430$	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	$-0.557896\pi$
$433$	−7.52786	−0.361766	−0.180883	+	0.983505i	$0.557896\pi$
$434$	−23.4164	−1.12402
$435$	0	0
$436$	−6.00000	−0.287348
$437$	25.8885	1.23842
$438$	0	0
$439$	10.4721	0.499808	0.249904	−	0.968271i	$-0.419601\pi$
$439$	10.4721	0.499808	0.249904	+	0.968271i	$0.419601\pi$
$440$	0	0
$441$	0	0
$442$	−20.0000	−0.951303
$443$	−8.00000	−0.380091	−0.190046	−	0.981775i	$-0.560864\pi$
$443$	−8.00000	−0.380091	−0.190046	+	0.981775i	$0.560864\pi$
$444$	0	0
$445$	0	0
$446$	−11.0557	−0.523504
$447$	0	0
$448$	13.0000	0.614192
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	0	0
$451$	4.94427	0.232817
$452$	−25.4164	−1.19549
$453$	0	0
$454$	−37.8885	−1.77820
$455$	0	0
$456$	0	0
$457$	10.9443	0.511951	0.255976	−	0.966683i	$-0.417603\pi$
$457$	10.9443	0.511951	0.255976	+	0.966683i	$0.417603\pi$
$458$	−26.5836	−1.24217
$459$	0	0
$460$	0	0
$461$	31.8885	1.48520	0.742599	−	0.669737i	$-0.233593\pi$
$461$	31.8885	1.48520	0.742599	+	0.669737i	$0.233593\pi$
$462$	0	0
$463$	−3.05573	−0.142012	−0.0710059	−	0.997476i	$-0.522621\pi$
$463$	−3.05573	−0.142012	−0.0710059	+	0.997476i	$0.522621\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	38.9443	1.80406
$467$	8.94427	0.413892	0.206946	−	0.978352i	$-0.433648\pi$
$467$	8.94427	0.413892	0.206946	+	0.978352i	$0.433648\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	0	0
$471$	0	0
$472$	−20.0000	−0.920575
$473$	22.1115	1.01669
$474$	0	0
$475$	0	0
$476$	6.00000	0.275010
$477$	0	0
$478$	3.41641	0.156263
$479$	−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
$482$	−2.36068	−0.107526
$483$	0	0
$484$	−14.6656	−0.666620
$485$	0	0
$486$	0	0
$487$	3.05573	0.138468	0.0692341	−	0.997600i	$-0.477944\pi$
$487$	3.05573	0.138468	0.0692341	+	0.997600i	$0.477944\pi$
$488$	−4.47214	−0.202444
$489$	0	0
$490$	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
$494$	64.7214	2.91195
$495$	0	0
$496$	−10.4721	−0.470213
$497$	14.4721	0.649164
$498$	0	0
$499$	21.8885	0.979866	0.489933	−	0.871760i	$-0.337021\pi$
$499$	21.8885	0.979866	0.489933	+	0.871760i	$0.337021\pi$
$500$	0	0
$501$	0	0
$502$	2.11146	0.0942389
$503$	32.0000	1.42681	0.713405	−	0.700752i	$-0.247152\pi$
$503$	32.0000	1.42681	0.713405	+	0.700752i	$0.247152\pi$
$504$	0	0
$505$	0	0
$506$	22.1115	0.982974
$507$	0	0
$508$	−38.8328	−1.72293
$509$	−11.8885	−0.526950	−0.263475	−	0.964666i	$-0.584869\pi$
$509$	−11.8885	−0.526950	−0.263475	+	0.964666i	$0.584869\pi$
$510$	0	0
$511$	−3.52786	−0.156064
$512$	11.1803	0.494106
$513$	0	0
$514$	2.36068	0.104125
$515$	0	0
$516$	0	0
$517$	−12.2229	−0.537563
$518$	24.4721	1.07524
$519$	0	0
$520$	0	0
$521$	−15.8885	−0.696090	−0.348045	−	0.937478i	$-0.613154\pi$
$521$	−15.8885	−0.696090	−0.348045	+	0.937478i	$0.613154\pi$
$522$	0	0
$523$	−8.94427	−0.391106	−0.195553	−	0.980693i	$-0.562650\pi$
$523$	−8.94427	−0.391106	−0.195553	+	0.980693i	$0.562650\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	55.7771	2.43200
$527$	−20.9443	−0.912347
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	−19.4164	−0.841808
$533$	8.94427	0.387419
$534$	0	0
$535$	0	0
$536$	8.94427	0.386334
$537$	0	0
$538$	53.4164	2.30294
$539$	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	$0.328340\pi$
$542$	−23.4164	−1.00582
$543$	0	0
$544$	13.4164	0.575224
$545$	0	0
$546$	0	0
$547$	29.8885	1.27794	0.638971	−	0.769231i	$-0.279360\pi$
$547$	29.8885	1.27794	0.638971	+	0.769231i	$0.279360\pi$
$548$	37.4164	1.59835
$549$	0	0
$550$	0	0
$551$	12.9443	0.551445
$552$	0	0
$553$	4.94427	0.210252
$554$	−2.36068	−0.100296
$555$	0	0
$556$	−58.2492	−2.47032
$557$	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
$561$	0	0
$562$	−15.5279	−0.655003
$563$	−21.8885	−0.922492	−0.461246	−	0.887272i	$-0.652597\pi$
$563$	−21.8885	−0.922492	−0.461246	+	0.887272i	$0.652597\pi$
$564$	0	0
$565$	0	0
$566$	−26.8328	−1.12787
$567$	0	0
$568$	−32.3607	−1.35782
$569$	4.11146	0.172361	0.0861806	−	0.996280i	$-0.472534\pi$
$569$	4.11146	0.172361	0.0861806	+	0.996280i	$0.472534\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	33.1672	1.38679
$573$	0	0
$574$	−4.47214	−0.186663
$575$	0	0
$576$	0	0
$577$	34.3607	1.43045	0.715227	−	0.698892i	$-0.246323\pi$
$577$	34.3607	1.43045	0.715227	+	0.698892i	$0.246323\pi$
$578$	−29.0689	−1.20911
$579$	0	0
$580$	0	0
$581$	−0.944272	−0.0391750
$582$	0	0
$583$	−30.8328	−1.27696
$584$	7.88854	0.326430
$585$	0	0
$586$	51.3050	2.11939
$587$	−4.00000	−0.165098	−0.0825488	−	0.996587i	$-0.526306\pi$
$587$	−4.00000	−0.165098	−0.0825488	+	0.996587i	$0.526306\pi$
$588$	0	0
$589$	67.7771	2.79271
$590$	0	0
$591$	0	0
$592$	10.9443	0.449807
$593$	−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
$595$	0	0
$596$	−8.83282	−0.361806
$597$	0	0
$598$	40.0000	1.63572
$599$	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	$0.358711\pi$
$602$	−20.0000	−0.815139
$603$	0	0
$604$	−48.0000	−1.95309
$605$	0	0
$606$	0	0
$607$	14.8328	0.602045	0.301023	−	0.953617i	$-0.402672\pi$
$607$	14.8328	0.602045	0.301023	+	0.953617i	$0.402672\pi$
$608$	−43.4164	−1.76077
$609$	0	0
$610$	0	0
$611$	−22.1115	−0.894534
$612$	0	0
$613$	−10.9443	−0.442035	−0.221017	−	0.975270i	$-0.570938\pi$
$613$	−10.9443	−0.442035	−0.221017	+	0.975270i	$0.570938\pi$
$614$	73.6656	2.97290
$615$	0	0
$616$	−5.52786	−0.222724
$617$	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	$-0.418619\pi$
$619$	12.5836	0.505777	0.252889	+	0.967495i	$0.418619\pi$
$620$	0	0
$621$	0	0
$622$	22.1115	0.886589
$623$	−2.00000	−0.0801283
$624$	0	0
$625$	0	0
$626$	−21.0557	−0.841556
$627$	0	0
$628$	−25.4164	−1.01423
$629$	21.8885	0.872753
$630$	0	0
$631$	−22.8328	−0.908960	−0.454480	−	0.890757i	$-0.650175\pi$
$631$	−22.8328	−0.908960	−0.454480	+	0.890757i	$0.650175\pi$
$632$	−11.0557	−0.439773
$633$	0	0
$634$	−67.8885	−2.69620
$635$	0	0
$636$	0	0
$637$	4.47214	0.177192
$638$	11.0557	0.437700
$639$	0	0
$640$	0	0
$641$	36.8328	1.45481	0.727404	−	0.686209i	$-0.240726\pi$
$641$	36.8328	1.45481	0.727404	+	0.686209i	$0.240726\pi$
$642$	0	0
$643$	32.9443	1.29920	0.649598	−	0.760278i	$-0.274937\pi$
$643$	32.9443	1.29920	0.649598	+	0.760278i	$0.274937\pi$
$644$	−12.0000	−0.472866
$645$	0	0
$646$	−28.9443	−1.13880
$647$	33.8885	1.33230	0.666148	−	0.745820i	$-0.267942\pi$
$647$	33.8885	1.33230	0.666148	+	0.745820i	$0.267942\pi$
$648$	0	0
$649$	−22.1115	−0.867951
$650$	0	0
$651$	0	0
$652$	−2.83282	−0.110942
$653$	−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
$655$	0	0
$656$	−2.00000	−0.0780869
$657$	0	0
$658$	11.0557	0.430997
$659$	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	$0.500690\pi$
$662$	−37.8885	−1.47258
$663$	0	0
$664$	2.11146	0.0819404
$665$	0	0
$666$	0	0
$667$	8.00000	0.309761
$668$	−24.0000	−0.928588
$669$	0	0
$670$	0	0
$671$	−4.94427	−0.190872
$672$	0	0
$673$	44.8328	1.72818	0.864089	−	0.503339i	$-0.167895\pi$
$673$	44.8328	1.72818	0.864089	+	0.503339i	$0.167895\pi$
$674$	−26.5836	−1.02396
$675$	0	0
$676$	21.0000	0.807692
$677$	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
$681$	0	0
$682$	57.8885	2.21667
$683$	33.8885	1.29671	0.648355	−	0.761339i	$-0.275457\pi$
$683$	33.8885	1.29671	0.648355	+	0.761339i	$0.275457\pi$
$684$	0	0
$685$	0	0
$686$	−2.23607	−0.0853735
$687$	0	0
$688$	−8.94427	−0.340997
$689$	−55.7771	−2.12494
$690$	0	0
$691$	−0.360680	−0.0137209	−0.00686045	−	0.999976i	$-0.502184\pi$
$691$	−0.360680	−0.0137209	−0.00686045	+	0.999976i	$0.502184\pi$
$692$	44.8328	1.70429
$693$	0	0
$694$	−17.8885	−0.679040
$695$	0	0
$696$	0	0
$697$	−4.00000	−0.151511
$698$	53.4164	2.02184
$699$	0	0
$700$	0	0
$701$	34.0000	1.28416	0.642081	−	0.766637i	$-0.278071\pi$
$701$	34.0000	1.28416	0.642081	+	0.766637i	$0.278071\pi$
$702$	0	0
$703$	−70.8328	−2.67151
$704$	−32.1378	−1.21124
$705$	0	0
$706$	−62.3607	−2.34698
$707$	−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
$711$	0	0
$712$	4.47214	0.167600
$713$	41.8885	1.56874
$714$	0	0
$715$	0	0
$716$	7.41641	0.277164
$717$	0	0
$718$	−21.3050	−0.795094
$719$	−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
$722$	51.1803	1.90474
$723$	0	0
$724$	56.8328	2.11217
$725$	0	0
$726$	0	0
$727$	−14.8328	−0.550119	−0.275059	−	0.961427i	$-0.588698\pi$
$727$	−14.8328	−0.550119	−0.275059	+	0.961427i	$0.588698\pi$
$728$	−10.0000	−0.370625
$729$	0	0
$730$	0	0
$731$	−17.8885	−0.661632
$732$	0	0
$733$	−37.4164	−1.38201	−0.691003	−	0.722852i	$-0.742831\pi$
$733$	−37.4164	−1.38201	−0.691003	+	0.722852i	$0.742831\pi$
$734$	−46.8328	−1.72863
$735$	0	0
$736$	−26.8328	−0.989071
$737$	9.88854	0.364249
$738$	0	0
$739$	29.8885	1.09947	0.549734	−	0.835340i	$-0.314729\pi$
$739$	29.8885	1.09947	0.549734	+	0.835340i	$0.314729\pi$
$740$	0	0
$741$	0	0
$742$	27.8885	1.02382
$743$	−18.8328	−0.690909	−0.345455	−	0.938436i	$-0.612275\pi$
$743$	−18.8328	−0.690909	−0.345455	+	0.938436i	$0.612275\pi$
$744$	0	0
$745$	0	0
$746$	−13.4164	−0.491210
$747$	0	0
$748$	−14.8328	−0.542341
$749$	−4.94427	−0.180660
$750$	0	0
$751$	−3.05573	−0.111505	−0.0557526	−	0.998445i	$-0.517756\pi$
$751$	−3.05573	−0.111505	−0.0557526	+	0.998445i	$0.517756\pi$
$752$	4.94427	0.180299
$753$	0	0
$754$	20.0000	0.728357
$755$	0	0
$756$	0	0
$757$	3.88854	0.141332	0.0706658	−	0.997500i	$-0.477488\pi$
$757$	3.88854	0.141332	0.0706658	+	0.997500i	$0.477488\pi$
$758$	−4.72136	−0.171488
$759$	0	0
$760$	0	0
$761$	−7.88854	−0.285959	−0.142980	−	0.989726i	$-0.545668\pi$
$761$	−7.88854	−0.285959	−0.142980	+	0.989726i	$0.545668\pi$
$762$	0	0
$763$	2.00000	0.0724049
$764$	−82.2492	−2.97567
$765$	0	0
$766$	−17.8885	−0.646339
$767$	−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
$771$	0	0
$772$	42.0000	1.51161
$773$	−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
$775$	0	0
$776$	1.05573	0.0378984
$777$	0	0
$778$	−24.4721	−0.877369
$779$	12.9443	0.463777
$780$	0	0
$781$	−35.7771	−1.28020
$782$	−17.8885	−0.639693
$783$	0	0
$784$	−1.00000	−0.0357143
$785$	0	0
$786$	0	0
$787$	−48.9443	−1.74467	−0.872337	−	0.488904i	$-0.837397\pi$
$787$	−48.9443	−1.74467	−0.872337	+	0.488904i	$0.837397\pi$
$788$	73.4164	2.61535
$789$	0	0
$790$	0	0
$791$	8.47214	0.301234
$792$	0	0
$793$	−8.94427	−0.317620
$794$	−30.0000	−1.06466
$795$	0	0
$796$	1.75078	0.0620546
$797$	−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
$801$	0	0
$802$	−22.3607	−0.789583
$803$	8.72136	0.307770
$804$	0	0
$805$	0	0
$806$	104.721	3.68865
$807$	0	0
$808$	31.3050	1.10130
$809$	−21.0557	−0.740280	−0.370140	−	0.928976i	$-0.620690\pi$
$809$	−21.0557	−0.740280	−0.370140	+	0.928976i	$0.620690\pi$
$810$	0	0
$811$	28.5836	1.00371	0.501853	−	0.864953i	$-0.332652\pi$
$811$	28.5836	1.00371	0.501853	+	0.864953i	$0.332652\pi$
$812$	−6.00000	−0.210559
$813$	0	0
$814$	−60.4984	−2.12047
$815$	0	0
$816$	0	0
$817$	57.8885	2.02526
$818$	−53.4164	−1.86766
$819$	0	0
$820$	0	0
$821$	37.7771	1.31843	0.659215	−	0.751955i	$-0.270889\pi$
$821$	37.7771	1.31843	0.659215	+	0.751955i	$0.270889\pi$
$822$	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
$825$	0	0
$826$	20.0000	0.695889
$827$	−4.94427	−0.171929	−0.0859646	−	0.996298i	$-0.527397\pi$
$827$	−4.94427	−0.171929	−0.0859646	+	0.996298i	$0.527397\pi$
$828$	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
$831$	0	0
$832$	−58.1378	−2.01556
$833$	−2.00000	−0.0692959
$834$	0	0
$835$	0	0
$836$	48.0000	1.66011
$837$	0	0
$838$	−13.1672	−0.454853
$839$	−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
$842$	49.1935	1.69532
$843$	0	0
$844$	2.83282	0.0975095
$845$	0	0
$846$	0	0
$847$	4.88854	0.167972
$848$	12.4721	0.428295
$849$	0	0
$850$	0	0
$851$	−43.7771	−1.50066
$852$	0	0
$853$	−31.3050	−1.07186	−0.535931	−	0.844262i	$-0.680039\pi$
$853$	−31.3050	−1.07186	−0.535931	+	0.844262i	$0.680039\pi$
$854$	4.47214	0.153033
$855$	0	0
$856$	11.0557	0.377877
$857$	−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	$-0.249392\pi$
$859$	41.5279	1.41691	0.708456	+	0.705755i	$0.249392\pi$
$860$	0	0
$861$	0	0
$862$	21.3050	0.725650
$863$	−13.8885	−0.472772	−0.236386	−	0.971659i	$-0.575963\pi$
$863$	−13.8885	−0.472772	−0.236386	+	0.971659i	$0.575963\pi$
$864$	0	0
$865$	0	0
$866$	−16.8328	−0.572002
$867$	0	0
$868$	−31.4164	−1.06634
$869$	−12.2229	−0.414634
$870$	0	0
$871$	17.8885	0.606130
$872$	−4.47214	−0.151446
$873$	0	0
$874$	57.8885	1.95811
$875$	0	0
$876$	0	0
$877$	3.16718	0.106948	0.0534741	−	0.998569i	$-0.482971\pi$
$877$	3.16718	0.106948	0.0534741	+	0.998569i	$0.482971\pi$
$878$	23.4164	0.790265
$879$	0	0
$880$	0	0
$881$	−7.88854	−0.265772	−0.132886	−	0.991131i	$-0.542424\pi$
$881$	−7.88854	−0.265772	−0.132886	+	0.991131i	$0.542424\pi$
$882$	0	0
$883$	2.11146	0.0710562	0.0355281	−	0.999369i	$-0.488689\pi$
$883$	2.11146	0.0710562	0.0355281	+	0.999369i	$0.488689\pi$
$884$	−26.8328	−0.902485
$885$	0	0
$886$	−17.8885	−0.600977
$887$	22.8328	0.766651	0.383325	−	0.923613i	$-0.374779\pi$
$887$	22.8328	0.766651	0.383325	+	0.923613i	$0.374779\pi$
$888$	0	0
$889$	12.9443	0.434137
$890$	0	0
$891$	0	0
$892$	−14.8328	−0.496639
$893$	−32.0000	−1.07084
$894$	0	0
$895$	0	0
$896$	15.6525	0.522913
$897$	0	0
$898$	31.3050	1.04466
$899$	20.9443	0.698531
$900$	0	0
$901$	24.9443	0.831014
$902$	11.0557	0.368115
$903$	0	0
$904$	−18.9443	−0.630077
$905$	0	0
$906$	0	0
$907$	−18.1115	−0.601381	−0.300691	−	0.953722i	$-0.597217\pi$
$907$	−18.1115	−0.601381	−0.300691	+	0.953722i	$0.597217\pi$
$908$	−50.8328	−1.68695
$909$	0	0
$910$	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
$914$	24.4721	0.809466
$915$	0	0
$916$	−35.6656	−1.17843
$917$	4.00000	0.132092
$918$	0	0
$919$	−52.9443	−1.74647	−0.873235	−	0.487299i	$-0.837982\pi$
$919$	−52.9443	−1.74647	−0.873235	+	0.487299i	$0.837982\pi$
$920$	0	0
$921$	0	0
$922$	71.3050	2.34830
$923$	−64.7214	−2.13033
$924$	0	0
$925$	0	0
$926$	−6.83282	−0.224540
$927$	0	0
$928$	−13.4164	−0.440415
$929$	51.8885	1.70241	0.851204	−	0.524835i	$-0.175873\pi$
$929$	51.8885	1.70241	0.851204	+	0.524835i	$0.175873\pi$
$930$	0	0
$931$	6.47214	0.212116
$932$	52.2492	1.71148
$933$	0	0
$934$	20.0000	0.654420
$935$	0	0
$936$	0	0
$937$	43.5279	1.42199	0.710997	−	0.703195i	$-0.248244\pi$
$937$	43.5279	1.42199	0.710997	+	0.703195i	$0.248244\pi$
$938$	−8.94427	−0.292041
$939$	0	0
$940$	0	0
$941$	30.0000	0.977972	0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000	0.977972	0.488986	+	0.872292i	$0.337367\pi$
$942$	0	0
$943$	8.00000	0.260516
$944$	8.94427	0.291111
$945$	0	0
$946$	49.4427	1.60752
$947$	17.8885	0.581300	0.290650	−	0.956830i	$-0.406129\pi$
$947$	17.8885	0.581300	0.290650	+	0.956830i	$0.406129\pi$
$948$	0	0
$949$	15.7771	0.512146
$950$	0	0
$951$	0	0
$952$	4.47214	0.144943
$953$	−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
$955$	0	0
$956$	4.58359	0.148244
$957$	0	0
$958$	−40.0000	−1.29234
$959$	−12.4721	−0.402746
$960$	0	0
$961$	78.6656	2.53760
$962$	−109.443	−3.52857
$963$	0	0
$964$	−3.16718	−0.102008
$965$	0	0
$966$	0	0
$967$	9.88854	0.317994	0.158997	−	0.987279i	$-0.449174\pi$
$967$	9.88854	0.317994	0.158997	+	0.987279i	$0.449174\pi$
$968$	−10.9311	−0.351339
$969$	0	0
$970$	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
$974$	6.83282	0.218938
$975$	0	0
$976$	2.00000	0.0640184
$977$	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
$981$	0	0
$982$	−92.3607	−2.94735
$983$	30.8328	0.983414	0.491707	−	0.870761i	$-0.336373\pi$
$983$	30.8328	0.983414	0.491707	+	0.870761i	$0.336373\pi$
$984$	0	0
$985$	0	0
$986$	−8.94427	−0.284844
$987$	0	0
$988$	86.8328	2.76252
$989$	35.7771	1.13765
$990$	0	0
$991$	−12.9443	−0.411188	−0.205594	−	0.978637i	$-0.565913\pi$
$991$	−12.9443	−0.411188	−0.205594	+	0.978637i	$0.565913\pi$
$992$	−70.2492	−2.23042
$993$	0	0
$994$	32.3607	1.02642
$995$	0	0
$996$	0	0
$997$	−21.4164	−0.678264	−0.339132	−	0.940739i	$-0.610133\pi$
$997$	−21.4164	−0.678264	−0.339132	+	0.940739i	$0.610133\pi$
$998$	48.9443	1.54930
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.2.a.r.1.2		2
3.2	odd	2		525.2.a.g.1.1		2
5.2	odd	4		1575.2.d.d.1324.3		4
5.3	odd	4		1575.2.d.d.1324.2		4
5.4	even	2		315.2.a.d.1.1		2
12.11	even	2		8400.2.a.cx.1.2		2
15.2	even	4		525.2.d.c.274.1		4
15.8	even	4		525.2.d.c.274.4		4
15.14	odd	2		105.2.a.b.1.2	✓	2
20.19	odd	2		5040.2.a.bw.1.1		2
21.20	even	2		3675.2.a.y.1.1		2
35.34	odd	2		2205.2.a.w.1.1		2
60.59	even	2		1680.2.a.v.1.2		2
105.44	odd	6		735.2.i.k.361.1		4
105.59	even	6		735.2.i.i.226.1		4
105.74	odd	6		735.2.i.k.226.1		4
105.89	even	6		735.2.i.i.361.1		4
105.104	even	2		735.2.a.k.1.2		2
120.29	odd	2		6720.2.a.cx.1.2		2
120.59	even	2		6720.2.a.cs.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.a.b.1.2	✓	2	15.14	odd	2
315.2.a.d.1.1		2	5.4	even	2
525.2.a.g.1.1		2	3.2	odd	2
525.2.d.c.274.1		4	15.2	even	4
525.2.d.c.274.4		4	15.8	even	4
735.2.a.k.1.2		2	105.104	even	2
735.2.i.i.226.1		4	105.59	even	6
735.2.i.i.361.1		4	105.89	even	6
735.2.i.k.226.1		4	105.74	odd	6
735.2.i.k.361.1		4	105.44	odd	6
1575.2.a.r.1.2		2	1.1	even	1	trivial
1575.2.d.d.1324.2		4	5.3	odd	4
1575.2.d.d.1324.3		4	5.2	odd	4
1680.2.a.v.1.2		2	60.59	even	2
2205.2.a.w.1.1		2	35.34	odd	2
3675.2.a.y.1.1		2	21.20	even	2
5040.2.a.bw.1.1		2	20.19	odd	2
6720.2.a.cs.1.1		2	120.59	even	2
6720.2.a.cx.1.2		2	120.29	odd	2
8400.2.a.cx.1.2		2	12.11	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:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)

\(f(q)\)	\(=\)	\(q+2.23607 q^{2} +3.00000 q^{4} -1.00000 q^{7} +2.23607 q^{8} +O(q^{10})\)	\(q+2.23607 q^{2} +3.00000 q^{4} -1.00000 q^{7} +2.23607 q^{8} +2.47214 q^{11} +4.47214 q^{13} -2.23607 q^{14} -1.00000 q^{16} -2.00000 q^{17} +6.47214 q^{19} +5.52786 q^{22} +4.00000 q^{23} +10.0000 q^{26} -3.00000 q^{28} +2.00000 q^{29} +10.4721 q^{31} -6.70820 q^{32} -4.47214 q^{34} -10.9443 q^{37} +14.4721 q^{38} +2.00000 q^{41} +8.94427 q^{43} +7.41641 q^{44} +8.94427 q^{46} -4.94427 q^{47} +1.00000 q^{49} +13.4164 q^{52} -12.4721 q^{53} -2.23607 q^{56} +4.47214 q^{58} -8.94427 q^{59} -2.00000 q^{61} +23.4164 q^{62} -13.0000 q^{64} +4.00000 q^{67} -6.00000 q^{68} -14.4721 q^{71} +3.52786 q^{73} -24.4721 q^{74} +19.4164 q^{76} -2.47214 q^{77} -4.94427 q^{79} +4.47214 q^{82} +0.944272 q^{83} +20.0000 q^{86} +5.52786 q^{88} +2.00000 q^{89} -4.47214 q^{91} +12.0000 q^{92} -11.0557 q^{94} +0.472136 q^{97} +2.23607 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{4} - 2 q^{7}+O(q^{10}) \) 2 * q + 6 * q^4 - 2 * q^7	\( 2 q + 6 q^{4} - 2 q^{7} - 4 q^{11} - 2 q^{16} - 4 q^{17} + 4 q^{19} + 20 q^{22} + 8 q^{23} + 20 q^{26} - 6 q^{28} + 4 q^{29} + 12 q^{31} - 4 q^{37} + 20 q^{38} + 4 q^{41} - 12 q^{44} + 8 q^{47} + 2 q^{49} - 16 q^{53} - 4 q^{61} + 20 q^{62} - 26 q^{64} + 8 q^{67} - 12 q^{68} - 20 q^{71} + 16 q^{73} - 40 q^{74} + 12 q^{76} + 4 q^{77} + 8 q^{79} - 16 q^{83} + 40 q^{86} + 20 q^{88} + 4 q^{89} + 24 q^{92} - 40 q^{94} - 8 q^{97}+O(q^{100}) \) 2 * q + 6 * q^4 - 2 * q^7 - 4 * q^11 - 2 * q^16 - 4 * q^17 + 4 * q^19 + 20 * q^22 + 8 * q^23 + 20 * q^26 - 6 * q^28 + 4 * q^29 + 12 * q^31 - 4 * q^37 + 20 * q^38 + 4 * q^41 - 12 * q^44 + 8 * q^47 + 2 * q^49 - 16 * q^53 - 4 * q^61 + 20 * q^62 - 26 * q^64 + 8 * q^67 - 12 * q^68 - 20 * q^71 + 16 * q^73 - 40 * q^74 + 12 * q^76 + 4 * q^77 + 8 * q^79 - 16 * q^83 + 40 * q^86 + 20 * q^88 + 4 * q^89 + 24 * q^92 - 40 * q^94 - 8 * q^97

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