LMFDB - Embedded newform 315.2.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,2,Mod(1,315)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(315, 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("315.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	315.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:	$2.51528766367$
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	$\chi$	$=$	315.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.23607 q^{2} +3.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} +2.23607 q^{8} +2.23607 q^{10} -6.47214 q^{11} +4.47214 q^{13} +2.23607 q^{14} -1.00000 q^{16} +2.00000 q^{17} -2.47214 q^{19} +3.00000 q^{20} -14.4721 q^{22} -4.00000 q^{23} +1.00000 q^{25} +10.0000 q^{26} +3.00000 q^{28} +2.00000 q^{29} +1.52786 q^{31} -6.70820 q^{32} +4.47214 q^{34} +1.00000 q^{35} -6.94427 q^{37} -5.52786 q^{38} +2.23607 q^{40} +2.00000 q^{41} +8.94427 q^{43} -19.4164 q^{44} -8.94427 q^{46} -12.9443 q^{47} +1.00000 q^{49} +2.23607 q^{50} +13.4164 q^{52} +3.52786 q^{53} -6.47214 q^{55} +2.23607 q^{56} +4.47214 q^{58} +8.94427 q^{59} -2.00000 q^{61} +3.41641 q^{62} -13.0000 q^{64} +4.47214 q^{65} -4.00000 q^{67} +6.00000 q^{68} +2.23607 q^{70} -5.52786 q^{71} -12.4721 q^{73} -15.5279 q^{74} -7.41641 q^{76} -6.47214 q^{77} +12.9443 q^{79} -1.00000 q^{80} +4.47214 q^{82} +16.9443 q^{83} +2.00000 q^{85} +20.0000 q^{86} -14.4721 q^{88} +2.00000 q^{89} +4.47214 q^{91} -12.0000 q^{92} -28.9443 q^{94} -2.47214 q^{95} +8.47214 q^{97} +2.23607 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{4} + 2 q^{5} + 2 q^{7} - 4 q^{11} - 2 q^{16} + 4 q^{17} + 4 q^{19} + 6 q^{20} - 20 q^{22} - 8 q^{23} + 2 q^{25} + 20 q^{26} + 6 q^{28} + 4 q^{29} + 12 q^{31} + 2 q^{35} + 4 q^{37} - 20 q^{38}+ \cdots + 8 q^{97}+O(q^{100}) $ 2 * q + 6 * q^4 + 2 * q^5 + 2 * q^7 - 4 * q^11 - 2 * q^16 + 4 * q^17 + 4 * q^19 + 6 * q^20 - 20 * q^22 - 8 * q^23 + 2 * q^25 + 20 * q^26 + 6 * q^28 + 4 * q^29 + 12 * q^31 + 2 * q^35 + 4 * q^37 - 20 * q^38 + 4 * q^41 - 12 * q^44 - 8 * q^47 + 2 * q^49 + 16 * q^53 - 4 * q^55 - 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 - 2 * q^80 + 16 * q^83 + 4 * q^85 + 40 * q^86 - 20 * q^88 + 4 * q^89 - 24 * q^92 - 40 * q^94 + 4 * q^95 + 8 * q^97

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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.23607	0.790569
$9$	0	0
$10$	2.23607	0.707107
$11$	−6.47214	−1.95142	−0.975711	−	0.219061i	$-0.929701\pi$
$11$	−6.47214	−1.95142	−0.975711	+	0.219061i	$0.929701\pi$
$12$	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	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	−2.47214	−0.567147	−0.283573	−	0.958951i	$-0.591520\pi$
$19$	−2.47214	−0.567147	−0.283573	+	0.958951i	$0.591520\pi$
$20$	3.00000	0.670820
$21$	0	0
$22$	−14.4721	−3.08547
$23$	−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
$26$	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
$31$	1.52786	0.274412	0.137206	−	0.990543i	$-0.456188\pi$
$31$	1.52786	0.274412	0.137206	+	0.990543i	$0.456188\pi$
$32$	−6.70820	−1.18585
$33$	0	0
$34$	4.47214	0.766965
$35$	1.00000	0.169031
$36$	0	0
$37$	−6.94427	−1.14163	−0.570816	−	0.821078i	$-0.693373\pi$
$37$	−6.94427	−1.14163	−0.570816	+	0.821078i	$0.693373\pi$
$38$	−5.52786	−0.896738
$39$	0	0
$40$	2.23607	0.353553
$41$	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	$0.261113\pi$
$44$	−19.4164	−2.92713
$45$	0	0
$46$	−8.94427	−1.31876
$47$	−12.9443	−1.88812	−0.944058	−	0.329779i	$-0.893026\pi$
$47$	−12.9443	−1.88812	−0.944058	+	0.329779i	$0.893026\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	2.23607	0.316228
$51$	0	0
$52$	13.4164	1.86052
$53$	3.52786	0.484589	0.242295	−	0.970203i	$-0.422100\pi$
$53$	3.52786	0.484589	0.242295	+	0.970203i	$0.422100\pi$
$54$	0	0
$55$	−6.47214	−0.872703
$56$	2.23607	0.298807
$57$	0	0
$58$	4.47214	0.587220
$59$	8.94427	1.16445	0.582223	−	0.813029i	$-0.302183\pi$
$59$	8.94427	1.16445	0.582223	+	0.813029i	$0.302183\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	3.41641	0.433884
$63$	0	0
$64$	−13.0000	−1.62500
$65$	4.47214	0.554700
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	6.00000	0.727607
$69$	0	0
$70$	2.23607	0.267261
$71$	−5.52786	−0.656037	−0.328018	−	0.944671i	$-0.606381\pi$
$71$	−5.52786	−0.656037	−0.328018	+	0.944671i	$0.606381\pi$
$72$	0	0
$73$	−12.4721	−1.45975	−0.729877	−	0.683579i	$-0.760422\pi$
$73$	−12.4721	−1.45975	−0.729877	+	0.683579i	$0.760422\pi$
$74$	−15.5279	−1.80508
$75$	0	0
$76$	−7.41641	−0.850720
$77$	−6.47214	−0.737568
$78$	0	0
$79$	12.9443	1.45634	0.728172	−	0.685394i	$-0.240370\pi$
$79$	12.9443	1.45634	0.728172	+	0.685394i	$0.240370\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	4.47214	0.493865
$83$	16.9443	1.85988	0.929938	−	0.367717i	$-0.119860\pi$
$83$	16.9443	1.85988	0.929938	+	0.367717i	$0.119860\pi$
$84$	0	0
$85$	2.00000	0.216930
$86$	20.0000	2.15666
$87$	0	0
$88$	−14.4721	−1.54273
$89$	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
$94$	−28.9443	−2.98537
$95$	−2.47214	−0.253636
$96$	0	0
$97$	8.47214	0.860215	0.430108	−	0.902778i	$-0.358476\pi$
$97$	8.47214	0.860215	0.430108	+	0.902778i	$0.358476\pi$
$98$	2.23607	0.225877
$99$	0	0
$100$	3.00000	0.300000
$101$	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
$106$	7.88854	0.766203
$107$	12.9443	1.25137	0.625685	−	0.780076i	$-0.284820\pi$
$107$	12.9443	1.25137	0.625685	+	0.780076i	$0.284820\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	−14.4721	−1.37986
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	−0.472136	−0.0444148	−0.0222074	−	0.999753i	$-0.507069\pi$
$113$	−0.472136	−0.0444148	−0.0222074	+	0.999753i	$0.507069\pi$
$114$	0	0
$115$	−4.00000	−0.373002
$116$	6.00000	0.557086
$117$	0	0
$118$	20.0000	1.84115
$119$	2.00000	0.183340
$120$	0	0
$121$	30.8885	2.80805
$122$	−4.47214	−0.404888
$123$	0	0
$124$	4.58359	0.411619
$125$	1.00000	0.0894427
$126$	0	0
$127$	−4.94427	−0.438733	−0.219367	−	0.975643i	$-0.570399\pi$
$127$	−4.94427	−0.438733	−0.219367	+	0.975643i	$0.570399\pi$
$128$	−15.6525	−1.38350
$129$	0	0
$130$	10.0000	0.877058
$131$	−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
$133$	−2.47214	−0.214361
$134$	−8.94427	−0.772667
$135$	0	0
$136$	4.47214	0.383482
$137$	−3.52786	−0.301406	−0.150703	−	0.988579i	$-0.548154\pi$
$137$	−3.52786	−0.301406	−0.150703	+	0.988579i	$0.548154\pi$
$138$	0	0
$139$	7.41641	0.629052	0.314526	−	0.949249i	$-0.398155\pi$
$139$	7.41641	0.629052	0.314526	+	0.949249i	$0.398155\pi$
$140$	3.00000	0.253546
$141$	0	0
$142$	−12.3607	−1.03729
$143$	−28.9443	−2.42044
$144$	0	0
$145$	2.00000	0.166091
$146$	−27.8885	−2.30807
$147$	0	0
$148$	−20.8328	−1.71245
$149$	14.9443	1.22428	0.612141	−	0.790748i	$-0.290308\pi$
$149$	14.9443	1.22428	0.612141	+	0.790748i	$0.290308\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	−5.52786	−0.448369
$153$	0	0
$154$	−14.4721	−1.16620
$155$	1.52786	0.122721
$156$	0	0
$157$	−0.472136	−0.0376806	−0.0188403	−	0.999823i	$-0.505997\pi$
$157$	−0.472136	−0.0376806	−0.0188403	+	0.999823i	$0.505997\pi$
$158$	28.9443	2.30268
$159$	0	0
$160$	−6.70820	−0.530330
$161$	−4.00000	−0.315244
$162$	0	0
$163$	−16.9443	−1.32718	−0.663589	−	0.748097i	$-0.730968\pi$
$163$	−16.9443	−1.32718	−0.663589	+	0.748097i	$0.730968\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	37.8885	2.94072
$167$	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
$170$	4.47214	0.342997
$171$	0	0
$172$	26.8328	2.04598
$173$	2.94427	0.223849	0.111924	−	0.993717i	$-0.464299\pi$
$173$	2.94427	0.223849	0.111924	+	0.993717i	$0.464299\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	6.47214	0.487856
$177$	0	0
$178$	4.47214	0.335201
$179$	−6.47214	−0.483750	−0.241875	−	0.970307i	$-0.577762\pi$
$179$	−6.47214	−0.483750	−0.241875	+	0.970307i	$0.577762\pi$
$180$	0	0
$181$	1.05573	0.0784717	0.0392358	−	0.999230i	$-0.487508\pi$
$181$	1.05573	0.0784717	0.0392358	+	0.999230i	$0.487508\pi$
$182$	10.0000	0.741249
$183$	0	0
$184$	−8.94427	−0.659380
$185$	−6.94427	−0.510553
$186$	0	0
$187$	−12.9443	−0.946579
$188$	−38.8328	−2.83217
$189$	0	0
$190$	−5.52786	−0.401033
$191$	−0.583592	−0.0422272	−0.0211136	−	0.999777i	$-0.506721\pi$
$191$	−0.583592	−0.0422272	−0.0211136	+	0.999777i	$0.506721\pi$
$192$	0	0
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	18.9443	1.36012
$195$	0	0
$196$	3.00000	0.214286
$197$	−15.5279	−1.10631	−0.553157	−	0.833077i	$-0.686577\pi$
$197$	−15.5279	−1.10631	−0.553157	+	0.833077i	$0.686577\pi$
$198$	0	0
$199$	27.4164	1.94350	0.971749	−	0.236017i	$-0.0758423\pi$
$199$	27.4164	1.94350	0.971749	+	0.236017i	$0.0758423\pi$
$200$	2.23607	0.158114
$201$	0	0
$202$	31.3050	2.20261
$203$	2.00000	0.140372
$204$	0	0
$205$	2.00000	0.139686
$206$	0	0
$207$	0	0
$208$	−4.47214	−0.310087
$209$	16.0000	1.10674
$210$	0	0
$211$	−16.9443	−1.16649	−0.583246	−	0.812296i	$-0.698218\pi$
$211$	−16.9443	−1.16649	−0.583246	+	0.812296i	$0.698218\pi$
$212$	10.5836	0.726884
$213$	0	0
$214$	28.9443	1.97859
$215$	8.94427	0.609994
$216$	0	0
$217$	1.52786	0.103718
$218$	−4.47214	−0.302891
$219$	0	0
$220$	−19.4164	−1.30905
$221$	8.94427	0.601657
$222$	0	0
$223$	−12.9443	−0.866813	−0.433406	−	0.901199i	$-0.642688\pi$
$223$	−12.9443	−0.866813	−0.433406	+	0.901199i	$0.642688\pi$
$224$	−6.70820	−0.448211
$225$	0	0
$226$	−1.05573	−0.0702260
$227$	−0.944272	−0.0626735	−0.0313368	−	0.999509i	$-0.509976\pi$
$227$	−0.944272	−0.0626735	−0.0313368	+	0.999509i	$0.509976\pi$
$228$	0	0
$229$	23.8885	1.57860	0.789300	−	0.614008i	$-0.210444\pi$
$229$	23.8885	1.57860	0.789300	+	0.614008i	$0.210444\pi$
$230$	−8.94427	−0.589768
$231$	0	0
$232$	4.47214	0.293610
$233$	9.41641	0.616889	0.308445	−	0.951242i	$-0.400192\pi$
$233$	9.41641	0.616889	0.308445	+	0.951242i	$0.400192\pi$
$234$	0	0
$235$	−12.9443	−0.844391
$236$	26.8328	1.74667
$237$	0	0
$238$	4.47214	0.289886
$239$	10.4721	0.677386	0.338693	−	0.940897i	$-0.390015\pi$
$239$	10.4721	0.677386	0.338693	+	0.940897i	$0.390015\pi$
$240$	0	0
$241$	−18.9443	−1.22031	−0.610154	−	0.792283i	$-0.708892\pi$
$241$	−18.9443	−1.22031	−0.610154	+	0.792283i	$0.708892\pi$
$242$	69.0689	4.43992
$243$	0	0
$244$	−6.00000	−0.384111
$245$	1.00000	0.0638877
$246$	0	0
$247$	−11.0557	−0.703459
$248$	3.41641	0.216942
$249$	0	0
$250$	2.23607	0.141421
$251$	−16.9443	−1.06951	−0.534756	−	0.845006i	$-0.679597\pi$
$251$	−16.9443	−1.06951	−0.534756	+	0.845006i	$0.679597\pi$
$252$	0	0
$253$	25.8885	1.62760
$254$	−11.0557	−0.693698
$255$	0	0
$256$	−9.00000	−0.562500
$257$	−18.9443	−1.18171	−0.590856	−	0.806777i	$-0.701210\pi$
$257$	−18.9443	−1.18171	−0.590856	+	0.806777i	$0.701210\pi$
$258$	0	0
$259$	−6.94427	−0.431496
$260$	13.4164	0.832050
$261$	0	0
$262$	−8.94427	−0.552579
$263$	−7.05573	−0.435075	−0.217537	−	0.976052i	$-0.569802\pi$
$263$	−7.05573	−0.435075	−0.217537	+	0.976052i	$0.569802\pi$
$264$	0	0
$265$	3.52786	0.216715
$266$	−5.52786	−0.338935
$267$	0	0
$268$	−12.0000	−0.733017
$269$	−11.8885	−0.724857	−0.362429	−	0.932012i	$-0.618052\pi$
$269$	−11.8885	−0.724857	−0.362429	+	0.932012i	$0.618052\pi$
$270$	0	0
$271$	−1.52786	−0.0928111	−0.0464056	−	0.998923i	$-0.514777\pi$
$271$	−1.52786	−0.0928111	−0.0464056	+	0.998923i	$0.514777\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	−7.88854	−0.476564
$275$	−6.47214	−0.390284
$276$	0	0
$277$	18.9443	1.13825	0.569125	−	0.822251i	$-0.307282\pi$
$277$	18.9443	1.13825	0.569125	+	0.822251i	$0.307282\pi$
$278$	16.5836	0.994618
$279$	0	0
$280$	2.23607	0.133631
$281$	10.9443	0.652881	0.326440	−	0.945218i	$-0.394151\pi$
$281$	10.9443	0.652881	0.326440	+	0.945218i	$0.394151\pi$
$282$	0	0
$283$	12.0000	0.713326	0.356663	−	0.934233i	$-0.383914\pi$
$283$	12.0000	0.713326	0.356663	+	0.934233i	$0.383914\pi$
$284$	−16.5836	−0.984055
$285$	0	0
$286$	−64.7214	−3.82705
$287$	2.00000	0.118056
$288$	0	0
$289$	−13.0000	−0.764706
$290$	4.47214	0.262613
$291$	0	0
$292$	−37.4164	−2.18963
$293$	−5.05573	−0.295359	−0.147679	−	0.989035i	$-0.547180\pi$
$293$	−5.05573	−0.295359	−0.147679	+	0.989035i	$0.547180\pi$
$294$	0	0
$295$	8.94427	0.520756
$296$	−15.5279	−0.902539
$297$	0	0
$298$	33.4164	1.93576
$299$	−17.8885	−1.03452
$300$	0	0
$301$	8.94427	0.515539
$302$	−35.7771	−2.05874
$303$	0	0
$304$	2.47214	0.141787
$305$	−2.00000	−0.114520
$306$	0	0
$307$	−15.0557	−0.859276	−0.429638	−	0.903001i	$-0.641359\pi$
$307$	−15.0557	−0.859276	−0.429638	+	0.903001i	$0.641359\pi$
$308$	−19.4164	−1.10635
$309$	0	0
$310$	3.41641	0.194039
$311$	−25.8885	−1.46800	−0.734002	−	0.679147i	$-0.762350\pi$
$311$	−25.8885	−1.46800	−0.734002	+	0.679147i	$0.762350\pi$
$312$	0	0
$313$	−17.4164	−0.984434	−0.492217	−	0.870473i	$-0.663813\pi$
$313$	−17.4164	−0.984434	−0.492217	+	0.870473i	$0.663813\pi$
$314$	−1.05573	−0.0595782
$315$	0	0
$316$	38.8328	2.18452
$317$	−14.3607	−0.806576	−0.403288	−	0.915073i	$-0.632133\pi$
$317$	−14.3607	−0.806576	−0.403288	+	0.915073i	$0.632133\pi$
$318$	0	0
$319$	−12.9443	−0.724740
$320$	−13.0000	−0.726722
$321$	0	0
$322$	−8.94427	−0.498445
$323$	−4.94427	−0.275107
$324$	0	0
$325$	4.47214	0.248069
$326$	−37.8885	−2.09845
$327$	0	0
$328$	4.47214	0.246932
$329$	−12.9443	−0.713641
$330$	0	0
$331$	0.944272	0.0519019	0.0259509	−	0.999663i	$-0.491739\pi$
$331$	0.944272	0.0519019	0.0259509	+	0.999663i	$0.491739\pi$
$332$	50.8328	2.78981
$333$	0	0
$334$	17.8885	0.978818
$335$	−4.00000	−0.218543
$336$	0	0
$337$	−23.8885	−1.30129	−0.650646	−	0.759381i	$-0.725502\pi$
$337$	−23.8885	−1.30129	−0.650646	+	0.759381i	$0.725502\pi$
$338$	15.6525	0.851382
$339$	0	0
$340$	6.00000	0.325396
$341$	−9.88854	−0.535495
$342$	0	0
$343$	1.00000	0.0539949
$344$	20.0000	1.07833
$345$	0	0
$346$	6.58359	0.353936
$347$	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
$349$	−11.8885	−0.636379	−0.318190	−	0.948027i	$-0.603075\pi$
$349$	−11.8885	−0.636379	−0.318190	+	0.948027i	$0.603075\pi$
$350$	2.23607	0.119523
$351$	0	0
$352$	43.4164	2.31410
$353$	−7.88854	−0.419865	−0.209932	−	0.977716i	$-0.567324\pi$
$353$	−7.88854	−0.419865	−0.209932	+	0.977716i	$0.567324\pi$
$354$	0	0
$355$	−5.52786	−0.293389
$356$	6.00000	0.317999
$357$	0	0
$358$	−14.4721	−0.764876
$359$	−18.4721	−0.974922	−0.487461	−	0.873145i	$-0.662077\pi$
$359$	−18.4721	−0.974922	−0.487461	+	0.873145i	$0.662077\pi$
$360$	0	0
$361$	−12.8885	−0.678344
$362$	2.36068	0.124075
$363$	0	0
$364$	13.4164	0.703211
$365$	−12.4721	−0.652821
$366$	0	0
$367$	3.05573	0.159508	0.0797539	−	0.996815i	$-0.474587\pi$
$367$	3.05573	0.159508	0.0797539	+	0.996815i	$0.474587\pi$
$368$	4.00000	0.208514
$369$	0	0
$370$	−15.5279	−0.807255
$371$	3.52786	0.183158
$372$	0	0
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	−28.9443	−1.49667
$375$	0	0
$376$	−28.9443	−1.49269
$377$	8.94427	0.460653
$378$	0	0
$379$	−37.8885	−1.94620	−0.973102	−	0.230375i	$-0.926005\pi$
$379$	−37.8885	−1.94620	−0.973102	+	0.230375i	$0.926005\pi$
$380$	−7.41641	−0.380454
$381$	0	0
$382$	−1.30495	−0.0667671
$383$	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
$385$	−6.47214	−0.329851
$386$	−31.3050	−1.59338
$387$	0	0
$388$	25.4164	1.29032
$389$	6.94427	0.352089	0.176044	−	0.984382i	$-0.443670\pi$
$389$	6.94427	0.352089	0.176044	+	0.984382i	$0.443670\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	2.23607	0.112938
$393$	0	0
$394$	−34.7214	−1.74924
$395$	12.9443	0.651297
$396$	0	0
$397$	−13.4164	−0.673350	−0.336675	−	0.941621i	$-0.609302\pi$
$397$	−13.4164	−0.673350	−0.336675	+	0.941621i	$0.609302\pi$
$398$	61.3050	3.07294
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	−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
$403$	6.83282	0.340367
$404$	42.0000	2.08958
$405$	0	0
$406$	4.47214	0.221948
$407$	44.9443	2.22780
$408$	0	0
$409$	11.8885	0.587851	0.293925	−	0.955828i	$-0.405038\pi$
$409$	11.8885	0.587851	0.293925	+	0.955828i	$0.405038\pi$
$410$	4.47214	0.220863
$411$	0	0
$412$	0	0
$413$	8.94427	0.440119
$414$	0	0
$415$	16.9443	0.831762
$416$	−30.0000	−1.47087
$417$	0	0
$418$	35.7771	1.74991
$419$	29.8885	1.46015	0.730075	−	0.683367i	$-0.239485\pi$
$419$	29.8885	1.46015	0.730075	+	0.683367i	$0.239485\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	−37.8885	−1.84439
$423$	0	0
$424$	7.88854	0.383102
$425$	2.00000	0.0970143
$426$	0	0
$427$	−2.00000	−0.0967868
$428$	38.8328	1.87705
$429$	0	0
$430$	20.0000	0.964486
$431$	18.4721	0.889771	0.444886	−	0.895587i	$-0.353244\pi$
$431$	18.4721	0.889771	0.444886	+	0.895587i	$0.353244\pi$
$432$	0	0
$433$	16.4721	0.791600	0.395800	−	0.918337i	$-0.370467\pi$
$433$	16.4721	0.791600	0.395800	+	0.918337i	$0.370467\pi$
$434$	3.41641	0.163993
$435$	0	0
$436$	−6.00000	−0.287348
$437$	9.88854	0.473033
$438$	0	0
$439$	1.52786	0.0729210	0.0364605	−	0.999335i	$-0.488392\pi$
$439$	1.52786	0.0729210	0.0364605	+	0.999335i	$0.488392\pi$
$440$	−14.4721	−0.689932
$441$	0	0
$442$	20.0000	0.951303
$443$	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
$446$	−28.9443	−1.37055
$447$	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
$451$	−12.9443	−0.609522
$452$	−1.41641	−0.0666222
$453$	0	0
$454$	−2.11146	−0.0990955
$455$	4.47214	0.209657
$456$	0	0
$457$	6.94427	0.324839	0.162420	−	0.986722i	$-0.448070\pi$
$457$	6.94427	0.324839	0.162420	+	0.986722i	$0.448070\pi$
$458$	53.4164	2.49598
$459$	0	0
$460$	−12.0000	−0.559503
$461$	−3.88854	−0.181108	−0.0905538	−	0.995892i	$-0.528864\pi$
$461$	−3.88854	−0.181108	−0.0905538	+	0.995892i	$0.528864\pi$
$462$	0	0
$463$	20.9443	0.973363	0.486681	−	0.873580i	$-0.338207\pi$
$463$	20.9443	0.973363	0.486681	+	0.873580i	$0.338207\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	21.0557	0.975388
$467$	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
$470$	−28.9443	−1.33510
$471$	0	0
$472$	20.0000	0.920575
$473$	−57.8885	−2.66172
$474$	0	0
$475$	−2.47214	−0.113429
$476$	6.00000	0.275010
$477$	0	0
$478$	23.4164	1.07104
$479$	17.8885	0.817348	0.408674	−	0.912680i	$-0.365991\pi$
$479$	17.8885	0.817348	0.408674	+	0.912680i	$0.365991\pi$
$480$	0	0
$481$	−31.0557	−1.41602
$482$	−42.3607	−1.92948
$483$	0	0
$484$	92.6656	4.21207
$485$	8.47214	0.384700
$486$	0	0
$487$	−20.9443	−0.949076	−0.474538	−	0.880235i	$-0.657385\pi$
$487$	−20.9443	−0.949076	−0.474538	+	0.880235i	$0.657385\pi$
$488$	−4.47214	−0.202444
$489$	0	0
$490$	2.23607	0.101015
$491$	21.3050	0.961479	0.480740	−	0.876863i	$-0.340368\pi$
$491$	21.3050	0.961479	0.480740	+	0.876863i	$0.340368\pi$
$492$	0	0
$493$	4.00000	0.180151
$494$	−24.7214	−1.11227
$495$	0	0
$496$	−1.52786	−0.0686031
$497$	−5.52786	−0.247959
$498$	0	0
$499$	−13.8885	−0.621737	−0.310868	−	0.950453i	$-0.600620\pi$
$499$	−13.8885	−0.621737	−0.310868	+	0.950453i	$0.600620\pi$
$500$	3.00000	0.134164
$501$	0	0
$502$	−37.8885	−1.69105
$503$	−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
$506$	57.8885	2.57346
$507$	0	0
$508$	−14.8328	−0.658100
$509$	23.8885	1.05884	0.529421	−	0.848360i	$-0.322409\pi$
$509$	23.8885	1.05884	0.529421	+	0.848360i	$0.322409\pi$
$510$	0	0
$511$	−12.4721	−0.551735
$512$	11.1803	0.494106
$513$	0	0
$514$	−42.3607	−1.86845
$515$	0	0
$516$	0	0
$517$	83.7771	3.68451
$518$	−15.5279	−0.682255
$519$	0	0
$520$	10.0000	0.438529
$521$	19.8885	0.871333	0.435666	−	0.900108i	$-0.356513\pi$
$521$	19.8885	0.871333	0.435666	+	0.900108i	$0.356513\pi$
$522$	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
$526$	−15.7771	−0.687914
$527$	3.05573	0.133110
$528$	0	0
$529$	−7.00000	−0.304348
$530$	7.88854	0.342656
$531$	0	0
$532$	−7.41641	−0.321542
$533$	8.94427	0.387419
$534$	0	0
$535$	12.9443	0.559630
$536$	−8.94427	−0.386334
$537$	0	0
$538$	−26.5836	−1.14610
$539$	−6.47214	−0.278775
$540$	0	0
$541$	−11.8885	−0.511128	−0.255564	−	0.966792i	$-0.582261\pi$
$541$	−11.8885	−0.511128	−0.255564	+	0.966792i	$0.582261\pi$
$542$	−3.41641	−0.146747
$543$	0	0
$544$	−13.4164	−0.575224
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	5.88854	0.251776	0.125888	−	0.992044i	$-0.459822\pi$
$547$	5.88854	0.251776	0.125888	+	0.992044i	$0.459822\pi$
$548$	−10.5836	−0.452109
$549$	0	0
$550$	−14.4721	−0.617094
$551$	−4.94427	−0.210633
$552$	0	0
$553$	12.9443	0.550446
$554$	42.3607	1.79973
$555$	0	0
$556$	22.2492	0.943577
$557$	−20.4721	−0.867432	−0.433716	−	0.901050i	$-0.642798\pi$
$557$	−20.4721	−0.867432	−0.433716	+	0.901050i	$0.642798\pi$
$558$	0	0
$559$	40.0000	1.69182
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	24.4721	1.03229
$563$	−13.8885	−0.585332	−0.292666	−	0.956215i	$-0.594542\pi$
$563$	−13.8885	−0.585332	−0.292666	+	0.956215i	$0.594542\pi$
$564$	0	0
$565$	−0.472136	−0.0198629
$566$	26.8328	1.12787
$567$	0	0
$568$	−12.3607	−0.518643
$569$	39.8885	1.67221	0.836107	−	0.548566i	$-0.184826\pi$
$569$	39.8885	1.67221	0.836107	+	0.548566i	$0.184826\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	−86.8328	−3.63066
$573$	0	0
$574$	4.47214	0.186663
$575$	−4.00000	−0.166812
$576$	0	0
$577$	10.3607	0.431321	0.215660	−	0.976468i	$-0.430810\pi$
$577$	10.3607	0.431321	0.215660	+	0.976468i	$0.430810\pi$
$578$	−29.0689	−1.20911
$579$	0	0
$580$	6.00000	0.249136
$581$	16.9443	0.702967
$582$	0	0
$583$	−22.8328	−0.945639
$584$	−27.8885	−1.15404
$585$	0	0
$586$	−11.3050	−0.467003
$587$	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
$589$	−3.77709	−0.155632
$590$	20.0000	0.823387
$591$	0	0
$592$	6.94427	0.285408
$593$	−23.8885	−0.980985	−0.490492	−	0.871445i	$-0.663183\pi$
$593$	−23.8885	−0.980985	−0.490492	+	0.871445i	$0.663183\pi$
$594$	0	0
$595$	2.00000	0.0819920
$596$	44.8328	1.83642
$597$	0	0
$598$	−40.0000	−1.63572
$599$	−12.3607	−0.505044	−0.252522	−	0.967591i	$-0.581260\pi$
$599$	−12.3607	−0.505044	−0.252522	+	0.967591i	$0.581260\pi$
$600$	0	0
$601$	38.9443	1.58857	0.794285	−	0.607545i	$-0.207846\pi$
$601$	38.9443	1.58857	0.794285	+	0.607545i	$0.207846\pi$
$602$	20.0000	0.815139
$603$	0	0
$604$	−48.0000	−1.95309
$605$	30.8885	1.25580
$606$	0	0
$607$	38.8328	1.57618	0.788088	−	0.615563i	$-0.211071\pi$
$607$	38.8328	1.57618	0.788088	+	0.615563i	$0.211071\pi$
$608$	16.5836	0.672553
$609$	0	0
$610$	−4.47214	−0.181071
$611$	−57.8885	−2.34192
$612$	0	0
$613$	−6.94427	−0.280477	−0.140238	−	0.990118i	$-0.544787\pi$
$613$	−6.94427	−0.280477	−0.140238	+	0.990118i	$0.544787\pi$
$614$	−33.6656	−1.35863
$615$	0	0
$616$	−14.4721	−0.583099
$617$	−16.4721	−0.663143	−0.331572	−	0.943430i	$-0.607579\pi$
$617$	−16.4721	−0.663143	−0.331572	+	0.943430i	$0.607579\pi$
$618$	0	0
$619$	39.4164	1.58428	0.792140	−	0.610340i	$-0.208967\pi$
$619$	39.4164	1.58428	0.792140	+	0.610340i	$0.208967\pi$
$620$	4.58359	0.184081
$621$	0	0
$622$	−57.8885	−2.32112
$623$	2.00000	0.0801283
$624$	0	0
$625$	1.00000	0.0400000
$626$	−38.9443	−1.55653
$627$	0	0
$628$	−1.41641	−0.0565208
$629$	−13.8885	−0.553773
$630$	0	0
$631$	30.8328	1.22744	0.613718	−	0.789526i	$-0.289673\pi$
$631$	30.8328	1.22744	0.613718	+	0.789526i	$0.289673\pi$
$632$	28.9443	1.15134
$633$	0	0
$634$	−32.1115	−1.27531
$635$	−4.94427	−0.196207
$636$	0	0
$637$	4.47214	0.177192
$638$	−28.9443	−1.14591
$639$	0	0
$640$	−15.6525	−0.618718
$641$	−16.8328	−0.664856	−0.332428	−	0.943129i	$-0.607868\pi$
$641$	−16.8328	−0.664856	−0.332428	+	0.943129i	$0.607868\pi$
$642$	0	0
$643$	−15.0557	−0.593740	−0.296870	−	0.954918i	$-0.595943\pi$
$643$	−15.0557	−0.593740	−0.296870	+	0.954918i	$0.595943\pi$
$644$	−12.0000	−0.472866
$645$	0	0
$646$	−11.0557	−0.434982
$647$	1.88854	0.0742463	0.0371232	−	0.999311i	$-0.488181\pi$
$647$	1.88854	0.0742463	0.0371232	+	0.999311i	$0.488181\pi$
$648$	0	0
$649$	−57.8885	−2.27232
$650$	10.0000	0.392232
$651$	0	0
$652$	−50.8328	−1.99077
$653$	22.5836	0.883764	0.441882	−	0.897073i	$-0.354311\pi$
$653$	22.5836	0.883764	0.441882	+	0.897073i	$0.354311\pi$
$654$	0	0
$655$	−4.00000	−0.156293
$656$	−2.00000	−0.0780869
$657$	0	0
$658$	−28.9443	−1.12837
$659$	−21.3050	−0.829923	−0.414962	−	0.909839i	$-0.636205\pi$
$659$	−21.3050	−0.829923	−0.414962	+	0.909839i	$0.636205\pi$
$660$	0	0
$661$	−35.8885	−1.39590	−0.697951	−	0.716145i	$-0.745905\pi$
$661$	−35.8885	−1.39590	−0.697951	+	0.716145i	$0.745905\pi$
$662$	2.11146	0.0820641
$663$	0	0
$664$	37.8885	1.47036
$665$	−2.47214	−0.0958653
$666$	0	0
$667$	−8.00000	−0.309761
$668$	24.0000	0.928588
$669$	0	0
$670$	−8.94427	−0.345547
$671$	12.9443	0.499708
$672$	0	0
$673$	8.83282	0.340480	0.170240	−	0.985403i	$-0.445546\pi$
$673$	8.83282	0.340480	0.170240	+	0.985403i	$0.445546\pi$
$674$	−53.4164	−2.05752
$675$	0	0
$676$	21.0000	0.807692
$677$	−21.0557	−0.809237	−0.404619	−	0.914485i	$-0.632596\pi$
$677$	−21.0557	−0.809237	−0.404619	+	0.914485i	$0.632596\pi$
$678$	0	0
$679$	8.47214	0.325131
$680$	4.47214	0.171499
$681$	0	0
$682$	−22.1115	−0.846691
$683$	1.88854	0.0722631	0.0361316	−	0.999347i	$-0.488496\pi$
$683$	1.88854	0.0722631	0.0361316	+	0.999347i	$0.488496\pi$
$684$	0	0
$685$	−3.52786	−0.134793
$686$	2.23607	0.0853735
$687$	0	0
$688$	−8.94427	−0.340997
$689$	15.7771	0.601059
$690$	0	0
$691$	44.3607	1.68756	0.843780	−	0.536689i	$-0.180325\pi$
$691$	44.3607	1.68756	0.843780	+	0.536689i	$0.180325\pi$
$692$	8.83282	0.335773
$693$	0	0
$694$	17.8885	0.679040
$695$	7.41641	0.281320
$696$	0	0
$697$	4.00000	0.151511
$698$	−26.5836	−1.00620
$699$	0	0
$700$	3.00000	0.113389
$701$	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
$703$	17.1672	0.647473
$704$	84.1378	3.17106
$705$	0	0
$706$	−17.6393	−0.663865
$707$	14.0000	0.526524
$708$	0	0
$709$	25.7771	0.968079	0.484039	−	0.875046i	$-0.339169\pi$
$709$	25.7771	0.968079	0.484039	+	0.875046i	$0.339169\pi$
$710$	−12.3607	−0.463888
$711$	0	0
$712$	4.47214	0.167600
$713$	−6.11146	−0.228876
$714$	0	0
$715$	−28.9443	−1.08245
$716$	−19.4164	−0.725625
$717$	0	0
$718$	−41.3050	−1.54149
$719$	6.83282	0.254821	0.127411	−	0.991850i	$-0.459333\pi$
$719$	6.83282	0.254821	0.127411	+	0.991850i	$0.459333\pi$
$720$	0	0
$721$	0	0
$722$	−28.8197	−1.07256
$723$	0	0
$724$	3.16718	0.117707
$725$	2.00000	0.0742781
$726$	0	0
$727$	−38.8328	−1.44023	−0.720115	−	0.693855i	$-0.755911\pi$
$727$	−38.8328	−1.44023	−0.720115	+	0.693855i	$0.755911\pi$
$728$	10.0000	0.370625
$729$	0	0
$730$	−27.8885	−1.03220
$731$	17.8885	0.661632
$732$	0	0
$733$	10.5836	0.390914	0.195457	−	0.980712i	$-0.437381\pi$
$733$	10.5836	0.390914	0.195457	+	0.980712i	$0.437381\pi$
$734$	6.83282	0.252204
$735$	0	0
$736$	26.8328	0.989071
$737$	25.8885	0.953617
$738$	0	0
$739$	−5.88854	−0.216614	−0.108307	−	0.994118i	$-0.534543\pi$
$739$	−5.88854	−0.216614	−0.108307	+	0.994118i	$0.534543\pi$
$740$	−20.8328	−0.765830
$741$	0	0
$742$	7.88854	0.289598
$743$	−34.8328	−1.27789	−0.638946	−	0.769252i	$-0.720629\pi$
$743$	−34.8328	−1.27789	−0.638946	+	0.769252i	$0.720629\pi$
$744$	0	0
$745$	14.9443	0.547516
$746$	13.4164	0.491210
$747$	0	0
$748$	−38.8328	−1.41987
$749$	12.9443	0.472973
$750$	0	0
$751$	−20.9443	−0.764267	−0.382134	−	0.924107i	$-0.624811\pi$
$751$	−20.9443	−0.764267	−0.382134	+	0.924107i	$0.624811\pi$
$752$	12.9443	0.472029
$753$	0	0
$754$	20.0000	0.728357
$755$	−16.0000	−0.582300
$756$	0	0
$757$	31.8885	1.15901	0.579504	−	0.814969i	$-0.303246\pi$
$757$	31.8885	1.15901	0.579504	+	0.814969i	$0.303246\pi$
$758$	−84.7214	−3.07722
$759$	0	0
$760$	−5.52786	−0.200517
$761$	27.8885	1.01096	0.505479	−	0.862839i	$-0.331316\pi$
$761$	27.8885	1.01096	0.505479	+	0.862839i	$0.331316\pi$
$762$	0	0
$763$	−2.00000	−0.0724049
$764$	−1.75078	−0.0633409
$765$	0	0
$766$	17.8885	0.646339
$767$	40.0000	1.44432
$768$	0	0
$769$	−52.8328	−1.90520	−0.952600	−	0.304226i	$-0.901602\pi$
$769$	−52.8328	−1.90520	−0.952600	+	0.304226i	$0.901602\pi$
$770$	−14.4721	−0.521540
$771$	0	0
$772$	−42.0000	−1.51161
$773$	42.9443	1.54460	0.772299	−	0.635259i	$-0.219107\pi$
$773$	42.9443	1.54460	0.772299	+	0.635259i	$0.219107\pi$
$774$	0	0
$775$	1.52786	0.0548825
$776$	18.9443	0.680060
$777$	0	0
$778$	15.5279	0.556701
$779$	−4.94427	−0.177147
$780$	0	0
$781$	35.7771	1.28020
$782$	−17.8885	−0.639693
$783$	0	0
$784$	−1.00000	−0.0357143
$785$	−0.472136	−0.0168513
$786$	0	0
$787$	31.0557	1.10702	0.553509	−	0.832843i	$-0.313289\pi$
$787$	31.0557	1.10702	0.553509	+	0.832843i	$0.313289\pi$
$788$	−46.5836	−1.65947
$789$	0	0
$790$	28.9443	1.02979
$791$	−0.472136	−0.0167872
$792$	0	0
$793$	−8.94427	−0.317620
$794$	−30.0000	−1.06466
$795$	0	0
$796$	82.2492	2.91525
$797$	18.9443	0.671041	0.335520	−	0.942033i	$-0.391088\pi$
$797$	18.9443	0.671041	0.335520	+	0.942033i	$0.391088\pi$
$798$	0	0
$799$	−25.8885	−0.915871
$800$	−6.70820	−0.237171
$801$	0	0
$802$	−22.3607	−0.789583
$803$	80.7214	2.84859
$804$	0	0
$805$	−4.00000	−0.140981
$806$	15.2786	0.538167
$807$	0	0
$808$	31.3050	1.10130
$809$	−38.9443	−1.36921	−0.684604	−	0.728915i	$-0.740025\pi$
$809$	−38.9443	−1.36921	−0.684604	+	0.728915i	$0.740025\pi$
$810$	0	0
$811$	55.4164	1.94593	0.972967	−	0.230946i	$-0.0741820\pi$
$811$	55.4164	1.94593	0.972967	+	0.230946i	$0.0741820\pi$
$812$	6.00000	0.210559
$813$	0	0
$814$	100.498	3.52247
$815$	−16.9443	−0.593532
$816$	0	0
$817$	−22.1115	−0.773582
$818$	26.5836	0.929474
$819$	0	0
$820$	6.00000	0.209529
$821$	−33.7771	−1.17883	−0.589414	−	0.807831i	$-0.700641\pi$
$821$	−33.7771	−1.17883	−0.589414	+	0.807831i	$0.700641\pi$
$822$	0	0
$823$	44.9443	1.56666	0.783329	−	0.621607i	$-0.213520\pi$
$823$	44.9443	1.56666	0.783329	+	0.621607i	$0.213520\pi$
$824$	0	0
$825$	0	0
$826$	20.0000	0.695889
$827$	−12.9443	−0.450116	−0.225058	−	0.974345i	$-0.572257\pi$
$827$	−12.9443	−0.450116	−0.225058	+	0.974345i	$0.572257\pi$
$828$	0	0
$829$	−13.0557	−0.453444	−0.226722	−	0.973959i	$-0.572801\pi$
$829$	−13.0557	−0.453444	−0.226722	+	0.973959i	$0.572801\pi$
$830$	37.8885	1.31513
$831$	0	0
$832$	−58.1378	−2.01556
$833$	2.00000	0.0692959
$834$	0	0
$835$	8.00000	0.276851
$836$	48.0000	1.66011
$837$	0	0
$838$	66.8328	2.30870
$839$	−54.8328	−1.89304	−0.946520	−	0.322647i	$-0.895427\pi$
$839$	−54.8328	−1.89304	−0.946520	+	0.322647i	$0.895427\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	49.1935	1.69532
$843$	0	0
$844$	−50.8328	−1.74974
$845$	7.00000	0.240807
$846$	0	0
$847$	30.8885	1.06134
$848$	−3.52786	−0.121147
$849$	0	0
$850$	4.47214	0.153393
$851$	27.7771	0.952186
$852$	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
$856$	28.9443	0.989295
$857$	−36.8328	−1.25819	−0.629093	−	0.777330i	$-0.716573\pi$
$857$	−36.8328	−1.25819	−0.629093	+	0.777330i	$0.716573\pi$
$858$	0	0
$859$	50.4721	1.72209	0.861044	−	0.508531i	$-0.169811\pi$
$859$	50.4721	1.72209	0.861044	+	0.508531i	$0.169811\pi$
$860$	26.8328	0.914991
$861$	0	0
$862$	41.3050	1.40685
$863$	−21.8885	−0.745095	−0.372547	−	0.928013i	$-0.621516\pi$
$863$	−21.8885	−0.745095	−0.372547	+	0.928013i	$0.621516\pi$
$864$	0	0
$865$	2.94427	0.100108
$866$	36.8328	1.25163
$867$	0	0
$868$	4.58359	0.155577
$869$	−83.7771	−2.84194
$870$	0	0
$871$	−17.8885	−0.606130
$872$	−4.47214	−0.151446
$873$	0	0
$874$	22.1115	0.747931
$875$	1.00000	0.0338062
$876$	0	0
$877$	−56.8328	−1.91911	−0.959554	−	0.281525i	$-0.909160\pi$
$877$	−56.8328	−1.91911	−0.959554	+	0.281525i	$0.909160\pi$
$878$	3.41641	0.115298
$879$	0	0
$880$	6.47214	0.218176
$881$	27.8885	0.939589	0.469794	−	0.882776i	$-0.344328\pi$
$881$	27.8885	0.939589	0.469794	+	0.882776i	$0.344328\pi$
$882$	0	0
$883$	−37.8885	−1.27505	−0.637526	−	0.770429i	$-0.720042\pi$
$883$	−37.8885	−1.27505	−0.637526	+	0.770429i	$0.720042\pi$
$884$	26.8328	0.902485
$885$	0	0
$886$	17.8885	0.600977
$887$	30.8328	1.03526	0.517632	−	0.855603i	$-0.326814\pi$
$887$	30.8328	1.03526	0.517632	+	0.855603i	$0.326814\pi$
$888$	0	0
$889$	−4.94427	−0.165826
$890$	4.47214	0.149906
$891$	0	0
$892$	−38.8328	−1.30022
$893$	32.0000	1.07084
$894$	0	0
$895$	−6.47214	−0.216340
$896$	−15.6525	−0.522913
$897$	0	0
$898$	31.3050	1.04466
$899$	3.05573	0.101914
$900$	0	0
$901$	7.05573	0.235060
$902$	−28.9443	−0.963739
$903$	0	0
$904$	−1.05573	−0.0351130
$905$	1.05573	0.0350936
$906$	0	0
$907$	53.8885	1.78934	0.894670	−	0.446728i	$-0.147411\pi$
$907$	53.8885	1.78934	0.894670	+	0.446728i	$0.147411\pi$
$908$	−2.83282	−0.0940103
$909$	0	0
$910$	10.0000	0.331497
$911$	−46.2492	−1.53231	−0.766153	−	0.642659i	$-0.777831\pi$
$911$	−46.2492	−1.53231	−0.766153	+	0.642659i	$0.777831\pi$
$912$	0	0
$913$	−109.666	−3.62940
$914$	15.5279	0.513616
$915$	0	0
$916$	71.6656	2.36790
$917$	−4.00000	−0.132092
$918$	0	0
$919$	−35.0557	−1.15638	−0.578191	−	0.815902i	$-0.696241\pi$
$919$	−35.0557	−1.15638	−0.578191	+	0.815902i	$0.696241\pi$
$920$	−8.94427	−0.294884
$921$	0	0
$922$	−8.69505	−0.286356
$923$	−24.7214	−0.813713
$924$	0	0
$925$	−6.94427	−0.228326
$926$	46.8328	1.53902
$927$	0	0
$928$	−13.4164	−0.440415
$929$	16.1115	0.528600	0.264300	−	0.964441i	$-0.414859\pi$
$929$	16.1115	0.528600	0.264300	+	0.964441i	$0.414859\pi$
$930$	0	0
$931$	−2.47214	−0.0810210
$932$	28.2492	0.925334
$933$	0	0
$934$	20.0000	0.654420
$935$	−12.9443	−0.423323
$936$	0	0
$937$	−52.4721	−1.71419	−0.857095	−	0.515158i	$-0.827733\pi$
$937$	−52.4721	−1.71419	−0.857095	+	0.515158i	$0.827733\pi$
$938$	−8.94427	−0.292041
$939$	0	0
$940$	−38.8328	−1.26659
$941$	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
$946$	−129.443	−4.20855
$947$	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
$949$	−55.7771	−1.81060
$950$	−5.52786	−0.179348
$951$	0	0
$952$	4.47214	0.144943
$953$	33.4164	1.08246	0.541232	−	0.840873i	$-0.317958\pi$
$953$	33.4164	1.08246	0.541232	+	0.840873i	$0.317958\pi$
$954$	0	0
$955$	−0.583592	−0.0188846
$956$	31.4164	1.01608
$957$	0	0
$958$	40.0000	1.29234
$959$	−3.52786	−0.113921
$960$	0	0
$961$	−28.6656	−0.924698
$962$	−69.4427	−2.23892
$963$	0	0
$964$	−56.8328	−1.83046
$965$	−14.0000	−0.450676
$966$	0	0
$967$	25.8885	0.832519	0.416260	−	0.909246i	$-0.363341\pi$
$967$	25.8885	0.832519	0.416260	+	0.909246i	$0.363341\pi$
$968$	69.0689	2.21996
$969$	0	0
$970$	18.9443	0.608264
$971$	−40.9443	−1.31396	−0.656982	−	0.753906i	$-0.728167\pi$
$971$	−40.9443	−1.31396	−0.656982	+	0.753906i	$0.728167\pi$
$972$	0	0
$973$	7.41641	0.237759
$974$	−46.8328	−1.50062
$975$	0	0
$976$	2.00000	0.0640184
$977$	−30.5836	−0.978456	−0.489228	−	0.872156i	$-0.662721\pi$
$977$	−30.5836	−0.978456	−0.489228	+	0.872156i	$0.662721\pi$
$978$	0	0
$979$	−12.9443	−0.413701
$980$	3.00000	0.0958315
$981$	0	0
$982$	47.6393	1.52023
$983$	22.8328	0.728254	0.364127	−	0.931349i	$-0.381367\pi$
$983$	22.8328	0.728254	0.364127	+	0.931349i	$0.381367\pi$
$984$	0	0
$985$	−15.5279	−0.494759
$986$	8.94427	0.284844
$987$	0	0
$988$	−33.1672	−1.05519
$989$	−35.7771	−1.13765
$990$	0	0
$991$	4.94427	0.157060	0.0785300	−	0.996912i	$-0.474977\pi$
$991$	4.94427	0.157060	0.0785300	+	0.996912i	$0.474977\pi$
$992$	−10.2492	−0.325413
$993$	0	0
$994$	−12.3607	−0.392057
$995$	27.4164	0.869159
$996$	0	0
$997$	−5.41641	−0.171539	−0.0857697	−	0.996315i	$-0.527335\pi$
$997$	−5.41641	−0.171539	−0.0857697	+	0.996315i	$0.527335\pi$
$998$	−31.0557	−0.983052
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.a.d.1.2		2
3.2	odd	2		105.2.a.b.1.1	✓	2
4.3	odd	2		5040.2.a.bw.1.2		2
5.2	odd	4		1575.2.d.d.1324.4		4
5.3	odd	4		1575.2.d.d.1324.1		4
5.4	even	2		1575.2.a.r.1.1		2
7.6	odd	2		2205.2.a.w.1.2		2
12.11	even	2		1680.2.a.v.1.1		2
15.2	even	4		525.2.d.c.274.2		4
15.8	even	4		525.2.d.c.274.3		4
15.14	odd	2		525.2.a.g.1.2		2
21.2	odd	6		735.2.i.k.361.2		4
21.5	even	6		735.2.i.i.361.2		4
21.11	odd	6		735.2.i.k.226.2		4
21.17	even	6		735.2.i.i.226.2		4
21.20	even	2		735.2.a.k.1.1		2
24.5	odd	2		6720.2.a.cx.1.1		2
24.11	even	2		6720.2.a.cs.1.2		2
60.59	even	2		8400.2.a.cx.1.1		2
105.104	even	2		3675.2.a.y.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.a.b.1.1	✓	2	3.2	odd	2
315.2.a.d.1.2		2	1.1	even	1	trivial
525.2.a.g.1.2		2	15.14	odd	2
525.2.d.c.274.2		4	15.2	even	4
525.2.d.c.274.3		4	15.8	even	4
735.2.a.k.1.1		2	21.20	even	2
735.2.i.i.226.2		4	21.17	even	6
735.2.i.i.361.2		4	21.5	even	6
735.2.i.k.226.2		4	21.11	odd	6
735.2.i.k.361.2		4	21.2	odd	6
1575.2.a.r.1.1		2	5.4	even	2
1575.2.d.d.1324.1		4	5.3	odd	4
1575.2.d.d.1324.4		4	5.2	odd	4
1680.2.a.v.1.1		2	12.11	even	2
2205.2.a.w.1.2		2	7.6	odd	2
3675.2.a.y.1.2		2	105.104	even	2
5040.2.a.bw.1.2		2	4.3	odd	2
6720.2.a.cs.1.2		2	24.11	even	2
6720.2.a.cx.1.1		2	24.5	odd	2
8400.2.a.cx.1.1		2	60.59	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:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	315.a (trivial)

\(f(q)\)	\(=\)	\(q+2.23607 q^{2} +3.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} +2.23607 q^{8} +2.23607 q^{10} -6.47214 q^{11} +4.47214 q^{13} +2.23607 q^{14} -1.00000 q^{16} +2.00000 q^{17} -2.47214 q^{19} +3.00000 q^{20} -14.4721 q^{22} -4.00000 q^{23} +1.00000 q^{25} +10.0000 q^{26} +3.00000 q^{28} +2.00000 q^{29} +1.52786 q^{31} -6.70820 q^{32} +4.47214 q^{34} +1.00000 q^{35} -6.94427 q^{37} -5.52786 q^{38} +2.23607 q^{40} +2.00000 q^{41} +8.94427 q^{43} -19.4164 q^{44} -8.94427 q^{46} -12.9443 q^{47} +1.00000 q^{49} +2.23607 q^{50} +13.4164 q^{52} +3.52786 q^{53} -6.47214 q^{55} +2.23607 q^{56} +4.47214 q^{58} +8.94427 q^{59} -2.00000 q^{61} +3.41641 q^{62} -13.0000 q^{64} +4.47214 q^{65} -4.00000 q^{67} +6.00000 q^{68} +2.23607 q^{70} -5.52786 q^{71} -12.4721 q^{73} -15.5279 q^{74} -7.41641 q^{76} -6.47214 q^{77} +12.9443 q^{79} -1.00000 q^{80} +4.47214 q^{82} +16.9443 q^{83} +2.00000 q^{85} +20.0000 q^{86} -14.4721 q^{88} +2.00000 q^{89} +4.47214 q^{91} -12.0000 q^{92} -28.9443 q^{94} -2.47214 q^{95} +8.47214 q^{97} +2.23607 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{4} + 2 q^{5} + 2 q^{7} - 4 q^{11} - 2 q^{16} + 4 q^{17} + 4 q^{19} + 6 q^{20} - 20 q^{22} - 8 q^{23} + 2 q^{25} + 20 q^{26} + 6 q^{28} + 4 q^{29} + 12 q^{31} + 2 q^{35} + 4 q^{37} - 20 q^{38}+ \cdots + 8 q^{97}+O(q^{100}) \) 2 * q + 6 * q^4 + 2 * q^5 + 2 * q^7 - 4 * q^11 - 2 * q^16 + 4 * q^17 + 4 * q^19 + 6 * q^20 - 20 * q^22 - 8 * q^23 + 2 * q^25 + 20 * q^26 + 6 * q^28 + 4 * q^29 + 12 * q^31 + 2 * q^35 + 4 * q^37 - 20 * q^38 + 4 * q^41 - 12 * q^44 - 8 * q^47 + 2 * q^49 + 16 * q^53 - 4 * q^55 - 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 - 2 * q^80 + 16 * q^83 + 4 * q^85 + 40 * q^86 - 20 * q^88 + 4 * q^89 - 24 * q^92 - 40 * q^94 + 4 * q^95 + 8 * q^97

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