LMFDB - Embedded newform 2394.2.a.u.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2394,2,Mod(1,2394)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2394, 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("2394.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2394.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:	$19.1161862439$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	2394.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{10} -4.82843 q^{11} +5.65685 q^{13} +1.00000 q^{14} +1.00000 q^{16} -6.82843 q^{17} -1.00000 q^{19} -2.00000 q^{20} -4.82843 q^{22} -4.00000 q^{23} -1.00000 q^{25} +5.65685 q^{26} +1.00000 q^{28} -2.00000 q^{29} -4.82843 q^{31} +1.00000 q^{32} -6.82843 q^{34} -2.00000 q^{35} -8.48528 q^{37} -1.00000 q^{38} -2.00000 q^{40} +3.65685 q^{41} +9.65685 q^{43} -4.82843 q^{44} -4.00000 q^{46} +4.48528 q^{47} +1.00000 q^{49} -1.00000 q^{50} +5.65685 q^{52} +7.65685 q^{53} +9.65685 q^{55} +1.00000 q^{56} -2.00000 q^{58} -8.00000 q^{59} -13.3137 q^{61} -4.82843 q^{62} +1.00000 q^{64} -11.3137 q^{65} -6.82843 q^{67} -6.82843 q^{68} -2.00000 q^{70} +5.65685 q^{71} -7.65685 q^{73} -8.48528 q^{74} -1.00000 q^{76} -4.82843 q^{77} +7.65685 q^{79} -2.00000 q^{80} +3.65685 q^{82} -11.6569 q^{83} +13.6569 q^{85} +9.65685 q^{86} -4.82843 q^{88} -13.3137 q^{89} +5.65685 q^{91} -4.00000 q^{92} +4.48528 q^{94} +2.00000 q^{95} -18.4853 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 4 q^{5} + 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^5 + 2 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} - 4 q^{5} + 2 q^{7} + 2 q^{8} - 4 q^{10} - 4 q^{11} + 2 q^{14} + 2 q^{16} - 8 q^{17} - 2 q^{19} - 4 q^{20} - 4 q^{22} - 8 q^{23} - 2 q^{25} + 2 q^{28} - 4 q^{29} - 4 q^{31} + 2 q^{32} - 8 q^{34} - 4 q^{35} - 2 q^{38} - 4 q^{40} - 4 q^{41} + 8 q^{43} - 4 q^{44} - 8 q^{46} - 8 q^{47} + 2 q^{49} - 2 q^{50} + 4 q^{53} + 8 q^{55} + 2 q^{56} - 4 q^{58} - 16 q^{59} - 4 q^{61} - 4 q^{62} + 2 q^{64} - 8 q^{67} - 8 q^{68} - 4 q^{70} - 4 q^{73} - 2 q^{76} - 4 q^{77} + 4 q^{79} - 4 q^{80} - 4 q^{82} - 12 q^{83} + 16 q^{85} + 8 q^{86} - 4 q^{88} - 4 q^{89} - 8 q^{92} - 8 q^{94} + 4 q^{95} - 20 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^5 + 2 * q^7 + 2 * q^8 - 4 * q^10 - 4 * q^11 + 2 * q^14 + 2 * q^16 - 8 * q^17 - 2 * q^19 - 4 * q^20 - 4 * q^22 - 8 * q^23 - 2 * q^25 + 2 * q^28 - 4 * q^29 - 4 * q^31 + 2 * q^32 - 8 * q^34 - 4 * q^35 - 2 * q^38 - 4 * q^40 - 4 * q^41 + 8 * q^43 - 4 * q^44 - 8 * q^46 - 8 * q^47 + 2 * q^49 - 2 * q^50 + 4 * q^53 + 8 * q^55 + 2 * q^56 - 4 * q^58 - 16 * q^59 - 4 * q^61 - 4 * q^62 + 2 * q^64 - 8 * q^67 - 8 * q^68 - 4 * q^70 - 4 * q^73 - 2 * q^76 - 4 * q^77 + 4 * q^79 - 4 * q^80 - 4 * q^82 - 12 * q^83 + 16 * q^85 + 8 * q^86 - 4 * q^88 - 4 * q^89 - 8 * q^92 - 8 * q^94 + 4 * q^95 - 20 * q^97 + 2 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	−2.00000	−0.632456
$11$	−4.82843	−1.45583	−0.727913	−	0.685670i	$-0.759509\pi$
$11$	−4.82843	−1.45583	−0.727913	+	0.685670i	$0.759509\pi$
$12$	0	0
$13$	5.65685	1.56893	0.784465	−	0.620174i	$-0.212938\pi$
$13$	5.65685	1.56893	0.784465	+	0.620174i	$0.212938\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.82843	−1.65614	−0.828068	−	0.560627i	$-0.810560\pi$
$17$	−6.82843	−1.65614	−0.828068	+	0.560627i	$0.810560\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−2.00000	−0.447214
$21$	0	0
$22$	−4.82843	−1.02942
$23$	−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
$26$	5.65685	1.10940
$27$	0	0
$28$	1.00000	0.188982
$29$	−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
$31$	−4.82843	−0.867211	−0.433606	−	0.901103i	$-0.642759\pi$
$31$	−4.82843	−0.867211	−0.433606	+	0.901103i	$0.642759\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−6.82843	−1.17107
$35$	−2.00000	−0.338062
$36$	0	0
$37$	−8.48528	−1.39497	−0.697486	−	0.716599i	$-0.745698\pi$
$37$	−8.48528	−1.39497	−0.697486	+	0.716599i	$0.745698\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	−2.00000	−0.316228
$41$	3.65685	0.571105	0.285552	−	0.958363i	$-0.407823\pi$
$41$	3.65685	0.571105	0.285552	+	0.958363i	$0.407823\pi$
$42$	0	0
$43$	9.65685	1.47266	0.736328	−	0.676625i	$-0.236558\pi$
$43$	9.65685	1.47266	0.736328	+	0.676625i	$0.236558\pi$
$44$	−4.82843	−0.727913
$45$	0	0
$46$	−4.00000	−0.589768
$47$	4.48528	0.654246	0.327123	−	0.944982i	$-0.393921\pi$
$47$	4.48528	0.654246	0.327123	+	0.944982i	$0.393921\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	0	0
$52$	5.65685	0.784465
$53$	7.65685	1.05175	0.525875	−	0.850562i	$-0.323738\pi$
$53$	7.65685	1.05175	0.525875	+	0.850562i	$0.323738\pi$
$54$	0	0
$55$	9.65685	1.30213
$56$	1.00000	0.133631
$57$	0	0
$58$	−2.00000	−0.262613
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	−13.3137	−1.70465	−0.852323	−	0.523016i	$-0.824807\pi$
$61$	−13.3137	−1.70465	−0.852323	+	0.523016i	$0.824807\pi$
$62$	−4.82843	−0.613211
$63$	0	0
$64$	1.00000	0.125000
$65$	−11.3137	−1.40329
$66$	0	0
$67$	−6.82843	−0.834225	−0.417113	−	0.908855i	$-0.636958\pi$
$67$	−6.82843	−0.834225	−0.417113	+	0.908855i	$0.636958\pi$
$68$	−6.82843	−0.828068
$69$	0	0
$70$	−2.00000	−0.239046
$71$	5.65685	0.671345	0.335673	−	0.941979i	$-0.391036\pi$
$71$	5.65685	0.671345	0.335673	+	0.941979i	$0.391036\pi$
$72$	0	0
$73$	−7.65685	−0.896167	−0.448084	−	0.893992i	$-0.647893\pi$
$73$	−7.65685	−0.896167	−0.448084	+	0.893992i	$0.647893\pi$
$74$	−8.48528	−0.986394
$75$	0	0
$76$	−1.00000	−0.114708
$77$	−4.82843	−0.550250
$78$	0	0
$79$	7.65685	0.861463	0.430732	−	0.902480i	$-0.358256\pi$
$79$	7.65685	0.861463	0.430732	+	0.902480i	$0.358256\pi$
$80$	−2.00000	−0.223607
$81$	0	0
$82$	3.65685	0.403832
$83$	−11.6569	−1.27951	−0.639753	−	0.768581i	$-0.720963\pi$
$83$	−11.6569	−1.27951	−0.639753	+	0.768581i	$0.720963\pi$
$84$	0	0
$85$	13.6569	1.48129
$86$	9.65685	1.04133
$87$	0	0
$88$	−4.82843	−0.514712
$89$	−13.3137	−1.41125	−0.705625	−	0.708585i	$-0.749334\pi$
$89$	−13.3137	−1.41125	−0.705625	+	0.708585i	$0.749334\pi$
$90$	0	0
$91$	5.65685	0.592999
$92$	−4.00000	−0.417029
$93$	0	0
$94$	4.48528	0.462621
$95$	2.00000	0.205196
$96$	0	0
$97$	−18.4853	−1.87690	−0.938448	−	0.345421i	$-0.887736\pi$
$97$	−18.4853	−1.87690	−0.938448	+	0.345421i	$0.887736\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	−1.00000	−0.100000
$101$	5.31371	0.528734	0.264367	−	0.964422i	$-0.414837\pi$
$101$	5.31371	0.528734	0.264367	+	0.964422i	$0.414837\pi$
$102$	0	0
$103$	−7.17157	−0.706636	−0.353318	−	0.935503i	$-0.614947\pi$
$103$	−7.17157	−0.706636	−0.353318	+	0.935503i	$0.614947\pi$
$104$	5.65685	0.554700
$105$	0	0
$106$	7.65685	0.743699
$107$	−5.65685	−0.546869	−0.273434	−	0.961891i	$-0.588160\pi$
$107$	−5.65685	−0.546869	−0.273434	+	0.961891i	$0.588160\pi$
$108$	0	0
$109$	−10.8284	−1.03718	−0.518588	−	0.855024i	$-0.673542\pi$
$109$	−10.8284	−1.03718	−0.518588	+	0.855024i	$0.673542\pi$
$110$	9.65685	0.920745
$111$	0	0
$112$	1.00000	0.0944911
$113$	11.6569	1.09658	0.548292	−	0.836287i	$-0.315278\pi$
$113$	11.6569	1.09658	0.548292	+	0.836287i	$0.315278\pi$
$114$	0	0
$115$	8.00000	0.746004
$116$	−2.00000	−0.185695
$117$	0	0
$118$	−8.00000	−0.736460
$119$	−6.82843	−0.625961
$120$	0	0
$121$	12.3137	1.11943
$122$	−13.3137	−1.20537
$123$	0	0
$124$	−4.82843	−0.433606
$125$	12.0000	1.07331
$126$	0	0
$127$	4.34315	0.385392	0.192696	−	0.981259i	$-0.438277\pi$
$127$	4.34315	0.385392	0.192696	+	0.981259i	$0.438277\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−11.3137	−0.992278
$131$	9.31371	0.813742	0.406871	−	0.913486i	$-0.366620\pi$
$131$	9.31371	0.813742	0.406871	+	0.913486i	$0.366620\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	−6.82843	−0.589886
$135$	0	0
$136$	−6.82843	−0.585533
$137$	−19.3137	−1.65008	−0.825041	−	0.565073i	$-0.808848\pi$
$137$	−19.3137	−1.65008	−0.825041	+	0.565073i	$0.808848\pi$
$138$	0	0
$139$	−17.6569	−1.49763	−0.748817	−	0.662776i	$-0.769378\pi$
$139$	−17.6569	−1.49763	−0.748817	+	0.662776i	$0.769378\pi$
$140$	−2.00000	−0.169031
$141$	0	0
$142$	5.65685	0.474713
$143$	−27.3137	−2.28409
$144$	0	0
$145$	4.00000	0.332182
$146$	−7.65685	−0.633686
$147$	0	0
$148$	−8.48528	−0.697486
$149$	10.4853	0.858988	0.429494	−	0.903070i	$-0.358692\pi$
$149$	10.4853	0.858988	0.429494	+	0.903070i	$0.358692\pi$
$150$	0	0
$151$	7.65685	0.623106	0.311553	−	0.950229i	$-0.399151\pi$
$151$	7.65685	0.623106	0.311553	+	0.950229i	$0.399151\pi$
$152$	−1.00000	−0.0811107
$153$	0	0
$154$	−4.82843	−0.389086
$155$	9.65685	0.775657
$156$	0	0
$157$	7.65685	0.611083	0.305542	−	0.952179i	$-0.401162\pi$
$157$	7.65685	0.611083	0.305542	+	0.952179i	$0.401162\pi$
$158$	7.65685	0.609147
$159$	0	0
$160$	−2.00000	−0.158114
$161$	−4.00000	−0.315244
$162$	0	0
$163$	0.686292	0.0537545	0.0268772	−	0.999639i	$-0.491444\pi$
$163$	0.686292	0.0537545	0.0268772	+	0.999639i	$0.491444\pi$
$164$	3.65685	0.285552
$165$	0	0
$166$	−11.6569	−0.904747
$167$	−5.65685	−0.437741	−0.218870	−	0.975754i	$-0.570237\pi$
$167$	−5.65685	−0.437741	−0.218870	+	0.975754i	$0.570237\pi$
$168$	0	0
$169$	19.0000	1.46154
$170$	13.6569	1.04743
$171$	0	0
$172$	9.65685	0.736328
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	−4.82843	−0.363956
$177$	0	0
$178$	−13.3137	−0.997905
$179$	15.3137	1.14460	0.572300	−	0.820044i	$-0.306051\pi$
$179$	15.3137	1.14460	0.572300	+	0.820044i	$0.306051\pi$
$180$	0	0
$181$	9.65685	0.717788	0.358894	−	0.933378i	$-0.383154\pi$
$181$	9.65685	0.717788	0.358894	+	0.933378i	$0.383154\pi$
$182$	5.65685	0.419314
$183$	0	0
$184$	−4.00000	−0.294884
$185$	16.9706	1.24770
$186$	0	0
$187$	32.9706	2.41105
$188$	4.48528	0.327123
$189$	0	0
$190$	2.00000	0.145095
$191$	19.3137	1.39749	0.698745	−	0.715370i	$-0.253742\pi$
$191$	19.3137	1.39749	0.698745	+	0.715370i	$0.253742\pi$
$192$	0	0
$193$	6.97056	0.501752	0.250876	−	0.968019i	$-0.419281\pi$
$193$	6.97056	0.501752	0.250876	+	0.968019i	$0.419281\pi$
$194$	−18.4853	−1.32717
$195$	0	0
$196$	1.00000	0.0714286
$197$	−12.8284	−0.913988	−0.456994	−	0.889470i	$-0.651074\pi$
$197$	−12.8284	−0.913988	−0.456994	+	0.889470i	$0.651074\pi$
$198$	0	0
$199$	6.34315	0.449654	0.224827	−	0.974399i	$-0.427818\pi$
$199$	6.34315	0.449654	0.224827	+	0.974399i	$0.427818\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	5.31371	0.373871
$203$	−2.00000	−0.140372
$204$	0	0
$205$	−7.31371	−0.510812
$206$	−7.17157	−0.499667
$207$	0	0
$208$	5.65685	0.392232
$209$	4.82843	0.333989
$210$	0	0
$211$	−0.485281	−0.0334081	−0.0167041	−	0.999860i	$-0.505317\pi$
$211$	−0.485281	−0.0334081	−0.0167041	+	0.999860i	$0.505317\pi$
$212$	7.65685	0.525875
$213$	0	0
$214$	−5.65685	−0.386695
$215$	−19.3137	−1.31718
$216$	0	0
$217$	−4.82843	−0.327775
$218$	−10.8284	−0.733394
$219$	0	0
$220$	9.65685	0.651065
$221$	−38.6274	−2.59836
$222$	0	0
$223$	−3.17157	−0.212384	−0.106192	−	0.994346i	$-0.533866\pi$
$223$	−3.17157	−0.212384	−0.106192	+	0.994346i	$0.533866\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	11.6569	0.775402
$227$	−11.3137	−0.750917	−0.375459	−	0.926839i	$-0.622515\pi$
$227$	−11.3137	−0.750917	−0.375459	+	0.926839i	$0.622515\pi$
$228$	0	0
$229$	−7.65685	−0.505979	−0.252990	−	0.967469i	$-0.581414\pi$
$229$	−7.65685	−0.505979	−0.252990	+	0.967469i	$0.581414\pi$
$230$	8.00000	0.527504
$231$	0	0
$232$	−2.00000	−0.131306
$233$	21.6569	1.41879	0.709394	−	0.704812i	$-0.248969\pi$
$233$	21.6569	1.41879	0.709394	+	0.704812i	$0.248969\pi$
$234$	0	0
$235$	−8.97056	−0.585175
$236$	−8.00000	−0.520756
$237$	0	0
$238$	−6.82843	−0.442621
$239$	−12.9706	−0.838996	−0.419498	−	0.907756i	$-0.637794\pi$
$239$	−12.9706	−0.838996	−0.419498	+	0.907756i	$0.637794\pi$
$240$	0	0
$241$	22.4853	1.44840	0.724202	−	0.689588i	$-0.242208\pi$
$241$	22.4853	1.44840	0.724202	+	0.689588i	$0.242208\pi$
$242$	12.3137	0.791555
$243$	0	0
$244$	−13.3137	−0.852323
$245$	−2.00000	−0.127775
$246$	0	0
$247$	−5.65685	−0.359937
$248$	−4.82843	−0.306605
$249$	0	0
$250$	12.0000	0.758947
$251$	−2.68629	−0.169557	−0.0847786	−	0.996400i	$-0.527018\pi$
$251$	−2.68629	−0.169557	−0.0847786	+	0.996400i	$0.527018\pi$
$252$	0	0
$253$	19.3137	1.21424
$254$	4.34315	0.272513
$255$	0	0
$256$	1.00000	0.0625000
$257$	−14.9706	−0.933838	−0.466919	−	0.884300i	$-0.654636\pi$
$257$	−14.9706	−0.933838	−0.466919	+	0.884300i	$0.654636\pi$
$258$	0	0
$259$	−8.48528	−0.527250
$260$	−11.3137	−0.701646
$261$	0	0
$262$	9.31371	0.575403
$263$	26.6274	1.64192	0.820958	−	0.570988i	$-0.193440\pi$
$263$	26.6274	1.64192	0.820958	+	0.570988i	$0.193440\pi$
$264$	0	0
$265$	−15.3137	−0.940714
$266$	−1.00000	−0.0613139
$267$	0	0
$268$	−6.82843	−0.417113
$269$	−17.3137	−1.05564	−0.527818	−	0.849358i	$-0.676990\pi$
$269$	−17.3137	−1.05564	−0.527818	+	0.849358i	$0.676990\pi$
$270$	0	0
$271$	20.9706	1.27387	0.636935	−	0.770917i	$-0.280202\pi$
$271$	20.9706	1.27387	0.636935	+	0.770917i	$0.280202\pi$
$272$	−6.82843	−0.414034
$273$	0	0
$274$	−19.3137	−1.16678
$275$	4.82843	0.291165
$276$	0	0
$277$	27.6569	1.66174	0.830870	−	0.556467i	$-0.187843\pi$
$277$	27.6569	1.66174	0.830870	+	0.556467i	$0.187843\pi$
$278$	−17.6569	−1.05899
$279$	0	0
$280$	−2.00000	−0.119523
$281$	−14.9706	−0.893069	−0.446534	−	0.894766i	$-0.647342\pi$
$281$	−14.9706	−0.893069	−0.446534	+	0.894766i	$0.647342\pi$
$282$	0	0
$283$	14.3431	0.852612	0.426306	−	0.904579i	$-0.359815\pi$
$283$	14.3431	0.852612	0.426306	+	0.904579i	$0.359815\pi$
$284$	5.65685	0.335673
$285$	0	0
$286$	−27.3137	−1.61509
$287$	3.65685	0.215857
$288$	0	0
$289$	29.6274	1.74279
$290$	4.00000	0.234888
$291$	0	0
$292$	−7.65685	−0.448084
$293$	6.68629	0.390617	0.195309	−	0.980742i	$-0.437429\pi$
$293$	6.68629	0.390617	0.195309	+	0.980742i	$0.437429\pi$
$294$	0	0
$295$	16.0000	0.931556
$296$	−8.48528	−0.493197
$297$	0	0
$298$	10.4853	0.607396
$299$	−22.6274	−1.30858
$300$	0	0
$301$	9.65685	0.556612
$302$	7.65685	0.440602
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	26.6274	1.52468
$306$	0	0
$307$	0.686292	0.0391687	0.0195844	−	0.999808i	$-0.493766\pi$
$307$	0.686292	0.0391687	0.0195844	+	0.999808i	$0.493766\pi$
$308$	−4.82843	−0.275125
$309$	0	0
$310$	9.65685	0.548472
$311$	2.14214	0.121469	0.0607347	−	0.998154i	$-0.480656\pi$
$311$	2.14214	0.121469	0.0607347	+	0.998154i	$0.480656\pi$
$312$	0	0
$313$	−17.3137	−0.978629	−0.489314	−	0.872107i	$-0.662753\pi$
$313$	−17.3137	−0.978629	−0.489314	+	0.872107i	$0.662753\pi$
$314$	7.65685	0.432101
$315$	0	0
$316$	7.65685	0.430732
$317$	29.3137	1.64642	0.823211	−	0.567736i	$-0.192180\pi$
$317$	29.3137	1.64642	0.823211	+	0.567736i	$0.192180\pi$
$318$	0	0
$319$	9.65685	0.540680
$320$	−2.00000	−0.111803
$321$	0	0
$322$	−4.00000	−0.222911
$323$	6.82843	0.379944
$324$	0	0
$325$	−5.65685	−0.313786
$326$	0.686292	0.0380102
$327$	0	0
$328$	3.65685	0.201916
$329$	4.48528	0.247282
$330$	0	0
$331$	−0.485281	−0.0266735	−0.0133367	−	0.999911i	$-0.504245\pi$
$331$	−0.485281	−0.0266735	−0.0133367	+	0.999911i	$0.504245\pi$
$332$	−11.6569	−0.639753
$333$	0	0
$334$	−5.65685	−0.309529
$335$	13.6569	0.746154
$336$	0	0
$337$	−23.6569	−1.28867	−0.644335	−	0.764743i	$-0.722866\pi$
$337$	−23.6569	−1.28867	−0.644335	+	0.764743i	$0.722866\pi$
$338$	19.0000	1.03346
$339$	0	0
$340$	13.6569	0.740647
$341$	23.3137	1.26251
$342$	0	0
$343$	1.00000	0.0539949
$344$	9.65685	0.520663
$345$	0	0
$346$	2.00000	0.107521
$347$	−2.48528	−0.133417	−0.0667084	−	0.997773i	$-0.521250\pi$
$347$	−2.48528	−0.133417	−0.0667084	+	0.997773i	$0.521250\pi$
$348$	0	0
$349$	10.9706	0.587241	0.293620	−	0.955922i	$-0.405140\pi$
$349$	10.9706	0.587241	0.293620	+	0.955922i	$0.405140\pi$
$350$	−1.00000	−0.0534522
$351$	0	0
$352$	−4.82843	−0.257356
$353$	30.1421	1.60430	0.802152	−	0.597120i	$-0.203688\pi$
$353$	30.1421	1.60430	0.802152	+	0.597120i	$0.203688\pi$
$354$	0	0
$355$	−11.3137	−0.600469
$356$	−13.3137	−0.705625
$357$	0	0
$358$	15.3137	0.809355
$359$	−16.9706	−0.895672	−0.447836	−	0.894116i	$-0.647805\pi$
$359$	−16.9706	−0.895672	−0.447836	+	0.894116i	$0.647805\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	9.65685	0.507553
$363$	0	0
$364$	5.65685	0.296500
$365$	15.3137	0.801556
$366$	0	0
$367$	9.65685	0.504084	0.252042	−	0.967716i	$-0.418898\pi$
$367$	9.65685	0.504084	0.252042	+	0.967716i	$0.418898\pi$
$368$	−4.00000	−0.208514
$369$	0	0
$370$	16.9706	0.882258
$371$	7.65685	0.397524
$372$	0	0
$373$	−34.8284	−1.80335	−0.901674	−	0.432417i	$-0.857661\pi$
$373$	−34.8284	−1.80335	−0.901674	+	0.432417i	$0.857661\pi$
$374$	32.9706	1.70487
$375$	0	0
$376$	4.48528	0.231311
$377$	−11.3137	−0.582686
$378$	0	0
$379$	34.8284	1.78902	0.894508	−	0.447052i	$-0.147526\pi$
$379$	34.8284	1.78902	0.894508	+	0.447052i	$0.147526\pi$
$380$	2.00000	0.102598
$381$	0	0
$382$	19.3137	0.988175
$383$	2.34315	0.119729	0.0598646	−	0.998207i	$-0.480933\pi$
$383$	2.34315	0.119729	0.0598646	+	0.998207i	$0.480933\pi$
$384$	0	0
$385$	9.65685	0.492159
$386$	6.97056	0.354792
$387$	0	0
$388$	−18.4853	−0.938448
$389$	13.7990	0.699637	0.349818	−	0.936818i	$-0.386243\pi$
$389$	13.7990	0.699637	0.349818	+	0.936818i	$0.386243\pi$
$390$	0	0
$391$	27.3137	1.38131
$392$	1.00000	0.0505076
$393$	0	0
$394$	−12.8284	−0.646287
$395$	−15.3137	−0.770516
$396$	0	0
$397$	−9.31371	−0.467442	−0.233721	−	0.972304i	$-0.575090\pi$
$397$	−9.31371	−0.467442	−0.233721	+	0.972304i	$0.575090\pi$
$398$	6.34315	0.317953
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	−13.3137	−0.664855	−0.332427	−	0.943129i	$-0.607868\pi$
$401$	−13.3137	−0.664855	−0.332427	+	0.943129i	$0.607868\pi$
$402$	0	0
$403$	−27.3137	−1.36059
$404$	5.31371	0.264367
$405$	0	0
$406$	−2.00000	−0.0992583
$407$	40.9706	2.03084
$408$	0	0
$409$	−18.4853	−0.914038	−0.457019	−	0.889457i	$-0.651083\pi$
$409$	−18.4853	−0.914038	−0.457019	+	0.889457i	$0.651083\pi$
$410$	−7.31371	−0.361198
$411$	0	0
$412$	−7.17157	−0.353318
$413$	−8.00000	−0.393654
$414$	0	0
$415$	23.3137	1.14442
$416$	5.65685	0.277350
$417$	0	0
$418$	4.82843	0.236166
$419$	−14.9706	−0.731360	−0.365680	−	0.930741i	$-0.619164\pi$
$419$	−14.9706	−0.731360	−0.365680	+	0.930741i	$0.619164\pi$
$420$	0	0
$421$	−18.1421	−0.884194	−0.442097	−	0.896967i	$-0.645765\pi$
$421$	−18.1421	−0.884194	−0.442097	+	0.896967i	$0.645765\pi$
$422$	−0.485281	−0.0236231
$423$	0	0
$424$	7.65685	0.371850
$425$	6.82843	0.331227
$426$	0	0
$427$	−13.3137	−0.644296
$428$	−5.65685	−0.273434
$429$	0	0
$430$	−19.3137	−0.931390
$431$	−32.0000	−1.54139	−0.770693	−	0.637207i	$-0.780090\pi$
$431$	−32.0000	−1.54139	−0.770693	+	0.637207i	$0.780090\pi$
$432$	0	0
$433$	−25.7990	−1.23982	−0.619910	−	0.784673i	$-0.712831\pi$
$433$	−25.7990	−1.23982	−0.619910	+	0.784673i	$0.712831\pi$
$434$	−4.82843	−0.231772
$435$	0	0
$436$	−10.8284	−0.518588
$437$	4.00000	0.191346
$438$	0	0
$439$	13.7990	0.658590	0.329295	−	0.944227i	$-0.393189\pi$
$439$	13.7990	0.658590	0.329295	+	0.944227i	$0.393189\pi$
$440$	9.65685	0.460372
$441$	0	0
$442$	−38.6274	−1.83732
$443$	27.1716	1.29096	0.645480	−	0.763777i	$-0.276657\pi$
$443$	27.1716	1.29096	0.645480	+	0.763777i	$0.276657\pi$
$444$	0	0
$445$	26.6274	1.26226
$446$	−3.17157	−0.150178
$447$	0	0
$448$	1.00000	0.0472456
$449$	−2.68629	−0.126774	−0.0633870	−	0.997989i	$-0.520190\pi$
$449$	−2.68629	−0.126774	−0.0633870	+	0.997989i	$0.520190\pi$
$450$	0	0
$451$	−17.6569	−0.831429
$452$	11.6569	0.548292
$453$	0	0
$454$	−11.3137	−0.530979
$455$	−11.3137	−0.530395
$456$	0	0
$457$	−2.68629	−0.125659	−0.0628297	−	0.998024i	$-0.520012\pi$
$457$	−2.68629	−0.125659	−0.0628297	+	0.998024i	$0.520012\pi$
$458$	−7.65685	−0.357781
$459$	0	0
$460$	8.00000	0.373002
$461$	19.6569	0.915511	0.457755	−	0.889078i	$-0.348654\pi$
$461$	19.6569	0.915511	0.457755	+	0.889078i	$0.348654\pi$
$462$	0	0
$463$	−31.3137	−1.45527	−0.727636	−	0.685964i	$-0.759381\pi$
$463$	−31.3137	−1.45527	−0.727636	+	0.685964i	$0.759381\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	21.6569	1.00323
$467$	34.9706	1.61824	0.809122	−	0.587640i	$-0.199943\pi$
$467$	34.9706	1.61824	0.809122	+	0.587640i	$0.199943\pi$
$468$	0	0
$469$	−6.82843	−0.315307
$470$	−8.97056	−0.413781
$471$	0	0
$472$	−8.00000	−0.368230
$473$	−46.6274	−2.14393
$474$	0	0
$475$	1.00000	0.0458831
$476$	−6.82843	−0.312980
$477$	0	0
$478$	−12.9706	−0.593260
$479$	−3.51472	−0.160592	−0.0802958	−	0.996771i	$-0.525586\pi$
$479$	−3.51472	−0.160592	−0.0802958	+	0.996771i	$0.525586\pi$
$480$	0	0
$481$	−48.0000	−2.18861
$482$	22.4853	1.02418
$483$	0	0
$484$	12.3137	0.559714
$485$	36.9706	1.67875
$486$	0	0
$487$	−2.00000	−0.0906287	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	−13.3137	−0.602683
$489$	0	0
$490$	−2.00000	−0.0903508
$491$	−4.14214	−0.186932	−0.0934660	−	0.995622i	$-0.529795\pi$
$491$	−4.14214	−0.186932	−0.0934660	+	0.995622i	$0.529795\pi$
$492$	0	0
$493$	13.6569	0.615074
$494$	−5.65685	−0.254514
$495$	0	0
$496$	−4.82843	−0.216803
$497$	5.65685	0.253745
$498$	0	0
$499$	−39.3137	−1.75992	−0.879962	−	0.475045i	$-0.842432\pi$
$499$	−39.3137	−1.75992	−0.879962	+	0.475045i	$0.842432\pi$
$500$	12.0000	0.536656
$501$	0	0
$502$	−2.68629	−0.119895
$503$	−35.7990	−1.59620	−0.798099	−	0.602526i	$-0.794161\pi$
$503$	−35.7990	−1.59620	−0.798099	+	0.602526i	$0.794161\pi$
$504$	0	0
$505$	−10.6274	−0.472914
$506$	19.3137	0.858599
$507$	0	0
$508$	4.34315	0.192696
$509$	31.9411	1.41577	0.707883	−	0.706330i	$-0.249651\pi$
$509$	31.9411	1.41577	0.707883	+	0.706330i	$0.249651\pi$
$510$	0	0
$511$	−7.65685	−0.338719
$512$	1.00000	0.0441942
$513$	0	0
$514$	−14.9706	−0.660323
$515$	14.3431	0.632035
$516$	0	0
$517$	−21.6569	−0.952467
$518$	−8.48528	−0.372822
$519$	0	0
$520$	−11.3137	−0.496139
$521$	−15.6569	−0.685939	−0.342970	−	0.939346i	$-0.611433\pi$
$521$	−15.6569	−0.685939	−0.342970	+	0.939346i	$0.611433\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	9.31371	0.406871
$525$	0	0
$526$	26.6274	1.16101
$527$	32.9706	1.43622
$528$	0	0
$529$	−7.00000	−0.304348
$530$	−15.3137	−0.665185
$531$	0	0
$532$	−1.00000	−0.0433555
$533$	20.6863	0.896023
$534$	0	0
$535$	11.3137	0.489134
$536$	−6.82843	−0.294943
$537$	0	0
$538$	−17.3137	−0.746447
$539$	−4.82843	−0.207975
$540$	0	0
$541$	27.9411	1.20128	0.600641	−	0.799519i	$-0.294912\pi$
$541$	27.9411	1.20128	0.600641	+	0.799519i	$0.294912\pi$
$542$	20.9706	0.900763
$543$	0	0
$544$	−6.82843	−0.292766
$545$	21.6569	0.927678
$546$	0	0
$547$	39.1127	1.67234	0.836169	−	0.548472i	$-0.184790\pi$
$547$	39.1127	1.67234	0.836169	+	0.548472i	$0.184790\pi$
$548$	−19.3137	−0.825041
$549$	0	0
$550$	4.82843	0.205885
$551$	2.00000	0.0852029
$552$	0	0
$553$	7.65685	0.325603
$554$	27.6569	1.17503
$555$	0	0
$556$	−17.6569	−0.748817
$557$	−34.7696	−1.47323	−0.736617	−	0.676311i	$-0.763578\pi$
$557$	−34.7696	−1.47323	−0.736617	+	0.676311i	$0.763578\pi$
$558$	0	0
$559$	54.6274	2.31049
$560$	−2.00000	−0.0845154
$561$	0	0
$562$	−14.9706	−0.631495
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	0	0
$565$	−23.3137	−0.980815
$566$	14.3431	0.602887
$567$	0	0
$568$	5.65685	0.237356
$569$	9.02944	0.378534	0.189267	−	0.981926i	$-0.439389\pi$
$569$	9.02944	0.378534	0.189267	+	0.981926i	$0.439389\pi$
$570$	0	0
$571$	0.686292	0.0287204	0.0143602	−	0.999897i	$-0.495429\pi$
$571$	0.686292	0.0287204	0.0143602	+	0.999897i	$0.495429\pi$
$572$	−27.3137	−1.14204
$573$	0	0
$574$	3.65685	0.152634
$575$	4.00000	0.166812
$576$	0	0
$577$	26.2843	1.09423	0.547114	−	0.837058i	$-0.315726\pi$
$577$	26.2843	1.09423	0.547114	+	0.837058i	$0.315726\pi$
$578$	29.6274	1.23234
$579$	0	0
$580$	4.00000	0.166091
$581$	−11.6569	−0.483608
$582$	0	0
$583$	−36.9706	−1.53116
$584$	−7.65685	−0.316843
$585$	0	0
$586$	6.68629	0.276208
$587$	17.3137	0.714613	0.357307	−	0.933987i	$-0.383695\pi$
$587$	17.3137	0.714613	0.357307	+	0.933987i	$0.383695\pi$
$588$	0	0
$589$	4.82843	0.198952
$590$	16.0000	0.658710
$591$	0	0
$592$	−8.48528	−0.348743
$593$	−19.5147	−0.801373	−0.400687	−	0.916215i	$-0.631228\pi$
$593$	−19.5147	−0.801373	−0.400687	+	0.916215i	$0.631228\pi$
$594$	0	0
$595$	13.6569	0.559876
$596$	10.4853	0.429494
$597$	0	0
$598$	−22.6274	−0.925304
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	−43.4558	−1.77260	−0.886300	−	0.463111i	$-0.846733\pi$
$601$	−43.4558	−1.77260	−0.886300	+	0.463111i	$0.846733\pi$
$602$	9.65685	0.393584
$603$	0	0
$604$	7.65685	0.311553
$605$	−24.6274	−1.00125
$606$	0	0
$607$	−44.1421	−1.79167	−0.895837	−	0.444383i	$-0.853423\pi$
$607$	−44.1421	−1.79167	−0.895837	+	0.444383i	$0.853423\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	26.6274	1.07811
$611$	25.3726	1.02646
$612$	0	0
$613$	−49.3137	−1.99176	−0.995881	−	0.0906703i	$-0.971099\pi$
$613$	−49.3137	−1.99176	−0.995881	+	0.0906703i	$0.971099\pi$
$614$	0.686292	0.0276965
$615$	0	0
$616$	−4.82843	−0.194543
$617$	−36.9706	−1.48838	−0.744189	−	0.667969i	$-0.767164\pi$
$617$	−36.9706	−1.48838	−0.744189	+	0.667969i	$0.767164\pi$
$618$	0	0
$619$	41.6569	1.67433	0.837165	−	0.546950i	$-0.184211\pi$
$619$	41.6569	1.67433	0.837165	+	0.546950i	$0.184211\pi$
$620$	9.65685	0.387829
$621$	0	0
$622$	2.14214	0.0858918
$623$	−13.3137	−0.533402
$624$	0	0
$625$	−19.0000	−0.760000
$626$	−17.3137	−0.691995
$627$	0	0
$628$	7.65685	0.305542
$629$	57.9411	2.31026
$630$	0	0
$631$	26.6274	1.06002	0.530010	−	0.847991i	$-0.322188\pi$
$631$	26.6274	1.06002	0.530010	+	0.847991i	$0.322188\pi$
$632$	7.65685	0.304573
$633$	0	0
$634$	29.3137	1.16420
$635$	−8.68629	−0.344705
$636$	0	0
$637$	5.65685	0.224133
$638$	9.65685	0.382319
$639$	0	0
$640$	−2.00000	−0.0790569
$641$	−5.31371	−0.209879	−0.104939	−	0.994479i	$-0.533465\pi$
$641$	−5.31371	−0.209879	−0.104939	+	0.994479i	$0.533465\pi$
$642$	0	0
$643$	47.3137	1.86587	0.932935	−	0.360044i	$-0.117238\pi$
$643$	47.3137	1.86587	0.932935	+	0.360044i	$0.117238\pi$
$644$	−4.00000	−0.157622
$645$	0	0
$646$	6.82843	0.268661
$647$	−38.1421	−1.49952	−0.749761	−	0.661709i	$-0.769832\pi$
$647$	−38.1421	−1.49952	−0.749761	+	0.661709i	$0.769832\pi$
$648$	0	0
$649$	38.6274	1.51626
$650$	−5.65685	−0.221880
$651$	0	0
$652$	0.686292	0.0268772
$653$	−6.48528	−0.253789	−0.126894	−	0.991916i	$-0.540501\pi$
$653$	−6.48528	−0.253789	−0.126894	+	0.991916i	$0.540501\pi$
$654$	0	0
$655$	−18.6274	−0.727833
$656$	3.65685	0.142776
$657$	0	0
$658$	4.48528	0.174854
$659$	29.6569	1.15527	0.577634	−	0.816296i	$-0.303976\pi$
$659$	29.6569	1.15527	0.577634	+	0.816296i	$0.303976\pi$
$660$	0	0
$661$	8.68629	0.337858	0.168929	−	0.985628i	$-0.445969\pi$
$661$	8.68629	0.337858	0.168929	+	0.985628i	$0.445969\pi$
$662$	−0.485281	−0.0188610
$663$	0	0
$664$	−11.6569	−0.452374
$665$	2.00000	0.0775567
$666$	0	0
$667$	8.00000	0.309761
$668$	−5.65685	−0.218870
$669$	0	0
$670$	13.6569	0.527610
$671$	64.2843	2.48167
$672$	0	0
$673$	−29.3137	−1.12996	−0.564980	−	0.825104i	$-0.691116\pi$
$673$	−29.3137	−1.12996	−0.564980	+	0.825104i	$0.691116\pi$
$674$	−23.6569	−0.911228
$675$	0	0
$676$	19.0000	0.730769
$677$	13.3137	0.511687	0.255844	−	0.966718i	$-0.417647\pi$
$677$	13.3137	0.511687	0.255844	+	0.966718i	$0.417647\pi$
$678$	0	0
$679$	−18.4853	−0.709400
$680$	13.6569	0.523716
$681$	0	0
$682$	23.3137	0.892728
$683$	37.9411	1.45178	0.725888	−	0.687812i	$-0.241429\pi$
$683$	37.9411	1.45178	0.725888	+	0.687812i	$0.241429\pi$
$684$	0	0
$685$	38.6274	1.47588
$686$	1.00000	0.0381802
$687$	0	0
$688$	9.65685	0.368164
$689$	43.3137	1.65012
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	2.00000	0.0760286
$693$	0	0
$694$	−2.48528	−0.0943400
$695$	35.3137	1.33953
$696$	0	0
$697$	−24.9706	−0.945828
$698$	10.9706	0.415242
$699$	0	0
$700$	−1.00000	−0.0377964
$701$	21.1127	0.797416	0.398708	−	0.917078i	$-0.369459\pi$
$701$	21.1127	0.797416	0.398708	+	0.917078i	$0.369459\pi$
$702$	0	0
$703$	8.48528	0.320028
$704$	−4.82843	−0.181978
$705$	0	0
$706$	30.1421	1.13441
$707$	5.31371	0.199843
$708$	0	0
$709$	−26.9706	−1.01290	−0.506450	−	0.862269i	$-0.669043\pi$
$709$	−26.9706	−1.01290	−0.506450	+	0.862269i	$0.669043\pi$
$710$	−11.3137	−0.424596
$711$	0	0
$712$	−13.3137	−0.498952
$713$	19.3137	0.723304
$714$	0	0
$715$	54.6274	2.04295
$716$	15.3137	0.572300
$717$	0	0
$718$	−16.9706	−0.633336
$719$	−31.7990	−1.18590	−0.592951	−	0.805238i	$-0.702037\pi$
$719$	−31.7990	−1.18590	−0.592951	+	0.805238i	$0.702037\pi$
$720$	0	0
$721$	−7.17157	−0.267083
$722$	1.00000	0.0372161
$723$	0	0
$724$	9.65685	0.358894
$725$	2.00000	0.0742781
$726$	0	0
$727$	−19.3137	−0.716306	−0.358153	−	0.933663i	$-0.616593\pi$
$727$	−19.3137	−0.716306	−0.358153	+	0.933663i	$0.616593\pi$
$728$	5.65685	0.209657
$729$	0	0
$730$	15.3137	0.566786
$731$	−65.9411	−2.43892
$732$	0	0
$733$	10.9706	0.405207	0.202603	−	0.979261i	$-0.435060\pi$
$733$	10.9706	0.405207	0.202603	+	0.979261i	$0.435060\pi$
$734$	9.65685	0.356441
$735$	0	0
$736$	−4.00000	−0.147442
$737$	32.9706	1.21449
$738$	0	0
$739$	9.65685	0.355233	0.177617	−	0.984100i	$-0.443161\pi$
$739$	9.65685	0.355233	0.177617	+	0.984100i	$0.443161\pi$
$740$	16.9706	0.623850
$741$	0	0
$742$	7.65685	0.281092
$743$	53.6569	1.96848	0.984240	−	0.176840i	$-0.0565877\pi$
$743$	53.6569	1.96848	0.984240	+	0.176840i	$0.0565877\pi$
$744$	0	0
$745$	−20.9706	−0.768302
$746$	−34.8284	−1.27516
$747$	0	0
$748$	32.9706	1.20552
$749$	−5.65685	−0.206697
$750$	0	0
$751$	40.6274	1.48252	0.741258	−	0.671220i	$-0.234230\pi$
$751$	40.6274	1.48252	0.741258	+	0.671220i	$0.234230\pi$
$752$	4.48528	0.163561
$753$	0	0
$754$	−11.3137	−0.412021
$755$	−15.3137	−0.557323
$756$	0	0
$757$	−31.9411	−1.16092	−0.580460	−	0.814289i	$-0.697127\pi$
$757$	−31.9411	−1.16092	−0.580460	+	0.814289i	$0.697127\pi$
$758$	34.8284	1.26503
$759$	0	0
$760$	2.00000	0.0725476
$761$	−33.1716	−1.20247	−0.601234	−	0.799073i	$-0.705324\pi$
$761$	−33.1716	−1.20247	−0.601234	+	0.799073i	$0.705324\pi$
$762$	0	0
$763$	−10.8284	−0.392015
$764$	19.3137	0.698745
$765$	0	0
$766$	2.34315	0.0846613
$767$	−45.2548	−1.63406
$768$	0	0
$769$	6.68629	0.241114	0.120557	−	0.992706i	$-0.461532\pi$
$769$	6.68629	0.241114	0.120557	+	0.992706i	$0.461532\pi$
$770$	9.65685	0.348009
$771$	0	0
$772$	6.97056	0.250876
$773$	−10.0000	−0.359675	−0.179838	−	0.983696i	$-0.557557\pi$
$773$	−10.0000	−0.359675	−0.179838	+	0.983696i	$0.557557\pi$
$774$	0	0
$775$	4.82843	0.173442
$776$	−18.4853	−0.663583
$777$	0	0
$778$	13.7990	0.494718
$779$	−3.65685	−0.131020
$780$	0	0
$781$	−27.3137	−0.977361
$782$	27.3137	0.976736
$783$	0	0
$784$	1.00000	0.0357143
$785$	−15.3137	−0.546570
$786$	0	0
$787$	9.65685	0.344230	0.172115	−	0.985077i	$-0.444940\pi$
$787$	9.65685	0.344230	0.172115	+	0.985077i	$0.444940\pi$
$788$	−12.8284	−0.456994
$789$	0	0
$790$	−15.3137	−0.544837
$791$	11.6569	0.414470
$792$	0	0
$793$	−75.3137	−2.67447
$794$	−9.31371	−0.330531
$795$	0	0
$796$	6.34315	0.224827
$797$	−19.9411	−0.706351	−0.353175	−	0.935557i	$-0.614898\pi$
$797$	−19.9411	−0.706351	−0.353175	+	0.935557i	$0.614898\pi$
$798$	0	0
$799$	−30.6274	−1.08352
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	−13.3137	−0.470123
$803$	36.9706	1.30466
$804$	0	0
$805$	8.00000	0.281963
$806$	−27.3137	−0.962084
$807$	0	0
$808$	5.31371	0.186936
$809$	−1.65685	−0.0582519	−0.0291259	−	0.999576i	$-0.509272\pi$
$809$	−1.65685	−0.0582519	−0.0291259	+	0.999576i	$0.509272\pi$
$810$	0	0
$811$	8.68629	0.305017	0.152508	−	0.988302i	$-0.451265\pi$
$811$	8.68629	0.305017	0.152508	+	0.988302i	$0.451265\pi$
$812$	−2.00000	−0.0701862
$813$	0	0
$814$	40.9706	1.43602
$815$	−1.37258	−0.0480795
$816$	0	0
$817$	−9.65685	−0.337851
$818$	−18.4853	−0.646323
$819$	0	0
$820$	−7.31371	−0.255406
$821$	−18.4853	−0.645141	−0.322570	−	0.946545i	$-0.604547\pi$
$821$	−18.4853	−0.645141	−0.322570	+	0.946545i	$0.604547\pi$
$822$	0	0
$823$	−39.3137	−1.37039	−0.685195	−	0.728360i	$-0.740283\pi$
$823$	−39.3137	−1.37039	−0.685195	+	0.728360i	$0.740283\pi$
$824$	−7.17157	−0.249834
$825$	0	0
$826$	−8.00000	−0.278356
$827$	−23.3137	−0.810697	−0.405349	−	0.914162i	$-0.632850\pi$
$827$	−23.3137	−0.810697	−0.405349	+	0.914162i	$0.632850\pi$
$828$	0	0
$829$	29.9411	1.03990	0.519949	−	0.854197i	$-0.325951\pi$
$829$	29.9411	1.03990	0.519949	+	0.854197i	$0.325951\pi$
$830$	23.3137	0.809231
$831$	0	0
$832$	5.65685	0.196116
$833$	−6.82843	−0.236591
$834$	0	0
$835$	11.3137	0.391527
$836$	4.82843	0.166995
$837$	0	0
$838$	−14.9706	−0.517150
$839$	19.3137	0.666783	0.333392	−	0.942788i	$-0.391807\pi$
$839$	19.3137	0.666783	0.333392	+	0.942788i	$0.391807\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−18.1421	−0.625219
$843$	0	0
$844$	−0.485281	−0.0167041
$845$	−38.0000	−1.30724
$846$	0	0
$847$	12.3137	0.423104
$848$	7.65685	0.262937
$849$	0	0
$850$	6.82843	0.234213
$851$	33.9411	1.16349
$852$	0	0
$853$	1.31371	0.0449805	0.0224903	−	0.999747i	$-0.492841\pi$
$853$	1.31371	0.0449805	0.0224903	+	0.999747i	$0.492841\pi$
$854$	−13.3137	−0.455586
$855$	0	0
$856$	−5.65685	−0.193347
$857$	−29.3137	−1.00134	−0.500669	−	0.865639i	$-0.666912\pi$
$857$	−29.3137	−1.00134	−0.500669	+	0.865639i	$0.666912\pi$
$858$	0	0
$859$	−3.02944	−0.103363	−0.0516815	−	0.998664i	$-0.516458\pi$
$859$	−3.02944	−0.103363	−0.0516815	+	0.998664i	$0.516458\pi$
$860$	−19.3137	−0.658592
$861$	0	0
$862$	−32.0000	−1.08992
$863$	28.2843	0.962808	0.481404	−	0.876499i	$-0.340127\pi$
$863$	28.2843	0.962808	0.481404	+	0.876499i	$0.340127\pi$
$864$	0	0
$865$	−4.00000	−0.136004
$866$	−25.7990	−0.876685
$867$	0	0
$868$	−4.82843	−0.163887
$869$	−36.9706	−1.25414
$870$	0	0
$871$	−38.6274	−1.30884
$872$	−10.8284	−0.366697
$873$	0	0
$874$	4.00000	0.135302
$875$	12.0000	0.405674
$876$	0	0
$877$	−58.4264	−1.97292	−0.986460	−	0.164003i	$-0.947559\pi$
$877$	−58.4264	−1.97292	−0.986460	+	0.164003i	$0.947559\pi$
$878$	13.7990	0.465693
$879$	0	0
$880$	9.65685	0.325532
$881$	54.1421	1.82409	0.912047	−	0.410085i	$-0.134501\pi$
$881$	54.1421	1.82409	0.912047	+	0.410085i	$0.134501\pi$
$882$	0	0
$883$	36.0000	1.21150	0.605748	−	0.795656i	$-0.292874\pi$
$883$	36.0000	1.21150	0.605748	+	0.795656i	$0.292874\pi$
$884$	−38.6274	−1.29918
$885$	0	0
$886$	27.1716	0.912847
$887$	17.9411	0.602404	0.301202	−	0.953560i	$-0.402612\pi$
$887$	17.9411	0.602404	0.301202	+	0.953560i	$0.402612\pi$
$888$	0	0
$889$	4.34315	0.145664
$890$	26.6274	0.892553
$891$	0	0
$892$	−3.17157	−0.106192
$893$	−4.48528	−0.150094
$894$	0	0
$895$	−30.6274	−1.02376
$896$	1.00000	0.0334077
$897$	0	0
$898$	−2.68629	−0.0896427
$899$	9.65685	0.322074
$900$	0	0
$901$	−52.2843	−1.74184
$902$	−17.6569	−0.587909
$903$	0	0
$904$	11.6569	0.387701
$905$	−19.3137	−0.642009
$906$	0	0
$907$	60.0833	1.99503	0.997516	−	0.0704407i	$-0.0224405\pi$
$907$	60.0833	1.99503	0.997516	+	0.0704407i	$0.0224405\pi$
$908$	−11.3137	−0.375459
$909$	0	0
$910$	−11.3137	−0.375046
$911$	−30.6274	−1.01473	−0.507366	−	0.861731i	$-0.669381\pi$
$911$	−30.6274	−1.01473	−0.507366	+	0.861731i	$0.669381\pi$
$912$	0	0
$913$	56.2843	1.86274
$914$	−2.68629	−0.0888546
$915$	0	0
$916$	−7.65685	−0.252990
$917$	9.31371	0.307566
$918$	0	0
$919$	−10.6274	−0.350566	−0.175283	−	0.984518i	$-0.556084\pi$
$919$	−10.6274	−0.350566	−0.175283	+	0.984518i	$0.556084\pi$
$920$	8.00000	0.263752
$921$	0	0
$922$	19.6569	0.647364
$923$	32.0000	1.05329
$924$	0	0
$925$	8.48528	0.278994
$926$	−31.3137	−1.02903
$927$	0	0
$928$	−2.00000	−0.0656532
$929$	−41.4558	−1.36012	−0.680061	−	0.733155i	$-0.738047\pi$
$929$	−41.4558	−1.36012	−0.680061	+	0.733155i	$0.738047\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	21.6569	0.709394
$933$	0	0
$934$	34.9706	1.14427
$935$	−65.9411	−2.15651
$936$	0	0
$937$	−43.9411	−1.43549	−0.717747	−	0.696304i	$-0.754827\pi$
$937$	−43.9411	−1.43549	−0.717747	+	0.696304i	$0.754827\pi$
$938$	−6.82843	−0.222956
$939$	0	0
$940$	−8.97056	−0.292587
$941$	−30.6863	−1.00034	−0.500172	−	0.865926i	$-0.666730\pi$
$941$	−30.6863	−1.00034	−0.500172	+	0.865926i	$0.666730\pi$
$942$	0	0
$943$	−14.6274	−0.476334
$944$	−8.00000	−0.260378
$945$	0	0
$946$	−46.6274	−1.51599
$947$	45.7990	1.48827	0.744134	−	0.668031i	$-0.232863\pi$
$947$	45.7990	1.48827	0.744134	+	0.668031i	$0.232863\pi$
$948$	0	0
$949$	−43.3137	−1.40602
$950$	1.00000	0.0324443
$951$	0	0
$952$	−6.82843	−0.221311
$953$	7.65685	0.248030	0.124015	−	0.992280i	$-0.460423\pi$
$953$	7.65685	0.248030	0.124015	+	0.992280i	$0.460423\pi$
$954$	0	0
$955$	−38.6274	−1.24995
$956$	−12.9706	−0.419498
$957$	0	0
$958$	−3.51472	−0.113555
$959$	−19.3137	−0.623672
$960$	0	0
$961$	−7.68629	−0.247945
$962$	−48.0000	−1.54758
$963$	0	0
$964$	22.4853	0.724202
$965$	−13.9411	−0.448781
$966$	0	0
$967$	−43.3137	−1.39287	−0.696437	−	0.717617i	$-0.745233\pi$
$967$	−43.3137	−1.39287	−0.696437	+	0.717617i	$0.745233\pi$
$968$	12.3137	0.395778
$969$	0	0
$970$	36.9706	1.18705
$971$	−22.6274	−0.726148	−0.363074	−	0.931760i	$-0.618273\pi$
$971$	−22.6274	−0.726148	−0.363074	+	0.931760i	$0.618273\pi$
$972$	0	0
$973$	−17.6569	−0.566053
$974$	−2.00000	−0.0640841
$975$	0	0
$976$	−13.3137	−0.426161
$977$	−11.3726	−0.363841	−0.181921	−	0.983313i	$-0.558231\pi$
$977$	−11.3726	−0.363841	−0.181921	+	0.983313i	$0.558231\pi$
$978$	0	0
$979$	64.2843	2.05453
$980$	−2.00000	−0.0638877
$981$	0	0
$982$	−4.14214	−0.132181
$983$	−21.6569	−0.690746	−0.345373	−	0.938465i	$-0.612248\pi$
$983$	−21.6569	−0.690746	−0.345373	+	0.938465i	$0.612248\pi$
$984$	0	0
$985$	25.6569	0.817495
$986$	13.6569	0.434923
$987$	0	0
$988$	−5.65685	−0.179969
$989$	−38.6274	−1.22828
$990$	0	0
$991$	−54.0000	−1.71537	−0.857683	−	0.514178i	$-0.828097\pi$
$991$	−54.0000	−1.71537	−0.857683	+	0.514178i	$0.828097\pi$
$992$	−4.82843	−0.153303
$993$	0	0
$994$	5.65685	0.179425
$995$	−12.6863	−0.402182
$996$	0	0
$997$	38.2843	1.21248	0.606238	−	0.795284i	$-0.292678\pi$
$997$	38.2843	1.21248	0.606238	+	0.795284i	$0.292678\pi$
$998$	−39.3137	−1.24445
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2394.2.a.u.1.1	yes	2
3.2	odd	2		2394.2.a.t.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2394.2.a.t.1.2	✓	2	3.2	odd	2
2394.2.a.u.1.1	yes	2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2394.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{10} -4.82843 q^{11} +5.65685 q^{13} +1.00000 q^{14} +1.00000 q^{16} -6.82843 q^{17} -1.00000 q^{19} -2.00000 q^{20} -4.82843 q^{22} -4.00000 q^{23} -1.00000 q^{25} +5.65685 q^{26} +1.00000 q^{28} -2.00000 q^{29} -4.82843 q^{31} +1.00000 q^{32} -6.82843 q^{34} -2.00000 q^{35} -8.48528 q^{37} -1.00000 q^{38} -2.00000 q^{40} +3.65685 q^{41} +9.65685 q^{43} -4.82843 q^{44} -4.00000 q^{46} +4.48528 q^{47} +1.00000 q^{49} -1.00000 q^{50} +5.65685 q^{52} +7.65685 q^{53} +9.65685 q^{55} +1.00000 q^{56} -2.00000 q^{58} -8.00000 q^{59} -13.3137 q^{61} -4.82843 q^{62} +1.00000 q^{64} -11.3137 q^{65} -6.82843 q^{67} -6.82843 q^{68} -2.00000 q^{70} +5.65685 q^{71} -7.65685 q^{73} -8.48528 q^{74} -1.00000 q^{76} -4.82843 q^{77} +7.65685 q^{79} -2.00000 q^{80} +3.65685 q^{82} -11.6569 q^{83} +13.6569 q^{85} +9.65685 q^{86} -4.82843 q^{88} -13.3137 q^{89} +5.65685 q^{91} -4.00000 q^{92} +4.48528 q^{94} +2.00000 q^{95} -18.4853 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 4 q^{5} + 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^5 + 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - 4 q^{5} + 2 q^{7} + 2 q^{8} - 4 q^{10} - 4 q^{11} + 2 q^{14} + 2 q^{16} - 8 q^{17} - 2 q^{19} - 4 q^{20} - 4 q^{22} - 8 q^{23} - 2 q^{25} + 2 q^{28} - 4 q^{29} - 4 q^{31} + 2 q^{32} - 8 q^{34} - 4 q^{35} - 2 q^{38} - 4 q^{40} - 4 q^{41} + 8 q^{43} - 4 q^{44} - 8 q^{46} - 8 q^{47} + 2 q^{49} - 2 q^{50} + 4 q^{53} + 8 q^{55} + 2 q^{56} - 4 q^{58} - 16 q^{59} - 4 q^{61} - 4 q^{62} + 2 q^{64} - 8 q^{67} - 8 q^{68} - 4 q^{70} - 4 q^{73} - 2 q^{76} - 4 q^{77} + 4 q^{79} - 4 q^{80} - 4 q^{82} - 12 q^{83} + 16 q^{85} + 8 q^{86} - 4 q^{88} - 4 q^{89} - 8 q^{92} - 8 q^{94} + 4 q^{95} - 20 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^5 + 2 * q^7 + 2 * q^8 - 4 * q^10 - 4 * q^11 + 2 * q^14 + 2 * q^16 - 8 * q^17 - 2 * q^19 - 4 * q^20 - 4 * q^22 - 8 * q^23 - 2 * q^25 + 2 * q^28 - 4 * q^29 - 4 * q^31 + 2 * q^32 - 8 * q^34 - 4 * q^35 - 2 * q^38 - 4 * q^40 - 4 * q^41 + 8 * q^43 - 4 * q^44 - 8 * q^46 - 8 * q^47 + 2 * q^49 - 2 * q^50 + 4 * q^53 + 8 * q^55 + 2 * q^56 - 4 * q^58 - 16 * q^59 - 4 * q^61 - 4 * q^62 + 2 * q^64 - 8 * q^67 - 8 * q^68 - 4 * q^70 - 4 * q^73 - 2 * q^76 - 4 * q^77 + 4 * q^79 - 4 * q^80 - 4 * q^82 - 12 * q^83 + 16 * q^85 + 8 * q^86 - 4 * q^88 - 4 * q^89 - 8 * q^92 - 8 * q^94 + 4 * q^95 - 20 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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