LMFDB - Embedded newform 6534.2.a.cs.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6534,2,Mod(1,6534)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6534, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6534.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6534 = 2 \cdot 3^{3} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6534.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:	$52.1742526807$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.4752.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 3x^{2} + 4x + 1 $ x^4 - 2x^3 - 3x^2 + 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.21969$ of defining polynomial
Character	$\chi$	$=$	6534.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -2.11256 q^{5} -4.03957 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -2.11256 q^{5} -4.03957 q^{7} +1.00000 q^{8} -2.11256 q^{10} +3.15213 q^{13} -4.03957 q^{14} +1.00000 q^{16} +4.88418 q^{19} -2.11256 q^{20} -0.887440 q^{23} -0.537092 q^{25} +3.15213 q^{26} -4.03957 q^{28} +4.80059 q^{29} -2.65906 q^{31} +1.00000 q^{32} +8.53383 q^{35} +3.19615 q^{37} +4.88418 q^{38} -2.11256 q^{40} +6.99674 q^{41} -12.9967 q^{43} -0.887440 q^{46} -1.85847 q^{47} +9.31812 q^{49} -0.537092 q^{50} +3.15213 q^{52} +14.3149 q^{53} -4.03957 q^{56} +4.80059 q^{58} -8.23127 q^{59} -7.22512 q^{61} -2.65906 q^{62} +1.00000 q^{64} -6.65906 q^{65} +2.80059 q^{67} +8.53383 q^{70} -10.2251 q^{71} -7.81564 q^{73} +3.19615 q^{74} +4.88418 q^{76} -10.4201 q^{79} -2.11256 q^{80} +6.99674 q^{82} -16.6558 q^{83} -12.9967 q^{86} -13.3087 q^{89} -12.7332 q^{91} -0.887440 q^{92} -1.85847 q^{94} -10.3181 q^{95} +3.31812 q^{97} +9.31812 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{4} - 6 q^{7} + 4 q^{8}+O(q^{10}) $ 4 * q + 4 * q^2 + 4 * q^4 - 6 * q^7 + 4 * q^8	$ 4 q + 4 q^{2} + 4 q^{4} - 6 q^{7} + 4 q^{8} - 6 q^{13} - 6 q^{14} + 4 q^{16} - 6 q^{19} - 12 q^{23} + 10 q^{25} - 6 q^{26} - 6 q^{28} + 6 q^{29} - 2 q^{31} + 4 q^{32} - 12 q^{35} - 8 q^{37} - 6 q^{38} - 6 q^{41} - 18 q^{43} - 12 q^{46} - 12 q^{47} + 20 q^{49} + 10 q^{50} - 6 q^{52} + 6 q^{53} - 6 q^{56} + 6 q^{58} + 6 q^{59} - 12 q^{61} - 2 q^{62} + 4 q^{64} - 18 q^{65} - 2 q^{67} - 12 q^{70} - 24 q^{71} - 12 q^{73} - 8 q^{74} - 6 q^{76} - 30 q^{79} - 6 q^{82} - 24 q^{83} - 18 q^{86} - 24 q^{89} - 18 q^{91} - 12 q^{92} - 12 q^{94} - 24 q^{95} - 4 q^{97} + 20 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^4 - 6 * q^7 + 4 * q^8 - 6 * q^13 - 6 * q^14 + 4 * q^16 - 6 * q^19 - 12 * q^23 + 10 * q^25 - 6 * q^26 - 6 * q^28 + 6 * q^29 - 2 * q^31 + 4 * q^32 - 12 * q^35 - 8 * q^37 - 6 * q^38 - 6 * q^41 - 18 * q^43 - 12 * q^46 - 12 * q^47 + 20 * q^49 + 10 * q^50 - 6 * q^52 + 6 * q^53 - 6 * q^56 + 6 * q^58 + 6 * q^59 - 12 * q^61 - 2 * q^62 + 4 * q^64 - 18 * q^65 - 2 * q^67 - 12 * q^70 - 24 * q^71 - 12 * q^73 - 8 * q^74 - 6 * q^76 - 30 * q^79 - 6 * q^82 - 24 * q^83 - 18 * q^86 - 24 * q^89 - 18 * q^91 - 12 * q^92 - 12 * q^94 - 24 * q^95 - 4 * q^97 + 20 * 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
$5$	−2.11256	−0.944765	−0.472383	−	0.881394i	$-0.656606\pi$
$5$	−2.11256	−0.944765	−0.472383	+	0.881394i	$0.656606\pi$
$6$	0	0
$7$	−4.03957	−1.52681	−0.763407	−	0.645918i	$-0.776475\pi$
$7$	−4.03957	−1.52681	−0.763407	+	0.645918i	$0.776475\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−2.11256	−0.668050
$11$	0	0
$11$	0	0
$12$	0	0
$13$	3.15213	0.874243	0.437122	−	0.899402i	$-0.355998\pi$
$13$	3.15213	0.874243	0.437122	+	0.899402i	$0.355998\pi$
$14$	−4.03957	−1.07962
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	4.88418	1.12051	0.560254	−	0.828321i	$-0.310704\pi$
$19$	4.88418	1.12051	0.560254	+	0.828321i	$0.310704\pi$
$20$	−2.11256	−0.472383
$21$	0	0
$22$	0	0
$23$	−0.887440	−0.185044	−0.0925221	−	0.995711i	$-0.529493\pi$
$23$	−0.887440	−0.185044	−0.0925221	+	0.995711i	$0.529493\pi$
$24$	0	0
$25$	−0.537092	−0.107418
$26$	3.15213	0.618183
$27$	0	0
$28$	−4.03957	−0.763407
$29$	4.80059	0.891447	0.445723	−	0.895171i	$-0.352947\pi$
$29$	4.80059	0.891447	0.445723	+	0.895171i	$0.352947\pi$
$30$	0	0
$31$	−2.65906	−0.477581	−0.238791	−	0.971071i	$-0.576751\pi$
$31$	−2.65906	−0.477581	−0.238791	+	0.971071i	$0.576751\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0	0
$35$	8.53383	1.44248
$36$	0	0
$37$	3.19615	0.525444	0.262722	−	0.964872i	$-0.415380\pi$
$37$	3.19615	0.525444	0.262722	+	0.964872i	$0.415380\pi$
$38$	4.88418	0.792319
$39$	0	0
$40$	−2.11256	−0.334025
$41$	6.99674	1.09271	0.546354	−	0.837555i	$-0.316015\pi$
$41$	6.99674	1.09271	0.546354	+	0.837555i	$0.316015\pi$
$42$	0	0
$43$	−12.9967	−1.98198	−0.990992	−	0.133920i	$-0.957243\pi$
$43$	−12.9967	−1.98198	−0.990992	+	0.133920i	$0.957243\pi$
$44$	0	0
$45$	0	0
$46$	−0.887440	−0.130846
$47$	−1.85847	−0.271086	−0.135543	−	0.990771i	$-0.543278\pi$
$47$	−1.85847	−0.271086	−0.135543	+	0.990771i	$0.543278\pi$
$48$	0	0
$49$	9.31812	1.33116
$50$	−0.537092	−0.0759563
$51$	0	0
$52$	3.15213	0.437122
$53$	14.3149	1.96630	0.983149	−	0.182808i	$-0.0585186\pi$
$53$	14.3149	1.96630	0.983149	+	0.182808i	$0.0585186\pi$
$54$	0	0
$55$	0	0
$56$	−4.03957	−0.539810
$57$	0	0
$58$	4.80059	0.630348
$59$	−8.23127	−1.07162	−0.535810	−	0.844339i	$-0.679994\pi$
$59$	−8.23127	−1.07162	−0.535810	+	0.844339i	$0.679994\pi$
$60$	0	0
$61$	−7.22512	−0.925082	−0.462541	−	0.886598i	$-0.653062\pi$
$61$	−7.22512	−0.925082	−0.462541	+	0.886598i	$0.653062\pi$
$62$	−2.65906	−0.337701
$63$	0	0
$64$	1.00000	0.125000
$65$	−6.65906	−0.825955
$66$	0	0
$67$	2.80059	0.342146	0.171073	−	0.985258i	$-0.445277\pi$
$67$	2.80059	0.342146	0.171073	+	0.985258i	$0.445277\pi$
$68$	0	0
$69$	0	0
$70$	8.53383	1.01999
$71$	−10.2251	−1.21350	−0.606749	−	0.794893i	$-0.707527\pi$
$71$	−10.2251	−1.21350	−0.606749	+	0.794893i	$0.707527\pi$
$72$	0	0
$73$	−7.81564	−0.914752	−0.457376	−	0.889273i	$-0.651211\pi$
$73$	−7.81564	−0.914752	−0.457376	+	0.889273i	$0.651211\pi$
$74$	3.19615	0.371545
$75$	0	0
$76$	4.88418	0.560254
$77$	0	0
$78$	0	0
$79$	−10.4201	−1.17235	−0.586175	−	0.810184i	$-0.699367\pi$
$79$	−10.4201	−1.17235	−0.586175	+	0.810184i	$0.699367\pi$
$80$	−2.11256	−0.236191
$81$	0	0
$82$	6.99674	0.772661
$83$	−16.6558	−1.82821	−0.914106	−	0.405476i	$-0.867106\pi$
$83$	−16.6558	−1.82821	−0.914106	+	0.405476i	$0.867106\pi$
$84$	0	0
$85$	0	0
$86$	−12.9967	−1.40147
$87$	0	0
$88$	0	0
$89$	−13.3087	−1.41072	−0.705360	−	0.708849i	$-0.749215\pi$
$89$	−13.3087	−1.41072	−0.705360	+	0.708849i	$0.749215\pi$
$90$	0	0
$91$	−12.7332	−1.33481
$92$	−0.887440	−0.0925221
$93$	0	0
$94$	−1.85847	−0.191687
$95$	−10.3181	−1.05862
$96$	0	0
$97$	3.31812	0.336904	0.168452	−	0.985710i	$-0.446123\pi$
$97$	3.31812	0.336904	0.168452	+	0.985710i	$0.446123\pi$
$98$	9.31812	0.941272
$99$	0	0
$100$	−0.537092	−0.0537092
$101$	16.3923	1.63110	0.815548	−	0.578690i	$-0.196436\pi$
$101$	16.3923	1.63110	0.815548	+	0.578690i	$0.196436\pi$
$102$	0	0
$103$	12.7137	1.25272	0.626358	−	0.779535i	$-0.284545\pi$
$103$	12.7137	1.25272	0.626358	+	0.779535i	$0.284545\pi$
$104$	3.15213	0.309092
$105$	0	0
$106$	14.3149	1.39038
$107$	−15.3116	−1.48023	−0.740114	−	0.672481i	$-0.765229\pi$
$107$	−15.3116	−1.48023	−0.740114	+	0.672481i	$0.765229\pi$
$108$	0	0
$109$	−17.3911	−1.66577	−0.832883	−	0.553449i	$-0.813311\pi$
$109$	−17.3911	−1.66577	−0.832883	+	0.553449i	$0.813311\pi$
$110$	0	0
$111$	0	0
$112$	−4.03957	−0.381703
$113$	−3.03512	−0.285520	−0.142760	−	0.989757i	$-0.545598\pi$
$113$	−3.03512	−0.285520	−0.142760	+	0.989757i	$0.545598\pi$
$114$	0	0
$115$	1.87477	0.174823
$116$	4.80059	0.445723
$117$	0	0
$118$	−8.23127	−0.757750
$119$	0	0
$120$	0	0
$121$	0	0
$122$	−7.22512	−0.654132
$123$	0	0
$124$	−2.65906	−0.238791
$125$	11.6974	1.04625
$126$	0	0
$127$	16.7060	1.48242	0.741208	−	0.671276i	$-0.234253\pi$
$127$	16.7060	1.48242	0.741208	+	0.671276i	$0.234253\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−6.65906	−0.584038
$131$	5.19615	0.453990	0.226995	−	0.973896i	$-0.427110\pi$
$131$	5.19615	0.453990	0.226995	+	0.973896i	$0.427110\pi$
$132$	0	0
$133$	−19.7300	−1.71081
$134$	2.80059	0.241934
$135$	0	0
$136$	0	0
$137$	1.30871	0.111811	0.0559054	−	0.998436i	$-0.482195\pi$
$137$	1.30871	0.111811	0.0559054	+	0.998436i	$0.482195\pi$
$138$	0	0
$139$	3.69982	0.313815	0.156907	−	0.987613i	$-0.449848\pi$
$139$	3.69982	0.313815	0.156907	+	0.987613i	$0.449848\pi$
$140$	8.53383	0.721240
$141$	0	0
$142$	−10.2251	−0.858073
$143$	0	0
$144$	0	0
$145$	−10.1415	−0.842208
$146$	−7.81564	−0.646827
$147$	0	0
$148$	3.19615	0.262722
$149$	−12.8520	−1.05287	−0.526436	−	0.850214i	$-0.676472\pi$
$149$	−12.8520	−1.05287	−0.526436	+	0.850214i	$0.676472\pi$
$150$	0	0
$151$	−13.8079	−1.12367	−0.561837	−	0.827248i	$-0.689905\pi$
$151$	−13.8079	−1.12367	−0.561837	+	0.827248i	$0.689905\pi$
$152$	4.88418	0.396159
$153$	0	0
$154$	0	0
$155$	5.61742	0.451202
$156$	0	0
$157$	−22.5078	−1.79631	−0.898157	−	0.439675i	$-0.855094\pi$
$157$	−22.5078	−1.79631	−0.898157	+	0.439675i	$0.855094\pi$
$158$	−10.4201	−0.828977
$159$	0	0
$160$	−2.11256	−0.167013
$161$	3.58488	0.282528
$162$	0	0
$163$	5.65906	0.443252	0.221626	−	0.975132i	$-0.428864\pi$
$163$	5.65906	0.443252	0.221626	+	0.975132i	$0.428864\pi$
$164$	6.99674	0.546354
$165$	0	0
$166$	−16.6558	−1.29274
$167$	−11.0641	−0.856164	−0.428082	−	0.903740i	$-0.640811\pi$
$167$	−11.0641	−0.856164	−0.428082	+	0.903740i	$0.640811\pi$
$168$	0	0
$169$	−3.06408	−0.235699
$170$	0	0
$171$	0	0
$172$	−12.9967	−0.990992
$173$	0.118708	0.00902523	0.00451261	−	0.999990i	$-0.498564\pi$
$173$	0.118708	0.00902523	0.00451261	+	0.999990i	$0.498564\pi$
$174$	0	0
$175$	2.16962	0.164008
$176$	0	0
$177$	0	0
$178$	−13.3087	−0.997530
$179$	24.6236	1.84045	0.920226	−	0.391387i	$-0.128005\pi$
$179$	24.6236	1.84045	0.920226	+	0.391387i	$0.128005\pi$
$180$	0	0
$181$	−24.2570	−1.80301	−0.901503	−	0.432772i	$-0.857535\pi$
$181$	−24.2570	−1.80301	−0.901503	+	0.432772i	$0.857535\pi$
$182$	−12.7332	−0.943851
$183$	0	0
$184$	−0.887440	−0.0654230
$185$	−6.75206	−0.496422
$186$	0	0
$187$	0	0
$188$	−1.85847	−0.135543
$189$	0	0
$190$	−10.3181	−0.748555
$191$	16.3087	1.18006	0.590028	−	0.807382i	$-0.299116\pi$
$191$	16.3087	1.18006	0.590028	+	0.807382i	$0.299116\pi$
$192$	0	0
$193$	8.36665	0.602244	0.301122	−	0.953586i	$-0.402639\pi$
$193$	8.36665	0.602244	0.301122	+	0.953586i	$0.402639\pi$
$194$	3.31812	0.238227
$195$	0	0
$196$	9.31812	0.665580
$197$	−15.3116	−1.09091	−0.545453	−	0.838141i	$-0.683642\pi$
$197$	−15.3116	−1.09091	−0.545453	+	0.838141i	$0.683642\pi$
$198$	0	0
$199$	−13.7104	−0.971906	−0.485953	−	0.873985i	$-0.661527\pi$
$199$	−13.7104	−0.971906	−0.485953	+	0.873985i	$0.661527\pi$
$200$	−0.537092	−0.0379781
$201$	0	0
$202$	16.3923	1.15336
$203$	−19.3923	−1.36107
$204$	0	0
$205$	−14.7810	−1.03235
$206$	12.7137	0.885805
$207$	0	0
$208$	3.15213	0.218561
$209$	0	0
$210$	0	0
$211$	−26.9388	−1.85454	−0.927272	−	0.374389i	$-0.877853\pi$
$211$	−26.9388	−1.85454	−0.927272	+	0.374389i	$0.877853\pi$
$212$	14.3149	0.983149
$213$	0	0
$214$	−15.3116	−1.04668
$215$	27.4564	1.87251
$216$	0	0
$217$	10.7415	0.729178
$218$	−17.3911	−1.17787
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−11.6558	−0.780530	−0.390265	−	0.920703i	$-0.627617\pi$
$223$	−11.6558	−0.780530	−0.390265	+	0.920703i	$0.627617\pi$
$224$	−4.03957	−0.269905
$225$	0	0
$226$	−3.03512	−0.201893
$227$	−19.4531	−1.29115	−0.645575	−	0.763697i	$-0.723382\pi$
$227$	−19.4531	−1.29115	−0.645575	+	0.763697i	$0.723382\pi$
$228$	0	0
$229$	−8.77777	−0.580051	−0.290026	−	0.957019i	$-0.593664\pi$
$229$	−8.77777	−0.580051	−0.290026	+	0.957019i	$0.593664\pi$
$230$	1.87477	0.123619
$231$	0	0
$232$	4.80059	0.315174
$233$	18.3695	1.20343	0.601713	−	0.798713i	$-0.294485\pi$
$233$	18.3695	1.20343	0.601713	+	0.798713i	$0.294485\pi$
$234$	0	0
$235$	3.92614	0.256113
$236$	−8.23127	−0.535810
$237$	0	0
$238$	0	0
$239$	−6.07418	−0.392906	−0.196453	−	0.980513i	$-0.562942\pi$
$239$	−6.07418	−0.392906	−0.196453	+	0.980513i	$0.562942\pi$
$240$	0	0
$241$	1.62238	0.104507	0.0522533	−	0.998634i	$-0.483360\pi$
$241$	1.62238	0.104507	0.0522533	+	0.998634i	$0.483360\pi$
$242$	0	0
$243$	0	0
$244$	−7.22512	−0.462541
$245$	−19.6851	−1.25763
$246$	0	0
$247$	15.3956	0.979596
$248$	−2.65906	−0.168851
$249$	0	0
$250$	11.6974	0.739811
$251$	−7.68151	−0.484852	−0.242426	−	0.970170i	$-0.577943\pi$
$251$	−7.68151	−0.484852	−0.242426	+	0.970170i	$0.577943\pi$
$252$	0	0
$253$	0	0
$254$	16.7060	1.04823
$255$	0	0
$256$	1.00000	0.0625000
$257$	24.7941	1.54661	0.773306	−	0.634033i	$-0.218602\pi$
$257$	24.7941	1.54661	0.773306	+	0.634033i	$0.218602\pi$
$258$	0	0
$259$	−12.9111	−0.802255
$260$	−6.65906	−0.412977
$261$	0	0
$262$	5.19615	0.321019
$263$	1.31812	0.0812788	0.0406394	−	0.999174i	$-0.487061\pi$
$263$	1.31812	0.0812788	0.0406394	+	0.999174i	$0.487061\pi$
$264$	0	0
$265$	−30.2410	−1.85769
$266$	−19.7300	−1.20972
$267$	0	0
$268$	2.80059	0.171073
$269$	−7.97718	−0.486377	−0.243189	−	0.969979i	$-0.578193\pi$
$269$	−7.97718	−0.486377	−0.243189	+	0.969979i	$0.578193\pi$
$270$	0	0
$271$	−8.11821	−0.493146	−0.246573	−	0.969124i	$-0.579305\pi$
$271$	−8.11821	−0.493146	−0.246573	+	0.969124i	$0.579305\pi$
$272$	0	0
$273$	0	0
$274$	1.30871	0.0790622
$275$	0	0
$276$	0	0
$277$	−3.82786	−0.229994	−0.114997	−	0.993366i	$-0.536686\pi$
$277$	−3.82786	−0.229994	−0.114997	+	0.993366i	$0.536686\pi$
$278$	3.69982	0.221901
$279$	0	0
$280$	8.53383	0.509994
$281$	−3.34746	−0.199693	−0.0998464	−	0.995003i	$-0.531835\pi$
$281$	−3.34746	−0.199693	−0.0998464	+	0.995003i	$0.531835\pi$
$282$	0	0
$283$	−1.92494	−0.114426	−0.0572130	−	0.998362i	$-0.518221\pi$
$283$	−1.92494	−0.114426	−0.0572130	+	0.998362i	$0.518221\pi$
$284$	−10.2251	−0.606749
$285$	0	0
$286$	0	0
$287$	−28.2638	−1.66836
$288$	0	0
$289$	−17.0000	−1.00000
$290$	−10.1415	−0.595531
$291$	0	0
$292$	−7.81564	−0.457376
$293$	−6.61454	−0.386425	−0.193213	−	0.981157i	$-0.561891\pi$
$293$	−6.61454	−0.386425	−0.193213	+	0.981157i	$0.561891\pi$
$294$	0	0
$295$	17.3890	1.01243
$296$	3.19615	0.185773
$297$	0	0
$298$	−12.8520	−0.744494
$299$	−2.79733	−0.161774
$300$	0	0
$301$	52.5012	3.02612
$302$	−13.8079	−0.794557
$303$	0	0
$304$	4.88418	0.280127
$305$	15.2635	0.873985
$306$	0	0
$307$	−11.8792	−0.677983	−0.338992	−	0.940789i	$-0.610086\pi$
$307$	−11.8792	−0.677983	−0.338992	+	0.940789i	$0.610086\pi$
$308$	0	0
$309$	0	0
$310$	5.61742	0.319048
$311$	27.8259	1.57786	0.788930	−	0.614483i	$-0.210635\pi$
$311$	27.8259	1.57786	0.788930	+	0.614483i	$0.210635\pi$
$312$	0	0
$313$	−3.46291	−0.195735	−0.0978676	−	0.995199i	$-0.531202\pi$
$313$	−3.46291	−0.195735	−0.0978676	+	0.995199i	$0.531202\pi$
$314$	−22.5078	−1.27019
$315$	0	0
$316$	−10.4201	−0.586175
$317$	4.05793	0.227916	0.113958	−	0.993486i	$-0.463647\pi$
$317$	4.05793	0.227916	0.113958	+	0.993486i	$0.463647\pi$
$318$	0	0
$319$	0	0
$320$	−2.11256	−0.118096
$321$	0	0
$322$	3.58488	0.199777
$323$	0	0
$324$	0	0
$325$	−1.69298	−0.0939098
$326$	5.65906	0.313426
$327$	0	0
$328$	6.99674	0.386330
$329$	7.50743	0.413898
$330$	0	0
$331$	−16.9098	−0.929449	−0.464724	−	0.885455i	$-0.653847\pi$
$331$	−16.9098	−0.929449	−0.464724	+	0.885455i	$0.653847\pi$
$332$	−16.6558	−0.914106
$333$	0	0
$334$	−11.0641	−0.605400
$335$	−5.91641	−0.323248
$336$	0	0
$337$	−2.30187	−0.125391	−0.0626955	−	0.998033i	$-0.519970\pi$
$337$	−2.30187	−0.125391	−0.0626955	+	0.998033i	$0.519970\pi$
$338$	−3.06408	−0.166664
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−9.36421	−0.505620
$344$	−12.9967	−0.700737
$345$	0	0
$346$	0.118708	0.00638180
$347$	18.2508	0.979753	0.489877	−	0.871792i	$-0.337042\pi$
$347$	18.2508	0.979753	0.489877	+	0.871792i	$0.337042\pi$
$348$	0	0
$349$	16.1545	0.864731	0.432366	−	0.901698i	$-0.357679\pi$
$349$	16.1545	0.864731	0.432366	+	0.901698i	$0.357679\pi$
$350$	2.16962	0.115971
$351$	0	0
$352$	0	0
$353$	22.5595	1.20072	0.600360	−	0.799730i	$-0.295024\pi$
$353$	22.5595	1.20072	0.600360	+	0.799730i	$0.295024\pi$
$354$	0	0
$355$	21.6012	1.14647
$356$	−13.3087	−0.705360
$357$	0	0
$358$	24.6236	1.30140
$359$	−6.60117	−0.348397	−0.174198	−	0.984711i	$-0.555733\pi$
$359$	−6.60117	−0.348397	−0.174198	+	0.984711i	$0.555733\pi$
$360$	0	0
$361$	4.85521	0.255538
$362$	−24.2570	−1.27492
$363$	0	0
$364$	−12.7332	−0.667403
$365$	16.5110	0.864226
$366$	0	0
$367$	10.7365	0.560441	0.280220	−	0.959936i	$-0.409592\pi$
$367$	10.7365	0.560441	0.280220	+	0.959936i	$0.409592\pi$
$368$	−0.887440	−0.0462610
$369$	0	0
$370$	−6.75206	−0.351023
$371$	−57.8259	−3.00217
$372$	0	0
$373$	−8.89928	−0.460787	−0.230394	−	0.973097i	$-0.574001\pi$
$373$	−8.89928	−0.460787	−0.230394	+	0.973097i	$0.574001\pi$
$374$	0	0
$375$	0	0
$376$	−1.85847	−0.0958435
$377$	15.1321	0.779341
$378$	0	0
$379$	18.4531	0.947873	0.473937	−	0.880559i	$-0.342833\pi$
$379$	18.4531	0.947873	0.473937	+	0.880559i	$0.342833\pi$
$380$	−10.3181	−0.529308
$381$	0	0
$382$	16.3087	0.834426
$383$	8.57873	0.438353	0.219176	−	0.975685i	$-0.429663\pi$
$383$	8.57873	0.438353	0.219176	+	0.975685i	$0.429663\pi$
$384$	0	0
$385$	0	0
$386$	8.36665	0.425851
$387$	0	0
$388$	3.31812	0.168452
$389$	−0.372794	−0.0189014	−0.00945071	−	0.999955i	$-0.503008\pi$
$389$	−0.372794	−0.0189014	−0.00945071	+	0.999955i	$0.503008\pi$
$390$	0	0
$391$	0	0
$392$	9.31812	0.470636
$393$	0	0
$394$	−15.3116	−0.771387
$395$	22.0130	1.10760
$396$	0	0
$397$	3.84511	0.192981	0.0964903	−	0.995334i	$-0.469238\pi$
$397$	3.84511	0.192981	0.0964903	+	0.995334i	$0.469238\pi$
$398$	−13.7104	−0.687242
$399$	0	0
$400$	−0.537092	−0.0268546
$401$	30.3279	1.51450	0.757251	−	0.653123i	$-0.226542\pi$
$401$	30.3279	1.51450	0.757251	+	0.653123i	$0.226542\pi$
$402$	0	0
$403$	−8.38170	−0.417522
$404$	16.3923	0.815548
$405$	0	0
$406$	−19.3923	−0.962424
$407$	0	0
$408$	0	0
$409$	−32.0771	−1.58611	−0.793055	−	0.609151i	$-0.791510\pi$
$409$	−32.0771	−1.58611	−0.793055	+	0.609151i	$0.791510\pi$
$410$	−14.7810	−0.729983
$411$	0	0
$412$	12.7137	0.626358
$413$	33.2508	1.63616
$414$	0	0
$415$	35.1864	1.72723
$416$	3.15213	0.154546
$417$	0	0
$418$	0	0
$419$	9.67536	0.472672	0.236336	−	0.971671i	$-0.424053\pi$
$419$	9.67536	0.472672	0.236336	+	0.971671i	$0.424053\pi$
$420$	0	0
$421$	23.7166	1.15588	0.577939	−	0.816080i	$-0.303857\pi$
$421$	23.7166	1.15588	0.577939	+	0.816080i	$0.303857\pi$
$422$	−26.9388	−1.31136
$423$	0	0
$424$	14.3149	0.695191
$425$	0	0
$426$	0	0
$427$	29.1864	1.41243
$428$	−15.3116	−0.740114
$429$	0	0
$430$	27.4564	1.32406
$431$	−19.9935	−0.963052	−0.481526	−	0.876432i	$-0.659917\pi$
$431$	−19.9935	−0.963052	−0.481526	+	0.876432i	$0.659917\pi$
$432$	0	0
$433$	4.85521	0.233327	0.116663	−	0.993172i	$-0.462780\pi$
$433$	4.85521	0.233327	0.116663	+	0.993172i	$0.462780\pi$
$434$	10.7415	0.515607
$435$	0	0
$436$	−17.3911	−0.832883
$437$	−4.33442	−0.207343
$438$	0	0
$439$	−17.8659	−0.852691	−0.426346	−	0.904560i	$-0.640199\pi$
$439$	−17.8659	−0.852691	−0.426346	+	0.904560i	$0.640199\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−28.5595	−1.35690	−0.678451	−	0.734645i	$-0.737349\pi$
$443$	−28.5595	−1.35690	−0.678451	+	0.734645i	$0.737349\pi$
$444$	0	0
$445$	28.1154	1.33280
$446$	−11.6558	−0.551918
$447$	0	0
$448$	−4.03957	−0.190852
$449$	6.21566	0.293335	0.146668	−	0.989186i	$-0.453145\pi$
$449$	6.21566	0.293335	0.146668	+	0.989186i	$0.453145\pi$
$450$	0	0
$451$	0	0
$452$	−3.03512	−0.142760
$453$	0	0
$454$	−19.4531	−0.912980
$455$	26.8997	1.26108
$456$	0	0
$457$	−21.4274	−1.00233	−0.501165	−	0.865352i	$-0.667095\pi$
$457$	−21.4274	−1.00233	−0.501165	+	0.865352i	$0.667095\pi$
$458$	−8.77777	−0.410158
$459$	0	0
$460$	1.87477	0.0874116
$461$	−10.2736	−0.478489	−0.239245	−	0.970959i	$-0.576900\pi$
$461$	−10.2736	−0.478489	−0.239245	+	0.970959i	$0.576900\pi$
$462$	0	0
$463$	14.8647	0.690820	0.345410	−	0.938452i	$-0.387740\pi$
$463$	14.8647	0.690820	0.345410	+	0.938452i	$0.387740\pi$
$464$	4.80059	0.222862
$465$	0	0
$466$	18.3695	0.850950
$467$	13.6815	0.633105	0.316552	−	0.948575i	$-0.397475\pi$
$467$	13.6815	0.633105	0.316552	+	0.948575i	$0.397475\pi$
$468$	0	0
$469$	−11.3132	−0.522393
$470$	3.92614	0.181099
$471$	0	0
$472$	−8.23127	−0.378875
$473$	0	0
$474$	0	0
$475$	−2.62325	−0.120363
$476$	0	0
$477$	0	0
$478$	−6.07418	−0.277827
$479$	9.16681	0.418842	0.209421	−	0.977826i	$-0.432842\pi$
$479$	9.16681	0.418842	0.209421	+	0.977826i	$0.432842\pi$
$480$	0	0
$481$	10.0747	0.459366
$482$	1.62238	0.0738974
$483$	0	0
$484$	0	0
$485$	−7.00973	−0.318295
$486$	0	0
$487$	−19.3890	−0.878601	−0.439301	−	0.898340i	$-0.644774\pi$
$487$	−19.3890	−0.878601	−0.439301	+	0.898340i	$0.644774\pi$
$488$	−7.22512	−0.327066
$489$	0	0
$490$	−19.6851	−0.889282
$491$	−30.7517	−1.38780	−0.693902	−	0.720069i	$-0.744110\pi$
$491$	−30.7517	−1.38780	−0.693902	+	0.720069i	$0.744110\pi$
$492$	0	0
$493$	0	0
$494$	15.3956	0.692679
$495$	0	0
$496$	−2.65906	−0.119395
$497$	41.3051	1.85279
$498$	0	0
$499$	−20.3561	−0.911265	−0.455633	−	0.890168i	$-0.650587\pi$
$499$	−20.3561	−0.911265	−0.455633	+	0.890168i	$0.650587\pi$
$500$	11.6974	0.523125
$501$	0	0
$502$	−7.68151	−0.342842
$503$	23.6653	1.05518	0.527591	−	0.849499i	$-0.323095\pi$
$503$	23.6653	1.05518	0.527591	+	0.849499i	$0.323095\pi$
$504$	0	0
$505$	−34.6297	−1.54100
$506$	0	0
$507$	0	0
$508$	16.7060	0.741208
$509$	30.3728	1.34625	0.673125	−	0.739529i	$-0.264951\pi$
$509$	30.3728	1.34625	0.673125	+	0.739529i	$0.264951\pi$
$510$	0	0
$511$	31.5718	1.39666
$512$	1.00000	0.0441942
$513$	0	0
$514$	24.7941	1.09362
$515$	−26.8584	−1.18352
$516$	0	0
$517$	0	0
$518$	−12.9111	−0.567280
$519$	0	0
$520$	−6.65906	−0.292019
$521$	−37.6648	−1.65013	−0.825063	−	0.565040i	$-0.808861\pi$
$521$	−37.6648	−1.65013	−0.825063	+	0.565040i	$0.808861\pi$
$522$	0	0
$523$	42.7423	1.86899	0.934495	−	0.355977i	$-0.115852\pi$
$523$	42.7423	1.86899	0.934495	+	0.355977i	$0.115852\pi$
$524$	5.19615	0.226995
$525$	0	0
$526$	1.31812	0.0574728
$527$	0	0
$528$	0	0
$529$	−22.2124	−0.965759
$530$	−30.2410	−1.31358
$531$	0	0
$532$	−19.7300	−0.855403
$533$	22.0546	0.955292
$534$	0	0
$535$	32.3467	1.39847
$536$	2.80059	0.120967
$537$	0	0
$538$	−7.97718	−0.343921
$539$	0	0
$540$	0	0
$541$	31.4070	1.35029	0.675146	−	0.737684i	$-0.264081\pi$
$541$	31.4070	1.35029	0.675146	+	0.737684i	$0.264081\pi$
$542$	−8.11821	−0.348707
$543$	0	0
$544$	0	0
$545$	36.7398	1.57376
$546$	0	0
$547$	−2.36541	−0.101137	−0.0505687	−	0.998721i	$-0.516103\pi$
$547$	−2.36541	−0.101137	−0.0505687	+	0.998721i	$0.516103\pi$
$548$	1.30871	0.0559054
$549$	0	0
$550$	0	0
$551$	23.4469	0.998873
$552$	0	0
$553$	42.0926	1.78996
$554$	−3.82786	−0.162630
$555$	0	0
$556$	3.69982	0.156907
$557$	24.3279	1.03081	0.515403	−	0.856948i	$-0.327642\pi$
$557$	24.3279	1.03081	0.515403	+	0.856948i	$0.327642\pi$
$558$	0	0
$559$	−40.9674	−1.73274
$560$	8.53383	0.360620
$561$	0	0
$562$	−3.34746	−0.141204
$563$	16.2765	0.685974	0.342987	−	0.939340i	$-0.388561\pi$
$563$	16.2765	0.685974	0.342987	+	0.939340i	$0.388561\pi$
$564$	0	0
$565$	6.41186	0.269749
$566$	−1.92494	−0.0809114
$567$	0	0
$568$	−10.2251	−0.429036
$569$	−35.3890	−1.48359	−0.741793	−	0.670629i	$-0.766024\pi$
$569$	−35.3890	−1.48359	−0.741793	+	0.670629i	$0.766024\pi$
$570$	0	0
$571$	−22.8319	−0.955487	−0.477743	−	0.878499i	$-0.658545\pi$
$571$	−22.8319	−0.955487	−0.477743	+	0.878499i	$0.658545\pi$
$572$	0	0
$573$	0	0
$574$	−28.2638	−1.17971
$575$	0.476637	0.0198771
$576$	0	0
$577$	−14.6754	−0.610943	−0.305472	−	0.952201i	$-0.598814\pi$
$577$	−14.6754	−0.610943	−0.305472	+	0.952201i	$0.598814\pi$
$578$	−17.0000	−0.707107
$579$	0	0
$580$	−10.1415	−0.421104
$581$	67.2823	2.79134
$582$	0	0
$583$	0	0
$584$	−7.81564	−0.323414
$585$	0	0
$586$	−6.61454	−0.273244
$587$	38.5240	1.59006	0.795029	−	0.606572i	$-0.207456\pi$
$587$	38.5240	1.59006	0.795029	+	0.606572i	$0.207456\pi$
$588$	0	0
$589$	−12.9873	−0.535134
$590$	17.3890	0.715896
$591$	0	0
$592$	3.19615	0.131361
$593$	−3.52764	−0.144863	−0.0724313	−	0.997373i	$-0.523076\pi$
$593$	−3.52764	−0.144863	−0.0724313	+	0.997373i	$0.523076\pi$
$594$	0	0
$595$	0	0
$596$	−12.8520	−0.526436
$597$	0	0
$598$	−2.79733	−0.114391
$599$	−29.0543	−1.18713	−0.593563	−	0.804787i	$-0.702279\pi$
$599$	−29.0543	−1.18713	−0.593563	+	0.804787i	$0.702279\pi$
$600$	0	0
$601$	−3.48811	−0.142283	−0.0711416	−	0.997466i	$-0.522664\pi$
$601$	−3.48811	−0.142283	−0.0711416	+	0.997466i	$0.522664\pi$
$602$	52.5012	2.13979
$603$	0	0
$604$	−13.8079	−0.561837
$605$	0	0
$606$	0	0
$607$	−15.6129	−0.633709	−0.316854	−	0.948474i	$-0.602627\pi$
$607$	−15.6129	−0.633709	−0.316854	+	0.948474i	$0.602627\pi$
$608$	4.88418	0.198080
$609$	0	0
$610$	15.2635	0.618001
$611$	−5.85815	−0.236995
$612$	0	0
$613$	−34.5441	−1.39522	−0.697611	−	0.716477i	$-0.745753\pi$
$613$	−34.5441	−1.39522	−0.697611	+	0.716477i	$0.745753\pi$
$614$	−11.8792	−0.479406
$615$	0	0
$616$	0	0
$617$	2.74922	0.110679	0.0553397	−	0.998468i	$-0.482376\pi$
$617$	2.74922	0.110679	0.0553397	+	0.998468i	$0.482376\pi$
$618$	0	0
$619$	−6.57836	−0.264406	−0.132203	−	0.991223i	$-0.542205\pi$
$619$	−6.57836	−0.264406	−0.132203	+	0.991223i	$0.542205\pi$
$620$	5.61742	0.225601
$621$	0	0
$622$	27.8259	1.11572
$623$	53.7615	2.15391
$624$	0	0
$625$	−22.0261	−0.881043
$626$	−3.46291	−0.138406
$627$	0	0
$628$	−22.5078	−0.898157
$629$	0	0
$630$	0	0
$631$	18.2309	0.725760	0.362880	−	0.931836i	$-0.381793\pi$
$631$	18.2309	0.725760	0.362880	+	0.931836i	$0.381793\pi$
$632$	−10.4201	−0.414488
$633$	0	0
$634$	4.05793	0.161161
$635$	−35.2924	−1.40053
$636$	0	0
$637$	29.3719	1.16376
$638$	0	0
$639$	0	0
$640$	−2.11256	−0.0835063
$641$	−28.2120	−1.11431	−0.557154	−	0.830409i	$-0.688107\pi$
$641$	−28.2120	−1.11431	−0.557154	+	0.830409i	$0.688107\pi$
$642$	0	0
$643$	−5.55554	−0.219089	−0.109545	−	0.993982i	$-0.534939\pi$
$643$	−5.55554	−0.219089	−0.109545	+	0.993982i	$0.534939\pi$
$644$	3.58488	0.141264
$645$	0	0
$646$	0	0
$647$	29.5305	1.16096	0.580482	−	0.814273i	$-0.302864\pi$
$647$	29.5305	1.16096	0.580482	+	0.814273i	$0.302864\pi$
$648$	0	0
$649$	0	0
$650$	−1.69298	−0.0664042
$651$	0	0
$652$	5.65906	0.221626
$653$	−3.78718	−0.148204	−0.0741019	−	0.997251i	$-0.523609\pi$
$653$	−3.78718	−0.148204	−0.0741019	+	0.997251i	$0.523609\pi$
$654$	0	0
$655$	−10.9772	−0.428914
$656$	6.99674	0.273177
$657$	0	0
$658$	7.50743	0.292670
$659$	−25.7427	−1.00279	−0.501397	−	0.865218i	$-0.667180\pi$
$659$	−25.7427	−1.00279	−0.501397	+	0.865218i	$0.667180\pi$
$660$	0	0
$661$	9.31193	0.362192	0.181096	−	0.983465i	$-0.442036\pi$
$661$	9.31193	0.362192	0.181096	+	0.983465i	$0.442036\pi$
$662$	−16.9098	−0.657220
$663$	0	0
$664$	−16.6558	−0.646370
$665$	41.6808	1.61631
$666$	0	0
$667$	−4.26023	−0.164957
$668$	−11.0641	−0.428082
$669$	0	0
$670$	−5.91641	−0.228571
$671$	0	0
$672$	0	0
$673$	28.7480	1.10815	0.554077	−	0.832466i	$-0.313071\pi$
$673$	28.7480	1.10815	0.554077	+	0.832466i	$0.313071\pi$
$674$	−2.30187	−0.0886648
$675$	0	0
$676$	−3.06408	−0.117849
$677$	−35.5368	−1.36579	−0.682894	−	0.730517i	$-0.739279\pi$
$677$	−35.5368	−1.36579	−0.682894	+	0.730517i	$0.739279\pi$
$678$	0	0
$679$	−13.4038	−0.514390
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1.65286	−0.0632451	−0.0316226	−	0.999500i	$-0.510067\pi$
$683$	−1.65286	−0.0632451	−0.0316226	+	0.999500i	$0.510067\pi$
$684$	0	0
$685$	−2.76473	−0.105635
$686$	−9.36421	−0.357527
$687$	0	0
$688$	−12.9967	−0.495496
$689$	45.1223	1.71902
$690$	0	0
$691$	−14.5689	−0.554227	−0.277114	−	0.960837i	$-0.589378\pi$
$691$	−14.5689	−0.554227	−0.277114	+	0.960837i	$0.589378\pi$
$692$	0.118708	0.00451261
$693$	0	0
$694$	18.2508	0.692790
$695$	−7.81610	−0.296481
$696$	0	0
$697$	0	0
$698$	16.1545	0.611457
$699$	0	0
$700$	2.16962	0.0820039
$701$	38.6080	1.45820	0.729102	−	0.684405i	$-0.239938\pi$
$701$	38.6080	1.45820	0.729102	+	0.684405i	$0.239938\pi$
$702$	0	0
$703$	15.6106	0.588764
$704$	0	0
$705$	0	0
$706$	22.5595	0.849038
$707$	−66.2179	−2.49038
$708$	0	0
$709$	−14.1611	−0.531831	−0.265915	−	0.963996i	$-0.585674\pi$
$709$	−14.1611	−0.531831	−0.265915	+	0.963996i	$0.585674\pi$
$710$	21.6012	0.810678
$711$	0	0
$712$	−13.3087	−0.498765
$713$	2.35976	0.0883736
$714$	0	0
$715$	0	0
$716$	24.6236	0.920226
$717$	0	0
$718$	−6.60117	−0.246354
$719$	−3.75564	−0.140062	−0.0700309	−	0.997545i	$-0.522310\pi$
$719$	−3.75564	−0.140062	−0.0700309	+	0.997545i	$0.522310\pi$
$720$	0	0
$721$	−51.3578	−1.91267
$722$	4.85521	0.180692
$723$	0	0
$724$	−24.2570	−0.901503
$725$	−2.57836	−0.0957577
$726$	0	0
$727$	−2.52699	−0.0937209	−0.0468604	−	0.998901i	$-0.514922\pi$
$727$	−2.52699	−0.0937209	−0.0468604	+	0.998901i	$0.514922\pi$
$728$	−12.7332	−0.471925
$729$	0	0
$730$	16.5110	0.611100
$731$	0	0
$732$	0	0
$733$	17.9294	0.662237	0.331119	−	0.943589i	$-0.392574\pi$
$733$	17.9294	0.662237	0.331119	+	0.943589i	$0.392574\pi$
$734$	10.7365	0.396292
$735$	0	0
$736$	−0.887440	−0.0327115
$737$	0	0
$738$	0	0
$739$	−21.2594	−0.782038	−0.391019	−	0.920383i	$-0.627877\pi$
$739$	−21.2594	−0.782038	−0.391019	+	0.920383i	$0.627877\pi$
$740$	−6.75206	−0.248211
$741$	0	0
$742$	−57.8259	−2.12285
$743$	44.8031	1.64366	0.821832	−	0.569729i	$-0.192952\pi$
$743$	44.8031	1.64366	0.821832	+	0.569729i	$0.192952\pi$
$744$	0	0
$745$	27.1505	0.994718
$746$	−8.89928	−0.325826
$747$	0	0
$748$	0	0
$749$	61.8523	2.26003
$750$	0	0
$751$	9.59465	0.350114	0.175057	−	0.984558i	$-0.443989\pi$
$751$	9.59465	0.350114	0.175057	+	0.984558i	$0.443989\pi$
$752$	−1.85847	−0.0677716
$753$	0	0
$754$	15.1321	0.551077
$755$	29.1701	1.06161
$756$	0	0
$757$	35.8650	1.30354	0.651768	−	0.758419i	$-0.274028\pi$
$757$	35.8650	1.30354	0.651768	+	0.758419i	$0.274028\pi$
$758$	18.4531	0.670248
$759$	0	0
$760$	−10.3181	−0.374278
$761$	30.0644	1.08983	0.544917	−	0.838490i	$-0.316561\pi$
$761$	30.0644	1.08983	0.544917	+	0.838490i	$0.316561\pi$
$762$	0	0
$763$	70.2526	2.54331
$764$	16.3087	0.590028
$765$	0	0
$766$	8.57873	0.309962
$767$	−25.9460	−0.936856
$768$	0	0
$769$	48.1685	1.73700	0.868500	−	0.495688i	$-0.165084\pi$
$769$	48.1685	1.73700	0.868500	+	0.495688i	$0.165084\pi$
$770$	0	0
$771$	0	0
$772$	8.36665	0.301122
$773$	47.0995	1.69405	0.847025	−	0.531553i	$-0.178391\pi$
$773$	47.0995	1.69405	0.847025	+	0.531553i	$0.178391\pi$
$774$	0	0
$775$	1.42816	0.0513010
$776$	3.31812	0.119114
$777$	0	0
$778$	−0.372794	−0.0133653
$779$	34.1733	1.22439
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	9.31812	0.332790
$785$	47.5490	1.69710
$786$	0	0
$787$	−25.2195	−0.898977	−0.449489	−	0.893286i	$-0.648394\pi$
$787$	−25.2195	−0.898977	−0.449489	+	0.893286i	$0.648394\pi$
$788$	−15.3116	−0.545453
$789$	0	0
$790$	22.0130	0.783189
$791$	12.2606	0.435935
$792$	0	0
$793$	−22.7745	−0.808747
$794$	3.84511	0.136458
$795$	0	0
$796$	−13.7104	−0.485953
$797$	15.3826	0.544879	0.272439	−	0.962173i	$-0.412170\pi$
$797$	15.3826	0.544879	0.272439	+	0.962173i	$0.412170\pi$
$798$	0	0
$799$	0	0
$800$	−0.537092	−0.0189891
$801$	0	0
$802$	30.3279	1.07092
$803$	0	0
$804$	0	0
$805$	−7.57327	−0.266923
$806$	−8.38170	−0.295233
$807$	0	0
$808$	16.3923	0.576679
$809$	18.3214	0.644145	0.322073	−	0.946715i	$-0.395620\pi$
$809$	18.3214	0.644145	0.322073	+	0.946715i	$0.395620\pi$
$810$	0	0
$811$	−42.7122	−1.49983	−0.749914	−	0.661536i	$-0.769905\pi$
$811$	−42.7122	−1.49983	−0.749914	+	0.661536i	$0.769905\pi$
$812$	−19.3923	−0.680536
$813$	0	0
$814$	0	0
$815$	−11.9551	−0.418769
$816$	0	0
$817$	−63.4784	−2.22083
$818$	−32.0771	−1.12155
$819$	0	0
$820$	−14.7810	−0.516176
$821$	−48.6811	−1.69898	−0.849491	−	0.527603i	$-0.823091\pi$
$821$	−48.6811	−1.69898	−0.849491	+	0.527603i	$0.823091\pi$
$822$	0	0
$823$	52.8900	1.84363	0.921814	−	0.387633i	$-0.126707\pi$
$823$	52.8900	1.84363	0.921814	+	0.387633i	$0.126707\pi$
$824$	12.7137	0.442902
$825$	0	0
$826$	33.2508	1.15694
$827$	−50.3793	−1.75186	−0.875929	−	0.482439i	$-0.839751\pi$
$827$	−50.3793	−1.75186	−0.875929	+	0.482439i	$0.839751\pi$
$828$	0	0
$829$	−46.7256	−1.62285	−0.811424	−	0.584458i	$-0.801307\pi$
$829$	−46.7256	−1.62285	−0.811424	+	0.584458i	$0.801307\pi$
$830$	35.1864	1.22134
$831$	0	0
$832$	3.15213	0.109280
$833$	0	0
$834$	0	0
$835$	23.3735	0.808874
$836$	0	0
$837$	0	0
$838$	9.67536	0.334230
$839$	27.9161	0.963770	0.481885	−	0.876234i	$-0.339952\pi$
$839$	27.9161	0.963770	0.481885	+	0.876234i	$0.339952\pi$
$840$	0	0
$841$	−5.95436	−0.205323
$842$	23.7166	0.817329
$843$	0	0
$844$	−26.9388	−0.927272
$845$	6.47306	0.222680
$846$	0	0
$847$	0	0
$848$	14.3149	0.491574
$849$	0	0
$850$	0	0
$851$	−2.83639	−0.0972304
$852$	0	0
$853$	30.2815	1.03682	0.518409	−	0.855133i	$-0.326524\pi$
$853$	30.2815	1.03682	0.518409	+	0.855133i	$0.326524\pi$
$854$	29.1864	0.998737
$855$	0	0
$856$	−15.3116	−0.523340
$857$	−51.4755	−1.75837	−0.879184	−	0.476482i	$-0.841912\pi$
$857$	−51.4755	−1.75837	−0.879184	+	0.476482i	$0.841912\pi$
$858$	0	0
$859$	35.3413	1.20583	0.602914	−	0.797806i	$-0.294006\pi$
$859$	35.3413	1.20583	0.602914	+	0.797806i	$0.294006\pi$
$860$	27.4564	0.936255
$861$	0	0
$862$	−19.9935	−0.680981
$863$	−57.6974	−1.96404	−0.982021	−	0.188772i	$-0.939549\pi$
$863$	−57.6974	−1.96404	−0.982021	+	0.188772i	$0.939549\pi$
$864$	0	0
$865$	−0.250778	−0.00852672
$866$	4.85521	0.164987
$867$	0	0
$868$	10.7415	0.364589
$869$	0	0
$870$	0	0
$871$	8.82781	0.299119
$872$	−17.3911	−0.588937
$873$	0	0
$874$	−4.33442	−0.146614
$875$	−47.2526	−1.59743
$876$	0	0
$877$	−41.3279	−1.39554	−0.697771	−	0.716321i	$-0.745825\pi$
$877$	−41.3279	−1.39554	−0.697771	+	0.716321i	$0.745825\pi$
$878$	−17.8659	−0.602944
$879$	0	0
$880$	0	0
$881$	−42.7810	−1.44133	−0.720664	−	0.693285i	$-0.756163\pi$
$881$	−42.7810	−1.44133	−0.720664	+	0.693285i	$0.756163\pi$
$882$	0	0
$883$	−4.01598	−0.135148	−0.0675742	−	0.997714i	$-0.521526\pi$
$883$	−4.01598	−0.135148	−0.0675742	+	0.997714i	$0.521526\pi$
$884$	0	0
$885$	0	0
$886$	−28.5595	−0.959475
$887$	10.4665	0.351430	0.175715	−	0.984441i	$-0.443776\pi$
$887$	10.4665	0.351430	0.175715	+	0.984441i	$0.443776\pi$
$888$	0	0
$889$	−67.4849	−2.26337
$890$	28.1154	0.942432
$891$	0	0
$892$	−11.6558	−0.390265
$893$	−9.07712	−0.303754
$894$	0	0
$895$	−52.0188	−1.73880
$896$	−4.03957	−0.134953
$897$	0	0
$898$	6.21566	0.207419
$899$	−12.7651	−0.425738
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	−3.03512	−0.100946
$905$	51.2443	1.70342
$906$	0	0
$907$	−51.2472	−1.70164	−0.850818	−	0.525461i	$-0.823893\pi$
$907$	−51.2472	−1.70164	−0.850818	+	0.525461i	$0.823893\pi$
$908$	−19.4531	−0.645575
$909$	0	0
$910$	26.8997	0.891718
$911$	27.9997	0.927671	0.463835	−	0.885921i	$-0.346473\pi$
$911$	27.9997	0.927671	0.463835	+	0.885921i	$0.346473\pi$
$912$	0	0
$913$	0	0
$914$	−21.4274	−0.708755
$915$	0	0
$916$	−8.77777	−0.290026
$917$	−20.9902	−0.693158
$918$	0	0
$919$	−6.34333	−0.209247	−0.104624	−	0.994512i	$-0.533364\pi$
$919$	−6.34333	−0.209247	−0.104624	+	0.994512i	$0.533364\pi$
$920$	1.87477	0.0618094
$921$	0	0
$922$	−10.2736	−0.338343
$923$	−32.2309	−1.06089
$924$	0	0
$925$	−1.71663	−0.0564424
$926$	14.8647	0.488483
$927$	0	0
$928$	4.80059	0.157587
$929$	14.8679	0.487801	0.243900	−	0.969800i	$-0.421573\pi$
$929$	14.8679	0.487801	0.243900	+	0.969800i	$0.421573\pi$
$930$	0	0
$931$	45.5114	1.49158
$932$	18.3695	0.601713
$933$	0	0
$934$	13.6815	0.447673
$935$	0	0
$936$	0	0
$937$	11.1626	0.364668	0.182334	−	0.983237i	$-0.441635\pi$
$937$	11.1626	0.364668	0.182334	+	0.983237i	$0.441635\pi$
$938$	−11.3132	−0.369388
$939$	0	0
$940$	3.92614	0.128056
$941$	−32.2309	−1.05070	−0.525349	−	0.850887i	$-0.676065\pi$
$941$	−32.2309	−1.05070	−0.525349	+	0.850887i	$0.676065\pi$
$942$	0	0
$943$	−6.20919	−0.202199
$944$	−8.23127	−0.267905
$945$	0	0
$946$	0	0
$947$	28.3441	0.921059	0.460530	−	0.887644i	$-0.347659\pi$
$947$	28.3441	0.921059	0.460530	+	0.887644i	$0.347659\pi$
$948$	0	0
$949$	−24.6359	−0.799716
$950$	−2.62325	−0.0851096
$951$	0	0
$952$	0	0
$953$	−34.9837	−1.13323	−0.566617	−	0.823982i	$-0.691748\pi$
$953$	−34.9837	−1.13323	−0.566617	+	0.823982i	$0.691748\pi$
$954$	0	0
$955$	−34.4531	−1.11488
$956$	−6.07418	−0.196453
$957$	0	0
$958$	9.16681	0.296166
$959$	−5.28663	−0.170714
$960$	0	0
$961$	−23.9294	−0.771916
$962$	10.0747	0.324821
$963$	0	0
$964$	1.62238	0.0522533
$965$	−17.6750	−0.568980
$966$	0	0
$967$	−13.7684	−0.442762	−0.221381	−	0.975187i	$-0.571056\pi$
$967$	−13.7684	−0.442762	−0.221381	+	0.975187i	$0.571056\pi$
$968$	0	0
$969$	0	0
$970$	−7.00973	−0.225069
$971$	−8.11587	−0.260451	−0.130225	−	0.991484i	$-0.541570\pi$
$971$	−8.11587	−0.260451	−0.130225	+	0.991484i	$0.541570\pi$
$972$	0	0
$973$	−14.9457	−0.479137
$974$	−19.3890	−0.621265
$975$	0	0
$976$	−7.22512	−0.231270
$977$	−5.83281	−0.186608	−0.0933041	−	0.995638i	$-0.529743\pi$
$977$	−5.83281	−0.186608	−0.0933041	+	0.995638i	$0.529743\pi$
$978$	0	0
$979$	0	0
$980$	−19.6851	−0.628817
$981$	0	0
$982$	−30.7517	−0.981326
$983$	−26.9833	−0.860635	−0.430317	−	0.902678i	$-0.641598\pi$
$983$	−26.9833	−0.860635	−0.430317	+	0.902678i	$0.641598\pi$
$984$	0	0
$985$	32.3467	1.03065
$986$	0	0
$987$	0	0
$988$	15.3956	0.489798
$989$	11.5338	0.366754
$990$	0	0
$991$	37.4918	1.19097	0.595483	−	0.803368i	$-0.296961\pi$
$991$	37.4918	1.19097	0.595483	+	0.803368i	$0.296961\pi$
$992$	−2.65906	−0.0844253
$993$	0	0
$994$	41.3051	1.31012
$995$	28.9641	0.918223
$996$	0	0
$997$	−3.02974	−0.0959529	−0.0479764	−	0.998848i	$-0.515277\pi$
$997$	−3.02974	−0.0959529	−0.0479764	+	0.998848i	$0.515277\pi$
$998$	−20.3561	−0.644362
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6534.2.a.cs.1.2	yes	4
3.2	odd	2		6534.2.a.cq.1.3	✓	4
11.10	odd	2		6534.2.a.cr.1.2	yes	4
33.32	even	2		6534.2.a.ct.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6534.2.a.cq.1.3	✓	4	3.2	odd	2
6534.2.a.cr.1.2	yes	4	11.10	odd	2
6534.2.a.cs.1.2	yes	4	1.1	even	1	trivial
6534.2.a.ct.1.3	yes	4	33.32	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:	\( N \)	\(=\)	\( 6534 = 2 \cdot 3^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6534.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.11256 q^{5} -4.03957 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.11256 q^{5} -4.03957 q^{7} +1.00000 q^{8} -2.11256 q^{10} +3.15213 q^{13} -4.03957 q^{14} +1.00000 q^{16} +4.88418 q^{19} -2.11256 q^{20} -0.887440 q^{23} -0.537092 q^{25} +3.15213 q^{26} -4.03957 q^{28} +4.80059 q^{29} -2.65906 q^{31} +1.00000 q^{32} +8.53383 q^{35} +3.19615 q^{37} +4.88418 q^{38} -2.11256 q^{40} +6.99674 q^{41} -12.9967 q^{43} -0.887440 q^{46} -1.85847 q^{47} +9.31812 q^{49} -0.537092 q^{50} +3.15213 q^{52} +14.3149 q^{53} -4.03957 q^{56} +4.80059 q^{58} -8.23127 q^{59} -7.22512 q^{61} -2.65906 q^{62} +1.00000 q^{64} -6.65906 q^{65} +2.80059 q^{67} +8.53383 q^{70} -10.2251 q^{71} -7.81564 q^{73} +3.19615 q^{74} +4.88418 q^{76} -10.4201 q^{79} -2.11256 q^{80} +6.99674 q^{82} -16.6558 q^{83} -12.9967 q^{86} -13.3087 q^{89} -12.7332 q^{91} -0.887440 q^{92} -1.85847 q^{94} -10.3181 q^{95} +3.31812 q^{97} +9.31812 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{4} - 6 q^{7} + 4 q^{8}+O(q^{10}) \) 4 * q + 4 * q^2 + 4 * q^4 - 6 * q^7 + 4 * q^8	\( 4 q + 4 q^{2} + 4 q^{4} - 6 q^{7} + 4 q^{8} - 6 q^{13} - 6 q^{14} + 4 q^{16} - 6 q^{19} - 12 q^{23} + 10 q^{25} - 6 q^{26} - 6 q^{28} + 6 q^{29} - 2 q^{31} + 4 q^{32} - 12 q^{35} - 8 q^{37} - 6 q^{38} - 6 q^{41} - 18 q^{43} - 12 q^{46} - 12 q^{47} + 20 q^{49} + 10 q^{50} - 6 q^{52} + 6 q^{53} - 6 q^{56} + 6 q^{58} + 6 q^{59} - 12 q^{61} - 2 q^{62} + 4 q^{64} - 18 q^{65} - 2 q^{67} - 12 q^{70} - 24 q^{71} - 12 q^{73} - 8 q^{74} - 6 q^{76} - 30 q^{79} - 6 q^{82} - 24 q^{83} - 18 q^{86} - 24 q^{89} - 18 q^{91} - 12 q^{92} - 12 q^{94} - 24 q^{95} - 4 q^{97} + 20 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^4 - 6 * q^7 + 4 * q^8 - 6 * q^13 - 6 * q^14 + 4 * q^16 - 6 * q^19 - 12 * q^23 + 10 * q^25 - 6 * q^26 - 6 * q^28 + 6 * q^29 - 2 * q^31 + 4 * q^32 - 12 * q^35 - 8 * q^37 - 6 * q^38 - 6 * q^41 - 18 * q^43 - 12 * q^46 - 12 * q^47 + 20 * q^49 + 10 * q^50 - 6 * q^52 + 6 * q^53 - 6 * q^56 + 6 * q^58 + 6 * q^59 - 12 * q^61 - 2 * q^62 + 4 * q^64 - 18 * q^65 - 2 * q^67 - 12 * q^70 - 24 * q^71 - 12 * q^73 - 8 * q^74 - 6 * q^76 - 30 * q^79 - 6 * q^82 - 24 * q^83 - 18 * q^86 - 24 * q^89 - 18 * q^91 - 12 * q^92 - 12 * q^94 - 24 * q^95 - 4 * q^97 + 20 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more