LMFDB - Embedded newform 8281.2.a.w.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8281,2,Mod(1,8281)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8281, 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("8281.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8281 = 7^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8281.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:	$66.1241179138$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 91)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	8281.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.73205 q^{5} +1.73205 q^{6} -1.73205 q^{8} -2.00000 q^{9} +O(q^{10})$	\(q+1.73205 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.73205 q^{5} +1.73205 q^{6} -1.73205 q^{8} -2.00000 q^{9} +3.00000 q^{10} -5.19615 q^{11} +1.00000 q^{12} +1.73205 q^{15} -5.00000 q^{16} +6.00000 q^{17} -3.46410 q^{18} +1.73205 q^{19} +1.73205 q^{20} -9.00000 q^{22} -1.73205 q^{24} -2.00000 q^{25} -5.00000 q^{27} +3.00000 q^{29} +3.00000 q^{30} +1.73205 q^{31} -5.19615 q^{32} -5.19615 q^{33} +10.3923 q^{34} -2.00000 q^{36} +3.00000 q^{38} -3.00000 q^{40} +5.19615 q^{41} -11.0000 q^{43} -5.19615 q^{44} -3.46410 q^{45} -8.66025 q^{47} -5.00000 q^{48} -3.46410 q^{50} +6.00000 q^{51} -9.00000 q^{53} -8.66025 q^{54} -9.00000 q^{55} +1.73205 q^{57} +5.19615 q^{58} -3.46410 q^{59} +1.73205 q^{60} -7.00000 q^{61} +3.00000 q^{62} +1.00000 q^{64} -9.00000 q^{66} -8.66025 q^{67} +6.00000 q^{68} -1.73205 q^{71} +3.46410 q^{72} +8.66025 q^{73} -2.00000 q^{75} +1.73205 q^{76} -5.00000 q^{79} -8.66025 q^{80} +1.00000 q^{81} +9.00000 q^{82} -3.46410 q^{83} +10.3923 q^{85} -19.0526 q^{86} +3.00000 q^{87} +9.00000 q^{88} +6.92820 q^{89} -6.00000 q^{90} +1.73205 q^{93} -15.0000 q^{94} +3.00000 q^{95} -5.19615 q^{96} -5.19615 q^{97} +10.3923 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{4} - 4 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^4 - 4 * q^9	$ 2 q + 2 q^{3} + 2 q^{4} - 4 q^{9} + 6 q^{10} + 2 q^{12} - 10 q^{16} + 12 q^{17} - 18 q^{22} - 4 q^{25} - 10 q^{27} + 6 q^{29} + 6 q^{30} - 4 q^{36} + 6 q^{38} - 6 q^{40} - 22 q^{43} - 10 q^{48} + 12 q^{51} - 18 q^{53} - 18 q^{55} - 14 q^{61} + 6 q^{62} + 2 q^{64} - 18 q^{66} + 12 q^{68} - 4 q^{75} - 10 q^{79} + 2 q^{81} + 18 q^{82} + 6 q^{87} + 18 q^{88} - 12 q^{90} - 30 q^{94} + 6 q^{95}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^4 - 4 * q^9 + 6 * q^10 + 2 * q^12 - 10 * q^16 + 12 * q^17 - 18 * q^22 - 4 * q^25 - 10 * q^27 + 6 * q^29 + 6 * q^30 - 4 * q^36 + 6 * q^38 - 6 * q^40 - 22 * q^43 - 10 * q^48 + 12 * q^51 - 18 * q^53 - 18 * q^55 - 14 * q^61 + 6 * q^62 + 2 * q^64 - 18 * q^66 + 12 * q^68 - 4 * q^75 - 10 * q^79 + 2 * q^81 + 18 * q^82 + 6 * q^87 + 18 * q^88 - 12 * q^90 - 30 * q^94 + 6 * q^95

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 $

$2$	1.73205	1.22474	0.612372	−	0.790569i	$-0.290215\pi$
$2$	1.73205	1.22474	0.612372	+	0.790569i	$0.290215\pi$
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	1.00000	0.500000
$5$	1.73205	0.774597	0.387298	−	0.921954i	$-0.373408\pi$
$5$	1.73205	0.774597	0.387298	+	0.921954i	$0.373408\pi$
$6$	1.73205	0.707107
$7$	0	0
$7$	0	0
$8$	−1.73205	−0.612372
$9$	−2.00000	−0.666667
$10$	3.00000	0.948683
$11$	−5.19615	−1.56670	−0.783349	−	0.621582i	$-0.786490\pi$
$11$	−5.19615	−1.56670	−0.783349	+	0.621582i	$0.786490\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	0	0
$15$	1.73205	0.447214
$16$	−5.00000	−1.25000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	−3.46410	−0.816497
$19$	1.73205	0.397360	0.198680	−	0.980064i	$-0.436335\pi$
$19$	1.73205	0.397360	0.198680	+	0.980064i	$0.436335\pi$
$20$	1.73205	0.387298
$21$	0	0
$22$	−9.00000	−1.91881
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−1.73205	−0.353553
$25$	−2.00000	−0.400000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	3.00000	0.547723
$31$	1.73205	0.311086	0.155543	−	0.987829i	$-0.450287\pi$
$31$	1.73205	0.311086	0.155543	+	0.987829i	$0.450287\pi$
$32$	−5.19615	−0.918559
$33$	−5.19615	−0.904534
$34$	10.3923	1.78227
$35$	0	0
$36$	−2.00000	−0.333333
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	3.00000	0.486664
$39$	0	0
$40$	−3.00000	−0.474342
$41$	5.19615	0.811503	0.405751	−	0.913984i	$-0.367010\pi$
$41$	5.19615	0.811503	0.405751	+	0.913984i	$0.367010\pi$
$42$	0	0
$43$	−11.0000	−1.67748	−0.838742	−	0.544529i	$-0.816708\pi$
$43$	−11.0000	−1.67748	−0.838742	+	0.544529i	$0.816708\pi$
$44$	−5.19615	−0.783349
$45$	−3.46410	−0.516398
$46$	0	0
$47$	−8.66025	−1.26323	−0.631614	−	0.775283i	$-0.717607\pi$
$47$	−8.66025	−1.26323	−0.631614	+	0.775283i	$0.717607\pi$
$48$	−5.00000	−0.721688
$49$	0	0
$50$	−3.46410	−0.489898
$51$	6.00000	0.840168
$52$	0	0
$53$	−9.00000	−1.23625	−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000	−1.23625	−0.618123	+	0.786082i	$0.712106\pi$
$54$	−8.66025	−1.17851
$55$	−9.00000	−1.21356
$56$	0	0
$57$	1.73205	0.229416
$58$	5.19615	0.682288
$59$	−3.46410	−0.450988	−0.225494	−	0.974245i	$-0.572400\pi$
$59$	−3.46410	−0.450988	−0.225494	+	0.974245i	$0.572400\pi$
$60$	1.73205	0.223607
$61$	−7.00000	−0.896258	−0.448129	−	0.893969i	$-0.647910\pi$
$61$	−7.00000	−0.896258	−0.448129	+	0.893969i	$0.647910\pi$
$62$	3.00000	0.381000
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	−9.00000	−1.10782
$67$	−8.66025	−1.05802	−0.529009	−	0.848616i	$-0.677436\pi$
$67$	−8.66025	−1.05802	−0.529009	+	0.848616i	$0.677436\pi$
$68$	6.00000	0.727607
$69$	0	0
$70$	0	0
$71$	−1.73205	−0.205557	−0.102778	−	0.994704i	$-0.532773\pi$
$71$	−1.73205	−0.205557	−0.102778	+	0.994704i	$0.532773\pi$
$72$	3.46410	0.408248
$73$	8.66025	1.01361	0.506803	−	0.862062i	$-0.330827\pi$
$73$	8.66025	1.01361	0.506803	+	0.862062i	$0.330827\pi$
$74$	0	0
$75$	−2.00000	−0.230940
$76$	1.73205	0.198680
$77$	0	0
$78$	0	0
$79$	−5.00000	−0.562544	−0.281272	−	0.959628i	$-0.590756\pi$
$79$	−5.00000	−0.562544	−0.281272	+	0.959628i	$0.590756\pi$
$80$	−8.66025	−0.968246
$81$	1.00000	0.111111
$82$	9.00000	0.993884
$83$	−3.46410	−0.380235	−0.190117	−	0.981761i	$-0.560887\pi$
$83$	−3.46410	−0.380235	−0.190117	+	0.981761i	$0.560887\pi$
$84$	0	0
$85$	10.3923	1.12720
$86$	−19.0526	−2.05449
$87$	3.00000	0.321634
$88$	9.00000	0.959403
$89$	6.92820	0.734388	0.367194	−	0.930144i	$-0.380318\pi$
$89$	6.92820	0.734388	0.367194	+	0.930144i	$0.380318\pi$
$90$	−6.00000	−0.632456
$91$	0	0
$92$	0	0
$93$	1.73205	0.179605
$94$	−15.0000	−1.54713
$95$	3.00000	0.307794
$96$	−5.19615	−0.530330
$97$	−5.19615	−0.527589	−0.263795	−	0.964579i	$-0.584974\pi$
$97$	−5.19615	−0.527589	−0.263795	+	0.964579i	$0.584974\pi$
$98$	0	0
$99$	10.3923	1.04447
$100$	−2.00000	−0.200000
$101$	−9.00000	−0.895533	−0.447767	−	0.894150i	$-0.647781\pi$
$101$	−9.00000	−0.895533	−0.447767	+	0.894150i	$0.647781\pi$
$102$	10.3923	1.02899
$103$	−13.0000	−1.28093	−0.640464	−	0.767988i	$-0.721258\pi$
$103$	−13.0000	−1.28093	−0.640464	+	0.767988i	$0.721258\pi$
$104$	0	0
$105$	0	0
$106$	−15.5885	−1.51408
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	−5.00000	−0.481125
$109$	−5.19615	−0.497701	−0.248851	−	0.968542i	$-0.580053\pi$
$109$	−5.19615	−0.497701	−0.248851	+	0.968542i	$0.580053\pi$
$110$	−15.5885	−1.48630
$111$	0	0
$112$	0	0
$113$	15.0000	1.41108	0.705541	−	0.708669i	$-0.250704\pi$
$113$	15.0000	1.41108	0.705541	+	0.708669i	$0.250704\pi$
$114$	3.00000	0.280976
$115$	0	0
$116$	3.00000	0.278543
$117$	0	0
$118$	−6.00000	−0.552345
$119$	0	0
$120$	−3.00000	−0.273861
$121$	16.0000	1.45455
$122$	−12.1244	−1.09769
$123$	5.19615	0.468521
$124$	1.73205	0.155543
$125$	−12.1244	−1.08444
$126$	0	0
$127$	−13.0000	−1.15356	−0.576782	−	0.816898i	$-0.695692\pi$
$127$	−13.0000	−1.15356	−0.576782	+	0.816898i	$0.695692\pi$
$128$	12.1244	1.07165
$129$	−11.0000	−0.968496
$130$	0	0
$131$	−15.0000	−1.31056	−0.655278	−	0.755388i	$-0.727449\pi$
$131$	−15.0000	−1.31056	−0.655278	+	0.755388i	$0.727449\pi$
$132$	−5.19615	−0.452267
$133$	0	0
$134$	−15.0000	−1.29580
$135$	−8.66025	−0.745356
$136$	−10.3923	−0.891133
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	13.0000	1.10265	0.551323	−	0.834292i	$-0.314123\pi$
$139$	13.0000	1.10265	0.551323	+	0.834292i	$0.314123\pi$
$140$	0	0
$141$	−8.66025	−0.729325
$142$	−3.00000	−0.251754
$143$	0	0
$144$	10.0000	0.833333
$145$	5.19615	0.431517
$146$	15.0000	1.24141
$147$	0	0
$148$	0	0
$149$	−19.0526	−1.56085	−0.780423	−	0.625252i	$-0.784996\pi$
$149$	−19.0526	−1.56085	−0.780423	+	0.625252i	$0.784996\pi$
$150$	−3.46410	−0.282843
$151$	−12.1244	−0.986666	−0.493333	−	0.869841i	$-0.664222\pi$
$151$	−12.1244	−0.986666	−0.493333	+	0.869841i	$0.664222\pi$
$152$	−3.00000	−0.243332
$153$	−12.0000	−0.970143
$154$	0	0
$155$	3.00000	0.240966
$156$	0	0
$157$	−23.0000	−1.83560	−0.917800	−	0.397043i	$-0.870036\pi$
$157$	−23.0000	−1.83560	−0.917800	+	0.397043i	$0.870036\pi$
$158$	−8.66025	−0.688973
$159$	−9.00000	−0.713746
$160$	−9.00000	−0.711512
$161$	0	0
$162$	1.73205	0.136083
$163$	−12.1244	−0.949653	−0.474826	−	0.880079i	$-0.657489\pi$
$163$	−12.1244	−0.949653	−0.474826	+	0.880079i	$0.657489\pi$
$164$	5.19615	0.405751
$165$	−9.00000	−0.700649
$166$	−6.00000	−0.465690
$167$	−1.73205	−0.134030	−0.0670151	−	0.997752i	$-0.521348\pi$
$167$	−1.73205	−0.134030	−0.0670151	+	0.997752i	$0.521348\pi$
$168$	0	0
$169$	0	0
$170$	18.0000	1.38054
$171$	−3.46410	−0.264906
$172$	−11.0000	−0.838742
$173$	15.0000	1.14043	0.570214	−	0.821496i	$-0.306860\pi$
$173$	15.0000	1.14043	0.570214	+	0.821496i	$0.306860\pi$
$174$	5.19615	0.393919
$175$	0	0
$176$	25.9808	1.95837
$177$	−3.46410	−0.260378
$178$	12.0000	0.899438
$179$	−3.00000	−0.224231	−0.112115	−	0.993695i	$-0.535763\pi$
$179$	−3.00000	−0.224231	−0.112115	+	0.993695i	$0.535763\pi$
$180$	−3.46410	−0.258199
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	−7.00000	−0.517455
$184$	0	0
$185$	0	0
$186$	3.00000	0.219971
$187$	−31.1769	−2.27988
$188$	−8.66025	−0.631614
$189$	0	0
$190$	5.19615	0.376969
$191$	−15.0000	−1.08536	−0.542681	−	0.839939i	$-0.682591\pi$
$191$	−15.0000	−1.08536	−0.542681	+	0.839939i	$0.682591\pi$
$192$	1.00000	0.0721688
$193$	1.73205	0.124676	0.0623379	−	0.998055i	$-0.480144\pi$
$193$	1.73205	0.124676	0.0623379	+	0.998055i	$0.480144\pi$
$194$	−9.00000	−0.646162
$195$	0	0
$196$	0	0
$197$	22.5167	1.60425	0.802123	−	0.597159i	$-0.203704\pi$
$197$	22.5167	1.60425	0.802123	+	0.597159i	$0.203704\pi$
$198$	18.0000	1.27920
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	3.46410	0.244949
$201$	−8.66025	−0.610847
$202$	−15.5885	−1.09680
$203$	0	0
$204$	6.00000	0.420084
$205$	9.00000	0.628587
$206$	−22.5167	−1.56881
$207$	0	0
$208$	0	0
$209$	−9.00000	−0.622543
$210$	0	0
$211$	13.0000	0.894957	0.447478	−	0.894295i	$-0.352322\pi$
$211$	13.0000	0.894957	0.447478	+	0.894295i	$0.352322\pi$
$212$	−9.00000	−0.618123
$213$	−1.73205	−0.118678
$214$	0	0
$215$	−19.0526	−1.29937
$216$	8.66025	0.589256
$217$	0	0
$218$	−9.00000	−0.609557
$219$	8.66025	0.585206
$220$	−9.00000	−0.606780
$221$	0	0
$222$	0	0
$223$	−5.19615	−0.347960	−0.173980	−	0.984749i	$-0.555663\pi$
$223$	−5.19615	−0.347960	−0.173980	+	0.984749i	$0.555663\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	25.9808	1.72821
$227$	17.3205	1.14960	0.574801	−	0.818293i	$-0.305079\pi$
$227$	17.3205	1.14960	0.574801	+	0.818293i	$0.305079\pi$
$228$	1.73205	0.114708
$229$	12.1244	0.801200	0.400600	−	0.916253i	$-0.368802\pi$
$229$	12.1244	0.801200	0.400600	+	0.916253i	$0.368802\pi$
$230$	0	0
$231$	0	0
$232$	−5.19615	−0.341144
$233$	−3.00000	−0.196537	−0.0982683	−	0.995160i	$-0.531330\pi$
$233$	−3.00000	−0.196537	−0.0982683	+	0.995160i	$0.531330\pi$
$234$	0	0
$235$	−15.0000	−0.978492
$236$	−3.46410	−0.225494
$237$	−5.00000	−0.324785
$238$	0	0
$239$	10.3923	0.672222	0.336111	−	0.941822i	$-0.390888\pi$
$239$	10.3923	0.672222	0.336111	+	0.941822i	$0.390888\pi$
$240$	−8.66025	−0.559017
$241$	−6.92820	−0.446285	−0.223142	−	0.974786i	$-0.571631\pi$
$241$	−6.92820	−0.446285	−0.223142	+	0.974786i	$0.571631\pi$
$242$	27.7128	1.78145
$243$	16.0000	1.02640
$244$	−7.00000	−0.448129
$245$	0	0
$246$	9.00000	0.573819
$247$	0	0
$248$	−3.00000	−0.190500
$249$	−3.46410	−0.219529
$250$	−21.0000	−1.32816
$251$	3.00000	0.189358	0.0946792	−	0.995508i	$-0.469817\pi$
$251$	3.00000	0.189358	0.0946792	+	0.995508i	$0.469817\pi$
$252$	0	0
$253$	0	0
$254$	−22.5167	−1.41282
$255$	10.3923	0.650791
$256$	19.0000	1.18750
$257$	30.0000	1.87135	0.935674	−	0.352865i	$-0.114792\pi$
$257$	30.0000	1.87135	0.935674	+	0.352865i	$0.114792\pi$
$258$	−19.0526	−1.18616
$259$	0	0
$260$	0	0
$261$	−6.00000	−0.371391
$262$	−25.9808	−1.60510
$263$	3.00000	0.184988	0.0924940	−	0.995713i	$-0.470516\pi$
$263$	3.00000	0.184988	0.0924940	+	0.995713i	$0.470516\pi$
$264$	9.00000	0.553912
$265$	−15.5885	−0.957591
$266$	0	0
$267$	6.92820	0.423999
$268$	−8.66025	−0.529009
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	−15.0000	−0.912871
$271$	17.3205	1.05215	0.526073	−	0.850439i	$-0.323664\pi$
$271$	17.3205	1.05215	0.526073	+	0.850439i	$0.323664\pi$
$272$	−30.0000	−1.81902
$273$	0	0
$274$	0	0
$275$	10.3923	0.626680
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	22.5167	1.35046
$279$	−3.46410	−0.207390
$280$	0	0
$281$	6.92820	0.413302	0.206651	−	0.978415i	$-0.433744\pi$
$281$	6.92820	0.413302	0.206651	+	0.978415i	$0.433744\pi$
$282$	−15.0000	−0.893237
$283$	19.0000	1.12943	0.564716	−	0.825285i	$-0.308986\pi$
$283$	19.0000	1.12943	0.564716	+	0.825285i	$0.308986\pi$
$284$	−1.73205	−0.102778
$285$	3.00000	0.177705
$286$	0	0
$287$	0	0
$288$	10.3923	0.612372
$289$	19.0000	1.11765
$290$	9.00000	0.528498
$291$	−5.19615	−0.304604
$292$	8.66025	0.506803
$293$	25.9808	1.51781	0.758906	−	0.651200i	$-0.225734\pi$
$293$	25.9808	1.51781	0.758906	+	0.651200i	$0.225734\pi$
$294$	0	0
$295$	−6.00000	−0.349334
$296$	0	0
$297$	25.9808	1.50756
$298$	−33.0000	−1.91164
$299$	0	0
$300$	−2.00000	−0.115470
$301$	0	0
$302$	−21.0000	−1.20841
$303$	−9.00000	−0.517036
$304$	−8.66025	−0.496700
$305$	−12.1244	−0.694239
$306$	−20.7846	−1.18818
$307$	24.2487	1.38395	0.691974	−	0.721923i	$-0.256741\pi$
$307$	24.2487	1.38395	0.691974	+	0.721923i	$0.256741\pi$
$308$	0	0
$309$	−13.0000	−0.739544
$310$	5.19615	0.295122
$311$	−15.0000	−0.850572	−0.425286	−	0.905059i	$-0.639826\pi$
$311$	−15.0000	−0.850572	−0.425286	+	0.905059i	$0.639826\pi$
$312$	0	0
$313$	−19.0000	−1.07394	−0.536972	−	0.843600i	$-0.680432\pi$
$313$	−19.0000	−1.07394	−0.536972	+	0.843600i	$0.680432\pi$
$314$	−39.8372	−2.24814
$315$	0	0
$316$	−5.00000	−0.281272
$317$	5.19615	0.291845	0.145922	−	0.989296i	$-0.453385\pi$
$317$	5.19615	0.291845	0.145922	+	0.989296i	$0.453385\pi$
$318$	−15.5885	−0.874157
$319$	−15.5885	−0.872786
$320$	1.73205	0.0968246
$321$	0	0
$322$	0	0
$323$	10.3923	0.578243
$324$	1.00000	0.0555556
$325$	0	0
$326$	−21.0000	−1.16308
$327$	−5.19615	−0.287348
$328$	−9.00000	−0.496942
$329$	0	0
$330$	−15.5885	−0.858116
$331$	32.9090	1.80884	0.904420	−	0.426643i	$-0.140304\pi$
$331$	32.9090	1.80884	0.904420	+	0.426643i	$0.140304\pi$
$332$	−3.46410	−0.190117
$333$	0	0
$334$	−3.00000	−0.164153
$335$	−15.0000	−0.819538
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	0	0
$339$	15.0000	0.814688
$340$	10.3923	0.563602
$341$	−9.00000	−0.487377
$342$	−6.00000	−0.324443
$343$	0	0
$344$	19.0526	1.02725
$345$	0	0
$346$	25.9808	1.39673
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	3.00000	0.160817
$349$	−5.19615	−0.278144	−0.139072	−	0.990282i	$-0.544412\pi$
$349$	−5.19615	−0.278144	−0.139072	+	0.990282i	$0.544412\pi$
$350$	0	0
$351$	0	0
$352$	27.0000	1.43910
$353$	1.73205	0.0921878	0.0460939	−	0.998937i	$-0.485323\pi$
$353$	1.73205	0.0921878	0.0460939	+	0.998937i	$0.485323\pi$
$354$	−6.00000	−0.318896
$355$	−3.00000	−0.159223
$356$	6.92820	0.367194
$357$	0	0
$358$	−5.19615	−0.274625
$359$	−19.0526	−1.00556	−0.502778	−	0.864416i	$-0.667689\pi$
$359$	−19.0526	−1.00556	−0.502778	+	0.864416i	$0.667689\pi$
$360$	6.00000	0.316228
$361$	−16.0000	−0.842105
$362$	3.46410	0.182069
$363$	16.0000	0.839782
$364$	0	0
$365$	15.0000	0.785136
$366$	−12.1244	−0.633750
$367$	−23.0000	−1.20059	−0.600295	−	0.799779i	$-0.704950\pi$
$367$	−23.0000	−1.20059	−0.600295	+	0.799779i	$0.704950\pi$
$368$	0	0
$369$	−10.3923	−0.541002
$370$	0	0
$371$	0	0
$372$	1.73205	0.0898027
$373$	19.0000	0.983783	0.491891	−	0.870657i	$-0.336306\pi$
$373$	19.0000	0.983783	0.491891	+	0.870657i	$0.336306\pi$
$374$	−54.0000	−2.79227
$375$	−12.1244	−0.626099
$376$	15.0000	0.773566
$377$	0	0
$378$	0	0
$379$	−1.73205	−0.0889695	−0.0444847	−	0.999010i	$-0.514165\pi$
$379$	−1.73205	−0.0889695	−0.0444847	+	0.999010i	$0.514165\pi$
$380$	3.00000	0.153897
$381$	−13.0000	−0.666010
$382$	−25.9808	−1.32929
$383$	15.5885	0.796533	0.398266	−	0.917270i	$-0.369612\pi$
$383$	15.5885	0.796533	0.398266	+	0.917270i	$0.369612\pi$
$384$	12.1244	0.618718
$385$	0	0
$386$	3.00000	0.152696
$387$	22.0000	1.11832
$388$	−5.19615	−0.263795
$389$	−3.00000	−0.152106	−0.0760530	−	0.997104i	$-0.524232\pi$
$389$	−3.00000	−0.152106	−0.0760530	+	0.997104i	$0.524232\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−15.0000	−0.756650
$394$	39.0000	1.96479
$395$	−8.66025	−0.435745
$396$	10.3923	0.522233
$397$	36.3731	1.82551	0.912756	−	0.408505i	$-0.133950\pi$
$397$	36.3731	1.82551	0.912756	+	0.408505i	$0.133950\pi$
$398$	6.92820	0.347279
$399$	0	0
$400$	10.0000	0.500000
$401$	−6.92820	−0.345978	−0.172989	−	0.984924i	$-0.555343\pi$
$401$	−6.92820	−0.345978	−0.172989	+	0.984924i	$0.555343\pi$
$402$	−15.0000	−0.748132
$403$	0	0
$404$	−9.00000	−0.447767
$405$	1.73205	0.0860663
$406$	0	0
$407$	0	0
$408$	−10.3923	−0.514496
$409$	−6.92820	−0.342578	−0.171289	−	0.985221i	$-0.554793\pi$
$409$	−6.92820	−0.342578	−0.171289	+	0.985221i	$0.554793\pi$
$410$	15.5885	0.769859
$411$	0	0
$412$	−13.0000	−0.640464
$413$	0	0
$414$	0	0
$415$	−6.00000	−0.294528
$416$	0	0
$417$	13.0000	0.636613
$418$	−15.5885	−0.762456
$419$	−21.0000	−1.02592	−0.512959	−	0.858413i	$-0.671451\pi$
$419$	−21.0000	−1.02592	−0.512959	+	0.858413i	$0.671451\pi$
$420$	0	0
$421$	0	0		−	1.00000i	$-0.5\pi$
$421$	0	0			1.00000i	$0.5\pi$
$422$	22.5167	1.09609
$423$	17.3205	0.842152
$424$	15.5885	0.757042
$425$	−12.0000	−0.582086
$426$	−3.00000	−0.145350
$427$	0	0
$428$	0	0
$429$	0	0
$430$	−33.0000	−1.59140
$431$	32.9090	1.58517	0.792585	−	0.609762i	$-0.208735\pi$
$431$	32.9090	1.58517	0.792585	+	0.609762i	$0.208735\pi$
$432$	25.0000	1.20281
$433$	19.0000	0.913082	0.456541	−	0.889702i	$-0.349088\pi$
$433$	19.0000	0.913082	0.456541	+	0.889702i	$0.349088\pi$
$434$	0	0
$435$	5.19615	0.249136
$436$	−5.19615	−0.248851
$437$	0	0
$438$	15.0000	0.716728
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	15.5885	0.743151
$441$	0	0
$442$	0	0
$443$	15.0000	0.712672	0.356336	−	0.934358i	$-0.384026\pi$
$443$	15.0000	0.712672	0.356336	+	0.934358i	$0.384026\pi$
$444$	0	0
$445$	12.0000	0.568855
$446$	−9.00000	−0.426162
$447$	−19.0526	−0.901155
$448$	0	0
$449$	1.73205	0.0817405	0.0408703	−	0.999164i	$-0.486987\pi$
$449$	1.73205	0.0817405	0.0408703	+	0.999164i	$0.486987\pi$
$450$	6.92820	0.326599
$451$	−27.0000	−1.27138
$452$	15.0000	0.705541
$453$	−12.1244	−0.569652
$454$	30.0000	1.40797
$455$	0	0
$456$	−3.00000	−0.140488
$457$	34.6410	1.62044	0.810219	−	0.586127i	$-0.199348\pi$
$457$	34.6410	1.62044	0.810219	+	0.586127i	$0.199348\pi$
$458$	21.0000	0.981266
$459$	−30.0000	−1.40028
$460$	0	0
$461$	29.4449	1.37138	0.685692	−	0.727892i	$-0.259500\pi$
$461$	29.4449	1.37138	0.685692	+	0.727892i	$0.259500\pi$
$462$	0	0
$463$	24.2487	1.12693	0.563467	−	0.826139i	$-0.309467\pi$
$463$	24.2487	1.12693	0.563467	+	0.826139i	$0.309467\pi$
$464$	−15.0000	−0.696358
$465$	3.00000	0.139122
$466$	−5.19615	−0.240707
$467$	−21.0000	−0.971764	−0.485882	−	0.874024i	$-0.661502\pi$
$467$	−21.0000	−0.971764	−0.485882	+	0.874024i	$0.661502\pi$
$468$	0	0
$469$	0	0
$470$	−25.9808	−1.19840
$471$	−23.0000	−1.05978
$472$	6.00000	0.276172
$473$	57.1577	2.62811
$474$	−8.66025	−0.397779
$475$	−3.46410	−0.158944
$476$	0	0
$477$	18.0000	0.824163
$478$	18.0000	0.823301
$479$	−29.4449	−1.34537	−0.672685	−	0.739929i	$-0.734859\pi$
$479$	−29.4449	−1.34537	−0.672685	+	0.739929i	$0.734859\pi$
$480$	−9.00000	−0.410792
$481$	0	0
$482$	−12.0000	−0.546585
$483$	0	0
$484$	16.0000	0.727273
$485$	−9.00000	−0.408669
$486$	27.7128	1.25708
$487$	−24.2487	−1.09881	−0.549407	−	0.835555i	$-0.685146\pi$
$487$	−24.2487	−1.09881	−0.549407	+	0.835555i	$0.685146\pi$
$488$	12.1244	0.548844
$489$	−12.1244	−0.548282
$490$	0	0
$491$	27.0000	1.21849	0.609246	−	0.792981i	$-0.291472\pi$
$491$	27.0000	1.21849	0.609246	+	0.792981i	$0.291472\pi$
$492$	5.19615	0.234261
$493$	18.0000	0.810679
$494$	0	0
$495$	18.0000	0.809040
$496$	−8.66025	−0.388857
$497$	0	0
$498$	−6.00000	−0.268866
$499$	−1.73205	−0.0775372	−0.0387686	−	0.999248i	$-0.512344\pi$
$499$	−1.73205	−0.0775372	−0.0387686	+	0.999248i	$0.512344\pi$
$500$	−12.1244	−0.542218
$501$	−1.73205	−0.0773823
$502$	5.19615	0.231916
$503$	9.00000	0.401290	0.200645	−	0.979664i	$-0.435696\pi$
$503$	9.00000	0.401290	0.200645	+	0.979664i	$0.435696\pi$
$504$	0	0
$505$	−15.5885	−0.693677
$506$	0	0
$507$	0	0
$508$	−13.0000	−0.576782
$509$	−6.92820	−0.307087	−0.153544	−	0.988142i	$-0.549069\pi$
$509$	−6.92820	−0.307087	−0.153544	+	0.988142i	$0.549069\pi$
$510$	18.0000	0.797053
$511$	0	0
$512$	8.66025	0.382733
$513$	−8.66025	−0.382360
$514$	51.9615	2.29192
$515$	−22.5167	−0.992203
$516$	−11.0000	−0.484248
$517$	45.0000	1.97910
$518$	0	0
$519$	15.0000	0.658427
$520$	0	0
$521$	−39.0000	−1.70862	−0.854311	−	0.519763i	$-0.826020\pi$
$521$	−39.0000	−1.70862	−0.854311	+	0.519763i	$0.826020\pi$
$522$	−10.3923	−0.454859
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	−15.0000	−0.655278
$525$	0	0
$526$	5.19615	0.226563
$527$	10.3923	0.452696
$528$	25.9808	1.13067
$529$	−23.0000	−1.00000
$530$	−27.0000	−1.17281
$531$	6.92820	0.300658
$532$	0	0
$533$	0	0
$534$	12.0000	0.519291
$535$	0	0
$536$	15.0000	0.647901
$537$	−3.00000	−0.129460
$538$	10.3923	0.448044
$539$	0	0
$540$	−8.66025	−0.372678
$541$	−12.1244	−0.521267	−0.260633	−	0.965438i	$-0.583931\pi$
$541$	−12.1244	−0.521267	−0.260633	+	0.965438i	$0.583931\pi$
$542$	30.0000	1.28861
$543$	2.00000	0.0858282
$544$	−31.1769	−1.33670
$545$	−9.00000	−0.385518
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	0	0
$549$	14.0000	0.597505
$550$	18.0000	0.767523
$551$	5.19615	0.221364
$552$	0	0
$553$	0	0
$554$	17.3205	0.735878
$555$	0	0
$556$	13.0000	0.551323
$557$	15.5885	0.660504	0.330252	−	0.943893i	$-0.392866\pi$
$557$	15.5885	0.660504	0.330252	+	0.943893i	$0.392866\pi$
$558$	−6.00000	−0.254000
$559$	0	0
$560$	0	0
$561$	−31.1769	−1.31629
$562$	12.0000	0.506189
$563$	−36.0000	−1.51722	−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000	−1.51722	−0.758610	+	0.651546i	$0.774121\pi$
$564$	−8.66025	−0.364662
$565$	25.9808	1.09302
$566$	32.9090	1.38327
$567$	0	0
$568$	3.00000	0.125877
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	5.19615	0.217643
$571$	23.0000	0.962520	0.481260	−	0.876578i	$-0.340179\pi$
$571$	23.0000	0.962520	0.481260	+	0.876578i	$0.340179\pi$
$572$	0	0
$573$	−15.0000	−0.626634
$574$	0	0
$575$	0	0
$576$	−2.00000	−0.0833333
$577$	−15.5885	−0.648956	−0.324478	−	0.945893i	$-0.605189\pi$
$577$	−15.5885	−0.648956	−0.324478	+	0.945893i	$0.605189\pi$
$578$	32.9090	1.36883
$579$	1.73205	0.0719816
$580$	5.19615	0.215758
$581$	0	0
$582$	−9.00000	−0.373062
$583$	46.7654	1.93682
$584$	−15.0000	−0.620704
$585$	0	0
$586$	45.0000	1.85893
$587$	15.5885	0.643404	0.321702	−	0.946841i	$-0.395745\pi$
$587$	15.5885	0.643404	0.321702	+	0.946841i	$0.395745\pi$
$588$	0	0
$589$	3.00000	0.123613
$590$	−10.3923	−0.427844
$591$	22.5167	0.926212
$592$	0	0
$593$	5.19615	0.213380	0.106690	−	0.994292i	$-0.465975\pi$
$593$	5.19615	0.213380	0.106690	+	0.994292i	$0.465975\pi$
$594$	45.0000	1.84637
$595$	0	0
$596$	−19.0526	−0.780423
$597$	4.00000	0.163709
$598$	0	0
$599$	9.00000	0.367730	0.183865	−	0.982952i	$-0.441139\pi$
$599$	9.00000	0.367730	0.183865	+	0.982952i	$0.441139\pi$
$600$	3.46410	0.141421
$601$	−19.0000	−0.775026	−0.387513	−	0.921864i	$-0.626666\pi$
$601$	−19.0000	−0.775026	−0.387513	+	0.921864i	$0.626666\pi$
$602$	0	0
$603$	17.3205	0.705346
$604$	−12.1244	−0.493333
$605$	27.7128	1.12669
$606$	−15.5885	−0.633238
$607$	43.0000	1.74532	0.872658	−	0.488332i	$-0.162394\pi$
$607$	43.0000	1.74532	0.872658	+	0.488332i	$0.162394\pi$
$608$	−9.00000	−0.364998
$609$	0	0
$610$	−21.0000	−0.850265
$611$	0	0
$612$	−12.0000	−0.485071
$613$	−36.3731	−1.46909	−0.734547	−	0.678558i	$-0.762605\pi$
$613$	−36.3731	−1.46909	−0.734547	+	0.678558i	$0.762605\pi$
$614$	42.0000	1.69498
$615$	9.00000	0.362915
$616$	0	0
$617$	−43.3013	−1.74324	−0.871622	−	0.490179i	$-0.836931\pi$
$617$	−43.3013	−1.74324	−0.871622	+	0.490179i	$0.836931\pi$
$618$	−22.5167	−0.905753
$619$	19.0526	0.765787	0.382893	−	0.923792i	$-0.374928\pi$
$619$	19.0526	0.765787	0.382893	+	0.923792i	$0.374928\pi$
$620$	3.00000	0.120483
$621$	0	0
$622$	−25.9808	−1.04173
$623$	0	0
$624$	0	0
$625$	−11.0000	−0.440000
$626$	−32.9090	−1.31531
$627$	−9.00000	−0.359425
$628$	−23.0000	−0.917800
$629$	0	0
$630$	0	0
$631$	−46.7654	−1.86170	−0.930850	−	0.365401i	$-0.880932\pi$
$631$	−46.7654	−1.86170	−0.930850	+	0.365401i	$0.880932\pi$
$632$	8.66025	0.344486
$633$	13.0000	0.516704
$634$	9.00000	0.357436
$635$	−22.5167	−0.893546
$636$	−9.00000	−0.356873
$637$	0	0
$638$	−27.0000	−1.06894
$639$	3.46410	0.137038
$640$	21.0000	0.830098
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	0	0
$643$	−5.19615	−0.204916	−0.102458	−	0.994737i	$-0.532671\pi$
$643$	−5.19615	−0.204916	−0.102458	+	0.994737i	$0.532671\pi$
$644$	0	0
$645$	−19.0526	−0.750194
$646$	18.0000	0.708201
$647$	−9.00000	−0.353827	−0.176913	−	0.984226i	$-0.556611\pi$
$647$	−9.00000	−0.353827	−0.176913	+	0.984226i	$0.556611\pi$
$648$	−1.73205	−0.0680414
$649$	18.0000	0.706562
$650$	0	0
$651$	0	0
$652$	−12.1244	−0.474826
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	−9.00000	−0.351928
$655$	−25.9808	−1.01515
$656$	−25.9808	−1.01438
$657$	−17.3205	−0.675737
$658$	0	0
$659$	15.0000	0.584317	0.292159	−	0.956370i	$-0.405627\pi$
$659$	15.0000	0.584317	0.292159	+	0.956370i	$0.405627\pi$
$660$	−9.00000	−0.350325
$661$	−36.3731	−1.41475	−0.707374	−	0.706839i	$-0.750120\pi$
$661$	−36.3731	−1.41475	−0.707374	+	0.706839i	$0.750120\pi$
$662$	57.0000	2.21537
$663$	0	0
$664$	6.00000	0.232845
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−1.73205	−0.0670151
$669$	−5.19615	−0.200895
$670$	−25.9808	−1.00372
$671$	36.3731	1.40417
$672$	0	0
$673$	1.00000	0.0385472	0.0192736	−	0.999814i	$-0.493865\pi$
$673$	1.00000	0.0385472	0.0192736	+	0.999814i	$0.493865\pi$
$674$	−38.1051	−1.46775
$675$	10.0000	0.384900
$676$	0	0
$677$	−27.0000	−1.03769	−0.518847	−	0.854867i	$-0.673639\pi$
$677$	−27.0000	−1.03769	−0.518847	+	0.854867i	$0.673639\pi$
$678$	25.9808	0.997785
$679$	0	0
$680$	−18.0000	−0.690268
$681$	17.3205	0.663723
$682$	−15.5885	−0.596913
$683$	−24.2487	−0.927851	−0.463926	−	0.885874i	$-0.653559\pi$
$683$	−24.2487	−0.927851	−0.463926	+	0.885874i	$0.653559\pi$
$684$	−3.46410	−0.132453
$685$	0	0
$686$	0	0
$687$	12.1244	0.462573
$688$	55.0000	2.09686
$689$	0	0
$690$	0	0
$691$	31.1769	1.18603	0.593013	−	0.805193i	$-0.297938\pi$
$691$	31.1769	1.18603	0.593013	+	0.805193i	$0.297938\pi$
$692$	15.0000	0.570214
$693$	0	0
$694$	0	0
$695$	22.5167	0.854106
$696$	−5.19615	−0.196960
$697$	31.1769	1.18091
$698$	−9.00000	−0.340655
$699$	−3.00000	−0.113470
$700$	0	0
$701$	6.00000	0.226617	0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000	0.226617	0.113308	+	0.993560i	$0.463855\pi$
$702$	0	0
$703$	0	0
$704$	−5.19615	−0.195837
$705$	−15.0000	−0.564933
$706$	3.00000	0.112906
$707$	0	0
$708$	−3.46410	−0.130189
$709$	12.1244	0.455340	0.227670	−	0.973738i	$-0.426889\pi$
$709$	12.1244	0.455340	0.227670	+	0.973738i	$0.426889\pi$
$710$	−5.19615	−0.195008
$711$	10.0000	0.375029
$712$	−12.0000	−0.449719
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−3.00000	−0.112115
$717$	10.3923	0.388108
$718$	−33.0000	−1.23155
$719$	−15.0000	−0.559406	−0.279703	−	0.960087i	$-0.590236\pi$
$719$	−15.0000	−0.559406	−0.279703	+	0.960087i	$0.590236\pi$
$720$	17.3205	0.645497
$721$	0	0
$722$	−27.7128	−1.03136
$723$	−6.92820	−0.257663
$724$	2.00000	0.0743294
$725$	−6.00000	−0.222834
$726$	27.7128	1.02852
$727$	−32.0000	−1.18681	−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000	−1.18681	−0.593407	+	0.804902i	$0.702218\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	25.9808	0.961591
$731$	−66.0000	−2.44110
$732$	−7.00000	−0.258727
$733$	50.2295	1.85527	0.927634	−	0.373491i	$-0.121839\pi$
$733$	50.2295	1.85527	0.927634	+	0.373491i	$0.121839\pi$
$734$	−39.8372	−1.47042
$735$	0	0
$736$	0	0
$737$	45.0000	1.65760
$738$	−18.0000	−0.662589
$739$	−39.8372	−1.46543	−0.732717	−	0.680534i	$-0.761748\pi$
$739$	−39.8372	−1.46543	−0.732717	+	0.680534i	$0.761748\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.73205	−0.0635428	−0.0317714	−	0.999495i	$-0.510115\pi$
$743$	−1.73205	−0.0635428	−0.0317714	+	0.999495i	$0.510115\pi$
$744$	−3.00000	−0.109985
$745$	−33.0000	−1.20903
$746$	32.9090	1.20488
$747$	6.92820	0.253490
$748$	−31.1769	−1.13994
$749$	0	0
$750$	−21.0000	−0.766812
$751$	−20.0000	−0.729810	−0.364905	−	0.931045i	$-0.618899\pi$
$751$	−20.0000	−0.729810	−0.364905	+	0.931045i	$0.618899\pi$
$752$	43.3013	1.57903
$753$	3.00000	0.109326
$754$	0	0
$755$	−21.0000	−0.764268
$756$	0	0
$757$	−17.0000	−0.617876	−0.308938	−	0.951082i	$-0.599973\pi$
$757$	−17.0000	−0.617876	−0.308938	+	0.951082i	$0.599973\pi$
$758$	−3.00000	−0.108965
$759$	0	0
$760$	−5.19615	−0.188484
$761$	29.4449	1.06738	0.533688	−	0.845682i	$-0.320806\pi$
$761$	29.4449	1.06738	0.533688	+	0.845682i	$0.320806\pi$
$762$	−22.5167	−0.815693
$763$	0	0
$764$	−15.0000	−0.542681
$765$	−20.7846	−0.751469
$766$	27.0000	0.975550
$767$	0	0
$768$	19.0000	0.685603
$769$	19.0526	0.687053	0.343526	−	0.939143i	$-0.388379\pi$
$769$	19.0526	0.687053	0.343526	+	0.939143i	$0.388379\pi$
$770$	0	0
$771$	30.0000	1.08042
$772$	1.73205	0.0623379
$773$	13.8564	0.498380	0.249190	−	0.968455i	$-0.419836\pi$
$773$	13.8564	0.498380	0.249190	+	0.968455i	$0.419836\pi$
$774$	38.1051	1.36966
$775$	−3.46410	−0.124434
$776$	9.00000	0.323081
$777$	0	0
$778$	−5.19615	−0.186291
$779$	9.00000	0.322458
$780$	0	0
$781$	9.00000	0.322045
$782$	0	0
$783$	−15.0000	−0.536056
$784$	0	0
$785$	−39.8372	−1.42185
$786$	−25.9808	−0.926703
$787$	−31.1769	−1.11134	−0.555668	−	0.831404i	$-0.687538\pi$
$787$	−31.1769	−1.11134	−0.555668	+	0.831404i	$0.687538\pi$
$788$	22.5167	0.802123
$789$	3.00000	0.106803
$790$	−15.0000	−0.533676
$791$	0	0
$792$	−18.0000	−0.639602
$793$	0	0
$794$	63.0000	2.23579
$795$	−15.5885	−0.552866
$796$	4.00000	0.141776
$797$	−33.0000	−1.16892	−0.584460	−	0.811423i	$-0.698694\pi$
$797$	−33.0000	−1.16892	−0.584460	+	0.811423i	$0.698694\pi$
$798$	0	0
$799$	−51.9615	−1.83827
$800$	10.3923	0.367423
$801$	−13.8564	−0.489592
$802$	−12.0000	−0.423735
$803$	−45.0000	−1.58802
$804$	−8.66025	−0.305424
$805$	0	0
$806$	0	0
$807$	6.00000	0.211210
$808$	15.5885	0.548400
$809$	−21.0000	−0.738321	−0.369160	−	0.929366i	$-0.620355\pi$
$809$	−21.0000	−0.738321	−0.369160	+	0.929366i	$0.620355\pi$
$810$	3.00000	0.105409
$811$	3.46410	0.121641	0.0608205	−	0.998149i	$-0.480628\pi$
$811$	3.46410	0.121641	0.0608205	+	0.998149i	$0.480628\pi$
$812$	0	0
$813$	17.3205	0.607457
$814$	0	0
$815$	−21.0000	−0.735598
$816$	−30.0000	−1.05021
$817$	−19.0526	−0.666565
$818$	−12.0000	−0.419570
$819$	0	0
$820$	9.00000	0.314294
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	−32.0000	−1.11545	−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	22.5167	0.784405
$825$	10.3923	0.361814
$826$	0	0
$827$	−10.3923	−0.361376	−0.180688	−	0.983540i	$-0.557832\pi$
$827$	−10.3923	−0.361376	−0.180688	+	0.983540i	$0.557832\pi$
$828$	0	0
$829$	7.00000	0.243120	0.121560	−	0.992584i	$-0.461210\pi$
$829$	7.00000	0.243120	0.121560	+	0.992584i	$0.461210\pi$
$830$	−10.3923	−0.360722
$831$	10.0000	0.346896
$832$	0	0
$833$	0	0
$834$	22.5167	0.779688
$835$	−3.00000	−0.103819
$836$	−9.00000	−0.311272
$837$	−8.66025	−0.299342
$838$	−36.3731	−1.25649
$839$	−1.73205	−0.0597970	−0.0298985	−	0.999553i	$-0.509518\pi$
$839$	−1.73205	−0.0597970	−0.0298985	+	0.999553i	$0.509518\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	6.92820	0.238620
$844$	13.0000	0.447478
$845$	0	0
$846$	30.0000	1.03142
$847$	0	0
$848$	45.0000	1.54531
$849$	19.0000	0.652078
$850$	−20.7846	−0.712906
$851$	0	0
$852$	−1.73205	−0.0593391
$853$	41.5692	1.42330	0.711651	−	0.702533i	$-0.247948\pi$
$853$	41.5692	1.42330	0.711651	+	0.702533i	$0.247948\pi$
$854$	0	0
$855$	−6.00000	−0.205196
$856$	0	0
$857$	−33.0000	−1.12726	−0.563629	−	0.826028i	$-0.690595\pi$
$857$	−33.0000	−1.12726	−0.563629	+	0.826028i	$0.690595\pi$
$858$	0	0
$859$	29.0000	0.989467	0.494734	−	0.869045i	$-0.335266\pi$
$859$	29.0000	0.989467	0.494734	+	0.869045i	$0.335266\pi$
$860$	−19.0526	−0.649687
$861$	0	0
$862$	57.0000	1.94143
$863$	−1.73205	−0.0589597	−0.0294798	−	0.999565i	$-0.509385\pi$
$863$	−1.73205	−0.0589597	−0.0294798	+	0.999565i	$0.509385\pi$
$864$	25.9808	0.883883
$865$	25.9808	0.883372
$866$	32.9090	1.11829
$867$	19.0000	0.645274
$868$	0	0
$869$	25.9808	0.881337
$870$	9.00000	0.305129
$871$	0	0
$872$	9.00000	0.304778
$873$	10.3923	0.351726
$874$	0	0
$875$	0	0
$876$	8.66025	0.292603
$877$	1.73205	0.0584872	0.0292436	−	0.999572i	$-0.490690\pi$
$877$	1.73205	0.0584872	0.0292436	+	0.999572i	$0.490690\pi$
$878$	13.8564	0.467631
$879$	25.9808	0.876309
$880$	45.0000	1.51695
$881$	−33.0000	−1.11180	−0.555899	−	0.831250i	$-0.687626\pi$
$881$	−33.0000	−1.11180	−0.555899	+	0.831250i	$0.687626\pi$
$882$	0	0
$883$	44.0000	1.48072	0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000	1.48072	0.740359	+	0.672212i	$0.234656\pi$
$884$	0	0
$885$	−6.00000	−0.201688
$886$	25.9808	0.872841
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	0	0
$889$	0	0
$890$	20.7846	0.696702
$891$	−5.19615	−0.174078
$892$	−5.19615	−0.173980
$893$	−15.0000	−0.501956
$894$	−33.0000	−1.10369
$895$	−5.19615	−0.173688
$896$	0	0
$897$	0	0
$898$	3.00000	0.100111
$899$	5.19615	0.173301
$900$	4.00000	0.133333
$901$	−54.0000	−1.79900
$902$	−46.7654	−1.55712
$903$	0	0
$904$	−25.9808	−0.864107
$905$	3.46410	0.115151
$906$	−21.0000	−0.697678
$907$	29.0000	0.962929	0.481465	−	0.876466i	$-0.340105\pi$
$907$	29.0000	0.962929	0.481465	+	0.876466i	$0.340105\pi$
$908$	17.3205	0.574801
$909$	18.0000	0.597022
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	−8.66025	−0.286770
$913$	18.0000	0.595713
$914$	60.0000	1.98462
$915$	−12.1244	−0.400819
$916$	12.1244	0.400600
$917$	0	0
$918$	−51.9615	−1.71499
$919$	−47.0000	−1.55039	−0.775193	−	0.631724i	$-0.782348\pi$
$919$	−47.0000	−1.55039	−0.775193	+	0.631724i	$0.782348\pi$
$920$	0	0
$921$	24.2487	0.799022
$922$	51.0000	1.67960
$923$	0	0
$924$	0	0
$925$	0	0
$926$	42.0000	1.38021
$927$	26.0000	0.853952
$928$	−15.5885	−0.511716
$929$	46.7654	1.53432	0.767161	−	0.641455i	$-0.221669\pi$
$929$	46.7654	1.53432	0.767161	+	0.641455i	$0.221669\pi$
$930$	5.19615	0.170389
$931$	0	0
$932$	−3.00000	−0.0982683
$933$	−15.0000	−0.491078
$934$	−36.3731	−1.19016
$935$	−54.0000	−1.76599
$936$	0	0
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	0	0
$939$	−19.0000	−0.620042
$940$	−15.0000	−0.489246
$941$	5.19615	0.169390	0.0846949	−	0.996407i	$-0.473008\pi$
$941$	5.19615	0.169390	0.0846949	+	0.996407i	$0.473008\pi$
$942$	−39.8372	−1.29797
$943$	0	0
$944$	17.3205	0.563735
$945$	0	0
$946$	99.0000	3.21877
$947$	−38.1051	−1.23825	−0.619125	−	0.785292i	$-0.712513\pi$
$947$	−38.1051	−1.23825	−0.619125	+	0.785292i	$0.712513\pi$
$948$	−5.00000	−0.162392
$949$	0	0
$950$	−6.00000	−0.194666
$951$	5.19615	0.168497
$952$	0	0
$953$	57.0000	1.84641	0.923206	−	0.384307i	$-0.125559\pi$
$953$	57.0000	1.84641	0.923206	+	0.384307i	$0.125559\pi$
$954$	31.1769	1.00939
$955$	−25.9808	−0.840718
$956$	10.3923	0.336111
$957$	−15.5885	−0.503903
$958$	−51.0000	−1.64774
$959$	0	0
$960$	1.73205	0.0559017
$961$	−28.0000	−0.903226
$962$	0	0
$963$	0	0
$964$	−6.92820	−0.223142
$965$	3.00000	0.0965734
$966$	0	0
$967$	−10.3923	−0.334194	−0.167097	−	0.985940i	$-0.553439\pi$
$967$	−10.3923	−0.334194	−0.167097	+	0.985940i	$0.553439\pi$
$968$	−27.7128	−0.890724
$969$	10.3923	0.333849
$970$	−15.5885	−0.500515
$971$	−21.0000	−0.673922	−0.336961	−	0.941519i	$-0.609399\pi$
$971$	−21.0000	−0.673922	−0.336961	+	0.941519i	$0.609399\pi$
$972$	16.0000	0.513200
$973$	0	0
$974$	−42.0000	−1.34577
$975$	0	0
$976$	35.0000	1.12032
$977$	19.0526	0.609545	0.304773	−	0.952425i	$-0.401420\pi$
$977$	19.0526	0.609545	0.304773	+	0.952425i	$0.401420\pi$
$978$	−21.0000	−0.671506
$979$	−36.0000	−1.15056
$980$	0	0
$981$	10.3923	0.331801
$982$	46.7654	1.49234
$983$	−36.3731	−1.16012	−0.580060	−	0.814574i	$-0.696971\pi$
$983$	−36.3731	−1.16012	−0.580060	+	0.814574i	$0.696971\pi$
$984$	−9.00000	−0.286910
$985$	39.0000	1.24264
$986$	31.1769	0.992875
$987$	0	0
$988$	0	0
$989$	0	0
$990$	31.1769	0.990867
$991$	59.0000	1.87420	0.937098	−	0.349065i	$-0.113501\pi$
$991$	59.0000	1.87420	0.937098	+	0.349065i	$0.113501\pi$
$992$	−9.00000	−0.285750
$993$	32.9090	1.04433
$994$	0	0
$995$	6.92820	0.219639
$996$	−3.46410	−0.109764
$997$	−2.00000	−0.0633406	−0.0316703	−	0.999498i	$-0.510083\pi$
$997$	−2.00000	−0.0633406	−0.0316703	+	0.999498i	$0.510083\pi$
$998$	−3.00000	−0.0949633
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.w.1.2		2
7.3	odd	6		1183.2.e.e.170.1		4
7.5	odd	6		1183.2.e.e.508.1		4
7.6	odd	2		8281.2.a.s.1.2		2
13.2	odd	12		637.2.q.b.589.1		2
13.7	odd	12		637.2.q.b.491.1		2
13.12	even	2	inner	8281.2.a.w.1.1		2
91.2	odd	12		637.2.k.b.459.1		2
91.12	odd	6		1183.2.e.e.508.2		4
91.20	even	12		637.2.q.c.491.1		2
91.33	even	12		91.2.u.a.88.1	yes	2
91.38	odd	6		1183.2.e.e.170.2		4
91.41	even	12		637.2.q.c.589.1		2
91.46	odd	12		637.2.k.b.569.1		2
91.54	even	12		91.2.k.a.4.1	✓	2
91.59	even	12		91.2.k.a.23.1	yes	2
91.67	odd	12		637.2.u.a.30.1		2
91.72	odd	12		637.2.u.a.361.1		2
91.80	even	12		91.2.u.a.30.1	yes	2
91.90	odd	2		8281.2.a.s.1.1		2
273.59	odd	12		819.2.bm.a.478.1		2
273.80	odd	12		819.2.do.c.667.1		2
273.215	odd	12		819.2.do.c.361.1		2
273.236	odd	12		819.2.bm.a.550.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.k.a.4.1	✓	2	91.54	even	12
91.2.k.a.23.1	yes	2	91.59	even	12
91.2.u.a.30.1	yes	2	91.80	even	12
91.2.u.a.88.1	yes	2	91.33	even	12
637.2.k.b.459.1		2	91.2	odd	12
637.2.k.b.569.1		2	91.46	odd	12
637.2.q.b.491.1		2	13.7	odd	12
637.2.q.b.589.1		2	13.2	odd	12
637.2.q.c.491.1		2	91.20	even	12
637.2.q.c.589.1		2	91.41	even	12
637.2.u.a.30.1		2	91.67	odd	12
637.2.u.a.361.1		2	91.72	odd	12
819.2.bm.a.478.1		2	273.59	odd	12
819.2.bm.a.550.1		2	273.236	odd	12
819.2.do.c.361.1		2	273.215	odd	12
819.2.do.c.667.1		2	273.80	odd	12
1183.2.e.e.170.1		4	7.3	odd	6
1183.2.e.e.170.2		4	91.38	odd	6
1183.2.e.e.508.1		4	7.5	odd	6
1183.2.e.e.508.2		4	91.12	odd	6
8281.2.a.s.1.1		2	91.90	odd	2
8281.2.a.s.1.2		2	7.6	odd	2
8281.2.a.w.1.1		2	13.12	even	2	inner
8281.2.a.w.1.2		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:	\( N \)	\(=\)	\( 8281 = 7^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8281.a (trivial)

\(f(q)\)	\(=\)	\(q+1.73205 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.73205 q^{5} +1.73205 q^{6} -1.73205 q^{8} -2.00000 q^{9} +O(q^{10})\)	\(q+1.73205 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.73205 q^{5} +1.73205 q^{6} -1.73205 q^{8} -2.00000 q^{9} +3.00000 q^{10} -5.19615 q^{11} +1.00000 q^{12} +1.73205 q^{15} -5.00000 q^{16} +6.00000 q^{17} -3.46410 q^{18} +1.73205 q^{19} +1.73205 q^{20} -9.00000 q^{22} -1.73205 q^{24} -2.00000 q^{25} -5.00000 q^{27} +3.00000 q^{29} +3.00000 q^{30} +1.73205 q^{31} -5.19615 q^{32} -5.19615 q^{33} +10.3923 q^{34} -2.00000 q^{36} +3.00000 q^{38} -3.00000 q^{40} +5.19615 q^{41} -11.0000 q^{43} -5.19615 q^{44} -3.46410 q^{45} -8.66025 q^{47} -5.00000 q^{48} -3.46410 q^{50} +6.00000 q^{51} -9.00000 q^{53} -8.66025 q^{54} -9.00000 q^{55} +1.73205 q^{57} +5.19615 q^{58} -3.46410 q^{59} +1.73205 q^{60} -7.00000 q^{61} +3.00000 q^{62} +1.00000 q^{64} -9.00000 q^{66} -8.66025 q^{67} +6.00000 q^{68} -1.73205 q^{71} +3.46410 q^{72} +8.66025 q^{73} -2.00000 q^{75} +1.73205 q^{76} -5.00000 q^{79} -8.66025 q^{80} +1.00000 q^{81} +9.00000 q^{82} -3.46410 q^{83} +10.3923 q^{85} -19.0526 q^{86} +3.00000 q^{87} +9.00000 q^{88} +6.92820 q^{89} -6.00000 q^{90} +1.73205 q^{93} -15.0000 q^{94} +3.00000 q^{95} -5.19615 q^{96} -5.19615 q^{97} +10.3923 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{4} - 4 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^4 - 4 * q^9	\( 2 q + 2 q^{3} + 2 q^{4} - 4 q^{9} + 6 q^{10} + 2 q^{12} - 10 q^{16} + 12 q^{17} - 18 q^{22} - 4 q^{25} - 10 q^{27} + 6 q^{29} + 6 q^{30} - 4 q^{36} + 6 q^{38} - 6 q^{40} - 22 q^{43} - 10 q^{48} + 12 q^{51} - 18 q^{53} - 18 q^{55} - 14 q^{61} + 6 q^{62} + 2 q^{64} - 18 q^{66} + 12 q^{68} - 4 q^{75} - 10 q^{79} + 2 q^{81} + 18 q^{82} + 6 q^{87} + 18 q^{88} - 12 q^{90} - 30 q^{94} + 6 q^{95}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^4 - 4 * q^9 + 6 * q^10 + 2 * q^12 - 10 * q^16 + 12 * q^17 - 18 * q^22 - 4 * q^25 - 10 * q^27 + 6 * q^29 + 6 * q^30 - 4 * q^36 + 6 * q^38 - 6 * q^40 - 22 * q^43 - 10 * q^48 + 12 * q^51 - 18 * q^53 - 18 * q^55 - 14 * q^61 + 6 * q^62 + 2 * q^64 - 18 * q^66 + 12 * q^68 - 4 * q^75 - 10 * q^79 + 2 * q^81 + 18 * q^82 + 6 * q^87 + 18 * q^88 - 12 * q^90 - 30 * q^94 + 6 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more