LMFDB - Embedded newform 8281.2.a.bw.1.4

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:	$5$
Coefficient field:	5.5.746052.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 7x^{3} + 8x + 2 $ x^5 - x^4 - 7x^3 + 8x + 2
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			1.4
Root			$1.19566$ 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)

$f(q)$	$=$	\(q+0.195656 q^{2} -0.259788 q^{3} -1.96172 q^{4} -3.93251 q^{5} -0.0508292 q^{6} -0.775135 q^{8} -2.93251 q^{9} -0.769420 q^{10} -4.50627 q^{11} +0.509632 q^{12} +1.02162 q^{15} +3.77178 q^{16} -2.28141 q^{17} -0.573764 q^{18} +1.78768 q^{19} +7.71448 q^{20} -0.881681 q^{22} +1.74021 q^{23} +0.201371 q^{24} +10.4646 q^{25} +1.54120 q^{27} +1.65110 q^{29} +0.199886 q^{30} -5.60523 q^{31} +2.28824 q^{32} +1.17068 q^{33} -0.446372 q^{34} +5.75276 q^{36} -7.14407 q^{37} +0.349771 q^{38} +3.04823 q^{40} +8.11574 q^{41} +6.81353 q^{43} +8.84004 q^{44} +11.5321 q^{45} +0.340483 q^{46} -3.54543 q^{47} -0.979864 q^{48} +2.04747 q^{50} +0.592684 q^{51} +3.28965 q^{53} +0.301545 q^{54} +17.7210 q^{55} -0.464419 q^{57} +0.323048 q^{58} -4.50627 q^{59} -2.00413 q^{60} +7.54467 q^{61} -1.09670 q^{62} -7.09585 q^{64} +0.229050 q^{66} +12.6653 q^{67} +4.47548 q^{68} -0.452087 q^{69} -9.54869 q^{71} +2.27309 q^{72} -1.08004 q^{73} -1.39778 q^{74} -2.71859 q^{75} -3.50693 q^{76} +0.791698 q^{79} -14.8326 q^{80} +8.39714 q^{81} +1.58789 q^{82} +7.14643 q^{83} +8.97166 q^{85} +1.33311 q^{86} -0.428937 q^{87} +3.49297 q^{88} +11.2656 q^{89} +2.25633 q^{90} -3.41381 q^{92} +1.45617 q^{93} -0.693685 q^{94} -7.03008 q^{95} -0.594459 q^{96} +8.81353 q^{97} +13.2147 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 4 q^{2} + 8 q^{4} - 2 q^{5} + 5 q^{6} - 9 q^{8} + 3 q^{9} - 5 q^{10} - 11 q^{11} + 5 q^{12} + 10 q^{16} - 5 q^{17} - 9 q^{18} - 9 q^{19} - q^{20} + 8 q^{22} + 10 q^{23} + 9 q^{25} - 3 q^{29} - 13 q^{30}+ \cdots - 11 q^{99}+O(q^{100}) $ 5 * q - 4 * q^2 + 8 * q^4 - 2 * q^5 + 5 * q^6 - 9 * q^8 + 3 * q^9 - 5 * q^10 - 11 * q^11 + 5 * q^12 + 10 * q^16 - 5 * q^17 - 9 * q^18 - 9 * q^19 - q^20 + 8 * q^22 + 10 * q^23 + 9 * q^25 - 3 * q^29 - 13 * q^30 + 6 * q^31 - 22 * q^32 - 8 * q^33 + 22 * q^34 + 7 * q^36 - 4 * q^37 - 10 * q^38 + 28 * q^40 - 14 * q^41 + 2 * q^43 + 32 * q^45 - 3 * q^46 - q^47 - 23 * q^48 - 9 * q^50 - 8 * q^51 + 17 * q^53 - 23 * q^54 + 16 * q^57 + 27 * q^58 - 11 * q^59 + 29 * q^60 - 11 * q^61 - 23 * q^62 + 9 * q^64 + 21 * q^66 - 13 * q^67 - 32 * q^68 + 18 * q^69 - 15 * q^71 + 19 * q^72 - 33 * q^74 - 20 * q^75 - 8 * q^76 + 2 * q^79 - 55 * q^80 - 19 * q^81 + 34 * q^82 - 6 * q^83 + 22 * q^85 - 28 * q^86 - 8 * q^87 - 3 * q^88 + 4 * q^89 - 34 * q^90 + 21 * q^92 - 18 * q^93 + 20 * q^94 - 12 * q^95 + 37 * q^96 + 12 * q^97 - 11 * q^99

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 $

$2$	0.195656	0.138350	0.0691749	−	0.997605i	$-0.477963\pi$
$2$	0.195656	0.138350	0.0691749	+	0.997605i	$0.477963\pi$
$3$	−0.259788	−0.149989	−0.0749945	−	0.997184i	$-0.523894\pi$
$3$	−0.259788	−0.149989	−0.0749945	+	0.997184i	$0.523894\pi$
$4$	−1.96172	−0.980859
$5$	−3.93251	−1.75867	−0.879336	−	0.476202i	$-0.842013\pi$
$5$	−3.93251	−1.75867	−0.879336	+	0.476202i	$0.842013\pi$
$6$	−0.0508292	−0.0207509
$7$	0	0
$7$	0	0
$8$	−0.775135	−0.274052
$9$	−2.93251	−0.977503
$10$	−0.769420	−0.243312
$11$	−4.50627	−1.35869	−0.679346	−	0.733818i	$-0.737737\pi$
$11$	−4.50627	−1.35869	−0.679346	+	0.733818i	$0.737737\pi$
$12$	0.509632	0.147118
$13$	0	0
$13$	0	0
$14$	0	0
$15$	1.02162	0.263781
$16$	3.77178	0.942944
$17$	−2.28141	−0.553323	−0.276661	−	0.960967i	$-0.589228\pi$
$17$	−2.28141	−0.553323	−0.276661	+	0.960967i	$0.589228\pi$
$18$	−0.573764	−0.135237
$19$	1.78768	0.410123	0.205061	−	0.978749i	$-0.434261\pi$
$19$	1.78768	0.410123	0.205061	+	0.978749i	$0.434261\pi$
$20$	7.71448	1.72501
$21$	0	0
$22$	−0.881681	−0.187975
$23$	1.74021	0.362859	0.181430	−	0.983404i	$-0.441928\pi$
$23$	1.74021	0.362859	0.181430	+	0.983404i	$0.441928\pi$
$24$	0.201371	0.0411047
$25$	10.4646	2.09293
$26$	0	0
$27$	1.54120	0.296604
$28$	0	0
$29$	1.65110	0.306602	0.153301	−	0.988180i	$-0.451010\pi$
$29$	1.65110	0.306602	0.153301	+	0.988180i	$0.451010\pi$
$30$	0.199886	0.0364941
$31$	−5.60523	−1.00673	−0.503365	−	0.864074i	$-0.667905\pi$
$31$	−5.60523	−1.00673	−0.503365	+	0.864074i	$0.667905\pi$
$32$	2.28824	0.404508
$33$	1.17068	0.203789
$34$	−0.446372	−0.0765522
$35$	0	0
$36$	5.75276	0.958793
$37$	−7.14407	−1.17448	−0.587239	−	0.809414i	$-0.699785\pi$
$37$	−7.14407	−1.17448	−0.587239	+	0.809414i	$0.699785\pi$
$38$	0.349771	0.0567404
$39$	0	0
$40$	3.04823	0.481967
$41$	8.11574	1.26746	0.633732	−	0.773552i	$-0.281522\pi$
$41$	8.11574	1.26746	0.633732	+	0.773552i	$0.281522\pi$
$42$	0	0
$43$	6.81353	1.03905	0.519527	−	0.854454i	$-0.326108\pi$
$43$	6.81353	1.03905	0.519527	+	0.854454i	$0.326108\pi$
$44$	8.84004	1.33269
$45$	11.5321	1.71911
$46$	0.340483	0.0502015
$47$	−3.54543	−0.517154	−0.258577	−	0.965991i	$-0.583254\pi$
$47$	−3.54543	−0.517154	−0.258577	+	0.965991i	$0.583254\pi$
$48$	−0.979864	−0.141431
$49$	0	0
$50$	2.04747	0.289556
$51$	0.592684	0.0829923
$52$	0	0
$53$	3.28965	0.451869	0.225934	−	0.974143i	$-0.427457\pi$
$53$	3.28965	0.451869	0.225934	+	0.974143i	$0.427457\pi$
$54$	0.301545	0.0410351
$55$	17.7210	2.38949
$56$	0	0
$57$	−0.464419	−0.0615138
$58$	0.323048	0.0424183
$59$	−4.50627	−0.586667	−0.293333	−	0.956010i	$-0.594765\pi$
$59$	−4.50627	−0.586667	−0.293333	+	0.956010i	$0.594765\pi$
$60$	−2.00413	−0.258732
$61$	7.54467	0.965996	0.482998	−	0.875621i	$-0.339548\pi$
$61$	7.54467	0.965996	0.482998	+	0.875621i	$0.339548\pi$
$62$	−1.09670	−0.139281
$63$	0	0
$64$	−7.09585	−0.886981
$65$	0	0
$66$	0.229050	0.0281942
$67$	12.6653	1.54731	0.773653	−	0.633609i	$-0.218427\pi$
$67$	12.6653	1.54731	0.773653	+	0.633609i	$0.218427\pi$
$68$	4.47548	0.542732
$69$	−0.452087	−0.0544249
$70$	0	0
$71$	−9.54869	−1.13322	−0.566610	−	0.823986i	$-0.691746\pi$
$71$	−9.54869	−1.13322	−0.566610	+	0.823986i	$0.691746\pi$
$72$	2.27309	0.267886
$73$	−1.08004	−0.126409	−0.0632044	−	0.998001i	$-0.520132\pi$
$73$	−1.08004	−0.126409	−0.0632044	+	0.998001i	$0.520132\pi$
$74$	−1.39778	−0.162489
$75$	−2.71859	−0.313916
$76$	−3.50693	−0.402273
$77$	0	0
$78$	0	0
$79$	0.791698	0.0890730	0.0445365	−	0.999008i	$-0.485819\pi$
$79$	0.791698	0.0890730	0.0445365	+	0.999008i	$0.485819\pi$
$80$	−14.8326	−1.65833
$81$	8.39714	0.933016
$82$	1.58789	0.175354
$83$	7.14643	0.784422	0.392211	−	0.919875i	$-0.371710\pi$
$83$	7.14643	0.784422	0.392211	+	0.919875i	$0.371710\pi$
$84$	0	0
$85$	8.97166	0.973114
$86$	1.33311	0.143753
$87$	−0.428937	−0.0459869
$88$	3.49297	0.372352
$89$	11.2656	1.19415	0.597077	−	0.802184i	$-0.296329\pi$
$89$	11.2656	1.19415	0.597077	+	0.802184i	$0.296329\pi$
$90$	2.25633	0.237838
$91$	0	0
$92$	−3.41381	−0.355914
$93$	1.45617	0.150998
$94$	−0.693685	−0.0715482
$95$	−7.03008	−0.721271
$96$	−0.594459	−0.0606717
$97$	8.81353	0.894879	0.447439	−	0.894314i	$-0.352336\pi$
$97$	8.81353	0.894879	0.447439	+	0.894314i	$0.352336\pi$
$98$	0	0
$99$	13.2147	1.32813
$100$	−20.5287	−2.05287
$101$	−14.3171	−1.42461	−0.712303	−	0.701872i	$-0.752348\pi$
$101$	−14.3171	−1.42461	−0.712303	+	0.701872i	$0.752348\pi$
$102$	0.115962	0.0114820
$103$	−7.49214	−0.738223	−0.369111	−	0.929385i	$-0.620338\pi$
$103$	−7.49214	−0.738223	−0.369111	+	0.929385i	$0.620338\pi$
$104$	0	0
$105$	0	0
$106$	0.643641	0.0625160
$107$	10.9784	1.06132	0.530660	−	0.847585i	$-0.321944\pi$
$107$	10.9784	1.06132	0.530660	+	0.847585i	$0.321944\pi$
$108$	−3.02340	−0.290926
$109$	12.4463	1.19214	0.596068	−	0.802934i	$-0.296729\pi$
$109$	12.4463	1.19214	0.596068	+	0.802934i	$0.296729\pi$
$110$	3.46722	0.330586
$111$	1.85595	0.176159
$112$	0	0
$113$	−1.65110	−0.155323	−0.0776613	−	0.996980i	$-0.524745\pi$
$113$	−1.65110	−0.155323	−0.0776613	+	0.996980i	$0.524745\pi$
$114$	−0.0908666	−0.00851043
$115$	−6.84340	−0.638150
$116$	−3.23900	−0.300733
$117$	0	0
$118$	−0.881681	−0.0811653
$119$	0	0
$120$	−0.791894	−0.0722897
$121$	9.30650	0.846046
$122$	1.47616	0.133645
$123$	−2.10837	−0.190106
$124$	10.9959	0.987460
$125$	−21.4897	−1.92210
$126$	0	0
$127$	−4.49297	−0.398687	−0.199343	−	0.979930i	$-0.563881\pi$
$127$	−4.49297	−0.398687	−0.199343	+	0.979930i	$0.563881\pi$
$128$	−5.96483	−0.527222
$129$	−1.77008	−0.155847
$130$	0	0
$131$	−12.6567	−1.10582	−0.552911	−	0.833240i	$-0.686483\pi$
$131$	−12.6567	−1.10582	−0.552911	+	0.833240i	$0.686483\pi$
$132$	−2.29654	−0.199888
$133$	0	0
$134$	2.47804	0.214070
$135$	−6.06077	−0.521628
$136$	1.76840	0.151639
$137$	9.28641	0.793392	0.396696	−	0.917950i	$-0.370157\pi$
$137$	9.28641	0.793392	0.396696	+	0.917950i	$0.370157\pi$
$138$	−0.0884536	−0.00752967
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	0.921061	0.0775673
$142$	−1.86826	−0.156781
$143$	0	0
$144$	−11.0608	−0.921731
$145$	−6.49297	−0.539212
$146$	−0.211316	−0.0174887
$147$	0	0
$148$	14.0147	1.15200
$149$	15.1649	1.24235	0.621177	−	0.783670i	$-0.286655\pi$
$149$	15.1649	1.24235	0.621177	+	0.783670i	$0.286655\pi$
$150$	−0.531909	−0.0434302
$151$	−5.14159	−0.418416	−0.209208	−	0.977871i	$-0.567089\pi$
$151$	−5.14159	−0.418416	−0.209208	+	0.977871i	$0.567089\pi$
$152$	−1.38570	−0.112395
$153$	6.69025	0.540875
$154$	0	0
$155$	22.0426	1.77051
$156$	0	0
$157$	−10.7311	−0.856438	−0.428219	−	0.903675i	$-0.640859\pi$
$157$	−10.7311	−0.856438	−0.428219	+	0.903675i	$0.640859\pi$
$158$	0.154901	0.0123232
$159$	−0.854614	−0.0677753
$160$	−8.99853	−0.711397
$161$	0	0
$162$	1.64295	0.129083
$163$	−2.37239	−0.185820	−0.0929101	−	0.995675i	$-0.529617\pi$
$163$	−2.37239	−0.185820	−0.0929101	+	0.995675i	$0.529617\pi$
$164$	−15.9208	−1.24320
$165$	−4.60370	−0.358398
$166$	1.39824	0.108525
$167$	−12.0784	−0.934653	−0.467327	−	0.884085i	$-0.654783\pi$
$167$	−12.0784	−0.934653	−0.467327	+	0.884085i	$0.654783\pi$
$168$	0	0
$169$	0	0
$170$	1.75536	0.134630
$171$	−5.24240	−0.400896
$172$	−13.3662	−1.01917
$173$	−19.4097	−1.47569	−0.737846	−	0.674969i	$-0.764157\pi$
$173$	−19.4097	−1.47569	−0.737846	+	0.674969i	$0.764157\pi$
$174$	−0.0839242	−0.00636228
$175$	0	0
$176$	−16.9967	−1.28117
$177$	1.17068	0.0879935
$178$	2.20419	0.165211
$179$	14.6444	1.09457	0.547286	−	0.836945i	$-0.315661\pi$
$179$	14.6444	1.09457	0.547286	+	0.836945i	$0.315661\pi$
$180$	−22.6228	−1.68620
$181$	−9.44627	−0.702136	−0.351068	−	0.936350i	$-0.614181\pi$
$181$	−9.44627	−0.702136	−0.351068	+	0.936350i	$0.614181\pi$
$182$	0	0
$183$	−1.96002	−0.144889
$184$	−1.34890	−0.0994422
$185$	28.0941	2.06552
$186$	0.284910	0.0208906
$187$	10.2807	0.751796
$188$	6.95513	0.507255
$189$	0	0
$190$	−1.37548	−0.0997878
$191$	12.5537	0.908357	0.454179	−	0.890911i	$-0.349933\pi$
$191$	12.5537	0.908357	0.454179	+	0.890911i	$0.349933\pi$
$192$	1.84342	0.133037
$193$	9.36859	0.674366	0.337183	−	0.941439i	$-0.390526\pi$
$193$	9.36859	0.674366	0.337183	+	0.941439i	$0.390526\pi$
$194$	1.72442	0.123806
$195$	0	0
$196$	0	0
$197$	7.62276	0.543099	0.271550	−	0.962424i	$-0.412464\pi$
$197$	7.62276	0.543099	0.271550	+	0.962424i	$0.412464\pi$
$198$	2.58554	0.183746
$199$	13.5289	0.959036	0.479518	−	0.877532i	$-0.340812\pi$
$199$	13.5289	0.959036	0.479518	+	0.877532i	$0.340812\pi$
$200$	−8.11151	−0.573570
$201$	−3.29029	−0.232079
$202$	−2.80123	−0.197094
$203$	0	0
$204$	−1.16268	−0.0814038
$205$	−31.9152	−2.22906
$206$	−1.46588	−0.102133
$207$	−5.10319	−0.354696
$208$	0	0
$209$	−8.05579	−0.557231
$210$	0	0
$211$	−15.7995	−1.08768	−0.543840	−	0.839189i	$-0.683030\pi$
$211$	−15.7995	−1.08768	−0.543840	+	0.839189i	$0.683030\pi$
$212$	−6.45338	−0.443220
$213$	2.48064	0.169971
$214$	2.14799	0.146833
$215$	−26.7943	−1.82736
$216$	−1.19464	−0.0812847
$217$	0	0
$218$	2.43519	0.164932
$219$	0.280581	0.0189599
$220$	−34.7636	−2.34376
$221$	0	0
$222$	0.363128	0.0243715
$223$	−22.4737	−1.50495	−0.752474	−	0.658622i	$-0.771139\pi$
$223$	−22.4737	−1.50495	−0.752474	+	0.658622i	$0.771139\pi$
$224$	0	0
$225$	−30.6876	−2.04584
$226$	−0.323048	−0.0214889
$227$	9.20249	0.610791	0.305395	−	0.952226i	$-0.401211\pi$
$227$	9.20249	0.610791	0.305395	+	0.952226i	$0.401211\pi$
$228$	0.911060	0.0603364
$229$	−15.2922	−1.01054	−0.505269	−	0.862962i	$-0.668607\pi$
$229$	−15.2922	−1.01054	−0.505269	+	0.862962i	$0.668607\pi$
$230$	−1.33895	−0.0882880
$231$	0	0
$232$	−1.27983	−0.0840247
$233$	−8.05579	−0.527752	−0.263876	−	0.964557i	$-0.585001\pi$
$233$	−8.05579	−0.527752	−0.263876	+	0.964557i	$0.585001\pi$
$234$	0	0
$235$	13.9424	0.909504
$236$	8.84004	0.575438
$237$	−0.205674	−0.0133600
$238$	0	0
$239$	−21.7258	−1.40533	−0.702663	−	0.711523i	$-0.748006\pi$
$239$	−21.7258	−1.40533	−0.702663	+	0.711523i	$0.748006\pi$
$240$	3.85333	0.248731
$241$	20.4980	1.32039	0.660195	−	0.751094i	$-0.270473\pi$
$241$	20.4980	1.32039	0.660195	+	0.751094i	$0.270473\pi$
$242$	1.82088	0.117050
$243$	−6.80507	−0.436546
$244$	−14.8005	−0.947507
$245$	0	0
$246$	−0.412517	−0.0263011
$247$	0	0
$248$	4.34481	0.275896
$249$	−1.85656	−0.117655
$250$	−4.20460	−0.265922
$251$	2.60871	0.164660	0.0823301	−	0.996605i	$-0.473764\pi$
$251$	2.60871	0.164660	0.0823301	+	0.996605i	$0.473764\pi$
$252$	0	0
$253$	−7.84187	−0.493014
$254$	−0.879078	−0.0551583
$255$	−2.33073	−0.145956
$256$	13.0246	0.814040
$257$	−8.99676	−0.561202	−0.280601	−	0.959824i	$-0.590534\pi$
$257$	−8.99676	−0.561202	−0.280601	+	0.959824i	$0.590534\pi$
$258$	−0.346327	−0.0215614
$259$	0	0
$260$	0	0
$261$	−4.84187	−0.299704
$262$	−2.47637	−0.152990
$263$	1.43392	0.0884194	0.0442097	−	0.999022i	$-0.485923\pi$
$263$	1.43392	0.0884194	0.0442097	+	0.999022i	$0.485923\pi$
$264$	−0.907433	−0.0558487
$265$	−12.9366	−0.794689
$266$	0	0
$267$	−2.92668	−0.179110
$268$	−24.8457	−1.51769
$269$	−8.16832	−0.498031	−0.249016	−	0.968499i	$-0.580107\pi$
$269$	−8.16832	−0.498031	−0.249016	+	0.968499i	$0.580107\pi$
$270$	−1.18583	−0.0721672
$271$	0.212317	0.0128973	0.00644867	−	0.999979i	$-0.497947\pi$
$271$	0.212317	0.0128973	0.00644867	+	0.999979i	$0.497947\pi$
$272$	−8.60497	−0.521753
$273$	0	0
$274$	1.81695	0.109766
$275$	−47.1565	−2.84364
$276$	0.886867	0.0533831
$277$	22.9749	1.38043	0.690215	−	0.723604i	$-0.257516\pi$
$277$	22.9749	1.38043	0.690215	+	0.723604i	$0.257516\pi$
$278$	−0.782625	−0.0469387
$279$	16.4374	0.984081
$280$	0	0
$281$	0.345228	0.0205946	0.0102973	−	0.999947i	$-0.496722\pi$
$281$	0.345228	0.0205946	0.0102973	+	0.999947i	$0.496722\pi$
$282$	0.180211	0.0107314
$283$	−28.9715	−1.72217	−0.861087	−	0.508457i	$-0.830216\pi$
$283$	−28.9715	−1.72217	−0.861087	+	0.508457i	$0.830216\pi$
$284$	18.7318	1.11153
$285$	1.82633	0.108183
$286$	0	0
$287$	0	0
$288$	−6.71029	−0.395408
$289$	−11.7952	−0.693834
$290$	−1.27039	−0.0745999
$291$	−2.28965	−0.134222
$292$	2.11873	0.123989
$293$	−31.5427	−1.84274	−0.921372	−	0.388682i	$-0.872930\pi$
$293$	−31.5427	−1.84274	−0.921372	+	0.388682i	$0.872930\pi$
$294$	0	0
$295$	17.7210	1.03175
$296$	5.53762	0.321868
$297$	−6.94506	−0.402993
$298$	2.96710	0.171880
$299$	0	0
$300$	5.33311	0.307907
$301$	0	0
$302$	−1.00598	−0.0578879
$303$	3.71942	0.213675
$304$	6.74274	0.386723
$305$	−29.6695	−1.69887
$306$	1.30899	0.0748300
$307$	18.1941	1.03839	0.519197	−	0.854655i	$-0.326231\pi$
$307$	18.1941	1.03839	0.519197	+	0.854655i	$0.326231\pi$
$308$	0	0
$309$	1.94637	0.110725
$310$	4.31278	0.244949
$311$	−0.376623	−0.0213563	−0.0106782	−	0.999943i	$-0.503399\pi$
$311$	−0.376623	−0.0213563	−0.0106782	+	0.999943i	$0.503399\pi$
$312$	0	0
$313$	10.9883	0.621095	0.310548	−	0.950558i	$-0.399488\pi$
$313$	10.9883	0.621095	0.310548	+	0.950558i	$0.399488\pi$
$314$	−2.09961	−0.118488
$315$	0	0
$316$	−1.55309	−0.0873680
$317$	−26.1806	−1.47045	−0.735225	−	0.677823i	$-0.762923\pi$
$317$	−26.1806	−1.47045	−0.735225	+	0.677823i	$0.762923\pi$
$318$	−0.167211	−0.00937670
$319$	−7.44031	−0.416578
$320$	27.9045	1.55991
$321$	−2.85206	−0.159186
$322$	0	0
$323$	−4.07844	−0.226930
$324$	−16.4728	−0.915158
$325$	0	0
$326$	−0.464174	−0.0257082
$327$	−3.23340	−0.178807
$328$	−6.29079	−0.347351
$329$	0	0
$330$	−0.900743	−0.0495843
$331$	34.0932	1.87393	0.936967	−	0.349419i	$-0.113621\pi$
$331$	34.0932	1.87393	0.936967	+	0.349419i	$0.113621\pi$
$332$	−14.0193	−0.769408
$333$	20.9501	1.14806
$334$	−2.36321	−0.129309
$335$	−49.8062	−2.72120
$336$	0	0
$337$	14.7532	0.803657	0.401829	−	0.915715i	$-0.368375\pi$
$337$	14.7532	0.803657	0.401829	+	0.915715i	$0.368375\pi$
$338$	0	0
$339$	0.428937	0.0232967
$340$	−17.5999	−0.954487
$341$	25.2587	1.36784
$342$	−1.02571	−0.0554639
$343$	0	0
$344$	−5.28141	−0.284754
$345$	1.77784	0.0957155
$346$	−3.79763	−0.204162
$347$	29.9466	1.60762	0.803809	−	0.594888i	$-0.202804\pi$
$347$	29.9466	1.60762	0.803809	+	0.594888i	$0.202804\pi$
$348$	0.841454	0.0451066
$349$	13.4793	0.721532	0.360766	−	0.932656i	$-0.382515\pi$
$349$	13.4793	0.721532	0.360766	+	0.932656i	$0.382515\pi$
$350$	0	0
$351$	0	0
$352$	−10.3114	−0.549602
$353$	−0.163532	−0.00870392	−0.00435196	−	0.999991i	$-0.501385\pi$
$353$	−0.163532	−0.00870392	−0.00435196	+	0.999991i	$0.501385\pi$
$354$	0.229050	0.0121739
$355$	37.5503	1.99296
$356$	−22.1000	−1.17130
$357$	0	0
$358$	2.86527	0.151434
$359$	−8.92130	−0.470848	−0.235424	−	0.971893i	$-0.575648\pi$
$359$	−8.92130	−0.470848	−0.235424	+	0.971893i	$0.575648\pi$
$360$	−8.93895	−0.471124
$361$	−15.8042	−0.831799
$362$	−1.84822	−0.0971404
$363$	−2.41772	−0.126898
$364$	0	0
$365$	4.24726	0.222312
$366$	−0.383490	−0.0200453
$367$	36.6552	1.91339	0.956693	−	0.291098i	$-0.0940207\pi$
$367$	36.6552	1.91339	0.956693	+	0.291098i	$0.0940207\pi$
$368$	6.56369	0.342156
$369$	−23.7995	−1.23895
$370$	5.49679	0.285765
$371$	0	0
$372$	−2.85660	−0.148108
$373$	27.1274	1.40460	0.702302	−	0.711879i	$-0.252156\pi$
$373$	27.1274	1.40460	0.702302	+	0.711879i	$0.252156\pi$
$374$	2.01147	0.104011
$375$	5.58278	0.288294
$376$	2.74819	0.141727
$377$	0	0
$378$	0	0
$379$	15.8943	0.816434	0.408217	−	0.912885i	$-0.366151\pi$
$379$	15.8943	0.816434	0.408217	+	0.912885i	$0.366151\pi$
$380$	13.7910	0.707465
$381$	1.16722	0.0597986
$382$	2.45622	0.125671
$383$	−1.15079	−0.0588025	−0.0294013	−	0.999568i	$-0.509360\pi$
$383$	−1.15079	−0.0588025	−0.0294013	+	0.999568i	$0.509360\pi$
$384$	1.54959	0.0790774
$385$	0	0
$386$	1.83302	0.0932985
$387$	−19.9808	−1.01568
$388$	−17.2897	−0.877750
$389$	−14.3130	−0.725699	−0.362850	−	0.931848i	$-0.618196\pi$
$389$	−14.3130	−0.725699	−0.362850	+	0.931848i	$0.618196\pi$
$390$	0	0
$391$	−3.97013	−0.200778
$392$	0	0
$393$	3.28807	0.165861
$394$	1.49144	0.0751377
$395$	−3.11336	−0.156650
$396$	−25.9235	−1.30271
$397$	25.9176	1.30077	0.650383	−	0.759607i	$-0.274609\pi$
$397$	25.9176	1.30077	0.650383	+	0.759607i	$0.274609\pi$
$398$	2.64701	0.132682
$399$	0	0
$400$	39.4703	1.97351
$401$	4.29631	0.214548	0.107274	−	0.994230i	$-0.465788\pi$
$401$	4.29631	0.214548	0.107274	+	0.994230i	$0.465788\pi$
$402$	−0.643765	−0.0321081
$403$	0	0
$404$	28.0861	1.39734
$405$	−33.0219	−1.64087
$406$	0	0
$407$	32.1931	1.59575
$408$	−0.459410	−0.0227442
$409$	24.7071	1.22169	0.610844	−	0.791751i	$-0.290830\pi$
$409$	24.7071	1.22169	0.610844	+	0.791751i	$0.290830\pi$
$410$	−6.24441	−0.308389
$411$	−2.41250	−0.119000
$412$	14.6975	0.724093
$413$	0	0
$414$	−0.998471	−0.0490722
$415$	−28.1034	−1.37954
$416$	0	0
$417$	1.03915	0.0508876
$418$	−1.57617	−0.0770928
$419$	−24.9293	−1.21787	−0.608937	−	0.793218i	$-0.708404\pi$
$419$	−24.9293	−1.21787	−0.608937	+	0.793218i	$0.708404\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	−3.09127	−0.150480
$423$	10.3970	0.505520
$424$	−2.54993	−0.123835
$425$	−23.8741	−1.15806
$426$	0.485352	0.0235154
$427$	0	0
$428$	−21.5365	−1.04101
$429$	0	0
$430$	−5.24247	−0.252814
$431$	5.68851	0.274006	0.137003	−	0.990571i	$-0.456253\pi$
$431$	5.68851	0.274006	0.137003	+	0.990571i	$0.456253\pi$
$432$	5.81305	0.279681
$433$	12.2598	0.589169	0.294584	−	0.955625i	$-0.404819\pi$
$433$	12.2598	0.589169	0.294584	+	0.955625i	$0.404819\pi$
$434$	0	0
$435$	1.68680	0.0808758
$436$	−24.4161	−1.16932
$437$	3.11095	0.148817
$438$	0.0548975	0.00262310
$439$	5.02317	0.239743	0.119871	−	0.992789i	$-0.461752\pi$
$439$	5.02317	0.239743	0.119871	+	0.992789i	$0.461752\pi$
$440$	−13.7361	−0.654845
$441$	0	0
$442$	0	0
$443$	0.578803	0.0274997	0.0137499	−	0.999905i	$-0.495623\pi$
$443$	0.578803	0.0274997	0.0137499	+	0.999905i	$0.495623\pi$
$444$	−3.64085	−0.172787
$445$	−44.3022	−2.10012
$446$	−4.39711	−0.208209
$447$	−3.93966	−0.186339
$448$	0	0
$449$	7.36359	0.347509	0.173755	−	0.984789i	$-0.444410\pi$
$449$	7.36359	0.347509	0.173755	+	0.984789i	$0.444410\pi$
$450$	−6.00423	−0.283042
$451$	−36.5717	−1.72210
$452$	3.23900	0.152350
$453$	1.33572	0.0627578
$454$	1.80052	0.0845028
$455$	0	0
$456$	0.359988	0.0168580
$457$	−7.91824	−0.370399	−0.185200	−	0.982701i	$-0.559293\pi$
$457$	−7.91824	−0.370399	−0.185200	+	0.982701i	$0.559293\pi$
$458$	−2.99202	−0.139808
$459$	−3.51610	−0.164118
$460$	13.4248	0.625936
$461$	9.53600	0.444136	0.222068	−	0.975031i	$-0.428719\pi$
$461$	9.53600	0.444136	0.222068	+	0.975031i	$0.428719\pi$
$462$	0	0
$463$	−2.16049	−0.100406	−0.0502032	−	0.998739i	$-0.515987\pi$
$463$	−2.16049	−0.100406	−0.0502032	+	0.998739i	$0.515987\pi$
$464$	6.22758	0.289108
$465$	−5.72642	−0.265556
$466$	−1.57617	−0.0730145
$467$	−8.11900	−0.375702	−0.187851	−	0.982198i	$-0.560152\pi$
$467$	−8.11900	−0.375702	−0.187851	+	0.982198i	$0.560152\pi$
$468$	0	0
$469$	0	0
$470$	2.72792	0.125830
$471$	2.78783	0.128456
$472$	3.49297	0.160777
$473$	−30.7036	−1.41176
$474$	−0.0402414	−0.00184835
$475$	18.7074	0.858357
$476$	0	0
$477$	−9.64694	−0.441703
$478$	−4.25079	−0.194427
$479$	−14.5533	−0.664955	−0.332478	−	0.943111i	$-0.607885\pi$
$479$	−14.5533	−0.664955	−0.332478	+	0.943111i	$0.607885\pi$
$480$	2.33772	0.106702
$481$	0	0
$482$	4.01056	0.182676
$483$	0	0
$484$	−18.2567	−0.829852
$485$	−34.6593	−1.57380
$486$	−1.33146	−0.0603960
$487$	33.2590	1.50711	0.753554	−	0.657386i	$-0.228338\pi$
$487$	33.2590	1.50711	0.753554	+	0.657386i	$0.228338\pi$
$488$	−5.84814	−0.264733
$489$	0.616320	0.0278710
$490$	0	0
$491$	−22.5563	−1.01795	−0.508977	−	0.860780i	$-0.669976\pi$
$491$	−22.5563	−1.01795	−0.508977	+	0.860780i	$0.669976\pi$
$492$	4.13604	0.186467
$493$	−3.76684	−0.169650
$494$	0	0
$495$	−51.9669	−2.33574
$496$	−21.1417	−0.949290
$497$	0	0
$498$	−0.363248	−0.0162775
$499$	11.3854	0.509682	0.254841	−	0.966983i	$-0.417977\pi$
$499$	11.3854	0.509682	0.254841	+	0.966983i	$0.417977\pi$
$500$	42.1568	1.88531
$501$	3.13782	0.140188
$502$	0.510410	0.0227807
$503$	−8.81825	−0.393186	−0.196593	−	0.980485i	$-0.562988\pi$
$503$	−8.81825	−0.393186	−0.196593	+	0.980485i	$0.562988\pi$
$504$	0	0
$505$	56.3022	2.50541
$506$	−1.53431	−0.0682084
$507$	0	0
$508$	8.81394	0.391056
$509$	−19.2838	−0.854738	−0.427369	−	0.904077i	$-0.640559\pi$
$509$	−19.2838	−0.854738	−0.427369	+	0.904077i	$0.640559\pi$
$510$	−0.456023	−0.0201930
$511$	0	0
$512$	14.4780	0.639844
$513$	2.75517	0.121644
$514$	−1.76027	−0.0776423
$515$	29.4629	1.29829
$516$	3.47239	0.152864
$517$	15.9767	0.702653
$518$	0	0
$519$	5.04241	0.221337
$520$	0	0
$521$	25.1168	1.10039	0.550193	−	0.835037i	$-0.314554\pi$
$521$	25.1168	1.10039	0.550193	+	0.835037i	$0.314554\pi$
$522$	−0.947342	−0.0414640
$523$	−29.9648	−1.31027	−0.655134	−	0.755513i	$-0.727388\pi$
$523$	−29.9648	−1.31027	−0.655134	+	0.755513i	$0.727388\pi$
$524$	24.8289	1.08466
$525$	0	0
$526$	0.280556	0.0122328
$527$	12.7878	0.557046
$528$	4.41554	0.192162
$529$	−19.9717	−0.868333
$530$	−2.53113	−0.109945
$531$	13.2147	0.573469
$532$	0	0
$533$	0	0
$534$	−0.572623	−0.0247798
$535$	−43.1726	−1.86651
$536$	−9.81728	−0.424042
$537$	−3.80444	−0.164174
$538$	−1.59818	−0.0689026
$539$	0	0
$540$	11.8895	0.511644
$541$	−5.09973	−0.219255	−0.109627	−	0.993973i	$-0.534966\pi$
$541$	−5.09973	−0.219255	−0.109627	+	0.993973i	$0.534966\pi$
$542$	0.0415412	0.00178435
$543$	2.45403	0.105313
$544$	−5.22042	−0.223823
$545$	−48.9451	−2.09658
$546$	0	0
$547$	2.92025	0.124861	0.0624305	−	0.998049i	$-0.480115\pi$
$547$	2.92025	0.124861	0.0624305	+	0.998049i	$0.480115\pi$
$548$	−18.2173	−0.778206
$549$	−22.1248	−0.944265
$550$	−9.22647	−0.393418
$551$	2.95165	0.125744
$552$	0.350428	0.0149152
$553$	0	0
$554$	4.49519	0.190982
$555$	−7.29853	−0.309805
$556$	7.84687	0.332782
$557$	−25.9874	−1.10112	−0.550561	−	0.834795i	$-0.685586\pi$
$557$	−25.9874	−1.10112	−0.550561	+	0.834795i	$0.685586\pi$
$558$	3.21608	0.136148
$559$	0	0
$560$	0	0
$561$	−2.67079	−0.112761
$562$	0.0675460	0.00284925
$563$	3.65069	0.153858	0.0769291	−	0.997037i	$-0.475488\pi$
$563$	3.65069	0.153858	0.0769291	+	0.997037i	$0.475488\pi$
$564$	−1.80686	−0.0760826
$565$	6.49297	0.273161
$566$	−5.66845	−0.238263
$567$	0	0
$568$	7.40152	0.310561
$569$	−25.3533	−1.06286	−0.531432	−	0.847101i	$-0.678346\pi$
$569$	−25.3533	−1.06286	−0.531432	+	0.847101i	$0.678346\pi$
$570$	0.357334	0.0149671
$571$	27.7253	1.16027	0.580133	−	0.814522i	$-0.303000\pi$
$571$	27.7253	1.16027	0.580133	+	0.814522i	$0.303000\pi$
$572$	0	0
$573$	−3.26132	−0.136244
$574$	0	0
$575$	18.2107	0.759438
$576$	20.8086	0.867027
$577$	−39.5754	−1.64755	−0.823773	−	0.566920i	$-0.808135\pi$
$577$	−39.5754	−1.64755	−0.823773	+	0.566920i	$0.808135\pi$
$578$	−2.30780	−0.0959918
$579$	−2.43385	−0.101147
$580$	12.7374	0.528891
$581$	0	0
$582$	−0.447985	−0.0185696
$583$	−14.8241	−0.613951
$584$	0.837175	0.0346426
$585$	0	0
$586$	−6.17153	−0.254943
$587$	8.24177	0.340174	0.170087	−	0.985429i	$-0.445595\pi$
$587$	8.24177	0.340174	0.170087	+	0.985429i	$0.445595\pi$
$588$	0	0
$589$	−10.0204	−0.412882
$590$	3.46722	0.142743
$591$	−1.98031	−0.0814589
$592$	−26.9458	−1.10747
$593$	11.9230	0.489618	0.244809	−	0.969571i	$-0.421275\pi$
$593$	11.9230	0.489618	0.244809	+	0.969571i	$0.421275\pi$
$594$	−1.35884	−0.0557540
$595$	0	0
$596$	−29.7492	−1.21857
$597$	−3.51464	−0.143845
$598$	0	0
$599$	−35.6158	−1.45522	−0.727611	−	0.685990i	$-0.759369\pi$
$599$	−35.6158	−1.45522	−0.727611	+	0.685990i	$0.759369\pi$
$600$	2.10728	0.0860292
$601$	38.9252	1.58779	0.793896	−	0.608054i	$-0.208050\pi$
$601$	38.9252	1.58779	0.793896	+	0.608054i	$0.208050\pi$
$602$	0	0
$603$	−37.1410	−1.51250
$604$	10.0863	0.410408
$605$	−36.5979	−1.48792
$606$	0.727728	0.0295619
$607$	−13.6966	−0.555926	−0.277963	−	0.960592i	$-0.589659\pi$
$607$	−13.6966	−0.555926	−0.277963	+	0.960592i	$0.589659\pi$
$608$	4.09065	0.165898
$609$	0	0
$610$	−5.80502	−0.235039
$611$	0	0
$612$	−13.1244	−0.530522
$613$	−3.16112	−0.127676	−0.0638382	−	0.997960i	$-0.520334\pi$
$613$	−3.16112	−0.127676	−0.0638382	+	0.997960i	$0.520334\pi$
$614$	3.55979	0.143662
$615$	8.29120	0.334334
$616$	0	0
$617$	−20.9297	−0.842597	−0.421299	−	0.906922i	$-0.638426\pi$
$617$	−20.9297	−0.842597	−0.421299	+	0.906922i	$0.638426\pi$
$618$	0.380820	0.0153188
$619$	−30.9544	−1.24416	−0.622082	−	0.782952i	$-0.713713\pi$
$619$	−30.9544	−1.24416	−0.622082	+	0.782952i	$0.713713\pi$
$620$	−43.2414	−1.73662
$621$	2.68201	0.107625
$622$	−0.0736887	−0.00295465
$623$	0	0
$624$	0	0
$625$	32.1854	1.28742
$626$	2.14993	0.0859285
$627$	2.09280	0.0835784
$628$	21.0515	0.840045
$629$	16.2986	0.649866
$630$	0	0
$631$	15.1218	0.601988	0.300994	−	0.953626i	$-0.402682\pi$
$631$	15.1218	0.601988	0.300994	+	0.953626i	$0.402682\pi$
$632$	−0.613673	−0.0244106
$633$	4.10452	0.163140
$634$	−5.12240	−0.203437
$635$	17.6687	0.701159
$636$	1.67651	0.0664780
$637$	0	0
$638$	−1.45574	−0.0576335
$639$	28.0016	1.10773
$640$	23.4568	0.927210
$641$	−47.2414	−1.86592	−0.932962	−	0.359976i	$-0.882785\pi$
$641$	−47.2414	−1.86592	−0.932962	+	0.359976i	$0.882785\pi$
$642$	−0.558023	−0.0220234
$643$	−39.9249	−1.57448	−0.787241	−	0.616645i	$-0.788491\pi$
$643$	−39.9249	−1.57448	−0.787241	+	0.616645i	$0.788491\pi$
$644$	0	0
$645$	6.96085	0.274083
$646$	−0.797972	−0.0313958
$647$	−29.8278	−1.17265	−0.586327	−	0.810075i	$-0.699426\pi$
$647$	−29.8278	−1.17265	−0.586327	+	0.810075i	$0.699426\pi$
$648$	−6.50892	−0.255695
$649$	20.3065	0.797100
$650$	0	0
$651$	0	0
$652$	4.65397	0.182263
$653$	25.1549	0.984387	0.492194	−	0.870486i	$-0.336195\pi$
$653$	25.1549	0.984387	0.492194	+	0.870486i	$0.336195\pi$
$654$	−0.632635	−0.0247380
$655$	49.7727	1.94478
$656$	30.6107	1.19515
$657$	3.16722	0.123565
$658$	0	0
$659$	17.3155	0.674517	0.337258	−	0.941412i	$-0.390500\pi$
$659$	17.3155	0.674517	0.337258	+	0.941412i	$0.390500\pi$
$660$	9.03117	0.351538
$661$	9.20074	0.357867	0.178934	−	0.983861i	$-0.442735\pi$
$661$	9.20074	0.357867	0.178934	+	0.983861i	$0.442735\pi$
$662$	6.67055	0.259258
$663$	0	0
$664$	−5.53945	−0.214972
$665$	0	0
$666$	4.09901	0.158833
$667$	2.87327	0.111253
$668$	23.6944	0.916763
$669$	5.83840	0.225725
$670$	−9.74490	−0.376478
$671$	−33.9984	−1.31249
$672$	0	0
$673$	17.3609	0.669212	0.334606	−	0.942358i	$-0.391397\pi$
$673$	17.3609	0.669212	0.334606	+	0.942358i	$0.391397\pi$
$674$	2.88655	0.111186
$675$	16.1281	0.620770
$676$	0	0
$677$	49.9825	1.92098	0.960492	−	0.278307i	$-0.0897733\pi$
$677$	49.9825	1.92098	0.960492	+	0.278307i	$0.0897733\pi$
$678$	0.0839242	0.00322309
$679$	0	0
$680$	−6.95425	−0.266683
$681$	−2.39070	−0.0916118
$682$	4.94202	0.189240
$683$	−33.6153	−1.28626	−0.643128	−	0.765759i	$-0.722364\pi$
$683$	−33.6153	−1.28626	−0.643128	+	0.765759i	$0.722364\pi$
$684$	10.2841	0.393223
$685$	−36.5189	−1.39532
$686$	0	0
$687$	3.97274	0.151570
$688$	25.6991	0.979770
$689$	0	0
$690$	0.347845	0.0132422
$691$	15.1309	0.575607	0.287803	−	0.957689i	$-0.407075\pi$
$691$	15.1309	0.575607	0.287803	+	0.957689i	$0.407075\pi$
$692$	38.0764	1.44745
$693$	0	0
$694$	5.85924	0.222414
$695$	15.7300	0.596674
$696$	0.332484	0.0126028
$697$	−18.5153	−0.701317
$698$	2.63731	0.0998238
$699$	2.09280	0.0791570
$700$	0	0
$701$	2.02467	0.0764705	0.0382353	−	0.999269i	$-0.487826\pi$
$701$	2.02467	0.0764705	0.0382353	+	0.999269i	$0.487826\pi$
$702$	0	0
$703$	−12.7713	−0.481680
$704$	31.9758	1.20513
$705$	−3.62208	−0.136415
$706$	−0.0319960	−0.00120419
$707$	0	0
$708$	−2.29654	−0.0863093
$709$	30.4553	1.14377	0.571886	−	0.820333i	$-0.306212\pi$
$709$	30.4553	1.14377	0.571886	+	0.820333i	$0.306212\pi$
$710$	7.34695	0.275726
$711$	−2.32166	−0.0870691
$712$	−8.73238	−0.327260
$713$	−9.75429	−0.365301
$714$	0	0
$715$	0	0
$716$	−28.7282	−1.07362
$717$	5.64411	0.210783
$718$	−1.74551	−0.0651418
$719$	−24.4246	−0.910883	−0.455442	−	0.890266i	$-0.650519\pi$
$719$	−24.4246	−0.910883	−0.455442	+	0.890266i	$0.650519\pi$
$720$	43.4966	1.62102
$721$	0	0
$722$	−3.09219	−0.115079
$723$	−5.32514	−0.198044
$724$	18.5309	0.688697
$725$	17.2782	0.641695
$726$	−0.473043	−0.0175563
$727$	−3.09307	−0.114716	−0.0573578	−	0.998354i	$-0.518268\pi$
$727$	−3.09307	−0.114716	−0.0573578	+	0.998354i	$0.518268\pi$
$728$	0	0
$729$	−23.4236	−0.867539
$730$	0.831003	0.0307568
$731$	−15.5445	−0.574933
$732$	3.84501	0.142115
$733$	8.41427	0.310788	0.155394	−	0.987853i	$-0.450335\pi$
$733$	8.41427	0.310788	0.155394	+	0.987853i	$0.450335\pi$
$734$	7.17182	0.264717
$735$	0	0
$736$	3.98203	0.146779
$737$	−57.0731	−2.10231
$738$	−4.65652	−0.171409
$739$	7.22758	0.265871	0.132936	−	0.991125i	$-0.457560\pi$
$739$	7.22758	0.265871	0.132936	+	0.991125i	$0.457560\pi$
$740$	−55.1128	−2.02599
$741$	0	0
$742$	0	0
$743$	53.9092	1.97774	0.988869	−	0.148791i	$-0.0475383\pi$
$743$	53.9092	1.97774	0.988869	+	0.148791i	$0.0475383\pi$
$744$	−1.12873	−0.0413813
$745$	−59.6360	−2.18489
$746$	5.30765	0.194327
$747$	−20.9570	−0.766776
$748$	−20.1678	−0.737406
$749$	0	0
$750$	1.09231	0.0398854
$751$	29.2442	1.06714	0.533568	−	0.845757i	$-0.320851\pi$
$751$	29.2442	1.06714	0.533568	+	0.845757i	$0.320851\pi$
$752$	−13.3726	−0.487647
$753$	−0.677712	−0.0246972
$754$	0	0
$755$	20.2193	0.735857
$756$	0	0
$757$	−22.0597	−0.801773	−0.400887	−	0.916128i	$-0.631298\pi$
$757$	−22.0597	−0.801773	−0.400887	+	0.916128i	$0.631298\pi$
$758$	3.10981	0.112954
$759$	2.03723	0.0739467
$760$	5.44926	0.197666
$761$	−17.8161	−0.645833	−0.322917	−	0.946427i	$-0.604663\pi$
$761$	−17.8161	−0.645833	−0.322917	+	0.946427i	$0.604663\pi$
$762$	0.228374	0.00827313
$763$	0	0
$764$	−24.6269	−0.890971
$765$	−26.3095	−0.951222
$766$	−0.225159	−0.00813532
$767$	0	0
$768$	−3.38365	−0.122097
$769$	11.3069	0.407738	0.203869	−	0.978998i	$-0.434648\pi$
$769$	11.3069	0.407738	0.203869	+	0.978998i	$0.434648\pi$
$770$	0	0
$771$	2.33725	0.0841741
$772$	−18.3785	−0.661458
$773$	1.92821	0.0693528	0.0346764	−	0.999399i	$-0.488960\pi$
$773$	1.92821	0.0693528	0.0346764	+	0.999399i	$0.488960\pi$
$774$	−3.90936	−0.140519
$775$	−58.6567	−2.10701
$776$	−6.83168	−0.245243
$777$	0	0
$778$	−2.80043	−0.100400
$779$	14.5084	0.519816
$780$	0	0
$781$	43.0290	1.53970
$782$	−0.776782	−0.0277777
$783$	2.54467	0.0909392
$784$	0	0
$785$	42.2003	1.50619
$786$	0.643331	0.0229469
$787$	−5.53155	−0.197178	−0.0985892	−	0.995128i	$-0.531433\pi$
$787$	−5.53155	−0.197178	−0.0985892	+	0.995128i	$0.531433\pi$
$788$	−14.9537	−0.532704
$789$	−0.372516	−0.0132619
$790$	−0.609148	−0.0216725
$791$	0	0
$792$	−10.2432	−0.363975
$793$	0	0
$794$	5.07093	0.179961
$795$	3.36078	0.119195
$796$	−26.5398	−0.940679
$797$	−13.8038	−0.488955	−0.244477	−	0.969655i	$-0.578616\pi$
$797$	−13.8038	−0.488955	−0.244477	+	0.969655i	$0.578616\pi$
$798$	0	0
$799$	8.08857	0.286153
$800$	23.9456	0.846605
$801$	−33.0365	−1.16729
$802$	0.840601	0.0296826
$803$	4.86695	0.171751
$804$	6.45461	0.227637
$805$	0	0
$806$	0	0
$807$	2.12204	0.0746992
$808$	11.0977	0.390415
$809$	28.2996	0.994961	0.497480	−	0.867475i	$-0.334259\pi$
$809$	28.2996	0.994961	0.497480	+	0.867475i	$0.334259\pi$
$810$	−6.46093	−0.227014
$811$	12.2124	0.428837	0.214418	−	0.976742i	$-0.431214\pi$
$811$	12.2124	0.428837	0.214418	+	0.976742i	$0.431214\pi$
$812$	0	0
$813$	−0.0551575	−0.00193446
$814$	6.29879	0.220772
$815$	9.32946	0.326797
$816$	2.23547	0.0782571
$817$	12.1804	0.426140
$818$	4.83410	0.169020
$819$	0	0
$820$	62.6087	2.18639
$821$	41.4011	1.44491	0.722453	−	0.691420i	$-0.243014\pi$
$821$	41.4011	1.44491	0.722453	+	0.691420i	$0.243014\pi$
$822$	−0.472021	−0.0164636
$823$	47.1752	1.64443	0.822213	−	0.569180i	$-0.192739\pi$
$823$	47.1752	1.64443	0.822213	+	0.569180i	$0.192739\pi$
$824$	5.80742	0.202311
$825$	12.2507	0.426515
$826$	0	0
$827$	−21.1124	−0.734150	−0.367075	−	0.930191i	$-0.619641\pi$
$827$	−21.1124	−0.734150	−0.367075	+	0.930191i	$0.619641\pi$
$828$	10.0110	0.347907
$829$	−0.636752	−0.0221153	−0.0110577	−	0.999939i	$-0.503520\pi$
$829$	−0.636752	−0.0221153	−0.0110577	+	0.999939i	$0.503520\pi$
$830$	−5.49861	−0.190859
$831$	−5.96862	−0.207049
$832$	0	0
$833$	0	0
$834$	0.203317	0.00704029
$835$	47.4983	1.64375
$836$	15.8032	0.546565
$837$	−8.63877	−0.298600
$838$	−4.87757	−0.168493
$839$	26.9432	0.930183	0.465092	−	0.885263i	$-0.346021\pi$
$839$	26.9432	0.930183	0.465092	+	0.885263i	$0.346021\pi$
$840$	0	0
$841$	−26.2739	−0.905995
$842$	1.95656	0.0674276
$843$	−0.0896862	−0.00308896
$844$	30.9941	1.06686
$845$	0	0
$846$	2.03424	0.0699386
$847$	0	0
$848$	12.4078	0.426087
$849$	7.52645	0.258307
$850$	−4.67112	−0.160218
$851$	−12.4322	−0.426170
$852$	−4.86631	−0.166717
$853$	−6.74784	−0.231042	−0.115521	−	0.993305i	$-0.536854\pi$
$853$	−6.74784	−0.231042	−0.115521	+	0.993305i	$0.536854\pi$
$854$	0	0
$855$	20.6158	0.705045
$856$	−8.50973	−0.290856
$857$	−45.0268	−1.53809	−0.769043	−	0.639197i	$-0.779267\pi$
$857$	−45.0268	−1.53809	−0.769043	+	0.639197i	$0.779267\pi$
$858$	0	0
$859$	−36.7270	−1.25311	−0.626554	−	0.779378i	$-0.715535\pi$
$859$	−36.7270	−1.25311	−0.626554	+	0.779378i	$0.715535\pi$
$860$	52.5629	1.79238
$861$	0	0
$862$	1.11299	0.0379087
$863$	43.4275	1.47829	0.739144	−	0.673547i	$-0.235230\pi$
$863$	43.4275	1.47829	0.739144	+	0.673547i	$0.235230\pi$
$864$	3.52663	0.119978
$865$	76.3288	2.59526
$866$	2.39871	0.0815114
$867$	3.06425	0.104067
$868$	0	0
$869$	−3.56761	−0.121023
$870$	0.330033	0.0111892
$871$	0	0
$872$	−9.64754	−0.326707
$873$	−25.8458	−0.874747
$874$	0.608676	0.0205888
$875$	0	0
$876$	−0.550422	−0.0185970
$877$	−40.0081	−1.35098	−0.675488	−	0.737371i	$-0.736067\pi$
$877$	−40.0081	−1.35098	−0.675488	+	0.737371i	$0.736067\pi$
$878$	0.982815	0.0331684
$879$	8.19443	0.276391
$880$	66.8395	2.25316
$881$	35.4308	1.19370	0.596848	−	0.802355i	$-0.296420\pi$
$881$	35.4308	1.19370	0.596848	+	0.802355i	$0.296420\pi$
$882$	0	0
$883$	22.6654	0.762751	0.381375	−	0.924420i	$-0.375451\pi$
$883$	22.6654	0.762751	0.381375	+	0.924420i	$0.375451\pi$
$884$	0	0
$885$	−4.60370	−0.154752
$886$	0.113246	0.00380459
$887$	44.6881	1.50048	0.750240	−	0.661166i	$-0.229938\pi$
$887$	44.6881	1.50048	0.750240	+	0.661166i	$0.229938\pi$
$888$	−1.43861	−0.0482766
$889$	0	0
$890$	−8.66800	−0.290552
$891$	−37.8398	−1.26768
$892$	44.0870	1.47614
$893$	−6.33810	−0.212096
$894$	−0.770818	−0.0257800
$895$	−57.5892	−1.92499
$896$	0	0
$897$	0	0
$898$	1.44073	0.0480779
$899$	−9.25480	−0.308665
$900$	60.2005	2.00668
$901$	−7.50505	−0.250029
$902$	−7.15549	−0.238252
$903$	0	0
$904$	1.27983	0.0425664
$905$	37.1476	1.23483
$906$	0.261343	0.00868254
$907$	54.4748	1.80881	0.904403	−	0.426680i	$-0.140317\pi$
$907$	54.4748	1.80881	0.904403	+	0.426680i	$0.140317\pi$
$908$	−18.0527	−0.599100
$909$	41.9851	1.39256
$910$	0	0
$911$	−27.4793	−0.910431	−0.455215	−	0.890381i	$-0.650438\pi$
$911$	−27.4793	−0.910431	−0.455215	+	0.890381i	$0.650438\pi$
$912$	−1.75169	−0.0580041
$913$	−32.2038	−1.06579
$914$	−1.54925	−0.0512447
$915$	7.70779	0.254812
$916$	29.9990	0.991196
$917$	0	0
$918$	−0.687947	−0.0227056
$919$	−48.2880	−1.59287	−0.796437	−	0.604722i	$-0.793284\pi$
$919$	−48.2880	−1.59287	−0.796437	+	0.604722i	$0.793284\pi$
$920$	5.30456	0.174886
$921$	−4.72662	−0.155747
$922$	1.86578	0.0614461
$923$	0	0
$924$	0	0
$925$	−74.7601	−2.45810
$926$	−0.422713	−0.0138912
$927$	21.9708	0.721615
$928$	3.77812	0.124023
$929$	−43.3154	−1.42113	−0.710566	−	0.703631i	$-0.751561\pi$
$929$	−43.3154	−1.42113	−0.710566	+	0.703631i	$0.751561\pi$
$930$	−1.12041	−0.0367397
$931$	0	0
$932$	15.8032	0.517651
$933$	0.0978423	0.00320321
$934$	−1.58853	−0.0519784
$935$	−40.4288	−1.32216
$936$	0	0
$937$	−37.2211	−1.21596	−0.607980	−	0.793952i	$-0.708020\pi$
$937$	−37.2211	−1.21596	−0.607980	+	0.793952i	$0.708020\pi$
$938$	0	0
$939$	−2.85463	−0.0931574
$940$	−27.3511	−0.892095
$941$	−15.9751	−0.520773	−0.260386	−	0.965504i	$-0.583850\pi$
$941$	−15.9751	−0.520773	−0.260386	+	0.965504i	$0.583850\pi$
$942$	0.545456	0.0177719
$943$	14.1231	0.459911
$944$	−16.9967	−0.553194
$945$	0	0
$946$	−6.00736	−0.195316
$947$	−27.7572	−0.901988	−0.450994	−	0.892527i	$-0.648930\pi$
$947$	−27.7572	−0.901988	−0.450994	+	0.892527i	$0.648930\pi$
$948$	0.403474	0.0131042
$949$	0	0
$950$	3.66023	0.118754
$951$	6.80142	0.220551
$952$	0	0
$953$	12.0303	0.389700	0.194850	−	0.980833i	$-0.437578\pi$
$953$	12.0303	0.389700	0.194850	+	0.980833i	$0.437578\pi$
$954$	−1.88748	−0.0611096
$955$	−49.3677	−1.59750
$956$	42.6199	1.37843
$957$	1.93291	0.0624820
$958$	−2.84744	−0.0919965
$959$	0	0
$960$	−7.24926	−0.233969
$961$	0.418620	0.0135039
$962$	0	0
$963$	−32.1942	−1.03744
$964$	−40.2113	−1.29512
$965$	−36.8421	−1.18599
$966$	0	0
$967$	−5.40788	−0.173906	−0.0869528	−	0.996212i	$-0.527713\pi$
$967$	−5.40788	−0.173906	−0.0869528	+	0.996212i	$0.527713\pi$
$968$	−7.21380	−0.231860
$969$	1.05953	0.0340370
$970$	−6.78131	−0.217735
$971$	42.2752	1.35668	0.678338	−	0.734750i	$-0.262700\pi$
$971$	42.2752	1.35668	0.678338	+	0.734750i	$0.262700\pi$
$972$	13.3496	0.428190
$973$	0	0
$974$	6.50733	0.208508
$975$	0	0
$976$	28.4568	0.910881
$977$	10.8302	0.346488	0.173244	−	0.984879i	$-0.444575\pi$
$977$	10.8302	0.346488	0.173244	+	0.984879i	$0.444575\pi$
$978$	0.120587	0.00385594
$979$	−50.7660	−1.62249
$980$	0	0
$981$	−36.4988	−1.16532
$982$	−4.41329	−0.140834
$983$	−21.5610	−0.687688	−0.343844	−	0.939027i	$-0.611729\pi$
$983$	−21.5610	−0.687688	−0.343844	+	0.939027i	$0.611729\pi$
$984$	1.63427	0.0520988
$985$	−29.9766	−0.955134
$986$	−0.737005	−0.0234710
$987$	0	0
$988$	0	0
$989$	11.8570	0.377030
$990$	−10.1677	−0.323149
$991$	8.62624	0.274022	0.137011	−	0.990570i	$-0.456251\pi$
$991$	8.62624	0.274022	0.137011	+	0.990570i	$0.456251\pi$
$992$	−12.8261	−0.407230
$993$	−8.85703	−0.281069
$994$	0	0
$995$	−53.2024	−1.68663
$996$	3.64205	0.115403
$997$	21.6967	0.687142	0.343571	−	0.939127i	$-0.388363\pi$
$997$	21.6967	0.687142	0.343571	+	0.939127i	$0.388363\pi$
$998$	2.22763	0.0705144
$999$	−11.0104	−0.348354

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.bw.1.4		5
7.2	even	3		1183.2.e.f.508.2		10
7.4	even	3		1183.2.e.f.170.2		10
7.6	odd	2		8281.2.a.bx.1.4		5
13.12	even	2		637.2.a.l.1.2		5
39.38	odd	2		5733.2.a.bl.1.4		5
91.12	odd	6		637.2.e.m.508.4		10
91.25	even	6		91.2.e.c.79.4	yes	10
91.38	odd	6		637.2.e.m.79.4		10
91.51	even	6		91.2.e.c.53.4	✓	10
91.90	odd	2		637.2.a.k.1.2		5
273.116	odd	6		819.2.j.h.352.2		10
273.233	odd	6		819.2.j.h.235.2		10
273.272	even	2		5733.2.a.bm.1.4		5
364.51	odd	6		1456.2.r.p.417.3		10
364.207	odd	6		1456.2.r.p.625.3		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.e.c.53.4	✓	10	91.51	even	6
91.2.e.c.79.4	yes	10	91.25	even	6
637.2.a.k.1.2		5	91.90	odd	2
637.2.a.l.1.2		5	13.12	even	2
637.2.e.m.79.4		10	91.38	odd	6
637.2.e.m.508.4		10	91.12	odd	6
819.2.j.h.235.2		10	273.233	odd	6
819.2.j.h.352.2		10	273.116	odd	6
1183.2.e.f.170.2		10	7.4	even	3
1183.2.e.f.508.2		10	7.2	even	3
1456.2.r.p.417.3		10	364.51	odd	6
1456.2.r.p.625.3		10	364.207	odd	6
5733.2.a.bl.1.4		5	39.38	odd	2
5733.2.a.bm.1.4		5	273.272	even	2
8281.2.a.bw.1.4		5	1.1	even	1	trivial
8281.2.a.bx.1.4		5	7.6	odd	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:	\( N \)	\(=\)	\( 8281 = 7^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8281.a (trivial)

\(f(q)\)	\(=\)	\(q+0.195656 q^{2} -0.259788 q^{3} -1.96172 q^{4} -3.93251 q^{5} -0.0508292 q^{6} -0.775135 q^{8} -2.93251 q^{9} -0.769420 q^{10} -4.50627 q^{11} +0.509632 q^{12} +1.02162 q^{15} +3.77178 q^{16} -2.28141 q^{17} -0.573764 q^{18} +1.78768 q^{19} +7.71448 q^{20} -0.881681 q^{22} +1.74021 q^{23} +0.201371 q^{24} +10.4646 q^{25} +1.54120 q^{27} +1.65110 q^{29} +0.199886 q^{30} -5.60523 q^{31} +2.28824 q^{32} +1.17068 q^{33} -0.446372 q^{34} +5.75276 q^{36} -7.14407 q^{37} +0.349771 q^{38} +3.04823 q^{40} +8.11574 q^{41} +6.81353 q^{43} +8.84004 q^{44} +11.5321 q^{45} +0.340483 q^{46} -3.54543 q^{47} -0.979864 q^{48} +2.04747 q^{50} +0.592684 q^{51} +3.28965 q^{53} +0.301545 q^{54} +17.7210 q^{55} -0.464419 q^{57} +0.323048 q^{58} -4.50627 q^{59} -2.00413 q^{60} +7.54467 q^{61} -1.09670 q^{62} -7.09585 q^{64} +0.229050 q^{66} +12.6653 q^{67} +4.47548 q^{68} -0.452087 q^{69} -9.54869 q^{71} +2.27309 q^{72} -1.08004 q^{73} -1.39778 q^{74} -2.71859 q^{75} -3.50693 q^{76} +0.791698 q^{79} -14.8326 q^{80} +8.39714 q^{81} +1.58789 q^{82} +7.14643 q^{83} +8.97166 q^{85} +1.33311 q^{86} -0.428937 q^{87} +3.49297 q^{88} +11.2656 q^{89} +2.25633 q^{90} -3.41381 q^{92} +1.45617 q^{93} -0.693685 q^{94} -7.03008 q^{95} -0.594459 q^{96} +8.81353 q^{97} +13.2147 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 4 q^{2} + 8 q^{4} - 2 q^{5} + 5 q^{6} - 9 q^{8} + 3 q^{9} - 5 q^{10} - 11 q^{11} + 5 q^{12} + 10 q^{16} - 5 q^{17} - 9 q^{18} - 9 q^{19} - q^{20} + 8 q^{22} + 10 q^{23} + 9 q^{25} - 3 q^{29} - 13 q^{30}+ \cdots - 11 q^{99}+O(q^{100}) \) 5 * q - 4 * q^2 + 8 * q^4 - 2 * q^5 + 5 * q^6 - 9 * q^8 + 3 * q^9 - 5 * q^10 - 11 * q^11 + 5 * q^12 + 10 * q^16 - 5 * q^17 - 9 * q^18 - 9 * q^19 - q^20 + 8 * q^22 + 10 * q^23 + 9 * q^25 - 3 * q^29 - 13 * q^30 + 6 * q^31 - 22 * q^32 - 8 * q^33 + 22 * q^34 + 7 * q^36 - 4 * q^37 - 10 * q^38 + 28 * q^40 - 14 * q^41 + 2 * q^43 + 32 * q^45 - 3 * q^46 - q^47 - 23 * q^48 - 9 * q^50 - 8 * q^51 + 17 * q^53 - 23 * q^54 + 16 * q^57 + 27 * q^58 - 11 * q^59 + 29 * q^60 - 11 * q^61 - 23 * q^62 + 9 * q^64 + 21 * q^66 - 13 * q^67 - 32 * q^68 + 18 * q^69 - 15 * q^71 + 19 * q^72 - 33 * q^74 - 20 * q^75 - 8 * q^76 + 2 * q^79 - 55 * q^80 - 19 * q^81 + 34 * q^82 - 6 * q^83 + 22 * q^85 - 28 * q^86 - 8 * q^87 - 3 * q^88 + 4 * q^89 - 34 * q^90 + 21 * q^92 - 18 * q^93 + 20 * q^94 - 12 * q^95 + 37 * q^96 + 12 * q^97 - 11 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more