LMFDB - Embedded newform 6525.2.a.by.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6525,2,Mod(1,6525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6525, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6525.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6525 = 3^{2} \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6525.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:	$52.1023873189$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 12x^{6} + 23x^{5} + 36x^{4} - 62x^{3} - 15x^{2} + 14x + 2 $ x^8 - 2x^7 - 12x^6 + 23x^5 + 36x^4 - 62x^3 - 15x^2 + 14*x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2175)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.59587$ of defining polynomial
Character	$\chi$	$=$	6525.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.59587 q^{2} +4.73855 q^{4} -3.93826 q^{7} -7.10893 q^{8} +O(q^{10})$	$q-2.59587 q^{2} +4.73855 q^{4} -3.93826 q^{7} -7.10893 q^{8} -2.24645 q^{11} +2.09464 q^{13} +10.2232 q^{14} +8.97678 q^{16} -6.95310 q^{17} +4.29591 q^{19} +5.83149 q^{22} +5.19955 q^{23} -5.43743 q^{26} -18.6617 q^{28} -1.00000 q^{29} +2.25035 q^{31} -9.08471 q^{32} +18.0494 q^{34} -10.5280 q^{37} -11.1516 q^{38} +2.66629 q^{41} +6.03681 q^{43} -10.6449 q^{44} -13.4974 q^{46} +7.74829 q^{47} +8.50991 q^{49} +9.92559 q^{52} +1.41309 q^{53} +27.9968 q^{56} +2.59587 q^{58} -5.76800 q^{59} -10.9192 q^{61} -5.84162 q^{62} +5.62918 q^{64} -11.7471 q^{67} -32.9476 q^{68} +4.55691 q^{71} +12.3564 q^{73} +27.3294 q^{74} +20.3564 q^{76} +8.84710 q^{77} +14.4098 q^{79} -6.92135 q^{82} +3.81272 q^{83} -15.6708 q^{86} +15.9699 q^{88} +2.07340 q^{89} -8.24926 q^{91} +24.6383 q^{92} -20.1136 q^{94} +12.0068 q^{97} -22.0906 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{2} + 12 q^{4} - 2 q^{7} - 3 q^{8}+O(q^{10}) $ 8 * q - 2 * q^2 + 12 * q^4 - 2 * q^7 - 3 * q^8	$ 8 q - 2 q^{2} + 12 q^{4} - 2 q^{7} - 3 q^{8} - 6 q^{11} - 6 q^{13} - 9 q^{14} + 32 q^{16} - 12 q^{17} - 3 q^{22} - 14 q^{23} - 18 q^{26} - 14 q^{28} - 8 q^{29} + 8 q^{31} + 2 q^{32} - 13 q^{34} - 4 q^{37} - 26 q^{38} - 2 q^{41} - 2 q^{43} + 15 q^{44} + 24 q^{46} - 12 q^{47} + 38 q^{49} - 49 q^{52} - 4 q^{53} - 58 q^{56} + 2 q^{58} - 18 q^{59} + 12 q^{61} + 4 q^{62} + 21 q^{64} - 26 q^{67} - 81 q^{68} - 24 q^{71} + 14 q^{73} + 22 q^{74} + 26 q^{77} + 10 q^{79} - 48 q^{82} - 40 q^{83} - 8 q^{86} + 10 q^{88} - 34 q^{89} + 26 q^{91} + 18 q^{92} - 43 q^{94} - 30 q^{97} - 29 q^{98}+O(q^{100}) $ 8 * q - 2 * q^2 + 12 * q^4 - 2 * q^7 - 3 * q^8 - 6 * q^11 - 6 * q^13 - 9 * q^14 + 32 * q^16 - 12 * q^17 - 3 * q^22 - 14 * q^23 - 18 * q^26 - 14 * q^28 - 8 * q^29 + 8 * q^31 + 2 * q^32 - 13 * q^34 - 4 * q^37 - 26 * q^38 - 2 * q^41 - 2 * q^43 + 15 * q^44 + 24 * q^46 - 12 * q^47 + 38 * q^49 - 49 * q^52 - 4 * q^53 - 58 * q^56 + 2 * q^58 - 18 * q^59 + 12 * q^61 + 4 * q^62 + 21 * q^64 - 26 * q^67 - 81 * q^68 - 24 * q^71 + 14 * q^73 + 22 * q^74 + 26 * q^77 + 10 * q^79 - 48 * q^82 - 40 * q^83 - 8 * q^86 + 10 * q^88 - 34 * q^89 + 26 * q^91 + 18 * q^92 - 43 * q^94 - 30 * q^97 - 29 * 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 $

$2$	−2.59587	−1.83556	−0.917779	−	0.397090i	$-0.870020\pi$
$2$	−2.59587	−1.83556	−0.917779	+	0.397090i	$0.870020\pi$
$3$	0	0
$3$	0	0
$4$	4.73855	2.36928
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.93826	−1.48852	−0.744262	−	0.667888i	$-0.767198\pi$
$7$	−3.93826	−1.48852	−0.744262	+	0.667888i	$0.767198\pi$
$8$	−7.10893	−2.51339
$9$	0	0
$10$	0	0
$11$	−2.24645	−0.677329	−0.338665	−	0.940907i	$-0.609975\pi$
$11$	−2.24645	−0.677329	−0.338665	+	0.940907i	$0.609975\pi$
$12$	0	0
$13$	2.09464	0.580950	0.290475	−	0.956883i	$-0.406187\pi$
$13$	2.09464	0.580950	0.290475	+	0.956883i	$0.406187\pi$
$14$	10.2232	2.73227
$15$	0	0
$16$	8.97678	2.24420
$17$	−6.95310	−1.68637	−0.843187	−	0.537620i	$-0.819324\pi$
$17$	−6.95310	−1.68637	−0.843187	+	0.537620i	$0.819324\pi$
$18$	0	0
$19$	4.29591	0.985549	0.492774	−	0.870157i	$-0.335983\pi$
$19$	4.29591	0.985549	0.492774	+	0.870157i	$0.335983\pi$
$20$	0	0
$21$	0	0
$22$	5.83149	1.24328
$23$	5.19955	1.08418	0.542090	−	0.840320i	$-0.317633\pi$
$23$	5.19955	1.08418	0.542090	+	0.840320i	$0.317633\pi$
$24$	0	0
$25$	0	0
$26$	−5.43743	−1.06637
$27$	0	0
$28$	−18.6617	−3.52672
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	2.25035	0.404175	0.202087	−	0.979367i	$-0.435227\pi$
$31$	2.25035	0.404175	0.202087	+	0.979367i	$0.435227\pi$
$32$	−9.08471	−1.60596
$33$	0	0
$34$	18.0494	3.09544
$35$	0	0
$36$	0	0
$37$	−10.5280	−1.73079	−0.865397	−	0.501086i	$-0.832934\pi$
$37$	−10.5280	−1.73079	−0.865397	+	0.501086i	$0.832934\pi$
$38$	−11.1516	−1.80903
$39$	0	0
$40$	0	0
$41$	2.66629	0.416405	0.208202	−	0.978086i	$-0.433239\pi$
$41$	2.66629	0.416405	0.208202	+	0.978086i	$0.433239\pi$
$42$	0	0
$43$	6.03681	0.920605	0.460302	−	0.887762i	$-0.347741\pi$
$43$	6.03681	0.920605	0.460302	+	0.887762i	$0.347741\pi$
$44$	−10.6449	−1.60478
$45$	0	0
$46$	−13.4974	−1.99008
$47$	7.74829	1.13020	0.565102	−	0.825021i	$-0.308837\pi$
$47$	7.74829	1.13020	0.565102	+	0.825021i	$0.308837\pi$
$48$	0	0
$49$	8.50991	1.21570
$50$	0	0
$51$	0	0
$52$	9.92559	1.37643
$53$	1.41309	0.194103	0.0970513	−	0.995279i	$-0.469059\pi$
$53$	1.41309	0.194103	0.0970513	+	0.995279i	$0.469059\pi$
$54$	0	0
$55$	0	0
$56$	27.9968	3.74124
$57$	0	0
$58$	2.59587	0.340855
$59$	−5.76800	−0.750929	−0.375465	−	0.926837i	$-0.622517\pi$
$59$	−5.76800	−0.750929	−0.375465	+	0.926837i	$0.622517\pi$
$60$	0	0
$61$	−10.9192	−1.39806	−0.699030	−	0.715092i	$-0.746385\pi$
$61$	−10.9192	−1.39806	−0.699030	+	0.715092i	$0.746385\pi$
$62$	−5.84162	−0.741887
$63$	0	0
$64$	5.62918	0.703647
$65$	0	0
$66$	0	0
$67$	−11.7471	−1.43514	−0.717569	−	0.696487i	$-0.754745\pi$
$67$	−11.7471	−1.43514	−0.717569	+	0.696487i	$0.754745\pi$
$68$	−32.9476	−3.99549
$69$	0	0
$70$	0	0
$71$	4.55691	0.540806	0.270403	−	0.962747i	$-0.412843\pi$
$71$	4.55691	0.540806	0.270403	+	0.962747i	$0.412843\pi$
$72$	0	0
$73$	12.3564	1.44621	0.723103	−	0.690740i	$-0.242715\pi$
$73$	12.3564	1.44621	0.723103	+	0.690740i	$0.242715\pi$
$74$	27.3294	3.17698
$75$	0	0
$76$	20.3564	2.33504
$77$	8.84710	1.00822
$78$	0	0
$79$	14.4098	1.62122	0.810612	−	0.585584i	$-0.199135\pi$
$79$	14.4098	1.62122	0.810612	+	0.585584i	$0.199135\pi$
$80$	0	0
$81$	0	0
$82$	−6.92135	−0.764335
$83$	3.81272	0.418501	0.209250	−	0.977862i	$-0.432898\pi$
$83$	3.81272	0.418501	0.209250	+	0.977862i	$0.432898\pi$
$84$	0	0
$85$	0	0
$86$	−15.6708	−1.68982
$87$	0	0
$88$	15.9699	1.70239
$89$	2.07340	0.219780	0.109890	−	0.993944i	$-0.464950\pi$
$89$	2.07340	0.219780	0.109890	+	0.993944i	$0.464950\pi$
$90$	0	0
$91$	−8.24926	−0.864757
$92$	24.6383	2.56872
$93$	0	0
$94$	−20.1136	−2.07456
$95$	0	0
$96$	0	0
$97$	12.0068	1.21911	0.609553	−	0.792745i	$-0.291349\pi$
$97$	12.0068	1.21911	0.609553	+	0.792745i	$0.291349\pi$
$98$	−22.0906	−2.23149
$99$	0	0
$100$	0	0
$101$	15.8642	1.57855	0.789276	−	0.614039i	$-0.210456\pi$
$101$	15.8642	1.57855	0.789276	+	0.614039i	$0.210456\pi$
$102$	0	0
$103$	−10.3119	−1.01606	−0.508031	−	0.861339i	$-0.669627\pi$
$103$	−10.3119	−1.01606	−0.508031	+	0.861339i	$0.669627\pi$
$104$	−14.8907	−1.46015
$105$	0	0
$106$	−3.66820	−0.356287
$107$	−4.47645	−0.432755	−0.216378	−	0.976310i	$-0.569424\pi$
$107$	−4.47645	−0.432755	−0.216378	+	0.976310i	$0.569424\pi$
$108$	0	0
$109$	13.4783	1.29098	0.645492	−	0.763767i	$-0.276652\pi$
$109$	13.4783	1.29098	0.645492	+	0.763767i	$0.276652\pi$
$110$	0	0
$111$	0	0
$112$	−35.3529	−3.34054
$113$	11.4525	1.07736	0.538679	−	0.842511i	$-0.318923\pi$
$113$	11.4525	1.07736	0.538679	+	0.842511i	$0.318923\pi$
$114$	0	0
$115$	0	0
$116$	−4.73855	−0.439964
$117$	0	0
$118$	14.9730	1.37838
$119$	27.3831	2.51021
$120$	0	0
$121$	−5.95347	−0.541225
$122$	28.3449	2.56622
$123$	0	0
$124$	10.6634	0.957602
$125$	0	0
$126$	0	0
$127$	−3.23499	−0.287059	−0.143529	−	0.989646i	$-0.545845\pi$
$127$	−3.23499	−0.287059	−0.143529	+	0.989646i	$0.545845\pi$
$128$	3.55679	0.314378
$129$	0	0
$130$	0	0
$131$	13.3567	1.16698	0.583489	−	0.812121i	$-0.301687\pi$
$131$	13.3567	1.16698	0.583489	+	0.812121i	$0.301687\pi$
$132$	0	0
$133$	−16.9184	−1.46701
$134$	30.4940	2.63428
$135$	0	0
$136$	49.4291	4.23851
$137$	−4.35642	−0.372194	−0.186097	−	0.982531i	$-0.559584\pi$
$137$	−4.35642	−0.372194	−0.186097	+	0.982531i	$0.559584\pi$
$138$	0	0
$139$	−18.4171	−1.56212	−0.781061	−	0.624455i	$-0.785321\pi$
$139$	−18.4171	−1.56212	−0.781061	+	0.624455i	$0.785321\pi$
$140$	0	0
$141$	0	0
$142$	−11.8292	−0.992681
$143$	−4.70551	−0.393495
$144$	0	0
$145$	0	0
$146$	−32.0756	−2.65460
$147$	0	0
$148$	−49.8876	−4.10073
$149$	−17.6427	−1.44535	−0.722674	−	0.691189i	$-0.757087\pi$
$149$	−17.6427	−1.44535	−0.722674	+	0.691189i	$0.757087\pi$
$150$	0	0
$151$	4.02381	0.327453	0.163726	−	0.986506i	$-0.447649\pi$
$151$	4.02381	0.327453	0.163726	+	0.986506i	$0.447649\pi$
$152$	−30.5393	−2.47707
$153$	0	0
$154$	−22.9659	−1.85065
$155$	0	0
$156$	0	0
$157$	−7.70119	−0.614622	−0.307311	−	0.951609i	$-0.599429\pi$
$157$	−7.70119	−0.614622	−0.307311	+	0.951609i	$0.599429\pi$
$158$	−37.4059	−2.97585
$159$	0	0
$160$	0	0
$161$	−20.4772	−1.61383
$162$	0	0
$163$	−13.6829	−1.07173	−0.535866	−	0.844303i	$-0.680015\pi$
$163$	−13.6829	−1.07173	−0.535866	+	0.844303i	$0.680015\pi$
$164$	12.6344	0.986578
$165$	0	0
$166$	−9.89734	−0.768183
$167$	24.9607	1.93151	0.965757	−	0.259449i	$-0.0835408\pi$
$167$	24.9607	1.93151	0.965757	+	0.259449i	$0.0835408\pi$
$168$	0	0
$169$	−8.61246	−0.662497
$170$	0	0
$171$	0	0
$172$	28.6057	2.18117
$173$	−19.9187	−1.51439	−0.757194	−	0.653191i	$-0.773430\pi$
$173$	−19.9187	−1.51439	−0.757194	+	0.653191i	$0.773430\pi$
$174$	0	0
$175$	0	0
$176$	−20.1659	−1.52006
$177$	0	0
$178$	−5.38227	−0.403418
$179$	−10.8816	−0.813329	−0.406664	−	0.913578i	$-0.633308\pi$
$179$	−10.8816	−0.813329	−0.406664	+	0.913578i	$0.633308\pi$
$180$	0	0
$181$	−13.9372	−1.03594	−0.517972	−	0.855398i	$-0.673313\pi$
$181$	−13.9372	−1.03594	−0.517972	+	0.855398i	$0.673313\pi$
$182$	21.4140	1.58731
$183$	0	0
$184$	−36.9632	−2.72497
$185$	0	0
$186$	0	0
$187$	15.6198	1.14223
$188$	36.7157	2.67777
$189$	0	0
$190$	0	0
$191$	−13.3458	−0.965670	−0.482835	−	0.875711i	$-0.660393\pi$
$191$	−13.3458	−0.965670	−0.482835	+	0.875711i	$0.660393\pi$
$192$	0	0
$193$	15.6337	1.12534	0.562668	−	0.826683i	$-0.309775\pi$
$193$	15.6337	1.12534	0.562668	+	0.826683i	$0.309775\pi$
$194$	−31.1681	−2.23774
$195$	0	0
$196$	40.3247	2.88033
$197$	20.1140	1.43306	0.716530	−	0.697556i	$-0.245729\pi$
$197$	20.1140	1.43306	0.716530	+	0.697556i	$0.245729\pi$
$198$	0	0
$199$	−9.27232	−0.657297	−0.328649	−	0.944452i	$-0.606593\pi$
$199$	−9.27232	−0.657297	−0.328649	+	0.944452i	$0.606593\pi$
$200$	0	0
$201$	0	0
$202$	−41.1816	−2.89752
$203$	3.93826	0.276412
$204$	0	0
$205$	0	0
$206$	26.7684	1.86504
$207$	0	0
$208$	18.8032	1.30376
$209$	−9.65053	−0.667541
$210$	0	0
$211$	−0.228998	−0.0157649	−0.00788245	−	0.999969i	$-0.502509\pi$
$211$	−0.228998	−0.0157649	−0.00788245	+	0.999969i	$0.502509\pi$
$212$	6.69600	0.459883
$213$	0	0
$214$	11.6203	0.794348
$215$	0	0
$216$	0	0
$217$	−8.86247	−0.601624
$218$	−34.9879	−2.36968
$219$	0	0
$220$	0	0
$221$	−14.5643	−0.979699
$222$	0	0
$223$	−25.1117	−1.68160	−0.840802	−	0.541342i	$-0.817916\pi$
$223$	−25.1117	−1.68160	−0.840802	+	0.541342i	$0.817916\pi$
$224$	35.7780	2.39052
$225$	0	0
$226$	−29.7292	−1.97756
$227$	4.98706	0.331003	0.165501	−	0.986210i	$-0.447076\pi$
$227$	4.98706	0.331003	0.165501	+	0.986210i	$0.447076\pi$
$228$	0	0
$229$	1.60591	0.106122	0.0530608	−	0.998591i	$-0.483102\pi$
$229$	1.60591	0.106122	0.0530608	+	0.998591i	$0.483102\pi$
$230$	0	0
$231$	0	0
$232$	7.10893	0.466724
$233$	−6.65588	−0.436041	−0.218021	−	0.975944i	$-0.569960\pi$
$233$	−6.65588	−0.436041	−0.218021	+	0.975944i	$0.569960\pi$
$234$	0	0
$235$	0	0
$236$	−27.3320	−1.77916
$237$	0	0
$238$	−71.0831	−4.60763
$239$	−2.87227	−0.185792	−0.0928959	−	0.995676i	$-0.529612\pi$
$239$	−2.87227	−0.185792	−0.0928959	+	0.995676i	$0.529612\pi$
$240$	0	0
$241$	15.2768	0.984068	0.492034	−	0.870576i	$-0.336254\pi$
$241$	15.2768	0.984068	0.492034	+	0.870576i	$0.336254\pi$
$242$	15.4545	0.993450
$243$	0	0
$244$	−51.7412	−3.31239
$245$	0	0
$246$	0	0
$247$	8.99840	0.572554
$248$	−15.9976	−1.01585
$249$	0	0
$250$	0	0
$251$	−10.8782	−0.686627	−0.343314	−	0.939221i	$-0.611549\pi$
$251$	−10.8782	−0.686627	−0.343314	+	0.939221i	$0.611549\pi$
$252$	0	0
$253$	−11.6805	−0.734348
$254$	8.39761	0.526913
$255$	0	0
$256$	−20.4913	−1.28071
$257$	18.0748	1.12748	0.563739	−	0.825953i	$-0.309363\pi$
$257$	18.0748	1.12748	0.563739	+	0.825953i	$0.309363\pi$
$258$	0	0
$259$	41.4621	2.57633
$260$	0	0
$261$	0	0
$262$	−34.6722	−2.14206
$263$	18.4454	1.13739	0.568697	−	0.822547i	$-0.307448\pi$
$263$	18.4454	1.13739	0.568697	+	0.822547i	$0.307448\pi$
$264$	0	0
$265$	0	0
$266$	43.9180	2.69279
$267$	0	0
$268$	−55.6643	−3.40024
$269$	−11.6702	−0.711544	−0.355772	−	0.934573i	$-0.615782\pi$
$269$	−11.6702	−0.711544	−0.355772	+	0.934573i	$0.615782\pi$
$270$	0	0
$271$	20.2501	1.23011	0.615054	−	0.788485i	$-0.289134\pi$
$271$	20.2501	1.23011	0.615054	+	0.788485i	$0.289134\pi$
$272$	−62.4165	−3.78455
$273$	0	0
$274$	11.3087	0.683184
$275$	0	0
$276$	0	0
$277$	3.66952	0.220480	0.110240	−	0.993905i	$-0.464838\pi$
$277$	3.66952	0.220480	0.110240	+	0.993905i	$0.464838\pi$
$278$	47.8085	2.86737
$279$	0	0
$280$	0	0
$281$	−22.8056	−1.36047	−0.680233	−	0.732996i	$-0.738121\pi$
$281$	−22.8056	−1.36047	−0.680233	+	0.732996i	$0.738121\pi$
$282$	0	0
$283$	−2.38657	−0.141867	−0.0709335	−	0.997481i	$-0.522598\pi$
$283$	−2.38657	−0.141867	−0.0709335	+	0.997481i	$0.522598\pi$
$284$	21.5932	1.28132
$285$	0	0
$286$	12.2149	0.722282
$287$	−10.5006	−0.619828
$288$	0	0
$289$	31.3456	1.84386
$290$	0	0
$291$	0	0
$292$	58.5514	3.42646
$293$	−19.4843	−1.13828	−0.569142	−	0.822239i	$-0.692725\pi$
$293$	−19.4843	−1.13828	−0.569142	+	0.822239i	$0.692725\pi$
$294$	0	0
$295$	0	0
$296$	74.8430	4.35016
$297$	0	0
$298$	45.7982	2.65302
$299$	10.8912	0.629855
$300$	0	0
$301$	−23.7745	−1.37034
$302$	−10.4453	−0.601059
$303$	0	0
$304$	38.5634	2.21176
$305$	0	0
$306$	0	0
$307$	−0.527333	−0.0300965	−0.0150482	−	0.999887i	$-0.504790\pi$
$307$	−0.527333	−0.0300965	−0.0150482	+	0.999887i	$0.504790\pi$
$308$	41.9225	2.38875
$309$	0	0
$310$	0	0
$311$	7.89624	0.447755	0.223877	−	0.974617i	$-0.428129\pi$
$311$	7.89624	0.447755	0.223877	+	0.974617i	$0.428129\pi$
$312$	0	0
$313$	10.0064	0.565597	0.282799	−	0.959179i	$-0.408737\pi$
$313$	10.0064	0.565597	0.282799	+	0.959179i	$0.408737\pi$
$314$	19.9913	1.12818
$315$	0	0
$316$	68.2814	3.84113
$317$	−32.5401	−1.82763	−0.913817	−	0.406125i	$-0.866880\pi$
$317$	−32.5401	−1.82763	−0.913817	+	0.406125i	$0.866880\pi$
$318$	0	0
$319$	2.24645	0.125777
$320$	0	0
$321$	0	0
$322$	53.1562	2.96228
$323$	−29.8699	−1.66200
$324$	0	0
$325$	0	0
$326$	35.5192	1.96723
$327$	0	0
$328$	−18.9545	−1.04659
$329$	−30.5148	−1.68233
$330$	0	0
$331$	22.9411	1.26096	0.630480	−	0.776206i	$-0.282858\pi$
$331$	22.9411	1.26096	0.630480	+	0.776206i	$0.282858\pi$
$332$	18.0668	0.991544
$333$	0	0
$334$	−64.7947	−3.54541
$335$	0	0
$336$	0	0
$337$	16.0039	0.871789	0.435894	−	0.899998i	$-0.356432\pi$
$337$	16.0039	0.871789	0.435894	+	0.899998i	$0.356432\pi$
$338$	22.3569	1.21605
$339$	0	0
$340$	0	0
$341$	−5.05529	−0.273760
$342$	0	0
$343$	−5.94642	−0.321076
$344$	−42.9153	−2.31384
$345$	0	0
$346$	51.7063	2.77975
$347$	−22.9990	−1.23465	−0.617326	−	0.786707i	$-0.711784\pi$
$347$	−22.9990	−1.23465	−0.617326	+	0.786707i	$0.711784\pi$
$348$	0	0
$349$	24.3108	1.30133	0.650664	−	0.759366i	$-0.274491\pi$
$349$	24.3108	1.30133	0.650664	+	0.759366i	$0.274491\pi$
$350$	0	0
$351$	0	0
$352$	20.4083	1.08777
$353$	29.2685	1.55780	0.778902	−	0.627146i	$-0.215777\pi$
$353$	29.2685	1.55780	0.778902	+	0.627146i	$0.215777\pi$
$354$	0	0
$355$	0	0
$356$	9.82490	0.520719
$357$	0	0
$358$	28.2472	1.49291
$359$	−18.5090	−0.976868	−0.488434	−	0.872601i	$-0.662432\pi$
$359$	−18.5090	−0.976868	−0.488434	+	0.872601i	$0.662432\pi$
$360$	0	0
$361$	−0.545179	−0.0286936
$362$	36.1792	1.90154
$363$	0	0
$364$	−39.0896	−2.04885
$365$	0	0
$366$	0	0
$367$	0.896398	0.0467916	0.0233958	−	0.999726i	$-0.492552\pi$
$367$	0.896398	0.0467916	0.0233958	+	0.999726i	$0.492552\pi$
$368$	46.6752	2.43311
$369$	0	0
$370$	0	0
$371$	−5.56511	−0.288926
$372$	0	0
$373$	4.64559	0.240540	0.120270	−	0.992741i	$-0.461624\pi$
$373$	4.64559	0.240540	0.120270	+	0.992741i	$0.461624\pi$
$374$	−40.5469	−2.09663
$375$	0	0
$376$	−55.0821	−2.84064
$377$	−2.09464	−0.107880
$378$	0	0
$379$	−32.4863	−1.66871	−0.834356	−	0.551226i	$-0.814160\pi$
$379$	−32.4863	−1.66871	−0.834356	+	0.551226i	$0.814160\pi$
$380$	0	0
$381$	0	0
$382$	34.6441	1.77254
$383$	−9.48311	−0.484564	−0.242282	−	0.970206i	$-0.577896\pi$
$383$	−9.48311	−0.484564	−0.242282	+	0.970206i	$0.577896\pi$
$384$	0	0
$385$	0	0
$386$	−40.5830	−2.06562
$387$	0	0
$388$	56.8949	2.88840
$389$	−33.9509	−1.72138	−0.860689	−	0.509132i	$-0.829967\pi$
$389$	−33.9509	−1.72138	−0.860689	+	0.509132i	$0.829967\pi$
$390$	0	0
$391$	−36.1530	−1.82833
$392$	−60.4964	−3.05553
$393$	0	0
$394$	−52.2132	−2.63047
$395$	0	0
$396$	0	0
$397$	−25.0892	−1.25919	−0.629596	−	0.776923i	$-0.716780\pi$
$397$	−25.0892	−1.25919	−0.629596	+	0.776923i	$0.716780\pi$
$398$	24.0698	1.20651
$399$	0	0
$400$	0	0
$401$	4.85814	0.242604	0.121302	−	0.992616i	$-0.461293\pi$
$401$	4.85814	0.242604	0.121302	+	0.992616i	$0.461293\pi$
$402$	0	0
$403$	4.71368	0.234805
$404$	75.1736	3.74003
$405$	0	0
$406$	−10.2232	−0.507370
$407$	23.6506	1.17232
$408$	0	0
$409$	14.6604	0.724911	0.362456	−	0.932001i	$-0.381938\pi$
$409$	14.6604	0.724911	0.362456	+	0.932001i	$0.381938\pi$
$410$	0	0
$411$	0	0
$412$	−48.8635	−2.40733
$413$	22.7159	1.11778
$414$	0	0
$415$	0	0
$416$	−19.0292	−0.932985
$417$	0	0
$418$	25.0515	1.22531
$419$	18.4123	0.899502	0.449751	−	0.893154i	$-0.351513\pi$
$419$	18.4123	0.899502	0.449751	+	0.893154i	$0.351513\pi$
$420$	0	0
$421$	29.1489	1.42063	0.710316	−	0.703883i	$-0.248552\pi$
$421$	29.1489	1.42063	0.710316	+	0.703883i	$0.248552\pi$
$422$	0.594451	0.0289374
$423$	0	0
$424$	−10.0456	−0.487855
$425$	0	0
$426$	0	0
$427$	43.0027	2.08105
$428$	−21.2119	−1.02532
$429$	0	0
$430$	0	0
$431$	2.05315	0.0988966	0.0494483	−	0.998777i	$-0.484254\pi$
$431$	2.05315	0.0988966	0.0494483	+	0.998777i	$0.484254\pi$
$432$	0	0
$433$	−18.7820	−0.902603	−0.451302	−	0.892371i	$-0.649040\pi$
$433$	−18.7820	−0.902603	−0.451302	+	0.892371i	$0.649040\pi$
$434$	23.0058	1.10432
$435$	0	0
$436$	63.8675	3.05870
$437$	22.3368	1.06851
$438$	0	0
$439$	−33.7889	−1.61266	−0.806329	−	0.591467i	$-0.798549\pi$
$439$	−33.7889	−1.61266	−0.806329	+	0.591467i	$0.798549\pi$
$440$	0	0
$441$	0	0
$442$	37.8070	1.79830
$443$	−24.2962	−1.15435	−0.577173	−	0.816622i	$-0.695844\pi$
$443$	−24.2962	−1.15435	−0.577173	+	0.816622i	$0.695844\pi$
$444$	0	0
$445$	0	0
$446$	65.1868	3.08668
$447$	0	0
$448$	−22.1692	−1.04740
$449$	21.0976	0.995656	0.497828	−	0.867276i	$-0.334131\pi$
$449$	21.0976	0.995656	0.497828	+	0.867276i	$0.334131\pi$
$450$	0	0
$451$	−5.98968	−0.282043
$452$	54.2682	2.55256
$453$	0	0
$454$	−12.9458	−0.607575
$455$	0	0
$456$	0	0
$457$	8.70983	0.407429	0.203714	−	0.979030i	$-0.434699\pi$
$457$	8.70983	0.407429	0.203714	+	0.979030i	$0.434699\pi$
$458$	−4.16874	−0.194793
$459$	0	0
$460$	0	0
$461$	−29.5424	−1.37592	−0.687962	−	0.725746i	$-0.741495\pi$
$461$	−29.5424	−1.37592	−0.687962	+	0.725746i	$0.741495\pi$
$462$	0	0
$463$	37.3067	1.73379	0.866894	−	0.498493i	$-0.166113\pi$
$463$	37.3067	1.73379	0.866894	+	0.498493i	$0.166113\pi$
$464$	−8.97678	−0.416737
$465$	0	0
$466$	17.2778	0.800379
$467$	−30.2144	−1.39816	−0.699079	−	0.715045i	$-0.746406\pi$
$467$	−30.2144	−1.39816	−0.699079	+	0.715045i	$0.746406\pi$
$468$	0	0
$469$	46.2632	2.13624
$470$	0	0
$471$	0	0
$472$	41.0043	1.88738
$473$	−13.5614	−0.623553
$474$	0	0
$475$	0	0
$476$	129.756	5.94738
$477$	0	0
$478$	7.45605	0.341032
$479$	−39.1511	−1.78886	−0.894430	−	0.447209i	$-0.852418\pi$
$479$	−39.1511	−1.78886	−0.894430	+	0.447209i	$0.852418\pi$
$480$	0	0
$481$	−22.0524	−1.00551
$482$	−39.6567	−1.80632
$483$	0	0
$484$	−28.2108	−1.28231
$485$	0	0
$486$	0	0
$487$	−23.9182	−1.08384	−0.541919	−	0.840430i	$-0.682302\pi$
$487$	−23.9182	−1.08384	−0.541919	+	0.840430i	$0.682302\pi$
$488$	77.6239	3.51387
$489$	0	0
$490$	0	0
$491$	−4.40974	−0.199009	−0.0995044	−	0.995037i	$-0.531726\pi$
$491$	−4.40974	−0.199009	−0.0995044	+	0.995037i	$0.531726\pi$
$492$	0	0
$493$	6.95310	0.313152
$494$	−23.3587	−1.05096
$495$	0	0
$496$	20.2009	0.907047
$497$	−17.9463	−0.805002
$498$	0	0
$499$	−37.0446	−1.65834	−0.829171	−	0.558995i	$-0.811187\pi$
$499$	−37.0446	−1.65834	−0.829171	+	0.558995i	$0.811187\pi$
$500$	0	0
$501$	0	0
$502$	28.2385	1.26034
$503$	12.6691	0.564886	0.282443	−	0.959284i	$-0.408855\pi$
$503$	12.6691	0.564886	0.282443	+	0.959284i	$0.408855\pi$
$504$	0	0
$505$	0	0
$506$	30.3211	1.34794
$507$	0	0
$508$	−15.3292	−0.680121
$509$	11.7078	0.518939	0.259469	−	0.965751i	$-0.416452\pi$
$509$	11.7078	0.518939	0.259469	+	0.965751i	$0.416452\pi$
$510$	0	0
$511$	−48.6627	−2.15271
$512$	46.0793	2.03644
$513$	0	0
$514$	−46.9200	−2.06955
$515$	0	0
$516$	0	0
$517$	−17.4061	−0.765520
$518$	−107.630	−4.72900
$519$	0	0
$520$	0	0
$521$	−0.0559550	−0.00245143	−0.00122572	−	0.999999i	$-0.500390\pi$
$521$	−0.0559550	−0.00245143	−0.00122572	+	0.999999i	$0.500390\pi$
$522$	0	0
$523$	−0.256669	−0.0112234	−0.00561168	−	0.999984i	$-0.501786\pi$
$523$	−0.256669	−0.0112234	−0.00561168	+	0.999984i	$0.501786\pi$
$524$	63.2913	2.76489
$525$	0	0
$526$	−47.8819	−2.08775
$527$	−15.6469	−0.681590
$528$	0	0
$529$	4.03530	0.175448
$530$	0	0
$531$	0	0
$532$	−80.1688	−3.47576
$533$	5.58493	0.241910
$534$	0	0
$535$	0	0
$536$	83.5095	3.60706
$537$	0	0
$538$	30.2943	1.30608
$539$	−19.1171	−0.823430
$540$	0	0
$541$	1.88887	0.0812090	0.0406045	−	0.999175i	$-0.487072\pi$
$541$	1.88887	0.0812090	0.0406045	+	0.999175i	$0.487072\pi$
$542$	−52.5668	−2.25794
$543$	0	0
$544$	63.1669	2.70826
$545$	0	0
$546$	0	0
$547$	−0.193341	−0.00826668	−0.00413334	−	0.999991i	$-0.501316\pi$
$547$	−0.193341	−0.00826668	−0.00413334	+	0.999991i	$0.501316\pi$
$548$	−20.6431	−0.881831
$549$	0	0
$550$	0	0
$551$	−4.29591	−0.183012
$552$	0	0
$553$	−56.7494	−2.41323
$554$	−9.52561	−0.404704
$555$	0	0
$556$	−87.2706	−3.70110
$557$	26.4282	1.11980	0.559899	−	0.828561i	$-0.310840\pi$
$557$	26.4282	1.11980	0.559899	+	0.828561i	$0.310840\pi$
$558$	0	0
$559$	12.6450	0.534825
$560$	0	0
$561$	0	0
$562$	59.2003	2.49722
$563$	6.18961	0.260861	0.130430	−	0.991457i	$-0.458364\pi$
$563$	6.18961	0.260861	0.130430	+	0.991457i	$0.458364\pi$
$564$	0	0
$565$	0	0
$566$	6.19523	0.260405
$567$	0	0
$568$	−32.3948	−1.35926
$569$	15.9034	0.666705	0.333353	−	0.942802i	$-0.391820\pi$
$569$	15.9034	0.666705	0.333353	+	0.942802i	$0.391820\pi$
$570$	0	0
$571$	−16.8555	−0.705380	−0.352690	−	0.935740i	$-0.614733\pi$
$571$	−16.8555	−0.705380	−0.352690	+	0.935740i	$0.614733\pi$
$572$	−22.2973	−0.932297
$573$	0	0
$574$	27.2581	1.13773
$575$	0	0
$576$	0	0
$577$	−23.1237	−0.962650	−0.481325	−	0.876542i	$-0.659844\pi$
$577$	−23.1237	−0.962650	−0.481325	+	0.876542i	$0.659844\pi$
$578$	−81.3692	−3.38451
$579$	0	0
$580$	0	0
$581$	−15.0155	−0.622948
$582$	0	0
$583$	−3.17443	−0.131471
$584$	−87.8408	−3.63488
$585$	0	0
$586$	50.5787	2.08939
$587$	7.80902	0.322313	0.161156	−	0.986929i	$-0.448478\pi$
$587$	7.80902	0.322313	0.161156	+	0.986929i	$0.448478\pi$
$588$	0	0
$589$	9.66730	0.398334
$590$	0	0
$591$	0	0
$592$	−94.5077	−3.88424
$593$	−19.5804	−0.804069	−0.402034	−	0.915625i	$-0.631697\pi$
$593$	−19.5804	−0.804069	−0.402034	+	0.915625i	$0.631697\pi$
$594$	0	0
$595$	0	0
$596$	−83.6009	−3.42443
$597$	0	0
$598$	−28.2722	−1.15614
$599$	−30.2322	−1.23525	−0.617627	−	0.786471i	$-0.711906\pi$
$599$	−30.2322	−1.23525	−0.617627	+	0.786471i	$0.711906\pi$
$600$	0	0
$601$	33.1180	1.35091	0.675457	−	0.737400i	$-0.263947\pi$
$601$	33.1180	1.35091	0.675457	+	0.737400i	$0.263947\pi$
$602$	61.7157	2.51534
$603$	0	0
$604$	19.0670	0.775826
$605$	0	0
$606$	0	0
$607$	−23.4711	−0.952661	−0.476330	−	0.879266i	$-0.658033\pi$
$607$	−23.4711	−0.952661	−0.476330	+	0.879266i	$0.658033\pi$
$608$	−39.0271	−1.58276
$609$	0	0
$610$	0	0
$611$	16.2299	0.656592
$612$	0	0
$613$	24.1560	0.975650	0.487825	−	0.872941i	$-0.337790\pi$
$613$	24.1560	0.975650	0.487825	+	0.872941i	$0.337790\pi$
$614$	1.36889	0.0552439
$615$	0	0
$616$	−62.8935	−2.53405
$617$	47.7509	1.92238	0.961189	−	0.275890i	$-0.0889725\pi$
$617$	47.7509	1.92238	0.961189	+	0.275890i	$0.0889725\pi$
$618$	0	0
$619$	0.779735	0.0313402	0.0156701	−	0.999877i	$-0.495012\pi$
$619$	0.779735	0.0313402	0.0156701	+	0.999877i	$0.495012\pi$
$620$	0	0
$621$	0	0
$622$	−20.4976	−0.821880
$623$	−8.16558	−0.327147
$624$	0	0
$625$	0	0
$626$	−25.9754	−1.03819
$627$	0	0
$628$	−36.4925	−1.45621
$629$	73.2023	2.91877
$630$	0	0
$631$	17.8120	0.709083	0.354542	−	0.935040i	$-0.384637\pi$
$631$	17.8120	0.709083	0.354542	+	0.935040i	$0.384637\pi$
$632$	−102.438	−4.07476
$633$	0	0
$634$	84.4700	3.35473
$635$	0	0
$636$	0	0
$637$	17.8252	0.706262
$638$	−5.83149	−0.230871
$639$	0	0
$640$	0	0
$641$	−29.7647	−1.17563	−0.587817	−	0.808994i	$-0.700013\pi$
$641$	−29.7647	−1.17563	−0.587817	+	0.808994i	$0.700013\pi$
$642$	0	0
$643$	−41.4049	−1.63285	−0.816426	−	0.577451i	$-0.804048\pi$
$643$	−41.4049	−1.63285	−0.816426	+	0.577451i	$0.804048\pi$
$644$	−97.0322	−3.82361
$645$	0	0
$646$	77.5384	3.05071
$647$	−14.6074	−0.574275	−0.287137	−	0.957889i	$-0.592704\pi$
$647$	−14.6074	−0.574275	−0.287137	+	0.957889i	$0.592704\pi$
$648$	0	0
$649$	12.9575	0.508627
$650$	0	0
$651$	0	0
$652$	−64.8374	−2.53923
$653$	−30.5647	−1.19609	−0.598045	−	0.801462i	$-0.704056\pi$
$653$	−30.5647	−1.19609	−0.598045	+	0.801462i	$0.704056\pi$
$654$	0	0
$655$	0	0
$656$	23.9347	0.934493
$657$	0	0
$658$	79.2125	3.08802
$659$	−22.1893	−0.864371	−0.432186	−	0.901785i	$-0.642257\pi$
$659$	−22.1893	−0.864371	−0.432186	+	0.901785i	$0.642257\pi$
$660$	0	0
$661$	−9.85731	−0.383405	−0.191702	−	0.981453i	$-0.561401\pi$
$661$	−9.85731	−0.383405	−0.191702	+	0.981453i	$0.561401\pi$
$662$	−59.5523	−2.31457
$663$	0	0
$664$	−27.1044	−1.05185
$665$	0	0
$666$	0	0
$667$	−5.19955	−0.201327
$668$	118.277	4.57629
$669$	0	0
$670$	0	0
$671$	24.5294	0.946948
$672$	0	0
$673$	−25.3057	−0.975463	−0.487732	−	0.872994i	$-0.662176\pi$
$673$	−25.3057	−0.975463	−0.487732	+	0.872994i	$0.662176\pi$
$674$	−41.5441	−1.60022
$675$	0	0
$676$	−40.8106	−1.56964
$677$	3.67306	0.141167	0.0705836	−	0.997506i	$-0.477514\pi$
$677$	3.67306	0.141167	0.0705836	+	0.997506i	$0.477514\pi$
$678$	0	0
$679$	−47.2860	−1.81467
$680$	0	0
$681$	0	0
$682$	13.1229	0.502502
$683$	−17.3807	−0.665053	−0.332527	−	0.943094i	$-0.607901\pi$
$683$	−17.3807	−0.665053	−0.332527	+	0.943094i	$0.607901\pi$
$684$	0	0
$685$	0	0
$686$	15.4361	0.589354
$687$	0	0
$688$	54.1911	2.06602
$689$	2.95992	0.112764
$690$	0	0
$691$	−28.5585	−1.08642	−0.543209	−	0.839598i	$-0.682791\pi$
$691$	−28.5585	−1.08642	−0.543209	+	0.839598i	$0.682791\pi$
$692$	−94.3856	−3.58800
$693$	0	0
$694$	59.7025	2.26628
$695$	0	0
$696$	0	0
$697$	−18.5390	−0.702214
$698$	−63.1078	−2.38866
$699$	0	0
$700$	0	0
$701$	−21.5401	−0.813557	−0.406779	−	0.913527i	$-0.633348\pi$
$701$	−21.5401	−0.813557	−0.406779	+	0.913527i	$0.633348\pi$
$702$	0	0
$703$	−45.2274	−1.70578
$704$	−12.6457	−0.476601
$705$	0	0
$706$	−75.9772	−2.85944
$707$	−62.4776	−2.34971
$708$	0	0
$709$	−2.29948	−0.0863587	−0.0431793	−	0.999067i	$-0.513749\pi$
$709$	−2.29948	−0.0863587	−0.0431793	+	0.999067i	$0.513749\pi$
$710$	0	0
$711$	0	0
$712$	−14.7396	−0.552391
$713$	11.7008	0.438199
$714$	0	0
$715$	0	0
$716$	−51.5630	−1.92700
$717$	0	0
$718$	48.0470	1.79310
$719$	−13.6814	−0.510232	−0.255116	−	0.966910i	$-0.582114\pi$
$719$	−13.6814	−0.510232	−0.255116	+	0.966910i	$0.582114\pi$
$720$	0	0
$721$	40.6110	1.51243
$722$	1.41521	0.0526688
$723$	0	0
$724$	−66.0422	−2.45444
$725$	0	0
$726$	0	0
$727$	50.0451	1.85607	0.928035	−	0.372492i	$-0.121497\pi$
$727$	50.0451	1.85607	0.928035	+	0.372492i	$0.121497\pi$
$728$	58.6435	2.17347
$729$	0	0
$730$	0	0
$731$	−41.9745	−1.55248
$732$	0	0
$733$	38.7734	1.43213	0.716065	−	0.698034i	$-0.245942\pi$
$733$	38.7734	1.43213	0.716065	+	0.698034i	$0.245942\pi$
$734$	−2.32694	−0.0858887
$735$	0	0
$736$	−47.2364	−1.74116
$737$	26.3893	0.972062
$738$	0	0
$739$	−18.0707	−0.664742	−0.332371	−	0.943149i	$-0.607849\pi$
$739$	−18.0707	−0.664742	−0.332371	+	0.943149i	$0.607849\pi$
$740$	0	0
$741$	0	0
$742$	14.4463	0.530341
$743$	20.9864	0.769917	0.384959	−	0.922934i	$-0.374216\pi$
$743$	20.9864	0.769917	0.384959	+	0.922934i	$0.374216\pi$
$744$	0	0
$745$	0	0
$746$	−12.0594	−0.441525
$747$	0	0
$748$	74.0151	2.70626
$749$	17.6294	0.644166
$750$	0	0
$751$	−20.3274	−0.741758	−0.370879	−	0.928681i	$-0.620944\pi$
$751$	−20.3274	−0.741758	−0.370879	+	0.928681i	$0.620944\pi$
$752$	69.5547	2.53640
$753$	0	0
$754$	5.43743	0.198020
$755$	0	0
$756$	0	0
$757$	0.0654846	0.00238008	0.00119004	−	0.999999i	$-0.499621\pi$
$757$	0.0654846	0.00238008	0.00119004	+	0.999999i	$0.499621\pi$
$758$	84.3304	3.06302
$759$	0	0
$760$	0	0
$761$	28.8872	1.04716	0.523580	−	0.851976i	$-0.324596\pi$
$761$	28.8872	1.04716	0.523580	+	0.851976i	$0.324596\pi$
$762$	0	0
$763$	−53.0810	−1.92166
$764$	−63.2399	−2.28794
$765$	0	0
$766$	24.6169	0.889446
$767$	−12.0819	−0.436252
$768$	0	0
$769$	−10.0813	−0.363541	−0.181771	−	0.983341i	$-0.558183\pi$
$769$	−10.0813	−0.363541	−0.181771	+	0.983341i	$0.558183\pi$
$770$	0	0
$771$	0	0
$772$	74.0809	2.66623
$773$	20.5353	0.738602	0.369301	−	0.929310i	$-0.379597\pi$
$773$	20.5353	0.738602	0.369301	+	0.929310i	$0.379597\pi$
$774$	0	0
$775$	0	0
$776$	−85.3556	−3.06409
$777$	0	0
$778$	88.1321	3.15969
$779$	11.4541	0.410387
$780$	0	0
$781$	−10.2369	−0.366304
$782$	93.8485	3.35602
$783$	0	0
$784$	76.3916	2.72827
$785$	0	0
$786$	0	0
$787$	−46.0601	−1.64186	−0.820932	−	0.571026i	$-0.806545\pi$
$787$	−46.0601	−1.64186	−0.820932	+	0.571026i	$0.806545\pi$
$788$	95.3110	3.39531
$789$	0	0
$790$	0	0
$791$	−45.1029	−1.60367
$792$	0	0
$793$	−22.8718	−0.812203
$794$	65.1284	2.31132
$795$	0	0
$796$	−43.9374	−1.55732
$797$	−37.4745	−1.32742	−0.663708	−	0.747992i	$-0.731018\pi$
$797$	−37.4745	−1.32742	−0.663708	+	0.747992i	$0.731018\pi$
$798$	0	0
$799$	−53.8746	−1.90595
$800$	0	0
$801$	0	0
$802$	−12.6111	−0.445314
$803$	−27.7580	−0.979558
$804$	0	0
$805$	0	0
$806$	−12.2361	−0.430999
$807$	0	0
$808$	−112.778	−3.96751
$809$	−29.9284	−1.05223	−0.526113	−	0.850414i	$-0.676351\pi$
$809$	−29.9284	−1.05223	−0.526113	+	0.850414i	$0.676351\pi$
$810$	0	0
$811$	−49.1738	−1.72673	−0.863363	−	0.504583i	$-0.831646\pi$
$811$	−49.1738	−1.72673	−0.863363	+	0.504583i	$0.831646\pi$
$812$	18.6617	0.654896
$813$	0	0
$814$	−61.3940	−2.15186
$815$	0	0
$816$	0	0
$817$	25.9336	0.907301
$818$	−38.0566	−1.33062
$819$	0	0
$820$	0	0
$821$	−33.9715	−1.18561	−0.592807	−	0.805345i	$-0.701980\pi$
$821$	−33.9715	−1.18561	−0.592807	+	0.805345i	$0.701980\pi$
$822$	0	0
$823$	33.8272	1.17914	0.589572	−	0.807716i	$-0.299297\pi$
$823$	33.8272	1.17914	0.589572	+	0.807716i	$0.299297\pi$
$824$	73.3067	2.55376
$825$	0	0
$826$	−58.9676	−2.05174
$827$	−35.1083	−1.22083	−0.610417	−	0.792080i	$-0.708998\pi$
$827$	−35.1083	−1.22083	−0.610417	+	0.792080i	$0.708998\pi$
$828$	0	0
$829$	15.6701	0.544245	0.272122	−	0.962263i	$-0.412274\pi$
$829$	15.6701	0.544245	0.272122	+	0.962263i	$0.412274\pi$
$830$	0	0
$831$	0	0
$832$	11.7911	0.408784
$833$	−59.1702	−2.05013
$834$	0	0
$835$	0	0
$836$	−45.7296	−1.58159
$837$	0	0
$838$	−47.7961	−1.65109
$839$	16.5131	0.570095	0.285047	−	0.958513i	$-0.407991\pi$
$839$	16.5131	0.570095	0.285047	+	0.958513i	$0.407991\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−75.6669	−2.60765
$843$	0	0
$844$	−1.08512	−0.0373514
$845$	0	0
$846$	0	0
$847$	23.4463	0.805626
$848$	12.6850	0.435604
$849$	0	0
$850$	0	0
$851$	−54.7409	−1.87649
$852$	0	0
$853$	−16.3926	−0.561271	−0.280636	−	0.959814i	$-0.590545\pi$
$853$	−16.3926	−0.561271	−0.280636	+	0.959814i	$0.590545\pi$
$854$	−111.629	−3.81988
$855$	0	0
$856$	31.8228	1.08768
$857$	−9.83745	−0.336041	−0.168020	−	0.985784i	$-0.553737\pi$
$857$	−9.83745	−0.336041	−0.168020	+	0.985784i	$0.553737\pi$
$858$	0	0
$859$	39.8257	1.35883	0.679417	−	0.733752i	$-0.262233\pi$
$859$	39.8257	1.35883	0.679417	+	0.733752i	$0.262233\pi$
$860$	0	0
$861$	0	0
$862$	−5.32971	−0.181530
$863$	−15.6707	−0.533438	−0.266719	−	0.963774i	$-0.585940\pi$
$863$	−15.6707	−0.533438	−0.266719	+	0.963774i	$0.585940\pi$
$864$	0	0
$865$	0	0
$866$	48.7556	1.65678
$867$	0	0
$868$	−41.9953	−1.42541
$869$	−32.3708	−1.09810
$870$	0	0
$871$	−24.6060	−0.833744
$872$	−95.8162	−3.24474
$873$	0	0
$874$	−57.9834	−1.96132
$875$	0	0
$876$	0	0
$877$	−21.2819	−0.718640	−0.359320	−	0.933214i	$-0.616991\pi$
$877$	−21.2819	−0.718640	−0.359320	+	0.933214i	$0.616991\pi$
$878$	87.7118	2.96013
$879$	0	0
$880$	0	0
$881$	−29.1037	−0.980529	−0.490265	−	0.871574i	$-0.663100\pi$
$881$	−29.1037	−0.980529	−0.490265	+	0.871574i	$0.663100\pi$
$882$	0	0
$883$	−4.03715	−0.135861	−0.0679305	−	0.997690i	$-0.521640\pi$
$883$	−4.03715	−0.135861	−0.0679305	+	0.997690i	$0.521640\pi$
$884$	−69.0136	−2.32118
$885$	0	0
$886$	63.0698	2.11887
$887$	2.85498	0.0958608	0.0479304	−	0.998851i	$-0.484737\pi$
$887$	2.85498	0.0958608	0.0479304	+	0.998851i	$0.484737\pi$
$888$	0	0
$889$	12.7402	0.427293
$890$	0	0
$891$	0	0
$892$	−118.993	−3.98419
$893$	33.2859	1.11387
$894$	0	0
$895$	0	0
$896$	−14.0076	−0.467960
$897$	0	0
$898$	−54.7666	−1.82758
$899$	−2.25035	−0.0750534
$900$	0	0
$901$	−9.82535	−0.327330
$902$	15.5485	0.517707
$903$	0	0
$904$	−81.4150	−2.70782
$905$	0	0
$906$	0	0
$907$	−3.08757	−0.102521	−0.0512605	−	0.998685i	$-0.516324\pi$
$907$	−3.08757	−0.102521	−0.0512605	+	0.998685i	$0.516324\pi$
$908$	23.6314	0.784237
$909$	0	0
$910$	0	0
$911$	−34.1306	−1.13080	−0.565398	−	0.824818i	$-0.691277\pi$
$911$	−34.1306	−1.13080	−0.565398	+	0.824818i	$0.691277\pi$
$912$	0	0
$913$	−8.56508	−0.283463
$914$	−22.6096	−0.747860
$915$	0	0
$916$	7.60970	0.251432
$917$	−52.6021	−1.73707
$918$	0	0
$919$	16.9603	0.559468	0.279734	−	0.960078i	$-0.409754\pi$
$919$	16.9603	0.559468	0.279734	+	0.960078i	$0.409754\pi$
$920$	0	0
$921$	0	0
$922$	76.6882	2.52559
$923$	9.54511	0.314181
$924$	0	0
$925$	0	0
$926$	−96.8433	−3.18247
$927$	0	0
$928$	9.08471	0.298220
$929$	−5.63831	−0.184987	−0.0924935	−	0.995713i	$-0.529484\pi$
$929$	−5.63831	−0.184987	−0.0924935	+	0.995713i	$0.529484\pi$
$930$	0	0
$931$	36.5578	1.19813
$932$	−31.5392	−1.03310
$933$	0	0
$934$	78.4328	2.56640
$935$	0	0
$936$	0	0
$937$	32.0790	1.04798	0.523988	−	0.851726i	$-0.324444\pi$
$937$	32.0790	1.04798	0.523988	+	0.851726i	$0.324444\pi$
$938$	−120.093	−3.92119
$939$	0	0
$940$	0	0
$941$	−31.2028	−1.01718	−0.508592	−	0.861008i	$-0.669834\pi$
$941$	−31.2028	−1.01718	−0.508592	+	0.861008i	$0.669834\pi$
$942$	0	0
$943$	13.8635	0.451458
$944$	−51.7781	−1.68523
$945$	0	0
$946$	35.2036	1.14457
$947$	27.7302	0.901110	0.450555	−	0.892749i	$-0.351226\pi$
$947$	27.7302	0.901110	0.450555	+	0.892749i	$0.351226\pi$
$948$	0	0
$949$	25.8822	0.840173
$950$	0	0
$951$	0	0
$952$	−194.665	−6.30913
$953$	−23.9481	−0.775755	−0.387877	−	0.921711i	$-0.626792\pi$
$953$	−23.9481	−0.775755	−0.387877	+	0.921711i	$0.626792\pi$
$954$	0	0
$955$	0	0
$956$	−13.6104	−0.440192
$957$	0	0
$958$	101.631	3.28356
$959$	17.1567	0.554020
$960$	0	0
$961$	−25.9359	−0.836643
$962$	57.2453	1.84566
$963$	0	0
$964$	72.3902	2.33153
$965$	0	0
$966$	0	0
$967$	−43.6821	−1.40472	−0.702361	−	0.711821i	$-0.747871\pi$
$967$	−43.6821	−1.40472	−0.702361	+	0.711821i	$0.747871\pi$
$968$	42.3228	1.36031
$969$	0	0
$970$	0	0
$971$	−1.12406	−0.0360729	−0.0180365	−	0.999837i	$-0.505741\pi$
$971$	−1.12406	−0.0360729	−0.0180365	+	0.999837i	$0.505741\pi$
$972$	0	0
$973$	72.5315	2.32525
$974$	62.0887	1.98945
$975$	0	0
$976$	−98.0193	−3.13752
$977$	18.8819	0.604086	0.302043	−	0.953294i	$-0.402331\pi$
$977$	18.8819	0.604086	0.302043	+	0.953294i	$0.402331\pi$
$978$	0	0
$979$	−4.65778	−0.148863
$980$	0	0
$981$	0	0
$982$	11.4471	0.365292
$983$	23.5807	0.752109	0.376054	−	0.926598i	$-0.377281\pi$
$983$	23.5807	0.752109	0.376054	+	0.926598i	$0.377281\pi$
$984$	0	0
$985$	0	0
$986$	−18.0494	−0.574809
$987$	0	0
$988$	42.6394	1.35654
$989$	31.3887	0.998102
$990$	0	0
$991$	−31.3818	−0.996877	−0.498438	−	0.866925i	$-0.666093\pi$
$991$	−31.3818	−0.996877	−0.498438	+	0.866925i	$0.666093\pi$
$992$	−20.4438	−0.649090
$993$	0	0
$994$	46.5864	1.47763
$995$	0	0
$996$	0	0
$997$	45.3442	1.43606	0.718032	−	0.696010i	$-0.245043\pi$
$997$	45.3442	1.43606	0.718032	+	0.696010i	$0.245043\pi$
$998$	96.1629	3.04398
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6525.2.a.by.1.1		8
3.2	odd	2		2175.2.a.bd.1.8	yes	8
5.4	even	2		6525.2.a.bz.1.8		8
15.2	even	4		2175.2.c.p.349.15		16
15.8	even	4		2175.2.c.p.349.2		16
15.14	odd	2		2175.2.a.bc.1.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2175.2.a.bc.1.1	✓	8	15.14	odd	2
2175.2.a.bd.1.8	yes	8	3.2	odd	2
2175.2.c.p.349.2		16	15.8	even	4
2175.2.c.p.349.15		16	15.2	even	4
6525.2.a.by.1.1		8	1.1	even	1	trivial
6525.2.a.bz.1.8		8	5.4	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:	\( N \)	\(=\)	\( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6525.a (trivial)

\(f(q)\)	\(=\)	\(q-2.59587 q^{2} +4.73855 q^{4} -3.93826 q^{7} -7.10893 q^{8} +O(q^{10})\)	\(q-2.59587 q^{2} +4.73855 q^{4} -3.93826 q^{7} -7.10893 q^{8} -2.24645 q^{11} +2.09464 q^{13} +10.2232 q^{14} +8.97678 q^{16} -6.95310 q^{17} +4.29591 q^{19} +5.83149 q^{22} +5.19955 q^{23} -5.43743 q^{26} -18.6617 q^{28} -1.00000 q^{29} +2.25035 q^{31} -9.08471 q^{32} +18.0494 q^{34} -10.5280 q^{37} -11.1516 q^{38} +2.66629 q^{41} +6.03681 q^{43} -10.6449 q^{44} -13.4974 q^{46} +7.74829 q^{47} +8.50991 q^{49} +9.92559 q^{52} +1.41309 q^{53} +27.9968 q^{56} +2.59587 q^{58} -5.76800 q^{59} -10.9192 q^{61} -5.84162 q^{62} +5.62918 q^{64} -11.7471 q^{67} -32.9476 q^{68} +4.55691 q^{71} +12.3564 q^{73} +27.3294 q^{74} +20.3564 q^{76} +8.84710 q^{77} +14.4098 q^{79} -6.92135 q^{82} +3.81272 q^{83} -15.6708 q^{86} +15.9699 q^{88} +2.07340 q^{89} -8.24926 q^{91} +24.6383 q^{92} -20.1136 q^{94} +12.0068 q^{97} -22.0906 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 12 q^{4} - 2 q^{7} - 3 q^{8}+O(q^{10}) \) 8 * q - 2 * q^2 + 12 * q^4 - 2 * q^7 - 3 * q^8	\( 8 q - 2 q^{2} + 12 q^{4} - 2 q^{7} - 3 q^{8} - 6 q^{11} - 6 q^{13} - 9 q^{14} + 32 q^{16} - 12 q^{17} - 3 q^{22} - 14 q^{23} - 18 q^{26} - 14 q^{28} - 8 q^{29} + 8 q^{31} + 2 q^{32} - 13 q^{34} - 4 q^{37} - 26 q^{38} - 2 q^{41} - 2 q^{43} + 15 q^{44} + 24 q^{46} - 12 q^{47} + 38 q^{49} - 49 q^{52} - 4 q^{53} - 58 q^{56} + 2 q^{58} - 18 q^{59} + 12 q^{61} + 4 q^{62} + 21 q^{64} - 26 q^{67} - 81 q^{68} - 24 q^{71} + 14 q^{73} + 22 q^{74} + 26 q^{77} + 10 q^{79} - 48 q^{82} - 40 q^{83} - 8 q^{86} + 10 q^{88} - 34 q^{89} + 26 q^{91} + 18 q^{92} - 43 q^{94} - 30 q^{97} - 29 q^{98}+O(q^{100}) \) 8 * q - 2 * q^2 + 12 * q^4 - 2 * q^7 - 3 * q^8 - 6 * q^11 - 6 * q^13 - 9 * q^14 + 32 * q^16 - 12 * q^17 - 3 * q^22 - 14 * q^23 - 18 * q^26 - 14 * q^28 - 8 * q^29 + 8 * q^31 + 2 * q^32 - 13 * q^34 - 4 * q^37 - 26 * q^38 - 2 * q^41 - 2 * q^43 + 15 * q^44 + 24 * q^46 - 12 * q^47 + 38 * q^49 - 49 * q^52 - 4 * q^53 - 58 * q^56 + 2 * q^58 - 18 * q^59 + 12 * q^61 + 4 * q^62 + 21 * q^64 - 26 * q^67 - 81 * q^68 - 24 * q^71 + 14 * q^73 + 22 * q^74 + 26 * q^77 + 10 * q^79 - 48 * q^82 - 40 * q^83 - 8 * q^86 + 10 * q^88 - 34 * q^89 + 26 * q^91 + 18 * q^92 - 43 * q^94 - 30 * q^97 - 29 * q^98

Introduction

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