LMFDB - Embedded newform 6525.2.a.cf.1.8

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:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 20x^{10} + 148x^{8} - 502x^{6} + 792x^{4} - 496x^{2} + 45 $ x^12 - 20x^10 + 148x^8 - 502x^6 + 792x^4 - 496*x^2 + 45
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 1305)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$1.27263$ 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)

$f(q)$	$=$	$q+1.27263 q^{2} -0.380419 q^{4} +0.255813 q^{7} -3.02939 q^{8} -4.63446 q^{11} +5.02700 q^{13} +0.325555 q^{14} -3.09444 q^{16} -0.336444 q^{17} -2.91437 q^{19} -5.89795 q^{22} +8.65656 q^{23} +6.39750 q^{26} -0.0973162 q^{28} -1.00000 q^{29} -3.26943 q^{31} +2.12070 q^{32} -0.428167 q^{34} -3.86954 q^{37} -3.70891 q^{38} -5.71649 q^{41} +6.98619 q^{43} +1.76304 q^{44} +11.0166 q^{46} -0.336444 q^{47} -6.93456 q^{49} -1.91237 q^{52} +6.01484 q^{53} -0.774958 q^{56} -1.27263 q^{58} +13.2799 q^{59} -7.77792 q^{61} -4.16077 q^{62} +8.88775 q^{64} -11.2605 q^{67} +0.127989 q^{68} +13.9668 q^{71} +8.46426 q^{73} -4.92448 q^{74} +1.10868 q^{76} -1.18556 q^{77} -15.3102 q^{79} -7.27496 q^{82} +7.60521 q^{83} +8.89082 q^{86} +14.0396 q^{88} +13.0383 q^{89} +1.28597 q^{91} -3.29312 q^{92} -0.428167 q^{94} +11.5445 q^{97} -8.82511 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 16 q^{4} + 12 q^{11} + 16 q^{14} + 16 q^{16} - 20 q^{19} + 56 q^{26} - 12 q^{29} - 16 q^{31} + 4 q^{34} + 32 q^{41} + 68 q^{44} + 20 q^{46} - 4 q^{49} + 76 q^{56} + 44 q^{59} - 52 q^{61} + 36 q^{64}+ \cdots + 4 q^{94}+O(q^{100}) $ 12 * q + 16 * q^4 + 12 * q^11 + 16 * q^14 + 16 * q^16 - 20 * q^19 + 56 * q^26 - 12 * q^29 - 16 * q^31 + 4 * q^34 + 32 * q^41 + 68 * q^44 + 20 * q^46 - 4 * q^49 + 76 * q^56 + 44 * q^59 - 52 * q^61 + 36 * q^64 + 20 * q^71 + 20 * q^74 - 28 * q^76 - 4 * q^79 + 36 * q^86 + 68 * q^89 + 48 * q^91 + 4 * q^94

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 $

$2$	1.27263	0.899884	0.449942	−	0.893058i	$-0.351445\pi$
$2$	1.27263	0.899884	0.449942	+	0.893058i	$0.351445\pi$
$3$	0	0
$3$	0	0
$4$	−0.380419	−0.190209
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.255813	0.0966884	0.0483442	−	0.998831i	$-0.484606\pi$
$7$	0.255813	0.0966884	0.0483442	+	0.998831i	$0.484606\pi$
$8$	−3.02939	−1.07105
$9$	0	0
$10$	0	0
$11$	−4.63446	−1.39734	−0.698672	−	0.715442i	$-0.746225\pi$
$11$	−4.63446	−1.39734	−0.698672	+	0.715442i	$0.746225\pi$
$12$	0	0
$13$	5.02700	1.39424	0.697120	−	0.716955i	$-0.254465\pi$
$13$	5.02700	1.39424	0.697120	+	0.716955i	$0.254465\pi$
$14$	0.325555	0.0870083
$15$	0	0
$16$	−3.09444	−0.773611
$17$	−0.336444	−0.0815995	−0.0407998	−	0.999167i	$-0.512991\pi$
$17$	−0.336444	−0.0815995	−0.0407998	+	0.999167i	$0.512991\pi$
$18$	0	0
$19$	−2.91437	−0.668602	−0.334301	−	0.942466i	$-0.608500\pi$
$19$	−2.91437	−0.668602	−0.334301	+	0.942466i	$0.608500\pi$
$20$	0	0
$21$	0	0
$22$	−5.89795	−1.25745
$23$	8.65656	1.80502	0.902509	−	0.430671i	$-0.141723\pi$
$23$	8.65656	1.80502	0.902509	+	0.430671i	$0.141723\pi$
$24$	0	0
$25$	0	0
$26$	6.39750	1.25465
$27$	0	0
$28$	−0.0973162	−0.0183910
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−3.26943	−0.587207	−0.293603	−	0.955927i	$-0.594854\pi$
$31$	−3.26943	−0.587207	−0.293603	+	0.955927i	$0.594854\pi$
$32$	2.12070	0.374890
$33$	0	0
$34$	−0.428167	−0.0734301
$35$	0	0
$36$	0	0
$37$	−3.86954	−0.636148	−0.318074	−	0.948066i	$-0.603036\pi$
$37$	−3.86954	−0.636148	−0.318074	+	0.948066i	$0.603036\pi$
$38$	−3.70891	−0.601664
$39$	0	0
$40$	0	0
$41$	−5.71649	−0.892765	−0.446383	−	0.894842i	$-0.647288\pi$
$41$	−5.71649	−0.892765	−0.446383	+	0.894842i	$0.647288\pi$
$42$	0	0
$43$	6.98619	1.06538	0.532692	−	0.846309i	$-0.321180\pi$
$43$	6.98619	1.06538	0.532692	+	0.846309i	$0.321180\pi$
$44$	1.76304	0.265788
$45$	0	0
$46$	11.0166	1.62431
$47$	−0.336444	−0.0490753	−0.0245377	−	0.999699i	$-0.507811\pi$
$47$	−0.336444	−0.0490753	−0.0245377	+	0.999699i	$0.507811\pi$
$48$	0	0
$49$	−6.93456	−0.990651
$50$	0	0
$51$	0	0
$52$	−1.91237	−0.265197
$53$	6.01484	0.826202	0.413101	−	0.910685i	$-0.364446\pi$
$53$	6.01484	0.826202	0.413101	+	0.910685i	$0.364446\pi$
$54$	0	0
$55$	0	0
$56$	−0.774958	−0.103558
$57$	0	0
$58$	−1.27263	−0.167104
$59$	13.2799	1.72890	0.864451	−	0.502718i	$-0.167666\pi$
$59$	13.2799	1.72890	0.864451	+	0.502718i	$0.167666\pi$
$60$	0	0
$61$	−7.77792	−0.995861	−0.497930	−	0.867217i	$-0.665906\pi$
$61$	−7.77792	−0.995861	−0.497930	+	0.867217i	$0.665906\pi$
$62$	−4.16077	−0.528418
$63$	0	0
$64$	8.88775	1.11097
$65$	0	0
$66$	0	0
$67$	−11.2605	−1.37569	−0.687847	−	0.725856i	$-0.741444\pi$
$67$	−11.2605	−1.37569	−0.687847	+	0.725856i	$0.741444\pi$
$68$	0.127989	0.0155210
$69$	0	0
$70$	0	0
$71$	13.9668	1.65756	0.828778	−	0.559578i	$-0.189037\pi$
$71$	13.9668	1.65756	0.828778	+	0.559578i	$0.189037\pi$
$72$	0	0
$73$	8.46426	0.990667	0.495333	−	0.868703i	$-0.335046\pi$
$73$	8.46426	0.990667	0.495333	+	0.868703i	$0.335046\pi$
$74$	−4.92448	−0.572459
$75$	0	0
$76$	1.10868	0.127174
$77$	−1.18556	−0.135107
$78$	0	0
$79$	−15.3102	−1.72253	−0.861267	−	0.508153i	$-0.830328\pi$
$79$	−15.3102	−1.72253	−0.861267	+	0.508153i	$0.830328\pi$
$80$	0	0
$81$	0	0
$82$	−7.27496	−0.803385
$83$	7.60521	0.834781	0.417390	−	0.908727i	$-0.362945\pi$
$83$	7.60521	0.834781	0.417390	+	0.908727i	$0.362945\pi$
$84$	0	0
$85$	0	0
$86$	8.89082	0.958722
$87$	0	0
$88$	14.0396	1.49662
$89$	13.0383	1.38206	0.691029	−	0.722827i	$-0.257157\pi$
$89$	13.0383	1.38206	0.691029	+	0.722827i	$0.257157\pi$
$90$	0	0
$91$	1.28597	0.134807
$92$	−3.29312	−0.343331
$93$	0	0
$94$	−0.428167	−0.0441621
$95$	0	0
$96$	0	0
$97$	11.5445	1.17216	0.586081	−	0.810253i	$-0.300670\pi$
$97$	11.5445	1.17216	0.586081	+	0.810253i	$0.300670\pi$
$98$	−8.82511	−0.891471
$99$	0	0
$100$	0	0
$101$	8.19832	0.815763	0.407882	−	0.913035i	$-0.366268\pi$
$101$	8.19832	0.815763	0.407882	+	0.913035i	$0.366268\pi$
$102$	0	0
$103$	−1.73014	−0.170476	−0.0852380	−	0.996361i	$-0.527165\pi$
$103$	−1.73014	−0.170476	−0.0852380	+	0.996361i	$0.527165\pi$
$104$	−15.2287	−1.49330
$105$	0	0
$106$	7.65465	0.743486
$107$	5.38921	0.520995	0.260497	−	0.965475i	$-0.416113\pi$
$107$	5.38921	0.520995	0.260497	+	0.965475i	$0.416113\pi$
$108$	0	0
$109$	14.2391	1.36386	0.681929	−	0.731418i	$-0.261141\pi$
$109$	14.2391	1.36386	0.681929	+	0.731418i	$0.261141\pi$
$110$	0	0
$111$	0	0
$112$	−0.791600	−0.0747992
$113$	−9.89466	−0.930811	−0.465406	−	0.885098i	$-0.654091\pi$
$113$	−9.89466	−0.930811	−0.465406	+	0.885098i	$0.654091\pi$
$114$	0	0
$115$	0	0
$116$	0.380419	0.0353210
$117$	0	0
$118$	16.9004	1.55581
$119$	−0.0860668	−0.00788973
$120$	0	0
$121$	10.4783	0.952569
$122$	−9.89840	−0.896159
$123$	0	0
$124$	1.24375	0.111692
$125$	0	0
$126$	0	0
$127$	12.2226	1.08458	0.542291	−	0.840191i	$-0.317557\pi$
$127$	12.2226	1.08458	0.542291	+	0.840191i	$0.317557\pi$
$128$	7.06940	0.624852
$129$	0	0
$130$	0	0
$131$	12.0464	1.05250	0.526250	−	0.850330i	$-0.323598\pi$
$131$	12.0464	1.05250	0.526250	+	0.850330i	$0.323598\pi$
$132$	0	0
$133$	−0.745535	−0.0646461
$134$	−14.3305	−1.23796
$135$	0	0
$136$	1.01922	0.0873972
$137$	−5.47512	−0.467771	−0.233886	−	0.972264i	$-0.575144\pi$
$137$	−5.47512	−0.467771	−0.233886	+	0.972264i	$0.575144\pi$
$138$	0	0
$139$	2.71403	0.230201	0.115100	−	0.993354i	$-0.463281\pi$
$139$	2.71403	0.230201	0.115100	+	0.993354i	$0.463281\pi$
$140$	0	0
$141$	0	0
$142$	17.7745	1.49161
$143$	−23.2975	−1.94823
$144$	0	0
$145$	0	0
$146$	10.7719	0.891485
$147$	0	0
$148$	1.47204	0.121001
$149$	3.70395	0.303439	0.151720	−	0.988424i	$-0.451519\pi$
$149$	3.70395	0.303439	0.151720	+	0.988424i	$0.451519\pi$
$150$	0	0
$151$	15.3481	1.24901	0.624505	−	0.781021i	$-0.285301\pi$
$151$	15.3481	1.24901	0.624505	+	0.781021i	$0.285301\pi$
$152$	8.82875	0.716107
$153$	0	0
$154$	−1.50877	−0.121580
$155$	0	0
$156$	0	0
$157$	21.8231	1.74168	0.870838	−	0.491571i	$-0.163577\pi$
$157$	21.8231	1.74168	0.870838	+	0.491571i	$0.163577\pi$
$158$	−19.4842	−1.55008
$159$	0	0
$160$	0	0
$161$	2.21447	0.174524
$162$	0	0
$163$	−3.08019	−0.241259	−0.120630	−	0.992698i	$-0.538491\pi$
$163$	−3.08019	−0.241259	−0.120630	+	0.992698i	$0.538491\pi$
$164$	2.17466	0.169812
$165$	0	0
$166$	9.67861	0.751206
$167$	8.55017	0.661632	0.330816	−	0.943695i	$-0.392676\pi$
$167$	8.55017	0.661632	0.330816	+	0.943695i	$0.392676\pi$
$168$	0	0
$169$	12.2707	0.943903
$170$	0	0
$171$	0	0
$172$	−2.65768	−0.202646
$173$	5.76340	0.438183	0.219092	−	0.975704i	$-0.429691\pi$
$173$	5.76340	0.438183	0.219092	+	0.975704i	$0.429691\pi$
$174$	0	0
$175$	0	0
$176$	14.3411	1.08100
$177$	0	0
$178$	16.5929	1.24369
$179$	−14.1430	−1.05710	−0.528548	−	0.848903i	$-0.677264\pi$
$179$	−14.1430	−1.05710	−0.528548	+	0.848903i	$0.677264\pi$
$180$	0	0
$181$	−15.0322	−1.11734	−0.558668	−	0.829391i	$-0.688688\pi$
$181$	−15.0322	−1.11734	−0.558668	+	0.829391i	$0.688688\pi$
$182$	1.63657	0.121310
$183$	0	0
$184$	−26.2241	−1.93326
$185$	0	0
$186$	0	0
$187$	1.55924	0.114023
$188$	0.127989	0.00933459
$189$	0	0
$190$	0	0
$191$	12.2634	0.887346	0.443673	−	0.896189i	$-0.353675\pi$
$191$	12.2634	0.887346	0.443673	+	0.896189i	$0.353675\pi$
$192$	0	0
$193$	8.34203	0.600472	0.300236	−	0.953865i	$-0.402935\pi$
$193$	8.34203	0.600472	0.300236	+	0.953865i	$0.402935\pi$
$194$	14.6918	1.05481
$195$	0	0
$196$	2.63804	0.188431
$197$	6.69263	0.476830	0.238415	−	0.971163i	$-0.423372\pi$
$197$	6.69263	0.476830	0.238415	+	0.971163i	$0.423372\pi$
$198$	0	0
$199$	12.5583	0.890235	0.445117	−	0.895472i	$-0.353162\pi$
$199$	12.5583	0.890235	0.445117	+	0.895472i	$0.353162\pi$
$200$	0	0
$201$	0	0
$202$	10.4334	0.734092
$203$	−0.255813	−0.0179546
$204$	0	0
$205$	0	0
$206$	−2.20183	−0.153409
$207$	0	0
$208$	−15.5558	−1.07860
$209$	13.5065	0.934267
$210$	0	0
$211$	−13.6479	−0.939563	−0.469782	−	0.882783i	$-0.655667\pi$
$211$	−13.6479	−0.939563	−0.469782	+	0.882783i	$0.655667\pi$
$212$	−2.28816	−0.157151
$213$	0	0
$214$	6.85846	0.468835
$215$	0	0
$216$	0	0
$217$	−0.836364	−0.0567761
$218$	18.1211	1.22731
$219$	0	0
$220$	0	0
$221$	−1.69130	−0.113769
$222$	0	0
$223$	13.4721	0.902158	0.451079	−	0.892484i	$-0.351039\pi$
$223$	13.4721	0.902158	0.451079	+	0.892484i	$0.351039\pi$
$224$	0.542503	0.0362475
$225$	0	0
$226$	−12.5922	−0.837622
$227$	−1.54298	−0.102411	−0.0512057	−	0.998688i	$-0.516306\pi$
$227$	−1.54298	−0.102411	−0.0512057	+	0.998688i	$0.516306\pi$
$228$	0	0
$229$	7.89526	0.521734	0.260867	−	0.965375i	$-0.415992\pi$
$229$	7.89526	0.521734	0.260867	+	0.965375i	$0.415992\pi$
$230$	0	0
$231$	0	0
$232$	3.02939	0.198889
$233$	28.4782	1.86567	0.932836	−	0.360301i	$-0.117326\pi$
$233$	28.4782	1.86567	0.932836	+	0.360301i	$0.117326\pi$
$234$	0	0
$235$	0	0
$236$	−5.05194	−0.328853
$237$	0	0
$238$	−0.109531	−0.00709984
$239$	−8.88508	−0.574728	−0.287364	−	0.957821i	$-0.592779\pi$
$239$	−8.88508	−0.574728	−0.287364	+	0.957821i	$0.592779\pi$
$240$	0	0
$241$	−29.8901	−1.92539	−0.962695	−	0.270589i	$-0.912782\pi$
$241$	−29.8901	−1.92539	−0.962695	+	0.270589i	$0.912782\pi$
$242$	13.3349	0.857202
$243$	0	0
$244$	2.95887	0.189422
$245$	0	0
$246$	0	0
$247$	−14.6505	−0.932192
$248$	9.90437	0.628928
$249$	0	0
$250$	0	0
$251$	15.9551	1.00708	0.503539	−	0.863973i	$-0.332031\pi$
$251$	15.9551	1.00708	0.503539	+	0.863973i	$0.332031\pi$
$252$	0	0
$253$	−40.1185	−2.52223
$254$	15.5548	0.975998
$255$	0	0
$256$	−8.77878	−0.548674
$257$	−17.1593	−1.07037	−0.535185	−	0.844735i	$-0.679758\pi$
$257$	−17.1593	−1.07037	−0.535185	+	0.844735i	$0.679758\pi$
$258$	0	0
$259$	−0.989880	−0.0615081
$260$	0	0
$261$	0	0
$262$	15.3306	0.947128
$263$	1.72781	0.106542	0.0532708	−	0.998580i	$-0.483035\pi$
$263$	1.72781	0.106542	0.0532708	+	0.998580i	$0.483035\pi$
$264$	0	0
$265$	0	0
$266$	−0.948789	−0.0581740
$267$	0	0
$268$	4.28372	0.261670
$269$	4.97931	0.303594	0.151797	−	0.988412i	$-0.451494\pi$
$269$	4.97931	0.303594	0.151797	+	0.988412i	$0.451494\pi$
$270$	0	0
$271$	13.0881	0.795047	0.397524	−	0.917592i	$-0.369870\pi$
$271$	13.0881	0.795047	0.397524	+	0.917592i	$0.369870\pi$
$272$	1.04111	0.0631263
$273$	0	0
$274$	−6.96779	−0.420939
$275$	0	0
$276$	0	0
$277$	20.8147	1.25063	0.625316	−	0.780371i	$-0.284970\pi$
$277$	20.8147	1.25063	0.625316	+	0.780371i	$0.284970\pi$
$278$	3.45394	0.207154
$279$	0	0
$280$	0	0
$281$	−3.93338	−0.234646	−0.117323	−	0.993094i	$-0.537431\pi$
$281$	−3.93338	−0.234646	−0.117323	+	0.993094i	$0.537431\pi$
$282$	0	0
$283$	−5.47745	−0.325600	−0.162800	−	0.986659i	$-0.552053\pi$
$283$	−5.47745	−0.325600	−0.162800	+	0.986659i	$0.552053\pi$
$284$	−5.31324	−0.315282
$285$	0	0
$286$	−29.6490	−1.75318
$287$	−1.46235	−0.0863201
$288$	0	0
$289$	−16.8868	−0.993342
$290$	0	0
$291$	0	0
$292$	−3.21996	−0.188434
$293$	−11.9974	−0.700893	−0.350446	−	0.936583i	$-0.613970\pi$
$293$	−11.9974	−0.700893	−0.350446	+	0.936583i	$0.613970\pi$
$294$	0	0
$295$	0	0
$296$	11.7223	0.681347
$297$	0	0
$298$	4.71375	0.273060
$299$	43.5166	2.51663
$300$	0	0
$301$	1.78716	0.103010
$302$	19.5324	1.12396
$303$	0	0
$304$	9.01836	0.517238
$305$	0	0
$306$	0	0
$307$	−26.4807	−1.51133	−0.755666	−	0.654957i	$-0.772687\pi$
$307$	−26.4807	−1.51133	−0.755666	+	0.654957i	$0.772687\pi$
$308$	0.451009	0.0256986
$309$	0	0
$310$	0	0
$311$	1.71530	0.0972660	0.0486330	−	0.998817i	$-0.484514\pi$
$311$	1.71530	0.0972660	0.0486330	+	0.998817i	$0.484514\pi$
$312$	0	0
$313$	−33.3782	−1.88665	−0.943325	−	0.331871i	$-0.892320\pi$
$313$	−33.3782	−1.88665	−0.943325	+	0.331871i	$0.892320\pi$
$314$	27.7727	1.56731
$315$	0	0
$316$	5.82429	0.327642
$317$	−17.9712	−1.00936	−0.504682	−	0.863305i	$-0.668390\pi$
$317$	−17.9712	−1.00936	−0.504682	+	0.863305i	$0.668390\pi$
$318$	0	0
$319$	4.63446	0.259480
$320$	0	0
$321$	0	0
$322$	2.81819	0.157052
$323$	0.980521	0.0545577
$324$	0	0
$325$	0	0
$326$	−3.91994	−0.217105
$327$	0	0
$328$	17.3175	0.956196
$329$	−0.0860668	−0.00474501
$330$	0	0
$331$	6.49393	0.356939	0.178469	−	0.983945i	$-0.442885\pi$
$331$	6.49393	0.356939	0.178469	+	0.983945i	$0.442885\pi$
$332$	−2.89317	−0.158783
$333$	0	0
$334$	10.8812	0.595392
$335$	0	0
$336$	0	0
$337$	6.36156	0.346536	0.173268	−	0.984875i	$-0.444567\pi$
$337$	6.36156	0.346536	0.173268	+	0.984875i	$0.444567\pi$
$338$	15.6161	0.849403
$339$	0	0
$340$	0	0
$341$	15.1521	0.820530
$342$	0	0
$343$	−3.56465	−0.192473
$344$	−21.1639	−1.14108
$345$	0	0
$346$	7.33466	0.394314
$347$	21.6036	1.15974	0.579871	−	0.814708i	$-0.303103\pi$
$347$	21.6036	1.15974	0.579871	+	0.814708i	$0.303103\pi$
$348$	0	0
$349$	−4.12603	−0.220861	−0.110431	−	0.993884i	$-0.535223\pi$
$349$	−4.12603	−0.220861	−0.110431	+	0.993884i	$0.535223\pi$
$350$	0	0
$351$	0	0
$352$	−9.82830	−0.523850
$353$	7.07724	0.376684	0.188342	−	0.982104i	$-0.439689\pi$
$353$	7.07724	0.376684	0.188342	+	0.982104i	$0.439689\pi$
$354$	0	0
$355$	0	0
$356$	−4.96002	−0.262881
$357$	0	0
$358$	−17.9988	−0.951264
$359$	14.4359	0.761900	0.380950	−	0.924596i	$-0.375597\pi$
$359$	14.4359	0.761900	0.380950	+	0.924596i	$0.375597\pi$
$360$	0	0
$361$	−10.5064	−0.552971
$362$	−19.1304	−1.00547
$363$	0	0
$364$	−0.489209	−0.0256415
$365$	0	0
$366$	0	0
$367$	15.4145	0.804630	0.402315	−	0.915501i	$-0.368206\pi$
$367$	15.4145	0.804630	0.402315	+	0.915501i	$0.368206\pi$
$368$	−26.7873	−1.39638
$369$	0	0
$370$	0	0
$371$	1.53868	0.0798841
$372$	0	0
$373$	29.8574	1.54596	0.772979	−	0.634432i	$-0.218766\pi$
$373$	29.8574	1.54596	0.772979	+	0.634432i	$0.218766\pi$
$374$	1.98433	0.102607
$375$	0	0
$376$	1.01922	0.0525621
$377$	−5.02700	−0.258904
$378$	0	0
$379$	37.3451	1.91829	0.959144	−	0.282920i	$-0.0913030\pi$
$379$	37.3451	1.91829	0.959144	+	0.282920i	$0.0913030\pi$
$380$	0	0
$381$	0	0
$382$	15.6067	0.798508
$383$	−25.8159	−1.31913	−0.659566	−	0.751647i	$-0.729260\pi$
$383$	−25.8159	−1.31913	−0.659566	+	0.751647i	$0.729260\pi$
$384$	0	0
$385$	0	0
$386$	10.6163	0.540355
$387$	0	0
$388$	−4.39173	−0.222956
$389$	−20.6935	−1.04920	−0.524600	−	0.851349i	$-0.675785\pi$
$389$	−20.6935	−1.04920	−0.524600	+	0.851349i	$0.675785\pi$
$390$	0	0
$391$	−2.91245	−0.147289
$392$	21.0075	1.06104
$393$	0	0
$394$	8.51723	0.429092
$395$	0	0
$396$	0	0
$397$	2.38045	0.119471	0.0597357	−	0.998214i	$-0.480974\pi$
$397$	2.38045	0.119471	0.0597357	+	0.998214i	$0.480974\pi$
$398$	15.9820	0.801108
$399$	0	0
$400$	0	0
$401$	32.0936	1.60268	0.801339	−	0.598211i	$-0.204121\pi$
$401$	32.0936	1.60268	0.801339	+	0.598211i	$0.204121\pi$
$402$	0	0
$403$	−16.4354	−0.818707
$404$	−3.11879	−0.155166
$405$	0	0
$406$	−0.325555	−0.0161570
$407$	17.9332	0.888918
$408$	0	0
$409$	−31.5011	−1.55763	−0.778815	−	0.627253i	$-0.784179\pi$
$409$	−31.5011	−1.55763	−0.778815	+	0.627253i	$0.784179\pi$
$410$	0	0
$411$	0	0
$412$	0.658179	0.0324261
$413$	3.39719	0.167165
$414$	0	0
$415$	0	0
$416$	10.6608	0.522686
$417$	0	0
$418$	17.1888	0.840732
$419$	3.15481	0.154123	0.0770613	−	0.997026i	$-0.475446\pi$
$419$	3.15481	0.154123	0.0770613	+	0.997026i	$0.475446\pi$
$420$	0	0
$421$	−11.9631	−0.583048	−0.291524	−	0.956564i	$-0.594162\pi$
$421$	−11.9631	−0.583048	−0.291524	+	0.956564i	$0.594162\pi$
$422$	−17.3687	−0.845497
$423$	0	0
$424$	−18.2213	−0.884904
$425$	0	0
$426$	0	0
$427$	−1.98970	−0.0962882
$428$	−2.05016	−0.0990981
$429$	0	0
$430$	0	0
$431$	31.2658	1.50602	0.753011	−	0.658008i	$-0.228601\pi$
$431$	31.2658	1.50602	0.753011	+	0.658008i	$0.228601\pi$
$432$	0	0
$433$	−30.6459	−1.47275	−0.736373	−	0.676575i	$-0.763463\pi$
$433$	−30.6459	−1.47275	−0.736373	+	0.676575i	$0.763463\pi$
$434$	−1.06438	−0.0510919
$435$	0	0
$436$	−5.41682	−0.259419
$437$	−25.2284	−1.20684
$438$	0	0
$439$	0.717425	0.0342408	0.0171204	−	0.999853i	$-0.494550\pi$
$439$	0.717425	0.0342408	0.0171204	+	0.999853i	$0.494550\pi$
$440$	0	0
$441$	0	0
$442$	−2.15240	−0.102379
$443$	−27.0649	−1.28589	−0.642947	−	0.765911i	$-0.722288\pi$
$443$	−27.0649	−1.28589	−0.642947	+	0.765911i	$0.722288\pi$
$444$	0	0
$445$	0	0
$446$	17.1450	0.811838
$447$	0	0
$448$	2.27361	0.107418
$449$	30.3663	1.43307	0.716536	−	0.697550i	$-0.245727\pi$
$449$	30.3663	1.43307	0.716536	+	0.697550i	$0.245727\pi$
$450$	0	0
$451$	26.4929	1.24750
$452$	3.76411	0.177049
$453$	0	0
$454$	−1.96364	−0.0921584
$455$	0	0
$456$	0	0
$457$	−2.43167	−0.113749	−0.0568743	−	0.998381i	$-0.518113\pi$
$457$	−2.43167	−0.113749	−0.0568743	+	0.998381i	$0.518113\pi$
$458$	10.0477	0.469500
$459$	0	0
$460$	0	0
$461$	−8.28439	−0.385842	−0.192921	−	0.981214i	$-0.561796\pi$
$461$	−8.28439	−0.385842	−0.192921	+	0.981214i	$0.561796\pi$
$462$	0	0
$463$	−7.05654	−0.327945	−0.163973	−	0.986465i	$-0.552431\pi$
$463$	−7.05654	−0.327945	−0.163973	+	0.986465i	$0.552431\pi$
$464$	3.09444	0.143656
$465$	0	0
$466$	36.2422	1.67889
$467$	−6.07165	−0.280963	−0.140481	−	0.990083i	$-0.544865\pi$
$467$	−6.07165	−0.280963	−0.140481	+	0.990083i	$0.544865\pi$
$468$	0	0
$469$	−2.88060	−0.133014
$470$	0	0
$471$	0	0
$472$	−40.2301	−1.85174
$473$	−32.3773	−1.48871
$474$	0	0
$475$	0	0
$476$	0.0327414	0.00150070
$477$	0	0
$478$	−11.3074	−0.517188
$479$	29.7675	1.36011	0.680055	−	0.733161i	$-0.261956\pi$
$479$	29.7675	1.36011	0.680055	+	0.733161i	$0.261956\pi$
$480$	0	0
$481$	−19.4522	−0.886943
$482$	−38.0390	−1.73263
$483$	0	0
$484$	−3.98613	−0.181188
$485$	0	0
$486$	0	0
$487$	−41.9050	−1.89890	−0.949448	−	0.313923i	$-0.898357\pi$
$487$	−41.9050	−1.89890	−0.949448	+	0.313923i	$0.898357\pi$
$488$	23.5623	1.06662
$489$	0	0
$490$	0	0
$491$	7.22435	0.326030	0.163015	−	0.986624i	$-0.447878\pi$
$491$	7.22435	0.326030	0.163015	+	0.986624i	$0.447878\pi$
$492$	0	0
$493$	0.336444	0.0151527
$494$	−18.6447	−0.838864
$495$	0	0
$496$	10.1171	0.454270
$497$	3.57290	0.160266
$498$	0	0
$499$	−15.5607	−0.696593	−0.348296	−	0.937384i	$-0.613240\pi$
$499$	−15.5607	−0.696593	−0.348296	+	0.937384i	$0.613240\pi$
$500$	0	0
$501$	0	0
$502$	20.3049	0.906253
$503$	38.5096	1.71706	0.858529	−	0.512765i	$-0.171379\pi$
$503$	38.5096	1.71706	0.858529	+	0.512765i	$0.171379\pi$
$504$	0	0
$505$	0	0
$506$	−51.0560	−2.26971
$507$	0	0
$508$	−4.64971	−0.206298
$509$	−31.6646	−1.40351	−0.701754	−	0.712419i	$-0.747600\pi$
$509$	−31.6646	−1.40351	−0.701754	+	0.712419i	$0.747600\pi$
$510$	0	0
$511$	2.16527	0.0957860
$512$	−25.3109	−1.11860
$513$	0	0
$514$	−21.8375	−0.963209
$515$	0	0
$516$	0	0
$517$	1.55924	0.0685751
$518$	−1.25975	−0.0553502
$519$	0	0
$520$	0	0
$521$	−20.2023	−0.885079	−0.442539	−	0.896749i	$-0.645922\pi$
$521$	−20.2023	−0.885079	−0.442539	+	0.896749i	$0.645922\pi$
$522$	0	0
$523$	12.1052	0.529325	0.264663	−	0.964341i	$-0.414739\pi$
$523$	12.1052	0.529325	0.264663	+	0.964341i	$0.414739\pi$
$524$	−4.58268	−0.200195
$525$	0	0
$526$	2.19886	0.0958751
$527$	1.09998	0.0479158
$528$	0	0
$529$	51.9361	2.25809
$530$	0	0
$531$	0	0
$532$	0.283615	0.0122963
$533$	−28.7368	−1.24473
$534$	0	0
$535$	0	0
$536$	34.1125	1.47344
$537$	0	0
$538$	6.33681	0.273199
$539$	32.1380	1.38428
$540$	0	0
$541$	−14.4279	−0.620304	−0.310152	−	0.950687i	$-0.600380\pi$
$541$	−14.4279	−0.620304	−0.310152	+	0.950687i	$0.600380\pi$
$542$	16.6563	0.715450
$543$	0	0
$544$	−0.713495	−0.0305909
$545$	0	0
$546$	0	0
$547$	−41.9178	−1.79228	−0.896139	−	0.443774i	$-0.853639\pi$
$547$	−41.9178	−1.79228	−0.896139	+	0.443774i	$0.853639\pi$
$548$	2.08284	0.0889744
$549$	0	0
$550$	0	0
$551$	2.91437	0.124156
$552$	0	0
$553$	−3.91656	−0.166549
$554$	26.4893	1.12542
$555$	0	0
$556$	−1.03247	−0.0437863
$557$	3.60324	0.152674	0.0763372	−	0.997082i	$-0.475677\pi$
$557$	3.60324	0.152674	0.0763372	+	0.997082i	$0.475677\pi$
$558$	0	0
$559$	35.1196	1.48540
$560$	0	0
$561$	0	0
$562$	−5.00573	−0.211154
$563$	15.5829	0.656741	0.328370	−	0.944549i	$-0.393501\pi$
$563$	15.5829	0.656741	0.328370	+	0.944549i	$0.393501\pi$
$564$	0	0
$565$	0	0
$566$	−6.97075	−0.293003
$567$	0	0
$568$	−42.3109	−1.77532
$569$	−36.9718	−1.54994	−0.774970	−	0.631998i	$-0.782235\pi$
$569$	−36.9718	−1.54994	−0.774970	+	0.631998i	$0.782235\pi$
$570$	0	0
$571$	29.9692	1.25417	0.627086	−	0.778950i	$-0.284248\pi$
$571$	29.9692	1.25417	0.627086	+	0.778950i	$0.284248\pi$
$572$	8.86279	0.370572
$573$	0	0
$574$	−1.86103	−0.0776780
$575$	0	0
$576$	0	0
$577$	−7.12327	−0.296546	−0.148273	−	0.988946i	$-0.547371\pi$
$577$	−7.12327	−0.296546	−0.148273	+	0.988946i	$0.547371\pi$
$578$	−21.4906	−0.893892
$579$	0	0
$580$	0	0
$581$	1.94552	0.0807136
$582$	0	0
$583$	−27.8756	−1.15449
$584$	−25.6415	−1.06105
$585$	0	0
$586$	−15.2682	−0.630722
$587$	−7.87237	−0.324928	−0.162464	−	0.986715i	$-0.551944\pi$
$587$	−7.87237	−0.324928	−0.162464	+	0.986715i	$0.551944\pi$
$588$	0	0
$589$	9.52833	0.392608
$590$	0	0
$591$	0	0
$592$	11.9741	0.492131
$593$	−17.9188	−0.735836	−0.367918	−	0.929858i	$-0.619929\pi$
$593$	−17.9188	−0.735836	−0.367918	+	0.929858i	$0.619929\pi$
$594$	0	0
$595$	0	0
$596$	−1.40905	−0.0577170
$597$	0	0
$598$	55.3804	2.26467
$599$	−20.1934	−0.825078	−0.412539	−	0.910940i	$-0.635358\pi$
$599$	−20.1934	−0.825078	−0.412539	+	0.910940i	$0.635358\pi$
$600$	0	0
$601$	23.2527	0.948496	0.474248	−	0.880391i	$-0.342720\pi$
$601$	23.2527	0.948496	0.474248	+	0.880391i	$0.342720\pi$
$602$	2.27439	0.0926973
$603$	0	0
$604$	−5.83870	−0.237573
$605$	0	0
$606$	0	0
$607$	−17.6177	−0.715082	−0.357541	−	0.933897i	$-0.616385\pi$
$607$	−17.6177	−0.715082	−0.357541	+	0.933897i	$0.616385\pi$
$608$	−6.18050	−0.250652
$609$	0	0
$610$	0	0
$611$	−1.69130	−0.0684228
$612$	0	0
$613$	13.4161	0.541873	0.270937	−	0.962597i	$-0.412667\pi$
$613$	13.4161	0.541873	0.270937	+	0.962597i	$0.412667\pi$
$614$	−33.7001	−1.36002
$615$	0	0
$616$	3.59151	0.144706
$617$	7.99166	0.321732	0.160866	−	0.986976i	$-0.448571\pi$
$617$	7.99166	0.321732	0.160866	+	0.986976i	$0.448571\pi$
$618$	0	0
$619$	−15.0770	−0.605996	−0.302998	−	0.952991i	$-0.597988\pi$
$619$	−15.0770	−0.605996	−0.302998	+	0.952991i	$0.597988\pi$
$620$	0	0
$621$	0	0
$622$	2.18294	0.0875281
$623$	3.33538	0.133629
$624$	0	0
$625$	0	0
$626$	−42.4781	−1.69776
$627$	0	0
$628$	−8.30193	−0.331283
$629$	1.30188	0.0519094
$630$	0	0
$631$	10.5154	0.418611	0.209306	−	0.977850i	$-0.432880\pi$
$631$	10.5154	0.418611	0.209306	+	0.977850i	$0.432880\pi$
$632$	46.3805	1.84492
$633$	0	0
$634$	−22.8707	−0.908311
$635$	0	0
$636$	0	0
$637$	−34.8600	−1.38121
$638$	5.89795	0.233502
$639$	0	0
$640$	0	0
$641$	27.2479	1.07623	0.538113	−	0.842873i	$-0.319137\pi$
$641$	27.2479	1.07623	0.538113	+	0.842873i	$0.319137\pi$
$642$	0	0
$643$	36.3023	1.43162	0.715812	−	0.698293i	$-0.246057\pi$
$643$	36.3023	1.43162	0.715812	+	0.698293i	$0.246057\pi$
$644$	−0.842424	−0.0331962
$645$	0	0
$646$	1.24784	0.0490955
$647$	8.77892	0.345135	0.172567	−	0.984998i	$-0.444794\pi$
$647$	8.77892	0.345135	0.172567	+	0.984998i	$0.444794\pi$
$648$	0	0
$649$	−61.5454	−2.41587
$650$	0	0
$651$	0	0
$652$	1.17176	0.0458898
$653$	44.1466	1.72759	0.863794	−	0.503845i	$-0.168082\pi$
$653$	44.1466	1.72759	0.863794	+	0.503845i	$0.168082\pi$
$654$	0	0
$655$	0	0
$656$	17.6894	0.690653
$657$	0	0
$658$	−0.109531	−0.00426996
$659$	−22.7653	−0.886810	−0.443405	−	0.896321i	$-0.646230\pi$
$659$	−22.7653	−0.886810	−0.443405	+	0.896321i	$0.646230\pi$
$660$	0	0
$661$	29.3474	1.14148	0.570740	−	0.821131i	$-0.306656\pi$
$661$	29.3474	1.14148	0.570740	+	0.821131i	$0.306656\pi$
$662$	8.26435	0.321203
$663$	0	0
$664$	−23.0391	−0.894092
$665$	0	0
$666$	0	0
$667$	−8.65656	−0.335183
$668$	−3.25264	−0.125849
$669$	0	0
$670$	0	0
$671$	36.0465	1.39156
$672$	0	0
$673$	−42.2267	−1.62772	−0.813860	−	0.581060i	$-0.802638\pi$
$673$	−42.2267	−1.62772	−0.813860	+	0.581060i	$0.802638\pi$
$674$	8.09590	0.311842
$675$	0	0
$676$	−4.66802	−0.179539
$677$	−19.9803	−0.767907	−0.383953	−	0.923353i	$-0.625438\pi$
$677$	−19.9803	−0.767907	−0.383953	+	0.923353i	$0.625438\pi$
$678$	0	0
$679$	2.95323	0.113334
$680$	0	0
$681$	0	0
$682$	19.2829	0.738381
$683$	23.0491	0.881950	0.440975	−	0.897519i	$-0.354633\pi$
$683$	23.0491	0.881950	0.440975	+	0.897519i	$0.354633\pi$
$684$	0	0
$685$	0	0
$686$	−4.53647	−0.173203
$687$	0	0
$688$	−21.6184	−0.824193
$689$	30.2366	1.15192
$690$	0	0
$691$	4.01208	0.152627	0.0763134	−	0.997084i	$-0.475685\pi$
$691$	4.01208	0.152627	0.0763134	+	0.997084i	$0.475685\pi$
$692$	−2.19250	−0.0833465
$693$	0	0
$694$	27.4933	1.04363
$695$	0	0
$696$	0	0
$697$	1.92328	0.0728493
$698$	−5.25090	−0.198750
$699$	0	0
$700$	0	0
$701$	−4.49887	−0.169920	−0.0849599	−	0.996384i	$-0.527076\pi$
$701$	−4.49887	−0.169920	−0.0849599	+	0.996384i	$0.527076\pi$
$702$	0	0
$703$	11.2773	0.425330
$704$	−41.1900	−1.55240
$705$	0	0
$706$	9.00670	0.338971
$707$	2.09724	0.0788748
$708$	0	0
$709$	36.7777	1.38121	0.690607	−	0.723230i	$-0.257343\pi$
$709$	36.7777	1.38121	0.690607	+	0.723230i	$0.257343\pi$
$710$	0	0
$711$	0	0
$712$	−39.4981	−1.48025
$713$	−28.3020	−1.05992
$714$	0	0
$715$	0	0
$716$	5.38026	0.201070
$717$	0	0
$718$	18.3716	0.685621
$719$	25.4279	0.948300	0.474150	−	0.880444i	$-0.342755\pi$
$719$	25.4279	0.948300	0.474150	+	0.880444i	$0.342755\pi$
$720$	0	0
$721$	−0.442594	−0.0164831
$722$	−13.3708	−0.497609
$723$	0	0
$724$	5.71854	0.212528
$725$	0	0
$726$	0	0
$727$	11.6265	0.431205	0.215602	−	0.976481i	$-0.430829\pi$
$727$	11.6265	0.431205	0.215602	+	0.976481i	$0.430829\pi$
$728$	−3.89571	−0.144385
$729$	0	0
$730$	0	0
$731$	−2.35046	−0.0869349
$732$	0	0
$733$	25.6926	0.948978	0.474489	−	0.880261i	$-0.342633\pi$
$733$	25.6926	0.948978	0.474489	+	0.880261i	$0.342633\pi$
$734$	19.6169	0.724073
$735$	0	0
$736$	18.3580	0.676683
$737$	52.1866	1.92232
$738$	0	0
$739$	−33.3612	−1.22721	−0.613606	−	0.789612i	$-0.710282\pi$
$739$	−33.3612	−1.22721	−0.613606	+	0.789612i	$0.710282\pi$
$740$	0	0
$741$	0	0
$742$	1.95816	0.0718864
$743$	25.6430	0.940750	0.470375	−	0.882467i	$-0.344119\pi$
$743$	25.6430	0.940750	0.470375	+	0.882467i	$0.344119\pi$
$744$	0	0
$745$	0	0
$746$	37.9973	1.39118
$747$	0	0
$748$	−0.593162	−0.0216882
$749$	1.37863	0.0503741
$750$	0	0
$751$	46.6411	1.70196	0.850979	−	0.525200i	$-0.176009\pi$
$751$	46.6411	1.70196	0.850979	+	0.525200i	$0.176009\pi$
$752$	1.04111	0.0379652
$753$	0	0
$754$	−6.39750	−0.232983
$755$	0	0
$756$	0	0
$757$	2.69262	0.0978650	0.0489325	−	0.998802i	$-0.484418\pi$
$757$	2.69262	0.0978650	0.0489325	+	0.998802i	$0.484418\pi$
$758$	47.5264	1.72624
$759$	0	0
$760$	0	0
$761$	0.471501	0.0170919	0.00854595	−	0.999963i	$-0.497280\pi$
$761$	0.471501	0.0170919	0.00854595	+	0.999963i	$0.497280\pi$
$762$	0	0
$763$	3.64255	0.131869
$764$	−4.66521	−0.168781
$765$	0	0
$766$	−32.8540	−1.18706
$767$	66.7583	2.41050
$768$	0	0
$769$	9.04380	0.326128	0.163064	−	0.986616i	$-0.447862\pi$
$769$	9.04380	0.326128	0.163064	+	0.986616i	$0.447862\pi$
$770$	0	0
$771$	0	0
$772$	−3.17346	−0.114215
$773$	−9.91726	−0.356699	−0.178350	−	0.983967i	$-0.557076\pi$
$773$	−9.91726	−0.356699	−0.178350	+	0.983967i	$0.557076\pi$
$774$	0	0
$775$	0	0
$776$	−34.9726	−1.25544
$777$	0	0
$778$	−26.3351	−0.944159
$779$	16.6600	0.596905
$780$	0	0
$781$	−64.7287	−2.31617
$782$	−3.70646	−0.132543
$783$	0	0
$784$	21.4586	0.766379
$785$	0	0
$786$	0	0
$787$	8.53609	0.304279	0.152139	−	0.988359i	$-0.451384\pi$
$787$	8.53609	0.304279	0.152139	+	0.988359i	$0.451384\pi$
$788$	−2.54600	−0.0906976
$789$	0	0
$790$	0	0
$791$	−2.53119	−0.0899986
$792$	0	0
$793$	−39.0996	−1.38847
$794$	3.02943	0.107510
$795$	0	0
$796$	−4.77741	−0.169331
$797$	−12.5563	−0.444768	−0.222384	−	0.974959i	$-0.571384\pi$
$797$	−12.5563	−0.444768	−0.222384	+	0.974959i	$0.571384\pi$
$798$	0	0
$799$	0.113194	0.00400453
$800$	0	0
$801$	0	0
$802$	40.8432	1.44222
$803$	−39.2273	−1.38430
$804$	0	0
$805$	0	0
$806$	−20.9162	−0.736741
$807$	0	0
$808$	−24.8359	−0.873723
$809$	−16.4082	−0.576883	−0.288441	−	0.957498i	$-0.593137\pi$
$809$	−16.4082	−0.576883	−0.288441	+	0.957498i	$0.593137\pi$
$810$	0	0
$811$	29.4688	1.03479	0.517394	−	0.855747i	$-0.326902\pi$
$811$	29.4688	1.03479	0.517394	+	0.855747i	$0.326902\pi$
$812$	0.0973162	0.00341513
$813$	0	0
$814$	22.8223	0.799923
$815$	0	0
$816$	0	0
$817$	−20.3603	−0.712318
$818$	−40.0892	−1.40169
$819$	0	0
$820$	0	0
$821$	28.5209	0.995385	0.497693	−	0.867353i	$-0.334181\pi$
$821$	28.5209	0.995385	0.497693	+	0.867353i	$0.334181\pi$
$822$	0	0
$823$	16.7546	0.584028	0.292014	−	0.956414i	$-0.405675\pi$
$823$	16.7546	0.584028	0.292014	+	0.956414i	$0.405675\pi$
$824$	5.24127	0.182588
$825$	0	0
$826$	4.32336	0.150429
$827$	3.38369	0.117662	0.0588312	−	0.998268i	$-0.481263\pi$
$827$	3.38369	0.117662	0.0588312	+	0.998268i	$0.481263\pi$
$828$	0	0
$829$	−14.4434	−0.501639	−0.250819	−	0.968034i	$-0.580700\pi$
$829$	−14.4434	−0.501639	−0.250819	+	0.968034i	$0.580700\pi$
$830$	0	0
$831$	0	0
$832$	44.6787	1.54896
$833$	2.33309	0.0808367
$834$	0	0
$835$	0	0
$836$	−5.13814	−0.177706
$837$	0	0
$838$	4.01490	0.138692
$839$	−27.6642	−0.955075	−0.477538	−	0.878611i	$-0.658471\pi$
$839$	−27.6642	−0.955075	−0.477538	+	0.878611i	$0.658471\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−15.2246	−0.524675
$843$	0	0
$844$	5.19193	0.178714
$845$	0	0
$846$	0	0
$847$	2.68048	0.0921024
$848$	−18.6126	−0.639159
$849$	0	0
$850$	0	0
$851$	−33.4969	−1.14826
$852$	0	0
$853$	4.26068	0.145883	0.0729415	−	0.997336i	$-0.476761\pi$
$853$	4.26068	0.145883	0.0729415	+	0.997336i	$0.476761\pi$
$854$	−2.53214	−0.0866481
$855$	0	0
$856$	−16.3260	−0.558011
$857$	−28.7104	−0.980729	−0.490364	−	0.871518i	$-0.663136\pi$
$857$	−28.7104	−0.980729	−0.490364	+	0.871518i	$0.663136\pi$
$858$	0	0
$859$	−39.3734	−1.34340	−0.671701	−	0.740822i	$-0.734436\pi$
$859$	−39.3734	−1.34340	−0.671701	+	0.740822i	$0.734436\pi$
$860$	0	0
$861$	0	0
$862$	39.7898	1.35525
$863$	−22.1235	−0.753094	−0.376547	−	0.926398i	$-0.622889\pi$
$863$	−22.1235	−0.753094	−0.376547	+	0.926398i	$0.622889\pi$
$864$	0	0
$865$	0	0
$866$	−39.0008	−1.32530
$867$	0	0
$868$	0.318168	0.0107993
$869$	70.9546	2.40697
$870$	0	0
$871$	−56.6068	−1.91805
$872$	−43.1357	−1.46076
$873$	0	0
$874$	−32.1064	−1.08602
$875$	0	0
$876$	0	0
$877$	−32.3439	−1.09217	−0.546087	−	0.837728i	$-0.683883\pi$
$877$	−32.3439	−1.09217	−0.546087	+	0.837728i	$0.683883\pi$
$878$	0.913015	0.0308128
$879$	0	0
$880$	0	0
$881$	−54.6109	−1.83989	−0.919944	−	0.392051i	$-0.871766\pi$
$881$	−54.6109	−1.83989	−0.919944	+	0.392051i	$0.871766\pi$
$882$	0	0
$883$	47.5511	1.60022	0.800111	−	0.599852i	$-0.204774\pi$
$883$	47.5511	1.60022	0.800111	+	0.599852i	$0.204774\pi$
$884$	0.643403	0.0216400
$885$	0	0
$886$	−34.4436	−1.15715
$887$	−10.4630	−0.351312	−0.175656	−	0.984452i	$-0.556205\pi$
$887$	−10.4630	−0.351312	−0.175656	+	0.984452i	$0.556205\pi$
$888$	0	0
$889$	3.12671	0.104866
$890$	0	0
$891$	0	0
$892$	−5.12504	−0.171599
$893$	0.980521	0.0328119
$894$	0	0
$895$	0	0
$896$	1.80845	0.0604160
$897$	0	0
$898$	38.6449	1.28960
$899$	3.26943	0.109042
$900$	0	0
$901$	−2.02365	−0.0674177
$902$	33.7156	1.12261
$903$	0	0
$904$	29.9747	0.996945
$905$	0	0
$906$	0	0
$907$	10.7211	0.355989	0.177994	−	0.984032i	$-0.443039\pi$
$907$	10.7211	0.355989	0.177994	+	0.984032i	$0.443039\pi$
$908$	0.586980	0.0194796
$909$	0	0
$910$	0	0
$911$	−3.57634	−0.118489	−0.0592447	−	0.998243i	$-0.518869\pi$
$911$	−3.57634	−0.118489	−0.0592447	+	0.998243i	$0.518869\pi$
$912$	0	0
$913$	−35.2461	−1.16648
$914$	−3.09461	−0.102361
$915$	0	0
$916$	−3.00351	−0.0992386
$917$	3.08163	0.101765
$918$	0	0
$919$	−22.2776	−0.734870	−0.367435	−	0.930049i	$-0.619764\pi$
$919$	−22.2776	−0.734870	−0.367435	+	0.930049i	$0.619764\pi$
$920$	0	0
$921$	0	0
$922$	−10.5429	−0.347213
$923$	70.2112	2.31103
$924$	0	0
$925$	0	0
$926$	−8.98035	−0.295113
$927$	0	0
$928$	−2.12070	−0.0696153
$929$	−21.1779	−0.694826	−0.347413	−	0.937712i	$-0.612940\pi$
$929$	−21.1779	−0.694826	−0.347413	+	0.937712i	$0.612940\pi$
$930$	0	0
$931$	20.2099	0.662352
$932$	−10.8337	−0.354868
$933$	0	0
$934$	−7.72695	−0.252834
$935$	0	0
$936$	0	0
$937$	−34.4462	−1.12531	−0.562654	−	0.826693i	$-0.690220\pi$
$937$	−34.4462	−1.12531	−0.562654	+	0.826693i	$0.690220\pi$
$938$	−3.66593	−0.119697
$939$	0	0
$940$	0	0
$941$	0.296167	0.00965477	0.00482738	−	0.999988i	$-0.498463\pi$
$941$	0.296167	0.00965477	0.00482738	+	0.999988i	$0.498463\pi$
$942$	0	0
$943$	−49.4851	−1.61146
$944$	−41.0940	−1.33750
$945$	0	0
$946$	−41.2042	−1.33966
$947$	−46.9467	−1.52556	−0.762782	−	0.646656i	$-0.776167\pi$
$947$	−46.9467	−1.52556	−0.762782	+	0.646656i	$0.776167\pi$
$948$	0	0
$949$	42.5498	1.38123
$950$	0	0
$951$	0	0
$952$	0.260730	0.00845029
$953$	1.91375	0.0619926	0.0309963	−	0.999519i	$-0.490132\pi$
$953$	1.91375	0.0619926	0.0309963	+	0.999519i	$0.490132\pi$
$954$	0	0
$955$	0	0
$956$	3.38005	0.109319
$957$	0	0
$958$	37.8829	1.22394
$959$	−1.40061	−0.0452280
$960$	0	0
$961$	−20.3108	−0.655188
$962$	−24.7554	−0.798145
$963$	0	0
$964$	11.3707	0.366227
$965$	0	0
$966$	0	0
$967$	4.01326	0.129058	0.0645288	−	0.997916i	$-0.479446\pi$
$967$	4.01326	0.129058	0.0645288	+	0.997916i	$0.479446\pi$
$968$	−31.7427	−1.02025
$969$	0	0
$970$	0	0
$971$	−4.54751	−0.145937	−0.0729683	−	0.997334i	$-0.523247\pi$
$971$	−4.54751	−0.145937	−0.0729683	+	0.997334i	$0.523247\pi$
$972$	0	0
$973$	0.694284	0.0222577
$974$	−53.3295	−1.70879
$975$	0	0
$976$	24.0683	0.770409
$977$	44.2333	1.41515	0.707574	−	0.706639i	$-0.249789\pi$
$977$	44.2333	1.41515	0.707574	+	0.706639i	$0.249789\pi$
$978$	0	0
$979$	−60.4256	−1.93121
$980$	0	0
$981$	0	0
$982$	9.19391	0.293389
$983$	34.2083	1.09107	0.545537	−	0.838087i	$-0.316326\pi$
$983$	34.2083	1.09107	0.545537	+	0.838087i	$0.316326\pi$
$984$	0	0
$985$	0	0
$986$	0.428167	0.0136356
$987$	0	0
$988$	5.57334	0.177312
$989$	60.4764	1.92304
$990$	0	0
$991$	−1.76770	−0.0561528	−0.0280764	−	0.999606i	$-0.508938\pi$
$991$	−1.76770	−0.0561528	−0.0280764	+	0.999606i	$0.508938\pi$
$992$	−6.93347	−0.220138
$993$	0	0
$994$	4.54697	0.144221
$995$	0	0
$996$	0	0
$997$	23.7398	0.751848	0.375924	−	0.926650i	$-0.377325\pi$
$997$	23.7398	0.751848	0.375924	+	0.926650i	$0.377325\pi$
$998$	−19.8030	−0.626853
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6525.2.a.cf.1.8		12
3.2	odd	2		6525.2.a.ce.1.5		12
5.2	odd	4		1305.2.c.l.784.8	yes	12
5.3	odd	4		1305.2.c.l.784.5	yes	12
5.4	even	2	inner	6525.2.a.cf.1.5		12
15.2	even	4		1305.2.c.k.784.5	✓	12
15.8	even	4		1305.2.c.k.784.8	yes	12
15.14	odd	2		6525.2.a.ce.1.8		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1305.2.c.k.784.5	✓	12	15.2	even	4
1305.2.c.k.784.8	yes	12	15.8	even	4
1305.2.c.l.784.5	yes	12	5.3	odd	4
1305.2.c.l.784.8	yes	12	5.2	odd	4
6525.2.a.ce.1.5		12	3.2	odd	2
6525.2.a.ce.1.8		12	15.14	odd	2
6525.2.a.cf.1.5		12	5.4	even	2	inner
6525.2.a.cf.1.8		12	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6525.a (trivial)

\(f(q)\)	\(=\)	\(q+1.27263 q^{2} -0.380419 q^{4} +0.255813 q^{7} -3.02939 q^{8} -4.63446 q^{11} +5.02700 q^{13} +0.325555 q^{14} -3.09444 q^{16} -0.336444 q^{17} -2.91437 q^{19} -5.89795 q^{22} +8.65656 q^{23} +6.39750 q^{26} -0.0973162 q^{28} -1.00000 q^{29} -3.26943 q^{31} +2.12070 q^{32} -0.428167 q^{34} -3.86954 q^{37} -3.70891 q^{38} -5.71649 q^{41} +6.98619 q^{43} +1.76304 q^{44} +11.0166 q^{46} -0.336444 q^{47} -6.93456 q^{49} -1.91237 q^{52} +6.01484 q^{53} -0.774958 q^{56} -1.27263 q^{58} +13.2799 q^{59} -7.77792 q^{61} -4.16077 q^{62} +8.88775 q^{64} -11.2605 q^{67} +0.127989 q^{68} +13.9668 q^{71} +8.46426 q^{73} -4.92448 q^{74} +1.10868 q^{76} -1.18556 q^{77} -15.3102 q^{79} -7.27496 q^{82} +7.60521 q^{83} +8.89082 q^{86} +14.0396 q^{88} +13.0383 q^{89} +1.28597 q^{91} -3.29312 q^{92} -0.428167 q^{94} +11.5445 q^{97} -8.82511 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 16 q^{4} + 12 q^{11} + 16 q^{14} + 16 q^{16} - 20 q^{19} + 56 q^{26} - 12 q^{29} - 16 q^{31} + 4 q^{34} + 32 q^{41} + 68 q^{44} + 20 q^{46} - 4 q^{49} + 76 q^{56} + 44 q^{59} - 52 q^{61} + 36 q^{64}+ \cdots + 4 q^{94}+O(q^{100}) \) 12 * q + 16 * q^4 + 12 * q^11 + 16 * q^14 + 16 * q^16 - 20 * q^19 + 56 * q^26 - 12 * q^29 - 16 * q^31 + 4 * q^34 + 32 * q^41 + 68 * q^44 + 20 * q^46 - 4 * q^49 + 76 * q^56 + 44 * q^59 - 52 * q^61 + 36 * q^64 + 20 * q^71 + 20 * q^74 - 28 * q^76 - 4 * q^79 + 36 * q^86 + 68 * q^89 + 48 * q^91 + 4 * q^94

Introduction

L-functions

Modular forms

Varieties

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Database

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