LMFDB - Embedded newform 6525.2.a.ca.1.9

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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 2x^{8} - 12x^{7} + 21x^{6} + 48x^{5} - 68x^{4} - 73x^{3} + 66x^{2} + 40x - 10 $ x^9 - 2x^8 - 12x^7 + 21x^6 + 48x^5 - 68x^4 - 73x^3 + 66x^2 + 40x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-2.21081$ 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)

$f(q)$	$=$	$q+2.21081 q^{2} +2.88766 q^{4} +1.31646 q^{7} +1.96245 q^{8} +O(q^{10})$	$q+2.21081 q^{2} +2.88766 q^{4} +1.31646 q^{7} +1.96245 q^{8} -4.29189 q^{11} +2.97543 q^{13} +2.91044 q^{14} -1.43674 q^{16} -0.642441 q^{17} -5.07923 q^{19} -9.48853 q^{22} -8.84038 q^{23} +6.57809 q^{26} +3.80150 q^{28} +1.00000 q^{29} -6.27016 q^{31} -7.10123 q^{32} -1.42031 q^{34} -0.934054 q^{37} -11.2292 q^{38} -11.0792 q^{41} -2.03981 q^{43} -12.3935 q^{44} -19.5444 q^{46} -9.57178 q^{47} -5.26692 q^{49} +8.59203 q^{52} +13.0891 q^{53} +2.58349 q^{56} +2.21081 q^{58} +2.55069 q^{59} +8.46575 q^{61} -13.8621 q^{62} -12.8260 q^{64} +12.9808 q^{67} -1.85515 q^{68} -5.10419 q^{71} +16.0007 q^{73} -2.06501 q^{74} -14.6671 q^{76} -5.65012 q^{77} -3.30518 q^{79} -24.4939 q^{82} -5.24640 q^{83} -4.50962 q^{86} -8.42260 q^{88} +5.05088 q^{89} +3.91704 q^{91} -25.5280 q^{92} -21.1614 q^{94} +2.04414 q^{97} -11.6441 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 2 q^{2} + 10 q^{4} - q^{7} - 9 q^{8}+O(q^{10}) $ 9 * q - 2 * q^2 + 10 * q^4 - q^7 - 9 * q^8	$ 9 q - 2 q^{2} + 10 q^{4} - q^{7} - 9 q^{8} + 2 q^{11} - q^{13} - 3 q^{14} + 4 q^{16} - 12 q^{17} - q^{19} - 3 q^{22} - 16 q^{23} + 6 q^{26} + 4 q^{28} + 9 q^{29} + 5 q^{31} - 20 q^{32} + 3 q^{34} - 30 q^{38} - 10 q^{41} - 3 q^{43} - 13 q^{44} + 4 q^{46} - 26 q^{47} - 8 q^{49} + 9 q^{52} - 22 q^{53} + 22 q^{56} - 2 q^{58} + 4 q^{59} + 7 q^{61} - 28 q^{62} + 9 q^{64} - 5 q^{67} - 39 q^{68} + 10 q^{73} - 34 q^{74} - 2 q^{76} - 34 q^{77} + 10 q^{79} + 8 q^{82} - 46 q^{83} + 28 q^{86} - 2 q^{88} + 4 q^{89} - 21 q^{91} - 20 q^{92} + 5 q^{94} - 7 q^{97} - 51 q^{98}+O(q^{100}) $ 9 * q - 2 * q^2 + 10 * q^4 - q^7 - 9 * q^8 + 2 * q^11 - q^13 - 3 * q^14 + 4 * q^16 - 12 * q^17 - q^19 - 3 * q^22 - 16 * q^23 + 6 * q^26 + 4 * q^28 + 9 * q^29 + 5 * q^31 - 20 * q^32 + 3 * q^34 - 30 * q^38 - 10 * q^41 - 3 * q^43 - 13 * q^44 + 4 * q^46 - 26 * q^47 - 8 * q^49 + 9 * q^52 - 22 * q^53 + 22 * q^56 - 2 * q^58 + 4 * q^59 + 7 * q^61 - 28 * q^62 + 9 * q^64 - 5 * q^67 - 39 * q^68 + 10 * q^73 - 34 * q^74 - 2 * q^76 - 34 * q^77 + 10 * q^79 + 8 * q^82 - 46 * q^83 + 28 * q^86 - 2 * q^88 + 4 * q^89 - 21 * q^91 - 20 * q^92 + 5 * q^94 - 7 * q^97 - 51 * 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 $

$2$	2.21081	1.56328	0.781638	−	0.623733i	$-0.214385\pi$
$2$	2.21081	1.56328	0.781638	+	0.623733i	$0.214385\pi$
$3$	0	0
$3$	0	0
$4$	2.88766	1.44383
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.31646	0.497576	0.248788	−	0.968558i	$-0.419968\pi$
$7$	1.31646	0.497576	0.248788	+	0.968558i	$0.419968\pi$
$8$	1.96245	0.693830
$9$	0	0
$10$	0	0
$11$	−4.29189	−1.29405	−0.647027	−	0.762467i	$-0.723988\pi$
$11$	−4.29189	−1.29405	−0.647027	+	0.762467i	$0.723988\pi$
$12$	0	0
$13$	2.97543	0.825235	0.412617	−	0.910904i	$-0.364615\pi$
$13$	2.97543	0.825235	0.412617	+	0.910904i	$0.364615\pi$
$14$	2.91044	0.777849
$15$	0	0
$16$	−1.43674	−0.359184
$17$	−0.642441	−0.155815	−0.0779074	−	0.996961i	$-0.524824\pi$
$17$	−0.642441	−0.155815	−0.0779074	+	0.996961i	$0.524824\pi$
$18$	0	0
$19$	−5.07923	−1.16525	−0.582627	−	0.812740i	$-0.697975\pi$
$19$	−5.07923	−1.16525	−0.582627	+	0.812740i	$0.697975\pi$
$20$	0	0
$21$	0	0
$22$	−9.48853	−2.02296
$23$	−8.84038	−1.84335	−0.921674	−	0.387966i	$-0.873178\pi$
$23$	−8.84038	−1.84335	−0.921674	+	0.387966i	$0.873178\pi$
$24$	0	0
$25$	0	0
$26$	6.57809	1.29007
$27$	0	0
$28$	3.80150	0.718416
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−6.27016	−1.12615	−0.563077	−	0.826404i	$-0.690383\pi$
$31$	−6.27016	−1.12615	−0.563077	+	0.826404i	$0.690383\pi$
$32$	−7.10123	−1.25533
$33$	0	0
$34$	−1.42031	−0.243582
$35$	0	0
$36$	0	0
$37$	−0.934054	−0.153558	−0.0767788	−	0.997048i	$-0.524464\pi$
$37$	−0.934054	−0.153558	−0.0767788	+	0.997048i	$0.524464\pi$
$38$	−11.2292	−1.82161
$39$	0	0
$40$	0	0
$41$	−11.0792	−1.73028	−0.865139	−	0.501533i	$-0.832770\pi$
$41$	−11.0792	−1.73028	−0.865139	+	0.501533i	$0.832770\pi$
$42$	0	0
$43$	−2.03981	−0.311068	−0.155534	−	0.987831i	$-0.549710\pi$
$43$	−2.03981	−0.311068	−0.155534	+	0.987831i	$0.549710\pi$
$44$	−12.3935	−1.86839
$45$	0	0
$46$	−19.5444	−2.88166
$47$	−9.57178	−1.39619	−0.698094	−	0.716006i	$-0.745968\pi$
$47$	−9.57178	−1.39619	−0.698094	+	0.716006i	$0.745968\pi$
$48$	0	0
$49$	−5.26692	−0.752418
$50$	0	0
$51$	0	0
$52$	8.59203	1.19150
$53$	13.0891	1.79792	0.898960	−	0.438030i	$-0.144324\pi$
$53$	13.0891	1.79792	0.898960	+	0.438030i	$0.144324\pi$
$54$	0	0
$55$	0	0
$56$	2.58349	0.345233
$57$	0	0
$58$	2.21081	0.290293
$59$	2.55069	0.332071	0.166036	−	0.986120i	$-0.446903\pi$
$59$	2.55069	0.332071	0.166036	+	0.986120i	$0.446903\pi$
$60$	0	0
$61$	8.46575	1.08393	0.541964	−	0.840402i	$-0.317681\pi$
$61$	8.46575	1.08393	0.541964	+	0.840402i	$0.317681\pi$
$62$	−13.8621	−1.76049
$63$	0	0
$64$	−12.8260	−1.60325
$65$	0	0
$66$	0	0
$67$	12.9808	1.58586	0.792930	−	0.609313i	$-0.208555\pi$
$67$	12.9808	1.58586	0.792930	+	0.609313i	$0.208555\pi$
$68$	−1.85515	−0.224970
$69$	0	0
$70$	0	0
$71$	−5.10419	−0.605756	−0.302878	−	0.953029i	$-0.597947\pi$
$71$	−5.10419	−0.605756	−0.302878	+	0.953029i	$0.597947\pi$
$72$	0	0
$73$	16.0007	1.87274	0.936372	−	0.351009i	$-0.114161\pi$
$73$	16.0007	1.87274	0.936372	+	0.351009i	$0.114161\pi$
$74$	−2.06501	−0.240053
$75$	0	0
$76$	−14.6671	−1.68243
$77$	−5.65012	−0.643890
$78$	0	0
$79$	−3.30518	−0.371861	−0.185931	−	0.982563i	$-0.559530\pi$
$79$	−3.30518	−0.371861	−0.185931	+	0.982563i	$0.559530\pi$
$80$	0	0
$81$	0	0
$82$	−24.4939	−2.70490
$83$	−5.24640	−0.575867	−0.287934	−	0.957650i	$-0.592968\pi$
$83$	−5.24640	−0.575867	−0.287934	+	0.957650i	$0.592968\pi$
$84$	0	0
$85$	0	0
$86$	−4.50962	−0.486284
$87$	0	0
$88$	−8.42260	−0.897853
$89$	5.05088	0.535392	0.267696	−	0.963503i	$-0.413738\pi$
$89$	5.05088	0.535392	0.267696	+	0.963503i	$0.413738\pi$
$90$	0	0
$91$	3.91704	0.410617
$92$	−25.5280	−2.66148
$93$	0	0
$94$	−21.1614	−2.18263
$95$	0	0
$96$	0	0
$97$	2.04414	0.207551	0.103776	−	0.994601i	$-0.466908\pi$
$97$	2.04414	0.207551	0.103776	+	0.994601i	$0.466908\pi$
$98$	−11.6441	−1.17624
$99$	0	0
$100$	0	0
$101$	12.4871	1.24251	0.621255	−	0.783609i	$-0.286623\pi$
$101$	12.4871	1.24251	0.621255	+	0.783609i	$0.286623\pi$
$102$	0	0
$103$	10.6723	1.05157	0.525785	−	0.850617i	$-0.323772\pi$
$103$	10.6723	1.05157	0.525785	+	0.850617i	$0.323772\pi$
$104$	5.83912	0.572572
$105$	0	0
$106$	28.9374	2.81065
$107$	4.85956	0.469791	0.234896	−	0.972021i	$-0.424525\pi$
$107$	4.85956	0.469791	0.234896	+	0.972021i	$0.424525\pi$
$108$	0	0
$109$	15.3415	1.46945	0.734723	−	0.678367i	$-0.237312\pi$
$109$	15.3415	1.46945	0.734723	+	0.678367i	$0.237312\pi$
$110$	0	0
$111$	0	0
$112$	−1.89141	−0.178721
$113$	−4.32180	−0.406560	−0.203280	−	0.979121i	$-0.565160\pi$
$113$	−4.32180	−0.406560	−0.203280	+	0.979121i	$0.565160\pi$
$114$	0	0
$115$	0	0
$116$	2.88766	0.268113
$117$	0	0
$118$	5.63908	0.519119
$119$	−0.845750	−0.0775298
$120$	0	0
$121$	7.42032	0.674575
$122$	18.7161	1.69448
$123$	0	0
$124$	−18.1061	−1.62598
$125$	0	0
$126$	0	0
$127$	−21.5809	−1.91500	−0.957498	−	0.288440i	$-0.906863\pi$
$127$	−21.5809	−1.91500	−0.957498	+	0.288440i	$0.906863\pi$
$128$	−14.1533	−1.25098
$129$	0	0
$130$	0	0
$131$	−9.25658	−0.808751	−0.404376	−	0.914593i	$-0.632511\pi$
$131$	−9.25658	−0.808751	−0.404376	+	0.914593i	$0.632511\pi$
$132$	0	0
$133$	−6.68661	−0.579803
$134$	28.6981	2.47914
$135$	0	0
$136$	−1.26076	−0.108109
$137$	0.843767	0.0720879	0.0360439	−	0.999350i	$-0.488524\pi$
$137$	0.843767	0.0720879	0.0360439	+	0.999350i	$0.488524\pi$
$138$	0	0
$139$	−13.9777	−1.18557	−0.592785	−	0.805361i	$-0.701972\pi$
$139$	−13.9777	−1.18557	−0.592785	+	0.805361i	$0.701972\pi$
$140$	0	0
$141$	0	0
$142$	−11.2844	−0.946964
$143$	−12.7702	−1.06790
$144$	0	0
$145$	0	0
$146$	35.3745	2.92762
$147$	0	0
$148$	−2.69723	−0.221711
$149$	−19.6274	−1.60794	−0.803971	−	0.594669i	$-0.797283\pi$
$149$	−19.6274	−1.60794	−0.803971	+	0.594669i	$0.797283\pi$
$150$	0	0
$151$	−10.7559	−0.875301	−0.437650	−	0.899145i	$-0.644189\pi$
$151$	−10.7559	−0.875301	−0.437650	+	0.899145i	$0.644189\pi$
$152$	−9.96771	−0.808488
$153$	0	0
$154$	−12.4913	−1.00658
$155$	0	0
$156$	0	0
$157$	−23.5513	−1.87960	−0.939800	−	0.341724i	$-0.888989\pi$
$157$	−23.5513	−1.87960	−0.939800	+	0.341724i	$0.888989\pi$
$158$	−7.30710	−0.581322
$159$	0	0
$160$	0	0
$161$	−11.6380	−0.917206
$162$	0	0
$163$	−9.50148	−0.744213	−0.372107	−	0.928190i	$-0.621364\pi$
$163$	−9.50148	−0.744213	−0.372107	+	0.928190i	$0.621364\pi$
$164$	−31.9929	−2.49823
$165$	0	0
$166$	−11.5988	−0.900239
$167$	1.82786	0.141444	0.0707220	−	0.997496i	$-0.477470\pi$
$167$	1.82786	0.141444	0.0707220	+	0.997496i	$0.477470\pi$
$168$	0	0
$169$	−4.14683	−0.318987
$170$	0	0
$171$	0	0
$172$	−5.89027	−0.449129
$173$	−5.50734	−0.418715	−0.209358	−	0.977839i	$-0.567137\pi$
$173$	−5.50734	−0.418715	−0.209358	+	0.977839i	$0.567137\pi$
$174$	0	0
$175$	0	0
$176$	6.16631	0.464803
$177$	0	0
$178$	11.1665	0.836966
$179$	−9.36470	−0.699950	−0.349975	−	0.936759i	$-0.613810\pi$
$179$	−9.36470	−0.699950	−0.349975	+	0.936759i	$0.613810\pi$
$180$	0	0
$181$	25.6019	1.90297	0.951487	−	0.307688i	$-0.0995552\pi$
$181$	25.6019	1.90297	0.951487	+	0.307688i	$0.0995552\pi$
$182$	8.65981	0.641908
$183$	0	0
$184$	−17.3488	−1.27897
$185$	0	0
$186$	0	0
$187$	2.75729	0.201633
$188$	−27.6401	−2.01586
$189$	0	0
$190$	0	0
$191$	12.3658	0.894756	0.447378	−	0.894345i	$-0.352358\pi$
$191$	12.3658	0.894756	0.447378	+	0.894345i	$0.352358\pi$
$192$	0	0
$193$	15.7223	1.13172	0.565859	−	0.824502i	$-0.308545\pi$
$193$	15.7223	1.13172	0.565859	+	0.824502i	$0.308545\pi$
$194$	4.51920	0.324459
$195$	0	0
$196$	−15.2091	−1.08636
$197$	−21.6728	−1.54412	−0.772061	−	0.635549i	$-0.780774\pi$
$197$	−21.6728	−1.54412	−0.772061	+	0.635549i	$0.780774\pi$
$198$	0	0
$199$	−1.23732	−0.0877116	−0.0438558	−	0.999038i	$-0.513964\pi$
$199$	−1.23732	−0.0877116	−0.0438558	+	0.999038i	$0.513964\pi$
$200$	0	0
$201$	0	0
$202$	27.6065	1.94238
$203$	1.31646	0.0923976
$204$	0	0
$205$	0	0
$206$	23.5943	1.64389
$207$	0	0
$208$	−4.27490	−0.296411
$209$	21.7995	1.50790
$210$	0	0
$211$	−5.05346	−0.347895	−0.173947	−	0.984755i	$-0.555652\pi$
$211$	−5.05346	−0.347895	−0.173947	+	0.984755i	$0.555652\pi$
$212$	37.7968	2.59589
$213$	0	0
$214$	10.7435	0.734413
$215$	0	0
$216$	0	0
$217$	−8.25444	−0.560348
$218$	33.9170	2.29715
$219$	0	0
$220$	0	0
$221$	−1.91154	−0.128584
$222$	0	0
$223$	16.2248	1.08649	0.543245	−	0.839574i	$-0.317195\pi$
$223$	16.2248	1.08649	0.543245	+	0.839574i	$0.317195\pi$
$224$	−9.34851	−0.624624
$225$	0	0
$226$	−9.55465	−0.635566
$227$	−6.82690	−0.453117	−0.226559	−	0.973998i	$-0.572747\pi$
$227$	−6.82690	−0.453117	−0.226559	+	0.973998i	$0.572747\pi$
$228$	0	0
$229$	−5.87121	−0.387980	−0.193990	−	0.981003i	$-0.562143\pi$
$229$	−5.87121	−0.387980	−0.193990	+	0.981003i	$0.562143\pi$
$230$	0	0
$231$	0	0
$232$	1.96245	0.128841
$233$	14.1453	0.926692	0.463346	−	0.886178i	$-0.346649\pi$
$233$	14.1453	0.926692	0.463346	+	0.886178i	$0.346649\pi$
$234$	0	0
$235$	0	0
$236$	7.36553	0.479455
$237$	0	0
$238$	−1.86979	−0.121200
$239$	20.6920	1.33845	0.669226	−	0.743059i	$-0.266626\pi$
$239$	20.6920	1.33845	0.669226	+	0.743059i	$0.266626\pi$
$240$	0	0
$241$	−7.90845	−0.509428	−0.254714	−	0.967016i	$-0.581981\pi$
$241$	−7.90845	−0.509428	−0.254714	+	0.967016i	$0.581981\pi$
$242$	16.4049	1.05455
$243$	0	0
$244$	24.4462	1.56501
$245$	0	0
$246$	0	0
$247$	−15.1129	−0.961609
$248$	−12.3049	−0.781359
$249$	0	0
$250$	0	0
$251$	−11.8140	−0.745691	−0.372846	−	0.927893i	$-0.621618\pi$
$251$	−11.8140	−0.745691	−0.372846	+	0.927893i	$0.621618\pi$
$252$	0	0
$253$	37.9420	2.38539
$254$	−47.7112	−2.99367
$255$	0	0
$256$	−5.63818	−0.352386
$257$	4.35440	0.271620	0.135810	−	0.990735i	$-0.456636\pi$
$257$	4.35440	0.271620	0.135810	+	0.990735i	$0.456636\pi$
$258$	0	0
$259$	−1.22965	−0.0764066
$260$	0	0
$261$	0	0
$262$	−20.4645	−1.26430
$263$	−27.6727	−1.70637	−0.853187	−	0.521605i	$-0.825334\pi$
$263$	−27.6727	−1.70637	−0.853187	+	0.521605i	$0.825334\pi$
$264$	0	0
$265$	0	0
$266$	−14.7828	−0.906392
$267$	0	0
$268$	37.4842	2.28971
$269$	6.32037	0.385360	0.192680	−	0.981262i	$-0.438282\pi$
$269$	6.32037	0.385360	0.192680	+	0.981262i	$0.438282\pi$
$270$	0	0
$271$	4.37191	0.265575	0.132787	−	0.991145i	$-0.457607\pi$
$271$	4.37191	0.265575	0.132787	+	0.991145i	$0.457607\pi$
$272$	0.923018	0.0559662
$273$	0	0
$274$	1.86540	0.112693
$275$	0	0
$276$	0	0
$277$	4.01638	0.241321	0.120660	−	0.992694i	$-0.461499\pi$
$277$	4.01638	0.241321	0.120660	+	0.992694i	$0.461499\pi$
$278$	−30.9019	−1.85337
$279$	0	0
$280$	0	0
$281$	13.6990	0.817215	0.408607	−	0.912710i	$-0.366015\pi$
$281$	13.6990	0.817215	0.408607	+	0.912710i	$0.366015\pi$
$282$	0	0
$283$	14.8058	0.880114	0.440057	−	0.897970i	$-0.354958\pi$
$283$	14.8058	0.880114	0.440057	+	0.897970i	$0.354958\pi$
$284$	−14.7392	−0.874609
$285$	0	0
$286$	−28.2324	−1.66942
$287$	−14.5853	−0.860945
$288$	0	0
$289$	−16.5873	−0.975722
$290$	0	0
$291$	0	0
$292$	46.2047	2.70393
$293$	16.6890	0.974983	0.487492	−	0.873128i	$-0.337912\pi$
$293$	16.6890	0.974983	0.487492	+	0.873128i	$0.337912\pi$
$294$	0	0
$295$	0	0
$296$	−1.83303	−0.106543
$297$	0	0
$298$	−43.3924	−2.51366
$299$	−26.3039	−1.52119
$300$	0	0
$301$	−2.68533	−0.154780
$302$	−23.7792	−1.36834
$303$	0	0
$304$	7.29750	0.418540
$305$	0	0
$306$	0	0
$307$	−4.97073	−0.283695	−0.141847	−	0.989889i	$-0.545304\pi$
$307$	−4.97073	−0.283695	−0.141847	+	0.989889i	$0.545304\pi$
$308$	−16.3156	−0.929669
$309$	0	0
$310$	0	0
$311$	21.0786	1.19526	0.597629	−	0.801773i	$-0.296110\pi$
$311$	21.0786	1.19526	0.597629	+	0.801773i	$0.296110\pi$
$312$	0	0
$313$	−24.3744	−1.37772	−0.688860	−	0.724894i	$-0.741889\pi$
$313$	−24.3744	−1.37772	−0.688860	+	0.724894i	$0.741889\pi$
$314$	−52.0674	−2.93833
$315$	0	0
$316$	−9.54423	−0.536905
$317$	13.4102	0.753194	0.376597	−	0.926377i	$-0.377094\pi$
$317$	13.4102	0.753194	0.376597	+	0.926377i	$0.377094\pi$
$318$	0	0
$319$	−4.29189	−0.240300
$320$	0	0
$321$	0	0
$322$	−25.7294	−1.43385
$323$	3.26310	0.181564
$324$	0	0
$325$	0	0
$326$	−21.0059	−1.16341
$327$	0	0
$328$	−21.7423	−1.20052
$329$	−12.6009	−0.694710
$330$	0	0
$331$	23.4187	1.28721	0.643603	−	0.765360i	$-0.277439\pi$
$331$	23.4187	1.28721	0.643603	+	0.765360i	$0.277439\pi$
$332$	−15.1498	−0.831455
$333$	0	0
$334$	4.04104	0.221116
$335$	0	0
$336$	0	0
$337$	−22.7865	−1.24126	−0.620631	−	0.784103i	$-0.713123\pi$
$337$	−22.7865	−1.24126	−0.620631	+	0.784103i	$0.713123\pi$
$338$	−9.16784	−0.498665
$339$	0	0
$340$	0	0
$341$	26.9108	1.45730
$342$	0	0
$343$	−16.1490	−0.871962
$344$	−4.00301	−0.215828
$345$	0	0
$346$	−12.1757	−0.654567
$347$	−4.98513	−0.267616	−0.133808	−	0.991007i	$-0.542721\pi$
$347$	−4.98513	−0.267616	−0.133808	+	0.991007i	$0.542721\pi$
$348$	0	0
$349$	25.6783	1.37453	0.687264	−	0.726407i	$-0.258811\pi$
$349$	25.6783	1.37453	0.687264	+	0.726407i	$0.258811\pi$
$350$	0	0
$351$	0	0
$352$	30.4777	1.62447
$353$	−20.3160	−1.08131	−0.540656	−	0.841244i	$-0.681824\pi$
$353$	−20.3160	−1.08131	−0.540656	+	0.841244i	$0.681824\pi$
$354$	0	0
$355$	0	0
$356$	14.5852	0.773016
$357$	0	0
$358$	−20.7035	−1.09422
$359$	9.68907	0.511370	0.255685	−	0.966760i	$-0.417699\pi$
$359$	9.68907	0.511370	0.255685	+	0.966760i	$0.417699\pi$
$360$	0	0
$361$	6.79854	0.357818
$362$	56.6008	2.97487
$363$	0	0
$364$	11.3111	0.592862
$365$	0	0
$366$	0	0
$367$	−9.16319	−0.478315	−0.239157	−	0.970981i	$-0.576871\pi$
$367$	−9.16319	−0.478315	−0.239157	+	0.970981i	$0.576871\pi$
$368$	12.7013	0.662101
$369$	0	0
$370$	0	0
$371$	17.2313	0.894603
$372$	0	0
$373$	13.5773	0.703007	0.351504	−	0.936186i	$-0.385670\pi$
$373$	13.5773	0.703007	0.351504	+	0.936186i	$0.385670\pi$
$374$	6.09582	0.315208
$375$	0	0
$376$	−18.7841	−0.968717
$377$	2.97543	0.153242
$378$	0	0
$379$	0.0601424	0.00308931	0.00154465	−	0.999999i	$-0.499508\pi$
$379$	0.0601424	0.00308931	0.00154465	+	0.999999i	$0.499508\pi$
$380$	0	0
$381$	0	0
$382$	27.3383	1.39875
$383$	9.71642	0.496486	0.248243	−	0.968698i	$-0.420147\pi$
$383$	9.71642	0.496486	0.248243	+	0.968698i	$0.420147\pi$
$384$	0	0
$385$	0	0
$386$	34.7590	1.76919
$387$	0	0
$388$	5.90279	0.299669
$389$	−5.68991	−0.288490	−0.144245	−	0.989542i	$-0.546075\pi$
$389$	−5.68991	−0.288490	−0.144245	+	0.989542i	$0.546075\pi$
$390$	0	0
$391$	5.67943	0.287221
$392$	−10.3361	−0.522050
$393$	0	0
$394$	−47.9143	−2.41389
$395$	0	0
$396$	0	0
$397$	−6.80729	−0.341648	−0.170824	−	0.985302i	$-0.554643\pi$
$397$	−6.80729	−0.341648	−0.170824	+	0.985302i	$0.554643\pi$
$398$	−2.73548	−0.137117
$399$	0	0
$400$	0	0
$401$	−11.7255	−0.585543	−0.292771	−	0.956182i	$-0.594578\pi$
$401$	−11.7255	−0.585543	−0.292771	+	0.956182i	$0.594578\pi$
$402$	0	0
$403$	−18.6564	−0.929342
$404$	36.0584	1.79397
$405$	0	0
$406$	2.91044	0.144443
$407$	4.00886	0.198712
$408$	0	0
$409$	−22.2665	−1.10101	−0.550505	−	0.834832i	$-0.685565\pi$
$409$	−22.2665	−1.10101	−0.550505	+	0.834832i	$0.685565\pi$
$410$	0	0
$411$	0	0
$412$	30.8179	1.51829
$413$	3.35789	0.165231
$414$	0	0
$415$	0	0
$416$	−21.1292	−1.03594
$417$	0	0
$418$	48.1944	2.35727
$419$	28.2149	1.37839	0.689194	−	0.724577i	$-0.257965\pi$
$419$	28.2149	1.37839	0.689194	+	0.724577i	$0.257965\pi$
$420$	0	0
$421$	−34.5167	−1.68224	−0.841121	−	0.540847i	$-0.818104\pi$
$421$	−34.5167	−1.68224	−0.841121	+	0.540847i	$0.818104\pi$
$422$	−11.1722	−0.543855
$423$	0	0
$424$	25.6866	1.24745
$425$	0	0
$426$	0	0
$427$	11.1449	0.539337
$428$	14.0328	0.678299
$429$	0	0
$430$	0	0
$431$	−23.0678	−1.11114	−0.555568	−	0.831471i	$-0.687499\pi$
$431$	−23.0678	−1.11114	−0.555568	+	0.831471i	$0.687499\pi$
$432$	0	0
$433$	17.8519	0.857908	0.428954	−	0.903326i	$-0.358882\pi$
$433$	17.8519	0.857908	0.428954	+	0.903326i	$0.358882\pi$
$434$	−18.2490	−0.875978
$435$	0	0
$436$	44.3010	2.12163
$437$	44.9023	2.14797
$438$	0	0
$439$	−3.31655	−0.158290	−0.0791452	−	0.996863i	$-0.525219\pi$
$439$	−3.31655	−0.158290	−0.0791452	+	0.996863i	$0.525219\pi$
$440$	0	0
$441$	0	0
$442$	−4.22604	−0.201012
$443$	−29.8818	−1.41973	−0.709863	−	0.704340i	$-0.751243\pi$
$443$	−29.8818	−1.41973	−0.709863	+	0.704340i	$0.751243\pi$
$444$	0	0
$445$	0	0
$446$	35.8698	1.69848
$447$	0	0
$448$	−16.8849	−0.797738
$449$	−8.89198	−0.419639	−0.209819	−	0.977740i	$-0.567288\pi$
$449$	−8.89198	−0.419639	−0.209819	+	0.977740i	$0.567288\pi$
$450$	0	0
$451$	47.5506	2.23907
$452$	−12.4799	−0.587004
$453$	0	0
$454$	−15.0929	−0.708347
$455$	0	0
$456$	0	0
$457$	19.8853	0.930194	0.465097	−	0.885260i	$-0.346019\pi$
$457$	19.8853	0.930194	0.465097	+	0.885260i	$0.346019\pi$
$458$	−12.9801	−0.606520
$459$	0	0
$460$	0	0
$461$	13.5776	0.632372	0.316186	−	0.948697i	$-0.397598\pi$
$461$	13.5776	0.632372	0.316186	+	0.948697i	$0.397598\pi$
$462$	0	0
$463$	−29.7645	−1.38328	−0.691638	−	0.722244i	$-0.743111\pi$
$463$	−29.7645	−1.38328	−0.691638	+	0.722244i	$0.743111\pi$
$464$	−1.43674	−0.0666988
$465$	0	0
$466$	31.2726	1.44867
$467$	−6.56767	−0.303915	−0.151958	−	0.988387i	$-0.548558\pi$
$467$	−6.56767	−0.303915	−0.151958	+	0.988387i	$0.548558\pi$
$468$	0	0
$469$	17.0888	0.789086
$470$	0	0
$471$	0	0
$472$	5.00559	0.230401
$473$	8.75463	0.402538
$474$	0	0
$475$	0	0
$476$	−2.44224	−0.111940
$477$	0	0
$478$	45.7459	2.09237
$479$	−36.9147	−1.68668	−0.843338	−	0.537383i	$-0.819413\pi$
$479$	−36.9147	−1.68668	−0.843338	+	0.537383i	$0.819413\pi$
$480$	0	0
$481$	−2.77921	−0.126721
$482$	−17.4840	−0.796376
$483$	0	0
$484$	21.4274	0.973972
$485$	0	0
$486$	0	0
$487$	−24.4257	−1.10684	−0.553418	−	0.832904i	$-0.686677\pi$
$487$	−24.4257	−1.10684	−0.553418	+	0.832904i	$0.686677\pi$
$488$	16.6136	0.752062
$489$	0	0
$490$	0	0
$491$	0.828059	0.0373698	0.0186849	−	0.999825i	$-0.494052\pi$
$491$	0.828059	0.0373698	0.0186849	+	0.999825i	$0.494052\pi$
$492$	0	0
$493$	−0.642441	−0.0289341
$494$	−33.4116	−1.50326
$495$	0	0
$496$	9.00856	0.404496
$497$	−6.71948	−0.301410
$498$	0	0
$499$	32.7413	1.46570	0.732850	−	0.680390i	$-0.238189\pi$
$499$	32.7413	1.46570	0.732850	+	0.680390i	$0.238189\pi$
$500$	0	0
$501$	0	0
$502$	−26.1184	−1.16572
$503$	−3.04972	−0.135980	−0.0679901	−	0.997686i	$-0.521659\pi$
$503$	−3.04972	−0.135980	−0.0679901	+	0.997686i	$0.521659\pi$
$504$	0	0
$505$	0	0
$506$	83.8823	3.72902
$507$	0	0
$508$	−62.3183	−2.76493
$509$	−11.2666	−0.499384	−0.249692	−	0.968325i	$-0.580329\pi$
$509$	−11.2666	−0.499384	−0.249692	+	0.968325i	$0.580329\pi$
$510$	0	0
$511$	21.0644	0.931833
$512$	15.8416	0.700107
$513$	0	0
$514$	9.62673	0.424617
$515$	0	0
$516$	0	0
$517$	41.0810	1.80674
$518$	−2.71851	−0.119445
$519$	0	0
$520$	0	0
$521$	1.39366	0.0610573	0.0305287	−	0.999534i	$-0.490281\pi$
$521$	1.39366	0.0610573	0.0305287	+	0.999534i	$0.490281\pi$
$522$	0	0
$523$	10.6212	0.464434	0.232217	−	0.972664i	$-0.425402\pi$
$523$	10.6212	0.464434	0.232217	+	0.972664i	$0.425402\pi$
$524$	−26.7299	−1.16770
$525$	0	0
$526$	−61.1791	−2.66753
$527$	4.02821	0.175472
$528$	0	0
$529$	55.1524	2.39793
$530$	0	0
$531$	0	0
$532$	−19.3087	−0.837137
$533$	−32.9653	−1.42789
$534$	0	0
$535$	0	0
$536$	25.4742	1.10032
$537$	0	0
$538$	13.9731	0.602424
$539$	22.6051	0.973669
$540$	0	0
$541$	31.1071	1.33740	0.668698	−	0.743534i	$-0.266852\pi$
$541$	31.1071	1.33740	0.668698	+	0.743534i	$0.266852\pi$
$542$	9.66544	0.415166
$543$	0	0
$544$	4.56212	0.195599
$545$	0	0
$546$	0	0
$547$	17.0828	0.730408	0.365204	−	0.930927i	$-0.380999\pi$
$547$	17.0828	0.730408	0.365204	+	0.930927i	$0.380999\pi$
$548$	2.43651	0.104083
$549$	0	0
$550$	0	0
$551$	−5.07923	−0.216382
$552$	0	0
$553$	−4.35114	−0.185029
$554$	8.87942	0.377251
$555$	0	0
$556$	−40.3628	−1.71176
$557$	15.1831	0.643329	0.321665	−	0.946854i	$-0.395758\pi$
$557$	15.1831	0.643329	0.321665	+	0.946854i	$0.395758\pi$
$558$	0	0
$559$	−6.06930	−0.256704
$560$	0	0
$561$	0	0
$562$	30.2859	1.27753
$563$	34.9344	1.47231	0.736154	−	0.676814i	$-0.236640\pi$
$563$	34.9344	1.47231	0.736154	+	0.676814i	$0.236640\pi$
$564$	0	0
$565$	0	0
$566$	32.7328	1.37586
$567$	0	0
$568$	−10.0167	−0.420291
$569$	2.51836	0.105575	0.0527875	−	0.998606i	$-0.483189\pi$
$569$	2.51836	0.105575	0.0527875	+	0.998606i	$0.483189\pi$
$570$	0	0
$571$	18.9130	0.791485	0.395742	−	0.918362i	$-0.370487\pi$
$571$	18.9130	0.791485	0.395742	+	0.918362i	$0.370487\pi$
$572$	−36.8760	−1.54186
$573$	0	0
$574$	−32.2453	−1.34589
$575$	0	0
$576$	0	0
$577$	4.64049	0.193186	0.0965930	−	0.995324i	$-0.469205\pi$
$577$	4.64049	0.193186	0.0965930	+	0.995324i	$0.469205\pi$
$578$	−36.6712	−1.52532
$579$	0	0
$580$	0	0
$581$	−6.90669	−0.286538
$582$	0	0
$583$	−56.1768	−2.32661
$584$	31.4006	1.29937
$585$	0	0
$586$	36.8962	1.52417
$587$	−25.7669	−1.06351	−0.531756	−	0.846897i	$-0.678468\pi$
$587$	−25.7669	−1.06351	−0.531756	+	0.846897i	$0.678468\pi$
$588$	0	0
$589$	31.8476	1.31226
$590$	0	0
$591$	0	0
$592$	1.34199	0.0551554
$593$	3.02524	0.124232	0.0621158	−	0.998069i	$-0.480215\pi$
$593$	3.02524	0.124232	0.0621158	+	0.998069i	$0.480215\pi$
$594$	0	0
$595$	0	0
$596$	−56.6774	−2.32160
$597$	0	0
$598$	−58.1529	−2.37805
$599$	−0.521774	−0.0213191	−0.0106596	−	0.999943i	$-0.503393\pi$
$599$	−0.521774	−0.0213191	−0.0106596	+	0.999943i	$0.503393\pi$
$600$	0	0
$601$	−34.7237	−1.41641	−0.708204	−	0.706008i	$-0.750494\pi$
$601$	−34.7237	−1.41641	−0.708204	+	0.706008i	$0.750494\pi$
$602$	−5.93674	−0.241964
$603$	0	0
$604$	−31.0593	−1.26379
$605$	0	0
$606$	0	0
$607$	−14.0480	−0.570190	−0.285095	−	0.958499i	$-0.592025\pi$
$607$	−14.0480	−0.570190	−0.285095	+	0.958499i	$0.592025\pi$
$608$	36.0688	1.46278
$609$	0	0
$610$	0	0
$611$	−28.4801	−1.15218
$612$	0	0
$613$	15.3819	0.621269	0.310635	−	0.950529i	$-0.399458\pi$
$613$	15.3819	0.621269	0.310635	+	0.950529i	$0.399458\pi$
$614$	−10.9893	−0.443493
$615$	0	0
$616$	−11.0880	−0.446750
$617$	−15.6734	−0.630988	−0.315494	−	0.948928i	$-0.602170\pi$
$617$	−15.6734	−0.630988	−0.315494	+	0.948928i	$0.602170\pi$
$618$	0	0
$619$	36.1380	1.45251	0.726254	−	0.687427i	$-0.241260\pi$
$619$	36.1380	1.45251	0.726254	+	0.687427i	$0.241260\pi$
$620$	0	0
$621$	0	0
$622$	46.6007	1.86852
$623$	6.64930	0.266399
$624$	0	0
$625$	0	0
$626$	−53.8870	−2.15376
$627$	0	0
$628$	−68.0083	−2.71383
$629$	0.600075	0.0239265
$630$	0	0
$631$	−5.20523	−0.207217	−0.103608	−	0.994618i	$-0.533039\pi$
$631$	−5.20523	−0.207217	−0.103608	+	0.994618i	$0.533039\pi$
$632$	−6.48623	−0.258008
$633$	0	0
$634$	29.6474	1.17745
$635$	0	0
$636$	0	0
$637$	−15.6714	−0.620921
$638$	−9.48853	−0.375655
$639$	0	0
$640$	0	0
$641$	−28.6342	−1.13098	−0.565492	−	0.824754i	$-0.691314\pi$
$641$	−28.6342	−1.13098	−0.565492	+	0.824754i	$0.691314\pi$
$642$	0	0
$643$	−35.6717	−1.40676	−0.703378	−	0.710816i	$-0.748326\pi$
$643$	−35.6717	−1.40676	−0.703378	+	0.710816i	$0.748326\pi$
$644$	−33.6067	−1.32429
$645$	0	0
$646$	7.21409	0.283834
$647$	−6.14445	−0.241563	−0.120782	−	0.992679i	$-0.538540\pi$
$647$	−6.14445	−0.241563	−0.120782	+	0.992679i	$0.538540\pi$
$648$	0	0
$649$	−10.9473	−0.429718
$650$	0	0
$651$	0	0
$652$	−27.4371	−1.07452
$653$	−8.62982	−0.337711	−0.168855	−	0.985641i	$-0.554007\pi$
$653$	−8.62982	−0.337711	−0.168855	+	0.985641i	$0.554007\pi$
$654$	0	0
$655$	0	0
$656$	15.9178	0.621488
$657$	0	0
$658$	−27.8581	−1.08602
$659$	35.4336	1.38030	0.690148	−	0.723668i	$-0.257545\pi$
$659$	35.4336	1.38030	0.690148	+	0.723668i	$0.257545\pi$
$660$	0	0
$661$	12.7938	0.497622	0.248811	−	0.968552i	$-0.419960\pi$
$661$	12.7938	0.497622	0.248811	+	0.968552i	$0.419960\pi$
$662$	51.7741	2.01226
$663$	0	0
$664$	−10.2958	−0.399554
$665$	0	0
$666$	0	0
$667$	−8.84038	−0.342301
$668$	5.27824	0.204221
$669$	0	0
$670$	0	0
$671$	−36.3341	−1.40266
$672$	0	0
$673$	39.0602	1.50566	0.752830	−	0.658215i	$-0.228688\pi$
$673$	39.0602	1.50566	0.752830	+	0.658215i	$0.228688\pi$
$674$	−50.3766	−1.94043
$675$	0	0
$676$	−11.9747	−0.460564
$677$	−51.8790	−1.99387	−0.996936	−	0.0782187i	$-0.975077\pi$
$677$	−51.8790	−1.99387	−0.996936	+	0.0782187i	$0.975077\pi$
$678$	0	0
$679$	2.69104	0.103272
$680$	0	0
$681$	0	0
$682$	59.4946	2.27817
$683$	−27.2220	−1.04162	−0.520811	−	0.853672i	$-0.674370\pi$
$683$	−27.2220	−1.04162	−0.520811	+	0.853672i	$0.674370\pi$
$684$	0	0
$685$	0	0
$686$	−35.7022	−1.36312
$687$	0	0
$688$	2.93066	0.111730
$689$	38.9455	1.48371
$690$	0	0
$691$	21.7474	0.827309	0.413655	−	0.910434i	$-0.364252\pi$
$691$	21.7474	0.827309	0.413655	+	0.910434i	$0.364252\pi$
$692$	−15.9033	−0.604554
$693$	0	0
$694$	−11.0212	−0.418357
$695$	0	0
$696$	0	0
$697$	7.11772	0.269603
$698$	56.7698	2.14877
$699$	0	0
$700$	0	0
$701$	−35.4994	−1.34079	−0.670397	−	0.742003i	$-0.733876\pi$
$701$	−35.4994	−1.34079	−0.670397	+	0.742003i	$0.733876\pi$
$702$	0	0
$703$	4.74427	0.178934
$704$	55.0477	2.07469
$705$	0	0
$706$	−44.9148	−1.69039
$707$	16.4388	0.618243
$708$	0	0
$709$	33.4325	1.25558	0.627792	−	0.778381i	$-0.283959\pi$
$709$	33.4325	1.25558	0.627792	+	0.778381i	$0.283959\pi$
$710$	0	0
$711$	0	0
$712$	9.91208	0.371471
$713$	55.4306	2.07589
$714$	0	0
$715$	0	0
$716$	−27.0421	−1.01061
$717$	0	0
$718$	21.4207	0.799412
$719$	−44.5343	−1.66085	−0.830425	−	0.557130i	$-0.811902\pi$
$719$	−44.5343	−1.66085	−0.830425	+	0.557130i	$0.811902\pi$
$720$	0	0
$721$	14.0497	0.523237
$722$	15.0302	0.559368
$723$	0	0
$724$	73.9296	2.74757
$725$	0	0
$726$	0	0
$727$	−15.6700	−0.581170	−0.290585	−	0.956849i	$-0.593850\pi$
$727$	−15.6700	−0.581170	−0.290585	+	0.956849i	$0.593850\pi$
$728$	7.68698	0.284898
$729$	0	0
$730$	0	0
$731$	1.31046	0.0484689
$732$	0	0
$733$	−29.7350	−1.09829	−0.549143	−	0.835728i	$-0.685046\pi$
$733$	−29.7350	−1.09829	−0.549143	+	0.835728i	$0.685046\pi$
$734$	−20.2580	−0.747738
$735$	0	0
$736$	62.7776	2.31401
$737$	−55.7123	−2.05219
$738$	0	0
$739$	42.9178	1.57876	0.789379	−	0.613906i	$-0.210403\pi$
$739$	42.9178	1.57876	0.789379	+	0.613906i	$0.210403\pi$
$740$	0	0
$741$	0	0
$742$	38.0950	1.39851
$743$	−10.9064	−0.400117	−0.200058	−	0.979784i	$-0.564113\pi$
$743$	−10.9064	−0.400117	−0.200058	+	0.979784i	$0.564113\pi$
$744$	0	0
$745$	0	0
$746$	30.0168	1.09899
$747$	0	0
$748$	7.96211	0.291124
$749$	6.39743	0.233757
$750$	0	0
$751$	47.2977	1.72592	0.862960	−	0.505273i	$-0.168608\pi$
$751$	47.2977	1.72592	0.862960	+	0.505273i	$0.168608\pi$
$752$	13.7521	0.501488
$753$	0	0
$754$	6.57809	0.239560
$755$	0	0
$756$	0	0
$757$	−33.3406	−1.21179	−0.605893	−	0.795546i	$-0.707184\pi$
$757$	−33.3406	−1.21179	−0.605893	+	0.795546i	$0.707184\pi$
$758$	0.132963	0.00482944
$759$	0	0
$760$	0	0
$761$	45.9707	1.66644	0.833219	−	0.552944i	$-0.186496\pi$
$761$	45.9707	1.66644	0.833219	+	0.552944i	$0.186496\pi$
$762$	0	0
$763$	20.1965	0.731162
$764$	35.7082	1.29188
$765$	0	0
$766$	21.4811	0.776144
$767$	7.58939	0.274037
$768$	0	0
$769$	−9.21048	−0.332138	−0.166069	−	0.986114i	$-0.553108\pi$
$769$	−9.21048	−0.332138	−0.166069	+	0.986114i	$0.553108\pi$
$770$	0	0
$771$	0	0
$772$	45.4007	1.63401
$773$	−28.6595	−1.03081	−0.515405	−	0.856947i	$-0.672358\pi$
$773$	−28.6595	−1.03081	−0.515405	+	0.856947i	$0.672358\pi$
$774$	0	0
$775$	0	0
$776$	4.01152	0.144005
$777$	0	0
$778$	−12.5793	−0.450989
$779$	56.2737	2.01621
$780$	0	0
$781$	21.9066	0.783881
$782$	12.5561	0.449005
$783$	0	0
$784$	7.56718	0.270256
$785$	0	0
$786$	0	0
$787$	−20.8195	−0.742136	−0.371068	−	0.928606i	$-0.621008\pi$
$787$	−20.8195	−0.742136	−0.371068	+	0.928606i	$0.621008\pi$
$788$	−62.5837	−2.22945
$789$	0	0
$790$	0	0
$791$	−5.68948	−0.202295
$792$	0	0
$793$	25.1892	0.894496
$794$	−15.0496	−0.534090
$795$	0	0
$796$	−3.57298	−0.126641
$797$	−44.8656	−1.58922	−0.794610	−	0.607120i	$-0.792325\pi$
$797$	−44.8656	−1.58922	−0.794610	+	0.607120i	$0.792325\pi$
$798$	0	0
$799$	6.14931	0.217547
$800$	0	0
$801$	0	0
$802$	−25.9228	−0.915365
$803$	−68.6734	−2.42343
$804$	0	0
$805$	0	0
$806$	−41.2457	−1.45282
$807$	0	0
$808$	24.5052	0.862090
$809$	26.6975	0.938633	0.469317	−	0.883030i	$-0.344500\pi$
$809$	26.6975	0.938633	0.469317	+	0.883030i	$0.344500\pi$
$810$	0	0
$811$	−21.6414	−0.759934	−0.379967	−	0.925000i	$-0.624065\pi$
$811$	−21.6414	−0.759934	−0.379967	+	0.925000i	$0.624065\pi$
$812$	3.80150	0.133406
$813$	0	0
$814$	8.86281	0.310641
$815$	0	0
$816$	0	0
$817$	10.3606	0.362473
$818$	−49.2270	−1.72118
$819$	0	0
$820$	0	0
$821$	−36.0374	−1.25771	−0.628857	−	0.777521i	$-0.716477\pi$
$821$	−36.0374	−1.25771	−0.628857	+	0.777521i	$0.716477\pi$
$822$	0	0
$823$	23.4577	0.817684	0.408842	−	0.912605i	$-0.365933\pi$
$823$	23.4577	0.817684	0.408842	+	0.912605i	$0.365933\pi$
$824$	20.9438	0.729611
$825$	0	0
$826$	7.42364	0.258301
$827$	14.7623	0.513334	0.256667	−	0.966500i	$-0.417376\pi$
$827$	14.7623	0.513334	0.256667	+	0.966500i	$0.417376\pi$
$828$	0	0
$829$	5.63801	0.195816	0.0979081	−	0.995195i	$-0.468785\pi$
$829$	5.63801	0.195816	0.0979081	+	0.995195i	$0.468785\pi$
$830$	0	0
$831$	0	0
$832$	−38.1628	−1.32306
$833$	3.38369	0.117238
$834$	0	0
$835$	0	0
$836$	62.9495	2.17715
$837$	0	0
$838$	62.3776	2.15480
$839$	56.4632	1.94932	0.974662	−	0.223681i	$-0.0718073\pi$
$839$	56.4632	1.94932	0.974662	+	0.223681i	$0.0718073\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−76.3098	−2.62981
$843$	0	0
$844$	−14.5927	−0.502301
$845$	0	0
$846$	0	0
$847$	9.76858	0.335652
$848$	−18.8055	−0.645784
$849$	0	0
$850$	0	0
$851$	8.25740	0.283060
$852$	0	0
$853$	−38.2731	−1.31045	−0.655223	−	0.755436i	$-0.727425\pi$
$853$	−38.2731	−1.31045	−0.655223	+	0.755436i	$0.727425\pi$
$854$	24.6391	0.843133
$855$	0	0
$856$	9.53663	0.325955
$857$	−47.5891	−1.62561	−0.812806	−	0.582534i	$-0.802061\pi$
$857$	−47.5891	−1.62561	−0.812806	+	0.582534i	$0.802061\pi$
$858$	0	0
$859$	−19.4348	−0.663108	−0.331554	−	0.943436i	$-0.607573\pi$
$859$	−19.4348	−0.663108	−0.331554	+	0.943436i	$0.607573\pi$
$860$	0	0
$861$	0	0
$862$	−50.9984	−1.73701
$863$	−24.3695	−0.829546	−0.414773	−	0.909925i	$-0.636139\pi$
$863$	−24.3695	−0.829546	−0.414773	+	0.909925i	$0.636139\pi$
$864$	0	0
$865$	0	0
$866$	39.4671	1.34115
$867$	0	0
$868$	−23.8360	−0.809047
$869$	14.1855	0.481209
$870$	0	0
$871$	38.6235	1.30871
$872$	30.1068	1.01955
$873$	0	0
$874$	99.2703	3.35787
$875$	0	0
$876$	0	0
$877$	−23.4009	−0.790193	−0.395097	−	0.918640i	$-0.629289\pi$
$877$	−23.4009	−0.790193	−0.395097	+	0.918640i	$0.629289\pi$
$878$	−7.33225	−0.247451
$879$	0	0
$880$	0	0
$881$	−49.1434	−1.65568	−0.827841	−	0.560963i	$-0.810431\pi$
$881$	−49.1434	−1.65568	−0.827841	+	0.560963i	$0.810431\pi$
$882$	0	0
$883$	42.0245	1.41424	0.707118	−	0.707095i	$-0.249995\pi$
$883$	42.0245	1.41424	0.707118	+	0.707095i	$0.249995\pi$
$884$	−5.51987	−0.185653
$885$	0	0
$886$	−66.0628	−2.21942
$887$	−38.7280	−1.30036	−0.650179	−	0.759781i	$-0.725306\pi$
$887$	−38.7280	−1.30036	−0.650179	+	0.759781i	$0.725306\pi$
$888$	0	0
$889$	−28.4105	−0.952857
$890$	0	0
$891$	0	0
$892$	46.8516	1.56871
$893$	48.6173	1.62691
$894$	0	0
$895$	0	0
$896$	−18.6323	−0.622460
$897$	0	0
$898$	−19.6584	−0.656011
$899$	−6.27016	−0.209122
$900$	0	0
$901$	−8.40895	−0.280143
$902$	105.125	3.50029
$903$	0	0
$904$	−8.48129	−0.282084
$905$	0	0
$906$	0	0
$907$	−22.7366	−0.754956	−0.377478	−	0.926019i	$-0.623209\pi$
$907$	−22.7366	−0.754956	−0.377478	+	0.926019i	$0.623209\pi$
$908$	−19.7138	−0.654224
$909$	0	0
$910$	0	0
$911$	−5.74791	−0.190437	−0.0952183	−	0.995456i	$-0.530355\pi$
$911$	−5.74791	−0.190437	−0.0952183	+	0.995456i	$0.530355\pi$
$912$	0	0
$913$	22.5170	0.745203
$914$	43.9625	1.45415
$915$	0	0
$916$	−16.9541	−0.560178
$917$	−12.1859	−0.402415
$918$	0	0
$919$	52.1835	1.72137	0.860687	−	0.509135i	$-0.170035\pi$
$919$	52.1835	1.72137	0.860687	+	0.509135i	$0.170035\pi$
$920$	0	0
$921$	0	0
$922$	30.0175	0.988572
$923$	−15.1871	−0.499891
$924$	0	0
$925$	0	0
$926$	−65.8036	−2.16244
$927$	0	0
$928$	−7.10123	−0.233109
$929$	4.61899	0.151544	0.0757721	−	0.997125i	$-0.475858\pi$
$929$	4.61899	0.151544	0.0757721	+	0.997125i	$0.475858\pi$
$930$	0	0
$931$	26.7519	0.876758
$932$	40.8469	1.33799
$933$	0	0
$934$	−14.5198	−0.475104
$935$	0	0
$936$	0	0
$937$	22.8991	0.748082	0.374041	−	0.927412i	$-0.377972\pi$
$937$	22.8991	0.748082	0.374041	+	0.927412i	$0.377972\pi$
$938$	37.7800	1.23356
$939$	0	0
$940$	0	0
$941$	−24.1297	−0.786604	−0.393302	−	0.919409i	$-0.628667\pi$
$941$	−24.1297	−0.786604	−0.393302	+	0.919409i	$0.628667\pi$
$942$	0	0
$943$	97.9442	3.18950
$944$	−3.66466	−0.119275
$945$	0	0
$946$	19.3548	0.629278
$947$	13.6427	0.443329	0.221665	−	0.975123i	$-0.428851\pi$
$947$	13.6427	0.443329	0.221665	+	0.975123i	$0.428851\pi$
$948$	0	0
$949$	47.6090	1.54545
$950$	0	0
$951$	0	0
$952$	−1.65974	−0.0537925
$953$	45.3303	1.46839	0.734196	−	0.678937i	$-0.237559\pi$
$953$	45.3303	1.46839	0.734196	+	0.678937i	$0.237559\pi$
$954$	0	0
$955$	0	0
$956$	59.7514	1.93250
$957$	0	0
$958$	−81.6113	−2.63674
$959$	1.11079	0.0358692
$960$	0	0
$961$	8.31492	0.268223
$962$	−6.14429	−0.198100
$963$	0	0
$964$	−22.8369	−0.735528
$965$	0	0
$966$	0	0
$967$	45.2638	1.45558	0.727792	−	0.685798i	$-0.240547\pi$
$967$	45.2638	1.45558	0.727792	+	0.685798i	$0.240547\pi$
$968$	14.5620	0.468040
$969$	0	0
$970$	0	0
$971$	47.1089	1.51180	0.755899	−	0.654688i	$-0.227200\pi$
$971$	47.1089	1.51180	0.755899	+	0.654688i	$0.227200\pi$
$972$	0	0
$973$	−18.4011	−0.589912
$974$	−54.0006	−1.73029
$975$	0	0
$976$	−12.1630	−0.389330
$977$	−2.89922	−0.0927542	−0.0463771	−	0.998924i	$-0.514768\pi$
$977$	−2.89922	−0.0927542	−0.0463771	+	0.998924i	$0.514768\pi$
$978$	0	0
$979$	−21.6778	−0.692826
$980$	0	0
$981$	0	0
$982$	1.83068	0.0584193
$983$	37.0610	1.18206	0.591031	−	0.806649i	$-0.298721\pi$
$983$	37.0610	1.18206	0.591031	+	0.806649i	$0.298721\pi$
$984$	0	0
$985$	0	0
$986$	−1.42031	−0.0452320
$987$	0	0
$988$	−43.6408	−1.38840
$989$	18.0327	0.573406
$990$	0	0
$991$	17.4669	0.554854	0.277427	−	0.960747i	$-0.410518\pi$
$991$	17.4669	0.554854	0.277427	+	0.960747i	$0.410518\pi$
$992$	44.5259	1.41370
$993$	0	0
$994$	−14.8555	−0.471187
$995$	0	0
$996$	0	0
$997$	19.4726	0.616703	0.308352	−	0.951272i	$-0.400223\pi$
$997$	19.4726	0.616703	0.308352	+	0.951272i	$0.400223\pi$
$998$	72.3846	2.29129
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6525.2.a.ca.1.9	✓	9
3.2	odd	2		6525.2.a.cc.1.1	yes	9
5.4	even	2		6525.2.a.cd.1.1	yes	9
15.14	odd	2		6525.2.a.cb.1.9	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6525.2.a.ca.1.9	✓	9	1.1	even	1	trivial
6525.2.a.cb.1.9	yes	9	15.14	odd	2
6525.2.a.cc.1.1	yes	9	3.2	odd	2
6525.2.a.cd.1.1	yes	9	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)

\(f(q)\)	\(=\)	\(q+2.21081 q^{2} +2.88766 q^{4} +1.31646 q^{7} +1.96245 q^{8} +O(q^{10})\)	\(q+2.21081 q^{2} +2.88766 q^{4} +1.31646 q^{7} +1.96245 q^{8} -4.29189 q^{11} +2.97543 q^{13} +2.91044 q^{14} -1.43674 q^{16} -0.642441 q^{17} -5.07923 q^{19} -9.48853 q^{22} -8.84038 q^{23} +6.57809 q^{26} +3.80150 q^{28} +1.00000 q^{29} -6.27016 q^{31} -7.10123 q^{32} -1.42031 q^{34} -0.934054 q^{37} -11.2292 q^{38} -11.0792 q^{41} -2.03981 q^{43} -12.3935 q^{44} -19.5444 q^{46} -9.57178 q^{47} -5.26692 q^{49} +8.59203 q^{52} +13.0891 q^{53} +2.58349 q^{56} +2.21081 q^{58} +2.55069 q^{59} +8.46575 q^{61} -13.8621 q^{62} -12.8260 q^{64} +12.9808 q^{67} -1.85515 q^{68} -5.10419 q^{71} +16.0007 q^{73} -2.06501 q^{74} -14.6671 q^{76} -5.65012 q^{77} -3.30518 q^{79} -24.4939 q^{82} -5.24640 q^{83} -4.50962 q^{86} -8.42260 q^{88} +5.05088 q^{89} +3.91704 q^{91} -25.5280 q^{92} -21.1614 q^{94} +2.04414 q^{97} -11.6441 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 2 q^{2} + 10 q^{4} - q^{7} - 9 q^{8}+O(q^{10}) \) 9 * q - 2 * q^2 + 10 * q^4 - q^7 - 9 * q^8	\( 9 q - 2 q^{2} + 10 q^{4} - q^{7} - 9 q^{8} + 2 q^{11} - q^{13} - 3 q^{14} + 4 q^{16} - 12 q^{17} - q^{19} - 3 q^{22} - 16 q^{23} + 6 q^{26} + 4 q^{28} + 9 q^{29} + 5 q^{31} - 20 q^{32} + 3 q^{34} - 30 q^{38} - 10 q^{41} - 3 q^{43} - 13 q^{44} + 4 q^{46} - 26 q^{47} - 8 q^{49} + 9 q^{52} - 22 q^{53} + 22 q^{56} - 2 q^{58} + 4 q^{59} + 7 q^{61} - 28 q^{62} + 9 q^{64} - 5 q^{67} - 39 q^{68} + 10 q^{73} - 34 q^{74} - 2 q^{76} - 34 q^{77} + 10 q^{79} + 8 q^{82} - 46 q^{83} + 28 q^{86} - 2 q^{88} + 4 q^{89} - 21 q^{91} - 20 q^{92} + 5 q^{94} - 7 q^{97} - 51 q^{98}+O(q^{100}) \) 9 * q - 2 * q^2 + 10 * q^4 - q^7 - 9 * q^8 + 2 * q^11 - q^13 - 3 * q^14 + 4 * q^16 - 12 * q^17 - q^19 - 3 * q^22 - 16 * q^23 + 6 * q^26 + 4 * q^28 + 9 * q^29 + 5 * q^31 - 20 * q^32 + 3 * q^34 - 30 * q^38 - 10 * q^41 - 3 * q^43 - 13 * q^44 + 4 * q^46 - 26 * q^47 - 8 * q^49 + 9 * q^52 - 22 * q^53 + 22 * q^56 - 2 * q^58 + 4 * q^59 + 7 * q^61 - 28 * q^62 + 9 * q^64 - 5 * q^67 - 39 * q^68 + 10 * q^73 - 34 * q^74 - 2 * q^76 - 34 * q^77 + 10 * q^79 + 8 * q^82 - 46 * q^83 + 28 * q^86 - 2 * q^88 + 4 * q^89 - 21 * q^91 - 20 * q^92 + 5 * q^94 - 7 * q^97 - 51 * q^98

Introduction

L-functions

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Database

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