LMFDB - Embedded newform 6525.2.a.ca.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6525,2,Mod(1,6525)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [9,-2,0,10,0,0,-1,-9,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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			1.1
Root			$2.74387$ 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)

$f(q)$	$=$	$q-2.74387 q^{2} +5.52881 q^{4} +1.10432 q^{7} -9.68257 q^{8} -0.0961550 q^{11} -1.00817 q^{13} -3.03012 q^{14} +15.5101 q^{16} -1.92647 q^{17} +1.36951 q^{19} +0.263836 q^{22} -1.36474 q^{23} +2.76628 q^{26} +6.10559 q^{28} +1.00000 q^{29} +2.17445 q^{31} -23.1924 q^{32} +5.28599 q^{34} -6.61652 q^{37} -3.75776 q^{38} -5.07876 q^{41} +7.53382 q^{43} -0.531622 q^{44} +3.74467 q^{46} -5.77938 q^{47} -5.78047 q^{49} -5.57396 q^{52} +2.54747 q^{53} -10.6927 q^{56} -2.74387 q^{58} +12.6670 q^{59} +7.29508 q^{61} -5.96640 q^{62} +32.6168 q^{64} +2.77418 q^{67} -10.6511 q^{68} +6.05383 q^{71} -11.5699 q^{73} +18.1549 q^{74} +7.57176 q^{76} -0.106186 q^{77} -3.01211 q^{79} +13.9354 q^{82} +0.455950 q^{83} -20.6718 q^{86} +0.931027 q^{88} -7.57581 q^{89} -1.11334 q^{91} -7.54539 q^{92} +15.8578 q^{94} +10.7828 q^{97} +15.8608 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots - 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

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 $

$2$	−2.74387	−1.94021	−0.970103	−	0.242692i	$-0.921970\pi$
$2$	−2.74387	−1.94021	−0.970103	+	0.242692i	$0.921970\pi$
$3$	0	0
$3$	0	0
$4$	5.52881	2.76440
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.10432	0.417395	0.208697	−	0.977980i	$-0.433078\pi$
$7$	1.10432	0.417395	0.208697	+	0.977980i	$0.433078\pi$
$8$	−9.68257	−3.42331
$9$	0	0
$10$	0	0
$11$	−0.0961550	−0.0289918	−0.0144959	−	0.999895i	$-0.504614\pi$
$11$	−0.0961550	−0.0289918	−0.0144959	+	0.999895i	$0.504614\pi$
$12$	0	0
$13$	−1.00817	−0.279615	−0.139808	−	0.990179i	$-0.544648\pi$
$13$	−1.00817	−0.279615	−0.139808	+	0.990179i	$0.544648\pi$
$14$	−3.03012	−0.809832
$15$	0	0
$16$	15.5101	3.87752
$17$	−1.92647	−0.467239	−0.233619	−	0.972328i	$-0.575057\pi$
$17$	−1.92647	−0.467239	−0.233619	+	0.972328i	$0.575057\pi$
$18$	0	0
$19$	1.36951	0.314187	0.157094	−	0.987584i	$-0.449788\pi$
$19$	1.36951	0.314187	0.157094	+	0.987584i	$0.449788\pi$
$20$	0	0
$21$	0	0
$22$	0.263836	0.0562501
$23$	−1.36474	−0.284568	−0.142284	−	0.989826i	$-0.545445\pi$
$23$	−1.36474	−0.284568	−0.142284	+	0.989826i	$0.545445\pi$
$24$	0	0
$25$	0	0
$26$	2.76628	0.542512
$27$	0	0
$28$	6.10559	1.15385
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	2.17445	0.390543	0.195271	−	0.980749i	$-0.437441\pi$
$31$	2.17445	0.390543	0.195271	+	0.980749i	$0.437441\pi$
$32$	−23.1924	−4.09988
$33$	0	0
$34$	5.28599	0.906540
$35$	0	0
$36$	0	0
$37$	−6.61652	−1.08775	−0.543875	−	0.839166i	$-0.683043\pi$
$37$	−6.61652	−1.08775	−0.543875	+	0.839166i	$0.683043\pi$
$38$	−3.75776	−0.609589
$39$	0	0
$40$	0	0
$41$	−5.07876	−0.793169	−0.396584	−	0.917998i	$-0.629805\pi$
$41$	−5.07876	−0.793169	−0.396584	+	0.917998i	$0.629805\pi$
$42$	0	0
$43$	7.53382	1.14890	0.574448	−	0.818541i	$-0.305217\pi$
$43$	7.53382	1.14890	0.574448	+	0.818541i	$0.305217\pi$
$44$	−0.531622	−0.0801450
$45$	0	0
$46$	3.74467	0.552121
$47$	−5.77938	−0.843009	−0.421504	−	0.906826i	$-0.638498\pi$
$47$	−5.77938	−0.843009	−0.421504	+	0.906826i	$0.638498\pi$
$48$	0	0
$49$	−5.78047	−0.825782
$50$	0	0
$51$	0	0
$52$	−5.57396	−0.772970
$53$	2.54747	0.349922	0.174961	−	0.984575i	$-0.444020\pi$
$53$	2.54747	0.349922	0.174961	+	0.984575i	$0.444020\pi$
$54$	0	0
$55$	0	0
$56$	−10.6927	−1.42887
$57$	0	0
$58$	−2.74387	−0.360287
$59$	12.6670	1.64910	0.824548	−	0.565792i	$-0.191429\pi$
$59$	12.6670	1.64910	0.824548	+	0.565792i	$0.191429\pi$
$60$	0	0
$61$	7.29508	0.934040	0.467020	−	0.884247i	$-0.345328\pi$
$61$	7.29508	0.934040	0.467020	+	0.884247i	$0.345328\pi$
$62$	−5.96640	−0.757734
$63$	0	0
$64$	32.6168	4.07710
$65$	0	0
$66$	0	0
$67$	2.77418	0.338920	0.169460	−	0.985537i	$-0.445798\pi$
$67$	2.77418	0.338920	0.169460	+	0.985537i	$0.445798\pi$
$68$	−10.6511	−1.29164
$69$	0	0
$70$	0	0
$71$	6.05383	0.718457	0.359229	−	0.933250i	$-0.383040\pi$
$71$	6.05383	0.718457	0.359229	+	0.933250i	$0.383040\pi$
$72$	0	0
$73$	−11.5699	−1.35415	−0.677077	−	0.735912i	$-0.736754\pi$
$73$	−11.5699	−1.35415	−0.677077	+	0.735912i	$0.736754\pi$
$74$	18.1549	2.11046
$75$	0	0
$76$	7.57176	0.868541
$77$	−0.106186	−0.0121010
$78$	0	0
$79$	−3.01211	−0.338889	−0.169444	−	0.985540i	$-0.554197\pi$
$79$	−3.01211	−0.338889	−0.169444	+	0.985540i	$0.554197\pi$
$80$	0	0
$81$	0	0
$82$	13.9354	1.53891
$83$	0.455950	0.0500470	0.0250235	−	0.999687i	$-0.492034\pi$
$83$	0.455950	0.0500470	0.0250235	+	0.999687i	$0.492034\pi$
$84$	0	0
$85$	0	0
$86$	−20.6718	−2.22910
$87$	0	0
$88$	0.931027	0.0992478
$89$	−7.57581	−0.803034	−0.401517	−	0.915852i	$-0.631517\pi$
$89$	−7.57581	−0.803034	−0.401517	+	0.915852i	$0.631517\pi$
$90$	0	0
$91$	−1.11334	−0.116710
$92$	−7.54539	−0.786661
$93$	0	0
$94$	15.8578	1.63561
$95$	0	0
$96$	0	0
$97$	10.7828	1.09483	0.547415	−	0.836862i	$-0.315612\pi$
$97$	10.7828	1.09483	0.547415	+	0.836862i	$0.315612\pi$
$98$	15.8608	1.60219
$99$	0	0
$100$	0	0
$101$	−18.8731	−1.87794	−0.938971	−	0.343996i	$-0.888219\pi$
$101$	−18.8731	−1.87794	−0.938971	+	0.343996i	$0.888219\pi$
$102$	0	0
$103$	−12.0872	−1.19098	−0.595491	−	0.803362i	$-0.703043\pi$
$103$	−12.0872	−1.19098	−0.595491	+	0.803362i	$0.703043\pi$
$104$	9.76166	0.957209
$105$	0	0
$106$	−6.98991	−0.678920
$107$	10.8319	1.04716	0.523578	−	0.851977i	$-0.324597\pi$
$107$	10.8319	1.04716	0.523578	+	0.851977i	$0.324597\pi$
$108$	0	0
$109$	13.2131	1.26558	0.632790	−	0.774323i	$-0.281910\pi$
$109$	13.2131	1.26558	0.632790	+	0.774323i	$0.281910\pi$
$110$	0	0
$111$	0	0
$112$	17.1281	1.61846
$113$	−2.20594	−0.207517	−0.103758	−	0.994603i	$-0.533087\pi$
$113$	−2.20594	−0.207517	−0.103758	+	0.994603i	$0.533087\pi$
$114$	0	0
$115$	0	0
$116$	5.52881	0.513337
$117$	0	0
$118$	−34.7564	−3.19959
$119$	−2.12745	−0.195023
$120$	0	0
$121$	−10.9908	−0.999159
$122$	−20.0167	−1.81223
$123$	0	0
$124$	12.0221	1.07962
$125$	0	0
$126$	0	0
$127$	16.8893	1.49868	0.749340	−	0.662185i	$-0.230371\pi$
$127$	16.8893	1.49868	0.749340	+	0.662185i	$0.230371\pi$
$128$	−43.1113	−3.81054
$129$	0	0
$130$	0	0
$131$	−8.43970	−0.737380	−0.368690	−	0.929552i	$-0.620194\pi$
$131$	−8.43970	−0.737380	−0.368690	+	0.929552i	$0.620194\pi$
$132$	0	0
$133$	1.51238	0.131140
$134$	−7.61198	−0.657575
$135$	0	0
$136$	18.6532	1.59950
$137$	−8.80002	−0.751836	−0.375918	−	0.926653i	$-0.622673\pi$
$137$	−8.80002	−0.751836	−0.375918	+	0.926653i	$0.622673\pi$
$138$	0	0
$139$	1.93520	0.164142	0.0820709	−	0.996626i	$-0.473847\pi$
$139$	1.93520	0.164142	0.0820709	+	0.996626i	$0.473847\pi$
$140$	0	0
$141$	0	0
$142$	−16.6109	−1.39396
$143$	0.0969403	0.00810656
$144$	0	0
$145$	0	0
$146$	31.7463	2.62734
$147$	0	0
$148$	−36.5815	−3.00698
$149$	19.4823	1.59605	0.798025	−	0.602624i	$-0.205878\pi$
$149$	19.4823	1.59605	0.798025	+	0.602624i	$0.205878\pi$
$150$	0	0
$151$	−19.0946	−1.55390	−0.776949	−	0.629563i	$-0.783234\pi$
$151$	−19.0946	−1.55390	−0.776949	+	0.629563i	$0.783234\pi$
$152$	−13.2604	−1.07556
$153$	0	0
$154$	0.291361	0.0234785
$155$	0	0
$156$	0	0
$157$	−11.1697	−0.891438	−0.445719	−	0.895173i	$-0.647052\pi$
$157$	−11.1697	−0.891438	−0.445719	+	0.895173i	$0.647052\pi$
$158$	8.26483	0.657515
$159$	0	0
$160$	0	0
$161$	−1.50711	−0.118777
$162$	0	0
$163$	−6.15583	−0.482161	−0.241081	−	0.970505i	$-0.577502\pi$
$163$	−6.15583	−0.482161	−0.241081	+	0.970505i	$0.577502\pi$
$164$	−28.0795	−2.19264
$165$	0	0
$166$	−1.25106	−0.0971015
$167$	−14.4540	−1.11849	−0.559244	−	0.829003i	$-0.688908\pi$
$167$	−14.4540	−1.11849	−0.559244	+	0.829003i	$0.688908\pi$
$168$	0	0
$169$	−11.9836	−0.921815
$170$	0	0
$171$	0	0
$172$	41.6530	3.17601
$173$	−19.0331	−1.44706	−0.723531	−	0.690292i	$-0.757482\pi$
$173$	−19.0331	−1.44706	−0.723531	+	0.690292i	$0.757482\pi$
$174$	0	0
$175$	0	0
$176$	−1.49137	−0.112416
$177$	0	0
$178$	20.7870	1.55805
$179$	−3.15556	−0.235857	−0.117929	−	0.993022i	$-0.537625\pi$
$179$	−3.15556	−0.235857	−0.117929	+	0.993022i	$0.537625\pi$
$180$	0	0
$181$	−21.9375	−1.63060	−0.815301	−	0.579037i	$-0.803429\pi$
$181$	−21.9375	−1.63060	−0.815301	+	0.579037i	$0.803429\pi$
$182$	3.05487	0.226442
$183$	0	0
$184$	13.2142	0.974164
$185$	0	0
$186$	0	0
$187$	0.185240	0.0135461
$188$	−31.9530	−2.33042
$189$	0	0
$190$	0	0
$191$	22.8151	1.65084	0.825422	−	0.564517i	$-0.190937\pi$
$191$	22.8151	1.65084	0.825422	+	0.564517i	$0.190937\pi$
$192$	0	0
$193$	−7.27135	−0.523403	−0.261702	−	0.965149i	$-0.584284\pi$
$193$	−7.27135	−0.523403	−0.261702	+	0.965149i	$0.584284\pi$
$194$	−29.5866	−2.12420
$195$	0	0
$196$	−31.9591	−2.28279
$197$	13.6506	0.972563	0.486282	−	0.873802i	$-0.338353\pi$
$197$	13.6506	0.972563	0.486282	+	0.873802i	$0.338353\pi$
$198$	0	0
$199$	−9.81861	−0.696023	−0.348011	−	0.937490i	$-0.613143\pi$
$199$	−9.81861	−0.696023	−0.348011	+	0.937490i	$0.613143\pi$
$200$	0	0
$201$	0	0
$202$	51.7852	3.64360
$203$	1.10432	0.0775083
$204$	0	0
$205$	0	0
$206$	33.1655	2.31075
$207$	0	0
$208$	−15.6368	−1.08421
$209$	−0.131685	−0.00910886
$210$	0	0
$211$	−13.3349	−0.918009	−0.459004	−	0.888434i	$-0.651794\pi$
$211$	−13.3349	−0.918009	−0.459004	+	0.888434i	$0.651794\pi$
$212$	14.0845	0.967324
$213$	0	0
$214$	−29.7212	−2.03170
$215$	0	0
$216$	0	0
$217$	2.40130	0.163011
$218$	−36.2549	−2.45549
$219$	0	0
$220$	0	0
$221$	1.94221	0.130647
$222$	0	0
$223$	−8.60466	−0.576211	−0.288105	−	0.957599i	$-0.593025\pi$
$223$	−8.60466	−0.576211	−0.288105	+	0.957599i	$0.593025\pi$
$224$	−25.6120	−1.71127
$225$	0	0
$226$	6.05279	0.402626
$227$	14.6273	0.970846	0.485423	−	0.874279i	$-0.338666\pi$
$227$	14.6273	0.970846	0.485423	+	0.874279i	$0.338666\pi$
$228$	0	0
$229$	9.64412	0.637301	0.318651	−	0.947872i	$-0.396770\pi$
$229$	9.64412	0.637301	0.318651	+	0.947872i	$0.396770\pi$
$230$	0	0
$231$	0	0
$232$	−9.68257	−0.635692
$233$	17.9885	1.17847	0.589233	−	0.807963i	$-0.299430\pi$
$233$	17.9885	1.17847	0.589233	+	0.807963i	$0.299430\pi$
$234$	0	0
$235$	0	0
$236$	70.0331	4.55877
$237$	0	0
$238$	5.83744	0.378385
$239$	7.74493	0.500978	0.250489	−	0.968120i	$-0.419409\pi$
$239$	7.74493	0.500978	0.250489	+	0.968120i	$0.419409\pi$
$240$	0	0
$241$	5.30756	0.341890	0.170945	−	0.985281i	$-0.445318\pi$
$241$	5.30756	0.341890	0.170945	+	0.985281i	$0.445318\pi$
$242$	30.1572	1.93858
$243$	0	0
$244$	40.3331	2.58206
$245$	0	0
$246$	0	0
$247$	−1.38070	−0.0878517
$248$	−21.0543	−1.33695
$249$	0	0
$250$	0	0
$251$	−13.0470	−0.823518	−0.411759	−	0.911293i	$-0.635085\pi$
$251$	−13.0470	−0.823518	−0.411759	+	0.911293i	$0.635085\pi$
$252$	0	0
$253$	0.131227	0.00825015
$254$	−46.3419	−2.90775
$255$	0	0
$256$	53.0581	3.31613
$257$	−15.4865	−0.966024	−0.483012	−	0.875614i	$-0.660457\pi$
$257$	−15.4865	−0.966024	−0.483012	+	0.875614i	$0.660457\pi$
$258$	0	0
$259$	−7.30678	−0.454021
$260$	0	0
$261$	0	0
$262$	23.1574	1.43067
$263$	−5.90216	−0.363943	−0.181971	−	0.983304i	$-0.558248\pi$
$263$	−5.90216	−0.363943	−0.181971	+	0.983304i	$0.558248\pi$
$264$	0	0
$265$	0	0
$266$	−4.14978	−0.254439
$267$	0	0
$268$	15.3379	0.936911
$269$	−26.1012	−1.59142	−0.795710	−	0.605678i	$-0.792902\pi$
$269$	−26.1012	−1.59142	−0.795710	+	0.605678i	$0.792902\pi$
$270$	0	0
$271$	3.68372	0.223770	0.111885	−	0.993721i	$-0.464311\pi$
$271$	3.68372	0.223770	0.111885	+	0.993721i	$0.464311\pi$
$272$	−29.8798	−1.81173
$273$	0	0
$274$	24.1461	1.45872
$275$	0	0
$276$	0	0
$277$	24.6901	1.48349	0.741743	−	0.670684i	$-0.233999\pi$
$277$	24.6901	1.48349	0.741743	+	0.670684i	$0.233999\pi$
$278$	−5.30994	−0.318469
$279$	0	0
$280$	0	0
$281$	−22.8059	−1.36049	−0.680243	−	0.732986i	$-0.738126\pi$
$281$	−22.8059	−1.36049	−0.680243	+	0.732986i	$0.738126\pi$
$282$	0	0
$283$	−5.95693	−0.354103	−0.177051	−	0.984202i	$-0.556656\pi$
$283$	−5.95693	−0.354103	−0.177051	+	0.984202i	$0.556656\pi$
$284$	33.4704	1.98610
$285$	0	0
$286$	−0.265991	−0.0157284
$287$	−5.60859	−0.331065
$288$	0	0
$289$	−13.2887	−0.781688
$290$	0	0
$291$	0	0
$292$	−63.9677	−3.74343
$293$	16.9708	0.991443	0.495721	−	0.868482i	$-0.334904\pi$
$293$	16.9708	0.991443	0.495721	+	0.868482i	$0.334904\pi$
$294$	0	0
$295$	0	0
$296$	64.0650	3.72370
$297$	0	0
$298$	−53.4568	−3.09667
$299$	1.37589	0.0795697
$300$	0	0
$301$	8.31977	0.479543
$302$	52.3931	3.01488
$303$	0	0
$304$	21.2412	1.21827
$305$	0	0
$306$	0	0
$307$	2.13874	0.122064	0.0610321	−	0.998136i	$-0.480561\pi$
$307$	2.13874	0.122064	0.0610321	+	0.998136i	$0.480561\pi$
$308$	−0.587082	−0.0334521
$309$	0	0
$310$	0	0
$311$	−13.6680	−0.775044	−0.387522	−	0.921861i	$-0.626669\pi$
$311$	−13.6680	−0.775044	−0.387522	+	0.921861i	$0.626669\pi$
$312$	0	0
$313$	−5.07682	−0.286959	−0.143480	−	0.989653i	$-0.545829\pi$
$313$	−5.07682	−0.286959	−0.143480	+	0.989653i	$0.545829\pi$
$314$	30.6481	1.72957
$315$	0	0
$316$	−16.6534	−0.936825
$317$	−7.29420	−0.409683	−0.204842	−	0.978795i	$-0.565668\pi$
$317$	−7.29420	−0.409683	−0.204842	+	0.978795i	$0.565668\pi$
$318$	0	0
$319$	−0.0961550	−0.00538364
$320$	0	0
$321$	0	0
$322$	4.13532	0.230452
$323$	−2.63833	−0.146801
$324$	0	0
$325$	0	0
$326$	16.8908	0.935493
$327$	0	0
$328$	49.1755	2.71526
$329$	−6.38230	−0.351867
$330$	0	0
$331$	−15.6728	−0.861454	−0.430727	−	0.902482i	$-0.641743\pi$
$331$	−15.6728	−0.861454	−0.430727	+	0.902482i	$0.641743\pi$
$332$	2.52086	0.138350
$333$	0	0
$334$	39.6600	2.17010
$335$	0	0
$336$	0	0
$337$	27.1038	1.47644	0.738219	−	0.674562i	$-0.235667\pi$
$337$	27.1038	1.47644	0.738219	+	0.674562i	$0.235667\pi$
$338$	32.8814	1.78851
$339$	0	0
$340$	0	0
$341$	−0.209084	−0.0113225
$342$	0	0
$343$	−14.1138	−0.762072
$344$	−72.9467	−3.93302
$345$	0	0
$346$	52.2243	2.80760
$347$	19.8312	1.06460	0.532298	−	0.846557i	$-0.321329\pi$
$347$	19.8312	1.06460	0.532298	+	0.846557i	$0.321329\pi$
$348$	0	0
$349$	−25.5566	−1.36802	−0.684008	−	0.729475i	$-0.739765\pi$
$349$	−25.5566	−1.36802	−0.684008	+	0.729475i	$0.739765\pi$
$350$	0	0
$351$	0	0
$352$	2.23007	0.118863
$353$	7.94312	0.422770	0.211385	−	0.977403i	$-0.432203\pi$
$353$	7.94312	0.422770	0.211385	+	0.977403i	$0.432203\pi$
$354$	0	0
$355$	0	0
$356$	−41.8852	−2.21991
$357$	0	0
$358$	8.65843	0.457612
$359$	5.79924	0.306072	0.153036	−	0.988221i	$-0.451095\pi$
$359$	5.79924	0.306072	0.153036	+	0.988221i	$0.451095\pi$
$360$	0	0
$361$	−17.1244	−0.901286
$362$	60.1936	3.16371
$363$	0	0
$364$	−6.15546	−0.322634
$365$	0	0
$366$	0	0
$367$	−22.6318	−1.18137	−0.590686	−	0.806901i	$-0.701143\pi$
$367$	−22.6318	−1.18137	−0.590686	+	0.806901i	$0.701143\pi$
$368$	−21.1672	−1.10342
$369$	0	0
$370$	0	0
$371$	2.81323	0.146056
$372$	0	0
$373$	23.3987	1.21154	0.605769	−	0.795641i	$-0.292866\pi$
$373$	23.3987	1.21154	0.605769	+	0.795641i	$0.292866\pi$
$374$	−0.508274	−0.0262822
$375$	0	0
$376$	55.9592	2.88588
$377$	−1.00817	−0.0519233
$378$	0	0
$379$	27.4252	1.40874	0.704369	−	0.709834i	$-0.251230\pi$
$379$	27.4252	1.40874	0.704369	+	0.709834i	$0.251230\pi$
$380$	0	0
$381$	0	0
$382$	−62.6016	−3.20298
$383$	26.7931	1.36906	0.684531	−	0.728984i	$-0.260007\pi$
$383$	26.7931	1.36906	0.684531	+	0.728984i	$0.260007\pi$
$384$	0	0
$385$	0	0
$386$	19.9516	1.01551
$387$	0	0
$388$	59.6161	3.02655
$389$	25.7792	1.30706	0.653528	−	0.756902i	$-0.273288\pi$
$389$	25.7792	1.30706	0.653528	+	0.756902i	$0.273288\pi$
$390$	0	0
$391$	2.62914	0.132961
$392$	55.9698	2.82690
$393$	0	0
$394$	−37.4554	−1.88697
$395$	0	0
$396$	0	0
$397$	21.2065	1.06432	0.532161	−	0.846643i	$-0.321380\pi$
$397$	21.2065	1.06432	0.532161	+	0.846643i	$0.321380\pi$
$398$	26.9410	1.35043
$399$	0	0
$400$	0	0
$401$	20.5845	1.02794	0.513971	−	0.857807i	$-0.328174\pi$
$401$	20.5845	1.02794	0.513971	+	0.857807i	$0.328174\pi$
$402$	0	0
$403$	−2.19221	−0.109202
$404$	−104.346	−5.19139
$405$	0	0
$406$	−3.03012	−0.150382
$407$	0.636211	0.0315358
$408$	0	0
$409$	1.93199	0.0955309	0.0477654	−	0.998859i	$-0.484790\pi$
$409$	1.93199	0.0955309	0.0477654	+	0.998859i	$0.484790\pi$
$410$	0	0
$411$	0	0
$412$	−66.8275	−3.29236
$413$	13.9884	0.688324
$414$	0	0
$415$	0	0
$416$	23.3819	1.14639
$417$	0	0
$418$	0.361327	0.0176731
$419$	5.69653	0.278294	0.139147	−	0.990272i	$-0.455564\pi$
$419$	5.69653	0.278294	0.139147	+	0.990272i	$0.455564\pi$
$420$	0	0
$421$	−34.1352	−1.66365	−0.831825	−	0.555038i	$-0.812704\pi$
$421$	−34.1352	−1.66365	−0.831825	+	0.555038i	$0.812704\pi$
$422$	36.5891	1.78113
$423$	0	0
$424$	−24.6660	−1.19789
$425$	0	0
$426$	0	0
$427$	8.05613	0.389863
$428$	59.8873	2.89476
$429$	0	0
$430$	0	0
$431$	−15.1704	−0.730734	−0.365367	−	0.930864i	$-0.619056\pi$
$431$	−15.1704	−0.730734	−0.365367	+	0.930864i	$0.619056\pi$
$432$	0	0
$433$	6.09226	0.292775	0.146388	−	0.989227i	$-0.453235\pi$
$433$	6.09226	0.292775	0.146388	+	0.989227i	$0.453235\pi$
$434$	−6.58883	−0.316274
$435$	0	0
$436$	73.0524	3.49857
$437$	−1.86903	−0.0894077
$438$	0	0
$439$	−4.42241	−0.211070	−0.105535	−	0.994416i	$-0.533655\pi$
$439$	−4.42241	−0.211070	−0.105535	+	0.994416i	$0.533655\pi$
$440$	0	0
$441$	0	0
$442$	−5.32917	−0.253483
$443$	10.1506	0.482268	0.241134	−	0.970492i	$-0.422481\pi$
$443$	10.1506	0.482268	0.241134	+	0.970492i	$0.422481\pi$
$444$	0	0
$445$	0	0
$446$	23.6101	1.11797
$447$	0	0
$448$	36.0195	1.70176
$449$	15.6656	0.739307	0.369653	−	0.929170i	$-0.379476\pi$
$449$	15.6656	0.739307	0.369653	+	0.929170i	$0.379476\pi$
$450$	0	0
$451$	0.488348	0.0229954
$452$	−12.1962	−0.573660
$453$	0	0
$454$	−40.1353	−1.88364
$455$	0	0
$456$	0	0
$457$	−29.3357	−1.37226	−0.686132	−	0.727477i	$-0.740693\pi$
$457$	−29.3357	−1.37226	−0.686132	+	0.727477i	$0.740693\pi$
$458$	−26.4622	−1.23650
$459$	0	0
$460$	0	0
$461$	−0.433920	−0.0202097	−0.0101048	−	0.999949i	$-0.503217\pi$
$461$	−0.433920	−0.0202097	−0.0101048	+	0.999949i	$0.503217\pi$
$462$	0	0
$463$	9.63020	0.447553	0.223777	−	0.974640i	$-0.428161\pi$
$463$	9.63020	0.447553	0.223777	+	0.974640i	$0.428161\pi$
$464$	15.5101	0.720037
$465$	0	0
$466$	−49.3581	−2.28647
$467$	−13.4944	−0.624447	−0.312223	−	0.950009i	$-0.601074\pi$
$467$	−13.4944	−0.624447	−0.312223	+	0.950009i	$0.601074\pi$
$468$	0	0
$469$	3.06359	0.141463
$470$	0	0
$471$	0	0
$472$	−122.649	−5.64536
$473$	−0.724414	−0.0333086
$474$	0	0
$475$	0	0
$476$	−11.7623	−0.539122
$477$	0	0
$478$	−21.2510	−0.972000
$479$	7.97389	0.364336	0.182168	−	0.983267i	$-0.441688\pi$
$479$	7.97389	0.364336	0.182168	+	0.983267i	$0.441688\pi$
$480$	0	0
$481$	6.67057	0.304152
$482$	−14.5632	−0.663337
$483$	0	0
$484$	−60.7657	−2.76208
$485$	0	0
$486$	0	0
$487$	−1.61623	−0.0732385	−0.0366192	−	0.999329i	$-0.511659\pi$
$487$	−1.61623	−0.0732385	−0.0366192	+	0.999329i	$0.511659\pi$
$488$	−70.6352	−3.19750
$489$	0	0
$490$	0	0
$491$	−10.0585	−0.453932	−0.226966	−	0.973903i	$-0.572881\pi$
$491$	−10.0585	−0.453932	−0.226966	+	0.973903i	$0.572881\pi$
$492$	0	0
$493$	−1.92647	−0.0867641
$494$	3.78845	0.170450
$495$	0	0
$496$	33.7259	1.51434
$497$	6.68538	0.299880
$498$	0	0
$499$	20.3154	0.909440	0.454720	−	0.890634i	$-0.349739\pi$
$499$	20.3154	0.909440	0.454720	+	0.890634i	$0.349739\pi$
$500$	0	0
$501$	0	0
$502$	35.7992	1.59780
$503$	−40.7407	−1.81654	−0.908270	−	0.418385i	$-0.862596\pi$
$503$	−40.7407	−1.81654	−0.908270	+	0.418385i	$0.862596\pi$
$504$	0	0
$505$	0	0
$506$	−0.360068	−0.0160070
$507$	0	0
$508$	93.3775	4.14296
$509$	35.8671	1.58978	0.794891	−	0.606752i	$-0.207528\pi$
$509$	35.8671	1.58978	0.794891	+	0.606752i	$0.207528\pi$
$510$	0	0
$511$	−12.7769	−0.565217
$512$	−59.3618	−2.62344
$513$	0	0
$514$	42.4930	1.87429
$515$	0	0
$516$	0	0
$517$	0.555716	0.0244403
$518$	20.0488	0.880895
$519$	0	0
$520$	0	0
$521$	10.6032	0.464534	0.232267	−	0.972652i	$-0.425386\pi$
$521$	10.6032	0.464534	0.232267	+	0.972652i	$0.425386\pi$
$522$	0	0
$523$	−20.6859	−0.904530	−0.452265	−	0.891884i	$-0.649384\pi$
$523$	−20.6859	−0.904530	−0.452265	+	0.891884i	$0.649384\pi$
$524$	−46.6615	−2.03842
$525$	0	0
$526$	16.1947	0.706124
$527$	−4.18902	−0.182477
$528$	0	0
$529$	−21.1375	−0.919021
$530$	0	0
$531$	0	0
$532$	8.36167	0.362524
$533$	5.12024	0.221782
$534$	0	0
$535$	0	0
$536$	−26.8612	−1.16023
$537$	0	0
$538$	71.6183	3.08768
$539$	0.555821	0.0239409
$540$	0	0
$541$	1.77636	0.0763715	0.0381857	−	0.999271i	$-0.487842\pi$
$541$	1.77636	0.0763715	0.0381857	+	0.999271i	$0.487842\pi$
$542$	−10.1076	−0.434160
$543$	0	0
$544$	44.6797	1.91563
$545$	0	0
$546$	0	0
$547$	39.7930	1.70143	0.850713	−	0.525630i	$-0.176170\pi$
$547$	39.7930	1.70143	0.850713	+	0.525630i	$0.176170\pi$
$548$	−48.6536	−2.07838
$549$	0	0
$550$	0	0
$551$	1.36951	0.0583431
$552$	0	0
$553$	−3.32634	−0.141450
$554$	−67.7464	−2.87827
$555$	0	0
$556$	10.6994	0.453754
$557$	−10.6253	−0.450206	−0.225103	−	0.974335i	$-0.572272\pi$
$557$	−10.6253	−0.450206	−0.225103	+	0.974335i	$0.572272\pi$
$558$	0	0
$559$	−7.59535	−0.321249
$560$	0	0
$561$	0	0
$562$	62.5764	2.63963
$563$	−24.7572	−1.04339	−0.521696	−	0.853132i	$-0.674700\pi$
$563$	−24.7572	−1.04339	−0.521696	+	0.853132i	$0.674700\pi$
$564$	0	0
$565$	0	0
$566$	16.3450	0.687033
$567$	0	0
$568$	−58.6166	−2.45950
$569$	−33.7414	−1.41451	−0.707257	−	0.706957i	$-0.750067\pi$
$569$	−33.7414	−1.41451	−0.707257	+	0.706957i	$0.750067\pi$
$570$	0	0
$571$	−12.2259	−0.511639	−0.255819	−	0.966725i	$-0.582345\pi$
$571$	−12.2259	−0.511639	−0.255819	+	0.966725i	$0.582345\pi$
$572$	0.535964	0.0224098
$573$	0	0
$574$	15.3892	0.642334
$575$	0	0
$576$	0	0
$577$	−22.9772	−0.956555	−0.478278	−	0.878209i	$-0.658739\pi$
$577$	−22.9772	−0.956555	−0.478278	+	0.878209i	$0.658739\pi$
$578$	36.4624	1.51664
$579$	0	0
$580$	0	0
$581$	0.503516	0.0208893
$582$	0	0
$583$	−0.244952	−0.0101449
$584$	112.026	4.63568
$585$	0	0
$586$	−46.5655	−1.92360
$587$	−41.5985	−1.71695	−0.858477	−	0.512851i	$-0.828589\pi$
$587$	−41.5985	−1.71695	−0.858477	+	0.512851i	$0.828589\pi$
$588$	0	0
$589$	2.97793	0.122704
$590$	0	0
$591$	0	0
$592$	−102.623	−4.21777
$593$	−4.89980	−0.201211	−0.100605	−	0.994926i	$-0.532078\pi$
$593$	−4.89980	−0.201211	−0.100605	+	0.994926i	$0.532078\pi$
$594$	0	0
$595$	0	0
$596$	107.714	4.41212
$597$	0	0
$598$	−3.77525	−0.154382
$599$	−24.3309	−0.994133	−0.497066	−	0.867713i	$-0.665589\pi$
$599$	−24.3309	−0.994133	−0.497066	+	0.867713i	$0.665589\pi$
$600$	0	0
$601$	−35.2493	−1.43785	−0.718923	−	0.695089i	$-0.755365\pi$
$601$	−35.2493	−1.43785	−0.718923	+	0.695089i	$0.755365\pi$
$602$	−22.8283	−0.930414
$603$	0	0
$604$	−105.570	−4.29560
$605$	0	0
$606$	0	0
$607$	46.7359	1.89695	0.948476	−	0.316850i	$-0.102625\pi$
$607$	46.7359	1.89695	0.948476	+	0.316850i	$0.102625\pi$
$608$	−31.7623	−1.28813
$609$	0	0
$610$	0	0
$611$	5.82658	0.235718
$612$	0	0
$613$	0.159393	0.00643782	0.00321891	−	0.999995i	$-0.498975\pi$
$613$	0.159393	0.00643782	0.00321891	+	0.999995i	$0.498975\pi$
$614$	−5.86841	−0.236830
$615$	0	0
$616$	1.02815	0.0414255
$617$	18.5015	0.744841	0.372421	−	0.928064i	$-0.378528\pi$
$617$	18.5015	0.744841	0.372421	+	0.928064i	$0.378528\pi$
$618$	0	0
$619$	−36.4393	−1.46462	−0.732310	−	0.680971i	$-0.761558\pi$
$619$	−36.4393	−1.46462	−0.732310	+	0.680971i	$0.761558\pi$
$620$	0	0
$621$	0	0
$622$	37.5033	1.50375
$623$	−8.36614	−0.335182
$624$	0	0
$625$	0	0
$626$	13.9301	0.556760
$627$	0	0
$628$	−61.7550	−2.46429
$629$	12.7466	0.508239
$630$	0	0
$631$	−33.2812	−1.32490	−0.662451	−	0.749105i	$-0.730484\pi$
$631$	−33.2812	−1.32490	−0.662451	+	0.749105i	$0.730484\pi$
$632$	29.1650	1.16012
$633$	0	0
$634$	20.0143	0.794870
$635$	0	0
$636$	0	0
$637$	5.82769	0.230901
$638$	0.263836	0.0104454
$639$	0	0
$640$	0	0
$641$	−14.8916	−0.588185	−0.294092	−	0.955777i	$-0.595017\pi$
$641$	−14.8916	−0.588185	−0.294092	+	0.955777i	$0.595017\pi$
$642$	0	0
$643$	−27.7299	−1.09356	−0.546779	−	0.837277i	$-0.684146\pi$
$643$	−27.7299	−1.09356	−0.546779	+	0.837277i	$0.684146\pi$
$644$	−8.33254	−0.328348
$645$	0	0
$646$	7.23922	0.284823
$647$	−30.9301	−1.21599	−0.607994	−	0.793942i	$-0.708026\pi$
$647$	−30.9301	−1.21599	−0.607994	+	0.793942i	$0.708026\pi$
$648$	0	0
$649$	−1.21799	−0.0478103
$650$	0	0
$651$	0	0
$652$	−34.0344	−1.33289
$653$	−25.0243	−0.979278	−0.489639	−	0.871925i	$-0.662871\pi$
$653$	−25.0243	−0.979278	−0.489639	+	0.871925i	$0.662871\pi$
$654$	0	0
$655$	0	0
$656$	−78.7720	−3.07553
$657$	0	0
$658$	17.5122	0.682696
$659$	−20.2013	−0.786930	−0.393465	−	0.919340i	$-0.628724\pi$
$659$	−20.2013	−0.786930	−0.393465	+	0.919340i	$0.628724\pi$
$660$	0	0
$661$	−14.2973	−0.556099	−0.278049	−	0.960567i	$-0.589688\pi$
$661$	−14.2973	−0.556099	−0.278049	+	0.960567i	$0.589688\pi$
$662$	43.0040	1.67140
$663$	0	0
$664$	−4.41476	−0.171326
$665$	0	0
$666$	0	0
$667$	−1.36474	−0.0528430
$668$	−79.9136	−3.09195
$669$	0	0
$670$	0	0
$671$	−0.701459	−0.0270795
$672$	0	0
$673$	−33.6661	−1.29773	−0.648867	−	0.760902i	$-0.724757\pi$
$673$	−33.6661	−1.29773	−0.648867	+	0.760902i	$0.724757\pi$
$674$	−74.3692	−2.86459
$675$	0	0
$676$	−66.2550	−2.54827
$677$	−8.78111	−0.337486	−0.168743	−	0.985660i	$-0.553971\pi$
$677$	−8.78111	−0.337486	−0.168743	+	0.985660i	$0.553971\pi$
$678$	0	0
$679$	11.9077	0.456976
$680$	0	0
$681$	0	0
$682$	0.573699	0.0219681
$683$	−37.3359	−1.42862	−0.714310	−	0.699830i	$-0.753259\pi$
$683$	−37.3359	−1.42862	−0.714310	+	0.699830i	$0.753259\pi$
$684$	0	0
$685$	0	0
$686$	38.7263	1.47858
$687$	0	0
$688$	116.850	4.45487
$689$	−2.56828	−0.0978435
$690$	0	0
$691$	38.7767	1.47513	0.737567	−	0.675274i	$-0.235975\pi$
$691$	38.7767	1.47513	0.737567	+	0.675274i	$0.235975\pi$
$692$	−105.230	−4.00026
$693$	0	0
$694$	−54.4142	−2.06554
$695$	0	0
$696$	0	0
$697$	9.78410	0.370599
$698$	70.1240	2.65423
$699$	0	0
$700$	0	0
$701$	−20.8879	−0.788924	−0.394462	−	0.918912i	$-0.629069\pi$
$701$	−20.8879	−0.788924	−0.394462	+	0.918912i	$0.629069\pi$
$702$	0	0
$703$	−9.06140	−0.341757
$704$	−3.13627	−0.118203
$705$	0	0
$706$	−21.7949	−0.820261
$707$	−20.8420	−0.783843
$708$	0	0
$709$	−9.72205	−0.365119	−0.182560	−	0.983195i	$-0.558438\pi$
$709$	−9.72205	−0.365119	−0.182560	+	0.983195i	$0.558438\pi$
$710$	0	0
$711$	0	0
$712$	73.3533	2.74903
$713$	−2.96756	−0.111136
$714$	0	0
$715$	0	0
$716$	−17.4465	−0.652005
$717$	0	0
$718$	−15.9124	−0.593844
$719$	45.2454	1.68737	0.843684	−	0.536841i	$-0.180382\pi$
$719$	45.2454	1.68737	0.843684	+	0.536841i	$0.180382\pi$
$720$	0	0
$721$	−13.3481	−0.497110
$722$	46.9872	1.74868
$723$	0	0
$724$	−121.288	−4.50764
$725$	0	0
$726$	0	0
$727$	−19.4957	−0.723054	−0.361527	−	0.932362i	$-0.617744\pi$
$727$	−19.4957	−0.723054	−0.361527	+	0.932362i	$0.617744\pi$
$728$	10.7800	0.399534
$729$	0	0
$730$	0	0
$731$	−14.5137	−0.536809
$732$	0	0
$733$	−35.7113	−1.31903	−0.659513	−	0.751693i	$-0.729238\pi$
$733$	−35.7113	−1.31903	−0.659513	+	0.751693i	$0.729238\pi$
$734$	62.0988	2.29211
$735$	0	0
$736$	31.6517	1.16670
$737$	−0.266751	−0.00982590
$738$	0	0
$739$	−39.8367	−1.46542	−0.732708	−	0.680543i	$-0.761744\pi$
$739$	−39.8367	−1.46542	−0.732708	+	0.680543i	$0.761744\pi$
$740$	0	0
$741$	0	0
$742$	−7.71912	−0.283378
$743$	21.7579	0.798218	0.399109	−	0.916903i	$-0.369319\pi$
$743$	21.7579	0.798218	0.399109	+	0.916903i	$0.369319\pi$
$744$	0	0
$745$	0	0
$746$	−64.2028	−2.35063
$747$	0	0
$748$	1.02416	0.0374469
$749$	11.9619	0.437078
$750$	0	0
$751$	−36.0557	−1.31569	−0.657846	−	0.753152i	$-0.728532\pi$
$751$	−36.0557	−1.31569	−0.657846	+	0.753152i	$0.728532\pi$
$752$	−89.6386	−3.26878
$753$	0	0
$754$	2.76628	0.100742
$755$	0	0
$756$	0	0
$757$	7.57942	0.275479	0.137739	−	0.990469i	$-0.456016\pi$
$757$	7.57942	0.275479	0.137739	+	0.990469i	$0.456016\pi$
$758$	−75.2512	−2.73325
$759$	0	0
$760$	0	0
$761$	−17.3466	−0.628814	−0.314407	−	0.949288i	$-0.601806\pi$
$761$	−17.3466	−0.628814	−0.314407	+	0.949288i	$0.601806\pi$
$762$	0	0
$763$	14.5915	0.528247
$764$	126.140	4.56360
$765$	0	0
$766$	−73.5166	−2.65626
$767$	−12.7704	−0.461113
$768$	0	0
$769$	13.7054	0.494228	0.247114	−	0.968986i	$-0.420518\pi$
$769$	13.7054	0.494228	0.247114	+	0.968986i	$0.420518\pi$
$770$	0	0
$771$	0	0
$772$	−40.2019	−1.44690
$773$	15.1545	0.545069	0.272534	−	0.962146i	$-0.412138\pi$
$773$	15.1545	0.545069	0.272534	+	0.962146i	$0.412138\pi$
$774$	0	0
$775$	0	0
$776$	−104.405	−3.74794
$777$	0	0
$778$	−70.7346	−2.53596
$779$	−6.95542	−0.249204
$780$	0	0
$781$	−0.582105	−0.0208294
$782$	−7.21401	−0.257972
$783$	0	0
$784$	−89.6556	−3.20198
$785$	0	0
$786$	0	0
$787$	30.7331	1.09552	0.547759	−	0.836636i	$-0.315481\pi$
$787$	30.7331	1.09552	0.547759	+	0.836636i	$0.315481\pi$
$788$	75.4714	2.68856
$789$	0	0
$790$	0	0
$791$	−2.43606	−0.0866165
$792$	0	0
$793$	−7.35467	−0.261172
$794$	−58.1877	−2.06500
$795$	0	0
$796$	−54.2852	−1.92409
$797$	−20.4224	−0.723399	−0.361699	−	0.932295i	$-0.617803\pi$
$797$	−20.4224	−0.723399	−0.361699	+	0.932295i	$0.617803\pi$
$798$	0	0
$799$	11.1338	0.393886
$800$	0	0
$801$	0	0
$802$	−56.4812	−1.99442
$803$	1.11250	0.0392594
$804$	0	0
$805$	0	0
$806$	6.01514	0.211874
$807$	0	0
$808$	182.740	6.42877
$809$	41.0932	1.44476	0.722380	−	0.691497i	$-0.243048\pi$
$809$	41.0932	1.44476	0.722380	+	0.691497i	$0.243048\pi$
$810$	0	0
$811$	11.0882	0.389360	0.194680	−	0.980867i	$-0.437633\pi$
$811$	11.0882	0.389360	0.194680	+	0.980867i	$0.437633\pi$
$812$	6.10559	0.214264
$813$	0	0
$814$	−1.74568	−0.0611860
$815$	0	0
$816$	0	0
$817$	10.3176	0.360969
$818$	−5.30113	−0.185350
$819$	0	0
$820$	0	0
$821$	11.4735	0.400427	0.200213	−	0.979752i	$-0.435836\pi$
$821$	11.4735	0.400427	0.200213	+	0.979752i	$0.435836\pi$
$822$	0	0
$823$	12.7083	0.442985	0.221492	−	0.975162i	$-0.428907\pi$
$823$	12.7083	0.442985	0.221492	+	0.975162i	$0.428907\pi$
$824$	117.035	4.07710
$825$	0	0
$826$	−38.3823	−1.33549
$827$	−13.3335	−0.463650	−0.231825	−	0.972757i	$-0.574470\pi$
$827$	−13.3335	−0.463650	−0.231825	+	0.972757i	$0.574470\pi$
$828$	0	0
$829$	−26.2252	−0.910838	−0.455419	−	0.890277i	$-0.650510\pi$
$829$	−26.2252	−0.910838	−0.455419	+	0.890277i	$0.650510\pi$
$830$	0	0
$831$	0	0
$832$	−32.8832	−1.14002
$833$	11.1359	0.385837
$834$	0	0
$835$	0	0
$836$	−0.728062	−0.0251806
$837$	0	0
$838$	−15.6305	−0.539947
$839$	−12.8865	−0.444891	−0.222446	−	0.974945i	$-0.571404\pi$
$839$	−12.8865	−0.444891	−0.222446	+	0.974945i	$0.571404\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	93.6625	3.22782
$843$	0	0
$844$	−73.7258	−2.53775
$845$	0	0
$846$	0	0
$847$	−12.1373	−0.417044
$848$	39.5114	1.35683
$849$	0	0
$850$	0	0
$851$	9.02984	0.309539
$852$	0	0
$853$	−17.3965	−0.595645	−0.297822	−	0.954621i	$-0.596260\pi$
$853$	−17.3965	−0.595645	−0.297822	+	0.954621i	$0.596260\pi$
$854$	−22.1049	−0.756416
$855$	0	0
$856$	−104.880	−3.58474
$857$	5.74494	0.196243	0.0981217	−	0.995174i	$-0.468717\pi$
$857$	5.74494	0.196243	0.0981217	+	0.995174i	$0.468717\pi$
$858$	0	0
$859$	−36.5326	−1.24648	−0.623238	−	0.782032i	$-0.714183\pi$
$859$	−36.5326	−1.24648	−0.623238	+	0.782032i	$0.714183\pi$
$860$	0	0
$861$	0	0
$862$	41.6256	1.41777
$863$	5.99556	0.204091	0.102046	−	0.994780i	$-0.467461\pi$
$863$	5.99556	0.204091	0.102046	+	0.994780i	$0.467461\pi$
$864$	0	0
$865$	0	0
$866$	−16.7163	−0.568044
$867$	0	0
$868$	13.2763	0.450627
$869$	0.289629	0.00982500
$870$	0	0
$871$	−2.79684	−0.0947673
$872$	−127.936	−4.33247
$873$	0	0
$874$	5.12836	0.173469
$875$	0	0
$876$	0	0
$877$	−10.7265	−0.362209	−0.181105	−	0.983464i	$-0.557967\pi$
$877$	−10.7265	−0.362209	−0.181105	+	0.983464i	$0.557967\pi$
$878$	12.1345	0.409519
$879$	0	0
$880$	0	0
$881$	−35.4683	−1.19496	−0.597479	−	0.801885i	$-0.703831\pi$
$881$	−35.4683	−1.19496	−0.597479	+	0.801885i	$0.703831\pi$
$882$	0	0
$883$	31.2074	1.05021	0.525106	−	0.851037i	$-0.324026\pi$
$883$	31.2074	1.05021	0.525106	+	0.851037i	$0.324026\pi$
$884$	10.7381	0.361162
$885$	0	0
$886$	−27.8518	−0.935700
$887$	36.5271	1.22646	0.613230	−	0.789904i	$-0.289870\pi$
$887$	36.5271	1.22646	0.613230	+	0.789904i	$0.289870\pi$
$888$	0	0
$889$	18.6512	0.625542
$890$	0	0
$891$	0	0
$892$	−47.5735	−1.59288
$893$	−7.91492	−0.264863
$894$	0	0
$895$	0	0
$896$	−47.6088	−1.59050
$897$	0	0
$898$	−42.9844	−1.43441
$899$	2.17445	0.0725220
$900$	0	0
$901$	−4.90763	−0.163497
$902$	−1.33996	−0.0446158
$903$	0	0
$904$	21.3591	0.710394
$905$	0	0
$906$	0	0
$907$	−26.6054	−0.883416	−0.441708	−	0.897159i	$-0.645627\pi$
$907$	−26.6054	−0.883416	−0.441708	+	0.897159i	$0.645627\pi$
$908$	80.8713	2.68381
$909$	0	0
$910$	0	0
$911$	29.9283	0.991567	0.495784	−	0.868446i	$-0.334881\pi$
$911$	29.9283	0.991567	0.495784	+	0.868446i	$0.334881\pi$
$912$	0	0
$913$	−0.0438418	−0.00145095
$914$	80.4932	2.66248
$915$	0	0
$916$	53.3204	1.76176
$917$	−9.32015	−0.307779
$918$	0	0
$919$	−2.30814	−0.0761385	−0.0380693	−	0.999275i	$-0.512121\pi$
$919$	−2.30814	−0.0761385	−0.0380693	+	0.999275i	$0.512121\pi$
$920$	0	0
$921$	0	0
$922$	1.19062	0.0392109
$923$	−6.10327	−0.200892
$924$	0	0
$925$	0	0
$926$	−26.4240	−0.868346
$927$	0	0
$928$	−23.1924	−0.761329
$929$	13.6997	0.449473	0.224737	−	0.974420i	$-0.427848\pi$
$929$	13.6997	0.449473	0.224737	+	0.974420i	$0.427848\pi$
$930$	0	0
$931$	−7.91642	−0.259450
$932$	99.4549	3.25775
$933$	0	0
$934$	37.0269	1.21156
$935$	0	0
$936$	0	0
$937$	44.5601	1.45572	0.727858	−	0.685728i	$-0.240516\pi$
$937$	44.5601	1.45572	0.727858	+	0.685728i	$0.240516\pi$
$938$	−8.40608	−0.274468
$939$	0	0
$940$	0	0
$941$	−59.0327	−1.92441	−0.962206	−	0.272324i	$-0.912208\pi$
$941$	−59.0327	−1.92441	−0.962206	+	0.272324i	$0.912208\pi$
$942$	0	0
$943$	6.93119	0.225711
$944$	196.465	6.39440
$945$	0	0
$946$	1.98770	0.0646256
$947$	−40.4627	−1.31486	−0.657431	−	0.753515i	$-0.728357\pi$
$947$	−40.4627	−1.31486	−0.657431	+	0.753515i	$0.728357\pi$
$948$	0	0
$949$	11.6644	0.378642
$950$	0	0
$951$	0	0
$952$	20.5992	0.667624
$953$	50.2692	1.62838	0.814190	−	0.580599i	$-0.197182\pi$
$953$	50.2692	1.62838	0.814190	+	0.580599i	$0.197182\pi$
$954$	0	0
$955$	0	0
$956$	42.8202	1.38490
$957$	0	0
$958$	−21.8793	−0.706888
$959$	−9.71806	−0.313813
$960$	0	0
$961$	−26.2718	−0.847476
$962$	−18.3031	−0.590117
$963$	0	0
$964$	29.3444	0.945121
$965$	0	0
$966$	0	0
$967$	−24.4422	−0.786009	−0.393004	−	0.919537i	$-0.628564\pi$
$967$	−24.4422	−0.786009	−0.393004	+	0.919537i	$0.628564\pi$
$968$	106.419	3.42043
$969$	0	0
$970$	0	0
$971$	47.1924	1.51447	0.757237	−	0.653140i	$-0.226549\pi$
$971$	47.1924	1.51447	0.757237	+	0.653140i	$0.226549\pi$
$972$	0	0
$973$	2.13709	0.0685120
$974$	4.43473	0.142098
$975$	0	0
$976$	113.147	3.62176
$977$	−55.7755	−1.78442	−0.892209	−	0.451623i	$-0.850845\pi$
$977$	−55.7755	−1.78442	−0.892209	+	0.451623i	$0.850845\pi$
$978$	0	0
$979$	0.728451	0.0232814
$980$	0	0
$981$	0	0
$982$	27.5991	0.880722
$983$	−44.1469	−1.40807	−0.704033	−	0.710167i	$-0.748619\pi$
$983$	−44.1469	−1.40807	−0.704033	+	0.710167i	$0.748619\pi$
$984$	0	0
$985$	0	0
$986$	5.28599	0.168340
$987$	0	0
$988$	−7.63361	−0.242857
$989$	−10.2817	−0.326939
$990$	0	0
$991$	−25.7068	−0.816605	−0.408302	−	0.912847i	$-0.633879\pi$
$991$	−25.7068	−0.816605	−0.408302	+	0.912847i	$0.633879\pi$
$992$	−50.4308	−1.60118
$993$	0	0
$994$	−18.3438	−0.581830
$995$	0	0
$996$	0	0
$997$	−5.14864	−0.163059	−0.0815296	−	0.996671i	$-0.525981\pi$
$997$	−5.14864	−0.163059	−0.0815296	+	0.996671i	$0.525981\pi$
$998$	−55.7426	−1.76450
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6525.2.a.ca.1.1	✓	9	1.1	even	1	trivial
6525.2.a.cb.1.1	yes	9	15.14	odd	2
6525.2.a.cc.1.9	yes	9	3.2	odd	2
6525.2.a.cd.1.9	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-2.74387 q^{2} +5.52881 q^{4} +1.10432 q^{7} -9.68257 q^{8} -0.0961550 q^{11} -1.00817 q^{13} -3.03012 q^{14} +15.5101 q^{16} -1.92647 q^{17} +1.36951 q^{19} +0.263836 q^{22} -1.36474 q^{23} +2.76628 q^{26} +6.10559 q^{28} +1.00000 q^{29} +2.17445 q^{31} -23.1924 q^{32} +5.28599 q^{34} -6.61652 q^{37} -3.75776 q^{38} -5.07876 q^{41} +7.53382 q^{43} -0.531622 q^{44} +3.74467 q^{46} -5.77938 q^{47} -5.78047 q^{49} -5.57396 q^{52} +2.54747 q^{53} -10.6927 q^{56} -2.74387 q^{58} +12.6670 q^{59} +7.29508 q^{61} -5.96640 q^{62} +32.6168 q^{64} +2.77418 q^{67} -10.6511 q^{68} +6.05383 q^{71} -11.5699 q^{73} +18.1549 q^{74} +7.57176 q^{76} -0.106186 q^{77} -3.01211 q^{79} +13.9354 q^{82} +0.455950 q^{83} -20.6718 q^{86} +0.931027 q^{88} -7.57581 q^{89} -1.11334 q^{91} -7.54539 q^{92} +15.8578 q^{94} +10.7828 q^{97} +15.8608 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 2 q^{2} + 10 q^{4} - q^{7} - 9 q^{8} + 2 q^{11} - q^{13} - 3 q^{14} + 4 q^{16} - 12 q^{17} - q^{19} - 3 q^{22} - 16 q^{23} + 6 q^{26} + 4 q^{28} + 9 q^{29} + 5 q^{31} - 20 q^{32} + 3 q^{34} - 30 q^{38}+ \cdots - 51 q^{98}+O(q^{100}) \) 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

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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