LMFDB - Embedded newform 7605.2.a.cq.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7605,2,Mod(1,7605)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7605, 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("7605.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7605 = 3^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7605.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:	$60.7262307372$
Analytic rank:	$0$
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} - 3x^{8} - 9x^{7} + 29x^{6} + 17x^{5} - 83x^{4} + 17x^{3} + 70x^{2} - 48x + 8 $ x^9 - 3x^8 - 9x^7 + 29x^6 + 17x^5 - 83x^4 + 17x^3 + 70x^2 - 48x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2535)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.813006$ of defining polynomial
Character	$\chi$	$=$	7605.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.16585 q^{2} -0.640788 q^{4} -1.00000 q^{5} -0.957202 q^{7} +3.07877 q^{8} +O(q^{10})$	$q-1.16585 q^{2} -0.640788 q^{4} -1.00000 q^{5} -0.957202 q^{7} +3.07877 q^{8} +1.16585 q^{10} -5.05761 q^{11} +1.11596 q^{14} -2.30781 q^{16} +1.25739 q^{17} -2.44704 q^{19} +0.640788 q^{20} +5.89643 q^{22} +5.45555 q^{23} +1.00000 q^{25} +0.613364 q^{28} -10.2006 q^{29} +1.02783 q^{31} -3.46697 q^{32} -1.46593 q^{34} +0.957202 q^{35} +0.939148 q^{37} +2.85289 q^{38} -3.07877 q^{40} -7.55650 q^{41} +0.259367 q^{43} +3.24086 q^{44} -6.36037 q^{46} +0.115292 q^{47} -6.08376 q^{49} -1.16585 q^{50} +2.43698 q^{53} +5.05761 q^{55} -2.94700 q^{56} +11.8924 q^{58} -8.32438 q^{59} +13.1657 q^{61} -1.19830 q^{62} +8.65760 q^{64} +11.8247 q^{67} -0.805722 q^{68} -1.11596 q^{70} +0.977874 q^{71} -7.06112 q^{73} -1.09491 q^{74} +1.56804 q^{76} +4.84116 q^{77} -13.0206 q^{79} +2.30781 q^{80} +8.80977 q^{82} -3.76930 q^{83} -1.25739 q^{85} -0.302384 q^{86} -15.5712 q^{88} -17.9201 q^{89} -3.49585 q^{92} -0.134414 q^{94} +2.44704 q^{95} -8.42595 q^{97} +7.09277 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 10 q^{4} - 9 q^{5} - 10 q^{7} - 3 q^{8}+O(q^{10}) $ 9 * q + 10 * q^4 - 9 * q^5 - 10 * q^7 - 3 * q^8	$ 9 q + 10 q^{4} - 9 q^{5} - 10 q^{7} - 3 q^{8} - 11 q^{11} - 10 q^{14} + 8 q^{16} - 18 q^{17} + 10 q^{19} - 10 q^{20} + 17 q^{22} + 7 q^{23} + 9 q^{25} - 43 q^{28} + 9 q^{29} + 17 q^{31} - 11 q^{32} - 4 q^{34} + 10 q^{35} + 3 q^{37} - 21 q^{38} + 3 q^{40} - 4 q^{41} + 14 q^{43} - 13 q^{44} + 3 q^{46} + 2 q^{47} + 43 q^{49} - 5 q^{53} + 11 q^{55} - q^{56} - 14 q^{58} - 3 q^{59} + 48 q^{61} + 7 q^{62} + 41 q^{64} - 40 q^{67} - 40 q^{68} + 10 q^{70} - 12 q^{71} - 38 q^{73} + 4 q^{74} + 51 q^{76} - 8 q^{77} + 22 q^{79} - 8 q^{80} + 38 q^{82} - 11 q^{83} + 18 q^{85} - 56 q^{86} + 50 q^{88} - 2 q^{89} + 36 q^{92} + 42 q^{94} - 10 q^{95} - 23 q^{97} + 79 q^{98}+O(q^{100}) $ 9 * q + 10 * q^4 - 9 * q^5 - 10 * q^7 - 3 * q^8 - 11 * q^11 - 10 * q^14 + 8 * q^16 - 18 * q^17 + 10 * q^19 - 10 * q^20 + 17 * q^22 + 7 * q^23 + 9 * q^25 - 43 * q^28 + 9 * q^29 + 17 * q^31 - 11 * q^32 - 4 * q^34 + 10 * q^35 + 3 * q^37 - 21 * q^38 + 3 * q^40 - 4 * q^41 + 14 * q^43 - 13 * q^44 + 3 * q^46 + 2 * q^47 + 43 * q^49 - 5 * q^53 + 11 * q^55 - q^56 - 14 * q^58 - 3 * q^59 + 48 * q^61 + 7 * q^62 + 41 * q^64 - 40 * q^67 - 40 * q^68 + 10 * q^70 - 12 * q^71 - 38 * q^73 + 4 * q^74 + 51 * q^76 - 8 * q^77 + 22 * q^79 - 8 * q^80 + 38 * q^82 - 11 * q^83 + 18 * q^85 - 56 * q^86 + 50 * q^88 - 2 * q^89 + 36 * q^92 + 42 * q^94 - 10 * q^95 - 23 * q^97 + 79 * 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 $

$2$	−1.16585	−0.824382	−0.412191	−	0.911097i	$-0.635236\pi$
$2$	−1.16585	−0.824382	−0.412191	+	0.911097i	$0.635236\pi$
$3$	0	0
$3$	0	0
$4$	−0.640788	−0.320394
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−0.957202	−0.361788	−0.180894	−	0.983503i	$-0.557899\pi$
$7$	−0.957202	−0.361788	−0.180894	+	0.983503i	$0.557899\pi$
$8$	3.07877	1.08851
$9$	0	0
$10$	1.16585	0.368675
$11$	−5.05761	−1.52493	−0.762464	−	0.647031i	$-0.776011\pi$
$11$	−5.05761	−1.52493	−0.762464	+	0.647031i	$0.776011\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	1.11596	0.298252
$15$	0	0
$16$	−2.30781	−0.576954
$17$	1.25739	0.304962	0.152481	−	0.988306i	$-0.451274\pi$
$17$	1.25739	0.304962	0.152481	+	0.988306i	$0.451274\pi$
$18$	0	0
$19$	−2.44704	−0.561390	−0.280695	−	0.959797i	$-0.590565\pi$
$19$	−2.44704	−0.561390	−0.280695	+	0.959797i	$0.590565\pi$
$20$	0.640788	0.143285
$21$	0	0
$22$	5.89643	1.25712
$23$	5.45555	1.13756	0.568781	−	0.822489i	$-0.307415\pi$
$23$	5.45555	1.13756	0.568781	+	0.822489i	$0.307415\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0.613364	0.115915
$29$	−10.2006	−1.89421	−0.947103	−	0.320929i	$-0.896005\pi$
$29$	−10.2006	−1.89421	−0.947103	+	0.320929i	$0.896005\pi$
$30$	0	0
$31$	1.02783	0.184604	0.0923022	−	0.995731i	$-0.470577\pi$
$31$	1.02783	0.184604	0.0923022	+	0.995731i	$0.470577\pi$
$32$	−3.46697	−0.612879
$33$	0	0
$34$	−1.46593	−0.251405
$35$	0.957202	0.161797
$36$	0	0
$37$	0.939148	0.154395	0.0771975	−	0.997016i	$-0.475403\pi$
$37$	0.939148	0.154395	0.0771975	+	0.997016i	$0.475403\pi$
$38$	2.85289	0.462800
$39$	0	0
$40$	−3.07877	−0.486796
$41$	−7.55650	−1.18013	−0.590064	−	0.807357i	$-0.700897\pi$
$41$	−7.55650	−1.18013	−0.590064	+	0.807357i	$0.700897\pi$
$42$	0	0
$43$	0.259367	0.0395531	0.0197765	−	0.999804i	$-0.493705\pi$
$43$	0.259367	0.0395531	0.0197765	+	0.999804i	$0.493705\pi$
$44$	3.24086	0.488578
$45$	0	0
$46$	−6.36037	−0.937786
$47$	0.115292	0.0168171	0.00840855	−	0.999965i	$-0.497323\pi$
$47$	0.115292	0.0168171	0.00840855	+	0.999965i	$0.497323\pi$
$48$	0	0
$49$	−6.08376	−0.869109
$50$	−1.16585	−0.164876
$51$	0	0
$52$	0	0
$53$	2.43698	0.334745	0.167372	−	0.985894i	$-0.446472\pi$
$53$	2.43698	0.334745	0.167372	+	0.985894i	$0.446472\pi$
$54$	0	0
$55$	5.05761	0.681969
$56$	−2.94700	−0.393810
$57$	0	0
$58$	11.8924	1.56155
$59$	−8.32438	−1.08374	−0.541871	−	0.840461i	$-0.682284\pi$
$59$	−8.32438	−1.08374	−0.541871	+	0.840461i	$0.682284\pi$
$60$	0	0
$61$	13.1657	1.68569	0.842845	−	0.538156i	$-0.180879\pi$
$61$	13.1657	1.68569	0.842845	+	0.538156i	$0.180879\pi$
$62$	−1.19830	−0.152185
$63$	0	0
$64$	8.65760	1.08220
$65$	0	0
$66$	0	0
$67$	11.8247	1.44462	0.722311	−	0.691569i	$-0.243080\pi$
$67$	11.8247	1.44462	0.722311	+	0.691569i	$0.243080\pi$
$68$	−0.805722	−0.0977081
$69$	0	0
$70$	−1.11596	−0.133382
$71$	0.977874	0.116052	0.0580261	−	0.998315i	$-0.481519\pi$
$71$	0.977874	0.116052	0.0580261	+	0.998315i	$0.481519\pi$
$72$	0	0
$73$	−7.06112	−0.826442	−0.413221	−	0.910631i	$-0.635596\pi$
$73$	−7.06112	−0.826442	−0.413221	+	0.910631i	$0.635596\pi$
$74$	−1.09491	−0.127280
$75$	0	0
$76$	1.56804	0.179866
$77$	4.84116	0.551701
$78$	0	0
$79$	−13.0206	−1.46493	−0.732467	−	0.680803i	$-0.761631\pi$
$79$	−13.0206	−1.46493	−0.732467	+	0.680803i	$0.761631\pi$
$80$	2.30781	0.258022
$81$	0	0
$82$	8.80977	0.972876
$83$	−3.76930	−0.413734	−0.206867	−	0.978369i	$-0.566327\pi$
$83$	−3.76930	−0.413734	−0.206867	+	0.978369i	$0.566327\pi$
$84$	0	0
$85$	−1.25739	−0.136383
$86$	−0.302384	−0.0326069
$87$	0	0
$88$	−15.5712	−1.65990
$89$	−17.9201	−1.89952	−0.949762	−	0.312972i	$-0.898675\pi$
$89$	−17.9201	−1.89952	−0.949762	+	0.312972i	$0.898675\pi$
$90$	0	0
$91$	0	0
$92$	−3.49585	−0.364468
$93$	0	0
$94$	−0.134414	−0.0138637
$95$	2.44704	0.251061
$96$	0	0
$97$	−8.42595	−0.855525	−0.427763	−	0.903891i	$-0.640698\pi$
$97$	−8.42595	−0.855525	−0.427763	+	0.903891i	$0.640698\pi$
$98$	7.09277	0.716478
$99$	0	0
$100$	−0.640788	−0.0640788
$101$	−4.78493	−0.476118	−0.238059	−	0.971251i	$-0.576511\pi$
$101$	−4.78493	−0.476118	−0.238059	+	0.971251i	$0.576511\pi$
$102$	0	0
$103$	10.6406	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$103$	10.6406	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$104$	0	0
$105$	0	0
$106$	−2.84116	−0.275958
$107$	−16.5805	−1.60290	−0.801449	−	0.598063i	$-0.795937\pi$
$107$	−16.5805	−1.60290	−0.801449	+	0.598063i	$0.795937\pi$
$108$	0	0
$109$	12.8615	1.23191	0.615953	−	0.787783i	$-0.288771\pi$
$109$	12.8615	1.23191	0.615953	+	0.787783i	$0.288771\pi$
$110$	−5.89643	−0.562203
$111$	0	0
$112$	2.20905	0.208735
$113$	−13.2710	−1.24843	−0.624213	−	0.781254i	$-0.714580\pi$
$113$	−13.2710	−1.24843	−0.624213	+	0.781254i	$0.714580\pi$
$114$	0	0
$115$	−5.45555	−0.508733
$116$	6.53643	0.606892
$117$	0	0
$118$	9.70500	0.893418
$119$	−1.20358	−0.110332
$120$	0	0
$121$	14.5795	1.32541
$122$	−15.3492	−1.38965
$123$	0	0
$124$	−0.658624	−0.0591462
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	13.4321	1.19191	0.595953	−	0.803020i	$-0.296775\pi$
$127$	13.4321	1.19191	0.595953	+	0.803020i	$0.296775\pi$
$128$	−3.15955	−0.279268
$129$	0	0
$130$	0	0
$131$	−8.09375	−0.707154	−0.353577	−	0.935405i	$-0.615035\pi$
$131$	−8.09375	−0.707154	−0.353577	+	0.935405i	$0.615035\pi$
$132$	0	0
$133$	2.34231	0.203104
$134$	−13.7859	−1.19092
$135$	0	0
$136$	3.87122	0.331954
$137$	17.5189	1.49674	0.748372	−	0.663279i	$-0.230836\pi$
$137$	17.5189	1.49674	0.748372	+	0.663279i	$0.230836\pi$
$138$	0	0
$139$	−8.84347	−0.750094	−0.375047	−	0.927006i	$-0.622373\pi$
$139$	−8.84347	−0.750094	−0.375047	+	0.927006i	$0.622373\pi$
$140$	−0.613364	−0.0518387
$141$	0	0
$142$	−1.14006	−0.0956714
$143$	0	0
$144$	0	0
$145$	10.2006	0.847115
$146$	8.23223	0.681304
$147$	0	0
$148$	−0.601795	−0.0494672
$149$	−7.44717	−0.610096	−0.305048	−	0.952337i	$-0.598673\pi$
$149$	−7.44717	−0.610096	−0.305048	+	0.952337i	$0.598673\pi$
$150$	0	0
$151$	−18.9412	−1.54141	−0.770705	−	0.637192i	$-0.780096\pi$
$151$	−18.9412	−1.54141	−0.770705	+	0.637192i	$0.780096\pi$
$152$	−7.53388	−0.611078
$153$	0	0
$154$	−5.64408	−0.454813
$155$	−1.02783	−0.0825576
$156$	0	0
$157$	20.1480	1.60799	0.803993	−	0.594638i	$-0.202705\pi$
$157$	20.1480	1.60799	0.803993	+	0.594638i	$0.202705\pi$
$158$	15.1801	1.20766
$159$	0	0
$160$	3.46697	0.274088
$161$	−5.22207	−0.411557
$162$	0	0
$163$	−9.33184	−0.730926	−0.365463	−	0.930826i	$-0.619089\pi$
$163$	−9.33184	−0.730926	−0.365463	+	0.930826i	$0.619089\pi$
$164$	4.84212	0.378106
$165$	0	0
$166$	4.39444	0.341075
$167$	−0.308104	−0.0238418	−0.0119209	−	0.999929i	$-0.503795\pi$
$167$	−0.308104	−0.0238418	−0.0119209	+	0.999929i	$0.503795\pi$
$168$	0	0
$169$	0	0
$170$	1.46593	0.112432
$171$	0	0
$172$	−0.166199	−0.0126726
$173$	9.83812	0.747978	0.373989	−	0.927433i	$-0.377990\pi$
$173$	9.83812	0.747978	0.373989	+	0.927433i	$0.377990\pi$
$174$	0	0
$175$	−0.957202	−0.0723577
$176$	11.6720	0.879813
$177$	0	0
$178$	20.8922	1.56593
$179$	3.92634	0.293469	0.146734	−	0.989176i	$-0.453124\pi$
$179$	3.92634	0.293469	0.146734	+	0.989176i	$0.453124\pi$
$180$	0	0
$181$	21.3537	1.58721	0.793603	−	0.608436i	$-0.208203\pi$
$181$	21.3537	1.58721	0.793603	+	0.608436i	$0.208203\pi$
$182$	0	0
$183$	0	0
$184$	16.7964	1.23825
$185$	−0.939148	−0.0690475
$186$	0	0
$187$	−6.35940	−0.465046
$188$	−0.0738779	−0.00538810
$189$	0	0
$190$	−2.85289	−0.206970
$191$	−2.06081	−0.149115	−0.0745577	−	0.997217i	$-0.523754\pi$
$191$	−2.06081	−0.149115	−0.0745577	+	0.997217i	$0.523754\pi$
$192$	0	0
$193$	−10.6654	−0.767709	−0.383855	−	0.923394i	$-0.625404\pi$
$193$	−10.6654	−0.767709	−0.383855	+	0.923394i	$0.625404\pi$
$194$	9.82341	0.705280
$195$	0	0
$196$	3.89840	0.278457
$197$	17.2608	1.22978	0.614889	−	0.788613i	$-0.289201\pi$
$197$	17.2608	1.22978	0.614889	+	0.788613i	$0.289201\pi$
$198$	0	0
$199$	1.48323	0.105143	0.0525716	−	0.998617i	$-0.483258\pi$
$199$	1.48323	0.105143	0.0525716	+	0.998617i	$0.483258\pi$
$200$	3.07877	0.217702
$201$	0	0
$202$	5.57852	0.392503
$203$	9.76405	0.685302
$204$	0	0
$205$	7.55650	0.527769
$206$	−12.4053	−0.864319
$207$	0	0
$208$	0	0
$209$	12.3762	0.856080
$210$	0	0
$211$	4.95941	0.341420	0.170710	−	0.985321i	$-0.445394\pi$
$211$	4.95941	0.341420	0.170710	+	0.985321i	$0.445394\pi$
$212$	−1.56159	−0.107250
$213$	0	0
$214$	19.3304	1.32140
$215$	−0.259367	−0.0176887
$216$	0	0
$217$	−0.983845	−0.0667878
$218$	−14.9946	−1.01556
$219$	0	0
$220$	−3.24086	−0.218499
$221$	0	0
$222$	0	0
$223$	−28.1738	−1.88666	−0.943329	−	0.331860i	$-0.892324\pi$
$223$	−28.1738	−1.88666	−0.943329	+	0.331860i	$0.892324\pi$
$224$	3.31859	0.221732
$225$	0	0
$226$	15.4720	1.02918
$227$	−21.1618	−1.40456	−0.702280	−	0.711901i	$-0.747835\pi$
$227$	−21.1618	−1.40456	−0.702280	+	0.711901i	$0.747835\pi$
$228$	0	0
$229$	−24.2688	−1.60373	−0.801865	−	0.597506i	$-0.796159\pi$
$229$	−24.2688	−1.60373	−0.801865	+	0.597506i	$0.796159\pi$
$230$	6.36037	0.419391
$231$	0	0
$232$	−31.4053	−2.06186
$233$	24.5751	1.60997	0.804983	−	0.593298i	$-0.202174\pi$
$233$	24.5751	1.60997	0.804983	+	0.593298i	$0.202174\pi$
$234$	0	0
$235$	−0.115292	−0.00752083
$236$	5.33417	0.347225
$237$	0	0
$238$	1.40319	0.0909556
$239$	25.0729	1.62183	0.810916	−	0.585163i	$-0.198969\pi$
$239$	25.0729	1.62183	0.810916	+	0.585163i	$0.198969\pi$
$240$	0	0
$241$	1.78188	0.114781	0.0573904	−	0.998352i	$-0.481722\pi$
$241$	1.78188	0.114781	0.0573904	+	0.998352i	$0.481722\pi$
$242$	−16.9975	−1.09264
$243$	0	0
$244$	−8.43640	−0.540085
$245$	6.08376	0.388677
$246$	0	0
$247$	0	0
$248$	3.16446	0.200944
$249$	0	0
$250$	1.16585	0.0737350
$251$	17.7480	1.12024	0.560122	−	0.828410i	$-0.310754\pi$
$251$	17.7480	1.12024	0.560122	+	0.828410i	$0.310754\pi$
$252$	0	0
$253$	−27.5921	−1.73470
$254$	−15.6598	−0.982585
$255$	0	0
$256$	−13.6316	−0.851977
$257$	−27.8765	−1.73889	−0.869445	−	0.494031i	$-0.835523\pi$
$257$	−27.8765	−1.73889	−0.869445	+	0.494031i	$0.835523\pi$
$258$	0	0
$259$	−0.898955	−0.0558583
$260$	0	0
$261$	0	0
$262$	9.43612	0.582965
$263$	−11.9705	−0.738134	−0.369067	−	0.929403i	$-0.620323\pi$
$263$	−11.9705	−0.738134	−0.369067	+	0.929403i	$0.620323\pi$
$264$	0	0
$265$	−2.43698	−0.149702
$266$	−2.73079	−0.167436
$267$	0	0
$268$	−7.57715	−0.462848
$269$	17.0425	1.03910	0.519551	−	0.854440i	$-0.326099\pi$
$269$	17.0425	1.03910	0.519551	+	0.854440i	$0.326099\pi$
$270$	0	0
$271$	12.3515	0.750301	0.375151	−	0.926964i	$-0.377591\pi$
$271$	12.3515	0.750301	0.375151	+	0.926964i	$0.377591\pi$
$272$	−2.90183	−0.175949
$273$	0	0
$274$	−20.4245	−1.23389
$275$	−5.05761	−0.304986
$276$	0	0
$277$	−1.71328	−0.102941	−0.0514705	−	0.998675i	$-0.516391\pi$
$277$	−1.71328	−0.102941	−0.0514705	+	0.998675i	$0.516391\pi$
$278$	10.3102	0.618364
$279$	0	0
$280$	2.94700	0.176117
$281$	−0.221037	−0.0131860	−0.00659299	−	0.999978i	$-0.502099\pi$
$281$	−0.221037	−0.0131860	−0.00659299	+	0.999978i	$0.502099\pi$
$282$	0	0
$283$	9.44547	0.561475	0.280737	−	0.959785i	$-0.409421\pi$
$283$	9.44547	0.561475	0.280737	+	0.959785i	$0.409421\pi$
$284$	−0.626610	−0.0371825
$285$	0	0
$286$	0	0
$287$	7.23310	0.426956
$288$	0	0
$289$	−15.4190	−0.906998
$290$	−11.8924	−0.698347
$291$	0	0
$292$	4.52468	0.264787
$293$	−13.5832	−0.793539	−0.396769	−	0.917918i	$-0.629869\pi$
$293$	−13.5832	−0.793539	−0.396769	+	0.917918i	$0.629869\pi$
$294$	0	0
$295$	8.32438	0.484664
$296$	2.89142	0.168060
$297$	0	0
$298$	8.68230	0.502952
$299$	0	0
$300$	0	0
$301$	−0.248267	−0.0143098
$302$	22.0826	1.27071
$303$	0	0
$304$	5.64732	0.323896
$305$	−13.1657	−0.753864
$306$	0	0
$307$	8.65595	0.494021	0.247011	−	0.969013i	$-0.420552\pi$
$307$	8.65595	0.494021	0.247011	+	0.969013i	$0.420552\pi$
$308$	−3.10216	−0.176762
$309$	0	0
$310$	1.19830	0.0680590
$311$	8.06452	0.457297	0.228649	−	0.973509i	$-0.426569\pi$
$311$	8.06452	0.457297	0.228649	+	0.973509i	$0.426569\pi$
$312$	0	0
$313$	22.0529	1.24650	0.623251	−	0.782022i	$-0.285811\pi$
$313$	22.0529	1.24650	0.623251	+	0.782022i	$0.285811\pi$
$314$	−23.4896	−1.32560
$315$	0	0
$316$	8.34345	0.469356
$317$	−20.8086	−1.16873	−0.584364	−	0.811492i	$-0.698656\pi$
$317$	−20.8086	−1.16873	−0.584364	+	0.811492i	$0.698656\pi$
$318$	0	0
$319$	51.5908	2.88853
$320$	−8.65760	−0.483975
$321$	0	0
$322$	6.08816	0.339280
$323$	−3.07689	−0.171203
$324$	0	0
$325$	0	0
$326$	10.8795	0.602562
$327$	0	0
$328$	−23.2647	−1.28458
$329$	−0.110358	−0.00608423
$330$	0	0
$331$	−10.3654	−0.569732	−0.284866	−	0.958567i	$-0.591949\pi$
$331$	−10.3654	−0.569732	−0.284866	+	0.958567i	$0.591949\pi$
$332$	2.41532	0.132558
$333$	0	0
$334$	0.359204	0.0196548
$335$	−11.8247	−0.646054
$336$	0	0
$337$	15.4304	0.840546	0.420273	−	0.907398i	$-0.361934\pi$
$337$	15.4304	0.840546	0.420273	+	0.907398i	$0.361934\pi$
$338$	0	0
$339$	0	0
$340$	0.805722	0.0436964
$341$	−5.19839	−0.281509
$342$	0	0
$343$	12.5238	0.676222
$344$	0.798531	0.0430539
$345$	0	0
$346$	−11.4698	−0.616620
$347$	24.6928	1.32558	0.662789	−	0.748806i	$-0.269373\pi$
$347$	24.6928	1.32558	0.662789	+	0.748806i	$0.269373\pi$
$348$	0	0
$349$	−15.4777	−0.828505	−0.414252	−	0.910162i	$-0.635957\pi$
$349$	−15.4777	−0.828505	−0.414252	+	0.910162i	$0.635957\pi$
$350$	1.11596	0.0596504
$351$	0	0
$352$	17.5346	0.934596
$353$	14.6323	0.778796	0.389398	−	0.921069i	$-0.372683\pi$
$353$	14.6323	0.778796	0.389398	+	0.921069i	$0.372683\pi$
$354$	0	0
$355$	−0.977874	−0.0519002
$356$	11.4830	0.608596
$357$	0	0
$358$	−4.57754	−0.241930
$359$	18.1306	0.956898	0.478449	−	0.878115i	$-0.341199\pi$
$359$	18.1306	0.956898	0.478449	+	0.878115i	$0.341199\pi$
$360$	0	0
$361$	−13.0120	−0.684841
$362$	−24.8952	−1.30846
$363$	0	0
$364$	0	0
$365$	7.06112	0.369596
$366$	0	0
$367$	−9.19781	−0.480122	−0.240061	−	0.970758i	$-0.577167\pi$
$367$	−9.19781	−0.480122	−0.240061	+	0.970758i	$0.577167\pi$
$368$	−12.5904	−0.656320
$369$	0	0
$370$	1.09491	0.0569216
$371$	−2.33268	−0.121107
$372$	0	0
$373$	28.5940	1.48054	0.740270	−	0.672310i	$-0.234698\pi$
$373$	28.5940	1.48054	0.740270	+	0.672310i	$0.234698\pi$
$374$	7.41413	0.383375
$375$	0	0
$376$	0.354958	0.0183056
$377$	0	0
$378$	0	0
$379$	18.7115	0.961147	0.480573	−	0.876955i	$-0.340429\pi$
$379$	18.7115	0.961147	0.480573	+	0.876955i	$0.340429\pi$
$380$	−1.56804	−0.0804385
$381$	0	0
$382$	2.40261	0.122928
$383$	34.2009	1.74759	0.873793	−	0.486299i	$-0.161653\pi$
$383$	34.2009	1.74759	0.873793	+	0.486299i	$0.161653\pi$
$384$	0	0
$385$	−4.84116	−0.246728
$386$	12.4342	0.632886
$387$	0	0
$388$	5.39925	0.274105
$389$	−14.5536	−0.737896	−0.368948	−	0.929450i	$-0.620282\pi$
$389$	−14.5536	−0.737896	−0.368948	+	0.929450i	$0.620282\pi$
$390$	0	0
$391$	6.85977	0.346913
$392$	−18.7305	−0.946033
$393$	0	0
$394$	−20.1235	−1.01381
$395$	13.0206	0.655138
$396$	0	0
$397$	8.07706	0.405376	0.202688	−	0.979243i	$-0.435032\pi$
$397$	8.07706	0.405376	0.202688	+	0.979243i	$0.435032\pi$
$398$	−1.72923	−0.0866782
$399$	0	0
$400$	−2.30781	−0.115391
$401$	14.1755	0.707889	0.353944	−	0.935266i	$-0.384840\pi$
$401$	14.1755	0.707889	0.353944	+	0.935266i	$0.384840\pi$
$402$	0	0
$403$	0	0
$404$	3.06612	0.152545
$405$	0	0
$406$	−11.3834	−0.564951
$407$	−4.74985	−0.235441
$408$	0	0
$409$	−1.93570	−0.0957144	−0.0478572	−	0.998854i	$-0.515239\pi$
$409$	−1.93570	−0.0957144	−0.0478572	+	0.998854i	$0.515239\pi$
$410$	−8.80977	−0.435083
$411$	0	0
$412$	−6.81834	−0.335915
$413$	7.96812	0.392085
$414$	0	0
$415$	3.76930	0.185028
$416$	0	0
$417$	0	0
$418$	−14.4288	−0.705737
$419$	−26.1769	−1.27883	−0.639413	−	0.768863i	$-0.720823\pi$
$419$	−26.1769	−1.27883	−0.639413	+	0.768863i	$0.720823\pi$
$420$	0	0
$421$	12.6661	0.617309	0.308655	−	0.951174i	$-0.400121\pi$
$421$	12.6661	0.617309	0.308655	+	0.951174i	$0.400121\pi$
$422$	−5.78194	−0.281460
$423$	0	0
$424$	7.50289	0.364373
$425$	1.25739	0.0609925
$426$	0	0
$427$	−12.6022	−0.609863
$428$	10.6246	0.513559
$429$	0	0
$430$	0.302384	0.0145822
$431$	28.3470	1.36543	0.682713	−	0.730687i	$-0.260800\pi$
$431$	28.3470	1.36543	0.682713	+	0.730687i	$0.260800\pi$
$432$	0	0
$433$	−9.98885	−0.480033	−0.240017	−	0.970769i	$-0.577153\pi$
$433$	−9.98885	−0.480033	−0.240017	+	0.970769i	$0.577153\pi$
$434$	1.14702	0.0550586
$435$	0	0
$436$	−8.24148	−0.394695
$437$	−13.3500	−0.638616
$438$	0	0
$439$	19.3922	0.925540	0.462770	−	0.886478i	$-0.346856\pi$
$439$	19.3922	0.925540	0.462770	+	0.886478i	$0.346856\pi$
$440$	15.5712	0.742329
$441$	0	0
$442$	0	0
$443$	−5.82980	−0.276982	−0.138491	−	0.990364i	$-0.544225\pi$
$443$	−5.82980	−0.276982	−0.138491	+	0.990364i	$0.544225\pi$
$444$	0	0
$445$	17.9201	0.849493
$446$	32.8465	1.55533
$447$	0	0
$448$	−8.28707	−0.391527
$449$	−27.2523	−1.28612	−0.643058	−	0.765818i	$-0.722335\pi$
$449$	−27.2523	−1.28612	−0.643058	+	0.765818i	$0.722335\pi$
$450$	0	0
$451$	38.2179	1.79961
$452$	8.50387	0.399988
$453$	0	0
$454$	24.6716	1.15789
$455$	0	0
$456$	0	0
$457$	−23.7369	−1.11036	−0.555182	−	0.831729i	$-0.687352\pi$
$457$	−23.7369	−1.11036	−0.555182	+	0.831729i	$0.687352\pi$
$458$	28.2939	1.32209
$459$	0	0
$460$	3.49585	0.162995
$461$	17.3059	0.806014	0.403007	−	0.915197i	$-0.367965\pi$
$461$	17.3059	0.806014	0.403007	+	0.915197i	$0.367965\pi$
$462$	0	0
$463$	34.5377	1.60510	0.802552	−	0.596582i	$-0.203475\pi$
$463$	34.5377	1.60510	0.802552	+	0.596582i	$0.203475\pi$
$464$	23.5411	1.09287
$465$	0	0
$466$	−28.6509	−1.32723
$467$	−13.7597	−0.636724	−0.318362	−	0.947969i	$-0.603133\pi$
$467$	−13.7597	−0.636724	−0.318362	+	0.947969i	$0.603133\pi$
$468$	0	0
$469$	−11.3187	−0.522647
$470$	0.134414	0.00620004
$471$	0	0
$472$	−25.6289	−1.17966
$473$	−1.31178	−0.0603156
$474$	0	0
$475$	−2.44704	−0.112278
$476$	0.771238	0.0353497
$477$	0	0
$478$	−29.2313	−1.33701
$479$	−2.32088	−0.106044	−0.0530219	−	0.998593i	$-0.516885\pi$
$479$	−2.32088	−0.106044	−0.0530219	+	0.998593i	$0.516885\pi$
$480$	0	0
$481$	0	0
$482$	−2.07741	−0.0946232
$483$	0	0
$484$	−9.34235	−0.424652
$485$	8.42595	0.382603
$486$	0	0
$487$	−14.7251	−0.667257	−0.333629	−	0.942705i	$-0.608273\pi$
$487$	−14.7251	−0.667257	−0.333629	+	0.942705i	$0.608273\pi$
$488$	40.5340	1.83489
$489$	0	0
$490$	−7.09277	−0.320419
$491$	39.3204	1.77451	0.887253	−	0.461283i	$-0.152611\pi$
$491$	39.3204	1.77451	0.887253	+	0.461283i	$0.152611\pi$
$492$	0	0
$493$	−12.8262	−0.577662
$494$	0	0
$495$	0	0
$496$	−2.37205	−0.106508
$497$	−0.936023	−0.0419864
$498$	0	0
$499$	−21.0832	−0.943814	−0.471907	−	0.881648i	$-0.656434\pi$
$499$	−21.0832	−0.943814	−0.471907	+	0.881648i	$0.656434\pi$
$500$	0.640788	0.0286569
$501$	0	0
$502$	−20.6916	−0.923510
$503$	6.33407	0.282422	0.141211	−	0.989980i	$-0.454900\pi$
$503$	6.33407	0.282422	0.141211	+	0.989980i	$0.454900\pi$
$504$	0	0
$505$	4.78493	0.212927
$506$	32.1683	1.43006
$507$	0	0
$508$	−8.60712	−0.381879
$509$	11.0119	0.488092	0.244046	−	0.969764i	$-0.421525\pi$
$509$	11.0119	0.488092	0.244046	+	0.969764i	$0.421525\pi$
$510$	0	0
$511$	6.75892	0.298997
$512$	22.2116	0.981622
$513$	0	0
$514$	32.4999	1.43351
$515$	−10.6406	−0.468879
$516$	0	0
$517$	−0.583104	−0.0256449
$518$	1.04805	0.0460486
$519$	0	0
$520$	0	0
$521$	−24.2840	−1.06390	−0.531951	−	0.846775i	$-0.678541\pi$
$521$	−24.2840	−1.06390	−0.531951	+	0.846775i	$0.678541\pi$
$522$	0	0
$523$	−25.8888	−1.13204	−0.566018	−	0.824393i	$-0.691517\pi$
$523$	−25.8888	−1.13204	−0.566018	+	0.824393i	$0.691517\pi$
$524$	5.18638	0.226568
$525$	0	0
$526$	13.9559	0.608504
$527$	1.29239	0.0562974
$528$	0	0
$529$	6.76308	0.294047
$530$	2.84116	0.123412
$531$	0	0
$532$	−1.50093	−0.0650734
$533$	0	0
$534$	0	0
$535$	16.5805	0.716838
$536$	36.4056	1.57248
$537$	0	0
$538$	−19.8691	−0.856617
$539$	30.7693	1.32533
$540$	0	0
$541$	3.67904	0.158174	0.0790871	−	0.996868i	$-0.474799\pi$
$541$	3.67904	0.158174	0.0790871	+	0.996868i	$0.474799\pi$
$542$	−14.4000	−0.618535
$543$	0	0
$544$	−4.35934	−0.186905
$545$	−12.8615	−0.550925
$546$	0	0
$547$	33.0031	1.41111	0.705554	−	0.708656i	$-0.250698\pi$
$547$	33.0031	1.41111	0.705554	+	0.708656i	$0.250698\pi$
$548$	−11.2259	−0.479548
$549$	0	0
$550$	5.89643	0.251425
$551$	24.9613	1.06339
$552$	0	0
$553$	12.4634	0.529996
$554$	1.99743	0.0848627
$555$	0	0
$556$	5.66679	0.240326
$557$	25.6886	1.08846	0.544230	−	0.838936i	$-0.316822\pi$
$557$	25.6886	1.08846	0.544230	+	0.838936i	$0.316822\pi$
$558$	0	0
$559$	0	0
$560$	−2.20905	−0.0933492
$561$	0	0
$562$	0.257697	0.0108703
$563$	43.7560	1.84410	0.922048	−	0.387075i	$-0.126515\pi$
$563$	43.7560	1.84410	0.922048	+	0.387075i	$0.126515\pi$
$564$	0	0
$565$	13.2710	0.558313
$566$	−11.0120	−0.462870
$567$	0	0
$568$	3.01065	0.126324
$569$	24.5255	1.02816	0.514082	−	0.857741i	$-0.328133\pi$
$569$	24.5255	1.02816	0.514082	+	0.857741i	$0.328133\pi$
$570$	0	0
$571$	4.90263	0.205169	0.102584	−	0.994724i	$-0.467289\pi$
$571$	4.90263	0.205169	0.102584	+	0.994724i	$0.467289\pi$
$572$	0	0
$573$	0	0
$574$	−8.43273	−0.351975
$575$	5.45555	0.227512
$576$	0	0
$577$	−19.1780	−0.798391	−0.399195	−	0.916866i	$-0.630710\pi$
$577$	−19.1780	−0.798391	−0.399195	+	0.916866i	$0.630710\pi$
$578$	17.9762	0.747713
$579$	0	0
$580$	−6.53643	−0.271411
$581$	3.60798	0.149684
$582$	0	0
$583$	−12.3253	−0.510462
$584$	−21.7396	−0.899590
$585$	0	0
$586$	15.8360	0.654179
$587$	14.8326	0.612209	0.306104	−	0.951998i	$-0.400974\pi$
$587$	14.8326	0.612209	0.306104	+	0.951998i	$0.400974\pi$
$588$	0	0
$589$	−2.51515	−0.103635
$590$	−9.70500	−0.399549
$591$	0	0
$592$	−2.16738	−0.0890788
$593$	6.61437	0.271620	0.135810	−	0.990735i	$-0.456636\pi$
$593$	6.61437	0.271620	0.135810	+	0.990735i	$0.456636\pi$
$594$	0	0
$595$	1.20358	0.0493419
$596$	4.77206	0.195471
$597$	0	0
$598$	0	0
$599$	6.94677	0.283837	0.141919	−	0.989878i	$-0.454673\pi$
$599$	6.94677	0.283837	0.141919	+	0.989878i	$0.454673\pi$
$600$	0	0
$601$	18.3658	0.749157	0.374579	−	0.927195i	$-0.377787\pi$
$601$	18.3658	0.749157	0.374579	+	0.927195i	$0.377787\pi$
$602$	0.289442	0.0117968
$603$	0	0
$604$	12.1373	0.493859
$605$	−14.5795	−0.592740
$606$	0	0
$607$	−31.6416	−1.28429	−0.642147	−	0.766582i	$-0.721956\pi$
$607$	−31.6416	−1.28429	−0.642147	+	0.766582i	$0.721956\pi$
$608$	8.48382	0.344064
$609$	0	0
$610$	15.3492	0.621472
$611$	0	0
$612$	0	0
$613$	43.9923	1.77683	0.888417	−	0.459037i	$-0.151806\pi$
$613$	43.9923	1.77683	0.888417	+	0.459037i	$0.151806\pi$
$614$	−10.0916	−0.407262
$615$	0	0
$616$	14.9048	0.600532
$617$	−5.08640	−0.204771	−0.102385	−	0.994745i	$-0.532647\pi$
$617$	−5.08640	−0.204771	−0.102385	+	0.994745i	$0.532647\pi$
$618$	0	0
$619$	44.5174	1.78931	0.894653	−	0.446762i	$-0.147423\pi$
$619$	44.5174	1.78931	0.894653	+	0.446762i	$0.147423\pi$
$620$	0.658624	0.0264510
$621$	0	0
$622$	−9.40204	−0.376988
$623$	17.1531	0.687226
$624$	0	0
$625$	1.00000	0.0400000
$626$	−25.7104	−1.02759
$627$	0	0
$628$	−12.9106	−0.515189
$629$	1.18088	0.0470847
$630$	0	0
$631$	−25.4933	−1.01487	−0.507437	−	0.861689i	$-0.669407\pi$
$631$	−25.4933	−1.01487	−0.507437	+	0.861689i	$0.669407\pi$
$632$	−40.0875	−1.59459
$633$	0	0
$634$	24.2598	0.963478
$635$	−13.4321	−0.533036
$636$	0	0
$637$	0	0
$638$	−60.1472	−2.38125
$639$	0	0
$640$	3.15955	0.124892
$641$	−16.9617	−0.669948	−0.334974	−	0.942227i	$-0.608727\pi$
$641$	−16.9617	−0.669948	−0.334974	+	0.942227i	$0.608727\pi$
$642$	0	0
$643$	11.7358	0.462814	0.231407	−	0.972857i	$-0.425667\pi$
$643$	11.7358	0.462814	0.231407	+	0.972857i	$0.425667\pi$
$644$	3.34624	0.131860
$645$	0	0
$646$	3.58720	0.141137
$647$	−16.4828	−0.648007	−0.324004	−	0.946056i	$-0.605029\pi$
$647$	−16.4828	−0.648007	−0.324004	+	0.946056i	$0.605029\pi$
$648$	0	0
$649$	42.1015	1.65263
$650$	0	0
$651$	0	0
$652$	5.97973	0.234184
$653$	11.3795	0.445316	0.222658	−	0.974897i	$-0.428527\pi$
$653$	11.3795	0.445316	0.222658	+	0.974897i	$0.428527\pi$
$654$	0	0
$655$	8.09375	0.316249
$656$	17.4390	0.680879
$657$	0	0
$658$	0.128661	0.00501573
$659$	31.5912	1.23062	0.615310	−	0.788285i	$-0.289031\pi$
$659$	31.5912	1.23062	0.615310	+	0.788285i	$0.289031\pi$
$660$	0	0
$661$	41.2523	1.60453	0.802265	−	0.596968i	$-0.203628\pi$
$661$	41.2523	1.60453	0.802265	+	0.596968i	$0.203628\pi$
$662$	12.0845	0.469677
$663$	0	0
$664$	−11.6048	−0.450353
$665$	−2.34231	−0.0908311
$666$	0	0
$667$	−55.6500	−2.15478
$668$	0.197429	0.00763877
$669$	0	0
$670$	13.7859	0.532596
$671$	−66.5868	−2.57056
$672$	0	0
$673$	31.1149	1.19939	0.599695	−	0.800229i	$-0.295289\pi$
$673$	31.1149	1.19939	0.599695	+	0.800229i	$0.295289\pi$
$674$	−17.9895	−0.692931
$675$	0	0
$676$	0	0
$677$	33.9996	1.30671	0.653356	−	0.757051i	$-0.273360\pi$
$677$	33.9996	1.30671	0.653356	+	0.757051i	$0.273360\pi$
$678$	0	0
$679$	8.06534	0.309519
$680$	−3.87122	−0.148454
$681$	0	0
$682$	6.06056	0.232071
$683$	39.2385	1.50142	0.750710	−	0.660632i	$-0.229711\pi$
$683$	39.2385	1.50142	0.750710	+	0.660632i	$0.229711\pi$
$684$	0	0
$685$	−17.5189	−0.669365
$686$	−14.6009	−0.557465
$687$	0	0
$688$	−0.598571	−0.0228203
$689$	0	0
$690$	0	0
$691$	−3.87229	−0.147309	−0.0736544	−	0.997284i	$-0.523466\pi$
$691$	−3.87229	−0.147309	−0.0736544	+	0.997284i	$0.523466\pi$
$692$	−6.30415	−0.239648
$693$	0	0
$694$	−28.7881	−1.09278
$695$	8.84347	0.335452
$696$	0	0
$697$	−9.50148	−0.359894
$698$	18.0448	0.683005
$699$	0	0
$700$	0.613364	0.0231830
$701$	−43.7894	−1.65390	−0.826952	−	0.562273i	$-0.809927\pi$
$701$	−43.7894	−1.65390	−0.826952	+	0.562273i	$0.809927\pi$
$702$	0	0
$703$	−2.29814	−0.0866758
$704$	−43.7868	−1.65028
$705$	0	0
$706$	−17.0591	−0.642026
$707$	4.58014	0.172254
$708$	0	0
$709$	43.3413	1.62772	0.813859	−	0.581062i	$-0.197363\pi$
$709$	43.3413	1.62772	0.813859	+	0.581062i	$0.197363\pi$
$710$	1.14006	0.0427856
$711$	0	0
$712$	−55.1718	−2.06765
$713$	5.60741	0.209999
$714$	0	0
$715$	0	0
$716$	−2.51595	−0.0940256
$717$	0	0
$718$	−21.1376	−0.788850
$719$	−0.135145	−0.00504005	−0.00252003	−	0.999997i	$-0.500802\pi$
$719$	−0.135145	−0.00504005	−0.00252003	+	0.999997i	$0.500802\pi$
$720$	0	0
$721$	−10.1852	−0.379315
$722$	15.1701	0.564571
$723$	0	0
$724$	−13.6832	−0.508531
$725$	−10.2006	−0.378841
$726$	0	0
$727$	−7.02528	−0.260553	−0.130277	−	0.991478i	$-0.541587\pi$
$727$	−7.02528	−0.260553	−0.130277	+	0.991478i	$0.541587\pi$
$728$	0	0
$729$	0	0
$730$	−8.23223	−0.304688
$731$	0.326126	0.0120622
$732$	0	0
$733$	21.2839	0.786140	0.393070	−	0.919509i	$-0.371413\pi$
$733$	21.2839	0.786140	0.393070	+	0.919509i	$0.371413\pi$
$734$	10.7233	0.395804
$735$	0	0
$736$	−18.9142	−0.697188
$737$	−59.8050	−2.20294
$738$	0	0
$739$	9.68852	0.356398	0.178199	−	0.983994i	$-0.442973\pi$
$739$	9.68852	0.356398	0.178199	+	0.983994i	$0.442973\pi$
$740$	0.601795	0.0221224
$741$	0	0
$742$	2.71956	0.0998382
$743$	44.8866	1.64673	0.823364	−	0.567513i	$-0.192095\pi$
$743$	44.8866	1.64673	0.823364	+	0.567513i	$0.192095\pi$
$744$	0	0
$745$	7.44717	0.272843
$746$	−33.3364	−1.22053
$747$	0	0
$748$	4.07503	0.148998
$749$	15.8709	0.579910
$750$	0	0
$751$	3.50781	0.128002	0.0640009	−	0.997950i	$-0.479614\pi$
$751$	3.50781	0.128002	0.0640009	+	0.997950i	$0.479614\pi$
$752$	−0.266073	−0.00970269
$753$	0	0
$754$	0	0
$755$	18.9412	0.689340
$756$	0	0
$757$	−22.0135	−0.800095	−0.400048	−	0.916494i	$-0.631006\pi$
$757$	−22.0135	−0.800095	−0.400048	+	0.916494i	$0.631006\pi$
$758$	−21.8149	−0.792352
$759$	0	0
$760$	7.53388	0.273283
$761$	10.9054	0.395321	0.197661	−	0.980271i	$-0.436666\pi$
$761$	10.9054	0.395321	0.197661	+	0.980271i	$0.436666\pi$
$762$	0	0
$763$	−12.3110	−0.445689
$764$	1.32055	0.0477757
$765$	0	0
$766$	−39.8732	−1.44068
$767$	0	0
$768$	0	0
$769$	29.5569	1.06585	0.532925	−	0.846163i	$-0.321093\pi$
$769$	29.5569	1.06585	0.532925	+	0.846163i	$0.321093\pi$
$770$	5.64408	0.203398
$771$	0	0
$772$	6.83423	0.245969
$773$	−10.0137	−0.360169	−0.180085	−	0.983651i	$-0.557637\pi$
$773$	−10.0137	−0.360169	−0.180085	+	0.983651i	$0.557637\pi$
$774$	0	0
$775$	1.02783	0.0369209
$776$	−25.9416	−0.931247
$777$	0	0
$778$	16.9673	0.608309
$779$	18.4911	0.662512
$780$	0	0
$781$	−4.94571	−0.176971
$782$	−7.99748	−0.285989
$783$	0	0
$784$	14.0402	0.501436
$785$	−20.1480	−0.719114
$786$	0	0
$787$	29.9935	1.06915	0.534576	−	0.845121i	$-0.320471\pi$
$787$	29.9935	1.06915	0.534576	+	0.845121i	$0.320471\pi$
$788$	−11.0605	−0.394014
$789$	0	0
$790$	−15.1801	−0.540084
$791$	12.7030	0.451666
$792$	0	0
$793$	0	0
$794$	−9.41665	−0.334185
$795$	0	0
$796$	−0.950435	−0.0336873
$797$	−28.8344	−1.02137	−0.510684	−	0.859768i	$-0.670608\pi$
$797$	−28.8344	−1.02137	−0.510684	+	0.859768i	$0.670608\pi$
$798$	0	0
$799$	0.144967	0.00512858
$800$	−3.46697	−0.122576
$801$	0	0
$802$	−16.5265	−0.583571
$803$	35.7124	1.26026
$804$	0	0
$805$	5.22207	0.184054
$806$	0	0
$807$	0	0
$808$	−14.7317	−0.518259
$809$	−37.0934	−1.30414	−0.652068	−	0.758161i	$-0.726098\pi$
$809$	−37.0934	−1.30414	−0.652068	+	0.758161i	$0.726098\pi$
$810$	0	0
$811$	−19.6091	−0.688568	−0.344284	−	0.938866i	$-0.611878\pi$
$811$	−19.6091	−0.688568	−0.344284	+	0.938866i	$0.611878\pi$
$812$	−6.25669	−0.219567
$813$	0	0
$814$	5.53762	0.194094
$815$	9.33184	0.326880
$816$	0	0
$817$	−0.634682	−0.0222047
$818$	2.25674	0.0789052
$819$	0	0
$820$	−4.84212	−0.169094
$821$	−2.89699	−0.101106	−0.0505528	−	0.998721i	$-0.516098\pi$
$821$	−2.89699	−0.101106	−0.0505528	+	0.998721i	$0.516098\pi$
$822$	0	0
$823$	−28.6695	−0.999356	−0.499678	−	0.866211i	$-0.666548\pi$
$823$	−28.6695	−0.999356	−0.499678	+	0.866211i	$0.666548\pi$
$824$	32.7598	1.14124
$825$	0	0
$826$	−9.28965	−0.323228
$827$	−30.8087	−1.07132	−0.535662	−	0.844432i	$-0.679938\pi$
$827$	−30.8087	−1.07132	−0.535662	+	0.844432i	$0.679938\pi$
$828$	0	0
$829$	2.42720	0.0843003	0.0421501	−	0.999111i	$-0.486579\pi$
$829$	2.42720	0.0843003	0.0421501	+	0.999111i	$0.486579\pi$
$830$	−4.39444	−0.152533
$831$	0	0
$832$	0	0
$833$	−7.64968	−0.265046
$834$	0	0
$835$	0.308104	0.0106624
$836$	−7.93052	−0.274283
$837$	0	0
$838$	30.5184	1.05424
$839$	4.26940	0.147396	0.0736981	−	0.997281i	$-0.476520\pi$
$839$	4.26940	0.147396	0.0736981	+	0.997281i	$0.476520\pi$
$840$	0	0
$841$	75.0526	2.58802
$842$	−14.7668	−0.508899
$843$	0	0
$844$	−3.17793	−0.109389
$845$	0	0
$846$	0	0
$847$	−13.9555	−0.479516
$848$	−5.62409	−0.193132
$849$	0	0
$850$	−1.46593	−0.0502811
$851$	5.12357	0.175634
$852$	0	0
$853$	−21.5257	−0.737025	−0.368513	−	0.929623i	$-0.620133\pi$
$853$	−21.5257	−0.737025	−0.368513	+	0.929623i	$0.620133\pi$
$854$	14.6923	0.502760
$855$	0	0
$856$	−51.0475	−1.74477
$857$	−42.2276	−1.44247	−0.721234	−	0.692691i	$-0.756425\pi$
$857$	−42.2276	−1.44247	−0.721234	+	0.692691i	$0.756425\pi$
$858$	0	0
$859$	35.9388	1.22622	0.613108	−	0.789999i	$-0.289919\pi$
$859$	35.9388	1.22622	0.613108	+	0.789999i	$0.289919\pi$
$860$	0.166199	0.00566735
$861$	0	0
$862$	−33.0484	−1.12563
$863$	−23.5593	−0.801970	−0.400985	−	0.916085i	$-0.631332\pi$
$863$	−23.5593	−0.801970	−0.400985	+	0.916085i	$0.631332\pi$
$864$	0	0
$865$	−9.83812	−0.334506
$866$	11.6455	0.395731
$867$	0	0
$868$	0.630436	0.0213984
$869$	65.8532	2.23392
$870$	0	0
$871$	0	0
$872$	39.5975	1.34094
$873$	0	0
$874$	15.5641	0.526464
$875$	0.957202	0.0323593
$876$	0	0
$877$	−15.0874	−0.509464	−0.254732	−	0.967012i	$-0.581987\pi$
$877$	−15.0874	−0.509464	−0.254732	+	0.967012i	$0.581987\pi$
$878$	−22.6085	−0.762999
$879$	0	0
$880$	−11.6720	−0.393464
$881$	−1.03462	−0.0348573	−0.0174287	−	0.999848i	$-0.505548\pi$
$881$	−1.03462	−0.0348573	−0.0174287	+	0.999848i	$0.505548\pi$
$882$	0	0
$883$	2.06340	0.0694389	0.0347194	−	0.999397i	$-0.488946\pi$
$883$	2.06340	0.0694389	0.0347194	+	0.999397i	$0.488946\pi$
$884$	0	0
$885$	0	0
$886$	6.79669	0.228339
$887$	53.8193	1.80707	0.903537	−	0.428511i	$-0.140962\pi$
$887$	53.8193	1.80707	0.903537	+	0.428511i	$0.140962\pi$
$888$	0	0
$889$	−12.8572	−0.431217
$890$	−20.8922	−0.700307
$891$	0	0
$892$	18.0534	0.604474
$893$	−0.282125	−0.00944095
$894$	0	0
$895$	−3.92634	−0.131243
$896$	3.02433	0.101036
$897$	0	0
$898$	31.7722	1.06025
$899$	−10.4845	−0.349679
$900$	0	0
$901$	3.06424	0.102085
$902$	−44.5564	−1.48357
$903$	0	0
$904$	−40.8582	−1.35892
$905$	−21.3537	−0.709820
$906$	0	0
$907$	26.4675	0.878839	0.439420	−	0.898282i	$-0.355184\pi$
$907$	26.4675	0.878839	0.439420	+	0.898282i	$0.355184\pi$
$908$	13.5602	0.450013
$909$	0	0
$910$	0	0
$911$	4.13621	0.137039	0.0685193	−	0.997650i	$-0.478173\pi$
$911$	4.13621	0.137039	0.0685193	+	0.997650i	$0.478173\pi$
$912$	0	0
$913$	19.0637	0.630915
$914$	27.6737	0.915365
$915$	0	0
$916$	15.5512	0.513825
$917$	7.74735	0.255840
$918$	0	0
$919$	4.30934	0.142152	0.0710761	−	0.997471i	$-0.477357\pi$
$919$	4.30934	0.142152	0.0710761	+	0.997471i	$0.477357\pi$
$920$	−16.7964	−0.553761
$921$	0	0
$922$	−20.1761	−0.664463
$923$	0	0
$924$	0	0
$925$	0.939148	0.0308790
$926$	−40.2659	−1.32322
$927$	0	0
$928$	35.3652	1.16092
$929$	−43.7707	−1.43607	−0.718035	−	0.696007i	$-0.754958\pi$
$929$	−43.7707	−1.43607	−0.718035	+	0.696007i	$0.754958\pi$
$930$	0	0
$931$	14.8872	0.487909
$932$	−15.7474	−0.515824
$933$	0	0
$934$	16.0418	0.524904
$935$	6.35940	0.207975
$936$	0	0
$937$	2.17755	0.0711374	0.0355687	−	0.999367i	$-0.488676\pi$
$937$	2.17755	0.0711374	0.0355687	+	0.999367i	$0.488676\pi$
$938$	13.1959	0.430861
$939$	0	0
$940$	0.0738779	0.00240963
$941$	18.6557	0.608158	0.304079	−	0.952647i	$-0.401651\pi$
$941$	18.6557	0.608158	0.304079	+	0.952647i	$0.401651\pi$
$942$	0	0
$943$	−41.2249	−1.34247
$944$	19.2111	0.625269
$945$	0	0
$946$	1.52934	0.0497231
$947$	0.281211	0.00913813	0.00456907	−	0.999990i	$-0.498546\pi$
$947$	0.281211	0.00913813	0.00456907	+	0.999990i	$0.498546\pi$
$948$	0	0
$949$	0	0
$950$	2.85289	0.0925600
$951$	0	0
$952$	−3.70554	−0.120097
$953$	−29.2264	−0.946737	−0.473368	−	0.880865i	$-0.656962\pi$
$953$	−29.2264	−0.946737	−0.473368	+	0.880865i	$0.656962\pi$
$954$	0	0
$955$	2.06081	0.0666864
$956$	−16.0664	−0.519625
$957$	0	0
$958$	2.70580	0.0874206
$959$	−16.7692	−0.541505
$960$	0	0
$961$	−29.9436	−0.965921
$962$	0	0
$963$	0	0
$964$	−1.14181	−0.0367751
$965$	10.6654	0.343330
$966$	0	0
$967$	−13.1700	−0.423519	−0.211760	−	0.977322i	$-0.567919\pi$
$967$	−13.1700	−0.423519	−0.211760	+	0.977322i	$0.567919\pi$
$968$	44.8868	1.44272
$969$	0	0
$970$	−9.82341	−0.315411
$971$	27.3988	0.879270	0.439635	−	0.898176i	$-0.355108\pi$
$971$	27.3988	0.879270	0.439635	+	0.898176i	$0.355108\pi$
$972$	0	0
$973$	8.46499	0.271375
$974$	17.1673	0.550075
$975$	0	0
$976$	−30.3839	−0.972565
$977$	28.1077	0.899245	0.449622	−	0.893219i	$-0.351559\pi$
$977$	28.1077	0.899245	0.449622	+	0.893219i	$0.351559\pi$
$978$	0	0
$979$	90.6329	2.89664
$980$	−3.89840	−0.124530
$981$	0	0
$982$	−45.8418	−1.46287
$983$	37.5892	1.19891	0.599455	−	0.800409i	$-0.295384\pi$
$983$	37.5892	1.19891	0.599455	+	0.800409i	$0.295384\pi$
$984$	0	0
$985$	−17.2608	−0.549974
$986$	14.9534	0.476214
$987$	0	0
$988$	0	0
$989$	1.41499	0.0449941
$990$	0	0
$991$	53.2906	1.69283	0.846416	−	0.532522i	$-0.178756\pi$
$991$	53.2906	1.69283	0.846416	+	0.532522i	$0.178756\pi$
$992$	−3.56347	−0.113140
$993$	0	0
$994$	1.09126	0.0346128
$995$	−1.48323	−0.0470215
$996$	0	0
$997$	−13.2842	−0.420715	−0.210357	−	0.977625i	$-0.567463\pi$
$997$	−13.2842	−0.420715	−0.210357	+	0.977625i	$0.567463\pi$
$998$	24.5799	0.778063
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7605.2.a.cq.1.3		9
3.2	odd	2		2535.2.a.bn.1.7	yes	9
13.12	even	2		7605.2.a.cr.1.7		9
39.38	odd	2		2535.2.a.bm.1.3	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2535.2.a.bm.1.3	✓	9	39.38	odd	2
2535.2.a.bn.1.7	yes	9	3.2	odd	2
7605.2.a.cq.1.3		9	1.1	even	1	trivial
7605.2.a.cr.1.7		9	13.12	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:	\( N \)	\(=\)	\( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7605.a (trivial)

\(f(q)\)	\(=\)	\(q-1.16585 q^{2} -0.640788 q^{4} -1.00000 q^{5} -0.957202 q^{7} +3.07877 q^{8} +O(q^{10})\)	\(q-1.16585 q^{2} -0.640788 q^{4} -1.00000 q^{5} -0.957202 q^{7} +3.07877 q^{8} +1.16585 q^{10} -5.05761 q^{11} +1.11596 q^{14} -2.30781 q^{16} +1.25739 q^{17} -2.44704 q^{19} +0.640788 q^{20} +5.89643 q^{22} +5.45555 q^{23} +1.00000 q^{25} +0.613364 q^{28} -10.2006 q^{29} +1.02783 q^{31} -3.46697 q^{32} -1.46593 q^{34} +0.957202 q^{35} +0.939148 q^{37} +2.85289 q^{38} -3.07877 q^{40} -7.55650 q^{41} +0.259367 q^{43} +3.24086 q^{44} -6.36037 q^{46} +0.115292 q^{47} -6.08376 q^{49} -1.16585 q^{50} +2.43698 q^{53} +5.05761 q^{55} -2.94700 q^{56} +11.8924 q^{58} -8.32438 q^{59} +13.1657 q^{61} -1.19830 q^{62} +8.65760 q^{64} +11.8247 q^{67} -0.805722 q^{68} -1.11596 q^{70} +0.977874 q^{71} -7.06112 q^{73} -1.09491 q^{74} +1.56804 q^{76} +4.84116 q^{77} -13.0206 q^{79} +2.30781 q^{80} +8.80977 q^{82} -3.76930 q^{83} -1.25739 q^{85} -0.302384 q^{86} -15.5712 q^{88} -17.9201 q^{89} -3.49585 q^{92} -0.134414 q^{94} +2.44704 q^{95} -8.42595 q^{97} +7.09277 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 10 q^{4} - 9 q^{5} - 10 q^{7} - 3 q^{8}+O(q^{10}) \) 9 * q + 10 * q^4 - 9 * q^5 - 10 * q^7 - 3 * q^8	\( 9 q + 10 q^{4} - 9 q^{5} - 10 q^{7} - 3 q^{8} - 11 q^{11} - 10 q^{14} + 8 q^{16} - 18 q^{17} + 10 q^{19} - 10 q^{20} + 17 q^{22} + 7 q^{23} + 9 q^{25} - 43 q^{28} + 9 q^{29} + 17 q^{31} - 11 q^{32} - 4 q^{34} + 10 q^{35} + 3 q^{37} - 21 q^{38} + 3 q^{40} - 4 q^{41} + 14 q^{43} - 13 q^{44} + 3 q^{46} + 2 q^{47} + 43 q^{49} - 5 q^{53} + 11 q^{55} - q^{56} - 14 q^{58} - 3 q^{59} + 48 q^{61} + 7 q^{62} + 41 q^{64} - 40 q^{67} - 40 q^{68} + 10 q^{70} - 12 q^{71} - 38 q^{73} + 4 q^{74} + 51 q^{76} - 8 q^{77} + 22 q^{79} - 8 q^{80} + 38 q^{82} - 11 q^{83} + 18 q^{85} - 56 q^{86} + 50 q^{88} - 2 q^{89} + 36 q^{92} + 42 q^{94} - 10 q^{95} - 23 q^{97} + 79 q^{98}+O(q^{100}) \) 9 * q + 10 * q^4 - 9 * q^5 - 10 * q^7 - 3 * q^8 - 11 * q^11 - 10 * q^14 + 8 * q^16 - 18 * q^17 + 10 * q^19 - 10 * q^20 + 17 * q^22 + 7 * q^23 + 9 * q^25 - 43 * q^28 + 9 * q^29 + 17 * q^31 - 11 * q^32 - 4 * q^34 + 10 * q^35 + 3 * q^37 - 21 * q^38 + 3 * q^40 - 4 * q^41 + 14 * q^43 - 13 * q^44 + 3 * q^46 + 2 * q^47 + 43 * q^49 - 5 * q^53 + 11 * q^55 - q^56 - 14 * q^58 - 3 * q^59 + 48 * q^61 + 7 * q^62 + 41 * q^64 - 40 * q^67 - 40 * q^68 + 10 * q^70 - 12 * q^71 - 38 * q^73 + 4 * q^74 + 51 * q^76 - 8 * q^77 + 22 * q^79 - 8 * q^80 + 38 * q^82 - 11 * q^83 + 18 * q^85 - 56 * q^86 + 50 * q^88 - 2 * q^89 + 36 * q^92 + 42 * q^94 - 10 * q^95 - 23 * q^97 + 79 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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