LMFDB - Embedded newform 8415.2.a.by.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8415,2,Mod(1,8415)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8415, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8415.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8415 = 3^{2} \cdot 5 \cdot 11 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8415.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:	$67.1941133009$
Analytic rank:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - x^{10} - 21 x^{9} + 20 x^{8} + 161 x^{7} - 148 x^{6} - 536 x^{5} + 481 x^{4} + 689 x^{3} + \cdots + 61 $ x^11 - x^10 - 21x^9 + 20x^8 + 161x^7 - 148x^6 - 536x^5 + 481x^4 + 689x^3 - 587x^2 - 156*x + 61
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 935)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$1.23394$ of defining polynomial
Character	$\chi$	$=$	8415.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.23394 q^{2} -0.477390 q^{4} -1.00000 q^{5} -2.41473 q^{7} -3.05695 q^{8} +O(q^{10})$	\(q+1.23394 q^{2} -0.477390 q^{4} -1.00000 q^{5} -2.41473 q^{7} -3.05695 q^{8} -1.23394 q^{10} +1.00000 q^{11} +5.99070 q^{13} -2.97963 q^{14} -2.81732 q^{16} +1.00000 q^{17} -3.44044 q^{19} +0.477390 q^{20} +1.23394 q^{22} -8.43743 q^{23} +1.00000 q^{25} +7.39218 q^{26} +1.15277 q^{28} -3.64537 q^{29} +8.98705 q^{31} +2.63750 q^{32} +1.23394 q^{34} +2.41473 q^{35} +4.91313 q^{37} -4.24530 q^{38} +3.05695 q^{40} -11.0117 q^{41} -8.45703 q^{43} -0.477390 q^{44} -10.4113 q^{46} -4.79052 q^{47} -1.16908 q^{49} +1.23394 q^{50} -2.85990 q^{52} -1.90119 q^{53} -1.00000 q^{55} +7.38172 q^{56} -4.49817 q^{58} +6.48062 q^{59} +1.96375 q^{61} +11.0895 q^{62} +8.88916 q^{64} -5.99070 q^{65} +9.36951 q^{67} -0.477390 q^{68} +2.97963 q^{70} -4.63594 q^{71} -6.01487 q^{73} +6.06251 q^{74} +1.64243 q^{76} -2.41473 q^{77} +9.30990 q^{79} +2.81732 q^{80} -13.5878 q^{82} +6.34804 q^{83} -1.00000 q^{85} -10.4355 q^{86} -3.05695 q^{88} -2.98945 q^{89} -14.4659 q^{91} +4.02794 q^{92} -5.91122 q^{94} +3.44044 q^{95} +14.4946 q^{97} -1.44257 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + q^{2} + 21 q^{4} - 11 q^{5} - 2 q^{7}+O(q^{10}) $ 11 * q + q^2 + 21 * q^4 - 11 * q^5 - 2 * q^7	$ 11 q + q^{2} + 21 q^{4} - 11 q^{5} - 2 q^{7} - q^{10} + 11 q^{11} + 13 q^{13} + 29 q^{16} + 11 q^{17} + 4 q^{19} - 21 q^{20} + q^{22} + 7 q^{23} + 11 q^{25} - 4 q^{26} + 8 q^{28} - 9 q^{29} + 20 q^{31} + 26 q^{32} + q^{34} + 2 q^{35} - q^{37} + 9 q^{41} + 5 q^{43} + 21 q^{44} - 10 q^{46} + 43 q^{49} + q^{50} + 18 q^{52} - 14 q^{53} - 11 q^{55} + 6 q^{56} - 10 q^{58} - 13 q^{59} + 11 q^{61} + 26 q^{62} + 24 q^{64} - 13 q^{65} + 26 q^{67} + 21 q^{68} + 8 q^{71} + 42 q^{73} - 16 q^{74} - 70 q^{76} - 2 q^{77} - 27 q^{79} - 29 q^{80} + 57 q^{82} + 41 q^{83} - 11 q^{85} - 47 q^{86} - 37 q^{89} + 30 q^{91} + 47 q^{92} + 94 q^{94} - 4 q^{95} + 29 q^{97} + 69 q^{98}+O(q^{100}) $ 11 * q + q^2 + 21 * q^4 - 11 * q^5 - 2 * q^7 - q^10 + 11 * q^11 + 13 * q^13 + 29 * q^16 + 11 * q^17 + 4 * q^19 - 21 * q^20 + q^22 + 7 * q^23 + 11 * q^25 - 4 * q^26 + 8 * q^28 - 9 * q^29 + 20 * q^31 + 26 * q^32 + q^34 + 2 * q^35 - q^37 + 9 * q^41 + 5 * q^43 + 21 * q^44 - 10 * q^46 + 43 * q^49 + q^50 + 18 * q^52 - 14 * q^53 - 11 * q^55 + 6 * q^56 - 10 * q^58 - 13 * q^59 + 11 * q^61 + 26 * q^62 + 24 * q^64 - 13 * q^65 + 26 * q^67 + 21 * q^68 + 8 * q^71 + 42 * q^73 - 16 * q^74 - 70 * q^76 - 2 * q^77 - 27 * q^79 - 29 * q^80 + 57 * q^82 + 41 * q^83 - 11 * q^85 - 47 * q^86 - 37 * q^89 + 30 * q^91 + 47 * q^92 + 94 * q^94 - 4 * q^95 + 29 * q^97 + 69 * 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 $

$2$	1.23394	0.872528	0.436264	−	0.899819i	$-0.356301\pi$
$2$	1.23394	0.872528	0.436264	+	0.899819i	$0.356301\pi$
$3$	0	0
$3$	0	0
$4$	−0.477390	−0.238695
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.41473	−0.912682	−0.456341	−	0.889805i	$-0.650840\pi$
$7$	−2.41473	−0.912682	−0.456341	+	0.889805i	$0.650840\pi$
$8$	−3.05695	−1.08080
$9$	0	0
$10$	−1.23394	−0.390206
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	5.99070	1.66152	0.830761	−	0.556629i	$-0.187905\pi$
$13$	5.99070	1.66152	0.830761	+	0.556629i	$0.187905\pi$
$14$	−2.97963	−0.796341
$15$	0	0
$16$	−2.81732	−0.704330
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−3.44044	−0.789291	−0.394645	−	0.918834i	$-0.629133\pi$
$19$	−3.44044	−0.789291	−0.394645	+	0.918834i	$0.629133\pi$
$20$	0.477390	0.106748
$21$	0	0
$22$	1.23394	0.263077
$23$	−8.43743	−1.75933	−0.879663	−	0.475597i	$-0.842232\pi$
$23$	−8.43743	−1.75933	−0.879663	+	0.475597i	$0.842232\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	7.39218	1.44972
$27$	0	0
$28$	1.15277	0.217853
$29$	−3.64537	−0.676928	−0.338464	−	0.940979i	$-0.609907\pi$
$29$	−3.64537	−0.676928	−0.338464	+	0.940979i	$0.609907\pi$
$30$	0	0
$31$	8.98705	1.61412	0.807061	−	0.590468i	$-0.201057\pi$
$31$	8.98705	1.61412	0.807061	+	0.590468i	$0.201057\pi$
$32$	2.63750	0.466248
$33$	0	0
$34$	1.23394	0.211619
$35$	2.41473	0.408164
$36$	0	0
$37$	4.91313	0.807713	0.403856	−	0.914822i	$-0.367670\pi$
$37$	4.91313	0.807713	0.403856	+	0.914822i	$0.367670\pi$
$38$	−4.24530	−0.688678
$39$	0	0
$40$	3.05695	0.483347
$41$	−11.0117	−1.71974	−0.859868	−	0.510517i	$-0.829454\pi$
$41$	−11.0117	−1.71974	−0.859868	+	0.510517i	$0.829454\pi$
$42$	0	0
$43$	−8.45703	−1.28969	−0.644843	−	0.764315i	$-0.723077\pi$
$43$	−8.45703	−1.28969	−0.644843	+	0.764315i	$0.723077\pi$
$44$	−0.477390	−0.0719692
$45$	0	0
$46$	−10.4113	−1.53506
$47$	−4.79052	−0.698769	−0.349385	−	0.936979i	$-0.613609\pi$
$47$	−4.79052	−0.698769	−0.349385	+	0.936979i	$0.613609\pi$
$48$	0	0
$49$	−1.16908	−0.167011
$50$	1.23394	0.174506
$51$	0	0
$52$	−2.85990	−0.396597
$53$	−1.90119	−0.261148	−0.130574	−	0.991439i	$-0.541682\pi$
$53$	−1.90119	−0.261148	−0.130574	+	0.991439i	$0.541682\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	7.38172	0.986423
$57$	0	0
$58$	−4.49817	−0.590638
$59$	6.48062	0.843705	0.421853	−	0.906664i	$-0.361380\pi$
$59$	6.48062	0.843705	0.421853	+	0.906664i	$0.361380\pi$
$60$	0	0
$61$	1.96375	0.251432	0.125716	−	0.992066i	$-0.459877\pi$
$61$	1.96375	0.251432	0.125716	+	0.992066i	$0.459877\pi$
$62$	11.0895	1.40837
$63$	0	0
$64$	8.88916	1.11114
$65$	−5.99070	−0.743055
$66$	0	0
$67$	9.36951	1.14467	0.572334	−	0.820021i	$-0.306038\pi$
$67$	9.36951	1.14467	0.572334	+	0.820021i	$0.306038\pi$
$68$	−0.477390	−0.0578920
$69$	0	0
$70$	2.97963	0.356134
$71$	−4.63594	−0.550185	−0.275092	−	0.961418i	$-0.588708\pi$
$71$	−4.63594	−0.550185	−0.275092	+	0.961418i	$0.588708\pi$
$72$	0	0
$73$	−6.01487	−0.703987	−0.351994	−	0.936002i	$-0.614496\pi$
$73$	−6.01487	−0.703987	−0.351994	+	0.936002i	$0.614496\pi$
$74$	6.06251	0.704752
$75$	0	0
$76$	1.64243	0.188400
$77$	−2.41473	−0.275184
$78$	0	0
$79$	9.30990	1.04745	0.523723	−	0.851889i	$-0.324543\pi$
$79$	9.30990	1.04745	0.523723	+	0.851889i	$0.324543\pi$
$80$	2.81732	0.314986
$81$	0	0
$82$	−13.5878	−1.50052
$83$	6.34804	0.696788	0.348394	−	0.937348i	$-0.386727\pi$
$83$	6.34804	0.696788	0.348394	+	0.937348i	$0.386727\pi$
$84$	0	0
$85$	−1.00000	−0.108465
$86$	−10.4355	−1.12529
$87$	0	0
$88$	−3.05695	−0.325872
$89$	−2.98945	−0.316881	−0.158441	−	0.987368i	$-0.550647\pi$
$89$	−2.98945	−0.316881	−0.158441	+	0.987368i	$0.550647\pi$
$90$	0	0
$91$	−14.4659	−1.51644
$92$	4.02794	0.419942
$93$	0	0
$94$	−5.91122	−0.609696
$95$	3.44044	0.352982
$96$	0	0
$97$	14.4946	1.47171	0.735854	−	0.677140i	$-0.236781\pi$
$97$	14.4946	1.47171	0.735854	+	0.677140i	$0.236781\pi$
$98$	−1.44257	−0.145722
$99$	0	0
$100$	−0.477390	−0.0477390
$101$	−10.3233	−1.02721	−0.513605	−	0.858027i	$-0.671690\pi$
$101$	−10.3233	−1.02721	−0.513605	+	0.858027i	$0.671690\pi$
$102$	0	0
$103$	3.70840	0.365400	0.182700	−	0.983169i	$-0.441516\pi$
$103$	3.70840	0.365400	0.182700	+	0.983169i	$0.441516\pi$
$104$	−18.3133	−1.79577
$105$	0	0
$106$	−2.34595	−0.227859
$107$	20.2079	1.95358	0.976788	−	0.214209i	$-0.0687173\pi$
$107$	20.2079	1.95358	0.976788	+	0.214209i	$0.0687173\pi$
$108$	0	0
$109$	6.50276	0.622852	0.311426	−	0.950270i	$-0.399193\pi$
$109$	6.50276	0.622852	0.311426	+	0.950270i	$0.399193\pi$
$110$	−1.23394	−0.117652
$111$	0	0
$112$	6.80307	0.642829
$113$	2.73362	0.257157	0.128579	−	0.991699i	$-0.458959\pi$
$113$	2.73362	0.257157	0.128579	+	0.991699i	$0.458959\pi$
$114$	0	0
$115$	8.43743	0.786795
$116$	1.74026	0.161579
$117$	0	0
$118$	7.99671	0.736156
$119$	−2.41473	−0.221358
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.42315	0.219381
$123$	0	0
$124$	−4.29033	−0.385283
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	3.02996	0.268866	0.134433	−	0.990923i	$-0.457079\pi$
$127$	3.02996	0.268866	0.134433	+	0.990923i	$0.457079\pi$
$128$	5.69370	0.503256
$129$	0	0
$130$	−7.39218	−0.648337
$131$	4.39116	0.383658	0.191829	−	0.981428i	$-0.438558\pi$
$131$	4.39116	0.383658	0.191829	+	0.981428i	$0.438558\pi$
$132$	0	0
$133$	8.30773	0.720372
$134$	11.5614	0.998755
$135$	0	0
$136$	−3.05695	−0.262132
$137$	−19.9818	−1.70716	−0.853580	−	0.520961i	$-0.825574\pi$
$137$	−19.9818	−1.70716	−0.853580	+	0.520961i	$0.825574\pi$
$138$	0	0
$139$	17.9238	1.52028	0.760139	−	0.649761i	$-0.225131\pi$
$139$	17.9238	1.52028	0.760139	+	0.649761i	$0.225131\pi$
$140$	−1.15277	−0.0974266
$141$	0	0
$142$	−5.72048	−0.480052
$143$	5.99070	0.500968
$144$	0	0
$145$	3.64537	0.302731
$146$	−7.42199	−0.614248
$147$	0	0
$148$	−2.34548	−0.192797
$149$	20.5661	1.68484	0.842422	−	0.538819i	$-0.181129\pi$
$149$	20.5661	1.68484	0.842422	+	0.538819i	$0.181129\pi$
$150$	0	0
$151$	−13.3256	−1.08442	−0.542210	−	0.840243i	$-0.682412\pi$
$151$	−13.3256	−1.08442	−0.542210	+	0.840243i	$0.682412\pi$
$152$	10.5173	0.853062
$153$	0	0
$154$	−2.97963	−0.240106
$155$	−8.98705	−0.721858
$156$	0	0
$157$	−3.51594	−0.280603	−0.140301	−	0.990109i	$-0.544807\pi$
$157$	−3.51594	−0.280603	−0.140301	+	0.990109i	$0.544807\pi$
$158$	11.4879	0.913925
$159$	0	0
$160$	−2.63750	−0.208513
$161$	20.3741	1.60571
$162$	0	0
$163$	−16.0177	−1.25460	−0.627301	−	0.778777i	$-0.715840\pi$
$163$	−16.0177	−1.25460	−0.627301	+	0.778777i	$0.715840\pi$
$164$	5.25686	0.410492
$165$	0	0
$166$	7.83310	0.607967
$167$	−2.84725	−0.220327	−0.110163	−	0.993913i	$-0.535137\pi$
$167$	−2.84725	−0.220327	−0.110163	+	0.993913i	$0.535137\pi$
$168$	0	0
$169$	22.8885	1.76066
$170$	−1.23394	−0.0946390
$171$	0	0
$172$	4.03730	0.307841
$173$	19.0302	1.44684	0.723418	−	0.690410i	$-0.242570\pi$
$173$	19.0302	1.44684	0.723418	+	0.690410i	$0.242570\pi$
$174$	0	0
$175$	−2.41473	−0.182536
$176$	−2.81732	−0.212363
$177$	0	0
$178$	−3.68881	−0.276488
$179$	−1.32898	−0.0993324	−0.0496662	−	0.998766i	$-0.515816\pi$
$179$	−1.32898	−0.0993324	−0.0496662	+	0.998766i	$0.515816\pi$
$180$	0	0
$181$	15.4937	1.15164	0.575820	−	0.817577i	$-0.304683\pi$
$181$	15.4937	1.15164	0.575820	+	0.817577i	$0.304683\pi$
$182$	−17.8501	−1.32314
$183$	0	0
$184$	25.7928	1.90147
$185$	−4.91313	−0.361220
$186$	0	0
$187$	1.00000	0.0731272
$188$	2.28695	0.166793
$189$	0	0
$190$	4.24530	0.307986
$191$	15.7278	1.13803	0.569013	−	0.822329i	$-0.307326\pi$
$191$	15.7278	1.13803	0.569013	+	0.822329i	$0.307326\pi$
$192$	0	0
$193$	7.31329	0.526422	0.263211	−	0.964738i	$-0.415218\pi$
$193$	7.31329	0.526422	0.263211	+	0.964738i	$0.415218\pi$
$194$	17.8855	1.28411
$195$	0	0
$196$	0.558106	0.0398647
$197$	21.0661	1.50090	0.750449	−	0.660929i	$-0.229838\pi$
$197$	21.0661	1.50090	0.750449	+	0.660929i	$0.229838\pi$
$198$	0	0
$199$	2.78470	0.197402	0.0987011	−	0.995117i	$-0.468531\pi$
$199$	2.78470	0.197402	0.0987011	+	0.995117i	$0.468531\pi$
$200$	−3.05695	−0.216159
$201$	0	0
$202$	−12.7384	−0.896270
$203$	8.80258	0.617820
$204$	0	0
$205$	11.0117	0.769089
$206$	4.57595	0.318821
$207$	0	0
$208$	−16.8777	−1.17026
$209$	−3.44044	−0.237980
$210$	0	0
$211$	18.5625	1.27790	0.638949	−	0.769249i	$-0.279369\pi$
$211$	18.5625	1.27790	0.638949	+	0.769249i	$0.279369\pi$
$212$	0.907608	0.0623348
$213$	0	0
$214$	24.9354	1.70455
$215$	8.45703	0.576765
$216$	0	0
$217$	−21.7013	−1.47318
$218$	8.02403	0.543456
$219$	0	0
$220$	0.477390	0.0321856
$221$	5.99070	0.402978
$222$	0	0
$223$	−17.1138	−1.14603	−0.573013	−	0.819546i	$-0.694226\pi$
$223$	−17.1138	−1.14603	−0.573013	+	0.819546i	$0.694226\pi$
$224$	−6.36885	−0.425537
$225$	0	0
$226$	3.37312	0.224377
$227$	29.8924	1.98403	0.992013	−	0.126136i	$-0.0402577\pi$
$227$	29.8924	1.98403	0.992013	+	0.126136i	$0.0402577\pi$
$228$	0	0
$229$	−2.00284	−0.132351	−0.0661757	−	0.997808i	$-0.521080\pi$
$229$	−2.00284	−0.132351	−0.0661757	+	0.997808i	$0.521080\pi$
$230$	10.4113	0.686500
$231$	0	0
$232$	11.1437	0.731621
$233$	12.0095	0.786767	0.393383	−	0.919375i	$-0.371305\pi$
$233$	12.0095	0.786767	0.393383	+	0.919375i	$0.371305\pi$
$234$	0	0
$235$	4.79052	0.312499
$236$	−3.09378	−0.201388
$237$	0	0
$238$	−2.97963	−0.193141
$239$	−1.75736	−0.113674	−0.0568370	−	0.998383i	$-0.518102\pi$
$239$	−1.75736	−0.113674	−0.0568370	+	0.998383i	$0.518102\pi$
$240$	0	0
$241$	−18.9940	−1.22351	−0.611754	−	0.791048i	$-0.709536\pi$
$241$	−18.9940	−1.22351	−0.611754	+	0.791048i	$0.709536\pi$
$242$	1.23394	0.0793207
$243$	0	0
$244$	−0.937472	−0.0600155
$245$	1.16908	0.0746897
$246$	0	0
$247$	−20.6106	−1.31142
$248$	−27.4730	−1.74454
$249$	0	0
$250$	−1.23394	−0.0780413
$251$	1.48343	0.0936334	0.0468167	−	0.998903i	$-0.485092\pi$
$251$	1.48343	0.0936334	0.0468167	+	0.998903i	$0.485092\pi$
$252$	0	0
$253$	−8.43743	−0.530457
$254$	3.73880	0.234593
$255$	0	0
$256$	−10.7526	−0.672039
$257$	−2.58897	−0.161496	−0.0807478	−	0.996735i	$-0.525731\pi$
$257$	−2.58897	−0.161496	−0.0807478	+	0.996735i	$0.525731\pi$
$258$	0	0
$259$	−11.8639	−0.737185
$260$	2.85990	0.177364
$261$	0	0
$262$	5.41844	0.334752
$263$	−3.93842	−0.242853	−0.121427	−	0.992600i	$-0.538747\pi$
$263$	−3.93842	−0.242853	−0.121427	+	0.992600i	$0.538747\pi$
$264$	0	0
$265$	1.90119	0.116789
$266$	10.2512	0.628544
$267$	0	0
$268$	−4.47291	−0.273226
$269$	13.6803	0.834102	0.417051	−	0.908883i	$-0.363064\pi$
$269$	13.6803	0.834102	0.417051	+	0.908883i	$0.363064\pi$
$270$	0	0
$271$	−4.76224	−0.289285	−0.144643	−	0.989484i	$-0.546203\pi$
$271$	−4.76224	−0.289285	−0.144643	+	0.989484i	$0.546203\pi$
$272$	−2.81732	−0.170825
$273$	0	0
$274$	−24.6564	−1.48955
$275$	1.00000	0.0603023
$276$	0	0
$277$	−11.6893	−0.702344	−0.351172	−	0.936311i	$-0.614217\pi$
$277$	−11.6893	−0.702344	−0.351172	+	0.936311i	$0.614217\pi$
$278$	22.1169	1.32648
$279$	0	0
$280$	−7.38172	−0.441142
$281$	9.05001	0.539878	0.269939	−	0.962877i	$-0.412996\pi$
$281$	9.05001	0.539878	0.269939	+	0.962877i	$0.412996\pi$
$282$	0	0
$283$	−25.9889	−1.54488	−0.772441	−	0.635087i	$-0.780964\pi$
$283$	−25.9889	−1.54488	−0.772441	+	0.635087i	$0.780964\pi$
$284$	2.21315	0.131326
$285$	0	0
$286$	7.39218	0.437109
$287$	26.5902	1.56957
$288$	0	0
$289$	1.00000	0.0588235
$290$	4.49817	0.264142
$291$	0	0
$292$	2.87144	0.168038
$293$	12.6078	0.736556	0.368278	−	0.929716i	$-0.379947\pi$
$293$	12.6078	0.736556	0.368278	+	0.929716i	$0.379947\pi$
$294$	0	0
$295$	−6.48062	−0.377316
$296$	−15.0192	−0.872973
$297$	0	0
$298$	25.3774	1.47007
$299$	−50.5462	−2.92316
$300$	0	0
$301$	20.4215	1.17707
$302$	−16.4430	−0.946187
$303$	0	0
$304$	9.69281	0.555921
$305$	−1.96375	−0.112444
$306$	0	0
$307$	6.74829	0.385145	0.192573	−	0.981283i	$-0.438317\pi$
$307$	6.74829	0.385145	0.192573	+	0.981283i	$0.438317\pi$
$308$	1.15277	0.0656850
$309$	0	0
$310$	−11.0895	−0.629841
$311$	−5.47406	−0.310406	−0.155203	−	0.987883i	$-0.549603\pi$
$311$	−5.47406	−0.310406	−0.155203	+	0.987883i	$0.549603\pi$
$312$	0	0
$313$	−2.34332	−0.132452	−0.0662262	−	0.997805i	$-0.521096\pi$
$313$	−2.34332	−0.132452	−0.0662262	+	0.997805i	$0.521096\pi$
$314$	−4.33847	−0.244834
$315$	0	0
$316$	−4.44445	−0.250020
$317$	4.16233	0.233780	0.116890	−	0.993145i	$-0.462708\pi$
$317$	4.16233	0.233780	0.116890	+	0.993145i	$0.462708\pi$
$318$	0	0
$319$	−3.64537	−0.204101
$320$	−8.88916	−0.496919
$321$	0	0
$322$	25.1405	1.40102
$323$	−3.44044	−0.191431
$324$	0	0
$325$	5.99070	0.332305
$326$	−19.7649	−1.09468
$327$	0	0
$328$	33.6622	1.85868
$329$	11.5678	0.637754
$330$	0	0
$331$	14.2161	0.781389	0.390695	−	0.920520i	$-0.372235\pi$
$331$	14.2161	0.781389	0.390695	+	0.920520i	$0.372235\pi$
$332$	−3.03049	−0.166320
$333$	0	0
$334$	−3.51334	−0.192241
$335$	−9.36951	−0.511911
$336$	0	0
$337$	14.2128	0.774221	0.387110	−	0.922033i	$-0.373473\pi$
$337$	14.2128	0.774221	0.387110	+	0.922033i	$0.373473\pi$
$338$	28.2431	1.53622
$339$	0	0
$340$	0.477390	0.0258901
$341$	8.98705	0.486676
$342$	0	0
$343$	19.7261	1.06511
$344$	25.8527	1.39389
$345$	0	0
$346$	23.4821	1.26241
$347$	−1.24538	−0.0668553	−0.0334277	−	0.999441i	$-0.510642\pi$
$347$	−1.24538	−0.0668553	−0.0334277	+	0.999441i	$0.510642\pi$
$348$	0	0
$349$	17.4562	0.934411	0.467205	−	0.884149i	$-0.345261\pi$
$349$	17.4562	0.934411	0.467205	+	0.884149i	$0.345261\pi$
$350$	−2.97963	−0.159268
$351$	0	0
$352$	2.63750	0.140579
$353$	27.7611	1.47757	0.738787	−	0.673939i	$-0.235399\pi$
$353$	27.7611	1.47757	0.738787	+	0.673939i	$0.235399\pi$
$354$	0	0
$355$	4.63594	0.246050
$356$	1.42713	0.0756380
$357$	0	0
$358$	−1.63988	−0.0866703
$359$	−9.11940	−0.481303	−0.240652	−	0.970612i	$-0.577361\pi$
$359$	−9.11940	−0.481303	−0.240652	+	0.970612i	$0.577361\pi$
$360$	0	0
$361$	−7.16339	−0.377020
$362$	19.1183	1.00484
$363$	0	0
$364$	6.90589	0.361967
$365$	6.01487	0.314833
$366$	0	0
$367$	9.59212	0.500704	0.250352	−	0.968155i	$-0.419454\pi$
$367$	9.59212	0.500704	0.250352	+	0.968155i	$0.419454\pi$
$368$	23.7709	1.23915
$369$	0	0
$370$	−6.06251	−0.315175
$371$	4.59086	0.238345
$372$	0	0
$373$	18.4660	0.956134	0.478067	−	0.878323i	$-0.341338\pi$
$373$	18.4660	0.956134	0.478067	+	0.878323i	$0.341338\pi$
$374$	1.23394	0.0638056
$375$	0	0
$376$	14.6444	0.755227
$377$	−21.8383	−1.12473
$378$	0	0
$379$	−17.2393	−0.885525	−0.442762	−	0.896639i	$-0.646002\pi$
$379$	−17.2393	−0.885525	−0.442762	+	0.896639i	$0.646002\pi$
$380$	−1.64243	−0.0842549
$381$	0	0
$382$	19.4072	0.992959
$383$	3.53718	0.180742	0.0903708	−	0.995908i	$-0.471195\pi$
$383$	3.53718	0.180742	0.0903708	+	0.995908i	$0.471195\pi$
$384$	0	0
$385$	2.41473	0.123066
$386$	9.02417	0.459318
$387$	0	0
$388$	−6.91960	−0.351289
$389$	−14.6780	−0.744202	−0.372101	−	0.928192i	$-0.621363\pi$
$389$	−14.6780	−0.744202	−0.372101	+	0.928192i	$0.621363\pi$
$390$	0	0
$391$	−8.43743	−0.426699
$392$	3.57382	0.180505
$393$	0	0
$394$	25.9943	1.30957
$395$	−9.30990	−0.468432
$396$	0	0
$397$	22.1812	1.11324	0.556621	−	0.830767i	$-0.312098\pi$
$397$	22.1812	1.11324	0.556621	+	0.830767i	$0.312098\pi$
$398$	3.43616	0.172239
$399$	0	0
$400$	−2.81732	−0.140866
$401$	15.9486	0.796436	0.398218	−	0.917291i	$-0.369629\pi$
$401$	15.9486	0.796436	0.398218	+	0.917291i	$0.369629\pi$
$402$	0	0
$403$	53.8388	2.68190
$404$	4.92826	0.245190
$405$	0	0
$406$	10.8619	0.539065
$407$	4.91313	0.243535
$408$	0	0
$409$	6.30144	0.311586	0.155793	−	0.987790i	$-0.450207\pi$
$409$	6.30144	0.311586	0.155793	+	0.987790i	$0.450207\pi$
$410$	13.5878	0.671052
$411$	0	0
$412$	−1.77035	−0.0872190
$413$	−15.6490	−0.770035
$414$	0	0
$415$	−6.34804	−0.311613
$416$	15.8005	0.774682
$417$	0	0
$418$	−4.24530	−0.207644
$419$	−5.74293	−0.280561	−0.140280	−	0.990112i	$-0.544800\pi$
$419$	−5.74293	−0.280561	−0.140280	+	0.990112i	$0.544800\pi$
$420$	0	0
$421$	35.3133	1.72106	0.860532	−	0.509397i	$-0.170132\pi$
$421$	35.3133	1.72106	0.860532	+	0.509397i	$0.170132\pi$
$422$	22.9051	1.11500
$423$	0	0
$424$	5.81184	0.282248
$425$	1.00000	0.0485071
$426$	0	0
$427$	−4.74192	−0.229477
$428$	−9.64707	−0.466309
$429$	0	0
$430$	10.4355	0.503243
$431$	−23.1663	−1.11588	−0.557940	−	0.829881i	$-0.688408\pi$
$431$	−23.1663	−1.11588	−0.557940	+	0.829881i	$0.688408\pi$
$432$	0	0
$433$	12.8510	0.617581	0.308790	−	0.951130i	$-0.400076\pi$
$433$	12.8510	0.617581	0.308790	+	0.951130i	$0.400076\pi$
$434$	−26.7781	−1.28539
$435$	0	0
$436$	−3.10435	−0.148672
$437$	29.0285	1.38862
$438$	0	0
$439$	−25.3359	−1.20922	−0.604609	−	0.796522i	$-0.706671\pi$
$439$	−25.3359	−1.20922	−0.604609	+	0.796522i	$0.706671\pi$
$440$	3.05695	0.145734
$441$	0	0
$442$	7.39218	0.351610
$443$	31.3720	1.49053	0.745264	−	0.666770i	$-0.232323\pi$
$443$	31.3720	1.49053	0.745264	+	0.666770i	$0.232323\pi$
$444$	0	0
$445$	2.98945	0.141714
$446$	−21.1175	−0.999940
$447$	0	0
$448$	−21.4649	−1.01412
$449$	3.26014	0.153855	0.0769277	−	0.997037i	$-0.475489\pi$
$449$	3.26014	0.153855	0.0769277	+	0.997037i	$0.475489\pi$
$450$	0	0
$451$	−11.0117	−0.518520
$452$	−1.30500	−0.0613821
$453$	0	0
$454$	36.8854	1.73112
$455$	14.4659	0.678174
$456$	0	0
$457$	26.9473	1.26054	0.630271	−	0.776375i	$-0.282944\pi$
$457$	26.9473	1.26054	0.630271	+	0.776375i	$0.282944\pi$
$458$	−2.47139	−0.115480
$459$	0	0
$460$	−4.02794	−0.187804
$461$	−32.8151	−1.52835	−0.764175	−	0.645009i	$-0.776853\pi$
$461$	−32.8151	−1.52835	−0.764175	+	0.645009i	$0.776853\pi$
$462$	0	0
$463$	−22.2453	−1.03383	−0.516913	−	0.856038i	$-0.672919\pi$
$463$	−22.2453	−1.03383	−0.516913	+	0.856038i	$0.672919\pi$
$464$	10.2702	0.476780
$465$	0	0
$466$	14.8190	0.686476
$467$	−40.0742	−1.85441	−0.927207	−	0.374548i	$-0.877798\pi$
$467$	−40.0742	−1.85441	−0.927207	+	0.374548i	$0.877798\pi$
$468$	0	0
$469$	−22.6248	−1.04472
$470$	5.91122	0.272664
$471$	0	0
$472$	−19.8110	−0.911873
$473$	−8.45703	−0.388855
$474$	0	0
$475$	−3.44044	−0.157858
$476$	1.15277	0.0528370
$477$	0	0
$478$	−2.16848	−0.0991837
$479$	14.2030	0.648950	0.324475	−	0.945894i	$-0.394812\pi$
$479$	14.2030	0.648950	0.324475	+	0.945894i	$0.394812\pi$
$480$	0	0
$481$	29.4331	1.34203
$482$	−23.4374	−1.06755
$483$	0	0
$484$	−0.477390	−0.0216995
$485$	−14.4946	−0.658168
$486$	0	0
$487$	7.58673	0.343787	0.171894	−	0.985115i	$-0.445011\pi$
$487$	7.58673	0.343787	0.171894	+	0.985115i	$0.445011\pi$
$488$	−6.00308	−0.271747
$489$	0	0
$490$	1.44257	0.0651688
$491$	−21.4850	−0.969605	−0.484803	−	0.874624i	$-0.661109\pi$
$491$	−21.4850	−0.969605	−0.484803	+	0.874624i	$0.661109\pi$
$492$	0	0
$493$	−3.64537	−0.164179
$494$	−25.4323	−1.14425
$495$	0	0
$496$	−25.3194	−1.13687
$497$	11.1945	0.502144
$498$	0	0
$499$	−30.7518	−1.37664	−0.688321	−	0.725406i	$-0.741652\pi$
$499$	−30.7518	−1.37664	−0.688321	+	0.725406i	$0.741652\pi$
$500$	0.477390	0.0213495
$501$	0	0
$502$	1.83047	0.0816978
$503$	−25.1135	−1.11976	−0.559879	−	0.828575i	$-0.689152\pi$
$503$	−25.1135	−1.11976	−0.559879	+	0.828575i	$0.689152\pi$
$504$	0	0
$505$	10.3233	0.459383
$506$	−10.4113	−0.462838
$507$	0	0
$508$	−1.44647	−0.0641769
$509$	6.84134	0.303237	0.151619	−	0.988439i	$-0.451551\pi$
$509$	6.84134	0.303237	0.151619	+	0.988439i	$0.451551\pi$
$510$	0	0
$511$	14.5243	0.642516
$512$	−24.6555	−1.08963
$513$	0	0
$514$	−3.19464	−0.140909
$515$	−3.70840	−0.163412
$516$	0	0
$517$	−4.79052	−0.210687
$518$	−14.6393	−0.643215
$519$	0	0
$520$	18.3133	0.803091
$521$	34.5798	1.51497	0.757485	−	0.652852i	$-0.226428\pi$
$521$	34.5798	1.51497	0.757485	+	0.652852i	$0.226428\pi$
$522$	0	0
$523$	−4.62535	−0.202253	−0.101126	−	0.994874i	$-0.532245\pi$
$523$	−4.62535	−0.202253	−0.101126	+	0.994874i	$0.532245\pi$
$524$	−2.09630	−0.0915771
$525$	0	0
$526$	−4.85977	−0.211896
$527$	8.98705	0.391482
$528$	0	0
$529$	48.1903	2.09523
$530$	2.34595	0.101902
$531$	0	0
$532$	−3.96602	−0.171949
$533$	−65.9677	−2.85738
$534$	0	0
$535$	−20.2079	−0.873666
$536$	−28.6421	−1.23715
$537$	0	0
$538$	16.8807	0.727777
$539$	−1.16908	−0.0503558
$540$	0	0
$541$	33.1907	1.42698	0.713491	−	0.700665i	$-0.247113\pi$
$541$	33.1907	1.42698	0.713491	+	0.700665i	$0.247113\pi$
$542$	−5.87632	−0.252409
$543$	0	0
$544$	2.63750	0.113082
$545$	−6.50276	−0.278548
$546$	0	0
$547$	40.9416	1.75054	0.875268	−	0.483638i	$-0.160685\pi$
$547$	40.9416	1.75054	0.875268	+	0.483638i	$0.160685\pi$
$548$	9.53911	0.407490
$549$	0	0
$550$	1.23394	0.0526154
$551$	12.5417	0.534293
$552$	0	0
$553$	−22.4809	−0.955985
$554$	−14.4240	−0.612815
$555$	0	0
$556$	−8.55664	−0.362882
$557$	3.99055	0.169085	0.0845426	−	0.996420i	$-0.473057\pi$
$557$	3.99055	0.169085	0.0845426	+	0.996420i	$0.473057\pi$
$558$	0	0
$559$	−50.6636	−2.14284
$560$	−6.80307	−0.287482
$561$	0	0
$562$	11.1672	0.471059
$563$	−35.5510	−1.49829	−0.749147	−	0.662404i	$-0.769536\pi$
$563$	−35.5510	−1.49829	−0.749147	+	0.662404i	$0.769536\pi$
$564$	0	0
$565$	−2.73362	−0.115004
$566$	−32.0688	−1.34795
$567$	0	0
$568$	14.1718	0.594638
$569$	−32.8016	−1.37511	−0.687557	−	0.726131i	$-0.741317\pi$
$569$	−32.8016	−1.37511	−0.687557	+	0.726131i	$0.741317\pi$
$570$	0	0
$571$	−21.0268	−0.879946	−0.439973	−	0.898011i	$-0.645012\pi$
$571$	−21.0268	−0.879946	−0.439973	+	0.898011i	$0.645012\pi$
$572$	−2.85990	−0.119578
$573$	0	0
$574$	32.8108	1.36950
$575$	−8.43743	−0.351865
$576$	0	0
$577$	−8.25080	−0.343485	−0.171743	−	0.985142i	$-0.554940\pi$
$577$	−8.25080	−0.343485	−0.171743	+	0.985142i	$0.554940\pi$
$578$	1.23394	0.0513252
$579$	0	0
$580$	−1.74026	−0.0722604
$581$	−15.3288	−0.635946
$582$	0	0
$583$	−1.90119	−0.0787392
$584$	18.3872	0.760866
$585$	0	0
$586$	15.5573	0.642666
$587$	25.2181	1.04086	0.520430	−	0.853904i	$-0.325772\pi$
$587$	25.2181	1.04086	0.520430	+	0.853904i	$0.325772\pi$
$588$	0	0
$589$	−30.9194	−1.27401
$590$	−7.99671	−0.329219
$591$	0	0
$592$	−13.8418	−0.568896
$593$	−15.4984	−0.636443	−0.318222	−	0.948016i	$-0.603086\pi$
$593$	−15.4984	−0.636443	−0.318222	+	0.948016i	$0.603086\pi$
$594$	0	0
$595$	2.41473	0.0989943
$596$	−9.81806	−0.402163
$597$	0	0
$598$	−62.3710	−2.55054
$599$	24.2055	0.989010	0.494505	−	0.869175i	$-0.335349\pi$
$599$	24.2055	0.989010	0.494505	+	0.869175i	$0.335349\pi$
$600$	0	0
$601$	−0.482194	−0.0196691	−0.00983455	−	0.999952i	$-0.503130\pi$
$601$	−0.482194	−0.0196691	−0.00983455	+	0.999952i	$0.503130\pi$
$602$	25.1989	1.02703
$603$	0	0
$604$	6.36150	0.258846
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	−32.3043	−1.31119	−0.655595	−	0.755113i	$-0.727582\pi$
$607$	−32.3043	−1.31119	−0.655595	+	0.755113i	$0.727582\pi$
$608$	−9.07415	−0.368006
$609$	0	0
$610$	−2.42315	−0.0981104
$611$	−28.6986	−1.16102
$612$	0	0
$613$	−34.1168	−1.37796	−0.688981	−	0.724779i	$-0.741942\pi$
$613$	−34.1168	−1.37796	−0.688981	+	0.724779i	$0.741942\pi$
$614$	8.32699	0.336050
$615$	0	0
$616$	7.38172	0.297418
$617$	2.61004	0.105076	0.0525381	−	0.998619i	$-0.483269\pi$
$617$	2.61004	0.105076	0.0525381	+	0.998619i	$0.483269\pi$
$618$	0	0
$619$	−5.23893	−0.210570	−0.105285	−	0.994442i	$-0.533576\pi$
$619$	−5.23893	−0.210570	−0.105285	+	0.994442i	$0.533576\pi$
$620$	4.29033	0.172304
$621$	0	0
$622$	−6.75467	−0.270838
$623$	7.21872	0.289212
$624$	0	0
$625$	1.00000	0.0400000
$626$	−2.89152	−0.115568
$627$	0	0
$628$	1.67848	0.0669785
$629$	4.91313	0.195899
$630$	0	0
$631$	−34.7132	−1.38191	−0.690956	−	0.722897i	$-0.742810\pi$
$631$	−34.7132	−1.38191	−0.690956	+	0.722897i	$0.742810\pi$
$632$	−28.4599	−1.13207
$633$	0	0
$634$	5.13607	0.203979
$635$	−3.02996	−0.120240
$636$	0	0
$637$	−7.00360	−0.277493
$638$	−4.49817	−0.178084
$639$	0	0
$640$	−5.69370	−0.225063
$641$	7.06650	0.279110	0.139555	−	0.990214i	$-0.455433\pi$
$641$	7.06650	0.279110	0.139555	+	0.990214i	$0.455433\pi$
$642$	0	0
$643$	8.15696	0.321679	0.160839	−	0.986981i	$-0.448580\pi$
$643$	8.15696	0.321679	0.160839	+	0.986981i	$0.448580\pi$
$644$	−9.72640	−0.383274
$645$	0	0
$646$	−4.24530	−0.167029
$647$	−22.2146	−0.873348	−0.436674	−	0.899620i	$-0.643844\pi$
$647$	−22.2146	−0.873348	−0.436674	+	0.899620i	$0.643844\pi$
$648$	0	0
$649$	6.48062	0.254387
$650$	7.39218	0.289945
$651$	0	0
$652$	7.64668	0.299467
$653$	33.6327	1.31615	0.658074	−	0.752953i	$-0.271371\pi$
$653$	33.6327	1.31615	0.658074	+	0.752953i	$0.271371\pi$
$654$	0	0
$655$	−4.39116	−0.171577
$656$	31.0234	1.21126
$657$	0	0
$658$	14.2740	0.556458
$659$	30.8042	1.19996	0.599980	−	0.800015i	$-0.295175\pi$
$659$	30.8042	1.19996	0.599980	+	0.800015i	$0.295175\pi$
$660$	0	0
$661$	−19.3485	−0.752568	−0.376284	−	0.926504i	$-0.622798\pi$
$661$	−19.3485	−0.752568	−0.376284	+	0.926504i	$0.622798\pi$
$662$	17.5419	0.681784
$663$	0	0
$664$	−19.4057	−0.753085
$665$	−8.30773	−0.322160
$666$	0	0
$667$	30.7575	1.19094
$668$	1.35925	0.0525909
$669$	0	0
$670$	−11.5614	−0.446657
$671$	1.96375	0.0758096
$672$	0	0
$673$	−2.82236	−0.108794	−0.0543969	−	0.998519i	$-0.517324\pi$
$673$	−2.82236	−0.108794	−0.0543969	+	0.998519i	$0.517324\pi$
$674$	17.5378	0.675529
$675$	0	0
$676$	−10.9268	−0.420260
$677$	27.3518	1.05122	0.525608	−	0.850727i	$-0.323838\pi$
$677$	27.3518	1.05122	0.525608	+	0.850727i	$0.323838\pi$
$678$	0	0
$679$	−35.0007	−1.34320
$680$	3.05695	0.117229
$681$	0	0
$682$	11.0895	0.424639
$683$	−38.5727	−1.47594	−0.737972	−	0.674831i	$-0.764216\pi$
$683$	−38.5727	−1.47594	−0.737972	+	0.674831i	$0.764216\pi$
$684$	0	0
$685$	19.9818	0.763465
$686$	24.3409	0.929339
$687$	0	0
$688$	23.8262	0.908364
$689$	−11.3895	−0.433904
$690$	0	0
$691$	−32.1876	−1.22448	−0.612238	−	0.790674i	$-0.709730\pi$
$691$	−32.1876	−1.22448	−0.612238	+	0.790674i	$0.709730\pi$
$692$	−9.08480	−0.345352
$693$	0	0
$694$	−1.53672	−0.0583331
$695$	−17.9238	−0.679889
$696$	0	0
$697$	−11.0117	−0.417097
$698$	21.5400	0.815299
$699$	0	0
$700$	1.15277	0.0435705
$701$	−20.7377	−0.783253	−0.391626	−	0.920124i	$-0.628087\pi$
$701$	−20.7377	−0.783253	−0.391626	+	0.920124i	$0.628087\pi$
$702$	0	0
$703$	−16.9033	−0.637520
$704$	8.88916	0.335023
$705$	0	0
$706$	34.2556	1.28923
$707$	24.9281	0.937517
$708$	0	0
$709$	41.4231	1.55568	0.777839	−	0.628463i	$-0.216316\pi$
$709$	41.4231	1.55568	0.777839	+	0.628463i	$0.216316\pi$
$710$	5.72048	0.214686
$711$	0	0
$712$	9.13862	0.342484
$713$	−75.8277	−2.83977
$714$	0	0
$715$	−5.99070	−0.224040
$716$	0.634440	0.0237101
$717$	0	0
$718$	−11.2528	−0.419951
$719$	6.12396	0.228385	0.114193	−	0.993459i	$-0.463572\pi$
$719$	6.12396	0.228385	0.114193	+	0.993459i	$0.463572\pi$
$720$	0	0
$721$	−8.95479	−0.333494
$722$	−8.83919	−0.328961
$723$	0	0
$724$	−7.39654	−0.274890
$725$	−3.64537	−0.135386
$726$	0	0
$727$	−17.9206	−0.664637	−0.332319	−	0.943167i	$-0.607831\pi$
$727$	−17.9206	−0.664637	−0.332319	+	0.943167i	$0.607831\pi$
$728$	44.2217	1.63896
$729$	0	0
$730$	7.42199	0.274700
$731$	−8.45703	−0.312795
$732$	0	0
$733$	−41.4764	−1.53197	−0.765983	−	0.642861i	$-0.777747\pi$
$733$	−41.4764	−1.53197	−0.765983	+	0.642861i	$0.777747\pi$
$734$	11.8361	0.436879
$735$	0	0
$736$	−22.2537	−0.820283
$737$	9.36951	0.345130
$738$	0	0
$739$	0.839066	0.0308656	0.0154328	−	0.999881i	$-0.495087\pi$
$739$	0.839066	0.0308656	0.0154328	+	0.999881i	$0.495087\pi$
$740$	2.34548	0.0862214
$741$	0	0
$742$	5.66485	0.207963
$743$	38.2537	1.40339	0.701696	−	0.712477i	$-0.252427\pi$
$743$	38.2537	1.40339	0.701696	+	0.712477i	$0.252427\pi$
$744$	0	0
$745$	−20.5661	−0.753485
$746$	22.7860	0.834253
$747$	0	0
$748$	−0.477390	−0.0174551
$749$	−48.7967	−1.78299
$750$	0	0
$751$	21.0649	0.768668	0.384334	−	0.923194i	$-0.374431\pi$
$751$	21.0649	0.768668	0.384334	+	0.923194i	$0.374431\pi$
$752$	13.4964	0.492164
$753$	0	0
$754$	−26.9472	−0.981359
$755$	13.3256	0.484968
$756$	0	0
$757$	40.9997	1.49016	0.745080	−	0.666975i	$-0.232411\pi$
$757$	40.9997	1.49016	0.745080	+	0.666975i	$0.232411\pi$
$758$	−21.2723	−0.772645
$759$	0	0
$760$	−10.5173	−0.381501
$761$	−0.300937	−0.0109090	−0.00545448	−	0.999985i	$-0.501736\pi$
$761$	−0.300937	−0.0109090	−0.00545448	+	0.999985i	$0.501736\pi$
$762$	0	0
$763$	−15.7024	−0.568466
$764$	−7.50830	−0.271641
$765$	0	0
$766$	4.36468	0.157702
$767$	38.8235	1.40184
$768$	0	0
$769$	−9.51413	−0.343088	−0.171544	−	0.985176i	$-0.554876\pi$
$769$	−9.51413	−0.343088	−0.171544	+	0.985176i	$0.554876\pi$
$770$	2.97963	0.107379
$771$	0	0
$772$	−3.49129	−0.125654
$773$	16.5434	0.595025	0.297513	−	0.954718i	$-0.403843\pi$
$773$	16.5434	0.595025	0.297513	+	0.954718i	$0.403843\pi$
$774$	0	0
$775$	8.98705	0.322825
$776$	−44.3094	−1.59062
$777$	0	0
$778$	−18.1117	−0.649337
$779$	37.8850	1.35737
$780$	0	0
$781$	−4.63594	−0.165887
$782$	−10.4113	−0.372307
$783$	0	0
$784$	3.29367	0.117631
$785$	3.51594	0.125489
$786$	0	0
$787$	−20.7956	−0.741284	−0.370642	−	0.928776i	$-0.620862\pi$
$787$	−20.7956	−0.741284	−0.370642	+	0.928776i	$0.620862\pi$
$788$	−10.0567	−0.358256
$789$	0	0
$790$	−11.4879	−0.408720
$791$	−6.60095	−0.234703
$792$	0	0
$793$	11.7642	0.417760
$794$	27.3703	0.971335
$795$	0	0
$796$	−1.32939	−0.0471189
$797$	14.2666	0.505350	0.252675	−	0.967551i	$-0.418690\pi$
$797$	14.2666	0.505350	0.252675	+	0.967551i	$0.418690\pi$
$798$	0	0
$799$	−4.79052	−0.169476
$800$	2.63750	0.0932497
$801$	0	0
$802$	19.6796	0.694912
$803$	−6.01487	−0.212260
$804$	0	0
$805$	−20.3741	−0.718093
$806$	66.4339	2.34003
$807$	0	0
$808$	31.5580	1.11021
$809$	7.23800	0.254475	0.127237	−	0.991872i	$-0.459389\pi$
$809$	7.23800	0.254475	0.127237	+	0.991872i	$0.459389\pi$
$810$	0	0
$811$	−5.36765	−0.188484	−0.0942418	−	0.995549i	$-0.530043\pi$
$811$	−5.36765	−0.188484	−0.0942418	+	0.995549i	$0.530043\pi$
$812$	−4.20226	−0.147470
$813$	0	0
$814$	6.06251	0.212491
$815$	16.0177	0.561075
$816$	0	0
$817$	29.0959	1.01794
$818$	7.77560	0.271867
$819$	0	0
$820$	−5.25686	−0.183578
$821$	−48.7059	−1.69985	−0.849924	−	0.526905i	$-0.823352\pi$
$821$	−48.7059	−1.69985	−0.849924	+	0.526905i	$0.823352\pi$
$822$	0	0
$823$	42.8022	1.49199	0.745996	−	0.665950i	$-0.231974\pi$
$823$	42.8022	1.49199	0.745996	+	0.665950i	$0.231974\pi$
$824$	−11.3364	−0.394922
$825$	0	0
$826$	−19.3099	−0.671877
$827$	37.5847	1.30695	0.653474	−	0.756949i	$-0.273311\pi$
$827$	37.5847	1.30695	0.653474	+	0.756949i	$0.273311\pi$
$828$	0	0
$829$	43.2360	1.50165	0.750823	−	0.660503i	$-0.229657\pi$
$829$	43.2360	1.50165	0.750823	+	0.660503i	$0.229657\pi$
$830$	−7.83310	−0.271891
$831$	0	0
$832$	53.2523	1.84619
$833$	−1.16908	−0.0405062
$834$	0	0
$835$	2.84725	0.0985331
$836$	1.64243	0.0568046
$837$	0	0
$838$	−7.08644	−0.244797
$839$	−53.1197	−1.83389	−0.916947	−	0.399009i	$-0.869354\pi$
$839$	−53.1197	−1.83389	−0.916947	+	0.399009i	$0.869354\pi$
$840$	0	0
$841$	−15.7113	−0.541769
$842$	43.5745	1.50168
$843$	0	0
$844$	−8.86157	−0.305028
$845$	−22.8885	−0.787390
$846$	0	0
$847$	−2.41473	−0.0829711
$848$	5.35626	0.183935
$849$	0	0
$850$	1.23394	0.0423238
$851$	−41.4542	−1.42103
$852$	0	0
$853$	42.1260	1.44237	0.721183	−	0.692745i	$-0.243599\pi$
$853$	42.1260	1.44237	0.721183	+	0.692745i	$0.243599\pi$
$854$	−5.85125	−0.200226
$855$	0	0
$856$	−61.7747	−2.11142
$857$	−37.9248	−1.29549	−0.647744	−	0.761858i	$-0.724287\pi$
$857$	−37.9248	−1.29549	−0.647744	+	0.761858i	$0.724287\pi$
$858$	0	0
$859$	−51.3968	−1.75364	−0.876819	−	0.480821i	$-0.840339\pi$
$859$	−51.3968	−1.75364	−0.876819	+	0.480821i	$0.840339\pi$
$860$	−4.03730	−0.137671
$861$	0	0
$862$	−28.5858	−0.973636
$863$	−51.0886	−1.73908	−0.869538	−	0.493866i	$-0.835583\pi$
$863$	−51.0886	−1.73908	−0.869538	+	0.493866i	$0.835583\pi$
$864$	0	0
$865$	−19.0302	−0.647045
$866$	15.8574	0.538856
$867$	0	0
$868$	10.3600	0.351641
$869$	9.30990	0.315817
$870$	0	0
$871$	56.1300	1.90189
$872$	−19.8786	−0.673176
$873$	0	0
$874$	35.8194	1.21161
$875$	2.41473	0.0816328
$876$	0	0
$877$	16.9154	0.571194	0.285597	−	0.958350i	$-0.407808\pi$
$877$	16.9154	0.571194	0.285597	+	0.958350i	$0.407808\pi$
$878$	−31.2630	−1.05508
$879$	0	0
$880$	2.81732	0.0949718
$881$	−25.2394	−0.850337	−0.425168	−	0.905114i	$-0.639785\pi$
$881$	−25.2394	−0.850337	−0.425168	+	0.905114i	$0.639785\pi$
$882$	0	0
$883$	14.9184	0.502043	0.251022	−	0.967981i	$-0.419233\pi$
$883$	14.9184	0.502043	0.251022	+	0.967981i	$0.419233\pi$
$884$	−2.85990	−0.0961889
$885$	0	0
$886$	38.7112	1.30053
$887$	40.9742	1.37578	0.687890	−	0.725815i	$-0.258537\pi$
$887$	40.9742	1.37578	0.687890	+	0.725815i	$0.258537\pi$
$888$	0	0
$889$	−7.31655	−0.245389
$890$	3.68881	0.123649
$891$	0	0
$892$	8.16997	0.273551
$893$	16.4815	0.551532
$894$	0	0
$895$	1.32898	0.0444228
$896$	−13.7487	−0.459313
$897$	0	0
$898$	4.02282	0.134243
$899$	−32.7611	−1.09264
$900$	0	0
$901$	−1.90119	−0.0633378
$902$	−13.5878	−0.452423
$903$	0	0
$904$	−8.35654	−0.277934
$905$	−15.4937	−0.515029
$906$	0	0
$907$	−55.7531	−1.85125	−0.925626	−	0.378438i	$-0.876461\pi$
$907$	−55.7531	−1.85125	−0.925626	+	0.378438i	$0.876461\pi$
$908$	−14.2703	−0.473577
$909$	0	0
$910$	17.8501	0.591725
$911$	−25.8245	−0.855603	−0.427802	−	0.903873i	$-0.640712\pi$
$911$	−25.8245	−0.855603	−0.427802	+	0.903873i	$0.640712\pi$
$912$	0	0
$913$	6.34804	0.210089
$914$	33.2514	1.09986
$915$	0	0
$916$	0.956136	0.0315916
$917$	−10.6035	−0.350158
$918$	0	0
$919$	7.98018	0.263242	0.131621	−	0.991300i	$-0.457982\pi$
$919$	7.98018	0.263242	0.131621	+	0.991300i	$0.457982\pi$
$920$	−25.7928	−0.850364
$921$	0	0
$922$	−40.4918	−1.33353
$923$	−27.7725	−0.914145
$924$	0	0
$925$	4.91313	0.161543
$926$	−27.4494	−0.902043
$927$	0	0
$928$	−9.61466	−0.315617
$929$	7.44605	0.244297	0.122148	−	0.992512i	$-0.461022\pi$
$929$	7.44605	0.244297	0.122148	+	0.992512i	$0.461022\pi$
$930$	0	0
$931$	4.02214	0.131820
$932$	−5.73320	−0.187797
$933$	0	0
$934$	−49.4492	−1.61803
$935$	−1.00000	−0.0327035
$936$	0	0
$937$	4.35263	0.142194	0.0710971	−	0.997469i	$-0.477350\pi$
$937$	4.35263	0.142194	0.0710971	+	0.997469i	$0.477350\pi$
$938$	−27.9177	−0.911546
$939$	0	0
$940$	−2.28695	−0.0745919
$941$	46.4557	1.51441	0.757206	−	0.653177i	$-0.226564\pi$
$941$	46.4557	1.51441	0.757206	+	0.653177i	$0.226564\pi$
$942$	0	0
$943$	92.9103	3.02558
$944$	−18.2580	−0.594247
$945$	0	0
$946$	−10.4355	−0.339287
$947$	48.9668	1.59121	0.795604	−	0.605817i	$-0.207154\pi$
$947$	48.9668	1.59121	0.795604	+	0.605817i	$0.207154\pi$
$948$	0	0
$949$	−36.0333	−1.16969
$950$	−4.24530	−0.137736
$951$	0	0
$952$	7.38172	0.239243
$953$	2.48728	0.0805709	0.0402854	−	0.999188i	$-0.487173\pi$
$953$	2.48728	0.0805709	0.0402854	+	0.999188i	$0.487173\pi$
$954$	0	0
$955$	−15.7278	−0.508940
$956$	0.838944	0.0271334
$957$	0	0
$958$	17.5256	0.566227
$959$	48.2507	1.55810
$960$	0	0
$961$	49.7671	1.60539
$962$	36.3187	1.17096
$963$	0	0
$964$	9.06752	0.292045
$965$	−7.31329	−0.235423
$966$	0	0
$967$	32.7700	1.05381	0.526906	−	0.849923i	$-0.323352\pi$
$967$	32.7700	1.05381	0.526906	+	0.849923i	$0.323352\pi$
$968$	−3.05695	−0.0982542
$969$	0	0
$970$	−17.8855	−0.574270
$971$	8.27116	0.265434	0.132717	−	0.991154i	$-0.457630\pi$
$971$	8.27116	0.265434	0.132717	+	0.991154i	$0.457630\pi$
$972$	0	0
$973$	−43.2811	−1.38753
$974$	9.36158	0.299964
$975$	0	0
$976$	−5.53250	−0.177091
$977$	29.2283	0.935097	0.467548	−	0.883967i	$-0.345137\pi$
$977$	29.2283	0.935097	0.467548	+	0.883967i	$0.345137\pi$
$978$	0	0
$979$	−2.98945	−0.0955433
$980$	−0.558106	−0.0178280
$981$	0	0
$982$	−26.5112	−0.846008
$983$	9.09841	0.290194	0.145097	−	0.989417i	$-0.453651\pi$
$983$	9.09841	0.290194	0.145097	+	0.989417i	$0.453651\pi$
$984$	0	0
$985$	−21.0661	−0.671222
$986$	−4.49817	−0.143251
$987$	0	0
$988$	9.83931	0.313030
$989$	71.3556	2.26898
$990$	0	0
$991$	−32.1602	−1.02160	−0.510801	−	0.859699i	$-0.670651\pi$
$991$	−32.1602	−1.02160	−0.510801	+	0.859699i	$0.670651\pi$
$992$	23.7033	0.752582
$993$	0	0
$994$	13.8134	0.438135
$995$	−2.78470	−0.0882809
$996$	0	0
$997$	−61.8701	−1.95945	−0.979724	−	0.200354i	$-0.935791\pi$
$997$	−61.8701	−1.95945	−0.979724	+	0.200354i	$0.935791\pi$
$998$	−37.9460	−1.20116
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8415.2.a.by.1.7		11
3.2	odd	2		935.2.a.j.1.5	✓	11
15.14	odd	2		4675.2.a.bl.1.7		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
935.2.a.j.1.5	✓	11	3.2	odd	2
4675.2.a.bl.1.7		11	15.14	odd	2
8415.2.a.by.1.7		11	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8415 = 3^{2} \cdot 5 \cdot 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8415.a (trivial)

\(f(q)\)	\(=\)	\(q+1.23394 q^{2} -0.477390 q^{4} -1.00000 q^{5} -2.41473 q^{7} -3.05695 q^{8} +O(q^{10})\)	\(q+1.23394 q^{2} -0.477390 q^{4} -1.00000 q^{5} -2.41473 q^{7} -3.05695 q^{8} -1.23394 q^{10} +1.00000 q^{11} +5.99070 q^{13} -2.97963 q^{14} -2.81732 q^{16} +1.00000 q^{17} -3.44044 q^{19} +0.477390 q^{20} +1.23394 q^{22} -8.43743 q^{23} +1.00000 q^{25} +7.39218 q^{26} +1.15277 q^{28} -3.64537 q^{29} +8.98705 q^{31} +2.63750 q^{32} +1.23394 q^{34} +2.41473 q^{35} +4.91313 q^{37} -4.24530 q^{38} +3.05695 q^{40} -11.0117 q^{41} -8.45703 q^{43} -0.477390 q^{44} -10.4113 q^{46} -4.79052 q^{47} -1.16908 q^{49} +1.23394 q^{50} -2.85990 q^{52} -1.90119 q^{53} -1.00000 q^{55} +7.38172 q^{56} -4.49817 q^{58} +6.48062 q^{59} +1.96375 q^{61} +11.0895 q^{62} +8.88916 q^{64} -5.99070 q^{65} +9.36951 q^{67} -0.477390 q^{68} +2.97963 q^{70} -4.63594 q^{71} -6.01487 q^{73} +6.06251 q^{74} +1.64243 q^{76} -2.41473 q^{77} +9.30990 q^{79} +2.81732 q^{80} -13.5878 q^{82} +6.34804 q^{83} -1.00000 q^{85} -10.4355 q^{86} -3.05695 q^{88} -2.98945 q^{89} -14.4659 q^{91} +4.02794 q^{92} -5.91122 q^{94} +3.44044 q^{95} +14.4946 q^{97} -1.44257 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + q^{2} + 21 q^{4} - 11 q^{5} - 2 q^{7}+O(q^{10}) \) 11 * q + q^2 + 21 * q^4 - 11 * q^5 - 2 * q^7	\( 11 q + q^{2} + 21 q^{4} - 11 q^{5} - 2 q^{7} - q^{10} + 11 q^{11} + 13 q^{13} + 29 q^{16} + 11 q^{17} + 4 q^{19} - 21 q^{20} + q^{22} + 7 q^{23} + 11 q^{25} - 4 q^{26} + 8 q^{28} - 9 q^{29} + 20 q^{31} + 26 q^{32} + q^{34} + 2 q^{35} - q^{37} + 9 q^{41} + 5 q^{43} + 21 q^{44} - 10 q^{46} + 43 q^{49} + q^{50} + 18 q^{52} - 14 q^{53} - 11 q^{55} + 6 q^{56} - 10 q^{58} - 13 q^{59} + 11 q^{61} + 26 q^{62} + 24 q^{64} - 13 q^{65} + 26 q^{67} + 21 q^{68} + 8 q^{71} + 42 q^{73} - 16 q^{74} - 70 q^{76} - 2 q^{77} - 27 q^{79} - 29 q^{80} + 57 q^{82} + 41 q^{83} - 11 q^{85} - 47 q^{86} - 37 q^{89} + 30 q^{91} + 47 q^{92} + 94 q^{94} - 4 q^{95} + 29 q^{97} + 69 q^{98}+O(q^{100}) \) 11 * q + q^2 + 21 * q^4 - 11 * q^5 - 2 * q^7 - q^10 + 11 * q^11 + 13 * q^13 + 29 * q^16 + 11 * q^17 + 4 * q^19 - 21 * q^20 + q^22 + 7 * q^23 + 11 * q^25 - 4 * q^26 + 8 * q^28 - 9 * q^29 + 20 * q^31 + 26 * q^32 + q^34 + 2 * q^35 - q^37 + 9 * q^41 + 5 * q^43 + 21 * q^44 - 10 * q^46 + 43 * q^49 + q^50 + 18 * q^52 - 14 * q^53 - 11 * q^55 + 6 * q^56 - 10 * q^58 - 13 * q^59 + 11 * q^61 + 26 * q^62 + 24 * q^64 - 13 * q^65 + 26 * q^67 + 21 * q^68 + 8 * q^71 + 42 * q^73 - 16 * q^74 - 70 * q^76 - 2 * q^77 - 27 * q^79 - 29 * q^80 + 57 * q^82 + 41 * q^83 - 11 * q^85 - 47 * q^86 - 37 * q^89 + 30 * q^91 + 47 * q^92 + 94 * q^94 - 4 * q^95 + 29 * q^97 + 69 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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