LMFDB - Embedded newform 1305.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1305,2,Mod(1,1305)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1305, 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("1305.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1305 = 3^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1305.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:	$10.4204774638$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	1305.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.618034 q^{2} -1.61803 q^{4} +1.00000 q^{5} +0.236068 q^{7} +2.23607 q^{8} -0.618034 q^{10} -4.23607 q^{11} +1.00000 q^{13} -0.145898 q^{14} +1.85410 q^{16} +1.47214 q^{17} -6.47214 q^{19} -1.61803 q^{20} +2.61803 q^{22} +4.47214 q^{23} +1.00000 q^{25} -0.618034 q^{26} -0.381966 q^{28} +1.00000 q^{29} -8.00000 q^{31} -5.61803 q^{32} -0.909830 q^{34} +0.236068 q^{35} +4.00000 q^{38} +2.23607 q^{40} +6.00000 q^{41} -6.00000 q^{43} +6.85410 q^{44} -2.76393 q^{46} -8.23607 q^{47} -6.94427 q^{49} -0.618034 q^{50} -1.61803 q^{52} +6.47214 q^{53} -4.23607 q^{55} +0.527864 q^{56} -0.618034 q^{58} +6.00000 q^{59} +0.472136 q^{61} +4.94427 q^{62} -0.236068 q^{64} +1.00000 q^{65} -14.7082 q^{67} -2.38197 q^{68} -0.145898 q^{70} -2.47214 q^{71} -6.00000 q^{73} +10.4721 q^{76} -1.00000 q^{77} -6.00000 q^{79} +1.85410 q^{80} -3.70820 q^{82} -6.47214 q^{83} +1.47214 q^{85} +3.70820 q^{86} -9.47214 q^{88} -7.94427 q^{89} +0.236068 q^{91} -7.23607 q^{92} +5.09017 q^{94} -6.47214 q^{95} -11.4164 q^{97} +4.29180 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - q^{4} + 2 q^{5} - 4 q^{7} + q^{10} - 4 q^{11} + 2 q^{13} - 7 q^{14} - 3 q^{16} - 6 q^{17} - 4 q^{19} - q^{20} + 3 q^{22} + 2 q^{25} + q^{26} - 3 q^{28} + 2 q^{29} - 16 q^{31} - 9 q^{32}+ \cdots + 22 q^{98}+O(q^{100}) $ 2 * q + q^2 - q^4 + 2 * q^5 - 4 * q^7 + q^10 - 4 * q^11 + 2 * q^13 - 7 * q^14 - 3 * q^16 - 6 * q^17 - 4 * q^19 - q^20 + 3 * q^22 + 2 * q^25 + q^26 - 3 * q^28 + 2 * q^29 - 16 * q^31 - 9 * q^32 - 13 * q^34 - 4 * q^35 + 8 * q^38 + 12 * q^41 - 12 * q^43 + 7 * q^44 - 10 * q^46 - 12 * q^47 + 4 * q^49 + q^50 - q^52 + 4 * q^53 - 4 * q^55 + 10 * q^56 + q^58 + 12 * q^59 - 8 * q^61 - 8 * q^62 + 4 * q^64 + 2 * q^65 - 16 * q^67 - 7 * q^68 - 7 * q^70 + 4 * q^71 - 12 * q^73 + 12 * q^76 - 2 * q^77 - 12 * q^79 - 3 * q^80 + 6 * q^82 - 4 * q^83 - 6 * q^85 - 6 * q^86 - 10 * q^88 + 2 * q^89 - 4 * q^91 - 10 * q^92 - q^94 - 4 * q^95 + 4 * q^97 + 22 * 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 $

$2$	−0.618034	−0.437016	−0.218508	−	0.975835i	$-0.570119\pi$
$2$	−0.618034	−0.437016	−0.218508	+	0.975835i	$0.570119\pi$
$3$	0	0
$3$	0	0
$4$	−1.61803	−0.809017
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.236068	0.0892253	0.0446127	−	0.999004i	$-0.485795\pi$
$7$	0.236068	0.0892253	0.0446127	+	0.999004i	$0.485795\pi$
$8$	2.23607	0.790569
$9$	0	0
$10$	−0.618034	−0.195440
$11$	−4.23607	−1.27722	−0.638611	−	0.769529i	$-0.720491\pi$
$11$	−4.23607	−1.27722	−0.638611	+	0.769529i	$0.720491\pi$
$12$	0	0
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	−0.145898	−0.0389929
$15$	0	0
$16$	1.85410	0.463525
$17$	1.47214	0.357045	0.178523	−	0.983936i	$-0.442868\pi$
$17$	1.47214	0.357045	0.178523	+	0.983936i	$0.442868\pi$
$18$	0	0
$19$	−6.47214	−1.48481	−0.742405	−	0.669951i	$-0.766315\pi$
$19$	−6.47214	−1.48481	−0.742405	+	0.669951i	$0.766315\pi$
$20$	−1.61803	−0.361803
$21$	0	0
$22$	2.61803	0.558167
$23$	4.47214	0.932505	0.466252	−	0.884652i	$-0.345604\pi$
$23$	4.47214	0.932505	0.466252	+	0.884652i	$0.345604\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−0.618034	−0.121206
$27$	0	0
$28$	−0.381966	−0.0721848
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	−5.61803	−0.993137
$33$	0	0
$34$	−0.909830	−0.156035
$35$	0.236068	0.0399028
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	4.00000	0.648886
$39$	0	0
$40$	2.23607	0.353553
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	6.85410	1.03329
$45$	0	0
$46$	−2.76393	−0.407520
$47$	−8.23607	−1.20135	−0.600677	−	0.799492i	$-0.705102\pi$
$47$	−8.23607	−1.20135	−0.600677	+	0.799492i	$0.705102\pi$
$48$	0	0
$49$	−6.94427	−0.992039
$50$	−0.618034	−0.0874032
$51$	0	0
$52$	−1.61803	−0.224381
$53$	6.47214	0.889016	0.444508	−	0.895775i	$-0.353378\pi$
$53$	6.47214	0.889016	0.444508	+	0.895775i	$0.353378\pi$
$54$	0	0
$55$	−4.23607	−0.571191
$56$	0.527864	0.0705388
$57$	0	0
$58$	−0.618034	−0.0811518
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	0.472136	0.0604508	0.0302254	−	0.999543i	$-0.490377\pi$
$61$	0.472136	0.0604508	0.0302254	+	0.999543i	$0.490377\pi$
$62$	4.94427	0.627923
$63$	0	0
$64$	−0.236068	−0.0295085
$65$	1.00000	0.124035
$66$	0	0
$67$	−14.7082	−1.79689	−0.898447	−	0.439083i	$-0.855303\pi$
$67$	−14.7082	−1.79689	−0.898447	+	0.439083i	$0.855303\pi$
$68$	−2.38197	−0.288856
$69$	0	0
$70$	−0.145898	−0.0174382
$71$	−2.47214	−0.293389	−0.146694	−	0.989182i	$-0.546863\pi$
$71$	−2.47214	−0.293389	−0.146694	+	0.989182i	$0.546863\pi$
$72$	0	0
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	0	0
$75$	0	0
$76$	10.4721	1.20124
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	1.85410	0.207295
$81$	0	0
$82$	−3.70820	−0.409503
$83$	−6.47214	−0.710409	−0.355205	−	0.934789i	$-0.615589\pi$
$83$	−6.47214	−0.710409	−0.355205	+	0.934789i	$0.615589\pi$
$84$	0	0
$85$	1.47214	0.159676
$86$	3.70820	0.399866
$87$	0	0
$88$	−9.47214	−1.00973
$89$	−7.94427	−0.842091	−0.421046	−	0.907039i	$-0.638337\pi$
$89$	−7.94427	−0.842091	−0.421046	+	0.907039i	$0.638337\pi$
$90$	0	0
$91$	0.236068	0.0247466
$92$	−7.23607	−0.754412
$93$	0	0
$94$	5.09017	0.525011
$95$	−6.47214	−0.664027
$96$	0	0
$97$	−11.4164	−1.15916	−0.579580	−	0.814915i	$-0.696783\pi$
$97$	−11.4164	−1.15916	−0.579580	+	0.814915i	$0.696783\pi$
$98$	4.29180	0.433537
$99$	0	0
$100$	−1.61803	−0.161803
$101$	−15.4721	−1.53954	−0.769768	−	0.638324i	$-0.779628\pi$
$101$	−15.4721	−1.53954	−0.769768	+	0.638324i	$0.779628\pi$
$102$	0	0
$103$	16.9443	1.66957	0.834784	−	0.550577i	$-0.185592\pi$
$103$	16.9443	1.66957	0.834784	+	0.550577i	$0.185592\pi$
$104$	2.23607	0.219265
$105$	0	0
$106$	−4.00000	−0.388514
$107$	−4.47214	−0.432338	−0.216169	−	0.976356i	$-0.569356\pi$
$107$	−4.47214	−0.432338	−0.216169	+	0.976356i	$0.569356\pi$
$108$	0	0
$109$	−8.41641	−0.806146	−0.403073	−	0.915168i	$-0.632058\pi$
$109$	−8.41641	−0.806146	−0.403073	+	0.915168i	$0.632058\pi$
$110$	2.61803	0.249620
$111$	0	0
$112$	0.437694	0.0413582
$113$	−7.47214	−0.702919	−0.351460	−	0.936203i	$-0.614315\pi$
$113$	−7.47214	−0.702919	−0.351460	+	0.936203i	$0.614315\pi$
$114$	0	0
$115$	4.47214	0.417029
$116$	−1.61803	−0.150231
$117$	0	0
$118$	−3.70820	−0.341368
$119$	0.347524	0.0318575
$120$	0	0
$121$	6.94427	0.631297
$122$	−0.291796	−0.0264180
$123$	0	0
$124$	12.9443	1.16243
$125$	1.00000	0.0894427
$126$	0	0
$127$	−12.0000	−1.06483	−0.532414	−	0.846484i	$-0.678715\pi$
$127$	−12.0000	−1.06483	−0.532414	+	0.846484i	$0.678715\pi$
$128$	11.3820	1.00603
$129$	0	0
$130$	−0.618034	−0.0542052
$131$	−0.236068	−0.0206254	−0.0103127	−	0.999947i	$-0.503283\pi$
$131$	−0.236068	−0.0206254	−0.0103127	+	0.999947i	$0.503283\pi$
$132$	0	0
$133$	−1.52786	−0.132483
$134$	9.09017	0.785271
$135$	0	0
$136$	3.29180	0.282269
$137$	−10.9443	−0.935032	−0.467516	−	0.883985i	$-0.654851\pi$
$137$	−10.9443	−0.935032	−0.467516	+	0.883985i	$0.654851\pi$
$138$	0	0
$139$	12.2361	1.03785	0.518925	−	0.854820i	$-0.326332\pi$
$139$	12.2361	1.03785	0.518925	+	0.854820i	$0.326332\pi$
$140$	−0.381966	−0.0322820
$141$	0	0
$142$	1.52786	0.128216
$143$	−4.23607	−0.354238
$144$	0	0
$145$	1.00000	0.0830455
$146$	3.70820	0.306893
$147$	0	0
$148$	0	0
$149$	14.4721	1.18560	0.592802	−	0.805348i	$-0.298022\pi$
$149$	14.4721	1.18560	0.592802	+	0.805348i	$0.298022\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	−14.4721	−1.17385
$153$	0	0
$154$	0.618034	0.0498026
$155$	−8.00000	−0.642575
$156$	0	0
$157$	11.4164	0.911129	0.455564	−	0.890203i	$-0.349438\pi$
$157$	11.4164	0.911129	0.455564	+	0.890203i	$0.349438\pi$
$158$	3.70820	0.295009
$159$	0	0
$160$	−5.61803	−0.444145
$161$	1.05573	0.0832030
$162$	0	0
$163$	−5.05573	−0.395995	−0.197998	−	0.980203i	$-0.563444\pi$
$163$	−5.05573	−0.395995	−0.197998	+	0.980203i	$0.563444\pi$
$164$	−9.70820	−0.758083
$165$	0	0
$166$	4.00000	0.310460
$167$	−13.4164	−1.03819	−0.519096	−	0.854716i	$-0.673731\pi$
$167$	−13.4164	−1.03819	−0.519096	+	0.854716i	$0.673731\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	−0.909830	−0.0697808
$171$	0	0
$172$	9.70820	0.740244
$173$	23.8885	1.81621	0.908106	−	0.418740i	$-0.137528\pi$
$173$	23.8885	1.81621	0.908106	+	0.418740i	$0.137528\pi$
$174$	0	0
$175$	0.236068	0.0178451
$176$	−7.85410	−0.592025
$177$	0	0
$178$	4.90983	0.368007
$179$	2.94427	0.220065	0.110033	−	0.993928i	$-0.464904\pi$
$179$	2.94427	0.220065	0.110033	+	0.993928i	$0.464904\pi$
$180$	0	0
$181$	4.41641	0.328269	0.164135	−	0.986438i	$-0.447517\pi$
$181$	4.41641	0.328269	0.164135	+	0.986438i	$0.447517\pi$
$182$	−0.145898	−0.0108147
$183$	0	0
$184$	10.0000	0.737210
$185$	0	0
$186$	0	0
$187$	−6.23607	−0.456026
$188$	13.3262	0.971916
$189$	0	0
$190$	4.00000	0.290191
$191$	8.94427	0.647185	0.323592	−	0.946197i	$-0.395109\pi$
$191$	8.94427	0.647185	0.323592	+	0.946197i	$0.395109\pi$
$192$	0	0
$193$	20.0000	1.43963	0.719816	−	0.694165i	$-0.244226\pi$
$193$	20.0000	1.43963	0.719816	+	0.694165i	$0.244226\pi$
$194$	7.05573	0.506572
$195$	0	0
$196$	11.2361	0.802576
$197$	7.05573	0.502700	0.251350	−	0.967896i	$-0.419126\pi$
$197$	7.05573	0.502700	0.251350	+	0.967896i	$0.419126\pi$
$198$	0	0
$199$	20.1246	1.42660	0.713298	−	0.700861i	$-0.247201\pi$
$199$	20.1246	1.42660	0.713298	+	0.700861i	$0.247201\pi$
$200$	2.23607	0.158114
$201$	0	0
$202$	9.56231	0.672801
$203$	0.236068	0.0165687
$204$	0	0
$205$	6.00000	0.419058
$206$	−10.4721	−0.729628
$207$	0	0
$208$	1.85410	0.128559
$209$	27.4164	1.89643
$210$	0	0
$211$	16.9443	1.16649	0.583246	−	0.812296i	$-0.301782\pi$
$211$	16.9443	1.16649	0.583246	+	0.812296i	$0.301782\pi$
$212$	−10.4721	−0.719229
$213$	0	0
$214$	2.76393	0.188939
$215$	−6.00000	−0.409197
$216$	0	0
$217$	−1.88854	−0.128203
$218$	5.20163	0.352299
$219$	0	0
$220$	6.85410	0.462103
$221$	1.47214	0.0990266
$222$	0	0
$223$	−9.18034	−0.614761	−0.307381	−	0.951587i	$-0.599452\pi$
$223$	−9.18034	−0.614761	−0.307381	+	0.951587i	$0.599452\pi$
$224$	−1.32624	−0.0886130
$225$	0	0
$226$	4.61803	0.307187
$227$	−22.4721	−1.49153	−0.745764	−	0.666210i	$-0.767915\pi$
$227$	−22.4721	−1.49153	−0.745764	+	0.666210i	$0.767915\pi$
$228$	0	0
$229$	4.00000	0.264327	0.132164	−	0.991228i	$-0.457808\pi$
$229$	4.00000	0.264327	0.132164	+	0.991228i	$0.457808\pi$
$230$	−2.76393	−0.182248
$231$	0	0
$232$	2.23607	0.146805
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	−8.23607	−0.537262
$236$	−9.70820	−0.631950
$237$	0	0
$238$	−0.214782	−0.0139222
$239$	3.05573	0.197659	0.0988293	−	0.995104i	$-0.468490\pi$
$239$	3.05573	0.197659	0.0988293	+	0.995104i	$0.468490\pi$
$240$	0	0
$241$	−5.00000	−0.322078	−0.161039	−	0.986948i	$-0.551485\pi$
$241$	−5.00000	−0.322078	−0.161039	+	0.986948i	$0.551485\pi$
$242$	−4.29180	−0.275887
$243$	0	0
$244$	−0.763932	−0.0489057
$245$	−6.94427	−0.443653
$246$	0	0
$247$	−6.47214	−0.411812
$248$	−17.8885	−1.13592
$249$	0	0
$250$	−0.618034	−0.0390879
$251$	30.5967	1.93125	0.965625	−	0.259940i	$-0.0837028\pi$
$251$	30.5967	1.93125	0.965625	+	0.259940i	$0.0837028\pi$
$252$	0	0
$253$	−18.9443	−1.19102
$254$	7.41641	0.465347
$255$	0	0
$256$	−6.56231	−0.410144
$257$	−22.4721	−1.40177	−0.700887	−	0.713273i	$-0.747212\pi$
$257$	−22.4721	−1.40177	−0.700887	+	0.713273i	$0.747212\pi$
$258$	0	0
$259$	0	0
$260$	−1.61803	−0.100346
$261$	0	0
$262$	0.145898	0.00901361
$263$	7.05573	0.435075	0.217537	−	0.976052i	$-0.430198\pi$
$263$	7.05573	0.435075	0.217537	+	0.976052i	$0.430198\pi$
$264$	0	0
$265$	6.47214	0.397580
$266$	0.944272	0.0578970
$267$	0	0
$268$	23.7984	1.45372
$269$	0.527864	0.0321844	0.0160922	−	0.999871i	$-0.494877\pi$
$269$	0.527864	0.0321844	0.0160922	+	0.999871i	$0.494877\pi$
$270$	0	0
$271$	−12.4721	−0.757628	−0.378814	−	0.925473i	$-0.623668\pi$
$271$	−12.4721	−0.757628	−0.378814	+	0.925473i	$0.623668\pi$
$272$	2.72949	0.165500
$273$	0	0
$274$	6.76393	0.408624
$275$	−4.23607	−0.255445
$276$	0	0
$277$	2.05573	0.123517	0.0617584	−	0.998091i	$-0.480329\pi$
$277$	2.05573	0.123517	0.0617584	+	0.998091i	$0.480329\pi$
$278$	−7.56231	−0.453557
$279$	0	0
$280$	0.527864	0.0315459
$281$	13.5279	0.807005	0.403502	−	0.914979i	$-0.367793\pi$
$281$	13.5279	0.807005	0.403502	+	0.914979i	$0.367793\pi$
$282$	0	0
$283$	−12.9443	−0.769457	−0.384729	−	0.923030i	$-0.625705\pi$
$283$	−12.9443	−0.769457	−0.384729	+	0.923030i	$0.625705\pi$
$284$	4.00000	0.237356
$285$	0	0
$286$	2.61803	0.154808
$287$	1.41641	0.0836079
$288$	0	0
$289$	−14.8328	−0.872519
$290$	−0.618034	−0.0362922
$291$	0	0
$292$	9.70820	0.568130
$293$	9.00000	0.525786	0.262893	−	0.964825i	$-0.415323\pi$
$293$	9.00000	0.525786	0.262893	+	0.964825i	$0.415323\pi$
$294$	0	0
$295$	6.00000	0.349334
$296$	0	0
$297$	0	0
$298$	−8.94427	−0.518128
$299$	4.47214	0.258630
$300$	0	0
$301$	−1.41641	−0.0816404
$302$	7.41641	0.426766
$303$	0	0
$304$	−12.0000	−0.688247
$305$	0.472136	0.0270344
$306$	0	0
$307$	26.8328	1.53143	0.765715	−	0.643180i	$-0.222385\pi$
$307$	26.8328	1.53143	0.765715	+	0.643180i	$0.222385\pi$
$308$	1.61803	0.0921960
$309$	0	0
$310$	4.94427	0.280816
$311$	20.1246	1.14116	0.570581	−	0.821241i	$-0.306718\pi$
$311$	20.1246	1.14116	0.570581	+	0.821241i	$0.306718\pi$
$312$	0	0
$313$	1.58359	0.0895099	0.0447550	−	0.998998i	$-0.485749\pi$
$313$	1.58359	0.0895099	0.0447550	+	0.998998i	$0.485749\pi$
$314$	−7.05573	−0.398178
$315$	0	0
$316$	9.70820	0.546129
$317$	14.8885	0.836224	0.418112	−	0.908395i	$-0.362692\pi$
$317$	14.8885	0.836224	0.418112	+	0.908395i	$0.362692\pi$
$318$	0	0
$319$	−4.23607	−0.237174
$320$	−0.236068	−0.0131966
$321$	0	0
$322$	−0.652476	−0.0363611
$323$	−9.52786	−0.530145
$324$	0	0
$325$	1.00000	0.0554700
$326$	3.12461	0.173056
$327$	0	0
$328$	13.4164	0.740797
$329$	−1.94427	−0.107191
$330$	0	0
$331$	25.8885	1.42296	0.711482	−	0.702705i	$-0.248025\pi$
$331$	25.8885	1.42296	0.711482	+	0.702705i	$0.248025\pi$
$332$	10.4721	0.574733
$333$	0	0
$334$	8.29180	0.453707
$335$	−14.7082	−0.803595
$336$	0	0
$337$	5.05573	0.275403	0.137702	−	0.990474i	$-0.456029\pi$
$337$	5.05573	0.275403	0.137702	+	0.990474i	$0.456029\pi$
$338$	7.41641	0.403399
$339$	0	0
$340$	−2.38197	−0.129180
$341$	33.8885	1.83517
$342$	0	0
$343$	−3.29180	−0.177740
$344$	−13.4164	−0.723364
$345$	0	0
$346$	−14.7639	−0.793714
$347$	5.88854	0.316114	0.158057	−	0.987430i	$-0.449477\pi$
$347$	5.88854	0.316114	0.158057	+	0.987430i	$0.449477\pi$
$348$	0	0
$349$	−5.05573	−0.270627	−0.135313	−	0.990803i	$-0.543204\pi$
$349$	−5.05573	−0.270627	−0.135313	+	0.990803i	$0.543204\pi$
$350$	−0.145898	−0.00779858
$351$	0	0
$352$	23.7984	1.26846
$353$	−33.8885	−1.80371	−0.901853	−	0.432044i	$-0.857793\pi$
$353$	−33.8885	−1.80371	−0.901853	+	0.432044i	$0.857793\pi$
$354$	0	0
$355$	−2.47214	−0.131207
$356$	12.8541	0.681266
$357$	0	0
$358$	−1.81966	−0.0961720
$359$	35.7771	1.88824	0.944121	−	0.329598i	$-0.106913\pi$
$359$	35.7771	1.88824	0.944121	+	0.329598i	$0.106913\pi$
$360$	0	0
$361$	22.8885	1.20466
$362$	−2.72949	−0.143459
$363$	0	0
$364$	−0.381966	−0.0200205
$365$	−6.00000	−0.314054
$366$	0	0
$367$	−25.4164	−1.32673	−0.663363	−	0.748298i	$-0.730871\pi$
$367$	−25.4164	−1.32673	−0.663363	+	0.748298i	$0.730871\pi$
$368$	8.29180	0.432240
$369$	0	0
$370$	0	0
$371$	1.52786	0.0793227
$372$	0	0
$373$	24.8328	1.28579	0.642897	−	0.765952i	$-0.277732\pi$
$373$	24.8328	1.28579	0.642897	+	0.765952i	$0.277732\pi$
$374$	3.85410	0.199291
$375$	0	0
$376$	−18.4164	−0.949754
$377$	1.00000	0.0515026
$378$	0	0
$379$	−29.4164	−1.51102	−0.755510	−	0.655137i	$-0.772611\pi$
$379$	−29.4164	−1.51102	−0.755510	+	0.655137i	$0.772611\pi$
$380$	10.4721	0.537209
$381$	0	0
$382$	−5.52786	−0.282830
$383$	18.0000	0.919757	0.459879	−	0.887982i	$-0.347893\pi$
$383$	18.0000	0.919757	0.459879	+	0.887982i	$0.347893\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	−12.3607	−0.629142
$387$	0	0
$388$	18.4721	0.937781
$389$	10.5279	0.533784	0.266892	−	0.963726i	$-0.414003\pi$
$389$	10.5279	0.533784	0.266892	+	0.963726i	$0.414003\pi$
$390$	0	0
$391$	6.58359	0.332947
$392$	−15.5279	−0.784276
$393$	0	0
$394$	−4.36068	−0.219688
$395$	−6.00000	−0.301893
$396$	0	0
$397$	27.8885	1.39969	0.699843	−	0.714297i	$-0.253253\pi$
$397$	27.8885	1.39969	0.699843	+	0.714297i	$0.253253\pi$
$398$	−12.4377	−0.623445
$399$	0	0
$400$	1.85410	0.0927051
$401$	−34.3607	−1.71589	−0.857945	−	0.513741i	$-0.828259\pi$
$401$	−34.3607	−1.71589	−0.857945	+	0.513741i	$0.828259\pi$
$402$	0	0
$403$	−8.00000	−0.398508
$404$	25.0344	1.24551
$405$	0	0
$406$	−0.145898	−0.00724080
$407$	0	0
$408$	0	0
$409$	0.583592	0.0288568	0.0144284	−	0.999896i	$-0.495407\pi$
$409$	0.583592	0.0288568	0.0144284	+	0.999896i	$0.495407\pi$
$410$	−3.70820	−0.183135
$411$	0	0
$412$	−27.4164	−1.35071
$413$	1.41641	0.0696969
$414$	0	0
$415$	−6.47214	−0.317705
$416$	−5.61803	−0.275447
$417$	0	0
$418$	−16.9443	−0.828771
$419$	10.5836	0.517042	0.258521	−	0.966006i	$-0.416765\pi$
$419$	10.5836	0.517042	0.258521	+	0.966006i	$0.416765\pi$
$420$	0	0
$421$	9.41641	0.458928	0.229464	−	0.973317i	$-0.426303\pi$
$421$	9.41641	0.458928	0.229464	+	0.973317i	$0.426303\pi$
$422$	−10.4721	−0.509776
$423$	0	0
$424$	14.4721	0.702829
$425$	1.47214	0.0714091
$426$	0	0
$427$	0.111456	0.00539374
$428$	7.23607	0.349769
$429$	0	0
$430$	3.70820	0.178825
$431$	−3.52786	−0.169931	−0.0849656	−	0.996384i	$-0.527078\pi$
$431$	−3.52786	−0.169931	−0.0849656	+	0.996384i	$0.527078\pi$
$432$	0	0
$433$	23.3050	1.11996	0.559982	−	0.828505i	$-0.310808\pi$
$433$	23.3050	1.11996	0.559982	+	0.828505i	$0.310808\pi$
$434$	1.16718	0.0560266
$435$	0	0
$436$	13.6180	0.652186
$437$	−28.9443	−1.38459
$438$	0	0
$439$	−1.76393	−0.0841879	−0.0420939	−	0.999114i	$-0.513403\pi$
$439$	−1.76393	−0.0841879	−0.0420939	+	0.999114i	$0.513403\pi$
$440$	−9.47214	−0.451566
$441$	0	0
$442$	−0.909830	−0.0432762
$443$	−10.2361	−0.486330	−0.243165	−	0.969985i	$-0.578186\pi$
$443$	−10.2361	−0.486330	−0.243165	+	0.969985i	$0.578186\pi$
$444$	0	0
$445$	−7.94427	−0.376595
$446$	5.67376	0.268660
$447$	0	0
$448$	−0.0557281	−0.00263290
$449$	−24.0557	−1.13526	−0.567630	−	0.823284i	$-0.692140\pi$
$449$	−24.0557	−1.13526	−0.567630	+	0.823284i	$0.692140\pi$
$450$	0	0
$451$	−25.4164	−1.19681
$452$	12.0902	0.568674
$453$	0	0
$454$	13.8885	0.651822
$455$	0.236068	0.0110670
$456$	0	0
$457$	21.3607	0.999210	0.499605	−	0.866253i	$-0.333478\pi$
$457$	21.3607	0.999210	0.499605	+	0.866253i	$0.333478\pi$
$458$	−2.47214	−0.115515
$459$	0	0
$460$	−7.23607	−0.337383
$461$	−29.7771	−1.38686	−0.693429	−	0.720525i	$-0.743901\pi$
$461$	−29.7771	−1.38686	−0.693429	+	0.720525i	$0.743901\pi$
$462$	0	0
$463$	−12.2361	−0.568658	−0.284329	−	0.958727i	$-0.591771\pi$
$463$	−12.2361	−0.568658	−0.284329	+	0.958727i	$0.591771\pi$
$464$	1.85410	0.0860745
$465$	0	0
$466$	11.1246	0.515338
$467$	−24.0000	−1.11059	−0.555294	−	0.831654i	$-0.687394\pi$
$467$	−24.0000	−1.11059	−0.555294	+	0.831654i	$0.687394\pi$
$468$	0	0
$469$	−3.47214	−0.160328
$470$	5.09017	0.234792
$471$	0	0
$472$	13.4164	0.617540
$473$	25.4164	1.16865
$474$	0	0
$475$	−6.47214	−0.296962
$476$	−0.562306	−0.0257732
$477$	0	0
$478$	−1.88854	−0.0863800
$479$	−1.88854	−0.0862898	−0.0431449	−	0.999069i	$-0.513738\pi$
$479$	−1.88854	−0.0862898	−0.0431449	+	0.999069i	$0.513738\pi$
$480$	0	0
$481$	0	0
$482$	3.09017	0.140753
$483$	0	0
$484$	−11.2361	−0.510730
$485$	−11.4164	−0.518392
$486$	0	0
$487$	−24.9443	−1.13033	−0.565166	−	0.824977i	$-0.691188\pi$
$487$	−24.9443	−1.13033	−0.565166	+	0.824977i	$0.691188\pi$
$488$	1.05573	0.0477906
$489$	0	0
$490$	4.29180	0.193884
$491$	38.8328	1.75250	0.876250	−	0.481856i	$-0.160037\pi$
$491$	38.8328	1.75250	0.876250	+	0.481856i	$0.160037\pi$
$492$	0	0
$493$	1.47214	0.0663017
$494$	4.00000	0.179969
$495$	0	0
$496$	−14.8328	−0.666013
$497$	−0.583592	−0.0261777
$498$	0	0
$499$	14.1246	0.632304	0.316152	−	0.948708i	$-0.397609\pi$
$499$	14.1246	0.632304	0.316152	+	0.948708i	$0.397609\pi$
$500$	−1.61803	−0.0723607
$501$	0	0
$502$	−18.9098	−0.843987
$503$	−26.1246	−1.16484	−0.582419	−	0.812888i	$-0.697894\pi$
$503$	−26.1246	−1.16484	−0.582419	+	0.812888i	$0.697894\pi$
$504$	0	0
$505$	−15.4721	−0.688501
$506$	11.7082	0.520493
$507$	0	0
$508$	19.4164	0.861464
$509$	6.36068	0.281932	0.140966	−	0.990014i	$-0.454979\pi$
$509$	6.36068	0.281932	0.140966	+	0.990014i	$0.454979\pi$
$510$	0	0
$511$	−1.41641	−0.0626582
$512$	−18.7082	−0.826794
$513$	0	0
$514$	13.8885	0.612597
$515$	16.9443	0.746654
$516$	0	0
$517$	34.8885	1.53440
$518$	0	0
$519$	0	0
$520$	2.23607	0.0980581
$521$	3.52786	0.154559	0.0772793	−	0.997009i	$-0.475377\pi$
$521$	3.52786	0.154559	0.0772793	+	0.997009i	$0.475377\pi$
$522$	0	0
$523$	33.5410	1.46665	0.733323	−	0.679880i	$-0.237968\pi$
$523$	33.5410	1.46665	0.733323	+	0.679880i	$0.237968\pi$
$524$	0.381966	0.0166863
$525$	0	0
$526$	−4.36068	−0.190135
$527$	−11.7771	−0.513018
$528$	0	0
$529$	−3.00000	−0.130435
$530$	−4.00000	−0.173749
$531$	0	0
$532$	2.47214	0.107181
$533$	6.00000	0.259889
$534$	0	0
$535$	−4.47214	−0.193347
$536$	−32.8885	−1.42057
$537$	0	0
$538$	−0.326238	−0.0140651
$539$	29.4164	1.26705
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	7.70820	0.331096
$543$	0	0
$544$	−8.27051	−0.354595
$545$	−8.41641	−0.360519
$546$	0	0
$547$	−17.2918	−0.739344	−0.369672	−	0.929162i	$-0.620530\pi$
$547$	−17.2918	−0.739344	−0.369672	+	0.929162i	$0.620530\pi$
$548$	17.7082	0.756457
$549$	0	0
$550$	2.61803	0.111633
$551$	−6.47214	−0.275722
$552$	0	0
$553$	−1.41641	−0.0602318
$554$	−1.27051	−0.0539788
$555$	0	0
$556$	−19.7984	−0.839638
$557$	−4.58359	−0.194213	−0.0971065	−	0.995274i	$-0.530959\pi$
$557$	−4.58359	−0.194213	−0.0971065	+	0.995274i	$0.530959\pi$
$558$	0	0
$559$	−6.00000	−0.253773
$560$	0.437694	0.0184960
$561$	0	0
$562$	−8.36068	−0.352674
$563$	−3.18034	−0.134035	−0.0670177	−	0.997752i	$-0.521348\pi$
$563$	−3.18034	−0.134035	−0.0670177	+	0.997752i	$0.521348\pi$
$564$	0	0
$565$	−7.47214	−0.314355
$566$	8.00000	0.336265
$567$	0	0
$568$	−5.52786	−0.231944
$569$	−14.0557	−0.589247	−0.294623	−	0.955613i	$-0.595194\pi$
$569$	−14.0557	−0.589247	−0.294623	+	0.955613i	$0.595194\pi$
$570$	0	0
$571$	−22.8328	−0.955524	−0.477762	−	0.878489i	$-0.658552\pi$
$571$	−22.8328	−0.955524	−0.477762	+	0.878489i	$0.658552\pi$
$572$	6.85410	0.286584
$573$	0	0
$574$	−0.875388	−0.0365380
$575$	4.47214	0.186501
$576$	0	0
$577$	−46.9443	−1.95432	−0.977158	−	0.212515i	$-0.931835\pi$
$577$	−46.9443	−1.95432	−0.977158	+	0.212515i	$0.931835\pi$
$578$	9.16718	0.381305
$579$	0	0
$580$	−1.61803	−0.0671852
$581$	−1.52786	−0.0633865
$582$	0	0
$583$	−27.4164	−1.13547
$584$	−13.4164	−0.555175
$585$	0	0
$586$	−5.56231	−0.229777
$587$	10.9443	0.451718	0.225859	−	0.974160i	$-0.427481\pi$
$587$	10.9443	0.451718	0.225859	+	0.974160i	$0.427481\pi$
$588$	0	0
$589$	51.7771	2.13344
$590$	−3.70820	−0.152664
$591$	0	0
$592$	0	0
$593$	−3.52786	−0.144872	−0.0724360	−	0.997373i	$-0.523077\pi$
$593$	−3.52786	−0.144872	−0.0724360	+	0.997373i	$0.523077\pi$
$594$	0	0
$595$	0.347524	0.0142471
$596$	−23.4164	−0.959173
$597$	0	0
$598$	−2.76393	−0.113026
$599$	29.0689	1.18772	0.593861	−	0.804568i	$-0.297603\pi$
$599$	29.0689	1.18772	0.593861	+	0.804568i	$0.297603\pi$
$600$	0	0
$601$	−18.8328	−0.768207	−0.384103	−	0.923290i	$-0.625489\pi$
$601$	−18.8328	−0.768207	−0.384103	+	0.923290i	$0.625489\pi$
$602$	0.875388	0.0356782
$603$	0	0
$604$	19.4164	0.790042
$605$	6.94427	0.282325
$606$	0	0
$607$	3.41641	0.138668	0.0693339	−	0.997594i	$-0.477913\pi$
$607$	3.41641	0.138668	0.0693339	+	0.997594i	$0.477913\pi$
$608$	36.3607	1.47462
$609$	0	0
$610$	−0.291796	−0.0118145
$611$	−8.23607	−0.333196
$612$	0	0
$613$	−23.0000	−0.928961	−0.464481	−	0.885583i	$-0.653759\pi$
$613$	−23.0000	−0.928961	−0.464481	+	0.885583i	$0.653759\pi$
$614$	−16.5836	−0.669259
$615$	0	0
$616$	−2.23607	−0.0900937
$617$	9.05573	0.364570	0.182285	−	0.983246i	$-0.441651\pi$
$617$	9.05573	0.364570	0.182285	+	0.983246i	$0.441651\pi$
$618$	0	0
$619$	−27.3050	−1.09748	−0.548739	−	0.835994i	$-0.684892\pi$
$619$	−27.3050	−1.09748	−0.548739	+	0.835994i	$0.684892\pi$
$620$	12.9443	0.519854
$621$	0	0
$622$	−12.4377	−0.498706
$623$	−1.87539	−0.0751358
$624$	0	0
$625$	1.00000	0.0400000
$626$	−0.978714	−0.0391173
$627$	0	0
$628$	−18.4721	−0.737118
$629$	0	0
$630$	0	0
$631$	−7.87539	−0.313514	−0.156757	−	0.987637i	$-0.550104\pi$
$631$	−7.87539	−0.313514	−0.156757	+	0.987637i	$0.550104\pi$
$632$	−13.4164	−0.533676
$633$	0	0
$634$	−9.20163	−0.365443
$635$	−12.0000	−0.476205
$636$	0	0
$637$	−6.94427	−0.275142
$638$	2.61803	0.103649
$639$	0	0
$640$	11.3820	0.449912
$641$	13.9443	0.550766	0.275383	−	0.961335i	$-0.411195\pi$
$641$	13.9443	0.550766	0.275383	+	0.961335i	$0.411195\pi$
$642$	0	0
$643$	−21.5410	−0.849495	−0.424747	−	0.905312i	$-0.639637\pi$
$643$	−21.5410	−0.849495	−0.424747	+	0.905312i	$0.639637\pi$
$644$	−1.70820	−0.0673127
$645$	0	0
$646$	5.88854	0.231682
$647$	29.0557	1.14230	0.571149	−	0.820846i	$-0.306498\pi$
$647$	29.0557	1.14230	0.571149	+	0.820846i	$0.306498\pi$
$648$	0	0
$649$	−25.4164	−0.997681
$650$	−0.618034	−0.0242413
$651$	0	0
$652$	8.18034	0.320367
$653$	18.8885	0.739166	0.369583	−	0.929198i	$-0.379501\pi$
$653$	18.8885	0.739166	0.369583	+	0.929198i	$0.379501\pi$
$654$	0	0
$655$	−0.236068	−0.00922394
$656$	11.1246	0.434343
$657$	0	0
$658$	1.20163	0.0468443
$659$	−48.4853	−1.88872	−0.944359	−	0.328915i	$-0.893317\pi$
$659$	−48.4853	−1.88872	−0.944359	+	0.328915i	$0.893317\pi$
$660$	0	0
$661$	−26.4164	−1.02748	−0.513740	−	0.857946i	$-0.671740\pi$
$661$	−26.4164	−1.02748	−0.513740	+	0.857946i	$0.671740\pi$
$662$	−16.0000	−0.621858
$663$	0	0
$664$	−14.4721	−0.561628
$665$	−1.52786	−0.0592480
$666$	0	0
$667$	4.47214	0.173162
$668$	21.7082	0.839916
$669$	0	0
$670$	9.09017	0.351184
$671$	−2.00000	−0.0772091
$672$	0	0
$673$	1.58359	0.0610430	0.0305215	−	0.999534i	$-0.490283\pi$
$673$	1.58359	0.0610430	0.0305215	+	0.999534i	$0.490283\pi$
$674$	−3.12461	−0.120356
$675$	0	0
$676$	19.4164	0.746785
$677$	−0.0557281	−0.00214180	−0.00107090	−	0.999999i	$-0.500341\pi$
$677$	−0.0557281	−0.00214180	−0.00107090	+	0.999999i	$0.500341\pi$
$678$	0	0
$679$	−2.69505	−0.103426
$680$	3.29180	0.126235
$681$	0	0
$682$	−20.9443	−0.801998
$683$	−36.9443	−1.41363	−0.706817	−	0.707397i	$-0.749869\pi$
$683$	−36.9443	−1.41363	−0.706817	+	0.707397i	$0.749869\pi$
$684$	0	0
$685$	−10.9443	−0.418159
$686$	2.03444	0.0776754
$687$	0	0
$688$	−11.1246	−0.424122
$689$	6.47214	0.246569
$690$	0	0
$691$	−15.2918	−0.581727	−0.290864	−	0.956765i	$-0.593943\pi$
$691$	−15.2918	−0.581727	−0.290864	+	0.956765i	$0.593943\pi$
$692$	−38.6525	−1.46935
$693$	0	0
$694$	−3.63932	−0.138147
$695$	12.2361	0.464141
$696$	0	0
$697$	8.83282	0.334567
$698$	3.12461	0.118268
$699$	0	0
$700$	−0.381966	−0.0144370
$701$	25.5279	0.964174	0.482087	−	0.876123i	$-0.339879\pi$
$701$	25.5279	0.964174	0.482087	+	0.876123i	$0.339879\pi$
$702$	0	0
$703$	0	0
$704$	1.00000	0.0376889
$705$	0	0
$706$	20.9443	0.788248
$707$	−3.65248	−0.137365
$708$	0	0
$709$	14.0000	0.525781	0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000	0.525781	0.262891	+	0.964826i	$0.415324\pi$
$710$	1.52786	0.0573397
$711$	0	0
$712$	−17.7639	−0.665731
$713$	−35.7771	−1.33986
$714$	0	0
$715$	−4.23607	−0.158420
$716$	−4.76393	−0.178036
$717$	0	0
$718$	−22.1115	−0.825192
$719$	11.8885	0.443368	0.221684	−	0.975119i	$-0.428845\pi$
$719$	11.8885	0.443368	0.221684	+	0.975119i	$0.428845\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	−14.1459	−0.526456
$723$	0	0
$724$	−7.14590	−0.265575
$725$	1.00000	0.0371391
$726$	0	0
$727$	32.2492	1.19606	0.598029	−	0.801475i	$-0.295951\pi$
$727$	32.2492	1.19606	0.598029	+	0.801475i	$0.295951\pi$
$728$	0.527864	0.0195639
$729$	0	0
$730$	3.70820	0.137247
$731$	−8.83282	−0.326693
$732$	0	0
$733$	−12.4721	−0.460669	−0.230334	−	0.973112i	$-0.573982\pi$
$733$	−12.4721	−0.460669	−0.230334	+	0.973112i	$0.573982\pi$
$734$	15.7082	0.579800
$735$	0	0
$736$	−25.1246	−0.926105
$737$	62.3050	2.29503
$738$	0	0
$739$	16.5836	0.610037	0.305019	−	0.952346i	$-0.401337\pi$
$739$	16.5836	0.610037	0.305019	+	0.952346i	$0.401337\pi$
$740$	0	0
$741$	0	0
$742$	−0.944272	−0.0346653
$743$	−44.2361	−1.62286	−0.811432	−	0.584447i	$-0.801312\pi$
$743$	−44.2361	−1.62286	−0.811432	+	0.584447i	$0.801312\pi$
$744$	0	0
$745$	14.4721	0.530218
$746$	−15.3475	−0.561913
$747$	0	0
$748$	10.0902	0.368933
$749$	−1.05573	−0.0385755
$750$	0	0
$751$	−45.8885	−1.67450	−0.837248	−	0.546823i	$-0.815837\pi$
$751$	−45.8885	−1.67450	−0.837248	+	0.546823i	$0.815837\pi$
$752$	−15.2705	−0.556858
$753$	0	0
$754$	−0.618034	−0.0225075
$755$	−12.0000	−0.436725
$756$	0	0
$757$	24.8328	0.902564	0.451282	−	0.892381i	$-0.350967\pi$
$757$	24.8328	0.902564	0.451282	+	0.892381i	$0.350967\pi$
$758$	18.1803	0.660340
$759$	0	0
$760$	−14.4721	−0.524960
$761$	35.5279	1.28788	0.643942	−	0.765074i	$-0.277298\pi$
$761$	35.5279	1.28788	0.643942	+	0.765074i	$0.277298\pi$
$762$	0	0
$763$	−1.98684	−0.0719286
$764$	−14.4721	−0.523584
$765$	0	0
$766$	−11.1246	−0.401949
$767$	6.00000	0.216647
$768$	0	0
$769$	45.3050	1.63374	0.816869	−	0.576823i	$-0.195708\pi$
$769$	45.3050	1.63374	0.816869	+	0.576823i	$0.195708\pi$
$770$	0.618034	0.0222724
$771$	0	0
$772$	−32.3607	−1.16469
$773$	−32.8328	−1.18091	−0.590457	−	0.807069i	$-0.701053\pi$
$773$	−32.8328	−1.18091	−0.590457	+	0.807069i	$0.701053\pi$
$774$	0	0
$775$	−8.00000	−0.287368
$776$	−25.5279	−0.916397
$777$	0	0
$778$	−6.50658	−0.233272
$779$	−38.8328	−1.39133
$780$	0	0
$781$	10.4721	0.374722
$782$	−4.06888	−0.145503
$783$	0	0
$784$	−12.8754	−0.459835
$785$	11.4164	0.407469
$786$	0	0
$787$	−43.7771	−1.56048	−0.780242	−	0.625477i	$-0.784904\pi$
$787$	−43.7771	−1.56048	−0.780242	+	0.625477i	$0.784904\pi$
$788$	−11.4164	−0.406693
$789$	0	0
$790$	3.70820	0.131932
$791$	−1.76393	−0.0627182
$792$	0	0
$793$	0.472136	0.0167660
$794$	−17.2361	−0.611685
$795$	0	0
$796$	−32.5623	−1.15414
$797$	−46.7214	−1.65496	−0.827478	−	0.561499i	$-0.810225\pi$
$797$	−46.7214	−1.65496	−0.827478	+	0.561499i	$0.810225\pi$
$798$	0	0
$799$	−12.1246	−0.428938
$800$	−5.61803	−0.198627
$801$	0	0
$802$	21.2361	0.749872
$803$	25.4164	0.896926
$804$	0	0
$805$	1.05573	0.0372095
$806$	4.94427	0.174155
$807$	0	0
$808$	−34.5967	−1.21711
$809$	2.05573	0.0722756	0.0361378	−	0.999347i	$-0.488494\pi$
$809$	2.05573	0.0722756	0.0361378	+	0.999347i	$0.488494\pi$
$810$	0	0
$811$	2.81966	0.0990117	0.0495058	−	0.998774i	$-0.484235\pi$
$811$	2.81966	0.0990117	0.0495058	+	0.998774i	$0.484235\pi$
$812$	−0.381966	−0.0134044
$813$	0	0
$814$	0	0
$815$	−5.05573	−0.177094
$816$	0	0
$817$	38.8328	1.35859
$818$	−0.360680	−0.0126109
$819$	0	0
$820$	−9.70820	−0.339025
$821$	−48.9443	−1.70817	−0.854083	−	0.520136i	$-0.825881\pi$
$821$	−48.9443	−1.70817	−0.854083	+	0.520136i	$0.825881\pi$
$822$	0	0
$823$	−2.00000	−0.0697156	−0.0348578	−	0.999392i	$-0.511098\pi$
$823$	−2.00000	−0.0697156	−0.0348578	+	0.999392i	$0.511098\pi$
$824$	37.8885	1.31991
$825$	0	0
$826$	−0.875388	−0.0304587
$827$	41.8885	1.45661	0.728304	−	0.685254i	$-0.240309\pi$
$827$	41.8885	1.45661	0.728304	+	0.685254i	$0.240309\pi$
$828$	0	0
$829$	−19.8885	−0.690758	−0.345379	−	0.938463i	$-0.612250\pi$
$829$	−19.8885	−0.690758	−0.345379	+	0.938463i	$0.612250\pi$
$830$	4.00000	0.138842
$831$	0	0
$832$	−0.236068	−0.00818418
$833$	−10.2229	−0.354203
$834$	0	0
$835$	−13.4164	−0.464294
$836$	−44.3607	−1.53425
$837$	0	0
$838$	−6.54102	−0.225956
$839$	−18.7082	−0.645879	−0.322939	−	0.946420i	$-0.604671\pi$
$839$	−18.7082	−0.645879	−0.322939	+	0.946420i	$0.604671\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−5.81966	−0.200559
$843$	0	0
$844$	−27.4164	−0.943712
$845$	−12.0000	−0.412813
$846$	0	0
$847$	1.63932	0.0563277
$848$	12.0000	0.412082
$849$	0	0
$850$	−0.909830	−0.0312069
$851$	0	0
$852$	0	0
$853$	−11.3050	−0.387074	−0.193537	−	0.981093i	$-0.561996\pi$
$853$	−11.3050	−0.387074	−0.193537	+	0.981093i	$0.561996\pi$
$854$	−0.0688837	−0.00235715
$855$	0	0
$856$	−10.0000	−0.341793
$857$	40.7214	1.39102	0.695508	−	0.718519i	$-0.255180\pi$
$857$	40.7214	1.39102	0.695508	+	0.718519i	$0.255180\pi$
$858$	0	0
$859$	−16.8328	−0.574328	−0.287164	−	0.957881i	$-0.592713\pi$
$859$	−16.8328	−0.574328	−0.287164	+	0.957881i	$0.592713\pi$
$860$	9.70820	0.331047
$861$	0	0
$862$	2.18034	0.0742627
$863$	54.4721	1.85425	0.927127	−	0.374748i	$-0.122271\pi$
$863$	54.4721	1.85425	0.927127	+	0.374748i	$0.122271\pi$
$864$	0	0
$865$	23.8885	0.812235
$866$	−14.4033	−0.489442
$867$	0	0
$868$	3.05573	0.103718
$869$	25.4164	0.862193
$870$	0	0
$871$	−14.7082	−0.498368
$872$	−18.8197	−0.637314
$873$	0	0
$874$	17.8885	0.605089
$875$	0.236068	0.00798055
$876$	0	0
$877$	15.8885	0.536518	0.268259	−	0.963347i	$-0.413552\pi$
$877$	15.8885	0.536518	0.268259	+	0.963347i	$0.413552\pi$
$878$	1.09017	0.0367915
$879$	0	0
$880$	−7.85410	−0.264762
$881$	50.8885	1.71448	0.857239	−	0.514918i	$-0.172178\pi$
$881$	50.8885	1.71448	0.857239	+	0.514918i	$0.172178\pi$
$882$	0	0
$883$	38.8328	1.30683	0.653414	−	0.757001i	$-0.273336\pi$
$883$	38.8328	1.30683	0.653414	+	0.757001i	$0.273336\pi$
$884$	−2.38197	−0.0801142
$885$	0	0
$886$	6.32624	0.212534
$887$	17.1803	0.576859	0.288430	−	0.957501i	$-0.406867\pi$
$887$	17.1803	0.576859	0.288430	+	0.957501i	$0.406867\pi$
$888$	0	0
$889$	−2.83282	−0.0950096
$890$	4.90983	0.164578
$891$	0	0
$892$	14.8541	0.497352
$893$	53.3050	1.78378
$894$	0	0
$895$	2.94427	0.0984162
$896$	2.68692	0.0897636
$897$	0	0
$898$	14.8673	0.496127
$899$	−8.00000	−0.266815
$900$	0	0
$901$	9.52786	0.317419
$902$	15.7082	0.523026
$903$	0	0
$904$	−16.7082	−0.555707
$905$	4.41641	0.146806
$906$	0	0
$907$	28.8328	0.957378	0.478689	−	0.877985i	$-0.341112\pi$
$907$	28.8328	0.957378	0.478689	+	0.877985i	$0.341112\pi$
$908$	36.3607	1.20667
$909$	0	0
$910$	−0.145898	−0.00483647
$911$	26.2361	0.869240	0.434620	−	0.900614i	$-0.356883\pi$
$911$	26.2361	0.869240	0.434620	+	0.900614i	$0.356883\pi$
$912$	0	0
$913$	27.4164	0.907351
$914$	−13.2016	−0.436671
$915$	0	0
$916$	−6.47214	−0.213845
$917$	−0.0557281	−0.00184030
$918$	0	0
$919$	−45.5410	−1.50226	−0.751130	−	0.660155i	$-0.770491\pi$
$919$	−45.5410	−1.50226	−0.751130	+	0.660155i	$0.770491\pi$
$920$	10.0000	0.329690
$921$	0	0
$922$	18.4033	0.606079
$923$	−2.47214	−0.0813713
$924$	0	0
$925$	0	0
$926$	7.56231	0.248513
$927$	0	0
$928$	−5.61803	−0.184421
$929$	−26.9443	−0.884013	−0.442006	−	0.897012i	$-0.645733\pi$
$929$	−26.9443	−0.884013	−0.442006	+	0.897012i	$0.645733\pi$
$930$	0	0
$931$	44.9443	1.47299
$932$	29.1246	0.954008
$933$	0	0
$934$	14.8328	0.485345
$935$	−6.23607	−0.203941
$936$	0	0
$937$	36.5279	1.19331	0.596657	−	0.802497i	$-0.296495\pi$
$937$	36.5279	1.19331	0.596657	+	0.802497i	$0.296495\pi$
$938$	2.14590	0.0700661
$939$	0	0
$940$	13.3262	0.434654
$941$	13.4164	0.437362	0.218681	−	0.975796i	$-0.429825\pi$
$941$	13.4164	0.437362	0.218681	+	0.975796i	$0.429825\pi$
$942$	0	0
$943$	26.8328	0.873797
$944$	11.1246	0.362075
$945$	0	0
$946$	−15.7082	−0.510718
$947$	−9.54102	−0.310041	−0.155021	−	0.987911i	$-0.549544\pi$
$947$	−9.54102	−0.310041	−0.155021	+	0.987911i	$0.549544\pi$
$948$	0	0
$949$	−6.00000	−0.194768
$950$	4.00000	0.129777
$951$	0	0
$952$	0.777088	0.0251856
$953$	−46.3607	−1.50177	−0.750885	−	0.660433i	$-0.770373\pi$
$953$	−46.3607	−1.50177	−0.750885	+	0.660433i	$0.770373\pi$
$954$	0	0
$955$	8.94427	0.289430
$956$	−4.94427	−0.159909
$957$	0	0
$958$	1.16718	0.0377100
$959$	−2.58359	−0.0834285
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	0	0
$964$	8.09017	0.260567
$965$	20.0000	0.643823
$966$	0	0
$967$	−15.4164	−0.495758	−0.247879	−	0.968791i	$-0.579734\pi$
$967$	−15.4164	−0.495758	−0.247879	+	0.968791i	$0.579734\pi$
$968$	15.5279	0.499084
$969$	0	0
$970$	7.05573	0.226546
$971$	−45.8885	−1.47263	−0.736317	−	0.676637i	$-0.763437\pi$
$971$	−45.8885	−1.47263	−0.736317	+	0.676637i	$0.763437\pi$
$972$	0	0
$973$	2.88854	0.0926025
$974$	15.4164	0.493974
$975$	0	0
$976$	0.875388	0.0280205
$977$	31.0557	0.993561	0.496780	−	0.867876i	$-0.334515\pi$
$977$	31.0557	0.993561	0.496780	+	0.867876i	$0.334515\pi$
$978$	0	0
$979$	33.6525	1.07554
$980$	11.2361	0.358923
$981$	0	0
$982$	−24.0000	−0.765871
$983$	16.9443	0.540438	0.270219	−	0.962799i	$-0.412904\pi$
$983$	16.9443	0.540438	0.270219	+	0.962799i	$0.412904\pi$
$984$	0	0
$985$	7.05573	0.224814
$986$	−0.909830	−0.0289749
$987$	0	0
$988$	10.4721	0.333163
$989$	−26.8328	−0.853234
$990$	0	0
$991$	10.7082	0.340157	0.170079	−	0.985430i	$-0.445598\pi$
$991$	10.7082	0.340157	0.170079	+	0.985430i	$0.445598\pi$
$992$	44.9443	1.42698
$993$	0	0
$994$	0.360680	0.0114401
$995$	20.1246	0.637993
$996$	0	0
$997$	−4.94427	−0.156587	−0.0782933	−	0.996930i	$-0.524947\pi$
$997$	−4.94427	−0.156587	−0.0782933	+	0.996930i	$0.524947\pi$
$998$	−8.72949	−0.276327
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1305.2.a.l.1.1	yes	2
3.2	odd	2		1305.2.a.h.1.2	✓	2
5.4	even	2		6525.2.a.r.1.2		2
15.14	odd	2		6525.2.a.bb.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1305.2.a.h.1.2	✓	2	3.2	odd	2
1305.2.a.l.1.1	yes	2	1.1	even	1	trivial
6525.2.a.r.1.2		2	5.4	even	2
6525.2.a.bb.1.1		2	15.14	odd	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:	\( N \)	\(=\)	\( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1305.a (trivial)

\(f(q)\)	\(=\)	\(q-0.618034 q^{2} -1.61803 q^{4} +1.00000 q^{5} +0.236068 q^{7} +2.23607 q^{8} -0.618034 q^{10} -4.23607 q^{11} +1.00000 q^{13} -0.145898 q^{14} +1.85410 q^{16} +1.47214 q^{17} -6.47214 q^{19} -1.61803 q^{20} +2.61803 q^{22} +4.47214 q^{23} +1.00000 q^{25} -0.618034 q^{26} -0.381966 q^{28} +1.00000 q^{29} -8.00000 q^{31} -5.61803 q^{32} -0.909830 q^{34} +0.236068 q^{35} +4.00000 q^{38} +2.23607 q^{40} +6.00000 q^{41} -6.00000 q^{43} +6.85410 q^{44} -2.76393 q^{46} -8.23607 q^{47} -6.94427 q^{49} -0.618034 q^{50} -1.61803 q^{52} +6.47214 q^{53} -4.23607 q^{55} +0.527864 q^{56} -0.618034 q^{58} +6.00000 q^{59} +0.472136 q^{61} +4.94427 q^{62} -0.236068 q^{64} +1.00000 q^{65} -14.7082 q^{67} -2.38197 q^{68} -0.145898 q^{70} -2.47214 q^{71} -6.00000 q^{73} +10.4721 q^{76} -1.00000 q^{77} -6.00000 q^{79} +1.85410 q^{80} -3.70820 q^{82} -6.47214 q^{83} +1.47214 q^{85} +3.70820 q^{86} -9.47214 q^{88} -7.94427 q^{89} +0.236068 q^{91} -7.23607 q^{92} +5.09017 q^{94} -6.47214 q^{95} -11.4164 q^{97} +4.29180 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} + 2 q^{5} - 4 q^{7} + q^{10} - 4 q^{11} + 2 q^{13} - 7 q^{14} - 3 q^{16} - 6 q^{17} - 4 q^{19} - q^{20} + 3 q^{22} + 2 q^{25} + q^{26} - 3 q^{28} + 2 q^{29} - 16 q^{31} - 9 q^{32}+ \cdots + 22 q^{98}+O(q^{100}) \) 2 * q + q^2 - q^4 + 2 * q^5 - 4 * q^7 + q^10 - 4 * q^11 + 2 * q^13 - 7 * q^14 - 3 * q^16 - 6 * q^17 - 4 * q^19 - q^20 + 3 * q^22 + 2 * q^25 + q^26 - 3 * q^28 + 2 * q^29 - 16 * q^31 - 9 * q^32 - 13 * q^34 - 4 * q^35 + 8 * q^38 + 12 * q^41 - 12 * q^43 + 7 * q^44 - 10 * q^46 - 12 * q^47 + 4 * q^49 + q^50 - q^52 + 4 * q^53 - 4 * q^55 + 10 * q^56 + q^58 + 12 * q^59 - 8 * q^61 - 8 * q^62 + 4 * q^64 + 2 * q^65 - 16 * q^67 - 7 * q^68 - 7 * q^70 + 4 * q^71 - 12 * q^73 + 12 * q^76 - 2 * q^77 - 12 * q^79 - 3 * q^80 + 6 * q^82 - 4 * q^83 - 6 * q^85 - 6 * q^86 - 10 * q^88 + 2 * q^89 - 4 * q^91 - 10 * q^92 - q^94 - 4 * q^95 + 4 * q^97 + 22 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more