LMFDB - Embedded newform 7605.2.a.cn.1.4

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:	$5$
Coefficient field:	5.5.3352656.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 6x^{3} + 10x^{2} + 6x - 6 $ x^5 - 2x^4 - 6x^3 + 10x^2 + 6x - 6
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 585)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.81031$ 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)

$f(q)$	$=$	$q+1.81031 q^{2} +1.27724 q^{4} +1.00000 q^{5} +4.42503 q^{7} -1.30843 q^{8} +O(q^{10})$	$q+1.81031 q^{2} +1.27724 q^{4} +1.00000 q^{5} +4.42503 q^{7} -1.30843 q^{8} +1.81031 q^{10} -1.61471 q^{11} +8.01069 q^{14} -4.92314 q^{16} +1.25584 q^{17} +0.722761 q^{19} +1.27724 q^{20} -2.92314 q^{22} -1.30843 q^{23} +1.00000 q^{25} +5.65182 q^{28} -1.16064 q^{29} +1.64968 q^{31} -6.29558 q^{32} +2.27346 q^{34} +4.42503 q^{35} -4.92314 q^{37} +1.30843 q^{38} -1.30843 q^{40} +8.56517 q^{41} +7.15370 q^{43} -2.06237 q^{44} -2.36866 q^{46} +12.3482 q^{47} +12.5809 q^{49} +1.81031 q^{50} +13.4045 q^{53} -1.61471 q^{55} -5.78982 q^{56} -2.10112 q^{58} +4.28489 q^{59} +5.64968 q^{61} +2.98644 q^{62} -1.55070 q^{64} -0.218735 q^{67} +1.60400 q^{68} +8.01069 q^{70} +5.84150 q^{71} -9.91448 q^{73} -8.91243 q^{74} +0.923139 q^{76} -7.14515 q^{77} -14.1353 q^{79} -4.92314 q^{80} +15.5056 q^{82} +11.0342 q^{83} +1.25584 q^{85} +12.9505 q^{86} +2.11273 q^{88} -0.845279 q^{89} -1.67117 q^{92} +22.3541 q^{94} +0.722761 q^{95} -13.5545 q^{97} +22.7753 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{2} + 6 q^{4} + 5 q^{5} - q^{7} + 6 q^{8}+O(q^{10}) $ 5 * q + 2 * q^2 + 6 * q^4 + 5 * q^5 - q^7 + 6 * q^8	$ 5 q + 2 q^{2} + 6 q^{4} + 5 q^{5} - q^{7} + 6 q^{8} + 2 q^{10} + 8 q^{11} + 4 q^{14} + 4 q^{16} + 4 q^{19} + 6 q^{20} + 14 q^{22} + 6 q^{23} + 5 q^{25} + 2 q^{28} - 16 q^{29} - 9 q^{31} + 14 q^{32} - q^{35} + 4 q^{37} - 6 q^{38} + 6 q^{40} + 6 q^{41} + 15 q^{43} + 14 q^{44} + 16 q^{46} + 10 q^{47} + 10 q^{49} + 2 q^{50} + 20 q^{53} + 8 q^{55} + 2 q^{56} + 4 q^{58} + 12 q^{59} + 11 q^{61} + 22 q^{62} + 4 q^{64} - 5 q^{67} - 50 q^{68} + 4 q^{70} + 10 q^{71} - q^{73} + 26 q^{74} - 24 q^{76} - 42 q^{77} - 17 q^{79} + 4 q^{80} + 16 q^{82} + 16 q^{83} + 44 q^{86} + 20 q^{88} + 4 q^{89} + 34 q^{92} - 16 q^{94} + 4 q^{95} + 11 q^{97} + 30 q^{98}+O(q^{100}) $ 5 * q + 2 * q^2 + 6 * q^4 + 5 * q^5 - q^7 + 6 * q^8 + 2 * q^10 + 8 * q^11 + 4 * q^14 + 4 * q^16 + 4 * q^19 + 6 * q^20 + 14 * q^22 + 6 * q^23 + 5 * q^25 + 2 * q^28 - 16 * q^29 - 9 * q^31 + 14 * q^32 - q^35 + 4 * q^37 - 6 * q^38 + 6 * q^40 + 6 * q^41 + 15 * q^43 + 14 * q^44 + 16 * q^46 + 10 * q^47 + 10 * q^49 + 2 * q^50 + 20 * q^53 + 8 * q^55 + 2 * q^56 + 4 * q^58 + 12 * q^59 + 11 * q^61 + 22 * q^62 + 4 * q^64 - 5 * q^67 - 50 * q^68 + 4 * q^70 + 10 * q^71 - q^73 + 26 * q^74 - 24 * q^76 - 42 * q^77 - 17 * q^79 + 4 * q^80 + 16 * q^82 + 16 * q^83 + 44 * q^86 + 20 * q^88 + 4 * q^89 + 34 * q^92 - 16 * q^94 + 4 * q^95 + 11 * q^97 + 30 * 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 $

$2$	1.81031	1.28009	0.640043	−	0.768339i	$-0.278917\pi$
$2$	1.81031	1.28009	0.640043	+	0.768339i	$0.278917\pi$
$3$	0	0
$3$	0	0
$4$	1.27724	0.638619
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.42503	1.67250	0.836252	−	0.548346i	$-0.184742\pi$
$7$	4.42503	1.67250	0.836252	+	0.548346i	$0.184742\pi$
$8$	−1.30843	−0.462598
$9$	0	0
$10$	1.81031	0.572472
$11$	−1.61471	−0.486854	−0.243427	−	0.969919i	$-0.578272\pi$
$11$	−1.61471	−0.486854	−0.243427	+	0.969919i	$0.578272\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	8.01069	2.14095
$15$	0	0
$16$	−4.92314	−1.23078
$17$	1.25584	0.304585	0.152293	−	0.988335i	$-0.451334\pi$
$17$	1.25584	0.304585	0.152293	+	0.988335i	$0.451334\pi$
$18$	0	0
$19$	0.722761	0.165813	0.0829064	−	0.996557i	$-0.473580\pi$
$19$	0.722761	0.165813	0.0829064	+	0.996557i	$0.473580\pi$
$20$	1.27724	0.285599
$21$	0	0
$22$	−2.92314	−0.623215
$23$	−1.30843	−0.272826	−0.136413	−	0.990652i	$-0.543557\pi$
$23$	−1.30843	−0.272826	−0.136413	+	0.990652i	$0.543557\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	5.65182	1.06809
$29$	−1.16064	−0.215525	−0.107762	−	0.994177i	$-0.534369\pi$
$29$	−1.16064	−0.215525	−0.107762	+	0.994177i	$0.534369\pi$
$30$	0	0
$31$	1.64968	0.296291	0.148145	−	0.988966i	$-0.452670\pi$
$31$	1.64968	0.296291	0.148145	+	0.988966i	$0.452670\pi$
$32$	−6.29558	−1.11291
$33$	0	0
$34$	2.27346	0.389895
$35$	4.42503	0.747966
$36$	0	0
$37$	−4.92314	−0.809359	−0.404680	−	0.914459i	$-0.632617\pi$
$37$	−4.92314	−0.809359	−0.404680	+	0.914459i	$0.632617\pi$
$38$	1.30843	0.212255
$39$	0	0
$40$	−1.30843	−0.206880
$41$	8.56517	1.33765	0.668827	−	0.743418i	$-0.266797\pi$
$41$	8.56517	1.33765	0.668827	+	0.743418i	$0.266797\pi$
$42$	0	0
$43$	7.15370	1.09093	0.545465	−	0.838134i	$-0.316353\pi$
$43$	7.15370	1.09093	0.545465	+	0.838134i	$0.316353\pi$
$44$	−2.06237	−0.310915
$45$	0	0
$46$	−2.36866	−0.349240
$47$	12.3482	1.80117	0.900583	−	0.434685i	$-0.143140\pi$
$47$	12.3482	1.80117	0.900583	+	0.434685i	$0.143140\pi$
$48$	0	0
$49$	12.5809	1.79727
$50$	1.81031	0.256017
$51$	0	0
$52$	0	0
$53$	13.4045	1.84125	0.920627	−	0.390443i	$-0.127678\pi$
$53$	13.4045	1.84125	0.920627	+	0.390443i	$0.127678\pi$
$54$	0	0
$55$	−1.61471	−0.217728
$56$	−5.78982	−0.773697
$57$	0	0
$58$	−2.10112	−0.275890
$59$	4.28489	0.557845	0.278922	−	0.960314i	$-0.410023\pi$
$59$	4.28489	0.557845	0.278922	+	0.960314i	$0.410023\pi$
$60$	0	0
$61$	5.64968	0.723367	0.361684	−	0.932301i	$-0.382202\pi$
$61$	5.64968	0.723367	0.361684	+	0.932301i	$0.382202\pi$
$62$	2.98644	0.379278
$63$	0	0
$64$	−1.55070	−0.193837
$65$	0	0
$66$	0	0
$67$	−0.218735	−0.0267227	−0.0133613	−	0.999911i	$-0.504253\pi$
$67$	−0.218735	−0.0267227	−0.0133613	+	0.999911i	$0.504253\pi$
$68$	1.60400	0.194514
$69$	0	0
$70$	8.01069	0.957461
$71$	5.84150	0.693259	0.346629	−	0.938002i	$-0.387326\pi$
$71$	5.84150	0.693259	0.346629	+	0.938002i	$0.387326\pi$
$72$	0	0
$73$	−9.91448	−1.16040	−0.580201	−	0.814473i	$-0.697026\pi$
$73$	−9.91448	−1.16040	−0.580201	+	0.814473i	$0.697026\pi$
$74$	−8.91243	−1.03605
$75$	0	0
$76$	0.923139	0.105891
$77$	−7.14515	−0.814266
$78$	0	0
$79$	−14.1353	−1.59035	−0.795175	−	0.606379i	$-0.792621\pi$
$79$	−14.1353	−1.59035	−0.795175	+	0.606379i	$0.792621\pi$
$80$	−4.92314	−0.550424
$81$	0	0
$82$	15.5056	1.71231
$83$	11.0342	1.21116	0.605582	−	0.795783i	$-0.292940\pi$
$83$	11.0342	1.21116	0.605582	+	0.795783i	$0.292940\pi$
$84$	0	0
$85$	1.25584	0.136215
$86$	12.9505	1.39648
$87$	0	0
$88$	2.11273	0.225218
$89$	−0.845279	−0.0895994	−0.0447997	−	0.998996i	$-0.514265\pi$
$89$	−0.845279	−0.0895994	−0.0447997	+	0.998996i	$0.514265\pi$
$90$	0	0
$91$	0	0
$92$	−1.67117	−0.174232
$93$	0	0
$94$	22.3541	2.30565
$95$	0.722761	0.0741538
$96$	0	0
$97$	−13.5545	−1.37625	−0.688123	−	0.725594i	$-0.741565\pi$
$97$	−13.5545	−1.37625	−0.688123	+	0.725594i	$0.741565\pi$
$98$	22.7753	2.30066
$99$	0	0
$100$	1.27724	0.127724
$101$	17.9221	1.78332	0.891659	−	0.452708i	$-0.149542\pi$
$101$	17.9221	1.78332	0.891659	+	0.452708i	$0.149542\pi$
$102$	0	0
$103$	13.8134	1.36107	0.680535	−	0.732715i	$-0.261747\pi$
$103$	13.8134	1.36107	0.680535	+	0.732715i	$0.261747\pi$
$104$	0	0
$105$	0	0
$106$	24.2664	2.35696
$107$	−9.73132	−0.940762	−0.470381	−	0.882463i	$-0.655883\pi$
$107$	−9.73132	−0.940762	−0.470381	+	0.882463i	$0.655883\pi$
$108$	0	0
$109$	−4.36061	−0.417671	−0.208835	−	0.977951i	$-0.566967\pi$
$109$	−4.36061	−0.417671	−0.208835	+	0.977951i	$0.566967\pi$
$110$	−2.92314	−0.278710
$111$	0	0
$112$	−21.7850	−2.05849
$113$	19.1927	1.80550	0.902749	−	0.430167i	$-0.141545\pi$
$113$	19.1927	1.80550	0.902749	+	0.430167i	$0.141545\pi$
$114$	0	0
$115$	−1.30843	−0.122011
$116$	−1.48241	−0.137638
$117$	0	0
$118$	7.75699	0.714089
$119$	5.55712	0.509420
$120$	0	0
$121$	−8.39270	−0.762973
$122$	10.2277	0.925972
$123$	0	0
$124$	2.10703	0.189217
$125$	1.00000	0.0894427
$126$	0	0
$127$	−3.04076	−0.269824	−0.134912	−	0.990858i	$-0.543075\pi$
$127$	−3.04076	−0.269824	−0.134912	+	0.990858i	$0.543075\pi$
$128$	9.78390	0.864783
$129$	0	0
$130$	0	0
$131$	−15.4894	−1.35332	−0.676659	−	0.736297i	$-0.736573\pi$
$131$	−15.4894	−1.35332	−0.676659	+	0.736297i	$0.736573\pi$
$132$	0	0
$133$	3.19824	0.277323
$134$	−0.395979	−0.0342073
$135$	0	0
$136$	−1.64317	−0.140901
$137$	−8.69838	−0.743153	−0.371576	−	0.928402i	$-0.621183\pi$
$137$	−8.69838	−0.743153	−0.371576	+	0.928402i	$0.621183\pi$
$138$	0	0
$139$	2.02832	0.172039	0.0860197	−	0.996293i	$-0.472585\pi$
$139$	2.02832	0.172039	0.0860197	+	0.996293i	$0.472585\pi$
$140$	5.65182	0.477666
$141$	0	0
$142$	10.5750	0.887430
$143$	0	0
$144$	0	0
$145$	−1.16064	−0.0963856
$146$	−17.9483	−1.48541
$147$	0	0
$148$	−6.28802	−0.516872
$149$	5.61044	0.459625	0.229812	−	0.973235i	$-0.426189\pi$
$149$	5.61044	0.459625	0.229812	+	0.973235i	$0.426189\pi$
$150$	0	0
$151$	−15.5158	−1.26266	−0.631331	−	0.775514i	$-0.717491\pi$
$151$	−15.5158	−1.26266	−0.631331	+	0.775514i	$0.717491\pi$
$152$	−0.945680	−0.0767047
$153$	0	0
$154$	−12.9350	−1.04233
$155$	1.64968	0.132505
$156$	0	0
$157$	−14.4852	−1.15604	−0.578021	−	0.816022i	$-0.696175\pi$
$157$	−14.4852	−1.15604	−0.578021	+	0.816022i	$0.696175\pi$
$158$	−25.5894	−2.03579
$159$	0	0
$160$	−6.29558	−0.497709
$161$	−5.78982	−0.456302
$162$	0	0
$163$	−15.5340	−1.21672	−0.608358	−	0.793663i	$-0.708171\pi$
$163$	−15.5340	−1.21672	−0.608358	+	0.793663i	$0.708171\pi$
$164$	10.9398	0.854252
$165$	0	0
$166$	19.9754	1.55039
$167$	−20.3288	−1.57309	−0.786545	−	0.617533i	$-0.788132\pi$
$167$	−20.3288	−1.57309	−0.786545	+	0.617533i	$0.788132\pi$
$168$	0	0
$169$	0	0
$170$	2.27346	0.174366
$171$	0	0
$172$	9.13699	0.696689
$173$	19.1021	1.45231	0.726153	−	0.687533i	$-0.241306\pi$
$173$	19.1021	1.45231	0.726153	+	0.687533i	$0.241306\pi$
$174$	0	0
$175$	4.42503	0.334501
$176$	7.94946	0.599213
$177$	0	0
$178$	−1.53022	−0.114695
$179$	10.9429	0.817911	0.408955	−	0.912554i	$-0.365893\pi$
$179$	10.9429	0.817911	0.408955	+	0.912554i	$0.365893\pi$
$180$	0	0
$181$	−1.45122	−0.107869	−0.0539343	−	0.998544i	$-0.517176\pi$
$181$	−1.45122	−0.107869	−0.0539343	+	0.998544i	$0.517176\pi$
$182$	0	0
$183$	0	0
$184$	1.71198	0.126209
$185$	−4.92314	−0.361956
$186$	0	0
$187$	−2.02782	−0.148289
$188$	15.7716	1.15026
$189$	0	0
$190$	1.30843	0.0949232
$191$	−14.6797	−1.06219	−0.531094	−	0.847313i	$-0.678219\pi$
$191$	−14.6797	−1.06219	−0.531094	+	0.847313i	$0.678219\pi$
$192$	0	0
$193$	19.8338	1.42767	0.713835	−	0.700314i	$-0.246956\pi$
$193$	19.8338	1.42767	0.713835	+	0.700314i	$0.246956\pi$
$194$	−24.5378	−1.76171
$195$	0	0
$196$	16.0688	1.14777
$197$	−12.3917	−0.882871	−0.441436	−	0.897293i	$-0.645531\pi$
$197$	−12.3917	−0.882871	−0.441436	+	0.897293i	$0.645531\pi$
$198$	0	0
$199$	23.1127	1.63842	0.819208	−	0.573497i	$-0.194413\pi$
$199$	23.1127	1.63842	0.819208	+	0.573497i	$0.194413\pi$
$200$	−1.30843	−0.0925197
$201$	0	0
$202$	32.4447	2.28280
$203$	−5.13585	−0.360466
$204$	0	0
$205$	8.56517	0.598217
$206$	25.0065	1.74229
$207$	0	0
$208$	0	0
$209$	−1.16705	−0.0807267
$210$	0	0
$211$	−8.85383	−0.609523	−0.304762	−	0.952429i	$-0.598577\pi$
$211$	−8.85383	−0.609523	−0.304762	+	0.952429i	$0.598577\pi$
$212$	17.1208	1.17586
$213$	0	0
$214$	−17.6167	−1.20426
$215$	7.15370	0.487879
$216$	0	0
$217$	7.29987	0.495548
$218$	−7.89407	−0.534654
$219$	0	0
$220$	−2.06237	−0.139045
$221$	0	0
$222$	0	0
$223$	4.27532	0.286296	0.143148	−	0.989701i	$-0.454277\pi$
$223$	4.27532	0.286296	0.143148	+	0.989701i	$0.454277\pi$
$224$	−27.8581	−1.86135
$225$	0	0
$226$	34.7448	2.31119
$227$	21.4143	1.42132	0.710659	−	0.703537i	$-0.248397\pi$
$227$	21.4143	1.42132	0.710659	+	0.703537i	$0.248397\pi$
$228$	0	0
$229$	−0.661510	−0.0437138	−0.0218569	−	0.999761i	$-0.506958\pi$
$229$	−0.661510	−0.0437138	−0.0218569	+	0.999761i	$0.506958\pi$
$230$	−2.36866	−0.156185
$231$	0	0
$232$	1.51861	0.0997014
$233$	−13.3665	−0.875670	−0.437835	−	0.899055i	$-0.644255\pi$
$233$	−13.3665	−0.875670	−0.437835	+	0.899055i	$0.644255\pi$
$234$	0	0
$235$	12.3482	0.805506
$236$	5.47282	0.356250
$237$	0	0
$238$	10.0601	0.652101
$239$	4.44339	0.287419	0.143709	−	0.989620i	$-0.454097\pi$
$239$	4.44339	0.287419	0.143709	+	0.989620i	$0.454097\pi$
$240$	0	0
$241$	14.4226	0.929039	0.464519	−	0.885563i	$-0.346227\pi$
$241$	14.4226	0.929039	0.464519	+	0.885563i	$0.346227\pi$
$242$	−15.1934	−0.976671
$243$	0	0
$244$	7.21599	0.461956
$245$	12.5809	0.803762
$246$	0	0
$247$	0	0
$248$	−2.15848	−0.137064
$249$	0	0
$250$	1.81031	0.114494
$251$	3.36540	0.212422	0.106211	−	0.994344i	$-0.466128\pi$
$251$	3.36540	0.212422	0.106211	+	0.994344i	$0.466128\pi$
$252$	0	0
$253$	2.11273	0.132826
$254$	−5.50473	−0.345397
$255$	0	0
$256$	20.8133	1.30083
$257$	−23.8116	−1.48533	−0.742664	−	0.669665i	$-0.766438\pi$
$257$	−23.8116	−1.48533	−0.742664	+	0.669665i	$0.766438\pi$
$258$	0	0
$259$	−21.7850	−1.35366
$260$	0	0
$261$	0	0
$262$	−28.0407	−1.73236
$263$	−20.2963	−1.25152	−0.625762	−	0.780014i	$-0.715212\pi$
$263$	−20.2963	−1.25152	−0.625762	+	0.780014i	$0.715212\pi$
$264$	0	0
$265$	13.4045	0.823434
$266$	5.78982	0.354997
$267$	0	0
$268$	−0.279376	−0.0170656
$269$	−0.205576	−0.0125342	−0.00626711	−	0.999980i	$-0.501995\pi$
$269$	−0.205576	−0.0125342	−0.00626711	+	0.999980i	$0.501995\pi$
$270$	0	0
$271$	10.1985	0.619512	0.309756	−	0.950816i	$-0.399753\pi$
$271$	10.1985	0.619512	0.309756	+	0.950816i	$0.399753\pi$
$272$	−6.18266	−0.374879
$273$	0	0
$274$	−15.7468	−0.951299
$275$	−1.61471	−0.0973709
$276$	0	0
$277$	28.1858	1.69352	0.846760	−	0.531976i	$-0.178550\pi$
$277$	28.1858	1.69352	0.846760	+	0.531976i	$0.178550\pi$
$278$	3.67189	0.220225
$279$	0	0
$280$	−5.78982	−0.346008
$281$	17.2672	1.03008	0.515039	−	0.857167i	$-0.327778\pi$
$281$	17.2672	1.03008	0.515039	+	0.857167i	$0.327778\pi$
$282$	0	0
$283$	28.0197	1.66560	0.832798	−	0.553577i	$-0.186738\pi$
$283$	28.0197	1.66560	0.832798	+	0.553577i	$0.186738\pi$
$284$	7.46099	0.442728
$285$	0	0
$286$	0	0
$287$	37.9011	2.23723
$288$	0	0
$289$	−15.4229	−0.907228
$290$	−2.10112	−0.123382
$291$	0	0
$292$	−12.6631	−0.741055
$293$	−7.03495	−0.410986	−0.205493	−	0.978659i	$-0.565880\pi$
$293$	−7.03495	−0.410986	−0.205493	+	0.978659i	$0.565880\pi$
$294$	0	0
$295$	4.28489	0.249476
$296$	6.44156	0.374408
$297$	0	0
$298$	10.1567	0.588359
$299$	0	0
$300$	0	0
$301$	31.6553	1.82458
$302$	−28.0886	−1.61631
$303$	0	0
$304$	−3.55825	−0.204080
$305$	5.64968	0.323500
$306$	0	0
$307$	−17.9307	−1.02336	−0.511679	−	0.859177i	$-0.670976\pi$
$307$	−17.9307	−1.02336	−0.511679	+	0.859177i	$0.670976\pi$
$308$	−9.12606	−0.520006
$309$	0	0
$310$	2.98644	0.169618
$311$	8.59247	0.487234	0.243617	−	0.969871i	$-0.421666\pi$
$311$	8.59247	0.487234	0.243617	+	0.969871i	$0.421666\pi$
$312$	0	0
$313$	−21.8298	−1.23389	−0.616946	−	0.787006i	$-0.711630\pi$
$313$	−21.8298	−1.23389	−0.616946	+	0.787006i	$0.711630\pi$
$314$	−26.2227	−1.47983
$315$	0	0
$316$	−18.0542	−1.01563
$317$	−20.4269	−1.14729	−0.573645	−	0.819104i	$-0.694471\pi$
$317$	−20.4269	−1.14729	−0.573645	+	0.819104i	$0.694471\pi$
$318$	0	0
$319$	1.87409	0.104929
$320$	−1.55070	−0.0866867
$321$	0	0
$322$	−10.4814	−0.584105
$323$	0.907671	0.0505042
$324$	0	0
$325$	0	0
$326$	−28.1214	−1.55750
$327$	0	0
$328$	−11.2069	−0.618797
$329$	54.6410	3.01245
$330$	0	0
$331$	1.08491	0.0596323	0.0298162	−	0.999555i	$-0.490508\pi$
$331$	1.08491	0.0596323	0.0298162	+	0.999555i	$0.490508\pi$
$332$	14.0933	0.773473
$333$	0	0
$334$	−36.8015	−2.01369
$335$	−0.218735	−0.0119508
$336$	0	0
$337$	−15.3641	−0.836934	−0.418467	−	0.908232i	$-0.637432\pi$
$337$	−15.3641	−0.836934	−0.418467	+	0.908232i	$0.637432\pi$
$338$	0	0
$339$	0	0
$340$	1.60400	0.0869893
$341$	−2.66376	−0.144251
$342$	0	0
$343$	24.6955	1.33343
$344$	−9.36009	−0.504662
$345$	0	0
$346$	34.5808	1.85908
$347$	−31.7743	−1.70573	−0.852867	−	0.522128i	$-0.825138\pi$
$347$	−31.7743	−1.70573	−0.852867	+	0.522128i	$0.825138\pi$
$348$	0	0
$349$	−7.01456	−0.375481	−0.187740	−	0.982219i	$-0.560116\pi$
$349$	−7.01456	−0.375481	−0.187740	+	0.982219i	$0.560116\pi$
$350$	8.01069	0.428189
$351$	0	0
$352$	10.1656	0.541826
$353$	30.2070	1.60776	0.803878	−	0.594794i	$-0.202766\pi$
$353$	30.2070	1.60776	0.803878	+	0.594794i	$0.202766\pi$
$354$	0	0
$355$	5.84150	0.310035
$356$	−1.07962	−0.0572199
$357$	0	0
$358$	19.8101	1.04700
$359$	−16.5984	−0.876029	−0.438015	−	0.898968i	$-0.644318\pi$
$359$	−16.5984	−0.876029	−0.438015	+	0.898968i	$0.644318\pi$
$360$	0	0
$361$	−18.4776	−0.972506
$362$	−2.62717	−0.138081
$363$	0	0
$364$	0	0
$365$	−9.91448	−0.518947
$366$	0	0
$367$	20.3285	1.06114	0.530569	−	0.847641i	$-0.321978\pi$
$367$	20.3285	1.06114	0.530569	+	0.847641i	$0.321978\pi$
$368$	6.44156	0.335790
$369$	0	0
$370$	−8.91243	−0.463335
$371$	59.3154	3.07950
$372$	0	0
$373$	−20.7527	−1.07453	−0.537267	−	0.843412i	$-0.680543\pi$
$373$	−20.7527	−1.07453	−0.537267	+	0.843412i	$0.680543\pi$
$374$	−3.67099	−0.189822
$375$	0	0
$376$	−16.1567	−0.833216
$377$	0	0
$378$	0	0
$379$	0.865150	0.0444398	0.0222199	−	0.999753i	$-0.492927\pi$
$379$	0.865150	0.0444398	0.0222199	+	0.999753i	$0.492927\pi$
$380$	0.923139	0.0473560
$381$	0	0
$382$	−26.5749	−1.35969
$383$	2.82316	0.144257	0.0721284	−	0.997395i	$-0.477021\pi$
$383$	2.82316	0.144257	0.0721284	+	0.997395i	$0.477021\pi$
$384$	0	0
$385$	−7.14515	−0.364151
$386$	35.9055	1.82754
$387$	0	0
$388$	−17.3123	−0.878898
$389$	−13.4514	−0.682015	−0.341008	−	0.940061i	$-0.610768\pi$
$389$	−13.4514	−0.682015	−0.341008	+	0.940061i	$0.610768\pi$
$390$	0	0
$391$	−1.64317	−0.0830987
$392$	−16.4611	−0.831413
$393$	0	0
$394$	−22.4328	−1.13015
$395$	−14.1353	−0.711227
$396$	0	0
$397$	−8.04013	−0.403523	−0.201761	−	0.979435i	$-0.564667\pi$
$397$	−8.04013	−0.403523	−0.201761	+	0.979435i	$0.564667\pi$
$398$	41.8413	2.09731
$399$	0	0
$400$	−4.92314	−0.246157
$401$	34.3620	1.71596	0.857978	−	0.513686i	$-0.171720\pi$
$401$	34.3620	1.71596	0.857978	+	0.513686i	$0.171720\pi$
$402$	0	0
$403$	0	0
$404$	22.8908	1.13886
$405$	0	0
$406$	−9.29750	−0.461427
$407$	7.94946	0.394040
$408$	0	0
$409$	19.4668	0.962571	0.481286	−	0.876564i	$-0.340170\pi$
$409$	19.4668	0.962571	0.481286	+	0.876564i	$0.340170\pi$
$410$	15.5056	0.765769
$411$	0	0
$412$	17.6430	0.869206
$413$	18.9607	0.932997
$414$	0	0
$415$	11.0342	0.541649
$416$	0	0
$417$	0	0
$418$	−2.11273	−0.103337
$419$	24.4720	1.19554	0.597768	−	0.801669i	$-0.296055\pi$
$419$	24.4720	1.19554	0.597768	+	0.801669i	$0.296055\pi$
$420$	0	0
$421$	−27.6979	−1.34991	−0.674956	−	0.737858i	$-0.735837\pi$
$421$	−27.6979	−1.34991	−0.674956	+	0.737858i	$0.735837\pi$
$422$	−16.0282	−0.780242
$423$	0	0
$424$	−17.5388	−0.851761
$425$	1.25584	0.0609171
$426$	0	0
$427$	25.0000	1.20983
$428$	−12.4292	−0.600789
$429$	0	0
$430$	12.9505	0.624526
$431$	−4.36771	−0.210385	−0.105193	−	0.994452i	$-0.533546\pi$
$431$	−4.36771	−0.210385	−0.105193	+	0.994452i	$0.533546\pi$
$432$	0	0
$433$	−9.05717	−0.435260	−0.217630	−	0.976031i	$-0.569833\pi$
$433$	−9.05717	−0.435260	−0.217630	+	0.976031i	$0.569833\pi$
$434$	13.2151	0.634343
$435$	0	0
$436$	−5.56954	−0.266732
$437$	−0.945680	−0.0452380
$438$	0	0
$439$	−16.1793	−0.772195	−0.386097	−	0.922458i	$-0.626177\pi$
$439$	−16.1793	−0.772195	−0.386097	+	0.922458i	$0.626177\pi$
$440$	2.11273	0.100721
$441$	0	0
$442$	0	0
$443$	−13.6295	−0.647555	−0.323778	−	0.946133i	$-0.604953\pi$
$443$	−13.6295	−0.647555	−0.323778	+	0.946133i	$0.604953\pi$
$444$	0	0
$445$	−0.845279	−0.0400701
$446$	7.73967	0.366484
$447$	0	0
$448$	−6.86189	−0.324194
$449$	11.0543	0.521686	0.260843	−	0.965381i	$-0.416000\pi$
$449$	11.0543	0.521686	0.260843	+	0.965381i	$0.416000\pi$
$450$	0	0
$451$	−13.8303	−0.651243
$452$	24.5137	1.15303
$453$	0	0
$454$	38.7667	1.81941
$455$	0	0
$456$	0	0
$457$	−24.2352	−1.13367	−0.566836	−	0.823830i	$-0.691833\pi$
$457$	−24.2352	−1.13367	−0.566836	+	0.823830i	$0.691833\pi$
$458$	−1.19754	−0.0559574
$459$	0	0
$460$	−1.67117	−0.0779188
$461$	26.6958	1.24335	0.621674	−	0.783276i	$-0.286453\pi$
$461$	26.6958	1.24335	0.621674	+	0.783276i	$0.286453\pi$
$462$	0	0
$463$	−19.8137	−0.920819	−0.460410	−	0.887707i	$-0.652297\pi$
$463$	−19.8137	−0.920819	−0.460410	+	0.887707i	$0.652297\pi$
$464$	5.71397	0.265265
$465$	0	0
$466$	−24.1976	−1.12093
$467$	−22.4652	−1.03956	−0.519782	−	0.854299i	$-0.673987\pi$
$467$	−22.4652	−1.03956	−0.519782	+	0.854299i	$0.673987\pi$
$468$	0	0
$469$	−0.967907	−0.0446938
$470$	22.3541	1.03112
$471$	0	0
$472$	−5.60646	−0.258058
$473$	−11.5512	−0.531124
$474$	0	0
$475$	0.722761	0.0331626
$476$	7.09776	0.325325
$477$	0	0
$478$	8.04392	0.367921
$479$	−25.1683	−1.14997	−0.574985	−	0.818164i	$-0.694992\pi$
$479$	−25.1683	−1.14997	−0.574985	+	0.818164i	$0.694992\pi$
$480$	0	0
$481$	0	0
$482$	26.1094	1.18925
$483$	0	0
$484$	−10.7195	−0.487249
$485$	−13.5545	−0.615476
$486$	0	0
$487$	29.5542	1.33923	0.669613	−	0.742710i	$-0.266460\pi$
$487$	29.5542	1.33923	0.669613	+	0.742710i	$0.266460\pi$
$488$	−7.39218	−0.334628
$489$	0	0
$490$	22.7753	1.02888
$491$	15.4585	0.697630	0.348815	−	0.937192i	$-0.386584\pi$
$491$	15.4585	0.697630	0.348815	+	0.937192i	$0.386584\pi$
$492$	0	0
$493$	−1.45757	−0.0656457
$494$	0	0
$495$	0	0
$496$	−8.12159	−0.364670
$497$	25.8488	1.15948
$498$	0	0
$499$	1.38273	0.0618993	0.0309497	−	0.999521i	$-0.490147\pi$
$499$	1.38273	0.0618993	0.0309497	+	0.999521i	$0.490147\pi$
$500$	1.27724	0.0571198
$501$	0	0
$502$	6.09244	0.271919
$503$	−40.5920	−1.80991	−0.904953	−	0.425511i	$-0.860094\pi$
$503$	−40.5920	−1.80991	−0.904953	+	0.425511i	$0.860094\pi$
$504$	0	0
$505$	17.9221	0.797524
$506$	3.82471	0.170029
$507$	0	0
$508$	−3.88377	−0.172315
$509$	−6.37142	−0.282408	−0.141204	−	0.989980i	$-0.545097\pi$
$509$	−6.37142	−0.282408	−0.141204	+	0.989980i	$0.545097\pi$
$510$	0	0
$511$	−43.8718	−1.94078
$512$	18.1109	0.800396
$513$	0	0
$514$	−43.1065	−1.90135
$515$	13.8134	0.608689
$516$	0	0
$517$	−19.9387	−0.876905
$518$	−39.4377	−1.73280
$519$	0	0
$520$	0	0
$521$	−1.54817	−0.0678264	−0.0339132	−	0.999425i	$-0.510797\pi$
$521$	−1.54817	−0.0678264	−0.0339132	+	0.999425i	$0.510797\pi$
$522$	0	0
$523$	33.0971	1.44723	0.723617	−	0.690201i	$-0.242478\pi$
$523$	33.0971	1.44723	0.723617	+	0.690201i	$0.242478\pi$
$524$	−19.7837	−0.864255
$525$	0	0
$526$	−36.7427	−1.60206
$527$	2.07173	0.0902459
$528$	0	0
$529$	−21.2880	−0.925566
$530$	24.2664	1.05407
$531$	0	0
$532$	4.08491	0.177104
$533$	0	0
$534$	0	0
$535$	−9.73132	−0.420721
$536$	0.286198	0.0123619
$537$	0	0
$538$	−0.372158	−0.0160449
$539$	−20.3145	−0.875007
$540$	0	0
$541$	0.149945	0.00644662	0.00322331	−	0.999995i	$-0.498974\pi$
$541$	0.149945	0.00644662	0.00322331	+	0.999995i	$0.498974\pi$
$542$	18.4624	0.793029
$543$	0	0
$544$	−7.90622	−0.338977
$545$	−4.36061	−0.186788
$546$	0	0
$547$	−5.12882	−0.219293	−0.109646	−	0.993971i	$-0.534972\pi$
$547$	−5.12882	−0.219293	−0.109646	+	0.993971i	$0.534972\pi$
$548$	−11.1099	−0.474592
$549$	0	0
$550$	−2.92314	−0.124643
$551$	−0.838863	−0.0357368
$552$	0	0
$553$	−62.5493	−2.65987
$554$	51.0251	2.16785
$555$	0	0
$556$	2.59064	0.109868
$557$	−3.27399	−0.138723	−0.0693617	−	0.997592i	$-0.522096\pi$
$557$	−3.27399	−0.138723	−0.0693617	+	0.997592i	$0.522096\pi$
$558$	0	0
$559$	0	0
$560$	−21.7850	−0.920585
$561$	0	0
$562$	31.2591	1.31859
$563$	−7.77432	−0.327648	−0.163824	−	0.986490i	$-0.552383\pi$
$563$	−7.77432	−0.327648	−0.163824	+	0.986490i	$0.552383\pi$
$564$	0	0
$565$	19.1927	0.807443
$566$	50.7244	2.13210
$567$	0	0
$568$	−7.64317	−0.320700
$569$	−7.87472	−0.330125	−0.165063	−	0.986283i	$-0.552783\pi$
$569$	−7.87472	−0.330125	−0.165063	+	0.986283i	$0.552783\pi$
$570$	0	0
$571$	−11.4994	−0.481236	−0.240618	−	0.970620i	$-0.577350\pi$
$571$	−11.4994	−0.481236	−0.240618	+	0.970620i	$0.577350\pi$
$572$	0	0
$573$	0	0
$574$	68.6129	2.86385
$575$	−1.30843	−0.0545651
$576$	0	0
$577$	0.797701	0.0332087	0.0166044	−	0.999862i	$-0.494714\pi$
$577$	0.797701	0.0332087	0.0166044	+	0.999862i	$0.494714\pi$
$578$	−27.9202	−1.16133
$579$	0	0
$580$	−1.48241	−0.0615537
$581$	48.8268	2.02568
$582$	0	0
$583$	−21.6445	−0.896423
$584$	12.9724	0.536800
$585$	0	0
$586$	−12.7355	−0.526097
$587$	−18.5437	−0.765379	−0.382689	−	0.923877i	$-0.625002\pi$
$587$	−18.5437	−0.765379	−0.382689	+	0.923877i	$0.625002\pi$
$588$	0	0
$589$	1.19232	0.0491288
$590$	7.75699	0.319350
$591$	0	0
$592$	24.2373	0.996147
$593$	11.6068	0.476633	0.238316	−	0.971188i	$-0.423404\pi$
$593$	11.6068	0.476633	0.238316	+	0.971188i	$0.423404\pi$
$594$	0	0
$595$	5.55712	0.227820
$596$	7.16587	0.293525
$597$	0	0
$598$	0	0
$599$	35.2656	1.44091	0.720457	−	0.693500i	$-0.243932\pi$
$599$	35.2656	1.44091	0.720457	+	0.693500i	$0.243932\pi$
$600$	0	0
$601$	−1.63747	−0.0667938	−0.0333969	−	0.999442i	$-0.510633\pi$
$601$	−1.63747	−0.0667938	−0.0333969	+	0.999442i	$0.510633\pi$
$602$	57.3061	2.33562
$603$	0	0
$604$	−19.8174	−0.806360
$605$	−8.39270	−0.341212
$606$	0	0
$607$	28.2721	1.14753	0.573765	−	0.819020i	$-0.305482\pi$
$607$	28.2721	1.14753	0.573765	+	0.819020i	$0.305482\pi$
$608$	−4.55020	−0.184535
$609$	0	0
$610$	10.2277	0.414107
$611$	0	0
$612$	0	0
$613$	−20.9945	−0.847959	−0.423980	−	0.905672i	$-0.639367\pi$
$613$	−20.9945	−0.847959	−0.423980	+	0.905672i	$0.639367\pi$
$614$	−32.4602	−1.30999
$615$	0	0
$616$	9.34890	0.376678
$617$	−5.35694	−0.215662	−0.107831	−	0.994169i	$-0.534391\pi$
$617$	−5.35694	−0.215662	−0.107831	+	0.994169i	$0.534391\pi$
$618$	0	0
$619$	38.1621	1.53386	0.766931	−	0.641729i	$-0.221783\pi$
$619$	38.1621	1.53386	0.766931	+	0.641729i	$0.221783\pi$
$620$	2.10703	0.0846205
$621$	0	0
$622$	15.5551	0.623701
$623$	−3.74038	−0.149855
$624$	0	0
$625$	1.00000	0.0400000
$626$	−39.5188	−1.57949
$627$	0	0
$628$	−18.5010	−0.738270
$629$	−6.18266	−0.246519
$630$	0	0
$631$	−7.48325	−0.297903	−0.148952	−	0.988844i	$-0.547590\pi$
$631$	−7.48325	−0.297903	−0.148952	+	0.988844i	$0.547590\pi$
$632$	18.4950	0.735694
$633$	0	0
$634$	−36.9792	−1.46863
$635$	−3.04076	−0.120669
$636$	0	0
$637$	0	0
$638$	3.39270	0.134318
$639$	0	0
$640$	9.78390	0.386743
$641$	−45.4922	−1.79683	−0.898416	−	0.439145i	$-0.855282\pi$
$641$	−45.4922	−1.79683	−0.898416	+	0.439145i	$0.855282\pi$
$642$	0	0
$643$	30.7806	1.21387	0.606935	−	0.794752i	$-0.292399\pi$
$643$	30.7806	1.21387	0.606935	+	0.794752i	$0.292399\pi$
$644$	−7.39498	−0.291403
$645$	0	0
$646$	1.64317	0.0646496
$647$	−47.7652	−1.87784	−0.938922	−	0.344130i	$-0.888174\pi$
$647$	−47.7652	−1.87784	−0.938922	+	0.344130i	$0.888174\pi$
$648$	0	0
$649$	−6.91886	−0.271589
$650$	0	0
$651$	0	0
$652$	−19.8406	−0.777018
$653$	9.03547	0.353585	0.176793	−	0.984248i	$-0.443428\pi$
$653$	9.03547	0.353585	0.176793	+	0.984248i	$0.443428\pi$
$654$	0	0
$655$	−15.4894	−0.605222
$656$	−42.1675	−1.64636
$657$	0	0
$658$	98.9174	3.85620
$659$	−40.0775	−1.56120	−0.780598	−	0.625034i	$-0.785085\pi$
$659$	−40.0775	−1.56120	−0.780598	+	0.625034i	$0.785085\pi$
$660$	0	0
$661$	−18.9194	−0.735878	−0.367939	−	0.929850i	$-0.619936\pi$
$661$	−18.9194	−0.735878	−0.367939	+	0.929850i	$0.619936\pi$
$662$	1.96404	0.0763345
$663$	0	0
$664$	−14.4375	−0.560282
$665$	3.19824	0.124022
$666$	0	0
$667$	1.51861	0.0588007
$668$	−25.9647	−1.00461
$669$	0	0
$670$	−0.395979	−0.0152980
$671$	−9.12261	−0.352174
$672$	0	0
$673$	−23.0031	−0.886706	−0.443353	−	0.896347i	$-0.646211\pi$
$673$	−23.0031	−0.886706	−0.443353	+	0.896347i	$0.646211\pi$
$674$	−27.8138	−1.07135
$675$	0	0
$676$	0	0
$677$	41.0126	1.57624	0.788121	−	0.615520i	$-0.211054\pi$
$677$	41.0126	1.57624	0.788121	+	0.615520i	$0.211054\pi$
$678$	0	0
$679$	−59.9789	−2.30178
$680$	−1.64317	−0.0630127
$681$	0	0
$682$	−4.82224	−0.184653
$683$	2.52625	0.0966644	0.0483322	−	0.998831i	$-0.484609\pi$
$683$	2.52625	0.0966644	0.0483322	+	0.998831i	$0.484609\pi$
$684$	0	0
$685$	−8.69838	−0.332348
$686$	44.7066	1.70691
$687$	0	0
$688$	−35.2187	−1.34270
$689$	0	0
$690$	0	0
$691$	−37.7869	−1.43748	−0.718741	−	0.695278i	$-0.755281\pi$
$691$	−37.7869	−1.43748	−0.718741	+	0.695278i	$0.755281\pi$
$692$	24.3980	0.927471
$693$	0	0
$694$	−57.5215	−2.18349
$695$	2.02832	0.0769384
$696$	0	0
$697$	10.7565	0.407430
$698$	−12.6986	−0.480648
$699$	0	0
$700$	5.65182	0.213619
$701$	9.12075	0.344486	0.172243	−	0.985054i	$-0.444899\pi$
$701$	9.12075	0.344486	0.172243	+	0.985054i	$0.444899\pi$
$702$	0	0
$703$	−3.55825	−0.134202
$704$	2.50393	0.0943706
$705$	0	0
$706$	54.6842	2.05807
$707$	79.3059	2.98261
$708$	0	0
$709$	21.7403	0.816473	0.408236	−	0.912876i	$-0.366144\pi$
$709$	21.7403	0.816473	0.408236	+	0.912876i	$0.366144\pi$
$710$	10.5750	0.396871
$711$	0	0
$712$	1.10599	0.0414485
$713$	−2.15848	−0.0808357
$714$	0	0
$715$	0	0
$716$	13.9767	0.522334
$717$	0	0
$718$	−30.0483	−1.12139
$719$	14.3709	0.535945	0.267973	−	0.963427i	$-0.413646\pi$
$719$	14.3709	0.535945	0.267973	+	0.963427i	$0.413646\pi$
$720$	0	0
$721$	61.1245	2.27640
$722$	−33.4503	−1.24489
$723$	0	0
$724$	−1.85356	−0.0688869
$725$	−1.16064	−0.0431050
$726$	0	0
$727$	−38.5088	−1.42821	−0.714106	−	0.700038i	$-0.753166\pi$
$727$	−38.5088	−1.42821	−0.714106	+	0.700038i	$0.753166\pi$
$728$	0	0
$729$	0	0
$730$	−17.9483	−0.664297
$731$	8.98389	0.332281
$732$	0	0
$733$	1.95952	0.0723767	0.0361884	−	0.999345i	$-0.488478\pi$
$733$	1.95952	0.0723767	0.0361884	+	0.999345i	$0.488478\pi$
$734$	36.8010	1.35835
$735$	0	0
$736$	8.23730	0.303631
$737$	0.353194	0.0130101
$738$	0	0
$739$	39.4603	1.45157	0.725784	−	0.687922i	$-0.241477\pi$
$739$	39.4603	1.45157	0.725784	+	0.687922i	$0.241477\pi$
$740$	−6.28802	−0.231152
$741$	0	0
$742$	107.380	3.94203
$743$	33.4899	1.22862	0.614312	−	0.789063i	$-0.289434\pi$
$743$	33.4899	1.22862	0.614312	+	0.789063i	$0.289434\pi$
$744$	0	0
$745$	5.61044	0.205551
$746$	−37.5689	−1.37550
$747$	0	0
$748$	−2.59001	−0.0947000
$749$	−43.0613	−1.57343
$750$	0	0
$751$	−17.1750	−0.626725	−0.313362	−	0.949634i	$-0.601455\pi$
$751$	−17.1750	−0.626725	−0.313362	+	0.949634i	$0.601455\pi$
$752$	−60.7917	−2.21685
$753$	0	0
$754$	0	0
$755$	−15.5158	−0.564679
$756$	0	0
$757$	−42.3190	−1.53811	−0.769055	−	0.639183i	$-0.779273\pi$
$757$	−42.3190	−1.53811	−0.769055	+	0.639183i	$0.779273\pi$
$758$	1.56619	0.0568867
$759$	0	0
$760$	−0.945680	−0.0343034
$761$	12.4474	0.451219	0.225610	−	0.974218i	$-0.427563\pi$
$761$	12.4474	0.451219	0.225610	+	0.974218i	$0.427563\pi$
$762$	0	0
$763$	−19.2958	−0.698555
$764$	−18.7495	−0.678334
$765$	0	0
$766$	5.11081	0.184661
$767$	0	0
$768$	0	0
$769$	7.35494	0.265226	0.132613	−	0.991168i	$-0.457663\pi$
$769$	7.35494	0.265226	0.132613	+	0.991168i	$0.457663\pi$
$770$	−12.9350	−0.466144
$771$	0	0
$772$	25.3325	0.911738
$773$	12.8701	0.462907	0.231453	−	0.972846i	$-0.425652\pi$
$773$	12.8701	0.462907	0.231453	+	0.972846i	$0.425652\pi$
$774$	0	0
$775$	1.64968	0.0592582
$776$	17.7350	0.636649
$777$	0	0
$778$	−24.3513	−0.873038
$779$	6.19057	0.221800
$780$	0	0
$781$	−9.43235	−0.337516
$782$	−2.97465	−0.106373
$783$	0	0
$784$	−61.9374	−2.21205
$785$	−14.4852	−0.516997
$786$	0	0
$787$	4.96901	0.177126	0.0885631	−	0.996071i	$-0.471772\pi$
$787$	4.96901	0.177126	0.0885631	+	0.996071i	$0.471772\pi$
$788$	−15.8271	−0.563818
$789$	0	0
$790$	−25.5894	−0.910431
$791$	84.9283	3.01970
$792$	0	0
$793$	0	0
$794$	−14.5552	−0.516543
$795$	0	0
$796$	29.5204	1.04632
$797$	21.1244	0.748266	0.374133	−	0.927375i	$-0.377940\pi$
$797$	21.1244	0.748266	0.374133	+	0.927375i	$0.377940\pi$
$798$	0	0
$799$	15.5073	0.548608
$800$	−6.29558	−0.222582
$801$	0	0
$802$	62.2060	2.19657
$803$	16.0090	0.564947
$804$	0	0
$805$	−5.78982	−0.204064
$806$	0	0
$807$	0	0
$808$	−23.4498	−0.824960
$809$	11.2608	0.395910	0.197955	−	0.980211i	$-0.436570\pi$
$809$	11.2608	0.395910	0.197955	+	0.980211i	$0.436570\pi$
$810$	0	0
$811$	−30.9957	−1.08841	−0.544203	−	0.838954i	$-0.683168\pi$
$811$	−30.9957	−1.08841	−0.544203	+	0.838954i	$0.683168\pi$
$812$	−6.55970	−0.230200
$813$	0	0
$814$	14.3910	0.504405
$815$	−15.5340	−0.544132
$816$	0	0
$817$	5.17042	0.180890
$818$	35.2410	1.23217
$819$	0	0
$820$	10.9398	0.382033
$821$	43.8685	1.53102	0.765510	−	0.643424i	$-0.222487\pi$
$821$	43.8685	1.53102	0.765510	+	0.643424i	$0.222487\pi$
$822$	0	0
$823$	−6.08060	−0.211957	−0.105978	−	0.994368i	$-0.533797\pi$
$823$	−6.08060	−0.211957	−0.105978	+	0.994368i	$0.533797\pi$
$824$	−18.0738	−0.629629
$825$	0	0
$826$	34.3249	1.19432
$827$	11.0488	0.384205	0.192102	−	0.981375i	$-0.438469\pi$
$827$	11.0488	0.384205	0.192102	+	0.981375i	$0.438469\pi$
$828$	0	0
$829$	31.5605	1.09614	0.548070	−	0.836433i	$-0.315363\pi$
$829$	31.5605	1.09614	0.548070	+	0.836433i	$0.315363\pi$
$830$	19.9754	0.693357
$831$	0	0
$832$	0	0
$833$	15.7995	0.547421
$834$	0	0
$835$	−20.3288	−0.703507
$836$	−1.49060	−0.0515536
$837$	0	0
$838$	44.3020	1.53039
$839$	16.0683	0.554739	0.277369	−	0.960763i	$-0.410537\pi$
$839$	16.0683	0.554739	0.277369	+	0.960763i	$0.410537\pi$
$840$	0	0
$841$	−27.6529	−0.953549
$842$	−50.1419	−1.72800
$843$	0	0
$844$	−11.3085	−0.389253
$845$	0	0
$846$	0	0
$847$	−37.1379	−1.27607
$848$	−65.9924	−2.26619
$849$	0	0
$850$	2.27346	0.0779791
$851$	6.44156	0.220814
$852$	0	0
$853$	−11.5424	−0.395205	−0.197602	−	0.980282i	$-0.563315\pi$
$853$	−11.5424	−0.395205	−0.197602	+	0.980282i	$0.563315\pi$
$854$	45.2578	1.54869
$855$	0	0
$856$	12.7327	0.435195
$857$	−53.7559	−1.83627	−0.918134	−	0.396270i	$-0.870304\pi$
$857$	−53.7559	−1.83627	−0.918134	+	0.396270i	$0.870304\pi$
$858$	0	0
$859$	3.28910	0.112223	0.0561113	−	0.998425i	$-0.482130\pi$
$859$	3.28910	0.112223	0.0561113	+	0.998425i	$0.482130\pi$
$860$	9.13699	0.311569
$861$	0	0
$862$	−7.90693	−0.269311
$863$	32.7600	1.11516	0.557581	−	0.830123i	$-0.311730\pi$
$863$	32.7600	1.11516	0.557581	+	0.830123i	$0.311730\pi$
$864$	0	0
$865$	19.1021	0.649491
$866$	−16.3963	−0.557170
$867$	0	0
$868$	9.32368	0.316466
$869$	22.8245	0.774269
$870$	0	0
$871$	0	0
$872$	5.70553	0.193214
$873$	0	0
$874$	−1.71198	−0.0579085
$875$	4.42503	0.149593
$876$	0	0
$877$	8.92654	0.301428	0.150714	−	0.988577i	$-0.451843\pi$
$877$	8.92654	0.301428	0.150714	+	0.988577i	$0.451843\pi$
$878$	−29.2896	−0.988475
$879$	0	0
$880$	7.94946	0.267976
$881$	27.5224	0.927252	0.463626	−	0.886031i	$-0.346548\pi$
$881$	27.5224	0.927252	0.463626	+	0.886031i	$0.346548\pi$
$882$	0	0
$883$	43.2667	1.45604	0.728020	−	0.685555i	$-0.240441\pi$
$883$	43.2667	1.45604	0.728020	+	0.685555i	$0.240441\pi$
$884$	0	0
$885$	0	0
$886$	−24.6736	−0.828926
$887$	−29.3318	−0.984866	−0.492433	−	0.870350i	$-0.663892\pi$
$887$	−29.3318	−0.984866	−0.492433	+	0.870350i	$0.663892\pi$
$888$	0	0
$889$	−13.4554	−0.451281
$890$	−1.53022	−0.0512931
$891$	0	0
$892$	5.46060	0.182834
$893$	8.92478	0.298656
$894$	0	0
$895$	10.9429	0.365781
$896$	43.2940	1.44635
$897$	0	0
$898$	20.0118	0.667802
$899$	−1.91468	−0.0638580
$900$	0	0
$901$	16.8339	0.560819
$902$	−25.0372	−0.833647
$903$	0	0
$904$	−25.1122	−0.835220
$905$	−1.45122	−0.0482403
$906$	0	0
$907$	−35.3799	−1.17477	−0.587385	−	0.809308i	$-0.699842\pi$
$907$	−35.3799	−1.17477	−0.587385	+	0.809308i	$0.699842\pi$
$908$	27.3512	0.907681
$909$	0	0
$910$	0	0
$911$	−46.6105	−1.54428	−0.772138	−	0.635455i	$-0.780812\pi$
$911$	−46.6105	−1.54428	−0.772138	+	0.635455i	$0.780812\pi$
$912$	0	0
$913$	−17.8171	−0.589661
$914$	−43.8732	−1.45120
$915$	0	0
$916$	−0.844906	−0.0279165
$917$	−68.5412	−2.26343
$918$	0	0
$919$	−26.3865	−0.870410	−0.435205	−	0.900331i	$-0.643324\pi$
$919$	−26.3865	−0.870410	−0.435205	+	0.900331i	$0.643324\pi$
$920$	1.71198	0.0564422
$921$	0	0
$922$	48.3279	1.59159
$923$	0	0
$924$	0	0
$925$	−4.92314	−0.161872
$926$	−35.8690	−1.17873
$927$	0	0
$928$	7.30688	0.239860
$929$	5.52357	0.181223	0.0906113	−	0.995886i	$-0.471118\pi$
$929$	5.52357	0.181223	0.0906113	+	0.995886i	$0.471118\pi$
$930$	0	0
$931$	9.09297	0.298010
$932$	−17.0722	−0.559220
$933$	0	0
$934$	−40.6690	−1.33073
$935$	−2.02782	−0.0663167
$936$	0	0
$937$	7.15026	0.233589	0.116794	−	0.993156i	$-0.462738\pi$
$937$	7.15026	0.233589	0.116794	+	0.993156i	$0.462738\pi$
$938$	−1.75222	−0.0572119
$939$	0	0
$940$	15.7716	0.514411
$941$	19.8331	0.646541	0.323271	−	0.946307i	$-0.395218\pi$
$941$	19.8331	0.646541	0.323271	+	0.946307i	$0.395218\pi$
$942$	0	0
$943$	−11.2069	−0.364946
$944$	−21.0951	−0.686587
$945$	0	0
$946$	−20.9113	−0.679884
$947$	3.19122	0.103701	0.0518503	−	0.998655i	$-0.483488\pi$
$947$	3.19122	0.103701	0.0518503	+	0.998655i	$0.483488\pi$
$948$	0	0
$949$	0	0
$950$	1.30843	0.0424509
$951$	0	0
$952$	−7.27107	−0.235657
$953$	−40.9786	−1.32743	−0.663713	−	0.747987i	$-0.731020\pi$
$953$	−40.9786	−1.32743	−0.663713	+	0.747987i	$0.731020\pi$
$954$	0	0
$955$	−14.6797	−0.475025
$956$	5.67526	0.183551
$957$	0	0
$958$	−45.5626	−1.47206
$959$	−38.4906	−1.24293
$960$	0	0
$961$	−28.2786	−0.912212
$962$	0	0
$963$	0	0
$964$	18.4211	0.593302
$965$	19.8338	0.638474
$966$	0	0
$967$	41.6355	1.33891	0.669453	−	0.742854i	$-0.266529\pi$
$967$	41.6355	1.33891	0.669453	+	0.742854i	$0.266529\pi$
$968$	10.9812	0.352950
$969$	0	0
$970$	−24.5378	−0.787862
$971$	−21.1935	−0.680133	−0.340067	−	0.940401i	$-0.610450\pi$
$971$	−21.1935	−0.680133	−0.340067	+	0.940401i	$0.610450\pi$
$972$	0	0
$973$	8.97535	0.287737
$974$	53.5023	1.71433
$975$	0	0
$976$	−27.8141	−0.890309
$977$	16.2332	0.519346	0.259673	−	0.965697i	$-0.416385\pi$
$977$	16.2332	0.519346	0.259673	+	0.965697i	$0.416385\pi$
$978$	0	0
$979$	1.36488	0.0436219
$980$	16.0688	0.513298
$981$	0	0
$982$	27.9847	0.893027
$983$	−27.0181	−0.861743	−0.430872	−	0.902413i	$-0.641794\pi$
$983$	−27.0181	−0.861743	−0.430872	+	0.902413i	$0.641794\pi$
$984$	0	0
$985$	−12.3917	−0.394832
$986$	−2.63866	−0.0840321
$987$	0	0
$988$	0	0
$989$	−9.36009	−0.297634
$990$	0	0
$991$	−16.9905	−0.539722	−0.269861	−	0.962899i	$-0.586978\pi$
$991$	−16.9905	−0.539722	−0.269861	+	0.962899i	$0.586978\pi$
$992$	−10.3857	−0.329746
$993$	0	0
$994$	46.7945	1.48423
$995$	23.1127	0.732722
$996$	0	0
$997$	39.4255	1.24862	0.624309	−	0.781177i	$-0.285380\pi$
$997$	39.4255	1.24862	0.624309	+	0.781177i	$0.285380\pi$
$998$	2.50317	0.0792364
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7605.2.a.cn.1.4		5
3.2	odd	2		7605.2.a.cl.1.2		5
13.4	even	6		585.2.j.i.406.4	yes	10
13.10	even	6		585.2.j.i.451.4	yes	10
13.12	even	2		7605.2.a.cm.1.2		5
39.17	odd	6		585.2.j.h.406.2	✓	10
39.23	odd	6		585.2.j.h.451.2	yes	10
39.38	odd	2		7605.2.a.co.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
585.2.j.h.406.2	✓	10	39.17	odd	6
585.2.j.h.451.2	yes	10	39.23	odd	6
585.2.j.i.406.4	yes	10	13.4	even	6
585.2.j.i.451.4	yes	10	13.10	even	6
7605.2.a.cl.1.2		5	3.2	odd	2
7605.2.a.cm.1.2		5	13.12	even	2
7605.2.a.cn.1.4		5	1.1	even	1	trivial
7605.2.a.co.1.4		5	39.38	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:	\( N \)	\(=\)	\( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7605.a (trivial)

\(f(q)\)	\(=\)	\(q+1.81031 q^{2} +1.27724 q^{4} +1.00000 q^{5} +4.42503 q^{7} -1.30843 q^{8} +O(q^{10})\)	\(q+1.81031 q^{2} +1.27724 q^{4} +1.00000 q^{5} +4.42503 q^{7} -1.30843 q^{8} +1.81031 q^{10} -1.61471 q^{11} +8.01069 q^{14} -4.92314 q^{16} +1.25584 q^{17} +0.722761 q^{19} +1.27724 q^{20} -2.92314 q^{22} -1.30843 q^{23} +1.00000 q^{25} +5.65182 q^{28} -1.16064 q^{29} +1.64968 q^{31} -6.29558 q^{32} +2.27346 q^{34} +4.42503 q^{35} -4.92314 q^{37} +1.30843 q^{38} -1.30843 q^{40} +8.56517 q^{41} +7.15370 q^{43} -2.06237 q^{44} -2.36866 q^{46} +12.3482 q^{47} +12.5809 q^{49} +1.81031 q^{50} +13.4045 q^{53} -1.61471 q^{55} -5.78982 q^{56} -2.10112 q^{58} +4.28489 q^{59} +5.64968 q^{61} +2.98644 q^{62} -1.55070 q^{64} -0.218735 q^{67} +1.60400 q^{68} +8.01069 q^{70} +5.84150 q^{71} -9.91448 q^{73} -8.91243 q^{74} +0.923139 q^{76} -7.14515 q^{77} -14.1353 q^{79} -4.92314 q^{80} +15.5056 q^{82} +11.0342 q^{83} +1.25584 q^{85} +12.9505 q^{86} +2.11273 q^{88} -0.845279 q^{89} -1.67117 q^{92} +22.3541 q^{94} +0.722761 q^{95} -13.5545 q^{97} +22.7753 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{2} + 6 q^{4} + 5 q^{5} - q^{7} + 6 q^{8}+O(q^{10}) \) 5 * q + 2 * q^2 + 6 * q^4 + 5 * q^5 - q^7 + 6 * q^8	\( 5 q + 2 q^{2} + 6 q^{4} + 5 q^{5} - q^{7} + 6 q^{8} + 2 q^{10} + 8 q^{11} + 4 q^{14} + 4 q^{16} + 4 q^{19} + 6 q^{20} + 14 q^{22} + 6 q^{23} + 5 q^{25} + 2 q^{28} - 16 q^{29} - 9 q^{31} + 14 q^{32} - q^{35} + 4 q^{37} - 6 q^{38} + 6 q^{40} + 6 q^{41} + 15 q^{43} + 14 q^{44} + 16 q^{46} + 10 q^{47} + 10 q^{49} + 2 q^{50} + 20 q^{53} + 8 q^{55} + 2 q^{56} + 4 q^{58} + 12 q^{59} + 11 q^{61} + 22 q^{62} + 4 q^{64} - 5 q^{67} - 50 q^{68} + 4 q^{70} + 10 q^{71} - q^{73} + 26 q^{74} - 24 q^{76} - 42 q^{77} - 17 q^{79} + 4 q^{80} + 16 q^{82} + 16 q^{83} + 44 q^{86} + 20 q^{88} + 4 q^{89} + 34 q^{92} - 16 q^{94} + 4 q^{95} + 11 q^{97} + 30 q^{98}+O(q^{100}) \) 5 * q + 2 * q^2 + 6 * q^4 + 5 * q^5 - q^7 + 6 * q^8 + 2 * q^10 + 8 * q^11 + 4 * q^14 + 4 * q^16 + 4 * q^19 + 6 * q^20 + 14 * q^22 + 6 * q^23 + 5 * q^25 + 2 * q^28 - 16 * q^29 - 9 * q^31 + 14 * q^32 - q^35 + 4 * q^37 - 6 * q^38 + 6 * q^40 + 6 * q^41 + 15 * q^43 + 14 * q^44 + 16 * q^46 + 10 * q^47 + 10 * q^49 + 2 * q^50 + 20 * q^53 + 8 * q^55 + 2 * q^56 + 4 * q^58 + 12 * q^59 + 11 * q^61 + 22 * q^62 + 4 * q^64 - 5 * q^67 - 50 * q^68 + 4 * q^70 + 10 * q^71 - q^73 + 26 * q^74 - 24 * q^76 - 42 * q^77 - 17 * q^79 + 4 * q^80 + 16 * q^82 + 16 * q^83 + 44 * q^86 + 20 * q^88 + 4 * q^89 + 34 * q^92 - 16 * q^94 + 4 * q^95 + 11 * q^97 + 30 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more