LMFDB - Embedded newform 7605.2.a.bs.1.2

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:	$3$
Coefficient field:	3.3.564.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.571993$ 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)

$f(q)$	$=$	$q-0.571993 q^{2} -1.67282 q^{4} +1.00000 q^{5} -2.67282 q^{7} +2.10083 q^{8} +O(q^{10})$	$q-0.571993 q^{2} -1.67282 q^{4} +1.00000 q^{5} -2.67282 q^{7} +2.10083 q^{8} -0.571993 q^{10} -5.10083 q^{11} +1.52884 q^{14} +2.14399 q^{16} -5.34565 q^{17} +6.24482 q^{19} -1.67282 q^{20} +2.91764 q^{22} +2.42801 q^{23} +1.00000 q^{25} +4.47116 q^{28} -2.67282 q^{29} -0.244817 q^{31} -5.42801 q^{32} +3.05767 q^{34} -2.67282 q^{35} +3.32718 q^{37} -3.57199 q^{38} +2.10083 q^{40} -6.48963 q^{41} -10.9176 q^{43} +8.53279 q^{44} -1.38880 q^{46} -2.67282 q^{47} +0.143987 q^{49} -0.571993 q^{50} +4.20166 q^{53} -5.10083 q^{55} -5.61515 q^{56} +1.52884 q^{58} +0.899170 q^{59} -5.81681 q^{61} +0.140034 q^{62} -1.18319 q^{64} +2.18319 q^{67} +8.94233 q^{68} +1.52884 q^{70} -6.24482 q^{71} +10.9608 q^{73} -1.90312 q^{74} -10.4465 q^{76} +13.6336 q^{77} -3.63362 q^{79} +2.14399 q^{80} +3.71203 q^{82} -9.81681 q^{83} -5.34565 q^{85} +6.24482 q^{86} -10.7160 q^{88} -7.63362 q^{89} -4.06163 q^{92} +1.52884 q^{94} +6.24482 q^{95} -11.3456 q^{97} -0.0823593 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 5 q^{4} + 3 q^{5} + 2 q^{7} - 3 q^{8}+O(q^{10}) $ 3 * q - q^2 + 5 * q^4 + 3 * q^5 + 2 * q^7 - 3 * q^8	$ 3 q - q^{2} + 5 q^{4} + 3 q^{5} + 2 q^{7} - 3 q^{8} - q^{10} - 6 q^{11} - 4 q^{14} + 5 q^{16} + 4 q^{17} + 8 q^{19} + 5 q^{20} - 12 q^{22} + 8 q^{23} + 3 q^{25} + 22 q^{28} + 2 q^{29} + 10 q^{31} - 17 q^{32} - 8 q^{34} + 2 q^{35} + 20 q^{37} - 10 q^{38} - 3 q^{40} + 2 q^{41} - 12 q^{43} + 12 q^{44} + 8 q^{46} + 2 q^{47} - q^{49} - q^{50} - 6 q^{53} - 6 q^{55} - 24 q^{56} - 4 q^{58} + 12 q^{59} - 6 q^{61} + 4 q^{62} - 15 q^{64} + 18 q^{67} + 44 q^{68} - 4 q^{70} - 8 q^{71} + 20 q^{73} - 10 q^{74} - 2 q^{76} + 18 q^{77} + 12 q^{79} + 5 q^{80} + 14 q^{82} - 18 q^{83} + 4 q^{85} + 8 q^{86} - 30 q^{88} + 10 q^{92} - 4 q^{94} + 8 q^{95} - 14 q^{97} - 21 q^{98}+O(q^{100}) $ 3 * q - q^2 + 5 * q^4 + 3 * q^5 + 2 * q^7 - 3 * q^8 - q^10 - 6 * q^11 - 4 * q^14 + 5 * q^16 + 4 * q^17 + 8 * q^19 + 5 * q^20 - 12 * q^22 + 8 * q^23 + 3 * q^25 + 22 * q^28 + 2 * q^29 + 10 * q^31 - 17 * q^32 - 8 * q^34 + 2 * q^35 + 20 * q^37 - 10 * q^38 - 3 * q^40 + 2 * q^41 - 12 * q^43 + 12 * q^44 + 8 * q^46 + 2 * q^47 - q^49 - q^50 - 6 * q^53 - 6 * q^55 - 24 * q^56 - 4 * q^58 + 12 * q^59 - 6 * q^61 + 4 * q^62 - 15 * q^64 + 18 * q^67 + 44 * q^68 - 4 * q^70 - 8 * q^71 + 20 * q^73 - 10 * q^74 - 2 * q^76 + 18 * q^77 + 12 * q^79 + 5 * q^80 + 14 * q^82 - 18 * q^83 + 4 * q^85 + 8 * q^86 - 30 * q^88 + 10 * q^92 - 4 * q^94 + 8 * q^95 - 14 * q^97 - 21 * 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 $

$2$	−0.571993	−0.404460	−0.202230	−	0.979338i	$-0.564819\pi$
$2$	−0.571993	−0.404460	−0.202230	+	0.979338i	$0.564819\pi$
$3$	0	0
$3$	0	0
$4$	−1.67282	−0.836412
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.67282	−1.01023	−0.505116	−	0.863051i	$-0.668550\pi$
$7$	−2.67282	−1.01023	−0.505116	+	0.863051i	$0.668550\pi$
$8$	2.10083	0.742756
$9$	0	0
$10$	−0.571993	−0.180880
$11$	−5.10083	−1.53796	−0.768979	−	0.639274i	$-0.779235\pi$
$11$	−5.10083	−1.53796	−0.768979	+	0.639274i	$0.779235\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	1.52884	0.408599
$15$	0	0
$16$	2.14399	0.535997
$17$	−5.34565	−1.29651	−0.648255	−	0.761423i	$-0.724501\pi$
$17$	−5.34565	−1.29651	−0.648255	+	0.761423i	$0.724501\pi$
$18$	0	0
$19$	6.24482	1.43266	0.716330	−	0.697762i	$-0.245821\pi$
$19$	6.24482	1.43266	0.716330	+	0.697762i	$0.245821\pi$
$20$	−1.67282	−0.374055
$21$	0	0
$22$	2.91764	0.622043
$23$	2.42801	0.506274	0.253137	−	0.967430i	$-0.418538\pi$
$23$	2.42801	0.506274	0.253137	+	0.967430i	$0.418538\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	4.47116	0.844970
$29$	−2.67282	−0.496331	−0.248165	−	0.968718i	$-0.579828\pi$
$29$	−2.67282	−0.496331	−0.248165	+	0.968718i	$0.579828\pi$
$30$	0	0
$31$	−0.244817	−0.0439704	−0.0219852	−	0.999758i	$-0.506999\pi$
$31$	−0.244817	−0.0439704	−0.0219852	+	0.999758i	$0.506999\pi$
$32$	−5.42801	−0.959545
$33$	0	0
$34$	3.05767	0.524387
$35$	−2.67282	−0.451790
$36$	0	0
$37$	3.32718	0.546984	0.273492	−	0.961874i	$-0.411821\pi$
$37$	3.32718	0.546984	0.273492	+	0.961874i	$0.411821\pi$
$38$	−3.57199	−0.579454
$39$	0	0
$40$	2.10083	0.332170
$41$	−6.48963	−1.01351	−0.506755	−	0.862090i	$-0.669155\pi$
$41$	−6.48963	−1.01351	−0.506755	+	0.862090i	$0.669155\pi$
$42$	0	0
$43$	−10.9176	−1.66492	−0.832462	−	0.554082i	$-0.813070\pi$
$43$	−10.9176	−1.66492	−0.832462	+	0.554082i	$0.813070\pi$
$44$	8.53279	1.28637
$45$	0	0
$46$	−1.38880	−0.204768
$47$	−2.67282	−0.389871	−0.194936	−	0.980816i	$-0.562450\pi$
$47$	−2.67282	−0.389871	−0.194936	+	0.980816i	$0.562450\pi$
$48$	0	0
$49$	0.143987	0.0205695
$50$	−0.571993	−0.0808921
$51$	0	0
$52$	0	0
$53$	4.20166	0.577143	0.288571	−	0.957458i	$-0.406820\pi$
$53$	4.20166	0.577143	0.288571	+	0.957458i	$0.406820\pi$
$54$	0	0
$55$	−5.10083	−0.687796
$56$	−5.61515	−0.750356
$57$	0	0
$58$	1.52884	0.200746
$59$	0.899170	0.117062	0.0585310	−	0.998286i	$-0.481358\pi$
$59$	0.899170	0.117062	0.0585310	+	0.998286i	$0.481358\pi$
$60$	0	0
$61$	−5.81681	−0.744766	−0.372383	−	0.928079i	$-0.621459\pi$
$61$	−5.81681	−0.744766	−0.372383	+	0.928079i	$0.621459\pi$
$62$	0.140034	0.0177843
$63$	0	0
$64$	−1.18319	−0.147899
$65$	0	0
$66$	0	0
$67$	2.18319	0.266719	0.133360	−	0.991068i	$-0.457423\pi$
$67$	2.18319	0.266719	0.133360	+	0.991068i	$0.457423\pi$
$68$	8.94233	1.08442
$69$	0	0
$70$	1.52884	0.182731
$71$	−6.24482	−0.741123	−0.370562	−	0.928808i	$-0.620835\pi$
$71$	−6.24482	−0.741123	−0.370562	+	0.928808i	$0.620835\pi$
$72$	0	0
$73$	10.9608	1.28286	0.641432	−	0.767180i	$-0.278341\pi$
$73$	10.9608	1.28286	0.641432	+	0.767180i	$0.278341\pi$
$74$	−1.90312	−0.221233
$75$	0	0
$76$	−10.4465	−1.19829
$77$	13.6336	1.55370
$78$	0	0
$79$	−3.63362	−0.408814	−0.204407	−	0.978886i	$-0.565527\pi$
$79$	−3.63362	−0.408814	−0.204407	+	0.978886i	$0.565527\pi$
$80$	2.14399	0.239705
$81$	0	0
$82$	3.71203	0.409925
$83$	−9.81681	−1.07753	−0.538767	−	0.842455i	$-0.681110\pi$
$83$	−9.81681	−1.07753	−0.538767	+	0.842455i	$0.681110\pi$
$84$	0	0
$85$	−5.34565	−0.579817
$86$	6.24482	0.673396
$87$	0	0
$88$	−10.7160	−1.14233
$89$	−7.63362	−0.809162	−0.404581	−	0.914502i	$-0.632583\pi$
$89$	−7.63362	−0.809162	−0.404581	+	0.914502i	$0.632583\pi$
$90$	0	0
$91$	0	0
$92$	−4.06163	−0.423454
$93$	0	0
$94$	1.52884	0.157688
$95$	6.24482	0.640705
$96$	0	0
$97$	−11.3456	−1.15198	−0.575988	−	0.817458i	$-0.695383\pi$
$97$	−11.3456	−1.15198	−0.575988	+	0.817458i	$0.695383\pi$
$98$	−0.0823593	−0.00831955
$99$	0	0
$100$	−1.67282	−0.167282
$101$	−10.8560	−1.08021	−0.540107	−	0.841596i	$-0.681616\pi$
$101$	−10.8560	−1.08021	−0.540107	+	0.841596i	$0.681616\pi$
$102$	0	0
$103$	10.6297	1.04737	0.523686	−	0.851911i	$-0.324556\pi$
$103$	10.6297	1.04737	0.523686	+	0.851911i	$0.324556\pi$
$104$	0	0
$105$	0	0
$106$	−2.40332	−0.233431
$107$	−0.140034	−0.0135376	−0.00676878	−	0.999977i	$-0.502155\pi$
$107$	−0.140034	−0.0135376	−0.00676878	+	0.999977i	$0.502155\pi$
$108$	0	0
$109$	9.05767	0.867568	0.433784	−	0.901017i	$-0.357178\pi$
$109$	9.05767	0.867568	0.433784	+	0.901017i	$0.357178\pi$
$110$	2.91764	0.278186
$111$	0	0
$112$	−5.73050	−0.541481
$113$	−12.9793	−1.22099	−0.610493	−	0.792021i	$-0.709029\pi$
$113$	−12.9793	−1.22099	−0.610493	+	0.792021i	$0.709029\pi$
$114$	0	0
$115$	2.42801	0.226413
$116$	4.47116	0.415137
$117$	0	0
$118$	−0.514319	−0.0473469
$119$	14.2880	1.30978
$120$	0	0
$121$	15.0185	1.36532
$122$	3.32718	0.301228
$123$	0	0
$124$	0.409536	0.0367774
$125$	1.00000	0.0894427
$126$	0	0
$127$	12.0616	1.07030	0.535148	−	0.844758i	$-0.320256\pi$
$127$	12.0616	1.07030	0.535148	+	0.844758i	$0.320256\pi$
$128$	11.5328	1.01936
$129$	0	0
$130$	0	0
$131$	−7.63362	−0.666953	−0.333476	−	0.942758i	$-0.608222\pi$
$131$	−7.63362	−0.666953	−0.333476	+	0.942758i	$0.608222\pi$
$132$	0	0
$133$	−16.6913	−1.44732
$134$	−1.24877	−0.107877
$135$	0	0
$136$	−11.2303	−0.962990
$137$	3.43196	0.293212	0.146606	−	0.989195i	$-0.453165\pi$
$137$	3.43196	0.293212	0.146606	+	0.989195i	$0.453165\pi$
$138$	0	0
$139$	−0.942326	−0.0799270	−0.0399635	−	0.999201i	$-0.512724\pi$
$139$	−0.942326	−0.0799270	−0.0399635	+	0.999201i	$0.512724\pi$
$140$	4.47116	0.377882
$141$	0	0
$142$	3.57199	0.299755
$143$	0	0
$144$	0	0
$145$	−2.67282	−0.221966
$146$	−6.26950	−0.518868
$147$	0	0
$148$	−5.56578	−0.457504
$149$	11.3456	0.929472	0.464736	−	0.885449i	$-0.346149\pi$
$149$	11.3456	0.929472	0.464736	+	0.885449i	$0.346149\pi$
$150$	0	0
$151$	21.0224	1.71078	0.855390	−	0.517984i	$-0.173317\pi$
$151$	21.0224	1.71078	0.855390	+	0.517984i	$0.173317\pi$
$152$	13.1193	1.06412
$153$	0	0
$154$	−7.79834	−0.628408
$155$	−0.244817	−0.0196642
$156$	0	0
$157$	−9.83528	−0.784941	−0.392470	−	0.919765i	$-0.628379\pi$
$157$	−9.83528	−0.784941	−0.392470	+	0.919765i	$0.628379\pi$
$158$	2.07841	0.165349
$159$	0	0
$160$	−5.42801	−0.429122
$161$	−6.48963	−0.511455
$162$	0	0
$163$	12.3849	0.970056	0.485028	−	0.874499i	$-0.338809\pi$
$163$	12.3849	0.970056	0.485028	+	0.874499i	$0.338809\pi$
$164$	10.8560	0.847712
$165$	0	0
$166$	5.61515	0.435820
$167$	8.50811	0.658377	0.329188	−	0.944264i	$-0.393225\pi$
$167$	8.50811	0.658377	0.329188	+	0.944264i	$0.393225\pi$
$168$	0	0
$169$	0	0
$170$	3.05767	0.234513
$171$	0	0
$172$	18.2633	1.39256
$173$	21.5473	1.63821	0.819106	−	0.573643i	$-0.194470\pi$
$173$	21.5473	1.63821	0.819106	+	0.573643i	$0.194470\pi$
$174$	0	0
$175$	−2.67282	−0.202046
$176$	−10.9361	−0.824340
$177$	0	0
$178$	4.36638	0.327274
$179$	−2.56804	−0.191944	−0.0959722	−	0.995384i	$-0.530596\pi$
$179$	−2.56804	−0.191944	−0.0959722	+	0.995384i	$0.530596\pi$
$180$	0	0
$181$	−13.7305	−1.02058	−0.510290	−	0.860002i	$-0.670462\pi$
$181$	−13.7305	−1.02058	−0.510290	+	0.860002i	$0.670462\pi$
$182$	0	0
$183$	0	0
$184$	5.10083	0.376038
$185$	3.32718	0.244619
$186$	0	0
$187$	27.2672	1.99398
$188$	4.47116	0.326093
$189$	0	0
$190$	−3.57199	−0.259140
$191$	10.6913	0.773595	0.386797	−	0.922165i	$-0.373581\pi$
$191$	10.6913	0.773595	0.386797	+	0.922165i	$0.373581\pi$
$192$	0	0
$193$	−19.2593	−1.38632	−0.693159	−	0.720785i	$-0.743781\pi$
$193$	−19.2593	−1.38632	−0.693159	+	0.720785i	$0.743781\pi$
$194$	6.48963	0.465929
$195$	0	0
$196$	−0.240864	−0.0172046
$197$	17.1809	1.22409	0.612045	−	0.790823i	$-0.290347\pi$
$197$	17.1809	1.22409	0.612045	+	0.790823i	$0.290347\pi$
$198$	0	0
$199$	11.5473	0.818567	0.409283	−	0.912407i	$-0.365779\pi$
$199$	11.5473	0.818567	0.409283	+	0.912407i	$0.365779\pi$
$200$	2.10083	0.148551
$201$	0	0
$202$	6.20957	0.436904
$203$	7.14399	0.501410
$204$	0	0
$205$	−6.48963	−0.453256
$206$	−6.08010	−0.423621
$207$	0	0
$208$	0	0
$209$	−31.8538	−2.20337
$210$	0	0
$211$	24.1233	1.66071	0.830357	−	0.557232i	$-0.188137\pi$
$211$	24.1233	1.66071	0.830357	+	0.557232i	$0.188137\pi$
$212$	−7.02864	−0.482729
$213$	0	0
$214$	0.0800983	0.00547541
$215$	−10.9176	−0.744577
$216$	0	0
$217$	0.654353	0.0444203
$218$	−5.18093	−0.350897
$219$	0	0
$220$	8.53279	0.575281
$221$	0	0
$222$	0	0
$223$	5.73050	0.383743	0.191871	−	0.981420i	$-0.438544\pi$
$223$	5.73050	0.383743	0.191871	+	0.981420i	$0.438544\pi$
$224$	14.5081	0.969364
$225$	0	0
$226$	7.42405	0.493841
$227$	−2.95289	−0.195990	−0.0979951	−	0.995187i	$-0.531243\pi$
$227$	−2.95289	−0.195990	−0.0979951	+	0.995187i	$0.531243\pi$
$228$	0	0
$229$	25.2593	1.66918	0.834592	−	0.550869i	$-0.185703\pi$
$229$	25.2593	1.66918	0.834592	+	0.550869i	$0.185703\pi$
$230$	−1.38880	−0.0915750
$231$	0	0
$232$	−5.61515	−0.368653
$233$	−23.3456	−1.52942	−0.764712	−	0.644372i	$-0.777119\pi$
$233$	−23.3456	−1.52942	−0.764712	+	0.644372i	$0.777119\pi$
$234$	0	0
$235$	−2.67282	−0.174356
$236$	−1.50415	−0.0979120
$237$	0	0
$238$	−8.17262	−0.529753
$239$	3.79213	0.245292	0.122646	−	0.992450i	$-0.460862\pi$
$239$	3.79213	0.245292	0.122646	+	0.992450i	$0.460862\pi$
$240$	0	0
$241$	−3.43196	−0.221072	−0.110536	−	0.993872i	$-0.535257\pi$
$241$	−3.43196	−0.221072	−0.110536	+	0.993872i	$0.535257\pi$
$242$	−8.59046	−0.552216
$243$	0	0
$244$	9.73050	0.622931
$245$	0.143987	0.00919896
$246$	0	0
$247$	0	0
$248$	−0.514319	−0.0326593
$249$	0	0
$250$	−0.571993	−0.0361760
$251$	−25.4689	−1.60758	−0.803791	−	0.594911i	$-0.797187\pi$
$251$	−25.4689	−1.60758	−0.803791	+	0.594911i	$0.797187\pi$
$252$	0	0
$253$	−12.3849	−0.778629
$254$	−6.89917	−0.432892
$255$	0	0
$256$	−4.23030	−0.264394
$257$	9.26724	0.578075	0.289037	−	0.957318i	$-0.406665\pi$
$257$	9.26724	0.578075	0.289037	+	0.957318i	$0.406665\pi$
$258$	0	0
$259$	−8.89296	−0.552581
$260$	0	0
$261$	0	0
$262$	4.36638	0.269756
$263$	8.14794	0.502423	0.251212	−	0.967932i	$-0.419171\pi$
$263$	8.14794	0.502423	0.251212	+	0.967932i	$0.419171\pi$
$264$	0	0
$265$	4.20166	0.258106
$266$	9.54731	0.585383
$267$	0	0
$268$	−3.65209	−0.223087
$269$	−1.14399	−0.0697501	−0.0348750	−	0.999392i	$-0.511103\pi$
$269$	−1.14399	−0.0697501	−0.0348750	+	0.999392i	$0.511103\pi$
$270$	0	0
$271$	−3.18714	−0.193605	−0.0968026	−	0.995304i	$-0.530862\pi$
$271$	−3.18714	−0.193605	−0.0968026	+	0.995304i	$0.530862\pi$
$272$	−11.4610	−0.694925
$273$	0	0
$274$	−1.96306	−0.118593
$275$	−5.10083	−0.307592
$276$	0	0
$277$	−7.62571	−0.458185	−0.229092	−	0.973405i	$-0.573576\pi$
$277$	−7.62571	−0.458185	−0.229092	+	0.973405i	$0.573576\pi$
$278$	0.539004	0.0323273
$279$	0	0
$280$	−5.61515	−0.335569
$281$	20.2386	1.20733	0.603667	−	0.797237i	$-0.293706\pi$
$281$	20.2386	1.20733	0.603667	+	0.797237i	$0.293706\pi$
$282$	0	0
$283$	−19.8969	−1.18275	−0.591374	−	0.806397i	$-0.701414\pi$
$283$	−19.8969	−1.18275	−0.591374	+	0.806397i	$0.701414\pi$
$284$	10.4465	0.619884
$285$	0	0
$286$	0	0
$287$	17.3456	1.02388
$288$	0	0
$289$	11.5759	0.680938
$290$	1.52884	0.0897764
$291$	0	0
$292$	−18.3355	−1.07300
$293$	26.0185	1.52002	0.760008	−	0.649914i	$-0.225195\pi$
$293$	26.0185	1.52002	0.760008	+	0.649914i	$0.225195\pi$
$294$	0	0
$295$	0.899170	0.0523517
$296$	6.98983	0.406276
$297$	0	0
$298$	−6.48963	−0.375934
$299$	0	0
$300$	0	0
$301$	29.1809	1.68196
$302$	−12.0247	−0.691943
$303$	0	0
$304$	13.3888	0.767901
$305$	−5.81681	−0.333070
$306$	0	0
$307$	−2.39276	−0.136562	−0.0682809	−	0.997666i	$-0.521751\pi$
$307$	−2.39276	−0.136562	−0.0682809	+	0.997666i	$0.521751\pi$
$308$	−22.8066	−1.29953
$309$	0	0
$310$	0.140034	0.00795338
$311$	3.54731	0.201149	0.100575	−	0.994930i	$-0.467932\pi$
$311$	3.54731	0.201149	0.100575	+	0.994930i	$0.467932\pi$
$312$	0	0
$313$	4.97927	0.281445	0.140722	−	0.990049i	$-0.455057\pi$
$313$	4.97927	0.281445	0.140722	+	0.990049i	$0.455057\pi$
$314$	5.62571	0.317477
$315$	0	0
$316$	6.07841	0.341937
$317$	18.5944	1.04437	0.522183	−	0.852833i	$-0.325118\pi$
$317$	18.5944	1.04437	0.522183	+	0.852833i	$0.325118\pi$
$318$	0	0
$319$	13.6336	0.763336
$320$	−1.18319	−0.0661423
$321$	0	0
$322$	3.71203	0.206863
$323$	−33.3826	−1.85746
$324$	0	0
$325$	0	0
$326$	−7.08405	−0.392349
$327$	0	0
$328$	−13.6336	−0.752791
$329$	7.14399	0.393861
$330$	0	0
$331$	5.91990	0.325387	0.162694	−	0.986677i	$-0.447982\pi$
$331$	5.91990	0.325387	0.162694	+	0.986677i	$0.447982\pi$
$332$	16.4218	0.901263
$333$	0	0
$334$	−4.86658	−0.266287
$335$	2.18319	0.119280
$336$	0	0
$337$	−13.9216	−0.758358	−0.379179	−	0.925323i	$-0.623793\pi$
$337$	−13.9216	−0.758358	−0.379179	+	0.925323i	$0.623793\pi$
$338$	0	0
$339$	0	0
$340$	8.94233	0.484966
$341$	1.24877	0.0676247
$342$	0	0
$343$	18.3249	0.989452
$344$	−22.9361	−1.23663
$345$	0	0
$346$	−12.3249	−0.662592
$347$	−7.65831	−0.411119	−0.205560	−	0.978645i	$-0.565901\pi$
$347$	−7.65831	−0.411119	−0.205560	+	0.978645i	$0.565901\pi$
$348$	0	0
$349$	11.1809	0.598501	0.299251	−	0.954175i	$-0.403263\pi$
$349$	11.1809	0.598501	0.299251	+	0.954175i	$0.403263\pi$
$350$	1.52884	0.0817198
$351$	0	0
$352$	27.6873	1.47574
$353$	9.65209	0.513729	0.256864	−	0.966447i	$-0.417311\pi$
$353$	9.65209	0.513729	0.256864	+	0.966447i	$0.417311\pi$
$354$	0	0
$355$	−6.24482	−0.331440
$356$	12.7697	0.676793
$357$	0	0
$358$	1.46890	0.0776339
$359$	−9.51206	−0.502027	−0.251014	−	0.967984i	$-0.580764\pi$
$359$	−9.51206	−0.502027	−0.251014	+	0.967984i	$0.580764\pi$
$360$	0	0
$361$	19.9977	1.05251
$362$	7.85375	0.412784
$363$	0	0
$364$	0	0
$365$	10.9608	0.573714
$366$	0	0
$367$	−17.0040	−0.887599	−0.443800	−	0.896126i	$-0.646370\pi$
$367$	−17.0040	−0.887599	−0.443800	+	0.896126i	$0.646370\pi$
$368$	5.20561	0.271361
$369$	0	0
$370$	−1.90312	−0.0989386
$371$	−11.2303	−0.583048
$372$	0	0
$373$	−1.83528	−0.0950273	−0.0475136	−	0.998871i	$-0.515130\pi$
$373$	−1.83528	−0.0950273	−0.0475136	+	0.998871i	$0.515130\pi$
$374$	−15.5967	−0.806485
$375$	0	0
$376$	−5.61515	−0.289579
$377$	0	0
$378$	0	0
$379$	32.3681	1.66264	0.831318	−	0.555797i	$-0.187587\pi$
$379$	32.3681	1.66264	0.831318	+	0.555797i	$0.187587\pi$
$380$	−10.4465	−0.535893
$381$	0	0
$382$	−6.11535	−0.312888
$383$	−28.1417	−1.43798	−0.718988	−	0.695023i	$-0.755394\pi$
$383$	−28.1417	−1.43798	−0.718988	+	0.695023i	$0.755394\pi$
$384$	0	0
$385$	13.6336	0.694834
$386$	11.0162	0.560710
$387$	0	0
$388$	18.9793	0.963526
$389$	−31.9585	−1.62036	−0.810181	−	0.586180i	$-0.800631\pi$
$389$	−31.9585	−1.62036	−0.810181	+	0.586180i	$0.800631\pi$
$390$	0	0
$391$	−12.9793	−0.656390
$392$	0.302491	0.0152781
$393$	0	0
$394$	−9.82738	−0.495096
$395$	−3.63362	−0.182827
$396$	0	0
$397$	12.7098	0.637885	0.318942	−	0.947774i	$-0.396672\pi$
$397$	12.7098	0.637885	0.318942	+	0.947774i	$0.396672\pi$
$398$	−6.60498	−0.331078
$399$	0	0
$400$	2.14399	0.107199
$401$	0.979268	0.0489023	0.0244512	−	0.999701i	$-0.492216\pi$
$401$	0.979268	0.0489023	0.0244512	+	0.999701i	$0.492216\pi$
$402$	0	0
$403$	0	0
$404$	18.1602	0.903504
$405$	0	0
$406$	−4.08631	−0.202800
$407$	−16.9714	−0.841239
$408$	0	0
$409$	−7.79834	−0.385603	−0.192802	−	0.981238i	$-0.561757\pi$
$409$	−7.79834	−0.385603	−0.192802	+	0.981238i	$0.561757\pi$
$410$	3.71203	0.183324
$411$	0	0
$412$	−17.7816	−0.876035
$413$	−2.40332	−0.118260
$414$	0	0
$415$	−9.81681	−0.481888
$416$	0	0
$417$	0	0
$418$	18.2201	0.891176
$419$	−10.2017	−0.498384	−0.249192	−	0.968454i	$-0.580165\pi$
$419$	−10.2017	−0.498384	−0.249192	+	0.968454i	$0.580165\pi$
$420$	0	0
$421$	−31.1888	−1.52005	−0.760025	−	0.649893i	$-0.774814\pi$
$421$	−31.1888	−1.52005	−0.760025	+	0.649893i	$0.774814\pi$
$422$	−13.7983	−0.671693
$423$	0	0
$424$	8.82698	0.428676
$425$	−5.34565	−0.259302
$426$	0	0
$427$	15.5473	0.752387
$428$	0.234252	0.0113230
$429$	0	0
$430$	6.24482	0.301152
$431$	22.9361	1.10479	0.552397	−	0.833581i	$-0.313713\pi$
$431$	22.9361	1.10479	0.552397	+	0.833581i	$0.313713\pi$
$432$	0	0
$433$	16.2880	0.782750	0.391375	−	0.920231i	$-0.372000\pi$
$433$	16.2880	0.782750	0.391375	+	0.920231i	$0.372000\pi$
$434$	−0.374285	−0.0179663
$435$	0	0
$436$	−15.1519	−0.725644
$437$	15.1625	0.725319
$438$	0	0
$439$	−23.7569	−1.13385	−0.566927	−	0.823768i	$-0.691868\pi$
$439$	−23.7569	−1.13385	−0.566927	+	0.823768i	$0.691868\pi$
$440$	−10.7160	−0.510864
$441$	0	0
$442$	0	0
$443$	2.75292	0.130795	0.0653976	−	0.997859i	$-0.479168\pi$
$443$	2.75292	0.130795	0.0653976	+	0.997859i	$0.479168\pi$
$444$	0	0
$445$	−7.63362	−0.361868
$446$	−3.27781	−0.155209
$447$	0	0
$448$	3.16246	0.149412
$449$	27.1025	1.27905	0.639524	−	0.768772i	$-0.279132\pi$
$449$	27.1025	1.27905	0.639524	+	0.768772i	$0.279132\pi$
$450$	0	0
$451$	33.1025	1.55874
$452$	21.7120	1.02125
$453$	0	0
$454$	1.68903	0.0792703
$455$	0	0
$456$	0	0
$457$	40.1523	1.87824	0.939122	−	0.343583i	$-0.111641\pi$
$457$	40.1523	1.87824	0.939122	+	0.343583i	$0.111641\pi$
$458$	−14.4482	−0.675119
$459$	0	0
$460$	−4.06163	−0.189374
$461$	−2.00791	−0.0935175	−0.0467587	−	0.998906i	$-0.514889\pi$
$461$	−2.00791	−0.0935175	−0.0467587	+	0.998906i	$0.514889\pi$
$462$	0	0
$463$	2.67282	0.124217	0.0621083	−	0.998069i	$-0.480218\pi$
$463$	2.67282	0.124217	0.0621083	+	0.998069i	$0.480218\pi$
$464$	−5.73050	−0.266032
$465$	0	0
$466$	13.3536	0.618591
$467$	31.8890	1.47565	0.737824	−	0.674994i	$-0.235854\pi$
$467$	31.8890	1.47565	0.737824	+	0.674994i	$0.235854\pi$
$468$	0	0
$469$	−5.83528	−0.269448
$470$	1.52884	0.0705200
$471$	0	0
$472$	1.88900	0.0869484
$473$	55.6890	2.56058
$474$	0	0
$475$	6.24482	0.286532
$476$	−23.9013	−1.09551
$477$	0	0
$478$	−2.16907	−0.0992110
$479$	6.73445	0.307705	0.153852	−	0.988094i	$-0.450832\pi$
$479$	6.73445	0.307705	0.153852	+	0.988094i	$0.450832\pi$
$480$	0	0
$481$	0	0
$482$	1.96306	0.0894148
$483$	0	0
$484$	−25.1233	−1.14197
$485$	−11.3456	−0.515179
$486$	0	0
$487$	39.6521	1.79681	0.898404	−	0.439170i	$-0.144727\pi$
$487$	39.6521	1.79681	0.898404	+	0.439170i	$0.144727\pi$
$488$	−12.2201	−0.553179
$489$	0	0
$490$	−0.0823593	−0.00372062
$491$	31.0946	1.40328	0.701640	−	0.712531i	$-0.252451\pi$
$491$	31.0946	1.40328	0.701640	+	0.712531i	$0.252451\pi$
$492$	0	0
$493$	14.2880	0.643498
$494$	0	0
$495$	0	0
$496$	−0.524884	−0.0235680
$497$	16.6913	0.748707
$498$	0	0
$499$	13.1087	0.586828	0.293414	−	0.955986i	$-0.405209\pi$
$499$	13.1087	0.586828	0.293414	+	0.955986i	$0.405209\pi$
$500$	−1.67282	−0.0748110
$501$	0	0
$502$	14.5680	0.650203
$503$	8.59273	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$503$	8.59273	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$504$	0	0
$505$	−10.8560	−0.483086
$506$	7.08405	0.314924
$507$	0	0
$508$	−20.1770	−0.895209
$509$	−29.8353	−1.32243	−0.661213	−	0.750198i	$-0.729958\pi$
$509$	−29.8353	−1.32243	−0.661213	+	0.750198i	$0.729958\pi$
$510$	0	0
$511$	−29.2963	−1.29599
$512$	−20.6459	−0.912428
$513$	0	0
$514$	−5.30080	−0.233808
$515$	10.6297	0.468399
$516$	0	0
$517$	13.6336	0.599606
$518$	5.08671	0.223497
$519$	0	0
$520$	0	0
$521$	−10.0969	−0.442352	−0.221176	−	0.975234i	$-0.570990\pi$
$521$	−10.0969	−0.442352	−0.221176	+	0.975234i	$0.570990\pi$
$522$	0	0
$523$	16.1479	0.706100	0.353050	−	0.935604i	$-0.385145\pi$
$523$	16.1479	0.706100	0.353050	+	0.935604i	$0.385145\pi$
$524$	12.7697	0.557847
$525$	0	0
$526$	−4.66057	−0.203210
$527$	1.30871	0.0570081
$528$	0	0
$529$	−17.1048	−0.743686
$530$	−2.40332	−0.104394
$531$	0	0
$532$	27.9216	1.21055
$533$	0	0
$534$	0	0
$535$	−0.140034	−0.00605418
$536$	4.58651	0.198107
$537$	0	0
$538$	0.654353	0.0282111
$539$	−0.734451	−0.0316350
$540$	0	0
$541$	26.7776	1.15126	0.575630	−	0.817711i	$-0.304757\pi$
$541$	26.7776	1.15126	0.575630	+	0.817711i	$0.304757\pi$
$542$	1.82302	0.0783056
$543$	0	0
$544$	29.0162	1.24406
$545$	9.05767	0.387988
$546$	0	0
$547$	12.3865	0.529610	0.264805	−	0.964302i	$-0.414692\pi$
$547$	12.3865	0.529610	0.264805	+	0.964302i	$0.414692\pi$
$548$	−5.74106	−0.245246
$549$	0	0
$550$	2.91764	0.124409
$551$	−16.6913	−0.711073
$552$	0	0
$553$	9.71203	0.412997
$554$	4.36186	0.185318
$555$	0	0
$556$	1.57634	0.0668519
$557$	−10.4218	−0.441586	−0.220793	−	0.975321i	$-0.570864\pi$
$557$	−10.4218	−0.441586	−0.220793	+	0.975321i	$0.570864\pi$
$558$	0	0
$559$	0	0
$560$	−5.73050	−0.242158
$561$	0	0
$562$	−11.5763	−0.488319
$563$	9.78156	0.412244	0.206122	−	0.978526i	$-0.433916\pi$
$563$	9.78156	0.412244	0.206122	+	0.978526i	$0.433916\pi$
$564$	0	0
$565$	−12.9793	−0.546042
$566$	11.3809	0.478375
$567$	0	0
$568$	−13.1193	−0.550474
$569$	−30.2201	−1.26689	−0.633447	−	0.773786i	$-0.718361\pi$
$569$	−30.2201	−1.26689	−0.633447	+	0.773786i	$0.718361\pi$
$570$	0	0
$571$	39.8643	1.66827	0.834135	−	0.551561i	$-0.185967\pi$
$571$	39.8643	1.66827	0.834135	+	0.551561i	$0.185967\pi$
$572$	0	0
$573$	0	0
$574$	−9.92159	−0.414119
$575$	2.42801	0.101255
$576$	0	0
$577$	−8.46326	−0.352330	−0.176165	−	0.984361i	$-0.556369\pi$
$577$	−8.46326	−0.352330	−0.176165	+	0.984361i	$0.556369\pi$
$578$	−6.62136	−0.275412
$579$	0	0
$580$	4.47116	0.185655
$581$	26.2386	1.08856
$582$	0	0
$583$	−21.4320	−0.887621
$584$	23.0268	0.952855
$585$	0	0
$586$	−14.8824	−0.614786
$587$	9.32718	0.384974	0.192487	−	0.981300i	$-0.438345\pi$
$587$	9.32718	0.384974	0.192487	+	0.981300i	$0.438345\pi$
$588$	0	0
$589$	−1.52884	−0.0629946
$590$	−0.514319	−0.0211742
$591$	0	0
$592$	7.13342	0.293182
$593$	8.07841	0.331740	0.165870	−	0.986148i	$-0.446957\pi$
$593$	8.07841	0.331740	0.165870	+	0.986148i	$0.446957\pi$
$594$	0	0
$595$	14.2880	0.585750
$596$	−18.9793	−0.777421
$597$	0	0
$598$	0	0
$599$	−19.1440	−0.782202	−0.391101	−	0.920348i	$-0.627906\pi$
$599$	−19.1440	−0.782202	−0.391101	+	0.920348i	$0.627906\pi$
$600$	0	0
$601$	−4.28797	−0.174910	−0.0874550	−	0.996168i	$-0.527873\pi$
$601$	−4.28797	−0.174910	−0.0874550	+	0.996168i	$0.527873\pi$
$602$	−16.6913	−0.680286
$603$	0	0
$604$	−35.1668	−1.43092
$605$	15.0185	0.610588
$606$	0	0
$607$	29.2426	1.18692	0.593459	−	0.804864i	$-0.297762\pi$
$607$	29.2426	1.18692	0.593459	+	0.804864i	$0.297762\pi$
$608$	−33.8969	−1.37470
$609$	0	0
$610$	3.32718	0.134713
$611$	0	0
$612$	0	0
$613$	1.42405	0.0575170	0.0287585	−	0.999586i	$-0.490845\pi$
$613$	1.42405	0.0575170	0.0287585	+	0.999586i	$0.490845\pi$
$614$	1.36864	0.0552338
$615$	0	0
$616$	28.6419	1.15402
$617$	46.5266	1.87309	0.936545	−	0.350548i	$-0.114005\pi$
$617$	46.5266	1.87309	0.936545	+	0.350548i	$0.114005\pi$
$618$	0	0
$619$	−34.2818	−1.37790	−0.688950	−	0.724809i	$-0.741928\pi$
$619$	−34.2818	−1.37790	−0.688950	+	0.724809i	$0.741928\pi$
$620$	0.409536	0.0164473
$621$	0	0
$622$	−2.02904	−0.0813569
$623$	20.4033	0.817442
$624$	0	0
$625$	1.00000	0.0400000
$626$	−2.84811	−0.113833
$627$	0	0
$628$	16.4527	0.656534
$629$	−17.7859	−0.709171
$630$	0	0
$631$	2.32322	0.0924861	0.0462430	−	0.998930i	$-0.485275\pi$
$631$	2.32322	0.0924861	0.0462430	+	0.998930i	$0.485275\pi$
$632$	−7.63362	−0.303649
$633$	0	0
$634$	−10.6359	−0.422405
$635$	12.0616	0.478651
$636$	0	0
$637$	0	0
$638$	−7.79834	−0.308739
$639$	0	0
$640$	11.5328	0.455874
$641$	28.1312	1.11111	0.555557	−	0.831478i	$-0.312505\pi$
$641$	28.1312	1.11111	0.555557	+	0.831478i	$0.312505\pi$
$642$	0	0
$643$	−14.1338	−0.557383	−0.278692	−	0.960381i	$-0.589901\pi$
$643$	−14.1338	−0.557383	−0.278692	+	0.960381i	$0.589901\pi$
$644$	10.8560	0.427787
$645$	0	0
$646$	19.0946	0.751268
$647$	45.8969	1.80439	0.902197	−	0.431325i	$-0.141954\pi$
$647$	45.8969	1.80439	0.902197	+	0.431325i	$0.141954\pi$
$648$	0	0
$649$	−4.58651	−0.180036
$650$	0	0
$651$	0	0
$652$	−20.7177	−0.811367
$653$	−50.4482	−1.97419	−0.987095	−	0.160137i	$-0.948806\pi$
$653$	−50.4482	−1.97419	−0.987095	+	0.160137i	$0.948806\pi$
$654$	0	0
$655$	−7.63362	−0.298270
$656$	−13.9137	−0.543238
$657$	0	0
$658$	−4.08631	−0.159301
$659$	45.8722	1.78693	0.893464	−	0.449135i	$-0.148268\pi$
$659$	45.8722	1.78693	0.893464	+	0.449135i	$0.148268\pi$
$660$	0	0
$661$	−25.8432	−1.00518	−0.502592	−	0.864524i	$-0.667620\pi$
$661$	−25.8432	−1.00518	−0.502592	+	0.864524i	$0.667620\pi$
$662$	−3.38614	−0.131606
$663$	0	0
$664$	−20.6235	−0.800345
$665$	−16.6913	−0.647261
$666$	0	0
$667$	−6.48963	−0.251280
$668$	−14.2326	−0.550674
$669$	0	0
$670$	−1.24877	−0.0482442
$671$	29.6706	1.14542
$672$	0	0
$673$	4.97927	0.191937	0.0959683	−	0.995384i	$-0.469405\pi$
$673$	4.97927	0.191937	0.0959683	+	0.995384i	$0.469405\pi$
$674$	7.96306	0.306726
$675$	0	0
$676$	0	0
$677$	35.8353	1.37726	0.688631	−	0.725112i	$-0.258212\pi$
$677$	35.8353	1.37726	0.688631	+	0.725112i	$0.258212\pi$
$678$	0	0
$679$	30.3249	1.16376
$680$	−11.2303	−0.430662
$681$	0	0
$682$	−0.714288	−0.0273515
$683$	−40.4712	−1.54859	−0.774293	−	0.632827i	$-0.781894\pi$
$683$	−40.4712	−1.54859	−0.774293	+	0.632827i	$0.781894\pi$
$684$	0	0
$685$	3.43196	0.131128
$686$	−10.4817	−0.400194
$687$	0	0
$688$	−23.4073	−0.892394
$689$	0	0
$690$	0	0
$691$	12.2448	0.465815	0.232907	−	0.972499i	$-0.425176\pi$
$691$	12.2448	0.465815	0.232907	+	0.972499i	$0.425176\pi$
$692$	−36.0448	−1.37022
$693$	0	0
$694$	4.38050	0.166281
$695$	−0.942326	−0.0357445
$696$	0	0
$697$	34.6913	1.31403
$698$	−6.39542	−0.242070
$699$	0	0
$700$	4.47116	0.168994
$701$	−11.8353	−0.447012	−0.223506	−	0.974703i	$-0.571750\pi$
$701$	−11.8353	−0.447012	−0.223506	+	0.974703i	$0.571750\pi$
$702$	0	0
$703$	20.7776	0.783642
$704$	6.03525	0.227462
$705$	0	0
$706$	−5.52093	−0.207783
$707$	29.0162	1.09127
$708$	0	0
$709$	6.37429	0.239391	0.119696	−	0.992811i	$-0.461808\pi$
$709$	6.37429	0.239391	0.119696	+	0.992811i	$0.461808\pi$
$710$	3.57199	0.134055
$711$	0	0
$712$	−16.0369	−0.601010
$713$	−0.594417	−0.0222611
$714$	0	0
$715$	0	0
$716$	4.29588	0.160545
$717$	0	0
$718$	5.44083	0.203050
$719$	34.9009	1.30158	0.650791	−	0.759257i	$-0.274437\pi$
$719$	34.9009	1.30158	0.650791	+	0.759257i	$0.274437\pi$
$720$	0	0
$721$	−28.4112	−1.05809
$722$	−11.4386	−0.425700
$723$	0	0
$724$	22.9687	0.853625
$725$	−2.67282	−0.0992662
$726$	0	0
$727$	19.8106	0.734734	0.367367	−	0.930076i	$-0.380259\pi$
$727$	19.8106	0.734734	0.367367	+	0.930076i	$0.380259\pi$
$728$	0	0
$729$	0	0
$730$	−6.26950	−0.232045
$731$	58.3619	2.15859
$732$	0	0
$733$	12.6050	0.465576	0.232788	−	0.972528i	$-0.425215\pi$
$733$	12.6050	0.465576	0.232788	+	0.972528i	$0.425215\pi$
$734$	9.72615	0.358999
$735$	0	0
$736$	−13.1792	−0.485793
$737$	−11.1361	−0.410203
$738$	0	0
$739$	−37.1747	−1.36749	−0.683747	−	0.729719i	$-0.739651\pi$
$739$	−37.1747	−1.36749	−0.683747	+	0.729719i	$0.739651\pi$
$740$	−5.56578	−0.204602
$741$	0	0
$742$	6.42366	0.235820
$743$	27.9322	1.02473	0.512366	−	0.858767i	$-0.328769\pi$
$743$	27.9322	1.02473	0.512366	+	0.858767i	$0.328769\pi$
$744$	0	0
$745$	11.3456	0.415672
$746$	1.04977	0.0384348
$747$	0	0
$748$	−45.6133	−1.66779
$749$	0.374285	0.0136761
$750$	0	0
$751$	−13.6257	−0.497209	−0.248605	−	0.968605i	$-0.579972\pi$
$751$	−13.6257	−0.497209	−0.248605	+	0.968605i	$0.579972\pi$
$752$	−5.73050	−0.208970
$753$	0	0
$754$	0	0
$755$	21.0224	0.765084
$756$	0	0
$757$	−30.8560	−1.12148	−0.560740	−	0.827992i	$-0.689483\pi$
$757$	−30.8560	−1.12148	−0.560740	+	0.827992i	$0.689483\pi$
$758$	−18.5143	−0.672470
$759$	0	0
$760$	13.1193	0.475887
$761$	−17.0162	−0.616837	−0.308419	−	0.951251i	$-0.599800\pi$
$761$	−17.0162	−0.616837	−0.308419	+	0.951251i	$0.599800\pi$
$762$	0	0
$763$	−24.2096	−0.876445
$764$	−17.8847	−0.647044
$765$	0	0
$766$	16.0969	0.581604
$767$	0	0
$768$	0	0
$769$	−29.0162	−1.04635	−0.523176	−	0.852225i	$-0.675253\pi$
$769$	−29.0162	−1.04635	−0.523176	+	0.852225i	$0.675253\pi$
$770$	−7.79834	−0.281033
$771$	0	0
$772$	32.2175	1.15953
$773$	40.5160	1.45726	0.728630	−	0.684908i	$-0.240158\pi$
$773$	40.5160	1.45726	0.728630	+	0.684908i	$0.240158\pi$
$774$	0	0
$775$	−0.244817	−0.00879409
$776$	−23.8353	−0.855637
$777$	0	0
$778$	18.2801	0.655372
$779$	−40.5266	−1.45202
$780$	0	0
$781$	31.8538	1.13982
$782$	7.42405	0.265484
$783$	0	0
$784$	0.308705	0.0110252
$785$	−9.83528	−0.351036
$786$	0	0
$787$	34.0264	1.21291	0.606455	−	0.795118i	$-0.292591\pi$
$787$	34.0264	1.21291	0.606455	+	0.795118i	$0.292591\pi$
$788$	−28.7407	−1.02384
$789$	0	0
$790$	2.07841	0.0739464
$791$	34.6913	1.23348
$792$	0	0
$793$	0	0
$794$	−7.26990	−0.257999
$795$	0	0
$796$	−19.3166	−0.684659
$797$	14.7776	0.523450	0.261725	−	0.965143i	$-0.415709\pi$
$797$	14.7776	0.523450	0.261725	+	0.965143i	$0.415709\pi$
$798$	0	0
$799$	14.2880	0.505472
$800$	−5.42801	−0.191909
$801$	0	0
$802$	−0.560135	−0.0197790
$803$	−55.9092	−1.97299
$804$	0	0
$805$	−6.48963	−0.228730
$806$	0	0
$807$	0	0
$808$	−22.8066	−0.802335
$809$	36.0554	1.26764	0.633820	−	0.773480i	$-0.281486\pi$
$809$	36.0554	1.26764	0.633820	+	0.773480i	$0.281486\pi$
$810$	0	0
$811$	−5.31040	−0.186473	−0.0932366	−	0.995644i	$-0.529721\pi$
$811$	−5.31040	−0.186473	−0.0932366	+	0.995644i	$0.529721\pi$
$812$	−11.9506	−0.419385
$813$	0	0
$814$	9.70750	0.340248
$815$	12.3849	0.433822
$816$	0	0
$817$	−68.1787	−2.38527
$818$	4.46060	0.155961
$819$	0	0
$820$	10.8560	0.379108
$821$	13.6336	0.475817	0.237908	−	0.971288i	$-0.423538\pi$
$821$	13.6336	0.475817	0.237908	+	0.971288i	$0.423538\pi$
$822$	0	0
$823$	39.6459	1.38197	0.690984	−	0.722870i	$-0.257177\pi$
$823$	39.6459	1.38197	0.690984	+	0.722870i	$0.257177\pi$
$824$	22.3311	0.777942
$825$	0	0
$826$	1.37468	0.0478314
$827$	23.8952	0.830918	0.415459	−	0.909612i	$-0.363621\pi$
$827$	23.8952	0.830918	0.415459	+	0.909612i	$0.363621\pi$
$828$	0	0
$829$	−27.3562	−0.950121	−0.475060	−	0.879953i	$-0.657574\pi$
$829$	−27.3562	−0.950121	−0.475060	+	0.879953i	$0.657574\pi$
$830$	5.61515	0.194905
$831$	0	0
$832$	0	0
$833$	−0.769701	−0.0266686
$834$	0	0
$835$	8.50811	0.294435
$836$	53.2857	1.84292
$837$	0	0
$838$	5.83528	0.201576
$839$	−29.5411	−1.01987	−0.509936	−	0.860212i	$-0.670331\pi$
$839$	−29.5411	−1.01987	−0.509936	+	0.860212i	$0.670331\pi$
$840$	0	0
$841$	−21.8560	−0.753656
$842$	17.8398	0.614800
$843$	0	0
$844$	−40.3540	−1.38904
$845$	0	0
$846$	0	0
$847$	−40.1417	−1.37929
$848$	9.00830	0.309346
$849$	0	0
$850$	3.05767	0.104877
$851$	8.07841	0.276924
$852$	0	0
$853$	−5.61515	−0.192259	−0.0961295	−	0.995369i	$-0.530646\pi$
$853$	−5.61515	−0.192259	−0.0961295	+	0.995369i	$0.530646\pi$
$854$	−8.89296	−0.304311
$855$	0	0
$856$	−0.294187	−0.0100551
$857$	39.9216	1.36370	0.681848	−	0.731494i	$-0.261177\pi$
$857$	39.9216	1.36370	0.681848	+	0.731494i	$0.261177\pi$
$858$	0	0
$859$	28.6498	0.977520	0.488760	−	0.872418i	$-0.337449\pi$
$859$	28.6498	0.977520	0.488760	+	0.872418i	$0.337449\pi$
$860$	18.2633	0.622773
$861$	0	0
$862$	−13.1193	−0.446845
$863$	−38.5530	−1.31236	−0.656179	−	0.754605i	$-0.727828\pi$
$863$	−38.5530	−1.31236	−0.656179	+	0.754605i	$0.727828\pi$
$864$	0	0
$865$	21.5473	0.732630
$866$	−9.31661	−0.316591
$867$	0	0
$868$	−1.09462	−0.0371537
$869$	18.5345	0.628739
$870$	0	0
$871$	0	0
$872$	19.0286	0.644391
$873$	0	0
$874$	−8.67282	−0.293363
$875$	−2.67282	−0.0903579
$876$	0	0
$877$	27.3826	0.924644	0.462322	−	0.886712i	$-0.347016\pi$
$877$	27.3826	0.924644	0.462322	+	0.886712i	$0.347016\pi$
$878$	13.5888	0.458599
$879$	0	0
$880$	−10.9361	−0.368656
$881$	25.0841	0.845103	0.422552	−	0.906339i	$-0.361135\pi$
$881$	25.0841	0.845103	0.422552	+	0.906339i	$0.361135\pi$
$882$	0	0
$883$	−50.5513	−1.70119	−0.850593	−	0.525825i	$-0.823757\pi$
$883$	−50.5513	−1.70119	−0.850593	+	0.525825i	$0.823757\pi$
$884$	0	0
$885$	0	0
$886$	−1.57465	−0.0529015
$887$	19.6583	0.660061	0.330031	−	0.943970i	$-0.392941\pi$
$887$	19.6583	0.660061	0.330031	+	0.943970i	$0.392941\pi$
$888$	0	0
$889$	−32.2386	−1.08125
$890$	4.36638	0.146361
$891$	0	0
$892$	−9.58611	−0.320967
$893$	−16.6913	−0.558553
$894$	0	0
$895$	−2.56804	−0.0858401
$896$	−30.8251	−1.02979
$897$	0	0
$898$	−15.5025	−0.517324
$899$	0.654353	0.0218239
$900$	0	0
$901$	−22.4606	−0.748271
$902$	−18.9344	−0.630447
$903$	0	0
$904$	−27.2672	−0.906895
$905$	−13.7305	−0.456417
$906$	0	0
$907$	22.0537	0.732282	0.366141	−	0.930559i	$-0.380679\pi$
$907$	22.0537	0.732282	0.366141	+	0.930559i	$0.380679\pi$
$908$	4.93967	0.163929
$909$	0	0
$910$	0	0
$911$	14.0079	0.464103	0.232051	−	0.972704i	$-0.425456\pi$
$911$	14.0079	0.464103	0.232051	+	0.972704i	$0.425456\pi$
$912$	0	0
$913$	50.0739	1.65720
$914$	−22.9668	−0.759676
$915$	0	0
$916$	−42.2544	−1.39613
$917$	20.4033	0.673777
$918$	0	0
$919$	21.8353	0.720279	0.360140	−	0.932898i	$-0.382729\pi$
$919$	21.8353	0.720279	0.360140	+	0.932898i	$0.382729\pi$
$920$	5.10083	0.168169
$921$	0	0
$922$	1.14851	0.0378241
$923$	0	0
$924$	0	0
$925$	3.32718	0.109397
$926$	−1.52884	−0.0502407
$927$	0	0
$928$	14.5081	0.476252
$929$	−15.2224	−0.499431	−0.249715	−	0.968319i	$-0.580337\pi$
$929$	−15.2224	−0.499431	−0.249715	+	0.968319i	$0.580337\pi$
$930$	0	0
$931$	0.899170	0.0294691
$932$	39.0532	1.27923
$933$	0	0
$934$	−18.2403	−0.596841
$935$	27.2672	0.891734
$936$	0	0
$937$	−25.4610	−0.831774	−0.415887	−	0.909416i	$-0.636529\pi$
$937$	−25.4610	−0.831774	−0.415887	+	0.909416i	$0.636529\pi$
$938$	3.33774	0.108981
$939$	0	0
$940$	4.47116	0.145833
$941$	24.9299	0.812691	0.406346	−	0.913719i	$-0.366803\pi$
$941$	24.9299	0.812691	0.406346	+	0.913719i	$0.366803\pi$
$942$	0	0
$943$	−15.7569	−0.513114
$944$	1.92781	0.0627448
$945$	0	0
$946$	−31.8538	−1.03565
$947$	−16.6807	−0.542051	−0.271025	−	0.962572i	$-0.587363\pi$
$947$	−16.6807	−0.542051	−0.271025	+	0.962572i	$0.587363\pi$
$948$	0	0
$949$	0	0
$950$	−3.57199	−0.115891
$951$	0	0
$952$	30.0166	0.972844
$953$	−27.1730	−0.880221	−0.440110	−	0.897944i	$-0.645061\pi$
$953$	−27.1730	−0.880221	−0.440110	+	0.897944i	$0.645061\pi$
$954$	0	0
$955$	10.6913	0.345962
$956$	−6.34356	−0.205165
$957$	0	0
$958$	−3.85206	−0.124454
$959$	−9.17302	−0.296212
$960$	0	0
$961$	−30.9401	−0.998067
$962$	0	0
$963$	0	0
$964$	5.74106	0.184907
$965$	−19.2593	−0.619980
$966$	0	0
$967$	−6.75914	−0.217359	−0.108680	−	0.994077i	$-0.534662\pi$
$967$	−6.75914	−0.217359	−0.108680	+	0.994077i	$0.534662\pi$
$968$	31.5513	1.01410
$969$	0	0
$970$	6.48963	0.208370
$971$	−46.6419	−1.49681	−0.748405	−	0.663242i	$-0.769180\pi$
$971$	−46.6419	−1.49681	−0.748405	+	0.663242i	$0.769180\pi$
$972$	0	0
$973$	2.51867	0.0807449
$974$	−22.6807	−0.726737
$975$	0	0
$976$	−12.4712	−0.399192
$977$	−52.7467	−1.68752	−0.843758	−	0.536723i	$-0.819662\pi$
$977$	−52.7467	−1.68752	−0.843758	+	0.536723i	$0.819662\pi$
$978$	0	0
$979$	38.9378	1.24446
$980$	−0.240864	−0.00769412
$981$	0	0
$982$	−17.7859	−0.567571
$983$	20.9977	0.669724	0.334862	−	0.942267i	$-0.391310\pi$
$983$	20.9977	0.669724	0.334862	+	0.942267i	$0.391310\pi$
$984$	0	0
$985$	17.1809	0.547430
$986$	−8.17262	−0.260269
$987$	0	0
$988$	0	0
$989$	−26.5081	−0.842909
$990$	0	0
$991$	−32.3170	−1.02658	−0.513292	−	0.858214i	$-0.671574\pi$
$991$	−32.3170	−1.02658	−0.513292	+	0.858214i	$0.671574\pi$
$992$	1.32887	0.0421916
$993$	0	0
$994$	−9.54731	−0.302822
$995$	11.5473	0.366074
$996$	0	0
$997$	−21.3905	−0.677444	−0.338722	−	0.940887i	$-0.609995\pi$
$997$	−21.3905	−0.677444	−0.338722	+	0.940887i	$0.609995\pi$
$998$	−7.49811	−0.237348
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7605.2.a.bs.1.2		3
3.2	odd	2		845.2.a.k.1.2		3
13.5	odd	4		585.2.b.g.181.4		6
13.8	odd	4		585.2.b.g.181.3		6
13.12	even	2		7605.2.a.cc.1.2		3
15.14	odd	2		4225.2.a.bc.1.2		3
39.2	even	12		845.2.m.h.316.4		12
39.5	even	4		65.2.c.a.51.3	✓	6
39.8	even	4		65.2.c.a.51.4	yes	6
39.11	even	12		845.2.m.h.316.3		12
39.17	odd	6		845.2.e.k.146.2		6
39.20	even	12		845.2.m.h.361.4		12
39.23	odd	6		845.2.e.k.191.2		6
39.29	odd	6		845.2.e.i.191.2		6
39.32	even	12		845.2.m.h.361.3		12
39.35	odd	6		845.2.e.i.146.2		6
39.38	odd	2		845.2.a.i.1.2		3
156.47	odd	4		1040.2.k.d.961.3		6
156.83	odd	4		1040.2.k.d.961.4		6
195.8	odd	4		325.2.d.f.324.3		6
195.44	even	4		325.2.c.g.51.4		6
195.47	odd	4		325.2.d.e.324.4		6
195.83	odd	4		325.2.d.e.324.3		6
195.122	odd	4		325.2.d.f.324.4		6
195.164	even	4		325.2.c.g.51.3		6
195.194	odd	2		4225.2.a.be.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.c.a.51.3	✓	6	39.5	even	4
65.2.c.a.51.4	yes	6	39.8	even	4
325.2.c.g.51.3		6	195.164	even	4
325.2.c.g.51.4		6	195.44	even	4
325.2.d.e.324.3		6	195.83	odd	4
325.2.d.e.324.4		6	195.47	odd	4
325.2.d.f.324.3		6	195.8	odd	4
325.2.d.f.324.4		6	195.122	odd	4
585.2.b.g.181.3		6	13.8	odd	4
585.2.b.g.181.4		6	13.5	odd	4
845.2.a.i.1.2		3	39.38	odd	2
845.2.a.k.1.2		3	3.2	odd	2
845.2.e.i.146.2		6	39.35	odd	6
845.2.e.i.191.2		6	39.29	odd	6
845.2.e.k.146.2		6	39.17	odd	6
845.2.e.k.191.2		6	39.23	odd	6
845.2.m.h.316.3		12	39.11	even	12
845.2.m.h.316.4		12	39.2	even	12
845.2.m.h.361.3		12	39.32	even	12
845.2.m.h.361.4		12	39.20	even	12
1040.2.k.d.961.3		6	156.47	odd	4
1040.2.k.d.961.4		6	156.83	odd	4
4225.2.a.bc.1.2		3	15.14	odd	2
4225.2.a.be.1.2		3	195.194	odd	2
7605.2.a.bs.1.2		3	1.1	even	1	trivial
7605.2.a.cc.1.2		3	13.12	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7605.a (trivial)

\(f(q)\)	\(=\)	\(q-0.571993 q^{2} -1.67282 q^{4} +1.00000 q^{5} -2.67282 q^{7} +2.10083 q^{8} +O(q^{10})\)	\(q-0.571993 q^{2} -1.67282 q^{4} +1.00000 q^{5} -2.67282 q^{7} +2.10083 q^{8} -0.571993 q^{10} -5.10083 q^{11} +1.52884 q^{14} +2.14399 q^{16} -5.34565 q^{17} +6.24482 q^{19} -1.67282 q^{20} +2.91764 q^{22} +2.42801 q^{23} +1.00000 q^{25} +4.47116 q^{28} -2.67282 q^{29} -0.244817 q^{31} -5.42801 q^{32} +3.05767 q^{34} -2.67282 q^{35} +3.32718 q^{37} -3.57199 q^{38} +2.10083 q^{40} -6.48963 q^{41} -10.9176 q^{43} +8.53279 q^{44} -1.38880 q^{46} -2.67282 q^{47} +0.143987 q^{49} -0.571993 q^{50} +4.20166 q^{53} -5.10083 q^{55} -5.61515 q^{56} +1.52884 q^{58} +0.899170 q^{59} -5.81681 q^{61} +0.140034 q^{62} -1.18319 q^{64} +2.18319 q^{67} +8.94233 q^{68} +1.52884 q^{70} -6.24482 q^{71} +10.9608 q^{73} -1.90312 q^{74} -10.4465 q^{76} +13.6336 q^{77} -3.63362 q^{79} +2.14399 q^{80} +3.71203 q^{82} -9.81681 q^{83} -5.34565 q^{85} +6.24482 q^{86} -10.7160 q^{88} -7.63362 q^{89} -4.06163 q^{92} +1.52884 q^{94} +6.24482 q^{95} -11.3456 q^{97} -0.0823593 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 5 q^{4} + 3 q^{5} + 2 q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q - q^2 + 5 * q^4 + 3 * q^5 + 2 * q^7 - 3 * q^8	\( 3 q - q^{2} + 5 q^{4} + 3 q^{5} + 2 q^{7} - 3 q^{8} - q^{10} - 6 q^{11} - 4 q^{14} + 5 q^{16} + 4 q^{17} + 8 q^{19} + 5 q^{20} - 12 q^{22} + 8 q^{23} + 3 q^{25} + 22 q^{28} + 2 q^{29} + 10 q^{31} - 17 q^{32} - 8 q^{34} + 2 q^{35} + 20 q^{37} - 10 q^{38} - 3 q^{40} + 2 q^{41} - 12 q^{43} + 12 q^{44} + 8 q^{46} + 2 q^{47} - q^{49} - q^{50} - 6 q^{53} - 6 q^{55} - 24 q^{56} - 4 q^{58} + 12 q^{59} - 6 q^{61} + 4 q^{62} - 15 q^{64} + 18 q^{67} + 44 q^{68} - 4 q^{70} - 8 q^{71} + 20 q^{73} - 10 q^{74} - 2 q^{76} + 18 q^{77} + 12 q^{79} + 5 q^{80} + 14 q^{82} - 18 q^{83} + 4 q^{85} + 8 q^{86} - 30 q^{88} + 10 q^{92} - 4 q^{94} + 8 q^{95} - 14 q^{97} - 21 q^{98}+O(q^{100}) \) 3 * q - q^2 + 5 * q^4 + 3 * q^5 + 2 * q^7 - 3 * q^8 - q^10 - 6 * q^11 - 4 * q^14 + 5 * q^16 + 4 * q^17 + 8 * q^19 + 5 * q^20 - 12 * q^22 + 8 * q^23 + 3 * q^25 + 22 * q^28 + 2 * q^29 + 10 * q^31 - 17 * q^32 - 8 * q^34 + 2 * q^35 + 20 * q^37 - 10 * q^38 - 3 * q^40 + 2 * q^41 - 12 * q^43 + 12 * q^44 + 8 * q^46 + 2 * q^47 - q^49 - q^50 - 6 * q^53 - 6 * q^55 - 24 * q^56 - 4 * q^58 + 12 * q^59 - 6 * q^61 + 4 * q^62 - 15 * q^64 + 18 * q^67 + 44 * q^68 - 4 * q^70 - 8 * q^71 + 20 * q^73 - 10 * q^74 - 2 * q^76 + 18 * q^77 + 12 * q^79 + 5 * q^80 + 14 * q^82 - 18 * q^83 + 4 * q^85 + 8 * q^86 - 30 * q^88 + 10 * q^92 - 4 * q^94 + 8 * q^95 - 14 * q^97 - 21 * q^98

Introduction

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