LMFDB - Embedded newform 2325.2.a.bb.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2325,2,Mod(1,2325)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2325.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2325 = 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2325.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,1,6,7,0,1,2,-3,6,0,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$18.5652184699$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.75968016.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 9x^{4} + 9x^{3} + 14x^{2} - 6x - 6 $ x^6 - x^5 - 9x^4 + 9x^3 + 14x^2 - 6x - 6
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			1.2
Root			$-0.814748$ of defining polynomial
Character	$\chi$	$=$	2325.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.814748 q^{2} +1.00000 q^{3} -1.33619 q^{4} -0.814748 q^{6} -3.06134 q^{7} +2.71815 q^{8} +1.00000 q^{9} +6.25805 q^{11} -1.33619 q^{12} -1.92186 q^{13} +2.49422 q^{14} +0.457766 q^{16} +2.14965 q^{17} -0.814748 q^{18} -4.29544 q^{19} -3.06134 q^{21} -5.09873 q^{22} +2.53804 q^{23} +2.71815 q^{24} +1.56583 q^{26} +1.00000 q^{27} +4.09052 q^{28} -2.37777 q^{29} -1.00000 q^{31} -5.80926 q^{32} +6.25805 q^{33} -1.75142 q^{34} -1.33619 q^{36} -3.09873 q^{37} +3.49970 q^{38} -1.92186 q^{39} +9.38690 q^{41} +2.49422 q^{42} -6.79095 q^{43} -8.36192 q^{44} -2.06786 q^{46} +0.0447403 q^{47} +0.457766 q^{48} +2.37179 q^{49} +2.14965 q^{51} +2.56797 q^{52} +2.37358 q^{53} -0.814748 q^{54} -8.32118 q^{56} -4.29544 q^{57} +1.93728 q^{58} +9.75741 q^{59} -2.13211 q^{61} +0.814748 q^{62} -3.06134 q^{63} +3.81755 q^{64} -5.09873 q^{66} +9.68208 q^{67} -2.87233 q^{68} +2.53804 q^{69} -1.94137 q^{71} +2.71815 q^{72} -3.07607 q^{73} +2.52468 q^{74} +5.73951 q^{76} -19.1580 q^{77} +1.56583 q^{78} +13.1684 q^{79} +1.00000 q^{81} -7.64796 q^{82} +2.27194 q^{83} +4.09052 q^{84} +5.53291 q^{86} -2.37777 q^{87} +17.0103 q^{88} +18.5502 q^{89} +5.88347 q^{91} -3.39129 q^{92} -1.00000 q^{93} -0.0364520 q^{94} -5.80926 q^{96} +3.56816 q^{97} -1.93241 q^{98} +6.25805 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + q^{2} + 6 q^{3} + 7 q^{4} + q^{6} + 2 q^{7} - 3 q^{8} + 6 q^{9} + 7 q^{11} + 7 q^{12} + 4 q^{13} + 10 q^{14} + 17 q^{16} + q^{18} + 17 q^{19} + 2 q^{21} + 2 q^{22} - q^{23} - 3 q^{24} + 2 q^{26}+ \cdots + 7 q^{99}+O(q^{100}) $ 6 * q + q^2 + 6 * q^3 + 7 * q^4 + q^6 + 2 * q^7 - 3 * q^8 + 6 * q^9 + 7 * q^11 + 7 * q^12 + 4 * q^13 + 10 * q^14 + 17 * q^16 + q^18 + 17 * q^19 + 2 * q^21 + 2 * q^22 - q^23 - 3 * q^24 + 2 * q^26 + 6 * q^27 + 22 * q^28 - 8 * q^29 - 6 * q^31 - 35 * q^32 + 7 * q^33 - 13 * q^34 + 7 * q^36 + 14 * q^37 + 10 * q^38 + 4 * q^39 + 18 * q^41 + 10 * q^42 + 15 * q^43 + 4 * q^44 - 10 * q^46 - 13 * q^47 + 17 * q^48 + 18 * q^49 - 38 * q^52 - 13 * q^53 + q^54 + 32 * q^56 + 17 * q^57 + 23 * q^58 + 32 * q^59 - q^62 + 2 * q^63 + 29 * q^64 + 2 * q^66 + 9 * q^67 + 9 * q^68 - q^69 + q^71 - 3 * q^72 + 14 * q^73 + 20 * q^74 + 42 * q^76 - 14 * q^77 + 2 * q^78 + 13 * q^79 + 6 * q^81 - 32 * q^82 + 9 * q^83 + 22 * q^84 - 11 * q^86 - 8 * q^87 + 36 * q^88 + 34 * q^89 - 2 * q^91 - 46 * q^92 - 6 * q^93 + 19 * q^94 - 35 * q^96 + 30 * q^97 + 37 * q^98 + 7 * q^99

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 $

$2$	−0.814748	−0.576114	−0.288057	−	0.957613i	$-0.593009\pi$
$2$	−0.814748	−0.576114	−0.288057	+	0.957613i	$0.593009\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.33619	−0.668093
$5$	0	0
$5$	0	0
$6$	−0.814748	−0.332619
$7$	−3.06134	−1.15708	−0.578539	−	0.815655i	$-0.696377\pi$
$7$	−3.06134	−1.15708	−0.578539	+	0.815655i	$0.696377\pi$
$8$	2.71815	0.961011
$9$	1.00000	0.333333
$10$	0	0
$11$	6.25805	1.88687	0.943436	−	0.331553i	$-0.107573\pi$
$11$	6.25805	1.88687	0.943436	+	0.331553i	$0.107573\pi$
$12$	−1.33619	−0.385724
$13$	−1.92186	−0.533029	−0.266514	−	0.963831i	$-0.585872\pi$
$13$	−1.92186	−0.533029	−0.266514	+	0.963831i	$0.585872\pi$
$14$	2.49422	0.666608
$15$	0	0
$16$	0.457766	0.114442
$17$	2.14965	0.521366	0.260683	−	0.965424i	$-0.416052\pi$
$17$	2.14965	0.521366	0.260683	+	0.965424i	$0.416052\pi$
$18$	−0.814748	−0.192038
$19$	−4.29544	−0.985442	−0.492721	−	0.870187i	$-0.663998\pi$
$19$	−4.29544	−0.985442	−0.492721	+	0.870187i	$0.663998\pi$
$20$	0	0
$21$	−3.06134	−0.668039
$22$	−5.09873	−1.08705
$23$	2.53804	0.529217	0.264609	−	0.964356i	$-0.414757\pi$
$23$	2.53804	0.529217	0.264609	+	0.964356i	$0.414757\pi$
$24$	2.71815	0.554840
$25$	0	0
$26$	1.56583	0.307085
$27$	1.00000	0.192450
$28$	4.09052	0.773035
$29$	−2.37777	−0.441540	−0.220770	−	0.975326i	$-0.570857\pi$
$29$	−2.37777	−0.441540	−0.220770	+	0.975326i	$0.570857\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−5.80926	−1.02694
$33$	6.25805	1.08939
$34$	−1.75142	−0.300366
$35$	0	0
$36$	−1.33619	−0.222698
$37$	−3.09873	−0.509428	−0.254714	−	0.967016i	$-0.581981\pi$
$37$	−3.09873	−0.509428	−0.254714	+	0.967016i	$0.581981\pi$
$38$	3.49970	0.567727
$39$	−1.92186	−0.307744
$40$	0	0
$41$	9.38690	1.46599	0.732994	−	0.680235i	$-0.238122\pi$
$41$	9.38690	1.46599	0.732994	+	0.680235i	$0.238122\pi$
$42$	2.49422	0.384866
$43$	−6.79095	−1.03561	−0.517805	−	0.855499i	$-0.673251\pi$
$43$	−6.79095	−1.03561	−0.517805	+	0.855499i	$0.673251\pi$
$44$	−8.36192	−1.26061
$45$	0	0
$46$	−2.06786	−0.304889
$47$	0.0447403	0.00652604	0.00326302	−	0.999995i	$-0.498961\pi$
$47$	0.0447403	0.00652604	0.00326302	+	0.999995i	$0.498961\pi$
$48$	0.457766	0.0660728
$49$	2.37179	0.338827
$50$	0	0
$51$	2.14965	0.301011
$52$	2.56797	0.356113
$53$	2.37358	0.326036	0.163018	−	0.986623i	$-0.447877\pi$
$53$	2.37358	0.326036	0.163018	+	0.986623i	$0.447877\pi$
$54$	−0.814748	−0.110873
$55$	0	0
$56$	−8.32118	−1.11196
$57$	−4.29544	−0.568945
$58$	1.93728	0.254377
$59$	9.75741	1.27031	0.635153	−	0.772386i	$-0.280937\pi$
$59$	9.75741	1.27031	0.635153	+	0.772386i	$0.280937\pi$
$60$	0	0
$61$	−2.13211	−0.272989	−0.136494	−	0.990641i	$-0.543584\pi$
$61$	−2.13211	−0.272989	−0.136494	+	0.990641i	$0.543584\pi$
$62$	0.814748	0.103473
$63$	−3.06134	−0.385692
$64$	3.81755	0.477194
$65$	0	0
$66$	−5.09873	−0.627610
$67$	9.68208	1.18285	0.591427	−	0.806358i	$-0.298565\pi$
$67$	9.68208	1.18285	0.591427	+	0.806358i	$0.298565\pi$
$68$	−2.87233	−0.348321
$69$	2.53804	0.305544
$70$	0	0
$71$	−1.94137	−0.230399	−0.115199	−	0.993342i	$-0.536751\pi$
$71$	−1.94137	−0.230399	−0.115199	+	0.993342i	$0.536751\pi$
$72$	2.71815	0.320337
$73$	−3.07607	−0.360027	−0.180014	−	0.983664i	$-0.557614\pi$
$73$	−3.07607	−0.360027	−0.180014	+	0.983664i	$0.557614\pi$
$74$	2.52468	0.293489
$75$	0	0
$76$	5.73951	0.658367
$77$	−19.1580	−2.18326
$78$	1.56583	0.177296
$79$	13.1684	1.48156	0.740779	−	0.671749i	$-0.234456\pi$
$79$	13.1684	1.48156	0.740779	+	0.671749i	$0.234456\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−7.64796	−0.844575
$83$	2.27194	0.249377	0.124689	−	0.992196i	$-0.460207\pi$
$83$	2.27194	0.249377	0.124689	+	0.992196i	$0.460207\pi$
$84$	4.09052	0.446312
$85$	0	0
$86$	5.53291	0.596629
$87$	−2.37777	−0.254923
$88$	17.0103	1.81331
$89$	18.5502	1.96632	0.983160	−	0.182749i	$-0.0584997\pi$
$89$	18.5502	1.96632	0.983160	+	0.182749i	$0.0584997\pi$
$90$	0	0
$91$	5.88347	0.616756
$92$	−3.39129	−0.353566
$93$	−1.00000	−0.103695
$94$	−0.0364520	−0.00375974
$95$	0	0
$96$	−5.80926	−0.592906
$97$	3.56816	0.362291	0.181146	−	0.983456i	$-0.442019\pi$
$97$	3.56816	0.362291	0.181146	+	0.983456i	$0.442019\pi$
$98$	−1.93241	−0.195203
$99$	6.25805	0.628958
$100$	0	0
$101$	11.3485	1.12922	0.564608	−	0.825360i	$-0.309028\pi$
$101$	11.3485	1.12922	0.564608	+	0.825360i	$0.309028\pi$
$102$	−1.75142	−0.173416
$103$	−4.38784	−0.432347	−0.216174	−	0.976355i	$-0.569358\pi$
$103$	−4.38784	−0.432347	−0.216174	+	0.976355i	$0.569358\pi$
$104$	−5.22391	−0.512247
$105$	0	0
$106$	−1.93387	−0.187834
$107$	0.759270	0.0734014	0.0367007	−	0.999326i	$-0.488315\pi$
$107$	0.759270	0.0734014	0.0367007	+	0.999326i	$0.488315\pi$
$108$	−1.33619	−0.128575
$109$	17.2241	1.64977	0.824884	−	0.565303i	$-0.191241\pi$
$109$	17.2241	1.64977	0.824884	+	0.565303i	$0.191241\pi$
$110$	0	0
$111$	−3.09873	−0.294119
$112$	−1.40138	−0.132418
$113$	0.689986	0.0649084	0.0324542	−	0.999473i	$-0.489668\pi$
$113$	0.689986	0.0649084	0.0324542	+	0.999473i	$0.489668\pi$
$114$	3.49970	0.327777
$115$	0	0
$116$	3.17714	0.294990
$117$	−1.92186	−0.177676
$118$	−7.94982	−0.731841
$119$	−6.58080	−0.603261
$120$	0	0
$121$	28.1632	2.56029
$122$	1.73713	0.157272
$123$	9.38690	0.846388
$124$	1.33619	0.119993
$125$	0	0
$126$	2.49422	0.222203
$127$	12.9581	1.14985	0.574924	−	0.818207i	$-0.305032\pi$
$127$	12.9581	1.14985	0.574924	+	0.818207i	$0.305032\pi$
$128$	8.50819	0.752024
$129$	−6.79095	−0.597910
$130$	0	0
$131$	10.1273	0.884829	0.442415	−	0.896811i	$-0.354122\pi$
$131$	10.1273	0.884829	0.442415	+	0.896811i	$0.354122\pi$
$132$	−8.36192	−0.727812
$133$	13.1498	1.14023
$134$	−7.88846	−0.681459
$135$	0	0
$136$	5.84306	0.501039
$137$	−15.8094	−1.35069	−0.675345	−	0.737502i	$-0.736005\pi$
$137$	−15.8094	−1.35069	−0.675345	+	0.737502i	$0.736005\pi$
$138$	−2.06786	−0.176028
$139$	23.2476	1.97183	0.985917	−	0.167235i	$-0.0534838\pi$
$139$	23.2476	1.97183	0.985917	+	0.167235i	$0.0534838\pi$
$140$	0	0
$141$	0.0447403	0.00376781
$142$	1.58173	0.132736
$143$	−12.0271	−1.00576
$144$	0.457766	0.0381472
$145$	0	0
$146$	2.50622	0.207417
$147$	2.37179	0.195622
$148$	4.14048	0.340346
$149$	−11.4764	−0.940182	−0.470091	−	0.882618i	$-0.655779\pi$
$149$	−11.4764	−0.940182	−0.470091	+	0.882618i	$0.655779\pi$
$150$	0	0
$151$	−15.4597	−1.25809	−0.629047	−	0.777368i	$-0.716555\pi$
$151$	−15.4597	−1.25809	−0.629047	+	0.777368i	$0.716555\pi$
$152$	−11.6757	−0.947021
$153$	2.14965	0.173789
$154$	15.6089	1.25780
$155$	0	0
$156$	2.56797	0.205602
$157$	0.633105	0.0505273	0.0252637	−	0.999681i	$-0.491957\pi$
$157$	0.633105	0.0505273	0.0252637	+	0.999681i	$0.491957\pi$
$158$	−10.7289	−0.853546
$159$	2.37358	0.188237
$160$	0	0
$161$	−7.76979	−0.612345
$162$	−0.814748	−0.0640126
$163$	0.396238	0.0310358	0.0155179	−	0.999880i	$-0.495060\pi$
$163$	0.396238	0.0310358	0.0155179	+	0.999880i	$0.495060\pi$
$164$	−12.5426	−0.979416
$165$	0	0
$166$	−1.85106	−0.143670
$167$	−6.62529	−0.512680	−0.256340	−	0.966587i	$-0.582517\pi$
$167$	−6.62529	−0.512680	−0.256340	+	0.966587i	$0.582517\pi$
$168$	−8.32118	−0.641993
$169$	−9.30644	−0.715880
$170$	0	0
$171$	−4.29544	−0.328481
$172$	9.07397	0.691884
$173$	14.3869	1.09382	0.546909	−	0.837192i	$-0.315804\pi$
$173$	14.3869	1.09382	0.546909	+	0.837192i	$0.315804\pi$
$174$	1.93728	0.146865
$175$	0	0
$176$	2.86472	0.215937
$177$	9.75741	0.733412
$178$	−15.1137	−1.13282
$179$	−0.101643	−0.00759712	−0.00379856	−	0.999993i	$-0.501209\pi$
$179$	−0.101643	−0.00759712	−0.00379856	+	0.999993i	$0.501209\pi$
$180$	0	0
$181$	−19.6352	−1.45947	−0.729736	−	0.683729i	$-0.760357\pi$
$181$	−19.6352	−1.45947	−0.729736	+	0.683729i	$0.760357\pi$
$182$	−4.79355	−0.355321
$183$	−2.13211	−0.157610
$184$	6.89876	0.508584
$185$	0	0
$186$	0.814748	0.0597402
$187$	13.4526	0.983752
$188$	−0.0597813	−0.00436000
$189$	−3.06134	−0.222680
$190$	0	0
$191$	2.11276	0.152874	0.0764368	−	0.997074i	$-0.475646\pi$
$191$	2.11276	0.152874	0.0764368	+	0.997074i	$0.475646\pi$
$192$	3.81755	0.275508
$193$	17.9065	1.28894	0.644468	−	0.764631i	$-0.277079\pi$
$193$	17.9065	1.28894	0.644468	+	0.764631i	$0.277079\pi$
$194$	−2.90715	−0.208721
$195$	0	0
$196$	−3.16915	−0.226368
$197$	−3.23922	−0.230785	−0.115393	−	0.993320i	$-0.536813\pi$
$197$	−3.23922	−0.230785	−0.115393	+	0.993320i	$0.536813\pi$
$198$	−5.09873	−0.362351
$199$	−11.9427	−0.846596	−0.423298	−	0.905990i	$-0.639128\pi$
$199$	−11.9427	−0.846596	−0.423298	+	0.905990i	$0.639128\pi$
$200$	0	0
$201$	9.68208	0.682922
$202$	−9.24614	−0.650556
$203$	7.27915	0.510896
$204$	−2.87233	−0.201103
$205$	0	0
$206$	3.57498	0.249081
$207$	2.53804	0.176406
$208$	−0.879764	−0.0610006
$209$	−26.8811	−1.85940
$210$	0	0
$211$	−1.06298	−0.0731786	−0.0365893	−	0.999330i	$-0.511649\pi$
$211$	−1.06298	−0.0731786	−0.0365893	+	0.999330i	$0.511649\pi$
$212$	−3.17154	−0.217823
$213$	−1.94137	−0.133021
$214$	−0.618613	−0.0422875
$215$	0	0
$216$	2.71815	0.184947
$217$	3.06134	0.207817
$218$	−14.0333	−0.950453
$219$	−3.07607	−0.207862
$220$	0	0
$221$	−4.13133	−0.277903
$222$	2.52468	0.169446
$223$	−23.9213	−1.60189	−0.800943	−	0.598741i	$-0.795668\pi$
$223$	−23.9213	−1.60189	−0.800943	+	0.598741i	$0.795668\pi$
$224$	17.7841	1.18825
$225$	0	0
$226$	−0.562165	−0.0373946
$227$	22.7268	1.50843	0.754215	−	0.656628i	$-0.228018\pi$
$227$	22.7268	1.50843	0.754215	+	0.656628i	$0.228018\pi$
$228$	5.73951	0.380108
$229$	9.93641	0.656616	0.328308	−	0.944571i	$-0.393522\pi$
$229$	9.93641	0.656616	0.328308	+	0.944571i	$0.393522\pi$
$230$	0	0
$231$	−19.1580	−1.26050
$232$	−6.46313	−0.424325
$233$	−3.23150	−0.211703	−0.105851	−	0.994382i	$-0.533757\pi$
$233$	−3.23150	−0.211703	−0.105851	+	0.994382i	$0.533757\pi$
$234$	1.56583	0.102362
$235$	0	0
$236$	−13.0377	−0.848683
$237$	13.1684	0.855378
$238$	5.36169	0.347547
$239$	12.0594	0.780055	0.390028	−	0.920803i	$-0.372465\pi$
$239$	12.0594	0.780055	0.390028	+	0.920803i	$0.372465\pi$
$240$	0	0
$241$	4.37449	0.281786	0.140893	−	0.990025i	$-0.455003\pi$
$241$	4.37449	0.281786	0.140893	+	0.990025i	$0.455003\pi$
$242$	−22.9459	−1.47502
$243$	1.00000	0.0641500
$244$	2.84889	0.182382
$245$	0	0
$246$	−7.64796	−0.487616
$247$	8.25525	0.525269
$248$	−2.71815	−0.172603
$249$	2.27194	0.143978
$250$	0	0
$251$	−14.4610	−0.912771	−0.456385	−	0.889782i	$-0.650856\pi$
$251$	−14.4610	−0.912771	−0.456385	+	0.889782i	$0.650856\pi$
$252$	4.09052	0.257678
$253$	15.8832	0.998566
$254$	−10.5576	−0.662443
$255$	0	0
$256$	−14.5671	−0.910446
$257$	−6.75766	−0.421531	−0.210765	−	0.977537i	$-0.567596\pi$
$257$	−6.75766	−0.421531	−0.210765	+	0.977537i	$0.567596\pi$
$258$	5.53291	0.344464
$259$	9.48626	0.589448
$260$	0	0
$261$	−2.37777	−0.147180
$262$	−8.25122	−0.509762
$263$	21.5901	1.33130	0.665651	−	0.746263i	$-0.268154\pi$
$263$	21.5901	1.33130	0.665651	+	0.746263i	$0.268154\pi$
$264$	17.0103	1.04691
$265$	0	0
$266$	−10.7138	−0.656903
$267$	18.5502	1.13525
$268$	−12.9371	−0.790257
$269$	26.7921	1.63355	0.816773	−	0.576960i	$-0.195761\pi$
$269$	26.7921	1.63355	0.816773	+	0.576960i	$0.195761\pi$
$270$	0	0
$271$	14.5309	0.882691	0.441346	−	0.897337i	$-0.354501\pi$
$271$	14.5309	0.882691	0.441346	+	0.897337i	$0.354501\pi$
$272$	0.984036	0.0596659
$273$	5.88347	0.356084
$274$	12.8807	0.778151
$275$	0	0
$276$	−3.39129	−0.204132
$277$	28.4512	1.70947	0.854733	−	0.519068i	$-0.173721\pi$
$277$	28.4512	1.70947	0.854733	+	0.519068i	$0.173721\pi$
$278$	−18.9409	−1.13600
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	9.95480	0.593854	0.296927	−	0.954900i	$-0.404038\pi$
$281$	9.95480	0.593854	0.296927	+	0.954900i	$0.404038\pi$
$282$	−0.0364520	−0.00217069
$283$	−20.4848	−1.21770	−0.608848	−	0.793287i	$-0.708368\pi$
$283$	−20.4848	−1.21770	−0.608848	+	0.793287i	$0.708368\pi$
$284$	2.59403	0.153928
$285$	0	0
$286$	9.79906	0.579431
$287$	−28.7365	−1.69626
$288$	−5.80926	−0.342314
$289$	−12.3790	−0.728177
$290$	0	0
$291$	3.56816	0.209169
$292$	4.11021	0.240532
$293$	−4.84395	−0.282987	−0.141493	−	0.989939i	$-0.545190\pi$
$293$	−4.84395	−0.282987	−0.141493	+	0.989939i	$0.545190\pi$
$294$	−1.93241	−0.112701
$295$	0	0
$296$	−8.42282	−0.489566
$297$	6.25805	0.363129
$298$	9.35036	0.541652
$299$	−4.87776	−0.282088
$300$	0	0
$301$	20.7894	1.19828
$302$	12.5958	0.724805
$303$	11.3485	0.651953
$304$	−1.96631	−0.112775
$305$	0	0
$306$	−1.75142	−0.100122
$307$	−32.4859	−1.85407	−0.927035	−	0.374975i	$-0.877651\pi$
$307$	−32.4859	−1.85407	−0.927035	+	0.374975i	$0.877651\pi$
$308$	25.5987	1.45862
$309$	−4.38784	−0.249616
$310$	0	0
$311$	2.63183	0.149237	0.0746187	−	0.997212i	$-0.476226\pi$
$311$	2.63183	0.149237	0.0746187	+	0.997212i	$0.476226\pi$
$312$	−5.22391	−0.295746
$313$	−12.1880	−0.688904	−0.344452	−	0.938804i	$-0.611935\pi$
$313$	−12.1880	−0.688904	−0.344452	+	0.938804i	$0.611935\pi$
$314$	−0.515821	−0.0291095
$315$	0	0
$316$	−17.5954	−0.989819
$317$	−29.2725	−1.64411	−0.822054	−	0.569410i	$-0.807172\pi$
$317$	−29.2725	−1.64411	−0.822054	+	0.569410i	$0.807172\pi$
$318$	−1.93387	−0.108446
$319$	−14.8802	−0.833130
$320$	0	0
$321$	0.759270	0.0423783
$322$	6.33042	0.352780
$323$	−9.23369	−0.513776
$324$	−1.33619	−0.0742326
$325$	0	0
$326$	−0.322834	−0.0178801
$327$	17.2241	0.952494
$328$	25.5150	1.40883
$329$	−0.136965	−0.00755113
$330$	0	0
$331$	2.79790	0.153787	0.0768933	−	0.997039i	$-0.475500\pi$
$331$	2.79790	0.153787	0.0768933	+	0.997039i	$0.475500\pi$
$332$	−3.03573	−0.166607
$333$	−3.09873	−0.169809
$334$	5.39794	0.295362
$335$	0	0
$336$	−1.40138	−0.0764514
$337$	32.8136	1.78747	0.893735	−	0.448596i	$-0.148076\pi$
$337$	32.8136	1.78747	0.893735	+	0.448596i	$0.148076\pi$
$338$	7.58240	0.412428
$339$	0.689986	0.0374749
$340$	0	0
$341$	−6.25805	−0.338892
$342$	3.49970	0.189242
$343$	14.1685	0.765028
$344$	−18.4588	−0.995232
$345$	0	0
$346$	−11.7217	−0.630164
$347$	24.1159	1.29461	0.647306	−	0.762231i	$-0.275896\pi$
$347$	24.1159	1.29461	0.647306	+	0.762231i	$0.275896\pi$
$348$	3.17714	0.170312
$349$	3.70288	0.198211	0.0991053	−	0.995077i	$-0.468402\pi$
$349$	3.70288	0.198211	0.0991053	+	0.995077i	$0.468402\pi$
$350$	0	0
$351$	−1.92186	−0.102581
$352$	−36.3547	−1.93771
$353$	−17.4522	−0.928888	−0.464444	−	0.885602i	$-0.653746\pi$
$353$	−17.4522	−0.928888	−0.464444	+	0.885602i	$0.653746\pi$
$354$	−7.94982	−0.422528
$355$	0	0
$356$	−24.7865	−1.31368
$357$	−6.58080	−0.348293
$358$	0.0828130	0.00437680
$359$	−4.87364	−0.257221	−0.128610	−	0.991695i	$-0.541052\pi$
$359$	−4.87364	−0.257221	−0.128610	+	0.991695i	$0.541052\pi$
$360$	0	0
$361$	−0.549172	−0.0289038
$362$	15.9977	0.840821
$363$	28.1632	1.47818
$364$	−7.86142	−0.412050
$365$	0	0
$366$	1.73713	0.0908013
$367$	18.1098	0.945326	0.472663	−	0.881243i	$-0.343293\pi$
$367$	18.1098	0.945326	0.472663	+	0.881243i	$0.343293\pi$
$368$	1.16183	0.0605644
$369$	9.38690	0.488663
$370$	0	0
$371$	−7.26633	−0.377249
$372$	1.33619	0.0692780
$373$	−5.16154	−0.267254	−0.133627	−	0.991032i	$-0.542662\pi$
$373$	−5.16154	−0.267254	−0.133627	+	0.991032i	$0.542662\pi$
$374$	−10.9605	−0.566753
$375$	0	0
$376$	0.121611	0.00627160
$377$	4.56974	0.235354
$378$	2.49422	0.128289
$379$	−9.11288	−0.468097	−0.234049	−	0.972225i	$-0.575197\pi$
$379$	−9.11288	−0.468097	−0.234049	+	0.972225i	$0.575197\pi$
$380$	0	0
$381$	12.9581	0.663865
$382$	−1.72136	−0.0880726
$383$	−5.51110	−0.281604	−0.140802	−	0.990038i	$-0.544968\pi$
$383$	−5.51110	−0.281604	−0.140802	+	0.990038i	$0.544968\pi$
$384$	8.50819	0.434182
$385$	0	0
$386$	−14.5893	−0.742574
$387$	−6.79095	−0.345203
$388$	−4.76772	−0.242044
$389$	−25.8364	−1.30996	−0.654978	−	0.755648i	$-0.727322\pi$
$389$	−25.8364	−1.30996	−0.654978	+	0.755648i	$0.727322\pi$
$390$	0	0
$391$	5.45588	0.275916
$392$	6.44688	0.325617
$393$	10.1273	0.510856
$394$	2.63915	0.132958
$395$	0	0
$396$	−8.36192	−0.420202
$397$	26.4578	1.32788	0.663941	−	0.747785i	$-0.268883\pi$
$397$	26.4578	1.32788	0.663941	+	0.747785i	$0.268883\pi$
$398$	9.73029	0.487736
$399$	13.1498	0.658314
$400$	0	0
$401$	32.4069	1.61832	0.809162	−	0.587586i	$-0.199922\pi$
$401$	32.4069	1.61832	0.809162	+	0.587586i	$0.199922\pi$
$402$	−7.88846	−0.393440
$403$	1.92186	0.0957348
$404$	−15.1637	−0.754421
$405$	0	0
$406$	−5.93067	−0.294334
$407$	−19.3920	−0.961226
$408$	5.84306	0.289275
$409$	−31.1727	−1.54139	−0.770696	−	0.637203i	$-0.780091\pi$
$409$	−31.1727	−1.54139	−0.770696	+	0.637203i	$0.780091\pi$
$410$	0	0
$411$	−15.8094	−0.779821
$412$	5.86298	0.288848
$413$	−29.8707	−1.46984
$414$	−2.06786	−0.101630
$415$	0	0
$416$	11.1646	0.547390
$417$	23.2476	1.13844
$418$	21.9013	1.07123
$419$	33.2948	1.62656	0.813280	−	0.581873i	$-0.197680\pi$
$419$	33.2948	1.62656	0.813280	+	0.581873i	$0.197680\pi$
$420$	0	0
$421$	−30.5713	−1.48996	−0.744978	−	0.667089i	$-0.767540\pi$
$421$	−30.5713	−1.48996	−0.744978	+	0.667089i	$0.767540\pi$
$422$	0.866061	0.0421592
$423$	0.0447403	0.00217535
$424$	6.45174	0.313324
$425$	0	0
$426$	1.58173	0.0766350
$427$	6.52711	0.315869
$428$	−1.01453	−0.0490389
$429$	−12.0271	−0.580675
$430$	0	0
$431$	17.6780	0.851520	0.425760	−	0.904836i	$-0.360007\pi$
$431$	17.6780	0.851520	0.425760	+	0.904836i	$0.360007\pi$
$432$	0.457766	0.0220243
$433$	−6.19475	−0.297701	−0.148850	−	0.988860i	$-0.547557\pi$
$433$	−6.19475	−0.297701	−0.148850	+	0.988860i	$0.547557\pi$
$434$	−2.49422	−0.119726
$435$	0	0
$436$	−23.0146	−1.10220
$437$	−10.9020	−0.521513
$438$	2.50622	0.119752
$439$	−29.0218	−1.38513	−0.692567	−	0.721354i	$-0.743520\pi$
$439$	−29.0218	−1.38513	−0.692567	+	0.721354i	$0.743520\pi$
$440$	0	0
$441$	2.37179	0.112942
$442$	3.36599	0.160104
$443$	−9.05107	−0.430029	−0.215015	−	0.976611i	$-0.568980\pi$
$443$	−9.05107	−0.430029	−0.215015	+	0.976611i	$0.568980\pi$
$444$	4.14048	0.196499
$445$	0	0
$446$	19.4898	0.922868
$447$	−11.4764	−0.542814
$448$	−11.6868	−0.552150
$449$	−3.37540	−0.159295	−0.0796475	−	0.996823i	$-0.525379\pi$
$449$	−3.37540	−0.159295	−0.0796475	+	0.996823i	$0.525379\pi$
$450$	0	0
$451$	58.7437	2.76613
$452$	−0.921950	−0.0433649
$453$	−15.4597	−0.726361
$454$	−18.5166	−0.869027
$455$	0	0
$456$	−11.6757	−0.546763
$457$	31.4976	1.47340	0.736698	−	0.676221i	$-0.236384\pi$
$457$	31.4976	1.47340	0.736698	+	0.676221i	$0.236384\pi$
$458$	−8.09567	−0.378286
$459$	2.14965	0.100337
$460$	0	0
$461$	4.80633	0.223853	0.111926	−	0.993716i	$-0.464298\pi$
$461$	4.80633	0.223853	0.111926	+	0.993716i	$0.464298\pi$
$462$	15.6089	0.726194
$463$	39.6402	1.84224	0.921119	−	0.389282i	$-0.127277\pi$
$463$	39.6402	1.84224	0.921119	+	0.389282i	$0.127277\pi$
$464$	−1.08846	−0.0505305
$465$	0	0
$466$	2.63286	0.121965
$467$	38.2581	1.77037	0.885186	−	0.465237i	$-0.154031\pi$
$467$	38.2581	1.77037	0.885186	+	0.465237i	$0.154031\pi$
$468$	2.56797	0.118704
$469$	−29.6401	−1.36865
$470$	0	0
$471$	0.633105	0.0291720
$472$	26.5221	1.22078
$473$	−42.4981	−1.95406
$474$	−10.7289	−0.492795
$475$	0	0
$476$	8.79317	0.403034
$477$	2.37358	0.108679
$478$	−9.82534	−0.449400
$479$	10.4061	0.475468	0.237734	−	0.971330i	$-0.423595\pi$
$479$	10.4061	0.475468	0.237734	+	0.971330i	$0.423595\pi$
$480$	0	0
$481$	5.95534	0.271540
$482$	−3.56411	−0.162341
$483$	−7.76979	−0.353538
$484$	−37.6313	−1.71051
$485$	0	0
$486$	−0.814748	−0.0369577
$487$	−31.2648	−1.41674	−0.708372	−	0.705840i	$-0.750570\pi$
$487$	−31.2648	−1.41674	−0.708372	+	0.705840i	$0.750570\pi$
$488$	−5.79539	−0.262345
$489$	0.396238	0.0179185
$490$	0	0
$491$	37.1097	1.67474	0.837369	−	0.546637i	$-0.184092\pi$
$491$	37.1097	1.67474	0.837369	+	0.546637i	$0.184092\pi$
$492$	−12.5426	−0.565466
$493$	−5.11136	−0.230204
$494$	−6.72595	−0.302615
$495$	0	0
$496$	−0.457766	−0.0205543
$497$	5.94320	0.266589
$498$	−1.85106	−0.0829478
$499$	−15.4963	−0.693708	−0.346854	−	0.937919i	$-0.612750\pi$
$499$	−15.4963	−0.693708	−0.346854	+	0.937919i	$0.612750\pi$
$500$	0	0
$501$	−6.62529	−0.295996
$502$	11.7821	0.525860
$503$	−15.0939	−0.673003	−0.336501	−	0.941683i	$-0.609244\pi$
$503$	−15.0939	−0.673003	−0.336501	+	0.941683i	$0.609244\pi$
$504$	−8.32118	−0.370655
$505$	0	0
$506$	−12.9408	−0.575287
$507$	−9.30644	−0.413314
$508$	−17.3145	−0.768206
$509$	−34.0674	−1.51001	−0.755005	−	0.655719i	$-0.772366\pi$
$509$	−34.0674	−1.51001	−0.755005	+	0.655719i	$0.772366\pi$
$510$	0	0
$511$	9.41690	0.416579
$512$	−5.14784	−0.227504
$513$	−4.29544	−0.189648
$514$	5.50578	0.242850
$515$	0	0
$516$	9.07397	0.399459
$517$	0.279987	0.0123138
$518$	−7.72891	−0.339589
$519$	14.3869	0.631517
$520$	0	0
$521$	14.8298	0.649703	0.324852	−	0.945765i	$-0.394686\pi$
$521$	14.8298	0.649703	0.324852	+	0.945765i	$0.394686\pi$
$522$	1.93728	0.0847924
$523$	−1.32531	−0.0579519	−0.0289760	−	0.999580i	$-0.509225\pi$
$523$	−1.32531	−0.0579519	−0.0289760	+	0.999580i	$0.509225\pi$
$524$	−13.5320	−0.591148
$525$	0	0
$526$	−17.5905	−0.766981
$527$	−2.14965	−0.0936401
$528$	2.86472	0.124671
$529$	−16.5584	−0.719929
$530$	0	0
$531$	9.75741	0.423435
$532$	−17.5706	−0.761781
$533$	−18.0403	−0.781414
$534$	−15.1137	−0.654036
$535$	0	0
$536$	26.3174	1.13674
$537$	−0.101643	−0.00438620
$538$	−21.8288	−0.941108
$539$	14.8428	0.639324
$540$	0	0
$541$	−22.7998	−0.980238	−0.490119	−	0.871656i	$-0.663047\pi$
$541$	−22.7998	−0.980238	−0.490119	+	0.871656i	$0.663047\pi$
$542$	−11.8390	−0.508530
$543$	−19.6352	−0.842626
$544$	−12.4879	−0.535413
$545$	0	0
$546$	−4.79355	−0.205145
$547$	−44.3323	−1.89551	−0.947756	−	0.318995i	$-0.896655\pi$
$547$	−44.3323	−1.89551	−0.947756	+	0.318995i	$0.896655\pi$
$548$	21.1243	0.902387
$549$	−2.13211	−0.0909962
$550$	0	0
$551$	10.2136	0.435112
$552$	6.89876	0.293631
$553$	−40.3129	−1.71428
$554$	−23.1805	−0.984847
$555$	0	0
$556$	−31.0631	−1.31737
$557$	−42.1780	−1.78714	−0.893570	−	0.448924i	$-0.851807\pi$
$557$	−42.1780	−1.78714	−0.893570	+	0.448924i	$0.851807\pi$
$558$	0.814748	0.0344910
$559$	13.0513	0.552010
$560$	0	0
$561$	13.4526	0.567969
$562$	−8.11065	−0.342127
$563$	−40.8544	−1.72181	−0.860903	−	0.508768i	$-0.830101\pi$
$563$	−40.8544	−1.72181	−0.860903	+	0.508768i	$0.830101\pi$
$564$	−0.0597813	−0.00251725
$565$	0	0
$566$	16.6900	0.701531
$567$	−3.06134	−0.128564
$568$	−5.27694	−0.221416
$569$	−43.6669	−1.83061	−0.915306	−	0.402759i	$-0.868051\pi$
$569$	−43.6669	−1.83061	−0.915306	+	0.402759i	$0.868051\pi$
$570$	0	0
$571$	−25.7023	−1.07561	−0.537804	−	0.843070i	$-0.680746\pi$
$571$	−25.7023	−1.07561	−0.537804	+	0.843070i	$0.680746\pi$
$572$	16.0705	0.671940
$573$	2.11276	0.0882617
$574$	23.4130	0.977239
$575$	0	0
$576$	3.81755	0.159065
$577$	24.7113	1.02874	0.514372	−	0.857567i	$-0.328025\pi$
$577$	24.7113	1.02874	0.514372	+	0.857567i	$0.328025\pi$
$578$	10.0858	0.419513
$579$	17.9065	0.744168
$580$	0	0
$581$	−6.95517	−0.288549
$582$	−2.90715	−0.120505
$583$	14.8540	0.615189
$584$	−8.36123	−0.345990
$585$	0	0
$586$	3.94660	0.163032
$587$	17.8011	0.734729	0.367364	−	0.930077i	$-0.380260\pi$
$587$	17.8011	0.734729	0.367364	+	0.930077i	$0.380260\pi$
$588$	−3.16915	−0.130694
$589$	4.29544	0.176991
$590$	0	0
$591$	−3.23922	−0.133244
$592$	−1.41849	−0.0582997
$593$	28.1408	1.15560	0.577802	−	0.816177i	$-0.303910\pi$
$593$	28.1408	1.15560	0.577802	+	0.816177i	$0.303910\pi$
$594$	−5.09873	−0.209203
$595$	0	0
$596$	15.3346	0.628129
$597$	−11.9427	−0.488782
$598$	3.97414	0.162515
$599$	−4.56567	−0.186548	−0.0932742	−	0.995640i	$-0.529733\pi$
$599$	−4.56567	−0.186548	−0.0932742	+	0.995640i	$0.529733\pi$
$600$	0	0
$601$	−36.3342	−1.48210	−0.741051	−	0.671448i	$-0.765672\pi$
$601$	−36.3342	−1.48210	−0.741051	+	0.671448i	$0.765672\pi$
$602$	−16.9381	−0.690346
$603$	9.68208	0.394285
$604$	20.6571	0.840523
$605$	0	0
$606$	−9.24614	−0.375599
$607$	35.3618	1.43529	0.717646	−	0.696408i	$-0.245220\pi$
$607$	35.3618	1.43529	0.717646	+	0.696408i	$0.245220\pi$
$608$	24.9534	1.01199
$609$	7.27915	0.294966
$610$	0	0
$611$	−0.0859847	−0.00347857
$612$	−2.87233	−0.116107
$613$	17.7586	0.717264	0.358632	−	0.933479i	$-0.383243\pi$
$613$	17.7586	0.717264	0.358632	+	0.933479i	$0.383243\pi$
$614$	26.4678	1.06815
$615$	0	0
$616$	−52.0743	−2.09813
$617$	−25.6310	−1.03186	−0.515932	−	0.856629i	$-0.672554\pi$
$617$	−25.6310	−1.03186	−0.515932	+	0.856629i	$0.672554\pi$
$618$	3.57498	0.143807
$619$	10.8503	0.436110	0.218055	−	0.975936i	$-0.430029\pi$
$619$	10.8503	0.436110	0.218055	+	0.975936i	$0.430029\pi$
$620$	0	0
$621$	2.53804	0.101848
$622$	−2.14428	−0.0859777
$623$	−56.7885	−2.27518
$624$	−0.879764	−0.0352187
$625$	0	0
$626$	9.93011	0.396887
$627$	−26.8811	−1.07353
$628$	−0.845947	−0.0337569
$629$	−6.66118	−0.265599
$630$	0	0
$631$	−11.6128	−0.462299	−0.231150	−	0.972918i	$-0.574249\pi$
$631$	−11.6128	−0.462299	−0.231150	+	0.972918i	$0.574249\pi$
$632$	35.7936	1.42379
$633$	−1.06298	−0.0422497
$634$	23.8497	0.947193
$635$	0	0
$636$	−3.17154	−0.125760
$637$	−4.55826	−0.180605
$638$	12.1236	0.479978
$639$	−1.94137	−0.0767995
$640$	0	0
$641$	−27.2175	−1.07503	−0.537514	−	0.843255i	$-0.680636\pi$
$641$	−27.2175	−1.07503	−0.537514	+	0.843255i	$0.680636\pi$
$642$	−0.618613	−0.0244147
$643$	39.3223	1.55072	0.775360	−	0.631520i	$-0.217569\pi$
$643$	39.3223	1.55072	0.775360	+	0.631520i	$0.217569\pi$
$644$	10.3819	0.409104
$645$	0	0
$646$	7.52313	0.295993
$647$	−10.0918	−0.396748	−0.198374	−	0.980126i	$-0.563566\pi$
$647$	−10.0918	−0.396748	−0.198374	+	0.980126i	$0.563566\pi$
$648$	2.71815	0.106779
$649$	61.0623	2.39691
$650$	0	0
$651$	3.06134	0.119983
$652$	−0.529448	−0.0207348
$653$	−49.1010	−1.92147	−0.960736	−	0.277465i	$-0.910506\pi$
$653$	−49.1010	−1.92147	−0.960736	+	0.277465i	$0.910506\pi$
$654$	−14.0333	−0.548745
$655$	0	0
$656$	4.29701	0.167770
$657$	−3.07607	−0.120009
$658$	0.111592	0.00435031
$659$	20.5311	0.799780	0.399890	−	0.916563i	$-0.369048\pi$
$659$	20.5311	0.799780	0.399890	+	0.916563i	$0.369048\pi$
$660$	0	0
$661$	35.4798	1.38001	0.690003	−	0.723807i	$-0.257609\pi$
$661$	35.4798	1.38001	0.690003	+	0.723807i	$0.257609\pi$
$662$	−2.27958	−0.0885985
$663$	−4.13133	−0.160448
$664$	6.17547	0.239655
$665$	0	0
$666$	2.52468	0.0978295
$667$	−6.03486	−0.233671
$668$	8.85262	0.342518
$669$	−23.9213	−0.924849
$670$	0	0
$671$	−13.3428	−0.515095
$672$	17.7841	0.686037
$673$	−40.8488	−1.57461	−0.787304	−	0.616565i	$-0.788524\pi$
$673$	−40.8488	−1.57461	−0.787304	+	0.616565i	$0.788524\pi$
$674$	−26.7348	−1.02979
$675$	0	0
$676$	12.4351	0.478275
$677$	20.7335	0.796855	0.398427	−	0.917200i	$-0.369556\pi$
$677$	20.7335	0.796855	0.398427	+	0.917200i	$0.369556\pi$
$678$	−0.562165	−0.0215898
$679$	−10.9233	−0.419199
$680$	0	0
$681$	22.7268	0.870892
$682$	5.09873	0.195241
$683$	25.3363	0.969468	0.484734	−	0.874662i	$-0.338916\pi$
$683$	25.3363	0.969468	0.484734	+	0.874662i	$0.338916\pi$
$684$	5.73951	0.219456
$685$	0	0
$686$	−11.5438	−0.440743
$687$	9.93641	0.379098
$688$	−3.10867	−0.118517
$689$	−4.56170	−0.173787
$690$	0	0
$691$	−4.41093	−0.167800	−0.0838999	−	0.996474i	$-0.526738\pi$
$691$	−4.41093	−0.167800	−0.0838999	+	0.996474i	$0.526738\pi$
$692$	−19.2236	−0.730773
$693$	−19.1580	−0.727752
$694$	−19.6484	−0.745843
$695$	0	0
$696$	−6.46313	−0.244984
$697$	20.1785	0.764316
$698$	−3.01691	−0.114192
$699$	−3.23150	−0.122227
$700$	0	0
$701$	−19.8054	−0.748041	−0.374020	−	0.927420i	$-0.622021\pi$
$701$	−19.8054	−0.748041	−0.374020	+	0.927420i	$0.622021\pi$
$702$	1.56583	0.0590986
$703$	13.3104	0.502012
$704$	23.8904	0.900404
$705$	0	0
$706$	14.2192	0.535145
$707$	−34.7415	−1.30659
$708$	−13.0377	−0.489987
$709$	−20.0167	−0.751742	−0.375871	−	0.926672i	$-0.622657\pi$
$709$	−20.0167	−0.751742	−0.375871	+	0.926672i	$0.622657\pi$
$710$	0	0
$711$	13.1684	0.493853
$712$	50.4223	1.88965
$713$	−2.53804	−0.0950502
$714$	5.36169	0.200656
$715$	0	0
$716$	0.135813	0.00507558
$717$	12.0594	0.450365
$718$	3.97078	0.148188
$719$	30.3316	1.13118	0.565589	−	0.824687i	$-0.308649\pi$
$719$	30.3316	1.13118	0.565589	+	0.824687i	$0.308649\pi$
$720$	0	0
$721$	13.4327	0.500259
$722$	0.447436	0.0166519
$723$	4.37449	0.162689
$724$	26.2363	0.975063
$725$	0	0
$726$	−22.9459	−0.851602
$727$	−44.9071	−1.66551	−0.832756	−	0.553640i	$-0.813239\pi$
$727$	−44.9071	−1.66551	−0.832756	+	0.553640i	$0.813239\pi$
$728$	15.9922	0.592709
$729$	1.00000	0.0370370
$730$	0	0
$731$	−14.5981	−0.539932
$732$	2.84889	0.105298
$733$	43.7146	1.61463	0.807317	−	0.590117i	$-0.200919\pi$
$733$	43.7146	1.61463	0.807317	+	0.590117i	$0.200919\pi$
$734$	−14.7549	−0.544615
$735$	0	0
$736$	−14.7441	−0.543476
$737$	60.5910	2.23190
$738$	−7.64796	−0.281525
$739$	−3.82705	−0.140780	−0.0703902	−	0.997520i	$-0.522424\pi$
$739$	−3.82705	−0.140780	−0.0703902	+	0.997520i	$0.522424\pi$
$740$	0	0
$741$	8.25525	0.303264
$742$	5.92022	0.217338
$743$	0.238858	0.00876284	0.00438142	−	0.999990i	$-0.498605\pi$
$743$	0.238858	0.00876284	0.00438142	+	0.999990i	$0.498605\pi$
$744$	−2.71815	−0.0996522
$745$	0	0
$746$	4.20535	0.153969
$747$	2.27194	0.0831258
$748$	−17.9752	−0.657238
$749$	−2.32438	−0.0849310
$750$	0	0
$751$	0.229588	0.00837780	0.00418890	−	0.999991i	$-0.498667\pi$
$751$	0.229588	0.00837780	0.00418890	+	0.999991i	$0.498667\pi$
$752$	0.0204806	0.000746850 0
$753$	−14.4610	−0.526988
$754$	−3.72319	−0.135590
$755$	0	0
$756$	4.09052	0.148771
$757$	−14.5670	−0.529448	−0.264724	−	0.964324i	$-0.585281\pi$
$757$	−14.5670	−0.529448	−0.264724	+	0.964324i	$0.585281\pi$
$758$	7.42470	0.269677
$759$	15.8832	0.576522
$760$	0	0
$761$	−17.4223	−0.631558	−0.315779	−	0.948833i	$-0.602266\pi$
$761$	−17.4223	−0.631558	−0.315779	+	0.948833i	$0.602266\pi$
$762$	−10.5576	−0.382462
$763$	−52.7287	−1.90891
$764$	−2.82304	−0.102134
$765$	0	0
$766$	4.49016	0.162236
$767$	−18.7524	−0.677110
$768$	−14.5671	−0.525646
$769$	−5.67168	−0.204526	−0.102263	−	0.994757i	$-0.532608\pi$
$769$	−5.67168	−0.204526	−0.102263	+	0.994757i	$0.532608\pi$
$770$	0	0
$771$	−6.75766	−0.243371
$772$	−23.9264	−0.861130
$773$	−1.33846	−0.0481410	−0.0240705	−	0.999710i	$-0.507663\pi$
$773$	−1.33846	−0.0481410	−0.0240705	+	0.999710i	$0.507663\pi$
$774$	5.53291	0.198876
$775$	0	0
$776$	9.69879	0.348166
$777$	9.48626	0.340318
$778$	21.0501	0.754684
$779$	−40.3209	−1.44465
$780$	0	0
$781$	−12.1492	−0.434733
$782$	−4.44517	−0.158959
$783$	−2.37777	−0.0849744
$784$	1.08573	0.0387759
$785$	0	0
$786$	−8.25122	−0.294311
$787$	−9.06643	−0.323183	−0.161592	−	0.986858i	$-0.551663\pi$
$787$	−9.06643	−0.323183	−0.161592	+	0.986858i	$0.551663\pi$
$788$	4.32820	0.154186
$789$	21.5901	0.768628
$790$	0	0
$791$	−2.11228	−0.0751041
$792$	17.0103	0.604435
$793$	4.09762	0.145511
$794$	−21.5565	−0.765010
$795$	0	0
$796$	15.9577	0.565605
$797$	42.0326	1.48887	0.744435	−	0.667695i	$-0.232719\pi$
$797$	42.0326	1.48887	0.744435	+	0.667695i	$0.232719\pi$
$798$	−10.7138	−0.379263
$799$	0.0961758	0.00340246
$800$	0	0
$801$	18.5502	0.655440
$802$	−26.4035	−0.932338
$803$	−19.2502	−0.679325
$804$	−12.9371	−0.456255
$805$	0	0
$806$	−1.56583	−0.0551541
$807$	26.7921	0.943128
$808$	30.8468	1.08519
$809$	8.36057	0.293942	0.146971	−	0.989141i	$-0.453048\pi$
$809$	8.36057	0.293942	0.146971	+	0.989141i	$0.453048\pi$
$810$	0	0
$811$	−8.34955	−0.293192	−0.146596	−	0.989196i	$-0.546832\pi$
$811$	−8.34955	−0.293192	−0.146596	+	0.989196i	$0.546832\pi$
$812$	−9.72629	−0.341326
$813$	14.5309	0.509622
$814$	15.7996	0.553776
$815$	0	0
$816$	0.984036	0.0344481
$817$	29.1701	1.02053
$818$	25.3979	0.888017
$819$	5.88347	0.205585
$820$	0	0
$821$	−38.5533	−1.34552	−0.672760	−	0.739861i	$-0.734891\pi$
$821$	−38.5533	−1.34552	−0.672760	+	0.739861i	$0.734891\pi$
$822$	12.8807	0.449266
$823$	−13.0332	−0.454307	−0.227154	−	0.973859i	$-0.572942\pi$
$823$	−13.0332	−0.454307	−0.227154	+	0.973859i	$0.572942\pi$
$824$	−11.9268	−0.415490
$825$	0	0
$826$	24.3371	0.846796
$827$	−21.5679	−0.749988	−0.374994	−	0.927027i	$-0.622355\pi$
$827$	−21.5679	−0.749988	−0.374994	+	0.927027i	$0.622355\pi$
$828$	−3.39129	−0.117855
$829$	−36.3699	−1.26318	−0.631589	−	0.775304i	$-0.717597\pi$
$829$	−36.3699	−1.26318	−0.631589	+	0.775304i	$0.717597\pi$
$830$	0	0
$831$	28.4512	0.986961
$832$	−7.33681	−0.254358
$833$	5.09851	0.176653
$834$	−18.9409	−0.655870
$835$	0	0
$836$	35.9181	1.24226
$837$	−1.00000	−0.0345651
$838$	−27.1269	−0.937083
$839$	22.5371	0.778066	0.389033	−	0.921224i	$-0.372809\pi$
$839$	22.5371	0.778066	0.389033	+	0.921224i	$0.372809\pi$
$840$	0	0
$841$	−23.3462	−0.805042
$842$	24.9079	0.858384
$843$	9.95480	0.342862
$844$	1.42034	0.0488901
$845$	0	0
$846$	−0.0364520	−0.00125325
$847$	−86.2170	−2.96245
$848$	1.08654	0.0373121
$849$	−20.4848	−0.703037
$850$	0	0
$851$	−7.86469	−0.269598
$852$	2.59403	0.0888702
$853$	−13.9666	−0.478206	−0.239103	−	0.970994i	$-0.576853\pi$
$853$	−13.9666	−0.478206	−0.239103	+	0.970994i	$0.576853\pi$
$854$	−5.31794	−0.181976
$855$	0	0
$856$	2.06381	0.0705395
$857$	21.2802	0.726917	0.363459	−	0.931610i	$-0.381596\pi$
$857$	21.2802	0.726917	0.363459	+	0.931610i	$0.381596\pi$
$858$	9.79906	0.334535
$859$	−25.1971	−0.859715	−0.429857	−	0.902897i	$-0.641436\pi$
$859$	−25.1971	−0.859715	−0.429857	+	0.902897i	$0.641436\pi$
$860$	0	0
$861$	−28.7365	−0.979337
$862$	−14.4031	−0.490572
$863$	11.5544	0.393315	0.196658	−	0.980472i	$-0.436991\pi$
$863$	11.5544	0.393315	0.196658	+	0.980472i	$0.436991\pi$
$864$	−5.80926	−0.197635
$865$	0	0
$866$	5.04716	0.171509
$867$	−12.3790	−0.420413
$868$	−4.09052	−0.138841
$869$	82.4084	2.79551
$870$	0	0
$871$	−18.6076	−0.630496
$872$	46.8176	1.58544
$873$	3.56816	0.120764
$874$	8.88237	0.300451
$875$	0	0
$876$	4.11021	0.138871
$877$	−8.31898	−0.280912	−0.140456	−	0.990087i	$-0.544857\pi$
$877$	−8.31898	−0.280912	−0.140456	+	0.990087i	$0.544857\pi$
$878$	23.6454	0.797994
$879$	−4.84395	−0.163382
$880$	0	0
$881$	33.9387	1.14342	0.571712	−	0.820454i	$-0.306279\pi$
$881$	33.9387	1.14342	0.571712	+	0.820454i	$0.306279\pi$
$882$	−1.93241	−0.0650677
$883$	8.11857	0.273212	0.136606	−	0.990625i	$-0.456381\pi$
$883$	8.11857	0.273212	0.136606	+	0.990625i	$0.456381\pi$
$884$	5.52022	0.185665
$885$	0	0
$886$	7.37434	0.247746
$887$	24.3414	0.817306	0.408653	−	0.912690i	$-0.365999\pi$
$887$	24.3414	0.817306	0.408653	+	0.912690i	$0.365999\pi$
$888$	−8.42282	−0.282651
$889$	−39.6692	−1.33046
$890$	0	0
$891$	6.25805	0.209653
$892$	31.9632	1.07021
$893$	−0.192179	−0.00643103
$894$	9.35036	0.312723
$895$	0	0
$896$	−26.0464	−0.870150
$897$	−4.87776	−0.162864
$898$	2.75010	0.0917721
$899$	2.37777	0.0793029
$900$	0	0
$901$	5.10236	0.169984
$902$	−47.8613	−1.59361
$903$	20.7894	0.691827
$904$	1.87549	0.0623777
$905$	0	0
$906$	12.5958	0.418466
$907$	−41.2773	−1.37059	−0.685296	−	0.728265i	$-0.740327\pi$
$907$	−41.2773	−1.37059	−0.685296	+	0.728265i	$0.740327\pi$
$908$	−30.3672	−1.00777
$909$	11.3485	0.376405
$910$	0	0
$911$	−22.2389	−0.736808	−0.368404	−	0.929666i	$-0.620096\pi$
$911$	−22.2389	−0.736808	−0.368404	+	0.929666i	$0.620096\pi$
$912$	−1.96631	−0.0651110
$913$	14.2179	0.470544
$914$	−25.6626	−0.848844
$915$	0	0
$916$	−13.2769	−0.438681
$917$	−31.0032	−1.02382
$918$	−1.75142	−0.0578055
$919$	3.34964	0.110494	0.0552472	−	0.998473i	$-0.482405\pi$
$919$	3.34964	0.110494	0.0552472	+	0.998473i	$0.482405\pi$
$920$	0	0
$921$	−32.4859	−1.07045
$922$	−3.91594	−0.128965
$923$	3.73105	0.122809
$924$	25.5987	0.842134
$925$	0	0
$926$	−32.2968	−1.06134
$927$	−4.38784	−0.144116
$928$	13.8131	0.453436
$929$	−39.8614	−1.30781	−0.653905	−	0.756576i	$-0.726871\pi$
$929$	−39.8614	−1.30781	−0.653905	+	0.756576i	$0.726871\pi$
$930$	0	0
$931$	−10.1879	−0.333895
$932$	4.31789	0.141437
$933$	2.63183	0.0861622
$934$	−31.1707	−1.01994
$935$	0	0
$936$	−5.22391	−0.170749
$937$	26.1149	0.853135	0.426568	−	0.904456i	$-0.359723\pi$
$937$	26.1149	0.853135	0.426568	+	0.904456i	$0.359723\pi$
$938$	24.1492	0.788500
$939$	−12.1880	−0.397739
$940$	0	0
$941$	−13.4578	−0.438711	−0.219355	−	0.975645i	$-0.570395\pi$
$941$	−13.4578	−0.438711	−0.219355	+	0.975645i	$0.570395\pi$
$942$	−0.515821	−0.0168064
$943$	23.8243	0.775826
$944$	4.46661	0.145376
$945$	0	0
$946$	34.6252	1.12576
$947$	32.6693	1.06161	0.530804	−	0.847494i	$-0.321890\pi$
$947$	32.6693	1.06161	0.530804	+	0.847494i	$0.321890\pi$
$948$	−17.5954	−0.571472
$949$	5.91179	0.191905
$950$	0	0
$951$	−29.2725	−0.949226
$952$	−17.8876	−0.579740
$953$	46.0674	1.49227	0.746135	−	0.665794i	$-0.231907\pi$
$953$	46.0674	1.49227	0.746135	+	0.665794i	$0.231907\pi$
$954$	−1.93387	−0.0626113
$955$	0	0
$956$	−16.1136	−0.521149
$957$	−14.8802	−0.481008
$958$	−8.47836	−0.273923
$959$	48.3980	1.56285
$960$	0	0
$961$	1.00000	0.0322581
$962$	−4.85210	−0.156438
$963$	0.759270	0.0244671
$964$	−5.84513	−0.188259
$965$	0	0
$966$	6.33042	0.203678
$967$	21.1842	0.681239	0.340620	−	0.940201i	$-0.389363\pi$
$967$	21.1842	0.681239	0.340620	+	0.940201i	$0.389363\pi$
$968$	76.5518	2.46047
$969$	−9.23369	−0.296629
$970$	0	0
$971$	10.2935	0.330335	0.165167	−	0.986266i	$-0.447184\pi$
$971$	10.2935	0.330335	0.165167	+	0.986266i	$0.447184\pi$
$972$	−1.33619	−0.0428582
$973$	−71.1687	−2.28156
$974$	25.4729	0.816205
$975$	0	0
$976$	−0.976007	−0.0312412
$977$	−38.6604	−1.23686	−0.618428	−	0.785841i	$-0.712230\pi$
$977$	−38.6604	−1.23686	−0.618428	+	0.785841i	$0.712230\pi$
$978$	−0.322834	−0.0103231
$979$	116.088	3.71019
$980$	0	0
$981$	17.2241	0.549922
$982$	−30.2351	−0.964840
$983$	−29.1883	−0.930961	−0.465480	−	0.885058i	$-0.654118\pi$
$983$	−29.1883	−0.930961	−0.465480	+	0.885058i	$0.654118\pi$
$984$	25.5150	0.813389
$985$	0	0
$986$	4.16447	0.132624
$987$	−0.136965	−0.00435965
$988$	−11.0306	−0.350929
$989$	−17.2357	−0.548062
$990$	0	0
$991$	1.83184	0.0581903	0.0290951	−	0.999577i	$-0.490737\pi$
$991$	1.83184	0.0581903	0.0290951	+	0.999577i	$0.490737\pi$
$992$	5.80926	0.184444
$993$	2.79790	0.0887887
$994$	−4.84221	−0.153585
$995$	0	0
$996$	−3.03573	−0.0961908
$997$	−48.0354	−1.52130	−0.760648	−	0.649165i	$-0.775118\pi$
$997$	−48.0354	−1.52130	−0.760648	+	0.649165i	$0.775118\pi$
$998$	12.6255	0.399654
$999$	−3.09873	−0.0980395

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2325.2.a.bb.1.2	yes	6
3.2	odd	2		6975.2.a.ca.1.5		6
5.2	odd	4		2325.2.c.r.1024.5		12
5.3	odd	4		2325.2.c.r.1024.8		12
5.4	even	2		2325.2.a.y.1.5	✓	6
15.14	odd	2		6975.2.a.cc.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2325.2.a.y.1.5	✓	6	5.4	even	2
2325.2.a.bb.1.2	yes	6	1.1	even	1	trivial
2325.2.c.r.1024.5		12	5.2	odd	4
2325.2.c.r.1024.8		12	5.3	odd	4
6975.2.a.ca.1.5		6	3.2	odd	2
6975.2.a.cc.1.2		6	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2325 = 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2325.a (trivial)

\(f(q)\)	\(=\)	\(q-0.814748 q^{2} +1.00000 q^{3} -1.33619 q^{4} -0.814748 q^{6} -3.06134 q^{7} +2.71815 q^{8} +1.00000 q^{9} +6.25805 q^{11} -1.33619 q^{12} -1.92186 q^{13} +2.49422 q^{14} +0.457766 q^{16} +2.14965 q^{17} -0.814748 q^{18} -4.29544 q^{19} -3.06134 q^{21} -5.09873 q^{22} +2.53804 q^{23} +2.71815 q^{24} +1.56583 q^{26} +1.00000 q^{27} +4.09052 q^{28} -2.37777 q^{29} -1.00000 q^{31} -5.80926 q^{32} +6.25805 q^{33} -1.75142 q^{34} -1.33619 q^{36} -3.09873 q^{37} +3.49970 q^{38} -1.92186 q^{39} +9.38690 q^{41} +2.49422 q^{42} -6.79095 q^{43} -8.36192 q^{44} -2.06786 q^{46} +0.0447403 q^{47} +0.457766 q^{48} +2.37179 q^{49} +2.14965 q^{51} +2.56797 q^{52} +2.37358 q^{53} -0.814748 q^{54} -8.32118 q^{56} -4.29544 q^{57} +1.93728 q^{58} +9.75741 q^{59} -2.13211 q^{61} +0.814748 q^{62} -3.06134 q^{63} +3.81755 q^{64} -5.09873 q^{66} +9.68208 q^{67} -2.87233 q^{68} +2.53804 q^{69} -1.94137 q^{71} +2.71815 q^{72} -3.07607 q^{73} +2.52468 q^{74} +5.73951 q^{76} -19.1580 q^{77} +1.56583 q^{78} +13.1684 q^{79} +1.00000 q^{81} -7.64796 q^{82} +2.27194 q^{83} +4.09052 q^{84} +5.53291 q^{86} -2.37777 q^{87} +17.0103 q^{88} +18.5502 q^{89} +5.88347 q^{91} -3.39129 q^{92} -1.00000 q^{93} -0.0364520 q^{94} -5.80926 q^{96} +3.56816 q^{97} -1.93241 q^{98} +6.25805 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{2} + 6 q^{3} + 7 q^{4} + q^{6} + 2 q^{7} - 3 q^{8} + 6 q^{9} + 7 q^{11} + 7 q^{12} + 4 q^{13} + 10 q^{14} + 17 q^{16} + q^{18} + 17 q^{19} + 2 q^{21} + 2 q^{22} - q^{23} - 3 q^{24} + 2 q^{26}+ \cdots + 7 q^{99}+O(q^{100}) \) 6 * q + q^2 + 6 * q^3 + 7 * q^4 + q^6 + 2 * q^7 - 3 * q^8 + 6 * q^9 + 7 * q^11 + 7 * q^12 + 4 * q^13 + 10 * q^14 + 17 * q^16 + q^18 + 17 * q^19 + 2 * q^21 + 2 * q^22 - q^23 - 3 * q^24 + 2 * q^26 + 6 * q^27 + 22 * q^28 - 8 * q^29 - 6 * q^31 - 35 * q^32 + 7 * q^33 - 13 * q^34 + 7 * q^36 + 14 * q^37 + 10 * q^38 + 4 * q^39 + 18 * q^41 + 10 * q^42 + 15 * q^43 + 4 * q^44 - 10 * q^46 - 13 * q^47 + 17 * q^48 + 18 * q^49 - 38 * q^52 - 13 * q^53 + q^54 + 32 * q^56 + 17 * q^57 + 23 * q^58 + 32 * q^59 - q^62 + 2 * q^63 + 29 * q^64 + 2 * q^66 + 9 * q^67 + 9 * q^68 - q^69 + q^71 - 3 * q^72 + 14 * q^73 + 20 * q^74 + 42 * q^76 - 14 * q^77 + 2 * q^78 + 13 * q^79 + 6 * q^81 - 32 * q^82 + 9 * q^83 + 22 * q^84 - 11 * q^86 - 8 * q^87 + 36 * q^88 + 34 * q^89 - 2 * q^91 - 46 * q^92 - 6 * q^93 + 19 * q^94 - 35 * q^96 + 30 * q^97 + 37 * q^98 + 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more