LMFDB - Embedded newform 1058.4.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1058,4,Mod(1,1058)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1058, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1058.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1058 = 2 \cdot 23^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1058.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:	$62.4240207861$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{26}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 26 $ x^2 - 26
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$5.09902$ of defining polynomial
Character	$\chi$	$=$	1058.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000 q^{2} -4.00000 q^{3} +4.00000 q^{4} +10.1980 q^{5} -8.00000 q^{6} +20.3961 q^{7} +8.00000 q^{8} -11.0000 q^{9} +20.3961 q^{10} +71.3863 q^{11} -16.0000 q^{12} -42.0000 q^{13} +40.7922 q^{14} -40.7922 q^{15} +16.0000 q^{16} +20.3961 q^{17} -22.0000 q^{18} +91.7824 q^{19} +40.7922 q^{20} -81.5843 q^{21} +142.773 q^{22} -32.0000 q^{24} -21.0000 q^{25} -84.0000 q^{26} +152.000 q^{27} +81.5843 q^{28} -22.0000 q^{29} -81.5843 q^{30} +296.000 q^{31} +32.0000 q^{32} -285.545 q^{33} +40.7922 q^{34} +208.000 q^{35} -44.0000 q^{36} -275.347 q^{37} +183.565 q^{38} +168.000 q^{39} +81.5843 q^{40} -318.000 q^{41} -163.169 q^{42} +316.139 q^{43} +285.545 q^{44} -112.178 q^{45} -184.000 q^{47} -64.0000 q^{48} +73.0000 q^{49} -42.0000 q^{50} -81.5843 q^{51} -168.000 q^{52} +91.7824 q^{53} +304.000 q^{54} +728.000 q^{55} +163.169 q^{56} -367.129 q^{57} -44.0000 q^{58} -500.000 q^{59} -163.169 q^{60} +30.5941 q^{61} +592.000 q^{62} -224.357 q^{63} +64.0000 q^{64} -428.318 q^{65} -571.090 q^{66} -642.476 q^{67} +81.5843 q^{68} +416.000 q^{70} +224.000 q^{71} -88.0000 q^{72} -210.000 q^{73} -550.694 q^{74} +84.0000 q^{75} +367.129 q^{76} +1456.00 q^{77} +336.000 q^{78} +1080.99 q^{79} +163.169 q^{80} -311.000 q^{81} -636.000 q^{82} +1152.38 q^{83} -326.337 q^{84} +208.000 q^{85} +632.278 q^{86} +88.0000 q^{87} +571.090 q^{88} -815.843 q^{89} -224.357 q^{90} -856.635 q^{91} -1184.00 q^{93} -368.000 q^{94} +936.000 q^{95} -128.000 q^{96} +1244.16 q^{97} +146.000 q^{98} -785.249 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} - 8 q^{3} + 8 q^{4} - 16 q^{6} + 16 q^{8} - 22 q^{9} - 32 q^{12} - 84 q^{13} + 32 q^{16} - 44 q^{18} - 64 q^{24} - 42 q^{25} - 168 q^{26} + 304 q^{27} - 44 q^{29} + 592 q^{31} + 64 q^{32}+ \cdots + 292 q^{98}+O(q^{100}) $ 2 * q + 4 * q^2 - 8 * q^3 + 8 * q^4 - 16 * q^6 + 16 * q^8 - 22 * q^9 - 32 * q^12 - 84 * q^13 + 32 * q^16 - 44 * q^18 - 64 * q^24 - 42 * q^25 - 168 * q^26 + 304 * q^27 - 44 * q^29 + 592 * q^31 + 64 * q^32 + 416 * q^35 - 88 * q^36 + 336 * q^39 - 636 * q^41 - 368 * q^47 - 128 * q^48 + 146 * q^49 - 84 * q^50 - 336 * q^52 + 608 * q^54 + 1456 * q^55 - 88 * q^58 - 1000 * q^59 + 1184 * q^62 + 128 * q^64 + 832 * q^70 + 448 * q^71 - 176 * q^72 - 420 * q^73 + 168 * q^75 + 2912 * q^77 + 672 * q^78 - 622 * q^81 - 1272 * q^82 + 416 * q^85 + 176 * q^87 - 2368 * q^93 - 736 * q^94 + 1872 * q^95 - 256 * q^96 + 292 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	−4.00000	−0.769800	−0.384900	−	0.922958i	$-0.625764\pi$
$3$	−4.00000	−0.769800	−0.384900	+	0.922958i	$0.625764\pi$
$4$	4.00000	0.500000
$5$	10.1980	0.912140	0.456070	−	0.889944i	$-0.349257\pi$
$5$	10.1980	0.912140	0.456070	+	0.889944i	$0.349257\pi$
$6$	−8.00000	−0.544331
$7$	20.3961	1.10128	0.550642	−	0.834741i	$-0.314383\pi$
$7$	20.3961	1.10128	0.550642	+	0.834741i	$0.314383\pi$
$8$	8.00000	0.353553
$9$	−11.0000	−0.407407
$10$	20.3961	0.644981
$11$	71.3863	1.95671	0.978353	−	0.206942i	$-0.0663511\pi$
$11$	71.3863	1.95671	0.978353	+	0.206942i	$0.0663511\pi$
$12$	−16.0000	−0.384900
$13$	−42.0000	−0.896054	−0.448027	−	0.894020i	$-0.647873\pi$
$13$	−42.0000	−0.896054	−0.448027	+	0.894020i	$0.647873\pi$
$14$	40.7922	0.778726
$15$	−40.7922	−0.702166
$16$	16.0000	0.250000
$17$	20.3961	0.290987	0.145493	−	0.989359i	$-0.453523\pi$
$17$	20.3961	0.290987	0.145493	+	0.989359i	$0.453523\pi$
$18$	−22.0000	−0.288081
$19$	91.7824	1.10823	0.554114	−	0.832441i	$-0.313057\pi$
$19$	91.7824	1.10823	0.554114	+	0.832441i	$0.313057\pi$
$20$	40.7922	0.456070
$21$	−81.5843	−0.847769
$22$	142.773	1.38360
$23$	0	0
$23$	0	0
$24$	−32.0000	−0.272166
$25$	−21.0000	−0.168000
$26$	−84.0000	−0.633606
$27$	152.000	1.08342
$28$	81.5843	0.550642
$29$	−22.0000	−0.140872	−0.0704362	−	0.997516i	$-0.522439\pi$
$29$	−22.0000	−0.140872	−0.0704362	+	0.997516i	$0.522439\pi$
$30$	−81.5843	−0.496506
$31$	296.000	1.71494	0.857470	−	0.514533i	$-0.172035\pi$
$31$	296.000	1.71494	0.857470	+	0.514533i	$0.172035\pi$
$32$	32.0000	0.176777
$33$	−285.545	−1.50627
$34$	40.7922	0.205759
$35$	208.000	1.00453
$36$	−44.0000	−0.203704
$37$	−275.347	−1.22343	−0.611713	−	0.791080i	$-0.709519\pi$
$37$	−275.347	−1.22343	−0.611713	+	0.791080i	$0.709519\pi$
$38$	183.565	0.783635
$39$	168.000	0.689783
$40$	81.5843	0.322490
$41$	−318.000	−1.21130	−0.605649	−	0.795732i	$-0.707087\pi$
$41$	−318.000	−1.21130	−0.605649	+	0.795732i	$0.707087\pi$
$42$	−163.169	−0.599463
$43$	316.139	1.12118	0.560590	−	0.828093i	$-0.310574\pi$
$43$	316.139	1.12118	0.560590	+	0.828093i	$0.310574\pi$
$44$	285.545	0.978353
$45$	−112.178	−0.371613
$46$	0	0
$47$	−184.000	−0.571046	−0.285523	−	0.958372i	$-0.592167\pi$
$47$	−184.000	−0.571046	−0.285523	+	0.958372i	$0.592167\pi$
$48$	−64.0000	−0.192450
$49$	73.0000	0.212828
$50$	−42.0000	−0.118794
$51$	−81.5843	−0.224002
$52$	−168.000	−0.448027
$53$	91.7824	0.237873	0.118937	−	0.992902i	$-0.462051\pi$
$53$	91.7824	0.237873	0.118937	+	0.992902i	$0.462051\pi$
$54$	304.000	0.766096
$55$	728.000	1.78479
$56$	163.169	0.389363
$57$	−367.129	−0.853114
$58$	−44.0000	−0.0996118
$59$	−500.000	−1.10330	−0.551648	−	0.834077i	$-0.686001\pi$
$59$	−500.000	−1.10330	−0.551648	+	0.834077i	$0.686001\pi$
$60$	−163.169	−0.351083
$61$	30.5941	0.0642160	0.0321080	−	0.999484i	$-0.489778\pi$
$61$	30.5941	0.0642160	0.0321080	+	0.999484i	$0.489778\pi$
$62$	592.000	1.21265
$63$	−224.357	−0.448672
$64$	64.0000	0.125000
$65$	−428.318	−0.817327
$66$	−571.090	−1.06510
$67$	−642.476	−1.17151	−0.585754	−	0.810489i	$-0.699201\pi$
$67$	−642.476	−1.17151	−0.585754	+	0.810489i	$0.699201\pi$
$68$	81.5843	0.145493
$69$	0	0
$70$	416.000	0.710307
$71$	224.000	0.374421	0.187211	−	0.982320i	$-0.440055\pi$
$71$	224.000	0.374421	0.187211	+	0.982320i	$0.440055\pi$
$72$	−88.0000	−0.144040
$73$	−210.000	−0.336694	−0.168347	−	0.985728i	$-0.553843\pi$
$73$	−210.000	−0.336694	−0.168347	+	0.985728i	$0.553843\pi$
$74$	−550.694	−0.865093
$75$	84.0000	0.129326
$76$	367.129	0.554114
$77$	1456.00	2.15489
$78$	336.000	0.487750
$79$	1080.99	1.53951	0.769754	−	0.638341i	$-0.220379\pi$
$79$	1080.99	1.53951	0.769754	+	0.638341i	$0.220379\pi$
$80$	163.169	0.228035
$81$	−311.000	−0.426612
$82$	−636.000	−0.856518
$83$	1152.38	1.52398	0.761988	−	0.647591i	$-0.224224\pi$
$83$	1152.38	1.52398	0.761988	+	0.647591i	$0.224224\pi$
$84$	−326.337	−0.423885
$85$	208.000	0.265421
$86$	632.278	0.792795
$87$	88.0000	0.108444
$88$	571.090	0.691800
$89$	−815.843	−0.971676	−0.485838	−	0.874049i	$-0.661486\pi$
$89$	−815.843	−0.971676	−0.485838	+	0.874049i	$0.661486\pi$
$90$	−224.357	−0.262770
$91$	−856.635	−0.986811
$92$	0	0
$93$	−1184.00	−1.32016
$94$	−368.000	−0.403790
$95$	936.000	1.01086
$96$	−128.000	−0.136083
$97$	1244.16	1.30232	0.651162	−	0.758939i	$-0.274282\pi$
$97$	1244.16	1.30232	0.651162	+	0.758939i	$0.274282\pi$
$98$	146.000	0.150492
$99$	−785.249	−0.797177
$100$	−84.0000	−0.0840000
$101$	1934.00	1.90535	0.952674	−	0.303993i	$-0.0983200\pi$
$101$	1934.00	1.90535	0.952674	+	0.303993i	$0.0983200\pi$
$102$	−163.169	−0.158393
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	−336.000	−0.316803
$105$	−832.000	−0.773285
$106$	183.565	0.168202
$107$	132.575	0.119780	0.0598900	−	0.998205i	$-0.480925\pi$
$107$	132.575	0.119780	0.0598900	+	0.998205i	$0.480925\pi$
$108$	608.000	0.541711
$109$	−1193.17	−1.04849	−0.524243	−	0.851569i	$-0.675652\pi$
$109$	−1193.17	−1.04849	−0.524243	+	0.851569i	$0.675652\pi$
$110$	1456.00	1.26204
$111$	1101.39	0.941794
$112$	326.337	0.275321
$113$	2080.40	1.73193	0.865963	−	0.500109i	$-0.166707\pi$
$113$	2080.40	1.73193	0.865963	+	0.500109i	$0.166707\pi$
$114$	−734.259	−0.603242
$115$	0	0
$116$	−88.0000	−0.0704362
$117$	462.000	0.365059
$118$	−1000.00	−0.780148
$119$	416.000	0.320459
$120$	−326.337	−0.248253
$121$	3765.00	2.82870
$122$	61.1882	0.0454076
$123$	1272.00	0.932458
$124$	1184.00	0.857470
$125$	−1488.91	−1.06538
$126$	−448.714	−0.317259
$127$	−408.000	−0.285072	−0.142536	−	0.989790i	$-0.545526\pi$
$127$	−408.000	−0.285072	−0.142536	+	0.989790i	$0.545526\pi$
$128$	128.000	0.0883883
$129$	−1264.56	−0.863085
$130$	−856.635	−0.577938
$131$	196.000	0.130722	0.0653611	−	0.997862i	$-0.479180\pi$
$131$	196.000	0.130722	0.0653611	+	0.997862i	$0.479180\pi$
$132$	−1142.18	−0.753137
$133$	1872.00	1.22047
$134$	−1284.95	−0.828381
$135$	1550.10	0.988234
$136$	163.169	0.102879
$137$	2467.93	1.53904	0.769522	−	0.638620i	$-0.220494\pi$
$137$	2467.93	1.53904	0.769522	+	0.638620i	$0.220494\pi$
$138$	0	0
$139$	−2292.00	−1.39860	−0.699298	−	0.714830i	$-0.746504\pi$
$139$	−2292.00	−1.39860	−0.699298	+	0.714830i	$0.746504\pi$
$140$	832.000	0.502263
$141$	736.000	0.439591
$142$	448.000	0.264756
$143$	−2998.22	−1.75331
$144$	−176.000	−0.101852
$145$	−224.357	−0.128495
$146$	−420.000	−0.238078
$147$	−292.000	−0.163835
$148$	−1101.39	−0.611713
$149$	132.575	0.0728921	0.0364461	−	0.999336i	$-0.488396\pi$
$149$	132.575	0.0728921	0.0364461	+	0.999336i	$0.488396\pi$
$150$	168.000	0.0914476
$151$	−3072.00	−1.65560	−0.827801	−	0.561022i	$-0.810408\pi$
$151$	−3072.00	−1.65560	−0.827801	+	0.561022i	$0.810408\pi$
$152$	734.259	0.391817
$153$	−224.357	−0.118550
$154$	2912.00	1.52374
$155$	3018.62	1.56427
$156$	672.000	0.344891
$157$	−30.5941	−0.0155521	−0.00777604	−	0.999970i	$-0.502475\pi$
$157$	−30.5941	−0.0155521	−0.00777604	+	0.999970i	$0.502475\pi$
$158$	2161.98	1.08860
$159$	−367.129	−0.183115
$160$	326.337	0.161245
$161$	0	0
$162$	−622.000	−0.301660
$163$	3116.00	1.49732	0.748662	−	0.662951i	$-0.230696\pi$
$163$	3116.00	1.49732	0.748662	+	0.662951i	$0.230696\pi$
$164$	−1272.00	−0.605649
$165$	−2912.00	−1.37393
$166$	2304.76	1.07761
$167$	2112.00	0.978632	0.489316	−	0.872107i	$-0.337247\pi$
$167$	2112.00	0.978632	0.489316	+	0.872107i	$0.337247\pi$
$168$	−652.674	−0.299732
$169$	−433.000	−0.197087
$170$	416.000	0.187681
$171$	−1009.61	−0.451500
$172$	1264.56	0.560590
$173$	442.000	0.194246	0.0971232	−	0.995272i	$-0.469036\pi$
$173$	442.000	0.194246	0.0971232	+	0.995272i	$0.469036\pi$
$174$	176.000	0.0766812
$175$	−428.318	−0.185016
$176$	1142.18	0.489177
$177$	2000.00	0.849318
$178$	−1631.69	−0.687079
$179$	2764.00	1.15414	0.577070	−	0.816695i	$-0.304196\pi$
$179$	2764.00	1.15414	0.577070	+	0.816695i	$0.304196\pi$
$180$	−448.714	−0.185806
$181$	−4456.54	−1.83012	−0.915061	−	0.403314i	$-0.867858\pi$
$181$	−4456.54	−1.83012	−0.915061	+	0.403314i	$0.867858\pi$
$182$	−1713.27	−0.697781
$183$	−122.376	−0.0494335
$184$	0	0
$185$	−2808.00	−1.11594
$186$	−2368.00	−0.933496
$187$	1456.00	0.569376
$188$	−736.000	−0.285523
$189$	3100.20	1.19316
$190$	1872.00	0.714785
$191$	4018.03	1.52217	0.761084	−	0.648653i	$-0.224667\pi$
$191$	4018.03	1.52217	0.761084	+	0.648653i	$0.224667\pi$
$192$	−256.000	−0.0962250
$193$	−1666.00	−0.621354	−0.310677	−	0.950516i	$-0.600556\pi$
$193$	−1666.00	−0.621354	−0.310677	+	0.950516i	$0.600556\pi$
$194$	2488.32	0.920882
$195$	1713.27	0.629179
$196$	292.000	0.106414
$197$	−2190.00	−0.792036	−0.396018	−	0.918243i	$-0.629608\pi$
$197$	−2190.00	−0.792036	−0.396018	+	0.918243i	$0.629608\pi$
$198$	−1570.50	−0.563689
$199$	917.824	0.326949	0.163474	−	0.986548i	$-0.447730\pi$
$199$	917.824	0.326949	0.163474	+	0.986548i	$0.447730\pi$
$200$	−168.000	−0.0593970
$201$	2569.91	0.901827
$202$	3868.00	1.34728
$203$	−448.714	−0.155141
$204$	−326.337	−0.112001
$205$	−3242.98	−1.10487
$206$	0	0
$207$	0	0
$208$	−672.000	−0.224014
$209$	6552.00	2.16848
$210$	−1664.00	−0.546795
$211$	−1324.00	−0.431981	−0.215990	−	0.976396i	$-0.569298\pi$
$211$	−1324.00	−0.431981	−0.215990	+	0.976396i	$0.569298\pi$
$212$	367.129	0.118937
$213$	−896.000	−0.288230
$214$	265.149	0.0846973
$215$	3224.00	1.02267
$216$	1216.00	0.383048
$217$	6037.24	1.88864
$218$	−2386.34	−0.741392
$219$	840.000	0.259187
$220$	2912.00	0.892395
$221$	−856.635	−0.260740
$222$	2202.78	0.665949
$223$	−1928.00	−0.578962	−0.289481	−	0.957184i	$-0.593483\pi$
$223$	−1928.00	−0.578962	−0.289481	+	0.957184i	$0.593483\pi$
$224$	652.674	0.194681
$225$	231.000	0.0684444
$226$	4160.80	1.22466
$227$	3905.85	1.14203	0.571014	−	0.820940i	$-0.306550\pi$
$227$	3905.85	1.14203	0.571014	+	0.820940i	$0.306550\pi$
$228$	−1468.52	−0.426557
$229$	−1070.79	−0.308996	−0.154498	−	0.987993i	$-0.549376\pi$
$229$	−1070.79	−0.308996	−0.154498	+	0.987993i	$0.549376\pi$
$230$	0	0
$231$	−5824.00	−1.65884
$232$	−176.000	−0.0498059
$233$	762.000	0.214250	0.107125	−	0.994246i	$-0.465835\pi$
$233$	762.000	0.214250	0.107125	+	0.994246i	$0.465835\pi$
$234$	924.000	0.258136
$235$	−1876.44	−0.520874
$236$	−2000.00	−0.551648
$237$	−4323.97	−1.18511
$238$	832.000	0.226599
$239$	−136.000	−0.0368080	−0.0184040	−	0.999831i	$-0.505859\pi$
$239$	−136.000	−0.0368080	−0.0184040	+	0.999831i	$0.505859\pi$
$240$	−652.674	−0.175541
$241$	1448.12	0.387061	0.193531	−	0.981094i	$-0.438006\pi$
$241$	1448.12	0.387061	0.193531	+	0.981094i	$0.438006\pi$
$242$	7530.00	2.00019
$243$	−2860.00	−0.755017
$244$	122.376	0.0321080
$245$	744.457	0.194129
$246$	2544.00	0.659348
$247$	−3854.86	−0.993032
$248$	2368.00	0.606323
$249$	−4609.51	−1.17316
$250$	−2977.83	−0.753337
$251$	10.1980	0.00256452	0.00128226	−	0.999999i	$-0.499592\pi$
$251$	10.1980	0.00256452	0.00128226	+	0.999999i	$0.499592\pi$
$252$	−897.427	−0.224336
$253$	0	0
$254$	−816.000	−0.201576
$255$	−832.000	−0.204321
$256$	256.000	0.0625000
$257$	−2918.00	−0.708248	−0.354124	−	0.935198i	$-0.615221\pi$
$257$	−2918.00	−0.708248	−0.354124	+	0.935198i	$0.615221\pi$
$258$	−2529.11	−0.610294
$259$	−5616.00	−1.34734
$260$	−1713.27	−0.408664
$261$	242.000	0.0573924
$262$	392.000	0.0924345
$263$	−5262.19	−1.23377	−0.616883	−	0.787055i	$-0.711605\pi$
$263$	−5262.19	−1.23377	−0.616883	+	0.787055i	$0.711605\pi$
$264$	−2284.36	−0.532548
$265$	936.000	0.216974
$266$	3744.00	0.863005
$267$	3263.37	0.747997
$268$	−2569.91	−0.585754
$269$	−5238.00	−1.18724	−0.593618	−	0.804747i	$-0.702301\pi$
$269$	−5238.00	−1.18724	−0.593618	+	0.804747i	$0.702301\pi$
$270$	3100.20	0.698787
$271$	3048.00	0.683221	0.341610	−	0.939842i	$-0.389028\pi$
$271$	3048.00	0.683221	0.341610	+	0.939842i	$0.389028\pi$
$272$	326.337	0.0727467
$273$	3426.54	0.759647
$274$	4935.85	1.08827
$275$	−1499.11	−0.328727
$276$	0	0
$277$	5650.00	1.22554	0.612772	−	0.790260i	$-0.290054\pi$
$277$	5650.00	1.22554	0.612772	+	0.790260i	$0.290054\pi$
$278$	−4584.00	−0.988957
$279$	−3256.00	−0.698680
$280$	1664.00	0.355154
$281$	−4283.18	−0.909299	−0.454649	−	0.890671i	$-0.650235\pi$
$281$	−4283.18	−0.909299	−0.454649	+	0.890671i	$0.650235\pi$
$282$	1472.00	0.310838
$283$	3253.17	0.683326	0.341663	−	0.939823i	$-0.389010\pi$
$283$	3253.17	0.683326	0.341663	+	0.939823i	$0.389010\pi$
$284$	896.000	0.187211
$285$	−3744.00	−0.778159
$286$	−5996.45	−1.23978
$287$	−6485.95	−1.33398
$288$	−352.000	−0.0720201
$289$	−4497.00	−0.915327
$290$	−448.714	−0.0908599
$291$	−4976.64	−1.00253
$292$	−840.000	−0.168347
$293$	−4069.02	−0.811312	−0.405656	−	0.914026i	$-0.632957\pi$
$293$	−4069.02	−0.811312	−0.405656	+	0.914026i	$0.632957\pi$
$294$	−584.000	−0.115849
$295$	−5099.02	−1.00636
$296$	−2202.78	−0.432547
$297$	10850.7	2.11994
$298$	265.149	0.0515425
$299$	0	0
$300$	336.000	0.0646632
$301$	6448.00	1.23474
$302$	−6144.00	−1.17069
$303$	−7736.00	−1.46674
$304$	1468.52	0.277057
$305$	312.000	0.0585740
$306$	−448.714	−0.0838276
$307$	4492.00	0.835088	0.417544	−	0.908657i	$-0.362891\pi$
$307$	4492.00	0.835088	0.417544	+	0.908657i	$0.362891\pi$
$308$	5824.00	1.07745
$309$	0	0
$310$	6037.24	1.10610
$311$	−2880.00	−0.525112	−0.262556	−	0.964917i	$-0.584565\pi$
$311$	−2880.00	−0.525112	−0.262556	+	0.964917i	$0.584565\pi$
$312$	1344.00	0.243875
$313$	2263.96	0.408840	0.204420	−	0.978883i	$-0.434469\pi$
$313$	2263.96	0.408840	0.204420	+	0.978883i	$0.434469\pi$
$314$	−61.1882	−0.0109970
$315$	−2288.00	−0.409251
$316$	4323.97	0.769754
$317$	−8454.00	−1.49787	−0.748934	−	0.662645i	$-0.769434\pi$
$317$	−8454.00	−1.49787	−0.748934	+	0.662645i	$0.769434\pi$
$318$	−734.259	−0.129482
$319$	−1570.50	−0.275646
$320$	652.674	0.114018
$321$	−530.298	−0.0922067
$322$	0	0
$323$	1872.00	0.322479
$324$	−1244.00	−0.213306
$325$	882.000	0.150537
$326$	6232.00	1.05877
$327$	4772.68	0.807125
$328$	−2544.00	−0.428259
$329$	−3752.88	−0.628884
$330$	−5824.00	−0.971517
$331$	−10052.0	−1.66921	−0.834604	−	0.550850i	$-0.814304\pi$
$331$	−10052.0	−1.66921	−0.834604	+	0.550850i	$0.814304\pi$
$332$	4609.51	0.761988
$333$	3028.82	0.498433
$334$	4224.00	0.691997
$335$	−6552.00	−1.06858
$336$	−1305.35	−0.211942
$337$	−489.506	−0.0791249	−0.0395624	−	0.999217i	$-0.512596\pi$
$337$	−489.506	−0.0791249	−0.0395624	+	0.999217i	$0.512596\pi$
$338$	−866.000	−0.139362
$339$	−8321.60	−1.33324
$340$	832.000	0.132710
$341$	21130.3	3.35564
$342$	−2019.21	−0.319259
$343$	−5506.94	−0.866900
$344$	2529.11	0.396397
$345$	0	0
$346$	884.000	0.137353
$347$	−8604.00	−1.33109	−0.665543	−	0.746359i	$-0.731800\pi$
$347$	−8604.00	−1.33109	−0.665543	+	0.746359i	$0.731800\pi$
$348$	352.000	0.0542218
$349$	−1174.00	−0.180065	−0.0900326	−	0.995939i	$-0.528697\pi$
$349$	−1174.00	−0.180065	−0.0900326	+	0.995939i	$0.528697\pi$
$350$	−856.635	−0.130826
$351$	−6384.00	−0.970805
$352$	2284.36	0.345900
$353$	−2818.00	−0.424892	−0.212446	−	0.977173i	$-0.568143\pi$
$353$	−2818.00	−0.424892	−0.212446	+	0.977173i	$0.568143\pi$
$354$	4000.00	0.600558
$355$	2284.36	0.341525
$356$	−3263.37	−0.485838
$357$	−1664.00	−0.246690
$358$	5528.00	0.816100
$359$	1366.54	0.200900	0.100450	−	0.994942i	$-0.467972\pi$
$359$	1366.54	0.200900	0.100450	+	0.994942i	$0.467972\pi$
$360$	−897.427	−0.131385
$361$	1565.00	0.228167
$362$	−8913.09	−1.29409
$363$	−15060.0	−2.17753
$364$	−3426.54	−0.493405
$365$	−2141.59	−0.307112
$366$	−244.753	−0.0349548
$367$	3079.81	0.438051	0.219025	−	0.975719i	$-0.429712\pi$
$367$	3079.81	0.438051	0.219025	+	0.975719i	$0.429712\pi$
$368$	0	0
$369$	3498.00	0.493492
$370$	−5616.00	−0.789086
$371$	1872.00	0.261966
$372$	−4736.00	−0.660081
$373$	4415.75	0.612973	0.306486	−	0.951875i	$-0.400847\pi$
$373$	4415.75	0.612973	0.306486	+	0.951875i	$0.400847\pi$
$374$	2912.00	0.402609
$375$	5955.65	0.820130
$376$	−1472.00	−0.201895
$377$	924.000	0.126229
$378$	6200.41	0.843689
$379$	10.1980	0.00138216	0.000691079	−	1.00000i	$-0.499780\pi$
$379$	10.1980	0.00138216	0.000691079		1.00000i	$0.499780\pi$
$380$	3744.00	0.505429
$381$	1632.00	0.219449
$382$	8036.05	1.07634
$383$	−40.7922	−0.00544225	−0.00272113	−	0.999996i	$-0.500866\pi$
$383$	−40.7922	−0.00544225	−0.00272113	+	0.999996i	$0.500866\pi$
$384$	−512.000	−0.0680414
$385$	14848.3	1.96556
$386$	−3332.00	−0.439364
$387$	−3477.53	−0.456777
$388$	4976.64	0.651162
$389$	−14328.2	−1.86753	−0.933767	−	0.357881i	$-0.883499\pi$
$389$	−14328.2	−1.86753	−0.933767	+	0.357881i	$0.883499\pi$
$390$	3426.54	0.444897
$391$	0	0
$392$	584.000	0.0752461
$393$	−784.000	−0.100630
$394$	−4380.00	−0.560054
$395$	11024.0	1.40425
$396$	−3141.00	−0.398588
$397$	11046.0	1.39643	0.698215	−	0.715888i	$-0.253978\pi$
$397$	11046.0	1.39643	0.698215	+	0.715888i	$0.253978\pi$
$398$	1835.65	0.231188
$399$	−7488.00	−0.939521
$400$	−336.000	−0.0420000
$401$	−4425.95	−0.551175	−0.275588	−	0.961276i	$-0.588872\pi$
$401$	−4425.95	−0.551175	−0.275588	+	0.961276i	$0.588872\pi$
$402$	5139.81	0.637688
$403$	−12432.0	−1.53668
$404$	7736.00	0.952674
$405$	−3171.59	−0.389130
$406$	−897.427	−0.109701
$407$	−19656.0	−2.39389
$408$	−652.674	−0.0791966
$409$	−5194.00	−0.627938	−0.313969	−	0.949433i	$-0.601659\pi$
$409$	−5194.00	−0.627938	−0.313969	+	0.949433i	$0.601659\pi$
$410$	−6485.95	−0.781264
$411$	−9871.70	−1.18476
$412$	0	0
$413$	−10198.0	−1.21504
$414$	0	0
$415$	11752.0	1.39008
$416$	−1344.00	−0.158401
$417$	9168.00	1.07664
$418$	13104.0	1.53334
$419$	5272.39	0.614733	0.307366	−	0.951591i	$-0.400552\pi$
$419$	5272.39	0.614733	0.307366	+	0.951591i	$0.400552\pi$
$420$	−3328.00	−0.386642
$421$	−9616.75	−1.11328	−0.556641	−	0.830753i	$-0.687910\pi$
$421$	−9616.75	−1.11328	−0.556641	+	0.830753i	$0.687910\pi$
$422$	−2648.00	−0.305456
$423$	2024.00	0.232648
$424$	734.259	0.0841008
$425$	−428.318	−0.0488858
$426$	−1792.00	−0.203809
$427$	624.000	0.0707201
$428$	530.298	0.0598900
$429$	11992.9	1.34970
$430$	6448.00	0.723140
$431$	7607.74	0.850236	0.425118	−	0.905138i	$-0.360233\pi$
$431$	7607.74	0.850236	0.425118	+	0.905138i	$0.360233\pi$
$432$	2432.00	0.270856
$433$	−9932.89	−1.10241	−0.551206	−	0.834369i	$-0.685832\pi$
$433$	−9932.89	−1.10241	−0.551206	+	0.834369i	$0.685832\pi$
$434$	12074.5	1.33547
$435$	897.427	0.0989158
$436$	−4772.68	−0.524243
$437$	0	0
$438$	1680.00	0.183273
$439$	−6768.00	−0.735806	−0.367903	−	0.929864i	$-0.619924\pi$
$439$	−6768.00	−0.735806	−0.367903	+	0.929864i	$0.619924\pi$
$440$	5824.00	0.631019
$441$	−803.000	−0.0867077
$442$	−1713.27	−0.184371
$443$	−7292.00	−0.782062	−0.391031	−	0.920378i	$-0.627881\pi$
$443$	−7292.00	−0.782062	−0.391031	+	0.920378i	$0.627881\pi$
$444$	4405.55	0.470897
$445$	−8320.00	−0.886305
$446$	−3856.00	−0.409388
$447$	−530.298	−0.0561124
$448$	1305.35	0.137661
$449$	−10198.0	−1.07188	−0.535939	−	0.844257i	$-0.680042\pi$
$449$	−10198.0	−1.07188	−0.535939	+	0.844257i	$0.680042\pi$
$450$	462.000	0.0483975
$451$	−22700.8	−2.37016
$452$	8321.60	0.865963
$453$	12288.0	1.27448
$454$	7811.70	0.807536
$455$	−8736.00	−0.900110
$456$	−2937.04	−0.301621
$457$	10707.9	1.09605	0.548027	−	0.836461i	$-0.315379\pi$
$457$	10707.9	1.09605	0.548027	+	0.836461i	$0.315379\pi$
$458$	−2141.59	−0.218493
$459$	3100.20	0.315262
$460$	0	0
$461$	4794.00	0.484336	0.242168	−	0.970234i	$-0.422142\pi$
$461$	4794.00	0.484336	0.242168	+	0.970234i	$0.422142\pi$
$462$	−11648.0	−1.17297
$463$	9848.00	0.988500	0.494250	−	0.869320i	$-0.335443\pi$
$463$	9848.00	0.988500	0.494250	+	0.869320i	$0.335443\pi$
$464$	−352.000	−0.0352181
$465$	−12074.5	−1.20417
$466$	1524.00	0.151498
$467$	−5986.25	−0.593170	−0.296585	−	0.955006i	$-0.595848\pi$
$467$	−5986.25	−0.593170	−0.296585	+	0.955006i	$0.595848\pi$
$468$	1848.00	0.182530
$469$	−13104.0	−1.29016
$470$	−3752.88	−0.368314
$471$	122.376	0.0119720
$472$	−4000.00	−0.390074
$473$	22568.0	2.19382
$474$	−8647.94	−0.838002
$475$	−1927.43	−0.186182
$476$	1664.00	0.160230
$477$	−1009.61	−0.0969113
$478$	−272.000	−0.0260272
$479$	2651.49	0.252922	0.126461	−	0.991972i	$-0.459638\pi$
$479$	2651.49	0.252922	0.126461	+	0.991972i	$0.459638\pi$
$480$	−1305.35	−0.124127
$481$	11564.6	1.09626
$482$	2896.24	0.273693
$483$	0	0
$484$	15060.0	1.41435
$485$	12688.0	1.18790
$486$	−5720.00	−0.533878
$487$	9328.00	0.867951	0.433975	−	0.900925i	$-0.357110\pi$
$487$	9328.00	0.867951	0.433975	+	0.900925i	$0.357110\pi$
$488$	244.753	0.0227038
$489$	−12464.0	−1.15264
$490$	1488.91	0.137270
$491$	1332.00	0.122428	0.0612142	−	0.998125i	$-0.480503\pi$
$491$	1332.00	0.122428	0.0612142	+	0.998125i	$0.480503\pi$
$492$	5088.00	0.466229
$493$	−448.714	−0.0409920
$494$	−7709.72	−0.702179
$495$	−8008.00	−0.727137
$496$	4736.00	0.428735
$497$	4568.72	0.412344
$498$	−9219.03	−0.829547
$499$	−6308.00	−0.565902	−0.282951	−	0.959134i	$-0.591313\pi$
$499$	−6308.00	−0.565902	−0.282951	+	0.959134i	$0.591313\pi$
$500$	−5955.65	−0.532690
$501$	−8448.00	−0.753351
$502$	20.3961	0.00181339
$503$	1325.75	0.117519	0.0587595	−	0.998272i	$-0.481285\pi$
$503$	1325.75	0.117519	0.0587595	+	0.998272i	$0.481285\pi$
$504$	−1794.85	−0.158629
$505$	19723.0	1.73795
$506$	0	0
$507$	1732.00	0.151718
$508$	−1632.00	−0.142536
$509$	5034.00	0.438366	0.219183	−	0.975684i	$-0.429661\pi$
$509$	5034.00	0.438366	0.219183	+	0.975684i	$0.429661\pi$
$510$	−1664.00	−0.144477
$511$	−4283.18	−0.370796
$512$	512.000	0.0441942
$513$	13950.9	1.20068
$514$	−5836.00	−0.500807
$515$	0	0
$516$	−5058.23	−0.431543
$517$	−13135.1	−1.11737
$518$	−11232.0	−0.952714
$519$	−1768.00	−0.149531
$520$	−3426.54	−0.288969
$521$	7077.44	0.595141	0.297570	−	0.954700i	$-0.403824\pi$
$521$	7077.44	0.595141	0.297570	+	0.954700i	$0.403824\pi$
$522$	484.000	0.0405826
$523$	−13573.6	−1.13486	−0.567430	−	0.823422i	$-0.692062\pi$
$523$	−13573.6	−1.13486	−0.567430	+	0.823422i	$0.692062\pi$
$524$	784.000	0.0653611
$525$	1713.27	0.142425
$526$	−10524.4	−0.872404
$527$	6037.24	0.499025
$528$	−4568.72	−0.376568
$529$	0	0
$530$	1872.00	0.153424
$531$	5500.00	0.449491
$532$	7488.00	0.610237
$533$	13356.0	1.08539
$534$	6526.74	0.528914
$535$	1352.00	0.109256
$536$	−5139.81	−0.414190
$537$	−11056.0	−0.888457
$538$	−10476.0	−0.839503
$539$	5211.20	0.416442
$540$	6200.41	0.494117
$541$	−13334.0	−1.05966	−0.529828	−	0.848105i	$-0.677743\pi$
$541$	−13334.0	−1.05966	−0.529828	+	0.848105i	$0.677743\pi$
$542$	6096.00	0.483110
$543$	17826.2	1.40883
$544$	652.674	0.0514397
$545$	−12168.0	−0.956367
$546$	6853.08	0.537152
$547$	−1468.00	−0.114748	−0.0573740	−	0.998353i	$-0.518273\pi$
$547$	−1468.00	−0.114748	−0.0573740	+	0.998353i	$0.518273\pi$
$548$	9871.70	0.769522
$549$	−336.535	−0.0261621
$550$	−2998.22	−0.232445
$551$	−2019.21	−0.156119
$552$	0	0
$553$	22048.0	1.69544
$554$	11300.0	0.866590
$555$	11232.0	0.859048
$556$	−9168.00	−0.699298
$557$	18427.9	1.40182	0.700910	−	0.713250i	$-0.252778\pi$
$557$	18427.9	1.40182	0.700910	+	0.713250i	$0.252778\pi$
$558$	−6512.00	−0.494041
$559$	−13277.8	−1.00464
$560$	3328.00	0.251132
$561$	−5824.00	−0.438306
$562$	−8566.35	−0.642971
$563$	22507.1	1.68483	0.842416	−	0.538828i	$-0.181133\pi$
$563$	22507.1	1.68483	0.842416	+	0.538828i	$0.181133\pi$
$564$	2944.00	0.219796
$565$	21216.0	1.57976
$566$	6506.35	0.483184
$567$	−6343.18	−0.469821
$568$	1792.00	0.132378
$569$	−8382.79	−0.617618	−0.308809	−	0.951124i	$-0.599930\pi$
$569$	−8382.79	−0.617618	−0.308809	+	0.951124i	$0.599930\pi$
$570$	−7488.00	−0.550242
$571$	4089.41	0.299714	0.149857	−	0.988708i	$-0.452119\pi$
$571$	4089.41	0.299714	0.149857	+	0.988708i	$0.452119\pi$
$572$	−11992.9	−0.876657
$573$	−16072.1	−1.17177
$574$	−12971.9	−0.943270
$575$	0	0
$576$	−704.000	−0.0509259
$577$	−8378.00	−0.604473	−0.302236	−	0.953233i	$-0.597733\pi$
$577$	−8378.00	−0.604473	−0.302236	+	0.953233i	$0.597733\pi$
$578$	−8994.00	−0.647234
$579$	6664.00	0.478318
$580$	−897.427	−0.0642477
$581$	23504.0	1.67833
$582$	−9953.29	−0.708895
$583$	6552.00	0.465448
$584$	−1680.00	−0.119039
$585$	4711.49	0.332985
$586$	−8138.04	−0.573685
$587$	27228.0	1.91451	0.957257	−	0.289238i	$-0.0934020\pi$
$587$	27228.0	1.91451	0.957257	+	0.289238i	$0.0934020\pi$
$588$	−1168.00	−0.0819175
$589$	27167.6	1.90054
$590$	−10198.0	−0.711604
$591$	8760.00	0.609709
$592$	−4405.55	−0.305857
$593$	−13462.0	−0.932240	−0.466120	−	0.884722i	$-0.654348\pi$
$593$	−13462.0	−0.932240	−0.466120	+	0.884722i	$0.654348\pi$
$594$	21701.4	1.49902
$595$	4242.38	0.292304
$596$	530.298	0.0364461
$597$	−3671.29	−0.251685
$598$	0	0
$599$	−16224.0	−1.10667	−0.553334	−	0.832959i	$-0.686645\pi$
$599$	−16224.0	−1.10667	−0.553334	+	0.832959i	$0.686645\pi$
$600$	672.000	0.0457238
$601$	13118.0	0.890340	0.445170	−	0.895446i	$-0.353143\pi$
$601$	13118.0	0.890340	0.445170	+	0.895446i	$0.353143\pi$
$602$	12896.0	0.873093
$603$	7067.24	0.477281
$604$	−12288.0	−0.827801
$605$	38395.6	2.58017
$606$	−15472.0	−1.03714
$607$	1144.00	0.0764968	0.0382484	−	0.999268i	$-0.487822\pi$
$607$	1144.00	0.0764968	0.0382484	+	0.999268i	$0.487822\pi$
$608$	2937.04	0.195909
$609$	1794.85	0.119427
$610$	624.000	0.0414181
$611$	7728.00	0.511688
$612$	−897.427	−0.0592751
$613$	−27259.4	−1.79608	−0.898038	−	0.439917i	$-0.855008\pi$
$613$	−27259.4	−1.79608	−0.898038	+	0.439917i	$0.855008\pi$
$614$	8984.00	0.590496
$615$	12971.9	0.850533
$616$	11648.0	0.761869
$617$	−21334.3	−1.39204	−0.696018	−	0.718024i	$-0.745047\pi$
$617$	−21334.3	−1.39204	−0.696018	+	0.718024i	$0.745047\pi$
$618$	0	0
$619$	19692.4	1.27868	0.639342	−	0.768923i	$-0.279207\pi$
$619$	19692.4	1.27868	0.639342	+	0.768923i	$0.279207\pi$
$620$	12074.5	0.782133
$621$	0	0
$622$	−5760.00	−0.371310
$623$	−16640.0	−1.07009
$624$	2688.00	0.172446
$625$	−12559.0	−0.803776
$626$	4527.93	0.289093
$627$	−26208.0	−1.66929
$628$	−122.376	−0.00777604
$629$	−5616.00	−0.356001
$630$	−4576.00	−0.289384
$631$	5384.56	0.339709	0.169854	−	0.985469i	$-0.445670\pi$
$631$	5384.56	0.339709	0.169854	+	0.985469i	$0.445670\pi$
$632$	8647.94	0.544298
$633$	5296.00	0.332539
$634$	−16908.0	−1.05915
$635$	−4160.80	−0.260026
$636$	−1468.52	−0.0915574
$637$	−3066.00	−0.190705
$638$	−3141.00	−0.194911
$639$	−2464.00	−0.152542
$640$	1305.35	0.0806226
$641$	−10422.4	−0.642215	−0.321108	−	0.947043i	$-0.604055\pi$
$641$	−10422.4	−0.642215	−0.321108	+	0.947043i	$0.604055\pi$
$642$	−1060.60	−0.0652000
$643$	−18162.7	−1.11395	−0.556973	−	0.830531i	$-0.688037\pi$
$643$	−18162.7	−1.11395	−0.556973	+	0.830531i	$0.688037\pi$
$644$	0	0
$645$	−12896.0	−0.787255
$646$	3744.00	0.228027
$647$	−23824.0	−1.44763	−0.723816	−	0.689993i	$-0.757614\pi$
$647$	−23824.0	−1.44763	−0.723816	+	0.689993i	$0.757614\pi$
$648$	−2488.00	−0.150830
$649$	−35693.1	−2.15883
$650$	1764.00	0.106446
$651$	−24149.0	−1.45387
$652$	12464.0	0.748662
$653$	−5690.00	−0.340991	−0.170495	−	0.985358i	$-0.554537\pi$
$653$	−5690.00	−0.340991	−0.170495	+	0.985358i	$0.554537\pi$
$654$	9545.36	0.570724
$655$	1998.82	0.119237
$656$	−5088.00	−0.302825
$657$	2310.00	0.137172
$658$	−7505.76	−0.444688
$659$	−4109.81	−0.242937	−0.121468	−	0.992595i	$-0.538760\pi$
$659$	−4109.81	−0.242937	−0.121468	+	0.992595i	$0.538760\pi$
$660$	−11648.0	−0.686966
$661$	−20936.6	−1.23198	−0.615990	−	0.787754i	$-0.711244\pi$
$661$	−20936.6	−1.23198	−0.615990	+	0.787754i	$0.711244\pi$
$662$	−20104.0	−1.18031
$663$	3426.54	0.200718
$664$	9219.03	0.538807
$665$	19090.7	1.11324
$666$	6057.64	0.352445
$667$	0	0
$668$	8448.00	0.489316
$669$	7712.00	0.445685
$670$	−13104.0	−0.755600
$671$	2184.00	0.125652
$672$	−2610.70	−0.149866
$673$	18986.0	1.08745	0.543727	−	0.839262i	$-0.317013\pi$
$673$	18986.0	1.08745	0.543727	+	0.839262i	$0.317013\pi$
$674$	−979.012	−0.0559497
$675$	−3192.00	−0.182015
$676$	−1732.00	−0.0985435
$677$	−1091.19	−0.0619466	−0.0309733	−	0.999520i	$-0.509861\pi$
$677$	−1091.19	−0.0619466	−0.0309733	+	0.999520i	$0.509861\pi$
$678$	−16643.2	−0.942741
$679$	25376.0	1.43423
$680$	1664.00	0.0938404
$681$	−15623.4	−0.879133
$682$	42260.7	2.37279
$683$	−21396.0	−1.19868	−0.599338	−	0.800496i	$-0.704569\pi$
$683$	−21396.0	−1.19868	−0.599338	+	0.800496i	$0.704569\pi$
$684$	−4038.42	−0.225750
$685$	25168.0	1.40382
$686$	−11013.9	−0.612991
$687$	4283.18	0.237865
$688$	5058.23	0.280295
$689$	−3854.86	−0.213147
$690$	0	0
$691$	15700.0	0.864336	0.432168	−	0.901793i	$-0.357749\pi$
$691$	15700.0	0.864336	0.432168	+	0.901793i	$0.357749\pi$
$692$	1768.00	0.0971232
$693$	−16016.0	−0.877919
$694$	−17208.0	−0.941220
$695$	−23373.9	−1.27572
$696$	704.000	0.0383406
$697$	−6485.95	−0.352472
$698$	−2348.00	−0.127325
$699$	−3048.00	−0.164930
$700$	−1713.27	−0.0925079
$701$	−10657.0	−0.574190	−0.287095	−	0.957902i	$-0.592690\pi$
$701$	−10657.0	−0.574190	−0.287095	+	0.957902i	$0.592690\pi$
$702$	−12768.0	−0.686463
$703$	−25272.0	−1.35583
$704$	4568.72	0.244588
$705$	7505.76	0.400969
$706$	−5636.00	−0.300444
$707$	39446.0	2.09833
$708$	8000.00	0.424659
$709$	9188.43	0.486712	0.243356	−	0.969937i	$-0.421752\pi$
$709$	9188.43	0.486712	0.243356	+	0.969937i	$0.421752\pi$
$710$	4568.72	0.241494
$711$	−11890.9	−0.627207
$712$	−6526.74	−0.343539
$713$	0	0
$714$	−3328.00	−0.174436
$715$	−30576.0	−1.59927
$716$	11056.0	0.577070
$717$	544.000	0.0283348
$718$	2733.07	0.142058
$719$	−18600.0	−0.964761	−0.482380	−	0.875962i	$-0.660228\pi$
$719$	−18600.0	−0.964761	−0.482380	+	0.875962i	$0.660228\pi$
$720$	−1794.85	−0.0929032
$721$	0	0
$722$	3130.00	0.161339
$723$	−5792.49	−0.297960
$724$	−17826.2	−0.915061
$725$	462.000	0.0236666
$726$	−30120.0	−1.53975
$727$	−16786.0	−0.856337	−0.428169	−	0.903699i	$-0.640841\pi$
$727$	−16786.0	−0.856337	−0.428169	+	0.903699i	$0.640841\pi$
$728$	−6853.08	−0.348890
$729$	19837.0	1.00782
$730$	−4283.18	−0.217161
$731$	6448.00	0.326249
$732$	−489.506	−0.0247167
$733$	7108.03	0.358174	0.179087	−	0.983833i	$-0.442686\pi$
$733$	7108.03	0.358174	0.179087	+	0.983833i	$0.442686\pi$
$734$	6159.62	0.309749
$735$	−2977.83	−0.149441
$736$	0	0
$737$	−45864.0	−2.29230
$738$	6996.00	0.348952
$739$	−35364.0	−1.76033	−0.880166	−	0.474665i	$-0.842569\pi$
$739$	−35364.0	−1.76033	−0.880166	+	0.474665i	$0.842569\pi$
$740$	−11232.0	−0.557968
$741$	15419.4	0.764436
$742$	3744.00	0.185238
$743$	−7505.76	−0.370605	−0.185302	−	0.982682i	$-0.559326\pi$
$743$	−7505.76	−0.370605	−0.185302	+	0.982682i	$0.559326\pi$
$744$	−9472.00	−0.466748
$745$	1352.00	0.0664878
$746$	8831.50	0.433437
$747$	−12676.2	−0.620879
$748$	5824.00	0.284688
$749$	2704.00	0.131912
$750$	11911.3	0.579919
$751$	37936.7	1.84332	0.921658	−	0.388004i	$-0.126835\pi$
$751$	37936.7	1.84332	0.921658	+	0.388004i	$0.126835\pi$
$752$	−2944.00	−0.142761
$753$	−40.7922	−0.00197417
$754$	1848.00	0.0892575
$755$	−31328.4	−1.51014
$756$	12400.8	0.596578
$757$	−32440.0	−1.55753	−0.778765	−	0.627316i	$-0.784154\pi$
$757$	−32440.0	−1.55753	−0.778765	+	0.627316i	$0.784154\pi$
$758$	20.3961	0.000977334 0
$759$	0	0
$760$	7488.00	0.357393
$761$	9046.00	0.430903	0.215452	−	0.976515i	$-0.430878\pi$
$761$	9046.00	0.430903	0.215452	+	0.976515i	$0.430878\pi$
$762$	3264.00	0.155174
$763$	−24336.0	−1.15468
$764$	16072.1	0.761084
$765$	−2288.00	−0.108134
$766$	−81.5843	−0.00384825
$767$	21000.0	0.988613
$768$	−1024.00	−0.0481125
$769$	−40690.2	−1.90810	−0.954048	−	0.299655i	$-0.903128\pi$
$769$	−40690.2	−1.90810	−0.954048	+	0.299655i	$0.903128\pi$
$770$	29696.7	1.38986
$771$	11672.0	0.545210
$772$	−6664.00	−0.310677
$773$	−26076.4	−1.21333	−0.606664	−	0.794958i	$-0.707493\pi$
$773$	−26076.4	−1.21333	−0.606664	+	0.794958i	$0.707493\pi$
$774$	−6955.06	−0.322990
$775$	−6216.00	−0.288110
$776$	9953.29	0.460441
$777$	22464.0	1.03718
$778$	−28656.5	−1.32055
$779$	−29186.8	−1.34239
$780$	6853.08	0.314589
$781$	15990.5	0.732632
$782$	0	0
$783$	−3344.00	−0.152624
$784$	1168.00	0.0532070
$785$	−312.000	−0.0141857
$786$	−1568.00	−0.0711561
$787$	28870.6	1.30766	0.653829	−	0.756642i	$-0.273161\pi$
$787$	28870.6	1.30766	0.653829	+	0.756642i	$0.273161\pi$
$788$	−8760.00	−0.396018
$789$	21048.8	0.949753
$790$	22048.0	0.992953
$791$	42432.0	1.90734
$792$	−6281.99	−0.281845
$793$	−1284.95	−0.0575410
$794$	22092.0	0.987425
$795$	−3744.00	−0.167026
$796$	3671.29	0.163474
$797$	−24607.9	−1.09367	−0.546835	−	0.837240i	$-0.684168\pi$
$797$	−24607.9	−1.09367	−0.546835	+	0.837240i	$0.684168\pi$
$798$	−14976.0	−0.664342
$799$	−3752.88	−0.166167
$800$	−672.000	−0.0296985
$801$	8974.27	0.395868
$802$	−8851.90	−0.389740
$803$	−14991.1	−0.658811
$804$	10279.6	0.450913
$805$	0	0
$806$	−24864.0	−1.08660
$807$	20952.0	0.913935
$808$	15472.0	0.673642
$809$	−9594.00	−0.416943	−0.208472	−	0.978028i	$-0.566849\pi$
$809$	−9594.00	−0.416943	−0.208472	+	0.978028i	$0.566849\pi$
$810$	−6343.18	−0.275156
$811$	14060.0	0.608771	0.304386	−	0.952549i	$-0.401549\pi$
$811$	14060.0	0.608771	0.304386	+	0.952549i	$0.401549\pi$
$812$	−1794.85	−0.0775703
$813$	−12192.0	−0.525944
$814$	−39312.0	−1.69273
$815$	31777.1	1.36577
$816$	−1305.35	−0.0560004
$817$	29016.0	1.24252
$818$	−10388.0	−0.444019
$819$	9422.99	0.402034
$820$	−12971.9	−0.552437
$821$	−14850.0	−0.631265	−0.315633	−	0.948882i	$-0.602217\pi$
$821$	−14850.0	−0.631265	−0.315633	+	0.948882i	$0.602217\pi$
$822$	−19743.4	−0.837750
$823$	944.000	0.0399827	0.0199914	−	0.999800i	$-0.493636\pi$
$823$	944.000	0.0399827	0.0199914	+	0.999800i	$0.493636\pi$
$824$	0	0
$825$	5996.45	0.253054
$826$	−20396.1	−0.859165
$827$	15959.9	0.671078	0.335539	−	0.942026i	$-0.391082\pi$
$827$	15959.9	0.671078	0.335539	+	0.942026i	$0.391082\pi$
$828$	0	0
$829$	27482.0	1.15137	0.575687	−	0.817670i	$-0.304735\pi$
$829$	27482.0	1.15137	0.575687	+	0.817670i	$0.304735\pi$
$830$	23504.0	0.982935
$831$	−22600.0	−0.943424
$832$	−2688.00	−0.112007
$833$	1488.91	0.0619301
$834$	18336.0	0.761299
$835$	21538.3	0.892649
$836$	26208.0	1.08424
$837$	44992.0	1.85801
$838$	10544.8	0.434682
$839$	11748.1	0.483422	0.241711	−	0.970348i	$-0.422291\pi$
$839$	11748.1	0.483422	0.241711	+	0.970348i	$0.422291\pi$
$840$	−6656.00	−0.273397
$841$	−23905.0	−0.980155
$842$	−19233.5	−0.787209
$843$	17132.7	0.699978
$844$	−5296.00	−0.215990
$845$	−4415.75	−0.179771
$846$	4048.00	0.164507
$847$	76791.2	3.11520
$848$	1468.52	0.0594683
$849$	−13012.7	−0.526024
$850$	−856.635	−0.0345675
$851$	0	0
$852$	−3584.00	−0.144115
$853$	−12850.0	−0.515798	−0.257899	−	0.966172i	$-0.583030\pi$
$853$	−12850.0	−0.515798	−0.257899	+	0.966172i	$0.583030\pi$
$854$	1248.00	0.0500067
$855$	−10296.0	−0.411831
$856$	1060.60	0.0423486
$857$	25098.0	1.00039	0.500193	−	0.865914i	$-0.333262\pi$
$857$	25098.0	1.00039	0.500193	+	0.865914i	$0.333262\pi$
$858$	23985.8	0.954384
$859$	40052.0	1.59087	0.795435	−	0.606039i	$-0.207243\pi$
$859$	40052.0	1.59087	0.795435	+	0.606039i	$0.207243\pi$
$860$	12896.0	0.511337
$861$	25943.8	1.02690
$862$	15215.5	0.601208
$863$	11928.0	0.470491	0.235246	−	0.971936i	$-0.424411\pi$
$863$	11928.0	0.470491	0.235246	+	0.971936i	$0.424411\pi$
$864$	4864.00	0.191524
$865$	4507.53	0.177180
$866$	−19865.8	−0.779523
$867$	17988.0	0.704619
$868$	24149.0	0.944319
$869$	77168.0	3.01236
$870$	1794.85	0.0699440
$871$	26984.0	1.04973
$872$	−9545.36	−0.370696
$873$	−13685.8	−0.530576
$874$	0	0
$875$	−30368.0	−1.17329
$876$	3360.00	0.129593
$877$	3306.00	0.127293	0.0636463	−	0.997973i	$-0.479727\pi$
$877$	3306.00	0.127293	0.0636463	+	0.997973i	$0.479727\pi$
$878$	−13536.0	−0.520294
$879$	16276.1	0.624549
$880$	11648.0	0.446198
$881$	−28065.0	−1.07325	−0.536625	−	0.843821i	$-0.680301\pi$
$881$	−28065.0	−1.07325	−0.536625	+	0.843821i	$0.680301\pi$
$882$	−1606.00	−0.0613116
$883$	−6812.00	−0.259617	−0.129809	−	0.991539i	$-0.541436\pi$
$883$	−6812.00	−0.259617	−0.129809	+	0.991539i	$0.541436\pi$
$884$	−3426.54	−0.130370
$885$	20396.1	0.774697
$886$	−14584.0	−0.553001
$887$	25440.0	0.963012	0.481506	−	0.876443i	$-0.340090\pi$
$887$	25440.0	0.963012	0.481506	+	0.876443i	$0.340090\pi$
$888$	8811.11	0.332974
$889$	−8321.60	−0.313945
$890$	−16640.0	−0.626712
$891$	−22201.1	−0.834754
$892$	−7712.00	−0.289481
$893$	−16888.0	−0.632849
$894$	−1060.60	−0.0396774
$895$	28187.4	1.05274
$896$	2610.70	0.0973407
$897$	0	0
$898$	−20396.0	−0.757932
$899$	−6512.00	−0.241588
$900$	924.000	0.0342222
$901$	1872.00	0.0692179
$902$	−45401.7	−1.67595
$903$	−25792.0	−0.950503
$904$	16643.2	0.612328
$905$	−45448.0	−1.66933
$906$	24576.0	0.901195
$907$	−9514.77	−0.348327	−0.174164	−	0.984717i	$-0.555722\pi$
$907$	−9514.77	−0.348327	−0.174164	+	0.984717i	$0.555722\pi$
$908$	15623.4	0.571014
$909$	−21274.0	−0.776253
$910$	−17472.0	−0.636474
$911$	41628.4	1.51395	0.756976	−	0.653443i	$-0.226676\pi$
$911$	41628.4	1.51395	0.756976	+	0.653443i	$0.226676\pi$
$912$	−5874.07	−0.213278
$913$	82264.0	2.98197
$914$	21415.9	0.775027
$915$	−1248.00	−0.0450903
$916$	−4283.18	−0.154498
$917$	3997.63	0.143962
$918$	6200.41	0.222924
$919$	30043.4	1.07839	0.539195	−	0.842181i	$-0.318728\pi$
$919$	30043.4	1.07839	0.539195	+	0.842181i	$0.318728\pi$
$920$	0	0
$921$	−17968.0	−0.642851
$922$	9588.00	0.342477
$923$	−9408.00	−0.335502
$924$	−23296.0	−0.829418
$925$	5782.29	0.205536
$926$	19696.0	0.698975
$927$	0	0
$928$	−704.000	−0.0249029
$929$	16266.0	0.574457	0.287228	−	0.957862i	$-0.407266\pi$
$929$	16266.0	0.574457	0.287228	+	0.957862i	$0.407266\pi$
$930$	−24149.0	−0.851479
$931$	6700.11	0.235862
$932$	3048.00	0.107125
$933$	11520.0	0.404231
$934$	−11972.5	−0.419435
$935$	14848.3	0.519351
$936$	3696.00	0.129068
$937$	11013.9	0.384000	0.192000	−	0.981395i	$-0.438503\pi$
$937$	11013.9	0.384000	0.192000	+	0.981395i	$0.438503\pi$
$938$	−26208.0	−0.912283
$939$	−9055.86	−0.314725
$940$	−7505.76	−0.260437
$941$	42393.2	1.46863	0.734315	−	0.678809i	$-0.237504\pi$
$941$	42393.2	1.46863	0.734315	+	0.678809i	$0.237504\pi$
$942$	244.753	0.00846548
$943$	0	0
$944$	−8000.00	−0.275824
$945$	31616.0	1.08833
$946$	45136.0	1.55127
$947$	−3428.00	−0.117629	−0.0588147	−	0.998269i	$-0.518732\pi$
$947$	−3428.00	−0.117629	−0.0588147	+	0.998269i	$0.518732\pi$
$948$	−17295.9	−0.592557
$949$	8820.00	0.301696
$950$	−3854.86	−0.131651
$951$	33816.0	1.15306
$952$	3328.00	0.113299
$953$	−21191.5	−0.720316	−0.360158	−	0.932891i	$-0.617277\pi$
$953$	−21191.5	−0.720316	−0.360158	+	0.932891i	$0.617277\pi$
$954$	−2019.21	−0.0685266
$955$	40976.0	1.38843
$956$	−544.000	−0.0184040
$957$	6281.99	0.212192
$958$	5302.98	0.178843
$959$	50336.0	1.69493
$960$	−2610.70	−0.0877707
$961$	57825.0	1.94102
$962$	23129.2	0.775170
$963$	−1458.32	−0.0487993
$964$	5792.49	0.193531
$965$	−16989.9	−0.566762
$966$	0	0
$967$	−3568.00	−0.118655	−0.0593274	−	0.998239i	$-0.518896\pi$
$967$	−3568.00	−0.118655	−0.0593274	+	0.998239i	$0.518896\pi$
$968$	30120.0	1.00010
$969$	−7488.00	−0.248245
$970$	25376.0	0.839973
$971$	31195.8	1.03102	0.515510	−	0.856883i	$-0.327602\pi$
$971$	31195.8	1.03102	0.515510	+	0.856883i	$0.327602\pi$
$972$	−11440.0	−0.377508
$973$	−46747.8	−1.54025
$974$	18656.0	0.613734
$975$	−3528.00	−0.115884
$976$	489.506	0.0160540
$977$	−11401.4	−0.373350	−0.186675	−	0.982422i	$-0.559771\pi$
$977$	−11401.4	−0.373350	−0.186675	+	0.982422i	$0.559771\pi$
$978$	−24928.0	−0.815040
$979$	−58240.0	−1.90129
$980$	2977.83	0.0970645
$981$	13124.9	0.427161
$982$	2664.00	0.0865699
$983$	−34326.6	−1.11378	−0.556891	−	0.830585i	$-0.688006\pi$
$983$	−34326.6	−1.11378	−0.556891	+	0.830585i	$0.688006\pi$
$984$	10176.0	0.329674
$985$	−22333.7	−0.722448
$986$	−897.427	−0.0289857
$987$	15011.5	0.484115
$988$	−15419.4	−0.496516
$989$	0	0
$990$	−16016.0	−0.514164
$991$	−10376.0	−0.332598	−0.166299	−	0.986075i	$-0.553182\pi$
$991$	−10376.0	−0.332598	−0.166299	+	0.986075i	$0.553182\pi$
$992$	9472.00	0.303162
$993$	40208.0	1.28496
$994$	9137.44	0.291572
$995$	9360.00	0.298223
$996$	−18438.1	−0.586578
$997$	25922.0	0.823428	0.411714	−	0.911313i	$-0.364930\pi$
$997$	25922.0	0.823428	0.411714	+	0.911313i	$0.364930\pi$
$998$	−12616.0	−0.400153
$999$	−41852.8	−1.32549

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1058.4.a.i.1.2	yes	2
23.22	odd	2	inner	1058.4.a.i.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1058.4.a.i.1.1	✓	2	23.22	odd	2	inner
1058.4.a.i.1.2	yes	2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1058 = 2 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1058.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -4.00000 q^{3} +4.00000 q^{4} +10.1980 q^{5} -8.00000 q^{6} +20.3961 q^{7} +8.00000 q^{8} -11.0000 q^{9} +20.3961 q^{10} +71.3863 q^{11} -16.0000 q^{12} -42.0000 q^{13} +40.7922 q^{14} -40.7922 q^{15} +16.0000 q^{16} +20.3961 q^{17} -22.0000 q^{18} +91.7824 q^{19} +40.7922 q^{20} -81.5843 q^{21} +142.773 q^{22} -32.0000 q^{24} -21.0000 q^{25} -84.0000 q^{26} +152.000 q^{27} +81.5843 q^{28} -22.0000 q^{29} -81.5843 q^{30} +296.000 q^{31} +32.0000 q^{32} -285.545 q^{33} +40.7922 q^{34} +208.000 q^{35} -44.0000 q^{36} -275.347 q^{37} +183.565 q^{38} +168.000 q^{39} +81.5843 q^{40} -318.000 q^{41} -163.169 q^{42} +316.139 q^{43} +285.545 q^{44} -112.178 q^{45} -184.000 q^{47} -64.0000 q^{48} +73.0000 q^{49} -42.0000 q^{50} -81.5843 q^{51} -168.000 q^{52} +91.7824 q^{53} +304.000 q^{54} +728.000 q^{55} +163.169 q^{56} -367.129 q^{57} -44.0000 q^{58} -500.000 q^{59} -163.169 q^{60} +30.5941 q^{61} +592.000 q^{62} -224.357 q^{63} +64.0000 q^{64} -428.318 q^{65} -571.090 q^{66} -642.476 q^{67} +81.5843 q^{68} +416.000 q^{70} +224.000 q^{71} -88.0000 q^{72} -210.000 q^{73} -550.694 q^{74} +84.0000 q^{75} +367.129 q^{76} +1456.00 q^{77} +336.000 q^{78} +1080.99 q^{79} +163.169 q^{80} -311.000 q^{81} -636.000 q^{82} +1152.38 q^{83} -326.337 q^{84} +208.000 q^{85} +632.278 q^{86} +88.0000 q^{87} +571.090 q^{88} -815.843 q^{89} -224.357 q^{90} -856.635 q^{91} -1184.00 q^{93} -368.000 q^{94} +936.000 q^{95} -128.000 q^{96} +1244.16 q^{97} +146.000 q^{98} -785.249 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} - 8 q^{3} + 8 q^{4} - 16 q^{6} + 16 q^{8} - 22 q^{9} - 32 q^{12} - 84 q^{13} + 32 q^{16} - 44 q^{18} - 64 q^{24} - 42 q^{25} - 168 q^{26} + 304 q^{27} - 44 q^{29} + 592 q^{31} + 64 q^{32}+ \cdots + 292 q^{98}+O(q^{100}) \) 2 * q + 4 * q^2 - 8 * q^3 + 8 * q^4 - 16 * q^6 + 16 * q^8 - 22 * q^9 - 32 * q^12 - 84 * q^13 + 32 * q^16 - 44 * q^18 - 64 * q^24 - 42 * q^25 - 168 * q^26 + 304 * q^27 - 44 * q^29 + 592 * q^31 + 64 * q^32 + 416 * q^35 - 88 * q^36 + 336 * q^39 - 636 * q^41 - 368 * q^47 - 128 * q^48 + 146 * q^49 - 84 * q^50 - 336 * q^52 + 608 * q^54 + 1456 * q^55 - 88 * q^58 - 1000 * q^59 + 1184 * q^62 + 128 * q^64 + 832 * q^70 + 448 * q^71 - 176 * q^72 - 420 * q^73 + 168 * q^75 + 2912 * q^77 + 672 * q^78 - 622 * q^81 - 1272 * q^82 + 416 * q^85 + 176 * q^87 - 2368 * q^93 - 736 * q^94 + 1872 * q^95 - 256 * q^96 + 292 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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