LMFDB - Embedded newform 1400.4.g.l.449.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1400,4,Mod(449,1400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

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

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

chi := DirichletCharacter("1400.449");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1400 = 2^{3} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1400.g (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$82.6026740080$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.7807489600.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 63x^{4} + 1041x^{2} + 1600 $ x^6 + 63x^4 + 1041x^2 + 1600
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 280)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.2
Root			$1.30726i$ of defining polynomial
Character	$\chi$	$=$	1400.449
Dual form			1400.4.g.l.449.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.45643i q^{3} +7.00000i q^{7} +15.0531 q^{9} +O(q^{10})$	$q-3.45643i q^{3} +7.00000i q^{7} +15.0531 q^{9} -51.8821 q^{11} +5.91120i q^{13} -103.850i q^{17} -118.470 q^{19} +24.1950 q^{21} +80.7029i q^{23} -145.354i q^{27} +218.344 q^{29} +39.2738 q^{31} +179.327i q^{33} +159.803i q^{37} +20.4316 q^{39} +233.322 q^{41} +524.515i q^{43} -210.441i q^{47} -49.0000 q^{49} -358.949 q^{51} -348.028i q^{53} +409.485i q^{57} +85.1752 q^{59} -801.720 q^{61} +105.372i q^{63} +716.049i q^{67} +278.944 q^{69} -520.412 q^{71} -230.005i q^{73} -363.175i q^{77} +970.534 q^{79} -95.9705 q^{81} +1109.38i q^{83} -754.689i q^{87} -1127.45 q^{89} -41.3784 q^{91} -135.747i q^{93} +1136.37i q^{97} -780.987 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 50 q^{9}+O(q^{10}) $ 6 * q - 50 * q^9	$ 6 q - 50 q^{9} - 12 q^{11} + 304 q^{19} - 84 q^{21} + 936 q^{29} - 536 q^{31} + 1348 q^{39} - 148 q^{41} - 294 q^{49} - 732 q^{51} + 2176 q^{59} + 348 q^{61} + 3568 q^{69} - 2112 q^{71} - 724 q^{79} + 5158 q^{81} - 316 q^{89} + 112 q^{91} + 2544 q^{99}+O(q^{100}) $ 6 * q - 50 * q^9 - 12 * q^11 + 304 * q^19 - 84 * q^21 + 936 * q^29 - 536 * q^31 + 1348 * q^39 - 148 * q^41 - 294 * q^49 - 732 * q^51 + 2176 * q^59 + 348 * q^61 + 3568 * q^69 - 2112 * q^71 - 724 * q^79 + 5158 * q^81 - 316 * q^89 + 112 * q^91 + 2544 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times$.

$n$	$351$	$701$	$801$	$1177$
$\chi(n)$	$1$	$1$	$1$	$-1$

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	0
$2$	0	0
$3$	− 3.45643i	− 0.665190i	−0.943070	−	0.332595i	$-0.892076\pi$
$3$	− 3.45643i	− 0.665190i	0.943070	−	0.332595i	$-0.107924\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.00000i	0.377964i
$7$	7.00000i	0.377964i
$8$	0	0
$9$	15.0531	0.557522
$10$	0	0
$11$	−51.8821	−1.42210	−0.711048	−	0.703144i	$-0.751779\pi$
$11$	−51.8821	−1.42210	−0.711048	+	0.703144i	$0.751779\pi$
$12$	0	0
$13$	5.91120i	0.126113i	0.998010	+	0.0630566i	$0.0200849\pi$
$13$	5.91120i	0.126113i	−0.998010	+	0.0630566i	$0.979915\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 103.850i	− 1.48160i	−0.671724	−	0.740802i	$-0.734446\pi$
$17$	− 103.850i	− 1.48160i	0.671724	−	0.740802i	$-0.265554\pi$
$18$	0	0
$19$	−118.470	−1.43047	−0.715236	−	0.698883i	$-0.753681\pi$
$19$	−118.470	−1.43047	−0.715236	+	0.698883i	$0.753681\pi$
$20$	0	0
$21$	24.1950	0.251418
$22$	0	0
$23$	80.7029i	0.731640i	0.930686	+	0.365820i	$0.119211\pi$
$23$	80.7029i	0.731640i	−0.930686	+	0.365820i	$0.880789\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 145.354i	− 1.03605i
$28$	0	0
$29$	218.344	1.39812	0.699058	−	0.715065i	$-0.253603\pi$
$29$	218.344	1.39812	0.699058	+	0.715065i	$0.253603\pi$
$30$	0	0
$31$	39.2738	0.227541	0.113771	−	0.993507i	$-0.463707\pi$
$31$	39.2738	0.227541	0.113771	+	0.993507i	$0.463707\pi$
$32$	0	0
$33$	179.327i	0.945964i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	159.803i	0.710040i	0.934859	+	0.355020i	$0.115526\pi$
$37$	159.803i	0.710040i	−0.934859	+	0.355020i	$0.884474\pi$
$38$	0	0
$39$	20.4316	0.0838892
$40$	0	0
$41$	233.322	0.888750	0.444375	−	0.895841i	$-0.353426\pi$
$41$	233.322	0.888750	0.444375	+	0.895841i	$0.353426\pi$
$42$	0	0
$43$	524.515i	1.86018i	0.367329	+	0.930091i	$0.380272\pi$
$43$	524.515i	1.86018i	−0.367329	+	0.930091i	$0.619728\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 210.441i	− 0.653105i	−0.945179	−	0.326552i	$-0.894113\pi$
$47$	− 210.441i	− 0.653105i	0.945179	−	0.326552i	$-0.105887\pi$
$48$	0	0
$49$	−49.0000	−0.142857
$50$	0	0
$51$	−358.949	−0.985548
$52$	0	0
$53$	− 348.028i	− 0.901987i	−0.892527	−	0.450994i	$-0.851070\pi$
$53$	− 348.028i	− 0.901987i	0.892527	−	0.450994i	$-0.148930\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	409.485i	0.951536i
$58$	0	0
$59$	85.1752	0.187947	0.0939735	−	0.995575i	$-0.470043\pi$
$59$	85.1752	0.187947	0.0939735	+	0.995575i	$0.470043\pi$
$60$	0	0
$61$	−801.720	−1.68278	−0.841391	−	0.540427i	$-0.818263\pi$
$61$	−801.720	−1.68278	−0.841391	+	0.540427i	$0.818263\pi$
$62$	0	0
$63$	105.372i	0.210724i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	716.049i	1.30566i	0.757504	+	0.652831i	$0.226419\pi$
$67$	716.049i	1.30566i	−0.757504	+	0.652831i	$0.773581\pi$
$68$	0	0
$69$	278.944	0.486679
$70$	0	0
$71$	−520.412	−0.869880	−0.434940	−	0.900459i	$-0.643230\pi$
$71$	−520.412	−0.869880	−0.434940	+	0.900459i	$0.643230\pi$
$72$	0	0
$73$	− 230.005i	− 0.368768i	−0.982854	−	0.184384i	$-0.940971\pi$
$73$	− 230.005i	− 0.368768i	0.982854	−	0.184384i	$-0.0590289\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 363.175i	− 0.537502i
$78$	0	0
$79$	970.534	1.38220	0.691099	−	0.722760i	$-0.257127\pi$
$79$	970.534	1.38220	0.691099	+	0.722760i	$0.257127\pi$
$80$	0	0
$81$	−95.9705	−0.131647
$82$	0	0
$83$	1109.38i	1.46711i	0.679630	+	0.733555i	$0.262140\pi$
$83$	1109.38i	1.46711i	−0.679630	+	0.733555i	$0.737860\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 754.689i	− 0.930013i
$88$	0	0
$89$	−1127.45	−1.34280	−0.671401	−	0.741094i	$-0.734307\pi$
$89$	−1127.45	−1.34280	−0.671401	+	0.741094i	$0.734307\pi$
$90$	0	0
$91$	−41.3784	−0.0476663
$92$	0	0
$93$	− 135.747i	− 0.151358i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1136.37i	1.18950i	0.803912	+	0.594748i	$0.202748\pi$
$97$	1136.37i	1.18950i	−0.803912	+	0.594748i	$0.797252\pi$
$98$	0	0
$99$	−780.987	−0.792850
$100$	0	0
$101$	858.838	0.846115	0.423057	−	0.906103i	$-0.360957\pi$
$101$	858.838	0.846115	0.423057	+	0.906103i	$0.360957\pi$
$102$	0	0
$103$	348.379i	0.333270i	0.986019	+	0.166635i	$0.0532901\pi$
$103$	348.379i	0.333270i	−0.986019	+	0.166635i	$0.946710\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1860.54i	1.68098i	0.541828	+	0.840490i	$0.317733\pi$
$107$	1860.54i	1.68098i	−0.541828	+	0.840490i	$0.682267\pi$
$108$	0	0
$109$	−1213.61	−1.06645	−0.533224	−	0.845974i	$-0.679020\pi$
$109$	−1213.61	−1.06645	−0.533224	+	0.845974i	$0.679020\pi$
$110$	0	0
$111$	552.349	0.472312
$112$	0	0
$113$	267.531i	0.222718i	0.993780	+	0.111359i	$0.0355204\pi$
$113$	267.531i	0.222718i	−0.993780	+	0.111359i	$0.964480\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	88.9819i	0.0703109i
$118$	0	0
$119$	726.948	0.559994
$120$	0	0
$121$	1360.76	1.02236
$122$	0	0
$123$	− 806.460i	− 0.591188i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 712.388i	− 0.497750i	−0.968536	−	0.248875i	$-0.919939\pi$
$127$	− 712.388i	− 0.497750i	0.968536	−	0.248875i	$-0.0800608\pi$
$128$	0	0
$129$	1812.95	1.23737
$130$	0	0
$131$	746.364	0.497787	0.248894	−	0.968531i	$-0.419933\pi$
$131$	746.364	0.497787	0.248894	+	0.968531i	$0.419933\pi$
$132$	0	0
$133$	− 829.293i	− 0.540668i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2612.55i	1.62924i	0.579996	+	0.814619i	$0.303054\pi$
$137$	2612.55i	1.62924i	−0.579996	+	0.814619i	$0.696946\pi$
$138$	0	0
$139$	−637.320	−0.388898	−0.194449	−	0.980913i	$-0.562292\pi$
$139$	−637.320	−0.388898	−0.194449	+	0.980913i	$0.562292\pi$
$140$	0	0
$141$	−727.373	−0.434439
$142$	0	0
$143$	− 306.686i	− 0.179345i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	169.365i	0.0950271i
$148$	0	0
$149$	2151.92	1.18317	0.591585	−	0.806242i	$-0.298502\pi$
$149$	2151.92	1.18317	0.591585	+	0.806242i	$0.298502\pi$
$150$	0	0
$151$	1685.18	0.908201	0.454101	−	0.890950i	$-0.349961\pi$
$151$	1685.18	0.908201	0.454101	+	0.890950i	$0.349961\pi$
$152$	0	0
$153$	− 1563.26i	− 0.826027i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1139.31i	0.579150i	0.957155	+	0.289575i	$0.0935140\pi$
$157$	1139.31i	0.579150i	−0.957155	+	0.289575i	$0.906486\pi$
$158$	0	0
$159$	−1202.93	−0.599993
$160$	0	0
$161$	−564.920	−0.276534
$162$	0	0
$163$	1758.81i	0.845156i	0.906327	+	0.422578i	$0.138875\pi$
$163$	1758.81i	0.845156i	−0.906327	+	0.422578i	$0.861125\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2350.50i	1.08915i	0.838713	+	0.544573i	$0.183308\pi$
$167$	2350.50i	1.08915i	−0.838713	+	0.544573i	$0.816692\pi$
$168$	0	0
$169$	2162.06	0.984095
$170$	0	0
$171$	−1783.35	−0.797520
$172$	0	0
$173$	2403.94i	1.05646i	0.849100	+	0.528231i	$0.177145\pi$
$173$	2403.94i	1.05646i	−0.849100	+	0.528231i	$0.822855\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 294.402i	− 0.125020i
$178$	0	0
$179$	3361.51	1.40364	0.701819	−	0.712355i	$-0.252372\pi$
$179$	3361.51	1.40364	0.701819	+	0.712355i	$0.252372\pi$
$180$	0	0
$181$	−4681.04	−1.92231	−0.961156	−	0.276004i	$-0.910990\pi$
$181$	−4681.04	−1.92231	−0.961156	+	0.276004i	$0.910990\pi$
$182$	0	0
$183$	2771.09i	1.11937i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	5387.95i	2.10698i
$188$	0	0
$189$	1017.47	0.391589
$190$	0	0
$191$	2375.90	0.900075	0.450037	−	0.893010i	$-0.351411\pi$
$191$	2375.90	0.900075	0.450037	+	0.893010i	$0.351411\pi$
$192$	0	0
$193$	− 4940.87i	− 1.84275i	−0.388670	−	0.921377i	$-0.627065\pi$
$193$	− 4940.87i	− 1.84275i	0.388670	−	0.921377i	$-0.372935\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1335.30i	0.482926i	0.970410	+	0.241463i	$0.0776272\pi$
$197$	1335.30i	0.482926i	−0.970410	+	0.241463i	$0.922373\pi$
$198$	0	0
$199$	2235.61	0.796374	0.398187	−	0.917304i	$-0.369640\pi$
$199$	2235.61	0.796374	0.398187	+	0.917304i	$0.369640\pi$
$200$	0	0
$201$	2474.97	0.868513
$202$	0	0
$203$	1528.40i	0.528438i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1214.83i	0.407905i
$208$	0	0
$209$	6146.50	2.03427
$210$	0	0
$211$	−875.868	−0.285769	−0.142884	−	0.989739i	$-0.545638\pi$
$211$	−875.868	−0.285769	−0.142884	+	0.989739i	$0.545638\pi$
$212$	0	0
$213$	1798.77i	0.578636i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	274.917i	0.0860025i
$218$	0	0
$219$	−794.995	−0.245300
$220$	0	0
$221$	613.877	0.186850
$222$	0	0
$223$	3950.23i	1.18622i	0.805122	+	0.593110i	$0.202100\pi$
$223$	3950.23i	1.18622i	−0.805122	+	0.593110i	$0.797900\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 2254.32i	− 0.659138i	−0.944132	−	0.329569i	$-0.893097\pi$
$227$	− 2254.32i	− 0.659138i	0.944132	−	0.329569i	$-0.106903\pi$
$228$	0	0
$229$	3680.17	1.06198	0.530988	−	0.847379i	$-0.321821\pi$
$229$	3680.17	1.06198	0.530988	+	0.847379i	$0.321821\pi$
$230$	0	0
$231$	−1255.29	−0.357541
$232$	0	0
$233$	1109.74i	0.312024i	0.987755	+	0.156012i	$0.0498638\pi$
$233$	1109.74i	0.312024i	−0.987755	+	0.156012i	$0.950136\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 3354.58i	− 0.919424i
$238$	0	0
$239$	−1079.71	−0.292220	−0.146110	−	0.989268i	$-0.546675\pi$
$239$	−1079.71	−0.292220	−0.146110	+	0.989268i	$0.546675\pi$
$240$	0	0
$241$	81.1127	0.0216802	0.0108401	−	0.999941i	$-0.496549\pi$
$241$	81.1127	0.0216802	0.0108401	+	0.999941i	$0.496549\pi$
$242$	0	0
$243$	− 3592.83i	− 0.948478i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 700.302i	− 0.180402i
$248$	0	0
$249$	3834.49	0.975907
$250$	0	0
$251$	−2193.90	−0.551704	−0.275852	−	0.961200i	$-0.588960\pi$
$251$	−2193.90	−0.551704	−0.275852	+	0.961200i	$0.588960\pi$
$252$	0	0
$253$	− 4187.04i	− 1.04046i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 3933.07i	− 0.954623i	−0.878734	−	0.477312i	$-0.841611\pi$
$257$	− 3933.07i	− 0.954623i	0.878734	−	0.477312i	$-0.158389\pi$
$258$	0	0
$259$	−1118.62	−0.268370
$260$	0	0
$261$	3286.75	0.779481
$262$	0	0
$263$	− 497.852i	− 0.116726i	−0.998295	−	0.0583629i	$-0.981412\pi$
$263$	− 497.852i	− 0.116726i	0.998295	−	0.0583629i	$-0.0185880\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	3896.95i	0.893219i
$268$	0	0
$269$	7672.17	1.73896	0.869481	−	0.493967i	$-0.164454\pi$
$269$	7672.17	1.73896	0.869481	+	0.493967i	$0.164454\pi$
$270$	0	0
$271$	−3392.88	−0.760526	−0.380263	−	0.924878i	$-0.624167\pi$
$271$	−3392.88	−0.760526	−0.380263	+	0.924878i	$0.624167\pi$
$272$	0	0
$273$	143.021i	0.0317072i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2314.21i	0.501975i	0.967990	+	0.250988i	$0.0807553\pi$
$277$	2314.21i	0.501975i	−0.967990	+	0.250988i	$0.919245\pi$
$278$	0	0
$279$	591.192	0.126859
$280$	0	0
$281$	−3604.79	−0.765280	−0.382640	−	0.923897i	$-0.624985\pi$
$281$	−3604.79	−0.765280	−0.382640	+	0.923897i	$0.624985\pi$
$282$	0	0
$283$	2304.38i	0.484032i	0.970272	+	0.242016i	$0.0778087\pi$
$283$	2304.38i	0.484032i	−0.970272	+	0.242016i	$0.922191\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1633.25i	0.335916i
$288$	0	0
$289$	−5871.77	−1.19515
$290$	0	0
$291$	3927.79	0.791241
$292$	0	0
$293$	6680.29i	1.33197i	0.745966	+	0.665984i	$0.231988\pi$
$293$	6680.29i	1.33197i	−0.745966	+	0.665984i	$0.768012\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	7541.25i	1.47336i
$298$	0	0
$299$	−477.051	−0.0922694
$300$	0	0
$301$	−3671.61	−0.703083
$302$	0	0
$303$	− 2968.51i	− 0.562827i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 3532.96i	− 0.656798i	−0.944539	−	0.328399i	$-0.893491\pi$
$307$	− 3532.96i	− 0.656798i	0.944539	−	0.328399i	$-0.106509\pi$
$308$	0	0
$309$	1204.15	0.221688
$310$	0	0
$311$	−7713.50	−1.40641	−0.703204	−	0.710989i	$-0.748248\pi$
$311$	−7713.50	−1.40641	−0.703204	+	0.710989i	$0.748248\pi$
$312$	0	0
$313$	− 5776.97i	− 1.04324i	−0.853178	−	0.521619i	$-0.825328\pi$
$313$	− 5776.97i	− 1.04324i	0.853178	−	0.521619i	$-0.174672\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2879.80i	0.510239i	0.966910	+	0.255120i	$0.0821148\pi$
$317$	2879.80i	0.510239i	−0.966910	+	0.255120i	$0.917885\pi$
$318$	0	0
$319$	−11328.1	−1.98826
$320$	0	0
$321$	6430.81	1.11817
$322$	0	0
$323$	12303.1i	2.11939i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	4194.76i	0.709391i
$328$	0	0
$329$	1473.08	0.246850
$330$	0	0
$331$	−2784.80	−0.462437	−0.231218	−	0.972902i	$-0.574271\pi$
$331$	−2784.80	−0.462437	−0.231218	+	0.972902i	$0.574271\pi$
$332$	0	0
$333$	2405.53i	0.395863i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2009.52i	0.324824i	0.986723	+	0.162412i	$0.0519273\pi$
$337$	2009.52i	0.324824i	−0.986723	+	0.162412i	$0.948073\pi$
$338$	0	0
$339$	924.701	0.148150
$340$	0	0
$341$	−2037.61	−0.323586
$342$	0	0
$343$	− 343.000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5117.20i	0.791659i	0.918324	+	0.395829i	$0.129543\pi$
$347$	5117.20i	0.791659i	−0.918324	+	0.395829i	$0.870457\pi$
$348$	0	0
$349$	9330.98	1.43116	0.715581	−	0.698530i	$-0.246162\pi$
$349$	9330.98	1.43116	0.715581	+	0.698530i	$0.246162\pi$
$350$	0	0
$351$	859.214	0.130659
$352$	0	0
$353$	1549.68i	0.233657i	0.993152	+	0.116829i	$0.0372728\pi$
$353$	1549.68i	0.233657i	−0.993152	+	0.116829i	$0.962727\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 2512.64i	− 0.372502i
$358$	0	0
$359$	−9513.09	−1.39856	−0.699278	−	0.714850i	$-0.746495\pi$
$359$	−9513.09	−1.39856	−0.699278	+	0.714850i	$0.746495\pi$
$360$	0	0
$361$	7176.25	1.04625
$362$	0	0
$363$	− 4703.35i	− 0.680061i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 6441.14i	− 0.916144i	−0.888915	−	0.458072i	$-0.848540\pi$
$367$	− 6441.14i	− 0.916144i	0.888915	−	0.458072i	$-0.151460\pi$
$368$	0	0
$369$	3512.22	0.495498
$370$	0	0
$371$	2436.20	0.340919
$372$	0	0
$373$	7428.14i	1.03114i	0.856848	+	0.515569i	$0.172419\pi$
$373$	7428.14i	1.03114i	−0.856848	+	0.515569i	$0.827581\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1290.67i	0.176321i
$378$	0	0
$379$	10369.5	1.40539	0.702696	−	0.711491i	$-0.251980\pi$
$379$	10369.5	1.40539	0.702696	+	0.711491i	$0.251980\pi$
$380$	0	0
$381$	−2462.32	−0.331098
$382$	0	0
$383$	− 5924.21i	− 0.790374i	−0.918601	−	0.395187i	$-0.870680\pi$
$383$	− 5924.21i	− 0.790374i	0.918601	−	0.395187i	$-0.129320\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	7895.58i	1.03709i
$388$	0	0
$389$	−1330.48	−0.173413	−0.0867067	−	0.996234i	$-0.527634\pi$
$389$	−1330.48	−0.173413	−0.0867067	+	0.996234i	$0.527634\pi$
$390$	0	0
$391$	8380.97	1.08400
$392$	0	0
$393$	− 2579.75i	− 0.331123i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	10355.1i	1.30909i	0.756023	+	0.654544i	$0.227140\pi$
$397$	10355.1i	1.30909i	−0.756023	+	0.654544i	$0.772860\pi$
$398$	0	0
$399$	−2866.39	−0.359647
$400$	0	0
$401$	3844.38	0.478751	0.239376	−	0.970927i	$-0.423057\pi$
$401$	3844.38	0.478751	0.239376	+	0.970927i	$0.423057\pi$
$402$	0	0
$403$	232.155i	0.0286960i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 8290.93i	− 1.00975i
$408$	0	0
$409$	−8351.01	−1.00961	−0.504805	−	0.863233i	$-0.668436\pi$
$409$	−8351.01	−1.00961	−0.504805	+	0.863233i	$0.668436\pi$
$410$	0	0
$411$	9030.11	1.08375
$412$	0	0
$413$	596.227i	0.0710373i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	2202.85i	0.258691i
$418$	0	0
$419$	1966.13	0.229240	0.114620	−	0.993409i	$-0.463435\pi$
$419$	1966.13	0.229240	0.114620	+	0.993409i	$0.463435\pi$
$420$	0	0
$421$	12154.0	1.40700	0.703501	−	0.710694i	$-0.251619\pi$
$421$	12154.0	1.40700	0.703501	+	0.710694i	$0.251619\pi$
$422$	0	0
$423$	− 3167.78i	− 0.364120i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 5612.04i	− 0.636032i
$428$	0	0
$429$	−1060.04	−0.119299
$430$	0	0
$431$	−15045.0	−1.68143	−0.840713	−	0.541481i	$-0.817864\pi$
$431$	−15045.0	−1.68143	−0.840713	+	0.541481i	$0.817864\pi$
$432$	0	0
$433$	− 1945.75i	− 0.215950i	−0.994154	−	0.107975i	$-0.965563\pi$
$433$	− 1945.75i	− 0.215950i	0.994154	−	0.107975i	$-0.0344367\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 9560.91i	− 1.04659i
$438$	0	0
$439$	−8484.35	−0.922405	−0.461203	−	0.887295i	$-0.652582\pi$
$439$	−8484.35	−0.922405	−0.461203	+	0.887295i	$0.652582\pi$
$440$	0	0
$441$	−737.602	−0.0796460
$442$	0	0
$443$	14875.3i	1.59537i	0.603078	+	0.797683i	$0.293941\pi$
$443$	14875.3i	1.59537i	−0.603078	+	0.797683i	$0.706059\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 7437.97i	− 0.787033i
$448$	0	0
$449$	−17995.8	−1.89148	−0.945742	−	0.324918i	$-0.894663\pi$
$449$	−17995.8	−1.89148	−0.945742	+	0.324918i	$0.894663\pi$
$450$	0	0
$451$	−12105.2	−1.26389
$452$	0	0
$453$	− 5824.72i	− 0.604126i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4243.31i	0.434340i	0.976134	+	0.217170i	$0.0696827\pi$
$457$	4243.31i	0.434340i	−0.976134	+	0.217170i	$0.930317\pi$
$458$	0	0
$459$	−15094.9	−1.53501
$460$	0	0
$461$	−11234.8	−1.13504	−0.567522	−	0.823358i	$-0.692098\pi$
$461$	−11234.8	−1.13504	−0.567522	+	0.823358i	$0.692098\pi$
$462$	0	0
$463$	226.775i	0.0227627i	0.999935	+	0.0113814i	$0.00362288\pi$
$463$	226.775i	0.0227627i	−0.999935	+	0.0113814i	$0.996377\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 11886.0i	− 1.17777i	−0.808217	−	0.588885i	$-0.799567\pi$
$467$	− 11886.0i	− 1.17777i	0.808217	−	0.588885i	$-0.200433\pi$
$468$	0	0
$469$	−5012.35	−0.493494
$470$	0	0
$471$	3937.93	0.385245
$472$	0	0
$473$	− 27213.0i	− 2.64536i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 5238.90i	− 0.502878i
$478$	0	0
$479$	−5052.68	−0.481969	−0.240984	−	0.970529i	$-0.577470\pi$
$479$	−5052.68	−0.481969	−0.240984	+	0.970529i	$0.577470\pi$
$480$	0	0
$481$	−944.629	−0.0895455
$482$	0	0
$483$	1952.61i	0.183948i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	3987.21i	0.371001i	0.982644	+	0.185501i	$0.0593907\pi$
$487$	3987.21i	0.371001i	−0.982644	+	0.185501i	$0.940609\pi$
$488$	0	0
$489$	6079.19	0.562189
$490$	0	0
$491$	−4969.66	−0.456777	−0.228389	−	0.973570i	$-0.573346\pi$
$491$	−4969.66	−0.456777	−0.228389	+	0.973570i	$0.573346\pi$
$492$	0	0
$493$	− 22674.9i	− 2.07145i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 3642.88i	− 0.328784i
$498$	0	0
$499$	−11651.2	−1.04525	−0.522625	−	0.852562i	$-0.675047\pi$
$499$	−11651.2	−1.04525	−0.522625	+	0.852562i	$0.675047\pi$
$500$	0	0
$501$	8124.35	0.724490
$502$	0	0
$503$	− 1024.77i	− 0.0908398i	−0.998968	−	0.0454199i	$-0.985537\pi$
$503$	− 1024.77i	− 0.0908398i	0.998968	−	0.0454199i	$-0.0144626\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 7473.00i	− 0.654610i
$508$	0	0
$509$	6350.40	0.552999	0.276500	−	0.961014i	$-0.410826\pi$
$509$	6350.40	0.552999	0.276500	+	0.961014i	$0.410826\pi$
$510$	0	0
$511$	1610.03	0.139381
$512$	0	0
$513$	17220.1i	1.48204i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	10918.1i	0.928778i
$518$	0	0
$519$	8309.04	0.702749
$520$	0	0
$521$	11664.5	0.980864	0.490432	−	0.871479i	$-0.336839\pi$
$521$	11664.5	0.980864	0.490432	+	0.871479i	$0.336839\pi$
$522$	0	0
$523$	− 2475.61i	− 0.206980i	−0.994630	−	0.103490i	$-0.966999\pi$
$523$	− 2475.61i	− 0.206980i	0.994630	−	0.103490i	$-0.0330011\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 4078.57i	− 0.337126i
$528$	0	0
$529$	5654.05	0.464703
$530$	0	0
$531$	1282.15	0.104785
$532$	0	0
$533$	1379.21i	0.112083i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 11618.8i	− 0.933686i
$538$	0	0
$539$	2542.22	0.203157
$540$	0	0
$541$	−5689.60	−0.452153	−0.226077	−	0.974110i	$-0.572590\pi$
$541$	−5689.60	−0.452153	−0.226077	+	0.974110i	$0.572590\pi$
$542$	0	0
$543$	16179.7i	1.27870i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 13796.6i	− 1.07843i	−0.842169	−	0.539213i	$-0.818722\pi$
$547$	− 13796.6i	− 1.07843i	0.842169	−	0.539213i	$-0.181278\pi$
$548$	0	0
$549$	−12068.4	−0.938188
$550$	0	0
$551$	−25867.3	−1.99997
$552$	0	0
$553$	6793.74i	0.522422i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 14995.2i	− 1.14069i	−0.821404	−	0.570347i	$-0.806809\pi$
$557$	− 14995.2i	− 1.14069i	0.821404	−	0.570347i	$-0.193191\pi$
$558$	0	0
$559$	−3100.51	−0.234593
$560$	0	0
$561$	18623.1	1.40154
$562$	0	0
$563$	7110.77i	0.532297i	0.963932	+	0.266149i	$0.0857512\pi$
$563$	7110.77i	0.532297i	−0.963932	+	0.266149i	$0.914249\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 671.793i	− 0.0497578i
$568$	0	0
$569$	19858.3	1.46310	0.731549	−	0.681789i	$-0.238798\pi$
$569$	19858.3	1.46310	0.731549	+	0.681789i	$0.238798\pi$
$570$	0	0
$571$	−8680.50	−0.636196	−0.318098	−	0.948058i	$-0.603044\pi$
$571$	−8680.50	−0.636196	−0.318098	+	0.948058i	$0.603044\pi$
$572$	0	0
$573$	− 8212.14i	− 0.598721i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 4639.54i	− 0.334743i	−0.985894	−	0.167372i	$-0.946472\pi$
$577$	− 4639.54i	− 0.334743i	0.985894	−	0.167372i	$-0.0535279\pi$
$578$	0	0
$579$	−17077.8	−1.22578
$580$	0	0
$581$	−7765.65	−0.554516
$582$	0	0
$583$	18056.4i	1.28271i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	11286.4i	0.793597i	0.917906	+	0.396799i	$0.129879\pi$
$587$	11286.4i	0.793597i	−0.917906	+	0.396799i	$0.870121\pi$
$588$	0	0
$589$	−4652.78	−0.325492
$590$	0	0
$591$	4615.38	0.321237
$592$	0	0
$593$	2434.90i	0.168616i	0.996440	+	0.0843082i	$0.0268680\pi$
$593$	2434.90i	0.168616i	−0.996440	+	0.0843082i	$0.973132\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 7727.24i	− 0.529740i
$598$	0	0
$599$	5668.75	0.386676	0.193338	−	0.981132i	$-0.438069\pi$
$599$	5668.75	0.386676	0.193338	+	0.981132i	$0.438069\pi$
$600$	0	0
$601$	22312.6	1.51439	0.757195	−	0.653189i	$-0.226569\pi$
$601$	22312.6	1.51439	0.757195	+	0.653189i	$0.226569\pi$
$602$	0	0
$603$	10778.8i	0.727936i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 13583.7i	− 0.908314i	−0.890922	−	0.454157i	$-0.849941\pi$
$607$	− 13583.7i	− 0.908314i	0.890922	−	0.454157i	$-0.150059\pi$
$608$	0	0
$609$	5282.82	0.351512
$610$	0	0
$611$	1243.96	0.0823651
$612$	0	0
$613$	− 6448.94i	− 0.424911i	−0.977171	−	0.212455i	$-0.931854\pi$
$613$	− 6448.94i	− 0.424911i	0.977171	−	0.212455i	$-0.0681460\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	4075.91i	0.265948i	0.991120	+	0.132974i	$0.0424526\pi$
$617$	4075.91i	0.265948i	−0.991120	+	0.132974i	$0.957547\pi$
$618$	0	0
$619$	17730.9	1.15131	0.575657	−	0.817691i	$-0.304746\pi$
$619$	17730.9	1.15131	0.575657	+	0.817691i	$0.304746\pi$
$620$	0	0
$621$	11730.4	0.758014
$622$	0	0
$623$	− 7892.14i	− 0.507531i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 21244.9i	− 1.35318i
$628$	0	0
$629$	16595.5	1.05200
$630$	0	0
$631$	−24687.2	−1.55750	−0.778748	−	0.627337i	$-0.784145\pi$
$631$	−24687.2	−1.55750	−0.778748	+	0.627337i	$0.784145\pi$
$632$	0	0
$633$	3027.38i	0.190091i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 289.649i	− 0.0180162i
$638$	0	0
$639$	−7833.81	−0.484977
$640$	0	0
$641$	−3640.27	−0.224309	−0.112154	−	0.993691i	$-0.535775\pi$
$641$	−3640.27	−0.224309	−0.112154	+	0.993691i	$0.535775\pi$
$642$	0	0
$643$	− 18809.9i	− 1.15364i	−0.816871	−	0.576821i	$-0.804293\pi$
$643$	− 18809.9i	− 1.15364i	0.816871	−	0.576821i	$-0.195707\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	7914.91i	0.480939i	0.970657	+	0.240469i	$0.0773014\pi$
$647$	7914.91i	0.480939i	−0.970657	+	0.240469i	$0.922699\pi$
$648$	0	0
$649$	−4419.07	−0.267279
$650$	0	0
$651$	950.229	0.0572080
$652$	0	0
$653$	− 20629.9i	− 1.23631i	−0.786056	−	0.618155i	$-0.787880\pi$
$653$	− 20629.9i	− 1.23631i	0.786056	−	0.618155i	$-0.212120\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 3462.29i	− 0.205596i
$658$	0	0
$659$	−20033.4	−1.18420	−0.592101	−	0.805864i	$-0.701701\pi$
$659$	−20033.4	−1.18420	−0.592101	+	0.805864i	$0.701701\pi$
$660$	0	0
$661$	−33181.9	−1.95254	−0.976268	−	0.216565i	$-0.930515\pi$
$661$	−33181.9	−1.95254	−0.976268	+	0.216565i	$0.930515\pi$
$662$	0	0
$663$	− 2121.82i	− 0.124291i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	17621.0i	1.02292i
$668$	0	0
$669$	13653.7	0.789061
$670$	0	0
$671$	41594.9	2.39308
$672$	0	0
$673$	− 9881.92i	− 0.566003i	−0.959119	−	0.283001i	$-0.908670\pi$
$673$	− 9881.92i	− 0.566003i	0.959119	−	0.283001i	$-0.0913301\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 2059.42i	− 0.116913i	−0.998290	−	0.0584563i	$-0.981382\pi$
$677$	− 2059.42i	− 0.116913i	0.998290	−	0.0584563i	$-0.0186178\pi$
$678$	0	0
$679$	−7954.61	−0.449587
$680$	0	0
$681$	−7791.89	−0.438452
$682$	0	0
$683$	− 29350.2i	− 1.64429i	−0.569275	−	0.822147i	$-0.692776\pi$
$683$	− 29350.2i	− 1.64429i	0.569275	−	0.822147i	$-0.307224\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 12720.3i	− 0.706416i
$688$	0	0
$689$	2057.26	0.113753
$690$	0	0
$691$	−11343.4	−0.624491	−0.312246	−	0.950001i	$-0.601081\pi$
$691$	−11343.4	−0.624491	−0.312246	+	0.950001i	$0.601081\pi$
$692$	0	0
$693$	− 5466.91i	− 0.299669i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 24230.4i	− 1.31678i
$698$	0	0
$699$	3835.74	0.207555
$700$	0	0
$701$	−16939.1	−0.912667	−0.456334	−	0.889809i	$-0.650838\pi$
$701$	−16939.1	−0.912667	−0.456334	+	0.889809i	$0.650838\pi$
$702$	0	0
$703$	− 18932.0i	− 1.01569i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6011.87i	0.319801i
$708$	0	0
$709$	−20813.0	−1.10247	−0.551233	−	0.834351i	$-0.685843\pi$
$709$	−20813.0	−1.10247	−0.551233	+	0.834351i	$0.685843\pi$
$710$	0	0
$711$	14609.5	0.770606
$712$	0	0
$713$	3169.51i	0.166478i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	3731.94i	0.194382i
$718$	0	0
$719$	−15526.6	−0.805348	−0.402674	−	0.915344i	$-0.631919\pi$
$719$	−15526.6	−0.805348	−0.402674	+	0.915344i	$0.631919\pi$
$720$	0	0
$721$	−2438.65	−0.125964
$722$	0	0
$723$	− 280.360i	− 0.0144214i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 18301.0i	− 0.933628i	−0.884356	−	0.466814i	$-0.845402\pi$
$727$	− 18301.0i	− 0.933628i	0.884356	−	0.466814i	$-0.154598\pi$
$728$	0	0
$729$	−15009.6	−0.762565
$730$	0	0
$731$	54470.8	2.75605
$732$	0	0
$733$	31175.7i	1.57094i	0.618897	+	0.785472i	$0.287580\pi$
$733$	31175.7i	1.57094i	−0.618897	+	0.785472i	$0.712420\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 37150.2i	− 1.85678i
$738$	0	0
$739$	38874.4	1.93507	0.967537	−	0.252730i	$-0.0813284\pi$
$739$	38874.4	1.93507	0.967537	+	0.252730i	$0.0813284\pi$
$740$	0	0
$741$	−2420.55	−0.120001
$742$	0	0
$743$	15899.8i	0.785071i	0.919737	+	0.392536i	$0.128402\pi$
$743$	15899.8i	0.785071i	−0.919737	+	0.392536i	$0.871598\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	16699.6i	0.817947i
$748$	0	0
$749$	−13023.7	−0.635350
$750$	0	0
$751$	20419.5	0.992168	0.496084	−	0.868275i	$-0.334771\pi$
$751$	20419.5	0.992168	0.496084	+	0.868275i	$0.334771\pi$
$752$	0	0
$753$	7583.05i	0.366988i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	33358.9i	1.60165i	0.598897	+	0.800826i	$0.295606\pi$
$757$	33358.9i	1.60165i	−0.598897	+	0.800826i	$0.704394\pi$
$758$	0	0
$759$	−14472.2	−0.692105
$760$	0	0
$761$	−7773.61	−0.370293	−0.185147	−	0.982711i	$-0.559276\pi$
$761$	−7773.61	−0.370293	−0.185147	+	0.982711i	$0.559276\pi$
$762$	0	0
$763$	− 8495.28i	− 0.403080i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	503.488i	0.0237026i
$768$	0	0
$769$	−30704.0	−1.43981	−0.719906	−	0.694072i	$-0.755815\pi$
$769$	−30704.0	−1.43981	−0.719906	+	0.694072i	$0.755815\pi$
$770$	0	0
$771$	−13594.4	−0.635006
$772$	0	0
$773$	− 31749.5i	− 1.47730i	−0.674091	−	0.738648i	$-0.735464\pi$
$773$	− 31749.5i	− 1.47730i	0.674091	−	0.738648i	$-0.264536\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	3866.44i	0.178517i
$778$	0	0
$779$	−27641.7	−1.27133
$780$	0	0
$781$	27000.1	1.23705
$782$	0	0
$783$	− 31737.0i	− 1.44852i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	4536.07i	0.205455i	0.994710	+	0.102728i	$0.0327570\pi$
$787$	4536.07i	0.205455i	−0.994710	+	0.102728i	$0.967243\pi$
$788$	0	0
$789$	−1720.79	−0.0776448
$790$	0	0
$791$	−1872.71	−0.0841796
$792$	0	0
$793$	− 4739.13i	− 0.212221i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 237.980i	− 0.0105768i	−0.999986	−	0.00528840i	$-0.998317\pi$
$797$	− 237.980i	− 0.0105768i	0.999986	−	0.00528840i	$-0.00168336\pi$
$798$	0	0
$799$	−21854.2	−0.967643
$800$	0	0
$801$	−16971.6	−0.748642
$802$	0	0
$803$	11933.1i	0.524423i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 26518.3i	− 1.15674i
$808$	0	0
$809$	33008.4	1.43450	0.717251	−	0.696815i	$-0.245400\pi$
$809$	33008.4	1.43450	0.717251	+	0.696815i	$0.245400\pi$
$810$	0	0
$811$	−16028.1	−0.693988	−0.346994	−	0.937867i	$-0.612798\pi$
$811$	−16028.1	−0.693988	−0.346994	+	0.937867i	$0.612798\pi$
$812$	0	0
$813$	11727.2i	0.505894i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 62139.6i	− 2.66094i
$818$	0	0
$819$	−622.873	−0.0265750
$820$	0	0
$821$	6121.99	0.260242	0.130121	−	0.991498i	$-0.458463\pi$
$821$	6121.99	0.260242	0.130121	+	0.991498i	$0.458463\pi$
$822$	0	0
$823$	− 10372.7i	− 0.439331i	−0.975575	−	0.219666i	$-0.929503\pi$
$823$	− 10372.7i	− 0.439331i	0.975575	−	0.219666i	$-0.0704966\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 5018.82i	− 0.211030i	−0.994418	−	0.105515i	$-0.966351\pi$
$827$	− 5018.82i	− 0.211030i	0.994418	−	0.105515i	$-0.0336491\pi$
$828$	0	0
$829$	13231.4	0.554336	0.277168	−	0.960822i	$-0.410604\pi$
$829$	13231.4	0.554336	0.277168	+	0.960822i	$0.410604\pi$
$830$	0	0
$831$	7998.89	0.333909
$832$	0	0
$833$	5088.64i	0.211658i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 5708.59i	− 0.235744i
$838$	0	0
$839$	9522.56	0.391842	0.195921	−	0.980620i	$-0.437230\pi$
$839$	9522.56	0.391842	0.195921	+	0.980620i	$0.437230\pi$
$840$	0	0
$841$	23284.9	0.954730
$842$	0	0
$843$	12459.7i	0.509057i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	9525.29i	0.386414i
$848$	0	0
$849$	7964.92	0.321973
$850$	0	0
$851$	−12896.6	−0.519494
$852$	0	0
$853$	− 43594.1i	− 1.74986i	−0.484245	−	0.874932i	$-0.660906\pi$
$853$	− 43594.1i	− 1.74986i	0.484245	−	0.874932i	$-0.339094\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	31892.6i	1.27122i	0.772012	+	0.635608i	$0.219251\pi$
$857$	31892.6i	1.27122i	−0.772012	+	0.635608i	$0.780749\pi$
$858$	0	0
$859$	−13058.6	−0.518689	−0.259345	−	0.965785i	$-0.583507\pi$
$859$	−13058.6	−0.518689	−0.259345	+	0.965785i	$0.583507\pi$
$860$	0	0
$861$	5645.22	0.223448
$862$	0	0
$863$	21487.2i	0.847548i	0.905768	+	0.423774i	$0.139295\pi$
$863$	21487.2i	0.847548i	−0.905768	+	0.423774i	$0.860705\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	20295.4i	0.795002i
$868$	0	0
$869$	−50353.4	−1.96562
$870$	0	0
$871$	−4232.71	−0.164661
$872$	0	0
$873$	17105.9i	0.663171i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 2743.60i	− 0.105638i	−0.998604	−	0.0528192i	$-0.983179\pi$
$877$	− 2743.60i	− 0.105638i	0.998604	−	0.0528192i	$-0.0168207\pi$
$878$	0	0
$879$	23089.9	0.886012
$880$	0	0
$881$	−12554.8	−0.480115	−0.240057	−	0.970759i	$-0.577166\pi$
$881$	−12554.8	−0.480115	−0.240057	+	0.970759i	$0.577166\pi$
$882$	0	0
$883$	− 26353.2i	− 1.00437i	−0.864761	−	0.502184i	$-0.832530\pi$
$883$	− 26353.2i	− 1.00437i	0.864761	−	0.502184i	$-0.167470\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	14016.6i	0.530587i	0.964168	+	0.265293i	$0.0854688\pi$
$887$	14016.6i	0.530587i	−0.964168	+	0.265293i	$0.914531\pi$
$888$	0	0
$889$	4986.72	0.188132
$890$	0	0
$891$	4979.15	0.187214
$892$	0	0
$893$	24931.0i	0.934249i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1648.89i	0.0613767i
$898$	0	0
$899$	8575.18	0.318129
$900$	0	0
$901$	−36142.6	−1.33639
$902$	0	0
$903$	12690.6i	0.467684i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	30710.0i	1.12427i	0.827047	+	0.562134i	$0.190019\pi$
$907$	30710.0i	1.12427i	−0.827047	+	0.562134i	$0.809981\pi$
$908$	0	0
$909$	12928.2	0.471728
$910$	0	0
$911$	47099.3	1.71292	0.856459	−	0.516215i	$-0.172659\pi$
$911$	47099.3	1.71292	0.856459	+	0.516215i	$0.172659\pi$
$912$	0	0
$913$	− 57556.9i	− 2.08637i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5224.55i	0.188146i
$918$	0	0
$919$	−7922.89	−0.284387	−0.142194	−	0.989839i	$-0.545416\pi$
$919$	−7922.89	−0.284387	−0.142194	+	0.989839i	$0.545416\pi$
$920$	0	0
$921$	−12211.4	−0.436895
$922$	0	0
$923$	− 3076.26i	− 0.109703i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	5244.18i	0.185805i
$928$	0	0
$929$	−22388.1	−0.790668	−0.395334	−	0.918537i	$-0.629371\pi$
$929$	−22388.1	−0.790668	−0.395334	+	0.918537i	$0.629371\pi$
$930$	0	0
$931$	5805.05	0.204353
$932$	0	0
$933$	26661.2i	0.935528i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 15161.4i	− 0.528603i	−0.964440	−	0.264302i	$-0.914859\pi$
$937$	− 15161.4i	− 0.528603i	0.964440	−	0.264302i	$-0.0851414\pi$
$938$	0	0
$939$	−19967.7	−0.693952
$940$	0	0
$941$	32547.3	1.12754	0.563768	−	0.825933i	$-0.309351\pi$
$941$	32547.3	1.12754	0.563768	+	0.825933i	$0.309351\pi$
$942$	0	0
$943$	18829.7i	0.650245i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	16496.7i	0.566072i	0.959109	+	0.283036i	$0.0913416\pi$
$947$	16496.7i	0.566072i	−0.959109	+	0.283036i	$0.908658\pi$
$948$	0	0
$949$	1359.60	0.0465065
$950$	0	0
$951$	9953.83	0.339406
$952$	0	0
$953$	42608.4i	1.44829i	0.689647	+	0.724146i	$0.257766\pi$
$953$	42608.4i	1.44829i	−0.689647	+	0.724146i	$0.742234\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	39154.9i	1.32257i
$958$	0	0
$959$	−18287.9	−0.615794
$960$	0	0
$961$	−28248.6	−0.948225
$962$	0	0
$963$	28006.8i	0.937183i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 25980.4i	− 0.863983i	−0.901878	−	0.431992i	$-0.857811\pi$
$967$	− 25980.4i	− 0.863983i	0.901878	−	0.431992i	$-0.142189\pi$
$968$	0	0
$969$	42524.9	1.40980
$970$	0	0
$971$	475.566	0.0157174	0.00785872	−	0.999969i	$-0.497498\pi$
$971$	475.566	0.0157174	0.00785872	+	0.999969i	$0.497498\pi$
$972$	0	0
$973$	− 4461.24i	− 0.146990i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	17562.9i	0.575115i	0.957763	+	0.287557i	$0.0928432\pi$
$977$	17562.9i	0.575115i	−0.957763	+	0.287557i	$0.907157\pi$
$978$	0	0
$979$	58494.5	1.90959
$980$	0	0
$981$	−18268.6	−0.594569
$982$	0	0
$983$	20943.8i	0.679557i	0.940505	+	0.339779i	$0.110352\pi$
$983$	20943.8i	0.679557i	−0.940505	+	0.339779i	$0.889648\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 5091.61i	− 0.164202i
$988$	0	0
$989$	−42329.9	−1.36098
$990$	0	0
$991$	−55297.8	−1.77254	−0.886272	−	0.463164i	$-0.846714\pi$
$991$	−55297.8	−1.77254	−0.886272	+	0.463164i	$0.846714\pi$
$992$	0	0
$993$	9625.47i	0.307608i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 1485.25i	− 0.0471800i	−0.999722	−	0.0235900i	$-0.992490\pi$
$997$	− 1485.25i	− 0.0471800i	0.999722	−	0.0235900i	$-0.00750963\pi$
$998$	0	0
$999$	23228.0	0.735636

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1400.4.g.l.449.2		6
5.2	odd	4		1400.4.a.m.1.1		3
5.3	odd	4		280.4.a.f.1.3	✓	3
5.4	even	2	inner	1400.4.g.l.449.5		6
20.3	even	4		560.4.a.w.1.1		3
35.13	even	4		1960.4.a.r.1.1		3
40.3	even	4		2240.4.a.br.1.3		3
40.13	odd	4		2240.4.a.bz.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.4.a.f.1.3	✓	3	5.3	odd	4
560.4.a.w.1.1		3	20.3	even	4
1400.4.a.m.1.1		3	5.2	odd	4
1400.4.g.l.449.2		6	1.1	even	1	trivial
1400.4.g.l.449.5		6	5.4	even	2	inner
1960.4.a.r.1.1		3	35.13	even	4
2240.4.a.br.1.3		3	40.3	even	4
2240.4.a.bz.1.1		3	40.13	odd	4

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:	\( N \)	\(=\)	\( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1400.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.45643i q^{3} +7.00000i q^{7} +15.0531 q^{9} +O(q^{10})\)	\(q-3.45643i q^{3} +7.00000i q^{7} +15.0531 q^{9} -51.8821 q^{11} +5.91120i q^{13} -103.850i q^{17} -118.470 q^{19} +24.1950 q^{21} +80.7029i q^{23} -145.354i q^{27} +218.344 q^{29} +39.2738 q^{31} +179.327i q^{33} +159.803i q^{37} +20.4316 q^{39} +233.322 q^{41} +524.515i q^{43} -210.441i q^{47} -49.0000 q^{49} -358.949 q^{51} -348.028i q^{53} +409.485i q^{57} +85.1752 q^{59} -801.720 q^{61} +105.372i q^{63} +716.049i q^{67} +278.944 q^{69} -520.412 q^{71} -230.005i q^{73} -363.175i q^{77} +970.534 q^{79} -95.9705 q^{81} +1109.38i q^{83} -754.689i q^{87} -1127.45 q^{89} -41.3784 q^{91} -135.747i q^{93} +1136.37i q^{97} -780.987 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 50 q^{9}+O(q^{10}) \) 6 * q - 50 * q^9	\( 6 q - 50 q^{9} - 12 q^{11} + 304 q^{19} - 84 q^{21} + 936 q^{29} - 536 q^{31} + 1348 q^{39} - 148 q^{41} - 294 q^{49} - 732 q^{51} + 2176 q^{59} + 348 q^{61} + 3568 q^{69} - 2112 q^{71} - 724 q^{79} + 5158 q^{81} - 316 q^{89} + 112 q^{91} + 2544 q^{99}+O(q^{100}) \) 6 * q - 50 * q^9 - 12 * q^11 + 304 * q^19 - 84 * q^21 + 936 * q^29 - 536 * q^31 + 1348 * q^39 - 148 * q^41 - 294 * q^49 - 732 * q^51 + 2176 * q^59 + 348 * q^61 + 3568 * q^69 - 2112 * q^71 - 724 * q^79 + 5158 * q^81 - 316 * q^89 + 112 * q^91 + 2544 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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