LMFDB - Embedded newform 2300.4.c.f.1749.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2300,4,Mod(1749,2300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2300, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2300.1749");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2300 = 2^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2300.c (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:	$135.704393013$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 237x^{10} + 20424x^{8} + 761789x^{6} + 10924545x^{4} + 24554784x^{2} + 7840000 $ x^12 + 237x^10 + 20424x^8 + 761789x^6 + 10924545x^4 + 24554784*x^2 + 7840000
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 460)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1749.2
Root			$-8.72903i$ of defining polynomial
Character	$\chi$	$=$	2300.1749
Dual form			2300.4.c.f.1749.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.72903i q^{3} -5.16219i q^{7} -49.1960 q^{9} +O(q^{10})$	$q-8.72903i q^{3} -5.16219i q^{7} -49.1960 q^{9} +38.0098 q^{11} -39.8463i q^{13} -52.1600i q^{17} +35.3134 q^{19} -45.0609 q^{21} +23.0000i q^{23} +193.749i q^{27} -154.377 q^{29} +275.161 q^{31} -331.789i q^{33} -315.469i q^{37} -347.820 q^{39} +195.373 q^{41} -158.622i q^{43} -358.521i q^{47} +316.352 q^{49} -455.306 q^{51} -50.4171i q^{53} -308.251i q^{57} -408.817 q^{59} +218.668 q^{61} +253.959i q^{63} +228.419i q^{67} +200.768 q^{69} -850.928 q^{71} -675.795i q^{73} -196.214i q^{77} +933.178 q^{79} +362.952 q^{81} +663.096i q^{83} +1347.56i q^{87} +243.459 q^{89} -205.694 q^{91} -2401.88i q^{93} +585.232i q^{97} -1869.93 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 150 q^{9}+O(q^{10}) $ 12 * q - 150 * q^9	$ 12 q - 150 q^{9} + 234 q^{11} - 210 q^{19} + 88 q^{21} + 142 q^{29} + 792 q^{31} - 1436 q^{39} + 1384 q^{41} - 1916 q^{49} + 1090 q^{51} - 3078 q^{59} + 922 q^{61} - 46 q^{69} + 756 q^{71} - 3388 q^{79} + 996 q^{81} - 1376 q^{89} + 4726 q^{91} - 8154 q^{99}+O(q^{100}) $ 12 * q - 150 * q^9 + 234 * q^11 - 210 * q^19 + 88 * q^21 + 142 * q^29 + 792 * q^31 - 1436 * q^39 + 1384 * q^41 - 1916 * q^49 + 1090 * q^51 - 3078 * q^59 + 922 * q^61 - 46 * q^69 + 756 * q^71 - 3388 * q^79 + 996 * q^81 - 1376 * q^89 + 4726 * q^91 - 8154 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2300\mathbb{Z}\right)^\times$.

$n$	$277$	$1151$	$1201$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	− 8.72903i	− 1.67990i	−0.542662	−	0.839951i	$-0.682583\pi$
$3$	− 8.72903i	− 1.67990i	0.542662	−	0.839951i	$-0.317417\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 5.16219i	− 0.278732i	−0.990241	−	0.139366i	$-0.955494\pi$
$7$	− 5.16219i	− 0.278732i	0.990241	−	0.139366i	$-0.0445065\pi$
$8$	0	0
$9$	−49.1960	−1.82207
$10$	0	0
$11$	38.0098	1.04185	0.520927	−	0.853601i	$-0.325586\pi$
$11$	38.0098	1.04185	0.520927	+	0.853601i	$0.325586\pi$
$12$	0	0
$13$	− 39.8463i	− 0.850106i	−0.905169	−	0.425053i	$-0.860255\pi$
$13$	− 39.8463i	− 0.850106i	0.905169	−	0.425053i	$-0.139745\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 52.1600i	− 0.744157i	−0.928201	−	0.372078i	$-0.878645\pi$
$17$	− 52.1600i	− 0.744157i	0.928201	−	0.372078i	$-0.121355\pi$
$18$	0	0
$19$	35.3134	0.426392	0.213196	−	0.977009i	$-0.431613\pi$
$19$	35.3134	0.426392	0.213196	+	0.977009i	$0.431613\pi$
$20$	0	0
$21$	−45.0609	−0.468243
$22$	0	0
$23$	23.0000i	0.208514i
$23$	23.0000i	0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	193.749i	1.38100i
$28$	0	0
$29$	−154.377	−0.988518	−0.494259	−	0.869315i	$-0.664561\pi$
$29$	−154.377	−0.988518	−0.494259	+	0.869315i	$0.664561\pi$
$30$	0	0
$31$	275.161	1.59420	0.797101	−	0.603845i	$-0.206366\pi$
$31$	275.161	1.59420	0.797101	+	0.603845i	$0.206366\pi$
$32$	0	0
$33$	− 331.789i	− 1.75021i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 315.469i	− 1.40170i	−0.713310	−	0.700849i	$-0.752805\pi$
$37$	− 315.469i	− 1.40170i	0.713310	−	0.700849i	$-0.247195\pi$
$38$	0	0
$39$	−347.820	−1.42809
$40$	0	0
$41$	195.373	0.744200	0.372100	−	0.928193i	$-0.378638\pi$
$41$	195.373	0.744200	0.372100	+	0.928193i	$0.378638\pi$
$42$	0	0
$43$	− 158.622i	− 0.562550i	−0.959627	−	0.281275i	$-0.909243\pi$
$43$	− 158.622i	− 0.562550i	0.959627	−	0.281275i	$-0.0907572\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 358.521i	− 1.11267i	−0.830957	−	0.556337i	$-0.812206\pi$
$47$	− 358.521i	− 1.11267i	0.830957	−	0.556337i	$-0.187794\pi$
$48$	0	0
$49$	316.352	0.922308
$50$	0	0
$51$	−455.306	−1.25011
$52$	0	0
$53$	− 50.4171i	− 0.130666i	−0.997864	−	0.0653332i	$-0.979189\pi$
$53$	− 50.4171i	− 0.130666i	0.997864	−	0.0653332i	$-0.0208110\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 308.251i	− 0.716296i
$58$	0	0
$59$	−408.817	−0.902092	−0.451046	−	0.892501i	$-0.648949\pi$
$59$	−408.817	−0.902092	−0.451046	+	0.892501i	$0.648949\pi$
$60$	0	0
$61$	218.668	0.458977	0.229488	−	0.973311i	$-0.426295\pi$
$61$	218.668	0.458977	0.229488	+	0.973311i	$0.426295\pi$
$62$	0	0
$63$	253.959i	0.507870i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	228.419i	0.416505i	0.978075	+	0.208253i	$0.0667776\pi$
$67$	228.419i	0.416505i	−0.978075	+	0.208253i	$0.933222\pi$
$68$	0	0
$69$	200.768	0.350284
$70$	0	0
$71$	−850.928	−1.42235	−0.711173	−	0.703017i	$-0.751836\pi$
$71$	−850.928	−1.42235	−0.711173	+	0.703017i	$0.751836\pi$
$72$	0	0
$73$	− 675.795i	− 1.08350i	−0.840538	−	0.541752i	$-0.817761\pi$
$73$	− 675.795i	− 1.08350i	0.840538	−	0.541752i	$-0.182239\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 196.214i	− 0.290398i
$78$	0	0
$79$	933.178	1.32900	0.664498	−	0.747290i	$-0.268645\pi$
$79$	933.178	1.32900	0.664498	+	0.747290i	$0.268645\pi$
$80$	0	0
$81$	362.952	0.497877
$82$	0	0
$83$	663.096i	0.876919i	0.898751	+	0.438459i	$0.144476\pi$
$83$	663.096i	0.876919i	−0.898751	+	0.438459i	$0.855524\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1347.56i	1.66061i
$88$	0	0
$89$	243.459	0.289962	0.144981	−	0.989434i	$-0.453688\pi$
$89$	243.459	0.289962	0.144981	+	0.989434i	$0.453688\pi$
$90$	0	0
$91$	−205.694	−0.236952
$92$	0	0
$93$	− 2401.88i	− 2.67811i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	585.232i	0.612591i	0.951936	+	0.306296i	$0.0990896\pi$
$97$	585.232i	0.612591i	−0.951936	+	0.306296i	$0.900910\pi$
$98$	0	0
$99$	−1869.93	−1.89833
$100$	0	0
$101$	−443.740	−0.437166	−0.218583	−	0.975818i	$-0.570143\pi$
$101$	−443.740	−0.437166	−0.218583	+	0.975818i	$0.570143\pi$
$102$	0	0
$103$	− 213.096i	− 0.203854i	−0.994792	−	0.101927i	$-0.967499\pi$
$103$	− 213.096i	− 0.203854i	0.994792	−	0.101927i	$-0.0325008\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 685.864i	− 0.619672i	−0.950790	−	0.309836i	$-0.899726\pi$
$107$	− 685.864i	− 0.619672i	0.950790	−	0.309836i	$-0.100274\pi$
$108$	0	0
$109$	−275.012	−0.241664	−0.120832	−	0.992673i	$-0.538556\pi$
$109$	−275.012	−0.241664	−0.120832	+	0.992673i	$0.538556\pi$
$110$	0	0
$111$	−2753.74	−2.35472
$112$	0	0
$113$	− 1423.71i	− 1.18523i	−0.805485	−	0.592617i	$-0.798095\pi$
$113$	− 1423.71i	− 1.18523i	0.805485	−	0.592617i	$-0.201905\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1960.28i	1.54895i
$118$	0	0
$119$	−269.260	−0.207420
$120$	0	0
$121$	113.748	0.0854605
$122$	0	0
$123$	− 1705.42i	− 1.25018i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 1360.84i	− 0.950826i	−0.879763	−	0.475413i	$-0.842299\pi$
$127$	− 1360.84i	− 0.950826i	0.879763	−	0.475413i	$-0.157701\pi$
$128$	0	0
$129$	−1384.62	−0.945029
$130$	0	0
$131$	465.737	0.310623	0.155312	−	0.987866i	$-0.450362\pi$
$131$	465.737	0.310623	0.155312	+	0.987866i	$0.450362\pi$
$132$	0	0
$133$	− 182.294i	− 0.118849i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1532.93i	− 0.955963i	−0.878370	−	0.477982i	$-0.841369\pi$
$137$	− 1532.93i	− 0.955963i	0.878370	−	0.477982i	$-0.158631\pi$
$138$	0	0
$139$	−2391.08	−1.45905	−0.729527	−	0.683952i	$-0.760260\pi$
$139$	−2391.08	−1.45905	−0.729527	+	0.683952i	$0.760260\pi$
$140$	0	0
$141$	−3129.54	−1.86918
$142$	0	0
$143$	− 1514.55i	− 0.885686i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 2761.44i	− 1.54939i
$148$	0	0
$149$	1474.01	0.810442	0.405221	−	0.914219i	$-0.367195\pi$
$149$	1474.01	0.810442	0.405221	+	0.914219i	$0.367195\pi$
$150$	0	0
$151$	−2188.82	−1.17962	−0.589812	−	0.807541i	$-0.700798\pi$
$151$	−2188.82	−1.17962	−0.589812	+	0.807541i	$0.700798\pi$
$152$	0	0
$153$	2566.06i	1.35591i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	579.541i	0.294601i	0.989092	+	0.147301i	$0.0470585\pi$
$157$	579.541i	0.294601i	−0.989092	+	0.147301i	$0.952942\pi$
$158$	0	0
$159$	−440.092	−0.219507
$160$	0	0
$161$	118.730	0.0581197
$162$	0	0
$163$	517.107i	0.248484i	0.992252	+	0.124242i	$0.0396500\pi$
$163$	517.107i	0.248484i	−0.992252	+	0.124242i	$0.960350\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1503.58i	0.696708i	0.937363	+	0.348354i	$0.113259\pi$
$167$	1503.58i	0.696708i	−0.937363	+	0.348354i	$0.886741\pi$
$168$	0	0
$169$	609.272	0.277320
$170$	0	0
$171$	−1737.28	−0.776917
$172$	0	0
$173$	3366.76i	1.47960i	0.672829	+	0.739798i	$0.265079\pi$
$173$	3366.76i	1.47960i	−0.672829	+	0.739798i	$0.734921\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3568.58i	1.51543i
$178$	0	0
$179$	−2849.55	−1.18986	−0.594931	−	0.803777i	$-0.702820\pi$
$179$	−2849.55	−1.18986	−0.594931	+	0.803777i	$0.702820\pi$
$180$	0	0
$181$	−2659.06	−1.09197	−0.545984	−	0.837796i	$-0.683844\pi$
$181$	−2659.06	−1.09197	−0.545984	+	0.837796i	$0.683844\pi$
$182$	0	0
$183$	− 1908.76i	− 0.771036i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 1982.59i	− 0.775303i
$188$	0	0
$189$	1000.17	0.384930
$190$	0	0
$191$	850.736	0.322289	0.161144	−	0.986931i	$-0.448482\pi$
$191$	850.736	0.322289	0.161144	+	0.986931i	$0.448482\pi$
$192$	0	0
$193$	854.757i	0.318792i	0.987215	+	0.159396i	$0.0509546\pi$
$193$	854.757i	0.318792i	−0.987215	+	0.159396i	$0.949045\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 4662.14i	− 1.68611i	−0.537827	−	0.843055i	$-0.680755\pi$
$197$	− 4662.14i	− 1.68611i	0.537827	−	0.843055i	$-0.319245\pi$
$198$	0	0
$199$	4706.99	1.67673	0.838366	−	0.545108i	$-0.183511\pi$
$199$	4706.99	1.67673	0.838366	+	0.545108i	$0.183511\pi$
$200$	0	0
$201$	1993.88	0.699688
$202$	0	0
$203$	796.922i	0.275532i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 1131.51i	− 0.379928i
$208$	0	0
$209$	1342.26	0.444238
$210$	0	0
$211$	4307.70	1.40547	0.702735	−	0.711452i	$-0.251962\pi$
$211$	4307.70	1.40547	0.702735	+	0.711452i	$0.251962\pi$
$212$	0	0
$213$	7427.78i	2.38940i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 1420.43i	− 0.444356i
$218$	0	0
$219$	−5899.03	−1.82018
$220$	0	0
$221$	−2078.38	−0.632612
$222$	0	0
$223$	6129.19i	1.84054i	0.391283	+	0.920271i	$0.372032\pi$
$223$	6129.19i	1.84054i	−0.391283	+	0.920271i	$0.627968\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	380.409i	0.111228i	0.998452	+	0.0556138i	$0.0177116\pi$
$227$	380.409i	0.111228i	−0.998452	+	0.0556138i	$0.982288\pi$
$228$	0	0
$229$	2280.21	0.657994	0.328997	−	0.944331i	$-0.393289\pi$
$229$	2280.21	0.657994	0.328997	+	0.944331i	$0.393289\pi$
$230$	0	0
$231$	−1712.76	−0.487841
$232$	0	0
$233$	2162.63i	0.608062i	0.952662	+	0.304031i	$0.0983326\pi$
$233$	2162.63i	0.608062i	−0.952662	+	0.304031i	$0.901667\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 8145.73i	− 2.23258i
$238$	0	0
$239$	−3111.07	−0.842000	−0.421000	−	0.907061i	$-0.638321\pi$
$239$	−3111.07	−0.842000	−0.421000	+	0.907061i	$0.638321\pi$
$240$	0	0
$241$	−5330.64	−1.42480	−0.712400	−	0.701774i	$-0.752392\pi$
$241$	−5330.64	−1.42480	−0.712400	+	0.701774i	$0.752392\pi$
$242$	0	0
$243$	2063.01i	0.544618i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 1407.11i	− 0.362478i
$248$	0	0
$249$	5788.19	1.47314
$250$	0	0
$251$	6350.09	1.59687	0.798434	−	0.602082i	$-0.205662\pi$
$251$	6350.09	1.59687	0.798434	+	0.602082i	$0.205662\pi$
$252$	0	0
$253$	874.226i	0.217242i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 3024.25i	− 0.734038i	−0.930213	−	0.367019i	$-0.880378\pi$
$257$	− 3024.25i	− 0.734038i	0.930213	−	0.367019i	$-0.119622\pi$
$258$	0	0
$259$	−1628.51	−0.390698
$260$	0	0
$261$	7594.71	1.80115
$262$	0	0
$263$	3721.96i	0.872646i	0.899790	+	0.436323i	$0.143719\pi$
$263$	3721.96i	0.872646i	−0.899790	+	0.436323i	$0.856281\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 2125.16i	− 0.487107i
$268$	0	0
$269$	−2767.39	−0.627253	−0.313626	−	0.949546i	$-0.601544\pi$
$269$	−2767.39	−0.627253	−0.313626	+	0.949546i	$0.601544\pi$
$270$	0	0
$271$	−8043.73	−1.80303	−0.901516	−	0.432746i	$-0.857545\pi$
$271$	−8043.73	−1.80303	−0.901516	+	0.432746i	$0.857545\pi$
$272$	0	0
$273$	1795.51i	0.398056i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	4637.94i	1.00602i	0.864281	+	0.503009i	$0.167774\pi$
$277$	4637.94i	1.00602i	−0.864281	+	0.503009i	$0.832226\pi$
$278$	0	0
$279$	−13536.8	−2.90475
$280$	0	0
$281$	−3052.82	−0.648100	−0.324050	−	0.946040i	$-0.605045\pi$
$281$	−3052.82	−0.648100	−0.324050	+	0.946040i	$0.605045\pi$
$282$	0	0
$283$	− 3693.93i	− 0.775906i	−0.921679	−	0.387953i	$-0.873182\pi$
$283$	− 3693.93i	− 0.775906i	0.921679	−	0.387953i	$-0.126818\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 1008.56i	− 0.207432i
$288$	0	0
$289$	2192.33	0.446231
$290$	0	0
$291$	5108.51	1.02909
$292$	0	0
$293$	− 3468.45i	− 0.691566i	−0.938315	−	0.345783i	$-0.887613\pi$
$293$	− 3468.45i	− 0.691566i	0.938315	−	0.345783i	$-0.112387\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	7364.38i	1.43880i
$298$	0	0
$299$	916.465	0.177259
$300$	0	0
$301$	−818.837	−0.156801
$302$	0	0
$303$	3873.42i	0.734396i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	4702.66i	0.874251i	0.899401	+	0.437125i	$0.144003\pi$
$307$	4702.66i	0.874251i	−0.899401	+	0.437125i	$0.855997\pi$
$308$	0	0
$309$	−1860.12	−0.342455
$310$	0	0
$311$	8242.68	1.50289	0.751446	−	0.659794i	$-0.229357\pi$
$311$	8242.68	1.50289	0.751446	+	0.659794i	$0.229357\pi$
$312$	0	0
$313$	9561.64i	1.72670i	0.504610	+	0.863348i	$0.331636\pi$
$313$	9561.64i	1.72670i	−0.504610	+	0.863348i	$0.668364\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8811.57i	1.56122i	0.625018	+	0.780611i	$0.285092\pi$
$317$	8811.57i	1.56122i	−0.625018	+	0.780611i	$0.714908\pi$
$318$	0	0
$319$	−5867.83	−1.02989
$320$	0	0
$321$	−5986.92	−1.04099
$322$	0	0
$323$	− 1841.95i	− 0.317302i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2400.59i	0.405972i
$328$	0	0
$329$	−1850.75	−0.310138
$330$	0	0
$331$	7709.72	1.28026	0.640128	−	0.768268i	$-0.278881\pi$
$331$	7709.72	1.28026	0.640128	+	0.768268i	$0.278881\pi$
$332$	0	0
$333$	15519.8i	2.55400i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3874.65i	0.626308i	0.949702	+	0.313154i	$0.101386\pi$
$337$	3874.65i	0.626308i	−0.949702	+	0.313154i	$0.898614\pi$
$338$	0	0
$339$	−12427.6	−1.99108
$340$	0	0
$341$	10458.8	1.66093
$342$	0	0
$343$	− 3403.70i	− 0.535809i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2660.21i	0.411549i	0.978599	+	0.205775i	$0.0659714\pi$
$347$	2660.21i	0.411549i	−0.978599	+	0.205775i	$0.934029\pi$
$348$	0	0
$349$	−5871.11	−0.900496	−0.450248	−	0.892904i	$-0.648664\pi$
$349$	−5871.11	−0.900496	−0.450248	+	0.892904i	$0.648664\pi$
$350$	0	0
$351$	7720.19	1.17400
$352$	0	0
$353$	821.405i	0.123850i	0.998081	+	0.0619249i	$0.0197239\pi$
$353$	821.405i	0.123850i	−0.998081	+	0.0619249i	$0.980276\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2350.38i	0.348446i
$358$	0	0
$359$	−1539.67	−0.226352	−0.113176	−	0.993575i	$-0.536102\pi$
$359$	−1539.67	−0.226352	−0.113176	+	0.993575i	$0.536102\pi$
$360$	0	0
$361$	−5611.97	−0.818190
$362$	0	0
$363$	− 992.910i	− 0.143565i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 8566.05i	− 1.21838i	−0.793026	−	0.609188i	$-0.791495\pi$
$367$	− 8566.05i	− 1.21838i	0.793026	−	0.609188i	$-0.208505\pi$
$368$	0	0
$369$	−9611.59	−1.35599
$370$	0	0
$371$	−260.262	−0.0364209
$372$	0	0
$373$	6795.52i	0.943322i	0.881780	+	0.471661i	$0.156345\pi$
$373$	6795.52i	0.943322i	−0.881780	+	0.471661i	$0.843655\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6151.34i	0.840345i
$378$	0	0
$379$	3495.70	0.473778	0.236889	−	0.971537i	$-0.423872\pi$
$379$	3495.70	0.473778	0.236889	+	0.971537i	$0.423872\pi$
$380$	0	0
$381$	−11878.8	−1.59729
$382$	0	0
$383$	− 1622.75i	− 0.216498i	−0.994124	−	0.108249i	$-0.965476\pi$
$383$	− 1622.75i	− 0.216498i	0.994124	−	0.108249i	$-0.0345244\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	7803.57i	1.02501i
$388$	0	0
$389$	9773.83	1.27391	0.636957	−	0.770899i	$-0.280193\pi$
$389$	9773.83	1.27391	0.636957	+	0.770899i	$0.280193\pi$
$390$	0	0
$391$	1199.68	0.155167
$392$	0	0
$393$	− 4065.43i	− 0.521816i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	4147.13i	0.524279i	0.965030	+	0.262139i	$0.0844280\pi$
$397$	4147.13i	0.524279i	−0.965030	+	0.262139i	$0.915572\pi$
$398$	0	0
$399$	−1591.25	−0.199655
$400$	0	0
$401$	−5786.11	−0.720559	−0.360280	−	0.932844i	$-0.617319\pi$
$401$	−5786.11	−0.720559	−0.360280	+	0.932844i	$0.617319\pi$
$402$	0	0
$403$	− 10964.1i	− 1.35524i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 11990.9i	− 1.46036i
$408$	0	0
$409$	756.583	0.0914685	0.0457342	−	0.998954i	$-0.485437\pi$
$409$	756.583	0.0914685	0.0457342	+	0.998954i	$0.485437\pi$
$410$	0	0
$411$	−13381.0	−1.60593
$412$	0	0
$413$	2110.39i	0.251442i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	20871.8i	2.45107i
$418$	0	0
$419$	−8412.65	−0.980870	−0.490435	−	0.871478i	$-0.663162\pi$
$419$	−8412.65	−0.980870	−0.490435	+	0.871478i	$0.663162\pi$
$420$	0	0
$421$	4671.31	0.540774	0.270387	−	0.962752i	$-0.412848\pi$
$421$	4671.31	0.540774	0.270387	+	0.962752i	$0.412848\pi$
$422$	0	0
$423$	17637.8i	2.02737i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 1128.81i	− 0.127932i
$428$	0	0
$429$	−13220.6	−1.48787
$430$	0	0
$431$	6924.66	0.773896	0.386948	−	0.922102i	$-0.373529\pi$
$431$	6924.66	0.773896	0.386948	+	0.922102i	$0.373529\pi$
$432$	0	0
$433$	13654.8i	1.51549i	0.652550	+	0.757746i	$0.273699\pi$
$433$	13654.8i	1.51549i	−0.652550	+	0.757746i	$0.726301\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	812.207i	0.0889088i
$438$	0	0
$439$	−14879.9	−1.61772	−0.808859	−	0.588003i	$-0.799914\pi$
$439$	−14879.9	−1.61772	−0.808859	+	0.588003i	$0.799914\pi$
$440$	0	0
$441$	−15563.2	−1.68051
$442$	0	0
$443$	5404.18i	0.579594i	0.957088	+	0.289797i	$0.0935878\pi$
$443$	5404.18i	0.579594i	−0.957088	+	0.289797i	$0.906412\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 12866.7i	− 1.36146i
$448$	0	0
$449$	−10215.5	−1.07371	−0.536857	−	0.843673i	$-0.680388\pi$
$449$	−10215.5	−1.07371	−0.536857	+	0.843673i	$0.680388\pi$
$450$	0	0
$451$	7426.11	0.775348
$452$	0	0
$453$	19106.2i	1.98165i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	660.128i	0.0675700i	0.999429	+	0.0337850i	$0.0107561\pi$
$457$	660.128i	0.0675700i	−0.999429	+	0.0337850i	$0.989244\pi$
$458$	0	0
$459$	10106.0	1.02768
$460$	0	0
$461$	3308.55	0.334261	0.167131	−	0.985935i	$-0.446550\pi$
$461$	3308.55	0.334261	0.167131	+	0.985935i	$0.446550\pi$
$462$	0	0
$463$	3413.57i	0.342640i	0.985215	+	0.171320i	$0.0548032\pi$
$463$	3413.57i	0.342640i	−0.985215	+	0.171320i	$0.945197\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 11026.0i	− 1.09256i	−0.837604	−	0.546278i	$-0.816044\pi$
$467$	− 11026.0i	− 1.09256i	0.837604	−	0.546278i	$-0.183956\pi$
$468$	0	0
$469$	1179.14	0.116093
$470$	0	0
$471$	5058.83	0.494901
$472$	0	0
$473$	− 6029.20i	− 0.586095i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2480.32i	0.238084i
$478$	0	0
$479$	−13542.0	−1.29175	−0.645876	−	0.763443i	$-0.723508\pi$
$479$	−13542.0	−1.29175	−0.645876	+	0.763443i	$0.723508\pi$
$480$	0	0
$481$	−12570.3	−1.19159
$482$	0	0
$483$	− 1036.40i	− 0.0976354i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	1440.00i	0.133989i	0.997753	+	0.0669944i	$0.0213410\pi$
$487$	1440.00i	0.133989i	−0.997753	+	0.0669944i	$0.978659\pi$
$488$	0	0
$489$	4513.85	0.417430
$490$	0	0
$491$	14147.1	1.30031	0.650153	−	0.759804i	$-0.274705\pi$
$491$	14147.1	1.30031	0.650153	+	0.759804i	$0.274705\pi$
$492$	0	0
$493$	8052.29i	0.735612i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4392.65i	0.396454i
$498$	0	0
$499$	3711.47	0.332962	0.166481	−	0.986045i	$-0.446760\pi$
$499$	3711.47	0.332962	0.166481	+	0.986045i	$0.446760\pi$
$500$	0	0
$501$	13124.8	1.17040
$502$	0	0
$503$	− 21745.2i	− 1.92758i	−0.266665	−	0.963789i	$-0.585922\pi$
$503$	− 21745.2i	− 1.92758i	0.266665	−	0.963789i	$-0.414078\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 5318.36i	− 0.465871i
$508$	0	0
$509$	−2069.49	−0.180213	−0.0901064	−	0.995932i	$-0.528721\pi$
$509$	−2069.49	−0.180213	−0.0901064	+	0.995932i	$0.528721\pi$
$510$	0	0
$511$	−3488.58	−0.302007
$512$	0	0
$513$	6841.94i	0.588848i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 13627.3i	− 1.15924i
$518$	0	0
$519$	29388.6	2.48558
$520$	0	0
$521$	−4167.35	−0.350432	−0.175216	−	0.984530i	$-0.556062\pi$
$521$	−4167.35	−0.350432	−0.175216	+	0.984530i	$0.556062\pi$
$522$	0	0
$523$	11965.4i	1.00040i	0.865910	+	0.500200i	$0.166740\pi$
$523$	11965.4i	1.00040i	−0.865910	+	0.500200i	$0.833260\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 14352.4i	− 1.18634i
$528$	0	0
$529$	−529.000	−0.0434783
$530$	0	0
$531$	20112.1	1.64368
$532$	0	0
$533$	− 7784.91i	− 0.632649i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	24873.8i	1.99885i
$538$	0	0
$539$	12024.5	0.960911
$540$	0	0
$541$	13853.3	1.10092	0.550462	−	0.834860i	$-0.314452\pi$
$541$	13853.3	1.10092	0.550462	+	0.834860i	$0.314452\pi$
$542$	0	0
$543$	23211.0i	1.83440i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 13072.6i	− 1.02183i	−0.859630	−	0.510917i	$-0.829306\pi$
$547$	− 13072.6i	− 1.02183i	0.859630	−	0.510917i	$-0.170694\pi$
$548$	0	0
$549$	−10757.6	−0.836289
$550$	0	0
$551$	−5451.56	−0.421496
$552$	0	0
$553$	− 4817.24i	− 0.370434i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5378.57i	0.409152i	0.978851	+	0.204576i	$0.0655815\pi$
$557$	5378.57i	0.409152i	−0.978851	+	0.204576i	$0.934418\pi$
$558$	0	0
$559$	−6320.50	−0.478227
$560$	0	0
$561$	−17306.1	−1.30243
$562$	0	0
$563$	− 19049.9i	− 1.42603i	−0.701148	−	0.713016i	$-0.747329\pi$
$563$	− 19049.9i	− 1.42603i	0.701148	−	0.713016i	$-0.252671\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 1873.63i	− 0.138774i
$568$	0	0
$569$	−18403.2	−1.35589	−0.677945	−	0.735112i	$-0.737129\pi$
$569$	−18403.2	−1.35589	−0.677945	+	0.735112i	$0.737129\pi$
$570$	0	0
$571$	15486.9	1.13504	0.567518	−	0.823361i	$-0.307904\pi$
$571$	15486.9	1.13504	0.567518	+	0.823361i	$0.307904\pi$
$572$	0	0
$573$	− 7426.10i	− 0.541414i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 17193.6i	− 1.24052i	−0.784397	−	0.620260i	$-0.787027\pi$
$577$	− 17193.6i	− 1.24052i	0.784397	−	0.620260i	$-0.212973\pi$
$578$	0	0
$579$	7461.20	0.535539
$580$	0	0
$581$	3423.03	0.244425
$582$	0	0
$583$	− 1916.34i	− 0.136135i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	26790.1i	1.88372i	0.336003	+	0.941861i	$0.390925\pi$
$587$	26790.1i	1.88372i	−0.336003	+	0.941861i	$0.609075\pi$
$588$	0	0
$589$	9716.84	0.679755
$590$	0	0
$591$	−40696.0	−2.83250
$592$	0	0
$593$	− 23314.3i	− 1.61451i	−0.590204	−	0.807254i	$-0.700953\pi$
$593$	− 23314.3i	− 1.61451i	0.590204	−	0.807254i	$-0.299047\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 41087.4i	− 2.81675i
$598$	0	0
$599$	−4311.34	−0.294085	−0.147042	−	0.989130i	$-0.546975\pi$
$599$	−4311.34	−0.294085	−0.147042	+	0.989130i	$0.546975\pi$
$600$	0	0
$601$	28384.5	1.92650	0.963251	−	0.268602i	$-0.0865617\pi$
$601$	28384.5	1.92650	0.963251	+	0.268602i	$0.0865617\pi$
$602$	0	0
$603$	− 11237.3i	− 0.758903i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 3132.56i	− 0.209467i	−0.994500	−	0.104734i	$-0.966601\pi$
$607$	− 3132.56i	− 0.209467i	0.994500	−	0.104734i	$-0.0333990\pi$
$608$	0	0
$609$	6956.35	0.462867
$610$	0	0
$611$	−14285.7	−0.945890
$612$	0	0
$613$	− 24702.7i	− 1.62762i	−0.581129	−	0.813811i	$-0.697389\pi$
$613$	− 24702.7i	− 1.62762i	0.581129	−	0.813811i	$-0.302611\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	21754.9i	1.41948i	0.704464	+	0.709739i	$0.251187\pi$
$617$	21754.9i	1.41948i	−0.704464	+	0.709739i	$0.748813\pi$
$618$	0	0
$619$	−9291.07	−0.603295	−0.301647	−	0.953420i	$-0.597537\pi$
$619$	−9291.07	−0.603295	−0.301647	+	0.953420i	$0.597537\pi$
$620$	0	0
$621$	−4456.23	−0.287959
$622$	0	0
$623$	− 1256.78i	− 0.0808216i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 11716.6i	− 0.746277i
$628$	0	0
$629$	−16454.9	−1.04308
$630$	0	0
$631$	−13411.8	−0.846143	−0.423072	−	0.906096i	$-0.639048\pi$
$631$	−13411.8	−0.846143	−0.423072	+	0.906096i	$0.639048\pi$
$632$	0	0
$633$	− 37602.0i	− 2.36105i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 12605.4i	− 0.784060i
$638$	0	0
$639$	41862.2	2.59162
$640$	0	0
$641$	−28688.3	−1.76774	−0.883868	−	0.467736i	$-0.845070\pi$
$641$	−28688.3	−1.76774	−0.883868	+	0.467736i	$0.845070\pi$
$642$	0	0
$643$	− 223.993i	− 0.0137378i	−0.999976	−	0.00686891i	$-0.997814\pi$
$643$	− 223.993i	− 0.0137378i	0.999976	−	0.00686891i	$-0.00218646\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 19074.2i	− 1.15901i	−0.814967	−	0.579507i	$-0.803245\pi$
$647$	− 19074.2i	− 1.15901i	0.814967	−	0.579507i	$-0.196755\pi$
$648$	0	0
$649$	−15539.1	−0.939849
$650$	0	0
$651$	−12399.0	−0.746474
$652$	0	0
$653$	5587.94i	0.334874i	0.985883	+	0.167437i	$0.0535492\pi$
$653$	5587.94i	0.334874i	−0.985883	+	0.167437i	$0.946451\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	33246.4i	1.97422i
$658$	0	0
$659$	−17983.8	−1.06305	−0.531526	−	0.847042i	$-0.678381\pi$
$659$	−17983.8	−1.06305	−0.531526	+	0.847042i	$0.678381\pi$
$660$	0	0
$661$	−9991.31	−0.587923	−0.293961	−	0.955817i	$-0.594974\pi$
$661$	−9991.31	−0.587923	−0.293961	+	0.955817i	$0.594974\pi$
$662$	0	0
$663$	18142.3i	1.06273i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 3550.66i	− 0.206120i
$668$	0	0
$669$	53501.9	3.09193
$670$	0	0
$671$	8311.54	0.478187
$672$	0	0
$673$	4963.63i	0.284300i	0.989845	+	0.142150i	$0.0454015\pi$
$673$	4963.63i	0.284300i	−0.989845	+	0.142150i	$0.954599\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 5947.62i	− 0.337645i	−0.985646	−	0.168822i	$-0.946004\pi$
$677$	− 5947.62i	− 0.337645i	0.985646	−	0.168822i	$-0.0539964\pi$
$678$	0	0
$679$	3021.08	0.170749
$680$	0	0
$681$	3320.61	0.186851
$682$	0	0
$683$	− 7882.10i	− 0.441582i	−0.975321	−	0.220791i	$-0.929136\pi$
$683$	− 7882.10i	− 0.441582i	0.975321	−	0.220791i	$-0.0708638\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 19904.0i	− 1.10537i
$688$	0	0
$689$	−2008.93	−0.111080
$690$	0	0
$691$	5702.63	0.313948	0.156974	−	0.987603i	$-0.449826\pi$
$691$	5702.63	0.313948	0.156974	+	0.987603i	$0.449826\pi$
$692$	0	0
$693$	9652.94i	0.529127i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 10190.7i	− 0.553802i
$698$	0	0
$699$	18877.6	1.02148
$700$	0	0
$701$	−29520.0	−1.59052	−0.795261	−	0.606268i	$-0.792666\pi$
$701$	−29520.0	−1.59052	−0.795261	+	0.606268i	$0.792666\pi$
$702$	0	0
$703$	− 11140.3i	− 0.597672i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2290.67i	0.121852i
$708$	0	0
$709$	25193.5	1.33450	0.667251	−	0.744833i	$-0.267471\pi$
$709$	25193.5	1.33450	0.667251	+	0.744833i	$0.267471\pi$
$710$	0	0
$711$	−45908.6	−2.42153
$712$	0	0
$713$	6328.69i	0.332414i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	27156.6i	1.41448i
$718$	0	0
$719$	−22488.0	−1.16642	−0.583212	−	0.812320i	$-0.698204\pi$
$719$	−22488.0	−1.16642	−0.583212	+	0.812320i	$0.698204\pi$
$720$	0	0
$721$	−1100.04	−0.0568206
$722$	0	0
$723$	46531.3i	2.39352i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 13224.7i	− 0.674657i	−0.941387	−	0.337328i	$-0.890477\pi$
$727$	− 13224.7i	− 0.674657i	0.941387	−	0.337328i	$-0.109523\pi$
$728$	0	0
$729$	27807.8	1.41278
$730$	0	0
$731$	−8273.73	−0.418625
$732$	0	0
$733$	− 9962.01i	− 0.501985i	−0.967989	−	0.250993i	$-0.919243\pi$
$733$	− 9962.01i	− 0.501985i	0.967989	−	0.250993i	$-0.0807570\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8682.18i	0.433938i
$738$	0	0
$739$	−5841.02	−0.290751	−0.145376	−	0.989377i	$-0.546439\pi$
$739$	−5841.02	−0.290751	−0.145376	+	0.989377i	$0.546439\pi$
$740$	0	0
$741$	−12282.7	−0.608928
$742$	0	0
$743$	− 32537.0i	− 1.60655i	−0.595609	−	0.803274i	$-0.703089\pi$
$743$	− 32537.0i	− 1.60655i	0.595609	−	0.803274i	$-0.296911\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 32621.7i	− 1.59781i
$748$	0	0
$749$	−3540.56	−0.172723
$750$	0	0
$751$	−1296.30	−0.0629860	−0.0314930	−	0.999504i	$-0.510026\pi$
$751$	−1296.30	−0.0629860	−0.0314930	+	0.999504i	$0.510026\pi$
$752$	0	0
$753$	− 55430.1i	− 2.68258i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 10560.1i	− 0.507017i	−0.967333	−	0.253509i	$-0.918415\pi$
$757$	− 10560.1i	− 0.507017i	0.967333	−	0.253509i	$-0.0815846\pi$
$758$	0	0
$759$	7631.15	0.364945
$760$	0	0
$761$	10648.0	0.507213	0.253606	−	0.967307i	$-0.418383\pi$
$761$	10648.0	0.507213	0.253606	+	0.967307i	$0.418383\pi$
$762$	0	0
$763$	1419.66i	0.0673594i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	16289.8i	0.766874i
$768$	0	0
$769$	−34957.9	−1.63929	−0.819645	−	0.572871i	$-0.805829\pi$
$769$	−34957.9	−1.63929	−0.819645	+	0.572871i	$0.805829\pi$
$770$	0	0
$771$	−26398.8	−1.23311
$772$	0	0
$773$	34404.7i	1.60084i	0.599437	+	0.800422i	$0.295391\pi$
$773$	34404.7i	1.60084i	−0.599437	+	0.800422i	$0.704609\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	14215.3i	0.656335i
$778$	0	0
$779$	6899.29	0.317321
$780$	0	0
$781$	−32343.6	−1.48188
$782$	0	0
$783$	− 29910.4i	− 1.36515i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 28711.0i	− 1.30043i	−0.759752	−	0.650213i	$-0.774680\pi$
$787$	− 28711.0i	− 1.30043i	0.759752	−	0.650213i	$-0.225320\pi$
$788$	0	0
$789$	32489.1	1.46596
$790$	0	0
$791$	−7349.46	−0.330363
$792$	0	0
$793$	− 8713.11i	− 0.390179i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 13725.0i	− 0.609995i	−0.952353	−	0.304998i	$-0.901344\pi$
$797$	− 13725.0i	− 0.609995i	0.952353	−	0.304998i	$-0.0986557\pi$
$798$	0	0
$799$	−18700.5	−0.828003
$800$	0	0
$801$	−11977.2	−0.528331
$802$	0	0
$803$	− 25686.9i	− 1.12885i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	24156.7i	1.05372i
$808$	0	0
$809$	−2335.57	−0.101501	−0.0507504	−	0.998711i	$-0.516161\pi$
$809$	−2335.57	−0.101501	−0.0507504	+	0.998711i	$0.516161\pi$
$810$	0	0
$811$	26955.3	1.16711	0.583556	−	0.812073i	$-0.301661\pi$
$811$	26955.3	1.16711	0.583556	+	0.812073i	$0.301661\pi$
$812$	0	0
$813$	70214.0i	3.02892i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 5601.48i	− 0.239866i
$818$	0	0
$819$	10119.3	0.431743
$820$	0	0
$821$	7822.07	0.332512	0.166256	−	0.986083i	$-0.446832\pi$
$821$	7822.07	0.332512	0.166256	+	0.986083i	$0.446832\pi$
$822$	0	0
$823$	− 45298.9i	− 1.91861i	−0.282366	−	0.959307i	$-0.591119\pi$
$823$	− 45298.9i	− 1.91861i	0.282366	−	0.959307i	$-0.408881\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	35367.7i	1.48713i	0.668665	+	0.743563i	$0.266866\pi$
$827$	35367.7i	1.48713i	−0.668665	+	0.743563i	$0.733134\pi$
$828$	0	0
$829$	4234.85	0.177422	0.0887108	−	0.996057i	$-0.471725\pi$
$829$	4234.85	0.177422	0.0887108	+	0.996057i	$0.471725\pi$
$830$	0	0
$831$	40484.7	1.69001
$832$	0	0
$833$	− 16500.9i	− 0.686342i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	53312.2i	2.20160i
$838$	0	0
$839$	−23205.3	−0.954869	−0.477435	−	0.878667i	$-0.658433\pi$
$839$	−23205.3	−0.954869	−0.477435	+	0.878667i	$0.658433\pi$
$840$	0	0
$841$	−556.847	−0.0228319
$842$	0	0
$843$	26648.2i	1.08875i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 587.189i	− 0.0238206i
$848$	0	0
$849$	−32244.5	−1.30345
$850$	0	0
$851$	7255.79	0.292274
$852$	0	0
$853$	− 14766.9i	− 0.592742i	−0.955073	−	0.296371i	$-0.904224\pi$
$853$	− 14766.9i	− 0.592742i	0.955073	−	0.296371i	$-0.0957764\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	8867.95i	0.353469i	0.984259	+	0.176735i	$0.0565535\pi$
$857$	8867.95i	0.353469i	−0.984259	+	0.176735i	$0.943447\pi$
$858$	0	0
$859$	2456.28	0.0975638	0.0487819	−	0.998809i	$-0.484466\pi$
$859$	2456.28	0.0975638	0.0487819	+	0.998809i	$0.484466\pi$
$860$	0	0
$861$	−8803.71	−0.348466
$862$	0	0
$863$	5523.03i	0.217852i	0.994050	+	0.108926i	$0.0347411\pi$
$863$	5523.03i	0.217852i	−0.994050	+	0.108926i	$0.965259\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 19136.9i	− 0.749624i
$868$	0	0
$869$	35469.9	1.38462
$870$	0	0
$871$	9101.66	0.354074
$872$	0	0
$873$	− 28791.1i	− 1.11619i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 31016.4i	− 1.19424i	−0.802152	−	0.597120i	$-0.796312\pi$
$877$	− 31016.4i	− 1.19424i	0.802152	−	0.597120i	$-0.203688\pi$
$878$	0	0
$879$	−30276.2	−1.16176
$880$	0	0
$881$	−1126.17	−0.0430667	−0.0215334	−	0.999768i	$-0.506855\pi$
$881$	−1126.17	−0.0430667	−0.0215334	+	0.999768i	$0.506855\pi$
$882$	0	0
$883$	− 18161.5i	− 0.692168i	−0.938204	−	0.346084i	$-0.887511\pi$
$883$	− 18161.5i	− 0.692168i	0.938204	−	0.346084i	$-0.112489\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 2413.56i	− 0.0913634i	−0.998956	−	0.0456817i	$-0.985454\pi$
$887$	− 2413.56i	− 0.0913634i	0.998956	−	0.0456817i	$-0.0145460\pi$
$888$	0	0
$889$	−7024.91	−0.265026
$890$	0	0
$891$	13795.8	0.518715
$892$	0	0
$893$	− 12660.6i	− 0.474435i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 7999.85i	− 0.297778i
$898$	0	0
$899$	−42478.4	−1.57590
$900$	0	0
$901$	−2629.76	−0.0972362
$902$	0	0
$903$	7147.66i	0.263410i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	28064.7i	1.02742i	0.857963	+	0.513711i	$0.171730\pi$
$907$	28064.7i	1.02742i	−0.857963	+	0.513711i	$0.828270\pi$
$908$	0	0
$909$	21830.2	0.796548
$910$	0	0
$911$	47846.5	1.74009	0.870047	−	0.492969i	$-0.164088\pi$
$911$	47846.5	1.74009	0.870047	+	0.492969i	$0.164088\pi$
$912$	0	0
$913$	25204.2i	0.913622i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 2404.22i	− 0.0865806i
$918$	0	0
$919$	35931.0	1.28972	0.644861	−	0.764300i	$-0.276915\pi$
$919$	35931.0	1.28972	0.644861	+	0.764300i	$0.276915\pi$
$920$	0	0
$921$	41049.7	1.46866
$922$	0	0
$923$	33906.3i	1.20914i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	10483.4i	0.371437i
$928$	0	0
$929$	27780.1	0.981093	0.490546	−	0.871415i	$-0.336797\pi$
$929$	27780.1	0.981093	0.490546	+	0.871415i	$0.336797\pi$
$930$	0	0
$931$	11171.4	0.393265
$932$	0	0
$933$	− 71950.6i	− 2.52471i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	18978.7i	0.661694i	0.943684	+	0.330847i	$0.107334\pi$
$937$	18978.7i	0.661694i	−0.943684	+	0.330847i	$0.892666\pi$
$938$	0	0
$939$	83463.8	2.90068
$940$	0	0
$941$	53850.9	1.86556	0.932778	−	0.360450i	$-0.117377\pi$
$941$	53850.9	1.86556	0.932778	+	0.360450i	$0.117377\pi$
$942$	0	0
$943$	4493.59i	0.155176i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 34403.0i	− 1.18051i	−0.807215	−	0.590257i	$-0.799026\pi$
$947$	− 34403.0i	− 1.18051i	0.807215	−	0.590257i	$-0.200974\pi$
$948$	0	0
$949$	−26927.9	−0.921093
$950$	0	0
$951$	76916.5	2.62270
$952$	0	0
$953$	32692.9i	1.11126i	0.831431	+	0.555628i	$0.187522\pi$
$953$	32692.9i	1.11126i	−0.831431	+	0.555628i	$0.812478\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	51220.5i	1.73012i
$958$	0	0
$959$	−7913.27	−0.266458
$960$	0	0
$961$	45922.3	1.54148
$962$	0	0
$963$	33741.7i	1.12909i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 26256.8i	− 0.873177i	−0.899661	−	0.436588i	$-0.856187\pi$
$967$	− 26256.8i	− 0.873177i	0.899661	−	0.436588i	$-0.143813\pi$
$968$	0	0
$969$	−16078.4	−0.533037
$970$	0	0
$971$	46759.7	1.54541	0.772704	−	0.634767i	$-0.218904\pi$
$971$	46759.7	1.54541	0.772704	+	0.634767i	$0.218904\pi$
$972$	0	0
$973$	12343.2i	0.406685i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 33207.1i	− 1.08740i	−0.839279	−	0.543700i	$-0.817023\pi$
$977$	− 33207.1i	− 1.08740i	0.839279	−	0.543700i	$-0.182977\pi$
$978$	0	0
$979$	9253.83	0.302098
$980$	0	0
$981$	13529.5	0.440329
$982$	0	0
$983$	10045.9i	0.325956i	0.986630	+	0.162978i	$0.0521099\pi$
$983$	10045.9i	0.325956i	−0.986630	+	0.162978i	$0.947890\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	16155.3i	0.521001i
$988$	0	0
$989$	3648.31	0.117300
$990$	0	0
$991$	−17797.6	−0.570495	−0.285248	−	0.958454i	$-0.592076\pi$
$991$	−17797.6	−0.570495	−0.285248	+	0.958454i	$0.592076\pi$
$992$	0	0
$993$	− 67298.4i	− 2.15070i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	28689.0i	0.911324i	0.890153	+	0.455662i	$0.150597\pi$
$997$	28689.0i	0.911324i	−0.890153	+	0.455662i	$0.849403\pi$
$998$	0	0
$999$	61121.9	1.93575

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2300.4.c.f.1749.2		12
5.2	odd	4		2300.4.a.f.1.1		6
5.3	odd	4		460.4.a.c.1.6	✓	6
5.4	even	2	inner	2300.4.c.f.1749.11		12
20.3	even	4		1840.4.a.t.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.4.a.c.1.6	✓	6	5.3	odd	4
1840.4.a.t.1.1		6	20.3	even	4
2300.4.a.f.1.1		6	5.2	odd	4
2300.4.c.f.1749.2		12	1.1	even	1	trivial
2300.4.c.f.1749.11		12	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2300.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-8.72903i q^{3} -5.16219i q^{7} -49.1960 q^{9} +O(q^{10})\)	\(q-8.72903i q^{3} -5.16219i q^{7} -49.1960 q^{9} +38.0098 q^{11} -39.8463i q^{13} -52.1600i q^{17} +35.3134 q^{19} -45.0609 q^{21} +23.0000i q^{23} +193.749i q^{27} -154.377 q^{29} +275.161 q^{31} -331.789i q^{33} -315.469i q^{37} -347.820 q^{39} +195.373 q^{41} -158.622i q^{43} -358.521i q^{47} +316.352 q^{49} -455.306 q^{51} -50.4171i q^{53} -308.251i q^{57} -408.817 q^{59} +218.668 q^{61} +253.959i q^{63} +228.419i q^{67} +200.768 q^{69} -850.928 q^{71} -675.795i q^{73} -196.214i q^{77} +933.178 q^{79} +362.952 q^{81} +663.096i q^{83} +1347.56i q^{87} +243.459 q^{89} -205.694 q^{91} -2401.88i q^{93} +585.232i q^{97} -1869.93 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 150 q^{9}+O(q^{10}) \) 12 * q - 150 * q^9	\( 12 q - 150 q^{9} + 234 q^{11} - 210 q^{19} + 88 q^{21} + 142 q^{29} + 792 q^{31} - 1436 q^{39} + 1384 q^{41} - 1916 q^{49} + 1090 q^{51} - 3078 q^{59} + 922 q^{61} - 46 q^{69} + 756 q^{71} - 3388 q^{79} + 996 q^{81} - 1376 q^{89} + 4726 q^{91} - 8154 q^{99}+O(q^{100}) \) 12 * q - 150 * q^9 + 234 * q^11 - 210 * q^19 + 88 * q^21 + 142 * q^29 + 792 * q^31 - 1436 * q^39 + 1384 * q^41 - 1916 * q^49 + 1090 * q^51 - 3078 * q^59 + 922 * q^61 - 46 * q^69 + 756 * q^71 - 3388 * q^79 + 996 * q^81 - 1376 * q^89 + 4726 * q^91 - 8154 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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