LMFDB - Embedded newform 2100.4.k.e.1849.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2100,4,Mod(1849,2100)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2100.1849"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2100, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2100.k (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,-18,0,-32,0,0,0,0,0,0,0,-208] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$123.904011012$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 420)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1849.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2100.1849
Dual form			2100.4.k.e.1849.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} -7.00000i q^{7} -9.00000 q^{9} -16.0000 q^{11} -14.0000i q^{13} -130.000i q^{17} -104.000 q^{19} +21.0000 q^{21} -88.0000i q^{23} -27.0000i q^{27} -54.0000 q^{29} +28.0000 q^{31} -48.0000i q^{33} +266.000i q^{37} +42.0000 q^{39} +202.000 q^{41} +348.000i q^{43} -104.000i q^{47} -49.0000 q^{49} +390.000 q^{51} +402.000i q^{53} -312.000i q^{57} +100.000 q^{59} +310.000 q^{61} +63.0000i q^{63} +324.000i q^{67} +264.000 q^{69} -644.000 q^{71} -290.000i q^{73} +112.000i q^{77} -744.000 q^{79} +81.0000 q^{81} +1044.00i q^{83} -162.000i q^{87} -298.000 q^{89} -98.0000 q^{91} +84.0000i q^{93} +290.000i q^{97} +144.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{9} - 32 q^{11} - 208 q^{19} + 42 q^{21} - 108 q^{29} + 56 q^{31} + 84 q^{39} + 404 q^{41} - 98 q^{49} + 780 q^{51} + 200 q^{59} + 620 q^{61} + 528 q^{69} - 1288 q^{71} - 1488 q^{79} + 162 q^{81}+ \cdots + 288 q^{99}+O(q^{100}) $ 2 * q - 18 * q^9 - 32 * q^11 - 208 * q^19 + 42 * q^21 - 108 * q^29 + 56 * q^31 + 84 * q^39 + 404 * q^41 - 98 * q^49 + 780 * q^51 + 200 * q^59 + 620 * q^61 + 528 * q^69 - 1288 * q^71 - 1488 * q^79 + 162 * q^81 - 596 * q^89 - 196 * q^91 + 288 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times$.

$n$	$701$	$1051$	$1177$	$1501$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	3.00000i	0.577350i
$3$	3.00000i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 7.00000i	− 0.377964i
$7$	− 7.00000i	− 0.377964i
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	−16.0000	−0.438562	−0.219281	−	0.975662i	$-0.570371\pi$
$11$	−16.0000	−0.438562	−0.219281	+	0.975662i	$0.570371\pi$
$12$	0	0
$13$	− 14.0000i	− 0.298685i	−0.988786	−	0.149342i	$-0.952284\pi$
$13$	− 14.0000i	− 0.298685i	0.988786	−	0.149342i	$-0.0477157\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 130.000i	− 1.85468i	−0.374215	−	0.927342i	$-0.622088\pi$
$17$	− 130.000i	− 1.85468i	0.374215	−	0.927342i	$-0.377912\pi$
$18$	0	0
$19$	−104.000	−1.25575	−0.627875	−	0.778314i	$-0.716075\pi$
$19$	−104.000	−1.25575	−0.627875	+	0.778314i	$0.716075\pi$
$20$	0	0
$21$	21.0000	0.218218
$22$	0	0
$23$	− 88.0000i	− 0.797794i	−0.916996	−	0.398897i	$-0.869393\pi$
$23$	− 88.0000i	− 0.797794i	0.916996	−	0.398897i	$-0.130607\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 27.0000i	− 0.192450i
$28$	0	0
$29$	−54.0000	−0.345778	−0.172889	−	0.984941i	$-0.555310\pi$
$29$	−54.0000	−0.345778	−0.172889	+	0.984941i	$0.555310\pi$
$30$	0	0
$31$	28.0000	0.162224	0.0811121	−	0.996705i	$-0.474153\pi$
$31$	28.0000	0.162224	0.0811121	+	0.996705i	$0.474153\pi$
$32$	0	0
$33$	− 48.0000i	− 0.253204i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	266.000i	1.18190i	0.806710	+	0.590948i	$0.201246\pi$
$37$	266.000i	1.18190i	−0.806710	+	0.590948i	$0.798754\pi$
$38$	0	0
$39$	42.0000	0.172446
$40$	0	0
$41$	202.000	0.769441	0.384721	−	0.923033i	$-0.374298\pi$
$41$	202.000	0.769441	0.384721	+	0.923033i	$0.374298\pi$
$42$	0	0
$43$	348.000i	1.23417i	0.786895	+	0.617087i	$0.211687\pi$
$43$	348.000i	1.23417i	−0.786895	+	0.617087i	$0.788313\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 104.000i	− 0.322765i	−0.986892	−	0.161383i	$-0.948405\pi$
$47$	− 104.000i	− 0.322765i	0.986892	−	0.161383i	$-0.0515953\pi$
$48$	0	0
$49$	−49.0000	−0.142857
$50$	0	0
$51$	390.000	1.07080
$52$	0	0
$53$	402.000i	1.04187i	0.853597	+	0.520933i	$0.174416\pi$
$53$	402.000i	1.04187i	−0.853597	+	0.520933i	$0.825584\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 312.000i	− 0.725007i
$58$	0	0
$59$	100.000	0.220659	0.110330	−	0.993895i	$-0.464809\pi$
$59$	100.000	0.220659	0.110330	+	0.993895i	$0.464809\pi$
$60$	0	0
$61$	310.000	0.650679	0.325340	−	0.945597i	$-0.394521\pi$
$61$	310.000	0.650679	0.325340	+	0.945597i	$0.394521\pi$
$62$	0	0
$63$	63.0000i	0.125988i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	324.000i	0.590790i	0.955375	+	0.295395i	$0.0954512\pi$
$67$	324.000i	0.590790i	−0.955375	+	0.295395i	$0.904549\pi$
$68$	0	0
$69$	264.000	0.460607
$70$	0	0
$71$	−644.000	−1.07646	−0.538231	−	0.842798i	$-0.680907\pi$
$71$	−644.000	−1.07646	−0.538231	+	0.842798i	$0.680907\pi$
$72$	0	0
$73$	− 290.000i	− 0.464958i	−0.972601	−	0.232479i	$-0.925316\pi$
$73$	− 290.000i	− 0.464958i	0.972601	−	0.232479i	$-0.0746837\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	112.000i	0.165761i
$78$	0	0
$79$	−744.000	−1.05958	−0.529788	−	0.848130i	$-0.677729\pi$
$79$	−744.000	−1.05958	−0.529788	+	0.848130i	$0.677729\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	1044.00i	1.38065i	0.723500	+	0.690325i	$0.242532\pi$
$83$	1044.00i	1.38065i	−0.723500	+	0.690325i	$0.757468\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 162.000i	− 0.199635i
$88$	0	0
$89$	−298.000	−0.354921	−0.177460	−	0.984128i	$-0.556788\pi$
$89$	−298.000	−0.354921	−0.177460	+	0.984128i	$0.556788\pi$
$90$	0	0
$91$	−98.0000	−0.112892
$92$	0	0
$93$	84.0000i	0.0936602i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	290.000i	0.303557i	0.988415	+	0.151779i	$0.0485001\pi$
$97$	290.000i	0.303557i	−0.988415	+	0.151779i	$0.951500\pi$
$98$	0	0
$99$	144.000	0.146187
$100$	0	0
$101$	1518.00	1.49551	0.747756	−	0.663974i	$-0.231131\pi$
$101$	1518.00	1.49551	0.747756	+	0.663974i	$0.231131\pi$
$102$	0	0
$103$	− 616.000i	− 0.589284i	−0.955608	−	0.294642i	$-0.904800\pi$
$103$	− 616.000i	− 0.589284i	0.955608	−	0.294642i	$-0.0952004\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	52.0000i	0.0469816i	0.999724	+	0.0234908i	$0.00747804\pi$
$107$	52.0000i	0.0469816i	−0.999724	+	0.0234908i	$0.992522\pi$
$108$	0	0
$109$	1034.00	0.908617	0.454308	−	0.890844i	$-0.349886\pi$
$109$	1034.00	0.908617	0.454308	+	0.890844i	$0.349886\pi$
$110$	0	0
$111$	−798.000	−0.682368
$112$	0	0
$113$	662.000i	0.551113i	0.961285	+	0.275556i	$0.0888620\pi$
$113$	662.000i	0.551113i	−0.961285	+	0.275556i	$0.911138\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	126.000i	0.0995616i
$118$	0	0
$119$	−910.000	−0.701005
$120$	0	0
$121$	−1075.00	−0.807663
$122$	0	0
$123$	606.000i	0.444237i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1128.00i	0.788140i	0.919080	+	0.394070i	$0.128933\pi$
$127$	1128.00i	0.788140i	−0.919080	+	0.394070i	$0.871067\pi$
$128$	0	0
$129$	−1044.00	−0.712551
$130$	0	0
$131$	492.000	0.328139	0.164070	−	0.986449i	$-0.447538\pi$
$131$	492.000	0.328139	0.164070	+	0.986449i	$0.447538\pi$
$132$	0	0
$133$	728.000i	0.474629i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1258.00i	0.784512i	0.919856	+	0.392256i	$0.128305\pi$
$137$	1258.00i	0.784512i	−0.919856	+	0.392256i	$0.871695\pi$
$138$	0	0
$139$	2176.00	1.32781	0.663906	−	0.747816i	$-0.268897\pi$
$139$	2176.00	1.32781	0.663906	+	0.747816i	$0.268897\pi$
$140$	0	0
$141$	312.000	0.186349
$142$	0	0
$143$	224.000i	0.130992i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 147.000i	− 0.0824786i
$148$	0	0
$149$	714.000	0.392572	0.196286	−	0.980547i	$-0.437112\pi$
$149$	714.000	0.392572	0.196286	+	0.980547i	$0.437112\pi$
$150$	0	0
$151$	−952.000	−0.513064	−0.256532	−	0.966536i	$-0.582580\pi$
$151$	−952.000	−0.513064	−0.256532	+	0.966536i	$0.582580\pi$
$152$	0	0
$153$	1170.00i	0.618228i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 938.000i	− 0.476819i	−0.971165	−	0.238409i	$-0.923374\pi$
$157$	− 938.000i	− 0.476819i	0.971165	−	0.238409i	$-0.0766260\pi$
$158$	0	0
$159$	−1206.00	−0.601522
$160$	0	0
$161$	−616.000	−0.301538
$162$	0	0
$163$	− 2988.00i	− 1.43582i	−0.696137	−	0.717909i	$-0.745100\pi$
$163$	− 2988.00i	− 1.43582i	0.696137	−	0.717909i	$-0.254900\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3224.00i	1.49390i	0.664883	+	0.746948i	$0.268482\pi$
$167$	3224.00i	1.49390i	−0.664883	+	0.746948i	$0.731518\pi$
$168$	0	0
$169$	2001.00	0.910787
$170$	0	0
$171$	936.000	0.418583
$172$	0	0
$173$	2254.00i	0.990569i	0.868731	+	0.495285i	$0.164936\pi$
$173$	2254.00i	0.990569i	−0.868731	+	0.495285i	$0.835064\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	300.000i	0.127398i
$178$	0	0
$179$	2296.00	0.958721	0.479361	−	0.877618i	$-0.340869\pi$
$179$	2296.00	0.958721	0.479361	+	0.877618i	$0.340869\pi$
$180$	0	0
$181$	−4378.00	−1.79787	−0.898934	−	0.438084i	$-0.855657\pi$
$181$	−4378.00	−1.79787	−0.898934	+	0.438084i	$0.855657\pi$
$182$	0	0
$183$	930.000i	0.375670i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2080.00i	0.813394i
$188$	0	0
$189$	−189.000	−0.0727393
$190$	0	0
$191$	76.0000	0.0287915	0.0143957	−	0.999896i	$-0.495418\pi$
$191$	76.0000	0.0287915	0.0143957	+	0.999896i	$0.495418\pi$
$192$	0	0
$193$	3730.00i	1.39115i	0.718455	+	0.695573i	$0.244850\pi$
$193$	3730.00i	1.39115i	−0.718455	+	0.695573i	$0.755150\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 2298.00i	− 0.831095i	−0.909571	−	0.415548i	$-0.863590\pi$
$197$	− 2298.00i	− 0.831095i	0.909571	−	0.415548i	$-0.136410\pi$
$198$	0	0
$199$	2044.00	0.728117	0.364059	−	0.931376i	$-0.381391\pi$
$199$	2044.00	0.728117	0.364059	+	0.931376i	$0.381391\pi$
$200$	0	0
$201$	−972.000	−0.341093
$202$	0	0
$203$	378.000i	0.130692i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	792.000i	0.265931i
$208$	0	0
$209$	1664.00	0.550724
$210$	0	0
$211$	−2524.00	−0.823504	−0.411752	−	0.911296i	$-0.635083\pi$
$211$	−2524.00	−0.823504	−0.411752	+	0.911296i	$0.635083\pi$
$212$	0	0
$213$	− 1932.00i	− 0.621495i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 196.000i	− 0.0613150i
$218$	0	0
$219$	870.000	0.268444
$220$	0	0
$221$	−1820.00	−0.553966
$222$	0	0
$223$	− 4296.00i	− 1.29005i	−0.764161	−	0.645026i	$-0.776847\pi$
$223$	− 4296.00i	− 1.29005i	0.764161	−	0.645026i	$-0.223153\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4812.00i	1.40698i	0.710707	+	0.703488i	$0.248375\pi$
$227$	4812.00i	1.40698i	−0.710707	+	0.703488i	$0.751625\pi$
$228$	0	0
$229$	2402.00	0.693138	0.346569	−	0.938024i	$-0.387347\pi$
$229$	2402.00	0.693138	0.346569	+	0.938024i	$0.387347\pi$
$230$	0	0
$231$	−336.000	−0.0957021
$232$	0	0
$233$	6158.00i	1.73143i	0.500534	+	0.865717i	$0.333137\pi$
$233$	6158.00i	1.73143i	−0.500534	+	0.865717i	$0.666863\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 2232.00i	− 0.611747i
$238$	0	0
$239$	−2932.00	−0.793537	−0.396768	−	0.917919i	$-0.629868\pi$
$239$	−2932.00	−0.793537	−0.396768	+	0.917919i	$0.629868\pi$
$240$	0	0
$241$	−5670.00	−1.51551	−0.757753	−	0.652542i	$-0.773703\pi$
$241$	−5670.00	−1.51551	−0.757753	+	0.652542i	$0.773703\pi$
$242$	0	0
$243$	243.000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1456.00i	0.375073i
$248$	0	0
$249$	−3132.00	−0.797118
$250$	0	0
$251$	−5620.00	−1.41327	−0.706636	−	0.707577i	$-0.749788\pi$
$251$	−5620.00	−1.41327	−0.706636	+	0.707577i	$0.749788\pi$
$252$	0	0
$253$	1408.00i	0.349882i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 4666.00i	− 1.13252i	−0.824227	−	0.566259i	$-0.808390\pi$
$257$	− 4666.00i	− 1.13252i	0.824227	−	0.566259i	$-0.191610\pi$
$258$	0	0
$259$	1862.00	0.446714
$260$	0	0
$261$	486.000	0.115259
$262$	0	0
$263$	− 3648.00i	− 0.855305i	−0.903943	−	0.427653i	$-0.859341\pi$
$263$	− 3648.00i	− 0.855305i	0.903943	−	0.427653i	$-0.140659\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 894.000i	− 0.204914i
$268$	0	0
$269$	−1206.00	−0.273350	−0.136675	−	0.990616i	$-0.543642\pi$
$269$	−1206.00	−0.273350	−0.136675	+	0.990616i	$0.543642\pi$
$270$	0	0
$271$	−396.000	−0.0887649	−0.0443824	−	0.999015i	$-0.514132\pi$
$271$	−396.000	−0.0887649	−0.0443824	+	0.999015i	$0.514132\pi$
$272$	0	0
$273$	− 294.000i	− 0.0651783i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5914.00i	1.28281i	0.767203	+	0.641404i	$0.221648\pi$
$277$	5914.00i	1.28281i	−0.767203	+	0.641404i	$0.778352\pi$
$278$	0	0
$279$	−252.000	−0.0540747
$280$	0	0
$281$	−478.000	−0.101477	−0.0507386	−	0.998712i	$-0.516158\pi$
$281$	−478.000	−0.101477	−0.0507386	+	0.998712i	$0.516158\pi$
$282$	0	0
$283$	6260.00i	1.31491i	0.753496	+	0.657453i	$0.228366\pi$
$283$	6260.00i	1.31491i	−0.753496	+	0.657453i	$0.771634\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 1414.00i	− 0.290822i
$288$	0	0
$289$	−11987.0	−2.43985
$290$	0	0
$291$	−870.000	−0.175259
$292$	0	0
$293$	− 1146.00i	− 0.228498i	−0.993452	−	0.114249i	$-0.963554\pi$
$293$	− 1146.00i	− 0.228498i	0.993452	−	0.114249i	$-0.0364462\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	432.000i	0.0844013i
$298$	0	0
$299$	−1232.00	−0.238289
$300$	0	0
$301$	2436.00	0.466474
$302$	0	0
$303$	4554.00i	0.863434i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	3580.00i	0.665542i	0.943008	+	0.332771i	$0.107984\pi$
$307$	3580.00i	0.665542i	−0.943008	+	0.332771i	$0.892016\pi$
$308$	0	0
$309$	1848.00	0.340223
$310$	0	0
$311$	−5432.00	−0.990419	−0.495210	−	0.868773i	$-0.664909\pi$
$311$	−5432.00	−0.990419	−0.495210	+	0.868773i	$0.664909\pi$
$312$	0	0
$313$	7798.00i	1.40821i	0.710097	+	0.704104i	$0.248651\pi$
$313$	7798.00i	1.40821i	−0.710097	+	0.704104i	$0.751349\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 3130.00i	− 0.554569i	−0.960788	−	0.277284i	$-0.910566\pi$
$317$	− 3130.00i	− 0.554569i	0.960788	−	0.277284i	$-0.0894344\pi$
$318$	0	0
$319$	864.000	0.151645
$320$	0	0
$321$	−156.000	−0.0271248
$322$	0	0
$323$	13520.0i	2.32902i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	3102.00i	0.524590i
$328$	0	0
$329$	−728.000	−0.121994
$330$	0	0
$331$	−6020.00	−0.999665	−0.499833	−	0.866122i	$-0.666605\pi$
$331$	−6020.00	−0.999665	−0.499833	+	0.866122i	$0.666605\pi$
$332$	0	0
$333$	− 2394.00i	− 0.393965i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 2066.00i	− 0.333953i	−0.985961	−	0.166977i	$-0.946600\pi$
$337$	− 2066.00i	− 0.333953i	0.985961	−	0.166977i	$-0.0534004\pi$
$338$	0	0
$339$	−1986.00	−0.318185
$340$	0	0
$341$	−448.000	−0.0711453
$342$	0	0
$343$	343.000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 10644.0i	− 1.64669i	−0.567544	−	0.823343i	$-0.692107\pi$
$347$	− 10644.0i	− 1.64669i	0.567544	−	0.823343i	$-0.307893\pi$
$348$	0	0
$349$	−2150.00	−0.329762	−0.164881	−	0.986313i	$-0.552724\pi$
$349$	−2150.00	−0.329762	−0.164881	+	0.986313i	$0.552724\pi$
$350$	0	0
$351$	−378.000	−0.0574819
$352$	0	0
$353$	11250.0i	1.69625i	0.529794	+	0.848126i	$0.322269\pi$
$353$	11250.0i	1.69625i	−0.529794	+	0.848126i	$0.677731\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 2730.00i	− 0.404725i
$358$	0	0
$359$	7756.00	1.14024	0.570120	−	0.821562i	$-0.306897\pi$
$359$	7756.00	1.14024	0.570120	+	0.821562i	$0.306897\pi$
$360$	0	0
$361$	3957.00	0.576906
$362$	0	0
$363$	− 3225.00i	− 0.466305i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	9176.00i	1.30513i	0.757732	+	0.652566i	$0.226307\pi$
$367$	9176.00i	1.30513i	−0.757732	+	0.652566i	$0.773693\pi$
$368$	0	0
$369$	−1818.00	−0.256480
$370$	0	0
$371$	2814.00	0.393789
$372$	0	0
$373$	12590.0i	1.74768i	0.486212	+	0.873841i	$0.338378\pi$
$373$	12590.0i	1.74768i	−0.486212	+	0.873841i	$0.661622\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	756.000i	0.103278i
$378$	0	0
$379$	−6580.00	−0.891799	−0.445900	−	0.895083i	$-0.647116\pi$
$379$	−6580.00	−0.891799	−0.445900	+	0.895083i	$0.647116\pi$
$380$	0	0
$381$	−3384.00	−0.455033
$382$	0	0
$383$	2928.00i	0.390637i	0.980740	+	0.195318i	$0.0625740\pi$
$383$	2928.00i	0.390637i	−0.980740	+	0.195318i	$0.937426\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 3132.00i	− 0.411391i
$388$	0	0
$389$	10170.0	1.32555	0.662776	−	0.748818i	$-0.269378\pi$
$389$	10170.0	1.32555	0.662776	+	0.748818i	$0.269378\pi$
$390$	0	0
$391$	−11440.0	−1.47966
$392$	0	0
$393$	1476.00i	0.189451i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 9538.00i	− 1.20579i	−0.797821	−	0.602895i	$-0.794014\pi$
$397$	− 9538.00i	− 1.20579i	0.797821	−	0.602895i	$-0.205986\pi$
$398$	0	0
$399$	−2184.00	−0.274027
$400$	0	0
$401$	4626.00	0.576088	0.288044	−	0.957617i	$-0.406995\pi$
$401$	4626.00	0.576088	0.288044	+	0.957617i	$0.406995\pi$
$402$	0	0
$403$	− 392.000i	− 0.0484539i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 4256.00i	− 0.518334i
$408$	0	0
$409$	4934.00	0.596505	0.298253	−	0.954487i	$-0.403596\pi$
$409$	4934.00	0.596505	0.298253	+	0.954487i	$0.403596\pi$
$410$	0	0
$411$	−3774.00	−0.452938
$412$	0	0
$413$	− 700.000i	− 0.0834013i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	6528.00i	0.766613i
$418$	0	0
$419$	2196.00	0.256042	0.128021	−	0.991771i	$-0.459138\pi$
$419$	2196.00	0.256042	0.128021	+	0.991771i	$0.459138\pi$
$420$	0	0
$421$	10366.0	1.20002	0.600009	−	0.799993i	$-0.295163\pi$
$421$	10366.0	1.20002	0.600009	+	0.799993i	$0.295163\pi$
$422$	0	0
$423$	936.000i	0.107588i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 2170.00i	− 0.245934i
$428$	0	0
$429$	−672.000	−0.0756281
$430$	0	0
$431$	−10284.0	−1.14933	−0.574667	−	0.818387i	$-0.694868\pi$
$431$	−10284.0	−1.14933	−0.574667	+	0.818387i	$0.694868\pi$
$432$	0	0
$433$	− 16242.0i	− 1.80263i	−0.433160	−	0.901317i	$-0.642601\pi$
$433$	− 16242.0i	− 1.80263i	0.433160	−	0.901317i	$-0.357399\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	9152.00i	1.00183i
$438$	0	0
$439$	−10052.0	−1.09284	−0.546419	−	0.837512i	$-0.684009\pi$
$439$	−10052.0	−1.09284	−0.546419	+	0.837512i	$0.684009\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	14260.0i	1.52937i	0.644401	+	0.764687i	$0.277107\pi$
$443$	14260.0i	1.52937i	−0.644401	+	0.764687i	$0.722893\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	2142.00i	0.226651i
$448$	0	0
$449$	11710.0	1.23080	0.615400	−	0.788215i	$-0.288995\pi$
$449$	11710.0	1.23080	0.615400	+	0.788215i	$0.288995\pi$
$450$	0	0
$451$	−3232.00	−0.337448
$452$	0	0
$453$	− 2856.00i	− 0.296218i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 11266.0i	− 1.15318i	−0.817035	−	0.576588i	$-0.804384\pi$
$457$	− 11266.0i	− 1.15318i	0.817035	−	0.576588i	$-0.195616\pi$
$458$	0	0
$459$	−3510.00	−0.356934
$460$	0	0
$461$	−698.000	−0.0705187	−0.0352593	−	0.999378i	$-0.511226\pi$
$461$	−698.000	−0.0705187	−0.0352593	+	0.999378i	$0.511226\pi$
$462$	0	0
$463$	− 7624.00i	− 0.765264i	−0.923901	−	0.382632i	$-0.875018\pi$
$463$	− 7624.00i	− 0.765264i	0.923901	−	0.382632i	$-0.124982\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	18220.0i	1.80540i	0.430272	+	0.902699i	$0.358418\pi$
$467$	18220.0i	1.80540i	−0.430272	+	0.902699i	$0.641582\pi$
$468$	0	0
$469$	2268.00	0.223297
$470$	0	0
$471$	2814.00	0.275291
$472$	0	0
$473$	− 5568.00i	− 0.541262i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 3618.00i	− 0.347289i
$478$	0	0
$479$	−6432.00	−0.613540	−0.306770	−	0.951784i	$-0.599248\pi$
$479$	−6432.00	−0.613540	−0.306770	+	0.951784i	$0.599248\pi$
$480$	0	0
$481$	3724.00	0.353014
$482$	0	0
$483$	− 1848.00i	− 0.174093i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 3984.00i	− 0.370703i	−0.982672	−	0.185351i	$-0.940658\pi$
$487$	− 3984.00i	− 0.370703i	0.982672	−	0.185351i	$-0.0593423\pi$
$488$	0	0
$489$	8964.00	0.828970
$490$	0	0
$491$	11952.0	1.09855	0.549273	−	0.835643i	$-0.314905\pi$
$491$	11952.0	1.09855	0.549273	+	0.835643i	$0.314905\pi$
$492$	0	0
$493$	7020.00i	0.641308i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4508.00i	0.406864i
$498$	0	0
$499$	−18844.0	−1.69053	−0.845264	−	0.534349i	$-0.820557\pi$
$499$	−18844.0	−1.69053	−0.845264	+	0.534349i	$0.820557\pi$
$500$	0	0
$501$	−9672.00	−0.862501
$502$	0	0
$503$	13208.0i	1.17081i	0.810742	+	0.585403i	$0.199064\pi$
$503$	13208.0i	1.17081i	−0.810742	+	0.585403i	$0.800936\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	6003.00i	0.525843i
$508$	0	0
$509$	14714.0	1.28131	0.640655	−	0.767829i	$-0.278663\pi$
$509$	14714.0	1.28131	0.640655	+	0.767829i	$0.278663\pi$
$510$	0	0
$511$	−2030.00	−0.175738
$512$	0	0
$513$	2808.00i	0.241669i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1664.00i	0.141552i
$518$	0	0
$519$	−6762.00	−0.571905
$520$	0	0
$521$	−5670.00	−0.476789	−0.238395	−	0.971168i	$-0.576621\pi$
$521$	−5670.00	−0.476789	−0.238395	+	0.971168i	$0.576621\pi$
$522$	0	0
$523$	12668.0i	1.05915i	0.848265	+	0.529573i	$0.177648\pi$
$523$	12668.0i	1.05915i	−0.848265	+	0.529573i	$0.822352\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 3640.00i	− 0.300875i
$528$	0	0
$529$	4423.00	0.363524
$530$	0	0
$531$	−900.000	−0.0735531
$532$	0	0
$533$	− 2828.00i	− 0.229820i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	6888.00i	0.553518i
$538$	0	0
$539$	784.000	0.0626517
$540$	0	0
$541$	−7498.00	−0.595867	−0.297934	−	0.954587i	$-0.596297\pi$
$541$	−7498.00	−0.595867	−0.297934	+	0.954587i	$0.596297\pi$
$542$	0	0
$543$	− 13134.0i	− 1.03800i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 16396.0i	− 1.28161i	−0.767702	−	0.640807i	$-0.778600\pi$
$547$	− 16396.0i	− 1.28161i	0.767702	−	0.640807i	$-0.221400\pi$
$548$	0	0
$549$	−2790.00	−0.216893
$550$	0	0
$551$	5616.00	0.434210
$552$	0	0
$553$	5208.00i	0.400482i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	13126.0i	0.998504i	0.866457	+	0.499252i	$0.166392\pi$
$557$	13126.0i	0.998504i	−0.866457	+	0.499252i	$0.833608\pi$
$558$	0	0
$559$	4872.00	0.368629
$560$	0	0
$561$	−6240.00	−0.469613
$562$	0	0
$563$	− 12180.0i	− 0.911769i	−0.890039	−	0.455884i	$-0.849323\pi$
$563$	− 12180.0i	− 0.911769i	0.890039	−	0.455884i	$-0.150677\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 567.000i	− 0.0419961i
$568$	0	0
$569$	−1914.00	−0.141018	−0.0705088	−	0.997511i	$-0.522462\pi$
$569$	−1914.00	−0.141018	−0.0705088	+	0.997511i	$0.522462\pi$
$570$	0	0
$571$	340.000	0.0249187	0.0124593	−	0.999922i	$-0.496034\pi$
$571$	340.000	0.0249187	0.0124593	+	0.999922i	$0.496034\pi$
$572$	0	0
$573$	228.000i	0.0166228i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	9394.00i	0.677777i	0.940827	+	0.338889i	$0.110051\pi$
$577$	9394.00i	0.677777i	−0.940827	+	0.338889i	$0.889949\pi$
$578$	0	0
$579$	−11190.0	−0.803179
$580$	0	0
$581$	7308.00	0.521836
$582$	0	0
$583$	− 6432.00i	− 0.456923i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6108.00i	0.429479i	0.976671	+	0.214739i	$0.0688902\pi$
$587$	6108.00i	0.429479i	−0.976671	+	0.214739i	$0.931110\pi$
$588$	0	0
$589$	−2912.00	−0.203713
$590$	0	0
$591$	6894.00	0.479833
$592$	0	0
$593$	2274.00i	0.157474i	0.996895	+	0.0787369i	$0.0250887\pi$
$593$	2274.00i	0.157474i	−0.996895	+	0.0787369i	$0.974911\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	6132.00i	0.420379i
$598$	0	0
$599$	24140.0	1.64663	0.823317	−	0.567582i	$-0.192121\pi$
$599$	24140.0	1.64663	0.823317	+	0.567582i	$0.192121\pi$
$600$	0	0
$601$	−1438.00	−0.0975994	−0.0487997	−	0.998809i	$-0.515540\pi$
$601$	−1438.00	−0.0975994	−0.0487997	+	0.998809i	$0.515540\pi$
$602$	0	0
$603$	− 2916.00i	− 0.196930i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	17736.0i	1.18597i	0.805215	+	0.592984i	$0.202050\pi$
$607$	17736.0i	1.18597i	−0.805215	+	0.592984i	$0.797950\pi$
$608$	0	0
$609$	−1134.00	−0.0754548
$610$	0	0
$611$	−1456.00	−0.0964050
$612$	0	0
$613$	6374.00i	0.419973i	0.977704	+	0.209986i	$0.0673420\pi$
$613$	6374.00i	0.419973i	−0.977704	+	0.209986i	$0.932658\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 9854.00i	− 0.642961i	−0.946916	−	0.321481i	$-0.895819\pi$
$617$	− 9854.00i	− 0.642961i	0.946916	−	0.321481i	$-0.104181\pi$
$618$	0	0
$619$	8480.00	0.550630	0.275315	−	0.961354i	$-0.411218\pi$
$619$	8480.00	0.550630	0.275315	+	0.961354i	$0.411218\pi$
$620$	0	0
$621$	−2376.00	−0.153536
$622$	0	0
$623$	2086.00i	0.134147i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	4992.00i	0.317961i
$628$	0	0
$629$	34580.0	2.19204
$630$	0	0
$631$	−8880.00	−0.560233	−0.280117	−	0.959966i	$-0.590373\pi$
$631$	−8880.00	−0.560233	−0.280117	+	0.959966i	$0.590373\pi$
$632$	0	0
$633$	− 7572.00i	− 0.475450i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	686.000i	0.0426692i
$638$	0	0
$639$	5796.00	0.358820
$640$	0	0
$641$	−2118.00	−0.130509	−0.0652543	−	0.997869i	$-0.520786\pi$
$641$	−2118.00	−0.130509	−0.0652543	+	0.997869i	$0.520786\pi$
$642$	0	0
$643$	− 2732.00i	− 0.167558i	−0.996484	−	0.0837788i	$-0.973301\pi$
$643$	− 2732.00i	− 0.167558i	0.996484	−	0.0837788i	$-0.0266989\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	10312.0i	0.626594i	0.949655	+	0.313297i	$0.101434\pi$
$647$	10312.0i	0.626594i	−0.949655	+	0.313297i	$0.898566\pi$
$648$	0	0
$649$	−1600.00	−0.0967727
$650$	0	0
$651$	588.000	0.0354002
$652$	0	0
$653$	− 11774.0i	− 0.705593i	−0.935700	−	0.352796i	$-0.885231\pi$
$653$	− 11774.0i	− 0.705593i	0.935700	−	0.352796i	$-0.114769\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2610.00i	0.154986i
$658$	0	0
$659$	−656.000	−0.0387771	−0.0193886	−	0.999812i	$-0.506172\pi$
$659$	−656.000	−0.0387771	−0.0193886	+	0.999812i	$0.506172\pi$
$660$	0	0
$661$	25358.0	1.49215	0.746076	−	0.665861i	$-0.231936\pi$
$661$	25358.0	1.49215	0.746076	+	0.665861i	$0.231936\pi$
$662$	0	0
$663$	− 5460.00i	− 0.319832i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4752.00i	0.275859i
$668$	0	0
$669$	12888.0	0.744811
$670$	0	0
$671$	−4960.00	−0.285363
$672$	0	0
$673$	− 15622.0i	− 0.894775i	−0.894340	−	0.447388i	$-0.852354\pi$
$673$	− 15622.0i	− 0.894775i	0.894340	−	0.447388i	$-0.147646\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 10854.0i	− 0.616179i	−0.951357	−	0.308089i	$-0.900310\pi$
$677$	− 10854.0i	− 0.616179i	0.951357	−	0.308089i	$-0.0996896\pi$
$678$	0	0
$679$	2030.00	0.114734
$680$	0	0
$681$	−14436.0	−0.812318
$682$	0	0
$683$	6004.00i	0.336364i	0.985756	+	0.168182i	$0.0537896\pi$
$683$	6004.00i	0.336364i	−0.985756	+	0.168182i	$0.946210\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	7206.00i	0.400183i
$688$	0	0
$689$	5628.00	0.311190
$690$	0	0
$691$	26176.0	1.44107	0.720537	−	0.693417i	$-0.243895\pi$
$691$	26176.0	1.44107	0.720537	+	0.693417i	$0.243895\pi$
$692$	0	0
$693$	− 1008.00i	− 0.0552536i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 26260.0i	− 1.42707i
$698$	0	0
$699$	−18474.0	−0.999644
$700$	0	0
$701$	−20874.0	−1.12468	−0.562340	−	0.826906i	$-0.690099\pi$
$701$	−20874.0	−1.12468	−0.562340	+	0.826906i	$0.690099\pi$
$702$	0	0
$703$	− 27664.0i	− 1.48416i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 10626.0i	− 0.565250i
$708$	0	0
$709$	−22750.0	−1.20507	−0.602535	−	0.798093i	$-0.705842\pi$
$709$	−22750.0	−1.20507	−0.602535	+	0.798093i	$0.705842\pi$
$710$	0	0
$711$	6696.00	0.353192
$712$	0	0
$713$	− 2464.00i	− 0.129421i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 8796.00i	− 0.458149i
$718$	0	0
$719$	−11160.0	−0.578856	−0.289428	−	0.957200i	$-0.593465\pi$
$719$	−11160.0	−0.578856	−0.289428	+	0.957200i	$0.593465\pi$
$720$	0	0
$721$	−4312.00	−0.222729
$722$	0	0
$723$	− 17010.0i	− 0.874977i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	21264.0i	1.08478i	0.840125	+	0.542392i	$0.182481\pi$
$727$	21264.0i	1.08478i	−0.840125	+	0.542392i	$0.817519\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	45240.0	2.28900
$732$	0	0
$733$	− 27438.0i	− 1.38260i	−0.722568	−	0.691300i	$-0.757038\pi$
$733$	− 27438.0i	− 1.38260i	0.722568	−	0.691300i	$-0.242962\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 5184.00i	− 0.259098i
$738$	0	0
$739$	−35388.0	−1.76153	−0.880764	−	0.473556i	$-0.842970\pi$
$739$	−35388.0	−1.76153	−0.880764	+	0.473556i	$0.842970\pi$
$740$	0	0
$741$	−4368.00	−0.216549
$742$	0	0
$743$	17568.0i	0.867439i	0.901048	+	0.433720i	$0.142799\pi$
$743$	17568.0i	0.867439i	−0.901048	+	0.433720i	$0.857201\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 9396.00i	− 0.460216i
$748$	0	0
$749$	364.000	0.0177574
$750$	0	0
$751$	11752.0	0.571021	0.285510	−	0.958376i	$-0.407837\pi$
$751$	11752.0	0.571021	0.285510	+	0.958376i	$0.407837\pi$
$752$	0	0
$753$	− 16860.0i	− 0.815953i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	10834.0i	0.520170i	0.965586	+	0.260085i	$0.0837505\pi$
$757$	10834.0i	0.520170i	−0.965586	+	0.260085i	$0.916249\pi$
$758$	0	0
$759$	−4224.00	−0.202005
$760$	0	0
$761$	6394.00	0.304576	0.152288	−	0.988336i	$-0.451336\pi$
$761$	6394.00	0.304576	0.152288	+	0.988336i	$0.451336\pi$
$762$	0	0
$763$	− 7238.00i	− 0.343425i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 1400.00i	− 0.0659075i
$768$	0	0
$769$	7526.00	0.352919	0.176459	−	0.984308i	$-0.443536\pi$
$769$	7526.00	0.352919	0.176459	+	0.984308i	$0.443536\pi$
$770$	0	0
$771$	13998.0	0.653859
$772$	0	0
$773$	− 874.000i	− 0.0406670i	−0.999793	−	0.0203335i	$-0.993527\pi$
$773$	− 874.000i	− 0.0406670i	0.999793	−	0.0203335i	$-0.00647280\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	5586.00i	0.257911i
$778$	0	0
$779$	−21008.0	−0.966226
$780$	0	0
$781$	10304.0	0.472095
$782$	0	0
$783$	1458.00i	0.0665449i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	26668.0i	1.20789i	0.797025	+	0.603946i	$0.206406\pi$
$787$	26668.0i	1.20789i	−0.797025	+	0.603946i	$0.793594\pi$
$788$	0	0
$789$	10944.0	0.493811
$790$	0	0
$791$	4634.00	0.208301
$792$	0	0
$793$	− 4340.00i	− 0.194348i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 15582.0i	− 0.692525i	−0.938138	−	0.346263i	$-0.887451\pi$
$797$	− 15582.0i	− 0.692525i	0.938138	−	0.346263i	$-0.112549\pi$
$798$	0	0
$799$	−13520.0	−0.598627
$800$	0	0
$801$	2682.00	0.118307
$802$	0	0
$803$	4640.00i	0.203913i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 3618.00i	− 0.157819i
$808$	0	0
$809$	24286.0	1.05544	0.527720	−	0.849419i	$-0.323047\pi$
$809$	24286.0	1.05544	0.527720	+	0.849419i	$0.323047\pi$
$810$	0	0
$811$	−5280.00	−0.228614	−0.114307	−	0.993445i	$-0.536465\pi$
$811$	−5280.00	−0.228614	−0.114307	+	0.993445i	$0.536465\pi$
$812$	0	0
$813$	− 1188.00i	− 0.0512484i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 36192.0i	− 1.54981i
$818$	0	0
$819$	882.000	0.0376307
$820$	0	0
$821$	13598.0	0.578043	0.289022	−	0.957323i	$-0.406670\pi$
$821$	13598.0	0.578043	0.289022	+	0.957323i	$0.406670\pi$
$822$	0	0
$823$	24208.0i	1.02532i	0.858592	+	0.512660i	$0.171340\pi$
$823$	24208.0i	1.02532i	−0.858592	+	0.512660i	$0.828660\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 4428.00i	− 0.186187i	−0.995657	−	0.0930935i	$-0.970324\pi$
$827$	− 4428.00i	− 0.186187i	0.995657	−	0.0930935i	$-0.0296756\pi$
$828$	0	0
$829$	−28142.0	−1.17903	−0.589513	−	0.807759i	$-0.700680\pi$
$829$	−28142.0	−1.17903	−0.589513	+	0.807759i	$0.700680\pi$
$830$	0	0
$831$	−17742.0	−0.740630
$832$	0	0
$833$	6370.00i	0.264955i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 756.000i	− 0.0312201i
$838$	0	0
$839$	15072.0	0.620195	0.310097	−	0.950705i	$-0.399638\pi$
$839$	15072.0	0.620195	0.310097	+	0.950705i	$0.399638\pi$
$840$	0	0
$841$	−21473.0	−0.880438
$842$	0	0
$843$	− 1434.00i	− 0.0585879i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	7525.00i	0.305268i
$848$	0	0
$849$	−18780.0	−0.759161
$850$	0	0
$851$	23408.0	0.942909
$852$	0	0
$853$	− 4118.00i	− 0.165296i	−0.996579	−	0.0826481i	$-0.973662\pi$
$853$	− 4118.00i	− 0.165296i	0.996579	−	0.0826481i	$-0.0263377\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 14914.0i	− 0.594460i	−0.954806	−	0.297230i	$-0.903937\pi$
$857$	− 14914.0i	− 0.594460i	0.954806	−	0.297230i	$-0.0960629\pi$
$858$	0	0
$859$	−13064.0	−0.518903	−0.259452	−	0.965756i	$-0.583542\pi$
$859$	−13064.0	−0.518903	−0.259452	+	0.965756i	$0.583542\pi$
$860$	0	0
$861$	4242.00	0.167906
$862$	0	0
$863$	5232.00i	0.206372i	0.994662	+	0.103186i	$0.0329037\pi$
$863$	5232.00i	0.206372i	−0.994662	+	0.103186i	$0.967096\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 35961.0i	− 1.40865i
$868$	0	0
$869$	11904.0	0.464690
$870$	0	0
$871$	4536.00	0.176460
$872$	0	0
$873$	− 2610.00i	− 0.101186i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 30158.0i	− 1.16119i	−0.814193	−	0.580595i	$-0.802820\pi$
$877$	− 30158.0i	− 1.16119i	0.814193	−	0.580595i	$-0.197180\pi$
$878$	0	0
$879$	3438.00	0.131924
$880$	0	0
$881$	17154.0	0.655997	0.327998	−	0.944678i	$-0.393626\pi$
$881$	17154.0	0.655997	0.327998	+	0.944678i	$0.393626\pi$
$882$	0	0
$883$	15868.0i	0.604757i	0.953188	+	0.302379i	$0.0977807\pi$
$883$	15868.0i	0.604757i	−0.953188	+	0.302379i	$0.902219\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	12928.0i	0.489380i	0.969601	+	0.244690i	$0.0786861\pi$
$887$	12928.0i	0.489380i	−0.969601	+	0.244690i	$0.921314\pi$
$888$	0	0
$889$	7896.00	0.297889
$890$	0	0
$891$	−1296.00	−0.0487291
$892$	0	0
$893$	10816.0i	0.405312i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 3696.00i	− 0.137576i
$898$	0	0
$899$	−1512.00	−0.0560935
$900$	0	0
$901$	52260.0	1.93233
$902$	0	0
$903$	7308.00i	0.269319i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 13572.0i	− 0.496859i	−0.968650	−	0.248429i	$-0.920086\pi$
$907$	− 13572.0i	− 0.496859i	0.968650	−	0.248429i	$-0.0799144\pi$
$908$	0	0
$909$	−13662.0	−0.498504
$910$	0	0
$911$	25164.0	0.915171	0.457585	−	0.889166i	$-0.348714\pi$
$911$	25164.0	0.915171	0.457585	+	0.889166i	$0.348714\pi$
$912$	0	0
$913$	− 16704.0i	− 0.605500i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 3444.00i	− 0.124025i
$918$	0	0
$919$	36944.0	1.32608	0.663041	−	0.748583i	$-0.269265\pi$
$919$	36944.0	1.32608	0.663041	+	0.748583i	$0.269265\pi$
$920$	0	0
$921$	−10740.0	−0.384251
$922$	0	0
$923$	9016.00i	0.321522i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	5544.00i	0.196428i
$928$	0	0
$929$	38542.0	1.36116	0.680582	−	0.732672i	$-0.261727\pi$
$929$	38542.0	1.36116	0.680582	+	0.732672i	$0.261727\pi$
$930$	0	0
$931$	5096.00	0.179393
$932$	0	0
$933$	− 16296.0i	− 0.571819i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 55134.0i	− 1.92225i	−0.276112	−	0.961126i	$-0.589046\pi$
$937$	− 55134.0i	− 1.92225i	0.276112	−	0.961126i	$-0.410954\pi$
$938$	0	0
$939$	−23394.0	−0.813029
$940$	0	0
$941$	35286.0	1.22241	0.611207	−	0.791471i	$-0.290684\pi$
$941$	35286.0	1.22241	0.611207	+	0.791471i	$0.290684\pi$
$942$	0	0
$943$	− 17776.0i	− 0.613856i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 25740.0i	− 0.883250i	−0.897200	−	0.441625i	$-0.854402\pi$
$947$	− 25740.0i	− 0.883250i	0.897200	−	0.441625i	$-0.145598\pi$
$948$	0	0
$949$	−4060.00	−0.138876
$950$	0	0
$951$	9390.00	0.320180
$952$	0	0
$953$	− 34914.0i	− 1.18675i	−0.804925	−	0.593376i	$-0.797795\pi$
$953$	− 34914.0i	− 1.18675i	0.804925	−	0.593376i	$-0.202205\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2592.00i	0.0875522i
$958$	0	0
$959$	8806.00	0.296518
$960$	0	0
$961$	−29007.0	−0.973683
$962$	0	0
$963$	− 468.000i	− 0.0156605i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 3784.00i	− 0.125838i	−0.998019	−	0.0629189i	$-0.979959\pi$
$967$	− 3784.00i	− 0.125838i	0.998019	−	0.0629189i	$-0.0200410\pi$
$968$	0	0
$969$	−40560.0	−1.34466
$970$	0	0
$971$	13452.0	0.444588	0.222294	−	0.974980i	$-0.428645\pi$
$971$	13452.0	0.444588	0.222294	+	0.974980i	$0.428645\pi$
$972$	0	0
$973$	− 15232.0i	− 0.501866i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	21978.0i	0.719691i	0.933012	+	0.359846i	$0.117171\pi$
$977$	21978.0i	0.719691i	−0.933012	+	0.359846i	$0.882829\pi$
$978$	0	0
$979$	4768.00	0.155655
$980$	0	0
$981$	−9306.00	−0.302872
$982$	0	0
$983$	50176.0i	1.62804i	0.580835	+	0.814021i	$0.302726\pi$
$983$	50176.0i	1.62804i	−0.580835	+	0.814021i	$0.697274\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 2184.00i	− 0.0704331i
$988$	0	0
$989$	30624.0	0.984617
$990$	0	0
$991$	13480.0	0.432095	0.216048	−	0.976383i	$-0.430683\pi$
$991$	13480.0	0.432095	0.216048	+	0.976383i	$0.430683\pi$
$992$	0	0
$993$	− 18060.0i	− 0.577157i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	33910.0i	1.07717i	0.842571	+	0.538586i	$0.181041\pi$
$997$	33910.0i	1.07717i	−0.842571	+	0.538586i	$0.818959\pi$
$998$	0	0
$999$	7182.00	0.227456

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2100.4.k.e.1849.2		2
5.2	odd	4		420.4.a.e.1.1	✓	1
5.3	odd	4		2100.4.a.b.1.1		1
5.4	even	2	inner	2100.4.k.e.1849.1		2
15.2	even	4		1260.4.a.f.1.1		1
20.7	even	4		1680.4.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.4.a.e.1.1	✓	1	5.2	odd	4
1260.4.a.f.1.1		1	15.2	even	4
1680.4.a.g.1.1		1	20.7	even	4
2100.4.a.b.1.1		1	5.3	odd	4
2100.4.k.e.1849.1		2	5.4	even	2	inner
2100.4.k.e.1849.2		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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2100.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} -7.00000i q^{7} -9.00000 q^{9} -16.0000 q^{11} -14.0000i q^{13} -130.000i q^{17} -104.000 q^{19} +21.0000 q^{21} -88.0000i q^{23} -27.0000i q^{27} -54.0000 q^{29} +28.0000 q^{31} -48.0000i q^{33} +266.000i q^{37} +42.0000 q^{39} +202.000 q^{41} +348.000i q^{43} -104.000i q^{47} -49.0000 q^{49} +390.000 q^{51} +402.000i q^{53} -312.000i q^{57} +100.000 q^{59} +310.000 q^{61} +63.0000i q^{63} +324.000i q^{67} +264.000 q^{69} -644.000 q^{71} -290.000i q^{73} +112.000i q^{77} -744.000 q^{79} +81.0000 q^{81} +1044.00i q^{83} -162.000i q^{87} -298.000 q^{89} -98.0000 q^{91} +84.0000i q^{93} +290.000i q^{97} +144.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{9} - 32 q^{11} - 208 q^{19} + 42 q^{21} - 108 q^{29} + 56 q^{31} + 84 q^{39} + 404 q^{41} - 98 q^{49} + 780 q^{51} + 200 q^{59} + 620 q^{61} + 528 q^{69} - 1288 q^{71} - 1488 q^{79} + 162 q^{81}+ \cdots + 288 q^{99}+O(q^{100}) \) 2 * q - 18 * q^9 - 32 * q^11 - 208 * q^19 + 42 * q^21 - 108 * q^29 + 56 * q^31 + 84 * q^39 + 404 * q^41 - 98 * q^49 + 780 * q^51 + 200 * q^59 + 620 * q^61 + 528 * q^69 - 1288 * q^71 - 1488 * q^79 + 162 * q^81 - 596 * q^89 - 196 * q^91 + 288 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more