LMFDB - Embedded newform 2300.4.c.d.1749.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2300,4,Mod(1749,2300)] 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("2300.1749"); S:= CuspForms(chi, 4); N := Newforms(S);

sage: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")

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:traces = [10,0,0,0,0,0,0,0,-60,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$135.704393013$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 164x^{8} + 9308x^{6} + 204385x^{4} + 1216296x^{2} + 1089936 $ x^10 + 164x^8 + 9308x^6 + 204385x^4 + 1216296x^2 + 1089936
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
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			1749.3
Root			$-6.72467i$ of defining polynomial
Character	$\chi$	$=$	2300.1749
Dual form			2300.4.c.d.1749.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-5.72467i q^{3} -12.6137i q^{7} -5.77185 q^{9} -21.9786 q^{11} -37.2893i q^{13} +62.5817i q^{17} +137.633 q^{19} -72.2093 q^{21} +23.0000i q^{23} -121.524i q^{27} +71.8975 q^{29} -219.731 q^{31} +125.820i q^{33} -296.223i q^{37} -213.469 q^{39} -305.282 q^{41} -140.234i q^{43} -185.509i q^{47} +183.895 q^{49} +358.260 q^{51} -312.176i q^{53} -787.903i q^{57} +161.771 q^{59} +835.780 q^{61} +72.8043i q^{63} -24.5018i q^{67} +131.667 q^{69} -973.973 q^{71} +343.447i q^{73} +277.231i q^{77} -1214.62 q^{79} -851.526 q^{81} -749.055i q^{83} -411.590i q^{87} -811.662 q^{89} -470.356 q^{91} +1257.89i q^{93} +1026.90i q^{97} +126.857 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 60 q^{9} + 14 q^{11} + 26 q^{19} + 506 q^{29} - 196 q^{31} + 820 q^{39} - 1548 q^{41} + 1366 q^{49} - 1314 q^{51} + 1526 q^{59} + 674 q^{61} - 138 q^{69} - 3008 q^{71} - 1252 q^{79} - 1918 q^{81}+ \cdots - 222 q^{99}+O(q^{100}) $ 10 * q - 60 * q^9 + 14 * q^11 + 26 * q^19 + 506 * q^29 - 196 * q^31 + 820 * q^39 - 1548 * q^41 + 1366 * q^49 - 1314 * q^51 + 1526 * q^59 + 674 * q^61 - 138 * q^69 - 3008 * q^71 - 1252 * q^79 - 1918 * q^81 - 1292 * q^89 - 1534 * q^91 - 222 * q^99

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
$3$	− 5.72467i	− 1.10171i	−0.834600	−	0.550857i	$-0.814301\pi$
$3$	− 5.72467i	− 1.10171i	0.834600	−	0.550857i	$-0.185699\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 12.6137i	− 0.681076i	−0.940231	−	0.340538i	$-0.889391\pi$
$7$	− 12.6137i	− 0.681076i	0.940231	−	0.340538i	$-0.110609\pi$
$8$	0	0
$9$	−5.77185	−0.213772
$10$	0	0
$11$	−21.9786	−0.602435	−0.301217	−	0.953555i	$-0.597393\pi$
$11$	−21.9786	−0.602435	−0.301217	+	0.953555i	$0.597393\pi$
$12$	0	0
$13$	− 37.2893i	− 0.795554i	−0.917482	−	0.397777i	$-0.869782\pi$
$13$	− 37.2893i	− 0.795554i	0.917482	−	0.397777i	$-0.130218\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	62.5817i	0.892841i	0.894823	+	0.446420i	$0.147301\pi$
$17$	62.5817i	0.892841i	−0.894823	+	0.446420i	$0.852699\pi$
$18$	0	0
$19$	137.633	1.66185	0.830925	−	0.556384i	$-0.187812\pi$
$19$	137.633	1.66185	0.830925	+	0.556384i	$0.187812\pi$
$20$	0	0
$21$	−72.2093	−0.750350
$22$	0	0
$23$	23.0000i	0.208514i
$23$	23.0000i	0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 121.524i	− 0.866198i
$28$	0	0
$29$	71.8975	0.460380	0.230190	−	0.973146i	$-0.426065\pi$
$29$	71.8975	0.460380	0.230190	+	0.973146i	$0.426065\pi$
$30$	0	0
$31$	−219.731	−1.27306	−0.636531	−	0.771251i	$-0.719631\pi$
$31$	−219.731	−1.27306	−0.636531	+	0.771251i	$0.719631\pi$
$32$	0	0
$33$	125.820i	0.663710i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 296.223i	− 1.31618i	−0.752939	−	0.658091i	$-0.771364\pi$
$37$	− 296.223i	− 1.31618i	0.752939	−	0.658091i	$-0.228636\pi$
$38$	0	0
$39$	−213.469	−0.876473
$40$	0	0
$41$	−305.282	−1.16285	−0.581427	−	0.813599i	$-0.697505\pi$
$41$	−305.282	−1.16285	−0.581427	+	0.813599i	$0.697505\pi$
$42$	0	0
$43$	− 140.234i	− 0.497338i	−0.968588	−	0.248669i	$-0.920007\pi$
$43$	− 140.234i	− 0.497338i	0.968588	−	0.248669i	$-0.0799932\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 185.509i	− 0.575730i	−0.957671	−	0.287865i	$-0.907055\pi$
$47$	− 185.509i	− 0.575730i	0.957671	−	0.287865i	$-0.0929454\pi$
$48$	0	0
$49$	183.895	0.536136
$50$	0	0
$51$	358.260	0.983654
$52$	0	0
$53$	− 312.176i	− 0.809069i	−0.914523	−	0.404534i	$-0.867434\pi$
$53$	− 312.176i	− 0.809069i	0.914523	−	0.404534i	$-0.132566\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 787.903i	− 1.83088i
$58$	0	0
$59$	161.771	0.356964	0.178482	−	0.983943i	$-0.442881\pi$
$59$	161.771	0.356964	0.178482	+	0.983943i	$0.442881\pi$
$60$	0	0
$61$	835.780	1.75427	0.877137	−	0.480240i	$-0.159450\pi$
$61$	835.780	1.75427	0.877137	+	0.480240i	$0.159450\pi$
$62$	0	0
$63$	72.8043i	0.145595i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 24.5018i	− 0.0446772i	−0.999750	−	0.0223386i	$-0.992889\pi$
$67$	− 24.5018i	− 0.0446772i	0.999750	−	0.0223386i	$-0.00711119\pi$
$68$	0	0
$69$	131.667	0.229723
$70$	0	0
$71$	−973.973	−1.62802	−0.814009	−	0.580852i	$-0.802719\pi$
$71$	−973.973	−1.62802	−0.814009	+	0.580852i	$0.802719\pi$
$72$	0	0
$73$	343.447i	0.550650i	0.961351	+	0.275325i	$0.0887855\pi$
$73$	343.447i	0.550650i	−0.961351	+	0.275325i	$0.911214\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	277.231i	0.410304i
$78$	0	0
$79$	−1214.62	−1.72981	−0.864905	−	0.501935i	$-0.832622\pi$
$79$	−1214.62	−1.72981	−0.864905	+	0.501935i	$0.832622\pi$
$80$	0	0
$81$	−851.526	−1.16807
$82$	0	0
$83$	− 749.055i	− 0.990597i	−0.868723	−	0.495298i	$-0.835059\pi$
$83$	− 749.055i	− 0.990597i	0.868723	−	0.495298i	$-0.164941\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 411.590i	− 0.507207i
$88$	0	0
$89$	−811.662	−0.966696	−0.483348	−	0.875428i	$-0.660579\pi$
$89$	−811.662	−0.966696	−0.483348	+	0.875428i	$0.660579\pi$
$90$	0	0
$91$	−470.356	−0.541833
$92$	0	0
$93$	1257.89i	1.40255i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1026.90i	1.07491i	0.843293	+	0.537455i	$0.180614\pi$
$97$	1026.90i	1.07491i	−0.843293	+	0.537455i	$0.819386\pi$
$98$	0	0
$99$	126.857	0.128784
$100$	0	0
$101$	−788.330	−0.776651	−0.388326	−	0.921522i	$-0.626946\pi$
$101$	−788.330	−0.776651	−0.388326	+	0.921522i	$0.626946\pi$
$102$	0	0
$103$	− 471.764i	− 0.451304i	−0.974208	−	0.225652i	$-0.927549\pi$
$103$	− 471.764i	− 0.451304i	0.974208	−	0.225652i	$-0.0724512\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1402.85i	− 1.26746i	−0.773553	−	0.633732i	$-0.781522\pi$
$107$	− 1402.85i	− 1.26746i	0.773553	−	0.633732i	$-0.218478\pi$
$108$	0	0
$109$	2151.31	1.89044	0.945221	−	0.326432i	$-0.105846\pi$
$109$	2151.31	1.89044	0.945221	+	0.326432i	$0.105846\pi$
$110$	0	0
$111$	−1695.78	−1.45005
$112$	0	0
$113$	− 754.833i	− 0.628396i	−0.949358	−	0.314198i	$-0.898264\pi$
$113$	− 754.833i	− 0.628396i	0.949358	−	0.314198i	$-0.101736\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	215.228i	0.170067i
$118$	0	0
$119$	789.387	0.608092
$120$	0	0
$121$	−847.943	−0.637072
$122$	0	0
$123$	1747.64i	1.28113i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2161.28i	1.51010i	0.655667	+	0.755050i	$0.272388\pi$
$127$	2161.28i	1.51010i	−0.655667	+	0.755050i	$0.727612\pi$
$128$	0	0
$129$	−802.796	−0.547924
$130$	0	0
$131$	397.736	0.265270	0.132635	−	0.991165i	$-0.457656\pi$
$131$	397.736	0.265270	0.132635	+	0.991165i	$0.457656\pi$
$132$	0	0
$133$	− 1736.06i	− 1.13185i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1660.73i	− 1.03566i	−0.855483	−	0.517831i	$-0.826740\pi$
$137$	− 1660.73i	− 1.03566i	0.855483	−	0.517831i	$-0.173260\pi$
$138$	0	0
$139$	997.853	0.608897	0.304449	−	0.952529i	$-0.401528\pi$
$139$	997.853	0.608897	0.304449	+	0.952529i	$0.401528\pi$
$140$	0	0
$141$	−1061.98	−0.634289
$142$	0	0
$143$	819.566i	0.479270i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 1052.74i	− 0.590668i
$148$	0	0
$149$	−3014.50	−1.65743	−0.828715	−	0.559671i	$-0.810928\pi$
$149$	−3014.50	−1.65743	−0.828715	+	0.559671i	$0.810928\pi$
$150$	0	0
$151$	−684.908	−0.369120	−0.184560	−	0.982821i	$-0.559086\pi$
$151$	−684.908	−0.369120	−0.184560	+	0.982821i	$0.559086\pi$
$152$	0	0
$153$	− 361.212i	− 0.190864i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	727.598i	0.369864i	0.982751	+	0.184932i	$0.0592065\pi$
$157$	727.598i	0.369864i	−0.982751	+	0.184932i	$0.940794\pi$
$158$	0	0
$159$	−1787.10	−0.891362
$160$	0	0
$161$	290.115	0.142014
$162$	0	0
$163$	768.718i	0.369390i	0.982796	+	0.184695i	$0.0591298\pi$
$163$	768.718i	0.369390i	−0.982796	+	0.184695i	$0.940870\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 243.397i	− 0.112782i	−0.998409	−	0.0563910i	$-0.982041\pi$
$167$	− 243.397i	− 0.112782i	0.998409	−	0.0563910i	$-0.0179593\pi$
$168$	0	0
$169$	806.505	0.367094
$170$	0	0
$171$	−794.396	−0.355257
$172$	0	0
$173$	201.538i	0.0885702i	0.999019	+	0.0442851i	$0.0141010\pi$
$173$	201.538i	0.0885702i	−0.999019	+	0.0442851i	$0.985899\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 926.088i	− 0.393272i
$178$	0	0
$179$	722.564	0.301715	0.150857	−	0.988556i	$-0.451797\pi$
$179$	722.564	0.301715	0.150857	+	0.988556i	$0.451797\pi$
$180$	0	0
$181$	2117.76	0.869678	0.434839	−	0.900508i	$-0.356805\pi$
$181$	2117.76	0.869678	0.434839	+	0.900508i	$0.356805\pi$
$182$	0	0
$183$	− 4784.57i	− 1.93271i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 1375.46i	− 0.537878i
$188$	0	0
$189$	−1532.87	−0.589946
$190$	0	0
$191$	82.9624	0.0314290	0.0157145	−	0.999877i	$-0.494998\pi$
$191$	82.9624	0.0314290	0.0157145	+	0.999877i	$0.494998\pi$
$192$	0	0
$193$	− 1350.43i	− 0.503657i	−0.967772	−	0.251829i	$-0.918968\pi$
$193$	− 1350.43i	− 0.503657i	0.967772	−	0.251829i	$-0.0810320\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4317.69i	1.56154i	0.624820	+	0.780769i	$0.285172\pi$
$197$	4317.69i	1.56154i	−0.624820	+	0.780769i	$0.714828\pi$
$198$	0	0
$199$	−3026.27	−1.07802	−0.539011	−	0.842299i	$-0.681202\pi$
$199$	−3026.27	−1.07802	−0.539011	+	0.842299i	$0.681202\pi$
$200$	0	0
$201$	−140.265	−0.0492215
$202$	0	0
$203$	− 906.893i	− 0.313554i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 132.752i	− 0.0445746i
$208$	0	0
$209$	−3024.97	−1.00116
$210$	0	0
$211$	2292.01	0.747812	0.373906	−	0.927467i	$-0.378018\pi$
$211$	2292.01	0.747812	0.373906	+	0.927467i	$0.378018\pi$
$212$	0	0
$213$	5575.67i	1.79361i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	2771.62i	0.867051i
$218$	0	0
$219$	1966.12	0.606659
$220$	0	0
$221$	2333.63	0.710303
$222$	0	0
$223$	2997.29i	0.900061i	0.893013	+	0.450030i	$0.148587\pi$
$223$	2997.29i	0.900061i	−0.893013	+	0.450030i	$0.851413\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 5786.16i	− 1.69181i	−0.533333	−	0.845905i	$-0.679061\pi$
$227$	− 5786.16i	− 1.69181i	0.533333	−	0.845905i	$-0.320939\pi$
$228$	0	0
$229$	−4516.35	−1.30327	−0.651635	−	0.758532i	$-0.725917\pi$
$229$	−4516.35	−1.30327	−0.651635	+	0.758532i	$0.725917\pi$
$230$	0	0
$231$	1587.05	0.452037
$232$	0	0
$233$	− 2990.75i	− 0.840903i	−0.907315	−	0.420452i	$-0.861872\pi$
$233$	− 2990.75i	− 0.840903i	0.907315	−	0.420452i	$-0.138128\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	6953.28i	1.90575i
$238$	0	0
$239$	−4043.74	−1.09443	−0.547213	−	0.836994i	$-0.684311\pi$
$239$	−4043.74	−1.09443	−0.547213	+	0.836994i	$0.684311\pi$
$240$	0	0
$241$	3653.64	0.976564	0.488282	−	0.872686i	$-0.337624\pi$
$241$	3653.64	0.976564	0.488282	+	0.872686i	$0.337624\pi$
$242$	0	0
$243$	1593.55i	0.420684i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 5132.24i	− 1.32209i
$248$	0	0
$249$	−4288.09	−1.09135
$250$	0	0
$251$	−6799.45	−1.70987	−0.854935	−	0.518735i	$-0.826403\pi$
$251$	−6799.45	−1.70987	−0.854935	+	0.518735i	$0.826403\pi$
$252$	0	0
$253$	− 505.507i	− 0.125616i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	179.925i	0.0436710i	0.999762	+	0.0218355i	$0.00695100\pi$
$257$	179.925i	0.0436710i	−0.999762	+	0.0218355i	$0.993049\pi$
$258$	0	0
$259$	−3736.46	−0.896419
$260$	0	0
$261$	−414.981	−0.0984165
$262$	0	0
$263$	1148.48i	0.269271i	0.990895	+	0.134635i	$0.0429863\pi$
$263$	1148.48i	0.269271i	−0.990895	+	0.134635i	$0.957014\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	4646.50i	1.06502i
$268$	0	0
$269$	−2999.90	−0.679953	−0.339976	−	0.940434i	$-0.610419\pi$
$269$	−2999.90	−0.679953	−0.339976	+	0.940434i	$0.610419\pi$
$270$	0	0
$271$	5587.83	1.25253	0.626266	−	0.779609i	$-0.284582\pi$
$271$	5587.83	1.25253	0.626266	+	0.779609i	$0.284582\pi$
$272$	0	0
$273$	2692.64i	0.596944i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	6219.06i	1.34898i	0.738285	+	0.674489i	$0.235636\pi$
$277$	6219.06i	1.34898i	−0.738285	+	0.674489i	$0.764364\pi$
$278$	0	0
$279$	1268.26	0.272145
$280$	0	0
$281$	−6770.91	−1.43743	−0.718717	−	0.695303i	$-0.755270\pi$
$281$	−6770.91	−1.43743	−0.718717	+	0.695303i	$0.755270\pi$
$282$	0	0
$283$	7811.42i	1.64078i	0.571804	+	0.820390i	$0.306244\pi$
$283$	7811.42i	1.64078i	−0.571804	+	0.820390i	$0.693756\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3850.73i	0.791991i
$288$	0	0
$289$	996.531	0.202836
$290$	0	0
$291$	5878.68	1.18424
$292$	0	0
$293$	8415.02i	1.67785i	0.544246	+	0.838926i	$0.316816\pi$
$293$	8415.02i	1.67785i	−0.544246	+	0.838926i	$0.683184\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2670.93i	0.521828i
$298$	0	0
$299$	857.655	0.165884
$300$	0	0
$301$	−1768.87	−0.338725
$302$	0	0
$303$	4512.93i	0.855647i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 9633.23i	− 1.79087i	−0.445192	−	0.895435i	$-0.646865\pi$
$307$	− 9633.23i	− 1.79087i	0.445192	−	0.895435i	$-0.353135\pi$
$308$	0	0
$309$	−2700.69	−0.497207
$310$	0	0
$311$	−7000.64	−1.27643	−0.638216	−	0.769858i	$-0.720327\pi$
$311$	−7000.64	−1.27643	−0.638216	+	0.769858i	$0.720327\pi$
$312$	0	0
$313$	2954.79i	0.533593i	0.963753	+	0.266796i	$0.0859651\pi$
$313$	2954.79i	0.533593i	−0.963753	+	0.266796i	$0.914035\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 4015.68i	− 0.711492i	−0.934583	−	0.355746i	$-0.884227\pi$
$317$	− 4015.68i	− 0.711492i	0.934583	−	0.355746i	$-0.115773\pi$
$318$	0	0
$319$	−1580.20	−0.277349
$320$	0	0
$321$	−8030.85	−1.39638
$322$	0	0
$323$	8613.30i	1.48377i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 12315.5i	− 2.08272i
$328$	0	0
$329$	−2339.96	−0.392116
$330$	0	0
$331$	596.567	0.0990644	0.0495322	−	0.998773i	$-0.484227\pi$
$331$	596.567	0.0990644	0.0495322	+	0.998773i	$0.484227\pi$
$332$	0	0
$333$	1709.75i	0.281363i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2162.05i	0.349479i	0.984615	+	0.174739i	$0.0559083\pi$
$337$	2162.05i	0.349479i	−0.984615	+	0.174739i	$0.944092\pi$
$338$	0	0
$339$	−4321.17	−0.692312
$340$	0	0
$341$	4829.38	0.766937
$342$	0	0
$343$	− 6646.09i	− 1.04622i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 8014.61i	− 1.23990i	−0.784639	−	0.619952i	$-0.787152\pi$
$347$	− 8014.61i	− 1.23990i	0.784639	−	0.619952i	$-0.212848\pi$
$348$	0	0
$349$	−3264.77	−0.500743	−0.250371	−	0.968150i	$-0.580553\pi$
$349$	−3264.77	−0.500743	−0.250371	+	0.968150i	$0.580553\pi$
$350$	0	0
$351$	−4531.56	−0.689107
$352$	0	0
$353$	6659.27i	1.00407i	0.864847	+	0.502036i	$0.167416\pi$
$353$	6659.27i	1.00407i	−0.864847	+	0.502036i	$0.832584\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 4518.98i	− 0.669943i
$358$	0	0
$359$	−670.397	−0.0985577	−0.0492789	−	0.998785i	$-0.515692\pi$
$359$	−670.397	−0.0985577	−0.0492789	+	0.998785i	$0.515692\pi$
$360$	0	0
$361$	12083.8	1.76175
$362$	0	0
$363$	4854.19i	0.701871i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3528.04i	0.501804i	0.968013	+	0.250902i	$0.0807272\pi$
$367$	3528.04i	0.501804i	−0.968013	+	0.250902i	$0.919273\pi$
$368$	0	0
$369$	1762.04	0.248586
$370$	0	0
$371$	−3937.69	−0.551037
$372$	0	0
$373$	2718.66i	0.377391i	0.982036	+	0.188695i	$0.0604258\pi$
$373$	2718.66i	0.377391i	−0.982036	+	0.188695i	$0.939574\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 2681.01i	− 0.366258i
$378$	0	0
$379$	−10773.8	−1.46019	−0.730096	−	0.683345i	$-0.760525\pi$
$379$	−10773.8	−1.46019	−0.730096	+	0.683345i	$0.760525\pi$
$380$	0	0
$381$	12372.6	1.66370
$382$	0	0
$383$	6104.18i	0.814384i	0.913343	+	0.407192i	$0.133492\pi$
$383$	6104.18i	0.814384i	−0.913343	+	0.407192i	$0.866508\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	809.412i	0.106317i
$388$	0	0
$389$	5056.94	0.659118	0.329559	−	0.944135i	$-0.393100\pi$
$389$	5056.94	0.659118	0.329559	+	0.944135i	$0.393100\pi$
$390$	0	0
$391$	−1439.38	−0.186170
$392$	0	0
$393$	− 2276.91i	− 0.292251i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	8320.52i	1.05188i	0.850523	+	0.525938i	$0.176286\pi$
$397$	8320.52i	1.05188i	−0.850523	+	0.525938i	$0.823714\pi$
$398$	0	0
$399$	−9938.37	−1.24697
$400$	0	0
$401$	1253.78	0.156137	0.0780685	−	0.996948i	$-0.475125\pi$
$401$	1253.78	0.156137	0.0780685	+	0.996948i	$0.475125\pi$
$402$	0	0
$403$	8193.64i	1.01279i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6510.55i	0.792914i
$408$	0	0
$409$	11283.0	1.36408	0.682041	−	0.731314i	$-0.261093\pi$
$409$	11283.0	1.36408	0.682041	+	0.731314i	$0.261093\pi$
$410$	0	0
$411$	−9507.13	−1.14100
$412$	0	0
$413$	− 2040.54i	− 0.243119i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 5712.38i	− 0.670830i
$418$	0	0
$419$	−4178.10	−0.487144	−0.243572	−	0.969883i	$-0.578319\pi$
$419$	−4178.10	−0.487144	−0.243572	+	0.969883i	$0.578319\pi$
$420$	0	0
$421$	−11693.0	−1.35364	−0.676821	−	0.736147i	$-0.736643\pi$
$421$	−11693.0	−1.35364	−0.676821	+	0.736147i	$0.736643\pi$
$422$	0	0
$423$	1070.73i	0.123075i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 10542.3i	− 1.19479i
$428$	0	0
$429$	4691.74	0.528018
$430$	0	0
$431$	−11158.2	−1.24703	−0.623515	−	0.781812i	$-0.714296\pi$
$431$	−11158.2	−1.24703	−0.623515	+	0.781812i	$0.714296\pi$
$432$	0	0
$433$	13192.6i	1.46419i	0.681201	+	0.732096i	$0.261458\pi$
$433$	13192.6i	1.46419i	−0.681201	+	0.732096i	$0.738542\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3165.56i	0.346520i
$438$	0	0
$439$	1659.40	0.180407	0.0902037	−	0.995923i	$-0.471248\pi$
$439$	1659.40	0.180407	0.0902037	+	0.995923i	$0.471248\pi$
$440$	0	0
$441$	−1061.41	−0.114611
$442$	0	0
$443$	− 5144.57i	− 0.551752i	−0.961193	−	0.275876i	$-0.911032\pi$
$443$	− 5144.57i	− 0.551752i	0.961193	−	0.275876i	$-0.0889679\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	17257.0i	1.82601i
$448$	0	0
$449$	−5469.59	−0.574891	−0.287446	−	0.957797i	$-0.592806\pi$
$449$	−5469.59	−0.574891	−0.287446	+	0.957797i	$0.592806\pi$
$450$	0	0
$451$	6709.65	0.700544
$452$	0	0
$453$	3920.87i	0.406664i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6063.04i	0.620606i	0.950638	+	0.310303i	$0.100431\pi$
$457$	6063.04i	0.620606i	−0.950638	+	0.310303i	$0.899569\pi$
$458$	0	0
$459$	7605.19	0.773377
$460$	0	0
$461$	4025.39	0.406684	0.203342	−	0.979108i	$-0.434820\pi$
$461$	4025.39	0.406684	0.203342	+	0.979108i	$0.434820\pi$
$462$	0	0
$463$	4617.49i	0.463483i	0.972777	+	0.231742i	$0.0744424\pi$
$463$	4617.49i	0.463483i	−0.972777	+	0.231742i	$0.925558\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4012.00i	0.397544i	0.980046	+	0.198772i	$0.0636953\pi$
$467$	4012.00i	0.397544i	−0.980046	+	0.198772i	$0.936305\pi$
$468$	0	0
$469$	−309.058	−0.0304285
$470$	0	0
$471$	4165.26	0.407484
$472$	0	0
$473$	3082.15i	0.299614i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1801.83i	0.172956i
$478$	0	0
$479$	7317.02	0.697961	0.348980	−	0.937130i	$-0.386528\pi$
$479$	7317.02	0.697961	0.348980	+	0.937130i	$0.386528\pi$
$480$	0	0
$481$	−11046.0	−1.04709
$482$	0	0
$483$	− 1660.81i	− 0.156459i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 3708.67i	− 0.345084i	−0.985002	−	0.172542i	$-0.944802\pi$
$487$	− 3708.67i	− 0.345084i	0.985002	−	0.172542i	$-0.0551981\pi$
$488$	0	0
$489$	4400.66	0.406962
$490$	0	0
$491$	6689.69	0.614871	0.307435	−	0.951569i	$-0.400529\pi$
$491$	6689.69	0.614871	0.307435	+	0.951569i	$0.400529\pi$
$492$	0	0
$493$	4499.47i	0.411046i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	12285.4i	1.10880i
$498$	0	0
$499$	−17381.9	−1.55936	−0.779678	−	0.626180i	$-0.784617\pi$
$499$	−17381.9	−1.55936	−0.779678	+	0.626180i	$0.784617\pi$
$500$	0	0
$501$	−1393.37	−0.124253
$502$	0	0
$503$	− 9771.20i	− 0.866156i	−0.901357	−	0.433078i	$-0.857428\pi$
$503$	− 9771.20i	− 0.866156i	0.901357	−	0.433078i	$-0.142572\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 4616.97i	− 0.404432i
$508$	0	0
$509$	−5820.46	−0.506852	−0.253426	−	0.967355i	$-0.581557\pi$
$509$	−5820.46	−0.506852	−0.253426	+	0.967355i	$0.581557\pi$
$510$	0	0
$511$	4332.14	0.375035
$512$	0	0
$513$	− 16725.7i	− 1.43949i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	4077.22i	0.346840i
$518$	0	0
$519$	1153.74	0.0975789
$520$	0	0
$521$	−6357.64	−0.534613	−0.267307	−	0.963612i	$-0.586134\pi$
$521$	−6357.64	−0.534613	−0.267307	+	0.963612i	$0.586134\pi$
$522$	0	0
$523$	10556.2i	0.882580i	0.897364	+	0.441290i	$0.145479\pi$
$523$	10556.2i	0.882580i	−0.897364	+	0.441290i	$0.854521\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 13751.2i	− 1.13664i
$528$	0	0
$529$	−529.000	−0.0434783
$530$	0	0
$531$	−933.720	−0.0763089
$532$	0	0
$533$	11383.8i	0.925113i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 4136.44i	− 0.332403i
$538$	0	0
$539$	−4041.74	−0.322987
$540$	0	0
$541$	−16883.4	−1.34172	−0.670862	−	0.741582i	$-0.734076\pi$
$541$	−16883.4	−1.34172	−0.670862	+	0.741582i	$0.734076\pi$
$542$	0	0
$543$	− 12123.5i	− 0.958136i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 14318.4i	− 1.11922i	−0.828757	−	0.559608i	$-0.810952\pi$
$547$	− 14318.4i	− 1.11922i	0.828757	−	0.559608i	$-0.189048\pi$
$548$	0	0
$549$	−4824.00	−0.375015
$550$	0	0
$551$	9895.47	0.765084
$552$	0	0
$553$	15320.8i	1.17813i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 11764.7i	− 0.894946i	−0.894297	−	0.447473i	$-0.852324\pi$
$557$	− 11764.7i	− 0.894946i	0.894297	−	0.447473i	$-0.147676\pi$
$558$	0	0
$559$	−5229.25	−0.395660
$560$	0	0
$561$	−7874.03	−0.592588
$562$	0	0
$563$	21201.3i	1.58709i	0.608514	+	0.793543i	$0.291766\pi$
$563$	21201.3i	1.58709i	−0.608514	+	0.793543i	$0.708234\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	10740.9i	0.795546i
$568$	0	0
$569$	17206.5	1.26772	0.633861	−	0.773447i	$-0.281469\pi$
$569$	17206.5	1.26772	0.633861	+	0.773447i	$0.281469\pi$
$570$	0	0
$571$	12208.3	0.894746	0.447373	−	0.894348i	$-0.352360\pi$
$571$	12208.3	0.894746	0.447373	+	0.894348i	$0.352360\pi$
$572$	0	0
$573$	− 474.932i	− 0.0346258i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	12432.0i	0.896967i	0.893791	+	0.448484i	$0.148036\pi$
$577$	12432.0i	0.896967i	−0.893791	+	0.448484i	$0.851964\pi$
$578$	0	0
$579$	−7730.75	−0.554886
$580$	0	0
$581$	−9448.36	−0.674671
$582$	0	0
$583$	6861.17i	0.487411i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 18732.7i	− 1.31718i	−0.752504	−	0.658588i	$-0.771154\pi$
$587$	− 18732.7i	− 1.31718i	0.752504	−	0.658588i	$-0.228846\pi$
$588$	0	0
$589$	−30242.3	−2.11564
$590$	0	0
$591$	24717.4	1.72037
$592$	0	0
$593$	− 16320.1i	− 1.13016i	−0.825036	−	0.565081i	$-0.808845\pi$
$593$	− 16320.1i	− 1.13016i	0.825036	−	0.565081i	$-0.191155\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	17324.4i	1.18767i
$598$	0	0
$599$	7015.52	0.478542	0.239271	−	0.970953i	$-0.423092\pi$
$599$	7015.52	0.478542	0.239271	+	0.970953i	$0.423092\pi$
$600$	0	0
$601$	−4517.10	−0.306583	−0.153292	−	0.988181i	$-0.548987\pi$
$601$	−4517.10	−0.306583	−0.153292	+	0.988181i	$0.548987\pi$
$602$	0	0
$603$	141.421i	0.00955074i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 9861.96i	− 0.659448i	−0.944077	−	0.329724i	$-0.893044\pi$
$607$	− 9861.96i	− 0.659448i	0.944077	−	0.329724i	$-0.106956\pi$
$608$	0	0
$609$	−5191.67	−0.345447
$610$	0	0
$611$	−6917.52	−0.458024
$612$	0	0
$613$	− 20044.7i	− 1.32071i	−0.750953	−	0.660356i	$-0.770405\pi$
$613$	− 20044.7i	− 1.32071i	0.750953	−	0.660356i	$-0.229595\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 3604.26i	− 0.235174i	−0.993063	−	0.117587i	$-0.962484\pi$
$617$	− 3604.26i	− 0.235174i	0.993063	−	0.117587i	$-0.0375158\pi$
$618$	0	0
$619$	−14864.7	−0.965204	−0.482602	−	0.875840i	$-0.660308\pi$
$619$	−14864.7	−0.965204	−0.482602	+	0.875840i	$0.660308\pi$
$620$	0	0
$621$	2795.06	0.180615
$622$	0	0
$623$	10238.1i	0.658393i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	17317.0i	1.10299i
$628$	0	0
$629$	18538.1	1.17514
$630$	0	0
$631$	12677.7	0.799828	0.399914	−	0.916553i	$-0.369040\pi$
$631$	12677.7	0.799828	0.399914	+	0.916553i	$0.369040\pi$
$632$	0	0
$633$	− 13121.0i	− 0.823875i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 6857.31i	− 0.426525i
$638$	0	0
$639$	5621.62	0.348025
$640$	0	0
$641$	−19168.5	−1.18114	−0.590571	−	0.806986i	$-0.701097\pi$
$641$	−19168.5	−1.18114	−0.590571	+	0.806986i	$0.701097\pi$
$642$	0	0
$643$	− 31298.6i	− 1.91959i	−0.280707	−	0.959793i	$-0.590569\pi$
$643$	− 31298.6i	− 1.91959i	0.280707	−	0.959793i	$-0.409431\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 26340.3i	− 1.60053i	−0.599643	−	0.800267i	$-0.704691\pi$
$647$	− 26340.3i	− 1.60053i	0.599643	−	0.800267i	$-0.295309\pi$
$648$	0	0
$649$	−3555.50	−0.215047
$650$	0	0
$651$	15866.6	0.955242
$652$	0	0
$653$	− 27113.8i	− 1.62488i	−0.583046	−	0.812439i	$-0.698139\pi$
$653$	− 27113.8i	− 1.62488i	0.583046	−	0.812439i	$-0.301861\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 1982.33i	− 0.117714i
$658$	0	0
$659$	24125.5	1.42609	0.713046	−	0.701117i	$-0.247315\pi$
$659$	24125.5	1.42609	0.713046	+	0.701117i	$0.247315\pi$
$660$	0	0
$661$	1776.69	0.104546	0.0522732	−	0.998633i	$-0.483353\pi$
$661$	1776.69	0.104546	0.0522732	+	0.998633i	$0.483353\pi$
$662$	0	0
$663$	− 13359.3i	− 0.782550i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1653.64i	0.0959960i
$668$	0	0
$669$	17158.5	0.991609
$670$	0	0
$671$	−18369.2	−1.05684
$672$	0	0
$673$	14137.6i	0.809756i	0.914371	+	0.404878i	$0.132686\pi$
$673$	14137.6i	0.809756i	−0.914371	+	0.404878i	$0.867314\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 9057.23i	− 0.514177i	−0.966388	−	0.257088i	$-0.917237\pi$
$677$	− 9057.23i	− 0.514177i	0.966388	−	0.257088i	$-0.0827632\pi$
$678$	0	0
$679$	12953.0	0.732094
$680$	0	0
$681$	−33123.9	−1.86389
$682$	0	0
$683$	− 23206.6i	− 1.30011i	−0.759888	−	0.650054i	$-0.774746\pi$
$683$	− 23206.6i	− 1.30011i	0.759888	−	0.650054i	$-0.225254\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	25854.6i	1.43583i
$688$	0	0
$689$	−11640.8	−0.643658
$690$	0	0
$691$	2060.79	0.113453	0.0567265	−	0.998390i	$-0.481934\pi$
$691$	2060.79	0.113453	0.0567265	+	0.998390i	$0.481934\pi$
$692$	0	0
$693$	− 1600.13i	− 0.0877115i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 19105.0i	− 1.03824i
$698$	0	0
$699$	−17121.1	−0.926434
$700$	0	0
$701$	4057.66	0.218624	0.109312	−	0.994007i	$-0.465135\pi$
$701$	4057.66	0.218624	0.109312	+	0.994007i	$0.465135\pi$
$702$	0	0
$703$	− 40770.0i	− 2.18730i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	9943.76i	0.528958i
$708$	0	0
$709$	−18777.0	−0.994622	−0.497311	−	0.867572i	$-0.665679\pi$
$709$	−18777.0	−0.994622	−0.497311	+	0.867572i	$0.665679\pi$
$710$	0	0
$711$	7010.58	0.369785
$712$	0	0
$713$	− 5053.82i	− 0.265452i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	23149.1i	1.20574i
$718$	0	0
$719$	2217.63	0.115026	0.0575129	−	0.998345i	$-0.481683\pi$
$719$	2217.63	0.115026	0.0575129	+	0.998345i	$0.481683\pi$
$720$	0	0
$721$	−5950.69	−0.307372
$722$	0	0
$723$	− 20915.9i	− 1.07589i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 23162.5i	− 1.18164i	−0.806805	−	0.590818i	$-0.798805\pi$
$727$	− 23162.5i	− 1.18164i	0.806805	−	0.590818i	$-0.201195\pi$
$728$	0	0
$729$	−13868.6	−0.704600
$730$	0	0
$731$	8776.11	0.444044
$732$	0	0
$733$	− 11686.4i	− 0.588875i	−0.955671	−	0.294438i	$-0.904868\pi$
$733$	− 11686.4i	− 0.588875i	0.955671	−	0.294438i	$-0.0951324\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	538.514i	0.0269151i
$738$	0	0
$739$	7408.28	0.368766	0.184383	−	0.982854i	$-0.440971\pi$
$739$	7408.28	0.368766	0.184383	+	0.982854i	$0.440971\pi$
$740$	0	0
$741$	−29380.4	−1.45657
$742$	0	0
$743$	7008.68i	0.346061i	0.984916	+	0.173031i	$0.0553560\pi$
$743$	7008.68i	0.346061i	−0.984916	+	0.173031i	$0.944644\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	4323.43i	0.211762i
$748$	0	0
$749$	−17695.1	−0.863238
$750$	0	0
$751$	10916.9	0.530443	0.265222	−	0.964187i	$-0.414555\pi$
$751$	10916.9	0.530443	0.265222	+	0.964187i	$0.414555\pi$
$752$	0	0
$753$	38924.6i	1.88379i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	14124.7i	0.678165i	0.940757	+	0.339083i	$0.110117\pi$
$757$	14124.7i	0.678165i	−0.940757	+	0.339083i	$0.889883\pi$
$758$	0	0
$759$	−2893.86	−0.138393
$760$	0	0
$761$	6947.27	0.330931	0.165465	−	0.986216i	$-0.447087\pi$
$761$	6947.27	0.330931	0.165465	+	0.986216i	$0.447087\pi$
$762$	0	0
$763$	− 27136.0i	− 1.28753i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 6032.35i	− 0.283984i
$768$	0	0
$769$	−19409.1	−0.910158	−0.455079	−	0.890451i	$-0.650389\pi$
$769$	−19409.1	−0.910158	−0.455079	+	0.890451i	$0.650389\pi$
$770$	0	0
$771$	1030.01	0.0481129
$772$	0	0
$773$	32794.2i	1.52591i	0.646453	+	0.762954i	$0.276252\pi$
$773$	32794.2i	1.52591i	−0.646453	+	0.762954i	$0.723748\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	21390.0i	0.987597i
$778$	0	0
$779$	−42016.8	−1.93249
$780$	0	0
$781$	21406.5	0.980775
$782$	0	0
$783$	− 8737.29i	− 0.398781i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1234.92i	0.0559341i	0.999609	+	0.0279670i	$0.00890335\pi$
$787$	1234.92i	0.0559341i	−0.999609	+	0.0279670i	$0.991097\pi$
$788$	0	0
$789$	6574.66	0.296659
$790$	0	0
$791$	−9521.24	−0.427985
$792$	0	0
$793$	− 31165.7i	− 1.39562i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 19464.3i	− 0.865070i	−0.901617	−	0.432535i	$-0.857619\pi$
$797$	− 19464.3i	− 0.865070i	0.901617	−	0.432535i	$-0.142381\pi$
$798$	0	0
$799$	11609.5	0.514035
$800$	0	0
$801$	4684.79	0.206653
$802$	0	0
$803$	− 7548.48i	− 0.331731i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	17173.4i	0.749113i
$808$	0	0
$809$	33587.0	1.45965	0.729824	−	0.683635i	$-0.239602\pi$
$809$	33587.0	1.45965	0.729824	+	0.683635i	$0.239602\pi$
$810$	0	0
$811$	−2612.03	−0.113096	−0.0565479	−	0.998400i	$-0.518009\pi$
$811$	−2612.03	−0.113096	−0.0565479	+	0.998400i	$0.518009\pi$
$812$	0	0
$813$	− 31988.5i	− 1.37993i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 19300.9i	− 0.826502i
$818$	0	0
$819$	2714.83	0.115829
$820$	0	0
$821$	−17811.2	−0.757146	−0.378573	−	0.925572i	$-0.623585\pi$
$821$	−17811.2	−0.757146	−0.378573	+	0.925572i	$0.623585\pi$
$822$	0	0
$823$	− 30535.0i	− 1.29330i	−0.762788	−	0.646649i	$-0.776170\pi$
$823$	− 30535.0i	− 1.29330i	0.762788	−	0.646649i	$-0.223830\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 3344.05i	− 0.140609i	−0.997526	−	0.0703047i	$-0.977603\pi$
$827$	− 3344.05i	− 0.140609i	0.997526	−	0.0703047i	$-0.0223972\pi$
$828$	0	0
$829$	24905.6	1.04344	0.521718	−	0.853118i	$-0.325291\pi$
$829$	24905.6	1.04344	0.521718	+	0.853118i	$0.325291\pi$
$830$	0	0
$831$	35602.1	1.48619
$832$	0	0
$833$	11508.4i	0.478684i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	26702.7i	1.10272i
$838$	0	0
$839$	38233.9	1.57328	0.786639	−	0.617413i	$-0.211819\pi$
$839$	38233.9	1.57328	0.786639	+	0.617413i	$0.211819\pi$
$840$	0	0
$841$	−19219.7	−0.788050
$842$	0	0
$843$	38761.2i	1.58364i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	10695.7i	0.433894i
$848$	0	0
$849$	44717.8	1.80767
$850$	0	0
$851$	6813.12	0.274443
$852$	0	0
$853$	− 21297.4i	− 0.854875i	−0.904045	−	0.427438i	$-0.859416\pi$
$853$	− 21297.4i	− 0.854875i	0.904045	−	0.427438i	$-0.140584\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 2100.08i	− 0.0837074i	−0.999124	−	0.0418537i	$-0.986674\pi$
$857$	− 2100.08i	− 0.0837074i	0.999124	−	0.0418537i	$-0.0133263\pi$
$858$	0	0
$859$	16555.4	0.657582	0.328791	−	0.944403i	$-0.393359\pi$
$859$	16555.4	0.657582	0.328791	+	0.944403i	$0.393359\pi$
$860$	0	0
$861$	22044.2	0.872547
$862$	0	0
$863$	40225.0i	1.58664i	0.608802	+	0.793322i	$0.291650\pi$
$863$	40225.0i	1.58664i	−0.608802	+	0.793322i	$0.708350\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 5704.81i	− 0.223467i
$868$	0	0
$869$	26695.5	1.04210
$870$	0	0
$871$	−913.656	−0.0355431
$872$	0	0
$873$	− 5927.13i	− 0.229786i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	35335.8i	1.36055i	0.732955	+	0.680277i	$0.238141\pi$
$877$	35335.8i	1.36055i	−0.732955	+	0.680277i	$0.761859\pi$
$878$	0	0
$879$	48173.2	1.84851
$880$	0	0
$881$	28460.9	1.08839	0.544196	−	0.838958i	$-0.316835\pi$
$881$	28460.9	1.08839	0.544196	+	0.838958i	$0.316835\pi$
$882$	0	0
$883$	35713.3i	1.36110i	0.732703	+	0.680548i	$0.238258\pi$
$883$	35713.3i	1.36110i	−0.732703	+	0.680548i	$0.761742\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 354.759i	− 0.0134292i	−0.999977	−	0.00671458i	$-0.997863\pi$
$887$	− 354.759i	− 0.0134292i	0.999977	−	0.00671458i	$-0.00213733\pi$
$888$	0	0
$889$	27261.8	1.02849
$890$	0	0
$891$	18715.3	0.703688
$892$	0	0
$893$	− 25532.2i	− 0.956777i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 4909.79i	− 0.182757i
$898$	0	0
$899$	−15798.1	−0.586093
$900$	0	0
$901$	19536.5	0.722370
$902$	0	0
$903$	10126.2i	0.373178i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 41414.8i	− 1.51616i	−0.652162	−	0.758080i	$-0.726138\pi$
$907$	− 41414.8i	− 1.51616i	0.652162	−	0.758080i	$-0.273862\pi$
$908$	0	0
$909$	4550.12	0.166026
$910$	0	0
$911$	4521.49	0.164439	0.0822194	−	0.996614i	$-0.473799\pi$
$911$	4521.49	0.164439	0.0822194	+	0.996614i	$0.473799\pi$
$912$	0	0
$913$	16463.2i	0.596770i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 5016.92i	− 0.180669i
$918$	0	0
$919$	31097.0	1.11621	0.558104	−	0.829771i	$-0.311529\pi$
$919$	31097.0	1.11621	0.558104	+	0.829771i	$0.311529\pi$
$920$	0	0
$921$	−55147.0	−1.97303
$922$	0	0
$923$	36318.8i	1.29518i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	2722.95i	0.0964761i
$928$	0	0
$929$	−11243.2	−0.397070	−0.198535	−	0.980094i	$-0.563618\pi$
$929$	−11243.2	−0.397070	−0.198535	+	0.980094i	$0.563618\pi$
$930$	0	0
$931$	25310.0	0.890978
$932$	0	0
$933$	40076.4i	1.40626i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 31275.3i	− 1.09042i	−0.838300	−	0.545209i	$-0.816450\pi$
$937$	− 31275.3i	− 1.09042i	0.838300	−	0.545209i	$-0.183550\pi$
$938$	0	0
$939$	16915.2	0.587866
$940$	0	0
$941$	−19726.6	−0.683387	−0.341694	−	0.939811i	$-0.611001\pi$
$941$	−19726.6	−0.683387	−0.341694	+	0.939811i	$0.611001\pi$
$942$	0	0
$943$	− 7021.48i	− 0.242472i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 9073.85i	− 0.311363i	−0.987807	−	0.155681i	$-0.950243\pi$
$947$	− 9073.85i	− 0.311363i	0.987807	−	0.155681i	$-0.0497573\pi$
$948$	0	0
$949$	12806.9	0.438072
$950$	0	0
$951$	−22988.4	−0.783860
$952$	0	0
$953$	− 20332.8i	− 0.691125i	−0.938396	−	0.345563i	$-0.887688\pi$
$953$	− 20332.8i	− 0.691125i	0.938396	−	0.345563i	$-0.112312\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	9046.14i	0.305559i
$958$	0	0
$959$	−20947.9	−0.705364
$960$	0	0
$961$	18490.8	0.620686
$962$	0	0
$963$	8097.03i	0.270948i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 10303.5i	− 0.342646i	−0.985215	−	0.171323i	$-0.945196\pi$
$967$	− 10303.5i	− 0.342646i	0.985215	−	0.171323i	$-0.0548041\pi$
$968$	0	0
$969$	49308.3	1.63469
$970$	0	0
$971$	−937.151	−0.0309728	−0.0154864	−	0.999880i	$-0.504930\pi$
$971$	−937.151	−0.0309728	−0.0154864	+	0.999880i	$0.504930\pi$
$972$	0	0
$973$	− 12586.6i	− 0.414705i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 23339.0i	− 0.764258i	−0.924109	−	0.382129i	$-0.875191\pi$
$977$	− 23339.0i	− 0.764258i	0.924109	−	0.382129i	$-0.124809\pi$
$978$	0	0
$979$	17839.2	0.582372
$980$	0	0
$981$	−12417.0	−0.404124
$982$	0	0
$983$	− 26638.6i	− 0.864334i	−0.901794	−	0.432167i	$-0.857749\pi$
$983$	− 26638.6i	− 0.864334i	0.901794	−	0.432167i	$-0.142251\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	13395.5i	0.431999i
$988$	0	0
$989$	3225.39	0.103702
$990$	0	0
$991$	−24631.4	−0.789548	−0.394774	−	0.918778i	$-0.629177\pi$
$991$	−24631.4	−0.789548	−0.394774	+	0.918778i	$0.629177\pi$
$992$	0	0
$993$	− 3415.15i	− 0.109141i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 31438.3i	− 0.998656i	−0.866413	−	0.499328i	$-0.833580\pi$
$997$	− 31438.3i	− 0.998656i	0.866413	−	0.499328i	$-0.166420\pi$
$998$	0	0
$999$	−35998.2	−1.14007

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2300.4.c.d.1749.3		10
5.2	odd	4		2300.4.a.c.1.2		5
5.3	odd	4		460.4.a.b.1.4	✓	5
5.4	even	2	inner	2300.4.c.d.1749.8		10
20.3	even	4		1840.4.a.o.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.4.a.b.1.4	✓	5	5.3	odd	4
1840.4.a.o.1.2		5	20.3	even	4
2300.4.a.c.1.2		5	5.2	odd	4
2300.4.c.d.1749.3		10	1.1	even	1	trivial
2300.4.c.d.1749.8		10	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 $\F_{q}$
Belyi maps

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)

\(f(q)\)	\(=\)	\(q-5.72467i q^{3} -12.6137i q^{7} -5.77185 q^{9} -21.9786 q^{11} -37.2893i q^{13} +62.5817i q^{17} +137.633 q^{19} -72.2093 q^{21} +23.0000i q^{23} -121.524i q^{27} +71.8975 q^{29} -219.731 q^{31} +125.820i q^{33} -296.223i q^{37} -213.469 q^{39} -305.282 q^{41} -140.234i q^{43} -185.509i q^{47} +183.895 q^{49} +358.260 q^{51} -312.176i q^{53} -787.903i q^{57} +161.771 q^{59} +835.780 q^{61} +72.8043i q^{63} -24.5018i q^{67} +131.667 q^{69} -973.973 q^{71} +343.447i q^{73} +277.231i q^{77} -1214.62 q^{79} -851.526 q^{81} -749.055i q^{83} -411.590i q^{87} -811.662 q^{89} -470.356 q^{91} +1257.89i q^{93} +1026.90i q^{97} +126.857 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 60 q^{9} + 14 q^{11} + 26 q^{19} + 506 q^{29} - 196 q^{31} + 820 q^{39} - 1548 q^{41} + 1366 q^{49} - 1314 q^{51} + 1526 q^{59} + 674 q^{61} - 138 q^{69} - 3008 q^{71} - 1252 q^{79} - 1918 q^{81}+ \cdots - 222 q^{99}+O(q^{100}) \) 10 * q - 60 * q^9 + 14 * q^11 + 26 * q^19 + 506 * q^29 - 196 * q^31 + 820 * q^39 - 1548 * q^41 + 1366 * q^49 - 1314 * q^51 + 1526 * q^59 + 674 * q^61 - 138 * q^69 - 3008 * q^71 - 1252 * q^79 - 1918 * q^81 - 1292 * q^89 - 1534 * q^91 - 222 * q^99

Introduction

L-functions

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Database

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