LMFDB - Embedded newform 448.4.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,4,Mod(1,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("448.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$448 = 2^{6} \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	448.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$26.4328556826$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 56)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	448.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q-2.00000 q^{3} +16.0000 q^{5} +7.00000 q^{7} -23.0000 q^{9} +24.0000 q^{11} +68.0000 q^{13} -32.0000 q^{15} +54.0000 q^{17} -46.0000 q^{19} -14.0000 q^{21} -176.000 q^{23} +131.000 q^{25} +100.000 q^{27} +174.000 q^{29} +116.000 q^{31} -48.0000 q^{33} +112.000 q^{35} -74.0000 q^{37} -136.000 q^{39} -10.0000 q^{41} -480.000 q^{43} -368.000 q^{45} +572.000 q^{47} +49.0000 q^{49} -108.000 q^{51} +162.000 q^{53} +384.000 q^{55} +92.0000 q^{57} -86.0000 q^{59} +904.000 q^{61} -161.000 q^{63} +1088.00 q^{65} +660.000 q^{67} +352.000 q^{69} -1024.00 q^{71} +770.000 q^{73} -262.000 q^{75} +168.000 q^{77} +904.000 q^{79} +421.000 q^{81} +682.000 q^{83} +864.000 q^{85} -348.000 q^{87} -102.000 q^{89} +476.000 q^{91} -232.000 q^{93} -736.000 q^{95} -218.000 q^{97} -552.000 q^{99} +O(q^{100})

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
$3$	−2.00000	−0.384900	−0.192450	−	0.981307i	$-0.561643\pi$
$3$	−2.00000	−0.384900	−0.192450	+	0.981307i	$0.561643\pi$
$4$	0	0
$5$	16.0000	1.43108	0.715542	−	0.698570i	$-0.246180\pi$
$5$	16.0000	1.43108	0.715542	+	0.698570i	$0.246180\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	−23.0000	−0.851852
$10$	0	0
$11$	24.0000	0.657843	0.328921	−	0.944357i	$-0.393315\pi$
$11$	24.0000	0.657843	0.328921	+	0.944357i	$0.393315\pi$
$12$	0	0
$13$	68.0000	1.45075	0.725377	−	0.688352i	$-0.241665\pi$
$13$	68.0000	1.45075	0.725377	+	0.688352i	$0.241665\pi$
$14$	0	0
$15$	−32.0000	−0.550824
$16$	0	0
$17$	54.0000	0.770407	0.385204	−	0.922832i	$-0.374131\pi$
$17$	54.0000	0.770407	0.385204	+	0.922832i	$0.374131\pi$
$18$	0	0
$19$	−46.0000	−0.555428	−0.277714	−	0.960664i	$-0.589577\pi$
$19$	−46.0000	−0.555428	−0.277714	+	0.960664i	$0.589577\pi$
$20$	0	0
$21$	−14.0000	−0.145479
$22$	0	0
$23$	−176.000	−1.59559	−0.797794	−	0.602930i	$-0.794000\pi$
$23$	−176.000	−1.59559	−0.797794	+	0.602930i	$0.794000\pi$
$24$	0	0
$25$	131.000	1.04800
$26$	0	0
$27$	100.000	0.712778
$28$	0	0
$29$	174.000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	174.000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	116.000	0.672071	0.336036	−	0.941849i	$-0.390914\pi$
$31$	116.000	0.672071	0.336036	+	0.941849i	$0.390914\pi$
$32$	0	0
$33$	−48.0000	−0.253204
$34$	0	0
$35$	112.000	0.540899
$36$	0	0
$37$	−74.0000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−74.0000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	−136.000	−0.558396
$40$	0	0
$41$	−10.0000	−0.0380912	−0.0190456	−	0.999819i	$-0.506063\pi$
$41$	−10.0000	−0.0380912	−0.0190456	+	0.999819i	$0.506063\pi$
$42$	0	0
$43$	−480.000	−1.70231	−0.851155	−	0.524915i	$-0.824097\pi$
$43$	−480.000	−1.70231	−0.851155	+	0.524915i	$0.824097\pi$
$44$	0	0
$45$	−368.000	−1.21907
$46$	0	0
$47$	572.000	1.77521	0.887604	−	0.460607i	$-0.152368\pi$
$47$	572.000	1.77521	0.887604	+	0.460607i	$0.152368\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−108.000	−0.296530
$52$	0	0
$53$	162.000	0.419857	0.209928	−	0.977717i	$-0.432677\pi$
$53$	162.000	0.419857	0.209928	+	0.977717i	$0.432677\pi$
$54$	0	0
$55$	384.000	0.941428
$56$	0	0
$57$	92.0000	0.213784
$58$	0	0
$59$	−86.0000	−0.189767	−0.0948834	−	0.995488i	$-0.530248\pi$
$59$	−86.0000	−0.189767	−0.0948834	+	0.995488i	$0.530248\pi$
$60$	0	0
$61$	904.000	1.89746	0.948732	−	0.316081i	$-0.102367\pi$
$61$	904.000	1.89746	0.948732	+	0.316081i	$0.102367\pi$
$62$	0	0
$63$	−161.000	−0.321970
$64$	0	0
$65$	1088.00	2.07615
$66$	0	0
$67$	660.000	1.20346	0.601730	−	0.798699i	$-0.294478\pi$
$67$	660.000	1.20346	0.601730	+	0.798699i	$0.294478\pi$
$68$	0	0
$69$	352.000	0.614142
$70$	0	0
$71$	−1024.00	−1.71164	−0.855820	−	0.517274i	$-0.826947\pi$
$71$	−1024.00	−1.71164	−0.855820	+	0.517274i	$0.826947\pi$
$72$	0	0
$73$	770.000	1.23454	0.617272	−	0.786750i	$-0.288238\pi$
$73$	770.000	1.23454	0.617272	+	0.786750i	$0.288238\pi$
$74$	0	0
$75$	−262.000	−0.403375
$76$	0	0
$77$	168.000	0.248641
$78$	0	0
$79$	904.000	1.28744	0.643721	−	0.765260i	$-0.277390\pi$
$79$	904.000	1.28744	0.643721	+	0.765260i	$0.277390\pi$
$80$	0	0
$81$	421.000	0.577503
$82$	0	0
$83$	682.000	0.901918	0.450959	−	0.892545i	$-0.351082\pi$
$83$	682.000	0.901918	0.450959	+	0.892545i	$0.351082\pi$
$84$	0	0
$85$	864.000	1.10252
$86$	0	0
$87$	−348.000	−0.428845
$88$	0	0
$89$	−102.000	−0.121483	−0.0607415	−	0.998154i	$-0.519347\pi$
$89$	−102.000	−0.121483	−0.0607415	+	0.998154i	$0.519347\pi$
$90$	0	0
$91$	476.000	0.548334
$92$	0	0
$93$	−232.000	−0.258680
$94$	0	0
$95$	−736.000	−0.794863
$96$	0	0
$97$	−218.000	−0.228191	−0.114096	−	0.993470i	$-0.536397\pi$
$97$	−218.000	−0.228191	−0.114096	+	0.993470i	$0.536397\pi$
$98$	0	0
$99$	−552.000	−0.560385
$100$	0	0
$101$	80.0000	0.0788148	0.0394074	−	0.999223i	$-0.487453\pi$
$101$	80.0000	0.0788148	0.0394074	+	0.999223i	$0.487453\pi$
$102$	0	0
$103$	−108.000	−0.103316	−0.0516580	−	0.998665i	$-0.516451\pi$
$103$	−108.000	−0.103316	−0.0516580	+	0.998665i	$0.516451\pi$
$104$	0	0
$105$	−224.000	−0.208192
$106$	0	0
$107$	1492.00	1.34801	0.674005	−	0.738727i	$-0.264573\pi$
$107$	1492.00	1.34801	0.674005	+	0.738727i	$0.264573\pi$
$108$	0	0
$109$	1510.00	1.32690	0.663448	−	0.748222i	$-0.269092\pi$
$109$	1510.00	1.32690	0.663448	+	0.748222i	$0.269092\pi$
$110$	0	0
$111$	148.000	0.126554
$112$	0	0
$113$	−2138.00	−1.77988	−0.889939	−	0.456080i	$-0.849253\pi$
$113$	−2138.00	−1.77988	−0.889939	+	0.456080i	$0.849253\pi$
$114$	0	0
$115$	−2816.00	−2.28342
$116$	0	0
$117$	−1564.00	−1.23583
$118$	0	0
$119$	378.000	0.291187
$120$	0	0
$121$	−755.000	−0.567243
$122$	0	0
$123$	20.0000	0.0146613
$124$	0	0
$125$	96.0000	0.0686920
$126$	0	0
$127$	−1616.00	−1.12911	−0.564554	−	0.825396i	$-0.690952\pi$
$127$	−1616.00	−1.12911	−0.564554	+	0.825396i	$0.690952\pi$
$128$	0	0
$129$	960.000	0.655219
$130$	0	0
$131$	−414.000	−0.276117	−0.138059	−	0.990424i	$-0.544086\pi$
$131$	−414.000	−0.276117	−0.138059	+	0.990424i	$0.544086\pi$
$132$	0	0
$133$	−322.000	−0.209932
$134$	0	0
$135$	1600.00	1.02004
$136$	0	0
$137$	−926.000	−0.577471	−0.288735	−	0.957409i	$-0.593235\pi$
$137$	−926.000	−0.577471	−0.288735	+	0.957409i	$0.593235\pi$
$138$	0	0
$139$	494.000	0.301443	0.150721	−	0.988576i	$-0.451840\pi$
$139$	494.000	0.301443	0.150721	+	0.988576i	$0.451840\pi$
$140$	0	0
$141$	−1144.00	−0.683278
$142$	0	0
$143$	1632.00	0.954369
$144$	0	0
$145$	2784.00	1.59447
$146$	0	0
$147$	−98.0000	−0.0549857
$148$	0	0
$149$	474.000	0.260615	0.130307	−	0.991474i	$-0.458404\pi$
$149$	474.000	0.260615	0.130307	+	0.991474i	$0.458404\pi$
$150$	0	0
$151$	−88.0000	−0.0474261	−0.0237130	−	0.999719i	$-0.507549\pi$
$151$	−88.0000	−0.0474261	−0.0237130	+	0.999719i	$0.507549\pi$
$152$	0	0
$153$	−1242.00	−0.656273
$154$	0	0
$155$	1856.00	0.961790
$156$	0	0
$157$	−2972.00	−1.51077	−0.755387	−	0.655279i	$-0.772551\pi$
$157$	−2972.00	−1.51077	−0.755387	+	0.655279i	$0.772551\pi$
$158$	0	0
$159$	−324.000	−0.161603
$160$	0	0
$161$	−1232.00	−0.603076
$162$	0	0
$163$	−1480.00	−0.711181	−0.355591	−	0.934642i	$-0.615720\pi$
$163$	−1480.00	−0.711181	−0.355591	+	0.934642i	$0.615720\pi$
$164$	0	0
$165$	−768.000	−0.362356
$166$	0	0
$167$	−2500.00	−1.15842	−0.579209	−	0.815179i	$-0.696638\pi$
$167$	−2500.00	−1.15842	−0.579209	+	0.815179i	$0.696638\pi$
$168$	0	0
$169$	2427.00	1.10469
$170$	0	0
$171$	1058.00	0.473142
$172$	0	0
$173$	−980.000	−0.430682	−0.215341	−	0.976539i	$-0.569086\pi$
$173$	−980.000	−0.430682	−0.215341	+	0.976539i	$0.569086\pi$
$174$	0	0
$175$	917.000	0.396107
$176$	0	0
$177$	172.000	0.0730413
$178$	0	0
$179$	−2580.00	−1.07731	−0.538654	−	0.842527i	$-0.681067\pi$
$179$	−2580.00	−1.07731	−0.538654	+	0.842527i	$0.681067\pi$
$180$	0	0
$181$	−2388.00	−0.980655	−0.490328	−	0.871538i	$-0.663123\pi$
$181$	−2388.00	−0.980655	−0.490328	+	0.871538i	$0.663123\pi$
$182$	0	0
$183$	−1808.00	−0.730334
$184$	0	0
$185$	−1184.00	−0.470537
$186$	0	0
$187$	1296.00	0.506807
$188$	0	0
$189$	700.000	0.269405
$190$	0	0
$191$	1912.00	0.724332	0.362166	−	0.932114i	$-0.382037\pi$
$191$	1912.00	0.724332	0.362166	+	0.932114i	$0.382037\pi$
$192$	0	0
$193$	−1938.00	−0.722799	−0.361400	−	0.932411i	$-0.617701\pi$
$193$	−1938.00	−0.722799	−0.361400	+	0.932411i	$0.617701\pi$
$194$	0	0
$195$	−2176.00	−0.799111
$196$	0	0
$197$	122.000	0.0441225	0.0220613	−	0.999757i	$-0.492977\pi$
$197$	122.000	0.0441225	0.0220613	+	0.999757i	$0.492977\pi$
$198$	0	0
$199$	3900.00	1.38926	0.694632	−	0.719365i	$-0.255567\pi$
$199$	3900.00	1.38926	0.694632	+	0.719365i	$0.255567\pi$
$200$	0	0
$201$	−1320.00	−0.463212
$202$	0	0
$203$	1218.00	0.421117
$204$	0	0
$205$	−160.000	−0.0545116
$206$	0	0
$207$	4048.00	1.35921
$208$	0	0
$209$	−1104.00	−0.365384
$210$	0	0
$211$	−1164.00	−0.379778	−0.189889	−	0.981806i	$-0.560813\pi$
$211$	−1164.00	−0.379778	−0.189889	+	0.981806i	$0.560813\pi$
$212$	0	0
$213$	2048.00	0.658811
$214$	0	0
$215$	−7680.00	−2.43615
$216$	0	0
$217$	812.000	0.254019
$218$	0	0
$219$	−1540.00	−0.475176
$220$	0	0
$221$	3672.00	1.11767
$222$	0	0
$223$	−1480.00	−0.444431	−0.222216	−	0.974998i	$-0.571329\pi$
$223$	−1480.00	−0.444431	−0.222216	+	0.974998i	$0.571329\pi$
$224$	0	0
$225$	−3013.00	−0.892741
$226$	0	0
$227$	3906.00	1.14207	0.571036	−	0.820925i	$-0.306542\pi$
$227$	3906.00	1.14207	0.571036	+	0.820925i	$0.306542\pi$
$228$	0	0
$229$	−252.000	−0.0727189	−0.0363595	−	0.999339i	$-0.511576\pi$
$229$	−252.000	−0.0727189	−0.0363595	+	0.999339i	$0.511576\pi$
$230$	0	0
$231$	−336.000	−0.0957021
$232$	0	0
$233$	−2070.00	−0.582018	−0.291009	−	0.956720i	$-0.593991\pi$
$233$	−2070.00	−0.582018	−0.291009	+	0.956720i	$0.593991\pi$
$234$	0	0
$235$	9152.00	2.54047
$236$	0	0
$237$	−1808.00	−0.495537
$238$	0	0
$239$	5352.00	1.44850	0.724251	−	0.689536i	$-0.242186\pi$
$239$	5352.00	1.44850	0.724251	+	0.689536i	$0.242186\pi$
$240$	0	0
$241$	−6298.00	−1.68336	−0.841680	−	0.539976i	$-0.818433\pi$
$241$	−6298.00	−1.68336	−0.841680	+	0.539976i	$0.818433\pi$
$242$	0	0
$243$	−3542.00	−0.935059
$244$	0	0
$245$	784.000	0.204441
$246$	0	0
$247$	−3128.00	−0.805789
$248$	0	0
$249$	−1364.00	−0.347149
$250$	0	0
$251$	−1810.00	−0.455164	−0.227582	−	0.973759i	$-0.573082\pi$
$251$	−1810.00	−0.455164	−0.227582	+	0.973759i	$0.573082\pi$
$252$	0	0
$253$	−4224.00	−1.04965
$254$	0	0
$255$	−1728.00	−0.424359
$256$	0	0
$257$	−334.000	−0.0810675	−0.0405338	−	0.999178i	$-0.512906\pi$
$257$	−334.000	−0.0810675	−0.0405338	+	0.999178i	$0.512906\pi$
$258$	0	0
$259$	−518.000	−0.124274
$260$	0	0
$261$	−4002.00	−0.949110
$262$	0	0
$263$	3488.00	0.817792	0.408896	−	0.912581i	$-0.365914\pi$
$263$	3488.00	0.817792	0.408896	+	0.912581i	$0.365914\pi$
$264$	0	0
$265$	2592.00	0.600850
$266$	0	0
$267$	204.000	0.0467588
$268$	0	0
$269$	−3796.00	−0.860395	−0.430197	−	0.902735i	$-0.641556\pi$
$269$	−3796.00	−0.860395	−0.430197	+	0.902735i	$0.641556\pi$
$270$	0	0
$271$	−2416.00	−0.541556	−0.270778	−	0.962642i	$-0.587281\pi$
$271$	−2416.00	−0.541556	−0.270778	+	0.962642i	$0.587281\pi$
$272$	0	0
$273$	−952.000	−0.211054
$274$	0	0
$275$	3144.00	0.689419
$276$	0	0
$277$	−6734.00	−1.46067	−0.730337	−	0.683087i	$-0.760637\pi$
$277$	−6734.00	−1.46067	−0.730337	+	0.683087i	$0.760637\pi$
$278$	0	0
$279$	−2668.00	−0.572505
$280$	0	0
$281$	2250.00	0.477665	0.238832	−	0.971061i	$-0.423235\pi$
$281$	2250.00	0.477665	0.238832	+	0.971061i	$0.423235\pi$
$282$	0	0
$283$	3642.00	0.764998	0.382499	−	0.923956i	$-0.375064\pi$
$283$	3642.00	0.764998	0.382499	+	0.923956i	$0.375064\pi$
$284$	0	0
$285$	1472.00	0.305943
$286$	0	0
$287$	−70.0000	−0.0143971
$288$	0	0
$289$	−1997.00	−0.406473
$290$	0	0
$291$	436.000	0.0878308
$292$	0	0
$293$	7288.00	1.45314	0.726569	−	0.687093i	$-0.241114\pi$
$293$	7288.00	1.45314	0.726569	+	0.687093i	$0.241114\pi$
$294$	0	0
$295$	−1376.00	−0.271572
$296$	0	0
$297$	2400.00	0.468896
$298$	0	0
$299$	−11968.0	−2.31481
$300$	0	0
$301$	−3360.00	−0.643413
$302$	0	0
$303$	−160.000	−0.0303358
$304$	0	0
$305$	14464.0	2.71543
$306$	0	0
$307$	−3790.00	−0.704582	−0.352291	−	0.935890i	$-0.614597\pi$
$307$	−3790.00	−0.704582	−0.352291	+	0.935890i	$0.614597\pi$
$308$	0	0
$309$	216.000	0.0397664
$310$	0	0
$311$	−2760.00	−0.503232	−0.251616	−	0.967827i	$-0.580962\pi$
$311$	−2760.00	−0.503232	−0.251616	+	0.967827i	$0.580962\pi$
$312$	0	0
$313$	−6714.00	−1.21245	−0.606226	−	0.795292i	$-0.707317\pi$
$313$	−6714.00	−1.21245	−0.606226	+	0.795292i	$0.707317\pi$
$314$	0	0
$315$	−2576.00	−0.460766
$316$	0	0
$317$	6386.00	1.13146	0.565731	−	0.824590i	$-0.308594\pi$
$317$	6386.00	1.13146	0.565731	+	0.824590i	$0.308594\pi$
$318$	0	0
$319$	4176.00	0.732950
$320$	0	0
$321$	−2984.00	−0.518849
$322$	0	0
$323$	−2484.00	−0.427905
$324$	0	0
$325$	8908.00	1.52039
$326$	0	0
$327$	−3020.00	−0.510723
$328$	0	0
$329$	4004.00	0.670966
$330$	0	0
$331$	7168.00	1.19030	0.595149	−	0.803615i	$-0.297093\pi$
$331$	7168.00	1.19030	0.595149	+	0.803615i	$0.297093\pi$
$332$	0	0
$333$	1702.00	0.280087
$334$	0	0
$335$	10560.0	1.72225
$336$	0	0
$337$	2478.00	0.400550	0.200275	−	0.979740i	$-0.435816\pi$
$337$	2478.00	0.400550	0.200275	+	0.979740i	$0.435816\pi$
$338$	0	0
$339$	4276.00	0.685075
$340$	0	0
$341$	2784.00	0.442117
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	5632.00	0.878889
$346$	0	0
$347$	−2144.00	−0.331689	−0.165844	−	0.986152i	$-0.553035\pi$
$347$	−2144.00	−0.331689	−0.165844	+	0.986152i	$0.553035\pi$
$348$	0	0
$349$	−7652.00	−1.17365	−0.586823	−	0.809716i	$-0.699621\pi$
$349$	−7652.00	−1.17365	−0.586823	+	0.809716i	$0.699621\pi$
$350$	0	0
$351$	6800.00	1.03407
$352$	0	0
$353$	7442.00	1.12209	0.561045	−	0.827785i	$-0.310400\pi$
$353$	7442.00	1.12209	0.561045	+	0.827785i	$0.310400\pi$
$354$	0	0
$355$	−16384.0	−2.44950
$356$	0	0
$357$	−756.000	−0.112078
$358$	0	0
$359$	−2616.00	−0.384588	−0.192294	−	0.981337i	$-0.561593\pi$
$359$	−2616.00	−0.384588	−0.192294	+	0.981337i	$0.561593\pi$
$360$	0	0
$361$	−4743.00	−0.691500
$362$	0	0
$363$	1510.00	0.218332
$364$	0	0
$365$	12320.0	1.76673
$366$	0	0
$367$	6616.00	0.941015	0.470507	−	0.882396i	$-0.344071\pi$
$367$	6616.00	0.941015	0.470507	+	0.882396i	$0.344071\pi$
$368$	0	0
$369$	230.000	0.0324480
$370$	0	0
$371$	1134.00	0.158691
$372$	0	0
$373$	3770.00	0.523333	0.261666	−	0.965158i	$-0.415728\pi$
$373$	3770.00	0.523333	0.261666	+	0.965158i	$0.415728\pi$
$374$	0	0
$375$	−192.000	−0.0264396
$376$	0	0
$377$	11832.0	1.61639
$378$	0	0
$379$	−6232.00	−0.844634	−0.422317	−	0.906448i	$-0.638783\pi$
$379$	−6232.00	−0.844634	−0.422317	+	0.906448i	$0.638783\pi$
$380$	0	0
$381$	3232.00	0.434594
$382$	0	0
$383$	−12212.0	−1.62925	−0.814627	−	0.579986i	$-0.803058\pi$
$383$	−12212.0	−1.62925	−0.814627	+	0.579986i	$0.803058\pi$
$384$	0	0
$385$	2688.00	0.355826
$386$	0	0
$387$	11040.0	1.45012
$388$	0	0
$389$	−7706.00	−1.00440	−0.502198	−	0.864753i	$-0.667475\pi$
$389$	−7706.00	−1.00440	−0.502198	+	0.864753i	$0.667475\pi$
$390$	0	0
$391$	−9504.00	−1.22925
$392$	0	0
$393$	828.000	0.106278
$394$	0	0
$395$	14464.0	1.84244
$396$	0	0
$397$	9260.00	1.17065	0.585323	−	0.810801i	$-0.300968\pi$
$397$	9260.00	1.17065	0.585323	+	0.810801i	$0.300968\pi$
$398$	0	0
$399$	644.000	0.0808028
$400$	0	0
$401$	2638.00	0.328517	0.164259	−	0.986417i	$-0.447477\pi$
$401$	2638.00	0.328517	0.164259	+	0.986417i	$0.447477\pi$
$402$	0	0
$403$	7888.00	0.975011
$404$	0	0
$405$	6736.00	0.826456
$406$	0	0
$407$	−1776.00	−0.216297
$408$	0	0
$409$	−9122.00	−1.10282	−0.551411	−	0.834234i	$-0.685910\pi$
$409$	−9122.00	−1.10282	−0.551411	+	0.834234i	$0.685910\pi$
$410$	0	0
$411$	1852.00	0.222269
$412$	0	0
$413$	−602.000	−0.0717251
$414$	0	0
$415$	10912.0	1.29072
$416$	0	0
$417$	−988.000	−0.116025
$418$	0	0
$419$	−5370.00	−0.626114	−0.313057	−	0.949734i	$-0.601353\pi$
$419$	−5370.00	−0.626114	−0.313057	+	0.949734i	$0.601353\pi$
$420$	0	0
$421$	−8046.00	−0.931444	−0.465722	−	0.884931i	$-0.654205\pi$
$421$	−8046.00	−0.931444	−0.465722	+	0.884931i	$0.654205\pi$
$422$	0	0
$423$	−13156.0	−1.51221
$424$	0	0
$425$	7074.00	0.807387
$426$	0	0
$427$	6328.00	0.717174
$428$	0	0
$429$	−3264.00	−0.367337
$430$	0	0
$431$	5064.00	0.565950	0.282975	−	0.959127i	$-0.408679\pi$
$431$	5064.00	0.565950	0.282975	+	0.959127i	$0.408679\pi$
$432$	0	0
$433$	−7922.00	−0.879231	−0.439616	−	0.898186i	$-0.644885\pi$
$433$	−7922.00	−0.879231	−0.439616	+	0.898186i	$0.644885\pi$
$434$	0	0
$435$	−5568.00	−0.613713
$436$	0	0
$437$	8096.00	0.886234
$438$	0	0
$439$	1080.00	0.117416	0.0587080	−	0.998275i	$-0.481302\pi$
$439$	1080.00	0.117416	0.0587080	+	0.998275i	$0.481302\pi$
$440$	0	0
$441$	−1127.00	−0.121693
$442$	0	0
$443$	3740.00	0.401112	0.200556	−	0.979682i	$-0.435725\pi$
$443$	3740.00	0.401112	0.200556	+	0.979682i	$0.435725\pi$
$444$	0	0
$445$	−1632.00	−0.173852
$446$	0	0
$447$	−948.000	−0.100311
$448$	0	0
$449$	5458.00	0.573672	0.286836	−	0.957980i	$-0.407396\pi$
$449$	5458.00	0.573672	0.286836	+	0.957980i	$0.407396\pi$
$450$	0	0
$451$	−240.000	−0.0250580
$452$	0	0
$453$	176.000	0.0182543
$454$	0	0
$455$	7616.00	0.784711
$456$	0	0
$457$	−3674.00	−0.376067	−0.188033	−	0.982163i	$-0.560211\pi$
$457$	−3674.00	−0.376067	−0.188033	+	0.982163i	$0.560211\pi$
$458$	0	0
$459$	5400.00	0.549129
$460$	0	0
$461$	−6252.00	−0.631637	−0.315819	−	0.948820i	$-0.602279\pi$
$461$	−6252.00	−0.631637	−0.315819	+	0.948820i	$0.602279\pi$
$462$	0	0
$463$	−19472.0	−1.95452	−0.977258	−	0.212055i	$-0.931984\pi$
$463$	−19472.0	−1.95452	−0.977258	+	0.212055i	$0.931984\pi$
$464$	0	0
$465$	−3712.00	−0.370193
$466$	0	0
$467$	−2806.00	−0.278043	−0.139022	−	0.990289i	$-0.544396\pi$
$467$	−2806.00	−0.278043	−0.139022	+	0.990289i	$0.544396\pi$
$468$	0	0
$469$	4620.00	0.454865
$470$	0	0
$471$	5944.00	0.581497
$472$	0	0
$473$	−11520.0	−1.11985
$474$	0	0
$475$	−6026.00	−0.582088
$476$	0	0
$477$	−3726.00	−0.357656
$478$	0	0
$479$	−588.000	−0.0560885	−0.0280443	−	0.999607i	$-0.508928\pi$
$479$	−588.000	−0.0560885	−0.0280443	+	0.999607i	$0.508928\pi$
$480$	0	0
$481$	−5032.00	−0.477005
$482$	0	0
$483$	2464.00	0.232124
$484$	0	0
$485$	−3488.00	−0.326561
$486$	0	0
$487$	7136.00	0.663990	0.331995	−	0.943281i	$-0.392278\pi$
$487$	7136.00	0.663990	0.331995	+	0.943281i	$0.392278\pi$
$488$	0	0
$489$	2960.00	0.273734
$490$	0	0
$491$	−7140.00	−0.656260	−0.328130	−	0.944633i	$-0.606418\pi$
$491$	−7140.00	−0.656260	−0.328130	+	0.944633i	$0.606418\pi$
$492$	0	0
$493$	9396.00	0.858366
$494$	0	0
$495$	−8832.00	−0.801957
$496$	0	0
$497$	−7168.00	−0.646939
$498$	0	0
$499$	−12908.0	−1.15800	−0.578999	−	0.815328i	$-0.696557\pi$
$499$	−12908.0	−1.15800	−0.578999	+	0.815328i	$0.696557\pi$
$500$	0	0
$501$	5000.00	0.445875
$502$	0	0
$503$	12192.0	1.08074	0.540372	−	0.841426i	$-0.318283\pi$
$503$	12192.0	1.08074	0.540372	+	0.841426i	$0.318283\pi$
$504$	0	0
$505$	1280.00	0.112791
$506$	0	0
$507$	−4854.00	−0.425195
$508$	0	0
$509$	−420.000	−0.0365740	−0.0182870	−	0.999833i	$-0.505821\pi$
$509$	−420.000	−0.0365740	−0.0182870	+	0.999833i	$0.505821\pi$
$510$	0	0
$511$	5390.00	0.466614
$512$	0	0
$513$	−4600.00	−0.395897
$514$	0	0
$515$	−1728.00	−0.147854
$516$	0	0
$517$	13728.0	1.16781
$518$	0	0
$519$	1960.00	0.165770
$520$	0	0
$521$	18582.0	1.56256	0.781279	−	0.624183i	$-0.214568\pi$
$521$	18582.0	1.56256	0.781279	+	0.624183i	$0.214568\pi$
$522$	0	0
$523$	17106.0	1.43020	0.715099	−	0.699024i	$-0.246382\pi$
$523$	17106.0	1.43020	0.715099	+	0.699024i	$0.246382\pi$
$524$	0	0
$525$	−1834.00	−0.152462
$526$	0	0
$527$	6264.00	0.517769
$528$	0	0
$529$	18809.0	1.54590
$530$	0	0
$531$	1978.00	0.161653
$532$	0	0
$533$	−680.000	−0.0552609
$534$	0	0
$535$	23872.0	1.92912
$536$	0	0
$537$	5160.00	0.414656
$538$	0	0
$539$	1176.00	0.0939776
$540$	0	0
$541$	8786.00	0.698225	0.349112	−	0.937081i	$-0.386483\pi$
$541$	8786.00	0.698225	0.349112	+	0.937081i	$0.386483\pi$
$542$	0	0
$543$	4776.00	0.377454
$544$	0	0
$545$	24160.0	1.89890
$546$	0	0
$547$	−7952.00	−0.621578	−0.310789	−	0.950479i	$-0.600593\pi$
$547$	−7952.00	−0.621578	−0.310789	+	0.950479i	$0.600593\pi$
$548$	0	0
$549$	−20792.0	−1.61636
$550$	0	0
$551$	−8004.00	−0.618842
$552$	0	0
$553$	6328.00	0.486607
$554$	0	0
$555$	2368.00	0.181110
$556$	0	0
$557$	9042.00	0.687831	0.343915	−	0.939001i	$-0.388247\pi$
$557$	9042.00	0.687831	0.343915	+	0.939001i	$0.388247\pi$
$558$	0	0
$559$	−32640.0	−2.46963
$560$	0	0
$561$	−2592.00	−0.195070
$562$	0	0
$563$	−22818.0	−1.70811	−0.854053	−	0.520185i	$-0.825863\pi$
$563$	−22818.0	−1.70811	−0.854053	+	0.520185i	$0.825863\pi$
$564$	0	0
$565$	−34208.0	−2.54715
$566$	0	0
$567$	2947.00	0.218276
$568$	0	0
$569$	726.000	0.0534895	0.0267447	−	0.999642i	$-0.491486\pi$
$569$	726.000	0.0534895	0.0267447	+	0.999642i	$0.491486\pi$
$570$	0	0
$571$	−2504.00	−0.183519	−0.0917593	−	0.995781i	$-0.529249\pi$
$571$	−2504.00	−0.183519	−0.0917593	+	0.995781i	$0.529249\pi$
$572$	0	0
$573$	−3824.00	−0.278796
$574$	0	0
$575$	−23056.0	−1.67218
$576$	0	0
$577$	−16158.0	−1.16580	−0.582900	−	0.812544i	$-0.698082\pi$
$577$	−16158.0	−1.16580	−0.582900	+	0.812544i	$0.698082\pi$
$578$	0	0
$579$	3876.00	0.278206
$580$	0	0
$581$	4774.00	0.340893
$582$	0	0
$583$	3888.00	0.276200
$584$	0	0
$585$	−25024.0	−1.76857
$586$	0	0
$587$	2070.00	0.145550	0.0727752	−	0.997348i	$-0.476814\pi$
$587$	2070.00	0.145550	0.0727752	+	0.997348i	$0.476814\pi$
$588$	0	0
$589$	−5336.00	−0.373287
$590$	0	0
$591$	−244.000	−0.0169828
$592$	0	0
$593$	7002.00	0.484886	0.242443	−	0.970166i	$-0.422051\pi$
$593$	7002.00	0.484886	0.242443	+	0.970166i	$0.422051\pi$
$594$	0	0
$595$	6048.00	0.416712
$596$	0	0
$597$	−7800.00	−0.534728
$598$	0	0
$599$	13272.0	0.905308	0.452654	−	0.891686i	$-0.350477\pi$
$599$	13272.0	0.905308	0.452654	+	0.891686i	$0.350477\pi$
$600$	0	0
$601$	−25990.0	−1.76398	−0.881992	−	0.471265i	$-0.843798\pi$
$601$	−25990.0	−1.76398	−0.881992	+	0.471265i	$0.843798\pi$
$602$	0	0
$603$	−15180.0	−1.02517
$604$	0	0
$605$	−12080.0	−0.811772
$606$	0	0
$607$	−448.000	−0.0299568	−0.0149784	−	0.999888i	$-0.504768\pi$
$607$	−448.000	−0.0299568	−0.0149784	+	0.999888i	$0.504768\pi$
$608$	0	0
$609$	−2436.00	−0.162088
$610$	0	0
$611$	38896.0	2.57539
$612$	0	0
$613$	−11018.0	−0.725959	−0.362979	−	0.931797i	$-0.618240\pi$
$613$	−11018.0	−0.725959	−0.362979	+	0.931797i	$0.618240\pi$
$614$	0	0
$615$	320.000	0.0209815
$616$	0	0
$617$	−3786.00	−0.247032	−0.123516	−	0.992343i	$-0.539417\pi$
$617$	−3786.00	−0.247032	−0.123516	+	0.992343i	$0.539417\pi$
$618$	0	0
$619$	−342.000	−0.0222070	−0.0111035	−	0.999938i	$-0.503534\pi$
$619$	−342.000	−0.0222070	−0.0111035	+	0.999938i	$0.503534\pi$
$620$	0	0
$621$	−17600.0	−1.13730
$622$	0	0
$623$	−714.000	−0.0459162
$624$	0	0
$625$	−14839.0	−0.949696
$626$	0	0
$627$	2208.00	0.140636
$628$	0	0
$629$	−3996.00	−0.253308
$630$	0	0
$631$	−26992.0	−1.70291	−0.851454	−	0.524430i	$-0.824279\pi$
$631$	−26992.0	−1.70291	−0.851454	+	0.524430i	$0.824279\pi$
$632$	0	0
$633$	2328.00	0.146176
$634$	0	0
$635$	−25856.0	−1.61585
$636$	0	0
$637$	3332.00	0.207251
$638$	0	0
$639$	23552.0	1.45806
$640$	0	0
$641$	−16978.0	−1.04616	−0.523082	−	0.852283i	$-0.675218\pi$
$641$	−16978.0	−1.04616	−0.523082	+	0.852283i	$0.675218\pi$
$642$	0	0
$643$	22510.0	1.38057	0.690286	−	0.723537i	$-0.257485\pi$
$643$	22510.0	1.38057	0.690286	+	0.723537i	$0.257485\pi$
$644$	0	0
$645$	15360.0	0.937674
$646$	0	0
$647$	9404.00	0.571421	0.285711	−	0.958316i	$-0.407770\pi$
$647$	9404.00	0.571421	0.285711	+	0.958316i	$0.407770\pi$
$648$	0	0
$649$	−2064.00	−0.124837
$650$	0	0
$651$	−1624.00	−0.0977720
$652$	0	0
$653$	4182.00	0.250619	0.125310	−	0.992118i	$-0.460008\pi$
$653$	4182.00	0.250619	0.125310	+	0.992118i	$0.460008\pi$
$654$	0	0
$655$	−6624.00	−0.395147
$656$	0	0
$657$	−17710.0	−1.05165
$658$	0	0
$659$	6096.00	0.360344	0.180172	−	0.983635i	$-0.442335\pi$
$659$	6096.00	0.360344	0.180172	+	0.983635i	$0.442335\pi$
$660$	0	0
$661$	−6632.00	−0.390249	−0.195125	−	0.980778i	$-0.562511\pi$
$661$	−6632.00	−0.390249	−0.195125	+	0.980778i	$0.562511\pi$
$662$	0	0
$663$	−7344.00	−0.430192
$664$	0	0
$665$	−5152.00	−0.300430
$666$	0	0
$667$	−30624.0	−1.77776
$668$	0	0
$669$	2960.00	0.171062
$670$	0	0
$671$	21696.0	1.24823
$672$	0	0
$673$	−1990.00	−0.113980	−0.0569902	−	0.998375i	$-0.518150\pi$
$673$	−1990.00	−0.113980	−0.0569902	+	0.998375i	$0.518150\pi$
$674$	0	0
$675$	13100.0	0.746991
$676$	0	0
$677$	22124.0	1.25597	0.627987	−	0.778224i	$-0.283879\pi$
$677$	22124.0	1.25597	0.627987	+	0.778224i	$0.283879\pi$
$678$	0	0
$679$	−1526.00	−0.0862482
$680$	0	0
$681$	−7812.00	−0.439584
$682$	0	0
$683$	6428.00	0.360118	0.180059	−	0.983656i	$-0.442371\pi$
$683$	6428.00	0.360118	0.180059	+	0.983656i	$0.442371\pi$
$684$	0	0
$685$	−14816.0	−0.826409
$686$	0	0
$687$	504.000	0.0279895
$688$	0	0
$689$	11016.0	0.609109
$690$	0	0
$691$	−23838.0	−1.31236	−0.656180	−	0.754605i	$-0.727829\pi$
$691$	−23838.0	−1.31236	−0.656180	+	0.754605i	$0.727829\pi$
$692$	0	0
$693$	−3864.00	−0.211806
$694$	0	0
$695$	7904.00	0.431390
$696$	0	0
$697$	−540.000	−0.0293457
$698$	0	0
$699$	4140.00	0.224019
$700$	0	0
$701$	886.000	0.0477372	0.0238686	−	0.999715i	$-0.492402\pi$
$701$	886.000	0.0477372	0.0238686	+	0.999715i	$0.492402\pi$
$702$	0	0
$703$	3404.00	0.182623
$704$	0	0
$705$	−18304.0	−0.977828
$706$	0	0
$707$	560.000	0.0297892
$708$	0	0
$709$	−16602.0	−0.879409	−0.439705	−	0.898142i	$-0.644917\pi$
$709$	−16602.0	−0.879409	−0.439705	+	0.898142i	$0.644917\pi$
$710$	0	0
$711$	−20792.0	−1.09671
$712$	0	0
$713$	−20416.0	−1.07235
$714$	0	0
$715$	26112.0	1.36578
$716$	0	0
$717$	−10704.0	−0.557529
$718$	0	0
$719$	−27260.0	−1.41394	−0.706972	−	0.707241i	$-0.749940\pi$
$719$	−27260.0	−1.41394	−0.706972	+	0.707241i	$0.749940\pi$
$720$	0	0
$721$	−756.000	−0.0390498
$722$	0	0
$723$	12596.0	0.647926
$724$	0	0
$725$	22794.0	1.16765
$726$	0	0
$727$	3652.00	0.186307	0.0931535	−	0.995652i	$-0.470305\pi$
$727$	3652.00	0.186307	0.0931535	+	0.995652i	$0.470305\pi$
$728$	0	0
$729$	−4283.00	−0.217599
$730$	0	0
$731$	−25920.0	−1.31147
$732$	0	0
$733$	13432.0	0.676838	0.338419	−	0.940996i	$-0.390108\pi$
$733$	13432.0	0.676838	0.338419	+	0.940996i	$0.390108\pi$
$734$	0	0
$735$	−1568.00	−0.0786892
$736$	0	0
$737$	15840.0	0.791688
$738$	0	0
$739$	34360.0	1.71036	0.855178	−	0.518334i	$-0.173448\pi$
$739$	34360.0	1.71036	0.855178	+	0.518334i	$0.173448\pi$
$740$	0	0
$741$	6256.00	0.310148
$742$	0	0
$743$	5624.00	0.277691	0.138846	−	0.990314i	$-0.455661\pi$
$743$	5624.00	0.277691	0.138846	+	0.990314i	$0.455661\pi$
$744$	0	0
$745$	7584.00	0.372961
$746$	0	0
$747$	−15686.0	−0.768301
$748$	0	0
$749$	10444.0	0.509500
$750$	0	0
$751$	−10392.0	−0.504939	−0.252470	−	0.967605i	$-0.581243\pi$
$751$	−10392.0	−0.504939	−0.252470	+	0.967605i	$0.581243\pi$
$752$	0	0
$753$	3620.00	0.175193
$754$	0	0
$755$	−1408.00	−0.0678707
$756$	0	0
$757$	−2666.00	−0.128002	−0.0640009	−	0.997950i	$-0.520386\pi$
$757$	−2666.00	−0.128002	−0.0640009	+	0.997950i	$0.520386\pi$
$758$	0	0
$759$	8448.00	0.404009
$760$	0	0
$761$	−3138.00	−0.149478	−0.0747388	−	0.997203i	$-0.523812\pi$
$761$	−3138.00	−0.149478	−0.0747388	+	0.997203i	$0.523812\pi$
$762$	0	0
$763$	10570.0	0.501520
$764$	0	0
$765$	−19872.0	−0.939181
$766$	0	0
$767$	−5848.00	−0.275305
$768$	0	0
$769$	38878.0	1.82312	0.911558	−	0.411171i	$-0.134880\pi$
$769$	38878.0	1.82312	0.911558	+	0.411171i	$0.134880\pi$
$770$	0	0
$771$	668.000	0.0312029
$772$	0	0
$773$	−3976.00	−0.185002	−0.0925012	−	0.995713i	$-0.529486\pi$
$773$	−3976.00	−0.185002	−0.0925012	+	0.995713i	$0.529486\pi$
$774$	0	0
$775$	15196.0	0.704331
$776$	0	0
$777$	1036.00	0.0478331
$778$	0	0
$779$	460.000	0.0211569
$780$	0	0
$781$	−24576.0	−1.12599
$782$	0	0
$783$	17400.0	0.794157
$784$	0	0
$785$	−47552.0	−2.16204
$786$	0	0
$787$	23258.0	1.05344	0.526721	−	0.850038i	$-0.323421\pi$
$787$	23258.0	1.05344	0.526721	+	0.850038i	$0.323421\pi$
$788$	0	0
$789$	−6976.00	−0.314768
$790$	0	0
$791$	−14966.0	−0.672730
$792$	0	0
$793$	61472.0	2.75276
$794$	0	0
$795$	−5184.00	−0.231267
$796$	0	0
$797$	2604.00	0.115732	0.0578660	−	0.998324i	$-0.481570\pi$
$797$	2604.00	0.115732	0.0578660	+	0.998324i	$0.481570\pi$
$798$	0	0
$799$	30888.0	1.36763
$800$	0	0
$801$	2346.00	0.103485
$802$	0	0
$803$	18480.0	0.812136
$804$	0	0
$805$	−19712.0	−0.863052
$806$	0	0
$807$	7592.00	0.331166
$808$	0	0
$809$	39926.0	1.73513	0.867567	−	0.497320i	$-0.165683\pi$
$809$	39926.0	1.73513	0.867567	+	0.497320i	$0.165683\pi$
$810$	0	0
$811$	−17422.0	−0.754339	−0.377170	−	0.926144i	$-0.623103\pi$
$811$	−17422.0	−0.754339	−0.377170	+	0.926144i	$0.623103\pi$
$812$	0	0
$813$	4832.00	0.208445
$814$	0	0
$815$	−23680.0	−1.01776
$816$	0	0
$817$	22080.0	0.945510
$818$	0	0
$819$	−10948.0	−0.467099
$820$	0	0
$821$	−3870.00	−0.164511	−0.0822557	−	0.996611i	$-0.526212\pi$
$821$	−3870.00	−0.164511	−0.0822557	+	0.996611i	$0.526212\pi$
$822$	0	0
$823$	22120.0	0.936883	0.468442	−	0.883495i	$-0.344816\pi$
$823$	22120.0	0.936883	0.468442	+	0.883495i	$0.344816\pi$
$824$	0	0
$825$	−6288.00	−0.265358
$826$	0	0
$827$	−13724.0	−0.577062	−0.288531	−	0.957471i	$-0.593167\pi$
$827$	−13724.0	−0.577062	−0.288531	+	0.957471i	$0.593167\pi$
$828$	0	0
$829$	11320.0	0.474258	0.237129	−	0.971478i	$-0.423794\pi$
$829$	11320.0	0.474258	0.237129	+	0.971478i	$0.423794\pi$
$830$	0	0
$831$	13468.0	0.562214
$832$	0	0
$833$	2646.00	0.110058
$834$	0	0
$835$	−40000.0	−1.65779
$836$	0	0
$837$	11600.0	0.479038
$838$	0	0
$839$	−18876.0	−0.776725	−0.388362	−	0.921507i	$-0.626959\pi$
$839$	−18876.0	−0.776725	−0.388362	+	0.921507i	$0.626959\pi$
$840$	0	0
$841$	5887.00	0.241379
$842$	0	0
$843$	−4500.00	−0.183853
$844$	0	0
$845$	38832.0	1.58090
$846$	0	0
$847$	−5285.00	−0.214398
$848$	0	0
$849$	−7284.00	−0.294448
$850$	0	0
$851$	13024.0	0.524626
$852$	0	0
$853$	33748.0	1.35464	0.677321	−	0.735688i	$-0.263141\pi$
$853$	33748.0	1.35464	0.677321	+	0.735688i	$0.263141\pi$
$854$	0	0
$855$	16928.0	0.677106
$856$	0	0
$857$	3230.00	0.128745	0.0643726	−	0.997926i	$-0.479495\pi$
$857$	3230.00	0.128745	0.0643726	+	0.997926i	$0.479495\pi$
$858$	0	0
$859$	−43922.0	−1.74459	−0.872293	−	0.488984i	$-0.837368\pi$
$859$	−43922.0	−1.74459	−0.872293	+	0.488984i	$0.837368\pi$
$860$	0	0
$861$	140.000	0.00554145
$862$	0	0
$863$	−26968.0	−1.06373	−0.531866	−	0.846828i	$-0.678509\pi$
$863$	−26968.0	−1.06373	−0.531866	+	0.846828i	$0.678509\pi$
$864$	0	0
$865$	−15680.0	−0.616342
$866$	0	0
$867$	3994.00	0.156451
$868$	0	0
$869$	21696.0	0.846935
$870$	0	0
$871$	44880.0	1.74593
$872$	0	0
$873$	5014.00	0.194385
$874$	0	0
$875$	672.000	0.0259631
$876$	0	0
$877$	11486.0	0.442252	0.221126	−	0.975245i	$-0.429027\pi$
$877$	11486.0	0.442252	0.221126	+	0.975245i	$0.429027\pi$
$878$	0	0
$879$	−14576.0	−0.559313
$880$	0	0
$881$	22026.0	0.842310	0.421155	−	0.906989i	$-0.361625\pi$
$881$	22026.0	0.842310	0.421155	+	0.906989i	$0.361625\pi$
$882$	0	0
$883$	8428.00	0.321206	0.160603	−	0.987019i	$-0.448656\pi$
$883$	8428.00	0.321206	0.160603	+	0.987019i	$0.448656\pi$
$884$	0	0
$885$	2752.00	0.104528
$886$	0	0
$887$	18996.0	0.719079	0.359540	−	0.933130i	$-0.382934\pi$
$887$	18996.0	0.719079	0.359540	+	0.933130i	$0.382934\pi$
$888$	0	0
$889$	−11312.0	−0.426763
$890$	0	0
$891$	10104.0	0.379907
$892$	0	0
$893$	−26312.0	−0.985999
$894$	0	0
$895$	−41280.0	−1.54172
$896$	0	0
$897$	23936.0	0.890970
$898$	0	0
$899$	20184.0	0.748803
$900$	0	0
$901$	8748.00	0.323461
$902$	0	0
$903$	6720.00	0.247650
$904$	0	0
$905$	−38208.0	−1.40340
$906$	0	0
$907$	20948.0	0.766887	0.383444	−	0.923564i	$-0.374738\pi$
$907$	20948.0	0.766887	0.383444	+	0.923564i	$0.374738\pi$
$908$	0	0
$909$	−1840.00	−0.0671386
$910$	0	0
$911$	1392.00	0.0506246	0.0253123	−	0.999680i	$-0.491942\pi$
$911$	1392.00	0.0506246	0.0253123	+	0.999680i	$0.491942\pi$
$912$	0	0
$913$	16368.0	0.593321
$914$	0	0
$915$	−28928.0	−1.04517
$916$	0	0
$917$	−2898.00	−0.104362
$918$	0	0
$919$	−41432.0	−1.48718	−0.743588	−	0.668638i	$-0.766878\pi$
$919$	−41432.0	−1.48718	−0.743588	+	0.668638i	$0.766878\pi$
$920$	0	0
$921$	7580.00	0.271194
$922$	0	0
$923$	−69632.0	−2.48317
$924$	0	0
$925$	−9694.00	−0.344580
$926$	0	0
$927$	2484.00	0.0880100
$928$	0	0
$929$	−25674.0	−0.906713	−0.453357	−	0.891329i	$-0.649774\pi$
$929$	−25674.0	−0.906713	−0.453357	+	0.891329i	$0.649774\pi$
$930$	0	0
$931$	−2254.00	−0.0793468
$932$	0	0
$933$	5520.00	0.193694
$934$	0	0
$935$	20736.0	0.725283
$936$	0	0
$937$	−46238.0	−1.61209	−0.806046	−	0.591853i	$-0.798396\pi$
$937$	−46238.0	−1.61209	−0.806046	+	0.591853i	$0.798396\pi$
$938$	0	0
$939$	13428.0	0.466673
$940$	0	0
$941$	−17432.0	−0.603897	−0.301948	−	0.953324i	$-0.597637\pi$
$941$	−17432.0	−0.603897	−0.301948	+	0.953324i	$0.597637\pi$
$942$	0	0
$943$	1760.00	0.0607778
$944$	0	0
$945$	11200.0	0.385541
$946$	0	0
$947$	26880.0	0.922368	0.461184	−	0.887304i	$-0.347425\pi$
$947$	26880.0	0.922368	0.461184	+	0.887304i	$0.347425\pi$
$948$	0	0
$949$	52360.0	1.79102
$950$	0	0
$951$	−12772.0	−0.435500
$952$	0	0
$953$	30970.0	1.05269	0.526347	−	0.850270i	$-0.323561\pi$
$953$	30970.0	1.05269	0.526347	+	0.850270i	$0.323561\pi$
$954$	0	0
$955$	30592.0	1.03658
$956$	0	0
$957$	−8352.00	−0.282113
$958$	0	0
$959$	−6482.00	−0.218264
$960$	0	0
$961$	−16335.0	−0.548320
$962$	0	0
$963$	−34316.0	−1.14830
$964$	0	0
$965$	−31008.0	−1.03439
$966$	0	0
$967$	19344.0	0.643290	0.321645	−	0.946860i	$-0.395764\pi$
$967$	19344.0	0.643290	0.321645	+	0.946860i	$0.395764\pi$
$968$	0	0
$969$	4968.00	0.164701
$970$	0	0
$971$	50122.0	1.65653	0.828265	−	0.560336i	$-0.189328\pi$
$971$	50122.0	1.65653	0.828265	+	0.560336i	$0.189328\pi$
$972$	0	0
$973$	3458.00	0.113935
$974$	0	0
$975$	−17816.0	−0.585199
$976$	0	0
$977$	25850.0	0.846484	0.423242	−	0.906017i	$-0.360892\pi$
$977$	25850.0	0.846484	0.423242	+	0.906017i	$0.360892\pi$
$978$	0	0
$979$	−2448.00	−0.0799167
$980$	0	0
$981$	−34730.0	−1.13032
$982$	0	0
$983$	−32012.0	−1.03868	−0.519341	−	0.854567i	$-0.673823\pi$
$983$	−32012.0	−1.03868	−0.519341	+	0.854567i	$0.673823\pi$
$984$	0	0
$985$	1952.00	0.0631430
$986$	0	0
$987$	−8008.00	−0.258255
$988$	0	0
$989$	84480.0	2.71619
$990$	0	0
$991$	−18240.0	−0.584675	−0.292337	−	0.956315i	$-0.594433\pi$
$991$	−18240.0	−0.584675	−0.292337	+	0.956315i	$0.594433\pi$
$992$	0	0
$993$	−14336.0	−0.458146
$994$	0	0
$995$	62400.0	1.98815
$996$	0	0
$997$	23144.0	0.735183	0.367592	−	0.929987i	$-0.380182\pi$
$997$	23144.0	0.735183	0.367592	+	0.929987i	$0.380182\pi$
$998$	0	0
$999$	−7400.00	−0.234360

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.4.a.h.1.1		1
4.3	odd	2		448.4.a.l.1.1		1
8.3	odd	2		56.4.a.a.1.1	✓	1
8.5	even	2		112.4.a.d.1.1		1
24.5	odd	2		1008.4.a.u.1.1		1
24.11	even	2		504.4.a.g.1.1		1
40.3	even	4		1400.4.g.f.449.1		2
40.19	odd	2		1400.4.a.f.1.1		1
40.27	even	4		1400.4.g.f.449.2		2
56.3	even	6		392.4.i.d.177.1		2
56.11	odd	6		392.4.i.e.177.1		2
56.13	odd	2		784.4.a.i.1.1		1
56.19	even	6		392.4.i.d.361.1		2
56.27	even	2		392.4.a.c.1.1		1
56.51	odd	6		392.4.i.e.361.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.4.a.a.1.1	✓	1	8.3	odd	2
112.4.a.d.1.1		1	8.5	even	2
392.4.a.c.1.1		1	56.27	even	2
392.4.i.d.177.1		2	56.3	even	6
392.4.i.d.361.1		2	56.19	even	6
392.4.i.e.177.1		2	56.11	odd	6
392.4.i.e.361.1		2	56.51	odd	6
448.4.a.h.1.1		1	1.1	even	1	trivial
448.4.a.l.1.1		1	4.3	odd	2
504.4.a.g.1.1		1	24.11	even	2
784.4.a.i.1.1		1	56.13	odd	2
1008.4.a.u.1.1		1	24.5	odd	2
1400.4.a.f.1.1		1	40.19	odd	2
1400.4.g.f.449.1		2	40.3	even	4
1400.4.g.f.449.2		2	40.27	even	4

Overview	Random
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Classical	Maass
Hilbert	Bianchi

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

Introduction

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Fields

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Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	448.4.a.h.1.1

Level	$448$
Weight	$4$
Character	448.1
Self dual	yes
Analytic conductor	$26.433$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$