LMFDB - Embedded newform 1232.4.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1232,4,Mod(1,1232)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1232, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1232.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$1232 = 2^{4} \cdot 7 \cdot 11$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1232.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:	$72.6903531271$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.5925.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{3} - x^{2} - 18x + 12$ x^3 - x^2 - 18*x + 12
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 308)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.658440$ of defining polynomial
Character	$\chi$	$=$	1232.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.34156 q^{3} +3.97532 q^{5} +7.00000 q^{7} -25.2002 q^{9} +11.0000 q^{11} +26.5401 q^{13} -5.33313 q^{15} -90.7650 q^{17} +68.4055 q^{19} -9.39092 q^{21} +36.3842 q^{23} -109.197 q^{25} +70.0297 q^{27} +176.413 q^{29} -177.565 q^{31} -14.7572 q^{33} +27.8272 q^{35} -299.322 q^{37} -35.6051 q^{39} -45.3448 q^{41} +528.573 q^{43} -100.179 q^{45} -357.443 q^{47} +49.0000 q^{49} +121.767 q^{51} -742.250 q^{53} +43.7285 q^{55} -91.7701 q^{57} +877.898 q^{59} +199.823 q^{61} -176.402 q^{63} +105.505 q^{65} -998.808 q^{67} -48.8116 q^{69} +27.8840 q^{71} -161.353 q^{73} +146.494 q^{75} +77.0000 q^{77} +624.995 q^{79} +586.457 q^{81} -396.724 q^{83} -360.820 q^{85} -236.668 q^{87} -1405.84 q^{89} +185.781 q^{91} +238.214 q^{93} +271.934 q^{95} -193.990 q^{97} -277.202 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$3 q - 5 q^{3} + 9 q^{5} + 21 q^{7} - 36 q^{9} + 33 q^{11} - 94 q^{13} + 95 q^{15} - 62 q^{17} - 32 q^{19} - 35 q^{21} + 139 q^{23} - 18 q^{25} + 55 q^{27} - 262 q^{29} + 169 q^{31} - 55 q^{33} + 63 q^{35}+ \cdots - 396 q^{99}+O(q^{100})$ 3 * q - 5 * q^3 + 9 * q^5 + 21 * q^7 - 36 * q^9 + 33 * q^11 - 94 * q^13 + 95 * q^15 - 62 * q^17 - 32 * q^19 - 35 * q^21 + 139 * q^23 - 18 * q^25 + 55 * q^27 - 262 * q^29 + 169 * q^31 - 55 * q^33 + 63 * q^35 - 437 * q^37 - 60 * q^39 - 106 * q^41 + 100 * q^43 - 528 * q^45 + 72 * q^47 + 147 * q^49 + 290 * q^51 - 798 * q^53 + 99 * q^55 - 470 * q^57 + 925 * q^59 - 24 * q^61 - 252 * q^63 - 932 * q^65 + 269 * q^67 - 5 * q^69 + 117 * q^71 - 10 * q^73 + 530 * q^75 + 231 * q^77 - 258 * q^79 - 33 * q^81 - 1086 * q^83 + 374 * q^85 + 510 * q^87 + 669 * q^89 - 658 * q^91 - 1035 * q^93 - 1666 * q^95 - 629 * q^97 - 396 * q^99

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
$3$	−1.34156	−0.258183	−0.129092	−	0.991633i	$-0.541206\pi$
$3$	−1.34156	−0.258183	−0.129092	+	0.991633i	$0.541206\pi$
$4$	0	0
$5$	3.97532	0.355563	0.177782	−	0.984070i	$-0.443108\pi$
$5$	3.97532	0.355563	0.177782	+	0.984070i	$0.443108\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	−25.2002	−0.933341
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	26.5401	0.566223	0.283111	−	0.959087i	$-0.408633\pi$
$13$	26.5401	0.566223	0.283111	+	0.959087i	$0.408633\pi$
$14$	0	0
$15$	−5.33313	−0.0918006
$16$	0	0
$17$	−90.7650	−1.29493	−0.647463	−	0.762097i	$-0.724170\pi$
$17$	−90.7650	−1.29493	−0.647463	+	0.762097i	$0.724170\pi$
$18$	0	0
$19$	68.4055	0.825963	0.412981	−	0.910739i	$-0.364487\pi$
$19$	68.4055	0.825963	0.412981	+	0.910739i	$0.364487\pi$
$20$	0	0
$21$	−9.39092	−0.0975841
$22$	0	0
$23$	36.3842	0.329853	0.164927	−	0.986306i	$-0.447261\pi$
$23$	36.3842	0.329853	0.164927	+	0.986306i	$0.447261\pi$
$24$	0	0
$25$	−109.197	−0.873575
$26$	0	0
$27$	70.0297	0.499157
$28$	0	0
$29$	176.413	1.12962	0.564811	−	0.825221i	$-0.308949\pi$
$29$	176.413	1.12962	0.564811	+	0.825221i	$0.308949\pi$
$30$	0	0
$31$	−177.565	−1.02876	−0.514382	−	0.857561i	$-0.671978\pi$
$31$	−177.565	−1.02876	−0.514382	+	0.857561i	$0.671978\pi$
$32$	0	0
$33$	−14.7572	−0.0778452
$34$	0	0
$35$	27.8272	0.134390
$36$	0	0
$37$	−299.322	−1.32995	−0.664976	−	0.746865i	$-0.731558\pi$
$37$	−299.322	−1.32995	−0.664976	+	0.746865i	$0.731558\pi$
$38$	0	0
$39$	−35.6051	−0.146189
$40$	0	0
$41$	−45.3448	−0.172724	−0.0863619	−	0.996264i	$-0.527524\pi$
$41$	−45.3448	−0.172724	−0.0863619	+	0.996264i	$0.527524\pi$
$42$	0	0
$43$	528.573	1.87457	0.937286	−	0.348560i	$-0.113329\pi$
$43$	528.573	1.87457	0.937286	+	0.348560i	$0.113329\pi$
$44$	0	0
$45$	−100.179	−0.331862
$46$	0	0
$47$	−357.443	−1.10933	−0.554665	−	0.832074i	$-0.687154\pi$
$47$	−357.443	−1.10933	−0.554665	+	0.832074i	$0.687154\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	121.767	0.334328
$52$	0	0
$53$	−742.250	−1.92370	−0.961848	−	0.273585i	$-0.911790\pi$
$53$	−742.250	−1.92370	−0.961848	+	0.273585i	$0.911790\pi$
$54$	0	0
$55$	43.7285	0.107206
$56$	0	0
$57$	−91.7701	−0.213250
$58$	0	0
$59$	877.898	1.93716	0.968581	−	0.248699i	$-0.0800029\pi$
$59$	877.898	1.93716	0.968581	+	0.248699i	$0.0800029\pi$
$60$	0	0
$61$	199.823	0.419422	0.209711	−	0.977763i	$-0.432748\pi$
$61$	199.823	0.419422	0.209711	+	0.977763i	$0.432748\pi$
$62$	0	0
$63$	−176.402	−0.352770
$64$	0	0
$65$	105.505	0.201328
$66$	0	0
$67$	−998.808	−1.82125	−0.910625	−	0.413233i	$-0.864400\pi$
$67$	−998.808	−1.82125	−0.910625	+	0.413233i	$0.864400\pi$
$68$	0	0
$69$	−48.8116	−0.0851626
$70$	0	0
$71$	27.8840	0.0466088	0.0233044	−	0.999728i	$-0.492581\pi$
$71$	27.8840	0.0466088	0.0233044	+	0.999728i	$0.492581\pi$
$72$	0	0
$73$	−161.353	−0.258698	−0.129349	−	0.991599i	$-0.541289\pi$
$73$	−161.353	−0.258698	−0.129349	+	0.991599i	$0.541289\pi$
$74$	0	0
$75$	146.494	0.225542
$76$	0	0
$77$	77.0000	0.113961
$78$	0	0
$79$	624.995	0.890094	0.445047	−	0.895507i	$-0.353187\pi$
$79$	624.995	0.890094	0.445047	+	0.895507i	$0.353187\pi$
$80$	0	0
$81$	586.457	0.804467
$82$	0	0
$83$	−396.724	−0.524652	−0.262326	−	0.964979i	$-0.584490\pi$
$83$	−396.724	−0.524652	−0.262326	+	0.964979i	$0.584490\pi$
$84$	0	0
$85$	−360.820	−0.460428
$86$	0	0
$87$	−236.668	−0.291649
$88$	0	0
$89$	−1405.84	−1.67436	−0.837182	−	0.546924i	$-0.815799\pi$
$89$	−1405.84	−1.67436	−0.837182	+	0.546924i	$0.815799\pi$
$90$	0	0
$91$	185.781	0.214012
$92$	0	0
$93$	238.214	0.265610
$94$	0	0
$95$	271.934	0.293682
$96$	0	0
$97$	−193.990	−0.203059	−0.101530	−	0.994833i	$-0.532374\pi$
$97$	−193.990	−0.203059	−0.101530	+	0.994833i	$0.532374\pi$
$98$	0	0
$99$	−277.202	−0.281413
$100$	0	0
$101$	688.977	0.678770	0.339385	−	0.940647i	$-0.389781\pi$
$101$	688.977	0.678770	0.339385	+	0.940647i	$0.389781\pi$
$102$	0	0
$103$	−1307.38	−1.25068	−0.625341	−	0.780351i	$-0.715040\pi$
$103$	−1307.38	−1.25068	−0.625341	+	0.780351i	$0.715040\pi$
$104$	0	0
$105$	−37.3319	−0.0346973
$106$	0	0
$107$	−1291.60	−1.16695	−0.583474	−	0.812131i	$-0.698307\pi$
$107$	−1291.60	−1.16695	−0.583474	+	0.812131i	$0.698307\pi$
$108$	0	0
$109$	68.2555	0.0599788	0.0299894	−	0.999550i	$-0.490453\pi$
$109$	68.2555	0.0599788	0.0299894	+	0.999550i	$0.490453\pi$
$110$	0	0
$111$	401.558	0.343371
$112$	0	0
$113$	1511.56	1.25837	0.629185	−	0.777255i	$-0.283389\pi$
$113$	1511.56	1.25837	0.629185	+	0.777255i	$0.283389\pi$
$114$	0	0
$115$	144.639	0.117284
$116$	0	0
$117$	−668.816	−0.528479
$118$	0	0
$119$	−635.355	−0.489436
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	60.8328	0.0445944
$124$	0	0
$125$	−931.007	−0.666175
$126$	0	0
$127$	−733.984	−0.512839	−0.256420	−	0.966566i	$-0.582543\pi$
$127$	−733.984	−0.512839	−0.256420	+	0.966566i	$0.582543\pi$
$128$	0	0
$129$	−709.112	−0.483983
$130$	0	0
$131$	633.639	0.422605	0.211303	−	0.977421i	$-0.432229\pi$
$131$	633.639	0.422605	0.211303	+	0.977421i	$0.432229\pi$
$132$	0	0
$133$	478.838	0.312185
$134$	0	0
$135$	278.391	0.177482
$136$	0	0
$137$	1383.48	0.862762	0.431381	−	0.902170i	$-0.358027\pi$
$137$	1383.48	0.862762	0.431381	+	0.902170i	$0.358027\pi$
$138$	0	0
$139$	−2714.89	−1.65665	−0.828324	−	0.560250i	$-0.810705\pi$
$139$	−2714.89	−1.65665	−0.828324	+	0.560250i	$0.810705\pi$
$140$	0	0
$141$	479.532	0.286410
$142$	0	0
$143$	291.941	0.170723
$144$	0	0
$145$	701.297	0.401652
$146$	0	0
$147$	−65.7364	−0.0368833
$148$	0	0
$149$	397.290	0.218438	0.109219	−	0.994018i	$-0.465165\pi$
$149$	397.290	0.218438	0.109219	+	0.994018i	$0.465165\pi$
$150$	0	0
$151$	−1545.82	−0.833091	−0.416546	−	0.909115i	$-0.636759\pi$
$151$	−1545.82	−0.833091	−0.416546	+	0.909115i	$0.636759\pi$
$152$	0	0
$153$	2287.30	1.20861
$154$	0	0
$155$	−705.879	−0.365791
$156$	0	0
$157$	−3307.96	−1.68155	−0.840776	−	0.541383i	$-0.817901\pi$
$157$	−3307.96	−1.68155	−0.840776	+	0.541383i	$0.817901\pi$
$158$	0	0
$159$	995.773	0.496666
$160$	0	0
$161$	254.689	0.124673
$162$	0	0
$163$	2003.34	0.962661	0.481331	−	0.876539i	$-0.340154\pi$
$163$	2003.34	0.962661	0.481331	+	0.876539i	$0.340154\pi$
$164$	0	0
$165$	−58.6644	−0.0276789
$166$	0	0
$167$	−3805.24	−1.76322	−0.881612	−	0.471974i	$-0.843541\pi$
$167$	−3805.24	−1.76322	−0.881612	+	0.471974i	$0.843541\pi$
$168$	0	0
$169$	−1492.62	−0.679392
$170$	0	0
$171$	−1723.83	−0.770905
$172$	0	0
$173$	2090.48	0.918706	0.459353	−	0.888254i	$-0.348081\pi$
$173$	2090.48	0.918706	0.459353	+	0.888254i	$0.348081\pi$
$174$	0	0
$175$	−764.378	−0.330180
$176$	0	0
$177$	−1177.75	−0.500143
$178$	0	0
$179$	−3675.58	−1.53478	−0.767390	−	0.641180i	$-0.778445\pi$
$179$	−3675.58	−1.53478	−0.767390	+	0.641180i	$0.778445\pi$
$180$	0	0
$181$	−2919.03	−1.19873	−0.599365	−	0.800476i	$-0.704580\pi$
$181$	−2919.03	−1.19873	−0.599365	+	0.800476i	$0.704580\pi$
$182$	0	0
$183$	−268.075	−0.108288
$184$	0	0
$185$	−1189.90	−0.472882
$186$	0	0
$187$	−998.415	−0.390435
$188$	0	0
$189$	490.208	0.188663
$190$	0	0
$191$	−1612.84	−0.611002	−0.305501	−	0.952192i	$-0.598824\pi$
$191$	−1612.84	−0.611002	−0.305501	+	0.952192i	$0.598824\pi$
$192$	0	0
$193$	−4373.46	−1.63113	−0.815566	−	0.578664i	$-0.803574\pi$
$193$	−4373.46	−1.63113	−0.815566	+	0.578664i	$0.803574\pi$
$194$	0	0
$195$	−141.542	−0.0519796
$196$	0	0
$197$	−391.411	−0.141558	−0.0707788	−	0.997492i	$-0.522548\pi$
$197$	−391.411	−0.141558	−0.0707788	+	0.997492i	$0.522548\pi$
$198$	0	0
$199$	3093.54	1.10199	0.550993	−	0.834510i	$-0.314249\pi$
$199$	3093.54	1.10199	0.550993	+	0.834510i	$0.314249\pi$
$200$	0	0
$201$	1339.96	0.470217
$202$	0	0
$203$	1234.89	0.426957
$204$	0	0
$205$	−180.260	−0.0614143
$206$	0	0
$207$	−916.889	−0.307866
$208$	0	0
$209$	752.460	0.249037
$210$	0	0
$211$	−4160.19	−1.35734	−0.678671	−	0.734442i	$-0.737444\pi$
$211$	−4160.19	−1.35734	−0.678671	+	0.734442i	$0.737444\pi$
$212$	0	0
$213$	−37.4081	−0.0120336
$214$	0	0
$215$	2101.25	0.666530
$216$	0	0
$217$	−1242.96	−0.388836
$218$	0	0
$219$	216.465	0.0667914
$220$	0	0
$221$	−2408.91	−0.733216
$222$	0	0
$223$	112.785	0.0338683	0.0169341	−	0.999857i	$-0.494609\pi$
$223$	112.785	0.0338683	0.0169341	+	0.999857i	$0.494609\pi$
$224$	0	0
$225$	2751.78	0.815343
$226$	0	0
$227$	4854.90	1.41952	0.709759	−	0.704444i	$-0.248804\pi$
$227$	4854.90	1.41952	0.709759	+	0.704444i	$0.248804\pi$
$228$	0	0
$229$	−3903.82	−1.12651	−0.563257	−	0.826281i	$-0.690452\pi$
$229$	−3903.82	−1.12651	−0.563257	+	0.826281i	$0.690452\pi$
$230$	0	0
$231$	−103.300	−0.0294227
$232$	0	0
$233$	−3682.12	−1.03529	−0.517647	−	0.855594i	$-0.673192\pi$
$233$	−3682.12	−1.03529	−0.517647	+	0.855594i	$0.673192\pi$
$234$	0	0
$235$	−1420.95	−0.394437
$236$	0	0
$237$	−838.468	−0.229807
$238$	0	0
$239$	−845.999	−0.228967	−0.114484	−	0.993425i	$-0.536521\pi$
$239$	−845.999	−0.228967	−0.114484	+	0.993425i	$0.536521\pi$
$240$	0	0
$241$	−6552.21	−1.75131	−0.875653	−	0.482940i	$-0.839569\pi$
$241$	−6552.21	−1.75131	−0.875653	+	0.482940i	$0.839569\pi$
$242$	0	0
$243$	−2677.57	−0.706857
$244$	0	0
$245$	194.791	0.0507948
$246$	0	0
$247$	1815.49	0.467679
$248$	0	0
$249$	532.229	0.135456
$250$	0	0
$251$	−4081.83	−1.02646	−0.513232	−	0.858250i	$-0.671552\pi$
$251$	−4081.83	−1.02646	−0.513232	+	0.858250i	$0.671552\pi$
$252$	0	0
$253$	400.226	0.0994545
$254$	0	0
$255$	484.061	0.118875
$256$	0	0
$257$	−1570.18	−0.381109	−0.190555	−	0.981677i	$-0.561029\pi$
$257$	−1570.18	−0.381109	−0.190555	+	0.981677i	$0.561029\pi$
$258$	0	0
$259$	−2095.25	−0.502674
$260$	0	0
$261$	−4445.64	−1.05432
$262$	0	0
$263$	1148.34	0.269239	0.134620	−	0.990897i	$-0.457019\pi$
$263$	1148.34	0.269239	0.134620	+	0.990897i	$0.457019\pi$
$264$	0	0
$265$	−2950.68	−0.683996
$266$	0	0
$267$	1886.02	0.432293
$268$	0	0
$269$	−4581.61	−1.03846	−0.519230	−	0.854635i	$-0.673781\pi$
$269$	−4581.61	−1.03846	−0.519230	+	0.854635i	$0.673781\pi$
$270$	0	0
$271$	6763.70	1.51611	0.758055	−	0.652191i	$-0.226150\pi$
$271$	6763.70	1.51611	0.758055	+	0.652191i	$0.226150\pi$
$272$	0	0
$273$	−249.236	−0.0552544
$274$	0	0
$275$	−1201.17	−0.263393
$276$	0	0
$277$	8894.53	1.92932	0.964658	−	0.263505i	$-0.0848785\pi$
$277$	8894.53	1.92932	0.964658	+	0.263505i	$0.0848785\pi$
$278$	0	0
$279$	4474.68	0.960187
$280$	0	0
$281$	8945.28	1.89904	0.949521	−	0.313704i	$-0.101570\pi$
$281$	8945.28	1.89904	0.949521	+	0.313704i	$0.101570\pi$
$282$	0	0
$283$	4521.94	0.949828	0.474914	−	0.880032i	$-0.342479\pi$
$283$	4521.94	0.949828	0.474914	+	0.880032i	$0.342479\pi$
$284$	0	0
$285$	−364.815	−0.0758238
$286$	0	0
$287$	−317.414	−0.0652834
$288$	0	0
$289$	3325.28	0.676833
$290$	0	0
$291$	260.250	0.0524265
$292$	0	0
$293$	1544.13	0.307880	0.153940	−	0.988080i	$-0.450804\pi$
$293$	1544.13	0.307880	0.153940	+	0.988080i	$0.450804\pi$
$294$	0	0
$295$	3489.92	0.688784
$296$	0	0
$297$	770.327	0.150501
$298$	0	0
$299$	965.639	0.186770
$300$	0	0
$301$	3700.01	0.708522
$302$	0	0
$303$	−924.304	−0.175247
$304$	0	0
$305$	794.361	0.149131
$306$	0	0
$307$	6540.45	1.21591	0.607953	−	0.793973i	$-0.291991\pi$
$307$	6540.45	1.21591	0.607953	+	0.793973i	$0.291991\pi$
$308$	0	0
$309$	1753.93	0.322906
$310$	0	0
$311$	−4746.76	−0.865480	−0.432740	−	0.901519i	$-0.642453\pi$
$311$	−4746.76	−0.865480	−0.432740	+	0.901519i	$0.642453\pi$
$312$	0	0
$313$	−4244.13	−0.766429	−0.383214	−	0.923659i	$-0.625183\pi$
$313$	−4244.13	−0.766429	−0.383214	+	0.923659i	$0.625183\pi$
$314$	0	0
$315$	−701.252	−0.125432
$316$	0	0
$317$	−10090.0	−1.78773	−0.893863	−	0.448340i	$-0.852015\pi$
$317$	−10090.0	−1.78773	−0.893863	+	0.448340i	$0.852015\pi$
$318$	0	0
$319$	1940.54	0.340594
$320$	0	0
$321$	1732.76	0.301287
$322$	0	0
$323$	−6208.82	−1.06956
$324$	0	0
$325$	−2898.09	−0.494638
$326$	0	0
$327$	−91.5689	−0.0154855
$328$	0	0
$329$	−2502.10	−0.419287
$330$	0	0
$331$	6015.52	0.998921	0.499461	−	0.866337i	$-0.333532\pi$
$331$	6015.52	0.998921	0.499461	+	0.866337i	$0.333532\pi$
$332$	0	0
$333$	7542.97	1.24130
$334$	0	0
$335$	−3970.58	−0.647570
$336$	0	0
$337$	−2908.47	−0.470132	−0.235066	−	0.971979i	$-0.575531\pi$
$337$	−2908.47	−0.470132	−0.235066	+	0.971979i	$0.575531\pi$
$338$	0	0
$339$	−2027.85	−0.324890
$340$	0	0
$341$	−1953.22	−0.310184
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	−194.042	−0.0302807
$346$	0	0
$347$	−4305.75	−0.666123	−0.333061	−	0.942905i	$-0.608082\pi$
$347$	−4305.75	−0.666123	−0.333061	+	0.942905i	$0.608082\pi$
$348$	0	0
$349$	8192.82	1.25660	0.628298	−	0.777973i	$-0.283752\pi$
$349$	8192.82	1.25660	0.628298	+	0.777973i	$0.283752\pi$
$350$	0	0
$351$	1858.59	0.282634
$352$	0	0
$353$	−5527.00	−0.833350	−0.416675	−	0.909056i	$-0.636805\pi$
$353$	−5527.00	−0.833350	−0.416675	+	0.909056i	$0.636805\pi$
$354$	0	0
$355$	110.848	0.0165724
$356$	0	0
$357$	852.367	0.126364
$358$	0	0
$359$	2085.92	0.306660	0.153330	−	0.988175i	$-0.451000\pi$
$359$	2085.92	0.306660	0.153330	+	0.988175i	$0.451000\pi$
$360$	0	0
$361$	−2179.69	−0.317785
$362$	0	0
$363$	−162.329	−0.0234712
$364$	0	0
$365$	−641.429	−0.0919834
$366$	0	0
$367$	10067.4	1.43191	0.715957	−	0.698144i	$-0.245991\pi$
$367$	10067.4	1.43191	0.715957	+	0.698144i	$0.245991\pi$
$368$	0	0
$369$	1142.70	0.161210
$370$	0	0
$371$	−5195.75	−0.727089
$372$	0	0
$373$	−6711.62	−0.931674	−0.465837	−	0.884871i	$-0.654247\pi$
$373$	−6711.62	−0.931674	−0.465837	+	0.884871i	$0.654247\pi$
$374$	0	0
$375$	1249.00	0.171995
$376$	0	0
$377$	4682.01	0.639617
$378$	0	0
$379$	−2745.99	−0.372169	−0.186085	−	0.982534i	$-0.559580\pi$
$379$	−2745.99	−0.372169	−0.186085	+	0.982534i	$0.559580\pi$
$380$	0	0
$381$	984.684	0.132407
$382$	0	0
$383$	−1317.74	−0.175804	−0.0879022	−	0.996129i	$-0.528016\pi$
$383$	−1317.74	−0.175804	−0.0879022	+	0.996129i	$0.528016\pi$
$384$	0	0
$385$	306.100	0.0405202
$386$	0	0
$387$	−13320.2	−1.74962
$388$	0	0
$389$	−2950.37	−0.384549	−0.192275	−	0.981341i	$-0.561586\pi$
$389$	−2950.37	−0.384549	−0.192275	+	0.981341i	$0.561586\pi$
$390$	0	0
$391$	−3302.41	−0.427136
$392$	0	0
$393$	−850.064	−0.109110
$394$	0	0
$395$	2484.55	0.316485
$396$	0	0
$397$	−3089.13	−0.390526	−0.195263	−	0.980751i	$-0.562556\pi$
$397$	−3089.13	−0.390526	−0.195263	+	0.980751i	$0.562556\pi$
$398$	0	0
$399$	−642.390	−0.0806009
$400$	0	0
$401$	−2672.29	−0.332787	−0.166394	−	0.986059i	$-0.553212\pi$
$401$	−2672.29	−0.332787	−0.166394	+	0.986059i	$0.553212\pi$
$402$	0	0
$403$	−4712.60	−0.582509
$404$	0	0
$405$	2331.35	0.286039
$406$	0	0
$407$	−3292.54	−0.400995
$408$	0	0
$409$	10229.9	1.23676	0.618380	−	0.785879i	$-0.287789\pi$
$409$	10229.9	1.23676	0.618380	+	0.785879i	$0.287789\pi$
$410$	0	0
$411$	−1856.02	−0.222751
$412$	0	0
$413$	6145.28	0.732178
$414$	0	0
$415$	−1577.10	−0.186547
$416$	0	0
$417$	3642.19	0.427719
$418$	0	0
$419$	10442.4	1.21752	0.608762	−	0.793353i	$-0.291666\pi$
$419$	10442.4	1.21752	0.608762	+	0.793353i	$0.291666\pi$
$420$	0	0
$421$	8956.62	1.03686	0.518431	−	0.855120i	$-0.326516\pi$
$421$	8956.62	1.03686	0.518431	+	0.855120i	$0.326516\pi$
$422$	0	0
$423$	9007.65	1.03538
$424$	0	0
$425$	9911.25	1.13121
$426$	0	0
$427$	1398.76	0.158527
$428$	0	0
$429$	−391.656	−0.0440777
$430$	0	0
$431$	−13564.2	−1.51593	−0.757963	−	0.652298i	$-0.773805\pi$
$431$	−13564.2	−1.51593	−0.757963	+	0.652298i	$0.773805\pi$
$432$	0	0
$433$	−1384.33	−0.153641	−0.0768204	−	0.997045i	$-0.524477\pi$
$433$	−1384.33	−0.153641	−0.0768204	+	0.997045i	$0.524477\pi$
$434$	0	0
$435$	−940.832	−0.103700
$436$	0	0
$437$	2488.88	0.272447
$438$	0	0
$439$	4836.01	0.525763	0.262882	−	0.964828i	$-0.415327\pi$
$439$	4836.01	0.525763	0.262882	+	0.964828i	$0.415327\pi$
$440$	0	0
$441$	−1234.81	−0.133334
$442$	0	0
$443$	−300.302	−0.0322071	−0.0161036	−	0.999870i	$-0.505126\pi$
$443$	−300.302	−0.0322071	−0.0161036	+	0.999870i	$0.505126\pi$
$444$	0	0
$445$	−5588.65	−0.595343
$446$	0	0
$447$	−532.988	−0.0563970
$448$	0	0
$449$	7565.06	0.795139	0.397569	−	0.917572i	$-0.369854\pi$
$449$	7565.06	0.795139	0.397569	+	0.917572i	$0.369854\pi$
$450$	0	0
$451$	−498.793	−0.0520782
$452$	0	0
$453$	2073.81	0.215090
$454$	0	0
$455$	738.537	0.0760949
$456$	0	0
$457$	4061.45	0.415726	0.207863	−	0.978158i	$-0.433349\pi$
$457$	4061.45	0.415726	0.207863	+	0.978158i	$0.433349\pi$
$458$	0	0
$459$	−6356.25	−0.646371
$460$	0	0
$461$	17270.3	1.74481	0.872406	−	0.488781i	$-0.162558\pi$
$461$	17270.3	1.74481	0.872406	+	0.488781i	$0.162558\pi$
$462$	0	0
$463$	5459.60	0.548011	0.274005	−	0.961728i	$-0.411651\pi$
$463$	5459.60	0.548011	0.274005	+	0.961728i	$0.411651\pi$
$464$	0	0
$465$	946.979	0.0944410
$466$	0	0
$467$	5983.95	0.592942	0.296471	−	0.955042i	$-0.404190\pi$
$467$	5983.95	0.592942	0.296471	+	0.955042i	$0.404190\pi$
$468$	0	0
$469$	−6991.65	−0.688368
$470$	0	0
$471$	4437.82	0.434149
$472$	0	0
$473$	5814.30	0.565205
$474$	0	0
$475$	−7469.66	−0.721540
$476$	0	0
$477$	18704.9	1.79546
$478$	0	0
$479$	11143.3	1.06295	0.531473	−	0.847075i	$-0.321639\pi$
$479$	11143.3	1.06295	0.531473	+	0.847075i	$0.321639\pi$
$480$	0	0
$481$	−7944.03	−0.753049
$482$	0	0
$483$	−341.681	−0.0321885
$484$	0	0
$485$	−771.174	−0.0722004
$486$	0	0
$487$	1166.82	0.108571	0.0542853	−	0.998525i	$-0.482712\pi$
$487$	1166.82	0.108571	0.0542853	+	0.998525i	$0.482712\pi$
$488$	0	0
$489$	−2687.60	−0.248543
$490$	0	0
$491$	2037.98	0.187317	0.0936586	−	0.995604i	$-0.470144\pi$
$491$	2037.98	0.187317	0.0936586	+	0.995604i	$0.470144\pi$
$492$	0	0
$493$	−16012.1	−1.46278
$494$	0	0
$495$	−1101.97	−0.100060
$496$	0	0
$497$	195.188	0.0176165
$498$	0	0
$499$	2338.76	0.209814	0.104907	−	0.994482i	$-0.466546\pi$
$499$	2338.76	0.209814	0.104907	+	0.994482i	$0.466546\pi$
$500$	0	0
$501$	5104.96	0.455235
$502$	0	0
$503$	13576.7	1.20349	0.601746	−	0.798688i	$-0.294472\pi$
$503$	13576.7	1.20349	0.601746	+	0.798688i	$0.294472\pi$
$504$	0	0
$505$	2738.91	0.241346
$506$	0	0
$507$	2002.44	0.175408
$508$	0	0
$509$	18038.6	1.57082	0.785408	−	0.618978i	$-0.212453\pi$
$509$	18038.6	1.57082	0.785408	+	0.618978i	$0.212453\pi$
$510$	0	0
$511$	−1129.47	−0.0977785
$512$	0	0
$513$	4790.42	0.412285
$514$	0	0
$515$	−5197.27	−0.444697
$516$	0	0
$517$	−3931.88	−0.334475
$518$	0	0
$519$	−2804.50	−0.237195
$520$	0	0
$521$	13776.2	1.15844	0.579218	−	0.815173i	$-0.303358\pi$
$521$	13776.2	1.15844	0.579218	+	0.815173i	$0.303358\pi$
$522$	0	0
$523$	−16607.0	−1.38847	−0.694237	−	0.719747i	$-0.744258\pi$
$523$	−16607.0	−1.38847	−0.694237	+	0.719747i	$0.744258\pi$
$524$	0	0
$525$	1025.46	0.0852470
$526$	0	0
$527$	16116.7	1.33217
$528$	0	0
$529$	−10843.2	−0.891197
$530$	0	0
$531$	−22123.2	−1.80803
$532$	0	0
$533$	−1203.46	−0.0978001
$534$	0	0
$535$	−5134.52	−0.414924
$536$	0	0
$537$	4931.01	0.396255
$538$	0	0
$539$	539.000	0.0430730
$540$	0	0
$541$	−13219.0	−1.05052	−0.525259	−	0.850942i	$-0.676032\pi$
$541$	−13219.0	−1.05052	−0.525259	+	0.850942i	$0.676032\pi$
$542$	0	0
$543$	3916.06	0.309492
$544$	0	0
$545$	271.338	0.0213263
$546$	0	0
$547$	−3601.14	−0.281488	−0.140744	−	0.990046i	$-0.544949\pi$
$547$	−3601.14	−0.281488	−0.140744	+	0.990046i	$0.544949\pi$
$548$	0	0
$549$	−5035.59	−0.391464
$550$	0	0
$551$	12067.6	0.933025
$552$	0	0
$553$	4374.96	0.336424
$554$	0	0
$555$	1596.32	0.122090
$556$	0	0
$557$	1639.82	0.124742	0.0623711	−	0.998053i	$-0.480134\pi$
$557$	1639.82	0.124742	0.0623711	+	0.998053i	$0.480134\pi$
$558$	0	0
$559$	14028.4	1.06143
$560$	0	0
$561$	1339.43	0.100804
$562$	0	0
$563$	−2882.31	−0.215763	−0.107882	−	0.994164i	$-0.534407\pi$
$563$	−2882.31	−0.215763	−0.107882	+	0.994164i	$0.534407\pi$
$564$	0	0
$565$	6008.94	0.447430
$566$	0	0
$567$	4105.20	0.304060
$568$	0	0
$569$	11392.4	0.839358	0.419679	−	0.907673i	$-0.362143\pi$
$569$	11392.4	0.839358	0.419679	+	0.907673i	$0.362143\pi$
$570$	0	0
$571$	7530.31	0.551898	0.275949	−	0.961172i	$-0.411008\pi$
$571$	7530.31	0.551898	0.275949	+	0.961172i	$0.411008\pi$
$572$	0	0
$573$	2163.73	0.157750
$574$	0	0
$575$	−3973.04	−0.288152
$576$	0	0
$577$	−5251.28	−0.378880	−0.189440	−	0.981892i	$-0.560667\pi$
$577$	−5251.28	−0.378880	−0.189440	+	0.981892i	$0.560667\pi$
$578$	0	0
$579$	5867.26	0.421131
$580$	0	0
$581$	−2777.07	−0.198300
$582$	0	0
$583$	−8164.75	−0.580016
$584$	0	0
$585$	−2658.76	−0.187908
$586$	0	0
$587$	11620.8	0.817109	0.408554	−	0.912734i	$-0.366033\pi$
$587$	11620.8	0.817109	0.408554	+	0.912734i	$0.366033\pi$
$588$	0	0
$589$	−12146.4	−0.849720
$590$	0	0
$591$	525.101	0.0365478
$592$	0	0
$593$	−12353.8	−0.855500	−0.427750	−	0.903897i	$-0.640694\pi$
$593$	−12353.8	−0.855500	−0.427750	+	0.903897i	$0.640694\pi$
$594$	0	0
$595$	−2525.74	−0.174026
$596$	0	0
$597$	−4150.17	−0.284515
$598$	0	0
$599$	11302.9	0.770995	0.385497	−	0.922709i	$-0.374030\pi$
$599$	11302.9	0.770995	0.385497	+	0.922709i	$0.374030\pi$
$600$	0	0
$601$	−5324.36	−0.361373	−0.180687	−	0.983541i	$-0.557832\pi$
$601$	−5324.36	−0.361373	−0.180687	+	0.983541i	$0.557832\pi$
$602$	0	0
$603$	25170.2	1.69985
$604$	0	0
$605$	481.014	0.0323239
$606$	0	0
$607$	15278.1	1.02162	0.510808	−	0.859695i	$-0.329346\pi$
$607$	15278.1	1.02162	0.510808	+	0.859695i	$0.329346\pi$
$608$	0	0
$609$	−1656.68	−0.110233
$610$	0	0
$611$	−9486.58	−0.628128
$612$	0	0
$613$	−11973.6	−0.788925	−0.394462	−	0.918912i	$-0.629069\pi$
$613$	−11973.6	−0.788925	−0.394462	+	0.918912i	$0.629069\pi$
$614$	0	0
$615$	241.830	0.0158561
$616$	0	0
$617$	−27903.5	−1.82067	−0.910333	−	0.413877i	$-0.864174\pi$
$617$	−27903.5	−1.82067	−0.910333	+	0.413877i	$0.864174\pi$
$618$	0	0
$619$	6496.11	0.421810	0.210905	−	0.977507i	$-0.432359\pi$
$619$	6496.11	0.421810	0.210905	+	0.977507i	$0.432359\pi$
$620$	0	0
$621$	2547.97	0.164648
$622$	0	0
$623$	−9840.86	−0.632850
$624$	0	0
$625$	9948.55	0.636707
$626$	0	0
$627$	−1009.47	−0.0642972
$628$	0	0
$629$	27167.9	1.72219
$630$	0	0
$631$	10165.2	0.641316	0.320658	−	0.947195i	$-0.396096\pi$
$631$	10165.2	0.641316	0.320658	+	0.947195i	$0.396096\pi$
$632$	0	0
$633$	5581.14	0.350443
$634$	0	0
$635$	−2917.82	−0.182347
$636$	0	0
$637$	1300.46	0.0808890
$638$	0	0
$639$	−702.683	−0.0435019
$640$	0	0
$641$	−17202.6	−1.06000	−0.530001	−	0.847997i	$-0.677809\pi$
$641$	−17202.6	−1.06000	−0.530001	+	0.847997i	$0.677809\pi$
$642$	0	0
$643$	13202.1	0.809706	0.404853	−	0.914382i	$-0.367323\pi$
$643$	13202.1	0.809706	0.404853	+	0.914382i	$0.367323\pi$
$644$	0	0
$645$	−2818.95	−0.172087
$646$	0	0
$647$	14079.8	0.855537	0.427768	−	0.903888i	$-0.359300\pi$
$647$	14079.8	0.855537	0.427768	+	0.903888i	$0.359300\pi$
$648$	0	0
$649$	9656.88	0.584076
$650$	0	0
$651$	1667.50	0.100391
$652$	0	0
$653$	17934.3	1.07477	0.537384	−	0.843338i	$-0.319413\pi$
$653$	17934.3	1.07477	0.537384	+	0.843338i	$0.319413\pi$
$654$	0	0
$655$	2518.92	0.150263
$656$	0	0
$657$	4066.13	0.241453
$658$	0	0
$659$	−7268.92	−0.429677	−0.214838	−	0.976650i	$-0.568922\pi$
$659$	−7268.92	−0.429677	−0.214838	+	0.976650i	$0.568922\pi$
$660$	0	0
$661$	−19035.2	−1.12009	−0.560047	−	0.828461i	$-0.689217\pi$
$661$	−19035.2	−1.12009	−0.560047	+	0.828461i	$0.689217\pi$
$662$	0	0
$663$	3231.70	0.189304
$664$	0	0
$665$	1903.54	0.111001
$666$	0	0
$667$	6418.63	0.372609
$668$	0	0
$669$	−151.308	−0.00874423
$670$	0	0
$671$	2198.06	0.126461
$672$	0	0
$673$	24963.4	1.42982	0.714910	−	0.699217i	$-0.246468\pi$
$673$	24963.4	1.42982	0.714910	+	0.699217i	$0.246468\pi$
$674$	0	0
$675$	−7647.02	−0.436051
$676$	0	0
$677$	3759.96	0.213452	0.106726	−	0.994288i	$-0.465963\pi$
$677$	3759.96	0.213452	0.106726	+	0.994288i	$0.465963\pi$
$678$	0	0
$679$	−1357.93	−0.0767491
$680$	0	0
$681$	−6513.13	−0.366496
$682$	0	0
$683$	−10321.7	−0.578257	−0.289129	−	0.957290i	$-0.593366\pi$
$683$	−10321.7	−0.578257	−0.289129	+	0.957290i	$0.593366\pi$
$684$	0	0
$685$	5499.76	0.306767
$686$	0	0
$687$	5237.21	0.290847
$688$	0	0
$689$	−19699.4	−1.08924
$690$	0	0
$691$	19059.0	1.04926	0.524630	−	0.851330i	$-0.324204\pi$
$691$	19059.0	1.04926	0.524630	+	0.851330i	$0.324204\pi$
$692$	0	0
$693$	−1940.42	−0.106364
$694$	0	0
$695$	−10792.6	−0.589043
$696$	0	0
$697$	4115.72	0.223664
$698$	0	0
$699$	4939.78	0.267296
$700$	0	0
$701$	−9851.92	−0.530816	−0.265408	−	0.964136i	$-0.585507\pi$
$701$	−9851.92	−0.530816	−0.265408	+	0.964136i	$0.585507\pi$
$702$	0	0
$703$	−20475.3	−1.09849
$704$	0	0
$705$	1906.29	0.101837
$706$	0	0
$707$	4822.84	0.256551
$708$	0	0
$709$	−8151.11	−0.431765	−0.215882	−	0.976419i	$-0.569263\pi$
$709$	−8151.11	−0.431765	−0.215882	+	0.976419i	$0.569263\pi$
$710$	0	0
$711$	−15750.0	−0.830761
$712$	0	0
$713$	−6460.57	−0.339341
$714$	0	0
$715$	1160.56	0.0607027
$716$	0	0
$717$	1134.96	0.0591155
$718$	0	0
$719$	31381.4	1.62772	0.813858	−	0.581064i	$-0.197363\pi$
$719$	31381.4	1.62772	0.813858	+	0.581064i	$0.197363\pi$
$720$	0	0
$721$	−9151.69	−0.472714
$722$	0	0
$723$	8790.18	0.452158
$724$	0	0
$725$	−19263.7	−0.986808
$726$	0	0
$727$	16681.8	0.851024	0.425512	−	0.904953i	$-0.360094\pi$
$727$	16681.8	0.851024	0.425512	+	0.904953i	$0.360094\pi$
$728$	0	0
$729$	−12242.2	−0.621969
$730$	0	0
$731$	−47975.9	−2.42743
$732$	0	0
$733$	−3494.64	−0.176095	−0.0880475	−	0.996116i	$-0.528063\pi$
$733$	−3494.64	−0.176095	−0.0880475	+	0.996116i	$0.528063\pi$
$734$	0	0
$735$	−261.323	−0.0131144
$736$	0	0
$737$	−10986.9	−0.549128
$738$	0	0
$739$	−7824.53	−0.389486	−0.194743	−	0.980854i	$-0.562387\pi$
$739$	−7824.53	−0.389486	−0.194743	+	0.980854i	$0.562387\pi$
$740$	0	0
$741$	−2435.59	−0.120747
$742$	0	0
$743$	4185.29	0.206653	0.103327	−	0.994647i	$-0.467051\pi$
$743$	4185.29	0.206653	0.103327	+	0.994647i	$0.467051\pi$
$744$	0	0
$745$	1579.35	0.0776685
$746$	0	0
$747$	9997.53	0.489679
$748$	0	0
$749$	−9041.19	−0.441065
$750$	0	0
$751$	−20789.0	−1.01012	−0.505061	−	0.863084i	$-0.668530\pi$
$751$	−20789.0	−1.01012	−0.505061	+	0.863084i	$0.668530\pi$
$752$	0	0
$753$	5476.01	0.265016
$754$	0	0
$755$	−6145.11	−0.296217
$756$	0	0
$757$	15618.5	0.749886	0.374943	−	0.927048i	$-0.377662\pi$
$757$	15618.5	0.749886	0.374943	+	0.927048i	$0.377662\pi$
$758$	0	0
$759$	−536.927	−0.0256775
$760$	0	0
$761$	40789.7	1.94300	0.971502	−	0.237030i	$-0.0761739\pi$
$761$	40789.7	1.94300	0.971502	+	0.237030i	$0.0761739\pi$
$762$	0	0
$763$	477.789	0.0226699
$764$	0	0
$765$	9092.74	0.429737
$766$	0	0
$767$	23299.5	1.09687
$768$	0	0
$769$	−33304.6	−1.56176	−0.780881	−	0.624680i	$-0.785230\pi$
$769$	−33304.6	−1.56176	−0.780881	+	0.624680i	$0.785230\pi$
$770$	0	0
$771$	2106.49	0.0983961
$772$	0	0
$773$	33766.3	1.57114	0.785570	−	0.618773i	$-0.212370\pi$
$773$	33766.3	1.57114	0.785570	+	0.618773i	$0.212370\pi$
$774$	0	0
$775$	19389.6	0.898701
$776$	0	0
$777$	2810.91	0.129782
$778$	0	0
$779$	−3101.84	−0.142663
$780$	0	0
$781$	306.724	0.0140531
$782$	0	0
$783$	12354.1	0.563858
$784$	0	0
$785$	−13150.2	−0.597898
$786$	0	0
$787$	−12756.1	−0.577772	−0.288886	−	0.957363i	$-0.593285\pi$
$787$	−12756.1	−0.577772	−0.288886	+	0.957363i	$0.593285\pi$
$788$	0	0
$789$	−1540.57	−0.0695131
$790$	0	0
$791$	10580.9	0.475619
$792$	0	0
$793$	5303.33	0.237486
$794$	0	0
$795$	3958.52	0.176596
$796$	0	0
$797$	7458.38	0.331480	0.165740	−	0.986169i	$-0.446999\pi$
$797$	7458.38	0.331480	0.165740	+	0.986169i	$0.446999\pi$
$798$	0	0
$799$	32443.4	1.43650
$800$	0	0
$801$	35427.4	1.56275
$802$	0	0
$803$	−1774.88	−0.0780003
$804$	0	0
$805$	1012.47	0.0443291
$806$	0	0
$807$	6146.51	0.268113
$808$	0	0
$809$	20250.3	0.880054	0.440027	−	0.897985i	$-0.354969\pi$
$809$	20250.3	0.880054	0.440027	+	0.897985i	$0.354969\pi$
$810$	0	0
$811$	−41221.5	−1.78481	−0.892406	−	0.451232i	$-0.850985\pi$
$811$	−41221.5	−1.78481	−0.892406	+	0.451232i	$0.850985\pi$
$812$	0	0
$813$	−9073.92	−0.391434
$814$	0	0
$815$	7963.92	0.342287
$816$	0	0
$817$	36157.3	1.54833
$818$	0	0
$819$	−4681.71	−0.199746
$820$	0	0
$821$	12490.9	0.530982	0.265491	−	0.964113i	$-0.414466\pi$
$821$	12490.9	0.530982	0.265491	+	0.964113i	$0.414466\pi$
$822$	0	0
$823$	2680.97	0.113551	0.0567756	−	0.998387i	$-0.481918\pi$
$823$	2680.97	0.113551	0.0567756	+	0.998387i	$0.481918\pi$
$824$	0	0
$825$	1611.44	0.0680036
$826$	0	0
$827$	6241.34	0.262434	0.131217	−	0.991354i	$-0.458112\pi$
$827$	6241.34	0.262434	0.131217	+	0.991354i	$0.458112\pi$
$828$	0	0
$829$	−16631.4	−0.696781	−0.348390	−	0.937350i	$-0.613272\pi$
$829$	−16631.4	−0.696781	−0.348390	+	0.937350i	$0.613272\pi$
$830$	0	0
$831$	−11932.5	−0.498117
$832$	0	0
$833$	−4447.48	−0.184989
$834$	0	0
$835$	−15127.1	−0.626938
$836$	0	0
$837$	−12434.8	−0.513514
$838$	0	0
$839$	5876.83	0.241824	0.120912	−	0.992663i	$-0.461418\pi$
$839$	5876.83	0.241824	0.120912	+	0.992663i	$0.461418\pi$
$840$	0	0
$841$	6732.44	0.276044
$842$	0	0
$843$	−12000.6	−0.490301
$844$	0	0
$845$	−5933.66	−0.241567
$846$	0	0
$847$	847.000	0.0343604
$848$	0	0
$849$	−6066.45	−0.245230
$850$	0	0
$851$	−10890.6	−0.438689
$852$	0	0
$853$	−17826.7	−0.715562	−0.357781	−	0.933806i	$-0.616467\pi$
$853$	−17826.7	−0.715562	−0.357781	+	0.933806i	$0.616467\pi$
$854$	0	0
$855$	−6852.79	−0.274106
$856$	0	0
$857$	2226.14	0.0887323	0.0443661	−	0.999015i	$-0.485873\pi$
$857$	2226.14	0.0887323	0.0443661	+	0.999015i	$0.485873\pi$
$858$	0	0
$859$	−35149.2	−1.39613	−0.698065	−	0.716034i	$-0.745955\pi$
$859$	−35149.2	−1.39613	−0.698065	+	0.716034i	$0.745955\pi$
$860$	0	0
$861$	425.830	0.0168551
$862$	0	0
$863$	1629.90	0.0642901	0.0321450	−	0.999483i	$-0.489766\pi$
$863$	1629.90	0.0642901	0.0321450	+	0.999483i	$0.489766\pi$
$864$	0	0
$865$	8310.32	0.326658
$866$	0	0
$867$	−4461.07	−0.174747
$868$	0	0
$869$	6874.94	0.268373
$870$	0	0
$871$	−26508.4	−1.03123
$872$	0	0
$873$	4888.60	0.189523
$874$	0	0
$875$	−6517.05	−0.251790
$876$	0	0
$877$	2438.38	0.0938862	0.0469431	−	0.998898i	$-0.485052\pi$
$877$	2438.38	0.0938862	0.0469431	+	0.998898i	$0.485052\pi$
$878$	0	0
$879$	−2071.54	−0.0794894
$880$	0	0
$881$	9761.67	0.373302	0.186651	−	0.982426i	$-0.440237\pi$
$881$	9761.67	0.373302	0.186651	+	0.982426i	$0.440237\pi$
$882$	0	0
$883$	−25129.5	−0.957729	−0.478865	−	0.877889i	$-0.658951\pi$
$883$	−25129.5	−0.957729	−0.478865	+	0.877889i	$0.658951\pi$
$884$	0	0
$885$	−4681.94	−0.177833
$886$	0	0
$887$	8803.79	0.333261	0.166630	−	0.986019i	$-0.446711\pi$
$887$	8803.79	0.333261	0.166630	+	0.986019i	$0.446711\pi$
$888$	0	0
$889$	−5137.89	−0.193835
$890$	0	0
$891$	6451.02	0.242556
$892$	0	0
$893$	−24451.1	−0.916265
$894$	0	0
$895$	−14611.6	−0.545712
$896$	0	0
$897$	−1295.46	−0.0482210
$898$	0	0
$899$	−31324.8	−1.16211
$900$	0	0
$901$	67370.3	2.49104
$902$	0	0
$903$	−4963.79	−0.182929
$904$	0	0
$905$	−11604.1	−0.426225
$906$	0	0
$907$	6930.10	0.253705	0.126852	−	0.991922i	$-0.459513\pi$
$907$	6930.10	0.253705	0.126852	+	0.991922i	$0.459513\pi$
$908$	0	0
$909$	−17362.4	−0.633524
$910$	0	0
$911$	7556.80	0.274828	0.137414	−	0.990514i	$-0.456121\pi$
$911$	7556.80	0.274828	0.137414	+	0.990514i	$0.456121\pi$
$912$	0	0
$913$	−4363.96	−0.158189
$914$	0	0
$915$	−1065.68	−0.0385032
$916$	0	0
$917$	4435.47	0.159730
$918$	0	0
$919$	−3075.63	−0.110398	−0.0551989	−	0.998475i	$-0.517579\pi$
$919$	−3075.63	−0.110398	−0.0551989	+	0.998475i	$0.517579\pi$
$920$	0	0
$921$	−8774.41	−0.313927
$922$	0	0
$923$	740.044	0.0263910
$924$	0	0
$925$	32685.0	1.16181
$926$	0	0
$927$	32946.4	1.16731
$928$	0	0
$929$	7598.42	0.268349	0.134174	−	0.990958i	$-0.457162\pi$
$929$	7598.42	0.268349	0.134174	+	0.990958i	$0.457162\pi$
$930$	0	0
$931$	3351.87	0.117995
$932$	0	0
$933$	6368.07	0.223453
$934$	0	0
$935$	−3969.02	−0.138824
$936$	0	0
$937$	−43728.9	−1.52461	−0.762306	−	0.647217i	$-0.775933\pi$
$937$	−43728.9	−1.52461	−0.762306	+	0.647217i	$0.775933\pi$
$938$	0	0
$939$	5693.75	0.197879
$940$	0	0
$941$	20746.8	0.718731	0.359366	−	0.933197i	$-0.382993\pi$
$941$	20746.8	0.718731	0.359366	+	0.933197i	$0.382993\pi$
$942$	0	0
$943$	−1649.83	−0.0569735
$944$	0	0
$945$	1948.73	0.0670818
$946$	0	0
$947$	−6932.82	−0.237895	−0.118947	−	0.992901i	$-0.537952\pi$
$947$	−6932.82	−0.237895	−0.118947	+	0.992901i	$0.537952\pi$
$948$	0	0
$949$	−4282.32	−0.146480
$950$	0	0
$951$	13536.3	0.461561
$952$	0	0
$953$	−3570.59	−0.121367	−0.0606836	−	0.998157i	$-0.519328\pi$
$953$	−3570.59	−0.121367	−0.0606836	+	0.998157i	$0.519328\pi$
$954$	0	0
$955$	−6411.57	−0.217250
$956$	0	0
$957$	−2603.35	−0.0879356
$958$	0	0
$959$	9684.33	0.326093
$960$	0	0
$961$	1738.41	0.0583537
$962$	0	0
$963$	32548.6	1.08916
$964$	0	0
$965$	−17385.9	−0.579971
$966$	0	0
$967$	32362.2	1.07621	0.538107	−	0.842877i	$-0.319140\pi$
$967$	32362.2	1.07621	0.538107	+	0.842877i	$0.319140\pi$
$968$	0	0
$969$	8329.51	0.276143
$970$	0	0
$971$	5304.75	0.175322	0.0876609	−	0.996150i	$-0.472061\pi$
$971$	5304.75	0.175322	0.0876609	+	0.996150i	$0.472061\pi$
$972$	0	0
$973$	−19004.2	−0.626154
$974$	0	0
$975$	3887.97	0.127707
$976$	0	0
$977$	−46235.2	−1.51402	−0.757009	−	0.653404i	$-0.773340\pi$
$977$	−46235.2	−1.51402	−0.757009	+	0.653404i	$0.773340\pi$
$978$	0	0
$979$	−15464.2	−0.504840
$980$	0	0
$981$	−1720.05	−0.0559807
$982$	0	0
$983$	−39377.5	−1.27767	−0.638833	−	0.769345i	$-0.720583\pi$
$983$	−39377.5	−1.27767	−0.638833	+	0.769345i	$0.720583\pi$
$984$	0	0
$985$	−1555.98	−0.0503327
$986$	0	0
$987$	3356.72	0.108253
$988$	0	0
$989$	19231.7	0.618334
$990$	0	0
$991$	−38673.9	−1.23968	−0.619838	−	0.784730i	$-0.712802\pi$
$991$	−38673.9	−1.23968	−0.619838	+	0.784730i	$0.712802\pi$
$992$	0	0
$993$	−8070.18	−0.257905
$994$	0	0
$995$	12297.8	0.391826
$996$	0	0
$997$	51487.1	1.63552	0.817760	−	0.575560i	$-0.195216\pi$
$997$	51487.1	1.63552	0.817760	+	0.575560i	$0.195216\pi$
$998$	0	0
$999$	−20961.4	−0.663854

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1232.4.a.r.1.2		3
4.3	odd	2		308.4.a.c.1.2	✓	3
28.27	even	2		2156.4.a.e.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
308.4.a.c.1.2	✓	3	4.3	odd	2
1232.4.a.r.1.2		3	1.1	even	1	trivial
2156.4.a.e.1.2		3	28.27	even	2

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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 $\F_{q}$

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Fields

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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	1232.4.a.r.1.2

Level	$1232$
Weight	$4$
Character	1232.1
Self dual	yes
Analytic conductor	$72.690$
Analytic rank	$1$
Dimension	$3$
CM	no
Inner twists	$1$