LMFDB - Embedded newform 1232.4.a.y.1.1

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:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 42x^{3} + 18x^{2} + 368x + 352 $ x^5 - x^4 - 42x^3 + 18x^2 + 368*x + 352
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.44399$ 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)

$f(q)$	$=$	$q-8.26395 q^{3} -22.0150 q^{5} -7.00000 q^{7} +41.2928 q^{9} -11.0000 q^{11} -51.5769 q^{13} +181.931 q^{15} -26.5590 q^{17} -99.6432 q^{19} +57.8476 q^{21} -28.1455 q^{23} +359.660 q^{25} -118.115 q^{27} -43.9369 q^{29} +83.8402 q^{31} +90.9034 q^{33} +154.105 q^{35} +306.353 q^{37} +426.228 q^{39} +200.991 q^{41} +13.7546 q^{43} -909.062 q^{45} +266.533 q^{47} +49.0000 q^{49} +219.482 q^{51} +308.867 q^{53} +242.165 q^{55} +823.446 q^{57} +622.446 q^{59} -87.3303 q^{61} -289.050 q^{63} +1135.46 q^{65} -608.395 q^{67} +232.593 q^{69} +464.926 q^{71} -255.407 q^{73} -2972.21 q^{75} +77.0000 q^{77} -261.237 q^{79} -138.809 q^{81} -953.986 q^{83} +584.696 q^{85} +363.092 q^{87} -839.910 q^{89} +361.038 q^{91} -692.851 q^{93} +2193.64 q^{95} -349.146 q^{97} -454.221 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{3} - 24 q^{5} - 35 q^{7} + 63 q^{9} - 55 q^{11} - 50 q^{13} + 146 q^{15} + 222 q^{17} - 160 q^{19} + 14 q^{21} - 54 q^{23} + 125 q^{25} - 110 q^{27} + 14 q^{29} + 34 q^{31} + 22 q^{33} + 168 q^{35}+ \cdots - 693 q^{99}+O(q^{100}) $ 5 * q - 2 * q^3 - 24 * q^5 - 35 * q^7 + 63 * q^9 - 55 * q^11 - 50 * q^13 + 146 * q^15 + 222 * q^17 - 160 * q^19 + 14 * q^21 - 54 * q^23 + 125 * q^25 - 110 * q^27 + 14 * q^29 + 34 * q^31 + 22 * q^33 + 168 * q^35 + 1044 * q^37 + 124 * q^39 - 114 * q^41 - 672 * q^43 - 530 * q^45 + 292 * q^47 + 245 * q^49 - 768 * q^51 - 710 * q^53 + 264 * q^55 + 2012 * q^57 - 270 * q^59 + 138 * q^61 - 441 * q^63 - 196 * q^65 - 1942 * q^67 - 1306 * q^69 + 278 * q^71 - 338 * q^73 - 3960 * q^75 + 385 * q^77 - 576 * q^79 - 1439 * q^81 - 1644 * q^83 + 360 * q^85 + 1956 * q^87 - 3656 * q^89 + 350 * q^91 - 1442 * q^93 + 1276 * q^95 + 692 * q^97 - 693 * 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
$3$	−8.26395	−1.59040	−0.795199	−	0.606349i	$-0.792633\pi$
$3$	−8.26395	−1.59040	−0.795199	+	0.606349i	$0.792633\pi$
$4$	0	0
$5$	−22.0150	−1.96908	−0.984541	−	0.175155i	$-0.943957\pi$
$5$	−22.0150	−1.96908	−0.984541	+	0.175155i	$0.943957\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	41.2928	1.52936
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	−51.5769	−1.10037	−0.550186	−	0.835042i	$-0.685443\pi$
$13$	−51.5769	−1.10037	−0.550186	+	0.835042i	$0.685443\pi$
$14$	0	0
$15$	181.931	3.13162
$16$	0	0
$17$	−26.5590	−0.378912	−0.189456	−	0.981889i	$-0.560672\pi$
$17$	−26.5590	−0.378912	−0.189456	+	0.981889i	$0.560672\pi$
$18$	0	0
$19$	−99.6432	−1.20314	−0.601571	−	0.798819i	$-0.705458\pi$
$19$	−99.6432	−1.20314	−0.601571	+	0.798819i	$0.705458\pi$
$20$	0	0
$21$	57.8476	0.601114
$22$	0	0
$23$	−28.1455	−0.255163	−0.127582	−	0.991828i	$-0.540721\pi$
$23$	−28.1455	−0.255163	−0.127582	+	0.991828i	$0.540721\pi$
$24$	0	0
$25$	359.660	2.87728
$26$	0	0
$27$	−118.115	−0.841899
$28$	0	0
$29$	−43.9369	−0.281340	−0.140670	−	0.990057i	$-0.544926\pi$
$29$	−43.9369	−0.281340	−0.140670	+	0.990057i	$0.544926\pi$
$30$	0	0
$31$	83.8402	0.485747	0.242873	−	0.970058i	$-0.421910\pi$
$31$	83.8402	0.485747	0.242873	+	0.970058i	$0.421910\pi$
$32$	0	0
$33$	90.9034	0.479523
$34$	0	0
$35$	154.105	0.744243
$36$	0	0
$37$	306.353	1.36119	0.680596	−	0.732659i	$-0.261721\pi$
$37$	306.353	1.36119	0.680596	+	0.732659i	$0.261721\pi$
$38$	0	0
$39$	426.228	1.75003
$40$	0	0
$41$	200.991	0.765599	0.382800	−	0.923831i	$-0.374960\pi$
$41$	200.991	0.765599	0.382800	+	0.923831i	$0.374960\pi$
$42$	0	0
$43$	13.7546	0.0487805	0.0243903	−	0.999703i	$-0.492236\pi$
$43$	13.7546	0.0487805	0.0243903	+	0.999703i	$0.492236\pi$
$44$	0	0
$45$	−909.062	−3.01144
$46$	0	0
$47$	266.533	0.827189	0.413594	−	0.910461i	$-0.364273\pi$
$47$	266.533	0.827189	0.413594	+	0.910461i	$0.364273\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	219.482	0.602621
$52$	0	0
$53$	308.867	0.800493	0.400247	−	0.916407i	$-0.368924\pi$
$53$	308.867	0.800493	0.400247	+	0.916407i	$0.368924\pi$
$54$	0	0
$55$	242.165	0.593700
$56$	0	0
$57$	823.446	1.91348
$58$	0	0
$59$	622.446	1.37348	0.686742	−	0.726901i	$-0.259040\pi$
$59$	622.446	1.37348	0.686742	+	0.726901i	$0.259040\pi$
$60$	0	0
$61$	−87.3303	−0.183303	−0.0916516	−	0.995791i	$-0.529215\pi$
$61$	−87.3303	−0.183303	−0.0916516	+	0.995791i	$0.529215\pi$
$62$	0	0
$63$	−289.050	−0.578045
$64$	0	0
$65$	1135.46	2.16672
$66$	0	0
$67$	−608.395	−1.10936	−0.554681	−	0.832063i	$-0.687160\pi$
$67$	−608.395	−1.10936	−0.554681	+	0.832063i	$0.687160\pi$
$68$	0	0
$69$	232.593	0.405811
$70$	0	0
$71$	464.926	0.777135	0.388567	−	0.921420i	$-0.372970\pi$
$71$	464.926	0.777135	0.388567	+	0.921420i	$0.372970\pi$
$72$	0	0
$73$	−255.407	−0.409495	−0.204747	−	0.978815i	$-0.565637\pi$
$73$	−255.407	−0.409495	−0.204747	+	0.978815i	$0.565637\pi$
$74$	0	0
$75$	−2972.21	−4.57602
$76$	0	0
$77$	77.0000	0.113961
$78$	0	0
$79$	−261.237	−0.372043	−0.186022	−	0.982546i	$-0.559559\pi$
$79$	−261.237	−0.372043	−0.186022	+	0.982546i	$0.559559\pi$
$80$	0	0
$81$	−138.809	−0.190410
$82$	0	0
$83$	−953.986	−1.26161	−0.630804	−	0.775942i	$-0.717275\pi$
$83$	−953.986	−1.26161	−0.630804	+	0.775942i	$0.717275\pi$
$84$	0	0
$85$	584.696	0.746109
$86$	0	0
$87$	363.092	0.447443
$88$	0	0
$89$	−839.910	−1.00034	−0.500170	−	0.865927i	$-0.666729\pi$
$89$	−839.910	−1.00034	−0.500170	+	0.865927i	$0.666729\pi$
$90$	0	0
$91$	361.038	0.415902
$92$	0	0
$93$	−692.851	−0.772530
$94$	0	0
$95$	2193.64	2.36909
$96$	0	0
$97$	−349.146	−0.365468	−0.182734	−	0.983162i	$-0.558495\pi$
$97$	−349.146	−0.365468	−0.182734	+	0.983162i	$0.558495\pi$
$98$	0	0
$99$	−454.221	−0.461121
$100$	0	0
$101$	1492.44	1.47033	0.735163	−	0.677890i	$-0.237106\pi$
$101$	1492.44	1.47033	0.735163	+	0.677890i	$0.237106\pi$
$102$	0	0
$103$	−558.687	−0.534457	−0.267228	−	0.963633i	$-0.586108\pi$
$103$	−558.687	−0.534457	−0.267228	+	0.963633i	$0.586108\pi$
$104$	0	0
$105$	−1273.52	−1.18364
$106$	0	0
$107$	694.047	0.627066	0.313533	−	0.949577i	$-0.398487\pi$
$107$	694.047	0.627066	0.313533	+	0.949577i	$0.398487\pi$
$108$	0	0
$109$	−341.005	−0.299654	−0.149827	−	0.988712i	$-0.547872\pi$
$109$	−341.005	−0.299654	−0.149827	+	0.988712i	$0.547872\pi$
$110$	0	0
$111$	−2531.68	−2.16484
$112$	0	0
$113$	−990.910	−0.824929	−0.412464	−	0.910974i	$-0.635332\pi$
$113$	−990.910	−0.824929	−0.412464	+	0.910974i	$0.635332\pi$
$114$	0	0
$115$	619.624	0.502437
$116$	0	0
$117$	−2129.75	−1.68287
$118$	0	0
$119$	185.913	0.143215
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	−1660.98	−1.21761
$124$	0	0
$125$	−5166.05	−3.69652
$126$	0	0
$127$	666.090	0.465401	0.232700	−	0.972548i	$-0.425244\pi$
$127$	666.090	0.465401	0.232700	+	0.972548i	$0.425244\pi$
$128$	0	0
$129$	−113.668	−0.0775804
$130$	0	0
$131$	−30.4356	−0.0202990	−0.0101495	−	0.999948i	$-0.503231\pi$
$131$	−30.4356	−0.0202990	−0.0101495	+	0.999948i	$0.503231\pi$
$132$	0	0
$133$	697.502	0.454745
$134$	0	0
$135$	2600.31	1.65777
$136$	0	0
$137$	−2810.25	−1.75252	−0.876262	−	0.481836i	$-0.839970\pi$
$137$	−2810.25	−1.75252	−0.876262	+	0.481836i	$0.839970\pi$
$138$	0	0
$139$	−3110.49	−1.89804	−0.949021	−	0.315212i	$-0.897924\pi$
$139$	−3110.49	−1.89804	−0.949021	+	0.315212i	$0.897924\pi$
$140$	0	0
$141$	−2202.62	−1.31556
$142$	0	0
$143$	567.345	0.331775
$144$	0	0
$145$	967.270	0.553982
$146$	0	0
$147$	−404.933	−0.227200
$148$	0	0
$149$	1916.92	1.05396	0.526979	−	0.849878i	$-0.323325\pi$
$149$	1916.92	1.05396	0.526979	+	0.849878i	$0.323325\pi$
$150$	0	0
$151$	2289.28	1.23377	0.616883	−	0.787055i	$-0.288395\pi$
$151$	2289.28	1.23377	0.616883	+	0.787055i	$0.288395\pi$
$152$	0	0
$153$	−1096.70	−0.579494
$154$	0	0
$155$	−1845.74	−0.956475
$156$	0	0
$157$	280.036	0.142352	0.0711762	−	0.997464i	$-0.477325\pi$
$157$	280.036	0.142352	0.0711762	+	0.997464i	$0.477325\pi$
$158$	0	0
$159$	−2552.46	−1.27310
$160$	0	0
$161$	197.019	0.0964426
$162$	0	0
$163$	866.571	0.416411	0.208206	−	0.978085i	$-0.433238\pi$
$163$	866.571	0.416411	0.208206	+	0.978085i	$0.433238\pi$
$164$	0	0
$165$	−2001.24	−0.944220
$166$	0	0
$167$	1965.18	0.910600	0.455300	−	0.890338i	$-0.349532\pi$
$167$	1965.18	0.910600	0.455300	+	0.890338i	$0.349532\pi$
$168$	0	0
$169$	463.173	0.210820
$170$	0	0
$171$	−4114.55	−1.84004
$172$	0	0
$173$	3956.88	1.73894	0.869469	−	0.493988i	$-0.164461\pi$
$173$	3956.88	1.73894	0.869469	+	0.493988i	$0.164461\pi$
$174$	0	0
$175$	−2517.62	−1.08751
$176$	0	0
$177$	−5143.86	−2.18439
$178$	0	0
$179$	3143.58	1.31264	0.656318	−	0.754484i	$-0.272113\pi$
$179$	3143.58	1.31264	0.656318	+	0.754484i	$0.272113\pi$
$180$	0	0
$181$	683.772	0.280798	0.140399	−	0.990095i	$-0.455162\pi$
$181$	683.772	0.280798	0.140399	+	0.990095i	$0.455162\pi$
$182$	0	0
$183$	721.693	0.291525
$184$	0	0
$185$	−6744.36	−2.68030
$186$	0	0
$187$	292.149	0.114246
$188$	0	0
$189$	826.806	0.318208
$190$	0	0
$191$	−2739.68	−1.03789	−0.518944	−	0.854809i	$-0.673675\pi$
$191$	−2739.68	−1.03789	−0.518944	+	0.854809i	$0.673675\pi$
$192$	0	0
$193$	2651.93	0.989067	0.494534	−	0.869159i	$-0.335339\pi$
$193$	2651.93	0.989067	0.494534	+	0.869159i	$0.335339\pi$
$194$	0	0
$195$	−9383.42	−3.44595
$196$	0	0
$197$	−1879.52	−0.679749	−0.339874	−	0.940471i	$-0.610385\pi$
$197$	−1879.52	−0.679749	−0.339874	+	0.940471i	$0.610385\pi$
$198$	0	0
$199$	3119.39	1.11119	0.555597	−	0.831452i	$-0.312490\pi$
$199$	3119.39	1.11119	0.555597	+	0.831452i	$0.312490\pi$
$200$	0	0
$201$	5027.75	1.76433
$202$	0	0
$203$	307.558	0.106337
$204$	0	0
$205$	−4424.82	−1.50753
$206$	0	0
$207$	−1162.21	−0.390237
$208$	0	0
$209$	1096.08	0.362761
$210$	0	0
$211$	520.718	0.169894	0.0849472	−	0.996385i	$-0.472928\pi$
$211$	520.718	0.169894	0.0849472	+	0.996385i	$0.472928\pi$
$212$	0	0
$213$	−3842.12	−1.23595
$214$	0	0
$215$	−302.808	−0.0960528
$216$	0	0
$217$	−586.881	−0.183595
$218$	0	0
$219$	2110.67	0.651260
$220$	0	0
$221$	1369.83	0.416944
$222$	0	0
$223$	−2101.08	−0.630935	−0.315467	−	0.948936i	$-0.602161\pi$
$223$	−2101.08	−0.630935	−0.315467	+	0.948936i	$0.602161\pi$
$224$	0	0
$225$	14851.4	4.40041
$226$	0	0
$227$	6051.96	1.76953	0.884764	−	0.466040i	$-0.154320\pi$
$227$	6051.96	1.76953	0.884764	+	0.466040i	$0.154320\pi$
$228$	0	0
$229$	−2995.73	−0.864470	−0.432235	−	0.901761i	$-0.642275\pi$
$229$	−2995.73	−0.864470	−0.432235	+	0.901761i	$0.642275\pi$
$230$	0	0
$231$	−636.324	−0.181243
$232$	0	0
$233$	−65.3656	−0.0183787	−0.00918936	−	0.999958i	$-0.502925\pi$
$233$	−65.3656	−0.0183787	−0.00918936	+	0.999958i	$0.502925\pi$
$234$	0	0
$235$	−5867.73	−1.62880
$236$	0	0
$237$	2158.85	0.591697
$238$	0	0
$239$	1102.33	0.298343	0.149171	−	0.988811i	$-0.452339\pi$
$239$	1102.33	0.298343	0.149171	+	0.988811i	$0.452339\pi$
$240$	0	0
$241$	5297.43	1.41592	0.707962	−	0.706250i	$-0.249615\pi$
$241$	5297.43	1.41592	0.707962	+	0.706250i	$0.249615\pi$
$242$	0	0
$243$	4336.22	1.14473
$244$	0	0
$245$	−1078.74	−0.281297
$246$	0	0
$247$	5139.28	1.32391
$248$	0	0
$249$	7883.69	2.00646
$250$	0	0
$251$	−177.964	−0.0447530	−0.0223765	−	0.999750i	$-0.507123\pi$
$251$	−177.964	−0.0447530	−0.0223765	+	0.999750i	$0.507123\pi$
$252$	0	0
$253$	309.601	0.0769346
$254$	0	0
$255$	−4831.90	−1.18661
$256$	0	0
$257$	3496.69	0.848707	0.424354	−	0.905497i	$-0.360501\pi$
$257$	3496.69	0.848707	0.424354	+	0.905497i	$0.360501\pi$
$258$	0	0
$259$	−2144.47	−0.514482
$260$	0	0
$261$	−1814.28	−0.430272
$262$	0	0
$263$	−5747.94	−1.34766	−0.673828	−	0.738889i	$-0.735351\pi$
$263$	−5747.94	−1.34766	−0.673828	+	0.738889i	$0.735351\pi$
$264$	0	0
$265$	−6799.71	−1.57624
$266$	0	0
$267$	6940.97	1.59094
$268$	0	0
$269$	235.217	0.0533140	0.0266570	−	0.999645i	$-0.491514\pi$
$269$	235.217	0.0533140	0.0266570	+	0.999645i	$0.491514\pi$
$270$	0	0
$271$	1179.58	0.264406	0.132203	−	0.991223i	$-0.457795\pi$
$271$	1179.58	0.264406	0.132203	+	0.991223i	$0.457795\pi$
$272$	0	0
$273$	−2983.60	−0.661449
$274$	0	0
$275$	−3956.26	−0.867533
$276$	0	0
$277$	−3638.98	−0.789331	−0.394666	−	0.918825i	$-0.629140\pi$
$277$	−3638.98	−0.789331	−0.394666	+	0.918825i	$0.629140\pi$
$278$	0	0
$279$	3462.00	0.742883
$280$	0	0
$281$	3236.81	0.687160	0.343580	−	0.939123i	$-0.388360\pi$
$281$	3236.81	0.687160	0.343580	+	0.939123i	$0.388360\pi$
$282$	0	0
$283$	−8303.78	−1.74420	−0.872100	−	0.489328i	$-0.837242\pi$
$283$	−8303.78	−1.74420	−0.872100	+	0.489328i	$0.837242\pi$
$284$	0	0
$285$	−18128.2	−3.76779
$286$	0	0
$287$	−1406.94	−0.289369
$288$	0	0
$289$	−4207.62	−0.856426
$290$	0	0
$291$	2885.32	0.581239
$292$	0	0
$293$	−1894.16	−0.377672	−0.188836	−	0.982009i	$-0.560471\pi$
$293$	−1894.16	−0.377672	−0.188836	+	0.982009i	$0.560471\pi$
$294$	0	0
$295$	−13703.2	−2.70450
$296$	0	0
$297$	1299.27	0.253842
$298$	0	0
$299$	1451.66	0.280775
$300$	0	0
$301$	−96.2825	−0.0184373
$302$	0	0
$303$	−12333.4	−2.33840
$304$	0	0
$305$	1922.58	0.360939
$306$	0	0
$307$	−6596.30	−1.22629	−0.613144	−	0.789971i	$-0.710096\pi$
$307$	−6596.30	−1.22629	−0.613144	+	0.789971i	$0.710096\pi$
$308$	0	0
$309$	4616.96	0.849999
$310$	0	0
$311$	5242.26	0.955824	0.477912	−	0.878408i	$-0.341394\pi$
$311$	5242.26	0.955824	0.477912	+	0.878408i	$0.341394\pi$
$312$	0	0
$313$	5338.75	0.964103	0.482051	−	0.876143i	$-0.339892\pi$
$313$	5338.75	0.964103	0.482051	+	0.876143i	$0.339892\pi$
$314$	0	0
$315$	6363.43	1.13822
$316$	0	0
$317$	−5807.21	−1.02891	−0.514456	−	0.857517i	$-0.672006\pi$
$317$	−5807.21	−1.02891	−0.514456	+	0.857517i	$0.672006\pi$
$318$	0	0
$319$	483.306	0.0848273
$320$	0	0
$321$	−5735.56	−0.997283
$322$	0	0
$323$	2646.42	0.455885
$324$	0	0
$325$	−18550.1	−3.16608
$326$	0	0
$327$	2818.04	0.476570
$328$	0	0
$329$	−1865.73	−0.312648
$330$	0	0
$331$	1366.51	0.226919	0.113460	−	0.993543i	$-0.463807\pi$
$331$	1366.51	0.226919	0.113460	+	0.993543i	$0.463807\pi$
$332$	0	0
$333$	12650.2	2.08176
$334$	0	0
$335$	13393.8	2.18443
$336$	0	0
$337$	−3363.75	−0.543724	−0.271862	−	0.962336i	$-0.587639\pi$
$337$	−3363.75	−0.543724	−0.271862	+	0.962336i	$0.587639\pi$
$338$	0	0
$339$	8188.83	1.31196
$340$	0	0
$341$	−922.242	−0.146458
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−5120.54	−0.799075
$346$	0	0
$347$	−2984.97	−0.461791	−0.230896	−	0.972979i	$-0.574166\pi$
$347$	−2984.97	−0.461791	−0.230896	+	0.972979i	$0.574166\pi$
$348$	0	0
$349$	1286.08	0.197255	0.0986276	−	0.995124i	$-0.468555\pi$
$349$	1286.08	0.197255	0.0986276	+	0.995124i	$0.468555\pi$
$350$	0	0
$351$	6092.01	0.926403
$352$	0	0
$353$	8417.60	1.26919	0.634594	−	0.772846i	$-0.281167\pi$
$353$	8417.60	1.26919	0.634594	+	0.772846i	$0.281167\pi$
$354$	0	0
$355$	−10235.3	−1.53024
$356$	0	0
$357$	−1536.38	−0.227769
$358$	0	0
$359$	−7483.47	−1.10017	−0.550087	−	0.835108i	$-0.685405\pi$
$359$	−7483.47	−1.10017	−0.550087	+	0.835108i	$0.685405\pi$
$360$	0	0
$361$	3069.76	0.447553
$362$	0	0
$363$	−999.938	−0.144582
$364$	0	0
$365$	5622.79	0.806329
$366$	0	0
$367$	−8588.73	−1.22160	−0.610801	−	0.791784i	$-0.709153\pi$
$367$	−8588.73	−1.22160	−0.610801	+	0.791784i	$0.709153\pi$
$368$	0	0
$369$	8299.50	1.17088
$370$	0	0
$371$	−2162.07	−0.302558
$372$	0	0
$373$	11833.0	1.64260	0.821298	−	0.570500i	$-0.193251\pi$
$373$	11833.0	1.64260	0.821298	+	0.570500i	$0.193251\pi$
$374$	0	0
$375$	42691.9	5.87894
$376$	0	0
$377$	2266.13	0.309579
$378$	0	0
$379$	−5056.39	−0.685301	−0.342651	−	0.939463i	$-0.611325\pi$
$379$	−5056.39	−0.685301	−0.342651	+	0.939463i	$0.611325\pi$
$380$	0	0
$381$	−5504.53	−0.740172
$382$	0	0
$383$	−6457.09	−0.861467	−0.430733	−	0.902479i	$-0.641745\pi$
$383$	−6457.09	−0.861467	−0.430733	+	0.902479i	$0.641745\pi$
$384$	0	0
$385$	−1695.16	−0.224398
$386$	0	0
$387$	567.968	0.0746032
$388$	0	0
$389$	12444.5	1.62201	0.811004	−	0.585040i	$-0.198921\pi$
$389$	12444.5	1.62201	0.811004	+	0.585040i	$0.198921\pi$
$390$	0	0
$391$	747.518	0.0966844
$392$	0	0
$393$	251.518	0.0322835
$394$	0	0
$395$	5751.12	0.732584
$396$	0	0
$397$	−619.207	−0.0782799	−0.0391400	−	0.999234i	$-0.512462\pi$
$397$	−619.207	−0.0782799	−0.0391400	+	0.999234i	$0.512462\pi$
$398$	0	0
$399$	−5764.12	−0.723226
$400$	0	0
$401$	9731.89	1.21194	0.605969	−	0.795488i	$-0.292785\pi$
$401$	9731.89	1.21194	0.605969	+	0.795488i	$0.292785\pi$
$402$	0	0
$403$	−4324.22	−0.534502
$404$	0	0
$405$	3055.87	0.374932
$406$	0	0
$407$	−3369.88	−0.410415
$408$	0	0
$409$	4621.43	0.558717	0.279358	−	0.960187i	$-0.409878\pi$
$409$	4621.43	0.558717	0.279358	+	0.960187i	$0.409878\pi$
$410$	0	0
$411$	23223.7	2.78721
$412$	0	0
$413$	−4357.12	−0.519128
$414$	0	0
$415$	21002.0	2.48421
$416$	0	0
$417$	25704.9	3.01864
$418$	0	0
$419$	−186.428	−0.0217365	−0.0108682	−	0.999941i	$-0.503460\pi$
$419$	−186.428	−0.0217365	−0.0108682	+	0.999941i	$0.503460\pi$
$420$	0	0
$421$	2670.29	0.309126	0.154563	−	0.987983i	$-0.450603\pi$
$421$	2670.29	0.309126	0.154563	+	0.987983i	$0.450603\pi$
$422$	0	0
$423$	11005.9	1.26507
$424$	0	0
$425$	−9552.22	−1.09024
$426$	0	0
$427$	611.312	0.0692821
$428$	0	0
$429$	−4688.51	−0.527654
$430$	0	0
$431$	12514.9	1.39866	0.699328	−	0.714801i	$-0.253483\pi$
$431$	12514.9	1.39866	0.699328	+	0.714801i	$0.253483\pi$
$432$	0	0
$433$	−16651.2	−1.84805	−0.924025	−	0.382332i	$-0.875121\pi$
$433$	−16651.2	−1.84805	−0.924025	+	0.382332i	$0.875121\pi$
$434$	0	0
$435$	−7993.47	−0.881052
$436$	0	0
$437$	2804.51	0.306998
$438$	0	0
$439$	−6033.38	−0.655940	−0.327970	−	0.944688i	$-0.606364\pi$
$439$	−6033.38	−0.655940	−0.327970	+	0.944688i	$0.606364\pi$
$440$	0	0
$441$	2023.35	0.218481
$442$	0	0
$443$	−6320.03	−0.677819	−0.338910	−	0.940819i	$-0.610058\pi$
$443$	−6320.03	−0.677819	−0.338910	+	0.940819i	$0.610058\pi$
$444$	0	0
$445$	18490.6	1.96975
$446$	0	0
$447$	−15841.3	−1.67621
$448$	0	0
$449$	−17893.6	−1.88074	−0.940368	−	0.340159i	$-0.889519\pi$
$449$	−17893.6	−1.88074	−0.940368	+	0.340159i	$0.889519\pi$
$450$	0	0
$451$	−2210.90	−0.230837
$452$	0	0
$453$	−18918.5	−1.96218
$454$	0	0
$455$	−7948.25	−0.818945
$456$	0	0
$457$	6208.00	0.635444	0.317722	−	0.948184i	$-0.397082\pi$
$457$	6208.00	0.635444	0.317722	+	0.948184i	$0.397082\pi$
$458$	0	0
$459$	3137.02	0.319006
$460$	0	0
$461$	7981.28	0.806346	0.403173	−	0.915124i	$-0.367907\pi$
$461$	7981.28	0.806346	0.403173	+	0.915124i	$0.367907\pi$
$462$	0	0
$463$	7495.19	0.752334	0.376167	−	0.926552i	$-0.377242\pi$
$463$	7495.19	0.752334	0.376167	+	0.926552i	$0.377242\pi$
$464$	0	0
$465$	15253.1	1.52118
$466$	0	0
$467$	−1519.39	−0.150555	−0.0752773	−	0.997163i	$-0.523984\pi$
$467$	−1519.39	−0.150555	−0.0752773	+	0.997163i	$0.523984\pi$
$468$	0	0
$469$	4258.77	0.419300
$470$	0	0
$471$	−2314.20	−0.226397
$472$	0	0
$473$	−151.301	−0.0147079
$474$	0	0
$475$	−35837.7	−3.46178
$476$	0	0
$477$	12754.0	1.22425
$478$	0	0
$479$	16394.2	1.56382	0.781909	−	0.623393i	$-0.214246\pi$
$479$	16394.2	1.56382	0.781909	+	0.623393i	$0.214246\pi$
$480$	0	0
$481$	−15800.7	−1.49782
$482$	0	0
$483$	−1628.15	−0.153382
$484$	0	0
$485$	7686.45	0.719636
$486$	0	0
$487$	−2275.01	−0.211685	−0.105843	−	0.994383i	$-0.533754\pi$
$487$	−2275.01	−0.211685	−0.105843	+	0.994383i	$0.533754\pi$
$488$	0	0
$489$	−7161.29	−0.662259
$490$	0	0
$491$	14629.2	1.34462	0.672308	−	0.740272i	$-0.265303\pi$
$491$	14629.2	1.34462	0.672308	+	0.740272i	$0.265303\pi$
$492$	0	0
$493$	1166.92	0.106603
$494$	0	0
$495$	9999.68	0.907984
$496$	0	0
$497$	−3254.48	−0.293729
$498$	0	0
$499$	−3534.01	−0.317042	−0.158521	−	0.987356i	$-0.550673\pi$
$499$	−3534.01	−0.317042	−0.158521	+	0.987356i	$0.550673\pi$
$500$	0	0
$501$	−16240.1	−1.44822
$502$	0	0
$503$	9233.35	0.818479	0.409239	−	0.912427i	$-0.365794\pi$
$503$	9233.35	0.818479	0.409239	+	0.912427i	$0.365794\pi$
$504$	0	0
$505$	−32856.0	−2.89519
$506$	0	0
$507$	−3827.63	−0.335288
$508$	0	0
$509$	11565.3	1.00712	0.503560	−	0.863960i	$-0.332023\pi$
$509$	11565.3	1.00712	0.503560	+	0.863960i	$0.332023\pi$
$510$	0	0
$511$	1787.85	0.154775
$512$	0	0
$513$	11769.4	1.01292
$514$	0	0
$515$	12299.5	1.05239
$516$	0	0
$517$	−2931.87	−0.249407
$518$	0	0
$519$	−32699.5	−2.76560
$520$	0	0
$521$	−2440.24	−0.205200	−0.102600	−	0.994723i	$-0.532716\pi$
$521$	−2440.24	−0.205200	−0.102600	+	0.994723i	$0.532716\pi$
$522$	0	0
$523$	911.213	0.0761847	0.0380923	−	0.999274i	$-0.487872\pi$
$523$	911.213	0.0761847	0.0380923	+	0.999274i	$0.487872\pi$
$524$	0	0
$525$	20805.5	1.72957
$526$	0	0
$527$	−2226.71	−0.184055
$528$	0	0
$529$	−11374.8	−0.934892
$530$	0	0
$531$	25702.6	2.10056
$532$	0	0
$533$	−10366.5	−0.842445
$534$	0	0
$535$	−15279.4	−1.23474
$536$	0	0
$537$	−25978.3	−2.08761
$538$	0	0
$539$	−539.000	−0.0430730
$540$	0	0
$541$	12277.5	0.975692	0.487846	−	0.872930i	$-0.337783\pi$
$541$	12277.5	0.975692	0.487846	+	0.872930i	$0.337783\pi$
$542$	0	0
$543$	−5650.65	−0.446580
$544$	0	0
$545$	7507.22	0.590044
$546$	0	0
$547$	12539.2	0.980141	0.490071	−	0.871683i	$-0.336971\pi$
$547$	12539.2	0.980141	0.490071	+	0.871683i	$0.336971\pi$
$548$	0	0
$549$	−3606.11	−0.280337
$550$	0	0
$551$	4378.01	0.338493
$552$	0	0
$553$	1828.66	0.140619
$554$	0	0
$555$	55735.0	4.26274
$556$	0	0
$557$	−14212.8	−1.08118	−0.540588	−	0.841287i	$-0.681798\pi$
$557$	−14212.8	−1.08118	−0.540588	+	0.841287i	$0.681798\pi$
$558$	0	0
$559$	−709.421	−0.0536768
$560$	0	0
$561$	−2414.30	−0.181697
$562$	0	0
$563$	−4446.83	−0.332880	−0.166440	−	0.986052i	$-0.553227\pi$
$563$	−4446.83	−0.332880	−0.166440	+	0.986052i	$0.553227\pi$
$564$	0	0
$565$	21814.9	1.62435
$566$	0	0
$567$	971.661	0.0719681
$568$	0	0
$569$	11258.8	0.829511	0.414756	−	0.909933i	$-0.363867\pi$
$569$	11258.8	0.829511	0.414756	+	0.909933i	$0.363867\pi$
$570$	0	0
$571$	16450.6	1.20567	0.602835	−	0.797866i	$-0.294038\pi$
$571$	16450.6	1.20567	0.602835	+	0.797866i	$0.294038\pi$
$572$	0	0
$573$	22640.6	1.65065
$574$	0	0
$575$	−10122.8	−0.734176
$576$	0	0
$577$	10175.6	0.734173	0.367086	−	0.930187i	$-0.380355\pi$
$577$	10175.6	0.734173	0.367086	+	0.930187i	$0.380355\pi$
$578$	0	0
$579$	−21915.4	−1.57301
$580$	0	0
$581$	6677.90	0.476843
$582$	0	0
$583$	−3397.54	−0.241358
$584$	0	0
$585$	46886.5	3.31371
$586$	0	0
$587$	5123.98	0.360289	0.180144	−	0.983640i	$-0.442344\pi$
$587$	5123.98	0.360289	0.180144	+	0.983640i	$0.442344\pi$
$588$	0	0
$589$	−8354.11	−0.584423
$590$	0	0
$591$	15532.3	1.08107
$592$	0	0
$593$	−23816.7	−1.64930	−0.824650	−	0.565643i	$-0.808628\pi$
$593$	−23816.7	−1.64930	−0.824650	+	0.565643i	$0.808628\pi$
$594$	0	0
$595$	−4092.88	−0.282003
$596$	0	0
$597$	−25778.4	−1.76724
$598$	0	0
$599$	−11801.0	−0.804965	−0.402482	−	0.915428i	$-0.631853\pi$
$599$	−11801.0	−0.804965	−0.402482	+	0.915428i	$0.631853\pi$
$600$	0	0
$601$	−10944.5	−0.742820	−0.371410	−	0.928469i	$-0.621125\pi$
$601$	−10944.5	−0.742820	−0.371410	+	0.928469i	$0.621125\pi$
$602$	0	0
$603$	−25122.4	−1.69662
$604$	0	0
$605$	−2663.82	−0.179007
$606$	0	0
$607$	1280.36	0.0856150	0.0428075	−	0.999083i	$-0.486370\pi$
$607$	1280.36	0.0856150	0.0428075	+	0.999083i	$0.486370\pi$
$608$	0	0
$609$	−2541.64	−0.169118
$610$	0	0
$611$	−13747.0	−0.910216
$612$	0	0
$613$	−11029.9	−0.726744	−0.363372	−	0.931644i	$-0.618375\pi$
$613$	−11029.9	−0.726744	−0.363372	+	0.931644i	$0.618375\pi$
$614$	0	0
$615$	36566.5	2.39757
$616$	0	0
$617$	−20861.3	−1.36117	−0.680586	−	0.732668i	$-0.738275\pi$
$617$	−20861.3	−1.36117	−0.680586	+	0.732668i	$0.738275\pi$
$618$	0	0
$619$	16877.9	1.09593	0.547966	−	0.836501i	$-0.315402\pi$
$619$	16877.9	1.09593	0.547966	+	0.836501i	$0.315402\pi$
$620$	0	0
$621$	3324.42	0.214822
$622$	0	0
$623$	5879.37	0.378093
$624$	0	0
$625$	68773.0	4.40147
$626$	0	0
$627$	−9057.91	−0.576935
$628$	0	0
$629$	−8136.43	−0.515772
$630$	0	0
$631$	1332.58	0.0840719	0.0420359	−	0.999116i	$-0.486616\pi$
$631$	1332.58	0.0840719	0.0420359	+	0.999116i	$0.486616\pi$
$632$	0	0
$633$	−4303.19	−0.270200
$634$	0	0
$635$	−14664.0	−0.916412
$636$	0	0
$637$	−2527.27	−0.157196
$638$	0	0
$639$	19198.1	1.18852
$640$	0	0
$641$	20472.2	1.26147	0.630736	−	0.775998i	$-0.282753\pi$
$641$	20472.2	1.26147	0.630736	+	0.775998i	$0.282753\pi$
$642$	0	0
$643$	−27140.3	−1.66455	−0.832276	−	0.554361i	$-0.812963\pi$
$643$	−27140.3	−1.66455	−0.832276	+	0.554361i	$0.812963\pi$
$644$	0	0
$645$	2502.39	0.152762
$646$	0	0
$647$	−13662.1	−0.830159	−0.415079	−	0.909785i	$-0.636246\pi$
$647$	−13662.1	−0.830159	−0.415079	+	0.909785i	$0.636246\pi$
$648$	0	0
$649$	−6846.91	−0.414121
$650$	0	0
$651$	4849.96	0.291989
$652$	0	0
$653$	1607.56	0.0963379	0.0481689	−	0.998839i	$-0.484661\pi$
$653$	1607.56	0.0963379	0.0481689	+	0.998839i	$0.484661\pi$
$654$	0	0
$655$	670.040	0.0399705
$656$	0	0
$657$	−10546.5	−0.626267
$658$	0	0
$659$	−27361.6	−1.61738	−0.808692	−	0.588233i	$-0.799824\pi$
$659$	−27361.6	−1.61738	−0.808692	+	0.588233i	$0.799824\pi$
$660$	0	0
$661$	−5117.29	−0.301119	−0.150559	−	0.988601i	$-0.548107\pi$
$661$	−5117.29	−0.301119	−0.150559	+	0.988601i	$0.548107\pi$
$662$	0	0
$663$	−11320.2	−0.663107
$664$	0	0
$665$	−15355.5	−0.895431
$666$	0	0
$667$	1236.63	0.0717877
$668$	0	0
$669$	17363.2	1.00344
$670$	0	0
$671$	960.633	0.0552680
$672$	0	0
$673$	11605.6	0.664729	0.332365	−	0.943151i	$-0.392154\pi$
$673$	11605.6	0.664729	0.332365	+	0.943151i	$0.392154\pi$
$674$	0	0
$675$	−42481.3	−2.42238
$676$	0	0
$677$	32514.6	1.84584	0.922922	−	0.384986i	$-0.125794\pi$
$677$	32514.6	1.84584	0.922922	+	0.384986i	$0.125794\pi$
$678$	0	0
$679$	2444.02	0.138134
$680$	0	0
$681$	−50013.1	−2.81425
$682$	0	0
$683$	−7201.06	−0.403427	−0.201714	−	0.979445i	$-0.564651\pi$
$683$	−7201.06	−0.403427	−0.201714	+	0.979445i	$0.564651\pi$
$684$	0	0
$685$	61867.6	3.45086
$686$	0	0
$687$	24756.6	1.37485
$688$	0	0
$689$	−15930.4	−0.880841
$690$	0	0
$691$	32357.7	1.78140	0.890698	−	0.454596i	$-0.150216\pi$
$691$	32357.7	1.78140	0.890698	+	0.454596i	$0.150216\pi$
$692$	0	0
$693$	3179.55	0.174287
$694$	0	0
$695$	68477.3	3.73740
$696$	0	0
$697$	−5338.13	−0.290095
$698$	0	0
$699$	540.178	0.0292295
$700$	0	0
$701$	11077.3	0.596838	0.298419	−	0.954435i	$-0.403541\pi$
$701$	11077.3	0.596838	0.298419	+	0.954435i	$0.403541\pi$
$702$	0	0
$703$	−30526.0	−1.63771
$704$	0	0
$705$	48490.6	2.59044
$706$	0	0
$707$	−10447.1	−0.555731
$708$	0	0
$709$	−28594.0	−1.51463	−0.757314	−	0.653051i	$-0.773489\pi$
$709$	−28594.0	−1.51463	−0.757314	+	0.653051i	$0.773489\pi$
$710$	0	0
$711$	−10787.2	−0.568990
$712$	0	0
$713$	−2359.73	−0.123945
$714$	0	0
$715$	−12490.1	−0.653292
$716$	0	0
$717$	−9109.61	−0.474483
$718$	0	0
$719$	−18240.5	−0.946114	−0.473057	−	0.881032i	$-0.656850\pi$
$719$	−18240.5	−0.946114	−0.473057	+	0.881032i	$0.656850\pi$
$720$	0	0
$721$	3910.81	0.202006
$722$	0	0
$723$	−43777.7	−2.25188
$724$	0	0
$725$	−15802.4	−0.809496
$726$	0	0
$727$	9792.53	0.499566	0.249783	−	0.968302i	$-0.419641\pi$
$727$	9792.53	0.499566	0.249783	+	0.968302i	$0.419641\pi$
$728$	0	0
$729$	−32086.4	−1.63016
$730$	0	0
$731$	−365.309	−0.0184835
$732$	0	0
$733$	−22723.8	−1.14505	−0.572525	−	0.819887i	$-0.694036\pi$
$733$	−22723.8	−1.14505	−0.572525	+	0.819887i	$0.694036\pi$
$734$	0	0
$735$	8914.61	0.447375
$736$	0	0
$737$	6692.35	0.334485
$738$	0	0
$739$	24063.6	1.19783	0.598913	−	0.800814i	$-0.295599\pi$
$739$	24063.6	1.19783	0.598913	+	0.800814i	$0.295599\pi$
$740$	0	0
$741$	−42470.8	−2.10554
$742$	0	0
$743$	29211.6	1.44235	0.721177	−	0.692751i	$-0.243601\pi$
$743$	29211.6	1.44235	0.721177	+	0.692751i	$0.243601\pi$
$744$	0	0
$745$	−42200.9	−2.07533
$746$	0	0
$747$	−39392.8	−1.92946
$748$	0	0
$749$	−4858.33	−0.237008
$750$	0	0
$751$	−1880.93	−0.0913931	−0.0456965	−	0.998955i	$-0.514551\pi$
$751$	−1880.93	−0.0913931	−0.0456965	+	0.998955i	$0.514551\pi$
$752$	0	0
$753$	1470.69	0.0711751
$754$	0	0
$755$	−50398.4	−2.42939
$756$	0	0
$757$	−36218.7	−1.73896	−0.869480	−	0.493968i	$-0.835546\pi$
$757$	−36218.7	−1.73896	−0.869480	+	0.493968i	$0.835546\pi$
$758$	0	0
$759$	−2558.53	−0.122357
$760$	0	0
$761$	36966.4	1.76088	0.880441	−	0.474156i	$-0.157247\pi$
$761$	36966.4	1.76088	0.880441	+	0.474156i	$0.157247\pi$
$762$	0	0
$763$	2387.03	0.113259
$764$	0	0
$765$	24143.8	1.14107
$766$	0	0
$767$	−32103.8	−1.51135
$768$	0	0
$769$	−38975.5	−1.82769	−0.913845	−	0.406062i	$-0.866902\pi$
$769$	−38975.5	−1.82769	−0.913845	+	0.406062i	$0.866902\pi$
$770$	0	0
$771$	−28896.5	−1.34978
$772$	0	0
$773$	27341.9	1.27221	0.636105	−	0.771603i	$-0.280545\pi$
$773$	27341.9	1.27221	0.636105	+	0.771603i	$0.280545\pi$
$774$	0	0
$775$	30154.0	1.39763
$776$	0	0
$777$	17721.8	0.818231
$778$	0	0
$779$	−20027.4	−0.921125
$780$	0	0
$781$	−5114.19	−0.234315
$782$	0	0
$783$	5189.61	0.236860
$784$	0	0
$785$	−6165.00	−0.280303
$786$	0	0
$787$	18268.5	0.827446	0.413723	−	0.910403i	$-0.364228\pi$
$787$	18268.5	0.827446	0.413723	+	0.910403i	$0.364228\pi$
$788$	0	0
$789$	47500.7	2.14331
$790$	0	0
$791$	6936.37	0.311794
$792$	0	0
$793$	4504.22	0.201702
$794$	0	0
$795$	56192.4	2.50684
$796$	0	0
$797$	12717.6	0.565219	0.282610	−	0.959235i	$-0.408800\pi$
$797$	12717.6	0.565219	0.282610	+	0.959235i	$0.408800\pi$
$798$	0	0
$799$	−7078.86	−0.313432
$800$	0	0
$801$	−34682.2	−1.52988
$802$	0	0
$803$	2809.48	0.123467
$804$	0	0
$805$	−4337.37	−0.189903
$806$	0	0
$807$	−1943.82	−0.0847904
$808$	0	0
$809$	12502.0	0.543322	0.271661	−	0.962393i	$-0.412427\pi$
$809$	12502.0	0.543322	0.271661	+	0.962393i	$0.412427\pi$
$810$	0	0
$811$	23431.3	1.01453	0.507264	−	0.861791i	$-0.330657\pi$
$811$	23431.3	1.01453	0.507264	+	0.861791i	$0.330657\pi$
$812$	0	0
$813$	−9747.95	−0.420511
$814$	0	0
$815$	−19077.6	−0.819948
$816$	0	0
$817$	−1370.56	−0.0586899
$818$	0	0
$819$	14908.3	0.636065
$820$	0	0
$821$	33116.5	1.40776	0.703881	−	0.710317i	$-0.251449\pi$
$821$	33116.5	1.40776	0.703881	+	0.710317i	$0.251449\pi$
$822$	0	0
$823$	6383.39	0.270366	0.135183	−	0.990821i	$-0.456838\pi$
$823$	6383.39	0.270366	0.135183	+	0.990821i	$0.456838\pi$
$824$	0	0
$825$	32694.4	1.37972
$826$	0	0
$827$	27701.3	1.16477	0.582386	−	0.812912i	$-0.302119\pi$
$827$	27701.3	1.16477	0.582386	+	0.812912i	$0.302119\pi$
$828$	0	0
$829$	−13160.2	−0.551353	−0.275676	−	0.961251i	$-0.588902\pi$
$829$	−13160.2	−0.551353	−0.275676	+	0.961251i	$0.588902\pi$
$830$	0	0
$831$	30072.3	1.25535
$832$	0	0
$833$	−1301.39	−0.0541303
$834$	0	0
$835$	−43263.4	−1.79305
$836$	0	0
$837$	−9902.80	−0.408950
$838$	0	0
$839$	24842.9	1.02226	0.511128	−	0.859505i	$-0.329228\pi$
$839$	24842.9	1.02226	0.511128	+	0.859505i	$0.329228\pi$
$840$	0	0
$841$	−22458.6	−0.920848
$842$	0	0
$843$	−26748.8	−1.09286
$844$	0	0
$845$	−10196.7	−0.415123
$846$	0	0
$847$	−847.000	−0.0343604
$848$	0	0
$849$	68622.0	2.77397
$850$	0	0
$851$	−8622.47	−0.347326
$852$	0	0
$853$	−10131.6	−0.406681	−0.203340	−	0.979108i	$-0.565180\pi$
$853$	−10131.6	−0.406681	−0.203340	+	0.979108i	$0.565180\pi$
$854$	0	0
$855$	90581.8	3.62320
$856$	0	0
$857$	10115.6	0.403199	0.201599	−	0.979468i	$-0.435386\pi$
$857$	10115.6	0.403199	0.201599	+	0.979468i	$0.435386\pi$
$858$	0	0
$859$	27491.4	1.09196	0.545980	−	0.837798i	$-0.316157\pi$
$859$	27491.4	1.09196	0.545980	+	0.837798i	$0.316157\pi$
$860$	0	0
$861$	11626.9	0.460212
$862$	0	0
$863$	117.276	0.00462588	0.00231294	−	0.999997i	$-0.499264\pi$
$863$	117.276	0.00462588	0.00231294	+	0.999997i	$0.499264\pi$
$864$	0	0
$865$	−87110.8	−3.42411
$866$	0	0
$867$	34771.5	1.36206
$868$	0	0
$869$	2873.60	0.112175
$870$	0	0
$871$	31379.1	1.22071
$872$	0	0
$873$	−14417.2	−0.558933
$874$	0	0
$875$	36162.3	1.39715
$876$	0	0
$877$	11597.4	0.446543	0.223271	−	0.974756i	$-0.428326\pi$
$877$	11597.4	0.446543	0.223271	+	0.974756i	$0.428326\pi$
$878$	0	0
$879$	15653.2	0.600648
$880$	0	0
$881$	7524.18	0.287737	0.143868	−	0.989597i	$-0.454046\pi$
$881$	7524.18	0.287737	0.143868	+	0.989597i	$0.454046\pi$
$882$	0	0
$883$	−13467.4	−0.513266	−0.256633	−	0.966509i	$-0.582613\pi$
$883$	−13467.4	−0.513266	−0.256633	+	0.966509i	$0.582613\pi$
$884$	0	0
$885$	113242.	4.30124
$886$	0	0
$887$	12955.7	0.490427	0.245214	−	0.969469i	$-0.421142\pi$
$887$	12955.7	0.490427	0.245214	+	0.969469i	$0.421142\pi$
$888$	0	0
$889$	−4662.63	−0.175905
$890$	0	0
$891$	1526.90	0.0574107
$892$	0	0
$893$	−26558.2	−0.995226
$894$	0	0
$895$	−69205.8	−2.58469
$896$	0	0
$897$	−11996.4	−0.446543
$898$	0	0
$899$	−3683.68	−0.136660
$900$	0	0
$901$	−8203.20	−0.303317
$902$	0	0
$903$	795.673	0.0293226
$904$	0	0
$905$	−15053.2	−0.552913
$906$	0	0
$907$	−47843.5	−1.75151	−0.875753	−	0.482759i	$-0.839635\pi$
$907$	−47843.5	−1.75151	−0.875753	+	0.482759i	$0.839635\pi$
$908$	0	0
$909$	61626.9	2.24866
$910$	0	0
$911$	−16969.8	−0.617162	−0.308581	−	0.951198i	$-0.599854\pi$
$911$	−16969.8	−0.617162	−0.308581	+	0.951198i	$0.599854\pi$
$912$	0	0
$913$	10493.8	0.380389
$914$	0	0
$915$	−15888.1	−0.574036
$916$	0	0
$917$	213.049	0.00767231
$918$	0	0
$919$	−12095.2	−0.434149	−0.217075	−	0.976155i	$-0.569652\pi$
$919$	−12095.2	−0.434149	−0.217075	+	0.976155i	$0.569652\pi$
$920$	0	0
$921$	54511.4	1.95029
$922$	0	0
$923$	−23979.4	−0.855138
$924$	0	0
$925$	110183.	3.91653
$926$	0	0
$927$	−23069.8	−0.817379
$928$	0	0
$929$	−44544.3	−1.57314	−0.786571	−	0.617499i	$-0.788146\pi$
$929$	−44544.3	−1.57314	−0.786571	+	0.617499i	$0.788146\pi$
$930$	0	0
$931$	−4882.52	−0.171878
$932$	0	0
$933$	−43321.8	−1.52014
$934$	0	0
$935$	−6431.66	−0.224960
$936$	0	0
$937$	−49265.8	−1.71766	−0.858828	−	0.512264i	$-0.828807\pi$
$937$	−49265.8	−1.71766	−0.858828	+	0.512264i	$0.828807\pi$
$938$	0	0
$939$	−44119.2	−1.53331
$940$	0	0
$941$	−18403.1	−0.637538	−0.318769	−	0.947832i	$-0.603269\pi$
$941$	−18403.1	−0.637538	−0.318769	+	0.947832i	$0.603269\pi$
$942$	0	0
$943$	−5657.01	−0.195353
$944$	0	0
$945$	−18202.1	−0.626577
$946$	0	0
$947$	−17689.3	−0.606996	−0.303498	−	0.952832i	$-0.598155\pi$
$947$	−17689.3	−0.606996	−0.303498	+	0.952832i	$0.598155\pi$
$948$	0	0
$949$	13173.1	0.450597
$950$	0	0
$951$	47990.5	1.63638
$952$	0	0
$953$	−5298.19	−0.180090	−0.0900448	−	0.995938i	$-0.528701\pi$
$953$	−5298.19	−0.180090	−0.0900448	+	0.995938i	$0.528701\pi$
$954$	0	0
$955$	60314.1	2.04368
$956$	0	0
$957$	−3994.01	−0.134909
$958$	0	0
$959$	19671.7	0.662392
$960$	0	0
$961$	−22761.8	−0.764050
$962$	0	0
$963$	28659.1	0.959011
$964$	0	0
$965$	−58382.2	−1.94755
$966$	0	0
$967$	−33990.6	−1.13037	−0.565184	−	0.824965i	$-0.691195\pi$
$967$	−33990.6	−1.13037	−0.565184	+	0.824965i	$0.691195\pi$
$968$	0	0
$969$	−21869.9	−0.725039
$970$	0	0
$971$	−41991.0	−1.38780	−0.693900	−	0.720071i	$-0.744109\pi$
$971$	−41991.0	−1.38780	−0.693900	+	0.720071i	$0.744109\pi$
$972$	0	0
$973$	21773.4	0.717393
$974$	0	0
$975$	153297.	5.03533
$976$	0	0
$977$	−31233.4	−1.02277	−0.511384	−	0.859352i	$-0.670867\pi$
$977$	−31233.4	−1.02277	−0.511384	+	0.859352i	$0.670867\pi$
$978$	0	0
$979$	9239.00	0.301614
$980$	0	0
$981$	−14081.0	−0.458281
$982$	0	0
$983$	45702.5	1.48289	0.741447	−	0.671012i	$-0.234140\pi$
$983$	45702.5	1.48289	0.741447	+	0.671012i	$0.234140\pi$
$984$	0	0
$985$	41377.7	1.33848
$986$	0	0
$987$	15418.3	0.497235
$988$	0	0
$989$	−387.132	−0.0124470
$990$	0	0
$991$	4310.72	0.138178	0.0690890	−	0.997611i	$-0.477991\pi$
$991$	4310.72	0.138178	0.0690890	+	0.997611i	$0.477991\pi$
$992$	0	0
$993$	−11292.8	−0.360892
$994$	0	0
$995$	−68673.3	−2.18803
$996$	0	0
$997$	−6103.28	−0.193875	−0.0969373	−	0.995290i	$-0.530905\pi$
$997$	−6103.28	−0.193875	−0.0969373	+	0.995290i	$0.530905\pi$
$998$	0	0
$999$	−36184.9	−1.14599

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1232.4.a.y.1.1		5
4.3	odd	2		77.4.a.e.1.4	✓	5
12.11	even	2		693.4.a.o.1.2		5
20.19	odd	2		1925.4.a.r.1.2		5
28.27	even	2		539.4.a.h.1.4		5
44.43	even	2		847.4.a.f.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.a.e.1.4	✓	5	4.3	odd	2
539.4.a.h.1.4		5	28.27	even	2
693.4.a.o.1.2		5	12.11	even	2
847.4.a.f.1.2		5	44.43	even	2
1232.4.a.y.1.1		5	1.1	even	1	trivial
1925.4.a.r.1.2		5	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1232.a (trivial)

\(f(q)\)	\(=\)	\(q-8.26395 q^{3} -22.0150 q^{5} -7.00000 q^{7} +41.2928 q^{9} -11.0000 q^{11} -51.5769 q^{13} +181.931 q^{15} -26.5590 q^{17} -99.6432 q^{19} +57.8476 q^{21} -28.1455 q^{23} +359.660 q^{25} -118.115 q^{27} -43.9369 q^{29} +83.8402 q^{31} +90.9034 q^{33} +154.105 q^{35} +306.353 q^{37} +426.228 q^{39} +200.991 q^{41} +13.7546 q^{43} -909.062 q^{45} +266.533 q^{47} +49.0000 q^{49} +219.482 q^{51} +308.867 q^{53} +242.165 q^{55} +823.446 q^{57} +622.446 q^{59} -87.3303 q^{61} -289.050 q^{63} +1135.46 q^{65} -608.395 q^{67} +232.593 q^{69} +464.926 q^{71} -255.407 q^{73} -2972.21 q^{75} +77.0000 q^{77} -261.237 q^{79} -138.809 q^{81} -953.986 q^{83} +584.696 q^{85} +363.092 q^{87} -839.910 q^{89} +361.038 q^{91} -692.851 q^{93} +2193.64 q^{95} -349.146 q^{97} -454.221 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{3} - 24 q^{5} - 35 q^{7} + 63 q^{9} - 55 q^{11} - 50 q^{13} + 146 q^{15} + 222 q^{17} - 160 q^{19} + 14 q^{21} - 54 q^{23} + 125 q^{25} - 110 q^{27} + 14 q^{29} + 34 q^{31} + 22 q^{33} + 168 q^{35}+ \cdots - 693 q^{99}+O(q^{100}) \) 5 * q - 2 * q^3 - 24 * q^5 - 35 * q^7 + 63 * q^9 - 55 * q^11 - 50 * q^13 + 146 * q^15 + 222 * q^17 - 160 * q^19 + 14 * q^21 - 54 * q^23 + 125 * q^25 - 110 * q^27 + 14 * q^29 + 34 * q^31 + 22 * q^33 + 168 * q^35 + 1044 * q^37 + 124 * q^39 - 114 * q^41 - 672 * q^43 - 530 * q^45 + 292 * q^47 + 245 * q^49 - 768 * q^51 - 710 * q^53 + 264 * q^55 + 2012 * q^57 - 270 * q^59 + 138 * q^61 - 441 * q^63 - 196 * q^65 - 1942 * q^67 - 1306 * q^69 + 278 * q^71 - 338 * q^73 - 3960 * q^75 + 385 * q^77 - 576 * q^79 - 1439 * q^81 - 1644 * q^83 + 360 * q^85 + 1956 * q^87 - 3656 * q^89 + 350 * q^91 - 1442 * q^93 + 1276 * q^95 + 692 * q^97 - 693 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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