LMFDB - Embedded newform 1232.4.a.bc.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:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 144x^{5} + 354x^{4} + 5172x^{3} - 6504x^{2} - 34432x + 18816 $ x^7 - 3x^6 - 144x^5 + 354x^4 + 5172x^3 - 6504x^2 - 34432x + 18816
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 616)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$9.28858$ 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)

$f(q)$	$=$	$q-9.28858 q^{3} -8.74645 q^{5} -7.00000 q^{7} +59.2778 q^{9} -11.0000 q^{11} +55.3616 q^{13} +81.2421 q^{15} +29.3027 q^{17} -47.7550 q^{19} +65.0201 q^{21} +18.1011 q^{23} -48.4996 q^{25} -299.815 q^{27} -173.271 q^{29} +15.4825 q^{31} +102.174 q^{33} +61.2251 q^{35} -420.885 q^{37} -514.230 q^{39} +371.455 q^{41} -326.552 q^{43} -518.470 q^{45} -457.573 q^{47} +49.0000 q^{49} -272.180 q^{51} +720.569 q^{53} +96.2109 q^{55} +443.576 q^{57} -630.053 q^{59} +100.515 q^{61} -414.944 q^{63} -484.217 q^{65} -212.752 q^{67} -168.134 q^{69} -425.508 q^{71} -390.008 q^{73} +450.493 q^{75} +77.0000 q^{77} +503.207 q^{79} +1184.36 q^{81} +778.053 q^{83} -256.294 q^{85} +1609.44 q^{87} -549.010 q^{89} -387.531 q^{91} -143.810 q^{93} +417.687 q^{95} +895.729 q^{97} -652.056 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 3 q^{3} + 11 q^{5} - 49 q^{7} + 108 q^{9} - 77 q^{11} + 26 q^{13} - 67 q^{15} + 44 q^{17} - 34 q^{19} + 21 q^{21} + 29 q^{23} + 182 q^{25} - 99 q^{27} + 94 q^{29} + 173 q^{31} + 33 q^{33} - 77 q^{35}+ \cdots - 1188 q^{99}+O(q^{100}) $ 7 * q - 3 * q^3 + 11 * q^5 - 49 * q^7 + 108 * q^9 - 77 * q^11 + 26 * q^13 - 67 * q^15 + 44 * q^17 - 34 * q^19 + 21 * q^21 + 29 * q^23 + 182 * q^25 - 99 * q^27 + 94 * q^29 + 173 * q^31 + 33 * q^33 - 77 * q^35 + 255 * q^37 + 60 * q^39 + 508 * q^41 - 656 * q^43 + 466 * q^45 + 18 * q^47 + 343 * q^49 - 850 * q^51 + 1806 * q^53 - 121 * q^55 + 1154 * q^57 - 665 * q^59 + 608 * q^61 - 756 * q^63 + 588 * q^65 - 669 * q^67 + 2067 * q^69 - 1169 * q^71 + 380 * q^73 - 1254 * q^75 + 539 * q^77 + 110 * q^79 + 2847 * q^81 + 496 * q^83 + 3446 * q^85 + 582 * q^87 + 1321 * q^89 - 182 * q^91 + 1493 * q^93 + 50 * q^95 + 3927 * q^97 - 1188 * 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
$3$	−9.28858	−1.78759	−0.893794	−	0.448477i	$-0.851967\pi$
$3$	−9.28858	−1.78759	−0.893794	+	0.448477i	$0.851967\pi$
$4$	0	0
$5$	−8.74645	−0.782306	−0.391153	−	0.920326i	$-0.627924\pi$
$5$	−8.74645	−0.782306	−0.391153	+	0.920326i	$0.627924\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	59.2778	2.19547
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	55.3616	1.18112	0.590559	−	0.806994i	$-0.298907\pi$
$13$	55.3616	1.18112	0.590559	+	0.806994i	$0.298907\pi$
$14$	0	0
$15$	81.2421	1.39844
$16$	0	0
$17$	29.3027	0.418055	0.209028	−	0.977910i	$-0.432970\pi$
$17$	29.3027	0.418055	0.209028	+	0.977910i	$0.432970\pi$
$18$	0	0
$19$	−47.7550	−0.576619	−0.288309	−	0.957537i	$-0.593093\pi$
$19$	−47.7550	−0.576619	−0.288309	+	0.957537i	$0.593093\pi$
$20$	0	0
$21$	65.0201	0.675645
$22$	0	0
$23$	18.1011	0.164102	0.0820510	−	0.996628i	$-0.473853\pi$
$23$	18.1011	0.164102	0.0820510	+	0.996628i	$0.473853\pi$
$24$	0	0
$25$	−48.4996	−0.387997
$26$	0	0
$27$	−299.815	−2.13701
$28$	0	0
$29$	−173.271	−1.10950	−0.554751	−	0.832017i	$-0.687186\pi$
$29$	−173.271	−1.10950	−0.554751	+	0.832017i	$0.687186\pi$
$30$	0	0
$31$	15.4825	0.0897011	0.0448506	−	0.998994i	$-0.485719\pi$
$31$	15.4825	0.0897011	0.0448506	+	0.998994i	$0.485719\pi$
$32$	0	0
$33$	102.174	0.538978
$34$	0	0
$35$	61.2251	0.295684
$36$	0	0
$37$	−420.885	−1.87008	−0.935042	−	0.354536i	$-0.884639\pi$
$37$	−420.885	−1.87008	−0.935042	+	0.354536i	$0.884639\pi$
$38$	0	0
$39$	−514.230	−2.11135
$40$	0	0
$41$	371.455	1.41492	0.707458	−	0.706755i	$-0.249842\pi$
$41$	371.455	1.41492	0.707458	+	0.706755i	$0.249842\pi$
$42$	0	0
$43$	−326.552	−1.15811	−0.579055	−	0.815289i	$-0.696578\pi$
$43$	−326.552	−1.15811	−0.579055	+	0.815289i	$0.696578\pi$
$44$	0	0
$45$	−518.470	−1.71753
$46$	0	0
$47$	−457.573	−1.42008	−0.710041	−	0.704160i	$-0.751324\pi$
$47$	−457.573	−1.42008	−0.710041	+	0.704160i	$0.751324\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−272.180	−0.747311
$52$	0	0
$53$	720.569	1.86750	0.933752	−	0.357920i	$-0.116514\pi$
$53$	720.569	1.86750	0.933752	+	0.357920i	$0.116514\pi$
$54$	0	0
$55$	96.2109	0.235874
$56$	0	0
$57$	443.576	1.03076
$58$	0	0
$59$	−630.053	−1.39027	−0.695135	−	0.718879i	$-0.744655\pi$
$59$	−630.053	−1.39027	−0.695135	+	0.718879i	$0.744655\pi$
$60$	0	0
$61$	100.515	0.210978	0.105489	−	0.994420i	$-0.466359\pi$
$61$	100.515	0.210978	0.105489	+	0.994420i	$0.466359\pi$
$62$	0	0
$63$	−414.944	−0.829811
$64$	0	0
$65$	−484.217	−0.923996
$66$	0	0
$67$	−212.752	−0.387938	−0.193969	−	0.981008i	$-0.562136\pi$
$67$	−212.752	−0.387938	−0.193969	+	0.981008i	$0.562136\pi$
$68$	0	0
$69$	−168.134	−0.293347
$70$	0	0
$71$	−425.508	−0.711246	−0.355623	−	0.934629i	$-0.615731\pi$
$71$	−425.508	−0.711246	−0.355623	+	0.934629i	$0.615731\pi$
$72$	0	0
$73$	−390.008	−0.625301	−0.312650	−	0.949868i	$-0.601217\pi$
$73$	−390.008	−0.625301	−0.312650	+	0.949868i	$0.601217\pi$
$74$	0	0
$75$	450.493	0.693579
$76$	0	0
$77$	77.0000	0.113961
$78$	0	0
$79$	503.207	0.716648	0.358324	−	0.933597i	$-0.383348\pi$
$79$	503.207	0.716648	0.358324	+	0.933597i	$0.383348\pi$
$80$	0	0
$81$	1184.36	1.62463
$82$	0	0
$83$	778.053	1.02894	0.514472	−	0.857507i	$-0.327988\pi$
$83$	778.053	1.02894	0.514472	+	0.857507i	$0.327988\pi$
$84$	0	0
$85$	−256.294	−0.327047
$86$	0	0
$87$	1609.44	1.98333
$88$	0	0
$89$	−549.010	−0.653876	−0.326938	−	0.945046i	$-0.606017\pi$
$89$	−549.010	−0.653876	−0.326938	+	0.945046i	$0.606017\pi$
$90$	0	0
$91$	−387.531	−0.446421
$92$	0	0
$93$	−143.810	−0.160349
$94$	0	0
$95$	417.687	0.451092
$96$	0	0
$97$	895.729	0.937603	0.468802	−	0.883303i	$-0.344686\pi$
$97$	895.729	0.937603	0.468802	+	0.883303i	$0.344686\pi$
$98$	0	0
$99$	−652.056	−0.661960
$100$	0	0
$101$	−242.349	−0.238758	−0.119379	−	0.992849i	$-0.538090\pi$
$101$	−242.349	−0.238758	−0.119379	+	0.992849i	$0.538090\pi$
$102$	0	0
$103$	250.151	0.239302	0.119651	−	0.992816i	$-0.461822\pi$
$103$	250.151	0.239302	0.119651	+	0.992816i	$0.461822\pi$
$104$	0	0
$105$	−568.695	−0.528561
$106$	0	0
$107$	−1226.41	−1.10805	−0.554025	−	0.832500i	$-0.686909\pi$
$107$	−1226.41	−1.10805	−0.554025	+	0.832500i	$0.686909\pi$
$108$	0	0
$109$	−1320.88	−1.16071	−0.580357	−	0.814362i	$-0.697087\pi$
$109$	−1320.88	−1.16071	−0.580357	+	0.814362i	$0.697087\pi$
$110$	0	0
$111$	3909.43	3.34294
$112$	0	0
$113$	−2016.73	−1.67892	−0.839461	−	0.543419i	$-0.817129\pi$
$113$	−2016.73	−1.67892	−0.839461	+	0.543419i	$0.817129\pi$
$114$	0	0
$115$	−158.321	−0.128378
$116$	0	0
$117$	3281.71	2.59311
$118$	0	0
$119$	−205.119	−0.158010
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	−3450.29	−2.52929
$124$	0	0
$125$	1517.51	1.08584
$126$	0	0
$127$	1479.84	1.03397	0.516986	−	0.855994i	$-0.327054\pi$
$127$	1479.84	1.03397	0.516986	+	0.855994i	$0.327054\pi$
$128$	0	0
$129$	3033.21	2.07022
$130$	0	0
$131$	345.718	0.230577	0.115288	−	0.993332i	$-0.463221\pi$
$131$	345.718	0.230577	0.115288	+	0.993332i	$0.463221\pi$
$132$	0	0
$133$	334.285	0.217941
$134$	0	0
$135$	2622.32	1.67180
$136$	0	0
$137$	1614.99	1.00714	0.503570	−	0.863954i	$-0.332020\pi$
$137$	1614.99	1.00714	0.503570	+	0.863954i	$0.332020\pi$
$138$	0	0
$139$	2422.25	1.47808	0.739038	−	0.673664i	$-0.235280\pi$
$139$	2422.25	1.47808	0.739038	+	0.673664i	$0.235280\pi$
$140$	0	0
$141$	4250.21	2.53852
$142$	0	0
$143$	−608.977	−0.356120
$144$	0	0
$145$	1515.50	0.867970
$146$	0	0
$147$	−455.141	−0.255370
$148$	0	0
$149$	−2569.68	−1.41286	−0.706430	−	0.707782i	$-0.749696\pi$
$149$	−2569.68	−1.41286	−0.706430	+	0.707782i	$0.749696\pi$
$150$	0	0
$151$	637.585	0.343616	0.171808	−	0.985130i	$-0.445039\pi$
$151$	637.585	0.343616	0.171808	+	0.985130i	$0.445039\pi$
$152$	0	0
$153$	1737.00	0.917829
$154$	0	0
$155$	−135.417	−0.0701737
$156$	0	0
$157$	−1200.72	−0.610368	−0.305184	−	0.952293i	$-0.598718\pi$
$157$	−1200.72	−0.610368	−0.305184	+	0.952293i	$0.598718\pi$
$158$	0	0
$159$	−6693.06	−3.33833
$160$	0	0
$161$	−126.708	−0.0620248
$162$	0	0
$163$	3746.60	1.80035	0.900173	−	0.435533i	$-0.143440\pi$
$163$	3746.60	1.80035	0.900173	+	0.435533i	$0.143440\pi$
$164$	0	0
$165$	−893.663	−0.421646
$166$	0	0
$167$	3401.91	1.57633	0.788166	−	0.615463i	$-0.211031\pi$
$167$	3401.91	1.57633	0.788166	+	0.615463i	$0.211031\pi$
$168$	0	0
$169$	867.902	0.395040
$170$	0	0
$171$	−2830.81	−1.26595
$172$	0	0
$173$	−764.688	−0.336059	−0.168029	−	0.985782i	$-0.553740\pi$
$173$	−764.688	−0.336059	−0.168029	+	0.985782i	$0.553740\pi$
$174$	0	0
$175$	339.497	0.146649
$176$	0	0
$177$	5852.30	2.48523
$178$	0	0
$179$	−1527.53	−0.637837	−0.318918	−	0.947782i	$-0.603320\pi$
$179$	−1527.53	−0.637837	−0.318918	+	0.947782i	$0.603320\pi$
$180$	0	0
$181$	144.636	0.0593963	0.0296982	−	0.999559i	$-0.490545\pi$
$181$	144.636	0.0593963	0.0296982	+	0.999559i	$0.490545\pi$
$182$	0	0
$183$	−933.643	−0.377141
$184$	0	0
$185$	3681.25	1.46298
$186$	0	0
$187$	−322.329	−0.126048
$188$	0	0
$189$	2098.70	0.807716
$190$	0	0
$191$	−2719.77	−1.03034	−0.515172	−	0.857087i	$-0.672272\pi$
$191$	−2719.77	−1.03034	−0.515172	+	0.857087i	$0.672272\pi$
$192$	0	0
$193$	364.326	0.135880	0.0679398	−	0.997689i	$-0.478357\pi$
$193$	364.326	0.135880	0.0679398	+	0.997689i	$0.478357\pi$
$194$	0	0
$195$	4497.69	1.65172
$196$	0	0
$197$	−2416.38	−0.873907	−0.436953	−	0.899484i	$-0.643943\pi$
$197$	−2416.38	−0.873907	−0.436953	+	0.899484i	$0.643943\pi$
$198$	0	0
$199$	245.275	0.0873721	0.0436861	−	0.999045i	$-0.486090\pi$
$199$	245.275	0.0873721	0.0436861	+	0.999045i	$0.486090\pi$
$200$	0	0
$201$	1976.17	0.693473
$202$	0	0
$203$	1212.89	0.419352
$204$	0	0
$205$	−3248.91	−1.10690
$206$	0	0
$207$	1073.00	0.360282
$208$	0	0
$209$	525.305	0.173857
$210$	0	0
$211$	48.2331	0.0157370	0.00786850	−	0.999969i	$-0.497495\pi$
$211$	48.2331	0.0157370	0.00786850	+	0.999969i	$0.497495\pi$
$212$	0	0
$213$	3952.37	1.27142
$214$	0	0
$215$	2856.17	0.905997
$216$	0	0
$217$	−108.377	−0.0339038
$218$	0	0
$219$	3622.62	1.11778
$220$	0	0
$221$	1622.24	0.493772
$222$	0	0
$223$	3448.23	1.03547	0.517737	−	0.855540i	$-0.326775\pi$
$223$	3448.23	1.03547	0.517737	+	0.855540i	$0.326775\pi$
$224$	0	0
$225$	−2874.95	−0.851837
$226$	0	0
$227$	−3385.07	−0.989756	−0.494878	−	0.868962i	$-0.664787\pi$
$227$	−3385.07	−0.989756	−0.494878	+	0.868962i	$0.664787\pi$
$228$	0	0
$229$	−3732.54	−1.07709	−0.538543	−	0.842598i	$-0.681025\pi$
$229$	−3732.54	−1.07709	−0.538543	+	0.842598i	$0.681025\pi$
$230$	0	0
$231$	−715.221	−0.203715
$232$	0	0
$233$	−2063.10	−0.580077	−0.290038	−	0.957015i	$-0.593668\pi$
$233$	−2063.10	−0.580077	−0.290038	+	0.957015i	$0.593668\pi$
$234$	0	0
$235$	4002.14	1.11094
$236$	0	0
$237$	−4674.08	−1.28107
$238$	0	0
$239$	5326.36	1.44156	0.720782	−	0.693162i	$-0.243783\pi$
$239$	5326.36	1.44156	0.720782	+	0.693162i	$0.243783\pi$
$240$	0	0
$241$	7042.61	1.88238	0.941192	−	0.337873i	$-0.109708\pi$
$241$	7042.61	1.88238	0.941192	+	0.337873i	$0.109708\pi$
$242$	0	0
$243$	−2905.98	−0.767156
$244$	0	0
$245$	−428.576	−0.111758
$246$	0	0
$247$	−2643.79	−0.681055
$248$	0	0
$249$	−7227.01	−1.83933
$250$	0	0
$251$	2446.72	0.615280	0.307640	−	0.951503i	$-0.400461\pi$
$251$	2446.72	0.615280	0.307640	+	0.951503i	$0.400461\pi$
$252$	0	0
$253$	−199.113	−0.0494786
$254$	0	0
$255$	2380.61	0.584626
$256$	0	0
$257$	6353.12	1.54201	0.771006	−	0.636828i	$-0.219754\pi$
$257$	6353.12	1.54201	0.771006	+	0.636828i	$0.219754\pi$
$258$	0	0
$259$	2946.20	0.706826
$260$	0	0
$261$	−10271.1	−2.43588
$262$	0	0
$263$	4480.12	1.05040	0.525202	−	0.850978i	$-0.323990\pi$
$263$	4480.12	1.05040	0.525202	+	0.850978i	$0.323990\pi$
$264$	0	0
$265$	−6302.42	−1.46096
$266$	0	0
$267$	5099.53	1.16886
$268$	0	0
$269$	2147.58	0.486767	0.243384	−	0.969930i	$-0.421743\pi$
$269$	2147.58	0.486767	0.243384	+	0.969930i	$0.421743\pi$
$270$	0	0
$271$	−4857.16	−1.08875	−0.544375	−	0.838842i	$-0.683233\pi$
$271$	−4857.16	−1.08875	−0.544375	+	0.838842i	$0.683233\pi$
$272$	0	0
$273$	3599.61	0.798016
$274$	0	0
$275$	533.496	0.116985
$276$	0	0
$277$	6166.94	1.33767	0.668837	−	0.743409i	$-0.266792\pi$
$277$	6166.94	1.33767	0.668837	+	0.743409i	$0.266792\pi$
$278$	0	0
$279$	917.767	0.196936
$280$	0	0
$281$	−4176.04	−0.886555	−0.443277	−	0.896385i	$-0.646184\pi$
$281$	−4176.04	−0.886555	−0.443277	+	0.896385i	$0.646184\pi$
$282$	0	0
$283$	3721.52	0.781701	0.390850	−	0.920454i	$-0.372181\pi$
$283$	3721.52	0.781701	0.390850	+	0.920454i	$0.372181\pi$
$284$	0	0
$285$	−3879.72	−0.806368
$286$	0	0
$287$	−2600.19	−0.534788
$288$	0	0
$289$	−4054.35	−0.825230
$290$	0	0
$291$	−8320.05	−1.67605
$292$	0	0
$293$	−1322.03	−0.263597	−0.131798	−	0.991277i	$-0.542075\pi$
$293$	−1322.03	−0.263597	−0.131798	+	0.991277i	$0.542075\pi$
$294$	0	0
$295$	5510.73	1.08762
$296$	0	0
$297$	3297.96	0.644334
$298$	0	0
$299$	1002.11	0.193824
$300$	0	0
$301$	2285.86	0.437724
$302$	0	0
$303$	2251.07	0.426801
$304$	0	0
$305$	−879.150	−0.165049
$306$	0	0
$307$	2324.29	0.432098	0.216049	−	0.976383i	$-0.430683\pi$
$307$	2324.29	0.432098	0.216049	+	0.976383i	$0.430683\pi$
$308$	0	0
$309$	−2323.54	−0.427773
$310$	0	0
$311$	2485.31	0.453148	0.226574	−	0.973994i	$-0.427247\pi$
$311$	2485.31	0.453148	0.226574	+	0.973994i	$0.427247\pi$
$312$	0	0
$313$	526.047	0.0949966	0.0474983	−	0.998871i	$-0.484875\pi$
$313$	526.047	0.0949966	0.0474983	+	0.998871i	$0.484875\pi$
$314$	0	0
$315$	3629.29	0.649166
$316$	0	0
$317$	−3654.74	−0.647542	−0.323771	−	0.946135i	$-0.604951\pi$
$317$	−3654.74	−0.647542	−0.323771	+	0.946135i	$0.604951\pi$
$318$	0	0
$319$	1905.98	0.334527
$320$	0	0
$321$	11391.6	1.98074
$322$	0	0
$323$	−1399.35	−0.241058
$324$	0	0
$325$	−2685.01	−0.458270
$326$	0	0
$327$	12269.1	2.07488
$328$	0	0
$329$	3203.01	0.536741
$330$	0	0
$331$	2961.53	0.491783	0.245892	−	0.969297i	$-0.420919\pi$
$331$	2961.53	0.491783	0.245892	+	0.969297i	$0.420919\pi$
$332$	0	0
$333$	−24949.2	−4.10572
$334$	0	0
$335$	1860.83	0.303486
$336$	0	0
$337$	−901.796	−0.145768	−0.0728842	−	0.997340i	$-0.523220\pi$
$337$	−901.796	−0.145768	−0.0728842	+	0.997340i	$0.523220\pi$
$338$	0	0
$339$	18732.6	3.00122
$340$	0	0
$341$	−170.307	−0.0270459
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	1470.57	0.229487
$346$	0	0
$347$	−6729.48	−1.04109	−0.520544	−	0.853835i	$-0.674271\pi$
$347$	−6729.48	−1.04109	−0.520544	+	0.853835i	$0.674271\pi$
$348$	0	0
$349$	−3267.52	−0.501165	−0.250582	−	0.968095i	$-0.580622\pi$
$349$	−3267.52	−0.501165	−0.250582	+	0.968095i	$0.580622\pi$
$350$	0	0
$351$	−16598.2	−2.52407
$352$	0	0
$353$	−1267.11	−0.191052	−0.0955259	−	0.995427i	$-0.530453\pi$
$353$	−1267.11	−0.191052	−0.0955259	+	0.995427i	$0.530453\pi$
$354$	0	0
$355$	3721.68	0.556412
$356$	0	0
$357$	1905.26	0.282457
$358$	0	0
$359$	2786.30	0.409625	0.204812	−	0.978801i	$-0.434342\pi$
$359$	2786.30	0.409625	0.204812	+	0.978801i	$0.434342\pi$
$360$	0	0
$361$	−4578.46	−0.667511
$362$	0	0
$363$	−1123.92	−0.162508
$364$	0	0
$365$	3411.18	0.489177
$366$	0	0
$367$	2910.98	0.414038	0.207019	−	0.978337i	$-0.433624\pi$
$367$	2910.98	0.414038	0.207019	+	0.978337i	$0.433624\pi$
$368$	0	0
$369$	22019.0	3.10641
$370$	0	0
$371$	−5043.98	−0.705850
$372$	0	0
$373$	10693.8	1.48446	0.742229	−	0.670146i	$-0.233769\pi$
$373$	10693.8	1.48446	0.742229	+	0.670146i	$0.233769\pi$
$374$	0	0
$375$	−14095.5	−1.94103
$376$	0	0
$377$	−9592.53	−1.31045
$378$	0	0
$379$	−8880.72	−1.20362	−0.601810	−	0.798639i	$-0.705553\pi$
$379$	−8880.72	−1.20362	−0.601810	+	0.798639i	$0.705553\pi$
$380$	0	0
$381$	−13745.6	−1.84832
$382$	0	0
$383$	−10212.5	−1.36249	−0.681247	−	0.732054i	$-0.738562\pi$
$383$	−10212.5	−1.36249	−0.681247	+	0.732054i	$0.738562\pi$
$384$	0	0
$385$	−673.477	−0.0891521
$386$	0	0
$387$	−19357.3	−2.54260
$388$	0	0
$389$	−2659.78	−0.346675	−0.173337	−	0.984863i	$-0.555455\pi$
$389$	−2659.78	−0.346675	−0.173337	+	0.984863i	$0.555455\pi$
$390$	0	0
$391$	530.411	0.0686037
$392$	0	0
$393$	−3211.23	−0.412176
$394$	0	0
$395$	−4401.27	−0.560638
$396$	0	0
$397$	−12008.1	−1.51806	−0.759031	−	0.651055i	$-0.774327\pi$
$397$	−12008.1	−1.51806	−0.759031	+	0.651055i	$0.774327\pi$
$398$	0	0
$399$	−3105.04	−0.389589
$400$	0	0
$401$	−2141.85	−0.266731	−0.133365	−	0.991067i	$-0.542578\pi$
$401$	−2141.85	−0.266731	−0.133365	+	0.991067i	$0.542578\pi$
$402$	0	0
$403$	857.134	0.105948
$404$	0	0
$405$	−10358.9	−1.27096
$406$	0	0
$407$	4629.74	0.563852
$408$	0	0
$409$	−5357.72	−0.647732	−0.323866	−	0.946103i	$-0.604983\pi$
$409$	−5357.72	−0.647732	−0.323866	+	0.946103i	$0.604983\pi$
$410$	0	0
$411$	−15001.0	−1.80035
$412$	0	0
$413$	4410.37	0.525473
$414$	0	0
$415$	−6805.20	−0.804950
$416$	0	0
$417$	−22499.3	−2.64219
$418$	0	0
$419$	−7919.14	−0.923330	−0.461665	−	0.887054i	$-0.652748\pi$
$419$	−7919.14	−0.923330	−0.461665	+	0.887054i	$0.652748\pi$
$420$	0	0
$421$	15475.2	1.79148	0.895742	−	0.444574i	$-0.146645\pi$
$421$	15475.2	1.79148	0.895742	+	0.444574i	$0.146645\pi$
$422$	0	0
$423$	−27123.9	−3.11775
$424$	0	0
$425$	−1421.17	−0.162204
$426$	0	0
$427$	−703.606	−0.0797421
$428$	0	0
$429$	5656.53	0.636597
$430$	0	0
$431$	−11309.0	−1.26389	−0.631943	−	0.775015i	$-0.717742\pi$
$431$	−11309.0	−1.26389	−0.631943	+	0.775015i	$0.717742\pi$
$432$	0	0
$433$	12362.7	1.37209	0.686044	−	0.727560i	$-0.259346\pi$
$433$	12362.7	1.37209	0.686044	+	0.727560i	$0.259346\pi$
$434$	0	0
$435$	−14076.9	−1.55157
$436$	0	0
$437$	−864.420	−0.0946243
$438$	0	0
$439$	16462.3	1.78975	0.894876	−	0.446315i	$-0.147264\pi$
$439$	16462.3	1.78975	0.894876	+	0.446315i	$0.147264\pi$
$440$	0	0
$441$	2904.61	0.313639
$442$	0	0
$443$	−1638.96	−0.175777	−0.0878887	−	0.996130i	$-0.528012\pi$
$443$	−1638.96	−0.175777	−0.0878887	+	0.996130i	$0.528012\pi$
$444$	0	0
$445$	4801.89	0.511532
$446$	0	0
$447$	23868.7	2.52561
$448$	0	0
$449$	16895.8	1.77586	0.887932	−	0.459975i	$-0.152142\pi$
$449$	16895.8	1.77586	0.887932	+	0.459975i	$0.152142\pi$
$450$	0	0
$451$	−4086.01	−0.426613
$452$	0	0
$453$	−5922.27	−0.614244
$454$	0	0
$455$	3389.52	0.349238
$456$	0	0
$457$	13309.4	1.36233	0.681166	−	0.732129i	$-0.261473\pi$
$457$	13309.4	1.36233	0.681166	+	0.732129i	$0.261473\pi$
$458$	0	0
$459$	−8785.37	−0.893390
$460$	0	0
$461$	7377.21	0.745317	0.372658	−	0.927969i	$-0.378446\pi$
$461$	7377.21	0.745317	0.372658	+	0.927969i	$0.378446\pi$
$462$	0	0
$463$	−4085.80	−0.410115	−0.205058	−	0.978750i	$-0.565738\pi$
$463$	−4085.80	−0.410115	−0.205058	+	0.978750i	$0.565738\pi$
$464$	0	0
$465$	1257.83	0.125442
$466$	0	0
$467$	−8493.21	−0.841583	−0.420791	−	0.907157i	$-0.638248\pi$
$467$	−8493.21	−0.841583	−0.420791	+	0.907157i	$0.638248\pi$
$468$	0	0
$469$	1489.27	0.146627
$470$	0	0
$471$	11153.0	1.09109
$472$	0	0
$473$	3592.07	0.349183
$474$	0	0
$475$	2316.10	0.223726
$476$	0	0
$477$	42713.7	4.10006
$478$	0	0
$479$	9692.50	0.924554	0.462277	−	0.886735i	$-0.347032\pi$
$479$	9692.50	0.924554	0.462277	+	0.886735i	$0.347032\pi$
$480$	0	0
$481$	−23300.9	−2.20879
$482$	0	0
$483$	1176.94	0.110875
$484$	0	0
$485$	−7834.45	−0.733493
$486$	0	0
$487$	−12601.5	−1.17255	−0.586273	−	0.810114i	$-0.699405\pi$
$487$	−12601.5	−1.17255	−0.586273	+	0.810114i	$0.699405\pi$
$488$	0	0
$489$	−34800.6	−3.21828
$490$	0	0
$491$	14766.2	1.35720	0.678602	−	0.734506i	$-0.262586\pi$
$491$	14766.2	1.35720	0.678602	+	0.734506i	$0.262586\pi$
$492$	0	0
$493$	−5077.29	−0.463833
$494$	0	0
$495$	5703.17	0.517856
$496$	0	0
$497$	2978.56	0.268826
$498$	0	0
$499$	5673.56	0.508985	0.254492	−	0.967075i	$-0.418092\pi$
$499$	5673.56	0.508985	0.254492	+	0.967075i	$0.418092\pi$
$500$	0	0
$501$	−31598.9	−2.81783
$502$	0	0
$503$	16615.6	1.47287	0.736437	−	0.676507i	$-0.236507\pi$
$503$	16615.6	1.47287	0.736437	+	0.676507i	$0.236507\pi$
$504$	0	0
$505$	2119.69	0.186782
$506$	0	0
$507$	−8061.58	−0.706168
$508$	0	0
$509$	12755.6	1.11077	0.555384	−	0.831594i	$-0.312571\pi$
$509$	12755.6	1.11077	0.555384	+	0.831594i	$0.312571\pi$
$510$	0	0
$511$	2730.05	0.236341
$512$	0	0
$513$	14317.7	1.23224
$514$	0	0
$515$	−2187.93	−0.187207
$516$	0	0
$517$	5033.30	0.428171
$518$	0	0
$519$	7102.87	0.600735
$520$	0	0
$521$	−23207.6	−1.95152	−0.975761	−	0.218841i	$-0.929773\pi$
$521$	−23207.6	−1.95152	−0.975761	+	0.218841i	$0.929773\pi$
$522$	0	0
$523$	14250.1	1.19142	0.595709	−	0.803200i	$-0.296871\pi$
$523$	14250.1	1.19142	0.595709	+	0.803200i	$0.296871\pi$
$524$	0	0
$525$	−3153.45	−0.262148
$526$	0	0
$527$	453.678	0.0375000
$528$	0	0
$529$	−11839.3	−0.973071
$530$	0	0
$531$	−37348.2	−3.05230
$532$	0	0
$533$	20564.3	1.67118
$534$	0	0
$535$	10726.7	0.866835
$536$	0	0
$537$	14188.6	1.14019
$538$	0	0
$539$	−539.000	−0.0430730
$540$	0	0
$541$	−4669.48	−0.371084	−0.185542	−	0.982636i	$-0.559404\pi$
$541$	−4669.48	−0.371084	−0.185542	+	0.982636i	$0.559404\pi$
$542$	0	0
$543$	−1343.47	−0.106176
$544$	0	0
$545$	11553.1	0.908034
$546$	0	0
$547$	6726.83	0.525811	0.262905	−	0.964822i	$-0.415319\pi$
$547$	6726.83	0.525811	0.262905	+	0.964822i	$0.415319\pi$
$548$	0	0
$549$	5958.31	0.463196
$550$	0	0
$551$	8274.54	0.639759
$552$	0	0
$553$	−3522.45	−0.270867
$554$	0	0
$555$	−34193.6	−2.61520
$556$	0	0
$557$	3162.64	0.240584	0.120292	−	0.992739i	$-0.461617\pi$
$557$	3162.64	0.240584	0.120292	+	0.992739i	$0.461617\pi$
$558$	0	0
$559$	−18078.4	−1.36786
$560$	0	0
$561$	2993.98	0.225323
$562$	0	0
$563$	2871.42	0.214948	0.107474	−	0.994208i	$-0.465724\pi$
$563$	2871.42	0.214948	0.107474	+	0.994208i	$0.465724\pi$
$564$	0	0
$565$	17639.3	1.31343
$566$	0	0
$567$	−8290.49	−0.614053
$568$	0	0
$569$	20069.0	1.47862	0.739310	−	0.673365i	$-0.235152\pi$
$569$	20069.0	1.47862	0.739310	+	0.673365i	$0.235152\pi$
$570$	0	0
$571$	4823.62	0.353524	0.176762	−	0.984254i	$-0.443438\pi$
$571$	4823.62	0.353524	0.176762	+	0.984254i	$0.443438\pi$
$572$	0	0
$573$	25262.8	1.84183
$574$	0	0
$575$	−877.898	−0.0636711
$576$	0	0
$577$	8263.59	0.596218	0.298109	−	0.954532i	$-0.403644\pi$
$577$	8263.59	0.596218	0.298109	+	0.954532i	$0.403644\pi$
$578$	0	0
$579$	−3384.07	−0.242897
$580$	0	0
$581$	−5446.37	−0.388904
$582$	0	0
$583$	−7926.26	−0.563074
$584$	0	0
$585$	−28703.3	−2.02861
$586$	0	0
$587$	10010.8	0.703904	0.351952	−	0.936018i	$-0.385518\pi$
$587$	10010.8	0.703904	0.351952	+	0.936018i	$0.385518\pi$
$588$	0	0
$589$	−739.366	−0.0517233
$590$	0	0
$591$	22444.7	1.56219
$592$	0	0
$593$	5528.35	0.382837	0.191418	−	0.981509i	$-0.438691\pi$
$593$	5528.35	0.382837	0.191418	+	0.981509i	$0.438691\pi$
$594$	0	0
$595$	1794.06	0.123612
$596$	0	0
$597$	−2278.25	−0.156185
$598$	0	0
$599$	14773.4	1.00772	0.503862	−	0.863784i	$-0.331912\pi$
$599$	14773.4	1.00772	0.503862	+	0.863784i	$0.331912\pi$
$600$	0	0
$601$	5027.65	0.341235	0.170617	−	0.985337i	$-0.445424\pi$
$601$	5027.65	0.341235	0.170617	+	0.985337i	$0.445424\pi$
$602$	0	0
$603$	−12611.5	−0.851707
$604$	0	0
$605$	−1058.32	−0.0711187
$606$	0	0
$607$	3478.73	0.232615	0.116307	−	0.993213i	$-0.462894\pi$
$607$	3478.73	0.232615	0.116307	+	0.993213i	$0.462894\pi$
$608$	0	0
$609$	−11266.1	−0.749629
$610$	0	0
$611$	−25332.0	−1.67729
$612$	0	0
$613$	29008.0	1.91129	0.955646	−	0.294517i	$-0.0951589\pi$
$613$	29008.0	1.91129	0.955646	+	0.294517i	$0.0951589\pi$
$614$	0	0
$615$	30177.8	1.97868
$616$	0	0
$617$	−9703.04	−0.633111	−0.316556	−	0.948574i	$-0.602526\pi$
$617$	−9703.04	−0.633111	−0.316556	+	0.948574i	$0.602526\pi$
$618$	0	0
$619$	12393.0	0.804714	0.402357	−	0.915483i	$-0.368191\pi$
$619$	12393.0	0.804714	0.402357	+	0.915483i	$0.368191\pi$
$620$	0	0
$621$	−5426.99	−0.350689
$622$	0	0
$623$	3843.07	0.247142
$624$	0	0
$625$	−7210.33	−0.461461
$626$	0	0
$627$	−4879.34	−0.310785
$628$	0	0
$629$	−12333.1	−0.781799
$630$	0	0
$631$	2520.06	0.158989	0.0794944	−	0.996835i	$-0.474669\pi$
$631$	2520.06	0.158989	0.0794944	+	0.996835i	$0.474669\pi$
$632$	0	0
$633$	−448.017	−0.0281313
$634$	0	0
$635$	−12943.3	−0.808883
$636$	0	0
$637$	2712.72	0.168731
$638$	0	0
$639$	−25223.2	−1.56152
$640$	0	0
$641$	−23140.4	−1.42588	−0.712940	−	0.701225i	$-0.752637\pi$
$641$	−23140.4	−1.42588	−0.712940	+	0.701225i	$0.752637\pi$
$642$	0	0
$643$	439.589	0.0269606	0.0134803	−	0.999909i	$-0.495709\pi$
$643$	439.589	0.0269606	0.0134803	+	0.999909i	$0.495709\pi$
$644$	0	0
$645$	−26529.8	−1.61955
$646$	0	0
$647$	−10005.8	−0.607992	−0.303996	−	0.952673i	$-0.598321\pi$
$647$	−10005.8	−0.607992	−0.303996	+	0.952673i	$0.598321\pi$
$648$	0	0
$649$	6930.59	0.419182
$650$	0	0
$651$	1006.67	0.0606061
$652$	0	0
$653$	17176.4	1.02935	0.514673	−	0.857386i	$-0.327913\pi$
$653$	17176.4	1.02935	0.514673	+	0.857386i	$0.327913\pi$
$654$	0	0
$655$	−3023.81	−0.180382
$656$	0	0
$657$	−23118.8	−1.37283
$658$	0	0
$659$	23395.7	1.38295	0.691476	−	0.722400i	$-0.256961\pi$
$659$	23395.7	1.38295	0.691476	+	0.722400i	$0.256961\pi$
$660$	0	0
$661$	1884.94	0.110916	0.0554582	−	0.998461i	$-0.482338\pi$
$661$	1884.94	0.110916	0.0554582	+	0.998461i	$0.482338\pi$
$662$	0	0
$663$	−15068.3	−0.882662
$664$	0	0
$665$	−2923.81	−0.170497
$666$	0	0
$667$	−3136.39	−0.182071
$668$	0	0
$669$	−32029.2	−1.85100
$670$	0	0
$671$	−1105.67	−0.0636122
$672$	0	0
$673$	−27956.4	−1.60125	−0.800624	−	0.599168i	$-0.795498\pi$
$673$	−27956.4	−1.60125	−0.800624	+	0.599168i	$0.795498\pi$
$674$	0	0
$675$	14540.9	0.829155
$676$	0	0
$677$	−8612.92	−0.488953	−0.244477	−	0.969655i	$-0.578616\pi$
$677$	−8612.92	−0.488953	−0.244477	+	0.969655i	$0.578616\pi$
$678$	0	0
$679$	−6270.10	−0.354381
$680$	0	0
$681$	31442.5	1.76928
$682$	0	0
$683$	−30753.7	−1.72292	−0.861462	−	0.507821i	$-0.830451\pi$
$683$	−30753.7	−1.72292	−0.861462	+	0.507821i	$0.830451\pi$
$684$	0	0
$685$	−14125.5	−0.787892
$686$	0	0
$687$	34670.0	1.92539
$688$	0	0
$689$	39891.8	2.20574
$690$	0	0
$691$	−16576.8	−0.912608	−0.456304	−	0.889824i	$-0.650827\pi$
$691$	−16576.8	−0.912608	−0.456304	+	0.889824i	$0.650827\pi$
$692$	0	0
$693$	4564.39	0.250197
$694$	0	0
$695$	−21186.1	−1.15631
$696$	0	0
$697$	10884.6	0.591513
$698$	0	0
$699$	19163.2	1.03694
$700$	0	0
$701$	9831.71	0.529727	0.264863	−	0.964286i	$-0.414673\pi$
$701$	9831.71	0.529727	0.264863	+	0.964286i	$0.414673\pi$
$702$	0	0
$703$	20099.4	1.07833
$704$	0	0
$705$	−37174.2	−1.98590
$706$	0	0
$707$	1696.44	0.0902421
$708$	0	0
$709$	147.484	0.00781223	0.00390611	−	0.999992i	$-0.498757\pi$
$709$	147.484	0.00781223	0.00390611	+	0.999992i	$0.498757\pi$
$710$	0	0
$711$	29829.0	1.57338
$712$	0	0
$713$	280.250	0.0147201
$714$	0	0
$715$	5326.39	0.278595
$716$	0	0
$717$	−49474.3	−2.57692
$718$	0	0
$719$	−28988.4	−1.50359	−0.751797	−	0.659395i	$-0.770812\pi$
$719$	−28988.4	−1.50359	−0.751797	+	0.659395i	$0.770812\pi$
$720$	0	0
$721$	−1751.05	−0.0904475
$722$	0	0
$723$	−65415.9	−3.36493
$724$	0	0
$725$	8403.55	0.430483
$726$	0	0
$727$	38809.4	1.97986	0.989932	−	0.141541i	$-0.0452057\pi$
$727$	38809.4	1.97986	0.989932	+	0.141541i	$0.0452057\pi$
$728$	0	0
$729$	−4985.13	−0.253271
$730$	0	0
$731$	−9568.84	−0.484154
$732$	0	0
$733$	30441.7	1.53396	0.766979	−	0.641672i	$-0.221759\pi$
$733$	30441.7	1.53396	0.766979	+	0.641672i	$0.221759\pi$
$734$	0	0
$735$	3980.86	0.199777
$736$	0	0
$737$	2340.27	0.116968
$738$	0	0
$739$	29172.5	1.45214	0.726068	−	0.687622i	$-0.241346\pi$
$739$	29172.5	1.45214	0.726068	+	0.687622i	$0.241346\pi$
$740$	0	0
$741$	24557.1	1.21745
$742$	0	0
$743$	−29299.5	−1.44670	−0.723348	−	0.690483i	$-0.757398\pi$
$743$	−29299.5	−1.44670	−0.723348	+	0.690483i	$0.757398\pi$
$744$	0	0
$745$	22475.6	1.10529
$746$	0	0
$747$	46121.2	2.25902
$748$	0	0
$749$	8584.86	0.418804
$750$	0	0
$751$	3455.71	0.167910	0.0839552	−	0.996470i	$-0.473245\pi$
$751$	3455.71	0.167910	0.0839552	+	0.996470i	$0.473245\pi$
$752$	0	0
$753$	−22726.5	−1.09987
$754$	0	0
$755$	−5576.61	−0.268813
$756$	0	0
$757$	35734.4	1.71571	0.857853	−	0.513895i	$-0.171798\pi$
$757$	35734.4	1.71571	0.857853	+	0.513895i	$0.171798\pi$
$758$	0	0
$759$	1849.47	0.0884475
$760$	0	0
$761$	−1256.47	−0.0598516	−0.0299258	−	0.999552i	$-0.509527\pi$
$761$	−1256.47	−0.0598516	−0.0299258	+	0.999552i	$0.509527\pi$
$762$	0	0
$763$	9246.19	0.438709
$764$	0	0
$765$	−15192.6	−0.718023
$766$	0	0
$767$	−34880.7	−1.64207
$768$	0	0
$769$	−22119.9	−1.03728	−0.518638	−	0.854994i	$-0.673561\pi$
$769$	−22119.9	−1.03728	−0.518638	+	0.854994i	$0.673561\pi$
$770$	0	0
$771$	−59011.5	−2.75648
$772$	0	0
$773$	25255.2	1.17512	0.587559	−	0.809181i	$-0.300089\pi$
$773$	25255.2	1.17512	0.587559	+	0.809181i	$0.300089\pi$
$774$	0	0
$775$	−750.894	−0.0348038
$776$	0	0
$777$	−27366.0	−1.26351
$778$	0	0
$779$	−17738.9	−0.815867
$780$	0	0
$781$	4680.59	0.214449
$782$	0	0
$783$	51949.1	2.37102
$784$	0	0
$785$	10502.0	0.477494
$786$	0	0
$787$	−12267.9	−0.555658	−0.277829	−	0.960631i	$-0.589615\pi$
$787$	−12267.9	−0.555658	−0.277829	+	0.960631i	$0.589615\pi$
$788$	0	0
$789$	−41614.0	−1.87769
$790$	0	0
$791$	14117.1	0.634573
$792$	0	0
$793$	5564.67	0.249190
$794$	0	0
$795$	58540.5	2.61160
$796$	0	0
$797$	38661.9	1.71829	0.859144	−	0.511734i	$-0.170997\pi$
$797$	38661.9	1.71829	0.859144	+	0.511734i	$0.170997\pi$
$798$	0	0
$799$	−13408.1	−0.593673
$800$	0	0
$801$	−32544.1	−1.43557
$802$	0	0
$803$	4290.09	0.188535
$804$	0	0
$805$	1108.24	0.0485224
$806$	0	0
$807$	−19948.0	−0.870140
$808$	0	0
$809$	15080.1	0.655364	0.327682	−	0.944788i	$-0.393733\pi$
$809$	15080.1	0.655364	0.327682	+	0.944788i	$0.393733\pi$
$810$	0	0
$811$	11347.8	0.491339	0.245670	−	0.969354i	$-0.420992\pi$
$811$	11347.8	0.491339	0.245670	+	0.969354i	$0.420992\pi$
$812$	0	0
$813$	45116.1	1.94624
$814$	0	0
$815$	−32769.4	−1.40842
$816$	0	0
$817$	15594.5	0.667788
$818$	0	0
$819$	−22972.0	−0.980105
$820$	0	0
$821$	718.501	0.0305431	0.0152715	−	0.999883i	$-0.495139\pi$
$821$	718.501	0.0305431	0.0152715	+	0.999883i	$0.495139\pi$
$822$	0	0
$823$	7900.61	0.334627	0.167313	−	0.985904i	$-0.446491\pi$
$823$	7900.61	0.334627	0.167313	+	0.985904i	$0.446491\pi$
$824$	0	0
$825$	−4955.42	−0.209122
$826$	0	0
$827$	30490.8	1.28207	0.641034	−	0.767513i	$-0.278506\pi$
$827$	30490.8	1.28207	0.641034	+	0.767513i	$0.278506\pi$
$828$	0	0
$829$	−26792.2	−1.12248	−0.561238	−	0.827654i	$-0.689675\pi$
$829$	−26792.2	−1.12248	−0.561238	+	0.827654i	$0.689675\pi$
$830$	0	0
$831$	−57282.2	−2.39121
$832$	0	0
$833$	1435.83	0.0597222
$834$	0	0
$835$	−29754.6	−1.23317
$836$	0	0
$837$	−4641.88	−0.191693
$838$	0	0
$839$	−37993.6	−1.56339	−0.781696	−	0.623660i	$-0.785645\pi$
$839$	−37993.6	−1.56339	−0.781696	+	0.623660i	$0.785645\pi$
$840$	0	0
$841$	5633.68	0.230993
$842$	0	0
$843$	38789.5	1.58480
$844$	0	0
$845$	−7591.06	−0.309042
$846$	0	0
$847$	−847.000	−0.0343604
$848$	0	0
$849$	−34567.6	−1.39736
$850$	0	0
$851$	−7618.50	−0.306885
$852$	0	0
$853$	9117.36	0.365970	0.182985	−	0.983116i	$-0.441424\pi$
$853$	9117.36	0.365970	0.182985	+	0.983116i	$0.441424\pi$
$854$	0	0
$855$	24759.6	0.990361
$856$	0	0
$857$	−28933.9	−1.15328	−0.576640	−	0.816998i	$-0.695637\pi$
$857$	−28933.9	−1.15328	−0.576640	+	0.816998i	$0.695637\pi$
$858$	0	0
$859$	38347.8	1.52318	0.761590	−	0.648059i	$-0.224419\pi$
$859$	38347.8	1.52318	0.761590	+	0.648059i	$0.224419\pi$
$860$	0	0
$861$	24152.1	0.955981
$862$	0	0
$863$	29878.9	1.17855	0.589275	−	0.807932i	$-0.299413\pi$
$863$	29878.9	1.17855	0.589275	+	0.807932i	$0.299413\pi$
$864$	0	0
$865$	6688.31	0.262901
$866$	0	0
$867$	37659.2	1.47517
$868$	0	0
$869$	−5535.27	−0.216077
$870$	0	0
$871$	−11778.3	−0.458200
$872$	0	0
$873$	53096.8	2.05848
$874$	0	0
$875$	−10622.5	−0.410408
$876$	0	0
$877$	43604.3	1.67892	0.839460	−	0.543421i	$-0.182871\pi$
$877$	43604.3	1.67892	0.839460	+	0.543421i	$0.182871\pi$
$878$	0	0
$879$	12279.8	0.471202
$880$	0	0
$881$	−49410.2	−1.88953	−0.944763	−	0.327754i	$-0.893708\pi$
$881$	−49410.2	−1.88953	−0.944763	+	0.327754i	$0.893708\pi$
$882$	0	0
$883$	−2015.96	−0.0768316	−0.0384158	−	0.999262i	$-0.512231\pi$
$883$	−2015.96	−0.0768316	−0.0384158	+	0.999262i	$0.512231\pi$
$884$	0	0
$885$	−51186.9	−1.94421
$886$	0	0
$887$	25750.0	0.974745	0.487373	−	0.873194i	$-0.337955\pi$
$887$	25750.0	0.974745	0.487373	+	0.873194i	$0.337955\pi$
$888$	0	0
$889$	−10358.9	−0.390805
$890$	0	0
$891$	−13027.9	−0.489844
$892$	0	0
$893$	21851.4	0.818846
$894$	0	0
$895$	13360.4	0.498984
$896$	0	0
$897$	−9308.16	−0.346477
$898$	0	0
$899$	−2682.66	−0.0995235
$900$	0	0
$901$	21114.6	0.780720
$902$	0	0
$903$	−21232.4	−0.782471
$904$	0	0
$905$	−1265.05	−0.0464661
$906$	0	0
$907$	10039.2	0.367527	0.183763	−	0.982971i	$-0.441172\pi$
$907$	10039.2	0.367527	0.183763	+	0.982971i	$0.441172\pi$
$908$	0	0
$909$	−14365.9	−0.524187
$910$	0	0
$911$	−25190.3	−0.916129	−0.458064	−	0.888919i	$-0.651457\pi$
$911$	−25190.3	−0.916129	−0.458064	+	0.888919i	$0.651457\pi$
$912$	0	0
$913$	−8558.58	−0.310238
$914$	0	0
$915$	8166.06	0.295040
$916$	0	0
$917$	−2420.03	−0.0871498
$918$	0	0
$919$	42342.1	1.51985	0.759923	−	0.650014i	$-0.225237\pi$
$919$	42342.1	1.51985	0.759923	+	0.650014i	$0.225237\pi$
$920$	0	0
$921$	−21589.3	−0.772413
$922$	0	0
$923$	−23556.8	−0.840066
$924$	0	0
$925$	20412.8	0.725587
$926$	0	0
$927$	14828.4	0.525380
$928$	0	0
$929$	−42175.0	−1.48947	−0.744735	−	0.667360i	$-0.767424\pi$
$929$	−42175.0	−1.48947	−0.744735	+	0.667360i	$0.767424\pi$
$930$	0	0
$931$	−2340.00	−0.0823741
$932$	0	0
$933$	−23085.0	−0.810043
$934$	0	0
$935$	2819.24	0.0986084
$936$	0	0
$937$	−18274.8	−0.637151	−0.318576	−	0.947897i	$-0.603204\pi$
$937$	−18274.8	−0.637151	−0.318576	+	0.947897i	$0.603204\pi$
$938$	0	0
$939$	−4886.23	−0.169815
$940$	0	0
$941$	22386.5	0.775537	0.387769	−	0.921757i	$-0.373246\pi$
$941$	22386.5	0.775537	0.387769	+	0.921757i	$0.373246\pi$
$942$	0	0
$943$	6723.76	0.232191
$944$	0	0
$945$	−18356.2	−0.631881
$946$	0	0
$947$	−21655.6	−0.743098	−0.371549	−	0.928413i	$-0.621173\pi$
$947$	−21655.6	−0.743098	−0.371549	+	0.928413i	$0.621173\pi$
$948$	0	0
$949$	−21591.4	−0.738554
$950$	0	0
$951$	33947.4	1.15754
$952$	0	0
$953$	33141.8	1.12651	0.563257	−	0.826282i	$-0.309548\pi$
$953$	33141.8	1.12651	0.563257	+	0.826282i	$0.309548\pi$
$954$	0	0
$955$	23788.3	0.806044
$956$	0	0
$957$	−17703.8	−0.597997
$958$	0	0
$959$	−11305.0	−0.380663
$960$	0	0
$961$	−29551.3	−0.991954
$962$	0	0
$963$	−72698.8	−2.43270
$964$	0	0
$965$	−3186.56	−0.106299
$966$	0	0
$967$	36846.1	1.22533	0.612664	−	0.790344i	$-0.290098\pi$
$967$	36846.1	1.22533	0.612664	+	0.790344i	$0.290098\pi$
$968$	0	0
$969$	12998.0	0.430913
$970$	0	0
$971$	−47644.8	−1.57466	−0.787330	−	0.616532i	$-0.788537\pi$
$971$	−47644.8	−1.57466	−0.787330	+	0.616532i	$0.788537\pi$
$972$	0	0
$973$	−16955.8	−0.558660
$974$	0	0
$975$	24940.0	0.819199
$976$	0	0
$977$	25696.6	0.841459	0.420730	−	0.907186i	$-0.361774\pi$
$977$	25696.6	0.841459	0.420730	+	0.907186i	$0.361774\pi$
$978$	0	0
$979$	6039.12	0.197151
$980$	0	0
$981$	−78299.1	−2.54832
$982$	0	0
$983$	48634.8	1.57804	0.789018	−	0.614370i	$-0.210590\pi$
$983$	48634.8	1.57804	0.789018	+	0.614370i	$0.210590\pi$
$984$	0	0
$985$	21134.7	0.683663
$986$	0	0
$987$	−29751.4	−0.959472
$988$	0	0
$989$	−5910.96	−0.190048
$990$	0	0
$991$	−21519.7	−0.689804	−0.344902	−	0.938639i	$-0.612088\pi$
$991$	−21519.7	−0.689804	−0.344902	+	0.938639i	$0.612088\pi$
$992$	0	0
$993$	−27508.4	−0.879107
$994$	0	0
$995$	−2145.28	−0.0683518
$996$	0	0
$997$	−44134.6	−1.40196	−0.700981	−	0.713180i	$-0.747254\pi$
$997$	−44134.6	−1.40196	−0.700981	+	0.713180i	$0.747254\pi$
$998$	0	0
$999$	126188.	3.99640

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1232.4.a.bc.1.1		7
4.3	odd	2		616.4.a.k.1.7	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
616.4.a.k.1.7	✓	7	4.3	odd	2
1232.4.a.bc.1.1		7	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-9.28858 q^{3} -8.74645 q^{5} -7.00000 q^{7} +59.2778 q^{9} -11.0000 q^{11} +55.3616 q^{13} +81.2421 q^{15} +29.3027 q^{17} -47.7550 q^{19} +65.0201 q^{21} +18.1011 q^{23} -48.4996 q^{25} -299.815 q^{27} -173.271 q^{29} +15.4825 q^{31} +102.174 q^{33} +61.2251 q^{35} -420.885 q^{37} -514.230 q^{39} +371.455 q^{41} -326.552 q^{43} -518.470 q^{45} -457.573 q^{47} +49.0000 q^{49} -272.180 q^{51} +720.569 q^{53} +96.2109 q^{55} +443.576 q^{57} -630.053 q^{59} +100.515 q^{61} -414.944 q^{63} -484.217 q^{65} -212.752 q^{67} -168.134 q^{69} -425.508 q^{71} -390.008 q^{73} +450.493 q^{75} +77.0000 q^{77} +503.207 q^{79} +1184.36 q^{81} +778.053 q^{83} -256.294 q^{85} +1609.44 q^{87} -549.010 q^{89} -387.531 q^{91} -143.810 q^{93} +417.687 q^{95} +895.729 q^{97} -652.056 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 3 q^{3} + 11 q^{5} - 49 q^{7} + 108 q^{9} - 77 q^{11} + 26 q^{13} - 67 q^{15} + 44 q^{17} - 34 q^{19} + 21 q^{21} + 29 q^{23} + 182 q^{25} - 99 q^{27} + 94 q^{29} + 173 q^{31} + 33 q^{33} - 77 q^{35}+ \cdots - 1188 q^{99}+O(q^{100}) \) 7 * q - 3 * q^3 + 11 * q^5 - 49 * q^7 + 108 * q^9 - 77 * q^11 + 26 * q^13 - 67 * q^15 + 44 * q^17 - 34 * q^19 + 21 * q^21 + 29 * q^23 + 182 * q^25 - 99 * q^27 + 94 * q^29 + 173 * q^31 + 33 * q^33 - 77 * q^35 + 255 * q^37 + 60 * q^39 + 508 * q^41 - 656 * q^43 + 466 * q^45 + 18 * q^47 + 343 * q^49 - 850 * q^51 + 1806 * q^53 - 121 * q^55 + 1154 * q^57 - 665 * q^59 + 608 * q^61 - 756 * q^63 + 588 * q^65 - 669 * q^67 + 2067 * q^69 - 1169 * q^71 + 380 * q^73 - 1254 * q^75 + 539 * q^77 + 110 * q^79 + 2847 * q^81 + 496 * q^83 + 3446 * q^85 + 582 * q^87 + 1321 * q^89 - 182 * q^91 + 1493 * q^93 + 50 * q^95 + 3927 * q^97 - 1188 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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