LMFDB - Embedded newform 1232.4.a.q.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:	$0$
Dimension:	$3$
Coefficient field:	3.3.7636.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 16x - 18 $ x^3 - 16*x - 18
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 154)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.24586$ 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)

$f(q)$	$=$	$q-4.49172 q^{3} +21.9122 q^{5} +7.00000 q^{7} -6.82442 q^{9} +O(q^{10})$	$q-4.49172 q^{3} +21.9122 q^{5} +7.00000 q^{7} -6.82442 q^{9} -11.0000 q^{11} -67.9319 q^{13} -98.4236 q^{15} -74.9700 q^{17} -8.78628 q^{19} -31.4421 q^{21} +92.8741 q^{23} +355.145 q^{25} +151.930 q^{27} +90.6553 q^{29} -234.566 q^{31} +49.4090 q^{33} +153.385 q^{35} +89.9108 q^{37} +305.131 q^{39} +174.022 q^{41} +353.940 q^{43} -149.538 q^{45} +496.591 q^{47} +49.0000 q^{49} +336.745 q^{51} -87.1321 q^{53} -241.034 q^{55} +39.4655 q^{57} -407.713 q^{59} +744.878 q^{61} -47.7709 q^{63} -1488.54 q^{65} -395.095 q^{67} -417.165 q^{69} +809.184 q^{71} -965.337 q^{73} -1595.21 q^{75} -77.0000 q^{77} +647.685 q^{79} -498.168 q^{81} +138.086 q^{83} -1642.76 q^{85} -407.199 q^{87} +1251.26 q^{89} -475.523 q^{91} +1053.60 q^{93} -192.527 q^{95} +818.367 q^{97} +75.0686 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 6 q^{3} + 26 q^{5} + 21 q^{7} + 59 q^{9}+O(q^{10}) $ 3 * q - 6 * q^3 + 26 * q^5 + 21 * q^7 + 59 * q^9	$ 3 q - 6 q^{3} + 26 q^{5} + 21 q^{7} + 59 q^{9} - 33 q^{11} + 72 q^{13} - 12 q^{15} + 56 q^{17} + 48 q^{19} - 42 q^{21} + 244 q^{23} + 181 q^{25} - 36 q^{27} + 198 q^{29} - 374 q^{31} + 66 q^{33} + 182 q^{35} + 110 q^{37} + 256 q^{39} - 648 q^{41} + 500 q^{43} - 150 q^{45} + 326 q^{47} + 147 q^{49} + 1096 q^{51} + 206 q^{53} - 286 q^{55} - 984 q^{57} + 266 q^{59} + 308 q^{61} + 413 q^{63} - 1160 q^{65} - 256 q^{67} + 360 q^{69} + 484 q^{71} - 1028 q^{73} - 1098 q^{75} - 231 q^{77} + 1072 q^{79} - 1297 q^{81} - 912 q^{83} - 728 q^{85} + 2660 q^{87} + 3406 q^{89} + 504 q^{91} - 2972 q^{93} - 816 q^{95} + 2042 q^{97} - 649 q^{99}+O(q^{100}) $ 3 * q - 6 * q^3 + 26 * q^5 + 21 * q^7 + 59 * q^9 - 33 * q^11 + 72 * q^13 - 12 * q^15 + 56 * q^17 + 48 * q^19 - 42 * q^21 + 244 * q^23 + 181 * q^25 - 36 * q^27 + 198 * q^29 - 374 * q^31 + 66 * q^33 + 182 * q^35 + 110 * q^37 + 256 * q^39 - 648 * q^41 + 500 * q^43 - 150 * q^45 + 326 * q^47 + 147 * q^49 + 1096 * q^51 + 206 * q^53 - 286 * q^55 - 984 * q^57 + 266 * q^59 + 308 * q^61 + 413 * q^63 - 1160 * q^65 - 256 * q^67 + 360 * q^69 + 484 * q^71 - 1028 * q^73 - 1098 * q^75 - 231 * q^77 + 1072 * q^79 - 1297 * q^81 - 912 * q^83 - 728 * q^85 + 2660 * q^87 + 3406 * q^89 + 504 * q^91 - 2972 * q^93 - 816 * q^95 + 2042 * q^97 - 649 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−4.49172	−0.864433	−0.432216	−	0.901770i	$-0.642268\pi$
$3$	−4.49172	−0.864433	−0.432216	+	0.901770i	$0.642268\pi$
$4$	0	0
$5$	21.9122	1.95989	0.979944	−	0.199274i	$-0.0638585\pi$
$5$	21.9122	1.95989	0.979944	+	0.199274i	$0.0638585\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	−6.82442	−0.252756
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	−67.9319	−1.44930	−0.724650	−	0.689117i	$-0.757999\pi$
$13$	−67.9319	−1.44930	−0.724650	+	0.689117i	$0.757999\pi$
$14$	0	0
$15$	−98.4236	−1.69419
$16$	0	0
$17$	−74.9700	−1.06958	−0.534791	−	0.844984i	$-0.679610\pi$
$17$	−74.9700	−1.06958	−0.534791	+	0.844984i	$0.679610\pi$
$18$	0	0
$19$	−8.78628	−0.106090	−0.0530450	−	0.998592i	$-0.516893\pi$
$19$	−8.78628	−0.106090	−0.0530450	+	0.998592i	$0.516893\pi$
$20$	0	0
$21$	−31.4421	−0.326725
$22$	0	0
$23$	92.8741	0.841982	0.420991	−	0.907065i	$-0.361682\pi$
$23$	92.8741	0.841982	0.420991	+	0.907065i	$0.361682\pi$
$24$	0	0
$25$	355.145	2.84116
$26$	0	0
$27$	151.930	1.08292
$28$	0	0
$29$	90.6553	0.580492	0.290246	−	0.956952i	$-0.406263\pi$
$29$	90.6553	0.580492	0.290246	+	0.956952i	$0.406263\pi$
$30$	0	0
$31$	−234.566	−1.35901	−0.679504	−	0.733672i	$-0.737805\pi$
$31$	−234.566	−1.35901	−0.679504	+	0.733672i	$0.737805\pi$
$32$	0	0
$33$	49.4090	0.260636
$34$	0	0
$35$	153.385	0.740768
$36$	0	0
$37$	89.9108	0.399493	0.199746	−	0.979848i	$-0.435988\pi$
$37$	89.9108	0.399493	0.199746	+	0.979848i	$0.435988\pi$
$38$	0	0
$39$	305.131	1.25282
$40$	0	0
$41$	174.022	0.662870	0.331435	−	0.943478i	$-0.392467\pi$
$41$	174.022	0.662870	0.331435	+	0.943478i	$0.392467\pi$
$42$	0	0
$43$	353.940	1.25524	0.627620	−	0.778520i	$-0.284029\pi$
$43$	353.940	1.25524	0.627620	+	0.778520i	$0.284029\pi$
$44$	0	0
$45$	−149.538	−0.495374
$46$	0	0
$47$	496.591	1.54118	0.770588	−	0.637334i	$-0.219963\pi$
$47$	496.591	1.54118	0.770588	+	0.637334i	$0.219963\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	336.745	0.924582
$52$	0	0
$53$	−87.1321	−0.225821	−0.112911	−	0.993605i	$-0.536017\pi$
$53$	−87.1321	−0.225821	−0.112911	+	0.993605i	$0.536017\pi$
$54$	0	0
$55$	−241.034	−0.590928
$56$	0	0
$57$	39.4655	0.0917077
$58$	0	0
$59$	−407.713	−0.899655	−0.449828	−	0.893115i	$-0.648515\pi$
$59$	−407.713	−0.899655	−0.449828	+	0.893115i	$0.648515\pi$
$60$	0	0
$61$	744.878	1.56347	0.781736	−	0.623609i	$-0.214334\pi$
$61$	744.878	1.56347	0.781736	+	0.623609i	$0.214334\pi$
$62$	0	0
$63$	−47.7709	−0.0955328
$64$	0	0
$65$	−1488.54	−2.84047
$66$	0	0
$67$	−395.095	−0.720426	−0.360213	−	0.932870i	$-0.617296\pi$
$67$	−395.095	−0.720426	−0.360213	+	0.932870i	$0.617296\pi$
$68$	0	0
$69$	−417.165	−0.727837
$70$	0	0
$71$	809.184	1.35257	0.676285	−	0.736640i	$-0.263589\pi$
$71$	809.184	1.35257	0.676285	+	0.736640i	$0.263589\pi$
$72$	0	0
$73$	−965.337	−1.54773	−0.773864	−	0.633352i	$-0.781679\pi$
$73$	−965.337	−1.54773	−0.773864	+	0.633352i	$0.781679\pi$
$74$	0	0
$75$	−1595.21	−2.45599
$76$	0	0
$77$	−77.0000	−0.113961
$78$	0	0
$79$	647.685	0.922409	0.461204	−	0.887294i	$-0.347418\pi$
$79$	647.685	0.922409	0.461204	+	0.887294i	$0.347418\pi$
$80$	0	0
$81$	−498.168	−0.683358
$82$	0	0
$83$	138.086	0.182614	0.0913069	−	0.995823i	$-0.470896\pi$
$83$	138.086	0.182614	0.0913069	+	0.995823i	$0.470896\pi$
$84$	0	0
$85$	−1642.76	−2.09626
$86$	0	0
$87$	−407.199	−0.501796
$88$	0	0
$89$	1251.26	1.49027	0.745134	−	0.666915i	$-0.232386\pi$
$89$	1251.26	1.49027	0.745134	+	0.666915i	$0.232386\pi$
$90$	0	0
$91$	−475.523	−0.547784
$92$	0	0
$93$	1053.60	1.17477
$94$	0	0
$95$	−192.527	−0.207925
$96$	0	0
$97$	818.367	0.856625	0.428312	−	0.903631i	$-0.359108\pi$
$97$	818.367	0.856625	0.428312	+	0.903631i	$0.359108\pi$
$98$	0	0
$99$	75.0686	0.0762088
$100$	0	0
$101$	−272.285	−0.268251	−0.134126	−	0.990964i	$-0.542823\pi$
$101$	−272.285	−0.268251	−0.134126	+	0.990964i	$0.542823\pi$
$102$	0	0
$103$	1178.05	1.12696	0.563480	−	0.826130i	$-0.309462\pi$
$103$	1178.05	1.12696	0.563480	+	0.826130i	$0.309462\pi$
$104$	0	0
$105$	−688.965	−0.640344
$106$	0	0
$107$	227.878	0.205886	0.102943	−	0.994687i	$-0.467174\pi$
$107$	227.878	0.205886	0.102943	+	0.994687i	$0.467174\pi$
$108$	0	0
$109$	391.937	0.344411	0.172205	−	0.985061i	$-0.444911\pi$
$109$	391.937	0.344411	0.172205	+	0.985061i	$0.444911\pi$
$110$	0	0
$111$	−403.854	−0.345335
$112$	0	0
$113$	1395.63	1.16185	0.580926	−	0.813956i	$-0.302690\pi$
$113$	1395.63	1.16185	0.580926	+	0.813956i	$0.302690\pi$
$114$	0	0
$115$	2035.08	1.65019
$116$	0	0
$117$	463.595	0.366320
$118$	0	0
$119$	−524.790	−0.404264
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	−781.659	−0.573007
$124$	0	0
$125$	5042.98	3.60846
$126$	0	0
$127$	−1838.24	−1.28439	−0.642196	−	0.766541i	$-0.721976\pi$
$127$	−1838.24	−1.28439	−0.642196	+	0.766541i	$0.721976\pi$
$128$	0	0
$129$	−1589.80	−1.08507
$130$	0	0
$131$	−1213.17	−0.809121	−0.404561	−	0.914511i	$-0.632576\pi$
$131$	−1213.17	−0.809121	−0.404561	+	0.914511i	$0.632576\pi$
$132$	0	0
$133$	−61.5039	−0.0400983
$134$	0	0
$135$	3329.12	2.12241
$136$	0	0
$137$	−2451.12	−1.52856	−0.764281	−	0.644884i	$-0.776906\pi$
$137$	−2451.12	−1.52856	−0.764281	+	0.644884i	$0.776906\pi$
$138$	0	0
$139$	−1770.82	−1.08057	−0.540285	−	0.841482i	$-0.681684\pi$
$139$	−1770.82	−1.08057	−0.540285	+	0.841482i	$0.681684\pi$
$140$	0	0
$141$	−2230.55	−1.33224
$142$	0	0
$143$	747.251	0.436981
$144$	0	0
$145$	1986.46	1.13770
$146$	0	0
$147$	−220.094	−0.123490
$148$	0	0
$149$	1958.62	1.07689	0.538445	−	0.842661i	$-0.319012\pi$
$149$	1958.62	1.07689	0.538445	+	0.842661i	$0.319012\pi$
$150$	0	0
$151$	719.176	0.387587	0.193794	−	0.981042i	$-0.437921\pi$
$151$	719.176	0.387587	0.193794	+	0.981042i	$0.437921\pi$
$152$	0	0
$153$	511.626	0.270343
$154$	0	0
$155$	−5139.85	−2.66350
$156$	0	0
$157$	1003.31	0.510021	0.255010	−	0.966938i	$-0.417921\pi$
$157$	1003.31	0.510021	0.255010	+	0.966938i	$0.417921\pi$
$158$	0	0
$159$	391.374	0.195207
$160$	0	0
$161$	650.118	0.318239
$162$	0	0
$163$	27.4350	0.0131833	0.00659165	−	0.999978i	$-0.497902\pi$
$163$	27.4350	0.0131833	0.00659165	+	0.999978i	$0.497902\pi$
$164$	0	0
$165$	1082.66	0.510818
$166$	0	0
$167$	1408.66	0.652729	0.326365	−	0.945244i	$-0.394176\pi$
$167$	1408.66	0.652729	0.326365	+	0.945244i	$0.394176\pi$
$168$	0	0
$169$	2417.74	1.10047
$170$	0	0
$171$	59.9612	0.0268149
$172$	0	0
$173$	2228.38	0.979311	0.489655	−	0.871916i	$-0.337123\pi$
$173$	2228.38	0.979311	0.489655	+	0.871916i	$0.337123\pi$
$174$	0	0
$175$	2486.01	1.07386
$176$	0	0
$177$	1831.33	0.777691
$178$	0	0
$179$	3448.05	1.43977	0.719887	−	0.694091i	$-0.244193\pi$
$179$	3448.05	1.43977	0.719887	+	0.694091i	$0.244193\pi$
$180$	0	0
$181$	350.621	0.143986	0.0719930	−	0.997405i	$-0.477064\pi$
$181$	350.621	0.143986	0.0719930	+	0.997405i	$0.477064\pi$
$182$	0	0
$183$	−3345.79	−1.35152
$184$	0	0
$185$	1970.14	0.782961
$186$	0	0
$187$	824.670	0.322491
$188$	0	0
$189$	1063.51	0.409307
$190$	0	0
$191$	844.258	0.319834	0.159917	−	0.987130i	$-0.448877\pi$
$191$	844.258	0.319834	0.159917	+	0.987130i	$0.448877\pi$
$192$	0	0
$193$	−3043.55	−1.13513	−0.567564	−	0.823329i	$-0.692114\pi$
$193$	−3043.55	−1.13513	−0.567564	+	0.823329i	$0.692114\pi$
$194$	0	0
$195$	6686.10	2.45539
$196$	0	0
$197$	3008.69	1.08812	0.544062	−	0.839045i	$-0.316886\pi$
$197$	3008.69	1.08812	0.544062	+	0.839045i	$0.316886\pi$
$198$	0	0
$199$	3150.74	1.12236	0.561181	−	0.827693i	$-0.310347\pi$
$199$	3150.74	1.12236	0.561181	+	0.827693i	$0.310347\pi$
$200$	0	0
$201$	1774.66	0.622760
$202$	0	0
$203$	634.587	0.219405
$204$	0	0
$205$	3813.21	1.29915
$206$	0	0
$207$	−633.811	−0.212816
$208$	0	0
$209$	96.6491	0.0319873
$210$	0	0
$211$	−781.849	−0.255093	−0.127547	−	0.991833i	$-0.540710\pi$
$211$	−781.849	−0.255093	−0.127547	+	0.991833i	$0.540710\pi$
$212$	0	0
$213$	−3634.63	−1.16921
$214$	0	0
$215$	7755.61	2.46013
$216$	0	0
$217$	−1641.96	−0.513656
$218$	0	0
$219$	4336.03	1.33791
$220$	0	0
$221$	5092.85	1.55015
$222$	0	0
$223$	−3459.11	−1.03874	−0.519371	−	0.854549i	$-0.673834\pi$
$223$	−3459.11	−1.03874	−0.519371	+	0.854549i	$0.673834\pi$
$224$	0	0
$225$	−2423.66	−0.718120
$226$	0	0
$227$	−1946.25	−0.569061	−0.284531	−	0.958667i	$-0.591838\pi$
$227$	−1946.25	−0.569061	−0.284531	+	0.958667i	$0.591838\pi$
$228$	0	0
$229$	−5415.25	−1.56266	−0.781332	−	0.624116i	$-0.785459\pi$
$229$	−5415.25	−1.56266	−0.781332	+	0.624116i	$0.785459\pi$
$230$	0	0
$231$	345.863	0.0985112
$232$	0	0
$233$	−5011.16	−1.40898	−0.704489	−	0.709715i	$-0.748824\pi$
$233$	−5011.16	−1.40898	−0.704489	+	0.709715i	$0.748824\pi$
$234$	0	0
$235$	10881.4	3.02053
$236$	0	0
$237$	−2909.22	−0.797360
$238$	0	0
$239$	2741.59	0.742003	0.371001	−	0.928632i	$-0.379015\pi$
$239$	2741.59	0.742003	0.371001	+	0.928632i	$0.379015\pi$
$240$	0	0
$241$	−2981.76	−0.796980	−0.398490	−	0.917173i	$-0.630466\pi$
$241$	−2981.76	−0.796980	−0.398490	+	0.917173i	$0.630466\pi$
$242$	0	0
$243$	−1864.47	−0.492206
$244$	0	0
$245$	1073.70	0.279984
$246$	0	0
$247$	596.868	0.153756
$248$	0	0
$249$	−620.246	−0.157857
$250$	0	0
$251$	277.811	0.0698616	0.0349308	−	0.999390i	$-0.488879\pi$
$251$	277.811	0.0698616	0.0349308	+	0.999390i	$0.488879\pi$
$252$	0	0
$253$	−1021.61	−0.253867
$254$	0	0
$255$	7378.82	1.81208
$256$	0	0
$257$	7275.68	1.76593	0.882966	−	0.469438i	$-0.155543\pi$
$257$	7275.68	1.76593	0.882966	+	0.469438i	$0.155543\pi$
$258$	0	0
$259$	629.375	0.150994
$260$	0	0
$261$	−618.670	−0.146723
$262$	0	0
$263$	2959.44	0.693865	0.346933	−	0.937890i	$-0.387223\pi$
$263$	2959.44	0.693865	0.346933	+	0.937890i	$0.387223\pi$
$264$	0	0
$265$	−1909.26	−0.442584
$266$	0	0
$267$	−5620.34	−1.28824
$268$	0	0
$269$	4089.87	0.927003	0.463501	−	0.886096i	$-0.346593\pi$
$269$	4089.87	0.927003	0.463501	+	0.886096i	$0.346593\pi$
$270$	0	0
$271$	−5319.71	−1.19243	−0.596216	−	0.802824i	$-0.703330\pi$
$271$	−5319.71	−1.19243	−0.596216	+	0.802824i	$0.703330\pi$
$272$	0	0
$273$	2135.92	0.473523
$274$	0	0
$275$	−3906.59	−0.856642
$276$	0	0
$277$	2871.53	0.622865	0.311432	−	0.950268i	$-0.399191\pi$
$277$	2871.53	0.622865	0.311432	+	0.950268i	$0.399191\pi$
$278$	0	0
$279$	1600.77	0.343497
$280$	0	0
$281$	−3430.52	−0.728284	−0.364142	−	0.931343i	$-0.618638\pi$
$281$	−3430.52	−0.728284	−0.364142	+	0.931343i	$0.618638\pi$
$282$	0	0
$283$	2934.45	0.616379	0.308189	−	0.951325i	$-0.400277\pi$
$283$	2934.45	0.616379	0.308189	+	0.951325i	$0.400277\pi$
$284$	0	0
$285$	864.777	0.179737
$286$	0	0
$287$	1218.15	0.250541
$288$	0	0
$289$	707.501	0.144006
$290$	0	0
$291$	−3675.88	−0.740495
$292$	0	0
$293$	8753.59	1.74536	0.872679	−	0.488294i	$-0.162381\pi$
$293$	8753.59	1.74536	0.872679	+	0.488294i	$0.162381\pi$
$294$	0	0
$295$	−8933.88	−1.76322
$296$	0	0
$297$	−1671.23	−0.326514
$298$	0	0
$299$	−6309.11	−1.22028
$300$	0	0
$301$	2477.58	0.474436
$302$	0	0
$303$	1223.03	0.231885
$304$	0	0
$305$	16321.9	3.06423
$306$	0	0
$307$	5612.03	1.04331	0.521654	−	0.853157i	$-0.325315\pi$
$307$	5612.03	1.04331	0.521654	+	0.853157i	$0.325315\pi$
$308$	0	0
$309$	−5291.48	−0.974181
$310$	0	0
$311$	−6218.79	−1.13388	−0.566938	−	0.823761i	$-0.691872\pi$
$311$	−6218.79	−1.13388	−0.566938	+	0.823761i	$0.691872\pi$
$312$	0	0
$313$	−2105.36	−0.380197	−0.190099	−	0.981765i	$-0.560881\pi$
$313$	−2105.36	−0.380197	−0.190099	+	0.981765i	$0.560881\pi$
$314$	0	0
$315$	−1046.77	−0.187234
$316$	0	0
$317$	1348.16	0.238866	0.119433	−	0.992842i	$-0.461892\pi$
$317$	1348.16	0.238866	0.119433	+	0.992842i	$0.461892\pi$
$318$	0	0
$319$	−997.208	−0.175025
$320$	0	0
$321$	−1023.57	−0.177975
$322$	0	0
$323$	658.707	0.113472
$324$	0	0
$325$	−24125.7	−4.11769
$326$	0	0
$327$	−1760.47	−0.297720
$328$	0	0
$329$	3476.14	0.582510
$330$	0	0
$331$	1672.07	0.277660	0.138830	−	0.990316i	$-0.455666\pi$
$331$	1672.07	0.277660	0.138830	+	0.990316i	$0.455666\pi$
$332$	0	0
$333$	−613.588	−0.100974
$334$	0	0
$335$	−8657.41	−1.41195
$336$	0	0
$337$	4072.03	0.658212	0.329106	−	0.944293i	$-0.393253\pi$
$337$	4072.03	0.658212	0.329106	+	0.944293i	$0.393253\pi$
$338$	0	0
$339$	−6268.76	−1.00434
$340$	0	0
$341$	2580.22	0.409756
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	−9141.00	−1.42648
$346$	0	0
$347$	−6718.47	−1.03938	−0.519692	−	0.854353i	$-0.673953\pi$
$347$	−6718.47	−1.03938	−0.519692	+	0.854353i	$0.673953\pi$
$348$	0	0
$349$	−9994.20	−1.53289	−0.766443	−	0.642312i	$-0.777975\pi$
$349$	−9994.20	−1.53289	−0.766443	+	0.642312i	$0.777975\pi$
$350$	0	0
$351$	−10320.9	−1.56948
$352$	0	0
$353$	2202.03	0.332018	0.166009	−	0.986124i	$-0.446912\pi$
$353$	2202.03	0.332018	0.166009	+	0.986124i	$0.446912\pi$
$354$	0	0
$355$	17731.0	2.65088
$356$	0	0
$357$	2357.21	0.349459
$358$	0	0
$359$	696.345	0.102372	0.0511862	−	0.998689i	$-0.483700\pi$
$359$	696.345	0.102372	0.0511862	+	0.998689i	$0.483700\pi$
$360$	0	0
$361$	−6781.80	−0.988745
$362$	0	0
$363$	−543.499	−0.0785848
$364$	0	0
$365$	−21152.7	−3.03337
$366$	0	0
$367$	−12382.2	−1.76117	−0.880583	−	0.473892i	$-0.842849\pi$
$367$	−12382.2	−1.76117	−0.880583	+	0.473892i	$0.842849\pi$
$368$	0	0
$369$	−1187.60	−0.167544
$370$	0	0
$371$	−609.925	−0.0853524
$372$	0	0
$373$	10891.5	1.51191	0.755954	−	0.654625i	$-0.227173\pi$
$373$	10891.5	1.51191	0.755954	+	0.654625i	$0.227173\pi$
$374$	0	0
$375$	−22651.7	−3.11927
$376$	0	0
$377$	−6158.38	−0.841308
$378$	0	0
$379$	11368.4	1.54078	0.770389	−	0.637574i	$-0.220062\pi$
$379$	11368.4	1.54078	0.770389	+	0.637574i	$0.220062\pi$
$380$	0	0
$381$	8256.88	1.11027
$382$	0	0
$383$	2601.60	0.347090	0.173545	−	0.984826i	$-0.444478\pi$
$383$	2601.60	0.347090	0.173545	+	0.984826i	$0.444478\pi$
$384$	0	0
$385$	−1687.24	−0.223350
$386$	0	0
$387$	−2415.43	−0.317270
$388$	0	0
$389$	−360.043	−0.0469278	−0.0234639	−	0.999725i	$-0.507469\pi$
$389$	−360.043	−0.0469278	−0.0234639	+	0.999725i	$0.507469\pi$
$390$	0	0
$391$	−6962.77	−0.900569
$392$	0	0
$393$	5449.21	0.699431
$394$	0	0
$395$	14192.2	1.80782
$396$	0	0
$397$	10004.0	1.26471	0.632353	−	0.774681i	$-0.282089\pi$
$397$	10004.0	1.26471	0.632353	+	0.774681i	$0.282089\pi$
$398$	0	0
$399$	276.259	0.0346622
$400$	0	0
$401$	−3281.48	−0.408652	−0.204326	−	0.978903i	$-0.565500\pi$
$401$	−3281.48	−0.408652	−0.204326	+	0.978903i	$0.565500\pi$
$402$	0	0
$403$	15934.5	1.96961
$404$	0	0
$405$	−10916.0	−1.33931
$406$	0	0
$407$	−989.018	−0.120452
$408$	0	0
$409$	−9734.75	−1.17690	−0.588450	−	0.808533i	$-0.700262\pi$
$409$	−9734.75	−1.17690	−0.588450	+	0.808533i	$0.700262\pi$
$410$	0	0
$411$	11009.7	1.32134
$412$	0	0
$413$	−2853.99	−0.340038
$414$	0	0
$415$	3025.78	0.357902
$416$	0	0
$417$	7954.05	0.934080
$418$	0	0
$419$	−321.029	−0.0374303	−0.0187152	−	0.999825i	$-0.505958\pi$
$419$	−321.029	−0.0374303	−0.0187152	+	0.999825i	$0.505958\pi$
$420$	0	0
$421$	10411.2	1.20525	0.602626	−	0.798024i	$-0.294121\pi$
$421$	10411.2	1.20525	0.602626	+	0.798024i	$0.294121\pi$
$422$	0	0
$423$	−3388.94	−0.389541
$424$	0	0
$425$	−26625.2	−3.03885
$426$	0	0
$427$	5214.14	0.590937
$428$	0	0
$429$	−3356.44	−0.377740
$430$	0	0
$431$	4817.39	0.538389	0.269195	−	0.963086i	$-0.413243\pi$
$431$	4817.39	0.538389	0.269195	+	0.963086i	$0.413243\pi$
$432$	0	0
$433$	1665.64	0.184863	0.0924315	−	0.995719i	$-0.470536\pi$
$433$	1665.64	0.184863	0.0924315	+	0.995719i	$0.470536\pi$
$434$	0	0
$435$	−8922.62	−0.983464
$436$	0	0
$437$	−816.017	−0.0893259
$438$	0	0
$439$	−8756.95	−0.952042	−0.476021	−	0.879434i	$-0.657921\pi$
$439$	−8756.95	−0.952042	−0.476021	+	0.879434i	$0.657921\pi$
$440$	0	0
$441$	−334.396	−0.0361080
$442$	0	0
$443$	11923.9	1.27883	0.639415	−	0.768862i	$-0.279177\pi$
$443$	11923.9	1.27883	0.639415	+	0.768862i	$0.279177\pi$
$444$	0	0
$445$	27418.0	2.92076
$446$	0	0
$447$	−8797.59	−0.930898
$448$	0	0
$449$	744.354	0.0782366	0.0391183	−	0.999235i	$-0.487545\pi$
$449$	744.354	0.0782366	0.0391183	+	0.999235i	$0.487545\pi$
$450$	0	0
$451$	−1914.24	−0.199863
$452$	0	0
$453$	−3230.34	−0.335043
$454$	0	0
$455$	−10419.8	−1.07360
$456$	0	0
$457$	−13275.6	−1.35887	−0.679436	−	0.733735i	$-0.737775\pi$
$457$	−13275.6	−1.35887	−0.679436	+	0.733735i	$0.737775\pi$
$458$	0	0
$459$	−11390.2	−1.15828
$460$	0	0
$461$	13659.1	1.37998	0.689989	−	0.723820i	$-0.257615\pi$
$461$	13659.1	1.37998	0.689989	+	0.723820i	$0.257615\pi$
$462$	0	0
$463$	281.999	0.0283058	0.0141529	−	0.999900i	$-0.495495\pi$
$463$	281.999	0.0283058	0.0141529	+	0.999900i	$0.495495\pi$
$464$	0	0
$465$	23086.8	2.30242
$466$	0	0
$467$	2187.00	0.216708	0.108354	−	0.994112i	$-0.465442\pi$
$467$	2187.00	0.216708	0.108354	+	0.994112i	$0.465442\pi$
$468$	0	0
$469$	−2765.67	−0.272296
$470$	0	0
$471$	−4506.61	−0.440878
$472$	0	0
$473$	−3893.34	−0.378469
$474$	0	0
$475$	−3120.40	−0.301419
$476$	0	0
$477$	594.626	0.0570777
$478$	0	0
$479$	−16158.4	−1.54132	−0.770662	−	0.637244i	$-0.780074\pi$
$479$	−16158.4	−1.54132	−0.770662	+	0.637244i	$0.780074\pi$
$480$	0	0
$481$	−6107.81	−0.578985
$482$	0	0
$483$	−2920.15	−0.275096
$484$	0	0
$485$	17932.2	1.67889
$486$	0	0
$487$	−6625.69	−0.616507	−0.308253	−	0.951304i	$-0.599744\pi$
$487$	−6625.69	−0.616507	−0.308253	+	0.951304i	$0.599744\pi$
$488$	0	0
$489$	−123.231	−0.0113961
$490$	0	0
$491$	−10365.9	−0.952762	−0.476381	−	0.879239i	$-0.658052\pi$
$491$	−10365.9	−0.952762	−0.476381	+	0.879239i	$0.658052\pi$
$492$	0	0
$493$	−6796.43	−0.620884
$494$	0	0
$495$	1644.92	0.149361
$496$	0	0
$497$	5664.29	0.511223
$498$	0	0
$499$	5535.54	0.496603	0.248301	−	0.968683i	$-0.420128\pi$
$499$	5535.54	0.496603	0.248301	+	0.968683i	$0.420128\pi$
$500$	0	0
$501$	−6327.33	−0.564240
$502$	0	0
$503$	17237.2	1.52797	0.763984	−	0.645236i	$-0.223241\pi$
$503$	17237.2	1.52797	0.763984	+	0.645236i	$0.223241\pi$
$504$	0	0
$505$	−5966.36	−0.525742
$506$	0	0
$507$	−10859.8	−0.951285
$508$	0	0
$509$	−12784.0	−1.11324	−0.556621	−	0.830766i	$-0.687903\pi$
$509$	−12784.0	−1.11324	−0.556621	+	0.830766i	$0.687903\pi$
$510$	0	0
$511$	−6757.36	−0.584986
$512$	0	0
$513$	−1334.90	−0.114887
$514$	0	0
$515$	25813.7	2.20871
$516$	0	0
$517$	−5462.50	−0.464682
$518$	0	0
$519$	−10009.3	−0.846548
$520$	0	0
$521$	15176.9	1.27622	0.638111	−	0.769944i	$-0.279716\pi$
$521$	15176.9	1.27622	0.638111	+	0.769944i	$0.279716\pi$
$522$	0	0
$523$	18562.2	1.55195	0.775975	−	0.630764i	$-0.217258\pi$
$523$	18562.2	1.55195	0.775975	+	0.630764i	$0.217258\pi$
$524$	0	0
$525$	−11166.5	−0.928277
$526$	0	0
$527$	17585.4	1.45357
$528$	0	0
$529$	−3541.41	−0.291067
$530$	0	0
$531$	2782.40	0.227393
$532$	0	0
$533$	−11821.6	−0.960698
$534$	0	0
$535$	4993.31	0.403514
$536$	0	0
$537$	−15487.7	−1.24459
$538$	0	0
$539$	−539.000	−0.0430730
$540$	0	0
$541$	14315.4	1.13764	0.568822	−	0.822461i	$-0.307399\pi$
$541$	14315.4	1.13764	0.568822	+	0.822461i	$0.307399\pi$
$542$	0	0
$543$	−1574.89	−0.124466
$544$	0	0
$545$	8588.21	0.675006
$546$	0	0
$547$	−2207.98	−0.172589	−0.0862945	−	0.996270i	$-0.527503\pi$
$547$	−2207.98	−0.172589	−0.0862945	+	0.996270i	$0.527503\pi$
$548$	0	0
$549$	−5083.36	−0.395177
$550$	0	0
$551$	−796.523	−0.0615844
$552$	0	0
$553$	4533.80	0.348638
$554$	0	0
$555$	−8849.34	−0.676817
$556$	0	0
$557$	−8018.42	−0.609967	−0.304983	−	0.952358i	$-0.598651\pi$
$557$	−8018.42	−0.609967	−0.304983	+	0.952358i	$0.598651\pi$
$558$	0	0
$559$	−24043.8	−1.81922
$560$	0	0
$561$	−3704.19	−0.278772
$562$	0	0
$563$	18689.3	1.39904	0.699520	−	0.714613i	$-0.253397\pi$
$563$	18689.3	1.39904	0.699520	+	0.714613i	$0.253397\pi$
$564$	0	0
$565$	30581.2	2.27710
$566$	0	0
$567$	−3487.18	−0.258285
$568$	0	0
$569$	605.833	0.0446359	0.0223180	−	0.999751i	$-0.492895\pi$
$569$	605.833	0.0446359	0.0223180	+	0.999751i	$0.492895\pi$
$570$	0	0
$571$	−12505.9	−0.916560	−0.458280	−	0.888808i	$-0.651534\pi$
$571$	−12505.9	−0.916560	−0.458280	+	0.888808i	$0.651534\pi$
$572$	0	0
$573$	−3792.17	−0.276475
$574$	0	0
$575$	32983.7	2.39220
$576$	0	0
$577$	−18774.3	−1.35457	−0.677284	−	0.735722i	$-0.736843\pi$
$577$	−18774.3	−1.35457	−0.677284	+	0.735722i	$0.736843\pi$
$578$	0	0
$579$	13670.8	0.981242
$580$	0	0
$581$	966.604	0.0690215
$582$	0	0
$583$	958.454	0.0680876
$584$	0	0
$585$	10158.4	0.717945
$586$	0	0
$587$	−8854.87	−0.622623	−0.311311	−	0.950308i	$-0.600768\pi$
$587$	−8854.87	−0.622623	−0.311311	+	0.950308i	$0.600768\pi$
$588$	0	0
$589$	2060.96	0.144177
$590$	0	0
$591$	−13514.2	−0.940610
$592$	0	0
$593$	−15898.5	−1.10097	−0.550483	−	0.834846i	$-0.685556\pi$
$593$	−15898.5	−1.10097	−0.550483	+	0.834846i	$0.685556\pi$
$594$	0	0
$595$	−11499.3	−0.792312
$596$	0	0
$597$	−14152.3	−0.970206
$598$	0	0
$599$	26940.9	1.83769	0.918846	−	0.394617i	$-0.129123\pi$
$599$	26940.9	1.83769	0.918846	+	0.394617i	$0.129123\pi$
$600$	0	0
$601$	19104.1	1.29663	0.648313	−	0.761374i	$-0.275475\pi$
$601$	19104.1	1.29663	0.648313	+	0.761374i	$0.275475\pi$
$602$	0	0
$603$	2696.29	0.182092
$604$	0	0
$605$	2651.38	0.178172
$606$	0	0
$607$	4787.19	0.320109	0.160054	−	0.987108i	$-0.448833\pi$
$607$	4787.19	0.320109	0.160054	+	0.987108i	$0.448833\pi$
$608$	0	0
$609$	−2850.39	−0.189661
$610$	0	0
$611$	−33734.4	−2.23363
$612$	0	0
$613$	7874.18	0.518818	0.259409	−	0.965768i	$-0.416472\pi$
$613$	7874.18	0.518818	0.259409	+	0.965768i	$0.416472\pi$
$614$	0	0
$615$	−17127.9	−1.12303
$616$	0	0
$617$	6140.24	0.400643	0.200322	−	0.979730i	$-0.435801\pi$
$617$	6140.24	0.400643	0.200322	+	0.979730i	$0.435801\pi$
$618$	0	0
$619$	13225.7	0.858782	0.429391	−	0.903119i	$-0.358728\pi$
$619$	13225.7	0.858782	0.429391	+	0.903119i	$0.358728\pi$
$620$	0	0
$621$	14110.4	0.911802
$622$	0	0
$623$	8758.85	0.563268
$624$	0	0
$625$	66109.8	4.23102
$626$	0	0
$627$	−434.121	−0.0276509
$628$	0	0
$629$	−6740.61	−0.427290
$630$	0	0
$631$	18204.2	1.14849	0.574244	−	0.818684i	$-0.305296\pi$
$631$	18204.2	1.14849	0.574244	+	0.818684i	$0.305296\pi$
$632$	0	0
$633$	3511.85	0.220511
$634$	0	0
$635$	−40280.0	−2.51726
$636$	0	0
$637$	−3328.66	−0.207043
$638$	0	0
$639$	−5522.21	−0.341870
$640$	0	0
$641$	−14900.5	−0.918148	−0.459074	−	0.888398i	$-0.651819\pi$
$641$	−14900.5	−0.918148	−0.459074	+	0.888398i	$0.651819\pi$
$642$	0	0
$643$	−14768.4	−0.905766	−0.452883	−	0.891570i	$-0.649604\pi$
$643$	−14768.4	−0.905766	−0.452883	+	0.891570i	$0.649604\pi$
$644$	0	0
$645$	−34836.0	−2.12662
$646$	0	0
$647$	−12793.2	−0.777359	−0.388679	−	0.921373i	$-0.627069\pi$
$647$	−12793.2	−0.777359	−0.388679	+	0.921373i	$0.627069\pi$
$648$	0	0
$649$	4484.84	0.271256
$650$	0	0
$651$	7375.23	0.444021
$652$	0	0
$653$	−15427.1	−0.924516	−0.462258	−	0.886745i	$-0.652961\pi$
$653$	−15427.1	−0.924516	−0.462258	+	0.886745i	$0.652961\pi$
$654$	0	0
$655$	−26583.2	−1.58579
$656$	0	0
$657$	6587.86	0.391198
$658$	0	0
$659$	12659.7	0.748334	0.374167	−	0.927361i	$-0.377929\pi$
$659$	12659.7	0.748334	0.374167	+	0.927361i	$0.377929\pi$
$660$	0	0
$661$	−28149.1	−1.65639	−0.828193	−	0.560443i	$-0.810631\pi$
$661$	−28149.1	−1.65639	−0.828193	+	0.560443i	$0.810631\pi$
$662$	0	0
$663$	−22875.7	−1.34000
$664$	0	0
$665$	−1347.69	−0.0785881
$666$	0	0
$667$	8419.53	0.488764
$668$	0	0
$669$	15537.4	0.897922
$670$	0	0
$671$	−8193.66	−0.471405
$672$	0	0
$673$	−5611.40	−0.321402	−0.160701	−	0.987003i	$-0.551375\pi$
$673$	−5611.40	−0.321402	−0.160701	+	0.987003i	$0.551375\pi$
$674$	0	0
$675$	53957.1	3.07676
$676$	0	0
$677$	−25884.9	−1.46948	−0.734740	−	0.678349i	$-0.762696\pi$
$677$	−25884.9	−1.46948	−0.734740	+	0.678349i	$0.762696\pi$
$678$	0	0
$679$	5728.57	0.323774
$680$	0	0
$681$	8742.00	0.491915
$682$	0	0
$683$	11568.9	0.648129	0.324064	−	0.946035i	$-0.394951\pi$
$683$	11568.9	0.648129	0.324064	+	0.946035i	$0.394951\pi$
$684$	0	0
$685$	−53709.4	−2.99581
$686$	0	0
$687$	24323.8	1.35082
$688$	0	0
$689$	5919.05	0.327283
$690$	0	0
$691$	−35995.7	−1.98168	−0.990841	−	0.135037i	$-0.956885\pi$
$691$	−35995.7	−1.98168	−0.990841	+	0.135037i	$0.956885\pi$
$692$	0	0
$693$	525.480	0.0288042
$694$	0	0
$695$	−38802.6	−2.11780
$696$	0	0
$697$	−13046.4	−0.708994
$698$	0	0
$699$	22508.7	1.21797
$700$	0	0
$701$	11775.5	0.634456	0.317228	−	0.948349i	$-0.397248\pi$
$701$	11775.5	0.634456	0.317228	+	0.948349i	$0.397248\pi$
$702$	0	0
$703$	−789.981	−0.0423822
$704$	0	0
$705$	−48876.3	−2.61105
$706$	0	0
$707$	−1905.99	−0.101389
$708$	0	0
$709$	−11644.4	−0.616807	−0.308403	−	0.951256i	$-0.599795\pi$
$709$	−11644.4	−0.616807	−0.308403	+	0.951256i	$0.599795\pi$
$710$	0	0
$711$	−4420.07	−0.233144
$712$	0	0
$713$	−21785.1	−1.14426
$714$	0	0
$715$	16373.9	0.856433
$716$	0	0
$717$	−12314.5	−0.641411
$718$	0	0
$719$	3676.51	0.190696	0.0953481	−	0.995444i	$-0.469604\pi$
$719$	3676.51	0.190696	0.0953481	+	0.995444i	$0.469604\pi$
$720$	0	0
$721$	8246.36	0.425951
$722$	0	0
$723$	13393.3	0.688936
$724$	0	0
$725$	32195.8	1.64927
$726$	0	0
$727$	10957.6	0.559004	0.279502	−	0.960145i	$-0.409831\pi$
$727$	10957.6	0.559004	0.279502	+	0.960145i	$0.409831\pi$
$728$	0	0
$729$	21825.2	1.10884
$730$	0	0
$731$	−26534.9	−1.34258
$732$	0	0
$733$	−13054.5	−0.657817	−0.328908	−	0.944362i	$-0.606681\pi$
$733$	−13054.5	−0.657817	−0.328908	+	0.944362i	$0.606681\pi$
$734$	0	0
$735$	−4822.76	−0.242027
$736$	0	0
$737$	4346.05	0.217217
$738$	0	0
$739$	33449.5	1.66503	0.832517	−	0.553999i	$-0.186899\pi$
$739$	33449.5	1.66503	0.832517	+	0.553999i	$0.186899\pi$
$740$	0	0
$741$	−2680.97	−0.132912
$742$	0	0
$743$	−18455.6	−0.911266	−0.455633	−	0.890168i	$-0.650587\pi$
$743$	−18455.6	−0.911266	−0.455633	+	0.890168i	$0.650587\pi$
$744$	0	0
$745$	42917.7	2.11058
$746$	0	0
$747$	−942.358	−0.0461567
$748$	0	0
$749$	1595.15	0.0778176
$750$	0	0
$751$	21393.2	1.03948	0.519740	−	0.854325i	$-0.326029\pi$
$751$	21393.2	1.03948	0.519740	+	0.854325i	$0.326029\pi$
$752$	0	0
$753$	−1247.85	−0.0603906
$754$	0	0
$755$	15758.7	0.759628
$756$	0	0
$757$	5511.55	0.264625	0.132312	−	0.991208i	$-0.457760\pi$
$757$	5511.55	0.264625	0.132312	+	0.991208i	$0.457760\pi$
$758$	0	0
$759$	4588.81	0.219451
$760$	0	0
$761$	−28743.1	−1.36917	−0.684585	−	0.728933i	$-0.740016\pi$
$761$	−28743.1	−1.36917	−0.684585	+	0.728933i	$0.740016\pi$
$762$	0	0
$763$	2743.56	0.130175
$764$	0	0
$765$	11210.9	0.529843
$766$	0	0
$767$	27696.7	1.30387
$768$	0	0
$769$	−22186.7	−1.04041	−0.520204	−	0.854042i	$-0.674144\pi$
$769$	−22186.7	−1.04041	−0.520204	+	0.854042i	$0.674144\pi$
$770$	0	0
$771$	−32680.3	−1.52653
$772$	0	0
$773$	−8800.99	−0.409508	−0.204754	−	0.978813i	$-0.565639\pi$
$773$	−8800.99	−0.409508	−0.204754	+	0.978813i	$0.565639\pi$
$774$	0	0
$775$	−83304.8	−3.86116
$776$	0	0
$777$	−2826.98	−0.130524
$778$	0	0
$779$	−1529.01	−0.0703239
$780$	0	0
$781$	−8901.02	−0.407815
$782$	0	0
$783$	13773.3	0.628628
$784$	0	0
$785$	21984.8	0.999583
$786$	0	0
$787$	−2358.25	−0.106814	−0.0534069	−	0.998573i	$-0.517008\pi$
$787$	−2358.25	−0.106814	−0.0534069	+	0.998573i	$0.517008\pi$
$788$	0	0
$789$	−13293.0	−0.599800
$790$	0	0
$791$	9769.38	0.439139
$792$	0	0
$793$	−50600.9	−2.26594
$794$	0	0
$795$	8575.86	0.382584
$796$	0	0
$797$	−19640.7	−0.872911	−0.436456	−	0.899726i	$-0.643766\pi$
$797$	−19640.7	−0.872911	−0.436456	+	0.899726i	$0.643766\pi$
$798$	0	0
$799$	−37229.4	−1.64841
$800$	0	0
$801$	−8539.15	−0.376674
$802$	0	0
$803$	10618.7	0.466658
$804$	0	0
$805$	14245.5	0.623713
$806$	0	0
$807$	−18370.6	−0.801331
$808$	0	0
$809$	4996.27	0.217132	0.108566	−	0.994089i	$-0.465374\pi$
$809$	4996.27	0.217132	0.108566	+	0.994089i	$0.465374\pi$
$810$	0	0
$811$	−11336.5	−0.490851	−0.245425	−	0.969416i	$-0.578928\pi$
$811$	−11336.5	−0.490851	−0.245425	+	0.969416i	$0.578928\pi$
$812$	0	0
$813$	23894.7	1.03078
$814$	0	0
$815$	601.162	0.0258378
$816$	0	0
$817$	−3109.82	−0.133169
$818$	0	0
$819$	3245.17	0.138456
$820$	0	0
$821$	−20681.1	−0.879141	−0.439571	−	0.898208i	$-0.644869\pi$
$821$	−20681.1	−0.879141	−0.439571	+	0.898208i	$0.644869\pi$
$822$	0	0
$823$	31729.7	1.34390	0.671950	−	0.740597i	$-0.265457\pi$
$823$	31729.7	1.34390	0.671950	+	0.740597i	$0.265457\pi$
$824$	0	0
$825$	17547.3	0.740509
$826$	0	0
$827$	−8715.00	−0.366445	−0.183223	−	0.983071i	$-0.558653\pi$
$827$	−8715.00	−0.366445	−0.183223	+	0.983071i	$0.558653\pi$
$828$	0	0
$829$	25215.4	1.05642	0.528208	−	0.849115i	$-0.322864\pi$
$829$	25215.4	1.05642	0.528208	+	0.849115i	$0.322864\pi$
$830$	0	0
$831$	−12898.1	−0.538425
$832$	0	0
$833$	−3673.53	−0.152797
$834$	0	0
$835$	30867.0	1.27928
$836$	0	0
$837$	−35637.5	−1.47170
$838$	0	0
$839$	−36301.8	−1.49377	−0.746887	−	0.664951i	$-0.768452\pi$
$839$	−36301.8	−1.49377	−0.746887	+	0.664951i	$0.768452\pi$
$840$	0	0
$841$	−16170.6	−0.663029
$842$	0	0
$843$	15409.0	0.629552
$844$	0	0
$845$	52978.0	2.15680
$846$	0	0
$847$	847.000	0.0343604
$848$	0	0
$849$	−13180.8	−0.532818
$850$	0	0
$851$	8350.38	0.336366
$852$	0	0
$853$	10225.4	0.410446	0.205223	−	0.978715i	$-0.434208\pi$
$853$	10225.4	0.410446	0.205223	+	0.978715i	$0.434208\pi$
$854$	0	0
$855$	1313.88	0.0525542
$856$	0	0
$857$	35601.1	1.41903	0.709517	−	0.704689i	$-0.248913\pi$
$857$	35601.1	1.41903	0.709517	+	0.704689i	$0.248913\pi$
$858$	0	0
$859$	4706.77	0.186953	0.0934766	−	0.995621i	$-0.470202\pi$
$859$	4706.77	0.186953	0.0934766	+	0.995621i	$0.470202\pi$
$860$	0	0
$861$	−5471.61	−0.216576
$862$	0	0
$863$	−20872.8	−0.823313	−0.411657	−	0.911339i	$-0.635050\pi$
$863$	−20872.8	−0.823313	−0.411657	+	0.911339i	$0.635050\pi$
$864$	0	0
$865$	48828.8	1.91934
$866$	0	0
$867$	−3177.90	−0.124483
$868$	0	0
$869$	−7124.54	−0.278117
$870$	0	0
$871$	26839.6	1.04411
$872$	0	0
$873$	−5584.88	−0.216517
$874$	0	0
$875$	35300.9	1.36387
$876$	0	0
$877$	−26126.1	−1.00595	−0.502973	−	0.864302i	$-0.667761\pi$
$877$	−26126.1	−1.00595	−0.502973	+	0.864302i	$0.667761\pi$
$878$	0	0
$879$	−39318.7	−1.50874
$880$	0	0
$881$	12107.9	0.463024	0.231512	−	0.972832i	$-0.425633\pi$
$881$	12107.9	0.463024	0.231512	+	0.972832i	$0.425633\pi$
$882$	0	0
$883$	19773.7	0.753609	0.376805	−	0.926293i	$-0.377023\pi$
$883$	19773.7	0.753609	0.376805	+	0.926293i	$0.377023\pi$
$884$	0	0
$885$	40128.5	1.52419
$886$	0	0
$887$	−33053.8	−1.25123	−0.625613	−	0.780134i	$-0.715151\pi$
$887$	−33053.8	−1.25123	−0.625613	+	0.780134i	$0.715151\pi$
$888$	0	0
$889$	−12867.7	−0.485454
$890$	0	0
$891$	5479.85	0.206040
$892$	0	0
$893$	−4363.19	−0.163503
$894$	0	0
$895$	75554.5	2.82180
$896$	0	0
$897$	28338.8	1.05485
$898$	0	0
$899$	−21264.6	−0.788893
$900$	0	0
$901$	6532.30	0.241534
$902$	0	0
$903$	−11128.6	−0.410118
$904$	0	0
$905$	7682.88	0.282196
$906$	0	0
$907$	−30401.5	−1.11297	−0.556486	−	0.830857i	$-0.687851\pi$
$907$	−30401.5	−1.11297	−0.556486	+	0.830857i	$0.687851\pi$
$908$	0	0
$909$	1858.18	0.0678021
$910$	0	0
$911$	−2902.85	−0.105572	−0.0527858	−	0.998606i	$-0.516810\pi$
$911$	−2902.85	−0.105572	−0.0527858	+	0.998606i	$0.516810\pi$
$912$	0	0
$913$	−1518.95	−0.0550601
$914$	0	0
$915$	−73313.5	−2.64882
$916$	0	0
$917$	−8492.17	−0.305819
$918$	0	0
$919$	1945.42	0.0698299	0.0349149	−	0.999390i	$-0.488884\pi$
$919$	1945.42	0.0698299	0.0349149	+	0.999390i	$0.488884\pi$
$920$	0	0
$921$	−25207.7	−0.901870
$922$	0	0
$923$	−54969.4	−1.96028
$924$	0	0
$925$	31931.3	1.13502
$926$	0	0
$927$	−8039.51	−0.284846
$928$	0	0
$929$	−15919.5	−0.562220	−0.281110	−	0.959676i	$-0.590703\pi$
$929$	−15919.5	−0.562220	−0.281110	+	0.959676i	$0.590703\pi$
$930$	0	0
$931$	−430.528	−0.0151557
$932$	0	0
$933$	27933.1	0.980159
$934$	0	0
$935$	18070.3	0.632046
$936$	0	0
$937$	18286.1	0.637545	0.318772	−	0.947831i	$-0.396729\pi$
$937$	18286.1	0.637545	0.318772	+	0.947831i	$0.396729\pi$
$938$	0	0
$939$	9456.68	0.328655
$940$	0	0
$941$	5748.52	0.199146	0.0995730	−	0.995030i	$-0.468252\pi$
$941$	5748.52	0.199146	0.0995730	+	0.995030i	$0.468252\pi$
$942$	0	0
$943$	16162.1	0.558125
$944$	0	0
$945$	23303.8	0.802195
$946$	0	0
$947$	10097.3	0.346480	0.173240	−	0.984880i	$-0.444576\pi$
$947$	10097.3	0.346480	0.173240	+	0.984880i	$0.444576\pi$
$948$	0	0
$949$	65577.1	2.24312
$950$	0	0
$951$	−6055.58	−0.206483
$952$	0	0
$953$	19926.4	0.677313	0.338656	−	0.940910i	$-0.390028\pi$
$953$	19926.4	0.677313	0.338656	+	0.940910i	$0.390028\pi$
$954$	0	0
$955$	18499.6	0.626839
$956$	0	0
$957$	4479.19	0.151297
$958$	0	0
$959$	−17157.8	−0.577742
$960$	0	0
$961$	25230.0	0.846901
$962$	0	0
$963$	−1555.14	−0.0520390
$964$	0	0
$965$	−66691.0	−2.22472
$966$	0	0
$967$	13778.5	0.458209	0.229105	−	0.973402i	$-0.426420\pi$
$967$	13778.5	0.458209	0.229105	+	0.973402i	$0.426420\pi$
$968$	0	0
$969$	−2958.73	−0.0980889
$970$	0	0
$971$	−6359.76	−0.210190	−0.105095	−	0.994462i	$-0.533515\pi$
$971$	−6359.76	−0.210190	−0.105095	+	0.994462i	$0.533515\pi$
$972$	0	0
$973$	−12395.8	−0.408417
$974$	0	0
$975$	108366.	3.55947
$976$	0	0
$977$	−39891.7	−1.30629	−0.653146	−	0.757232i	$-0.726551\pi$
$977$	−39891.7	−1.30629	−0.653146	+	0.757232i	$0.726551\pi$
$978$	0	0
$979$	−13763.9	−0.449332
$980$	0	0
$981$	−2674.74	−0.0870519
$982$	0	0
$983$	24225.5	0.786035	0.393018	−	0.919531i	$-0.371431\pi$
$983$	24225.5	0.786035	0.393018	+	0.919531i	$0.371431\pi$
$984$	0	0
$985$	65927.1	2.13260
$986$	0	0
$987$	−15613.8	−0.503540
$988$	0	0
$989$	32871.8	1.05689
$990$	0	0
$991$	45365.2	1.45416	0.727081	−	0.686552i	$-0.240877\pi$
$991$	45365.2	1.45416	0.727081	+	0.686552i	$0.240877\pi$
$992$	0	0
$993$	−7510.48	−0.240018
$994$	0	0
$995$	69039.7	2.19970
$996$	0	0
$997$	51909.1	1.64892	0.824462	−	0.565918i	$-0.191478\pi$
$997$	51909.1	1.64892	0.824462	+	0.565918i	$0.191478\pi$
$998$	0	0
$999$	13660.1	0.432620

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1232.4.a.q.1.2		3
4.3	odd	2		154.4.a.i.1.2	✓	3
12.11	even	2		1386.4.a.bc.1.1		3
28.27	even	2		1078.4.a.p.1.2		3
44.43	even	2		1694.4.a.s.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
154.4.a.i.1.2	✓	3	4.3	odd	2
1078.4.a.p.1.2		3	28.27	even	2
1232.4.a.q.1.2		3	1.1	even	1	trivial
1386.4.a.bc.1.1		3	12.11	even	2
1694.4.a.s.1.2		3	44.43	even	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)

\(f(q)\)	\(=\)	\(q-4.49172 q^{3} +21.9122 q^{5} +7.00000 q^{7} -6.82442 q^{9} +O(q^{10})\)	\(q-4.49172 q^{3} +21.9122 q^{5} +7.00000 q^{7} -6.82442 q^{9} -11.0000 q^{11} -67.9319 q^{13} -98.4236 q^{15} -74.9700 q^{17} -8.78628 q^{19} -31.4421 q^{21} +92.8741 q^{23} +355.145 q^{25} +151.930 q^{27} +90.6553 q^{29} -234.566 q^{31} +49.4090 q^{33} +153.385 q^{35} +89.9108 q^{37} +305.131 q^{39} +174.022 q^{41} +353.940 q^{43} -149.538 q^{45} +496.591 q^{47} +49.0000 q^{49} +336.745 q^{51} -87.1321 q^{53} -241.034 q^{55} +39.4655 q^{57} -407.713 q^{59} +744.878 q^{61} -47.7709 q^{63} -1488.54 q^{65} -395.095 q^{67} -417.165 q^{69} +809.184 q^{71} -965.337 q^{73} -1595.21 q^{75} -77.0000 q^{77} +647.685 q^{79} -498.168 q^{81} +138.086 q^{83} -1642.76 q^{85} -407.199 q^{87} +1251.26 q^{89} -475.523 q^{91} +1053.60 q^{93} -192.527 q^{95} +818.367 q^{97} +75.0686 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 6 q^{3} + 26 q^{5} + 21 q^{7} + 59 q^{9}+O(q^{10}) \) 3 * q - 6 * q^3 + 26 * q^5 + 21 * q^7 + 59 * q^9	\( 3 q - 6 q^{3} + 26 q^{5} + 21 q^{7} + 59 q^{9} - 33 q^{11} + 72 q^{13} - 12 q^{15} + 56 q^{17} + 48 q^{19} - 42 q^{21} + 244 q^{23} + 181 q^{25} - 36 q^{27} + 198 q^{29} - 374 q^{31} + 66 q^{33} + 182 q^{35} + 110 q^{37} + 256 q^{39} - 648 q^{41} + 500 q^{43} - 150 q^{45} + 326 q^{47} + 147 q^{49} + 1096 q^{51} + 206 q^{53} - 286 q^{55} - 984 q^{57} + 266 q^{59} + 308 q^{61} + 413 q^{63} - 1160 q^{65} - 256 q^{67} + 360 q^{69} + 484 q^{71} - 1028 q^{73} - 1098 q^{75} - 231 q^{77} + 1072 q^{79} - 1297 q^{81} - 912 q^{83} - 728 q^{85} + 2660 q^{87} + 3406 q^{89} + 504 q^{91} - 2972 q^{93} - 816 q^{95} + 2042 q^{97} - 649 q^{99}+O(q^{100}) \) 3 * q - 6 * q^3 + 26 * q^5 + 21 * q^7 + 59 * q^9 - 33 * q^11 + 72 * q^13 - 12 * q^15 + 56 * q^17 + 48 * q^19 - 42 * q^21 + 244 * q^23 + 181 * q^25 - 36 * q^27 + 198 * q^29 - 374 * q^31 + 66 * q^33 + 182 * q^35 + 110 * q^37 + 256 * q^39 - 648 * q^41 + 500 * q^43 - 150 * q^45 + 326 * q^47 + 147 * q^49 + 1096 * q^51 + 206 * q^53 - 286 * q^55 - 984 * q^57 + 266 * q^59 + 308 * q^61 + 413 * q^63 - 1160 * q^65 - 256 * q^67 + 360 * q^69 + 484 * q^71 - 1028 * q^73 - 1098 * q^75 - 231 * q^77 + 1072 * q^79 - 1297 * q^81 - 912 * q^83 - 728 * q^85 + 2660 * q^87 + 3406 * q^89 + 504 * q^91 - 2972 * q^93 - 816 * q^95 + 2042 * q^97 - 649 * q^99

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