LMFDB - Embedded newform 1232.4.a.q.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:	$3$
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			1.1
Root			$-3.22881$ 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)

$f(q)$	$=$	$q-8.45761 q^{3} -3.76558 q^{5} +7.00000 q^{7} +44.5312 q^{9} +O(q^{10})$	$q-8.45761 q^{3} -3.76558 q^{5} +7.00000 q^{7} +44.5312 q^{9} -11.0000 q^{11} +66.3055 q^{13} +31.8478 q^{15} +9.79421 q^{17} +92.0424 q^{19} -59.2033 q^{21} +17.7232 q^{23} -110.820 q^{25} -148.272 q^{27} -150.661 q^{29} +198.393 q^{31} +93.0337 q^{33} -26.3591 q^{35} +268.175 q^{37} -560.786 q^{39} -467.722 q^{41} +184.412 q^{43} -167.686 q^{45} -616.089 q^{47} +49.0000 q^{49} -82.8356 q^{51} +38.9725 q^{53} +41.4214 q^{55} -778.459 q^{57} +17.9910 q^{59} +177.206 q^{61} +311.718 q^{63} -249.679 q^{65} +47.0747 q^{67} -149.896 q^{69} +63.4492 q^{71} -787.401 q^{73} +937.276 q^{75} -77.0000 q^{77} +735.738 q^{79} +51.6835 q^{81} -701.349 q^{83} -36.8809 q^{85} +1274.23 q^{87} +1061.71 q^{89} +464.138 q^{91} -1677.93 q^{93} -346.594 q^{95} +850.496 q^{97} -489.843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−8.45761	−1.62767	−0.813834	−	0.581098i	$-0.802623\pi$
$3$	−8.45761	−1.62767	−0.813834	+	0.581098i	$0.802623\pi$
$4$	0	0
$5$	−3.76558	−0.336804	−0.168402	−	0.985718i	$-0.553861\pi$
$5$	−3.76558	−0.336804	−0.168402	+	0.985718i	$0.553861\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	44.5312	1.64930
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	66.3055	1.41460	0.707301	−	0.706913i	$-0.249913\pi$
$13$	66.3055	1.41460	0.707301	+	0.706913i	$0.249913\pi$
$14$	0	0
$15$	31.8478	0.548205
$16$	0	0
$17$	9.79421	0.139732	0.0698660	−	0.997556i	$-0.477743\pi$
$17$	9.79421	0.139732	0.0698660	+	0.997556i	$0.477743\pi$
$18$	0	0
$19$	92.0424	1.11137	0.555684	−	0.831394i	$-0.312457\pi$
$19$	92.0424	1.11137	0.555684	+	0.831394i	$0.312457\pi$
$20$	0	0
$21$	−59.2033	−0.615201
$22$	0	0
$23$	17.7232	0.160675	0.0803377	−	0.996768i	$-0.474400\pi$
$23$	17.7232	0.160675	0.0803377	+	0.996768i	$0.474400\pi$
$24$	0	0
$25$	−110.820	−0.886563
$26$	0	0
$27$	−148.272	−1.05685
$28$	0	0
$29$	−150.661	−0.964725	−0.482363	−	0.875972i	$-0.660221\pi$
$29$	−150.661	−0.964725	−0.482363	+	0.875972i	$0.660221\pi$
$30$	0	0
$31$	198.393	1.14943	0.574716	−	0.818353i	$-0.305112\pi$
$31$	198.393	1.14943	0.574716	+	0.818353i	$0.305112\pi$
$32$	0	0
$33$	93.0337	0.490760
$34$	0	0
$35$	−26.3591	−0.127300
$36$	0	0
$37$	268.175	1.19156	0.595779	−	0.803148i	$-0.296843\pi$
$37$	268.175	1.19156	0.595779	+	0.803148i	$0.296843\pi$
$38$	0	0
$39$	−560.786	−2.30250
$40$	0	0
$41$	−467.722	−1.78161	−0.890803	−	0.454389i	$-0.849858\pi$
$41$	−467.722	−1.78161	−0.890803	+	0.454389i	$0.849858\pi$
$42$	0	0
$43$	184.412	0.654012	0.327006	−	0.945022i	$-0.393960\pi$
$43$	184.412	0.654012	0.327006	+	0.945022i	$0.393960\pi$
$44$	0	0
$45$	−167.686	−0.555492
$46$	0	0
$47$	−616.089	−1.91204	−0.956019	−	0.293305i	$-0.905245\pi$
$47$	−616.089	−1.91204	−0.956019	+	0.293305i	$0.905245\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−82.8356	−0.227437
$52$	0	0
$53$	38.9725	0.101005	0.0505027	−	0.998724i	$-0.483918\pi$
$53$	38.9725	0.101005	0.0505027	+	0.998724i	$0.483918\pi$
$54$	0	0
$55$	41.4214	0.101550
$56$	0	0
$57$	−778.459	−1.80894
$58$	0	0
$59$	17.9910	0.0396988	0.0198494	−	0.999803i	$-0.493681\pi$
$59$	17.9910	0.0396988	0.0198494	+	0.999803i	$0.493681\pi$
$60$	0	0
$61$	177.206	0.371949	0.185975	−	0.982555i	$-0.440456\pi$
$61$	177.206	0.371949	0.185975	+	0.982555i	$0.440456\pi$
$62$	0	0
$63$	311.718	0.623378
$64$	0	0
$65$	−249.679	−0.476444
$66$	0	0
$67$	47.0747	0.0858372	0.0429186	−	0.999079i	$-0.486334\pi$
$67$	47.0747	0.0858372	0.0429186	+	0.999079i	$0.486334\pi$
$68$	0	0
$69$	−149.896	−0.261526
$70$	0	0
$71$	63.4492	0.106057	0.0530284	−	0.998593i	$-0.483113\pi$
$71$	63.4492	0.106057	0.0530284	+	0.998593i	$0.483113\pi$
$72$	0	0
$73$	−787.401	−1.26244	−0.631221	−	0.775603i	$-0.717446\pi$
$73$	−787.401	−1.26244	−0.631221	+	0.775603i	$0.717446\pi$
$74$	0	0
$75$	937.276	1.44303
$76$	0	0
$77$	−77.0000	−0.113961
$78$	0	0
$79$	735.738	1.04781	0.523905	−	0.851776i	$-0.324475\pi$
$79$	735.738	1.04781	0.523905	+	0.851776i	$0.324475\pi$
$80$	0	0
$81$	51.6835	0.0708964
$82$	0	0
$83$	−701.349	−0.927507	−0.463753	−	0.885964i	$-0.653498\pi$
$83$	−701.349	−0.927507	−0.463753	+	0.885964i	$0.653498\pi$
$84$	0	0
$85$	−36.8809	−0.0470623
$86$	0	0
$87$	1274.23	1.57025
$88$	0	0
$89$	1061.71	1.26450	0.632250	−	0.774764i	$-0.282131\pi$
$89$	1061.71	1.26450	0.632250	+	0.774764i	$0.282131\pi$
$90$	0	0
$91$	464.138	0.534669
$92$	0	0
$93$	−1677.93	−1.87089
$94$	0	0
$95$	−346.594	−0.374313
$96$	0	0
$97$	850.496	0.890256	0.445128	−	0.895467i	$-0.353158\pi$
$97$	850.496	0.890256	0.445128	+	0.895467i	$0.353158\pi$
$98$	0	0
$99$	−489.843	−0.497283
$100$	0	0
$101$	417.727	0.411538	0.205769	−	0.978601i	$-0.434030\pi$
$101$	417.727	0.411538	0.205769	+	0.978601i	$0.434030\pi$
$102$	0	0
$103$	−1949.59	−1.86504	−0.932521	−	0.361116i	$-0.882396\pi$
$103$	−1949.59	−1.86504	−0.932521	+	0.361116i	$0.882396\pi$
$104$	0	0
$105$	222.935	0.207202
$106$	0	0
$107$	547.307	0.494487	0.247244	−	0.968953i	$-0.420475\pi$
$107$	547.307	0.494487	0.247244	+	0.968953i	$0.420475\pi$
$108$	0	0
$109$	630.292	0.553863	0.276931	−	0.960890i	$-0.410683\pi$
$109$	630.292	0.553863	0.276931	+	0.960890i	$0.410683\pi$
$110$	0	0
$111$	−2268.12	−1.93946
$112$	0	0
$113$	−823.960	−0.685944	−0.342972	−	0.939346i	$-0.611434\pi$
$113$	−823.960	−0.685944	−0.342972	+	0.939346i	$0.611434\pi$
$114$	0	0
$115$	−66.7381	−0.0541162
$116$	0	0
$117$	2952.66	2.33311
$118$	0	0
$119$	68.5595	0.0528138
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	3955.81	2.89986
$124$	0	0
$125$	888.002	0.635402
$126$	0	0
$127$	1064.02	0.743440	0.371720	−	0.928345i	$-0.378768\pi$
$127$	1064.02	0.743440	0.371720	+	0.928345i	$0.378768\pi$
$128$	0	0
$129$	−1559.68	−1.06451
$130$	0	0
$131$	1145.55	0.764025	0.382013	−	0.924157i	$-0.375231\pi$
$131$	1145.55	0.764025	0.382013	+	0.924157i	$0.375231\pi$
$132$	0	0
$133$	644.297	0.420057
$134$	0	0
$135$	558.330	0.355951
$136$	0	0
$137$	1391.42	0.867713	0.433857	−	0.900982i	$-0.357152\pi$
$137$	1391.42	0.867713	0.433857	+	0.900982i	$0.357152\pi$
$138$	0	0
$139$	−2475.27	−1.51043	−0.755215	−	0.655478i	$-0.772467\pi$
$139$	−2475.27	−1.51043	−0.755215	+	0.655478i	$0.772467\pi$
$140$	0	0
$141$	5210.64	3.11216
$142$	0	0
$143$	−729.360	−0.426519
$144$	0	0
$145$	567.326	0.324923
$146$	0	0
$147$	−414.423	−0.232524
$148$	0	0
$149$	2097.08	1.15302	0.576508	−	0.817092i	$-0.304415\pi$
$149$	2097.08	1.15302	0.576508	+	0.817092i	$0.304415\pi$
$150$	0	0
$151$	3460.50	1.86498	0.932489	−	0.361198i	$-0.117632\pi$
$151$	3460.50	1.86498	0.932489	+	0.361198i	$0.117632\pi$
$152$	0	0
$153$	436.148	0.230460
$154$	0	0
$155$	−747.065	−0.387134
$156$	0	0
$157$	3025.39	1.53791	0.768956	−	0.639301i	$-0.220776\pi$
$157$	3025.39	1.53791	0.768956	+	0.639301i	$0.220776\pi$
$158$	0	0
$159$	−329.614	−0.164403
$160$	0	0
$161$	124.062	0.0607296
$162$	0	0
$163$	341.089	0.163903	0.0819514	−	0.996636i	$-0.473885\pi$
$163$	341.089	0.163903	0.0819514	+	0.996636i	$0.473885\pi$
$164$	0	0
$165$	−350.326	−0.165290
$166$	0	0
$167$	−3592.71	−1.66474	−0.832372	−	0.554218i	$-0.813018\pi$
$167$	−3592.71	−1.66474	−0.832372	+	0.554218i	$0.813018\pi$
$168$	0	0
$169$	2199.41	1.00110
$170$	0	0
$171$	4098.76	1.83298
$172$	0	0
$173$	625.667	0.274963	0.137482	−	0.990504i	$-0.456099\pi$
$173$	625.667	0.274963	0.137482	+	0.990504i	$0.456099\pi$
$174$	0	0
$175$	−775.743	−0.335089
$176$	0	0
$177$	−152.161	−0.0646164
$178$	0	0
$179$	771.642	0.322208	0.161104	−	0.986937i	$-0.448495\pi$
$179$	771.642	0.322208	0.161104	+	0.986937i	$0.448495\pi$
$180$	0	0
$181$	−1683.70	−0.691427	−0.345714	−	0.938340i	$-0.612363\pi$
$181$	−1683.70	−0.691427	−0.345714	+	0.938340i	$0.612363\pi$
$182$	0	0
$183$	−1498.74	−0.605410
$184$	0	0
$185$	−1009.83	−0.401322
$186$	0	0
$187$	−107.736	−0.0421308
$188$	0	0
$189$	−1037.90	−0.399451
$190$	0	0
$191$	1282.51	0.485861	0.242931	−	0.970044i	$-0.421891\pi$
$191$	1282.51	0.485861	0.242931	+	0.970044i	$0.421891\pi$
$192$	0	0
$193$	−4060.17	−1.51429	−0.757145	−	0.653247i	$-0.773406\pi$
$193$	−4060.17	−1.51429	−0.757145	+	0.653247i	$0.773406\pi$
$194$	0	0
$195$	2111.69	0.775492
$196$	0	0
$197$	3811.26	1.37838	0.689190	−	0.724580i	$-0.257966\pi$
$197$	3811.26	1.37838	0.689190	+	0.724580i	$0.257966\pi$
$198$	0	0
$199$	3427.62	1.22099	0.610497	−	0.792019i	$-0.290970\pi$
$199$	3427.62	1.22099	0.610497	+	0.792019i	$0.290970\pi$
$200$	0	0
$201$	−398.140	−0.139714
$202$	0	0
$203$	−1054.63	−0.364632
$204$	0	0
$205$	1761.25	0.600053
$206$	0	0
$207$	789.233	0.265003
$208$	0	0
$209$	−1012.47	−0.335090
$210$	0	0
$211$	4476.05	1.46040	0.730199	−	0.683235i	$-0.239427\pi$
$211$	4476.05	1.46040	0.730199	+	0.683235i	$0.239427\pi$
$212$	0	0
$213$	−536.628	−0.172625
$214$	0	0
$215$	−694.417	−0.220274
$216$	0	0
$217$	1388.75	0.434445
$218$	0	0
$219$	6659.53	2.05484
$220$	0	0
$221$	649.410	0.197665
$222$	0	0
$223$	3392.64	1.01878	0.509390	−	0.860536i	$-0.329871\pi$
$223$	3392.64	1.01878	0.509390	+	0.860536i	$0.329871\pi$
$224$	0	0
$225$	−4934.96	−1.46221
$226$	0	0
$227$	−3617.46	−1.05771	−0.528853	−	0.848713i	$-0.677378\pi$
$227$	−3617.46	−1.05771	−0.528853	+	0.848713i	$0.677378\pi$
$228$	0	0
$229$	751.458	0.216846	0.108423	−	0.994105i	$-0.465420\pi$
$229$	751.458	0.216846	0.108423	+	0.994105i	$0.465420\pi$
$230$	0	0
$231$	651.236	0.185490
$232$	0	0
$233$	5212.22	1.46551	0.732755	−	0.680492i	$-0.238234\pi$
$233$	5212.22	1.46551	0.732755	+	0.680492i	$0.238234\pi$
$234$	0	0
$235$	2319.93	0.643982
$236$	0	0
$237$	−6222.59	−1.70549
$238$	0	0
$239$	1798.87	0.486859	0.243429	−	0.969919i	$-0.421728\pi$
$239$	1798.87	0.486859	0.243429	+	0.969919i	$0.421728\pi$
$240$	0	0
$241$	−2390.97	−0.639071	−0.319536	−	0.947574i	$-0.603527\pi$
$241$	−2390.97	−0.639071	−0.319536	+	0.947574i	$0.603527\pi$
$242$	0	0
$243$	3566.22	0.941453
$244$	0	0
$245$	−184.514	−0.0481149
$246$	0	0
$247$	6102.91	1.57214
$248$	0	0
$249$	5931.74	1.50967
$250$	0	0
$251$	−4722.30	−1.18753	−0.593763	−	0.804640i	$-0.702359\pi$
$251$	−4722.30	−1.18753	−0.593763	+	0.804640i	$0.702359\pi$
$252$	0	0
$253$	−194.955	−0.0484455
$254$	0	0
$255$	311.925	0.0766019
$256$	0	0
$257$	2129.08	0.516764	0.258382	−	0.966043i	$-0.416811\pi$
$257$	2129.08	0.516764	0.258382	+	0.966043i	$0.416811\pi$
$258$	0	0
$259$	1877.22	0.450367
$260$	0	0
$261$	−6709.11	−1.59112
$262$	0	0
$263$	−1423.01	−0.333636	−0.166818	−	0.985988i	$-0.553349\pi$
$263$	−1423.01	−0.333636	−0.166818	+	0.985988i	$0.553349\pi$
$264$	0	0
$265$	−146.754	−0.0340190
$266$	0	0
$267$	−8979.49	−2.05819
$268$	0	0
$269$	−8139.46	−1.84488	−0.922438	−	0.386144i	$-0.873807\pi$
$269$	−8139.46	−1.84488	−0.922438	+	0.386144i	$0.873807\pi$
$270$	0	0
$271$	−483.799	−0.108445	−0.0542227	−	0.998529i	$-0.517268\pi$
$271$	−483.799	−0.108445	−0.0542227	+	0.998529i	$0.517268\pi$
$272$	0	0
$273$	−3925.50	−0.870264
$274$	0	0
$275$	1219.02	0.267309
$276$	0	0
$277$	5991.53	1.29963	0.649813	−	0.760094i	$-0.274847\pi$
$277$	5991.53	1.29963	0.649813	+	0.760094i	$0.274847\pi$
$278$	0	0
$279$	8834.67	1.89576
$280$	0	0
$281$	295.741	0.0627844	0.0313922	−	0.999507i	$-0.490006\pi$
$281$	295.741	0.0627844	0.0313922	+	0.999507i	$0.490006\pi$
$282$	0	0
$283$	−989.383	−0.207819	−0.103909	−	0.994587i	$-0.533135\pi$
$283$	−989.383	−0.207819	−0.103909	+	0.994587i	$0.533135\pi$
$284$	0	0
$285$	2931.35	0.609257
$286$	0	0
$287$	−3274.05	−0.673384
$288$	0	0
$289$	−4817.07	−0.980475
$290$	0	0
$291$	−7193.17	−1.44904
$292$	0	0
$293$	5757.40	1.14795	0.573977	−	0.818871i	$-0.305400\pi$
$293$	5757.40	1.14795	0.573977	+	0.818871i	$0.305400\pi$
$294$	0	0
$295$	−67.7466	−0.0133707
$296$	0	0
$297$	1630.99	0.318652
$298$	0	0
$299$	1175.14	0.227292
$300$	0	0
$301$	1290.88	0.247193
$302$	0	0
$303$	−3532.97	−0.669847
$304$	0	0
$305$	−667.285	−0.125274
$306$	0	0
$307$	1969.13	0.366072	0.183036	−	0.983106i	$-0.441407\pi$
$307$	1969.13	0.366072	0.183036	+	0.983106i	$0.441407\pi$
$308$	0	0
$309$	16488.9	3.03567
$310$	0	0
$311$	−7512.95	−1.36984	−0.684920	−	0.728618i	$-0.740163\pi$
$311$	−7512.95	−1.36984	−0.684920	+	0.728618i	$0.740163\pi$
$312$	0	0
$313$	3980.96	0.718904	0.359452	−	0.933164i	$-0.382964\pi$
$313$	3980.96	0.718904	0.359452	+	0.933164i	$0.382964\pi$
$314$	0	0
$315$	−1173.80	−0.209956
$316$	0	0
$317$	10298.5	1.82467	0.912337	−	0.409441i	$-0.134276\pi$
$317$	10298.5	1.82467	0.912337	+	0.409441i	$0.134276\pi$
$318$	0	0
$319$	1657.27	0.290876
$320$	0	0
$321$	−4628.91	−0.804861
$322$	0	0
$323$	901.483	0.155294
$324$	0	0
$325$	−7348.00	−1.25413
$326$	0	0
$327$	−5330.77	−0.901505
$328$	0	0
$329$	−4312.62	−0.722682
$330$	0	0
$331$	−11025.7	−1.83089	−0.915446	−	0.402441i	$-0.868162\pi$
$331$	−11025.7	−1.83089	−0.915446	+	0.402441i	$0.868162\pi$
$332$	0	0
$333$	11942.1	1.96524
$334$	0	0
$335$	−177.264	−0.0289103
$336$	0	0
$337$	−6558.43	−1.06012	−0.530060	−	0.847960i	$-0.677831\pi$
$337$	−6558.43	−1.06012	−0.530060	+	0.847960i	$0.677831\pi$
$338$	0	0
$339$	6968.73	1.11649
$340$	0	0
$341$	−2182.32	−0.346567
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	564.445	0.0880831
$346$	0	0
$347$	10897.7	1.68593	0.842967	−	0.537965i	$-0.180807\pi$
$347$	10897.7	1.68593	0.842967	+	0.537965i	$0.180807\pi$
$348$	0	0
$349$	1340.02	0.205528	0.102764	−	0.994706i	$-0.467231\pi$
$349$	1340.02	0.205528	0.102764	+	0.994706i	$0.467231\pi$
$350$	0	0
$351$	−9831.23	−1.49502
$352$	0	0
$353$	3879.04	0.584874	0.292437	−	0.956285i	$-0.405534\pi$
$353$	3879.04	0.584874	0.292437	+	0.956285i	$0.405534\pi$
$354$	0	0
$355$	−238.923	−0.0357204
$356$	0	0
$357$	−579.849	−0.0859633
$358$	0	0
$359$	−1081.83	−0.159044	−0.0795221	−	0.996833i	$-0.525339\pi$
$359$	−1081.83	−0.159044	−0.0795221	+	0.996833i	$0.525339\pi$
$360$	0	0
$361$	1612.81	0.235137
$362$	0	0
$363$	−1023.37	−0.147970
$364$	0	0
$365$	2965.02	0.425196
$366$	0	0
$367$	10847.4	1.54286	0.771428	−	0.636317i	$-0.219543\pi$
$367$	10847.4	1.54286	0.771428	+	0.636317i	$0.219543\pi$
$368$	0	0
$369$	−20828.2	−2.93841
$370$	0	0
$371$	272.807	0.0381764
$372$	0	0
$373$	−10510.1	−1.45896	−0.729480	−	0.684003i	$-0.760238\pi$
$373$	−10510.1	−1.45896	−0.729480	+	0.684003i	$0.760238\pi$
$374$	0	0
$375$	−7510.37	−1.03422
$376$	0	0
$377$	−9989.64	−1.36470
$378$	0	0
$379$	−10057.8	−1.36315	−0.681575	−	0.731749i	$-0.738705\pi$
$379$	−10057.8	−1.36315	−0.681575	+	0.731749i	$0.738705\pi$
$380$	0	0
$381$	−8999.10	−1.21007
$382$	0	0
$383$	8812.40	1.17570	0.587849	−	0.808971i	$-0.299975\pi$
$383$	8812.40	1.17570	0.587849	+	0.808971i	$0.299975\pi$
$384$	0	0
$385$	289.950	0.0383824
$386$	0	0
$387$	8212.06	1.07866
$388$	0	0
$389$	7952.29	1.03650	0.518248	−	0.855230i	$-0.326584\pi$
$389$	7952.29	1.03650	0.518248	+	0.855230i	$0.326584\pi$
$390$	0	0
$391$	173.584	0.0224515
$392$	0	0
$393$	−9688.63	−1.24358
$394$	0	0
$395$	−2770.49	−0.352907
$396$	0	0
$397$	−4886.59	−0.617761	−0.308880	−	0.951101i	$-0.599954\pi$
$397$	−4886.59	−0.617761	−0.308880	+	0.951101i	$0.599954\pi$
$398$	0	0
$399$	−5449.21	−0.683714
$400$	0	0
$401$	666.216	0.0829657	0.0414829	−	0.999139i	$-0.486792\pi$
$401$	666.216	0.0829657	0.0414829	+	0.999139i	$0.486792\pi$
$402$	0	0
$403$	13154.5	1.62599
$404$	0	0
$405$	−194.618	−0.0238782
$406$	0	0
$407$	−2949.92	−0.359268
$408$	0	0
$409$	−10700.8	−1.29370	−0.646850	−	0.762618i	$-0.723914\pi$
$409$	−10700.8	−1.29370	−0.646850	+	0.762618i	$0.723914\pi$
$410$	0	0
$411$	−11768.1	−1.41235
$412$	0	0
$413$	125.937	0.0150047
$414$	0	0
$415$	2640.99	0.312388
$416$	0	0
$417$	20934.9	2.45848
$418$	0	0
$419$	−8197.12	−0.955741	−0.477871	−	0.878430i	$-0.658591\pi$
$419$	−8197.12	−0.955741	−0.477871	+	0.878430i	$0.658591\pi$
$420$	0	0
$421$	−15969.1	−1.84866	−0.924331	−	0.381591i	$-0.875376\pi$
$421$	−15969.1	−1.84866	−0.924331	+	0.381591i	$0.875376\pi$
$422$	0	0
$423$	−27435.1	−3.15353
$424$	0	0
$425$	−1085.40	−0.123881
$426$	0	0
$427$	1240.44	0.140584
$428$	0	0
$429$	6168.64	0.694230
$430$	0	0
$431$	−1792.61	−0.200341	−0.100170	−	0.994970i	$-0.531939\pi$
$431$	−1792.61	−0.200341	−0.100170	+	0.994970i	$0.531939\pi$
$432$	0	0
$433$	4755.42	0.527785	0.263892	−	0.964552i	$-0.414994\pi$
$433$	4755.42	0.527785	0.263892	+	0.964552i	$0.414994\pi$
$434$	0	0
$435$	−4798.23	−0.528867
$436$	0	0
$437$	1631.28	0.178569
$438$	0	0
$439$	14613.2	1.58872	0.794360	−	0.607447i	$-0.207806\pi$
$439$	14613.2	1.58872	0.794360	+	0.607447i	$0.207806\pi$
$440$	0	0
$441$	2182.03	0.235615
$442$	0	0
$443$	17263.5	1.85150	0.925748	−	0.378140i	$-0.123436\pi$
$443$	17263.5	1.85150	0.925748	+	0.378140i	$0.123436\pi$
$444$	0	0
$445$	−3997.94	−0.425889
$446$	0	0
$447$	−17736.3	−1.87673
$448$	0	0
$449$	−13354.5	−1.40365	−0.701825	−	0.712349i	$-0.747631\pi$
$449$	−13354.5	−1.40365	−0.701825	+	0.712349i	$0.747631\pi$
$450$	0	0
$451$	5144.94	0.537175
$452$	0	0
$453$	−29267.6	−3.03556
$454$	0	0
$455$	−1747.75	−0.180079
$456$	0	0
$457$	14639.0	1.49843	0.749214	−	0.662328i	$-0.230432\pi$
$457$	14639.0	1.49843	0.749214	+	0.662328i	$0.230432\pi$
$458$	0	0
$459$	−1452.21	−0.147676
$460$	0	0
$461$	5358.57	0.541374	0.270687	−	0.962667i	$-0.412749\pi$
$461$	5358.57	0.541374	0.270687	+	0.962667i	$0.412749\pi$
$462$	0	0
$463$	−6928.11	−0.695414	−0.347707	−	0.937603i	$-0.613040\pi$
$463$	−6928.11	−0.695414	−0.347707	+	0.937603i	$0.613040\pi$
$464$	0	0
$465$	6318.39	0.630125
$466$	0	0
$467$	5349.51	0.530076	0.265038	−	0.964238i	$-0.414615\pi$
$467$	5349.51	0.530076	0.265038	+	0.964238i	$0.414615\pi$
$468$	0	0
$469$	329.523	0.0324434
$470$	0	0
$471$	−25587.6	−2.50321
$472$	0	0
$473$	−2028.53	−0.197192
$474$	0	0
$475$	−10200.2	−0.985297
$476$	0	0
$477$	1735.49	0.166588
$478$	0	0
$479$	13780.4	1.31449	0.657247	−	0.753675i	$-0.271721\pi$
$479$	13780.4	1.31449	0.657247	+	0.753675i	$0.271721\pi$
$480$	0	0
$481$	17781.4	1.68558
$482$	0	0
$483$	−1049.27	−0.0988477
$484$	0	0
$485$	−3202.62	−0.299842
$486$	0	0
$487$	641.654	0.0597045	0.0298523	−	0.999554i	$-0.490496\pi$
$487$	641.654	0.0597045	0.0298523	+	0.999554i	$0.490496\pi$
$488$	0	0
$489$	−2884.80	−0.266779
$490$	0	0
$491$	8777.54	0.806772	0.403386	−	0.915030i	$-0.367833\pi$
$491$	8777.54	0.806772	0.403386	+	0.915030i	$0.367833\pi$
$492$	0	0
$493$	−1475.61	−0.134803
$494$	0	0
$495$	1844.54	0.167487
$496$	0	0
$497$	444.144	0.0400857
$498$	0	0
$499$	1562.74	0.140196	0.0700979	−	0.997540i	$-0.477669\pi$
$499$	1562.74	0.140196	0.0700979	+	0.997540i	$0.477669\pi$
$500$	0	0
$501$	30385.7	2.70965
$502$	0	0
$503$	−6806.53	−0.603356	−0.301678	−	0.953410i	$-0.597547\pi$
$503$	−6806.53	−0.603356	−0.301678	+	0.953410i	$0.597547\pi$
$504$	0	0
$505$	−1572.98	−0.138608
$506$	0	0
$507$	−18601.8	−1.62946
$508$	0	0
$509$	−12461.4	−1.08515	−0.542574	−	0.840008i	$-0.682550\pi$
$509$	−12461.4	−1.08515	−0.542574	+	0.840008i	$0.682550\pi$
$510$	0	0
$511$	−5511.81	−0.477158
$512$	0	0
$513$	−13647.3	−1.17455
$514$	0	0
$515$	7341.36	0.628154
$516$	0	0
$517$	6776.98	0.576501
$518$	0	0
$519$	−5291.65	−0.447549
$520$	0	0
$521$	17200.0	1.44635	0.723173	−	0.690667i	$-0.242683\pi$
$521$	17200.0	1.44635	0.723173	+	0.690667i	$0.242683\pi$
$522$	0	0
$523$	6285.07	0.525482	0.262741	−	0.964866i	$-0.415374\pi$
$523$	6285.07	0.525482	0.262741	+	0.964866i	$0.415374\pi$
$524$	0	0
$525$	6560.93	0.545414
$526$	0	0
$527$	1943.10	0.160613
$528$	0	0
$529$	−11852.9	−0.974183
$530$	0	0
$531$	801.160	0.0654753
$532$	0	0
$533$	−31012.5	−2.52026
$534$	0	0
$535$	−2060.93	−0.166545
$536$	0	0
$537$	−6526.24	−0.524447
$538$	0	0
$539$	−539.000	−0.0430730
$540$	0	0
$541$	−9363.67	−0.744133	−0.372066	−	0.928206i	$-0.621351\pi$
$541$	−9363.67	−0.744133	−0.372066	+	0.928206i	$0.621351\pi$
$542$	0	0
$543$	14240.1	1.12541
$544$	0	0
$545$	−2373.42	−0.186543
$546$	0	0
$547$	5288.84	0.413408	0.206704	−	0.978403i	$-0.433726\pi$
$547$	5288.84	0.413408	0.206704	+	0.978403i	$0.433726\pi$
$548$	0	0
$549$	7891.19	0.613457
$550$	0	0
$551$	−13867.2	−1.07216
$552$	0	0
$553$	5150.17	0.396035
$554$	0	0
$555$	8540.79	0.653218
$556$	0	0
$557$	7215.91	0.548919	0.274460	−	0.961599i	$-0.411501\pi$
$557$	7215.91	0.548919	0.274460	+	0.961599i	$0.411501\pi$
$558$	0	0
$559$	12227.5	0.925166
$560$	0	0
$561$	911.192	0.0685750
$562$	0	0
$563$	19616.9	1.46848	0.734239	−	0.678891i	$-0.237539\pi$
$563$	19616.9	1.46848	0.734239	+	0.678891i	$0.237539\pi$
$564$	0	0
$565$	3102.69	0.231029
$566$	0	0
$567$	361.784	0.0267963
$568$	0	0
$569$	15765.2	1.16153	0.580767	−	0.814070i	$-0.302753\pi$
$569$	15765.2	1.16153	0.580767	+	0.814070i	$0.302753\pi$
$570$	0	0
$571$	16401.6	1.20208	0.601039	−	0.799220i	$-0.294754\pi$
$571$	16401.6	1.20208	0.601039	+	0.799220i	$0.294754\pi$
$572$	0	0
$573$	−10847.0	−0.790821
$574$	0	0
$575$	−1964.09	−0.142449
$576$	0	0
$577$	26752.5	1.93019	0.965095	−	0.261899i	$-0.0843488\pi$
$577$	26752.5	1.93019	0.965095	+	0.261899i	$0.0843488\pi$
$578$	0	0
$579$	34339.4	2.46476
$580$	0	0
$581$	−4909.44	−0.350565
$582$	0	0
$583$	−428.697	−0.0304542
$584$	0	0
$585$	−11118.5	−0.785800
$586$	0	0
$587$	16635.7	1.16972	0.584862	−	0.811133i	$-0.301149\pi$
$587$	16635.7	1.16972	0.584862	+	0.811133i	$0.301149\pi$
$588$	0	0
$589$	18260.6	1.27744
$590$	0	0
$591$	−32234.1	−2.24355
$592$	0	0
$593$	20625.4	1.42830	0.714150	−	0.699993i	$-0.246813\pi$
$593$	20625.4	1.42830	0.714150	+	0.699993i	$0.246813\pi$
$594$	0	0
$595$	−258.167	−0.0177879
$596$	0	0
$597$	−28989.5	−1.98737
$598$	0	0
$599$	−16354.6	−1.11558	−0.557790	−	0.829982i	$-0.688350\pi$
$599$	−16354.6	−1.11558	−0.557790	+	0.829982i	$0.688350\pi$
$600$	0	0
$601$	17108.4	1.16118	0.580588	−	0.814198i	$-0.302823\pi$
$601$	17108.4	1.16118	0.580588	+	0.814198i	$0.302823\pi$
$602$	0	0
$603$	2096.29	0.141571
$604$	0	0
$605$	−455.636	−0.0306186
$606$	0	0
$607$	−14693.6	−0.982527	−0.491264	−	0.871011i	$-0.663465\pi$
$607$	−14693.6	−0.982527	−0.491264	+	0.871011i	$0.663465\pi$
$608$	0	0
$609$	8919.62	0.593500
$610$	0	0
$611$	−40850.0	−2.70477
$612$	0	0
$613$	19907.8	1.31170	0.655848	−	0.754893i	$-0.272311\pi$
$613$	19907.8	1.31170	0.655848	+	0.754893i	$0.272311\pi$
$614$	0	0
$615$	−14895.9	−0.976686
$616$	0	0
$617$	−18505.7	−1.20747	−0.603736	−	0.797185i	$-0.706322\pi$
$617$	−18505.7	−1.20747	−0.603736	+	0.797185i	$0.706322\pi$
$618$	0	0
$619$	10787.2	0.700443	0.350221	−	0.936667i	$-0.386106\pi$
$619$	10787.2	0.700443	0.350221	+	0.936667i	$0.386106\pi$
$620$	0	0
$621$	−2627.85	−0.169810
$622$	0	0
$623$	7431.94	0.477936
$624$	0	0
$625$	10508.7	0.672557
$626$	0	0
$627$	8563.05	0.545415
$628$	0	0
$629$	2626.56	0.166499
$630$	0	0
$631$	−9110.66	−0.574786	−0.287393	−	0.957813i	$-0.592789\pi$
$631$	−9110.66	−0.574786	−0.287393	+	0.957813i	$0.592789\pi$
$632$	0	0
$633$	−37856.7	−2.37704
$634$	0	0
$635$	−4006.67	−0.250394
$636$	0	0
$637$	3248.97	0.202086
$638$	0	0
$639$	2825.47	0.174920
$640$	0	0
$641$	1077.56	0.0663980	0.0331990	−	0.999449i	$-0.489430\pi$
$641$	1077.56	0.0663980	0.0331990	+	0.999449i	$0.489430\pi$
$642$	0	0
$643$	7098.48	0.435361	0.217680	−	0.976020i	$-0.430151\pi$
$643$	7098.48	0.435361	0.217680	+	0.976020i	$0.430151\pi$
$644$	0	0
$645$	5873.11	0.358533
$646$	0	0
$647$	−10469.7	−0.636179	−0.318089	−	0.948061i	$-0.603041\pi$
$647$	−10469.7	−0.636179	−0.318089	+	0.948061i	$0.603041\pi$
$648$	0	0
$649$	−197.901	−0.0119696
$650$	0	0
$651$	−11745.5	−0.707132
$652$	0	0
$653$	953.510	0.0571420	0.0285710	−	0.999592i	$-0.490904\pi$
$653$	953.510	0.0571420	0.0285710	+	0.999592i	$0.490904\pi$
$654$	0	0
$655$	−4313.67	−0.257327
$656$	0	0
$657$	−35063.9	−2.08215
$658$	0	0
$659$	30548.6	1.80577	0.902887	−	0.429879i	$-0.141444\pi$
$659$	30548.6	1.80577	0.902887	+	0.429879i	$0.141444\pi$
$660$	0	0
$661$	−10484.6	−0.616952	−0.308476	−	0.951232i	$-0.599819\pi$
$661$	−10484.6	−0.616952	−0.308476	+	0.951232i	$0.599819\pi$
$662$	0	0
$663$	−5492.45	−0.321733
$664$	0	0
$665$	−2426.15	−0.141477
$666$	0	0
$667$	−2670.19	−0.155008
$668$	0	0
$669$	−28693.6	−1.65823
$670$	0	0
$671$	−1949.27	−0.112147
$672$	0	0
$673$	7708.88	0.441539	0.220769	−	0.975326i	$-0.429143\pi$
$673$	7708.88	0.441539	0.220769	+	0.975326i	$0.429143\pi$
$674$	0	0
$675$	16431.5	0.936963
$676$	0	0
$677$	−9212.28	−0.522979	−0.261489	−	0.965206i	$-0.584214\pi$
$677$	−9212.28	−0.522979	−0.261489	+	0.965206i	$0.584214\pi$
$678$	0	0
$679$	5953.47	0.336485
$680$	0	0
$681$	30595.1	1.72159
$682$	0	0
$683$	20394.2	1.14255	0.571277	−	0.820758i	$-0.306448\pi$
$683$	20394.2	1.14255	0.571277	+	0.820758i	$0.306448\pi$
$684$	0	0
$685$	−5239.49	−0.292249
$686$	0	0
$687$	−6355.54	−0.352953
$688$	0	0
$689$	2584.09	0.142882
$690$	0	0
$691$	−20835.2	−1.14705	−0.573524	−	0.819189i	$-0.694424\pi$
$691$	−20835.2	−1.14705	−0.573524	+	0.819189i	$0.694424\pi$
$692$	0	0
$693$	−3428.90	−0.187955
$694$	0	0
$695$	9320.84	0.508719
$696$	0	0
$697$	−4580.97	−0.248948
$698$	0	0
$699$	−44082.9	−2.38536
$700$	0	0
$701$	−17737.6	−0.955691	−0.477845	−	0.878444i	$-0.658582\pi$
$701$	−17737.6	−0.955691	−0.477845	+	0.878444i	$0.658582\pi$
$702$	0	0
$703$	24683.4	1.32426
$704$	0	0
$705$	−19621.1	−1.04819
$706$	0	0
$707$	2924.09	0.155547
$708$	0	0
$709$	17372.9	0.920242	0.460121	−	0.887856i	$-0.347806\pi$
$709$	17372.9	0.920242	0.460121	+	0.887856i	$0.347806\pi$
$710$	0	0
$711$	32763.3	1.72816
$712$	0	0
$713$	3516.15	0.184686
$714$	0	0
$715$	2746.47	0.143653
$716$	0	0
$717$	−15214.1	−0.792444
$718$	0	0
$719$	−1903.28	−0.0987211	−0.0493606	−	0.998781i	$-0.515718\pi$
$719$	−1903.28	−0.0987211	−0.0493606	+	0.998781i	$0.515718\pi$
$720$	0	0
$721$	−13647.2	−0.704919
$722$	0	0
$723$	20221.9	1.04020
$724$	0	0
$725$	16696.3	0.855290
$726$	0	0
$727$	−15929.5	−0.812643	−0.406322	−	0.913730i	$-0.633189\pi$
$727$	−15929.5	−0.812643	−0.406322	+	0.913730i	$0.633189\pi$
$728$	0	0
$729$	−31557.2	−1.60327
$730$	0	0
$731$	1806.17	0.0913864
$732$	0	0
$733$	−3099.88	−0.156203	−0.0781013	−	0.996945i	$-0.524886\pi$
$733$	−3099.88	−0.156203	−0.0781013	+	0.996945i	$0.524886\pi$
$734$	0	0
$735$	1560.54	0.0783150
$736$	0	0
$737$	−517.822	−0.0258809
$738$	0	0
$739$	−29280.0	−1.45749	−0.728743	−	0.684787i	$-0.759895\pi$
$739$	−29280.0	−1.45749	−0.728743	+	0.684787i	$0.759895\pi$
$740$	0	0
$741$	−51616.1	−2.55893
$742$	0	0
$743$	16811.4	0.830083	0.415041	−	0.909803i	$-0.363767\pi$
$743$	16811.4	0.830083	0.415041	+	0.909803i	$0.363767\pi$
$744$	0	0
$745$	−7896.73	−0.388341
$746$	0	0
$747$	−31231.9	−1.52974
$748$	0	0
$749$	3831.15	0.186899
$750$	0	0
$751$	16206.1	0.787442	0.393721	−	0.919230i	$-0.371188\pi$
$751$	16206.1	0.787442	0.393721	+	0.919230i	$0.371188\pi$
$752$	0	0
$753$	39939.4	1.93290
$754$	0	0
$755$	−13030.8	−0.628132
$756$	0	0
$757$	29338.2	1.40861	0.704303	−	0.709900i	$-0.251260\pi$
$757$	29338.2	1.40861	0.704303	+	0.709900i	$0.251260\pi$
$758$	0	0
$759$	1648.85	0.0788532
$760$	0	0
$761$	19500.4	0.928893	0.464447	−	0.885601i	$-0.346253\pi$
$761$	19500.4	0.928893	0.464447	+	0.885601i	$0.346253\pi$
$762$	0	0
$763$	4412.04	0.209340
$764$	0	0
$765$	−1642.35	−0.0776200
$766$	0	0
$767$	1192.90	0.0561579
$768$	0	0
$769$	19718.8	0.924680	0.462340	−	0.886703i	$-0.347010\pi$
$769$	19718.8	0.924680	0.462340	+	0.886703i	$0.347010\pi$
$770$	0	0
$771$	−18006.9	−0.841120
$772$	0	0
$773$	−18493.1	−0.860481	−0.430240	−	0.902714i	$-0.641571\pi$
$773$	−18493.1	−0.860481	−0.430240	+	0.902714i	$0.641571\pi$
$774$	0	0
$775$	−21986.0	−1.01904
$776$	0	0
$777$	−15876.8	−0.733047
$778$	0	0
$779$	−43050.2	−1.98002
$780$	0	0
$781$	−697.941	−0.0319773
$782$	0	0
$783$	22338.8	1.01957
$784$	0	0
$785$	−11392.4	−0.517975
$786$	0	0
$787$	30418.9	1.37779	0.688893	−	0.724863i	$-0.258097\pi$
$787$	30418.9	1.37779	0.688893	+	0.724863i	$0.258097\pi$
$788$	0	0
$789$	12035.2	0.543049
$790$	0	0
$791$	−5767.72	−0.259262
$792$	0	0
$793$	11749.7	0.526160
$794$	0	0
$795$	1241.19	0.0553716
$796$	0	0
$797$	−33956.0	−1.50914	−0.754569	−	0.656221i	$-0.772154\pi$
$797$	−33956.0	−1.50914	−0.754569	+	0.656221i	$0.772154\pi$
$798$	0	0
$799$	−6034.10	−0.267173
$800$	0	0
$801$	47279.0	2.08554
$802$	0	0
$803$	8661.41	0.380641
$804$	0	0
$805$	−467.167	−0.0204540
$806$	0	0
$807$	68840.4	3.00285
$808$	0	0
$809$	13051.3	0.567192	0.283596	−	0.958944i	$-0.408473\pi$
$809$	13051.3	0.567192	0.283596	+	0.958944i	$0.408473\pi$
$810$	0	0
$811$	16452.7	0.712369	0.356185	−	0.934416i	$-0.384077\pi$
$811$	16452.7	0.712369	0.356185	+	0.934416i	$0.384077\pi$
$812$	0	0
$813$	4091.79	0.176513
$814$	0	0
$815$	−1284.40	−0.0552031
$816$	0	0
$817$	16973.7	0.726847
$818$	0	0
$819$	20668.6	0.881831
$820$	0	0
$821$	36774.2	1.56325	0.781626	−	0.623748i	$-0.214391\pi$
$821$	36774.2	1.56325	0.781626	+	0.623748i	$0.214391\pi$
$822$	0	0
$823$	−7431.73	−0.314768	−0.157384	−	0.987538i	$-0.550306\pi$
$823$	−7431.73	−0.314768	−0.157384	+	0.987538i	$0.550306\pi$
$824$	0	0
$825$	−10310.0	−0.435090
$826$	0	0
$827$	−37702.2	−1.58529	−0.792644	−	0.609685i	$-0.791296\pi$
$827$	−37702.2	−1.58529	−0.792644	+	0.609685i	$0.791296\pi$
$828$	0	0
$829$	−368.015	−0.0154182	−0.00770910	−	0.999970i	$-0.502454\pi$
$829$	−368.015	−0.0154182	−0.00770910	+	0.999970i	$0.502454\pi$
$830$	0	0
$831$	−50674.0	−2.11536
$832$	0	0
$833$	479.916	0.0199617
$834$	0	0
$835$	13528.7	0.560693
$836$	0	0
$837$	−29416.1	−1.21478
$838$	0	0
$839$	18824.1	0.774591	0.387295	−	0.921956i	$-0.373409\pi$
$839$	18824.1	0.774591	0.387295	+	0.921956i	$0.373409\pi$
$840$	0	0
$841$	−1690.28	−0.0693052
$842$	0	0
$843$	−2501.26	−0.102192
$844$	0	0
$845$	−8282.08	−0.337174
$846$	0	0
$847$	847.000	0.0343604
$848$	0	0
$849$	8367.82	0.338260
$850$	0	0
$851$	4752.91	0.191454
$852$	0	0
$853$	11655.2	0.467839	0.233919	−	0.972256i	$-0.424845\pi$
$853$	11655.2	0.467839	0.233919	+	0.972256i	$0.424845\pi$
$854$	0	0
$855$	−15434.2	−0.617356
$856$	0	0
$857$	−11513.4	−0.458916	−0.229458	−	0.973319i	$-0.573695\pi$
$857$	−11513.4	−0.458916	−0.229458	+	0.973319i	$0.573695\pi$
$858$	0	0
$859$	4463.89	0.177306	0.0886532	−	0.996063i	$-0.471744\pi$
$859$	4463.89	0.177306	0.0886532	+	0.996063i	$0.471744\pi$
$860$	0	0
$861$	27690.7	1.09605
$862$	0	0
$863$	27841.8	1.09820	0.549099	−	0.835757i	$-0.314971\pi$
$863$	27841.8	1.09820	0.549099	+	0.835757i	$0.314971\pi$
$864$	0	0
$865$	−2356.00	−0.0926087
$866$	0	0
$867$	40740.9	1.59589
$868$	0	0
$869$	−8093.12	−0.315927
$870$	0	0
$871$	3121.31	0.121425
$872$	0	0
$873$	37873.6	1.46830
$874$	0	0
$875$	6216.01	0.240159
$876$	0	0
$877$	−26754.3	−1.03014	−0.515068	−	0.857149i	$-0.672233\pi$
$877$	−26754.3	−1.03014	−0.515068	+	0.857149i	$0.672233\pi$
$878$	0	0
$879$	−48693.8	−1.86849
$880$	0	0
$881$	−30728.2	−1.17509	−0.587547	−	0.809190i	$-0.699906\pi$
$881$	−30728.2	−1.17509	−0.587547	+	0.809190i	$0.699906\pi$
$882$	0	0
$883$	19018.0	0.724808	0.362404	−	0.932021i	$-0.381956\pi$
$883$	19018.0	0.724808	0.362404	+	0.932021i	$0.381956\pi$
$884$	0	0
$885$	572.974	0.0217631
$886$	0	0
$887$	−16405.4	−0.621013	−0.310507	−	0.950571i	$-0.600499\pi$
$887$	−16405.4	−0.621013	−0.310507	+	0.950571i	$0.600499\pi$
$888$	0	0
$889$	7448.17	0.280994
$890$	0	0
$891$	−568.518	−0.0213761
$892$	0	0
$893$	−56706.3	−2.12498
$894$	0	0
$895$	−2905.68	−0.108521
$896$	0	0
$897$	−9938.90	−0.369956
$898$	0	0
$899$	−29890.1	−1.10889
$900$	0	0
$901$	381.705	0.0141137
$902$	0	0
$903$	−10917.8	−0.402348
$904$	0	0
$905$	6340.11	0.232876
$906$	0	0
$907$	−11723.3	−0.429179	−0.214590	−	0.976704i	$-0.568841\pi$
$907$	−11723.3	−0.429179	−0.214590	+	0.976704i	$0.568841\pi$
$908$	0	0
$909$	18601.9	0.678751
$910$	0	0
$911$	8342.82	0.303414	0.151707	−	0.988426i	$-0.451523\pi$
$911$	8342.82	0.303414	0.151707	+	0.988426i	$0.451523\pi$
$912$	0	0
$913$	7714.84	0.279654
$914$	0	0
$915$	5643.63	0.203905
$916$	0	0
$917$	8018.86	0.288774
$918$	0	0
$919$	−47105.5	−1.69082	−0.845411	−	0.534116i	$-0.820645\pi$
$919$	−47105.5	−1.69082	−0.845411	+	0.534116i	$0.820645\pi$
$920$	0	0
$921$	−16654.1	−0.595844
$922$	0	0
$923$	4207.03	0.150028
$924$	0	0
$925$	−29719.2	−1.05639
$926$	0	0
$927$	−86817.7	−3.07602
$928$	0	0
$929$	12164.4	0.429603	0.214801	−	0.976658i	$-0.431090\pi$
$929$	12164.4	0.429603	0.214801	+	0.976658i	$0.431090\pi$
$930$	0	0
$931$	4510.08	0.158767
$932$	0	0
$933$	63541.6	2.22964
$934$	0	0
$935$	405.690	0.0141898
$936$	0	0
$937$	−49355.4	−1.72078	−0.860390	−	0.509636i	$-0.829780\pi$
$937$	−49355.4	−1.72078	−0.860390	+	0.509636i	$0.829780\pi$
$938$	0	0
$939$	−33669.4	−1.17014
$940$	0	0
$941$	21129.9	0.732003	0.366001	−	0.930614i	$-0.380727\pi$
$941$	21129.9	0.732003	0.366001	+	0.930614i	$0.380727\pi$
$942$	0	0
$943$	−8289.51	−0.286261
$944$	0	0
$945$	3908.31	0.134537
$946$	0	0
$947$	8292.15	0.284539	0.142270	−	0.989828i	$-0.454560\pi$
$947$	8292.15	0.284539	0.142270	+	0.989828i	$0.454560\pi$
$948$	0	0
$949$	−52209.0	−1.78585
$950$	0	0
$951$	−87100.7	−2.96996
$952$	0	0
$953$	42775.4	1.45397	0.726984	−	0.686655i	$-0.240922\pi$
$953$	42775.4	1.45397	0.726984	+	0.686655i	$0.240922\pi$
$954$	0	0
$955$	−4829.42	−0.163640
$956$	0	0
$957$	−14016.5	−0.473449
$958$	0	0
$959$	9739.91	0.327965
$960$	0	0
$961$	9568.72	0.321195
$962$	0	0
$963$	24372.2	0.815559
$964$	0	0
$965$	15288.9	0.510019
$966$	0	0
$967$	−5274.27	−0.175397	−0.0876986	−	0.996147i	$-0.527951\pi$
$967$	−5274.27	−0.175397	−0.0876986	+	0.996147i	$0.527951\pi$
$968$	0	0
$969$	−7624.39	−0.252766
$970$	0	0
$971$	9409.86	0.310996	0.155498	−	0.987836i	$-0.450302\pi$
$971$	9409.86	0.310996	0.155498	+	0.987836i	$0.450302\pi$
$972$	0	0
$973$	−17326.9	−0.570889
$974$	0	0
$975$	62146.5	2.04131
$976$	0	0
$977$	−13380.0	−0.438140	−0.219070	−	0.975709i	$-0.570302\pi$
$977$	−13380.0	−0.438140	−0.219070	+	0.975709i	$0.570302\pi$
$978$	0	0
$979$	−11678.8	−0.381261
$980$	0	0
$981$	28067.6	0.913487
$982$	0	0
$983$	−43791.6	−1.42089	−0.710444	−	0.703753i	$-0.751506\pi$
$983$	−43791.6	−1.42089	−0.710444	+	0.703753i	$0.751506\pi$
$984$	0	0
$985$	−14351.6	−0.464244
$986$	0	0
$987$	36474.5	1.17629
$988$	0	0
$989$	3268.36	0.105084
$990$	0	0
$991$	52705.7	1.68946	0.844729	−	0.535195i	$-0.179762\pi$
$991$	52705.7	1.68946	0.844729	+	0.535195i	$0.179762\pi$
$992$	0	0
$993$	93250.7	2.98008
$994$	0	0
$995$	−12907.0	−0.411236
$996$	0	0
$997$	33356.3	1.05958	0.529792	−	0.848128i	$-0.322270\pi$
$997$	33356.3	1.05958	0.529792	+	0.848128i	$0.322270\pi$
$998$	0	0
$999$	−39762.7	−1.25930

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1232.4.a.q.1.1		3
4.3	odd	2		154.4.a.i.1.3	✓	3
12.11	even	2		1386.4.a.bc.1.3		3
28.27	even	2		1078.4.a.p.1.1		3
44.43	even	2		1694.4.a.s.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
154.4.a.i.1.3	✓	3	4.3	odd	2
1078.4.a.p.1.1		3	28.27	even	2
1232.4.a.q.1.1		3	1.1	even	1	trivial
1386.4.a.bc.1.3		3	12.11	even	2
1694.4.a.s.1.3		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)

\(f(q)\)	\(=\)	\(q-8.45761 q^{3} -3.76558 q^{5} +7.00000 q^{7} +44.5312 q^{9} +O(q^{10})\)	\(q-8.45761 q^{3} -3.76558 q^{5} +7.00000 q^{7} +44.5312 q^{9} -11.0000 q^{11} +66.3055 q^{13} +31.8478 q^{15} +9.79421 q^{17} +92.0424 q^{19} -59.2033 q^{21} +17.7232 q^{23} -110.820 q^{25} -148.272 q^{27} -150.661 q^{29} +198.393 q^{31} +93.0337 q^{33} -26.3591 q^{35} +268.175 q^{37} -560.786 q^{39} -467.722 q^{41} +184.412 q^{43} -167.686 q^{45} -616.089 q^{47} +49.0000 q^{49} -82.8356 q^{51} +38.9725 q^{53} +41.4214 q^{55} -778.459 q^{57} +17.9910 q^{59} +177.206 q^{61} +311.718 q^{63} -249.679 q^{65} +47.0747 q^{67} -149.896 q^{69} +63.4492 q^{71} -787.401 q^{73} +937.276 q^{75} -77.0000 q^{77} +735.738 q^{79} +51.6835 q^{81} -701.349 q^{83} -36.8809 q^{85} +1274.23 q^{87} +1061.71 q^{89} +464.138 q^{91} -1677.93 q^{93} -346.594 q^{95} +850.496 q^{97} -489.843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 6 q^{3} + 26 q^{5} + 21 q^{7} + 59 q^{9}+O(q^{10}) \) 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

Introduction

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