LMFDB - Embedded newform 2448.4.a.bs.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2448,4,Mod(1,2448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2448, 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("2448.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2448 = 2^{4} \cdot 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2448.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:	$144.436675694$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.1506848.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 17x^{2} + 18 $ x^4 - 17*x^2 + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 153)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.06515$ of defining polynomial
Character	$\chi$	$=$	2448.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+20.6288 q^{5} +5.24213 q^{7} -5.21023 q^{11} -14.8007 q^{13} -17.0000 q^{17} -26.2601 q^{19} -165.443 q^{23} +300.546 q^{25} -42.3983 q^{29} -263.956 q^{31} +108.139 q^{35} -322.217 q^{37} +321.529 q^{41} -385.799 q^{43} -309.308 q^{47} -315.520 q^{49} +192.701 q^{53} -107.481 q^{55} +587.082 q^{59} -241.163 q^{61} -305.320 q^{65} -205.396 q^{67} -933.035 q^{71} -869.875 q^{73} -27.3127 q^{77} -102.161 q^{79} -298.886 q^{83} -350.689 q^{85} -666.125 q^{89} -77.5871 q^{91} -541.714 q^{95} +1351.60 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 22 q^{5} + 24 q^{7} - 50 q^{11} + 26 q^{13} - 68 q^{17} - 34 q^{19} - 382 q^{23} + 138 q^{25} + 540 q^{29} + 356 q^{31} - 304 q^{35} - 404 q^{37} - 114 q^{41} - 570 q^{43} - 496 q^{47} - 224 q^{49}+ \cdots - 76 q^{97}+O(q^{100}) $ 4 * q + 22 * q^5 + 24 * q^7 - 50 * q^11 + 26 * q^13 - 68 * q^17 - 34 * q^19 - 382 * q^23 + 138 * q^25 + 540 * q^29 + 356 * q^31 - 304 * q^35 - 404 * q^37 - 114 * q^41 - 570 * q^43 - 496 * q^47 - 224 * q^49 + 92 * q^53 + 482 * q^55 + 48 * q^59 - 1036 * q^61 - 342 * q^65 - 812 * q^67 - 1044 * q^71 - 1212 * q^73 - 564 * q^77 - 488 * q^79 - 1708 * q^83 - 374 * q^85 - 8 * q^89 - 716 * q^91 - 1010 * q^95 - 76 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	20.6288	1.84509	0.922547	−	0.385885i	$-0.126104\pi$
$5$	20.6288	1.84509	0.922547	+	0.385885i	$0.126104\pi$
$6$	0	0
$7$	5.24213	0.283049	0.141524	−	0.989935i	$-0.454800\pi$
$7$	5.24213	0.283049	0.141524	+	0.989935i	$0.454800\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.21023	−0.142813	−0.0714065	−	0.997447i	$-0.522749\pi$
$11$	−5.21023	−0.142813	−0.0714065	+	0.997447i	$0.522749\pi$
$12$	0	0
$13$	−14.8007	−0.315767	−0.157883	−	0.987458i	$-0.550467\pi$
$13$	−14.8007	−0.315767	−0.157883	+	0.987458i	$0.550467\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	0	0
$19$	−26.2601	−0.317078	−0.158539	−	0.987353i	$-0.550678\pi$
$19$	−26.2601	−0.317078	−0.158539	+	0.987353i	$0.550678\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−165.443	−1.49988	−0.749941	−	0.661504i	$-0.769918\pi$
$23$	−165.443	−1.49988	−0.749941	+	0.661504i	$0.769918\pi$
$24$	0	0
$25$	300.546	2.40437
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−42.3983	−0.271488	−0.135744	−	0.990744i	$-0.543343\pi$
$29$	−42.3983	−0.271488	−0.135744	+	0.990744i	$0.543343\pi$
$30$	0	0
$31$	−263.956	−1.52929	−0.764644	−	0.644452i	$-0.777085\pi$
$31$	−263.956	−1.52929	−0.764644	+	0.644452i	$0.777085\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	108.139	0.522251
$36$	0	0
$37$	−322.217	−1.43168	−0.715840	−	0.698264i	$-0.753956\pi$
$37$	−322.217	−1.43168	−0.715840	+	0.698264i	$0.753956\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	321.529	1.22474	0.612371	−	0.790571i	$-0.290216\pi$
$41$	321.529	1.22474	0.612371	+	0.790571i	$0.290216\pi$
$42$	0	0
$43$	−385.799	−1.36823	−0.684114	−	0.729375i	$-0.739811\pi$
$43$	−385.799	−1.36823	−0.684114	+	0.729375i	$0.739811\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−309.308	−0.959942	−0.479971	−	0.877284i	$-0.659353\pi$
$47$	−309.308	−0.959942	−0.479971	+	0.877284i	$0.659353\pi$
$48$	0	0
$49$	−315.520	−0.919884
$50$	0	0
$51$	0	0
$52$	0	0
$53$	192.701	0.499426	0.249713	−	0.968320i	$-0.419664\pi$
$53$	192.701	0.499426	0.249713	+	0.968320i	$0.419664\pi$
$54$	0	0
$55$	−107.481	−0.263504
$56$	0	0
$57$	0	0
$58$	0	0
$59$	587.082	1.29545	0.647725	−	0.761874i	$-0.275721\pi$
$59$	587.082	1.29545	0.647725	+	0.761874i	$0.275721\pi$
$60$	0	0
$61$	−241.163	−0.506192	−0.253096	−	0.967441i	$-0.581449\pi$
$61$	−241.163	−0.506192	−0.253096	+	0.967441i	$0.581449\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−305.320	−0.582619
$66$	0	0
$67$	−205.396	−0.374525	−0.187262	−	0.982310i	$-0.559961\pi$
$67$	−205.396	−0.374525	−0.187262	+	0.982310i	$0.559961\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−933.035	−1.55959	−0.779795	−	0.626035i	$-0.784677\pi$
$71$	−933.035	−1.55959	−0.779795	+	0.626035i	$0.784677\pi$
$72$	0	0
$73$	−869.875	−1.39467	−0.697337	−	0.716744i	$-0.745632\pi$
$73$	−869.875	−1.39467	−0.697337	+	0.716744i	$0.745632\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−27.3127	−0.0404230
$78$	0	0
$79$	−102.161	−0.145494	−0.0727470	−	0.997350i	$-0.523177\pi$
$79$	−102.161	−0.145494	−0.0727470	+	0.997350i	$0.523177\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−298.886	−0.395265	−0.197633	−	0.980276i	$-0.563325\pi$
$83$	−298.886	−0.395265	−0.197633	+	0.980276i	$0.563325\pi$
$84$	0	0
$85$	−350.689	−0.447501
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−666.125	−0.793361	−0.396680	−	0.917957i	$-0.629838\pi$
$89$	−666.125	−0.793361	−0.396680	+	0.917957i	$0.629838\pi$
$90$	0	0
$91$	−77.5871	−0.0893773
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−541.714	−0.585038
$96$	0	0
$97$	1351.60	1.41478	0.707391	−	0.706822i	$-0.249872\pi$
$97$	1351.60	1.41478	0.707391	+	0.706822i	$0.249872\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1628.22	−1.60410	−0.802050	−	0.597256i	$-0.796257\pi$
$101$	−1628.22	−1.60410	−0.802050	+	0.597256i	$0.796257\pi$
$102$	0	0
$103$	−48.4374	−0.0463367	−0.0231684	−	0.999732i	$-0.507375\pi$
$103$	−48.4374	−0.0463367	−0.0231684	+	0.999732i	$0.507375\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1493.56	−1.34942	−0.674711	−	0.738082i	$-0.735732\pi$
$107$	−1493.56	−1.34942	−0.674711	+	0.738082i	$0.735732\pi$
$108$	0	0
$109$	694.642	0.610410	0.305205	−	0.952287i	$-0.401275\pi$
$109$	694.642	0.610410	0.305205	+	0.952287i	$0.401275\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1302.15	1.08404	0.542018	−	0.840367i	$-0.317660\pi$
$113$	1302.15	1.08404	0.542018	+	0.840367i	$0.317660\pi$
$114$	0	0
$115$	−3412.89	−2.76742
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−89.1163	−0.0686494
$120$	0	0
$121$	−1303.85	−0.979604
$122$	0	0
$123$	0	0
$124$	0	0
$125$	3621.31	2.59120
$126$	0	0
$127$	2100.00	1.46728	0.733642	−	0.679536i	$-0.237819\pi$
$127$	2100.00	1.46728	0.733642	+	0.679536i	$0.237819\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	963.258	0.642444	0.321222	−	0.947004i	$-0.395906\pi$
$131$	963.258	0.642444	0.321222	+	0.947004i	$0.395906\pi$
$132$	0	0
$133$	−137.659	−0.0897484
$134$	0	0
$135$	0	0
$136$	0	0
$137$	311.771	0.194426	0.0972130	−	0.995264i	$-0.469007\pi$
$137$	311.771	0.194426	0.0972130	+	0.995264i	$0.469007\pi$
$138$	0	0
$139$	1039.05	0.634035	0.317018	−	0.948420i	$-0.397319\pi$
$139$	1039.05	0.634035	0.317018	+	0.948420i	$0.397319\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	77.1149	0.0450956
$144$	0	0
$145$	−874.624	−0.500921
$146$	0	0
$147$	0	0
$148$	0	0
$149$	375.216	0.206301	0.103151	−	0.994666i	$-0.467108\pi$
$149$	375.216	0.206301	0.103151	+	0.994666i	$0.467108\pi$
$150$	0	0
$151$	2602.60	1.40262	0.701312	−	0.712854i	$-0.252598\pi$
$151$	2602.60	1.40262	0.701312	+	0.712854i	$0.252598\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−5445.10	−2.82168
$156$	0	0
$157$	−3134.77	−1.59351	−0.796756	−	0.604301i	$-0.793453\pi$
$157$	−3134.77	−1.59351	−0.796756	+	0.604301i	$0.793453\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−867.276	−0.424540
$162$	0	0
$163$	52.9329	0.0254357	0.0127179	−	0.999919i	$-0.495952\pi$
$163$	52.9329	0.0254357	0.0127179	+	0.999919i	$0.495952\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3554.79	1.64717	0.823586	−	0.567192i	$-0.191970\pi$
$167$	3554.79	1.64717	0.823586	+	0.567192i	$0.191970\pi$
$168$	0	0
$169$	−1977.94	−0.900291
$170$	0	0
$171$	0	0
$172$	0	0
$173$	134.253	0.0590006	0.0295003	−	0.999565i	$-0.490608\pi$
$173$	134.253	0.0590006	0.0295003	+	0.999565i	$0.490608\pi$
$174$	0	0
$175$	1575.50	0.680554
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2368.80	0.989119	0.494559	−	0.869144i	$-0.335329\pi$
$179$	2368.80	0.989119	0.494559	+	0.869144i	$0.335329\pi$
$180$	0	0
$181$	1320.79	0.542396	0.271198	−	0.962524i	$-0.412580\pi$
$181$	1320.79	0.542396	0.271198	+	0.962524i	$0.412580\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−6646.95	−2.64158
$186$	0	0
$187$	88.5739	0.0346373
$188$	0	0
$189$	0	0
$190$	0	0
$191$	549.534	0.208183	0.104091	−	0.994568i	$-0.466807\pi$
$191$	549.534	0.208183	0.104091	+	0.994568i	$0.466807\pi$
$192$	0	0
$193$	4191.18	1.56315	0.781575	−	0.623811i	$-0.214417\pi$
$193$	4191.18	1.56315	0.781575	+	0.623811i	$0.214417\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3963.36	−1.43339	−0.716695	−	0.697386i	$-0.754346\pi$
$197$	−3963.36	−1.43339	−0.716695	+	0.697386i	$0.754346\pi$
$198$	0	0
$199$	4603.54	1.63988	0.819940	−	0.572450i	$-0.194007\pi$
$199$	4603.54	1.63988	0.819940	+	0.572450i	$0.194007\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−222.257	−0.0768444
$204$	0	0
$205$	6632.75	2.25976
$206$	0	0
$207$	0	0
$208$	0	0
$209$	136.821	0.0452829
$210$	0	0
$211$	3972.74	1.29618	0.648091	−	0.761563i	$-0.275568\pi$
$211$	3972.74	1.29618	0.648091	+	0.761563i	$0.275568\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−7958.56	−2.52451
$216$	0	0
$217$	−1383.69	−0.432863
$218$	0	0
$219$	0	0
$220$	0	0
$221$	251.611	0.0765847
$222$	0	0
$223$	−3347.58	−1.00525	−0.502624	−	0.864505i	$-0.667632\pi$
$223$	−3347.58	−1.00525	−0.502624	+	0.864505i	$0.667632\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2809.86	−0.821572	−0.410786	−	0.911732i	$-0.634746\pi$
$227$	−2809.86	−0.821572	−0.410786	+	0.911732i	$0.634746\pi$
$228$	0	0
$229$	5978.12	1.72509	0.862544	−	0.505981i	$-0.168870\pi$
$229$	5978.12	1.72509	0.862544	+	0.505981i	$0.168870\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2877.85	−0.809160	−0.404580	−	0.914503i	$-0.632582\pi$
$233$	−2877.85	−0.809160	−0.404580	+	0.914503i	$0.632582\pi$
$234$	0	0
$235$	−6380.65	−1.77118
$236$	0	0
$237$	0	0
$238$	0	0
$239$	836.275	0.226335	0.113168	−	0.993576i	$-0.463900\pi$
$239$	836.275	0.226335	0.113168	+	0.993576i	$0.463900\pi$
$240$	0	0
$241$	1336.21	0.357148	0.178574	−	0.983926i	$-0.442852\pi$
$241$	1336.21	0.357148	0.178574	+	0.983926i	$0.442852\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−6508.79	−1.69727
$246$	0	0
$247$	388.667	0.100123
$248$	0	0
$249$	0	0
$250$	0	0
$251$	5582.36	1.40381	0.701903	−	0.712273i	$-0.252334\pi$
$251$	5582.36	1.40381	0.701903	+	0.712273i	$0.252334\pi$
$252$	0	0
$253$	861.998	0.214203
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4627.76	1.12324	0.561618	−	0.827397i	$-0.310179\pi$
$257$	4627.76	1.12324	0.561618	+	0.827397i	$0.310179\pi$
$258$	0	0
$259$	−1689.11	−0.405235
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−399.519	−0.0936708	−0.0468354	−	0.998903i	$-0.514914\pi$
$263$	−399.519	−0.0936708	−0.0468354	+	0.998903i	$0.514914\pi$
$264$	0	0
$265$	3975.20	0.921488
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−2204.10	−0.499577	−0.249789	−	0.968300i	$-0.580361\pi$
$269$	−2204.10	−0.499577	−0.249789	+	0.968300i	$0.580361\pi$
$270$	0	0
$271$	2363.70	0.529833	0.264916	−	0.964271i	$-0.414656\pi$
$271$	2363.70	0.529833	0.264916	+	0.964271i	$0.414656\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1565.92	−0.343376
$276$	0	0
$277$	−3988.65	−0.865179	−0.432589	−	0.901591i	$-0.642400\pi$
$277$	−3988.65	−0.865179	−0.432589	+	0.901591i	$0.642400\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1494.96	−0.317373	−0.158686	−	0.987329i	$-0.550726\pi$
$281$	−1494.96	−0.317373	−0.158686	+	0.987329i	$0.550726\pi$
$282$	0	0
$283$	−5973.29	−1.25468	−0.627341	−	0.778744i	$-0.715857\pi$
$283$	−5973.29	−1.25468	−0.627341	+	0.778744i	$0.715857\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1685.50	0.346661
$288$	0	0
$289$	289.000	0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−5706.31	−1.13777	−0.568884	−	0.822417i	$-0.692625\pi$
$293$	−5706.31	−1.13777	−0.568884	+	0.822417i	$0.692625\pi$
$294$	0	0
$295$	12110.8	2.39023
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2448.67	0.473613
$300$	0	0
$301$	−2022.41	−0.387275
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−4974.89	−0.933973
$306$	0	0
$307$	−4684.64	−0.870900	−0.435450	−	0.900213i	$-0.643411\pi$
$307$	−4684.64	−0.870900	−0.435450	+	0.900213i	$0.643411\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−4086.74	−0.745138	−0.372569	−	0.928005i	$-0.621523\pi$
$311$	−4086.74	−0.745138	−0.372569	+	0.928005i	$0.621523\pi$
$312$	0	0
$313$	6893.79	1.24492	0.622460	−	0.782651i	$-0.286133\pi$
$313$	6893.79	1.24492	0.622460	+	0.782651i	$0.286133\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4907.72	0.869542	0.434771	−	0.900541i	$-0.356829\pi$
$317$	4907.72	0.869542	0.434771	+	0.900541i	$0.356829\pi$
$318$	0	0
$319$	220.905	0.0387721
$320$	0	0
$321$	0	0
$322$	0	0
$323$	446.422	0.0769027
$324$	0	0
$325$	−4448.29	−0.759220
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1621.44	−0.271710
$330$	0	0
$331$	8884.92	1.47541	0.737703	−	0.675126i	$-0.235911\pi$
$331$	8884.92	1.47541	0.737703	+	0.675126i	$0.235911\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−4237.07	−0.691033
$336$	0	0
$337$	−5398.17	−0.872572	−0.436286	−	0.899808i	$-0.643706\pi$
$337$	−5398.17	−0.872572	−0.436286	+	0.899808i	$0.643706\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1375.27	0.218402
$342$	0	0
$343$	−3452.05	−0.543420
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−6120.24	−0.946835	−0.473417	−	0.880838i	$-0.656980\pi$
$347$	−6120.24	−0.946835	−0.473417	+	0.880838i	$0.656980\pi$
$348$	0	0
$349$	−4211.65	−0.645972	−0.322986	−	0.946404i	$-0.604687\pi$
$349$	−4211.65	−0.645972	−0.322986	+	0.946404i	$0.604687\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−982.908	−0.148201	−0.0741005	−	0.997251i	$-0.523609\pi$
$353$	−982.908	−0.148201	−0.0741005	+	0.997251i	$0.523609\pi$
$354$	0	0
$355$	−19247.4	−2.87759
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−12216.9	−1.79605	−0.898027	−	0.439940i	$-0.855000\pi$
$359$	−12216.9	−1.79605	−0.898027	+	0.439940i	$0.855000\pi$
$360$	0	0
$361$	−6169.41	−0.899462
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−17944.5	−2.57330
$366$	0	0
$367$	3054.02	0.434383	0.217192	−	0.976129i	$-0.430310\pi$
$367$	3054.02	0.434383	0.217192	+	0.976129i	$0.430310\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1010.17	0.141362
$372$	0	0
$373$	5547.11	0.770023	0.385012	−	0.922912i	$-0.374197\pi$
$373$	5547.11	0.770023	0.385012	+	0.922912i	$0.374197\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	627.523	0.0857269
$378$	0	0
$379$	−3793.70	−0.514168	−0.257084	−	0.966389i	$-0.582762\pi$
$379$	−3793.70	−0.514168	−0.257084	+	0.966389i	$0.582762\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−10645.4	−1.42024	−0.710122	−	0.704079i	$-0.751360\pi$
$383$	−10645.4	−1.42024	−0.710122	+	0.704079i	$0.751360\pi$
$384$	0	0
$385$	−563.428	−0.0745843
$386$	0	0
$387$	0	0
$388$	0	0
$389$	551.008	0.0718180	0.0359090	−	0.999355i	$-0.488567\pi$
$389$	551.008	0.0718180	0.0359090	+	0.999355i	$0.488567\pi$
$390$	0	0
$391$	2812.54	0.363775
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2107.46	−0.268450
$396$	0	0
$397$	−12508.7	−1.58135	−0.790674	−	0.612238i	$-0.790269\pi$
$397$	−12508.7	−1.58135	−0.790674	+	0.612238i	$0.790269\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−9724.10	−1.21097	−0.605484	−	0.795857i	$-0.707021\pi$
$401$	−9724.10	−1.21097	−0.605484	+	0.795857i	$0.707021\pi$
$402$	0	0
$403$	3906.73	0.482898
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1678.83	0.204463
$408$	0	0
$409$	−1698.84	−0.205385	−0.102693	−	0.994713i	$-0.532746\pi$
$409$	−1698.84	−0.205385	−0.102693	+	0.994713i	$0.532746\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	3077.56	0.366675
$414$	0	0
$415$	−6165.66	−0.729302
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−2663.86	−0.310592	−0.155296	−	0.987868i	$-0.549633\pi$
$419$	−2663.86	−0.310592	−0.155296	+	0.987868i	$0.549633\pi$
$420$	0	0
$421$	776.756	0.0899211	0.0449606	−	0.998989i	$-0.485684\pi$
$421$	776.756	0.0899211	0.0449606	+	0.998989i	$0.485684\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−5109.29	−0.583146
$426$	0	0
$427$	−1264.21	−0.143277
$428$	0	0
$429$	0	0
$430$	0	0
$431$	5951.70	0.665158	0.332579	−	0.943075i	$-0.392081\pi$
$431$	5951.70	0.665158	0.332579	+	0.943075i	$0.392081\pi$
$432$	0	0
$433$	154.417	0.0171382	0.00856908	−	0.999963i	$-0.497272\pi$
$433$	154.417	0.0171382	0.00856908	+	0.999963i	$0.497272\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4344.56	0.475580
$438$	0	0
$439$	12859.5	1.39806	0.699031	−	0.715091i	$-0.253615\pi$
$439$	12859.5	1.39806	0.699031	+	0.715091i	$0.253615\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	16428.4	1.76193	0.880967	−	0.473177i	$-0.156893\pi$
$443$	16428.4	1.76193	0.880967	+	0.473177i	$0.156893\pi$
$444$	0	0
$445$	−13741.3	−1.46383
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−700.731	−0.0736516	−0.0368258	−	0.999322i	$-0.511725\pi$
$449$	−700.731	−0.0736516	−0.0368258	+	0.999322i	$0.511725\pi$
$450$	0	0
$451$	−1675.24	−0.174909
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1600.53	−0.164910
$456$	0	0
$457$	3127.81	0.320159	0.160080	−	0.987104i	$-0.448825\pi$
$457$	3127.81	0.320159	0.160080	+	0.987104i	$0.448825\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−12198.4	−1.23240	−0.616200	−	0.787590i	$-0.711329\pi$
$461$	−12198.4	−1.23240	−0.616200	+	0.787590i	$0.711329\pi$
$462$	0	0
$463$	5054.14	0.507313	0.253657	−	0.967294i	$-0.418367\pi$
$463$	5054.14	0.507313	0.253657	+	0.967294i	$0.418367\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−8366.25	−0.829002	−0.414501	−	0.910049i	$-0.636044\pi$
$467$	−8366.25	−0.829002	−0.414501	+	0.910049i	$0.636044\pi$
$468$	0	0
$469$	−1076.71	−0.106009
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2010.10	0.195401
$474$	0	0
$475$	−7892.38	−0.762373
$476$	0	0
$477$	0	0
$478$	0	0
$479$	3942.17	0.376038	0.188019	−	0.982165i	$-0.439793\pi$
$479$	3942.17	0.376038	0.188019	+	0.982165i	$0.439793\pi$
$480$	0	0
$481$	4769.03	0.452077
$482$	0	0
$483$	0	0
$484$	0	0
$485$	27881.8	2.61041
$486$	0	0
$487$	−160.598	−0.0149433	−0.00747165	−	0.999972i	$-0.502378\pi$
$487$	−160.598	−0.0149433	−0.00747165	+	0.999972i	$0.502378\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−17380.5	−1.59750	−0.798748	−	0.601666i	$-0.794504\pi$
$491$	−17380.5	−1.59750	−0.798748	+	0.601666i	$0.794504\pi$
$492$	0	0
$493$	720.770	0.0658456
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−4891.09	−0.441440
$498$	0	0
$499$	−17648.5	−1.58328	−0.791640	−	0.610988i	$-0.790772\pi$
$499$	−17648.5	−1.58328	−0.791640	+	0.610988i	$0.790772\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2367.22	−0.209839	−0.104920	−	0.994481i	$-0.533459\pi$
$503$	−2367.22	−0.209839	−0.104920	+	0.994481i	$0.533459\pi$
$504$	0	0
$505$	−33588.2	−2.95972
$506$	0	0
$507$	0	0
$508$	0	0
$509$	13983.5	1.21770	0.608850	−	0.793286i	$-0.291631\pi$
$509$	13983.5	1.21770	0.608850	+	0.793286i	$0.291631\pi$
$510$	0	0
$511$	−4560.00	−0.394760
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−999.205	−0.0854956
$516$	0	0
$517$	1611.57	0.137092
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−14012.1	−1.17828	−0.589139	−	0.808032i	$-0.700533\pi$
$521$	−14012.1	−1.17828	−0.589139	+	0.808032i	$0.700533\pi$
$522$	0	0
$523$	6665.75	0.557310	0.278655	−	0.960391i	$-0.410111\pi$
$523$	6665.75	0.557310	0.278655	+	0.960391i	$0.410111\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4487.26	0.370907
$528$	0	0
$529$	15204.5	1.24965
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−4758.85	−0.386733
$534$	0	0
$535$	−30810.4	−2.48981
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1643.93	0.131371
$540$	0	0
$541$	−2658.37	−0.211261	−0.105631	−	0.994405i	$-0.533686\pi$
$541$	−2658.37	−0.211261	−0.105631	+	0.994405i	$0.533686\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	14329.6	1.12626
$546$	0	0
$547$	20140.6	1.57431	0.787156	−	0.616755i	$-0.211553\pi$
$547$	20140.6	1.57431	0.787156	+	0.616755i	$0.211553\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1113.38	0.0860829
$552$	0	0
$553$	−535.543	−0.0411819
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−10449.2	−0.794874	−0.397437	−	0.917629i	$-0.630100\pi$
$557$	−10449.2	−0.794874	−0.397437	+	0.917629i	$0.630100\pi$
$558$	0	0
$559$	5710.08	0.432041
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−6606.44	−0.494544	−0.247272	−	0.968946i	$-0.579534\pi$
$563$	−6606.44	−0.494544	−0.247272	+	0.968946i	$0.579534\pi$
$564$	0	0
$565$	26861.8	2.00015
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−6016.94	−0.443310	−0.221655	−	0.975125i	$-0.571146\pi$
$569$	−6016.94	−0.443310	−0.221655	+	0.975125i	$0.571146\pi$
$570$	0	0
$571$	10560.4	0.773974	0.386987	−	0.922085i	$-0.373516\pi$
$571$	10560.4	0.773974	0.386987	+	0.922085i	$0.373516\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−49723.4	−3.60627
$576$	0	0
$577$	−9374.56	−0.676374	−0.338187	−	0.941079i	$-0.609814\pi$
$577$	−9374.56	−0.676374	−0.338187	+	0.941079i	$0.609814\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1566.80	−0.111879
$582$	0	0
$583$	−1004.02	−0.0713246
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−16742.5	−1.17724	−0.588618	−	0.808411i	$-0.700328\pi$
$587$	−16742.5	−1.17724	−0.588618	+	0.808411i	$0.700328\pi$
$588$	0	0
$589$	6931.52	0.484904
$590$	0	0
$591$	0	0
$592$	0	0
$593$	16820.2	1.16480	0.582398	−	0.812904i	$-0.302115\pi$
$593$	16820.2	1.16480	0.582398	+	0.812904i	$0.302115\pi$
$594$	0	0
$595$	−1838.36	−0.126665
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−16523.6	−1.12711	−0.563553	−	0.826080i	$-0.690566\pi$
$599$	−16523.6	−1.12711	−0.563553	+	0.826080i	$0.690566\pi$
$600$	0	0
$601$	1133.13	0.0769075	0.0384537	−	0.999260i	$-0.487757\pi$
$601$	1133.13	0.0769075	0.0384537	+	0.999260i	$0.487757\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−26896.9	−1.80746
$606$	0	0
$607$	−210.365	−0.0140667	−0.00703333	−	0.999975i	$-0.502239\pi$
$607$	−210.365	−0.0140667	−0.00703333	+	0.999975i	$0.502239\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4577.97	0.303118
$612$	0	0
$613$	−4813.64	−0.317163	−0.158582	−	0.987346i	$-0.550692\pi$
$613$	−4813.64	−0.317163	−0.158582	+	0.987346i	$0.550692\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	19782.2	1.29076	0.645381	−	0.763861i	$-0.276699\pi$
$617$	19782.2	1.29076	0.645381	+	0.763861i	$0.276699\pi$
$618$	0	0
$619$	−6340.53	−0.411709	−0.205854	−	0.978583i	$-0.565997\pi$
$619$	−6340.53	−0.411709	−0.205854	+	0.978583i	$0.565997\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−3491.92	−0.224560
$624$	0	0
$625$	37134.8	2.37663
$626$	0	0
$627$	0	0
$628$	0	0
$629$	5477.69	0.347233
$630$	0	0
$631$	4591.83	0.289696	0.144848	−	0.989454i	$-0.453731\pi$
$631$	4591.83	0.289696	0.144848	+	0.989454i	$0.453731\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	43320.5	2.70728
$636$	0	0
$637$	4669.91	0.290469
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−3018.43	−0.185992	−0.0929961	−	0.995666i	$-0.529644\pi$
$641$	−3018.43	−0.185992	−0.0929961	+	0.995666i	$0.529644\pi$
$642$	0	0
$643$	25834.1	1.58444	0.792222	−	0.610233i	$-0.208924\pi$
$643$	25834.1	1.58444	0.792222	+	0.610233i	$0.208924\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−4123.41	−0.250553	−0.125277	−	0.992122i	$-0.539982\pi$
$647$	−4123.41	−0.250553	−0.125277	+	0.992122i	$0.539982\pi$
$648$	0	0
$649$	−3058.83	−0.185007
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−18013.5	−1.07951	−0.539757	−	0.841821i	$-0.681484\pi$
$653$	−18013.5	−1.07951	−0.539757	+	0.841821i	$0.681484\pi$
$654$	0	0
$655$	19870.8	1.18537
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−5886.59	−0.347965	−0.173982	−	0.984749i	$-0.555664\pi$
$659$	−5886.59	−0.347965	−0.173982	+	0.984749i	$0.555664\pi$
$660$	0	0
$661$	−1099.06	−0.0646722	−0.0323361	−	0.999477i	$-0.510295\pi$
$661$	−1099.06	−0.0646722	−0.0323361	+	0.999477i	$0.510295\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2839.74	−0.165594
$666$	0	0
$667$	7014.51	0.407201
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1256.51	0.0722909
$672$	0	0
$673$	−5682.62	−0.325481	−0.162741	−	0.986669i	$-0.552033\pi$
$673$	−5682.62	−0.325481	−0.162741	+	0.986669i	$0.552033\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−4491.71	−0.254993	−0.127497	−	0.991839i	$-0.540694\pi$
$677$	−4491.71	−0.254993	−0.127497	+	0.991839i	$0.540694\pi$
$678$	0	0
$679$	7085.25	0.400452
$680$	0	0
$681$	0	0
$682$	0	0
$683$	11533.2	0.646131	0.323065	−	0.946377i	$-0.395287\pi$
$683$	11533.2	0.646131	0.323065	+	0.946377i	$0.395287\pi$
$684$	0	0
$685$	6431.45	0.358734
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−2852.11	−0.157702
$690$	0	0
$691$	−28818.2	−1.58654	−0.793268	−	0.608873i	$-0.791622\pi$
$691$	−28818.2	−1.58654	−0.793268	+	0.608873i	$0.791622\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	21434.3	1.16985
$696$	0	0
$697$	−5466.00	−0.297044
$698$	0	0
$699$	0	0
$700$	0	0
$701$	16494.8	0.888730	0.444365	−	0.895846i	$-0.353429\pi$
$701$	16494.8	0.888730	0.444365	+	0.895846i	$0.353429\pi$
$702$	0	0
$703$	8461.45	0.453954
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−8535.36	−0.454038
$708$	0	0
$709$	26363.9	1.39650	0.698248	−	0.715856i	$-0.253963\pi$
$709$	26363.9	1.39650	0.698248	+	0.715856i	$0.253963\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	43669.8	2.29375
$714$	0	0
$715$	1590.79	0.0832056
$716$	0	0
$717$	0	0
$718$	0	0
$719$	15410.9	0.799348	0.399674	−	0.916657i	$-0.369123\pi$
$719$	15410.9	0.799348	0.399674	+	0.916657i	$0.369123\pi$
$720$	0	0
$721$	−253.915	−0.0131155
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−12742.6	−0.652759
$726$	0	0
$727$	3399.02	0.173401	0.0867005	−	0.996234i	$-0.472368\pi$
$727$	3399.02	0.173401	0.0867005	+	0.996234i	$0.472368\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	6558.58	0.331844
$732$	0	0
$733$	−1731.67	−0.0872585	−0.0436293	−	0.999048i	$-0.513892\pi$
$733$	−1731.67	−0.0872585	−0.0436293	+	0.999048i	$0.513892\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1070.16	0.0534870
$738$	0	0
$739$	34116.1	1.69821	0.849107	−	0.528221i	$-0.177141\pi$
$739$	34116.1	1.69821	0.849107	+	0.528221i	$0.177141\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−16893.3	−0.834126	−0.417063	−	0.908878i	$-0.636941\pi$
$743$	−16893.3	−0.834126	−0.417063	+	0.908878i	$0.636941\pi$
$744$	0	0
$745$	7740.24	0.380645
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−7829.45	−0.381952
$750$	0	0
$751$	−32492.2	−1.57877	−0.789385	−	0.613898i	$-0.789601\pi$
$751$	−32492.2	−1.57877	−0.789385	+	0.613898i	$0.789601\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	53688.4	2.58797
$756$	0	0
$757$	31568.5	1.51569	0.757845	−	0.652434i	$-0.226252\pi$
$757$	31568.5	1.51569	0.757845	+	0.652434i	$0.226252\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	21878.0	1.04215	0.521076	−	0.853510i	$-0.325531\pi$
$761$	21878.0	1.04215	0.521076	+	0.853510i	$0.325531\pi$
$762$	0	0
$763$	3641.41	0.172776
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8689.20	−0.409060
$768$	0	0
$769$	12510.4	0.586653	0.293327	−	0.956012i	$-0.405238\pi$
$769$	12510.4	0.586653	0.293327	+	0.956012i	$0.405238\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	3483.57	0.162090	0.0810448	−	0.996710i	$-0.474174\pi$
$773$	3483.57	0.162090	0.0810448	+	0.996710i	$0.474174\pi$
$774$	0	0
$775$	−79331.1	−3.67698
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−8443.39	−0.388338
$780$	0	0
$781$	4861.33	0.222730
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−64666.4	−2.94018
$786$	0	0
$787$	21827.4	0.988643	0.494322	−	0.869279i	$-0.335416\pi$
$787$	21827.4	0.988643	0.494322	+	0.869279i	$0.335416\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	6826.05	0.306835
$792$	0	0
$793$	3569.37	0.159839
$794$	0	0
$795$	0	0
$796$	0	0
$797$	31873.4	1.41658	0.708289	−	0.705922i	$-0.249467\pi$
$797$	31873.4	1.41658	0.708289	+	0.705922i	$0.249467\pi$
$798$	0	0
$799$	5258.24	0.232820
$800$	0	0
$801$	0	0
$802$	0	0
$803$	4532.25	0.199178
$804$	0	0
$805$	−17890.8	−0.783316
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−19243.8	−0.836312	−0.418156	−	0.908375i	$-0.637323\pi$
$809$	−19243.8	−0.836312	−0.418156	+	0.908375i	$0.637323\pi$
$810$	0	0
$811$	−8969.81	−0.388376	−0.194188	−	0.980964i	$-0.562207\pi$
$811$	−8969.81	−0.388376	−0.194188	+	0.980964i	$0.562207\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1091.94	0.0469313
$816$	0	0
$817$	10131.1	0.433835
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3644.00	0.154904	0.0774522	−	0.996996i	$-0.475321\pi$
$821$	3644.00	0.154904	0.0774522	+	0.996996i	$0.475321\pi$
$822$	0	0
$823$	7219.82	0.305793	0.152896	−	0.988242i	$-0.451140\pi$
$823$	7219.82	0.305793	0.152896	+	0.988242i	$0.451140\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−5356.59	−0.225232	−0.112616	−	0.993639i	$-0.535923\pi$
$827$	−5356.59	−0.225232	−0.112616	+	0.993639i	$0.535923\pi$
$828$	0	0
$829$	−14161.8	−0.593319	−0.296659	−	0.954983i	$-0.595873\pi$
$829$	−14161.8	−0.593319	−0.296659	+	0.954983i	$0.595873\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	5363.84	0.223105
$834$	0	0
$835$	73330.9	3.03919
$836$	0	0
$837$	0	0
$838$	0	0
$839$	12912.5	0.531333	0.265667	−	0.964065i	$-0.414408\pi$
$839$	12912.5	0.531333	0.265667	+	0.964065i	$0.414408\pi$
$840$	0	0
$841$	−22591.4	−0.926294
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−40802.5	−1.66112
$846$	0	0
$847$	−6834.97	−0.277276
$848$	0	0
$849$	0	0
$850$	0	0
$851$	53308.7	2.14735
$852$	0	0
$853$	−6526.03	−0.261954	−0.130977	−	0.991385i	$-0.541811\pi$
$853$	−6526.03	−0.261954	−0.130977	+	0.991385i	$0.541811\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−19431.8	−0.774536	−0.387268	−	0.921967i	$-0.626581\pi$
$857$	−19431.8	−0.774536	−0.387268	+	0.921967i	$0.626581\pi$
$858$	0	0
$859$	−8373.86	−0.332610	−0.166305	−	0.986074i	$-0.553184\pi$
$859$	−8373.86	−0.332610	−0.166305	+	0.986074i	$0.553184\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−13066.9	−0.515412	−0.257706	−	0.966223i	$-0.582967\pi$
$863$	−13066.9	−0.515412	−0.257706	+	0.966223i	$0.582967\pi$
$864$	0	0
$865$	2769.48	0.108862
$866$	0	0
$867$	0	0
$868$	0	0
$869$	532.283	0.0207785
$870$	0	0
$871$	3040.00	0.118262
$872$	0	0
$873$	0	0
$874$	0	0
$875$	18983.4	0.733435
$876$	0	0
$877$	−40889.1	−1.57438	−0.787188	−	0.616713i	$-0.788464\pi$
$877$	−40889.1	−1.57438	−0.787188	+	0.616713i	$0.788464\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	16437.8	0.628608	0.314304	−	0.949322i	$-0.398229\pi$
$881$	16437.8	0.628608	0.314304	+	0.949322i	$0.398229\pi$
$882$	0	0
$883$	37873.1	1.44341	0.721705	−	0.692201i	$-0.243359\pi$
$883$	37873.1	1.44341	0.721705	+	0.692201i	$0.243359\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−35145.6	−1.33041	−0.665205	−	0.746661i	$-0.731656\pi$
$887$	−35145.6	−1.33041	−0.665205	+	0.746661i	$0.731656\pi$
$888$	0	0
$889$	11008.5	0.415313
$890$	0	0
$891$	0	0
$892$	0	0
$893$	8122.47	0.304376
$894$	0	0
$895$	48865.4	1.82502
$896$	0	0
$897$	0	0
$898$	0	0
$899$	11191.3	0.415184
$900$	0	0
$901$	−3275.93	−0.121129
$902$	0	0
$903$	0	0
$904$	0	0
$905$	27246.3	1.00077
$906$	0	0
$907$	−4884.66	−0.178823	−0.0894116	−	0.995995i	$-0.528499\pi$
$907$	−4884.66	−0.178823	−0.0894116	+	0.995995i	$0.528499\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−44687.5	−1.62521	−0.812603	−	0.582818i	$-0.801950\pi$
$911$	−44687.5	−1.62521	−0.812603	+	0.582818i	$0.801950\pi$
$912$	0	0
$913$	1557.27	0.0564491
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5049.52	0.181843
$918$	0	0
$919$	−43322.8	−1.55505	−0.777524	−	0.628853i	$-0.783525\pi$
$919$	−43322.8	−1.55505	−0.777524	+	0.628853i	$0.783525\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	13809.5	0.492467
$924$	0	0
$925$	−96841.2	−3.44229
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−31035.2	−1.09605	−0.548025	−	0.836462i	$-0.684620\pi$
$929$	−31035.2	−1.09605	−0.548025	+	0.836462i	$0.684620\pi$
$930$	0	0
$931$	8285.59	0.291675
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1827.17	0.0639090
$936$	0	0
$937$	−28794.9	−1.00394	−0.501968	−	0.864886i	$-0.667390\pi$
$937$	−28794.9	−1.00394	−0.501968	+	0.864886i	$0.667390\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	11949.5	0.413966	0.206983	−	0.978344i	$-0.433635\pi$
$941$	11949.5	0.413966	0.206983	+	0.978344i	$0.433635\pi$
$942$	0	0
$943$	−53194.8	−1.83697
$944$	0	0
$945$	0	0
$946$	0	0
$947$	20143.1	0.691197	0.345599	−	0.938382i	$-0.387676\pi$
$947$	20143.1	0.691197	0.345599	+	0.938382i	$0.387676\pi$
$948$	0	0
$949$	12874.7	0.440391
$950$	0	0
$951$	0	0
$952$	0	0
$953$	36544.6	1.24218	0.621090	−	0.783739i	$-0.286690\pi$
$953$	36544.6	1.24218	0.621090	+	0.783739i	$0.286690\pi$
$954$	0	0
$955$	11336.2	0.384117
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1634.34	0.0550320
$960$	0	0
$961$	39881.9	1.33872
$962$	0	0
$963$	0	0
$964$	0	0
$965$	86459.0	2.88416
$966$	0	0
$967$	40645.1	1.35166	0.675831	−	0.737056i	$-0.263785\pi$
$967$	40645.1	1.35166	0.675831	+	0.737056i	$0.263785\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−711.265	−0.0235073	−0.0117536	−	0.999931i	$-0.503741\pi$
$971$	−711.265	−0.0235073	−0.0117536	+	0.999931i	$0.503741\pi$
$972$	0	0
$973$	5446.83	0.179463
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−57865.1	−1.89485	−0.947425	−	0.319979i	$-0.896324\pi$
$977$	−57865.1	−1.89485	−0.947425	+	0.319979i	$0.896324\pi$
$978$	0	0
$979$	3470.67	0.113302
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−4983.77	−0.161707	−0.0808534	−	0.996726i	$-0.525765\pi$
$983$	−4983.77	−0.161707	−0.0808534	+	0.996726i	$0.525765\pi$
$984$	0	0
$985$	−81759.4	−2.64474
$986$	0	0
$987$	0	0
$988$	0	0
$989$	63827.8	2.05218
$990$	0	0
$991$	−20463.6	−0.655952	−0.327976	−	0.944686i	$-0.606367\pi$
$991$	−20463.6	−0.655952	−0.327976	+	0.944686i	$0.606367\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	94965.3	3.02573
$996$	0	0
$997$	37140.8	1.17980	0.589900	−	0.807476i	$-0.299167\pi$
$997$	37140.8	1.17980	0.589900	+	0.807476i	$0.299167\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2448.4.a.bs.1.4		4
3.2	odd	2		2448.4.a.bo.1.1		4
4.3	odd	2		153.4.a.i.1.1	yes	4
12.11	even	2		153.4.a.h.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
153.4.a.h.1.4	✓	4	12.11	even	2
153.4.a.i.1.1	yes	4	4.3	odd	2
2448.4.a.bo.1.1		4	3.2	odd	2
2448.4.a.bs.1.4		4	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:	\( N \)	\(=\)	\( 2448 = 2^{4} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2448.a (trivial)

\(f(q)\)	\(=\)	\(q+20.6288 q^{5} +5.24213 q^{7} -5.21023 q^{11} -14.8007 q^{13} -17.0000 q^{17} -26.2601 q^{19} -165.443 q^{23} +300.546 q^{25} -42.3983 q^{29} -263.956 q^{31} +108.139 q^{35} -322.217 q^{37} +321.529 q^{41} -385.799 q^{43} -309.308 q^{47} -315.520 q^{49} +192.701 q^{53} -107.481 q^{55} +587.082 q^{59} -241.163 q^{61} -305.320 q^{65} -205.396 q^{67} -933.035 q^{71} -869.875 q^{73} -27.3127 q^{77} -102.161 q^{79} -298.886 q^{83} -350.689 q^{85} -666.125 q^{89} -77.5871 q^{91} -541.714 q^{95} +1351.60 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 22 q^{5} + 24 q^{7} - 50 q^{11} + 26 q^{13} - 68 q^{17} - 34 q^{19} - 382 q^{23} + 138 q^{25} + 540 q^{29} + 356 q^{31} - 304 q^{35} - 404 q^{37} - 114 q^{41} - 570 q^{43} - 496 q^{47} - 224 q^{49}+ \cdots - 76 q^{97}+O(q^{100}) \) 4 * q + 22 * q^5 + 24 * q^7 - 50 * q^11 + 26 * q^13 - 68 * q^17 - 34 * q^19 - 382 * q^23 + 138 * q^25 + 540 * q^29 + 356 * q^31 - 304 * q^35 - 404 * q^37 - 114 * q^41 - 570 * q^43 - 496 * q^47 - 224 * q^49 + 92 * q^53 + 482 * q^55 + 48 * q^59 - 1036 * q^61 - 342 * q^65 - 812 * q^67 - 1044 * q^71 - 1212 * q^73 - 564 * q^77 - 488 * q^79 - 1708 * q^83 - 374 * q^85 - 8 * q^89 - 716 * q^91 - 1010 * q^95 - 76 * q^97

Introduction

L-functions

Modular forms

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Database

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