LMFDB - Embedded newform 1344.4.a.w.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,4,Mod(1,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, 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("1344.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.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:	$79.2985670477$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 21)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1344.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{3} +4.00000 q^{5} -7.00000 q^{7} +9.00000 q^{9} -62.0000 q^{11} +62.0000 q^{13} +12.0000 q^{15} +84.0000 q^{17} -100.000 q^{19} -21.0000 q^{21} -42.0000 q^{23} -109.000 q^{25} +27.0000 q^{27} +10.0000 q^{29} -48.0000 q^{31} -186.000 q^{33} -28.0000 q^{35} +246.000 q^{37} +186.000 q^{39} -248.000 q^{41} -68.0000 q^{43} +36.0000 q^{45} +324.000 q^{47} +49.0000 q^{49} +252.000 q^{51} -258.000 q^{53} -248.000 q^{55} -300.000 q^{57} -120.000 q^{59} -622.000 q^{61} -63.0000 q^{63} +248.000 q^{65} -904.000 q^{67} -126.000 q^{69} -678.000 q^{71} -642.000 q^{73} -327.000 q^{75} +434.000 q^{77} +740.000 q^{79} +81.0000 q^{81} -468.000 q^{83} +336.000 q^{85} +30.0000 q^{87} +200.000 q^{89} -434.000 q^{91} -144.000 q^{93} -400.000 q^{95} -1266.00 q^{97} -558.000 q^{99} +O(q^{100})$

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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	4.00000	0.357771	0.178885	−	0.983870i	$-0.442751\pi$
$5$	4.00000	0.357771	0.178885	+	0.983870i	$0.442751\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−62.0000	−1.69943	−0.849714	−	0.527244i	$-0.823225\pi$
$11$	−62.0000	−1.69943	−0.849714	+	0.527244i	$0.823225\pi$
$12$	0	0
$13$	62.0000	1.32275	0.661373	−	0.750057i	$-0.269974\pi$
$13$	62.0000	1.32275	0.661373	+	0.750057i	$0.269974\pi$
$14$	0	0
$15$	12.0000	0.206559
$16$	0	0
$17$	84.0000	1.19841	0.599206	−	0.800595i	$-0.295483\pi$
$17$	84.0000	1.19841	0.599206	+	0.800595i	$0.295483\pi$
$18$	0	0
$19$	−100.000	−1.20745	−0.603726	−	0.797192i	$-0.706318\pi$
$19$	−100.000	−1.20745	−0.603726	+	0.797192i	$0.706318\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	0	0
$23$	−42.0000	−0.380765	−0.190383	−	0.981710i	$-0.560973\pi$
$23$	−42.0000	−0.380765	−0.190383	+	0.981710i	$0.560973\pi$
$24$	0	0
$25$	−109.000	−0.872000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	10.0000	0.0640329	0.0320164	−	0.999487i	$-0.489807\pi$
$29$	10.0000	0.0640329	0.0320164	+	0.999487i	$0.489807\pi$
$30$	0	0
$31$	−48.0000	−0.278099	−0.139049	−	0.990285i	$-0.544405\pi$
$31$	−48.0000	−0.278099	−0.139049	+	0.990285i	$0.544405\pi$
$32$	0	0
$33$	−186.000	−0.981165
$34$	0	0
$35$	−28.0000	−0.135225
$36$	0	0
$37$	246.000	1.09303	0.546516	−	0.837449i	$-0.315954\pi$
$37$	246.000	1.09303	0.546516	+	0.837449i	$0.315954\pi$
$38$	0	0
$39$	186.000	0.763688
$40$	0	0
$41$	−248.000	−0.944661	−0.472330	−	0.881422i	$-0.656587\pi$
$41$	−248.000	−0.944661	−0.472330	+	0.881422i	$0.656587\pi$
$42$	0	0
$43$	−68.0000	−0.241161	−0.120580	−	0.992704i	$-0.538476\pi$
$43$	−68.0000	−0.241161	−0.120580	+	0.992704i	$0.538476\pi$
$44$	0	0
$45$	36.0000	0.119257
$46$	0	0
$47$	324.000	1.00554	0.502769	−	0.864421i	$-0.332315\pi$
$47$	324.000	1.00554	0.502769	+	0.864421i	$0.332315\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	252.000	0.691903
$52$	0	0
$53$	−258.000	−0.668661	−0.334330	−	0.942456i	$-0.608510\pi$
$53$	−258.000	−0.668661	−0.334330	+	0.942456i	$0.608510\pi$
$54$	0	0
$55$	−248.000	−0.608006
$56$	0	0
$57$	−300.000	−0.697122
$58$	0	0
$59$	−120.000	−0.264791	−0.132396	−	0.991197i	$-0.542267\pi$
$59$	−120.000	−0.264791	−0.132396	+	0.991197i	$0.542267\pi$
$60$	0	0
$61$	−622.000	−1.30556	−0.652778	−	0.757549i	$-0.726397\pi$
$61$	−622.000	−1.30556	−0.652778	+	0.757549i	$0.726397\pi$
$62$	0	0
$63$	−63.0000	−0.125988
$64$	0	0
$65$	248.000	0.473240
$66$	0	0
$67$	−904.000	−1.64838	−0.824188	−	0.566316i	$-0.808368\pi$
$67$	−904.000	−1.64838	−0.824188	+	0.566316i	$0.808368\pi$
$68$	0	0
$69$	−126.000	−0.219835
$70$	0	0
$71$	−678.000	−1.13329	−0.566646	−	0.823961i	$-0.691759\pi$
$71$	−678.000	−1.13329	−0.566646	+	0.823961i	$0.691759\pi$
$72$	0	0
$73$	−642.000	−1.02932	−0.514660	−	0.857394i	$-0.672082\pi$
$73$	−642.000	−1.02932	−0.514660	+	0.857394i	$0.672082\pi$
$74$	0	0
$75$	−327.000	−0.503449
$76$	0	0
$77$	434.000	0.642323
$78$	0	0
$79$	740.000	1.05388	0.526940	−	0.849903i	$-0.323339\pi$
$79$	740.000	1.05388	0.526940	+	0.849903i	$0.323339\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−468.000	−0.618912	−0.309456	−	0.950914i	$-0.600147\pi$
$83$	−468.000	−0.618912	−0.309456	+	0.950914i	$0.600147\pi$
$84$	0	0
$85$	336.000	0.428757
$86$	0	0
$87$	30.0000	0.0369694
$88$	0	0
$89$	200.000	0.238202	0.119101	−	0.992882i	$-0.461999\pi$
$89$	200.000	0.238202	0.119101	+	0.992882i	$0.461999\pi$
$90$	0	0
$91$	−434.000	−0.499951
$92$	0	0
$93$	−144.000	−0.160560
$94$	0	0
$95$	−400.000	−0.431991
$96$	0	0
$97$	−1266.00	−1.32518	−0.662592	−	0.748981i	$-0.730544\pi$
$97$	−1266.00	−1.32518	−0.662592	+	0.748981i	$0.730544\pi$
$98$	0	0
$99$	−558.000	−0.566476
$100$	0	0
$101$	−232.000	−0.228563	−0.114281	−	0.993448i	$-0.536457\pi$
$101$	−232.000	−0.228563	−0.114281	+	0.993448i	$0.536457\pi$
$102$	0	0
$103$	−1792.00	−1.71428	−0.857141	−	0.515082i	$-0.827761\pi$
$103$	−1792.00	−1.71428	−0.857141	+	0.515082i	$0.827761\pi$
$104$	0	0
$105$	−84.0000	−0.0780720
$106$	0	0
$107$	1906.00	1.72206	0.861028	−	0.508558i	$-0.169821\pi$
$107$	1906.00	1.72206	0.861028	+	0.508558i	$0.169821\pi$
$108$	0	0
$109$	90.0000	0.0790866	0.0395433	−	0.999218i	$-0.487410\pi$
$109$	90.0000	0.0790866	0.0395433	+	0.999218i	$0.487410\pi$
$110$	0	0
$111$	738.000	0.631062
$112$	0	0
$113$	458.000	0.381283	0.190642	−	0.981660i	$-0.438943\pi$
$113$	458.000	0.381283	0.190642	+	0.981660i	$0.438943\pi$
$114$	0	0
$115$	−168.000	−0.136227
$116$	0	0
$117$	558.000	0.440916
$118$	0	0
$119$	−588.000	−0.452957
$120$	0	0
$121$	2513.00	1.88805
$122$	0	0
$123$	−744.000	−0.545400
$124$	0	0
$125$	−936.000	−0.669747
$126$	0	0
$127$	804.000	0.561760	0.280880	−	0.959743i	$-0.409374\pi$
$127$	804.000	0.561760	0.280880	+	0.959743i	$0.409374\pi$
$128$	0	0
$129$	−204.000	−0.139234
$130$	0	0
$131$	−812.000	−0.541563	−0.270782	−	0.962641i	$-0.587282\pi$
$131$	−812.000	−0.541563	−0.270782	+	0.962641i	$0.587282\pi$
$132$	0	0
$133$	700.000	0.456374
$134$	0	0
$135$	108.000	0.0688530
$136$	0	0
$137$	414.000	0.258178	0.129089	−	0.991633i	$-0.458795\pi$
$137$	414.000	0.258178	0.129089	+	0.991633i	$0.458795\pi$
$138$	0	0
$139$	1620.00	0.988537	0.494268	−	0.869309i	$-0.335436\pi$
$139$	1620.00	0.988537	0.494268	+	0.869309i	$0.335436\pi$
$140$	0	0
$141$	972.000	0.580547
$142$	0	0
$143$	−3844.00	−2.24791
$144$	0	0
$145$	40.0000	0.0229091
$146$	0	0
$147$	147.000	0.0824786
$148$	0	0
$149$	−2370.00	−1.30307	−0.651537	−	0.758617i	$-0.725875\pi$
$149$	−2370.00	−1.30307	−0.651537	+	0.758617i	$0.725875\pi$
$150$	0	0
$151$	−568.000	−0.306114	−0.153057	−	0.988217i	$-0.548912\pi$
$151$	−568.000	−0.306114	−0.153057	+	0.988217i	$0.548912\pi$
$152$	0	0
$153$	756.000	0.399470
$154$	0	0
$155$	−192.000	−0.0994956
$156$	0	0
$157$	266.000	0.135217	0.0676086	−	0.997712i	$-0.478463\pi$
$157$	266.000	0.135217	0.0676086	+	0.997712i	$0.478463\pi$
$158$	0	0
$159$	−774.000	−0.386052
$160$	0	0
$161$	294.000	0.143916
$162$	0	0
$163$	272.000	0.130704	0.0653518	−	0.997862i	$-0.479183\pi$
$163$	272.000	0.130704	0.0653518	+	0.997862i	$0.479183\pi$
$164$	0	0
$165$	−744.000	−0.351032
$166$	0	0
$167$	−1876.00	−0.869277	−0.434638	−	0.900605i	$-0.643124\pi$
$167$	−1876.00	−0.869277	−0.434638	+	0.900605i	$0.643124\pi$
$168$	0	0
$169$	1647.00	0.749659
$170$	0	0
$171$	−900.000	−0.402484
$172$	0	0
$173$	152.000	0.0667997	0.0333998	−	0.999442i	$-0.489367\pi$
$173$	152.000	0.0667997	0.0333998	+	0.999442i	$0.489367\pi$
$174$	0	0
$175$	763.000	0.329585
$176$	0	0
$177$	−360.000	−0.152877
$178$	0	0
$179$	−610.000	−0.254713	−0.127356	−	0.991857i	$-0.540649\pi$
$179$	−610.000	−0.254713	−0.127356	+	0.991857i	$0.540649\pi$
$180$	0	0
$181$	−1042.00	−0.427907	−0.213954	−	0.976844i	$-0.568634\pi$
$181$	−1042.00	−0.427907	−0.213954	+	0.976844i	$0.568634\pi$
$182$	0	0
$183$	−1866.00	−0.753763
$184$	0	0
$185$	984.000	0.391055
$186$	0	0
$187$	−5208.00	−2.03661
$188$	0	0
$189$	−189.000	−0.0727393
$190$	0	0
$191$	−2038.00	−0.772065	−0.386033	−	0.922485i	$-0.626155\pi$
$191$	−2038.00	−0.772065	−0.386033	+	0.922485i	$0.626155\pi$
$192$	0	0
$193$	−2602.00	−0.970446	−0.485223	−	0.874390i	$-0.661262\pi$
$193$	−2602.00	−0.970446	−0.485223	+	0.874390i	$0.661262\pi$
$194$	0	0
$195$	744.000	0.273225
$196$	0	0
$197$	−2354.00	−0.851348	−0.425674	−	0.904877i	$-0.639963\pi$
$197$	−2354.00	−0.851348	−0.425674	+	0.904877i	$0.639963\pi$
$198$	0	0
$199$	1680.00	0.598452	0.299226	−	0.954182i	$-0.403271\pi$
$199$	1680.00	0.598452	0.299226	+	0.954182i	$0.403271\pi$
$200$	0	0
$201$	−2712.00	−0.951690
$202$	0	0
$203$	−70.0000	−0.0242022
$204$	0	0
$205$	−992.000	−0.337972
$206$	0	0
$207$	−378.000	−0.126922
$208$	0	0
$209$	6200.00	2.05198
$210$	0	0
$211$	668.000	0.217948	0.108974	−	0.994045i	$-0.465243\pi$
$211$	668.000	0.217948	0.108974	+	0.994045i	$0.465243\pi$
$212$	0	0
$213$	−2034.00	−0.654307
$214$	0	0
$215$	−272.000	−0.0862802
$216$	0	0
$217$	336.000	0.105111
$218$	0	0
$219$	−1926.00	−0.594279
$220$	0	0
$221$	5208.00	1.58519
$222$	0	0
$223$	−1832.00	−0.550134	−0.275067	−	0.961425i	$-0.588700\pi$
$223$	−1832.00	−0.550134	−0.275067	+	0.961425i	$0.588700\pi$
$224$	0	0
$225$	−981.000	−0.290667
$226$	0	0
$227$	−4944.00	−1.44557	−0.722786	−	0.691072i	$-0.757139\pi$
$227$	−4944.00	−1.44557	−0.722786	+	0.691072i	$0.757139\pi$
$228$	0	0
$229$	5470.00	1.57846	0.789231	−	0.614096i	$-0.210479\pi$
$229$	5470.00	1.57846	0.789231	+	0.614096i	$0.210479\pi$
$230$	0	0
$231$	1302.00	0.370846
$232$	0	0
$233$	−2802.00	−0.787833	−0.393917	−	0.919146i	$-0.628880\pi$
$233$	−2802.00	−0.787833	−0.393917	+	0.919146i	$0.628880\pi$
$234$	0	0
$235$	1296.00	0.359752
$236$	0	0
$237$	2220.00	0.608458
$238$	0	0
$239$	−1170.00	−0.316657	−0.158328	−	0.987386i	$-0.550610\pi$
$239$	−1170.00	−0.316657	−0.158328	+	0.987386i	$0.550610\pi$
$240$	0	0
$241$	−2338.00	−0.624912	−0.312456	−	0.949932i	$-0.601152\pi$
$241$	−2338.00	−0.624912	−0.312456	+	0.949932i	$0.601152\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	196.000	0.0511101
$246$	0	0
$247$	−6200.00	−1.59715
$248$	0	0
$249$	−1404.00	−0.357329
$250$	0	0
$251$	−2792.00	−0.702109	−0.351055	−	0.936355i	$-0.614177\pi$
$251$	−2792.00	−0.702109	−0.351055	+	0.936355i	$0.614177\pi$
$252$	0	0
$253$	2604.00	0.647083
$254$	0	0
$255$	1008.00	0.247543
$256$	0	0
$257$	7024.00	1.70484	0.852422	−	0.522854i	$-0.175133\pi$
$257$	7024.00	1.70484	0.852422	+	0.522854i	$0.175133\pi$
$258$	0	0
$259$	−1722.00	−0.413127
$260$	0	0
$261$	90.0000	0.0213443
$262$	0	0
$263$	2438.00	0.571610	0.285805	−	0.958288i	$-0.407739\pi$
$263$	2438.00	0.571610	0.285805	+	0.958288i	$0.407739\pi$
$264$	0	0
$265$	−1032.00	−0.239227
$266$	0	0
$267$	600.000	0.137526
$268$	0	0
$269$	6780.00	1.53674	0.768372	−	0.640004i	$-0.221067\pi$
$269$	6780.00	1.53674	0.768372	+	0.640004i	$0.221067\pi$
$270$	0	0
$271$	−1928.00	−0.432168	−0.216084	−	0.976375i	$-0.569329\pi$
$271$	−1928.00	−0.432168	−0.216084	+	0.976375i	$0.569329\pi$
$272$	0	0
$273$	−1302.00	−0.288647
$274$	0	0
$275$	6758.00	1.48190
$276$	0	0
$277$	−5554.00	−1.20472	−0.602360	−	0.798224i	$-0.705773\pi$
$277$	−5554.00	−1.20472	−0.602360	+	0.798224i	$0.705773\pi$
$278$	0	0
$279$	−432.000	−0.0926995
$280$	0	0
$281$	1942.00	0.412278	0.206139	−	0.978523i	$-0.433910\pi$
$281$	1942.00	0.412278	0.206139	+	0.978523i	$0.433910\pi$
$282$	0	0
$283$	−4828.00	−1.01412	−0.507058	−	0.861912i	$-0.669267\pi$
$283$	−4828.00	−1.01412	−0.507058	+	0.861912i	$0.669267\pi$
$284$	0	0
$285$	−1200.00	−0.249410
$286$	0	0
$287$	1736.00	0.357048
$288$	0	0
$289$	2143.00	0.436190
$290$	0	0
$291$	−3798.00	−0.765095
$292$	0	0
$293$	6152.00	1.22663	0.613317	−	0.789837i	$-0.289835\pi$
$293$	6152.00	1.22663	0.613317	+	0.789837i	$0.289835\pi$
$294$	0	0
$295$	−480.000	−0.0947345
$296$	0	0
$297$	−1674.00	−0.327055
$298$	0	0
$299$	−2604.00	−0.503656
$300$	0	0
$301$	476.000	0.0911501
$302$	0	0
$303$	−696.000	−0.131961
$304$	0	0
$305$	−2488.00	−0.467090
$306$	0	0
$307$	−5884.00	−1.09387	−0.546934	−	0.837176i	$-0.684205\pi$
$307$	−5884.00	−1.09387	−0.546934	+	0.837176i	$0.684205\pi$
$308$	0	0
$309$	−5376.00	−0.989741
$310$	0	0
$311$	9132.00	1.66504	0.832521	−	0.553993i	$-0.186897\pi$
$311$	9132.00	1.66504	0.832521	+	0.553993i	$0.186897\pi$
$312$	0	0
$313$	−9382.00	−1.69426	−0.847128	−	0.531389i	$-0.821670\pi$
$313$	−9382.00	−1.69426	−0.847128	+	0.531389i	$0.821670\pi$
$314$	0	0
$315$	−252.000	−0.0450749
$316$	0	0
$317$	−3114.00	−0.551734	−0.275867	−	0.961196i	$-0.588965\pi$
$317$	−3114.00	−0.551734	−0.275867	+	0.961196i	$0.588965\pi$
$318$	0	0
$319$	−620.000	−0.108819
$320$	0	0
$321$	5718.00	0.994229
$322$	0	0
$323$	−8400.00	−1.44702
$324$	0	0
$325$	−6758.00	−1.15344
$326$	0	0
$327$	270.000	0.0456607
$328$	0	0
$329$	−2268.00	−0.380057
$330$	0	0
$331$	−1532.00	−0.254400	−0.127200	−	0.991877i	$-0.540599\pi$
$331$	−1532.00	−0.254400	−0.127200	+	0.991877i	$0.540599\pi$
$332$	0	0
$333$	2214.00	0.364344
$334$	0	0
$335$	−3616.00	−0.589741
$336$	0	0
$337$	−4166.00	−0.673402	−0.336701	−	0.941612i	$-0.609311\pi$
$337$	−4166.00	−0.673402	−0.336701	+	0.941612i	$0.609311\pi$
$338$	0	0
$339$	1374.00	0.220134
$340$	0	0
$341$	2976.00	0.472608
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−504.000	−0.0786506
$346$	0	0
$347$	11366.0	1.75838	0.879191	−	0.476469i	$-0.158083\pi$
$347$	11366.0	1.75838	0.879191	+	0.476469i	$0.158083\pi$
$348$	0	0
$349$	−9310.00	−1.42795	−0.713973	−	0.700174i	$-0.753106\pi$
$349$	−9310.00	−1.42795	−0.713973	+	0.700174i	$0.753106\pi$
$350$	0	0
$351$	1674.00	0.254563
$352$	0	0
$353$	−8572.00	−1.29247	−0.646234	−	0.763139i	$-0.723657\pi$
$353$	−8572.00	−1.29247	−0.646234	+	0.763139i	$0.723657\pi$
$354$	0	0
$355$	−2712.00	−0.405459
$356$	0	0
$357$	−1764.00	−0.261515
$358$	0	0
$359$	−4790.00	−0.704196	−0.352098	−	0.935963i	$-0.614532\pi$
$359$	−4790.00	−0.704196	−0.352098	+	0.935963i	$0.614532\pi$
$360$	0	0
$361$	3141.00	0.457938
$362$	0	0
$363$	7539.00	1.09007
$364$	0	0
$365$	−2568.00	−0.368261
$366$	0	0
$367$	5424.00	0.771473	0.385736	−	0.922609i	$-0.373947\pi$
$367$	5424.00	0.771473	0.385736	+	0.922609i	$0.373947\pi$
$368$	0	0
$369$	−2232.00	−0.314887
$370$	0	0
$371$	1806.00	0.252730
$372$	0	0
$373$	−1838.00	−0.255142	−0.127571	−	0.991829i	$-0.540718\pi$
$373$	−1838.00	−0.255142	−0.127571	+	0.991829i	$0.540718\pi$
$374$	0	0
$375$	−2808.00	−0.386679
$376$	0	0
$377$	620.000	0.0846993
$378$	0	0
$379$	4260.00	0.577365	0.288683	−	0.957425i	$-0.406783\pi$
$379$	4260.00	0.577365	0.288683	+	0.957425i	$0.406783\pi$
$380$	0	0
$381$	2412.00	0.324332
$382$	0	0
$383$	9048.00	1.20713	0.603566	−	0.797313i	$-0.293746\pi$
$383$	9048.00	1.20713	0.603566	+	0.797313i	$0.293746\pi$
$384$	0	0
$385$	1736.00	0.229805
$386$	0	0
$387$	−612.000	−0.0803868
$388$	0	0
$389$	11490.0	1.49760	0.748800	−	0.662796i	$-0.230631\pi$
$389$	11490.0	1.49760	0.748800	+	0.662796i	$0.230631\pi$
$390$	0	0
$391$	−3528.00	−0.456314
$392$	0	0
$393$	−2436.00	−0.312672
$394$	0	0
$395$	2960.00	0.377048
$396$	0	0
$397$	1866.00	0.235899	0.117949	−	0.993020i	$-0.462368\pi$
$397$	1866.00	0.235899	0.117949	+	0.993020i	$0.462368\pi$
$398$	0	0
$399$	2100.00	0.263487
$400$	0	0
$401$	13662.0	1.70137	0.850683	−	0.525679i	$-0.176189\pi$
$401$	13662.0	1.70137	0.850683	+	0.525679i	$0.176189\pi$
$402$	0	0
$403$	−2976.00	−0.367854
$404$	0	0
$405$	324.000	0.0397523
$406$	0	0
$407$	−15252.0	−1.85753
$408$	0	0
$409$	−13210.0	−1.59705	−0.798524	−	0.601963i	$-0.794385\pi$
$409$	−13210.0	−1.59705	−0.798524	+	0.601963i	$0.794385\pi$
$410$	0	0
$411$	1242.00	0.149059
$412$	0	0
$413$	840.000	0.100082
$414$	0	0
$415$	−1872.00	−0.221429
$416$	0	0
$417$	4860.00	0.570732
$418$	0	0
$419$	−6960.00	−0.811499	−0.405750	−	0.913984i	$-0.632990\pi$
$419$	−6960.00	−0.811499	−0.405750	+	0.913984i	$0.632990\pi$
$420$	0	0
$421$	−8162.00	−0.944873	−0.472437	−	0.881365i	$-0.656625\pi$
$421$	−8162.00	−0.944873	−0.472437	+	0.881365i	$0.656625\pi$
$422$	0	0
$423$	2916.00	0.335179
$424$	0	0
$425$	−9156.00	−1.04501
$426$	0	0
$427$	4354.00	0.493454
$428$	0	0
$429$	−11532.0	−1.29783
$430$	0	0
$431$	16602.0	1.85543	0.927715	−	0.373290i	$-0.121770\pi$
$431$	16602.0	1.85543	0.927715	+	0.373290i	$0.121770\pi$
$432$	0	0
$433$	7738.00	0.858810	0.429405	−	0.903112i	$-0.358723\pi$
$433$	7738.00	0.858810	0.429405	+	0.903112i	$0.358723\pi$
$434$	0	0
$435$	120.000	0.0132266
$436$	0	0
$437$	4200.00	0.459756
$438$	0	0
$439$	−840.000	−0.0913235	−0.0456617	−	0.998957i	$-0.514540\pi$
$439$	−840.000	−0.0913235	−0.0456617	+	0.998957i	$0.514540\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	−6618.00	−0.709776	−0.354888	−	0.934909i	$-0.615481\pi$
$443$	−6618.00	−0.709776	−0.354888	+	0.934909i	$0.615481\pi$
$444$	0	0
$445$	800.000	0.0852217
$446$	0	0
$447$	−7110.00	−0.752330
$448$	0	0
$449$	3090.00	0.324780	0.162390	−	0.986727i	$-0.448080\pi$
$449$	3090.00	0.324780	0.162390	+	0.986727i	$0.448080\pi$
$450$	0	0
$451$	15376.0	1.60538
$452$	0	0
$453$	−1704.00	−0.176735
$454$	0	0
$455$	−1736.00	−0.178868
$456$	0	0
$457$	5914.00	0.605351	0.302675	−	0.953094i	$-0.402120\pi$
$457$	5914.00	0.605351	0.302675	+	0.953094i	$0.402120\pi$
$458$	0	0
$459$	2268.00	0.230634
$460$	0	0
$461$	15968.0	1.61324	0.806620	−	0.591070i	$-0.201294\pi$
$461$	15968.0	1.61324	0.806620	+	0.591070i	$0.201294\pi$
$462$	0	0
$463$	−1172.00	−0.117640	−0.0588202	−	0.998269i	$-0.518734\pi$
$463$	−1172.00	−0.117640	−0.0588202	+	0.998269i	$0.518734\pi$
$464$	0	0
$465$	−576.000	−0.0574438
$466$	0	0
$467$	−5304.00	−0.525567	−0.262784	−	0.964855i	$-0.584641\pi$
$467$	−5304.00	−0.525567	−0.262784	+	0.964855i	$0.584641\pi$
$468$	0	0
$469$	6328.00	0.623027
$470$	0	0
$471$	798.000	0.0780677
$472$	0	0
$473$	4216.00	0.409835
$474$	0	0
$475$	10900.0	1.05290
$476$	0	0
$477$	−2322.00	−0.222887
$478$	0	0
$479$	5740.00	0.547531	0.273765	−	0.961796i	$-0.411731\pi$
$479$	5740.00	0.547531	0.273765	+	0.961796i	$0.411731\pi$
$480$	0	0
$481$	15252.0	1.44580
$482$	0	0
$483$	882.000	0.0830898
$484$	0	0
$485$	−5064.00	−0.474112
$486$	0	0
$487$	8944.00	0.832220	0.416110	−	0.909314i	$-0.363393\pi$
$487$	8944.00	0.832220	0.416110	+	0.909314i	$0.363393\pi$
$488$	0	0
$489$	816.000	0.0754617
$490$	0	0
$491$	5558.00	0.510853	0.255427	−	0.966828i	$-0.417784\pi$
$491$	5558.00	0.510853	0.255427	+	0.966828i	$0.417784\pi$
$492$	0	0
$493$	840.000	0.0767377
$494$	0	0
$495$	−2232.00	−0.202669
$496$	0	0
$497$	4746.00	0.428344
$498$	0	0
$499$	19820.0	1.77809	0.889043	−	0.457823i	$-0.151371\pi$
$499$	19820.0	1.77809	0.889043	+	0.457823i	$0.151371\pi$
$500$	0	0
$501$	−5628.00	−0.501877
$502$	0	0
$503$	1848.00	0.163814	0.0819068	−	0.996640i	$-0.473899\pi$
$503$	1848.00	0.163814	0.0819068	+	0.996640i	$0.473899\pi$
$504$	0	0
$505$	−928.000	−0.0817732
$506$	0	0
$507$	4941.00	0.432816
$508$	0	0
$509$	−340.000	−0.0296075	−0.0148038	−	0.999890i	$-0.504712\pi$
$509$	−340.000	−0.0296075	−0.0148038	+	0.999890i	$0.504712\pi$
$510$	0	0
$511$	4494.00	0.389047
$512$	0	0
$513$	−2700.00	−0.232374
$514$	0	0
$515$	−7168.00	−0.613320
$516$	0	0
$517$	−20088.0	−1.70884
$518$	0	0
$519$	456.000	0.0385668
$520$	0	0
$521$	10212.0	0.858725	0.429363	−	0.903132i	$-0.358738\pi$
$521$	10212.0	0.858725	0.429363	+	0.903132i	$0.358738\pi$
$522$	0	0
$523$	9332.00	0.780229	0.390115	−	0.920766i	$-0.372435\pi$
$523$	9332.00	0.780229	0.390115	+	0.920766i	$0.372435\pi$
$524$	0	0
$525$	2289.00	0.190286
$526$	0	0
$527$	−4032.00	−0.333276
$528$	0	0
$529$	−10403.0	−0.855018
$530$	0	0
$531$	−1080.00	−0.0882637
$532$	0	0
$533$	−15376.0	−1.24955
$534$	0	0
$535$	7624.00	0.616101
$536$	0	0
$537$	−1830.00	−0.147058
$538$	0	0
$539$	−3038.00	−0.242775
$540$	0	0
$541$	8998.00	0.715073	0.357536	−	0.933899i	$-0.383617\pi$
$541$	8998.00	0.715073	0.357536	+	0.933899i	$0.383617\pi$
$542$	0	0
$543$	−3126.00	−0.247052
$544$	0	0
$545$	360.000	0.0282949
$546$	0	0
$547$	3416.00	0.267016	0.133508	−	0.991048i	$-0.457376\pi$
$547$	3416.00	0.267016	0.133508	+	0.991048i	$0.457376\pi$
$548$	0	0
$549$	−5598.00	−0.435185
$550$	0	0
$551$	−1000.00	−0.0773166
$552$	0	0
$553$	−5180.00	−0.398329
$554$	0	0
$555$	2952.00	0.225776
$556$	0	0
$557$	526.000	0.0400132	0.0200066	−	0.999800i	$-0.493631\pi$
$557$	526.000	0.0400132	0.0200066	+	0.999800i	$0.493631\pi$
$558$	0	0
$559$	−4216.00	−0.318994
$560$	0	0
$561$	−15624.0	−1.17584
$562$	0	0
$563$	6712.00	0.502446	0.251223	−	0.967929i	$-0.419167\pi$
$563$	6712.00	0.502446	0.251223	+	0.967929i	$0.419167\pi$
$564$	0	0
$565$	1832.00	0.136412
$566$	0	0
$567$	−567.000	−0.0419961
$568$	0	0
$569$	4190.00	0.308706	0.154353	−	0.988016i	$-0.450671\pi$
$569$	4190.00	0.308706	0.154353	+	0.988016i	$0.450671\pi$
$570$	0	0
$571$	−3032.00	−0.222216	−0.111108	−	0.993808i	$-0.535440\pi$
$571$	−3032.00	−0.222216	−0.111108	+	0.993808i	$0.535440\pi$
$572$	0	0
$573$	−6114.00	−0.445752
$574$	0	0
$575$	4578.00	0.332027
$576$	0	0
$577$	5434.00	0.392063	0.196032	−	0.980598i	$-0.437195\pi$
$577$	5434.00	0.392063	0.196032	+	0.980598i	$0.437195\pi$
$578$	0	0
$579$	−7806.00	−0.560287
$580$	0	0
$581$	3276.00	0.233927
$582$	0	0
$583$	15996.0	1.13634
$584$	0	0
$585$	2232.00	0.157747
$586$	0	0
$587$	−464.000	−0.0326258	−0.0163129	−	0.999867i	$-0.505193\pi$
$587$	−464.000	−0.0326258	−0.0163129	+	0.999867i	$0.505193\pi$
$588$	0	0
$589$	4800.00	0.335790
$590$	0	0
$591$	−7062.00	−0.491526
$592$	0	0
$593$	11748.0	0.813546	0.406773	−	0.913529i	$-0.366654\pi$
$593$	11748.0	0.813546	0.406773	+	0.913529i	$0.366654\pi$
$594$	0	0
$595$	−2352.00	−0.162055
$596$	0	0
$597$	5040.00	0.345517
$598$	0	0
$599$	7650.00	0.521821	0.260910	−	0.965363i	$-0.415977\pi$
$599$	7650.00	0.521821	0.260910	+	0.965363i	$0.415977\pi$
$600$	0	0
$601$	−22878.0	−1.55277	−0.776384	−	0.630261i	$-0.782948\pi$
$601$	−22878.0	−1.55277	−0.776384	+	0.630261i	$0.782948\pi$
$602$	0	0
$603$	−8136.00	−0.549459
$604$	0	0
$605$	10052.0	0.675491
$606$	0	0
$607$	704.000	0.0470749	0.0235375	−	0.999723i	$-0.492507\pi$
$607$	704.000	0.0470749	0.0235375	+	0.999723i	$0.492507\pi$
$608$	0	0
$609$	−210.000	−0.0139731
$610$	0	0
$611$	20088.0	1.33007
$612$	0	0
$613$	−24958.0	−1.64444	−0.822222	−	0.569167i	$-0.807266\pi$
$613$	−24958.0	−1.64444	−0.822222	+	0.569167i	$0.807266\pi$
$614$	0	0
$615$	−2976.00	−0.195128
$616$	0	0
$617$	−8826.00	−0.575886	−0.287943	−	0.957648i	$-0.592971\pi$
$617$	−8826.00	−0.575886	−0.287943	+	0.957648i	$0.592971\pi$
$618$	0	0
$619$	−21220.0	−1.37787	−0.688937	−	0.724821i	$-0.741922\pi$
$619$	−21220.0	−1.37787	−0.688937	+	0.724821i	$0.741922\pi$
$620$	0	0
$621$	−1134.00	−0.0732783
$622$	0	0
$623$	−1400.00	−0.0900318
$624$	0	0
$625$	9881.00	0.632384
$626$	0	0
$627$	18600.0	1.18471
$628$	0	0
$629$	20664.0	1.30990
$630$	0	0
$631$	−3268.00	−0.206176	−0.103088	−	0.994672i	$-0.532872\pi$
$631$	−3268.00	−0.206176	−0.103088	+	0.994672i	$0.532872\pi$
$632$	0	0
$633$	2004.00	0.125832
$634$	0	0
$635$	3216.00	0.200981
$636$	0	0
$637$	3038.00	0.188964
$638$	0	0
$639$	−6102.00	−0.377764
$640$	0	0
$641$	13062.0	0.804864	0.402432	−	0.915450i	$-0.368165\pi$
$641$	13062.0	0.804864	0.402432	+	0.915450i	$0.368165\pi$
$642$	0	0
$643$	28012.0	1.71802	0.859009	−	0.511961i	$-0.171081\pi$
$643$	28012.0	1.71802	0.859009	+	0.511961i	$0.171081\pi$
$644$	0	0
$645$	−816.000	−0.0498139
$646$	0	0
$647$	3844.00	0.233575	0.116788	−	0.993157i	$-0.462740\pi$
$647$	3844.00	0.233575	0.116788	+	0.993157i	$0.462740\pi$
$648$	0	0
$649$	7440.00	0.449993
$650$	0	0
$651$	1008.00	0.0606861
$652$	0	0
$653$	28482.0	1.70687	0.853436	−	0.521198i	$-0.174515\pi$
$653$	28482.0	1.70687	0.853436	+	0.521198i	$0.174515\pi$
$654$	0	0
$655$	−3248.00	−0.193756
$656$	0	0
$657$	−5778.00	−0.343107
$658$	0	0
$659$	9330.00	0.551510	0.275755	−	0.961228i	$-0.411072\pi$
$659$	9330.00	0.551510	0.275755	+	0.961228i	$0.411072\pi$
$660$	0	0
$661$	−8782.00	−0.516763	−0.258381	−	0.966043i	$-0.583189\pi$
$661$	−8782.00	−0.516763	−0.258381	+	0.966043i	$0.583189\pi$
$662$	0	0
$663$	15624.0	0.915212
$664$	0	0
$665$	2800.00	0.163277
$666$	0	0
$667$	−420.000	−0.0243815
$668$	0	0
$669$	−5496.00	−0.317620
$670$	0	0
$671$	38564.0	2.21870
$672$	0	0
$673$	−10562.0	−0.604956	−0.302478	−	0.953156i	$-0.597814\pi$
$673$	−10562.0	−0.604956	−0.302478	+	0.953156i	$0.597814\pi$
$674$	0	0
$675$	−2943.00	−0.167816
$676$	0	0
$677$	26016.0	1.47692	0.738461	−	0.674296i	$-0.235553\pi$
$677$	26016.0	1.47692	0.738461	+	0.674296i	$0.235553\pi$
$678$	0	0
$679$	8862.00	0.500872
$680$	0	0
$681$	−14832.0	−0.834601
$682$	0	0
$683$	−8898.00	−0.498496	−0.249248	−	0.968440i	$-0.580183\pi$
$683$	−8898.00	−0.498496	−0.249248	+	0.968440i	$0.580183\pi$
$684$	0	0
$685$	1656.00	0.0923686
$686$	0	0
$687$	16410.0	0.911325
$688$	0	0
$689$	−15996.0	−0.884469
$690$	0	0
$691$	−30572.0	−1.68309	−0.841544	−	0.540189i	$-0.818353\pi$
$691$	−30572.0	−1.68309	−0.841544	+	0.540189i	$0.818353\pi$
$692$	0	0
$693$	3906.00	0.214108
$694$	0	0
$695$	6480.00	0.353670
$696$	0	0
$697$	−20832.0	−1.13209
$698$	0	0
$699$	−8406.00	−0.454856
$700$	0	0
$701$	30618.0	1.64968	0.824840	−	0.565366i	$-0.191265\pi$
$701$	30618.0	1.64968	0.824840	+	0.565366i	$0.191265\pi$
$702$	0	0
$703$	−24600.0	−1.31978
$704$	0	0
$705$	3888.00	0.207703
$706$	0	0
$707$	1624.00	0.0863887
$708$	0	0
$709$	8130.00	0.430647	0.215323	−	0.976543i	$-0.430919\pi$
$709$	8130.00	0.430647	0.215323	+	0.976543i	$0.430919\pi$
$710$	0	0
$711$	6660.00	0.351293
$712$	0	0
$713$	2016.00	0.105890
$714$	0	0
$715$	−15376.0	−0.804237
$716$	0	0
$717$	−3510.00	−0.182822
$718$	0	0
$719$	−27840.0	−1.44403	−0.722014	−	0.691878i	$-0.756784\pi$
$719$	−27840.0	−1.44403	−0.722014	+	0.691878i	$0.756784\pi$
$720$	0	0
$721$	12544.0	0.647938
$722$	0	0
$723$	−7014.00	−0.360793
$724$	0	0
$725$	−1090.00	−0.0558367
$726$	0	0
$727$	14624.0	0.746044	0.373022	−	0.927822i	$-0.378322\pi$
$727$	14624.0	0.746044	0.373022	+	0.927822i	$0.378322\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−5712.00	−0.289010
$732$	0	0
$733$	20862.0	1.05124	0.525618	−	0.850721i	$-0.323834\pi$
$733$	20862.0	1.05124	0.525618	+	0.850721i	$0.323834\pi$
$734$	0	0
$735$	588.000	0.0295084
$736$	0	0
$737$	56048.0	2.80130
$738$	0	0
$739$	13920.0	0.692903	0.346452	−	0.938068i	$-0.387386\pi$
$739$	13920.0	0.692903	0.346452	+	0.938068i	$0.387386\pi$
$740$	0	0
$741$	−18600.0	−0.922116
$742$	0	0
$743$	25578.0	1.26294	0.631471	−	0.775400i	$-0.282452\pi$
$743$	25578.0	1.26294	0.631471	+	0.775400i	$0.282452\pi$
$744$	0	0
$745$	−9480.00	−0.466202
$746$	0	0
$747$	−4212.00	−0.206304
$748$	0	0
$749$	−13342.0	−0.650876
$750$	0	0
$751$	33472.0	1.62638	0.813189	−	0.581999i	$-0.197729\pi$
$751$	33472.0	1.62638	0.813189	+	0.581999i	$0.197729\pi$
$752$	0	0
$753$	−8376.00	−0.405363
$754$	0	0
$755$	−2272.00	−0.109519
$756$	0	0
$757$	−25934.0	−1.24516	−0.622581	−	0.782556i	$-0.713916\pi$
$757$	−25934.0	−1.24516	−0.622581	+	0.782556i	$0.713916\pi$
$758$	0	0
$759$	7812.00	0.373594
$760$	0	0
$761$	26952.0	1.28385	0.641925	−	0.766768i	$-0.278136\pi$
$761$	26952.0	1.28385	0.641925	+	0.766768i	$0.278136\pi$
$762$	0	0
$763$	−630.000	−0.0298919
$764$	0	0
$765$	3024.00	0.142919
$766$	0	0
$767$	−7440.00	−0.350251
$768$	0	0
$769$	23450.0	1.09965	0.549824	−	0.835281i	$-0.314695\pi$
$769$	23450.0	1.09965	0.549824	+	0.835281i	$0.314695\pi$
$770$	0	0
$771$	21072.0	0.984293
$772$	0	0
$773$	−39568.0	−1.84109	−0.920545	−	0.390637i	$-0.872255\pi$
$773$	−39568.0	−1.84109	−0.920545	+	0.390637i	$0.872255\pi$
$774$	0	0
$775$	5232.00	0.242502
$776$	0	0
$777$	−5166.00	−0.238519
$778$	0	0
$779$	24800.0	1.14063
$780$	0	0
$781$	42036.0	1.92595
$782$	0	0
$783$	270.000	0.0123231
$784$	0	0
$785$	1064.00	0.0483768
$786$	0	0
$787$	12356.0	0.559649	0.279825	−	0.960051i	$-0.409724\pi$
$787$	12356.0	0.559649	0.279825	+	0.960051i	$0.409724\pi$
$788$	0	0
$789$	7314.00	0.330019
$790$	0	0
$791$	−3206.00	−0.144112
$792$	0	0
$793$	−38564.0	−1.72692
$794$	0	0
$795$	−3096.00	−0.138118
$796$	0	0
$797$	21736.0	0.966033	0.483017	−	0.875611i	$-0.339541\pi$
$797$	21736.0	0.966033	0.483017	+	0.875611i	$0.339541\pi$
$798$	0	0
$799$	27216.0	1.20505
$800$	0	0
$801$	1800.00	0.0794006
$802$	0	0
$803$	39804.0	1.74926
$804$	0	0
$805$	1176.00	0.0514889
$806$	0	0
$807$	20340.0	0.887239
$808$	0	0
$809$	−38310.0	−1.66490	−0.832452	−	0.554097i	$-0.813064\pi$
$809$	−38310.0	−1.66490	−0.832452	+	0.554097i	$0.813064\pi$
$810$	0	0
$811$	−2132.00	−0.0923115	−0.0461558	−	0.998934i	$-0.514697\pi$
$811$	−2132.00	−0.0923115	−0.0461558	+	0.998934i	$0.514697\pi$
$812$	0	0
$813$	−5784.00	−0.249513
$814$	0	0
$815$	1088.00	0.0467619
$816$	0	0
$817$	6800.00	0.291190
$818$	0	0
$819$	−3906.00	−0.166650
$820$	0	0
$821$	−5002.00	−0.212632	−0.106316	−	0.994332i	$-0.533906\pi$
$821$	−5002.00	−0.212632	−0.106316	+	0.994332i	$0.533906\pi$
$822$	0	0
$823$	−3612.00	−0.152985	−0.0764923	−	0.997070i	$-0.524372\pi$
$823$	−3612.00	−0.152985	−0.0764923	+	0.997070i	$0.524372\pi$
$824$	0	0
$825$	20274.0	0.855576
$826$	0	0
$827$	27666.0	1.16329	0.581645	−	0.813443i	$-0.302409\pi$
$827$	27666.0	1.16329	0.581645	+	0.813443i	$0.302409\pi$
$828$	0	0
$829$	−12890.0	−0.540034	−0.270017	−	0.962856i	$-0.587029\pi$
$829$	−12890.0	−0.540034	−0.270017	+	0.962856i	$0.587029\pi$
$830$	0	0
$831$	−16662.0	−0.695546
$832$	0	0
$833$	4116.00	0.171202
$834$	0	0
$835$	−7504.00	−0.311002
$836$	0	0
$837$	−1296.00	−0.0535201
$838$	0	0
$839$	−9340.00	−0.384330	−0.192165	−	0.981363i	$-0.561551\pi$
$839$	−9340.00	−0.384330	−0.192165	+	0.981363i	$0.561551\pi$
$840$	0	0
$841$	−24289.0	−0.995900
$842$	0	0
$843$	5826.00	0.238029
$844$	0	0
$845$	6588.00	0.268206
$846$	0	0
$847$	−17591.0	−0.713617
$848$	0	0
$849$	−14484.0	−0.585500
$850$	0	0
$851$	−10332.0	−0.416188
$852$	0	0
$853$	33082.0	1.32791	0.663954	−	0.747773i	$-0.268877\pi$
$853$	33082.0	1.32791	0.663954	+	0.747773i	$0.268877\pi$
$854$	0	0
$855$	−3600.00	−0.143997
$856$	0	0
$857$	7544.00	0.300698	0.150349	−	0.988633i	$-0.451960\pi$
$857$	7544.00	0.300698	0.150349	+	0.988633i	$0.451960\pi$
$858$	0	0
$859$	−8180.00	−0.324910	−0.162455	−	0.986716i	$-0.551941\pi$
$859$	−8180.00	−0.324910	−0.162455	+	0.986716i	$0.551941\pi$
$860$	0	0
$861$	5208.00	0.206142
$862$	0	0
$863$	10518.0	0.414875	0.207437	−	0.978248i	$-0.433488\pi$
$863$	10518.0	0.414875	0.207437	+	0.978248i	$0.433488\pi$
$864$	0	0
$865$	608.000	0.0238990
$866$	0	0
$867$	6429.00	0.251834
$868$	0	0
$869$	−45880.0	−1.79099
$870$	0	0
$871$	−56048.0	−2.18038
$872$	0	0
$873$	−11394.0	−0.441728
$874$	0	0
$875$	6552.00	0.253141
$876$	0	0
$877$	−14134.0	−0.544209	−0.272104	−	0.962268i	$-0.587720\pi$
$877$	−14134.0	−0.544209	−0.272104	+	0.962268i	$0.587720\pi$
$878$	0	0
$879$	18456.0	0.708197
$880$	0	0
$881$	6492.00	0.248265	0.124132	−	0.992266i	$-0.460385\pi$
$881$	6492.00	0.248265	0.124132	+	0.992266i	$0.460385\pi$
$882$	0	0
$883$	−38228.0	−1.45694	−0.728468	−	0.685080i	$-0.759767\pi$
$883$	−38228.0	−1.45694	−0.728468	+	0.685080i	$0.759767\pi$
$884$	0	0
$885$	−1440.00	−0.0546950
$886$	0	0
$887$	−43076.0	−1.63061	−0.815305	−	0.579032i	$-0.803431\pi$
$887$	−43076.0	−1.63061	−0.815305	+	0.579032i	$0.803431\pi$
$888$	0	0
$889$	−5628.00	−0.212325
$890$	0	0
$891$	−5022.00	−0.188825
$892$	0	0
$893$	−32400.0	−1.21414
$894$	0	0
$895$	−2440.00	−0.0911287
$896$	0	0
$897$	−7812.00	−0.290786
$898$	0	0
$899$	−480.000	−0.0178074
$900$	0	0
$901$	−21672.0	−0.801331
$902$	0	0
$903$	1428.00	0.0526255
$904$	0	0
$905$	−4168.00	−0.153093
$906$	0	0
$907$	32236.0	1.18013	0.590065	−	0.807355i	$-0.299102\pi$
$907$	32236.0	1.18013	0.590065	+	0.807355i	$0.299102\pi$
$908$	0	0
$909$	−2088.00	−0.0761877
$910$	0	0
$911$	−46518.0	−1.69178	−0.845889	−	0.533359i	$-0.820930\pi$
$911$	−46518.0	−1.69178	−0.845889	+	0.533359i	$0.820930\pi$
$912$	0	0
$913$	29016.0	1.05180
$914$	0	0
$915$	−7464.00	−0.269675
$916$	0	0
$917$	5684.00	0.204692
$918$	0	0
$919$	17840.0	0.640356	0.320178	−	0.947357i	$-0.396257\pi$
$919$	17840.0	0.640356	0.320178	+	0.947357i	$0.396257\pi$
$920$	0	0
$921$	−17652.0	−0.631545
$922$	0	0
$923$	−42036.0	−1.49906
$924$	0	0
$925$	−26814.0	−0.953123
$926$	0	0
$927$	−16128.0	−0.571427
$928$	0	0
$929$	7000.00	0.247215	0.123607	−	0.992331i	$-0.460554\pi$
$929$	7000.00	0.247215	0.123607	+	0.992331i	$0.460554\pi$
$930$	0	0
$931$	−4900.00	−0.172493
$932$	0	0
$933$	27396.0	0.961313
$934$	0	0
$935$	−20832.0	−0.728641
$936$	0	0
$937$	36114.0	1.25912	0.629559	−	0.776953i	$-0.283236\pi$
$937$	36114.0	1.25912	0.629559	+	0.776953i	$0.283236\pi$
$938$	0	0
$939$	−28146.0	−0.978179
$940$	0	0
$941$	4748.00	0.164485	0.0822425	−	0.996612i	$-0.473792\pi$
$941$	4748.00	0.164485	0.0822425	+	0.996612i	$0.473792\pi$
$942$	0	0
$943$	10416.0	0.359694
$944$	0	0
$945$	−756.000	−0.0260240
$946$	0	0
$947$	−42694.0	−1.46501	−0.732507	−	0.680759i	$-0.761650\pi$
$947$	−42694.0	−1.46501	−0.732507	+	0.680759i	$0.761650\pi$
$948$	0	0
$949$	−39804.0	−1.36153
$950$	0	0
$951$	−9342.00	−0.318544
$952$	0	0
$953$	−16742.0	−0.569073	−0.284537	−	0.958665i	$-0.591840\pi$
$953$	−16742.0	−0.569073	−0.284537	+	0.958665i	$0.591840\pi$
$954$	0	0
$955$	−8152.00	−0.276223
$956$	0	0
$957$	−1860.00	−0.0628268
$958$	0	0
$959$	−2898.00	−0.0975822
$960$	0	0
$961$	−27487.0	−0.922661
$962$	0	0
$963$	17154.0	0.574019
$964$	0	0
$965$	−10408.0	−0.347197
$966$	0	0
$967$	−9956.00	−0.331089	−0.165545	−	0.986202i	$-0.552938\pi$
$967$	−9956.00	−0.331089	−0.165545	+	0.986202i	$0.552938\pi$
$968$	0	0
$969$	−25200.0	−0.835439
$970$	0	0
$971$	26388.0	0.872123	0.436061	−	0.899917i	$-0.356373\pi$
$971$	26388.0	0.872123	0.436061	+	0.899917i	$0.356373\pi$
$972$	0	0
$973$	−11340.0	−0.373632
$974$	0	0
$975$	−20274.0	−0.665936
$976$	0	0
$977$	−786.000	−0.0257383	−0.0128692	−	0.999917i	$-0.504096\pi$
$977$	−786.000	−0.0257383	−0.0128692	+	0.999917i	$0.504096\pi$
$978$	0	0
$979$	−12400.0	−0.404807
$980$	0	0
$981$	810.000	0.0263622
$982$	0	0
$983$	51888.0	1.68359	0.841796	−	0.539796i	$-0.181499\pi$
$983$	51888.0	1.68359	0.841796	+	0.539796i	$0.181499\pi$
$984$	0	0
$985$	−9416.00	−0.304588
$986$	0	0
$987$	−6804.00	−0.219426
$988$	0	0
$989$	2856.00	0.0918256
$990$	0	0
$991$	−51928.0	−1.66453	−0.832264	−	0.554379i	$-0.812956\pi$
$991$	−51928.0	−1.66453	−0.832264	+	0.554379i	$0.812956\pi$
$992$	0	0
$993$	−4596.00	−0.146878
$994$	0	0
$995$	6720.00	0.214109
$996$	0	0
$997$	386.000	0.0122615	0.00613076	−	0.999981i	$-0.498049\pi$
$997$	386.000	0.0122615	0.00613076	+	0.999981i	$0.498049\pi$
$998$	0	0
$999$	6642.00	0.210354

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.a.w.1.1		1
4.3	odd	2		1344.4.a.i.1.1		1
8.3	odd	2		336.4.a.h.1.1		1
8.5	even	2		21.4.a.b.1.1	✓	1
24.5	odd	2		63.4.a.a.1.1		1
24.11	even	2		1008.4.a.m.1.1		1
40.13	odd	4		525.4.d.b.274.1		2
40.29	even	2		525.4.a.b.1.1		1
40.37	odd	4		525.4.d.b.274.2		2
56.5	odd	6		147.4.e.b.67.1		2
56.13	odd	2		147.4.a.g.1.1		1
56.27	even	2		2352.4.a.l.1.1		1
56.37	even	6		147.4.e.c.67.1		2
56.45	odd	6		147.4.e.b.79.1		2
56.53	even	6		147.4.e.c.79.1		2
120.29	odd	2		1575.4.a.k.1.1		1
168.5	even	6		441.4.e.n.361.1		2
168.53	odd	6		441.4.e.m.226.1		2
168.101	even	6		441.4.e.n.226.1		2
168.125	even	2		441.4.a.b.1.1		1
168.149	odd	6		441.4.e.m.361.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.4.a.b.1.1	✓	1	8.5	even	2
63.4.a.a.1.1		1	24.5	odd	2
147.4.a.g.1.1		1	56.13	odd	2
147.4.e.b.67.1		2	56.5	odd	6
147.4.e.b.79.1		2	56.45	odd	6
147.4.e.c.67.1		2	56.37	even	6
147.4.e.c.79.1		2	56.53	even	6
336.4.a.h.1.1		1	8.3	odd	2
441.4.a.b.1.1		1	168.125	even	2
441.4.e.m.226.1		2	168.53	odd	6
441.4.e.m.361.1		2	168.149	odd	6
441.4.e.n.226.1		2	168.101	even	6
441.4.e.n.361.1		2	168.5	even	6
525.4.a.b.1.1		1	40.29	even	2
525.4.d.b.274.1		2	40.13	odd	4
525.4.d.b.274.2		2	40.37	odd	4
1008.4.a.m.1.1		1	24.11	even	2
1344.4.a.i.1.1		1	4.3	odd	2
1344.4.a.w.1.1		1	1.1	even	1	trivial
1575.4.a.k.1.1		1	120.29	odd	2
2352.4.a.l.1.1		1	56.27	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:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more