LMFDB - Embedded newform 1040.4.k.d.961.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1040,4,Mod(961,1040)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1040, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1040.961"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 1040 = 2^{4} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1040.k (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [14,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$61.3619864060$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 84x^{12} + 2674x^{10} + 40048x^{8} + 278769x^{6} + 727552x^{4} + 339456x^{2} + 9216 $ x^14 + 84x^12 + 2674x^10 + 40048x^8 + 278769x^6 + 727552x^4 + 339456x^2 + 9216
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{9}\cdot 5^{4} $
Twist minimal:	no (minimal twist has level 65)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			961.8
Root			$5.37688i$ of defining polynomial
Character	$\chi$	$=$	1040.961
Dual form			1040.4.k.d.961.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.936724 q^{3} +5.00000i q^{5} +0.201771i q^{7} -26.1225 q^{9} -10.4218i q^{11} +(39.9303 + 24.5473i) q^{13} +4.68362i q^{15} -54.7775 q^{17} -129.099i q^{19} +0.189004i q^{21} +175.846 q^{23} -25.0000 q^{25} -49.7612 q^{27} +107.045 q^{29} -22.2028i q^{31} -9.76239i q^{33} -1.00886 q^{35} +205.591i q^{37} +(37.4037 + 22.9940i) q^{39} +285.238i q^{41} -321.835 q^{43} -130.613i q^{45} +379.907i q^{47} +342.959 q^{49} -51.3115 q^{51} +506.192 q^{53} +52.1092 q^{55} -120.930i q^{57} -678.186i q^{59} -186.634 q^{61} -5.27077i q^{63} +(-122.736 + 199.652i) q^{65} +925.768i q^{67} +164.719 q^{69} +376.232i q^{71} +671.223i q^{73} -23.4181 q^{75} +2.10282 q^{77} +593.491 q^{79} +658.696 q^{81} +425.750i q^{83} -273.888i q^{85} +100.271 q^{87} +141.676i q^{89} +(-4.95293 + 8.05678i) q^{91} -20.7979i q^{93} +645.496 q^{95} -551.467i q^{97} +272.245i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 158 q^{9} - 4 q^{13} - 100 q^{17} + 532 q^{23} - 350 q^{25} + 48 q^{27} + 588 q^{29} - 540 q^{35} + 260 q^{39} - 728 q^{43} - 1302 q^{49} - 176 q^{51} + 1040 q^{53} + 40 q^{55} - 1460 q^{61} - 30 q^{65}+ \cdots + 3120 q^{95}+O(q^{100}) $ 14 * q + 158 * q^9 - 4 * q^13 - 100 * q^17 + 532 * q^23 - 350 * q^25 + 48 * q^27 + 588 * q^29 - 540 * q^35 + 260 * q^39 - 728 * q^43 - 1302 * q^49 - 176 * q^51 + 1040 * q^53 + 40 * q^55 - 1460 * q^61 - 30 * q^65 + 4160 * q^69 - 3568 * q^77 + 1024 * q^79 - 2890 * q^81 - 5256 * q^87 - 916 * q^91 + 3120 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times$.

$n$	$261$	$417$	$561$	$911$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	0.936724			0.180273			0.0901363	−	0.995929i	$-0.471270\pi$
$3$	0.936724			0.180273			0.0901363	+	0.995929i	$0.471270\pi$
$4$		0			0
$5$			5.00000i			0.447214i
$5$			5.00000i			0.447214i
$6$		0			0
$7$			0.201771i			0.0108946i	0.999985	+	0.00544731i	$0.00173394\pi$
$7$			0.201771i			0.0108946i	−0.999985	+	0.00544731i	$0.998266\pi$
$8$		0			0
$9$	−26.1225			−0.967502
$10$		0			0
$11$		−	10.4218i		−	0.285664i	−0.989747	−	0.142832i	$-0.954379\pi$
$11$		−	10.4218i		−	0.285664i	0.989747	−	0.142832i	$-0.0456208\pi$
$12$		0			0
$13$	39.9303	+	24.5473i	0.851898	+	0.523707i
$13$	39.9303	+	24.5473i	0.851898	+	0.523707i
$14$		0			0
$15$			4.68362i			0.0806204i
$16$		0			0
$17$	−54.7775			−0.781500			−0.390750	−	0.920497i	$-0.627784\pi$
$17$	−54.7775			−0.781500			−0.390750	+	0.920497i	$0.627784\pi$
$18$		0			0
$19$		−	129.099i		−	1.55881i	−0.626521	−	0.779405i	$-0.715522\pi$
$19$		−	129.099i		−	1.55881i	0.626521	−	0.779405i	$-0.284478\pi$
$20$		0			0
$21$			0.189004i			0.00196400i
$22$		0			0
$23$	175.846			1.59419			0.797097	−	0.603852i	$-0.206368\pi$
$23$	175.846			1.59419			0.797097	+	0.603852i	$0.206368\pi$
$24$		0			0
$25$	−25.0000			−0.200000
$26$		0			0
$27$	−49.7612			−0.354687
$28$		0			0
$29$	107.045			0.685438			0.342719	−	0.939438i	$-0.388652\pi$
$29$	107.045			0.685438			0.342719	+	0.939438i	$0.388652\pi$
$30$		0			0
$31$		−	22.2028i		−	0.128637i	−0.997929	−	0.0643184i	$-0.979513\pi$
$31$		−	22.2028i		−	0.128637i	0.997929	−	0.0643184i	$-0.0204873\pi$
$32$		0			0
$33$		−	9.76239i		−	0.0514974i
$34$		0			0
$35$	−1.00886			−0.00487222
$36$		0			0
$37$			205.591i			0.913486i	0.889599	+	0.456743i	$0.150984\pi$
$37$			205.591i			0.913486i	−0.889599	+	0.456743i	$0.849016\pi$
$38$		0			0
$39$	37.4037	+	22.9940i	0.153574	+	0.0944101i
$40$		0			0
$41$			285.238i			1.08651i	0.839569	+	0.543253i	$0.182807\pi$
$41$			285.238i			1.08651i	−0.839569	+	0.543253i	$0.817193\pi$
$42$		0			0
$43$	−321.835			−1.14138			−0.570690	−	0.821166i	$-0.693324\pi$
$43$	−321.835			−1.14138			−0.570690	+	0.821166i	$0.693324\pi$
$44$		0			0
$45$		−	130.613i		−	0.432680i
$46$		0			0
$47$			379.907i			1.17905i	0.807752	+	0.589523i	$0.200684\pi$
$47$			379.907i			1.17905i	−0.807752	+	0.589523i	$0.799316\pi$
$48$		0			0
$49$	342.959			0.999881
$50$		0			0
$51$	−51.3115			−0.140883
$52$		0			0
$53$	506.192			1.31190			0.655951	−	0.754804i	$-0.272268\pi$
$53$	506.192			1.31190			0.655951	+	0.754804i	$0.272268\pi$
$54$		0			0
$55$	52.1092			0.127753
$56$		0			0
$57$		−	120.930i		−	0.281011i
$58$		0			0
$59$		−	678.186i		−	1.49648i	−0.663428	−	0.748240i	$-0.730899\pi$
$59$		−	678.186i		−	1.49648i	0.663428	−	0.748240i	$-0.269101\pi$
$60$		0			0
$61$	−186.634			−0.391738			−0.195869	−	0.980630i	$-0.562753\pi$
$61$	−186.634			−0.391738			−0.195869	+	0.980630i	$0.562753\pi$
$62$		0			0
$63$		−	5.27077i		−	0.0105406i
$64$		0			0
$65$	−122.736	+	199.652i	−0.234209	+	0.380981i
$66$		0			0
$67$			925.768i			1.68807i	0.536289	+	0.844034i	$0.319826\pi$
$67$			925.768i			1.68807i	−0.536289	+	0.844034i	$0.680174\pi$
$68$		0			0
$69$	164.719			0.287390
$70$		0			0
$71$			376.232i			0.628881i	0.949277	+	0.314441i	$0.101817\pi$
$71$			376.232i			0.628881i	−0.949277	+	0.314441i	$0.898183\pi$
$72$		0			0
$73$			671.223i			1.07617i	0.842889	+	0.538087i	$0.180853\pi$
$73$			671.223i			1.07617i	−0.842889	+	0.538087i	$0.819147\pi$
$74$		0			0
$75$	−23.4181			−0.0360545
$76$		0			0
$77$	2.10282			0.00311220
$78$		0			0
$79$	593.491			0.845227			0.422613	−	0.906310i	$-0.361113\pi$
$79$	593.491			0.845227			0.422613	+	0.906310i	$0.361113\pi$
$80$		0			0
$81$	658.696			0.903561
$82$		0			0
$83$			425.750i			0.563037i	0.959556	+	0.281519i	$0.0908381\pi$
$83$			425.750i			0.563037i	−0.959556	+	0.281519i	$0.909162\pi$
$84$		0			0
$85$		−	273.888i		−	0.349498i
$86$		0			0
$87$	100.271			0.123566
$88$		0			0
$89$			141.676i			0.168737i	0.996435	+	0.0843685i	$0.0268873\pi$
$89$			141.676i			0.168737i	−0.996435	+	0.0843685i	$0.973113\pi$
$90$		0			0
$91$	−4.95293	+	8.05678i	−0.00570559	+	0.00928110i
$92$		0			0
$93$		−	20.7979i		−	0.0231897i
$94$		0			0
$95$	645.496			0.697121
$96$		0			0
$97$		−	551.467i		−	0.577247i	−0.957443	−	0.288623i	$-0.906802\pi$
$97$		−	551.467i		−	0.577247i	0.957443	−	0.288623i	$-0.0931976\pi$
$98$		0			0
$99$			272.245i			0.276380i
$100$		0			0
$101$	−1386.39			−1.36585			−0.682924	−	0.730489i	$-0.739292\pi$
$101$	−1386.39			−1.36585			−0.682924	+	0.730489i	$0.739292\pi$
$102$		0			0
$103$	−1156.61			−1.10645			−0.553223	−	0.833033i	$-0.686602\pi$
$103$	−1156.61			−1.10645			−0.553223	+	0.833033i	$0.686602\pi$
$104$		0			0
$105$	−0.945019			−0.000878328
$106$		0			0
$107$	1145.31			1.03478			0.517390	−	0.855750i	$-0.326904\pi$
$107$	1145.31			1.03478			0.517390	+	0.855750i	$0.326904\pi$
$108$		0			0
$109$			2033.95i			1.78731i	0.448750	+	0.893657i	$0.351869\pi$
$109$			2033.95i			1.78731i	−0.448750	+	0.893657i	$0.648131\pi$
$110$		0			0
$111$			192.582i			0.164677i
$112$		0			0
$113$	−1218.91			−1.01474			−0.507370	−	0.861728i	$-0.669382\pi$
$113$	−1218.91			−1.01474			−0.507370	+	0.861728i	$0.669382\pi$
$114$		0			0
$115$			879.231i			0.712945i
$116$		0			0
$117$	−1043.08	−	641.238i	−0.824213	−	0.506687i
$118$		0			0
$119$		−	11.0525i		−	0.00851414i
$120$		0			0
$121$	1222.39			0.918396
$122$		0			0
$123$			267.190i			0.195867i
$124$		0			0
$125$		−	125.000i		−	0.0894427i
$126$		0			0
$127$	1219.03			0.851746			0.425873	−	0.904783i	$-0.359967\pi$
$127$	1219.03			0.851746			0.425873	+	0.904783i	$0.359967\pi$
$128$		0			0
$129$	−301.470			−0.205760
$130$		0			0
$131$	−807.392			−0.538490			−0.269245	−	0.963072i	$-0.586774\pi$
$131$	−807.392			−0.538490			−0.269245	+	0.963072i	$0.586774\pi$
$132$		0			0
$133$	26.0485			0.0169826
$134$		0			0
$135$		−	248.806i		−	0.158621i
$136$		0			0
$137$			24.8642i			0.0155057i	0.999970	+	0.00775287i	$0.00246784\pi$
$137$			24.8642i			0.0155057i	−0.999970	+	0.00775287i	$0.997532\pi$
$138$		0			0
$139$	1351.45			0.824665			0.412332	−	0.911033i	$-0.364714\pi$
$139$	1351.45			0.824665			0.412332	+	0.911033i	$0.364714\pi$
$140$		0			0
$141$			355.868i			0.212550i
$142$		0			0
$143$	255.828	−	416.147i	0.149604	−	0.243357i
$144$		0			0
$145$			535.223i			0.306537i
$146$		0			0
$147$	321.258			0.180251
$148$		0			0
$149$			318.875i			0.175324i	0.996150	+	0.0876621i	$0.0279396\pi$
$149$			318.875i			0.175324i	−0.996150	+	0.0876621i	$0.972060\pi$
$150$		0			0
$151$		−	472.671i		−	0.254738i	−0.991855	−	0.127369i	$-0.959347\pi$
$151$		−	472.671i		−	0.254738i	0.991855	−	0.127369i	$-0.0406533\pi$
$152$		0			0
$153$	1430.93			0.756103
$154$		0			0
$155$	111.014			0.0575281
$156$		0			0
$157$	2157.92			1.09695			0.548474	−	0.836168i	$-0.315209\pi$
$157$	2157.92			1.09695			0.548474	+	0.836168i	$0.315209\pi$
$158$		0			0
$159$	474.162			0.236500
$160$		0			0
$161$			35.4807i			0.0173681i
$162$		0			0
$163$			2482.96i			1.19313i	0.802564	+	0.596567i	$0.203469\pi$
$163$			2482.96i			1.19313i	−0.802564	+	0.596567i	$0.796531\pi$
$164$		0			0
$165$	48.8119			0.0230303
$166$		0			0
$167$			1563.31i			0.724389i	0.932103	+	0.362194i	$0.117972\pi$
$167$			1563.31i			0.724389i	−0.932103	+	0.362194i	$0.882028\pi$
$168$		0			0
$169$	991.862	+	1960.36i	0.451462	+	0.892290i
$170$		0			0
$171$			3372.40i			1.50815i
$172$		0			0
$173$	2785.12			1.22398			0.611991	−	0.790865i	$-0.290369\pi$
$173$	2785.12			1.22398			0.611991	+	0.790865i	$0.290369\pi$
$174$		0			0
$175$		−	5.04428i		−	0.00217892i
$176$		0			0
$177$		−	635.274i		−	0.269774i
$178$		0			0
$179$	3352.21			1.39975			0.699877	−	0.714264i	$-0.253238\pi$
$179$	3352.21			1.39975			0.699877	+	0.714264i	$0.253238\pi$
$180$		0			0
$181$	3474.25			1.42674			0.713368	−	0.700789i	$-0.247169\pi$
$181$	3474.25			1.42674			0.713368	+	0.700789i	$0.247169\pi$
$182$		0			0
$183$	−174.825			−0.0706197
$184$		0			0
$185$	−1027.96			−0.408524
$186$		0			0
$187$			570.882i			0.223246i
$188$		0			0
$189$		−	10.0404i		−	0.00386418i
$190$		0			0
$191$	823.112			0.311824			0.155912	−	0.987771i	$-0.450168\pi$
$191$	823.112			0.311824			0.155912	+	0.987771i	$0.450168\pi$
$192$		0			0
$193$		−	186.060i		−	0.0693933i	−0.999398	−	0.0346967i	$-0.988953\pi$
$193$		−	186.060i		−	0.0693933i	0.999398	−	0.0346967i	$-0.0110465\pi$
$194$		0			0
$195$	−114.970	+	187.019i	−0.0422215	+	0.0686804i
$196$		0			0
$197$			3035.00i			1.09764i	0.835942	+	0.548819i	$0.184922\pi$
$197$			3035.00i			1.09764i	−0.835942	+	0.548819i	$0.815078\pi$
$198$		0			0
$199$	−2690.11			−0.958276			−0.479138	−	0.877740i	$-0.659051\pi$
$199$	−2690.11			−0.958276			−0.479138	+	0.877740i	$0.659051\pi$
$200$		0			0
$201$			867.190i			0.304313i
$202$		0			0
$203$			21.5985i			0.00746758i
$204$		0			0
$205$	−1426.19			−0.485900
$206$		0			0
$207$	−4593.55			−1.54239
$208$		0			0
$209$	−1345.45			−0.445295
$210$		0			0
$211$	−1411.57			−0.460554			−0.230277	−	0.973125i	$-0.573963\pi$
$211$	−1411.57			−0.460554			−0.230277	+	0.973125i	$0.573963\pi$
$212$		0			0
$213$			352.426i			0.113370i
$214$		0			0
$215$		−	1609.17i		−	0.510441i
$216$		0			0
$217$	4.47988			0.00140145
$218$		0			0
$219$			628.751i			0.194005i
$220$		0			0
$221$	−2187.28	−	1344.64i	−0.665759	−	0.409277i
$222$		0			0
$223$		−	474.236i		−	0.142409i	−0.997462	−	0.0712044i	$-0.977316\pi$
$223$		−	474.236i		−	0.142409i	0.997462	−	0.0712044i	$-0.0226843\pi$
$224$		0			0
$225$	653.064			0.193500
$226$		0			0
$227$		−	1561.74i		−	0.456635i	−0.973587	−	0.228318i	$-0.926678\pi$
$227$		−	1561.74i		−	0.456635i	0.973587	−	0.228318i	$-0.0733225\pi$
$228$		0			0
$229$		−	2928.82i		−	0.845162i	−0.906325	−	0.422581i	$-0.861124\pi$
$229$		−	2928.82i		−	0.845162i	0.906325	−	0.422581i	$-0.138876\pi$
$230$		0			0
$231$	1.96977			0.000561044
$232$		0			0
$233$	2785.91			0.783310			0.391655	−	0.920112i	$-0.371903\pi$
$233$	2785.91			0.783310			0.391655	+	0.920112i	$0.371903\pi$
$234$		0			0
$235$	−1899.54			−0.527285
$236$		0			0
$237$	555.937			0.152371
$238$		0			0
$239$		−	2488.61i		−	0.673534i	−0.941588	−	0.336767i	$-0.890667\pi$
$239$		−	2488.61i		−	0.673534i	0.941588	−	0.336767i	$-0.109333\pi$
$240$		0			0
$241$			4840.64i			1.29383i	0.762562	+	0.646915i	$0.223941\pi$
$241$			4840.64i			1.29383i	−0.762562	+	0.646915i	$0.776059\pi$
$242$		0			0
$243$	1960.57			0.517574
$244$		0			0
$245$			1714.80i			0.447161i
$246$		0			0
$247$	3169.03	−	5154.97i	0.816359	−	1.32795i
$248$		0			0
$249$			398.810i			0.101500i
$250$		0			0
$251$	−8.77945			−0.00220779			−0.00110389	−	0.999999i	$-0.500351\pi$
$251$	−8.77945			−0.00220779			−0.00110389	+	0.999999i	$0.500351\pi$
$252$		0			0
$253$		−	1832.64i		−	0.455403i
$254$		0			0
$255$		−	256.557i		−	0.0630049i
$256$		0			0
$257$	1600.04			0.388357			0.194179	−	0.980966i	$-0.437796\pi$
$257$	1600.04			0.388357			0.194179	+	0.980966i	$0.437796\pi$
$258$		0			0
$259$	−41.4824			−0.00995208
$260$		0			0
$261$	−2796.28			−0.663162
$262$		0			0
$263$	3050.24			0.715155			0.357578	−	0.933883i	$-0.383603\pi$
$263$	3050.24			0.715155			0.357578	+	0.933883i	$0.383603\pi$
$264$		0			0
$265$			2530.96i			0.586700i
$266$		0			0
$267$			132.711i			0.0304187i
$268$		0			0
$269$	−6168.58			−1.39816			−0.699080	−	0.715043i	$-0.746407\pi$
$269$	−6168.58			−1.39816			−0.699080	+	0.715043i	$0.746407\pi$
$270$		0			0
$271$			2424.93i			0.543556i	0.962360	+	0.271778i	$0.0876117\pi$
$271$			2424.93i			0.543556i	−0.962360	+	0.271778i	$0.912388\pi$
$272$		0			0
$273$	−4.63953	+	7.54699i	−0.00102856	+	0.00167313i
$274$		0			0
$275$			260.546i			0.0571328i
$276$		0			0
$277$	4442.72			0.963673			0.481836	−	0.876261i	$-0.339970\pi$
$277$	4442.72			0.963673			0.481836	+	0.876261i	$0.339970\pi$
$278$		0			0
$279$			579.993i			0.124456i
$280$		0			0
$281$		−	357.622i		−	0.0759215i	−0.999279	−	0.0379608i	$-0.987914\pi$
$281$		−	357.622i		−	0.0759215i	0.999279	−	0.0379608i	$-0.0120862\pi$
$282$		0			0
$283$	−1573.71			−0.330556			−0.165278	−	0.986247i	$-0.552852\pi$
$283$	−1573.71			−0.330556			−0.165278	+	0.986247i	$0.552852\pi$
$284$		0			0
$285$	604.652			0.125672
$286$		0			0
$287$	−57.5528			−0.0118371
$288$		0			0
$289$	−1912.42			−0.389257
$290$		0			0
$291$		−	516.572i		−	0.104062i
$292$		0			0
$293$			4370.32i			0.871389i	0.900095	+	0.435694i	$0.143497\pi$
$293$			4370.32i			0.871389i	−0.900095	+	0.435694i	$0.856503\pi$
$294$		0			0
$295$	3390.93			0.669246
$296$		0			0
$297$			518.603i			0.101321i
$298$		0			0
$299$	7021.59	+	4316.54i	1.35809	+	0.834890i
$300$		0			0
$301$		−	64.9369i		−	0.0124349i
$302$		0			0
$303$	−1298.66			−0.246225
$304$		0			0
$305$		−	933.170i		−	0.175191i
$306$		0			0
$307$		−	3998.97i		−	0.743430i	−0.928347	−	0.371715i	$-0.878770\pi$
$307$		−	3998.97i		−	0.743430i	0.928347	−	0.371715i	$-0.121230\pi$
$308$		0			0
$309$	−1083.42			−0.199462
$310$		0			0
$311$	−2116.08			−0.385825			−0.192913	−	0.981216i	$-0.561793\pi$
$311$	−2116.08			−0.385825			−0.192913	+	0.981216i	$0.561793\pi$
$312$		0			0
$313$	1729.75			0.312368			0.156184	−	0.987728i	$-0.450081\pi$
$313$	1729.75			0.312368			0.156184	+	0.987728i	$0.450081\pi$
$314$		0			0
$315$	26.3539			0.00471388
$316$		0			0
$317$			4636.84i			0.821548i	0.911737	+	0.410774i	$0.134741\pi$
$317$			4636.84i			0.821548i	−0.911737	+	0.410774i	$0.865259\pi$
$318$		0			0
$319$		−	1115.60i		−	0.195805i
$320$		0			0
$321$	1072.84			0.186543
$322$		0			0
$323$			7071.73i			1.21821i
$324$		0			0
$325$	−998.258	−	613.682i	−0.170380	−	0.104741i
$326$		0			0
$327$			1905.25i			0.322204i
$328$		0			0
$329$	−76.6543			−0.0128453
$330$		0			0
$331$			2278.33i			0.378333i	0.981945	+	0.189167i	$0.0605787\pi$
$331$			2278.33i			0.378333i	−0.981945	+	0.189167i	$0.939421\pi$
$332$		0			0
$333$		−	5370.57i		−	0.883800i
$334$		0			0
$335$	−4628.84			−0.754927
$336$		0			0
$337$	−419.703			−0.0678418			−0.0339209	−	0.999425i	$-0.510799\pi$
$337$	−419.703			−0.0678418			−0.0339209	+	0.999425i	$0.510799\pi$
$338$		0			0
$339$	−1141.79			−0.182930
$340$		0			0
$341$	−231.394			−0.0367469
$342$		0			0
$343$			138.407i			0.0217879i
$344$		0			0
$345$			823.597i			0.128525i
$346$		0			0
$347$	495.311			0.0766273			0.0383137	−	0.999266i	$-0.487801\pi$
$347$	495.311			0.0766273			0.0383137	+	0.999266i	$0.487801\pi$
$348$		0			0
$349$		−	4453.77i		−	0.683109i	−0.939862	−	0.341554i	$-0.889047\pi$
$349$		−	4453.77i		−	0.683109i	0.939862	−	0.341554i	$-0.110953\pi$
$350$		0			0
$351$	−1986.98	−	1221.50i	−0.302157	−	0.185752i
$352$		0			0
$353$		−	11893.7i		−	1.79330i	−0.442736	−	0.896652i	$-0.645992\pi$
$353$		−	11893.7i		−	1.79330i	0.442736	−	0.896652i	$-0.354008\pi$
$354$		0			0
$355$	−1881.16			−0.281244
$356$		0			0
$357$		−	10.3532i		−	0.00153487i
$358$		0			0
$359$			5141.43i			0.755861i	0.925834	+	0.377931i	$0.123364\pi$
$359$			5141.43i			0.755861i	−0.925834	+	0.377931i	$0.876636\pi$
$360$		0			0
$361$	−9807.58			−1.42989
$362$		0			0
$363$	1145.04			0.165562
$364$		0			0
$365$	−3356.12			−0.481280
$366$		0			0
$367$	−9410.74			−1.33852			−0.669260	−	0.743028i	$-0.733389\pi$
$367$	−9410.74			−1.33852			−0.669260	+	0.743028i	$0.733389\pi$
$368$		0			0
$369$		−	7451.15i		−	1.05120i
$370$		0			0
$371$			102.135i			0.0142927i
$372$		0			0
$373$	−4992.80			−0.693076			−0.346538	−	0.938036i	$-0.612643\pi$
$373$	−4992.80			−0.693076			−0.346538	+	0.938036i	$0.612643\pi$
$374$		0			0
$375$		−	117.091i		−	0.0161241i
$376$		0			0
$377$	4274.33	+	2627.66i	0.583923	+	0.358969i
$378$		0			0
$379$		−	2436.78i		−	0.330261i	−0.986272	−	0.165131i	$-0.947195\pi$
$379$		−	2436.78i		−	0.330261i	0.986272	−	0.165131i	$-0.0528046\pi$
$380$		0			0
$381$	1141.90			0.153547
$382$		0			0
$383$		−	10395.4i		−	1.38690i	−0.720506	−	0.693449i	$-0.756090\pi$
$383$		−	10395.4i		−	1.38690i	0.720506	−	0.693449i	$-0.243910\pi$
$384$		0			0
$385$			10.5141i			0.00139182i
$386$		0			0
$387$	8407.14			1.10429
$388$		0			0
$389$	13872.3			1.80811			0.904056	−	0.427413i	$-0.140575\pi$
$389$	13872.3			1.80811			0.904056	+	0.427413i	$0.140575\pi$
$390$		0			0
$391$	−9632.42			−1.24586
$392$		0			0
$393$	−756.304			−0.0970750
$394$		0			0
$395$			2967.45i			0.377997i
$396$		0			0
$397$		−	2216.16i		−	0.280166i	−0.990140	−	0.140083i	$-0.955263\pi$
$397$		−	2216.16i		−	0.280166i	0.990140	−	0.140083i	$-0.0447369\pi$
$398$		0			0
$399$	24.4002			0.00306150
$400$		0			0
$401$		−	12127.7i		−	1.51030i	−0.655554	−	0.755148i	$-0.727565\pi$
$401$		−	12127.7i		−	1.51030i	0.655554	−	0.755148i	$-0.272435\pi$
$402$		0			0
$403$	545.018	−	886.565i	0.0673680	−	0.109585i
$404$		0			0
$405$			3293.48i			0.404085i
$406$		0			0
$407$	2142.64			0.260950
$408$		0			0
$409$			2755.96i			0.333187i	0.986026	+	0.166594i	$0.0532768\pi$
$409$			2755.96i			0.333187i	−0.986026	+	0.166594i	$0.946723\pi$
$410$		0			0
$411$			23.2909i			0.00279526i
$412$		0			0
$413$	136.838			0.0163036
$414$		0			0
$415$	−2128.75			−0.251798
$416$		0			0
$417$	1265.94			0.148665
$418$		0			0
$419$	−417.932			−0.0487287			−0.0243643	−	0.999703i	$-0.507756\pi$
$419$	−417.932			−0.0487287			−0.0243643	+	0.999703i	$0.507756\pi$
$420$		0			0
$421$			7623.50i			0.882534i	0.897376	+	0.441267i	$0.145471\pi$
$421$			7623.50i			0.882534i	−0.897376	+	0.441267i	$0.854529\pi$
$422$		0			0
$423$		−	9924.15i		−	1.14073i
$424$		0			0
$425$	1369.44			0.156300
$426$		0			0
$427$		−	37.6573i		−	0.00426783i
$428$		0			0
$429$	239.640	−	389.815i	0.0269695	−	0.0438705i
$430$		0			0
$431$		−	6348.22i		−	0.709473i	−0.934966	−	0.354736i	$-0.884571\pi$
$431$		−	6348.22i		−	0.709473i	0.934966	−	0.354736i	$-0.115429\pi$
$432$		0			0
$433$	−4286.27			−0.475716			−0.237858	−	0.971300i	$-0.576445\pi$
$433$	−4286.27			−0.475716			−0.237858	+	0.971300i	$0.576445\pi$
$434$		0			0
$435$			501.357i			0.0552603i
$436$		0			0
$437$		−	22701.6i		−	2.48504i
$438$		0			0
$439$	898.674			0.0977024			0.0488512	−	0.998806i	$-0.484444\pi$
$439$	898.674			0.0977024			0.0488512	+	0.998806i	$0.484444\pi$
$440$		0			0
$441$	−8958.97			−0.967387
$442$		0			0
$443$	−4981.83			−0.534297			−0.267149	−	0.963655i	$-0.586081\pi$
$443$	−4981.83			−0.534297			−0.267149	+	0.963655i	$0.586081\pi$
$444$		0			0
$445$	−708.378			−0.0754615
$446$		0			0
$447$			298.698i			0.0316062i
$448$		0			0
$449$		−	4494.76i		−	0.472429i	−0.971701	−	0.236215i	$-0.924093\pi$
$449$		−	4494.76i		−	0.472429i	0.971701	−	0.236215i	$-0.0759068\pi$
$450$		0			0
$451$	2972.71			0.310375
$452$		0			0
$453$		−	442.763i		−	0.0459223i
$454$		0			0
$455$	−40.2839	−	24.7647i	−0.00415064	−	0.00255162i
$456$		0			0
$457$			2924.71i			0.299370i	0.988734	+	0.149685i	$0.0478259\pi$
$457$			2924.71i			0.299370i	−0.988734	+	0.149685i	$0.952174\pi$
$458$		0			0
$459$	2725.80			0.277188
$460$		0			0
$461$		−	10122.2i		−	1.02264i	−0.859390	−	0.511320i	$-0.829157\pi$
$461$		−	10122.2i		−	1.02264i	0.859390	−	0.511320i	$-0.170843\pi$
$462$		0			0
$463$			5225.27i			0.524490i	0.965001	+	0.262245i	$0.0844628\pi$
$463$			5225.27i			0.524490i	−0.965001	+	0.262245i	$0.915537\pi$
$464$		0			0
$465$	103.989			0.0103707
$466$		0			0
$467$	5596.68			0.554569			0.277284	−	0.960788i	$-0.410566\pi$
$467$	5596.68			0.554569			0.277284	+	0.960788i	$0.410566\pi$
$468$		0			0
$469$	−186.793			−0.0183909
$470$		0			0
$471$	2021.38			0.197750
$472$		0			0
$473$			3354.11i			0.326051i
$474$		0			0
$475$			3227.48i			0.311762i
$476$		0			0
$477$	−13223.0			−1.26927
$478$		0			0
$479$		−	14342.8i		−	1.36814i	−0.729418	−	0.684069i	$-0.760209\pi$
$479$		−	14342.8i		−	1.36814i	0.729418	−	0.684069i	$-0.239791\pi$
$480$		0			0
$481$	−5046.71	+	8209.33i	−0.478399	+	0.778198i
$482$		0			0
$483$			33.2356i			0.00313100i
$484$		0			0
$485$	2757.33			0.258153
$486$		0			0
$487$		−	19369.6i		−	1.80230i	−0.433512	−	0.901148i	$-0.642726\pi$
$487$		−	19369.6i		−	1.80230i	0.433512	−	0.901148i	$-0.357274\pi$
$488$		0			0
$489$			2325.85i			0.215089i
$490$		0			0
$491$	−12486.5			−1.14768			−0.573838	−	0.818969i	$-0.694546\pi$
$491$	−12486.5			−1.14768			−0.573838	+	0.818969i	$0.694546\pi$
$492$		0			0
$493$	−5863.64			−0.535670
$494$		0			0
$495$	−1361.22			−0.123601
$496$		0			0
$497$	−75.9128			−0.00685142
$498$		0			0
$499$		−	15364.4i		−	1.37837i	−0.724585	−	0.689186i	$-0.757968\pi$
$499$		−	15364.4i		−	1.37837i	0.724585	−	0.689186i	$-0.242032\pi$
$500$		0			0
$501$			1464.40i			0.130588i
$502$		0			0
$503$	−15778.4			−1.39866			−0.699328	−	0.714800i	$-0.746517\pi$
$503$	−15778.4			−1.39866			−0.699328	+	0.714800i	$0.746517\pi$
$504$		0			0
$505$		−	6931.94i		−	0.610826i
$506$		0			0
$507$	929.101	+	1836.32i	0.0813863	+	0.160856i
$508$		0			0
$509$		−	6252.54i		−	0.544478i	−0.962230	−	0.272239i	$-0.912236\pi$
$509$		−	6252.54i		−	0.544478i	0.962230	−	0.272239i	$-0.0877641\pi$
$510$		0			0
$511$	−135.433			−0.0117245
$512$		0			0
$513$			6424.13i			0.552889i
$514$		0			0
$515$		−	5783.04i		−	0.494817i
$516$		0			0
$517$	3959.33			0.336811
$518$		0			0
$519$	2608.89			0.220651
$520$		0			0
$521$	−13016.7			−1.09457			−0.547287	−	0.836945i	$-0.684340\pi$
$521$	−13016.7			−1.09457			−0.547287	+	0.836945i	$0.684340\pi$
$522$		0			0
$523$	16344.2			1.36650			0.683251	−	0.730183i	$-0.260565\pi$
$523$	16344.2			1.36650			0.683251	+	0.730183i	$0.260565\pi$
$524$		0			0
$525$		−	4.72510i		−	0.000392800i
$526$		0			0
$527$			1216.21i			0.100530i
$528$		0			0
$529$	18754.9			1.54145
$530$		0			0
$531$			17715.9i			1.44785i
$532$		0			0
$533$	−7001.82	+	11389.7i	−0.569010	+	0.925592i
$534$		0			0
$535$			5726.56i			0.462768i
$536$		0			0
$537$	3140.10			0.252337
$538$		0			0
$539$		−	3574.26i		−	0.285630i
$540$		0			0
$541$			21385.8i			1.69953i	0.527158	+	0.849767i	$0.323258\pi$
$541$			21385.8i			1.69953i	−0.527158	+	0.849767i	$0.676742\pi$
$542$		0			0
$543$	3254.42			0.257202
$544$		0			0
$545$	−10169.8			−0.799312
$546$		0			0
$547$	22836.7			1.78505			0.892527	−	0.450993i	$-0.148930\pi$
$547$	22836.7			1.78505			0.892527	+	0.450993i	$0.148930\pi$
$548$		0			0
$549$	4875.35			0.379007
$550$		0			0
$551$		−	13819.4i		−	1.06847i
$552$		0			0
$553$			119.749i			0.00920842i
$554$		0			0
$555$	−962.912			−0.0736456
$556$		0			0
$557$		−	14375.7i		−	1.09357i	−0.837274	−	0.546784i	$-0.815852\pi$
$557$		−	14375.7i		−	1.09357i	0.837274	−	0.546784i	$-0.184148\pi$
$558$		0			0
$559$	−12851.0	−	7900.17i	−0.972340	−	0.597749i
$560$		0			0
$561$			534.760i			0.0402452i
$562$		0			0
$563$	−2088.93			−0.156373			−0.0781866	−	0.996939i	$-0.524913\pi$
$563$	−2088.93			−0.156373			−0.0781866	+	0.996939i	$0.524913\pi$
$564$		0			0
$565$		−	6094.57i		−	0.453806i
$566$		0			0
$567$			132.906i			0.00984395i
$568$		0			0
$569$	7775.20			0.572853			0.286426	−	0.958102i	$-0.407533\pi$
$569$	7775.20			0.572853			0.286426	+	0.958102i	$0.407533\pi$
$570$		0			0
$571$	−21491.9			−1.57515			−0.787573	−	0.616221i	$-0.788663\pi$
$571$	−21491.9			−1.57515			−0.787573	+	0.616221i	$0.788663\pi$
$572$		0			0
$573$	771.029			0.0562133
$574$		0			0
$575$	−4396.15			−0.318839
$576$		0			0
$577$		−	4348.75i		−	0.313762i	−0.987617	−	0.156881i	$-0.949856\pi$
$577$		−	4348.75i		−	0.313762i	0.987617	−	0.156881i	$-0.0501440\pi$
$578$		0			0
$579$		−	174.287i		−	0.0125097i
$580$		0			0
$581$	−85.9040			−0.00613407
$582$		0			0
$583$		−	5275.45i		−	0.374763i
$584$		0			0
$585$	3206.19	−	5215.41i	0.226598	−	0.368599i
$586$		0			0
$587$			5800.67i			0.407870i	0.978984	+	0.203935i	$0.0653731\pi$
$587$			5800.67i			0.407870i	−0.978984	+	0.203935i	$0.934627\pi$
$588$		0			0
$589$	−2866.36			−0.200520
$590$		0			0
$591$			2842.95i			0.197874i
$592$		0			0
$593$			4577.14i			0.316966i	0.987362	+	0.158483i	$0.0506603\pi$
$593$			4577.14i			0.316966i	−0.987362	+	0.158483i	$0.949340\pi$
$594$		0			0
$595$	55.2626			0.00380764
$596$		0			0
$597$	−2519.89			−0.172751
$598$		0			0
$599$	−17537.7			−1.19628			−0.598140	−	0.801392i	$-0.704093\pi$
$599$	−17537.7			−1.19628			−0.598140	+	0.801392i	$0.704093\pi$
$600$		0			0
$601$	13316.0			0.903780			0.451890	−	0.892074i	$-0.350750\pi$
$601$	13316.0			0.903780			0.451890	+	0.892074i	$0.350750\pi$
$602$		0			0
$603$		−	24183.4i		−	1.63321i
$604$		0			0
$605$			6111.93i			0.410719i
$606$		0			0
$607$	−2103.20			−0.140636			−0.0703182	−	0.997525i	$-0.522401\pi$
$607$	−2103.20			−0.140636			−0.0703182	+	0.997525i	$0.522401\pi$
$608$		0			0
$609$			20.2319i			0.00134620i
$610$		0			0
$611$	−9325.69	+	15169.8i	−0.617475	+	1.00443i
$612$		0			0
$613$			6984.21i			0.460178i	0.973170	+	0.230089i	$0.0739018\pi$
$613$			6984.21i			0.460178i	−0.973170	+	0.230089i	$0.926098\pi$
$614$		0			0
$615$	−1335.95			−0.0875945
$616$		0			0
$617$		−	13605.0i		−	0.887713i	−0.896098	−	0.443856i	$-0.853610\pi$
$617$		−	13605.0i		−	0.887713i	0.896098	−	0.443856i	$-0.146390\pi$
$618$		0			0
$619$			24505.8i			1.59123i	0.605801	+	0.795616i	$0.292853\pi$
$619$			24505.8i			1.59123i	−0.605801	+	0.795616i	$0.707147\pi$
$620$		0			0
$621$	−8750.31			−0.565439
$622$		0			0
$623$	−28.5860			−0.00183832
$624$		0			0
$625$	625.000			0.0400000
$626$		0			0
$627$	−1260.32			−0.0802746
$628$		0			0
$629$		−	11261.8i		−	0.713890i
$630$		0			0
$631$		−	18694.2i		−	1.17940i	−0.807621	−	0.589702i	$-0.799245\pi$
$631$		−	18694.2i		−	1.17940i	0.807621	−	0.589702i	$-0.200755\pi$
$632$		0			0
$633$	−1322.26			−0.0830252
$634$		0			0
$635$			6095.17i			0.380912i
$636$		0			0
$637$	13694.5	+	8418.72i	0.851797	+	0.523645i
$638$		0			0
$639$		−	9828.15i		−	0.608444i
$640$		0			0
$641$	−22658.3			−1.39617			−0.698087	−	0.716013i	$-0.745965\pi$
$641$	−22658.3			−1.39617			−0.698087	+	0.716013i	$0.745965\pi$
$642$		0			0
$643$			18187.4i			1.11546i	0.830022	+	0.557730i	$0.188328\pi$
$643$			18187.4i			1.11546i	−0.830022	+	0.557730i	$0.811672\pi$
$644$		0			0
$645$		−	1507.35i		−	0.0920185i
$646$		0			0
$647$	−4755.08			−0.288936			−0.144468	−	0.989509i	$-0.546147\pi$
$647$	−4755.08			−0.288936			−0.144468	+	0.989509i	$0.546147\pi$
$648$		0			0
$649$	−7067.94			−0.427490
$650$		0			0
$651$	4.19641			0.000252643
$652$		0			0
$653$	731.494			0.0438370			0.0219185	−	0.999760i	$-0.493023\pi$
$653$	731.494			0.0438370			0.0219185	+	0.999760i	$0.493023\pi$
$654$		0			0
$655$		−	4036.96i		−	0.240820i
$656$		0			0
$657$		−	17534.1i		−	1.04120i
$658$		0			0
$659$	−16005.3			−0.946099			−0.473050	−	0.881036i	$-0.656847\pi$
$659$	−16005.3			−0.946099			−0.473050	+	0.881036i	$0.656847\pi$
$660$		0			0
$661$			12332.1i			0.725664i	0.931855	+	0.362832i	$0.118190\pi$
$661$			12332.1i			0.725664i	−0.931855	+	0.362832i	$0.881810\pi$
$662$		0			0
$663$	−2048.88	−	1259.56i	−0.120018	−	0.0737815i
$664$		0			0
$665$			130.242i			0.00759486i
$666$		0			0
$667$	18823.4			1.09272
$668$		0			0
$669$		−	444.228i		−	0.0256724i
$670$		0			0
$671$			1945.07i			0.111905i
$672$		0			0
$673$	12183.0			0.697799			0.348899	−	0.937160i	$-0.386556\pi$
$673$	12183.0			0.697799			0.348899	+	0.937160i	$0.386556\pi$
$674$		0			0
$675$	1244.03			0.0709374
$676$		0			0
$677$	25830.0			1.46637			0.733183	−	0.680032i	$-0.238034\pi$
$677$	25830.0			1.46637			0.733183	+	0.680032i	$0.238034\pi$
$678$		0			0
$679$	111.270			0.00628888
$680$		0			0
$681$		−	1462.92i		−	0.0823189i
$682$		0			0
$683$			6832.74i			0.382793i	0.981513	+	0.191396i	$0.0613016\pi$
$683$			6832.74i			0.382793i	−0.981513	+	0.191396i	$0.938698\pi$
$684$		0			0
$685$	−124.321			−0.00693438
$686$		0			0
$687$		−	2743.50i		−	0.152360i
$688$		0			0
$689$	20212.4	+	12425.6i	1.11761	+	0.687052i
$690$		0			0
$691$			26068.7i			1.43517i	0.696472	+	0.717584i	$0.254752\pi$
$691$			26068.7i			1.43517i	−0.696472	+	0.717584i	$0.745248\pi$
$692$		0			0
$693$	−54.9311			−0.00301106
$694$		0			0
$695$			6757.24i			0.368801i
$696$		0			0
$697$		−	15624.6i		−	0.849104i
$698$		0			0
$699$	2609.63			0.141209
$700$		0			0
$701$	−31003.6			−1.67045			−0.835227	−	0.549905i	$-0.814664\pi$
$701$	−31003.6			−1.67045			−0.835227	+	0.549905i	$0.814664\pi$
$702$		0			0
$703$	26541.7			1.42395
$704$		0			0
$705$	−1779.34			−0.0950552
$706$		0			0
$707$		−	279.733i		−	0.0148804i
$708$		0			0
$709$			19868.2i			1.05242i	0.850355	+	0.526209i	$0.176387\pi$
$709$			19868.2i			1.05242i	−0.850355	+	0.526209i	$0.823613\pi$
$710$		0			0
$711$	−15503.5			−0.817759
$712$		0			0
$713$		−	3904.27i		−	0.205072i
$714$		0			0
$715$	2080.74	+	1279.14i	0.108832	+	0.0669050i
$716$		0			0
$717$		−	2331.14i		−	0.121420i
$718$		0			0
$719$	−2878.38			−0.149298			−0.0746491	−	0.997210i	$-0.523784\pi$
$719$	−2878.38			−0.149298			−0.0746491	+	0.997210i	$0.523784\pi$
$720$		0			0
$721$		−	233.370i		−	0.0120543i
$722$		0			0
$723$			4534.35i			0.233242i
$724$		0			0
$725$	−2676.12			−0.137088
$726$		0			0
$727$	22798.9			1.16309			0.581543	−	0.813516i	$-0.302449\pi$
$727$	22798.9			1.16309			0.581543	+	0.813516i	$0.302449\pi$
$728$		0			0
$729$	−15948.3			−0.810257
$730$		0			0
$731$	17629.3			0.891989
$732$		0			0
$733$		−	27940.5i		−	1.40792i	−0.710240	−	0.703960i	$-0.751414\pi$
$733$		−	27940.5i		−	1.40792i	0.710240	−	0.703960i	$-0.248586\pi$
$734$		0			0
$735$			1606.29i			0.0806108i
$736$		0			0
$737$	9648.21			0.482220
$738$		0			0
$739$			30468.8i			1.51666i	0.651871	+	0.758330i	$0.273984\pi$
$739$			30468.8i			1.51666i	−0.651871	+	0.758330i	$0.726016\pi$
$740$		0			0
$741$	2968.51	−	4828.79i	0.147167	−	0.239393i
$742$		0			0
$743$		−	17528.7i		−	0.865501i	−0.901514	−	0.432750i	$-0.857543\pi$
$743$		−	17528.7i		−	0.865501i	0.901514	−	0.432750i	$-0.142457\pi$
$744$		0			0
$745$	−1594.38			−0.0784073
$746$		0			0
$747$		−	11121.7i		−	0.544740i
$748$		0			0
$749$			231.091i			0.0112735i
$750$		0			0
$751$	4371.31			0.212399			0.106199	−	0.994345i	$-0.466132\pi$
$751$	4371.31			0.212399			0.106199	+	0.994345i	$0.466132\pi$
$752$		0			0
$753$	−8.22393			−0.000398003
$754$		0			0
$755$	2363.36			0.113922
$756$		0			0
$757$	2920.51			0.140221			0.0701107	−	0.997539i	$-0.477665\pi$
$757$	2920.51			0.140221			0.0701107	+	0.997539i	$0.477665\pi$
$758$		0			0
$759$		−	1716.68i		−	0.0820968i
$760$		0			0
$761$			40352.4i			1.92217i	0.276246	+	0.961087i	$0.410909\pi$
$761$			40352.4i			1.92217i	−0.276246	+	0.961087i	$0.589091\pi$
$762$		0			0
$763$	−410.393			−0.0194721
$764$		0			0
$765$			7154.64i			0.338139i
$766$		0			0
$767$	16647.6	−	27080.2i	0.783717	−	1.27485i
$768$		0			0
$769$		−	27282.1i		−	1.27935i	−0.768646	−	0.639674i	$-0.779069\pi$
$769$		−	27282.1i		−	1.27935i	0.768646	−	0.639674i	$-0.220931\pi$
$770$		0			0
$771$	1498.80			0.0700102
$772$		0			0
$773$			10121.3i			0.470942i	0.971881	+	0.235471i	$0.0756633\pi$
$773$			10121.3i			0.470942i	−0.971881	+	0.235471i	$0.924337\pi$
$774$		0			0
$775$			555.070i			0.0257273i
$776$		0			0
$777$	−38.8575			−0.00179409
$778$		0			0
$779$	36824.0			1.69365
$780$		0			0
$781$	3921.03			0.179649
$782$		0			0
$783$	−5326.67			−0.243116
$784$		0			0
$785$			10789.6i			0.490570i
$786$		0			0
$787$			17973.2i			0.814072i	0.913412	+	0.407036i	$0.133438\pi$
$787$			17973.2i			0.814072i	−0.913412	+	0.407036i	$0.866562\pi$
$788$		0			0
$789$	2857.23			0.128923
$790$		0			0
$791$		−	245.941i		−	0.0110552i
$792$		0			0
$793$	−7452.35	−	4581.36i	−0.333721	−	0.205156i
$794$		0			0
$795$			2370.81i			0.105766i
$796$		0			0
$797$	−1050.61			−0.0466931			−0.0233465	−	0.999727i	$-0.507432\pi$
$797$	−1050.61			−0.0466931			−0.0233465	+	0.999727i	$0.507432\pi$
$798$		0			0
$799$		−	20810.4i		−	0.921425i
$800$		0			0
$801$		−	3700.93i		−	0.163253i
$802$		0			0
$803$	6995.38			0.307424
$804$		0			0
$805$	−177.403			−0.00776726
$806$		0			0
$807$	−5778.26			−0.252050
$808$		0			0
$809$	−10131.7			−0.440310			−0.220155	−	0.975465i	$-0.570656\pi$
$809$	−10131.7			−0.440310			−0.220155	+	0.975465i	$0.570656\pi$
$810$		0			0
$811$		−	3463.42i		−	0.149959i	−0.997185	−	0.0749797i	$-0.976111\pi$
$811$		−	3463.42i		−	0.149959i	0.997185	−	0.0749797i	$-0.0238892\pi$
$812$		0			0
$813$			2271.49i			0.0979884i
$814$		0			0
$815$	−12414.8			−0.533585
$816$		0			0
$817$			41548.6i			1.77919i
$818$		0			0
$819$	129.383	−	210.464i	0.00552016	−	0.00897948i
$820$		0			0
$821$		−	9876.00i		−	0.419823i	−0.977720	−	0.209912i	$-0.932682\pi$
$821$		−	9876.00i		−	0.419823i	0.977720	−	0.209912i	$-0.0673176\pi$
$822$		0			0
$823$	7199.95			0.304951			0.152475	−	0.988307i	$-0.451276\pi$
$823$	7199.95			0.304951			0.152475	+	0.988307i	$0.451276\pi$
$824$		0			0
$825$			244.060i			0.0102995i
$826$		0			0
$827$			16864.4i			0.709110i	0.935035	+	0.354555i	$0.115368\pi$
$827$			16864.4i			0.709110i	−0.935035	+	0.354555i	$0.884632\pi$
$828$		0			0
$829$	26061.1			1.09184			0.545922	−	0.837836i	$-0.316180\pi$
$829$	26061.1			1.09184			0.545922	+	0.837836i	$0.316180\pi$
$830$		0			0
$831$	4161.61			0.173724
$832$		0			0
$833$	−18786.5			−0.781407
$834$		0			0
$835$	−7816.57			−0.323957
$836$		0			0
$837$			1104.84i			0.0456258i
$838$		0			0
$839$			35491.9i			1.46045i	0.683208	+	0.730224i	$0.260584\pi$
$839$			35491.9i			1.46045i	−0.683208	+	0.730224i	$0.739416\pi$
$840$		0			0
$841$	−12930.4			−0.530175
$842$		0			0
$843$		−	334.993i		−	0.0136866i
$844$		0			0
$845$	−9801.81	+	4959.31i	−0.399044	+	0.201900i
$846$		0			0
$847$			246.642i			0.0100056i
$848$		0			0
$849$	−1474.13			−0.0595903
$850$		0			0
$851$			36152.4i			1.45627i
$852$		0			0
$853$		−	10763.1i		−	0.432029i	−0.976390	−	0.216014i	$-0.930694\pi$
$853$		−	10763.1i		−	0.432029i	0.976390	−	0.216014i	$-0.0693058\pi$
$854$		0			0
$855$	−16862.0			−0.674465
$856$		0			0
$857$	−3344.32			−0.133302			−0.0666511	−	0.997776i	$-0.521231\pi$
$857$	−3344.32			−0.133302			−0.0666511	+	0.997776i	$0.521231\pi$
$858$		0			0
$859$	31361.2			1.24567			0.622836	−	0.782353i	$-0.285980\pi$
$859$	31361.2			1.24567			0.622836	+	0.782353i	$0.285980\pi$
$860$		0			0
$861$	−53.9111			−0.00213390
$862$		0			0
$863$		−	9576.65i		−	0.377744i	−0.982002	−	0.188872i	$-0.939517\pi$
$863$		−	9576.65i		−	0.377744i	0.982002	−	0.188872i	$-0.0604831\pi$
$864$		0			0
$865$			13925.6i			0.547382i
$866$		0			0
$867$	−1791.41			−0.0701725
$868$		0			0
$869$		−	6185.26i		−	0.241451i
$870$		0			0
$871$	−22725.1	+	36966.2i	−0.884054	+	1.43806i
$872$		0			0
$873$			14405.7i			0.558487i
$874$		0			0
$875$	25.2214			0.000974444
$876$		0			0
$877$			28993.5i			1.11635i	0.829722	+	0.558177i	$0.188499\pi$
$877$			28993.5i			1.11635i	−0.829722	+	0.558177i	$0.811501\pi$
$878$		0			0
$879$			4093.79i			0.157088i
$880$		0			0
$881$	23068.0			0.882156			0.441078	−	0.897469i	$-0.354596\pi$
$881$	23068.0			0.882156			0.441078	+	0.897469i	$0.354596\pi$
$882$		0			0
$883$	29675.0			1.13097			0.565483	−	0.824760i	$-0.308690\pi$
$883$	29675.0			1.13097			0.565483	+	0.824760i	$0.308690\pi$
$884$		0			0
$885$	3176.37			0.120647
$886$		0			0
$887$	5118.12			0.193742			0.0968712	−	0.995297i	$-0.469117\pi$
$887$	5118.12			0.193742			0.0968712	+	0.995297i	$0.469117\pi$
$888$		0			0
$889$			245.966i			0.00927944i
$890$		0			0
$891$		−	6864.82i		−	0.258115i
$892$		0			0
$893$	49045.7			1.83791
$894$		0			0
$895$			16761.0i			0.625989i
$896$		0			0
$897$	6577.30	+	4043.41i	0.244827	+	0.150508i
$898$		0			0
$899$		−	2376.69i		−	0.0881725i
$900$		0			0
$901$	−27727.9			−1.02525
$902$		0			0
$903$		−	60.8280i		−	0.00224167i
$904$		0			0
$905$			17371.3i			0.638056i
$906$		0			0
$907$	−37154.2			−1.36018			−0.680091	−	0.733128i	$-0.738060\pi$
$907$	−37154.2			−1.36018			−0.680091	+	0.733128i	$0.738060\pi$
$908$		0			0
$909$	36216.0			1.32146
$910$		0			0
$911$	−7888.56			−0.286893			−0.143447	−	0.989658i	$-0.545819\pi$
$911$	−7888.56			−0.286893			−0.143447	+	0.989658i	$0.545819\pi$
$912$		0			0
$913$	4437.09			0.160839
$914$		0			0
$915$		−	874.123i		−	0.0315821i
$916$		0			0
$917$		−	162.908i		−	0.00586664i
$918$		0			0
$919$	32292.7			1.15913			0.579563	−	0.814927i	$-0.303223\pi$
$919$	32292.7			1.15913			0.579563	+	0.814927i	$0.303223\pi$
$920$		0			0
$921$		−	3745.93i		−	0.134020i
$922$		0			0
$923$	−9235.48	+	15023.1i	−0.329350	+	0.535743i
$924$		0			0
$925$		−	5139.78i		−	0.182697i
$926$		0			0
$927$	30213.5			1.07049
$928$		0			0
$929$		−	7931.61i		−	0.280116i	−0.990143	−	0.140058i	$-0.955271\pi$
$929$		−	7931.61i		−	0.280116i	0.990143	−	0.140058i	$-0.0447289\pi$
$930$		0			0
$931$		−	44275.7i		−	1.55862i
$932$		0			0
$933$	−1982.18			−0.0695537
$934$		0			0
$935$	−2854.41			−0.0998388
$936$		0			0
$937$	35368.6			1.23313			0.616565	−	0.787304i	$-0.288524\pi$
$937$	35368.6			1.23313			0.616565	+	0.787304i	$0.288524\pi$
$938$		0			0
$939$	1620.30			0.0563115
$940$		0			0
$941$			21319.4i			0.738569i	0.929316	+	0.369285i	$0.120397\pi$
$941$			21319.4i			0.738569i	−0.929316	+	0.369285i	$0.879603\pi$
$942$		0			0
$943$			50158.0i			1.73210i
$944$		0			0
$945$	50.2018			0.00172811
$946$		0			0
$947$		−	16623.5i		−	0.570422i	−0.958465	−	0.285211i	$-0.907936\pi$
$947$		−	16623.5i		−	0.570422i	0.958465	−	0.285211i	$-0.0920637\pi$
$948$		0			0
$949$	−16476.7	+	26802.2i	−0.563600	+	0.916792i
$950$		0			0
$951$			4343.44i			0.148103i
$952$		0			0
$953$	25047.2			0.851374			0.425687	−	0.904870i	$-0.360032\pi$
$953$	25047.2			0.851374			0.425687	+	0.904870i	$0.360032\pi$
$954$		0			0
$955$			4115.56i			0.139452i
$956$		0			0
$957$		−	1045.01i		−	0.0352982i
$958$		0			0
$959$	−5.01687			−0.000168929
$960$		0			0
$961$	29298.0			0.983453
$962$		0			0
$963$	−29918.4			−1.00115
$964$		0			0
$965$	930.301			0.0310336
$966$		0			0
$967$		−	13656.8i		−	0.454161i	−0.973876	−	0.227081i	$-0.927082\pi$
$967$		−	13656.8i		−	0.454161i	0.973876	−	0.227081i	$-0.0729181\pi$
$968$		0			0
$969$			6624.26i			0.219610i
$970$		0			0
$971$	−2265.67			−0.0748803			−0.0374401	−	0.999299i	$-0.511920\pi$
$971$	−2265.67			−0.0748803			−0.0374401	+	0.999299i	$0.511920\pi$
$972$		0			0
$973$			272.683i			0.00898440i
$974$		0			0
$975$	−935.093	−	574.851i	−0.0307148	−	0.0188820i
$976$		0			0
$977$			20144.9i			0.659666i	0.944039	+	0.329833i	$0.106992\pi$
$977$			20144.9i			0.659666i	−0.944039	+	0.329833i	$0.893008\pi$
$978$		0			0
$979$	1476.52			0.0482020
$980$		0			0
$981$		−	53132.0i		−	1.72923i
$982$		0			0
$983$		−	10526.6i		−	0.341552i	−0.985310	−	0.170776i	$-0.945373\pi$
$983$		−	10526.6i		−	0.341552i	0.985310	−	0.170776i	$-0.0546274\pi$
$984$		0			0
$985$	−15175.0			−0.490878
$986$		0			0
$987$	−71.8039			−0.00231565
$988$		0			0
$989$	−56593.4			−1.81958
$990$		0			0
$991$	−55992.7			−1.79482			−0.897410	−	0.441198i	$-0.854554\pi$
$991$	−55992.7			−1.79482			−0.897410	+	0.441198i	$0.854554\pi$
$992$		0			0
$993$			2134.17i			0.0682032i
$994$		0			0
$995$		−	13450.6i		−	0.428554i
$996$		0			0
$997$	48578.5			1.54312			0.771562	−	0.636154i	$-0.219476\pi$
$997$	48578.5			1.54312			0.771562	+	0.636154i	$0.219476\pi$
$998$		0			0
$999$		−	10230.5i		−	0.324002i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1040.4.k.d.961.8		14
4.3	odd	2		65.4.c.a.51.14	yes	14
12.11	even	2		585.4.b.e.181.1		14
13.12	even	2	inner	1040.4.k.d.961.7		14
20.3	even	4		325.4.d.d.324.13		14
20.7	even	4		325.4.d.c.324.2		14
20.19	odd	2		325.4.c.e.51.1		14
52.31	even	4		845.4.a.l.1.7		7
52.47	even	4		845.4.a.i.1.1		7
52.51	odd	2		65.4.c.a.51.1	✓	14
156.155	even	2		585.4.b.e.181.14		14
260.103	even	4		325.4.d.c.324.1		14
260.207	even	4		325.4.d.d.324.14		14
260.259	odd	2		325.4.c.e.51.14		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.4.c.a.51.1	✓	14	52.51	odd	2
65.4.c.a.51.14	yes	14	4.3	odd	2
325.4.c.e.51.1		14	20.19	odd	2
325.4.c.e.51.14		14	260.259	odd	2
325.4.d.c.324.1		14	260.103	even	4
325.4.d.c.324.2		14	20.7	even	4
325.4.d.d.324.13		14	20.3	even	4
325.4.d.d.324.14		14	260.207	even	4
585.4.b.e.181.1		14	12.11	even	2
585.4.b.e.181.14		14	156.155	even	2
845.4.a.i.1.1		7	52.47	even	4
845.4.a.l.1.7		7	52.31	even	4
1040.4.k.d.961.7		14	13.12	even	2	inner
1040.4.k.d.961.8		14	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1040.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.936724 q^{3} +5.00000i q^{5} +0.201771i q^{7} -26.1225 q^{9} -10.4218i q^{11} +(39.9303 + 24.5473i) q^{13} +4.68362i q^{15} -54.7775 q^{17} -129.099i q^{19} +0.189004i q^{21} +175.846 q^{23} -25.0000 q^{25} -49.7612 q^{27} +107.045 q^{29} -22.2028i q^{31} -9.76239i q^{33} -1.00886 q^{35} +205.591i q^{37} +(37.4037 + 22.9940i) q^{39} +285.238i q^{41} -321.835 q^{43} -130.613i q^{45} +379.907i q^{47} +342.959 q^{49} -51.3115 q^{51} +506.192 q^{53} +52.1092 q^{55} -120.930i q^{57} -678.186i q^{59} -186.634 q^{61} -5.27077i q^{63} +(-122.736 + 199.652i) q^{65} +925.768i q^{67} +164.719 q^{69} +376.232i q^{71} +671.223i q^{73} -23.4181 q^{75} +2.10282 q^{77} +593.491 q^{79} +658.696 q^{81} +425.750i q^{83} -273.888i q^{85} +100.271 q^{87} +141.676i q^{89} +(-4.95293 + 8.05678i) q^{91} -20.7979i q^{93} +645.496 q^{95} -551.467i q^{97} +272.245i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 158 q^{9} - 4 q^{13} - 100 q^{17} + 532 q^{23} - 350 q^{25} + 48 q^{27} + 588 q^{29} - 540 q^{35} + 260 q^{39} - 728 q^{43} - 1302 q^{49} - 176 q^{51} + 1040 q^{53} + 40 q^{55} - 1460 q^{61} - 30 q^{65}+ \cdots + 3120 q^{95}+O(q^{100}) \) 14 * q + 158 * q^9 - 4 * q^13 - 100 * q^17 + 532 * q^23 - 350 * q^25 + 48 * q^27 + 588 * q^29 - 540 * q^35 + 260 * q^39 - 728 * q^43 - 1302 * q^49 - 176 * q^51 + 1040 * q^53 + 40 * q^55 - 1460 * q^61 - 30 * q^65 + 4160 * q^69 - 3568 * q^77 + 1024 * q^79 - 2890 * q^81 - 5256 * q^87 - 916 * q^91 + 3120 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more