LMFDB - Embedded newform 1014.4.b.q.337.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1014,4,Mod(337,1014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1014, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1014.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1014 = 2 \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1014.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$59.8279367458$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 845x^{10} + 287958x^{8} + 50362537x^{6} + 4731667920x^{4} + 224458698240x^{2} + 4178851762176 $ x^12 + 845x^10 + 287958x^8 + 50362537x^6 + 4731667920x^4 + 224458698240*x^2 + 4178851762176
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 13^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.1
Root			$-7.49280i$ of defining polynomial
Character	$\chi$	$=$	1014.337
Dual form			1014.4.b.q.337.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{2} +3.00000 q^{3} -4.00000 q^{4} -13.0357i q^{5} -6.00000i q^{6} -14.8081i q^{7} +8.00000i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.00000i q^{2} +3.00000 q^{3} -4.00000 q^{4} -13.0357i q^{5} -6.00000i q^{6} -14.8081i q^{7} +8.00000i q^{8} +9.00000 q^{9} -26.0714 q^{10} -39.1957i q^{11} -12.0000 q^{12} -29.6162 q^{14} -39.1070i q^{15} +16.0000 q^{16} -8.45847 q^{17} -18.0000i q^{18} -33.2592i q^{19} +52.1427i q^{20} -44.4243i q^{21} -78.3914 q^{22} -173.056 q^{23} +24.0000i q^{24} -44.9290 q^{25} +27.0000 q^{27} +59.2324i q^{28} +142.438 q^{29} -78.2141 q^{30} -268.140i q^{31} -32.0000i q^{32} -117.587i q^{33} +16.9169i q^{34} -193.034 q^{35} -36.0000 q^{36} +201.704i q^{37} -66.5184 q^{38} +104.285 q^{40} -165.301i q^{41} -88.8486 q^{42} -365.241 q^{43} +156.783i q^{44} -117.321i q^{45} +346.111i q^{46} +175.160i q^{47} +48.0000 q^{48} +123.720 q^{49} +89.8579i q^{50} -25.3754 q^{51} +711.650 q^{53} -54.0000i q^{54} -510.942 q^{55} +118.465 q^{56} -99.7776i q^{57} -284.875i q^{58} -190.212i q^{59} +156.428i q^{60} -634.767 q^{61} -536.280 q^{62} -133.273i q^{63} -64.0000 q^{64} -235.174 q^{66} -432.263i q^{67} +33.8339 q^{68} -519.167 q^{69} +386.067i q^{70} +801.141i q^{71} +72.0000i q^{72} +171.241i q^{73} +403.408 q^{74} -134.787 q^{75} +133.037i q^{76} -580.414 q^{77} +604.457 q^{79} -208.571i q^{80} +81.0000 q^{81} -330.602 q^{82} +1235.02i q^{83} +177.697i q^{84} +110.262i q^{85} +730.481i q^{86} +427.313 q^{87} +313.566 q^{88} -1065.12i q^{89} -234.642 q^{90} +692.222 q^{92} -804.421i q^{93} +350.320 q^{94} -433.556 q^{95} -96.0000i q^{96} +1519.11i q^{97} -247.440i q^{98} -352.761i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 36 q^{3} - 48 q^{4} + 108 q^{9}+O(q^{10}) $ 12 * q + 36 * q^3 - 48 * q^4 + 108 * q^9	$ 12 q + 36 q^{3} - 48 q^{4} + 108 q^{9} + 12 q^{10} - 144 q^{12} - 4 q^{14} + 192 q^{16} - 102 q^{17} + 256 q^{22} - 444 q^{23} - 370 q^{25} + 324 q^{27} + 658 q^{29} + 36 q^{30} + 1688 q^{35} - 432 q^{36} - 852 q^{38} - 48 q^{40} - 12 q^{42} - 982 q^{43} + 576 q^{48} - 2266 q^{49} - 306 q^{51} + 4604 q^{53} - 658 q^{55} + 16 q^{56} + 690 q^{61} - 1156 q^{62} - 768 q^{64} + 768 q^{66} + 408 q^{68} - 1332 q^{69} + 1880 q^{74} - 1110 q^{75} - 6582 q^{77} + 6200 q^{79} + 972 q^{81} + 1284 q^{82} + 1974 q^{87} - 1024 q^{88} + 108 q^{90} + 1776 q^{92} - 2564 q^{94} + 1330 q^{95}+O(q^{100}) $ 12 * q + 36 * q^3 - 48 * q^4 + 108 * q^9 + 12 * q^10 - 144 * q^12 - 4 * q^14 + 192 * q^16 - 102 * q^17 + 256 * q^22 - 444 * q^23 - 370 * q^25 + 324 * q^27 + 658 * q^29 + 36 * q^30 + 1688 * q^35 - 432 * q^36 - 852 * q^38 - 48 * q^40 - 12 * q^42 - 982 * q^43 + 576 * q^48 - 2266 * q^49 - 306 * q^51 + 4604 * q^53 - 658 * q^55 + 16 * q^56 + 690 * q^61 - 1156 * q^62 - 768 * q^64 + 768 * q^66 + 408 * q^68 - 1332 * q^69 + 1880 * q^74 - 1110 * q^75 - 6582 * q^77 + 6200 * q^79 + 972 * q^81 + 1284 * q^82 + 1974 * q^87 - 1024 * q^88 + 108 * q^90 + 1776 * q^92 - 2564 * q^94 + 1330 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times$.

$n$	$677$	$847$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 2.00000i	− 0.707107i
$2$	− 2.00000i	− 0.707107i
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−4.00000	−0.500000
$5$	− 13.0357i	− 1.16595i	−0.812491	−	0.582973i	$-0.801889\pi$
$5$	− 13.0357i	− 1.16595i	0.812491	−	0.582973i	$-0.198111\pi$
$6$	− 6.00000i	− 0.408248i
$7$	− 14.8081i	− 0.799562i	−0.916611	−	0.399781i	$-0.869086\pi$
$7$	− 14.8081i	− 0.799562i	0.916611	−	0.399781i	$-0.130914\pi$
$8$	8.00000i	0.353553i
$9$	9.00000	0.333333
$10$	−26.0714	−0.824449
$11$	− 39.1957i	− 1.07436i	−0.843468	−	0.537179i	$-0.819490\pi$
$11$	− 39.1957i	− 1.07436i	0.843468	−	0.537179i	$-0.180510\pi$
$12$	−12.0000	−0.288675
$13$	0	0
$13$	0	0
$14$	−29.6162	−0.565376
$15$	− 39.1070i	− 0.673160i
$16$	16.0000	0.250000
$17$	−8.45847	−0.120675	−0.0603376	−	0.998178i	$-0.519218\pi$
$17$	−8.45847	−0.120675	−0.0603376	+	0.998178i	$0.519218\pi$
$18$	− 18.0000i	− 0.235702i
$19$	− 33.2592i	− 0.401588i	−0.979633	−	0.200794i	$-0.935648\pi$
$19$	− 33.2592i	− 0.401588i	0.979633	−	0.200794i	$-0.0643523\pi$
$20$	52.1427i	0.582973i
$21$	− 44.4243i	− 0.461628i
$22$	−78.3914	−0.759686
$23$	−173.056	−1.56890	−0.784448	−	0.620195i	$-0.787053\pi$
$23$	−173.056	−1.56890	−0.784448	+	0.620195i	$0.787053\pi$
$24$	24.0000i	0.204124i
$25$	−44.9290	−0.359432
$26$	0	0
$27$	27.0000	0.192450
$28$	59.2324i	0.399781i
$29$	142.438	0.912068	0.456034	−	0.889962i	$-0.349269\pi$
$29$	142.438	0.912068	0.456034	+	0.889962i	$0.349269\pi$
$30$	−78.2141	−0.475996
$31$	− 268.140i	− 1.55353i	−0.629791	−	0.776764i	$-0.716860\pi$
$31$	− 268.140i	− 1.55353i	0.629791	−	0.776764i	$-0.283140\pi$
$32$	− 32.0000i	− 0.176777i
$33$	− 117.587i	− 0.620281i
$34$	16.9169i	0.0853303i
$35$	−193.034	−0.932247
$36$	−36.0000	−0.166667
$37$	201.704i	0.896215i	0.893980	+	0.448107i	$0.147902\pi$
$37$	201.704i	0.896215i	−0.893980	+	0.448107i	$0.852098\pi$
$38$	−66.5184	−0.283966
$39$	0	0
$40$	104.285	0.412224
$41$	− 165.301i	− 0.629651i	−0.949150	−	0.314826i	$-0.898054\pi$
$41$	− 165.301i	− 0.629651i	0.949150	−	0.314826i	$-0.101946\pi$
$42$	−88.8486	−0.326420
$43$	−365.241	−1.29532	−0.647659	−	0.761930i	$-0.724252\pi$
$43$	−365.241	−1.29532	−0.647659	+	0.761930i	$0.724252\pi$
$44$	156.783i	0.537179i
$45$	− 117.321i	− 0.388649i
$46$	346.111i	1.10938i
$47$	175.160i	0.543611i	0.962352	+	0.271806i	$0.0876208\pi$
$47$	175.160i	0.543611i	−0.962352	+	0.271806i	$0.912379\pi$
$48$	48.0000	0.144338
$49$	123.720	0.360700
$50$	89.8579i	0.254157i
$51$	−25.3754	−0.0696719
$52$	0	0
$53$	711.650	1.84439	0.922195	−	0.386725i	$-0.126394\pi$
$53$	711.650	1.84439	0.922195	+	0.386725i	$0.126394\pi$
$54$	− 54.0000i	− 0.136083i
$55$	−510.942	−1.25264
$56$	118.465	0.282688
$57$	− 99.7776i	− 0.231857i
$58$	− 284.875i	− 0.644930i
$59$	− 190.212i	− 0.419719i	−0.977732	−	0.209860i	$-0.932699\pi$
$59$	− 190.212i	− 0.419719i	0.977732	−	0.209860i	$-0.0673007\pi$
$60$	156.428i	0.336580i
$61$	−634.767	−1.33235	−0.666177	−	0.745793i	$-0.732071\pi$
$61$	−634.767	−1.33235	−0.666177	+	0.745793i	$0.732071\pi$
$62$	−536.280	−1.09851
$63$	− 133.273i	− 0.266521i
$64$	−64.0000	−0.125000
$65$	0	0
$66$	−235.174	−0.438605
$67$	− 432.263i	− 0.788199i	−0.919068	−	0.394099i	$-0.871057\pi$
$67$	− 432.263i	− 0.788199i	0.919068	−	0.394099i	$-0.128943\pi$
$68$	33.8339	0.0603376
$69$	−519.167	−0.905802
$70$	386.067i	0.659198i
$71$	801.141i	1.33913i	0.742755	+	0.669563i	$0.233519\pi$
$71$	801.141i	1.33913i	−0.742755	+	0.669563i	$0.766481\pi$
$72$	72.0000i	0.117851i
$73$	171.241i	0.274552i	0.990533	+	0.137276i	$0.0438347\pi$
$73$	171.241i	0.274552i	−0.990533	+	0.137276i	$0.956165\pi$
$74$	403.408	0.633720
$75$	−134.787	−0.207518
$76$	133.037i	0.200794i
$77$	−580.414	−0.859017
$78$	0	0
$79$	604.457	0.860844	0.430422	−	0.902628i	$-0.358365\pi$
$79$	604.457	0.860844	0.430422	+	0.902628i	$0.358365\pi$
$80$	− 208.571i	− 0.291487i
$81$	81.0000	0.111111
$82$	−330.602	−0.445231
$83$	1235.02i	1.63326i	0.577161	+	0.816631i	$0.304161\pi$
$83$	1235.02i	1.63326i	−0.577161	+	0.816631i	$0.695839\pi$
$84$	177.697i	0.230814i
$85$	110.262i	0.140701i
$86$	730.481i	0.915928i
$87$	427.313	0.526583
$88$	313.566	0.379843
$89$	− 1065.12i	− 1.26857i	−0.773100	−	0.634284i	$-0.781295\pi$
$89$	− 1065.12i	− 1.26857i	0.773100	−	0.634284i	$-0.218705\pi$
$90$	−234.642	−0.274816
$91$	0	0
$92$	692.222	0.784448
$93$	− 804.421i	− 0.896930i
$94$	350.320	0.384391
$95$	−433.556	−0.468231
$96$	− 96.0000i	− 0.102062i
$97$	1519.11i	1.59013i	0.606525	+	0.795064i	$0.292563\pi$
$97$	1519.11i	1.59013i	−0.606525	+	0.795064i	$0.707437\pi$
$98$	− 247.440i	− 0.255053i
$99$	− 352.761i	− 0.358120i
$100$	179.716	0.179716
$101$	1503.76	1.48149	0.740743	−	0.671789i	$-0.234474\pi$
$101$	1503.76	1.48149	0.740743	+	0.671789i	$0.234474\pi$
$102$	50.7508i	0.0492655i
$103$	−1414.55	−1.35320	−0.676602	−	0.736349i	$-0.736548\pi$
$103$	−1414.55	−1.35320	−0.676602	+	0.736349i	$0.736548\pi$
$104$	0	0
$105$	−579.101	−0.538233
$106$	− 1423.30i	− 1.30418i
$107$	169.234	0.152901	0.0764507	−	0.997073i	$-0.475641\pi$
$107$	169.234	0.152901	0.0764507	+	0.997073i	$0.475641\pi$
$108$	−108.000	−0.0962250
$109$	− 47.3538i	− 0.0416116i	−0.999784	−	0.0208058i	$-0.993377\pi$
$109$	− 47.3538i	− 0.0416116i	0.999784	−	0.0208058i	$-0.00662317\pi$
$110$	1021.88i	0.885754i
$111$	605.112i	0.517430i
$112$	− 236.930i	− 0.199891i
$113$	−2271.29	−1.89084	−0.945418	−	0.325859i	$-0.894347\pi$
$113$	−2271.29	−1.89084	−0.945418	+	0.325859i	$0.894347\pi$
$114$	−199.555	−0.163948
$115$	2255.90i	1.82925i
$116$	−569.750	−0.456034
$117$	0	0
$118$	−380.423	−0.296786
$119$	125.254i	0.0964874i
$120$	312.856	0.237998
$121$	−205.302	−0.154247
$122$	1269.53i	0.942117i
$123$	− 495.903i	− 0.363529i
$124$	1072.56i	0.776764i
$125$	− 1043.78i	− 0.746869i
$126$	−266.546	−0.188459
$127$	−2137.49	−1.49348	−0.746739	−	0.665117i	$-0.768382\pi$
$127$	−2137.49	−1.49348	−0.746739	+	0.665117i	$0.768382\pi$
$128$	128.000i	0.0883883i
$129$	−1095.72	−0.747852
$130$	0	0
$131$	1.67293	0.00111576	0.000557879	−	1.00000i	$-0.499822\pi$
$131$	1.67293	0.00111576	0.000557879		1.00000i	$0.499822\pi$
$132$	470.348i	0.310141i
$133$	−492.506	−0.321095
$134$	−864.526	−0.557341
$135$	− 351.963i	− 0.224387i
$136$	− 67.6677i	− 0.0426652i
$137$	1080.09i	0.673563i	0.941583	+	0.336782i	$0.109338\pi$
$137$	1080.09i	0.673563i	−0.941583	+	0.336782i	$0.890662\pi$
$138$	1038.33i	0.640499i
$139$	−1373.60	−0.838182	−0.419091	−	0.907944i	$-0.637651\pi$
$139$	−1373.60	−0.838182	−0.419091	+	0.907944i	$0.637651\pi$
$140$	772.135	0.466124
$141$	525.480i	0.313854i
$142$	1602.28	0.946906
$143$	0	0
$144$	144.000	0.0833333
$145$	− 1856.77i	− 1.06342i
$146$	342.483	0.194138
$147$	371.160	0.208250
$148$	− 806.816i	− 0.448107i
$149$	978.562i	0.538033i	0.963136	+	0.269016i	$0.0866986\pi$
$149$	978.562i	0.538033i	−0.963136	+	0.269016i	$0.913301\pi$
$150$	269.574i	0.146737i
$151$	− 3183.95i	− 1.71593i	−0.513705	−	0.857967i	$-0.671728\pi$
$151$	− 3183.95i	− 1.71593i	0.513705	−	0.857967i	$-0.328272\pi$
$152$	266.074	0.141983
$153$	−76.1262	−0.0402251
$154$	1160.83i	0.607417i
$155$	−3495.39	−1.81133
$156$	0	0
$157$	−647.596	−0.329196	−0.164598	−	0.986361i	$-0.552633\pi$
$157$	−647.596	−0.329196	−0.164598	+	0.986361i	$0.552633\pi$
$158$	− 1208.91i	− 0.608709i
$159$	2134.95	1.06486
$160$	−417.142	−0.206112
$161$	2562.63i	1.25443i
$162$	− 162.000i	− 0.0785674i
$163$	− 2777.80i	− 1.33481i	−0.744695	−	0.667405i	$-0.767405\pi$
$163$	− 2777.80i	− 1.33481i	0.744695	−	0.667405i	$-0.232595\pi$
$164$	661.204i	0.314826i
$165$	−1532.83	−0.723215
$166$	2470.03	1.15489
$167$	1787.24i	0.828150i	0.910243	+	0.414075i	$0.135895\pi$
$167$	1787.24i	0.828150i	−0.910243	+	0.414075i	$0.864105\pi$
$168$	355.394	0.163210
$169$	0	0
$170$	220.524	0.0994906
$171$	− 299.333i	− 0.133863i
$172$	1460.96	0.647659
$173$	−1986.85	−0.873164	−0.436582	−	0.899664i	$-0.643811\pi$
$173$	−1986.85	−0.873164	−0.436582	+	0.899664i	$0.643811\pi$
$174$	− 854.625i	− 0.372350i
$175$	665.313i	0.287388i
$176$	− 627.131i	− 0.268590i
$177$	− 570.635i	− 0.242325i
$178$	−2130.24	−0.897013
$179$	−598.991	−0.250115	−0.125058	−	0.992149i	$-0.539912\pi$
$179$	−598.991	−0.250115	−0.125058	+	0.992149i	$0.539912\pi$
$180$	469.284i	0.194324i
$181$	2442.79	1.00316	0.501578	−	0.865112i	$-0.332753\pi$
$181$	2442.79	1.00316	0.501578	+	0.865112i	$0.332753\pi$
$182$	0	0
$183$	−1904.30	−0.769235
$184$	− 1384.44i	− 0.554688i
$185$	2629.35	1.04494
$186$	−1608.84	−0.634226
$187$	331.536i	0.129649i
$188$	− 700.640i	− 0.271806i
$189$	− 399.819i	− 0.153876i
$190$	867.112i	0.331089i
$191$	3581.69	1.35687	0.678435	−	0.734660i	$-0.262658\pi$
$191$	3581.69	1.35687	0.678435	+	0.734660i	$0.262658\pi$
$192$	−192.000	−0.0721688
$193$	− 592.014i	− 0.220798i	−0.993887	−	0.110399i	$-0.964787\pi$
$193$	− 592.014i	− 0.220798i	0.993887	−	0.110399i	$-0.0352129\pi$
$194$	3038.22	1.12439
$195$	0	0
$196$	−494.880	−0.180350
$197$	592.735i	0.214369i	0.994239	+	0.107184i	$0.0341835\pi$
$197$	592.735i	0.214369i	−0.994239	+	0.107184i	$0.965817\pi$
$198$	−705.522	−0.253229
$199$	−5175.84	−1.84375	−0.921874	−	0.387490i	$-0.873342\pi$
$199$	−5175.84	−1.84375	−0.921874	+	0.387490i	$0.873342\pi$
$200$	− 359.432i	− 0.127078i
$201$	− 1296.79i	− 0.455067i
$202$	− 3007.53i	− 1.04757i
$203$	− 2109.23i	− 0.729256i
$204$	101.502	0.0348360
$205$	−2154.81	−0.734140
$206$	2829.11i	0.956860i
$207$	−1557.50	−0.522965
$208$	0	0
$209$	−1303.62	−0.431450
$210$	1158.20i	0.380588i
$211$	5929.71	1.93468	0.967341	−	0.253477i	$-0.0815742\pi$
$211$	5929.71	1.93468	0.967341	+	0.253477i	$0.0815742\pi$
$212$	−2846.60	−0.922195
$213$	2403.42i	0.773145i
$214$	− 338.468i	− 0.108118i
$215$	4761.16i	1.51027i
$216$	216.000i	0.0680414i
$217$	−3970.65	−1.24214
$218$	−94.7075	−0.0294239
$219$	513.724i	0.158513i
$220$	2043.77	0.626322
$221$	0	0
$222$	1210.22	0.365878
$223$	612.048i	0.183793i	0.995769	+	0.0918964i	$0.0292928\pi$
$223$	612.048i	0.183793i	−0.995769	+	0.0918964i	$0.970707\pi$
$224$	−473.859	−0.141344
$225$	−404.361	−0.119811
$226$	4542.57i	1.33702i
$227$	3666.54i	1.07206i	0.844200	+	0.536028i	$0.180076\pi$
$227$	3666.54i	1.07206i	−0.844200	+	0.536028i	$0.819924\pi$
$228$	399.110i	0.115929i
$229$	555.460i	0.160287i	0.996783	+	0.0801437i	$0.0255379\pi$
$229$	555.460i	0.160287i	−0.996783	+	0.0801437i	$0.974462\pi$
$230$	4511.80	1.29347
$231$	−1741.24	−0.495954
$232$	1139.50i	0.322465i
$233$	−4041.62	−1.13638	−0.568188	−	0.822899i	$-0.692355\pi$
$233$	−4041.62	−1.13638	−0.568188	+	0.822899i	$0.692355\pi$
$234$	0	0
$235$	2283.33	0.633822
$236$	760.847i	0.209860i
$237$	1813.37	0.497009
$238$	250.508	0.0682269
$239$	− 1080.67i	− 0.292480i	−0.989249	−	0.146240i	$-0.953283\pi$
$239$	− 1080.67i	− 0.292480i	0.989249	−	0.146240i	$-0.0467171\pi$
$240$	− 625.713i	− 0.168290i
$241$	− 871.746i	− 0.233005i	−0.993190	−	0.116502i	$-0.962832\pi$
$241$	− 871.746i	− 0.233005i	0.993190	−	0.116502i	$-0.0371682\pi$
$242$	410.604i	0.109069i
$243$	243.000	0.0641500
$244$	2539.07	0.666177
$245$	− 1612.78i	− 0.420557i
$246$	−991.807	−0.257054
$247$	0	0
$248$	2145.12	0.549255
$249$	3705.05i	0.942964i
$250$	−2087.56	−0.528116
$251$	4341.09	1.09166	0.545831	−	0.837895i	$-0.316214\pi$
$251$	4341.09	1.09166	0.545831	+	0.837895i	$0.316214\pi$
$252$	533.092i	0.133260i
$253$	6783.03i	1.68556i
$254$	4274.98i	1.05605i
$255$	330.786i	0.0812337i
$256$	256.000	0.0625000
$257$	2590.04	0.628647	0.314324	−	0.949316i	$-0.398222\pi$
$257$	2590.04	0.628647	0.314324	+	0.949316i	$0.398222\pi$
$258$	2191.44i	0.528811i
$259$	2986.86	0.716580
$260$	0	0
$261$	1281.94	0.304023
$262$	− 3.34585i	0 0.000788960i
$263$	710.492	0.166581	0.0832905	−	0.996525i	$-0.473457\pi$
$263$	710.492	0.166581	0.0832905	+	0.996525i	$0.473457\pi$
$264$	940.697	0.219303
$265$	− 9276.84i	− 2.15046i
$266$	985.011i	0.227048i
$267$	− 3195.36i	− 0.732408i
$268$	1729.05i	0.394099i
$269$	1054.90	0.239102	0.119551	−	0.992828i	$-0.461855\pi$
$269$	1054.90	0.239102	0.119551	+	0.992828i	$0.461855\pi$
$270$	−703.927	−0.158665
$271$	− 2931.34i	− 0.657070i	−0.944492	−	0.328535i	$-0.893445\pi$
$271$	− 2931.34i	− 0.657070i	0.944492	−	0.328535i	$-0.106555\pi$
$272$	−135.335	−0.0301688
$273$	0	0
$274$	2160.18	0.476281
$275$	1761.02i	0.386159i
$276$	2076.67	0.452901
$277$	2326.39	0.504619	0.252310	−	0.967647i	$-0.418810\pi$
$277$	2326.39	0.504619	0.252310	+	0.967647i	$0.418810\pi$
$278$	2747.20i	0.592684i
$279$	− 2413.26i	− 0.517843i
$280$	− 1544.27i	− 0.329599i
$281$	− 3844.91i	− 0.816257i	−0.912925	−	0.408128i	$-0.866182\pi$
$281$	− 3844.91i	− 0.816257i	0.912925	−	0.408128i	$-0.133818\pi$
$282$	1050.96	0.221928
$283$	−100.318	−0.0210717	−0.0105358	−	0.999944i	$-0.503354\pi$
$283$	−100.318	−0.0210717	−0.0105358	+	0.999944i	$0.503354\pi$
$284$	− 3204.57i	− 0.669563i
$285$	−1300.67	−0.270333
$286$	0	0
$287$	−2447.80	−0.503445
$288$	− 288.000i	− 0.0589256i
$289$	−4841.45	−0.985437
$290$	−3713.54	−0.751954
$291$	4557.34i	0.918061i
$292$	− 684.966i	− 0.137276i
$293$	7655.79i	1.52647i	0.646121	+	0.763235i	$0.276390\pi$
$293$	7655.79i	1.52647i	−0.646121	+	0.763235i	$0.723610\pi$
$294$	− 742.320i	− 0.147255i
$295$	−2479.54	−0.489371
$296$	−1613.63	−0.316860
$297$	− 1058.28i	− 0.206760i
$298$	1957.12	0.380447
$299$	0	0
$300$	539.148	0.103759
$301$	5408.52i	1.03569i
$302$	−6367.89	−1.21335
$303$	4511.29	0.855336
$304$	− 532.147i	− 0.100397i
$305$	8274.62i	1.55345i
$306$	152.252i	0.0284434i
$307$	− 3623.38i	− 0.673606i	−0.941575	−	0.336803i	$-0.890654\pi$
$307$	− 3623.38i	− 0.673606i	0.941575	−	0.336803i	$-0.109346\pi$
$308$	2321.66	0.429508
$309$	−4243.66	−0.781273
$310$	6990.78i	1.28081i
$311$	−1040.58	−0.189730	−0.0948650	−	0.995490i	$-0.530242\pi$
$311$	−1040.58	−0.189730	−0.0948650	+	0.995490i	$0.530242\pi$
$312$	0	0
$313$	5064.24	0.914530	0.457265	−	0.889330i	$-0.348829\pi$
$313$	5064.24	0.914530	0.457265	+	0.889330i	$0.348829\pi$
$314$	1295.19i	0.232777i
$315$	−1737.30	−0.310749
$316$	−2417.83	−0.430422
$317$	− 3711.80i	− 0.657651i	−0.944391	−	0.328826i	$-0.893347\pi$
$317$	− 3711.80i	− 0.657651i	0.944391	−	0.328826i	$-0.106653\pi$
$318$	− 4269.90i	− 0.752969i
$319$	− 5582.94i	− 0.979889i
$320$	834.284i	0.145743i
$321$	507.702	0.0882777
$322$	5125.25	0.887016
$323$	281.322i	0.0484618i
$324$	−324.000	−0.0555556
$325$	0	0
$326$	−5555.60	−0.943853
$327$	− 142.061i	− 0.0240245i
$328$	1322.41	0.222615
$329$	2593.79	0.434651
$330$	3065.65i	0.511390i
$331$	7799.77i	1.29521i	0.761976	+	0.647605i	$0.224229\pi$
$331$	7799.77i	1.29521i	−0.761976	+	0.647605i	$0.775771\pi$
$332$	− 4940.07i	− 0.816631i
$333$	1815.34i	0.298738i
$334$	3574.49	0.585591
$335$	−5634.84	−0.918998
$336$	− 710.789i	− 0.115407i
$337$	4299.73	0.695019	0.347510	−	0.937676i	$-0.387027\pi$
$337$	4299.73	0.695019	0.347510	+	0.937676i	$0.387027\pi$
$338$	0	0
$339$	−6813.86	−1.09168
$340$	− 441.048i	− 0.0703505i
$341$	−10509.9	−1.66905
$342$	−598.665	−0.0946553
$343$	− 6911.24i	− 1.08796i
$344$	− 2921.93i	− 0.457964i
$345$	6767.69i	1.05612i
$346$	3973.70i	0.617420i
$347$	6579.84	1.01794	0.508969	−	0.860785i	$-0.330027\pi$
$347$	6579.84	1.01794	0.508969	+	0.860785i	$0.330027\pi$
$348$	−1709.25	−0.263291
$349$	− 5890.88i	− 0.903529i	−0.892137	−	0.451765i	$-0.850795\pi$
$349$	− 5890.88i	− 0.903529i	0.892137	−	0.451765i	$-0.149205\pi$
$350$	1330.63	0.203214
$351$	0	0
$352$	−1254.26	−0.189922
$353$	− 2703.67i	− 0.407653i	−0.979007	−	0.203827i	$-0.934662\pi$
$353$	− 2703.67i	− 0.407653i	0.979007	−	0.203827i	$-0.0653379\pi$
$354$	−1141.27	−0.171350
$355$	10443.4	1.56135
$356$	4260.48i	0.634284i
$357$	375.762i	0.0557070i
$358$	1197.98i	0.176858i
$359$	− 7582.18i	− 1.11469i	−0.830282	−	0.557343i	$-0.811821\pi$
$359$	− 7582.18i	− 1.11469i	0.830282	−	0.557343i	$-0.188179\pi$
$360$	938.569	0.137408
$361$	5752.83	0.838727
$362$	− 4885.59i	− 0.709339i
$363$	−615.906	−0.0890543
$364$	0	0
$365$	2232.25	0.320113
$366$	3808.60i	0.543931i
$367$	5766.13	0.820135	0.410068	−	0.912055i	$-0.365505\pi$
$367$	5766.13	0.820135	0.410068	+	0.912055i	$0.365505\pi$
$368$	−2768.89	−0.392224
$369$	− 1487.71i	− 0.209884i
$370$	− 5258.70i	− 0.738883i
$371$	− 10538.2i	− 1.47471i
$372$	3217.68i	0.448465i
$373$	−8704.42	−1.20831	−0.604153	−	0.796868i	$-0.706488\pi$
$373$	−8704.42	−1.20831	−0.604153	+	0.796868i	$0.706488\pi$
$374$	663.071	0.0916754
$375$	− 3131.34i	− 0.431205i
$376$	−1401.28	−0.192196
$377$	0	0
$378$	−799.638	−0.108807
$379$	− 11893.2i	− 1.61190i	−0.591983	−	0.805950i	$-0.701655\pi$
$379$	− 11893.2i	− 1.61190i	0.591983	−	0.805950i	$-0.298345\pi$
$380$	1734.22	0.234115
$381$	−6412.47	−0.862260
$382$	− 7163.39i	− 0.959452i
$383$	− 13553.4i	− 1.80821i	−0.427310	−	0.904105i	$-0.640539\pi$
$383$	− 13553.4i	− 1.80821i	0.427310	−	0.904105i	$-0.359461\pi$
$384$	384.000i	0.0510310i
$385$	7566.09i	1.00157i
$386$	−1184.03	−0.156128
$387$	−3287.17	−0.431773
$388$	− 6076.45i	− 0.795064i
$389$	−9612.16	−1.25284	−0.626421	−	0.779485i	$-0.715481\pi$
$389$	−9612.16	−1.25284	−0.626421	+	0.779485i	$0.715481\pi$
$390$	0	0
$391$	1463.79	0.189327
$392$	989.761i	0.127527i
$393$	5.01878	0.000644183 0
$394$	1185.47	0.151581
$395$	− 7879.51i	− 1.00370i
$396$	1411.04i	0.179060i
$397$	− 12295.5i	− 1.55439i	−0.629262	−	0.777194i	$-0.716643\pi$
$397$	− 12295.5i	− 1.55439i	0.629262	−	0.777194i	$-0.283357\pi$
$398$	10351.7i	1.30373i
$399$	−1477.52	−0.185384
$400$	−718.863	−0.0898579
$401$	− 3206.28i	− 0.399287i	−0.979869	−	0.199643i	$-0.936022\pi$
$401$	− 3206.28i	− 0.399287i	0.979869	−	0.199643i	$-0.0639784\pi$
$402$	−2593.58	−0.321781
$403$	0	0
$404$	−6015.05	−0.740743
$405$	− 1055.89i	− 0.129550i
$406$	−4218.46	−0.515662
$407$	7905.93	0.962856
$408$	− 203.003i	− 0.0246327i
$409$	− 10626.4i	− 1.28469i	−0.766414	−	0.642347i	$-0.777961\pi$
$409$	− 10626.4i	− 1.28469i	0.766414	−	0.642347i	$-0.222039\pi$
$410$	4309.62i	0.519115i
$411$	3240.26i	0.388882i
$412$	5658.21	0.676602
$413$	−2816.67	−0.335592
$414$	3115.00i	0.369792i
$415$	16099.3	1.90430
$416$	0	0
$417$	−4120.80	−0.483924
$418$	2607.23i	0.305081i
$419$	−4264.01	−0.497162	−0.248581	−	0.968611i	$-0.579964\pi$
$419$	−4264.01	−0.497162	−0.248581	+	0.968611i	$0.579964\pi$
$420$	2316.40	0.269117
$421$	− 9165.42i	− 1.06103i	−0.847675	−	0.530517i	$-0.821998\pi$
$421$	− 9165.42i	− 1.06103i	0.847675	−	0.530517i	$-0.178002\pi$
$422$	− 11859.4i	− 1.36803i
$423$	1576.44i	0.181204i
$424$	5693.20i	0.652090i
$425$	380.030	0.0433745
$426$	4806.85	0.546696
$427$	9399.70i	1.06530i
$428$	−676.935	−0.0764507
$429$	0	0
$430$	9522.32	1.06792
$431$	16237.5i	1.81469i	0.420389	+	0.907344i	$0.361894\pi$
$431$	16237.5i	1.81469i	−0.420389	+	0.907344i	$0.638106\pi$
$432$	432.000	0.0481125
$433$	15002.1	1.66502	0.832509	−	0.554012i	$-0.186904\pi$
$433$	15002.1	1.66502	0.832509	+	0.554012i	$0.186904\pi$
$434$	7941.30i	0.878328i
$435$	− 5570.31i	− 0.613968i
$436$	189.415i	0.0208058i
$437$	5755.69i	0.630050i
$438$	1027.45	0.112085
$439$	9365.84	1.01824	0.509119	−	0.860696i	$-0.329971\pi$
$439$	9365.84	1.01824	0.509119	+	0.860696i	$0.329971\pi$
$440$	− 4087.54i	− 0.442877i
$441$	1113.48	0.120233
$442$	0	0
$443$	−223.813	−0.0240038	−0.0120019	−	0.999928i	$-0.503820\pi$
$443$	−223.813	−0.0240038	−0.0120019	+	0.999928i	$0.503820\pi$
$444$	− 2420.45i	− 0.258715i
$445$	−13884.6	−1.47908
$446$	1224.10	0.129961
$447$	2935.69i	0.310633i
$448$	947.719i	0.0999453i
$449$	9170.14i	0.963844i	0.876214	+	0.481922i	$0.160061\pi$
$449$	9170.14i	0.963844i	−0.876214	+	0.481922i	$0.839939\pi$
$450$	808.721i	0.0847189i
$451$	−6479.09	−0.676471
$452$	9085.14	0.945418
$453$	− 9551.84i	− 0.990695i
$454$	7333.08	0.758058
$455$	0	0
$456$	798.221	0.0819739
$457$	− 13411.3i	− 1.37277i	−0.727241	−	0.686383i	$-0.759198\pi$
$457$	− 13411.3i	− 1.37277i	0.727241	−	0.686383i	$-0.240802\pi$
$458$	1110.92	0.113340
$459$	−228.379	−0.0232240
$460$	− 9023.59i	− 0.914624i
$461$	− 19257.8i	− 1.94561i	−0.231636	−	0.972803i	$-0.574408\pi$
$461$	− 19257.8i	− 1.94561i	0.231636	−	0.972803i	$-0.425592\pi$
$462$	3482.48i	0.350692i
$463$	− 10610.0i	− 1.06499i	−0.846435	−	0.532493i	$-0.821255\pi$
$463$	− 10610.0i	− 1.06499i	0.846435	−	0.532493i	$-0.178745\pi$
$464$	2279.00	0.228017
$465$	−10486.2	−1.04577
$466$	8083.24i	0.803539i
$467$	−1712.12	−0.169652	−0.0848260	−	0.996396i	$-0.527033\pi$
$467$	−1712.12	−0.169652	−0.0848260	+	0.996396i	$0.527033\pi$
$468$	0	0
$469$	−6400.99	−0.630214
$470$	− 4566.66i	− 0.448180i
$471$	−1942.79	−0.190061
$472$	1521.69	0.148393
$473$	14315.9i	1.39164i
$474$	− 3626.74i	− 0.351438i
$475$	1494.30i	0.144344i
$476$	− 501.015i	− 0.0482437i
$477$	6404.85	0.614797
$478$	−2161.34	−0.206814
$479$	− 190.036i	− 0.0181273i	−0.999959	−	0.00906363i	$-0.997115\pi$
$479$	− 190.036i	− 0.0181273i	0.999959	−	0.00906363i	$-0.00288508\pi$
$480$	−1251.43	−0.118999
$481$	0	0
$482$	−1743.49	−0.164759
$483$	7687.88i	0.724245i
$484$	821.209	0.0771233
$485$	19802.7	1.85401
$486$	− 486.000i	− 0.0453609i
$487$	− 4896.18i	− 0.455580i	−0.973710	−	0.227790i	$-0.926850\pi$
$487$	− 4896.18i	− 0.455580i	0.973710	−	0.227790i	$-0.0731499\pi$
$488$	− 5078.14i	− 0.471058i
$489$	− 8333.40i	− 0.770653i
$490$	−3225.55	−0.297379
$491$	−6375.34	−0.585978	−0.292989	−	0.956116i	$-0.594650\pi$
$491$	−6375.34	−0.585978	−0.292989	+	0.956116i	$0.594650\pi$
$492$	1983.61i	0.181765i
$493$	−1204.80	−0.110064
$494$	0	0
$495$	−4598.48	−0.417548
$496$	− 4290.24i	− 0.388382i
$497$	11863.4	1.07072
$498$	7410.10	0.666776
$499$	− 11804.2i	− 1.05897i	−0.848318	−	0.529487i	$-0.822384\pi$
$499$	− 11804.2i	− 1.05897i	0.848318	−	0.529487i	$-0.177616\pi$
$500$	4175.12i	0.373434i
$501$	5361.73i	0.478133i
$502$	− 8682.18i	− 0.771921i
$503$	−18089.7	−1.60354	−0.801770	−	0.597633i	$-0.796108\pi$
$503$	−18089.7	−1.60354	−0.801770	+	0.597633i	$0.796108\pi$
$504$	1066.18	0.0942293
$505$	− 19602.6i	− 1.72733i
$506$	13566.1	1.19187
$507$	0	0
$508$	8549.96	0.746739
$509$	11002.1i	0.958073i	0.877795	+	0.479036i	$0.159014\pi$
$509$	11002.1i	0.958073i	−0.877795	+	0.479036i	$0.840986\pi$
$510$	661.571	0.0574409
$511$	2535.76	0.219522
$512$	− 512.000i	− 0.0441942i
$513$	− 897.998i	− 0.0772857i
$514$	− 5180.08i	− 0.444521i
$515$	18439.7i	1.57776i
$516$	4382.89	0.373926
$517$	6865.52	0.584033
$518$	− 5973.71i	− 0.506698i
$519$	−5960.55	−0.504122
$520$	0	0
$521$	8833.33	0.742793	0.371397	−	0.928474i	$-0.378879\pi$
$521$	8833.33	0.742793	0.371397	+	0.928474i	$0.378879\pi$
$522$	− 2563.88i	− 0.214977i
$523$	−8429.90	−0.704806	−0.352403	−	0.935848i	$-0.614635\pi$
$523$	−8429.90	−0.704806	−0.352403	+	0.935848i	$0.614635\pi$
$524$	−6.69171	−0.000557879 0
$525$	1995.94i	0.165924i
$526$	− 1420.98i	− 0.117791i
$527$	2268.06i	0.187473i
$528$	− 1881.39i	− 0.155070i
$529$	17781.2	1.46143
$530$	−18553.7	−1.52061
$531$	− 1711.90i	− 0.139906i
$532$	1970.02	0.160548
$533$	0	0
$534$	−6390.73	−0.517891
$535$	− 2206.08i	− 0.178275i
$536$	3458.10	0.278670
$537$	−1796.97	−0.144404
$538$	− 2109.80i	− 0.169071i
$539$	− 4849.29i	− 0.387521i
$540$	1407.85i	0.112193i
$541$	− 13663.3i	− 1.08583i	−0.839789	−	0.542913i	$-0.817321\pi$
$541$	− 13663.3i	− 1.08583i	0.839789	−	0.542913i	$-0.182679\pi$
$542$	−5862.67	−0.464619
$543$	7328.38	0.579173
$544$	270.671i	0.0213326i
$545$	−617.288	−0.0485169
$546$	0	0
$547$	2196.18	0.171667	0.0858336	−	0.996309i	$-0.472645\pi$
$547$	2196.18	0.171667	0.0858336	+	0.996309i	$0.472645\pi$
$548$	− 4320.35i	− 0.336782i
$549$	−5712.91	−0.444118
$550$	3522.04	0.273055
$551$	− 4737.36i	− 0.366276i
$552$	− 4153.33i	− 0.320249i
$553$	− 8950.86i	− 0.688299i
$554$	− 4652.79i	− 0.356820i
$555$	7888.05	0.603296
$556$	5494.40	0.419091
$557$	− 13571.1i	− 1.03237i	−0.856478	−	0.516183i	$-0.827352\pi$
$557$	− 13571.1i	− 1.03237i	0.856478	−	0.516183i	$-0.172648\pi$
$558$	−4826.52	−0.366170
$559$	0	0
$560$	−3088.54	−0.233062
$561$	994.607i	0.0748526i
$562$	−7689.82	−0.577181
$563$	−2943.72	−0.220361	−0.110180	−	0.993912i	$-0.535143\pi$
$563$	−2943.72	−0.220361	−0.110180	+	0.993912i	$0.535143\pi$
$564$	− 2101.92i	− 0.156927i
$565$	29607.7i	2.20461i
$566$	200.636i	0.0148999i
$567$	− 1199.46i	− 0.0888403i
$568$	−6409.13	−0.473453
$569$	8640.09	0.636575	0.318288	−	0.947994i	$-0.396892\pi$
$569$	8640.09	0.636575	0.318288	+	0.947994i	$0.396892\pi$
$570$	2601.34i	0.191154i
$571$	21379.4	1.56690	0.783450	−	0.621455i	$-0.213458\pi$
$571$	21379.4	1.56690	0.783450	+	0.621455i	$0.213458\pi$
$572$	0	0
$573$	10745.1	0.783389
$574$	4895.59i	0.355990i
$575$	7775.21	0.563911
$576$	−576.000	−0.0416667
$577$	12426.4i	0.896566i	0.893892	+	0.448283i	$0.147964\pi$
$577$	12426.4i	0.896566i	−0.893892	+	0.448283i	$0.852036\pi$
$578$	9682.91i	0.696810i
$579$	− 1776.04i	− 0.127478i
$580$	7427.08i	0.531712i
$581$	18288.3	1.30589
$582$	9114.67	0.649167
$583$	− 27893.6i	− 1.98154i
$584$	−1369.93	−0.0970688
$585$	0	0
$586$	15311.6	1.07938
$587$	9338.84i	0.656652i	0.944564	+	0.328326i	$0.106485\pi$
$587$	9338.84i	0.656652i	−0.944564	+	0.328326i	$0.893515\pi$
$588$	−1484.64	−0.104125
$589$	−8918.12	−0.623879
$590$	4959.08i	0.346037i
$591$	1778.20i	0.123766i
$592$	3227.27i	0.224054i
$593$	− 8286.99i	− 0.573871i	−0.957950	−	0.286936i	$-0.907363\pi$
$593$	− 8286.99i	− 0.573871i	0.957950	−	0.286936i	$-0.0926366\pi$
$594$	−2116.57	−0.146202
$595$	1632.77	0.112499
$596$	− 3914.25i	− 0.269016i
$597$	−15527.5	−1.06449
$598$	0	0
$599$	−557.702	−0.0380419	−0.0190209	−	0.999819i	$-0.506055\pi$
$599$	−557.702	−0.0380419	−0.0190209	+	0.999819i	$0.506055\pi$
$600$	− 1078.30i	− 0.0733687i
$601$	−21343.7	−1.44863	−0.724315	−	0.689469i	$-0.757844\pi$
$601$	−21343.7	−1.44863	−0.724315	+	0.689469i	$0.757844\pi$
$602$	10817.0	0.732342
$603$	− 3890.37i	− 0.262733i
$604$	12735.8i	0.857967i
$605$	2676.25i	0.179843i
$606$	− 9022.58i	− 0.604814i
$607$	−17694.8	−1.18321	−0.591606	−	0.806227i	$-0.701506\pi$
$607$	−17694.8	−1.18321	−0.591606	+	0.806227i	$0.701506\pi$
$608$	−1064.29	−0.0709915
$609$	− 6327.69i	− 0.421036i
$610$	16549.2	1.09846
$611$	0	0
$612$	304.505	0.0201125
$613$	2125.39i	0.140039i	0.997546	+	0.0700193i	$0.0223061\pi$
$613$	2125.39i	0.140039i	−0.997546	+	0.0700193i	$0.977694\pi$
$614$	−7246.75	−0.476311
$615$	−6464.44	−0.423856
$616$	− 4643.31i	− 0.303708i
$617$	− 20370.8i	− 1.32917i	−0.747214	−	0.664583i	$-0.768609\pi$
$617$	− 20370.8i	− 1.32917i	0.747214	−	0.664583i	$-0.231391\pi$
$618$	8487.32i	0.552443i
$619$	− 6596.80i	− 0.428349i	−0.976795	−	0.214174i	$-0.931294\pi$
$619$	− 6596.80i	− 0.428349i	0.976795	−	0.214174i	$-0.0687061\pi$
$620$	13981.6	0.905666
$621$	−4672.50	−0.301934
$622$	2081.16i	0.134159i
$623$	−15772.4	−1.01430
$624$	0	0
$625$	−19222.5	−1.23024
$626$	− 10128.5i	− 0.646670i
$627$	−3910.85	−0.249098
$628$	2590.38	0.164598
$629$	− 1706.11i	− 0.108151i
$630$	3474.61i	0.219733i
$631$	4278.02i	0.269898i	0.990853	+	0.134949i	$0.0430870\pi$
$631$	4278.02i	0.269898i	−0.990853	+	0.134949i	$0.956913\pi$
$632$	4835.65i	0.304354i
$633$	17789.1	1.11699
$634$	−7423.60	−0.465030
$635$	27863.7i	1.74132i
$636$	−8539.80	−0.532430
$637$	0	0
$638$	−11165.9	−0.692886
$639$	7210.27i	0.446376i
$640$	1668.57	0.103056
$641$	10092.0	0.621859	0.310929	−	0.950433i	$-0.399360\pi$
$641$	10092.0	0.621859	0.310929	+	0.950433i	$0.399360\pi$
$642$	− 1015.40i	− 0.0624217i
$643$	− 18585.7i	− 1.13989i	−0.821683	−	0.569945i	$-0.806965\pi$
$643$	− 18585.7i	− 1.13989i	0.821683	−	0.569945i	$-0.193035\pi$
$644$	− 10250.5i	− 0.627215i
$645$	14283.5i	0.871956i
$646$	562.644	0.0342677
$647$	756.243	0.0459521	0.0229760	−	0.999736i	$-0.492686\pi$
$647$	756.243	0.0459521	0.0229760	+	0.999736i	$0.492686\pi$
$648$	648.000i	0.0392837i
$649$	−7455.48	−0.450929
$650$	0	0
$651$	−11911.9	−0.717152
$652$	11111.2i	0.667405i
$653$	16258.3	0.974331	0.487165	−	0.873310i	$-0.338031\pi$
$653$	16258.3	0.974331	0.487165	+	0.873310i	$0.338031\pi$
$654$	−284.123	−0.0169879
$655$	− 21.8077i	− 0.00130091i
$656$	− 2644.82i	− 0.157413i
$657$	1541.17i	0.0915174i
$658$	− 5187.58i	− 0.307345i
$659$	11145.1	0.658806	0.329403	−	0.944189i	$-0.393152\pi$
$659$	11145.1	0.658806	0.329403	+	0.944189i	$0.393152\pi$
$660$	6131.31	0.361607
$661$	29266.9i	1.72217i	0.508465	+	0.861083i	$0.330213\pi$
$661$	29266.9i	1.72217i	−0.508465	+	0.861083i	$0.669787\pi$
$662$	15599.5	0.915851
$663$	0	0
$664$	−9880.13	−0.577445
$665$	6420.14i	0.374380i
$666$	3630.67	0.211240
$667$	−24649.6	−1.43094
$668$	− 7148.97i	− 0.414075i
$669$	1836.14i	0.106113i
$670$	11269.7i	0.649829i
$671$	24880.1i	1.43143i
$672$	−1421.58	−0.0816050
$673$	28022.1	1.60501	0.802504	−	0.596646i	$-0.203500\pi$
$673$	28022.1	1.60501	0.802504	+	0.596646i	$0.203500\pi$
$674$	− 8599.47i	− 0.491453i
$675$	−1213.08	−0.0691727
$676$	0	0
$677$	19931.2	1.13149	0.565745	−	0.824580i	$-0.308589\pi$
$677$	19931.2	1.13149	0.565745	+	0.824580i	$0.308589\pi$
$678$	13627.7i	0.771931i
$679$	22495.2	1.27141
$680$	−882.095	−0.0497453
$681$	10999.6i	0.618952i
$682$	21019.9i	1.18019i
$683$	− 7710.87i	− 0.431989i	−0.976395	−	0.215994i	$-0.930701\pi$
$683$	− 7710.87i	− 0.431989i	0.976395	−	0.215994i	$-0.0692993\pi$
$684$	1197.33i	0.0669314i
$685$	14079.7	0.785339
$686$	−13822.5	−0.769307
$687$	1666.38i	0.0925419i
$688$	−5843.85	−0.323830
$689$	0	0
$690$	13535.4	0.746787
$691$	4634.43i	0.255140i	0.991830	+	0.127570i	$0.0407178\pi$
$691$	4634.43i	0.255140i	−0.991830	+	0.127570i	$0.959282\pi$
$692$	7947.40	0.436582
$693$	−5223.72	−0.286339
$694$	− 13159.7i	− 0.719790i
$695$	17905.8i	0.977275i
$696$	3418.50i	0.186175i
$697$	1398.19i	0.0759833i
$698$	−11781.8	−0.638892
$699$	−12124.9	−0.656087
$700$	− 2661.25i	− 0.143694i
$701$	−4795.63	−0.258386	−0.129193	−	0.991619i	$-0.541239\pi$
$701$	−4795.63	−0.258386	−0.129193	+	0.991619i	$0.541239\pi$
$702$	0	0
$703$	6708.51	0.359910
$704$	2508.52i	0.134295i
$705$	6849.99	0.365937
$706$	−5407.33	−0.288254
$707$	− 22267.9i	− 1.18454i
$708$	2282.54i	0.121163i
$709$	− 7044.68i	− 0.373157i	−0.982440	−	0.186579i	$-0.940260\pi$
$709$	− 7044.68i	− 0.373157i	0.982440	−	0.186579i	$-0.0597399\pi$
$710$	− 20886.8i	− 1.10404i
$711$	5440.11	0.286948
$712$	8520.97	0.448507
$713$	46403.2i	2.43732i
$714$	751.523	0.0393908
$715$	0	0
$716$	2395.96	0.125058
$717$	− 3242.01i	− 0.168863i
$718$	−15164.4	−0.788202
$719$	−26750.1	−1.38750	−0.693750	−	0.720216i	$-0.744043\pi$
$719$	−26750.1	−1.38750	−0.693750	+	0.720216i	$0.744043\pi$
$720$	− 1877.14i	− 0.0971622i
$721$	20946.8i	1.08197i
$722$	− 11505.7i	− 0.593069i
$723$	− 2615.24i	− 0.134525i
$724$	−9771.17	−0.501578
$725$	−6399.57	−0.327826
$726$	1231.81i	0.0629709i
$727$	34868.5	1.77882	0.889410	−	0.457110i	$-0.151115\pi$
$727$	34868.5	1.77882	0.889410	+	0.457110i	$0.151115\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	− 4464.50i	− 0.226354i
$731$	3089.38	0.156313
$732$	7617.21	0.384618
$733$	20936.5i	1.05499i	0.849559	+	0.527494i	$0.176868\pi$
$733$	20936.5i	1.05499i	−0.849559	+	0.527494i	$0.823132\pi$
$734$	− 11532.3i	− 0.579923i
$735$	− 4838.33i	− 0.242809i
$736$	5537.78i	0.277344i
$737$	−16942.8	−0.846808
$738$	−2975.42	−0.148410
$739$	8221.72i	0.409257i	0.978840	+	0.204628i	$0.0655986\pi$
$739$	8221.72i	0.409257i	−0.978840	+	0.204628i	$0.934401\pi$
$740$	−10517.4	−0.522469
$741$	0	0
$742$	−21076.4	−1.04277
$743$	19672.3i	0.971339i	0.874142	+	0.485670i	$0.161424\pi$
$743$	19672.3i	0.971339i	−0.874142	+	0.485670i	$0.838576\pi$
$744$	6435.36	0.317113
$745$	12756.2	0.627318
$746$	17408.8i	0.854401i
$747$	11115.2i	0.544420i
$748$	− 1326.14i	− 0.0648243i
$749$	− 2506.03i	− 0.122254i
$750$	−6262.68	−0.304908
$751$	−29414.5	−1.42923	−0.714613	−	0.699520i	$-0.753397\pi$
$751$	−29414.5	−1.42923	−0.714613	+	0.699520i	$0.753397\pi$
$752$	2802.56i	0.135903i
$753$	13023.3	0.630271
$754$	0	0
$755$	−41504.9	−2.00069
$756$	1599.28i	0.0769379i
$757$	−16086.0	−0.772334	−0.386167	−	0.922429i	$-0.626201\pi$
$757$	−16086.0	−0.772334	−0.386167	+	0.922429i	$0.626201\pi$
$758$	−23786.3	−1.13979
$759$	20349.1i	0.973156i
$760$	− 3468.45i	− 0.165545i
$761$	9449.81i	0.450139i	0.974343	+	0.225069i	$0.0722608\pi$
$761$	9449.81i	0.450139i	−0.974343	+	0.225069i	$0.927739\pi$
$762$	12824.9i	0.609710i
$763$	−701.219	−0.0332711
$764$	−14326.8	−0.678435
$765$	992.357i	0.0469003i
$766$	−27106.7	−1.27860
$767$	0	0
$768$	768.000	0.0360844
$769$	− 29699.8i	− 1.39272i	−0.717691	−	0.696361i	$-0.754801\pi$
$769$	− 29699.8i	− 1.39272i	0.717691	−	0.696361i	$-0.245199\pi$
$770$	15132.2	0.708215
$771$	7770.13	0.362950
$772$	2368.05i	0.110399i
$773$	2974.73i	0.138413i	0.997602	+	0.0692067i	$0.0220468\pi$
$773$	2974.73i	0.138413i	−0.997602	+	0.0692067i	$0.977953\pi$
$774$	6574.33i	0.305309i
$775$	12047.3i	0.558388i
$776$	−12152.9	−0.562195
$777$	8960.57	0.413717
$778$	19224.3i	0.885894i
$779$	−5497.78	−0.252861
$780$	0	0
$781$	31401.3	1.43870
$782$	− 2927.57i	− 0.133874i
$783$	3845.81	0.175528
$784$	1979.52	0.0901750
$785$	8441.85i	0.383825i
$786$	− 10.0376i	0 0.000455506i
$787$	− 28152.0i	− 1.27511i	−0.770406	−	0.637554i	$-0.779946\pi$
$787$	− 28152.0i	− 1.27511i	0.770406	−	0.637554i	$-0.220054\pi$
$788$	− 2370.94i	− 0.107184i
$789$	2131.48	0.0961756
$790$	−15759.0	−0.709722
$791$	33633.4i	1.51184i
$792$	2822.09	0.126614
$793$	0	0
$794$	−24590.9	−1.09912
$795$	− 27830.5i	− 1.24157i
$796$	20703.4	0.921874
$797$	−42721.2	−1.89870	−0.949350	−	0.314220i	$-0.898257\pi$
$797$	−42721.2	−1.89870	−0.949350	+	0.314220i	$0.898257\pi$
$798$	2955.03i	0.131087i
$799$	− 1481.59i	− 0.0656004i
$800$	1437.73i	0.0635391i
$801$	− 9586.09i	− 0.422856i
$802$	−6412.56	−0.282338
$803$	6711.93	0.294967
$804$	5187.15i	0.227533i
$805$	33405.6	1.46260
$806$	0	0
$807$	3164.70	0.138046
$808$	12030.1i	0.523784i
$809$	−6644.27	−0.288752	−0.144376	−	0.989523i	$-0.546117\pi$
$809$	−6644.27	−0.288752	−0.144376	+	0.989523i	$0.546117\pi$
$810$	−2111.78	−0.0916054
$811$	16122.1i	0.698058i	0.937112	+	0.349029i	$0.113489\pi$
$811$	16122.1i	0.698058i	−0.937112	+	0.349029i	$0.886511\pi$
$812$	8436.92i	0.364628i
$813$	− 8794.01i	− 0.379360i
$814$	− 15811.9i	− 0.680842i
$815$	−36210.5	−1.55632
$816$	−406.006	−0.0174180
$817$	12147.6i	0.520185i
$818$	−21252.7	−0.908415
$819$	0	0
$820$	8619.25	0.367070
$821$	− 8389.53i	− 0.356634i	−0.983973	−	0.178317i	$-0.942935\pi$
$821$	− 8389.53i	− 0.356634i	0.983973	−	0.178317i	$-0.0570652\pi$
$822$	6480.53	0.274981
$823$	−22647.4	−0.959221	−0.479610	−	0.877482i	$-0.659222\pi$
$823$	−22647.4	−0.959221	−0.479610	+	0.877482i	$0.659222\pi$
$824$	− 11316.4i	− 0.478430i
$825$	5283.06i	0.222949i
$826$	5633.35i	0.237299i
$827$	9837.48i	0.413643i	0.978379	+	0.206821i	$0.0663119\pi$
$827$	9837.48i	0.413643i	−0.978379	+	0.206821i	$0.933688\pi$
$828$	6230.00	0.261483
$829$	−24156.5	−1.01205	−0.506024	−	0.862519i	$-0.668885\pi$
$829$	−24156.5	−1.01205	−0.506024	+	0.862519i	$0.668885\pi$
$830$	− 32198.6i	− 1.34654i
$831$	6979.18	0.291342
$832$	0	0
$833$	−1046.48	−0.0435276
$834$	8241.60i	0.342186i
$835$	23297.9	0.965579
$836$	5214.47	0.215725
$837$	− 7239.78i	− 0.298977i
$838$	8528.03i	0.351546i
$839$	8737.00i	0.359517i	0.983711	+	0.179758i	$0.0575316\pi$
$839$	8737.00i	0.359517i	−0.983711	+	0.179758i	$0.942468\pi$
$840$	− 4632.81i	− 0.190294i
$841$	−4100.55	−0.168131
$842$	−18330.8	−0.750264
$843$	− 11534.7i	− 0.471266i
$844$	−23718.8	−0.967341
$845$	0	0
$846$	3152.88	0.128130
$847$	3040.14i	0.123330i
$848$	11386.4	0.461098
$849$	−300.954	−0.0121657
$850$	− 760.060i	− 0.0306704i
$851$	− 34906.0i	− 1.40607i
$852$	− 9613.70i	− 0.386573i
$853$	13145.6i	0.527665i	0.964568	+	0.263833i	$0.0849867\pi$
$853$	13145.6i	0.527665i	−0.964568	+	0.263833i	$0.915013\pi$
$854$	18799.4	0.753281
$855$	−3902.01	−0.156077
$856$	1353.87i	0.0540588i
$857$	−4344.97	−0.173187	−0.0865936	−	0.996244i	$-0.527598\pi$
$857$	−4344.97	−0.173187	−0.0865936	+	0.996244i	$0.527598\pi$
$858$	0	0
$859$	8258.30	0.328020	0.164010	−	0.986459i	$-0.447557\pi$
$859$	8258.30	0.328020	0.164010	+	0.986459i	$0.447557\pi$
$860$	− 19044.6i	− 0.755136i
$861$	−7343.39	−0.290664
$862$	32474.9	1.28318
$863$	− 44634.3i	− 1.76057i	−0.474449	−	0.880283i	$-0.657352\pi$
$863$	− 44634.3i	− 1.76057i	0.474449	−	0.880283i	$-0.342648\pi$
$864$	− 864.000i	− 0.0340207i
$865$	25899.9i	1.01806i
$866$	− 30004.1i	− 1.17735i
$867$	−14524.4	−0.568943
$868$	15882.6	0.621072
$869$	− 23692.1i	− 0.924856i
$870$	−11140.6	−0.434141
$871$	0	0
$872$	378.830	0.0147119
$873$	13672.0i	0.530043i
$874$	11511.4	0.445513
$875$	−15456.4	−0.597168
$876$	− 2054.90i	− 0.0792564i
$877$	46745.1i	1.79985i	0.436042	+	0.899926i	$0.356380\pi$
$877$	46745.1i	1.79985i	−0.436042	+	0.899926i	$0.643620\pi$
$878$	− 18731.7i	− 0.720004i
$879$	22967.4i	0.881308i
$880$	−8175.08	−0.313161
$881$	50235.6	1.92109	0.960545	−	0.278123i	$-0.0897124\pi$
$881$	50235.6	1.92109	0.960545	+	0.278123i	$0.0897124\pi$
$882$	− 2226.96i	− 0.0850178i
$883$	−20035.3	−0.763581	−0.381791	−	0.924249i	$-0.624692\pi$
$883$	−20035.3	−0.763581	−0.381791	+	0.924249i	$0.624692\pi$
$884$	0	0
$885$	−7438.62	−0.282538
$886$	447.627i	0.0169733i
$887$	12435.1	0.470722	0.235361	−	0.971908i	$-0.424373\pi$
$887$	12435.1	0.470722	0.235361	+	0.971908i	$0.424373\pi$
$888$	−4840.90	−0.182939
$889$	31652.2i	1.19413i
$890$	27769.2i	1.04587i
$891$	− 3174.85i	− 0.119373i
$892$	− 2448.19i	− 0.0918964i
$893$	5825.68	0.218308
$894$	5871.37	0.219651
$895$	7808.25i	0.291621i
$896$	1895.44	0.0706720
$897$	0	0
$898$	18340.3	0.681540
$899$	− 38193.2i	− 1.41692i
$900$	1617.44	0.0599053
$901$	−6019.47	−0.222572
$902$	12958.2i	0.478337i
$903$	16225.6i	0.597955i
$904$	− 18170.3i	− 0.668512i
$905$	− 31843.5i	− 1.16963i
$906$	−19103.7	−0.700527
$907$	16893.0	0.618438	0.309219	−	0.950991i	$-0.399932\pi$
$907$	16893.0	0.618438	0.309219	+	0.950991i	$0.399932\pi$
$908$	− 14666.2i	− 0.536028i
$909$	13533.9	0.493829
$910$	0	0
$911$	−20165.3	−0.733376	−0.366688	−	0.930344i	$-0.619508\pi$
$911$	−20165.3	−0.733376	−0.366688	+	0.930344i	$0.619508\pi$
$912$	− 1596.44i	− 0.0579643i
$913$	48407.3	1.75471
$914$	−26822.6	−0.970692
$915$	24823.9i	0.896887i
$916$	− 2221.84i	− 0.0801437i
$917$	− 24.7729i	0 0.000892118i
$918$	456.757i	0.0164218i
$919$	−21687.4	−0.778456	−0.389228	−	0.921141i	$-0.627258\pi$
$919$	−21687.4	−0.778456	−0.389228	+	0.921141i	$0.627258\pi$
$920$	−18047.2	−0.646737
$921$	− 10870.1i	− 0.388907i
$922$	−38515.6	−1.37575
$923$	0	0
$924$	6964.97	0.247977
$925$	− 9062.36i	− 0.322128i
$926$	−21220.0	−0.753058
$927$	−12731.0	−0.451068
$928$	− 4558.00i	− 0.161232i
$929$	37257.4i	1.31580i	0.753107	+	0.657898i	$0.228554\pi$
$929$	37257.4i	1.31580i	−0.753107	+	0.657898i	$0.771446\pi$
$930$	20972.3i	0.739473i
$931$	− 4114.83i	− 0.144853i
$932$	16166.5	0.568188
$933$	−3121.75	−0.109541
$934$	3424.24i	0.119962i
$935$	4321.79	0.151163
$936$	0	0
$937$	41917.1	1.46144	0.730722	−	0.682675i	$-0.239184\pi$
$937$	41917.1	1.46144	0.730722	+	0.682675i	$0.239184\pi$
$938$	12802.0i	0.445629i
$939$	15192.7	0.528004
$940$	−9133.32	−0.316911
$941$	22543.9i	0.780988i	0.920606	+	0.390494i	$0.127696\pi$
$941$	22543.9i	0.780988i	−0.920606	+	0.390494i	$0.872304\pi$
$942$	3885.58i	0.134394i
$943$	28606.3i	0.987856i
$944$	− 3043.39i	− 0.104930i
$945$	−5211.91	−0.179411
$946$	28631.7	0.984036
$947$	40950.6i	1.40519i	0.711590	+	0.702595i	$0.247975\pi$
$947$	40950.6i	1.40519i	−0.711590	+	0.702595i	$0.752025\pi$
$948$	−7253.48	−0.248504
$949$	0	0
$950$	2988.60	0.102066
$951$	− 11135.4i	− 0.379695i
$952$	−1002.03	−0.0341135
$953$	24992.5	0.849514	0.424757	−	0.905307i	$-0.360360\pi$
$953$	24992.5	0.849514	0.424757	+	0.905307i	$0.360360\pi$
$954$	− 12809.7i	− 0.434727i
$955$	− 46689.8i	− 1.58204i
$956$	4322.68i	0.146240i
$957$	− 16748.8i	− 0.565739i
$958$	−380.072	−0.0128179
$959$	15994.1	0.538556
$960$	2502.85i	0.0841450i
$961$	−42108.2	−1.41345
$962$	0	0
$963$	1523.10	0.0509671
$964$	3486.98i	0.116502i
$965$	−7717.30	−0.257439
$966$	15375.8	0.512119
$967$	− 14911.7i	− 0.495893i	−0.968774	−	0.247946i	$-0.920244\pi$
$967$	− 14911.7i	− 0.495893i	0.968774	−	0.247946i	$-0.0797557\pi$
$968$	− 1642.42i	− 0.0545344i
$969$	843.965i	0.0279794i
$970$	− 39605.3i	− 1.31098i
$971$	−9264.44	−0.306189	−0.153095	−	0.988212i	$-0.548924\pi$
$971$	−9264.44	−0.306189	−0.153095	+	0.988212i	$0.548924\pi$
$972$	−972.000	−0.0320750
$973$	20340.4i	0.670179i
$974$	−9792.37	−0.322143
$975$	0	0
$976$	−10156.3	−0.333089
$977$	− 26928.7i	− 0.881809i	−0.897554	−	0.440904i	$-0.854658\pi$
$977$	− 26928.7i	− 0.881809i	0.897554	−	0.440904i	$-0.145342\pi$
$978$	−16666.8	−0.544934
$979$	−41748.1	−1.36290
$980$	6451.10i	0.210278i
$981$	− 426.184i	− 0.0138705i
$982$	12750.7i	0.414349i
$983$	3632.98i	0.117878i	0.998262	+	0.0589389i	$0.0187717\pi$
$983$	3632.98i	0.117878i	−0.998262	+	0.0589389i	$0.981228\pi$
$984$	3967.23	0.128527
$985$	7726.70	0.249942
$986$	2409.61i	0.0778271i
$987$	7781.37	0.250946
$988$	0	0
$989$	63206.9	2.03222
$990$	9196.96i	0.295251i
$991$	−33953.7	−1.08837	−0.544186	−	0.838965i	$-0.683161\pi$
$991$	−33953.7	−1.08837	−0.544186	+	0.838965i	$0.683161\pi$
$992$	−8580.49	−0.274628
$993$	23399.3i	0.747789i
$994$	− 23726.8i	− 0.757110i
$995$	67470.6i	2.14971i
$996$	− 14820.2i	− 0.471482i
$997$	57600.9	1.82973	0.914864	−	0.403762i	$-0.132298\pi$
$997$	57600.9	1.82973	0.914864	+	0.403762i	$0.132298\pi$
$998$	−23608.4	−0.748808
$999$	5446.01i	0.172477i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1014.4.b.q.337.1		12
13.5	odd	4		1014.4.a.bc.1.1	✓	6
13.8	odd	4		1014.4.a.be.1.6	yes	6
13.12	even	2	inner	1014.4.b.q.337.12		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1014.4.a.bc.1.1	✓	6	13.5	odd	4
1014.4.a.be.1.6	yes	6	13.8	odd	4
1014.4.b.q.337.1		12	1.1	even	1	trivial
1014.4.b.q.337.12		12	13.12	even	2	inner

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:	\( N \)	\(=\)	\( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1014.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} +3.00000 q^{3} -4.00000 q^{4} -13.0357i q^{5} -6.00000i q^{6} -14.8081i q^{7} +8.00000i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.00000i q^{2} +3.00000 q^{3} -4.00000 q^{4} -13.0357i q^{5} -6.00000i q^{6} -14.8081i q^{7} +8.00000i q^{8} +9.00000 q^{9} -26.0714 q^{10} -39.1957i q^{11} -12.0000 q^{12} -29.6162 q^{14} -39.1070i q^{15} +16.0000 q^{16} -8.45847 q^{17} -18.0000i q^{18} -33.2592i q^{19} +52.1427i q^{20} -44.4243i q^{21} -78.3914 q^{22} -173.056 q^{23} +24.0000i q^{24} -44.9290 q^{25} +27.0000 q^{27} +59.2324i q^{28} +142.438 q^{29} -78.2141 q^{30} -268.140i q^{31} -32.0000i q^{32} -117.587i q^{33} +16.9169i q^{34} -193.034 q^{35} -36.0000 q^{36} +201.704i q^{37} -66.5184 q^{38} +104.285 q^{40} -165.301i q^{41} -88.8486 q^{42} -365.241 q^{43} +156.783i q^{44} -117.321i q^{45} +346.111i q^{46} +175.160i q^{47} +48.0000 q^{48} +123.720 q^{49} +89.8579i q^{50} -25.3754 q^{51} +711.650 q^{53} -54.0000i q^{54} -510.942 q^{55} +118.465 q^{56} -99.7776i q^{57} -284.875i q^{58} -190.212i q^{59} +156.428i q^{60} -634.767 q^{61} -536.280 q^{62} -133.273i q^{63} -64.0000 q^{64} -235.174 q^{66} -432.263i q^{67} +33.8339 q^{68} -519.167 q^{69} +386.067i q^{70} +801.141i q^{71} +72.0000i q^{72} +171.241i q^{73} +403.408 q^{74} -134.787 q^{75} +133.037i q^{76} -580.414 q^{77} +604.457 q^{79} -208.571i q^{80} +81.0000 q^{81} -330.602 q^{82} +1235.02i q^{83} +177.697i q^{84} +110.262i q^{85} +730.481i q^{86} +427.313 q^{87} +313.566 q^{88} -1065.12i q^{89} -234.642 q^{90} +692.222 q^{92} -804.421i q^{93} +350.320 q^{94} -433.556 q^{95} -96.0000i q^{96} +1519.11i q^{97} -247.440i q^{98} -352.761i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 36 q^{3} - 48 q^{4} + 108 q^{9}+O(q^{10}) \) 12 * q + 36 * q^3 - 48 * q^4 + 108 * q^9	\( 12 q + 36 q^{3} - 48 q^{4} + 108 q^{9} + 12 q^{10} - 144 q^{12} - 4 q^{14} + 192 q^{16} - 102 q^{17} + 256 q^{22} - 444 q^{23} - 370 q^{25} + 324 q^{27} + 658 q^{29} + 36 q^{30} + 1688 q^{35} - 432 q^{36} - 852 q^{38} - 48 q^{40} - 12 q^{42} - 982 q^{43} + 576 q^{48} - 2266 q^{49} - 306 q^{51} + 4604 q^{53} - 658 q^{55} + 16 q^{56} + 690 q^{61} - 1156 q^{62} - 768 q^{64} + 768 q^{66} + 408 q^{68} - 1332 q^{69} + 1880 q^{74} - 1110 q^{75} - 6582 q^{77} + 6200 q^{79} + 972 q^{81} + 1284 q^{82} + 1974 q^{87} - 1024 q^{88} + 108 q^{90} + 1776 q^{92} - 2564 q^{94} + 1330 q^{95}+O(q^{100}) \) 12 * q + 36 * q^3 - 48 * q^4 + 108 * q^9 + 12 * q^10 - 144 * q^12 - 4 * q^14 + 192 * q^16 - 102 * q^17 + 256 * q^22 - 444 * q^23 - 370 * q^25 + 324 * q^27 + 658 * q^29 + 36 * q^30 + 1688 * q^35 - 432 * q^36 - 852 * q^38 - 48 * q^40 - 12 * q^42 - 982 * q^43 + 576 * q^48 - 2266 * q^49 - 306 * q^51 + 4604 * q^53 - 658 * q^55 + 16 * q^56 + 690 * q^61 - 1156 * q^62 - 768 * q^64 + 768 * q^66 + 408 * q^68 - 1332 * q^69 + 1880 * q^74 - 1110 * q^75 - 6582 * q^77 + 6200 * q^79 + 972 * q^81 + 1284 * q^82 + 1974 * q^87 - 1024 * q^88 + 108 * q^90 + 1776 * q^92 - 2564 * q^94 + 1330 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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