LMFDB - Embedded newform 2254.4.a.u.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2254,4,Mod(1,2254)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2254, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2254.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2254 = 2 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2254.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$132.990305153$
Analytic rank:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 234 x^{9} - 105 x^{8} + 18997 x^{7} + 16513 x^{6} - 621598 x^{5} - 743169 x^{4} + \cdots - 12103441 $ x^11 - 234x^9 - 105x^8 + 18997x^7 + 16513x^6 - 621598x^5 - 743169x^4 + 7229916x^3 + 7324254x^2 - 26794327*x - 12103441
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 7 $
Twist minimal:	no (minimal twist has level 322)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-5.89611$ of defining polynomial
Character	$\chi$	$=$	2254.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} -5.89611 q^{3} +4.00000 q^{4} -12.9445 q^{5} +11.7922 q^{6} -8.00000 q^{8} +7.76415 q^{9} +25.8891 q^{10} +12.8116 q^{11} -23.5845 q^{12} -21.0208 q^{13} +76.3225 q^{15} +16.0000 q^{16} +58.8720 q^{17} -15.5283 q^{18} +152.626 q^{19} -51.7782 q^{20} -25.6232 q^{22} +23.0000 q^{23} +47.1689 q^{24} +42.5611 q^{25} +42.0417 q^{26} +113.417 q^{27} +209.234 q^{29} -152.645 q^{30} +238.911 q^{31} -32.0000 q^{32} -75.5386 q^{33} -117.744 q^{34} +31.0566 q^{36} -364.105 q^{37} -305.252 q^{38} +123.941 q^{39} +103.556 q^{40} -235.402 q^{41} -458.046 q^{43} +51.2464 q^{44} -100.503 q^{45} -46.0000 q^{46} +164.072 q^{47} -94.3378 q^{48} -85.1222 q^{50} -347.116 q^{51} -84.0833 q^{52} -245.524 q^{53} -226.833 q^{54} -165.840 q^{55} -899.899 q^{57} -418.467 q^{58} +389.626 q^{59} +305.290 q^{60} +386.464 q^{61} -477.822 q^{62} +64.0000 q^{64} +272.105 q^{65} +151.077 q^{66} +573.894 q^{67} +235.488 q^{68} -135.611 q^{69} +724.966 q^{71} -62.1132 q^{72} +1239.25 q^{73} +728.210 q^{74} -250.945 q^{75} +610.503 q^{76} -247.882 q^{78} +35.5640 q^{79} -207.113 q^{80} -878.350 q^{81} +470.805 q^{82} -1195.88 q^{83} -762.070 q^{85} +916.091 q^{86} -1233.67 q^{87} -102.493 q^{88} -88.3557 q^{89} +201.007 q^{90} +92.0000 q^{92} -1408.65 q^{93} -328.144 q^{94} -1975.67 q^{95} +188.676 q^{96} -1759.37 q^{97} +99.4712 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - 22 q^{2} + 44 q^{4} + 23 q^{5} - 88 q^{8} + 171 q^{9} - 46 q^{10} - 48 q^{11} + 77 q^{13} + 104 q^{15} + 176 q^{16} + 97 q^{17} - 342 q^{18} + 138 q^{19} + 92 q^{20} + 96 q^{22} + 253 q^{23} + 30 q^{25}+ \cdots - 2545 q^{99}+O(q^{100}) $ 11 * q - 22 * q^2 + 44 * q^4 + 23 * q^5 - 88 * q^8 + 171 * q^9 - 46 * q^10 - 48 * q^11 + 77 * q^13 + 104 * q^15 + 176 * q^16 + 97 * q^17 - 342 * q^18 + 138 * q^19 + 92 * q^20 + 96 * q^22 + 253 * q^23 + 30 * q^25 - 154 * q^26 + 315 * q^27 + 137 * q^29 - 208 * q^30 + 234 * q^31 - 352 * q^32 + 28 * q^33 - 194 * q^34 + 684 * q^36 + 87 * q^37 - 276 * q^38 + 129 * q^39 - 184 * q^40 + 612 * q^41 + 187 * q^43 - 192 * q^44 + 597 * q^45 - 506 * q^46 + 793 * q^47 - 60 * q^50 - 1257 * q^51 + 308 * q^52 + 575 * q^53 - 630 * q^54 - 549 * q^55 + 39 * q^57 - 274 * q^58 + 1241 * q^59 + 416 * q^60 + 187 * q^61 - 468 * q^62 + 704 * q^64 - 937 * q^65 - 56 * q^66 + 2275 * q^67 + 388 * q^68 + 115 * q^71 - 1368 * q^72 + 2385 * q^73 - 174 * q^74 + 196 * q^75 + 552 * q^76 - 258 * q^78 - 1225 * q^79 + 368 * q^80 + 3635 * q^81 - 1224 * q^82 - 597 * q^83 - 2493 * q^85 - 374 * q^86 + 1428 * q^87 + 384 * q^88 + 602 * q^89 - 1194 * q^90 + 1012 * q^92 - 1360 * q^93 - 1586 * q^94 + 4835 * q^95 - 1039 * q^97 - 2545 * q^99

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	−5.89611	−1.13471	−0.567354	−	0.823474i	$-0.692033\pi$
$3$	−5.89611	−1.13471	−0.567354	+	0.823474i	$0.692033\pi$
$4$	4.00000	0.500000
$5$	−12.9445	−1.15779	−0.578897	−	0.815400i	$-0.696517\pi$
$5$	−12.9445	−1.15779	−0.578897	+	0.815400i	$0.696517\pi$
$6$	11.7922	0.802359
$7$	0	0
$7$	0	0
$8$	−8.00000	−0.353553
$9$	7.76415	0.287561
$10$	25.8891	0.818685
$11$	12.8116	0.351167	0.175584	−	0.984464i	$-0.443819\pi$
$11$	12.8116	0.351167	0.175584	+	0.984464i	$0.443819\pi$
$12$	−23.5845	−0.567354
$13$	−21.0208	−0.448472	−0.224236	−	0.974535i	$-0.571989\pi$
$13$	−21.0208	−0.448472	−0.224236	+	0.974535i	$0.571989\pi$
$14$	0	0
$15$	76.3225	1.31376
$16$	16.0000	0.250000
$17$	58.8720	0.839915	0.419957	−	0.907544i	$-0.362045\pi$
$17$	58.8720	0.839915	0.419957	+	0.907544i	$0.362045\pi$
$18$	−15.5283	−0.203336
$19$	152.626	1.84288	0.921441	−	0.388519i	$-0.127013\pi$
$19$	152.626	1.84288	0.921441	+	0.388519i	$0.127013\pi$
$20$	−51.7782	−0.578897
$21$	0	0
$22$	−25.6232	−0.248313
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	47.1689	0.401180
$25$	42.5611	0.340489
$26$	42.0417	0.317117
$27$	113.417	0.808410
$28$	0	0
$29$	209.234	1.33978	0.669891	−	0.742459i	$-0.266341\pi$
$29$	209.234	1.33978	0.669891	+	0.742459i	$0.266341\pi$
$30$	−152.645	−0.928968
$31$	238.911	1.38418	0.692092	−	0.721810i	$-0.256689\pi$
$31$	238.911	1.38418	0.692092	+	0.721810i	$0.256689\pi$
$32$	−32.0000	−0.176777
$33$	−75.5386	−0.398472
$34$	−117.744	−0.593909
$35$	0	0
$36$	31.0566	0.143781
$37$	−364.105	−1.61780	−0.808898	−	0.587948i	$-0.799936\pi$
$37$	−364.105	−1.61780	−0.808898	+	0.587948i	$0.799936\pi$
$38$	−305.252	−1.30311
$39$	123.941	0.508884
$40$	103.556	0.409342
$41$	−235.402	−0.896675	−0.448338	−	0.893864i	$-0.647984\pi$
$41$	−235.402	−0.896675	−0.448338	+	0.893864i	$0.647984\pi$
$42$	0	0
$43$	−458.046	−1.62445	−0.812224	−	0.583345i	$-0.801744\pi$
$43$	−458.046	−1.62445	−0.812224	+	0.583345i	$0.801744\pi$
$44$	51.2464	0.175584
$45$	−100.503	−0.332937
$46$	−46.0000	−0.147442
$47$	164.072	0.509200	0.254600	−	0.967046i	$-0.418056\pi$
$47$	164.072	0.509200	0.254600	+	0.967046i	$0.418056\pi$
$48$	−94.3378	−0.283677
$49$	0	0
$50$	−85.1222	−0.240762
$51$	−347.116	−0.953057
$52$	−84.0833	−0.224236
$53$	−245.524	−0.636328	−0.318164	−	0.948036i	$-0.603066\pi$
$53$	−245.524	−0.636328	−0.318164	+	0.948036i	$0.603066\pi$
$54$	−226.833	−0.571632
$55$	−165.840	−0.406580
$56$	0	0
$57$	−899.899	−2.09113
$58$	−418.467	−0.947370
$59$	389.626	0.859746	0.429873	−	0.902889i	$-0.358558\pi$
$59$	389.626	0.859746	0.429873	+	0.902889i	$0.358558\pi$
$60$	305.290	0.656879
$61$	386.464	0.811174	0.405587	−	0.914057i	$-0.367067\pi$
$61$	386.464	0.811174	0.405587	+	0.914057i	$0.367067\pi$
$62$	−477.822	−0.978765
$63$	0	0
$64$	64.0000	0.125000
$65$	272.105	0.519238
$66$	151.077	0.281763
$67$	573.894	1.04645	0.523226	−	0.852194i	$-0.324728\pi$
$67$	573.894	1.04645	0.523226	+	0.852194i	$0.324728\pi$
$68$	235.488	0.419957
$69$	−135.611	−0.236603
$70$	0	0
$71$	724.966	1.21180	0.605899	−	0.795542i	$-0.292814\pi$
$71$	724.966	1.21180	0.605899	+	0.795542i	$0.292814\pi$
$72$	−62.1132	−0.101668
$73$	1239.25	1.98689	0.993445	−	0.114312i	$-0.0364663\pi$
$73$	1239.25	1.98689	0.993445	+	0.114312i	$0.0364663\pi$
$74$	728.210	1.14396
$75$	−250.945	−0.386355
$76$	610.503	0.921441
$77$	0	0
$78$	−247.882	−0.359835
$79$	35.5640	0.0506490	0.0253245	−	0.999679i	$-0.491938\pi$
$79$	35.5640	0.0506490	0.0253245	+	0.999679i	$0.491938\pi$
$80$	−207.113	−0.289449
$81$	−878.350	−1.20487
$82$	470.805	0.634045
$83$	−1195.88	−1.58151	−0.790755	−	0.612132i	$-0.790312\pi$
$83$	−1195.88	−1.58151	−0.790755	+	0.612132i	$0.790312\pi$
$84$	0	0
$85$	−762.070	−0.972449
$86$	916.091	1.14866
$87$	−1233.67	−1.52026
$88$	−102.493	−0.124156
$89$	−88.3557	−0.105232	−0.0526162	−	0.998615i	$-0.516756\pi$
$89$	−88.3557	−0.105232	−0.0526162	+	0.998615i	$0.516756\pi$
$90$	201.007	0.235422
$91$	0	0
$92$	92.0000	0.104257
$93$	−1408.65	−1.57064
$94$	−328.144	−0.360059
$95$	−1975.67	−2.13368
$96$	188.676	0.200590
$97$	−1759.37	−1.84162	−0.920811	−	0.390009i	$-0.872472\pi$
$97$	−1759.37	−1.84162	−0.920811	+	0.390009i	$0.872472\pi$
$98$	0	0
$99$	99.4712	0.100982
$100$	170.244	0.170244
$101$	158.671	0.156320	0.0781601	−	0.996941i	$-0.475095\pi$
$101$	158.671	0.156320	0.0781601	+	0.996941i	$0.475095\pi$
$102$	694.231	0.673913
$103$	109.309	0.104568	0.0522841	−	0.998632i	$-0.483350\pi$
$103$	109.309	0.104568	0.0522841	+	0.998632i	$0.483350\pi$
$104$	168.167	0.158559
$105$	0	0
$106$	491.049	0.449952
$107$	489.965	0.442679	0.221340	−	0.975197i	$-0.428957\pi$
$107$	489.965	0.442679	0.221340	+	0.975197i	$0.428957\pi$
$108$	453.667	0.404205
$109$	−2055.49	−1.80624	−0.903120	−	0.429388i	$-0.858729\pi$
$109$	−2055.49	−1.80624	−0.903120	+	0.429388i	$0.858729\pi$
$110$	331.680	0.287495
$111$	2146.80	1.83573
$112$	0	0
$113$	−1677.71	−1.39668	−0.698342	−	0.715764i	$-0.746079\pi$
$113$	−1677.71	−1.39668	−0.698342	+	0.715764i	$0.746079\pi$
$114$	1799.80	1.47865
$115$	−297.724	−0.241417
$116$	836.934	0.669891
$117$	−163.209	−0.128963
$118$	−779.253	−0.607933
$119$	0	0
$120$	−610.580	−0.464484
$121$	−1166.86	−0.876681
$122$	−772.927	−0.573586
$123$	1387.96	1.01746
$124$	955.644	0.692092
$125$	1067.13	0.763579
$126$	0	0
$127$	−658.313	−0.459967	−0.229983	−	0.973195i	$-0.573867\pi$
$127$	−658.313	−0.459967	−0.229983	+	0.973195i	$0.573867\pi$
$128$	−128.000	−0.0883883
$129$	2700.69	1.84327
$130$	−544.210	−0.367157
$131$	2061.93	1.37520	0.687602	−	0.726088i	$-0.258664\pi$
$131$	2061.93	1.37520	0.687602	+	0.726088i	$0.258664\pi$
$132$	−302.155	−0.199236
$133$	0	0
$134$	−1147.79	−0.739953
$135$	−1468.13	−0.935973
$136$	−470.976	−0.296955
$137$	140.764	0.0877831	0.0438916	−	0.999036i	$-0.486024\pi$
$137$	140.764	0.0877831	0.0438916	+	0.999036i	$0.486024\pi$
$138$	271.221	0.167303
$139$	333.739	0.203650	0.101825	−	0.994802i	$-0.467532\pi$
$139$	333.739	0.203650	0.101825	+	0.994802i	$0.467532\pi$
$140$	0	0
$141$	−967.388	−0.577793
$142$	−1449.93	−0.856870
$143$	−269.310	−0.157489
$144$	124.226	0.0718903
$145$	−2708.43	−1.55119
$146$	−2478.50	−1.40494
$147$	0	0
$148$	−1456.42	−0.808898
$149$	316.025	0.173757	0.0868783	−	0.996219i	$-0.472311\pi$
$149$	316.025	0.173757	0.0868783	+	0.996219i	$0.472311\pi$
$150$	501.890	0.273194
$151$	1485.50	0.800585	0.400292	−	0.916387i	$-0.368909\pi$
$151$	1485.50	0.800585	0.400292	+	0.916387i	$0.368909\pi$
$152$	−1221.01	−0.651557
$153$	457.091	0.241527
$154$	0	0
$155$	−3092.59	−1.60260
$156$	495.765	0.254442
$157$	−16.5978	−0.00843724	−0.00421862	−	0.999991i	$-0.501343\pi$
$157$	−16.5978	−0.00843724	−0.00421862	+	0.999991i	$0.501343\pi$
$158$	−71.1281	−0.0358142
$159$	1447.64	0.722046
$160$	414.225	0.204671
$161$	0	0
$162$	1756.70	0.851972
$163$	1952.79	0.938372	0.469186	−	0.883099i	$-0.344547\pi$
$163$	1952.79	0.938372	0.469186	+	0.883099i	$0.344547\pi$
$164$	−941.610	−0.448338
$165$	977.813	0.461349
$166$	2391.77	1.11830
$167$	−2373.55	−1.09983	−0.549913	−	0.835222i	$-0.685339\pi$
$167$	−2373.55	−1.09983	−0.549913	+	0.835222i	$0.685339\pi$
$168$	0	0
$169$	−1755.12	−0.798873
$170$	1524.14	0.687625
$171$	1185.01	0.529941
$172$	−1832.18	−0.812224
$173$	−1663.75	−0.731172	−0.365586	−	0.930778i	$-0.619131\pi$
$173$	−1663.75	−0.731172	−0.365586	+	0.930778i	$0.619131\pi$
$174$	2467.33	1.07499
$175$	0	0
$176$	204.986	0.0877919
$177$	−2297.28	−0.975561
$178$	176.711	0.0744105
$179$	−256.393	−0.107060	−0.0535299	−	0.998566i	$-0.517047\pi$
$179$	−256.393	−0.107060	−0.0535299	+	0.998566i	$0.517047\pi$
$180$	−402.013	−0.166468
$181$	229.505	0.0942485	0.0471243	−	0.998889i	$-0.484994\pi$
$181$	229.505	0.0942485	0.0471243	+	0.998889i	$0.484994\pi$
$182$	0	0
$183$	−2278.63	−0.920445
$184$	−184.000	−0.0737210
$185$	4713.17	1.87308
$186$	2817.29	1.11061
$187$	754.244	0.294951
$188$	656.289	0.254600
$189$	0	0
$190$	3951.34	1.50874
$191$	1495.82	0.566668	0.283334	−	0.959021i	$-0.408560\pi$
$191$	1495.82	0.566668	0.283334	+	0.959021i	$0.408560\pi$
$192$	−377.351	−0.141838
$193$	495.110	0.184657	0.0923286	−	0.995729i	$-0.470569\pi$
$193$	495.110	0.184657	0.0923286	+	0.995729i	$0.470569\pi$
$194$	3518.75	1.30222
$195$	−1604.36	−0.589183
$196$	0	0
$197$	1979.85	0.716035	0.358017	−	0.933715i	$-0.383453\pi$
$197$	1979.85	0.716035	0.358017	+	0.933715i	$0.383453\pi$
$198$	−198.942	−0.0714051
$199$	−2023.26	−0.720728	−0.360364	−	0.932812i	$-0.617348\pi$
$199$	−2023.26	−0.720728	−0.360364	+	0.932812i	$0.617348\pi$
$200$	−340.489	−0.120381
$201$	−3383.74	−1.18742
$202$	−317.342	−0.110535
$203$	0	0
$204$	−1388.46	−0.476529
$205$	3047.18	1.03817
$206$	−218.618	−0.0739409
$207$	178.575	0.0599606
$208$	−336.333	−0.112118
$209$	1955.38	0.647160
$210$	0	0
$211$	−222.612	−0.0726316	−0.0363158	−	0.999340i	$-0.511562\pi$
$211$	−222.612	−0.0726316	−0.0363158	+	0.999340i	$0.511562\pi$
$212$	−982.098	−0.318164
$213$	−4274.48	−1.37504
$214$	−979.930	−0.313022
$215$	5929.19	1.88078
$216$	−907.334	−0.285816
$217$	0	0
$218$	4110.98	1.27720
$219$	−7306.74	−2.25454
$220$	−663.361	−0.203290
$221$	−1237.54	−0.376678
$222$	−4293.61	−1.29805
$223$	4542.87	1.36418	0.682092	−	0.731266i	$-0.261070\pi$
$223$	4542.87	1.36418	0.682092	+	0.731266i	$0.261070\pi$
$224$	0	0
$225$	330.451	0.0979114
$226$	3355.41	0.987605
$227$	−1788.28	−0.522874	−0.261437	−	0.965220i	$-0.584196\pi$
$227$	−1788.28	−0.522874	−0.261437	+	0.965220i	$0.584196\pi$
$228$	−3599.59	−1.04557
$229$	−5393.71	−1.55645	−0.778224	−	0.627987i	$-0.783879\pi$
$229$	−5393.71	−1.55645	−0.778224	+	0.627987i	$0.783879\pi$
$230$	595.449	0.170708
$231$	0	0
$232$	−1673.87	−0.473685
$233$	5619.96	1.58015	0.790077	−	0.613008i	$-0.210041\pi$
$233$	5619.96	1.58015	0.790077	+	0.613008i	$0.210041\pi$
$234$	326.418	0.0911906
$235$	−2123.84	−0.589549
$236$	1558.51	0.429873
$237$	−209.690	−0.0574718
$238$	0	0
$239$	1759.52	0.476208	0.238104	−	0.971240i	$-0.423474\pi$
$239$	1759.52	0.476208	0.238104	+	0.971240i	$0.423474\pi$
$240$	1221.16	0.328440
$241$	−4671.17	−1.24853	−0.624267	−	0.781211i	$-0.714602\pi$
$241$	−4671.17	−1.24853	−0.624267	+	0.781211i	$0.714602\pi$
$242$	2333.73	0.619907
$243$	2116.60	0.558765
$244$	1545.85	0.405587
$245$	0	0
$246$	−2775.92	−0.719456
$247$	−3208.32	−0.826480
$248$	−1911.29	−0.489383
$249$	7051.07	1.79455
$250$	−2134.27	−0.539932
$251$	−6737.86	−1.69438	−0.847191	−	0.531288i	$-0.821708\pi$
$251$	−6737.86	−1.69438	−0.847191	+	0.531288i	$0.821708\pi$
$252$	0	0
$253$	294.667	0.0732235
$254$	1316.63	0.325246
$255$	4493.25	1.10344
$256$	256.000	0.0625000
$257$	720.298	0.174829	0.0874143	−	0.996172i	$-0.472140\pi$
$257$	720.298	0.174829	0.0874143	+	0.996172i	$0.472140\pi$
$258$	−5401.38	−1.30339
$259$	0	0
$260$	1088.42	0.259619
$261$	1624.52	0.385270
$262$	−4123.86	−0.972415
$263$	8283.70	1.94219	0.971093	−	0.238702i	$-0.0767218\pi$
$263$	8283.70	1.94219	0.971093	+	0.238702i	$0.0767218\pi$
$264$	604.309	0.140881
$265$	3178.20	0.736737
$266$	0	0
$267$	520.955	0.119408
$268$	2295.57	0.523226
$269$	4673.70	1.05933	0.529666	−	0.848206i	$-0.322317\pi$
$269$	4673.70	1.05933	0.529666	+	0.848206i	$0.322317\pi$
$270$	2936.26	0.661833
$271$	5874.48	1.31679	0.658393	−	0.752674i	$-0.271237\pi$
$271$	5874.48	1.31679	0.658393	+	0.752674i	$0.271237\pi$
$272$	941.951	0.209979
$273$	0	0
$274$	−281.528	−0.0620721
$275$	545.276	0.119569
$276$	−542.442	−0.118301
$277$	3572.67	0.774950	0.387475	−	0.921880i	$-0.373347\pi$
$277$	3572.67	0.774950	0.387475	+	0.921880i	$0.373347\pi$
$278$	−667.478	−0.144002
$279$	1854.94	0.398037
$280$	0	0
$281$	791.003	0.167926	0.0839631	−	0.996469i	$-0.473242\pi$
$281$	791.003	0.167926	0.0839631	+	0.996469i	$0.473242\pi$
$282$	1934.78	0.408561
$283$	−5126.83	−1.07689	−0.538443	−	0.842662i	$-0.680987\pi$
$283$	−5126.83	−1.07689	−0.538443	+	0.842662i	$0.680987\pi$
$284$	2899.86	0.605899
$285$	11648.8	2.42110
$286$	538.621	0.111361
$287$	0	0
$288$	−248.453	−0.0508341
$289$	−1447.09	−0.294544
$290$	5416.87	1.09686
$291$	10373.5	2.08970
$292$	4956.99	0.993445
$293$	2443.73	0.487249	0.243625	−	0.969870i	$-0.421664\pi$
$293$	2443.73	0.487249	0.243625	+	0.969870i	$0.421664\pi$
$294$	0	0
$295$	−5043.54	−0.995410
$296$	2912.84	0.571978
$297$	1453.05	0.283887
$298$	−632.049	−0.122865
$299$	−483.479	−0.0935128
$300$	−1003.78	−0.193178
$301$	0	0
$302$	−2971.00	−0.566099
$303$	−935.541	−0.177378
$304$	2442.01	0.460720
$305$	−5002.59	−0.939173
$306$	−914.181	−0.170785
$307$	−3867.30	−0.718952	−0.359476	−	0.933154i	$-0.617045\pi$
$307$	−3867.30	−0.718952	−0.359476	+	0.933154i	$0.617045\pi$
$308$	0	0
$309$	−644.498	−0.118654
$310$	6185.19	1.13321
$311$	9233.41	1.68353	0.841767	−	0.539841i	$-0.181516\pi$
$311$	9233.41	1.68353	0.841767	+	0.539841i	$0.181516\pi$
$312$	−991.530	−0.179918
$313$	−2273.51	−0.410563	−0.205282	−	0.978703i	$-0.565811\pi$
$313$	−2273.51	−0.410563	−0.205282	+	0.978703i	$0.565811\pi$
$314$	33.1955	0.00596603
$315$	0	0
$316$	142.256	0.0253245
$317$	−6166.98	−1.09266	−0.546329	−	0.837571i	$-0.683975\pi$
$317$	−6166.98	−1.09266	−0.546329	+	0.837571i	$0.683975\pi$
$318$	−2895.28	−0.510564
$319$	2680.62	0.470488
$320$	−828.451	−0.144724
$321$	−2888.89	−0.502312
$322$	0	0
$323$	8985.38	1.54786
$324$	−3513.40	−0.602435
$325$	−894.670	−0.152700
$326$	−3905.59	−0.663529
$327$	12119.4	2.04955
$328$	1883.22	0.317023
$329$	0	0
$330$	−1955.63	−0.326223
$331$	4566.40	0.758285	0.379142	−	0.925338i	$-0.376219\pi$
$331$	4566.40	0.758285	0.379142	+	0.925338i	$0.376219\pi$
$332$	−4783.54	−0.790755
$333$	−2826.97	−0.465215
$334$	4747.11	0.777695
$335$	−7428.79	−1.21158
$336$	0	0
$337$	−4102.09	−0.663071	−0.331536	−	0.943443i	$-0.607567\pi$
$337$	−4102.09	−0.663071	−0.331536	+	0.943443i	$0.607567\pi$
$338$	3510.25	0.564889
$339$	9891.94	1.58483
$340$	−3048.28	−0.486224
$341$	3060.83	0.486080
$342$	−2370.02	−0.374725
$343$	0	0
$344$	3664.37	0.574329
$345$	1755.42	0.273938
$346$	3327.51	0.517017
$347$	−3075.49	−0.475795	−0.237897	−	0.971290i	$-0.576458\pi$
$347$	−3075.49	−0.475795	−0.237897	+	0.971290i	$0.576458\pi$
$348$	−4934.66	−0.760131
$349$	3383.09	0.518890	0.259445	−	0.965758i	$-0.416460\pi$
$349$	3383.09	0.518890	0.259445	+	0.965758i	$0.416460\pi$
$350$	0	0
$351$	−2384.11	−0.362549
$352$	−409.971	−0.0620782
$353$	11394.3	1.71801	0.859003	−	0.511971i	$-0.171085\pi$
$353$	11394.3	1.71801	0.859003	+	0.511971i	$0.171085\pi$
$354$	4594.56	0.689826
$355$	−9384.35	−1.40301
$356$	−353.423	−0.0526162
$357$	0	0
$358$	512.786	0.0757028
$359$	−9614.23	−1.41342	−0.706712	−	0.707501i	$-0.749823\pi$
$359$	−9614.23	−1.41342	−0.706712	+	0.707501i	$0.749823\pi$
$360$	804.027	0.117711
$361$	16435.6	2.39621
$362$	−459.010	−0.0666438
$363$	6879.96	0.994777
$364$	0	0
$365$	−16041.5	−2.30041
$366$	4557.27	0.650853
$367$	8435.35	1.19979	0.599893	−	0.800080i	$-0.295210\pi$
$367$	8435.35	1.19979	0.599893	+	0.800080i	$0.295210\pi$
$368$	368.000	0.0521286
$369$	−1827.70	−0.257849
$370$	−9426.34	−1.32447
$371$	0	0
$372$	−5634.59	−0.785322
$373$	−6006.02	−0.833726	−0.416863	−	0.908969i	$-0.636871\pi$
$373$	−6006.02	−0.833726	−0.416863	+	0.908969i	$0.636871\pi$
$374$	−1508.49	−0.208562
$375$	−6291.94	−0.866438
$376$	−1312.58	−0.180029
$377$	−4398.27	−0.600855
$378$	0	0
$379$	−1668.89	−0.226188	−0.113094	−	0.993584i	$-0.536076\pi$
$379$	−1668.89	−0.226188	−0.113094	+	0.993584i	$0.536076\pi$
$380$	−7902.68	−1.06684
$381$	3881.49	0.521928
$382$	−2991.64	−0.400695
$383$	6067.78	0.809528	0.404764	−	0.914421i	$-0.367354\pi$
$383$	6067.78	0.809528	0.404764	+	0.914421i	$0.367354\pi$
$384$	754.702	0.100295
$385$	0	0
$386$	−990.221	−0.130572
$387$	−3556.34	−0.467128
$388$	−7037.50	−0.920811
$389$	7672.25	0.999996	0.499998	−	0.866026i	$-0.333334\pi$
$389$	7672.25	0.999996	0.499998	+	0.866026i	$0.333334\pi$
$390$	3208.72	0.416616
$391$	1354.05	0.175134
$392$	0	0
$393$	−12157.4	−1.56045
$394$	−3959.71	−0.506313
$395$	−460.360	−0.0586411
$396$	397.885	0.0504911
$397$	240.873	0.0304510	0.0152255	−	0.999884i	$-0.495153\pi$
$397$	240.873	0.0304510	0.0152255	+	0.999884i	$0.495153\pi$
$398$	4046.51	0.509632
$399$	0	0
$400$	680.978	0.0851222
$401$	10924.7	1.36048	0.680239	−	0.732991i	$-0.261876\pi$
$401$	10924.7	1.36048	0.680239	+	0.732991i	$0.261876\pi$
$402$	6767.48	0.839630
$403$	−5022.11	−0.620767
$404$	634.684	0.0781601
$405$	11369.8	1.39499
$406$	0	0
$407$	−4664.77	−0.568118
$408$	2776.93	0.336957
$409$	7689.39	0.929623	0.464812	−	0.885410i	$-0.346122\pi$
$409$	7689.39	0.929623	0.464812	+	0.885410i	$0.346122\pi$
$410$	−6094.35	−0.734094
$411$	−829.961	−0.0996082
$412$	437.236	0.0522841
$413$	0	0
$414$	−357.151	−0.0423986
$415$	15480.2	1.83106
$416$	672.667	0.0792793
$417$	−1967.76	−0.231083
$418$	−3910.76	−0.457611
$419$	6179.10	0.720450	0.360225	−	0.932865i	$-0.382700\pi$
$419$	6179.10	0.720450	0.360225	+	0.932865i	$0.382700\pi$
$420$	0	0
$421$	2730.71	0.316120	0.158060	−	0.987430i	$-0.449476\pi$
$421$	2730.71	0.316120	0.158060	+	0.987430i	$0.449476\pi$
$422$	445.224	0.0513583
$423$	1273.88	0.146426
$424$	1964.20	0.224976
$425$	2505.66	0.285982
$426$	8548.96	0.972297
$427$	0	0
$428$	1959.86	0.221340
$429$	1587.89	0.178704
$430$	−11858.4	−1.32991
$431$	−7518.72	−0.840288	−0.420144	−	0.907458i	$-0.638020\pi$
$431$	−7518.72	−0.840288	−0.420144	+	0.907458i	$0.638020\pi$
$432$	1814.67	0.202102
$433$	11944.3	1.32565	0.662827	−	0.748772i	$-0.269356\pi$
$433$	11944.3	1.32565	0.662827	+	0.748772i	$0.269356\pi$
$434$	0	0
$435$	15969.2	1.76015
$436$	−8221.96	−0.903120
$437$	3510.39	0.384267
$438$	14613.5	1.59420
$439$	−9551.57	−1.03843	−0.519216	−	0.854643i	$-0.673776\pi$
$439$	−9551.57	−1.03843	−0.519216	+	0.854643i	$0.673776\pi$
$440$	1326.72	0.143748
$441$	0	0
$442$	2475.08	0.266351
$443$	7896.15	0.846857	0.423428	−	0.905930i	$-0.360827\pi$
$443$	7896.15	0.846857	0.423428	+	0.905930i	$0.360827\pi$
$444$	8587.22	0.917863
$445$	1143.72	0.121837
$446$	−9085.74	−0.964624
$447$	−1863.32	−0.197163
$448$	0	0
$449$	−445.645	−0.0468403	−0.0234201	−	0.999726i	$-0.507456\pi$
$449$	−445.645	−0.0468403	−0.0234201	+	0.999726i	$0.507456\pi$
$450$	−660.902	−0.0692338
$451$	−3015.88	−0.314883
$452$	−6710.82	−0.698342
$453$	−8758.68	−0.908430
$454$	3576.57	0.369728
$455$	0	0
$456$	7199.19	0.739327
$457$	−4431.13	−0.453566	−0.226783	−	0.973945i	$-0.572821\pi$
$457$	−4431.13	−0.453566	−0.226783	+	0.973945i	$0.572821\pi$
$458$	10787.4	1.10057
$459$	6677.07	0.678995
$460$	−1190.90	−0.120708
$461$	−763.727	−0.0771590	−0.0385795	−	0.999256i	$-0.512283\pi$
$461$	−763.727	−0.0771590	−0.0385795	+	0.999256i	$0.512283\pi$
$462$	0	0
$463$	−14826.9	−1.48826	−0.744132	−	0.668033i	$-0.767136\pi$
$463$	−14826.9	−1.48826	−0.744132	+	0.668033i	$0.767136\pi$
$464$	3347.74	0.334946
$465$	18234.3	1.81848
$466$	−11239.9	−1.11734
$467$	−6593.35	−0.653328	−0.326664	−	0.945141i	$-0.605924\pi$
$467$	−6593.35	−0.653328	−0.326664	+	0.945141i	$0.605924\pi$
$468$	−652.836	−0.0644815
$469$	0	0
$470$	4247.68	0.416874
$471$	97.8623	0.00957379
$472$	−3117.01	−0.303966
$473$	−5868.30	−0.570454
$474$	419.379	0.0406387
$475$	6495.92	0.627481
$476$	0	0
$477$	−1906.29	−0.182983
$478$	−3519.03	−0.336730
$479$	−15862.5	−1.51311	−0.756553	−	0.653932i	$-0.773118\pi$
$479$	−15862.5	−1.51311	−0.756553	+	0.653932i	$0.773118\pi$
$480$	−2442.32	−0.232242
$481$	7653.79	0.725536
$482$	9342.35	0.882847
$483$	0	0
$484$	−4667.45	−0.438341
$485$	22774.3	2.13222
$486$	−4233.20	−0.395107
$487$	7245.32	0.674162	0.337081	−	0.941476i	$-0.390560\pi$
$487$	7245.32	0.674162	0.337081	+	0.941476i	$0.390560\pi$
$488$	−3091.71	−0.286793
$489$	−11513.9	−1.06478
$490$	0	0
$491$	19714.8	1.81205	0.906023	−	0.423227i	$-0.139103\pi$
$491$	19714.8	1.81205	0.906023	+	0.423227i	$0.139103\pi$
$492$	5551.84	0.508732
$493$	12318.0	1.12530
$494$	6416.64	0.584410
$495$	−1287.61	−0.116917
$496$	3822.58	0.346046
$497$	0	0
$498$	−14102.1	−1.26894
$499$	−12798.2	−1.14815	−0.574074	−	0.818803i	$-0.694638\pi$
$499$	−12798.2	−1.14815	−0.574074	+	0.818803i	$0.694638\pi$
$500$	4268.53	0.381789
$501$	13994.7	1.24798
$502$	13475.7	1.19811
$503$	9450.70	0.837745	0.418873	−	0.908045i	$-0.362425\pi$
$503$	9450.70	0.837745	0.418873	+	0.908045i	$0.362425\pi$
$504$	0	0
$505$	−2053.92	−0.180987
$506$	−589.334	−0.0517768
$507$	10348.4	0.906487
$508$	−2633.25	−0.229983
$509$	5686.25	0.495164	0.247582	−	0.968867i	$-0.420364\pi$
$509$	5686.25	0.495164	0.247582	+	0.968867i	$0.420364\pi$
$510$	−8986.51	−0.780253
$511$	0	0
$512$	−512.000	−0.0441942
$513$	17310.3	1.48980
$514$	−1440.60	−0.123622
$515$	−1414.95	−0.121069
$516$	10802.8	0.921637
$517$	2102.03	0.178814
$518$	0	0
$519$	9809.67	0.829667
$520$	−2176.84	−0.183578
$521$	−5996.66	−0.504258	−0.252129	−	0.967694i	$-0.581131\pi$
$521$	−5996.66	−0.504258	−0.252129	+	0.967694i	$0.581131\pi$
$522$	−3249.04	−0.272427
$523$	15839.5	1.32430	0.662152	−	0.749369i	$-0.269643\pi$
$523$	15839.5	1.32430	0.662152	+	0.749369i	$0.269643\pi$
$524$	8247.72	0.687602
$525$	0	0
$526$	−16567.4	−1.37333
$527$	14065.2	1.16260
$528$	−1208.62	−0.0996181
$529$	529.000	0.0434783
$530$	−6356.40	−0.520952
$531$	3025.12	0.247230
$532$	0	0
$533$	4948.36	0.402133
$534$	−1041.91	−0.0844342
$535$	−6342.37	−0.512532
$536$	−4591.15	−0.369977
$537$	1511.72	0.121482
$538$	−9347.40	−0.749061
$539$	0	0
$540$	−5872.51	−0.467986
$541$	−18060.1	−1.43524	−0.717620	−	0.696435i	$-0.754768\pi$
$541$	−18060.1	−1.43524	−0.717620	+	0.696435i	$0.754768\pi$
$542$	−11749.0	−0.931108
$543$	−1353.19	−0.106944
$544$	−1883.90	−0.148477
$545$	26607.4	2.09126
$546$	0	0
$547$	−7336.84	−0.573493	−0.286746	−	0.958007i	$-0.592574\pi$
$547$	−7336.84	−0.573493	−0.286746	+	0.958007i	$0.592574\pi$
$548$	563.057	0.0438916
$549$	3000.56	0.233262
$550$	−1090.55	−0.0845478
$551$	31934.4	2.46906
$552$	1084.88	0.0836517
$553$	0	0
$554$	−7145.35	−0.547973
$555$	−27789.4	−2.12539
$556$	1334.96	0.101825
$557$	−11409.2	−0.867908	−0.433954	−	0.900935i	$-0.642882\pi$
$557$	−11409.2	−0.867908	−0.433954	+	0.900935i	$0.642882\pi$
$558$	−3709.88	−0.281455
$559$	9628.50	0.728519
$560$	0	0
$561$	−4447.11	−0.334683
$562$	−1582.01	−0.118742
$563$	3108.20	0.232673	0.116337	−	0.993210i	$-0.462885\pi$
$563$	3108.20	0.232673	0.116337	+	0.993210i	$0.462885\pi$
$564$	−3869.55	−0.288896
$565$	21717.1	1.61707
$566$	10253.7	0.761473
$567$	0	0
$568$	−5799.73	−0.428435
$569$	2619.60	0.193004	0.0965019	−	0.995333i	$-0.469235\pi$
$569$	2619.60	0.193004	0.0965019	+	0.995333i	$0.469235\pi$
$570$	−23297.5	−1.71198
$571$	7686.35	0.563334	0.281667	−	0.959512i	$-0.409113\pi$
$571$	7686.35	0.563334	0.281667	+	0.959512i	$0.409113\pi$
$572$	−1077.24	−0.0787443
$573$	−8819.51	−0.643002
$574$	0	0
$575$	978.906	0.0709968
$576$	496.906	0.0359451
$577$	12401.1	0.894738	0.447369	−	0.894349i	$-0.352361\pi$
$577$	12401.1	0.894738	0.447369	+	0.894349i	$0.352361\pi$
$578$	2894.19	0.208274
$579$	−2919.23	−0.209532
$580$	−10833.7	−0.775597
$581$	0	0
$582$	−20746.9	−1.47764
$583$	−3145.56	−0.223458
$584$	−9913.98	−0.702472
$585$	2112.66	0.149313
$586$	−4887.45	−0.344537
$587$	5716.81	0.401973	0.200986	−	0.979594i	$-0.435585\pi$
$587$	5716.81	0.401973	0.200986	+	0.979594i	$0.435585\pi$
$588$	0	0
$589$	36464.0	2.55089
$590$	10087.1	0.703861
$591$	−11673.4	−0.812490
$592$	−5825.68	−0.404449
$593$	−15544.0	−1.07642	−0.538210	−	0.842811i	$-0.680899\pi$
$593$	−15544.0	−1.07642	−0.538210	+	0.842811i	$0.680899\pi$
$594$	−2906.10	−0.200739
$595$	0	0
$596$	1264.10	0.0868783
$597$	11929.4	0.817816
$598$	966.958	0.0661235
$599$	−15021.9	−1.02467	−0.512337	−	0.858785i	$-0.671220\pi$
$599$	−15021.9	−1.02467	−0.512337	+	0.858785i	$0.671220\pi$
$600$	2007.56	0.136597
$601$	11378.9	0.772307	0.386153	−	0.922435i	$-0.373804\pi$
$601$	11378.9	0.772307	0.386153	+	0.922435i	$0.373804\pi$
$602$	0	0
$603$	4455.80	0.300919
$604$	5942.00	0.400292
$605$	15104.5	1.01502
$606$	1871.08	0.125425
$607$	−9434.98	−0.630896	−0.315448	−	0.948943i	$-0.602155\pi$
$607$	−9434.98	−0.630896	−0.315448	+	0.948943i	$0.602155\pi$
$608$	−4884.02	−0.325779
$609$	0	0
$610$	10005.2	0.664095
$611$	−3448.93	−0.228362
$612$	1828.36	0.120763
$613$	−20154.7	−1.32796	−0.663980	−	0.747750i	$-0.731134\pi$
$613$	−20154.7	−1.32796	−0.663980	+	0.747750i	$0.731134\pi$
$614$	7734.60	0.508376
$615$	−17966.5	−1.17801
$616$	0	0
$617$	28318.9	1.84777	0.923886	−	0.382668i	$-0.124995\pi$
$617$	28318.9	1.84777	0.923886	+	0.382668i	$0.124995\pi$
$618$	1289.00	0.0839013
$619$	3720.57	0.241587	0.120794	−	0.992678i	$-0.461456\pi$
$619$	3720.57	0.241587	0.120794	+	0.992678i	$0.461456\pi$
$620$	−12370.4	−0.801300
$621$	2608.59	0.168565
$622$	−18466.8	−1.19044
$623$	0	0
$624$	1983.06	0.127221
$625$	−19133.7	−1.22456
$626$	4547.02	0.290312
$627$	−11529.1	−0.734337
$628$	−66.3911	−0.00421862
$629$	−21435.6	−1.35881
$630$	0	0
$631$	12468.1	0.786606	0.393303	−	0.919409i	$-0.371332\pi$
$631$	12468.1	0.786606	0.393303	+	0.919409i	$0.371332\pi$
$632$	−284.512	−0.0179071
$633$	1312.55	0.0824156
$634$	12334.0	0.772625
$635$	8521.55	0.532547
$636$	5790.56	0.361023
$637$	0	0
$638$	−5361.23	−0.332685
$639$	5628.74	0.348466
$640$	1656.90	0.102336
$641$	1319.33	0.0812957	0.0406479	−	0.999174i	$-0.487058\pi$
$641$	1319.33	0.0812957	0.0406479	+	0.999174i	$0.487058\pi$
$642$	5777.78	0.355188
$643$	9473.96	0.581052	0.290526	−	0.956867i	$-0.406170\pi$
$643$	9473.96	0.581052	0.290526	+	0.956867i	$0.406170\pi$
$644$	0	0
$645$	−34959.2	−2.13413
$646$	−17970.8	−1.09450
$647$	12271.0	0.745631	0.372816	−	0.927905i	$-0.378392\pi$
$647$	12271.0	0.745631	0.372816	+	0.927905i	$0.378392\pi$
$648$	7026.80	0.425986
$649$	4991.74	0.301915
$650$	1789.34	0.107975
$651$	0	0
$652$	7811.17	0.469186
$653$	−29239.6	−1.75227	−0.876136	−	0.482064i	$-0.839887\pi$
$653$	−29239.6	−1.75227	−0.876136	+	0.482064i	$0.839887\pi$
$654$	−24238.8	−1.44925
$655$	−26690.7	−1.59220
$656$	−3766.44	−0.224169
$657$	9621.70	0.571352
$658$	0	0
$659$	2743.35	0.162164	0.0810819	−	0.996707i	$-0.474162\pi$
$659$	2743.35	0.162164	0.0810819	+	0.996707i	$0.474162\pi$
$660$	3911.25	0.230675
$661$	−25028.3	−1.47275	−0.736376	−	0.676572i	$-0.763465\pi$
$661$	−25028.3	−1.47275	−0.736376	+	0.676572i	$0.763465\pi$
$662$	−9132.81	−0.536188
$663$	7296.66	0.427419
$664$	9567.08	0.559148
$665$	0	0
$666$	5653.93	0.328957
$667$	4812.37	0.279364
$668$	−9494.22	−0.549913
$669$	−26785.3	−1.54795
$670$	14857.6	0.856714
$671$	4951.22	0.284858
$672$	0	0
$673$	−17201.1	−0.985219	−0.492609	−	0.870251i	$-0.663957\pi$
$673$	−17201.1	−0.985219	−0.492609	+	0.870251i	$0.663957\pi$
$674$	8204.18	0.468862
$675$	4827.14	0.275255
$676$	−7020.50	−0.399437
$677$	12366.8	0.702060	0.351030	−	0.936364i	$-0.385832\pi$
$677$	12366.8	0.702060	0.351030	+	0.936364i	$0.385832\pi$
$678$	−19783.9	−1.12064
$679$	0	0
$680$	6096.56	0.343813
$681$	10543.9	0.593310
$682$	−6121.66	−0.343711
$683$	4091.25	0.229206	0.114603	−	0.993411i	$-0.463440\pi$
$683$	4091.25	0.229206	0.114603	+	0.993411i	$0.463440\pi$
$684$	4740.04	0.264971
$685$	−1822.13	−0.101635
$686$	0	0
$687$	31801.9	1.76611
$688$	−7328.73	−0.406112
$689$	5161.13	0.285375
$690$	−3510.83	−0.193703
$691$	8816.73	0.485389	0.242695	−	0.970103i	$-0.421969\pi$
$691$	8816.73	0.485389	0.242695	+	0.970103i	$0.421969\pi$
$692$	−6655.01	−0.365586
$693$	0	0
$694$	6150.98	0.336438
$695$	−4320.10	−0.235785
$696$	9869.32	0.537494
$697$	−13858.6	−0.753130
$698$	−6766.18	−0.366911
$699$	−33135.9	−1.79301
$700$	0	0
$701$	−3370.35	−0.181593	−0.0907963	−	0.995869i	$-0.528941\pi$
$701$	−3370.35	−0.181593	−0.0907963	+	0.995869i	$0.528941\pi$
$702$	4768.23	0.256361
$703$	−55571.8	−2.98141
$704$	819.942	0.0438959
$705$	12522.4	0.668965
$706$	−22788.5	−1.21481
$707$	0	0
$708$	−9189.13	−0.487780
$709$	−27565.3	−1.46013	−0.730067	−	0.683375i	$-0.760511\pi$
$709$	−27565.3	−1.46013	−0.730067	+	0.683375i	$0.760511\pi$
$710$	18768.7	0.992080
$711$	276.125	0.0145647
$712$	706.845	0.0372053
$713$	5494.95	0.288622
$714$	0	0
$715$	3486.10	0.182340
$716$	−1025.57	−0.0535299
$717$	−10374.3	−0.540357
$718$	19228.5	0.999442
$719$	8370.38	0.434162	0.217081	−	0.976154i	$-0.430346\pi$
$719$	8370.38	0.434162	0.217081	+	0.976154i	$0.430346\pi$
$720$	−1608.05	−0.0832342
$721$	0	0
$722$	−32871.2	−1.69438
$723$	27541.8	1.41672
$724$	918.020	0.0471243
$725$	8905.22	0.456181
$726$	−13759.9	−0.703414
$727$	−15995.5	−0.816012	−0.408006	−	0.912979i	$-0.633776\pi$
$727$	−15995.5	−0.816012	−0.408006	+	0.912979i	$0.633776\pi$
$728$	0	0
$729$	11235.7	0.570835
$730$	32083.0	1.62664
$731$	−26966.0	−1.36440
$732$	−9114.53	−0.460222
$733$	−15834.5	−0.797899	−0.398950	−	0.916973i	$-0.630625\pi$
$733$	−15834.5	−0.797899	−0.398950	+	0.916973i	$0.630625\pi$
$734$	−16870.7	−0.848377
$735$	0	0
$736$	−736.000	−0.0368605
$737$	7352.49	0.367480
$738$	3655.40	0.182327
$739$	11264.3	0.560711	0.280355	−	0.959896i	$-0.409548\pi$
$739$	11264.3	0.560711	0.280355	+	0.959896i	$0.409548\pi$
$740$	18852.7	0.936538
$741$	18916.6	0.937813
$742$	0	0
$743$	7326.72	0.361765	0.180883	−	0.983505i	$-0.442105\pi$
$743$	7326.72	0.361765	0.180883	+	0.983505i	$0.442105\pi$
$744$	11269.2	0.555306
$745$	−4090.79	−0.201175
$746$	12012.0	0.589533
$747$	−9285.03	−0.454781
$748$	3016.98	0.147475
$749$	0	0
$750$	12583.9	0.612664
$751$	−28755.5	−1.39721	−0.698603	−	0.715510i	$-0.746195\pi$
$751$	−28755.5	−1.39721	−0.698603	+	0.715510i	$0.746195\pi$
$752$	2625.15	0.127300
$753$	39727.2	1.92263
$754$	8796.53	0.424868
$755$	−19229.1	−0.926913
$756$	0	0
$757$	15928.3	0.764760	0.382380	−	0.924005i	$-0.375104\pi$
$757$	15928.3	0.764760	0.382380	+	0.924005i	$0.375104\pi$
$758$	3337.78	0.159939
$759$	−1737.39	−0.0830872
$760$	15805.4	0.754369
$761$	20406.9	0.972074	0.486037	−	0.873938i	$-0.338442\pi$
$761$	20406.9	0.972074	0.486037	+	0.873938i	$0.338442\pi$
$762$	−7762.97	−0.369059
$763$	0	0
$764$	5983.27	0.283334
$765$	−5916.83	−0.279638
$766$	−12135.6	−0.572422
$767$	−8190.27	−0.385572
$768$	−1509.40	−0.0709192
$769$	20579.9	0.965059	0.482530	−	0.875880i	$-0.339718\pi$
$769$	20579.9	0.965059	0.482530	+	0.875880i	$0.339718\pi$
$770$	0	0
$771$	−4246.96	−0.198379
$772$	1980.44	0.0923286
$773$	31392.9	1.46071	0.730353	−	0.683070i	$-0.239356\pi$
$773$	31392.9	1.46071	0.730353	+	0.683070i	$0.239356\pi$
$774$	7112.67	0.330310
$775$	10168.3	0.471299
$776$	14075.0	0.651112
$777$	0	0
$778$	−15344.5	−0.707104
$779$	−35928.5	−1.65247
$780$	−6417.45	−0.294592
$781$	9287.97	0.425544
$782$	−2708.11	−0.123839
$783$	23730.6	1.08309
$784$	0	0
$785$	214.850	0.00976859
$786$	24314.7	1.10341
$787$	−37409.7	−1.69443	−0.847213	−	0.531254i	$-0.821721\pi$
$787$	−37409.7	−1.69443	−0.847213	+	0.531254i	$0.821721\pi$
$788$	7919.42	0.358017
$789$	−48841.6	−2.20381
$790$	920.720	0.0414655
$791$	0	0
$792$	−795.769	−0.0357026
$793$	−8123.79	−0.363788
$794$	−481.745	−0.0215321
$795$	−18739.0	−0.835981
$796$	−8093.03	−0.360364
$797$	18800.9	0.835587	0.417793	−	0.908542i	$-0.362804\pi$
$797$	18800.9	0.835587	0.417793	+	0.908542i	$0.362804\pi$
$798$	0	0
$799$	9659.25	0.427684
$800$	−1361.96	−0.0601905
$801$	−686.007	−0.0302607
$802$	−21849.3	−0.962003
$803$	15876.7	0.697731
$804$	−13535.0	−0.593708
$805$	0	0
$806$	10044.2	0.438948
$807$	−27556.7	−1.20203
$808$	−1269.37	−0.0552675
$809$	22375.5	0.972413	0.486207	−	0.873844i	$-0.338380\pi$
$809$	22375.5	0.972413	0.486207	+	0.873844i	$0.338380\pi$
$810$	−22739.7	−0.986408
$811$	5013.76	0.217086	0.108543	−	0.994092i	$-0.465381\pi$
$811$	5013.76	0.217086	0.108543	+	0.994092i	$0.465381\pi$
$812$	0	0
$813$	−34636.6	−1.49417
$814$	9329.53	0.401720
$815$	−25278.0	−1.08644
$816$	−5553.85	−0.238264
$817$	−69909.6	−2.99367
$818$	−15378.8	−0.657343
$819$	0	0
$820$	12188.7	0.519083
$821$	11210.4	0.476547	0.238274	−	0.971198i	$-0.423419\pi$
$821$	11210.4	0.476547	0.238274	+	0.971198i	$0.423419\pi$
$822$	1659.92	0.0704336
$823$	−12289.5	−0.520515	−0.260257	−	0.965539i	$-0.583807\pi$
$823$	−12289.5	−0.520515	−0.260257	+	0.965539i	$0.583807\pi$
$824$	−874.472	−0.0369705
$825$	−3215.01	−0.135675
$826$	0	0
$827$	−8887.19	−0.373686	−0.186843	−	0.982390i	$-0.559826\pi$
$827$	−8887.19	−0.373686	−0.186843	+	0.982390i	$0.559826\pi$
$828$	714.302	0.0299803
$829$	16836.7	0.705382	0.352691	−	0.935740i	$-0.385267\pi$
$829$	16836.7	0.705382	0.352691	+	0.935740i	$0.385267\pi$
$830$	−30960.4	−1.29476
$831$	−21064.9	−0.879342
$832$	−1345.33	−0.0560589
$833$	0	0
$834$	3935.53	0.163401
$835$	30724.6	1.27337
$836$	7821.52	0.323580
$837$	27096.5	1.11899
$838$	−12358.2	−0.509435
$839$	15226.8	0.626566	0.313283	−	0.949660i	$-0.398571\pi$
$839$	15226.8	0.626566	0.313283	+	0.949660i	$0.398571\pi$
$840$	0	0
$841$	19389.7	0.795018
$842$	−5461.42	−0.223531
$843$	−4663.84	−0.190547
$844$	−890.449	−0.0363158
$845$	22719.3	0.924931
$846$	−2547.76	−0.103539
$847$	0	0
$848$	−3928.39	−0.159082
$849$	30228.4	1.22195
$850$	−5011.31	−0.202220
$851$	−8374.41	−0.337334
$852$	−17097.9	−0.687518
$853$	−32570.6	−1.30738	−0.653691	−	0.756762i	$-0.726780\pi$
$853$	−32570.6	−1.30738	−0.653691	+	0.756762i	$0.726780\pi$
$854$	0	0
$855$	−15339.4	−0.613563
$856$	−3919.72	−0.156511
$857$	26573.4	1.05919	0.529597	−	0.848249i	$-0.322343\pi$
$857$	26573.4	1.05919	0.529597	+	0.848249i	$0.322343\pi$
$858$	−3175.77	−0.126362
$859$	10254.7	0.407317	0.203658	−	0.979042i	$-0.434717\pi$
$859$	10254.7	0.407317	0.203658	+	0.979042i	$0.434717\pi$
$860$	23716.8	0.940389
$861$	0	0
$862$	15037.4	0.594173
$863$	25341.8	0.999587	0.499794	−	0.866144i	$-0.333409\pi$
$863$	25341.8	0.999587	0.499794	+	0.866144i	$0.333409\pi$
$864$	−3629.34	−0.142908
$865$	21536.5	0.846547
$866$	−23888.7	−0.937379
$867$	8532.22	0.334221
$868$	0	0
$869$	455.632	0.0177863
$870$	−31938.5	−1.24461
$871$	−12063.7	−0.469304
$872$	16443.9	0.638602
$873$	−13660.0	−0.529579
$874$	−7020.78	−0.271718
$875$	0	0
$876$	−29227.0	−1.12727
$877$	12594.8	0.484943	0.242471	−	0.970159i	$-0.422042\pi$
$877$	12594.8	0.484943	0.242471	+	0.970159i	$0.422042\pi$
$878$	19103.1	0.734282
$879$	−14408.5	−0.552885
$880$	−2653.44	−0.101645
$881$	−40740.9	−1.55800	−0.778999	−	0.627025i	$-0.784272\pi$
$881$	−40740.9	−1.55800	−0.778999	+	0.627025i	$0.784272\pi$
$882$	0	0
$883$	4433.94	0.168985	0.0844926	−	0.996424i	$-0.473073\pi$
$883$	4433.94	0.168985	0.0844926	+	0.996424i	$0.473073\pi$
$884$	−4950.15	−0.188339
$885$	29737.3	1.12950
$886$	−15792.3	−0.598818
$887$	−19202.6	−0.726900	−0.363450	−	0.931614i	$-0.618401\pi$
$887$	−19202.6	−0.726900	−0.363450	+	0.931614i	$0.618401\pi$
$888$	−17174.4	−0.649027
$889$	0	0
$890$	−2287.45	−0.0861521
$891$	−11253.1	−0.423111
$892$	18171.5	0.682092
$893$	25041.6	0.938395
$894$	3726.63	0.139415
$895$	3318.89	0.123953
$896$	0	0
$897$	2850.65	0.106110
$898$	891.290	0.0331211
$899$	49988.2	1.85451
$900$	1321.80	0.0489557
$901$	−14454.5	−0.534461
$902$	6031.76	0.222656
$903$	0	0
$904$	13421.6	0.493802
$905$	−2970.84	−0.109120
$906$	17517.4	0.642357
$907$	25546.0	0.935215	0.467608	−	0.883936i	$-0.345116\pi$
$907$	25546.0	0.935215	0.467608	+	0.883936i	$0.345116\pi$
$908$	−7153.13	−0.261437
$909$	1231.94	0.0449516
$910$	0	0
$911$	−47365.7	−1.72261	−0.861304	−	0.508089i	$-0.830352\pi$
$911$	−47365.7	−1.72261	−0.861304	+	0.508089i	$0.830352\pi$
$912$	−14398.4	−0.522783
$913$	−15321.2	−0.555375
$914$	8862.26	0.320719
$915$	29495.9	1.06569
$916$	−21574.8	−0.778224
$917$	0	0
$918$	−13354.1	−0.480122
$919$	6661.75	0.239120	0.119560	−	0.992827i	$-0.461852\pi$
$919$	6661.75	0.239120	0.119560	+	0.992827i	$0.461852\pi$
$920$	2381.80	0.0853538
$921$	22802.0	0.815801
$922$	1527.45	0.0545597
$923$	−15239.4	−0.543457
$924$	0	0
$925$	−15496.7	−0.550842
$926$	29653.9	1.05236
$927$	848.691	0.0300698
$928$	−6695.48	−0.236842
$929$	−12168.4	−0.429743	−0.214872	−	0.976642i	$-0.568933\pi$
$929$	−12168.4	−0.429743	−0.214872	+	0.976642i	$0.568933\pi$
$930$	−36468.6	−1.28586
$931$	0	0
$932$	22479.8	0.790077
$933$	−54441.3	−1.91032
$934$	13186.7	0.461972
$935$	−9763.34	−0.341492
$936$	1305.67	0.0455953
$937$	19003.8	0.662569	0.331285	−	0.943531i	$-0.392518\pi$
$937$	19003.8	0.662569	0.331285	+	0.943531i	$0.392518\pi$
$938$	0	0
$939$	13404.9	0.465869
$940$	−8495.35	−0.294774
$941$	20231.2	0.700868	0.350434	−	0.936587i	$-0.386034\pi$
$941$	20231.2	0.700868	0.350434	+	0.936587i	$0.386034\pi$
$942$	−195.725	−0.00676969
$943$	−5414.26	−0.186970
$944$	6234.02	0.214937
$945$	0	0
$946$	11736.6	0.403372
$947$	−3117.17	−0.106963	−0.0534817	−	0.998569i	$-0.517032\pi$
$947$	−3117.17	−0.106963	−0.0534817	+	0.998569i	$0.517032\pi$
$948$	−838.759	−0.0287359
$949$	−26050.0	−0.891064
$950$	−12991.8	−0.443696
$951$	36361.2	1.23985
$952$	0	0
$953$	47116.8	1.60153	0.800767	−	0.598976i	$-0.204425\pi$
$953$	47116.8	1.60153	0.800767	+	0.598976i	$0.204425\pi$
$954$	3812.58	0.129389
$955$	−19362.7	−0.656085
$956$	7038.07	0.238104
$957$	−15805.2	−0.533866
$958$	31725.1	1.06993
$959$	0	0
$960$	4884.64	0.164220
$961$	27287.5	0.915963
$962$	−15307.6	−0.513031
$963$	3804.16	0.127297
$964$	−18684.7	−0.624267
$965$	−6408.98	−0.213795
$966$	0	0
$967$	−640.862	−0.0213120	−0.0106560	−	0.999943i	$-0.503392\pi$
$967$	−640.862	−0.0213120	−0.0106560	+	0.999943i	$0.503392\pi$
$968$	9334.90	0.309954
$969$	−52978.8	−1.75637
$970$	−45548.6	−1.50771
$971$	−32681.9	−1.08013	−0.540067	−	0.841622i	$-0.681601\pi$
$971$	−32681.9	−1.08013	−0.540067	+	0.841622i	$0.681601\pi$
$972$	8466.40	0.279383
$973$	0	0
$974$	−14490.6	−0.476704
$975$	5275.08	0.173269
$976$	6183.42	0.202793
$977$	−35650.5	−1.16741	−0.583706	−	0.811965i	$-0.698398\pi$
$977$	−35650.5	−1.16741	−0.583706	+	0.811965i	$0.698398\pi$
$978$	23027.8	0.752911
$979$	−1131.98	−0.0369542
$980$	0	0
$981$	−15959.1	−0.519404
$982$	−39429.5	−1.28131
$983$	15011.9	0.487086	0.243543	−	0.969890i	$-0.421690\pi$
$983$	15011.9	0.487086	0.243543	+	0.969890i	$0.421690\pi$
$984$	−11103.7	−0.359728
$985$	−25628.3	−0.829021
$986$	−24636.0	−0.795709
$987$	0	0
$988$	−12833.3	−0.413240
$989$	−10535.0	−0.338721
$990$	2575.22	0.0826725
$991$	−39792.6	−1.27553	−0.637767	−	0.770229i	$-0.720142\pi$
$991$	−39792.6	−1.27553	−0.637767	+	0.770229i	$0.720142\pi$
$992$	−7645.15	−0.244691
$993$	−26924.0	−0.860431
$994$	0	0
$995$	26190.1	0.834455
$996$	28204.3	0.897276
$997$	−25568.6	−0.812202	−0.406101	−	0.913828i	$-0.633112\pi$
$997$	−25568.6	−0.812202	−0.406101	+	0.913828i	$0.633112\pi$
$998$	25596.4	0.811864
$999$	−41295.6	−1.30784

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2254.4.a.u.1.3		11
7.3	odd	6		322.4.e.d.93.3	✓	22
7.5	odd	6		322.4.e.d.277.3	yes	22
7.6	odd	2		2254.4.a.r.1.9		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.4.e.d.93.3	✓	22	7.3	odd	6
322.4.e.d.277.3	yes	22	7.5	odd	6
2254.4.a.r.1.9		11	7.6	odd	2
2254.4.a.u.1.3		11	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2254 = 2 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2254.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -5.89611 q^{3} +4.00000 q^{4} -12.9445 q^{5} +11.7922 q^{6} -8.00000 q^{8} +7.76415 q^{9} +25.8891 q^{10} +12.8116 q^{11} -23.5845 q^{12} -21.0208 q^{13} +76.3225 q^{15} +16.0000 q^{16} +58.8720 q^{17} -15.5283 q^{18} +152.626 q^{19} -51.7782 q^{20} -25.6232 q^{22} +23.0000 q^{23} +47.1689 q^{24} +42.5611 q^{25} +42.0417 q^{26} +113.417 q^{27} +209.234 q^{29} -152.645 q^{30} +238.911 q^{31} -32.0000 q^{32} -75.5386 q^{33} -117.744 q^{34} +31.0566 q^{36} -364.105 q^{37} -305.252 q^{38} +123.941 q^{39} +103.556 q^{40} -235.402 q^{41} -458.046 q^{43} +51.2464 q^{44} -100.503 q^{45} -46.0000 q^{46} +164.072 q^{47} -94.3378 q^{48} -85.1222 q^{50} -347.116 q^{51} -84.0833 q^{52} -245.524 q^{53} -226.833 q^{54} -165.840 q^{55} -899.899 q^{57} -418.467 q^{58} +389.626 q^{59} +305.290 q^{60} +386.464 q^{61} -477.822 q^{62} +64.0000 q^{64} +272.105 q^{65} +151.077 q^{66} +573.894 q^{67} +235.488 q^{68} -135.611 q^{69} +724.966 q^{71} -62.1132 q^{72} +1239.25 q^{73} +728.210 q^{74} -250.945 q^{75} +610.503 q^{76} -247.882 q^{78} +35.5640 q^{79} -207.113 q^{80} -878.350 q^{81} +470.805 q^{82} -1195.88 q^{83} -762.070 q^{85} +916.091 q^{86} -1233.67 q^{87} -102.493 q^{88} -88.3557 q^{89} +201.007 q^{90} +92.0000 q^{92} -1408.65 q^{93} -328.144 q^{94} -1975.67 q^{95} +188.676 q^{96} -1759.37 q^{97} +99.4712 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 22 q^{2} + 44 q^{4} + 23 q^{5} - 88 q^{8} + 171 q^{9} - 46 q^{10} - 48 q^{11} + 77 q^{13} + 104 q^{15} + 176 q^{16} + 97 q^{17} - 342 q^{18} + 138 q^{19} + 92 q^{20} + 96 q^{22} + 253 q^{23} + 30 q^{25}+ \cdots - 2545 q^{99}+O(q^{100}) \) 11 * q - 22 * q^2 + 44 * q^4 + 23 * q^5 - 88 * q^8 + 171 * q^9 - 46 * q^10 - 48 * q^11 + 77 * q^13 + 104 * q^15 + 176 * q^16 + 97 * q^17 - 342 * q^18 + 138 * q^19 + 92 * q^20 + 96 * q^22 + 253 * q^23 + 30 * q^25 - 154 * q^26 + 315 * q^27 + 137 * q^29 - 208 * q^30 + 234 * q^31 - 352 * q^32 + 28 * q^33 - 194 * q^34 + 684 * q^36 + 87 * q^37 - 276 * q^38 + 129 * q^39 - 184 * q^40 + 612 * q^41 + 187 * q^43 - 192 * q^44 + 597 * q^45 - 506 * q^46 + 793 * q^47 - 60 * q^50 - 1257 * q^51 + 308 * q^52 + 575 * q^53 - 630 * q^54 - 549 * q^55 + 39 * q^57 - 274 * q^58 + 1241 * q^59 + 416 * q^60 + 187 * q^61 - 468 * q^62 + 704 * q^64 - 937 * q^65 - 56 * q^66 + 2275 * q^67 + 388 * q^68 + 115 * q^71 - 1368 * q^72 + 2385 * q^73 - 174 * q^74 + 196 * q^75 + 552 * q^76 - 258 * q^78 - 1225 * q^79 + 368 * q^80 + 3635 * q^81 - 1224 * q^82 - 597 * q^83 - 2493 * q^85 - 374 * q^86 + 1428 * q^87 + 384 * q^88 + 602 * q^89 - 1194 * q^90 + 1012 * q^92 - 1360 * q^93 - 1586 * q^94 + 4835 * q^95 - 1039 * q^97 - 2545 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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