LMFDB - Embedded newform 1275.4.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1275,4,Mod(1,1275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1275, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1275.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1275 = 3 \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1275.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:	$75.2274352573$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.5912.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 14x - 10 $ x^3 - x^2 - 14*x - 10
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 51)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.75985$ of defining polynomial
Character	$\chi$	$=$	1275.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.75985 q^{2} -3.00000 q^{3} +14.6562 q^{4} +14.2795 q^{6} +31.5852 q^{7} -31.6823 q^{8} +9.00000 q^{9} -7.18910 q^{11} -43.9685 q^{12} -84.3331 q^{13} -150.341 q^{14} +33.5537 q^{16} -17.0000 q^{17} -42.8386 q^{18} -37.0838 q^{19} -94.7557 q^{21} +34.2190 q^{22} -150.218 q^{23} +95.0469 q^{24} +401.413 q^{26} -27.0000 q^{27} +462.918 q^{28} -11.5846 q^{29} -53.2865 q^{31} +93.7478 q^{32} +21.5673 q^{33} +80.9174 q^{34} +131.905 q^{36} +99.2134 q^{37} +176.513 q^{38} +252.999 q^{39} +118.249 q^{41} +451.023 q^{42} +456.016 q^{43} -105.365 q^{44} +715.014 q^{46} -571.014 q^{47} -100.661 q^{48} +654.627 q^{49} +51.0000 q^{51} -1236.00 q^{52} -462.867 q^{53} +128.516 q^{54} -1000.69 q^{56} +111.251 q^{57} +55.1407 q^{58} +48.0674 q^{59} +59.5236 q^{61} +253.636 q^{62} +284.267 q^{63} -714.655 q^{64} -102.657 q^{66} +740.787 q^{67} -249.155 q^{68} +450.653 q^{69} -930.437 q^{71} -285.141 q^{72} +697.419 q^{73} -472.241 q^{74} -543.506 q^{76} -227.070 q^{77} -1204.24 q^{78} +1036.04 q^{79} +81.0000 q^{81} -562.849 q^{82} +22.2043 q^{83} -1388.75 q^{84} -2170.57 q^{86} +34.7537 q^{87} +227.767 q^{88} -369.726 q^{89} -2663.68 q^{91} -2201.62 q^{92} +159.860 q^{93} +2717.94 q^{94} -281.244 q^{96} -1139.56 q^{97} -3115.93 q^{98} -64.7019 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 5 q^{2} - 9 q^{3} + 13 q^{4} + 15 q^{6} + 8 q^{7} - 33 q^{8} + 27 q^{9} + 34 q^{11} - 39 q^{12} - 36 q^{13} - 104 q^{14} - 79 q^{16} - 51 q^{17} - 45 q^{18} - 142 q^{19} - 24 q^{21} + 248 q^{22} - 110 q^{23}+ \cdots + 306 q^{99}+O(q^{100}) $ 3 * q - 5 * q^2 - 9 * q^3 + 13 * q^4 + 15 * q^6 + 8 * q^7 - 33 * q^8 + 27 * q^9 + 34 * q^11 - 39 * q^12 - 36 * q^13 - 104 * q^14 - 79 * q^16 - 51 * q^17 - 45 * q^18 - 142 * q^19 - 24 * q^21 + 248 * q^22 - 110 * q^23 + 99 * q^24 + 154 * q^26 - 81 * q^27 + 472 * q^28 + 90 * q^29 - 148 * q^31 + 151 * q^32 - 102 * q^33 + 85 * q^34 + 117 * q^36 - 110 * q^37 - 80 * q^38 + 108 * q^39 + 720 * q^41 + 312 * q^42 + 146 * q^43 - 192 * q^44 + 748 * q^46 - 500 * q^47 + 237 * q^48 + 379 * q^49 + 153 * q^51 - 1218 * q^52 - 610 * q^53 + 135 * q^54 - 1368 * q^56 + 426 * q^57 - 1006 * q^58 - 216 * q^59 - 18 * q^61 + 904 * q^62 + 72 * q^63 + 377 * q^64 - 744 * q^66 + 1404 * q^67 - 221 * q^68 + 330 * q^69 - 960 * q^71 - 297 * q^72 + 794 * q^73 - 1874 * q^74 - 392 * q^76 - 48 * q^77 - 462 * q^78 - 276 * q^79 + 243 * q^81 - 382 * q^82 + 1552 * q^83 - 1416 * q^84 + 16 * q^86 - 270 * q^87 - 1724 * q^88 + 1394 * q^89 - 3968 * q^91 - 2244 * q^92 + 444 * q^93 + 3960 * q^94 - 453 * q^96 - 402 * q^97 - 4109 * q^98 + 306 * 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 $

$2$	−4.75985	−1.68286	−0.841430	−	0.540366i	$-0.818286\pi$
$2$	−4.75985	−1.68286	−0.841430	+	0.540366i	$0.818286\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	14.6562	1.83202
$5$	0	0
$5$	0	0
$6$	14.2795	0.971600
$7$	31.5852	1.70544	0.852721	−	0.522366i	$-0.174951\pi$
$7$	31.5852	1.70544	0.852721	+	0.522366i	$0.174951\pi$
$8$	−31.6823	−1.40017
$9$	9.00000	0.333333
$10$	0	0
$11$	−7.18910	−0.197054	−0.0985271	−	0.995134i	$-0.531413\pi$
$11$	−7.18910	−0.197054	−0.0985271	+	0.995134i	$0.531413\pi$
$12$	−43.9685	−1.05772
$13$	−84.3331	−1.79921	−0.899607	−	0.436700i	$-0.856147\pi$
$13$	−84.3331	−1.79921	−0.899607	+	0.436700i	$0.856147\pi$
$14$	−150.341	−2.87002
$15$	0	0
$16$	33.5537	0.524277
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	−42.8386	−0.560954
$19$	−37.0838	−0.447769	−0.223885	−	0.974616i	$-0.571874\pi$
$19$	−37.0838	−0.447769	−0.223885	+	0.974616i	$0.571874\pi$
$20$	0	0
$21$	−94.7557	−0.984638
$22$	34.2190	0.331615
$23$	−150.218	−1.36185	−0.680925	−	0.732353i	$-0.738422\pi$
$23$	−150.218	−1.36185	−0.680925	+	0.732353i	$0.738422\pi$
$24$	95.0469	0.808390
$25$	0	0
$26$	401.413	3.02783
$27$	−27.0000	−0.192450
$28$	462.918	3.12440
$29$	−11.5846	−0.0741792	−0.0370896	−	0.999312i	$-0.511809\pi$
$29$	−11.5846	−0.0741792	−0.0370896	+	0.999312i	$0.511809\pi$
$30$	0	0
$31$	−53.2865	−0.308727	−0.154364	−	0.988014i	$-0.549333\pi$
$31$	−53.2865	−0.308727	−0.154364	+	0.988014i	$0.549333\pi$
$32$	93.7478	0.517889
$33$	21.5673	0.113769
$34$	80.9174	0.408154
$35$	0	0
$36$	131.905	0.610673
$37$	99.2134	0.440827	0.220413	−	0.975407i	$-0.429259\pi$
$37$	99.2134	0.440827	0.220413	+	0.975407i	$0.429259\pi$
$38$	176.513	0.753533
$39$	252.999	1.03878
$40$	0	0
$41$	118.249	0.450425	0.225213	−	0.974310i	$-0.427692\pi$
$41$	118.249	0.450425	0.225213	+	0.974310i	$0.427692\pi$
$42$	451.023	1.65701
$43$	456.016	1.61725	0.808626	−	0.588323i	$-0.200211\pi$
$43$	456.016	1.61725	0.808626	+	0.588323i	$0.200211\pi$
$44$	−105.365	−0.361007
$45$	0	0
$46$	715.014	2.29180
$47$	−571.014	−1.77215	−0.886073	−	0.463545i	$-0.846577\pi$
$47$	−571.014	−1.77215	−0.886073	+	0.463545i	$0.846577\pi$
$48$	−100.661	−0.302691
$49$	654.627	1.90853
$50$	0	0
$51$	51.0000	0.140028
$52$	−1236.00	−3.29620
$53$	−462.867	−1.19962	−0.599809	−	0.800143i	$-0.704757\pi$
$53$	−462.867	−1.19962	−0.599809	+	0.800143i	$0.704757\pi$
$54$	128.516	0.323867
$55$	0	0
$56$	−1000.69	−2.38792
$57$	111.251	0.258520
$58$	55.1407	0.124833
$59$	48.0674	0.106065	0.0530325	−	0.998593i	$-0.483111\pi$
$59$	48.0674	0.106065	0.0530325	+	0.998593i	$0.483111\pi$
$60$	0	0
$61$	59.5236	0.124938	0.0624689	−	0.998047i	$-0.480103\pi$
$61$	59.5236	0.124938	0.0624689	+	0.998047i	$0.480103\pi$
$62$	253.636	0.519545
$63$	284.267	0.568481
$64$	−714.655	−1.39581
$65$	0	0
$66$	−102.657	−0.191458
$67$	740.787	1.35077	0.675384	−	0.737466i	$-0.263978\pi$
$67$	740.787	1.35077	0.675384	+	0.737466i	$0.263978\pi$
$68$	−249.155	−0.444330
$69$	450.653	0.786265
$70$	0	0
$71$	−930.437	−1.55525	−0.777623	−	0.628730i	$-0.783575\pi$
$71$	−930.437	−1.55525	−0.777623	+	0.628730i	$0.783575\pi$
$72$	−285.141	−0.466724
$73$	697.419	1.11817	0.559087	−	0.829109i	$-0.311152\pi$
$73$	697.419	1.11817	0.559087	+	0.829109i	$0.311152\pi$
$74$	−472.241	−0.741850
$75$	0	0
$76$	−543.506	−0.820322
$77$	−227.070	−0.336065
$78$	−1204.24	−1.74812
$79$	1036.04	1.47549	0.737747	−	0.675077i	$-0.235890\pi$
$79$	1036.04	1.47549	0.737747	+	0.675077i	$0.235890\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	−562.849	−0.758003
$83$	22.2043	0.0293643	0.0146822	−	0.999892i	$-0.495326\pi$
$83$	22.2043	0.0293643	0.0146822	+	0.999892i	$0.495326\pi$
$84$	−1388.75	−1.80388
$85$	0	0
$86$	−2170.57	−2.72161
$87$	34.7537	0.0428274
$88$	227.767	0.275910
$89$	−369.726	−0.440346	−0.220173	−	0.975461i	$-0.570662\pi$
$89$	−369.726	−0.440346	−0.220173	+	0.975461i	$0.570662\pi$
$90$	0	0
$91$	−2663.68	−3.06846
$92$	−2201.62	−2.49494
$93$	159.860	0.178244
$94$	2717.94	2.98228
$95$	0	0
$96$	−281.244	−0.299003
$97$	−1139.56	−1.19283	−0.596415	−	0.802676i	$-0.703409\pi$
$97$	−1139.56	−1.19283	−0.596415	+	0.802676i	$0.703409\pi$
$98$	−3115.93	−3.21180
$99$	−64.7019	−0.0656847
$100$	0	0
$101$	703.083	0.692667	0.346334	−	0.938111i	$-0.387427\pi$
$101$	703.083	0.692667	0.346334	+	0.938111i	$0.387427\pi$
$102$	−242.752	−0.235648
$103$	897.160	0.858250	0.429125	−	0.903245i	$-0.358822\pi$
$103$	897.160	0.858250	0.429125	+	0.903245i	$0.358822\pi$
$104$	2671.87	2.51921
$105$	0	0
$106$	2203.18	2.01879
$107$	1901.21	1.71773	0.858864	−	0.512203i	$-0.171171\pi$
$107$	1901.21	1.71773	0.858864	+	0.512203i	$0.171171\pi$
$108$	−395.716	−0.352572
$109$	584.555	0.513671	0.256836	−	0.966455i	$-0.417320\pi$
$109$	584.555	0.513671	0.256836	+	0.966455i	$0.417320\pi$
$110$	0	0
$111$	−297.640	−0.254511
$112$	1059.80	0.894124
$113$	63.4225	0.0527990	0.0263995	−	0.999651i	$-0.491596\pi$
$113$	63.4225	0.0527990	0.0263995	+	0.999651i	$0.491596\pi$
$114$	−529.540	−0.435052
$115$	0	0
$116$	−169.785	−0.135898
$117$	−758.998	−0.599738
$118$	−228.793	−0.178493
$119$	−536.949	−0.413631
$120$	0	0
$121$	−1279.32	−0.961170
$122$	−283.323	−0.210253
$123$	−354.748	−0.260053
$124$	−780.976	−0.565594
$125$	0	0
$126$	−1353.07	−0.956674
$127$	−175.543	−0.122653	−0.0613266	−	0.998118i	$-0.519533\pi$
$127$	−175.543	−0.122653	−0.0613266	+	0.998118i	$0.519533\pi$
$128$	2651.67	1.83107
$129$	−1368.05	−0.933721
$130$	0	0
$131$	1865.42	1.24414	0.622070	−	0.782961i	$-0.286292\pi$
$131$	1865.42	1.24414	0.622070	+	0.782961i	$0.286292\pi$
$132$	316.094	0.208428
$133$	−1171.30	−0.763644
$134$	−3526.03	−2.27315
$135$	0	0
$136$	538.599	0.339592
$137$	−1057.57	−0.659519	−0.329760	−	0.944065i	$-0.606968\pi$
$137$	−1057.57	−0.659519	−0.329760	+	0.944065i	$0.606968\pi$
$138$	−2145.04	−1.32317
$139$	904.833	0.552136	0.276068	−	0.961138i	$-0.410968\pi$
$139$	904.833	0.552136	0.276068	+	0.961138i	$0.410968\pi$
$140$	0	0
$141$	1713.04	1.02315
$142$	4428.74	2.61726
$143$	606.279	0.354543
$144$	301.983	0.174759
$145$	0	0
$146$	−3319.61	−1.88173
$147$	−1963.88	−1.10189
$148$	1454.09	0.807603
$149$	809.001	0.444805	0.222402	−	0.974955i	$-0.428610\pi$
$149$	809.001	0.444805	0.222402	+	0.974955i	$0.428610\pi$
$150$	0	0
$151$	−352.121	−0.189769	−0.0948847	−	0.995488i	$-0.530248\pi$
$151$	−352.121	−0.189769	−0.0948847	+	0.995488i	$0.530248\pi$
$152$	1174.90	0.626954
$153$	−153.000	−0.0808452
$154$	1080.82	0.565550
$155$	0	0
$156$	3708.00	1.90306
$157$	−537.882	−0.273424	−0.136712	−	0.990611i	$-0.543654\pi$
$157$	−537.882	−0.273424	−0.136712	+	0.990611i	$0.543654\pi$
$158$	−4931.41	−2.48305
$159$	1388.60	0.692599
$160$	0	0
$161$	−4744.66	−2.32256
$162$	−385.548	−0.186985
$163$	−1922.74	−0.923933	−0.461966	−	0.886897i	$-0.652856\pi$
$163$	−1922.74	−0.923933	−0.461966	+	0.886897i	$0.652856\pi$
$164$	1733.08	0.825188
$165$	0	0
$166$	−105.689	−0.0494160
$167$	2971.76	1.37702	0.688509	−	0.725228i	$-0.258266\pi$
$167$	2971.76	1.37702	0.688509	+	0.725228i	$0.258266\pi$
$168$	3002.08	1.37866
$169$	4915.07	2.23717
$170$	0	0
$171$	−333.754	−0.149256
$172$	6683.45	2.96284
$173$	−988.564	−0.434446	−0.217223	−	0.976122i	$-0.569700\pi$
$173$	−988.564	−0.434446	−0.217223	+	0.976122i	$0.569700\pi$
$174$	−165.422	−0.0720725
$175$	0	0
$176$	−241.221	−0.103311
$177$	−144.202	−0.0612367
$178$	1759.84	0.741042
$179$	−1937.65	−0.809089	−0.404545	−	0.914518i	$-0.632570\pi$
$179$	−1937.65	−0.809089	−0.404545	+	0.914518i	$0.632570\pi$
$180$	0	0
$181$	−2180.07	−0.895267	−0.447634	−	0.894217i	$-0.647733\pi$
$181$	−2180.07	−0.895267	−0.447634	+	0.894217i	$0.647733\pi$
$182$	12678.7	5.16379
$183$	−178.571	−0.0721329
$184$	4759.24	1.90683
$185$	0	0
$186$	−760.907	−0.299959
$187$	122.215	0.0477927
$188$	−8368.86	−3.24661
$189$	−852.801	−0.328213
$190$	0	0
$191$	1675.78	0.634845	0.317423	−	0.948284i	$-0.397183\pi$
$191$	1675.78	0.634845	0.317423	+	0.948284i	$0.397183\pi$
$192$	2143.97	0.805872
$193$	257.961	0.0962094	0.0481047	−	0.998842i	$-0.484682\pi$
$193$	257.961	0.0962094	0.0481047	+	0.998842i	$0.484682\pi$
$194$	5424.12	2.00737
$195$	0	0
$196$	9594.32	3.49647
$197$	693.466	0.250799	0.125399	−	0.992106i	$-0.459979\pi$
$197$	693.466	0.250799	0.125399	+	0.992106i	$0.459979\pi$
$198$	307.971	0.110538
$199$	−240.295	−0.0855984	−0.0427992	−	0.999084i	$-0.513628\pi$
$199$	−240.295	−0.0855984	−0.0427992	+	0.999084i	$0.513628\pi$
$200$	0	0
$201$	−2222.36	−0.779866
$202$	−3346.57	−1.16566
$203$	−365.901	−0.126508
$204$	747.464	0.256534
$205$	0	0
$206$	−4270.34	−1.44432
$207$	−1351.96	−0.453950
$208$	−2829.69	−0.943287
$209$	266.599	0.0882348
$210$	0	0
$211$	268.114	0.0874774	0.0437387	−	0.999043i	$-0.486073\pi$
$211$	268.114	0.0874774	0.0437387	+	0.999043i	$0.486073\pi$
$212$	−6783.86	−2.19772
$213$	2791.31	0.897922
$214$	−9049.47	−2.89070
$215$	0	0
$216$	855.422	0.269463
$217$	−1683.07	−0.526516
$218$	−2782.39	−0.864437
$219$	−2092.26	−0.645578
$220$	0	0
$221$	1433.66	0.436374
$222$	1416.72	0.428307
$223$	5524.43	1.65894	0.829468	−	0.558554i	$-0.188643\pi$
$223$	5524.43	1.65894	0.829468	+	0.558554i	$0.188643\pi$
$224$	2961.05	0.883229
$225$	0	0
$226$	−301.882	−0.0888534
$227$	384.400	0.112394	0.0561972	−	0.998420i	$-0.482102\pi$
$227$	384.400	0.112394	0.0561972	+	0.998420i	$0.482102\pi$
$228$	1630.52	0.473613
$229$	1395.48	0.402690	0.201345	−	0.979520i	$-0.435469\pi$
$229$	1395.48	0.402690	0.201345	+	0.979520i	$0.435469\pi$
$230$	0	0
$231$	681.209	0.194027
$232$	367.025	0.103864
$233$	−3409.39	−0.958613	−0.479307	−	0.877648i	$-0.659112\pi$
$233$	−3409.39	−0.958613	−0.479307	+	0.877648i	$0.659112\pi$
$234$	3612.71	1.00928
$235$	0	0
$236$	704.483	0.194313
$237$	−3108.13	−0.851877
$238$	2555.80	0.696083
$239$	−1509.18	−0.408456	−0.204228	−	0.978923i	$-0.565468\pi$
$239$	−1509.18	−0.408456	−0.204228	+	0.978923i	$0.565468\pi$
$240$	0	0
$241$	3406.91	0.910615	0.455307	−	0.890334i	$-0.349529\pi$
$241$	3406.91	0.910615	0.455307	+	0.890334i	$0.349529\pi$
$242$	6089.35	1.61751
$243$	−243.000	−0.0641500
$244$	872.387	0.228889
$245$	0	0
$246$	1688.55	0.437633
$247$	3127.39	0.805633
$248$	1688.24	0.432271
$249$	−66.6129	−0.0169535
$250$	0	0
$251$	3394.43	0.853605	0.426802	−	0.904345i	$-0.359640\pi$
$251$	3394.43	0.853605	0.426802	+	0.904345i	$0.359640\pi$
$252$	4166.26	1.04147
$253$	1079.93	0.268358
$254$	835.560	0.206408
$255$	0	0
$256$	−6904.30	−1.68562
$257$	−1778.91	−0.431772	−0.215886	−	0.976419i	$-0.569264\pi$
$257$	−1778.91	−0.431772	−0.215886	+	0.976419i	$0.569264\pi$
$258$	6511.71	1.57132
$259$	3133.68	0.751804
$260$	0	0
$261$	−104.261	−0.0247264
$262$	−8879.11	−2.09372
$263$	4316.88	1.01213	0.506065	−	0.862495i	$-0.331100\pi$
$263$	4316.88	1.01213	0.506065	+	0.862495i	$0.331100\pi$
$264$	−683.302	−0.159297
$265$	0	0
$266$	5575.22	1.28511
$267$	1109.18	0.254234
$268$	10857.1	2.47463
$269$	6546.31	1.48378	0.741888	−	0.670524i	$-0.233931\pi$
$269$	6546.31	1.48378	0.741888	+	0.670524i	$0.233931\pi$
$270$	0	0
$271$	3785.20	0.848466	0.424233	−	0.905553i	$-0.360544\pi$
$271$	3785.20	0.848466	0.424233	+	0.905553i	$0.360544\pi$
$272$	−570.413	−0.127156
$273$	7991.04	1.77157
$274$	5033.86	1.10988
$275$	0	0
$276$	6604.85	1.44045
$277$	3521.06	0.763755	0.381878	−	0.924213i	$-0.375278\pi$
$277$	3521.06	0.763755	0.381878	+	0.924213i	$0.375278\pi$
$278$	−4306.87	−0.929168
$279$	−479.579	−0.102909
$280$	0	0
$281$	−2922.30	−0.620391	−0.310195	−	0.950673i	$-0.600394\pi$
$281$	−2922.30	−0.620391	−0.310195	+	0.950673i	$0.600394\pi$
$282$	−8153.81	−1.72182
$283$	−735.075	−0.154402	−0.0772008	−	0.997016i	$-0.524598\pi$
$283$	−735.075	−0.154402	−0.0772008	+	0.997016i	$0.524598\pi$
$284$	−13636.6	−2.84924
$285$	0	0
$286$	−2885.80	−0.596646
$287$	3734.93	0.768175
$288$	843.731	0.172630
$289$	289.000	0.0588235
$290$	0	0
$291$	3418.67	0.688681
$292$	10221.5	2.04852
$293$	8702.92	1.73526	0.867628	−	0.497213i	$-0.165643\pi$
$293$	8702.92	1.73526	0.867628	+	0.497213i	$0.165643\pi$
$294$	9347.78	1.85433
$295$	0	0
$296$	−3143.31	−0.617234
$297$	194.106	0.0379231
$298$	−3850.72	−0.748545
$299$	12668.3	2.45026
$300$	0	0
$301$	14403.4	2.75813
$302$	1676.04	0.319355
$303$	−2109.25	−0.399912
$304$	−1244.30	−0.234755
$305$	0	0
$306$	728.257	0.136051
$307$	−2516.95	−0.467916	−0.233958	−	0.972247i	$-0.575168\pi$
$307$	−2516.95	−0.467916	−0.233958	+	0.972247i	$0.575168\pi$
$308$	−3327.97	−0.615677
$309$	−2691.48	−0.495511
$310$	0	0
$311$	6593.31	1.20216	0.601081	−	0.799188i	$-0.294737\pi$
$311$	6593.31	1.20216	0.601081	+	0.799188i	$0.294737\pi$
$312$	−8015.60	−1.45447
$313$	−4392.99	−0.793312	−0.396656	−	0.917967i	$-0.629829\pi$
$313$	−4392.99	−0.793312	−0.396656	+	0.917967i	$0.629829\pi$
$314$	2560.24	0.460135
$315$	0	0
$316$	15184.4	2.70314
$317$	−2601.23	−0.460882	−0.230441	−	0.973086i	$-0.574017\pi$
$317$	−2601.23	−0.460882	−0.230441	+	0.973086i	$0.574017\pi$
$318$	−6609.54	−1.16555
$319$	83.2825	0.0146173
$320$	0	0
$321$	−5703.63	−0.991731
$322$	22583.9	3.90854
$323$	630.425	0.108600
$324$	1187.15	0.203558
$325$	0	0
$326$	9151.97	1.55485
$327$	−1753.66	−0.296568
$328$	−3746.41	−0.630674
$329$	−18035.6	−3.02229
$330$	0	0
$331$	−4670.49	−0.775568	−0.387784	−	0.921750i	$-0.626759\pi$
$331$	−4670.49	−0.775568	−0.387784	+	0.921750i	$0.626759\pi$
$332$	325.430	0.0537960
$333$	892.921	0.146942
$334$	−14145.1	−2.31733
$335$	0	0
$336$	−3179.41	−0.516223
$337$	−1801.67	−0.291226	−0.145613	−	0.989342i	$-0.546515\pi$
$337$	−1801.67	−0.291226	−0.145613	+	0.989342i	$0.546515\pi$
$338$	−23395.0	−3.76485
$339$	−190.268	−0.0304835
$340$	0	0
$341$	383.082	0.0608360
$342$	1588.62	0.251178
$343$	9842.83	1.54945
$344$	−14447.7	−2.26443
$345$	0	0
$346$	4705.41	0.731112
$347$	−168.340	−0.0260431	−0.0130216	−	0.999915i	$-0.504145\pi$
$347$	−168.340	−0.0260431	−0.0130216	+	0.999915i	$0.504145\pi$
$348$	509.355	0.0784606
$349$	−4447.85	−0.682200	−0.341100	−	0.940027i	$-0.610799\pi$
$349$	−4447.85	−0.682200	−0.341100	+	0.940027i	$0.610799\pi$
$350$	0	0
$351$	2276.99	0.346259
$352$	−673.963	−0.102052
$353$	10509.5	1.58460	0.792298	−	0.610135i	$-0.208885\pi$
$353$	10509.5	1.58460	0.792298	+	0.610135i	$0.208885\pi$
$354$	686.380	0.103053
$355$	0	0
$356$	−5418.76	−0.806723
$357$	1610.85	0.238810
$358$	9222.93	1.36158
$359$	8342.99	1.22654	0.613268	−	0.789875i	$-0.289855\pi$
$359$	8342.99	1.22654	0.613268	+	0.789875i	$0.289855\pi$
$360$	0	0
$361$	−5483.79	−0.799503
$362$	10376.8	1.50661
$363$	3837.95	0.554932
$364$	−39039.3	−5.62147
$365$	0	0
$366$	849.969	0.121390
$367$	−352.402	−0.0501232	−0.0250616	−	0.999686i	$-0.507978\pi$
$367$	−352.402	−0.0501232	−0.0250616	+	0.999686i	$0.507978\pi$
$368$	−5040.36	−0.713987
$369$	1064.24	0.150142
$370$	0	0
$371$	−14619.8	−2.04588
$372$	2342.93	0.326546
$373$	−12563.2	−1.74397	−0.871983	−	0.489537i	$-0.837166\pi$
$373$	−12563.2	−1.74397	−0.871983	+	0.489537i	$0.837166\pi$
$374$	−581.724	−0.0804284
$375$	0	0
$376$	18091.0	2.48131
$377$	976.961	0.133464
$378$	4059.21	0.552336
$379$	−1770.57	−0.239969	−0.119984	−	0.992776i	$-0.538284\pi$
$379$	−1770.57	−0.239969	−0.119984	+	0.992776i	$0.538284\pi$
$380$	0	0
$381$	526.630	0.0708139
$382$	−7976.47	−1.06836
$383$	−4330.57	−0.577759	−0.288880	−	0.957365i	$-0.593283\pi$
$383$	−4330.57	−0.577759	−0.288880	+	0.957365i	$0.593283\pi$
$384$	−7955.00	−1.05717
$385$	0	0
$386$	−1227.85	−0.161907
$387$	4104.15	0.539084
$388$	−16701.5	−2.18529
$389$	10295.5	1.34191	0.670957	−	0.741496i	$-0.265883\pi$
$389$	10295.5	1.34191	0.670957	+	0.741496i	$0.265883\pi$
$390$	0	0
$391$	2553.70	0.330297
$392$	−20740.1	−2.67228
$393$	−5596.26	−0.718305
$394$	−3300.79	−0.422060
$395$	0	0
$396$	−948.282	−0.120336
$397$	−93.1792	−0.0117797	−0.00588983	−	0.999983i	$-0.501875\pi$
$397$	−93.1792	−0.0117797	−0.00588983	+	0.999983i	$0.501875\pi$
$398$	1143.77	0.144050
$399$	3513.90	0.440890
$400$	0	0
$401$	−13320.9	−1.65889	−0.829443	−	0.558591i	$-0.811342\pi$
$401$	−13320.9	−1.65889	−0.829443	+	0.558591i	$0.811342\pi$
$402$	10578.1	1.31241
$403$	4493.82	0.555466
$404$	10304.5	1.26898
$405$	0	0
$406$	1741.63	0.212896
$407$	−713.255	−0.0868667
$408$	−1615.80	−0.196063
$409$	9272.21	1.12098	0.560491	−	0.828161i	$-0.310613\pi$
$409$	9272.21	1.12098	0.560491	+	0.828161i	$0.310613\pi$
$410$	0	0
$411$	3172.70	0.380774
$412$	13148.9	1.57233
$413$	1518.22	0.180888
$414$	6435.12	0.763935
$415$	0	0
$416$	−7906.05	−0.931793
$417$	−2714.50	−0.318776
$418$	−1268.97	−0.148487
$419$	11325.4	1.32049	0.660244	−	0.751051i	$-0.270453\pi$
$419$	11325.4	1.32049	0.660244	+	0.751051i	$0.270453\pi$
$420$	0	0
$421$	6934.54	0.802776	0.401388	−	0.915908i	$-0.368528\pi$
$421$	6934.54	0.802776	0.401388	+	0.915908i	$0.368528\pi$
$422$	−1276.18	−0.147212
$423$	−5139.12	−0.590715
$424$	14664.7	1.67967
$425$	0	0
$426$	−13286.2	−1.51108
$427$	1880.07	0.213074
$428$	27864.4	3.14691
$429$	−1818.84	−0.204695
$430$	0	0
$431$	−2776.55	−0.310306	−0.155153	−	0.987890i	$-0.549587\pi$
$431$	−2776.55	−0.310306	−0.155153	+	0.987890i	$0.549587\pi$
$432$	−905.950	−0.100897
$433$	3252.22	0.360951	0.180476	−	0.983579i	$-0.442236\pi$
$433$	3252.22	0.360951	0.180476	+	0.983579i	$0.442236\pi$
$434$	8011.15	0.886054
$435$	0	0
$436$	8567.32	0.941056
$437$	5570.65	0.609795
$438$	9958.82	1.08642
$439$	−13345.7	−1.45093	−0.725464	−	0.688260i	$-0.758375\pi$
$439$	−13345.7	−1.45093	−0.725464	+	0.688260i	$0.758375\pi$
$440$	0	0
$441$	5891.65	0.636178
$442$	−6824.02	−0.734356
$443$	−11639.6	−1.24834	−0.624169	−	0.781290i	$-0.714562\pi$
$443$	−11639.6	−1.24834	−0.624169	+	0.781290i	$0.714562\pi$
$444$	−4362.26	−0.466270
$445$	0	0
$446$	−26295.4	−2.79176
$447$	−2427.00	−0.256808
$448$	−22572.6	−2.38048
$449$	−6937.72	−0.729201	−0.364601	−	0.931164i	$-0.618794\pi$
$449$	−6937.72	−0.729201	−0.364601	+	0.931164i	$0.618794\pi$
$450$	0	0
$451$	−850.106	−0.0887582
$452$	929.531	0.0967289
$453$	1056.36	0.109563
$454$	−1829.69	−0.189144
$455$	0	0
$456$	−3524.70	−0.361972
$457$	10285.8	1.05284	0.526422	−	0.850224i	$-0.323533\pi$
$457$	10285.8	1.05284	0.526422	+	0.850224i	$0.323533\pi$
$458$	−6642.28	−0.677671
$459$	459.000	0.0466760
$460$	0	0
$461$	−625.833	−0.0632277	−0.0316138	−	0.999500i	$-0.510065\pi$
$461$	−625.833	−0.0632277	−0.0316138	+	0.999500i	$0.510065\pi$
$462$	−3242.45	−0.326520
$463$	6055.97	0.607872	0.303936	−	0.952692i	$-0.401699\pi$
$463$	6055.97	0.607872	0.303936	+	0.952692i	$0.401699\pi$
$464$	−388.705	−0.0388904
$465$	0	0
$466$	16228.2	1.61321
$467$	815.966	0.0808531	0.0404265	−	0.999183i	$-0.487128\pi$
$467$	815.966	0.0808531	0.0404265	+	0.999183i	$0.487128\pi$
$468$	−11124.0	−1.09873
$469$	23397.9	2.30366
$470$	0	0
$471$	1613.65	0.157862
$472$	−1522.88	−0.148509
$473$	−3278.35	−0.318686
$474$	14794.2	1.43359
$475$	0	0
$476$	−7869.61	−0.757779
$477$	−4165.81	−0.399872
$478$	7183.49	0.687375
$479$	−16219.1	−1.54712	−0.773559	−	0.633724i	$-0.781525\pi$
$479$	−16219.1	−1.54712	−0.773559	+	0.633724i	$0.781525\pi$
$480$	0	0
$481$	−8366.97	−0.793142
$482$	−16216.4	−1.53244
$483$	14234.0	1.34093
$484$	−18749.9	−1.76088
$485$	0	0
$486$	1156.64	0.107956
$487$	−2725.13	−0.253568	−0.126784	−	0.991930i	$-0.540465\pi$
$487$	−2725.13	−0.253568	−0.126784	+	0.991930i	$0.540465\pi$
$488$	−1885.84	−0.174935
$489$	5768.23	0.533433
$490$	0	0
$491$	−8344.13	−0.766935	−0.383468	−	0.923554i	$-0.625270\pi$
$491$	−8344.13	−0.766935	−0.383468	+	0.923554i	$0.625270\pi$
$492$	−5199.24	−0.476423
$493$	196.937	0.0179911
$494$	−14885.9	−1.35577
$495$	0	0
$496$	−1787.96	−0.161858
$497$	−29388.1	−2.65238
$498$	317.067	0.0285304
$499$	13762.0	1.23461	0.617306	−	0.786723i	$-0.288224\pi$
$499$	13762.0	1.23461	0.617306	+	0.786723i	$0.288224\pi$
$500$	0	0
$501$	−8915.29	−0.795022
$502$	−16157.0	−1.43650
$503$	11909.5	1.05570	0.527852	−	0.849336i	$-0.322997\pi$
$503$	11909.5	1.05570	0.527852	+	0.849336i	$0.322997\pi$
$504$	−9006.24	−0.795972
$505$	0	0
$506$	−5140.31	−0.451610
$507$	−14745.2	−1.29163
$508$	−2572.79	−0.224703
$509$	742.666	0.0646721	0.0323360	−	0.999477i	$-0.489705\pi$
$509$	742.666	0.0646721	0.0323360	+	0.999477i	$0.489705\pi$
$510$	0	0
$511$	22028.1	1.90698
$512$	11650.1	1.00560
$513$	1001.26	0.0861732
$514$	8467.35	0.726612
$515$	0	0
$516$	−20050.3	−1.71060
$517$	4105.07	0.349209
$518$	−14915.8	−1.26518
$519$	2965.69	0.250827
$520$	0	0
$521$	−4815.73	−0.404953	−0.202477	−	0.979287i	$-0.564899\pi$
$521$	−4815.73	−0.404953	−0.202477	+	0.979287i	$0.564899\pi$
$522$	496.266	0.0416111
$523$	16249.5	1.35858	0.679291	−	0.733869i	$-0.262287\pi$
$523$	16249.5	1.35858	0.679291	+	0.733869i	$0.262287\pi$
$524$	27339.9	2.27929
$525$	0	0
$526$	−20547.7	−1.70327
$527$	905.871	0.0748773
$528$	723.663	0.0596466
$529$	10398.4	0.854637
$530$	0	0
$531$	432.606	0.0353550
$532$	−17166.8	−1.39901
$533$	−9972.33	−0.810412
$534$	−5279.51	−0.427841
$535$	0	0
$536$	−23469.8	−1.89131
$537$	5812.96	0.467128
$538$	−31159.4	−2.49699
$539$	−4706.18	−0.376085
$540$	0	0
$541$	−5458.04	−0.433751	−0.216876	−	0.976199i	$-0.569587\pi$
$541$	−5458.04	−0.433751	−0.216876	+	0.976199i	$0.569587\pi$
$542$	−18017.0	−1.42785
$543$	6540.21	0.516883
$544$	−1593.71	−0.125606
$545$	0	0
$546$	−38036.2	−2.98131
$547$	−17237.0	−1.34735	−0.673677	−	0.739026i	$-0.735286\pi$
$547$	−17237.0	−1.34735	−0.673677	+	0.739026i	$0.735286\pi$
$548$	−15499.9	−1.20825
$549$	535.712	0.0416460
$550$	0	0
$551$	429.600	0.0332152
$552$	−14277.7	−1.10091
$553$	32723.7	2.51637
$554$	−16759.7	−1.28529
$555$	0	0
$556$	13261.4	1.01152
$557$	8452.71	0.643003	0.321502	−	0.946909i	$-0.395812\pi$
$557$	8452.71	0.643003	0.321502	+	0.946909i	$0.395812\pi$
$558$	2282.72	0.173182
$559$	−38457.3	−2.90978
$560$	0	0
$561$	−366.644	−0.0275931
$562$	13909.7	1.04403
$563$	8547.32	0.639834	0.319917	−	0.947446i	$-0.396345\pi$
$563$	8547.32	0.639834	0.319917	+	0.947446i	$0.396345\pi$
$564$	25106.6	1.87443
$565$	0	0
$566$	3498.85	0.259836
$567$	2558.40	0.189494
$568$	29478.4	2.17762
$569$	19464.8	1.43411	0.717055	−	0.697017i	$-0.245490\pi$
$569$	19464.8	1.43411	0.717055	+	0.697017i	$0.245490\pi$
$570$	0	0
$571$	−3839.06	−0.281366	−0.140683	−	0.990055i	$-0.544930\pi$
$571$	−3839.06	−0.281366	−0.140683	+	0.990055i	$0.544930\pi$
$572$	8885.72	0.649529
$573$	−5027.35	−0.366528
$574$	−17777.7	−1.29273
$575$	0	0
$576$	−6431.90	−0.465270
$577$	18797.7	1.35625	0.678127	−	0.734945i	$-0.262792\pi$
$577$	18797.7	1.35625	0.678127	+	0.734945i	$0.262792\pi$
$578$	−1375.60	−0.0989918
$579$	−773.882	−0.0555465
$580$	0	0
$581$	701.328	0.0500791
$582$	−16272.4	−1.15895
$583$	3327.60	0.236390
$584$	−22095.8	−1.56564
$585$	0	0
$586$	−41424.6	−2.92020
$587$	−4217.92	−0.296580	−0.148290	−	0.988944i	$-0.547377\pi$
$587$	−4217.92	−0.296580	−0.148290	+	0.988944i	$0.547377\pi$
$588$	−28783.0	−2.01869
$589$	1976.07	0.138238
$590$	0	0
$591$	−2080.40	−0.144799
$592$	3328.98	0.231115
$593$	3011.92	0.208575	0.104287	−	0.994547i	$-0.466744\pi$
$593$	3011.92	0.208575	0.104287	+	0.994547i	$0.466744\pi$
$594$	−923.914	−0.0638193
$595$	0	0
$596$	11856.8	0.814891
$597$	720.886	0.0494203
$598$	−60299.3	−4.12345
$599$	15137.1	1.03253	0.516266	−	0.856428i	$-0.327322\pi$
$599$	15137.1	1.03253	0.516266	+	0.856428i	$0.327322\pi$
$600$	0	0
$601$	−18980.1	−1.28821	−0.644104	−	0.764938i	$-0.722770\pi$
$601$	−18980.1	−1.28821	−0.644104	+	0.764938i	$0.722770\pi$
$602$	−68558.0	−4.64155
$603$	6667.08	0.450256
$604$	−5160.74	−0.347661
$605$	0	0
$606$	10039.7	0.672996
$607$	4593.63	0.307166	0.153583	−	0.988136i	$-0.450919\pi$
$607$	4593.63	0.307166	0.153583	+	0.988136i	$0.450919\pi$
$608$	−3476.53	−0.231894
$609$	1097.70	0.0730397
$610$	0	0
$611$	48155.3	3.18847
$612$	−2242.39	−0.148110
$613$	25654.3	1.69032	0.845160	−	0.534513i	$-0.179505\pi$
$613$	25654.3	1.69032	0.845160	+	0.534513i	$0.179505\pi$
$614$	11980.3	0.787437
$615$	0	0
$616$	7194.09	0.470549
$617$	7170.97	0.467897	0.233948	−	0.972249i	$-0.424835\pi$
$617$	7170.97	0.467897	0.233948	+	0.972249i	$0.424835\pi$
$618$	12811.0	0.833876
$619$	−13560.6	−0.880525	−0.440263	−	0.897869i	$-0.645115\pi$
$619$	−13560.6	−0.880525	−0.440263	+	0.897869i	$0.645115\pi$
$620$	0	0
$621$	4055.88	0.262088
$622$	−31383.2	−2.02307
$623$	−11677.9	−0.750985
$624$	8489.07	0.544607
$625$	0	0
$626$	20910.0	1.33503
$627$	−799.798	−0.0509424
$628$	−7883.28	−0.500919
$629$	−1686.63	−0.106916
$630$	0	0
$631$	−1414.98	−0.0892700	−0.0446350	−	0.999003i	$-0.514212\pi$
$631$	−1414.98	−0.0892700	−0.0446350	+	0.999003i	$0.514212\pi$
$632$	−32824.3	−2.06595
$633$	−804.342	−0.0505051
$634$	12381.4	0.775600
$635$	0	0
$636$	20351.6	1.26886
$637$	−55206.7	−3.43386
$638$	−396.412	−0.0245989
$639$	−8373.93	−0.518416
$640$	0	0
$641$	−21708.4	−1.33764	−0.668822	−	0.743422i	$-0.733201\pi$
$641$	−21708.4	−1.33764	−0.668822	+	0.743422i	$0.733201\pi$
$642$	27148.4	1.66895
$643$	−7537.23	−0.462270	−0.231135	−	0.972922i	$-0.574244\pi$
$643$	−7537.23	−0.462270	−0.231135	+	0.972922i	$0.574244\pi$
$644$	−69538.5	−4.25497
$645$	0	0
$646$	−3000.73	−0.182759
$647$	32667.3	1.98498	0.992491	−	0.122316i	$-0.0390320\pi$
$647$	32667.3	1.98498	0.992491	+	0.122316i	$0.0390320\pi$
$648$	−2566.27	−0.155575
$649$	−345.561	−0.0209006
$650$	0	0
$651$	5049.20	0.303984
$652$	−28180.1	−1.69266
$653$	25845.8	1.54889	0.774443	−	0.632644i	$-0.218030\pi$
$653$	25845.8	1.54889	0.774443	+	0.632644i	$0.218030\pi$
$654$	8347.17	0.499083
$655$	0	0
$656$	3967.70	0.236148
$657$	6276.77	0.372725
$658$	85846.7	5.08610
$659$	−15741.2	−0.930485	−0.465243	−	0.885183i	$-0.654033\pi$
$659$	−15741.2	−0.930485	−0.465243	+	0.885183i	$0.654033\pi$
$660$	0	0
$661$	23495.2	1.38254	0.691269	−	0.722598i	$-0.257052\pi$
$661$	23495.2	1.38254	0.691269	+	0.722598i	$0.257052\pi$
$662$	22230.8	1.30517
$663$	−4300.99	−0.251940
$664$	−703.483	−0.0411151
$665$	0	0
$666$	−4250.17	−0.247283
$667$	1740.21	0.101021
$668$	43554.7	2.52272
$669$	−16573.3	−0.957788
$670$	0	0
$671$	−427.921	−0.0246195
$672$	−8883.15	−0.509933
$673$	7057.34	0.404221	0.202110	−	0.979363i	$-0.435220\pi$
$673$	7057.34	0.404221	0.202110	+	0.979363i	$0.435220\pi$
$674$	8575.67	0.490092
$675$	0	0
$676$	72036.0	4.09855
$677$	−20756.4	−1.17833	−0.589167	−	0.808011i	$-0.700544\pi$
$677$	−20756.4	−1.17833	−0.589167	+	0.808011i	$0.700544\pi$
$678$	905.645	0.0512995
$679$	−35993.2	−2.03430
$680$	0	0
$681$	−1153.20	−0.0648909
$682$	−1823.41	−0.102378
$683$	7013.65	0.392928	0.196464	−	0.980511i	$-0.437054\pi$
$683$	7013.65	0.392928	0.196464	+	0.980511i	$0.437054\pi$
$684$	−4891.56	−0.273441
$685$	0	0
$686$	−46850.4	−2.60751
$687$	−4186.44	−0.232493
$688$	15301.0	0.847888
$689$	39035.0	2.15837
$690$	0	0
$691$	−4897.47	−0.269622	−0.134811	−	0.990871i	$-0.543043\pi$
$691$	−4897.47	−0.269622	−0.134811	+	0.990871i	$0.543043\pi$
$692$	−14488.5	−0.795913
$693$	−2043.63	−0.112022
$694$	801.273	0.0438269
$695$	0	0
$696$	−1101.08	−0.0599658
$697$	−2010.24	−0.109244
$698$	21171.1	1.14805
$699$	10228.2	0.553456
$700$	0	0
$701$	17525.5	0.944264	0.472132	−	0.881528i	$-0.343485\pi$
$701$	17525.5	0.944264	0.472132	+	0.881528i	$0.343485\pi$
$702$	−10838.1	−0.582706
$703$	−3679.21	−0.197388
$704$	5137.73	0.275050
$705$	0	0
$706$	−50023.4	−2.66665
$707$	22207.1	1.18130
$708$	−2113.45	−0.112187
$709$	32564.3	1.72493	0.862466	−	0.506115i	$-0.168919\pi$
$709$	32564.3	1.72493	0.862466	+	0.506115i	$0.168919\pi$
$710$	0	0
$711$	9324.40	0.491832
$712$	11713.8	0.616561
$713$	8004.58	0.420440
$714$	−7667.39	−0.401884
$715$	0	0
$716$	−28398.5	−1.48227
$717$	4527.55	0.235822
$718$	−39711.4	−2.06409
$719$	14498.2	0.752007	0.376004	−	0.926618i	$-0.377298\pi$
$719$	14498.2	0.752007	0.376004	+	0.926618i	$0.377298\pi$
$720$	0	0
$721$	28337.0	1.46370
$722$	26102.0	1.34545
$723$	−10220.7	−0.525744
$724$	−31951.5	−1.64015
$725$	0	0
$726$	−18268.1	−0.933872
$727$	25787.6	1.31556	0.657778	−	0.753212i	$-0.271497\pi$
$727$	25787.6	1.31556	0.657778	+	0.753212i	$0.271497\pi$
$728$	84391.6	4.29637
$729$	729.000	0.0370370
$730$	0	0
$731$	−7752.28	−0.392241
$732$	−2617.16	−0.132149
$733$	4177.45	0.210502	0.105251	−	0.994446i	$-0.466435\pi$
$733$	4177.45	0.210502	0.105251	+	0.994446i	$0.466435\pi$
$734$	1677.38	0.0843504
$735$	0	0
$736$	−14082.6	−0.705287
$737$	−5325.59	−0.266175
$738$	−5065.64	−0.252668
$739$	14115.5	0.702636	0.351318	−	0.936256i	$-0.385734\pi$
$739$	14115.5	0.702636	0.351318	+	0.936256i	$0.385734\pi$
$740$	0	0
$741$	−9382.18	−0.465132
$742$	69587.9	3.44293
$743$	−17992.0	−0.888376	−0.444188	−	0.895934i	$-0.646508\pi$
$743$	−17992.0	−0.888376	−0.444188	+	0.895934i	$0.646508\pi$
$744$	−5064.72	−0.249572
$745$	0	0
$746$	59799.0	2.93485
$747$	199.839	0.00978810
$748$	1791.20	0.0875571
$749$	60050.2	2.92949
$750$	0	0
$751$	2055.99	0.0998989	0.0499495	−	0.998752i	$-0.484094\pi$
$751$	2055.99	0.0998989	0.0499495	+	0.998752i	$0.484094\pi$
$752$	−19159.6	−0.929095
$753$	−10183.3	−0.492829
$754$	−4650.19	−0.224602
$755$	0	0
$756$	−12498.8	−0.601292
$757$	−12132.4	−0.582508	−0.291254	−	0.956646i	$-0.594073\pi$
$757$	−12132.4	−0.582508	−0.291254	+	0.956646i	$0.594073\pi$
$758$	8427.65	0.403834
$759$	−3239.79	−0.154937
$760$	0	0
$761$	9319.01	0.443908	0.221954	−	0.975057i	$-0.428757\pi$
$761$	9319.01	0.443908	0.221954	+	0.975057i	$0.428757\pi$
$762$	−2506.68	−0.119170
$763$	18463.3	0.876037
$764$	24560.5	1.16305
$765$	0	0
$766$	20612.9	0.972288
$767$	−4053.67	−0.190834
$768$	20712.9	0.973193
$769$	38790.8	1.81903	0.909514	−	0.415672i	$-0.136454\pi$
$769$	38790.8	1.81903	0.909514	+	0.415672i	$0.136454\pi$
$770$	0	0
$771$	5336.73	0.249284
$772$	3780.71	0.176257
$773$	−12857.4	−0.598252	−0.299126	−	0.954214i	$-0.596695\pi$
$773$	−12857.4	−0.598252	−0.299126	+	0.954214i	$0.596695\pi$
$774$	−19535.1	−0.907204
$775$	0	0
$776$	36103.8	1.67017
$777$	−9401.04	−0.434054
$778$	−49005.2	−2.25825
$779$	−4385.14	−0.201687
$780$	0	0
$781$	6689.00	0.306468
$782$	−12155.2	−0.555844
$783$	312.783	0.0142758
$784$	21965.2	1.00060
$785$	0	0
$786$	26637.3	1.20881
$787$	26862.0	1.21668	0.608339	−	0.793677i	$-0.291836\pi$
$787$	26862.0	1.21668	0.608339	+	0.793677i	$0.291836\pi$
$788$	10163.5	0.459468
$789$	−12950.6	−0.584353
$790$	0	0
$791$	2003.22	0.0900457
$792$	2049.91	0.0919700
$793$	−5019.81	−0.224790
$794$	443.519	0.0198235
$795$	0	0
$796$	−3521.81	−0.156818
$797$	12471.5	0.554285	0.277142	−	0.960829i	$-0.410613\pi$
$797$	12471.5	0.554285	0.277142	+	0.960829i	$0.410613\pi$
$798$	−16725.7	−0.741957
$799$	9707.23	0.429809
$800$	0	0
$801$	−3327.53	−0.146782
$802$	63405.4	2.79168
$803$	−5013.82	−0.220341
$804$	−32571.3	−1.42873
$805$	0	0
$806$	−21389.9	−0.934772
$807$	−19638.9	−0.856658
$808$	−22275.3	−0.969854
$809$	−24760.8	−1.07607	−0.538037	−	0.842921i	$-0.680834\pi$
$809$	−24760.8	−1.07607	−0.538037	+	0.842921i	$0.680834\pi$
$810$	0	0
$811$	11237.3	0.486556	0.243278	−	0.969957i	$-0.421777\pi$
$811$	11237.3	0.486556	0.243278	+	0.969957i	$0.421777\pi$
$812$	−5362.70	−0.231766
$813$	−11355.6	−0.489862
$814$	3394.99	0.146185
$815$	0	0
$816$	1711.24	0.0734134
$817$	−16910.8	−0.724156
$818$	−44134.3	−1.88645
$819$	−23973.1	−1.02282
$820$	0	0
$821$	39976.4	1.69937	0.849687	−	0.527287i	$-0.176791\pi$
$821$	39976.4	1.69937	0.849687	+	0.527287i	$0.176791\pi$
$822$	−15101.6	−0.640789
$823$	−36877.0	−1.56191	−0.780955	−	0.624588i	$-0.785267\pi$
$823$	−36877.0	−1.56191	−0.780955	+	0.624588i	$0.785267\pi$
$824$	−28424.1	−1.20170
$825$	0	0
$826$	−7226.49	−0.304409
$827$	14311.9	0.601781	0.300890	−	0.953659i	$-0.402716\pi$
$827$	14311.9	0.601781	0.300890	+	0.953659i	$0.402716\pi$
$828$	−19814.5	−0.831646
$829$	−12629.7	−0.529130	−0.264565	−	0.964368i	$-0.585228\pi$
$829$	−12629.7	−0.529130	−0.264565	+	0.964368i	$0.585228\pi$
$830$	0	0
$831$	−10563.2	−0.440954
$832$	60269.1	2.51136
$833$	−11128.7	−0.462888
$834$	12920.6	0.536455
$835$	0	0
$836$	3907.32	0.161648
$837$	1438.74	0.0594146
$838$	−53907.4	−2.22220
$839$	−18683.1	−0.768786	−0.384393	−	0.923170i	$-0.625589\pi$
$839$	−18683.1	−0.768786	−0.384393	+	0.923170i	$0.625589\pi$
$840$	0	0
$841$	−24254.8	−0.994497
$842$	−33007.3	−1.35096
$843$	8766.90	0.358183
$844$	3929.52	0.160260
$845$	0	0
$846$	24461.4	0.994092
$847$	−40407.5	−1.63922
$848$	−15530.9	−0.628932
$849$	2205.22	0.0891438
$850$	0	0
$851$	−14903.6	−0.600340
$852$	40909.9	1.64501
$853$	35648.7	1.43094	0.715468	−	0.698646i	$-0.246214\pi$
$853$	35648.7	1.43094	0.715468	+	0.698646i	$0.246214\pi$
$854$	−8948.83	−0.358575
$855$	0	0
$856$	−60234.7	−2.40512
$857$	49860.8	1.98741	0.993707	−	0.112010i	$-0.0357290\pi$
$857$	49860.8	1.98741	0.993707	+	0.112010i	$0.0357290\pi$
$858$	8657.39	0.344474
$859$	21487.0	0.853466	0.426733	−	0.904378i	$-0.359664\pi$
$859$	21487.0	0.853466	0.426733	+	0.904378i	$0.359664\pi$
$860$	0	0
$861$	−11204.8	−0.443506
$862$	13216.0	0.522201
$863$	15067.6	0.594330	0.297165	−	0.954826i	$-0.403959\pi$
$863$	15067.6	0.594330	0.297165	+	0.954826i	$0.403959\pi$
$864$	−2531.19	−0.0996677
$865$	0	0
$866$	−15480.1	−0.607431
$867$	−867.000	−0.0339618
$868$	−24667.3	−0.964588
$869$	−7448.23	−0.290752
$870$	0	0
$871$	−62472.8	−2.43032
$872$	−18520.0	−0.719229
$873$	−10256.0	−0.397610
$874$	−26515.4	−1.02620
$875$	0	0
$876$	−30664.4	−1.18271
$877$	24852.6	0.956912	0.478456	−	0.878111i	$-0.341197\pi$
$877$	24852.6	0.956912	0.478456	+	0.878111i	$0.341197\pi$
$878$	63523.7	2.44171
$879$	−26108.8	−1.00185
$880$	0	0
$881$	1840.51	0.0703842	0.0351921	−	0.999381i	$-0.488796\pi$
$881$	1840.51	0.0703842	0.0351921	+	0.999381i	$0.488796\pi$
$882$	−28043.3	−1.07060
$883$	−49803.7	−1.89811	−0.949054	−	0.315115i	$-0.897957\pi$
$883$	−49803.7	−1.89811	−0.949054	+	0.315115i	$0.897957\pi$
$884$	21012.0	0.799445
$885$	0	0
$886$	55402.7	2.10078
$887$	−36314.7	−1.37467	−0.687333	−	0.726342i	$-0.741219\pi$
$887$	−36314.7	−1.37467	−0.687333	+	0.726342i	$0.741219\pi$
$888$	9429.93	0.356360
$889$	−5544.58	−0.209178
$890$	0	0
$891$	−582.317	−0.0218949
$892$	80966.9	3.03921
$893$	21175.4	0.793512
$894$	11552.2	0.432172
$895$	0	0
$896$	83753.6	3.12278
$897$	−38005.0	−1.41466
$898$	33022.5	1.22714
$899$	617.300	0.0229011
$900$	0	0
$901$	7868.75	0.290950
$902$	4046.38	0.149368
$903$	−43210.2	−1.59241
$904$	−2009.37	−0.0739278
$905$	0	0
$906$	−5028.12	−0.184380
$907$	−33679.5	−1.23297	−0.616487	−	0.787365i	$-0.711445\pi$
$907$	−33679.5	−1.23297	−0.616487	+	0.787365i	$0.711445\pi$
$908$	5633.83	0.205909
$909$	6327.75	0.230889
$910$	0	0
$911$	32437.2	1.17968	0.589841	−	0.807519i	$-0.299190\pi$
$911$	32437.2	1.17968	0.589841	+	0.807519i	$0.299190\pi$
$912$	3732.90	0.135536
$913$	−159.629	−0.00578636
$914$	−48958.8	−1.77179
$915$	0	0
$916$	20452.4	0.737736
$917$	58919.7	2.12181
$918$	−2184.77	−0.0785492
$919$	−3511.45	−0.126042	−0.0630208	−	0.998012i	$-0.520073\pi$
$919$	−3511.45	−0.126042	−0.0630208	+	0.998012i	$0.520073\pi$
$920$	0	0
$921$	7550.86	0.270151
$922$	2978.87	0.106403
$923$	78466.6	2.79822
$924$	9983.90	0.355461
$925$	0	0
$926$	−28825.5	−1.02296
$927$	8074.44	0.286083
$928$	−1086.03	−0.0384166
$929$	13527.2	0.477733	0.238866	−	0.971052i	$-0.423224\pi$
$929$	13527.2	0.477733	0.238866	+	0.971052i	$0.423224\pi$
$930$	0	0
$931$	−24276.1	−0.854583
$932$	−49968.6	−1.75620
$933$	−19779.9	−0.694069
$934$	−3883.87	−0.136064
$935$	0	0
$936$	24046.8	0.839737
$937$	8862.80	0.309002	0.154501	−	0.987993i	$-0.450623\pi$
$937$	8862.80	0.309002	0.154501	+	0.987993i	$0.450623\pi$
$938$	−111371.	−3.87674
$939$	13179.0	0.458019
$940$	0	0
$941$	22824.0	0.790693	0.395346	−	0.918532i	$-0.370625\pi$
$941$	22824.0	0.790693	0.395346	+	0.918532i	$0.370625\pi$
$942$	−7680.71	−0.265659
$943$	−17763.1	−0.613412
$944$	1612.84	0.0556074
$945$	0	0
$946$	15604.4	0.536305
$947$	26308.4	0.902754	0.451377	−	0.892333i	$-0.350933\pi$
$947$	26308.4	0.902754	0.451377	+	0.892333i	$0.350933\pi$
$948$	−45553.3	−1.56066
$949$	−58815.5	−2.01184
$950$	0	0
$951$	7803.68	0.266090
$952$	17011.8	0.579155
$953$	−11947.5	−0.406106	−0.203053	−	0.979168i	$-0.565086\pi$
$953$	−11947.5	−0.406106	−0.203053	+	0.979168i	$0.565086\pi$
$954$	19828.6	0.672930
$955$	0	0
$956$	−22118.9	−0.748300
$957$	−249.848	−0.00843932
$958$	77200.5	2.60358
$959$	−33403.5	−1.12477
$960$	0	0
$961$	−26951.5	−0.904688
$962$	39825.5	1.33475
$963$	17110.9	0.572576
$964$	49932.2	1.66826
$965$	0	0
$966$	−67751.6	−2.25660
$967$	−21505.3	−0.715163	−0.357581	−	0.933882i	$-0.616399\pi$
$967$	−21505.3	−0.715163	−0.357581	+	0.933882i	$0.616399\pi$
$968$	40531.7	1.34580
$969$	−1891.27	−0.0627002
$970$	0	0
$971$	45685.3	1.50990	0.754950	−	0.655783i	$-0.227661\pi$
$971$	45685.3	1.50990	0.754950	+	0.655783i	$0.227661\pi$
$972$	−3561.45	−0.117524
$973$	28579.4	0.941636
$974$	12971.2	0.426719
$975$	0	0
$976$	1997.24	0.0655020
$977$	−31710.3	−1.03839	−0.519193	−	0.854657i	$-0.673767\pi$
$977$	−31710.3	−1.03839	−0.519193	+	0.854657i	$0.673767\pi$
$978$	−27455.9	−0.897693
$979$	2657.99	0.0867721
$980$	0	0
$981$	5260.99	0.171224
$982$	39716.8	1.29065
$983$	−46038.9	−1.49381	−0.746904	−	0.664932i	$-0.768460\pi$
$983$	−46038.9	−1.49381	−0.746904	+	0.664932i	$0.768460\pi$
$984$	11239.2	0.364120
$985$	0	0
$986$	−937.392	−0.0302765
$987$	54106.8	1.74492
$988$	45835.6	1.47593
$989$	−68501.8	−2.20246
$990$	0	0
$991$	14394.9	0.461420	0.230710	−	0.973023i	$-0.425895\pi$
$991$	14394.9	0.461420	0.230710	+	0.973023i	$0.425895\pi$
$992$	−4995.50	−0.159886
$993$	14011.5	0.447775
$994$	139883.	4.46359
$995$	0	0
$996$	−976.289	−0.0310591
$997$	33473.4	1.06330	0.531652	−	0.846963i	$-0.321572\pi$
$997$	33473.4	1.06330	0.531652	+	0.846963i	$0.321572\pi$
$998$	−65505.0	−2.07768
$999$	−2678.76	−0.0848371

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1275.4.a.q.1.1		3
5.4	even	2		51.4.a.e.1.3	✓	3
15.14	odd	2		153.4.a.f.1.1		3
20.19	odd	2		816.4.a.s.1.1		3
35.34	odd	2		2499.4.a.n.1.3		3
60.59	even	2		2448.4.a.bd.1.3		3
85.84	even	2		867.4.a.k.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.4.a.e.1.3	✓	3	5.4	even	2
153.4.a.f.1.1		3	15.14	odd	2
816.4.a.s.1.1		3	20.19	odd	2
867.4.a.k.1.3		3	85.84	even	2
1275.4.a.q.1.1		3	1.1	even	1	trivial
2448.4.a.bd.1.3		3	60.59	even	2
2499.4.a.n.1.3		3	35.34	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1275 = 3 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1275.a (trivial)

\(f(q)\)	\(=\)	\(q-4.75985 q^{2} -3.00000 q^{3} +14.6562 q^{4} +14.2795 q^{6} +31.5852 q^{7} -31.6823 q^{8} +9.00000 q^{9} -7.18910 q^{11} -43.9685 q^{12} -84.3331 q^{13} -150.341 q^{14} +33.5537 q^{16} -17.0000 q^{17} -42.8386 q^{18} -37.0838 q^{19} -94.7557 q^{21} +34.2190 q^{22} -150.218 q^{23} +95.0469 q^{24} +401.413 q^{26} -27.0000 q^{27} +462.918 q^{28} -11.5846 q^{29} -53.2865 q^{31} +93.7478 q^{32} +21.5673 q^{33} +80.9174 q^{34} +131.905 q^{36} +99.2134 q^{37} +176.513 q^{38} +252.999 q^{39} +118.249 q^{41} +451.023 q^{42} +456.016 q^{43} -105.365 q^{44} +715.014 q^{46} -571.014 q^{47} -100.661 q^{48} +654.627 q^{49} +51.0000 q^{51} -1236.00 q^{52} -462.867 q^{53} +128.516 q^{54} -1000.69 q^{56} +111.251 q^{57} +55.1407 q^{58} +48.0674 q^{59} +59.5236 q^{61} +253.636 q^{62} +284.267 q^{63} -714.655 q^{64} -102.657 q^{66} +740.787 q^{67} -249.155 q^{68} +450.653 q^{69} -930.437 q^{71} -285.141 q^{72} +697.419 q^{73} -472.241 q^{74} -543.506 q^{76} -227.070 q^{77} -1204.24 q^{78} +1036.04 q^{79} +81.0000 q^{81} -562.849 q^{82} +22.2043 q^{83} -1388.75 q^{84} -2170.57 q^{86} +34.7537 q^{87} +227.767 q^{88} -369.726 q^{89} -2663.68 q^{91} -2201.62 q^{92} +159.860 q^{93} +2717.94 q^{94} -281.244 q^{96} -1139.56 q^{97} -3115.93 q^{98} -64.7019 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 5 q^{2} - 9 q^{3} + 13 q^{4} + 15 q^{6} + 8 q^{7} - 33 q^{8} + 27 q^{9} + 34 q^{11} - 39 q^{12} - 36 q^{13} - 104 q^{14} - 79 q^{16} - 51 q^{17} - 45 q^{18} - 142 q^{19} - 24 q^{21} + 248 q^{22} - 110 q^{23}+ \cdots + 306 q^{99}+O(q^{100}) \) 3 * q - 5 * q^2 - 9 * q^3 + 13 * q^4 + 15 * q^6 + 8 * q^7 - 33 * q^8 + 27 * q^9 + 34 * q^11 - 39 * q^12 - 36 * q^13 - 104 * q^14 - 79 * q^16 - 51 * q^17 - 45 * q^18 - 142 * q^19 - 24 * q^21 + 248 * q^22 - 110 * q^23 + 99 * q^24 + 154 * q^26 - 81 * q^27 + 472 * q^28 + 90 * q^29 - 148 * q^31 + 151 * q^32 - 102 * q^33 + 85 * q^34 + 117 * q^36 - 110 * q^37 - 80 * q^38 + 108 * q^39 + 720 * q^41 + 312 * q^42 + 146 * q^43 - 192 * q^44 + 748 * q^46 - 500 * q^47 + 237 * q^48 + 379 * q^49 + 153 * q^51 - 1218 * q^52 - 610 * q^53 + 135 * q^54 - 1368 * q^56 + 426 * q^57 - 1006 * q^58 - 216 * q^59 - 18 * q^61 + 904 * q^62 + 72 * q^63 + 377 * q^64 - 744 * q^66 + 1404 * q^67 - 221 * q^68 + 330 * q^69 - 960 * q^71 - 297 * q^72 + 794 * q^73 - 1874 * q^74 - 392 * q^76 - 48 * q^77 - 462 * q^78 - 276 * q^79 + 243 * q^81 - 382 * q^82 + 1552 * q^83 - 1416 * q^84 + 16 * q^86 - 270 * q^87 - 1724 * q^88 + 1394 * q^89 - 3968 * q^91 - 2244 * q^92 + 444 * q^93 + 3960 * q^94 - 453 * q^96 - 402 * q^97 - 4109 * q^98 + 306 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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