LMFDB - Embedded newform 816.4.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [816,4,Mod(1,816)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(816, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("816.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 816 = 2^{4} \cdot 3 \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	816.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:	$48.1455585647$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
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	$\chi$	$=$	816.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -7.65616 q^{5} +31.5852 q^{7} +9.00000 q^{9} +7.18910 q^{11} +84.3331 q^{13} +22.9685 q^{15} +17.0000 q^{17} +37.0838 q^{19} -94.7557 q^{21} -150.218 q^{23} -66.3832 q^{25} -27.0000 q^{27} -11.5846 q^{29} +53.2865 q^{31} -21.5673 q^{33} -241.822 q^{35} -99.2134 q^{37} -252.999 q^{39} +118.249 q^{41} +456.016 q^{43} -68.9054 q^{45} -571.014 q^{47} +654.627 q^{49} -51.0000 q^{51} +462.867 q^{53} -55.0409 q^{55} -111.251 q^{57} -48.0674 q^{59} +59.5236 q^{61} +284.267 q^{63} -645.668 q^{65} +740.787 q^{67} +450.653 q^{69} +930.437 q^{71} -697.419 q^{73} +199.150 q^{75} +227.070 q^{77} -1036.04 q^{79} +81.0000 q^{81} +22.2043 q^{83} -130.155 q^{85} +34.7537 q^{87} -369.726 q^{89} +2663.68 q^{91} -159.860 q^{93} -283.920 q^{95} +1139.56 q^{97} +64.7019 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 9 q^{3} + 8 q^{5} + 8 q^{7} + 27 q^{9} - 34 q^{11} + 36 q^{13} - 24 q^{15} + 51 q^{17} + 142 q^{19} - 24 q^{21} - 110 q^{23} - 193 q^{25} - 81 q^{27} + 90 q^{29} + 148 q^{31} + 102 q^{33} - 416 q^{35}+ \cdots - 306 q^{99}+O(q^{100}) $ 3 * q - 9 * q^3 + 8 * q^5 + 8 * q^7 + 27 * q^9 - 34 * q^11 + 36 * q^13 - 24 * q^15 + 51 * q^17 + 142 * q^19 - 24 * q^21 - 110 * q^23 - 193 * q^25 - 81 * q^27 + 90 * q^29 + 148 * q^31 + 102 * q^33 - 416 * q^35 + 110 * q^37 - 108 * q^39 + 720 * q^41 + 146 * q^43 + 72 * q^45 - 500 * q^47 + 379 * q^49 - 153 * q^51 + 610 * q^53 - 430 * q^55 - 426 * q^57 + 216 * q^59 - 18 * q^61 + 72 * q^63 - 966 * q^65 + 1404 * q^67 + 330 * q^69 + 960 * q^71 - 794 * q^73 + 579 * q^75 + 48 * q^77 + 276 * q^79 + 243 * q^81 + 1552 * q^83 + 136 * q^85 - 270 * q^87 + 1394 * q^89 + 3968 * q^91 - 444 * q^93 + 602 * q^95 + 402 * q^97 - 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 $

$2$	0	0
$2$	0	0
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	−7.65616	−0.684788	−0.342394	−	0.939557i	$-0.611238\pi$
$5$	−7.65616	−0.684788	−0.342394	+	0.939557i	$0.611238\pi$
$6$	0	0
$7$	31.5852	1.70544	0.852721	−	0.522366i	$-0.174951\pi$
$7$	31.5852	1.70544	0.852721	+	0.522366i	$0.174951\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	7.18910	0.197054	0.0985271	−	0.995134i	$-0.468587\pi$
$11$	7.18910	0.197054	0.0985271	+	0.995134i	$0.468587\pi$
$12$	0	0
$13$	84.3331	1.79921	0.899607	−	0.436700i	$-0.143853\pi$
$13$	84.3331	1.79921	0.899607	+	0.436700i	$0.143853\pi$
$14$	0	0
$15$	22.9685	0.395362
$16$	0	0
$17$	17.0000	0.242536
$17$	17.0000	0.242536
$18$	0	0
$19$	37.0838	0.447769	0.223885	−	0.974616i	$-0.428126\pi$
$19$	37.0838	0.447769	0.223885	+	0.974616i	$0.428126\pi$
$20$	0	0
$21$	−94.7557	−0.984638
$22$	0	0
$23$	−150.218	−1.36185	−0.680925	−	0.732353i	$-0.738422\pi$
$23$	−150.218	−1.36185	−0.680925	+	0.732353i	$0.738422\pi$
$24$	0	0
$25$	−66.3832	−0.531066
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−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	$-0.450667\pi$
$31$	53.2865	0.308727	0.154364	+	0.988014i	$0.450667\pi$
$32$	0	0
$33$	−21.5673	−0.113769
$34$	0	0
$35$	−241.822	−1.16787
$36$	0	0
$37$	−99.2134	−0.440827	−0.220413	−	0.975407i	$-0.570741\pi$
$37$	−99.2134	−0.440827	−0.220413	+	0.975407i	$0.570741\pi$
$38$	0	0
$39$	−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	$0.427692\pi$
$42$	0	0
$43$	456.016	1.61725	0.808626	−	0.588323i	$-0.200211\pi$
$43$	456.016	1.61725	0.808626	+	0.588323i	$0.200211\pi$
$44$	0	0
$45$	−68.9054	−0.228263
$46$	0	0
$47$	−571.014	−1.77215	−0.886073	−	0.463545i	$-0.846577\pi$
$47$	−571.014	−1.77215	−0.886073	+	0.463545i	$0.846577\pi$
$48$	0	0
$49$	654.627	1.90853
$50$	0	0
$51$	−51.0000	−0.140028
$52$	0	0
$53$	462.867	1.19962	0.599809	−	0.800143i	$-0.295243\pi$
$53$	462.867	1.19962	0.599809	+	0.800143i	$0.295243\pi$
$54$	0	0
$55$	−55.0409	−0.134940
$56$	0	0
$57$	−111.251	−0.258520
$58$	0	0
$59$	−48.0674	−0.106065	−0.0530325	−	0.998593i	$-0.516889\pi$
$59$	−48.0674	−0.106065	−0.0530325	+	0.998593i	$0.516889\pi$
$60$	0	0
$61$	59.5236	0.124938	0.0624689	−	0.998047i	$-0.480103\pi$
$61$	59.5236	0.124938	0.0624689	+	0.998047i	$0.480103\pi$
$62$	0	0
$63$	284.267	0.568481
$64$	0	0
$65$	−645.668	−1.23208
$66$	0	0
$67$	740.787	1.35077	0.675384	−	0.737466i	$-0.263978\pi$
$67$	740.787	1.35077	0.675384	+	0.737466i	$0.263978\pi$
$68$	0	0
$69$	450.653	0.786265
$70$	0	0
$71$	930.437	1.55525	0.777623	−	0.628730i	$-0.216425\pi$
$71$	930.437	1.55525	0.777623	+	0.628730i	$0.216425\pi$
$72$	0	0
$73$	−697.419	−1.11817	−0.559087	−	0.829109i	$-0.688848\pi$
$73$	−697.419	−1.11817	−0.559087	+	0.829109i	$0.688848\pi$
$74$	0	0
$75$	199.150	0.306611
$76$	0	0
$77$	227.070	0.336065
$78$	0	0
$79$	−1036.04	−1.47549	−0.737747	−	0.675077i	$-0.764110\pi$
$79$	−1036.04	−1.47549	−0.737747	+	0.675077i	$0.764110\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	22.2043	0.0293643	0.0146822	−	0.999892i	$-0.495326\pi$
$83$	22.2043	0.0293643	0.0146822	+	0.999892i	$0.495326\pi$
$84$	0	0
$85$	−130.155	−0.166085
$86$	0	0
$87$	34.7537	0.0428274
$88$	0	0
$89$	−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
$92$	0	0
$93$	−159.860	−0.178244
$94$	0	0
$95$	−283.920	−0.306627
$96$	0	0
$97$	1139.56	1.19283	0.596415	−	0.802676i	$-0.296591\pi$
$97$	1139.56	1.19283	0.596415	+	0.802676i	$0.296591\pi$
$98$	0	0
$99$	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	$0.387427\pi$
$102$	0	0
$103$	897.160	0.858250	0.429125	−	0.903245i	$-0.358822\pi$
$103$	897.160	0.858250	0.429125	+	0.903245i	$0.358822\pi$
$104$	0	0
$105$	725.465	0.674268
$106$	0	0
$107$	1901.21	1.71773	0.858864	−	0.512203i	$-0.171171\pi$
$107$	1901.21	1.71773	0.858864	+	0.512203i	$0.171171\pi$
$108$	0	0
$109$	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
$112$	0	0
$113$	−63.4225	−0.0527990	−0.0263995	−	0.999651i	$-0.508404\pi$
$113$	−63.4225	−0.0527990	−0.0263995	+	0.999651i	$0.508404\pi$
$114$	0	0
$115$	1150.09	0.932578
$116$	0	0
$117$	758.998	0.599738
$118$	0	0
$119$	536.949	0.413631
$120$	0	0
$121$	−1279.32	−0.961170
$122$	0	0
$123$	−354.748	−0.260053
$124$	0	0
$125$	1465.26	1.04845
$126$	0	0
$127$	−175.543	−0.122653	−0.0613266	−	0.998118i	$-0.519533\pi$
$127$	−175.543	−0.122653	−0.0613266	+	0.998118i	$0.519533\pi$
$128$	0	0
$129$	−1368.05	−0.933721
$130$	0	0
$131$	−1865.42	−1.24414	−0.622070	−	0.782961i	$-0.713708\pi$
$131$	−1865.42	−1.24414	−0.622070	+	0.782961i	$0.713708\pi$
$132$	0	0
$133$	1171.30	0.763644
$134$	0	0
$135$	206.716	0.131787
$136$	0	0
$137$	1057.57	0.659519	0.329760	−	0.944065i	$-0.393032\pi$
$137$	1057.57	0.659519	0.329760	+	0.944065i	$0.393032\pi$
$138$	0	0
$139$	−904.833	−0.552136	−0.276068	−	0.961138i	$-0.589032\pi$
$139$	−904.833	−0.552136	−0.276068	+	0.961138i	$0.589032\pi$
$140$	0	0
$141$	1713.04	1.02315
$142$	0	0
$143$	606.279	0.354543
$144$	0	0
$145$	88.6932	0.0507970
$146$	0	0
$147$	−1963.88	−1.10189
$148$	0	0
$149$	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	$-0.469752\pi$
$151$	352.121	0.189769	0.0948847	+	0.995488i	$0.469752\pi$
$152$	0	0
$153$	153.000	0.0808452
$154$	0	0
$155$	−407.970	−0.211413
$156$	0	0
$157$	537.882	0.273424	0.136712	−	0.990611i	$-0.456346\pi$
$157$	537.882	0.273424	0.136712	+	0.990611i	$0.456346\pi$
$158$	0	0
$159$	−1388.60	−0.692599
$160$	0	0
$161$	−4744.66	−2.32256
$162$	0	0
$163$	−1922.74	−0.923933	−0.461966	−	0.886897i	$-0.652856\pi$
$163$	−1922.74	−0.923933	−0.461966	+	0.886897i	$0.652856\pi$
$164$	0	0
$165$	165.123	0.0779078
$166$	0	0
$167$	2971.76	1.37702	0.688509	−	0.725228i	$-0.258266\pi$
$167$	2971.76	1.37702	0.688509	+	0.725228i	$0.258266\pi$
$168$	0	0
$169$	4915.07	2.23717
$170$	0	0
$171$	333.754	0.149256
$172$	0	0
$173$	988.564	0.434446	0.217223	−	0.976122i	$-0.430300\pi$
$173$	988.564	0.434446	0.217223	+	0.976122i	$0.430300\pi$
$174$	0	0
$175$	−2096.73	−0.905702
$176$	0	0
$177$	144.202	0.0612367
$178$	0	0
$179$	1937.65	0.809089	0.404545	−	0.914518i	$-0.367430\pi$
$179$	1937.65	0.809089	0.404545	+	0.914518i	$0.367430\pi$
$180$	0	0
$181$	−2180.07	−0.895267	−0.447634	−	0.894217i	$-0.647733\pi$
$181$	−2180.07	−0.895267	−0.447634	+	0.894217i	$0.647733\pi$
$182$	0	0
$183$	−178.571	−0.0721329
$184$	0	0
$185$	759.594	0.301873
$186$	0	0
$187$	122.215	0.0477927
$188$	0	0
$189$	−852.801	−0.328213
$190$	0	0
$191$	−1675.78	−0.634845	−0.317423	−	0.948284i	$-0.602817\pi$
$191$	−1675.78	−0.634845	−0.317423	+	0.948284i	$0.602817\pi$
$192$	0	0
$193$	−257.961	−0.0962094	−0.0481047	−	0.998842i	$-0.515318\pi$
$193$	−257.961	−0.0962094	−0.0481047	+	0.998842i	$0.515318\pi$
$194$	0	0
$195$	1937.00	0.711342
$196$	0	0
$197$	−693.466	−0.250799	−0.125399	−	0.992106i	$-0.540021\pi$
$197$	−693.466	−0.250799	−0.125399	+	0.992106i	$0.540021\pi$
$198$	0	0
$199$	240.295	0.0855984	0.0427992	−	0.999084i	$-0.486372\pi$
$199$	240.295	0.0855984	0.0427992	+	0.999084i	$0.486372\pi$
$200$	0	0
$201$	−2222.36	−0.779866
$202$	0	0
$203$	−365.901	−0.126508
$204$	0	0
$205$	−905.335	−0.308446
$206$	0	0
$207$	−1351.96	−0.453950
$208$	0	0
$209$	266.599	0.0882348
$210$	0	0
$211$	−268.114	−0.0874774	−0.0437387	−	0.999043i	$-0.513927\pi$
$211$	−268.114	−0.0874774	−0.0437387	+	0.999043i	$0.513927\pi$
$212$	0	0
$213$	−2791.31	−0.897922
$214$	0	0
$215$	−3491.33	−1.10747
$216$	0	0
$217$	1683.07	0.526516
$218$	0	0
$219$	2092.26	0.645578
$220$	0	0
$221$	1433.66	0.436374
$222$	0	0
$223$	5524.43	1.65894	0.829468	−	0.558554i	$-0.188643\pi$
$223$	5524.43	1.65894	0.829468	+	0.558554i	$0.188643\pi$
$224$	0	0
$225$	−597.449	−0.177022
$226$	0	0
$227$	384.400	0.112394	0.0561972	−	0.998420i	$-0.482102\pi$
$227$	384.400	0.112394	0.0561972	+	0.998420i	$0.482102\pi$
$228$	0	0
$229$	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
$232$	0	0
$233$	3409.39	0.958613	0.479307	−	0.877648i	$-0.340888\pi$
$233$	3409.39	0.958613	0.479307	+	0.877648i	$0.340888\pi$
$234$	0	0
$235$	4371.77	1.21354
$236$	0	0
$237$	3108.13	0.851877
$238$	0	0
$239$	1509.18	0.408456	0.204228	−	0.978923i	$-0.434532\pi$
$239$	1509.18	0.408456	0.204228	+	0.978923i	$0.434532\pi$
$240$	0	0
$241$	3406.91	0.910615	0.455307	−	0.890334i	$-0.349529\pi$
$241$	3406.91	0.910615	0.455307	+	0.890334i	$0.349529\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−5011.93	−1.30694
$246$	0	0
$247$	3127.39	0.805633
$248$	0	0
$249$	−66.6129	−0.0169535
$250$	0	0
$251$	−3394.43	−0.853605	−0.426802	−	0.904345i	$-0.640360\pi$
$251$	−3394.43	−0.853605	−0.426802	+	0.904345i	$0.640360\pi$
$252$	0	0
$253$	−1079.93	−0.268358
$254$	0	0
$255$	390.464	0.0958894
$256$	0	0
$257$	1778.91	0.431772	0.215886	−	0.976419i	$-0.430736\pi$
$257$	1778.91	0.431772	0.215886	+	0.976419i	$0.430736\pi$
$258$	0	0
$259$	−3133.68	−0.751804
$260$	0	0
$261$	−104.261	−0.0247264
$262$	0	0
$263$	4316.88	1.01213	0.506065	−	0.862495i	$-0.331100\pi$
$263$	4316.88	1.01213	0.506065	+	0.862495i	$0.331100\pi$
$264$	0	0
$265$	−3543.79	−0.821483
$266$	0	0
$267$	1109.18	0.254234
$268$	0	0
$269$	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	$-0.639456\pi$
$271$	−3785.20	−0.848466	−0.424233	+	0.905553i	$0.639456\pi$
$272$	0	0
$273$	−7991.04	−1.77157
$274$	0	0
$275$	−477.236	−0.104649
$276$	0	0
$277$	−3521.06	−0.763755	−0.381878	−	0.924213i	$-0.624722\pi$
$277$	−3521.06	−0.763755	−0.381878	+	0.924213i	$0.624722\pi$
$278$	0	0
$279$	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	$0.600394\pi$
$282$	0	0
$283$	−735.075	−0.154402	−0.0772008	−	0.997016i	$-0.524598\pi$
$283$	−735.075	−0.154402	−0.0772008	+	0.997016i	$0.524598\pi$
$284$	0	0
$285$	851.759	0.177031
$286$	0	0
$287$	3734.93	0.768175
$288$	0	0
$289$	289.000	0.0588235
$290$	0	0
$291$	−3418.67	−0.688681
$292$	0	0
$293$	−8702.92	−1.73526	−0.867628	−	0.497213i	$-0.834357\pi$
$293$	−8702.92	−1.73526	−0.867628	+	0.497213i	$0.834357\pi$
$294$	0	0
$295$	368.011	0.0726320
$296$	0	0
$297$	−194.106	−0.0379231
$298$	0	0
$299$	−12668.3	−2.45026
$300$	0	0
$301$	14403.4	2.75813
$302$	0	0
$303$	−2109.25	−0.399912
$304$	0	0
$305$	−455.722	−0.0855559
$306$	0	0
$307$	−2516.95	−0.467916	−0.233958	−	0.972247i	$-0.575168\pi$
$307$	−2516.95	−0.467916	−0.233958	+	0.972247i	$0.575168\pi$
$308$	0	0
$309$	−2691.48	−0.495511
$310$	0	0
$311$	−6593.31	−1.20216	−0.601081	−	0.799188i	$-0.705263\pi$
$311$	−6593.31	−1.20216	−0.601081	+	0.799188i	$0.705263\pi$
$312$	0	0
$313$	4392.99	0.793312	0.396656	−	0.917967i	$-0.370171\pi$
$313$	4392.99	0.793312	0.396656	+	0.917967i	$0.370171\pi$
$314$	0	0
$315$	−2176.39	−0.389289
$316$	0	0
$317$	2601.23	0.460882	0.230441	−	0.973086i	$-0.425983\pi$
$317$	2601.23	0.460882	0.230441	+	0.973086i	$0.425983\pi$
$318$	0	0
$319$	−83.2825	−0.0146173
$320$	0	0
$321$	−5703.63	−0.991731
$322$	0	0
$323$	630.425	0.108600
$324$	0	0
$325$	−5598.30	−0.955502
$326$	0	0
$327$	−1753.66	−0.296568
$328$	0	0
$329$	−18035.6	−3.02229
$330$	0	0
$331$	4670.49	0.775568	0.387784	−	0.921750i	$-0.373241\pi$
$331$	4670.49	0.775568	0.387784	+	0.921750i	$0.373241\pi$
$332$	0	0
$333$	−892.921	−0.146942
$334$	0	0
$335$	−5671.58	−0.924989
$336$	0	0
$337$	1801.67	0.291226	0.145613	−	0.989342i	$-0.453485\pi$
$337$	1801.67	0.291226	0.145613	+	0.989342i	$0.453485\pi$
$338$	0	0
$339$	190.268	0.0304835
$340$	0	0
$341$	383.082	0.0608360
$342$	0	0
$343$	9842.83	1.54945
$344$	0	0
$345$	−3450.27	−0.538424
$346$	0	0
$347$	−168.340	−0.0260431	−0.0130216	−	0.999915i	$-0.504145\pi$
$347$	−168.340	−0.0260431	−0.0130216	+	0.999915i	$0.504145\pi$
$348$	0	0
$349$	−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
$352$	0	0
$353$	−10509.5	−1.58460	−0.792298	−	0.610135i	$-0.791115\pi$
$353$	−10509.5	−1.58460	−0.792298	+	0.610135i	$0.791115\pi$
$354$	0	0
$355$	−7123.57	−1.06501
$356$	0	0
$357$	−1610.85	−0.238810
$358$	0	0
$359$	−8342.99	−1.22654	−0.613268	−	0.789875i	$-0.710145\pi$
$359$	−8342.99	−1.22654	−0.613268	+	0.789875i	$0.710145\pi$
$360$	0	0
$361$	−5483.79	−0.799503
$362$	0	0
$363$	3837.95	0.554932
$364$	0	0
$365$	5339.55	0.765712
$366$	0	0
$367$	−352.402	−0.0501232	−0.0250616	−	0.999686i	$-0.507978\pi$
$367$	−352.402	−0.0501232	−0.0250616	+	0.999686i	$0.507978\pi$
$368$	0	0
$369$	1064.24	0.150142
$370$	0	0
$371$	14619.8	2.04588
$372$	0	0
$373$	12563.2	1.74397	0.871983	−	0.489537i	$-0.162834\pi$
$373$	12563.2	1.74397	0.871983	+	0.489537i	$0.162834\pi$
$374$	0	0
$375$	−4395.78	−0.605326
$376$	0	0
$377$	−976.961	−0.133464
$378$	0	0
$379$	1770.57	0.239969	0.119984	−	0.992776i	$-0.461716\pi$
$379$	1770.57	0.239969	0.119984	+	0.992776i	$0.461716\pi$
$380$	0	0
$381$	526.630	0.0708139
$382$	0	0
$383$	−4330.57	−0.577759	−0.288880	−	0.957365i	$-0.593283\pi$
$383$	−4330.57	−0.577759	−0.288880	+	0.957365i	$0.593283\pi$
$384$	0	0
$385$	−1738.48	−0.230133
$386$	0	0
$387$	4104.15	0.539084
$388$	0	0
$389$	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
$392$	0	0
$393$	5596.26	0.718305
$394$	0	0
$395$	7932.12	1.01040
$396$	0	0
$397$	93.1792	0.0117797	0.00588983	−	0.999983i	$-0.498125\pi$
$397$	93.1792	0.0117797	0.00588983	+	0.999983i	$0.498125\pi$
$398$	0	0
$399$	−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	$0.811342\pi$
$402$	0	0
$403$	4493.82	0.555466
$404$	0	0
$405$	−620.149	−0.0760875
$406$	0	0
$407$	−713.255	−0.0868667
$408$	0	0
$409$	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
$412$	0	0
$413$	−1518.22	−0.180888
$414$	0	0
$415$	−170.000	−0.0201083
$416$	0	0
$417$	2714.50	0.318776
$418$	0	0
$419$	−11325.4	−1.32049	−0.660244	−	0.751051i	$-0.729547\pi$
$419$	−11325.4	−1.32049	−0.660244	+	0.751051i	$0.729547\pi$
$420$	0	0
$421$	6934.54	0.802776	0.401388	−	0.915908i	$-0.368528\pi$
$421$	6934.54	0.802776	0.401388	+	0.915908i	$0.368528\pi$
$422$	0	0
$423$	−5139.12	−0.590715
$424$	0	0
$425$	−1128.52	−0.128802
$426$	0	0
$427$	1880.07	0.213074
$428$	0	0
$429$	−1818.84	−0.204695
$430$	0	0
$431$	2776.55	0.310306	0.155153	−	0.987890i	$-0.450413\pi$
$431$	2776.55	0.310306	0.155153	+	0.987890i	$0.450413\pi$
$432$	0	0
$433$	−3252.22	−0.360951	−0.180476	−	0.983579i	$-0.557764\pi$
$433$	−3252.22	−0.360951	−0.180476	+	0.983579i	$0.557764\pi$
$434$	0	0
$435$	−266.080	−0.0293277
$436$	0	0
$437$	−5570.65	−0.609795
$438$	0	0
$439$	13345.7	1.45093	0.725464	−	0.688260i	$-0.241625\pi$
$439$	13345.7	1.45093	0.725464	+	0.688260i	$0.241625\pi$
$440$	0	0
$441$	5891.65	0.636178
$442$	0	0
$443$	−11639.6	−1.24834	−0.624169	−	0.781290i	$-0.714562\pi$
$443$	−11639.6	−1.24834	−0.624169	+	0.781290i	$0.714562\pi$
$444$	0	0
$445$	2830.68	0.301544
$446$	0	0
$447$	−2427.00	−0.256808
$448$	0	0
$449$	−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
$452$	0	0
$453$	−1056.36	−0.109563
$454$	0	0
$455$	−20393.6	−2.10124
$456$	0	0
$457$	−10285.8	−1.05284	−0.526422	−	0.850224i	$-0.676467\pi$
$457$	−10285.8	−1.05284	−0.526422	+	0.850224i	$0.676467\pi$
$458$	0	0
$459$	−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	$0.510065\pi$
$462$	0	0
$463$	6055.97	0.607872	0.303936	−	0.952692i	$-0.401699\pi$
$463$	6055.97	0.607872	0.303936	+	0.952692i	$0.401699\pi$
$464$	0	0
$465$	1223.91	0.122059
$466$	0	0
$467$	815.966	0.0808531	0.0404265	−	0.999183i	$-0.487128\pi$
$467$	815.966	0.0808531	0.0404265	+	0.999183i	$0.487128\pi$
$468$	0	0
$469$	23397.9	2.30366
$470$	0	0
$471$	−1613.65	−0.157862
$472$	0	0
$473$	3278.35	0.318686
$474$	0	0
$475$	−2461.74	−0.237795
$476$	0	0
$477$	4165.81	0.399872
$478$	0	0
$479$	16219.1	1.54712	0.773559	−	0.633724i	$-0.218475\pi$
$479$	16219.1	1.54712	0.773559	+	0.633724i	$0.218475\pi$
$480$	0	0
$481$	−8366.97	−0.793142
$482$	0	0
$483$	14234.0	1.34093
$484$	0	0
$485$	−8724.63	−0.816835
$486$	0	0
$487$	−2725.13	−0.253568	−0.126784	−	0.991930i	$-0.540465\pi$
$487$	−2725.13	−0.253568	−0.126784	+	0.991930i	$0.540465\pi$
$488$	0	0
$489$	5768.23	0.533433
$490$	0	0
$491$	8344.13	0.766935	0.383468	−	0.923554i	$-0.374730\pi$
$491$	8344.13	0.766935	0.383468	+	0.923554i	$0.374730\pi$
$492$	0	0
$493$	−196.937	−0.0179911
$494$	0	0
$495$	−495.368	−0.0449801
$496$	0	0
$497$	29388.1	2.65238
$498$	0	0
$499$	−13762.0	−1.23461	−0.617306	−	0.786723i	$-0.711776\pi$
$499$	−13762.0	−1.23461	−0.617306	+	0.786723i	$0.711776\pi$
$500$	0	0
$501$	−8915.29	−0.795022
$502$	0	0
$503$	11909.5	1.05570	0.527852	−	0.849336i	$-0.322997\pi$
$503$	11909.5	1.05570	0.527852	+	0.849336i	$0.322997\pi$
$504$	0	0
$505$	−5382.92	−0.474330
$506$	0	0
$507$	−14745.2	−1.29163
$508$	0	0
$509$	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
$512$	0	0
$513$	−1001.26	−0.0861732
$514$	0	0
$515$	−6868.80	−0.587719
$516$	0	0
$517$	−4105.07	−0.349209
$518$	0	0
$519$	−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	$0.564899\pi$
$522$	0	0
$523$	16249.5	1.35858	0.679291	−	0.733869i	$-0.262287\pi$
$523$	16249.5	1.35858	0.679291	+	0.733869i	$0.262287\pi$
$524$	0	0
$525$	6290.19	0.522908
$526$	0	0
$527$	905.871	0.0748773
$528$	0	0
$529$	10398.4	0.854637
$530$	0	0
$531$	−432.606	−0.0353550
$532$	0	0
$533$	9972.33	0.810412
$534$	0	0
$535$	−14556.0	−1.17628
$536$	0	0
$537$	−5812.96	−0.467128
$538$	0	0
$539$	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	$0.569587\pi$
$542$	0	0
$543$	6540.21	0.516883
$544$	0	0
$545$	−4475.44	−0.351756
$546$	0	0
$547$	−17237.0	−1.34735	−0.673677	−	0.739026i	$-0.735286\pi$
$547$	−17237.0	−1.34735	−0.673677	+	0.739026i	$0.735286\pi$
$548$	0	0
$549$	535.712	0.0416460
$550$	0	0
$551$	−429.600	−0.0332152
$552$	0	0
$553$	−32723.7	−2.51637
$554$	0	0
$555$	−2278.78	−0.174286
$556$	0	0
$557$	−8452.71	−0.643003	−0.321502	−	0.946909i	$-0.604188\pi$
$557$	−8452.71	−0.643003	−0.321502	+	0.946909i	$0.604188\pi$
$558$	0	0
$559$	38457.3	2.90978
$560$	0	0
$561$	−366.644	−0.0275931
$562$	0	0
$563$	8547.32	0.639834	0.319917	−	0.947446i	$-0.396345\pi$
$563$	8547.32	0.639834	0.319917	+	0.947446i	$0.396345\pi$
$564$	0	0
$565$	485.573	0.0361561
$566$	0	0
$567$	2558.40	0.189494
$568$	0	0
$569$	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	$-0.455070\pi$
$571$	3839.06	0.281366	0.140683	+	0.990055i	$0.455070\pi$
$572$	0	0
$573$	5027.35	0.366528
$574$	0	0
$575$	9971.94	0.723232
$576$	0	0
$577$	−18797.7	−1.35625	−0.678127	−	0.734945i	$-0.737208\pi$
$577$	−18797.7	−1.35625	−0.678127	+	0.734945i	$0.737208\pi$
$578$	0	0
$579$	773.882	0.0555465
$580$	0	0
$581$	701.328	0.0500791
$582$	0	0
$583$	3327.60	0.236390
$584$	0	0
$585$	−5811.01	−0.410693
$586$	0	0
$587$	−4217.92	−0.296580	−0.148290	−	0.988944i	$-0.547377\pi$
$587$	−4217.92	−0.296580	−0.148290	+	0.988944i	$0.547377\pi$
$588$	0	0
$589$	1976.07	0.138238
$590$	0	0
$591$	2080.40	0.144799
$592$	0	0
$593$	−3011.92	−0.208575	−0.104287	−	0.994547i	$-0.533256\pi$
$593$	−3011.92	−0.208575	−0.104287	+	0.994547i	$0.533256\pi$
$594$	0	0
$595$	−4110.97	−0.283249
$596$	0	0
$597$	−720.886	−0.0494203
$598$	0	0
$599$	−15137.1	−1.03253	−0.516266	−	0.856428i	$-0.672678\pi$
$599$	−15137.1	−1.03253	−0.516266	+	0.856428i	$0.672678\pi$
$600$	0	0
$601$	−18980.1	−1.28821	−0.644104	−	0.764938i	$-0.722770\pi$
$601$	−18980.1	−1.28821	−0.644104	+	0.764938i	$0.722770\pi$
$602$	0	0
$603$	6667.08	0.450256
$604$	0	0
$605$	9794.65	0.658197
$606$	0	0
$607$	4593.63	0.307166	0.153583	−	0.988136i	$-0.450919\pi$
$607$	4593.63	0.307166	0.153583	+	0.988136i	$0.450919\pi$
$608$	0	0
$609$	1097.70	0.0730397
$610$	0	0
$611$	−48155.3	−3.18847
$612$	0	0
$613$	−25654.3	−1.69032	−0.845160	−	0.534513i	$-0.820495\pi$
$613$	−25654.3	−1.69032	−0.845160	+	0.534513i	$0.820495\pi$
$614$	0	0
$615$	2716.01	0.178081
$616$	0	0
$617$	−7170.97	−0.467897	−0.233948	−	0.972249i	$-0.575165\pi$
$617$	−7170.97	−0.467897	−0.233948	+	0.972249i	$0.575165\pi$
$618$	0	0
$619$	13560.6	0.880525	0.440263	−	0.897869i	$-0.354885\pi$
$619$	13560.6	0.880525	0.440263	+	0.897869i	$0.354885\pi$
$620$	0	0
$621$	4055.88	0.262088
$622$	0	0
$623$	−11677.9	−0.750985
$624$	0	0
$625$	−2920.36	−0.186903
$626$	0	0
$627$	−799.798	−0.0509424
$628$	0	0
$629$	−1686.63	−0.106916
$630$	0	0
$631$	1414.98	0.0892700	0.0446350	−	0.999003i	$-0.485788\pi$
$631$	1414.98	0.0892700	0.0446350	+	0.999003i	$0.485788\pi$
$632$	0	0
$633$	804.342	0.0505051
$634$	0	0
$635$	1343.99	0.0839914
$636$	0	0
$637$	55206.7	3.43386
$638$	0	0
$639$	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	$0.733201\pi$
$642$	0	0
$643$	−7537.23	−0.462270	−0.231135	−	0.972922i	$-0.574244\pi$
$643$	−7537.23	−0.462270	−0.231135	+	0.972922i	$0.574244\pi$
$644$	0	0
$645$	10474.0	0.639401
$646$	0	0
$647$	32667.3	1.98498	0.992491	−	0.122316i	$-0.0390320\pi$
$647$	32667.3	1.98498	0.992491	+	0.122316i	$0.0390320\pi$
$648$	0	0
$649$	−345.561	−0.0209006
$650$	0	0
$651$	−5049.20	−0.303984
$652$	0	0
$653$	−25845.8	−1.54889	−0.774443	−	0.632644i	$-0.781970\pi$
$653$	−25845.8	−1.54889	−0.774443	+	0.632644i	$0.781970\pi$
$654$	0	0
$655$	14281.9	0.851972
$656$	0	0
$657$	−6276.77	−0.372725
$658$	0	0
$659$	15741.2	0.930485	0.465243	−	0.885183i	$-0.345967\pi$
$659$	15741.2	0.930485	0.465243	+	0.885183i	$0.345967\pi$
$660$	0	0
$661$	23495.2	1.38254	0.691269	−	0.722598i	$-0.257052\pi$
$661$	23495.2	1.38254	0.691269	+	0.722598i	$0.257052\pi$
$662$	0	0
$663$	−4300.99	−0.251940
$664$	0	0
$665$	−8967.67	−0.522934
$666$	0	0
$667$	1740.21	0.101021
$668$	0	0
$669$	−16573.3	−0.957788
$670$	0	0
$671$	427.921	0.0246195
$672$	0	0
$673$	−7057.34	−0.404221	−0.202110	−	0.979363i	$-0.564780\pi$
$673$	−7057.34	−0.404221	−0.202110	+	0.979363i	$0.564780\pi$
$674$	0	0
$675$	1792.35	0.102204
$676$	0	0
$677$	20756.4	1.17833	0.589167	−	0.808011i	$-0.299456\pi$
$677$	20756.4	1.17833	0.589167	+	0.808011i	$0.299456\pi$
$678$	0	0
$679$	35993.2	2.03430
$680$	0	0
$681$	−1153.20	−0.0648909
$682$	0	0
$683$	7013.65	0.392928	0.196464	−	0.980511i	$-0.437054\pi$
$683$	7013.65	0.392928	0.196464	+	0.980511i	$0.437054\pi$
$684$	0	0
$685$	−8096.91	−0.451631
$686$	0	0
$687$	−4186.44	−0.232493
$688$	0	0
$689$	39035.0	2.15837
$690$	0	0
$691$	4897.47	0.269622	0.134811	−	0.990871i	$-0.456957\pi$
$691$	4897.47	0.269622	0.134811	+	0.990871i	$0.456957\pi$
$692$	0	0
$693$	2043.63	0.112022
$694$	0	0
$695$	6927.54	0.378096
$696$	0	0
$697$	2010.24	0.109244
$698$	0	0
$699$	−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	$0.343485\pi$
$702$	0	0
$703$	−3679.21	−0.197388
$704$	0	0
$705$	−13115.3	−0.700640
$706$	0	0
$707$	22207.1	1.18130
$708$	0	0
$709$	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
$712$	0	0
$713$	−8004.58	−0.420440
$714$	0	0
$715$	−4641.77	−0.242786
$716$	0	0
$717$	−4527.55	−0.235822
$718$	0	0
$719$	−14498.2	−0.752007	−0.376004	−	0.926618i	$-0.622702\pi$
$719$	−14498.2	−0.752007	−0.376004	+	0.926618i	$0.622702\pi$
$720$	0	0
$721$	28337.0	1.46370
$722$	0	0
$723$	−10220.7	−0.525744
$724$	0	0
$725$	769.020	0.0393941
$726$	0	0
$727$	25787.6	1.31556	0.657778	−	0.753212i	$-0.271497\pi$
$727$	25787.6	1.31556	0.657778	+	0.753212i	$0.271497\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	7752.28	0.392241
$732$	0	0
$733$	−4177.45	−0.210502	−0.105251	−	0.994446i	$-0.533565\pi$
$733$	−4177.45	−0.210502	−0.105251	+	0.994446i	$0.533565\pi$
$734$	0	0
$735$	15035.8	0.754563
$736$	0	0
$737$	5325.59	0.266175
$738$	0	0
$739$	−14115.5	−0.702636	−0.351318	−	0.936256i	$-0.614266\pi$
$739$	−14115.5	−0.702636	−0.351318	+	0.936256i	$0.614266\pi$
$740$	0	0
$741$	−9382.18	−0.465132
$742$	0	0
$743$	−17992.0	−0.888376	−0.444188	−	0.895934i	$-0.646508\pi$
$743$	−17992.0	−0.888376	−0.444188	+	0.895934i	$0.646508\pi$
$744$	0	0
$745$	−6193.84	−0.304597
$746$	0	0
$747$	199.839	0.00978810
$748$	0	0
$749$	60050.2	2.92949
$750$	0	0
$751$	−2055.99	−0.0998989	−0.0499495	−	0.998752i	$-0.515906\pi$
$751$	−2055.99	−0.0998989	−0.0499495	+	0.998752i	$0.515906\pi$
$752$	0	0
$753$	10183.3	0.492829
$754$	0	0
$755$	−2695.89	−0.129952
$756$	0	0
$757$	12132.4	0.582508	0.291254	−	0.956646i	$-0.405927\pi$
$757$	12132.4	0.582508	0.291254	+	0.956646i	$0.405927\pi$
$758$	0	0
$759$	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	$0.428757\pi$
$762$	0	0
$763$	18463.3	0.876037
$764$	0	0
$765$	−1171.39	−0.0553618
$766$	0	0
$767$	−4053.67	−0.190834
$768$	0	0
$769$	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
$772$	0	0
$773$	12857.4	0.598252	0.299126	−	0.954214i	$-0.403305\pi$
$773$	12857.4	0.598252	0.299126	+	0.954214i	$0.403305\pi$
$774$	0	0
$775$	−3537.33	−0.163954
$776$	0	0
$777$	9401.04	0.434054
$778$	0	0
$779$	4385.14	0.201687
$780$	0	0
$781$	6689.00	0.306468
$782$	0	0
$783$	312.783	0.0142758
$784$	0	0
$785$	−4118.11	−0.187238
$786$	0	0
$787$	26862.0	1.21668	0.608339	−	0.793677i	$-0.291836\pi$
$787$	26862.0	1.21668	0.608339	+	0.793677i	$0.291836\pi$
$788$	0	0
$789$	−12950.6	−0.584353
$790$	0	0
$791$	−2003.22	−0.0900457
$792$	0	0
$793$	5019.81	0.224790
$794$	0	0
$795$	10631.4	0.474284
$796$	0	0
$797$	−12471.5	−0.554285	−0.277142	−	0.960829i	$-0.589387\pi$
$797$	−12471.5	−0.554285	−0.277142	+	0.960829i	$0.589387\pi$
$798$	0	0
$799$	−9707.23	−0.429809
$800$	0	0
$801$	−3327.53	−0.146782
$802$	0	0
$803$	−5013.82	−0.220341
$804$	0	0
$805$	36325.9	1.59046
$806$	0	0
$807$	−19638.9	−0.856658
$808$	0	0
$809$	−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	$-0.578223\pi$
$811$	−11237.3	−0.486556	−0.243278	+	0.969957i	$0.578223\pi$
$812$	0	0
$813$	11355.6	0.489862
$814$	0	0
$815$	14720.8	0.632698
$816$	0	0
$817$	16910.8	0.724156
$818$	0	0
$819$	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	$0.176791\pi$
$822$	0	0
$823$	−36877.0	−1.56191	−0.780955	−	0.624588i	$-0.785267\pi$
$823$	−36877.0	−1.56191	−0.780955	+	0.624588i	$0.785267\pi$
$824$	0	0
$825$	1431.71	0.0604190
$826$	0	0
$827$	14311.9	0.601781	0.300890	−	0.953659i	$-0.402716\pi$
$827$	14311.9	0.601781	0.300890	+	0.953659i	$0.402716\pi$
$828$	0	0
$829$	−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
$832$	0	0
$833$	11128.7	0.462888
$834$	0	0
$835$	−22752.3	−0.942965
$836$	0	0
$837$	−1438.74	−0.0594146
$838$	0	0
$839$	18683.1	0.768786	0.384393	−	0.923170i	$-0.374411\pi$
$839$	18683.1	0.768786	0.384393	+	0.923170i	$0.374411\pi$
$840$	0	0
$841$	−24254.8	−0.994497
$842$	0	0
$843$	8766.90	0.358183
$844$	0	0
$845$	−37630.6	−1.53199
$846$	0	0
$847$	−40407.5	−1.63922
$848$	0	0
$849$	2205.22	0.0891438
$850$	0	0
$851$	14903.6	0.600340
$852$	0	0
$853$	−35648.7	−1.43094	−0.715468	−	0.698646i	$-0.753786\pi$
$853$	−35648.7	−1.43094	−0.715468	+	0.698646i	$0.753786\pi$
$854$	0	0
$855$	−2555.28	−0.102209
$856$	0	0
$857$	−49860.8	−1.98741	−0.993707	−	0.112010i	$-0.964271\pi$
$857$	−49860.8	−1.98741	−0.993707	+	0.112010i	$0.964271\pi$
$858$	0	0
$859$	−21487.0	−0.853466	−0.426733	−	0.904378i	$-0.640336\pi$
$859$	−21487.0	−0.853466	−0.426733	+	0.904378i	$0.640336\pi$
$860$	0	0
$861$	−11204.8	−0.443506
$862$	0	0
$863$	15067.6	0.594330	0.297165	−	0.954826i	$-0.403959\pi$
$863$	15067.6	0.594330	0.297165	+	0.954826i	$0.403959\pi$
$864$	0	0
$865$	−7568.60	−0.297503
$866$	0	0
$867$	−867.000	−0.0339618
$868$	0	0
$869$	−7448.23	−0.290752
$870$	0	0
$871$	62472.8	2.43032
$872$	0	0
$873$	10256.0	0.397610
$874$	0	0
$875$	46280.6	1.78808
$876$	0	0
$877$	−24852.6	−0.956912	−0.478456	−	0.878111i	$-0.658803\pi$
$877$	−24852.6	−0.956912	−0.478456	+	0.878111i	$0.658803\pi$
$878$	0	0
$879$	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	$0.488796\pi$
$882$	0	0
$883$	−49803.7	−1.89811	−0.949054	−	0.315115i	$-0.897957\pi$
$883$	−49803.7	−1.89811	−0.949054	+	0.315115i	$0.897957\pi$
$884$	0	0
$885$	−1104.03	−0.0419341
$886$	0	0
$887$	−36314.7	−1.37467	−0.687333	−	0.726342i	$-0.741219\pi$
$887$	−36314.7	−1.37467	−0.687333	+	0.726342i	$0.741219\pi$
$888$	0	0
$889$	−5544.58	−0.209178
$890$	0	0
$891$	582.317	0.0218949
$892$	0	0
$893$	−21175.4	−0.793512
$894$	0	0
$895$	−14835.0	−0.554054
$896$	0	0
$897$	38005.0	1.41466
$898$	0	0
$899$	−617.300	−0.0229011
$900$	0	0
$901$	7868.75	0.290950
$902$	0	0
$903$	−43210.2	−1.59241
$904$	0	0
$905$	16691.0	0.613068
$906$	0	0
$907$	−33679.5	−1.23297	−0.616487	−	0.787365i	$-0.711445\pi$
$907$	−33679.5	−1.23297	−0.616487	+	0.787365i	$0.711445\pi$
$908$	0	0
$909$	6327.75	0.230889
$910$	0	0
$911$	−32437.2	−1.17968	−0.589841	−	0.807519i	$-0.700810\pi$
$911$	−32437.2	−1.17968	−0.589841	+	0.807519i	$0.700810\pi$
$912$	0	0
$913$	159.629	0.00578636
$914$	0	0
$915$	1367.17	0.0493957
$916$	0	0
$917$	−58919.7	−2.12181
$918$	0	0
$919$	3511.45	0.126042	0.0630208	−	0.998012i	$-0.479927\pi$
$919$	3511.45	0.126042	0.0630208	+	0.998012i	$0.479927\pi$
$920$	0	0
$921$	7550.86	0.270151
$922$	0	0
$923$	78466.6	2.79822
$924$	0	0
$925$	6586.11	0.234108
$926$	0	0
$927$	8074.44	0.286083
$928$	0	0
$929$	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
$932$	0	0
$933$	19779.9	0.694069
$934$	0	0
$935$	−935.695	−0.0327278
$936$	0	0
$937$	−8862.80	−0.309002	−0.154501	−	0.987993i	$-0.549377\pi$
$937$	−8862.80	−0.309002	−0.154501	+	0.987993i	$0.549377\pi$
$938$	0	0
$939$	−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	$0.370625\pi$
$942$	0	0
$943$	−17763.1	−0.613412
$944$	0	0
$945$	6529.18	0.224756
$946$	0	0
$947$	26308.4	0.902754	0.451377	−	0.892333i	$-0.350933\pi$
$947$	26308.4	0.902754	0.451377	+	0.892333i	$0.350933\pi$
$948$	0	0
$949$	−58815.5	−2.01184
$950$	0	0
$951$	−7803.68	−0.266090
$952$	0	0
$953$	11947.5	0.406106	0.203053	−	0.979168i	$-0.434914\pi$
$953$	11947.5	0.406106	0.203053	+	0.979168i	$0.434914\pi$
$954$	0	0
$955$	12830.1	0.434734
$956$	0	0
$957$	249.848	0.00843932
$958$	0	0
$959$	33403.5	1.12477
$960$	0	0
$961$	−26951.5	−0.904688
$962$	0	0
$963$	17110.9	0.572576
$964$	0	0
$965$	1974.99	0.0658830
$966$	0	0
$967$	−21505.3	−0.715163	−0.357581	−	0.933882i	$-0.616399\pi$
$967$	−21505.3	−0.715163	−0.357581	+	0.933882i	$0.616399\pi$
$968$	0	0
$969$	−1891.27	−0.0627002
$970$	0	0
$971$	−45685.3	−1.50990	−0.754950	−	0.655783i	$-0.772339\pi$
$971$	−45685.3	−1.50990	−0.754950	+	0.655783i	$0.772339\pi$
$972$	0	0
$973$	−28579.4	−0.941636
$974$	0	0
$975$	16794.9	0.551659
$976$	0	0
$977$	31710.3	1.03839	0.519193	−	0.854657i	$-0.326233\pi$
$977$	31710.3	1.03839	0.519193	+	0.854657i	$0.326233\pi$
$978$	0	0
$979$	−2657.99	−0.0867721
$980$	0	0
$981$	5260.99	0.171224
$982$	0	0
$983$	−46038.9	−1.49381	−0.746904	−	0.664932i	$-0.768460\pi$
$983$	−46038.9	−1.49381	−0.746904	+	0.664932i	$0.768460\pi$
$984$	0	0
$985$	5309.28	0.171744
$986$	0	0
$987$	54106.8	1.74492
$988$	0	0
$989$	−68501.8	−2.20246
$990$	0	0
$991$	−14394.9	−0.461420	−0.230710	−	0.973023i	$-0.574105\pi$
$991$	−14394.9	−0.461420	−0.230710	+	0.973023i	$0.574105\pi$
$992$	0	0
$993$	−14011.5	−0.447775
$994$	0	0
$995$	−1839.74	−0.0586167
$996$	0	0
$997$	−33473.4	−1.06330	−0.531652	−	0.846963i	$-0.678428\pi$
$997$	−33473.4	−1.06330	−0.531652	+	0.846963i	$0.678428\pi$
$998$	0	0
$999$	2678.76	0.0848371

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	816.4.a.s.1.1		3
3.2	odd	2		2448.4.a.bd.1.3		3
4.3	odd	2		51.4.a.e.1.3	✓	3
12.11	even	2		153.4.a.f.1.1		3
20.19	odd	2		1275.4.a.q.1.1		3
28.27	even	2		2499.4.a.n.1.3		3
68.67	odd	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	4.3	odd	2
153.4.a.f.1.1		3	12.11	even	2
816.4.a.s.1.1		3	1.1	even	1	trivial
867.4.a.k.1.3		3	68.67	odd	2
1275.4.a.q.1.1		3	20.19	odd	2
2448.4.a.bd.1.3		3	3.2	odd	2
2499.4.a.n.1.3		3	28.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 816 = 2^{4} \cdot 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	816.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -7.65616 q^{5} +31.5852 q^{7} +9.00000 q^{9} +7.18910 q^{11} +84.3331 q^{13} +22.9685 q^{15} +17.0000 q^{17} +37.0838 q^{19} -94.7557 q^{21} -150.218 q^{23} -66.3832 q^{25} -27.0000 q^{27} -11.5846 q^{29} +53.2865 q^{31} -21.5673 q^{33} -241.822 q^{35} -99.2134 q^{37} -252.999 q^{39} +118.249 q^{41} +456.016 q^{43} -68.9054 q^{45} -571.014 q^{47} +654.627 q^{49} -51.0000 q^{51} +462.867 q^{53} -55.0409 q^{55} -111.251 q^{57} -48.0674 q^{59} +59.5236 q^{61} +284.267 q^{63} -645.668 q^{65} +740.787 q^{67} +450.653 q^{69} +930.437 q^{71} -697.419 q^{73} +199.150 q^{75} +227.070 q^{77} -1036.04 q^{79} +81.0000 q^{81} +22.2043 q^{83} -130.155 q^{85} +34.7537 q^{87} -369.726 q^{89} +2663.68 q^{91} -159.860 q^{93} -283.920 q^{95} +1139.56 q^{97} +64.7019 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 9 q^{3} + 8 q^{5} + 8 q^{7} + 27 q^{9} - 34 q^{11} + 36 q^{13} - 24 q^{15} + 51 q^{17} + 142 q^{19} - 24 q^{21} - 110 q^{23} - 193 q^{25} - 81 q^{27} + 90 q^{29} + 148 q^{31} + 102 q^{33} - 416 q^{35}+ \cdots - 306 q^{99}+O(q^{100}) \) 3 * q - 9 * q^3 + 8 * q^5 + 8 * q^7 + 27 * q^9 - 34 * q^11 + 36 * q^13 - 24 * q^15 + 51 * q^17 + 142 * q^19 - 24 * q^21 - 110 * q^23 - 193 * q^25 - 81 * q^27 + 90 * q^29 + 148 * q^31 + 102 * q^33 - 416 * q^35 + 110 * q^37 - 108 * q^39 + 720 * q^41 + 146 * q^43 + 72 * q^45 - 500 * q^47 + 379 * q^49 - 153 * q^51 + 610 * q^53 - 430 * q^55 - 426 * q^57 + 216 * q^59 - 18 * q^61 + 72 * q^63 - 966 * q^65 + 1404 * q^67 + 330 * q^69 + 960 * q^71 - 794 * q^73 + 579 * q^75 + 48 * q^77 + 276 * q^79 + 243 * q^81 + 1552 * q^83 + 136 * q^85 - 270 * q^87 + 1394 * q^89 + 3968 * q^91 - 444 * q^93 + 602 * q^95 + 402 * q^97 - 306 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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