LMFDB - Embedded newform 560.4.a.u.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [560,4,Mod(1,560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(560, 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("560.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 560 = 2^{4} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	560.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:	$33.0410696032$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.14360.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 17x - 14 $ x^3 - 17*x - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.62456$ of defining polynomial
Character	$\chi$	$=$	560.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.38660 q^{3} +5.00000 q^{5} -7.00000 q^{7} +43.3350 q^{9} +30.1117 q^{11} +88.9295 q^{13} +41.9330 q^{15} -4.73699 q^{17} -124.818 q^{19} -58.7062 q^{21} -20.2680 q^{23} +25.0000 q^{25} +136.995 q^{27} +134.088 q^{29} +2.03767 q^{31} +252.534 q^{33} -35.0000 q^{35} -141.137 q^{37} +745.816 q^{39} +95.2784 q^{41} +298.646 q^{43} +216.675 q^{45} +129.054 q^{47} +49.0000 q^{49} -39.7272 q^{51} +388.429 q^{53} +150.558 q^{55} -1046.80 q^{57} -838.501 q^{59} +389.422 q^{61} -303.345 q^{63} +444.647 q^{65} -697.794 q^{67} -169.979 q^{69} +523.450 q^{71} +66.4684 q^{73} +209.665 q^{75} -210.782 q^{77} +526.982 q^{79} -21.1236 q^{81} -70.0265 q^{83} -23.6850 q^{85} +1124.54 q^{87} -9.27925 q^{89} -622.506 q^{91} +17.0891 q^{93} -624.089 q^{95} -4.19493 q^{97} +1304.89 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{3} + 15 q^{5} - 21 q^{7} + 81 q^{9} + 74 q^{11} + 44 q^{13} - 10 q^{15} - 52 q^{17} - 168 q^{19} + 14 q^{21} + 124 q^{23} + 75 q^{25} - 170 q^{27} + 332 q^{29} - 320 q^{31} - 106 q^{33} - 105 q^{35}+ \cdots + 3488 q^{99}+O(q^{100}) $ 3 * q - 2 * q^3 + 15 * q^5 - 21 * q^7 + 81 * q^9 + 74 * q^11 + 44 * q^13 - 10 * q^15 - 52 * q^17 - 168 * q^19 + 14 * q^21 + 124 * q^23 + 75 * q^25 - 170 * q^27 + 332 * q^29 - 320 * q^31 - 106 * q^33 - 105 * q^35 - 54 * q^37 + 982 * q^39 + 362 * q^41 + 16 * q^43 + 405 * q^45 + 730 * q^47 + 147 * q^49 + 1178 * q^51 + 110 * q^53 + 370 * q^55 - 956 * q^57 + 180 * q^59 + 1222 * q^61 - 567 * q^63 + 220 * q^65 - 204 * q^67 - 716 * q^69 + 136 * q^71 + 310 * q^73 - 50 * q^75 - 518 * q^77 + 1034 * q^79 + 2283 * q^81 + 1660 * q^83 - 260 * q^85 + 1574 * q^87 + 242 * q^89 - 308 * q^91 + 1376 * q^93 - 840 * q^95 + 100 * q^97 + 3488 * 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
$3$	8.38660	1.61400	0.807001	−	0.590551i	$-0.201089\pi$
$3$	8.38660	1.61400	0.807001	+	0.590551i	$0.201089\pi$
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	43.3350	1.60500
$10$	0	0
$11$	30.1117	0.825364	0.412682	−	0.910875i	$-0.364592\pi$
$11$	30.1117	0.825364	0.412682	+	0.910875i	$0.364592\pi$
$12$	0	0
$13$	88.9295	1.89728	0.948639	−	0.316362i	$-0.102461\pi$
$13$	88.9295	1.89728	0.948639	+	0.316362i	$0.102461\pi$
$14$	0	0
$15$	41.9330	0.721803
$16$	0	0
$17$	−4.73699	−0.0675817	−0.0337909	−	0.999429i	$-0.510758\pi$
$17$	−4.73699	−0.0675817	−0.0337909	+	0.999429i	$0.510758\pi$
$18$	0	0
$19$	−124.818	−1.50711	−0.753557	−	0.657382i	$-0.771664\pi$
$19$	−124.818	−1.50711	−0.753557	+	0.657382i	$0.771664\pi$
$20$	0	0
$21$	−58.7062	−0.610035
$22$	0	0
$23$	−20.2680	−0.183746	−0.0918731	−	0.995771i	$-0.529285\pi$
$23$	−20.2680	−0.183746	−0.0918731	+	0.995771i	$0.529285\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	136.995	0.976470
$28$	0	0
$29$	134.088	0.858603	0.429301	−	0.903161i	$-0.358760\pi$
$29$	134.088	0.858603	0.429301	+	0.903161i	$0.358760\pi$
$30$	0	0
$31$	2.03767	0.0118057	0.00590284	−	0.999983i	$-0.498121\pi$
$31$	2.03767	0.0118057	0.00590284	+	0.999983i	$0.498121\pi$
$32$	0	0
$33$	252.534	1.33214
$34$	0	0
$35$	−35.0000	−0.169031
$36$	0	0
$37$	−141.137	−0.627104	−0.313552	−	0.949571i	$-0.601519\pi$
$37$	−141.137	−0.627104	−0.313552	+	0.949571i	$0.601519\pi$
$38$	0	0
$39$	745.816	3.06221
$40$	0	0
$41$	95.2784	0.362927	0.181463	−	0.983398i	$-0.441917\pi$
$41$	95.2784	0.362927	0.181463	+	0.983398i	$0.441917\pi$
$42$	0	0
$43$	298.646	1.05914	0.529571	−	0.848266i	$-0.322353\pi$
$43$	298.646	1.05914	0.529571	+	0.848266i	$0.322353\pi$
$44$	0	0
$45$	216.675	0.717778
$46$	0	0
$47$	129.054	0.400519	0.200260	−	0.979743i	$-0.435821\pi$
$47$	129.054	0.400519	0.200260	+	0.979743i	$0.435821\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−39.7272	−0.109077
$52$	0	0
$53$	388.429	1.00669	0.503347	−	0.864084i	$-0.332102\pi$
$53$	388.429	1.00669	0.503347	+	0.864084i	$0.332102\pi$
$54$	0	0
$55$	150.558	0.369114
$56$	0	0
$57$	−1046.80	−2.43248
$58$	0	0
$59$	−838.501	−1.85023	−0.925114	−	0.379688i	$-0.876031\pi$
$59$	−838.501	−1.85023	−0.925114	+	0.379688i	$0.876031\pi$
$60$	0	0
$61$	389.422	0.817384	0.408692	−	0.912672i	$-0.365985\pi$
$61$	389.422	0.817384	0.408692	+	0.912672i	$0.365985\pi$
$62$	0	0
$63$	−303.345	−0.606633
$64$	0	0
$65$	444.647	0.848488
$66$	0	0
$67$	−697.794	−1.27237	−0.636187	−	0.771534i	$-0.719490\pi$
$67$	−697.794	−1.27237	−0.636187	+	0.771534i	$0.719490\pi$
$68$	0	0
$69$	−169.979	−0.296567
$70$	0	0
$71$	523.450	0.874959	0.437479	−	0.899228i	$-0.355871\pi$
$71$	523.450	0.874959	0.437479	+	0.899228i	$0.355871\pi$
$72$	0	0
$73$	66.4684	0.106569	0.0532845	−	0.998579i	$-0.483031\pi$
$73$	66.4684	0.106569	0.0532845	+	0.998579i	$0.483031\pi$
$74$	0	0
$75$	209.665	0.322800
$76$	0	0
$77$	−210.782	−0.311958
$78$	0	0
$79$	526.982	0.750508	0.375254	−	0.926922i	$-0.377556\pi$
$79$	526.982	0.750508	0.375254	+	0.926922i	$0.377556\pi$
$80$	0	0
$81$	−21.1236	−0.0289762
$82$	0	0
$83$	−70.0265	−0.0926074	−0.0463037	−	0.998927i	$-0.514744\pi$
$83$	−70.0265	−0.0926074	−0.0463037	+	0.998927i	$0.514744\pi$
$84$	0	0
$85$	−23.6850	−0.0302235
$86$	0	0
$87$	1124.54	1.38579
$88$	0	0
$89$	−9.27925	−0.0110517	−0.00552584	−	0.999985i	$-0.501759\pi$
$89$	−9.27925	−0.0110517	−0.00552584	+	0.999985i	$0.501759\pi$
$90$	0	0
$91$	−622.506	−0.717103
$92$	0	0
$93$	17.0891	0.0190544
$94$	0	0
$95$	−624.089	−0.674002
$96$	0	0
$97$	−4.19493	−0.00439104	−0.00219552	−	0.999998i	$-0.500699\pi$
$97$	−4.19493	−0.00439104	−0.00219552	+	0.999998i	$0.500699\pi$
$98$	0	0
$99$	1304.89	1.32471
$100$	0	0
$101$	−865.844	−0.853016	−0.426508	−	0.904484i	$-0.640256\pi$
$101$	−865.844	−0.853016	−0.426508	+	0.904484i	$0.640256\pi$
$102$	0	0
$103$	1166.12	1.11554	0.557771	−	0.829995i	$-0.311657\pi$
$103$	1166.12	1.11554	0.557771	+	0.829995i	$0.311657\pi$
$104$	0	0
$105$	−293.531	−0.272816
$106$	0	0
$107$	−56.9652	−0.0514676	−0.0257338	−	0.999669i	$-0.508192\pi$
$107$	−56.9652	−0.0514676	−0.0257338	+	0.999669i	$0.508192\pi$
$108$	0	0
$109$	−1358.89	−1.19411	−0.597055	−	0.802200i	$-0.703663\pi$
$109$	−1358.89	−1.19411	−0.597055	+	0.802200i	$0.703663\pi$
$110$	0	0
$111$	−1183.66	−1.01215
$112$	0	0
$113$	436.038	0.363000	0.181500	−	0.983391i	$-0.441905\pi$
$113$	436.038	0.363000	0.181500	+	0.983391i	$0.441905\pi$
$114$	0	0
$115$	−101.340	−0.0821738
$116$	0	0
$117$	3853.76	3.04513
$118$	0	0
$119$	33.1590	0.0255435
$120$	0	0
$121$	−424.288	−0.318774
$122$	0	0
$123$	799.062	0.585764
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−1186.69	−0.829144	−0.414572	−	0.910017i	$-0.636069\pi$
$127$	−1186.69	−0.829144	−0.414572	+	0.910017i	$0.636069\pi$
$128$	0	0
$129$	2504.62	1.70946
$130$	0	0
$131$	−1034.56	−0.689997	−0.344999	−	0.938603i	$-0.612121\pi$
$131$	−1034.56	−0.689997	−0.344999	+	0.938603i	$0.612121\pi$
$132$	0	0
$133$	873.725	0.569636
$134$	0	0
$135$	684.975	0.436691
$136$	0	0
$137$	646.219	0.402994	0.201497	−	0.979489i	$-0.435419\pi$
$137$	646.219	0.402994	0.201497	+	0.979489i	$0.435419\pi$
$138$	0	0
$139$	−506.484	−0.309061	−0.154530	−	0.987988i	$-0.549386\pi$
$139$	−506.484	−0.309061	−0.154530	+	0.987988i	$0.549386\pi$
$140$	0	0
$141$	1082.32	0.646439
$142$	0	0
$143$	2677.81	1.56594
$144$	0	0
$145$	670.439	0.383979
$146$	0	0
$147$	410.943	0.230572
$148$	0	0
$149$	−1828.12	−1.00513	−0.502567	−	0.864538i	$-0.667611\pi$
$149$	−1828.12	−1.00513	−0.502567	+	0.864538i	$0.667611\pi$
$150$	0	0
$151$	−2975.17	−1.60342	−0.801708	−	0.597716i	$-0.796075\pi$
$151$	−2975.17	−1.60342	−0.801708	+	0.597716i	$0.796075\pi$
$152$	0	0
$153$	−205.278	−0.108469
$154$	0	0
$155$	10.1883	0.00527966
$156$	0	0
$157$	−2131.74	−1.08364	−0.541820	−	0.840495i	$-0.682264\pi$
$157$	−2131.74	−1.08364	−0.541820	+	0.840495i	$0.682264\pi$
$158$	0	0
$159$	3257.59	1.62481
$160$	0	0
$161$	141.876	0.0694495
$162$	0	0
$163$	593.939	0.285404	0.142702	−	0.989766i	$-0.454421\pi$
$163$	593.939	0.285404	0.142702	+	0.989766i	$0.454421\pi$
$164$	0	0
$165$	1262.67	0.595751
$166$	0	0
$167$	2936.30	1.36059	0.680293	−	0.732941i	$-0.261853\pi$
$167$	2936.30	1.36059	0.680293	+	0.732941i	$0.261853\pi$
$168$	0	0
$169$	5711.45	2.59966
$170$	0	0
$171$	−5408.98	−2.41892
$172$	0	0
$173$	2347.31	1.03158	0.515788	−	0.856716i	$-0.327499\pi$
$173$	2347.31	1.03158	0.515788	+	0.856716i	$0.327499\pi$
$174$	0	0
$175$	−175.000	−0.0755929
$176$	0	0
$177$	−7032.17	−2.98627
$178$	0	0
$179$	−3036.56	−1.26795	−0.633975	−	0.773354i	$-0.718578\pi$
$179$	−3036.56	−1.26795	−0.633975	+	0.773354i	$0.718578\pi$
$180$	0	0
$181$	−899.776	−0.369502	−0.184751	−	0.982785i	$-0.559148\pi$
$181$	−899.776	−0.369502	−0.184751	+	0.982785i	$0.559148\pi$
$182$	0	0
$183$	3265.93	1.31926
$184$	0	0
$185$	−705.687	−0.280449
$186$	0	0
$187$	−142.639	−0.0557796
$188$	0	0
$189$	−958.964	−0.369071
$190$	0	0
$191$	−416.168	−0.157659	−0.0788294	−	0.996888i	$-0.525118\pi$
$191$	−416.168	−0.157659	−0.0788294	+	0.996888i	$0.525118\pi$
$192$	0	0
$193$	−5181.05	−1.93233	−0.966166	−	0.257922i	$-0.916962\pi$
$193$	−5181.05	−1.93233	−0.966166	+	0.257922i	$0.916962\pi$
$194$	0	0
$195$	3729.08	1.36946
$196$	0	0
$197$	1452.34	0.525255	0.262627	−	0.964897i	$-0.415411\pi$
$197$	1452.34	0.525255	0.262627	+	0.964897i	$0.415411\pi$
$198$	0	0
$199$	1277.23	0.454978	0.227489	−	0.973781i	$-0.426948\pi$
$199$	1277.23	0.454978	0.227489	+	0.973781i	$0.426948\pi$
$200$	0	0
$201$	−5852.12	−2.05361
$202$	0	0
$203$	−938.615	−0.324521
$204$	0	0
$205$	476.392	0.162306
$206$	0	0
$207$	−878.312	−0.294913
$208$	0	0
$209$	−3758.47	−1.24392
$210$	0	0
$211$	3259.09	1.06334	0.531670	−	0.846951i	$-0.321564\pi$
$211$	3259.09	1.06334	0.531670	+	0.846951i	$0.321564\pi$
$212$	0	0
$213$	4389.96	1.41218
$214$	0	0
$215$	1493.23	0.473663
$216$	0	0
$217$	−14.2637	−0.00446213
$218$	0	0
$219$	557.444	0.172002
$220$	0	0
$221$	−421.258	−0.128221
$222$	0	0
$223$	−4373.35	−1.31328	−0.656639	−	0.754205i	$-0.728023\pi$
$223$	−4373.35	−1.31328	−0.656639	+	0.754205i	$0.728023\pi$
$224$	0	0
$225$	1083.37	0.321000
$226$	0	0
$227$	61.1145	0.0178692	0.00893461	−	0.999960i	$-0.497156\pi$
$227$	61.1145	0.0178692	0.00893461	+	0.999960i	$0.497156\pi$
$228$	0	0
$229$	3019.41	0.871302	0.435651	−	0.900116i	$-0.356518\pi$
$229$	3019.41	0.871302	0.435651	+	0.900116i	$0.356518\pi$
$230$	0	0
$231$	−1767.74	−0.503501
$232$	0	0
$233$	−3531.17	−0.992851	−0.496426	−	0.868079i	$-0.665354\pi$
$233$	−3531.17	−0.992851	−0.496426	+	0.868079i	$0.665354\pi$
$234$	0	0
$235$	645.268	0.179118
$236$	0	0
$237$	4419.58	1.21132
$238$	0	0
$239$	−2282.62	−0.617785	−0.308893	−	0.951097i	$-0.599958\pi$
$239$	−2282.62	−0.617785	−0.308893	+	0.951097i	$0.599958\pi$
$240$	0	0
$241$	−2215.68	−0.592217	−0.296109	−	0.955154i	$-0.595689\pi$
$241$	−2215.68	−0.592217	−0.296109	+	0.955154i	$0.595689\pi$
$242$	0	0
$243$	−3876.02	−1.02324
$244$	0	0
$245$	245.000	0.0638877
$246$	0	0
$247$	−11100.0	−2.85941
$248$	0	0
$249$	−587.284	−0.149468
$250$	0	0
$251$	3082.55	0.775174	0.387587	−	0.921833i	$-0.373309\pi$
$251$	3082.55	0.775174	0.387587	+	0.921833i	$0.373309\pi$
$252$	0	0
$253$	−610.302	−0.151658
$254$	0	0
$255$	−198.636	−0.0487807
$256$	0	0
$257$	−6032.40	−1.46417	−0.732083	−	0.681215i	$-0.761452\pi$
$257$	−6032.40	−1.46417	−0.732083	+	0.681215i	$0.761452\pi$
$258$	0	0
$259$	987.962	0.237023
$260$	0	0
$261$	5810.69	1.37806
$262$	0	0
$263$	−5923.81	−1.38889	−0.694445	−	0.719546i	$-0.744350\pi$
$263$	−5923.81	−1.38889	−0.694445	+	0.719546i	$0.744350\pi$
$264$	0	0
$265$	1942.14	0.450207
$266$	0	0
$267$	−77.8213	−0.0178374
$268$	0	0
$269$	3252.80	0.737273	0.368637	−	0.929574i	$-0.379825\pi$
$269$	3252.80	0.737273	0.368637	+	0.929574i	$0.379825\pi$
$270$	0	0
$271$	6246.26	1.40012	0.700061	−	0.714083i	$-0.253156\pi$
$271$	6246.26	1.40012	0.700061	+	0.714083i	$0.253156\pi$
$272$	0	0
$273$	−5220.71	−1.15741
$274$	0	0
$275$	752.792	0.165073
$276$	0	0
$277$	−1572.17	−0.341020	−0.170510	−	0.985356i	$-0.554541\pi$
$277$	−1572.17	−0.341020	−0.170510	+	0.985356i	$0.554541\pi$
$278$	0	0
$279$	88.3024	0.0189481
$280$	0	0
$281$	−7846.03	−1.66567	−0.832837	−	0.553518i	$-0.813285\pi$
$281$	−7846.03	−1.66567	−0.832837	+	0.553518i	$0.813285\pi$
$282$	0	0
$283$	−6265.58	−1.31608	−0.658039	−	0.752984i	$-0.728614\pi$
$283$	−6265.58	−1.31608	−0.658039	+	0.752984i	$0.728614\pi$
$284$	0	0
$285$	−5233.98	−1.08784
$286$	0	0
$287$	−666.949	−0.137173
$288$	0	0
$289$	−4890.56	−0.995433
$290$	0	0
$291$	−35.1812	−0.00708714
$292$	0	0
$293$	−7264.99	−1.44855	−0.724276	−	0.689511i	$-0.757826\pi$
$293$	−7264.99	−1.44855	−0.724276	+	0.689511i	$0.757826\pi$
$294$	0	0
$295$	−4192.50	−0.827448
$296$	0	0
$297$	4125.14	0.805943
$298$	0	0
$299$	−1802.42	−0.348617
$300$	0	0
$301$	−2090.52	−0.400318
$302$	0	0
$303$	−7261.48	−1.37677
$304$	0	0
$305$	1947.11	0.365545
$306$	0	0
$307$	−1328.32	−0.246943	−0.123471	−	0.992348i	$-0.539403\pi$
$307$	−1328.32	−0.246943	−0.123471	+	0.992348i	$0.539403\pi$
$308$	0	0
$309$	9779.75	1.80049
$310$	0	0
$311$	−4868.68	−0.887709	−0.443855	−	0.896099i	$-0.646389\pi$
$311$	−4868.68	−0.887709	−0.443855	+	0.896099i	$0.646389\pi$
$312$	0	0
$313$	7733.39	1.39654	0.698270	−	0.715835i	$-0.253954\pi$
$313$	7733.39	1.39654	0.698270	+	0.715835i	$0.253954\pi$
$314$	0	0
$315$	−1516.72	−0.271294
$316$	0	0
$317$	−8175.03	−1.44844	−0.724220	−	0.689569i	$-0.757800\pi$
$317$	−8175.03	−1.44844	−0.724220	+	0.689569i	$0.757800\pi$
$318$	0	0
$319$	4037.61	0.708660
$320$	0	0
$321$	−477.744	−0.0830688
$322$	0	0
$323$	591.261	0.101853
$324$	0	0
$325$	2223.24	0.379455
$326$	0	0
$327$	−11396.5	−1.92729
$328$	0	0
$329$	−903.375	−0.151382
$330$	0	0
$331$	2040.76	0.338884	0.169442	−	0.985540i	$-0.445803\pi$
$331$	2040.76	0.338884	0.169442	+	0.985540i	$0.445803\pi$
$332$	0	0
$333$	−6116.19	−1.00650
$334$	0	0
$335$	−3488.97	−0.569023
$336$	0	0
$337$	7349.73	1.18803	0.594013	−	0.804455i	$-0.297543\pi$
$337$	7349.73	1.18803	0.594013	+	0.804455i	$0.297543\pi$
$338$	0	0
$339$	3656.87	0.585882
$340$	0	0
$341$	61.3576	0.00974399
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−849.896	−0.132629
$346$	0	0
$347$	12069.9	1.86728	0.933642	−	0.358207i	$-0.116612\pi$
$347$	12069.9	1.86728	0.933642	+	0.358207i	$0.116612\pi$
$348$	0	0
$349$	−4484.96	−0.687892	−0.343946	−	0.938989i	$-0.611764\pi$
$349$	−4484.96	−0.687892	−0.343946	+	0.938989i	$0.611764\pi$
$350$	0	0
$351$	12182.9	1.85263
$352$	0	0
$353$	−12762.5	−1.92430	−0.962151	−	0.272517i	$-0.912144\pi$
$353$	−12762.5	−1.92430	−0.962151	+	0.272517i	$0.912144\pi$
$354$	0	0
$355$	2617.25	0.391293
$356$	0	0
$357$	278.091	0.0412272
$358$	0	0
$359$	2419.42	0.355689	0.177844	−	0.984059i	$-0.443088\pi$
$359$	2419.42	0.355689	0.177844	+	0.984059i	$0.443088\pi$
$360$	0	0
$361$	8720.49	1.27139
$362$	0	0
$363$	−3558.33	−0.514501
$364$	0	0
$365$	332.342	0.0476591
$366$	0	0
$367$	7129.74	1.01409	0.507043	−	0.861921i	$-0.330739\pi$
$367$	7129.74	1.01409	0.507043	+	0.861921i	$0.330739\pi$
$368$	0	0
$369$	4128.89	0.582497
$370$	0	0
$371$	−2719.00	−0.380495
$372$	0	0
$373$	11596.9	1.60983	0.804914	−	0.593391i	$-0.202211\pi$
$373$	11596.9	1.60983	0.804914	+	0.593391i	$0.202211\pi$
$374$	0	0
$375$	1048.32	0.144361
$376$	0	0
$377$	11924.4	1.62901
$378$	0	0
$379$	12770.8	1.73085	0.865424	−	0.501040i	$-0.167049\pi$
$379$	12770.8	1.73085	0.865424	+	0.501040i	$0.167049\pi$
$380$	0	0
$381$	−9952.25	−1.33824
$382$	0	0
$383$	−7470.10	−0.996617	−0.498308	−	0.867000i	$-0.666045\pi$
$383$	−7470.10	−0.996617	−0.498308	+	0.867000i	$0.666045\pi$
$384$	0	0
$385$	−1053.91	−0.139512
$386$	0	0
$387$	12941.8	1.69992
$388$	0	0
$389$	8749.77	1.14044	0.570220	−	0.821492i	$-0.306858\pi$
$389$	8749.77	1.14044	0.570220	+	0.821492i	$0.306858\pi$
$390$	0	0
$391$	96.0092	0.0124179
$392$	0	0
$393$	−8676.41	−1.11366
$394$	0	0
$395$	2634.91	0.335637
$396$	0	0
$397$	5375.25	0.679537	0.339769	−	0.940509i	$-0.389651\pi$
$397$	5375.25	0.679537	0.339769	+	0.940509i	$0.389651\pi$
$398$	0	0
$399$	7327.58	0.919393
$400$	0	0
$401$	7361.33	0.916727	0.458363	−	0.888765i	$-0.348436\pi$
$401$	7361.33	0.916727	0.458363	+	0.888765i	$0.348436\pi$
$402$	0	0
$403$	181.209	0.0223987
$404$	0	0
$405$	−105.618	−0.0129585
$406$	0	0
$407$	−4249.88	−0.517589
$408$	0	0
$409$	−2612.45	−0.315837	−0.157919	−	0.987452i	$-0.550478\pi$
$409$	−2612.45	−0.315837	−0.157919	+	0.987452i	$0.550478\pi$
$410$	0	0
$411$	5419.57	0.650433
$412$	0	0
$413$	5869.51	0.699321
$414$	0	0
$415$	−350.133	−0.0414153
$416$	0	0
$417$	−4247.68	−0.498824
$418$	0	0
$419$	−4398.21	−0.512808	−0.256404	−	0.966570i	$-0.582538\pi$
$419$	−4398.21	−0.512808	−0.256404	+	0.966570i	$0.582538\pi$
$420$	0	0
$421$	9723.32	1.12562	0.562810	−	0.826587i	$-0.309720\pi$
$421$	9723.32	1.12562	0.562810	+	0.826587i	$0.309720\pi$
$422$	0	0
$423$	5592.54	0.642833
$424$	0	0
$425$	−118.425	−0.0135163
$426$	0	0
$427$	−2725.96	−0.308942
$428$	0	0
$429$	22457.7	2.52744
$430$	0	0
$431$	14314.5	1.59978	0.799892	−	0.600144i	$-0.204890\pi$
$431$	14314.5	1.59978	0.799892	+	0.600144i	$0.204890\pi$
$432$	0	0
$433$	−2373.62	−0.263438	−0.131719	−	0.991287i	$-0.542050\pi$
$433$	−2373.62	−0.263438	−0.131719	+	0.991287i	$0.542050\pi$
$434$	0	0
$435$	5622.70	0.619742
$436$	0	0
$437$	2529.80	0.276927
$438$	0	0
$439$	9533.46	1.03646	0.518231	−	0.855240i	$-0.326591\pi$
$439$	9533.46	1.03646	0.518231	+	0.855240i	$0.326591\pi$
$440$	0	0
$441$	2123.41	0.229286
$442$	0	0
$443$	−6647.94	−0.712987	−0.356493	−	0.934298i	$-0.616028\pi$
$443$	−6647.94	−0.712987	−0.356493	+	0.934298i	$0.616028\pi$
$444$	0	0
$445$	−46.3963	−0.00494246
$446$	0	0
$447$	−15331.7	−1.62229
$448$	0	0
$449$	−768.256	−0.0807489	−0.0403744	−	0.999185i	$-0.512855\pi$
$449$	−768.256	−0.0807489	−0.0403744	+	0.999185i	$0.512855\pi$
$450$	0	0
$451$	2868.99	0.299547
$452$	0	0
$453$	−24951.5	−2.58791
$454$	0	0
$455$	−3112.53	−0.320698
$456$	0	0
$457$	−3323.50	−0.340190	−0.170095	−	0.985428i	$-0.554407\pi$
$457$	−3323.50	−0.340190	−0.170095	+	0.985428i	$0.554407\pi$
$458$	0	0
$459$	−648.944	−0.0659915
$460$	0	0
$461$	−18840.7	−1.90347	−0.951733	−	0.306926i	$-0.900700\pi$
$461$	−18840.7	−1.90347	−0.951733	+	0.306926i	$0.900700\pi$
$462$	0	0
$463$	10759.1	1.07995	0.539977	−	0.841679i	$-0.318433\pi$
$463$	10759.1	1.07995	0.539977	+	0.841679i	$0.318433\pi$
$464$	0	0
$465$	85.4456	0.00852138
$466$	0	0
$467$	−7441.70	−0.737390	−0.368695	−	0.929550i	$-0.620195\pi$
$467$	−7441.70	−0.737390	−0.368695	+	0.929550i	$0.620195\pi$
$468$	0	0
$469$	4884.56	0.480913
$470$	0	0
$471$	−17878.0	−1.74899
$472$	0	0
$473$	8992.73	0.874178
$474$	0	0
$475$	−3120.45	−0.301423
$476$	0	0
$477$	16832.6	1.61574
$478$	0	0
$479$	−5691.97	−0.542949	−0.271475	−	0.962446i	$-0.587511\pi$
$479$	−5691.97	−0.542949	−0.271475	+	0.962446i	$0.587511\pi$
$480$	0	0
$481$	−12551.3	−1.18979
$482$	0	0
$483$	1189.85	0.112092
$484$	0	0
$485$	−20.9746	−0.00196373
$486$	0	0
$487$	2020.25	0.187980	0.0939899	−	0.995573i	$-0.470038\pi$
$487$	2020.25	0.187980	0.0939899	+	0.995573i	$0.470038\pi$
$488$	0	0
$489$	4981.12	0.460642
$490$	0	0
$491$	−7636.02	−0.701851	−0.350925	−	0.936403i	$-0.614133\pi$
$491$	−7636.02	−0.701851	−0.350925	+	0.936403i	$0.614133\pi$
$492$	0	0
$493$	−635.173	−0.0580259
$494$	0	0
$495$	6524.44	0.592428
$496$	0	0
$497$	−3664.15	−0.330703
$498$	0	0
$499$	−6284.56	−0.563799	−0.281900	−	0.959444i	$-0.590964\pi$
$499$	−6284.56	−0.563799	−0.281900	+	0.959444i	$0.590964\pi$
$500$	0	0
$501$	24625.6	2.19599
$502$	0	0
$503$	−11310.9	−1.00264	−0.501319	−	0.865262i	$-0.667152\pi$
$503$	−11310.9	−1.00264	−0.501319	+	0.865262i	$0.667152\pi$
$504$	0	0
$505$	−4329.22	−0.381481
$506$	0	0
$507$	47899.7	4.19586
$508$	0	0
$509$	10712.7	0.932876	0.466438	−	0.884554i	$-0.345537\pi$
$509$	10712.7	0.932876	0.466438	+	0.884554i	$0.345537\pi$
$510$	0	0
$511$	−465.279	−0.0402793
$512$	0	0
$513$	−17099.4	−1.47165
$514$	0	0
$515$	5830.58	0.498886
$516$	0	0
$517$	3886.02	0.330574
$518$	0	0
$519$	19685.9	1.66496
$520$	0	0
$521$	17721.9	1.49023	0.745116	−	0.666935i	$-0.232394\pi$
$521$	17721.9	1.49023	0.745116	+	0.666935i	$0.232394\pi$
$522$	0	0
$523$	−237.193	−0.0198312	−0.00991562	−	0.999951i	$-0.503156\pi$
$523$	−237.193	−0.0198312	−0.00991562	+	0.999951i	$0.503156\pi$
$524$	0	0
$525$	−1467.65	−0.122007
$526$	0	0
$527$	−9.65243	−0.000797849 0
$528$	0	0
$529$	−11756.2	−0.966237
$530$	0	0
$531$	−36336.4	−2.96962
$532$	0	0
$533$	8473.06	0.688572
$534$	0	0
$535$	−284.826	−0.0230170
$536$	0	0
$537$	−25466.4	−2.04647
$538$	0	0
$539$	1475.47	0.117909
$540$	0	0
$541$	−5352.94	−0.425399	−0.212699	−	0.977118i	$-0.568226\pi$
$541$	−5352.94	−0.425399	−0.212699	+	0.977118i	$0.568226\pi$
$542$	0	0
$543$	−7546.06	−0.596376
$544$	0	0
$545$	−6794.45	−0.534022
$546$	0	0
$547$	192.162	0.0150206	0.00751030	−	0.999972i	$-0.497609\pi$
$547$	192.162	0.0150206	0.00751030	+	0.999972i	$0.497609\pi$
$548$	0	0
$549$	16875.6	1.31190
$550$	0	0
$551$	−16736.5	−1.29401
$552$	0	0
$553$	−3688.87	−0.283665
$554$	0	0
$555$	−5918.31	−0.452646
$556$	0	0
$557$	−4850.62	−0.368990	−0.184495	−	0.982833i	$-0.559065\pi$
$557$	−4850.62	−0.368990	−0.184495	+	0.982833i	$0.559065\pi$
$558$	0	0
$559$	26558.4	2.00949
$560$	0	0
$561$	−1196.25	−0.0900283
$562$	0	0
$563$	−9699.11	−0.726055	−0.363027	−	0.931778i	$-0.618257\pi$
$563$	−9699.11	−0.726055	−0.363027	+	0.931778i	$0.618257\pi$
$564$	0	0
$565$	2180.19	0.162338
$566$	0	0
$567$	147.865	0.0109520
$568$	0	0
$569$	3109.53	0.229100	0.114550	−	0.993417i	$-0.463457\pi$
$569$	3109.53	0.229100	0.114550	+	0.993417i	$0.463457\pi$
$570$	0	0
$571$	14476.2	1.06097	0.530483	−	0.847695i	$-0.322010\pi$
$571$	14476.2	1.06097	0.530483	+	0.847695i	$0.322010\pi$
$572$	0	0
$573$	−3490.23	−0.254461
$574$	0	0
$575$	−506.699	−0.0367492
$576$	0	0
$577$	−2208.23	−0.159323	−0.0796617	−	0.996822i	$-0.525384\pi$
$577$	−2208.23	−0.159323	−0.0796617	+	0.996822i	$0.525384\pi$
$578$	0	0
$579$	−43451.4	−3.11879
$580$	0	0
$581$	490.186	0.0350023
$582$	0	0
$583$	11696.2	0.830889
$584$	0	0
$585$	19268.8	1.36182
$586$	0	0
$587$	23988.7	1.68675	0.843374	−	0.537327i	$-0.180566\pi$
$587$	23988.7	1.68675	0.843374	+	0.537327i	$0.180566\pi$
$588$	0	0
$589$	−254.338	−0.0177925
$590$	0	0
$591$	12180.2	0.847762
$592$	0	0
$593$	−15869.4	−1.09895	−0.549474	−	0.835511i	$-0.685172\pi$
$593$	−15869.4	−1.09895	−0.549474	+	0.835511i	$0.685172\pi$
$594$	0	0
$595$	165.795	0.0114234
$596$	0	0
$597$	10711.6	0.734335
$598$	0	0
$599$	15236.6	1.03932	0.519660	−	0.854373i	$-0.326059\pi$
$599$	15236.6	1.03932	0.519660	+	0.854373i	$0.326059\pi$
$600$	0	0
$601$	12258.8	0.832026	0.416013	−	0.909359i	$-0.363427\pi$
$601$	12258.8	0.832026	0.416013	+	0.909359i	$0.363427\pi$
$602$	0	0
$603$	−30238.9	−2.04216
$604$	0	0
$605$	−2121.44	−0.142560
$606$	0	0
$607$	−23487.2	−1.57054	−0.785269	−	0.619155i	$-0.787475\pi$
$607$	−23487.2	−1.57054	−0.785269	+	0.619155i	$0.787475\pi$
$608$	0	0
$609$	−7871.78	−0.523778
$610$	0	0
$611$	11476.7	0.759896
$612$	0	0
$613$	−22305.3	−1.46966	−0.734830	−	0.678251i	$-0.762738\pi$
$613$	−22305.3	−1.46966	−0.734830	+	0.678251i	$0.762738\pi$
$614$	0	0
$615$	3995.31	0.261962
$616$	0	0
$617$	3285.91	0.214402	0.107201	−	0.994237i	$-0.465811\pi$
$617$	3285.91	0.214402	0.107201	+	0.994237i	$0.465811\pi$
$618$	0	0
$619$	11613.1	0.754069	0.377035	−	0.926199i	$-0.376944\pi$
$619$	11613.1	0.754069	0.377035	+	0.926199i	$0.376944\pi$
$620$	0	0
$621$	−2776.61	−0.179423
$622$	0	0
$623$	64.9548	0.00417714
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−31520.8	−2.00769
$628$	0	0
$629$	668.567	0.0423808
$630$	0	0
$631$	−6890.91	−0.434743	−0.217372	−	0.976089i	$-0.569748\pi$
$631$	−6890.91	−0.434743	−0.217372	+	0.976089i	$0.569748\pi$
$632$	0	0
$633$	27332.7	1.71623
$634$	0	0
$635$	−5933.43	−0.370804
$636$	0	0
$637$	4357.55	0.271040
$638$	0	0
$639$	22683.7	1.40431
$640$	0	0
$641$	18769.3	1.15654	0.578269	−	0.815846i	$-0.303728\pi$
$641$	18769.3	1.15654	0.578269	+	0.815846i	$0.303728\pi$
$642$	0	0
$643$	3142.30	0.192722	0.0963609	−	0.995346i	$-0.469280\pi$
$643$	3142.30	0.192722	0.0963609	+	0.995346i	$0.469280\pi$
$644$	0	0
$645$	12523.1	0.764492
$646$	0	0
$647$	−19038.1	−1.15683	−0.578413	−	0.815744i	$-0.696328\pi$
$647$	−19038.1	−1.15683	−0.578413	+	0.815744i	$0.696328\pi$
$648$	0	0
$649$	−25248.7	−1.52711
$650$	0	0
$651$	−119.624	−0.00720188
$652$	0	0
$653$	−20538.6	−1.23084	−0.615420	−	0.788199i	$-0.711014\pi$
$653$	−20538.6	−1.23084	−0.615420	+	0.788199i	$0.711014\pi$
$654$	0	0
$655$	−5172.78	−0.308576
$656$	0	0
$657$	2880.41	0.171043
$658$	0	0
$659$	937.046	0.0553902	0.0276951	−	0.999616i	$-0.491183\pi$
$659$	937.046	0.0553902	0.0276951	+	0.999616i	$0.491183\pi$
$660$	0	0
$661$	21116.5	1.24257	0.621283	−	0.783586i	$-0.286612\pi$
$661$	21116.5	1.24257	0.621283	+	0.783586i	$0.286612\pi$
$662$	0	0
$663$	−3532.92	−0.206949
$664$	0	0
$665$	4368.62	0.254749
$666$	0	0
$667$	−2717.69	−0.157765
$668$	0	0
$669$	−36677.5	−2.11963
$670$	0	0
$671$	11726.2	0.674640
$672$	0	0
$673$	13825.9	0.791903	0.395952	−	0.918271i	$-0.370415\pi$
$673$	13825.9	0.791903	0.395952	+	0.918271i	$0.370415\pi$
$674$	0	0
$675$	3424.87	0.195294
$676$	0	0
$677$	−16928.4	−0.961021	−0.480510	−	0.876989i	$-0.659549\pi$
$677$	−16928.4	−0.961021	−0.480510	+	0.876989i	$0.659549\pi$
$678$	0	0
$679$	29.3645	0.00165966
$680$	0	0
$681$	512.543	0.0288409
$682$	0	0
$683$	13817.3	0.774091	0.387045	−	0.922061i	$-0.373496\pi$
$683$	13817.3	0.774091	0.387045	+	0.922061i	$0.373496\pi$
$684$	0	0
$685$	3231.09	0.180224
$686$	0	0
$687$	25322.6	1.40628
$688$	0	0
$689$	34542.8	1.90998
$690$	0	0
$691$	23671.6	1.30320	0.651600	−	0.758563i	$-0.274098\pi$
$691$	23671.6	1.30320	0.651600	+	0.758563i	$0.274098\pi$
$692$	0	0
$693$	−9134.22	−0.500693
$694$	0	0
$695$	−2532.42	−0.138216
$696$	0	0
$697$	−451.333	−0.0245272
$698$	0	0
$699$	−29614.5	−1.60246
$700$	0	0
$701$	−17009.7	−0.916472	−0.458236	−	0.888831i	$-0.651519\pi$
$701$	−17009.7	−0.916472	−0.458236	+	0.888831i	$0.651519\pi$
$702$	0	0
$703$	17616.5	0.945117
$704$	0	0
$705$	5411.60	0.289096
$706$	0	0
$707$	6060.91	0.322410
$708$	0	0
$709$	22038.9	1.16740	0.583701	−	0.811969i	$-0.301604\pi$
$709$	22038.9	1.16740	0.583701	+	0.811969i	$0.301604\pi$
$710$	0	0
$711$	22836.8	1.20456
$712$	0	0
$713$	−41.2994	−0.00216925
$714$	0	0
$715$	13389.1	0.700312
$716$	0	0
$717$	−19143.4	−0.997106
$718$	0	0
$719$	−7287.44	−0.377991	−0.188996	−	0.981978i	$-0.560523\pi$
$719$	−7287.44	−0.377991	−0.188996	+	0.981978i	$0.560523\pi$
$720$	0	0
$721$	−8162.82	−0.421636
$722$	0	0
$723$	−18582.0	−0.955839
$724$	0	0
$725$	3352.20	0.171721
$726$	0	0
$727$	29676.7	1.51396	0.756980	−	0.653438i	$-0.226674\pi$
$727$	29676.7	1.51396	0.756980	+	0.653438i	$0.226674\pi$
$728$	0	0
$729$	−31936.3	−1.62253
$730$	0	0
$731$	−1414.68	−0.0715786
$732$	0	0
$733$	23111.8	1.16460	0.582300	−	0.812974i	$-0.302153\pi$
$733$	23111.8	1.16460	0.582300	+	0.812974i	$0.302153\pi$
$734$	0	0
$735$	2054.72	0.103115
$736$	0	0
$737$	−21011.7	−1.05017
$738$	0	0
$739$	31171.4	1.55164	0.775818	−	0.630957i	$-0.217338\pi$
$739$	31171.4	1.55164	0.775818	+	0.630957i	$0.217338\pi$
$740$	0	0
$741$	−93091.1	−4.61510
$742$	0	0
$743$	31324.4	1.54668	0.773338	−	0.633993i	$-0.218585\pi$
$743$	31324.4	1.54668	0.773338	+	0.633993i	$0.218585\pi$
$744$	0	0
$745$	−9140.58	−0.449510
$746$	0	0
$747$	−3034.60	−0.148635
$748$	0	0
$749$	398.756	0.0194529
$750$	0	0
$751$	−4032.20	−0.195922	−0.0979608	−	0.995190i	$-0.531232\pi$
$751$	−4032.20	−0.195922	−0.0979608	+	0.995190i	$0.531232\pi$
$752$	0	0
$753$	25852.1	1.25113
$754$	0	0
$755$	−14875.8	−0.717069
$756$	0	0
$757$	34263.7	1.64509	0.822546	−	0.568699i	$-0.192553\pi$
$757$	34263.7	1.64509	0.822546	+	0.568699i	$0.192553\pi$
$758$	0	0
$759$	−5118.36	−0.244775
$760$	0	0
$761$	7265.88	0.346108	0.173054	−	0.984912i	$-0.444637\pi$
$761$	7265.88	0.346108	0.173054	+	0.984912i	$0.444637\pi$
$762$	0	0
$763$	9512.22	0.451331
$764$	0	0
$765$	−1026.39	−0.0485087
$766$	0	0
$767$	−74567.5	−3.51040
$768$	0	0
$769$	38116.2	1.78739	0.893695	−	0.448674i	$-0.148104\pi$
$769$	38116.2	1.78739	0.893695	+	0.448674i	$0.148104\pi$
$770$	0	0
$771$	−50591.3	−2.36317
$772$	0	0
$773$	16158.2	0.751838	0.375919	−	0.926652i	$-0.377327\pi$
$773$	16158.2	0.751838	0.375919	+	0.926652i	$0.377327\pi$
$774$	0	0
$775$	50.9417	0.00236114
$776$	0	0
$777$	8285.64	0.382555
$778$	0	0
$779$	−11892.4	−0.546972
$780$	0	0
$781$	15761.9	0.722160
$782$	0	0
$783$	18369.3	0.838400
$784$	0	0
$785$	−10658.7	−0.484618
$786$	0	0
$787$	−5092.49	−0.230658	−0.115329	−	0.993327i	$-0.536792\pi$
$787$	−5092.49	−0.230658	−0.115329	+	0.993327i	$0.536792\pi$
$788$	0	0
$789$	−49680.6	−2.24167
$790$	0	0
$791$	−3052.26	−0.137201
$792$	0	0
$793$	34631.1	1.55080
$794$	0	0
$795$	16288.0	0.726635
$796$	0	0
$797$	−34666.2	−1.54070	−0.770350	−	0.637621i	$-0.779919\pi$
$797$	−34666.2	−1.54070	−0.770350	+	0.637621i	$0.779919\pi$
$798$	0	0
$799$	−611.326	−0.0270678
$800$	0	0
$801$	−402.116	−0.0177379
$802$	0	0
$803$	2001.47	0.0879582
$804$	0	0
$805$	709.379	0.0310588
$806$	0	0
$807$	27279.9	1.18996
$808$	0	0
$809$	15126.2	0.657365	0.328683	−	0.944440i	$-0.393395\pi$
$809$	15126.2	0.657365	0.328683	+	0.944440i	$0.393395\pi$
$810$	0	0
$811$	−29416.5	−1.27368	−0.636840	−	0.770996i	$-0.719759\pi$
$811$	−29416.5	−1.27368	−0.636840	+	0.770996i	$0.719759\pi$
$812$	0	0
$813$	52384.9	2.25980
$814$	0	0
$815$	2969.69	0.127637
$816$	0	0
$817$	−37276.3	−1.59625
$818$	0	0
$819$	−26976.3	−1.15095
$820$	0	0
$821$	−15334.4	−0.651856	−0.325928	−	0.945395i	$-0.605677\pi$
$821$	−15334.4	−0.651856	−0.325928	+	0.945395i	$0.605677\pi$
$822$	0	0
$823$	11003.7	0.466056	0.233028	−	0.972470i	$-0.425137\pi$
$823$	11003.7	0.466056	0.233028	+	0.972470i	$0.425137\pi$
$824$	0	0
$825$	6313.36	0.266428
$826$	0	0
$827$	3261.59	0.137142	0.0685711	−	0.997646i	$-0.478156\pi$
$827$	3261.59	0.137142	0.0685711	+	0.997646i	$0.478156\pi$
$828$	0	0
$829$	5163.30	0.216319	0.108160	−	0.994134i	$-0.465504\pi$
$829$	5163.30	0.216319	0.108160	+	0.994134i	$0.465504\pi$
$830$	0	0
$831$	−13185.1	−0.550406
$832$	0	0
$833$	−232.113	−0.00965453
$834$	0	0
$835$	14681.5	0.608472
$836$	0	0
$837$	279.150	0.0115279
$838$	0	0
$839$	5641.70	0.232149	0.116075	−	0.993241i	$-0.462969\pi$
$839$	5641.70	0.232149	0.116075	+	0.993241i	$0.462969\pi$
$840$	0	0
$841$	−6409.46	−0.262801
$842$	0	0
$843$	−65801.4	−2.68840
$844$	0	0
$845$	28557.3	1.16260
$846$	0	0
$847$	2970.01	0.120485
$848$	0	0
$849$	−52546.8	−2.12415
$850$	0	0
$851$	2860.57	0.115228
$852$	0	0
$853$	−7799.52	−0.313072	−0.156536	−	0.987672i	$-0.550033\pi$
$853$	−7799.52	−0.313072	−0.156536	+	0.987672i	$0.550033\pi$
$854$	0	0
$855$	−27044.9	−1.08177
$856$	0	0
$857$	−21540.0	−0.858568	−0.429284	−	0.903170i	$-0.641234\pi$
$857$	−21540.0	−0.858568	−0.429284	+	0.903170i	$0.641234\pi$
$858$	0	0
$859$	−4447.97	−0.176674	−0.0883370	−	0.996091i	$-0.528155\pi$
$859$	−4447.97	−0.176674	−0.0883370	+	0.996091i	$0.528155\pi$
$860$	0	0
$861$	−5593.43	−0.221398
$862$	0	0
$863$	9425.21	0.371770	0.185885	−	0.982571i	$-0.440485\pi$
$863$	9425.21	0.371770	0.185885	+	0.982571i	$0.440485\pi$
$864$	0	0
$865$	11736.5	0.461335
$866$	0	0
$867$	−41015.2	−1.60663
$868$	0	0
$869$	15868.3	0.619442
$870$	0	0
$871$	−62054.5	−2.41405
$872$	0	0
$873$	−181.787	−0.00704761
$874$	0	0
$875$	−875.000	−0.0338062
$876$	0	0
$877$	22346.1	0.860403	0.430201	−	0.902733i	$-0.358443\pi$
$877$	22346.1	0.860403	0.430201	+	0.902733i	$0.358443\pi$
$878$	0	0
$879$	−60928.6	−2.33796
$880$	0	0
$881$	12074.9	0.461762	0.230881	−	0.972982i	$-0.425839\pi$
$881$	12074.9	0.461762	0.230881	+	0.972982i	$0.425839\pi$
$882$	0	0
$883$	30499.6	1.16239	0.581196	−	0.813764i	$-0.302585\pi$
$883$	30499.6	1.16239	0.581196	+	0.813764i	$0.302585\pi$
$884$	0	0
$885$	−35160.8	−1.33550
$886$	0	0
$887$	−23344.2	−0.883675	−0.441838	−	0.897095i	$-0.645673\pi$
$887$	−23344.2	−0.883675	−0.441838	+	0.897095i	$0.645673\pi$
$888$	0	0
$889$	8306.80	0.313387
$890$	0	0
$891$	−636.068	−0.0239159
$892$	0	0
$893$	−16108.2	−0.603628
$894$	0	0
$895$	−15182.8	−0.567044
$896$	0	0
$897$	−15116.2	−0.562669
$898$	0	0
$899$	273.227	0.0101364
$900$	0	0
$901$	−1839.98	−0.0680341
$902$	0	0
$903$	−17532.4	−0.646113
$904$	0	0
$905$	−4498.88	−0.165246
$906$	0	0
$907$	15092.5	0.552523	0.276262	−	0.961082i	$-0.410904\pi$
$907$	15092.5	0.552523	0.276262	+	0.961082i	$0.410904\pi$
$908$	0	0
$909$	−37521.3	−1.36909
$910$	0	0
$911$	15207.8	0.553081	0.276541	−	0.961002i	$-0.410812\pi$
$911$	15207.8	0.553081	0.276541	+	0.961002i	$0.410812\pi$
$912$	0	0
$913$	−2108.62	−0.0764348
$914$	0	0
$915$	16329.6	0.589990
$916$	0	0
$917$	7241.90	0.260794
$918$	0	0
$919$	−24818.1	−0.890831	−0.445415	−	0.895324i	$-0.646944\pi$
$919$	−24818.1	−0.890831	−0.445415	+	0.895324i	$0.646944\pi$
$920$	0	0
$921$	−11140.1	−0.398566
$922$	0	0
$923$	46550.1	1.66004
$924$	0	0
$925$	−3528.44	−0.125421
$926$	0	0
$927$	50533.7	1.79045
$928$	0	0
$929$	39906.4	1.40935	0.704675	−	0.709530i	$-0.251093\pi$
$929$	39906.4	1.40935	0.704675	+	0.709530i	$0.251093\pi$
$930$	0	0
$931$	−6116.07	−0.215302
$932$	0	0
$933$	−40831.7	−1.43276
$934$	0	0
$935$	−713.194	−0.0249454
$936$	0	0
$937$	16923.0	0.590020	0.295010	−	0.955494i	$-0.404677\pi$
$937$	16923.0	0.590020	0.295010	+	0.955494i	$0.404677\pi$
$938$	0	0
$939$	64856.8	2.25402
$940$	0	0
$941$	−53014.1	−1.83657	−0.918285	−	0.395921i	$-0.870426\pi$
$941$	−53014.1	−1.83657	−0.918285	+	0.395921i	$0.870426\pi$
$942$	0	0
$943$	−1931.10	−0.0666864
$944$	0	0
$945$	−4794.82	−0.165054
$946$	0	0
$947$	25798.9	0.885271	0.442636	−	0.896702i	$-0.354043\pi$
$947$	25798.9	0.885271	0.442636	+	0.896702i	$0.354043\pi$
$948$	0	0
$949$	5911.00	0.202191
$950$	0	0
$951$	−68560.6	−2.33778
$952$	0	0
$953$	−17942.7	−0.609885	−0.304943	−	0.952371i	$-0.598637\pi$
$953$	−17942.7	−0.609885	−0.304943	+	0.952371i	$0.598637\pi$
$954$	0	0
$955$	−2080.84	−0.0705072
$956$	0	0
$957$	33861.8	1.14378
$958$	0	0
$959$	−4523.53	−0.152317
$960$	0	0
$961$	−29786.8	−0.999861
$962$	0	0
$963$	−2468.59	−0.0826055
$964$	0	0
$965$	−25905.2	−0.864165
$966$	0	0
$967$	−19668.3	−0.654073	−0.327036	−	0.945012i	$-0.606050\pi$
$967$	−19668.3	−0.654073	−0.327036	+	0.945012i	$0.606050\pi$
$968$	0	0
$969$	4958.67	0.164392
$970$	0	0
$971$	−6332.97	−0.209304	−0.104652	−	0.994509i	$-0.533373\pi$
$971$	−6332.97	−0.209304	−0.104652	+	0.994509i	$0.533373\pi$
$972$	0	0
$973$	3545.39	0.116814
$974$	0	0
$975$	18645.4	0.612441
$976$	0	0
$977$	−11334.1	−0.371145	−0.185573	−	0.982631i	$-0.559414\pi$
$977$	−11334.1	−0.371145	−0.185573	+	0.982631i	$0.559414\pi$
$978$	0	0
$979$	−279.414	−0.00912166
$980$	0	0
$981$	−58887.4	−1.91655
$982$	0	0
$983$	37654.3	1.22175	0.610877	−	0.791725i	$-0.290817\pi$
$983$	37654.3	1.22175	0.610877	+	0.791725i	$0.290817\pi$
$984$	0	0
$985$	7261.72	0.234901
$986$	0	0
$987$	−7576.24	−0.244331
$988$	0	0
$989$	−6052.95	−0.194613
$990$	0	0
$991$	53441.5	1.71304	0.856522	−	0.516111i	$-0.172621\pi$
$991$	53441.5	1.71304	0.856522	+	0.516111i	$0.172621\pi$
$992$	0	0
$993$	17115.1	0.546959
$994$	0	0
$995$	6386.16	0.203472
$996$	0	0
$997$	37919.3	1.20453	0.602266	−	0.798296i	$-0.294265\pi$
$997$	37919.3	1.20453	0.602266	+	0.798296i	$0.294265\pi$
$998$	0	0
$999$	−19335.1	−0.612348

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.4.a.u.1.3		3
4.3	odd	2		35.4.a.c.1.1	✓	3
8.3	odd	2		2240.4.a.bt.1.3		3
8.5	even	2		2240.4.a.bv.1.1		3
12.11	even	2		315.4.a.p.1.3		3
20.3	even	4		175.4.b.e.99.6		6
20.7	even	4		175.4.b.e.99.1		6
20.19	odd	2		175.4.a.f.1.3		3
28.3	even	6		245.4.e.n.226.3		6
28.11	odd	6		245.4.e.m.226.3		6
28.19	even	6		245.4.e.n.116.3		6
28.23	odd	6		245.4.e.m.116.3		6
28.27	even	2		245.4.a.l.1.1		3
60.59	even	2		1575.4.a.ba.1.1		3
84.83	odd	2		2205.4.a.bm.1.3		3
140.139	even	2		1225.4.a.y.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.4.a.c.1.1	✓	3	4.3	odd	2
175.4.a.f.1.3		3	20.19	odd	2
175.4.b.e.99.1		6	20.7	even	4
175.4.b.e.99.6		6	20.3	even	4
245.4.a.l.1.1		3	28.27	even	2
245.4.e.m.116.3		6	28.23	odd	6
245.4.e.m.226.3		6	28.11	odd	6
245.4.e.n.116.3		6	28.19	even	6
245.4.e.n.226.3		6	28.3	even	6
315.4.a.p.1.3		3	12.11	even	2
560.4.a.u.1.3		3	1.1	even	1	trivial
1225.4.a.y.1.3		3	140.139	even	2
1575.4.a.ba.1.1		3	60.59	even	2
2205.4.a.bm.1.3		3	84.83	odd	2
2240.4.a.bt.1.3		3	8.3	odd	2
2240.4.a.bv.1.1		3	8.5	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:	\( N \)	\(=\)	\( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	560.a (trivial)

\(f(q)\)	\(=\)	\(q+8.38660 q^{3} +5.00000 q^{5} -7.00000 q^{7} +43.3350 q^{9} +30.1117 q^{11} +88.9295 q^{13} +41.9330 q^{15} -4.73699 q^{17} -124.818 q^{19} -58.7062 q^{21} -20.2680 q^{23} +25.0000 q^{25} +136.995 q^{27} +134.088 q^{29} +2.03767 q^{31} +252.534 q^{33} -35.0000 q^{35} -141.137 q^{37} +745.816 q^{39} +95.2784 q^{41} +298.646 q^{43} +216.675 q^{45} +129.054 q^{47} +49.0000 q^{49} -39.7272 q^{51} +388.429 q^{53} +150.558 q^{55} -1046.80 q^{57} -838.501 q^{59} +389.422 q^{61} -303.345 q^{63} +444.647 q^{65} -697.794 q^{67} -169.979 q^{69} +523.450 q^{71} +66.4684 q^{73} +209.665 q^{75} -210.782 q^{77} +526.982 q^{79} -21.1236 q^{81} -70.0265 q^{83} -23.6850 q^{85} +1124.54 q^{87} -9.27925 q^{89} -622.506 q^{91} +17.0891 q^{93} -624.089 q^{95} -4.19493 q^{97} +1304.89 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{3} + 15 q^{5} - 21 q^{7} + 81 q^{9} + 74 q^{11} + 44 q^{13} - 10 q^{15} - 52 q^{17} - 168 q^{19} + 14 q^{21} + 124 q^{23} + 75 q^{25} - 170 q^{27} + 332 q^{29} - 320 q^{31} - 106 q^{33} - 105 q^{35}+ \cdots + 3488 q^{99}+O(q^{100}) \) 3 * q - 2 * q^3 + 15 * q^5 - 21 * q^7 + 81 * q^9 + 74 * q^11 + 44 * q^13 - 10 * q^15 - 52 * q^17 - 168 * q^19 + 14 * q^21 + 124 * q^23 + 75 * q^25 - 170 * q^27 + 332 * q^29 - 320 * q^31 - 106 * q^33 - 105 * q^35 - 54 * q^37 + 982 * q^39 + 362 * q^41 + 16 * q^43 + 405 * q^45 + 730 * q^47 + 147 * q^49 + 1178 * q^51 + 110 * q^53 + 370 * q^55 - 956 * q^57 + 180 * q^59 + 1222 * q^61 - 567 * q^63 + 220 * q^65 - 204 * q^67 - 716 * q^69 + 136 * q^71 + 310 * q^73 - 50 * q^75 - 518 * q^77 + 1034 * q^79 + 2283 * q^81 + 1660 * q^83 - 260 * q^85 + 1574 * q^87 + 242 * q^89 - 308 * q^91 + 1376 * q^93 - 840 * q^95 + 100 * q^97 + 3488 * q^99

Introduction

L-functions

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