LMFDB - Embedded newform 1472.4.a.bd.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1472,4,Mod(1,1472)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1472.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1472, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1472 = 2^{6} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1472.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,1,0,20,0,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$86.8508115285$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.167313.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 16x^{2} + 4x + 24 $ x^4 - x^3 - 16x^2 + 4x + 24
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 184)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.19877$ of defining polynomial
Character	$\chi$	$=$	1472.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-7.81151 q^{3} +3.75144 q^{5} -31.3950 q^{7} +34.0197 q^{9} +11.5069 q^{11} +48.7341 q^{13} -29.3045 q^{15} -46.4231 q^{17} -77.9077 q^{19} +245.243 q^{21} +23.0000 q^{23} -110.927 q^{25} -54.8345 q^{27} +213.000 q^{29} -207.799 q^{31} -89.8865 q^{33} -117.777 q^{35} -197.651 q^{37} -380.687 q^{39} -383.696 q^{41} -437.399 q^{43} +127.623 q^{45} +156.104 q^{47} +642.648 q^{49} +362.634 q^{51} -12.0337 q^{53} +43.1676 q^{55} +608.577 q^{57} -559.360 q^{59} +37.7365 q^{61} -1068.05 q^{63} +182.823 q^{65} -30.4246 q^{67} -179.665 q^{69} +398.612 q^{71} -877.484 q^{73} +866.505 q^{75} -361.260 q^{77} -632.926 q^{79} -490.192 q^{81} +622.703 q^{83} -174.154 q^{85} -1663.85 q^{87} -780.748 q^{89} -1530.01 q^{91} +1623.23 q^{93} -292.267 q^{95} +964.791 q^{97} +391.462 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{3} + 20 q^{5} + 10 q^{7} + 23 q^{9} - 30 q^{11} + 153 q^{13} + 136 q^{15} - 68 q^{17} - 120 q^{19} + 426 q^{21} + 92 q^{23} - 76 q^{25} + 43 q^{27} + 315 q^{29} - 249 q^{31} - 504 q^{33} + 224 q^{35}+ \cdots - 1470 q^{99}+O(q^{100}) $ 4 * q + q^3 + 20 * q^5 + 10 * q^7 + 23 * q^9 - 30 * q^11 + 153 * q^13 + 136 * q^15 - 68 * q^17 - 120 * q^19 + 426 * q^21 + 92 * q^23 - 76 * q^25 + 43 * q^27 + 315 * q^29 - 249 * q^31 - 504 * q^33 + 224 * q^35 + 348 * q^37 + 61 * q^39 - 929 * q^41 + 50 * q^43 + 1028 * q^45 - 205 * q^47 + 456 * q^49 + 890 * q^51 + 578 * q^53 - 1128 * q^55 + 366 * q^57 - 664 * q^59 + 182 * q^61 - 624 * q^63 + 848 * q^65 + 1078 * q^67 + 23 * q^69 + 739 * q^71 + 921 * q^73 + 2883 * q^75 - 528 * q^77 - 1754 * q^79 + 788 * q^81 + 1020 * q^83 + 1072 * q^85 - 2121 * q^87 - 430 * q^89 + 1334 * q^91 - 1035 * q^93 - 744 * q^95 + 48 * q^97 - 1470 * 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
$3$	−7.81151	−1.50333	−0.751663	−	0.659547i	$-0.770748\pi$
$3$	−7.81151	−1.50333	−0.751663	+	0.659547i	$0.770748\pi$
$4$	0	0
$5$	3.75144	0.335539	0.167770	−	0.985826i	$-0.446343\pi$
$5$	3.75144	0.335539	0.167770	+	0.985826i	$0.446343\pi$
$6$	0	0
$7$	−31.3950	−1.69517	−0.847586	−	0.530658i	$-0.821945\pi$
$7$	−31.3950	−1.69517	−0.847586	+	0.530658i	$0.821945\pi$
$8$	0	0
$9$	34.0197	1.25999
$10$	0	0
$11$	11.5069	0.315406	0.157703	−	0.987487i	$-0.449591\pi$
$11$	11.5069	0.315406	0.157703	+	0.987487i	$0.449591\pi$
$12$	0	0
$13$	48.7341	1.03972	0.519862	−	0.854251i	$-0.325984\pi$
$13$	48.7341	1.03972	0.519862	+	0.854251i	$0.325984\pi$
$14$	0	0
$15$	−29.3045	−0.504425
$16$	0	0
$17$	−46.4231	−0.662309	−0.331154	−	0.943577i	$-0.607438\pi$
$17$	−46.4231	−0.662309	−0.331154	+	0.943577i	$0.607438\pi$
$18$	0	0
$19$	−77.9077	−0.940698	−0.470349	−	0.882481i	$-0.655872\pi$
$19$	−77.9077	−0.940698	−0.470349	+	0.882481i	$0.655872\pi$
$20$	0	0
$21$	245.243	2.54840
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	−110.927	−0.887413
$26$	0	0
$27$	−54.8345	−0.390849
$28$	0	0
$29$	213.000	1.36390	0.681950	−	0.731399i	$-0.261132\pi$
$29$	213.000	1.36390	0.681950	+	0.731399i	$0.261132\pi$
$30$	0	0
$31$	−207.799	−1.20393	−0.601965	−	0.798522i	$-0.705615\pi$
$31$	−207.799	−1.20393	−0.601965	+	0.798522i	$0.705615\pi$
$32$	0	0
$33$	−89.8865	−0.474158
$34$	0	0
$35$	−117.777	−0.568797
$36$	0	0
$37$	−197.651	−0.878204	−0.439102	−	0.898437i	$-0.644703\pi$
$37$	−197.651	−0.878204	−0.439102	+	0.898437i	$0.644703\pi$
$38$	0	0
$39$	−380.687	−1.56304
$40$	0	0
$41$	−383.696	−1.46154	−0.730771	−	0.682623i	$-0.760839\pi$
$41$	−383.696	−1.46154	−0.730771	+	0.682623i	$0.760839\pi$
$42$	0	0
$43$	−437.399	−1.55123	−0.775613	−	0.631209i	$-0.782559\pi$
$43$	−437.399	−1.55123	−0.775613	+	0.631209i	$0.782559\pi$
$44$	0	0
$45$	127.623	0.422776
$46$	0	0
$47$	156.104	0.484470	0.242235	−	0.970218i	$-0.422120\pi$
$47$	156.104	0.484470	0.242235	+	0.970218i	$0.422120\pi$
$48$	0	0
$49$	642.648	1.87361
$50$	0	0
$51$	362.634	0.995666
$52$	0	0
$53$	−12.0337	−0.0311879	−0.0155939	−	0.999878i	$-0.504964\pi$
$53$	−12.0337	−0.0311879	−0.0155939	+	0.999878i	$0.504964\pi$
$54$	0	0
$55$	43.1676	0.105831
$56$	0	0
$57$	608.577	1.41418
$58$	0	0
$59$	−559.360	−1.23428	−0.617140	−	0.786854i	$-0.711709\pi$
$59$	−559.360	−1.23428	−0.617140	+	0.786854i	$0.711709\pi$
$60$	0	0
$61$	37.7365	0.0792075	0.0396037	−	0.999215i	$-0.487390\pi$
$61$	37.7365	0.0792075	0.0396037	+	0.999215i	$0.487390\pi$
$62$	0	0
$63$	−1068.05	−2.13590
$64$	0	0
$65$	182.823	0.348868
$66$	0	0
$67$	−30.4246	−0.0554771	−0.0277385	−	0.999615i	$-0.508831\pi$
$67$	−30.4246	−0.0554771	−0.0277385	+	0.999615i	$0.508831\pi$
$68$	0	0
$69$	−179.665	−0.313465
$70$	0	0
$71$	398.612	0.666290	0.333145	−	0.942876i	$-0.391890\pi$
$71$	398.612	0.666290	0.333145	+	0.942876i	$0.391890\pi$
$72$	0	0
$73$	−877.484	−1.40687	−0.703436	−	0.710758i	$-0.748352\pi$
$73$	−877.484	−1.40687	−0.703436	+	0.710758i	$0.748352\pi$
$74$	0	0
$75$	866.505	1.33407
$76$	0	0
$77$	−361.260	−0.534668
$78$	0	0
$79$	−632.926	−0.901389	−0.450694	−	0.892678i	$-0.648824\pi$
$79$	−632.926	−0.901389	−0.450694	+	0.892678i	$0.648824\pi$
$80$	0	0
$81$	−490.192	−0.672416
$82$	0	0
$83$	622.703	0.823501	0.411750	−	0.911297i	$-0.364918\pi$
$83$	622.703	0.823501	0.411750	+	0.911297i	$0.364918\pi$
$84$	0	0
$85$	−174.154	−0.222231
$86$	0	0
$87$	−1663.85	−2.05039
$88$	0	0
$89$	−780.748	−0.929878	−0.464939	−	0.885343i	$-0.653924\pi$
$89$	−780.748	−0.929878	−0.464939	+	0.885343i	$0.653924\pi$
$90$	0	0
$91$	−1530.01	−1.76251
$92$	0	0
$93$	1623.23	1.80990
$94$	0	0
$95$	−292.267	−0.315641
$96$	0	0
$97$	964.791	1.00989	0.504947	−	0.863150i	$-0.331512\pi$
$97$	964.791	1.00989	0.504947	+	0.863150i	$0.331512\pi$
$98$	0	0
$99$	391.462	0.397408
$100$	0	0
$101$	913.863	0.900324	0.450162	−	0.892947i	$-0.351366\pi$
$101$	913.863	0.900324	0.450162	+	0.892947i	$0.351366\pi$
$102$	0	0
$103$	1233.53	1.18004	0.590018	−	0.807390i	$-0.299121\pi$
$103$	1233.53	1.18004	0.590018	+	0.807390i	$0.299121\pi$
$104$	0	0
$105$	920.014	0.855087
$106$	0	0
$107$	638.848	0.577194	0.288597	−	0.957451i	$-0.406811\pi$
$107$	638.848	0.577194	0.288597	+	0.957451i	$0.406811\pi$
$108$	0	0
$109$	−1851.20	−1.62672	−0.813361	−	0.581760i	$-0.802364\pi$
$109$	−1851.20	−1.62672	−0.813361	+	0.581760i	$0.802364\pi$
$110$	0	0
$111$	1543.95	1.32023
$112$	0	0
$113$	1159.21	0.965042	0.482521	−	0.875885i	$-0.339721\pi$
$113$	1159.21	0.965042	0.482521	+	0.875885i	$0.339721\pi$
$114$	0	0
$115$	86.2832	0.0699648
$116$	0	0
$117$	1657.92	1.31004
$118$	0	0
$119$	1457.45	1.12273
$120$	0	0
$121$	−1198.59	−0.900519
$122$	0	0
$123$	2997.24	2.19717
$124$	0	0
$125$	−885.066	−0.633302
$126$	0	0
$127$	−2524.72	−1.76404	−0.882020	−	0.471212i	$-0.843817\pi$
$127$	−2524.72	−1.76404	−0.882020	+	0.471212i	$0.843817\pi$
$128$	0	0
$129$	3416.75	2.33200
$130$	0	0
$131$	−399.064	−0.266156	−0.133078	−	0.991106i	$-0.542486\pi$
$131$	−399.064	−0.266156	−0.133078	+	0.991106i	$0.542486\pi$
$132$	0	0
$133$	2445.92	1.59464
$134$	0	0
$135$	−205.709	−0.131145
$136$	0	0
$137$	728.680	0.454418	0.227209	−	0.973846i	$-0.427040\pi$
$137$	728.680	0.454418	0.227209	+	0.973846i	$0.427040\pi$
$138$	0	0
$139$	−1142.70	−0.697283	−0.348642	−	0.937256i	$-0.613357\pi$
$139$	−1142.70	−0.697283	−0.348642	+	0.937256i	$0.613357\pi$
$140$	0	0
$141$	−1219.41	−0.728316
$142$	0	0
$143$	560.779	0.327935
$144$	0	0
$145$	799.058	0.457642
$146$	0	0
$147$	−5020.05	−2.81664
$148$	0	0
$149$	−640.678	−0.352258	−0.176129	−	0.984367i	$-0.556358\pi$
$149$	−640.678	−0.352258	−0.176129	+	0.984367i	$0.556358\pi$
$150$	0	0
$151$	−39.3746	−0.0212203	−0.0106101	−	0.999944i	$-0.503377\pi$
$151$	−39.3746	−0.0212203	−0.0106101	+	0.999944i	$0.503377\pi$
$152$	0	0
$153$	−1579.30	−0.834502
$154$	0	0
$155$	−779.547	−0.403966
$156$	0	0
$157$	3500.50	1.77943	0.889715	−	0.456517i	$-0.150903\pi$
$157$	3500.50	1.77943	0.889715	+	0.456517i	$0.150903\pi$
$158$	0	0
$159$	94.0015	0.0468856
$160$	0	0
$161$	−722.086	−0.353468
$162$	0	0
$163$	2831.67	1.36069	0.680347	−	0.732890i	$-0.261829\pi$
$163$	2831.67	1.36069	0.680347	+	0.732890i	$0.261829\pi$
$164$	0	0
$165$	−337.204	−0.159099
$166$	0	0
$167$	3956.82	1.83346	0.916730	−	0.399508i	$-0.130819\pi$
$167$	3956.82	1.83346	0.916730	+	0.399508i	$0.130819\pi$
$168$	0	0
$169$	178.010	0.0810239
$170$	0	0
$171$	−2650.40	−1.18527
$172$	0	0
$173$	3074.04	1.35095	0.675477	−	0.737381i	$-0.263938\pi$
$173$	3074.04	1.35095	0.675477	+	0.737381i	$0.263938\pi$
$174$	0	0
$175$	3482.55	1.50432
$176$	0	0
$177$	4369.45	1.85552
$178$	0	0
$179$	−2236.25	−0.933773	−0.466887	−	0.884317i	$-0.654624\pi$
$179$	−2236.25	−0.933773	−0.466887	+	0.884317i	$0.654624\pi$
$180$	0	0
$181$	−1496.39	−0.614506	−0.307253	−	0.951628i	$-0.599410\pi$
$181$	−1496.39	−0.614506	−0.307253	+	0.951628i	$0.599410\pi$
$182$	0	0
$183$	−294.779	−0.119075
$184$	0	0
$185$	−741.475	−0.294672
$186$	0	0
$187$	−534.187	−0.208896
$188$	0	0
$189$	1721.53	0.662555
$190$	0	0
$191$	4891.23	1.85297	0.926485	−	0.376333i	$-0.122815\pi$
$191$	4891.23	1.85297	0.926485	+	0.376333i	$0.122815\pi$
$192$	0	0
$193$	2496.29	0.931020	0.465510	−	0.885043i	$-0.345871\pi$
$193$	2496.29	0.931020	0.465510	+	0.885043i	$0.345871\pi$
$194$	0	0
$195$	−1428.13	−0.524462
$196$	0	0
$197$	−3009.45	−1.08840	−0.544198	−	0.838957i	$-0.683166\pi$
$197$	−3009.45	−1.08840	−0.544198	+	0.838957i	$0.683166\pi$
$198$	0	0
$199$	−4856.74	−1.73008	−0.865039	−	0.501705i	$-0.832706\pi$
$199$	−4856.74	−1.73008	−0.865039	+	0.501705i	$0.832706\pi$
$200$	0	0
$201$	237.662	0.0834001
$202$	0	0
$203$	−6687.14	−2.31205
$204$	0	0
$205$	−1439.41	−0.490405
$206$	0	0
$207$	782.453	0.262726
$208$	0	0
$209$	−896.478	−0.296702
$210$	0	0
$211$	747.857	0.244003	0.122001	−	0.992530i	$-0.461069\pi$
$211$	747.857	0.244003	0.122001	+	0.992530i	$0.461069\pi$
$212$	0	0
$213$	−3113.76	−1.00165
$214$	0	0
$215$	−1640.88	−0.520497
$216$	0	0
$217$	6523.86	2.04087
$218$	0	0
$219$	6854.47	2.11499
$220$	0	0
$221$	−2262.38	−0.688617
$222$	0	0
$223$	−3721.68	−1.11759	−0.558794	−	0.829306i	$-0.688736\pi$
$223$	−3721.68	−1.11759	−0.558794	+	0.829306i	$0.688736\pi$
$224$	0	0
$225$	−3773.69	−1.11813
$226$	0	0
$227$	5991.25	1.75178	0.875888	−	0.482514i	$-0.160276\pi$
$227$	5991.25	1.75178	0.875888	+	0.482514i	$0.160276\pi$
$228$	0	0
$229$	2389.83	0.689627	0.344813	−	0.938671i	$-0.387942\pi$
$229$	2389.83	0.689627	0.344813	+	0.938671i	$0.387942\pi$
$230$	0	0
$231$	2821.99	0.803780
$232$	0	0
$233$	−2345.92	−0.659597	−0.329799	−	0.944051i	$-0.606981\pi$
$233$	−2345.92	−0.659597	−0.329799	+	0.944051i	$0.606981\pi$
$234$	0	0
$235$	585.615	0.162559
$236$	0	0
$237$	4944.11	1.35508
$238$	0	0
$239$	4849.51	1.31251	0.656253	−	0.754541i	$-0.272140\pi$
$239$	4849.51	1.31251	0.656253	+	0.754541i	$0.272140\pi$
$240$	0	0
$241$	3746.89	1.00149	0.500743	−	0.865596i	$-0.333060\pi$
$241$	3746.89	1.00149	0.500743	+	0.865596i	$0.333060\pi$
$242$	0	0
$243$	5309.67	1.40171
$244$	0	0
$245$	2410.86	0.628669
$246$	0	0
$247$	−3796.76	−0.978065
$248$	0	0
$249$	−4864.25	−1.23799
$250$	0	0
$251$	5670.90	1.42607	0.713036	−	0.701127i	$-0.247320\pi$
$251$	5670.90	1.42607	0.713036	+	0.701127i	$0.247320\pi$
$252$	0	0
$253$	264.659	0.0657667
$254$	0	0
$255$	1360.40	0.334085
$256$	0	0
$257$	4028.82	0.977864	0.488932	−	0.872322i	$-0.337387\pi$
$257$	4028.82	0.977864	0.488932	+	0.872322i	$0.337387\pi$
$258$	0	0
$259$	6205.25	1.48871
$260$	0	0
$261$	7246.20	1.71850
$262$	0	0
$263$	−6801.68	−1.59471	−0.797357	−	0.603508i	$-0.793769\pi$
$263$	−6801.68	−1.59471	−0.797357	+	0.603508i	$0.793769\pi$
$264$	0	0
$265$	−45.1438	−0.0104648
$266$	0	0
$267$	6098.82	1.39791
$268$	0	0
$269$	1840.94	0.417263	0.208632	−	0.977994i	$-0.433099\pi$
$269$	1840.94	0.417263	0.208632	+	0.977994i	$0.433099\pi$
$270$	0	0
$271$	5021.45	1.12558	0.562789	−	0.826601i	$-0.309728\pi$
$271$	5021.45	1.12558	0.562789	+	0.826601i	$0.309728\pi$
$272$	0	0
$273$	11951.7	2.64963
$274$	0	0
$275$	−1276.42	−0.279896
$276$	0	0
$277$	−959.651	−0.208158	−0.104079	−	0.994569i	$-0.533190\pi$
$277$	−959.651	−0.208158	−0.104079	+	0.994569i	$0.533190\pi$
$278$	0	0
$279$	−7069.27	−1.51694
$280$	0	0
$281$	3635.74	0.771851	0.385925	−	0.922530i	$-0.373882\pi$
$281$	3635.74	0.771851	0.385925	+	0.922530i	$0.373882\pi$
$282$	0	0
$283$	3294.45	0.691996	0.345998	−	0.938235i	$-0.387540\pi$
$283$	3294.45	0.691996	0.345998	+	0.938235i	$0.387540\pi$
$284$	0	0
$285$	2283.04	0.474512
$286$	0	0
$287$	12046.1	2.47756
$288$	0	0
$289$	−2757.90	−0.561347
$290$	0	0
$291$	−7536.48	−1.51820
$292$	0	0
$293$	713.186	0.142200	0.0711002	−	0.997469i	$-0.477349\pi$
$293$	713.186	0.142200	0.0711002	+	0.997469i	$0.477349\pi$
$294$	0	0
$295$	−2098.41	−0.414149
$296$	0	0
$297$	−630.977	−0.123276
$298$	0	0
$299$	1120.88	0.216797
$300$	0	0
$301$	13732.1	2.62959
$302$	0	0
$303$	−7138.65	−1.35348
$304$	0	0
$305$	141.566	0.0265772
$306$	0	0
$307$	1588.95	0.295395	0.147697	−	0.989033i	$-0.452814\pi$
$307$	1588.95	0.295395	0.147697	+	0.989033i	$0.452814\pi$
$308$	0	0
$309$	−9635.76	−1.77398
$310$	0	0
$311$	2724.74	0.496804	0.248402	−	0.968657i	$-0.420095\pi$
$311$	2724.74	0.496804	0.248402	+	0.968657i	$0.420095\pi$
$312$	0	0
$313$	−3641.80	−0.657656	−0.328828	−	0.944390i	$-0.606654\pi$
$313$	−3641.80	−0.657656	−0.328828	+	0.944390i	$0.606654\pi$
$314$	0	0
$315$	−4006.73	−0.716678
$316$	0	0
$317$	9030.62	1.60003	0.800016	−	0.599978i	$-0.204824\pi$
$317$	9030.62	1.60003	0.800016	+	0.599978i	$0.204824\pi$
$318$	0	0
$319$	2450.97	0.430182
$320$	0	0
$321$	−4990.37	−0.867711
$322$	0	0
$323$	3616.72	0.623032
$324$	0	0
$325$	−5405.91	−0.922664
$326$	0	0
$327$	14460.7	2.44549
$328$	0	0
$329$	−4900.88	−0.821259
$330$	0	0
$331$	−4751.67	−0.789049	−0.394525	−	0.918885i	$-0.629091\pi$
$331$	−4751.67	−0.789049	−0.394525	+	0.918885i	$0.629091\pi$
$332$	0	0
$333$	−6724.02	−1.10653
$334$	0	0
$335$	−114.136	−0.0186147
$336$	0	0
$337$	3159.87	0.510769	0.255384	−	0.966840i	$-0.417798\pi$
$337$	3159.87	0.510769	0.255384	+	0.966840i	$0.417798\pi$
$338$	0	0
$339$	−9055.21	−1.45077
$340$	0	0
$341$	−2391.13	−0.379727
$342$	0	0
$343$	−9407.44	−1.48092
$344$	0	0
$345$	−674.002	−0.105180
$346$	0	0
$347$	6117.73	0.946446	0.473223	−	0.880943i	$-0.343090\pi$
$347$	6117.73	0.946446	0.473223	+	0.880943i	$0.343090\pi$
$348$	0	0
$349$	4063.16	0.623198	0.311599	−	0.950214i	$-0.399135\pi$
$349$	4063.16	0.623198	0.311599	+	0.950214i	$0.399135\pi$
$350$	0	0
$351$	−2672.31	−0.406374
$352$	0	0
$353$	6253.73	0.942925	0.471462	−	0.881886i	$-0.343726\pi$
$353$	6253.73	0.942925	0.471462	+	0.881886i	$0.343726\pi$
$354$	0	0
$355$	1495.37	0.223566
$356$	0	0
$357$	−11384.9	−1.68782
$358$	0	0
$359$	631.825	0.0928871	0.0464436	−	0.998921i	$-0.485211\pi$
$359$	631.825	0.0928871	0.0464436	+	0.998921i	$0.485211\pi$
$360$	0	0
$361$	−789.386	−0.115088
$362$	0	0
$363$	9362.80	1.35377
$364$	0	0
$365$	−3291.83	−0.472061
$366$	0	0
$367$	−11790.6	−1.67702	−0.838510	−	0.544886i	$-0.816573\pi$
$367$	−11790.6	−1.67702	−0.838510	+	0.544886i	$0.816573\pi$
$368$	0	0
$369$	−13053.2	−1.84153
$370$	0	0
$371$	377.799	0.0528688
$372$	0	0
$373$	−270.355	−0.0375294	−0.0187647	−	0.999824i	$-0.505973\pi$
$373$	−270.355	−0.0375294	−0.0187647	+	0.999824i	$0.505973\pi$
$374$	0	0
$375$	6913.70	0.952059
$376$	0	0
$377$	10380.4	1.41808
$378$	0	0
$379$	2658.36	0.360293	0.180146	−	0.983640i	$-0.442343\pi$
$379$	2658.36	0.360293	0.180146	+	0.983640i	$0.442343\pi$
$380$	0	0
$381$	19721.9	2.65193
$382$	0	0
$383$	−1697.17	−0.226426	−0.113213	−	0.993571i	$-0.536114\pi$
$383$	−1697.17	−0.226426	−0.113213	+	0.993571i	$0.536114\pi$
$384$	0	0
$385$	−1355.25	−0.179402
$386$	0	0
$387$	−14880.2	−1.95453
$388$	0	0
$389$	11725.4	1.52828	0.764142	−	0.645048i	$-0.223163\pi$
$389$	11725.4	1.52828	0.764142	+	0.645048i	$0.223163\pi$
$390$	0	0
$391$	−1067.73	−0.138101
$392$	0	0
$393$	3117.29	0.400119
$394$	0	0
$395$	−2374.39	−0.302451
$396$	0	0
$397$	−4556.25	−0.576000	−0.288000	−	0.957630i	$-0.592990\pi$
$397$	−4556.25	−0.576000	−0.288000	+	0.957630i	$0.592990\pi$
$398$	0	0
$399$	−19106.3	−2.39727
$400$	0	0
$401$	−8709.84	−1.08466	−0.542330	−	0.840166i	$-0.682458\pi$
$401$	−8709.84	−1.08466	−0.542330	+	0.840166i	$0.682458\pi$
$402$	0	0
$403$	−10126.9	−1.25175
$404$	0	0
$405$	−1838.93	−0.225622
$406$	0	0
$407$	−2274.35	−0.276991
$408$	0	0
$409$	−10137.7	−1.22561	−0.612805	−	0.790234i	$-0.709959\pi$
$409$	−10137.7	−1.22561	−0.612805	+	0.790234i	$0.709959\pi$
$410$	0	0
$411$	−5692.09	−0.683139
$412$	0	0
$413$	17561.1	2.09232
$414$	0	0
$415$	2336.04	0.276317
$416$	0	0
$417$	8926.20	1.04824
$418$	0	0
$419$	−13645.4	−1.59099	−0.795494	−	0.605962i	$-0.792788\pi$
$419$	−13645.4	−1.59099	−0.795494	+	0.605962i	$0.792788\pi$
$420$	0	0
$421$	−6845.15	−0.792428	−0.396214	−	0.918158i	$-0.629676\pi$
$421$	−6845.15	−0.792428	−0.396214	+	0.918158i	$0.629676\pi$
$422$	0	0
$423$	5310.60	0.610426
$424$	0	0
$425$	5149.56	0.587741
$426$	0	0
$427$	−1184.74	−0.134270
$428$	0	0
$429$	−4380.53	−0.492993
$430$	0	0
$431$	3120.80	0.348778	0.174389	−	0.984677i	$-0.444205\pi$
$431$	3120.80	0.348778	0.174389	+	0.984677i	$0.444205\pi$
$432$	0	0
$433$	−16153.2	−1.79278	−0.896392	−	0.443262i	$-0.853821\pi$
$433$	−16153.2	−1.79278	−0.896392	+	0.443262i	$0.853821\pi$
$434$	0	0
$435$	−6241.85	−0.687985
$436$	0	0
$437$	−1791.88	−0.196149
$438$	0	0
$439$	2925.81	0.318090	0.159045	−	0.987271i	$-0.449159\pi$
$439$	2925.81	0.318090	0.159045	+	0.987271i	$0.449159\pi$
$440$	0	0
$441$	21862.7	2.36073
$442$	0	0
$443$	−10398.5	−1.11523	−0.557615	−	0.830100i	$-0.688283\pi$
$443$	−10398.5	−1.11523	−0.557615	+	0.830100i	$0.688283\pi$
$444$	0	0
$445$	−2928.93	−0.312011
$446$	0	0
$447$	5004.66	0.529558
$448$	0	0
$449$	14520.4	1.52619	0.763094	−	0.646288i	$-0.223679\pi$
$449$	14520.4	1.52619	0.763094	+	0.646288i	$0.223679\pi$
$450$	0	0
$451$	−4415.16	−0.460979
$452$	0	0
$453$	307.575	0.0319010
$454$	0	0
$455$	−5739.74	−0.591391
$456$	0	0
$457$	−10207.9	−1.04487	−0.522435	−	0.852679i	$-0.674976\pi$
$457$	−10207.9	−1.04487	−0.522435	+	0.852679i	$0.674976\pi$
$458$	0	0
$459$	2545.59	0.258862
$460$	0	0
$461$	745.399	0.0753074	0.0376537	−	0.999291i	$-0.488012\pi$
$461$	745.399	0.0753074	0.0376537	+	0.999291i	$0.488012\pi$
$462$	0	0
$463$	−6284.87	−0.630848	−0.315424	−	0.948951i	$-0.602147\pi$
$463$	−6284.87	−0.630848	−0.315424	+	0.948951i	$0.602147\pi$
$464$	0	0
$465$	6089.44	0.607293
$466$	0	0
$467$	−16886.2	−1.67324	−0.836619	−	0.547785i	$-0.815471\pi$
$467$	−16886.2	−1.67324	−0.836619	+	0.547785i	$0.815471\pi$
$468$	0	0
$469$	955.183	0.0940431
$470$	0	0
$471$	−27344.2	−2.67506
$472$	0	0
$473$	−5033.11	−0.489266
$474$	0	0
$475$	8642.04	0.834788
$476$	0	0
$477$	−409.384	−0.0392964
$478$	0	0
$479$	−9966.58	−0.950699	−0.475349	−	0.879797i	$-0.657678\pi$
$479$	−9966.58	−0.950699	−0.475349	+	0.879797i	$0.657678\pi$
$480$	0	0
$481$	−9632.32	−0.913089
$482$	0	0
$483$	5640.58	0.531377
$484$	0	0
$485$	3619.36	0.338859
$486$	0	0
$487$	−5453.97	−0.507480	−0.253740	−	0.967272i	$-0.581661\pi$
$487$	−5453.97	−0.507480	−0.253740	+	0.967272i	$0.581661\pi$
$488$	0	0
$489$	−22119.6	−2.04557
$490$	0	0
$491$	12259.1	1.12678	0.563389	−	0.826192i	$-0.309497\pi$
$491$	12259.1	1.12678	0.563389	+	0.826192i	$0.309497\pi$
$492$	0	0
$493$	−9888.11	−0.903323
$494$	0	0
$495$	1468.55	0.133346
$496$	0	0
$497$	−12514.4	−1.12948
$498$	0	0
$499$	−305.167	−0.0273770	−0.0136885	−	0.999906i	$-0.504357\pi$
$499$	−305.167	−0.0273770	−0.0136885	+	0.999906i	$0.504357\pi$
$500$	0	0
$501$	−30908.7	−2.75629
$502$	0	0
$503$	12300.1	1.09033	0.545163	−	0.838330i	$-0.316468\pi$
$503$	12300.1	1.09033	0.545163	+	0.838330i	$0.316468\pi$
$504$	0	0
$505$	3428.31	0.302094
$506$	0	0
$507$	−1390.52	−0.121805
$508$	0	0
$509$	−6855.35	−0.596971	−0.298485	−	0.954414i	$-0.596481\pi$
$509$	−6855.35	−0.596971	−0.298485	+	0.954414i	$0.596481\pi$
$510$	0	0
$511$	27548.6	2.38489
$512$	0	0
$513$	4272.03	0.367670
$514$	0	0
$515$	4627.53	0.395948
$516$	0	0
$517$	1796.27	0.152805
$518$	0	0
$519$	−24012.9	−2.03093
$520$	0	0
$521$	−19811.8	−1.66597	−0.832986	−	0.553293i	$-0.813371\pi$
$521$	−19811.8	−1.66597	−0.832986	+	0.553293i	$0.813371\pi$
$522$	0	0
$523$	−154.428	−0.0129114	−0.00645572	−	0.999979i	$-0.502055\pi$
$523$	−154.428	−0.0129114	−0.00645572	+	0.999979i	$0.502055\pi$
$524$	0	0
$525$	−27203.9	−2.26148
$526$	0	0
$527$	9646.67	0.797373
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−19029.3	−1.55518
$532$	0	0
$533$	−18699.0	−1.51960
$534$	0	0
$535$	2396.60	0.193671
$536$	0	0
$537$	17468.5	1.40377
$538$	0	0
$539$	7394.90	0.590948
$540$	0	0
$541$	6393.60	0.508100	0.254050	−	0.967191i	$-0.418237\pi$
$541$	6393.60	0.508100	0.254050	+	0.967191i	$0.418237\pi$
$542$	0	0
$543$	11689.0	0.923802
$544$	0	0
$545$	−6944.67	−0.545829
$546$	0	0
$547$	23211.0	1.81431	0.907157	−	0.420793i	$-0.138248\pi$
$547$	23211.0	1.81431	0.907157	+	0.420793i	$0.138248\pi$
$548$	0	0
$549$	1283.78	0.0998006
$550$	0	0
$551$	−16594.3	−1.28302
$552$	0	0
$553$	19870.7	1.52801
$554$	0	0
$555$	5792.04	0.442988
$556$	0	0
$557$	−14698.2	−1.11810	−0.559052	−	0.829132i	$-0.688835\pi$
$557$	−14698.2	−1.11810	−0.559052	+	0.829132i	$0.688835\pi$
$558$	0	0
$559$	−21316.2	−1.61284
$560$	0	0
$561$	4172.80	0.314039
$562$	0	0
$563$	18682.6	1.39854	0.699271	−	0.714856i	$-0.253508\pi$
$563$	18682.6	1.39854	0.699271	+	0.714856i	$0.253508\pi$
$564$	0	0
$565$	4348.73	0.323809
$566$	0	0
$567$	15389.6	1.13986
$568$	0	0
$569$	−7458.00	−0.549483	−0.274741	−	0.961518i	$-0.588592\pi$
$569$	−7458.00	−0.549483	−0.274741	+	0.961518i	$0.588592\pi$
$570$	0	0
$571$	−1508.50	−0.110558	−0.0552792	−	0.998471i	$-0.517605\pi$
$571$	−1508.50	−0.110558	−0.0552792	+	0.998471i	$0.517605\pi$
$572$	0	0
$573$	−38207.9	−2.78562
$574$	0	0
$575$	−2551.31	−0.185038
$576$	0	0
$577$	−20029.2	−1.44511	−0.722554	−	0.691314i	$-0.757032\pi$
$577$	−20029.2	−1.44511	−0.722554	+	0.691314i	$0.757032\pi$
$578$	0	0
$579$	−19499.8	−1.39963
$580$	0	0
$581$	−19549.8	−1.39598
$582$	0	0
$583$	−138.471	−0.00983685
$584$	0	0
$585$	6219.59	0.439570
$586$	0	0
$587$	3763.95	0.264659	0.132330	−	0.991206i	$-0.457754\pi$
$587$	3763.95	0.264659	0.132330	+	0.991206i	$0.457754\pi$
$588$	0	0
$589$	16189.2	1.13253
$590$	0	0
$591$	23508.3	1.63621
$592$	0	0
$593$	22874.3	1.58404	0.792020	−	0.610495i	$-0.209029\pi$
$593$	22874.3	1.58404	0.792020	+	0.610495i	$0.209029\pi$
$594$	0	0
$595$	5467.55	0.376719
$596$	0	0
$597$	37938.5	2.60087
$598$	0	0
$599$	10605.7	0.723432	0.361716	−	0.932288i	$-0.382191\pi$
$599$	10605.7	0.723432	0.361716	+	0.932288i	$0.382191\pi$
$600$	0	0
$601$	4268.95	0.289740	0.144870	−	0.989451i	$-0.453724\pi$
$601$	4268.95	0.289740	0.144870	+	0.989451i	$0.453724\pi$
$602$	0	0
$603$	−1035.04	−0.0699005
$604$	0	0
$605$	−4496.45	−0.302160
$606$	0	0
$607$	16509.1	1.10393	0.551964	−	0.833868i	$-0.313879\pi$
$607$	16509.1	1.10393	0.551964	+	0.833868i	$0.313879\pi$
$608$	0	0
$609$	52236.7	3.47576
$610$	0	0
$611$	7607.57	0.503714
$612$	0	0
$613$	11429.3	0.753057	0.376529	−	0.926405i	$-0.377118\pi$
$613$	11429.3	0.753057	0.376529	+	0.926405i	$0.377118\pi$
$614$	0	0
$615$	11244.0	0.737238
$616$	0	0
$617$	−20123.2	−1.31301	−0.656507	−	0.754320i	$-0.727967\pi$
$617$	−20123.2	−1.31301	−0.656507	+	0.754320i	$0.727967\pi$
$618$	0	0
$619$	−16879.9	−1.09606	−0.548028	−	0.836460i	$-0.684621\pi$
$619$	−16879.9	−1.09606	−0.548028	+	0.836460i	$0.684621\pi$
$620$	0	0
$621$	−1261.19	−0.0814975
$622$	0	0
$623$	24511.6	1.57630
$624$	0	0
$625$	10545.6	0.674916
$626$	0	0
$627$	7002.85	0.446040
$628$	0	0
$629$	9175.55	0.581642
$630$	0	0
$631$	−459.538	−0.0289919	−0.0144960	−	0.999895i	$-0.504614\pi$
$631$	−459.538	−0.0289919	−0.0144960	+	0.999895i	$0.504614\pi$
$632$	0	0
$633$	−5841.90	−0.366816
$634$	0	0
$635$	−9471.37	−0.591905
$636$	0	0
$637$	31318.8	1.94803
$638$	0	0
$639$	13560.7	0.839518
$640$	0	0
$641$	13339.7	0.821973	0.410986	−	0.911641i	$-0.365184\pi$
$641$	13339.7	0.821973	0.410986	+	0.911641i	$0.365184\pi$
$642$	0	0
$643$	23541.6	1.44384	0.721920	−	0.691976i	$-0.243260\pi$
$643$	23541.6	1.44384	0.721920	+	0.691976i	$0.243260\pi$
$644$	0	0
$645$	12817.7	0.782477
$646$	0	0
$647$	24094.0	1.46404	0.732018	−	0.681285i	$-0.238579\pi$
$647$	24094.0	1.46404	0.732018	+	0.681285i	$0.238579\pi$
$648$	0	0
$649$	−6436.51	−0.389299
$650$	0	0
$651$	−50961.2	−3.06809
$652$	0	0
$653$	−21867.4	−1.31047	−0.655235	−	0.755425i	$-0.727431\pi$
$653$	−21867.4	−1.31047	−0.655235	+	0.755425i	$0.727431\pi$
$654$	0	0
$655$	−1497.07	−0.0893057
$656$	0	0
$657$	−29851.7	−1.77264
$658$	0	0
$659$	10425.9	0.616290	0.308145	−	0.951339i	$-0.400292\pi$
$659$	10425.9	0.616290	0.308145	+	0.951339i	$0.400292\pi$
$660$	0	0
$661$	−10419.3	−0.613108	−0.306554	−	0.951853i	$-0.599176\pi$
$661$	−10419.3	−0.613108	−0.306554	+	0.951853i	$0.599176\pi$
$662$	0	0
$663$	17672.6	1.03522
$664$	0	0
$665$	9175.71	0.535066
$666$	0	0
$667$	4899.00	0.284393
$668$	0	0
$669$	29072.0	1.68010
$670$	0	0
$671$	434.230	0.0249825
$672$	0	0
$673$	−30039.8	−1.72058	−0.860289	−	0.509806i	$-0.829717\pi$
$673$	−30039.8	−1.72058	−0.860289	+	0.509806i	$0.829717\pi$
$674$	0	0
$675$	6082.61	0.346844
$676$	0	0
$677$	5832.80	0.331126	0.165563	−	0.986199i	$-0.447056\pi$
$677$	5832.80	0.331126	0.165563	+	0.986199i	$0.447056\pi$
$678$	0	0
$679$	−30289.6	−1.71194
$680$	0	0
$681$	−46800.7	−2.63349
$682$	0	0
$683$	31858.1	1.78480	0.892399	−	0.451247i	$-0.149021\pi$
$683$	31858.1	1.78480	0.892399	+	0.451247i	$0.149021\pi$
$684$	0	0
$685$	2733.60	0.152475
$686$	0	0
$687$	−18668.2	−1.03673
$688$	0	0
$689$	−586.452	−0.0324268
$690$	0	0
$691$	29510.2	1.62463	0.812315	−	0.583219i	$-0.198207\pi$
$691$	29510.2	1.62463	0.812315	+	0.583219i	$0.198207\pi$
$692$	0	0
$693$	−12290.0	−0.673675
$694$	0	0
$695$	−4286.77	−0.233966
$696$	0	0
$697$	17812.3	0.967991
$698$	0	0
$699$	18325.2	0.991590
$700$	0	0
$701$	−24576.4	−1.32416	−0.662082	−	0.749431i	$-0.730327\pi$
$701$	−24576.4	−1.32416	−0.662082	+	0.749431i	$0.730327\pi$
$702$	0	0
$703$	15398.5	0.826125
$704$	0	0
$705$	−4574.53	−0.244379
$706$	0	0
$707$	−28690.7	−1.52620
$708$	0	0
$709$	−5175.41	−0.274142	−0.137071	−	0.990561i	$-0.543769\pi$
$709$	−5175.41	−0.274142	−0.137071	+	0.990561i	$0.543769\pi$
$710$	0	0
$711$	−21531.9	−1.13574
$712$	0	0
$713$	−4779.38	−0.251037
$714$	0	0
$715$	2103.73	0.110035
$716$	0	0
$717$	−37882.0	−1.97312
$718$	0	0
$719$	8032.65	0.416644	0.208322	−	0.978060i	$-0.433200\pi$
$719$	8032.65	0.416644	0.208322	+	0.978060i	$0.433200\pi$
$720$	0	0
$721$	−38726.8	−2.00036
$722$	0	0
$723$	−29268.9	−1.50556
$724$	0	0
$725$	−23627.4	−1.21034
$726$	0	0
$727$	13611.5	0.694393	0.347196	−	0.937792i	$-0.387134\pi$
$727$	13611.5	0.694393	0.347196	+	0.937792i	$0.387134\pi$
$728$	0	0
$729$	−28241.4	−1.43481
$730$	0	0
$731$	20305.4	1.02739
$732$	0	0
$733$	26923.2	1.35666	0.678329	−	0.734759i	$-0.262705\pi$
$733$	26923.2	1.35666	0.678329	+	0.734759i	$0.262705\pi$
$734$	0	0
$735$	−18832.4	−0.945095
$736$	0	0
$737$	−350.094	−0.0174978
$738$	0	0
$739$	14164.4	0.705068	0.352534	−	0.935799i	$-0.385320\pi$
$739$	14164.4	0.705068	0.352534	+	0.935799i	$0.385320\pi$
$740$	0	0
$741$	29658.4	1.47035
$742$	0	0
$743$	−18114.8	−0.894438	−0.447219	−	0.894425i	$-0.647585\pi$
$743$	−18114.8	−0.894438	−0.447219	+	0.894425i	$0.647585\pi$
$744$	0	0
$745$	−2403.47	−0.118196
$746$	0	0
$747$	21184.2	1.03760
$748$	0	0
$749$	−20056.7	−0.978444
$750$	0	0
$751$	24952.2	1.21241	0.606204	−	0.795309i	$-0.292692\pi$
$751$	24952.2	1.21241	0.606204	+	0.795309i	$0.292692\pi$
$752$	0	0
$753$	−44298.3	−2.14385
$754$	0	0
$755$	−147.712	−0.00712023
$756$	0	0
$757$	25869.3	1.24205	0.621027	−	0.783790i	$-0.286716\pi$
$757$	25869.3	1.24205	0.621027	+	0.783790i	$0.286716\pi$
$758$	0	0
$759$	−2067.39	−0.0988688
$760$	0	0
$761$	39174.5	1.86606	0.933031	−	0.359796i	$-0.117154\pi$
$761$	39174.5	1.86606	0.933031	+	0.359796i	$0.117154\pi$
$762$	0	0
$763$	58118.4	2.75757
$764$	0	0
$765$	−5924.65	−0.280008
$766$	0	0
$767$	−27259.9	−1.28331
$768$	0	0
$769$	−14544.5	−0.682038	−0.341019	−	0.940056i	$-0.610772\pi$
$769$	−14544.5	−0.682038	−0.341019	+	0.940056i	$0.610772\pi$
$770$	0	0
$771$	−31471.2	−1.47005
$772$	0	0
$773$	−39420.3	−1.83422	−0.917109	−	0.398637i	$-0.869483\pi$
$773$	−39420.3	−1.83422	−0.917109	+	0.398637i	$0.869483\pi$
$774$	0	0
$775$	23050.5	1.06838
$776$	0	0
$777$	−48472.4	−2.23801
$778$	0	0
$779$	29892.8	1.37487
$780$	0	0
$781$	4586.80	0.210152
$782$	0	0
$783$	−11679.8	−0.533078
$784$	0	0
$785$	13131.9	0.597069
$786$	0	0
$787$	−9491.24	−0.429894	−0.214947	−	0.976626i	$-0.568958\pi$
$787$	−9491.24	−0.429894	−0.214947	+	0.976626i	$0.568958\pi$
$788$	0	0
$789$	53131.4	2.39737
$790$	0	0
$791$	−36393.6	−1.63591
$792$	0	0
$793$	1839.05	0.0823539
$794$	0	0
$795$	352.642	0.0157320
$796$	0	0
$797$	12185.9	0.541591	0.270795	−	0.962637i	$-0.412713\pi$
$797$	12185.9	0.541591	0.270795	+	0.962637i	$0.412713\pi$
$798$	0	0
$799$	−7246.81	−0.320868
$800$	0	0
$801$	−26560.8	−1.17164
$802$	0	0
$803$	−10097.1	−0.443736
$804$	0	0
$805$	−2708.86	−0.118602
$806$	0	0
$807$	−14380.5	−0.627283
$808$	0	0
$809$	2498.29	0.108573	0.0542863	−	0.998525i	$-0.482712\pi$
$809$	2498.29	0.108573	0.0542863	+	0.998525i	$0.482712\pi$
$810$	0	0
$811$	18005.8	0.779615	0.389808	−	0.920896i	$-0.372541\pi$
$811$	18005.8	0.779615	0.389808	+	0.920896i	$0.372541\pi$
$812$	0	0
$813$	−39225.1	−1.69211
$814$	0	0
$815$	10622.8	0.456567
$816$	0	0
$817$	34076.7	1.45923
$818$	0	0
$819$	−52050.4	−2.22074
$820$	0	0
$821$	39469.4	1.67782	0.838912	−	0.544267i	$-0.183192\pi$
$821$	39469.4	1.67782	0.838912	+	0.544267i	$0.183192\pi$
$822$	0	0
$823$	12730.6	0.539198	0.269599	−	0.962973i	$-0.413109\pi$
$823$	12730.6	0.539198	0.269599	+	0.962973i	$0.413109\pi$
$824$	0	0
$825$	9970.80	0.420774
$826$	0	0
$827$	−10536.3	−0.443027	−0.221513	−	0.975157i	$-0.571100\pi$
$827$	−10536.3	−0.443027	−0.221513	+	0.975157i	$0.571100\pi$
$828$	0	0
$829$	−21400.9	−0.896602	−0.448301	−	0.893883i	$-0.647971\pi$
$829$	−21400.9	−0.896602	−0.448301	+	0.893883i	$0.647971\pi$
$830$	0	0
$831$	7496.32	0.312930
$832$	0	0
$833$	−29833.7	−1.24091
$834$	0	0
$835$	14843.8	0.615198
$836$	0	0
$837$	11394.6	0.470554
$838$	0	0
$839$	13113.2	0.539591	0.269795	−	0.962918i	$-0.413044\pi$
$839$	13113.2	0.539591	0.269795	+	0.962918i	$0.413044\pi$
$840$	0	0
$841$	20980.0	0.860224
$842$	0	0
$843$	−28400.6	−1.16034
$844$	0	0
$845$	667.793	0.0271867
$846$	0	0
$847$	37629.8	1.52653
$848$	0	0
$849$	−25734.7	−1.04030
$850$	0	0
$851$	−4545.96	−0.183118
$852$	0	0
$853$	962.614	0.0386392	0.0193196	−	0.999813i	$-0.493850\pi$
$853$	962.614	0.0386392	0.0193196	+	0.999813i	$0.493850\pi$
$854$	0	0
$855$	−9942.82	−0.397704
$856$	0	0
$857$	−27608.2	−1.10044	−0.550221	−	0.835019i	$-0.685457\pi$
$857$	−27608.2	−1.10044	−0.550221	+	0.835019i	$0.685457\pi$
$858$	0	0
$859$	−14922.2	−0.592713	−0.296356	−	0.955077i	$-0.595772\pi$
$859$	−14922.2	−0.592713	−0.296356	+	0.955077i	$0.595772\pi$
$860$	0	0
$861$	−94098.5	−3.72459
$862$	0	0
$863$	−25684.7	−1.01311	−0.506557	−	0.862206i	$-0.669082\pi$
$863$	−25684.7	−1.01311	−0.506557	+	0.862206i	$0.669082\pi$
$864$	0	0
$865$	11532.1	0.453299
$866$	0	0
$867$	21543.4	0.843888
$868$	0	0
$869$	−7283.03	−0.284304
$870$	0	0
$871$	−1482.72	−0.0576808
$872$	0	0
$873$	32821.9	1.27246
$874$	0	0
$875$	27786.7	1.07356
$876$	0	0
$877$	−24651.7	−0.949179	−0.474590	−	0.880207i	$-0.657404\pi$
$877$	−24651.7	−0.949179	−0.474590	+	0.880207i	$0.657404\pi$
$878$	0	0
$879$	−5571.06	−0.213774
$880$	0	0
$881$	−2333.94	−0.0892535	−0.0446267	−	0.999004i	$-0.514210\pi$
$881$	−2333.94	−0.0892535	−0.0446267	+	0.999004i	$0.514210\pi$
$882$	0	0
$883$	−19670.6	−0.749681	−0.374841	−	0.927089i	$-0.622303\pi$
$883$	−19670.6	−0.749681	−0.374841	+	0.927089i	$0.622303\pi$
$884$	0	0
$885$	16391.7	0.622602
$886$	0	0
$887$	21451.1	0.812016	0.406008	−	0.913870i	$-0.366921\pi$
$887$	21451.1	0.812016	0.406008	+	0.913870i	$0.366921\pi$
$888$	0	0
$889$	79263.8	2.99035
$890$	0	0
$891$	−5640.60	−0.212084
$892$	0	0
$893$	−12161.7	−0.455739
$894$	0	0
$895$	−8389.18	−0.313318
$896$	0	0
$897$	−8755.79	−0.325917
$898$	0	0
$899$	−44261.2	−1.64204
$900$	0	0
$901$	558.642	0.0206560
$902$	0	0
$903$	−107269.	−3.95314
$904$	0	0
$905$	−5613.61	−0.206191
$906$	0	0
$907$	2902.17	0.106246	0.0531229	−	0.998588i	$-0.483082\pi$
$907$	2902.17	0.106246	0.0531229	+	0.998588i	$0.483082\pi$
$908$	0	0
$909$	31089.3	1.13440
$910$	0	0
$911$	−8321.85	−0.302651	−0.151326	−	0.988484i	$-0.548354\pi$
$911$	−8321.85	−0.302651	−0.151326	+	0.988484i	$0.548354\pi$
$912$	0	0
$913$	7165.40	0.259737
$914$	0	0
$915$	−1105.85	−0.0399543
$916$	0	0
$917$	12528.6	0.451180
$918$	0	0
$919$	25339.9	0.909559	0.454780	−	0.890604i	$-0.349718\pi$
$919$	25339.9	0.909559	0.454780	+	0.890604i	$0.349718\pi$
$920$	0	0
$921$	−12412.1	−0.444074
$922$	0	0
$923$	19426.0	0.692757
$924$	0	0
$925$	21924.7	0.779330
$926$	0	0
$927$	41964.5	1.48683
$928$	0	0
$929$	−2468.08	−0.0871638	−0.0435819	−	0.999050i	$-0.513877\pi$
$929$	−2468.08	−0.0871638	−0.0435819	+	0.999050i	$0.513877\pi$
$930$	0	0
$931$	−50067.2	−1.76250
$932$	0	0
$933$	−21284.3	−0.746858
$934$	0	0
$935$	−2003.97	−0.0700929
$936$	0	0
$937$	511.833	0.0178451	0.00892255	−	0.999960i	$-0.497160\pi$
$937$	511.833	0.0178451	0.00892255	+	0.999960i	$0.497160\pi$
$938$	0	0
$939$	28447.9	0.988672
$940$	0	0
$941$	−12986.8	−0.449900	−0.224950	−	0.974370i	$-0.572222\pi$
$941$	−12986.8	−0.449900	−0.224950	+	0.974370i	$0.572222\pi$
$942$	0	0
$943$	−8825.00	−0.304752
$944$	0	0
$945$	6458.23	0.222313
$946$	0	0
$947$	8230.45	0.282422	0.141211	−	0.989980i	$-0.454900\pi$
$947$	8230.45	0.282422	0.141211	+	0.989980i	$0.454900\pi$
$948$	0	0
$949$	−42763.3	−1.46276
$950$	0	0
$951$	−70542.8	−2.40537
$952$	0	0
$953$	21749.1	0.739269	0.369634	−	0.929177i	$-0.379483\pi$
$953$	21749.1	0.739269	0.369634	+	0.929177i	$0.379483\pi$
$954$	0	0
$955$	18349.2	0.621744
$956$	0	0
$957$	−19145.8	−0.646705
$958$	0	0
$959$	−22876.9	−0.770317
$960$	0	0
$961$	13389.5	0.449448
$962$	0	0
$963$	21733.4	0.727259
$964$	0	0
$965$	9364.69	0.312394
$966$	0	0
$967$	1523.85	0.0506760	0.0253380	−	0.999679i	$-0.491934\pi$
$967$	1523.85	0.0506760	0.0253380	+	0.999679i	$0.491934\pi$
$968$	0	0
$969$	−28252.0	−0.936621
$970$	0	0
$971$	−36956.9	−1.22143	−0.610713	−	0.791852i	$-0.709117\pi$
$971$	−36956.9	−1.22143	−0.610713	+	0.791852i	$0.709117\pi$
$972$	0	0
$973$	35875.0	1.18202
$974$	0	0
$975$	42228.3	1.38706
$976$	0	0
$977$	10253.2	0.335751	0.167875	−	0.985808i	$-0.446309\pi$
$977$	10253.2	0.335751	0.167875	+	0.985808i	$0.446309\pi$
$978$	0	0
$979$	−8984.01	−0.293289
$980$	0	0
$981$	−62977.2	−2.04965
$982$	0	0
$983$	−1973.81	−0.0640434	−0.0320217	−	0.999487i	$-0.510195\pi$
$983$	−1973.81	−0.0640434	−0.0320217	+	0.999487i	$0.510195\pi$
$984$	0	0
$985$	−11289.8	−0.365200
$986$	0	0
$987$	38283.3	1.23462
$988$	0	0
$989$	−10060.2	−0.323453
$990$	0	0
$991$	41195.2	1.32049	0.660247	−	0.751049i	$-0.270452\pi$
$991$	41195.2	1.32049	0.660247	+	0.751049i	$0.270452\pi$
$992$	0	0
$993$	37117.7	1.18620
$994$	0	0
$995$	−18219.8	−0.580509
$996$	0	0
$997$	34156.8	1.08501	0.542505	−	0.840052i	$-0.317476\pi$
$997$	34156.8	1.08501	0.542505	+	0.840052i	$0.317476\pi$
$998$	0	0
$999$	10838.1	0.343245

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1472.4.a.bd.1.1		4
4.3	odd	2		1472.4.a.ba.1.4		4
8.3	odd	2		184.4.a.e.1.1	✓	4
8.5	even	2		368.4.a.n.1.4		4
24.11	even	2		1656.4.a.n.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
184.4.a.e.1.1	✓	4	8.3	odd	2
368.4.a.n.1.4		4	8.5	even	2
1472.4.a.ba.1.4		4	4.3	odd	2
1472.4.a.bd.1.1		4	1.1	even	1	trivial
1656.4.a.n.1.3		4	24.11	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1472 = 2^{6} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1472.a (trivial)

\(f(q)\)	\(=\)	\(q-7.81151 q^{3} +3.75144 q^{5} -31.3950 q^{7} +34.0197 q^{9} +11.5069 q^{11} +48.7341 q^{13} -29.3045 q^{15} -46.4231 q^{17} -77.9077 q^{19} +245.243 q^{21} +23.0000 q^{23} -110.927 q^{25} -54.8345 q^{27} +213.000 q^{29} -207.799 q^{31} -89.8865 q^{33} -117.777 q^{35} -197.651 q^{37} -380.687 q^{39} -383.696 q^{41} -437.399 q^{43} +127.623 q^{45} +156.104 q^{47} +642.648 q^{49} +362.634 q^{51} -12.0337 q^{53} +43.1676 q^{55} +608.577 q^{57} -559.360 q^{59} +37.7365 q^{61} -1068.05 q^{63} +182.823 q^{65} -30.4246 q^{67} -179.665 q^{69} +398.612 q^{71} -877.484 q^{73} +866.505 q^{75} -361.260 q^{77} -632.926 q^{79} -490.192 q^{81} +622.703 q^{83} -174.154 q^{85} -1663.85 q^{87} -780.748 q^{89} -1530.01 q^{91} +1623.23 q^{93} -292.267 q^{95} +964.791 q^{97} +391.462 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{3} + 20 q^{5} + 10 q^{7} + 23 q^{9} - 30 q^{11} + 153 q^{13} + 136 q^{15} - 68 q^{17} - 120 q^{19} + 426 q^{21} + 92 q^{23} - 76 q^{25} + 43 q^{27} + 315 q^{29} - 249 q^{31} - 504 q^{33} + 224 q^{35}+ \cdots - 1470 q^{99}+O(q^{100}) \) 4 * q + q^3 + 20 * q^5 + 10 * q^7 + 23 * q^9 - 30 * q^11 + 153 * q^13 + 136 * q^15 - 68 * q^17 - 120 * q^19 + 426 * q^21 + 92 * q^23 - 76 * q^25 + 43 * q^27 + 315 * q^29 - 249 * q^31 - 504 * q^33 + 224 * q^35 + 348 * q^37 + 61 * q^39 - 929 * q^41 + 50 * q^43 + 1028 * q^45 - 205 * q^47 + 456 * q^49 + 890 * q^51 + 578 * q^53 - 1128 * q^55 + 366 * q^57 - 664 * q^59 + 182 * q^61 - 624 * q^63 + 848 * q^65 + 1078 * q^67 + 23 * q^69 + 739 * q^71 + 921 * q^73 + 2883 * q^75 - 528 * q^77 - 1754 * q^79 + 788 * q^81 + 1020 * q^83 + 1072 * q^85 - 2121 * q^87 - 430 * q^89 + 1334 * q^91 - 1035 * q^93 - 744 * q^95 + 48 * q^97 - 1470 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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