LMFDB - Embedded newform 448.4.a.r.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [448,4,Mod(1,448)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(448, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("448.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	448.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,0,-22,0,-14] 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:	$26.4328556826$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{57}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 14 $ x^2 - x - 14
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 56)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.27492$ of defining polynomial
Character	$\chi$	$=$	448.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+6.54983 q^{3} -3.45017 q^{5} -7.00000 q^{7} +15.9003 q^{9} -27.2990 q^{11} -58.7492 q^{13} -22.5980 q^{15} +49.2990 q^{17} -157.148 q^{19} -45.8488 q^{21} +82.1993 q^{23} -113.096 q^{25} -72.7010 q^{27} +194.096 q^{29} -115.698 q^{31} -178.804 q^{33} +24.1512 q^{35} -327.498 q^{37} -384.797 q^{39} -136.103 q^{41} +311.698 q^{43} -54.8588 q^{45} +355.698 q^{47} +49.0000 q^{49} +322.900 q^{51} -677.588 q^{53} +94.1861 q^{55} -1029.29 q^{57} -197.148 q^{59} -61.0449 q^{61} -111.302 q^{63} +202.694 q^{65} +1017.39 q^{67} +538.392 q^{69} +279.807 q^{71} +629.794 q^{73} -740.762 q^{75} +191.093 q^{77} -20.2060 q^{79} -905.488 q^{81} -260.158 q^{83} -170.090 q^{85} +1271.30 q^{87} -909.974 q^{89} +411.244 q^{91} -757.801 q^{93} +542.186 q^{95} +100.488 q^{97} -434.063 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 22 q^{5} - 14 q^{7} + 62 q^{9} + 36 q^{11} - 42 q^{13} + 136 q^{15} + 8 q^{17} - 118 q^{19} + 14 q^{21} + 104 q^{23} + 106 q^{25} - 236 q^{27} + 56 q^{29} - 20 q^{31} - 720 q^{33} + 154 q^{35}+ \cdots + 2484 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 22 * q^5 - 14 * q^7 + 62 * q^9 + 36 * q^11 - 42 * q^13 + 136 * q^15 + 8 * q^17 - 118 * q^19 + 14 * q^21 + 104 * q^23 + 106 * q^25 - 236 * q^27 + 56 * q^29 - 20 * q^31 - 720 * q^33 + 154 * q^35 - 504 * q^37 - 528 * q^39 - 544 * q^41 + 412 * q^43 - 910 * q^45 + 500 * q^47 + 98 * q^49 + 676 * q^51 - 268 * q^53 - 1080 * q^55 - 1364 * q^57 - 198 * q^59 + 346 * q^61 - 434 * q^63 - 108 * q^65 + 1008 * q^67 + 352 * q^69 + 1224 * q^71 + 716 * q^73 - 2614 * q^75 - 252 * q^77 - 584 * q^79 - 754 * q^81 - 1230 * q^83 + 596 * q^85 + 2452 * q^87 + 596 * q^89 + 294 * q^91 - 1576 * q^93 - 184 * q^95 - 856 * q^97 + 2484 * 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
$3$	6.54983	1.26052	0.630258	−	0.776386i	$-0.282949\pi$
$3$	6.54983	1.26052	0.630258	+	0.776386i	$0.282949\pi$
$4$	0	0
$5$	−3.45017	−0.308592	−0.154296	−	0.988025i	$-0.549311\pi$
$5$	−3.45017	−0.308592	−0.154296	+	0.988025i	$0.549311\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	15.9003	0.588901
$10$	0	0
$11$	−27.2990	−0.748269	−0.374135	−	0.927374i	$-0.622060\pi$
$11$	−27.2990	−0.748269	−0.374135	+	0.927374i	$0.622060\pi$
$12$	0	0
$13$	−58.7492	−1.25339	−0.626696	−	0.779264i	$-0.715593\pi$
$13$	−58.7492	−1.25339	−0.626696	+	0.779264i	$0.715593\pi$
$14$	0	0
$15$	−22.5980	−0.388985
$16$	0	0
$17$	49.2990	0.703339	0.351670	−	0.936124i	$-0.385614\pi$
$17$	49.2990	0.703339	0.351670	+	0.936124i	$0.385614\pi$
$18$	0	0
$19$	−157.148	−1.89748	−0.948742	−	0.316052i	$-0.897643\pi$
$19$	−157.148	−1.89748	−0.948742	+	0.316052i	$0.897643\pi$
$20$	0	0
$21$	−45.8488	−0.476430
$22$	0	0
$23$	82.1993	0.745206	0.372603	−	0.927991i	$-0.378465\pi$
$23$	82.1993	0.745206	0.372603	+	0.927991i	$0.378465\pi$
$24$	0	0
$25$	−113.096	−0.904771
$26$	0	0
$27$	−72.7010	−0.518197
$28$	0	0
$29$	194.096	1.24285	0.621427	−	0.783472i	$-0.286553\pi$
$29$	194.096	1.24285	0.621427	+	0.783472i	$0.286553\pi$
$30$	0	0
$31$	−115.698	−0.670320	−0.335160	−	0.942161i	$-0.608790\pi$
$31$	−115.698	−0.670320	−0.335160	+	0.942161i	$0.608790\pi$
$32$	0	0
$33$	−178.804	−0.943205
$34$	0	0
$35$	24.1512	0.116637
$36$	0	0
$37$	−327.498	−1.45515	−0.727573	−	0.686030i	$-0.759352\pi$
$37$	−327.498	−1.45515	−0.727573	+	0.686030i	$0.759352\pi$
$38$	0	0
$39$	−384.797	−1.57992
$40$	0	0
$41$	−136.103	−0.518432	−0.259216	−	0.965819i	$-0.583464\pi$
$41$	−136.103	−0.518432	−0.259216	+	0.965819i	$0.583464\pi$
$42$	0	0
$43$	311.698	1.10543	0.552715	−	0.833371i	$-0.313592\pi$
$43$	311.698	1.10543	0.552715	+	0.833371i	$0.313592\pi$
$44$	0	0
$45$	−54.8588	−0.181730
$46$	0	0
$47$	355.698	1.10391	0.551956	−	0.833873i	$-0.313882\pi$
$47$	355.698	1.10391	0.551956	+	0.833873i	$0.313882\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	322.900	0.886570
$52$	0	0
$53$	−677.588	−1.75611	−0.878055	−	0.478559i	$-0.841159\pi$
$53$	−677.588	−1.75611	−0.878055	+	0.478559i	$0.841159\pi$
$54$	0	0
$55$	94.1861	0.230910
$56$	0	0
$57$	−1029.29	−2.39181
$58$	0	0
$59$	−197.148	−0.435025	−0.217512	−	0.976058i	$-0.569794\pi$
$59$	−197.148	−0.435025	−0.217512	+	0.976058i	$0.569794\pi$
$60$	0	0
$61$	−61.0449	−0.128131	−0.0640655	−	0.997946i	$-0.520407\pi$
$61$	−61.0449	−0.128131	−0.0640655	+	0.997946i	$0.520407\pi$
$62$	0	0
$63$	−111.302	−0.222584
$64$	0	0
$65$	202.694	0.386787
$66$	0	0
$67$	1017.39	1.85513	0.927566	−	0.373660i	$-0.121897\pi$
$67$	1017.39	1.85513	0.927566	+	0.373660i	$0.121897\pi$
$68$	0	0
$69$	538.392	0.939345
$70$	0	0
$71$	279.807	0.467704	0.233852	−	0.972272i	$-0.424867\pi$
$71$	279.807	0.467704	0.233852	+	0.972272i	$0.424867\pi$
$72$	0	0
$73$	629.794	1.00975	0.504875	−	0.863192i	$-0.331538\pi$
$73$	629.794	1.00975	0.504875	+	0.863192i	$0.331538\pi$
$74$	0	0
$75$	−740.762	−1.14048
$76$	0	0
$77$	191.093	0.282819
$78$	0	0
$79$	−20.2060	−0.0287766	−0.0143883	−	0.999896i	$-0.504580\pi$
$79$	−20.2060	−0.0287766	−0.0143883	+	0.999896i	$0.504580\pi$
$80$	0	0
$81$	−905.488	−1.24210
$82$	0	0
$83$	−260.158	−0.344049	−0.172024	−	0.985093i	$-0.555031\pi$
$83$	−260.158	−0.344049	−0.172024	+	0.985093i	$0.555031\pi$
$84$	0	0
$85$	−170.090	−0.217045
$86$	0	0
$87$	1271.30	1.56664
$88$	0	0
$89$	−909.974	−1.08379	−0.541893	−	0.840447i	$-0.682292\pi$
$89$	−909.974	−1.08379	−0.541893	+	0.840447i	$0.682292\pi$
$90$	0	0
$91$	411.244	0.473737
$92$	0	0
$93$	−757.801	−0.844949
$94$	0	0
$95$	542.186	0.585549
$96$	0	0
$97$	100.488	0.105186	0.0525931	−	0.998616i	$-0.483251\pi$
$97$	100.488	0.105186	0.0525931	+	0.998616i	$0.483251\pi$
$98$	0	0
$99$	−434.063	−0.440657
$100$	0	0
$101$	−1868.25	−1.84057	−0.920285	−	0.391249i	$-0.872043\pi$
$101$	−1868.25	−1.84057	−0.920285	+	0.391249i	$0.872043\pi$
$102$	0	0
$103$	−7.10629	−0.00679809	−0.00339905	−	0.999994i	$-0.501082\pi$
$103$	−7.10629	−0.00679809	−0.00339905	+	0.999994i	$0.501082\pi$
$104$	0	0
$105$	158.186	0.147023
$106$	0	0
$107$	−1078.01	−0.973970	−0.486985	−	0.873410i	$-0.661903\pi$
$107$	−1078.01	−0.973970	−0.486985	+	0.873410i	$0.661903\pi$
$108$	0	0
$109$	−457.698	−0.402197	−0.201099	−	0.979571i	$-0.564451\pi$
$109$	−457.698	−0.402197	−0.201099	+	0.979571i	$0.564451\pi$
$110$	0	0
$111$	−2145.06	−1.83423
$112$	0	0
$113$	1931.25	1.60776	0.803880	−	0.594791i	$-0.202765\pi$
$113$	1931.25	1.60776	0.803880	+	0.594791i	$0.202765\pi$
$114$	0	0
$115$	−283.601	−0.229965
$116$	0	0
$117$	−934.131	−0.738124
$118$	0	0
$119$	−345.093	−0.265837
$120$	0	0
$121$	−585.764	−0.440093
$122$	0	0
$123$	−891.452	−0.653492
$124$	0	0
$125$	821.472	0.587797
$126$	0	0
$127$	−1616.00	−1.12911	−0.564554	−	0.825396i	$-0.690952\pi$
$127$	−1616.00	−1.12911	−0.564554	+	0.825396i	$0.690952\pi$
$128$	0	0
$129$	2041.57	1.39341
$130$	0	0
$131$	507.650	0.338577	0.169288	−	0.985567i	$-0.445853\pi$
$131$	507.650	0.338577	0.169288	+	0.985567i	$0.445853\pi$
$132$	0	0
$133$	1100.03	0.717181
$134$	0	0
$135$	250.830	0.159911
$136$	0	0
$137$	−2763.36	−1.72328	−0.861642	−	0.507516i	$-0.830564\pi$
$137$	−2763.36	−1.72328	−0.861642	+	0.507516i	$0.830564\pi$
$138$	0	0
$139$	1227.55	0.749063	0.374531	−	0.927214i	$-0.377804\pi$
$139$	1227.55	0.749063	0.374531	+	0.927214i	$0.377804\pi$
$140$	0	0
$141$	2329.76	1.39150
$142$	0	0
$143$	1603.79	0.937874
$144$	0	0
$145$	−669.665	−0.383535
$146$	0	0
$147$	320.942	0.180074
$148$	0	0
$149$	4.61126	0.00253536	0.00126768	−	0.999999i	$-0.499596\pi$
$149$	4.61126	0.00253536	0.00126768	+	0.999999i	$0.499596\pi$
$150$	0	0
$151$	−74.3656	−0.0400780	−0.0200390	−	0.999799i	$-0.506379\pi$
$151$	−74.3656	−0.0400780	−0.0200390	+	0.999799i	$0.506379\pi$
$152$	0	0
$153$	783.871	0.414197
$154$	0	0
$155$	399.176	0.206855
$156$	0	0
$157$	3241.65	1.64785	0.823923	−	0.566702i	$-0.191781\pi$
$157$	3241.65	1.64785	0.823923	+	0.566702i	$0.191781\pi$
$158$	0	0
$159$	−4438.09	−2.21361
$160$	0	0
$161$	−575.395	−0.281662
$162$	0	0
$163$	−943.093	−0.453183	−0.226591	−	0.973990i	$-0.572758\pi$
$163$	−943.093	−0.453183	−0.226591	+	0.973990i	$0.572758\pi$
$164$	0	0
$165$	616.903	0.291066
$166$	0	0
$167$	2325.46	1.07754	0.538771	−	0.842452i	$-0.318889\pi$
$167$	2325.46	1.07754	0.538771	+	0.842452i	$0.318889\pi$
$168$	0	0
$169$	1254.47	0.570990
$170$	0	0
$171$	−2498.70	−1.11743
$172$	0	0
$173$	−2920.14	−1.28332	−0.641659	−	0.766990i	$-0.721753\pi$
$173$	−2920.14	−1.28332	−0.641659	+	0.766990i	$0.721753\pi$
$174$	0	0
$175$	791.675	0.341971
$176$	0	0
$177$	−1291.29	−0.548356
$178$	0	0
$179$	3049.14	1.27320	0.636602	−	0.771192i	$-0.280339\pi$
$179$	3049.14	1.27320	0.636602	+	0.771192i	$0.280339\pi$
$180$	0	0
$181$	3046.00	1.25087	0.625434	−	0.780277i	$-0.284922\pi$
$181$	3046.00	1.25087	0.625434	+	0.780277i	$0.284922\pi$
$182$	0	0
$183$	−399.834	−0.161511
$184$	0	0
$185$	1129.92	0.449047
$186$	0	0
$187$	−1345.81	−0.526287
$188$	0	0
$189$	508.907	0.195860
$190$	0	0
$191$	2056.55	0.779093	0.389547	−	0.921007i	$-0.372632\pi$
$191$	2056.55	0.779093	0.389547	+	0.921007i	$0.372632\pi$
$192$	0	0
$193$	−718.455	−0.267956	−0.133978	−	0.990984i	$-0.542775\pi$
$193$	−718.455	−0.267956	−0.133978	+	0.990984i	$0.542775\pi$
$194$	0	0
$195$	1327.61	0.487551
$196$	0	0
$197$	1200.16	0.434050	0.217025	−	0.976166i	$-0.430365\pi$
$197$	1200.16	0.434050	0.217025	+	0.976166i	$0.430365\pi$
$198$	0	0
$199$	950.335	0.338530	0.169265	−	0.985571i	$-0.445861\pi$
$199$	950.335	0.338530	0.169265	+	0.985571i	$0.445861\pi$
$200$	0	0
$201$	6663.73	2.33842
$202$	0	0
$203$	−1358.67	−0.469755
$204$	0	0
$205$	469.578	0.159984
$206$	0	0
$207$	1307.00	0.438853
$208$	0	0
$209$	4289.98	1.41983
$210$	0	0
$211$	2558.56	0.834779	0.417390	−	0.908728i	$-0.362945\pi$
$211$	2558.56	0.834779	0.417390	+	0.908728i	$0.362945\pi$
$212$	0	0
$213$	1832.69	0.589549
$214$	0	0
$215$	−1075.41	−0.341127
$216$	0	0
$217$	809.884	0.253357
$218$	0	0
$219$	4125.05	1.27281
$220$	0	0
$221$	−2896.28	−0.881559
$222$	0	0
$223$	−419.821	−0.126068	−0.0630342	−	0.998011i	$-0.520078\pi$
$223$	−419.821	−0.126068	−0.0630342	+	0.998011i	$0.520078\pi$
$224$	0	0
$225$	−1798.27	−0.532821
$226$	0	0
$227$	4425.95	1.29410	0.647050	−	0.762448i	$-0.276003\pi$
$227$	4425.95	1.29410	0.647050	+	0.762448i	$0.276003\pi$
$228$	0	0
$229$	682.254	0.196876	0.0984380	−	0.995143i	$-0.468615\pi$
$229$	682.254	0.196876	0.0984380	+	0.995143i	$0.468615\pi$
$230$	0	0
$231$	1251.63	0.356498
$232$	0	0
$233$	4280.74	1.20361	0.601804	−	0.798644i	$-0.294449\pi$
$233$	4280.74	1.20361	0.601804	+	0.798644i	$0.294449\pi$
$234$	0	0
$235$	−1227.22	−0.340658
$236$	0	0
$237$	−132.346	−0.0362733
$238$	0	0
$239$	−4828.95	−1.30694	−0.653470	−	0.756952i	$-0.726688\pi$
$239$	−4828.95	−1.30694	−0.653470	+	0.756952i	$0.726688\pi$
$240$	0	0
$241$	−959.139	−0.256363	−0.128182	−	0.991751i	$-0.540914\pi$
$241$	−959.139	−0.256363	−0.128182	+	0.991751i	$0.540914\pi$
$242$	0	0
$243$	−3967.87	−1.04749
$244$	0	0
$245$	−169.058	−0.0440846
$246$	0	0
$247$	9232.31	2.37829
$248$	0	0
$249$	−1703.99	−0.433679
$250$	0	0
$251$	−5226.44	−1.31430	−0.657151	−	0.753759i	$-0.728239\pi$
$251$	−5226.44	−1.31430	−0.657151	+	0.753759i	$0.728239\pi$
$252$	0	0
$253$	−2243.96	−0.557615
$254$	0	0
$255$	−1114.06	−0.273589
$256$	0	0
$257$	−3288.78	−0.798242	−0.399121	−	0.916898i	$-0.630685\pi$
$257$	−3288.78	−0.798242	−0.399121	+	0.916898i	$0.630685\pi$
$258$	0	0
$259$	2292.49	0.549993
$260$	0	0
$261$	3086.20	0.731919
$262$	0	0
$263$	6681.35	1.56650	0.783250	−	0.621707i	$-0.213560\pi$
$263$	6681.35	1.56650	0.783250	+	0.621707i	$0.213560\pi$
$264$	0	0
$265$	2337.79	0.541922
$266$	0	0
$267$	−5960.18	−1.36613
$268$	0	0
$269$	299.875	0.0679692	0.0339846	−	0.999422i	$-0.489180\pi$
$269$	299.875	0.0679692	0.0339846	+	0.999422i	$0.489180\pi$
$270$	0	0
$271$	−126.405	−0.0283342	−0.0141671	−	0.999900i	$-0.504510\pi$
$271$	−126.405	−0.0283342	−0.0141671	+	0.999900i	$0.504510\pi$
$272$	0	0
$273$	2693.58	0.597154
$274$	0	0
$275$	3087.42	0.677012
$276$	0	0
$277$	1439.53	0.312249	0.156124	−	0.987737i	$-0.450100\pi$
$277$	1439.53	0.312249	0.156124	+	0.987737i	$0.450100\pi$
$278$	0	0
$279$	−1839.63	−0.394752
$280$	0	0
$281$	−1024.45	−0.217485	−0.108742	−	0.994070i	$-0.534682\pi$
$281$	−1024.45	−0.217485	−0.108742	+	0.994070i	$0.534682\pi$
$282$	0	0
$283$	−147.168	−0.0309124	−0.0154562	−	0.999881i	$-0.504920\pi$
$283$	−147.168	−0.0309124	−0.0154562	+	0.999881i	$0.504920\pi$
$284$	0	0
$285$	3551.23	0.738094
$286$	0	0
$287$	952.721	0.195949
$288$	0	0
$289$	−2482.61	−0.505314
$290$	0	0
$291$	658.182	0.132589
$292$	0	0
$293$	−4611.20	−0.919418	−0.459709	−	0.888070i	$-0.652046\pi$
$293$	−4611.20	−0.919418	−0.459709	+	0.888070i	$0.652046\pi$
$294$	0	0
$295$	680.193	0.134245
$296$	0	0
$297$	1984.66	0.387751
$298$	0	0
$299$	−4829.14	−0.934035
$300$	0	0
$301$	−2181.88	−0.417813
$302$	0	0
$303$	−12236.7	−2.32007
$304$	0	0
$305$	210.615	0.0395402
$306$	0	0
$307$	−1136.28	−0.211242	−0.105621	−	0.994406i	$-0.533683\pi$
$307$	−1136.28	−0.211242	−0.105621	+	0.994406i	$0.533683\pi$
$308$	0	0
$309$	−46.5450	−0.00856911
$310$	0	0
$311$	−8798.71	−1.60427	−0.802137	−	0.597140i	$-0.796304\pi$
$311$	−8798.71	−1.60427	−0.802137	+	0.597140i	$0.796304\pi$
$312$	0	0
$313$	−8738.34	−1.57802	−0.789010	−	0.614381i	$-0.789406\pi$
$313$	−8738.34	−1.57802	−0.789010	+	0.614381i	$0.789406\pi$
$314$	0	0
$315$	384.011	0.0686876
$316$	0	0
$317$	−613.475	−0.108695	−0.0543473	−	0.998522i	$-0.517308\pi$
$317$	−613.475	−0.108695	−0.0543473	+	0.998522i	$0.517308\pi$
$318$	0	0
$319$	−5298.64	−0.929990
$320$	0	0
$321$	−7060.76	−1.22771
$322$	0	0
$323$	−7747.23	−1.33457
$324$	0	0
$325$	6644.32	1.13403
$326$	0	0
$327$	−2997.84	−0.506976
$328$	0	0
$329$	−2489.88	−0.417239
$330$	0	0
$331$	2920.52	0.484974	0.242487	−	0.970155i	$-0.422037\pi$
$331$	2920.52	0.484974	0.242487	+	0.970155i	$0.422037\pi$
$332$	0	0
$333$	−5207.33	−0.856937
$334$	0	0
$335$	−3510.16	−0.572479
$336$	0	0
$337$	5181.86	0.837609	0.418804	−	0.908077i	$-0.362449\pi$
$337$	5181.86	0.837609	0.418804	+	0.908077i	$0.362449\pi$
$338$	0	0
$339$	12649.4	2.02661
$340$	0	0
$341$	3158.43	0.501580
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−1857.54	−0.289874
$346$	0	0
$347$	6366.41	0.984919	0.492459	−	0.870335i	$-0.336098\pi$
$347$	6366.41	0.984919	0.492459	+	0.870335i	$0.336098\pi$
$348$	0	0
$349$	−2114.58	−0.324330	−0.162165	−	0.986764i	$-0.551848\pi$
$349$	−2114.58	−0.324330	−0.162165	+	0.986764i	$0.551848\pi$
$350$	0	0
$351$	4271.12	0.649503
$352$	0	0
$353$	−8424.62	−1.27025	−0.635124	−	0.772411i	$-0.719051\pi$
$353$	−8424.62	−1.27025	−0.635124	+	0.772411i	$0.719051\pi$
$354$	0	0
$355$	−965.381	−0.144330
$356$	0	0
$357$	−2260.30	−0.335092
$358$	0	0
$359$	597.090	0.0877805	0.0438903	−	0.999036i	$-0.486025\pi$
$359$	597.090	0.0877805	0.0438903	+	0.999036i	$0.486025\pi$
$360$	0	0
$361$	17836.4	2.60044
$362$	0	0
$363$	−3836.66	−0.554745
$364$	0	0
$365$	−2172.89	−0.311601
$366$	0	0
$367$	7563.58	1.07579	0.537896	−	0.843011i	$-0.319219\pi$
$367$	7563.58	1.07579	0.537896	+	0.843011i	$0.319219\pi$
$368$	0	0
$369$	−2164.08	−0.305305
$370$	0	0
$371$	4743.12	0.663748
$372$	0	0
$373$	−9760.01	−1.35484	−0.677419	−	0.735598i	$-0.736901\pi$
$373$	−9760.01	−1.35484	−0.677419	+	0.735598i	$0.736901\pi$
$374$	0	0
$375$	5380.50	0.740928
$376$	0	0
$377$	−11403.0	−1.55778
$378$	0	0
$379$	−9275.91	−1.25718	−0.628590	−	0.777737i	$-0.716368\pi$
$379$	−9275.91	−1.25718	−0.628590	+	0.777737i	$0.716368\pi$
$380$	0	0
$381$	−10584.5	−1.42326
$382$	0	0
$383$	−8008.24	−1.06841	−0.534206	−	0.845354i	$-0.679389\pi$
$383$	−8008.24	−1.06841	−0.534206	+	0.845354i	$0.679389\pi$
$384$	0	0
$385$	−659.303	−0.0872758
$386$	0	0
$387$	4956.10	0.650988
$388$	0	0
$389$	208.861	0.0272228	0.0136114	−	0.999907i	$-0.495667\pi$
$389$	208.861	0.0272228	0.0136114	+	0.999907i	$0.495667\pi$
$390$	0	0
$391$	4052.35	0.524133
$392$	0	0
$393$	3325.02	0.426781
$394$	0	0
$395$	69.7139	0.00888022
$396$	0	0
$397$	4004.51	0.506249	0.253124	−	0.967434i	$-0.418542\pi$
$397$	4004.51	0.506249	0.253124	+	0.967434i	$0.418542\pi$
$398$	0	0
$399$	7205.05	0.904019
$400$	0	0
$401$	−5085.30	−0.633286	−0.316643	−	0.948545i	$-0.602556\pi$
$401$	−5085.30	−0.633286	−0.316643	+	0.948545i	$0.602556\pi$
$402$	0	0
$403$	6797.14	0.840173
$404$	0	0
$405$	3124.08	0.383301
$406$	0	0
$407$	8940.38	1.08884
$408$	0	0
$409$	−15143.3	−1.83078	−0.915390	−	0.402569i	$-0.868117\pi$
$409$	−15143.3	−1.83078	−0.915390	+	0.402569i	$0.868117\pi$
$410$	0	0
$411$	−18099.6	−2.17223
$412$	0	0
$413$	1380.03	0.164424
$414$	0	0
$415$	897.587	0.106171
$416$	0	0
$417$	8040.27	0.944206
$418$	0	0
$419$	−8407.07	−0.980220	−0.490110	−	0.871661i	$-0.663043\pi$
$419$	−8407.07	−0.980220	−0.490110	+	0.871661i	$0.663043\pi$
$420$	0	0
$421$	−10020.9	−1.16006	−0.580032	−	0.814593i	$-0.696960\pi$
$421$	−10020.9	−1.16006	−0.580032	+	0.814593i	$0.696960\pi$
$422$	0	0
$423$	5655.71	0.650095
$424$	0	0
$425$	−5575.54	−0.636361
$426$	0	0
$427$	427.314	0.0484290
$428$	0	0
$429$	10504.6	1.18221
$430$	0	0
$431$	15424.5	1.72384	0.861918	−	0.507048i	$-0.169263\pi$
$431$	15424.5	1.72384	0.861918	+	0.507048i	$0.169263\pi$
$432$	0	0
$433$	10209.8	1.13314	0.566571	−	0.824013i	$-0.308270\pi$
$433$	10209.8	1.13314	0.566571	+	0.824013i	$0.308270\pi$
$434$	0	0
$435$	−4386.19	−0.483452
$436$	0	0
$437$	−12917.4	−1.41402
$438$	0	0
$439$	−13469.1	−1.46434	−0.732168	−	0.681124i	$-0.761491\pi$
$439$	−13469.1	−1.46434	−0.732168	+	0.681124i	$0.761491\pi$
$440$	0	0
$441$	779.116	0.0841287
$442$	0	0
$443$	17987.5	1.92915	0.964573	−	0.263817i	$-0.0849816\pi$
$443$	17987.5	1.92915	0.964573	+	0.263817i	$0.0849816\pi$
$444$	0	0
$445$	3139.56	0.334448
$446$	0	0
$447$	30.2030	0.00319587
$448$	0	0
$449$	−16413.6	−1.72518	−0.862591	−	0.505903i	$-0.831160\pi$
$449$	−16413.6	−1.72518	−0.862591	+	0.505903i	$0.831160\pi$
$450$	0	0
$451$	3715.48	0.387927
$452$	0	0
$453$	−487.082	−0.0505190
$454$	0	0
$455$	−1418.86	−0.146192
$456$	0	0
$457$	283.139	0.0289818	0.0144909	−	0.999895i	$-0.495387\pi$
$457$	283.139	0.0289818	0.0144909	+	0.999895i	$0.495387\pi$
$458$	0	0
$459$	−3584.09	−0.364468
$460$	0	0
$461$	−13769.3	−1.39111	−0.695555	−	0.718473i	$-0.744841\pi$
$461$	−13769.3	−1.39111	−0.695555	+	0.718473i	$0.744841\pi$
$462$	0	0
$463$	−12715.4	−1.27632	−0.638161	−	0.769903i	$-0.720304\pi$
$463$	−12715.4	−1.27632	−0.638161	+	0.769903i	$0.720304\pi$
$464$	0	0
$465$	2614.54	0.260745
$466$	0	0
$467$	8052.37	0.797900	0.398950	−	0.916973i	$-0.369375\pi$
$467$	8052.37	0.797900	0.398950	+	0.916973i	$0.369375\pi$
$468$	0	0
$469$	−7121.72	−0.701174
$470$	0	0
$471$	21232.3	2.07714
$472$	0	0
$473$	−8509.04	−0.827158
$474$	0	0
$475$	17772.8	1.71679
$476$	0	0
$477$	−10773.9	−1.03418
$478$	0	0
$479$	−5407.58	−0.515822	−0.257911	−	0.966169i	$-0.583034\pi$
$479$	−5407.58	−0.515822	−0.257911	+	0.966169i	$0.583034\pi$
$480$	0	0
$481$	19240.3	1.82387
$482$	0	0
$483$	−3768.74	−0.355039
$484$	0	0
$485$	−346.702	−0.0324596
$486$	0	0
$487$	−2672.17	−0.248639	−0.124320	−	0.992242i	$-0.539675\pi$
$487$	−2672.17	−0.248639	−0.124320	+	0.992242i	$0.539675\pi$
$488$	0	0
$489$	−6177.10	−0.571244
$490$	0	0
$491$	6537.75	0.600905	0.300453	−	0.953797i	$-0.402862\pi$
$491$	6537.75	0.600905	0.300453	+	0.953797i	$0.402862\pi$
$492$	0	0
$493$	9568.76	0.874148
$494$	0	0
$495$	1497.59	0.135983
$496$	0	0
$497$	−1958.65	−0.176776
$498$	0	0
$499$	−14621.8	−1.31175	−0.655875	−	0.754870i	$-0.727700\pi$
$499$	−14621.8	−1.31175	−0.655875	+	0.754870i	$0.727700\pi$
$500$	0	0
$501$	15231.4	1.35826
$502$	0	0
$503$	2928.33	0.259578	0.129789	−	0.991542i	$-0.458570\pi$
$503$	2928.33	0.259578	0.129789	+	0.991542i	$0.458570\pi$
$504$	0	0
$505$	6445.76	0.567986
$506$	0	0
$507$	8216.54	0.719742
$508$	0	0
$509$	2262.06	0.196982	0.0984910	−	0.995138i	$-0.468598\pi$
$509$	2262.06	0.196982	0.0984910	+	0.995138i	$0.468598\pi$
$510$	0	0
$511$	−4408.56	−0.381650
$512$	0	0
$513$	11424.8	0.983270
$514$	0	0
$515$	24.5179	0.00209784
$516$	0	0
$517$	−9710.19	−0.826023
$518$	0	0
$519$	−19126.4	−1.61764
$520$	0	0
$521$	−6284.94	−0.528499	−0.264250	−	0.964454i	$-0.585124\pi$
$521$	−6284.94	−0.528499	−0.264250	+	0.964454i	$0.585124\pi$
$522$	0	0
$523$	−23595.8	−1.97279	−0.986397	−	0.164378i	$-0.947438\pi$
$523$	−23595.8	−1.97279	−0.986397	+	0.164378i	$0.947438\pi$
$524$	0	0
$525$	5185.34	0.431060
$526$	0	0
$527$	−5703.78	−0.471462
$528$	0	0
$529$	−5410.27	−0.444667
$530$	0	0
$531$	−3134.72	−0.256187
$532$	0	0
$533$	7995.94	0.649798
$534$	0	0
$535$	3719.30	0.300560
$536$	0	0
$537$	19971.4	1.60490
$538$	0	0
$539$	−1337.65	−0.106896
$540$	0	0
$541$	11156.9	0.886642	0.443321	−	0.896363i	$-0.353800\pi$
$541$	11156.9	0.886642	0.443321	+	0.896363i	$0.353800\pi$
$542$	0	0
$543$	19950.8	1.57674
$544$	0	0
$545$	1579.13	0.124115
$546$	0	0
$547$	4578.08	0.357851	0.178926	−	0.983863i	$-0.442738\pi$
$547$	4578.08	0.357851	0.178926	+	0.983863i	$0.442738\pi$
$548$	0	0
$549$	−970.634	−0.0754565
$550$	0	0
$551$	−30501.8	−2.35830
$552$	0	0
$553$	141.442	0.0108765
$554$	0	0
$555$	7400.81	0.566031
$556$	0	0
$557$	19499.5	1.48334	0.741669	−	0.670766i	$-0.234035\pi$
$557$	19499.5	1.48334	0.741669	+	0.670766i	$0.234035\pi$
$558$	0	0
$559$	−18312.0	−1.38554
$560$	0	0
$561$	−8814.86	−0.663393
$562$	0	0
$563$	−4797.53	−0.359133	−0.179566	−	0.983746i	$-0.557469\pi$
$563$	−4797.53	−0.359133	−0.179566	+	0.983746i	$0.557469\pi$
$564$	0	0
$565$	−6663.14	−0.496142
$566$	0	0
$567$	6338.42	0.469468
$568$	0	0
$569$	−9461.07	−0.697063	−0.348531	−	0.937297i	$-0.613320\pi$
$569$	−9461.07	−0.697063	−0.348531	+	0.937297i	$0.613320\pi$
$570$	0	0
$571$	8146.88	0.597086	0.298543	−	0.954396i	$-0.403499\pi$
$571$	8146.88	0.597086	0.298543	+	0.954396i	$0.403499\pi$
$572$	0	0
$573$	13470.1	0.982060
$574$	0	0
$575$	−9296.45	−0.674241
$576$	0	0
$577$	231.608	0.0167105	0.00835526	−	0.999965i	$-0.497340\pi$
$577$	231.608	0.0167105	0.00835526	+	0.999965i	$0.497340\pi$
$578$	0	0
$579$	−4705.76	−0.337763
$580$	0	0
$581$	1821.10	0.130038
$582$	0	0
$583$	18497.5	1.31404
$584$	0	0
$585$	3222.91	0.227779
$586$	0	0
$587$	−12579.0	−0.884481	−0.442241	−	0.896896i	$-0.645816\pi$
$587$	−12579.0	−0.884481	−0.442241	+	0.896896i	$0.645816\pi$
$588$	0	0
$589$	18181.6	1.27192
$590$	0	0
$591$	7860.85	0.547127
$592$	0	0
$593$	−447.349	−0.0309788	−0.0154894	−	0.999880i	$-0.504931\pi$
$593$	−447.349	−0.0309788	−0.0154894	+	0.999880i	$0.504931\pi$
$594$	0	0
$595$	1190.63	0.0820353
$596$	0	0
$597$	6224.54	0.426723
$598$	0	0
$599$	−10640.9	−0.725833	−0.362916	−	0.931822i	$-0.618219\pi$
$599$	−10640.9	−0.725833	−0.362916	+	0.931822i	$0.618219\pi$
$600$	0	0
$601$	−9179.29	−0.623013	−0.311507	−	0.950244i	$-0.600834\pi$
$601$	−9179.29	−0.623013	−0.311507	+	0.950244i	$0.600834\pi$
$602$	0	0
$603$	16176.8	1.09249
$604$	0	0
$605$	2020.98	0.135809
$606$	0	0
$607$	−5404.92	−0.361415	−0.180707	−	0.983537i	$-0.557839\pi$
$607$	−5404.92	−0.361415	−0.180707	+	0.983537i	$0.557839\pi$
$608$	0	0
$609$	−8899.09	−0.592134
$610$	0	0
$611$	−20896.9	−1.38363
$612$	0	0
$613$	10971.0	0.722864	0.361432	−	0.932398i	$-0.382288\pi$
$613$	10971.0	0.722864	0.361432	+	0.932398i	$0.382288\pi$
$614$	0	0
$615$	3075.66	0.201663
$616$	0	0
$617$	−5008.36	−0.326789	−0.163395	−	0.986561i	$-0.552244\pi$
$617$	−5008.36	−0.326789	−0.163395	+	0.986561i	$0.552244\pi$
$618$	0	0
$619$	−26422.1	−1.71566	−0.857830	−	0.513934i	$-0.828188\pi$
$619$	−26422.1	−1.71566	−0.857830	+	0.513934i	$0.828188\pi$
$620$	0	0
$621$	−5975.97	−0.386164
$622$	0	0
$623$	6369.81	0.409633
$624$	0	0
$625$	11302.8	0.723381
$626$	0	0
$627$	28098.7	1.78972
$628$	0	0
$629$	−16145.3	−1.02346
$630$	0	0
$631$	−119.528	−0.00754093	−0.00377047	−	0.999993i	$-0.501200\pi$
$631$	−119.528	−0.00754093	−0.00377047	+	0.999993i	$0.501200\pi$
$632$	0	0
$633$	16758.1	1.05225
$634$	0	0
$635$	5575.47	0.348434
$636$	0	0
$637$	−2878.71	−0.179056
$638$	0	0
$639$	4449.03	0.275432
$640$	0	0
$641$	23746.6	1.46324	0.731618	−	0.681714i	$-0.238765\pi$
$641$	23746.6	1.46324	0.731618	+	0.681714i	$0.238765\pi$
$642$	0	0
$643$	−18138.5	−1.11246	−0.556231	−	0.831028i	$-0.687753\pi$
$643$	−18138.5	−1.11246	−0.556231	+	0.831028i	$0.687753\pi$
$644$	0	0
$645$	−7043.75	−0.429996
$646$	0	0
$647$	8176.52	0.496835	0.248417	−	0.968653i	$-0.420090\pi$
$647$	8176.52	0.496835	0.248417	+	0.968653i	$0.420090\pi$
$648$	0	0
$649$	5381.94	0.325516
$650$	0	0
$651$	5304.60	0.319361
$652$	0	0
$653$	−24315.5	−1.45718	−0.728589	−	0.684951i	$-0.759824\pi$
$653$	−24315.5	−1.45718	−0.728589	+	0.684951i	$0.759824\pi$
$654$	0	0
$655$	−1751.47	−0.104482
$656$	0	0
$657$	10013.9	0.594643
$658$	0	0
$659$	−13443.8	−0.794680	−0.397340	−	0.917671i	$-0.630067\pi$
$659$	−13443.8	−0.794680	−0.397340	+	0.917671i	$0.630067\pi$
$660$	0	0
$661$	20267.3	1.19260	0.596299	−	0.802762i	$-0.296637\pi$
$661$	20267.3	1.19260	0.596299	+	0.802762i	$0.296637\pi$
$662$	0	0
$663$	−18970.1	−1.11122
$664$	0	0
$665$	−3795.30	−0.221317
$666$	0	0
$667$	15954.6	0.926183
$668$	0	0
$669$	−2749.75	−0.158911
$670$	0	0
$671$	1666.46	0.0958765
$672$	0	0
$673$	20471.1	1.17252	0.586258	−	0.810125i	$-0.300601\pi$
$673$	20471.1	1.17252	0.586258	+	0.810125i	$0.300601\pi$
$674$	0	0
$675$	8222.22	0.468849
$676$	0	0
$677$	−17693.5	−1.00446	−0.502229	−	0.864735i	$-0.667486\pi$
$677$	−17693.5	−1.00446	−0.502229	+	0.864735i	$0.667486\pi$
$678$	0	0
$679$	−703.419	−0.0397566
$680$	0	0
$681$	28989.2	1.63123
$682$	0	0
$683$	−16613.3	−0.930734	−0.465367	−	0.885118i	$-0.654078\pi$
$683$	−16613.3	−0.930734	−0.465367	+	0.885118i	$0.654078\pi$
$684$	0	0
$685$	9534.06	0.531792
$686$	0	0
$687$	4468.65	0.248165
$688$	0	0
$689$	39807.7	2.20109
$690$	0	0
$691$	−8207.34	−0.451841	−0.225920	−	0.974146i	$-0.572539\pi$
$691$	−8207.34	−0.451841	−0.225920	+	0.974146i	$0.572539\pi$
$692$	0	0
$693$	3038.44	0.166553
$694$	0	0
$695$	−4235.26	−0.231155
$696$	0	0
$697$	−6709.74	−0.364634
$698$	0	0
$699$	28038.1	1.51717
$700$	0	0
$701$	21752.7	1.17202	0.586011	−	0.810303i	$-0.300698\pi$
$701$	21752.7	1.17202	0.586011	+	0.810303i	$0.300698\pi$
$702$	0	0
$703$	51465.7	2.76112
$704$	0	0
$705$	−8038.06	−0.429406
$706$	0	0
$707$	13077.7	0.695670
$708$	0	0
$709$	14636.1	0.775277	0.387638	−	0.921811i	$-0.373291\pi$
$709$	14636.1	0.775277	0.387638	+	0.921811i	$0.373291\pi$
$710$	0	0
$711$	−321.281	−0.0169465
$712$	0	0
$713$	−9510.27	−0.499527
$714$	0	0
$715$	−5533.35	−0.289421
$716$	0	0
$717$	−31628.8	−1.64742
$718$	0	0
$719$	33477.7	1.73645	0.868224	−	0.496173i	$-0.165262\pi$
$719$	33477.7	1.73645	0.868224	+	0.496173i	$0.165262\pi$
$720$	0	0
$721$	49.7440	0.00256944
$722$	0	0
$723$	−6282.20	−0.323150
$724$	0	0
$725$	−21951.6	−1.12450
$726$	0	0
$727$	−20073.4	−1.02404	−0.512022	−	0.858972i	$-0.671103\pi$
$727$	−20073.4	−1.02404	−0.512022	+	0.858972i	$0.671103\pi$
$728$	0	0
$729$	−1540.72	−0.0782767
$730$	0	0
$731$	15366.4	0.777492
$732$	0	0
$733$	19982.3	1.00691	0.503455	−	0.864022i	$-0.332062\pi$
$733$	19982.3	1.00691	0.503455	+	0.864022i	$0.332062\pi$
$734$	0	0
$735$	−1107.30	−0.0555694
$736$	0	0
$737$	−27773.7	−1.38814
$738$	0	0
$739$	11224.8	0.558744	0.279372	−	0.960183i	$-0.409874\pi$
$739$	11224.8	0.558744	0.279372	+	0.960183i	$0.409874\pi$
$740$	0	0
$741$	60470.1	2.99787
$742$	0	0
$743$	−7894.51	−0.389800	−0.194900	−	0.980823i	$-0.562438\pi$
$743$	−7894.51	−0.389800	−0.194900	+	0.980823i	$0.562438\pi$
$744$	0	0
$745$	−15.9096	−0.000782393 0
$746$	0	0
$747$	−4136.59	−0.202611
$748$	0	0
$749$	7546.05	0.368126
$750$	0	0
$751$	2613.52	0.126989	0.0634945	−	0.997982i	$-0.479775\pi$
$751$	2613.52	0.126989	0.0634945	+	0.997982i	$0.479775\pi$
$752$	0	0
$753$	−34232.3	−1.65670
$754$	0	0
$755$	256.574	0.0123678
$756$	0	0
$757$	−12055.8	−0.578831	−0.289415	−	0.957204i	$-0.593461\pi$
$757$	−12055.8	−0.578831	−0.289415	+	0.957204i	$0.593461\pi$
$758$	0	0
$759$	−14697.6	−0.702883
$760$	0	0
$761$	1196.38	0.0569893	0.0284946	−	0.999594i	$-0.490929\pi$
$761$	1196.38	0.0569893	0.0284946	+	0.999594i	$0.490929\pi$
$762$	0	0
$763$	3203.88	0.152016
$764$	0	0
$765$	−2704.48	−0.127818
$766$	0	0
$767$	11582.3	0.545256
$768$	0	0
$769$	5216.18	0.244604	0.122302	−	0.992493i	$-0.460972\pi$
$769$	5216.18	0.244604	0.122302	+	0.992493i	$0.460972\pi$
$770$	0	0
$771$	−21540.9	−1.00620
$772$	0	0
$773$	−17623.0	−0.819994	−0.409997	−	0.912087i	$-0.634470\pi$
$773$	−17623.0	−0.819994	−0.409997	+	0.912087i	$0.634470\pi$
$774$	0	0
$775$	13085.0	0.606486
$776$	0	0
$777$	15015.4	0.693276
$778$	0	0
$779$	21388.3	0.983716
$780$	0	0
$781$	−7638.46	−0.349969
$782$	0	0
$783$	−14111.0	−0.644043
$784$	0	0
$785$	−11184.2	−0.508512
$786$	0	0
$787$	−4779.24	−0.216470	−0.108235	−	0.994125i	$-0.534520\pi$
$787$	−4779.24	−0.216470	−0.108235	+	0.994125i	$0.534520\pi$
$788$	0	0
$789$	43761.7	1.97460
$790$	0	0
$791$	−13518.8	−0.607676
$792$	0	0
$793$	3586.34	0.160598
$794$	0	0
$795$	15312.1	0.683102
$796$	0	0
$797$	−37903.6	−1.68459	−0.842293	−	0.539020i	$-0.818795\pi$
$797$	−37903.6	−1.68459	−0.842293	+	0.539020i	$0.818795\pi$
$798$	0	0
$799$	17535.5	0.776424
$800$	0	0
$801$	−14468.9	−0.638243
$802$	0	0
$803$	−17192.8	−0.755565
$804$	0	0
$805$	1985.21	0.0869186
$806$	0	0
$807$	1964.13	0.0856763
$808$	0	0
$809$	−19848.0	−0.862570	−0.431285	−	0.902216i	$-0.641940\pi$
$809$	−19848.0	−0.862570	−0.431285	+	0.902216i	$0.641940\pi$
$810$	0	0
$811$	7098.27	0.307342	0.153671	−	0.988122i	$-0.450890\pi$
$811$	7098.27	0.307342	0.153671	+	0.988122i	$0.450890\pi$
$812$	0	0
$813$	−827.934	−0.0357158
$814$	0	0
$815$	3253.83	0.139849
$816$	0	0
$817$	−48982.6	−2.09753
$818$	0	0
$819$	6538.92	0.278985
$820$	0	0
$821$	−17294.5	−0.735179	−0.367589	−	0.929988i	$-0.619817\pi$
$821$	−17294.5	−0.735179	−0.367589	+	0.929988i	$0.619817\pi$
$822$	0	0
$823$	−17003.5	−0.720176	−0.360088	−	0.932918i	$-0.617253\pi$
$823$	−17003.5	−0.720176	−0.360088	+	0.932918i	$0.617253\pi$
$824$	0	0
$825$	20222.1	0.853385
$826$	0	0
$827$	24445.1	1.02786	0.513929	−	0.857833i	$-0.328189\pi$
$827$	24445.1	1.02786	0.513929	+	0.857833i	$0.328189\pi$
$828$	0	0
$829$	−19644.3	−0.823011	−0.411506	−	0.911407i	$-0.634997\pi$
$829$	−19644.3	−0.823011	−0.411506	+	0.911407i	$0.634997\pi$
$830$	0	0
$831$	9428.67	0.393595
$832$	0	0
$833$	2415.65	0.100477
$834$	0	0
$835$	−8023.22	−0.332521
$836$	0	0
$837$	8411.34	0.347358
$838$	0	0
$839$	23132.6	0.951879	0.475939	−	0.879478i	$-0.342108\pi$
$839$	23132.6	0.951879	0.475939	+	0.879478i	$0.342108\pi$
$840$	0	0
$841$	13284.4	0.544688
$842$	0	0
$843$	−6709.95	−0.274143
$844$	0	0
$845$	−4328.11	−0.176203
$846$	0	0
$847$	4100.35	0.166340
$848$	0	0
$849$	−963.924	−0.0389656
$850$	0	0
$851$	−26920.1	−1.08438
$852$	0	0
$853$	45519.8	1.82716	0.913581	−	0.406658i	$-0.133306\pi$
$853$	45519.8	1.82716	0.913581	+	0.406658i	$0.133306\pi$
$854$	0	0
$855$	8620.94	0.344830
$856$	0	0
$857$	15295.3	0.609659	0.304829	−	0.952407i	$-0.401401\pi$
$857$	15295.3	0.609659	0.304829	+	0.952407i	$0.401401\pi$
$858$	0	0
$859$	6495.13	0.257987	0.128994	−	0.991645i	$-0.458825\pi$
$859$	6495.13	0.257987	0.128994	+	0.991645i	$0.458825\pi$
$860$	0	0
$861$	6240.16	0.246997
$862$	0	0
$863$	−34002.5	−1.34120	−0.670601	−	0.741818i	$-0.733964\pi$
$863$	−34002.5	−1.34120	−0.670601	+	0.741818i	$0.733964\pi$
$864$	0	0
$865$	10075.0	0.396022
$866$	0	0
$867$	−16260.7	−0.636957
$868$	0	0
$869$	551.603	0.0215326
$870$	0	0
$871$	−59770.7	−2.32521
$872$	0	0
$873$	1597.80	0.0619442
$874$	0	0
$875$	−5750.30	−0.222167
$876$	0	0
$877$	−2063.66	−0.0794581	−0.0397291	−	0.999210i	$-0.512649\pi$
$877$	−2063.66	−0.0794581	−0.0397291	+	0.999210i	$0.512649\pi$
$878$	0	0
$879$	−30202.6	−1.15894
$880$	0	0
$881$	−1681.78	−0.0643141	−0.0321570	−	0.999483i	$-0.510238\pi$
$881$	−1681.78	−0.0643141	−0.0321570	+	0.999483i	$0.510238\pi$
$882$	0	0
$883$	11207.2	0.427125	0.213563	−	0.976929i	$-0.431493\pi$
$883$	11207.2	0.427125	0.213563	+	0.976929i	$0.431493\pi$
$884$	0	0
$885$	4455.15	0.169218
$886$	0	0
$887$	−8759.49	−0.331584	−0.165792	−	0.986161i	$-0.553018\pi$
$887$	−8759.49	−0.331584	−0.165792	+	0.986161i	$0.553018\pi$
$888$	0	0
$889$	11312.0	0.426763
$890$	0	0
$891$	24718.9	0.929423
$892$	0	0
$893$	−55897.1	−2.09465
$894$	0	0
$895$	−10520.0	−0.392901
$896$	0	0
$897$	−31630.1	−1.17737
$898$	0	0
$899$	−22456.5	−0.833110
$900$	0	0
$901$	−33404.4	−1.23514
$902$	0	0
$903$	−14291.0	−0.526660
$904$	0	0
$905$	−10509.2	−0.386008
$906$	0	0
$907$	27925.3	1.02232	0.511161	−	0.859485i	$-0.329216\pi$
$907$	27925.3	1.02232	0.511161	+	0.859485i	$0.329216\pi$
$908$	0	0
$909$	−29705.8	−1.08391
$910$	0	0
$911$	−23965.3	−0.871577	−0.435789	−	0.900049i	$-0.643530\pi$
$911$	−23965.3	−0.871577	−0.435789	+	0.900049i	$0.643530\pi$
$912$	0	0
$913$	7102.05	0.257441
$914$	0	0
$915$	1379.49	0.0498411
$916$	0	0
$917$	−3553.55	−0.127970
$918$	0	0
$919$	19192.2	0.688893	0.344446	−	0.938806i	$-0.388067\pi$
$919$	19192.2	0.688893	0.344446	+	0.938806i	$0.388067\pi$
$920$	0	0
$921$	−7442.47	−0.266273
$922$	0	0
$923$	−16438.4	−0.586217
$924$	0	0
$925$	37038.9	1.31657
$926$	0	0
$927$	−112.992	−0.00400341
$928$	0	0
$929$	−28616.5	−1.01063	−0.505317	−	0.862934i	$-0.668624\pi$
$929$	−28616.5	−1.01063	−0.505317	+	0.862934i	$0.668624\pi$
$930$	0	0
$931$	−7700.24	−0.271069
$932$	0	0
$933$	−57630.1	−2.02221
$934$	0	0
$935$	4643.28	0.162408
$936$	0	0
$937$	−27352.7	−0.953656	−0.476828	−	0.878997i	$-0.658213\pi$
$937$	−27352.7	−0.953656	−0.476828	+	0.878997i	$0.658213\pi$
$938$	0	0
$939$	−57234.7	−1.98912
$940$	0	0
$941$	34254.4	1.18668	0.593338	−	0.804953i	$-0.297810\pi$
$941$	34254.4	1.18668	0.593338	+	0.804953i	$0.297810\pi$
$942$	0	0
$943$	−11187.6	−0.386339
$944$	0	0
$945$	−1755.81	−0.0604409
$946$	0	0
$947$	−1099.15	−0.0377166	−0.0188583	−	0.999822i	$-0.506003\pi$
$947$	−1099.15	−0.0377166	−0.0188583	+	0.999822i	$0.506003\pi$
$948$	0	0
$949$	−36999.9	−1.26561
$950$	0	0
$951$	−4018.16	−0.137011
$952$	0	0
$953$	11037.5	0.375174	0.187587	−	0.982248i	$-0.439933\pi$
$953$	11037.5	0.375174	0.187587	+	0.982248i	$0.439933\pi$
$954$	0	0
$955$	−7095.44	−0.240422
$956$	0	0
$957$	−34705.2	−1.17227
$958$	0	0
$959$	19343.5	0.651340
$960$	0	0
$961$	−16405.0	−0.550671
$962$	0	0
$963$	−17140.7	−0.573572
$964$	0	0
$965$	2478.79	0.0826892
$966$	0	0
$967$	−11405.9	−0.379307	−0.189653	−	0.981851i	$-0.560736\pi$
$967$	−11405.9	−0.379307	−0.189653	+	0.981851i	$0.560736\pi$
$968$	0	0
$969$	−50743.1	−1.68225
$970$	0	0
$971$	−27331.4	−0.903302	−0.451651	−	0.892195i	$-0.649165\pi$
$971$	−27331.4	−0.903302	−0.451651	+	0.892195i	$0.649165\pi$
$972$	0	0
$973$	−8592.87	−0.283119
$974$	0	0
$975$	43519.2	1.42947
$976$	0	0
$977$	−4661.46	−0.152644	−0.0763221	−	0.997083i	$-0.524318\pi$
$977$	−4661.46	−0.152644	−0.0763221	+	0.997083i	$0.524318\pi$
$978$	0	0
$979$	24841.4	0.810964
$980$	0	0
$981$	−7277.54	−0.236854
$982$	0	0
$983$	7066.31	0.229278	0.114639	−	0.993407i	$-0.463429\pi$
$983$	7066.31	0.229278	0.114639	+	0.993407i	$0.463429\pi$
$984$	0	0
$985$	−4140.75	−0.133944
$986$	0	0
$987$	−16308.3	−0.525937
$988$	0	0
$989$	25621.3	0.823773
$990$	0	0
$991$	22127.8	0.709296	0.354648	−	0.935000i	$-0.384601\pi$
$991$	22127.8	0.709296	0.354648	+	0.935000i	$0.384601\pi$
$992$	0	0
$993$	19128.9	0.611318
$994$	0	0
$995$	−3278.81	−0.104468
$996$	0	0
$997$	45937.0	1.45922	0.729609	−	0.683865i	$-0.239702\pi$
$997$	45937.0	1.45922	0.729609	+	0.683865i	$0.239702\pi$
$998$	0	0
$999$	23809.5	0.754052

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.4.a.r.1.2		2
4.3	odd	2		448.4.a.s.1.1		2
8.3	odd	2		56.4.a.c.1.2	✓	2
8.5	even	2		112.4.a.h.1.1		2
24.5	odd	2		1008.4.a.x.1.2		2
24.11	even	2		504.4.a.i.1.2		2
40.3	even	4		1400.4.g.h.449.3		4
40.19	odd	2		1400.4.a.i.1.1		2
40.27	even	4		1400.4.g.h.449.2		4
56.3	even	6		392.4.i.i.177.2		4
56.11	odd	6		392.4.i.l.177.1		4
56.13	odd	2		784.4.a.t.1.2		2
56.19	even	6		392.4.i.i.361.2		4
56.27	even	2		392.4.a.h.1.1		2
56.51	odd	6		392.4.i.l.361.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.4.a.c.1.2	✓	2	8.3	odd	2
112.4.a.h.1.1		2	8.5	even	2
392.4.a.h.1.1		2	56.27	even	2
392.4.i.i.177.2		4	56.3	even	6
392.4.i.i.361.2		4	56.19	even	6
392.4.i.l.177.1		4	56.11	odd	6
392.4.i.l.361.1		4	56.51	odd	6
448.4.a.r.1.2		2	1.1	even	1	trivial
448.4.a.s.1.1		2	4.3	odd	2
504.4.a.i.1.2		2	24.11	even	2
784.4.a.t.1.2		2	56.13	odd	2
1008.4.a.x.1.2		2	24.5	odd	2
1400.4.a.i.1.1		2	40.19	odd	2
1400.4.g.h.449.2		4	40.27	even	4
1400.4.g.h.449.3		4	40.3	even	4

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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	448.a (trivial)

\(f(q)\)	\(=\)	\(q+6.54983 q^{3} -3.45017 q^{5} -7.00000 q^{7} +15.9003 q^{9} -27.2990 q^{11} -58.7492 q^{13} -22.5980 q^{15} +49.2990 q^{17} -157.148 q^{19} -45.8488 q^{21} +82.1993 q^{23} -113.096 q^{25} -72.7010 q^{27} +194.096 q^{29} -115.698 q^{31} -178.804 q^{33} +24.1512 q^{35} -327.498 q^{37} -384.797 q^{39} -136.103 q^{41} +311.698 q^{43} -54.8588 q^{45} +355.698 q^{47} +49.0000 q^{49} +322.900 q^{51} -677.588 q^{53} +94.1861 q^{55} -1029.29 q^{57} -197.148 q^{59} -61.0449 q^{61} -111.302 q^{63} +202.694 q^{65} +1017.39 q^{67} +538.392 q^{69} +279.807 q^{71} +629.794 q^{73} -740.762 q^{75} +191.093 q^{77} -20.2060 q^{79} -905.488 q^{81} -260.158 q^{83} -170.090 q^{85} +1271.30 q^{87} -909.974 q^{89} +411.244 q^{91} -757.801 q^{93} +542.186 q^{95} +100.488 q^{97} -434.063 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 22 q^{5} - 14 q^{7} + 62 q^{9} + 36 q^{11} - 42 q^{13} + 136 q^{15} + 8 q^{17} - 118 q^{19} + 14 q^{21} + 104 q^{23} + 106 q^{25} - 236 q^{27} + 56 q^{29} - 20 q^{31} - 720 q^{33} + 154 q^{35}+ \cdots + 2484 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 22 * q^5 - 14 * q^7 + 62 * q^9 + 36 * q^11 - 42 * q^13 + 136 * q^15 + 8 * q^17 - 118 * q^19 + 14 * q^21 + 104 * q^23 + 106 * q^25 - 236 * q^27 + 56 * q^29 - 20 * q^31 - 720 * q^33 + 154 * q^35 - 504 * q^37 - 528 * q^39 - 544 * q^41 + 412 * q^43 - 910 * q^45 + 500 * q^47 + 98 * q^49 + 676 * q^51 - 268 * q^53 - 1080 * q^55 - 1364 * q^57 - 198 * q^59 + 346 * q^61 - 434 * q^63 - 108 * q^65 + 1008 * q^67 + 352 * q^69 + 1224 * q^71 + 716 * q^73 - 2614 * q^75 - 252 * q^77 - 584 * q^79 - 754 * q^81 - 1230 * q^83 + 596 * q^85 + 2452 * q^87 + 596 * q^89 + 294 * q^91 - 1576 * q^93 - 184 * q^95 - 856 * q^97 + 2484 * q^99

Introduction

L-functions

Modular forms

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Database

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