LMFDB - Embedded newform 448.4.a.u.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,4,Mod(1,448)]

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

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$26.4328556826$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.2981.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 11x + 14 $ x^3 - x^2 - 11*x + 14
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 224)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.09300$ 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)

$f(q)$	$=$	$q-8.18600 q^{3} -16.4526 q^{5} +7.00000 q^{7} +40.0106 q^{9} +O(q^{10})$	$q-8.18600 q^{3} -16.4526 q^{5} +7.00000 q^{7} +40.0106 q^{9} -22.4774 q^{11} +75.2178 q^{13} +134.681 q^{15} +53.8388 q^{17} +96.0176 q^{19} -57.3020 q^{21} -174.681 q^{23} +145.688 q^{25} -106.505 q^{27} -139.228 q^{29} -125.004 q^{31} +184.000 q^{33} -115.168 q^{35} +343.665 q^{37} -615.733 q^{39} +445.656 q^{41} -94.1250 q^{43} -658.279 q^{45} -240.694 q^{47} +49.0000 q^{49} -440.724 q^{51} -419.362 q^{53} +369.812 q^{55} -786.000 q^{57} -520.544 q^{59} -540.977 q^{61} +280.074 q^{63} -1237.53 q^{65} +193.514 q^{67} +1429.94 q^{69} -296.239 q^{71} +92.8520 q^{73} -1192.60 q^{75} -157.342 q^{77} +122.540 q^{79} -208.438 q^{81} +1225.87 q^{83} -885.788 q^{85} +1139.72 q^{87} +1300.81 q^{89} +526.525 q^{91} +1023.29 q^{93} -1579.74 q^{95} -1376.17 q^{97} -899.334 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 8 q^{3} - 10 q^{5} + 21 q^{7} + 31 q^{9}+O(q^{10}) $ 3 * q - 8 * q^3 - 10 * q^5 + 21 * q^7 + 31 * q^9	$ 3 q - 8 q^{3} - 10 q^{5} + 21 q^{7} + 31 q^{9} - 24 q^{11} - 58 q^{13} - 8 q^{15} + 174 q^{17} - 64 q^{19} - 56 q^{21} - 112 q^{23} + 373 q^{25} - 104 q^{27} - 314 q^{29} - 280 q^{31} + 552 q^{33} - 70 q^{35} - 178 q^{37} - 648 q^{39} + 318 q^{41} - 632 q^{43} - 714 q^{45} - 664 q^{47} + 147 q^{49} - 896 q^{51} - 434 q^{53} - 808 q^{55} - 256 q^{57} - 816 q^{59} - 450 q^{61} + 217 q^{63} - 1604 q^{65} - 408 q^{67} + 1304 q^{69} + 184 q^{71} + 6 q^{73} - 2096 q^{75} - 168 q^{77} - 552 q^{79} - 1381 q^{81} + 1736 q^{83} + 988 q^{85} - 96 q^{87} - 218 q^{89} - 406 q^{91} + 2768 q^{93} - 3832 q^{95} - 1090 q^{97} - 824 q^{99}+O(q^{100}) $ 3 * q - 8 * q^3 - 10 * q^5 + 21 * q^7 + 31 * q^9 - 24 * q^11 - 58 * q^13 - 8 * q^15 + 174 * q^17 - 64 * q^19 - 56 * q^21 - 112 * q^23 + 373 * q^25 - 104 * q^27 - 314 * q^29 - 280 * q^31 + 552 * q^33 - 70 * q^35 - 178 * q^37 - 648 * q^39 + 318 * q^41 - 632 * q^43 - 714 * q^45 - 664 * q^47 + 147 * q^49 - 896 * q^51 - 434 * q^53 - 808 * q^55 - 256 * q^57 - 816 * q^59 - 450 * q^61 + 217 * q^63 - 1604 * q^65 - 408 * q^67 + 1304 * q^69 + 184 * q^71 + 6 * q^73 - 2096 * q^75 - 168 * q^77 - 552 * q^79 - 1381 * q^81 + 1736 * q^83 + 988 * q^85 - 96 * q^87 - 218 * q^89 - 406 * q^91 + 2768 * q^93 - 3832 * q^95 - 1090 * q^97 - 824 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−8.18600	−1.57540	−0.787698	−	0.616061i	$-0.788727\pi$
$3$	−8.18600	−1.57540	−0.787698	+	0.616061i	$0.788727\pi$
$4$	0	0
$5$	−16.4526	−1.47157	−0.735783	−	0.677218i	$-0.763186\pi$
$5$	−16.4526	−1.47157	−0.735783	+	0.677218i	$0.763186\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	40.0106	1.48187
$10$	0	0
$11$	−22.4774	−0.616108	−0.308054	−	0.951369i	$-0.599678\pi$
$11$	−22.4774	−0.616108	−0.308054	+	0.951369i	$0.599678\pi$
$12$	0	0
$13$	75.2178	1.60474	0.802372	−	0.596824i	$-0.203571\pi$
$13$	75.2178	1.60474	0.802372	+	0.596824i	$0.203571\pi$
$14$	0	0
$15$	134.681	2.31830
$16$	0	0
$17$	53.8388	0.768107	0.384054	−	0.923311i	$-0.374528\pi$
$17$	53.8388	0.768107	0.384054	+	0.923311i	$0.374528\pi$
$18$	0	0
$19$	96.0176	1.15937	0.579683	−	0.814842i	$-0.303176\pi$
$19$	96.0176	1.15937	0.579683	+	0.814842i	$0.303176\pi$
$20$	0	0
$21$	−57.3020	−0.595444
$22$	0	0
$23$	−174.681	−1.58363	−0.791815	−	0.610760i	$-0.790864\pi$
$23$	−174.681	−1.58363	−0.791815	+	0.610760i	$0.790864\pi$
$24$	0	0
$25$	145.688	1.16551
$26$	0	0
$27$	−106.505	−0.759143
$28$	0	0
$29$	−139.228	−0.891514	−0.445757	−	0.895154i	$-0.647066\pi$
$29$	−139.228	−0.891514	−0.445757	+	0.895154i	$0.647066\pi$
$30$	0	0
$31$	−125.004	−0.724240	−0.362120	−	0.932131i	$-0.617947\pi$
$31$	−125.004	−0.724240	−0.362120	+	0.932131i	$0.617947\pi$
$32$	0	0
$33$	184.000	0.970615
$34$	0	0
$35$	−115.168	−0.556200
$36$	0	0
$37$	343.665	1.52698	0.763489	−	0.645821i	$-0.223485\pi$
$37$	343.665	1.52698	0.763489	+	0.645821i	$0.223485\pi$
$38$	0	0
$39$	−615.733	−2.52811
$40$	0	0
$41$	445.656	1.69756	0.848778	−	0.528750i	$-0.177339\pi$
$41$	445.656	1.69756	0.848778	+	0.528750i	$0.177339\pi$
$42$	0	0
$43$	−94.1250	−0.333812	−0.166906	−	0.985973i	$-0.553378\pi$
$43$	−94.1250	−0.333812	−0.166906	+	0.985973i	$0.553378\pi$
$44$	0	0
$45$	−658.279	−2.18068
$46$	0	0
$47$	−240.694	−0.746998	−0.373499	−	0.927631i	$-0.621842\pi$
$47$	−240.694	−0.746998	−0.373499	+	0.927631i	$0.621842\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−440.724	−1.21007
$52$	0	0
$53$	−419.362	−1.08686	−0.543432	−	0.839453i	$-0.682875\pi$
$53$	−419.362	−1.08686	−0.543432	+	0.839453i	$0.682875\pi$
$54$	0	0
$55$	369.812	0.906644
$56$	0	0
$57$	−786.000	−1.82646
$58$	0	0
$59$	−520.544	−1.14863	−0.574314	−	0.818635i	$-0.694731\pi$
$59$	−520.544	−1.14863	−0.574314	+	0.818635i	$0.694731\pi$
$60$	0	0
$61$	−540.977	−1.13549	−0.567746	−	0.823204i	$-0.692184\pi$
$61$	−540.977	−1.13549	−0.567746	+	0.823204i	$0.692184\pi$
$62$	0	0
$63$	280.074	0.560096
$64$	0	0
$65$	−1237.53	−2.36149
$66$	0	0
$67$	193.514	0.352857	0.176429	−	0.984313i	$-0.443546\pi$
$67$	193.514	0.352857	0.176429	+	0.984313i	$0.443546\pi$
$68$	0	0
$69$	1429.94	2.49485
$70$	0	0
$71$	−296.239	−0.495170	−0.247585	−	0.968866i	$-0.579637\pi$
$71$	−296.239	−0.495170	−0.247585	+	0.968866i	$0.579637\pi$
$72$	0	0
$73$	92.8520	0.148870	0.0744349	−	0.997226i	$-0.476285\pi$
$73$	92.8520	0.148870	0.0744349	+	0.997226i	$0.476285\pi$
$74$	0	0
$75$	−1192.60	−1.83613
$76$	0	0
$77$	−157.342	−0.232867
$78$	0	0
$79$	122.540	0.174517	0.0872585	−	0.996186i	$-0.472189\pi$
$79$	122.540	0.174517	0.0872585	+	0.996186i	$0.472189\pi$
$80$	0	0
$81$	−208.438	−0.285923
$82$	0	0
$83$	1225.87	1.62117	0.810585	−	0.585621i	$-0.199149\pi$
$83$	1225.87	1.62117	0.810585	+	0.585621i	$0.199149\pi$
$84$	0	0
$85$	−885.788	−1.13032
$86$	0	0
$87$	1139.72	1.40449
$88$	0	0
$89$	1300.81	1.54928	0.774638	−	0.632405i	$-0.217932\pi$
$89$	1300.81	1.54928	0.774638	+	0.632405i	$0.217932\pi$
$90$	0	0
$91$	526.525	0.606536
$92$	0	0
$93$	1023.29	1.14097
$94$	0	0
$95$	−1579.74	−1.70608
$96$	0	0
$97$	−1376.17	−1.44051	−0.720254	−	0.693710i	$-0.755975\pi$
$97$	−1376.17	−1.44051	−0.720254	+	0.693710i	$0.755975\pi$
$98$	0	0
$99$	−899.334	−0.912995
$100$	0	0
$101$	191.393	0.188558	0.0942790	−	0.995546i	$-0.469945\pi$
$101$	191.393	0.188558	0.0942790	+	0.995546i	$0.469945\pi$
$102$	0	0
$103$	−1303.30	−1.24678	−0.623388	−	0.781913i	$-0.714244\pi$
$103$	−1303.30	−1.24678	−0.623388	+	0.781913i	$0.714244\pi$
$104$	0	0
$105$	942.767	0.876235
$106$	0	0
$107$	−426.508	−0.385347	−0.192673	−	0.981263i	$-0.561716\pi$
$107$	−426.508	−0.385347	−0.192673	+	0.981263i	$0.561716\pi$
$108$	0	0
$109$	37.4577	0.0329156	0.0164578	−	0.999865i	$-0.494761\pi$
$109$	37.4577	0.0329156	0.0164578	+	0.999865i	$0.494761\pi$
$110$	0	0
$111$	−2813.24	−2.40559
$112$	0	0
$113$	−1584.38	−1.31899	−0.659494	−	0.751710i	$-0.729230\pi$
$113$	−1584.38	−1.31899	−0.659494	+	0.751710i	$0.729230\pi$
$114$	0	0
$115$	2873.96	2.33042
$116$	0	0
$117$	3009.51	2.37803
$118$	0	0
$119$	376.872	0.290317
$120$	0	0
$121$	−825.767	−0.620411
$122$	0	0
$123$	−3648.14	−2.67432
$124$	0	0
$125$	−340.375	−0.243552
$126$	0	0
$127$	1868.17	1.30530	0.652652	−	0.757658i	$-0.273656\pi$
$127$	1868.17	1.30530	0.652652	+	0.757658i	$0.273656\pi$
$128$	0	0
$129$	770.507	0.525887
$130$	0	0
$131$	−1601.71	−1.06826	−0.534129	−	0.845403i	$-0.679360\pi$
$131$	−1601.71	−1.06826	−0.534129	+	0.845403i	$0.679360\pi$
$132$	0	0
$133$	672.123	0.438199
$134$	0	0
$135$	1752.28	1.11713
$136$	0	0
$137$	−584.927	−0.364771	−0.182386	−	0.983227i	$-0.558382\pi$
$137$	−584.927	−0.364771	−0.182386	+	0.983227i	$0.558382\pi$
$138$	0	0
$139$	2529.58	1.54357	0.771785	−	0.635884i	$-0.219364\pi$
$139$	2529.58	1.54357	0.771785	+	0.635884i	$0.219364\pi$
$140$	0	0
$141$	1970.32	1.17682
$142$	0	0
$143$	−1690.70	−0.988696
$144$	0	0
$145$	2290.66	1.31192
$146$	0	0
$147$	−401.114	−0.225057
$148$	0	0
$149$	857.567	0.471507	0.235754	−	0.971813i	$-0.424244\pi$
$149$	857.567	0.471507	0.235754	+	0.971813i	$0.424244\pi$
$150$	0	0
$151$	−26.3850	−0.0142198	−0.00710988	−	0.999975i	$-0.502263\pi$
$151$	−26.3850	−0.0142198	−0.00710988	+	0.999975i	$0.502263\pi$
$152$	0	0
$153$	2154.12	1.13824
$154$	0	0
$155$	2056.65	1.06577
$156$	0	0
$157$	−2517.90	−1.27994	−0.639970	−	0.768400i	$-0.721053\pi$
$157$	−2517.90	−1.27994	−0.639970	+	0.768400i	$0.721053\pi$
$158$	0	0
$159$	3432.90	1.71224
$160$	0	0
$161$	−1222.77	−0.598556
$162$	0	0
$163$	−3353.11	−1.61126	−0.805632	−	0.592416i	$-0.798174\pi$
$163$	−3353.11	−1.61126	−0.805632	+	0.592416i	$0.798174\pi$
$164$	0	0
$165$	−3027.28	−1.42832
$166$	0	0
$167$	2701.45	1.25176	0.625882	−	0.779917i	$-0.284739\pi$
$167$	2701.45	1.25176	0.625882	+	0.779917i	$0.284739\pi$
$168$	0	0
$169$	3460.72	1.57520
$170$	0	0
$171$	3841.72	1.71803
$172$	0	0
$173$	763.286	0.335442	0.167721	−	0.985834i	$-0.446359\pi$
$173$	763.286	0.335442	0.167721	+	0.985834i	$0.446359\pi$
$174$	0	0
$175$	1019.82	0.440520
$176$	0	0
$177$	4261.17	1.80954
$178$	0	0
$179$	−935.044	−0.390438	−0.195219	−	0.980760i	$-0.562542\pi$
$179$	−935.044	−0.390438	−0.195219	+	0.980760i	$0.562542\pi$
$180$	0	0
$181$	−3076.33	−1.26332	−0.631662	−	0.775244i	$-0.717627\pi$
$181$	−3076.33	−1.26332	−0.631662	+	0.775244i	$0.717627\pi$
$182$	0	0
$183$	4428.43	1.78885
$184$	0	0
$185$	−5654.18	−2.24705
$186$	0	0
$187$	−1210.16	−0.473237
$188$	0	0
$189$	−745.534	−0.286929
$190$	0	0
$191$	−1245.11	−0.471692	−0.235846	−	0.971790i	$-0.575786\pi$
$191$	−1245.11	−0.471692	−0.235846	+	0.971790i	$0.575786\pi$
$192$	0	0
$193$	−414.781	−0.154697	−0.0773487	−	0.997004i	$-0.524645\pi$
$193$	−414.781	−0.154697	−0.0773487	+	0.997004i	$0.524645\pi$
$194$	0	0
$195$	10130.4	3.72028
$196$	0	0
$197$	−1883.00	−0.681007	−0.340504	−	0.940243i	$-0.610598\pi$
$197$	−1883.00	−0.681007	−0.340504	+	0.940243i	$0.610598\pi$
$198$	0	0
$199$	1272.49	0.453290	0.226645	−	0.973977i	$-0.427224\pi$
$199$	1272.49	0.453290	0.226645	+	0.973977i	$0.427224\pi$
$200$	0	0
$201$	−1584.10	−0.555890
$202$	0	0
$203$	−974.593	−0.336961
$204$	0	0
$205$	−7332.20	−2.49806
$206$	0	0
$207$	−6989.09	−2.34674
$208$	0	0
$209$	−2158.23	−0.714295
$210$	0	0
$211$	907.243	0.296005	0.148003	−	0.988987i	$-0.452716\pi$
$211$	907.243	0.296005	0.148003	+	0.988987i	$0.452716\pi$
$212$	0	0
$213$	2425.01	0.780090
$214$	0	0
$215$	1548.60	0.491227
$216$	0	0
$217$	−875.031	−0.273737
$218$	0	0
$219$	−760.086	−0.234529
$220$	0	0
$221$	4049.64	1.23262
$222$	0	0
$223$	−2844.17	−0.854079	−0.427040	−	0.904233i	$-0.640443\pi$
$223$	−2844.17	−0.854079	−0.427040	+	0.904233i	$0.640443\pi$
$224$	0	0
$225$	5829.07	1.72713
$226$	0	0
$227$	−733.521	−0.214474	−0.107237	−	0.994234i	$-0.534200\pi$
$227$	−733.521	−0.214474	−0.107237	+	0.994234i	$0.534200\pi$
$228$	0	0
$229$	−2750.90	−0.793820	−0.396910	−	0.917858i	$-0.629917\pi$
$229$	−2750.90	−0.793820	−0.396910	+	0.917858i	$0.629917\pi$
$230$	0	0
$231$	1288.00	0.366858
$232$	0	0
$233$	2000.51	0.562478	0.281239	−	0.959638i	$-0.409255\pi$
$233$	2000.51	0.562478	0.281239	+	0.959638i	$0.409255\pi$
$234$	0	0
$235$	3960.05	1.09926
$236$	0	0
$237$	−1003.11	−0.274933
$238$	0	0
$239$	−162.403	−0.0439538	−0.0219769	−	0.999758i	$-0.506996\pi$
$239$	−162.403	−0.0439538	−0.0219769	+	0.999758i	$0.506996\pi$
$240$	0	0
$241$	942.142	0.251820	0.125910	−	0.992042i	$-0.459815\pi$
$241$	942.142	0.251820	0.125910	+	0.992042i	$0.459815\pi$
$242$	0	0
$243$	4581.90	1.20958
$244$	0	0
$245$	−806.178	−0.210224
$246$	0	0
$247$	7222.24	1.86049
$248$	0	0
$249$	−10035.0	−2.55399
$250$	0	0
$251$	1083.92	0.272575	0.136288	−	0.990669i	$-0.456483\pi$
$251$	1083.92	0.272575	0.136288	+	0.990669i	$0.456483\pi$
$252$	0	0
$253$	3926.38	0.975688
$254$	0	0
$255$	7251.06	1.78070
$256$	0	0
$257$	−1864.68	−0.452590	−0.226295	−	0.974059i	$-0.572661\pi$
$257$	−1864.68	−0.452590	−0.226295	+	0.974059i	$0.572661\pi$
$258$	0	0
$259$	2405.65	0.577143
$260$	0	0
$261$	−5570.58	−1.32111
$262$	0	0
$263$	−4325.69	−1.01420	−0.507098	−	0.861888i	$-0.669282\pi$
$263$	−4325.69	−1.01420	−0.507098	+	0.861888i	$0.669282\pi$
$264$	0	0
$265$	6899.60	1.59939
$266$	0	0
$267$	−10648.4	−2.44073
$268$	0	0
$269$	2158.42	0.489224	0.244612	−	0.969621i	$-0.421339\pi$
$269$	2158.42	0.489224	0.244612	+	0.969621i	$0.421339\pi$
$270$	0	0
$271$	−5113.43	−1.14619	−0.573097	−	0.819488i	$-0.694258\pi$
$271$	−5113.43	−1.14619	−0.573097	+	0.819488i	$0.694258\pi$
$272$	0	0
$273$	−4310.13	−0.955535
$274$	0	0
$275$	−3274.69	−0.718078
$276$	0	0
$277$	−8329.29	−1.80671	−0.903355	−	0.428894i	$-0.858903\pi$
$277$	−8329.29	−1.80671	−0.903355	+	0.428894i	$0.858903\pi$
$278$	0	0
$279$	−5001.50	−1.07323
$280$	0	0
$281$	1568.66	0.333019	0.166510	−	0.986040i	$-0.446750\pi$
$281$	1568.66	0.333019	0.166510	+	0.986040i	$0.446750\pi$
$282$	0	0
$283$	−8797.57	−1.84792	−0.923960	−	0.382489i	$-0.875067\pi$
$283$	−8797.57	−1.84792	−0.923960	+	0.382489i	$0.875067\pi$
$284$	0	0
$285$	12931.8	2.68776
$286$	0	0
$287$	3119.59	0.641616
$288$	0	0
$289$	−2014.38	−0.410011
$290$	0	0
$291$	11265.4	2.26937
$292$	0	0
$293$	3849.34	0.767511	0.383756	−	0.923435i	$-0.374630\pi$
$293$	3849.34	0.767511	0.383756	+	0.923435i	$0.374630\pi$
$294$	0	0
$295$	8564.30	1.69028
$296$	0	0
$297$	2393.95	0.467714
$298$	0	0
$299$	−13139.1	−2.54132
$300$	0	0
$301$	−658.875	−0.126169
$302$	0	0
$303$	−1566.75	−0.297054
$304$	0	0
$305$	8900.47	1.67095
$306$	0	0
$307$	−6118.60	−1.13748	−0.568741	−	0.822516i	$-0.692569\pi$
$307$	−6118.60	−1.13748	−0.568741	+	0.822516i	$0.692569\pi$
$308$	0	0
$309$	10668.8	1.96417
$310$	0	0
$311$	−9347.02	−1.70425	−0.852124	−	0.523340i	$-0.824686\pi$
$311$	−9347.02	−1.70425	−0.852124	+	0.523340i	$0.824686\pi$
$312$	0	0
$313$	−1753.54	−0.316664	−0.158332	−	0.987386i	$-0.550612\pi$
$313$	−1753.54	−0.316664	−0.158332	+	0.987386i	$0.550612\pi$
$314$	0	0
$315$	−4607.95	−0.824218
$316$	0	0
$317$	−3101.24	−0.549473	−0.274737	−	0.961519i	$-0.588591\pi$
$317$	−3101.24	−0.549473	−0.274737	+	0.961519i	$0.588591\pi$
$318$	0	0
$319$	3129.47	0.549269
$320$	0	0
$321$	3491.40	0.607074
$322$	0	0
$323$	5169.47	0.890518
$324$	0	0
$325$	10958.3	1.87034
$326$	0	0
$327$	−306.629	−0.0518551
$328$	0	0
$329$	−1684.86	−0.282339
$330$	0	0
$331$	4621.64	0.767457	0.383729	−	0.923446i	$-0.374640\pi$
$331$	4621.64	0.767457	0.383729	+	0.923446i	$0.374640\pi$
$332$	0	0
$333$	13750.2	2.26279
$334$	0	0
$335$	−3183.80	−0.519253
$336$	0	0
$337$	−1513.42	−0.244632	−0.122316	−	0.992491i	$-0.539032\pi$
$337$	−1513.42	−0.244632	−0.122316	+	0.992491i	$0.539032\pi$
$338$	0	0
$339$	12969.7	2.07793
$340$	0	0
$341$	2809.77	0.446210
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	−23526.2	−3.67133
$346$	0	0
$347$	868.637	0.134383	0.0671915	−	0.997740i	$-0.478596\pi$
$347$	868.637	0.134383	0.0671915	+	0.997740i	$0.478596\pi$
$348$	0	0
$349$	2436.12	0.373646	0.186823	−	0.982394i	$-0.440181\pi$
$349$	2436.12	0.373646	0.186823	+	0.982394i	$0.440181\pi$
$350$	0	0
$351$	−8011.06	−1.21823
$352$	0	0
$353$	−6426.93	−0.969040	−0.484520	−	0.874780i	$-0.661006\pi$
$353$	−6426.93	−0.969040	−0.484520	+	0.874780i	$0.661006\pi$
$354$	0	0
$355$	4873.90	0.728676
$356$	0	0
$357$	−3085.07	−0.457365
$358$	0	0
$359$	7767.84	1.14198	0.570990	−	0.820957i	$-0.306560\pi$
$359$	7767.84	1.14198	0.570990	+	0.820957i	$0.306560\pi$
$360$	0	0
$361$	2360.39	0.344130
$362$	0	0
$363$	6759.73	0.977393
$364$	0	0
$365$	−1527.66	−0.219072
$366$	0	0
$367$	−3873.66	−0.550963	−0.275481	−	0.961306i	$-0.588837\pi$
$367$	−3873.66	−0.550963	−0.275481	+	0.961306i	$0.588837\pi$
$368$	0	0
$369$	17831.0	2.51556
$370$	0	0
$371$	−2935.53	−0.410796
$372$	0	0
$373$	−9950.56	−1.38129	−0.690644	−	0.723195i	$-0.742673\pi$
$373$	−9950.56	−1.38129	−0.690644	+	0.723195i	$0.742673\pi$
$374$	0	0
$375$	2786.31	0.383691
$376$	0	0
$377$	−10472.4	−1.43065
$378$	0	0
$379$	−4880.15	−0.661415	−0.330708	−	0.943733i	$-0.607287\pi$
$379$	−4880.15	−0.661415	−0.330708	+	0.943733i	$0.607287\pi$
$380$	0	0
$381$	−15292.9	−2.05637
$382$	0	0
$383$	4309.91	0.575003	0.287501	−	0.957780i	$-0.407175\pi$
$383$	4309.91	0.575003	0.287501	+	0.957780i	$0.407175\pi$
$384$	0	0
$385$	2588.68	0.342679
$386$	0	0
$387$	−3766.00	−0.494668
$388$	0	0
$389$	−7920.80	−1.03239	−0.516196	−	0.856471i	$-0.672652\pi$
$389$	−7920.80	−1.03239	−0.516196	+	0.856471i	$0.672652\pi$
$390$	0	0
$391$	−9404.62	−1.21640
$392$	0	0
$393$	13111.6	1.68293
$394$	0	0
$395$	−2016.10	−0.256813
$396$	0	0
$397$	−1547.10	−0.195583	−0.0977916	−	0.995207i	$-0.531178\pi$
$397$	−1547.10	−0.195583	−0.0977916	+	0.995207i	$0.531178\pi$
$398$	0	0
$399$	−5502.00	−0.690337
$400$	0	0
$401$	2506.91	0.312193	0.156096	−	0.987742i	$-0.450109\pi$
$401$	2506.91	0.312193	0.156096	+	0.987742i	$0.450109\pi$
$402$	0	0
$403$	−9402.56	−1.16222
$404$	0	0
$405$	3429.34	0.420754
$406$	0	0
$407$	−7724.69	−0.940783
$408$	0	0
$409$	12602.0	1.52354	0.761772	−	0.647845i	$-0.224330\pi$
$409$	12602.0	1.52354	0.761772	+	0.647845i	$0.224330\pi$
$410$	0	0
$411$	4788.21	0.574659
$412$	0	0
$413$	−3643.81	−0.434140
$414$	0	0
$415$	−20168.8	−2.38566
$416$	0	0
$417$	−20707.1	−2.43173
$418$	0	0
$419$	−15018.3	−1.75105	−0.875525	−	0.483172i	$-0.839484\pi$
$419$	−15018.3	−1.75105	−0.875525	+	0.483172i	$0.839484\pi$
$420$	0	0
$421$	4399.14	0.509266	0.254633	−	0.967038i	$-0.418045\pi$
$421$	4399.14	0.509266	0.254633	+	0.967038i	$0.418045\pi$
$422$	0	0
$423$	−9630.33	−1.10696
$424$	0	0
$425$	7843.68	0.895233
$426$	0	0
$427$	−3786.84	−0.429175
$428$	0	0
$429$	13840.1	1.55759
$430$	0	0
$431$	13463.2	1.50463	0.752317	−	0.658801i	$-0.228936\pi$
$431$	13463.2	1.50463	0.752317	+	0.658801i	$0.228936\pi$
$432$	0	0
$433$	−12885.5	−1.43011	−0.715055	−	0.699069i	$-0.753598\pi$
$433$	−12885.5	−1.43011	−0.715055	+	0.699069i	$0.753598\pi$
$434$	0	0
$435$	−18751.3	−2.06680
$436$	0	0
$437$	−16772.5	−1.83601
$438$	0	0
$439$	11350.1	1.23397	0.616984	−	0.786975i	$-0.288354\pi$
$439$	11350.1	1.23397	0.616984	+	0.786975i	$0.288354\pi$
$440$	0	0
$441$	1960.52	0.211696
$442$	0	0
$443$	−12705.9	−1.36269	−0.681347	−	0.731960i	$-0.738606\pi$
$443$	−12705.9	−1.36269	−0.681347	+	0.731960i	$0.738606\pi$
$444$	0	0
$445$	−21401.7	−2.27986
$446$	0	0
$447$	−7020.04	−0.742811
$448$	0	0
$449$	−11277.6	−1.18535	−0.592674	−	0.805443i	$-0.701928\pi$
$449$	−11277.6	−1.18535	−0.592674	+	0.805443i	$0.701928\pi$
$450$	0	0
$451$	−10017.2	−1.04588
$452$	0	0
$453$	215.988	0.0224018
$454$	0	0
$455$	−8662.70	−0.892558
$456$	0	0
$457$	−1859.13	−0.190298	−0.0951492	−	0.995463i	$-0.530333\pi$
$457$	−1859.13	−0.190298	−0.0951492	+	0.995463i	$0.530333\pi$
$458$	0	0
$459$	−5734.09	−0.583103
$460$	0	0
$461$	−2279.21	−0.230268	−0.115134	−	0.993350i	$-0.536730\pi$
$461$	−2279.21	−0.230268	−0.115134	+	0.993350i	$0.536730\pi$
$462$	0	0
$463$	−117.584	−0.0118025	−0.00590127	−	0.999983i	$-0.501878\pi$
$463$	−117.584	−0.0118025	−0.00590127	+	0.999983i	$0.501878\pi$
$464$	0	0
$465$	−16835.7	−1.67901
$466$	0	0
$467$	3951.45	0.391544	0.195772	−	0.980649i	$-0.437279\pi$
$467$	3951.45	0.391544	0.195772	+	0.980649i	$0.437279\pi$
$468$	0	0
$469$	1354.59	0.133368
$470$	0	0
$471$	20611.6	2.01641
$472$	0	0
$473$	2115.69	0.205664
$474$	0	0
$475$	13988.6	1.35125
$476$	0	0
$477$	−16778.9	−1.61060
$478$	0	0
$479$	−8331.54	−0.794734	−0.397367	−	0.917660i	$-0.630076\pi$
$479$	−8331.54	−0.794734	−0.397367	+	0.917660i	$0.630076\pi$
$480$	0	0
$481$	25849.7	2.45041
$482$	0	0
$483$	10009.6	0.942963
$484$	0	0
$485$	22641.7	2.11980
$486$	0	0
$487$	6369.75	0.592692	0.296346	−	0.955081i	$-0.404232\pi$
$487$	6369.75	0.592692	0.296346	+	0.955081i	$0.404232\pi$
$488$	0	0
$489$	27448.6	2.53838
$490$	0	0
$491$	17725.7	1.62923	0.814614	−	0.580004i	$-0.196949\pi$
$491$	17725.7	1.62923	0.814614	+	0.580004i	$0.196949\pi$
$492$	0	0
$493$	−7495.85	−0.684779
$494$	0	0
$495$	14796.4	1.34353
$496$	0	0
$497$	−2073.67	−0.187157
$498$	0	0
$499$	3869.06	0.347100	0.173550	−	0.984825i	$-0.444476\pi$
$499$	3869.06	0.347100	0.173550	+	0.984825i	$0.444476\pi$
$500$	0	0
$501$	−22114.1	−1.97203
$502$	0	0
$503$	−2583.83	−0.229040	−0.114520	−	0.993421i	$-0.536533\pi$
$503$	−2583.83	−0.229040	−0.114520	+	0.993421i	$0.536533\pi$
$504$	0	0
$505$	−3148.92	−0.277475
$506$	0	0
$507$	−28329.5	−2.48157
$508$	0	0
$509$	11425.7	0.994957	0.497479	−	0.867476i	$-0.334259\pi$
$509$	11425.7	0.994957	0.497479	+	0.867476i	$0.334259\pi$
$510$	0	0
$511$	649.964	0.0562675
$512$	0	0
$513$	−10226.3	−0.880125
$514$	0	0
$515$	21442.7	1.83471
$516$	0	0
$517$	5410.18	0.460231
$518$	0	0
$519$	−6248.26	−0.528455
$520$	0	0
$521$	9041.35	0.760285	0.380143	−	0.924928i	$-0.375875\pi$
$521$	9041.35	0.760285	0.380143	+	0.924928i	$0.375875\pi$
$522$	0	0
$523$	13583.0	1.13564	0.567822	−	0.823152i	$-0.307786\pi$
$523$	13583.0	1.13564	0.567822	+	0.823152i	$0.307786\pi$
$524$	0	0
$525$	−8348.23	−0.693993
$526$	0	0
$527$	−6730.09	−0.556294
$528$	0	0
$529$	18346.5	1.50789
$530$	0	0
$531$	−20827.3	−1.70212
$532$	0	0
$533$	33521.3	2.72414
$534$	0	0
$535$	7017.17	0.567063
$536$	0	0
$537$	7654.27	0.615095
$538$	0	0
$539$	−1101.39	−0.0880155
$540$	0	0
$541$	−513.919	−0.0408412	−0.0204206	−	0.999791i	$-0.506501\pi$
$541$	−513.919	−0.0408412	−0.0204206	+	0.999791i	$0.506501\pi$
$542$	0	0
$543$	25182.8	1.99024
$544$	0	0
$545$	−616.277	−0.0484375
$546$	0	0
$547$	14341.5	1.12102	0.560510	−	0.828147i	$-0.310605\pi$
$547$	14341.5	1.12102	0.560510	+	0.828147i	$0.310605\pi$
$548$	0	0
$549$	−21644.8	−1.68266
$550$	0	0
$551$	−13368.3	−1.03359
$552$	0	0
$553$	857.781	0.0659612
$554$	0	0
$555$	46285.1	3.53999
$556$	0	0
$557$	−13163.3	−1.00134	−0.500671	−	0.865638i	$-0.666913\pi$
$557$	−13163.3	−1.00134	−0.500671	+	0.865638i	$0.666913\pi$
$558$	0	0
$559$	−7079.88	−0.535683
$560$	0	0
$561$	9906.34	0.745536
$562$	0	0
$563$	16016.0	1.19893	0.599463	−	0.800403i	$-0.295381\pi$
$563$	16016.0	1.19893	0.599463	+	0.800403i	$0.295381\pi$
$564$	0	0
$565$	26067.1	1.94098
$566$	0	0
$567$	−1459.06	−0.108069
$568$	0	0
$569$	16543.8	1.21889	0.609447	−	0.792827i	$-0.291392\pi$
$569$	16543.8	1.21889	0.609447	+	0.792827i	$0.291392\pi$
$570$	0	0
$571$	7008.76	0.513674	0.256837	−	0.966455i	$-0.417320\pi$
$571$	7008.76	0.513674	0.256837	+	0.966455i	$0.417320\pi$
$572$	0	0
$573$	10192.5	0.743102
$574$	0	0
$575$	−25449.0	−1.84573
$576$	0	0
$577$	−9663.21	−0.697200	−0.348600	−	0.937272i	$-0.613343\pi$
$577$	−9663.21	−0.697200	−0.348600	+	0.937272i	$0.613343\pi$
$578$	0	0
$579$	3395.40	0.243710
$580$	0	0
$581$	8581.12	0.612745
$582$	0	0
$583$	9426.17	0.669626
$584$	0	0
$585$	−49514.3	−3.49943
$586$	0	0
$587$	−25908.4	−1.82173	−0.910863	−	0.412708i	$-0.864583\pi$
$587$	−25908.4	−1.82173	−0.910863	+	0.412708i	$0.864583\pi$
$588$	0	0
$589$	−12002.6	−0.839660
$590$	0	0
$591$	15414.3	1.07286
$592$	0	0
$593$	7381.86	0.511192	0.255596	−	0.966784i	$-0.417728\pi$
$593$	7381.86	0.511192	0.255596	+	0.966784i	$0.417728\pi$
$594$	0	0
$595$	−6200.52	−0.427221
$596$	0	0
$597$	−10416.6	−0.714112
$598$	0	0
$599$	−13420.6	−0.915445	−0.457722	−	0.889095i	$-0.651335\pi$
$599$	−13420.6	−0.915445	−0.457722	+	0.889095i	$0.651335\pi$
$600$	0	0
$601$	−7451.64	−0.505755	−0.252877	−	0.967498i	$-0.581377\pi$
$601$	−7451.64	−0.505755	−0.252877	+	0.967498i	$0.581377\pi$
$602$	0	0
$603$	7742.59	0.522890
$604$	0	0
$605$	13586.0	0.912975
$606$	0	0
$607$	−8757.50	−0.585595	−0.292797	−	0.956175i	$-0.594586\pi$
$607$	−8757.50	−0.585595	−0.292797	+	0.956175i	$0.594586\pi$
$608$	0	0
$609$	7978.02	0.530847
$610$	0	0
$611$	−18104.5	−1.19874
$612$	0	0
$613$	26967.2	1.77683	0.888415	−	0.459042i	$-0.151807\pi$
$613$	26967.2	1.77683	0.888415	+	0.459042i	$0.151807\pi$
$614$	0	0
$615$	60021.4	3.93544
$616$	0	0
$617$	−1347.62	−0.0879302	−0.0439651	−	0.999033i	$-0.513999\pi$
$617$	−1347.62	−0.0879302	−0.0439651	+	0.999033i	$0.513999\pi$
$618$	0	0
$619$	14652.9	0.951451	0.475725	−	0.879594i	$-0.342186\pi$
$619$	14652.9	0.951451	0.475725	+	0.879594i	$0.342186\pi$
$620$	0	0
$621$	18604.4	1.20220
$622$	0	0
$623$	9105.67	0.585572
$624$	0	0
$625$	−12611.0	−0.807102
$626$	0	0
$627$	17667.2	1.12530
$628$	0	0
$629$	18502.5	1.17288
$630$	0	0
$631$	17739.5	1.11917	0.559587	−	0.828771i	$-0.310960\pi$
$631$	17739.5	1.11917	0.559587	+	0.828771i	$0.310960\pi$
$632$	0	0
$633$	−7426.69	−0.466326
$634$	0	0
$635$	−30736.3	−1.92084
$636$	0	0
$637$	3685.67	0.229249
$638$	0	0
$639$	−11852.7	−0.733780
$640$	0	0
$641$	−2280.50	−0.140521	−0.0702607	−	0.997529i	$-0.522383\pi$
$641$	−2280.50	−0.140521	−0.0702607	+	0.997529i	$0.522383\pi$
$642$	0	0
$643$	−9267.10	−0.568365	−0.284183	−	0.958770i	$-0.591722\pi$
$643$	−9267.10	−0.568365	−0.284183	+	0.958770i	$0.591722\pi$
$644$	0	0
$645$	−12676.9	−0.773877
$646$	0	0
$647$	−8082.65	−0.491131	−0.245566	−	0.969380i	$-0.578974\pi$
$647$	−8082.65	−0.491131	−0.245566	+	0.969380i	$0.578974\pi$
$648$	0	0
$649$	11700.5	0.707679
$650$	0	0
$651$	7163.00	0.431245
$652$	0	0
$653$	−16903.5	−1.01299	−0.506497	−	0.862242i	$-0.669060\pi$
$653$	−16903.5	−1.01299	−0.506497	+	0.862242i	$0.669060\pi$
$654$	0	0
$655$	26352.2	1.57201
$656$	0	0
$657$	3715.06	0.220606
$658$	0	0
$659$	−29675.3	−1.75415	−0.877076	−	0.480352i	$-0.840509\pi$
$659$	−29675.3	−1.75415	−0.877076	+	0.480352i	$0.840509\pi$
$660$	0	0
$661$	22575.5	1.32842	0.664210	−	0.747546i	$-0.268768\pi$
$661$	22575.5	1.32842	0.664210	+	0.747546i	$0.268768\pi$
$662$	0	0
$663$	−33150.3	−1.94186
$664$	0	0
$665$	−11058.2	−0.644839
$666$	0	0
$667$	24320.4	1.41183
$668$	0	0
$669$	23282.4	1.34551
$670$	0	0
$671$	12159.7	0.699585
$672$	0	0
$673$	−4900.58	−0.280688	−0.140344	−	0.990103i	$-0.544821\pi$
$673$	−4900.58	−0.280688	−0.140344	+	0.990103i	$0.544821\pi$
$674$	0	0
$675$	−15516.5	−0.884786
$676$	0	0
$677$	−9606.03	−0.545332	−0.272666	−	0.962109i	$-0.587905\pi$
$677$	−9606.03	−0.545332	−0.272666	+	0.962109i	$0.587905\pi$
$678$	0	0
$679$	−9633.22	−0.544461
$680$	0	0
$681$	6004.60	0.337881
$682$	0	0
$683$	−3198.29	−0.179179	−0.0895895	−	0.995979i	$-0.528556\pi$
$683$	−3198.29	−0.179179	−0.0895895	+	0.995979i	$0.528556\pi$
$684$	0	0
$685$	9623.57	0.536785
$686$	0	0
$687$	22518.9	1.25058
$688$	0	0
$689$	−31543.5	−1.74414
$690$	0	0
$691$	33371.5	1.83721	0.918604	−	0.395180i	$-0.129318\pi$
$691$	33371.5	1.83721	0.918604	+	0.395180i	$0.129318\pi$
$692$	0	0
$693$	−6295.34	−0.345080
$694$	0	0
$695$	−41618.2	−2.27146
$696$	0	0
$697$	23993.6	1.30391
$698$	0	0
$699$	−16376.1	−0.886126
$700$	0	0
$701$	−9497.01	−0.511694	−0.255847	−	0.966717i	$-0.582354\pi$
$701$	−9497.01	−0.511694	−0.255847	+	0.966717i	$0.582354\pi$
$702$	0	0
$703$	32997.9	1.77033
$704$	0	0
$705$	−32417.0	−1.73176
$706$	0	0
$707$	1339.75	0.0712682
$708$	0	0
$709$	−33909.7	−1.79620	−0.898099	−	0.439793i	$-0.855052\pi$
$709$	−33909.7	−1.79620	−0.898099	+	0.439793i	$0.855052\pi$
$710$	0	0
$711$	4902.90	0.258612
$712$	0	0
$713$	21835.9	1.14693
$714$	0	0
$715$	27816.4	1.45493
$716$	0	0
$717$	1329.43	0.0692447
$718$	0	0
$719$	9750.54	0.505750	0.252875	−	0.967499i	$-0.418624\pi$
$719$	9750.54	0.505750	0.252875	+	0.967499i	$0.418624\pi$
$720$	0	0
$721$	−9123.09	−0.471237
$722$	0	0
$723$	−7712.37	−0.396717
$724$	0	0
$725$	−20283.8	−1.03907
$726$	0	0
$727$	23314.3	1.18938	0.594691	−	0.803955i	$-0.297275\pi$
$727$	23314.3	1.18938	0.594691	+	0.803955i	$0.297275\pi$
$728$	0	0
$729$	−31879.6	−1.61965
$730$	0	0
$731$	−5067.58	−0.256404
$732$	0	0
$733$	17472.5	0.880438	0.440219	−	0.897890i	$-0.354901\pi$
$733$	17472.5	0.880438	0.440219	+	0.897890i	$0.354901\pi$
$734$	0	0
$735$	6599.37	0.331186
$736$	0	0
$737$	−4349.68	−0.217398
$738$	0	0
$739$	−9323.31	−0.464091	−0.232046	−	0.972705i	$-0.574542\pi$
$739$	−9323.31	−0.464091	−0.232046	+	0.972705i	$0.574542\pi$
$740$	0	0
$741$	−59121.2	−2.93100
$742$	0	0
$743$	−3899.59	−0.192547	−0.0962733	−	0.995355i	$-0.530692\pi$
$743$	−3899.59	−0.192547	−0.0962733	+	0.995355i	$0.530692\pi$
$744$	0	0
$745$	−14109.2	−0.693854
$746$	0	0
$747$	49047.9	2.40237
$748$	0	0
$749$	−2985.56	−0.145647
$750$	0	0
$751$	9221.69	0.448075	0.224037	−	0.974581i	$-0.428076\pi$
$751$	9221.69	0.448075	0.224037	+	0.974581i	$0.428076\pi$
$752$	0	0
$753$	−8872.96	−0.429414
$754$	0	0
$755$	434.103	0.0209253
$756$	0	0
$757$	−17799.9	−0.854620	−0.427310	−	0.904105i	$-0.640539\pi$
$757$	−17799.9	−0.854620	−0.427310	+	0.904105i	$0.640539\pi$
$758$	0	0
$759$	−32141.3	−1.53710
$760$	0	0
$761$	13865.1	0.660458	0.330229	−	0.943901i	$-0.392874\pi$
$761$	13865.1	0.660458	0.330229	+	0.943901i	$0.392874\pi$
$762$	0	0
$763$	262.204	0.0124409
$764$	0	0
$765$	−35440.9	−1.67499
$766$	0	0
$767$	−39154.2	−1.84325
$768$	0	0
$769$	4256.76	0.199613	0.0998067	−	0.995007i	$-0.468178\pi$
$769$	4256.76	0.199613	0.0998067	+	0.995007i	$0.468178\pi$
$770$	0	0
$771$	15264.3	0.713008
$772$	0	0
$773$	11189.4	0.520641	0.260320	−	0.965522i	$-0.416172\pi$
$773$	11189.4	0.520641	0.260320	+	0.965522i	$0.416172\pi$
$774$	0	0
$775$	−18211.7	−0.844106
$776$	0	0
$777$	−19692.7	−0.909229
$778$	0	0
$779$	42790.8	1.96809
$780$	0	0
$781$	6658.68	0.305078
$782$	0	0
$783$	14828.4	0.676787
$784$	0	0
$785$	41426.1	1.88352
$786$	0	0
$787$	10389.8	0.470594	0.235297	−	0.971923i	$-0.424394\pi$
$787$	10389.8	0.470594	0.235297	+	0.971923i	$0.424394\pi$
$788$	0	0
$789$	35410.1	1.59776
$790$	0	0
$791$	−11090.6	−0.498531
$792$	0	0
$793$	−40691.1	−1.82217
$794$	0	0
$795$	−56480.1	−2.51968
$796$	0	0
$797$	7012.53	0.311664	0.155832	−	0.987784i	$-0.450194\pi$
$797$	7012.53	0.311664	0.155832	+	0.987784i	$0.450194\pi$
$798$	0	0
$799$	−12958.7	−0.573774
$800$	0	0
$801$	52046.2	2.29583
$802$	0	0
$803$	−2087.07	−0.0917199
$804$	0	0
$805$	20117.7	0.880815
$806$	0	0
$807$	−17668.8	−0.770722
$808$	0	0
$809$	−5591.94	−0.243019	−0.121509	−	0.992590i	$-0.538773\pi$
$809$	−5591.94	−0.243019	−0.121509	+	0.992590i	$0.538773\pi$
$810$	0	0
$811$	13436.4	0.581771	0.290885	−	0.956758i	$-0.406050\pi$
$811$	13436.4	0.581771	0.290885	+	0.956758i	$0.406050\pi$
$812$	0	0
$813$	41858.5	1.80571
$814$	0	0
$815$	55167.4	2.37108
$816$	0	0
$817$	−9037.66	−0.387011
$818$	0	0
$819$	21066.6	0.898810
$820$	0	0
$821$	−20767.9	−0.882833	−0.441417	−	0.897302i	$-0.645524\pi$
$821$	−20767.9	−0.882833	−0.441417	+	0.897302i	$0.645524\pi$
$822$	0	0
$823$	−23838.8	−1.00968	−0.504842	−	0.863212i	$-0.668449\pi$
$823$	−23838.8	−1.00968	−0.504842	+	0.863212i	$0.668449\pi$
$824$	0	0
$825$	26806.6	1.13126
$826$	0	0
$827$	14325.4	0.602350	0.301175	−	0.953569i	$-0.402621\pi$
$827$	14325.4	0.602350	0.301175	+	0.953569i	$0.402621\pi$
$828$	0	0
$829$	40610.8	1.70141	0.850707	−	0.525639i	$-0.176174\pi$
$829$	40610.8	1.70141	0.850707	+	0.525639i	$0.176174\pi$
$830$	0	0
$831$	68183.6	2.84628
$832$	0	0
$833$	2638.10	0.109730
$834$	0	0
$835$	−44445.9	−1.84205
$836$	0	0
$837$	13313.6	0.549802
$838$	0	0
$839$	−12846.3	−0.528609	−0.264304	−	0.964439i	$-0.585142\pi$
$839$	−12846.3	−0.528609	−0.264304	+	0.964439i	$0.585142\pi$
$840$	0	0
$841$	−5004.67	−0.205202
$842$	0	0
$843$	−12841.1	−0.524637
$844$	0	0
$845$	−56937.9	−2.31801
$846$	0	0
$847$	−5780.37	−0.234493
$848$	0	0
$849$	72016.9	2.91121
$850$	0	0
$851$	−60031.7	−2.41817
$852$	0	0
$853$	−3824.15	−0.153501	−0.0767505	−	0.997050i	$-0.524455\pi$
$853$	−3824.15	−0.153501	−0.0767505	+	0.997050i	$0.524455\pi$
$854$	0	0
$855$	−63206.4	−2.52820
$856$	0	0
$857$	20983.5	0.836385	0.418192	−	0.908358i	$-0.362664\pi$
$857$	20983.5	0.836385	0.418192	+	0.908358i	$0.362664\pi$
$858$	0	0
$859$	−37690.6	−1.49707	−0.748536	−	0.663094i	$-0.769243\pi$
$859$	−37690.6	−1.49707	−0.748536	+	0.663094i	$0.769243\pi$
$860$	0	0
$861$	−25537.0	−1.01080
$862$	0	0
$863$	743.664	0.0293333	0.0146666	−	0.999892i	$-0.495331\pi$
$863$	743.664	0.0293333	0.0146666	+	0.999892i	$0.495331\pi$
$864$	0	0
$865$	−12558.0	−0.493626
$866$	0	0
$867$	16489.8	0.645930
$868$	0	0
$869$	−2754.38	−0.107521
$870$	0	0
$871$	14555.7	0.566246
$872$	0	0
$873$	−55061.6	−2.13465
$874$	0	0
$875$	−2382.62	−0.0920541
$876$	0	0
$877$	−4208.23	−0.162032	−0.0810158	−	0.996713i	$-0.525816\pi$
$877$	−4208.23	−0.162032	−0.0810158	+	0.996713i	$0.525816\pi$
$878$	0	0
$879$	−31510.7	−1.20913
$880$	0	0
$881$	−8042.31	−0.307551	−0.153775	−	0.988106i	$-0.549143\pi$
$881$	−8042.31	−0.307551	−0.153775	+	0.988106i	$0.549143\pi$
$882$	0	0
$883$	28873.1	1.10040	0.550202	−	0.835031i	$-0.314551\pi$
$883$	28873.1	1.10040	0.550202	+	0.835031i	$0.314551\pi$
$884$	0	0
$885$	−70107.4	−2.66286
$886$	0	0
$887$	−13477.7	−0.510188	−0.255094	−	0.966916i	$-0.582106\pi$
$887$	−13477.7	−0.510188	−0.255094	+	0.966916i	$0.582106\pi$
$888$	0	0
$889$	13077.2	0.493359
$890$	0	0
$891$	4685.14	0.176159
$892$	0	0
$893$	−23110.9	−0.866044
$894$	0	0
$895$	15383.9	0.574556
$896$	0	0
$897$	107557.	4.00359
$898$	0	0
$899$	17404.1	0.645671
$900$	0	0
$901$	−22577.9	−0.834829
$902$	0	0
$903$	5393.55	0.198766
$904$	0	0
$905$	50613.6	1.85906
$906$	0	0
$907$	−31250.7	−1.14406	−0.572030	−	0.820233i	$-0.693844\pi$
$907$	−31250.7	−1.14406	−0.572030	+	0.820233i	$0.693844\pi$
$908$	0	0
$909$	7657.76	0.279419
$910$	0	0
$911$	38010.9	1.38239	0.691194	−	0.722669i	$-0.257085\pi$
$911$	38010.9	1.38239	0.691194	+	0.722669i	$0.257085\pi$
$912$	0	0
$913$	−27554.5	−0.998816
$914$	0	0
$915$	−72859.3	−2.63241
$916$	0	0
$917$	−11211.9	−0.403764
$918$	0	0
$919$	50958.3	1.82912	0.914560	−	0.404451i	$-0.132537\pi$
$919$	50958.3	1.82912	0.914560	+	0.404451i	$0.132537\pi$
$920$	0	0
$921$	50086.9	1.79199
$922$	0	0
$923$	−22282.4	−0.794621
$924$	0	0
$925$	50067.9	1.77970
$926$	0	0
$927$	−52145.8	−1.84756
$928$	0	0
$929$	14021.0	0.495171	0.247585	−	0.968866i	$-0.420363\pi$
$929$	14021.0	0.495171	0.247585	+	0.968866i	$0.420363\pi$
$930$	0	0
$931$	4704.86	0.165624
$932$	0	0
$933$	76514.7	2.68487
$934$	0	0
$935$	19910.2	0.696400
$936$	0	0
$937$	−43816.2	−1.52766	−0.763828	−	0.645420i	$-0.776683\pi$
$937$	−43816.2	−1.52766	−0.763828	+	0.645420i	$0.776683\pi$
$938$	0	0
$939$	14354.4	0.498871
$940$	0	0
$941$	39384.2	1.36439	0.682194	−	0.731171i	$-0.261026\pi$
$941$	39384.2	1.36439	0.682194	+	0.731171i	$0.261026\pi$
$942$	0	0
$943$	−77847.7	−2.68830
$944$	0	0
$945$	12266.0	0.422235
$946$	0	0
$947$	−44794.8	−1.53710	−0.768551	−	0.639789i	$-0.779022\pi$
$947$	−44794.8	−1.53710	−0.768551	+	0.639789i	$0.779022\pi$
$948$	0	0
$949$	6984.12	0.238898
$950$	0	0
$951$	25386.8	0.865639
$952$	0	0
$953$	750.874	0.0255228	0.0127614	−	0.999919i	$-0.495938\pi$
$953$	750.874	0.0255228	0.0127614	+	0.999919i	$0.495938\pi$
$954$	0	0
$955$	20485.3	0.694126
$956$	0	0
$957$	−25617.9	−0.865317
$958$	0	0
$959$	−4094.49	−0.137871
$960$	0	0
$961$	−14164.9	−0.475476
$962$	0	0
$963$	−17064.8	−0.571035
$964$	0	0
$965$	6824.23	0.227647
$966$	0	0
$967$	−8540.91	−0.284030	−0.142015	−	0.989864i	$-0.545358\pi$
$967$	−8540.91	−0.284030	−0.142015	+	0.989864i	$0.545358\pi$
$968$	0	0
$969$	−42317.3	−1.40292
$970$	0	0
$971$	−9800.24	−0.323898	−0.161949	−	0.986799i	$-0.551778\pi$
$971$	−9800.24	−0.323898	−0.161949	+	0.986799i	$0.551778\pi$
$972$	0	0
$973$	17707.1	0.583414
$974$	0	0
$975$	−89705.0	−2.94652
$976$	0	0
$977$	−24644.8	−0.807019	−0.403509	−	0.914976i	$-0.632210\pi$
$977$	−24644.8	−0.807019	−0.403509	+	0.914976i	$0.632210\pi$
$978$	0	0
$979$	−29238.8	−0.954522
$980$	0	0
$981$	1498.71	0.0487768
$982$	0	0
$983$	−2466.73	−0.0800371	−0.0400186	−	0.999199i	$-0.512742\pi$
$983$	−2466.73	−0.0800371	−0.0400186	+	0.999199i	$0.512742\pi$
$984$	0	0
$985$	30980.3	1.00215
$986$	0	0
$987$	13792.3	0.444795
$988$	0	0
$989$	16441.9	0.528635
$990$	0	0
$991$	30650.0	0.982471	0.491236	−	0.871027i	$-0.336545\pi$
$991$	30650.0	0.982471	0.491236	+	0.871027i	$0.336545\pi$
$992$	0	0
$993$	−37832.8	−1.20905
$994$	0	0
$995$	−20935.9	−0.667046
$996$	0	0
$997$	−7889.02	−0.250599	−0.125300	−	0.992119i	$-0.539989\pi$
$997$	−7889.02	−0.250599	−0.125300	+	0.992119i	$0.539989\pi$
$998$	0	0
$999$	−36602.0	−1.15919

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.4.a.u.1.1		3
4.3	odd	2		448.4.a.x.1.3		3
8.3	odd	2		224.4.a.e.1.1	✓	3
8.5	even	2		224.4.a.h.1.3	yes	3
24.5	odd	2		2016.4.a.t.1.1		3
24.11	even	2		2016.4.a.s.1.1		3
56.13	odd	2		1568.4.a.v.1.1		3
56.27	even	2		1568.4.a.y.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.4.a.e.1.1	✓	3	8.3	odd	2
224.4.a.h.1.3	yes	3	8.5	even	2
448.4.a.u.1.1		3	1.1	even	1	trivial
448.4.a.x.1.3		3	4.3	odd	2
1568.4.a.v.1.1		3	56.13	odd	2
1568.4.a.y.1.3		3	56.27	even	2
2016.4.a.s.1.1		3	24.11	even	2
2016.4.a.t.1.1		3	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-8.18600 q^{3} -16.4526 q^{5} +7.00000 q^{7} +40.0106 q^{9} +O(q^{10})\)	\(q-8.18600 q^{3} -16.4526 q^{5} +7.00000 q^{7} +40.0106 q^{9} -22.4774 q^{11} +75.2178 q^{13} +134.681 q^{15} +53.8388 q^{17} +96.0176 q^{19} -57.3020 q^{21} -174.681 q^{23} +145.688 q^{25} -106.505 q^{27} -139.228 q^{29} -125.004 q^{31} +184.000 q^{33} -115.168 q^{35} +343.665 q^{37} -615.733 q^{39} +445.656 q^{41} -94.1250 q^{43} -658.279 q^{45} -240.694 q^{47} +49.0000 q^{49} -440.724 q^{51} -419.362 q^{53} +369.812 q^{55} -786.000 q^{57} -520.544 q^{59} -540.977 q^{61} +280.074 q^{63} -1237.53 q^{65} +193.514 q^{67} +1429.94 q^{69} -296.239 q^{71} +92.8520 q^{73} -1192.60 q^{75} -157.342 q^{77} +122.540 q^{79} -208.438 q^{81} +1225.87 q^{83} -885.788 q^{85} +1139.72 q^{87} +1300.81 q^{89} +526.525 q^{91} +1023.29 q^{93} -1579.74 q^{95} -1376.17 q^{97} -899.334 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 8 q^{3} - 10 q^{5} + 21 q^{7} + 31 q^{9}+O(q^{10}) \) 3 * q - 8 * q^3 - 10 * q^5 + 21 * q^7 + 31 * q^9	\( 3 q - 8 q^{3} - 10 q^{5} + 21 q^{7} + 31 q^{9} - 24 q^{11} - 58 q^{13} - 8 q^{15} + 174 q^{17} - 64 q^{19} - 56 q^{21} - 112 q^{23} + 373 q^{25} - 104 q^{27} - 314 q^{29} - 280 q^{31} + 552 q^{33} - 70 q^{35} - 178 q^{37} - 648 q^{39} + 318 q^{41} - 632 q^{43} - 714 q^{45} - 664 q^{47} + 147 q^{49} - 896 q^{51} - 434 q^{53} - 808 q^{55} - 256 q^{57} - 816 q^{59} - 450 q^{61} + 217 q^{63} - 1604 q^{65} - 408 q^{67} + 1304 q^{69} + 184 q^{71} + 6 q^{73} - 2096 q^{75} - 168 q^{77} - 552 q^{79} - 1381 q^{81} + 1736 q^{83} + 988 q^{85} - 96 q^{87} - 218 q^{89} - 406 q^{91} + 2768 q^{93} - 3832 q^{95} - 1090 q^{97} - 824 q^{99}+O(q^{100}) \) 3 * q - 8 * q^3 - 10 * q^5 + 21 * q^7 + 31 * q^9 - 24 * q^11 - 58 * q^13 - 8 * q^15 + 174 * q^17 - 64 * q^19 - 56 * q^21 - 112 * q^23 + 373 * q^25 - 104 * q^27 - 314 * q^29 - 280 * q^31 + 552 * q^33 - 70 * q^35 - 178 * q^37 - 648 * q^39 + 318 * q^41 - 632 * q^43 - 714 * q^45 - 664 * q^47 + 147 * q^49 - 896 * q^51 - 434 * q^53 - 808 * q^55 - 256 * q^57 - 816 * q^59 - 450 * q^61 + 217 * q^63 - 1604 * q^65 - 408 * q^67 + 1304 * q^69 + 184 * q^71 + 6 * q^73 - 2096 * q^75 - 168 * q^77 - 552 * q^79 - 1381 * q^81 + 1736 * q^83 + 988 * q^85 - 96 * q^87 - 218 * q^89 - 406 * q^91 + 2768 * q^93 - 3832 * q^95 - 1090 * q^97 - 824 * q^99

Introduction

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