LMFDB - Embedded newform 224.6.a.i.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [224,6,Mod(1,224)] 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("224.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(224, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 224 = 2^{5} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	224.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$35.9259756381$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 229x^{3} - 272x^{2} + 7973x - 13998 $ x^5 - 229x^3 - 272x^2 + 7973*x - 13998
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.24605$ of defining polynomial
Character	$\chi$	$=$	224.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.49210 q^{3} -99.5911 q^{5} +49.0000 q^{7} -236.789 q^{9} +292.685 q^{11} -163.156 q^{13} -248.190 q^{15} -1509.24 q^{17} -2117.22 q^{19} +122.113 q^{21} +4585.65 q^{23} +6793.38 q^{25} -1195.68 q^{27} +800.137 q^{29} +1276.43 q^{31} +729.399 q^{33} -4879.96 q^{35} +9375.25 q^{37} -406.599 q^{39} +6583.50 q^{41} -4935.97 q^{43} +23582.1 q^{45} +25847.4 q^{47} +2401.00 q^{49} -3761.16 q^{51} -6214.40 q^{53} -29148.8 q^{55} -5276.32 q^{57} -2803.61 q^{59} -828.062 q^{61} -11602.7 q^{63} +16248.8 q^{65} +62870.3 q^{67} +11427.9 q^{69} -60730.6 q^{71} +26058.6 q^{73} +16929.7 q^{75} +14341.6 q^{77} +55359.0 q^{79} +54560.1 q^{81} -118671. q^{83} +150306. q^{85} +1994.02 q^{87} +98218.5 q^{89} -7994.62 q^{91} +3180.98 q^{93} +210856. q^{95} +106682. q^{97} -69304.7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 10 q^{3} + 36 q^{5} + 245 q^{7} + 637 q^{9} + 116 q^{11} + 40 q^{13} + 16 q^{15} - 402 q^{17} - 3582 q^{19} - 490 q^{21} + 472 q^{23} + 9615 q^{25} + 356 q^{27} + 4754 q^{29} - 10500 q^{31} + 15864 q^{33}+ \cdots - 260236 q^{99}+O(q^{100}) $ 5 * q - 10 * q^3 + 36 * q^5 + 245 * q^7 + 637 * q^9 + 116 * q^11 + 40 * q^13 + 16 * q^15 - 402 * q^17 - 3582 * q^19 - 490 * q^21 + 472 * q^23 + 9615 * q^25 + 356 * q^27 + 4754 * q^29 - 10500 * q^31 + 15864 * q^33 + 1764 * q^35 + 19642 * q^37 + 10872 * q^39 + 23398 * q^41 + 22044 * q^43 + 49476 * q^45 + 16004 * q^47 + 12005 * q^49 - 45676 * q^51 + 54246 * q^53 + 53456 * q^55 + 109556 * q^57 - 74366 * q^59 + 68316 * q^61 + 31213 * q^63 + 152568 * q^65 - 26560 * q^67 + 214720 * q^69 + 93072 * q^71 + 136098 * q^73 - 124510 * q^75 + 5684 * q^77 + 96080 * q^79 + 104801 * q^81 - 145894 * q^83 + 117352 * q^85 + 168876 * q^87 + 188554 * q^89 + 1960 * q^91 + 86296 * q^93 + 74736 * q^95 - 88146 * q^97 - 260236 * 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
$3$	2.49210	0.159868	0.0799340	−	0.996800i	$-0.474529\pi$
$3$	2.49210	0.159868	0.0799340	+	0.996800i	$0.474529\pi$
$4$	0	0
$5$	−99.5911	−1.78154	−0.890769	−	0.454455i	$-0.849834\pi$
$5$	−99.5911	−1.78154	−0.890769	+	0.454455i	$0.849834\pi$
$6$	0	0
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	0	0
$9$	−236.789	−0.974442
$10$	0	0
$11$	292.685	0.729321	0.364660	−	0.931141i	$-0.381185\pi$
$11$	292.685	0.729321	0.364660	+	0.931141i	$0.381185\pi$
$12$	0	0
$13$	−163.156	−0.267759	−0.133879	−	0.990998i	$-0.542743\pi$
$13$	−163.156	−0.267759	−0.133879	+	0.990998i	$0.542743\pi$
$14$	0	0
$15$	−248.190	−0.284811
$16$	0	0
$17$	−1509.24	−1.26659	−0.633293	−	0.773912i	$-0.718297\pi$
$17$	−1509.24	−1.26659	−0.633293	+	0.773912i	$0.718297\pi$
$18$	0	0
$19$	−2117.22	−1.34550	−0.672748	−	0.739871i	$-0.734886\pi$
$19$	−2117.22	−1.34550	−0.672748	+	0.739871i	$0.734886\pi$
$20$	0	0
$21$	122.113	0.0604244
$22$	0	0
$23$	4585.65	1.80751	0.903756	−	0.428048i	$-0.140799\pi$
$23$	4585.65	1.80751	0.903756	+	0.428048i	$0.140799\pi$
$24$	0	0
$25$	6793.38	2.17388
$26$	0	0
$27$	−1195.68	−0.315650
$28$	0	0
$29$	800.137	0.176673	0.0883364	−	0.996091i	$-0.471845\pi$
$29$	800.137	0.176673	0.0883364	+	0.996091i	$0.471845\pi$
$30$	0	0
$31$	1276.43	0.238557	0.119278	−	0.992861i	$-0.461942\pi$
$31$	1276.43	0.238557	0.119278	+	0.992861i	$0.461942\pi$
$32$	0	0
$33$	729.399	0.116595
$34$	0	0
$35$	−4879.96	−0.673358
$36$	0	0
$37$	9375.25	1.12584	0.562922	−	0.826510i	$-0.309677\pi$
$37$	9375.25	1.12584	0.562922	+	0.826510i	$0.309677\pi$
$38$	0	0
$39$	−406.599	−0.0428060
$40$	0	0
$41$	6583.50	0.611642	0.305821	−	0.952089i	$-0.401069\pi$
$41$	6583.50	0.611642	0.305821	+	0.952089i	$0.401069\pi$
$42$	0	0
$43$	−4935.97	−0.407100	−0.203550	−	0.979065i	$-0.565248\pi$
$43$	−4935.97	−0.407100	−0.203550	+	0.979065i	$0.565248\pi$
$44$	0	0
$45$	23582.1	1.73601
$46$	0	0
$47$	25847.4	1.70676	0.853379	−	0.521291i	$-0.174549\pi$
$47$	25847.4	1.70676	0.853379	+	0.521291i	$0.174549\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	−3761.16	−0.202487
$52$	0	0
$53$	−6214.40	−0.303885	−0.151943	−	0.988389i	$-0.548553\pi$
$53$	−6214.40	−0.303885	−0.151943	+	0.988389i	$0.548553\pi$
$54$	0	0
$55$	−29148.8	−1.29931
$56$	0	0
$57$	−5276.32	−0.215102
$58$	0	0
$59$	−2803.61	−0.104855	−0.0524273	−	0.998625i	$-0.516696\pi$
$59$	−2803.61	−0.104855	−0.0524273	+	0.998625i	$0.516696\pi$
$60$	0	0
$61$	−828.062	−0.0284930	−0.0142465	−	0.999899i	$-0.504535\pi$
$61$	−828.062	−0.0284930	−0.0142465	+	0.999899i	$0.504535\pi$
$62$	0	0
$63$	−11602.7	−0.368305
$64$	0	0
$65$	16248.8	0.477022
$66$	0	0
$67$	62870.3	1.71103	0.855517	−	0.517774i	$-0.173239\pi$
$67$	62870.3	1.71103	0.855517	+	0.517774i	$0.173239\pi$
$68$	0	0
$69$	11427.9	0.288963
$70$	0	0
$71$	−60730.6	−1.42976	−0.714878	−	0.699249i	$-0.753518\pi$
$71$	−60730.6	−1.42976	−0.714878	+	0.699249i	$0.753518\pi$
$72$	0	0
$73$	26058.6	0.572327	0.286163	−	0.958181i	$-0.407620\pi$
$73$	26058.6	0.572327	0.286163	+	0.958181i	$0.407620\pi$
$74$	0	0
$75$	16929.7	0.347534
$76$	0	0
$77$	14341.6	0.275657
$78$	0	0
$79$	55359.0	0.997976	0.498988	−	0.866609i	$-0.333705\pi$
$79$	55359.0	0.997976	0.498988	+	0.866609i	$0.333705\pi$
$80$	0	0
$81$	54560.1	0.923980
$82$	0	0
$83$	−118671.	−1.89081	−0.945405	−	0.325898i	$-0.894333\pi$
$83$	−118671.	−1.89081	−0.945405	+	0.325898i	$0.894333\pi$
$84$	0	0
$85$	150306.	2.25647
$86$	0	0
$87$	1994.02	0.0282443
$88$	0	0
$89$	98218.5	1.31437	0.657186	−	0.753728i	$-0.271747\pi$
$89$	98218.5	1.31437	0.657186	+	0.753728i	$0.271747\pi$
$90$	0	0
$91$	−7994.62	−0.101203
$92$	0	0
$93$	3180.98	0.0381376
$94$	0	0
$95$	210856.	2.39705
$96$	0	0
$97$	106682.	1.15123	0.575615	−	0.817721i	$-0.304763\pi$
$97$	106682.	1.15123	0.575615	+	0.817721i	$0.304763\pi$
$98$	0	0
$99$	−69304.7	−0.710681
$100$	0	0
$101$	−72914.8	−0.711234	−0.355617	−	0.934632i	$-0.615729\pi$
$101$	−72914.8	−0.711234	−0.355617	+	0.934632i	$0.615729\pi$
$102$	0	0
$103$	116173.	1.07897	0.539487	−	0.841994i	$-0.318618\pi$
$103$	116173.	1.07897	0.539487	+	0.841994i	$0.318618\pi$
$104$	0	0
$105$	−12161.3	−0.107648
$106$	0	0
$107$	27014.4	0.228106	0.114053	−	0.993475i	$-0.463617\pi$
$107$	27014.4	0.228106	0.114053	+	0.993475i	$0.463617\pi$
$108$	0	0
$109$	65745.1	0.530026	0.265013	−	0.964245i	$-0.414624\pi$
$109$	65745.1	0.530026	0.265013	+	0.964245i	$0.414624\pi$
$110$	0	0
$111$	23364.0	0.179987
$112$	0	0
$113$	−237242.	−1.74782	−0.873908	−	0.486091i	$-0.838422\pi$
$113$	−237242.	−1.74782	−0.873908	+	0.486091i	$0.838422\pi$
$114$	0	0
$115$	−456690.	−3.22015
$116$	0	0
$117$	38633.5	0.260915
$118$	0	0
$119$	−73952.5	−0.478725
$120$	0	0
$121$	−75386.5	−0.468091
$122$	0	0
$123$	16406.7	0.0977820
$124$	0	0
$125$	−365338.	−2.09131
$126$	0	0
$127$	277543.	1.52694	0.763469	−	0.645845i	$-0.223495\pi$
$127$	277543.	1.52694	0.763469	+	0.645845i	$0.223495\pi$
$128$	0	0
$129$	−12300.9	−0.0650823
$130$	0	0
$131$	46802.2	0.238280	0.119140	−	0.992877i	$-0.461986\pi$
$131$	46802.2	0.238280	0.119140	+	0.992877i	$0.461986\pi$
$132$	0	0
$133$	−103744.	−0.508550
$134$	0	0
$135$	119079.	0.562343
$136$	0	0
$137$	−303785.	−1.38282	−0.691410	−	0.722463i	$-0.743010\pi$
$137$	−303785.	−1.38282	−0.691410	+	0.722463i	$0.743010\pi$
$138$	0	0
$139$	−118250.	−0.519116	−0.259558	−	0.965727i	$-0.583577\pi$
$139$	−118250.	−0.519116	−0.259558	+	0.965727i	$0.583577\pi$
$140$	0	0
$141$	64414.1	0.272856
$142$	0	0
$143$	−47753.2	−0.195282
$144$	0	0
$145$	−79686.5	−0.314749
$146$	0	0
$147$	5983.52	0.0228383
$148$	0	0
$149$	−192791.	−0.711413	−0.355707	−	0.934598i	$-0.615760\pi$
$149$	−192791.	−0.711413	−0.355707	+	0.934598i	$0.615760\pi$
$150$	0	0
$151$	197879.	0.706248	0.353124	−	0.935576i	$-0.385119\pi$
$151$	197879.	0.706248	0.353124	+	0.935576i	$0.385119\pi$
$152$	0	0
$153$	357371.	1.23422
$154$	0	0
$155$	−127121.	−0.424999
$156$	0	0
$157$	431399.	1.39679	0.698394	−	0.715714i	$-0.253899\pi$
$157$	431399.	1.39679	0.698394	+	0.715714i	$0.253899\pi$
$158$	0	0
$159$	−15486.9	−0.0485815
$160$	0	0
$161$	224697.	0.683175
$162$	0	0
$163$	−220103.	−0.648868	−0.324434	−	0.945908i	$-0.605174\pi$
$163$	−220103.	−0.648868	−0.324434	+	0.945908i	$0.605174\pi$
$164$	0	0
$165$	−72641.6	−0.207719
$166$	0	0
$167$	−350625.	−0.972862	−0.486431	−	0.873719i	$-0.661702\pi$
$167$	−350625.	−0.972862	−0.486431	+	0.873719i	$0.661702\pi$
$168$	0	0
$169$	−344673.	−0.928305
$170$	0	0
$171$	501336.	1.31111
$172$	0	0
$173$	−618823.	−1.57200	−0.785998	−	0.618229i	$-0.787850\pi$
$173$	−618823.	−1.57200	−0.785998	+	0.618229i	$0.787850\pi$
$174$	0	0
$175$	332876.	0.821650
$176$	0	0
$177$	−6986.85	−0.0167629
$178$	0	0
$179$	−233031.	−0.543602	−0.271801	−	0.962353i	$-0.587619\pi$
$179$	−233031.	−0.543602	−0.271801	+	0.962353i	$0.587619\pi$
$180$	0	0
$181$	−336718.	−0.763960	−0.381980	−	0.924171i	$-0.624758\pi$
$181$	−336718.	−0.763960	−0.381980	+	0.924171i	$0.624758\pi$
$182$	0	0
$183$	−2063.61	−0.00455512
$184$	0	0
$185$	−933691.	−2.00574
$186$	0	0
$187$	−441731.	−0.923748
$188$	0	0
$189$	−58588.4	−0.119305
$190$	0	0
$191$	186039.	0.368996	0.184498	−	0.982833i	$-0.440934\pi$
$191$	186039.	0.368996	0.184498	+	0.982833i	$0.440934\pi$
$192$	0	0
$193$	210636.	0.407043	0.203521	−	0.979071i	$-0.434761\pi$
$193$	210636.	0.407043	0.203521	+	0.979071i	$0.434761\pi$
$194$	0	0
$195$	40493.6	0.0762606
$196$	0	0
$197$	579772.	1.06437	0.532184	−	0.846629i	$-0.321371\pi$
$197$	579772.	1.06437	0.532184	+	0.846629i	$0.321371\pi$
$198$	0	0
$199$	162586.	0.291038	0.145519	−	0.989355i	$-0.453515\pi$
$199$	162586.	0.291038	0.145519	+	0.989355i	$0.453515\pi$
$200$	0	0
$201$	156679.	0.273540
$202$	0	0
$203$	39206.7	0.0667760
$204$	0	0
$205$	−655658.	−1.08966
$206$	0	0
$207$	−1.08583e6	−1.76132
$208$	0	0
$209$	−619679.	−0.981299
$210$	0	0
$211$	663978.	1.02671	0.513355	−	0.858176i	$-0.328402\pi$
$211$	663978.	1.02671	0.513355	+	0.858176i	$0.328402\pi$
$212$	0	0
$213$	−151346.	−0.228572
$214$	0	0
$215$	491579.	0.725265
$216$	0	0
$217$	62545.0	0.0901661
$218$	0	0
$219$	64940.5	0.0914967
$220$	0	0
$221$	246240.	0.339139
$222$	0	0
$223$	635216.	0.855380	0.427690	−	0.903926i	$-0.359328\pi$
$223$	635216.	0.855380	0.427690	+	0.903926i	$0.359328\pi$
$224$	0	0
$225$	−1.60860e6	−2.11832
$226$	0	0
$227$	833129.	1.07312	0.536559	−	0.843863i	$-0.319724\pi$
$227$	833129.	1.07312	0.536559	+	0.843863i	$0.319724\pi$
$228$	0	0
$229$	−259565.	−0.327082	−0.163541	−	0.986537i	$-0.552292\pi$
$229$	−259565.	−0.327082	−0.163541	+	0.986537i	$0.552292\pi$
$230$	0	0
$231$	35740.5	0.0440688
$232$	0	0
$233$	1.11572e6	1.34638	0.673189	−	0.739471i	$-0.264924\pi$
$233$	1.11572e6	1.34638	0.673189	+	0.739471i	$0.264924\pi$
$234$	0	0
$235$	−2.57417e6	−3.04066
$236$	0	0
$237$	137960.	0.159544
$238$	0	0
$239$	662116.	0.749789	0.374895	−	0.927067i	$-0.377679\pi$
$239$	662116.	0.749789	0.374895	+	0.927067i	$0.377679\pi$
$240$	0	0
$241$	918955.	1.01918	0.509591	−	0.860417i	$-0.329797\pi$
$241$	918955.	1.01918	0.509591	+	0.860417i	$0.329797\pi$
$242$	0	0
$243$	426519.	0.463365
$244$	0	0
$245$	−239118.	−0.254506
$246$	0	0
$247$	345437.	0.360268
$248$	0	0
$249$	−295738.	−0.302280
$250$	0	0
$251$	−861484.	−0.863104	−0.431552	−	0.902088i	$-0.642034\pi$
$251$	−861484.	−0.863104	−0.431552	+	0.902088i	$0.642034\pi$
$252$	0	0
$253$	1.34215e6	1.31826
$254$	0	0
$255$	374578.	0.360738
$256$	0	0
$257$	2.01631e6	1.90425	0.952124	−	0.305713i	$-0.0988947\pi$
$257$	2.01631e6	1.90425	0.952124	+	0.305713i	$0.0988947\pi$
$258$	0	0
$259$	459387.	0.425529
$260$	0	0
$261$	−189464.	−0.172157
$262$	0	0
$263$	−1.34088e6	−1.19536	−0.597682	−	0.801733i	$-0.703911\pi$
$263$	−1.34088e6	−1.19536	−0.597682	+	0.801733i	$0.703911\pi$
$264$	0	0
$265$	618899.	0.541384
$266$	0	0
$267$	244770.	0.210126
$268$	0	0
$269$	1.24957e6	1.05288	0.526440	−	0.850212i	$-0.323526\pi$
$269$	1.24957e6	1.05288	0.526440	+	0.850212i	$0.323526\pi$
$270$	0	0
$271$	−1.18725e6	−0.982016	−0.491008	−	0.871155i	$-0.663371\pi$
$271$	−1.18725e6	−0.982016	−0.491008	+	0.871155i	$0.663371\pi$
$272$	0	0
$273$	−19923.4	−0.0161792
$274$	0	0
$275$	1.98832e6	1.58546
$276$	0	0
$277$	805766.	0.630971	0.315486	−	0.948930i	$-0.397833\pi$
$277$	805766.	0.630971	0.315486	+	0.948930i	$0.397833\pi$
$278$	0	0
$279$	−302245.	−0.232460
$280$	0	0
$281$	1.21309e6	0.916490	0.458245	−	0.888826i	$-0.348478\pi$
$281$	1.21309e6	0.916490	0.458245	+	0.888826i	$0.348478\pi$
$282$	0	0
$283$	−196007.	−0.145481	−0.0727403	−	0.997351i	$-0.523174\pi$
$283$	−196007.	−0.145481	−0.0727403	+	0.997351i	$0.523174\pi$
$284$	0	0
$285$	525474.	0.383212
$286$	0	0
$287$	322592.	0.231179
$288$	0	0
$289$	857935.	0.604241
$290$	0	0
$291$	265862.	0.184045
$292$	0	0
$293$	1.96904e6	1.33994	0.669969	−	0.742389i	$-0.266307\pi$
$293$	1.96904e6	1.33994	0.669969	+	0.742389i	$0.266307\pi$
$294$	0	0
$295$	279214.	0.186802
$296$	0	0
$297$	−349958.	−0.230210
$298$	0	0
$299$	−748174.	−0.483977
$300$	0	0
$301$	−241863.	−0.153869
$302$	0	0
$303$	−181711.	−0.113703
$304$	0	0
$305$	82467.5	0.0507614
$306$	0	0
$307$	−445961.	−0.270054	−0.135027	−	0.990842i	$-0.543112\pi$
$307$	−445961.	−0.270054	−0.135027	+	0.990842i	$0.543112\pi$
$308$	0	0
$309$	289513.	0.172493
$310$	0	0
$311$	−1.76254e6	−1.03333	−0.516664	−	0.856188i	$-0.672826\pi$
$311$	−1.76254e6	−1.03333	−0.516664	+	0.856188i	$0.672826\pi$
$312$	0	0
$313$	−1.43435e6	−0.827549	−0.413774	−	0.910379i	$-0.635790\pi$
$313$	−1.43435e6	−0.827549	−0.413774	+	0.910379i	$0.635790\pi$
$314$	0	0
$315$	1.15552e6	0.656149
$316$	0	0
$317$	−983837.	−0.549889	−0.274944	−	0.961460i	$-0.588659\pi$
$317$	−983837.	−0.549889	−0.274944	+	0.961460i	$0.588659\pi$
$318$	0	0
$319$	234188.	0.128851
$320$	0	0
$321$	67322.6	0.0364668
$322$	0	0
$323$	3.19539e6	1.70419
$324$	0	0
$325$	−1.10838e6	−0.582075
$326$	0	0
$327$	163843.	0.0847341
$328$	0	0
$329$	1.26652e6	0.645094
$330$	0	0
$331$	606501.	0.304272	0.152136	−	0.988360i	$-0.451385\pi$
$331$	606501.	0.304272	0.152136	+	0.988360i	$0.451385\pi$
$332$	0	0
$333$	−2.21996e6	−1.09707
$334$	0	0
$335$	−6.26132e6	−3.04827
$336$	0	0
$337$	1.50022e6	0.719580	0.359790	−	0.933033i	$-0.382848\pi$
$337$	1.50022e6	0.719580	0.359790	+	0.933033i	$0.382848\pi$
$338$	0	0
$339$	−591230.	−0.279420
$340$	0	0
$341$	373591.	0.173985
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	−1.13811e6	−0.514799
$346$	0	0
$347$	3.03086e6	1.35127	0.675635	−	0.737236i	$-0.263870\pi$
$347$	3.03086e6	1.35127	0.675635	+	0.737236i	$0.263870\pi$
$348$	0	0
$349$	294660.	0.129496	0.0647482	−	0.997902i	$-0.479376\pi$
$349$	294660.	0.129496	0.0647482	+	0.997902i	$0.479376\pi$
$350$	0	0
$351$	195082.	0.0845180
$352$	0	0
$353$	−834926.	−0.356625	−0.178312	−	0.983974i	$-0.557064\pi$
$353$	−834926.	−0.356625	−0.178312	+	0.983974i	$0.557064\pi$
$354$	0	0
$355$	6.04823e6	2.54717
$356$	0	0
$357$	−184297.	−0.0765327
$358$	0	0
$359$	−1.59728e6	−0.654099	−0.327050	−	0.945007i	$-0.606054\pi$
$359$	−1.59728e6	−0.654099	−0.327050	+	0.945007i	$0.606054\pi$
$360$	0	0
$361$	2.00653e6	0.810361
$362$	0	0
$363$	−187870.	−0.0748328
$364$	0	0
$365$	−2.59521e6	−1.01962
$366$	0	0
$367$	−1.37342e6	−0.532277	−0.266139	−	0.963935i	$-0.585748\pi$
$367$	−1.37342e6	−0.532277	−0.266139	+	0.963935i	$0.585748\pi$
$368$	0	0
$369$	−1.55890e6	−0.596010
$370$	0	0
$371$	−304506.	−0.114858
$372$	0	0
$373$	4.62695e6	1.72196	0.860980	−	0.508640i	$-0.169851\pi$
$373$	4.62695e6	1.72196	0.860980	+	0.508640i	$0.169851\pi$
$374$	0	0
$375$	−910456.	−0.334334
$376$	0	0
$377$	−130547.	−0.0473056
$378$	0	0
$379$	216405.	0.0773874	0.0386937	−	0.999251i	$-0.487680\pi$
$379$	216405.	0.0773874	0.0386937	+	0.999251i	$0.487680\pi$
$380$	0	0
$381$	691664.	0.244108
$382$	0	0
$383$	4.14346e6	1.44333	0.721665	−	0.692242i	$-0.243377\pi$
$383$	4.14346e6	1.44333	0.721665	+	0.692242i	$0.243377\pi$
$384$	0	0
$385$	−1.42829e6	−0.491094
$386$	0	0
$387$	1.16879e6	0.396696
$388$	0	0
$389$	3.41596e6	1.14456	0.572280	−	0.820059i	$-0.306059\pi$
$389$	3.41596e6	1.14456	0.572280	+	0.820059i	$0.306059\pi$
$390$	0	0
$391$	−6.92082e6	−2.28937
$392$	0	0
$393$	116636.	0.0380934
$394$	0	0
$395$	−5.51326e6	−1.77793
$396$	0	0
$397$	5.12761e6	1.63282	0.816411	−	0.577471i	$-0.195960\pi$
$397$	5.12761e6	1.63282	0.816411	+	0.577471i	$0.195960\pi$
$398$	0	0
$399$	−258540.	−0.0813008
$400$	0	0
$401$	65538.5	0.0203533	0.0101767	−	0.999948i	$-0.496761\pi$
$401$	65538.5	0.0203533	0.0101767	+	0.999948i	$0.496761\pi$
$402$	0	0
$403$	−208256.	−0.0638757
$404$	0	0
$405$	−5.43370e6	−1.64611
$406$	0	0
$407$	2.74399e6	0.821102
$408$	0	0
$409$	5.28095e6	1.56100	0.780501	−	0.625154i	$-0.214964\pi$
$409$	5.28095e6	1.56100	0.780501	+	0.625154i	$0.214964\pi$
$410$	0	0
$411$	−757062.	−0.221069
$412$	0	0
$413$	−137377.	−0.0396313
$414$	0	0
$415$	1.18185e7	3.36855
$416$	0	0
$417$	−294691.	−0.0829901
$418$	0	0
$419$	−6.18612e6	−1.72141	−0.860703	−	0.509107i	$-0.829976\pi$
$419$	−6.18612e6	−1.72141	−0.860703	+	0.509107i	$0.829976\pi$
$420$	0	0
$421$	−3.52682e6	−0.969792	−0.484896	−	0.874572i	$-0.661143\pi$
$421$	−3.52682e6	−0.969792	−0.484896	+	0.874572i	$0.661143\pi$
$422$	0	0
$423$	−6.12039e6	−1.66314
$424$	0	0
$425$	−1.02528e7	−2.75341
$426$	0	0
$427$	−40575.0	−0.0107693
$428$	0	0
$429$	−119005.	−0.0312193
$430$	0	0
$431$	4.38322e6	1.13658	0.568290	−	0.822828i	$-0.307605\pi$
$431$	4.38322e6	1.13658	0.568290	+	0.822828i	$0.307605\pi$
$432$	0	0
$433$	−3.55834e6	−0.912070	−0.456035	−	0.889962i	$-0.650731\pi$
$433$	−3.55834e6	−0.912070	−0.456035	+	0.889962i	$0.650731\pi$
$434$	0	0
$435$	−198586.	−0.0503183
$436$	0	0
$437$	−9.70884e6	−2.43200
$438$	0	0
$439$	512004.	0.126798	0.0633990	−	0.997988i	$-0.479806\pi$
$439$	512004.	0.126798	0.0633990	+	0.997988i	$0.479806\pi$
$440$	0	0
$441$	−568531.	−0.139206
$442$	0	0
$443$	4.17109e6	1.00981	0.504906	−	0.863174i	$-0.331527\pi$
$443$	4.17109e6	1.00981	0.504906	+	0.863174i	$0.331527\pi$
$444$	0	0
$445$	−9.78169e6	−2.34161
$446$	0	0
$447$	−480454.	−0.113732
$448$	0	0
$449$	−2.70167e6	−0.632435	−0.316217	−	0.948687i	$-0.602413\pi$
$449$	−2.70167e6	−0.632435	−0.316217	+	0.948687i	$0.602413\pi$
$450$	0	0
$451$	1.92689e6	0.446083
$452$	0	0
$453$	493133.	0.112907
$454$	0	0
$455$	796193.	0.180298
$456$	0	0
$457$	−2.36210e6	−0.529063	−0.264531	−	0.964377i	$-0.585217\pi$
$457$	−2.36210e6	−0.529063	−0.264531	+	0.964377i	$0.585217\pi$
$458$	0	0
$459$	1.80456e6	0.399798
$460$	0	0
$461$	5.98494e6	1.31162	0.655809	−	0.754927i	$-0.272328\pi$
$461$	5.98494e6	1.31162	0.655809	+	0.754927i	$0.272328\pi$
$462$	0	0
$463$	−5.00684e6	−1.08545	−0.542727	−	0.839909i	$-0.682608\pi$
$463$	−5.00684e6	−1.08545	−0.542727	+	0.839909i	$0.682608\pi$
$464$	0	0
$465$	−316797.	−0.0679437
$466$	0	0
$467$	−1.65286e6	−0.350706	−0.175353	−	0.984506i	$-0.556107\pi$
$467$	−1.65286e6	−0.350706	−0.175353	+	0.984506i	$0.556107\pi$
$468$	0	0
$469$	3.08065e6	0.646710
$470$	0	0
$471$	1.07509e6	0.223302
$472$	0	0
$473$	−1.44468e6	−0.296907
$474$	0	0
$475$	−1.43831e7	−2.92495
$476$	0	0
$477$	1.47151e6	0.296119
$478$	0	0
$479$	5.95428e6	1.18574	0.592871	−	0.805297i	$-0.297994\pi$
$479$	5.95428e6	1.18574	0.592871	+	0.805297i	$0.297994\pi$
$480$	0	0
$481$	−1.52962e6	−0.301455
$482$	0	0
$483$	559966.	0.109218
$484$	0	0
$485$	−1.06246e7	−2.05096
$486$	0	0
$487$	−6.14901e6	−1.17485	−0.587426	−	0.809278i	$-0.699858\pi$
$487$	−6.14901e6	−1.17485	−0.587426	+	0.809278i	$0.699858\pi$
$488$	0	0
$489$	−548517.	−0.103733
$490$	0	0
$491$	6.26140e6	1.17211	0.586054	−	0.810272i	$-0.300681\pi$
$491$	6.26140e6	1.17211	0.586054	+	0.810272i	$0.300681\pi$
$492$	0	0
$493$	−1.20760e6	−0.223771
$494$	0	0
$495$	6.90213e6	1.26611
$496$	0	0
$497$	−2.97580e6	−0.540397
$498$	0	0
$499$	7.10012e6	1.27648	0.638240	−	0.769837i	$-0.279663\pi$
$499$	7.10012e6	1.27648	0.638240	+	0.769837i	$0.279663\pi$
$500$	0	0
$501$	−873790.	−0.155530
$502$	0	0
$503$	−5.65833e6	−0.997168	−0.498584	−	0.866841i	$-0.666146\pi$
$503$	−5.65833e6	−0.997168	−0.498584	+	0.866841i	$0.666146\pi$
$504$	0	0
$505$	7.26166e6	1.26709
$506$	0	0
$507$	−858959.	−0.148406
$508$	0	0
$509$	607346.	0.103906	0.0519532	−	0.998650i	$-0.483455\pi$
$509$	607346.	0.103906	0.0519532	+	0.998650i	$0.483455\pi$
$510$	0	0
$511$	1.27687e6	0.216319
$512$	0	0
$513$	2.53152e6	0.424706
$514$	0	0
$515$	−1.15698e7	−1.92223
$516$	0	0
$517$	7.56514e6	1.24477
$518$	0	0
$519$	−1.54217e6	−0.251312
$520$	0	0
$521$	502953.	0.0811770	0.0405885	−	0.999176i	$-0.487077\pi$
$521$	502953.	0.0811770	0.0405885	+	0.999176i	$0.487077\pi$
$522$	0	0
$523$	9.59744e6	1.53427	0.767134	−	0.641487i	$-0.221682\pi$
$523$	9.59744e6	1.53427	0.767134	+	0.641487i	$0.221682\pi$
$524$	0	0
$525$	829557.	0.131355
$526$	0	0
$527$	−1.92643e6	−0.302153
$528$	0	0
$529$	1.45918e7	2.26710
$530$	0	0
$531$	663865.	0.102175
$532$	0	0
$533$	−1.07414e6	−0.163772
$534$	0	0
$535$	−2.69040e6	−0.406380
$536$	0	0
$537$	−580735.	−0.0869046
$538$	0	0
$539$	702737.	0.104189
$540$	0	0
$541$	9.46010e6	1.38964	0.694821	−	0.719183i	$-0.255484\pi$
$541$	9.46010e6	1.38964	0.694821	+	0.719183i	$0.255484\pi$
$542$	0	0
$543$	−839134.	−0.122133
$544$	0	0
$545$	−6.54762e6	−0.944261
$546$	0	0
$547$	4.26708e6	0.609765	0.304882	−	0.952390i	$-0.401383\pi$
$547$	4.26708e6	0.609765	0.304882	+	0.952390i	$0.401383\pi$
$548$	0	0
$549$	196076.	0.0277648
$550$	0	0
$551$	−1.69407e6	−0.237713
$552$	0	0
$553$	2.71259e6	0.377200
$554$	0	0
$555$	−2.32685e6	−0.320653
$556$	0	0
$557$	−1.19790e7	−1.63600	−0.818001	−	0.575216i	$-0.804918\pi$
$557$	−1.19790e7	−1.63600	−0.818001	+	0.575216i	$0.804918\pi$
$558$	0	0
$559$	805331.	0.109005
$560$	0	0
$561$	−1.10083e6	−0.147678
$562$	0	0
$563$	5.63496e6	0.749238	0.374619	−	0.927179i	$-0.377774\pi$
$563$	5.63496e6	0.749238	0.374619	+	0.927179i	$0.377774\pi$
$564$	0	0
$565$	2.36272e7	3.11380
$566$	0	0
$567$	2.67344e6	0.349232
$568$	0	0
$569$	−4.23993e6	−0.549008	−0.274504	−	0.961586i	$-0.588514\pi$
$569$	−4.23993e6	−0.549008	−0.274504	+	0.961586i	$0.588514\pi$
$570$	0	0
$571$	−1.26701e7	−1.62626	−0.813129	−	0.582084i	$-0.802238\pi$
$571$	−1.26701e7	−1.62626	−0.813129	+	0.582084i	$0.802238\pi$
$572$	0	0
$573$	463628.	0.0589906
$574$	0	0
$575$	3.11520e7	3.92932
$576$	0	0
$577$	6.16763e6	0.771221	0.385610	−	0.922662i	$-0.373991\pi$
$577$	6.16763e6	0.771221	0.385610	+	0.922662i	$0.373991\pi$
$578$	0	0
$579$	524926.	0.0650731
$580$	0	0
$581$	−5.81486e6	−0.714659
$582$	0	0
$583$	−1.81886e6	−0.221630
$584$	0	0
$585$	−3.84755e6	−0.464831
$586$	0	0
$587$	−9.30640e6	−1.11477	−0.557387	−	0.830253i	$-0.688196\pi$
$587$	−9.30640e6	−1.11477	−0.557387	+	0.830253i	$0.688196\pi$
$588$	0	0
$589$	−2.70248e6	−0.320978
$590$	0	0
$591$	1.44485e6	0.170158
$592$	0	0
$593$	−9.23152e6	−1.07804	−0.539022	−	0.842292i	$-0.681206\pi$
$593$	−9.23152e6	−1.07804	−0.539022	+	0.842292i	$0.681206\pi$
$594$	0	0
$595$	7.36501e6	0.852867
$596$	0	0
$597$	405180.	0.0465277
$598$	0	0
$599$	−8.33587e6	−0.949257	−0.474629	−	0.880186i	$-0.657418\pi$
$599$	−8.33587e6	−0.949257	−0.474629	+	0.880186i	$0.657418\pi$
$600$	0	0
$601$	6.13490e6	0.692821	0.346411	−	0.938083i	$-0.387400\pi$
$601$	6.13490e6	0.692821	0.346411	+	0.938083i	$0.387400\pi$
$602$	0	0
$603$	−1.48870e7	−1.66730
$604$	0	0
$605$	7.50782e6	0.833922
$606$	0	0
$607$	−2.03253e6	−0.223905	−0.111953	−	0.993714i	$-0.535710\pi$
$607$	−2.03253e6	−0.223905	−0.111953	+	0.993714i	$0.535710\pi$
$608$	0	0
$609$	97706.9	0.0106753
$610$	0	0
$611$	−4.21714e6	−0.456999
$612$	0	0
$613$	1.12990e7	1.21448	0.607238	−	0.794520i	$-0.292277\pi$
$613$	1.12990e7	1.21448	0.607238	+	0.794520i	$0.292277\pi$
$614$	0	0
$615$	−1.63396e6	−0.174202
$616$	0	0
$617$	4.42817e6	0.468286	0.234143	−	0.972202i	$-0.424772\pi$
$617$	4.42817e6	0.468286	0.234143	+	0.972202i	$0.424772\pi$
$618$	0	0
$619$	−8.33489e6	−0.874325	−0.437163	−	0.899382i	$-0.644017\pi$
$619$	−8.33489e6	−0.874325	−0.437163	+	0.899382i	$0.644017\pi$
$620$	0	0
$621$	−5.48297e6	−0.570541
$622$	0	0
$623$	4.81271e6	0.496786
$624$	0	0
$625$	1.51551e7	1.55188
$626$	0	0
$627$	−1.54430e6	−0.156878
$628$	0	0
$629$	−1.41495e7	−1.42598
$630$	0	0
$631$	6.07865e6	0.607762	0.303881	−	0.952710i	$-0.401718\pi$
$631$	6.07865e6	0.607762	0.303881	+	0.952710i	$0.401718\pi$
$632$	0	0
$633$	1.65470e6	0.164138
$634$	0	0
$635$	−2.76408e7	−2.72030
$636$	0	0
$637$	−391737.	−0.0382512
$638$	0	0
$639$	1.43804e7	1.39321
$640$	0	0
$641$	1.56516e7	1.50457	0.752287	−	0.658836i	$-0.228951\pi$
$641$	1.56516e7	1.50457	0.752287	+	0.658836i	$0.228951\pi$
$642$	0	0
$643$	1.01616e7	0.969245	0.484623	−	0.874723i	$-0.338957\pi$
$643$	1.01616e7	0.969245	0.484623	+	0.874723i	$0.338957\pi$
$644$	0	0
$645$	1.22506e6	0.115947
$646$	0	0
$647$	1.29888e7	1.21986	0.609930	−	0.792455i	$-0.291198\pi$
$647$	1.29888e7	1.21986	0.609930	+	0.792455i	$0.291198\pi$
$648$	0	0
$649$	−820573.	−0.0764726
$650$	0	0
$651$	155868.	0.0144147
$652$	0	0
$653$	−6.43248e6	−0.590331	−0.295166	−	0.955446i	$-0.595375\pi$
$653$	−6.43248e6	−0.590331	−0.295166	+	0.955446i	$0.595375\pi$
$654$	0	0
$655$	−4.66108e6	−0.424506
$656$	0	0
$657$	−6.17041e6	−0.557700
$658$	0	0
$659$	1.44738e7	1.29828	0.649140	−	0.760669i	$-0.275129\pi$
$659$	1.44738e7	1.29828	0.649140	+	0.760669i	$0.275129\pi$
$660$	0	0
$661$	−5.72612e6	−0.509750	−0.254875	−	0.966974i	$-0.582034\pi$
$661$	−5.72612e6	−0.509750	−0.254875	+	0.966974i	$0.582034\pi$
$662$	0	0
$663$	613654.	0.0542175
$664$	0	0
$665$	1.03320e7	0.906002
$666$	0	0
$667$	3.66915e6	0.319338
$668$	0	0
$669$	1.58302e6	0.136748
$670$	0	0
$671$	−242361.	−0.0207805
$672$	0	0
$673$	−6.63951e6	−0.565065	−0.282533	−	0.959258i	$-0.591175\pi$
$673$	−6.63951e6	−0.565065	−0.282533	+	0.959258i	$0.591175\pi$
$674$	0	0
$675$	−8.12271e6	−0.686186
$676$	0	0
$677$	1.60011e7	1.34177	0.670885	−	0.741562i	$-0.265915\pi$
$677$	1.60011e7	1.34177	0.670885	+	0.741562i	$0.265915\pi$
$678$	0	0
$679$	5.22742e6	0.435124
$680$	0	0
$681$	2.07624e6	0.171557
$682$	0	0
$683$	−2.15548e7	−1.76804	−0.884020	−	0.467450i	$-0.845173\pi$
$683$	−2.15548e7	−1.76804	−0.884020	+	0.467450i	$0.845173\pi$
$684$	0	0
$685$	3.02543e7	2.46355
$686$	0	0
$687$	−646859.	−0.0522899
$688$	0	0
$689$	1.01391e6	0.0813679
$690$	0	0
$691$	5.75267e6	0.458325	0.229163	−	0.973388i	$-0.426401\pi$
$691$	5.75267e6	0.458325	0.229163	+	0.973388i	$0.426401\pi$
$692$	0	0
$693$	−3.39593e6	−0.268612
$694$	0	0
$695$	1.17767e7	0.924826
$696$	0	0
$697$	−9.93606e6	−0.774697
$698$	0	0
$699$	2.78049e6	0.215243
$700$	0	0
$701$	2.26190e7	1.73851	0.869257	−	0.494360i	$-0.164598\pi$
$701$	2.26190e7	1.73851	0.869257	+	0.494360i	$0.164598\pi$
$702$	0	0
$703$	−1.98495e7	−1.51482
$704$	0	0
$705$	−6.41507e6	−0.486103
$706$	0	0
$707$	−3.57283e6	−0.268821
$708$	0	0
$709$	−1.02453e7	−0.765434	−0.382717	−	0.923866i	$-0.625012\pi$
$709$	−1.02453e7	−0.765434	−0.382717	+	0.923866i	$0.625012\pi$
$710$	0	0
$711$	−1.31084e7	−0.972470
$712$	0	0
$713$	5.85325e6	0.431195
$714$	0	0
$715$	4.75579e6	0.347902
$716$	0	0
$717$	1.65006e6	0.119867
$718$	0	0
$719$	−953623.	−0.0687946	−0.0343973	−	0.999408i	$-0.510951\pi$
$719$	−953623.	−0.0687946	−0.0343973	+	0.999408i	$0.510951\pi$
$720$	0	0
$721$	5.69246e6	0.407814
$722$	0	0
$723$	2.29012e6	0.162935
$724$	0	0
$725$	5.43563e6	0.384065
$726$	0	0
$727$	2.15807e6	0.151436	0.0757182	−	0.997129i	$-0.475875\pi$
$727$	2.15807e6	0.151436	0.0757182	+	0.997129i	$0.475875\pi$
$728$	0	0
$729$	−1.21952e7	−0.849903
$730$	0	0
$731$	7.44954e6	0.515628
$732$	0	0
$733$	−1.47257e7	−1.01232	−0.506158	−	0.862441i	$-0.668935\pi$
$733$	−1.47257e7	−1.01232	−0.506158	+	0.862441i	$0.668935\pi$
$734$	0	0
$735$	−595905.	−0.0406873
$736$	0	0
$737$	1.84012e7	1.24789
$738$	0	0
$739$	2.65235e7	1.78657	0.893286	−	0.449489i	$-0.148394\pi$
$739$	2.65235e7	1.78657	0.893286	+	0.449489i	$0.148394\pi$
$740$	0	0
$741$	860861.	0.0575954
$742$	0	0
$743$	3.35953e6	0.223258	0.111629	−	0.993750i	$-0.464393\pi$
$743$	3.35953e6	0.223258	0.111629	+	0.993750i	$0.464393\pi$
$744$	0	0
$745$	1.92003e7	1.26741
$746$	0	0
$747$	2.80999e7	1.84248
$748$	0	0
$749$	1.32371e6	0.0862160
$750$	0	0
$751$	−2.35689e7	−1.52489	−0.762446	−	0.647051i	$-0.776002\pi$
$751$	−2.35689e7	−1.52489	−0.762446	+	0.647051i	$0.776002\pi$
$752$	0	0
$753$	−2.14690e6	−0.137983
$754$	0	0
$755$	−1.97070e7	−1.25821
$756$	0	0
$757$	1.80124e7	1.14244	0.571219	−	0.820798i	$-0.306471\pi$
$757$	1.80124e7	1.14244	0.571219	+	0.820798i	$0.306471\pi$
$758$	0	0
$759$	3.34477e6	0.210747
$760$	0	0
$761$	−1.04944e7	−0.656898	−0.328449	−	0.944522i	$-0.606526\pi$
$761$	−1.04944e7	−0.656898	−0.328449	+	0.944522i	$0.606526\pi$
$762$	0	0
$763$	3.22151e6	0.200331
$764$	0	0
$765$	−3.55910e7	−2.19880
$766$	0	0
$767$	457424.	0.0280757
$768$	0	0
$769$	−2.03672e7	−1.24198	−0.620991	−	0.783818i	$-0.713270\pi$
$769$	−2.03672e7	−1.24198	−0.620991	+	0.783818i	$0.713270\pi$
$770$	0	0
$771$	5.02482e6	0.304428
$772$	0	0
$773$	−1.41705e7	−0.852978	−0.426489	−	0.904493i	$-0.640250\pi$
$773$	−1.41705e7	−0.852978	−0.426489	+	0.904493i	$0.640250\pi$
$774$	0	0
$775$	8.67126e6	0.518594
$776$	0	0
$777$	1.14484e6	0.0680285
$778$	0	0
$779$	−1.39387e7	−0.822962
$780$	0	0
$781$	−1.77749e7	−1.04275
$782$	0	0
$783$	−956709.	−0.0557668
$784$	0	0
$785$	−4.29635e7	−2.48843
$786$	0	0
$787$	−1.94138e6	−0.111731	−0.0558655	−	0.998438i	$-0.517792\pi$
$787$	−1.94138e6	−0.111731	−0.0558655	+	0.998438i	$0.517792\pi$
$788$	0	0
$789$	−3.34160e6	−0.191100
$790$	0	0
$791$	−1.16249e7	−0.660612
$792$	0	0
$793$	135103.	0.00762925
$794$	0	0
$795$	1.54236e6	0.0865499
$796$	0	0
$797$	−2.44578e7	−1.36387	−0.681933	−	0.731415i	$-0.738860\pi$
$797$	−2.44578e7	−1.36387	−0.681933	+	0.731415i	$0.738860\pi$
$798$	0	0
$799$	−3.90098e7	−2.16176
$800$	0	0
$801$	−2.32571e7	−1.28078
$802$	0	0
$803$	7.62696e6	0.417410
$804$	0	0
$805$	−2.23778e7	−1.21710
$806$	0	0
$807$	3.11404e6	0.168322
$808$	0	0
$809$	−2.18414e7	−1.17330	−0.586651	−	0.809840i	$-0.699554\pi$
$809$	−2.18414e7	−1.17330	−0.586651	+	0.809840i	$0.699554\pi$
$810$	0	0
$811$	−8.22026e6	−0.438867	−0.219434	−	0.975627i	$-0.570421\pi$
$811$	−8.22026e6	−0.438867	−0.219434	+	0.975627i	$0.570421\pi$
$812$	0	0
$813$	−2.95874e6	−0.156993
$814$	0	0
$815$	2.19203e7	1.15598
$816$	0	0
$817$	1.04506e7	0.547752
$818$	0	0
$819$	1.89304e6	0.0986167
$820$	0	0
$821$	5.77947e6	0.299247	0.149624	−	0.988743i	$-0.452194\pi$
$821$	5.77947e6	0.299247	0.149624	+	0.988743i	$0.452194\pi$
$822$	0	0
$823$	2.01035e7	1.03460	0.517300	−	0.855804i	$-0.326937\pi$
$823$	2.01035e7	1.03460	0.517300	+	0.855804i	$0.326937\pi$
$824$	0	0
$825$	4.95508e6	0.253464
$826$	0	0
$827$	1.33274e7	0.677615	0.338808	−	0.940856i	$-0.389976\pi$
$827$	1.33274e7	0.677615	0.338808	+	0.940856i	$0.389976\pi$
$828$	0	0
$829$	7.21926e6	0.364843	0.182422	−	0.983220i	$-0.441606\pi$
$829$	7.21926e6	0.364843	0.182422	+	0.983220i	$0.441606\pi$
$830$	0	0
$831$	2.00805e6	0.100872
$832$	0	0
$833$	−3.62367e6	−0.180941
$834$	0	0
$835$	3.49191e7	1.73319
$836$	0	0
$837$	−1.52620e6	−0.0753005
$838$	0	0
$839$	−3.63257e7	−1.78159	−0.890797	−	0.454401i	$-0.849853\pi$
$839$	−3.63257e7	−1.78159	−0.890797	+	0.454401i	$0.849853\pi$
$840$	0	0
$841$	−1.98709e7	−0.968787
$842$	0	0
$843$	3.02314e6	0.146517
$844$	0	0
$845$	3.43264e7	1.65381
$846$	0	0
$847$	−3.69394e6	−0.176922
$848$	0	0
$849$	−488468.	−0.0232577
$850$	0	0
$851$	4.29916e7	2.03498
$852$	0	0
$853$	−2.39547e7	−1.12725	−0.563623	−	0.826032i	$-0.690593\pi$
$853$	−2.39547e7	−1.12725	−0.563623	+	0.826032i	$0.690593\pi$
$854$	0	0
$855$	−4.99286e7	−2.33579
$856$	0	0
$857$	−3.22646e6	−0.150063	−0.0750316	−	0.997181i	$-0.523906\pi$
$857$	−3.22646e6	−0.150063	−0.0750316	+	0.997181i	$0.523906\pi$
$858$	0	0
$859$	−8.71503e6	−0.402983	−0.201491	−	0.979490i	$-0.564579\pi$
$859$	−8.71503e6	−0.402983	−0.201491	+	0.979490i	$0.564579\pi$
$860$	0	0
$861$	803929.	0.0369581
$862$	0	0
$863$	7.23457e6	0.330663	0.165332	−	0.986238i	$-0.447131\pi$
$863$	7.23457e6	0.330663	0.165332	+	0.986238i	$0.447131\pi$
$864$	0	0
$865$	6.16293e7	2.80057
$866$	0	0
$867$	2.13806e6	0.0965987
$868$	0	0
$869$	1.62027e7	0.727845
$870$	0	0
$871$	−1.02576e7	−0.458144
$872$	0	0
$873$	−2.52612e7	−1.12181
$874$	0	0
$875$	−1.79015e7	−0.790443
$876$	0	0
$877$	−2.33000e7	−1.02295	−0.511477	−	0.859297i	$-0.670902\pi$
$877$	−2.33000e7	−1.02295	−0.511477	+	0.859297i	$0.670902\pi$
$878$	0	0
$879$	4.90702e6	0.214213
$880$	0	0
$881$	1.69404e7	0.735333	0.367666	−	0.929958i	$-0.380157\pi$
$881$	1.69404e7	0.735333	0.367666	+	0.929958i	$0.380157\pi$
$882$	0	0
$883$	−3.40145e7	−1.46812	−0.734060	−	0.679084i	$-0.762377\pi$
$883$	−3.40145e7	−1.46812	−0.734060	+	0.679084i	$0.762377\pi$
$884$	0	0
$885$	695828.	0.0298637
$886$	0	0
$887$	−2.67616e7	−1.14210	−0.571049	−	0.820916i	$-0.693463\pi$
$887$	−2.67616e7	−1.14210	−0.571049	+	0.820916i	$0.693463\pi$
$888$	0	0
$889$	1.35996e7	0.577128
$890$	0	0
$891$	1.59689e7	0.673878
$892$	0	0
$893$	−5.47247e7	−2.29644
$894$	0	0
$895$	2.32078e7	0.968448
$896$	0	0
$897$	−1.86452e6	−0.0773724
$898$	0	0
$899$	1.02132e6	0.0421465
$900$	0	0
$901$	9.37900e6	0.384897
$902$	0	0
$903$	−602745.	−0.0245988
$904$	0	0
$905$	3.35341e7	1.36102
$906$	0	0
$907$	1.76702e7	0.713221	0.356610	−	0.934253i	$-0.383932\pi$
$907$	1.76702e7	0.713221	0.356610	+	0.934253i	$0.383932\pi$
$908$	0	0
$909$	1.72655e7	0.693056
$910$	0	0
$911$	4.31389e7	1.72216	0.861079	−	0.508471i	$-0.169789\pi$
$911$	4.31389e7	1.72216	0.861079	+	0.508471i	$0.169789\pi$
$912$	0	0
$913$	−3.47331e7	−1.37901
$914$	0	0
$915$	205517.	0.00811512
$916$	0	0
$917$	2.29331e6	0.0900615
$918$	0	0
$919$	−1.25403e7	−0.489800	−0.244900	−	0.969548i	$-0.578755\pi$
$919$	−1.25403e7	−0.489800	−0.244900	+	0.969548i	$0.578755\pi$
$920$	0	0
$921$	−1.11138e6	−0.0431731
$922$	0	0
$923$	9.90854e6	0.382829
$924$	0	0
$925$	6.36896e7	2.44745
$926$	0	0
$927$	−2.75084e7	−1.05140
$928$	0	0
$929$	−7.84854e6	−0.298366	−0.149183	−	0.988810i	$-0.547664\pi$
$929$	−7.84854e6	−0.298366	−0.149183	+	0.988810i	$0.547664\pi$
$930$	0	0
$931$	−5.08345e6	−0.192214
$932$	0	0
$933$	−4.39242e6	−0.165196
$934$	0	0
$935$	4.39924e7	1.64569
$936$	0	0
$937$	1.03129e7	0.383736	0.191868	−	0.981421i	$-0.438545\pi$
$937$	1.03129e7	0.383736	0.191868	+	0.981421i	$0.438545\pi$
$938$	0	0
$939$	−3.57453e6	−0.132299
$940$	0	0
$941$	9.04037e6	0.332822	0.166411	−	0.986056i	$-0.446782\pi$
$941$	9.04037e6	0.332822	0.166411	+	0.986056i	$0.446782\pi$
$942$	0	0
$943$	3.01896e7	1.10555
$944$	0	0
$945$	5.83488e6	0.212546
$946$	0	0
$947$	−4.69853e7	−1.70250	−0.851251	−	0.524759i	$-0.824155\pi$
$947$	−4.69853e7	−1.70250	−0.851251	+	0.524759i	$0.824155\pi$
$948$	0	0
$949$	−4.25161e6	−0.153245
$950$	0	0
$951$	−2.45181e6	−0.0879096
$952$	0	0
$953$	−2.63875e7	−0.941165	−0.470582	−	0.882356i	$-0.655956\pi$
$953$	−2.63875e7	−0.941165	−0.470582	+	0.882356i	$0.655956\pi$
$954$	0	0
$955$	−1.85278e7	−0.657380
$956$	0	0
$957$	583619.	0.0205992
$958$	0	0
$959$	−1.48855e7	−0.522657
$960$	0	0
$961$	−2.69999e7	−0.943091
$962$	0	0
$963$	−6.39674e6	−0.222276
$964$	0	0
$965$	−2.09775e7	−0.725162
$966$	0	0
$967$	4.08697e6	0.140551	0.0702757	−	0.997528i	$-0.477612\pi$
$967$	4.08697e6	0.140551	0.0702757	+	0.997528i	$0.477612\pi$
$968$	0	0
$969$	7.96321e6	0.272445
$970$	0	0
$971$	1.58015e7	0.537837	0.268919	−	0.963163i	$-0.413334\pi$
$971$	1.58015e7	0.537837	0.268919	+	0.963163i	$0.413334\pi$
$972$	0	0
$973$	−5.79426e6	−0.196208
$974$	0	0
$975$	−2.76218e6	−0.0930552
$976$	0	0
$977$	2.78593e7	0.933757	0.466879	−	0.884321i	$-0.345378\pi$
$977$	2.78593e7	0.933757	0.466879	+	0.884321i	$0.345378\pi$
$978$	0	0
$979$	2.87471e7	0.958599
$980$	0	0
$981$	−1.55677e7	−0.516479
$982$	0	0
$983$	−2.89667e7	−0.956128	−0.478064	−	0.878325i	$-0.658661\pi$
$983$	−2.89667e7	−0.956128	−0.478064	+	0.878325i	$0.658661\pi$
$984$	0	0
$985$	−5.77401e7	−1.89621
$986$	0	0
$987$	3.15629e6	0.103130
$988$	0	0
$989$	−2.26346e7	−0.735839
$990$	0	0
$991$	−3.98714e7	−1.28967	−0.644833	−	0.764324i	$-0.723073\pi$
$991$	−3.98714e7	−1.28967	−0.644833	+	0.764324i	$0.723073\pi$
$992$	0	0
$993$	1.51146e6	0.0486433
$994$	0	0
$995$	−1.61921e7	−0.518496
$996$	0	0
$997$	−1.42645e7	−0.454484	−0.227242	−	0.973838i	$-0.572971\pi$
$997$	−1.42645e7	−0.454484	−0.227242	+	0.973838i	$0.572971\pi$
$998$	0	0
$999$	−1.12098e7	−0.355373

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.6.a.i.1.3	✓	5
4.3	odd	2		224.6.a.j.1.3	yes	5
8.3	odd	2		448.6.a.be.1.3		5
8.5	even	2		448.6.a.bf.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.6.a.i.1.3	✓	5	1.1	even	1	trivial
224.6.a.j.1.3	yes	5	4.3	odd	2
448.6.a.be.1.3		5	8.3	odd	2
448.6.a.bf.1.3		5	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	224.a (trivial)

\(f(q)\)	\(=\)	\(q+2.49210 q^{3} -99.5911 q^{5} +49.0000 q^{7} -236.789 q^{9} +292.685 q^{11} -163.156 q^{13} -248.190 q^{15} -1509.24 q^{17} -2117.22 q^{19} +122.113 q^{21} +4585.65 q^{23} +6793.38 q^{25} -1195.68 q^{27} +800.137 q^{29} +1276.43 q^{31} +729.399 q^{33} -4879.96 q^{35} +9375.25 q^{37} -406.599 q^{39} +6583.50 q^{41} -4935.97 q^{43} +23582.1 q^{45} +25847.4 q^{47} +2401.00 q^{49} -3761.16 q^{51} -6214.40 q^{53} -29148.8 q^{55} -5276.32 q^{57} -2803.61 q^{59} -828.062 q^{61} -11602.7 q^{63} +16248.8 q^{65} +62870.3 q^{67} +11427.9 q^{69} -60730.6 q^{71} +26058.6 q^{73} +16929.7 q^{75} +14341.6 q^{77} +55359.0 q^{79} +54560.1 q^{81} -118671. q^{83} +150306. q^{85} +1994.02 q^{87} +98218.5 q^{89} -7994.62 q^{91} +3180.98 q^{93} +210856. q^{95} +106682. q^{97} -69304.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 10 q^{3} + 36 q^{5} + 245 q^{7} + 637 q^{9} + 116 q^{11} + 40 q^{13} + 16 q^{15} - 402 q^{17} - 3582 q^{19} - 490 q^{21} + 472 q^{23} + 9615 q^{25} + 356 q^{27} + 4754 q^{29} - 10500 q^{31} + 15864 q^{33}+ \cdots - 260236 q^{99}+O(q^{100}) \) 5 * q - 10 * q^3 + 36 * q^5 + 245 * q^7 + 637 * q^9 + 116 * q^11 + 40 * q^13 + 16 * q^15 - 402 * q^17 - 3582 * q^19 - 490 * q^21 + 472 * q^23 + 9615 * q^25 + 356 * q^27 + 4754 * q^29 - 10500 * q^31 + 15864 * q^33 + 1764 * q^35 + 19642 * q^37 + 10872 * q^39 + 23398 * q^41 + 22044 * q^43 + 49476 * q^45 + 16004 * q^47 + 12005 * q^49 - 45676 * q^51 + 54246 * q^53 + 53456 * q^55 + 109556 * q^57 - 74366 * q^59 + 68316 * q^61 + 31213 * q^63 + 152568 * q^65 - 26560 * q^67 + 214720 * q^69 + 93072 * q^71 + 136098 * q^73 - 124510 * q^75 + 5684 * q^77 + 96080 * q^79 + 104801 * q^81 - 145894 * q^83 + 117352 * q^85 + 168876 * q^87 + 188554 * q^89 + 1960 * q^91 + 86296 * q^93 + 74736 * q^95 - 88146 * q^97 - 260236 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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