LMFDB - Embedded newform 224.6.a.i.1.5

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			1.5
Root			$14.6594$ 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)

$f(q)$	$=$	$q+27.3187 q^{3} +52.2232 q^{5} +49.0000 q^{7} +503.312 q^{9} +142.342 q^{11} +219.279 q^{13} +1426.67 q^{15} -1555.97 q^{17} +372.596 q^{19} +1338.62 q^{21} +3653.43 q^{23} -397.737 q^{25} +7111.38 q^{27} +4492.29 q^{29} -9151.19 q^{31} +3888.60 q^{33} +2558.94 q^{35} +3668.75 q^{37} +5990.41 q^{39} -17679.9 q^{41} +11082.6 q^{43} +26284.5 q^{45} -19707.8 q^{47} +2401.00 q^{49} -42507.0 q^{51} +22932.9 q^{53} +7433.55 q^{55} +10178.9 q^{57} -50657.1 q^{59} +23242.6 q^{61} +24662.3 q^{63} +11451.4 q^{65} -34909.7 q^{67} +99807.0 q^{69} +36659.7 q^{71} +28419.8 q^{73} -10865.7 q^{75} +6974.75 q^{77} -15866.1 q^{79} +71968.9 q^{81} +26106.6 q^{83} -81257.6 q^{85} +122724. q^{87} +98204.8 q^{89} +10744.7 q^{91} -249999. q^{93} +19458.2 q^{95} -123942. q^{97} +71642.3 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	27.3187	1.75250	0.876248	−	0.481861i	$-0.160039\pi$
$3$	27.3187	1.75250	0.876248	+	0.481861i	$0.160039\pi$
$4$	0	0
$5$	52.2232	0.934197	0.467099	−	0.884205i	$-0.345299\pi$
$5$	52.2232	0.934197	0.467099	+	0.884205i	$0.345299\pi$
$6$	0	0
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	0	0
$9$	503.312	2.07124
$10$	0	0
$11$	142.342	0.354692	0.177346	−	0.984149i	$-0.443249\pi$
$11$	142.342	0.354692	0.177346	+	0.984149i	$0.443249\pi$
$12$	0	0
$13$	219.279	0.359864	0.179932	−	0.983679i	$-0.442412\pi$
$13$	219.279	0.359864	0.179932	+	0.983679i	$0.442412\pi$
$14$	0	0
$15$	1426.67	1.63718
$16$	0	0
$17$	−1555.97	−1.30581	−0.652903	−	0.757442i	$-0.726449\pi$
$17$	−1555.97	−1.30581	−0.652903	+	0.757442i	$0.726449\pi$
$18$	0	0
$19$	372.596	0.236785	0.118393	−	0.992967i	$-0.462226\pi$
$19$	372.596	0.236785	0.118393	+	0.992967i	$0.462226\pi$
$20$	0	0
$21$	1338.62	0.662381
$22$	0	0
$23$	3653.43	1.44006	0.720031	−	0.693942i	$-0.244128\pi$
$23$	3653.43	1.44006	0.720031	+	0.693942i	$0.244128\pi$
$24$	0	0
$25$	−397.737	−0.127276
$26$	0	0
$27$	7111.38	1.87735
$28$	0	0
$29$	4492.29	0.991911	0.495955	−	0.868348i	$-0.334818\pi$
$29$	4492.29	0.991911	0.495955	+	0.868348i	$0.334818\pi$
$30$	0	0
$31$	−9151.19	−1.71030	−0.855152	−	0.518378i	$-0.826536\pi$
$31$	−9151.19	−1.71030	−0.855152	+	0.518378i	$0.826536\pi$
$32$	0	0
$33$	3888.60	0.621596
$34$	0	0
$35$	2558.94	0.353093
$36$	0	0
$37$	3668.75	0.440569	0.220284	−	0.975436i	$-0.429302\pi$
$37$	3668.75	0.440569	0.220284	+	0.975436i	$0.429302\pi$
$38$	0	0
$39$	5990.41	0.630659
$40$	0	0
$41$	−17679.9	−1.64255	−0.821277	−	0.570530i	$-0.806738\pi$
$41$	−17679.9	−1.64255	−0.821277	+	0.570530i	$0.806738\pi$
$42$	0	0
$43$	11082.6	0.914050	0.457025	−	0.889454i	$-0.348915\pi$
$43$	11082.6	0.914050	0.457025	+	0.889454i	$0.348915\pi$
$44$	0	0
$45$	26284.5	1.93495
$46$	0	0
$47$	−19707.8	−1.30135	−0.650673	−	0.759358i	$-0.725513\pi$
$47$	−19707.8	−1.30135	−0.650673	+	0.759358i	$0.725513\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	−42507.0	−2.28842
$52$	0	0
$53$	22932.9	1.12142	0.560711	−	0.828012i	$-0.310528\pi$
$53$	22932.9	1.12142	0.560711	+	0.828012i	$0.310528\pi$
$54$	0	0
$55$	7433.55	0.331352
$56$	0	0
$57$	10178.9	0.414965
$58$	0	0
$59$	−50657.1	−1.89457	−0.947284	−	0.320395i	$-0.896184\pi$
$59$	−50657.1	−1.89457	−0.947284	+	0.320395i	$0.896184\pi$
$60$	0	0
$61$	23242.6	0.799759	0.399880	−	0.916568i	$-0.369052\pi$
$61$	23242.6	0.799759	0.399880	+	0.916568i	$0.369052\pi$
$62$	0	0
$63$	24662.3	0.782856
$64$	0	0
$65$	11451.4	0.336183
$66$	0	0
$67$	−34909.7	−0.950076	−0.475038	−	0.879965i	$-0.657566\pi$
$67$	−34909.7	−0.950076	−0.475038	+	0.879965i	$0.657566\pi$
$68$	0	0
$69$	99807.0	2.52370
$70$	0	0
$71$	36659.7	0.863064	0.431532	−	0.902098i	$-0.357973\pi$
$71$	36659.7	0.863064	0.431532	+	0.902098i	$0.357973\pi$
$72$	0	0
$73$	28419.8	0.624186	0.312093	−	0.950052i	$-0.398970\pi$
$73$	28419.8	0.624186	0.312093	+	0.950052i	$0.398970\pi$
$74$	0	0
$75$	−10865.7	−0.223050
$76$	0	0
$77$	6974.75	0.134061
$78$	0	0
$79$	−15866.1	−0.286024	−0.143012	−	0.989721i	$-0.545679\pi$
$79$	−15866.1	−0.286024	−0.143012	+	0.989721i	$0.545679\pi$
$80$	0	0
$81$	71968.9	1.21880
$82$	0	0
$83$	26106.6	0.415964	0.207982	−	0.978133i	$-0.433310\pi$
$83$	26106.6	0.415964	0.207982	+	0.978133i	$0.433310\pi$
$84$	0	0
$85$	−81257.6	−1.21988
$86$	0	0
$87$	122724.	1.73832
$88$	0	0
$89$	98204.8	1.31419	0.657094	−	0.753809i	$-0.271785\pi$
$89$	98204.8	1.31419	0.657094	+	0.753809i	$0.271785\pi$
$90$	0	0
$91$	10744.7	0.136016
$92$	0	0
$93$	−249999.	−2.99730
$94$	0	0
$95$	19458.2	0.221204
$96$	0	0
$97$	−123942.	−1.33749	−0.668744	−	0.743493i	$-0.733168\pi$
$97$	−123942.	−1.33749	−0.668744	+	0.743493i	$0.733168\pi$
$98$	0	0
$99$	71642.3	0.734652
$100$	0	0
$101$	115136.	1.12307	0.561534	−	0.827454i	$-0.310211\pi$
$101$	115136.	1.12307	0.561534	+	0.827454i	$0.310211\pi$
$102$	0	0
$103$	42730.9	0.396871	0.198435	−	0.980114i	$-0.436414\pi$
$103$	42730.9	0.396871	0.198435	+	0.980114i	$0.436414\pi$
$104$	0	0
$105$	69906.8	0.618794
$106$	0	0
$107$	81647.8	0.689422	0.344711	−	0.938709i	$-0.387977\pi$
$107$	81647.8	0.689422	0.344711	+	0.938709i	$0.387977\pi$
$108$	0	0
$109$	117518.	0.947412	0.473706	−	0.880683i	$-0.342916\pi$
$109$	117518.	0.947412	0.473706	+	0.880683i	$0.342916\pi$
$110$	0	0
$111$	100225.	0.772094
$112$	0	0
$113$	247972.	1.82687	0.913433	−	0.406989i	$-0.133421\pi$
$113$	247972.	1.82687	0.913433	+	0.406989i	$0.133421\pi$
$114$	0	0
$115$	190794.	1.34530
$116$	0	0
$117$	110365.	0.745364
$118$	0	0
$119$	−76242.4	−0.493548
$120$	0	0
$121$	−140790.	−0.874194
$122$	0	0
$123$	−482991.	−2.87857
$124$	0	0
$125$	−183969.	−1.05310
$126$	0	0
$127$	−118457.	−0.651704	−0.325852	−	0.945421i	$-0.605651\pi$
$127$	−118457.	−0.651704	−0.325852	+	0.945421i	$0.605651\pi$
$128$	0	0
$129$	302762.	1.60187
$130$	0	0
$131$	−282312.	−1.43731	−0.718656	−	0.695366i	$-0.755242\pi$
$131$	−282312.	−1.43731	−0.718656	+	0.695366i	$0.755242\pi$
$132$	0	0
$133$	18257.2	0.0894964
$134$	0	0
$135$	371379.	1.75381
$136$	0	0
$137$	−268767.	−1.22342	−0.611708	−	0.791084i	$-0.709517\pi$
$137$	−268767.	−1.22342	−0.611708	+	0.791084i	$0.709517\pi$
$138$	0	0
$139$	−241896.	−1.06192	−0.530961	−	0.847396i	$-0.678169\pi$
$139$	−241896.	−1.06192	−0.530961	+	0.847396i	$0.678169\pi$
$140$	0	0
$141$	−538391.	−2.28060
$142$	0	0
$143$	31212.5	0.127641
$144$	0	0
$145$	234602.	0.926640
$146$	0	0
$147$	65592.2	0.250357
$148$	0	0
$149$	−319446.	−1.17878	−0.589389	−	0.807850i	$-0.700631\pi$
$149$	−319446.	−1.17878	−0.589389	+	0.807850i	$0.700631\pi$
$150$	0	0
$151$	−154017.	−0.549701	−0.274851	−	0.961487i	$-0.588628\pi$
$151$	−154017.	−0.549701	−0.274851	+	0.961487i	$0.588628\pi$
$152$	0	0
$153$	−783137.	−2.70464
$154$	0	0
$155$	−477904.	−1.59776
$156$	0	0
$157$	−148158.	−0.479706	−0.239853	−	0.970809i	$-0.577099\pi$
$157$	−148158.	−0.479706	−0.239853	+	0.970809i	$0.577099\pi$
$158$	0	0
$159$	626497.	1.96529
$160$	0	0
$161$	179018.	0.544292
$162$	0	0
$163$	−433962.	−1.27933	−0.639666	−	0.768653i	$-0.720927\pi$
$163$	−433962.	−1.27933	−0.639666	+	0.768653i	$0.720927\pi$
$164$	0	0
$165$	203075.	0.580693
$166$	0	0
$167$	−568533.	−1.57748	−0.788741	−	0.614726i	$-0.789267\pi$
$167$	−568533.	−1.57748	−0.788741	+	0.614726i	$0.789267\pi$
$168$	0	0
$169$	−323210.	−0.870498
$170$	0	0
$171$	187532.	0.490439
$172$	0	0
$173$	373172.	0.947969	0.473984	−	0.880533i	$-0.342815\pi$
$173$	373172.	0.947969	0.473984	+	0.880533i	$0.342815\pi$
$174$	0	0
$175$	−19489.1	−0.0481058
$176$	0	0
$177$	−1.38389e6	−3.32022
$178$	0	0
$179$	149898.	0.349674	0.174837	−	0.984597i	$-0.444060\pi$
$179$	149898.	0.349674	0.174837	+	0.984597i	$0.444060\pi$
$180$	0	0
$181$	115243.	0.261468	0.130734	−	0.991417i	$-0.458267\pi$
$181$	115243.	0.261468	0.130734	+	0.991417i	$0.458267\pi$
$182$	0	0
$183$	634957.	1.40157
$184$	0	0
$185$	191594.	0.411578
$186$	0	0
$187$	−221479.	−0.463158
$188$	0	0
$189$	348457.	0.709570
$190$	0	0
$191$	18013.1	0.0357278	0.0178639	−	0.999840i	$-0.494313\pi$
$191$	18013.1	0.0357278	0.0178639	+	0.999840i	$0.494313\pi$
$192$	0	0
$193$	873827.	1.68862	0.844310	−	0.535855i	$-0.180011\pi$
$193$	873827.	1.68862	0.844310	+	0.535855i	$0.180011\pi$
$194$	0	0
$195$	312838.	0.589160
$196$	0	0
$197$	−148925.	−0.273402	−0.136701	−	0.990612i	$-0.543650\pi$
$197$	−148925.	−0.273402	−0.136701	+	0.990612i	$0.543650\pi$
$198$	0	0
$199$	−764586.	−1.36865	−0.684327	−	0.729175i	$-0.739904\pi$
$199$	−764586.	−1.36865	−0.684327	+	0.729175i	$0.739904\pi$
$200$	0	0
$201$	−953686.	−1.66500
$202$	0	0
$203$	220122.	0.374907
$204$	0	0
$205$	−923300.	−1.53447
$206$	0	0
$207$	1.83881e6	2.98272
$208$	0	0
$209$	53036.1	0.0839858
$210$	0	0
$211$	852579.	1.31834	0.659172	−	0.751992i	$-0.270907\pi$
$211$	852579.	1.31834	0.659172	+	0.751992i	$0.270907\pi$
$212$	0	0
$213$	1.00150e6	1.51252
$214$	0	0
$215$	578768.	0.853903
$216$	0	0
$217$	−448408.	−0.646434
$218$	0	0
$219$	776392.	1.09388
$220$	0	0
$221$	−341191.	−0.469912
$222$	0	0
$223$	−276149.	−0.371862	−0.185931	−	0.982563i	$-0.559530\pi$
$223$	−276149.	−0.371862	−0.185931	+	0.982563i	$0.559530\pi$
$224$	0	0
$225$	−200186.	−0.263619
$226$	0	0
$227$	−1.01983e6	−1.31360	−0.656802	−	0.754063i	$-0.728091\pi$
$227$	−1.01983e6	−1.31360	−0.656802	+	0.754063i	$0.728091\pi$
$228$	0	0
$229$	1.07254e6	1.35153	0.675765	−	0.737117i	$-0.263813\pi$
$229$	1.07254e6	1.35153	0.675765	+	0.737117i	$0.263813\pi$
$230$	0	0
$231$	190541.	0.234941
$232$	0	0
$233$	−376769.	−0.454658	−0.227329	−	0.973818i	$-0.572999\pi$
$233$	−376769.	−0.454658	−0.227329	+	0.973818i	$0.572999\pi$
$234$	0	0
$235$	−1.02920e6	−1.21571
$236$	0	0
$237$	−433441.	−0.501256
$238$	0	0
$239$	−286565.	−0.324510	−0.162255	−	0.986749i	$-0.551877\pi$
$239$	−286565.	−0.324510	−0.162255	+	0.986749i	$0.551877\pi$
$240$	0	0
$241$	809660.	0.897966	0.448983	−	0.893540i	$-0.351786\pi$
$241$	809660.	0.897966	0.448983	+	0.893540i	$0.351786\pi$
$242$	0	0
$243$	238032.	0.258594
$244$	0	0
$245$	125388.	0.133457
$246$	0	0
$247$	81702.4	0.0852104
$248$	0	0
$249$	713199.	0.728975
$250$	0	0
$251$	−672198.	−0.673462	−0.336731	−	0.941601i	$-0.609321\pi$
$251$	−672198.	−0.673462	−0.336731	+	0.941601i	$0.609321\pi$
$252$	0	0
$253$	520036.	0.510778
$254$	0	0
$255$	−2.21985e6	−2.13783
$256$	0	0
$257$	593485.	0.560501	0.280251	−	0.959927i	$-0.409582\pi$
$257$	593485.	0.560501	0.280251	+	0.959927i	$0.409582\pi$
$258$	0	0
$259$	179769.	0.166519
$260$	0	0
$261$	2.26102e6	2.05449
$262$	0	0
$263$	1.58789e6	1.41556	0.707782	−	0.706430i	$-0.249696\pi$
$263$	1.58789e6	1.41556	0.707782	+	0.706430i	$0.249696\pi$
$264$	0	0
$265$	1.19763e6	1.04763
$266$	0	0
$267$	2.68283e6	2.30311
$268$	0	0
$269$	−1.77916e6	−1.49911	−0.749556	−	0.661941i	$-0.769733\pi$
$269$	−1.77916e6	−1.49911	−0.749556	+	0.661941i	$0.769733\pi$
$270$	0	0
$271$	−1.07288e6	−0.887414	−0.443707	−	0.896172i	$-0.646337\pi$
$271$	−1.07288e6	−0.887414	−0.443707	+	0.896172i	$0.646337\pi$
$272$	0	0
$273$	293530.	0.238367
$274$	0	0
$275$	−56614.7	−0.0451437
$276$	0	0
$277$	−1.99945e6	−1.56571	−0.782856	−	0.622202i	$-0.786238\pi$
$277$	−1.99945e6	−1.56571	−0.782856	+	0.622202i	$0.786238\pi$
$278$	0	0
$279$	−4.60590e6	−3.54245
$280$	0	0
$281$	26939.8	0.0203530	0.0101765	−	0.999948i	$-0.496761\pi$
$281$	26939.8	0.0203530	0.0101765	+	0.999948i	$0.496761\pi$
$282$	0	0
$283$	462556.	0.343320	0.171660	−	0.985156i	$-0.445087\pi$
$283$	462556.	0.343320	0.171660	+	0.985156i	$0.445087\pi$
$284$	0	0
$285$	531572.	0.387659
$286$	0	0
$287$	−866314.	−0.620827
$288$	0	0
$289$	1.00118e6	0.705127
$290$	0	0
$291$	−3.38594e6	−2.34394
$292$	0	0
$293$	−1.74537e6	−1.18773	−0.593867	−	0.804563i	$-0.702400\pi$
$293$	−1.74537e6	−1.18773	−0.593867	+	0.804563i	$0.702400\pi$
$294$	0	0
$295$	−2.64548e6	−1.76990
$296$	0	0
$297$	1.01225e6	0.665879
$298$	0	0
$299$	801119.	0.518226
$300$	0	0
$301$	543047.	0.345479
$302$	0	0
$303$	3.14535e6	1.96817
$304$	0	0
$305$	1.21380e6	0.747133
$306$	0	0
$307$	1.61327e6	0.976924	0.488462	−	0.872585i	$-0.337558\pi$
$307$	1.61327e6	0.976924	0.488462	+	0.872585i	$0.337558\pi$
$308$	0	0
$309$	1.16735e6	0.695514
$310$	0	0
$311$	−985033.	−0.577497	−0.288749	−	0.957405i	$-0.593239\pi$
$311$	−985033.	−0.577497	−0.288749	+	0.957405i	$0.593239\pi$
$312$	0	0
$313$	1.06442e6	0.614116	0.307058	−	0.951691i	$-0.400655\pi$
$313$	1.06442e6	0.614116	0.307058	+	0.951691i	$0.400655\pi$
$314$	0	0
$315$	1.28794e6	0.731341
$316$	0	0
$317$	180539.	0.100907	0.0504536	−	0.998726i	$-0.483933\pi$
$317$	180539.	0.100907	0.0504536	+	0.998726i	$0.483933\pi$
$318$	0	0
$319$	639441.	0.351823
$320$	0	0
$321$	2.23051e6	1.20821
$322$	0	0
$323$	−579748.	−0.309195
$324$	0	0
$325$	−87215.2	−0.0458020
$326$	0	0
$327$	3.21045e6	1.66034
$328$	0	0
$329$	−965681.	−0.491863
$330$	0	0
$331$	−192466.	−0.0965571	−0.0482786	−	0.998834i	$-0.515374\pi$
$331$	−192466.	−0.0965571	−0.0482786	+	0.998834i	$0.515374\pi$
$332$	0	0
$333$	1.84652e6	0.912524
$334$	0	0
$335$	−1.82309e6	−0.887559
$336$	0	0
$337$	4.10014e6	1.96664	0.983318	−	0.181892i	$-0.0582222\pi$
$337$	4.10014e6	1.96664	0.983318	+	0.181892i	$0.0582222\pi$
$338$	0	0
$339$	6.77428e6	3.20158
$340$	0	0
$341$	−1.30260e6	−0.606631
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	5.21224e6	2.35764
$346$	0	0
$347$	−2.68642e6	−1.19770	−0.598852	−	0.800860i	$-0.704376\pi$
$347$	−2.68642e6	−1.19770	−0.598852	+	0.800860i	$0.704376\pi$
$348$	0	0
$349$	3.01244e6	1.32390	0.661948	−	0.749549i	$-0.269730\pi$
$349$	3.01244e6	1.32390	0.661948	+	0.749549i	$0.269730\pi$
$350$	0	0
$351$	1.55937e6	0.675588
$352$	0	0
$353$	−4.40559e6	−1.88177	−0.940886	−	0.338724i	$-0.890005\pi$
$353$	−4.40559e6	−1.88177	−0.940886	+	0.338724i	$0.890005\pi$
$354$	0	0
$355$	1.91449e6	0.806272
$356$	0	0
$357$	−2.08284e6	−0.864941
$358$	0	0
$359$	1.08572e6	0.444611	0.222306	−	0.974977i	$-0.428642\pi$
$359$	1.08572e6	0.444611	0.222306	+	0.974977i	$0.428642\pi$
$360$	0	0
$361$	−2.33727e6	−0.943933
$362$	0	0
$363$	−3.84619e6	−1.53202
$364$	0	0
$365$	1.48417e6	0.583113
$366$	0	0
$367$	3.46765e6	1.34391	0.671955	−	0.740592i	$-0.265455\pi$
$367$	3.46765e6	1.34391	0.671955	+	0.740592i	$0.265455\pi$
$368$	0	0
$369$	−8.89849e6	−3.40213
$370$	0	0
$371$	1.12371e6	0.423858
$372$	0	0
$373$	2.77398e6	1.03236	0.516180	−	0.856480i	$-0.327354\pi$
$373$	2.77398e6	1.03236	0.516180	+	0.856480i	$0.327354\pi$
$374$	0	0
$375$	−5.02578e6	−1.84555
$376$	0	0
$377$	985063.	0.356953
$378$	0	0
$379$	3.31407e6	1.18512	0.592562	−	0.805525i	$-0.298116\pi$
$379$	3.31407e6	1.18512	0.592562	+	0.805525i	$0.298116\pi$
$380$	0	0
$381$	−3.23609e6	−1.14211
$382$	0	0
$383$	3.13229e6	1.09110	0.545551	−	0.838078i	$-0.316321\pi$
$383$	3.13229e6	1.09110	0.545551	+	0.838078i	$0.316321\pi$
$384$	0	0
$385$	364244.	0.125239
$386$	0	0
$387$	5.57800e6	1.89322
$388$	0	0
$389$	−2.78999e6	−0.934823	−0.467411	−	0.884040i	$-0.654813\pi$
$389$	−2.78999e6	−0.934823	−0.467411	+	0.884040i	$0.654813\pi$
$390$	0	0
$391$	−5.68462e6	−1.88044
$392$	0	0
$393$	−7.71240e6	−2.51888
$394$	0	0
$395$	−828578.	−0.267203
$396$	0	0
$397$	−397948.	−0.126721	−0.0633607	−	0.997991i	$-0.520182\pi$
$397$	−397948.	−0.126721	−0.0633607	+	0.997991i	$0.520182\pi$
$398$	0	0
$399$	498764.	0.156842
$400$	0	0
$401$	741928.	0.230410	0.115205	−	0.993342i	$-0.463248\pi$
$401$	741928.	0.230410	0.115205	+	0.993342i	$0.463248\pi$
$402$	0	0
$403$	−2.00666e6	−0.615476
$404$	0	0
$405$	3.75844e6	1.13860
$406$	0	0
$407$	522216.	0.156266
$408$	0	0
$409$	2.16438e6	0.639771	0.319886	−	0.947456i	$-0.396356\pi$
$409$	2.16438e6	0.639771	0.319886	+	0.947456i	$0.396356\pi$
$410$	0	0
$411$	−7.34235e6	−2.14403
$412$	0	0
$413$	−2.48220e6	−0.716079
$414$	0	0
$415$	1.36337e6	0.388592
$416$	0	0
$417$	−6.60830e6	−1.86101
$418$	0	0
$419$	4.07575e6	1.13415	0.567077	−	0.823665i	$-0.308074\pi$
$419$	4.07575e6	1.13415	0.567077	+	0.823665i	$0.308074\pi$
$420$	0	0
$421$	−2.95488e6	−0.812522	−0.406261	−	0.913757i	$-0.633168\pi$
$421$	−2.95488e6	−0.812522	−0.406261	+	0.913757i	$0.633168\pi$
$422$	0	0
$423$	−9.91915e6	−2.69540
$424$	0	0
$425$	618866.	0.166198
$426$	0	0
$427$	1.13889e6	0.302281
$428$	0	0
$429$	852686.	0.223690
$430$	0	0
$431$	−1.59736e6	−0.414200	−0.207100	−	0.978320i	$-0.566403\pi$
$431$	−1.59736e6	−0.414200	−0.207100	+	0.978320i	$0.566403\pi$
$432$	0	0
$433$	−2.52435e6	−0.647039	−0.323519	−	0.946222i	$-0.604866\pi$
$433$	−2.52435e6	−0.647039	−0.323519	+	0.946222i	$0.604866\pi$
$434$	0	0
$435$	6.40901e6	1.62393
$436$	0	0
$437$	1.36125e6	0.340985
$438$	0	0
$439$	−297743.	−0.0737360	−0.0368680	−	0.999320i	$-0.511738\pi$
$439$	−297743.	−0.0737360	−0.0368680	+	0.999320i	$0.511738\pi$
$440$	0	0
$441$	1.20845e6	0.295892
$442$	0	0
$443$	1.09616e6	0.265377	0.132689	−	0.991158i	$-0.457639\pi$
$443$	1.09616e6	0.265377	0.132689	+	0.991158i	$0.457639\pi$
$444$	0	0
$445$	5.12857e6	1.22771
$446$	0	0
$447$	−8.72685e6	−2.06580
$448$	0	0
$449$	−1.14561e6	−0.268177	−0.134088	−	0.990969i	$-0.542811\pi$
$449$	−1.14561e6	−0.268177	−0.134088	+	0.990969i	$0.542811\pi$
$450$	0	0
$451$	−2.51659e6	−0.582600
$452$	0	0
$453$	−4.20755e6	−0.963349
$454$	0	0
$455$	561120.	0.127065
$456$	0	0
$457$	8.50626e6	1.90523	0.952617	−	0.304173i	$-0.0983800\pi$
$457$	8.50626e6	1.90523	0.952617	+	0.304173i	$0.0983800\pi$
$458$	0	0
$459$	−1.10651e7	−2.45145
$460$	0	0
$461$	4.07074e6	0.892115	0.446058	−	0.895004i	$-0.352828\pi$
$461$	4.07074e6	0.892115	0.446058	+	0.895004i	$0.352828\pi$
$462$	0	0
$463$	4.48385e6	0.972072	0.486036	−	0.873939i	$-0.338442\pi$
$463$	4.48385e6	0.972072	0.486036	+	0.873939i	$0.338442\pi$
$464$	0	0
$465$	−1.30557e7	−2.80007
$466$	0	0
$467$	229548.	0.0487059	0.0243529	−	0.999703i	$-0.492247\pi$
$467$	229548.	0.0487059	0.0243529	+	0.999703i	$0.492247\pi$
$468$	0	0
$469$	−1.71057e6	−0.359095
$470$	0	0
$471$	−4.04748e6	−0.840683
$472$	0	0
$473$	1.57752e6	0.324206
$474$	0	0
$475$	−148195.	−0.0301371
$476$	0	0
$477$	1.15424e7	2.32274
$478$	0	0
$479$	−2.50619e6	−0.499085	−0.249542	−	0.968364i	$-0.580280\pi$
$479$	−2.50619e6	−0.499085	−0.249542	+	0.968364i	$0.580280\pi$
$480$	0	0
$481$	804478.	0.158545
$482$	0	0
$483$	4.89054e6	0.953870
$484$	0	0
$485$	−6.47266e6	−1.24948
$486$	0	0
$487$	8.24551e6	1.57542	0.787708	−	0.616049i	$-0.211268\pi$
$487$	8.24551e6	1.57542	0.787708	+	0.616049i	$0.211268\pi$
$488$	0	0
$489$	−1.18553e7	−2.24202
$490$	0	0
$491$	1.08374e6	0.202872	0.101436	−	0.994842i	$-0.467656\pi$
$491$	1.08374e6	0.202872	0.101436	+	0.994842i	$0.467656\pi$
$492$	0	0
$493$	−6.98986e6	−1.29524
$494$	0	0
$495$	3.74139e6	0.686310
$496$	0	0
$497$	1.79632e6	0.326207
$498$	0	0
$499$	−3.19035e6	−0.573571	−0.286785	−	0.957995i	$-0.592587\pi$
$499$	−3.19035e6	−0.573571	−0.286785	+	0.957995i	$0.592587\pi$
$500$	0	0
$501$	−1.55316e7	−2.76453
$502$	0	0
$503$	5.20859e6	0.917910	0.458955	−	0.888459i	$-0.348224\pi$
$503$	5.20859e6	0.917910	0.458955	+	0.888459i	$0.348224\pi$
$504$	0	0
$505$	6.01274e6	1.04917
$506$	0	0
$507$	−8.82968e6	−1.52554
$508$	0	0
$509$	−6.19140e6	−1.05924	−0.529620	−	0.848235i	$-0.677666\pi$
$509$	−6.19140e6	−1.05924	−0.529620	+	0.848235i	$0.677666\pi$
$510$	0	0
$511$	1.39257e6	0.235920
$512$	0	0
$513$	2.64967e6	0.444528
$514$	0	0
$515$	2.23154e6	0.370755
$516$	0	0
$517$	−2.80524e6	−0.461577
$518$	0	0
$519$	1.01946e7	1.66131
$520$	0	0
$521$	2.49044e6	0.401959	0.200979	−	0.979595i	$-0.435588\pi$
$521$	2.49044e6	0.401959	0.200979	+	0.979595i	$0.435588\pi$
$522$	0	0
$523$	1.62094e6	0.259128	0.129564	−	0.991571i	$-0.458642\pi$
$523$	1.62094e6	0.259128	0.129564	+	0.991571i	$0.458642\pi$
$524$	0	0
$525$	−532418.	−0.0843051
$526$	0	0
$527$	1.42390e7	2.23332
$528$	0	0
$529$	6.91121e6	1.07378
$530$	0	0
$531$	−2.54963e7	−3.92411
$532$	0	0
$533$	−3.87682e6	−0.591095
$534$	0	0
$535$	4.26391e6	0.644056
$536$	0	0
$537$	4.09501e6	0.612801
$538$	0	0
$539$	341763.	0.0506702
$540$	0	0
$541$	215848.	0.0317070	0.0158535	−	0.999874i	$-0.494953\pi$
$541$	215848.	0.0317070	0.0158535	+	0.999874i	$0.494953\pi$
$542$	0	0
$543$	3.14829e6	0.458221
$544$	0	0
$545$	6.13718e6	0.885070
$546$	0	0
$547$	8.41491e6	1.20249	0.601244	−	0.799065i	$-0.294672\pi$
$547$	8.41491e6	1.20249	0.601244	+	0.799065i	$0.294672\pi$
$548$	0	0
$549$	1.16983e7	1.65649
$550$	0	0
$551$	1.67381e6	0.234870
$552$	0	0
$553$	−777439.	−0.108107
$554$	0	0
$555$	5.23409e6	0.721288
$556$	0	0
$557$	2.81520e6	0.384478	0.192239	−	0.981348i	$-0.438425\pi$
$557$	2.81520e6	0.384478	0.192239	+	0.981348i	$0.438425\pi$
$558$	0	0
$559$	2.43018e6	0.328933
$560$	0	0
$561$	−6.05053e6	−0.811683
$562$	0	0
$563$	3.14460e6	0.418114	0.209057	−	0.977903i	$-0.432961\pi$
$563$	3.14460e6	0.418114	0.209057	+	0.977903i	$0.432961\pi$
$564$	0	0
$565$	1.29499e7	1.70665
$566$	0	0
$567$	3.52647e6	0.460663
$568$	0	0
$569$	2.38913e6	0.309356	0.154678	−	0.987965i	$-0.450566\pi$
$569$	2.38913e6	0.309356	0.154678	+	0.987965i	$0.450566\pi$
$570$	0	0
$571$	1.59235e6	0.204384	0.102192	−	0.994765i	$-0.467414\pi$
$571$	1.59235e6	0.204384	0.102192	+	0.994765i	$0.467414\pi$
$572$	0	0
$573$	492096.	0.0626128
$574$	0	0
$575$	−1.45310e6	−0.183285
$576$	0	0
$577$	1.33877e7	1.67405	0.837023	−	0.547168i	$-0.184294\pi$
$577$	1.33877e7	1.67405	0.837023	+	0.547168i	$0.184294\pi$
$578$	0	0
$579$	2.38718e7	2.95930
$580$	0	0
$581$	1.27922e6	0.157220
$582$	0	0
$583$	3.26431e6	0.397759
$584$	0	0
$585$	5.76364e6	0.696317
$586$	0	0
$587$	−2.10664e6	−0.252346	−0.126173	−	0.992008i	$-0.540269\pi$
$587$	−2.10664e6	−0.252346	−0.126173	+	0.992008i	$0.540269\pi$
$588$	0	0
$589$	−3.40970e6	−0.404975
$590$	0	0
$591$	−4.06844e6	−0.479136
$592$	0	0
$593$	−2.06143e6	−0.240731	−0.120365	−	0.992730i	$-0.538407\pi$
$593$	−2.06143e6	−0.240731	−0.120365	+	0.992730i	$0.538407\pi$
$594$	0	0
$595$	−3.98162e6	−0.461071
$596$	0	0
$597$	−2.08875e7	−2.39856
$598$	0	0
$599$	−1.01049e7	−1.15071	−0.575355	−	0.817904i	$-0.695136\pi$
$599$	−1.01049e7	−1.15071	−0.575355	+	0.817904i	$0.695136\pi$
$600$	0	0
$601$	7.52826e6	0.850176	0.425088	−	0.905152i	$-0.360243\pi$
$601$	7.52826e6	0.850176	0.425088	+	0.905152i	$0.360243\pi$
$602$	0	0
$603$	−1.75704e7	−1.96784
$604$	0	0
$605$	−7.35249e6	−0.816669
$606$	0	0
$607$	2.03624e6	0.224314	0.112157	−	0.993691i	$-0.464224\pi$
$607$	2.03624e6	0.224314	0.112157	+	0.993691i	$0.464224\pi$
$608$	0	0
$609$	6.01345e6	0.657023
$610$	0	0
$611$	−4.32149e6	−0.468307
$612$	0	0
$613$	1.08503e7	1.16625	0.583123	−	0.812384i	$-0.301831\pi$
$613$	1.08503e7	1.16625	0.583123	+	0.812384i	$0.301831\pi$
$614$	0	0
$615$	−2.52234e7	−2.68915
$616$	0	0
$617$	−1.55938e7	−1.64907	−0.824533	−	0.565814i	$-0.808562\pi$
$617$	−1.55938e7	−1.64907	−0.824533	+	0.565814i	$0.808562\pi$
$618$	0	0
$619$	−7.72263e6	−0.810099	−0.405050	−	0.914295i	$-0.632746\pi$
$619$	−7.72263e6	−0.810099	−0.405050	+	0.914295i	$0.632746\pi$
$620$	0	0
$621$	2.59809e7	2.70349
$622$	0	0
$623$	4.81203e6	0.496716
$624$	0	0
$625$	−8.36450e6	−0.856525
$626$	0	0
$627$	1.44888e6	0.147185
$628$	0	0
$629$	−5.70845e6	−0.575297
$630$	0	0
$631$	−1.01204e7	−1.01187	−0.505936	−	0.862571i	$-0.668853\pi$
$631$	−1.01204e7	−1.01187	−0.505936	+	0.862571i	$0.668853\pi$
$632$	0	0
$633$	2.32914e7	2.31039
$634$	0	0
$635$	−6.18619e6	−0.608820
$636$	0	0
$637$	526488.	0.0514091
$638$	0	0
$639$	1.84512e7	1.78761
$640$	0	0
$641$	1.11080e7	1.06780	0.533902	−	0.845546i	$-0.320725\pi$
$641$	1.11080e7	1.06780	0.533902	+	0.845546i	$0.320725\pi$
$642$	0	0
$643$	−1.36445e7	−1.30146	−0.650730	−	0.759309i	$-0.725537\pi$
$643$	−1.36445e7	−1.30146	−0.650730	+	0.759309i	$0.725537\pi$
$644$	0	0
$645$	1.58112e7	1.49646
$646$	0	0
$647$	1.88459e7	1.76993	0.884965	−	0.465658i	$-0.154182\pi$
$647$	1.88459e7	1.76993	0.884965	+	0.465658i	$0.154182\pi$
$648$	0	0
$649$	−7.21063e6	−0.671988
$650$	0	0
$651$	−1.22499e7	−1.13287
$652$	0	0
$653$	−1.89196e6	−0.173632	−0.0868158	−	0.996224i	$-0.527669\pi$
$653$	−1.89196e6	−0.173632	−0.0868158	+	0.996224i	$0.527669\pi$
$654$	0	0
$655$	−1.47432e7	−1.34273
$656$	0	0
$657$	1.43040e7	1.29284
$658$	0	0
$659$	−7.03246e6	−0.630803	−0.315402	−	0.948958i	$-0.602139\pi$
$659$	−7.03246e6	−0.630803	−0.315402	+	0.948958i	$0.602139\pi$
$660$	0	0
$661$	1.49186e7	1.32808	0.664040	−	0.747697i	$-0.268840\pi$
$661$	1.49186e7	1.32808	0.664040	+	0.747697i	$0.268840\pi$
$662$	0	0
$663$	−9.32088e6	−0.823518
$664$	0	0
$665$	953451.	0.0836073
$666$	0	0
$667$	1.64123e7	1.42841
$668$	0	0
$669$	−7.54404e6	−0.651686
$670$	0	0
$671$	3.30839e6	0.283668
$672$	0	0
$673$	3.40310e6	0.289625	0.144813	−	0.989459i	$-0.453742\pi$
$673$	3.40310e6	0.289625	0.144813	+	0.989459i	$0.453742\pi$
$674$	0	0
$675$	−2.82846e6	−0.238941
$676$	0	0
$677$	9.76032e6	0.818450	0.409225	−	0.912433i	$-0.365799\pi$
$677$	9.76032e6	0.818450	0.409225	+	0.912433i	$0.365799\pi$
$678$	0	0
$679$	−6.07317e6	−0.505523
$680$	0	0
$681$	−2.78605e7	−2.30209
$682$	0	0
$683$	−3.43814e6	−0.282015	−0.141007	−	0.990009i	$-0.545034\pi$
$683$	−3.43814e6	−0.282015	−0.141007	+	0.990009i	$0.545034\pi$
$684$	0	0
$685$	−1.40359e7	−1.14291
$686$	0	0
$687$	2.93005e7	2.36855
$688$	0	0
$689$	5.02869e6	0.403559
$690$	0	0
$691$	−6.45739e6	−0.514472	−0.257236	−	0.966349i	$-0.582812\pi$
$691$	−6.45739e6	−0.514472	−0.257236	+	0.966349i	$0.582812\pi$
$692$	0	0
$693$	3.51047e6	0.277672
$694$	0	0
$695$	−1.26326e7	−0.992044
$696$	0	0
$697$	2.75093e7	2.14486
$698$	0	0
$699$	−1.02928e7	−0.796787
$700$	0	0
$701$	5.90115e6	0.453567	0.226784	−	0.973945i	$-0.427179\pi$
$701$	5.90115e6	0.453567	0.226784	+	0.973945i	$0.427179\pi$
$702$	0	0
$703$	1.36696e6	0.104320
$704$	0	0
$705$	−2.81165e7	−2.13053
$706$	0	0
$707$	5.64164e6	0.424480
$708$	0	0
$709$	1.36046e7	1.01641	0.508205	−	0.861236i	$-0.330309\pi$
$709$	1.36046e7	1.01641	0.508205	+	0.861236i	$0.330309\pi$
$710$	0	0
$711$	−7.98559e6	−0.592424
$712$	0	0
$713$	−3.34332e7	−2.46294
$714$	0	0
$715$	1.63002e6	0.119241
$716$	0	0
$717$	−7.82858e6	−0.568703
$718$	0	0
$719$	1.94690e7	1.40450	0.702248	−	0.711932i	$-0.252180\pi$
$719$	1.94690e7	1.40450	0.702248	+	0.711932i	$0.252180\pi$
$720$	0	0
$721$	2.09381e6	0.150003
$722$	0	0
$723$	2.21189e7	1.57368
$724$	0	0
$725$	−1.78675e6	−0.126246
$726$	0	0
$727$	1.13596e6	0.0797129	0.0398564	−	0.999205i	$-0.487310\pi$
$727$	1.13596e6	0.0797129	0.0398564	+	0.999205i	$0.487310\pi$
$728$	0	0
$729$	−1.09857e7	−0.765613
$730$	0	0
$731$	−1.72442e7	−1.19357
$732$	0	0
$733$	2.50171e7	1.71980	0.859898	−	0.510466i	$-0.170527\pi$
$733$	2.50171e7	1.71980	0.859898	+	0.510466i	$0.170527\pi$
$734$	0	0
$735$	3.42544e6	0.233882
$736$	0	0
$737$	−4.96911e6	−0.336984
$738$	0	0
$739$	2.83852e7	1.91197	0.955985	−	0.293416i	$-0.0947921\pi$
$739$	2.83852e7	1.91197	0.955985	+	0.293416i	$0.0947921\pi$
$740$	0	0
$741$	2.23200e6	0.149331
$742$	0	0
$743$	−1.77193e7	−1.17754	−0.588769	−	0.808301i	$-0.700387\pi$
$743$	−1.77193e7	−1.17754	−0.588769	+	0.808301i	$0.700387\pi$
$744$	0	0
$745$	−1.66825e7	−1.10121
$746$	0	0
$747$	1.31398e7	0.861561
$748$	0	0
$749$	4.00074e6	0.260577
$750$	0	0
$751$	−528823.	−0.0342145	−0.0171073	−	0.999854i	$-0.505446\pi$
$751$	−528823.	−0.0342145	−0.0171073	+	0.999854i	$0.505446\pi$
$752$	0	0
$753$	−1.83636e7	−1.18024
$754$	0	0
$755$	−8.04327e6	−0.513529
$756$	0	0
$757$	1.09456e6	0.0694222	0.0347111	−	0.999397i	$-0.488949\pi$
$757$	1.09456e6	0.0694222	0.0347111	+	0.999397i	$0.488949\pi$
$758$	0	0
$759$	1.42067e7	0.895136
$760$	0	0
$761$	−1.33654e6	−0.0836607	−0.0418304	−	0.999125i	$-0.513319\pi$
$761$	−1.33654e6	−0.0836607	−0.0418304	+	0.999125i	$0.513319\pi$
$762$	0	0
$763$	5.75839e6	0.358088
$764$	0	0
$765$	−4.08979e7	−2.52666
$766$	0	0
$767$	−1.11080e7	−0.681786
$768$	0	0
$769$	−2.94041e6	−0.179305	−0.0896523	−	0.995973i	$-0.528576\pi$
$769$	−2.94041e6	−0.179305	−0.0896523	+	0.995973i	$0.528576\pi$
$770$	0	0
$771$	1.62132e7	0.982276
$772$	0	0
$773$	−6.93399e6	−0.417383	−0.208691	−	0.977982i	$-0.566920\pi$
$773$	−6.93399e6	−0.417383	−0.208691	+	0.977982i	$0.566920\pi$
$774$	0	0
$775$	3.63977e6	0.217680
$776$	0	0
$777$	4.91105e6	0.291824
$778$	0	0
$779$	−6.58746e6	−0.388933
$780$	0	0
$781$	5.21821e6	0.306122
$782$	0	0
$783$	3.19464e7	1.86216
$784$	0	0
$785$	−7.73727e6	−0.448140
$786$	0	0
$787$	7.39059e6	0.425346	0.212673	−	0.977123i	$-0.431783\pi$
$787$	7.39059e6	0.425346	0.212673	+	0.977123i	$0.431783\pi$
$788$	0	0
$789$	4.33790e7	2.48077
$790$	0	0
$791$	1.21506e7	0.690491
$792$	0	0
$793$	5.09660e6	0.287804
$794$	0	0
$795$	3.27177e7	1.83597
$796$	0	0
$797$	−3.35937e7	−1.87332	−0.936659	−	0.350242i	$-0.886099\pi$
$797$	−3.35937e7	−1.87332	−0.936659	+	0.350242i	$0.886099\pi$
$798$	0	0
$799$	3.06647e7	1.69930
$800$	0	0
$801$	4.94276e7	2.72200
$802$	0	0
$803$	4.04533e6	0.221394
$804$	0	0
$805$	9.34890e6	0.508476
$806$	0	0
$807$	−4.86043e7	−2.62719
$808$	0	0
$809$	4.75845e6	0.255619	0.127810	−	0.991799i	$-0.459205\pi$
$809$	4.75845e6	0.255619	0.127810	+	0.991799i	$0.459205\pi$
$810$	0	0
$811$	−7.12488e6	−0.380387	−0.190193	−	0.981747i	$-0.560912\pi$
$811$	−7.12488e6	−0.380387	−0.190193	+	0.981747i	$0.560912\pi$
$812$	0	0
$813$	−2.93096e7	−1.55519
$814$	0	0
$815$	−2.26629e7	−1.19515
$816$	0	0
$817$	4.12933e6	0.216434
$818$	0	0
$819$	5.40791e6	0.281721
$820$	0	0
$821$	4.59860e6	0.238105	0.119052	−	0.992888i	$-0.462014\pi$
$821$	4.59860e6	0.238105	0.119052	+	0.992888i	$0.462014\pi$
$822$	0	0
$823$	−2.21796e7	−1.14144	−0.570722	−	0.821144i	$-0.693336\pi$
$823$	−2.21796e7	−1.14144	−0.570722	+	0.821144i	$0.693336\pi$
$824$	0	0
$825$	−1.54664e6	−0.0791141
$826$	0	0
$827$	−3.75259e6	−0.190795	−0.0953975	−	0.995439i	$-0.530412\pi$
$827$	−3.75259e6	−0.190795	−0.0953975	+	0.995439i	$0.530412\pi$
$828$	0	0
$829$	9.26844e6	0.468404	0.234202	−	0.972188i	$-0.424752\pi$
$829$	9.26844e6	0.468404	0.234202	+	0.972188i	$0.424752\pi$
$830$	0	0
$831$	−5.46225e7	−2.74390
$832$	0	0
$833$	−3.73588e6	−0.186544
$834$	0	0
$835$	−2.96906e7	−1.47368
$836$	0	0
$837$	−6.50775e7	−3.21083
$838$	0	0
$839$	2.88364e7	1.41428	0.707141	−	0.707073i	$-0.249985\pi$
$839$	2.88364e7	1.41428	0.707141	+	0.707073i	$0.249985\pi$
$840$	0	0
$841$	−330491.	−0.0161127
$842$	0	0
$843$	735960.	0.0356685
$844$	0	0
$845$	−1.68791e7	−0.813217
$846$	0	0
$847$	−6.89870e6	−0.330414
$848$	0	0
$849$	1.26364e7	0.601666
$850$	0	0
$851$	1.34035e7	0.634446
$852$	0	0
$853$	3.97102e7	1.86866	0.934329	−	0.356411i	$-0.116000\pi$
$853$	3.97102e7	1.86866	0.934329	+	0.356411i	$0.116000\pi$
$854$	0	0
$855$	9.79353e6	0.458167
$856$	0	0
$857$	−2.42053e7	−1.12579	−0.562897	−	0.826527i	$-0.690313\pi$
$857$	−2.42053e7	−1.12579	−0.562897	+	0.826527i	$0.690313\pi$
$858$	0	0
$859$	−1.20350e7	−0.556497	−0.278248	−	0.960509i	$-0.589754\pi$
$859$	−1.20350e7	−0.556497	−0.278248	+	0.960509i	$0.589754\pi$
$860$	0	0
$861$	−2.36666e7	−1.08800
$862$	0	0
$863$	1.84727e7	0.844313	0.422157	−	0.906523i	$-0.361273\pi$
$863$	1.84727e7	0.844313	0.422157	+	0.906523i	$0.361273\pi$
$864$	0	0
$865$	1.94882e7	0.885590
$866$	0	0
$867$	2.73509e7	1.23573
$868$	0	0
$869$	−2.25841e6	−0.101450
$870$	0	0
$871$	−7.65494e6	−0.341898
$872$	0	0
$873$	−6.23815e7	−2.77026
$874$	0	0
$875$	−9.01446e6	−0.398034
$876$	0	0
$877$	−1.26802e7	−0.556706	−0.278353	−	0.960479i	$-0.589788\pi$
$877$	−1.26802e7	−0.556706	−0.278353	+	0.960479i	$0.589788\pi$
$878$	0	0
$879$	−4.76813e7	−2.08150
$880$	0	0
$881$	−2.82766e7	−1.22740	−0.613701	−	0.789538i	$-0.710320\pi$
$881$	−2.82766e7	−1.22740	−0.613701	+	0.789538i	$0.710320\pi$
$882$	0	0
$883$	−2.31253e7	−0.998127	−0.499064	−	0.866565i	$-0.666323\pi$
$883$	−2.31253e7	−0.998127	−0.499064	+	0.866565i	$0.666323\pi$
$884$	0	0
$885$	−7.22710e7	−3.10174
$886$	0	0
$887$	−5.07655e6	−0.216650	−0.108325	−	0.994116i	$-0.534549\pi$
$887$	−5.07655e6	−0.216650	−0.108325	+	0.994116i	$0.534549\pi$
$888$	0	0
$889$	−5.80438e6	−0.246321
$890$	0	0
$891$	1.02442e7	0.432298
$892$	0	0
$893$	−7.34304e6	−0.308140
$894$	0	0
$895$	7.82814e6	0.326664
$896$	0	0
$897$	2.18855e7	0.908188
$898$	0	0
$899$	−4.11098e7	−1.69647
$900$	0	0
$901$	−3.56828e7	−1.46436
$902$	0	0
$903$	1.48353e7	0.605450
$904$	0	0
$905$	6.01836e6	0.244262
$906$	0	0
$907$	1.81858e6	0.0734029	0.0367015	−	0.999326i	$-0.488315\pi$
$907$	1.81858e6	0.0734029	0.0367015	+	0.999326i	$0.488315\pi$
$908$	0	0
$909$	5.79490e7	2.32614
$910$	0	0
$911$	−2.78317e7	−1.11108	−0.555539	−	0.831491i	$-0.687488\pi$
$911$	−2.78317e7	−1.11108	−0.555539	+	0.831491i	$0.687488\pi$
$912$	0	0
$913$	3.71607e6	0.147539
$914$	0	0
$915$	3.31595e7	1.30935
$916$	0	0
$917$	−1.38333e7	−0.543253
$918$	0	0
$919$	−3.51945e7	−1.37463	−0.687316	−	0.726359i	$-0.741211\pi$
$919$	−3.51945e7	−1.37463	−0.687316	+	0.726359i	$0.741211\pi$
$920$	0	0
$921$	4.40724e7	1.71206
$922$	0	0
$923$	8.03868e6	0.310585
$924$	0	0
$925$	−1.45920e6	−0.0560738
$926$	0	0
$927$	2.15070e7	0.822015
$928$	0	0
$929$	1.41933e6	0.0539565	0.0269783	−	0.999636i	$-0.491412\pi$
$929$	1.41933e6	0.0539565	0.0269783	+	0.999636i	$0.491412\pi$
$930$	0	0
$931$	894604.	0.0338265
$932$	0	0
$933$	−2.69098e7	−1.01206
$934$	0	0
$935$	−1.15664e7	−0.432681
$936$	0	0
$937$	−4.77633e6	−0.177724	−0.0888618	−	0.996044i	$-0.528323\pi$
$937$	−4.77633e6	−0.177724	−0.0888618	+	0.996044i	$0.528323\pi$
$938$	0	0
$939$	2.90785e7	1.07624
$940$	0	0
$941$	7.25077e6	0.266938	0.133469	−	0.991053i	$-0.457388\pi$
$941$	7.25077e6	0.266938	0.133469	+	0.991053i	$0.457388\pi$
$942$	0	0
$943$	−6.45922e7	−2.36538
$944$	0	0
$945$	1.81976e7	0.662878
$946$	0	0
$947$	−5.13887e6	−0.186206	−0.0931028	−	0.995656i	$-0.529679\pi$
$947$	−5.13887e6	−0.186206	−0.0931028	+	0.995656i	$0.529679\pi$
$948$	0	0
$949$	6.23185e6	0.224622
$950$	0	0
$951$	4.93208e6	0.176839
$952$	0	0
$953$	1.22957e7	0.438553	0.219276	−	0.975663i	$-0.429630\pi$
$953$	1.22957e7	0.438553	0.219276	+	0.975663i	$0.429630\pi$
$954$	0	0
$955$	940704.	0.0333768
$956$	0	0
$957$	1.74687e7	0.616568
$958$	0	0
$959$	−1.31696e7	−0.462407
$960$	0	0
$961$	5.51151e7	1.92514
$962$	0	0
$963$	4.10943e7	1.42796
$964$	0	0
$965$	4.56340e7	1.57750
$966$	0	0
$967$	−1.91017e7	−0.656911	−0.328455	−	0.944520i	$-0.606528\pi$
$967$	−1.91017e7	−0.656911	−0.328455	+	0.944520i	$0.606528\pi$
$968$	0	0
$969$	−1.58380e7	−0.541864
$970$	0	0
$971$	5.17949e6	0.176295	0.0881473	−	0.996107i	$-0.471905\pi$
$971$	5.17949e6	0.176295	0.0881473	+	0.996107i	$0.471905\pi$
$972$	0	0
$973$	−1.18529e7	−0.401369
$974$	0	0
$975$	−2.38261e6	−0.0802677
$976$	0	0
$977$	−3.66564e6	−0.122861	−0.0614304	−	0.998111i	$-0.519566\pi$
$977$	−3.66564e6	−0.122861	−0.0614304	+	0.998111i	$0.519566\pi$
$978$	0	0
$979$	1.39787e7	0.466132
$980$	0	0
$981$	5.91483e7	1.96232
$982$	0	0
$983$	4.05592e7	1.33877	0.669384	−	0.742916i	$-0.266558\pi$
$983$	4.05592e7	1.33877	0.669384	+	0.742916i	$0.266558\pi$
$984$	0	0
$985$	−7.77734e6	−0.255411
$986$	0	0
$987$	−2.63811e7	−0.861987
$988$	0	0
$989$	4.04895e7	1.31629
$990$	0	0
$991$	−2.01285e6	−0.0651068	−0.0325534	−	0.999470i	$-0.510364\pi$
$991$	−2.01285e6	−0.0651068	−0.0325534	+	0.999470i	$0.510364\pi$
$992$	0	0
$993$	−5.25792e6	−0.169216
$994$	0	0
$995$	−3.99291e7	−1.27859
$996$	0	0
$997$	−1.05641e6	−0.0336585	−0.0168293	−	0.999858i	$-0.505357\pi$
$997$	−1.05641e6	−0.0336585	−0.0168293	+	0.999858i	$0.505357\pi$
$998$	0	0
$999$	2.60898e7	0.827099

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.6.a.i.1.5	✓	5
4.3	odd	2		224.6.a.j.1.1	yes	5
8.3	odd	2		448.6.a.be.1.5		5
8.5	even	2		448.6.a.bf.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.6.a.i.1.5	✓	5	1.1	even	1	trivial
224.6.a.j.1.1	yes	5	4.3	odd	2
448.6.a.be.1.5		5	8.3	odd	2
448.6.a.bf.1.1		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)

\(f(q)\)	\(=\)	\(q+27.3187 q^{3} +52.2232 q^{5} +49.0000 q^{7} +503.312 q^{9} +142.342 q^{11} +219.279 q^{13} +1426.67 q^{15} -1555.97 q^{17} +372.596 q^{19} +1338.62 q^{21} +3653.43 q^{23} -397.737 q^{25} +7111.38 q^{27} +4492.29 q^{29} -9151.19 q^{31} +3888.60 q^{33} +2558.94 q^{35} +3668.75 q^{37} +5990.41 q^{39} -17679.9 q^{41} +11082.6 q^{43} +26284.5 q^{45} -19707.8 q^{47} +2401.00 q^{49} -42507.0 q^{51} +22932.9 q^{53} +7433.55 q^{55} +10178.9 q^{57} -50657.1 q^{59} +23242.6 q^{61} +24662.3 q^{63} +11451.4 q^{65} -34909.7 q^{67} +99807.0 q^{69} +36659.7 q^{71} +28419.8 q^{73} -10865.7 q^{75} +6974.75 q^{77} -15866.1 q^{79} +71968.9 q^{81} +26106.6 q^{83} -81257.6 q^{85} +122724. q^{87} +98204.8 q^{89} +10744.7 q^{91} -249999. q^{93} +19458.2 q^{95} -123942. q^{97} +71642.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 10 q^{3} + 36 q^{5} + 245 q^{7} + 637 q^{9} + 116 q^{11} + 40 q^{13} + 16 q^{15} - 402 q^{17} - 3582 q^{19} - 490 q^{21} + 472 q^{23} + 9615 q^{25} + 356 q^{27} + 4754 q^{29} - 10500 q^{31} + 15864 q^{33}+ \cdots - 260236 q^{99}+O(q^{100}) \) 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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