LMFDB - Embedded newform 1216.3.e.p.1025.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1216,3,Mod(1025,1216)] 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("1216.1025"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1216, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1216.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [20,0,0,0,0,0,0,0,-52,0,0,0,0,0,0,0,56] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$33.1336001462$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 8 x^{19} - 196 x^{18} + 1676 x^{17} + 16346 x^{16} - 161824 x^{15} - 667200 x^{14} + \cdots + 1135285065792 $ x^20 - 8x^19 - 196x^18 + 1676x^17 + 16346x^16 - 161824x^15 - 667200x^14 + 8850712x^13 + 9918758x^12 - 288078272x^11 + 224256138x^10 + 5383493580x^9 - 11116604703x^8 - 53755955160x^7 + 166723956672x^6 + 262105513072x^5 - 1053812084224x^4 - 855143026944x^3 + 3318150008320x^2 + 3262428344448*x + 1135285065792
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{27} $
Twist minimal:	no (minimal twist has level 608)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1025.5
Root			$-6.67959 - 1.97398i$ of defining polynomial
Character	$\chi$	$=$	1216.1025
Dual form			1216.3.e.p.1025.15

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.94796i q^{3} +7.51133 q^{5} -11.4311 q^{7} -6.58642 q^{9} +14.4687 q^{11} -23.9902i q^{13} -29.6545i q^{15} +26.8400 q^{17} +(-13.2008 + 13.6652i) q^{19} +45.1296i q^{21} +0.456250 q^{23} +31.4201 q^{25} -9.52875i q^{27} +7.85645i q^{29} -16.0928i q^{31} -57.1220i q^{33} -85.8628 q^{35} +12.5477i q^{37} -94.7126 q^{39} -48.5303i q^{41} +30.5237 q^{43} -49.4728 q^{45} -44.3643 q^{47} +81.6702 q^{49} -105.963i q^{51} -48.7157i q^{53} +108.679 q^{55} +(53.9497 + 52.1164i) q^{57} +43.7553i q^{59} -3.64259 q^{61} +75.2900 q^{63} -180.199i q^{65} -75.4361i q^{67} -1.80126i q^{69} -19.4733i q^{71} -83.0871 q^{73} -124.045i q^{75} -165.394 q^{77} +73.9522i q^{79} -96.8969 q^{81} -4.27148 q^{83} +201.604 q^{85} +31.0170 q^{87} -105.422i q^{89} +274.235i q^{91} -63.5337 q^{93} +(-99.1559 + 102.644i) q^{95} +177.628i q^{97} -95.2971 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 52 q^{9} + 56 q^{17} + 36 q^{25} - 64 q^{45} + 332 q^{49} + 88 q^{57} + 32 q^{61} - 152 q^{73} - 360 q^{77} - 476 q^{81} + 552 q^{85} - 336 q^{93}+O(q^{100}) $ 20 * q - 52 * q^9 + 56 * q^17 + 36 * q^25 - 64 * q^45 + 332 * q^49 + 88 * q^57 + 32 * q^61 - 152 * q^73 - 360 * q^77 - 476 * q^81 + 552 * q^85 - 336 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$.

$n$	$191$	$705$	$837$
$\chi(n)$	$1$	$-1$	$1$

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
$3$		−	3.94796i		−	1.31599i	−0.753023	−	0.657994i	$-0.771405\pi$
$3$		−	3.94796i		−	1.31599i	0.753023	−	0.657994i	$-0.228595\pi$
$4$		0			0
$5$	7.51133			1.50227			0.751133	−	0.660151i	$-0.229508\pi$
$5$	7.51133			1.50227			0.751133	+	0.660151i	$0.229508\pi$
$6$		0			0
$7$	−11.4311			−1.63301			−0.816507	−	0.577335i	$-0.804093\pi$
$7$	−11.4311			−1.63301			−0.816507	+	0.577335i	$0.804093\pi$
$8$		0			0
$9$	−6.58642			−0.731824
$10$		0			0
$11$	14.4687			1.31534			0.657670	−	0.753307i	$-0.271542\pi$
$11$	14.4687			1.31534			0.657670	+	0.753307i	$0.271542\pi$
$12$		0			0
$13$		−	23.9902i		−	1.84540i	−0.385516	−	0.922701i	$-0.625976\pi$
$13$		−	23.9902i		−	1.84540i	0.385516	−	0.922701i	$-0.374024\pi$
$14$		0			0
$15$		−	29.6545i		−	1.97696i
$16$		0			0
$17$	26.8400			1.57882			0.789412	−	0.613864i	$-0.210386\pi$
$17$	26.8400			1.57882			0.789412	+	0.613864i	$0.210386\pi$
$18$		0			0
$19$	−13.2008	+	13.6652i	−0.694781	+	0.719221i
$19$	−13.2008	+	13.6652i	−0.694781	+	0.719221i
$20$		0			0
$21$			45.1296i			2.14903i
$22$		0			0
$23$	0.456250			0.0198369			0.00991847	−	0.999951i	$-0.496843\pi$
$23$	0.456250			0.0198369			0.00991847	+	0.999951i	$0.496843\pi$
$24$		0			0
$25$	31.4201			1.25680
$26$		0			0
$27$		−	9.52875i		−	0.352916i
$28$		0			0
$29$			7.85645i			0.270912i	0.990783	+	0.135456i	$0.0432500\pi$
$29$			7.85645i			0.270912i	−0.990783	+	0.135456i	$0.956750\pi$
$30$		0			0
$31$		−	16.0928i		−	0.519122i	−0.965727	−	0.259561i	$-0.916422\pi$
$31$		−	16.0928i		−	0.519122i	0.965727	−	0.259561i	$-0.0835778\pi$
$32$		0			0
$33$		−	57.1220i		−	1.73097i
$34$		0			0
$35$	−85.8628			−2.45322
$36$		0			0
$37$			12.5477i			0.339126i	0.985519	+	0.169563i	$0.0542356\pi$
$37$			12.5477i			0.339126i	−0.985519	+	0.169563i	$0.945764\pi$
$38$		0			0
$39$	−94.7126			−2.42853
$40$		0			0
$41$		−	48.5303i		−	1.18367i	−0.806061	−	0.591833i	$-0.798404\pi$
$41$		−	48.5303i		−	1.18367i	0.806061	−	0.591833i	$-0.201596\pi$
$42$		0			0
$43$	30.5237			0.709853			0.354927	−	0.934894i	$-0.384506\pi$
$43$	30.5237			0.709853			0.354927	+	0.934894i	$0.384506\pi$
$44$		0			0
$45$	−49.4728			−1.09939
$46$		0			0
$47$	−44.3643			−0.943921			−0.471961	−	0.881620i	$-0.656454\pi$
$47$	−44.3643			−0.943921			−0.471961	+	0.881620i	$0.656454\pi$
$48$		0			0
$49$	81.6702			1.66674
$50$		0			0
$51$		−	105.963i		−	2.07771i
$52$		0			0
$53$		−	48.7157i		−	0.919164i	−0.888135	−	0.459582i	$-0.847999\pi$
$53$		−	48.7157i		−	0.919164i	0.888135	−	0.459582i	$-0.152001\pi$
$54$		0			0
$55$	108.679			1.97599
$56$		0			0
$57$	53.9497	+	52.1164i	0.946486	+	0.914324i
$58$		0			0
$59$			43.7553i			0.741615i	0.928710	+	0.370807i	$0.120919\pi$
$59$			43.7553i			0.741615i	−0.928710	+	0.370807i	$0.879081\pi$
$60$		0			0
$61$	−3.64259			−0.0597146			−0.0298573	−	0.999554i	$-0.509505\pi$
$61$	−3.64259			−0.0597146			−0.0298573	+	0.999554i	$0.509505\pi$
$62$		0			0
$63$	75.2900			1.19508
$64$		0			0
$65$		−	180.199i		−	2.77229i
$66$		0			0
$67$		−	75.4361i		−	1.12591i	−0.826487	−	0.562956i	$-0.809664\pi$
$67$		−	75.4361i		−	1.12591i	0.826487	−	0.562956i	$-0.190336\pi$
$68$		0			0
$69$		−	1.80126i		−	0.0261052i
$70$		0			0
$71$		−	19.4733i		−	0.274272i	−0.990552	−	0.137136i	$-0.956210\pi$
$71$		−	19.4733i		−	0.274272i	0.990552	−	0.137136i	$-0.0437897\pi$
$72$		0			0
$73$	−83.0871			−1.13818			−0.569090	−	0.822275i	$-0.692704\pi$
$73$	−83.0871			−1.13818			−0.569090	+	0.822275i	$0.692704\pi$
$74$		0			0
$75$		−	124.045i		−	1.65394i
$76$		0			0
$77$	−165.394			−2.14797
$78$		0			0
$79$			73.9522i			0.936104i	0.883701	+	0.468052i	$0.155044\pi$
$79$			73.9522i			0.936104i	−0.883701	+	0.468052i	$0.844956\pi$
$80$		0			0
$81$	−96.8969			−1.19626
$82$		0			0
$83$	−4.27148			−0.0514637			−0.0257318	−	0.999669i	$-0.508192\pi$
$83$	−4.27148			−0.0514637			−0.0257318	+	0.999669i	$0.508192\pi$
$84$		0			0
$85$	201.604			2.37181
$86$		0			0
$87$	31.0170			0.356517
$88$		0			0
$89$		−	105.422i		−	1.18452i	−0.805747	−	0.592260i	$-0.798236\pi$
$89$		−	105.422i		−	1.18452i	0.805747	−	0.592260i	$-0.201764\pi$
$90$		0			0
$91$			274.235i			3.01357i
$92$		0			0
$93$	−63.5337			−0.683158
$94$		0			0
$95$	−99.1559	+	102.644i	−1.04375	+	1.08046i
$96$		0			0
$97$			177.628i			1.83122i	0.402067	+	0.915610i	$0.368292\pi$
$97$			177.628i			1.83122i	−0.402067	+	0.915610i	$0.631708\pi$
$98$		0			0
$99$	−95.2971			−0.962597
$100$		0			0
$101$	−91.6325			−0.907253			−0.453626	−	0.891192i	$-0.649870\pi$
$101$	−91.6325			−0.907253			−0.453626	+	0.891192i	$0.649870\pi$
$102$		0			0
$103$		−	193.589i		−	1.87951i	−0.341852	−	0.939754i	$-0.611054\pi$
$103$		−	193.589i		−	1.87951i	0.341852	−	0.939754i	$-0.388946\pi$
$104$		0			0
$105$			338.983i			3.22841i
$106$		0			0
$107$		−	47.9868i		−	0.448475i	−0.974535	−	0.224238i	$-0.928011\pi$
$107$		−	47.9868i		−	0.448475i	0.974535	−	0.224238i	$-0.0719892\pi$
$108$		0			0
$109$			186.636i			1.71225i	0.516766	+	0.856127i	$0.327136\pi$
$109$			186.636i			1.71225i	−0.516766	+	0.856127i	$0.672864\pi$
$110$		0			0
$111$	49.5377			0.446285
$112$		0			0
$113$			10.9318i			0.0967415i	0.998829	+	0.0483708i	$0.0154029\pi$
$113$			10.9318i			0.0967415i	−0.998829	+	0.0483708i	$0.984597\pi$
$114$		0			0
$115$	3.42704			0.0298004
$116$		0			0
$117$			158.010i			1.35051i
$118$		0			0
$119$	−306.811			−2.57824
$120$		0			0
$121$	88.3441			0.730117
$122$		0			0
$123$	−191.596			−1.55769
$124$		0			0
$125$	48.2236			0.385789
$126$		0			0
$127$		−	44.8528i		−	0.353171i	−0.984285	−	0.176586i	$-0.943495\pi$
$127$		−	44.8528i		−	0.353171i	0.984285	−	0.176586i	$-0.0565053\pi$
$128$		0			0
$129$		−	120.506i		−	0.934158i
$130$		0			0
$131$	101.922			0.778030			0.389015	−	0.921231i	$-0.372815\pi$
$131$	101.922			0.778030			0.389015	+	0.921231i	$0.372815\pi$
$132$		0			0
$133$	150.900	−	156.208i	1.13459	−	1.17450i
$134$		0			0
$135$		−	71.5736i		−	0.530175i
$136$		0			0
$137$	141.158			1.03035			0.515174	−	0.857086i	$-0.327727\pi$
$137$	141.158			1.03035			0.515174	+	0.857086i	$0.327727\pi$
$138$		0			0
$139$	14.0165			0.100838			0.0504192	−	0.998728i	$-0.483944\pi$
$139$	14.0165			0.100838			0.0504192	+	0.998728i	$0.483944\pi$
$140$		0			0
$141$			175.149i			1.24219i
$142$		0			0
$143$		−	347.108i		−	2.42733i
$144$		0			0
$145$			59.0124i			0.406982i
$146$		0			0
$147$		−	322.431i		−	2.19341i
$148$		0			0
$149$	−202.051			−1.35605			−0.678025	−	0.735039i	$-0.737164\pi$
$149$	−202.051			−1.35605			−0.678025	+	0.735039i	$0.737164\pi$
$150$		0			0
$151$		−	8.83096i		−	0.0584831i	−0.999572	−	0.0292416i	$-0.990691\pi$
$151$		−	8.83096i		−	0.0584831i	0.999572	−	0.0292416i	$-0.00930921\pi$
$152$		0			0
$153$	−176.779			−1.15542
$154$		0			0
$155$		−	120.878i		−	0.779859i
$156$		0			0
$157$	115.182			0.733644			0.366822	−	0.930291i	$-0.380446\pi$
$157$	115.182			0.733644			0.366822	+	0.930291i	$0.380446\pi$
$158$		0			0
$159$	−192.328			−1.20961
$160$		0			0
$161$	−5.21544			−0.0323940
$162$		0			0
$163$	−22.2426			−0.136458			−0.0682289	−	0.997670i	$-0.521735\pi$
$163$	−22.2426			−0.136458			−0.0682289	+	0.997670i	$0.521735\pi$
$164$		0			0
$165$		−	429.062i		−	2.60038i
$166$		0			0
$167$		−	59.2364i		−	0.354709i	−0.984147	−	0.177355i	$-0.943246\pi$
$167$		−	59.2364i		−	0.354709i	0.984147	−	0.177355i	$-0.0567539\pi$
$168$		0			0
$169$	−406.531			−2.40551
$170$		0			0
$171$	86.9462	−	90.0047i	0.508457	−	0.526343i
$172$		0			0
$173$			219.813i			1.27059i	0.772268	+	0.635296i	$0.219122\pi$
$173$			219.813i			1.27059i	−0.772268	+	0.635296i	$0.780878\pi$
$174$		0			0
$175$	−359.167			−2.05238
$176$		0			0
$177$	172.744			0.975956
$178$		0			0
$179$			155.441i			0.868384i	0.900820	+	0.434192i	$0.142966\pi$
$179$			155.441i			0.868384i	−0.900820	+	0.434192i	$0.857034\pi$
$180$		0			0
$181$		−	63.4574i		−	0.350593i	−0.984516	−	0.175297i	$-0.943912\pi$
$181$		−	63.4574i		−	0.350593i	0.984516	−	0.175297i	$-0.0560885\pi$
$182$		0			0
$183$			14.3808i			0.0785836i
$184$		0			0
$185$			94.2496i			0.509457i
$186$		0			0
$187$	388.341			2.07669
$188$		0			0
$189$			108.924i			0.576318i
$190$		0			0
$191$	212.981			1.11509			0.557543	−	0.830148i	$-0.311744\pi$
$191$	212.981			1.11509			0.557543	+	0.830148i	$0.311744\pi$
$192$		0			0
$193$		−	94.4626i		−	0.489443i	−0.969593	−	0.244722i	$-0.921303\pi$
$193$		−	94.4626i		−	0.489443i	0.969593	−	0.244722i	$-0.0786966\pi$
$194$		0			0
$195$	−711.417			−3.64829
$196$		0			0
$197$	−0.440257			−0.00223481			−0.00111740	−	0.999999i	$-0.500356\pi$
$197$	−0.440257			−0.00223481			−0.00111740	+	0.999999i	$0.500356\pi$
$198$		0			0
$199$	79.7631			0.400820			0.200410	−	0.979712i	$-0.435773\pi$
$199$	79.7631			0.400820			0.200410	+	0.979712i	$0.435773\pi$
$200$		0			0
$201$	−297.819			−1.48169
$202$		0			0
$203$		−	89.8079i		−	0.442404i
$204$		0			0
$205$		−	364.527i		−	1.77818i
$206$		0			0
$207$	−3.00505			−0.0145171
$208$		0			0
$209$	−190.999	+	197.718i	−0.913873	+	0.946020i
$210$		0			0
$211$		−	180.350i		−	0.854737i	−0.904077	−	0.427369i	$-0.859441\pi$
$211$		−	180.350i		−	0.854737i	0.904077	−	0.427369i	$-0.140559\pi$
$212$		0			0
$213$	−76.8799			−0.360939
$214$		0			0
$215$	229.274			1.06639
$216$		0			0
$217$			183.958i			0.847733i
$218$		0			0
$219$			328.025i			1.49783i
$220$		0			0
$221$		−	643.898i		−	2.91356i
$222$		0			0
$223$			12.4050i			0.0556276i	0.999613	+	0.0278138i	$0.00885456\pi$
$223$			12.4050i			0.0556276i	−0.999613	+	0.0278138i	$0.991145\pi$
$224$		0			0
$225$	−206.946			−0.919759
$226$		0			0
$227$		−	63.0473i		−	0.277742i	−0.990311	−	0.138871i	$-0.955653\pi$
$227$		−	63.0473i		−	0.277742i	0.990311	−	0.138871i	$-0.0443473\pi$
$228$		0			0
$229$	−118.956			−0.519457			−0.259728	−	0.965682i	$-0.583633\pi$
$229$	−118.956			−0.519457			−0.259728	+	0.965682i	$0.583633\pi$
$230$		0			0
$231$			652.968i			2.82670i
$232$		0			0
$233$	−133.263			−0.571943			−0.285971	−	0.958238i	$-0.592316\pi$
$233$	−133.263			−0.571943			−0.285971	+	0.958238i	$0.592316\pi$
$234$		0			0
$235$	−333.235			−1.41802
$236$		0			0
$237$	291.961			1.23190
$238$		0			0
$239$	155.773			0.651768			0.325884	−	0.945410i	$-0.394338\pi$
$239$	155.773			0.651768			0.325884	+	0.945410i	$0.394338\pi$
$240$		0			0
$241$			206.792i			0.858059i	0.903290	+	0.429030i	$0.141144\pi$
$241$			206.792i			0.858059i	−0.903290	+	0.429030i	$0.858856\pi$
$242$		0			0
$243$			296.787i			1.22134i
$244$		0			0
$245$	613.452			2.50388
$246$		0			0
$247$	327.831	+	316.691i	1.32725	+	1.28215i
$248$		0			0
$249$			16.8637i			0.0677255i
$250$		0			0
$251$	61.1735			0.243719			0.121860	−	0.992547i	$-0.461114\pi$
$251$	61.1735			0.243719			0.121860	+	0.992547i	$0.461114\pi$
$252$		0			0
$253$	6.60135			0.0260923
$254$		0			0
$255$		−	795.926i		−	3.12128i
$256$		0			0
$257$		−	350.224i		−	1.36274i	−0.731939	−	0.681370i	$-0.761385\pi$
$257$		−	350.224i		−	1.36274i	0.731939	−	0.681370i	$-0.238615\pi$
$258$		0			0
$259$		−	143.434i		−	0.553797i
$260$		0			0
$261$		−	51.7459i		−	0.198260i
$262$		0			0
$263$	94.5466			0.359493			0.179746	−	0.983713i	$-0.442472\pi$
$263$	94.5466			0.359493			0.179746	+	0.983713i	$0.442472\pi$
$264$		0			0
$265$		−	365.920i		−	1.38083i
$266$		0			0
$267$	−416.203			−1.55881
$268$		0			0
$269$			69.4288i			0.258100i	0.991638	+	0.129050i	$0.0411927\pi$
$269$			69.4288i			0.258100i	−0.991638	+	0.129050i	$0.958807\pi$
$270$		0			0
$271$	−170.473			−0.629053			−0.314527	−	0.949249i	$-0.601846\pi$
$271$	−170.473			−0.629053			−0.314527	+	0.949249i	$0.601846\pi$
$272$		0			0
$273$	1082.67			3.96582
$274$		0			0
$275$	454.609			1.65312
$276$		0			0
$277$	350.498			1.26534			0.632668	−	0.774423i	$-0.281960\pi$
$277$	350.498			1.26534			0.632668	+	0.774423i	$0.281960\pi$
$278$		0			0
$279$			105.994i			0.379906i
$280$		0			0
$281$		−	5.69102i		−	0.0202527i	−0.999949	−	0.0101264i	$-0.996777\pi$
$281$		−	5.69102i		−	0.0202527i	0.999949	−	0.0101264i	$-0.00322338\pi$
$282$		0			0
$283$	496.812			1.75552			0.877759	−	0.479102i	$-0.159038\pi$
$283$	496.812			1.75552			0.877759	+	0.479102i	$0.159038\pi$
$284$		0			0
$285$	405.234	+	391.464i	1.42187	+	1.37356i
$286$		0			0
$287$			554.755i			1.93294i
$288$		0			0
$289$	431.386			1.49268
$290$		0			0
$291$	701.270			2.40986
$292$		0			0
$293$			169.911i			0.579901i	0.957042	+	0.289950i	$0.0936388\pi$
$293$			169.911i			0.579901i	−0.957042	+	0.289950i	$0.906361\pi$
$294$		0			0
$295$			328.660i			1.11410i
$296$		0			0
$297$		−	137.869i		−	0.464205i
$298$		0			0
$299$		−	10.9455i		−	0.0366071i
$300$		0			0
$301$	−348.919			−1.15920
$302$		0			0
$303$			361.762i			1.19393i
$304$		0			0
$305$	−27.3607			−0.0897072
$306$		0			0
$307$			17.6615i			0.0575293i	0.999586	+	0.0287647i	$0.00915734\pi$
$307$			17.6615i			0.0575293i	−0.999586	+	0.0287647i	$0.990843\pi$
$308$		0			0
$309$	−764.283			−2.47341
$310$		0			0
$311$	11.4249			0.0367359			0.0183680	−	0.999831i	$-0.494153\pi$
$311$	11.4249			0.0367359			0.0183680	+	0.999831i	$0.494153\pi$
$312$		0			0
$313$	150.411			0.480545			0.240273	−	0.970705i	$-0.422763\pi$
$313$	150.411			0.480545			0.240273	+	0.970705i	$0.422763\pi$
$314$		0			0
$315$	565.528			1.79533
$316$		0			0
$317$			472.665i			1.49106i	0.666474	+	0.745528i	$0.267803\pi$
$317$			472.665i			1.49106i	−0.666474	+	0.745528i	$0.732197\pi$
$318$		0			0
$319$			113.673i			0.356341i
$320$		0			0
$321$	−189.450			−0.590188
$322$		0			0
$323$	−354.311	+	366.774i	−1.09694	+	1.13552i
$324$		0			0
$325$		−	753.776i		−	2.31931i
$326$		0			0
$327$	736.831			2.25330
$328$		0			0
$329$	507.133			1.54144
$330$		0			0
$331$			461.495i			1.39424i	0.716953	+	0.697122i	$0.245536\pi$
$331$			461.495i			1.39424i	−0.716953	+	0.697122i	$0.754464\pi$
$332$		0			0
$333$		−	82.6440i		−	0.248180i
$334$		0			0
$335$		−	566.626i		−	1.69142i
$336$		0			0
$337$			212.915i			0.631795i	0.948793	+	0.315898i	$0.102306\pi$
$337$			212.915i			0.631795i	−0.948793	+	0.315898i	$0.897694\pi$
$338$		0			0
$339$	43.1583			0.127311
$340$		0			0
$341$		−	232.842i		−	0.682821i
$342$		0			0
$343$	−373.456			−1.08879
$344$		0			0
$345$		−	13.5298i		−	0.0392169i
$346$		0			0
$347$	4.69058			0.0135175			0.00675876	−	0.999977i	$-0.497849\pi$
$347$	4.69058			0.0135175			0.00675876	+	0.999977i	$0.497849\pi$
$348$		0			0
$349$	425.832			1.22015			0.610074	−	0.792345i	$-0.291140\pi$
$349$	425.832			1.22015			0.610074	+	0.792345i	$0.291140\pi$
$350$		0			0
$351$	−228.597			−0.651273
$352$		0			0
$353$	−203.358			−0.576084			−0.288042	−	0.957618i	$-0.593004\pi$
$353$	−203.358			−0.576084			−0.288042	+	0.957618i	$0.593004\pi$
$354$		0			0
$355$		−	146.270i		−	0.412030i
$356$		0			0
$357$			1211.28i			3.39294i
$358$		0			0
$359$	420.539			1.17142			0.585708	−	0.810522i	$-0.300816\pi$
$359$	420.539			1.17142			0.585708	+	0.810522i	$0.300816\pi$
$360$		0			0
$361$	−12.4755	−	360.784i	−0.0345582	−	0.999403i
$362$		0			0
$363$		−	348.779i		−	0.960825i
$364$		0			0
$365$	−624.095			−1.70985
$366$		0			0
$367$	−105.125			−0.286445			−0.143223	−	0.989691i	$-0.545746\pi$
$367$	−105.125			−0.286445			−0.143223	+	0.989691i	$0.545746\pi$
$368$		0			0
$369$			319.641i			0.866235i
$370$		0			0
$371$			556.874i			1.50101i
$372$		0			0
$373$			301.800i			0.809115i	0.914512	+	0.404558i	$0.132575\pi$
$373$			301.800i			0.809115i	−0.914512	+	0.404558i	$0.867425\pi$
$374$		0			0
$375$		−	190.385i		−	0.507693i
$376$		0			0
$377$	188.478			0.499942
$378$		0			0
$379$			541.837i			1.42965i	0.699303	+	0.714825i	$0.253494\pi$
$379$			541.837i			1.42965i	−0.699303	+	0.714825i	$0.746506\pi$
$380$		0			0
$381$	−177.077			−0.464769
$382$		0			0
$383$		−	547.453i		−	1.42938i	−0.699440	−	0.714691i	$-0.746567\pi$
$383$		−	547.453i		−	1.42938i	0.699440	−	0.714691i	$-0.253433\pi$
$384$		0			0
$385$	−1242.33			−3.22682
$386$		0			0
$387$	−201.042			−0.519487
$388$		0			0
$389$	727.065			1.86906			0.934531	−	0.355881i	$-0.115819\pi$
$389$	727.065			1.86906			0.934531	+	0.355881i	$0.115819\pi$
$390$		0			0
$391$	12.2457			0.0313190
$392$		0			0
$393$		−	402.384i		−	1.02388i
$394$		0			0
$395$			555.479i			1.40628i
$396$		0			0
$397$	30.7449			0.0774431			0.0387215	−	0.999250i	$-0.487671\pi$
$397$	30.7449			0.0774431			0.0387215	+	0.999250i	$0.487671\pi$
$398$		0			0
$399$	−616.705	−	595.749i	−1.54563	−	1.49310i
$400$		0			0
$401$		−	424.410i		−	1.05838i	−0.848504	−	0.529189i	$-0.822496\pi$
$401$		−	424.410i		−	1.05838i	0.848504	−	0.529189i	$-0.177504\pi$
$402$		0			0
$403$	−386.069			−0.957988
$404$		0			0
$405$	−727.825			−1.79710
$406$		0			0
$407$			181.549i			0.446065i
$408$		0			0
$409$			221.456i			0.541458i	0.962656	+	0.270729i	$0.0872648\pi$
$409$			221.456i			0.541458i	−0.962656	+	0.270729i	$0.912735\pi$
$410$		0			0
$411$		−	557.285i		−	1.35592i
$412$		0			0
$413$		−	500.171i		−	1.21107i
$414$		0			0
$415$	−32.0845			−0.0773121
$416$		0			0
$417$		−	55.3368i		−	0.132702i
$418$		0			0
$419$	292.225			0.697433			0.348717	−	0.937228i	$-0.386618\pi$
$419$	292.225			0.697433			0.348717	+	0.937228i	$0.386618\pi$
$420$		0			0
$421$		−	427.036i		−	1.01434i	−0.861847	−	0.507169i	$-0.830692\pi$
$421$		−	427.036i		−	1.01434i	0.861847	−	0.507169i	$-0.169308\pi$
$422$		0			0
$423$	292.202			0.690784
$424$		0			0
$425$	843.316			1.98427
$426$		0			0
$427$	41.6388			0.0975148
$428$		0			0
$429$	−1370.37			−3.19434
$430$		0			0
$431$			173.465i			0.402472i	0.979543	+	0.201236i	$0.0644958\pi$
$431$			173.465i			0.402472i	−0.979543	+	0.201236i	$0.935504\pi$
$432$		0			0
$433$		−	6.22957i		−	0.0143870i	−0.999974	−	0.00719350i	$-0.997710\pi$
$433$		−	6.22957i		−	0.0143870i	0.999974	−	0.00719350i	$-0.00228978\pi$
$434$		0			0
$435$	232.979			0.535584
$436$		0			0
$437$	−6.02288	+	6.23474i	−0.0137823	+	0.0142671i
$438$		0			0
$439$		−	312.023i		−	0.710758i	−0.934722	−	0.355379i	$-0.884352\pi$
$439$		−	312.023i		−	0.710758i	0.934722	−	0.355379i	$-0.115648\pi$
$440$		0			0
$441$	−537.914			−1.21976
$442$		0			0
$443$	779.260			1.75905			0.879526	−	0.475850i	$-0.157860\pi$
$443$	779.260			1.75905			0.879526	+	0.475850i	$0.157860\pi$
$444$		0			0
$445$		−	791.861i		−	1.77946i
$446$		0			0
$447$			797.691i			1.78454i
$448$		0			0
$449$		−	257.015i		−	0.572416i	−0.958167	−	0.286208i	$-0.907605\pi$
$449$		−	257.015i		−	0.572416i	0.958167	−	0.286208i	$-0.0923949\pi$
$450$		0			0
$451$		−	702.172i		−	1.55692i
$452$		0			0
$453$	−34.8643			−0.0769631
$454$		0			0
$455$			2059.87i			4.52718i
$456$		0			0
$457$	536.202			1.17331			0.586654	−	0.809838i	$-0.300445\pi$
$457$	536.202			1.17331			0.586654	+	0.809838i	$0.300445\pi$
$458$		0			0
$459$		−	255.752i		−	0.557193i
$460$		0			0
$461$	549.116			1.19114			0.595570	−	0.803303i	$-0.296926\pi$
$461$	549.116			1.19114			0.595570	+	0.803303i	$0.296926\pi$
$462$		0			0
$463$	531.777			1.14855			0.574273	−	0.818664i	$-0.305285\pi$
$463$	531.777			1.14855			0.574273	+	0.818664i	$0.305285\pi$
$464$		0			0
$465$	−477.222			−1.02628
$466$		0			0
$467$	−883.914			−1.89275			−0.946375	−	0.323070i	$-0.895285\pi$
$467$	−883.914			−1.89275			−0.946375	+	0.323070i	$0.895285\pi$
$468$		0			0
$469$			862.318i			1.83863i
$470$		0			0
$471$		−	454.734i		−	0.965466i
$472$		0			0
$473$	441.639			0.933697
$474$		0			0
$475$	−414.772	+	429.362i	−0.873204	+	0.903920i
$476$		0			0
$477$			320.862i			0.672666i
$478$		0			0
$479$	−434.287			−0.906653			−0.453327	−	0.891345i	$-0.649763\pi$
$479$	−434.287			−0.906653			−0.453327	+	0.891345i	$0.649763\pi$
$480$		0			0
$481$	301.021			0.625823
$482$		0			0
$483$			20.5904i			0.0426301i
$484$		0			0
$485$			1334.23i			2.75098i
$486$		0			0
$487$			420.157i			0.862746i	0.902174	+	0.431373i	$0.141971\pi$
$487$			420.157i			0.862746i	−0.902174	+	0.431373i	$0.858029\pi$
$488$		0			0
$489$			87.8131i			0.179577i
$490$		0			0
$491$	−148.458			−0.302359			−0.151179	−	0.988506i	$-0.548307\pi$
$491$	−148.458			−0.302359			−0.151179	+	0.988506i	$0.548307\pi$
$492$		0			0
$493$			210.867i			0.427722i
$494$		0			0
$495$	−715.808			−1.44608
$496$		0			0
$497$			222.601i			0.447890i
$498$		0			0
$499$	662.516			1.32769			0.663844	−	0.747871i	$-0.268924\pi$
$499$	662.516			1.32769			0.663844	+	0.747871i	$0.268924\pi$
$500$		0			0
$501$	−233.863			−0.466793
$502$		0			0
$503$	483.994			0.962214			0.481107	−	0.876662i	$-0.340235\pi$
$503$	483.994			0.962214			0.481107	+	0.876662i	$0.340235\pi$
$504$		0			0
$505$	−688.282			−1.36294
$506$		0			0
$507$			1604.97i			3.16562i
$508$		0			0
$509$		−	481.859i		−	0.946678i	−0.880880	−	0.473339i	$-0.843049\pi$
$509$		−	481.859i		−	0.946678i	0.880880	−	0.473339i	$-0.156951\pi$
$510$		0			0
$511$	949.778			1.85866
$512$		0			0
$513$	130.212	+	125.787i	0.253825	+	0.245200i
$514$		0			0
$515$		−	1454.11i		−	2.82352i
$516$		0			0
$517$	−641.895			−1.24158
$518$		0			0
$519$	867.812			1.67208
$520$		0			0
$521$		−	11.8425i		−	0.0227303i	−0.999935	−	0.0113651i	$-0.996382\pi$
$521$		−	11.8425i		−	0.0227303i	0.999935	−	0.0113651i	$-0.00361771\pi$
$522$		0			0
$523$		−	133.860i		−	0.255946i	−0.991778	−	0.127973i	$-0.959153\pi$
$523$		−	133.860i		−	0.255946i	0.991778	−	0.127973i	$-0.0408470\pi$
$524$		0			0
$525$			1417.98i			2.70091i
$526$		0			0
$527$		−	431.930i		−	0.819601i
$528$		0			0
$529$	−528.792			−0.999606
$530$		0			0
$531$		−	288.190i		−	0.542731i
$532$		0			0
$533$	−1164.25			−2.18434
$534$		0			0
$535$		−	360.445i		−	0.673729i
$536$		0			0
$537$	613.674			1.14278
$538$		0			0
$539$	1181.66			2.19233
$540$		0			0
$541$	926.200			1.71202			0.856008	−	0.516963i	$-0.172938\pi$
$541$	926.200			1.71202			0.856008	+	0.516963i	$0.172938\pi$
$542$		0			0
$543$	−250.528			−0.461377
$544$		0			0
$545$			1401.88i			2.57226i
$546$		0			0
$547$		−	408.806i		−	0.747360i	−0.927558	−	0.373680i	$-0.878096\pi$
$547$		−	408.806i		−	0.747360i	0.927558	−	0.373680i	$-0.121904\pi$
$548$		0			0
$549$	23.9916			0.0437005
$550$		0			0
$551$	−107.360	−	103.712i	−0.194846	−	0.188225i
$552$		0			0
$553$		−	845.355i		−	1.52867i
$554$		0			0
$555$	372.094			0.670439
$556$		0			0
$557$	−165.552			−0.297220			−0.148610	−	0.988896i	$-0.547480\pi$
$557$	−165.552			−0.297220			−0.148610	+	0.988896i	$0.547480\pi$
$558$		0			0
$559$		−	732.270i		−	1.30996i
$560$		0			0
$561$		−	1533.15i		−	2.73290i
$562$		0			0
$563$		−	318.882i		−	0.566397i	−0.959061	−	0.283199i	$-0.908604\pi$
$563$		−	318.882i		−	0.566397i	0.959061	−	0.283199i	$-0.0913956\pi$
$564$		0			0
$565$			82.1123i			0.145332i
$566$		0			0
$567$	1107.64			1.95351
$568$		0			0
$569$			601.270i			1.05671i	0.849022	+	0.528357i	$0.177192\pi$
$569$			601.270i			1.05671i	−0.849022	+	0.528357i	$0.822808\pi$
$570$		0			0
$571$	−649.481			−1.13745			−0.568723	−	0.822529i	$-0.692562\pi$
$571$	−649.481			−1.13745			−0.568723	+	0.822529i	$0.692562\pi$
$572$		0			0
$573$		−	840.843i		−	1.46744i
$574$		0			0
$575$	14.3354			0.0249312
$576$		0			0
$577$	335.898			0.582146			0.291073	−	0.956701i	$-0.405988\pi$
$577$	335.898			0.582146			0.291073	+	0.956701i	$0.405988\pi$
$578$		0			0
$579$	−372.935			−0.644102
$580$		0			0
$581$	48.8278			0.0840409
$582$		0			0
$583$		−	704.854i		−	1.20901i
$584$		0			0
$585$			1186.86i			2.02883i
$586$		0			0
$587$	−651.573			−1.11000			−0.555002	−	0.831849i	$-0.687283\pi$
$587$	−651.573			−1.11000			−0.555002	+	0.831849i	$0.687283\pi$
$588$		0			0
$589$	219.911	+	212.438i	0.373363	+	0.360676i
$590$		0			0
$591$			1.73812i			0.00294098i
$592$		0			0
$593$	−110.514			−0.186364			−0.0931822	−	0.995649i	$-0.529704\pi$
$593$	−110.514			−0.186364			−0.0931822	+	0.995649i	$0.529704\pi$
$594$		0			0
$595$	−2304.56			−3.87321
$596$		0			0
$597$		−	314.902i		−	0.527474i
$598$		0			0
$599$			767.215i			1.28083i	0.768031	+	0.640413i	$0.221237\pi$
$599$			767.215i			1.28083i	−0.768031	+	0.640413i	$0.778763\pi$
$600$		0			0
$601$			708.739i			1.17927i	0.807671	+	0.589633i	$0.200728\pi$
$601$			708.739i			1.17927i	−0.807671	+	0.589633i	$0.799272\pi$
$602$		0			0
$603$			496.853i			0.823969i
$604$		0			0
$605$	663.582			1.09683
$606$		0			0
$607$			779.388i			1.28400i	0.766705	+	0.642000i	$0.221895\pi$
$607$			779.388i			1.28400i	−0.766705	+	0.642000i	$0.778105\pi$
$608$		0			0
$609$	−354.558			−0.582198
$610$		0			0
$611$			1064.31i			1.74191i
$612$		0			0
$613$	744.842			1.21508			0.607538	−	0.794291i	$-0.292157\pi$
$613$	744.842			1.21508			0.607538	+	0.794291i	$0.292157\pi$
$614$		0			0
$615$	−1439.14			−2.34007
$616$		0			0
$617$	304.897			0.494161			0.247080	−	0.968995i	$-0.420529\pi$
$617$	304.897			0.494161			0.247080	+	0.968995i	$0.420529\pi$
$618$		0			0
$619$	764.804			1.23555			0.617774	−	0.786356i	$-0.288035\pi$
$619$	764.804			1.23555			0.617774	+	0.786356i	$0.288035\pi$
$620$		0			0
$621$		−	4.34749i		−	0.00700078i
$622$		0			0
$623$			1205.09i			1.93434i
$624$		0			0
$625$	−423.279			−0.677247
$626$		0			0
$627$	780.584	+	754.059i	1.24495	+	1.20265i
$628$		0			0
$629$			336.779i			0.535420i
$630$		0			0
$631$	−682.979			−1.08238			−0.541188	−	0.840902i	$-0.682025\pi$
$631$	−682.979			−1.08238			−0.541188	+	0.840902i	$0.682025\pi$
$632$		0			0
$633$	−712.014			−1.12482
$634$		0			0
$635$		−	336.904i		−	0.530557i
$636$		0			0
$637$		−	1959.29i		−	3.07580i
$638$		0			0
$639$			128.259i			0.200719i
$640$		0			0
$641$		−	1040.26i		−	1.62287i	−0.584445	−	0.811433i	$-0.698688\pi$
$641$		−	1040.26i		−	1.62287i	0.584445	−	0.811433i	$-0.301312\pi$
$642$		0			0
$643$	1097.65			1.70707			0.853535	−	0.521036i	$-0.174454\pi$
$643$	1097.65			1.70707			0.853535	+	0.521036i	$0.174454\pi$
$644$		0			0
$645$		−	905.163i		−	1.40335i
$646$		0			0
$647$	1024.20			1.58299			0.791497	−	0.611173i	$-0.209302\pi$
$647$	1024.20			1.58299			0.791497	+	0.611173i	$0.209302\pi$
$648$		0			0
$649$			633.083i			0.975475i
$650$		0			0
$651$	726.260			1.11561
$652$		0			0
$653$	996.126			1.52546			0.762731	−	0.646716i	$-0.223858\pi$
$653$	996.126			1.52546			0.762731	+	0.646716i	$0.223858\pi$
$654$		0			0
$655$	765.569			1.16881
$656$		0			0
$657$	547.246			0.832947
$658$		0			0
$659$			506.836i			0.769098i	0.923105	+	0.384549i	$0.125643\pi$
$659$			506.836i			0.769098i	−0.923105	+	0.384549i	$0.874357\pi$
$660$		0			0
$661$		−	891.671i		−	1.34897i	−0.738288	−	0.674486i	$-0.764365\pi$
$661$		−	891.671i		−	1.34897i	0.738288	−	0.674486i	$-0.235635\pi$
$662$		0			0
$663$	−2542.08			−3.83422
$664$		0			0
$665$	1133.46	−	1173.33i	1.70445	−	1.76441i
$666$		0			0
$667$			3.58450i			0.00537407i
$668$		0			0
$669$	48.9743			0.0732053
$670$		0			0
$671$	−52.7036			−0.0785449
$672$		0			0
$673$			493.557i			0.733368i	0.930346	+	0.366684i	$0.119507\pi$
$673$			493.557i			0.733368i	−0.930346	+	0.366684i	$0.880493\pi$
$674$		0			0
$675$		−	299.394i		−	0.443547i
$676$		0			0
$677$			689.996i			1.01920i	0.860412	+	0.509598i	$0.170206\pi$
$677$			689.996i			1.01920i	−0.860412	+	0.509598i	$0.829794\pi$
$678$		0			0
$679$		−	2030.49i		−	2.99041i
$680$		0			0
$681$	−248.909			−0.365505
$682$		0			0
$683$			1097.59i			1.60701i	0.595300	+	0.803503i	$0.297033\pi$
$683$			1097.59i			1.60701i	−0.595300	+	0.803503i	$0.702967\pi$
$684$		0			0
$685$	1060.28			1.54786
$686$		0			0
$687$			469.632i			0.683599i
$688$		0			0
$689$	−1168.70			−1.69623
$690$		0			0
$691$	−952.346			−1.37821			−0.689107	−	0.724659i	$-0.741997\pi$
$691$	−952.346			−1.37821			−0.689107	+	0.724659i	$0.741997\pi$
$692$		0			0
$693$	1089.35			1.57193
$694$		0			0
$695$	105.283			0.151486
$696$		0			0
$697$		−	1302.55i		−	1.86880i
$698$		0			0
$699$			526.116i			0.752670i
$700$		0			0
$701$	−563.636			−0.804046			−0.402023	−	0.915630i	$-0.631693\pi$
$701$	−563.636			−0.804046			−0.402023	+	0.915630i	$0.631693\pi$
$702$		0			0
$703$	−171.466	−	165.640i	−0.243906	−	0.235618i
$704$		0			0
$705$			1315.60i			1.86610i
$706$		0			0
$707$	1047.46			1.48156
$708$		0			0
$709$	−111.158			−0.156781			−0.0783904	−	0.996923i	$-0.524978\pi$
$709$	−111.158			−0.156781			−0.0783904	+	0.996923i	$0.524978\pi$
$710$		0			0
$711$		−	487.080i		−	0.685063i
$712$		0			0
$713$		−	7.34232i		−	0.0102978i
$714$		0			0
$715$		−	2607.24i		−	3.64650i
$716$		0			0
$717$		−	614.985i		−	0.857719i
$718$		0			0
$719$	183.695			0.255486			0.127743	−	0.991807i	$-0.459227\pi$
$719$	183.695			0.255486			0.127743	+	0.991807i	$0.459227\pi$
$720$		0			0
$721$			2212.94i			3.06926i
$722$		0			0
$723$	816.408			1.12920
$724$		0			0
$725$			246.851i			0.340484i
$726$		0			0
$727$	−559.944			−0.770212			−0.385106	−	0.922872i	$-0.625835\pi$
$727$	−559.944			−0.770212			−0.385106	+	0.922872i	$0.625835\pi$
$728$		0			0
$729$	299.631			0.411016
$730$		0			0
$731$	819.256			1.12073
$732$		0			0
$733$	607.258			0.828456			0.414228	−	0.910173i	$-0.364052\pi$
$733$	607.258			0.828456			0.414228	+	0.910173i	$0.364052\pi$
$734$		0			0
$735$		−	2421.88i		−	3.29508i
$736$		0			0
$737$		−	1091.46i		−	1.48096i
$738$		0			0
$739$	−1340.34			−1.81372			−0.906862	−	0.421428i	$-0.861529\pi$
$739$	−1340.34			−1.81372			−0.906862	+	0.421428i	$0.861529\pi$
$740$		0			0
$741$	1250.29	−	1294.27i	1.68729	−	1.74665i
$742$		0			0
$743$			1336.88i			1.79930i	0.436609	+	0.899651i	$0.356179\pi$
$743$			1336.88i			1.79930i	−0.436609	+	0.899651i	$0.643821\pi$
$744$		0			0
$745$	−1517.67			−2.03715
$746$		0			0
$747$	28.1338			0.0376623
$748$		0			0
$749$			548.543i			0.732367i
$750$		0			0
$751$		−	136.252i		−	0.181428i	−0.995877	−	0.0907138i	$-0.971085\pi$
$751$		−	136.252i		−	0.181428i	0.995877	−	0.0907138i	$-0.0289149\pi$
$752$		0			0
$753$		−	241.511i		−	0.320732i
$754$		0			0
$755$		−	66.3322i		−	0.0878573i
$756$		0			0
$757$	−152.519			−0.201478			−0.100739	−	0.994913i	$-0.532121\pi$
$757$	−152.519			−0.201478			−0.100739	+	0.994913i	$0.532121\pi$
$758$		0			0
$759$		−	26.0619i		−	0.0343371i
$760$		0			0
$761$	−961.843			−1.26392			−0.631960	−	0.775001i	$-0.717749\pi$
$761$	−961.843			−1.26392			−0.631960	+	0.775001i	$0.717749\pi$
$762$		0			0
$763$		−	2133.45i		−	2.79614i
$764$		0			0
$765$	−1327.85			−1.73575
$766$		0			0
$767$	1049.70			1.36858
$768$		0			0
$769$	848.788			1.10376			0.551878	−	0.833925i	$-0.313912\pi$
$769$	848.788			1.10376			0.551878	+	0.833925i	$0.313912\pi$
$770$		0			0
$771$	−1382.67			−1.79335
$772$		0			0
$773$		−	991.416i		−	1.28256i	−0.767308	−	0.641278i	$-0.778404\pi$
$773$		−	991.416i		−	1.28256i	0.767308	−	0.641278i	$-0.221596\pi$
$774$		0			0
$775$		−	505.637i		−	0.652434i
$776$		0			0
$777$	−566.270			−0.728791
$778$		0			0
$779$	663.177	+	640.641i	0.851318	+	0.822389i
$780$		0			0
$781$		−	281.754i		−	0.360761i
$782$		0			0
$783$	74.8621			0.0956094
$784$		0			0
$785$	865.171			1.10213
$786$		0			0
$787$		−	598.126i		−	0.760008i	−0.924985	−	0.380004i	$-0.875923\pi$
$787$		−	598.126i		−	0.760008i	0.924985	−	0.380004i	$-0.124077\pi$
$788$		0			0
$789$		−	373.266i		−	0.473088i
$790$		0			0
$791$		−	124.962i		−	0.157980i
$792$		0			0
$793$			87.3865i			0.110197i
$794$		0			0
$795$	−1444.64			−1.81716
$796$		0			0
$797$			144.141i			0.180855i	0.995903	+	0.0904275i	$0.0288233\pi$
$797$			144.141i			0.180855i	−0.995903	+	0.0904275i	$0.971177\pi$
$798$		0			0
$799$	−1190.74			−1.49028
$800$		0			0
$801$			694.354i			0.866860i
$802$		0			0
$803$	−1202.17			−1.49709
$804$		0			0
$805$	−39.1749			−0.0486644
$806$		0			0
$807$	274.102			0.339656
$808$		0			0
$809$	−271.105			−0.335111			−0.167556	−	0.985863i	$-0.553587\pi$
$809$	−271.105			−0.335111			−0.167556	+	0.985863i	$0.553587\pi$
$810$		0			0
$811$			1150.68i			1.41884i	0.704787	+	0.709419i	$0.251043\pi$
$811$			1150.68i			1.41884i	−0.704787	+	0.709419i	$0.748957\pi$
$812$		0			0
$813$			673.023i			0.827827i
$814$		0			0
$815$	−167.072			−0.204996
$816$		0			0
$817$	−402.938	+	417.112i	−0.493193	+	0.510541i
$818$		0			0
$819$		−	1806.22i		−	2.20540i
$820$		0			0
$821$	−851.160			−1.03674			−0.518368	−	0.855158i	$-0.673460\pi$
$821$	−851.160			−1.03674			−0.518368	+	0.855158i	$0.673460\pi$
$822$		0			0
$823$	−1074.72			−1.30586			−0.652928	−	0.757420i	$-0.726460\pi$
$823$	−1074.72			−1.30586			−0.652928	+	0.757420i	$0.726460\pi$
$824$		0			0
$825$		−	1794.78i		−	2.17549i
$826$		0			0
$827$			703.353i			0.850487i	0.905079	+	0.425243i	$0.139811\pi$
$827$			703.353i			0.850487i	−0.905079	+	0.425243i	$0.860189\pi$
$828$		0			0
$829$		−	62.9506i		−	0.0759356i	−0.999279	−	0.0379678i	$-0.987912\pi$
$829$		−	62.9506i		−	0.0759356i	0.999279	−	0.0379678i	$-0.0120884\pi$
$830$		0			0
$831$		−	1383.75i		−	1.66517i
$832$		0			0
$833$	2192.03			2.63148
$834$		0			0
$835$		−	444.945i		−	0.532868i
$836$		0			0
$837$	−153.344			−0.183207
$838$		0			0
$839$			745.845i			0.888969i	0.895787	+	0.444484i	$0.146613\pi$
$839$			745.845i			0.888969i	−0.895787	+	0.444484i	$0.853387\pi$
$840$		0			0
$841$	779.276			0.926607
$842$		0			0
$843$	−22.4679			−0.0266524
$844$		0			0
$845$	−3053.59			−3.61372
$846$		0			0
$847$	−1009.87			−1.19229
$848$		0			0
$849$		−	1961.39i		−	2.31024i
$850$		0			0
$851$			5.72486i			0.00672722i
$852$		0			0
$853$	63.4665			0.0744039			0.0372019	−	0.999308i	$-0.488156\pi$
$853$	63.4665			0.0744039			0.0372019	+	0.999308i	$0.488156\pi$
$854$		0			0
$855$	653.082	−	676.055i	0.763839	−	0.790708i
$856$		0			0
$857$		−	304.235i		−	0.355000i	−0.984121	−	0.177500i	$-0.943199\pi$
$857$		−	304.235i		−	0.355000i	0.984121	−	0.177500i	$-0.0568009\pi$
$858$		0			0
$859$	−1336.81			−1.55624			−0.778119	−	0.628117i	$-0.783826\pi$
$859$	−1336.81			−1.55624			−0.778119	+	0.628117i	$0.783826\pi$
$860$		0			0
$861$	2190.15			2.54373
$862$		0			0
$863$		−	752.833i		−	0.872344i	−0.899863	−	0.436172i	$-0.856334\pi$
$863$		−	752.833i		−	0.872344i	0.899863	−	0.436172i	$-0.143666\pi$
$864$		0			0
$865$			1651.09i			1.90877i
$866$		0			0
$867$		−	1703.09i		−	1.96435i
$868$		0			0
$869$			1069.99i			1.23129i
$870$		0			0
$871$	−1809.73			−2.07776
$872$		0			0
$873$		−	1169.93i		−	1.34013i
$874$		0			0
$875$	−551.249			−0.629999
$876$		0			0
$877$			1209.77i			1.37945i	0.724074	+	0.689723i	$0.242268\pi$
$877$			1209.77i			1.37945i	−0.724074	+	0.689723i	$0.757732\pi$
$878$		0			0
$879$	670.802			0.763142
$880$		0			0
$881$	−1126.67			−1.27885			−0.639425	−	0.768854i	$-0.720827\pi$
$881$	−1126.67			−1.27885			−0.639425	+	0.768854i	$0.720827\pi$
$882$		0			0
$883$	−509.573			−0.577092			−0.288546	−	0.957466i	$-0.593172\pi$
$883$	−509.573			−0.577092			−0.288546	+	0.957466i	$0.593172\pi$
$884$		0			0
$885$	1297.54			1.46615
$886$		0			0
$887$		−	1167.27i		−	1.31597i	−0.753030	−	0.657986i	$-0.771409\pi$
$887$		−	1167.27i		−	1.31597i	0.753030	−	0.657986i	$-0.228591\pi$
$888$		0			0
$889$			512.717i			0.576734i
$890$		0			0
$891$	−1401.97			−1.57348
$892$		0			0
$893$	585.646	−	606.247i	0.655819	−	0.678888i
$894$		0			0
$895$			1167.57i			1.30454i
$896$		0			0
$897$	−43.2126			−0.0481745
$898$		0			0
$899$	126.432			0.140636
$900$		0			0
$901$		−	1307.53i		−	1.45120i
$902$		0			0
$903$			1377.52i			1.52549i
$904$		0			0
$905$		−	476.650i		−	0.526685i
$906$		0			0
$907$			83.2264i			0.0917601i	0.998947	+	0.0458800i	$0.0146092\pi$
$907$			83.2264i			0.0917601i	−0.998947	+	0.0458800i	$0.985391\pi$
$908$		0			0
$909$	603.530			0.663949
$910$		0			0
$911$		−	300.362i		−	0.329706i	−0.986318	−	0.164853i	$-0.947285\pi$
$911$		−	300.362i		−	0.329706i	0.986318	−	0.164853i	$-0.0527149\pi$
$912$		0			0
$913$	−61.8029			−0.0676922
$914$		0			0
$915$			108.019i			0.118054i
$916$		0			0
$917$	−1165.08			−1.27053
$918$		0			0
$919$	−285.792			−0.310981			−0.155491	−	0.987837i	$-0.549696\pi$
$919$	−285.792			−0.310981			−0.155491	+	0.987837i	$0.549696\pi$
$920$		0			0
$921$	69.7270			0.0757079
$922$		0			0
$923$	−467.169			−0.506142
$924$		0			0
$925$			394.249i			0.426215i
$926$		0			0
$927$			1275.06i			1.37547i
$928$		0			0
$929$	−1456.23			−1.56752			−0.783760	−	0.621064i	$-0.786701\pi$
$929$	−1456.23			−1.56752			−0.783760	+	0.621064i	$0.786701\pi$
$930$		0			0
$931$	−1078.11	+	1116.04i	−1.15802	+	1.19875i
$932$		0			0
$933$		−	45.1050i		−	0.0483441i
$934$		0			0
$935$	2916.96			3.11974
$936$		0			0
$937$	−756.375			−0.807230			−0.403615	−	0.914929i	$-0.632247\pi$
$937$	−756.375			−0.807230			−0.403615	+	0.914929i	$0.632247\pi$
$938$		0			0
$939$		−	593.816i		−	0.632392i
$940$		0			0
$941$			710.309i			0.754845i	0.926041	+	0.377422i	$0.123189\pi$
$941$			710.309i			0.754845i	−0.926041	+	0.377422i	$0.876811\pi$
$942$		0			0
$943$		−	22.1419i		−	0.0234803i
$944$		0			0
$945$			818.165i			0.865783i
$946$		0			0
$947$	−134.816			−0.142361			−0.0711805	−	0.997463i	$-0.522677\pi$
$947$	−134.816			−0.142361			−0.0711805	+	0.997463i	$0.522677\pi$
$948$		0			0
$949$			1993.28i			2.10040i
$950$		0			0
$951$	1866.06			1.96221
$952$		0			0
$953$			665.726i			0.698558i	0.937019	+	0.349279i	$0.113573\pi$
$953$			665.726i			0.698558i	−0.937019	+	0.349279i	$0.886427\pi$
$954$		0			0
$955$	1599.77			1.67516
$956$		0			0
$957$	448.776			0.468941
$958$		0			0
$959$	−1613.59			−1.68257
$960$		0			0
$961$	702.023			0.730513
$962$		0			0
$963$			316.061i			0.328205i
$964$		0			0
$965$		−	709.540i		−	0.735274i
$966$		0			0
$967$	−241.670			−0.249917			−0.124959	−	0.992162i	$-0.539880\pi$
$967$	−241.670			−0.249917			−0.124959	+	0.992162i	$0.539880\pi$
$968$		0			0
$969$	1448.01	+	1398.81i	1.49433	+	1.44356i
$970$		0			0
$971$		−	917.585i		−	0.944990i	−0.881333	−	0.472495i	$-0.843354\pi$
$971$		−	917.585i		−	0.944990i	0.881333	−	0.472495i	$-0.156646\pi$
$972$		0			0
$973$	−160.225			−0.164671
$974$		0			0
$975$	−2975.88			−3.05218
$976$		0			0
$977$		−	282.297i		−	0.288943i	−0.989509	−	0.144472i	$-0.953852\pi$
$977$		−	282.297i		−	0.288943i	0.989509	−	0.144472i	$-0.0461482\pi$
$978$		0			0
$979$		−	1525.33i		−	1.55804i
$980$		0			0
$981$		−	1229.26i		−	1.25307i
$982$		0			0
$983$		−	1007.79i		−	1.02521i	−0.858623	−	0.512607i	$-0.828680\pi$
$983$		−	1007.79i		−	1.02521i	0.858623	−	0.512607i	$-0.171320\pi$
$984$		0			0
$985$	−3.30692			−0.00335728
$986$		0			0
$987$		−	2002.14i		−	2.02851i
$988$		0			0
$989$	13.9264			0.0140813
$990$		0			0
$991$			1698.36i			1.71379i	0.515492	+	0.856894i	$0.327609\pi$
$991$			1698.36i			1.71379i	−0.515492	+	0.856894i	$0.672391\pi$
$992$		0			0
$993$	1821.96			1.83481
$994$		0			0
$995$	599.127			0.602138
$996$		0			0
$997$	−910.114			−0.912852			−0.456426	−	0.889761i	$-0.650871\pi$
$997$	−910.114			−0.912852			−0.456426	+	0.889761i	$0.650871\pi$
$998$		0			0
$999$	119.563			0.119683

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.3.e.p.1025.5		20
4.3	odd	2	inner	1216.3.e.p.1025.16		20
8.3	odd	2		608.3.e.b.417.6	yes	20
8.5	even	2		608.3.e.b.417.15	yes	20
19.18	odd	2	inner	1216.3.e.p.1025.15		20
76.75	even	2	inner	1216.3.e.p.1025.6		20
152.37	odd	2		608.3.e.b.417.5	✓	20
152.75	even	2		608.3.e.b.417.16	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
608.3.e.b.417.5	✓	20	152.37	odd	2
608.3.e.b.417.6	yes	20	8.3	odd	2
608.3.e.b.417.15	yes	20	8.5	even	2
608.3.e.b.417.16	yes	20	152.75	even	2
1216.3.e.p.1025.5		20	1.1	even	1	trivial
1216.3.e.p.1025.6		20	76.75	even	2	inner
1216.3.e.p.1025.15		20	19.18	odd	2	inner
1216.3.e.p.1025.16		20	4.3	odd	2	inner

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:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1216.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.94796i q^{3} +7.51133 q^{5} -11.4311 q^{7} -6.58642 q^{9} +14.4687 q^{11} -23.9902i q^{13} -29.6545i q^{15} +26.8400 q^{17} +(-13.2008 + 13.6652i) q^{19} +45.1296i q^{21} +0.456250 q^{23} +31.4201 q^{25} -9.52875i q^{27} +7.85645i q^{29} -16.0928i q^{31} -57.1220i q^{33} -85.8628 q^{35} +12.5477i q^{37} -94.7126 q^{39} -48.5303i q^{41} +30.5237 q^{43} -49.4728 q^{45} -44.3643 q^{47} +81.6702 q^{49} -105.963i q^{51} -48.7157i q^{53} +108.679 q^{55} +(53.9497 + 52.1164i) q^{57} +43.7553i q^{59} -3.64259 q^{61} +75.2900 q^{63} -180.199i q^{65} -75.4361i q^{67} -1.80126i q^{69} -19.4733i q^{71} -83.0871 q^{73} -124.045i q^{75} -165.394 q^{77} +73.9522i q^{79} -96.8969 q^{81} -4.27148 q^{83} +201.604 q^{85} +31.0170 q^{87} -105.422i q^{89} +274.235i q^{91} -63.5337 q^{93} +(-99.1559 + 102.644i) q^{95} +177.628i q^{97} -95.2971 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 52 q^{9} + 56 q^{17} + 36 q^{25} - 64 q^{45} + 332 q^{49} + 88 q^{57} + 32 q^{61} - 152 q^{73} - 360 q^{77} - 476 q^{81} + 552 q^{85} - 336 q^{93}+O(q^{100}) \) 20 * q - 52 * q^9 + 56 * q^17 + 36 * q^25 - 64 * q^45 + 332 * q^49 + 88 * q^57 + 32 * q^61 - 152 * q^73 - 360 * q^77 - 476 * q^81 + 552 * q^85 - 336 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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