LMFDB - Embedded newform 624.6.c.d.337.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [624,6,Mod(337,624)] 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("624.337"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(624, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	624.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,54,0,0,0,0,0,486,0,0,0,530] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$100.079503563$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} - 2946x^{3} + 131769x^{2} - 1332936x + 6741792 $ x^6 - 2x^5 + 2x^4 - 2946x^3 + 131769x^2 - 1332936*x + 6741792
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{15}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 78)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.5
Root			$6.10758 - 6.10758i$ of defining polynomial
Character	$\chi$	$=$	624.337
Dual form			624.6.c.d.337.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000 q^{3} +37.1158i q^{5} -176.407i q^{7} +81.0000 q^{9} +179.404i q^{11} +(-554.740 - 252.104i) q^{13} +334.042i q^{15} -933.141 q^{17} +2335.97i q^{19} -1587.66i q^{21} +2792.36 q^{23} +1747.42 q^{25} +729.000 q^{27} +1503.58 q^{29} +737.236i q^{31} +1614.63i q^{33} +6547.48 q^{35} -3775.41i q^{37} +(-4992.66 - 2268.94i) q^{39} +368.848i q^{41} +20180.6 q^{43} +3006.38i q^{45} -20526.5i q^{47} -14312.3 q^{49} -8398.27 q^{51} -25081.9 q^{53} -6658.72 q^{55} +21023.7i q^{57} -35326.3i q^{59} +31741.4 q^{61} -14288.9i q^{63} +(9357.06 - 20589.6i) q^{65} +46661.3i q^{67} +25131.2 q^{69} -58973.4i q^{71} +3411.20i q^{73} +15726.8 q^{75} +31648.1 q^{77} +64977.3 q^{79} +6561.00 q^{81} -12233.4i q^{83} -34634.3i q^{85} +13532.2 q^{87} -61754.7i q^{89} +(-44472.9 + 97859.9i) q^{91} +6635.12i q^{93} -86701.4 q^{95} -28030.7i q^{97} +14531.7i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 54 q^{3} + 486 q^{9} + 530 q^{13} - 836 q^{17} + 416 q^{23} + 718 q^{25} + 4374 q^{27} + 18788 q^{29} - 6112 q^{35} + 4770 q^{39} + 24200 q^{43} - 3038 q^{49} - 7524 q^{51} - 42396 q^{53} - 124656 q^{55}+ \cdots - 567168 q^{95}+O(q^{100}) $ 6 * q + 54 * q^3 + 486 * q^9 + 530 * q^13 - 836 * q^17 + 416 * q^23 + 718 * q^25 + 4374 * q^27 + 18788 * q^29 - 6112 * q^35 + 4770 * q^39 + 24200 * q^43 - 3038 * q^49 - 7524 * q^51 - 42396 * q^53 - 124656 * q^55 - 3196 * q^61 - 17168 * q^65 + 3744 * q^69 + 6462 * q^75 - 114024 * q^77 + 169328 * q^79 + 39366 * q^81 + 169092 * q^87 - 236152 * q^91 - 567168 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/624\mathbb{Z}\right)^\times$.

$n$	$79$	$145$	$209$	$469$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	9.00000			0.577350
$3$	9.00000			0.577350
$4$		0			0
$5$			37.1158i			0.663948i	0.943289	+	0.331974i	$0.107715\pi$
$5$			37.1158i			0.663948i	−0.943289	+	0.331974i	$0.892285\pi$
$6$		0			0
$7$		−	176.407i		−	1.36072i	−0.732876	−	0.680362i	$-0.761823\pi$
$7$		−	176.407i		−	1.36072i	0.732876	−	0.680362i	$-0.238177\pi$
$8$		0			0
$9$	81.0000			0.333333
$10$		0			0
$11$			179.404i			0.447044i	0.974699	+	0.223522i	$0.0717554\pi$
$11$			179.404i			0.447044i	−0.974699	+	0.223522i	$0.928245\pi$
$12$		0			0
$13$	−554.740	−	252.104i	−0.910397	−	0.413735i
$13$	−554.740	−	252.104i	−0.910397	−	0.413735i
$14$		0			0
$15$			334.042i			0.383330i
$16$		0			0
$17$	−933.141			−0.783114			−0.391557	−	0.920154i	$-0.628063\pi$
$17$	−933.141			−0.783114			−0.391557	+	0.920154i	$0.628063\pi$
$18$		0			0
$19$			2335.97i			1.48451i	0.670117	+	0.742256i	$0.266244\pi$
$19$			2335.97i			1.48451i	−0.670117	+	0.742256i	$0.733756\pi$
$20$		0			0
$21$		−	1587.66i		−	0.785614i
$22$		0			0
$23$	2792.36			1.10066			0.550328	−	0.834948i	$-0.314503\pi$
$23$	2792.36			1.10066			0.550328	+	0.834948i	$0.314503\pi$
$24$		0			0
$25$	1747.42			0.559174
$26$		0			0
$27$	729.000			0.192450
$28$		0			0
$29$	1503.58			0.331996			0.165998	−	0.986126i	$-0.446916\pi$
$29$	1503.58			0.331996			0.165998	+	0.986126i	$0.446916\pi$
$30$		0			0
$31$			737.236i			0.137785i	0.997624	+	0.0688926i	$0.0219466\pi$
$31$			737.236i			0.137785i	−0.997624	+	0.0688926i	$0.978053\pi$
$32$		0			0
$33$			1614.63i			0.258101i
$34$		0			0
$35$	6547.48			0.903450
$36$		0			0
$37$		−	3775.41i		−	0.453377i	−0.973967	−	0.226689i	$-0.927210\pi$
$37$		−	3775.41i		−	0.453377i	0.973967	−	0.226689i	$-0.0727900\pi$
$38$		0			0
$39$	−4992.66	−	2268.94i	−0.525618	−	0.238870i
$40$		0			0
$41$			368.848i			0.0342680i	0.999853	+	0.0171340i	$0.00545418\pi$
$41$			368.848i			0.0342680i	−0.999853	+	0.0171340i	$0.994546\pi$
$42$		0			0
$43$	20180.6			1.66442			0.832211	−	0.554459i	$-0.187075\pi$
$43$	20180.6			1.66442			0.832211	+	0.554459i	$0.187075\pi$
$44$		0			0
$45$			3006.38i			0.221316i
$46$		0			0
$47$		−	20526.5i		−	1.35541i	−0.735335	−	0.677704i	$-0.762975\pi$
$47$		−	20526.5i		−	1.35541i	0.735335	−	0.677704i	$-0.237025\pi$
$48$		0			0
$49$	−14312.3			−0.851570
$50$		0			0
$51$	−8398.27			−0.452131
$52$		0			0
$53$	−25081.9			−1.22651			−0.613253	−	0.789886i	$-0.710140\pi$
$53$	−25081.9			−1.22651			−0.613253	+	0.789886i	$0.710140\pi$
$54$		0			0
$55$	−6658.72			−0.296814
$56$		0			0
$57$			21023.7i			0.857083i
$58$		0			0
$59$		−	35326.3i		−	1.32120i	−0.750738	−	0.660600i	$-0.770302\pi$
$59$		−	35326.3i		−	1.32120i	0.750738	−	0.660600i	$-0.229698\pi$
$60$		0			0
$61$	31741.4			1.09220			0.546099	−	0.837721i	$-0.316112\pi$
$61$	31741.4			1.09220			0.546099	+	0.837721i	$0.316112\pi$
$62$		0			0
$63$		−	14288.9i		−	0.453575i
$64$		0			0
$65$	9357.06	−	20589.6i	0.274698	−	0.604456i
$66$		0			0
$67$			46661.3i			1.26990i	0.772552	+	0.634951i	$0.218980\pi$
$67$			46661.3i			1.26990i	−0.772552	+	0.634951i	$0.781020\pi$
$68$		0			0
$69$	25131.2			0.635465
$70$		0			0
$71$		−	58973.4i		−	1.38839i	−0.719789	−	0.694193i	$-0.755762\pi$
$71$		−	58973.4i		−	1.38839i	0.719789	−	0.694193i	$-0.244238\pi$
$72$		0			0
$73$			3411.20i			0.0749204i	0.999298	+	0.0374602i	$0.0119267\pi$
$73$			3411.20i			0.0749204i	−0.999298	+	0.0374602i	$0.988073\pi$
$74$		0			0
$75$	15726.8			0.322839
$76$		0			0
$77$	31648.1			0.608303
$78$		0			0
$79$	64977.3			1.17137			0.585685	−	0.810539i	$-0.300826\pi$
$79$	64977.3			1.17137			0.585685	+	0.810539i	$0.300826\pi$
$80$		0			0
$81$	6561.00			0.111111
$82$		0			0
$83$		−	12233.4i		−	0.194918i	−0.995240	−	0.0974591i	$-0.968928\pi$
$83$		−	12233.4i		−	0.194918i	0.995240	−	0.0974591i	$-0.0310715\pi$
$84$		0			0
$85$		−	34634.3i		−	0.519947i
$86$		0			0
$87$	13532.2			0.191678
$88$		0			0
$89$		−	61754.7i		−	0.826409i	−0.910638	−	0.413204i	$-0.864410\pi$
$89$		−	61754.7i		−	0.826409i	0.910638	−	0.413204i	$-0.135590\pi$
$90$		0			0
$91$	−44472.9	+	97859.9i	−0.562979	+	1.23880i
$92$		0			0
$93$			6635.12i			0.0795503i
$94$		0			0
$95$	−86701.4			−0.985638
$96$		0			0
$97$		−	28030.7i		−	0.302485i	−0.988497	−	0.151243i	$-0.951673\pi$
$97$		−	28030.7i		−	0.302485i	0.988497	−	0.151243i	$-0.0483274\pi$
$98$		0			0
$99$			14531.7i			0.149015i
$100$		0			0
$101$	102600.			1.00080			0.500398	−	0.865796i	$-0.333187\pi$
$101$	102600.			1.00080			0.500398	+	0.865796i	$0.333187\pi$
$102$		0			0
$103$	203803.			1.89286			0.946430	−	0.322909i	$-0.104661\pi$
$103$	203803.			1.89286			0.946430	+	0.322909i	$0.104661\pi$
$104$		0			0
$105$	58927.3			0.521607
$106$		0			0
$107$	147356.			1.24425			0.622125	−	0.782918i	$-0.286269\pi$
$107$	147356.			1.24425			0.622125	+	0.782918i	$0.286269\pi$
$108$		0			0
$109$			7846.65i			0.0632584i	0.999500	+	0.0316292i	$0.0100696\pi$
$109$			7846.65i			0.0632584i	−0.999500	+	0.0316292i	$0.989930\pi$
$110$		0			0
$111$		−	33978.7i		−	0.261758i
$112$		0			0
$113$	−192673.			−1.41947			−0.709733	−	0.704471i	$-0.751184\pi$
$113$	−192673.			−1.41947			−0.709733	+	0.704471i	$0.751184\pi$
$114$		0			0
$115$			103641.i			0.730778i
$116$		0			0
$117$	−44933.9	−	20420.5i	−0.303466	−	0.137912i
$118$		0			0
$119$			164612.i			1.06560i
$120$		0			0
$121$	128865.			0.800152
$122$		0			0
$123$			3319.64i			0.0197846i
$124$		0			0
$125$			180844.i			1.03521i
$126$		0			0
$127$	−255644.			−1.40646			−0.703229	−	0.710964i	$-0.748259\pi$
$127$	−255644.			−1.40646			−0.703229	+	0.710964i	$0.748259\pi$
$128$		0			0
$129$	181626.			0.960955
$130$		0			0
$131$	134257.			0.683530			0.341765	−	0.939786i	$-0.388975\pi$
$131$	134257.			0.683530			0.341765	+	0.939786i	$0.388975\pi$
$132$		0			0
$133$	412081.			2.02001
$134$		0			0
$135$			27057.4i			0.127777i
$136$		0			0
$137$		−	74373.4i		−	0.338545i	−0.985569	−	0.169272i	$-0.945858\pi$
$137$		−	74373.4i		−	0.338545i	0.985569	−	0.169272i	$-0.0541418\pi$
$138$		0			0
$139$	−36348.4			−0.159569			−0.0797844	−	0.996812i	$-0.525423\pi$
$139$	−36348.4			−0.159569			−0.0797844	+	0.996812i	$0.525423\pi$
$140$		0			0
$141$		−	184738.i		−	0.782545i
$142$		0			0
$143$	45228.5	−	99522.5i	0.184958	−	0.406988i
$144$		0			0
$145$			55806.7i			0.220428i
$146$		0			0
$147$	−128811.			−0.491654
$148$		0			0
$149$			221253.i			0.816440i	0.912884	+	0.408220i	$0.133850\pi$
$149$			221253.i			0.816440i	−0.912884	+	0.408220i	$0.866150\pi$
$150$		0			0
$151$		−	482448.i		−	1.72190i	−0.508690	−	0.860950i	$-0.669870\pi$
$151$		−	482448.i		−	1.72190i	0.508690	−	0.860950i	$-0.330130\pi$
$152$		0			0
$153$	−75584.4			−0.261038
$154$		0			0
$155$	−27363.1			−0.0914821
$156$		0			0
$157$	−110726.			−0.358508			−0.179254	−	0.983803i	$-0.557368\pi$
$157$	−110726.			−0.358508			−0.179254	+	0.983803i	$0.557368\pi$
$158$		0			0
$159$	−225737.			−0.708124
$160$		0			0
$161$		−	492591.i		−	1.49769i
$162$		0			0
$163$		−	161064.i		−	0.474819i	−0.971410	−	0.237410i	$-0.923702\pi$
$163$		−	161064.i		−	0.474819i	0.971410	−	0.237410i	$-0.0762984\pi$
$164$		0			0
$165$	−59928.5			−0.171365
$166$		0			0
$167$		−	466578.i		−	1.29459i	−0.762238	−	0.647296i	$-0.775900\pi$
$167$		−	466578.i		−	1.29459i	0.762238	−	0.647296i	$-0.224100\pi$
$168$		0			0
$169$	244180.	+	279705.i	0.657647	+	0.753326i
$170$		0			0
$171$			189214.i			0.494837i
$172$		0			0
$173$	−240767.			−0.611620			−0.305810	−	0.952093i	$-0.598927\pi$
$173$	−240767.			−0.611620			−0.305810	+	0.952093i	$0.598927\pi$
$174$		0			0
$175$		−	308256.i		−	0.760881i
$176$		0			0
$177$		−	317937.i		−	0.762795i
$178$		0			0
$179$	266435.			0.621526			0.310763	−	0.950487i	$-0.399415\pi$
$179$	266435.			0.621526			0.310763	+	0.950487i	$0.399415\pi$
$180$		0			0
$181$	64307.2			0.145903			0.0729513	−	0.997336i	$-0.476758\pi$
$181$	64307.2			0.145903			0.0729513	+	0.997336i	$0.476758\pi$
$182$		0			0
$183$	285673.			0.630581
$184$		0			0
$185$	140127.			0.301019
$186$		0			0
$187$		−	167409.i		−	0.350086i
$188$		0			0
$189$		−	128601.i		−	0.261871i
$190$		0			0
$191$	925387.			1.83544			0.917720	−	0.397228i	$-0.130028\pi$
$191$	925387.			1.83544			0.917720	+	0.397228i	$0.130028\pi$
$192$		0			0
$193$		−	632306.i		−	1.22190i	−0.791671	−	0.610948i	$-0.790789\pi$
$193$		−	632306.i		−	1.22190i	0.791671	−	0.610948i	$-0.209211\pi$
$194$		0			0
$195$	84213.5	−	185307.i	0.158597	−	0.348983i
$196$		0			0
$197$			100048.i			0.183671i	0.995774	+	0.0918357i	$0.0292735\pi$
$197$			100048.i			0.183671i	−0.995774	+	0.0918357i	$0.970727\pi$
$198$		0			0
$199$	723076.			1.29435			0.647174	−	0.762342i	$-0.275951\pi$
$199$	723076.			1.29435			0.647174	+	0.762342i	$0.275951\pi$
$200$		0			0
$201$			419952.i			0.733178i
$202$		0			0
$203$		−	265242.i		−	0.451754i
$204$		0			0
$205$	−13690.1			−0.0227521
$206$		0			0
$207$	226181.			0.366886
$208$		0			0
$209$	−419082.			−0.663641
$210$		0			0
$211$	−259548.			−0.401340			−0.200670	−	0.979659i	$-0.564312\pi$
$211$	−259548.			−0.401340			−0.200670	+	0.979659i	$0.564312\pi$
$212$		0			0
$213$		−	530760.i		−	0.801585i
$214$		0			0
$215$			749020.i			1.10509i
$216$		0			0
$217$	130053.			0.187488
$218$		0			0
$219$			30700.8i			0.0432553i
$220$		0			0
$221$	517651.	+	235249.i	0.712945	+	0.324002i
$222$		0			0
$223$			1.07196e6i			1.44349i	0.692156	+	0.721747i	$0.256661\pi$
$223$			1.07196e6i			1.44349i	−0.692156	+	0.721747i	$0.743339\pi$
$224$		0			0
$225$	141541.			0.186391
$226$		0			0
$227$		−	65210.0i		−	0.0839942i	−0.999118	−	0.0419971i	$-0.986628\pi$
$227$		−	65210.0i		−	0.0839942i	0.999118	−	0.0419971i	$-0.0133720\pi$
$228$		0			0
$229$			191854.i			0.241759i	0.992667	+	0.120879i	$0.0385715\pi$
$229$			191854.i			0.241759i	−0.992667	+	0.120879i	$0.961429\pi$
$230$		0			0
$231$	284832.			0.351204
$232$		0			0
$233$	353920.			0.427087			0.213543	−	0.976934i	$-0.431500\pi$
$233$	353920.			0.427087			0.213543	+	0.976934i	$0.431500\pi$
$234$		0			0
$235$	761857.			0.899920
$236$		0			0
$237$	584796.			0.676291
$238$		0			0
$239$			1.15374e6i			1.30651i	0.757138	+	0.653255i	$0.226597\pi$
$239$			1.15374e6i			1.30651i	−0.757138	+	0.653255i	$0.773403\pi$
$240$		0			0
$241$			209753.i			0.232630i	0.993212	+	0.116315i	$0.0371083\pi$
$241$			209753.i			0.232630i	−0.993212	+	0.116315i	$0.962892\pi$
$242$		0			0
$243$	59049.0			0.0641500
$244$		0			0
$245$		−	531214.i		−	0.565398i
$246$		0			0
$247$	588909.	−	1.29586e6i	0.614194	−	1.35150i
$248$		0			0
$249$		−	110101.i		−	0.112536i
$250$		0			0
$251$	−427156.			−0.427959			−0.213979	−	0.976838i	$-0.568643\pi$
$251$	−427156.			−0.427959			−0.213979	+	0.976838i	$0.568643\pi$
$252$		0			0
$253$			500960.i			0.492042i
$254$		0			0
$255$		−	311708.i		−	0.300191i
$256$		0			0
$257$	207722.			0.196177			0.0980887	−	0.995178i	$-0.468727\pi$
$257$	207722.			0.196177			0.0980887	+	0.995178i	$0.468727\pi$
$258$		0			0
$259$	−666008.			−0.616922
$260$		0			0
$261$	121790.			0.110665
$262$		0			0
$263$	831712.			0.741453			0.370726	−	0.928742i	$-0.379109\pi$
$263$	831712.			0.741453			0.370726	+	0.928742i	$0.379109\pi$
$264$		0			0
$265$		−	930933.i		−	0.814336i
$266$		0			0
$267$		−	555792.i		−	0.477127i
$268$		0			0
$269$	1.67365e6			1.41021			0.705104	−	0.709104i	$-0.250901\pi$
$269$	1.67365e6			1.41021			0.705104	+	0.709104i	$0.250901\pi$
$270$		0			0
$271$		−	69792.6i		−	0.0577279i	−0.999583	−	0.0288640i	$-0.990811\pi$
$271$		−	69792.6i		−	0.0577279i	0.999583	−	0.0288640i	$-0.00918896\pi$
$272$		0			0
$273$	−400256.	+	880739.i	−0.325036	+	0.715221i
$274$		0			0
$275$			313493.i			0.249975i
$276$		0			0
$277$	1.61380e6			1.26372			0.631859	−	0.775083i	$-0.282292\pi$
$277$	1.61380e6			1.26372			0.631859	+	0.775083i	$0.282292\pi$
$278$		0			0
$279$			59716.1i			0.0459284i
$280$		0			0
$281$			1.05655e6i			0.798224i	0.916902	+	0.399112i	$0.130682\pi$
$281$			1.05655e6i			0.798224i	−0.916902	+	0.399112i	$0.869318\pi$
$282$		0			0
$283$	512741.			0.380567			0.190284	−	0.981729i	$-0.439059\pi$
$283$	512741.			0.380567			0.190284	+	0.981729i	$0.439059\pi$
$284$		0			0
$285$	−780313.			−0.569058
$286$		0			0
$287$	65067.4			0.0466293
$288$		0			0
$289$	−549105.			−0.386732
$290$		0			0
$291$		−	252276.i		−	0.174640i
$292$		0			0
$293$			2.71268e6i			1.84599i	0.384809	+	0.922996i	$0.374267\pi$
$293$			2.71268e6i			1.84599i	−0.384809	+	0.922996i	$0.625733\pi$
$294$		0			0
$295$	1.31117e6			0.877207
$296$		0			0
$297$			130785.i			0.0860336i
$298$		0			0
$299$	−1.54903e6	−	703967.i	−1.00204	−	0.455380i
$300$		0			0
$301$		−	3.56000e6i		−	2.26482i
$302$		0			0
$303$	923403.			0.577810
$304$		0			0
$305$			1.17811e6i			0.725162i
$306$		0			0
$307$		−	2.44475e6i		−	1.48043i	−0.672371	−	0.740215i	$-0.734724\pi$
$307$		−	2.44475e6i		−	1.48043i	0.672371	−	0.740215i	$-0.265276\pi$
$308$		0			0
$309$	1.83423e6			1.09284
$310$		0			0
$311$	−1.99183e6			−1.16775			−0.583877	−	0.811842i	$-0.698465\pi$
$311$	−1.99183e6			−1.16775			−0.583877	+	0.811842i	$0.698465\pi$
$312$		0			0
$313$	1.66396e6			0.960025			0.480013	−	0.877262i	$-0.340632\pi$
$313$	1.66396e6			0.960025			0.480013	+	0.877262i	$0.340632\pi$
$314$		0			0
$315$	530346.			0.301150
$316$		0			0
$317$		−	3.20482e6i		−	1.79125i	−0.444813	−	0.895624i	$-0.646730\pi$
$317$		−	3.20482e6i		−	1.79125i	0.444813	−	0.895624i	$-0.353270\pi$
$318$		0			0
$319$			269749.i			0.148417i
$320$		0			0
$321$	1.32620e6			0.718369
$322$		0			0
$323$		−	2.17979e6i		−	1.16254i
$324$		0			0
$325$	−969362.	−	440532.i	−0.509070	−	0.231350i
$326$		0			0
$327$			70619.9i			0.0365222i
$328$		0			0
$329$	−3.62101e6			−1.84434
$330$		0			0
$331$			3.53386e6i			1.77288i	0.462844	+	0.886440i	$0.346829\pi$
$331$			3.53386e6i			1.77288i	−0.462844	+	0.886440i	$0.653171\pi$
$332$		0			0
$333$		−	305808.i		−	0.151126i
$334$		0			0
$335$	−1.73187e6			−0.843148
$336$		0			0
$337$	1.10242e6			0.528777			0.264389	−	0.964416i	$-0.414830\pi$
$337$	1.10242e6			0.528777			0.264389	+	0.964416i	$0.414830\pi$
$338$		0			0
$339$	−1.73406e6			−0.819529
$340$		0			0
$341$	−132263.			−0.0615960
$342$		0			0
$343$		−	440075.i		−	0.201972i
$344$		0			0
$345$			932766.i			0.421915i
$346$		0			0
$347$	−3.16432e6			−1.41077			−0.705386	−	0.708823i	$-0.749226\pi$
$347$	−3.16432e6			−1.41077			−0.705386	+	0.708823i	$0.749226\pi$
$348$		0			0
$349$		−	1.97526e6i		−	0.868080i	−0.900894	−	0.434040i	$-0.857088\pi$
$349$		−	1.97526e6i		−	0.868080i	0.900894	−	0.434040i	$-0.142912\pi$
$350$		0			0
$351$	−404405.	−	183784.i	−0.175206	−	0.0796233i
$352$		0			0
$353$		−	2.24015e6i		−	0.956843i	−0.878130	−	0.478422i	$-0.841209\pi$
$353$		−	2.24015e6i		−	0.956843i	0.878130	−	0.478422i	$-0.158791\pi$
$354$		0			0
$355$	2.18884e6			0.921815
$356$		0			0
$357$			1.48151e6i			0.615226i
$358$		0			0
$359$		−	2.13770e6i		−	0.875407i	−0.899119	−	0.437704i	$-0.855792\pi$
$359$		−	2.13770e6i		−	0.875407i	0.899119	−	0.437704i	$-0.144208\pi$
$360$		0			0
$361$	−2.98066e6			−1.20377
$362$		0			0
$363$	1.15979e6			0.461968
$364$		0			0
$365$	−126609.			−0.0497432
$366$		0			0
$367$	−1.70502e6			−0.660791			−0.330395	−	0.943843i	$-0.607182\pi$
$367$	−1.70502e6			−0.660791			−0.330395	+	0.943843i	$0.607182\pi$
$368$		0			0
$369$			29876.7i			0.0114227i
$370$		0			0
$371$			4.42461e6i			1.66894i
$372$		0			0
$373$	−2.31503e6			−0.861558			−0.430779	−	0.902458i	$-0.641761\pi$
$373$	−2.31503e6			−0.861558			−0.430779	+	0.902458i	$0.641761\pi$
$374$		0			0
$375$			1.62759e6i			0.597679i
$376$		0			0
$377$	−834097.	−	379060.i	−0.302248	−	0.137358i
$378$		0			0
$379$			4.32354e6i			1.54612i	0.634336	+	0.773058i	$0.281274\pi$
$379$			4.32354e6i			1.54612i	−0.634336	+	0.773058i	$0.718726\pi$
$380$		0			0
$381$	−2.30080e6			−0.812018
$382$		0			0
$383$		−	2.00351e6i		−	0.697901i	−0.937141	−	0.348951i	$-0.886538\pi$
$383$		−	2.00351e6i		−	0.697901i	0.937141	−	0.348951i	$-0.113462\pi$
$384$		0			0
$385$			1.17464e6i			0.403882i
$386$		0			0
$387$	1.63463e6			0.554807
$388$		0			0
$389$	1.70696e6			0.571940			0.285970	−	0.958239i	$-0.407684\pi$
$389$	1.70696e6			0.571940			0.285970	+	0.958239i	$0.407684\pi$
$390$		0			0
$391$	−2.60567e6			−0.861940
$392$		0			0
$393$	1.20831e6			0.394636
$394$		0			0
$395$			2.41168e6i			0.777728i
$396$		0			0
$397$			1.22427e6i			0.389853i	0.980818	+	0.194927i	$0.0624469\pi$
$397$			1.22427e6i			0.389853i	−0.980818	+	0.194927i	$0.937553\pi$
$398$		0			0
$399$	3.70873e6			1.16625
$400$		0			0
$401$		−	4.15598e6i		−	1.29066i	−0.763903	−	0.645332i	$-0.776719\pi$
$401$		−	4.15598e6i		−	1.29066i	0.763903	−	0.645332i	$-0.223281\pi$
$402$		0			0
$403$	185860.	−	408974.i	0.0570065	−	0.125439i
$404$		0			0
$405$			243517.i			0.0737720i
$406$		0			0
$407$	677323.			0.202680
$408$		0			0
$409$			6.52428e6i			1.92852i	0.264958	+	0.964260i	$0.414642\pi$
$409$			6.52428e6i			1.92852i	−0.264958	+	0.964260i	$0.585358\pi$
$410$		0			0
$411$		−	669361.i		−	0.195459i
$412$		0			0
$413$	−6.23180e6			−1.79779
$414$		0			0
$415$	454053.			0.129416
$416$		0			0
$417$	−327136.			−0.0921271
$418$		0			0
$419$	−4.77950e6			−1.32999			−0.664994	−	0.746849i	$-0.731566\pi$
$419$	−4.77950e6			−1.32999			−0.664994	+	0.746849i	$0.731566\pi$
$420$		0			0
$421$			3.41558e6i			0.939203i	0.882878	+	0.469602i	$0.155602\pi$
$421$			3.41558e6i			0.939203i	−0.882878	+	0.469602i	$0.844398\pi$
$422$		0			0
$423$		−	1.66265e6i		−	0.451803i
$424$		0			0
$425$	−1.63059e6			−0.437897
$426$		0			0
$427$		−	5.59940e6i		−	1.48618i
$428$		0			0
$429$	407057.	−	895702.i	0.106785	−	0.234974i
$430$		0			0
$431$			6.21413e6i			1.61134i	0.592365	+	0.805670i	$0.298194\pi$
$431$			6.21413e6i			1.61134i	−0.592365	+	0.805670i	$0.701806\pi$
$432$		0			0
$433$	−5.83362e6			−1.49526			−0.747632	−	0.664113i	$-0.768809\pi$
$433$	−5.83362e6			−1.49526			−0.747632	+	0.664113i	$0.768809\pi$
$434$		0			0
$435$			502260.i			0.127264i
$436$		0			0
$437$			6.52287e6i			1.63394i
$438$		0			0
$439$	766922.			0.189928			0.0949642	−	0.995481i	$-0.469726\pi$
$439$	766922.			0.189928			0.0949642	+	0.995481i	$0.469726\pi$
$440$		0			0
$441$	−1.15930e6			−0.283857
$442$		0			0
$443$	−1.02042e6			−0.247042			−0.123521	−	0.992342i	$-0.539419\pi$
$443$	−1.02042e6			−0.247042			−0.123521	+	0.992342i	$0.539419\pi$
$444$		0			0
$445$	2.29207e6			0.548692
$446$		0			0
$447$			1.99128e6i			0.471372i
$448$		0			0
$449$			4.18123e6i			0.978787i	0.872063	+	0.489394i	$0.162782\pi$
$449$			4.18123e6i			0.978787i	−0.872063	+	0.489394i	$0.837218\pi$
$450$		0			0
$451$	−66172.8			−0.0153193
$452$		0			0
$453$		−	4.34203e6i		−	0.994139i
$454$		0			0
$455$	−3.63215e6	−	1.65065e6i	−0.822498	−	0.373789i
$456$		0			0
$457$			3.58333e6i			0.802594i	0.915948	+	0.401297i	$0.131441\pi$
$457$			3.58333e6i			0.802594i	−0.915948	+	0.401297i	$0.868559\pi$
$458$		0			0
$459$	−680260.			−0.150710
$460$		0			0
$461$		−	1.67459e6i		−	0.366992i	−0.983020	−	0.183496i	$-0.941259\pi$
$461$		−	1.67459e6i		−	0.366992i	0.983020	−	0.183496i	$-0.0587414\pi$
$462$		0			0
$463$		−	6.62362e6i		−	1.43596i	−0.696063	−	0.717981i	$-0.745066\pi$
$463$		−	6.62362e6i		−	1.43596i	0.696063	−	0.717981i	$-0.254934\pi$
$464$		0			0
$465$	−246268.			−0.0528172
$466$		0			0
$467$	−3.80744e6			−0.807868			−0.403934	−	0.914788i	$-0.632358\pi$
$467$	−3.80744e6			−0.807868			−0.403934	+	0.914788i	$0.632358\pi$
$468$		0			0
$469$	8.23138e6			1.72799
$470$		0			0
$471$	−996531.			−0.206985
$472$		0			0
$473$			3.62048e6i			0.744070i
$474$		0			0
$475$			4.08192e6i			0.830099i
$476$		0			0
$477$	−2.03163e6			−0.408836
$478$		0			0
$479$			4.85878e6i			0.967583i	0.875183	+	0.483792i	$0.160741\pi$
$479$			4.85878e6i			0.967583i	−0.875183	+	0.483792i	$0.839259\pi$
$480$		0			0
$481$	−951798.	+	2.09437e6i	−0.187578	+	0.412754i
$482$		0			0
$483$		−	4.43332e6i		−	0.864692i
$484$		0			0
$485$	1.04038e6			0.200834
$486$		0			0
$487$			71255.1i			0.0136142i	0.999977	+	0.00680712i	$0.00216679\pi$
$487$			71255.1i			0.0136142i	−0.999977	+	0.00680712i	$0.997833\pi$
$488$		0			0
$489$		−	1.44957e6i		−	0.274137i
$490$		0			0
$491$	6.78617e6			1.27034			0.635172	−	0.772371i	$-0.280929\pi$
$491$	6.78617e6			1.27034			0.635172	+	0.772371i	$0.280929\pi$
$492$		0			0
$493$	−1.40305e6			−0.259990
$494$		0			0
$495$	−539356.			−0.0989379
$496$		0			0
$497$	−1.04033e7			−1.88921
$498$		0			0
$499$		−	167487.i		−	0.0301114i	−0.999887	−	0.0150557i	$-0.995207\pi$
$499$		−	167487.i		−	0.0301114i	0.999887	−	0.0150557i	$-0.00479256\pi$
$500$		0			0
$501$		−	4.19920e6i		−	0.747433i
$502$		0			0
$503$	−1.00972e7			−1.77942			−0.889712	−	0.456522i	$-0.849095\pi$
$503$	−1.00972e7			−1.77942			−0.889712	+	0.456522i	$0.849095\pi$
$504$		0			0
$505$			3.80809e6i			0.664476i
$506$		0			0
$507$	2.19762e6	+	2.51734e6i	0.379693	+	0.434933i
$508$		0			0
$509$			9.79146e6i			1.67515i	0.546324	+	0.837574i	$0.316027\pi$
$509$			9.79146e6i			1.67515i	−0.546324	+	0.837574i	$0.683973\pi$
$510$		0			0
$511$	601758.			0.101946
$512$		0			0
$513$			1.70292e6i			0.285694i
$514$		0			0
$515$			7.56433e6i			1.25676i
$516$		0			0
$517$	3.68253e6			0.605927
$518$		0			0
$519$	−2.16690e6			−0.353119
$520$		0			0
$521$	−9.42352e6			−1.52096			−0.760481	−	0.649360i	$-0.775037\pi$
$521$	−9.42352e6			−1.52096			−0.760481	+	0.649360i	$0.775037\pi$
$522$		0			0
$523$	−5.16320e6			−0.825400			−0.412700	−	0.910867i	$-0.635414\pi$
$523$	−5.16320e6			−0.825400			−0.412700	+	0.910867i	$0.635414\pi$
$524$		0			0
$525$		−	2.77431e6i		−	0.439295i
$526$		0			0
$527$		−	687945.i		−	0.107901i
$528$		0			0
$529$	1.36094e6			0.211445
$530$		0			0
$531$		−	2.86143e6i		−	0.440400i
$532$		0			0
$533$	92988.3	−	204615.i	0.0141779	−	0.0311975i
$534$		0			0
$535$			5.46923e6i			0.826117i
$536$		0			0
$537$	2.39792e6			0.358838
$538$		0			0
$539$		−	2.56769e6i		−	0.380689i
$540$		0			0
$541$		−	1.03918e7i		−	1.52650i	−0.646104	−	0.763250i	$-0.723603\pi$
$541$		−	1.03918e7i		−	1.52650i	0.646104	−	0.763250i	$-0.276397\pi$
$542$		0			0
$543$	578765.			0.0842369
$544$		0			0
$545$	−291235.			−0.0420002
$546$		0			0
$547$	7.33920e6			1.04877			0.524385	−	0.851481i	$-0.324295\pi$
$547$	7.33920e6			1.04877			0.524385	+	0.851481i	$0.324295\pi$
$548$		0			0
$549$	2.57105e6			0.364066
$550$		0			0
$551$			3.51233e6i			0.492851i
$552$		0			0
$553$		−	1.14624e7i		−	1.59391i
$554$		0			0
$555$	1.26115e6			0.173793
$556$		0			0
$557$		−	7.97266e6i		−	1.08884i	−0.838812	−	0.544421i	$-0.816749\pi$
$557$		−	7.97266e6i		−	1.08884i	0.838812	−	0.544421i	$-0.183251\pi$
$558$		0			0
$559$	−1.11950e7	−	5.08763e6i	−1.51529	−	0.688630i
$560$		0			0
$561$		−	1.50668e6i		−	0.202122i
$562$		0			0
$563$	−5.23721e6			−0.696352			−0.348176	−	0.937429i	$-0.613199\pi$
$563$	−5.23721e6			−0.696352			−0.348176	+	0.937429i	$0.613199\pi$
$564$		0			0
$565$		−	7.15122e6i		−	0.942451i
$566$		0			0
$567$		−	1.15740e6i		−	0.151192i
$568$		0			0
$569$	−8.15975e6			−1.05657			−0.528283	−	0.849069i	$-0.677164\pi$
$569$	−8.15975e6			−1.05657			−0.528283	+	0.849069i	$0.677164\pi$
$570$		0			0
$571$	6.61371e6			0.848897			0.424448	−	0.905452i	$-0.360468\pi$
$571$	6.61371e6			0.848897			0.424448	+	0.905452i	$0.360468\pi$
$572$		0			0
$573$	8.32849e6			1.05969
$574$		0			0
$575$	4.87942e6			0.615458
$576$		0			0
$577$			4.82867e6i			0.603793i	0.953341	+	0.301896i	$0.0976196\pi$
$577$			4.82867e6i			0.603793i	−0.953341	+	0.301896i	$0.902380\pi$
$578$		0			0
$579$		−	5.69075e6i		−	0.705462i
$580$		0			0
$581$	−2.15806e6			−0.265230
$582$		0			0
$583$		−	4.49978e6i		−	0.548302i
$584$		0			0
$585$	757922.	−	1.66776e6i	0.0915661	−	0.201485i
$586$		0			0
$587$		−	7.52814e6i		−	0.901763i	−0.892584	−	0.450881i	$-0.851110\pi$
$587$		−	7.52814e6i		−	0.901763i	0.892584	−	0.450881i	$-0.148890\pi$
$588$		0			0
$589$	−1.72216e6			−0.204544
$590$		0			0
$591$			900430.i			0.106043i
$592$		0			0
$593$		−	8.22445e6i		−	0.960439i	−0.877148	−	0.480220i	$-0.840557\pi$
$593$		−	8.22445e6i		−	0.960439i	0.877148	−	0.480220i	$-0.159443\pi$
$594$		0			0
$595$	−6.10972e6			−0.707504
$596$		0			0
$597$	6.50769e6			0.747293
$598$		0			0
$599$	1.25959e6			0.143437			0.0717185	−	0.997425i	$-0.477152\pi$
$599$	1.25959e6			0.143437			0.0717185	+	0.997425i	$0.477152\pi$
$600$		0			0
$601$	6.56541e6			0.741439			0.370720	−	0.928745i	$-0.379111\pi$
$601$	6.56541e6			0.741439			0.370720	+	0.928745i	$0.379111\pi$
$602$		0			0
$603$			3.77957e6i			0.423301i
$604$		0			0
$605$			4.78294e6i			0.531259i
$606$		0			0
$607$	7.92228e6			0.872727			0.436364	−	0.899770i	$-0.356266\pi$
$607$	7.92228e6			0.872727			0.436364	+	0.899770i	$0.356266\pi$
$608$		0			0
$609$		−	2.38718e6i		−	0.260821i
$610$		0			0
$611$	−5.17482e6	+	1.13869e7i	−0.560780	+	1.23396i
$612$		0			0
$613$		−	9.74465e6i		−	1.04741i	−0.851901	−	0.523703i	$-0.824550\pi$
$613$		−	9.74465e6i		−	1.04741i	0.851901	−	0.523703i	$-0.175450\pi$
$614$		0			0
$615$	−123211.			−0.0131360
$616$		0			0
$617$			1.14334e7i			1.20910i	0.796569	+	0.604548i	$0.206646\pi$
$617$			1.14334e7i			1.20910i	−0.796569	+	0.604548i	$0.793354\pi$
$618$		0			0
$619$			1.07067e7i			1.12313i	0.827432	+	0.561566i	$0.189801\pi$
$619$			1.07067e7i			1.12313i	−0.827432	+	0.561566i	$0.810199\pi$
$620$		0			0
$621$	2.03563e6			0.211822
$622$		0			0
$623$	−1.08939e7			−1.12451
$624$		0			0
$625$	−1.25148e6			−0.128151
$626$		0			0
$627$	−3.77174e6			−0.383154
$628$		0			0
$629$			3.52299e6i			0.355046i
$630$		0			0
$631$			475976.i			0.0475895i	0.999717	+	0.0237948i	$0.00757483\pi$
$631$			475976.i			0.0475895i	−0.999717	+	0.0237948i	$0.992425\pi$
$632$		0			0
$633$	−2.33593e6			−0.231714
$634$		0			0
$635$		−	9.48844e6i		−	0.933814i
$636$		0			0
$637$	7.93963e6	+	3.60820e6i	0.775267	+	0.352324i
$638$		0			0
$639$		−	4.77684e6i		−	0.462795i
$640$		0			0
$641$	−1.90742e7			−1.83358			−0.916791	−	0.399368i	$-0.869230\pi$
$641$	−1.90742e7			−1.83358			−0.916791	+	0.399368i	$0.869230\pi$
$642$		0			0
$643$			1.16474e7i			1.11097i	0.831528	+	0.555483i	$0.187466\pi$
$643$			1.16474e7i			1.11097i	−0.831528	+	0.555483i	$0.812534\pi$
$644$		0			0
$645$			6.74118e6i			0.638024i
$646$		0			0
$647$	−1.83045e6			−0.171908			−0.0859540	−	0.996299i	$-0.527394\pi$
$647$	−1.83045e6			−0.171908			−0.0859540	+	0.996299i	$0.527394\pi$
$648$		0			0
$649$	6.33768e6			0.590634
$650$		0			0
$651$	1.17048e6			0.108246
$652$		0			0
$653$	1.75829e7			1.61365			0.806824	−	0.590792i	$-0.201185\pi$
$653$	1.75829e7			1.61365			0.806824	+	0.590792i	$0.201185\pi$
$654$		0			0
$655$			4.98304e6i			0.453828i
$656$		0			0
$657$			276307.i			0.0249735i
$658$		0			0
$659$	1.84370e7			1.65378			0.826888	−	0.562367i	$-0.190109\pi$
$659$	1.84370e7			1.65378			0.826888	+	0.562367i	$0.190109\pi$
$660$		0			0
$661$			1.08046e7i			0.961845i	0.876763	+	0.480922i	$0.159698\pi$
$661$			1.08046e7i			0.961845i	−0.876763	+	0.480922i	$0.840302\pi$
$662$		0			0
$663$	4.65886e6	+	2.11724e6i	0.411619	+	0.187062i
$664$		0			0
$665$			1.52947e7i			1.34118i
$666$		0			0
$667$	4.19855e6			0.365413
$668$		0			0
$669$			9.64761e6i			0.833402i
$670$		0			0
$671$			5.69453e6i			0.488260i
$672$		0			0
$673$	−1.57676e7			−1.34193			−0.670964	−	0.741490i	$-0.734119\pi$
$673$	−1.57676e7			−1.34193			−0.670964	+	0.741490i	$0.734119\pi$
$674$		0			0
$675$	1.27387e6			0.107613
$676$		0			0
$677$	2.86244e6			0.240029			0.120015	−	0.992772i	$-0.461706\pi$
$677$	2.86244e6			0.240029			0.120015	+	0.992772i	$0.461706\pi$
$678$		0			0
$679$	−4.94480e6			−0.411599
$680$		0			0
$681$		−	586890.i		−	0.0484941i
$682$		0			0
$683$		−	4.91047e6i		−	0.402783i	−0.979511	−	0.201391i	$-0.935454\pi$
$683$		−	4.91047e6i		−	0.402783i	0.979511	−	0.201391i	$-0.0645463\pi$
$684$		0			0
$685$	2.76043e6			0.224776
$686$		0			0
$687$			1.72669e6i			0.139580i
$688$		0			0
$689$	1.39139e7	+	6.32325e6i	1.11661	+	0.507449i
$690$		0			0
$691$		−	1.79749e7i		−	1.43209i	−0.698052	−	0.716047i	$-0.745949\pi$
$691$		−	1.79749e7i		−	1.43209i	0.698052	−	0.716047i	$-0.254051\pi$
$692$		0			0
$693$	2.56349e6			0.202768
$694$		0			0
$695$		−	1.34910e6i		−	0.105945i
$696$		0			0
$697$		−	344188.i		−	0.0268357i
$698$		0			0
$699$	3.18528e6			0.246579
$700$		0			0
$701$	−8.74640e6			−0.672255			−0.336128	−	0.941816i	$-0.609117\pi$
$701$	−8.74640e6			−0.672255			−0.336128	+	0.941816i	$0.609117\pi$
$702$		0			0
$703$	8.81925e6			0.673044
$704$		0			0
$705$	6.85671e6			0.519569
$706$		0			0
$707$		−	1.80994e7i		−	1.36181i
$708$		0			0
$709$		−	2.43397e6i		−	0.181844i	−0.995858	−	0.0909221i	$-0.971019\pi$
$709$		−	2.43397e6i		−	0.181844i	0.995858	−	0.0909221i	$-0.0289814\pi$
$710$		0			0
$711$	5.26316e6			0.390457
$712$		0			0
$713$			2.05863e6i			0.151654i
$714$		0			0
$715$	3.69386e6	+	1.67869e6i	0.270218	+	0.122802i
$716$		0			0
$717$			1.03836e7i			0.754313i
$718$		0			0
$719$	−6.81620e6			−0.491723			−0.245861	−	0.969305i	$-0.579071\pi$
$719$	−6.81620e6			−0.491723			−0.245861	+	0.969305i	$0.579071\pi$
$720$		0			0
$721$		−	3.59523e7i		−	2.57566i
$722$		0			0
$723$			1.88778e6i			0.134309i
$724$		0			0
$725$	2.62739e6			0.185643
$726$		0			0
$727$	4.33605e6			0.304269			0.152135	−	0.988360i	$-0.451385\pi$
$727$	4.33605e6			0.304269			0.152135	+	0.988360i	$0.451385\pi$
$728$		0			0
$729$	531441.			0.0370370
$730$		0			0
$731$	−1.88314e7			−1.30343
$732$		0			0
$733$			2.01689e7i			1.38651i	0.720693	+	0.693254i	$0.243824\pi$
$733$			2.01689e7i			1.38651i	−0.720693	+	0.693254i	$0.756176\pi$
$734$		0			0
$735$		−	4.78093e6i		−	0.326433i
$736$		0			0
$737$	−8.37122e6			−0.567702
$738$		0			0
$739$		−	2.05423e7i		−	1.38369i	−0.722048	−	0.691843i	$-0.756799\pi$
$739$		−	2.05423e7i		−	1.38369i	0.722048	−	0.691843i	$-0.243201\pi$
$740$		0			0
$741$	5.30018e6	−	1.16627e7i	0.354605	−	0.780286i
$742$		0			0
$743$		−	1.48776e7i		−	0.988689i	−0.869266	−	0.494345i	$-0.835408\pi$
$743$		−	1.48776e7i		−	0.988689i	0.869266	−	0.494345i	$-0.164592\pi$
$744$		0			0
$745$	−8.21200e6			−0.542074
$746$		0			0
$747$		−	990907.i		−	0.0649728i
$748$		0			0
$749$		−	2.59946e7i		−	1.69308i
$750$		0			0
$751$	2.94950e7			1.90831			0.954154	−	0.299315i	$-0.0967580\pi$
$751$	2.94950e7			1.90831			0.954154	+	0.299315i	$0.0967580\pi$
$752$		0			0
$753$	−3.84440e6			−0.247082
$754$		0			0
$755$	1.79064e7			1.14325
$756$		0			0
$757$	2.01715e7			1.27938			0.639690	−	0.768633i	$-0.279063\pi$
$757$	2.01715e7			1.27938			0.639690	+	0.768633i	$0.279063\pi$
$758$		0			0
$759$			4.50864e6i			0.284080i
$760$		0			0
$761$			2.06956e7i			1.29544i	0.761880	+	0.647719i	$0.224277\pi$
$761$			2.06956e7i			1.29544i	−0.761880	+	0.647719i	$0.775723\pi$
$762$		0			0
$763$	1.38420e6			0.0860772
$764$		0			0
$765$		−	2.80538e6i		−	0.173316i
$766$		0			0
$767$	−8.90592e6	+	1.95969e7i	−0.546626	+	1.20282i
$768$		0			0
$769$		−	1.69130e7i		−	1.03135i	−0.856785	−	0.515674i	$-0.827542\pi$
$769$		−	1.69130e7i		−	1.03135i	0.856785	−	0.515674i	$-0.172458\pi$
$770$		0			0
$771$	1.86950e6			0.113263
$772$		0			0
$773$		−	3.10969e7i		−	1.87184i	−0.352212	−	0.935920i	$-0.614570\pi$
$773$		−	3.10969e7i		−	1.87184i	0.352212	−	0.935920i	$-0.385430\pi$
$774$		0			0
$775$			1.28826e6i			0.0770458i
$776$		0			0
$777$	−5.99407e6			−0.356180
$778$		0			0
$779$	−861619.			−0.0508712
$780$		0			0
$781$	1.05800e7			0.620669
$782$		0			0
$783$	1.09611e6			0.0638926
$784$		0			0
$785$		−	4.10967e6i		−	0.238031i
$786$		0			0
$787$			628913.i			0.0361954i	0.999836	+	0.0180977i	$0.00576099\pi$
$787$			628913.i			0.0361954i	−0.999836	+	0.0180977i	$0.994239\pi$
$788$		0			0
$789$	7.48541e6			0.428078
$790$		0			0
$791$			3.39888e7i			1.93150i
$792$		0			0
$793$	−1.76082e7	−	8.00215e6i	−0.994335	−	0.451881i
$794$		0			0
$795$		−	8.37840e6i		−	0.470157i
$796$		0			0
$797$	−2.43866e7			−1.35990			−0.679948	−	0.733260i	$-0.737998\pi$
$797$	−2.43866e7			−1.35990			−0.679948	+	0.733260i	$0.737998\pi$
$798$		0			0
$799$			1.91541e7i			1.06144i
$800$		0			0
$801$		−	5.00213e6i		−	0.275470i
$802$		0			0
$803$	−611982.			−0.0334927
$804$		0			0
$805$	1.82829e7			0.994388
$806$		0			0
$807$	1.50628e7			0.814183
$808$		0			0
$809$	−2.86033e7			−1.53654			−0.768272	−	0.640123i	$-0.778883\pi$
$809$	−2.86033e7			−1.53654			−0.768272	+	0.640123i	$0.778883\pi$
$810$		0			0
$811$		−	5.97923e6i		−	0.319222i	−0.987180	−	0.159611i	$-0.948976\pi$
$811$		−	5.97923e6i		−	0.319222i	0.987180	−	0.159611i	$-0.0510240\pi$
$812$		0			0
$813$		−	628133.i		−	0.0333292i
$814$		0			0
$815$	5.97800e6			0.315255
$816$		0			0
$817$			4.71414e7i			2.47085i
$818$		0			0
$819$	−3.60231e6	+	7.92665e6i	−0.187660	+	0.412933i
$820$		0			0
$821$		−	1.46642e7i		−	0.759279i	−0.925134	−	0.379640i	$-0.876048\pi$
$821$		−	1.46642e7i		−	0.759279i	0.925134	−	0.379640i	$-0.123952\pi$
$822$		0			0
$823$	−1.80149e7			−0.927113			−0.463557	−	0.886067i	$-0.653427\pi$
$823$	−1.80149e7			−0.927113			−0.463557	+	0.886067i	$0.653427\pi$
$824$		0			0
$825$			2.82144e6i			0.144323i
$826$		0			0
$827$		−	9.93027e6i		−	0.504891i	−0.967611	−	0.252445i	$-0.918765\pi$
$827$		−	9.93027e6i		−	0.504891i	0.967611	−	0.252445i	$-0.0812348\pi$
$828$		0			0
$829$	2.97991e7			1.50597			0.752986	−	0.658037i	$-0.228613\pi$
$829$	2.97991e7			1.50597			0.752986	+	0.658037i	$0.228613\pi$
$830$		0			0
$831$	1.45242e7			0.729608
$832$		0			0
$833$	1.33554e7			0.666877
$834$		0			0
$835$	1.73174e7			0.859542
$836$		0			0
$837$			537445.i			0.0265168i
$838$		0			0
$839$			1.35283e7i			0.663496i	0.943368	+	0.331748i	$0.107638\pi$
$839$			1.35283e7i			0.663496i	−0.943368	+	0.331748i	$0.892362\pi$
$840$		0			0
$841$	−1.82504e7			−0.889779
$842$		0			0
$843$			9.50897e6i			0.460855i
$844$		0			0
$845$	−1.03815e7	+	9.06293e6i	−0.500169	+	0.436643i
$846$		0			0
$847$		−	2.27327e7i		−	1.08879i
$848$		0			0
$849$	4.61467e6			0.219721
$850$		0			0
$851$		−	1.05423e7i		−	0.499013i
$852$		0			0
$853$		−	9.03698e6i		−	0.425257i	−0.977133	−	0.212628i	$-0.931798\pi$
$853$		−	9.03698e6i		−	0.425257i	0.977133	−	0.212628i	$-0.0682023\pi$
$854$		0			0
$855$	−7.02282e6			−0.328546
$856$		0			0
$857$	−9.55364e6			−0.444341			−0.222171	−	0.975008i	$-0.571314\pi$
$857$	−9.55364e6			−0.444341			−0.222171	+	0.975008i	$0.571314\pi$
$858$		0			0
$859$	−3.73649e6			−0.172775			−0.0863874	−	0.996262i	$-0.527532\pi$
$859$	−3.73649e6			−0.172775			−0.0863874	+	0.996262i	$0.527532\pi$
$860$		0			0
$861$	585606.			0.0269214
$862$		0			0
$863$		−	3.44318e7i		−	1.57374i	−0.617120	−	0.786869i	$-0.711701\pi$
$863$		−	3.44318e7i		−	1.57374i	0.617120	−	0.786869i	$-0.288299\pi$
$864$		0			0
$865$		−	8.93626e6i		−	0.406084i
$866$		0			0
$867$	−4.94194e6			−0.223280
$868$		0			0
$869$			1.16572e7i			0.523653i
$870$		0			0
$871$	1.17635e7	−	2.58849e7i	0.525403	−	1.15612i
$872$		0			0
$873$		−	2.27048e6i		−	0.100828i
$874$		0			0
$875$	3.19020e7			1.40863
$876$		0			0
$877$		−	2.12195e7i		−	0.931613i	−0.884887	−	0.465807i	$-0.845764\pi$
$877$		−	2.12195e7i		−	0.931613i	0.884887	−	0.465807i	$-0.154236\pi$
$878$		0			0
$879$			2.44141e7i			1.06578i
$880$		0			0
$881$	3.83954e7			1.66663			0.833315	−	0.552798i	$-0.186440\pi$
$881$	3.83954e7			1.66663			0.833315	+	0.552798i	$0.186440\pi$
$882$		0			0
$883$	−3.58049e7			−1.54540			−0.772699	−	0.634773i	$-0.781094\pi$
$883$	−3.58049e7			−1.54540			−0.772699	+	0.634773i	$0.781094\pi$
$884$		0			0
$885$	1.18005e7			0.506456
$886$		0			0
$887$	−2.52692e7			−1.07841			−0.539203	−	0.842176i	$-0.681274\pi$
$887$	−2.52692e7			−1.07841			−0.539203	+	0.842176i	$0.681274\pi$
$888$		0			0
$889$			4.50973e7i			1.91380i
$890$		0			0
$891$			1.17707e6i			0.0496715i
$892$		0			0
$893$	4.79493e7			2.01212
$894$		0			0
$895$			9.88896e6i			0.412661i
$896$		0			0
$897$	−1.39413e7	−	6.33570e6i	−0.578525	−	0.262914i
$898$		0			0
$899$			1.10850e6i			0.0457441i
$900$		0			0
$901$	2.34049e7			0.960495
$902$		0			0
$903$		−	3.20400e7i		−	1.30759i
$904$		0			0
$905$			2.38681e6i			0.0968717i
$906$		0			0
$907$	2.49639e7			1.00761			0.503806	−	0.863817i	$-0.331933\pi$
$907$	2.49639e7			1.00761			0.503806	+	0.863817i	$0.331933\pi$
$908$		0			0
$909$	8.31063e6			0.333599
$910$		0			0
$911$	807347.			0.0322303			0.0161151	−	0.999870i	$-0.494870\pi$
$911$	807347.			0.0322303			0.0161151	+	0.999870i	$0.494870\pi$
$912$		0			0
$913$	2.19472e6			0.0871370
$914$		0			0
$915$			1.06030e7i			0.418673i
$916$		0			0
$917$		−	2.36838e7i		−	0.930095i
$918$		0			0
$919$	2.92665e6			0.114309			0.0571547	−	0.998365i	$-0.481797\pi$
$919$	2.92665e6			0.114309			0.0571547	+	0.998365i	$0.481797\pi$
$920$		0			0
$921$		−	2.20027e7i		−	0.854726i
$922$		0			0
$923$	−1.48674e7	+	3.27149e7i	−0.574423	+	1.26398i
$924$		0			0
$925$		−	6.59722e6i		−	0.253517i
$926$		0			0
$927$	1.65081e7			0.630953
$928$		0			0
$929$			2.03515e7i			0.773672i	0.922148	+	0.386836i	$0.126432\pi$
$929$			2.03515e7i			0.773672i	−0.922148	+	0.386836i	$0.873568\pi$
$930$		0			0
$931$		−	3.34332e7i		−	1.26417i
$932$		0			0
$933$	−1.79265e7			−0.674203
$934$		0			0
$935$	6.21352e6			0.232439
$936$		0			0
$937$	−659937.			−0.0245558			−0.0122779	−	0.999925i	$-0.503908\pi$
$937$	−659937.			−0.0245558			−0.0122779	+	0.999925i	$0.503908\pi$
$938$		0			0
$939$	1.49757e7			0.554271
$940$		0			0
$941$		−	3.40074e7i		−	1.25199i	−0.779828	−	0.625994i	$-0.784694\pi$
$941$		−	3.40074e7i		−	1.25199i	0.779828	−	0.625994i	$-0.215306\pi$
$942$		0			0
$943$			1.02996e6i			0.0377173i
$944$		0			0
$945$	4.77311e6			0.173869
$946$		0			0
$947$			1.44758e7i			0.524527i	0.964996	+	0.262263i	$0.0844689\pi$
$947$			1.44758e7i			0.524527i	−0.964996	+	0.262263i	$0.915531\pi$
$948$		0			0
$949$	859978.	−	1.89233e6i	0.0309972	−	0.0682073i
$950$		0			0
$951$		−	2.88434e7i		−	1.03418i
$952$		0			0
$953$	1.88172e7			0.671156			0.335578	−	0.942012i	$-0.391068\pi$
$953$	1.88172e7			0.671156			0.335578	+	0.942012i	$0.391068\pi$
$954$		0			0
$955$			3.43465e7i			1.21864i
$956$		0			0
$957$			2.42774e6i			0.0856883i
$958$		0			0
$959$	−1.31200e7			−0.460666
$960$		0			0
$961$	2.80856e7			0.981015
$962$		0			0
$963$	1.19358e7			0.414750
$964$		0			0
$965$	2.34685e7			0.811275
$966$		0			0
$967$		−	4.30702e7i		−	1.48119i	−0.671952	−	0.740595i	$-0.734544\pi$
$967$		−	4.30702e7i		−	1.48119i	0.671952	−	0.740595i	$-0.265456\pi$
$968$		0			0
$969$		−	1.96181e7i		−	0.671194i
$970$		0			0
$971$	−2.85474e7			−0.971668			−0.485834	−	0.874051i	$-0.661484\pi$
$971$	−2.85474e7			−0.971668			−0.485834	+	0.874051i	$0.661484\pi$
$972$		0			0
$973$			6.41210e6i			0.217129i
$974$		0			0
$975$	−8.72426e6	−	3.96479e6i	−0.293912	−	0.133570i
$976$		0			0
$977$			4.23484e7i			1.41939i	0.704510	+	0.709694i	$0.251167\pi$
$977$			4.23484e7i			1.41939i	−0.704510	+	0.709694i	$0.748833\pi$
$978$		0			0
$979$	1.10790e7			0.369441
$980$		0			0
$981$			635579.i			0.0210861i
$982$		0			0
$983$			1.39064e7i			0.459020i	0.973306	+	0.229510i	$0.0737124\pi$
$983$			1.39064e7i			0.459020i	−0.973306	+	0.229510i	$0.926288\pi$
$984$		0			0
$985$	−3.71335e6			−0.121948
$986$		0			0
$987$	−3.25891e7			−1.06483
$988$		0			0
$989$	5.63516e7			1.83196
$990$		0			0
$991$	3.49486e6			0.113043			0.0565217	−	0.998401i	$-0.481999\pi$
$991$	3.49486e6			0.113043			0.0565217	+	0.998401i	$0.481999\pi$
$992$		0			0
$993$			3.18047e7i			1.02357i
$994$		0			0
$995$			2.68376e7i			0.859380i
$996$		0			0
$997$	77983.9			0.00248466			0.00124233	−	0.999999i	$-0.499605\pi$
$997$	77983.9			0.00248466			0.00124233	+	0.999999i	$0.499605\pi$
$998$		0			0
$999$		−	2.75227e6i		−	0.0872525i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.6.c.d.337.5		6
4.3	odd	2		78.6.b.a.25.6	yes	6
12.11	even	2		234.6.b.c.181.1		6
13.12	even	2	inner	624.6.c.d.337.2		6
52.31	even	4		1014.6.a.q.1.3		3
52.47	even	4		1014.6.a.o.1.1		3
52.51	odd	2		78.6.b.a.25.1	✓	6
156.155	even	2		234.6.b.c.181.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.6.b.a.25.1	✓	6	52.51	odd	2
78.6.b.a.25.6	yes	6	4.3	odd	2
234.6.b.c.181.1		6	12.11	even	2
234.6.b.c.181.6		6	156.155	even	2
624.6.c.d.337.2		6	13.12	even	2	inner
624.6.c.d.337.5		6	1.1	even	1	trivial
1014.6.a.o.1.1		3	52.47	even	4
1014.6.a.q.1.3		3	52.31	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	624.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+9.00000 q^{3} +37.1158i q^{5} -176.407i q^{7} +81.0000 q^{9} +179.404i q^{11} +(-554.740 - 252.104i) q^{13} +334.042i q^{15} -933.141 q^{17} +2335.97i q^{19} -1587.66i q^{21} +2792.36 q^{23} +1747.42 q^{25} +729.000 q^{27} +1503.58 q^{29} +737.236i q^{31} +1614.63i q^{33} +6547.48 q^{35} -3775.41i q^{37} +(-4992.66 - 2268.94i) q^{39} +368.848i q^{41} +20180.6 q^{43} +3006.38i q^{45} -20526.5i q^{47} -14312.3 q^{49} -8398.27 q^{51} -25081.9 q^{53} -6658.72 q^{55} +21023.7i q^{57} -35326.3i q^{59} +31741.4 q^{61} -14288.9i q^{63} +(9357.06 - 20589.6i) q^{65} +46661.3i q^{67} +25131.2 q^{69} -58973.4i q^{71} +3411.20i q^{73} +15726.8 q^{75} +31648.1 q^{77} +64977.3 q^{79} +6561.00 q^{81} -12233.4i q^{83} -34634.3i q^{85} +13532.2 q^{87} -61754.7i q^{89} +(-44472.9 + 97859.9i) q^{91} +6635.12i q^{93} -86701.4 q^{95} -28030.7i q^{97} +14531.7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 54 q^{3} + 486 q^{9} + 530 q^{13} - 836 q^{17} + 416 q^{23} + 718 q^{25} + 4374 q^{27} + 18788 q^{29} - 6112 q^{35} + 4770 q^{39} + 24200 q^{43} - 3038 q^{49} - 7524 q^{51} - 42396 q^{53} - 124656 q^{55}+ \cdots - 567168 q^{95}+O(q^{100}) \) 6 * q + 54 * q^3 + 486 * q^9 + 530 * q^13 - 836 * q^17 + 416 * q^23 + 718 * q^25 + 4374 * q^27 + 18788 * q^29 - 6112 * q^35 + 4770 * q^39 + 24200 * q^43 - 3038 * q^49 - 7524 * q^51 - 42396 * q^53 - 124656 * q^55 - 3196 * q^61 - 17168 * q^65 + 3744 * q^69 + 6462 * q^75 - 114024 * q^77 + 169328 * q^79 + 39366 * q^81 + 169092 * q^87 - 236152 * q^91 - 567168 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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