LMFDB - Embedded newform 234.6.b.c.181.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [234,6,Mod(181,234)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(234, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("234.181");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 234 = 2 \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	234.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$37.5298138362$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{11}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 78)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			181.1
Root			$6.10758 + 6.10758i$ of defining polynomial
Character	$\chi$	$=$	234.181
Dual form			234.6.b.c.181.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000i q^{2} -16.0000 q^{4} -37.1158i q^{5} +176.407i q^{7} +64.0000i q^{8} +O(q^{10})$	\(q-4.00000i q^{2} -16.0000 q^{4} -37.1158i q^{5} +176.407i q^{7} +64.0000i q^{8} -148.463 q^{10} +179.404i q^{11} +(-554.740 - 252.104i) q^{13} +705.627 q^{14} +256.000 q^{16} +933.141 q^{17} -2335.97i q^{19} +593.853i q^{20} +717.615 q^{22} +2792.36 q^{23} +1747.42 q^{25} +(-1008.42 + 2218.96i) q^{26} -2822.51i q^{28} -1503.58 q^{29} -737.236i q^{31} -1024.00i q^{32} -3732.56i q^{34} +6547.48 q^{35} -3775.41i q^{37} -9343.88 q^{38} +2375.41 q^{40} -368.848i q^{41} -20180.6 q^{43} -2870.46i q^{44} -11169.4i q^{46} -20526.5i q^{47} -14312.3 q^{49} -6989.67i q^{50} +(8875.84 + 4033.67i) q^{52} +25081.9 q^{53} +6658.72 q^{55} -11290.0 q^{56} +6014.33i q^{58} -35326.3i q^{59} +31741.4 q^{61} -2948.94 q^{62} -4096.00 q^{64} +(-9357.06 + 20589.6i) q^{65} -46661.3i q^{67} -14930.3 q^{68} -26189.9i q^{70} -58973.4i q^{71} +3411.20i q^{73} -15101.6 q^{74} +37375.5i q^{76} -31648.1 q^{77} -64977.3 q^{79} -9501.64i q^{80} -1475.39 q^{82} -12233.4i q^{83} -34634.3i q^{85} +80722.5i q^{86} -11481.8 q^{88} +61754.7i q^{89} +(44472.9 - 97859.9i) q^{91} -44677.8 q^{92} -82106.0 q^{94} -86701.4 q^{95} -28030.7i q^{97} +57249.4i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 96 q^{4}+O(q^{10}) $ 6 * q - 96 * q^4	$ 6 q - 96 q^{4} + 320 q^{10} + 530 q^{13} + 1360 q^{14} + 1536 q^{16} + 836 q^{17} - 1296 q^{22} + 416 q^{23} + 718 q^{25} + 1360 q^{26} - 18788 q^{29} - 6112 q^{35} - 528 q^{38} - 5120 q^{40} - 24200 q^{43} - 3038 q^{49} - 8480 q^{52} + 42396 q^{53} + 124656 q^{55} - 21760 q^{56} - 3196 q^{61} + 59344 q^{62} - 24576 q^{64} + 17168 q^{65} - 13376 q^{68} + 62240 q^{74} + 114024 q^{77} - 169328 q^{79} + 145120 q^{82} + 20736 q^{88} + 236152 q^{91} - 6656 q^{92} - 32688 q^{94} - 567168 q^{95}+O(q^{100}) $ 6 * q - 96 * q^4 + 320 * q^10 + 530 * q^13 + 1360 * q^14 + 1536 * q^16 + 836 * q^17 - 1296 * q^22 + 416 * q^23 + 718 * q^25 + 1360 * q^26 - 18788 * q^29 - 6112 * q^35 - 528 * q^38 - 5120 * q^40 - 24200 * q^43 - 3038 * q^49 - 8480 * q^52 + 42396 * q^53 + 124656 * q^55 - 21760 * q^56 - 3196 * q^61 + 59344 * q^62 - 24576 * q^64 + 17168 * q^65 - 13376 * q^68 + 62240 * q^74 + 114024 * q^77 - 169328 * q^79 + 145120 * q^82 + 20736 * q^88 + 236152 * q^91 - 6656 * q^92 - 32688 * q^94 - 567168 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/234\mathbb{Z}\right)^\times$.

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

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 $

$2$		−	4.00000i		−	0.707107i
$2$		−	4.00000i		−	0.707107i
$3$		0			0
$3$		0			0
$4$	−16.0000			−0.500000
$5$		−	37.1158i		−	0.663948i	−0.943289	−	0.331974i	$-0.892285\pi$
$5$		−	37.1158i		−	0.663948i	0.943289	−	0.331974i	$-0.107715\pi$
$6$		0			0
$7$			176.407i			1.36072i	0.732876	+	0.680362i	$0.238177\pi$
$7$			176.407i			1.36072i	−0.732876	+	0.680362i	$0.761823\pi$
$8$			64.0000i			0.353553i
$9$		0			0
$10$	−148.463			−0.469482
$11$			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
$14$	705.627			0.962177
$15$		0			0
$16$	256.000			0.250000
$17$	933.141			0.783114			0.391557	−	0.920154i	$-0.371937\pi$
$17$	933.141			0.783114			0.391557	+	0.920154i	$0.371937\pi$
$18$		0			0
$19$		−	2335.97i		−	1.48451i	−0.670117	−	0.742256i	$-0.733756\pi$
$19$		−	2335.97i		−	1.48451i	0.670117	−	0.742256i	$-0.266244\pi$
$20$			593.853i			0.331974i
$21$		0			0
$22$	717.615			0.316108
$23$	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
$26$	−1008.42	+	2218.96i	−0.292555	+	0.643748i
$27$		0			0
$28$		−	2822.51i		−	0.680362i
$29$	−1503.58			−0.331996			−0.165998	−	0.986126i	$-0.553084\pi$
$29$	−1503.58			−0.331996			−0.165998	+	0.986126i	$0.553084\pi$
$30$		0			0
$31$		−	737.236i		−	0.137785i	−0.997624	−	0.0688926i	$-0.978053\pi$
$31$		−	737.236i		−	0.137785i	0.997624	−	0.0688926i	$-0.0219466\pi$
$32$		−	1024.00i		−	0.176777i
$33$		0			0
$34$		−	3732.56i		−	0.553745i
$35$	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	$-0.0727900\pi$
$38$	−9343.88			−1.04971
$39$		0			0
$40$	2375.41			0.234741
$41$		−	368.848i		−	0.0342680i	−0.999853	−	0.0171340i	$-0.994546\pi$
$41$		−	368.848i		−	0.0342680i	0.999853	−	0.0171340i	$-0.00545418\pi$
$42$		0			0
$43$	−20180.6			−1.66442			−0.832211	−	0.554459i	$-0.812925\pi$
$43$	−20180.6			−1.66442			−0.832211	+	0.554459i	$0.812925\pi$
$44$		−	2870.46i		−	0.223522i
$45$		0			0
$46$		−	11169.4i		−	0.778282i
$47$		−	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
$50$		−	6989.67i		−	0.395395i
$51$		0			0
$52$	8875.84	+	4033.67i	0.455199	+	0.206867i
$53$	25081.9			1.22651			0.613253	−	0.789886i	$-0.289860\pi$
$53$	25081.9			1.22651			0.613253	+	0.789886i	$0.289860\pi$
$54$		0			0
$55$	6658.72			0.296814
$56$	−11290.0			−0.481089
$57$		0			0
$58$			6014.33i			0.234756i
$59$		−	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	$0.316112\pi$
$62$	−2948.94			−0.0974288
$63$		0			0
$64$	−4096.00			−0.125000
$65$	−9357.06	+	20589.6i	−0.274698	+	0.604456i
$66$		0			0
$67$		−	46661.3i		−	1.26990i	−0.772552	−	0.634951i	$-0.781020\pi$
$67$		−	46661.3i		−	1.26990i	0.772552	−	0.634951i	$-0.218980\pi$
$68$	−14930.3			−0.391557
$69$		0			0
$70$		−	26189.9i		−	0.638835i
$71$		−	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	$0.988073\pi$
$74$	−15101.6			−0.320586
$75$		0			0
$76$			37375.5i			0.742256i
$77$	−31648.1			−0.608303
$78$		0			0
$79$	−64977.3			−1.17137			−0.585685	−	0.810539i	$-0.699174\pi$
$79$	−64977.3			−1.17137			−0.585685	+	0.810539i	$0.699174\pi$
$80$		−	9501.64i		−	0.165987i
$81$		0			0
$82$	−1475.39			−0.0242311
$83$		−	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
$86$			80722.5i			1.17692i
$87$		0			0
$88$	−11481.8			−0.158054
$89$			61754.7i			0.826409i	0.910638	+	0.413204i	$0.135590\pi$
$89$			61754.7i			0.826409i	−0.910638	+	0.413204i	$0.864410\pi$
$90$		0			0
$91$	44472.9	−	97859.9i	0.562979	−	1.23880i
$92$	−44677.8			−0.550328
$93$		0			0
$94$	−82106.0			−0.958418
$95$	−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	$-0.0483274\pi$
$98$			57249.4i			0.602151i
$99$		0			0
$100$	−27958.7			−0.279587
$101$	−102600.			−1.00080			−0.500398	−	0.865796i	$-0.666813\pi$
$101$	−102600.			−1.00080			−0.500398	+	0.865796i	$0.666813\pi$
$102$		0			0
$103$	−203803.			−1.89286			−0.946430	−	0.322909i	$-0.895339\pi$
$103$	−203803.			−1.89286			−0.946430	+	0.322909i	$0.895339\pi$
$104$	16134.7	−	35503.4i	0.146277	−	0.321874i
$105$		0			0
$106$		−	100327.i		−	0.867271i
$107$	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	$0.989930\pi$
$110$		−	26634.9i		−	0.209879i
$111$		0			0
$112$			45160.1i			0.340181i
$113$	192673.			1.41947			0.709733	−	0.704471i	$-0.248816\pi$
$113$	192673.			1.41947			0.709733	+	0.704471i	$0.248816\pi$
$114$		0			0
$115$		−	103641.i		−	0.730778i
$116$	24057.3			0.165998
$117$		0			0
$118$	−141305.			−0.934229
$119$			164612.i			1.06560i
$120$		0			0
$121$	128865.			0.800152
$122$		−	126966.i		−	0.772301i
$123$		0			0
$124$			11795.8i			0.0688926i
$125$		−	180844.i		−	1.03521i
$126$		0			0
$127$	255644.			1.40646			0.703229	−	0.710964i	$-0.251741\pi$
$127$	255644.			1.40646			0.703229	+	0.710964i	$0.251741\pi$
$128$			16384.0i			0.0883883i
$129$		0			0
$130$	82358.5	+	37428.2i	0.427415	+	0.194241i
$131$	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
$134$	−186645.			−0.897956
$135$		0			0
$136$			59721.0i			0.276873i
$137$			74373.4i			0.338545i	0.985569	+	0.169272i	$0.0541418\pi$
$137$			74373.4i			0.338545i	−0.985569	+	0.169272i	$0.945858\pi$
$138$		0			0
$139$	36348.4			0.159569			0.0797844	−	0.996812i	$-0.474577\pi$
$139$	36348.4			0.159569			0.0797844	+	0.996812i	$0.474577\pi$
$140$	−104760.			−0.451725
$141$		0			0
$142$	−235893.			−0.981737
$143$	45228.5	−	99522.5i	0.184958	−	0.406988i
$144$		0			0
$145$			55806.7i			0.220428i
$146$	13644.8			0.0529767
$147$		0			0
$148$			60406.6i			0.226689i
$149$		−	221253.i		−	0.816440i	−0.912884	−	0.408220i	$-0.866150\pi$
$149$		−	221253.i		−	0.816440i	0.912884	−	0.408220i	$-0.133850\pi$
$150$		0			0
$151$			482448.i			1.72190i	0.508690	+	0.860950i	$0.330130\pi$
$151$			482448.i			1.72190i	−0.508690	+	0.860950i	$0.669870\pi$
$152$	149502.			0.524854
$153$		0			0
$154$			126592.i			0.430135i
$155$	−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	$0.557368\pi$
$158$			259909.i			0.828283i
$159$		0			0
$160$	−38006.6			−0.117370
$161$			492591.i			1.49769i
$162$		0			0
$163$			161064.i			0.474819i	0.971410	+	0.237410i	$0.0762984\pi$
$163$			161064.i			0.474819i	−0.971410	+	0.237410i	$0.923702\pi$
$164$			5901.58i			0.0171340i
$165$		0			0
$166$	−48933.7			−0.137828
$167$		−	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
$170$	−138537.			−0.367658
$171$		0			0
$172$	322890.			0.832211
$173$	240767.			0.611620			0.305810	−	0.952093i	$-0.401073\pi$
$173$	240767.			0.611620			0.305810	+	0.952093i	$0.401073\pi$
$174$		0			0
$175$			308256.i			0.760881i
$176$			45927.4i			0.111761i
$177$		0			0
$178$	247019.			0.584359
$179$	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	$0.476758\pi$
$182$	−391439.	−	177892.i	−0.875964	−	0.398086i
$183$		0			0
$184$			178711.i			0.389141i
$185$	−140127.			−0.301019
$186$		0			0
$187$			167409.i			0.350086i
$188$			328424.i			0.677704i
$189$		0			0
$190$			346806.i			0.696951i
$191$	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	$-0.209211\pi$
$194$	−112123.			−0.213889
$195$		0			0
$196$	228997.			0.425785
$197$		−	100048.i		−	0.183671i	−0.995774	−	0.0918357i	$-0.970727\pi$
$197$		−	100048.i		−	0.183671i	0.995774	−	0.0918357i	$-0.0292735\pi$
$198$		0			0
$199$	−723076.			−1.29435			−0.647174	−	0.762342i	$-0.724049\pi$
$199$	−723076.			−1.29435			−0.647174	+	0.762342i	$0.724049\pi$
$200$			111835.i			0.197698i
$201$		0			0
$202$			410401.i			0.707669i
$203$		−	265242.i		−	0.451754i
$204$		0			0
$205$	−13690.1			−0.0227521
$206$			815214.i			1.33845i
$207$		0			0
$208$	−142013.	−	64538.7i	−0.227599	−	0.103434i
$209$	419082.			0.663641
$210$		0			0
$211$	259548.			0.401340			0.200670	−	0.979659i	$-0.435688\pi$
$211$	259548.			0.401340			0.200670	+	0.979659i	$0.435688\pi$
$212$	−401310.			−0.613253
$213$		0			0
$214$		−	589423.i		−	0.879818i
$215$			749020.i			1.10509i
$216$		0			0
$217$	130053.			0.187488
$218$	31386.6			0.0447304
$219$		0			0
$220$	−106539.			−0.148407
$221$	−517651.	−	235249.i	−0.712945	−	0.324002i
$222$		0			0
$223$		−	1.07196e6i		−	1.44349i	−0.692156	−	0.721747i	$-0.743339\pi$
$223$		−	1.07196e6i		−	1.44349i	0.692156	−	0.721747i	$-0.256661\pi$
$224$	180641.			0.240544
$225$		0			0
$226$		−	770692.i		−	1.00371i
$227$		−	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	$0.961429\pi$
$230$	−414563.			−0.516738
$231$		0			0
$232$		−	96229.3i		−	0.117378i
$233$	−353920.			−0.427087			−0.213543	−	0.976934i	$-0.568500\pi$
$233$	−353920.			−0.427087			−0.213543	+	0.976934i	$0.568500\pi$
$234$		0			0
$235$	−761857.			−0.899920
$236$			565221.i			0.660600i
$237$		0			0
$238$	658450.			0.753495
$239$			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	$0.962892\pi$
$242$		−	515461.i		−	0.565793i
$243$		0			0
$244$	−507862.			−0.546099
$245$			531214.i			0.565398i
$246$		0			0
$247$	−588909.	+	1.29586e6i	−0.614194	+	1.35150i
$248$	47183.1			0.0487144
$249$		0			0
$250$	−723375.			−0.732004
$251$	−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
$254$		−	1.02258e6i		−	0.994515i
$255$		0			0
$256$	65536.0			0.0625000
$257$	−207722.			−0.196177			−0.0980887	−	0.995178i	$-0.531273\pi$
$257$	−207722.			−0.196177			−0.0980887	+	0.995178i	$0.531273\pi$
$258$		0			0
$259$	666008.			0.616922
$260$	149713.	−	329434.i	0.137349	−	0.302228i
$261$		0			0
$262$		−	537026.i		−	0.483328i
$263$	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
$266$		−	1.64832e6i		−	1.42836i
$267$		0			0
$268$			746581.i			0.634951i
$269$	−1.67365e6			−1.41021			−0.705104	−	0.709104i	$-0.749099\pi$
$269$	−1.67365e6			−1.41021			−0.705104	+	0.709104i	$0.749099\pi$
$270$		0			0
$271$			69792.6i			0.0577279i	0.999583	+	0.0288640i	$0.00918896\pi$
$271$			69792.6i			0.0577279i	−0.999583	+	0.0288640i	$0.990811\pi$
$272$	238884.			0.195779
$273$		0			0
$274$	297494.			0.239387
$275$			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	$0.282292\pi$
$278$		−	145394.i		−	0.112832i
$279$		0			0
$280$			419039.i			0.319418i
$281$		−	1.05655e6i		−	0.798224i	−0.916902	−	0.399112i	$-0.869318\pi$
$281$		−	1.05655e6i		−	0.798224i	0.916902	−	0.399112i	$-0.130682\pi$
$282$		0			0
$283$	−512741.			−0.380567			−0.190284	−	0.981729i	$-0.560941\pi$
$283$	−512741.			−0.380567			−0.190284	+	0.981729i	$0.560941\pi$
$284$			943574.i			0.694193i
$285$		0			0
$286$	−398090.	−	180914.i	−0.287784	−	0.130785i
$287$	65067.4			0.0466293
$288$		0			0
$289$	−549105.			−0.386732
$290$	223227.			0.155866
$291$		0			0
$292$		−	54579.2i		−	0.0374602i
$293$		−	2.71268e6i		−	1.84599i	−0.384809	−	0.922996i	$-0.625733\pi$
$293$		−	2.71268e6i		−	1.84599i	0.384809	−	0.922996i	$-0.374267\pi$
$294$		0			0
$295$	−1.31117e6			−0.877207
$296$	241626.			0.160293
$297$		0			0
$298$	−885014.			−0.577310
$299$	−1.54903e6	−	703967.i	−1.00204	−	0.455380i
$300$		0			0
$301$		−	3.56000e6i		−	2.26482i
$302$	1.92979e6			1.21757
$303$		0			0
$304$		−	598009.i		−	0.371128i
$305$		−	1.17811e6i		−	0.725162i
$306$		0			0
$307$			2.44475e6i			1.48043i	0.672371	+	0.740215i	$0.265276\pi$
$307$			2.44475e6i			1.48043i	−0.672371	+	0.740215i	$0.734724\pi$
$308$	506369.			0.304152
$309$		0			0
$310$			109452.i			0.0646876i
$311$	−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	$0.340632\pi$
$314$			442903.i			0.253504i
$315$		0			0
$316$	1.03964e6			0.585685
$317$			3.20482e6i			1.79125i	0.444813	+	0.895624i	$0.353270\pi$
$317$			3.20482e6i			1.79125i	−0.444813	+	0.895624i	$0.646730\pi$
$318$		0			0
$319$		−	269749.i		−	0.148417i
$320$			152026.i			0.0829935i
$321$		0			0
$322$	1.97037e6			1.05903
$323$		−	2.17979e6i		−	1.16254i
$324$		0			0
$325$	−969362.	−	440532.i	−0.509070	−	0.231350i
$326$	644254.			0.335748
$327$		0			0
$328$	23606.3			0.0121156
$329$	3.62101e6			1.84434
$330$		0			0
$331$		−	3.53386e6i		−	1.77288i	−0.462844	−	0.886440i	$-0.653171\pi$
$331$		−	3.53386e6i		−	1.77288i	0.462844	−	0.886440i	$-0.346829\pi$
$332$			195735.i			0.0974591i
$333$		0			0
$334$	−1.86631e6			−0.915415
$335$	−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	$0.414830\pi$
$338$	1.11882e6	−	976719.i	0.532682	−	0.465027i
$339$		0			0
$340$			554148.i			0.259973i
$341$	132263.			0.0615960
$342$		0			0
$343$			440075.i			0.201972i
$344$		−	1.29156e6i		−	0.588462i
$345$		0			0
$346$		−	963068.i		−	0.432481i
$347$	−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	$-0.142912\pi$
$350$	1.23302e6			0.538024
$351$		0			0
$352$	183710.			0.0790269
$353$			2.24015e6i			0.956843i	0.878130	+	0.478422i	$0.158791\pi$
$353$			2.24015e6i			0.956843i	−0.878130	+	0.478422i	$0.841209\pi$
$354$		0			0
$355$	−2.18884e6			−0.921815
$356$		−	988075.i		−	0.413204i
$357$		0			0
$358$		−	1.06574e6i		−	0.439485i
$359$		−	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
$362$		−	257229.i		−	0.103169i
$363$		0			0
$364$	−711567.	+	1.56576e6i	−0.281490	+	0.619400i
$365$	126609.			0.0497432
$366$		0			0
$367$	1.70502e6			0.660791			0.330395	−	0.943843i	$-0.392818\pi$
$367$	1.70502e6			0.660791			0.330395	+	0.943843i	$0.392818\pi$
$368$	714844.			0.275164
$369$		0			0
$370$			560509.i			0.212852i
$371$			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	$0.641761\pi$
$374$	669636.			0.247548
$375$		0			0
$376$	1.31370e6			0.479209
$377$	834097.	+	379060.i	0.302248	+	0.137358i
$378$		0			0
$379$		−	4.32354e6i		−	1.54612i	−0.634336	−	0.773058i	$-0.718726\pi$
$379$		−	4.32354e6i		−	1.54612i	0.634336	−	0.773058i	$-0.281274\pi$
$380$	1.38722e6			0.492819
$381$		0			0
$382$		−	3.70155e6i		−	1.29785i
$383$		−	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
$386$	−2.52922e6			−0.864011
$387$		0			0
$388$			448490.i			0.151243i
$389$	−1.70696e6			−0.571940			−0.285970	−	0.958239i	$-0.592316\pi$
$389$	−1.70696e6			−0.571940			−0.285970	+	0.958239i	$0.592316\pi$
$390$		0			0
$391$	2.60567e6			0.861940
$392$		−	915990.i		−	0.301076i
$393$		0			0
$394$	−400191.			−0.129875
$395$			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	$0.937553\pi$
$398$			2.89230e6i			0.915243i
$399$		0			0
$400$	447339.			0.139793
$401$			4.15598e6i			1.29066i	0.763903	+	0.645332i	$0.223281\pi$
$401$			4.15598e6i			1.29066i	−0.763903	+	0.645332i	$0.776719\pi$
$402$		0			0
$403$	−185860.	+	408974.i	−0.0570065	+	0.125439i
$404$	1.64161e6			0.500398
$405$		0			0
$406$	−1.06097e6			−0.319439
$407$	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	$0.585358\pi$
$410$			54760.4i			0.0160882i
$411$		0			0
$412$	3.26086e6			0.946430
$413$	6.23180e6			1.79779
$414$		0			0
$415$	−454053.			−0.129416
$416$	−258155.	+	568054.i	−0.0731387	+	0.160937i
$417$		0			0
$418$		−	1.67633e6i		−	0.469265i
$419$	−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	$0.844398\pi$
$422$		−	1.03819e6i		−	0.283790i
$423$		0			0
$424$			1.60524e6i			0.433636i
$425$	1.63059e6			0.437897
$426$		0			0
$427$			5.59940e6i			1.48618i
$428$	−2.35769e6			−0.622125
$429$		0			0
$430$	2.99608e6			0.781416
$431$			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	$0.768809\pi$
$434$		−	520214.i		−	0.132574i
$435$		0			0
$436$		−	125546.i		−	0.0316292i
$437$		−	6.52287e6i		−	1.63394i
$438$		0			0
$439$	−766922.			−0.189928			−0.0949642	−	0.995481i	$-0.530274\pi$
$439$	−766922.			−0.189928			−0.0949642	+	0.995481i	$0.530274\pi$
$440$			426158.i			0.104939i
$441$		0			0
$442$	−940996.	+	2.07060e6i	−0.229104	+	0.504128i
$443$	−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
$446$	−4.28783e6			−1.02071
$447$		0			0
$448$		−	722562.i		−	0.170091i
$449$		−	4.18123e6i		−	0.978787i	−0.872063	−	0.489394i	$-0.837218\pi$
$449$		−	4.18123e6i		−	0.978787i	0.872063	−	0.489394i	$-0.162782\pi$
$450$		0			0
$451$	66172.8			0.0153193
$452$	−3.08277e6			−0.709733
$453$		0			0
$454$	−260840.			−0.0593929
$455$	−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	$0.868559\pi$
$458$	767417.			0.170949
$459$		0			0
$460$			1.65825e6i			0.365389i
$461$			1.67459e6i			0.366992i	0.983020	+	0.183496i	$0.0587414\pi$
$461$			1.67459e6i			0.366992i	−0.983020	+	0.183496i	$0.941259\pi$
$462$		0			0
$463$			6.62362e6i			1.43596i	0.696063	+	0.717981i	$0.254934\pi$
$463$			6.62362e6i			1.43596i	−0.696063	+	0.717981i	$0.745066\pi$
$464$	−384917.			−0.0829989
$465$		0			0
$466$			1.41568e6i			0.301996i
$467$	−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
$470$			3.04743e6i			0.636339i
$471$		0			0
$472$	2.26089e6			0.467115
$473$		−	3.62048e6i		−	0.744070i
$474$		0			0
$475$		−	4.08192e6i		−	0.830099i
$476$		−	2.63380e6i		−	0.532801i
$477$		0			0
$478$	4.61495e6			0.923841
$479$			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
$482$	839013.			0.164494
$483$		0			0
$484$	−2.06184e6			−0.400076
$485$	−1.04038e6			−0.200834
$486$		0			0
$487$		−	71255.1i		−	0.0136142i	−0.999977	−	0.00680712i	$-0.997833\pi$
$487$		−	71255.1i		−	0.0136142i	0.999977	−	0.00680712i	$-0.00216679\pi$
$488$			2.03145e6i			0.386150i
$489$		0			0
$490$	2.12486e6			0.399797
$491$	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
$494$	5.18343e6	+	2.35563e6i	0.955651	+	0.434301i
$495$		0			0
$496$		−	188732.i		−	0.0344463i
$497$	1.04033e7			1.88921
$498$		0			0
$499$			167487.i			0.0301114i	0.999887	+	0.0150557i	$0.00479256\pi$
$499$			167487.i			0.0301114i	−0.999887	+	0.0150557i	$0.995207\pi$
$500$			2.89350e6i			0.517605i
$501$		0			0
$502$			1.70862e6i			0.302613i
$503$	−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
$506$	2.00384e6			0.347926
$507$		0			0
$508$	−4.09031e6			−0.703229
$509$		−	9.79146e6i		−	1.67515i	−0.546324	−	0.837574i	$-0.683973\pi$
$509$		−	9.79146e6i		−	1.67515i	0.546324	−	0.837574i	$-0.316027\pi$
$510$		0			0
$511$	−601758.			−0.101946
$512$		−	262144.i		−	0.0441942i
$513$		0			0
$514$			830887.i			0.138718i
$515$			7.56433e6i			1.25676i
$516$		0			0
$517$	3.68253e6			0.605927
$518$		−	2.66403e6i		−	0.436229i
$519$		0			0
$520$	−1.31774e6	−	598852.i	−0.213708	−	0.0971205i
$521$	9.42352e6			1.52096			0.760481	−	0.649360i	$-0.224963\pi$
$521$	9.42352e6			1.52096			0.760481	+	0.649360i	$0.224963\pi$
$522$		0			0
$523$	5.16320e6			0.825400			0.412700	−	0.910867i	$-0.364586\pi$
$523$	5.16320e6			0.825400			0.412700	+	0.910867i	$0.364586\pi$
$524$	−2.14811e6			−0.341765
$525$		0			0
$526$		−	3.32685e6i		−	0.524286i
$527$		−	687945.i		−	0.107901i
$528$		0			0
$529$	1.36094e6			0.211445
$530$	−3.72373e6			−0.575823
$531$		0			0
$532$	−6.59330e6			−1.01001
$533$	−92988.3	+	204615.i	−0.0141779	+	0.0311975i
$534$		0			0
$535$		−	5.46923e6i		−	0.826117i
$536$	2.98633e6			0.448978
$537$		0			0
$538$			6.69458e6i			0.997167i
$539$		−	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	$-0.276397\pi$
$542$	279170.			0.0408198
$543$		0			0
$544$		−	955536.i		−	0.138436i
$545$	291235.			0.0420002
$546$		0			0
$547$	−7.33920e6			−1.04877			−0.524385	−	0.851481i	$-0.675705\pi$
$547$	−7.33920e6			−1.04877			−0.524385	+	0.851481i	$0.675705\pi$
$548$		−	1.18997e6i		−	0.169272i
$549$		0			0
$550$	1.25397e6			0.176759
$551$			3.51233e6i			0.492851i
$552$		0			0
$553$		−	1.14624e7i		−	1.59391i
$554$		−	6.45520e6i		−	0.893584i
$555$		0			0
$556$	−581574.			−0.0797844
$557$			7.97266e6i			1.08884i	0.838812	+	0.544421i	$0.183251\pi$
$557$			7.97266e6i			1.08884i	−0.838812	+	0.544421i	$0.816749\pi$
$558$		0			0
$559$	1.11950e7	+	5.08763e6i	1.51529	+	0.688630i
$560$	1.67615e6			0.225862
$561$		0			0
$562$	−4.22621e6			−0.564430
$563$	−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
$566$			2.05096e6i			0.269102i
$567$		0			0
$568$	3.77430e6			0.490868
$569$	8.15975e6			1.05657			0.528283	−	0.849069i	$-0.322836\pi$
$569$	8.15975e6			1.05657			0.528283	+	0.849069i	$0.322836\pi$
$570$		0			0
$571$	−6.61371e6			−0.848897			−0.424448	−	0.905452i	$-0.639532\pi$
$571$	−6.61371e6			−0.848897			−0.424448	+	0.905452i	$0.639532\pi$
$572$	−723656.	+	1.59236e6i	−0.0924788	+	0.203494i
$573$		0			0
$574$		−	260269.i		−	0.0329719i
$575$	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	$0.902380\pi$
$578$			2.19642e6i			0.273461i
$579$		0			0
$580$		−	892907.i		−	0.110214i
$581$	2.15806e6			0.265230
$582$		0			0
$583$			4.49978e6i			0.548302i
$584$	−218317.			−0.0264883
$585$		0			0
$586$	−1.08507e7			−1.30531
$587$		−	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
$590$			5.24466e6i			0.620279i
$591$		0			0
$592$		−	966505.i		−	0.113344i
$593$			8.22445e6i			0.960439i	0.877148	+	0.480220i	$0.159443\pi$
$593$			8.22445e6i			0.960439i	−0.877148	+	0.480220i	$0.840557\pi$
$594$		0			0
$595$	6.10972e6			0.707504
$596$			3.54006e6i			0.408220i
$597$		0			0
$598$	−2.81587e6	+	6.19614e6i	−0.322002	+	0.708546i
$599$	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	$0.379111\pi$
$602$	−1.42400e7			−1.60147
$603$		0			0
$604$		−	7.71916e6i		−	0.860950i
$605$		−	4.78294e6i		−	0.531259i
$606$		0			0
$607$	−7.92228e6			−0.872727			−0.436364	−	0.899770i	$-0.643734\pi$
$607$	−7.92228e6			−0.872727			−0.436364	+	0.899770i	$0.643734\pi$
$608$	−2.39203e6			−0.262427
$609$		0			0
$610$	−4.71243e6			−0.512767
$611$	−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	$-0.175450\pi$
$614$	9.77898e6			1.04682
$615$		0			0
$616$		−	2.02548e6i		−	0.215068i
$617$		−	1.14334e7i		−	1.20910i	−0.796569	−	0.604548i	$-0.793354\pi$
$617$		−	1.14334e7i		−	1.20910i	0.796569	−	0.604548i	$-0.206646\pi$
$618$		0			0
$619$		−	1.07067e7i		−	1.12313i	−0.827432	−	0.561566i	$-0.810199\pi$
$619$		−	1.07067e7i		−	1.12313i	0.827432	−	0.561566i	$-0.189801\pi$
$620$	437810.			0.0457411
$621$		0			0
$622$			7.96732e6i			0.825726i
$623$	−1.08939e7			−1.12451
$624$		0			0
$625$	−1.25148e6			−0.128151
$626$		−	6.65585e6i		−	0.678840i
$627$		0			0
$628$	1.77161e6			0.179254
$629$		−	3.52299e6i		−	0.355046i
$630$		0			0
$631$		−	475976.i		−	0.0475895i	−0.999717	−	0.0237948i	$-0.992425\pi$
$631$		−	475976.i		−	0.0475895i	0.999717	−	0.0237948i	$-0.00757483\pi$
$632$		−	4.15855e6i		−	0.414142i
$633$		0			0
$634$	1.28193e7			1.26660
$635$		−	9.48844e6i		−	0.933814i
$636$		0			0
$637$	7.93963e6	+	3.60820e6i	0.775267	+	0.352324i
$638$	−1.07899e6			−0.104946
$639$		0			0
$640$	608105.			0.0586852
$641$	1.90742e7			1.83358			0.916791	−	0.399368i	$-0.130770\pi$
$641$	1.90742e7			1.83358			0.916791	+	0.399368i	$0.130770\pi$
$642$		0			0
$643$		−	1.16474e7i		−	1.11097i	−0.831528	−	0.555483i	$-0.812534\pi$
$643$		−	1.16474e7i		−	1.11097i	0.831528	−	0.555483i	$-0.187466\pi$
$644$		−	7.88146e6i		−	0.748845i
$645$		0			0
$646$	−8.71916e6			−0.822041
$647$	−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
$650$	−1.76213e6	+	3.87745e6i	−0.163589	+	0.359967i
$651$		0			0
$652$		−	2.57702e6i		−	0.237410i
$653$	−1.75829e7			−1.61365			−0.806824	−	0.590792i	$-0.798815\pi$
$653$	−1.75829e7			−1.61365			−0.806824	+	0.590792i	$0.798815\pi$
$654$		0			0
$655$		−	4.98304e6i		−	0.453828i
$656$		−	94425.2i		−	0.00856699i
$657$		0			0
$658$		−	1.44840e7i		−	1.30414i
$659$	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	$0.840302\pi$
$662$	−1.41354e7			−1.25362
$663$		0			0
$664$	782939.			0.0689140
$665$		−	1.52947e7i		−	1.34118i
$666$		0			0
$667$	−4.19855e6			−0.365413
$668$			7.46525e6i			0.647296i
$669$		0			0
$670$			6.92749e6i			0.596196i
$671$			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	$0.734119\pi$
$674$		−	4.40968e6i		−	0.373902i
$675$		0			0
$676$	−3.90688e6	−	4.47528e6i	−0.328823	−	0.376663i
$677$	−2.86244e6			−0.240029			−0.120015	−	0.992772i	$-0.538294\pi$
$677$	−2.86244e6			−0.240029			−0.120015	+	0.992772i	$0.538294\pi$
$678$		0			0
$679$	4.94480e6			0.411599
$680$	2.21659e6			0.183829
$681$		0			0
$682$		−	529052.i		−	0.0435549i
$683$		−	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
$686$	1.76030e6			0.142816
$687$		0			0
$688$	−5.16624e6			−0.416106
$689$	−1.39139e7	−	6.32325e6i	−1.11661	−	0.507449i
$690$		0			0
$691$			1.79749e7i			1.43209i	0.698052	+	0.716047i	$0.254051\pi$
$691$			1.79749e7i			1.43209i	−0.698052	+	0.716047i	$0.745949\pi$
$692$	−3.85227e6			−0.305810
$693$		0			0
$694$			1.26573e7i			0.997566i
$695$		−	1.34910e6i		−	0.105945i
$696$		0			0
$697$		−	344188.i		−	0.0268357i
$698$	−7.90102e6			−0.613825
$699$		0			0
$700$		−	4.93210e6i		−	0.380440i
$701$	8.74640e6			0.672255			0.336128	−	0.941816i	$-0.390883\pi$
$701$	8.74640e6			0.672255			0.336128	+	0.941816i	$0.390883\pi$
$702$		0			0
$703$	−8.81925e6			−0.673044
$704$		−	734838.i		−	0.0558805i
$705$		0			0
$706$	8.96061e6			0.676590
$707$		−	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	$-0.0289814\pi$
$710$			8.75538e6i			0.651822i
$711$		0			0
$712$	−3.95230e6			−0.292180
$713$		−	2.05863e6i		−	0.151654i
$714$		0			0
$715$	−3.69386e6	−	1.67869e6i	−0.270218	−	0.122802i
$716$	−4.26297e6			−0.310763
$717$		0			0
$718$	−8.55079e6			−0.619006
$719$	−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
$722$			1.19226e7i			0.851196i
$723$		0			0
$724$	−1.02892e6			−0.0729513
$725$	−2.62739e6			−0.185643
$726$		0			0
$727$	−4.33605e6			−0.304269			−0.152135	−	0.988360i	$-0.548615\pi$
$727$	−4.33605e6			−0.304269			−0.152135	+	0.988360i	$0.548615\pi$
$728$	6.26303e6	+	2.84627e6i	0.437982	+	0.199043i
$729$		0			0
$730$		−	506437.i		−	0.0351737i
$731$	−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	$0.756176\pi$
$734$		−	6.82008e6i		−	0.467250i
$735$		0			0
$736$		−	2.85938e6i		−	0.194570i
$737$	8.37122e6			0.567702
$738$		0			0
$739$			2.05423e7i			1.38369i	0.722048	+	0.691843i	$0.243201\pi$
$739$			2.05423e7i			1.38369i	−0.722048	+	0.691843i	$0.756799\pi$
$740$	2.24204e6			0.150509
$741$		0			0
$742$	1.76984e7			1.18012
$743$		−	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
$746$			9.26012e6i			0.609213i
$747$		0			0
$748$		−	2.67855e6i		−	0.175043i
$749$			2.59946e7i			1.69308i
$750$		0			0
$751$	−2.94950e7			−1.90831			−0.954154	−	0.299315i	$-0.903242\pi$
$751$	−2.94950e7			−1.90831			−0.954154	+	0.299315i	$0.903242\pi$
$752$		−	5.25478e6i		−	0.338852i
$753$		0			0
$754$	1.51624e6	−	3.33639e6i	0.0971269	−	0.213722i
$755$	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	$0.279063\pi$
$758$	−1.72942e7			−1.09327
$759$		0			0
$760$		−	5.54889e6i		−	0.348476i
$761$		−	2.06956e7i		−	1.29544i	−0.761880	−	0.647719i	$-0.775723\pi$
$761$		−	2.06956e7i		−	1.29544i	0.761880	−	0.647719i	$-0.224277\pi$
$762$		0			0
$763$	−1.38420e6			−0.0860772
$764$	−1.48062e7			−0.917720
$765$		0			0
$766$	−8.01403e6			−0.493491
$767$	−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	$-0.172458\pi$
$770$	4.69857e6			0.285587
$771$		0			0
$772$			1.01169e7i			0.610948i
$773$			3.10969e7i			1.87184i	0.352212	+	0.935920i	$0.385430\pi$
$773$			3.10969e7i			1.87184i	−0.352212	+	0.935920i	$0.614570\pi$
$774$		0			0
$775$		−	1.28826e6i		−	0.0770458i
$776$	1.79396e6			0.106945
$777$		0			0
$778$			6.82786e6i			0.404423i
$779$	−861619.			−0.0508712
$780$		0			0
$781$	1.05800e7			0.620669
$782$		−	1.04227e7i		−	0.609483i
$783$		0			0
$784$	−3.66396e6			−0.212893
$785$			4.10967e6i

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	234.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.00000i q^{2} -16.0000 q^{4} -37.1158i q^{5} +176.407i q^{7} +64.0000i q^{8} +O(q^{10})\)	\(q-4.00000i q^{2} -16.0000 q^{4} -37.1158i q^{5} +176.407i q^{7} +64.0000i q^{8} -148.463 q^{10} +179.404i q^{11} +(-554.740 - 252.104i) q^{13} +705.627 q^{14} +256.000 q^{16} +933.141 q^{17} -2335.97i q^{19} +593.853i q^{20} +717.615 q^{22} +2792.36 q^{23} +1747.42 q^{25} +(-1008.42 + 2218.96i) q^{26} -2822.51i q^{28} -1503.58 q^{29} -737.236i q^{31} -1024.00i q^{32} -3732.56i q^{34} +6547.48 q^{35} -3775.41i q^{37} -9343.88 q^{38} +2375.41 q^{40} -368.848i q^{41} -20180.6 q^{43} -2870.46i q^{44} -11169.4i q^{46} -20526.5i q^{47} -14312.3 q^{49} -6989.67i q^{50} +(8875.84 + 4033.67i) q^{52} +25081.9 q^{53} +6658.72 q^{55} -11290.0 q^{56} +6014.33i q^{58} -35326.3i q^{59} +31741.4 q^{61} -2948.94 q^{62} -4096.00 q^{64} +(-9357.06 + 20589.6i) q^{65} -46661.3i q^{67} -14930.3 q^{68} -26189.9i q^{70} -58973.4i q^{71} +3411.20i q^{73} -15101.6 q^{74} +37375.5i q^{76} -31648.1 q^{77} -64977.3 q^{79} -9501.64i q^{80} -1475.39 q^{82} -12233.4i q^{83} -34634.3i q^{85} +80722.5i q^{86} -11481.8 q^{88} +61754.7i q^{89} +(44472.9 - 97859.9i) q^{91} -44677.8 q^{92} -82106.0 q^{94} -86701.4 q^{95} -28030.7i q^{97} +57249.4i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 96 q^{4}+O(q^{10}) \) 6 * q - 96 * q^4	\( 6 q - 96 q^{4} + 320 q^{10} + 530 q^{13} + 1360 q^{14} + 1536 q^{16} + 836 q^{17} - 1296 q^{22} + 416 q^{23} + 718 q^{25} + 1360 q^{26} - 18788 q^{29} - 6112 q^{35} - 528 q^{38} - 5120 q^{40} - 24200 q^{43} - 3038 q^{49} - 8480 q^{52} + 42396 q^{53} + 124656 q^{55} - 21760 q^{56} - 3196 q^{61} + 59344 q^{62} - 24576 q^{64} + 17168 q^{65} - 13376 q^{68} + 62240 q^{74} + 114024 q^{77} - 169328 q^{79} + 145120 q^{82} + 20736 q^{88} + 236152 q^{91} - 6656 q^{92} - 32688 q^{94} - 567168 q^{95}+O(q^{100}) \) 6 * q - 96 * q^4 + 320 * q^10 + 530 * q^13 + 1360 * q^14 + 1536 * q^16 + 836 * q^17 - 1296 * q^22 + 416 * q^23 + 718 * q^25 + 1360 * q^26 - 18788 * q^29 - 6112 * q^35 - 528 * q^38 - 5120 * q^40 - 24200 * q^43 - 3038 * q^49 - 8480 * q^52 + 42396 * q^53 + 124656 * q^55 - 21760 * q^56 - 3196 * q^61 + 59344 * q^62 - 24576 * q^64 + 17168 * q^65 - 13376 * q^68 + 62240 * q^74 + 114024 * q^77 - 169328 * q^79 + 145120 * q^82 + 20736 * q^88 + 236152 * q^91 - 6656 * q^92 - 32688 * q^94 - 567168 * q^95

Introduction

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