LMFDB - Embedded newform 432.6.i.d.289.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [432,6,Mod(145,432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(432, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

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

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

chi := DirichletCharacter("432.145");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 432 = 2^{4} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	432.i (of order $3$, degree $2$, not 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:	$69.2858101592$
Analytic rank:	$0$
Dimension:	$10$
Relative dimension:	$5$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 175x^{8} + 8800x^{6} + 124623x^{4} + 498609x^{2} + 442368 $ x^10 + 175x^8 + 8800x^6 + 124623x^4 + 498609x^2 + 442368
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{16} $
Twist minimal:	no (minimal twist has level 36)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			289.1
Root			$-3.71922i$ of defining polynomial
Character	$\chi$	$=$	432.289
Dual form			432.6.i.d.145.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-40.7270 - 70.5412i) q^{5} +(-89.6312 + 155.246i) q^{7} +O(q^{10})$	$q+(-40.7270 - 70.5412i) q^{5} +(-89.6312 + 155.246i) q^{7} +(250.250 - 433.446i) q^{11} +(275.245 + 476.739i) q^{13} -753.636 q^{17} +2570.83 q^{19} +(1372.72 + 2377.63i) q^{23} +(-1754.87 + 3039.53i) q^{25} +(1954.86 - 3385.92i) q^{29} +(-1552.42 - 2688.87i) q^{31} +14601.6 q^{35} -9568.10 q^{37} +(1113.47 + 1928.59i) q^{41} +(7143.42 - 12372.8i) q^{43} +(3236.07 - 5605.04i) q^{47} +(-7664.00 - 13274.4i) q^{49} -13692.2 q^{53} -40767.8 q^{55} +(-2854.22 - 4943.65i) q^{59} +(5899.59 - 10218.4i) q^{61} +(22419.8 - 38832.3i) q^{65} +(-1771.66 - 3068.60i) q^{67} -58429.0 q^{71} -60181.3 q^{73} +(44860.5 + 77700.6i) q^{77} +(-27811.7 + 48171.3i) q^{79} +(-19990.3 + 34624.2i) q^{83} +(30693.3 + 53162.4i) q^{85} -103171. q^{89} -98682.3 q^{91} +(-104702. - 181349. i) q^{95} +(82996.9 - 143755. i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 21 q^{5} - 29 q^{7}+O(q^{10}) $ 10 * q + 21 * q^5 - 29 * q^7	$ 10 q + 21 q^{5} - 29 q^{7} + 177 q^{11} - 181 q^{13} - 2280 q^{17} + 832 q^{19} + 399 q^{23} - 4778 q^{25} + 6033 q^{29} - 2759 q^{31} + 37146 q^{35} - 15172 q^{37} + 18435 q^{41} - 1469 q^{43} - 25155 q^{47} - 4056 q^{49} - 116844 q^{53} - 14778 q^{55} - 90537 q^{59} + 1403 q^{61} + 148407 q^{65} - 13907 q^{67} + 229368 q^{71} + 15200 q^{73} + 211983 q^{77} - 29993 q^{79} - 228951 q^{83} - 49662 q^{85} - 598332 q^{89} - 124930 q^{91} - 394764 q^{95} + 40541 q^{97}+O(q^{100}) $ 10 * q + 21 * q^5 - 29 * q^7 + 177 * q^11 - 181 * q^13 - 2280 * q^17 + 832 * q^19 + 399 * q^23 - 4778 * q^25 + 6033 * q^29 - 2759 * q^31 + 37146 * q^35 - 15172 * q^37 + 18435 * q^41 - 1469 * q^43 - 25155 * q^47 - 4056 * q^49 - 116844 * q^53 - 14778 * q^55 - 90537 * q^59 + 1403 * q^61 + 148407 * q^65 - 13907 * q^67 + 229368 * q^71 + 15200 * q^73 + 211983 * q^77 - 29993 * q^79 - 228951 * q^83 - 49662 * q^85 - 598332 * q^89 - 124930 * q^91 - 394764 * q^95 + 40541 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/432\mathbb{Z}\right)^\times$.

$n$	$271$	$325$	$353$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−40.7270	−	70.5412i	−0.728546	−	1.26188i	−0.957498	−	0.288441i	$-0.906863\pi$
$5$	−40.7270	−	70.5412i	−0.728546	−	1.26188i	0.228951	−	0.973438i	$-0.426470\pi$
$6$		0			0
$7$	−89.6312	+	155.246i	−0.691376	+	1.19750i	0.280012	+	0.959997i	$0.409662\pi$
$7$	−89.6312	+	155.246i	−0.691376	+	1.19750i	−0.971387	+	0.237501i	$0.923672\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	250.250	−	433.446i	0.623581	−	1.08007i	−0.365232	−	0.930916i	$-0.619010\pi$
$11$	250.250	−	433.446i	0.623581	−	1.08007i	0.988813	−	0.149158i	$-0.0476562\pi$
$12$		0			0
$13$	275.245	+	476.739i	0.451712	+	0.782388i	0.998493	−	0.0548877i	$-0.0174801\pi$
$13$	275.245	+	476.739i	0.451712	+	0.782388i	−0.546780	+	0.837276i	$0.684147\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−753.636			−0.632469			−0.316234	−	0.948681i	$-0.602419\pi$
$17$	−753.636			−0.632469			−0.316234	+	0.948681i	$0.602419\pi$
$18$		0			0
$19$	2570.83			1.63376			0.816882	−	0.576805i	$-0.195701\pi$
$19$	2570.83			1.63376			0.816882	+	0.576805i	$0.195701\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1372.72	+	2377.63i	0.541083	+	0.937183i	0.998842	+	0.0481071i	$0.0153189\pi$
$23$	1372.72	+	2377.63i	0.541083	+	0.937183i	−0.457759	+	0.889076i	$0.651348\pi$
$24$		0			0
$25$	−1754.87	+	3039.53i	−0.561560	+	0.972650i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1954.86	−	3385.92i	0.431639	−	0.747621i	−0.565376	−	0.824834i	$-0.691269\pi$
$29$	1954.86	−	3385.92i	0.431639	−	0.747621i	0.997015	+	0.0772128i	$0.0246021\pi$
$30$		0			0
$31$	−1552.42	−	2688.87i	−0.290138	−	0.502534i	0.683704	−	0.729759i	$-0.260368\pi$
$31$	−1552.42	−	2688.87i	−0.290138	−	0.502534i	−0.973842	+	0.227226i	$0.927035\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	14601.6			2.01480
$36$		0			0
$37$	−9568.10			−1.14900			−0.574502	−	0.818503i	$-0.694804\pi$
$37$	−9568.10			−1.14900			−0.574502	+	0.818503i	$0.694804\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1113.47	+	1928.59i	0.103447	+	0.179176i	0.913103	−	0.407730i	$-0.133679\pi$
$41$	1113.47	+	1928.59i	0.103447	+	0.179176i	−0.809656	+	0.586905i	$0.800346\pi$
$42$		0			0
$43$	7143.42	−	12372.8i	0.589162	−	1.02046i	−0.405180	−	0.914237i	$-0.632791\pi$
$43$	7143.42	−	12372.8i	0.589162	−	1.02046i	0.994342	−	0.106222i	$-0.0338754\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3236.07	−	5605.04i	0.213685	−	0.370112i	−0.739180	−	0.673508i	$-0.764787\pi$
$47$	3236.07	−	5605.04i	0.213685	−	0.370112i	0.952865	+	0.303395i	$0.0981202\pi$
$48$		0			0
$49$	−7664.00	−	13274.4i	−0.456000	−	0.789816i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−13692.2			−0.669553			−0.334777	−	0.942298i	$-0.608661\pi$
$53$	−13692.2			−0.669553			−0.334777	+	0.942298i	$0.608661\pi$
$54$		0			0
$55$	−40767.8			−1.81723
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−2854.22	−	4943.65i	−0.106747	−	0.184892i	0.807703	−	0.589589i	$-0.200710\pi$
$59$	−2854.22	−	4943.65i	−0.106747	−	0.184892i	−0.914451	+	0.404697i	$0.867377\pi$
$60$		0			0
$61$	5899.59	−	10218.4i	0.203000	−	0.351607i	−0.746493	−	0.665393i	$-0.768264\pi$
$61$	5899.59	−	10218.4i	0.203000	−	0.351607i	0.949494	+	0.313786i	$0.101597\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	22419.8	−	38832.3i	0.658186	−	1.14001i
$66$		0			0
$67$	−1771.66	−	3068.60i	−0.0482162	−	0.0835129i	0.840910	−	0.541175i	$-0.182020\pi$
$67$	−1771.66	−	3068.60i	−0.0482162	−	0.0835129i	−0.889126	+	0.457662i	$0.848687\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−58429.0			−1.37557			−0.687785	−	0.725914i	$-0.741417\pi$
$71$	−58429.0			−1.37557			−0.687785	+	0.725914i	$0.741417\pi$
$72$		0			0
$73$	−60181.3			−1.32176			−0.660882	−	0.750490i	$-0.729818\pi$
$73$	−60181.3			−1.32176			−0.660882	+	0.750490i	$0.729818\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	44860.5	+	77700.6i	0.862258	+	1.49347i
$78$		0			0
$79$	−27811.7	+	48171.3i	−0.501372	+	0.868401i	0.498627	+	0.866817i	$0.333838\pi$
$79$	−27811.7	+	48171.3i	−0.501372	+	0.868401i	−0.999999	+	0.00158440i	$0.999496\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−19990.3	+	34624.2i	−0.318511	+	0.551677i	−0.980178	−	0.198121i	$-0.936516\pi$
$83$	−19990.3	+	34624.2i	−0.318511	+	0.551677i	0.661667	+	0.749798i	$0.269849\pi$
$84$		0			0
$85$	30693.3	+	53162.4i	0.460783	+	0.798099i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−103171.			−1.38065			−0.690327	−	0.723498i	$-0.742533\pi$
$89$	−103171.			−1.38065			−0.690327	+	0.723498i	$0.742533\pi$
$90$		0			0
$91$	−98682.3			−1.24921
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−104702.	−	181349.i	−1.19027	−	2.06161i
$96$		0			0
$97$	82996.9	−	143755.i	0.895638	−	1.55129i	0.0626259	−	0.998037i	$-0.480053\pi$
$97$	82996.9	−	143755.i	0.895638	−	1.55129i	0.833013	−	0.553254i	$-0.186614\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	82524.4	−	142936.i	0.804969	−	1.39425i	−0.111344	−	0.993782i	$-0.535515\pi$
$101$	82524.4	−	142936.i	0.804969	−	1.39425i	0.916312	−	0.400464i	$-0.131151\pi$
$102$		0			0
$103$	36430.5	+	63099.4i	0.338354	+	0.586047i	0.984123	−	0.177486i	$-0.0567964\pi$
$103$	36430.5	+	63099.4i	0.338354	+	0.586047i	−0.645769	+	0.763533i	$0.723463\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	123024.			1.03880			0.519399	−	0.854532i	$-0.326156\pi$
$107$	123024.			1.03880			0.519399	+	0.854532i	$0.326156\pi$
$108$		0			0
$109$	24274.5			0.195697			0.0978486	−	0.995201i	$-0.468804\pi$
$109$	24274.5			0.195697			0.0978486	+	0.995201i	$0.468804\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−100686.	−	174394.i	−0.741778	−	1.28480i	−0.951685	−	0.307075i	$-0.900650\pi$
$113$	−100686.	−	174394.i	−0.741778	−	1.28480i	0.209908	−	0.977721i	$-0.432684\pi$
$114$		0			0
$115$	111814.	−	193667.i	0.788408	−	1.36556i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	67549.3	−	116999.i	0.437274	−	0.757380i
$120$		0			0
$121$	−44725.0	−	77465.9i	−0.277707	−	0.481002i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	31339.2			0.179396
$126$		0			0
$127$	104163.			0.573067			0.286534	−	0.958070i	$-0.407497\pi$
$127$	104163.			0.573067			0.286534	+	0.958070i	$0.407497\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−89424.9	−	154888.i	−0.455282	−	0.788571i	0.543423	−	0.839459i	$-0.317128\pi$
$131$	−89424.9	−	154888.i	−0.455282	−	0.788571i	−0.998704	+	0.0508883i	$0.983795\pi$
$132$		0			0
$133$	−230426.	+	399110.i	−1.12954	+	1.95643i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	67651.3	−	117176.i	0.307946	−	0.533378i	−0.669967	−	0.742391i	$-0.733692\pi$
$137$	67651.3	−	117176.i	0.307946	−	0.533378i	0.977913	+	0.209013i	$0.0670250\pi$
$138$		0			0
$139$	−113083.	−	195866.i	−0.496434	−	0.859848i	0.503558	−	0.863961i	$-0.332024\pi$
$139$	−113083.	−	195866.i	−0.496434	−	0.859848i	−0.999992	+	0.00411320i	$0.998691\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	275521.			1.12672
$144$		0			0
$145$	−318462.			−1.25788
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−171449.	−	296958.i	−0.632657	−	1.09579i	−0.987006	−	0.160680i	$-0.948631\pi$
$149$	−171449.	−	296958.i	−0.632657	−	1.09579i	0.354350	−	0.935113i	$-0.384702\pi$
$150$		0			0
$151$	−121970.	+	211258.i	−0.435322	+	0.754000i	−0.997322	−	0.0731373i	$-0.976699\pi$
$151$	−121970.	+	211258.i	−0.435322	+	0.754000i	0.562000	+	0.827137i	$0.310032\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−126451.	+	219019.i	−0.422758	+	0.732238i
$156$		0			0
$157$	−174438.	−	302136.i	−0.564797	−	0.978257i	−0.997069	−	0.0765136i	$-0.975621\pi$
$157$	−174438.	−	302136.i	−0.564797	−	0.978257i	0.432272	−	0.901743i	$-0.357712\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−492156.			−1.49637
$162$		0			0
$163$	−303629.			−0.895107			−0.447553	−	0.894257i	$-0.647705\pi$
$163$	−303629.			−0.895107			−0.447553	+	0.894257i	$0.647705\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	16387.7	+	28384.4i	0.0454703	+	0.0787568i	0.887865	−	0.460104i	$-0.152188\pi$
$167$	16387.7	+	28384.4i	0.0454703	+	0.0787568i	−0.842395	+	0.538861i	$0.818855\pi$
$168$		0			0
$169$	34126.4	−	59108.7i	0.0919123	−	0.159197i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−177651.	+	307701.i	−0.451287	+	0.781652i	−0.998466	−	0.0553636i	$-0.982368\pi$
$173$	−177651.	+	307701.i	−0.451287	+	0.781652i	0.547179	+	0.837015i	$0.315702\pi$
$174$		0			0
$175$	−314583.	−	544873.i	−0.776497	−	1.34493i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−459272.			−1.07137			−0.535683	−	0.844419i	$-0.679946\pi$
$179$	−459272.			−1.07137			−0.535683	+	0.844419i	$0.679946\pi$
$180$		0			0
$181$	−190088.			−0.431279			−0.215640	−	0.976473i	$-0.569184\pi$
$181$	−190088.			−0.431279			−0.215640	+	0.976473i	$0.569184\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	389680.	+	674945.i	0.837103	+	1.44990i
$186$		0			0
$187$	−188598.	+	326661.i	−0.394396	+	0.683113i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	344280.	−	596310.i	0.682854	−	1.18274i	−0.291252	−	0.956646i	$-0.594072\pi$
$191$	344280.	−	596310.i	0.682854	−	1.18274i	0.974106	−	0.226092i	$-0.0725948\pi$
$192$		0			0
$193$	−172138.	−	298152.i	−0.332648	−	0.576163i	0.650382	−	0.759607i	$-0.274609\pi$
$193$	−172138.	−	298152.i	−0.332648	−	0.576163i	−0.983030	+	0.183444i	$0.941275\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	596238.			1.09460			0.547298	−	0.836938i	$-0.315656\pi$
$197$	596238.			1.09460			0.547298	+	0.836938i	$0.315656\pi$
$198$		0			0
$199$	49436.6			0.0884945			0.0442473	−	0.999021i	$-0.485911\pi$
$199$	49436.6			0.0884945			0.0442473	+	0.999021i	$0.485911\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	350433.	+	606968.i	0.596849	+	1.03377i
$204$		0			0
$205$	90696.5	−	157091.i	0.150732	−	0.261076i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	643351.	−	1.11432e6i	1.01878	−	1.76459i
$210$		0			0
$211$	−75999.1	−	131634.i	−0.117518	−	0.203546i	0.801266	−	0.598309i	$-0.204160\pi$
$211$	−75999.1	−	131634.i	−0.117518	−	0.203546i	−0.918783	+	0.394762i	$0.870827\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1.16372e6			−1.71693
$216$		0			0
$217$	556580.			0.802377
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−207435.	−	359288.i	−0.285694	−	0.494836i
$222$		0			0
$223$	−281057.	+	486805.i	−0.378471	+	0.655531i	−0.990840	−	0.135041i	$-0.956883\pi$
$223$	−281057.	+	486805.i	−0.378471	+	0.655531i	0.612369	+	0.790572i	$0.290217\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	630493.	−	1.09205e6i	0.812111	−	1.40662i	−0.0992724	−	0.995060i	$-0.531652\pi$
$227$	630493.	−	1.09205e6i	0.812111	−	1.40662i	0.911384	−	0.411558i	$-0.135015\pi$
$228$		0			0
$229$	−69803.7	−	120904.i	−0.0879609	−	0.152353i	0.818688	−	0.574238i	$-0.194702\pi$
$229$	−69803.7	−	120904.i	−0.0879609	−	0.152353i	−0.906649	+	0.421886i	$0.861368\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.18799e6			1.43358			0.716790	−	0.697289i	$-0.245610\pi$
$233$	1.18799e6			1.43358			0.716790	+	0.697289i	$0.245610\pi$
$234$		0			0
$235$	−527181.			−0.622716
$236$		0			0
$237$		0			0
$238$		0			0
$239$	164952.	+	285705.i	0.186794	+	0.323536i	0.944179	−	0.329432i	$-0.106857\pi$
$239$	164952.	+	285705.i	0.186794	+	0.323536i	−0.757386	+	0.652968i	$0.773524\pi$
$240$		0			0
$241$	−84372.3	+	146137.i	−0.0935745	+	0.162076i	−0.909013	−	0.416768i	$-0.863163\pi$
$241$	−84372.3	+	146137.i	−0.0935745	+	0.162076i	0.815438	+	0.578844i	$0.196496\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−624263.	+	1.08126e6i	−0.664435	+	1.15083i
$246$		0			0
$247$	707609.	+	1.22561e6i	0.737991	+	1.27824i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−224796.			−0.225218			−0.112609	−	0.993639i	$-0.535921\pi$
$251$	−224796.			−0.225218			−0.112609	+	0.993639i	$0.535921\pi$
$252$		0			0
$253$	1.37410e6			1.34964
$254$		0			0
$255$		0			0
$256$		0			0
$257$	347968.	+	602698.i	0.328629	+	0.569203i	0.982240	−	0.187628i	$-0.0600799\pi$
$257$	347968.	+	602698.i	0.328629	+	0.569203i	−0.653611	+	0.756831i	$0.726747\pi$
$258$		0			0
$259$	857600.	−	1.48541e6i	0.794393	−	1.37593i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	241366.	−	418057.i	0.215172	−	0.372689i	−0.738154	−	0.674633i	$-0.764302\pi$
$263$	241366.	−	418057.i	0.215172	−	0.372689i	0.953326	+	0.301944i	$0.0976354\pi$
$264$		0			0
$265$	557644.	+	965867.i	0.487800	+	0.844895i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−2.23733e6			−1.88516			−0.942581	−	0.333978i	$-0.891609\pi$
$269$	−2.23733e6			−1.88516			−0.942581	+	0.333978i	$0.891609\pi$
$270$		0			0
$271$	1.74480e6			1.44318			0.721592	−	0.692318i	$-0.243411\pi$
$271$	1.74480e6			1.44318			0.721592	+	0.692318i	$0.243411\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	878315.	+	1.52129e6i	0.700356	+	1.21305i
$276$		0			0
$277$	378971.	−	656397.i	0.296761	−	0.514005i	−0.678632	−	0.734478i	$-0.737427\pi$
$277$	378971.	−	656397.i	0.296761	−	0.514005i	0.975393	+	0.220474i	$0.0707603\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	215216.	−	372764.i	0.162595	−	0.281623i	−0.773203	−	0.634158i	$-0.781347\pi$
$281$	215216.	−	372764.i	0.162595	−	0.281623i	0.935799	+	0.352535i	$0.114680\pi$
$282$		0			0
$283$	478649.	+	829044.i	0.355264	+	0.615335i	0.987163	−	0.159716i	$-0.0510577\pi$
$283$	478649.	+	829044.i	0.355264	+	0.615335i	−0.631899	+	0.775051i	$0.717724\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−399206.			−0.286083
$288$		0			0
$289$	−851890.			−0.599983
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−283147.	−	490425.i	−0.192683	−	0.333737i	0.753456	−	0.657499i	$-0.228386\pi$
$293$	−283147.	−	490425.i	−0.192683	−	0.333737i	−0.946138	+	0.323762i	$0.895052\pi$
$294$		0			0
$295$	−232487.	+	402680.i	−0.155541	+	0.269404i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−755673.	+	1.30886e6i	−0.488828	+	0.846674i
$300$		0			0
$301$	1.28055e6	+	2.21797e6i	0.814665	+	1.41104i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−961090.			−0.591581
$306$		0			0
$307$	2.00565e6			1.21453			0.607265	−	0.794499i	$-0.292267\pi$
$307$	2.00565e6			1.21453			0.607265	+	0.794499i	$0.292267\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	1.05835e6	+	1.83312e6i	0.620481	+	1.07471i	0.989396	+	0.145242i	$0.0463961\pi$
$311$	1.05835e6	+	1.83312e6i	0.620481	+	1.07471i	−0.368915	+	0.929463i	$0.620271\pi$
$312$		0			0
$313$	−606669.	+	1.05078e6i	−0.350018	+	0.606249i	−0.986252	−	0.165247i	$-0.947158\pi$
$313$	−606669.	+	1.05078e6i	−0.350018	+	0.606249i	0.636234	+	0.771496i	$0.280491\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.28378e6	+	2.22357e6i	−0.717533	+	1.24280i	0.244441	+	0.969664i	$0.421396\pi$
$317$	−1.28378e6	+	2.22357e6i	−0.717533	+	1.24280i	−0.961974	+	0.273140i	$0.911938\pi$
$318$		0			0
$319$	−978409.	−	1.69465e6i	−0.538324	−	0.932404i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−1.93747e6			−1.03330
$324$		0			0
$325$	−1.93208e6			−1.01465
$326$		0			0
$327$		0			0
$328$		0			0
$329$	580105.	+	1.00477e6i	0.295473	+	0.511773i
$330$		0			0
$331$	721129.	−	1.24903e6i	0.361778	−	0.626619i	−0.626475	−	0.779441i	$-0.715503\pi$
$331$	721129.	−	1.24903e6i	0.361778	−	0.626619i	0.988254	+	0.152823i	$0.0488363\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−144309.	+	249950.i	−0.0702555	+	0.121686i
$336$		0			0
$337$	−790934.	−	1.36994e6i	−0.379372	−	0.657092i	0.611599	−	0.791168i	$-0.290527\pi$
$337$	−790934.	−	1.36994e6i	−0.379372	−	0.657092i	−0.990971	+	0.134076i	$0.957193\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.55397e6			−0.723698
$342$		0			0
$343$	−265129.			−0.121681
$344$		0			0
$345$		0			0
$346$		0			0
$347$	824985.	+	1.42892e6i	0.367809	+	0.637064i	0.989223	−	0.146419i	$-0.0467747\pi$
$347$	824985.	+	1.42892e6i	0.367809	+	0.637064i	−0.621414	+	0.783483i	$0.713441\pi$
$348$		0			0
$349$	−325183.	+	563234.i	−0.142911	+	0.247528i	−0.928591	−	0.371104i	$-0.878979\pi$
$349$	−325183.	+	563234.i	−0.142911	+	0.247528i	0.785681	+	0.618632i	$0.212313\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	590817.	−	1.02333e6i	0.252358	−	0.437096i	−0.711817	−	0.702365i	$-0.752127\pi$
$353$	590817.	−	1.02333e6i	0.252358	−	0.437096i	0.964174	+	0.265269i	$0.0854607\pi$
$354$		0			0
$355$	2.37964e6	+	4.12165e6i	1.00217	+	1.73580i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	985030.			0.403379			0.201690	−	0.979449i	$-0.435357\pi$
$359$	985030.			0.403379			0.201690	+	0.979449i	$0.435357\pi$
$360$		0			0
$361$	4.13306e6			1.66918
$362$		0			0
$363$		0			0
$364$		0			0
$365$	2.45100e6	+	4.24526e6i	0.962967	+	1.66791i
$366$		0			0
$367$	−598436.	+	1.03652e6i	−0.231928	+	0.401711i	−0.958375	−	0.285511i	$-0.907837\pi$
$367$	−598436.	+	1.03652e6i	−0.231928	+	0.401711i	0.726448	+	0.687222i	$0.241170\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1.22725e6	−	2.12566e6i	0.462913	−	0.801788i
$372$		0			0
$373$	1.23230e6	+	2.13441e6i	0.458611	+	0.794337i	0.998888	−	0.0471500i	$-0.0150139\pi$
$373$	1.23230e6	+	2.13441e6i	0.458611	+	0.794337i	−0.540277	+	0.841487i	$0.681681\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	2.15227e6			0.779906
$378$		0			0
$379$	−5.33540e6			−1.90796			−0.953979	−	0.299875i	$-0.903055\pi$
$379$	−5.33540e6			−1.90796			−0.953979	+	0.299875i	$0.903055\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−826117.	−	1.43088e6i	−0.287770	−	0.498432i	0.685507	−	0.728066i	$-0.259580\pi$
$383$	−826117.	−	1.43088e6i	−0.287770	−	0.498432i	−0.973277	+	0.229634i	$0.926247\pi$
$384$		0			0
$385$	3.65406e6	−	6.32902e6i	1.25639	−	2.17613i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	2.83443e6	−	4.90938e6i	0.949712	−	1.64495i	0.203683	−	0.979037i	$-0.434709\pi$
$389$	2.83443e6	−	4.90938e6i	0.949712	−	1.64495i	0.746029	−	0.665913i	$-0.231958\pi$
$390$		0			0
$391$	−1.03453e6	−	1.79187e6i	−0.342218	−	0.592739i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4.53074e6			1.46109
$396$		0			0
$397$	418875.			0.133385			0.0666927	−	0.997774i	$-0.478755\pi$
$397$	418875.			0.133385			0.0666927	+	0.997774i	$0.478755\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1.43256e6	+	2.48126e6i	0.444889	+	0.770570i	0.998044	−	0.0625082i	$-0.0199100\pi$
$401$	1.43256e6	+	2.48126e6i	0.444889	+	0.770570i	−0.553156	+	0.833078i	$0.686577\pi$
$402$		0			0
$403$	854592.	−	1.48020e6i	0.262118	−	0.454001i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−2.39442e6	+	4.14726e6i	−0.716497	+	1.24101i
$408$		0			0
$409$	−2.65238e6	−	4.59406e6i	−0.784021	−	1.35796i	−0.929582	−	0.368615i	$-0.879832\pi$
$409$	−2.65238e6	−	4.59406e6i	−0.784021	−	1.35796i	0.145561	−	0.989349i	$-0.453501\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	1.02331e6			0.295210
$414$		0			0
$415$	3.25658e6			0.928200
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−2.42136e6	−	4.19393e6i	−0.673791	−	1.16704i	−0.976821	−	0.214059i	$-0.931332\pi$
$419$	−2.42136e6	−	4.19393e6i	−0.673791	−	1.16704i	0.303030	−	0.952981i	$-0.402002\pi$
$420$		0			0
$421$	−2.28969e6	+	3.96586e6i	−0.629610	+	1.09052i	0.358019	+	0.933714i	$0.383452\pi$
$421$	−2.28969e6	+	3.96586e6i	−0.629610	+	1.09052i	−0.987630	+	0.156803i	$0.949881\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.32254e6	−	2.29070e6i	0.355169	−	0.615171i
$426$		0			0
$427$	1.05757e6	+	1.83177e6i	0.280699	+	0.486185i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−3.09717e6			−0.803105			−0.401552	−	0.915836i	$-0.631529\pi$
$431$	−3.09717e6			−0.803105			−0.401552	+	0.915836i	$0.631529\pi$
$432$		0			0
$433$	−992453.			−0.254384			−0.127192	−	0.991878i	$-0.540596\pi$
$433$	−992453.			−0.254384			−0.127192	+	0.991878i	$0.540596\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	3.52904e6	+	6.11248e6i	0.884002	+	1.53114i
$438$		0			0
$439$	−442686.	+	766755.i	−0.109631	+	0.189887i	−0.915621	−	0.402043i	$-0.868300\pi$
$439$	−442686.	+	766755.i	−0.109631	+	0.189887i	0.805990	+	0.591930i	$0.201634\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	270389.	−	468327.i	0.0654605	−	0.113381i	−0.831438	−	0.555618i	$-0.812482\pi$
$443$	270389.	−	468327.i	0.0654605	−	0.113381i	0.896898	+	0.442237i	$0.145815\pi$
$444$		0			0
$445$	4.20186e6	+	7.27784e6i	1.00587	+	1.74222i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−2.22193e6			−0.520132			−0.260066	−	0.965591i	$-0.583744\pi$
$449$	−2.22193e6			−0.520132			−0.260066	+	0.965591i	$0.583744\pi$
$450$		0			0
$451$	1.11458e6			0.258031
$452$		0			0
$453$		0			0
$454$		0			0
$455$	4.01903e6	+	6.96117e6i	0.910108	+	1.57635i
$456$		0			0
$457$	288892.	−	500376.i	0.0647061	−	0.112074i	−0.831857	−	0.554989i	$-0.812722\pi$
$457$	288892.	−	500376.i	0.0647061	−	0.112074i	0.896564	+	0.442915i	$0.146056\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−1.70719e6	+	2.95694e6i	−0.374137	+	0.648023i	−0.990197	−	0.139675i	$-0.955394\pi$
$461$	−1.70719e6	+	2.95694e6i	−0.374137	+	0.648023i	0.616061	+	0.787699i	$0.288728\pi$
$462$		0			0
$463$	−927775.	−	1.60695e6i	−0.201136	−	0.348378i	0.747759	−	0.663971i	$-0.231130\pi$
$463$	−927775.	−	1.60695e6i	−0.201136	−	0.348378i	−0.948895	+	0.315593i	$0.897797\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−3.73338e6			−0.792154			−0.396077	−	0.918217i	$-0.629629\pi$
$467$	−3.73338e6			−0.792154			−0.396077	+	0.918217i	$0.629629\pi$
$468$		0			0
$469$	635184.			0.133342
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−3.57528e6	−	6.19258e6i	−0.734781	−	1.27268i
$474$		0			0
$475$	−4.51148e6	+	7.81411e6i	−0.917455	+	1.58908i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	1.99246e6	−	3.45105e6i	0.396782	−	0.687246i	−0.596545	−	0.802579i	$-0.703460\pi$
$479$	1.99246e6	−	3.45105e6i	0.396782	−	0.687246i	0.993327	+	0.115334i	$0.0367937\pi$
$480$		0			0
$481$	−2.63358e6	−	4.56149e6i	−0.519019	−	0.898967i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1.35209e7			−2.61006
$486$		0			0
$487$	−4.19007e6			−0.800570			−0.400285	−	0.916391i	$-0.631089\pi$
$487$	−4.19007e6			−0.800570			−0.400285	+	0.916391i	$0.631089\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−2.09606e6	−	3.63049e6i	−0.392374	−	0.679612i	0.600388	−	0.799709i	$-0.295013\pi$
$491$	−2.09606e6	−	3.63049e6i	−0.392374	−	0.679612i	−0.992762	+	0.120097i	$0.961680\pi$
$492$		0			0
$493$	−1.47325e6	+	2.55175e6i	−0.272998	+	0.472847i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	5.23706e6	−	9.07086e6i	0.951036	−	1.64724i
$498$		0			0
$499$	−1.85169e6	−	3.20722e6i	−0.332903	−	0.576604i	0.650177	−	0.759783i	$-0.274695\pi$
$499$	−1.85169e6	−	3.20722e6i	−0.332903	−	0.576604i	−0.983080	+	0.183178i	$0.941361\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	4.03820e6			0.711653			0.355827	−	0.934552i	$-0.384199\pi$
$503$	4.03820e6			0.711653			0.355827	+	0.934552i	$0.384199\pi$
$504$		0			0
$505$	−1.34439e7			−2.34583
$506$		0			0
$507$		0			0
$508$		0			0
$509$	83980.3	+	145458.i	0.0143676	+	0.0248853i	0.873120	−	0.487506i	$-0.162093\pi$
$509$	83980.3	+	145458.i	0.0143676	+	0.0248853i	−0.858752	+	0.512391i	$0.828760\pi$
$510$		0			0
$511$	5.39412e6	−	9.34289e6i	0.913836	−	1.58281i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	2.96741e6	−	5.13970e6i	0.493014	−	0.853925i
$516$		0			0
$517$	−1.61965e6	−	2.80532e6i	−0.266499	−	0.461590i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−3.98731e6			−0.643555			−0.321777	−	0.946815i	$-0.604280\pi$
$521$	−3.98731e6			−0.643555			−0.321777	+	0.946815i	$0.604280\pi$
$522$		0			0
$523$	4.41694e6			0.706102			0.353051	−	0.935604i	$-0.385144\pi$
$523$	4.41694e6			0.706102			0.353051	+	0.935604i	$0.385144\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1.16996e6	+	2.02643e6i	0.183503	+	0.317837i
$528$		0			0
$529$	−550576.	+	953626.i	−0.0855418	+	0.148163i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−612955.	+	1.06167e6i	−0.0934567	+	0.161872i
$534$		0			0
$535$	−5.01040e6	−	8.67827e6i	−0.756812	−	1.31084i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−7.67167e6			−1.13741
$540$		0			0
$541$	9.24640e6			1.35825			0.679125	−	0.734023i	$-0.262360\pi$
$541$	9.24640e6			1.35825			0.679125	+	0.734023i	$0.262360\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−988628.	−	1.71235e6i	−0.142574	−	0.246946i
$546$		0			0
$547$	762888.	−	1.32136e6i	0.109017	−	0.188822i	−0.806356	−	0.591431i	$-0.798563\pi$
$547$	762888.	−	1.32136e6i	0.109017	−	0.188822i	0.915372	+	0.402609i	$0.131897\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	5.02561e6	−	8.70461e6i	0.705196	−	1.22144i
$552$		0			0
$553$	−4.98559e6	−	8.63529e6i	−0.693272	−	1.20078i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−5.86988e6			−0.801662			−0.400831	−	0.916152i	$-0.631279\pi$
$557$	−5.86988e6			−0.801662			−0.400831	+	0.916152i	$0.631279\pi$
$558$		0			0
$559$	7.86477e6			1.06453
$560$		0			0
$561$		0			0
$562$		0			0
$563$	106686.	+	184785.i	0.0141852	+	0.0245695i	0.873031	−	0.487665i	$-0.162151\pi$
$563$	106686.	+	184785.i	0.0141852	+	0.0245695i	−0.858846	+	0.512234i	$0.828818\pi$
$564$		0			0
$565$	−8.20129e6	+	1.42050e7i	−1.08084	+	1.87207i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	5.64514e6	−	9.77767e6i	0.730961	−	1.26606i	−0.225512	−	0.974240i	$-0.572405\pi$
$569$	5.64514e6	−	9.77767e6i	0.730961	−	1.26606i	0.956473	−	0.291821i	$-0.0942612\pi$
$570$		0			0
$571$	−232342.	−	402427.i	−0.0298220	−	0.0516532i	0.850729	−	0.525604i	$-0.176161\pi$
$571$	−232342.	−	402427.i	−0.0298220	−	0.0516532i	−0.880551	+	0.473951i	$0.842827\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−9.63583e6			−1.21540
$576$		0			0
$577$	8.78227e6			1.09816			0.549082	−	0.835769i	$-0.314978\pi$
$577$	8.78227e6			1.09816			0.549082	+	0.835769i	$0.314978\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−3.58351e6	−	6.20682e6i	−0.440421	−	0.762832i
$582$		0			0
$583$	−3.42649e6	+	5.93485e6i	−0.417521	+	0.723167i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−5.02083e6	+	8.69633e6i	−0.601423	+	1.04170i	0.391183	+	0.920313i	$0.372066\pi$
$587$	−5.02083e6	+	8.69633e6i	−0.601423	+	1.04170i	−0.992606	+	0.121382i	$0.961267\pi$
$588$		0			0
$589$	−3.99100e6	−	6.91262e6i	−0.474017	−	0.821021i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1.15945e7			1.35399			0.676995	−	0.735988i	$-0.263282\pi$
$593$	1.15945e7			1.35399			0.676995	+	0.735988i	$0.263282\pi$
$594$		0			0
$595$	−1.10043e7			−1.27430
$596$		0			0
$597$		0			0
$598$		0			0
$599$	744422.	+	1.28938e6i	0.0847720	+	0.146829i	0.905294	−	0.424786i	$-0.139650\pi$
$599$	744422.	+	1.28938e6i	0.0847720	+	0.146829i	−0.820522	+	0.571615i	$0.806317\pi$
$600$		0			0
$601$	4.75503e6	−	8.23596e6i	0.536992	−	0.930097i	−0.462073	−	0.886842i	$-0.652894\pi$
$601$	4.75503e6	−	8.23596e6i	0.536992	−	0.930097i	0.999064	−	0.0432545i	$-0.0137726\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−3.64303e6	+	6.30991e6i	−0.404645	+	0.700865i
$606$		0			0
$607$	3.70203e6	+	6.41210e6i	0.407819	+	0.706364i	0.994645	−	0.103349i	$-0.0329560\pi$
$607$	3.70203e6	+	6.41210e6i	0.407819	+	0.706364i	−0.586826	+	0.809713i	$0.699623\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	3.56285e6			0.386096
$612$		0			0
$613$	−7.14368e6			−0.767840			−0.383920	−	0.923366i	$-0.625426\pi$
$613$	−7.14368e6			−0.767840			−0.383920	+	0.923366i	$0.625426\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−502021.	−	869525.i	−0.0530895	−	0.0919537i	0.838259	−	0.545272i	$-0.183574\pi$
$617$	−502021.	−	869525.i	−0.0530895	−	0.0919537i	−0.891349	+	0.453318i	$0.850240\pi$
$618$		0			0
$619$	−3.74328e6	+	6.48355e6i	−0.392668	+	0.680121i	−0.992801	−	0.119779i	$-0.961781\pi$
$619$	−3.74328e6	+	6.48355e6i	−0.392668	+	0.680121i	0.600132	+	0.799901i	$0.295115\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	9.24738e6	−	1.60169e7i	0.954550	−	1.65333i
$624$		0			0
$625$	4.20763e6	+	7.28783e6i	0.430861	+	0.746274i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	7.21086e6			0.726709
$630$		0			0
$631$	−1.34648e7			−1.34626			−0.673128	−	0.739526i	$-0.735050\pi$
$631$	−1.34648e7			−1.34626			−0.673128	+	0.739526i	$0.735050\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−4.24226e6	−	7.34781e6i	−0.417506	−	0.723142i
$636$		0			0
$637$	4.21896e6	−	7.30746e6i	0.411962	−	0.713539i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−8.13083e6	+	1.40830e7i	−0.781610	+	1.35379i	0.149394	+	0.988778i	$0.452268\pi$
$641$	−8.13083e6	+	1.40830e7i	−0.781610	+	1.35379i	−0.931004	+	0.365010i	$0.881065\pi$
$642$		0			0
$643$	−5.68437e6	−	9.84561e6i	−0.542194	−	0.939108i	−0.998778	−	0.0494272i	$-0.984260\pi$
$643$	−5.68437e6	−	9.84561e6i	−0.542194	−	0.939108i	0.456584	−	0.889680i	$-0.349073\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.18311e7			−1.11113			−0.555563	−	0.831474i	$-0.687497\pi$
$647$	−1.18311e7			−1.11113			−0.555563	+	0.831474i	$0.687497\pi$
$648$		0			0
$649$	−2.85707e6			−0.266262
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−2.67300e6	−	4.62977e6i	−0.245310	−	0.424890i	0.716909	−	0.697167i	$-0.245556\pi$
$653$	−2.67300e6	−	4.62977e6i	−0.245310	−	0.424890i	−0.962219	+	0.272278i	$0.912223\pi$
$654$		0			0
$655$	−7.28401e6	+	1.26163e7i	−0.663387	+	1.14902i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	6.18681e6	−	1.07159e7i	0.554949	−	0.961200i	−0.442958	−	0.896542i	$-0.646071\pi$
$659$	6.18681e6	−	1.07159e7i	0.554949	−	0.961200i	0.997907	−	0.0646581i	$-0.0205957\pi$
$660$		0			0
$661$	1.04915e7	+	1.81718e7i	0.933974	+	1.61769i	0.776454	+	0.630174i	$0.217016\pi$
$661$	1.04915e7	+	1.81718e7i	0.933974	+	1.61769i	0.157520	+	0.987516i	$0.449650\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	3.75383e7			3.29170
$666$		0			0
$667$	1.07339e7			0.934210
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−2.95275e6	−	5.11431e6i	−0.253175	−	0.438511i
$672$		0			0
$673$	6.30874e6	−	1.09271e7i	0.536915	−	0.929963i	−0.462154	−	0.886800i	$-0.652923\pi$
$673$	6.30874e6	−	1.09271e7i	0.536915	−	0.929963i	0.999068	−	0.0431633i	$-0.0137436\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	3.99728e6	−	6.92349e6i	0.335192	−	0.580569i	−0.648330	−	0.761359i	$-0.724532\pi$
$677$	3.99728e6	−	6.92349e6i	0.335192	−	0.580569i	0.983522	+	0.180791i	$0.0578656\pi$
$678$		0			0
$679$	1.48782e7	+	2.57698e7i	1.23845	+	2.14505i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.23985e7			1.01699			0.508496	−	0.861064i	$-0.330202\pi$
$683$	1.23985e7			1.01699			0.508496	+	0.861064i	$0.330202\pi$
$684$		0			0
$685$	−1.10209e7			−0.897412
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−3.76873e6	−	6.52763e6i	−0.302445	−	0.523851i
$690$		0			0
$691$	1.03172e7	−	1.78699e7i	0.821990	−	1.42373i	−0.0822079	−	0.996615i	$-0.526197\pi$
$691$	1.03172e7	−	1.78699e7i	0.821990	−	1.42373i	0.904198	−	0.427113i	$-0.140470\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−9.21108e6	+	1.59541e7i	−0.723350	+	1.25288i
$696$		0			0
$697$	−839150.	−	1.45345e6i	−0.0654271	−	0.113323i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−2.28362e7			−1.75521			−0.877603	−	0.479388i	$-0.840859\pi$
$701$	−2.28362e7			−1.75521			−0.877603	+	0.479388i	$0.840859\pi$
$702$		0			0
$703$	−2.45980e7			−1.87720
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.47935e7	+	2.56231e7i	1.11307	+	1.92790i
$708$		0			0
$709$	8.91237e6	−	1.54367e7i	0.665852	−	1.15329i	−0.313202	−	0.949687i	$-0.601402\pi$
$709$	8.91237e6	−	1.54367e7i	0.665852	−	1.15329i	0.979054	−	0.203603i	$-0.0652651\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	4.26209e6	−	7.38215e6i	0.313978	−	0.543825i
$714$		0			0
$715$	−1.12211e7	−	1.94356e7i	−0.820865	−	1.42178i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−1.10161e7			−0.794702			−0.397351	−	0.917667i	$-0.630071\pi$
$719$	−1.10161e7			−0.794702			−0.397351	+	0.917667i	$0.630071\pi$
$720$		0			0
$721$	−1.30612e7			−0.935720
$722$		0			0
$723$		0			0
$724$		0			0
$725$	6.86107e6	+	1.18837e7i	0.484782	+	0.839667i
$726$		0			0
$727$	1.30016e7	−	2.25195e7i	0.912351	−	1.58024i	0.101616	−	0.994824i	$-0.467599\pi$
$727$	1.30016e7	−	2.25195e7i	0.912351	−	1.58024i	0.810734	−	0.585414i	$-0.199068\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−5.38353e6	+	9.32456e6i	−0.372627	+	0.645409i
$732$		0			0
$733$	−1.33355e7	−	2.30978e7i	−0.916750	−	1.58786i	−0.804319	−	0.594198i	$-0.797470\pi$
$733$	−1.33355e7	−	2.30978e7i	−0.916750	−	1.58786i	−0.112431	−	0.993660i	$-0.535864\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.77343e6			−0.120267
$738$		0			0
$739$	−1.31065e6			−0.0882829			−0.0441414	−	0.999025i	$-0.514055\pi$
$739$	−1.31065e6			−0.0882829			−0.0441414	+	0.999025i	$0.514055\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1.02502e7	−	1.77539e7i	−0.681178	−	1.17983i	−0.974622	−	0.223858i	$-0.928135\pi$
$743$	−1.02502e7	−	1.77539e7i	−0.681178	−	1.17983i	0.293444	−	0.955976i	$-0.405199\pi$
$744$		0			0
$745$	−1.39652e7	+	2.41884e7i	−0.921839	+	1.59667i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1.10268e7	+	1.90990e7i	−0.718199	+	1.24396i
$750$		0			0
$751$	−6.78701e6	−	1.17554e7i	−0.439115	−	0.760570i	0.558506	−	0.829500i	$-0.311375\pi$
$751$	−6.78701e6	−	1.17554e7i	−0.439115	−	0.760570i	−0.997621	+	0.0689306i	$0.978041\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1.98699e7			1.26861
$756$		0			0
$757$	−470115.			−0.0298171			−0.0149085	−	0.999889i	$-0.504746\pi$
$757$	−470115.			−0.0298171			−0.0149085	+	0.999889i	$0.504746\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1.02690e6	−	1.77865e6i	−0.0642789	−	0.111334i	0.832095	−	0.554633i	$-0.187141\pi$
$761$	−1.02690e6	−	1.77865e6i	−0.0642789	−	0.111334i	−0.896374	+	0.443299i	$0.853808\pi$
$762$		0			0
$763$	−2.17575e6	+	3.76852e6i	−0.135300	+	0.234347i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1.57122e6	−	2.72143e6i	0.0964381	−	0.167036i
$768$		0			0
$769$	−1.25600e6	−	2.17546e6i	−0.0765905	−	0.132659i	0.825186	−	0.564861i	$-0.191070\pi$
$769$	−1.25600e6	−	2.17546e6i	−0.0765905	−	0.132659i	−0.901777	+	0.432202i	$0.857737\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	8.03339e6			0.483560			0.241780	−	0.970331i	$-0.422269\pi$
$773$	8.03339e6			0.483560			0.241780	+	0.970331i	$0.422269\pi$
$774$		0			0
$775$	1.08972e7			0.651719
$776$		0			0
$777$		0			0
$778$		0			0
$779$	2.86254e6	+	4.95806e6i	0.169008	+	0.292731i
$780$		0			0
$781$	−1.46219e7	+	2.53258e7i	−0.857780	+	1.48572i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1.42087e7	+	2.46102e7i	−0.822962	+	1.42541i
$786$		0			0
$787$	1.57883e7	+	2.73461e7i	0.908653	+	1.57383i	0.815938	+	0.578140i	$0.196221\pi$
$787$	1.57883e7	+	2.73461e7i	0.908653	+	1.57383i	0.0927148	+	0.995693i	$0.470446\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	3.60985e7			2.05139
$792$		0			0
$793$	6.49534e6			0.366791
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1.33523e7	−	2.31268e7i	−0.744577	−	1.28965i	−0.950392	−	0.311055i	$-0.899318\pi$
$797$	−1.33523e7	−	2.31268e7i	−0.744577	−	1.28965i	0.205815	−	0.978591i	$-0.434016\pi$
$798$		0			0
$799$	−2.43882e6	+	4.22416e6i	−0.135149	+	0.234085i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1.50604e7	+	2.60853e7i	−0.824228	+	1.42760i
$804$		0			0
$805$	2.00440e7	+	3.47173e7i	1.09017	+	1.88823i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1.27526e7			−0.685057			−0.342529	−	0.939507i	$-0.611283\pi$
$809$	−1.27526e7			−0.685057			−0.342529	+	0.939507i	$0.611283\pi$
$810$		0			0
$811$	2.34690e7			1.25297			0.626487	−	0.779432i	$-0.284492\pi$
$811$	2.34690e7			1.25297			0.626487	+	0.779432i	$0.284492\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1.23659e7	+	2.14184e7i	0.652127	+	1.12952i
$816$		0			0
$817$	1.83645e7	−	3.18082e7i	0.962552	−	1.66719i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−7.90381e6	+	1.36898e7i	−0.409240	+	0.708825i	−0.994805	−	0.101801i	$-0.967540\pi$
$821$	−7.90381e6	+	1.36898e7i	−0.409240	+	0.708825i	0.585564	+	0.810626i	$0.300873\pi$
$822$		0			0
$823$	8.12717e6	+	1.40767e7i	0.418254	+	0.724437i	0.995764	−	0.0919468i	$-0.0293090\pi$
$823$	8.12717e6	+	1.40767e7i	0.418254	+	0.724437i	−0.577510	+	0.816383i	$0.695976\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.39257e7			−0.708031			−0.354016	−	0.935239i	$-0.615184\pi$
$827$	−1.39257e7			−0.708031			−0.354016	+	0.935239i	$0.615184\pi$
$828$		0			0
$829$	2.52491e7			1.27603			0.638013	−	0.770026i	$-0.279757\pi$
$829$	2.52491e7			1.27603			0.638013	+	0.770026i	$0.279757\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	5.77586e6	+	1.00041e7i	0.288406	+	0.499534i
$834$		0			0
$835$	1.33485e6	−	2.31202e6i	0.0662544	−	0.114756i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−2.49554e6	+	4.32240e6i	−0.122394	+	0.211992i	−0.920711	−	0.390245i	$-0.872390\pi$
$839$	−2.49554e6	+	4.32240e6i	−0.122394	+	0.211992i	0.798317	+	0.602237i	$0.205724\pi$
$840$		0			0
$841$	2.61262e6	+	4.52518e6i	0.127375	+	0.220621i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−5.55946e6			−0.267850
$846$		0			0
$847$	1.60350e7			0.767999
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1.31344e7	−	2.27494e7i	−0.621707	−	1.07683i
$852$		0			0
$853$	−3.76286e6	+	6.51746e6i	−0.177070	+	0.306694i	−0.940876	−	0.338752i	$-0.889995\pi$
$853$	−3.76286e6	+	6.51746e6i	−0.177070	+	0.306694i	0.763806	+	0.645446i	$0.223329\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1.86940e7	−	3.23790e7i	0.869461	−	1.50595i	0.00691285	−	0.999976i	$-0.497800\pi$
$857$	1.86940e7	−	3.23790e7i	0.869461	−	1.50595i	0.862548	−	0.505975i	$-0.168867\pi$
$858$		0			0
$859$	−4.21798e6	−	7.30575e6i	−0.195039	−	0.337817i	0.751874	−	0.659306i	$-0.229150\pi$
$859$	−4.21798e6	−	7.30575e6i	−0.195039	−	0.337817i	−0.946913	+	0.321489i	$0.895817\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1.10157e7			−0.503482			−0.251741	−	0.967795i	$-0.581003\pi$
$863$	−1.10157e7			−0.503482			−0.251741	+	0.967795i	$0.581003\pi$
$864$		0			0
$865$	2.89408e7			1.31513
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1.39198e7	+	2.41097e7i	0.625292	+	1.08304i
$870$		0			0
$871$	975282.	−	1.68924e6i	0.0435597	−	0.0754476i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2.80897e6	+	4.86527e6i	−0.124030	+	0.214826i
$876$		0			0
$877$	7.42191e6	+	1.28551e7i	0.325849	+	0.564387i	0.981684	−	0.190517i	$-0.0610166\pi$
$877$	7.42191e6	+	1.28551e7i	0.325849	+	0.564387i	−0.655835	+	0.754904i	$0.727683\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−2.90839e7			−1.26245			−0.631224	−	0.775600i	$-0.717447\pi$
$881$	−2.90839e7			−1.26245			−0.631224	+	0.775600i	$0.717447\pi$
$882$		0			0
$883$	3.79255e7			1.63693			0.818464	−	0.574558i	$-0.194826\pi$
$883$	3.79255e7			1.63693			0.818464	+	0.574558i	$0.194826\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−7.65765e6	−	1.32634e7i	−0.326803	−	0.566040i	0.655072	−	0.755566i	$-0.272638\pi$
$887$	−7.65765e6	−	1.32634e7i	−0.326803	−	0.566040i	−0.981876	+	0.189526i	$0.939305\pi$
$888$		0			0
$889$	−9.33629e6	+	1.61709e7i	−0.396205	+	0.686247i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	8.31938e6	−	1.44096e7i	0.349110	−	0.604676i
$894$		0			0
$895$	1.87048e7	+	3.23976e7i	0.780540	+	1.35193i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1.21390e7			−0.500940
$900$		0			0
$901$	1.03190e7			0.423472
$902$		0			0
$903$		0			0
$904$		0			0
$905$	7.74171e6	+	1.34090e7i	0.314207	+	0.544222i
$906$		0			0
$907$	−1.34234e7	+	2.32500e7i	−0.541806	+	0.938436i	0.456994	+	0.889470i	$0.348926\pi$
$907$	−1.34234e7	+	2.32500e7i	−0.541806	+	0.938436i	−0.998800	+	0.0489660i	$0.984407\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−2.38433e6	+	4.12978e6i	−0.0951854	+	0.164866i	−0.909686	−	0.415297i	$-0.863678\pi$
$911$	−2.38433e6	+	4.12978e6i	−0.0951854	+	0.164866i	0.814501	+	0.580163i	$0.197011\pi$
$912$		0			0
$913$	1.00052e7	+	1.73295e7i	0.397235	+	0.688031i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	3.20610e7			1.25908
$918$		0			0
$919$	2.23134e7			0.871518			0.435759	−	0.900063i	$-0.356480\pi$
$919$	2.23134e7			0.871518			0.435759	+	0.900063i	$0.356480\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−1.60823e7	−	2.78554e7i	−0.621362	−	1.07623i
$924$		0			0
$925$	1.67908e7	−	2.90825e7i	0.645234	−	1.11758i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1.15737e7	−	2.00463e7i	0.439980	−	0.762068i	−0.557707	−	0.830038i	$-0.688319\pi$
$929$	1.15737e7	−	2.00463e7i	0.439980	−	0.762068i	0.997687	+	0.0679696i	$0.0216521\pi$
$930$		0			0
$931$	−1.97028e7	−	3.41263e7i	−0.744997	−	1.29037i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	3.07240e7			1.14934
$936$		0			0
$937$	−3.45453e6			−0.128541			−0.0642703	−	0.997933i	$-0.520472\pi$
$937$	−3.45453e6			−0.128541			−0.0642703	+	0.997933i	$0.520472\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1.77260e7	+	3.07024e7i	0.652586	+	1.13031i	0.982493	+	0.186298i	$0.0596491\pi$
$941$	1.77260e7	+	3.07024e7i	0.652586	+	1.13031i	−0.329908	+	0.944013i	$0.607018\pi$
$942$		0			0
$943$	−3.05697e6	+	5.29483e6i	−0.111947	+	0.193898i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1.95128e7	+	3.37971e7i	−0.707039	+	1.22463i	0.258911	+	0.965901i	$0.416636\pi$
$947$	−1.95128e7	+	3.37971e7i	−0.707039	+	1.22463i	−0.965950	+	0.258727i	$0.916697\pi$
$948$		0			0
$949$	−1.65646e7	−	2.86908e7i	−0.597057	−	1.03413i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	2.07434e6			0.0739858			0.0369929	−	0.999316i	$-0.488222\pi$
$953$	2.07434e6			0.0739858			0.0369929	+	0.999316i	$0.488222\pi$
$954$		0			0
$955$	−5.60859e7			−1.98996
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1.21273e7	+	2.10052e7i	0.425813	+	0.737530i
$960$		0			0
$961$	9.49457e6	−	1.64451e7i	0.331640	−	0.574417i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1.40214e7	+	2.42857e7i	−0.484698	+	0.839522i
$966$		0			0
$967$	−5.74657e6	−	9.95336e6i	−0.197625	−	0.342297i	0.750133	−	0.661287i	$-0.229990\pi$
$967$	−5.74657e6	−	9.95336e6i	−0.197625	−	0.342297i	−0.947758	+	0.318990i	$0.896656\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	4.36234e7			1.48481			0.742405	−	0.669951i	$-0.233685\pi$
$971$	4.36234e7			1.48481			0.742405	+	0.669951i	$0.233685\pi$
$972$		0			0
$973$	4.05431e7			1.37289
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.51800e7	+	2.62925e7i	0.508785	+	0.881241i	0.999948	+	0.0101736i	$0.00323841\pi$
$977$	1.51800e7	+	2.62925e7i	0.508785	+	0.881241i	−0.491164	+	0.871067i	$0.663428\pi$
$978$		0			0
$979$	−2.58187e7	+	4.47193e7i	−0.860950	+	1.49121i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	3.14034e6	−	5.43924e6i	0.103656	−	0.179537i	−0.809532	−	0.587075i	$-0.800279\pi$
$983$	3.14034e6	−	5.43924e6i	0.103656	−	0.179537i	0.913188	+	0.407538i	$0.133613\pi$
$984$		0			0
$985$	−2.42830e7	−	4.20593e7i	−0.797464	−	1.38125i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	3.92238e7			1.27514
$990$		0			0
$991$	−1.30907e7			−0.423428			−0.211714	−	0.977332i	$-0.567904\pi$
$991$	−1.30907e7			−0.423428			−0.211714	+	0.977332i	$0.567904\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−2.01341e6	−	3.48732e6i	−0.0644723	−	0.111669i
$996$		0			0
$997$	1.69254e6	−	2.93156e6i	0.0539262	−	0.0934029i	−0.837802	−	0.545974i	$-0.816160\pi$
$997$	1.69254e6	−	2.93156e6i	0.0539262	−	0.0934029i	0.891728	+	0.452571i	$0.149493\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.6.i.d.289.1		10
3.2	odd	2		144.6.i.d.97.3		10
4.3	odd	2		108.6.e.a.73.1		10
9.4	even	3	inner	432.6.i.d.145.1		10
9.5	odd	6		144.6.i.d.49.3		10
12.11	even	2		36.6.e.a.25.3	yes	10
36.7	odd	6		324.6.a.d.1.5		5
36.11	even	6		324.6.a.e.1.1		5
36.23	even	6		36.6.e.a.13.3	✓	10
36.31	odd	6		108.6.e.a.37.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.6.e.a.13.3	✓	10	36.23	even	6
36.6.e.a.25.3	yes	10	12.11	even	2
108.6.e.a.37.1		10	36.31	odd	6
108.6.e.a.73.1		10	4.3	odd	2
144.6.i.d.49.3		10	9.5	odd	6
144.6.i.d.97.3		10	3.2	odd	2
324.6.a.d.1.5		5	36.7	odd	6
324.6.a.e.1.1		5	36.11	even	6
432.6.i.d.145.1		10	9.4	even	3	inner
432.6.i.d.289.1		10	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	432.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-40.7270 - 70.5412i) q^{5} +(-89.6312 + 155.246i) q^{7} +O(q^{10})\)	\(q+(-40.7270 - 70.5412i) q^{5} +(-89.6312 + 155.246i) q^{7} +(250.250 - 433.446i) q^{11} +(275.245 + 476.739i) q^{13} -753.636 q^{17} +2570.83 q^{19} +(1372.72 + 2377.63i) q^{23} +(-1754.87 + 3039.53i) q^{25} +(1954.86 - 3385.92i) q^{29} +(-1552.42 - 2688.87i) q^{31} +14601.6 q^{35} -9568.10 q^{37} +(1113.47 + 1928.59i) q^{41} +(7143.42 - 12372.8i) q^{43} +(3236.07 - 5605.04i) q^{47} +(-7664.00 - 13274.4i) q^{49} -13692.2 q^{53} -40767.8 q^{55} +(-2854.22 - 4943.65i) q^{59} +(5899.59 - 10218.4i) q^{61} +(22419.8 - 38832.3i) q^{65} +(-1771.66 - 3068.60i) q^{67} -58429.0 q^{71} -60181.3 q^{73} +(44860.5 + 77700.6i) q^{77} +(-27811.7 + 48171.3i) q^{79} +(-19990.3 + 34624.2i) q^{83} +(30693.3 + 53162.4i) q^{85} -103171. q^{89} -98682.3 q^{91} +(-104702. - 181349. i) q^{95} +(82996.9 - 143755. i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 21 q^{5} - 29 q^{7}+O(q^{10}) \) 10 * q + 21 * q^5 - 29 * q^7	\( 10 q + 21 q^{5} - 29 q^{7} + 177 q^{11} - 181 q^{13} - 2280 q^{17} + 832 q^{19} + 399 q^{23} - 4778 q^{25} + 6033 q^{29} - 2759 q^{31} + 37146 q^{35} - 15172 q^{37} + 18435 q^{41} - 1469 q^{43} - 25155 q^{47} - 4056 q^{49} - 116844 q^{53} - 14778 q^{55} - 90537 q^{59} + 1403 q^{61} + 148407 q^{65} - 13907 q^{67} + 229368 q^{71} + 15200 q^{73} + 211983 q^{77} - 29993 q^{79} - 228951 q^{83} - 49662 q^{85} - 598332 q^{89} - 124930 q^{91} - 394764 q^{95} + 40541 q^{97}+O(q^{100}) \) 10 * q + 21 * q^5 - 29 * q^7 + 177 * q^11 - 181 * q^13 - 2280 * q^17 + 832 * q^19 + 399 * q^23 - 4778 * q^25 + 6033 * q^29 - 2759 * q^31 + 37146 * q^35 - 15172 * q^37 + 18435 * q^41 - 1469 * q^43 - 25155 * q^47 - 4056 * q^49 - 116844 * q^53 - 14778 * q^55 - 90537 * q^59 + 1403 * q^61 + 148407 * q^65 - 13907 * q^67 + 229368 * q^71 + 15200 * q^73 + 211983 * q^77 - 29993 * q^79 - 228951 * q^83 - 49662 * q^85 - 598332 * q^89 - 124930 * q^91 - 394764 * q^95 + 40541 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more