LMFDB - Embedded newform 400.6.n.e.143.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,6,Mod(143,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 3]))

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

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

chi := DirichletCharacter("400.143");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	400.n (of order $4$, degree $2$, 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:	$64.1535279252$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 164 x^{14} + 20144 x^{12} - 1089726 x^{10} + 44145944 x^{8} - 59360064 x^{6} + 76134209 x^{4} + \cdots + 38416 $ x^16 - 164x^14 + 20144x^12 - 1089726x^10 + 44145944x^8 - 59360064x^6 + 76134209x^4 - 1724996*x^2 + 38416
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{68} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			143.1
Root			$7.36491 - 4.25213i$ of defining polynomial
Character	$\chi$	$=$	400.143
Dual form			400.6.n.e.207.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-18.0085 + 18.0085i) q^{3} +(25.8552 + 25.8552i) q^{7} -405.614i q^{9} +O(q^{10})$	$q+(-18.0085 + 18.0085i) q^{3} +(25.8552 + 25.8552i) q^{7} -405.614i q^{9} -718.133i q^{11} +(347.119 + 347.119i) q^{13} +(-954.912 + 954.912i) q^{17} +496.431 q^{19} -931.229 q^{21} +(2252.48 - 2252.48i) q^{23} +(2928.44 + 2928.44i) q^{27} +5785.37i q^{29} +5851.12i q^{31} +(12932.5 + 12932.5i) q^{33} +(-9546.43 + 9546.43i) q^{37} -12502.2 q^{39} +6006.90 q^{41} +(11908.0 - 11908.0i) q^{43} +(2443.88 + 2443.88i) q^{47} -15470.0i q^{49} -34393.1i q^{51} +(22741.0 + 22741.0i) q^{53} +(-8939.98 + 8939.98i) q^{57} -6231.45 q^{59} -42507.9 q^{61} +(10487.2 - 10487.2i) q^{63} +(24167.2 + 24167.2i) q^{67} +81127.7i q^{69} -21553.1i q^{71} +(-36717.5 - 36717.5i) q^{73} +(18567.5 - 18567.5i) q^{77} -91428.3 q^{79} -6909.70 q^{81} +(-74086.8 + 74086.8i) q^{83} +(-104186. - 104186. i) q^{87} +56955.3i q^{89} +17949.6i q^{91} +(-105370. - 105370. i) q^{93} +(71785.9 - 71785.9i) q^{97} -291285. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 4576 q^{21} - 12288 q^{41} - 277504 q^{61} - 508016 q^{81}+O(q^{100}) $ 16 * q - 4576 * q^21 - 12288 * q^41 - 277504 * q^61 - 508016 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$	$-1$

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
$3$	−18.0085	+	18.0085i	−1.15525	+	1.15525i	−0.169763	+	0.985485i	$0.554300\pi$
$3$	−18.0085	+	18.0085i	−1.15525	+	1.15525i	−0.985485	+	0.169763i	$0.945700\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	25.8552	+	25.8552i	0.199436	+	0.199436i	0.799758	−	0.600322i	$-0.204961\pi$
$7$	25.8552	+	25.8552i	0.199436	+	0.199436i	−0.600322	+	0.799758i	$0.704961\pi$
$8$		0			0
$9$		−	405.614i		−	1.66919i
$10$		0			0
$11$		−	718.133i		−	1.78947i	−0.446602	−	0.894733i	$-0.647366\pi$
$11$		−	718.133i		−	1.78947i	0.446602	−	0.894733i	$-0.352634\pi$
$12$		0			0
$13$	347.119	+	347.119i	0.569665	+	0.569665i	0.932034	−	0.362370i	$-0.118032\pi$
$13$	347.119	+	347.119i	0.569665	+	0.569665i	−0.362370	+	0.932034i	$0.618032\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−954.912	+	954.912i	−0.801385	+	0.801385i	−0.983312	−	0.181927i	$-0.941767\pi$
$17$	−954.912	+	954.912i	−0.801385	+	0.801385i	0.181927	+	0.983312i	$0.441767\pi$
$18$		0			0
$19$	496.431			0.315482			0.157741	−	0.987481i	$-0.449579\pi$
$19$	496.431			0.315482			0.157741	+	0.987481i	$0.449579\pi$
$20$		0			0
$21$	−931.229			−0.460795
$22$		0			0
$23$	2252.48	−	2252.48i	0.887854	−	0.887854i	−0.106463	−	0.994317i	$-0.533953\pi$
$23$	2252.48	−	2252.48i	0.887854	−	0.887854i	0.994317	+	0.106463i	$0.0339526\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	2928.44	+	2928.44i	0.773086	+	0.773086i
$28$		0			0
$29$			5785.37i			1.27743i	0.769444	+	0.638714i	$0.220533\pi$
$29$			5785.37i			1.27743i	−0.769444	+	0.638714i	$0.779467\pi$
$30$		0			0
$31$			5851.12i			1.09354i	0.837283	+	0.546770i	$0.184143\pi$
$31$			5851.12i			1.09354i	−0.837283	+	0.546770i	$0.815857\pi$
$32$		0			0
$33$	12932.5	+	12932.5i	2.06728	+	2.06728i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−9546.43	+	9546.43i	−1.14640	+	1.14640i	−0.159147	+	0.987255i	$0.550874\pi$
$37$	−9546.43	+	9546.43i	−1.14640	+	1.14640i	−0.987255	+	0.159147i	$0.949126\pi$
$38$		0			0
$39$	−12502.2			−1.31621
$40$		0			0
$41$	6006.90			0.558073			0.279036	−	0.960281i	$-0.409985\pi$
$41$	6006.90			0.558073			0.279036	+	0.960281i	$0.409985\pi$
$42$		0			0
$43$	11908.0	−	11908.0i	0.982129	−	0.982129i	−0.0177144	−	0.999843i	$-0.505639\pi$
$43$	11908.0	−	11908.0i	0.982129	−	0.982129i	0.999843	+	0.0177144i	$0.00563896\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2443.88	+	2443.88i	0.161374	+	0.161374i	0.783175	−	0.621801i	$-0.213599\pi$
$47$	2443.88	+	2443.88i	0.161374	+	0.161374i	−0.621801	+	0.783175i	$0.713599\pi$
$48$		0			0
$49$		−	15470.0i		−	0.920451i
$50$		0			0
$51$		−	34393.1i		−	1.85160i
$52$		0			0
$53$	22741.0	+	22741.0i	1.11204	+	1.11204i	0.992875	+	0.119163i	$0.0380210\pi$
$53$	22741.0	+	22741.0i	1.11204	+	1.11204i	0.119163	+	0.992875i	$0.461979\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−8939.98	+	8939.98i	−0.364460	+	0.364460i
$58$		0			0
$59$	−6231.45			−0.233055			−0.116528	−	0.993187i	$-0.537176\pi$
$59$	−6231.45			−0.233055			−0.116528	+	0.993187i	$0.537176\pi$
$60$		0			0
$61$	−42507.9			−1.46267			−0.731333	−	0.682020i	$-0.761102\pi$
$61$	−42507.9			−1.46267			−0.731333	+	0.682020i	$0.761102\pi$
$62$		0			0
$63$	10487.2	−	10487.2i	0.332897	−	0.332897i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	24167.2	+	24167.2i	0.657718	+	0.657718i	0.954840	−	0.297122i	$-0.0960267\pi$
$67$	24167.2	+	24167.2i	0.657718	+	0.657718i	−0.297122	+	0.954840i	$0.596027\pi$
$68$		0			0
$69$			81127.7i			2.05138i
$70$		0			0
$71$		−	21553.1i		−	0.507417i	−0.967281	−	0.253708i	$-0.918350\pi$
$71$		−	21553.1i		−	0.507417i	0.967281	−	0.253708i	$-0.0816503\pi$
$72$		0			0
$73$	−36717.5	−	36717.5i	−0.806430	−	0.806430i	0.177662	−	0.984092i	$-0.443147\pi$
$73$	−36717.5	−	36717.5i	−0.806430	−	0.806430i	−0.984092	+	0.177662i	$0.943147\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	18567.5	−	18567.5i	0.356883	−	0.356883i
$78$		0			0
$79$	−91428.3			−1.64821			−0.824106	−	0.566436i	$-0.808322\pi$
$79$	−91428.3			−1.64821			−0.824106	+	0.566436i	$0.808322\pi$
$80$		0			0
$81$	−6909.70			−0.117016
$82$		0			0
$83$	−74086.8	+	74086.8i	−1.18044	+	1.18044i	−0.200816	+	0.979629i	$0.564359\pi$
$83$	−74086.8	+	74086.8i	−1.18044	+	1.18044i	−0.979629	+	0.200816i	$0.935641\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−104186.	−	104186.i	−1.47575	−	1.47575i
$88$		0			0
$89$			56955.3i			0.762182i	0.924537	+	0.381091i	$0.124452\pi$
$89$			56955.3i			0.762182i	−0.924537	+	0.381091i	$0.875548\pi$
$90$		0			0
$91$			17949.6i			0.227223i
$92$		0			0
$93$	−105370.	−	105370.i	−1.26331	−	1.26331i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	71785.9	−	71785.9i	0.774658	−	0.774658i	−0.204259	−	0.978917i	$-0.565478\pi$
$97$	71785.9	−	71785.9i	0.774658	−	0.774658i	0.978917	+	0.204259i	$0.0654784\pi$
$98$		0			0
$99$	−291285.			−2.98697
$100$		0			0
$101$	45986.1			0.448562			0.224281	−	0.974524i	$-0.427997\pi$
$101$	45986.1			0.448562			0.224281	+	0.974524i	$0.427997\pi$
$102$		0			0
$103$	−15768.2	+	15768.2i	−0.146450	+	0.146450i	−0.776530	−	0.630080i	$-0.783022\pi$
$103$	−15768.2	+	15768.2i	−0.146450	+	0.146450i	0.630080	+	0.776530i	$0.283022\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	91757.0	+	91757.0i	0.774783	+	0.774783i	0.978938	−	0.204156i	$-0.0654449\pi$
$107$	91757.0	+	91757.0i	0.774783	+	0.774783i	−0.204156	+	0.978938i	$0.565445\pi$
$108$		0			0
$109$		−	27083.3i		−	0.218341i	−0.994023	−	0.109170i	$-0.965181\pi$
$109$		−	27083.3i		−	0.218341i	0.994023	−	0.109170i	$-0.0348194\pi$
$110$		0			0
$111$		−	343834.i		−	2.64876i
$112$		0			0
$113$	−110557.	−	110557.i	−0.814495	−	0.814495i	0.170809	−	0.985304i	$-0.445362\pi$
$113$	−110557.	−	110557.i	−0.814495	−	0.814495i	−0.985304	+	0.170809i	$0.945362\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	140796.	−	140796.i	0.950882	−	0.950882i
$118$		0			0
$119$	−49378.9			−0.319650
$120$		0			0
$121$	−354664.			−2.20219
$122$		0			0
$123$	−108175.	+	108175.i	−0.644712	+	0.644712i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−209336.	−	209336.i	−1.15169	−	1.15169i	−0.986214	−	0.165476i	$-0.947084\pi$
$127$	−209336.	−	209336.i	−1.15169	−	1.15169i	−0.165476	−	0.986214i	$-0.552916\pi$
$128$		0			0
$129$			428892.i			2.26920i
$130$		0			0
$131$		−	198728.i		−	1.01177i	−0.862602	−	0.505883i	$-0.831167\pi$
$131$		−	198728.i		−	1.01177i	0.862602	−	0.505883i	$-0.168833\pi$
$132$		0			0
$133$	12835.3	+	12835.3i	0.0629184	+	0.0629184i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−140973.	+	140973.i	−0.641705	+	0.641705i	−0.950974	−	0.309270i	$-0.899915\pi$
$137$	−140973.	+	140973.i	−0.641705	+	0.641705i	0.309270	+	0.950974i	$0.399915\pi$
$138$		0			0
$139$	−161518.			−0.709061			−0.354530	−	0.935044i	$-0.615359\pi$
$139$	−161518.			−0.709061			−0.354530	+	0.935044i	$0.615359\pi$
$140$		0			0
$141$	−88021.3			−0.372855
$142$		0			0
$143$	249277.	−	249277.i	1.01940	−	1.01940i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	278592.	+	278592.i	1.06335	+	1.06335i
$148$		0			0
$149$			163066.i			0.601726i	0.953667	+	0.300863i	$0.0972747\pi$
$149$			163066.i			0.601726i	−0.953667	+	0.300863i	$0.902725\pi$
$150$		0			0
$151$			105255.i			0.375665i	0.982201	+	0.187833i	$0.0601462\pi$
$151$			105255.i			0.375665i	−0.982201	+	0.187833i	$0.939854\pi$
$152$		0			0
$153$	387326.	+	387326.i	1.33767	+	1.33767i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−364901.	+	364901.i	−1.18148	+	1.18148i	−0.202117	+	0.979361i	$0.564782\pi$
$157$	−364901.	+	364901.i	−1.18148	+	1.18148i	−0.979361	+	0.202117i	$0.935218\pi$
$158$		0			0
$159$	−819063.			−2.56936
$160$		0			0
$161$	116477.			0.354139
$162$		0			0
$163$	−148154.	+	148154.i	−0.436761	+	0.436761i	−0.890921	−	0.454159i	$-0.849940\pi$
$163$	−148154.	+	148154.i	−0.436761	+	0.436761i	0.454159	+	0.890921i	$0.349940\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−57451.6	−	57451.6i	−0.159408	−	0.159408i	0.622896	−	0.782305i	$-0.285956\pi$
$167$	−57451.6	−	57451.6i	−0.159408	−	0.159408i	−0.782305	+	0.622896i	$0.785956\pi$
$168$		0			0
$169$		−	130310.i		−	0.350964i
$170$		0			0
$171$		−	201359.i		−	0.526601i
$172$		0			0
$173$	348083.	+	348083.i	0.884234	+	0.884234i	0.993962	−	0.109728i	$-0.0349980\pi$
$173$	348083.	+	348083.i	0.884234	+	0.884234i	−0.109728	+	0.993962i	$0.534998\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	112219.	−	112219.i	0.269237	−	0.269237i
$178$		0			0
$179$	−297598.			−0.694221			−0.347111	−	0.937824i	$-0.612837\pi$
$179$	−297598.			−0.694221			−0.347111	+	0.937824i	$0.612837\pi$
$180$		0			0
$181$	472596.			1.07224			0.536122	−	0.844140i	$-0.319889\pi$
$181$	472596.			1.07224			0.536122	+	0.844140i	$0.319889\pi$
$182$		0			0
$183$	765505.	−	765505.i	1.68974	−	1.68974i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	685754.	+	685754.i	1.43405	+	1.43405i
$188$		0			0
$189$			151431.i			0.308362i
$190$		0			0
$191$			92884.6i			0.184230i	0.995748	+	0.0921150i	$0.0293628\pi$
$191$			92884.6i			0.184230i	−0.995748	+	0.0921150i	$0.970637\pi$
$192$		0			0
$193$	128368.	+	128368.i	0.248063	+	0.248063i	0.820175	−	0.572112i	$-0.193876\pi$
$193$	128368.	+	128368.i	0.248063	+	0.248063i	−0.572112	+	0.820175i	$0.693876\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−424580.	+	424580.i	−0.779460	+	0.779460i	−0.979739	−	0.200279i	$-0.935815\pi$
$197$	−424580.	+	424580.i	−0.779460	+	0.779460i	0.200279	+	0.979739i	$0.435815\pi$
$198$		0			0
$199$	529220.			0.947334			0.473667	−	0.880704i	$-0.342930\pi$
$199$	529220.			0.947334			0.473667	+	0.880704i	$0.342930\pi$
$200$		0			0
$201$	−870432.			−1.51965
$202$		0			0
$203$	−149582.	+	149582.i	−0.254765	+	0.254765i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−913638.	−	913638.i	−1.48200	−	1.48200i
$208$		0			0
$209$		−	356503.i		−	0.564544i
$210$		0			0
$211$		−	800518.i		−	1.23784i	−0.785454	−	0.618920i	$-0.787570\pi$
$211$		−	800518.i		−	1.23784i	0.785454	−	0.618920i	$-0.212430\pi$
$212$		0			0
$213$	388140.	+	388140.i	0.586192	+	0.586192i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−151282.	+	151282.i	−0.218091	+	0.218091i
$218$		0			0
$219$	1.32246e6			1.86325
$220$		0			0
$221$	−662936.			−0.913042
$222$		0			0
$223$	−382704.	+	382704.i	−0.515349	+	0.515349i	−0.916160	−	0.400812i	$-0.868728\pi$
$223$	−382704.	+	382704.i	−0.515349	+	0.515349i	0.400812	+	0.916160i	$0.368728\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	228572.	+	228572.i	0.294414	+	0.294414i	0.838821	−	0.544407i	$-0.183245\pi$
$227$	228572.	+	228572.i	0.294414	+	0.294414i	−0.544407	+	0.838821i	$0.683245\pi$
$228$		0			0
$229$			1.18494e6i			1.49317i	0.665290	+	0.746585i	$0.268308\pi$
$229$			1.18494e6i			1.49317i	−0.665290	+	0.746585i	$0.731692\pi$
$230$		0			0
$231$			668746.i			0.824577i
$232$		0			0
$233$	−821962.	−	821962.i	−0.991886	−	0.991886i	0.00808137	−	0.999967i	$-0.497428\pi$
$233$	−821962.	−	821962.i	−0.991886	−	0.991886i	−0.999967	+	0.00808137i	$0.997428\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	1.64649e6	−	1.64649e6i	1.90409	−	1.90409i
$238$		0			0
$239$	130401.			0.147668			0.0738342	−	0.997271i	$-0.476476\pi$
$239$	130401.			0.147668			0.0738342	+	0.997271i	$0.476476\pi$
$240$		0			0
$241$	1.19162e6			1.32158			0.660791	−	0.750570i	$-0.270221\pi$
$241$	1.19162e6			1.32158			0.660791	+	0.750570i	$0.270221\pi$
$242$		0			0
$243$	−587179.	+	587179.i	−0.637903	+	0.637903i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	172320.	+	172320.i	0.179719	+	0.179719i
$248$		0			0
$249$		−	2.66839e6i		−	2.72741i
$250$		0			0
$251$			1.88579e6i			1.88934i	0.328027	+	0.944668i	$0.393616\pi$
$251$			1.88579e6i			1.88934i	−0.328027	+	0.944668i	$0.606384\pi$
$252$		0			0
$253$	−1.61758e6	−	1.61758e6i	−1.58878	−	1.58878i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−332877.	+	332877.i	−0.314377	+	0.314377i	−0.846602	−	0.532226i	$-0.821356\pi$
$257$	−332877.	+	332877.i	−0.314377	+	0.314377i	0.532226	+	0.846602i	$0.321356\pi$
$258$		0			0
$259$	−493650.			−0.457267
$260$		0			0
$261$	2.34663e6			2.13228
$262$		0			0
$263$	−509422.	+	509422.i	−0.454138	+	0.454138i	−0.896725	−	0.442587i	$-0.854061\pi$
$263$	−509422.	+	509422.i	−0.454138	+	0.454138i	0.442587	+	0.896725i	$0.354061\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−1.02568e6	−	1.02568e6i	−0.880509	−	0.880509i
$268$		0			0
$269$		−	529322.i		−	0.446005i	−0.974818	−	0.223002i	$-0.928414\pi$
$269$		−	529322.i		−	0.446005i	0.974818	−	0.223002i	$-0.0715857\pi$
$270$		0			0
$271$			82763.3i			0.0684565i	0.999414	+	0.0342282i	$0.0108973\pi$
$271$			82763.3i			0.0684565i	−0.999414	+	0.0342282i	$0.989103\pi$
$272$		0			0
$273$	−323247.	−	323247.i	−0.262499	−	0.262499i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−211750.	+	211750.i	−0.165815	+	0.165815i	−0.785137	−	0.619322i	$-0.787407\pi$
$277$	−211750.	+	211750.i	−0.165815	+	0.165815i	0.619322	+	0.785137i	$0.287407\pi$
$278$		0			0
$279$	2.37330e6			1.82533
$280$		0			0
$281$	−570365.			−0.430911			−0.215455	−	0.976514i	$-0.569124\pi$
$281$	−570365.			−0.430911			−0.215455	+	0.976514i	$0.569124\pi$
$282$		0			0
$283$	300723.	−	300723.i	0.223203	−	0.223203i	−0.586643	−	0.809846i	$-0.699551\pi$
$283$	300723.	−	300723.i	0.223203	−	0.223203i	0.809846	+	0.586643i	$0.199551\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	155310.	+	155310.i	0.111300	+	0.111300i
$288$		0			0
$289$		−	403858.i		−	0.284436i
$290$		0			0
$291$			2.58552e6i			1.78984i
$292$		0			0
$293$	−1.22943e6	−	1.22943e6i	−0.836631	−	0.836631i	0.151783	−	0.988414i	$-0.451499\pi$
$293$	−1.22943e6	−	1.22943e6i	−0.836631	−	0.836631i	−0.988414	+	0.151783i	$0.951499\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	2.10301e6	−	2.10301e6i	1.38341	−	1.38341i
$298$		0			0
$299$	1.56376e6			1.01156
$300$		0			0
$301$	615769.			0.391743
$302$		0			0
$303$	−828141.	+	828141.i	−0.518201	+	0.518201i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−315561.	−	315561.i	−0.191090	−	0.191090i	0.605077	−	0.796167i	$-0.293142\pi$
$307$	−315561.	−	315561.i	−0.191090	−	0.191090i	−0.796167	+	0.605077i	$0.793142\pi$
$308$		0			0
$309$		−	567923.i		−	0.338371i
$310$		0			0
$311$		−	914944.i		−	0.536406i	−0.963362	−	0.268203i	$-0.913570\pi$
$311$		−	914944.i		−	0.536406i	0.963362	−	0.268203i	$-0.0864298\pi$
$312$		0			0
$313$	172079.	+	172079.i	0.0992810	+	0.0992810i	0.755003	−	0.655722i	$-0.227636\pi$
$313$	172079.	+	172079.i	0.0992810	+	0.0992810i	−0.655722	+	0.755003i	$0.727636\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.45057e6	+	1.45057e6i	−0.810756	+	0.810756i	−0.984747	−	0.173991i	$-0.944333\pi$
$317$	−1.45057e6	+	1.45057e6i	−0.810756	+	0.810756i	0.173991	+	0.984747i	$0.444333\pi$
$318$		0			0
$319$	4.15467e6			2.28591
$320$		0			0
$321$	−3.30482e6			−1.79013
$322$		0			0
$323$	−474048.	+	474048.i	−0.252823	+	0.252823i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	487730.	+	487730.i	0.252238	+	0.252238i
$328$		0			0
$329$			126374.i			0.0643677i
$330$		0			0
$331$		−	681355.i		−	0.341825i	−0.985286	−	0.170912i	$-0.945329\pi$
$331$		−	681355.i		−	0.341825i	0.985286	−	0.170912i	$-0.0546715\pi$
$332$		0			0
$333$	3.87217e6	+	3.87217e6i	1.91357	+	1.91357i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−1.72585e6	+	1.72585e6i	−0.827804	+	0.827804i	−0.987213	−	0.159409i	$-0.949041\pi$
$337$	−1.72585e6	+	1.72585e6i	−0.827804	+	0.827804i	0.159409	+	0.987213i	$0.449041\pi$
$338$		0			0
$339$	3.98192e6			1.88189
$340$		0			0
$341$	4.20188e6			1.95685
$342$		0			0
$343$	834529.	−	834529.i	0.383006	−	0.383006i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	618528.	+	618528.i	0.275763	+	0.275763i	0.831415	−	0.555652i	$-0.187531\pi$
$347$	618528.	+	618528.i	0.275763	+	0.275763i	−0.555652	+	0.831415i	$0.687531\pi$
$348$		0			0
$349$		−	2.81393e6i		−	1.23666i	−0.785918	−	0.618330i	$-0.787809\pi$
$349$		−	2.81393e6i		−	1.23666i	0.785918	−	0.618330i	$-0.212191\pi$
$350$		0			0
$351$			2.03304e6i			0.880800i
$352$		0			0
$353$	791647.	+	791647.i	0.338139	+	0.338139i	0.855666	−	0.517528i	$-0.173148\pi$
$353$	791647.	+	791647.i	0.338139	+	0.338139i	−0.517528	+	0.855666i	$0.673148\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	889242.	−	889242.i	0.369274	−	0.369274i
$358$		0			0
$359$	−3.62213e6			−1.48329			−0.741647	−	0.670790i	$-0.765955\pi$
$359$	−3.62213e6			−1.48329			−0.741647	+	0.670790i	$0.765955\pi$
$360$		0			0
$361$	−2.22966e6			−0.900471
$362$		0			0
$363$	6.38698e6	−	6.38698e6i	2.54407	−	2.54407i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	2.63566e6	+	2.63566e6i	1.02146	+	1.02146i	0.999765	+	0.0217004i	$0.00690799\pi$
$367$	2.63566e6	+	2.63566e6i	1.02146	+	1.02146i	0.0217004	+	0.999765i	$0.493092\pi$
$368$		0			0
$369$		−	2.43648e6i		−	0.931532i
$370$		0			0
$371$			1.17595e6i			0.443560i
$372$		0			0
$373$	423297.	+	423297.i	0.157534	+	0.157534i	0.781473	−	0.623939i	$-0.214469\pi$
$373$	423297.	+	423297.i	0.157534	+	0.157534i	−0.623939	+	0.781473i	$0.714469\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−2.00821e6	+	2.00821e6i	−0.727706	+	0.727706i
$378$		0			0
$379$	−3.21572e6			−1.14995			−0.574976	−	0.818170i	$-0.694989\pi$
$379$	−3.21572e6			−1.14995			−0.574976	+	0.818170i	$0.694989\pi$
$380$		0			0
$381$	7.53968e6			2.66097
$382$		0			0
$383$	−2.16953e6	+	2.16953e6i	−0.755734	+	0.755734i	−0.975543	−	0.219809i	$-0.929456\pi$
$383$	−2.16953e6	+	2.16953e6i	−0.755734	+	0.755734i	0.219809	+	0.975543i	$0.429456\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−4.83006e6	−	4.83006e6i	−1.63936	−	1.63936i
$388$		0			0
$389$			2.07341e6i			0.694722i	0.937732	+	0.347361i	$0.112922\pi$
$389$			2.07341e6i			0.694722i	−0.937732	+	0.347361i	$0.887078\pi$
$390$		0			0
$391$			4.30184e6i			1.42303i
$392$		0			0
$393$	3.57879e6	+	3.57879e6i	1.16884	+	1.16884i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−3.88030e6	+	3.88030e6i	−1.23563	+	1.23563i	−0.273863	+	0.961769i	$0.588301\pi$
$397$	−3.88030e6	+	3.88030e6i	−1.23563	+	1.23563i	−0.961769	+	0.273863i	$0.911699\pi$
$398$		0			0
$399$	−462290.			−0.145373
$400$		0			0
$401$	−1.98524e6			−0.616528			−0.308264	−	0.951301i	$-0.599748\pi$
$401$	−1.98524e6			−0.616528			−0.308264	+	0.951301i	$0.599748\pi$
$402$		0			0
$403$	−2.03103e6	+	2.03103e6i	−0.622951	+	0.622951i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	6.85561e6	+	6.85561e6i	2.05145	+	2.05145i
$408$		0			0
$409$		−	469288.i		−	0.138717i	−0.997592	−	0.0693587i	$-0.977905\pi$
$409$		−	469288.i		−	0.138717i	0.997592	−	0.0693587i	$-0.0220953\pi$
$410$		0			0
$411$		−	5.07744e6i		−	1.48266i
$412$		0			0
$413$	−161115.	−	161115.i	−0.0464795	−	0.0464795i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	2.90870e6	−	2.90870e6i	0.819141	−	0.819141i
$418$		0			0
$419$	3.83339e6			1.06671			0.533357	−	0.845890i	$-0.320930\pi$
$419$	3.83339e6			1.06671			0.533357	+	0.845890i	$0.320930\pi$
$420$		0			0
$421$	−4.64377e6			−1.27693			−0.638463	−	0.769653i	$-0.720429\pi$
$421$	−4.64377e6			−1.27693			−0.638463	+	0.769653i	$0.720429\pi$
$422$		0			0
$423$	991272.	−	991272.i	0.269365	−	0.269365i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−1.09905e6	−	1.09905e6i	−0.291708	−	0.291708i
$428$		0			0
$429$			8.97824e6i			2.35531i
$430$		0			0
$431$			1.19804e6i			0.310654i	0.987863	+	0.155327i	$0.0496432\pi$
$431$			1.19804e6i			0.310654i	−0.987863	+	0.155327i	$0.950357\pi$
$432$		0			0
$433$	−547634.	−	547634.i	−0.140369	−	0.140369i	0.633431	−	0.773799i	$-0.281646\pi$
$433$	−547634.	−	547634.i	−0.140369	−	0.140369i	−0.773799	+	0.633431i	$0.781646\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.11820e6	−	1.11820e6i	0.280102	−	0.280102i
$438$		0			0
$439$	−538104.			−0.133262			−0.0666308	−	0.997778i	$-0.521225\pi$
$439$	−538104.			−0.133262			−0.0666308	+	0.997778i	$0.521225\pi$
$440$		0			0
$441$	−6.27486e6			−1.53641
$442$		0			0
$443$	1.34605e6	−	1.34605e6i	0.325877	−	0.325877i	−0.525139	−	0.851016i	$-0.675987\pi$
$443$	1.34605e6	−	1.34605e6i	0.325877	−	0.325877i	0.851016	+	0.525139i	$0.175987\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−2.93659e6	−	2.93659e6i	−0.695143	−	0.695143i
$448$		0			0
$449$		−	1.96430e6i		−	0.459824i	−0.973211	−	0.229912i	$-0.926156\pi$
$449$		−	1.96430e6i		−	0.459824i	0.973211	−	0.229912i	$-0.0738438\pi$
$450$		0			0
$451$		−	4.31375e6i		−	0.998652i
$452$		0			0
$453$	−1.89549e6	−	1.89549e6i	−0.433986	−	0.433986i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−2.59374e6	+	2.59374e6i	−0.580946	+	0.580946i	−0.935163	−	0.354217i	$-0.884747\pi$
$457$	−2.59374e6	+	2.59374e6i	−0.580946	+	0.580946i	0.354217	+	0.935163i	$0.384747\pi$
$458$		0			0
$459$	−5.59282e6			−1.23908
$460$		0			0
$461$	4.04207e6			0.885832			0.442916	−	0.896563i	$-0.353944\pi$
$461$	4.04207e6			0.885832			0.442916	+	0.896563i	$0.353944\pi$
$462$		0			0
$463$	−5.00350e6	+	5.00350e6i	−1.08473	+	1.08473i	−0.0886690	+	0.996061i	$0.528261\pi$
$463$	−5.00350e6	+	5.00350e6i	−1.08473	+	1.08473i	−0.996061	+	0.0886690i	$0.971739\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−5.19575e6	−	5.19575e6i	−1.10244	−	1.10244i	−0.994115	−	0.108327i	$-0.965451\pi$
$467$	−5.19575e6	−	5.19575e6i	−1.10244	−	1.10244i	−0.108327	−	0.994115i	$-0.534549\pi$
$468$		0			0
$469$			1.24970e6i			0.262345i
$470$		0			0
$471$		−	1.31427e7i		−	2.72980i
$472$		0			0
$473$	−8.55154e6	−	8.55154e6i	−1.75748	−	1.75748i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	9.22407e6	−	9.22407e6i	1.85621	−	1.85621i
$478$		0			0
$479$	−5.05599e6			−1.00686			−0.503428	−	0.864037i	$-0.667928\pi$
$479$	−5.05599e6			−1.00686			−0.503428	+	0.864037i	$0.667928\pi$
$480$		0			0
$481$	−6.62749e6			−1.30613
$482$		0			0
$483$	−2.09757e6	+	2.09757e6i	−0.409119	+	0.409119i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	2.56621e6	+	2.56621e6i	0.490309	+	0.490309i	0.908403	−	0.418095i	$-0.137302\pi$
$487$	2.56621e6	+	2.56621e6i	0.490309	+	0.490309i	−0.418095	+	0.908403i	$0.637302\pi$
$488$		0			0
$489$		−	5.33607e6i		−	1.00914i
$490$		0			0
$491$			3.35413e6i			0.627880i	0.949443	+	0.313940i	$0.101649\pi$
$491$			3.35413e6i			0.627880i	−0.949443	+	0.313940i	$0.898351\pi$
$492$		0			0
$493$	−5.52452e6	−	5.52452e6i	−1.02371	−	1.02371i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	557261.	−	557261.i	0.101197	−	0.101197i
$498$		0			0
$499$	−7.14131e6			−1.28389			−0.641943	−	0.766753i	$-0.721871\pi$
$499$	−7.14131e6			−1.28389			−0.641943	+	0.766753i	$0.721871\pi$
$500$		0			0
$501$	2.06924e6			0.368312
$502$		0			0
$503$	1.59517e6	−	1.59517e6i	0.281117	−	0.281117i	−0.552437	−	0.833555i	$-0.686302\pi$
$503$	1.59517e6	−	1.59517e6i	0.281117	−	0.281117i	0.833555	+	0.552437i	$0.186302\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	2.34670e6	+	2.34670e6i	0.405450	+	0.405450i
$508$		0			0
$509$			1.07312e7i			1.83593i	0.396664	+	0.917964i	$0.370168\pi$
$509$			1.07312e7i			1.83593i	−0.396664	+	0.917964i	$0.629832\pi$
$510$		0			0
$511$		−	1.89868e6i		−	0.321662i
$512$		0			0
$513$	1.45377e6	+	1.45377e6i	0.243895	+	0.243895i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	1.75503e6	−	1.75503e6i	0.288774	−	0.288774i
$518$		0			0
$519$	−1.25369e7			−2.04302
$520$		0			0
$521$	6.35667e6			1.02597			0.512986	−	0.858397i	$-0.328539\pi$
$521$	6.35667e6			1.02597			0.512986	+	0.858397i	$0.328539\pi$
$522$		0			0
$523$	−613469.	+	613469.i	−0.0980705	+	0.0980705i	−0.754440	−	0.656369i	$-0.772091\pi$
$523$	−613469.	+	613469.i	−0.0980705	+	0.0980705i	0.656369	+	0.754440i	$0.272091\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−5.58730e6	−	5.58730e6i	−0.876346	−	0.876346i
$528$		0			0
$529$		−	3.71099e6i		−	0.576568i
$530$		0			0
$531$			2.52756e6i			0.389015i
$532$		0			0
$533$	2.08511e6	+	2.08511e6i	0.317914	+	0.317914i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	5.35931e6	−	5.35931e6i	0.801998	−	0.801998i
$538$		0			0
$539$	−1.11095e7			−1.64711
$540$		0			0
$541$	6.42729e6			0.944136			0.472068	−	0.881562i	$-0.343508\pi$
$541$	6.42729e6			0.944136			0.472068	+	0.881562i	$0.343508\pi$
$542$		0			0
$543$	−8.51076e6	+	8.51076e6i	−1.23871	+	1.23871i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−803376.	−	803376.i	−0.114802	−	0.114802i	0.647372	−	0.762174i	$-0.275868\pi$
$547$	−803376.	−	803376.i	−0.114802	−	0.114802i	−0.762174	+	0.647372i	$0.775868\pi$
$548$		0			0
$549$			1.72418e7i			2.44147i
$550$		0			0
$551$			2.87204e6i			0.403005i
$552$		0			0
$553$	−2.36390e6	−	2.36390e6i	−0.328712	−	0.328712i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−1.01353e7	+	1.01353e7i	−1.38421	+	1.38421i	−0.547210	+	0.836995i	$0.684310\pi$
$557$	−1.01353e7	+	1.01353e7i	−1.38421	+	1.38421i	−0.836995	+	0.547210i	$0.815690\pi$
$558$		0			0
$559$	8.26699e6			1.11897
$560$		0			0
$561$	−2.46988e7			−3.31337
$562$		0			0
$563$	−4.56483e6	+	4.56483e6i	−0.606951	+	0.606951i	−0.942148	−	0.335197i	$-0.891197\pi$
$563$	−4.56483e6	+	4.56483e6i	−0.606951	+	0.606951i	0.335197	+	0.942148i	$0.391197\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−178652.	−	178652.i	−0.0233372	−	0.0233372i
$568$		0			0
$569$			4.99167e6i			0.646346i	0.946340	+	0.323173i	$0.104750\pi$
$569$			4.99167e6i			0.646346i	−0.946340	+	0.323173i	$0.895250\pi$
$570$		0			0
$571$		−	6.47116e6i		−	0.830600i	−0.909684	−	0.415300i	$-0.863677\pi$
$571$		−	6.47116e6i		−	0.830600i	0.909684	−	0.415300i	$-0.136323\pi$
$572$		0			0
$573$	−1.67272e6	−	1.67272e6i	−0.212831	−	0.212831i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	2.83549e6	−	2.83549e6i	0.354559	−	0.354559i	−0.507244	−	0.861803i	$-0.669336\pi$
$577$	2.83549e6	−	2.83549e6i	0.354559	−	0.354559i	0.861803	+	0.507244i	$0.169336\pi$
$578$		0			0
$579$	−4.62342e6			−0.573149
$580$		0			0
$581$	−3.83106e6			−0.470846
$582$		0			0
$583$	1.63310e7	−	1.63310e7i	1.98995	−	1.98995i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	6.57105e6	+	6.57105e6i	0.787117	+	0.787117i	0.981021	−	0.193904i	$-0.0621149\pi$
$587$	6.57105e6	+	6.57105e6i	0.787117	+	0.787117i	−0.193904	+	0.981021i	$0.562115\pi$
$588$		0			0
$589$			2.90467e6i			0.344992i
$590$		0			0
$591$		−	1.52921e7i		−	1.80094i
$592$		0			0
$593$	4.04776e6	+	4.04776e6i	0.472691	+	0.472691i	0.902785	−	0.430093i	$-0.141519\pi$
$593$	4.04776e6	+	4.04776e6i	0.472691	+	0.472691i	−0.430093	+	0.902785i	$0.641519\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−9.53047e6	+	9.53047e6i	−1.09441	+	1.09441i
$598$		0			0
$599$	−37502.4			−0.00427063			−0.00213531	−	0.999998i	$-0.500680\pi$
$599$	−37502.4			−0.00427063			−0.00213531	+	0.999998i	$0.500680\pi$
$600$		0			0
$601$	−1.04123e6			−0.117587			−0.0587936	−	0.998270i	$-0.518725\pi$
$601$	−1.04123e6			−0.117587			−0.0587936	+	0.998270i	$0.518725\pi$
$602$		0			0
$603$	9.80257e6	−	9.80257e6i	1.09786	−	1.09786i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−175134.	−	175134.i	−0.0192929	−	0.0192929i	0.697395	−	0.716687i	$-0.254343\pi$
$607$	−175134.	−	175134.i	−0.0192929	−	0.0192929i	−0.716687	+	0.697395i	$0.754343\pi$
$608$		0			0
$609$		−	5.38750e6i		−	0.588633i
$610$		0			0
$611$			1.69663e6i			0.183859i
$612$		0			0
$613$	1.18265e7	+	1.18265e7i	1.27118	+	1.27118i	0.945471	+	0.325705i	$0.105602\pi$
$613$	1.18265e7	+	1.18265e7i	1.27118	+	1.27118i	0.325705	+	0.945471i	$0.394398\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−3.51651e6	+	3.51651e6i	−0.371876	+	0.371876i	−0.868160	−	0.496284i	$-0.834697\pi$
$617$	−3.51651e6	+	3.51651e6i	−0.371876	+	0.371876i	0.496284	+	0.868160i	$0.334697\pi$
$618$		0			0
$619$	7.86187e6			0.824707			0.412353	−	0.911024i	$-0.364707\pi$
$619$	7.86187e6			0.824707			0.412353	+	0.911024i	$0.364707\pi$
$620$		0			0
$621$	1.31925e7			1.37277
$622$		0			0
$623$	−1.47259e6	+	1.47259e6i	−0.152006	+	0.152006i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	6.42010e6	+	6.42010e6i	0.652188	+	0.652188i
$628$		0			0
$629$		−	1.82320e7i		−	1.83742i
$630$		0			0
$631$		−	8.54772e6i		−	0.854627i	−0.904103	−	0.427314i	$-0.859460\pi$
$631$		−	8.54772e6i		−	0.854627i	0.904103	−	0.427314i	$-0.140540\pi$
$632$		0			0
$633$	1.44161e7	+	1.44161e7i	1.43001	+	1.43001i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	5.36993e6	−	5.36993e6i	0.524349	−	0.524349i
$638$		0			0
$639$	−8.74226e6			−0.846977
$640$		0			0
$641$	−7.78193e6			−0.748070			−0.374035	−	0.927415i	$-0.622026\pi$
$641$	−7.78193e6			−0.748070			−0.374035	+	0.927415i	$0.622026\pi$
$642$		0			0
$643$	−9.36459e6	+	9.36459e6i	−0.893226	+	0.893226i	−0.994825	−	0.101599i	$-0.967604\pi$
$643$	−9.36459e6	+	9.36459e6i	−0.893226	+	0.893226i	0.101599	+	0.994825i	$0.467604\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	666707.	+	666707.i	0.0626144	+	0.0626144i	0.737721	−	0.675106i	$-0.235902\pi$
$647$	666707.	+	666707.i	0.0626144	+	0.0626144i	−0.675106	+	0.737721i	$0.735902\pi$
$648$		0			0
$649$			4.47501e6i			0.417044i
$650$		0			0
$651$		−	5.44873e6i		−	0.503898i
$652$		0			0
$653$	4.83197e6	+	4.83197e6i	0.443447	+	0.443447i	0.893169	−	0.449722i	$-0.148477\pi$
$653$	4.83197e6	+	4.83197e6i	0.443447	+	0.443447i	−0.449722	+	0.893169i	$0.648477\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−1.48932e7	+	1.48932e7i	−1.34609	+	1.34609i
$658$		0			0
$659$	−7.66026e6			−0.687116			−0.343558	−	0.939131i	$-0.611632\pi$
$659$	−7.66026e6			−0.687116			−0.343558	+	0.939131i	$0.611632\pi$
$660$		0			0
$661$	1.51315e7			1.34704			0.673519	−	0.739170i	$-0.264782\pi$
$661$	1.51315e7			1.34704			0.673519	+	0.739170i	$0.264782\pi$
$662$		0			0
$663$	1.19385e7	−	1.19385e7i	1.05479	−	1.05479i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	1.30314e7	+	1.30314e7i	1.13417	+	1.13417i
$668$		0			0
$669$		−	1.37839e7i		−	1.19071i
$670$		0			0
$671$			3.05263e7i			2.61739i
$672$		0			0
$673$	−3.10276e6	−	3.10276e6i	−0.264065	−	0.264065i	0.562638	−	0.826703i	$-0.309786\pi$
$673$	−3.10276e6	−	3.10276e6i	−0.264065	−	0.264065i	−0.826703	+	0.562638i	$0.809786\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	8.36156e6	−	8.36156e6i	0.701157	−	0.701157i	−0.263502	−	0.964659i	$-0.584877\pi$
$677$	8.36156e6	−	8.36156e6i	0.701157	−	0.701157i	0.964659	+	0.263502i	$0.0848774\pi$
$678$		0			0
$679$	3.71208e6			0.308989
$680$		0			0
$681$	−8.23248e6			−0.680242
$682$		0			0
$683$	8.68455e6	−	8.68455e6i	0.712354	−	0.712354i	−0.254673	−	0.967027i	$-0.581968\pi$
$683$	8.68455e6	−	8.68455e6i	0.712354	−	0.712354i	0.967027	+	0.254673i	$0.0819680\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−2.13391e7	−	2.13391e7i	−1.72498	−	1.72498i
$688$		0			0
$689$			1.57876e7i			1.26698i
$690$		0			0
$691$		−	9.43251e6i		−	0.751505i	−0.926720	−	0.375753i	$-0.877384\pi$
$691$		−	9.43251e6i		−	0.751505i	0.926720	−	0.375753i	$-0.122616\pi$
$692$		0			0
$693$	−7.53124e6	−	7.53124e6i	−0.595708	−	0.595708i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−5.73606e6	+	5.73606e6i	−0.447231	+	0.447231i
$698$		0			0
$699$	2.96046e7			2.29175
$700$		0			0
$701$	1.61178e7			1.23882			0.619412	−	0.785066i	$-0.287371\pi$
$701$	1.61178e7			1.23882			0.619412	+	0.785066i	$0.287371\pi$
$702$		0			0
$703$	−4.73914e6	+	4.73914e6i	−0.361669	+	0.361669i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.18898e6	+	1.18898e6i	0.0894594	+	0.0894594i
$708$		0			0
$709$			1.60369e7i			1.19813i	0.800700	+	0.599065i	$0.204461\pi$
$709$			1.60369e7i			1.19813i	−0.800700	+	0.599065i	$0.795539\pi$
$710$		0			0
$711$			3.70846e7i			2.75119i
$712$		0			0
$713$	1.31795e7	+	1.31795e7i	0.970903	+	0.970903i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−2.34834e6	+	2.34834e6i	−0.170594	+	0.170594i
$718$		0			0
$719$	−1.35776e7			−0.979492			−0.489746	−	0.871865i	$-0.662910\pi$
$719$	−1.35776e7			−0.979492			−0.489746	+	0.871865i	$0.662910\pi$
$720$		0			0
$721$	−815379.			−0.0584146
$722$		0			0
$723$	−2.14593e7	+	2.14593e7i	−1.52676	+	1.52676i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−1.48867e7	−	1.48867e7i	−1.04463	−	1.04463i	−0.998957	−	0.0456711i	$-0.985457\pi$
$727$	−1.48867e7	−	1.48867e7i	−1.04463	−	1.04463i	−0.0456711	−	0.998957i	$-0.514543\pi$
$728$		0			0
$729$		−	2.28275e7i		−	1.59089i
$730$		0			0
$731$			2.27422e7i			1.57413i
$732$		0			0
$733$	2.95902e6	+	2.95902e6i	0.203418	+	0.203418i	0.801463	−	0.598045i	$-0.204056\pi$
$733$	2.95902e6	+	2.95902e6i	0.203418	+	0.203418i	−0.598045	+	0.801463i	$0.704056\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1.73553e7	−	1.73553e7i	1.17696	−	1.17696i
$738$		0			0
$739$	1.59695e7			1.07567			0.537837	−	0.843049i	$-0.319241\pi$
$739$	1.59695e7			1.07567			0.537837	+	0.843049i	$0.319241\pi$
$740$		0			0
$741$	−6.20647e6			−0.415240
$742$		0			0
$743$	1.52849e7	−	1.52849e7i	1.01576	−	1.01576i	0.0158837	−	0.999874i	$-0.494944\pi$
$743$	1.52849e7	−	1.52849e7i	1.01576	−	1.01576i	0.999874	−	0.0158837i	$-0.00505614\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	3.00507e7	+	3.00507e7i	1.97039	+	1.97039i
$748$		0			0
$749$			4.74479e6i			0.309039i
$750$		0			0
$751$		−	2.39979e7i		−	1.55265i	−0.630334	−	0.776324i	$-0.717082\pi$
$751$		−	2.39979e7i		−	1.55265i	0.630334	−	0.776324i	$-0.282918\pi$
$752$		0			0
$753$	−3.39603e7	−	3.39603e7i	−2.18265	−	2.18265i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	743830.	−	743830.i	0.0471774	−	0.0471774i	−0.683125	−	0.730302i	$-0.739379\pi$
$757$	743830.	−	743830.i	0.0471774	−	0.0471774i	0.730302	+	0.683125i	$0.239379\pi$
$758$		0			0
$759$	5.82605e7			3.67088
$760$		0			0
$761$	8.32696e6			0.521224			0.260612	−	0.965444i	$-0.416076\pi$
$761$	8.32696e6			0.521224			0.260612	+	0.965444i	$0.416076\pi$
$762$		0			0
$763$	700244.	−	700244.i	0.0435450	−	0.0435450i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2.16305e6	−	2.16305e6i	−0.132763	−	0.132763i
$768$		0			0
$769$		−	2.09516e7i		−	1.27762i	−0.769366	−	0.638808i	$-0.779428\pi$
$769$		−	2.09516e7i		−	1.27762i	0.769366	−	0.638808i	$-0.220572\pi$
$770$		0			0
$771$		−	1.19892e7i		−	0.726366i
$772$		0			0
$773$	−2.05377e7	−	2.05377e7i	−1.23624	−	1.23624i	−0.961526	−	0.274715i	$-0.911417\pi$
$773$	−2.05377e7	−	2.05377e7i	−1.23624	−	1.23624i	−0.274715	−	0.961526i	$-0.588583\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	8.88991e6	−	8.88991e6i	0.528257	−	0.528257i
$778$		0			0
$779$	2.98201e6			0.176062
$780$		0			0
$781$	−1.54780e7			−0.908004
$782$		0			0
$783$	−1.69421e7	+	1.69421e7i	−0.987561	+	0.987561i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−2.44430e7	−	2.44430e7i	−1.40675	−	1.40675i	−0.775919	−	0.630833i	$-0.782713\pi$
$787$	−2.44430e7	−	2.44430e7i	−1.40675	−	1.40675i	−0.630833	−	0.775919i	$-0.717287\pi$
$788$		0			0
$789$		−	1.83479e7i		−	1.04928i
$790$		0			0
$791$		−	5.71693e6i		−	0.324879i
$792$		0			0
$793$	−1.47553e7	−	1.47553e7i	−0.833230	−	0.833230i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1.21183e7	−	1.21183e7i	0.675764	−	0.675764i	−0.283275	−	0.959039i	$-0.591421\pi$
$797$	1.21183e7	−	1.21183e7i	0.675764	−	0.675764i	0.959039	+	0.283275i	$0.0914209\pi$
$798$		0			0
$799$	−4.66738e6			−0.258646
$800$		0			0
$801$	2.31019e7			1.27223
$802$		0			0
$803$	−2.63681e7	+	2.63681e7i	−1.44308	+	1.44308i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	9.53231e6	+	9.53231e6i	0.515246	+	0.515246i
$808$		0			0
$809$		−	3.29221e7i		−	1.76854i	−0.466973	−	0.884272i	$-0.654655\pi$
$809$		−	3.29221e7i		−	1.76854i	0.466973	−	0.884272i	$-0.345345\pi$
$810$		0			0
$811$			2.09508e7i			1.11853i	0.828988	+	0.559266i	$0.188917\pi$
$811$			2.09508e7i			1.11853i	−0.828988	+	0.559266i	$0.811083\pi$
$812$		0			0
$813$	−1.49045e6	−	1.49045e6i	−0.0790842	−	0.0790842i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	5.91150e6	−	5.91150e6i	0.309844	−	0.309844i
$818$		0			0
$819$	7.28063e6			0.379280
$820$		0			0
$821$	2.58643e7			1.33919			0.669595	−	0.742727i	$-0.266468\pi$
$821$	2.58643e7			1.33919			0.669595	+	0.742727i	$0.266468\pi$
$822$		0			0
$823$	−54332.0	+	54332.0i	−0.00279612	+	0.00279612i	−0.708503	−	0.705707i	$-0.750629\pi$
$823$	−54332.0	+	54332.0i	−0.00279612	+	0.00279612i	0.705707	+	0.708503i	$0.250629\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.91047e7	−	1.91047e7i	−0.971353	−	0.971353i	0.0282475	−	0.999601i	$-0.491007\pi$
$827$	−1.91047e7	−	1.91047e7i	−0.971353	−	0.971353i	−0.999601	+	0.0282475i	$0.991007\pi$
$828$		0			0
$829$		−	2.99530e7i		−	1.51375i	−0.653560	−	0.756874i	$-0.726725\pi$
$829$		−	2.99530e7i		−	1.51375i	0.653560	−	0.756874i	$-0.273275\pi$
$830$		0			0
$831$		−	7.62660e6i		−	0.383114i
$832$		0			0
$833$	1.47725e7	+	1.47725e7i	0.737635	+	0.737635i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−1.71347e7	+	1.71347e7i	−0.845400	+	0.845400i
$838$		0			0
$839$	4.91308e6			0.240962			0.120481	−	0.992716i	$-0.461556\pi$
$839$	4.91308e6			0.240962			0.120481	+	0.992716i	$0.461556\pi$
$840$		0			0
$841$	−1.29594e7			−0.631821
$842$		0			0
$843$	1.02714e7	−	1.02714e7i	0.497808	−	0.497808i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−9.16991e6	−	9.16991e6i	−0.439194	−	0.439194i
$848$		0			0
$849$			1.08311e7i			0.515710i
$850$		0			0
$851$			4.30063e7i			2.03567i
$852$		0			0
$853$	5.25356e6	+	5.25356e6i	0.247218	+	0.247218i	0.819828	−	0.572610i	$-0.194069\pi$
$853$	5.25356e6	+	5.25356e6i	0.247218	+	0.247218i	−0.572610	+	0.819828i	$0.694069\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−642435.	+	642435.i	−0.0298797	+	0.0298797i	−0.721889	−	0.692009i	$-0.756726\pi$
$857$	−642435.	+	642435.i	−0.0298797	+	0.0298797i	0.692009	+	0.721889i	$0.256726\pi$
$858$		0			0
$859$	1.75592e7			0.811937			0.405969	−	0.913887i	$-0.366934\pi$
$859$	1.75592e7			0.811937			0.405969	+	0.913887i	$0.366934\pi$
$860$		0			0
$861$	−5.59380e6			−0.257157
$862$		0			0
$863$	7.00501e6	−	7.00501e6i	0.320171	−	0.320171i	−0.528662	−	0.848833i	$-0.677306\pi$
$863$	7.00501e6	−	7.00501e6i	0.320171	−	0.320171i	0.848833	+	0.528662i	$0.177306\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	7.27289e6	+	7.27289e6i	0.328594	+	0.328594i
$868$		0			0
$869$			6.56577e7i			2.94942i
$870$		0			0
$871$			1.67778e7i			0.749357i
$872$		0			0
$873$	−2.91174e7	−	2.91174e7i	−1.29306	−	1.29306i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	2.38052e7	−	2.38052e7i	1.04514	−	1.04514i	0.0462053	−	0.998932i	$-0.485287\pi$
$877$	2.38052e7	−	2.38052e7i	1.04514	−	1.04514i	0.998932	−	0.0462053i	$-0.0147128\pi$
$878$		0			0
$879$	4.42804e7			1.93303
$880$		0			0
$881$	4.51811e7			1.96118			0.980589	−	0.196076i	$-0.0628200\pi$
$881$	4.51811e7			1.96118			0.980589	+	0.196076i	$0.0628200\pi$
$882$		0			0
$883$	8.94089e6	−	8.94089e6i	0.385904	−	0.385904i	−0.487320	−	0.873224i	$-0.662025\pi$
$883$	8.94089e6	−	8.94089e6i	0.385904	−	0.385904i	0.873224	+	0.487320i	$0.162025\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1.42264e7	+	1.42264e7i	0.607136	+	0.607136i	0.942197	−	0.335060i	$-0.108757\pi$
$887$	1.42264e7	+	1.42264e7i	0.607136	+	0.607136i	−0.335060	+	0.942197i	$0.608757\pi$
$888$		0			0
$889$		−	1.08249e7i		−	0.459376i
$890$		0			0
$891$			4.96208e6i			0.209397i
$892$		0			0
$893$	1.21322e6	+	1.21322e6i	0.0509107	+	0.0509107i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−2.81609e7	+	2.81609e7i	−1.16860	+	1.16860i
$898$		0			0
$899$	−3.38509e7			−1.39692
$900$		0			0
$901$	−4.34313e7			−1.78234
$902$		0			0
$903$	−1.10891e7	+	1.10891e7i	−0.452560	+	0.452560i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	1.51023e7	+	1.51023e7i	0.609570	+	0.609570i	0.942834	−	0.333264i	$-0.108150\pi$
$907$	1.51023e7	+	1.51023e7i	0.609570	+	0.609570i	−0.333264	+	0.942834i	$0.608150\pi$
$908$		0			0
$909$		−	1.86526e7i		−	0.748738i
$910$		0			0
$911$			3.55577e7i			1.41951i	0.704450	+	0.709754i	$0.251194\pi$
$911$			3.55577e7i			1.41951i	−0.704450	+	0.709754i	$0.748806\pi$
$912$		0			0
$913$	5.32042e7	+	5.32042e7i	2.11236	+	2.11236i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	5.13815e6	−	5.13815e6i	0.201782	−	0.201782i
$918$		0			0
$919$	−1.09806e7			−0.428883			−0.214441	−	0.976737i	$-0.568793\pi$
$919$	−1.09806e7			−0.428883			−0.214441	+	0.976737i	$0.568793\pi$
$920$		0			0
$921$	1.13656e7			0.441513
$922$		0			0
$923$	7.48150e6	−	7.48150e6i	0.289057	−	0.289057i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	6.39579e6	+	6.39579e6i	0.244453	+	0.244453i
$928$		0			0
$929$			4.09868e7i			1.55813i	0.626940	+	0.779067i	$0.284307\pi$
$929$			4.09868e7i			1.55813i	−0.626940	+	0.779067i	$0.715693\pi$
$930$		0			0
$931$		−	7.67979e6i		−	0.290386i
$932$		0			0
$933$	1.64768e7	+	1.64768e7i	0.619682	+	0.619682i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	2.75880e7	−	2.75880e7i	1.02653	−	1.02653i	0.0268922	−	0.999638i	$-0.491439\pi$
$937$	2.75880e7	−	2.75880e7i	1.02653	−	1.02653i	0.999638	−	0.0268922i	$-0.00856108\pi$
$938$		0			0
$939$	−6.19777e6			−0.229388
$940$		0			0
$941$	−4.64453e6			−0.170989			−0.0854945	−	0.996339i	$-0.527247\pi$
$941$	−4.64453e6			−0.170989			−0.0854945	+	0.996339i	$0.527247\pi$
$942$		0			0
$943$	1.35304e7	−	1.35304e7i	0.495487	−	0.495487i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1.45390e7	−	1.45390e7i	−0.526817	−	0.526817i	0.392805	−	0.919622i	$-0.371505\pi$
$947$	−1.45390e7	−	1.45390e7i	−0.526817	−	0.526817i	−0.919622	+	0.392805i	$0.871505\pi$
$948$		0			0
$949$		−	2.54907e7i		−	0.918789i
$950$		0			0
$951$		−	5.22452e7i		−	1.87325i
$952$		0			0
$953$	2.15849e7	+	2.15849e7i	0.769872	+	0.769872i	0.978084	−	0.208212i	$-0.0667643\pi$
$953$	2.15849e7	+	2.15849e7i	0.769872	+	0.769872i	−0.208212	+	0.978084i	$0.566764\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−7.48194e7	+	7.48194e7i	−2.64079	+	2.64079i
$958$		0			0
$959$	−7.28978e6			−0.255958
$960$		0			0
$961$	−5.60640e6			−0.195828
$962$		0			0
$963$	3.72180e7	−	3.72180e7i	1.29326	−	1.29326i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	2.07269e7	+	2.07269e7i	0.712802	+	0.712802i	0.967121	−	0.254319i	$-0.0818512\pi$
$967$	2.07269e7	+	2.07269e7i	0.712802	+	0.712802i	−0.254319	+	0.967121i	$0.581851\pi$
$968$		0			0
$969$		−	1.70738e7i		−	0.584145i
$970$		0			0
$971$		−	1.77651e7i		−	0.604671i	−0.953202	−	0.302335i	$-0.902234\pi$
$971$		−	1.77651e7i		−	0.604671i	0.953202	−	0.302335i	$-0.0977663\pi$
$972$		0			0
$973$	−4.17608e6	−	4.17608e6i	−0.141412	−	0.141412i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−2.91253e6	+	2.91253e6i	−0.0976189	+	0.0976189i	−0.754230	−	0.656611i	$-0.771989\pi$
$977$	−2.91253e6	+	2.91253e6i	−0.0976189	+	0.0976189i	0.656611	+	0.754230i	$0.271989\pi$
$978$		0			0
$979$	4.09015e7			1.36390
$980$		0			0
$981$	−1.09854e7			−0.364453
$982$		0			0
$983$	2.33654e7	−	2.33654e7i	0.771240	−	0.771240i	−0.207084	−	0.978323i	$-0.566397\pi$
$983$	2.33654e7	−	2.33654e7i	0.771240	−	0.771240i	0.978323	+	0.207084i	$0.0663972\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−2.27581e6	−	2.27581e6i	−0.0743606	−	0.0743606i
$988$		0			0
$989$		−	5.36452e7i		−	1.74397i
$990$		0			0
$991$		−	4.77844e7i		−	1.54562i	−0.634638	−	0.772809i	$-0.718851\pi$
$991$		−	4.77844e7i		−	1.54562i	0.634638	−	0.772809i	$-0.281149\pi$
$992$		0			0
$993$	1.22702e7	+	1.22702e7i	0.394892	+	0.394892i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−2.10004e6	+	2.10004e6i	−0.0669098	+	0.0669098i	−0.739770	−	0.672860i	$-0.765066\pi$
$997$	−2.10004e6	+	2.10004e6i	−0.0669098	+	0.0669098i	0.672860	+	0.739770i	$0.265066\pi$
$998$		0			0
$999$	−5.59124e7			−1.77253

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.6.n.e.143.1	✓	16
4.3	odd	2	inner	400.6.n.e.143.8	yes	16
5.2	odd	4	inner	400.6.n.e.207.7	yes	16
5.3	odd	4	inner	400.6.n.e.207.1	yes	16
5.4	even	2	inner	400.6.n.e.143.7	yes	16
20.3	even	4	inner	400.6.n.e.207.8	yes	16
20.7	even	4	inner	400.6.n.e.207.2	yes	16
20.19	odd	2	inner	400.6.n.e.143.2	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.6.n.e.143.1	✓	16	1.1	even	1	trivial
400.6.n.e.143.2	yes	16	20.19	odd	2	inner
400.6.n.e.143.7	yes	16	5.4	even	2	inner
400.6.n.e.143.8	yes	16	4.3	odd	2	inner
400.6.n.e.207.1	yes	16	5.3	odd	4	inner
400.6.n.e.207.2	yes	16	20.7	even	4	inner
400.6.n.e.207.7	yes	16	5.2	odd	4	inner
400.6.n.e.207.8	yes	16	20.3	even	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	400.n (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-18.0085 + 18.0085i) q^{3} +(25.8552 + 25.8552i) q^{7} -405.614i q^{9} +O(q^{10})\)	\(q+(-18.0085 + 18.0085i) q^{3} +(25.8552 + 25.8552i) q^{7} -405.614i q^{9} -718.133i q^{11} +(347.119 + 347.119i) q^{13} +(-954.912 + 954.912i) q^{17} +496.431 q^{19} -931.229 q^{21} +(2252.48 - 2252.48i) q^{23} +(2928.44 + 2928.44i) q^{27} +5785.37i q^{29} +5851.12i q^{31} +(12932.5 + 12932.5i) q^{33} +(-9546.43 + 9546.43i) q^{37} -12502.2 q^{39} +6006.90 q^{41} +(11908.0 - 11908.0i) q^{43} +(2443.88 + 2443.88i) q^{47} -15470.0i q^{49} -34393.1i q^{51} +(22741.0 + 22741.0i) q^{53} +(-8939.98 + 8939.98i) q^{57} -6231.45 q^{59} -42507.9 q^{61} +(10487.2 - 10487.2i) q^{63} +(24167.2 + 24167.2i) q^{67} +81127.7i q^{69} -21553.1i q^{71} +(-36717.5 - 36717.5i) q^{73} +(18567.5 - 18567.5i) q^{77} -91428.3 q^{79} -6909.70 q^{81} +(-74086.8 + 74086.8i) q^{83} +(-104186. - 104186. i) q^{87} +56955.3i q^{89} +17949.6i q^{91} +(-105370. - 105370. i) q^{93} +(71785.9 - 71785.9i) q^{97} -291285. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 4576 q^{21} - 12288 q^{41} - 277504 q^{61} - 508016 q^{81}+O(q^{100}) \) 16 * q - 4576 * q^21 - 12288 * q^41 - 277504 * q^61 - 508016 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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