LMFDB - Embedded newform 1296.5.e.b.161.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1296,5,Mod(161,1296)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1296, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("1296.161");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	1296.e (of order $2$, degree $1$, 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:	$133.967472157$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 1 $ x^4 + 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{3} $
Twist minimal:	no (minimal twist has level 162)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.4
Root			$1.93185i$ of defining polynomial
Character	$\chi$	$=$	1296.161
Dual form			1296.5.e.b.161.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+45.9483i q^{5} +80.5500 q^{7} +O(q^{10})$	$q+45.9483i q^{5} +80.5500 q^{7} +113.110i q^{11} +70.2539 q^{13} +256.030i q^{17} -192.435 q^{19} +543.928i q^{23} -1486.25 q^{25} -822.977i q^{29} -615.408 q^{31} +3701.14i q^{35} -2355.20 q^{37} +502.171i q^{41} +164.719 q^{43} +1041.23i q^{47} +4087.30 q^{49} -182.004i q^{53} -5197.21 q^{55} -1811.76i q^{59} -2859.85 q^{61} +3228.05i q^{65} +8895.85 q^{67} +3556.66i q^{71} -8008.40 q^{73} +9111.00i q^{77} +4026.40 q^{79} +1846.98i q^{83} -11764.1 q^{85} +4204.70i q^{89} +5658.95 q^{91} -8842.05i q^{95} +13186.8 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 52 q^{7}+O(q^{10}) $ 4 * q + 52 * q^7	$ 4 q + 52 q^{7} + 572 q^{13} + 124 q^{19} - 1892 q^{25} - 3584 q^{31} - 9400 q^{37} - 3020 q^{43} + 9324 q^{49} - 11124 q^{55} - 4144 q^{61} + 12076 q^{67} - 1792 q^{73} + 15004 q^{79} - 24048 q^{85} - 12220 q^{91} + 46304 q^{97}+O(q^{100}) $ 4 * q + 52 * q^7 + 572 * q^13 + 124 * q^19 - 1892 * q^25 - 3584 * q^31 - 9400 * q^37 - 3020 * q^43 + 9324 * q^49 - 11124 * q^55 - 4144 * q^61 + 12076 * q^67 - 1792 * q^73 + 15004 * q^79 - 24048 * q^85 - 12220 * q^91 + 46304 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1217$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	45.9483i	1.83793i	0.394335	+	0.918967i	$0.370975\pi$
$5$	45.9483i	1.83793i	−0.394335	+	0.918967i	$0.629025\pi$
$6$	0	0
$7$	80.5500	1.64388	0.821939	−	0.569576i	$-0.192893\pi$
$7$	80.5500	1.64388	0.821939	+	0.569576i	$0.192893\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	113.110i	0.934792i	0.884048	+	0.467396i	$0.154808\pi$
$11$	113.110i	0.934792i	−0.884048	+	0.467396i	$0.845192\pi$
$12$	0	0
$13$	70.2539	0.415703	0.207852	−	0.978160i	$-0.433353\pi$
$13$	70.2539	0.415703	0.207852	+	0.978160i	$0.433353\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	256.030i	0.885916i	0.896542	+	0.442958i	$0.146071\pi$
$17$	256.030i	0.885916i	−0.896542	+	0.442958i	$0.853929\pi$
$18$	0	0
$19$	−192.435	−0.533060	−0.266530	−	0.963827i	$-0.585877\pi$
$19$	−192.435	−0.533060	−0.266530	+	0.963827i	$0.585877\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	543.928i	1.02822i	0.857724	+	0.514110i	$0.171878\pi$
$23$	543.928i	1.02822i	−0.857724	+	0.514110i	$0.828122\pi$
$24$	0	0
$25$	−1486.25	−2.37800
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 822.977i	− 0.978569i	−0.872124	−	0.489285i	$-0.837258\pi$
$29$	− 822.977i	− 0.978569i	0.872124	−	0.489285i	$-0.162742\pi$
$30$	0	0
$31$	−615.408	−0.640383	−0.320191	−	0.947353i	$-0.603747\pi$
$31$	−615.408	−0.640383	−0.320191	+	0.947353i	$0.603747\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3701.14i	3.02134i
$36$	0	0
$37$	−2355.20	−1.72038	−0.860189	−	0.509976i	$-0.829654\pi$
$37$	−2355.20	−1.72038	−0.860189	+	0.509976i	$0.829654\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	502.171i	0.298733i	0.988782	+	0.149367i	$0.0477235\pi$
$41$	502.171i	0.298733i	−0.988782	+	0.149367i	$0.952277\pi$
$42$	0	0
$43$	164.719	0.0890854	0.0445427	−	0.999007i	$-0.485817\pi$
$43$	164.719	0.0890854	0.0445427	+	0.999007i	$0.485817\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1041.23i	0.471356i	0.971831	+	0.235678i	$0.0757310\pi$
$47$	1041.23i	0.471356i	−0.971831	+	0.235678i	$0.924269\pi$
$48$	0	0
$49$	4087.30	1.70233
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 182.004i	− 0.0647930i	−0.999475	−	0.0323965i	$-0.989686\pi$
$53$	− 182.004i	− 0.0647930i	0.999475	−	0.0323965i	$-0.0103139\pi$
$54$	0	0
$55$	−5197.21	−1.71809
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 1811.76i	− 0.520473i	−0.965545	−	0.260236i	$-0.916200\pi$
$59$	− 1811.76i	− 0.520473i	0.965545	−	0.260236i	$-0.0838005\pi$
$60$	0	0
$61$	−2859.85	−0.768570	−0.384285	−	0.923214i	$-0.625552\pi$
$61$	−2859.85	−0.768570	−0.384285	+	0.923214i	$0.625552\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3228.05i	0.764035i
$66$	0	0
$67$	8895.85	1.98170	0.990850	−	0.134970i	$-0.0430938\pi$
$67$	8895.85	1.98170	0.990850	+	0.134970i	$0.0430938\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3556.66i	0.705547i	0.935709	+	0.352773i	$0.114761\pi$
$71$	3556.66i	0.705547i	−0.935709	+	0.352773i	$0.885239\pi$
$72$	0	0
$73$	−8008.40	−1.50280	−0.751398	−	0.659849i	$-0.770620\pi$
$73$	−8008.40	−1.50280	−0.751398	+	0.659849i	$0.770620\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	9111.00i	1.53668i
$78$	0	0
$79$	4026.40	0.645152	0.322576	−	0.946544i	$-0.395451\pi$
$79$	4026.40	0.645152	0.322576	+	0.946544i	$0.395451\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1846.98i	0.268106i	0.990974	+	0.134053i	$0.0427993\pi$
$83$	1846.98i	0.268106i	−0.990974	+	0.134053i	$0.957201\pi$
$84$	0	0
$85$	−11764.1	−1.62825
$86$	0	0
$87$	0	0
$88$	0	0
$89$	4204.70i	0.530830i	0.964134	+	0.265415i	$0.0855089\pi$
$89$	4204.70i	0.530830i	−0.964134	+	0.265415i	$0.914491\pi$
$90$	0	0
$91$	5658.95	0.683365
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 8842.05i	− 0.979728i
$96$	0	0
$97$	13186.8	1.40151	0.700755	−	0.713402i	$-0.252847\pi$
$97$	13186.8	1.40151	0.700755	+	0.713402i	$0.252847\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	9990.28i	0.979344i	0.871907	+	0.489672i	$0.162883\pi$
$101$	9990.28i	0.979344i	−0.871907	+	0.489672i	$0.837117\pi$
$102$	0	0
$103$	3278.99	0.309076	0.154538	−	0.987987i	$-0.450611\pi$
$103$	3278.99	0.309076	0.154538	+	0.987987i	$0.450611\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 20923.5i	− 1.82754i	−0.406236	−	0.913768i	$-0.633159\pi$
$107$	− 20923.5i	− 1.82754i	0.406236	−	0.913768i	$-0.366841\pi$
$108$	0	0
$109$	3340.77	0.281186	0.140593	−	0.990067i	$-0.455099\pi$
$109$	3340.77	0.281186	0.140593	+	0.990067i	$0.455099\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 8872.38i	− 0.694838i	−0.937710	−	0.347419i	$-0.887058\pi$
$113$	− 8872.38i	− 0.694838i	0.937710	−	0.347419i	$-0.112942\pi$
$114$	0	0
$115$	−24992.6	−1.88980
$116$	0	0
$117$	0	0
$118$	0	0
$119$	20623.2i	1.45634i
$120$	0	0
$121$	1847.16	0.126163
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 39573.0i	− 2.53267i
$126$	0	0
$127$	15509.1	0.961569	0.480784	−	0.876839i	$-0.340352\pi$
$127$	15509.1	0.961569	0.480784	+	0.876839i	$0.340352\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 13596.5i	− 0.792289i	−0.918188	−	0.396145i	$-0.870348\pi$
$131$	− 13596.5i	− 0.792289i	0.918188	−	0.396145i	$-0.129652\pi$
$132$	0	0
$133$	−15500.6	−0.876285
$134$	0	0
$135$	0	0
$136$	0	0
$137$	21664.7i	1.15428i	0.816644	+	0.577142i	$0.195832\pi$
$137$	21664.7i	1.15428i	−0.816644	+	0.577142i	$0.804168\pi$
$138$	0	0
$139$	−15438.2	−0.799040	−0.399520	−	0.916725i	$-0.630823\pi$
$139$	−15438.2	−0.799040	−0.399520	+	0.916725i	$0.630823\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	7946.41i	0.388596i
$144$	0	0
$145$	37814.4	1.79855
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 20472.0i	− 0.922120i	−0.887369	−	0.461060i	$-0.847469\pi$
$149$	− 20472.0i	− 0.922120i	0.887369	−	0.461060i	$-0.152531\pi$
$150$	0	0
$151$	32004.7	1.40365	0.701827	−	0.712347i	$-0.252368\pi$
$151$	32004.7	1.40365	0.701827	+	0.712347i	$0.252368\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 28277.0i	− 1.17698i
$156$	0	0
$157$	−24207.2	−0.982075	−0.491037	−	0.871139i	$-0.663382\pi$
$157$	−24207.2	−0.982075	−0.491037	+	0.871139i	$0.663382\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	43813.4i	1.69027i
$162$	0	0
$163$	−21091.0	−0.793821	−0.396911	−	0.917857i	$-0.629918\pi$
$163$	−21091.0	−0.793821	−0.396911	+	0.917857i	$0.629918\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12967.4i	0.464964i	0.972601	+	0.232482i	$0.0746848\pi$
$167$	12967.4i	0.464964i	−0.972601	+	0.232482i	$0.925315\pi$
$168$	0	0
$169$	−23625.4	−0.827191
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 41778.6i	− 1.39592i	−0.716135	−	0.697962i	$-0.754090\pi$
$173$	− 41778.6i	− 1.39592i	0.716135	−	0.697962i	$-0.245910\pi$
$174$	0	0
$175$	−119717.	−3.90914
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 13288.6i	− 0.414736i	−0.978263	−	0.207368i	$-0.933510\pi$
$179$	− 13288.6i	− 0.414736i	0.978263	−	0.207368i	$-0.0664897\pi$
$180$	0	0
$181$	−28910.7	−0.882475	−0.441237	−	0.897390i	$-0.645460\pi$
$181$	−28910.7	−0.882475	−0.441237	+	0.897390i	$0.645460\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 108217.i	− 3.16194i
$186$	0	0
$187$	−28959.5	−0.828148
$188$	0	0
$189$	0	0
$190$	0	0
$191$	50359.0i	1.38042i	0.723610	+	0.690209i	$0.242482\pi$
$191$	50359.0i	1.38042i	−0.723610	+	0.690209i	$0.757518\pi$
$192$	0	0
$193$	−3466.02	−0.0930500	−0.0465250	−	0.998917i	$-0.514815\pi$
$193$	−3466.02	−0.0930500	−0.0465250	+	0.998917i	$0.514815\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 7813.76i	− 0.201339i	−0.994920	−	0.100669i	$-0.967902\pi$
$197$	− 7813.76i	− 0.201339i	0.994920	−	0.100669i	$-0.0320984\pi$
$198$	0	0
$199$	−29153.3	−0.736176	−0.368088	−	0.929791i	$-0.619987\pi$
$199$	−29153.3	−0.736176	−0.368088	+	0.929791i	$0.619987\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 66290.8i	− 1.60865i
$204$	0	0
$205$	−23073.9	−0.549052
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 21766.2i	− 0.498300i
$210$	0	0
$211$	47158.0	1.05923	0.529615	−	0.848238i	$-0.322336\pi$
$211$	47158.0	1.05923	0.529615	+	0.848238i	$0.322336\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	7568.56i	0.163733i
$216$	0	0
$217$	−49571.1	−1.05271
$218$	0	0
$219$	0	0
$220$	0	0
$221$	17987.1i	0.368278i
$222$	0	0
$223$	−10053.3	−0.202161	−0.101081	−	0.994878i	$-0.532230\pi$
$223$	−10053.3	−0.202161	−0.101081	+	0.994878i	$0.532230\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	29951.8i	0.581260i	0.956835	+	0.290630i	$0.0938649\pi$
$227$	29951.8i	0.581260i	−0.956835	+	0.290630i	$0.906135\pi$
$228$	0	0
$229$	−20133.4	−0.383926	−0.191963	−	0.981402i	$-0.561485\pi$
$229$	−20133.4	−0.383926	−0.191963	+	0.981402i	$0.561485\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	19079.1i	0.351436i	0.984441	+	0.175718i	$0.0562246\pi$
$233$	19079.1i	0.351436i	−0.984441	+	0.175718i	$0.943775\pi$
$234$	0	0
$235$	−47842.6	−0.866321
$236$	0	0
$237$	0	0
$238$	0	0
$239$	90259.6i	1.58015i	0.613012	+	0.790073i	$0.289958\pi$
$239$	90259.6i	1.58015i	−0.613012	+	0.790073i	$0.710042\pi$
$240$	0	0
$241$	30161.0	0.519291	0.259646	−	0.965704i	$-0.416394\pi$
$241$	30161.0	0.519291	0.259646	+	0.965704i	$0.416394\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	187805.i	3.12877i
$246$	0	0
$247$	−13519.3	−0.221595
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 16436.6i	− 0.260894i	−0.991455	−	0.130447i	$-0.958359\pi$
$251$	− 16436.6i	− 0.260894i	0.991455	−	0.130447i	$-0.0416412\pi$
$252$	0	0
$253$	−61523.6	−0.961172
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 23244.0i	− 0.351920i	−0.984397	−	0.175960i	$-0.943697\pi$
$257$	− 23244.0i	− 0.351920i	0.984397	−	0.175960i	$-0.0563030\pi$
$258$	0	0
$259$	−189711.	−2.82809
$260$	0	0
$261$	0	0
$262$	0	0
$263$	30701.7i	0.443865i	0.975062	+	0.221932i	$0.0712364\pi$
$263$	30701.7i	0.443865i	−0.975062	+	0.221932i	$0.928764\pi$
$264$	0	0
$265$	8362.76	0.119085
$266$	0	0
$267$	0	0
$268$	0	0
$269$	98285.6i	1.35827i	0.734014	+	0.679134i	$0.237644\pi$
$269$	98285.6i	1.35827i	−0.734014	+	0.679134i	$0.762356\pi$
$270$	0	0
$271$	−111736.	−1.52144	−0.760722	−	0.649078i	$-0.775155\pi$
$271$	−111736.	−1.52144	−0.760722	+	0.649078i	$0.775155\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 168110.i	− 2.22294i
$276$	0	0
$277$	−64408.5	−0.839429	−0.419714	−	0.907656i	$-0.637870\pi$
$277$	−64408.5	−0.839429	−0.419714	+	0.907656i	$0.637870\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	124863.i	1.58133i	0.612251	+	0.790664i	$0.290264\pi$
$281$	124863.i	1.58133i	−0.612251	+	0.790664i	$0.709736\pi$
$282$	0	0
$283$	125798.	1.57072	0.785362	−	0.619037i	$-0.212477\pi$
$283$	125798.	1.57072	0.785362	+	0.619037i	$0.212477\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	40449.8i	0.491081i
$288$	0	0
$289$	17969.8	0.215153
$290$	0	0
$291$	0	0
$292$	0	0
$293$	70372.5i	0.819725i	0.912147	+	0.409862i	$0.134423\pi$
$293$	70372.5i	0.819725i	−0.912147	+	0.409862i	$0.865577\pi$
$294$	0	0
$295$	83247.6	0.956594
$296$	0	0
$297$	0	0
$298$	0	0
$299$	38213.1i	0.427434i
$300$	0	0
$301$	13268.1	0.146446
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 131405.i	− 1.41258i
$306$	0	0
$307$	26790.2	0.284249	0.142125	−	0.989849i	$-0.454607\pi$
$307$	26790.2	0.284249	0.142125	+	0.989849i	$0.454607\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 127879.i	− 1.32215i	−0.750321	−	0.661074i	$-0.770101\pi$
$311$	− 127879.i	− 1.32215i	0.750321	−	0.661074i	$-0.229899\pi$
$312$	0	0
$313$	9564.22	0.0976249	0.0488125	−	0.998808i	$-0.484456\pi$
$313$	9564.22	0.0976249	0.0488125	+	0.998808i	$0.484456\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	38993.9i	0.388042i	0.980997	+	0.194021i	$0.0621529\pi$
$317$	38993.9i	0.388042i	−0.980997	+	0.194021i	$0.937847\pi$
$318$	0	0
$319$	93086.8	0.914759
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 49269.0i	− 0.472246i
$324$	0	0
$325$	−104415.	−0.988542
$326$	0	0
$327$	0	0
$328$	0	0
$329$	83870.7i	0.774851i
$330$	0	0
$331$	−50434.2	−0.460330	−0.230165	−	0.973152i	$-0.573927\pi$
$331$	−50434.2	−0.460330	−0.230165	+	0.973152i	$0.573927\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	408749.i	3.64223i
$336$	0	0
$337$	−127374.	−1.12156	−0.560778	−	0.827966i	$-0.689498\pi$
$337$	−127374.	−1.12156	−0.560778	+	0.827966i	$0.689498\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 69608.7i	− 0.598625i
$342$	0	0
$343$	135831.	1.15455
$344$	0	0
$345$	0	0
$346$	0	0
$347$	193817.i	1.60965i	0.593510	+	0.804827i	$0.297742\pi$
$347$	193817.i	1.60965i	−0.593510	+	0.804827i	$0.702258\pi$
$348$	0	0
$349$	104069.	0.854417	0.427209	−	0.904153i	$-0.359497\pi$
$349$	104069.	0.854417	0.427209	+	0.904153i	$0.359497\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	160774.i	1.29022i	0.764088	+	0.645112i	$0.223189\pi$
$353$	160774.i	1.29022i	−0.764088	+	0.645112i	$0.776811\pi$
$354$	0	0
$355$	−163423.	−1.29675
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 21776.0i	− 0.168962i	−0.996425	−	0.0844811i	$-0.973077\pi$
$359$	− 21776.0i	− 0.168962i	0.996425	−	0.0844811i	$-0.0269233\pi$
$360$	0	0
$361$	−93289.9	−0.715847
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 367973.i	− 2.76204i
$366$	0	0
$367$	251965.	1.87072	0.935360	−	0.353698i	$-0.115076\pi$
$367$	251965.	1.87072	0.935360	+	0.353698i	$0.115076\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 14660.4i	− 0.106512i
$372$	0	0
$373$	128026.	0.920196	0.460098	−	0.887868i	$-0.347814\pi$
$373$	128026.	0.920196	0.460098	+	0.887868i	$0.347814\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 57817.3i	− 0.406795i
$378$	0	0
$379$	−92997.5	−0.647430	−0.323715	−	0.946155i	$-0.604932\pi$
$379$	−92997.5	−0.647430	−0.323715	+	0.946155i	$0.604932\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 278293.i	− 1.89716i	−0.316537	−	0.948580i	$-0.602520\pi$
$383$	− 278293.i	− 1.89716i	0.316537	−	0.948580i	$-0.397480\pi$
$384$	0	0
$385$	−418635.	−2.82432
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 138923.i	− 0.918071i	−0.888418	−	0.459035i	$-0.848195\pi$
$389$	− 138923.i	− 0.918071i	0.888418	−	0.459035i	$-0.151805\pi$
$390$	0	0
$391$	−139262.	−0.910916
$392$	0	0
$393$	0	0
$394$	0	0
$395$	185006.i	1.18575i
$396$	0	0
$397$	−1760.70	−0.0111713	−0.00558564	−	0.999984i	$-0.501778\pi$
$397$	−1760.70	−0.0111713	−0.00558564	+	0.999984i	$0.501778\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	41057.9i	0.255334i	0.991817	+	0.127667i	$0.0407489\pi$
$401$	41057.9i	0.255334i	−0.991817	+	0.127667i	$0.959251\pi$
$402$	0	0
$403$	−43234.8	−0.266209
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 266396.i	− 1.60820i
$408$	0	0
$409$	−166714.	−0.996608	−0.498304	−	0.867002i	$-0.666044\pi$
$409$	−166714.	−0.996608	−0.498304	+	0.867002i	$0.666044\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 145938.i	− 0.855593i
$414$	0	0
$415$	−84865.9	−0.492762
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 331708.i	− 1.88942i	−0.327912	−	0.944708i	$-0.606345\pi$
$419$	− 331708.i	− 1.88942i	0.327912	−	0.944708i	$-0.393655\pi$
$420$	0	0
$421$	−187827.	−1.05973	−0.529863	−	0.848083i	$-0.677757\pi$
$421$	−187827.	−1.05973	−0.529863	+	0.848083i	$0.677757\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 380524.i	− 2.10671i
$426$	0	0
$427$	−230361.	−1.26343
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 6952.01i	− 0.0374245i	−0.999825	−	0.0187122i	$-0.994043\pi$
$431$	− 6952.01i	− 0.0374245i	0.999825	−	0.0187122i	$-0.00595664\pi$
$432$	0	0
$433$	−37941.0	−0.202364	−0.101182	−	0.994868i	$-0.532262\pi$
$433$	−37941.0	−0.202364	−0.101182	+	0.994868i	$0.532262\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 104671.i	− 0.548102i
$438$	0	0
$439$	−248405.	−1.28893	−0.644467	−	0.764632i	$-0.722921\pi$
$439$	−248405.	−1.28893	−0.644467	+	0.764632i	$0.722921\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 162849.i	− 0.829806i	−0.909866	−	0.414903i	$-0.863816\pi$
$443$	− 162849.i	− 0.829806i	0.909866	−	0.414903i	$-0.136184\pi$
$444$	0	0
$445$	−193199.	−0.975630
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 139250.i	− 0.690723i	−0.938470	−	0.345361i	$-0.887756\pi$
$449$	− 139250.i	− 0.690723i	0.938470	−	0.345361i	$-0.112244\pi$
$450$	0	0
$451$	−56800.5	−0.279254
$452$	0	0
$453$	0	0
$454$	0	0
$455$	260019.i	1.25598i
$456$	0	0
$457$	250539.	1.19962	0.599810	−	0.800143i	$-0.295243\pi$
$457$	250539.	1.19962	0.599810	+	0.800143i	$0.295243\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	78186.5i	0.367900i	0.982936	+	0.183950i	$0.0588885\pi$
$461$	78186.5i	0.367900i	−0.982936	+	0.183950i	$0.941112\pi$
$462$	0	0
$463$	−352591.	−1.64479	−0.822393	−	0.568920i	$-0.807361\pi$
$463$	−352591.	−1.64479	−0.822393	+	0.568920i	$0.807361\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	38384.6i	0.176005i	0.996120	+	0.0880023i	$0.0280483\pi$
$467$	38384.6i	0.176005i	−0.996120	+	0.0880023i	$0.971952\pi$
$468$	0	0
$469$	716560.	3.25767
$470$	0	0
$471$	0	0
$472$	0	0
$473$	18631.3i	0.0832764i
$474$	0	0
$475$	286006.	1.26762
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 52021.2i	− 0.226730i	−0.993553	−	0.113365i	$-0.963837\pi$
$479$	− 52021.2i	− 0.226730i	0.993553	−	0.113365i	$-0.0361630\pi$
$480$	0	0
$481$	−165462.	−0.715166
$482$	0	0
$483$	0	0
$484$	0	0
$485$	605912.i	2.57588i
$486$	0	0
$487$	253420.	1.06852	0.534261	−	0.845320i	$-0.320590\pi$
$487$	253420.	1.06852	0.534261	+	0.845320i	$0.320590\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	182553.i	0.757228i	0.925555	+	0.378614i	$0.123599\pi$
$491$	182553.i	0.757228i	−0.925555	+	0.378614i	$0.876401\pi$
$492$	0	0
$493$	210707.	0.866930
$494$	0	0
$495$	0	0
$496$	0	0
$497$	286489.i	1.15983i
$498$	0	0
$499$	−34543.2	−0.138727	−0.0693636	−	0.997591i	$-0.522097\pi$
$499$	−34543.2	−0.138727	−0.0693636	+	0.997591i	$0.522097\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 466084.i	− 1.84216i	−0.389368	−	0.921082i	$-0.627307\pi$
$503$	− 466084.i	− 1.84216i	0.389368	−	0.921082i	$-0.372693\pi$
$504$	0	0
$505$	−459037.	−1.79997
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 157911.i	− 0.609506i	−0.952431	−	0.304753i	$-0.901426\pi$
$509$	− 157911.i	− 0.609506i	0.952431	−	0.304753i	$-0.0985739\pi$
$510$	0	0
$511$	−645077.	−2.47041
$512$	0	0
$513$	0	0
$514$	0	0
$515$	150664.i	0.568061i
$516$	0	0
$517$	−117773.	−0.440620
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 39855.1i	− 0.146828i	−0.997302	−	0.0734139i	$-0.976611\pi$
$521$	− 39855.1i	− 0.146828i	0.997302	−	0.0734139i	$-0.0233894\pi$
$522$	0	0
$523$	250487.	0.915762	0.457881	−	0.889014i	$-0.348609\pi$
$523$	250487.	0.915762	0.457881	+	0.889014i	$0.348609\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 157563.i	− 0.567325i
$528$	0	0
$529$	−16016.8	−0.0572352
$530$	0	0
$531$	0	0
$532$	0	0
$533$	35279.4i	0.124184i
$534$	0	0
$535$	961398.	3.35889
$536$	0	0
$537$	0	0
$538$	0	0
$539$	462314.i	1.59133i
$540$	0	0
$541$	−324441.	−1.10852	−0.554258	−	0.832345i	$-0.686998\pi$
$541$	−324441.	−1.10852	−0.554258	+	0.832345i	$0.686998\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	153503.i	0.516801i
$546$	0	0
$547$	−14074.4	−0.0470387	−0.0235193	−	0.999723i	$-0.507487\pi$
$547$	−14074.4	−0.0470387	−0.0235193	+	0.999723i	$0.507487\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	158369.i	0.521636i
$552$	0	0
$553$	324326.	1.06055
$554$	0	0
$555$	0	0
$556$	0	0
$557$	94500.4i	0.304595i	0.988335	+	0.152298i	$0.0486672\pi$
$557$	94500.4i	0.304595i	−0.988335	+	0.152298i	$0.951333\pi$
$558$	0	0
$559$	11572.1	0.0370331
$560$	0	0
$561$	0	0
$562$	0	0
$563$	107077.i	0.337815i	0.985632	+	0.168908i	$0.0540239\pi$
$563$	107077.i	0.337815i	−0.985632	+	0.168908i	$0.945976\pi$
$564$	0	0
$565$	407671.	1.27707
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 237865.i	− 0.734694i	−0.930084	−	0.367347i	$-0.880266\pi$
$569$	− 237865.i	− 0.734694i	0.930084	−	0.367347i	$-0.119734\pi$
$570$	0	0
$571$	−244568.	−0.750115	−0.375057	−	0.927002i	$-0.622377\pi$
$571$	−244568.	−0.750115	−0.375057	+	0.927002i	$0.622377\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 808413.i	− 2.44511i
$576$	0	0
$577$	456171.	1.37018	0.685088	−	0.728461i	$-0.259764\pi$
$577$	456171.	1.37018	0.685088	+	0.728461i	$0.259764\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	148775.i	0.440734i
$582$	0	0
$583$	20586.4	0.0605680
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 597028.i	− 1.73268i	−0.499454	−	0.866341i	$-0.666466\pi$
$587$	− 597028.i	− 1.73268i	0.499454	−	0.866341i	$-0.333534\pi$
$588$	0	0
$589$	118426.	0.341362
$590$	0	0
$591$	0	0
$592$	0	0
$593$	597802.i	1.70000i	0.526785	+	0.849999i	$0.323397\pi$
$593$	597802.i	1.70000i	−0.526785	+	0.849999i	$0.676603\pi$
$594$	0	0
$595$	−947601.	−2.67665
$596$	0	0
$597$	0	0
$598$	0	0
$599$	400872.i	1.11725i	0.829419	+	0.558627i	$0.188672\pi$
$599$	400872.i	1.11725i	−0.829419	+	0.558627i	$0.811328\pi$
$600$	0	0
$601$	77232.3	0.213821	0.106910	−	0.994269i	$-0.465904\pi$
$601$	77232.3	0.213821	0.106910	+	0.994269i	$0.465904\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	84873.7i	0.231880i
$606$	0	0
$607$	600706.	1.63036	0.815182	−	0.579204i	$-0.196637\pi$
$607$	600706.	1.63036	0.815182	+	0.579204i	$0.196637\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	73150.1i	0.195944i
$612$	0	0
$613$	211376.	0.562516	0.281258	−	0.959632i	$-0.409248\pi$
$613$	211376.	0.562516	0.281258	+	0.959632i	$0.409248\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	378327.i	0.993796i	0.867809	+	0.496898i	$0.165528\pi$
$617$	378327.i	0.993796i	−0.867809	+	0.496898i	$0.834472\pi$
$618$	0	0
$619$	491923.	1.28385	0.641927	−	0.766765i	$-0.278135\pi$
$619$	491923.	1.28385	0.641927	+	0.766765i	$0.278135\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	338689.i	0.872619i
$624$	0	0
$625$	889407.	2.27688
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 603000.i	− 1.52411i
$630$	0	0
$631$	64515.4	0.162033	0.0810167	−	0.996713i	$-0.474183\pi$
$631$	64515.4	0.162033	0.0810167	+	0.996713i	$0.474183\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	712619.i	1.76730i
$636$	0	0
$637$	287149.	0.707665
$638$	0	0
$639$	0	0
$640$	0	0
$641$	429607.i	1.04558i	0.852463	+	0.522788i	$0.175108\pi$
$641$	429607.i	1.04558i	−0.852463	+	0.522788i	$0.824892\pi$
$642$	0	0
$643$	−500950.	−1.21164	−0.605818	−	0.795603i	$-0.707154\pi$
$643$	−500950.	−1.21164	−0.605818	+	0.795603i	$0.707154\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	175906.i	0.420216i	0.977678	+	0.210108i	$0.0673815\pi$
$647$	175906.i	0.420216i	−0.977678	+	0.210108i	$0.932618\pi$
$648$	0	0
$649$	204929.	0.486534
$650$	0	0
$651$	0	0
$652$	0	0
$653$	52557.8i	0.123257i	0.998099	+	0.0616284i	$0.0196294\pi$
$653$	52557.8i	0.123257i	−0.998099	+	0.0616284i	$0.980371\pi$
$654$	0	0
$655$	624735.	1.45617
$656$	0	0
$657$	0	0
$658$	0	0
$659$	381919.i	0.879428i	0.898138	+	0.439714i	$0.144920\pi$
$659$	381919.i	0.879428i	−0.898138	+	0.439714i	$0.855080\pi$
$660$	0	0
$661$	711240.	1.62785	0.813923	−	0.580972i	$-0.197328\pi$
$661$	711240.	1.62785	0.813923	+	0.580972i	$0.197328\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 712227.i	− 1.61055i
$666$	0	0
$667$	447640.	1.00618
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 323477.i	− 0.718454i
$672$	0	0
$673$	−170774.	−0.377044	−0.188522	−	0.982069i	$-0.560370\pi$
$673$	−170774.	−0.377044	−0.188522	+	0.982069i	$0.560370\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 258544.i	− 0.564102i	−0.959399	−	0.282051i	$-0.908985\pi$
$677$	− 258544.i	− 0.564102i	0.959399	−	0.282051i	$-0.0910148\pi$
$678$	0	0
$679$	1.06220e6	2.30391
$680$	0	0
$681$	0	0
$682$	0	0
$683$	402556.i	0.862948i	0.902125	+	0.431474i	$0.142006\pi$
$683$	402556.i	0.862948i	−0.902125	+	0.431474i	$0.857994\pi$
$684$	0	0
$685$	−995459.	−2.12150
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 12786.5i	− 0.0269347i
$690$	0	0
$691$	−79523.0	−0.166547	−0.0832735	−	0.996527i	$-0.526538\pi$
$691$	−79523.0	−0.166547	−0.0832735	+	0.996527i	$0.526538\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 709362.i	− 1.46858i
$696$	0	0
$697$	−128571.	−0.264653
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 371798.i	− 0.756609i	−0.925681	−	0.378305i	$-0.876507\pi$
$701$	− 371798.i	− 0.756609i	0.925681	−	0.378305i	$-0.123493\pi$
$702$	0	0
$703$	453221.	0.917064
$704$	0	0
$705$	0	0
$706$	0	0
$707$	804717.i	1.60992i
$708$	0	0
$709$	−933206.	−1.85646	−0.928228	−	0.372011i	$-0.878668\pi$
$709$	−933206.	−1.85646	−0.928228	+	0.372011i	$0.878668\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 334738.i	− 0.658454i
$714$	0	0
$715$	−365124.	−0.714214
$716$	0	0
$717$	0	0
$718$	0	0
$719$	180424.i	0.349010i	0.984656	+	0.174505i	$0.0558325\pi$
$719$	180424.i	0.349010i	−0.984656	+	0.174505i	$0.944168\pi$
$720$	0	0
$721$	264122.	0.508083
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.22315e6i	2.32704i
$726$	0	0
$727$	455249.	0.861350	0.430675	−	0.902507i	$-0.358275\pi$
$727$	455249.	0.861350	0.430675	+	0.902507i	$0.358275\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	42173.0i	0.0789222i
$732$	0	0
$733$	550038.	1.02373	0.511864	−	0.859066i	$-0.328955\pi$
$733$	550038.	1.02373	0.511864	+	0.859066i	$0.328955\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.00621e6i	1.85248i
$738$	0	0
$739$	−663313.	−1.21459	−0.607295	−	0.794477i	$-0.707745\pi$
$739$	−663313.	−1.21459	−0.607295	+	0.794477i	$0.707745\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	322338.i	0.583893i	0.956435	+	0.291947i	$0.0943030\pi$
$743$	322338.i	0.583893i	−0.956435	+	0.291947i	$0.905697\pi$
$744$	0	0
$745$	940654.	1.69480
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 1.68538e6i	− 3.00425i
$750$	0	0
$751$	−328672.	−0.582751	−0.291375	−	0.956609i	$-0.594113\pi$
$751$	−328672.	−0.582751	−0.291375	+	0.956609i	$0.594113\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.47056e6i	2.57982i
$756$	0	0
$757$	530625.	0.925968	0.462984	−	0.886367i	$-0.346779\pi$
$757$	530625.	0.925968	0.462984	+	0.886367i	$0.346779\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 1.08627e6i	− 1.87573i	−0.347002	−	0.937864i	$-0.612800\pi$
$761$	− 1.08627e6i	− 1.87573i	0.347002	−	0.937864i	$-0.387200\pi$
$762$	0	0
$763$	269099.	0.462235
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 127283.i	− 0.216362i
$768$	0	0
$769$	−640074.	−1.08237	−0.541187	−	0.840902i	$-0.682025\pi$
$769$	−640074.	−1.08237	−0.541187	+	0.840902i	$0.682025\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	924647.i	1.54745i	0.633521	+	0.773726i	$0.281609\pi$
$773$	924647.i	1.54745i	−0.633521	+	0.773726i	$0.718391\pi$
$774$	0	0
$775$	914650.	1.52283
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 96635.0i	− 0.159243i
$780$	0	0
$781$	−402293.	−0.659540
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 1.11228e6i	− 1.80499i
$786$	0	0
$787$	−453454.	−0.732123	−0.366062	−	0.930591i	$-0.619294\pi$
$787$	−453454.	−0.732123	−0.366062	+	0.930591i	$0.619294\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 714670.i	− 1.14223i
$792$	0	0
$793$	−200915.	−0.319497
$794$	0	0
$795$	0	0
$796$	0	0
$797$	957183.i	1.50688i	0.657518	+	0.753439i	$0.271606\pi$
$797$	957183.i	1.50688i	−0.657518	+	0.753439i	$0.728394\pi$
$798$	0	0
$799$	−266585.	−0.417582
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 905829.i	− 1.40480i
$804$	0	0
$805$	−2.01315e6	−3.10660
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 464039.i	− 0.709018i	−0.935053	−	0.354509i	$-0.884648\pi$
$809$	− 464039.i	− 0.709018i	0.935053	−	0.354509i	$-0.115352\pi$
$810$	0	0
$811$	732128.	1.11313	0.556564	−	0.830804i	$-0.312119\pi$
$811$	732128.	1.11313	0.556564	+	0.830804i	$0.312119\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 969098.i	− 1.45899i
$816$	0	0
$817$	−31697.6	−0.0474879
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.26636e6i	1.87876i	0.342876	+	0.939381i	$0.388599\pi$
$821$	1.26636e6i	1.87876i	−0.342876	+	0.939381i	$0.611401\pi$
$822$	0	0
$823$	172258.	0.254320	0.127160	−	0.991882i	$-0.459414\pi$
$823$	172258.	0.254320	0.127160	+	0.991882i	$0.459414\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	135383.i	0.197949i	0.995090	+	0.0989746i	$0.0315563\pi$
$827$	135383.i	0.197949i	−0.995090	+	0.0989746i	$0.968444\pi$
$828$	0	0
$829$	290151.	0.422197	0.211099	−	0.977465i	$-0.432296\pi$
$829$	290151.	0.422197	0.211099	+	0.977465i	$0.432296\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.04647e6i	1.50812i
$834$	0	0
$835$	−595830.	−0.854574
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 534767.i	− 0.759697i	−0.925049	−	0.379849i	$-0.875976\pi$
$839$	− 534767.i	− 0.759697i	0.925049	−	0.379849i	$-0.124024\pi$
$840$	0	0
$841$	29990.2	0.0424021
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 1.08555e6i	− 1.52032i
$846$	0	0
$847$	148788.	0.207397
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 1.28106e6i	− 1.76893i
$852$	0	0
$853$	−939343.	−1.29100	−0.645500	−	0.763760i	$-0.723351\pi$
$853$	−939343.	−1.29100	−0.645500	+	0.763760i	$0.723351\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 671520.i	− 0.914318i	−0.889385	−	0.457159i	$-0.848867\pi$
$857$	− 671520.i	− 0.914318i	0.889385	−	0.457159i	$-0.151133\pi$
$858$	0	0
$859$	1.14276e6	1.54871	0.774356	−	0.632751i	$-0.218074\pi$
$859$	1.14276e6	1.54871	0.774356	+	0.632751i	$0.218074\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1.09963e6i	− 1.47647i	−0.674545	−	0.738234i	$-0.735660\pi$
$863$	− 1.09963e6i	− 1.47647i	0.674545	−	0.738234i	$-0.264340\pi$
$864$	0	0
$865$	1.91966e6	2.56562
$866$	0	0
$867$	0	0
$868$	0	0
$869$	455425.i	0.603084i
$870$	0	0
$871$	624968.	0.823799
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 3.18760e6i	− 4.16340i
$876$	0	0
$877$	779731.	1.01379	0.506893	−	0.862009i	$-0.330794\pi$
$877$	779731.	1.01379	0.506893	+	0.862009i	$0.330794\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	622180.i	0.801611i	0.916163	+	0.400806i	$0.131270\pi$
$881$	622180.i	0.801611i	−0.916163	+	0.400806i	$0.868730\pi$
$882$	0	0
$883$	1.39076e6	1.78373	0.891865	−	0.452301i	$-0.149397\pi$
$883$	1.39076e6	1.78373	0.891865	+	0.452301i	$0.149397\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	957517.i	1.21702i	0.793545	+	0.608512i	$0.208233\pi$
$887$	957517.i	1.21702i	−0.793545	+	0.608512i	$0.791767\pi$
$888$	0	0
$889$	1.24926e6	1.58070
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 200368.i	− 0.251261i
$894$	0	0
$895$	610587.	0.762257
$896$	0	0
$897$	0	0
$898$	0	0
$899$	506466.i	0.626659i
$900$	0	0
$901$	46598.3	0.0574012
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 1.32840e6i	− 1.62193i
$906$	0	0
$907$	−219571.	−0.266907	−0.133454	−	0.991055i	$-0.542607\pi$
$907$	−219571.	−0.266907	−0.133454	+	0.991055i	$0.542607\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1.52237e6i	1.83436i	0.398475	+	0.917179i	$0.369540\pi$
$911$	1.52237e6i	1.83436i	−0.398475	+	0.917179i	$0.630460\pi$
$912$	0	0
$913$	−208912.	−0.250624
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 1.09520e6i	− 1.30243i
$918$	0	0
$919$	239088.	0.283091	0.141546	−	0.989932i	$-0.454793\pi$
$919$	239088.	0.283091	0.141546	+	0.989932i	$0.454793\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	249869.i	0.293298i
$924$	0	0
$925$	3.50041e6	4.09106
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 36734.8i	− 0.0425643i	−0.999774	−	0.0212822i	$-0.993225\pi$
$929$	− 36734.8i	− 0.0425643i	0.999774	−	0.0212822i	$-0.00677483\pi$
$930$	0	0
$931$	−786538.	−0.907445
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 1.33064e6i	− 1.52208i
$936$	0	0
$937$	−849585.	−0.967671	−0.483835	−	0.875159i	$-0.660757\pi$
$937$	−849585.	−0.967671	−0.483835	+	0.875159i	$0.660757\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 115282.i	− 0.130191i	−0.997879	−	0.0650956i	$-0.979265\pi$
$941$	− 115282.i	− 0.130191i	0.997879	−	0.0650956i	$-0.0207352\pi$
$942$	0	0
$943$	−273145.	−0.307163
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 51123.6i	− 0.0570061i	−0.999594	−	0.0285031i	$-0.990926\pi$
$947$	− 51123.6i	− 0.0570061i	0.999594	−	0.0285031i	$-0.00907404\pi$
$948$	0	0
$949$	−562621.	−0.624717
$950$	0	0
$951$	0	0
$952$	0	0
$953$	407821.i	0.449038i	0.974470	+	0.224519i	$0.0720812\pi$
$953$	407821.i	0.449038i	−0.974470	+	0.224519i	$0.927919\pi$
$954$	0	0
$955$	−2.31391e6	−2.53712
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.74509e6i	1.89750i
$960$	0	0
$961$	−544794.	−0.589910
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 159258.i	− 0.171020i
$966$	0	0
$967$	−1.27909e6	−1.36788	−0.683941	−	0.729538i	$-0.739735\pi$
$967$	−1.27909e6	−1.36788	−0.683941	+	0.729538i	$0.739735\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	806048.i	0.854914i	0.904036	+	0.427457i	$0.140590\pi$
$971$	806048.i	0.854914i	−0.904036	+	0.427457i	$0.859410\pi$
$972$	0	0
$973$	−1.24355e6	−1.31352
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.36715e6i	1.43228i	0.697956	+	0.716141i	$0.254093\pi$
$977$	1.36715e6i	1.43228i	−0.697956	+	0.716141i	$0.745907\pi$
$978$	0	0
$979$	−475593.	−0.496216
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.44380e6i	1.49417i	0.664731	+	0.747083i	$0.268546\pi$
$983$	1.44380e6i	1.49417i	−0.664731	+	0.747083i	$0.731454\pi$
$984$	0	0
$985$	359029.	0.370047
$986$	0	0
$987$	0	0
$988$	0	0
$989$	89595.3i	0.0915994i
$990$	0	0
$991$	476821.	0.485521	0.242761	−	0.970086i	$-0.421947\pi$
$991$	476821.	0.485521	0.242761	+	0.970086i	$0.421947\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 1.33955e6i	− 1.35304i
$996$	0	0
$997$	−1.02549e6	−1.03167	−0.515834	−	0.856689i	$-0.672518\pi$
$997$	−1.02549e6	−1.03167	−0.515834	+	0.856689i	$0.672518\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.5.e.b.161.4		4
3.2	odd	2	inner	1296.5.e.b.161.1		4
4.3	odd	2		162.5.b.a.161.4	yes	4
12.11	even	2		162.5.b.a.161.1	✓	4
36.7	odd	6		162.5.d.d.53.3		8
36.11	even	6		162.5.d.d.53.2		8
36.23	even	6		162.5.d.d.107.3		8
36.31	odd	6		162.5.d.d.107.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.5.b.a.161.1	✓	4	12.11	even	2
162.5.b.a.161.4	yes	4	4.3	odd	2
162.5.d.d.53.2		8	36.11	even	6
162.5.d.d.53.3		8	36.7	odd	6
162.5.d.d.107.2		8	36.31	odd	6
162.5.d.d.107.3		8	36.23	even	6
1296.5.e.b.161.1		4	3.2	odd	2	inner
1296.5.e.b.161.4		4	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:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	1296.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+45.9483i q^{5} +80.5500 q^{7} +O(q^{10})\)	\(q+45.9483i q^{5} +80.5500 q^{7} +113.110i q^{11} +70.2539 q^{13} +256.030i q^{17} -192.435 q^{19} +543.928i q^{23} -1486.25 q^{25} -822.977i q^{29} -615.408 q^{31} +3701.14i q^{35} -2355.20 q^{37} +502.171i q^{41} +164.719 q^{43} +1041.23i q^{47} +4087.30 q^{49} -182.004i q^{53} -5197.21 q^{55} -1811.76i q^{59} -2859.85 q^{61} +3228.05i q^{65} +8895.85 q^{67} +3556.66i q^{71} -8008.40 q^{73} +9111.00i q^{77} +4026.40 q^{79} +1846.98i q^{83} -11764.1 q^{85} +4204.70i q^{89} +5658.95 q^{91} -8842.05i q^{95} +13186.8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 52 q^{7}+O(q^{10}) \) 4 * q + 52 * q^7	\( 4 q + 52 q^{7} + 572 q^{13} + 124 q^{19} - 1892 q^{25} - 3584 q^{31} - 9400 q^{37} - 3020 q^{43} + 9324 q^{49} - 11124 q^{55} - 4144 q^{61} + 12076 q^{67} - 1792 q^{73} + 15004 q^{79} - 24048 q^{85} - 12220 q^{91} + 46304 q^{97}+O(q^{100}) \) 4 * q + 52 * q^7 + 572 * q^13 + 124 * q^19 - 1892 * q^25 - 3584 * q^31 - 9400 * q^37 - 3020 * q^43 + 9324 * q^49 - 11124 * q^55 - 4144 * q^61 + 12076 * q^67 - 1792 * q^73 + 15004 * q^79 - 24048 * q^85 - 12220 * q^91 + 46304 * q^97

Introduction

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