LMFDB - Embedded newform 108.5.c.b.53.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [108,5,Mod(53,108)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(108, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("108.53");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 108 = 2^{2} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	108.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$11.1639560131$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 6 $ x^2 + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			53.2
Root			$2.44949i$ of defining polynomial
Character	$\chi$	$=$	108.53
Dual form			108.5.c.b.53.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+44.0908i q^{5} -31.0000 q^{7} +O(q^{10})$	$q+44.0908i q^{5} -31.0000 q^{7} -220.454i q^{11} -241.000 q^{13} +220.454i q^{17} -271.000 q^{19} +220.454i q^{23} -1319.00 q^{25} +440.908i q^{29} -778.000 q^{31} -1366.82i q^{35} +1079.00 q^{37} -2204.54i q^{41} -298.000 q^{43} +3306.81i q^{47} -1440.00 q^{49} +3086.36i q^{53} +9720.00 q^{55} +2865.90i q^{59} -2641.00 q^{61} -10625.9i q^{65} +5609.00 q^{67} -4409.08i q^{71} +7199.00 q^{73} +6834.08i q^{77} +329.000 q^{79} +1322.72i q^{83} -9720.00 q^{85} +8156.80i q^{89} +7471.00 q^{91} -11948.6i q^{95} -15961.0 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 62 q^{7}+O(q^{10}) $ 2 * q - 62 * q^7	$ 2 q - 62 q^{7} - 482 q^{13} - 542 q^{19} - 2638 q^{25} - 1556 q^{31} + 2158 q^{37} - 596 q^{43} - 2880 q^{49} + 19440 q^{55} - 5282 q^{61} + 11218 q^{67} + 14398 q^{73} + 658 q^{79} - 19440 q^{85} + 14942 q^{91} - 31922 q^{97}+O(q^{100}) $ 2 * q - 62 * q^7 - 482 * q^13 - 542 * q^19 - 2638 * q^25 - 1556 * q^31 + 2158 * q^37 - 596 * q^43 - 2880 * q^49 + 19440 * q^55 - 5282 * q^61 + 11218 * q^67 + 14398 * q^73 + 658 * q^79 - 19440 * q^85 + 14942 * q^91 - 31922 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$.

$n$	$29$	$55$
$\chi(n)$	$-1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	44.0908i	1.76363i	0.471593	+	0.881816i	$0.343679\pi$
$5$	44.0908i	1.76363i	−0.471593	+	0.881816i	$0.656321\pi$
$6$	0	0
$7$	−31.0000	−0.632653	−0.316327	−	0.948650i	$-0.602450\pi$
$7$	−31.0000	−0.632653	−0.316327	+	0.948650i	$0.602450\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 220.454i	− 1.82193i	−0.412479	−	0.910967i	$-0.635337\pi$
$11$	− 220.454i	− 1.82193i	0.412479	−	0.910967i	$-0.364663\pi$
$12$	0	0
$13$	−241.000	−1.42604	−0.713018	−	0.701146i	$-0.752672\pi$
$13$	−241.000	−1.42604	−0.713018	+	0.701146i	$0.752672\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	220.454i	0.762817i	0.924407	+	0.381408i	$0.124561\pi$
$17$	220.454i	0.762817i	−0.924407	+	0.381408i	$0.875439\pi$
$18$	0	0
$19$	−271.000	−0.750693	−0.375346	−	0.926885i	$-0.622476\pi$
$19$	−271.000	−0.750693	−0.375346	+	0.926885i	$0.622476\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	220.454i	0.416737i	0.978050	+	0.208369i	$0.0668154\pi$
$23$	220.454i	0.416737i	−0.978050	+	0.208369i	$0.933185\pi$
$24$	0	0
$25$	−1319.00	−2.11040
$26$	0	0
$27$	0	0
$28$	0	0
$29$	440.908i	0.524267i	0.965032	+	0.262133i	$0.0844260\pi$
$29$	440.908i	0.524267i	−0.965032	+	0.262133i	$0.915574\pi$
$30$	0	0
$31$	−778.000	−0.809573	−0.404787	−	0.914411i	$-0.632654\pi$
$31$	−778.000	−0.809573	−0.404787	+	0.914411i	$0.632654\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 1366.82i	− 1.11577i
$36$	0	0
$37$	1079.00	0.788167	0.394083	−	0.919075i	$-0.371062\pi$
$37$	1079.00	0.788167	0.394083	+	0.919075i	$0.371062\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 2204.54i	− 1.31145i	−0.755002	−	0.655723i	$-0.772364\pi$
$41$	− 2204.54i	− 1.31145i	0.755002	−	0.655723i	$-0.227636\pi$
$42$	0	0
$43$	−298.000	−0.161168	−0.0805841	−	0.996748i	$-0.525679\pi$
$43$	−298.000	−0.161168	−0.0805841	+	0.996748i	$0.525679\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3306.81i	1.49697i	0.663151	+	0.748486i	$0.269219\pi$
$47$	3306.81i	1.49697i	−0.663151	+	0.748486i	$0.730781\pi$
$48$	0	0
$49$	−1440.00	−0.599750
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3086.36i	1.09874i	0.835580	+	0.549369i	$0.185132\pi$
$53$	3086.36i	1.09874i	−0.835580	+	0.549369i	$0.814868\pi$
$54$	0	0
$55$	9720.00	3.21322
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2865.90i	0.823299i	0.911342	+	0.411649i	$0.135047\pi$
$59$	2865.90i	0.823299i	−0.911342	+	0.411649i	$0.864953\pi$
$60$	0	0
$61$	−2641.00	−0.709755	−0.354878	−	0.934913i	$-0.615478\pi$
$61$	−2641.00	−0.709755	−0.354878	+	0.934913i	$0.615478\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 10625.9i	− 2.51500i
$66$	0	0
$67$	5609.00	1.24950	0.624749	−	0.780825i	$-0.285201\pi$
$67$	5609.00	1.24950	0.624749	+	0.780825i	$0.285201\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 4409.08i	− 0.874644i	−0.899305	−	0.437322i	$-0.855927\pi$
$71$	− 4409.08i	− 0.874644i	0.899305	−	0.437322i	$-0.144073\pi$
$72$	0	0
$73$	7199.00	1.35091	0.675455	−	0.737401i	$-0.263947\pi$
$73$	7199.00	1.35091	0.675455	+	0.737401i	$0.263947\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6834.08i	1.15265i
$78$	0	0
$79$	329.000	0.0527159	0.0263580	−	0.999653i	$-0.491609\pi$
$79$	329.000	0.0527159	0.0263580	+	0.999653i	$0.491609\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1322.72i	0.192005i	0.995381	+	0.0960026i	$0.0306057\pi$
$83$	1322.72i	0.192005i	−0.995381	+	0.0960026i	$0.969394\pi$
$84$	0	0
$85$	−9720.00	−1.34533
$86$	0	0
$87$	0	0
$88$	0	0
$89$	8156.80i	1.02977i	0.857260	+	0.514885i	$0.172165\pi$
$89$	8156.80i	1.02977i	−0.857260	+	0.514885i	$0.827835\pi$
$90$	0	0
$91$	7471.00	0.902186
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 11948.6i	− 1.32395i
$96$	0	0
$97$	−15961.0	−1.69635	−0.848177	−	0.529712i	$-0.822300\pi$
$97$	−15961.0	−1.69635	−0.848177	+	0.529712i	$0.822300\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	19400.0i	1.90177i	0.309544	+	0.950885i	$0.399824\pi$
$101$	19400.0i	1.90177i	−0.309544	+	0.950885i	$0.600176\pi$
$102$	0	0
$103$	1649.00	0.155434	0.0777170	−	0.996975i	$-0.475237\pi$
$103$	1649.00	0.155434	0.0777170	+	0.996975i	$0.475237\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 4188.63i	− 0.365851i	−0.983127	−	0.182925i	$-0.941443\pi$
$107$	− 4188.63i	− 0.365851i	0.983127	−	0.182925i	$-0.0585567\pi$
$108$	0	0
$109$	−3238.00	−0.272536	−0.136268	−	0.990672i	$-0.543511\pi$
$109$	−3238.00	−0.272536	−0.136268	+	0.990672i	$0.543511\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 13888.6i	− 1.08768i	−0.839188	−	0.543841i	$-0.816970\pi$
$113$	− 13888.6i	− 1.08768i	0.839188	−	0.543841i	$-0.183030\pi$
$114$	0	0
$115$	−9720.00	−0.734972
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 6834.08i	− 0.482598i
$120$	0	0
$121$	−33959.0	−2.31945
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 30599.0i	− 1.95834i
$126$	0	0
$127$	−13858.0	−0.859198	−0.429599	−	0.903020i	$-0.641345\pi$
$127$	−13858.0	−0.859198	−0.429599	+	0.903020i	$0.641345\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	26895.4i	1.56724i	0.621241	+	0.783620i	$0.286629\pi$
$131$	26895.4i	1.56724i	−0.621241	+	0.783620i	$0.713371\pi$
$132$	0	0
$133$	8401.00	0.474928
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5070.44i	0.270150i	0.990835	+	0.135075i	$0.0431275\pi$
$137$	5070.44i	0.270150i	−0.990835	+	0.135075i	$0.956872\pi$
$138$	0	0
$139$	25409.0	1.31510	0.657549	−	0.753412i	$-0.271593\pi$
$139$	25409.0	1.31510	0.657549	+	0.753412i	$0.271593\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	53129.4i	2.59814i
$144$	0	0
$145$	−19440.0	−0.924614
$146$	0	0
$147$	0	0
$148$	0	0
$149$	6172.71i	0.278038i	0.990290	+	0.139019i	$0.0443949\pi$
$149$	6172.71i	0.278038i	−0.990290	+	0.139019i	$0.955605\pi$
$150$	0	0
$151$	12569.0	0.551248	0.275624	−	0.961266i	$-0.411116\pi$
$151$	12569.0	0.551248	0.275624	+	0.961266i	$0.411116\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 34302.7i	− 1.42779i
$156$	0	0
$157$	21002.0	0.852043	0.426021	−	0.904713i	$-0.359915\pi$
$157$	21002.0	0.852043	0.426021	+	0.904713i	$0.359915\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 6834.08i	− 0.263650i
$162$	0	0
$163$	−16831.0	−0.633483	−0.316741	−	0.948512i	$-0.602589\pi$
$163$	−16831.0	−0.633483	−0.316741	+	0.948512i	$0.602589\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 29761.3i	− 1.06713i	−0.845758	−	0.533567i	$-0.820851\pi$
$167$	− 29761.3i	− 1.06713i	0.845758	−	0.533567i	$-0.179149\pi$
$168$	0	0
$169$	29520.0	1.03358
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 25131.8i	− 0.839713i	−0.907591	−	0.419856i	$-0.862080\pi$
$173$	− 25131.8i	− 0.839713i	0.907591	−	0.419856i	$-0.137920\pi$
$174$	0	0
$175$	40889.0	1.33515
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 22045.4i	− 0.688037i	−0.938963	−	0.344019i	$-0.888212\pi$
$179$	− 22045.4i	− 0.688037i	0.938963	−	0.344019i	$-0.111788\pi$
$180$	0	0
$181$	10079.0	0.307652	0.153826	−	0.988098i	$-0.450840\pi$
$181$	10079.0	0.307652	0.153826	+	0.988098i	$0.450840\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	47574.0i	1.39004i
$186$	0	0
$187$	48600.0	1.38980
$188$	0	0
$189$	0	0
$190$	0	0
$191$	37256.7i	1.02126i	0.859799	+	0.510632i	$0.170589\pi$
$191$	37256.7i	1.02126i	−0.859799	+	0.510632i	$0.829411\pi$
$192$	0	0
$193$	6599.00	0.177159	0.0885796	−	0.996069i	$-0.471767\pi$
$193$	6599.00	0.177159	0.0885796	+	0.996069i	$0.471767\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 2424.99i	− 0.0624854i	−0.999512	−	0.0312427i	$-0.990054\pi$
$197$	− 2424.99i	− 0.0624854i	0.999512	−	0.0312427i	$-0.00994647\pi$
$198$	0	0
$199$	−57871.0	−1.46135	−0.730676	−	0.682724i	$-0.760795\pi$
$199$	−57871.0	−1.46135	−0.730676	+	0.682724i	$0.760795\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 13668.2i	− 0.331679i
$204$	0	0
$205$	97200.0	2.31291
$206$	0	0
$207$	0	0
$208$	0	0
$209$	59743.1i	1.36771i
$210$	0	0
$211$	−1231.00	−0.0276499	−0.0138249	−	0.999904i	$-0.504401\pi$
$211$	−1231.00	−0.0276499	−0.0138249	+	0.999904i	$0.504401\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 13139.1i	− 0.284241i
$216$	0	0
$217$	24118.0	0.512179
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 53129.4i	− 1.08780i
$222$	0	0
$223$	−60058.0	−1.20771	−0.603853	−	0.797096i	$-0.706369\pi$
$223$	−60058.0	−1.20771	−0.603853	+	0.797096i	$0.706369\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	37036.3i	0.718746i	0.933194	+	0.359373i	$0.117009\pi$
$227$	37036.3i	0.718746i	−0.933194	+	0.359373i	$0.882991\pi$
$228$	0	0
$229$	−3118.00	−0.0594573	−0.0297286	−	0.999558i	$-0.509464\pi$
$229$	−3118.00	−0.0594573	−0.0297286	+	0.999558i	$0.509464\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	36154.5i	0.665963i	0.942933	+	0.332982i	$0.108055\pi$
$233$	36154.5i	0.665963i	−0.942933	+	0.332982i	$0.891945\pi$
$234$	0	0
$235$	−145800.	−2.64011
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 71427.1i	− 1.25045i	−0.780443	−	0.625226i	$-0.785007\pi$
$239$	− 71427.1i	− 1.25045i	0.780443	−	0.625226i	$-0.214993\pi$
$240$	0	0
$241$	−54121.0	−0.931819	−0.465910	−	0.884832i	$-0.654273\pi$
$241$	−54121.0	−0.931819	−0.465910	+	0.884832i	$0.654273\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 63490.8i	− 1.05774i
$246$	0	0
$247$	65311.0	1.07051
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 76277.1i	− 1.21073i	−0.795949	−	0.605364i	$-0.793027\pi$
$251$	− 76277.1i	− 1.21073i	0.795949	−	0.605364i	$-0.206973\pi$
$252$	0	0
$253$	48600.0	0.759268
$254$	0	0
$255$	0	0
$256$	0	0
$257$	25131.8i	0.380502i	0.981735	+	0.190251i	$0.0609302\pi$
$257$	25131.8i	0.380502i	−0.981735	+	0.190251i	$0.939070\pi$
$258$	0	0
$259$	−33449.0	−0.498636
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 41445.4i	− 0.599190i	−0.954066	−	0.299595i	$-0.903148\pi$
$263$	− 41445.4i	− 0.599190i	0.954066	−	0.299595i	$-0.0968515\pi$
$264$	0	0
$265$	−136080.	−1.93777
$266$	0	0
$267$	0	0
$268$	0	0
$269$	91047.5i	1.25824i	0.777308	+	0.629120i	$0.216585\pi$
$269$	91047.5i	1.25824i	−0.777308	+	0.629120i	$0.783415\pi$
$270$	0	0
$271$	−67231.0	−0.915442	−0.457721	−	0.889096i	$-0.651334\pi$
$271$	−67231.0	−0.915442	−0.457721	+	0.889096i	$0.651334\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	290779.i	3.84501i
$276$	0	0
$277$	34442.0	0.448879	0.224439	−	0.974488i	$-0.427945\pi$
$277$	34442.0	0.448879	0.224439	+	0.974488i	$0.427945\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 68340.8i	− 0.865500i	−0.901514	−	0.432750i	$-0.857543\pi$
$281$	− 68340.8i	− 0.865500i	0.901514	−	0.432750i	$-0.142457\pi$
$282$	0	0
$283$	−83578.0	−1.04356	−0.521782	−	0.853079i	$-0.674733\pi$
$283$	−83578.0	−1.04356	−0.521782	+	0.853079i	$0.674733\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	68340.8i	0.829690i
$288$	0	0
$289$	34921.0	0.418110
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 89724.8i	− 1.04515i	−0.852594	−	0.522573i	$-0.824972\pi$
$293$	− 89724.8i	− 1.04515i	0.852594	−	0.522573i	$-0.175028\pi$
$294$	0	0
$295$	−126360.	−1.45200
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 53129.4i	− 0.594282i
$300$	0	0
$301$	9238.00	0.101964
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 116444.i	− 1.25175i
$306$	0	0
$307$	98102.0	1.04088	0.520441	−	0.853898i	$-0.325768\pi$
$307$	98102.0	1.04088	0.520441	+	0.853898i	$0.325768\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 95456.6i	− 0.986928i	−0.869766	−	0.493464i	$-0.835731\pi$
$311$	− 95456.6i	− 0.986928i	0.869766	−	0.493464i	$-0.164269\pi$
$312$	0	0
$313$	30839.0	0.314783	0.157392	−	0.987536i	$-0.449692\pi$
$313$	30839.0	0.314783	0.157392	+	0.987536i	$0.449692\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7495.44i	0.0745896i	0.999304	+	0.0372948i	$0.0118741\pi$
$317$	7495.44i	0.0745896i	−0.999304	+	0.0372948i	$0.988126\pi$
$318$	0	0
$319$	97200.0	0.955179
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 59743.1i	− 0.572641i
$324$	0	0
$325$	317879.	3.00951
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 102511.i	− 0.947064i
$330$	0	0
$331$	−191551.	−1.74835	−0.874175	−	0.485611i	$-0.838597\pi$
$331$	−191551.	−1.74835	−0.874175	+	0.485611i	$0.838597\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	247305.i	2.20366i
$336$	0	0
$337$	−155401.	−1.36834	−0.684170	−	0.729323i	$-0.739835\pi$
$337$	−155401.	−1.36834	−0.684170	+	0.729323i	$0.739835\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	171513.i	1.47499i
$342$	0	0
$343$	119071.	1.01209
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 193118.i	− 1.60385i	−0.597426	−	0.801924i	$-0.703810\pi$
$347$	− 193118.i	− 1.60385i	0.597426	−	0.801924i	$-0.296190\pi$
$348$	0	0
$349$	207239.	1.70146	0.850728	−	0.525606i	$-0.176162\pi$
$349$	207239.	1.70146	0.850728	+	0.525606i	$0.176162\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	115518.i	0.927043i	0.886086	+	0.463522i	$0.153414\pi$
$353$	115518.i	0.927043i	−0.886086	+	0.463522i	$0.846586\pi$
$354$	0	0
$355$	194400.	1.54255
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 94574.8i	− 0.733815i	−0.930257	−	0.366907i	$-0.880417\pi$
$359$	− 94574.8i	− 0.733815i	0.930257	−	0.366907i	$-0.119583\pi$
$360$	0	0
$361$	−56880.0	−0.436461
$362$	0	0
$363$	0	0
$364$	0	0
$365$	317410.i	2.38251i
$366$	0	0
$367$	62249.0	0.462168	0.231084	−	0.972934i	$-0.425773\pi$
$367$	62249.0	0.462168	0.231084	+	0.972934i	$0.425773\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 95677.1i	− 0.695120i
$372$	0	0
$373$	34679.0	0.249258	0.124629	−	0.992203i	$-0.460226\pi$
$373$	34679.0	0.249258	0.124629	+	0.992203i	$0.460226\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 106259.i	− 0.747623i
$378$	0	0
$379$	198089.	1.37906	0.689528	−	0.724259i	$-0.257818\pi$
$379$	198089.	1.37906	0.689528	+	0.724259i	$0.257818\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 8818.16i	− 0.0601147i	−0.999548	−	0.0300573i	$-0.990431\pi$
$383$	− 8818.16i	− 0.0601147i	0.999548	−	0.0300573i	$-0.00956899\pi$
$384$	0	0
$385$	−301320.	−2.03286
$386$	0	0
$387$	0	0
$388$	0	0
$389$	62388.5i	0.412292i	0.978521	+	0.206146i	$0.0660923\pi$
$389$	62388.5i	0.412292i	−0.978521	+	0.206146i	$0.933908\pi$
$390$	0	0
$391$	−48600.0	−0.317894
$392$	0	0
$393$	0	0
$394$	0	0
$395$	14505.9i	0.0929715i
$396$	0	0
$397$	202682.	1.28598	0.642990	−	0.765875i	$-0.277694\pi$
$397$	202682.	1.28598	0.642990	+	0.765875i	$0.277694\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 40563.6i	− 0.252259i	−0.992014	−	0.126130i	$-0.959744\pi$
$401$	− 40563.6i	− 0.252259i	0.992014	−	0.126130i	$-0.0402555\pi$
$402$	0	0
$403$	187498.	1.15448
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 237870.i	− 1.43599i
$408$	0	0
$409$	−105001.	−0.627692	−0.313846	−	0.949474i	$-0.601618\pi$
$409$	−105001.	−0.627692	−0.313846	+	0.949474i	$0.601618\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 88843.0i	− 0.520862i
$414$	0	0
$415$	−58320.0	−0.338627
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 19179.5i	− 0.109247i	−0.998507	−	0.0546235i	$-0.982604\pi$
$419$	− 19179.5i	− 0.109247i	0.998507	−	0.0546235i	$-0.0173959\pi$
$420$	0	0
$421$	−252121.	−1.42248	−0.711238	−	0.702951i	$-0.751865\pi$
$421$	−252121.	−1.42248	−0.711238	+	0.702951i	$0.751865\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 290779.i	− 1.60985i
$426$	0	0
$427$	81871.0	0.449029
$428$	0	0
$429$	0	0
$430$	0	0
$431$	334870.i	1.80269i	0.433100	+	0.901346i	$0.357420\pi$
$431$	334870.i	1.80269i	−0.433100	+	0.901346i	$0.642580\pi$
$432$	0	0
$433$	−22678.0	−0.120956	−0.0604782	−	0.998170i	$-0.519263\pi$
$433$	−22678.0	−0.120956	−0.0604782	+	0.998170i	$0.519263\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 59743.1i	− 0.312842i
$438$	0	0
$439$	−124858.	−0.647869	−0.323935	−	0.946079i	$-0.605006\pi$
$439$	−124858.	−0.647869	−0.323935	+	0.946079i	$0.605006\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	88622.5i	0.451582i	0.974176	+	0.225791i	$0.0724967\pi$
$443$	88622.5i	0.451582i	−0.974176	+	0.225791i	$0.927503\pi$
$444$	0	0
$445$	−359640.	−1.81613
$446$	0	0
$447$	0	0
$448$	0	0
$449$	73411.2i	0.364141i	0.983285	+	0.182071i	$0.0582799\pi$
$449$	73411.2i	0.364141i	−0.983285	+	0.182071i	$0.941720\pi$
$450$	0	0
$451$	−486000.	−2.38937
$452$	0	0
$453$	0	0
$454$	0	0
$455$	329402.i	1.59112i
$456$	0	0
$457$	33002.0	0.158018	0.0790092	−	0.996874i	$-0.474824\pi$
$457$	33002.0	0.158018	0.0790092	+	0.996874i	$0.474824\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	182316.i	0.857871i	0.903335	+	0.428935i	$0.141111\pi$
$461$	182316.i	0.857871i	−0.903335	+	0.428935i	$0.858889\pi$
$462$	0	0
$463$	63929.0	0.298219	0.149110	−	0.988821i	$-0.452359\pi$
$463$	63929.0	0.298219	0.149110	+	0.988821i	$0.452359\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	337074.i	1.54558i	0.634661	+	0.772791i	$0.281140\pi$
$467$	337074.i	1.54558i	−0.634661	+	0.772791i	$0.718860\pi$
$468$	0	0
$469$	−173879.	−0.790499
$470$	0	0
$471$	0	0
$472$	0	0
$473$	65695.3i	0.293638i
$474$	0	0
$475$	357449.	1.58426
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 133595.i	− 0.582264i	−0.956683	−	0.291132i	$-0.905968\pi$
$479$	− 133595.i	− 0.582264i	0.956683	−	0.291132i	$-0.0940318\pi$
$480$	0	0
$481$	−260039.	−1.12395
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 703734.i	− 2.99175i
$486$	0	0
$487$	319769.	1.34827	0.674137	−	0.738606i	$-0.264516\pi$
$487$	319769.	1.34827	0.674137	+	0.738606i	$0.264516\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 233902.i	− 0.970221i	−0.874453	−	0.485110i	$-0.838779\pi$
$491$	− 233902.i	− 0.970221i	0.874453	−	0.485110i	$-0.161221\pi$
$492$	0	0
$493$	−97200.0	−0.399919
$494$	0	0
$495$	0	0
$496$	0	0
$497$	136682.i	0.553346i
$498$	0	0
$499$	36302.0	0.145791	0.0728953	−	0.997340i	$-0.476776\pi$
$499$	36302.0	0.145791	0.0728953	+	0.997340i	$0.476776\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 457001.i	− 1.80626i	−0.429362	−	0.903132i	$-0.641262\pi$
$503$	− 457001.i	− 1.80626i	0.429362	−	0.903132i	$-0.358738\pi$
$504$	0	0
$505$	−855360.	−3.35402
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 140429.i	− 0.542028i	−0.962575	−	0.271014i	$-0.912641\pi$
$509$	− 140429.i	− 0.542028i	0.962575	−	0.271014i	$-0.0873590\pi$
$510$	0	0
$511$	−223169.	−0.854657
$512$	0	0
$513$	0	0
$514$	0	0
$515$	72705.8i	0.274129i
$516$	0	0
$517$	729000.	2.72738
$518$	0	0
$519$	0	0
$520$	0	0
$521$	300479.i	1.10698i	0.832857	+	0.553488i	$0.186704\pi$
$521$	300479.i	1.10698i	−0.832857	+	0.553488i	$0.813296\pi$
$522$	0	0
$523$	−195391.	−0.714334	−0.357167	−	0.934041i	$-0.616257\pi$
$523$	−195391.	−0.714334	−0.357167	+	0.934041i	$0.616257\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 171513.i	− 0.617556i
$528$	0	0
$529$	231241.	0.826330
$530$	0	0
$531$	0	0
$532$	0	0
$533$	531294.i	1.87017i
$534$	0	0
$535$	184680.	0.645227
$536$	0	0
$537$	0	0
$538$	0	0
$539$	317454.i	1.09271i
$540$	0	0
$541$	60119.0	0.205408	0.102704	−	0.994712i	$-0.467251\pi$
$541$	60119.0	0.205408	0.102704	+	0.994712i	$0.467251\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 142766.i	− 0.480653i
$546$	0	0
$547$	311369.	1.04064	0.520320	−	0.853971i	$-0.325813\pi$
$547$	311369.	1.04064	0.520320	+	0.853971i	$0.325813\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 119486.i	− 0.393563i
$552$	0	0
$553$	−10199.0	−0.0333509
$554$	0	0
$555$	0	0
$556$	0	0
$557$	256829.i	0.827816i	0.910319	+	0.413908i	$0.135836\pi$
$557$	256829.i	0.827816i	−0.910319	+	0.413908i	$0.864164\pi$
$558$	0	0
$559$	71818.0	0.229832
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 41004.5i	− 0.129364i	−0.997906	−	0.0646821i	$-0.979397\pi$
$563$	− 41004.5i	− 0.129364i	0.997906	−	0.0646821i	$-0.0206033\pi$
$564$	0	0
$565$	612360.	1.91827
$566$	0	0
$567$	0	0
$568$	0	0
$569$	30643.1i	0.0946473i	0.998880	+	0.0473237i	$0.0150692\pi$
$569$	30643.1i	0.0946473i	−0.998880	+	0.0473237i	$0.984931\pi$
$570$	0	0
$571$	−140911.	−0.432188	−0.216094	−	0.976373i	$-0.569332\pi$
$571$	−140911.	−0.432188	−0.216094	+	0.976373i	$0.569332\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 290779.i	− 0.879483i
$576$	0	0
$577$	−128161.	−0.384950	−0.192475	−	0.981302i	$-0.561651\pi$
$577$	−128161.	−0.384950	−0.192475	+	0.981302i	$0.561651\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 41004.5i	− 0.121473i
$582$	0	0
$583$	680400.	2.00183
$584$	0	0
$585$	0	0
$586$	0	0
$587$	308415.i	0.895075i	0.894265	+	0.447538i	$0.147699\pi$
$587$	308415.i	0.895075i	−0.894265	+	0.447538i	$0.852301\pi$
$588$	0	0
$589$	210838.	0.607741
$590$	0	0
$591$	0	0
$592$	0	0
$593$	428122.i	1.21747i	0.793374	+	0.608735i	$0.208323\pi$
$593$	428122.i	1.21747i	−0.793374	+	0.608735i	$0.791677\pi$
$594$	0	0
$595$	301320.	0.851126
$596$	0	0
$597$	0	0
$598$	0	0
$599$	684730.i	1.90838i	0.299194	+	0.954192i	$0.403282\pi$
$599$	684730.i	1.90838i	−0.299194	+	0.954192i	$0.596718\pi$
$600$	0	0
$601$	−373798.	−1.03488	−0.517438	−	0.855721i	$-0.673114\pi$
$601$	−373798.	−1.03488	−0.517438	+	0.855721i	$0.673114\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 1.49728e6i	− 4.09065i
$606$	0	0
$607$	195329.	0.530138	0.265069	−	0.964229i	$-0.414605\pi$
$607$	195329.	0.530138	0.265069	+	0.964229i	$0.414605\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 796941.i	− 2.13474i
$612$	0	0
$613$	−322201.	−0.857444	−0.428722	−	0.903436i	$-0.641036\pi$
$613$	−322201.	−0.857444	−0.428722	+	0.903436i	$0.641036\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	675692.i	1.77492i	0.460887	+	0.887459i	$0.347531\pi$
$617$	675692.i	1.77492i	−0.460887	+	0.887459i	$0.652469\pi$
$618$	0	0
$619$	249449.	0.651029	0.325515	−	0.945537i	$-0.394462\pi$
$619$	249449.	0.651029	0.325515	+	0.945537i	$0.394462\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 252861.i	− 0.651487i
$624$	0	0
$625$	524761.	1.34339
$626$	0	0
$627$	0	0
$628$	0	0
$629$	237870.i	0.601227i
$630$	0	0
$631$	−357751.	−0.898508	−0.449254	−	0.893404i	$-0.648310\pi$
$631$	−357751.	−0.898508	−0.449254	+	0.893404i	$0.648310\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 611011.i	− 1.51531i
$636$	0	0
$637$	347040.	0.855265
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 881.816i	− 0.00214616i	−0.999999	−	0.00107308i	$-0.999658\pi$
$641$	− 881.816i	− 0.00214616i	0.999999	−	0.00107308i	$-0.000341572\pi$
$642$	0	0
$643$	−527098.	−1.27488	−0.637440	−	0.770500i	$-0.720007\pi$
$643$	−527098.	−1.27488	−0.637440	+	0.770500i	$0.720007\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 22486.3i	− 0.0537168i	−0.999639	−	0.0268584i	$-0.991450\pi$
$647$	− 22486.3i	− 0.0537168i	0.999639	−	0.0268584i	$-0.00855031\pi$
$648$	0	0
$649$	631800.	1.50000
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 61286.2i	− 0.143726i	−0.997414	−	0.0718632i	$-0.977105\pi$
$653$	− 61286.2i	− 0.143726i	0.997414	−	0.0718632i	$-0.0228945\pi$
$654$	0	0
$655$	−1.18584e6	−2.76403
$656$	0	0
$657$	0	0
$658$	0	0
$659$	330240.i	0.760430i	0.924898	+	0.380215i	$0.124150\pi$
$659$	330240.i	0.760430i	−0.924898	+	0.380215i	$0.875850\pi$
$660$	0	0
$661$	−367801.	−0.841802	−0.420901	−	0.907107i	$-0.638286\pi$
$661$	−367801.	−0.841802	−0.420901	+	0.907107i	$0.638286\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	370407.i	0.837598i
$666$	0	0
$667$	−97200.0	−0.218481
$668$	0	0
$669$	0	0
$670$	0	0
$671$	582219.i	1.29313i
$672$	0	0
$673$	93479.0	0.206388	0.103194	−	0.994661i	$-0.467094\pi$
$673$	93479.0	0.206388	0.103194	+	0.994661i	$0.467094\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	401006.i	0.874930i	0.899235	+	0.437465i	$0.144124\pi$
$677$	401006.i	0.874930i	−0.899235	+	0.437465i	$0.855876\pi$
$678$	0	0
$679$	494791.	1.07320
$680$	0	0
$681$	0	0
$682$	0	0
$683$	616390.i	1.32134i	0.750677	+	0.660669i	$0.229727\pi$
$683$	616390.i	1.32134i	−0.750677	+	0.660669i	$0.770273\pi$
$684$	0	0
$685$	−223560.	−0.476445
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 743812.i	− 1.56684i
$690$	0	0
$691$	−19738.0	−0.0413378	−0.0206689	−	0.999786i	$-0.506580\pi$
$691$	−19738.0	−0.0413378	−0.0206689	+	0.999786i	$0.506580\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.12030e6i	2.31935i
$696$	0	0
$697$	486000.	1.00039
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 215825.i	− 0.439202i	−0.975590	−	0.219601i	$-0.929524\pi$
$701$	− 215825.i	− 0.439202i	0.975590	−	0.219601i	$-0.0704756\pi$
$702$	0	0
$703$	−292409.	−0.591671
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 601399.i	− 1.20316i
$708$	0	0
$709$	−688081.	−1.36882	−0.684411	−	0.729096i	$-0.739941\pi$
$709$	−688081.	−1.36882	−0.684411	+	0.729096i	$0.739941\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 171513.i	− 0.337379i
$714$	0	0
$715$	−2.34252e6	−4.58217
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 665330.i	− 1.28700i	−0.765445	−	0.643502i	$-0.777481\pi$
$719$	− 665330.i	− 1.28700i	0.765445	−	0.643502i	$-0.222519\pi$
$720$	0	0
$721$	−51119.0	−0.0983358
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 581558.i	− 1.10641i
$726$	0	0
$727$	−355258.	−0.672164	−0.336082	−	0.941833i	$-0.609102\pi$
$727$	−355258.	−0.672164	−0.336082	+	0.941833i	$0.609102\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 65695.3i	− 0.122942i
$732$	0	0
$733$	482522.	0.898068	0.449034	−	0.893515i	$-0.351768\pi$
$733$	482522.	0.898068	0.449034	+	0.893515i	$0.351768\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 1.23653e6i	− 2.27650i
$738$	0	0
$739$	571142.	1.04582	0.522908	−	0.852389i	$-0.324847\pi$
$739$	571142.	1.04582	0.522908	+	0.852389i	$0.324847\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	180552.i	0.327058i	0.986539	+	0.163529i	$0.0522877\pi$
$743$	180552.i	0.327058i	−0.986539	+	0.163529i	$0.947712\pi$
$744$	0	0
$745$	−272160.	−0.490356
$746$	0	0
$747$	0	0
$748$	0	0
$749$	129847.i	0.231457i
$750$	0	0
$751$	−835711.	−1.48175	−0.740877	−	0.671640i	$-0.765590\pi$
$751$	−835711.	−1.48175	−0.740877	+	0.671640i	$0.765590\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	554177.i	0.972199i
$756$	0	0
$757$	406079.	0.708629	0.354314	−	0.935126i	$-0.384714\pi$
$757$	406079.	0.708629	0.354314	+	0.935126i	$0.384714\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	330020.i	0.569863i	0.958548	+	0.284932i	$0.0919709\pi$
$761$	330020.i	0.569863i	−0.958548	+	0.284932i	$0.908029\pi$
$762$	0	0
$763$	100378.	0.172421
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 690683.i	− 1.17405i
$768$	0	0
$769$	358319.	0.605923	0.302961	−	0.953003i	$-0.402025\pi$
$769$	358319.	0.605923	0.302961	+	0.953003i	$0.402025\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	503958.i	0.843403i	0.906735	+	0.421702i	$0.138567\pi$
$773$	503958.i	0.843403i	−0.906735	+	0.421702i	$0.861433\pi$
$774$	0	0
$775$	1.02618e6	1.70852
$776$	0	0
$777$	0	0
$778$	0	0
$779$	597431.i	0.984493i
$780$	0	0
$781$	−972000.	−1.59354
$782$	0	0
$783$	0	0
$784$	0	0
$785$	925995.i	1.50269i
$786$	0	0
$787$	182369.	0.294443	0.147222	−	0.989104i	$-0.452967\pi$
$787$	182369.	0.294443	0.147222	+	0.989104i	$0.452967\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	430547.i	0.688125i
$792$	0	0
$793$	636481.	1.01214
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.19839e6i	1.88660i	0.331935	+	0.943302i	$0.392299\pi$
$797$	1.19839e6i	1.88660i	−0.331935	+	0.943302i	$0.607701\pi$
$798$	0	0
$799$	−729000.	−1.14192
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 1.58705e6i	− 2.46127i
$804$	0	0
$805$	301320.	0.464982
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 485881.i	− 0.742391i	−0.928555	−	0.371195i	$-0.878948\pi$
$809$	− 485881.i	− 0.742391i	0.928555	−	0.371195i	$-0.121052\pi$
$810$	0	0
$811$	664742.	1.01067	0.505337	−	0.862922i	$-0.331368\pi$
$811$	664742.	1.01067	0.505337	+	0.862922i	$0.331368\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 742093.i	− 1.11723i
$816$	0	0
$817$	80758.0	0.120988
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.03988e6i	1.54276i	0.636376	+	0.771379i	$0.280433\pi$
$821$	1.03988e6i	1.54276i	−0.636376	+	0.771379i	$0.719567\pi$
$822$	0	0
$823$	−745711.	−1.10096	−0.550479	−	0.834849i	$-0.685555\pi$
$823$	−745711.	−1.10096	−0.550479	+	0.834849i	$0.685555\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	383811.i	0.561185i	0.959827	+	0.280592i	$0.0905309\pi$
$827$	383811.i	0.561185i	−0.959827	+	0.280592i	$0.909469\pi$
$828$	0	0
$829$	−459001.	−0.667889	−0.333945	−	0.942593i	$-0.608380\pi$
$829$	−459001.	−0.667889	−0.333945	+	0.942593i	$0.608380\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 317454.i	− 0.457500i
$834$	0	0
$835$	1.31220e6	1.88203
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 1.08331e6i	− 1.53897i	−0.638667	−	0.769484i	$-0.720514\pi$
$839$	− 1.08331e6i	− 1.53897i	0.638667	−	0.769484i	$-0.279486\pi$
$840$	0	0
$841$	512881.	0.725145
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.30156e6i	1.82285i
$846$	0	0
$847$	1.05273e6	1.46740
$848$	0	0
$849$	0	0
$850$	0	0
$851$	237870.i	0.328458i
$852$	0	0
$853$	1.11984e6	1.53907	0.769533	−	0.638606i	$-0.220489\pi$
$853$	1.11984e6	1.53907	0.769533	+	0.638606i	$0.220489\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1.07405e6i	− 1.46239i	−0.682167	−	0.731196i	$-0.738963\pi$
$857$	− 1.07405e6i	− 1.46239i	0.682167	−	0.731196i	$-0.261037\pi$
$858$	0	0
$859$	−128311.	−0.173891	−0.0869456	−	0.996213i	$-0.527711\pi$
$859$	−128311.	−0.173891	−0.0869456	+	0.996213i	$0.527711\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 218029.i	− 0.292747i	−0.989229	−	0.146374i	$-0.953240\pi$
$863$	− 218029.i	− 0.292747i	0.989229	−	0.146374i	$-0.0467602\pi$
$864$	0	0
$865$	1.10808e6	1.48094
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 72529.4i	− 0.0960449i
$870$	0	0
$871$	−1.35177e6	−1.78183
$872$	0	0
$873$	0	0
$874$	0	0
$875$	948570.i	1.23895i
$876$	0	0
$877$	817679.	1.06312	0.531562	−	0.847020i	$-0.321605\pi$
$877$	817679.	1.06312	0.531562	+	0.847020i	$0.321605\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 1.15099e6i	− 1.48293i	−0.670993	−	0.741464i	$-0.734132\pi$
$881$	− 1.15099e6i	− 1.48293i	0.670993	−	0.741464i	$-0.265868\pi$
$882$	0	0
$883$	−757351.	−0.971350	−0.485675	−	0.874139i	$-0.661426\pi$
$883$	−757351.	−0.971350	−0.485675	+	0.874139i	$0.661426\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 811712.i	− 1.03170i	−0.856678	−	0.515851i	$-0.827476\pi$
$887$	− 811712.i	− 1.03170i	0.856678	−	0.515851i	$-0.172524\pi$
$888$	0	0
$889$	429598.	0.543574
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 896146.i	− 1.12377i
$894$	0	0
$895$	972000.	1.21345
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 343027.i	− 0.424432i
$900$	0	0
$901$	−680400.	−0.838136
$902$	0	0
$903$	0	0
$904$	0	0
$905$	444391.i	0.542586i
$906$	0	0
$907$	706289.	0.858554	0.429277	−	0.903173i	$-0.358768\pi$
$907$	706289.	0.858554	0.429277	+	0.903173i	$0.358768\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	284386.i	0.342666i	0.985213	+	0.171333i	$0.0548074\pi$
$911$	284386.i	0.342666i	−0.985213	+	0.171333i	$0.945193\pi$
$912$	0	0
$913$	291600.	0.349821
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 833757.i	− 0.991519i
$918$	0	0
$919$	1.47042e6	1.74105	0.870524	−	0.492125i	$-0.163780\pi$
$919$	1.47042e6	1.74105	0.870524	+	0.492125i	$0.163780\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1.06259e6i	1.24727i
$924$	0	0
$925$	−1.42320e6	−1.66335
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.34918e6i	1.56328i	0.623727	+	0.781642i	$0.285618\pi$
$929$	1.34918e6i	1.56328i	−0.623727	+	0.781642i	$0.714382\pi$
$930$	0	0
$931$	390240.	0.450228
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.14281e6i	2.45110i
$936$	0	0
$937$	183959.	0.209528	0.104764	−	0.994497i	$-0.466591\pi$
$937$	183959.	0.209528	0.104764	+	0.994497i	$0.466591\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 109566.i	− 0.123736i	−0.998084	−	0.0618679i	$-0.980294\pi$
$941$	− 109566.i	− 0.123736i	0.998084	−	0.0618679i	$-0.0197057\pi$
$942$	0	0
$943$	486000.	0.546529
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.53899e6i	1.71607i	0.513589	+	0.858037i	$0.328316\pi$
$947$	1.53899e6i	1.71607i	−0.513589	+	0.858037i	$0.671684\pi$
$948$	0	0
$949$	−1.73496e6	−1.92645
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 929214.i	− 1.02313i	−0.859245	−	0.511564i	$-0.829066\pi$
$953$	− 929214.i	− 1.02313i	0.859245	−	0.511564i	$-0.170934\pi$
$954$	0	0
$955$	−1.64268e6	−1.80113
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 157184.i	− 0.170911i
$960$	0	0
$961$	−318237.	−0.344591
$962$	0	0
$963$	0	0
$964$	0	0
$965$	290955.i	0.312444i
$966$	0	0
$967$	−561031.	−0.599976	−0.299988	−	0.953943i	$-0.596983\pi$
$967$	−561031.	−0.599976	−0.299988	+	0.953943i	$0.596983\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 994027.i	− 1.05429i	−0.849775	−	0.527145i	$-0.823263\pi$
$971$	− 994027.i	− 1.05429i	0.849775	−	0.527145i	$-0.176737\pi$
$972$	0	0
$973$	−787679.	−0.832000
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 548049.i	− 0.574156i	−0.957907	−	0.287078i	$-0.907316\pi$
$977$	− 548049.i	− 0.574156i	0.957907	−	0.287078i	$-0.0926839\pi$
$978$	0	0
$979$	1.79820e6	1.87617
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.47374e6i	1.52515i	0.646900	+	0.762575i	$0.276065\pi$
$983$	1.47374e6i	1.52515i	−0.646900	+	0.762575i	$0.723935\pi$
$984$	0	0
$985$	106920.	0.110201
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 65695.3i	− 0.0671648i
$990$	0	0
$991$	−905671.	−0.922196	−0.461098	−	0.887349i	$-0.652544\pi$
$991$	−905671.	−0.922196	−0.461098	+	0.887349i	$0.652544\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 2.55158e6i	− 2.57729i
$996$	0	0
$997$	−770518.	−0.775162	−0.387581	−	0.921836i	$-0.626689\pi$
$997$	−770518.	−0.775162	−0.387581	+	0.921836i	$0.626689\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.5.c.b.53.2	yes	2
3.2	odd	2	inner	108.5.c.b.53.1	✓	2
4.3	odd	2		432.5.e.f.161.2		2
9.2	odd	6		324.5.g.e.53.2		4
9.4	even	3		324.5.g.e.269.2		4
9.5	odd	6		324.5.g.e.269.1		4
9.7	even	3		324.5.g.e.53.1		4
12.11	even	2		432.5.e.f.161.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.5.c.b.53.1	✓	2	3.2	odd	2	inner
108.5.c.b.53.2	yes	2	1.1	even	1	trivial
324.5.g.e.53.1		4	9.7	even	3
324.5.g.e.53.2		4	9.2	odd	6
324.5.g.e.269.1		4	9.5	odd	6
324.5.g.e.269.2		4	9.4	even	3
432.5.e.f.161.1		2	12.11	even	2
432.5.e.f.161.2		2	4.3	odd	2

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:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	108.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+44.0908i q^{5} -31.0000 q^{7} +O(q^{10})\)	\(q+44.0908i q^{5} -31.0000 q^{7} -220.454i q^{11} -241.000 q^{13} +220.454i q^{17} -271.000 q^{19} +220.454i q^{23} -1319.00 q^{25} +440.908i q^{29} -778.000 q^{31} -1366.82i q^{35} +1079.00 q^{37} -2204.54i q^{41} -298.000 q^{43} +3306.81i q^{47} -1440.00 q^{49} +3086.36i q^{53} +9720.00 q^{55} +2865.90i q^{59} -2641.00 q^{61} -10625.9i q^{65} +5609.00 q^{67} -4409.08i q^{71} +7199.00 q^{73} +6834.08i q^{77} +329.000 q^{79} +1322.72i q^{83} -9720.00 q^{85} +8156.80i q^{89} +7471.00 q^{91} -11948.6i q^{95} -15961.0 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 62 q^{7}+O(q^{10}) \) 2 * q - 62 * q^7	\( 2 q - 62 q^{7} - 482 q^{13} - 542 q^{19} - 2638 q^{25} - 1556 q^{31} + 2158 q^{37} - 596 q^{43} - 2880 q^{49} + 19440 q^{55} - 5282 q^{61} + 11218 q^{67} + 14398 q^{73} + 658 q^{79} - 19440 q^{85} + 14942 q^{91} - 31922 q^{97}+O(q^{100}) \) 2 * q - 62 * q^7 - 482 * q^13 - 542 * q^19 - 2638 * q^25 - 1556 * q^31 + 2158 * q^37 - 596 * q^43 - 2880 * q^49 + 19440 * q^55 - 5282 * q^61 + 11218 * q^67 + 14398 * q^73 + 658 * q^79 - 19440 * q^85 + 14942 * q^91 - 31922 * q^97

Introduction

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