LMFDB - Embedded newform 4600.2.e.v.4049.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4600,2,Mod(4049,4600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4600, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4600.4049");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4600 = 2^{3} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4600.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:	$36.7311849298$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 18x^{8} + 103x^{6} + 239x^{4} + 197x^{2} + 4 $ x^10 + 18x^8 + 103x^6 + 239x^4 + 197x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.8
Root			$1.64975i$ of defining polynomial
Character	$\chi$	$=$	4600.4049
Dual form			4600.2.e.v.4049.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.51466i q^{3} +3.49880i q^{7} +0.705809 q^{9} +O(q^{10})$	$q+1.51466i q^{3} +3.49880i q^{7} +0.705809 q^{9} -4.35556 q^{11} +3.67906i q^{13} +5.18556i q^{17} -2.08538 q^{19} -5.29949 q^{21} +1.00000i q^{23} +5.61304i q^{27} +1.24902 q^{29} +4.82185 q^{31} -6.59718i q^{33} -1.04471i q^{37} -5.57253 q^{39} +9.05578 q^{41} +10.4964i q^{43} -12.8597i q^{47} -5.24162 q^{49} -7.85436 q^{51} +9.37352i q^{53} -3.15864i q^{57} -14.1570 q^{59} -9.29949 q^{61} +2.46949i q^{63} -4.44274i q^{67} -1.51466 q^{69} +2.45044 q^{71} -13.2197i q^{73} -15.2392i q^{77} -7.72666 q^{79} -6.38441 q^{81} -2.97086i q^{83} +1.89184i q^{87} +14.6241 q^{89} -12.8723 q^{91} +7.30345i q^{93} -9.37991i q^{97} -3.07419 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 8 q^{9}+O(q^{10}) $ 10 * q - 8 * q^9	$ 10 q - 8 q^{9} - 8 q^{11} - 8 q^{19} - 12 q^{21} + 22 q^{29} + 8 q^{31} - 62 q^{39} - 16 q^{41} + 4 q^{49} - 10 q^{51} - 46 q^{59} - 52 q^{61} - 6 q^{69} - 4 q^{71} - 86 q^{79} - 6 q^{81} - 30 q^{89} - 38 q^{91} - 74 q^{99}+O(q^{100}) $ 10 * q - 8 * q^9 - 8 * q^11 - 8 * q^19 - 12 * q^21 + 22 * q^29 + 8 * q^31 - 62 * q^39 - 16 * q^41 + 4 * q^49 - 10 * q^51 - 46 * q^59 - 52 * q^61 - 6 * q^69 - 4 * q^71 - 86 * q^79 - 6 * q^81 - 30 * q^89 - 38 * q^91 - 74 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times$.

$n$	$1151$	$1201$	$2301$	$2577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	1.51466i	0.874489i	0.899343	+	0.437244i	$0.144046\pi$
$3$	1.51466i	0.874489i	−0.899343	+	0.437244i	$0.855954\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.49880i	1.32242i	0.750199	+	0.661212i	$0.229957\pi$
$7$	3.49880i	1.32242i	−0.750199	+	0.661212i	$0.770043\pi$
$8$	0	0
$9$	0.705809	0.235270
$10$	0	0
$11$	−4.35556	−1.31325	−0.656625	−	0.754217i	$-0.728016\pi$
$11$	−4.35556	−1.31325	−0.656625	+	0.754217i	$0.728016\pi$
$12$	0	0
$13$	3.67906i	1.02039i	0.860059	+	0.510194i	$0.170427\pi$
$13$	3.67906i	1.02039i	−0.860059	+	0.510194i	$0.829573\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.18556i	1.25768i	0.777533	+	0.628842i	$0.216471\pi$
$17$	5.18556i	1.25768i	−0.777533	+	0.628842i	$0.783529\pi$
$18$	0	0
$19$	−2.08538	−0.478419	−0.239209	−	0.970968i	$-0.576888\pi$
$19$	−2.08538	−0.478419	−0.239209	+	0.970968i	$0.576888\pi$
$20$	0	0
$21$	−5.29949	−1.15644
$22$	0	0
$23$	1.00000i	0.208514i
$23$	1.00000i	0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.61304i	1.08023i
$28$	0	0
$29$	1.24902	0.231937	0.115968	−	0.993253i	$-0.463003\pi$
$29$	1.24902	0.231937	0.115968	+	0.993253i	$0.463003\pi$
$30$	0	0
$31$	4.82185	0.866029	0.433015	−	0.901387i	$-0.357450\pi$
$31$	4.82185	0.866029	0.433015	+	0.901387i	$0.357450\pi$
$32$	0	0
$33$	− 6.59718i	− 1.14842i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 1.04471i	− 0.171749i	−0.996306	−	0.0858745i	$-0.972632\pi$
$37$	− 1.04471i	− 0.171749i	0.996306	−	0.0858745i	$-0.0273684\pi$
$38$	0	0
$39$	−5.57253	−0.892318
$40$	0	0
$41$	9.05578	1.41427	0.707137	−	0.707076i	$-0.249986\pi$
$41$	9.05578	1.41427	0.707137	+	0.707076i	$0.249986\pi$
$42$	0	0
$43$	10.4964i	1.60069i	0.599541	+	0.800344i	$0.295350\pi$
$43$	10.4964i	1.60069i	−0.599541	+	0.800344i	$0.704650\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 12.8597i	− 1.87577i	−0.346940	−	0.937887i	$-0.612779\pi$
$47$	− 12.8597i	− 1.87577i	0.346940	−	0.937887i	$-0.387221\pi$
$48$	0	0
$49$	−5.24162	−0.748804
$50$	0	0
$51$	−7.85436	−1.09983
$52$	0	0
$53$	9.37352i	1.28755i	0.765214	+	0.643776i	$0.222633\pi$
$53$	9.37352i	1.28755i	−0.765214	+	0.643776i	$0.777367\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 3.15864i	− 0.418372i
$58$	0	0
$59$	−14.1570	−1.84309	−0.921543	−	0.388276i	$-0.873071\pi$
$59$	−14.1570	−1.84309	−0.921543	+	0.388276i	$0.873071\pi$
$60$	0	0
$61$	−9.29949	−1.19068	−0.595339	−	0.803475i	$-0.702982\pi$
$61$	−9.29949	−1.19068	−0.595339	+	0.803475i	$0.702982\pi$
$62$	0	0
$63$	2.46949i	0.311126i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 4.44274i	− 0.542767i	−0.962471	−	0.271384i	$-0.912519\pi$
$67$	− 4.44274i	− 0.542767i	0.962471	−	0.271384i	$-0.0874812\pi$
$68$	0	0
$69$	−1.51466	−0.182343
$70$	0	0
$71$	2.45044	0.290813	0.145407	−	0.989372i	$-0.453551\pi$
$71$	2.45044	0.290813	0.145407	+	0.989372i	$0.453551\pi$
$72$	0	0
$73$	− 13.2197i	− 1.54725i	−0.633643	−	0.773625i	$-0.718441\pi$
$73$	− 13.2197i	− 1.54725i	0.633643	−	0.773625i	$-0.281559\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 15.2392i	− 1.73667i
$78$	0	0
$79$	−7.72666	−0.869318	−0.434659	−	0.900595i	$-0.643131\pi$
$79$	−7.72666	−0.869318	−0.434659	+	0.900595i	$0.643131\pi$
$80$	0	0
$81$	−6.38441	−0.709379
$82$	0	0
$83$	− 2.97086i	− 0.326095i	−0.986618	−	0.163047i	$-0.947868\pi$
$83$	− 2.97086i	− 0.326095i	0.986618	−	0.163047i	$-0.0521323\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.89184i	0.202826i
$88$	0	0
$89$	14.6241	1.55015	0.775076	−	0.631868i	$-0.217712\pi$
$89$	14.6241	1.55015	0.775076	+	0.631868i	$0.217712\pi$
$90$	0	0
$91$	−12.8723	−1.34939
$92$	0	0
$93$	7.30345i	0.757333i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 9.37991i	− 0.952385i	−0.879341	−	0.476193i	$-0.842017\pi$
$97$	− 9.37991i	− 0.952385i	0.879341	−	0.476193i	$-0.157983\pi$
$98$	0	0
$99$	−3.07419	−0.308968
$100$	0	0
$101$	−1.02674	−0.102165	−0.0510824	−	0.998694i	$-0.516267\pi$
$101$	−1.02674	−0.102165	−0.0510824	+	0.998694i	$0.516267\pi$
$102$	0	0
$103$	− 1.71127i	− 0.168617i	−0.996440	−	0.0843084i	$-0.973132\pi$
$103$	− 1.71127i	− 0.168617i	0.996440	−	0.0843084i	$-0.0268681\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 7.58418i	− 0.733191i	−0.930380	−	0.366595i	$-0.880523\pi$
$107$	− 7.58418i	− 0.733191i	0.930380	−	0.366595i	$-0.119477\pi$
$108$	0	0
$109$	8.95951	0.858165	0.429083	−	0.903265i	$-0.358837\pi$
$109$	8.95951	0.858165	0.429083	+	0.903265i	$0.358837\pi$
$110$	0	0
$111$	1.58238	0.150193
$112$	0	0
$113$	18.8103i	1.76952i	0.466047	+	0.884760i	$0.345678\pi$
$113$	18.8103i	1.76952i	−0.466047	+	0.884760i	$0.654322\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.59672i	0.240066i
$118$	0	0
$119$	−18.1433	−1.66319
$120$	0	0
$121$	7.97086	0.724624
$122$	0	0
$123$	13.7164i	1.23677i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	15.0385i	1.33445i	0.744854	+	0.667227i	$0.232519\pi$
$127$	15.0385i	1.33445i	−0.744854	+	0.667227i	$0.767481\pi$
$128$	0	0
$129$	−15.8985	−1.39978
$130$	0	0
$131$	20.3503	1.77801	0.889007	−	0.457893i	$-0.151396\pi$
$131$	20.3503	1.77801	0.889007	+	0.457893i	$0.151396\pi$
$132$	0	0
$133$	− 7.29633i	− 0.632672i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1.51494i	− 0.129430i	−0.997904	−	0.0647152i	$-0.979386\pi$
$137$	− 1.51494i	− 0.129430i	0.997904	−	0.0647152i	$-0.0206139\pi$
$138$	0	0
$139$	−13.3658	−1.13367	−0.566837	−	0.823830i	$-0.691833\pi$
$139$	−13.3658	−1.13367	−0.566837	+	0.823830i	$0.691833\pi$
$140$	0	0
$141$	19.4780	1.64034
$142$	0	0
$143$	− 16.0244i	− 1.34002i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 7.93927i	− 0.654820i
$148$	0	0
$149$	10.6307	0.870901	0.435450	−	0.900213i	$-0.356589\pi$
$149$	10.6307	0.870901	0.435450	+	0.900213i	$0.356589\pi$
$150$	0	0
$151$	−22.2918	−1.81408	−0.907041	−	0.421042i	$-0.861664\pi$
$151$	−22.2918	−1.81408	−0.907041	+	0.421042i	$0.861664\pi$
$152$	0	0
$153$	3.66002i	0.295895i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.95590i	0.235906i	0.993019	+	0.117953i	$0.0376332\pi$
$157$	2.95590i	0.235906i	−0.993019	+	0.117953i	$0.962367\pi$
$158$	0	0
$159$	−14.1977	−1.12595
$160$	0	0
$161$	−3.49880	−0.275744
$162$	0	0
$163$	4.40543i	0.345060i	0.985004	+	0.172530i	$0.0551941\pi$
$163$	4.40543i	0.345060i	−0.985004	+	0.172530i	$0.944806\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 14.9252i	− 1.15495i	−0.816409	−	0.577474i	$-0.804038\pi$
$167$	− 14.9252i	− 1.15495i	0.816409	−	0.577474i	$-0.195962\pi$
$168$	0	0
$169$	−0.535514	−0.0411934
$170$	0	0
$171$	−1.47188	−0.112557
$172$	0	0
$173$	18.2696i	1.38901i	0.719487	+	0.694506i	$0.244377\pi$
$173$	18.2696i	1.38901i	−0.719487	+	0.694506i	$0.755623\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 21.4430i	− 1.61176i
$178$	0	0
$179$	6.20356	0.463676	0.231838	−	0.972754i	$-0.425526\pi$
$179$	6.20356	0.463676	0.231838	+	0.972754i	$0.425526\pi$
$180$	0	0
$181$	5.74043	0.426683	0.213341	−	0.976978i	$-0.431565\pi$
$181$	5.74043	0.426683	0.213341	+	0.976978i	$0.431565\pi$
$182$	0	0
$183$	− 14.0856i	− 1.04123i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 22.5860i	− 1.65165i
$188$	0	0
$189$	−19.6389	−1.42852
$190$	0	0
$191$	−13.4671	−0.974445	−0.487222	−	0.873278i	$-0.661990\pi$
$191$	−13.4671	−0.974445	−0.487222	+	0.873278i	$0.661990\pi$
$192$	0	0
$193$	− 0.295578i	− 0.0212762i	−0.999943	−	0.0106381i	$-0.996614\pi$
$193$	− 0.295578i	− 0.0212762i	0.999943	−	0.0106381i	$-0.00338628\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 11.6206i	− 0.827932i	−0.910292	−	0.413966i	$-0.864143\pi$
$197$	− 11.6206i	− 0.827932i	0.910292	−	0.413966i	$-0.135857\pi$
$198$	0	0
$199$	−4.46949	−0.316833	−0.158417	−	0.987372i	$-0.550639\pi$
$199$	−4.46949	−0.316833	−0.158417	+	0.987372i	$0.550639\pi$
$200$	0	0
$201$	6.72924	0.474644
$202$	0	0
$203$	4.37007i	0.306719i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0.705809i	0.0490571i
$208$	0	0
$209$	9.08299	0.628283
$210$	0	0
$211$	−15.0878	−1.03869	−0.519343	−	0.854566i	$-0.673823\pi$
$211$	−15.0878	−1.03869	−0.519343	+	0.854566i	$0.673823\pi$
$212$	0	0
$213$	3.71158i	0.254313i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	16.8707i	1.14526i
$218$	0	0
$219$	20.0234	1.35305
$220$	0	0
$221$	−19.0780	−1.28333
$222$	0	0
$223$	− 25.2033i	− 1.68774i	−0.536548	−	0.843870i	$-0.680272\pi$
$223$	− 25.2033i	− 1.68774i	0.536548	−	0.843870i	$-0.319728\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	20.1385i	1.33664i	0.743875	+	0.668319i	$0.232986\pi$
$227$	20.1385i	1.33664i	−0.743875	+	0.668319i	$0.767014\pi$
$228$	0	0
$229$	−9.72411	−0.642587	−0.321294	−	0.946980i	$-0.604118\pi$
$229$	−9.72411	−0.642587	−0.321294	+	0.946980i	$0.604118\pi$
$230$	0	0
$231$	23.0822	1.51870
$232$	0	0
$233$	− 6.37957i	− 0.417940i	−0.977922	−	0.208970i	$-0.932989\pi$
$233$	− 6.37957i	− 0.417940i	0.977922	−	0.208970i	$-0.0670110\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 11.7033i	− 0.760208i
$238$	0	0
$239$	3.07706	0.199039	0.0995194	−	0.995036i	$-0.468269\pi$
$239$	3.07706	0.199039	0.0995194	+	0.995036i	$0.468269\pi$
$240$	0	0
$241$	−28.6341	−1.84448	−0.922242	−	0.386612i	$-0.873645\pi$
$241$	−28.6341	−1.84448	−0.922242	+	0.386612i	$0.873645\pi$
$242$	0	0
$243$	7.16891i	0.459886i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 7.67225i	− 0.488173i
$248$	0	0
$249$	4.49984	0.285166
$250$	0	0
$251$	0.728308	0.0459704	0.0229852	−	0.999736i	$-0.492683\pi$
$251$	0.728308	0.0459704	0.0229852	+	0.999736i	$0.492683\pi$
$252$	0	0
$253$	− 4.35556i	− 0.273831i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 29.1681i	− 1.81945i	−0.415206	−	0.909727i	$-0.636290\pi$
$257$	− 29.1681i	− 1.81945i	0.415206	−	0.909727i	$-0.363710\pi$
$258$	0	0
$259$	3.65523	0.227125
$260$	0	0
$261$	0.881568	0.0545677
$262$	0	0
$263$	2.79631i	0.172428i	0.996277	+	0.0862139i	$0.0274769\pi$
$263$	2.79631i	0.172428i	−0.996277	+	0.0862139i	$0.972523\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	22.1505i	1.35559i
$268$	0	0
$269$	0.662454	0.0403905	0.0201953	−	0.999796i	$-0.493571\pi$
$269$	0.662454	0.0403905	0.0201953	+	0.999796i	$0.493571\pi$
$270$	0	0
$271$	14.5266	0.882428	0.441214	−	0.897402i	$-0.354548\pi$
$271$	14.5266	0.882428	0.441214	+	0.897402i	$0.354548\pi$
$272$	0	0
$273$	− 19.4972i	− 1.18002i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.15432i	0.0693562i	0.999399	+	0.0346781i	$0.0110406\pi$
$277$	1.15432i	0.0693562i	−0.999399	+	0.0346781i	$0.988959\pi$
$278$	0	0
$279$	3.40330	0.203750
$280$	0	0
$281$	−14.2977	−0.852930	−0.426465	−	0.904504i	$-0.640241\pi$
$281$	−14.2977	−0.852930	−0.426465	+	0.904504i	$0.640241\pi$
$282$	0	0
$283$	31.2654i	1.85854i	0.369407	+	0.929268i	$0.379561\pi$
$283$	31.2654i	1.85854i	−0.369407	+	0.929268i	$0.620439\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	31.6844i	1.87027i
$288$	0	0
$289$	−9.89006	−0.581768
$290$	0	0
$291$	14.2074	0.832850
$292$	0	0
$293$	4.59477i	0.268429i	0.990952	+	0.134215i	$0.0428512\pi$
$293$	4.59477i	0.268429i	−0.990952	+	0.134215i	$0.957149\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 24.4479i	− 1.41861i
$298$	0	0
$299$	−3.67906	−0.212766
$300$	0	0
$301$	−36.7249	−2.11679
$302$	0	0
$303$	− 1.55517i	− 0.0893420i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 9.96103i	− 0.568506i	−0.958749	−	0.284253i	$-0.908254\pi$
$307$	− 9.96103i	− 0.568506i	0.958749	−	0.284253i	$-0.0917455\pi$
$308$	0	0
$309$	2.59200	0.147453
$310$	0	0
$311$	−12.7520	−0.723102	−0.361551	−	0.932352i	$-0.617753\pi$
$311$	−12.7520	−0.723102	−0.361551	+	0.932352i	$0.617753\pi$
$312$	0	0
$313$	− 19.5585i	− 1.10551i	−0.833343	−	0.552756i	$-0.813576\pi$
$313$	− 19.5585i	− 1.10551i	0.833343	−	0.552756i	$-0.186424\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 32.5493i	− 1.82815i	−0.405542	−	0.914076i	$-0.632917\pi$
$317$	− 32.5493i	− 1.82815i	0.405542	−	0.914076i	$-0.367083\pi$
$318$	0	0
$319$	−5.44017	−0.304591
$320$	0	0
$321$	11.4874	0.641167
$322$	0	0
$323$	− 10.8139i	− 0.601700i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	13.5706i	0.750456i
$328$	0	0
$329$	44.9934	2.48057
$330$	0	0
$331$	23.9704	1.31753	0.658767	−	0.752347i	$-0.271078\pi$
$331$	23.9704	1.31753	0.658767	+	0.752347i	$0.271078\pi$
$332$	0	0
$333$	− 0.737364i	− 0.0404073i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 5.88246i	− 0.320438i	−0.987082	−	0.160219i	$-0.948780\pi$
$337$	− 5.88246i	− 0.320438i	0.987082	−	0.160219i	$-0.0512200\pi$
$338$	0	0
$339$	−28.4911	−1.54743
$340$	0	0
$341$	−21.0018	−1.13731
$342$	0	0
$343$	6.15221i	0.332188i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 25.7454i	− 1.38209i	−0.722814	−	0.691043i	$-0.757151\pi$
$347$	− 25.7454i	− 1.38209i	0.722814	−	0.691043i	$-0.242849\pi$
$348$	0	0
$349$	17.7758	0.951517	0.475758	−	0.879576i	$-0.342174\pi$
$349$	17.7758	0.951517	0.475758	+	0.879576i	$0.342174\pi$
$350$	0	0
$351$	−20.6507	−1.10225
$352$	0	0
$353$	20.4426i	1.08805i	0.839068	+	0.544026i	$0.183101\pi$
$353$	20.4426i	1.08805i	−0.839068	+	0.544026i	$0.816899\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 27.4809i	− 1.45444i
$358$	0	0
$359$	14.5323	0.766987	0.383494	−	0.923543i	$-0.374721\pi$
$359$	14.5323	0.766987	0.383494	+	0.923543i	$0.374721\pi$
$360$	0	0
$361$	−14.6512	−0.771115
$362$	0	0
$363$	12.0731i	0.633675i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	12.2205i	0.637906i	0.947771	+	0.318953i	$0.103331\pi$
$367$	12.2205i	0.637906i	−0.947771	+	0.318953i	$0.896669\pi$
$368$	0	0
$369$	6.39165	0.332736
$370$	0	0
$371$	−32.7961	−1.70269
$372$	0	0
$373$	− 1.78846i	− 0.0926029i	−0.998928	−	0.0463015i	$-0.985257\pi$
$373$	− 1.78846i	− 0.0926029i	0.998928	−	0.0463015i	$-0.0147435\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.59522i	0.236666i
$378$	0	0
$379$	−7.02119	−0.360654	−0.180327	−	0.983607i	$-0.557716\pi$
$379$	−7.02119	−0.360654	−0.180327	+	0.983607i	$0.557716\pi$
$380$	0	0
$381$	−22.7782	−1.16697
$382$	0	0
$383$	13.0896i	0.668846i	0.942423	+	0.334423i	$0.108541\pi$
$383$	13.0896i	0.668846i	−0.942423	+	0.334423i	$0.891459\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	7.40846i	0.376593i
$388$	0	0
$389$	21.5395	1.09210	0.546048	−	0.837754i	$-0.316132\pi$
$389$	21.5395	1.09210	0.546048	+	0.837754i	$0.316132\pi$
$390$	0	0
$391$	−5.18556	−0.262245
$392$	0	0
$393$	30.8238i	1.55485i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	31.2045i	1.56611i	0.621953	+	0.783055i	$0.286340\pi$
$397$	31.2045i	1.56611i	−0.621953	+	0.783055i	$0.713660\pi$
$398$	0	0
$399$	11.0515	0.553265
$400$	0	0
$401$	13.7592	0.687100	0.343550	−	0.939134i	$-0.388371\pi$
$401$	13.7592	0.687100	0.343550	+	0.939134i	$0.388371\pi$
$402$	0	0
$403$	17.7399i	0.883687i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.55028i	0.225549i
$408$	0	0
$409$	−21.9730	−1.08649	−0.543247	−	0.839573i	$-0.682805\pi$
$409$	−21.9730	−1.08649	−0.543247	+	0.839573i	$0.682805\pi$
$410$	0	0
$411$	2.29462	0.113185
$412$	0	0
$413$	− 49.5326i	− 2.43734i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 20.2447i	− 0.991385i
$418$	0	0
$419$	13.3024	0.649867	0.324934	−	0.945737i	$-0.394658\pi$
$419$	13.3024	0.649867	0.324934	+	0.945737i	$0.394658\pi$
$420$	0	0
$421$	−7.32759	−0.357125	−0.178562	−	0.983929i	$-0.557145\pi$
$421$	−7.32759	−0.357125	−0.178562	+	0.983929i	$0.557145\pi$
$422$	0	0
$423$	− 9.07646i	− 0.441313i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 32.5371i	− 1.57458i
$428$	0	0
$429$	24.2715	1.17184
$430$	0	0
$431$	29.4301	1.41760	0.708800	−	0.705409i	$-0.249237\pi$
$431$	29.4301	1.41760	0.708800	+	0.705409i	$0.249237\pi$
$432$	0	0
$433$	10.5491i	0.506960i	0.967341	+	0.253480i	$0.0815752\pi$
$433$	10.5491i	0.506960i	−0.967341	+	0.253480i	$0.918425\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 2.08538i	− 0.0997572i
$438$	0	0
$439$	−1.04921	−0.0500761	−0.0250380	−	0.999686i	$-0.507971\pi$
$439$	−1.04921	−0.0500761	−0.0250380	+	0.999686i	$0.507971\pi$
$440$	0	0
$441$	−3.69958	−0.176171
$442$	0	0
$443$	9.57098i	0.454731i	0.973810	+	0.227365i	$0.0730112\pi$
$443$	9.57098i	0.454731i	−0.973810	+	0.227365i	$0.926989\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	16.1019i	0.761593i
$448$	0	0
$449$	−8.10079	−0.382300	−0.191150	−	0.981561i	$-0.561222\pi$
$449$	−8.10079	−0.382300	−0.191150	+	0.981561i	$0.561222\pi$
$450$	0	0
$451$	−39.4429	−1.85730
$452$	0	0
$453$	− 33.7645i	− 1.58639i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 34.5867i	− 1.61790i	−0.587880	−	0.808948i	$-0.700037\pi$
$457$	− 34.5867i	− 1.61790i	0.587880	−	0.808948i	$-0.299963\pi$
$458$	0	0
$459$	−29.1068	−1.35859
$460$	0	0
$461$	−23.3497	−1.08750	−0.543752	−	0.839246i	$-0.682997\pi$
$461$	−23.3497	−1.08750	−0.543752	+	0.839246i	$0.682997\pi$
$462$	0	0
$463$	36.7512i	1.70797i	0.520295	+	0.853987i	$0.325822\pi$
$463$	36.7512i	1.70797i	−0.520295	+	0.853987i	$0.674178\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 29.5786i	− 1.36874i	−0.729137	−	0.684368i	$-0.760078\pi$
$467$	− 29.5786i	− 1.36874i	0.729137	−	0.684368i	$-0.239922\pi$
$468$	0	0
$469$	15.5443	0.717768
$470$	0	0
$471$	−4.47717	−0.206297
$472$	0	0
$473$	− 45.7177i	− 2.10210i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	6.61591i	0.302922i
$478$	0	0
$479$	−7.23409	−0.330534	−0.165267	−	0.986249i	$-0.552849\pi$
$479$	−7.23409	−0.330534	−0.165267	+	0.986249i	$0.552849\pi$
$480$	0	0
$481$	3.84355	0.175251
$482$	0	0
$483$	− 5.29949i	− 0.241135i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 8.99593i	− 0.407645i	−0.979008	−	0.203822i	$-0.934664\pi$
$487$	− 8.99593i	− 0.407645i	0.979008	−	0.203822i	$-0.0653365\pi$
$488$	0	0
$489$	−6.67272	−0.301751
$490$	0	0
$491$	9.71762	0.438550	0.219275	−	0.975663i	$-0.429631\pi$
$491$	9.71762	0.438550	0.219275	+	0.975663i	$0.429631\pi$
$492$	0	0
$493$	6.47686i	0.291703i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.57360i	0.384578i
$498$	0	0
$499$	−33.3758	−1.49410	−0.747052	−	0.664765i	$-0.768532\pi$
$499$	−33.3758	−1.49410	−0.747052	+	0.664765i	$0.768532\pi$
$500$	0	0
$501$	22.6066	1.00999
$502$	0	0
$503$	22.1147i	0.986046i	0.870016	+	0.493023i	$0.164108\pi$
$503$	22.1147i	0.986046i	−0.870016	+	0.493023i	$0.835892\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 0.811121i	− 0.0360232i
$508$	0	0
$509$	−10.8182	−0.479507	−0.239753	−	0.970834i	$-0.577067\pi$
$509$	−10.8182	−0.479507	−0.239753	+	0.970834i	$0.577067\pi$
$510$	0	0
$511$	46.2532	2.04612
$512$	0	0
$513$	− 11.7053i	− 0.516802i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	56.0110i	2.46336i
$518$	0	0
$519$	−27.6722	−1.21467
$520$	0	0
$521$	−3.20820	−0.140554	−0.0702770	−	0.997528i	$-0.522388\pi$
$521$	−3.20820	−0.140554	−0.0702770	+	0.997528i	$0.522388\pi$
$522$	0	0
$523$	23.7159i	1.03702i	0.855071	+	0.518512i	$0.173514\pi$
$523$	23.7159i	1.03702i	−0.855071	+	0.518512i	$0.826486\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	25.0040i	1.08919i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−9.99214	−0.433622
$532$	0	0
$533$	33.3168i	1.44311i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	9.39628i	0.405479i
$538$	0	0
$539$	22.8302	0.983366
$540$	0	0
$541$	−6.46616	−0.278002	−0.139001	−	0.990292i	$-0.544389\pi$
$541$	−6.46616	−0.278002	−0.139001	+	0.990292i	$0.544389\pi$
$542$	0	0
$543$	8.69479i	0.373129i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 7.48526i	− 0.320047i	−0.987113	−	0.160023i	$-0.948843\pi$
$547$	− 7.48526i	− 0.320047i	0.987113	−	0.160023i	$-0.0511570\pi$
$548$	0	0
$549$	−6.56366	−0.280130
$550$	0	0
$551$	−2.60468	−0.110963
$552$	0	0
$553$	− 27.0341i	− 1.14961i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	22.2922i	0.944553i	0.881451	+	0.472276i	$0.156568\pi$
$557$	22.2922i	0.944553i	−0.881451	+	0.472276i	$0.843432\pi$
$558$	0	0
$559$	−38.6170	−1.63332
$560$	0	0
$561$	34.2101	1.44435
$562$	0	0
$563$	− 9.93402i	− 0.418669i	−0.977844	−	0.209335i	$-0.932870\pi$
$563$	− 9.93402i	− 0.418669i	0.977844	−	0.209335i	$-0.0671298\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 22.3378i	− 0.938099i
$568$	0	0
$569$	26.8239	1.12452	0.562258	−	0.826962i	$-0.309933\pi$
$569$	26.8239	1.12452	0.562258	+	0.826962i	$0.309933\pi$
$570$	0	0
$571$	30.0260	1.25655	0.628274	−	0.777992i	$-0.283762\pi$
$571$	30.0260	1.25655	0.628274	+	0.777992i	$0.283762\pi$
$572$	0	0
$573$	− 20.3981i	− 0.852141i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	23.5058i	0.978559i	0.872127	+	0.489279i	$0.162740\pi$
$577$	23.5058i	0.978559i	−0.872127	+	0.489279i	$0.837260\pi$
$578$	0	0
$579$	0.447701	0.0186058
$580$	0	0
$581$	10.3945	0.431235
$582$	0	0
$583$	− 40.8269i	− 1.69088i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5.05879i	0.208799i	0.994535	+	0.104399i	$0.0332920\pi$
$587$	5.05879i	0.208799i	−0.994535	+	0.104399i	$0.966708\pi$
$588$	0	0
$589$	−10.0554	−0.414325
$590$	0	0
$591$	17.6012	0.724017
$592$	0	0
$593$	15.0522i	0.618120i	0.951043	+	0.309060i	$0.100014\pi$
$593$	15.0522i	0.618120i	−0.951043	+	0.309060i	$0.899986\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 6.76975i	− 0.277067i
$598$	0	0
$599$	15.9863	0.653181	0.326590	−	0.945166i	$-0.394100\pi$
$599$	15.9863	0.653181	0.326590	+	0.945166i	$0.394100\pi$
$600$	0	0
$601$	32.7941	1.33770	0.668851	−	0.743397i	$-0.266787\pi$
$601$	32.7941	1.33770	0.668851	+	0.743397i	$0.266787\pi$
$602$	0	0
$603$	− 3.13573i	− 0.127697i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	38.4884i	1.56220i	0.624409	+	0.781098i	$0.285340\pi$
$607$	38.4884i	1.56220i	−0.624409	+	0.781098i	$0.714660\pi$
$608$	0	0
$609$	−6.61916	−0.268222
$610$	0	0
$611$	47.3115	1.91402
$612$	0	0
$613$	6.47626i	0.261574i	0.991411	+	0.130787i	$0.0417503\pi$
$613$	6.47626i	0.261574i	−0.991411	+	0.130787i	$0.958250\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 33.8193i	− 1.36152i	−0.732509	−	0.680758i	$-0.761651\pi$
$617$	− 33.8193i	− 1.36152i	0.732509	−	0.680758i	$-0.238349\pi$
$618$	0	0
$619$	−2.02686	−0.0814663	−0.0407331	−	0.999170i	$-0.512969\pi$
$619$	−2.02686	−0.0814663	−0.0407331	+	0.999170i	$0.512969\pi$
$620$	0	0
$621$	−5.61304	−0.225243
$622$	0	0
$623$	51.1669i	2.04996i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	13.7576i	0.549427i
$628$	0	0
$629$	5.41740	0.216006
$630$	0	0
$631$	−3.45649	−0.137601	−0.0688003	−	0.997630i	$-0.521917\pi$
$631$	−3.45649	−0.137601	−0.0688003	+	0.997630i	$0.521917\pi$
$632$	0	0
$633$	− 22.8529i	− 0.908319i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 19.2843i	− 0.764071i
$638$	0	0
$639$	1.72954	0.0684195
$640$	0	0
$641$	47.2630	1.86678	0.933388	−	0.358869i	$-0.116837\pi$
$641$	47.2630	1.86678	0.933388	+	0.358869i	$0.116837\pi$
$642$	0	0
$643$	− 23.6052i	− 0.930899i	−0.885075	−	0.465449i	$-0.845893\pi$
$643$	− 23.6052i	− 0.930899i	0.885075	−	0.465449i	$-0.154107\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	20.3090i	0.798430i	0.916857	+	0.399215i	$0.130717\pi$
$647$	20.3090i	0.798430i	−0.916857	+	0.399215i	$0.869283\pi$
$648$	0	0
$649$	61.6617	2.42043
$650$	0	0
$651$	−25.5533	−1.00151
$652$	0	0
$653$	− 15.8982i	− 0.622146i	−0.950386	−	0.311073i	$-0.899312\pi$
$653$	− 15.8982i	− 0.622146i	0.950386	−	0.311073i	$-0.100688\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 9.33059i	− 0.364021i
$658$	0	0
$659$	−13.6170	−0.530441	−0.265221	−	0.964188i	$-0.585445\pi$
$659$	−13.6170	−0.530441	−0.265221	+	0.964188i	$0.585445\pi$
$660$	0	0
$661$	27.8049	1.08148	0.540742	−	0.841188i	$-0.318143\pi$
$661$	27.8049	1.08148	0.540742	+	0.841188i	$0.318143\pi$
$662$	0	0
$663$	− 28.8967i	− 1.12225i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.24902i	0.0483622i
$668$	0	0
$669$	38.1744	1.47591
$670$	0	0
$671$	40.5045	1.56366
$672$	0	0
$673$	− 4.92071i	− 0.189679i	−0.995493	−	0.0948396i	$-0.969766\pi$
$673$	− 4.92071i	− 0.189679i	0.995493	−	0.0948396i	$-0.0302338\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	48.4096i	1.86053i	0.366886	+	0.930266i	$0.380424\pi$
$677$	48.4096i	1.86053i	−0.366886	+	0.930266i	$0.619576\pi$
$678$	0	0
$679$	32.8184	1.25946
$680$	0	0
$681$	−30.5029	−1.16887
$682$	0	0
$683$	− 36.1788i	− 1.38434i	−0.721732	−	0.692172i	$-0.756654\pi$
$683$	− 36.1788i	− 1.38434i	0.721732	−	0.692172i	$-0.243346\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 14.7287i	− 0.561935i
$688$	0	0
$689$	−34.4858	−1.31380
$690$	0	0
$691$	32.1568	1.22330	0.611651	−	0.791127i	$-0.290506\pi$
$691$	32.1568	1.22330	0.611651	+	0.791127i	$0.290506\pi$
$692$	0	0
$693$	− 10.7560i	− 0.408586i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	46.9593i	1.77871i
$698$	0	0
$699$	9.66287	0.365483
$700$	0	0
$701$	17.9408	0.677614	0.338807	−	0.940856i	$-0.389977\pi$
$701$	17.9408	0.677614	0.338807	+	0.940856i	$0.389977\pi$
$702$	0	0
$703$	2.17861i	0.0821680i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 3.59238i	− 0.135105i
$708$	0	0
$709$	−45.4153	−1.70561	−0.852804	−	0.522231i	$-0.825100\pi$
$709$	−45.4153	−1.70561	−0.852804	+	0.522231i	$0.825100\pi$
$710$	0	0
$711$	−5.45355	−0.204524
$712$	0	0
$713$	4.82185i	0.180580i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	4.66070i	0.174057i
$718$	0	0
$719$	−5.90426	−0.220192	−0.110096	−	0.993921i	$-0.535116\pi$
$719$	−5.90426	−0.220192	−0.110096	+	0.993921i	$0.535116\pi$
$720$	0	0
$721$	5.98741	0.222983
$722$	0	0
$723$	− 43.3709i	− 1.61298i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	34.8037i	1.29080i	0.763846	+	0.645398i	$0.223309\pi$
$727$	34.8037i	1.29080i	−0.763846	+	0.645398i	$0.776691\pi$
$728$	0	0
$729$	−30.0117	−1.11154
$730$	0	0
$731$	−54.4298	−2.01316
$732$	0	0
$733$	− 21.6346i	− 0.799093i	−0.916713	−	0.399547i	$-0.869168\pi$
$733$	− 21.6346i	− 0.799093i	0.916713	−	0.399547i	$-0.130832\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	19.3506i	0.712789i
$738$	0	0
$739$	4.53418	0.166792	0.0833962	−	0.996516i	$-0.473423\pi$
$739$	4.53418	0.166792	0.0833962	+	0.996516i	$0.473423\pi$
$740$	0	0
$741$	11.6208	0.426902
$742$	0	0
$743$	17.9137i	0.657192i	0.944471	+	0.328596i	$0.106575\pi$
$743$	17.9137i	0.657192i	−0.944471	+	0.328596i	$0.893425\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 2.09686i	− 0.0767201i
$748$	0	0
$749$	26.5356	0.969588
$750$	0	0
$751$	−25.1812	−0.918875	−0.459438	−	0.888210i	$-0.651949\pi$
$751$	−25.1812	−0.918875	−0.459438	+	0.888210i	$0.651949\pi$
$752$	0	0
$753$	1.10314i	0.0402006i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 30.7079i	− 1.11610i	−0.829809	−	0.558048i	$-0.811550\pi$
$757$	− 30.7079i	− 1.11610i	0.829809	−	0.558048i	$-0.188450\pi$
$758$	0	0
$759$	6.59718	0.239462
$760$	0	0
$761$	9.58720	0.347536	0.173768	−	0.984787i	$-0.444406\pi$
$761$	9.58720	0.347536	0.173768	+	0.984787i	$0.444406\pi$
$762$	0	0
$763$	31.3476i	1.13486i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 52.0846i	− 1.88066i
$768$	0	0
$769$	6.31547	0.227742	0.113871	−	0.993496i	$-0.463675\pi$
$769$	6.31547	0.227742	0.113871	+	0.993496i	$0.463675\pi$
$770$	0	0
$771$	44.1797	1.59109
$772$	0	0
$773$	47.4787i	1.70769i	0.520528	+	0.853845i	$0.325735\pi$
$773$	47.4787i	1.70769i	−0.520528	+	0.853845i	$0.674265\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	5.53642i	0.198618i
$778$	0	0
$779$	−18.8847	−0.676616
$780$	0	0
$781$	−10.6730	−0.381910
$782$	0	0
$783$	7.01078i	0.250545i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	47.0588i	1.67747i	0.544543	+	0.838733i	$0.316703\pi$
$787$	47.0588i	1.67747i	−0.544543	+	0.838733i	$0.683297\pi$
$788$	0	0
$789$	−4.23546	−0.150786
$790$	0	0
$791$	−65.8134	−2.34005
$792$	0	0
$793$	− 34.2134i	− 1.21495i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	22.8819i	0.810517i	0.914202	+	0.405259i	$0.132819\pi$
$797$	22.8819i	0.810517i	−0.914202	+	0.405259i	$0.867181\pi$
$798$	0	0
$799$	66.6846	2.35913
$800$	0	0
$801$	10.3218	0.364704
$802$	0	0
$803$	57.5792i	2.03193i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.00339i	0.0353211i
$808$	0	0
$809$	−21.6918	−0.762643	−0.381322	−	0.924442i	$-0.624531\pi$
$809$	−21.6918	−0.762643	−0.381322	+	0.924442i	$0.624531\pi$
$810$	0	0
$811$	5.84992	0.205418	0.102709	−	0.994711i	$-0.467249\pi$
$811$	5.84992	0.205418	0.102709	+	0.994711i	$0.467249\pi$
$812$	0	0
$813$	22.0028i	0.771674i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 21.8890i	− 0.765799i
$818$	0	0
$819$	−9.08540	−0.317469
$820$	0	0
$821$	5.98613	0.208917	0.104459	−	0.994529i	$-0.466689\pi$
$821$	5.98613	0.208917	0.104459	+	0.994529i	$0.466689\pi$
$822$	0	0
$823$	27.4906i	0.958261i	0.877744	+	0.479131i	$0.159048\pi$
$823$	27.4906i	0.958261i	−0.877744	+	0.479131i	$0.840952\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.85519i	0.0992848i	0.998767	+	0.0496424i	$0.0158082\pi$
$827$	2.85519i	0.0992848i	−0.998767	+	0.0496424i	$0.984192\pi$
$828$	0	0
$829$	7.63982	0.265342	0.132671	−	0.991160i	$-0.457645\pi$
$829$	7.63982	0.265342	0.132671	+	0.991160i	$0.457645\pi$
$830$	0	0
$831$	−1.74840	−0.0606512
$832$	0	0
$833$	− 27.1808i	− 0.941758i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	27.0652i	0.935510i
$838$	0	0
$839$	8.41078	0.290372	0.145186	−	0.989404i	$-0.453622\pi$
$839$	8.41078	0.290372	0.145186	+	0.989404i	$0.453622\pi$
$840$	0	0
$841$	−27.4400	−0.946205
$842$	0	0
$843$	− 21.6561i	− 0.745877i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	27.8885i	0.958260i
$848$	0	0
$849$	−47.3564	−1.62527
$850$	0	0
$851$	1.04471	0.0358121
$852$	0	0
$853$	35.6430i	1.22039i	0.792250	+	0.610197i	$0.208910\pi$
$853$	35.6430i	1.22039i	−0.792250	+	0.610197i	$0.791090\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 2.66046i	− 0.0908794i	−0.998967	−	0.0454397i	$-0.985531\pi$
$857$	− 2.66046i	− 0.0908794i	0.998967	−	0.0454397i	$-0.0144689\pi$
$858$	0	0
$859$	−14.0909	−0.480774	−0.240387	−	0.970677i	$-0.577274\pi$
$859$	−14.0909	−0.480774	−0.240387	+	0.970677i	$0.577274\pi$
$860$	0	0
$861$	−47.9910	−1.63553
$862$	0	0
$863$	48.1228i	1.63812i	0.573708	+	0.819060i	$0.305504\pi$
$863$	48.1228i	1.63812i	−0.573708	+	0.819060i	$0.694496\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 14.9801i	− 0.508750i
$868$	0	0
$869$	33.6539	1.14163
$870$	0	0
$871$	16.3451	0.553834
$872$	0	0
$873$	− 6.62042i	− 0.224067i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	26.3172i	0.888668i	0.895861	+	0.444334i	$0.146560\pi$
$877$	26.3172i	0.888668i	−0.895861	+	0.444334i	$0.853440\pi$
$878$	0	0
$879$	−6.95951	−0.234738
$880$	0	0
$881$	7.59556	0.255901	0.127951	−	0.991781i	$-0.459160\pi$
$881$	7.59556	0.255901	0.127951	+	0.991781i	$0.459160\pi$
$882$	0	0
$883$	11.8772i	0.399699i	0.979827	+	0.199849i	$0.0640453\pi$
$883$	11.8772i	0.399699i	−0.979827	+	0.199849i	$0.935955\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	35.9421i	1.20682i	0.797432	+	0.603409i	$0.206191\pi$
$887$	35.9421i	1.20682i	−0.797432	+	0.603409i	$0.793809\pi$
$888$	0	0
$889$	−52.6169	−1.76471
$890$	0	0
$891$	27.8076	0.931591
$892$	0	0
$893$	26.8173i	0.897406i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 5.57253i	− 0.186061i
$898$	0	0
$899$	6.02258	0.200864
$900$	0	0
$901$	−48.6070	−1.61933
$902$	0	0
$903$	− 55.6257i	− 1.85111i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 56.9912i	− 1.89236i	−0.323637	−	0.946181i	$-0.604906\pi$
$907$	− 56.9912i	− 1.89236i	0.323637	−	0.946181i	$-0.395094\pi$
$908$	0	0
$909$	−0.724685	−0.0240363
$910$	0	0
$911$	−36.5339	−1.21042	−0.605212	−	0.796065i	$-0.706912\pi$
$911$	−36.5339	−1.21042	−0.605212	+	0.796065i	$0.706912\pi$
$912$	0	0
$913$	12.9398i	0.428243i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	71.2017i	2.35129i
$918$	0	0
$919$	1.65041	0.0544419	0.0272210	−	0.999629i	$-0.491334\pi$
$919$	1.65041	0.0544419	0.0272210	+	0.999629i	$0.491334\pi$
$920$	0	0
$921$	15.0876	0.497152
$922$	0	0
$923$	9.01531i	0.296743i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 1.20783i	− 0.0396704i
$928$	0	0
$929$	−3.24062	−0.106321	−0.0531606	−	0.998586i	$-0.516930\pi$
$929$	−3.24062	−0.106321	−0.0531606	+	0.998586i	$0.516930\pi$
$930$	0	0
$931$	10.9308	0.358242
$932$	0	0
$933$	− 19.3150i	− 0.632344i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	4.26930i	0.139472i	0.997565	+	0.0697360i	$0.0222157\pi$
$937$	4.26930i	0.139472i	−0.997565	+	0.0697360i	$0.977784\pi$
$938$	0	0
$939$	29.6244	0.966757
$940$	0	0
$941$	−57.0889	−1.86105	−0.930523	−	0.366234i	$-0.880647\pi$
$941$	−57.0889	−1.86105	−0.930523	+	0.366234i	$0.880647\pi$
$942$	0	0
$943$	9.05578i	0.294897i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	50.1371i	1.62924i	0.579997	+	0.814618i	$0.303054\pi$
$947$	50.1371i	1.62924i	−0.579997	+	0.814618i	$0.696946\pi$
$948$	0	0
$949$	48.6362	1.57880
$950$	0	0
$951$	49.3011	1.59870
$952$	0	0
$953$	6.17257i	0.199949i	0.994990	+	0.0999745i	$0.0318761\pi$
$953$	6.17257i	0.199949i	−0.994990	+	0.0999745i	$0.968124\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 8.24000i	− 0.266361i
$958$	0	0
$959$	5.30049	0.171162
$960$	0	0
$961$	−7.74979	−0.249993
$962$	0	0
$963$	− 5.35298i	− 0.172497i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 7.04066i	− 0.226412i	−0.993572	−	0.113206i	$-0.963888\pi$
$967$	− 7.04066i	− 0.226412i	0.993572	−	0.113206i	$-0.0361121\pi$
$968$	0	0
$969$	16.3793	0.526180
$970$	0	0
$971$	−5.91622	−0.189861	−0.0949303	−	0.995484i	$-0.530263\pi$
$971$	−5.91622	−0.189861	−0.0949303	+	0.995484i	$0.530263\pi$
$972$	0	0
$973$	− 46.7644i	− 1.49920i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	14.7695i	0.472517i	0.971690	+	0.236259i	$0.0759213\pi$
$977$	14.7695i	0.472517i	−0.971690	+	0.236259i	$0.924079\pi$
$978$	0	0
$979$	−63.6961	−2.03574
$980$	0	0
$981$	6.32370	0.201900
$982$	0	0
$983$	− 22.8892i	− 0.730054i	−0.930997	−	0.365027i	$-0.881060\pi$
$983$	− 22.8892i	− 0.730054i	0.930997	−	0.365027i	$-0.118940\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	68.1497i	2.16923i
$988$	0	0
$989$	−10.4964	−0.333766
$990$	0	0
$991$	22.4611	0.713500	0.356750	−	0.934200i	$-0.383885\pi$
$991$	22.4611	0.713500	0.356750	+	0.934200i	$0.383885\pi$
$992$	0	0
$993$	36.3070i	1.15217i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	43.9293i	1.39125i	0.718403	+	0.695627i	$0.244873\pi$
$997$	43.9293i	1.39125i	−0.718403	+	0.695627i	$0.755127\pi$
$998$	0	0
$999$	5.86398	0.185528

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.e.v.4049.8		10
5.2	odd	4		4600.2.a.bg.1.4	yes	5
5.3	odd	4		4600.2.a.bc.1.2	✓	5
5.4	even	2	inner	4600.2.e.v.4049.3		10
20.3	even	4		9200.2.a.cw.1.4		5
20.7	even	4		9200.2.a.cs.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4600.2.a.bc.1.2	✓	5	5.3	odd	4
4600.2.a.bg.1.4	yes	5	5.2	odd	4
4600.2.e.v.4049.3		10	5.4	even	2	inner
4600.2.e.v.4049.8		10	1.1	even	1	trivial
9200.2.a.cs.1.2		5	20.7	even	4
9200.2.a.cw.1.4		5	20.3	even	4

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:	\( N \)	\(=\)	\( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4600.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.51466i q^{3} +3.49880i q^{7} +0.705809 q^{9} +O(q^{10})\)	\(q+1.51466i q^{3} +3.49880i q^{7} +0.705809 q^{9} -4.35556 q^{11} +3.67906i q^{13} +5.18556i q^{17} -2.08538 q^{19} -5.29949 q^{21} +1.00000i q^{23} +5.61304i q^{27} +1.24902 q^{29} +4.82185 q^{31} -6.59718i q^{33} -1.04471i q^{37} -5.57253 q^{39} +9.05578 q^{41} +10.4964i q^{43} -12.8597i q^{47} -5.24162 q^{49} -7.85436 q^{51} +9.37352i q^{53} -3.15864i q^{57} -14.1570 q^{59} -9.29949 q^{61} +2.46949i q^{63} -4.44274i q^{67} -1.51466 q^{69} +2.45044 q^{71} -13.2197i q^{73} -15.2392i q^{77} -7.72666 q^{79} -6.38441 q^{81} -2.97086i q^{83} +1.89184i q^{87} +14.6241 q^{89} -12.8723 q^{91} +7.30345i q^{93} -9.37991i q^{97} -3.07419 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 8 q^{9}+O(q^{10}) \) 10 * q - 8 * q^9	\( 10 q - 8 q^{9} - 8 q^{11} - 8 q^{19} - 12 q^{21} + 22 q^{29} + 8 q^{31} - 62 q^{39} - 16 q^{41} + 4 q^{49} - 10 q^{51} - 46 q^{59} - 52 q^{61} - 6 q^{69} - 4 q^{71} - 86 q^{79} - 6 q^{81} - 30 q^{89} - 38 q^{91} - 74 q^{99}+O(q^{100}) \) 10 * q - 8 * q^9 - 8 * q^11 - 8 * q^19 - 12 * q^21 + 22 * q^29 + 8 * q^31 - 62 * q^39 - 16 * q^41 + 4 * q^49 - 10 * q^51 - 46 * q^59 - 52 * q^61 - 6 * q^69 - 4 * q^71 - 86 * q^79 - 6 * q^81 - 30 * q^89 - 38 * q^91 - 74 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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