LMFDB - Embedded newform 9200.2.a.cw.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9200,2,Mod(1,9200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9200, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("9200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9200 = 2^{4} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9200.a (trivial)

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:	yes
Analytic conductor:	$73.4623698596$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.791953.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 7x^{3} + 7x^{2} + 9x + 2 $ x^5 - 2x^4 - 7x^3 + 7x^2 + 9x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4600)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.514659$ of defining polynomial
Character	$\chi$	$=$	9200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.51466 q^{3} -3.49880 q^{7} -0.705809 q^{9} +O(q^{10})$	$q+1.51466 q^{3} -3.49880 q^{7} -0.705809 q^{9} +4.35556 q^{11} -3.67906 q^{13} +5.18556 q^{17} -2.08538 q^{19} -5.29949 q^{21} +1.00000 q^{23} -5.61304 q^{27} -1.24902 q^{29} -4.82185 q^{31} +6.59718 q^{33} -1.04471 q^{37} -5.57253 q^{39} +9.05578 q^{41} +10.4964 q^{43} +12.8597 q^{47} +5.24162 q^{49} +7.85436 q^{51} -9.37352 q^{53} -3.15864 q^{57} -14.1570 q^{59} -9.29949 q^{61} +2.46949 q^{63} +4.44274 q^{67} +1.51466 q^{69} -2.45044 q^{71} +13.2197 q^{73} -15.2392 q^{77} -7.72666 q^{79} -6.38441 q^{81} -2.97086 q^{83} -1.89184 q^{87} -14.6241 q^{89} +12.8723 q^{91} -7.30345 q^{93} -9.37991 q^{97} -3.07419 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{3} - q^{7} + 4 q^{9}+O(q^{10}) $ 5 * q + 3 * q^3 - q^7 + 4 * q^9	$ 5 q + 3 q^{3} - q^{7} + 4 q^{9} + 4 q^{11} + q^{13} - 5 q^{17} - 4 q^{19} - 6 q^{21} + 5 q^{23} + 6 q^{27} - 11 q^{29} - 4 q^{31} - 13 q^{33} - 6 q^{37} - 31 q^{39} - 8 q^{41} + 3 q^{43} + 2 q^{47} - 2 q^{49} + 5 q^{51} - 18 q^{53} - 27 q^{57} - 23 q^{59} - 26 q^{61} + 5 q^{63} + 3 q^{67} + 3 q^{69} + 2 q^{71} - 4 q^{73} - 15 q^{77} - 43 q^{79} - 3 q^{81} + 30 q^{83} + 27 q^{87} + 15 q^{89} + 19 q^{91} + 15 q^{93} - 8 q^{97} - 37 q^{99}+O(q^{100}) $ 5 * q + 3 * q^3 - q^7 + 4 * q^9 + 4 * q^11 + q^13 - 5 * q^17 - 4 * q^19 - 6 * q^21 + 5 * q^23 + 6 * q^27 - 11 * q^29 - 4 * q^31 - 13 * q^33 - 6 * q^37 - 31 * q^39 - 8 * q^41 + 3 * q^43 + 2 * q^47 - 2 * q^49 + 5 * q^51 - 18 * q^53 - 27 * q^57 - 23 * q^59 - 26 * q^61 + 5 * q^63 + 3 * q^67 + 3 * q^69 + 2 * q^71 - 4 * q^73 - 15 * q^77 - 43 * q^79 - 3 * q^81 + 30 * q^83 + 27 * q^87 + 15 * q^89 + 19 * q^91 + 15 * q^93 - 8 * q^97 - 37 * q^99

Display 100 coefficients 10 coefficients

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
$3$	1.51466	0.874489	0.437244	−	0.899343i	$-0.355954\pi$
$3$	1.51466	0.874489	0.437244	+	0.899343i	$0.355954\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.49880	−1.32242	−0.661212	−	0.750199i	$-0.729957\pi$
$7$	−3.49880	−1.32242	−0.661212	+	0.750199i	$0.729957\pi$
$8$	0	0
$9$	−0.705809	−0.235270
$10$	0	0
$11$	4.35556	1.31325	0.656625	−	0.754217i	$-0.271984\pi$
$11$	4.35556	1.31325	0.656625	+	0.754217i	$0.271984\pi$
$12$	0	0
$13$	−3.67906	−1.02039	−0.510194	−	0.860059i	$-0.670427\pi$
$13$	−3.67906	−1.02039	−0.510194	+	0.860059i	$0.670427\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.18556	1.25768	0.628842	−	0.777533i	$-0.283529\pi$
$17$	5.18556	1.25768	0.628842	+	0.777533i	$0.283529\pi$
$18$	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
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.61304	−1.08023
$28$	0	0
$29$	−1.24902	−0.231937	−0.115968	−	0.993253i	$-0.536997\pi$
$29$	−1.24902	−0.231937	−0.115968	+	0.993253i	$0.536997\pi$
$30$	0	0
$31$	−4.82185	−0.866029	−0.433015	−	0.901387i	$-0.642550\pi$
$31$	−4.82185	−0.866029	−0.433015	+	0.901387i	$0.642550\pi$
$32$	0	0
$33$	6.59718	1.14842
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1.04471	−0.171749	−0.0858745	−	0.996306i	$-0.527368\pi$
$37$	−1.04471	−0.171749	−0.0858745	+	0.996306i	$0.527368\pi$
$38$	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
$43$	10.4964	1.60069	0.800344	−	0.599541i	$-0.204650\pi$
$43$	10.4964	1.60069	0.800344	+	0.599541i	$0.204650\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.8597	1.87577	0.937887	−	0.346940i	$-0.112779\pi$
$47$	12.8597	1.87577	0.937887	+	0.346940i	$0.112779\pi$
$48$	0	0
$49$	5.24162	0.748804
$50$	0	0
$51$	7.85436	1.09983
$52$	0	0
$53$	−9.37352	−1.28755	−0.643776	−	0.765214i	$-0.722633\pi$
$53$	−9.37352	−1.28755	−0.643776	+	0.765214i	$0.722633\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−3.15864	−0.418372
$58$	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
$63$	2.46949	0.311126
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.44274	0.542767	0.271384	−	0.962471i	$-0.412519\pi$
$67$	4.44274	0.542767	0.271384	+	0.962471i	$0.412519\pi$
$68$	0	0
$69$	1.51466	0.182343
$70$	0	0
$71$	−2.45044	−0.290813	−0.145407	−	0.989372i	$-0.546449\pi$
$71$	−2.45044	−0.290813	−0.145407	+	0.989372i	$0.546449\pi$
$72$	0	0
$73$	13.2197	1.54725	0.773625	−	0.633643i	$-0.218441\pi$
$73$	13.2197	1.54725	0.773625	+	0.633643i	$0.218441\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−15.2392	−1.73667
$78$	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
$83$	−2.97086	−0.326095	−0.163047	−	0.986618i	$-0.552132\pi$
$83$	−2.97086	−0.326095	−0.163047	+	0.986618i	$0.552132\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−1.89184	−0.202826
$88$	0	0
$89$	−14.6241	−1.55015	−0.775076	−	0.631868i	$-0.782288\pi$
$89$	−14.6241	−1.55015	−0.775076	+	0.631868i	$0.782288\pi$
$90$	0	0
$91$	12.8723	1.34939
$92$	0	0
$93$	−7.30345	−0.757333
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−9.37991	−0.952385	−0.476193	−	0.879341i	$-0.657983\pi$
$97$	−9.37991	−0.952385	−0.476193	+	0.879341i	$0.657983\pi$
$98$	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
$103$	−1.71127	−0.168617	−0.0843084	−	0.996440i	$-0.526868\pi$
$103$	−1.71127	−0.168617	−0.0843084	+	0.996440i	$0.526868\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.58418	0.733191	0.366595	−	0.930380i	$-0.380523\pi$
$107$	7.58418	0.733191	0.366595	+	0.930380i	$0.380523\pi$
$108$	0	0
$109$	−8.95951	−0.858165	−0.429083	−	0.903265i	$-0.641163\pi$
$109$	−8.95951	−0.858165	−0.429083	+	0.903265i	$0.641163\pi$
$110$	0	0
$111$	−1.58238	−0.150193
$112$	0	0
$113$	−18.8103	−1.76952	−0.884760	−	0.466047i	$-0.845678\pi$
$113$	−18.8103	−1.76952	−0.884760	+	0.466047i	$0.845678\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.59672	0.240066
$118$	0	0
$119$	−18.1433	−1.66319
$120$	0	0
$121$	7.97086	0.724624
$122$	0	0
$123$	13.7164	1.23677
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−15.0385	−1.33445	−0.667227	−	0.744854i	$-0.732519\pi$
$127$	−15.0385	−1.33445	−0.667227	+	0.744854i	$0.732519\pi$
$128$	0	0
$129$	15.8985	1.39978
$130$	0	0
$131$	−20.3503	−1.77801	−0.889007	−	0.457893i	$-0.848604\pi$
$131$	−20.3503	−1.77801	−0.889007	+	0.457893i	$0.848604\pi$
$132$	0	0
$133$	7.29633	0.632672
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1.51494	−0.129430	−0.0647152	−	0.997904i	$-0.520614\pi$
$137$	−1.51494	−0.129430	−0.0647152	+	0.997904i	$0.520614\pi$
$138$	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
$143$	−16.0244	−1.34002
$144$	0	0
$145$	0	0
$146$	0	0
$147$	7.93927	0.654820
$148$	0	0
$149$	−10.6307	−0.870901	−0.435450	−	0.900213i	$-0.643411\pi$
$149$	−10.6307	−0.870901	−0.435450	+	0.900213i	$0.643411\pi$
$150$	0	0
$151$	22.2918	1.81408	0.907041	−	0.421042i	$-0.138336\pi$
$151$	22.2918	1.81408	0.907041	+	0.421042i	$0.138336\pi$
$152$	0	0
$153$	−3.66002	−0.295895
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.95590	0.235906	0.117953	−	0.993019i	$-0.462367\pi$
$157$	2.95590	0.235906	0.117953	+	0.993019i	$0.462367\pi$
$158$	0	0
$159$	−14.1977	−1.12595
$160$	0	0
$161$	−3.49880	−0.275744
$162$	0	0
$163$	4.40543	0.345060	0.172530	−	0.985004i	$-0.444806\pi$
$163$	4.40543	0.345060	0.172530	+	0.985004i	$0.444806\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	14.9252	1.15495	0.577474	−	0.816409i	$-0.304038\pi$
$167$	14.9252	1.15495	0.577474	+	0.816409i	$0.304038\pi$
$168$	0	0
$169$	0.535514	0.0411934
$170$	0	0
$171$	1.47188	0.112557
$172$	0	0
$173$	−18.2696	−1.38901	−0.694506	−	0.719487i	$-0.744377\pi$
$173$	−18.2696	−1.38901	−0.694506	+	0.719487i	$0.744377\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−21.4430	−1.61176
$178$	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
$183$	−14.0856	−1.04123
$184$	0	0
$185$	0	0
$186$	0	0
$187$	22.5860	1.65165
$188$	0	0
$189$	19.6389	1.42852
$190$	0	0
$191$	13.4671	0.974445	0.487222	−	0.873278i	$-0.338010\pi$
$191$	13.4671	0.974445	0.487222	+	0.873278i	$0.338010\pi$
$192$	0	0
$193$	0.295578	0.0212762	0.0106381	−	0.999943i	$-0.496614\pi$
$193$	0.295578	0.0212762	0.0106381	+	0.999943i	$0.496614\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−11.6206	−0.827932	−0.413966	−	0.910292i	$-0.635857\pi$
$197$	−11.6206	−0.827932	−0.413966	+	0.910292i	$0.635857\pi$
$198$	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
$203$	4.37007	0.306719
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−0.705809	−0.0490571
$208$	0	0
$209$	−9.08299	−0.628283
$210$	0	0
$211$	15.0878	1.03869	0.519343	−	0.854566i	$-0.326177\pi$
$211$	15.0878	1.03869	0.519343	+	0.854566i	$0.326177\pi$
$212$	0	0
$213$	−3.71158	−0.254313
$214$	0	0
$215$	0	0
$216$	0	0
$217$	16.8707	1.14526
$218$	0	0
$219$	20.0234	1.35305
$220$	0	0
$221$	−19.0780	−1.28333
$222$	0	0
$223$	−25.2033	−1.68774	−0.843870	−	0.536548i	$-0.819728\pi$
$223$	−25.2033	−1.68774	−0.843870	+	0.536548i	$0.819728\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−20.1385	−1.33664	−0.668319	−	0.743875i	$-0.732986\pi$
$227$	−20.1385	−1.33664	−0.668319	+	0.743875i	$0.732986\pi$
$228$	0	0
$229$	9.72411	0.642587	0.321294	−	0.946980i	$-0.395882\pi$
$229$	9.72411	0.642587	0.321294	+	0.946980i	$0.395882\pi$
$230$	0	0
$231$	−23.0822	−1.51870
$232$	0	0
$233$	6.37957	0.417940	0.208970	−	0.977922i	$-0.432989\pi$
$233$	6.37957	0.417940	0.208970	+	0.977922i	$0.432989\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−11.7033	−0.760208
$238$	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
$243$	7.16891	0.459886
$244$	0	0
$245$	0	0
$246$	0	0
$247$	7.67225	0.488173
$248$	0	0
$249$	−4.49984	−0.285166
$250$	0	0
$251$	−0.728308	−0.0459704	−0.0229852	−	0.999736i	$-0.507317\pi$
$251$	−0.728308	−0.0459704	−0.0229852	+	0.999736i	$0.507317\pi$
$252$	0	0
$253$	4.35556	0.273831
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−29.1681	−1.81945	−0.909727	−	0.415206i	$-0.863710\pi$
$257$	−29.1681	−1.81945	−0.909727	+	0.415206i	$0.863710\pi$
$258$	0	0
$259$	3.65523	0.227125
$260$	0	0
$261$	0.881568	0.0545677
$262$	0	0
$263$	2.79631	0.172428	0.0862139	−	0.996277i	$-0.472523\pi$
$263$	2.79631	0.172428	0.0862139	+	0.996277i	$0.472523\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−22.1505	−1.35559
$268$	0	0
$269$	−0.662454	−0.0403905	−0.0201953	−	0.999796i	$-0.506429\pi$
$269$	−0.662454	−0.0403905	−0.0201953	+	0.999796i	$0.506429\pi$
$270$	0	0
$271$	−14.5266	−0.882428	−0.441214	−	0.897402i	$-0.645452\pi$
$271$	−14.5266	−0.882428	−0.441214	+	0.897402i	$0.645452\pi$
$272$	0	0
$273$	19.4972	1.18002
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.15432	0.0693562	0.0346781	−	0.999399i	$-0.488959\pi$
$277$	1.15432	0.0693562	0.0346781	+	0.999399i	$0.488959\pi$
$278$	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
$283$	31.2654	1.85854	0.929268	−	0.369407i	$-0.120439\pi$
$283$	31.2654	1.85854	0.929268	+	0.369407i	$0.120439\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−31.6844	−1.87027
$288$	0	0
$289$	9.89006	0.581768
$290$	0	0
$291$	−14.2074	−0.832850
$292$	0	0
$293$	−4.59477	−0.268429	−0.134215	−	0.990952i	$-0.542851\pi$
$293$	−4.59477	−0.268429	−0.134215	+	0.990952i	$0.542851\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−24.4479	−1.41861
$298$	0	0
$299$	−3.67906	−0.212766
$300$	0	0
$301$	−36.7249	−2.11679
$302$	0	0
$303$	−1.55517	−0.0893420
$304$	0	0
$305$	0	0
$306$	0	0
$307$	9.96103	0.568506	0.284253	−	0.958749i	$-0.408254\pi$
$307$	9.96103	0.568506	0.284253	+	0.958749i	$0.408254\pi$
$308$	0	0
$309$	−2.59200	−0.147453
$310$	0	0
$311$	12.7520	0.723102	0.361551	−	0.932352i	$-0.382247\pi$
$311$	12.7520	0.723102	0.361551	+	0.932352i	$0.382247\pi$
$312$	0	0
$313$	19.5585	1.10551	0.552756	−	0.833343i	$-0.313576\pi$
$313$	19.5585	1.10551	0.552756	+	0.833343i	$0.313576\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−32.5493	−1.82815	−0.914076	−	0.405542i	$-0.867083\pi$
$317$	−32.5493	−1.82815	−0.914076	+	0.405542i	$0.867083\pi$
$318$	0	0
$319$	−5.44017	−0.304591
$320$	0	0
$321$	11.4874	0.641167
$322$	0	0
$323$	−10.8139	−0.601700
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−13.5706	−0.750456
$328$	0	0
$329$	−44.9934	−2.48057
$330$	0	0
$331$	−23.9704	−1.31753	−0.658767	−	0.752347i	$-0.728922\pi$
$331$	−23.9704	−1.31753	−0.658767	+	0.752347i	$0.728922\pi$
$332$	0	0
$333$	0.737364	0.0404073
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−5.88246	−0.320438	−0.160219	−	0.987082i	$-0.551220\pi$
$337$	−5.88246	−0.320438	−0.160219	+	0.987082i	$0.551220\pi$
$338$	0	0
$339$	−28.4911	−1.54743
$340$	0	0
$341$	−21.0018	−1.13731
$342$	0	0
$343$	6.15221	0.332188
$344$	0	0
$345$	0	0
$346$	0	0
$347$	25.7454	1.38209	0.691043	−	0.722814i	$-0.257151\pi$
$347$	25.7454	1.38209	0.691043	+	0.722814i	$0.257151\pi$
$348$	0	0
$349$	−17.7758	−0.951517	−0.475758	−	0.879576i	$-0.657826\pi$
$349$	−17.7758	−0.951517	−0.475758	+	0.879576i	$0.657826\pi$
$350$	0	0
$351$	20.6507	1.10225
$352$	0	0
$353$	−20.4426	−1.08805	−0.544026	−	0.839068i	$-0.683101\pi$
$353$	−20.4426	−1.08805	−0.544026	+	0.839068i	$0.683101\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−27.4809	−1.45444
$358$	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
$363$	12.0731	0.633675
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−12.2205	−0.637906	−0.318953	−	0.947771i	$-0.603331\pi$
$367$	−12.2205	−0.637906	−0.318953	+	0.947771i	$0.603331\pi$
$368$	0	0
$369$	−6.39165	−0.332736
$370$	0	0
$371$	32.7961	1.70269
$372$	0	0
$373$	1.78846	0.0926029	0.0463015	−	0.998928i	$-0.485257\pi$
$373$	1.78846	0.0926029	0.0463015	+	0.998928i	$0.485257\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.59522	0.236666
$378$	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
$383$	13.0896	0.668846	0.334423	−	0.942423i	$-0.391459\pi$
$383$	13.0896	0.668846	0.334423	+	0.942423i	$0.391459\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−7.40846	−0.376593
$388$	0	0
$389$	−21.5395	−1.09210	−0.546048	−	0.837754i	$-0.683868\pi$
$389$	−21.5395	−1.09210	−0.546048	+	0.837754i	$0.683868\pi$
$390$	0	0
$391$	5.18556	0.262245
$392$	0	0
$393$	−30.8238	−1.55485
$394$	0	0
$395$	0	0
$396$	0	0
$397$	31.2045	1.56611	0.783055	−	0.621953i	$-0.213660\pi$
$397$	31.2045	1.56611	0.783055	+	0.621953i	$0.213660\pi$
$398$	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
$403$	17.7399	0.883687
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4.55028	−0.225549
$408$	0	0
$409$	21.9730	1.08649	0.543247	−	0.839573i	$-0.317195\pi$
$409$	21.9730	1.08649	0.543247	+	0.839573i	$0.317195\pi$
$410$	0	0
$411$	−2.29462	−0.113185
$412$	0	0
$413$	49.5326	2.43734
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−20.2447	−0.991385
$418$	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
$423$	−9.07646	−0.441313
$424$	0	0
$425$	0	0
$426$	0	0
$427$	32.5371	1.57458
$428$	0	0
$429$	−24.2715	−1.17184
$430$	0	0
$431$	−29.4301	−1.41760	−0.708800	−	0.705409i	$-0.750763\pi$
$431$	−29.4301	−1.41760	−0.708800	+	0.705409i	$0.750763\pi$
$432$	0	0
$433$	−10.5491	−0.506960	−0.253480	−	0.967341i	$-0.581575\pi$
$433$	−10.5491	−0.506960	−0.253480	+	0.967341i	$0.581575\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.08538	−0.0997572
$438$	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
$443$	9.57098	0.454731	0.227365	−	0.973810i	$-0.426989\pi$
$443$	9.57098	0.454731	0.227365	+	0.973810i	$0.426989\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−16.1019	−0.761593
$448$	0	0
$449$	8.10079	0.382300	0.191150	−	0.981561i	$-0.438778\pi$
$449$	8.10079	0.382300	0.191150	+	0.981561i	$0.438778\pi$
$450$	0	0
$451$	39.4429	1.85730
$452$	0	0
$453$	33.7645	1.58639
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−34.5867	−1.61790	−0.808948	−	0.587880i	$-0.799963\pi$
$457$	−34.5867	−1.61790	−0.808948	+	0.587880i	$0.799963\pi$
$458$	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
$463$	36.7512	1.70797	0.853987	−	0.520295i	$-0.174178\pi$
$463$	36.7512	1.70797	0.853987	+	0.520295i	$0.174178\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	29.5786	1.36874	0.684368	−	0.729137i	$-0.260078\pi$
$467$	29.5786	1.36874	0.684368	+	0.729137i	$0.260078\pi$
$468$	0	0
$469$	−15.5443	−0.717768
$470$	0	0
$471$	4.47717	0.206297
$472$	0	0
$473$	45.7177	2.10210
$474$	0	0
$475$	0	0
$476$	0	0
$477$	6.61591	0.302922
$478$	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
$483$	−5.29949	−0.241135
$484$	0	0
$485$	0	0
$486$	0	0
$487$	8.99593	0.407645	0.203822	−	0.979008i	$-0.434664\pi$
$487$	8.99593	0.407645	0.203822	+	0.979008i	$0.434664\pi$
$488$	0	0
$489$	6.67272	0.301751
$490$	0	0
$491$	−9.71762	−0.438550	−0.219275	−	0.975663i	$-0.570369\pi$
$491$	−9.71762	−0.438550	−0.219275	+	0.975663i	$0.570369\pi$
$492$	0	0
$493$	−6.47686	−0.291703
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.57360	0.384578
$498$	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
$503$	22.1147	0.986046	0.493023	−	0.870016i	$-0.335892\pi$
$503$	22.1147	0.986046	0.493023	+	0.870016i	$0.335892\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0.811121	0.0360232
$508$	0	0
$509$	10.8182	0.479507	0.239753	−	0.970834i	$-0.422933\pi$
$509$	10.8182	0.479507	0.239753	+	0.970834i	$0.422933\pi$
$510$	0	0
$511$	−46.2532	−2.04612
$512$	0	0
$513$	11.7053	0.516802
$514$	0	0
$515$	0	0
$516$	0	0
$517$	56.0110	2.46336
$518$	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
$523$	23.7159	1.03702	0.518512	−	0.855071i	$-0.326486\pi$
$523$	23.7159	1.03702	0.518512	+	0.855071i	$0.326486\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−25.0040	−1.08919
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	9.99214	0.433622
$532$	0	0
$533$	−33.3168	−1.44311
$534$	0	0
$535$	0	0
$536$	0	0
$537$	9.39628	0.405479
$538$	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
$543$	8.69479	0.373129
$544$	0	0
$545$	0	0
$546$	0	0
$547$	7.48526	0.320047	0.160023	−	0.987113i	$-0.448843\pi$
$547$	7.48526	0.320047	0.160023	+	0.987113i	$0.448843\pi$
$548$	0	0
$549$	6.56366	0.280130
$550$	0	0
$551$	2.60468	0.110963
$552$	0	0
$553$	27.0341	1.14961
$554$	0	0
$555$	0	0
$556$	0	0
$557$	22.2922	0.944553	0.472276	−	0.881451i	$-0.343432\pi$
$557$	22.2922	0.944553	0.472276	+	0.881451i	$0.343432\pi$
$558$	0	0
$559$	−38.6170	−1.63332
$560$	0	0
$561$	34.2101	1.44435
$562$	0	0
$563$	−9.93402	−0.418669	−0.209335	−	0.977844i	$-0.567130\pi$
$563$	−9.93402	−0.418669	−0.209335	+	0.977844i	$0.567130\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	22.3378	0.938099
$568$	0	0
$569$	−26.8239	−1.12452	−0.562258	−	0.826962i	$-0.690067\pi$
$569$	−26.8239	−1.12452	−0.562258	+	0.826962i	$0.690067\pi$
$570$	0	0
$571$	−30.0260	−1.25655	−0.628274	−	0.777992i	$-0.716238\pi$
$571$	−30.0260	−1.25655	−0.628274	+	0.777992i	$0.716238\pi$
$572$	0	0
$573$	20.3981	0.852141
$574$	0	0
$575$	0	0
$576$	0	0
$577$	23.5058	0.978559	0.489279	−	0.872127i	$-0.337260\pi$
$577$	23.5058	0.978559	0.489279	+	0.872127i	$0.337260\pi$
$578$	0	0
$579$	0.447701	0.0186058
$580$	0	0
$581$	10.3945	0.431235
$582$	0	0
$583$	−40.8269	−1.69088
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−5.05879	−0.208799	−0.104399	−	0.994535i	$-0.533292\pi$
$587$	−5.05879	−0.208799	−0.104399	+	0.994535i	$0.533292\pi$
$588$	0	0
$589$	10.0554	0.414325
$590$	0	0
$591$	−17.6012	−0.724017
$592$	0	0
$593$	−15.0522	−0.618120	−0.309060	−	0.951043i	$-0.600014\pi$
$593$	−15.0522	−0.618120	−0.309060	+	0.951043i	$0.600014\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−6.76975	−0.277067
$598$	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
$603$	−3.13573	−0.127697
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−38.4884	−1.56220	−0.781098	−	0.624409i	$-0.785340\pi$
$607$	−38.4884	−1.56220	−0.781098	+	0.624409i	$0.785340\pi$
$608$	0	0
$609$	6.61916	0.268222
$610$	0	0
$611$	−47.3115	−1.91402
$612$	0	0
$613$	−6.47626	−0.261574	−0.130787	−	0.991411i	$-0.541750\pi$
$613$	−6.47626	−0.261574	−0.130787	+	0.991411i	$0.541750\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−33.8193	−1.36152	−0.680758	−	0.732509i	$-0.738349\pi$
$617$	−33.8193	−1.36152	−0.680758	+	0.732509i	$0.738349\pi$
$618$	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
$623$	51.1669	2.04996
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−13.7576	−0.549427
$628$	0	0
$629$	−5.41740	−0.216006
$630$	0	0
$631$	3.45649	0.137601	0.0688003	−	0.997630i	$-0.478083\pi$
$631$	3.45649	0.137601	0.0688003	+	0.997630i	$0.478083\pi$
$632$	0	0
$633$	22.8529	0.908319
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−19.2843	−0.764071
$638$	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
$643$	−23.6052	−0.930899	−0.465449	−	0.885075i	$-0.654107\pi$
$643$	−23.6052	−0.930899	−0.465449	+	0.885075i	$0.654107\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−20.3090	−0.798430	−0.399215	−	0.916857i	$-0.630717\pi$
$647$	−20.3090	−0.798430	−0.399215	+	0.916857i	$0.630717\pi$
$648$	0	0
$649$	−61.6617	−2.42043
$650$	0	0
$651$	25.5533	1.00151
$652$	0	0
$653$	15.8982	0.622146	0.311073	−	0.950386i	$-0.399312\pi$
$653$	15.8982	0.622146	0.311073	+	0.950386i	$0.399312\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−9.33059	−0.364021
$658$	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
$663$	−28.8967	−1.12225
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.24902	−0.0483622
$668$	0	0
$669$	−38.1744	−1.47591
$670$	0	0
$671$	−40.5045	−1.56366
$672$	0	0
$673$	4.92071	0.189679	0.0948396	−	0.995493i	$-0.469766\pi$
$673$	4.92071	0.189679	0.0948396	+	0.995493i	$0.469766\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	48.4096	1.86053	0.930266	−	0.366886i	$-0.119576\pi$
$677$	48.4096	1.86053	0.930266	+	0.366886i	$0.119576\pi$
$678$	0	0
$679$	32.8184	1.25946
$680$	0	0
$681$	−30.5029	−1.16887
$682$	0	0
$683$	−36.1788	−1.38434	−0.692172	−	0.721732i	$-0.743346\pi$
$683$	−36.1788	−1.38434	−0.692172	+	0.721732i	$0.743346\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	14.7287	0.561935
$688$	0	0
$689$	34.4858	1.31380
$690$	0	0
$691$	−32.1568	−1.22330	−0.611651	−	0.791127i	$-0.709494\pi$
$691$	−32.1568	−1.22330	−0.611651	+	0.791127i	$0.709494\pi$
$692$	0	0
$693$	10.7560	0.408586
$694$	0	0
$695$	0	0
$696$	0	0
$697$	46.9593	1.77871
$698$	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
$703$	2.17861	0.0821680
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.59238	0.135105
$708$	0	0
$709$	45.4153	1.70561	0.852804	−	0.522231i	$-0.174900\pi$
$709$	45.4153	1.70561	0.852804	+	0.522231i	$0.174900\pi$
$710$	0	0
$711$	5.45355	0.204524
$712$	0	0
$713$	−4.82185	−0.180580
$714$	0	0
$715$	0	0
$716$	0	0
$717$	4.66070	0.174057
$718$	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
$723$	−43.3709	−1.61298
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−34.8037	−1.29080	−0.645398	−	0.763846i	$-0.723309\pi$
$727$	−34.8037	−1.29080	−0.645398	+	0.763846i	$0.723309\pi$
$728$	0	0
$729$	30.0117	1.11154
$730$	0	0
$731$	54.4298	2.01316
$732$	0	0
$733$	21.6346	0.799093	0.399547	−	0.916713i	$-0.369168\pi$
$733$	21.6346	0.799093	0.399547	+	0.916713i	$0.369168\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	19.3506	0.712789
$738$	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
$743$	17.9137	0.657192	0.328596	−	0.944471i	$-0.393425\pi$
$743$	17.9137	0.657192	0.328596	+	0.944471i	$0.393425\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2.09686	0.0767201
$748$	0	0
$749$	−26.5356	−0.969588
$750$	0	0
$751$	25.1812	0.918875	0.459438	−	0.888210i	$-0.348051\pi$
$751$	25.1812	0.918875	0.459438	+	0.888210i	$0.348051\pi$
$752$	0	0
$753$	−1.10314	−0.0402006
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−30.7079	−1.11610	−0.558048	−	0.829809i	$-0.688450\pi$
$757$	−30.7079	−1.11610	−0.558048	+	0.829809i	$0.688450\pi$
$758$	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
$763$	31.3476	1.13486
$764$	0	0
$765$	0	0
$766$	0	0
$767$	52.0846	1.88066
$768$	0	0
$769$	−6.31547	−0.227742	−0.113871	−	0.993496i	$-0.536325\pi$
$769$	−6.31547	−0.227742	−0.113871	+	0.993496i	$0.536325\pi$
$770$	0	0
$771$	−44.1797	−1.59109
$772$	0	0
$773$	−47.4787	−1.70769	−0.853845	−	0.520528i	$-0.825735\pi$
$773$	−47.4787	−1.70769	−0.853845	+	0.520528i	$0.825735\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	5.53642	0.198618
$778$	0	0
$779$	−18.8847	−0.676616
$780$	0	0
$781$	−10.6730	−0.381910
$782$	0	0
$783$	7.01078	0.250545
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−47.0588	−1.67747	−0.838733	−	0.544543i	$-0.816703\pi$
$787$	−47.0588	−1.67747	−0.838733	+	0.544543i	$0.816703\pi$
$788$	0	0
$789$	4.23546	0.150786
$790$	0	0
$791$	65.8134	2.34005
$792$	0	0
$793$	34.2134	1.21495
$794$	0	0
$795$	0	0
$796$	0	0
$797$	22.8819	0.810517	0.405259	−	0.914202i	$-0.367181\pi$
$797$	22.8819	0.810517	0.405259	+	0.914202i	$0.367181\pi$
$798$	0	0
$799$	66.6846	2.35913
$800$	0	0
$801$	10.3218	0.364704
$802$	0	0
$803$	57.5792	2.03193
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−1.00339	−0.0353211
$808$	0	0
$809$	21.6918	0.762643	0.381322	−	0.924442i	$-0.375469\pi$
$809$	21.6918	0.762643	0.381322	+	0.924442i	$0.375469\pi$
$810$	0	0
$811$	−5.84992	−0.205418	−0.102709	−	0.994711i	$-0.532751\pi$
$811$	−5.84992	−0.205418	−0.102709	+	0.994711i	$0.532751\pi$
$812$	0	0
$813$	−22.0028	−0.771674
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−21.8890	−0.765799
$818$	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
$823$	27.4906	0.958261	0.479131	−	0.877744i	$-0.340952\pi$
$823$	27.4906	0.958261	0.479131	+	0.877744i	$0.340952\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−2.85519	−0.0992848	−0.0496424	−	0.998767i	$-0.515808\pi$
$827$	−2.85519	−0.0992848	−0.0496424	+	0.998767i	$0.515808\pi$
$828$	0	0
$829$	−7.63982	−0.265342	−0.132671	−	0.991160i	$-0.542355\pi$
$829$	−7.63982	−0.265342	−0.132671	+	0.991160i	$0.542355\pi$
$830$	0	0
$831$	1.74840	0.0606512
$832$	0	0
$833$	27.1808	0.941758
$834$	0	0
$835$	0	0
$836$	0	0
$837$	27.0652	0.935510
$838$	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
$843$	−21.6561	−0.745877
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−27.8885	−0.958260
$848$	0	0
$849$	47.3564	1.62527
$850$	0	0
$851$	−1.04471	−0.0358121
$852$	0	0
$853$	−35.6430	−1.22039	−0.610197	−	0.792250i	$-0.708910\pi$
$853$	−35.6430	−1.22039	−0.610197	+	0.792250i	$0.708910\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2.66046	−0.0908794	−0.0454397	−	0.998967i	$-0.514469\pi$
$857$	−2.66046	−0.0908794	−0.0454397	+	0.998967i	$0.514469\pi$
$858$	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
$863$	48.1228	1.63812	0.819060	−	0.573708i	$-0.194496\pi$
$863$	48.1228	1.63812	0.819060	+	0.573708i	$0.194496\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	14.9801	0.508750
$868$	0	0
$869$	−33.6539	−1.14163
$870$	0	0
$871$	−16.3451	−0.553834
$872$	0	0
$873$	6.62042	0.224067
$874$	0	0
$875$	0	0
$876$	0	0
$877$	26.3172	0.888668	0.444334	−	0.895861i	$-0.353440\pi$
$877$	26.3172	0.888668	0.444334	+	0.895861i	$0.353440\pi$
$878$	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
$883$	11.8772	0.399699	0.199849	−	0.979827i	$-0.435955\pi$
$883$	11.8772	0.399699	0.199849	+	0.979827i	$0.435955\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−35.9421	−1.20682	−0.603409	−	0.797432i	$-0.706191\pi$
$887$	−35.9421	−1.20682	−0.603409	+	0.797432i	$0.706191\pi$
$888$	0	0
$889$	52.6169	1.76471
$890$	0	0
$891$	−27.8076	−0.931591
$892$	0	0
$893$	−26.8173	−0.897406
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−5.57253	−0.186061
$898$	0	0
$899$	6.02258	0.200864
$900$	0	0
$901$	−48.6070	−1.61933
$902$	0	0
$903$	−55.6257	−1.85111
$904$	0	0
$905$	0	0
$906$	0	0
$907$	56.9912	1.89236	0.946181	−	0.323637i	$-0.104906\pi$
$907$	56.9912	1.89236	0.946181	+	0.323637i	$0.104906\pi$
$908$	0	0
$909$	0.724685	0.0240363
$910$	0	0
$911$	36.5339	1.21042	0.605212	−	0.796065i	$-0.293088\pi$
$911$	36.5339	1.21042	0.605212	+	0.796065i	$0.293088\pi$
$912$	0	0
$913$	−12.9398	−0.428243
$914$	0	0
$915$	0	0
$916$	0	0
$917$	71.2017	2.35129
$918$	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
$923$	9.01531	0.296743
$924$	0	0
$925$	0	0
$926$	0	0
$927$	1.20783	0.0396704
$928$	0	0
$929$	3.24062	0.106321	0.0531606	−	0.998586i	$-0.483070\pi$
$929$	3.24062	0.106321	0.0531606	+	0.998586i	$0.483070\pi$
$930$	0	0
$931$	−10.9308	−0.358242
$932$	0	0
$933$	19.3150	0.632344
$934$	0	0
$935$	0	0
$936$	0	0
$937$	4.26930	0.139472	0.0697360	−	0.997565i	$-0.477784\pi$
$937$	4.26930	0.139472	0.0697360	+	0.997565i	$0.477784\pi$
$938$	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
$943$	9.05578	0.294897
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−50.1371	−1.62924	−0.814618	−	0.579997i	$-0.803054\pi$
$947$	−50.1371	−1.62924	−0.814618	+	0.579997i	$0.803054\pi$
$948$	0	0
$949$	−48.6362	−1.57880
$950$	0	0
$951$	−49.3011	−1.59870
$952$	0	0
$953$	−6.17257	−0.199949	−0.0999745	−	0.994990i	$-0.531876\pi$
$953$	−6.17257	−0.199949	−0.0999745	+	0.994990i	$0.531876\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−8.24000	−0.266361
$958$	0	0
$959$	5.30049	0.171162
$960$	0	0
$961$	−7.74979	−0.249993
$962$	0	0
$963$	−5.35298	−0.172497
$964$	0	0
$965$	0	0
$966$	0	0
$967$	7.04066	0.226412	0.113206	−	0.993572i	$-0.463888\pi$
$967$	7.04066	0.226412	0.113206	+	0.993572i	$0.463888\pi$
$968$	0	0
$969$	−16.3793	−0.526180
$970$	0	0
$971$	5.91622	0.189861	0.0949303	−	0.995484i	$-0.469737\pi$
$971$	5.91622	0.189861	0.0949303	+	0.995484i	$0.469737\pi$
$972$	0	0
$973$	46.7644	1.49920
$974$	0	0
$975$	0	0
$976$	0	0
$977$	14.7695	0.472517	0.236259	−	0.971690i	$-0.424079\pi$
$977$	14.7695	0.472517	0.236259	+	0.971690i	$0.424079\pi$
$978$	0	0
$979$	−63.6961	−2.03574
$980$	0	0
$981$	6.32370	0.201900
$982$	0	0
$983$	−22.8892	−0.730054	−0.365027	−	0.930997i	$-0.618940\pi$
$983$	−22.8892	−0.730054	−0.365027	+	0.930997i	$0.618940\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−68.1497	−2.16923
$988$	0	0
$989$	10.4964	0.333766
$990$	0	0
$991$	−22.4611	−0.713500	−0.356750	−	0.934200i	$-0.616115\pi$
$991$	−22.4611	−0.713500	−0.356750	+	0.934200i	$0.616115\pi$
$992$	0	0
$993$	−36.3070	−1.15217
$994$	0	0
$995$	0	0
$996$	0	0
$997$	43.9293	1.39125	0.695627	−	0.718403i	$-0.255127\pi$
$997$	43.9293	1.39125	0.695627	+	0.718403i	$0.255127\pi$
$998$	0	0
$999$	5.86398	0.185528

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4600.2.a.bc.1.2	✓	5	4.3	odd	2
4600.2.a.bg.1.4	yes	5	20.19	odd	2
4600.2.e.v.4049.3		10	20.3	even	4
4600.2.e.v.4049.8		10	20.7	even	4
9200.2.a.cs.1.2		5	5.4	even	2
9200.2.a.cw.1.4		5	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:	\( N \)	\(=\)	\( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9200.a (trivial)

\(f(q)\)	\(=\)	\(q+1.51466 q^{3} -3.49880 q^{7} -0.705809 q^{9} +O(q^{10})\)	\(q+1.51466 q^{3} -3.49880 q^{7} -0.705809 q^{9} +4.35556 q^{11} -3.67906 q^{13} +5.18556 q^{17} -2.08538 q^{19} -5.29949 q^{21} +1.00000 q^{23} -5.61304 q^{27} -1.24902 q^{29} -4.82185 q^{31} +6.59718 q^{33} -1.04471 q^{37} -5.57253 q^{39} +9.05578 q^{41} +10.4964 q^{43} +12.8597 q^{47} +5.24162 q^{49} +7.85436 q^{51} -9.37352 q^{53} -3.15864 q^{57} -14.1570 q^{59} -9.29949 q^{61} +2.46949 q^{63} +4.44274 q^{67} +1.51466 q^{69} -2.45044 q^{71} +13.2197 q^{73} -15.2392 q^{77} -7.72666 q^{79} -6.38441 q^{81} -2.97086 q^{83} -1.89184 q^{87} -14.6241 q^{89} +12.8723 q^{91} -7.30345 q^{93} -9.37991 q^{97} -3.07419 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{3} - q^{7} + 4 q^{9}+O(q^{10}) \) 5 * q + 3 * q^3 - q^7 + 4 * q^9	\( 5 q + 3 q^{3} - q^{7} + 4 q^{9} + 4 q^{11} + q^{13} - 5 q^{17} - 4 q^{19} - 6 q^{21} + 5 q^{23} + 6 q^{27} - 11 q^{29} - 4 q^{31} - 13 q^{33} - 6 q^{37} - 31 q^{39} - 8 q^{41} + 3 q^{43} + 2 q^{47} - 2 q^{49} + 5 q^{51} - 18 q^{53} - 27 q^{57} - 23 q^{59} - 26 q^{61} + 5 q^{63} + 3 q^{67} + 3 q^{69} + 2 q^{71} - 4 q^{73} - 15 q^{77} - 43 q^{79} - 3 q^{81} + 30 q^{83} + 27 q^{87} + 15 q^{89} + 19 q^{91} + 15 q^{93} - 8 q^{97} - 37 q^{99}+O(q^{100}) \) 5 * q + 3 * q^3 - q^7 + 4 * q^9 + 4 * q^11 + q^13 - 5 * q^17 - 4 * q^19 - 6 * q^21 + 5 * q^23 + 6 * q^27 - 11 * q^29 - 4 * q^31 - 13 * q^33 - 6 * q^37 - 31 * q^39 - 8 * q^41 + 3 * q^43 + 2 * q^47 - 2 * q^49 + 5 * q^51 - 18 * q^53 - 27 * q^57 - 23 * q^59 - 26 * q^61 + 5 * q^63 + 3 * q^67 + 3 * q^69 + 2 * q^71 - 4 * q^73 - 15 * q^77 - 43 * q^79 - 3 * q^81 + 30 * q^83 + 27 * q^87 + 15 * q^89 + 19 * q^91 + 15 * q^93 - 8 * q^97 - 37 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more