LMFDB - Embedded newform 9200.2.a.dc.1.3

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:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 7x^{5} + 24x^{4} + x^{3} - 35x^{2} + 17x - 2 $ x^7 - 3x^6 - 7x^5 + 24x^4 + x^3 - 35x^2 + 17*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 920)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.189375$ 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)

$f(q)$	$=$	$q+0.189375 q^{3} -1.65661 q^{7} -2.96414 q^{9} +O(q^{10})$	$q+0.189375 q^{3} -1.65661 q^{7} -2.96414 q^{9} -0.0759321 q^{11} +1.24146 q^{13} -5.17593 q^{17} -0.792262 q^{19} -0.313721 q^{21} -1.00000 q^{23} -1.12946 q^{27} -2.85401 q^{29} -8.90112 q^{31} -0.0143797 q^{33} -6.92065 q^{37} +0.235102 q^{39} +4.64279 q^{41} -1.75255 q^{43} +10.0033 q^{47} -4.25565 q^{49} -0.980193 q^{51} +11.1284 q^{53} -0.150035 q^{57} +12.4581 q^{59} -12.0098 q^{61} +4.91041 q^{63} -6.96067 q^{67} -0.189375 q^{69} +7.71650 q^{71} +7.49476 q^{73} +0.125790 q^{77} +15.6615 q^{79} +8.67852 q^{81} +6.68518 q^{83} -0.540478 q^{87} -3.03333 q^{89} -2.05661 q^{91} -1.68565 q^{93} +3.21747 q^{97} +0.225073 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 3 q^{3} + 4 q^{7} + 2 q^{9}+O(q^{10}) $ 7 * q + 3 * q^3 + 4 * q^7 + 2 * q^9	$ 7 q + 3 q^{3} + 4 q^{7} + 2 q^{9} + 7 q^{11} + 7 q^{13} + 7 q^{19} - 6 q^{21} - 7 q^{23} - 11 q^{29} + 10 q^{31} + 19 q^{33} + 19 q^{37} + 24 q^{39} - 16 q^{41} + 6 q^{43} + 6 q^{47} - 17 q^{49} + 7 q^{51} + 15 q^{53} + 8 q^{57} + 11 q^{59} + 5 q^{61} + 13 q^{63} + 9 q^{67} - 3 q^{69} + 14 q^{71} + 10 q^{73} + 6 q^{77} + 32 q^{79} - 5 q^{81} + q^{83} + 10 q^{87} - 24 q^{89} + 7 q^{91} + 26 q^{93} - 7 q^{97} + 61 q^{99}+O(q^{100}) $ 7 * q + 3 * q^3 + 4 * q^7 + 2 * q^9 + 7 * q^11 + 7 * q^13 + 7 * q^19 - 6 * q^21 - 7 * q^23 - 11 * q^29 + 10 * q^31 + 19 * q^33 + 19 * q^37 + 24 * q^39 - 16 * q^41 + 6 * q^43 + 6 * q^47 - 17 * q^49 + 7 * q^51 + 15 * q^53 + 8 * q^57 + 11 * q^59 + 5 * q^61 + 13 * q^63 + 9 * q^67 - 3 * q^69 + 14 * q^71 + 10 * q^73 + 6 * q^77 + 32 * q^79 - 5 * q^81 + q^83 + 10 * q^87 - 24 * q^89 + 7 * q^91 + 26 * q^93 - 7 * q^97 + 61 * 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
$3$	0.189375	0.109336	0.0546680	−	0.998505i	$-0.482590\pi$
$3$	0.189375	0.109336	0.0546680	+	0.998505i	$0.482590\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.65661	−0.626139	−0.313070	−	0.949730i	$-0.601357\pi$
$7$	−1.65661	−0.626139	−0.313070	+	0.949730i	$0.601357\pi$
$8$	0	0
$9$	−2.96414	−0.988046
$10$	0	0
$11$	−0.0759321	−0.0228944	−0.0114472	−	0.999934i	$-0.503644\pi$
$11$	−0.0759321	−0.0228944	−0.0114472	+	0.999934i	$0.503644\pi$
$12$	0	0
$13$	1.24146	0.344319	0.172159	−	0.985069i	$-0.444926\pi$
$13$	1.24146	0.344319	0.172159	+	0.985069i	$0.444926\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.17593	−1.25535	−0.627673	−	0.778477i	$-0.715993\pi$
$17$	−5.17593	−1.25535	−0.627673	+	0.778477i	$0.715993\pi$
$18$	0	0
$19$	−0.792262	−0.181757	−0.0908787	−	0.995862i	$-0.528968\pi$
$19$	−0.792262	−0.181757	−0.0908787	+	0.995862i	$0.528968\pi$
$20$	0	0
$21$	−0.313721	−0.0684595
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.12946	−0.217365
$28$	0	0
$29$	−2.85401	−0.529976	−0.264988	−	0.964252i	$-0.585368\pi$
$29$	−2.85401	−0.529976	−0.264988	+	0.964252i	$0.585368\pi$
$30$	0	0
$31$	−8.90112	−1.59869	−0.799345	−	0.600873i	$-0.794820\pi$
$31$	−8.90112	−1.59869	−0.799345	+	0.600873i	$0.794820\pi$
$32$	0	0
$33$	−0.0143797	−0.00250318
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.92065	−1.13775	−0.568874	−	0.822425i	$-0.692621\pi$
$37$	−6.92065	−1.13775	−0.568874	+	0.822425i	$0.692621\pi$
$38$	0	0
$39$	0.235102	0.0376464
$40$	0	0
$41$	4.64279	0.725082	0.362541	−	0.931968i	$-0.381909\pi$
$41$	4.64279	0.725082	0.362541	+	0.931968i	$0.381909\pi$
$42$	0	0
$43$	−1.75255	−0.267262	−0.133631	−	0.991031i	$-0.542664\pi$
$43$	−1.75255	−0.267262	−0.133631	+	0.991031i	$0.542664\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.0033	1.45913	0.729563	−	0.683914i	$-0.239724\pi$
$47$	10.0033	1.45913	0.729563	+	0.683914i	$0.239724\pi$
$48$	0	0
$49$	−4.25565	−0.607950
$50$	0	0
$51$	−0.980193	−0.137254
$52$	0	0
$53$	11.1284	1.52860	0.764301	−	0.644859i	$-0.223084\pi$
$53$	11.1284	1.52860	0.764301	+	0.644859i	$0.223084\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−0.150035	−0.0198726
$58$	0	0
$59$	12.4581	1.62191	0.810955	−	0.585108i	$-0.198948\pi$
$59$	12.4581	1.62191	0.810955	+	0.585108i	$0.198948\pi$
$60$	0	0
$61$	−12.0098	−1.53770	−0.768849	−	0.639430i	$-0.779170\pi$
$61$	−12.0098	−1.53770	−0.768849	+	0.639430i	$0.779170\pi$
$62$	0	0
$63$	4.91041	0.618654
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−6.96067	−0.850381	−0.425191	−	0.905104i	$-0.639793\pi$
$67$	−6.96067	−0.850381	−0.425191	+	0.905104i	$0.639793\pi$
$68$	0	0
$69$	−0.189375	−0.0227981
$70$	0	0
$71$	7.71650	0.915779	0.457890	−	0.889009i	$-0.348605\pi$
$71$	7.71650	0.915779	0.457890	+	0.889009i	$0.348605\pi$
$72$	0	0
$73$	7.49476	0.877196	0.438598	−	0.898683i	$-0.355475\pi$
$73$	7.49476	0.877196	0.438598	+	0.898683i	$0.355475\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.125790	0.0143351
$78$	0	0
$79$	15.6615	1.76206	0.881029	−	0.473063i	$-0.156852\pi$
$79$	15.6615	1.76206	0.881029	+	0.473063i	$0.156852\pi$
$80$	0	0
$81$	8.67852	0.964280
$82$	0	0
$83$	6.68518	0.733793	0.366897	−	0.930262i	$-0.380420\pi$
$83$	6.68518	0.733793	0.366897	+	0.930262i	$0.380420\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−0.540478	−0.0579454
$88$	0	0
$89$	−3.03333	−0.321532	−0.160766	−	0.986993i	$-0.551396\pi$
$89$	−3.03333	−0.321532	−0.160766	+	0.986993i	$0.551396\pi$
$90$	0	0
$91$	−2.05661	−0.215592
$92$	0	0
$93$	−1.68565	−0.174794
$94$	0	0
$95$	0	0
$96$	0	0
$97$	3.21747	0.326684	0.163342	−	0.986569i	$-0.447773\pi$
$97$	3.21747	0.326684	0.163342	+	0.986569i	$0.447773\pi$
$98$	0	0
$99$	0.225073	0.0226207
$100$	0	0
$101$	−16.6059	−1.65235	−0.826173	−	0.563417i	$-0.809486\pi$
$101$	−16.6059	−1.65235	−0.826173	+	0.563417i	$0.809486\pi$
$102$	0	0
$103$	7.30241	0.719528	0.359764	−	0.933043i	$-0.382857\pi$
$103$	7.30241	0.719528	0.359764	+	0.933043i	$0.382857\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.356936	0.0345063	0.0172532	−	0.999851i	$-0.494508\pi$
$107$	0.356936	0.0345063	0.0172532	+	0.999851i	$0.494508\pi$
$108$	0	0
$109$	14.0359	1.34440	0.672198	−	0.740371i	$-0.265350\pi$
$109$	14.0359	1.34440	0.672198	+	0.740371i	$0.265350\pi$
$110$	0	0
$111$	−1.31060	−0.124397
$112$	0	0
$113$	−1.17924	−0.110933	−0.0554667	−	0.998461i	$-0.517665\pi$
$113$	−1.17924	−0.110933	−0.0554667	+	0.998461i	$0.517665\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−3.67986	−0.340203
$118$	0	0
$119$	8.57448	0.786022
$120$	0	0
$121$	−10.9942	−0.999476
$122$	0	0
$123$	0.879230	0.0792775
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−19.0097	−1.68684	−0.843420	−	0.537255i	$-0.819461\pi$
$127$	−19.0097	−1.68684	−0.843420	+	0.537255i	$0.819461\pi$
$128$	0	0
$129$	−0.331890	−0.0292213
$130$	0	0
$131$	−4.29043	−0.374857	−0.187428	−	0.982278i	$-0.560015\pi$
$131$	−4.29043	−0.374857	−0.187428	+	0.982278i	$0.560015\pi$
$132$	0	0
$133$	1.31247	0.113805
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12.2499	1.04658	0.523290	−	0.852155i	$-0.324704\pi$
$137$	12.2499	1.04658	0.523290	+	0.852155i	$0.324704\pi$
$138$	0	0
$139$	−2.61437	−0.221748	−0.110874	−	0.993834i	$-0.535365\pi$
$139$	−2.61437	−0.221748	−0.110874	+	0.993834i	$0.535365\pi$
$140$	0	0
$141$	1.89437	0.159535
$142$	0	0
$143$	−0.0942666	−0.00788297
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−0.805915	−0.0664707
$148$	0	0
$149$	−19.0404	−1.55985	−0.779927	−	0.625870i	$-0.784744\pi$
$149$	−19.0404	−1.55985	−0.779927	+	0.625870i	$0.784744\pi$
$150$	0	0
$151$	18.3252	1.49128	0.745640	−	0.666349i	$-0.232144\pi$
$151$	18.3252	1.49128	0.745640	+	0.666349i	$0.232144\pi$
$152$	0	0
$153$	15.3422	1.24034
$154$	0	0
$155$	0	0
$156$	0	0
$157$	4.37467	0.349137	0.174569	−	0.984645i	$-0.444147\pi$
$157$	4.37467	0.349137	0.174569	+	0.984645i	$0.444147\pi$
$158$	0	0
$159$	2.10744	0.167131
$160$	0	0
$161$	1.65661	0.130559
$162$	0	0
$163$	11.8050	0.924639	0.462319	−	0.886714i	$-0.347017\pi$
$163$	11.8050	0.924639	0.462319	+	0.886714i	$0.347017\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.19760	0.402202	0.201101	−	0.979570i	$-0.435548\pi$
$167$	5.19760	0.402202	0.201101	+	0.979570i	$0.435548\pi$
$168$	0	0
$169$	−11.4588	−0.881444
$170$	0	0
$171$	2.34837	0.179585
$172$	0	0
$173$	2.61834	0.199069	0.0995345	−	0.995034i	$-0.468265\pi$
$173$	2.61834	0.199069	0.0995345	+	0.995034i	$0.468265\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2.35926	0.177333
$178$	0	0
$179$	−0.627563	−0.0469062	−0.0234531	−	0.999725i	$-0.507466\pi$
$179$	−0.627563	−0.0469062	−0.0234531	+	0.999725i	$0.507466\pi$
$180$	0	0
$181$	−6.15848	−0.457756	−0.228878	−	0.973455i	$-0.573506\pi$
$181$	−6.15848	−0.457756	−0.228878	+	0.973455i	$0.573506\pi$
$182$	0	0
$183$	−2.27436	−0.168126
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0.393019	0.0287404
$188$	0	0
$189$	1.87107	0.136101
$190$	0	0
$191$	19.1343	1.38451	0.692254	−	0.721654i	$-0.256618\pi$
$191$	19.1343	1.38451	0.692254	+	0.721654i	$0.256618\pi$
$192$	0	0
$193$	0.810900	0.0583698	0.0291849	−	0.999574i	$-0.490709\pi$
$193$	0.810900	0.0583698	0.0291849	+	0.999574i	$0.490709\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.15110	0.438248	0.219124	−	0.975697i	$-0.429680\pi$
$197$	6.15110	0.438248	0.219124	+	0.975697i	$0.429680\pi$
$198$	0	0
$199$	−7.03061	−0.498387	−0.249193	−	0.968454i	$-0.580166\pi$
$199$	−7.03061	−0.498387	−0.249193	+	0.968454i	$0.580166\pi$
$200$	0	0
$201$	−1.31818	−0.0929772
$202$	0	0
$203$	4.72797	0.331838
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.96414	0.206022
$208$	0	0
$209$	0.0601581	0.00416122
$210$	0	0
$211$	−8.12120	−0.559086	−0.279543	−	0.960133i	$-0.590183\pi$
$211$	−8.12120	−0.559086	−0.279543	+	0.960133i	$0.590183\pi$
$212$	0	0
$213$	1.46131	0.100128
$214$	0	0
$215$	0	0
$216$	0	0
$217$	14.7457	1.00100
$218$	0	0
$219$	1.41932	0.0959090
$220$	0	0
$221$	−6.42570	−0.432240
$222$	0	0
$223$	2.86268	0.191699	0.0958495	−	0.995396i	$-0.469443\pi$
$223$	2.86268	0.191699	0.0958495	+	0.995396i	$0.469443\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	27.4500	1.82192	0.910960	−	0.412496i	$-0.135343\pi$
$227$	27.4500	1.82192	0.910960	+	0.412496i	$0.135343\pi$
$228$	0	0
$229$	1.11319	0.0735613	0.0367807	−	0.999323i	$-0.488290\pi$
$229$	1.11319	0.0735613	0.0367807	+	0.999323i	$0.488290\pi$
$230$	0	0
$231$	0.0238215	0.00156734
$232$	0	0
$233$	−21.2957	−1.39513	−0.697564	−	0.716523i	$-0.745733\pi$
$233$	−21.2957	−1.39513	−0.697564	+	0.716523i	$0.745733\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	2.96590	0.192656
$238$	0	0
$239$	−7.68036	−0.496801	−0.248401	−	0.968657i	$-0.579905\pi$
$239$	−7.68036	−0.496801	−0.248401	+	0.968657i	$0.579905\pi$
$240$	0	0
$241$	8.42476	0.542687	0.271343	−	0.962483i	$-0.412532\pi$
$241$	8.42476	0.542687	0.271343	+	0.962483i	$0.412532\pi$
$242$	0	0
$243$	5.03188	0.322795
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−0.983561	−0.0625825
$248$	0	0
$249$	1.26601	0.0802300
$250$	0	0
$251$	2.56210	0.161718	0.0808590	−	0.996726i	$-0.474234\pi$
$251$	2.56210	0.161718	0.0808590	+	0.996726i	$0.474234\pi$
$252$	0	0
$253$	0.0759321	0.00477381
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4.39999	−0.274464	−0.137232	−	0.990539i	$-0.543821\pi$
$257$	−4.39999	−0.274464	−0.137232	+	0.990539i	$0.543821\pi$
$258$	0	0
$259$	11.4648	0.712388
$260$	0	0
$261$	8.45966	0.523640
$262$	0	0
$263$	24.8510	1.53238	0.766191	−	0.642613i	$-0.222150\pi$
$263$	24.8510	1.53238	0.766191	+	0.642613i	$0.222150\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−0.574437	−0.0351550
$268$	0	0
$269$	−10.8742	−0.663012	−0.331506	−	0.943453i	$-0.607557\pi$
$269$	−10.8742	−0.663012	−0.331506	+	0.943453i	$0.607557\pi$
$270$	0	0
$271$	2.60290	0.158115	0.0790575	−	0.996870i	$-0.474809\pi$
$271$	2.60290	0.158115	0.0790575	+	0.996870i	$0.474809\pi$
$272$	0	0
$273$	−0.389472	−0.0235719
$274$	0	0
$275$	0	0
$276$	0	0
$277$	12.1918	0.732537	0.366269	−	0.930509i	$-0.380635\pi$
$277$	12.1918	0.732537	0.366269	+	0.930509i	$0.380635\pi$
$278$	0	0
$279$	26.3842	1.57958
$280$	0	0
$281$	5.85068	0.349022	0.174511	−	0.984655i	$-0.444165\pi$
$281$	5.85068	0.349022	0.174511	+	0.984655i	$0.444165\pi$
$282$	0	0
$283$	10.7223	0.637375	0.318688	−	0.947860i	$-0.396758\pi$
$283$	10.7223	0.637375	0.318688	+	0.947860i	$0.396758\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−7.69129	−0.454002
$288$	0	0
$289$	9.79021	0.575895
$290$	0	0
$291$	0.609309	0.0357183
$292$	0	0
$293$	2.72596	0.159252	0.0796261	−	0.996825i	$-0.474627\pi$
$293$	2.72596	0.159252	0.0796261	+	0.996825i	$0.474627\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0.0857623	0.00497643
$298$	0	0
$299$	−1.24146	−0.0717955
$300$	0	0
$301$	2.90329	0.167343
$302$	0	0
$303$	−3.14474	−0.180661
$304$	0	0
$305$	0	0
$306$	0	0
$307$	22.4979	1.28402	0.642012	−	0.766695i	$-0.278100\pi$
$307$	22.4979	1.28402	0.642012	+	0.766695i	$0.278100\pi$
$308$	0	0
$309$	1.38290	0.0786703
$310$	0	0
$311$	−17.9597	−1.01840	−0.509200	−	0.860648i	$-0.670058\pi$
$311$	−17.9597	−1.01840	−0.509200	+	0.860648i	$0.670058\pi$
$312$	0	0
$313$	14.4728	0.818052	0.409026	−	0.912523i	$-0.365869\pi$
$313$	14.4728	0.818052	0.409026	+	0.912523i	$0.365869\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−7.47689	−0.419944	−0.209972	−	0.977707i	$-0.567337\pi$
$317$	−7.47689	−0.419944	−0.209972	+	0.977707i	$0.567337\pi$
$318$	0	0
$319$	0.216711	0.0121335
$320$	0	0
$321$	0.0675949	0.00377278
$322$	0	0
$323$	4.10069	0.228168
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2.65806	0.146991
$328$	0	0
$329$	−16.5715	−0.913616
$330$	0	0
$331$	27.3565	1.50365	0.751826	−	0.659362i	$-0.229173\pi$
$331$	27.3565	1.50365	0.751826	+	0.659362i	$0.229173\pi$
$332$	0	0
$333$	20.5137	1.12415
$334$	0	0
$335$	0	0
$336$	0	0
$337$	5.65271	0.307923	0.153961	−	0.988077i	$-0.450797\pi$
$337$	5.65271	0.307923	0.153961	+	0.988077i	$0.450797\pi$
$338$	0	0
$339$	−0.223319	−0.0121290
$340$	0	0
$341$	0.675881	0.0366010
$342$	0	0
$343$	18.6462	1.00680
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−15.5293	−0.833658	−0.416829	−	0.908985i	$-0.636859\pi$
$347$	−15.5293	−0.833658	−0.416829	+	0.908985i	$0.636859\pi$
$348$	0	0
$349$	11.0559	0.591810	0.295905	−	0.955217i	$-0.404379\pi$
$349$	11.0559	0.591810	0.295905	+	0.955217i	$0.404379\pi$
$350$	0	0
$351$	−1.40218	−0.0748428
$352$	0	0
$353$	6.85381	0.364792	0.182396	−	0.983225i	$-0.441615\pi$
$353$	6.85381	0.364792	0.182396	+	0.983225i	$0.441615\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.62380	0.0859404
$358$	0	0
$359$	−16.3104	−0.860832	−0.430416	−	0.902631i	$-0.641633\pi$
$359$	−16.3104	−0.860832	−0.430416	+	0.902631i	$0.641633\pi$
$360$	0	0
$361$	−18.3723	−0.966964
$362$	0	0
$363$	−2.08204	−0.109279
$364$	0	0
$365$	0	0
$366$	0	0
$367$	31.6200	1.65055	0.825275	−	0.564731i	$-0.191020\pi$
$367$	31.6200	1.65055	0.825275	+	0.564731i	$0.191020\pi$
$368$	0	0
$369$	−13.7619	−0.716414
$370$	0	0
$371$	−18.4354	−0.957118
$372$	0	0
$373$	28.2458	1.46251	0.731256	−	0.682104i	$-0.238935\pi$
$373$	28.2458	1.46251	0.731256	+	0.682104i	$0.238935\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3.54313	−0.182481
$378$	0	0
$379$	−2.98772	−0.153469	−0.0767345	−	0.997052i	$-0.524449\pi$
$379$	−2.98772	−0.153469	−0.0767345	+	0.997052i	$0.524449\pi$
$380$	0	0
$381$	−3.59997	−0.184432
$382$	0	0
$383$	−12.5176	−0.639621	−0.319810	−	0.947482i	$-0.603619\pi$
$383$	−12.5176	−0.639621	−0.319810	+	0.947482i	$0.603619\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5.19480	0.264067
$388$	0	0
$389$	−19.4861	−0.987982	−0.493991	−	0.869467i	$-0.664462\pi$
$389$	−19.4861	−0.987982	−0.493991	+	0.869467i	$0.664462\pi$
$390$	0	0
$391$	5.17593	0.261758
$392$	0	0
$393$	−0.812503	−0.0409853
$394$	0	0
$395$	0	0
$396$	0	0
$397$	17.3203	0.869281	0.434640	−	0.900604i	$-0.356876\pi$
$397$	17.3203	0.869281	0.434640	+	0.900604i	$0.356876\pi$
$398$	0	0
$399$	0.248549	0.0124430
$400$	0	0
$401$	17.3727	0.867553	0.433777	−	0.901020i	$-0.357181\pi$
$401$	17.3727	0.867553	0.433777	+	0.901020i	$0.357181\pi$
$402$	0	0
$403$	−11.0504	−0.550459
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.525499	0.0260480
$408$	0	0
$409$	−12.6874	−0.627350	−0.313675	−	0.949530i	$-0.601560\pi$
$409$	−12.6874	−0.627350	−0.313675	+	0.949530i	$0.601560\pi$
$410$	0	0
$411$	2.31983	0.114429
$412$	0	0
$413$	−20.6383	−1.01554
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−0.495098	−0.0242450
$418$	0	0
$419$	4.23468	0.206878	0.103439	−	0.994636i	$-0.467015\pi$
$419$	4.23468	0.206878	0.103439	+	0.994636i	$0.467015\pi$
$420$	0	0
$421$	4.17837	0.203641	0.101821	−	0.994803i	$-0.467533\pi$
$421$	4.17837	0.203641	0.101821	+	0.994803i	$0.467533\pi$
$422$	0	0
$423$	−29.6510	−1.44168
$424$	0	0
$425$	0	0
$426$	0	0
$427$	19.8956	0.962813
$428$	0	0
$429$	−0.0178518	−0.000861892 0
$430$	0	0
$431$	21.9197	1.05583	0.527917	−	0.849296i	$-0.322973\pi$
$431$	21.9197	1.05583	0.527917	+	0.849296i	$0.322973\pi$
$432$	0	0
$433$	−20.2195	−0.971688	−0.485844	−	0.874046i	$-0.661488\pi$
$433$	−20.2195	−0.971688	−0.485844	+	0.874046i	$0.661488\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.792262	0.0378990
$438$	0	0
$439$	16.3149	0.778666	0.389333	−	0.921097i	$-0.372706\pi$
$439$	16.3149	0.778666	0.389333	+	0.921097i	$0.372706\pi$
$440$	0	0
$441$	12.6143	0.600682
$442$	0	0
$443$	−28.0868	−1.33444	−0.667222	−	0.744859i	$-0.732517\pi$
$443$	−28.0868	−1.33444	−0.667222	+	0.744859i	$0.732517\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−3.60579	−0.170548
$448$	0	0
$449$	−24.5071	−1.15656	−0.578282	−	0.815837i	$-0.696277\pi$
$449$	−24.5071	−1.15656	−0.578282	+	0.815837i	$0.696277\pi$
$450$	0	0
$451$	−0.352537	−0.0166003
$452$	0	0
$453$	3.47033	0.163050
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−2.87128	−0.134313	−0.0671564	−	0.997742i	$-0.521393\pi$
$457$	−2.87128	−0.134313	−0.0671564	+	0.997742i	$0.521393\pi$
$458$	0	0
$459$	5.84601	0.272868
$460$	0	0
$461$	15.2264	0.709162	0.354581	−	0.935025i	$-0.384623\pi$
$461$	15.2264	0.709162	0.354581	+	0.935025i	$0.384623\pi$
$462$	0	0
$463$	36.6480	1.70318	0.851588	−	0.524212i	$-0.175640\pi$
$463$	36.6480	1.70318	0.851588	+	0.524212i	$0.175640\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−14.9765	−0.693031	−0.346516	−	0.938044i	$-0.612635\pi$
$467$	−14.9765	−0.693031	−0.346516	+	0.938044i	$0.612635\pi$
$468$	0	0
$469$	11.5311	0.532457
$470$	0	0
$471$	0.828456	0.0381732
$472$	0	0
$473$	0.133075	0.00611879
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−32.9861	−1.51033
$478$	0	0
$479$	−4.03398	−0.184317	−0.0921587	−	0.995744i	$-0.529377\pi$
$479$	−4.03398	−0.184317	−0.0921587	+	0.995744i	$0.529377\pi$
$480$	0	0
$481$	−8.59171	−0.391748
$482$	0	0
$483$	0.313721	0.0142748
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−4.28846	−0.194329	−0.0971643	−	0.995268i	$-0.530977\pi$
$487$	−4.28846	−0.194329	−0.0971643	+	0.995268i	$0.530977\pi$
$488$	0	0
$489$	2.23558	0.101096
$490$	0	0
$491$	36.8754	1.66416	0.832081	−	0.554654i	$-0.187149\pi$
$491$	36.8754	1.66416	0.832081	+	0.554654i	$0.187149\pi$
$492$	0	0
$493$	14.7721	0.665303
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−12.7832	−0.573405
$498$	0	0
$499$	5.74707	0.257274	0.128637	−	0.991692i	$-0.458940\pi$
$499$	5.74707	0.257274	0.128637	+	0.991692i	$0.458940\pi$
$500$	0	0
$501$	0.984298	0.0439752
$502$	0	0
$503$	−18.7850	−0.837583	−0.418792	−	0.908082i	$-0.637546\pi$
$503$	−18.7850	−0.837583	−0.418792	+	0.908082i	$0.637546\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−2.17001	−0.0963735
$508$	0	0
$509$	−20.2669	−0.898313	−0.449157	−	0.893453i	$-0.648275\pi$
$509$	−20.2669	−0.898313	−0.449157	+	0.893453i	$0.648275\pi$
$510$	0	0
$511$	−12.4159	−0.549247
$512$	0	0
$513$	0.894829	0.0395077
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−0.759568	−0.0334058
$518$	0	0
$519$	0.495850	0.0217654
$520$	0	0
$521$	6.47593	0.283716	0.141858	−	0.989887i	$-0.454692\pi$
$521$	6.47593	0.283716	0.141858	+	0.989887i	$0.454692\pi$
$522$	0	0
$523$	−0.414671	−0.0181323	−0.00906615	−	0.999959i	$-0.502886\pi$
$523$	−0.414671	−0.0181323	−0.00906615	+	0.999959i	$0.502886\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	46.0716	2.00691
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−36.9276	−1.60252
$532$	0	0
$533$	5.76384	0.249659
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−0.118845	−0.00512854
$538$	0	0
$539$	0.323140	0.0139186
$540$	0	0
$541$	28.5708	1.22835	0.614177	−	0.789169i	$-0.289488\pi$
$541$	28.5708	1.22835	0.614177	+	0.789169i	$0.289488\pi$
$542$	0	0
$543$	−1.16626	−0.0500492
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−39.0974	−1.67168	−0.835841	−	0.548971i	$-0.815020\pi$
$547$	−39.0974	−1.67168	−0.835841	+	0.548971i	$0.815020\pi$
$548$	0	0
$549$	35.5987	1.51932
$550$	0	0
$551$	2.26112	0.0963270
$552$	0	0
$553$	−25.9450	−1.10329
$554$	0	0
$555$	0	0
$556$	0	0
$557$	33.9136	1.43696	0.718482	−	0.695546i	$-0.244837\pi$
$557$	33.9136	1.43696	0.718482	+	0.695546i	$0.244837\pi$
$558$	0	0
$559$	−2.17572	−0.0920232
$560$	0	0
$561$	0.0744281	0.00314236
$562$	0	0
$563$	25.6390	1.08056	0.540278	−	0.841486i	$-0.318319\pi$
$563$	25.6390	1.08056	0.540278	+	0.841486i	$0.318319\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−14.3769	−0.603773
$568$	0	0
$569$	−13.6257	−0.571218	−0.285609	−	0.958346i	$-0.592196\pi$
$569$	−13.6257	−0.571218	−0.285609	+	0.958346i	$0.592196\pi$
$570$	0	0
$571$	−0.354294	−0.0148267	−0.00741337	−	0.999973i	$-0.502360\pi$
$571$	−0.354294	−0.0148267	−0.00741337	+	0.999973i	$0.502360\pi$
$572$	0	0
$573$	3.62356	0.151376
$574$	0	0
$575$	0	0
$576$	0	0
$577$	18.5870	0.773788	0.386894	−	0.922124i	$-0.373548\pi$
$577$	18.5870	0.773788	0.386894	+	0.922124i	$0.373548\pi$
$578$	0	0
$579$	0.153564	0.00638192
$580$	0	0
$581$	−11.0747	−0.459457
$582$	0	0
$583$	−0.845002	−0.0349964
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7.75193	0.319956	0.159978	−	0.987121i	$-0.448858\pi$
$587$	7.75193	0.319956	0.159978	+	0.987121i	$0.448858\pi$
$588$	0	0
$589$	7.05202	0.290573
$590$	0	0
$591$	1.16487	0.0479162
$592$	0	0
$593$	23.3828	0.960215	0.480107	−	0.877210i	$-0.340598\pi$
$593$	23.3828	0.960215	0.480107	+	0.877210i	$0.340598\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1.33142	−0.0544916
$598$	0	0
$599$	25.6075	1.04629	0.523146	−	0.852243i	$-0.324758\pi$
$599$	25.6075	1.04629	0.523146	+	0.852243i	$0.324758\pi$
$600$	0	0
$601$	4.03750	0.164693	0.0823466	−	0.996604i	$-0.473759\pi$
$601$	4.03750	0.164693	0.0823466	+	0.996604i	$0.473759\pi$
$602$	0	0
$603$	20.6324	0.840215
$604$	0	0
$605$	0	0
$606$	0	0
$607$	45.8176	1.85968	0.929839	−	0.367967i	$-0.119946\pi$
$607$	45.8176	1.85968	0.929839	+	0.367967i	$0.119946\pi$
$608$	0	0
$609$	0.895361	0.0362819
$610$	0	0
$611$	12.4186	0.502405
$612$	0	0
$613$	29.1714	1.17822	0.589112	−	0.808052i	$-0.299478\pi$
$613$	29.1714	1.17822	0.589112	+	0.808052i	$0.299478\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−17.0562	−0.686659	−0.343329	−	0.939215i	$-0.611555\pi$
$617$	−17.0562	−0.686659	−0.343329	+	0.939215i	$0.611555\pi$
$618$	0	0
$619$	−36.0620	−1.44946	−0.724728	−	0.689035i	$-0.758034\pi$
$619$	−36.0620	−1.44946	−0.724728	+	0.689035i	$0.758034\pi$
$620$	0	0
$621$	1.12946	0.0453237
$622$	0	0
$623$	5.02503	0.201324
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0.0113925	0.000454971 0
$628$	0	0
$629$	35.8208	1.42827
$630$	0	0
$631$	−17.4958	−0.696496	−0.348248	−	0.937402i	$-0.613223\pi$
$631$	−17.4958	−0.696496	−0.348248	+	0.937402i	$0.613223\pi$
$632$	0	0
$633$	−1.53795	−0.0611282
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−5.28322	−0.209329
$638$	0	0
$639$	−22.8727	−0.904832
$640$	0	0
$641$	−39.9200	−1.57675	−0.788373	−	0.615197i	$-0.789076\pi$
$641$	−39.9200	−1.57675	−0.788373	+	0.615197i	$0.789076\pi$
$642$	0	0
$643$	41.8187	1.64917	0.824584	−	0.565740i	$-0.191409\pi$
$643$	41.8187	1.64917	0.824584	+	0.565740i	$0.191409\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−38.8026	−1.52549	−0.762743	−	0.646702i	$-0.776148\pi$
$647$	−38.8026	−1.52549	−0.762743	+	0.646702i	$0.776148\pi$
$648$	0	0
$649$	−0.945972	−0.0371327
$650$	0	0
$651$	2.79247	0.109445
$652$	0	0
$653$	39.8995	1.56139	0.780694	−	0.624914i	$-0.214866\pi$
$653$	39.8995	1.56139	0.780694	+	0.624914i	$0.214866\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−22.2155	−0.866709
$658$	0	0
$659$	−22.9449	−0.893808	−0.446904	−	0.894582i	$-0.647473\pi$
$659$	−22.9449	−0.893808	−0.446904	+	0.894582i	$0.647473\pi$
$660$	0	0
$661$	19.2759	0.749745	0.374873	−	0.927076i	$-0.377686\pi$
$661$	19.2759	0.749745	0.374873	+	0.927076i	$0.377686\pi$
$662$	0	0
$663$	−1.21687	−0.0472593
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.85401	0.110508
$668$	0	0
$669$	0.542121	0.0209596
$670$	0	0
$671$	0.911930	0.0352047
$672$	0	0
$673$	−38.0041	−1.46495	−0.732476	−	0.680793i	$-0.761635\pi$
$673$	−38.0041	−1.46495	−0.732476	+	0.680793i	$0.761635\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	41.0387	1.57725	0.788623	−	0.614878i	$-0.210795\pi$
$677$	41.0387	1.57725	0.788623	+	0.614878i	$0.210795\pi$
$678$	0	0
$679$	−5.33008	−0.204550
$680$	0	0
$681$	5.19835	0.199201
$682$	0	0
$683$	−27.6705	−1.05878	−0.529390	−	0.848378i	$-0.677579\pi$
$683$	−27.6705	−1.05878	−0.529390	+	0.848378i	$0.677579\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0.210810	0.00804290
$688$	0	0
$689$	13.8155	0.526327
$690$	0	0
$691$	−33.8078	−1.28611	−0.643056	−	0.765820i	$-0.722334\pi$
$691$	−33.8078	−1.28611	−0.643056	+	0.765820i	$0.722334\pi$
$692$	0	0
$693$	−0.372858	−0.0141637
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−24.0307	−0.910229
$698$	0	0
$699$	−4.03288	−0.152538
$700$	0	0
$701$	−46.4878	−1.75582	−0.877909	−	0.478827i	$-0.841062\pi$
$701$	−46.4878	−1.75582	−0.877909	+	0.478827i	$0.841062\pi$
$702$	0	0
$703$	5.48297	0.206794
$704$	0	0
$705$	0	0
$706$	0	0
$707$	27.5094	1.03460
$708$	0	0
$709$	−13.5298	−0.508122	−0.254061	−	0.967188i	$-0.581766\pi$
$709$	−13.5298	−0.508122	−0.254061	+	0.967188i	$0.581766\pi$
$710$	0	0
$711$	−46.4229	−1.74099
$712$	0	0
$713$	8.90112	0.333350
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−1.45447	−0.0543182
$718$	0	0
$719$	2.64955	0.0988115	0.0494057	−	0.998779i	$-0.484267\pi$
$719$	2.64955	0.0988115	0.0494057	+	0.998779i	$0.484267\pi$
$720$	0	0
$721$	−12.0972	−0.450525
$722$	0	0
$723$	1.59544	0.0593351
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−46.8796	−1.73867	−0.869334	−	0.494226i	$-0.835452\pi$
$727$	−46.8796	−1.73867	−0.869334	+	0.494226i	$0.835452\pi$
$728$	0	0
$729$	−25.0826	−0.928987
$730$	0	0
$731$	9.07108	0.335506
$732$	0	0
$733$	36.1389	1.33482	0.667410	−	0.744691i	$-0.267403\pi$
$733$	36.1389	1.33482	0.667410	+	0.744691i	$0.267403\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0.528538	0.0194690
$738$	0	0
$739$	−22.6473	−0.833095	−0.416547	−	0.909114i	$-0.636760\pi$
$739$	−22.6473	−0.833095	−0.416547	+	0.909114i	$0.636760\pi$
$740$	0	0
$741$	−0.186262	−0.00684252
$742$	0	0
$743$	−7.04015	−0.258278	−0.129139	−	0.991627i	$-0.541221\pi$
$743$	−7.04015	−0.258278	−0.129139	+	0.991627i	$0.541221\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−19.8158	−0.725021
$748$	0	0
$749$	−0.591304	−0.0216058
$750$	0	0
$751$	5.94539	0.216950	0.108475	−	0.994099i	$-0.465403\pi$
$751$	5.94539	0.216950	0.108475	+	0.994099i	$0.465403\pi$
$752$	0	0
$753$	0.485198	0.0176816
$754$	0	0
$755$	0	0
$756$	0	0
$757$	43.9519	1.59746	0.798729	−	0.601691i	$-0.205506\pi$
$757$	43.9519	1.59746	0.798729	+	0.601691i	$0.205506\pi$
$758$	0	0
$759$	0.0143797	0.000521949 0
$760$	0	0
$761$	22.0290	0.798549	0.399274	−	0.916831i	$-0.369262\pi$
$761$	22.0290	0.798549	0.399274	+	0.916831i	$0.369262\pi$
$762$	0	0
$763$	−23.2520	−0.841779
$764$	0	0
$765$	0	0
$766$	0	0
$767$	15.4663	0.558455
$768$	0	0
$769$	−17.6958	−0.638128	−0.319064	−	0.947733i	$-0.603368\pi$
$769$	−17.6958	−0.638128	−0.319064	+	0.947733i	$0.603368\pi$
$770$	0	0
$771$	−0.833250	−0.0300088
$772$	0	0
$773$	1.63758	0.0588997	0.0294498	−	0.999566i	$-0.490624\pi$
$773$	1.63758	0.0588997	0.0294498	+	0.999566i	$0.490624\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	2.17115	0.0778896
$778$	0	0
$779$	−3.67831	−0.131789
$780$	0	0
$781$	−0.585930	−0.0209662
$782$	0	0
$783$	3.22349	0.115198
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−27.6760	−0.986543	−0.493272	−	0.869875i	$-0.664199\pi$
$787$	−27.6760	−0.986543	−0.493272	+	0.869875i	$0.664199\pi$
$788$	0	0
$789$	4.70618	0.167544
$790$	0	0
$791$	1.95354	0.0694598
$792$	0	0
$793$	−14.9097	−0.529459
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−9.29831	−0.329363	−0.164681	−	0.986347i	$-0.552660\pi$
$797$	−9.29831	−0.329363	−0.164681	+	0.986347i	$0.552660\pi$
$798$	0	0
$799$	−51.7761	−1.83171
$800$	0	0
$801$	8.99119	0.317688
$802$	0	0
$803$	−0.569093	−0.0200829
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−2.05931	−0.0724911
$808$	0	0
$809$	48.8800	1.71853	0.859264	−	0.511532i	$-0.170922\pi$
$809$	48.8800	1.71853	0.859264	+	0.511532i	$0.170922\pi$
$810$	0	0
$811$	−44.0615	−1.54721	−0.773604	−	0.633669i	$-0.781548\pi$
$811$	−44.0615	−1.54721	−0.773604	+	0.633669i	$0.781548\pi$
$812$	0	0
$813$	0.492926	0.0172877
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1.38848	0.0485768
$818$	0	0
$819$	6.09608	0.213014
$820$	0	0
$821$	30.1832	1.05340	0.526700	−	0.850051i	$-0.323429\pi$
$821$	30.1832	1.05340	0.526700	+	0.850051i	$0.323429\pi$
$822$	0	0
$823$	32.0171	1.11605	0.558023	−	0.829826i	$-0.311560\pi$
$823$	32.0171	1.11605	0.558023	+	0.829826i	$0.311560\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	20.5192	0.713523	0.356761	−	0.934196i	$-0.383881\pi$
$827$	20.5192	0.713523	0.356761	+	0.934196i	$0.383881\pi$
$828$	0	0
$829$	38.0551	1.32171	0.660854	−	0.750515i	$-0.270194\pi$
$829$	38.0551	1.32171	0.660854	+	0.750515i	$0.270194\pi$
$830$	0	0
$831$	2.30884	0.0800926
$832$	0	0
$833$	22.0269	0.763188
$834$	0	0
$835$	0	0
$836$	0	0
$837$	10.0535	0.347499
$838$	0	0
$839$	−24.6946	−0.852553	−0.426277	−	0.904593i	$-0.640175\pi$
$839$	−24.6946	−0.852553	−0.426277	+	0.904593i	$0.640175\pi$
$840$	0	0
$841$	−20.8547	−0.719126
$842$	0	0
$843$	1.10797	0.0381607
$844$	0	0
$845$	0	0
$846$	0	0
$847$	18.2131	0.625811
$848$	0	0
$849$	2.03054	0.0696880
$850$	0	0
$851$	6.92065	0.237237
$852$	0	0
$853$	−23.4770	−0.803837	−0.401919	−	0.915675i	$-0.631657\pi$
$853$	−23.4770	−0.803837	−0.401919	+	0.915675i	$0.631657\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−20.9632	−0.716090	−0.358045	−	0.933704i	$-0.616557\pi$
$857$	−20.9632	−0.716090	−0.358045	+	0.933704i	$0.616557\pi$
$858$	0	0
$859$	−35.7273	−1.21900	−0.609500	−	0.792786i	$-0.708630\pi$
$859$	−35.7273	−1.21900	−0.609500	+	0.792786i	$0.708630\pi$
$860$	0	0
$861$	−1.45654	−0.0496388
$862$	0	0
$863$	19.5880	0.666784	0.333392	−	0.942788i	$-0.391807\pi$
$863$	19.5880	0.666784	0.333392	+	0.942788i	$0.391807\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	1.85403	0.0629660
$868$	0	0
$869$	−1.18921	−0.0403412
$870$	0	0
$871$	−8.64139	−0.292802
$872$	0	0
$873$	−9.53702	−0.322779
$874$	0	0
$875$	0	0
$876$	0	0
$877$	49.5791	1.67417	0.837084	−	0.547074i	$-0.184259\pi$
$877$	49.5791	1.67417	0.837084	+	0.547074i	$0.184259\pi$
$878$	0	0
$879$	0.516230	0.0174120
$880$	0	0
$881$	21.4406	0.722353	0.361177	−	0.932497i	$-0.382375\pi$
$881$	21.4406	0.722353	0.361177	+	0.932497i	$0.382375\pi$
$882$	0	0
$883$	52.7646	1.77567	0.887836	−	0.460161i	$-0.152208\pi$
$883$	52.7646	1.77567	0.887836	+	0.460161i	$0.152208\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−30.1462	−1.01221	−0.506105	−	0.862472i	$-0.668915\pi$
$887$	−30.1462	−1.01221	−0.506105	+	0.862472i	$0.668915\pi$
$888$	0	0
$889$	31.4917	1.05620
$890$	0	0
$891$	−0.658978	−0.0220766
$892$	0	0
$893$	−7.92520	−0.265207
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−0.235102	−0.00784982
$898$	0	0
$899$	25.4039	0.847266
$900$	0	0
$901$	−57.5998	−1.91893
$902$	0	0
$903$	0.549812	0.0182966
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−20.3987	−0.677329	−0.338665	−	0.940907i	$-0.609975\pi$
$907$	−20.3987	−0.677329	−0.338665	+	0.940907i	$0.609975\pi$
$908$	0	0
$909$	49.2221	1.63259
$910$	0	0
$911$	57.7921	1.91474	0.957369	−	0.288869i	$-0.0932792\pi$
$911$	57.7921	1.91474	0.957369	+	0.288869i	$0.0932792\pi$
$912$	0	0
$913$	−0.507619	−0.0167998
$914$	0	0
$915$	0	0
$916$	0	0
$917$	7.10757	0.234713
$918$	0	0
$919$	26.0255	0.858502	0.429251	−	0.903185i	$-0.358778\pi$
$919$	26.0255	0.858502	0.429251	+	0.903185i	$0.358778\pi$
$920$	0	0
$921$	4.26055	0.140390
$922$	0	0
$923$	9.57972	0.315320
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−21.6454	−0.710927
$928$	0	0
$929$	26.9839	0.885313	0.442656	−	0.896691i	$-0.354036\pi$
$929$	26.9839	0.885313	0.442656	+	0.896691i	$0.354036\pi$
$930$	0	0
$931$	3.37159	0.110499
$932$	0	0
$933$	−3.40112	−0.111348
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−35.8201	−1.17019	−0.585096	−	0.810964i	$-0.698943\pi$
$937$	−35.8201	−1.17019	−0.585096	+	0.810964i	$0.698943\pi$
$938$	0	0
$939$	2.74079	0.0894424
$940$	0	0
$941$	−32.0416	−1.04453	−0.522263	−	0.852785i	$-0.674912\pi$
$941$	−32.0416	−1.04453	−0.522263	+	0.852785i	$0.674912\pi$
$942$	0	0
$943$	−4.64279	−0.151190
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−8.53256	−0.277271	−0.138635	−	0.990343i	$-0.544272\pi$
$947$	−8.53256	−0.277271	−0.138635	+	0.990343i	$0.544272\pi$
$948$	0	0
$949$	9.30445	0.302035
$950$	0	0
$951$	−1.41594	−0.0459150
$952$	0	0
$953$	−42.4054	−1.37365	−0.686823	−	0.726825i	$-0.740995\pi$
$953$	−42.4054	−1.37365	−0.686823	+	0.726825i	$0.740995\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0.0410397	0.00132662
$958$	0	0
$959$	−20.2933	−0.655305
$960$	0	0
$961$	48.2300	1.55581
$962$	0	0
$963$	−1.05801	−0.0340938
$964$	0	0
$965$	0	0
$966$	0	0
$967$	10.0988	0.324755	0.162377	−	0.986729i	$-0.448084\pi$
$967$	10.0988	0.324755	0.162377	+	0.986729i	$0.448084\pi$
$968$	0	0
$969$	0.776570	0.0249470
$970$	0	0
$971$	15.7943	0.506863	0.253431	−	0.967353i	$-0.418441\pi$
$971$	15.7943	0.506863	0.253431	+	0.967353i	$0.418441\pi$
$972$	0	0
$973$	4.33099	0.138845
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−27.0055	−0.863984	−0.431992	−	0.901877i	$-0.642189\pi$
$977$	−27.0055	−0.863984	−0.431992	+	0.901877i	$0.642189\pi$
$978$	0	0
$979$	0.230327	0.00736128
$980$	0	0
$981$	−41.6043	−1.32832
$982$	0	0
$983$	16.0647	0.512384	0.256192	−	0.966626i	$-0.417532\pi$
$983$	16.0647	0.512384	0.256192	+	0.966626i	$0.417532\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−3.13823	−0.0998910
$988$	0	0
$989$	1.75255	0.0557279
$990$	0	0
$991$	−51.4433	−1.63415	−0.817075	−	0.576532i	$-0.804406\pi$
$991$	−51.4433	−1.63415	−0.817075	+	0.576532i	$0.804406\pi$
$992$	0	0
$993$	5.18066	0.164403
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−39.2639	−1.24350	−0.621751	−	0.783215i	$-0.713578\pi$
$997$	−39.2639	−1.24350	−0.621751	+	0.783215i	$0.713578\pi$
$998$	0	0
$999$	7.81660	0.247306

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9200.2.a.dc.1.3		7
4.3	odd	2		4600.2.a.bh.1.5		7
5.2	odd	4		1840.2.e.g.369.7		14
5.3	odd	4		1840.2.e.g.369.8		14
5.4	even	2		9200.2.a.cz.1.5		7
20.3	even	4		920.2.e.b.369.7	✓	14
20.7	even	4		920.2.e.b.369.8	yes	14
20.19	odd	2		4600.2.a.bi.1.3		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.e.b.369.7	✓	14	20.3	even	4
920.2.e.b.369.8	yes	14	20.7	even	4
1840.2.e.g.369.7		14	5.2	odd	4
1840.2.e.g.369.8		14	5.3	odd	4
4600.2.a.bh.1.5		7	4.3	odd	2
4600.2.a.bi.1.3		7	20.19	odd	2
9200.2.a.cz.1.5		7	5.4	even	2
9200.2.a.dc.1.3		7	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)

\(f(q)\)	\(=\)	\(q+0.189375 q^{3} -1.65661 q^{7} -2.96414 q^{9} +O(q^{10})\)	\(q+0.189375 q^{3} -1.65661 q^{7} -2.96414 q^{9} -0.0759321 q^{11} +1.24146 q^{13} -5.17593 q^{17} -0.792262 q^{19} -0.313721 q^{21} -1.00000 q^{23} -1.12946 q^{27} -2.85401 q^{29} -8.90112 q^{31} -0.0143797 q^{33} -6.92065 q^{37} +0.235102 q^{39} +4.64279 q^{41} -1.75255 q^{43} +10.0033 q^{47} -4.25565 q^{49} -0.980193 q^{51} +11.1284 q^{53} -0.150035 q^{57} +12.4581 q^{59} -12.0098 q^{61} +4.91041 q^{63} -6.96067 q^{67} -0.189375 q^{69} +7.71650 q^{71} +7.49476 q^{73} +0.125790 q^{77} +15.6615 q^{79} +8.67852 q^{81} +6.68518 q^{83} -0.540478 q^{87} -3.03333 q^{89} -2.05661 q^{91} -1.68565 q^{93} +3.21747 q^{97} +0.225073 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 3 q^{3} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) 7 * q + 3 * q^3 + 4 * q^7 + 2 * q^9	\( 7 q + 3 q^{3} + 4 q^{7} + 2 q^{9} + 7 q^{11} + 7 q^{13} + 7 q^{19} - 6 q^{21} - 7 q^{23} - 11 q^{29} + 10 q^{31} + 19 q^{33} + 19 q^{37} + 24 q^{39} - 16 q^{41} + 6 q^{43} + 6 q^{47} - 17 q^{49} + 7 q^{51} + 15 q^{53} + 8 q^{57} + 11 q^{59} + 5 q^{61} + 13 q^{63} + 9 q^{67} - 3 q^{69} + 14 q^{71} + 10 q^{73} + 6 q^{77} + 32 q^{79} - 5 q^{81} + q^{83} + 10 q^{87} - 24 q^{89} + 7 q^{91} + 26 q^{93} - 7 q^{97} + 61 q^{99}+O(q^{100}) \) 7 * q + 3 * q^3 + 4 * q^7 + 2 * q^9 + 7 * q^11 + 7 * q^13 + 7 * q^19 - 6 * q^21 - 7 * q^23 - 11 * q^29 + 10 * q^31 + 19 * q^33 + 19 * q^37 + 24 * q^39 - 16 * q^41 + 6 * q^43 + 6 * q^47 - 17 * q^49 + 7 * q^51 + 15 * q^53 + 8 * q^57 + 11 * q^59 + 5 * q^61 + 13 * q^63 + 9 * q^67 - 3 * q^69 + 14 * q^71 + 10 * q^73 + 6 * q^77 + 32 * q^79 - 5 * q^81 + q^83 + 10 * q^87 - 24 * q^89 + 7 * q^91 + 26 * q^93 - 7 * q^97 + 61 * q^99

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