LMFDB - Embedded newform 7700.2.e.j.1849.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7700,2,Mod(1849,7700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7700, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("7700.1849");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7700 = 2^{2} \cdot 5^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7700.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:	$61.4848095564$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 308)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1849.4
Root			$-1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	7700.1849
Dual form			7700.2.e.j.1849.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.44949i q^{3} +1.00000i q^{7} -3.00000 q^{9} +O(q^{10})$	$q+2.44949i q^{3} +1.00000i q^{7} -3.00000 q^{9} -1.00000 q^{11} -0.449490i q^{13} +0.449490i q^{17} +4.89898 q^{19} -2.44949 q^{21} -0.898979i q^{23} -2.89898 q^{29} +6.44949 q^{31} -2.44949i q^{33} -4.00000i q^{37} +1.10102 q^{39} -9.34847 q^{41} +2.89898i q^{43} +6.44949i q^{47} -1.00000 q^{49} -1.10102 q^{51} +9.79796i q^{53} +12.0000i q^{57} -7.34847 q^{59} +5.34847 q^{61} -3.00000i q^{63} +14.6969i q^{67} +2.20204 q^{69} -12.8990 q^{71} +12.4495i q^{73} -1.00000i q^{77} +10.8990 q^{79} -9.00000 q^{81} -16.8990i q^{83} -7.10102i q^{87} -6.00000 q^{89} +0.449490 q^{91} +15.7980i q^{93} -5.10102i q^{97} +3.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{9}+O(q^{10}) $ 4 * q - 12 * q^9	$ 4 q - 12 q^{9} - 4 q^{11} + 8 q^{29} + 16 q^{31} + 24 q^{39} - 8 q^{41} - 4 q^{49} - 24 q^{51} - 8 q^{61} + 48 q^{69} - 32 q^{71} + 24 q^{79} - 36 q^{81} - 24 q^{89} - 8 q^{91} + 12 q^{99}+O(q^{100}) $ 4 * q - 12 * q^9 - 4 * q^11 + 8 * q^29 + 16 * q^31 + 24 * q^39 - 8 * q^41 - 4 * q^49 - 24 * q^51 - 8 * q^61 + 48 * q^69 - 32 * q^71 + 24 * q^79 - 36 * q^81 - 24 * q^89 - 8 * q^91 + 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/7700\mathbb{Z}\right)^\times$.

$n$	$2201$	$3851$	$5601$	$6777$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	2.44949i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$3$	2.44949i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	0	0
$9$	−3.00000	−1.00000
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	− 0.449490i	− 0.124666i	−0.998055	−	0.0623330i	$-0.980146\pi$
$13$	− 0.449490i	− 0.124666i	0.998055	−	0.0623330i	$-0.0198541\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.449490i	0.109017i	0.998513	+	0.0545086i	$0.0173592\pi$
$17$	0.449490i	0.109017i	−0.998513	+	0.0545086i	$0.982641\pi$
$18$	0	0
$19$	4.89898	1.12390	0.561951	−	0.827170i	$-0.310051\pi$
$19$	4.89898	1.12390	0.561951	+	0.827170i	$0.310051\pi$
$20$	0	0
$21$	−2.44949	−0.534522
$22$	0	0
$23$	− 0.898979i	− 0.187450i	−0.995598	−	0.0937251i	$-0.970123\pi$
$23$	− 0.898979i	− 0.187450i	0.995598	−	0.0937251i	$-0.0298775\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2.89898	−0.538327	−0.269163	−	0.963095i	$-0.586747\pi$
$29$	−2.89898	−0.538327	−0.269163	+	0.963095i	$0.586747\pi$
$30$	0	0
$31$	6.44949	1.15836	0.579181	−	0.815199i	$-0.303372\pi$
$31$	6.44949	1.15836	0.579181	+	0.815199i	$0.303372\pi$
$32$	0	0
$33$	− 2.44949i	− 0.426401i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 4.00000i	− 0.657596i	−0.944400	−	0.328798i	$-0.893356\pi$
$37$	− 4.00000i	− 0.657596i	0.944400	−	0.328798i	$-0.106644\pi$
$38$	0	0
$39$	1.10102	0.176304
$40$	0	0
$41$	−9.34847	−1.45999	−0.729993	−	0.683455i	$-0.760477\pi$
$41$	−9.34847	−1.45999	−0.729993	+	0.683455i	$0.760477\pi$
$42$	0	0
$43$	2.89898i	0.442090i	0.975264	+	0.221045i	$0.0709468\pi$
$43$	2.89898i	0.442090i	−0.975264	+	0.221045i	$0.929053\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.44949i	0.940755i	0.882465	+	0.470377i	$0.155882\pi$
$47$	6.44949i	0.940755i	−0.882465	+	0.470377i	$0.844118\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	−1.10102	−0.154174
$52$	0	0
$53$	9.79796i	1.34585i	0.739709	+	0.672927i	$0.234963\pi$
$53$	9.79796i	1.34585i	−0.739709	+	0.672927i	$0.765037\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	12.0000i	1.58944i
$58$	0	0
$59$	−7.34847	−0.956689	−0.478345	−	0.878172i	$-0.658763\pi$
$59$	−7.34847	−0.956689	−0.478345	+	0.878172i	$0.658763\pi$
$60$	0	0
$61$	5.34847	0.684801	0.342401	−	0.939554i	$-0.388760\pi$
$61$	5.34847	0.684801	0.342401	+	0.939554i	$0.388760\pi$
$62$	0	0
$63$	− 3.00000i	− 0.377964i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	14.6969i	1.79552i	0.440488	+	0.897758i	$0.354805\pi$
$67$	14.6969i	1.79552i	−0.440488	+	0.897758i	$0.645195\pi$
$68$	0	0
$69$	2.20204	0.265095
$70$	0	0
$71$	−12.8990	−1.53083	−0.765414	−	0.643539i	$-0.777466\pi$
$71$	−12.8990	−1.53083	−0.765414	+	0.643539i	$0.777466\pi$
$72$	0	0
$73$	12.4495i	1.45710i	0.684991	+	0.728551i	$0.259806\pi$
$73$	12.4495i	1.45710i	−0.684991	+	0.728551i	$0.740194\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1.00000i	− 0.113961i
$78$	0	0
$79$	10.8990	1.22623	0.613115	−	0.789993i	$-0.289916\pi$
$79$	10.8990	1.22623	0.613115	+	0.789993i	$0.289916\pi$
$80$	0	0
$81$	−9.00000	−1.00000
$82$	0	0
$83$	− 16.8990i	− 1.85490i	−0.373942	−	0.927452i	$-0.621994\pi$
$83$	− 16.8990i	− 1.85490i	0.373942	−	0.927452i	$-0.378006\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 7.10102i	− 0.761309i
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	0.449490	0.0471193
$92$	0	0
$93$	15.7980i	1.63817i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 5.10102i	− 0.517930i	−0.965887	−	0.258965i	$-0.916619\pi$
$97$	− 5.10102i	− 0.517930i	0.965887	−	0.258965i	$-0.0833815\pi$
$98$	0	0
$99$	3.00000	0.301511
$100$	0	0
$101$	−8.44949	−0.840756	−0.420378	−	0.907349i	$-0.638102\pi$
$101$	−8.44949	−0.840756	−0.420378	+	0.907349i	$0.638102\pi$
$102$	0	0
$103$	14.4495i	1.42375i	0.702306	+	0.711875i	$0.252154\pi$
$103$	14.4495i	1.42375i	−0.702306	+	0.711875i	$0.747846\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.00000i	0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$	4.00000i	0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$	0	0
$109$	−14.8990	−1.42706	−0.713532	−	0.700623i	$-0.752906\pi$
$109$	−14.8990	−1.42706	−0.713532	+	0.700623i	$0.752906\pi$
$110$	0	0
$111$	9.79796	0.929981
$112$	0	0
$113$	− 10.0000i	− 0.940721i	−0.882474	−	0.470360i	$-0.844124\pi$
$113$	− 10.0000i	− 0.940721i	0.882474	−	0.470360i	$-0.155876\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.34847i	0.124666i
$118$	0	0
$119$	−0.449490	−0.0412047
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	− 22.8990i	− 2.06473i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1.79796i	0.159543i	0.996813	+	0.0797715i	$0.0254191\pi$
$127$	1.79796i	0.159543i	−0.996813	+	0.0797715i	$0.974581\pi$
$128$	0	0
$129$	−7.10102	−0.625210
$130$	0	0
$131$	9.79796	0.856052	0.428026	−	0.903767i	$-0.359209\pi$
$131$	9.79796	0.856052	0.428026	+	0.903767i	$0.359209\pi$
$132$	0	0
$133$	4.89898i	0.424795i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 9.79796i	− 0.837096i	−0.908195	−	0.418548i	$-0.862539\pi$
$137$	− 9.79796i	− 0.837096i	0.908195	−	0.418548i	$-0.137461\pi$
$138$	0	0
$139$	−22.6969	−1.92513	−0.962565	−	0.271052i	$-0.912628\pi$
$139$	−22.6969	−1.92513	−0.962565	+	0.271052i	$0.912628\pi$
$140$	0	0
$141$	−15.7980	−1.33043
$142$	0	0
$143$	0.449490i	0.0375882i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 2.44949i	− 0.202031i
$148$	0	0
$149$	−11.7980	−0.966526	−0.483263	−	0.875475i	$-0.660549\pi$
$149$	−11.7980	−0.966526	−0.483263	+	0.875475i	$0.660549\pi$
$150$	0	0
$151$	8.69694	0.707747	0.353873	−	0.935293i	$-0.384864\pi$
$151$	8.69694	0.707747	0.353873	+	0.935293i	$0.384864\pi$
$152$	0	0
$153$	− 1.34847i	− 0.109017i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 10.0000i	− 0.798087i	−0.916932	−	0.399043i	$-0.869342\pi$
$157$	− 10.0000i	− 0.798087i	0.916932	−	0.399043i	$-0.130658\pi$
$158$	0	0
$159$	−24.0000	−1.90332
$160$	0	0
$161$	0.898979	0.0708495
$162$	0	0
$163$	8.89898i	0.697022i	0.937305	+	0.348511i	$0.113313\pi$
$163$	8.89898i	0.697022i	−0.937305	+	0.348511i	$0.886687\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 13.7980i	− 1.06772i	−0.845573	−	0.533859i	$-0.820741\pi$
$167$	− 13.7980i	− 1.06772i	0.845573	−	0.533859i	$-0.179259\pi$
$168$	0	0
$169$	12.7980	0.984458
$170$	0	0
$171$	−14.6969	−1.12390
$172$	0	0
$173$	10.2474i	0.779099i	0.921006	+	0.389550i	$0.127369\pi$
$173$	10.2474i	0.779099i	−0.921006	+	0.389550i	$0.872631\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 18.0000i	− 1.35296i
$178$	0	0
$179$	15.5959	1.16569	0.582847	−	0.812582i	$-0.301939\pi$
$179$	15.5959	1.16569	0.582847	+	0.812582i	$0.301939\pi$
$180$	0	0
$181$	9.10102	0.676474	0.338237	−	0.941061i	$-0.390170\pi$
$181$	9.10102	0.676474	0.338237	+	0.941061i	$0.390170\pi$
$182$	0	0
$183$	13.1010i	0.968455i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 0.449490i	− 0.0328699i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1.79796	−0.130096	−0.0650479	−	0.997882i	$-0.520720\pi$
$191$	−1.79796	−0.130096	−0.0650479	+	0.997882i	$0.520720\pi$
$192$	0	0
$193$	21.5959i	1.55451i	0.629187	+	0.777254i	$0.283388\pi$
$193$	21.5959i	1.55451i	−0.629187	+	0.777254i	$0.716612\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	25.5959i	1.82363i	0.410597	+	0.911817i	$0.365320\pi$
$197$	25.5959i	1.82363i	−0.410597	+	0.911817i	$0.634680\pi$
$198$	0	0
$199$	−26.0454	−1.84631	−0.923155	−	0.384428i	$-0.874399\pi$
$199$	−26.0454	−1.84631	−0.923155	+	0.384428i	$0.874399\pi$
$200$	0	0
$201$	−36.0000	−2.53924
$202$	0	0
$203$	− 2.89898i	− 0.203468i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.69694i	0.187450i
$208$	0	0
$209$	−4.89898	−0.338869
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	0	0
$213$	− 31.5959i	− 2.16492i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	6.44949i	0.437820i
$218$	0	0
$219$	−30.4949	−2.06065
$220$	0	0
$221$	0.202041	0.0135908
$222$	0	0
$223$	− 13.1464i	− 0.880350i	−0.897912	−	0.440175i	$-0.854916\pi$
$223$	− 13.1464i	− 0.880350i	0.897912	−	0.440175i	$-0.145084\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 1.79796i	− 0.119335i	−0.998218	−	0.0596674i	$-0.980996\pi$
$227$	− 1.79796i	− 0.119335i	0.998218	−	0.0596674i	$-0.0190040\pi$
$228$	0	0
$229$	5.10102	0.337085	0.168542	−	0.985694i	$-0.446094\pi$
$229$	5.10102	0.337085	0.168542	+	0.985694i	$0.446094\pi$
$230$	0	0
$231$	2.44949	0.161165
$232$	0	0
$233$	− 1.10102i	− 0.0721303i	−0.999349	−	0.0360651i	$-0.988518\pi$
$233$	− 1.10102i	− 0.0721303i	0.999349	−	0.0360651i	$-0.0114824\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	26.6969i	1.73415i
$238$	0	0
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	−1.34847	−0.0868625	−0.0434313	−	0.999056i	$-0.513829\pi$
$241$	−1.34847	−0.0868625	−0.0434313	+	0.999056i	$0.513829\pi$
$242$	0	0
$243$	− 22.0454i	− 1.41421i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 2.20204i	− 0.140113i
$248$	0	0
$249$	41.3939	2.62323
$250$	0	0
$251$	2.44949	0.154610	0.0773052	−	0.997007i	$-0.475368\pi$
$251$	2.44949	0.154610	0.0773052	+	0.997007i	$0.475368\pi$
$252$	0	0
$253$	0.898979i	0.0565184i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.79796i	0.236910i	0.992959	+	0.118455i	$0.0377942\pi$
$257$	3.79796i	0.236910i	−0.992959	+	0.118455i	$0.962206\pi$
$258$	0	0
$259$	4.00000	0.248548
$260$	0	0
$261$	8.69694	0.538327
$262$	0	0
$263$	19.5959i	1.20834i	0.796857	+	0.604168i	$0.206494\pi$
$263$	19.5959i	1.20834i	−0.796857	+	0.604168i	$0.793506\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 14.6969i	− 0.899438i
$268$	0	0
$269$	13.1010	0.798783	0.399392	−	0.916780i	$-0.369221\pi$
$269$	13.1010	0.798783	0.399392	+	0.916780i	$0.369221\pi$
$270$	0	0
$271$	1.30306	0.0791554	0.0395777	−	0.999216i	$-0.487399\pi$
$271$	1.30306	0.0791554	0.0395777	+	0.999216i	$0.487399\pi$
$272$	0	0
$273$	1.10102i	0.0666368i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 15.7980i	− 0.949207i	−0.880200	−	0.474604i	$-0.842591\pi$
$277$	− 15.7980i	− 0.949207i	0.880200	−	0.474604i	$-0.157409\pi$
$278$	0	0
$279$	−19.3485	−1.15836
$280$	0	0
$281$	−5.10102	−0.304301	−0.152151	−	0.988357i	$-0.548620\pi$
$281$	−5.10102	−0.304301	−0.152151	+	0.988357i	$0.548620\pi$
$282$	0	0
$283$	− 24.0000i	− 1.42665i	−0.700832	−	0.713326i	$-0.747188\pi$
$283$	− 24.0000i	− 1.42665i	0.700832	−	0.713326i	$-0.252812\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 9.34847i	− 0.551823i
$288$	0	0
$289$	16.7980	0.988115
$290$	0	0
$291$	12.4949	0.732464
$292$	0	0
$293$	− 10.6515i	− 0.622269i	−0.950366	−	0.311135i	$-0.899291\pi$
$293$	− 10.6515i	− 0.622269i	0.950366	−	0.311135i	$-0.100709\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−0.404082	−0.0233687
$300$	0	0
$301$	−2.89898	−0.167094
$302$	0	0
$303$	− 20.6969i	− 1.18901i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1.79796i	0.102615i	0.998683	+	0.0513075i	$0.0163388\pi$
$307$	1.79796i	0.102615i	−0.998683	+	0.0513075i	$0.983661\pi$
$308$	0	0
$309$	−35.3939	−2.01349
$310$	0	0
$311$	−1.55051	−0.0879214	−0.0439607	−	0.999033i	$-0.513998\pi$
$311$	−1.55051	−0.0879214	−0.0439607	+	0.999033i	$0.513998\pi$
$312$	0	0
$313$	5.59592i	0.316300i	0.987415	+	0.158150i	$0.0505530\pi$
$313$	5.59592i	0.316300i	−0.987415	+	0.158150i	$0.949447\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	13.5959i	0.763623i	0.924240	+	0.381811i	$0.124700\pi$
$317$	13.5959i	0.763623i	−0.924240	+	0.381811i	$0.875300\pi$
$318$	0	0
$319$	2.89898	0.162312
$320$	0	0
$321$	−9.79796	−0.546869
$322$	0	0
$323$	2.20204i	0.122525i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 36.4949i	− 2.01817i
$328$	0	0
$329$	−6.44949	−0.355572
$330$	0	0
$331$	−32.0000	−1.75888	−0.879440	−	0.476011i	$-0.842082\pi$
$331$	−32.0000	−1.75888	−0.879440	+	0.476011i	$0.842082\pi$
$332$	0	0
$333$	12.0000i	0.657596i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 20.6969i	− 1.12743i	−0.825968	−	0.563717i	$-0.809371\pi$
$337$	− 20.6969i	− 1.12743i	0.825968	−	0.563717i	$-0.190629\pi$
$338$	0	0
$339$	24.4949	1.33038
$340$	0	0
$341$	−6.44949	−0.349259
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 26.4949i	− 1.42232i	−0.703030	−	0.711160i	$-0.748170\pi$
$347$	− 26.4949i	− 1.42232i	0.703030	−	0.711160i	$-0.251830\pi$
$348$	0	0
$349$	−1.75255	−0.0938119	−0.0469060	−	0.998899i	$-0.514936\pi$
$349$	−1.75255	−0.0938119	−0.0469060	+	0.998899i	$0.514936\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	25.5959i	1.36233i	0.732128	+	0.681167i	$0.238527\pi$
$353$	25.5959i	1.36233i	−0.732128	+	0.681167i	$0.761473\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 1.10102i	− 0.0582722i
$358$	0	0
$359$	24.6969	1.30345	0.651727	−	0.758453i	$-0.274045\pi$
$359$	24.6969	1.30345	0.651727	+	0.758453i	$0.274045\pi$
$360$	0	0
$361$	5.00000	0.263158
$362$	0	0
$363$	2.44949i	0.128565i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 21.1464i	− 1.10383i	−0.833899	−	0.551917i	$-0.813896\pi$
$367$	− 21.1464i	− 1.10383i	0.833899	−	0.551917i	$-0.186104\pi$
$368$	0	0
$369$	28.0454	1.45999
$370$	0	0
$371$	−9.79796	−0.508685
$372$	0	0
$373$	4.20204i	0.217573i	0.994065	+	0.108787i	$0.0346966\pi$
$373$	4.20204i	0.217573i	−0.994065	+	0.108787i	$0.965303\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.30306i	0.0671111i
$378$	0	0
$379$	3.10102	0.159289	0.0796444	−	0.996823i	$-0.474622\pi$
$379$	3.10102	0.159289	0.0796444	+	0.996823i	$0.474622\pi$
$380$	0	0
$381$	−4.40408	−0.225628
$382$	0	0
$383$	27.3485i	1.39744i	0.715395	+	0.698721i	$0.246247\pi$
$383$	27.3485i	1.39744i	−0.715395	+	0.698721i	$0.753753\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 8.69694i	− 0.442090i
$388$	0	0
$389$	13.5959	0.689340	0.344670	−	0.938724i	$-0.387991\pi$
$389$	13.5959	0.689340	0.344670	+	0.938724i	$0.387991\pi$
$390$	0	0
$391$	0.404082	0.0204353
$392$	0	0
$393$	24.0000i	1.21064i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	25.5959i	1.28462i	0.766444	+	0.642311i	$0.222024\pi$
$397$	25.5959i	1.28462i	−0.766444	+	0.642311i	$0.777976\pi$
$398$	0	0
$399$	−12.0000	−0.600751
$400$	0	0
$401$	−27.7980	−1.38816	−0.694082	−	0.719896i	$-0.744189\pi$
$401$	−27.7980	−1.38816	−0.694082	+	0.719896i	$0.744189\pi$
$402$	0	0
$403$	− 2.89898i	− 0.144408i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.00000i	0.198273i
$408$	0	0
$409$	11.5505	0.571136	0.285568	−	0.958358i	$-0.407818\pi$
$409$	11.5505	0.571136	0.285568	+	0.958358i	$0.407818\pi$
$410$	0	0
$411$	24.0000	1.18383
$412$	0	0
$413$	− 7.34847i	− 0.361595i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 55.5959i	− 2.72254i
$418$	0	0
$419$	−18.4495	−0.901317	−0.450658	−	0.892697i	$-0.648811\pi$
$419$	−18.4495	−0.901317	−0.450658	+	0.892697i	$0.648811\pi$
$420$	0	0
$421$	−5.79796	−0.282575	−0.141288	−	0.989969i	$-0.545124\pi$
$421$	−5.79796	−0.282575	−0.141288	+	0.989969i	$0.545124\pi$
$422$	0	0
$423$	− 19.3485i	− 0.940755i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	5.34847i	0.258831i
$428$	0	0
$429$	−1.10102	−0.0531578
$430$	0	0
$431$	−9.10102	−0.438381	−0.219190	−	0.975682i	$-0.570342\pi$
$431$	−9.10102	−0.438381	−0.219190	+	0.975682i	$0.570342\pi$
$432$	0	0
$433$	21.1010i	1.01405i	0.861931	+	0.507025i	$0.169255\pi$
$433$	21.1010i	1.01405i	−0.861931	+	0.507025i	$0.830745\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 4.40408i	− 0.210676i
$438$	0	0
$439$	24.8990	1.18836	0.594182	−	0.804331i	$-0.297476\pi$
$439$	24.8990	1.18836	0.594182	+	0.804331i	$0.297476\pi$
$440$	0	0
$441$	3.00000	0.142857
$442$	0	0
$443$	24.0000i	1.14027i	0.821549	+	0.570137i	$0.193110\pi$
$443$	24.0000i	1.14027i	−0.821549	+	0.570137i	$0.806890\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 28.8990i	− 1.36687i
$448$	0	0
$449$	−19.5959	−0.924789	−0.462394	−	0.886674i	$-0.653010\pi$
$449$	−19.5959	−0.924789	−0.462394	+	0.886674i	$0.653010\pi$
$450$	0	0
$451$	9.34847	0.440202
$452$	0	0
$453$	21.3031i	1.00091i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 19.3939i	− 0.907207i	−0.891204	−	0.453604i	$-0.850138\pi$
$457$	− 19.3939i	− 0.907207i	0.891204	−	0.453604i	$-0.149862\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−37.3485	−1.73949	−0.869746	−	0.493500i	$-0.835717\pi$
$461$	−37.3485	−1.73949	−0.869746	+	0.493500i	$0.835717\pi$
$462$	0	0
$463$	− 24.8990i	− 1.15715i	−0.815628	−	0.578577i	$-0.803608\pi$
$463$	− 24.8990i	− 1.15715i	0.815628	−	0.578577i	$-0.196392\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 7.34847i	− 0.340047i	−0.985440	−	0.170023i	$-0.945616\pi$
$467$	− 7.34847i	− 0.340047i	0.985440	−	0.170023i	$-0.0543843\pi$
$468$	0	0
$469$	−14.6969	−0.678642
$470$	0	0
$471$	24.4949	1.12867
$472$	0	0
$473$	− 2.89898i	− 0.133295i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 29.3939i	− 1.34585i
$478$	0	0
$479$	−0.898979	−0.0410754	−0.0205377	−	0.999789i	$-0.506538\pi$
$479$	−0.898979	−0.0410754	−0.0205377	+	0.999789i	$0.506538\pi$
$480$	0	0
$481$	−1.79796	−0.0819799
$482$	0	0
$483$	2.20204i	0.100196i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 36.8990i	− 1.67205i	−0.548690	−	0.836026i	$-0.684873\pi$
$487$	− 36.8990i	− 1.67205i	0.548690	−	0.836026i	$-0.315127\pi$
$488$	0	0
$489$	−21.7980	−0.985738
$490$	0	0
$491$	32.6969	1.47559	0.737796	−	0.675024i	$-0.235867\pi$
$491$	32.6969	1.47559	0.737796	+	0.675024i	$0.235867\pi$
$492$	0	0
$493$	− 1.30306i	− 0.0586869i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 12.8990i	− 0.578598i
$498$	0	0
$499$	42.6969	1.91138	0.955689	−	0.294379i	$-0.0951128\pi$
$499$	42.6969	1.91138	0.955689	+	0.294379i	$0.0951128\pi$
$500$	0	0
$501$	33.7980	1.50998
$502$	0	0
$503$	− 24.4949i	− 1.09217i	−0.837729	−	0.546087i	$-0.816117\pi$
$503$	− 24.4949i	− 1.09217i	0.837729	−	0.546087i	$-0.183883\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	31.3485i	1.39223i
$508$	0	0
$509$	24.6969	1.09467	0.547336	−	0.836913i	$-0.315642\pi$
$509$	24.6969	1.09467	0.547336	+	0.836913i	$0.315642\pi$
$510$	0	0
$511$	−12.4495	−0.550733
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 6.44949i	− 0.283648i
$518$	0	0
$519$	−25.1010	−1.10181
$520$	0	0
$521$	−9.10102	−0.398723	−0.199361	−	0.979926i	$-0.563887\pi$
$521$	−9.10102	−0.398723	−0.199361	+	0.979926i	$0.563887\pi$
$522$	0	0
$523$	25.3939i	1.11040i	0.831718	+	0.555198i	$0.187358\pi$
$523$	25.3939i	1.11040i	−0.831718	+	0.555198i	$0.812642\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.89898i	0.126282i
$528$	0	0
$529$	22.1918	0.964862
$530$	0	0
$531$	22.0454	0.956689
$532$	0	0
$533$	4.20204i	0.182011i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	38.2020i	1.64854i
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−17.5959	−0.756508	−0.378254	−	0.925702i	$-0.623475\pi$
$541$	−17.5959	−0.756508	−0.378254	+	0.925702i	$0.623475\pi$
$542$	0	0
$543$	22.2929i	0.956678i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	12.0000i	0.513083i	0.966533	+	0.256541i	$0.0825830\pi$
$547$	12.0000i	0.513083i	−0.966533	+	0.256541i	$0.917417\pi$
$548$	0	0
$549$	−16.0454	−0.684801
$550$	0	0
$551$	−14.2020	−0.605027
$552$	0	0
$553$	10.8990i	0.463472i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	43.7980i	1.85578i	0.372855	+	0.927890i	$0.378379\pi$
$557$	43.7980i	1.85578i	−0.372855	+	0.927890i	$0.621621\pi$
$558$	0	0
$559$	1.30306	0.0551136
$560$	0	0
$561$	1.10102	0.0464851
$562$	0	0
$563$	− 26.2929i	− 1.10811i	−0.832479	−	0.554056i	$-0.813079\pi$
$563$	− 26.2929i	− 1.10811i	0.832479	−	0.554056i	$-0.186921\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 9.00000i	− 0.377964i
$568$	0	0
$569$	8.69694	0.364595	0.182297	−	0.983243i	$-0.441647\pi$
$569$	8.69694	0.364595	0.182297	+	0.983243i	$0.441647\pi$
$570$	0	0
$571$	−10.2020	−0.426942	−0.213471	−	0.976949i	$-0.568477\pi$
$571$	−10.2020	−0.426942	−0.213471	+	0.976949i	$0.568477\pi$
$572$	0	0
$573$	− 4.40408i	− 0.183983i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	22.4949i	0.936475i	0.883603	+	0.468237i	$0.155111\pi$
$577$	22.4949i	0.936475i	−0.883603	+	0.468237i	$0.844889\pi$
$578$	0	0
$579$	−52.8990	−2.19841
$580$	0	0
$581$	16.8990	0.701088
$582$	0	0
$583$	− 9.79796i	− 0.405790i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 21.5505i	− 0.889485i	−0.895659	−	0.444742i	$-0.853295\pi$
$587$	− 21.5505i	− 0.889485i	0.895659	−	0.444742i	$-0.146705\pi$
$588$	0	0
$589$	31.5959	1.30189
$590$	0	0
$591$	−62.6969	−2.57901
$592$	0	0
$593$	− 0.853572i	− 0.0350520i	−0.999846	−	0.0175260i	$-0.994421\pi$
$593$	− 0.853572i	− 0.0350520i	0.999846	−	0.0175260i	$-0.00557898\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 63.7980i	− 2.61108i
$598$	0	0
$599$	7.10102	0.290140	0.145070	−	0.989421i	$-0.453659\pi$
$599$	7.10102	0.290140	0.145070	+	0.989421i	$0.453659\pi$
$600$	0	0
$601$	6.65153	0.271322	0.135661	−	0.990755i	$-0.456684\pi$
$601$	6.65153	0.271322	0.135661	+	0.990755i	$0.456684\pi$
$602$	0	0
$603$	− 44.0908i	− 1.79552i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 15.5959i	− 0.633019i	−0.948589	−	0.316509i	$-0.897489\pi$
$607$	− 15.5959i	− 0.633019i	0.948589	−	0.316509i	$-0.102511\pi$
$608$	0	0
$609$	7.10102	0.287748
$610$	0	0
$611$	2.89898	0.117280
$612$	0	0
$613$	− 16.6969i	− 0.674383i	−0.941436	−	0.337191i	$-0.890523\pi$
$613$	− 16.6969i	− 0.674383i	0.941436	−	0.337191i	$-0.109477\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	29.3939i	1.18335i	0.806176	+	0.591676i	$0.201534\pi$
$617$	29.3939i	1.18335i	−0.806176	+	0.591676i	$0.798466\pi$
$618$	0	0
$619$	−5.55051	−0.223094	−0.111547	−	0.993759i	$-0.535581\pi$
$619$	−5.55051	−0.223094	−0.111547	+	0.993759i	$0.535581\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 6.00000i	− 0.240385i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 12.0000i	− 0.479234i
$628$	0	0
$629$	1.79796	0.0716893
$630$	0	0
$631$	−2.20204	−0.0876619	−0.0438309	−	0.999039i	$-0.513956\pi$
$631$	−2.20204	−0.0876619	−0.0438309	+	0.999039i	$0.513956\pi$
$632$	0	0
$633$	− 48.9898i	− 1.94717i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0.449490i	0.0178094i
$638$	0	0
$639$	38.6969	1.53083
$640$	0	0
$641$	−7.59592	−0.300021	−0.150010	−	0.988684i	$-0.547931\pi$
$641$	−7.59592	−0.300021	−0.150010	+	0.988684i	$0.547931\pi$
$642$	0	0
$643$	4.24745i	0.167503i	0.996487	+	0.0837515i	$0.0266902\pi$
$643$	4.24745i	0.167503i	−0.996487	+	0.0837515i	$0.973310\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 0.247449i	− 0.00972821i	−0.999988	−	0.00486411i	$-0.998452\pi$
$647$	− 0.247449i	− 0.00972821i	0.999988	−	0.00486411i	$-0.00154830\pi$
$648$	0	0
$649$	7.34847	0.288453
$650$	0	0
$651$	−15.7980	−0.619171
$652$	0	0
$653$	15.3939i	0.602409i	0.953560	+	0.301204i	$0.0973887\pi$
$653$	15.3939i	0.602409i	−0.953560	+	0.301204i	$0.902611\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 37.3485i	− 1.45710i
$658$	0	0
$659$	−5.10102	−0.198708	−0.0993538	−	0.995052i	$-0.531678\pi$
$659$	−5.10102	−0.198708	−0.0993538	+	0.995052i	$0.531678\pi$
$660$	0	0
$661$	−29.5959	−1.15115	−0.575574	−	0.817750i	$-0.695221\pi$
$661$	−29.5959	−1.15115	−0.575574	+	0.817750i	$0.695221\pi$
$662$	0	0
$663$	0.494897i	0.0192202i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.60612i	0.100909i
$668$	0	0
$669$	32.2020	1.24500
$670$	0	0
$671$	−5.34847	−0.206475
$672$	0	0
$673$	7.30306i	0.281512i	0.990044	+	0.140756i	$0.0449534\pi$
$673$	7.30306i	0.281512i	−0.990044	+	0.140756i	$0.955047\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 28.9444i	− 1.11242i	−0.831041	−	0.556212i	$-0.812254\pi$
$677$	− 28.9444i	− 1.11242i	0.831041	−	0.556212i	$-0.187746\pi$
$678$	0	0
$679$	5.10102	0.195759
$680$	0	0
$681$	4.40408	0.168765
$682$	0	0
$683$	− 26.2020i	− 1.00259i	−0.865276	−	0.501297i	$-0.832857\pi$
$683$	− 26.2020i	− 1.00259i	0.865276	−	0.501297i	$-0.167143\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	12.4949i	0.476710i
$688$	0	0
$689$	4.40408	0.167782
$690$	0	0
$691$	−41.1464	−1.56528	−0.782642	−	0.622472i	$-0.786128\pi$
$691$	−41.1464	−1.56528	−0.782642	+	0.622472i	$0.786128\pi$
$692$	0	0
$693$	3.00000i	0.113961i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 4.20204i	− 0.159164i
$698$	0	0
$699$	2.69694	0.102008
$700$	0	0
$701$	−20.6969	−0.781713	−0.390856	−	0.920452i	$-0.627821\pi$
$701$	−20.6969	−0.781713	−0.390856	+	0.920452i	$0.627821\pi$
$702$	0	0
$703$	− 19.5959i	− 0.739074i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 8.44949i	− 0.317776i
$708$	0	0
$709$	−3.79796	−0.142635	−0.0713177	−	0.997454i	$-0.522720\pi$
$709$	−3.79796	−0.142635	−0.0713177	+	0.997454i	$0.522720\pi$
$710$	0	0
$711$	−32.6969	−1.22623
$712$	0	0
$713$	− 5.79796i	− 0.217135i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 39.1918i	− 1.46365i
$718$	0	0
$719$	−45.1464	−1.68368	−0.841839	−	0.539729i	$-0.818527\pi$
$719$	−45.1464	−1.68368	−0.841839	+	0.539729i	$0.818527\pi$
$720$	0	0
$721$	−14.4495	−0.538127
$722$	0	0
$723$	− 3.30306i	− 0.122842i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.55051i	0.0575052i	0.999587	+	0.0287526i	$0.00915351\pi$
$727$	1.55051i	0.0575052i	−0.999587	+	0.0287526i	$0.990846\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	−1.30306	−0.0481955
$732$	0	0
$733$	45.3485i	1.67498i	0.546450	+	0.837492i	$0.315979\pi$
$733$	45.3485i	1.67498i	−0.546450	+	0.837492i	$0.684021\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 14.6969i	− 0.541369i
$738$	0	0
$739$	41.3939	1.52270	0.761349	−	0.648342i	$-0.224537\pi$
$739$	41.3939	1.52270	0.761349	+	0.648342i	$0.224537\pi$
$740$	0	0
$741$	5.39388	0.198149
$742$	0	0
$743$	− 19.5959i	− 0.718905i	−0.933163	−	0.359452i	$-0.882964\pi$
$743$	− 19.5959i	− 0.718905i	0.933163	−	0.359452i	$-0.117036\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	50.6969i	1.85490i
$748$	0	0
$749$	−4.00000	−0.146157
$750$	0	0
$751$	−12.8990	−0.470690	−0.235345	−	0.971912i	$-0.575622\pi$
$751$	−12.8990	−0.470690	−0.235345	+	0.971912i	$0.575622\pi$
$752$	0	0
$753$	6.00000i	0.218652i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	22.0000i	0.799604i	0.916602	+	0.399802i	$0.130921\pi$
$757$	22.0000i	0.799604i	−0.916602	+	0.399802i	$0.869079\pi$
$758$	0	0
$759$	−2.20204	−0.0799290
$760$	0	0
$761$	49.8434	1.80682	0.903410	−	0.428777i	$-0.141055\pi$
$761$	49.8434	1.80682	0.903410	+	0.428777i	$0.141055\pi$
$762$	0	0
$763$	− 14.8990i	− 0.539379i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.30306i	0.119267i
$768$	0	0
$769$	−39.1464	−1.41166	−0.705828	−	0.708383i	$-0.749425\pi$
$769$	−39.1464	−1.41166	−0.705828	+	0.708383i	$0.749425\pi$
$770$	0	0
$771$	−9.30306	−0.335042
$772$	0	0
$773$	3.30306i	0.118803i	0.998234	+	0.0594014i	$0.0189192\pi$
$773$	3.30306i	0.118803i	−0.998234	+	0.0594014i	$0.981081\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	9.79796i	0.351500i
$778$	0	0
$779$	−45.7980	−1.64088
$780$	0	0
$781$	12.8990	0.461562
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 10.6969i	− 0.381305i	−0.981658	−	0.190652i	$-0.938940\pi$
$787$	− 10.6969i	− 0.381305i	0.981658	−	0.190652i	$-0.0610603\pi$
$788$	0	0
$789$	−48.0000	−1.70885
$790$	0	0
$791$	10.0000	0.355559
$792$	0	0
$793$	− 2.40408i	− 0.0853715i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	31.7980i	1.12634i	0.826341	+	0.563171i	$0.190419\pi$
$797$	31.7980i	1.12634i	−0.826341	+	0.563171i	$0.809581\pi$
$798$	0	0
$799$	−2.89898	−0.102559
$800$	0	0
$801$	18.0000	0.635999
$802$	0	0
$803$	− 12.4495i	− 0.439333i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	32.0908i	1.12965i
$808$	0	0
$809$	25.5959	0.899905	0.449952	−	0.893053i	$-0.351441\pi$
$809$	25.5959	0.899905	0.449952	+	0.893053i	$0.351441\pi$
$810$	0	0
$811$	−23.5959	−0.828565	−0.414282	−	0.910148i	$-0.635967\pi$
$811$	−23.5959	−0.828565	−0.414282	+	0.910148i	$0.635967\pi$
$812$	0	0
$813$	3.19184i	0.111943i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	14.2020i	0.496867i
$818$	0	0
$819$	−1.34847	−0.0471193
$820$	0	0
$821$	−15.7980	−0.551353	−0.275676	−	0.961251i	$-0.588902\pi$
$821$	−15.7980	−0.551353	−0.275676	+	0.961251i	$0.588902\pi$
$822$	0	0
$823$	35.1918i	1.22671i	0.789807	+	0.613355i	$0.210181\pi$
$823$	35.1918i	1.22671i	−0.789807	+	0.613355i	$0.789819\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 52.6969i	− 1.83245i	−0.400662	−	0.916226i	$-0.631220\pi$
$827$	− 52.6969i	− 1.83245i	0.400662	−	0.916226i	$-0.368780\pi$
$828$	0	0
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	0	0
$831$	38.6969	1.34238
$832$	0	0
$833$	− 0.449490i	− 0.0155739i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−9.55051	−0.329720	−0.164860	−	0.986317i	$-0.552717\pi$
$839$	−9.55051	−0.329720	−0.164860	+	0.986317i	$0.552717\pi$
$840$	0	0
$841$	−20.5959	−0.710204
$842$	0	0
$843$	− 12.4949i	− 0.430347i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.00000i	0.0343604i
$848$	0	0
$849$	58.7878	2.01759
$850$	0	0
$851$	−3.59592	−0.123266
$852$	0	0
$853$	43.1464i	1.47731i	0.674086	+	0.738653i	$0.264538\pi$
$853$	43.1464i	1.47731i	−0.674086	+	0.738653i	$0.735462\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 8.85357i	− 0.302432i	−0.988501	−	0.151216i	$-0.951681\pi$
$857$	− 8.85357i	− 0.302432i	0.988501	−	0.151216i	$-0.0483189\pi$
$858$	0	0
$859$	−23.8434	−0.813525	−0.406763	−	0.913534i	$-0.633342\pi$
$859$	−23.8434	−0.813525	−0.406763	+	0.913534i	$0.633342\pi$
$860$	0	0
$861$	22.8990	0.780395
$862$	0	0
$863$	− 32.0000i	− 1.08929i	−0.838666	−	0.544646i	$-0.816664\pi$
$863$	− 32.0000i	− 1.08929i	0.838666	−	0.544646i	$-0.183336\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	41.1464i	1.39741i
$868$	0	0
$869$	−10.8990	−0.369723
$870$	0	0
$871$	6.60612	0.223840
$872$	0	0
$873$	15.3031i	0.517930i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 36.6969i	− 1.23917i	−0.784931	−	0.619584i	$-0.787301\pi$
$877$	− 36.6969i	− 1.23917i	0.784931	−	0.619584i	$-0.212699\pi$
$878$	0	0
$879$	26.0908	0.880021
$880$	0	0
$881$	−20.6969	−0.697298	−0.348649	−	0.937253i	$-0.613359\pi$
$881$	−20.6969	−0.697298	−0.348649	+	0.937253i	$0.613359\pi$
$882$	0	0
$883$	− 29.7980i	− 1.00278i	−0.865221	−	0.501391i	$-0.832822\pi$
$883$	− 29.7980i	− 1.00278i	0.865221	−	0.501391i	$-0.167178\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	30.2929i	1.01713i	0.861022	+	0.508567i	$0.169825\pi$
$887$	30.2929i	1.01713i	−0.861022	+	0.508567i	$0.830175\pi$
$888$	0	0
$889$	−1.79796	−0.0603016
$890$	0	0
$891$	9.00000	0.301511
$892$	0	0
$893$	31.5959i	1.05732i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 0.989795i	− 0.0330483i
$898$	0	0
$899$	−18.6969	−0.623578
$900$	0	0
$901$	−4.40408	−0.146721
$902$	0	0
$903$	− 7.10102i	− 0.236307i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 3.10102i	− 0.102968i	−0.998674	−	0.0514838i	$-0.983605\pi$
$907$	− 3.10102i	− 0.102968i	0.998674	−	0.0514838i	$-0.0163951\pi$
$908$	0	0
$909$	25.3485	0.840756
$910$	0	0
$911$	36.4949	1.20913	0.604565	−	0.796556i	$-0.293347\pi$
$911$	36.4949	1.20913	0.604565	+	0.796556i	$0.293347\pi$
$912$	0	0
$913$	16.8990i	0.559275i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	9.79796i	0.323557i
$918$	0	0
$919$	−4.40408	−0.145277	−0.0726386	−	0.997358i	$-0.523142\pi$
$919$	−4.40408	−0.145277	−0.0726386	+	0.997358i	$0.523142\pi$
$920$	0	0
$921$	−4.40408	−0.145119
$922$	0	0
$923$	5.79796i	0.190842i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 43.3485i	− 1.42375i
$928$	0	0
$929$	52.2929	1.71567	0.857836	−	0.513923i	$-0.171808\pi$
$929$	52.2929	1.71567	0.857836	+	0.513923i	$0.171808\pi$
$930$	0	0
$931$	−4.89898	−0.160558
$932$	0	0
$933$	− 3.79796i	− 0.124340i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 32.9444i	− 1.07625i	−0.842866	−	0.538123i	$-0.819134\pi$
$937$	− 32.9444i	− 1.07625i	0.842866	−	0.538123i	$-0.180866\pi$
$938$	0	0
$939$	−13.7071	−0.447316
$940$	0	0
$941$	43.5505	1.41971	0.709853	−	0.704350i	$-0.248761\pi$
$941$	43.5505	1.41971	0.709853	+	0.704350i	$0.248761\pi$
$942$	0	0
$943$	8.40408i	0.273675i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 0.404082i	− 0.0131309i	−0.999978	−	0.00656545i	$-0.997910\pi$
$947$	− 0.404082i	− 0.0131309i	0.999978	−	0.00656545i	$-0.00208986\pi$
$948$	0	0
$949$	5.59592	0.181651
$950$	0	0
$951$	−33.3031	−1.07993
$952$	0	0
$953$	− 43.7980i	− 1.41876i	−0.704829	−	0.709378i	$-0.748976\pi$
$953$	− 43.7980i	− 1.41876i	0.704829	−	0.709378i	$-0.251024\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	7.10102i	0.229543i
$958$	0	0
$959$	9.79796	0.316393
$960$	0	0
$961$	10.5959	0.341804
$962$	0	0
$963$	− 12.0000i	− 0.386695i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	38.8990i	1.25091i	0.780262	+	0.625453i	$0.215086\pi$
$967$	38.8990i	1.25091i	−0.780262	+	0.625453i	$0.784914\pi$
$968$	0	0
$969$	−5.39388	−0.173276
$970$	0	0
$971$	−38.0454	−1.22094	−0.610468	−	0.792041i	$-0.709018\pi$
$971$	−38.0454	−1.22094	−0.610468	+	0.792041i	$0.709018\pi$
$972$	0	0
$973$	− 22.6969i	− 0.727630i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 4.20204i	− 0.134435i	−0.997738	−	0.0672176i	$-0.978588\pi$
$977$	− 4.20204i	− 0.134435i	0.997738	−	0.0672176i	$-0.0214122\pi$
$978$	0	0
$979$	6.00000	0.191761
$980$	0	0
$981$	44.6969	1.42706
$982$	0	0
$983$	19.3485i	0.617120i	0.951205	+	0.308560i	$0.0998471\pi$
$983$	19.3485i	0.617120i	−0.951205	+	0.308560i	$0.900153\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 15.7980i	− 0.502855i
$988$	0	0
$989$	2.60612	0.0828699
$990$	0	0
$991$	60.0908	1.90885	0.954424	−	0.298455i	$-0.0964711\pi$
$991$	60.0908	1.90885	0.954424	+	0.298455i	$0.0964711\pi$
$992$	0	0
$993$	− 78.3837i	− 2.48743i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	20.9444i	0.663315i	0.943400	+	0.331658i	$0.107608\pi$
$997$	20.9444i	0.663315i	−0.943400	+	0.331658i	$0.892392\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7700.2.e.j.1849.4		4
5.2	odd	4		308.2.a.b.1.2	✓	2
5.3	odd	4		7700.2.a.s.1.1		2
5.4	even	2	inner	7700.2.e.j.1849.1		4
15.2	even	4		2772.2.a.n.1.1		2
20.7	even	4		1232.2.a.n.1.1		2
35.2	odd	12		2156.2.i.e.1145.1		4
35.12	even	12		2156.2.i.i.1145.2		4
35.17	even	12		2156.2.i.i.177.2		4
35.27	even	4		2156.2.a.c.1.1		2
35.32	odd	12		2156.2.i.e.177.1		4
40.27	even	4		4928.2.a.bq.1.2		2
40.37	odd	4		4928.2.a.bp.1.1		2
55.32	even	4		3388.2.a.h.1.2		2
140.27	odd	4		8624.2.a.bj.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
308.2.a.b.1.2	✓	2	5.2	odd	4
1232.2.a.n.1.1		2	20.7	even	4
2156.2.a.c.1.1		2	35.27	even	4
2156.2.i.e.177.1		4	35.32	odd	12
2156.2.i.e.1145.1		4	35.2	odd	12
2156.2.i.i.177.2		4	35.17	even	12
2156.2.i.i.1145.2		4	35.12	even	12
2772.2.a.n.1.1		2	15.2	even	4
3388.2.a.h.1.2		2	55.32	even	4
4928.2.a.bp.1.1		2	40.37	odd	4
4928.2.a.bq.1.2		2	40.27	even	4
7700.2.a.s.1.1		2	5.3	odd	4
7700.2.e.j.1849.1		4	5.4	even	2	inner
7700.2.e.j.1849.4		4	1.1	even	1	trivial
8624.2.a.bj.1.2		2	140.27	odd	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:	\( N \)	\(=\)	\( 7700 = 2^{2} \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7700.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.44949i q^{3} +1.00000i q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q+2.44949i q^{3} +1.00000i q^{7} -3.00000 q^{9} -1.00000 q^{11} -0.449490i q^{13} +0.449490i q^{17} +4.89898 q^{19} -2.44949 q^{21} -0.898979i q^{23} -2.89898 q^{29} +6.44949 q^{31} -2.44949i q^{33} -4.00000i q^{37} +1.10102 q^{39} -9.34847 q^{41} +2.89898i q^{43} +6.44949i q^{47} -1.00000 q^{49} -1.10102 q^{51} +9.79796i q^{53} +12.0000i q^{57} -7.34847 q^{59} +5.34847 q^{61} -3.00000i q^{63} +14.6969i q^{67} +2.20204 q^{69} -12.8990 q^{71} +12.4495i q^{73} -1.00000i q^{77} +10.8990 q^{79} -9.00000 q^{81} -16.8990i q^{83} -7.10102i q^{87} -6.00000 q^{89} +0.449490 q^{91} +15.7980i q^{93} -5.10102i q^{97} +3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{9}+O(q^{10}) \) 4 * q - 12 * q^9	\( 4 q - 12 q^{9} - 4 q^{11} + 8 q^{29} + 16 q^{31} + 24 q^{39} - 8 q^{41} - 4 q^{49} - 24 q^{51} - 8 q^{61} + 48 q^{69} - 32 q^{71} + 24 q^{79} - 36 q^{81} - 24 q^{89} - 8 q^{91} + 12 q^{99}+O(q^{100}) \) 4 * q - 12 * q^9 - 4 * q^11 + 8 * q^29 + 16 * q^31 + 24 * q^39 - 8 * q^41 - 4 * q^49 - 24 * q^51 - 8 * q^61 + 48 * q^69 - 32 * q^71 + 24 * q^79 - 36 * q^81 - 24 * q^89 - 8 * q^91 + 12 * q^99

Introduction

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