LMFDB - Embedded newform 9300.2.g.l.3349.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9300,2,Mod(3349,9300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9300, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("9300.3349");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9300.g (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:	$74.2608738798$
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 1860)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3349.3
Root			$-1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	9300.3349
Dual form			9300.2.g.l.3349.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -2.44949i q^{7} -1.00000 q^{9} -0.449490i q^{13} +2.00000i q^{17} -4.89898 q^{19} +2.44949 q^{21} -6.89898i q^{23} -1.00000i q^{27} +4.44949 q^{29} +1.00000 q^{31} -8.44949i q^{37} +0.449490 q^{39} -11.7980 q^{41} +12.8990i q^{43} -0.898979i q^{47} +1.00000 q^{49} -2.00000 q^{51} +6.00000i q^{53} -4.89898i q^{57} -7.34847 q^{59} +2.89898 q^{61} +2.44949i q^{63} -1.55051i q^{67} +6.89898 q^{69} +6.44949 q^{71} +7.55051i q^{73} -6.89898 q^{79} +1.00000 q^{81} +10.8990i q^{83} +4.44949i q^{87} -0.449490 q^{89} -1.10102 q^{91} +1.00000i q^{93} -14.8990i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9} + 8 q^{29} + 4 q^{31} - 8 q^{39} - 8 q^{41} + 4 q^{49} - 8 q^{51} - 8 q^{61} + 8 q^{69} + 16 q^{71} - 8 q^{79} + 4 q^{81} + 8 q^{89} - 24 q^{91}+O(q^{100}) $ 4 * q - 4 * q^9 + 8 * q^29 + 4 * q^31 - 8 * q^39 - 8 * q^41 + 4 * q^49 - 8 * q^51 - 8 * q^61 + 8 * q^69 + 16 * q^71 - 8 * q^79 + 4 * q^81 + 8 * q^89 - 24 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/9300\mathbb{Z}\right)^\times$.

$n$	$1801$	$2977$	$3101$	$4651$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.44949i	− 0.925820i	−0.886405	−	0.462910i	$-0.846805\pi$
$7$	− 2.44949i	− 0.925820i	0.886405	−	0.462910i	$-0.153195\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	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
$17$	2.00000i	0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$	2.00000i	0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$	0	0
$19$	−4.89898	−1.12390	−0.561951	−	0.827170i	$-0.689949\pi$
$19$	−4.89898	−1.12390	−0.561951	+	0.827170i	$0.689949\pi$
$20$	0	0
$21$	2.44949	0.534522
$22$	0	0
$23$	− 6.89898i	− 1.43854i	−0.694732	−	0.719268i	$-0.744477\pi$
$23$	− 6.89898i	− 1.43854i	0.694732	−	0.719268i	$-0.255523\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	4.44949	0.826250	0.413125	−	0.910674i	$-0.364437\pi$
$29$	4.44949	0.826250	0.413125	+	0.910674i	$0.364437\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 8.44949i	− 1.38909i	−0.719450	−	0.694544i	$-0.755606\pi$
$37$	− 8.44949i	− 1.38909i	0.719450	−	0.694544i	$-0.244394\pi$
$38$	0	0
$39$	0.449490	0.0719760
$40$	0	0
$41$	−11.7980	−1.84253	−0.921266	−	0.388934i	$-0.872844\pi$
$41$	−11.7980	−1.84253	−0.921266	+	0.388934i	$0.872844\pi$
$42$	0	0
$43$	12.8990i	1.96708i	0.180702	+	0.983538i	$0.442163\pi$
$43$	12.8990i	1.96708i	−0.180702	+	0.983538i	$0.557837\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 0.898979i	− 0.131130i	−0.997848	−	0.0655648i	$-0.979115\pi$
$47$	− 0.898979i	− 0.131130i	0.997848	−	0.0655648i	$-0.0208849\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.00000	−0.280056
$52$	0	0
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 4.89898i	− 0.648886i
$58$	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
$61$	2.89898	0.371176	0.185588	−	0.982628i	$-0.440581\pi$
$61$	2.89898	0.371176	0.185588	+	0.982628i	$0.440581\pi$
$62$	0	0
$63$	2.44949i	0.308607i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 1.55051i	− 0.189425i	−0.995505	−	0.0947125i	$-0.969807\pi$
$67$	− 1.55051i	− 0.189425i	0.995505	−	0.0947125i	$-0.0301932\pi$
$68$	0	0
$69$	6.89898	0.830540
$70$	0	0
$71$	6.44949	0.765414	0.382707	−	0.923870i	$-0.374992\pi$
$71$	6.44949	0.765414	0.382707	+	0.923870i	$0.374992\pi$
$72$	0	0
$73$	7.55051i	0.883720i	0.897084	+	0.441860i	$0.145681\pi$
$73$	7.55051i	0.883720i	−0.897084	+	0.441860i	$0.854319\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−6.89898	−0.776196	−0.388098	−	0.921618i	$-0.626868\pi$
$79$	−6.89898	−0.776196	−0.388098	+	0.921618i	$0.626868\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.8990i	1.19632i	0.801377	+	0.598159i	$0.204101\pi$
$83$	10.8990i	1.19632i	−0.801377	+	0.598159i	$0.795899\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	4.44949i	0.477035i
$88$	0	0
$89$	−0.449490	−0.0476458	−0.0238229	−	0.999716i	$-0.507584\pi$
$89$	−0.449490	−0.0476458	−0.0238229	+	0.999716i	$0.507584\pi$
$90$	0	0
$91$	−1.10102	−0.115418
$92$	0	0
$93$	1.00000i	0.103695i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 14.8990i	− 1.51276i	−0.654131	−	0.756381i	$-0.726966\pi$
$97$	− 14.8990i	− 1.51276i	0.654131	−	0.756381i	$-0.273034\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	0	0
$103$	10.4495i	1.02962i	0.857305	+	0.514809i	$0.172137\pi$
$103$	10.4495i	1.02962i	−0.857305	+	0.514809i	$0.827863\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.89898i	0.473602i	0.971558	+	0.236801i	$0.0760990\pi$
$107$	4.89898i	0.473602i	−0.971558	+	0.236801i	$0.923901\pi$
$108$	0	0
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	0	0
$111$	8.44949	0.801990
$112$	0	0
$113$	− 3.79796i	− 0.357282i	−0.983914	−	0.178641i	$-0.942830\pi$
$113$	− 3.79796i	− 0.357282i	0.983914	−	0.178641i	$-0.0571701\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0.449490i	0.0415553i
$118$	0	0
$119$	4.89898	0.449089
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	− 11.7980i	− 1.06379i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0.898979i	0.0797715i	0.999204	+	0.0398858i	$0.0126994\pi$
$127$	0.898979i	0.0797715i	−0.999204	+	0.0398858i	$0.987301\pi$
$128$	0	0
$129$	−12.8990	−1.13569
$130$	0	0
$131$	−12.2474	−1.07006	−0.535032	−	0.844832i	$-0.679701\pi$
$131$	−12.2474	−1.07006	−0.535032	+	0.844832i	$0.679701\pi$
$132$	0	0
$133$	12.0000i	1.04053i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 5.10102i	− 0.435810i	−0.975970	−	0.217905i	$-0.930078\pi$
$137$	− 5.10102i	− 0.435810i	0.975970	−	0.217905i	$-0.0699222\pi$
$138$	0	0
$139$	5.79796	0.491776	0.245888	−	0.969298i	$-0.420920\pi$
$139$	5.79796	0.491776	0.245888	+	0.969298i	$0.420920\pi$
$140$	0	0
$141$	0.898979	0.0757077
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.00000i	0.0824786i
$148$	0	0
$149$	−19.7980	−1.62191	−0.810956	−	0.585107i	$-0.801052\pi$
$149$	−19.7980	−1.62191	−0.810956	+	0.585107i	$0.801052\pi$
$150$	0	0
$151$	−1.10102	−0.0895998	−0.0447999	−	0.998996i	$-0.514265\pi$
$151$	−1.10102	−0.0895998	−0.0447999	+	0.998996i	$0.514265\pi$
$152$	0	0
$153$	− 2.00000i	− 0.161690i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.89898i	0.231364i	0.993286	+	0.115682i	$0.0369053\pi$
$157$	2.89898i	0.231364i	−0.993286	+	0.115682i	$0.963095\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	−16.8990	−1.33183
$162$	0	0
$163$	21.1464i	1.65632i	0.560495	+	0.828158i	$0.310611\pi$
$163$	21.1464i	1.65632i	−0.560495	+	0.828158i	$0.689389\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.00000i	0.619059i	0.950890	+	0.309529i	$0.100171\pi$
$167$	8.00000i	0.619059i	−0.950890	+	0.309529i	$0.899829\pi$
$168$	0	0
$169$	12.7980	0.984458
$170$	0	0
$171$	4.89898	0.374634
$172$	0	0
$173$	19.7980i	1.50521i	0.658472	+	0.752605i	$0.271203\pi$
$173$	19.7980i	1.50521i	−0.658472	+	0.752605i	$0.728797\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 7.34847i	− 0.552345i
$178$	0	0
$179$	−23.5959	−1.76364	−0.881821	−	0.471585i	$-0.843682\pi$
$179$	−23.5959	−1.76364	−0.881821	+	0.471585i	$0.843682\pi$
$180$	0	0
$181$	6.89898	0.512797	0.256399	−	0.966571i	$-0.417464\pi$
$181$	6.89898	0.512797	0.256399	+	0.966571i	$0.417464\pi$
$182$	0	0
$183$	2.89898i	0.214299i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−2.44949	−0.178174
$190$	0	0
$191$	6.44949	0.466669	0.233334	−	0.972397i	$-0.425036\pi$
$191$	6.44949	0.466669	0.233334	+	0.972397i	$0.425036\pi$
$192$	0	0
$193$	− 20.6969i	− 1.48980i	−0.667177	−	0.744899i	$-0.732498\pi$
$193$	− 20.6969i	− 1.48980i	0.667177	−	0.744899i	$-0.267502\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.69694i	0.619631i	0.950797	+	0.309816i	$0.100267\pi$
$197$	8.69694i	0.619631i	−0.950797	+	0.309816i	$0.899733\pi$
$198$	0	0
$199$	4.69694	0.332957	0.166479	−	0.986045i	$-0.446760\pi$
$199$	4.69694	0.332957	0.166479	+	0.986045i	$0.446760\pi$
$200$	0	0
$201$	1.55051	0.109365
$202$	0	0
$203$	− 10.8990i	− 0.764958i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	6.89898i	0.479512i
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−28.4949	−1.96167	−0.980835	−	0.194841i	$-0.937581\pi$
$211$	−28.4949	−1.96167	−0.980835	+	0.194841i	$0.937581\pi$
$212$	0	0
$213$	6.44949i	0.441912i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 2.44949i	− 0.166282i
$218$	0	0
$219$	−7.55051	−0.510216
$220$	0	0
$221$	0.898979	0.0604719
$222$	0	0
$223$	− 9.79796i	− 0.656120i	−0.944657	−	0.328060i	$-0.893605\pi$
$223$	− 9.79796i	− 0.656120i	0.944657	−	0.328060i	$-0.106395\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.20204i	0.146155i	0.997326	+	0.0730773i	$0.0232820\pi$
$227$	2.20204i	0.146155i	−0.997326	+	0.0730773i	$0.976718\pi$
$228$	0	0
$229$	−0.202041	−0.0133512	−0.00667562	−	0.999978i	$-0.502125\pi$
$229$	−0.202041	−0.0133512	−0.00667562	+	0.999978i	$0.502125\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	21.7980i	1.42803i	0.700129	+	0.714016i	$0.253126\pi$
$233$	21.7980i	1.42803i	−0.700129	+	0.714016i	$0.746874\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 6.89898i	− 0.448137i
$238$	0	0
$239$	−7.10102	−0.459327	−0.229663	−	0.973270i	$-0.573763\pi$
$239$	−7.10102	−0.459327	−0.229663	+	0.973270i	$0.573763\pi$
$240$	0	0
$241$	−15.7980	−1.01764	−0.508818	−	0.860874i	$-0.669917\pi$
$241$	−15.7980	−1.01764	−0.508818	+	0.860874i	$0.669917\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.20204i	0.140113i
$248$	0	0
$249$	−10.8990	−0.690695
$250$	0	0
$251$	9.79796	0.618442	0.309221	−	0.950990i	$-0.399932\pi$
$251$	9.79796	0.618442	0.309221	+	0.950990i	$0.399932\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	16.0000i	0.998053i	0.866587	+	0.499026i	$0.166309\pi$
$257$	16.0000i	0.998053i	−0.866587	+	0.499026i	$0.833691\pi$
$258$	0	0
$259$	−20.6969	−1.28605
$260$	0	0
$261$	−4.44949	−0.275417
$262$	0	0
$263$	29.3939i	1.81250i	0.422738	+	0.906252i	$0.361069\pi$
$263$	29.3939i	1.81250i	−0.422738	+	0.906252i	$0.638931\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 0.449490i	− 0.0275083i
$268$	0	0
$269$	32.0454	1.95384	0.976921	−	0.213599i	$-0.0685185\pi$
$269$	32.0454	1.95384	0.976921	+	0.213599i	$0.0685185\pi$
$270$	0	0
$271$	1.79796	0.109218	0.0546091	−	0.998508i	$-0.482609\pi$
$271$	1.79796	0.109218	0.0546091	+	0.998508i	$0.482609\pi$
$272$	0	0
$273$	− 1.10102i	− 0.0666368i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 23.1464i	− 1.39073i	−0.718655	−	0.695367i	$-0.755242\pi$
$277$	− 23.1464i	− 1.39073i	0.718655	−	0.695367i	$-0.244758\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	5.10102	0.304301	0.152151	−	0.988357i	$-0.451380\pi$
$281$	5.10102	0.304301	0.152151	+	0.988357i	$0.451380\pi$
$282$	0	0
$283$	14.4495i	0.858933i	0.903083	+	0.429467i	$0.141298\pi$
$283$	14.4495i	0.858933i	−0.903083	+	0.429467i	$0.858702\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	28.8990i	1.70585i
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	14.8990	0.873394
$292$	0	0
$293$	2.20204i	0.128645i	0.997929	+	0.0643223i	$0.0204886\pi$
$293$	2.20204i	0.128645i	−0.997929	+	0.0643223i	$0.979511\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3.10102	−0.179337
$300$	0	0
$301$	31.5959	1.82116
$302$	0	0
$303$	2.00000i	0.114897i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 17.5505i	− 1.00166i	−0.865546	−	0.500830i	$-0.833028\pi$
$307$	− 17.5505i	− 1.00166i	0.865546	−	0.500830i	$-0.166972\pi$
$308$	0	0
$309$	−10.4495	−0.594451
$310$	0	0
$311$	19.3485	1.09715	0.548576	−	0.836101i	$-0.315170\pi$
$311$	19.3485	1.09715	0.548576	+	0.836101i	$0.315170\pi$
$312$	0	0
$313$	− 1.34847i	− 0.0762200i	−0.999274	−	0.0381100i	$-0.987866\pi$
$313$	− 1.34847i	− 0.0762200i	0.999274	−	0.0381100i	$-0.0121337\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	23.5959i	1.32528i	0.748939	+	0.662639i	$0.230564\pi$
$317$	23.5959i	1.32528i	−0.748939	+	0.662639i	$0.769436\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−4.89898	−0.273434
$322$	0	0
$323$	− 9.79796i	− 0.545173i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	4.00000i	0.221201i
$328$	0	0
$329$	−2.20204	−0.121402
$330$	0	0
$331$	−10.8990	−0.599062	−0.299531	−	0.954087i	$-0.596830\pi$
$331$	−10.8990	−0.599062	−0.299531	+	0.954087i	$0.596830\pi$
$332$	0	0
$333$	8.44949i	0.463029i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	15.1464i	0.825079i	0.910940	+	0.412539i	$0.135358\pi$
$337$	15.1464i	0.825079i	−0.910940	+	0.412539i	$0.864642\pi$
$338$	0	0
$339$	3.79796	0.206277
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 19.5959i	− 1.05808i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4.00000i	0.214731i	0.994220	+	0.107366i	$0.0342415\pi$
$347$	4.00000i	0.214731i	−0.994220	+	0.107366i	$0.965758\pi$
$348$	0	0
$349$	−33.7980	−1.80916	−0.904582	−	0.426300i	$-0.859817\pi$
$349$	−33.7980	−1.80916	−0.904582	+	0.426300i	$0.859817\pi$
$350$	0	0
$351$	−0.449490	−0.0239920
$352$	0	0
$353$	− 3.79796i	− 0.202145i	−0.994879	−	0.101072i	$-0.967773\pi$
$353$	− 3.79796i	− 0.202145i	0.994879	−	0.101072i	$-0.0322274\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.89898i	0.259281i
$358$	0	0
$359$	−14.4495	−0.762615	−0.381307	−	0.924448i	$-0.624526\pi$
$359$	−14.4495	−0.762615	−0.381307	+	0.924448i	$0.624526\pi$
$360$	0	0
$361$	5.00000	0.263158
$362$	0	0
$363$	− 11.0000i	− 0.577350i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	27.5959i	1.44050i	0.693717	+	0.720248i	$0.255972\pi$
$367$	27.5959i	1.44050i	−0.693717	+	0.720248i	$0.744028\pi$
$368$	0	0
$369$	11.7980	0.614177
$370$	0	0
$371$	14.6969	0.763027
$372$	0	0
$373$	32.6969i	1.69298i	0.532402	+	0.846492i	$0.321289\pi$
$373$	32.6969i	1.69298i	−0.532402	+	0.846492i	$0.678711\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 2.00000i	− 0.103005i
$378$	0	0
$379$	−7.10102	−0.364755	−0.182377	−	0.983229i	$-0.558379\pi$
$379$	−7.10102	−0.364755	−0.182377	+	0.983229i	$0.558379\pi$
$380$	0	0
$381$	−0.898979	−0.0460561
$382$	0	0
$383$	− 1.10102i	− 0.0562595i	−0.999604	−	0.0281298i	$-0.991045\pi$
$383$	− 1.10102i	− 0.0562595i	0.999604	−	0.0281298i	$-0.00895516\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 12.8990i	− 0.655692i
$388$	0	0
$389$	21.3485	1.08241	0.541205	−	0.840891i	$-0.317968\pi$
$389$	21.3485	1.08241	0.541205	+	0.840891i	$0.317968\pi$
$390$	0	0
$391$	13.7980	0.697793
$392$	0	0
$393$	− 12.2474i	− 0.617802i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 7.79796i	− 0.391368i	−0.980667	−	0.195684i	$-0.937307\pi$
$397$	− 7.79796i	− 0.391368i	0.980667	−	0.195684i	$-0.0626927\pi$
$398$	0	0
$399$	−12.0000	−0.600751
$400$	0	0
$401$	−33.8434	−1.69006	−0.845029	−	0.534721i	$-0.820417\pi$
$401$	−33.8434	−1.69006	−0.845029	+	0.534721i	$0.820417\pi$
$402$	0	0
$403$	− 0.449490i	− 0.0223907i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−28.6969	−1.41897	−0.709486	−	0.704719i	$-0.751073\pi$
$409$	−28.6969	−1.41897	−0.709486	+	0.704719i	$0.751073\pi$
$410$	0	0
$411$	5.10102	0.251615
$412$	0	0
$413$	18.0000i	0.885722i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	5.79796i	0.283927i
$418$	0	0
$419$	−21.5505	−1.05281	−0.526406	−	0.850234i	$-0.676461\pi$
$419$	−21.5505	−1.05281	−0.526406	+	0.850234i	$0.676461\pi$
$420$	0	0
$421$	−13.7980	−0.672471	−0.336236	−	0.941778i	$-0.609154\pi$
$421$	−13.7980	−0.672471	−0.336236	+	0.941778i	$0.609154\pi$
$422$	0	0
$423$	0.898979i	0.0437099i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 7.10102i	− 0.343642i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−22.4495	−1.08135	−0.540677	−	0.841230i	$-0.681832\pi$
$431$	−22.4495	−1.08135	−0.540677	+	0.841230i	$0.681832\pi$
$432$	0	0
$433$	3.14643i	0.151208i	0.997138	+	0.0756038i	$0.0240884\pi$
$433$	3.14643i	0.151208i	−0.997138	+	0.0756038i	$0.975912\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	33.7980i	1.61678i
$438$	0	0
$439$	−24.0000	−1.14546	−0.572729	−	0.819745i	$-0.694115\pi$
$439$	−24.0000	−1.14546	−0.572729	+	0.819745i	$0.694115\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 19.7980i	− 0.936411i
$448$	0	0
$449$	−31.1464	−1.46989	−0.734945	−	0.678126i	$-0.762792\pi$
$449$	−31.1464	−1.46989	−0.734945	+	0.678126i	$0.762792\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	− 1.10102i	− 0.0517305i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 9.34847i	− 0.437303i	−0.975803	−	0.218651i	$-0.929834\pi$
$457$	− 9.34847i	− 0.437303i	0.975803	−	0.218651i	$-0.0701658\pi$
$458$	0	0
$459$	2.00000	0.0933520
$460$	0	0
$461$	8.44949	0.393532	0.196766	−	0.980450i	$-0.436956\pi$
$461$	8.44949	0.393532	0.196766	+	0.980450i	$0.436956\pi$
$462$	0	0
$463$	− 30.6969i	− 1.42661i	−0.700855	−	0.713304i	$-0.747198\pi$
$463$	− 30.6969i	− 1.42661i	0.700855	−	0.713304i	$-0.252802\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	28.8990i	1.33729i	0.743584	+	0.668643i	$0.233124\pi$
$467$	28.8990i	1.33729i	−0.743584	+	0.668643i	$0.766876\pi$
$468$	0	0
$469$	−3.79796	−0.175373
$470$	0	0
$471$	−2.89898	−0.133578
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 6.00000i	− 0.274721i
$478$	0	0
$479$	−4.65153	−0.212534	−0.106267	−	0.994338i	$-0.533890\pi$
$479$	−4.65153	−0.212534	−0.106267	+	0.994338i	$0.533890\pi$
$480$	0	0
$481$	−3.79796	−0.173172
$482$	0	0
$483$	− 16.8990i	− 0.768930i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 23.1010i	− 1.04681i	−0.852085	−	0.523404i	$-0.824662\pi$
$487$	− 23.1010i	− 1.04681i	0.852085	−	0.523404i	$-0.175338\pi$
$488$	0	0
$489$	−21.1464	−0.956275
$490$	0	0
$491$	−20.8990	−0.943158	−0.471579	−	0.881824i	$-0.656316\pi$
$491$	−20.8990	−0.943158	−0.471579	+	0.881824i	$0.656316\pi$
$492$	0	0
$493$	8.89898i	0.400790i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 15.7980i	− 0.708635i
$498$	0	0
$499$	5.79796	0.259552	0.129776	−	0.991543i	$-0.458574\pi$
$499$	5.79796	0.259552	0.129776	+	0.991543i	$0.458574\pi$
$500$	0	0
$501$	−8.00000	−0.357414
$502$	0	0
$503$	− 36.8990i	− 1.64524i	−0.568589	−	0.822622i	$-0.692510\pi$
$503$	− 36.8990i	− 1.64524i	0.568589	−	0.822622i	$-0.307490\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	12.7980i	0.568377i
$508$	0	0
$509$	18.2474	0.808804	0.404402	−	0.914581i	$-0.367480\pi$
$509$	18.2474	0.808804	0.404402	+	0.914581i	$0.367480\pi$
$510$	0	0
$511$	18.4949	0.818166
$512$	0	0
$513$	4.89898i	0.216295i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−19.7980	−0.869034
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	0	0
$523$	20.8990i	0.913849i	0.889506	+	0.456924i	$0.151049\pi$
$523$	20.8990i	0.913849i	−0.889506	+	0.456924i	$0.848951\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.00000i	0.0871214i
$528$	0	0
$529$	−24.5959	−1.06939
$530$	0	0
$531$	7.34847	0.318896
$532$	0	0
$533$	5.30306i	0.229701i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 23.5959i	− 1.01824i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−39.5959	−1.70236	−0.851181	−	0.524873i	$-0.824113\pi$
$541$	−39.5959	−1.70236	−0.851181	+	0.524873i	$0.824113\pi$
$542$	0	0
$543$	6.89898i	0.296064i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 22.9444i	− 0.981031i	−0.871432	−	0.490516i	$-0.836808\pi$
$547$	− 22.9444i	− 0.981031i	0.871432	−	0.490516i	$-0.163192\pi$
$548$	0	0
$549$	−2.89898	−0.123725
$550$	0	0
$551$	−21.7980	−0.928624
$552$	0	0
$553$	16.8990i	0.718618i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 17.5959i	− 0.745563i	−0.927919	−	0.372781i	$-0.878404\pi$
$557$	− 17.5959i	− 0.745563i	0.927919	−	0.372781i	$-0.121596\pi$
$558$	0	0
$559$	5.79796	0.245228
$560$	0	0
$561$	0	0
$562$	0	0
$563$	8.89898i	0.375047i	0.982260	+	0.187524i	$0.0600461\pi$
$563$	8.89898i	0.375047i	−0.982260	+	0.187524i	$0.939954\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 2.44949i	− 0.102869i
$568$	0	0
$569$	35.5505	1.49035	0.745177	−	0.666866i	$-0.232365\pi$
$569$	35.5505	1.49035	0.745177	+	0.666866i	$0.232365\pi$
$570$	0	0
$571$	−29.7980	−1.24701	−0.623503	−	0.781821i	$-0.714291\pi$
$571$	−29.7980	−1.24701	−0.623503	+	0.781821i	$0.714291\pi$
$572$	0	0
$573$	6.44949i	0.269431i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	41.5959i	1.73166i	0.500338	+	0.865830i	$0.333209\pi$
$577$	41.5959i	1.73166i	−0.500338	+	0.865830i	$0.666791\pi$
$578$	0	0
$579$	20.6969	0.860135
$580$	0	0
$581$	26.6969	1.10758
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 23.5959i	− 0.973908i	−0.873428	−	0.486954i	$-0.838108\pi$
$587$	− 23.5959i	− 0.973908i	0.873428	−	0.486954i	$-0.161892\pi$
$588$	0	0
$589$	−4.89898	−0.201859
$590$	0	0
$591$	−8.69694	−0.357744
$592$	0	0
$593$	− 5.59592i	− 0.229797i	−0.993377	−	0.114898i	$-0.963346\pi$
$593$	− 5.59592i	− 0.229797i	0.993377	−	0.114898i	$-0.0366543\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4.69694i	0.192233i
$598$	0	0
$599$	−38.9444	−1.59122	−0.795612	−	0.605806i	$-0.792851\pi$
$599$	−38.9444	−1.59122	−0.795612	+	0.605806i	$0.792851\pi$
$600$	0	0
$601$	32.6969	1.33374	0.666868	−	0.745176i	$-0.267634\pi$
$601$	32.6969	1.33374	0.666868	+	0.745176i	$0.267634\pi$
$602$	0	0
$603$	1.55051i	0.0631417i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	10.9444i	0.444219i	0.975022	+	0.222109i	$0.0712942\pi$
$607$	10.9444i	0.444219i	−0.975022	+	0.222109i	$0.928706\pi$
$608$	0	0
$609$	10.8990	0.441649
$610$	0	0
$611$	−0.404082	−0.0163474
$612$	0	0
$613$	8.04541i	0.324951i	0.986713	+	0.162475i	$0.0519478\pi$
$613$	8.04541i	0.324951i	−0.986713	+	0.162475i	$0.948052\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.00000i	0.241551i	0.992680	+	0.120775i	$0.0385381\pi$
$617$	6.00000i	0.241551i	−0.992680	+	0.120775i	$0.961462\pi$
$618$	0	0
$619$	2.49490	0.100278	0.0501392	−	0.998742i	$-0.484034\pi$
$619$	2.49490	0.100278	0.0501392	+	0.998742i	$0.484034\pi$
$620$	0	0
$621$	−6.89898	−0.276847
$622$	0	0
$623$	1.10102i	0.0441115i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	16.8990	0.673806
$630$	0	0
$631$	−35.5959	−1.41705	−0.708526	−	0.705685i	$-0.750639\pi$
$631$	−35.5959	−1.41705	−0.708526	+	0.705685i	$0.750639\pi$
$632$	0	0
$633$	− 28.4949i	− 1.13257i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 0.449490i	− 0.0178094i
$638$	0	0
$639$	−6.44949	−0.255138
$640$	0	0
$641$	0.0454077	0.00179350	0.000896748	−	1.00000i	$-0.499715\pi$
$641$	0.0454077	0.00179350	0.000896748		1.00000i	$0.499715\pi$
$642$	0	0
$643$	− 28.4949i	− 1.12373i	−0.827229	−	0.561865i	$-0.810084\pi$
$643$	− 28.4949i	− 1.12373i	0.827229	−	0.561865i	$-0.189916\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 19.5959i	− 0.770395i	−0.922834	−	0.385198i	$-0.874133\pi$
$647$	− 19.5959i	− 0.770395i	0.922834	−	0.385198i	$-0.125867\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	2.44949	0.0960031
$652$	0	0
$653$	− 4.00000i	− 0.156532i	−0.996933	−	0.0782660i	$-0.975062\pi$
$653$	− 4.00000i	− 0.156532i	0.996933	−	0.0782660i	$-0.0249384\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 7.55051i	− 0.294573i
$658$	0	0
$659$	−15.3485	−0.597891	−0.298946	−	0.954270i	$-0.596635\pi$
$659$	−15.3485	−0.597891	−0.298946	+	0.954270i	$0.596635\pi$
$660$	0	0
$661$	−19.3939	−0.754334	−0.377167	−	0.926145i	$-0.623102\pi$
$661$	−19.3939	−0.754334	−0.377167	+	0.926145i	$0.623102\pi$
$662$	0	0
$663$	0.898979i	0.0349135i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 30.6969i	− 1.18859i
$668$	0	0
$669$	9.79796	0.378811
$670$	0	0
$671$	0	0
$672$	0	0
$673$	24.9444i	0.961535i	0.876848	+	0.480768i	$0.159642\pi$
$673$	24.9444i	0.961535i	−0.876848	+	0.480768i	$0.840358\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 27.7980i	− 1.06836i	−0.845370	−	0.534181i	$-0.820620\pi$
$677$	− 27.7980i	− 1.06836i	0.845370	−	0.534181i	$-0.179380\pi$
$678$	0	0
$679$	−36.4949	−1.40055
$680$	0	0
$681$	−2.20204	−0.0843824
$682$	0	0
$683$	35.5959i	1.36204i	0.732265	+	0.681020i	$0.238463\pi$
$683$	35.5959i	1.36204i	−0.732265	+	0.681020i	$0.761537\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 0.202041i	− 0.00770835i
$688$	0	0
$689$	2.69694	0.102745
$690$	0	0
$691$	13.3939	0.509527	0.254764	−	0.967003i	$-0.418002\pi$
$691$	13.3939	0.509527	0.254764	+	0.967003i	$0.418002\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 23.5959i	− 0.893759i
$698$	0	0
$699$	−21.7980	−0.824475
$700$	0	0
$701$	15.3031	0.577989	0.288994	−	0.957331i	$-0.406679\pi$
$701$	15.3031	0.577989	0.288994	+	0.957331i	$0.406679\pi$
$702$	0	0
$703$	41.3939i	1.56120i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 4.89898i	− 0.184245i
$708$	0	0
$709$	−11.7980	−0.443082	−0.221541	−	0.975151i	$-0.571109\pi$
$709$	−11.7980	−0.443082	−0.221541	+	0.975151i	$0.571109\pi$
$710$	0	0
$711$	6.89898	0.258732
$712$	0	0
$713$	− 6.89898i	− 0.258369i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 7.10102i	− 0.265192i
$718$	0	0
$719$	1.79796	0.0670526	0.0335263	−	0.999438i	$-0.489326\pi$
$719$	1.79796	0.0670526	0.0335263	+	0.999438i	$0.489326\pi$
$720$	0	0
$721$	25.5959	0.953242
$722$	0	0
$723$	− 15.7980i	− 0.587532i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 17.1464i	− 0.635926i	−0.948103	−	0.317963i	$-0.897001\pi$
$727$	− 17.1464i	− 0.635926i	0.948103	−	0.317963i	$-0.102999\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−25.7980	−0.954172
$732$	0	0
$733$	− 49.5959i	− 1.83187i	−0.401330	−	0.915934i	$-0.631452\pi$
$733$	− 49.5959i	− 1.83187i	0.401330	−	0.915934i	$-0.368548\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	11.3031	0.415790	0.207895	−	0.978151i	$-0.433339\pi$
$739$	11.3031	0.415790	0.207895	+	0.978151i	$0.433339\pi$
$740$	0	0
$741$	−2.20204	−0.0808940
$742$	0	0
$743$	− 17.7980i	− 0.652944i	−0.945207	−	0.326472i	$-0.894140\pi$
$743$	− 17.7980i	− 0.652944i	0.945207	−	0.326472i	$-0.105860\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 10.8990i	− 0.398773i
$748$	0	0
$749$	12.0000	0.438470
$750$	0	0
$751$	13.7980	0.503495	0.251747	−	0.967793i	$-0.418995\pi$
$751$	13.7980	0.503495	0.251747	+	0.967793i	$0.418995\pi$
$752$	0	0
$753$	9.79796i	0.357057i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 7.55051i	− 0.274428i	−0.990541	−	0.137214i	$-0.956185\pi$
$757$	− 7.55051i	− 0.274428i	0.990541	−	0.137214i	$-0.0438148\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	25.8434	0.936821	0.468411	−	0.883511i	$-0.344827\pi$
$761$	25.8434	0.936821	0.468411	+	0.883511i	$0.344827\pi$
$762$	0	0
$763$	− 9.79796i	− 0.354710i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.30306i	0.119267i
$768$	0	0
$769$	−14.2020	−0.512139	−0.256069	−	0.966658i	$-0.582428\pi$
$769$	−14.2020	−0.512139	−0.256069	+	0.966658i	$0.582428\pi$
$770$	0	0
$771$	−16.0000	−0.576226
$772$	0	0
$773$	10.0000i	0.359675i	0.983696	+	0.179838i	$0.0575572\pi$
$773$	10.0000i	0.359675i	−0.983696	+	0.179838i	$0.942443\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 20.6969i	− 0.742499i
$778$	0	0
$779$	57.7980	2.07083
$780$	0	0
$781$	0	0
$782$	0	0
$783$	− 4.44949i	− 0.159012i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	31.1918i	1.11187i	0.831226	+	0.555934i	$0.187639\pi$
$787$	31.1918i	1.11187i	−0.831226	+	0.555934i	$0.812361\pi$
$788$	0	0
$789$	−29.3939	−1.04645
$790$	0	0
$791$	−9.30306	−0.330779
$792$	0	0
$793$	− 1.30306i	− 0.0462731i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 22.4949i	− 0.796810i	−0.917210	−	0.398405i	$-0.869564\pi$
$797$	− 22.4949i	− 0.796810i	0.917210	−	0.398405i	$-0.130436\pi$
$798$	0	0
$799$	1.79796	0.0636072
$800$	0	0
$801$	0.449490	0.0158819
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	32.0454i	1.12805i
$808$	0	0
$809$	2.65153	0.0932229	0.0466114	−	0.998913i	$-0.485158\pi$
$809$	2.65153	0.0932229	0.0466114	+	0.998913i	$0.485158\pi$
$810$	0	0
$811$	−22.2929	−0.782808	−0.391404	−	0.920219i	$-0.628011\pi$
$811$	−22.2929	−0.782808	−0.391404	+	0.920219i	$0.628011\pi$
$812$	0	0
$813$	1.79796i	0.0630572i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 63.1918i	− 2.21080i
$818$	0	0
$819$	1.10102	0.0384728
$820$	0	0
$821$	27.6413	0.964689	0.482344	−	0.875982i	$-0.339785\pi$
$821$	27.6413	0.964689	0.482344	+	0.875982i	$0.339785\pi$
$822$	0	0
$823$	24.4949i	0.853838i	0.904290	+	0.426919i	$0.140401\pi$
$823$	24.4949i	0.853838i	−0.904290	+	0.426919i	$0.859599\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	46.0908i	1.60273i	0.598173	+	0.801367i	$0.295894\pi$
$827$	46.0908i	1.60273i	−0.598173	+	0.801367i	$0.704106\pi$
$828$	0	0
$829$	6.89898	0.239611	0.119806	−	0.992797i	$-0.461773\pi$
$829$	6.89898	0.239611	0.119806	+	0.992797i	$0.461773\pi$
$830$	0	0
$831$	23.1464	0.802941
$832$	0	0
$833$	2.00000i	0.0692959i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 1.00000i	− 0.0345651i
$838$	0	0
$839$	9.55051	0.329720	0.164860	−	0.986317i	$-0.447283\pi$
$839$	9.55051	0.329720	0.164860	+	0.986317i	$0.447283\pi$
$840$	0	0
$841$	−9.20204	−0.317312
$842$	0	0
$843$	5.10102i	0.175688i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	26.9444i	0.925820i
$848$	0	0
$849$	−14.4495	−0.495905
$850$	0	0
$851$	−58.2929	−1.99825
$852$	0	0
$853$	− 39.3939i	− 1.34882i	−0.738357	−	0.674410i	$-0.764398\pi$
$853$	− 39.3939i	− 1.34882i	0.738357	−	0.674410i	$-0.235602\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 45.1918i	− 1.54372i	−0.635790	−	0.771862i	$-0.719326\pi$
$857$	− 45.1918i	− 1.54372i	0.635790	−	0.771862i	$-0.280674\pi$
$858$	0	0
$859$	8.40408	0.286744	0.143372	−	0.989669i	$-0.454206\pi$
$859$	8.40408	0.286744	0.143372	+	0.989669i	$0.454206\pi$
$860$	0	0
$861$	−28.8990	−0.984875
$862$	0	0
$863$	56.0000i	1.90626i	0.302558	+	0.953131i	$0.402160\pi$
$863$	56.0000i	1.90626i	−0.302558	+	0.953131i	$0.597840\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	13.0000i	0.441503i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−0.696938	−0.0236149
$872$	0	0
$873$	14.8990i	0.504254i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 35.7980i	− 1.20881i	−0.796677	−	0.604406i	$-0.793411\pi$
$877$	− 35.7980i	− 1.20881i	0.796677	−	0.604406i	$-0.206589\pi$
$878$	0	0
$879$	−2.20204	−0.0742730
$880$	0	0
$881$	26.2474	0.884299	0.442150	−	0.896941i	$-0.354216\pi$
$881$	26.2474	0.884299	0.442150	+	0.896941i	$0.354216\pi$
$882$	0	0
$883$	− 37.3939i	− 1.25840i	−0.777242	−	0.629202i	$-0.783382\pi$
$883$	− 37.3939i	− 1.25840i	0.777242	−	0.629202i	$-0.216618\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	20.0000i	0.671534i	0.941945	+	0.335767i	$0.108996\pi$
$887$	20.0000i	0.671534i	−0.941945	+	0.335767i	$0.891004\pi$
$888$	0	0
$889$	2.20204	0.0738541
$890$	0	0
$891$	0	0
$892$	0	0
$893$	4.40408i	0.147377i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 3.10102i	− 0.103540i
$898$	0	0
$899$	4.44949	0.148399
$900$	0	0
$901$	−12.0000	−0.399778
$902$	0	0
$903$	31.5959i	1.05145i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	2.85357i	0.0947513i	0.998877	+	0.0473756i	$0.0150858\pi$
$907$	2.85357i	0.0947513i	−0.998877	+	0.0473756i	$0.984914\pi$
$908$	0	0
$909$	−2.00000	−0.0663358
$910$	0	0
$911$	−25.7980	−0.854725	−0.427362	−	0.904080i	$-0.640557\pi$
$911$	−25.7980	−0.854725	−0.427362	+	0.904080i	$0.640557\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	30.0000i	0.990687i
$918$	0	0
$919$	−54.6969	−1.80429	−0.902143	−	0.431438i	$-0.858006\pi$
$919$	−54.6969	−1.80429	−0.902143	+	0.431438i	$0.858006\pi$
$920$	0	0
$921$	17.5505	0.578309
$922$	0	0
$923$	− 2.89898i	− 0.0954211i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 10.4495i	− 0.343206i
$928$	0	0
$929$	−12.9444	−0.424692	−0.212346	−	0.977195i	$-0.568110\pi$
$929$	−12.9444	−0.424692	−0.212346	+	0.977195i	$0.568110\pi$
$930$	0	0
$931$	−4.89898	−0.160558
$932$	0	0
$933$	19.3485i	0.633440i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	26.4949i	0.865551i	0.901502	+	0.432775i	$0.142466\pi$
$937$	26.4949i	0.865551i	−0.901502	+	0.432775i	$0.857534\pi$
$938$	0	0
$939$	1.34847	0.0440056
$940$	0	0
$941$	22.2474	0.725246	0.362623	−	0.931936i	$-0.381881\pi$
$941$	22.2474	0.725246	0.362623	+	0.931936i	$0.381881\pi$
$942$	0	0
$943$	81.3939i	2.65055i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	42.4949i	1.38090i	0.723381	+	0.690449i	$0.242587\pi$
$947$	42.4949i	1.38090i	−0.723381	+	0.690449i	$0.757413\pi$
$948$	0	0
$949$	3.39388	0.110170
$950$	0	0
$951$	−23.5959	−0.765150
$952$	0	0
$953$	− 11.3031i	− 0.366142i	−0.983100	−	0.183071i	$-0.941396\pi$
$953$	− 11.3031i	− 0.366142i	0.983100	−	0.183071i	$-0.0586038\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−12.4949	−0.403481
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	− 4.89898i	− 0.157867i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 29.3939i	− 0.945243i	−0.881265	−	0.472622i	$-0.843308\pi$
$967$	− 29.3939i	− 0.945243i	0.881265	−	0.472622i	$-0.156692\pi$
$968$	0	0
$969$	9.79796	0.314756
$970$	0	0
$971$	−26.4495	−0.848805	−0.424402	−	0.905474i	$-0.639516\pi$
$971$	−26.4495	−0.848805	−0.424402	+	0.905474i	$0.639516\pi$
$972$	0	0
$973$	− 14.2020i	− 0.455297i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	53.1918i	1.70176i	0.525362	+	0.850879i	$0.323930\pi$
$977$	53.1918i	1.70176i	−0.525362	+	0.850879i	$0.676070\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−4.00000	−0.127710
$982$	0	0
$983$	− 25.7980i	− 0.822827i	−0.911449	−	0.411414i	$-0.865035\pi$
$983$	− 25.7980i	− 0.822827i	0.911449	−	0.411414i	$-0.134965\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 2.20204i	− 0.0700917i
$988$	0	0
$989$	88.9898	2.82971
$990$	0	0
$991$	−21.3939	−0.679599	−0.339799	−	0.940498i	$-0.610359\pi$
$991$	−21.3939	−0.679599	−0.339799	+	0.940498i	$0.610359\pi$
$992$	0	0
$993$	− 10.8990i	− 0.345869i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	40.2929i	1.27609i	0.770000	+	0.638044i	$0.220256\pi$
$997$	40.2929i	1.27609i	−0.770000	+	0.638044i	$0.779744\pi$
$998$	0	0
$999$	−8.44949	−0.267330

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9300.2.g.l.3349.3		4
5.2	odd	4		9300.2.a.p.1.2		2
5.3	odd	4		1860.2.a.d.1.1	✓	2
5.4	even	2	inner	9300.2.g.l.3349.2		4
15.8	even	4		5580.2.a.g.1.1		2
20.3	even	4		7440.2.a.bj.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1860.2.a.d.1.1	✓	2	5.3	odd	4
5580.2.a.g.1.1		2	15.8	even	4
7440.2.a.bj.1.2		2	20.3	even	4
9300.2.a.p.1.2		2	5.2	odd	4
9300.2.g.l.3349.2		4	5.4	even	2	inner
9300.2.g.l.3349.3		4	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:	\( N \)	\(=\)	\( 9300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9300.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -2.44949i q^{7} -1.00000 q^{9} -0.449490i q^{13} +2.00000i q^{17} -4.89898 q^{19} +2.44949 q^{21} -6.89898i q^{23} -1.00000i q^{27} +4.44949 q^{29} +1.00000 q^{31} -8.44949i q^{37} +0.449490 q^{39} -11.7980 q^{41} +12.8990i q^{43} -0.898979i q^{47} +1.00000 q^{49} -2.00000 q^{51} +6.00000i q^{53} -4.89898i q^{57} -7.34847 q^{59} +2.89898 q^{61} +2.44949i q^{63} -1.55051i q^{67} +6.89898 q^{69} +6.44949 q^{71} +7.55051i q^{73} -6.89898 q^{79} +1.00000 q^{81} +10.8990i q^{83} +4.44949i q^{87} -0.449490 q^{89} -1.10102 q^{91} +1.00000i q^{93} -14.8990i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9} + 8 q^{29} + 4 q^{31} - 8 q^{39} - 8 q^{41} + 4 q^{49} - 8 q^{51} - 8 q^{61} + 8 q^{69} + 16 q^{71} - 8 q^{79} + 4 q^{81} + 8 q^{89} - 24 q^{91}+O(q^{100}) \) 4 * q - 4 * q^9 + 8 * q^29 + 4 * q^31 - 8 * q^39 - 8 * q^41 + 4 * q^49 - 8 * q^51 - 8 * q^61 + 8 * q^69 + 16 * q^71 - 8 * q^79 + 4 * q^81 + 8 * q^89 - 24 * q^91

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