LMFDB - Embedded newform 3800.2.d.q.3649.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,2,Mod(3649,3800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, 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("3800.3649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3800 = 2^{3} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3800.d (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:	$30.3431527681$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 24x^{10} + 194x^{8} + 618x^{6} + 733x^{4} + 286x^{2} + 9 $ x^12 + 24x^10 + 194x^8 + 618x^6 + 733x^4 + 286*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3649.4
Root			$-1.08999i$ of defining polynomial
Character	$\chi$	$=$	3800.3649
Dual form			3800.2.d.q.3649.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.08999i q^{3} -4.19727i q^{7} +1.81192 q^{9} +O(q^{10})$	$q-1.08999i q^{3} -4.19727i q^{7} +1.81192 q^{9} -6.43052 q^{11} +2.24614i q^{13} +7.84744i q^{17} +1.00000 q^{19} -4.57497 q^{21} -0.859601i q^{23} -5.24494i q^{27} -8.38284 q^{29} +1.24541 q^{31} +7.00919i q^{33} +6.79977i q^{37} +2.44827 q^{39} +5.92480 q^{41} +6.81073i q^{43} +6.00919i q^{47} -10.6171 q^{49} +8.55363 q^{51} +13.7594i q^{53} -1.08999i q^{57} -6.88976 q^{59} -1.31884 q^{61} -7.60513i q^{63} -4.73266i q^{67} -0.936955 q^{69} +10.2546 q^{71} -9.86150i q^{73} +26.9906i q^{77} -6.47056 q^{79} -0.281158 q^{81} +5.42133i q^{83} +9.13720i q^{87} -3.10288 q^{89} +9.42766 q^{91} -1.35748i q^{93} -6.76032i q^{97} -11.6516 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^9	$ 12 q - 12 q^{9} + 6 q^{11} + 12 q^{19} + 30 q^{21} - 18 q^{29} + 10 q^{31} - 24 q^{39} + 6 q^{41} - 44 q^{49} + 66 q^{51} + 18 q^{61} + 22 q^{69} + 38 q^{71} + 32 q^{79} + 52 q^{81} - 28 q^{89} + 84 q^{91} + 12 q^{99}+O(q^{100}) $ 12 * q - 12 * q^9 + 6 * q^11 + 12 * q^19 + 30 * q^21 - 18 * q^29 + 10 * q^31 - 24 * q^39 + 6 * q^41 - 44 * q^49 + 66 * q^51 + 18 * q^61 + 22 * q^69 + 38 * q^71 + 32 * q^79 + 52 * q^81 - 28 * q^89 + 84 * q^91 + 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times$.

$n$	$401$	$951$	$1901$	$1977$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	− 1.08999i	− 0.629305i	−0.949207	−	0.314653i	$-0.898112\pi$
$3$	− 1.08999i	− 0.629305i	0.949207	−	0.314653i	$-0.101888\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 4.19727i	− 1.58642i	−0.608949	−	0.793209i	$-0.708409\pi$
$7$	− 4.19727i	− 1.58642i	0.608949	−	0.793209i	$-0.291591\pi$
$8$	0	0
$9$	1.81192	0.603975
$10$	0	0
$11$	−6.43052	−1.93887	−0.969437	−	0.245341i	$-0.921100\pi$
$11$	−6.43052	−1.93887	−0.969437	+	0.245341i	$0.921100\pi$
$12$	0	0
$13$	2.24614i	0.622967i	0.950251	+	0.311484i	$0.100826\pi$
$13$	2.24614i	0.622967i	−0.950251	+	0.311484i	$0.899174\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.84744i	1.90328i	0.307208	+	0.951642i	$0.400605\pi$
$17$	7.84744i	1.90328i	−0.307208	+	0.951642i	$0.599395\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−4.57497	−0.998341
$22$	0	0
$23$	− 0.859601i	− 0.179239i	−0.995976	−	0.0896195i	$-0.971435\pi$
$23$	− 0.859601i	− 0.179239i	0.995976	−	0.0896195i	$-0.0285651\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 5.24494i	− 1.00939i
$28$	0	0
$29$	−8.38284	−1.55665	−0.778327	−	0.627859i	$-0.783932\pi$
$29$	−8.38284	−1.55665	−0.778327	+	0.627859i	$0.783932\pi$
$30$	0	0
$31$	1.24541	0.223681	0.111841	−	0.993726i	$-0.464325\pi$
$31$	1.24541	0.223681	0.111841	+	0.993726i	$0.464325\pi$
$32$	0	0
$33$	7.00919i	1.22014i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.79977i	1.11787i	0.829210	+	0.558937i	$0.188791\pi$
$37$	6.79977i	1.11787i	−0.829210	+	0.558937i	$0.811209\pi$
$38$	0	0
$39$	2.44827	0.392037
$40$	0	0
$41$	5.92480	0.925298	0.462649	−	0.886542i	$-0.346899\pi$
$41$	5.92480	0.925298	0.462649	+	0.886542i	$0.346899\pi$
$42$	0	0
$43$	6.81073i	1.03863i	0.854584	+	0.519313i	$0.173812\pi$
$43$	6.81073i	1.03863i	−0.854584	+	0.519313i	$0.826188\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.00919i	0.876531i	0.898846	+	0.438265i	$0.144407\pi$
$47$	6.00919i	0.876531i	−0.898846	+	0.438265i	$0.855593\pi$
$48$	0	0
$49$	−10.6171	−1.51672
$50$	0	0
$51$	8.55363	1.19775
$52$	0	0
$53$	13.7594i	1.88999i	0.327082	+	0.944996i	$0.393935\pi$
$53$	13.7594i	1.88999i	−0.327082	+	0.944996i	$0.606065\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 1.08999i	− 0.144373i
$58$	0	0
$59$	−6.88976	−0.896970	−0.448485	−	0.893790i	$-0.648036\pi$
$59$	−6.88976	−0.896970	−0.448485	+	0.893790i	$0.648036\pi$
$60$	0	0
$61$	−1.31884	−0.168860	−0.0844301	−	0.996429i	$-0.526907\pi$
$61$	−1.31884	−0.168860	−0.0844301	+	0.996429i	$0.526907\pi$
$62$	0	0
$63$	− 7.60513i	− 0.958156i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 4.73266i	− 0.578187i	−0.957301	−	0.289093i	$-0.906646\pi$
$67$	− 4.73266i	− 0.578187i	0.957301	−	0.289093i	$-0.0933538\pi$
$68$	0	0
$69$	−0.936955	−0.112796
$70$	0	0
$71$	10.2546	1.21700	0.608498	−	0.793555i	$-0.291772\pi$
$71$	10.2546	1.21700	0.608498	+	0.793555i	$0.291772\pi$
$72$	0	0
$73$	− 9.86150i	− 1.15420i	−0.816673	−	0.577100i	$-0.804184\pi$
$73$	− 9.86150i	− 1.15420i	0.816673	−	0.577100i	$-0.195816\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	26.9906i	3.07586i
$78$	0	0
$79$	−6.47056	−0.727995	−0.363997	−	0.931400i	$-0.618588\pi$
$79$	−6.47056	−0.727995	−0.363997	+	0.931400i	$0.618588\pi$
$80$	0	0
$81$	−0.281158	−0.0312397
$82$	0	0
$83$	5.42133i	0.595068i	0.954711	+	0.297534i	$0.0961641\pi$
$83$	5.42133i	0.595068i	−0.954711	+	0.297534i	$0.903836\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	9.13720i	0.979611i
$88$	0	0
$89$	−3.10288	−0.328905	−0.164452	−	0.986385i	$-0.552586\pi$
$89$	−3.10288	−0.328905	−0.164452	+	0.986385i	$0.552586\pi$
$90$	0	0
$91$	9.42766	0.988287
$92$	0	0
$93$	− 1.35748i	− 0.140764i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 6.76032i	− 0.686406i	−0.939261	−	0.343203i	$-0.888488\pi$
$97$	− 6.76032i	− 0.686406i	0.939261	−	0.343203i	$-0.111512\pi$
$98$	0	0
$99$	−11.6516	−1.17103
$100$	0	0
$101$	−3.05247	−0.303732	−0.151866	−	0.988401i	$-0.548528\pi$
$101$	−3.05247	−0.303732	−0.151866	+	0.988401i	$0.548528\pi$
$102$	0	0
$103$	7.06366i	0.696003i	0.937494	+	0.348002i	$0.113140\pi$
$103$	7.06366i	0.696003i	−0.937494	+	0.348002i	$0.886860\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.05281i	0.488474i	0.969716	+	0.244237i	$0.0785374\pi$
$107$	5.05281i	0.488474i	−0.969716	+	0.244237i	$0.921463\pi$
$108$	0	0
$109$	6.39334	0.612371	0.306185	−	0.951972i	$-0.400947\pi$
$109$	6.39334	0.612371	0.306185	+	0.951972i	$0.400947\pi$
$110$	0	0
$111$	7.41167	0.703485
$112$	0	0
$113$	− 19.5837i	− 1.84228i	−0.389231	−	0.921140i	$-0.627259\pi$
$113$	− 19.5837i	− 1.84228i	0.389231	−	0.921140i	$-0.372741\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.06984i	0.376257i
$118$	0	0
$119$	32.9378	3.01941
$120$	0	0
$121$	30.3515	2.75923
$122$	0	0
$123$	− 6.45796i	− 0.582295i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 6.40323i	− 0.568195i	−0.958795	−	0.284098i	$-0.908306\pi$
$127$	− 6.40323i	− 0.568195i	0.958795	−	0.284098i	$-0.0916940\pi$
$128$	0	0
$129$	7.42362	0.653613
$130$	0	0
$131$	12.1913	1.06516	0.532581	−	0.846379i	$-0.321222\pi$
$131$	12.1913	1.06516	0.532581	+	0.846379i	$0.321222\pi$
$132$	0	0
$133$	− 4.19727i	− 0.363949i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.71228i	0.829776i	0.909872	+	0.414888i	$0.136179\pi$
$137$	9.71228i	0.829776i	−0.909872	+	0.414888i	$0.863821\pi$
$138$	0	0
$139$	−20.0687	−1.70220	−0.851102	−	0.525000i	$-0.824065\pi$
$139$	−20.0687	−1.70220	−0.851102	+	0.525000i	$0.824065\pi$
$140$	0	0
$141$	6.54995	0.551605
$142$	0	0
$143$	− 14.4438i	− 1.20786i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	11.5725i	0.954481i
$148$	0	0
$149$	14.8593	1.21732	0.608659	−	0.793432i	$-0.291708\pi$
$149$	14.8593	1.21732	0.608659	+	0.793432i	$0.291708\pi$
$150$	0	0
$151$	−1.29571	−0.105444	−0.0527218	−	0.998609i	$-0.516790\pi$
$151$	−1.29571	−0.105444	−0.0527218	+	0.998609i	$0.516790\pi$
$152$	0	0
$153$	14.2190i	1.14954i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 4.05355i	− 0.323508i	−0.986831	−	0.161754i	$-0.948285\pi$
$157$	− 4.05355i	− 0.323508i	0.986831	−	0.161754i	$-0.0517151\pi$
$158$	0	0
$159$	14.9975	1.18938
$160$	0	0
$161$	−3.60797	−0.284348
$162$	0	0
$163$	8.09764i	0.634256i	0.948383	+	0.317128i	$0.102719\pi$
$163$	8.09764i	0.634256i	−0.948383	+	0.317128i	$0.897281\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	18.6059i	1.43976i	0.694096	+	0.719882i	$0.255804\pi$
$167$	18.6059i	1.43976i	−0.694096	+	0.719882i	$0.744196\pi$
$168$	0	0
$169$	7.95485	0.611911
$170$	0	0
$171$	1.81192	0.138561
$172$	0	0
$173$	8.35972i	0.635578i	0.948161	+	0.317789i	$0.102940\pi$
$173$	8.35972i	0.635578i	−0.948161	+	0.317789i	$0.897060\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	7.50976i	0.564468i
$178$	0	0
$179$	−14.5971	−1.09104	−0.545521	−	0.838097i	$-0.683668\pi$
$179$	−14.5971	−1.09104	−0.545521	+	0.838097i	$0.683668\pi$
$180$	0	0
$181$	0.131566	0.00977923	0.00488961	−	0.999988i	$-0.498444\pi$
$181$	0.131566	0.00977923	0.00488961	+	0.999988i	$0.498444\pi$
$182$	0	0
$183$	1.43752i	0.106265i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 50.4631i	− 3.69023i
$188$	0	0
$189$	−22.0144	−1.60131
$190$	0	0
$191$	−22.5544	−1.63198	−0.815989	−	0.578067i	$-0.803807\pi$
$191$	−22.5544	−1.63198	−0.815989	+	0.578067i	$0.803807\pi$
$192$	0	0
$193$	4.93863i	0.355490i	0.984077	+	0.177745i	$0.0568803\pi$
$193$	4.93863i	0.355490i	−0.984077	+	0.177745i	$0.943120\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 26.2422i	− 1.86968i	−0.355069	−	0.934840i	$-0.615543\pi$
$197$	− 26.2422i	− 1.86968i	0.355069	−	0.934840i	$-0.384457\pi$
$198$	0	0
$199$	−15.1880	−1.07665	−0.538323	−	0.842738i	$-0.680942\pi$
$199$	−15.1880	−1.07665	−0.538323	+	0.842738i	$0.680942\pi$
$200$	0	0
$201$	−5.15855	−0.363856
$202$	0	0
$203$	35.1850i	2.46950i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 1.55753i	− 0.108256i
$208$	0	0
$209$	−6.43052	−0.444808
$210$	0	0
$211$	20.3632	1.40186	0.700929	−	0.713231i	$-0.252769\pi$
$211$	20.3632	1.40186	0.700929	+	0.713231i	$0.252769\pi$
$212$	0	0
$213$	− 11.1774i	− 0.765863i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 5.22730i	− 0.354852i
$218$	0	0
$219$	−10.7489	−0.726345
$220$	0	0
$221$	−17.6265	−1.18568
$222$	0	0
$223$	15.3764i	1.02968i	0.857285	+	0.514841i	$0.172149\pi$
$223$	15.3764i	1.02968i	−0.857285	+	0.514841i	$0.827851\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	20.0187i	1.32868i	0.747428	+	0.664342i	$0.231288\pi$
$227$	20.0187i	1.32868i	−0.747428	+	0.664342i	$0.768712\pi$
$228$	0	0
$229$	12.6175	0.833789	0.416895	−	0.908955i	$-0.363118\pi$
$229$	12.6175	0.833789	0.416895	+	0.908955i	$0.363118\pi$
$230$	0	0
$231$	29.4195	1.93566
$232$	0	0
$233$	21.9230i	1.43622i	0.695927	+	0.718112i	$0.254993\pi$
$233$	21.9230i	1.43622i	−0.695927	+	0.718112i	$0.745007\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	7.05284i	0.458131i
$238$	0	0
$239$	−3.66866	−0.237306	−0.118653	−	0.992936i	$-0.537858\pi$
$239$	−3.66866	−0.237306	−0.118653	+	0.992936i	$0.537858\pi$
$240$	0	0
$241$	−6.03281	−0.388608	−0.194304	−	0.980941i	$-0.562245\pi$
$241$	−6.03281	−0.388608	−0.194304	+	0.980941i	$0.562245\pi$
$242$	0	0
$243$	− 15.4284i	− 0.989731i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.24614i	0.142919i
$248$	0	0
$249$	5.90918	0.374479
$250$	0	0
$251$	0.145763	0.00920048	0.00460024	−	0.999989i	$-0.498536\pi$
$251$	0.145763	0.00920048	0.00460024	+	0.999989i	$0.498536\pi$
$252$	0	0
$253$	5.52768i	0.347522i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.7514i	0.920164i	0.887876	+	0.460082i	$0.152180\pi$
$257$	14.7514i	0.920164i	−0.887876	+	0.460082i	$0.847820\pi$
$258$	0	0
$259$	28.5404	1.77342
$260$	0	0
$261$	−15.1891	−0.940180
$262$	0	0
$263$	30.2967i	1.86817i	0.357046	+	0.934087i	$0.383784\pi$
$263$	30.2967i	1.86817i	−0.357046	+	0.934087i	$0.616216\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	3.38211i	0.206981i
$268$	0	0
$269$	−20.3460	−1.24052	−0.620259	−	0.784397i	$-0.712973\pi$
$269$	−20.3460	−1.24052	−0.620259	+	0.784397i	$0.712973\pi$
$270$	0	0
$271$	1.01349	0.0615651	0.0307825	−	0.999526i	$-0.490200\pi$
$271$	1.01349	0.0615651	0.0307825	+	0.999526i	$0.490200\pi$
$272$	0	0
$273$	− 10.2760i	− 0.621934i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	10.4346i	0.626953i	0.949596	+	0.313476i	$0.101494\pi$
$277$	10.4346i	0.626953i	−0.949596	+	0.313476i	$0.898506\pi$
$278$	0	0
$279$	2.25658	0.135098
$280$	0	0
$281$	−8.12732	−0.484835	−0.242418	−	0.970172i	$-0.577940\pi$
$281$	−8.12732	−0.484835	−0.242418	+	0.970172i	$0.577940\pi$
$282$	0	0
$283$	11.6397i	0.691907i	0.938252	+	0.345953i	$0.112444\pi$
$283$	11.6397i	0.691907i	−0.938252	+	0.345953i	$0.887556\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 24.8680i	− 1.46791i
$288$	0	0
$289$	−44.5824	−2.62249
$290$	0	0
$291$	−7.36867	−0.431959
$292$	0	0
$293$	− 2.21896i	− 0.129633i	−0.997897	−	0.0648164i	$-0.979354\pi$
$293$	− 2.21896i	− 0.129633i	0.997897	−	0.0648164i	$-0.0206462\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	33.7277i	1.95708i
$298$	0	0
$299$	1.93078	0.111660
$300$	0	0
$301$	28.5864	1.64770
$302$	0	0
$303$	3.32716i	0.191140i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 17.6712i	− 1.00855i	−0.863544	−	0.504274i	$-0.831760\pi$
$307$	− 17.6712i	− 1.00855i	0.863544	−	0.504274i	$-0.168240\pi$
$308$	0	0
$309$	7.69931	0.437999
$310$	0	0
$311$	12.3628	0.701031	0.350516	−	0.936557i	$-0.386006\pi$
$311$	12.3628	0.701031	0.350516	+	0.936557i	$0.386006\pi$
$312$	0	0
$313$	8.35545i	0.472278i	0.971719	+	0.236139i	$0.0758821\pi$
$313$	8.35545i	0.472278i	−0.971719	+	0.236139i	$0.924118\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	19.3526i	1.08695i	0.839425	+	0.543475i	$0.182892\pi$
$317$	19.3526i	1.08695i	−0.839425	+	0.543475i	$0.817108\pi$
$318$	0	0
$319$	53.9060	3.01816
$320$	0	0
$321$	5.50751	0.307399
$322$	0	0
$323$	7.84744i	0.436643i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 6.96867i	− 0.385368i
$328$	0	0
$329$	25.2222	1.39054
$330$	0	0
$331$	33.8929	1.86292	0.931461	−	0.363840i	$-0.118535\pi$
$331$	33.8929	1.86292	0.931461	+	0.363840i	$0.118535\pi$
$332$	0	0
$333$	12.3207i	0.675168i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	28.6728i	1.56191i	0.624588	+	0.780954i	$0.285267\pi$
$337$	28.6728i	1.56191i	−0.624588	+	0.780954i	$0.714733\pi$
$338$	0	0
$339$	−21.3460	−1.15936
$340$	0	0
$341$	−8.00860	−0.433690
$342$	0	0
$343$	15.1818i	0.819738i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 15.9590i	− 0.856725i	−0.903607	−	0.428362i	$-0.859091\pi$
$347$	− 15.9590i	− 0.856725i	0.903607	−	0.428362i	$-0.140909\pi$
$348$	0	0
$349$	−29.4645	−1.57720	−0.788599	−	0.614907i	$-0.789193\pi$
$349$	−29.4645	−1.57720	−0.788599	+	0.614907i	$0.789193\pi$
$350$	0	0
$351$	11.7809	0.628817
$352$	0	0
$353$	− 28.3655i	− 1.50974i	−0.655873	−	0.754871i	$-0.727699\pi$
$353$	− 28.3655i	− 1.50974i	0.655873	−	0.754871i	$-0.272301\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 35.9019i	− 1.90013i
$358$	0	0
$359$	−0.0301128	−0.00158929	−0.000794647	−	1.00000i	$-0.500253\pi$
$359$	−0.0301128	−0.00158929	−0.000794647		1.00000i	$0.500253\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	− 33.0829i	− 1.73640i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.31478i	0.0686310i	0.999411	+	0.0343155i	$0.0109251\pi$
$367$	1.31478i	0.0686310i	−0.999411	+	0.0343155i	$0.989075\pi$
$368$	0	0
$369$	10.7353	0.558857
$370$	0	0
$371$	57.7517	2.99832
$372$	0	0
$373$	− 3.63316i	− 0.188118i	−0.995567	−	0.0940589i	$-0.970016\pi$
$373$	− 3.63316i	− 0.188118i	0.995567	−	0.0940589i	$-0.0299842\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 18.8290i	− 0.969745i
$378$	0	0
$379$	−24.4908	−1.25801	−0.629004	−	0.777402i	$-0.716537\pi$
$379$	−24.4908	−1.25801	−0.629004	+	0.777402i	$0.716537\pi$
$380$	0	0
$381$	−6.97945	−0.357568
$382$	0	0
$383$	3.23421i	0.165261i	0.996580	+	0.0826303i	$0.0263321\pi$
$383$	3.23421i	0.165261i	−0.996580	+	0.0826303i	$0.973668\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	12.3405i	0.627304i
$388$	0	0
$389$	−22.9240	−1.16229	−0.581147	−	0.813799i	$-0.697396\pi$
$389$	−22.9240	−1.16229	−0.581147	+	0.813799i	$0.697396\pi$
$390$	0	0
$391$	6.74567	0.341143
$392$	0	0
$393$	− 13.2884i	− 0.670312i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	14.1567i	0.710507i	0.934770	+	0.355253i	$0.115605\pi$
$397$	14.1567i	0.710507i	−0.934770	+	0.355253i	$0.884395\pi$
$398$	0	0
$399$	−4.57497	−0.229035
$400$	0	0
$401$	−6.06391	−0.302817	−0.151409	−	0.988471i	$-0.548381\pi$
$401$	−6.06391	−0.302817	−0.151409	+	0.988471i	$0.548381\pi$
$402$	0	0
$403$	2.79736i	0.139346i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 43.7260i	− 2.16742i
$408$	0	0
$409$	−3.95218	−0.195423	−0.0977113	−	0.995215i	$-0.531152\pi$
$409$	−3.95218	−0.195423	−0.0977113	+	0.995215i	$0.531152\pi$
$410$	0	0
$411$	10.5863	0.522183
$412$	0	0
$413$	28.9182i	1.42297i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	21.8746i	1.07121i
$418$	0	0
$419$	12.6137	0.616221	0.308111	−	0.951351i	$-0.400303\pi$
$419$	12.6137	0.616221	0.308111	+	0.951351i	$0.400303\pi$
$420$	0	0
$421$	−9.15060	−0.445973	−0.222986	−	0.974822i	$-0.571581\pi$
$421$	−9.15060	−0.445973	−0.222986	+	0.974822i	$0.571581\pi$
$422$	0	0
$423$	10.8882i	0.529402i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	5.53553i	0.267883i
$428$	0	0
$429$	−15.7436	−0.760110
$430$	0	0
$431$	−27.2560	−1.31288	−0.656439	−	0.754379i	$-0.727938\pi$
$431$	−27.2560	−1.31288	−0.656439	+	0.754379i	$0.727938\pi$
$432$	0	0
$433$	2.33860i	0.112386i	0.998420	+	0.0561929i	$0.0178962\pi$
$433$	2.33860i	0.112386i	−0.998420	+	0.0561929i	$0.982104\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 0.859601i	− 0.0411203i
$438$	0	0
$439$	−37.6773	−1.79824	−0.899119	−	0.437703i	$-0.855792\pi$
$439$	−37.6773	−1.79824	−0.899119	+	0.437703i	$0.855792\pi$
$440$	0	0
$441$	−19.2373	−0.916062
$442$	0	0
$443$	6.39870i	0.304011i	0.988380	+	0.152006i	$0.0485732\pi$
$443$	6.39870i	0.304011i	−0.988380	+	0.152006i	$0.951427\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 16.1964i	− 0.766065i
$448$	0	0
$449$	30.1784	1.42421	0.712103	−	0.702075i	$-0.247743\pi$
$449$	30.1784	1.42421	0.712103	+	0.702075i	$0.247743\pi$
$450$	0	0
$451$	−38.0995	−1.79404
$452$	0	0
$453$	1.41231i	0.0663562i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 7.21464i	− 0.337487i	−0.985660	−	0.168743i	$-0.946029\pi$
$457$	− 7.21464i	− 0.337487i	0.985660	−	0.168743i	$-0.0539709\pi$
$458$	0	0
$459$	41.1594	1.92116
$460$	0	0
$461$	−2.06925	−0.0963748	−0.0481874	−	0.998838i	$-0.515344\pi$
$461$	−2.06925	−0.0963748	−0.0481874	+	0.998838i	$0.515344\pi$
$462$	0	0
$463$	− 4.01966i	− 0.186809i	−0.995628	−	0.0934047i	$-0.970225\pi$
$463$	− 4.01966i	− 0.186809i	0.995628	−	0.0934047i	$-0.0297750\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	21.5515i	0.997285i	0.866808	+	0.498643i	$0.166168\pi$
$467$	21.5515i	0.997285i	−0.866808	+	0.498643i	$0.833832\pi$
$468$	0	0
$469$	−19.8643	−0.917246
$470$	0	0
$471$	−4.41832	−0.203585
$472$	0	0
$473$	− 43.7965i	− 2.01377i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	24.9309i	1.14151i
$478$	0	0
$479$	31.8315	1.45442	0.727209	−	0.686416i	$-0.240817\pi$
$479$	31.8315	1.45442	0.727209	+	0.686416i	$0.240817\pi$
$480$	0	0
$481$	−15.2732	−0.696400
$482$	0	0
$483$	3.93265i	0.178942i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	3.54933i	0.160835i	0.996761	+	0.0804177i	$0.0256254\pi$
$487$	3.54933i	0.160835i	−0.996761	+	0.0804177i	$0.974375\pi$
$488$	0	0
$489$	8.82634	0.399141
$490$	0	0
$491$	−16.5336	−0.746152	−0.373076	−	0.927801i	$-0.621697\pi$
$491$	−16.5336	−0.746152	−0.373076	+	0.927801i	$0.621697\pi$
$492$	0	0
$493$	− 65.7839i	− 2.96276i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 43.0413i	− 1.93067i
$498$	0	0
$499$	32.8826	1.47203	0.736013	−	0.676968i	$-0.236706\pi$
$499$	32.8826	1.47203	0.736013	+	0.676968i	$0.236706\pi$
$500$	0	0
$501$	20.2802	0.906051
$502$	0	0
$503$	− 17.2150i	− 0.767580i	−0.923420	−	0.383790i	$-0.874619\pi$
$503$	− 17.2150i	− 0.767580i	0.923420	−	0.383790i	$-0.125381\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 8.67070i	− 0.385079i
$508$	0	0
$509$	−28.4368	−1.26044	−0.630221	−	0.776416i	$-0.717036\pi$
$509$	−28.4368	−1.26044	−0.630221	+	0.776416i	$0.717036\pi$
$510$	0	0
$511$	−41.3913	−1.83105
$512$	0	0
$513$	− 5.24494i	− 0.231570i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 38.6422i	− 1.69948i
$518$	0	0
$519$	9.11201	0.399973
$520$	0	0
$521$	25.1924	1.10370	0.551850	−	0.833944i	$-0.313922\pi$
$521$	25.1924	1.10370	0.551850	+	0.833944i	$0.313922\pi$
$522$	0	0
$523$	− 17.2647i	− 0.754934i	−0.926023	−	0.377467i	$-0.876795\pi$
$523$	− 17.2647i	− 0.754934i	0.926023	−	0.377467i	$-0.123205\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.77325i	0.425730i
$528$	0	0
$529$	22.2611	0.967873
$530$	0	0
$531$	−12.4837	−0.541747
$532$	0	0
$533$	13.3079i	0.576431i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	15.9107i	0.686598i
$538$	0	0
$539$	68.2732	2.94073
$540$	0	0
$541$	−9.71009	−0.417469	−0.208735	−	0.977972i	$-0.566935\pi$
$541$	−9.71009	−0.417469	−0.208735	+	0.977972i	$0.566935\pi$
$542$	0	0
$543$	− 0.143406i	− 0.00615412i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 5.60962i	− 0.239850i	−0.992783	−	0.119925i	$-0.961735\pi$
$547$	− 5.60962i	− 0.239850i	0.992783	−	0.119925i	$-0.0382654\pi$
$548$	0	0
$549$	−2.38964	−0.101987
$550$	0	0
$551$	−8.38284	−0.357121
$552$	0	0
$553$	27.1587i	1.15490i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	17.5323i	0.742868i	0.928459	+	0.371434i	$0.121134\pi$
$557$	17.5323i	0.742868i	−0.928459	+	0.371434i	$0.878866\pi$
$558$	0	0
$559$	−15.2979	−0.647030
$560$	0	0
$561$	−55.0042	−2.32228
$562$	0	0
$563$	− 8.85461i	− 0.373177i	−0.982438	−	0.186589i	$-0.940257\pi$
$563$	− 8.85461i	− 0.373177i	0.982438	−	0.186589i	$-0.0597432\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.18009i	0.0495593i
$568$	0	0
$569$	−33.0220	−1.38435	−0.692177	−	0.721727i	$-0.743348\pi$
$569$	−33.0220	−1.38435	−0.692177	+	0.721727i	$0.743348\pi$
$570$	0	0
$571$	12.9266	0.540963	0.270482	−	0.962725i	$-0.412817\pi$
$571$	12.9266	0.540963	0.270482	+	0.962725i	$0.412817\pi$
$572$	0	0
$573$	24.5840i	1.02701i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 3.22539i	− 0.134275i	−0.997744	−	0.0671374i	$-0.978613\pi$
$577$	− 3.22539i	− 0.134275i	0.997744	−	0.0671374i	$-0.0213866\pi$
$578$	0	0
$579$	5.38305	0.223712
$580$	0	0
$581$	22.7548	0.944026
$582$	0	0
$583$	− 88.4797i	− 3.66446i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7.17382i	0.296095i	0.988980	+	0.148048i	$0.0472989\pi$
$587$	7.17382i	0.296095i	−0.988980	+	0.148048i	$0.952701\pi$
$588$	0	0
$589$	1.24541	0.0513161
$590$	0	0
$591$	−28.6037	−1.17660
$592$	0	0
$593$	− 27.2576i	− 1.11933i	−0.828718	−	0.559667i	$-0.810929\pi$
$593$	− 27.2576i	− 1.11933i	0.828718	−	0.559667i	$-0.189071\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	16.5547i	0.677540i
$598$	0	0
$599$	3.53091	0.144269	0.0721344	−	0.997395i	$-0.477019\pi$
$599$	3.53091	0.144269	0.0721344	+	0.997395i	$0.477019\pi$
$600$	0	0
$601$	23.6488	0.964655	0.482327	−	0.875991i	$-0.339792\pi$
$601$	23.6488	0.964655	0.482327	+	0.875991i	$0.339792\pi$
$602$	0	0
$603$	− 8.57523i	− 0.349210i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	28.6598i	1.16326i	0.813452	+	0.581632i	$0.197586\pi$
$607$	28.6598i	1.16326i	−0.813452	+	0.581632i	$0.802414\pi$
$608$	0	0
$609$	38.3513	1.55407
$610$	0	0
$611$	−13.4975	−0.546050
$612$	0	0
$613$	4.64372i	0.187558i	0.995593	+	0.0937790i	$0.0298947\pi$
$613$	4.64372i	0.187558i	−0.995593	+	0.0937790i	$0.970105\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 4.49990i	− 0.181159i	−0.995889	−	0.0905796i	$-0.971128\pi$
$617$	− 4.49990i	− 0.181159i	0.995889	−	0.0905796i	$-0.0288719\pi$
$618$	0	0
$619$	−20.5767	−0.827049	−0.413525	−	0.910493i	$-0.635702\pi$
$619$	−20.5767	−0.827049	−0.413525	+	0.910493i	$0.635702\pi$
$620$	0	0
$621$	−4.50856	−0.180922
$622$	0	0
$623$	13.0236i	0.521780i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	7.00919i	0.279920i
$628$	0	0
$629$	−53.3608	−2.12763
$630$	0	0
$631$	−32.5361	−1.29524	−0.647620	−	0.761963i	$-0.724236\pi$
$631$	−32.5361	−1.29524	−0.647620	+	0.761963i	$0.724236\pi$
$632$	0	0
$633$	− 22.1956i	− 0.882196i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 23.8474i	− 0.944869i
$638$	0	0
$639$	18.5806	0.735035
$640$	0	0
$641$	13.0922	0.517113	0.258556	−	0.965996i	$-0.416753\pi$
$641$	13.0922	0.517113	0.258556	+	0.965996i	$0.416753\pi$
$642$	0	0
$643$	− 16.8450i	− 0.664304i	−0.943226	−	0.332152i	$-0.892225\pi$
$643$	− 16.8450i	− 0.664304i	0.943226	−	0.332152i	$-0.107775\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 46.9300i	− 1.84501i	−0.385987	−	0.922504i	$-0.626139\pi$
$647$	− 46.9300i	− 1.84501i	0.385987	−	0.922504i	$-0.373861\pi$
$648$	0	0
$649$	44.3047	1.73911
$650$	0	0
$651$	−5.69770	−0.223310
$652$	0	0
$653$	− 14.1069i	− 0.552046i	−0.961151	−	0.276023i	$-0.910983\pi$
$653$	− 14.1069i	− 0.552046i	0.961151	−	0.276023i	$-0.0890167\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 17.8683i	− 0.697108i
$658$	0	0
$659$	−30.5841	−1.19139	−0.595693	−	0.803212i	$-0.703122\pi$
$659$	−30.5841	−1.19139	−0.595693	+	0.803212i	$0.703122\pi$
$660$	0	0
$661$	−42.8953	−1.66843	−0.834216	−	0.551437i	$-0.814080\pi$
$661$	−42.8953	−1.66843	−0.834216	+	0.551437i	$0.814080\pi$
$662$	0	0
$663$	19.2127i	0.746158i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	7.20589i	0.279013i
$668$	0	0
$669$	16.7602	0.647985
$670$	0	0
$671$	8.48082	0.327399
$672$	0	0
$673$	− 19.0609i	− 0.734744i	−0.930074	−	0.367372i	$-0.880258\pi$
$673$	− 19.0609i	− 0.734744i	0.930074	−	0.367372i	$-0.119742\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	48.2121i	1.85294i	0.376366	+	0.926471i	$0.377173\pi$
$677$	48.2121i	1.85294i	−0.376366	+	0.926471i	$0.622827\pi$
$678$	0	0
$679$	−28.3749	−1.08893
$680$	0	0
$681$	21.8201	0.836149
$682$	0	0
$683$	− 2.59451i	− 0.0992763i	−0.998767	−	0.0496381i	$-0.984193\pi$
$683$	− 2.59451i	− 0.0992763i	0.998767	−	0.0496381i	$-0.0158068\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 13.7530i	− 0.524708i
$688$	0	0
$689$	−30.9054	−1.17740
$690$	0	0
$691$	−15.0799	−0.573667	−0.286834	−	0.957980i	$-0.592603\pi$
$691$	−15.0799	−0.573667	−0.286834	+	0.957980i	$0.592603\pi$
$692$	0	0
$693$	48.9049i	1.85774i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	46.4945i	1.76111i
$698$	0	0
$699$	23.8958	0.903824
$700$	0	0
$701$	27.2335	1.02860	0.514298	−	0.857612i	$-0.328053\pi$
$701$	27.2335	1.02860	0.514298	+	0.857612i	$0.328053\pi$
$702$	0	0
$703$	6.79977i	0.256458i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	12.8120i	0.481846i
$708$	0	0
$709$	29.9175	1.12357	0.561787	−	0.827282i	$-0.310114\pi$
$709$	29.9175	1.12357	0.561787	+	0.827282i	$0.310114\pi$
$710$	0	0
$711$	−11.7242	−0.439690
$712$	0	0
$713$	− 1.07055i	− 0.0400925i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	3.99880i	0.149338i
$718$	0	0
$719$	−31.7693	−1.18479	−0.592397	−	0.805646i	$-0.701818\pi$
$719$	−31.7693	−1.18479	−0.592397	+	0.805646i	$0.701818\pi$
$720$	0	0
$721$	29.6481	1.10415
$722$	0	0
$723$	6.57570i	0.244553i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	23.7272i	0.879993i	0.898000	+	0.439996i	$0.145020\pi$
$727$	23.7272i	0.879993i	−0.898000	+	0.439996i	$0.854980\pi$
$728$	0	0
$729$	−17.6602	−0.654082
$730$	0	0
$731$	−53.4468	−1.97680
$732$	0	0
$733$	37.4907i	1.38475i	0.721538	+	0.692374i	$0.243435\pi$
$733$	37.4907i	1.38475i	−0.721538	+	0.692374i	$0.756565\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	30.4335i	1.12103i
$738$	0	0
$739$	−7.10315	−0.261294	−0.130647	−	0.991429i	$-0.541705\pi$
$739$	−7.10315	−0.261294	−0.130647	+	0.991429i	$0.541705\pi$
$740$	0	0
$741$	2.44827	0.0899394
$742$	0	0
$743$	− 10.6047i	− 0.389049i	−0.980898	−	0.194525i	$-0.937684\pi$
$743$	− 10.6047i	− 0.389049i	0.980898	−	0.194525i	$-0.0623164\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	9.82303i	0.359406i
$748$	0	0
$749$	21.2080	0.774923
$750$	0	0
$751$	32.1380	1.17273	0.586365	−	0.810047i	$-0.300558\pi$
$751$	32.1380	1.17273	0.586365	+	0.810047i	$0.300558\pi$
$752$	0	0
$753$	− 0.158880i	− 0.00578991i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 36.2721i	− 1.31833i	−0.751997	−	0.659167i	$-0.770909\pi$
$757$	− 36.2721i	− 1.31833i	0.751997	−	0.659167i	$-0.229091\pi$
$758$	0	0
$759$	6.02510	0.218697
$760$	0	0
$761$	31.2638	1.13331	0.566656	−	0.823954i	$-0.308237\pi$
$761$	31.2638	1.13331	0.566656	+	0.823954i	$0.308237\pi$
$762$	0	0
$763$	− 26.8346i	− 0.971476i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 15.4754i	− 0.558783i
$768$	0	0
$769$	−48.5188	−1.74963	−0.874816	−	0.484456i	$-0.839018\pi$
$769$	−48.5188	−1.74963	−0.874816	+	0.484456i	$0.839018\pi$
$770$	0	0
$771$	16.0788	0.579064
$772$	0	0
$773$	26.4968i	0.953023i	0.879168	+	0.476512i	$0.158099\pi$
$773$	26.4968i	0.953023i	−0.879168	+	0.476512i	$0.841901\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 31.1088i	− 1.11602i
$778$	0	0
$779$	5.92480	0.212278
$780$	0	0
$781$	−65.9424	−2.35960
$782$	0	0
$783$	43.9675i	1.57127i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	16.9510i	0.604238i	0.953270	+	0.302119i	$0.0976940\pi$
$787$	16.9510i	0.604238i	−0.953270	+	0.302119i	$0.902306\pi$
$788$	0	0
$789$	33.0230	1.17565
$790$	0	0
$791$	−82.1981	−2.92263
$792$	0	0
$793$	− 2.96230i	− 0.105194i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 53.4354i	− 1.89278i	−0.323025	−	0.946390i	$-0.604700\pi$
$797$	− 53.4354i	− 1.89278i	0.323025	−	0.946390i	$-0.395300\pi$
$798$	0	0
$799$	−47.1568	−1.66829
$800$	0	0
$801$	−5.62218	−0.198650
$802$	0	0
$803$	63.4145i	2.23785i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	22.1769i	0.780665i
$808$	0	0
$809$	3.88632	0.136636	0.0683179	−	0.997664i	$-0.478237\pi$
$809$	3.88632	0.136636	0.0683179	+	0.997664i	$0.478237\pi$
$810$	0	0
$811$	10.8136	0.379716	0.189858	−	0.981812i	$-0.439197\pi$
$811$	10.8136	0.379716	0.189858	+	0.981812i	$0.439197\pi$
$812$	0	0
$813$	− 1.10469i	− 0.0387432i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	6.81073i	0.238277i
$818$	0	0
$819$	17.0822	0.596900
$820$	0	0
$821$	−43.7227	−1.52593	−0.762966	−	0.646438i	$-0.776258\pi$
$821$	−43.7227	−1.52593	−0.762966	+	0.646438i	$0.776258\pi$
$822$	0	0
$823$	− 4.51329i	− 0.157323i	−0.996901	−	0.0786617i	$-0.974935\pi$
$823$	− 4.51329i	− 0.157323i	0.996901	−	0.0786617i	$-0.0250647\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 9.56467i	− 0.332596i	−0.986076	−	0.166298i	$-0.946819\pi$
$827$	− 9.56467i	− 0.332596i	0.986076	−	0.166298i	$-0.0531814\pi$
$828$	0	0
$829$	−20.0719	−0.697126	−0.348563	−	0.937285i	$-0.613330\pi$
$829$	−20.0719	−0.697126	−0.348563	+	0.937285i	$0.613330\pi$
$830$	0	0
$831$	11.3736	0.394545
$832$	0	0
$833$	− 83.3168i	− 2.88675i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 6.53208i	− 0.225782i
$838$	0	0
$839$	−12.4591	−0.430135	−0.215067	−	0.976599i	$-0.568997\pi$
$839$	−12.4591	−0.430135	−0.215067	+	0.976599i	$0.568997\pi$
$840$	0	0
$841$	41.2720	1.42317
$842$	0	0
$843$	8.85869i	0.305110i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 127.394i	− 4.37730i
$848$	0	0
$849$	12.6871	0.435421
$850$	0	0
$851$	5.84508	0.200367
$852$	0	0
$853$	− 22.0112i	− 0.753649i	−0.926285	−	0.376824i	$-0.877016\pi$
$853$	− 22.0112i	− 0.753649i	0.926285	−	0.376824i	$-0.122984\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 15.8261i	− 0.540608i	−0.962775	−	0.270304i	$-0.912876\pi$
$857$	− 15.8261i	− 0.540608i	0.962775	−	0.270304i	$-0.0871242\pi$
$858$	0	0
$859$	23.8728	0.814530	0.407265	−	0.913310i	$-0.366483\pi$
$859$	23.8728	0.814530	0.407265	+	0.913310i	$0.366483\pi$
$860$	0	0
$861$	−27.1058	−0.923763
$862$	0	0
$863$	29.6871i	1.01056i	0.862956	+	0.505280i	$0.168611\pi$
$863$	29.6871i	1.01056i	−0.862956	+	0.505280i	$0.831389\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	48.5943i	1.65035i
$868$	0	0
$869$	41.6090	1.41149
$870$	0	0
$871$	10.6302	0.360192
$872$	0	0
$873$	− 12.2492i	− 0.414572i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	20.3979i	0.688790i	0.938825	+	0.344395i	$0.111916\pi$
$877$	20.3979i	0.688790i	−0.938825	+	0.344395i	$0.888084\pi$
$878$	0	0
$879$	−2.41864	−0.0815786
$880$	0	0
$881$	−43.2919	−1.45854	−0.729270	−	0.684226i	$-0.760140\pi$
$881$	−43.2919	−1.45854	−0.729270	+	0.684226i	$0.760140\pi$
$882$	0	0
$883$	9.74875i	0.328071i	0.986454	+	0.164036i	$0.0524512\pi$
$883$	9.74875i	0.328071i	−0.986454	+	0.164036i	$0.947549\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 10.5431i	− 0.354002i	−0.984211	−	0.177001i	$-0.943360\pi$
$887$	− 10.5431i	− 0.354002i	0.984211	−	0.177001i	$-0.0566395\pi$
$888$	0	0
$889$	−26.8761	−0.901395
$890$	0	0
$891$	1.80799	0.0605699
$892$	0	0
$893$	6.00919i	0.201090i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 2.10453i	− 0.0702683i
$898$	0	0
$899$	−10.4400	−0.348195
$900$	0	0
$901$	−107.976	−3.59719
$902$	0	0
$903$	− 31.1589i	− 1.03690i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 48.2385i	− 1.60173i	−0.598844	−	0.800866i	$-0.704373\pi$
$907$	− 48.2385i	− 1.60173i	0.598844	−	0.800866i	$-0.295627\pi$
$908$	0	0
$909$	−5.53084	−0.183447
$910$	0	0
$911$	−9.86236	−0.326755	−0.163377	−	0.986564i	$-0.552239\pi$
$911$	−9.86236	−0.326755	−0.163377	+	0.986564i	$0.552239\pi$
$912$	0	0
$913$	− 34.8619i	− 1.15376i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 51.1703i	− 1.68979i
$918$	0	0
$919$	−38.0526	−1.25524	−0.627620	−	0.778520i	$-0.715971\pi$
$919$	−38.0526	−1.25524	−0.627620	+	0.778520i	$0.715971\pi$
$920$	0	0
$921$	−19.2614	−0.634684
$922$	0	0
$923$	23.0333i	0.758149i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	12.7988i	0.420368i
$928$	0	0
$929$	−47.3243	−1.55266	−0.776330	−	0.630327i	$-0.782921\pi$
$929$	−47.3243	−1.55266	−0.776330	+	0.630327i	$0.782921\pi$
$930$	0	0
$931$	−10.6171	−0.347960
$932$	0	0
$933$	− 13.4753i	− 0.441163i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	19.0466i	0.622224i	0.950373	+	0.311112i	$0.100701\pi$
$937$	19.0466i	0.622224i	−0.950373	+	0.311112i	$0.899299\pi$
$938$	0	0
$939$	9.10735	0.297207
$940$	0	0
$941$	37.7416	1.23034	0.615171	−	0.788393i	$-0.289087\pi$
$941$	37.7416	1.23034	0.615171	+	0.788393i	$0.289087\pi$
$942$	0	0
$943$	− 5.09296i	− 0.165850i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 4.72833i	− 0.153650i	−0.997045	−	0.0768251i	$-0.975522\pi$
$947$	− 4.72833i	− 0.153650i	0.997045	−	0.0768251i	$-0.0244783\pi$
$948$	0	0
$949$	22.1503	0.719030
$950$	0	0
$951$	21.0941	0.684024
$952$	0	0
$953$	32.9799i	1.06832i	0.845382	+	0.534162i	$0.179373\pi$
$953$	32.9799i	1.06832i	−0.845382	+	0.534162i	$0.820627\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 58.7569i	− 1.89934i
$958$	0	0
$959$	40.7650	1.31637
$960$	0	0
$961$	−29.4490	−0.949967
$962$	0	0
$963$	9.15531i	0.295026i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 3.71736i	− 0.119542i	−0.998212	−	0.0597712i	$-0.980963\pi$
$967$	− 3.71736i	− 0.119542i	0.998212	−	0.0597712i	$-0.0190371\pi$
$968$	0	0
$969$	8.55363	0.274782
$970$	0	0
$971$	18.6887	0.599749	0.299875	−	0.953979i	$-0.403055\pi$
$971$	18.6887	0.599749	0.299875	+	0.953979i	$0.403055\pi$
$972$	0	0
$973$	84.2337i	2.70041i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	7.44705i	0.238252i	0.992879	+	0.119126i	$0.0380093\pi$
$977$	7.44705i	0.238252i	−0.992879	+	0.119126i	$0.961991\pi$
$978$	0	0
$979$	19.9531	0.637705
$980$	0	0
$981$	11.5842	0.369856
$982$	0	0
$983$	− 13.7703i	− 0.439204i	−0.975590	−	0.219602i	$-0.929524\pi$
$983$	− 13.7703i	− 0.439204i	0.975590	−	0.219602i	$-0.0704759\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 27.4919i	− 0.875077i
$988$	0	0
$989$	5.85450	0.186162
$990$	0	0
$991$	32.0927	1.01946	0.509728	−	0.860335i	$-0.329746\pi$
$991$	32.0927	1.01946	0.509728	+	0.860335i	$0.329746\pi$
$992$	0	0
$993$	− 36.9429i	− 1.17235i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 22.8068i	− 0.722299i	−0.932508	−	0.361149i	$-0.882384\pi$
$997$	− 22.8068i	− 0.722299i	0.932508	−	0.361149i	$-0.117616\pi$
$998$	0	0
$999$	35.6644	1.12837

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.d.q.3649.4		12
5.2	odd	4		3800.2.a.bc.1.2	yes	6
5.3	odd	4		3800.2.a.ba.1.5	✓	6
5.4	even	2	inner	3800.2.d.q.3649.9		12
20.3	even	4		7600.2.a.cl.1.2		6
20.7	even	4		7600.2.a.ch.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3800.2.a.ba.1.5	✓	6	5.3	odd	4
3800.2.a.bc.1.2	yes	6	5.2	odd	4
3800.2.d.q.3649.4		12	1.1	even	1	trivial
3800.2.d.q.3649.9		12	5.4	even	2	inner
7600.2.a.ch.1.5		6	20.7	even	4
7600.2.a.cl.1.2		6	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3800.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.08999i q^{3} -4.19727i q^{7} +1.81192 q^{9} +O(q^{10})\)	\(q-1.08999i q^{3} -4.19727i q^{7} +1.81192 q^{9} -6.43052 q^{11} +2.24614i q^{13} +7.84744i q^{17} +1.00000 q^{19} -4.57497 q^{21} -0.859601i q^{23} -5.24494i q^{27} -8.38284 q^{29} +1.24541 q^{31} +7.00919i q^{33} +6.79977i q^{37} +2.44827 q^{39} +5.92480 q^{41} +6.81073i q^{43} +6.00919i q^{47} -10.6171 q^{49} +8.55363 q^{51} +13.7594i q^{53} -1.08999i q^{57} -6.88976 q^{59} -1.31884 q^{61} -7.60513i q^{63} -4.73266i q^{67} -0.936955 q^{69} +10.2546 q^{71} -9.86150i q^{73} +26.9906i q^{77} -6.47056 q^{79} -0.281158 q^{81} +5.42133i q^{83} +9.13720i q^{87} -3.10288 q^{89} +9.42766 q^{91} -1.35748i q^{93} -6.76032i q^{97} -11.6516 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^9	\( 12 q - 12 q^{9} + 6 q^{11} + 12 q^{19} + 30 q^{21} - 18 q^{29} + 10 q^{31} - 24 q^{39} + 6 q^{41} - 44 q^{49} + 66 q^{51} + 18 q^{61} + 22 q^{69} + 38 q^{71} + 32 q^{79} + 52 q^{81} - 28 q^{89} + 84 q^{91} + 12 q^{99}+O(q^{100}) \) 12 * q - 12 * q^9 + 6 * q^11 + 12 * q^19 + 30 * q^21 - 18 * q^29 + 10 * q^31 - 24 * q^39 + 6 * q^41 - 44 * q^49 + 66 * q^51 + 18 * q^61 + 22 * q^69 + 38 * q^71 + 32 * q^79 + 52 * q^81 - 28 * q^89 + 84 * q^91 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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