LMFDB - Embedded newform 1904.2.a.m.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1904,2,Mod(1,1904)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1904, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1904.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1904 = 2^{4} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1904.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$15.2035165449$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 952)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.11491$ of defining polynomial
Character	$\chi$	$=$	1904.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.47283 q^{3} -1.11491 q^{5} -1.00000 q^{7} -0.830760 q^{9} +O(q^{10})$	$q+1.47283 q^{3} -1.11491 q^{5} -1.00000 q^{7} -0.830760 q^{9} +2.22982 q^{11} -1.64207 q^{15} -1.00000 q^{17} -4.94567 q^{19} -1.47283 q^{21} -9.17548 q^{23} -3.75698 q^{25} -5.64207 q^{27} +0.945668 q^{29} +1.11491 q^{31} +3.28415 q^{33} +1.11491 q^{35} -4.94567 q^{37} +10.2709 q^{41} +11.2166 q^{43} +0.926221 q^{45} -9.89134 q^{47} +1.00000 q^{49} -1.47283 q^{51} +7.83076 q^{53} -2.48604 q^{55} -7.28415 q^{57} -13.7438 q^{59} +0.904539 q^{61} +0.830760 q^{63} -9.91302 q^{67} -13.5140 q^{69} -14.6894 q^{71} +0.756981 q^{73} -5.53341 q^{75} -2.22982 q^{77} -8.00000 q^{79} -5.81756 q^{81} +11.9736 q^{83} +1.11491 q^{85} +1.39281 q^{87} +11.1755 q^{89} +1.64207 q^{93} +5.51396 q^{95} +10.0606 q^{97} -1.85244 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{3} + 3 q^{5} - 3 q^{7} + 2 q^{9}+O(q^{10}) $ 3 * q - q^3 + 3 * q^5 - 3 * q^7 + 2 * q^9	$ 3 q - q^{3} + 3 q^{5} - 3 q^{7} + 2 q^{9} - 6 q^{11} - 4 q^{15} - 3 q^{17} - 4 q^{19} + q^{21} - 4 q^{23} - 4 q^{25} - 16 q^{27} - 8 q^{29} - 3 q^{31} + 8 q^{33} - 3 q^{35} - 4 q^{37} + 9 q^{41} + q^{43} - 8 q^{47} + 3 q^{49} + q^{51} + 19 q^{53} - 22 q^{55} - 20 q^{57} - 14 q^{59} + q^{61} - 2 q^{63} - 7 q^{67} - 26 q^{69} - 6 q^{71} - 5 q^{73} + 6 q^{75} + 6 q^{77} - 24 q^{79} + 7 q^{81} - 4 q^{83} - 3 q^{85} + 24 q^{87} + 10 q^{89} + 4 q^{93} + 2 q^{95} + 13 q^{97}+O(q^{100}) $ 3 * q - q^3 + 3 * q^5 - 3 * q^7 + 2 * q^9 - 6 * q^11 - 4 * q^15 - 3 * q^17 - 4 * q^19 + q^21 - 4 * q^23 - 4 * q^25 - 16 * q^27 - 8 * q^29 - 3 * q^31 + 8 * q^33 - 3 * q^35 - 4 * q^37 + 9 * q^41 + q^43 - 8 * q^47 + 3 * q^49 + q^51 + 19 * q^53 - 22 * q^55 - 20 * q^57 - 14 * q^59 + q^61 - 2 * q^63 - 7 * q^67 - 26 * q^69 - 6 * q^71 - 5 * q^73 + 6 * q^75 + 6 * q^77 - 24 * q^79 + 7 * q^81 - 4 * q^83 - 3 * q^85 + 24 * q^87 + 10 * q^89 + 4 * q^93 + 2 * q^95 + 13 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.47283	0.850341	0.425171	−	0.905113i	$-0.360214\pi$
$3$	1.47283	0.850341	0.425171	+	0.905113i	$0.360214\pi$
$4$	0	0
$5$	−1.11491	−0.498602	−0.249301	−	0.968426i	$-0.580201\pi$
$5$	−1.11491	−0.498602	−0.249301	+	0.968426i	$0.580201\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−0.830760	−0.276920
$10$	0	0
$11$	2.22982	0.672315	0.336157	−	0.941806i	$-0.390873\pi$
$11$	2.22982	0.672315	0.336157	+	0.941806i	$0.390873\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	−1.64207	−0.423982
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−4.94567	−1.13461	−0.567307	−	0.823506i	$-0.692015\pi$
$19$	−4.94567	−1.13461	−0.567307	+	0.823506i	$0.692015\pi$
$20$	0	0
$21$	−1.47283	−0.321399
$22$	0	0
$23$	−9.17548	−1.91322	−0.956610	−	0.291371i	$-0.905889\pi$
$23$	−9.17548	−1.91322	−0.956610	+	0.291371i	$0.905889\pi$
$24$	0	0
$25$	−3.75698	−0.751396
$26$	0	0
$27$	−5.64207	−1.08582
$28$	0	0
$29$	0.945668	0.175606	0.0878031	−	0.996138i	$-0.472015\pi$
$29$	0.945668	0.175606	0.0878031	+	0.996138i	$0.472015\pi$
$30$	0	0
$31$	1.11491	0.200243	0.100122	−	0.994975i	$-0.468077\pi$
$31$	1.11491	0.200243	0.100122	+	0.994975i	$0.468077\pi$
$32$	0	0
$33$	3.28415	0.571697
$34$	0	0
$35$	1.11491	0.188454
$36$	0	0
$37$	−4.94567	−0.813063	−0.406531	−	0.913637i	$-0.633262\pi$
$37$	−4.94567	−0.813063	−0.406531	+	0.913637i	$0.633262\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	10.2709	1.60405	0.802026	−	0.597289i	$-0.203756\pi$
$41$	10.2709	1.60405	0.802026	+	0.597289i	$0.203756\pi$
$42$	0	0
$43$	11.2166	1.71052	0.855259	−	0.518201i	$-0.173398\pi$
$43$	11.2166	1.71052	0.855259	+	0.518201i	$0.173398\pi$
$44$	0	0
$45$	0.926221	0.138073
$46$	0	0
$47$	−9.89134	−1.44280	−0.721400	−	0.692519i	$-0.756501\pi$
$47$	−9.89134	−1.44280	−0.721400	+	0.692519i	$0.756501\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.47283	−0.206238
$52$	0	0
$53$	7.83076	1.07564	0.537819	−	0.843060i	$-0.319248\pi$
$53$	7.83076	1.07564	0.537819	+	0.843060i	$0.319248\pi$
$54$	0	0
$55$	−2.48604	−0.335217
$56$	0	0
$57$	−7.28415	−0.964809
$58$	0	0
$59$	−13.7438	−1.78929	−0.894644	−	0.446780i	$-0.852571\pi$
$59$	−13.7438	−1.78929	−0.894644	+	0.446780i	$0.852571\pi$
$60$	0	0
$61$	0.904539	0.115814	0.0579072	−	0.998322i	$-0.481557\pi$
$61$	0.904539	0.115814	0.0579072	+	0.998322i	$0.481557\pi$
$62$	0	0
$63$	0.830760	0.104666
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−9.91302	−1.21107	−0.605534	−	0.795820i	$-0.707040\pi$
$67$	−9.91302	−1.21107	−0.605534	+	0.795820i	$0.707040\pi$
$68$	0	0
$69$	−13.5140	−1.62689
$70$	0	0
$71$	−14.6894	−1.74332	−0.871658	−	0.490114i	$-0.836955\pi$
$71$	−14.6894	−1.74332	−0.871658	+	0.490114i	$0.836955\pi$
$72$	0	0
$73$	0.756981	0.0885979	0.0442990	−	0.999018i	$-0.485895\pi$
$73$	0.756981	0.0885979	0.0442990	+	0.999018i	$0.485895\pi$
$74$	0	0
$75$	−5.53341	−0.638943
$76$	0	0
$77$	−2.22982	−0.254111
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	−5.81756	−0.646395
$82$	0	0
$83$	11.9736	1.31427	0.657136	−	0.753772i	$-0.271768\pi$
$83$	11.9736	1.31427	0.657136	+	0.753772i	$0.271768\pi$
$84$	0	0
$85$	1.11491	0.120929
$86$	0	0
$87$	1.39281	0.149325
$88$	0	0
$89$	11.1755	1.18460	0.592299	−	0.805718i	$-0.298220\pi$
$89$	11.1755	1.18460	0.592299	+	0.805718i	$0.298220\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1.64207	0.170275
$94$	0	0
$95$	5.51396	0.565721
$96$	0	0
$97$	10.0606	1.02150	0.510748	−	0.859730i	$-0.329368\pi$
$97$	10.0606	1.02150	0.510748	+	0.859730i	$0.329368\pi$
$98$	0	0
$99$	−1.85244	−0.186177
$100$	0	0
$101$	−18.6894	−1.85967	−0.929835	−	0.367978i	$-0.880050\pi$
$101$	−18.6894	−1.85967	−0.929835	+	0.367978i	$0.880050\pi$
$102$	0	0
$103$	−12.1212	−1.19433	−0.597166	−	0.802118i	$-0.703707\pi$
$103$	−12.1212	−1.19433	−0.597166	+	0.802118i	$0.703707\pi$
$104$	0	0
$105$	1.64207	0.160250
$106$	0	0
$107$	1.89134	0.182842	0.0914212	−	0.995812i	$-0.470859\pi$
$107$	1.89134	0.182842	0.0914212	+	0.995812i	$0.470859\pi$
$108$	0	0
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	−7.28415	−0.691381
$112$	0	0
$113$	−3.85244	−0.362407	−0.181204	−	0.983446i	$-0.557999\pi$
$113$	−3.85244	−0.362407	−0.181204	+	0.983446i	$0.557999\pi$
$114$	0	0
$115$	10.2298	0.953935
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	−6.02792	−0.547993
$122$	0	0
$123$	15.1274	1.36399
$124$	0	0
$125$	9.76322	0.873249
$126$	0	0
$127$	21.2361	1.88440	0.942198	−	0.335057i	$-0.108756\pi$
$127$	21.2361	1.88440	0.942198	+	0.335057i	$0.108756\pi$
$128$	0	0
$129$	16.5202	1.45452
$130$	0	0
$131$	4.60719	0.402532	0.201266	−	0.979537i	$-0.435494\pi$
$131$	4.60719	0.402532	0.201266	+	0.979537i	$0.435494\pi$
$132$	0	0
$133$	4.94567	0.428844
$134$	0	0
$135$	6.29039	0.541391
$136$	0	0
$137$	0.379608	0.0324321	0.0162160	−	0.999869i	$-0.494838\pi$
$137$	0.379608	0.0324321	0.0162160	+	0.999869i	$0.494838\pi$
$138$	0	0
$139$	−6.06058	−0.514051	−0.257026	−	0.966405i	$-0.582742\pi$
$139$	−6.06058	−0.514051	−0.257026	+	0.966405i	$0.582742\pi$
$140$	0	0
$141$	−14.5683	−1.22687
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−1.05433	−0.0875575
$146$	0	0
$147$	1.47283	0.121477
$148$	0	0
$149$	−10.1887	−0.834690	−0.417345	−	0.908748i	$-0.637039\pi$
$149$	−10.1887	−0.834690	−0.417345	+	0.908748i	$0.637039\pi$
$150$	0	0
$151$	8.18869	0.666386	0.333193	−	0.942859i	$-0.391874\pi$
$151$	8.18869	0.666386	0.333193	+	0.942859i	$0.391874\pi$
$152$	0	0
$153$	0.830760	0.0671630
$154$	0	0
$155$	−1.24302	−0.0998417
$156$	0	0
$157$	−14.2687	−1.13877	−0.569383	−	0.822072i	$-0.692818\pi$
$157$	−14.2687	−1.13877	−0.569383	+	0.822072i	$0.692818\pi$
$158$	0	0
$159$	11.5334	0.914659
$160$	0	0
$161$	9.17548	0.723129
$162$	0	0
$163$	−15.7438	−1.23315	−0.616574	−	0.787297i	$-0.711480\pi$
$163$	−15.7438	−1.23315	−0.616574	+	0.787297i	$0.711480\pi$
$164$	0	0
$165$	−3.66152	−0.285049
$166$	0	0
$167$	0.648317	0.0501683	0.0250841	−	0.999685i	$-0.492015\pi$
$167$	0.648317	0.0501683	0.0250841	+	0.999685i	$0.492015\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	4.10866	0.314197
$172$	0	0
$173$	15.9325	1.21132	0.605661	−	0.795722i	$-0.292909\pi$
$173$	15.9325	1.21132	0.605661	+	0.795722i	$0.292909\pi$
$174$	0	0
$175$	3.75698	0.284001
$176$	0	0
$177$	−20.2423	−1.52150
$178$	0	0
$179$	17.0257	1.27256	0.636280	−	0.771458i	$-0.280472\pi$
$179$	17.0257	1.27256	0.636280	+	0.771458i	$0.280472\pi$
$180$	0	0
$181$	5.63511	0.418855	0.209427	−	0.977824i	$-0.432840\pi$
$181$	5.63511	0.418855	0.209427	+	0.977824i	$0.432840\pi$
$182$	0	0
$183$	1.33224	0.0984817
$184$	0	0
$185$	5.51396	0.405395
$186$	0	0
$187$	−2.22982	−0.163060
$188$	0	0
$189$	5.64207	0.410400
$190$	0	0
$191$	−2.88509	−0.208758	−0.104379	−	0.994538i	$-0.533285\pi$
$191$	−2.88509	−0.208758	−0.104379	+	0.994538i	$0.533285\pi$
$192$	0	0
$193$	17.5264	1.26158	0.630791	−	0.775953i	$-0.282731\pi$
$193$	17.5264	1.26158	0.630791	+	0.775953i	$0.282731\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	14.9193	1.06295	0.531477	−	0.847073i	$-0.321637\pi$
$197$	14.9193	1.06295	0.531477	+	0.847073i	$0.321637\pi$
$198$	0	0
$199$	16.6483	1.18017	0.590084	−	0.807342i	$-0.299095\pi$
$199$	16.6483	1.18017	0.590084	+	0.807342i	$0.299095\pi$
$200$	0	0
$201$	−14.6002	−1.02982
$202$	0	0
$203$	−0.945668	−0.0663729
$204$	0	0
$205$	−11.4512	−0.799783
$206$	0	0
$207$	7.62263	0.529809
$208$	0	0
$209$	−11.0279	−0.762818
$210$	0	0
$211$	−7.40530	−0.509802	−0.254901	−	0.966967i	$-0.582043\pi$
$211$	−7.40530	−0.509802	−0.254901	+	0.966967i	$0.582043\pi$
$212$	0	0
$213$	−21.6351	−1.48241
$214$	0	0
$215$	−12.5055	−0.852867
$216$	0	0
$217$	−1.11491	−0.0756849
$218$	0	0
$219$	1.11491	0.0753385
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−27.1491	−1.81804	−0.909018	−	0.416756i	$-0.863167\pi$
$223$	−27.1491	−1.81804	−0.909018	+	0.416756i	$0.863167\pi$
$224$	0	0
$225$	3.12115	0.208077
$226$	0	0
$227$	4.50772	0.299188	0.149594	−	0.988748i	$-0.452203\pi$
$227$	4.50772	0.299188	0.149594	+	0.988748i	$0.452203\pi$
$228$	0	0
$229$	10.2298	0.676005	0.338003	−	0.941145i	$-0.390249\pi$
$229$	10.2298	0.676005	0.338003	+	0.941145i	$0.390249\pi$
$230$	0	0
$231$	−3.28415	−0.216081
$232$	0	0
$233$	11.5529	0.756853	0.378426	−	0.925631i	$-0.376465\pi$
$233$	11.5529	0.756853	0.378426	+	0.925631i	$0.376465\pi$
$234$	0	0
$235$	11.0279	0.719382
$236$	0	0
$237$	−11.7827	−0.765367
$238$	0	0
$239$	30.7570	1.98950	0.994752	−	0.102317i	$-0.0326256\pi$
$239$	30.7570	1.98950	0.994752	+	0.102317i	$0.0326256\pi$
$240$	0	0
$241$	−0.967349	−0.0623124	−0.0311562	−	0.999515i	$-0.509919\pi$
$241$	−0.967349	−0.0623124	−0.0311562	+	0.999515i	$0.509919\pi$
$242$	0	0
$243$	8.35793	0.536161
$244$	0	0
$245$	−1.11491	−0.0712288
$246$	0	0
$247$	0	0
$248$	0	0
$249$	17.6351	1.11758
$250$	0	0
$251$	−0.0264075	−0.00166683	−0.000833413	−	1.00000i	$-0.500265\pi$
$251$	−0.0264075	−0.00166683	−0.000833413		1.00000i	$0.500265\pi$
$252$	0	0
$253$	−20.4596	−1.28629
$254$	0	0
$255$	1.64207	0.102831
$256$	0	0
$257$	14.7159	0.917950	0.458975	−	0.888449i	$-0.348217\pi$
$257$	14.7159	0.917950	0.458975	+	0.888449i	$0.348217\pi$
$258$	0	0
$259$	4.94567	0.307309
$260$	0	0
$261$	−0.785623	−0.0486289
$262$	0	0
$263$	−24.9193	−1.53659	−0.768294	−	0.640097i	$-0.778894\pi$
$263$	−24.9193	−1.53659	−0.768294	+	0.640097i	$0.778894\pi$
$264$	0	0
$265$	−8.73057	−0.536315
$266$	0	0
$267$	16.4596	1.00731
$268$	0	0
$269$	−23.9861	−1.46246	−0.731229	−	0.682133i	$-0.761053\pi$
$269$	−23.9861	−1.46246	−0.731229	+	0.682133i	$0.761053\pi$
$270$	0	0
$271$	15.0668	0.915244	0.457622	−	0.889147i	$-0.348701\pi$
$271$	15.0668	0.915244	0.457622	+	0.889147i	$0.348701\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−8.37737	−0.505175
$276$	0	0
$277$	0.350966	0.0210875	0.0105437	−	0.999944i	$-0.496644\pi$
$277$	0.350966	0.0210875	0.0105437	+	0.999944i	$0.496644\pi$
$278$	0	0
$279$	−0.926221	−0.0554514
$280$	0	0
$281$	5.30583	0.316519	0.158260	−	0.987398i	$-0.449412\pi$
$281$	5.30583	0.316519	0.158260	+	0.987398i	$0.449412\pi$
$282$	0	0
$283$	−11.6134	−0.690347	−0.345173	−	0.938539i	$-0.612180\pi$
$283$	−11.6134	−0.690347	−0.345173	+	0.938539i	$0.612180\pi$
$284$	0	0
$285$	8.12115	0.481055
$286$	0	0
$287$	−10.2709	−0.606275
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	14.8176	0.868621
$292$	0	0
$293$	20.2423	1.18257	0.591284	−	0.806463i	$-0.298621\pi$
$293$	20.2423	1.18257	0.591284	+	0.806463i	$0.298621\pi$
$294$	0	0
$295$	15.3230	0.892142
$296$	0	0
$297$	−12.5808	−0.730011
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−11.2166	−0.646515
$302$	0	0
$303$	−27.5264	−1.58135
$304$	0	0
$305$	−1.00848	−0.0577453
$306$	0	0
$307$	1.20189	0.0685955	0.0342978	−	0.999412i	$-0.489081\pi$
$307$	1.20189	0.0685955	0.0342978	+	0.999412i	$0.489081\pi$
$308$	0	0
$309$	−17.8524	−1.01559
$310$	0	0
$311$	10.2081	0.578850	0.289425	−	0.957201i	$-0.406536\pi$
$311$	10.2081	0.578850	0.289425	+	0.957201i	$0.406536\pi$
$312$	0	0
$313$	18.7375	1.05911	0.529554	−	0.848276i	$-0.322359\pi$
$313$	18.7375	1.05911	0.529554	+	0.848276i	$0.322359\pi$
$314$	0	0
$315$	−0.926221	−0.0521866
$316$	0	0
$317$	−10.4985	−0.589656	−0.294828	−	0.955550i	$-0.595262\pi$
$317$	−10.4985	−0.589656	−0.294828	+	0.955550i	$0.595262\pi$
$318$	0	0
$319$	2.10866	0.118063
$320$	0	0
$321$	2.78562	0.155478
$322$	0	0
$323$	4.94567	0.275184
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−20.6197	−1.14027
$328$	0	0
$329$	9.89134	0.545327
$330$	0	0
$331$	−27.0062	−1.48440	−0.742199	−	0.670180i	$-0.766217\pi$
$331$	−27.0062	−1.48440	−0.742199	+	0.670180i	$0.766217\pi$
$332$	0	0
$333$	4.10866	0.225153
$334$	0	0
$335$	11.0521	0.603841
$336$	0	0
$337$	−10.4985	−0.571891	−0.285946	−	0.958246i	$-0.592308\pi$
$337$	−10.4985	−0.571891	−0.285946	+	0.958246i	$0.592308\pi$
$338$	0	0
$339$	−5.67401	−0.308170
$340$	0	0
$341$	2.48604	0.134626
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	15.0668	0.811170
$346$	0	0
$347$	−15.0668	−0.808829	−0.404415	−	0.914576i	$-0.632525\pi$
$347$	−15.0668	−0.808829	−0.404415	+	0.914576i	$0.632525\pi$
$348$	0	0
$349$	7.86493	0.421000	0.210500	−	0.977594i	$-0.432491\pi$
$349$	7.86493	0.421000	0.210500	+	0.977594i	$0.432491\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	29.4053	1.56509	0.782543	−	0.622596i	$-0.213922\pi$
$353$	29.4053	1.56509	0.782543	+	0.622596i	$0.213922\pi$
$354$	0	0
$355$	16.3774	0.869221
$356$	0	0
$357$	1.47283	0.0779506
$358$	0	0
$359$	5.91302	0.312077	0.156039	−	0.987751i	$-0.450128\pi$
$359$	5.91302	0.312077	0.156039	+	0.987751i	$0.450128\pi$
$360$	0	0
$361$	5.45963	0.287349
$362$	0	0
$363$	−8.87813	−0.465981
$364$	0	0
$365$	−0.843964	−0.0441751
$366$	0	0
$367$	15.6762	0.818293	0.409147	−	0.912469i	$-0.365826\pi$
$367$	15.6762	0.818293	0.409147	+	0.912469i	$0.365826\pi$
$368$	0	0
$369$	−8.53269	−0.444194
$370$	0	0
$371$	−7.83076	−0.406553
$372$	0	0
$373$	−9.81131	−0.508011	−0.254005	−	0.967203i	$-0.581748\pi$
$373$	−9.81131	−0.508011	−0.254005	+	0.967203i	$0.581748\pi$
$374$	0	0
$375$	14.3796	0.742560
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−11.6615	−0.599012	−0.299506	−	0.954094i	$-0.596822\pi$
$379$	−11.6615	−0.599012	−0.299506	+	0.954094i	$0.596822\pi$
$380$	0	0
$381$	31.2772	1.60238
$382$	0	0
$383$	27.6087	1.41074	0.705369	−	0.708840i	$-0.250781\pi$
$383$	27.6087	1.41074	0.705369	+	0.708840i	$0.250781\pi$
$384$	0	0
$385$	2.48604	0.126700
$386$	0	0
$387$	−9.31832	−0.473677
$388$	0	0
$389$	11.8308	0.599843	0.299922	−	0.953964i	$-0.403039\pi$
$389$	11.8308	0.599843	0.299922	+	0.953964i	$0.403039\pi$
$390$	0	0
$391$	9.17548	0.464024
$392$	0	0
$393$	6.78562	0.342289
$394$	0	0
$395$	8.91926	0.448777
$396$	0	0
$397$	−22.7111	−1.13984	−0.569919	−	0.821701i	$-0.693026\pi$
$397$	−22.7111	−1.13984	−0.569919	+	0.821701i	$0.693026\pi$
$398$	0	0
$399$	7.28415	0.364663
$400$	0	0
$401$	37.6087	1.87809	0.939045	−	0.343795i	$-0.111713\pi$
$401$	37.6087	1.87809	0.939045	+	0.343795i	$0.111713\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	6.48604	0.322294
$406$	0	0
$407$	−11.0279	−0.546634
$408$	0	0
$409$	−22.5808	−1.11655	−0.558274	−	0.829657i	$-0.688536\pi$
$409$	−22.5808	−1.11655	−0.558274	+	0.829657i	$0.688536\pi$
$410$	0	0
$411$	0.559099	0.0275783
$412$	0	0
$413$	13.7438	0.676287
$414$	0	0
$415$	−13.3494	−0.655299
$416$	0	0
$417$	−8.92622	−0.437119
$418$	0	0
$419$	−24.2904	−1.18666	−0.593332	−	0.804958i	$-0.702188\pi$
$419$	−24.2904	−1.18666	−0.593332	+	0.804958i	$0.702188\pi$
$420$	0	0
$421$	11.5676	0.563769	0.281885	−	0.959448i	$-0.409040\pi$
$421$	11.5676	0.563769	0.281885	+	0.959448i	$0.409040\pi$
$422$	0	0
$423$	8.21733	0.399540
$424$	0	0
$425$	3.75698	0.182240
$426$	0	0
$427$	−0.904539	−0.0437737
$428$	0	0
$429$	0	0
$430$	0	0
$431$	22.9457	1.10525	0.552627	−	0.833429i	$-0.313625\pi$
$431$	22.9457	1.10525	0.552627	+	0.833429i	$0.313625\pi$
$432$	0	0
$433$	19.8913	0.955917	0.477958	−	0.878383i	$-0.341377\pi$
$433$	19.8913	0.955917	0.477958	+	0.878383i	$0.341377\pi$
$434$	0	0
$435$	−1.55286	−0.0744538
$436$	0	0
$437$	45.3789	2.17077
$438$	0	0
$439$	−19.8936	−0.949468	−0.474734	−	0.880129i	$-0.657456\pi$
$439$	−19.8936	−0.949468	−0.474734	+	0.880129i	$0.657456\pi$
$440$	0	0
$441$	−0.830760	−0.0395600
$442$	0	0
$443$	15.7827	0.749857	0.374929	−	0.927054i	$-0.377667\pi$
$443$	15.7827	0.749857	0.374929	+	0.927054i	$0.377667\pi$
$444$	0	0
$445$	−12.4596	−0.590643
$446$	0	0
$447$	−15.0062	−0.709771
$448$	0	0
$449$	−41.2702	−1.94766	−0.973831	−	0.227273i	$-0.927019\pi$
$449$	−41.2702	−1.94766	−0.973831	+	0.227273i	$0.927019\pi$
$450$	0	0
$451$	22.9023	1.07843
$452$	0	0
$453$	12.0606	0.566655
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−36.9923	−1.73043	−0.865214	−	0.501403i	$-0.832817\pi$
$457$	−36.9923	−1.73043	−0.865214	+	0.501403i	$0.832817\pi$
$458$	0	0
$459$	5.64207	0.263349
$460$	0	0
$461$	10.3649	0.482741	0.241370	−	0.970433i	$-0.422403\pi$
$461$	10.3649	0.482741	0.241370	+	0.970433i	$0.422403\pi$
$462$	0	0
$463$	−12.2709	−0.570279	−0.285140	−	0.958486i	$-0.592040\pi$
$463$	−12.2709	−0.570279	−0.285140	+	0.958486i	$0.592040\pi$
$464$	0	0
$465$	−1.83076	−0.0848995
$466$	0	0
$467$	−8.86341	−0.410150	−0.205075	−	0.978746i	$-0.565744\pi$
$467$	−8.86341	−0.410150	−0.205075	+	0.978746i	$0.565744\pi$
$468$	0	0
$469$	9.91302	0.457741
$470$	0	0
$471$	−21.0154	−0.968340
$472$	0	0
$473$	25.0110	1.15001
$474$	0	0
$475$	18.5808	0.852545
$476$	0	0
$477$	−6.50548	−0.297866
$478$	0	0
$479$	4.56606	0.208629	0.104314	−	0.994544i	$-0.466735\pi$
$479$	4.56606	0.208629	0.104314	+	0.994544i	$0.466735\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	13.5140	0.614907
$484$	0	0
$485$	−11.2166	−0.509320
$486$	0	0
$487$	−23.8649	−1.08142	−0.540712	−	0.841208i	$-0.681845\pi$
$487$	−23.8649	−1.08142	−0.540712	+	0.841208i	$0.681845\pi$
$488$	0	0
$489$	−23.1880	−1.04860
$490$	0	0
$491$	7.00624	0.316187	0.158094	−	0.987424i	$-0.449465\pi$
$491$	7.00624	0.316187	0.158094	+	0.987424i	$0.449465\pi$
$492$	0	0
$493$	−0.945668	−0.0425907
$494$	0	0
$495$	2.06530	0.0928284
$496$	0	0
$497$	14.6894	0.658912
$498$	0	0
$499$	−10.2687	−0.459691	−0.229845	−	0.973227i	$-0.573822\pi$
$499$	−10.2687	−0.459691	−0.229845	+	0.973227i	$0.573822\pi$
$500$	0	0
$501$	0.954863	0.0426601
$502$	0	0
$503$	−8.53413	−0.380518	−0.190259	−	0.981734i	$-0.560933\pi$
$503$	−8.53413	−0.380518	−0.190259	+	0.981734i	$0.560933\pi$
$504$	0	0
$505$	20.8370	0.927234
$506$	0	0
$507$	−19.1468	−0.850341
$508$	0	0
$509$	−23.8216	−1.05587	−0.527936	−	0.849284i	$-0.677034\pi$
$509$	−23.8216	−1.05587	−0.527936	+	0.849284i	$0.677034\pi$
$510$	0	0
$511$	−0.756981	−0.0334869
$512$	0	0
$513$	27.9038	1.23198
$514$	0	0
$515$	13.5140	0.595496
$516$	0	0
$517$	−22.0558	−0.970015
$518$	0	0
$519$	23.4659	1.03004
$520$	0	0
$521$	−3.48532	−0.152695	−0.0763473	−	0.997081i	$-0.524326\pi$
$521$	−3.48532	−0.152695	−0.0763473	+	0.997081i	$0.524326\pi$
$522$	0	0
$523$	23.2144	1.01509	0.507547	−	0.861624i	$-0.330552\pi$
$523$	23.2144	1.01509	0.507547	+	0.861624i	$0.330552\pi$
$524$	0	0
$525$	5.53341	0.241498
$526$	0	0
$527$	−1.11491	−0.0485661
$528$	0	0
$529$	61.1895	2.66041
$530$	0	0
$531$	11.4178	0.495490
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−2.10866	−0.0911655
$536$	0	0
$537$	25.0760	1.08211
$538$	0	0
$539$	2.22982	0.0960449
$540$	0	0
$541$	−9.08074	−0.390411	−0.195206	−	0.980762i	$-0.562537\pi$
$541$	−9.08074	−0.390411	−0.195206	+	0.980762i	$0.562537\pi$
$542$	0	0
$543$	8.29959	0.356169
$544$	0	0
$545$	15.6087	0.668603
$546$	0	0
$547$	−5.34945	−0.228726	−0.114363	−	0.993439i	$-0.536483\pi$
$547$	−5.34945	−0.228726	−0.114363	+	0.993439i	$0.536483\pi$
$548$	0	0
$549$	−0.751455	−0.0320713
$550$	0	0
$551$	−4.67696	−0.199245
$552$	0	0
$553$	8.00000	0.340195
$554$	0	0
$555$	8.12115	0.344724
$556$	0	0
$557$	−13.9472	−0.590961	−0.295481	−	0.955349i	$-0.595480\pi$
$557$	−13.9472	−0.590961	−0.295481	+	0.955349i	$0.595480\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−3.28415	−0.138657
$562$	0	0
$563$	−8.43322	−0.355418	−0.177709	−	0.984083i	$-0.556869\pi$
$563$	−8.43322	−0.355418	−0.177709	+	0.984083i	$0.556869\pi$
$564$	0	0
$565$	4.29512	0.180697
$566$	0	0
$567$	5.81756	0.244314
$568$	0	0
$569$	−5.89357	−0.247071	−0.123536	−	0.992340i	$-0.539423\pi$
$569$	−5.89357	−0.247071	−0.123536	+	0.992340i	$0.539423\pi$
$570$	0	0
$571$	−40.2857	−1.68590	−0.842951	−	0.537990i	$-0.819184\pi$
$571$	−40.2857	−1.68590	−0.842951	+	0.537990i	$0.819184\pi$
$572$	0	0
$573$	−4.24926	−0.177515
$574$	0	0
$575$	34.4721	1.43759
$576$	0	0
$577$	27.1616	1.13075	0.565375	−	0.824834i	$-0.308731\pi$
$577$	27.1616	1.13075	0.565375	+	0.824834i	$0.308731\pi$
$578$	0	0
$579$	25.8135	1.07277
$580$	0	0
$581$	−11.9736	−0.496748
$582$	0	0
$583$	17.4611	0.723167
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−10.2423	−0.422745	−0.211373	−	0.977406i	$-0.567793\pi$
$587$	−10.2423	−0.422745	−0.211373	+	0.977406i	$0.567793\pi$
$588$	0	0
$589$	−5.51396	−0.227199
$590$	0	0
$591$	21.9736	0.903873
$592$	0	0
$593$	27.9736	1.14874	0.574369	−	0.818597i	$-0.305248\pi$
$593$	27.9736	1.14874	0.574369	+	0.818597i	$0.305248\pi$
$594$	0	0
$595$	−1.11491	−0.0457068
$596$	0	0
$597$	24.5202	1.00355
$598$	0	0
$599$	−6.00223	−0.245245	−0.122622	−	0.992453i	$-0.539130\pi$
$599$	−6.00223	−0.245245	−0.122622	+	0.992453i	$0.539130\pi$
$600$	0	0
$601$	−19.8913	−0.811385	−0.405692	−	0.914010i	$-0.632970\pi$
$601$	−19.8913	−0.811385	−0.405692	+	0.914010i	$0.632970\pi$
$602$	0	0
$603$	8.23534	0.335369
$604$	0	0
$605$	6.72058	0.273230
$606$	0	0
$607$	29.4784	1.19649	0.598245	−	0.801313i	$-0.295865\pi$
$607$	29.4784	1.19649	0.598245	+	0.801313i	$0.295865\pi$
$608$	0	0
$609$	−1.39281	−0.0564396
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−36.4115	−1.47065	−0.735324	−	0.677716i	$-0.762970\pi$
$613$	−36.4115	−1.47065	−0.735324	+	0.677716i	$0.762970\pi$
$614$	0	0
$615$	−16.8656	−0.680088
$616$	0	0
$617$	−7.51396	−0.302501	−0.151250	−	0.988495i	$-0.548330\pi$
$617$	−7.51396	−0.302501	−0.151250	+	0.988495i	$0.548330\pi$
$618$	0	0
$619$	−21.7438	−0.873956	−0.436978	−	0.899472i	$-0.643951\pi$
$619$	−21.7438	−0.873956	−0.436978	+	0.899472i	$0.643951\pi$
$620$	0	0
$621$	51.7688	2.07741
$622$	0	0
$623$	−11.1755	−0.447736
$624$	0	0
$625$	7.89981	0.315993
$626$	0	0
$627$	−16.2423	−0.648655
$628$	0	0
$629$	4.94567	0.197197
$630$	0	0
$631$	9.57454	0.381156	0.190578	−	0.981672i	$-0.438964\pi$
$631$	9.57454	0.381156	0.190578	+	0.981672i	$0.438964\pi$
$632$	0	0
$633$	−10.9068	−0.433505
$634$	0	0
$635$	−23.6762	−0.939563
$636$	0	0
$637$	0	0
$638$	0	0
$639$	12.2034	0.482759
$640$	0	0
$641$	22.8759	0.903544	0.451772	−	0.892134i	$-0.350792\pi$
$641$	22.8759	0.903544	0.451772	+	0.892134i	$0.350792\pi$
$642$	0	0
$643$	−28.2834	−1.11539	−0.557695	−	0.830046i	$-0.688314\pi$
$643$	−28.2834	−1.11539	−0.557695	+	0.830046i	$0.688314\pi$
$644$	0	0
$645$	−18.4185	−0.725228
$646$	0	0
$647$	10.0125	0.393631	0.196816	−	0.980440i	$-0.436940\pi$
$647$	10.0125	0.393631	0.196816	+	0.980440i	$0.436940\pi$
$648$	0	0
$649$	−30.6461	−1.20296
$650$	0	0
$651$	−1.64207	−0.0643579
$652$	0	0
$653$	−2.45963	−0.0962528	−0.0481264	−	0.998841i	$-0.515325\pi$
$653$	−2.45963	−0.0962528	−0.0481264	+	0.998841i	$0.515325\pi$
$654$	0	0
$655$	−5.13659	−0.200703
$656$	0	0
$657$	−0.628870	−0.0245346
$658$	0	0
$659$	−33.0187	−1.28623	−0.643114	−	0.765771i	$-0.722358\pi$
$659$	−33.0187	−1.28623	−0.643114	+	0.765771i	$0.722358\pi$
$660$	0	0
$661$	5.97359	0.232346	0.116173	−	0.993229i	$-0.462937\pi$
$661$	5.97359	0.232346	0.116173	+	0.993229i	$0.462937\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−5.51396	−0.213822
$666$	0	0
$667$	−8.67696	−0.335973
$668$	0	0
$669$	−39.9861	−1.54595
$670$	0	0
$671$	2.01696	0.0778637
$672$	0	0
$673$	6.29512	0.242659	0.121329	−	0.992612i	$-0.461284\pi$
$673$	6.29512	0.242659	0.121329	+	0.992612i	$0.461284\pi$
$674$	0	0
$675$	21.1972	0.815879
$676$	0	0
$677$	46.5544	1.78923	0.894615	−	0.446838i	$-0.147450\pi$
$677$	46.5544	1.78923	0.894615	+	0.446838i	$0.147450\pi$
$678$	0	0
$679$	−10.0606	−0.386089
$680$	0	0
$681$	6.63912	0.254412
$682$	0	0
$683$	−43.9861	−1.68308	−0.841540	−	0.540194i	$-0.818351\pi$
$683$	−43.9861	−1.68308	−0.841540	+	0.540194i	$0.818351\pi$
$684$	0	0
$685$	−0.423228	−0.0161707
$686$	0	0
$687$	15.0668	0.574835
$688$	0	0
$689$	0	0
$690$	0	0
$691$	9.17772	0.349137	0.174568	−	0.984645i	$-0.444147\pi$
$691$	9.17772	0.349137	0.174568	+	0.984645i	$0.444147\pi$
$692$	0	0
$693$	1.85244	0.0703684
$694$	0	0
$695$	6.75698	0.256307
$696$	0	0
$697$	−10.2709	−0.389040
$698$	0	0
$699$	17.0154	0.643583
$700$	0	0
$701$	−7.59622	−0.286905	−0.143453	−	0.989657i	$-0.545820\pi$
$701$	−7.59622	−0.286905	−0.143453	+	0.989657i	$0.545820\pi$
$702$	0	0
$703$	24.4596	0.922512
$704$	0	0
$705$	16.2423	0.611720
$706$	0	0
$707$	18.6894	0.702889
$708$	0	0
$709$	7.63511	0.286743	0.143371	−	0.989669i	$-0.454206\pi$
$709$	7.63511	0.286743	0.143371	+	0.989669i	$0.454206\pi$
$710$	0	0
$711$	6.64608	0.249248
$712$	0	0
$713$	−10.2298	−0.383110
$714$	0	0
$715$	0	0
$716$	0	0
$717$	45.2999	1.69176
$718$	0	0
$719$	13.6204	0.507955	0.253977	−	0.967210i	$-0.418261\pi$
$719$	13.6204	0.507955	0.253977	+	0.967210i	$0.418261\pi$
$720$	0	0
$721$	12.1212	0.451415
$722$	0	0
$723$	−1.42474	−0.0529868
$724$	0	0
$725$	−3.55286	−0.131950
$726$	0	0
$727$	−10.9457	−0.405952	−0.202976	−	0.979184i	$-0.565061\pi$
$727$	−10.9457	−0.405952	−0.202976	+	0.979184i	$0.565061\pi$
$728$	0	0
$729$	29.7625	1.10232
$730$	0	0
$731$	−11.2166	−0.414861
$732$	0	0
$733$	7.28415	0.269046	0.134523	−	0.990910i	$-0.457050\pi$
$733$	7.28415	0.269046	0.134523	+	0.990910i	$0.457050\pi$
$734$	0	0
$735$	−1.64207	−0.0605688
$736$	0	0
$737$	−22.1042	−0.814218
$738$	0	0
$739$	50.3268	1.85130	0.925651	−	0.378380i	$-0.123519\pi$
$739$	50.3268	1.85130	0.925651	+	0.378380i	$0.123519\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−14.9317	−0.547793	−0.273896	−	0.961759i	$-0.588313\pi$
$743$	−14.9317	−0.547793	−0.273896	+	0.961759i	$0.588313\pi$
$744$	0	0
$745$	11.3594	0.416178
$746$	0	0
$747$	−9.94719	−0.363948
$748$	0	0
$749$	−1.89134	−0.0691079
$750$	0	0
$751$	−29.9347	−1.09233	−0.546166	−	0.837677i	$-0.683913\pi$
$751$	−29.9347	−1.09233	−0.546166	+	0.837677i	$0.683913\pi$
$752$	0	0
$753$	−0.0388938	−0.00141737
$754$	0	0
$755$	−9.12963	−0.332261
$756$	0	0
$757$	−6.48827	−0.235820	−0.117910	−	0.993024i	$-0.537619\pi$
$757$	−6.48827	−0.235820	−0.117910	+	0.993024i	$0.537619\pi$
$758$	0	0
$759$	−30.1336	−1.09378
$760$	0	0
$761$	−28.2159	−1.02283	−0.511413	−	0.859335i	$-0.670878\pi$
$761$	−28.2159	−1.02283	−0.511413	+	0.859335i	$0.670878\pi$
$762$	0	0
$763$	14.0000	0.506834
$764$	0	0
$765$	−0.926221	−0.0334876
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−21.8260	−0.787067	−0.393533	−	0.919310i	$-0.628747\pi$
$769$	−21.8260	−0.787067	−0.393533	+	0.919310i	$0.628747\pi$
$770$	0	0
$771$	21.6740	0.780570
$772$	0	0
$773$	26.5683	0.955595	0.477798	−	0.878470i	$-0.341435\pi$
$773$	26.5683	0.955595	0.477798	+	0.878470i	$0.341435\pi$
$774$	0	0
$775$	−4.18869	−0.150462
$776$	0	0
$777$	7.28415	0.261317
$778$	0	0
$779$	−50.7967	−1.81998
$780$	0	0
$781$	−32.7547	−1.17206
$782$	0	0
$783$	−5.33553	−0.190676
$784$	0	0
$785$	15.9083	0.567791
$786$	0	0
$787$	−4.98903	−0.177840	−0.0889199	−	0.996039i	$-0.528342\pi$
$787$	−4.98903	−0.177840	−0.0889199	+	0.996039i	$0.528342\pi$
$788$	0	0
$789$	−36.7019	−1.30662
$790$	0	0
$791$	3.85244	0.136977
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−12.8587	−0.456051
$796$	0	0
$797$	−47.0963	−1.66824	−0.834118	−	0.551587i	$-0.814023\pi$
$797$	−47.0963	−1.66824	−0.834118	+	0.551587i	$0.814023\pi$
$798$	0	0
$799$	9.89134	0.349930
$800$	0	0
$801$	−9.28415	−0.328039
$802$	0	0
$803$	1.68793	0.0595657
$804$	0	0
$805$	−10.2298	−0.360554
$806$	0	0
$807$	−35.3275	−1.24359
$808$	0	0
$809$	−10.2423	−0.360100	−0.180050	−	0.983657i	$-0.557626\pi$
$809$	−10.2423	−0.360100	−0.180050	+	0.983657i	$0.557626\pi$
$810$	0	0
$811$	2.98680	0.104881	0.0524403	−	0.998624i	$-0.483300\pi$
$811$	2.98680	0.104881	0.0524403	+	0.998624i	$0.483300\pi$
$812$	0	0
$813$	22.1909	0.778270
$814$	0	0
$815$	17.5529	0.614850
$816$	0	0
$817$	−55.4736	−1.94078
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−52.7283	−1.84023	−0.920116	−	0.391646i	$-0.871906\pi$
$821$	−52.7283	−1.84023	−0.920116	+	0.391646i	$0.871906\pi$
$822$	0	0
$823$	6.68945	0.233179	0.116590	−	0.993180i	$-0.462804\pi$
$823$	6.68945	0.233179	0.116590	+	0.993180i	$0.462804\pi$
$824$	0	0
$825$	−12.3385	−0.429571
$826$	0	0
$827$	−41.5140	−1.44358	−0.721791	−	0.692111i	$-0.756681\pi$
$827$	−41.5140	−1.44358	−0.721791	+	0.692111i	$0.756681\pi$
$828$	0	0
$829$	−48.4596	−1.68307	−0.841536	−	0.540201i	$-0.818348\pi$
$829$	−48.4596	−1.68307	−0.841536	+	0.540201i	$0.818348\pi$
$830$	0	0
$831$	0.516914	0.0179316
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	−0.722813	−0.0250140
$836$	0	0
$837$	−6.29039	−0.217428
$838$	0	0
$839$	9.64903	0.333122	0.166561	−	0.986031i	$-0.446734\pi$
$839$	9.64903	0.333122	0.166561	+	0.986031i	$0.446734\pi$
$840$	0	0
$841$	−28.1057	−0.969162
$842$	0	0
$843$	7.81460	0.269149
$844$	0	0
$845$	14.4938	0.498602
$846$	0	0
$847$	6.02792	0.207122
$848$	0	0
$849$	−17.1047	−0.587030
$850$	0	0
$851$	45.3789	1.55557
$852$	0	0
$853$	−35.3619	−1.21077	−0.605385	−	0.795933i	$-0.706981\pi$
$853$	−35.3619	−1.21077	−0.605385	+	0.795933i	$0.706981\pi$
$854$	0	0
$855$	−4.58078	−0.156659
$856$	0	0
$857$	29.0451	0.992163	0.496081	−	0.868276i	$-0.334772\pi$
$857$	29.0451	0.992163	0.496081	+	0.868276i	$0.334772\pi$
$858$	0	0
$859$	43.7996	1.49442	0.747212	−	0.664586i	$-0.231392\pi$
$859$	43.7996	1.49442	0.747212	+	0.664586i	$0.231392\pi$
$860$	0	0
$861$	−15.1274	−0.515540
$862$	0	0
$863$	3.39754	0.115654	0.0578268	−	0.998327i	$-0.481583\pi$
$863$	3.39754	0.115654	0.0578268	+	0.998327i	$0.481583\pi$
$864$	0	0
$865$	−17.7632	−0.603968
$866$	0	0
$867$	1.47283	0.0500201
$868$	0	0
$869$	−17.8385	−0.605130
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−8.35793	−0.282873
$874$	0	0
$875$	−9.76322	−0.330057
$876$	0	0
$877$	−29.0529	−0.981047	−0.490523	−	0.871428i	$-0.663194\pi$
$877$	−29.0529	−0.981047	−0.490523	+	0.871428i	$0.663194\pi$
$878$	0	0
$879$	29.8135	1.00559
$880$	0	0
$881$	−11.4923	−0.387185	−0.193592	−	0.981082i	$-0.562014\pi$
$881$	−11.4923	−0.387185	−0.193592	+	0.981082i	$0.562014\pi$
$882$	0	0
$883$	−25.1927	−0.847802	−0.423901	−	0.905709i	$-0.639340\pi$
$883$	−25.1927	−0.847802	−0.423901	+	0.905709i	$0.639340\pi$
$884$	0	0
$885$	22.5683	0.758625
$886$	0	0
$887$	35.1957	1.18176	0.590878	−	0.806761i	$-0.298781\pi$
$887$	35.1957	1.18176	0.590878	+	0.806761i	$0.298781\pi$
$888$	0	0
$889$	−21.2361	−0.712235
$890$	0	0
$891$	−12.9721	−0.434581
$892$	0	0
$893$	48.9193	1.63702
$894$	0	0
$895$	−18.9821	−0.634501
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.05433	0.0351639
$900$	0	0
$901$	−7.83076	−0.260880
$902$	0	0
$903$	−16.5202	−0.549758
$904$	0	0
$905$	−6.28263	−0.208842
$906$	0	0
$907$	25.1227	0.834184	0.417092	−	0.908864i	$-0.363049\pi$
$907$	25.1227	0.834184	0.417092	+	0.908864i	$0.363049\pi$
$908$	0	0
$909$	15.5264	0.514980
$910$	0	0
$911$	−22.5110	−0.745823	−0.372912	−	0.927867i	$-0.621641\pi$
$911$	−22.5110	−0.745823	−0.372912	+	0.927867i	$0.621641\pi$
$912$	0	0
$913$	26.6989	0.883605
$914$	0	0
$915$	−1.48532	−0.0491032
$916$	0	0
$917$	−4.60719	−0.152143
$918$	0	0
$919$	−13.5676	−0.447553	−0.223777	−	0.974640i	$-0.571839\pi$
$919$	−13.5676	−0.447553	−0.223777	+	0.974640i	$0.571839\pi$
$920$	0	0
$921$	1.77018	0.0583296
$922$	0	0
$923$	0	0
$924$	0	0
$925$	18.5808	0.610932
$926$	0	0
$927$	10.0698	0.330735
$928$	0	0
$929$	−11.9519	−0.392129	−0.196065	−	0.980591i	$-0.562816\pi$
$929$	−11.9519	−0.392129	−0.196065	+	0.980591i	$0.562816\pi$
$930$	0	0
$931$	−4.94567	−0.162088
$932$	0	0
$933$	15.0349	0.492220
$934$	0	0
$935$	2.48604	0.0813021
$936$	0	0
$937$	53.0140	1.73189	0.865946	−	0.500138i	$-0.166717\pi$
$937$	53.0140	1.73189	0.865946	+	0.500138i	$0.166717\pi$
$938$	0	0
$939$	27.5973	0.900603
$940$	0	0
$941$	−27.1204	−0.884101	−0.442050	−	0.896990i	$-0.645749\pi$
$941$	−27.1204	−0.884101	−0.442050	+	0.896990i	$0.645749\pi$
$942$	0	0
$943$	−94.2409	−3.06890
$944$	0	0
$945$	−6.29039	−0.204626
$946$	0	0
$947$	44.5808	1.44868	0.724340	−	0.689443i	$-0.242144\pi$
$947$	44.5808	1.44868	0.724340	+	0.689443i	$0.242144\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−15.4626	−0.501409
$952$	0	0
$953$	−51.8991	−1.68118	−0.840588	−	0.541675i	$-0.817791\pi$
$953$	−51.8991	−1.68118	−0.840588	+	0.541675i	$0.817791\pi$
$954$	0	0
$955$	3.21661	0.104087
$956$	0	0
$957$	3.10571	0.100393
$958$	0	0
$959$	−0.379608	−0.0122582
$960$	0	0
$961$	−29.7570	−0.959903
$962$	0	0
$963$	−1.57125	−0.0506327
$964$	0	0
$965$	−19.5404	−0.629027
$966$	0	0
$967$	−37.2680	−1.19846	−0.599229	−	0.800578i	$-0.704526\pi$
$967$	−37.2680	−1.19846	−0.599229	+	0.800578i	$0.704526\pi$
$968$	0	0
$969$	7.28415	0.234001
$970$	0	0
$971$	3.05433	0.0980182	0.0490091	−	0.998798i	$-0.484394\pi$
$971$	3.05433	0.0980182	0.0490091	+	0.998798i	$0.484394\pi$
$972$	0	0
$973$	6.06058	0.194293
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−47.3961	−1.51634	−0.758168	−	0.652059i	$-0.773905\pi$
$977$	−47.3961	−1.51634	−0.758168	+	0.652059i	$0.773905\pi$
$978$	0	0
$979$	24.9193	0.796423
$980$	0	0
$981$	11.6306	0.371338
$982$	0	0
$983$	51.1957	1.63289	0.816445	−	0.577423i	$-0.195942\pi$
$983$	51.1957	1.63289	0.816445	+	0.577423i	$0.195942\pi$
$984$	0	0
$985$	−16.6336	−0.529990
$986$	0	0
$987$	14.5683	0.463714
$988$	0	0
$989$	−102.918	−3.27260
$990$	0	0
$991$	−16.9332	−0.537900	−0.268950	−	0.963154i	$-0.586677\pi$
$991$	−16.9332	−0.537900	−0.268950	+	0.963154i	$0.586677\pi$
$992$	0	0
$993$	−39.7757	−1.26224
$994$	0	0
$995$	−18.5613	−0.588434
$996$	0	0
$997$	−53.3069	−1.68825	−0.844123	−	0.536150i	$-0.819878\pi$
$997$	−53.3069	−1.68825	−0.844123	+	0.536150i	$0.819878\pi$
$998$	0	0
$999$	27.9038	0.882838

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1904.2.a.m.1.3		3
4.3	odd	2		952.2.a.f.1.1	✓	3
8.3	odd	2		7616.2.a.bc.1.3		3
8.5	even	2		7616.2.a.be.1.1		3
12.11	even	2		8568.2.a.x.1.3		3
28.27	even	2		6664.2.a.j.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
952.2.a.f.1.1	✓	3	4.3	odd	2
1904.2.a.m.1.3		3	1.1	even	1	trivial
6664.2.a.j.1.3		3	28.27	even	2
7616.2.a.bc.1.3		3	8.3	odd	2
7616.2.a.be.1.1		3	8.5	even	2
8568.2.a.x.1.3		3	12.11	even	2

Overview	Random
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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:	\( N \)	\(=\)	\( 1904 = 2^{4} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1904.a (trivial)

\(f(q)\)	\(=\)	\(q+1.47283 q^{3} -1.11491 q^{5} -1.00000 q^{7} -0.830760 q^{9} +O(q^{10})\)	\(q+1.47283 q^{3} -1.11491 q^{5} -1.00000 q^{7} -0.830760 q^{9} +2.22982 q^{11} -1.64207 q^{15} -1.00000 q^{17} -4.94567 q^{19} -1.47283 q^{21} -9.17548 q^{23} -3.75698 q^{25} -5.64207 q^{27} +0.945668 q^{29} +1.11491 q^{31} +3.28415 q^{33} +1.11491 q^{35} -4.94567 q^{37} +10.2709 q^{41} +11.2166 q^{43} +0.926221 q^{45} -9.89134 q^{47} +1.00000 q^{49} -1.47283 q^{51} +7.83076 q^{53} -2.48604 q^{55} -7.28415 q^{57} -13.7438 q^{59} +0.904539 q^{61} +0.830760 q^{63} -9.91302 q^{67} -13.5140 q^{69} -14.6894 q^{71} +0.756981 q^{73} -5.53341 q^{75} -2.22982 q^{77} -8.00000 q^{79} -5.81756 q^{81} +11.9736 q^{83} +1.11491 q^{85} +1.39281 q^{87} +11.1755 q^{89} +1.64207 q^{93} +5.51396 q^{95} +10.0606 q^{97} -1.85244 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} + 3 q^{5} - 3 q^{7} + 2 q^{9}+O(q^{10}) \) 3 * q - q^3 + 3 * q^5 - 3 * q^7 + 2 * q^9	\( 3 q - q^{3} + 3 q^{5} - 3 q^{7} + 2 q^{9} - 6 q^{11} - 4 q^{15} - 3 q^{17} - 4 q^{19} + q^{21} - 4 q^{23} - 4 q^{25} - 16 q^{27} - 8 q^{29} - 3 q^{31} + 8 q^{33} - 3 q^{35} - 4 q^{37} + 9 q^{41} + q^{43} - 8 q^{47} + 3 q^{49} + q^{51} + 19 q^{53} - 22 q^{55} - 20 q^{57} - 14 q^{59} + q^{61} - 2 q^{63} - 7 q^{67} - 26 q^{69} - 6 q^{71} - 5 q^{73} + 6 q^{75} + 6 q^{77} - 24 q^{79} + 7 q^{81} - 4 q^{83} - 3 q^{85} + 24 q^{87} + 10 q^{89} + 4 q^{93} + 2 q^{95} + 13 q^{97}+O(q^{100}) \) 3 * q - q^3 + 3 * q^5 - 3 * q^7 + 2 * q^9 - 6 * q^11 - 4 * q^15 - 3 * q^17 - 4 * q^19 + q^21 - 4 * q^23 - 4 * q^25 - 16 * q^27 - 8 * q^29 - 3 * q^31 + 8 * q^33 - 3 * q^35 - 4 * q^37 + 9 * q^41 + q^43 - 8 * q^47 + 3 * q^49 + q^51 + 19 * q^53 - 22 * q^55 - 20 * q^57 - 14 * q^59 + q^61 - 2 * q^63 - 7 * q^67 - 26 * q^69 - 6 * q^71 - 5 * q^73 + 6 * q^75 + 6 * q^77 - 24 * q^79 + 7 * q^81 - 4 * q^83 - 3 * q^85 + 24 * q^87 + 10 * q^89 + 4 * q^93 + 2 * q^95 + 13 * q^97

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