LMFDB - Embedded newform 7600.2.a.cb.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7600,2,Mod(1,7600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7600, 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("7600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7600 = 2^{4} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7600.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:	$60.6863055362$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.257.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 3 $ x^3 - x^2 - 4*x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 950)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.91223$ of defining polynomial
Character	$\chi$	$=$	7600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.25561 q^{3} +4.22547 q^{7} +2.08777 q^{9} +O(q^{10})$	$q+2.25561 q^{3} +4.22547 q^{7} +2.08777 q^{9} +5.13770 q^{11} -3.16784 q^{13} +6.48108 q^{17} +1.00000 q^{19} +9.53101 q^{21} +7.56885 q^{23} -2.05763 q^{27} +0.832162 q^{29} +4.51122 q^{31} +11.5886 q^{33} -0.137699 q^{37} -7.14540 q^{39} -11.6489 q^{41} -2.51122 q^{43} -5.96216 q^{47} +10.8546 q^{49} +14.6188 q^{51} -0.225470 q^{53} +2.25561 q^{57} -5.39331 q^{59} +14.4509 q^{61} +8.82181 q^{63} -4.11021 q^{67} +17.0724 q^{69} -3.82446 q^{71} -4.70655 q^{73} +21.7092 q^{77} -10.6265 q^{79} -10.9045 q^{81} -12.0999 q^{83} +1.87703 q^{87} -10.0000 q^{89} -13.3856 q^{91} +10.1755 q^{93} -3.93972 q^{97} +10.7263 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{3} + 2 q^{7} + 13 q^{9}+O(q^{10}) $ 3 * q + 2 * q^3 + 2 * q^7 + 13 * q^9	$ 3 q + 2 q^{3} + 2 q^{7} + 13 q^{9} - 2 q^{11} + 2 q^{13} + 4 q^{17} + 3 q^{19} - 11 q^{21} + 14 q^{23} - 7 q^{27} + 14 q^{29} + 4 q^{31} - 4 q^{33} + 17 q^{37} - 29 q^{39} - 8 q^{41} + 2 q^{43} + 13 q^{47} + 25 q^{49} + 11 q^{51} + 10 q^{53} + 2 q^{57} + 6 q^{59} + 22 q^{61} + 2 q^{63} + 8 q^{69} + 2 q^{71} + 12 q^{73} + 50 q^{77} - 24 q^{79} - q^{81} + 12 q^{83} - 21 q^{87} - 30 q^{89} + 7 q^{91} + 44 q^{93} - 24 q^{99}+O(q^{100}) $ 3 * q + 2 * q^3 + 2 * q^7 + 13 * q^9 - 2 * q^11 + 2 * q^13 + 4 * q^17 + 3 * q^19 - 11 * q^21 + 14 * q^23 - 7 * q^27 + 14 * q^29 + 4 * q^31 - 4 * q^33 + 17 * q^37 - 29 * q^39 - 8 * q^41 + 2 * q^43 + 13 * q^47 + 25 * q^49 + 11 * q^51 + 10 * q^53 + 2 * q^57 + 6 * q^59 + 22 * q^61 + 2 * q^63 + 8 * q^69 + 2 * q^71 + 12 * q^73 + 50 * q^77 - 24 * q^79 - q^81 + 12 * q^83 - 21 * q^87 - 30 * q^89 + 7 * q^91 + 44 * q^93 - 24 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	2.25561	1.30228	0.651138	−	0.758959i	$-0.274292\pi$
$3$	2.25561	1.30228	0.651138	+	0.758959i	$0.274292\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.22547	1.59708	0.798539	−	0.601943i	$-0.205607\pi$
$7$	4.22547	1.59708	0.798539	+	0.601943i	$0.205607\pi$
$8$	0	0
$9$	2.08777	0.695924
$10$	0	0
$11$	5.13770	1.54907	0.774537	−	0.632528i	$-0.217983\pi$
$11$	5.13770	1.54907	0.774537	+	0.632528i	$0.217983\pi$
$12$	0	0
$13$	−3.16784	−0.878600	−0.439300	−	0.898340i	$-0.644774\pi$
$13$	−3.16784	−0.878600	−0.439300	+	0.898340i	$0.644774\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.48108	1.57189	0.785946	−	0.618295i	$-0.212176\pi$
$17$	6.48108	1.57189	0.785946	+	0.618295i	$0.212176\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	9.53101	2.07984
$22$	0	0
$23$	7.56885	1.57821	0.789107	−	0.614256i	$-0.210544\pi$
$23$	7.56885	1.57821	0.789107	+	0.614256i	$0.210544\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−2.05763	−0.395991
$28$	0	0
$29$	0.832162	0.154529	0.0772643	−	0.997011i	$-0.475381\pi$
$29$	0.832162	0.154529	0.0772643	+	0.997011i	$0.475381\pi$
$30$	0	0
$31$	4.51122	0.810239	0.405119	−	0.914264i	$-0.367230\pi$
$31$	4.51122	0.810239	0.405119	+	0.914264i	$0.367230\pi$
$32$	0	0
$33$	11.5886	2.01732
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−0.137699	−0.0226376	−0.0113188	−	0.999936i	$-0.503603\pi$
$37$	−0.137699	−0.0226376	−0.0113188	+	0.999936i	$0.503603\pi$
$38$	0	0
$39$	−7.14540	−1.14418
$40$	0	0
$41$	−11.6489	−1.81926	−0.909628	−	0.415425i	$-0.863633\pi$
$41$	−11.6489	−1.81926	−0.909628	+	0.415425i	$0.863633\pi$
$42$	0	0
$43$	−2.51122	−0.382957	−0.191479	−	0.981497i	$-0.561328\pi$
$43$	−2.51122	−0.382957	−0.191479	+	0.981497i	$0.561328\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.96216	−0.869670	−0.434835	−	0.900510i	$-0.643193\pi$
$47$	−5.96216	−0.869670	−0.434835	+	0.900510i	$0.643193\pi$
$48$	0	0
$49$	10.8546	1.55066
$50$	0	0
$51$	14.6188	2.04704
$52$	0	0
$53$	−0.225470	−0.0309707	−0.0154853	−	0.999880i	$-0.504929\pi$
$53$	−0.225470	−0.0309707	−0.0154853	+	0.999880i	$0.504929\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.25561	0.298763
$58$	0	0
$59$	−5.39331	−0.702149	−0.351074	−	0.936348i	$-0.614184\pi$
$59$	−5.39331	−0.702149	−0.351074	+	0.936348i	$0.614184\pi$
$60$	0	0
$61$	14.4509	1.85025	0.925127	−	0.379659i	$-0.123959\pi$
$61$	14.4509	1.85025	0.925127	+	0.379659i	$0.123959\pi$
$62$	0	0
$63$	8.82181	1.11144
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−4.11021	−0.502142	−0.251071	−	0.967969i	$-0.580783\pi$
$67$	−4.11021	−0.502142	−0.251071	+	0.967969i	$0.580783\pi$
$68$	0	0
$69$	17.0724	2.05527
$70$	0	0
$71$	−3.82446	−0.453880	−0.226940	−	0.973909i	$-0.572872\pi$
$71$	−3.82446	−0.453880	−0.226940	+	0.973909i	$0.572872\pi$
$72$	0	0
$73$	−4.70655	−0.550860	−0.275430	−	0.961321i	$-0.588820\pi$
$73$	−4.70655	−0.550860	−0.275430	+	0.961321i	$0.588820\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	21.7092	2.47399
$78$	0	0
$79$	−10.6265	−1.19557	−0.597786	−	0.801655i	$-0.703953\pi$
$79$	−10.6265	−1.19557	−0.597786	+	0.801655i	$0.703953\pi$
$80$	0	0
$81$	−10.9045	−1.21161
$82$	0	0
$83$	−12.0999	−1.32813	−0.664066	−	0.747674i	$-0.731171\pi$
$83$	−12.0999	−1.32813	−0.664066	+	0.747674i	$0.731171\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.87703	0.201239
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	−13.3856	−1.40319
$92$	0	0
$93$	10.1755	1.05515
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−3.93972	−0.400018	−0.200009	−	0.979794i	$-0.564097\pi$
$97$	−3.93972	−0.400018	−0.200009	+	0.979794i	$0.564097\pi$
$98$	0	0
$99$	10.7263	1.07804
$100$	0	0
$101$	3.19798	0.318211	0.159105	−	0.987262i	$-0.449139\pi$
$101$	3.19798	0.318211	0.159105	+	0.987262i	$0.449139\pi$
$102$	0	0
$103$	−10.6868	−1.05300	−0.526499	−	0.850176i	$-0.676496\pi$
$103$	−10.6868	−1.05300	−0.526499	+	0.850176i	$0.676496\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.8168	1.04570	0.522848	−	0.852426i	$-0.324870\pi$
$107$	10.8168	1.04570	0.522848	+	0.852426i	$0.324870\pi$
$108$	0	0
$109$	9.24791	0.885789	0.442894	−	0.896574i	$-0.353952\pi$
$109$	9.24791	0.885789	0.442894	+	0.896574i	$0.353952\pi$
$110$	0	0
$111$	−0.310596	−0.0294804
$112$	0	0
$113$	−17.6489	−1.66027	−0.830135	−	0.557562i	$-0.811737\pi$
$113$	−17.6489	−1.66027	−0.830135	+	0.557562i	$0.811737\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−6.61372	−0.611439
$118$	0	0
$119$	27.3856	2.51043
$120$	0	0
$121$	15.3960	1.39963
$122$	0	0
$123$	−26.2754	−2.36917
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−12.7866	−1.13463	−0.567314	−	0.823501i	$-0.692018\pi$
$127$	−12.7866	−1.13463	−0.567314	+	0.823501i	$0.692018\pi$
$128$	0	0
$129$	−5.66432	−0.498716
$130$	0	0
$131$	2.11526	0.184811	0.0924057	−	0.995721i	$-0.470544\pi$
$131$	2.11526	0.184811	0.0924057	+	0.995721i	$0.470544\pi$
$132$	0	0
$133$	4.22547	0.366395
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.36317	0.458206	0.229103	−	0.973402i	$-0.426421\pi$
$137$	5.36317	0.458206	0.229103	+	0.973402i	$0.426421\pi$
$138$	0	0
$139$	10.1601	0.861771	0.430886	−	0.902407i	$-0.358201\pi$
$139$	10.1601	0.861771	0.430886	+	0.902407i	$0.358201\pi$
$140$	0	0
$141$	−13.4483	−1.13255
$142$	0	0
$143$	−16.2754	−1.36102
$144$	0	0
$145$	0	0
$146$	0	0
$147$	24.4837	2.01938
$148$	0	0
$149$	−5.93972	−0.486601	−0.243301	−	0.969951i	$-0.578230\pi$
$149$	−5.93972	−0.486601	−0.243301	+	0.969951i	$0.578230\pi$
$150$	0	0
$151$	15.2978	1.24492	0.622460	−	0.782652i	$-0.286133\pi$
$151$	15.2978	1.24492	0.622460	+	0.782652i	$0.286133\pi$
$152$	0	0
$153$	13.5310	1.09392
$154$	0	0
$155$	0	0
$156$	0	0
$157$	12.7866	1.02048	0.510242	−	0.860031i	$-0.329556\pi$
$157$	12.7866	1.02048	0.510242	+	0.860031i	$0.329556\pi$
$158$	0	0
$159$	−0.508572	−0.0403324
$160$	0	0
$161$	31.9819	2.52053
$162$	0	0
$163$	−11.4734	−0.898664	−0.449332	−	0.893365i	$-0.648338\pi$
$163$	−11.4734	−0.898664	−0.449332	+	0.893365i	$0.648338\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	19.4131	1.50223	0.751115	−	0.660171i	$-0.229516\pi$
$167$	19.4131	1.50223	0.751115	+	0.660171i	$0.229516\pi$
$168$	0	0
$169$	−2.96480	−0.228062
$170$	0	0
$171$	2.08777	0.159656
$172$	0	0
$173$	−9.78662	−0.744063	−0.372031	−	0.928220i	$-0.621339\pi$
$173$	−9.78662	−0.744063	−0.372031	+	0.928220i	$0.621339\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−12.1652	−0.914392
$178$	0	0
$179$	6.82446	0.510084	0.255042	−	0.966930i	$-0.417911\pi$
$179$	6.82446	0.510084	0.255042	+	0.966930i	$0.417911\pi$
$180$	0	0
$181$	−0.137699	−0.0102351	−0.00511755	−	0.999987i	$-0.501629\pi$
$181$	−0.137699	−0.0102351	−0.00511755	+	0.999987i	$0.501629\pi$
$182$	0	0
$183$	32.5957	2.40954
$184$	0	0
$185$	0	0
$186$	0	0
$187$	33.2978	2.43498
$188$	0	0
$189$	−8.69446	−0.632429
$190$	0	0
$191$	−19.3779	−1.40214	−0.701068	−	0.713095i	$-0.747293\pi$
$191$	−19.3779	−1.40214	−0.701068	+	0.713095i	$0.747293\pi$
$192$	0	0
$193$	5.42851	0.390752	0.195376	−	0.980728i	$-0.437407\pi$
$193$	5.42851	0.390752	0.195376	+	0.980728i	$0.437407\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	15.6489	1.11494	0.557470	−	0.830197i	$-0.311772\pi$
$197$	15.6489	1.11494	0.557470	+	0.830197i	$0.311772\pi$
$198$	0	0
$199$	−18.0499	−1.27953	−0.639763	−	0.768572i	$-0.720967\pi$
$199$	−18.0499	−1.27953	−0.639763	+	0.768572i	$0.720967\pi$
$200$	0	0
$201$	−9.27102	−0.653927
$202$	0	0
$203$	3.51628	0.246794
$204$	0	0
$205$	0	0
$206$	0	0
$207$	15.8020	1.09832
$208$	0	0
$209$	5.13770	0.355382
$210$	0	0
$211$	7.50857	0.516911	0.258456	−	0.966023i	$-0.416786\pi$
$211$	7.50857	0.516911	0.258456	+	0.966023i	$0.416786\pi$
$212$	0	0
$213$	−8.62648	−0.591077
$214$	0	0
$215$	0	0
$216$	0	0
$217$	19.0620	1.29401
$218$	0	0
$219$	−10.6161	−0.717372
$220$	0	0
$221$	−20.5310	−1.38106
$222$	0	0
$223$	27.8091	1.86223	0.931116	−	0.364723i	$-0.118836\pi$
$223$	27.8091	1.86223	0.931116	+	0.364723i	$0.118836\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	8.91223	0.591525	0.295763	−	0.955261i	$-0.404426\pi$
$227$	8.91223	0.591525	0.295763	+	0.955261i	$0.404426\pi$
$228$	0	0
$229$	−13.6489	−0.901946	−0.450973	−	0.892538i	$-0.648923\pi$
$229$	−13.6489	−0.901946	−0.450973	+	0.892538i	$0.648923\pi$
$230$	0	0
$231$	48.9674	3.22182
$232$	0	0
$233$	−23.2754	−1.52482	−0.762411	−	0.647093i	$-0.775985\pi$
$233$	−23.2754	−1.52482	−0.762411	+	0.647093i	$0.775985\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−23.9692	−1.55697
$238$	0	0
$239$	−3.72898	−0.241208	−0.120604	−	0.992701i	$-0.538483\pi$
$239$	−3.72898	−0.241208	−0.120604	+	0.992701i	$0.538483\pi$
$240$	0	0
$241$	−1.48878	−0.0959009	−0.0479505	−	0.998850i	$-0.515269\pi$
$241$	−1.48878	−0.0959009	−0.0479505	+	0.998850i	$0.515269\pi$
$242$	0	0
$243$	−18.4234	−1.18186
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−3.16784	−0.201565
$248$	0	0
$249$	−27.2925	−1.72959
$250$	0	0
$251$	8.78662	0.554606	0.277303	−	0.960782i	$-0.410559\pi$
$251$	8.78662	0.554606	0.277303	+	0.960782i	$0.410559\pi$
$252$	0	0
$253$	38.8865	2.44477
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2.74175	−0.171025	−0.0855127	−	0.996337i	$-0.527253\pi$
$257$	−2.74175	−0.171025	−0.0855127	+	0.996337i	$0.527253\pi$
$258$	0	0
$259$	−0.581844	−0.0361540
$260$	0	0
$261$	1.73736	0.107540
$262$	0	0
$263$	2.74704	0.169390	0.0846948	−	0.996407i	$-0.473008\pi$
$263$	2.74704	0.169390	0.0846948	+	0.996407i	$0.473008\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−22.5561	−1.38041
$268$	0	0
$269$	−9.74175	−0.593965	−0.296982	−	0.954883i	$-0.595980\pi$
$269$	−9.74175	−0.593965	−0.296982	+	0.954883i	$0.595980\pi$
$270$	0	0
$271$	−26.3555	−1.60098	−0.800490	−	0.599346i	$-0.795427\pi$
$271$	−26.3555	−1.60098	−0.800490	+	0.599346i	$0.795427\pi$
$272$	0	0
$273$	−30.1927	−1.82734
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0.962158	0.0578104	0.0289052	−	0.999582i	$-0.490798\pi$
$277$	0.962158	0.0578104	0.0289052	+	0.999582i	$0.490798\pi$
$278$	0	0
$279$	9.41839	0.563864
$280$	0	0
$281$	−14.6714	−0.875219	−0.437610	−	0.899165i	$-0.644175\pi$
$281$	−14.6714	−0.875219	−0.437610	+	0.899165i	$0.644175\pi$
$282$	0	0
$283$	−26.1601	−1.55506	−0.777529	−	0.628847i	$-0.783527\pi$
$283$	−26.1601	−1.55506	−0.777529	+	0.628847i	$0.783527\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−49.2221	−2.90549
$288$	0	0
$289$	25.0044	1.47085
$290$	0	0
$291$	−8.88647	−0.520934
$292$	0	0
$293$	22.4657	1.31246	0.656229	−	0.754562i	$-0.272150\pi$
$293$	22.4657	1.31246	0.656229	+	0.754562i	$0.272150\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−10.5715	−0.613420
$298$	0	0
$299$	−23.9769	−1.38662
$300$	0	0
$301$	−10.6111	−0.611612
$302$	0	0
$303$	7.21338	0.414398
$304$	0	0
$305$	0	0
$306$	0	0
$307$	17.2204	0.982821	0.491410	−	0.870928i	$-0.336482\pi$
$307$	17.2204	0.982821	0.491410	+	0.870928i	$0.336482\pi$
$308$	0	0
$309$	−24.1051	−1.37129
$310$	0	0
$311$	7.87439	0.446516	0.223258	−	0.974759i	$-0.428331\pi$
$311$	7.87439	0.446516	0.223258	+	0.974759i	$0.428331\pi$
$312$	0	0
$313$	25.1678	1.42257	0.711285	−	0.702904i	$-0.248113\pi$
$313$	25.1678	1.42257	0.711285	+	0.702904i	$0.248113\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	23.9045	1.34261	0.671306	−	0.741180i	$-0.265734\pi$
$317$	23.9045	1.34261	0.671306	+	0.741180i	$0.265734\pi$
$318$	0	0
$319$	4.27540	0.239376
$320$	0	0
$321$	24.3984	1.36178
$322$	0	0
$323$	6.48108	0.360617
$324$	0	0
$325$	0	0
$326$	0	0
$327$	20.8597	1.15354
$328$	0	0
$329$	−25.1929	−1.38893
$330$	0	0
$331$	−30.7565	−1.69053	−0.845264	−	0.534348i	$-0.820557\pi$
$331$	−30.7565	−1.69053	−0.845264	+	0.534348i	$0.820557\pi$
$332$	0	0
$333$	−0.287484	−0.0157540
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−13.4734	−0.733942	−0.366971	−	0.930232i	$-0.619605\pi$
$337$	−13.4734	−0.733942	−0.366971	+	0.930232i	$0.619605\pi$
$338$	0	0
$339$	−39.8091	−2.16213
$340$	0	0
$341$	23.1773	1.25512
$342$	0	0
$343$	16.2875	0.879441
$344$	0	0
$345$	0	0
$346$	0	0
$347$	28.9468	1.55394	0.776971	−	0.629536i	$-0.216755\pi$
$347$	28.9468	1.55394	0.776971	+	0.629536i	$0.216755\pi$
$348$	0	0
$349$	27.9243	1.49475	0.747377	−	0.664400i	$-0.231313\pi$
$349$	27.9243	1.49475	0.747377	+	0.664400i	$0.231313\pi$
$350$	0	0
$351$	6.51825	0.347918
$352$	0	0
$353$	−28.3099	−1.50678	−0.753392	−	0.657571i	$-0.771584\pi$
$353$	−28.3099	−1.50678	−0.753392	+	0.657571i	$0.771584\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	61.7712	3.26928
$358$	0	0
$359$	−2.60163	−0.137309	−0.0686545	−	0.997640i	$-0.521871\pi$
$359$	−2.60163	−0.137309	−0.0686545	+	0.997640i	$0.521871\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	34.7272	1.82271
$364$	0	0
$365$	0	0
$366$	0	0
$367$	11.6489	0.608069	0.304034	−	0.952661i	$-0.401666\pi$
$367$	11.6489	0.608069	0.304034	+	0.952661i	$0.401666\pi$
$368$	0	0
$369$	−24.3203	−1.26606
$370$	0	0
$371$	−0.952717	−0.0494626
$372$	0	0
$373$	12.8064	0.663091	0.331545	−	0.943439i	$-0.392430\pi$
$373$	12.8064	0.663091	0.331545	+	0.943439i	$0.392430\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.63615	−0.135769
$378$	0	0
$379$	20.9122	1.07419	0.537095	−	0.843522i	$-0.319522\pi$
$379$	20.9122	1.07419	0.537095	+	0.843522i	$0.319522\pi$
$380$	0	0
$381$	−28.8416	−1.47760
$382$	0	0
$383$	10.3511	0.528916	0.264458	−	0.964397i	$-0.414807\pi$
$383$	10.3511	0.528916	0.264458	+	0.964397i	$0.414807\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−5.24285	−0.266509
$388$	0	0
$389$	6.56620	0.332920	0.166460	−	0.986048i	$-0.446766\pi$
$389$	6.56620	0.332920	0.166460	+	0.986048i	$0.446766\pi$
$390$	0	0
$391$	49.0543	2.48078
$392$	0	0
$393$	4.77121	0.240676
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−23.5284	−1.18085	−0.590427	−	0.807091i	$-0.701041\pi$
$397$	−23.5284	−1.18085	−0.590427	+	0.807091i	$0.701041\pi$
$398$	0	0
$399$	9.53101	0.477147
$400$	0	0
$401$	6.22041	0.310633	0.155316	−	0.987865i	$-0.450360\pi$
$401$	6.22041	0.310633	0.155316	+	0.987865i	$0.450360\pi$
$402$	0	0
$403$	−14.2908	−0.711876
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−0.707457	−0.0350674
$408$	0	0
$409$	−34.4905	−1.70545	−0.852723	−	0.522363i	$-0.825051\pi$
$409$	−34.4905	−1.70545	−0.852723	+	0.522363i	$0.825051\pi$
$410$	0	0
$411$	12.0972	0.596711
$412$	0	0
$413$	−22.7893	−1.12139
$414$	0	0
$415$	0	0
$416$	0	0
$417$	22.9173	1.12226
$418$	0	0
$419$	10.6265	0.519138	0.259569	−	0.965725i	$-0.416420\pi$
$419$	10.6265	0.519138	0.259569	+	0.965725i	$0.416420\pi$
$420$	0	0
$421$	−1.98021	−0.0965095	−0.0482548	−	0.998835i	$-0.515366\pi$
$421$	−1.98021	−0.0965095	−0.0482548	+	0.998835i	$0.515366\pi$
$422$	0	0
$423$	−12.4476	−0.605224
$424$	0	0
$425$	0	0
$426$	0	0
$427$	61.0620	2.95500
$428$	0	0
$429$	−36.7109	−1.77242
$430$	0	0
$431$	−15.0774	−0.726254	−0.363127	−	0.931740i	$-0.618291\pi$
$431$	−15.0774	−0.726254	−0.363127	+	0.931740i	$0.618291\pi$
$432$	0	0
$433$	−13.5337	−0.650386	−0.325193	−	0.945648i	$-0.605429\pi$
$433$	−13.5337	−0.650386	−0.325193	+	0.945648i	$0.605429\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.56885	0.362067
$438$	0	0
$439$	−39.1773	−1.86983	−0.934915	−	0.354872i	$-0.884524\pi$
$439$	−39.1773	−1.86983	−0.934915	+	0.354872i	$0.884524\pi$
$440$	0	0
$441$	22.6619	1.07914
$442$	0	0
$443$	19.3132	0.917600	0.458800	−	0.888540i	$-0.348279\pi$
$443$	19.3132	0.917600	0.458800	+	0.888540i	$0.348279\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−13.3977	−0.633689
$448$	0	0
$449$	41.4131	1.95440	0.977202	−	0.212310i	$-0.0680985\pi$
$449$	41.4131	1.95440	0.977202	+	0.212310i	$0.0680985\pi$
$450$	0	0
$451$	−59.8486	−2.81816
$452$	0	0
$453$	34.5059	1.62123
$454$	0	0
$455$	0	0
$456$	0	0
$457$	37.8392	1.77004	0.885021	−	0.465551i	$-0.154144\pi$
$457$	37.8392	1.77004	0.885021	+	0.465551i	$0.154144\pi$
$458$	0	0
$459$	−13.3357	−0.622456
$460$	0	0
$461$	1.58864	0.0739903	0.0369952	−	0.999315i	$-0.488221\pi$
$461$	1.58864	0.0739903	0.0369952	+	0.999315i	$0.488221\pi$
$462$	0	0
$463$	40.3581	1.87560	0.937800	−	0.347175i	$-0.112859\pi$
$463$	40.3581	1.87560	0.937800	+	0.347175i	$0.112859\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−39.5130	−1.82844	−0.914221	−	0.405217i	$-0.867196\pi$
$467$	−39.5130	−1.82844	−0.914221	+	0.405217i	$0.867196\pi$
$468$	0	0
$469$	−17.3676	−0.801959
$470$	0	0
$471$	28.8416	1.32895
$472$	0	0
$473$	−12.9019	−0.593229
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−0.470730	−0.0215532
$478$	0	0
$479$	−7.61107	−0.347759	−0.173879	−	0.984767i	$-0.555630\pi$
$479$	−7.61107	−0.347759	−0.173879	+	0.984767i	$0.555630\pi$
$480$	0	0
$481$	0.436209	0.0198894
$482$	0	0
$483$	72.1388	3.28243
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−20.3907	−0.923989	−0.461995	−	0.886883i	$-0.652866\pi$
$487$	−20.3907	−0.923989	−0.461995	+	0.886883i	$0.652866\pi$
$488$	0	0
$489$	−25.8794	−1.17031
$490$	0	0
$491$	−25.0224	−1.12925	−0.564623	−	0.825349i	$-0.690979\pi$
$491$	−25.0224	−1.12925	−0.564623	+	0.825349i	$0.690979\pi$
$492$	0	0
$493$	5.39331	0.242902
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−16.1601	−0.724881
$498$	0	0
$499$	−22.6111	−1.01221	−0.506105	−	0.862472i	$-0.668915\pi$
$499$	−22.6111	−1.01221	−0.506105	+	0.862472i	$0.668915\pi$
$500$	0	0
$501$	43.7884	1.95632
$502$	0	0
$503$	11.5035	0.512916	0.256458	−	0.966555i	$-0.417444\pi$
$503$	11.5035	0.512916	0.256458	+	0.966555i	$0.417444\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−6.68744	−0.296999
$508$	0	0
$509$	9.11526	0.404027	0.202013	−	0.979383i	$-0.435252\pi$
$509$	9.11526	0.404027	0.202013	+	0.979383i	$0.435252\pi$
$510$	0	0
$511$	−19.8874	−0.879766
$512$	0	0
$513$	−2.05763	−0.0908467
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−30.6318	−1.34718
$518$	0	0
$519$	−22.0748	−0.968975
$520$	0	0
$521$	15.4888	0.678576	0.339288	−	0.940683i	$-0.389814\pi$
$521$	15.4888	0.678576	0.339288	+	0.940683i	$0.389814\pi$
$522$	0	0
$523$	4.34073	0.189807	0.0949035	−	0.995486i	$-0.469746\pi$
$523$	4.34073	0.189807	0.0949035	+	0.995486i	$0.469746\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	29.2376	1.27361
$528$	0	0
$529$	34.2875	1.49076
$530$	0	0
$531$	−11.2600	−0.488642
$532$	0	0
$533$	36.9019	1.59840
$534$	0	0
$535$	0	0
$536$	0	0
$537$	15.3933	0.664270
$538$	0	0
$539$	55.7677	2.40208
$540$	0	0
$541$	19.5491	0.840480	0.420240	−	0.907413i	$-0.361946\pi$
$541$	19.5491	0.840480	0.420240	+	0.907413i	$0.361946\pi$
$542$	0	0
$543$	−0.310596	−0.0133289
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−41.8038	−1.78740	−0.893700	−	0.448665i	$-0.851900\pi$
$547$	−41.8038	−1.78740	−0.893700	+	0.448665i	$0.851900\pi$
$548$	0	0
$549$	30.1703	1.28763
$550$	0	0
$551$	0.832162	0.0354513
$552$	0	0
$553$	−44.9019	−1.90942
$554$	0	0
$555$	0	0
$556$	0	0
$557$	9.76418	0.413722	0.206861	−	0.978370i	$-0.433675\pi$
$557$	9.76418	0.413722	0.206861	+	0.978370i	$0.433675\pi$
$558$	0	0
$559$	7.95513	0.336466
$560$	0	0
$561$	75.1069	3.17102
$562$	0	0
$563$	11.4509	0.482600	0.241300	−	0.970451i	$-0.422426\pi$
$563$	11.4509	0.482600	0.241300	+	0.970451i	$0.422426\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−46.0767	−1.93504
$568$	0	0
$569$	−13.4338	−0.563174	−0.281587	−	0.959536i	$-0.590861\pi$
$569$	−13.4338	−0.563174	−0.281587	+	0.959536i	$0.590861\pi$
$570$	0	0
$571$	−37.6335	−1.57491	−0.787457	−	0.616370i	$-0.788603\pi$
$571$	−37.6335	−1.57491	−0.787457	+	0.616370i	$0.788603\pi$
$572$	0	0
$573$	−43.7090	−1.82597
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−1.56885	−0.0653121	−0.0326560	−	0.999467i	$-0.510397\pi$
$577$	−1.56885	−0.0653121	−0.0326560	+	0.999467i	$0.510397\pi$
$578$	0	0
$579$	12.2446	0.508868
$580$	0	0
$581$	−51.1276	−2.12113
$582$	0	0
$583$	−1.15840	−0.0479759
$584$	0	0
$585$	0	0
$586$	0	0
$587$	25.8693	1.06774	0.533871	−	0.845566i	$-0.320737\pi$
$587$	25.8693	1.06774	0.533871	+	0.845566i	$0.320737\pi$
$588$	0	0
$589$	4.51122	0.185881
$590$	0	0
$591$	35.2978	1.45196
$592$	0	0
$593$	28.9243	1.18778	0.593890	−	0.804547i	$-0.297592\pi$
$593$	28.9243	1.18778	0.593890	+	0.804547i	$0.297592\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−40.7136	−1.66630
$598$	0	0
$599$	−41.3581	−1.68985	−0.844923	−	0.534887i	$-0.820354\pi$
$599$	−41.3581	−1.68985	−0.844923	+	0.534887i	$0.820354\pi$
$600$	0	0
$601$	2.68147	0.109379	0.0546897	−	0.998503i	$-0.482583\pi$
$601$	2.68147	0.109379	0.0546897	+	0.998503i	$0.482583\pi$
$602$	0	0
$603$	−8.58117	−0.349452
$604$	0	0
$605$	0	0
$606$	0	0
$607$	3.31324	0.134480	0.0672401	−	0.997737i	$-0.478581\pi$
$607$	3.31324	0.134480	0.0672401	+	0.997737i	$0.478581\pi$
$608$	0	0
$609$	7.93134	0.321394
$610$	0	0
$611$	18.8871	0.764092
$612$	0	0
$613$	10.0603	0.406331	0.203165	−	0.979144i	$-0.434877\pi$
$613$	10.0603	0.406331	0.203165	+	0.979144i	$0.434877\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	27.6265	1.11220	0.556100	−	0.831115i	$-0.312297\pi$
$617$	27.6265	1.11220	0.556100	+	0.831115i	$0.312297\pi$
$618$	0	0
$619$	−16.6714	−0.670078	−0.335039	−	0.942204i	$-0.608750\pi$
$619$	−16.6714	−0.670078	−0.335039	+	0.942204i	$0.608750\pi$
$620$	0	0
$621$	−15.5739	−0.624959
$622$	0	0
$623$	−42.2547	−1.69290
$624$	0	0
$625$	0	0
$626$	0	0
$627$	11.5886	0.462806
$628$	0	0
$629$	−0.892439	−0.0355839
$630$	0	0
$631$	0.709194	0.0282326	0.0141163	−	0.999900i	$-0.495506\pi$
$631$	0.709194	0.0282326	0.0141163	+	0.999900i	$0.495506\pi$
$632$	0	0
$633$	16.9364	0.673162
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−34.3856	−1.36241
$638$	0	0
$639$	−7.98459	−0.315866
$640$	0	0
$641$	−2.43553	−0.0961978	−0.0480989	−	0.998843i	$-0.515316\pi$
$641$	−2.43553	−0.0961978	−0.0480989	+	0.998843i	$0.515316\pi$
$642$	0	0
$643$	7.70390	0.303812	0.151906	−	0.988395i	$-0.451459\pi$
$643$	7.70390	0.303812	0.151906	+	0.988395i	$0.451459\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−10.0499	−0.395103	−0.197552	−	0.980292i	$-0.563299\pi$
$647$	−10.0499	−0.395103	−0.197552	+	0.980292i	$0.563299\pi$
$648$	0	0
$649$	−27.7092	−1.08768
$650$	0	0
$651$	42.9964	1.68516
$652$	0	0
$653$	7.09283	0.277564	0.138782	−	0.990323i	$-0.455681\pi$
$653$	7.09283	0.277564	0.138782	+	0.990323i	$0.455681\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−9.82620	−0.383356
$658$	0	0
$659$	1.16346	0.0453218	0.0226609	−	0.999743i	$-0.492786\pi$
$659$	1.16346	0.0453218	0.0226609	+	0.999743i	$0.492786\pi$
$660$	0	0
$661$	1.79432	0.0697909	0.0348955	−	0.999391i	$-0.488890\pi$
$661$	1.79432	0.0697909	0.0348955	+	0.999391i	$0.488890\pi$
$662$	0	0
$663$	−46.3099	−1.79853
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.29851	0.243879
$668$	0	0
$669$	62.7263	2.42514
$670$	0	0
$671$	74.2446	2.86618
$672$	0	0
$673$	25.2824	0.974566	0.487283	−	0.873244i	$-0.337988\pi$
$673$	25.2824	0.974566	0.487283	+	0.873244i	$0.337988\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	4.89682	0.188200	0.0941001	−	0.995563i	$-0.470003\pi$
$677$	4.89682	0.188200	0.0941001	+	0.995563i	$0.470003\pi$
$678$	0	0
$679$	−16.6472	−0.638860
$680$	0	0
$681$	20.1025	0.770330
$682$	0	0
$683$	10.1980	0.390215	0.195107	−	0.980782i	$-0.437494\pi$
$683$	10.1980	0.390215	0.195107	+	0.980782i	$0.437494\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−30.7866	−1.17458
$688$	0	0
$689$	0.714253	0.0272109
$690$	0	0
$691$	39.9846	1.52109	0.760543	−	0.649288i	$-0.224933\pi$
$691$	39.9846	1.52109	0.760543	+	0.649288i	$0.224933\pi$
$692$	0	0
$693$	45.3238	1.72171
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−75.4975	−2.85967
$698$	0	0
$699$	−52.5002	−1.98574
$700$	0	0
$701$	4.91729	0.185723	0.0928617	−	0.995679i	$-0.470399\pi$
$701$	4.91729	0.185723	0.0928617	+	0.995679i	$0.470399\pi$
$702$	0	0
$703$	−0.137699	−0.00519342
$704$	0	0
$705$	0	0
$706$	0	0
$707$	13.5130	0.508207
$708$	0	0
$709$	−36.8865	−1.38530	−0.692650	−	0.721274i	$-0.743557\pi$
$709$	−36.8865	−1.38530	−0.692650	+	0.721274i	$0.743557\pi$
$710$	0	0
$711$	−22.1857	−0.832027
$712$	0	0
$713$	34.1447	1.27873
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−8.41113	−0.314119
$718$	0	0
$719$	23.8891	0.890914	0.445457	−	0.895303i	$-0.353041\pi$
$719$	23.8891	0.890914	0.445457	+	0.895303i	$0.353041\pi$
$720$	0	0
$721$	−45.1566	−1.68172
$722$	0	0
$723$	−3.35811	−0.124889
$724$	0	0
$725$	0	0
$726$	0	0
$727$	19.6894	0.730240	0.365120	−	0.930961i	$-0.381028\pi$
$727$	19.6894	0.730240	0.365120	+	0.930961i	$0.381028\pi$
$728$	0	0
$729$	−8.84251	−0.327500
$730$	0	0
$731$	−16.2754	−0.601967
$732$	0	0
$733$	−1.76418	−0.0651615	−0.0325808	−	0.999469i	$-0.510373\pi$
$733$	−1.76418	−0.0651615	−0.0325808	+	0.999469i	$0.510373\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−21.1170	−0.777855
$738$	0	0
$739$	−28.1755	−1.03645	−0.518227	−	0.855243i	$-0.673408\pi$
$739$	−28.1755	−1.03645	−0.518227	+	0.855243i	$0.673408\pi$
$740$	0	0
$741$	−7.14540	−0.262493
$742$	0	0
$743$	−5.42851	−0.199153	−0.0995763	−	0.995030i	$-0.531749\pi$
$743$	−5.42851	−0.199153	−0.0995763	+	0.995030i	$0.531749\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−25.2617	−0.924278
$748$	0	0
$749$	45.7059	1.67006
$750$	0	0
$751$	−25.8640	−0.943792	−0.471896	−	0.881654i	$-0.656430\pi$
$751$	−25.8640	−0.943792	−0.471896	+	0.881654i	$0.656430\pi$
$752$	0	0
$753$	19.8192	0.722251
$754$	0	0
$755$	0	0
$756$	0	0
$757$	3.76947	0.137004	0.0685019	−	0.997651i	$-0.478178\pi$
$757$	3.76947	0.137004	0.0685019	+	0.997651i	$0.478178\pi$
$758$	0	0
$759$	87.7127	3.18377
$760$	0	0
$761$	18.3605	0.665568	0.332784	−	0.943003i	$-0.392012\pi$
$761$	18.3605	0.665568	0.332784	+	0.943003i	$0.392012\pi$
$762$	0	0
$763$	39.0767	1.41467
$764$	0	0
$765$	0	0
$766$	0	0
$767$	17.0851	0.616908
$768$	0	0
$769$	−38.1300	−1.37500	−0.687501	−	0.726183i	$-0.741292\pi$
$769$	−38.1300	−1.37500	−0.687501	+	0.726183i	$0.741292\pi$
$770$	0	0
$771$	−6.18431	−0.222722
$772$	0	0
$773$	29.1575	1.04872	0.524361	−	0.851496i	$-0.324304\pi$
$773$	29.1575	1.04872	0.524361	+	0.851496i	$0.324304\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−1.31241	−0.0470825
$778$	0	0
$779$	−11.6489	−0.417366
$780$	0	0
$781$	−19.6489	−0.703094
$782$	0	0
$783$	−1.71228	−0.0611920
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−47.0165	−1.67596	−0.837978	−	0.545704i	$-0.816262\pi$
$787$	−47.0165	−1.67596	−0.837978	+	0.545704i	$0.816262\pi$
$788$	0	0
$789$	6.19624	0.220592
$790$	0	0
$791$	−74.5750	−2.65158
$792$	0	0
$793$	−45.7782	−1.62563
$794$	0	0
$795$	0	0
$796$	0	0
$797$	42.6359	1.51024	0.755121	−	0.655586i	$-0.227578\pi$
$797$	42.6359	1.51024	0.755121	+	0.655586i	$0.227578\pi$
$798$	0	0
$799$	−38.6412	−1.36703
$800$	0	0
$801$	−20.8777	−0.737678
$802$	0	0
$803$	−24.1808	−0.853323
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−21.9736	−0.773506
$808$	0	0
$809$	49.4630	1.73903	0.869514	−	0.493909i	$-0.164432\pi$
$809$	49.4630	1.73903	0.869514	+	0.493909i	$0.164432\pi$
$810$	0	0
$811$	16.7816	0.589280	0.294640	−	0.955608i	$-0.404800\pi$
$811$	16.7816	0.589280	0.294640	+	0.955608i	$0.404800\pi$
$812$	0	0
$813$	−59.4476	−2.08492
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−2.51122	−0.0878564
$818$	0	0
$819$	−27.9461	−0.976515
$820$	0	0
$821$	11.5337	0.402527	0.201264	−	0.979537i	$-0.435495\pi$
$821$	11.5337	0.402527	0.201264	+	0.979537i	$0.435495\pi$
$822$	0	0
$823$	32.6309	1.13744	0.568720	−	0.822531i	$-0.307439\pi$
$823$	32.6309	1.13744	0.568720	+	0.822531i	$0.307439\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	6.64650	0.231122	0.115561	−	0.993300i	$-0.463133\pi$
$827$	6.64650	0.231122	0.115561	+	0.993300i	$0.463133\pi$
$828$	0	0
$829$	−30.9217	−1.07395	−0.536977	−	0.843597i	$-0.680434\pi$
$829$	−30.9217	−1.07395	−0.536977	+	0.843597i	$0.680434\pi$
$830$	0	0
$831$	2.17025	0.0752852
$832$	0	0
$833$	70.3495	2.43747
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−9.28243	−0.320848
$838$	0	0
$839$	48.7109	1.68169	0.840844	−	0.541277i	$-0.182059\pi$
$839$	48.7109	1.68169	0.840844	+	0.541277i	$0.182059\pi$
$840$	0	0
$841$	−28.3075	−0.976121
$842$	0	0
$843$	−33.0928	−1.13978
$844$	0	0
$845$	0	0
$846$	0	0
$847$	65.0551	2.23532
$848$	0	0
$849$	−59.0070	−2.02512
$850$	0	0
$851$	−1.04222	−0.0357270
$852$	0	0
$853$	−46.1447	−1.57997	−0.789983	−	0.613129i	$-0.789911\pi$
$853$	−46.1447	−1.57997	−0.789983	+	0.613129i	$0.789911\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	8.40607	0.287146	0.143573	−	0.989640i	$-0.454141\pi$
$857$	8.40607	0.287146	0.143573	+	0.989640i	$0.454141\pi$
$858$	0	0
$859$	8.49581	0.289873	0.144937	−	0.989441i	$-0.453702\pi$
$859$	8.49581	0.289873	0.144937	+	0.989441i	$0.453702\pi$
$860$	0	0
$861$	−111.026	−3.78375
$862$	0	0
$863$	3.37352	0.114836	0.0574179	−	0.998350i	$-0.481713\pi$
$863$	3.37352	0.114836	0.0574179	+	0.998350i	$0.481713\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	56.4001	1.91545
$868$	0	0
$869$	−54.5957	−1.85203
$870$	0	0
$871$	13.0205	0.441182
$872$	0	0
$873$	−8.22524	−0.278382
$874$	0	0
$875$	0	0
$876$	0	0
$877$	13.2780	0.448368	0.224184	−	0.974547i	$-0.428028\pi$
$877$	13.2780	0.448368	0.224184	+	0.974547i	$0.428028\pi$
$878$	0	0
$879$	50.6738	1.70918
$880$	0	0
$881$	−26.6489	−0.897825	−0.448912	−	0.893576i	$-0.648188\pi$
$881$	−26.6489	−0.897825	−0.448912	+	0.893576i	$0.648188\pi$
$882$	0	0
$883$	−47.7884	−1.60821	−0.804103	−	0.594490i	$-0.797354\pi$
$883$	−47.7884	−1.60821	−0.804103	+	0.594490i	$0.797354\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−36.3304	−1.21985	−0.609927	−	0.792457i	$-0.708801\pi$
$887$	−36.3304	−1.21985	−0.609927	+	0.792457i	$0.708801\pi$
$888$	0	0
$889$	−54.0295	−1.81209
$890$	0	0
$891$	−56.0242	−1.87688
$892$	0	0
$893$	−5.96216	−0.199516
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−54.0825	−1.80576
$898$	0	0
$899$	3.75406	0.125205
$900$	0	0
$901$	−1.46129	−0.0486826
$902$	0	0
$903$	−23.9344	−0.796488
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−39.3873	−1.30784	−0.653918	−	0.756566i	$-0.726876\pi$
$907$	−39.3873	−1.30784	−0.653918	+	0.756566i	$0.726876\pi$
$908$	0	0
$909$	6.67664	0.221450
$910$	0	0
$911$	20.5561	0.681054	0.340527	−	0.940235i	$-0.389395\pi$
$911$	20.5561	0.681054	0.340527	+	0.940235i	$0.389395\pi$
$912$	0	0
$913$	−62.1654	−2.05738
$914$	0	0
$915$	0	0
$916$	0	0
$917$	8.93799	0.295158
$918$	0	0
$919$	−12.4054	−0.409216	−0.204608	−	0.978844i	$-0.565592\pi$
$919$	−12.4054	−0.409216	−0.204608	+	0.978844i	$0.565592\pi$
$920$	0	0
$921$	38.8425	1.27990
$922$	0	0
$923$	12.1153	0.398779
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−22.3115	−0.732806
$928$	0	0
$929$	−31.3330	−1.02800	−0.514002	−	0.857789i	$-0.671838\pi$
$929$	−31.3330	−1.02800	−0.514002	+	0.857789i	$0.671838\pi$
$930$	0	0
$931$	10.8546	0.355745
$932$	0	0
$933$	17.7615	0.581487
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−7.27299	−0.237598	−0.118799	−	0.992918i	$-0.537904\pi$
$937$	−7.27299	−0.237598	−0.118799	+	0.992918i	$0.537904\pi$
$938$	0	0
$939$	56.7688	1.85258
$940$	0	0
$941$	6.72128	0.219107	0.109554	−	0.993981i	$-0.465058\pi$
$941$	6.72128	0.219107	0.109554	+	0.993981i	$0.465058\pi$
$942$	0	0
$943$	−88.1689	−2.87117
$944$	0	0
$945$	0	0
$946$	0	0
$947$	10.0757	0.327416	0.163708	−	0.986509i	$-0.447655\pi$
$947$	10.0757	0.327416	0.163708	+	0.986509i	$0.447655\pi$
$948$	0	0
$949$	14.9096	0.483986
$950$	0	0
$951$	53.9193	1.74845
$952$	0	0
$953$	8.14473	0.263834	0.131917	−	0.991261i	$-0.457887\pi$
$953$	8.14473	0.263834	0.131917	+	0.991261i	$0.457887\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	9.64363	0.311734
$958$	0	0
$959$	22.6619	0.731791
$960$	0	0
$961$	−10.6489	−0.343513
$962$	0	0
$963$	22.5829	0.727724
$964$	0	0
$965$	0	0
$966$	0	0
$967$	47.1773	1.51712	0.758560	−	0.651604i	$-0.225903\pi$
$967$	47.1773	1.51712	0.758560	+	0.651604i	$0.225903\pi$
$968$	0	0
$969$	14.6188	0.469623
$970$	0	0
$971$	−0.230528	−0.00739801	−0.00369901	−	0.999993i	$-0.501177\pi$
$971$	−0.230528	−0.00739801	−0.00369901	+	0.999993i	$0.501177\pi$
$972$	0	0
$973$	42.9313	1.37632
$974$	0	0
$975$	0	0
$976$	0	0
$977$	5.80905	0.185848	0.0929240	−	0.995673i	$-0.470379\pi$
$977$	5.80905	0.185848	0.0929240	+	0.995673i	$0.470379\pi$
$978$	0	0
$979$	−51.3770	−1.64202
$980$	0	0
$981$	19.3075	0.616441
$982$	0	0
$983$	−22.3511	−0.712889	−0.356444	−	0.934317i	$-0.616011\pi$
$983$	−22.3511	−0.712889	−0.356444	+	0.934317i	$0.616011\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−56.8254	−1.80877
$988$	0	0
$989$	−19.0070	−0.604388
$990$	0	0
$991$	34.9415	1.10995	0.554976	−	0.831866i	$-0.312727\pi$
$991$	34.9415	1.10995	0.554976	+	0.831866i	$0.312727\pi$
$992$	0	0
$993$	−69.3746	−2.20154
$994$	0	0
$995$	0	0
$996$	0	0
$997$	9.35282	0.296207	0.148103	−	0.988972i	$-0.452683\pi$
$997$	9.35282	0.296207	0.148103	+	0.988972i	$0.452683\pi$
$998$	0	0
$999$	0.283334	0.00896430

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7600.2.a.cb.1.2		3
4.3	odd	2		950.2.a.k.1.2	✓	3
5.4	even	2		7600.2.a.bm.1.2		3
12.11	even	2		8550.2.a.co.1.1		3
20.3	even	4		950.2.b.g.799.5		6
20.7	even	4		950.2.b.g.799.2		6
20.19	odd	2		950.2.a.m.1.2	yes	3
60.59	even	2		8550.2.a.cj.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.a.k.1.2	✓	3	4.3	odd	2
950.2.a.m.1.2	yes	3	20.19	odd	2
950.2.b.g.799.2		6	20.7	even	4
950.2.b.g.799.5		6	20.3	even	4
7600.2.a.bm.1.2		3	5.4	even	2
7600.2.a.cb.1.2		3	1.1	even	1	trivial
8550.2.a.cj.1.3		3	60.59	even	2
8550.2.a.co.1.1		3	12.11	even	2

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:	\( N \)	\(=\)	\( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7600.a (trivial)

\(f(q)\)	\(=\)	\(q+2.25561 q^{3} +4.22547 q^{7} +2.08777 q^{9} +O(q^{10})\)	\(q+2.25561 q^{3} +4.22547 q^{7} +2.08777 q^{9} +5.13770 q^{11} -3.16784 q^{13} +6.48108 q^{17} +1.00000 q^{19} +9.53101 q^{21} +7.56885 q^{23} -2.05763 q^{27} +0.832162 q^{29} +4.51122 q^{31} +11.5886 q^{33} -0.137699 q^{37} -7.14540 q^{39} -11.6489 q^{41} -2.51122 q^{43} -5.96216 q^{47} +10.8546 q^{49} +14.6188 q^{51} -0.225470 q^{53} +2.25561 q^{57} -5.39331 q^{59} +14.4509 q^{61} +8.82181 q^{63} -4.11021 q^{67} +17.0724 q^{69} -3.82446 q^{71} -4.70655 q^{73} +21.7092 q^{77} -10.6265 q^{79} -10.9045 q^{81} -12.0999 q^{83} +1.87703 q^{87} -10.0000 q^{89} -13.3856 q^{91} +10.1755 q^{93} -3.93972 q^{97} +10.7263 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} + 2 q^{7} + 13 q^{9}+O(q^{10}) \) 3 * q + 2 * q^3 + 2 * q^7 + 13 * q^9	\( 3 q + 2 q^{3} + 2 q^{7} + 13 q^{9} - 2 q^{11} + 2 q^{13} + 4 q^{17} + 3 q^{19} - 11 q^{21} + 14 q^{23} - 7 q^{27} + 14 q^{29} + 4 q^{31} - 4 q^{33} + 17 q^{37} - 29 q^{39} - 8 q^{41} + 2 q^{43} + 13 q^{47} + 25 q^{49} + 11 q^{51} + 10 q^{53} + 2 q^{57} + 6 q^{59} + 22 q^{61} + 2 q^{63} + 8 q^{69} + 2 q^{71} + 12 q^{73} + 50 q^{77} - 24 q^{79} - q^{81} + 12 q^{83} - 21 q^{87} - 30 q^{89} + 7 q^{91} + 44 q^{93} - 24 q^{99}+O(q^{100}) \) 3 * q + 2 * q^3 + 2 * q^7 + 13 * q^9 - 2 * q^11 + 2 * q^13 + 4 * q^17 + 3 * q^19 - 11 * q^21 + 14 * q^23 - 7 * q^27 + 14 * q^29 + 4 * q^31 - 4 * q^33 + 17 * q^37 - 29 * q^39 - 8 * q^41 + 2 * q^43 + 13 * q^47 + 25 * q^49 + 11 * q^51 + 10 * q^53 + 2 * q^57 + 6 * q^59 + 22 * q^61 + 2 * q^63 + 8 * q^69 + 2 * q^71 + 12 * q^73 + 50 * q^77 - 24 * q^79 - q^81 + 12 * q^83 - 21 * q^87 - 30 * q^89 + 7 * q^91 + 44 * q^93 - 24 * q^99

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