LMFDB - Embedded newform 7600.2.a.ci.1.1

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:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 12x^{4} + 16x^{3} + 33x^{2} - 4x - 7 $ x^6 - 2x^5 - 12x^4 + 16x^3 + 33x^2 - 4*x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3800)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.30105$ 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)

$f(q)$	$=$	$q-3.30105 q^{3} +1.63094 q^{7} +7.89694 q^{9} +O(q^{10})$	$q-3.30105 q^{3} +1.63094 q^{7} +7.89694 q^{9} +4.67473 q^{11} +4.75288 q^{13} +1.41425 q^{17} -1.00000 q^{19} -5.38382 q^{21} -1.96494 q^{23} -16.1650 q^{27} +6.85936 q^{29} -5.08277 q^{31} -15.4315 q^{33} -10.6081 q^{37} -15.6895 q^{39} -4.52536 q^{41} -7.83425 q^{43} -10.9382 q^{47} -4.34003 q^{49} -4.66852 q^{51} +1.55831 q^{53} +3.30105 q^{57} +6.81879 q^{59} -0.109000 q^{61} +12.8794 q^{63} -10.5615 q^{67} +6.48638 q^{69} -12.7070 q^{71} -0.519837 q^{73} +7.62422 q^{77} -0.840485 q^{79} +29.6708 q^{81} +11.9407 q^{83} -22.6431 q^{87} -7.44352 q^{89} +7.75167 q^{91} +16.7785 q^{93} -8.85637 q^{97} +36.9161 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} - 2 q^{7} + 10 q^{9}+O(q^{10}) $ 6 * q - 2 * q^3 - 2 * q^7 + 10 * q^9	$ 6 q - 2 q^{3} - 2 q^{7} + 10 q^{9} - 3 q^{11} - 3 q^{13} - 2 q^{17} - 6 q^{19} + 11 q^{21} - 4 q^{23} - 20 q^{27} + 7 q^{29} - 5 q^{31} - 16 q^{33} - 8 q^{39} + 11 q^{41} + 7 q^{43} - 20 q^{47} - 2 q^{49} - 13 q^{51} - 7 q^{53} + 2 q^{57} + 4 q^{59} + 13 q^{61} + q^{63} - 25 q^{67} + 7 q^{69} - 29 q^{71} - 19 q^{73} + 24 q^{77} - 28 q^{79} + 38 q^{81} + 15 q^{83} - 57 q^{87} - 12 q^{89} - 27 q^{93} - 13 q^{97} - 10 q^{99}+O(q^{100}) $ 6 * q - 2 * q^3 - 2 * q^7 + 10 * q^9 - 3 * q^11 - 3 * q^13 - 2 * q^17 - 6 * q^19 + 11 * q^21 - 4 * q^23 - 20 * q^27 + 7 * q^29 - 5 * q^31 - 16 * q^33 - 8 * q^39 + 11 * q^41 + 7 * q^43 - 20 * q^47 - 2 * q^49 - 13 * q^51 - 7 * q^53 + 2 * q^57 + 4 * q^59 + 13 * q^61 + q^63 - 25 * q^67 + 7 * q^69 - 29 * q^71 - 19 * q^73 + 24 * q^77 - 28 * q^79 + 38 * q^81 + 15 * q^83 - 57 * q^87 - 12 * q^89 - 27 * q^93 - 13 * q^97 - 10 * 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
$3$	−3.30105	−1.90586	−0.952931	−	0.303186i	$-0.901950\pi$
$3$	−3.30105	−1.90586	−0.952931	+	0.303186i	$0.901950\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.63094	0.616438	0.308219	−	0.951315i	$-0.400267\pi$
$7$	1.63094	0.616438	0.308219	+	0.951315i	$0.400267\pi$
$8$	0	0
$9$	7.89694	2.63231
$10$	0	0
$11$	4.67473	1.40949	0.704743	−	0.709463i	$-0.251062\pi$
$11$	4.67473	1.40949	0.704743	+	0.709463i	$0.251062\pi$
$12$	0	0
$13$	4.75288	1.31821	0.659106	−	0.752050i	$-0.270935\pi$
$13$	4.75288	1.31821	0.659106	+	0.752050i	$0.270935\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.41425	0.343007	0.171503	−	0.985184i	$-0.445138\pi$
$17$	1.41425	0.343007	0.171503	+	0.985184i	$0.445138\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−5.38382	−1.17485
$22$	0	0
$23$	−1.96494	−0.409719	−0.204860	−	0.978791i	$-0.565674\pi$
$23$	−1.96494	−0.409719	−0.204860	+	0.978791i	$0.565674\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−16.1650	−3.11096
$28$	0	0
$29$	6.85936	1.27375	0.636876	−	0.770967i	$-0.280227\pi$
$29$	6.85936	1.27375	0.636876	+	0.770967i	$0.280227\pi$
$30$	0	0
$31$	−5.08277	−0.912893	−0.456446	−	0.889751i	$-0.650878\pi$
$31$	−5.08277	−0.912893	−0.456446	+	0.889751i	$0.650878\pi$
$32$	0	0
$33$	−15.4315	−2.68629
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−10.6081	−1.74397	−0.871983	−	0.489537i	$-0.837166\pi$
$37$	−10.6081	−1.74397	−0.871983	+	0.489537i	$0.837166\pi$
$38$	0	0
$39$	−15.6895	−2.51233
$40$	0	0
$41$	−4.52536	−0.706742	−0.353371	−	0.935483i	$-0.614965\pi$
$41$	−4.52536	−0.706742	−0.353371	+	0.935483i	$0.614965\pi$
$42$	0	0
$43$	−7.83425	−1.19471	−0.597356	−	0.801976i	$-0.703782\pi$
$43$	−7.83425	−1.19471	−0.597356	+	0.801976i	$0.703782\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10.9382	−1.59550	−0.797751	−	0.602987i	$-0.793977\pi$
$47$	−10.9382	−1.59550	−0.797751	+	0.602987i	$0.793977\pi$
$48$	0	0
$49$	−4.34003	−0.620004
$50$	0	0
$51$	−4.66852	−0.653723
$52$	0	0
$53$	1.55831	0.214050	0.107025	−	0.994256i	$-0.465867\pi$
$53$	1.55831	0.214050	0.107025	+	0.994256i	$0.465867\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.30105	0.437235
$58$	0	0
$59$	6.81879	0.887731	0.443865	−	0.896093i	$-0.353607\pi$
$59$	6.81879	0.887731	0.443865	+	0.896093i	$0.353607\pi$
$60$	0	0
$61$	−0.109000	−0.0139561	−0.00697803	−	0.999976i	$-0.502221\pi$
$61$	−0.109000	−0.0139561	−0.00697803	+	0.999976i	$0.502221\pi$
$62$	0	0
$63$	12.8794	1.62266
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−10.5615	−1.29030	−0.645148	−	0.764057i	$-0.723204\pi$
$67$	−10.5615	−1.29030	−0.645148	+	0.764057i	$0.723204\pi$
$68$	0	0
$69$	6.48638	0.780868
$70$	0	0
$71$	−12.7070	−1.50804	−0.754021	−	0.656850i	$-0.771888\pi$
$71$	−12.7070	−1.50804	−0.754021	+	0.656850i	$0.771888\pi$
$72$	0	0
$73$	−0.519837	−0.0608423	−0.0304211	−	0.999537i	$-0.509685\pi$
$73$	−0.519837	−0.0608423	−0.0304211	+	0.999537i	$0.509685\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7.62422	0.868861
$78$	0	0
$79$	−0.840485	−0.0945620	−0.0472810	−	0.998882i	$-0.515056\pi$
$79$	−0.840485	−0.0945620	−0.0472810	+	0.998882i	$0.515056\pi$
$80$	0	0
$81$	29.6708	3.29676
$82$	0	0
$83$	11.9407	1.31067	0.655333	−	0.755340i	$-0.272528\pi$
$83$	11.9407	1.31067	0.655333	+	0.755340i	$0.272528\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−22.6431	−2.42759
$88$	0	0
$89$	−7.44352	−0.789011	−0.394506	−	0.918894i	$-0.629084\pi$
$89$	−7.44352	−0.789011	−0.394506	+	0.918894i	$0.629084\pi$
$90$	0	0
$91$	7.75167	0.812596
$92$	0	0
$93$	16.7785	1.73985
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−8.85637	−0.899228	−0.449614	−	0.893223i	$-0.648438\pi$
$97$	−8.85637	−0.899228	−0.449614	+	0.893223i	$0.648438\pi$
$98$	0	0
$99$	36.9161	3.71021
$100$	0	0
$101$	−12.9534	−1.28891	−0.644455	−	0.764642i	$-0.722916\pi$
$101$	−12.9534	−1.28891	−0.644455	+	0.764642i	$0.722916\pi$
$102$	0	0
$103$	−3.16232	−0.311592	−0.155796	−	0.987789i	$-0.549794\pi$
$103$	−3.16232	−0.311592	−0.155796	+	0.987789i	$0.549794\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−18.8458	−1.82189	−0.910946	−	0.412527i	$-0.864646\pi$
$107$	−18.8458	−1.82189	−0.910946	+	0.412527i	$0.864646\pi$
$108$	0	0
$109$	13.4693	1.29013	0.645063	−	0.764130i	$-0.276831\pi$
$109$	13.4693	1.29013	0.645063	+	0.764130i	$0.276831\pi$
$110$	0	0
$111$	35.0180	3.32376
$112$	0	0
$113$	−4.43245	−0.416970	−0.208485	−	0.978026i	$-0.566853\pi$
$113$	−4.43245	−0.416970	−0.208485	+	0.978026i	$0.566853\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	37.5332	3.46994
$118$	0	0
$119$	2.30656	0.211442
$120$	0	0
$121$	10.8531	0.986649
$122$	0	0
$123$	14.9384	1.34695
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−4.48637	−0.398101	−0.199051	−	0.979989i	$-0.563786\pi$
$127$	−4.48637	−0.398101	−0.199051	+	0.979989i	$0.563786\pi$
$128$	0	0
$129$	25.8613	2.27696
$130$	0	0
$131$	11.1275	0.972211	0.486105	−	0.873900i	$-0.338417\pi$
$131$	11.1275	0.972211	0.486105	+	0.873900i	$0.338417\pi$
$132$	0	0
$133$	−1.63094	−0.141421
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−19.6851	−1.68181	−0.840904	−	0.541184i	$-0.817976\pi$
$137$	−19.6851	−1.68181	−0.840904	+	0.541184i	$0.817976\pi$
$138$	0	0
$139$	−9.15699	−0.776686	−0.388343	−	0.921515i	$-0.626952\pi$
$139$	−9.15699	−0.776686	−0.388343	+	0.921515i	$0.626952\pi$
$140$	0	0
$141$	36.1076	3.04081
$142$	0	0
$143$	22.2185	1.85800
$144$	0	0
$145$	0	0
$146$	0	0
$147$	14.3267	1.18164
$148$	0	0
$149$	2.68165	0.219689	0.109845	−	0.993949i	$-0.464965\pi$
$149$	2.68165	0.219689	0.109845	+	0.993949i	$0.464965\pi$
$150$	0	0
$151$	−18.6183	−1.51513	−0.757567	−	0.652758i	$-0.773612\pi$
$151$	−18.6183	−1.51513	−0.757567	+	0.652758i	$0.773612\pi$
$152$	0	0
$153$	11.1683	0.902900
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−6.33933	−0.505934	−0.252967	−	0.967475i	$-0.581406\pi$
$157$	−6.33933	−0.505934	−0.252967	+	0.967475i	$0.581406\pi$
$158$	0	0
$159$	−5.14406	−0.407950
$160$	0	0
$161$	−3.20471	−0.252566
$162$	0	0
$163$	−13.4585	−1.05415	−0.527074	−	0.849819i	$-0.676711\pi$
$163$	−13.4585	−1.05415	−0.527074	+	0.849819i	$0.676711\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	13.1325	1.01622	0.508112	−	0.861291i	$-0.330344\pi$
$167$	13.1325	1.01622	0.508112	+	0.861291i	$0.330344\pi$
$168$	0	0
$169$	9.58987	0.737682
$170$	0	0
$171$	−7.89694	−0.603894
$172$	0	0
$173$	0.857255	0.0651759	0.0325879	−	0.999469i	$-0.489625\pi$
$173$	0.857255	0.0651759	0.0325879	+	0.999469i	$0.489625\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−22.5092	−1.69189
$178$	0	0
$179$	−8.93269	−0.667661	−0.333830	−	0.942633i	$-0.608341\pi$
$179$	−8.93269	−0.667661	−0.333830	+	0.942633i	$0.608341\pi$
$180$	0	0
$181$	−6.95157	−0.516706	−0.258353	−	0.966051i	$-0.583180\pi$
$181$	−6.95157	−0.516706	−0.258353	+	0.966051i	$0.583180\pi$
$182$	0	0
$183$	0.359816	0.0265983
$184$	0	0
$185$	0	0
$186$	0	0
$187$	6.61125	0.483463
$188$	0	0
$189$	−26.3642	−1.91772
$190$	0	0
$191$	−9.36170	−0.677389	−0.338695	−	0.940896i	$-0.609985\pi$
$191$	−9.36170	−0.677389	−0.338695	+	0.940896i	$0.609985\pi$
$192$	0	0
$193$	−26.7785	−1.92756	−0.963779	−	0.266703i	$-0.914066\pi$
$193$	−26.7785	−1.92756	−0.963779	+	0.266703i	$0.914066\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.92419	0.279587	0.139793	−	0.990181i	$-0.455356\pi$
$197$	3.92419	0.279587	0.139793	+	0.990181i	$0.455356\pi$
$198$	0	0
$199$	7.52186	0.533211	0.266605	−	0.963806i	$-0.414098\pi$
$199$	7.52186	0.533211	0.266605	+	0.963806i	$0.414098\pi$
$200$	0	0
$201$	34.8642	2.45913
$202$	0	0
$203$	11.1872	0.785189
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−15.5170	−1.07851
$208$	0	0
$209$	−4.67473	−0.323358
$210$	0	0
$211$	−9.97579	−0.686761	−0.343381	−	0.939196i	$-0.611572\pi$
$211$	−9.97579	−0.686761	−0.343381	+	0.939196i	$0.611572\pi$
$212$	0	0
$213$	41.9464	2.87412
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−8.28970	−0.562742
$218$	0	0
$219$	1.71601	0.115957
$220$	0	0
$221$	6.72177	0.452155
$222$	0	0
$223$	21.3953	1.43273	0.716367	−	0.697724i	$-0.245804\pi$
$223$	21.3953	1.43273	0.716367	+	0.697724i	$0.245804\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	20.2964	1.34712	0.673560	−	0.739132i	$-0.264764\pi$
$227$	20.2964	1.34712	0.673560	+	0.739132i	$0.264764\pi$
$228$	0	0
$229$	19.9525	1.31850	0.659249	−	0.751925i	$-0.270874\pi$
$229$	19.9525	1.31850	0.659249	+	0.751925i	$0.270874\pi$
$230$	0	0
$231$	−25.1679	−1.65593
$232$	0	0
$233$	13.2102	0.865431	0.432715	−	0.901531i	$-0.357555\pi$
$233$	13.2102	0.865431	0.432715	+	0.901531i	$0.357555\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	2.77448	0.180222
$238$	0	0
$239$	−26.0537	−1.68527	−0.842636	−	0.538484i	$-0.818997\pi$
$239$	−26.0537	−1.68527	−0.842636	+	0.538484i	$0.818997\pi$
$240$	0	0
$241$	2.12098	0.136624	0.0683122	−	0.997664i	$-0.478239\pi$
$241$	2.12098	0.136624	0.0683122	+	0.997664i	$0.478239\pi$
$242$	0	0
$243$	−49.4497	−3.17220
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−4.75288	−0.302419
$248$	0	0
$249$	−39.4170	−2.49795
$250$	0	0
$251$	14.8911	0.939921	0.469960	−	0.882688i	$-0.344268\pi$
$251$	14.8911	0.939921	0.469960	+	0.882688i	$0.344268\pi$
$252$	0	0
$253$	−9.18559	−0.577493
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7.91006	−0.493416	−0.246708	−	0.969090i	$-0.579349\pi$
$257$	−7.91006	−0.493416	−0.246708	+	0.969090i	$0.579349\pi$
$258$	0	0
$259$	−17.3012	−1.07505
$260$	0	0
$261$	54.1679	3.35291
$262$	0	0
$263$	17.1828	1.05954	0.529768	−	0.848143i	$-0.322279\pi$
$263$	17.1828	1.05954	0.529768	+	0.848143i	$0.322279\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	24.5714	1.50375
$268$	0	0
$269$	−12.9237	−0.787972	−0.393986	−	0.919116i	$-0.628904\pi$
$269$	−12.9237	−0.787972	−0.393986	+	0.919116i	$0.628904\pi$
$270$	0	0
$271$	20.9897	1.27503	0.637515	−	0.770438i	$-0.279962\pi$
$271$	20.9897	1.27503	0.637515	+	0.770438i	$0.279962\pi$
$272$	0	0
$273$	−25.5887	−1.54870
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−6.27361	−0.376945	−0.188472	−	0.982078i	$-0.560354\pi$
$277$	−6.27361	−0.376945	−0.188472	+	0.982078i	$0.560354\pi$
$278$	0	0
$279$	−40.1383	−2.40302
$280$	0	0
$281$	−28.7635	−1.71588	−0.857942	−	0.513747i	$-0.828257\pi$
$281$	−28.7635	−1.71588	−0.857942	+	0.513747i	$0.828257\pi$
$282$	0	0
$283$	28.0145	1.66529	0.832644	−	0.553808i	$-0.186826\pi$
$283$	28.0145	1.66529	0.832644	+	0.553808i	$0.186826\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−7.38059	−0.435663
$288$	0	0
$289$	−14.9999	−0.882346
$290$	0	0
$291$	29.2353	1.71380
$292$	0	0
$293$	−22.5521	−1.31751	−0.658753	−	0.752359i	$-0.728916\pi$
$293$	−22.5521	−1.31751	−0.658753	+	0.752359i	$0.728916\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−75.5673	−4.38486
$298$	0	0
$299$	−9.33914	−0.540097
$300$	0	0
$301$	−12.7772	−0.736466
$302$	0	0
$303$	42.7598	2.45649
$304$	0	0
$305$	0	0
$306$	0	0
$307$	14.5977	0.833137	0.416568	−	0.909104i	$-0.363233\pi$
$307$	14.5977	0.833137	0.416568	+	0.909104i	$0.363233\pi$
$308$	0	0
$309$	10.4390	0.593852
$310$	0	0
$311$	−27.1608	−1.54015	−0.770074	−	0.637955i	$-0.779781\pi$
$311$	−27.1608	−1.54015	−0.770074	+	0.637955i	$0.779781\pi$
$312$	0	0
$313$	28.3854	1.60444	0.802219	−	0.597030i	$-0.203653\pi$
$313$	28.3854	1.60444	0.802219	+	0.597030i	$0.203653\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−20.9910	−1.17897	−0.589485	−	0.807779i	$-0.700669\pi$
$317$	−20.9910	−1.17897	−0.589485	+	0.807779i	$0.700669\pi$
$318$	0	0
$319$	32.0657	1.79533
$320$	0	0
$321$	62.2109	3.47227
$322$	0	0
$323$	−1.41425	−0.0786911
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−44.4629	−2.45880
$328$	0	0
$329$	−17.8396	−0.983528
$330$	0	0
$331$	1.81631	0.0998332	0.0499166	−	0.998753i	$-0.484104\pi$
$331$	1.81631	0.0998332	0.0499166	+	0.998753i	$0.484104\pi$
$332$	0	0
$333$	−83.7717	−4.59066
$334$	0	0
$335$	0	0
$336$	0	0
$337$	25.6967	1.39979	0.699895	−	0.714245i	$-0.253230\pi$
$337$	25.6967	1.39979	0.699895	+	0.714245i	$0.253230\pi$
$338$	0	0
$339$	14.6317	0.794687
$340$	0	0
$341$	−23.7606	−1.28671
$342$	0	0
$343$	−18.4949	−0.998632
$344$	0	0
$345$	0	0
$346$	0	0
$347$	11.2851	0.605814	0.302907	−	0.953020i	$-0.402043\pi$
$347$	11.2851	0.605814	0.302907	+	0.953020i	$0.402043\pi$
$348$	0	0
$349$	3.62702	0.194150	0.0970750	−	0.995277i	$-0.469051\pi$
$349$	3.62702	0.194150	0.0970750	+	0.995277i	$0.469051\pi$
$350$	0	0
$351$	−76.8305	−4.10091
$352$	0	0
$353$	−31.6766	−1.68597	−0.842987	−	0.537935i	$-0.819205\pi$
$353$	−31.6766	−1.68597	−0.842987	+	0.537935i	$0.819205\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−7.61408	−0.402980
$358$	0	0
$359$	23.4772	1.23908	0.619539	−	0.784966i	$-0.287320\pi$
$359$	23.4772	1.23908	0.619539	+	0.784966i	$0.287320\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−35.8268	−1.88042
$364$	0	0
$365$	0	0
$366$	0	0
$367$	32.2584	1.68387	0.841937	−	0.539576i	$-0.181416\pi$
$367$	32.2584	1.68387	0.841937	+	0.539576i	$0.181416\pi$
$368$	0	0
$369$	−35.7365	−1.86037
$370$	0	0
$371$	2.54151	0.131949
$372$	0	0
$373$	−21.5626	−1.11647	−0.558235	−	0.829683i	$-0.688521\pi$
$373$	−21.5626	−1.11647	−0.558235	+	0.829683i	$0.688521\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	32.6017	1.67907
$378$	0	0
$379$	−12.3881	−0.636335	−0.318168	−	0.948035i	$-0.603067\pi$
$379$	−12.3881	−0.636335	−0.318168	+	0.948035i	$0.603067\pi$
$380$	0	0
$381$	14.8097	0.758726
$382$	0	0
$383$	−6.64117	−0.339348	−0.169674	−	0.985500i	$-0.554271\pi$
$383$	−6.64117	−0.339348	−0.169674	+	0.985500i	$0.554271\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−61.8666	−3.14485
$388$	0	0
$389$	−10.0607	−0.510095	−0.255048	−	0.966928i	$-0.582091\pi$
$389$	−10.0607	−0.510095	−0.255048	+	0.966928i	$0.582091\pi$
$390$	0	0
$391$	−2.77893	−0.140536
$392$	0	0
$393$	−36.7323	−1.85290
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−6.64223	−0.333364	−0.166682	−	0.986011i	$-0.553305\pi$
$397$	−6.64223	−0.333364	−0.166682	+	0.986011i	$0.553305\pi$
$398$	0	0
$399$	5.38382	0.269528
$400$	0	0
$401$	−16.4296	−0.820456	−0.410228	−	0.911983i	$-0.634551\pi$
$401$	−16.4296	−0.820456	−0.410228	+	0.911983i	$0.634551\pi$
$402$	0	0
$403$	−24.1578	−1.20339
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−49.5902	−2.45809
$408$	0	0
$409$	32.6210	1.61300	0.806501	−	0.591232i	$-0.201358\pi$
$409$	32.6210	1.61300	0.806501	+	0.591232i	$0.201358\pi$
$410$	0	0
$411$	64.9814	3.20530
$412$	0	0
$413$	11.1211	0.547231
$414$	0	0
$415$	0	0
$416$	0	0
$417$	30.2277	1.48026
$418$	0	0
$419$	17.1286	0.836786	0.418393	−	0.908266i	$-0.362594\pi$
$419$	17.1286	0.836786	0.418393	+	0.908266i	$0.362594\pi$
$420$	0	0
$421$	18.5724	0.905162	0.452581	−	0.891723i	$-0.350503\pi$
$421$	18.5724	0.905162	0.452581	+	0.891723i	$0.350503\pi$
$422$	0	0
$423$	−86.3783	−4.19986
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−0.177773	−0.00860305
$428$	0	0
$429$	−73.3442	−3.54109
$430$	0	0
$431$	−2.21969	−0.106919	−0.0534593	−	0.998570i	$-0.517025\pi$
$431$	−2.21969	−0.106919	−0.0534593	+	0.998570i	$0.517025\pi$
$432$	0	0
$433$	−21.0400	−1.01112	−0.505560	−	0.862791i	$-0.668714\pi$
$433$	−21.0400	−1.01112	−0.505560	+	0.862791i	$0.668714\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.96494	0.0939960
$438$	0	0
$439$	−3.27039	−0.156087	−0.0780435	−	0.996950i	$-0.524867\pi$
$439$	−3.27039	−0.156087	−0.0780435	+	0.996950i	$0.524867\pi$
$440$	0	0
$441$	−34.2729	−1.63204
$442$	0	0
$443$	−27.6697	−1.31463	−0.657314	−	0.753617i	$-0.728307\pi$
$443$	−27.6697	−1.31463	−0.657314	+	0.753617i	$0.728307\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−8.85226	−0.418697
$448$	0	0
$449$	7.69551	0.363174	0.181587	−	0.983375i	$-0.441877\pi$
$449$	7.69551	0.363174	0.181587	+	0.983375i	$0.441877\pi$
$450$	0	0
$451$	−21.1548	−0.996143
$452$	0	0
$453$	61.4598	2.88764
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−8.06460	−0.377246	−0.188623	−	0.982050i	$-0.560402\pi$
$457$	−8.06460	−0.377246	−0.188623	+	0.982050i	$0.560402\pi$
$458$	0	0
$459$	−22.8614	−1.06708
$460$	0	0
$461$	−3.10933	−0.144816	−0.0724079	−	0.997375i	$-0.523068\pi$
$461$	−3.10933	−0.144816	−0.0724079	+	0.997375i	$0.523068\pi$
$462$	0	0
$463$	12.4068	0.576592	0.288296	−	0.957541i	$-0.406911\pi$
$463$	12.4068	0.576592	0.288296	+	0.957541i	$0.406911\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.63093	−0.0754706	−0.0377353	−	0.999288i	$-0.512014\pi$
$467$	−1.63093	−0.0754706	−0.0377353	+	0.999288i	$0.512014\pi$
$468$	0	0
$469$	−17.2252	−0.795388
$470$	0	0
$471$	20.9265	0.964240
$472$	0	0
$473$	−36.6230	−1.68393
$474$	0	0
$475$	0	0
$476$	0	0
$477$	12.3059	0.563447
$478$	0	0
$479$	−15.4392	−0.705433	−0.352716	−	0.935730i	$-0.614742\pi$
$479$	−15.4392	−0.705433	−0.352716	+	0.935730i	$0.614742\pi$
$480$	0	0
$481$	−50.4192	−2.29892
$482$	0	0
$483$	10.5789	0.481357
$484$	0	0
$485$	0	0
$486$	0	0
$487$	38.2792	1.73460	0.867298	−	0.497789i	$-0.165855\pi$
$487$	38.2792	1.73460	0.867298	+	0.497789i	$0.165855\pi$
$488$	0	0
$489$	44.4271	2.00906
$490$	0	0
$491$	19.3511	0.873301	0.436651	−	0.899631i	$-0.356165\pi$
$491$	19.3511	0.873301	0.436651	+	0.899631i	$0.356165\pi$
$492$	0	0
$493$	9.70086	0.436905
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−20.7244	−0.929615
$498$	0	0
$499$	−18.1775	−0.813738	−0.406869	−	0.913487i	$-0.633379\pi$
$499$	−18.1775	−0.813738	−0.406869	+	0.913487i	$0.633379\pi$
$500$	0	0
$501$	−43.3511	−1.93678
$502$	0	0
$503$	−19.4368	−0.866643	−0.433321	−	0.901240i	$-0.642658\pi$
$503$	−19.4368	−0.866643	−0.433321	+	0.901240i	$0.642658\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−31.6566	−1.40592
$508$	0	0
$509$	−0.155404	−0.00688816	−0.00344408	−	0.999994i	$-0.501096\pi$
$509$	−0.155404	−0.00688816	−0.00344408	+	0.999994i	$0.501096\pi$
$510$	0	0
$511$	−0.847823	−0.0375055
$512$	0	0
$513$	16.1650	0.713704
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−51.1332	−2.24884
$518$	0	0
$519$	−2.82984	−0.124216
$520$	0	0
$521$	−12.7626	−0.559140	−0.279570	−	0.960125i	$-0.590192\pi$
$521$	−12.7626	−0.559140	−0.279570	+	0.960125i	$0.590192\pi$
$522$	0	0
$523$	9.49811	0.415323	0.207662	−	0.978201i	$-0.433415\pi$
$523$	9.49811	0.415323	0.207662	+	0.978201i	$0.433415\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−7.18832	−0.313128
$528$	0	0
$529$	−19.1390	−0.832130
$530$	0	0
$531$	53.8476	2.33679
$532$	0	0
$533$	−21.5085	−0.931636
$534$	0	0
$535$	0	0
$536$	0	0
$537$	29.4873	1.27247
$538$	0	0
$539$	−20.2885	−0.873887
$540$	0	0
$541$	−0.840655	−0.0361426	−0.0180713	−	0.999837i	$-0.505753\pi$
$541$	−0.840655	−0.0361426	−0.0180713	+	0.999837i	$0.505753\pi$
$542$	0	0
$543$	22.9475	0.984771
$544$	0	0
$545$	0	0
$546$	0	0
$547$	30.9252	1.32227	0.661134	−	0.750268i	$-0.270075\pi$
$547$	30.9252	1.32227	0.661134	+	0.750268i	$0.270075\pi$
$548$	0	0
$549$	−0.860769	−0.0367367
$550$	0	0
$551$	−6.85936	−0.292219
$552$	0	0
$553$	−1.37078	−0.0582916
$554$	0	0
$555$	0	0
$556$	0	0
$557$	25.4896	1.08003	0.540014	−	0.841656i	$-0.318419\pi$
$557$	25.4896	1.08003	0.540014	+	0.841656i	$0.318419\pi$
$558$	0	0
$559$	−37.2352	−1.57488
$560$	0	0
$561$	−21.8241	−0.921414
$562$	0	0
$563$	10.2674	0.432718	0.216359	−	0.976314i	$-0.430582\pi$
$563$	10.2674	0.432718	0.216359	+	0.976314i	$0.430582\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	48.3913	2.03225
$568$	0	0
$569$	33.6002	1.40859	0.704296	−	0.709906i	$-0.251263\pi$
$569$	33.6002	1.40859	0.704296	+	0.709906i	$0.251263\pi$
$570$	0	0
$571$	−8.89143	−0.372095	−0.186047	−	0.982541i	$-0.559568\pi$
$571$	−8.89143	−0.372095	−0.186047	+	0.982541i	$0.559568\pi$
$572$	0	0
$573$	30.9035	1.29101
$574$	0	0
$575$	0	0
$576$	0	0
$577$	31.3264	1.30413	0.652067	−	0.758161i	$-0.273902\pi$
$577$	31.3264	1.30413	0.652067	+	0.758161i	$0.273902\pi$
$578$	0	0
$579$	88.3971	3.67366
$580$	0	0
$581$	19.4746	0.807944
$582$	0	0
$583$	7.28468	0.301701
$584$	0	0
$585$	0	0
$586$	0	0
$587$	38.7544	1.59957	0.799784	−	0.600289i	$-0.204948\pi$
$587$	38.7544	1.59957	0.799784	+	0.600289i	$0.204948\pi$
$588$	0	0
$589$	5.08277	0.209432
$590$	0	0
$591$	−12.9539	−0.532854
$592$	0	0
$593$	43.7426	1.79629	0.898146	−	0.439696i	$-0.144914\pi$
$593$	43.7426	1.79629	0.898146	+	0.439696i	$0.144914\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−24.8300	−1.01623
$598$	0	0
$599$	−31.2288	−1.27597	−0.637987	−	0.770047i	$-0.720233\pi$
$599$	−31.2288	−1.27597	−0.637987	+	0.770047i	$0.720233\pi$
$600$	0	0
$601$	−45.1869	−1.84321	−0.921606	−	0.388127i	$-0.873122\pi$
$601$	−45.1869	−1.84321	−0.921606	+	0.388127i	$0.873122\pi$
$602$	0	0
$603$	−83.4038	−3.39646
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−31.3958	−1.27432	−0.637159	−	0.770733i	$-0.719890\pi$
$607$	−31.3958	−1.27432	−0.637159	+	0.770733i	$0.719890\pi$
$608$	0	0
$609$	−36.9296	−1.49646
$610$	0	0
$611$	−51.9880	−2.10321
$612$	0	0
$613$	20.7340	0.837439	0.418719	−	0.908116i	$-0.362479\pi$
$613$	20.7340	0.837439	0.418719	+	0.908116i	$0.362479\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	10.4584	0.421038	0.210519	−	0.977590i	$-0.432485\pi$
$617$	10.4584	0.421038	0.210519	+	0.977590i	$0.432485\pi$
$618$	0	0
$619$	−13.9795	−0.561883	−0.280942	−	0.959725i	$-0.590647\pi$
$619$	−13.9795	−0.561883	−0.280942	+	0.959725i	$0.590647\pi$
$620$	0	0
$621$	31.7634	1.27462
$622$	0	0
$623$	−12.1399	−0.486377
$624$	0	0
$625$	0	0
$626$	0	0
$627$	15.4315	0.616276
$628$	0	0
$629$	−15.0026	−0.598192
$630$	0	0
$631$	−13.2899	−0.529061	−0.264531	−	0.964377i	$-0.585217\pi$
$631$	−13.2899	−0.529061	−0.264531	+	0.964377i	$0.585217\pi$
$632$	0	0
$633$	32.9306	1.30887
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−20.6276	−0.817297
$638$	0	0
$639$	−100.346	−3.96964
$640$	0	0
$641$	−14.8444	−0.586319	−0.293160	−	0.956064i	$-0.594707\pi$
$641$	−14.8444	−0.586319	−0.293160	+	0.956064i	$0.594707\pi$
$642$	0	0
$643$	19.8963	0.784634	0.392317	−	0.919830i	$-0.371674\pi$
$643$	19.8963	0.784634	0.392317	+	0.919830i	$0.371674\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	9.90793	0.389521	0.194760	−	0.980851i	$-0.437607\pi$
$647$	9.90793	0.389521	0.194760	+	0.980851i	$0.437607\pi$
$648$	0	0
$649$	31.8760	1.25124
$650$	0	0
$651$	27.3647	1.07251
$652$	0	0
$653$	−5.31852	−0.208130	−0.104065	−	0.994571i	$-0.533185\pi$
$653$	−5.31852	−0.208130	−0.104065	+	0.994571i	$0.533185\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−4.10512	−0.160156
$658$	0	0
$659$	6.86569	0.267449	0.133725	−	0.991019i	$-0.457306\pi$
$659$	6.86569	0.267449	0.133725	+	0.991019i	$0.457306\pi$
$660$	0	0
$661$	48.7637	1.89669	0.948344	−	0.317243i	$-0.102757\pi$
$661$	48.7637	1.89669	0.948344	+	0.317243i	$0.102757\pi$
$662$	0	0
$663$	−22.1889	−0.861746
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−13.4783	−0.521880
$668$	0	0
$669$	−70.6269	−2.73059
$670$	0	0
$671$	−0.509548	−0.0196709
$672$	0	0
$673$	25.8294	0.995651	0.497825	−	0.867277i	$-0.334132\pi$
$673$	25.8294	0.995651	0.497825	+	0.867277i	$0.334132\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−3.58131	−0.137641	−0.0688205	−	0.997629i	$-0.521924\pi$
$677$	−3.58131	−0.137641	−0.0688205	+	0.997629i	$0.521924\pi$
$678$	0	0
$679$	−14.4442	−0.554318
$680$	0	0
$681$	−66.9995	−2.56743
$682$	0	0
$683$	36.1449	1.38304	0.691522	−	0.722355i	$-0.256940\pi$
$683$	36.1449	1.38304	0.691522	+	0.722355i	$0.256940\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−65.8642	−2.51288
$688$	0	0
$689$	7.40645	0.282163
$690$	0	0
$691$	24.0392	0.914493	0.457246	−	0.889340i	$-0.348836\pi$
$691$	24.0392	0.914493	0.457246	+	0.889340i	$0.348836\pi$
$692$	0	0
$693$	60.2080	2.28711
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−6.40000	−0.242417
$698$	0	0
$699$	−43.6076	−1.64939
$700$	0	0
$701$	−14.5851	−0.550872	−0.275436	−	0.961319i	$-0.588822\pi$
$701$	−14.5851	−0.550872	−0.275436	+	0.961319i	$0.588822\pi$
$702$	0	0
$703$	10.6081	0.400093
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−21.1262	−0.794533
$708$	0	0
$709$	46.5636	1.74873	0.874366	−	0.485268i	$-0.161278\pi$
$709$	46.5636	1.74873	0.874366	+	0.485268i	$0.161278\pi$
$710$	0	0
$711$	−6.63726	−0.248917
$712$	0	0
$713$	9.98736	0.374030
$714$	0	0
$715$	0	0
$716$	0	0
$717$	86.0045	3.21190
$718$	0	0
$719$	−4.36765	−0.162886	−0.0814429	−	0.996678i	$-0.525953\pi$
$719$	−4.36765	−0.162886	−0.0814429	+	0.996678i	$0.525953\pi$
$720$	0	0
$721$	−5.15756	−0.192077
$722$	0	0
$723$	−7.00147	−0.260387
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−49.3720	−1.83111	−0.915553	−	0.402198i	$-0.868246\pi$
$727$	−49.3720	−1.83111	−0.915553	+	0.402198i	$0.868246\pi$
$728$	0	0
$729$	74.2236	2.74902
$730$	0	0
$731$	−11.0796	−0.409794
$732$	0	0
$733$	28.3530	1.04724	0.523621	−	0.851952i	$-0.324581\pi$
$733$	28.3530	1.04724	0.523621	+	0.851952i	$0.324581\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−49.3724	−1.81865
$738$	0	0
$739$	−25.4558	−0.936408	−0.468204	−	0.883620i	$-0.655099\pi$
$739$	−25.4558	−0.936408	−0.468204	+	0.883620i	$0.655099\pi$
$740$	0	0
$741$	15.6895	0.576368
$742$	0	0
$743$	−7.99036	−0.293138	−0.146569	−	0.989200i	$-0.546823\pi$
$743$	−7.99036	−0.293138	−0.146569	+	0.989200i	$0.546823\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	94.2952	3.45008
$748$	0	0
$749$	−30.7364	−1.12308
$750$	0	0
$751$	50.6911	1.84974	0.924872	−	0.380279i	$-0.124172\pi$
$751$	50.6911	1.84974	0.924872	+	0.380279i	$0.124172\pi$
$752$	0	0
$753$	−49.1564	−1.79136
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0.449092	0.0163225	0.00816127	−	0.999967i	$-0.497402\pi$
$757$	0.449092	0.0163225	0.00816127	+	0.999967i	$0.497402\pi$
$758$	0	0
$759$	30.3221	1.10062
$760$	0	0
$761$	32.9995	1.19623	0.598115	−	0.801410i	$-0.295917\pi$
$761$	32.9995	1.19623	0.598115	+	0.801410i	$0.295917\pi$
$762$	0	0
$763$	21.9676	0.795282
$764$	0	0
$765$	0	0
$766$	0	0
$767$	32.4089	1.17022
$768$	0	0
$769$	7.87810	0.284091	0.142046	−	0.989860i	$-0.454632\pi$
$769$	7.87810	0.284091	0.142046	+	0.989860i	$0.454632\pi$
$770$	0	0
$771$	26.1115	0.940383
$772$	0	0
$773$	−17.7424	−0.638149	−0.319074	−	0.947730i	$-0.603372\pi$
$773$	−17.7424	−0.638149	−0.319074	+	0.947730i	$0.603372\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	57.1123	2.04889
$778$	0	0
$779$	4.52536	0.162138
$780$	0	0
$781$	−59.4018	−2.12556
$782$	0	0
$783$	−110.882	−3.96259
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−47.7349	−1.70156	−0.850782	−	0.525519i	$-0.823871\pi$
$787$	−47.7349	−1.70156	−0.850782	+	0.525519i	$0.823871\pi$
$788$	0	0
$789$	−56.7213	−2.01933
$790$	0	0
$791$	−7.22907	−0.257036
$792$	0	0
$793$	−0.518066	−0.0183971
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−17.5233	−0.620706	−0.310353	−	0.950621i	$-0.600447\pi$
$797$	−17.5233	−0.620706	−0.310353	+	0.950621i	$0.600447\pi$
$798$	0	0
$799$	−15.4694	−0.547268
$800$	0	0
$801$	−58.7810	−2.07692
$802$	0	0
$803$	−2.43010	−0.0857563
$804$	0	0
$805$	0	0
$806$	0	0
$807$	42.6618	1.50177
$808$	0	0
$809$	41.8643	1.47187	0.735936	−	0.677051i	$-0.236742\pi$
$809$	41.8643	1.47187	0.735936	+	0.677051i	$0.236742\pi$
$810$	0	0
$811$	29.8960	1.04979	0.524895	−	0.851167i	$-0.324105\pi$
$811$	29.8960	1.04979	0.524895	+	0.851167i	$0.324105\pi$
$812$	0	0
$813$	−69.2879	−2.43003
$814$	0	0
$815$	0	0
$816$	0	0
$817$	7.83425	0.274086
$818$	0	0
$819$	61.2144	2.13901
$820$	0	0
$821$	46.3602	1.61798	0.808992	−	0.587820i	$-0.200014\pi$
$821$	46.3602	1.61798	0.808992	+	0.587820i	$0.200014\pi$
$822$	0	0
$823$	37.3940	1.30347	0.651737	−	0.758445i	$-0.274040\pi$
$823$	37.3940	1.30347	0.651737	+	0.758445i	$0.274040\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	9.68450	0.336763	0.168382	−	0.985722i	$-0.446146\pi$
$827$	9.68450	0.336763	0.168382	+	0.985722i	$0.446146\pi$
$828$	0	0
$829$	16.8732	0.586031	0.293016	−	0.956108i	$-0.405341\pi$
$829$	16.8732	0.586031	0.293016	+	0.956108i	$0.405341\pi$
$830$	0	0
$831$	20.7095	0.718405
$832$	0	0
$833$	−6.13790	−0.212665
$834$	0	0
$835$	0	0
$836$	0	0
$837$	82.1632	2.83997
$838$	0	0
$839$	35.3160	1.21924	0.609622	−	0.792692i	$-0.291321\pi$
$839$	35.3160	1.21924	0.609622	+	0.792692i	$0.291321\pi$
$840$	0	0
$841$	18.0508	0.622442
$842$	0	0
$843$	94.9496	3.27024
$844$	0	0
$845$	0	0
$846$	0	0
$847$	17.7008	0.608208
$848$	0	0
$849$	−92.4773	−3.17381
$850$	0	0
$851$	20.8444	0.714536
$852$	0	0
$853$	−18.3492	−0.628265	−0.314133	−	0.949379i	$-0.601714\pi$
$853$	−18.3492	−0.628265	−0.314133	+	0.949379i	$0.601714\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	25.4209	0.868361	0.434181	−	0.900826i	$-0.357038\pi$
$857$	25.4209	0.868361	0.434181	+	0.900826i	$0.357038\pi$
$858$	0	0
$859$	−27.2820	−0.930849	−0.465425	−	0.885088i	$-0.654098\pi$
$859$	−27.2820	−0.930849	−0.465425	+	0.885088i	$0.654098\pi$
$860$	0	0
$861$	24.3637	0.830313
$862$	0	0
$863$	3.63233	0.123646	0.0618230	−	0.998087i	$-0.480309\pi$
$863$	3.63233	0.123646	0.0618230	+	0.998087i	$0.480309\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	49.5154	1.68163
$868$	0	0
$869$	−3.92904	−0.133284
$870$	0	0
$871$	−50.1977	−1.70088
$872$	0	0
$873$	−69.9382	−2.36705
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−57.1986	−1.93146	−0.965731	−	0.259547i	$-0.916427\pi$
$877$	−57.1986	−1.93146	−0.965731	+	0.259547i	$0.916427\pi$
$878$	0	0
$879$	74.4455	2.51099
$880$	0	0
$881$	−48.3846	−1.63012	−0.815059	−	0.579378i	$-0.803296\pi$
$881$	−48.3846	−1.63012	−0.815059	+	0.579378i	$0.803296\pi$
$882$	0	0
$883$	17.6993	0.595630	0.297815	−	0.954624i	$-0.403742\pi$
$883$	17.6993	0.595630	0.297815	+	0.954624i	$0.403742\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−40.1640	−1.34858	−0.674288	−	0.738468i	$-0.735549\pi$
$887$	−40.1640	−1.34858	−0.674288	+	0.738468i	$0.735549\pi$
$888$	0	0
$889$	−7.31701	−0.245405
$890$	0	0
$891$	138.703	4.64673
$892$	0	0
$893$	10.9382	0.366033
$894$	0	0
$895$	0	0
$896$	0	0
$897$	30.8290	1.02935
$898$	0	0
$899$	−34.8646	−1.16280
$900$	0	0
$901$	2.20384	0.0734206
$902$	0	0
$903$	42.1782	1.40360
$904$	0	0
$905$	0	0
$906$	0	0
$907$	12.5765	0.417596	0.208798	−	0.977959i	$-0.433045\pi$
$907$	12.5765	0.417596	0.208798	+	0.977959i	$0.433045\pi$
$908$	0	0
$909$	−102.292	−3.39281
$910$	0	0
$911$	29.7278	0.984927	0.492464	−	0.870333i	$-0.336096\pi$
$911$	29.7278	0.984927	0.492464	+	0.870333i	$0.336096\pi$
$912$	0	0
$913$	55.8197	1.84736
$914$	0	0
$915$	0	0
$916$	0	0
$917$	18.1482	0.599308
$918$	0	0
$919$	−22.1146	−0.729492	−0.364746	−	0.931107i	$-0.618844\pi$
$919$	−22.1146	−0.729492	−0.364746	+	0.931107i	$0.618844\pi$
$920$	0	0
$921$	−48.1879	−1.58784
$922$	0	0
$923$	−60.3948	−1.98792
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−24.9726	−0.820208
$928$	0	0
$929$	−23.4746	−0.770175	−0.385088	−	0.922880i	$-0.625829\pi$
$929$	−23.4746	−0.770175	−0.385088	+	0.922880i	$0.625829\pi$
$930$	0	0
$931$	4.34003	0.142239
$932$	0	0
$933$	89.6592	2.93531
$934$	0	0
$935$	0	0
$936$	0	0
$937$	7.29393	0.238282	0.119141	−	0.992877i	$-0.461986\pi$
$937$	7.29393	0.238282	0.119141	+	0.992877i	$0.461986\pi$
$938$	0	0
$939$	−93.7016	−3.05784
$940$	0	0
$941$	−36.3844	−1.18610	−0.593048	−	0.805167i	$-0.702076\pi$
$941$	−36.3844	−1.18610	−0.593048	+	0.805167i	$0.702076\pi$
$942$	0	0
$943$	8.89207	0.289566
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−6.77953	−0.220305	−0.110152	−	0.993915i	$-0.535134\pi$
$947$	−6.77953	−0.220305	−0.110152	+	0.993915i	$0.535134\pi$
$948$	0	0
$949$	−2.47072	−0.0802030
$950$	0	0
$951$	69.2922	2.24695
$952$	0	0
$953$	9.00668	0.291755	0.145877	−	0.989303i	$-0.453399\pi$
$953$	9.00668	0.291755	0.145877	+	0.989303i	$0.453399\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−105.850	−3.42166
$958$	0	0
$959$	−32.1052	−1.03673
$960$	0	0
$961$	−5.16544	−0.166627
$962$	0	0
$963$	−148.824	−4.79579
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0.251168	0.00807702	0.00403851	−	0.999992i	$-0.498714\pi$
$967$	0.251168	0.00807702	0.00403851	+	0.999992i	$0.498714\pi$
$968$	0	0
$969$	4.66852	0.149974
$970$	0	0
$971$	9.76919	0.313508	0.156754	−	0.987638i	$-0.449897\pi$
$971$	9.76919	0.313508	0.156754	+	0.987638i	$0.449897\pi$
$972$	0	0
$973$	−14.9345	−0.478779
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−37.8438	−1.21073	−0.605365	−	0.795948i	$-0.706973\pi$
$977$	−37.8438	−1.21073	−0.605365	+	0.795948i	$0.706973\pi$
$978$	0	0
$979$	−34.7965	−1.11210
$980$	0	0
$981$	106.366	3.39601
$982$	0	0
$983$	32.0062	1.02084	0.510420	−	0.859925i	$-0.329490\pi$
$983$	32.0062	1.02084	0.510420	+	0.859925i	$0.329490\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	58.8894	1.87447
$988$	0	0
$989$	15.3939	0.489496
$990$	0	0
$991$	14.8408	0.471432	0.235716	−	0.971822i	$-0.424256\pi$
$991$	14.8408	0.471432	0.235716	+	0.971822i	$0.424256\pi$
$992$	0	0
$993$	−5.99572	−0.190268
$994$	0	0
$995$	0	0
$996$	0	0
$997$	46.6666	1.47795	0.738973	−	0.673735i	$-0.235311\pi$
$997$	46.6666	1.47795	0.738973	+	0.673735i	$0.235311\pi$
$998$	0	0
$999$	171.481	5.42541

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7600.2.a.ci.1.1		6
4.3	odd	2		3800.2.a.bd.1.6	yes	6
5.4	even	2		7600.2.a.cm.1.6		6
20.3	even	4		3800.2.d.p.3649.12		12
20.7	even	4		3800.2.d.p.3649.1		12
20.19	odd	2		3800.2.a.bb.1.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3800.2.a.bb.1.1	✓	6	20.19	odd	2
3800.2.a.bd.1.6	yes	6	4.3	odd	2
3800.2.d.p.3649.1		12	20.7	even	4
3800.2.d.p.3649.12		12	20.3	even	4
7600.2.a.ci.1.1		6	1.1	even	1	trivial
7600.2.a.cm.1.6		6	5.4	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)

\(f(q)\)	\(=\)	\(q-3.30105 q^{3} +1.63094 q^{7} +7.89694 q^{9} +O(q^{10})\)	\(q-3.30105 q^{3} +1.63094 q^{7} +7.89694 q^{9} +4.67473 q^{11} +4.75288 q^{13} +1.41425 q^{17} -1.00000 q^{19} -5.38382 q^{21} -1.96494 q^{23} -16.1650 q^{27} +6.85936 q^{29} -5.08277 q^{31} -15.4315 q^{33} -10.6081 q^{37} -15.6895 q^{39} -4.52536 q^{41} -7.83425 q^{43} -10.9382 q^{47} -4.34003 q^{49} -4.66852 q^{51} +1.55831 q^{53} +3.30105 q^{57} +6.81879 q^{59} -0.109000 q^{61} +12.8794 q^{63} -10.5615 q^{67} +6.48638 q^{69} -12.7070 q^{71} -0.519837 q^{73} +7.62422 q^{77} -0.840485 q^{79} +29.6708 q^{81} +11.9407 q^{83} -22.6431 q^{87} -7.44352 q^{89} +7.75167 q^{91} +16.7785 q^{93} -8.85637 q^{97} +36.9161 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 2 q^{7} + 10 q^{9}+O(q^{10}) \) 6 * q - 2 * q^3 - 2 * q^7 + 10 * q^9	\( 6 q - 2 q^{3} - 2 q^{7} + 10 q^{9} - 3 q^{11} - 3 q^{13} - 2 q^{17} - 6 q^{19} + 11 q^{21} - 4 q^{23} - 20 q^{27} + 7 q^{29} - 5 q^{31} - 16 q^{33} - 8 q^{39} + 11 q^{41} + 7 q^{43} - 20 q^{47} - 2 q^{49} - 13 q^{51} - 7 q^{53} + 2 q^{57} + 4 q^{59} + 13 q^{61} + q^{63} - 25 q^{67} + 7 q^{69} - 29 q^{71} - 19 q^{73} + 24 q^{77} - 28 q^{79} + 38 q^{81} + 15 q^{83} - 57 q^{87} - 12 q^{89} - 27 q^{93} - 13 q^{97} - 10 q^{99}+O(q^{100}) \) 6 * q - 2 * q^3 - 2 * q^7 + 10 * q^9 - 3 * q^11 - 3 * q^13 - 2 * q^17 - 6 * q^19 + 11 * q^21 - 4 * q^23 - 20 * q^27 + 7 * q^29 - 5 * q^31 - 16 * q^33 - 8 * q^39 + 11 * q^41 + 7 * q^43 - 20 * q^47 - 2 * q^49 - 13 * q^51 - 7 * q^53 + 2 * q^57 + 4 * q^59 + 13 * q^61 + q^63 - 25 * q^67 + 7 * q^69 - 29 * q^71 - 19 * q^73 + 24 * q^77 - 28 * q^79 + 38 * q^81 + 15 * q^83 - 57 * q^87 - 12 * q^89 - 27 * q^93 - 13 * q^97 - 10 * q^99

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