LMFDB - Embedded newform 7600.2.a.cb.1.3

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			1.3
Root			$0.713538$ 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)

$f(q)$	$=$	$q+2.77733 q^{3} -4.69527 q^{7} +4.71354 q^{9} +O(q^{10})$	$q+2.77733 q^{3} -4.69527 q^{7} +4.71354 q^{9} -6.40880 q^{11} -1.06379 q^{13} -1.91794 q^{17} +1.00000 q^{19} -13.0403 q^{21} +1.79560 q^{23} +4.75905 q^{27} +2.93621 q^{29} +5.55465 q^{31} -17.7993 q^{33} +11.4088 q^{37} -2.95449 q^{39} -1.14585 q^{41} -3.55465 q^{43} +10.8359 q^{47} +15.0455 q^{49} -5.32674 q^{51} +8.69527 q^{53} +2.77733 q^{57} +5.63148 q^{59} -3.39053 q^{61} -22.1313 q^{63} -8.82284 q^{67} +4.98696 q^{69} +1.42708 q^{71} +12.6132 q^{73} +30.0910 q^{77} +1.96345 q^{79} -0.923174 q^{81} +16.2447 q^{83} +8.15482 q^{87} -10.0000 q^{89} +4.99477 q^{91} +15.4271 q^{93} +14.9452 q^{97} -30.2081 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.77733	1.60349	0.801745	−	0.597666i	$-0.203905\pi$
$3$	2.77733	1.60349	0.801745	+	0.597666i	$0.203905\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.69527	−1.77464	−0.887322	−	0.461151i	$-0.847437\pi$
$7$	−4.69527	−1.77464	−0.887322	+	0.461151i	$0.847437\pi$
$8$	0	0
$9$	4.71354	1.57118
$10$	0	0
$11$	−6.40880	−1.93233	−0.966163	−	0.257931i	$-0.916959\pi$
$11$	−6.40880	−1.93233	−0.966163	+	0.257931i	$0.916959\pi$
$12$	0	0
$13$	−1.06379	−0.295042	−0.147521	−	0.989059i	$-0.547129\pi$
$13$	−1.06379	−0.295042	−0.147521	+	0.989059i	$0.547129\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.91794	−0.465169	−0.232584	−	0.972576i	$-0.574718\pi$
$17$	−1.91794	−0.465169	−0.232584	+	0.972576i	$0.574718\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−13.0403	−2.84562
$22$	0	0
$23$	1.79560	0.374408	0.187204	−	0.982321i	$-0.440057\pi$
$23$	1.79560	0.374408	0.187204	+	0.982321i	$0.440057\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.75905	0.915880
$28$	0	0
$29$	2.93621	0.545241	0.272620	−	0.962122i	$-0.412110\pi$
$29$	2.93621	0.545241	0.272620	+	0.962122i	$0.412110\pi$
$30$	0	0
$31$	5.55465	0.997645	0.498822	−	0.866704i	$-0.333766\pi$
$31$	5.55465	0.997645	0.498822	+	0.866704i	$0.333766\pi$
$32$	0	0
$33$	−17.7993	−3.09847
$34$	0	0
$35$	0	0
$36$	0	0
$37$	11.4088	1.87560	0.937798	−	0.347182i	$-0.112861\pi$
$37$	11.4088	1.87560	0.937798	+	0.347182i	$0.112861\pi$
$38$	0	0
$39$	−2.95449	−0.473096
$40$	0	0
$41$	−1.14585	−0.178951	−0.0894757	−	0.995989i	$-0.528519\pi$
$41$	−1.14585	−0.178951	−0.0894757	+	0.995989i	$0.528519\pi$
$42$	0	0
$43$	−3.55465	−0.542079	−0.271040	−	0.962568i	$-0.587367\pi$
$43$	−3.55465	−0.542079	−0.271040	+	0.962568i	$0.587367\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.8359	1.58058	0.790288	−	0.612736i	$-0.209931\pi$
$47$	10.8359	1.58058	0.790288	+	0.612736i	$0.209931\pi$
$48$	0	0
$49$	15.0455	2.14936
$50$	0	0
$51$	−5.32674	−0.745893
$52$	0	0
$53$	8.69527	1.19439	0.597193	−	0.802097i	$-0.296283\pi$
$53$	8.69527	1.19439	0.597193	+	0.802097i	$0.296283\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.77733	0.367866
$58$	0	0
$59$	5.63148	0.733156	0.366578	−	0.930387i	$-0.380529\pi$
$59$	5.63148	0.733156	0.366578	+	0.930387i	$0.380529\pi$
$60$	0	0
$61$	−3.39053	−0.434113	−0.217056	−	0.976159i	$-0.569646\pi$
$61$	−3.39053	−0.434113	−0.217056	+	0.976159i	$0.569646\pi$
$62$	0	0
$63$	−22.1313	−2.78828
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−8.82284	−1.07788	−0.538941	−	0.842344i	$-0.681175\pi$
$67$	−8.82284	−1.07788	−0.538941	+	0.842344i	$0.681175\pi$
$68$	0	0
$69$	4.98696	0.600360
$70$	0	0
$71$	1.42708	0.169363	0.0846814	−	0.996408i	$-0.473013\pi$
$71$	1.42708	0.169363	0.0846814	+	0.996408i	$0.473013\pi$
$72$	0	0
$73$	12.6132	1.47626	0.738132	−	0.674656i	$-0.235708\pi$
$73$	12.6132	1.47626	0.738132	+	0.674656i	$0.235708\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	30.0910	3.42919
$78$	0	0
$79$	1.96345	0.220906	0.110453	−	0.993881i	$-0.464770\pi$
$79$	1.96345	0.220906	0.110453	+	0.993881i	$0.464770\pi$
$80$	0	0
$81$	−0.923174	−0.102575
$82$	0	0
$83$	16.2447	1.78309	0.891543	−	0.452937i	$-0.149624\pi$
$83$	16.2447	1.78309	0.891543	+	0.452937i	$0.149624\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	8.15482	0.874288
$88$	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
$91$	4.99477	0.523594
$92$	0	0
$93$	15.4271	1.59971
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14.9452	1.51745	0.758727	−	0.651409i	$-0.225822\pi$
$97$	14.9452	1.51745	0.758727	+	0.651409i	$0.225822\pi$
$98$	0	0
$99$	−30.2081	−3.03603
$100$	0	0
$101$	10.5364	1.04841	0.524204	−	0.851592i	$-0.324363\pi$
$101$	10.5364	1.04841	0.524204	+	0.851592i	$0.324363\pi$
$102$	0	0
$103$	−16.9817	−1.67326	−0.836630	−	0.547769i	$-0.815477\pi$
$103$	−16.9817	−1.67326	−0.836630	+	0.547769i	$0.815477\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.79036	−0.173081	−0.0865405	−	0.996248i	$-0.527581\pi$
$107$	−1.79036	−0.173081	−0.0865405	+	0.996248i	$0.527581\pi$
$108$	0	0
$109$	2.41404	0.231223	0.115611	−	0.993295i	$-0.463117\pi$
$109$	2.41404	0.231223	0.115611	+	0.993295i	$0.463117\pi$
$110$	0	0
$111$	31.6860	3.00750
$112$	0	0
$113$	−7.14585	−0.672225	−0.336112	−	0.941822i	$-0.609112\pi$
$113$	−7.14585	−0.672225	−0.336112	+	0.941822i	$0.609112\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−5.01420	−0.463563
$118$	0	0
$119$	9.00523	0.825508
$120$	0	0
$121$	30.0728	2.73389
$122$	0	0
$123$	−3.18239	−0.286947
$124$	0	0
$125$	0	0
$126$	0	0
$127$	9.26295	0.821954	0.410977	−	0.911646i	$-0.365188\pi$
$127$	9.26295	0.821954	0.410977	+	0.911646i	$0.365188\pi$
$128$	0	0
$129$	−9.87242	−0.869219
$130$	0	0
$131$	−11.5181	−1.00634	−0.503171	−	0.864187i	$-0.667833\pi$
$131$	−11.5181	−1.00634	−0.503171	+	0.864187i	$0.667833\pi$
$132$	0	0
$133$	−4.69527	−0.407131
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−15.1041	−1.29043	−0.645214	−	0.764002i	$-0.723232\pi$
$137$	−15.1041	−1.29043	−0.645214	+	0.764002i	$0.723232\pi$
$138$	0	0
$139$	0.700500	0.0594156	0.0297078	−	0.999559i	$-0.490542\pi$
$139$	0.700500	0.0594156	0.0297078	+	0.999559i	$0.490542\pi$
$140$	0	0
$141$	30.0948	2.53444
$142$	0	0
$143$	6.81761	0.570117
$144$	0	0
$145$	0	0
$146$	0	0
$147$	41.7863	3.44648
$148$	0	0
$149$	12.9452	1.06051	0.530255	−	0.847838i	$-0.322096\pi$
$149$	12.9452	1.06051	0.530255	+	0.847838i	$0.322096\pi$
$150$	0	0
$151$	−5.70830	−0.464535	−0.232268	−	0.972652i	$-0.574615\pi$
$151$	−5.70830	−0.464535	−0.232268	+	0.972652i	$0.574615\pi$
$152$	0	0
$153$	−9.04028	−0.730863
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−9.26295	−0.739264	−0.369632	−	0.929178i	$-0.620516\pi$
$157$	−9.26295	−0.739264	−0.369632	+	0.929178i	$0.620516\pi$
$158$	0	0
$159$	24.1496	1.91519
$160$	0	0
$161$	−8.43081	−0.664441
$162$	0	0
$163$	4.28123	0.335332	0.167666	−	0.985844i	$-0.446377\pi$
$163$	4.28123	0.335332	0.167666	+	0.985844i	$0.446377\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−15.2264	−1.17825	−0.589127	−	0.808040i	$-0.700528\pi$
$167$	−15.2264	−1.17825	−0.589127	+	0.808040i	$0.700528\pi$
$168$	0	0
$169$	−11.8684	−0.912950
$170$	0	0
$171$	4.71354	0.360453
$172$	0	0
$173$	12.2630	0.932335	0.466168	−	0.884696i	$-0.345634\pi$
$173$	12.2630	0.932335	0.466168	+	0.884696i	$0.345634\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	15.6404	1.17561
$178$	0	0
$179$	1.57292	0.117566	0.0587829	−	0.998271i	$-0.481278\pi$
$179$	1.57292	0.117566	0.0587829	+	0.998271i	$0.481278\pi$
$180$	0	0
$181$	11.4088	0.848010	0.424005	−	0.905660i	$-0.360624\pi$
$181$	11.4088	0.848010	0.424005	+	0.905660i	$0.360624\pi$
$182$	0	0
$183$	−9.41661	−0.696096
$184$	0	0
$185$	0	0
$186$	0	0
$187$	12.2917	0.898858
$188$	0	0
$189$	−22.3450	−1.62536
$190$	0	0
$191$	6.35805	0.460053	0.230026	−	0.973184i	$-0.426119\pi$
$191$	6.35805	0.460053	0.230026	+	0.973184i	$0.426119\pi$
$192$	0	0
$193$	−14.4998	−1.04372	−0.521860	−	0.853031i	$-0.674762\pi$
$193$	−14.4998	−1.04372	−0.521860	+	0.853031i	$0.674762\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5.14585	0.366627	0.183313	−	0.983055i	$-0.441318\pi$
$197$	5.14585	0.366627	0.183313	+	0.983055i	$0.441318\pi$
$198$	0	0
$199$	−3.87766	−0.274880	−0.137440	−	0.990510i	$-0.543887\pi$
$199$	−3.87766	−0.274880	−0.137440	+	0.990510i	$0.543887\pi$
$200$	0	0
$201$	−24.5039	−1.72837
$202$	0	0
$203$	−13.7863	−0.967608
$204$	0	0
$205$	0	0
$206$	0	0
$207$	8.46362	0.588262
$208$	0	0
$209$	−6.40880	−0.443306
$210$	0	0
$211$	−17.1496	−1.18063	−0.590313	−	0.807174i	$-0.700996\pi$
$211$	−17.1496	−1.18063	−0.590313	+	0.807174i	$0.700996\pi$
$212$	0	0
$213$	3.96345	0.271571
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−26.0806	−1.77046
$218$	0	0
$219$	35.0310	2.36717
$220$	0	0
$221$	2.04028	0.137244
$222$	0	0
$223$	7.84635	0.525430	0.262715	−	0.964873i	$-0.415382\pi$
$223$	7.84635	0.525430	0.262715	+	0.964873i	$0.415382\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6.28646	0.417247	0.208624	−	0.977996i	$-0.433102\pi$
$227$	6.28646	0.417247	0.208624	+	0.977996i	$0.433102\pi$
$228$	0	0
$229$	−3.14585	−0.207884	−0.103942	−	0.994583i	$-0.533146\pi$
$229$	−3.14585	−0.207884	−0.103942	+	0.994583i	$0.533146\pi$
$230$	0	0
$231$	83.5726	5.49867
$232$	0	0
$233$	−0.182394	−0.0119490	−0.00597451	−	0.999982i	$-0.501902\pi$
$233$	−0.182394	−0.0119490	−0.00597451	+	0.999982i	$0.501902\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	5.45315	0.354220
$238$	0	0
$239$	11.5039	0.744126	0.372063	−	0.928208i	$-0.378651\pi$
$239$	11.5039	0.744126	0.372063	+	0.928208i	$0.378651\pi$
$240$	0	0
$241$	−0.445349	−0.0286874	−0.0143437	−	0.999897i	$-0.504566\pi$
$241$	−0.445349	−0.0286874	−0.0143437	+	0.999897i	$0.504566\pi$
$242$	0	0
$243$	−16.8411	−1.08036
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.06379	−0.0676872
$248$	0	0
$249$	45.1168	2.85916
$250$	0	0
$251$	−13.2630	−0.837150	−0.418575	−	0.908182i	$-0.637470\pi$
$251$	−13.2630	−0.837150	−0.418575	+	0.908182i	$0.637470\pi$
$252$	0	0
$253$	−11.5076	−0.723479
$254$	0	0
$255$	0	0
$256$	0	0
$257$	23.4816	1.46474	0.732370	−	0.680907i	$-0.238414\pi$
$257$	23.4816	1.46474	0.732370	+	0.680907i	$0.238414\pi$
$258$	0	0
$259$	−53.5674	−3.32851
$260$	0	0
$261$	13.8399	0.856671
$262$	0	0
$263$	27.9269	1.72205	0.861023	−	0.508565i	$-0.169824\pi$
$263$	27.9269	1.72205	0.861023	+	0.508565i	$0.169824\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−27.7733	−1.69970
$268$	0	0
$269$	16.4816	1.00490	0.502449	−	0.864607i	$-0.332432\pi$
$269$	16.4816	1.00490	0.502449	+	0.864607i	$0.332432\pi$
$270$	0	0
$271$	1.46736	0.0891356	0.0445678	−	0.999006i	$-0.485809\pi$
$271$	1.46736	0.0891356	0.0445678	+	0.999006i	$0.485809\pi$
$272$	0	0
$273$	13.8721	0.839577
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−15.8359	−0.951486	−0.475743	−	0.879584i	$-0.657821\pi$
$277$	−15.8359	−0.951486	−0.475743	+	0.879584i	$0.657821\pi$
$278$	0	0
$279$	26.1821	1.56748
$280$	0	0
$281$	−6.25515	−0.373151	−0.186576	−	0.982441i	$-0.559739\pi$
$281$	−6.25515	−0.373151	−0.186576	+	0.982441i	$0.559739\pi$
$282$	0	0
$283$	−16.7005	−0.992742	−0.496371	−	0.868111i	$-0.665334\pi$
$283$	−16.7005	−0.992742	−0.496371	+	0.868111i	$0.665334\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.38006	0.317575
$288$	0	0
$289$	−13.3215	−0.783618
$290$	0	0
$291$	41.5076	2.43322
$292$	0	0
$293$	−0.644516	−0.0376530	−0.0188265	−	0.999823i	$-0.505993\pi$
$293$	−0.644516	−0.0376530	−0.0188265	+	0.999823i	$0.505993\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−30.4998	−1.76978
$298$	0	0
$299$	−1.91014	−0.110466
$300$	0	0
$301$	16.6900	0.961997
$302$	0	0
$303$	29.2630	1.68111
$304$	0	0
$305$	0	0
$306$	0	0
$307$	26.6457	1.52075	0.760375	−	0.649485i	$-0.225015\pi$
$307$	26.6457	1.52075	0.760375	+	0.649485i	$0.225015\pi$
$308$	0	0
$309$	−47.1638	−2.68305
$310$	0	0
$311$	−11.5494	−0.654907	−0.327454	−	0.944867i	$-0.606191\pi$
$311$	−11.5494	−0.654907	−0.327454	+	0.944867i	$0.606191\pi$
$312$	0	0
$313$	23.0638	1.30364	0.651821	−	0.758373i	$-0.274005\pi$
$313$	23.0638	1.30364	0.651821	+	0.758373i	$0.274005\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	13.9232	0.782003	0.391002	−	0.920390i	$-0.372129\pi$
$317$	13.9232	0.782003	0.391002	+	0.920390i	$0.372129\pi$
$318$	0	0
$319$	−18.8176	−1.05358
$320$	0	0
$321$	−4.97242	−0.277534
$322$	0	0
$323$	−1.91794	−0.106717
$324$	0	0
$325$	0	0
$326$	0	0
$327$	6.70457	0.370763
$328$	0	0
$329$	−50.8773	−2.80496
$330$	0	0
$331$	0.735546	0.0404292	0.0202146	−	0.999796i	$-0.493565\pi$
$331$	0.735546	0.0404292	0.0202146	+	0.999796i	$0.493565\pi$
$332$	0	0
$333$	53.7758	2.94690
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2.28123	0.124266	0.0621332	−	0.998068i	$-0.480210\pi$
$337$	2.28123	0.124266	0.0621332	+	0.998068i	$0.480210\pi$
$338$	0	0
$339$	−19.8463	−1.07791
$340$	0	0
$341$	−35.5987	−1.92778
$342$	0	0
$343$	−37.7758	−2.03970
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2.56246	−0.137560	−0.0687799	−	0.997632i	$-0.521911\pi$
$347$	−2.56246	−0.137560	−0.0687799	+	0.997632i	$0.521911\pi$
$348$	0	0
$349$	−5.67176	−0.303602	−0.151801	−	0.988411i	$-0.548507\pi$
$349$	−5.67176	−0.303602	−0.151801	+	0.988411i	$0.548507\pi$
$350$	0	0
$351$	−5.06262	−0.270223
$352$	0	0
$353$	23.6665	1.25964	0.629821	−	0.776740i	$-0.283128\pi$
$353$	23.6665	1.25964	0.629821	+	0.776740i	$0.283128\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	25.0105	1.32369
$358$	0	0
$359$	−31.9724	−1.68744	−0.843720	−	0.536784i	$-0.819639\pi$
$359$	−31.9724	−1.68744	−0.843720	+	0.536784i	$0.819639\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	83.5218	4.38376
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.14585	0.0598128	0.0299064	−	0.999553i	$-0.490479\pi$
$367$	1.14585	0.0598128	0.0299064	+	0.999553i	$0.490479\pi$
$368$	0	0
$369$	−5.40100	−0.281165
$370$	0	0
$371$	−40.8266	−2.11961
$372$	0	0
$373$	−32.8579	−1.70132	−0.850658	−	0.525719i	$-0.823796\pi$
$373$	−32.8579	−1.70132	−0.850658	+	0.525719i	$0.823796\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3.12351	−0.160869
$378$	0	0
$379$	18.2865	0.939312	0.469656	−	0.882849i	$-0.344378\pi$
$379$	18.2865	0.939312	0.469656	+	0.882849i	$0.344378\pi$
$380$	0	0
$381$	25.7262	1.31800
$382$	0	0
$383$	20.8542	1.06560	0.532799	−	0.846242i	$-0.321140\pi$
$383$	20.8542	1.06560	0.532799	+	0.846242i	$0.321140\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−16.7550	−0.851704
$388$	0	0
$389$	−24.9086	−1.26292	−0.631459	−	0.775409i	$-0.717544\pi$
$389$	−24.9086	−1.26292	−0.631459	+	0.775409i	$0.717544\pi$
$390$	0	0
$391$	−3.44385	−0.174163
$392$	0	0
$393$	−31.9895	−1.61366
$394$	0	0
$395$	0	0
$396$	0	0
$397$	24.7445	1.24189	0.620946	−	0.783853i	$-0.286749\pi$
$397$	24.7445	1.24189	0.620946	+	0.783853i	$0.286749\pi$
$398$	0	0
$399$	−13.0403	−0.652831
$400$	0	0
$401$	15.6457	0.781308	0.390654	−	0.920538i	$-0.372249\pi$
$401$	15.6457	0.781308	0.390654	+	0.920538i	$0.372249\pi$
$402$	0	0
$403$	−5.90897	−0.294347
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−73.1168	−3.62426
$408$	0	0
$409$	30.5804	1.51210	0.756052	−	0.654512i	$-0.227126\pi$
$409$	30.5804	1.51210	0.756052	+	0.654512i	$0.227126\pi$
$410$	0	0
$411$	−41.9489	−2.06919
$412$	0	0
$413$	−26.4413	−1.30109
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1.94552	0.0952723
$418$	0	0
$419$	−1.96345	−0.0959210	−0.0479605	−	0.998849i	$-0.515272\pi$
$419$	−1.96345	−0.0959210	−0.0479605	+	0.998849i	$0.515272\pi$
$420$	0	0
$421$	−25.5949	−1.24742	−0.623710	−	0.781656i	$-0.714376\pi$
$421$	−25.5949	−1.24742	−0.623710	+	0.781656i	$0.714376\pi$
$422$	0	0
$423$	51.0753	2.48337
$424$	0	0
$425$	0	0
$426$	0	0
$427$	15.9194	0.770396
$428$	0	0
$429$	18.9347	0.914177
$430$	0	0
$431$	15.3540	0.739575	0.369788	−	0.929116i	$-0.379430\pi$
$431$	15.3540	0.739575	0.369788	+	0.929116i	$0.379430\pi$
$432$	0	0
$433$	−16.6640	−0.800819	−0.400409	−	0.916336i	$-0.631132\pi$
$433$	−16.6640	−0.800819	−0.400409	+	0.916336i	$0.631132\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.79560	0.0858951
$438$	0	0
$439$	19.5987	0.935393	0.467697	−	0.883889i	$-0.345084\pi$
$439$	19.5987	0.935393	0.467697	+	0.883889i	$0.345084\pi$
$440$	0	0
$441$	70.9176	3.37703
$442$	0	0
$443$	13.0183	0.618517	0.309258	−	0.950978i	$-0.399919\pi$
$443$	13.0183	0.618517	0.309258	+	0.950978i	$0.399919\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	35.9530	1.70052
$448$	0	0
$449$	6.77359	0.319666	0.159833	−	0.987144i	$-0.448905\pi$
$449$	6.77359	0.319666	0.159833	+	0.987144i	$0.448905\pi$
$450$	0	0
$451$	7.34352	0.345793
$452$	0	0
$453$	−15.8538	−0.744877
$454$	0	0
$455$	0	0
$456$	0	0
$457$	27.3189	1.27793	0.638963	−	0.769237i	$-0.279364\pi$
$457$	27.3189	1.27793	0.638963	+	0.769237i	$0.279364\pi$
$458$	0	0
$459$	−9.12758	−0.426039
$460$	0	0
$461$	−27.7993	−1.29474	−0.647372	−	0.762174i	$-0.724132\pi$
$461$	−27.7993	−1.29474	−0.647372	+	0.762174i	$0.724132\pi$
$462$	0	0
$463$	38.2369	1.77702	0.888509	−	0.458859i	$-0.151742\pi$
$463$	38.2369	1.77702	0.888509	+	0.458859i	$0.151742\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	23.4711	1.08611	0.543056	−	0.839696i	$-0.317267\pi$
$467$	23.4711	1.08611	0.543056	+	0.839696i	$0.317267\pi$
$468$	0	0
$469$	41.4256	1.91286
$470$	0	0
$471$	−25.7262	−1.18540
$472$	0	0
$473$	22.7811	1.04747
$474$	0	0
$475$	0	0
$476$	0	0
$477$	40.9855	1.87660
$478$	0	0
$479$	19.6900	0.899660	0.449830	−	0.893114i	$-0.351484\pi$
$479$	19.6900	0.899660	0.449830	+	0.893114i	$0.351484\pi$
$480$	0	0
$481$	−12.1365	−0.553379
$482$	0	0
$483$	−23.4151	−1.06542
$484$	0	0
$485$	0	0
$486$	0	0
$487$	16.3357	0.740242	0.370121	−	0.928984i	$-0.379316\pi$
$487$	16.3357	0.740242	0.370121	+	0.928984i	$0.379316\pi$
$488$	0	0
$489$	11.8904	0.537701
$490$	0	0
$491$	−27.1093	−1.22343	−0.611713	−	0.791080i	$-0.709519\pi$
$491$	−27.1093	−1.22343	−0.611713	+	0.791080i	$0.709519\pi$
$492$	0	0
$493$	−5.63148	−0.253629
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−6.70050	−0.300558
$498$	0	0
$499$	4.69003	0.209955	0.104977	−	0.994475i	$-0.466523\pi$
$499$	4.69003	0.209955	0.104977	+	0.994475i	$0.466523\pi$
$500$	0	0
$501$	−42.2887	−1.88932
$502$	0	0
$503$	5.19136	0.231471	0.115736	−	0.993280i	$-0.463077\pi$
$503$	5.19136	0.231471	0.115736	+	0.993280i	$0.463077\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−32.9623	−1.46391
$508$	0	0
$509$	−4.51811	−0.200262	−0.100131	−	0.994974i	$-0.531926\pi$
$509$	−4.51811	−0.200262	−0.100131	+	0.994974i	$0.531926\pi$
$510$	0	0
$511$	−59.2223	−2.61984
$512$	0	0
$513$	4.75905	0.210117
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−69.4450	−3.05419
$518$	0	0
$519$	34.0582	1.49499
$520$	0	0
$521$	14.4453	0.632862	0.316431	−	0.948615i	$-0.397515\pi$
$521$	14.4453	0.632862	0.316431	+	0.948615i	$0.397515\pi$
$522$	0	0
$523$	−18.2134	−0.796415	−0.398208	−	0.917295i	$-0.630368\pi$
$523$	−18.2134	−0.796415	−0.398208	+	0.917295i	$0.630368\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−10.6535	−0.464073
$528$	0	0
$529$	−19.7758	−0.859819
$530$	0	0
$531$	26.5442	1.15192
$532$	0	0
$533$	1.21894	0.0527981
$534$	0	0
$535$	0	0
$536$	0	0
$537$	4.36852	0.188516
$538$	0	0
$539$	−96.4237	−4.15326
$540$	0	0
$541$	37.3905	1.60754	0.803772	−	0.594937i	$-0.202823\pi$
$541$	37.3905	1.60754	0.803772	+	0.594937i	$0.202823\pi$
$542$	0	0
$543$	31.6860	1.35977
$544$	0	0
$545$	0	0
$546$	0	0
$547$	29.5621	1.26399	0.631993	−	0.774974i	$-0.282237\pi$
$547$	29.5621	1.26399	0.631993	+	0.774974i	$0.282237\pi$
$548$	0	0
$549$	−15.9814	−0.682069
$550$	0	0
$551$	2.93621	0.125087
$552$	0	0
$553$	−9.21894	−0.392029
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−14.3723	−0.608972	−0.304486	−	0.952517i	$-0.598485\pi$
$557$	−14.3723	−0.608972	−0.304486	+	0.952517i	$0.598485\pi$
$558$	0	0
$559$	3.78139	0.159936
$560$	0	0
$561$	34.1380	1.44131
$562$	0	0
$563$	−6.39053	−0.269329	−0.134664	−	0.990891i	$-0.542996\pi$
$563$	−6.39053	−0.269329	−0.134664	+	0.990891i	$0.542996\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.33455	0.182034
$568$	0	0
$569$	−44.9086	−1.88267	−0.941334	−	0.337476i	$-0.890427\pi$
$569$	−44.9086	−1.88267	−0.941334	+	0.337476i	$0.890427\pi$
$570$	0	0
$571$	−12.4193	−0.519730	−0.259865	−	0.965645i	$-0.583678\pi$
$571$	−12.4193	−0.519730	−0.259865	+	0.965645i	$0.583678\pi$
$572$	0	0
$573$	17.6584	0.737690
$574$	0	0
$575$	0	0
$576$	0	0
$577$	4.20440	0.175032	0.0875158	−	0.996163i	$-0.472107\pi$
$577$	4.20440	0.175032	0.0875158	+	0.996163i	$0.472107\pi$
$578$	0	0
$579$	−40.2708	−1.67360
$580$	0	0
$581$	−76.2731	−3.16434
$582$	0	0
$583$	−55.7262	−2.30795
$584$	0	0
$585$	0	0
$586$	0	0
$587$	24.7915	1.02326	0.511628	−	0.859207i	$-0.329043\pi$
$587$	24.7915	1.02326	0.511628	+	0.859207i	$0.329043\pi$
$588$	0	0
$589$	5.55465	0.228875
$590$	0	0
$591$	14.2917	0.587882
$592$	0	0
$593$	−4.67176	−0.191846	−0.0959231	−	0.995389i	$-0.530580\pi$
$593$	−4.67176	−0.191846	−0.0959231	+	0.995389i	$0.530580\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−10.7695	−0.440767
$598$	0	0
$599$	−39.2369	−1.60318	−0.801588	−	0.597877i	$-0.796011\pi$
$599$	−39.2369	−1.60318	−0.801588	+	0.597877i	$0.796011\pi$
$600$	0	0
$601$	−42.4267	−1.73062	−0.865311	−	0.501235i	$-0.832879\pi$
$601$	−42.4267	−1.73062	−0.865311	+	0.501235i	$0.832879\pi$
$602$	0	0
$603$	−41.5868	−1.69355
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−2.98173	−0.121025	−0.0605123	−	0.998167i	$-0.519273\pi$
$607$	−2.98173	−0.121025	−0.0605123	+	0.998167i	$0.519273\pi$
$608$	0	0
$609$	−38.2890	−1.55155
$610$	0	0
$611$	−11.5271	−0.466336
$612$	0	0
$613$	28.9452	1.16908	0.584542	−	0.811363i	$-0.301274\pi$
$613$	28.9452	1.16908	0.584542	+	0.811363i	$0.301274\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	15.0365	0.605349	0.302674	−	0.953094i	$-0.402121\pi$
$617$	15.0365	0.605349	0.302674	+	0.953094i	$0.402121\pi$
$618$	0	0
$619$	−8.25515	−0.331803	−0.165901	−	0.986142i	$-0.553053\pi$
$619$	−8.25515	−0.331803	−0.165901	+	0.986142i	$0.553053\pi$
$620$	0	0
$621$	8.54535	0.342913
$622$	0	0
$623$	46.9527	1.88112
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−17.7993	−0.710837
$628$	0	0
$629$	−21.8814	−0.872468
$630$	0	0
$631$	9.09103	0.361908	0.180954	−	0.983492i	$-0.442081\pi$
$631$	9.09103	0.361908	0.180954	+	0.983492i	$0.442081\pi$
$632$	0	0
$633$	−47.6300	−1.89312
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−16.0052	−0.634150
$638$	0	0
$639$	6.72658	0.266099
$640$	0	0
$641$	30.1171	1.18955	0.594777	−	0.803891i	$-0.297240\pi$
$641$	30.1171	1.18955	0.594777	+	0.803891i	$0.297240\pi$
$642$	0	0
$643$	−35.3174	−1.39278	−0.696392	−	0.717662i	$-0.745212\pi$
$643$	−35.3174	−1.39278	−0.696392	+	0.717662i	$0.745212\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	4.12234	0.162066	0.0810330	−	0.996711i	$-0.474178\pi$
$647$	4.12234	0.162066	0.0810330	+	0.996711i	$0.474178\pi$
$648$	0	0
$649$	−36.0910	−1.41670
$650$	0	0
$651$	−72.4342	−2.83892
$652$	0	0
$653$	−8.62741	−0.337617	−0.168808	−	0.985649i	$-0.553992\pi$
$653$	−8.62741	−0.337617	−0.168808	+	0.985649i	$0.553992\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	59.4528	2.31948
$658$	0	0
$659$	37.3853	1.45632	0.728162	−	0.685405i	$-0.240375\pi$
$659$	37.3853	1.45632	0.728162	+	0.685405i	$0.240375\pi$
$660$	0	0
$661$	−12.8997	−0.501739	−0.250869	−	0.968021i	$-0.580716\pi$
$661$	−12.8997	−0.501739	−0.250869	+	0.968021i	$0.580716\pi$
$662$	0	0
$663$	5.66652	0.220070
$664$	0	0
$665$	0	0
$666$	0	0
$667$	5.27226	0.204143
$668$	0	0
$669$	21.7919	0.842522
$670$	0	0
$671$	21.7292	0.838848
$672$	0	0
$673$	−10.4349	−0.402235	−0.201118	−	0.979567i	$-0.564457\pi$
$673$	−10.4349	−0.402235	−0.201118	+	0.979567i	$0.564457\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−12.4401	−0.478112	−0.239056	−	0.971006i	$-0.576838\pi$
$677$	−12.4401	−0.478112	−0.239056	+	0.971006i	$0.576838\pi$
$678$	0	0
$679$	−70.1716	−2.69294
$680$	0	0
$681$	17.4596	0.669052
$682$	0	0
$683$	17.5364	0.671011	0.335505	−	0.942038i	$-0.391093\pi$
$683$	17.5364	0.671011	0.335505	+	0.942038i	$0.391093\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−8.73705	−0.333339
$688$	0	0
$689$	−9.24992	−0.352394
$690$	0	0
$691$	25.2734	0.961446	0.480723	−	0.876872i	$-0.340374\pi$
$691$	25.2734	0.961446	0.480723	+	0.876872i	$0.340374\pi$
$692$	0	0
$693$	141.835	5.38787
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2.19767	0.0832426
$698$	0	0
$699$	−0.506567	−0.0191601
$700$	0	0
$701$	−16.0545	−0.606370	−0.303185	−	0.952932i	$-0.598050\pi$
$701$	−16.0545	−0.606370	−0.303185	+	0.952932i	$0.598050\pi$
$702$	0	0
$703$	11.4088	0.430291
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−49.4711	−1.86055
$708$	0	0
$709$	13.5076	0.507290	0.253645	−	0.967297i	$-0.418371\pi$
$709$	13.5076	0.507290	0.253645	+	0.967297i	$0.418371\pi$
$710$	0	0
$711$	9.25482	0.347083
$712$	0	0
$713$	9.97392	0.373526
$714$	0	0
$715$	0	0
$716$	0	0
$717$	31.9501	1.19320
$718$	0	0
$719$	−0.803402	−0.0299619	−0.0149809	−	0.999888i	$-0.504769\pi$
$719$	−0.803402	−0.0299619	−0.0149809	+	0.999888i	$0.504769\pi$
$720$	0	0
$721$	79.7337	2.96944
$722$	0	0
$723$	−1.23688	−0.0460000
$724$	0	0
$725$	0	0
$726$	0	0
$727$	51.6860	1.91693	0.958463	−	0.285217i	$-0.0920655\pi$
$727$	51.6860	1.91693	0.958463	+	0.285217i	$0.0920655\pi$
$728$	0	0
$729$	−44.0037	−1.62977
$730$	0	0
$731$	6.81761	0.252158
$732$	0	0
$733$	22.3723	0.826338	0.413169	−	0.910654i	$-0.364422\pi$
$733$	22.3723	0.826338	0.413169	+	0.910654i	$0.364422\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	56.5438	2.08282
$738$	0	0
$739$	−33.4271	−1.22963	−0.614817	−	0.788669i	$-0.710770\pi$
$739$	−33.4271	−1.22963	−0.614817	+	0.788669i	$0.710770\pi$
$740$	0	0
$741$	−2.95449	−0.108536
$742$	0	0
$743$	14.4998	0.531947	0.265974	−	0.963980i	$-0.414307\pi$
$743$	14.4998	0.531947	0.265974	+	0.963980i	$0.414307\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	76.5699	2.80155
$748$	0	0
$749$	8.40623	0.307157
$750$	0	0
$751$	26.6169	0.971266	0.485633	−	0.874163i	$-0.338589\pi$
$751$	26.6169	0.971266	0.485633	+	0.874163i	$0.338589\pi$
$752$	0	0
$753$	−36.8355	−1.34236
$754$	0	0
$755$	0	0
$756$	0	0
$757$	31.0362	1.12803	0.564015	−	0.825764i	$-0.309256\pi$
$757$	31.0362	1.12803	0.564015	+	0.825764i	$0.309256\pi$
$758$	0	0
$759$	−31.9605	−1.16009
$760$	0	0
$761$	−27.8083	−1.00805	−0.504025	−	0.863689i	$-0.668148\pi$
$761$	−27.8083	−1.00805	−0.504025	+	0.863689i	$0.668148\pi$
$762$	0	0
$763$	−11.3345	−0.410338
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5.99070	−0.216312
$768$	0	0
$769$	−19.2279	−0.693376	−0.346688	−	0.937980i	$-0.612694\pi$
$769$	−19.2279	−0.693376	−0.346688	+	0.937980i	$0.612694\pi$
$770$	0	0
$771$	65.2159	2.34869
$772$	0	0
$773$	−6.00373	−0.215939	−0.107970	−	0.994154i	$-0.534435\pi$
$773$	−6.00373	−0.215939	−0.107970	+	0.994154i	$0.534435\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−148.774	−5.33724
$778$	0	0
$779$	−1.14585	−0.0410543
$780$	0	0
$781$	−9.14585	−0.327264
$782$	0	0
$783$	13.9736	0.499375
$784$	0	0
$785$	0	0
$786$	0	0
$787$	22.2797	0.794187	0.397093	−	0.917778i	$-0.370019\pi$
$787$	22.2797	0.794187	0.397093	+	0.917778i	$0.370019\pi$
$788$	0	0
$789$	77.5621	2.76128
$790$	0	0
$791$	33.5517	1.19296
$792$	0	0
$793$	3.60680	0.128081
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−26.6259	−0.943138	−0.471569	−	0.881829i	$-0.656312\pi$
$797$	−26.6259	−0.943138	−0.471569	+	0.881829i	$0.656312\pi$
$798$	0	0
$799$	−20.7826	−0.735234
$800$	0	0
$801$	−47.1354	−1.66545
$802$	0	0
$803$	−80.8355	−2.85262
$804$	0	0
$805$	0	0
$806$	0	0
$807$	45.7747	1.61134
$808$	0	0
$809$	0.651250	0.0228967	0.0114484	−	0.999934i	$-0.496356\pi$
$809$	0.651250	0.0228967	0.0114484	+	0.999934i	$0.496356\pi$
$810$	0	0
$811$	13.0780	0.459230	0.229615	−	0.973281i	$-0.426253\pi$
$811$	13.0780	0.459230	0.229615	+	0.973281i	$0.426253\pi$
$812$	0	0
$813$	4.07533	0.142928
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−3.55465	−0.124362
$818$	0	0
$819$	23.5430	0.822660
$820$	0	0
$821$	14.6640	0.511776	0.255888	−	0.966706i	$-0.417632\pi$
$821$	14.6640	0.511776	0.255888	+	0.966706i	$0.417632\pi$
$822$	0	0
$823$	−18.2850	−0.637374	−0.318687	−	0.947860i	$-0.603242\pi$
$823$	−18.2850	−0.637374	−0.318687	+	0.947860i	$0.603242\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	40.1910	1.39758	0.698790	−	0.715327i	$-0.253722\pi$
$827$	40.1910	1.39758	0.698790	+	0.715327i	$0.253722\pi$
$828$	0	0
$829$	28.3760	0.985539	0.492769	−	0.870160i	$-0.335985\pi$
$829$	28.3760	0.985539	0.492769	+	0.870160i	$0.335985\pi$
$830$	0	0
$831$	−43.9814	−1.52570
$832$	0	0
$833$	−28.8564	−0.999815
$834$	0	0
$835$	0	0
$836$	0	0
$837$	26.4349	0.913723
$838$	0	0
$839$	−6.93471	−0.239413	−0.119706	−	0.992809i	$-0.538195\pi$
$839$	−6.93471	−0.239413	−0.119706	+	0.992809i	$0.538195\pi$
$840$	0	0
$841$	−20.3787	−0.702712
$842$	0	0
$843$	−17.3726	−0.598344
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−141.200	−4.85167
$848$	0	0
$849$	−46.3827	−1.59185
$850$	0	0
$851$	20.4856	0.702238
$852$	0	0
$853$	−21.9739	−0.752373	−0.376186	−	0.926544i	$-0.622765\pi$
$853$	−21.9739	−0.752373	−0.376186	+	0.926544i	$0.622765\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−13.6091	−0.464879	−0.232440	−	0.972611i	$-0.574671\pi$
$857$	−13.6091	−0.464879	−0.232440	+	0.972611i	$0.574671\pi$
$858$	0	0
$859$	−5.17192	−0.176464	−0.0882319	−	0.996100i	$-0.528122\pi$
$859$	−5.17192	−0.176464	−0.0882319	+	0.996100i	$0.528122\pi$
$860$	0	0
$861$	14.9422	0.509228
$862$	0	0
$863$	15.9635	0.543402	0.271701	−	0.962382i	$-0.412414\pi$
$863$	15.9635	0.543402	0.271701	+	0.962382i	$0.412414\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−36.9982	−1.25652
$868$	0	0
$869$	−12.5834	−0.426862
$870$	0	0
$871$	9.38563	0.318020
$872$	0	0
$873$	70.4447	2.38419
$874$	0	0
$875$	0	0
$876$	0	0
$877$	15.8866	0.536453	0.268227	−	0.963356i	$-0.413562\pi$
$877$	15.8866	0.536453	0.268227	+	0.963356i	$0.413562\pi$
$878$	0	0
$879$	−1.79003	−0.0603762
$880$	0	0
$881$	−16.1458	−0.543967	−0.271984	−	0.962302i	$-0.587680\pi$
$881$	−16.1458	−0.543967	−0.271984	+	0.962302i	$0.587680\pi$
$882$	0	0
$883$	38.2887	1.28852	0.644259	−	0.764808i	$-0.277166\pi$
$883$	38.2887	1.28852	0.644259	+	0.764808i	$0.277166\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	19.2809	0.647389	0.323695	−	0.946162i	$-0.395075\pi$
$887$	19.2809	0.647389	0.323695	+	0.946162i	$0.395075\pi$
$888$	0	0
$889$	−43.4920	−1.45868
$890$	0	0
$891$	5.91644	0.198208
$892$	0	0
$893$	10.8359	0.362609
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−5.30507	−0.177131
$898$	0	0
$899$	16.3096	0.543957
$900$	0	0
$901$	−16.6770	−0.555591
$902$	0	0
$903$	46.3537	1.54255
$904$	0	0
$905$	0	0
$906$	0	0
$907$	43.0205	1.42847	0.714236	−	0.699905i	$-0.246774\pi$
$907$	43.0205	1.42847	0.714236	+	0.699905i	$0.246774\pi$
$908$	0	0
$909$	49.6636	1.64724
$910$	0	0
$911$	25.7733	0.853906	0.426953	−	0.904274i	$-0.359587\pi$
$911$	25.7733	0.853906	0.426953	+	0.904274i	$0.359587\pi$
$912$	0	0
$913$	−104.109	−3.44550
$914$	0	0
$915$	0	0
$916$	0	0
$917$	54.0806	1.78590
$918$	0	0
$919$	29.5897	0.976074	0.488037	−	0.872823i	$-0.337713\pi$
$919$	29.5897	0.976074	0.488037	+	0.872823i	$0.337713\pi$
$920$	0	0
$921$	74.0037	2.43851
$922$	0	0
$923$	−1.51811	−0.0499691
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−80.0440	−2.62899
$928$	0	0
$929$	−1.42334	−0.0466983	−0.0233491	−	0.999727i	$-0.507433\pi$
$929$	−1.42334	−0.0466983	−0.0233491	+	0.999727i	$0.507433\pi$
$930$	0	0
$931$	15.0455	0.493097
$932$	0	0
$933$	−32.0765	−1.05014
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−28.2276	−0.922155	−0.461077	−	0.887360i	$-0.652537\pi$
$937$	−28.2276	−0.922155	−0.461077	+	0.887360i	$0.652537\pi$
$938$	0	0
$939$	64.0556	2.09038
$940$	0	0
$941$	−15.8672	−0.517256	−0.258628	−	0.965977i	$-0.583270\pi$
$941$	−15.8672	−0.517256	−0.258628	+	0.965977i	$0.583270\pi$
$942$	0	0
$943$	−2.05748	−0.0670009
$944$	0	0
$945$	0	0
$946$	0	0
$947$	43.6718	1.41914	0.709571	−	0.704634i	$-0.248889\pi$
$947$	43.6718	1.41914	0.709571	+	0.704634i	$0.248889\pi$
$948$	0	0
$949$	−13.4178	−0.435559
$950$	0	0
$951$	38.6692	1.25393
$952$	0	0
$953$	−16.0261	−0.519136	−0.259568	−	0.965725i	$-0.583580\pi$
$953$	−16.0261	−0.519136	−0.259568	+	0.965725i	$0.583580\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−52.2626	−1.68941
$958$	0	0
$959$	70.9176	2.29005
$960$	0	0
$961$	−0.145848	−0.00470478
$962$	0	0
$963$	−8.43895	−0.271941
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−11.5987	−0.372988	−0.186494	−	0.982456i	$-0.559712\pi$
$967$	−11.5987	−0.372988	−0.186494	+	0.982456i	$0.559712\pi$
$968$	0	0
$969$	−5.32674	−0.171120
$970$	0	0
$971$	27.0362	0.867633	0.433817	−	0.901001i	$-0.357167\pi$
$971$	27.0362	0.867633	0.433817	+	0.901001i	$0.357167\pi$
$972$	0	0
$973$	−3.28903	−0.105442
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−14.1537	−0.452815	−0.226408	−	0.974033i	$-0.572698\pi$
$977$	−14.1537	−0.452815	−0.226408	+	0.974033i	$0.572698\pi$
$978$	0	0
$979$	64.0880	2.04826
$980$	0	0
$981$	11.3787	0.363293
$982$	0	0
$983$	−32.8542	−1.04788	−0.523942	−	0.851754i	$-0.675539\pi$
$983$	−32.8542	−1.04788	−0.523942	+	0.851754i	$0.675539\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−141.303	−4.49772
$988$	0	0
$989$	−6.38273	−0.202959
$990$	0	0
$991$	−47.9709	−1.52385	−0.761923	−	0.647667i	$-0.775745\pi$
$991$	−47.9709	−1.52385	−0.761923	+	0.647667i	$0.775745\pi$
$992$	0	0
$993$	2.04285	0.0648279
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−44.1716	−1.39893	−0.699464	−	0.714668i	$-0.746578\pi$
$997$	−44.1716	−1.39893	−0.699464	+	0.714668i	$0.746578\pi$
$998$	0	0
$999$	54.2951	1.71782

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7600.2.a.cb.1.3		3
4.3	odd	2		950.2.a.k.1.1	✓	3
5.4	even	2		7600.2.a.bm.1.1		3
12.11	even	2		8550.2.a.co.1.3		3
20.3	even	4		950.2.b.g.799.4		6
20.7	even	4		950.2.b.g.799.3		6
20.19	odd	2		950.2.a.m.1.3	yes	3
60.59	even	2		8550.2.a.cj.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.a.k.1.1	✓	3	4.3	odd	2
950.2.a.m.1.3	yes	3	20.19	odd	2
950.2.b.g.799.3		6	20.7	even	4
950.2.b.g.799.4		6	20.3	even	4
7600.2.a.bm.1.1		3	5.4	even	2
7600.2.a.cb.1.3		3	1.1	even	1	trivial
8550.2.a.cj.1.1		3	60.59	even	2
8550.2.a.co.1.3		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)

\(f(q)\)	\(=\)	\(q+2.77733 q^{3} -4.69527 q^{7} +4.71354 q^{9} +O(q^{10})\)	\(q+2.77733 q^{3} -4.69527 q^{7} +4.71354 q^{9} -6.40880 q^{11} -1.06379 q^{13} -1.91794 q^{17} +1.00000 q^{19} -13.0403 q^{21} +1.79560 q^{23} +4.75905 q^{27} +2.93621 q^{29} +5.55465 q^{31} -17.7993 q^{33} +11.4088 q^{37} -2.95449 q^{39} -1.14585 q^{41} -3.55465 q^{43} +10.8359 q^{47} +15.0455 q^{49} -5.32674 q^{51} +8.69527 q^{53} +2.77733 q^{57} +5.63148 q^{59} -3.39053 q^{61} -22.1313 q^{63} -8.82284 q^{67} +4.98696 q^{69} +1.42708 q^{71} +12.6132 q^{73} +30.0910 q^{77} +1.96345 q^{79} -0.923174 q^{81} +16.2447 q^{83} +8.15482 q^{87} -10.0000 q^{89} +4.99477 q^{91} +15.4271 q^{93} +14.9452 q^{97} -30.2081 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} + 2 q^{7} + 13 q^{9}+O(q^{10}) \) 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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