LMFDB - Embedded newform 7600.2.a.bk.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:	$1$
Dimension:	$3$
Coefficient field:	3.3.993.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x + 3 $ x^3 - x^2 - 6*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			$0.480031$ 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)

$f(q)$	$=$	$q-0.519969 q^{3} -4.76957 q^{7} -2.72963 q^{9} +O(q^{10})$	$q-0.519969 q^{3} -4.76957 q^{7} -2.72963 q^{9} -0.960061 q^{11} +2.24960 q^{13} -0.249601 q^{17} -1.00000 q^{19} +2.48003 q^{21} +9.01917 q^{23} +2.97923 q^{27} +6.24960 q^{29} -2.96006 q^{31} +0.499202 q^{33} -0.0399387 q^{37} -1.16972 q^{39} +4.96006 q^{43} -9.49920 q^{47} +15.7488 q^{49} +0.129785 q^{51} -6.84945 q^{53} +0.519969 q^{57} +14.5583 q^{59} +7.53914 q^{61} +13.0192 q^{63} -5.72963 q^{67} -4.68969 q^{69} +9.61902 q^{71} -12.0591 q^{73} +4.57908 q^{77} -6.07988 q^{79} +6.63979 q^{81} -7.45926 q^{83} -3.24960 q^{87} +4.07988 q^{89} -10.7296 q^{91} +1.53914 q^{93} -18.4193 q^{97} +2.62061 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{3} - 2 q^{7} + 5 q^{9}+O(q^{10}) $ 3 * q - 2 * q^3 - 2 * q^7 + 5 * q^9	$ 3 q - 2 q^{3} - 2 q^{7} + 5 q^{9} - 2 q^{11} - 6 q^{13} + 12 q^{17} - 3 q^{19} + 7 q^{21} + 2 q^{23} - 17 q^{27} + 6 q^{29} - 8 q^{31} - 24 q^{33} - q^{37} + 11 q^{39} + 14 q^{43} - 3 q^{47} + 9 q^{49} - 15 q^{51} - 10 q^{53} + 2 q^{57} - 6 q^{59} - 2 q^{61} + 14 q^{63} - 4 q^{67} + 6 q^{71} - 12 q^{73} - 10 q^{77} - 20 q^{79} + 23 q^{81} + 4 q^{83} + 3 q^{87} + 14 q^{89} - 19 q^{91} - 20 q^{93} - 28 q^{97} + 36 q^{99}+O(q^{100}) $ 3 * q - 2 * q^3 - 2 * q^7 + 5 * q^9 - 2 * q^11 - 6 * q^13 + 12 * q^17 - 3 * q^19 + 7 * q^21 + 2 * q^23 - 17 * q^27 + 6 * q^29 - 8 * q^31 - 24 * q^33 - q^37 + 11 * q^39 + 14 * q^43 - 3 * q^47 + 9 * q^49 - 15 * q^51 - 10 * q^53 + 2 * q^57 - 6 * q^59 - 2 * q^61 + 14 * q^63 - 4 * q^67 + 6 * q^71 - 12 * q^73 - 10 * q^77 - 20 * q^79 + 23 * q^81 + 4 * q^83 + 3 * q^87 + 14 * q^89 - 19 * q^91 - 20 * q^93 - 28 * q^97 + 36 * 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
$3$	−0.519969	−0.300204	−0.150102	−	0.988670i	$-0.547960\pi$
$3$	−0.519969	−0.300204	−0.150102	+	0.988670i	$0.547960\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.76957	−1.80273	−0.901364	−	0.433062i	$-0.857433\pi$
$7$	−4.76957	−1.80273	−0.901364	+	0.433062i	$0.857433\pi$
$8$	0	0
$9$	−2.72963	−0.909877
$10$	0	0
$11$	−0.960061	−0.289469	−0.144735	−	0.989471i	$-0.546233\pi$
$11$	−0.960061	−0.289469	−0.144735	+	0.989471i	$0.546233\pi$
$12$	0	0
$13$	2.24960	0.623927	0.311964	−	0.950094i	$-0.399013\pi$
$13$	2.24960	0.623927	0.311964	+	0.950094i	$0.399013\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.249601	−0.0605372	−0.0302686	−	0.999542i	$-0.509636\pi$
$17$	−0.249601	−0.0605372	−0.0302686	+	0.999542i	$0.509636\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	2.48003	0.541187
$22$	0	0
$23$	9.01917	1.88063	0.940314	−	0.340309i	$-0.110532\pi$
$23$	9.01917	1.88063	0.940314	+	0.340309i	$0.110532\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	2.97923	0.573354
$28$	0	0
$29$	6.24960	1.16052	0.580261	−	0.814431i	$-0.302951\pi$
$29$	6.24960	1.16052	0.580261	+	0.814431i	$0.302951\pi$
$30$	0	0
$31$	−2.96006	−0.531643	−0.265821	−	0.964022i	$-0.585643\pi$
$31$	−2.96006	−0.531643	−0.265821	+	0.964022i	$0.585643\pi$
$32$	0	0
$33$	0.499202	0.0869000
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−0.0399387	−0.00656589	−0.00328294	−	0.999995i	$-0.501045\pi$
$37$	−0.0399387	−0.00656589	−0.00328294	+	0.999995i	$0.501045\pi$
$38$	0	0
$39$	−1.16972	−0.187306
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	4.96006	0.756402	0.378201	−	0.925723i	$-0.376543\pi$
$43$	4.96006	0.756402	0.378201	+	0.925723i	$0.376543\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.49920	−1.38560	−0.692801	−	0.721129i	$-0.743623\pi$
$47$	−9.49920	−1.38560	−0.692801	+	0.721129i	$0.743623\pi$
$48$	0	0
$49$	15.7488	2.24983
$50$	0	0
$51$	0.129785	0.0181735
$52$	0	0
$53$	−6.84945	−0.940844	−0.470422	−	0.882442i	$-0.655898\pi$
$53$	−6.84945	−0.940844	−0.470422	+	0.882442i	$0.655898\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0.519969	0.0688716
$58$	0	0
$59$	14.5583	1.89533	0.947665	−	0.319265i	$-0.103436\pi$
$59$	14.5583	1.89533	0.947665	+	0.319265i	$0.103436\pi$
$60$	0	0
$61$	7.53914	0.965288	0.482644	−	0.875817i	$-0.339676\pi$
$61$	7.53914	0.965288	0.482644	+	0.875817i	$0.339676\pi$
$62$	0	0
$63$	13.0192	1.64026
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−5.72963	−0.699986	−0.349993	−	0.936752i	$-0.613816\pi$
$67$	−5.72963	−0.699986	−0.349993	+	0.936752i	$0.613816\pi$
$68$	0	0
$69$	−4.68969	−0.564573
$70$	0	0
$71$	9.61902	1.14157	0.570784	−	0.821100i	$-0.306639\pi$
$71$	9.61902	1.14157	0.570784	+	0.821100i	$0.306639\pi$
$72$	0	0
$73$	−12.0591	−1.41141	−0.705706	−	0.708505i	$-0.749370\pi$
$73$	−12.0591	−1.41141	−0.705706	+	0.708505i	$0.749370\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.57908	0.521835
$78$	0	0
$79$	−6.07988	−0.684040	−0.342020	−	0.939693i	$-0.611111\pi$
$79$	−6.07988	−0.684040	−0.342020	+	0.939693i	$0.611111\pi$
$80$	0	0
$81$	6.63979	0.737754
$82$	0	0
$83$	−7.45926	−0.818761	−0.409380	−	0.912364i	$-0.634255\pi$
$83$	−7.45926	−0.818761	−0.409380	+	0.912364i	$0.634255\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−3.24960	−0.348394
$88$	0	0
$89$	4.07988	0.432466	0.216233	−	0.976342i	$-0.430623\pi$
$89$	4.07988	0.432466	0.216233	+	0.976342i	$0.430623\pi$
$90$	0	0
$91$	−10.7296	−1.12477
$92$	0	0
$93$	1.53914	0.159602
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−18.4193	−1.87020	−0.935100	−	0.354385i	$-0.884690\pi$
$97$	−18.4193	−1.87020	−0.935100	+	0.354385i	$0.884690\pi$
$98$	0	0
$99$	2.62061	0.263382
$100$	0	0
$101$	5.53914	0.551165	0.275583	−	0.961277i	$-0.411129\pi$
$101$	5.53914	0.551165	0.275583	+	0.961277i	$0.411129\pi$
$102$	0	0
$103$	−6.57908	−0.648256	−0.324128	−	0.946013i	$-0.605071\pi$
$103$	−6.57908	−0.648256	−0.324128	+	0.946013i	$0.605071\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.9086	1.44126	0.720632	−	0.693317i	$-0.243852\pi$
$107$	14.9086	1.44126	0.720632	+	0.693317i	$0.243852\pi$
$108$	0	0
$109$	4.76957	0.456842	0.228421	−	0.973562i	$-0.426644\pi$
$109$	4.76957	0.456842	0.228421	+	0.973562i	$0.426644\pi$
$110$	0	0
$111$	0.0207669	0.00197111
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−6.14058	−0.567697
$118$	0	0
$119$	1.19049	0.109132
$120$	0	0
$121$	−10.0783	−0.916207
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	17.9585	1.59356	0.796778	−	0.604272i	$-0.206536\pi$
$127$	17.9585	1.59356	0.796778	+	0.604272i	$0.206536\pi$
$128$	0	0
$129$	−2.57908	−0.227075
$130$	0	0
$131$	6.96006	0.608103	0.304052	−	0.952656i	$-0.401660\pi$
$131$	6.96006	0.608103	0.304052	+	0.952656i	$0.401660\pi$
$132$	0	0
$133$	4.76957	0.413574
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−7.80951	−0.667211	−0.333606	−	0.942713i	$-0.608265\pi$
$137$	−7.80951	−0.667211	−0.333606	+	0.942713i	$0.608265\pi$
$138$	0	0
$139$	16.0383	1.36035	0.680177	−	0.733048i	$-0.261903\pi$
$139$	16.0383	1.36035	0.680177	+	0.733048i	$0.261903\pi$
$140$	0	0
$141$	4.93929	0.415964
$142$	0	0
$143$	−2.15975	−0.180608
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−8.18890	−0.675409
$148$	0	0
$149$	−8.41932	−0.689738	−0.344869	−	0.938651i	$-0.612077\pi$
$149$	−8.41932	−0.689738	−0.344869	+	0.938651i	$0.612077\pi$
$150$	0	0
$151$	−20.0000	−1.62758	−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000	−1.62758	−0.813788	+	0.581161i	$0.802599\pi$
$152$	0	0
$153$	0.681319	0.0550814
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−4.96006	−0.395856	−0.197928	−	0.980217i	$-0.563421\pi$
$157$	−4.96006	−0.395856	−0.197928	+	0.980217i	$0.563421\pi$
$158$	0	0
$159$	3.56150	0.282446
$160$	0	0
$161$	−43.0176	−3.39026
$162$	0	0
$163$	−21.4593	−1.68082	−0.840410	−	0.541952i	$-0.817686\pi$
$163$	−21.4593	−1.68082	−0.840410	+	0.541952i	$0.817686\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−11.0399	−0.854296	−0.427148	−	0.904182i	$-0.640482\pi$
$167$	−11.0399	−0.854296	−0.427148	+	0.904182i	$0.640482\pi$
$168$	0	0
$169$	−7.93929	−0.610715
$170$	0	0
$171$	2.72963	0.208740
$172$	0	0
$173$	−16.9585	−1.28933	−0.644664	−	0.764466i	$-0.723003\pi$
$173$	−16.9585	−1.28933	−0.644664	+	0.764466i	$0.723003\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−7.56988	−0.568987
$178$	0	0
$179$	−2.53914	−0.189784	−0.0948922	−	0.995488i	$-0.530251\pi$
$179$	−2.53914	−0.189784	−0.0948922	+	0.995488i	$0.530251\pi$
$180$	0	0
$181$	−3.88018	−0.288412	−0.144206	−	0.989548i	$-0.546063\pi$
$181$	−3.88018	−0.288412	−0.144206	+	0.989548i	$0.546063\pi$
$182$	0	0
$183$	−3.92012	−0.289784
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0.239632	0.0175237
$188$	0	0
$189$	−14.2097	−1.03360
$190$	0	0
$191$	−2.05911	−0.148992	−0.0744960	−	0.997221i	$-0.523735\pi$
$191$	−2.05911	−0.148992	−0.0744960	+	0.997221i	$0.523735\pi$
$192$	0	0
$193$	8.46086	0.609026	0.304513	−	0.952508i	$-0.401506\pi$
$193$	8.46086	0.609026	0.304513	+	0.952508i	$0.401506\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−13.9201	−0.991768	−0.495884	−	0.868389i	$-0.665156\pi$
$197$	−13.9201	−0.991768	−0.495884	+	0.868389i	$0.665156\pi$
$198$	0	0
$199$	9.15055	0.648665	0.324333	−	0.945943i	$-0.394860\pi$
$199$	9.15055	0.648665	0.324333	+	0.945943i	$0.394860\pi$
$200$	0	0
$201$	2.97923	0.210139
$202$	0	0
$203$	−29.8079	−2.09211
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−24.6190	−1.71114
$208$	0	0
$209$	0.960061	0.0664088
$210$	0	0
$211$	−16.5200	−1.13728	−0.568641	−	0.822586i	$-0.692531\pi$
$211$	−16.5200	−1.13728	−0.568641	+	0.822586i	$0.692531\pi$
$212$	0	0
$213$	−5.00160	−0.342704
$214$	0	0
$215$	0	0
$216$	0	0
$217$	14.1182	0.958407
$218$	0	0
$219$	6.27037	0.423712
$220$	0	0
$221$	−0.561503	−0.0377708
$222$	0	0
$223$	9.03994	0.605359	0.302680	−	0.953092i	$-0.402119\pi$
$223$	9.03994	0.605359	0.302680	+	0.953092i	$0.402119\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0.350246	0.0232466	0.0116233	−	0.999932i	$-0.496300\pi$
$227$	0.350246	0.0232466	0.0116233	+	0.999932i	$0.496300\pi$
$228$	0	0
$229$	−8.07988	−0.533933	−0.266967	−	0.963706i	$-0.586021\pi$
$229$	−8.07988	−0.533933	−0.266967	+	0.963706i	$0.586021\pi$
$230$	0	0
$231$	−2.38098	−0.156657
$232$	0	0
$233$	4.92012	0.322328	0.161164	−	0.986928i	$-0.448475\pi$
$233$	4.92012	0.322328	0.161164	+	0.986928i	$0.448475\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	3.16135	0.205352
$238$	0	0
$239$	−2.98083	−0.192814	−0.0964069	−	0.995342i	$-0.530735\pi$
$239$	−2.98083	−0.192814	−0.0964069	+	0.995342i	$0.530735\pi$
$240$	0	0
$241$	20.0383	1.29078	0.645392	−	0.763852i	$-0.276694\pi$
$241$	20.0383	1.29078	0.645392	+	0.763852i	$0.276694\pi$
$242$	0	0
$243$	−12.3902	−0.794831
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2.24960	−0.143139
$248$	0	0
$249$	3.87859	0.245796
$250$	0	0
$251$	14.8802	0.939229	0.469614	−	0.882872i	$-0.344393\pi$
$251$	14.8802	0.939229	0.469614	+	0.882872i	$0.344393\pi$
$252$	0	0
$253$	−8.65896	−0.544384
$254$	0	0
$255$	0	0
$256$	0	0
$257$	11.0399	0.688652	0.344326	−	0.938850i	$-0.388107\pi$
$257$	11.0399	0.688652	0.344326	+	0.938850i	$0.388107\pi$
$258$	0	0
$259$	0.190491	0.0118365
$260$	0	0
$261$	−17.0591	−1.05593
$262$	0	0
$263$	−3.84025	−0.236800	−0.118400	−	0.992966i	$-0.537776\pi$
$263$	−3.84025	−0.236800	−0.118400	+	0.992966i	$0.537776\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−2.12141	−0.129828
$268$	0	0
$269$	23.1981	1.41441	0.707207	−	0.707007i	$-0.249955\pi$
$269$	23.1981	1.41441	0.707207	+	0.707007i	$0.249955\pi$
$270$	0	0
$271$	−23.9792	−1.45663	−0.728317	−	0.685240i	$-0.759697\pi$
$271$	−23.9792	−1.45663	−0.728317	+	0.685240i	$0.759697\pi$
$272$	0	0
$273$	5.57908	0.337661
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−24.6590	−1.48161	−0.740807	−	0.671718i	$-0.765556\pi$
$277$	−24.6590	−1.48161	−0.740807	+	0.671718i	$0.765556\pi$
$278$	0	0
$279$	8.07988	0.483730
$280$	0	0
$281$	−18.9984	−1.13335	−0.566675	−	0.823941i	$-0.691770\pi$
$281$	−18.9984	−1.13335	−0.566675	+	0.823941i	$0.691770\pi$
$282$	0	0
$283$	29.0367	1.72606	0.863028	−	0.505156i	$-0.168565\pi$
$283$	29.0367	1.72606	0.863028	+	0.505156i	$0.168565\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−16.9377	−0.996335
$290$	0	0
$291$	9.57748	0.561442
$292$	0	0
$293$	−6.24960	−0.365106	−0.182553	−	0.983196i	$-0.558436\pi$
$293$	−6.24960	−0.365106	−0.182553	+	0.983196i	$0.558436\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−2.86025	−0.165968
$298$	0	0
$299$	20.2895	1.17337
$300$	0	0
$301$	−23.6574	−1.36359
$302$	0	0
$303$	−2.88018	−0.165462
$304$	0	0
$305$	0	0
$306$	0	0
$307$	12.5391	0.715647	0.357823	−	0.933789i	$-0.383519\pi$
$307$	12.5391	0.715647	0.357823	+	0.933789i	$0.383519\pi$
$308$	0	0
$309$	3.42092	0.194609
$310$	0	0
$311$	−7.84785	−0.445011	−0.222505	−	0.974931i	$-0.571424\pi$
$311$	−7.84785	−0.445011	−0.222505	+	0.974931i	$0.571424\pi$
$312$	0	0
$313$	−10.9884	−0.621103	−0.310552	−	0.950557i	$-0.600514\pi$
$313$	−10.9884	−0.621103	−0.310552	+	0.950557i	$0.600514\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	21.5567	1.21075	0.605373	−	0.795942i	$-0.293024\pi$
$317$	21.5567	1.21075	0.605373	+	0.795942i	$0.293024\pi$
$318$	0	0
$319$	−6.00000	−0.335936
$320$	0	0
$321$	−7.75199	−0.432674
$322$	0	0
$323$	0.249601	0.0138882
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−2.48003	−0.137146
$328$	0	0
$329$	45.3071	2.49786
$330$	0	0
$331$	−33.4876	−1.84065	−0.920324	−	0.391158i	$-0.872075\pi$
$331$	−33.4876	−1.84065	−0.920324	+	0.391158i	$0.872075\pi$
$332$	0	0
$333$	0.109018	0.00597415
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−5.69890	−0.310439	−0.155219	−	0.987880i	$-0.549608\pi$
$337$	−5.69890	−0.310439	−0.155219	+	0.987880i	$0.549608\pi$
$338$	0	0
$339$	3.11982	0.169445
$340$	0	0
$341$	2.84184	0.153894
$342$	0	0
$343$	−41.7280	−2.25310
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−0.239632	−0.0128641	−0.00643207	−	0.999979i	$-0.502047\pi$
$347$	−0.239632	−0.0128641	−0.00643207	+	0.999979i	$0.502047\pi$
$348$	0	0
$349$	−17.2380	−0.922731	−0.461365	−	0.887210i	$-0.652640\pi$
$349$	−17.2380	−0.922731	−0.461365	+	0.887210i	$0.652640\pi$
$350$	0	0
$351$	6.70209	0.357731
$352$	0	0
$353$	34.7280	1.84839	0.924193	−	0.381925i	$-0.124739\pi$
$353$	34.7280	1.84839	0.924193	+	0.381925i	$0.124739\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−0.619019	−0.0327619
$358$	0	0
$359$	−26.6689	−1.40753	−0.703766	−	0.710432i	$-0.748500\pi$
$359$	−26.6689	−1.40753	−0.703766	+	0.710432i	$0.748500\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	5.24040	0.275050
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−15.2380	−0.795419	−0.397710	−	0.917511i	$-0.630195\pi$
$367$	−15.2380	−0.795419	−0.397710	+	0.917511i	$0.630195\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	32.6689	1.69609
$372$	0	0
$373$	26.6382	1.37927	0.689637	−	0.724156i	$-0.257770\pi$
$373$	26.6382	1.37927	0.689637	+	0.724156i	$0.257770\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	14.0591	0.724081
$378$	0	0
$379$	−11.9693	−0.614820	−0.307410	−	0.951577i	$-0.599462\pi$
$379$	−11.9693	−0.614820	−0.307410	+	0.951577i	$0.599462\pi$
$380$	0	0
$381$	−9.33785	−0.478393
$382$	0	0
$383$	−9.84025	−0.502813	−0.251407	−	0.967882i	$-0.580893\pi$
$383$	−9.84025	−0.502813	−0.251407	+	0.967882i	$0.580893\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−13.5391	−0.688233
$388$	0	0
$389$	−8.65896	−0.439027	−0.219513	−	0.975610i	$-0.570447\pi$
$389$	−8.65896	−0.439027	−0.219513	+	0.975610i	$0.570447\pi$
$390$	0	0
$391$	−2.25120	−0.113848
$392$	0	0
$393$	−3.61902	−0.182555
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−28.9984	−1.45539	−0.727694	−	0.685902i	$-0.759408\pi$
$397$	−28.9984	−1.45539	−0.727694	+	0.685902i	$0.759408\pi$
$398$	0	0
$399$	−2.48003	−0.124157
$400$	0	0
$401$	22.6174	1.12946	0.564730	−	0.825276i	$-0.308980\pi$
$401$	22.6174	1.12946	0.564730	+	0.825276i	$0.308980\pi$
$402$	0	0
$403$	−6.65896	−0.331706
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.0383436	0.00190062
$408$	0	0
$409$	5.34104	0.264098	0.132049	−	0.991243i	$-0.457844\pi$
$409$	5.34104	0.264098	0.132049	+	0.991243i	$0.457844\pi$
$410$	0	0
$411$	4.06071	0.200300
$412$	0	0
$413$	−69.4369	−3.41677
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−8.33945	−0.408384
$418$	0	0
$419$	22.1566	1.08242	0.541210	−	0.840888i	$-0.317967\pi$
$419$	22.1566	1.08242	0.541210	+	0.840888i	$0.317967\pi$
$420$	0	0
$421$	29.2787	1.42696	0.713479	−	0.700676i	$-0.247118\pi$
$421$	29.2787	1.42696	0.713479	+	0.700676i	$0.247118\pi$
$422$	0	0
$423$	25.9293	1.26073
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−35.9585	−1.74015
$428$	0	0
$429$	1.12301	0.0542193
$430$	0	0
$431$	−21.3794	−1.02981	−0.514904	−	0.857248i	$-0.672173\pi$
$431$	−21.3794	−1.02981	−0.514904	+	0.857248i	$0.672173\pi$
$432$	0	0
$433$	36.1182	1.73573	0.867865	−	0.496799i	$-0.165491\pi$
$433$	36.1182	1.73573	0.867865	+	0.496799i	$0.165491\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−9.01917	−0.431445
$438$	0	0
$439$	−20.9984	−1.00220	−0.501100	−	0.865390i	$-0.667071\pi$
$439$	−20.9984	−1.00220	−0.501100	+	0.865390i	$0.667071\pi$
$440$	0	0
$441$	−42.9884	−2.04707
$442$	0	0
$443$	−31.4976	−1.49650	−0.748248	−	0.663419i	$-0.769105\pi$
$443$	−31.4976	−1.49650	−0.748248	+	0.663419i	$0.769105\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	4.37779	0.207062
$448$	0	0
$449$	18.9601	0.894781	0.447390	−	0.894339i	$-0.352353\pi$
$449$	18.9601	0.894781	0.447390	+	0.894339i	$0.352353\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	10.3994	0.488606
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6.32948	0.296081	0.148040	−	0.988981i	$-0.452703\pi$
$457$	6.32948	0.296081	0.148040	+	0.988981i	$0.452703\pi$
$458$	0	0
$459$	−0.743620	−0.0347092
$460$	0	0
$461$	−6.49920	−0.302698	−0.151349	−	0.988480i	$-0.548362\pi$
$461$	−6.49920	−0.302698	−0.151349	+	0.988480i	$0.548362\pi$
$462$	0	0
$463$	28.4177	1.32068	0.660342	−	0.750965i	$-0.270411\pi$
$463$	28.4177	1.32068	0.660342	+	0.750965i	$0.270411\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	17.5775	0.813389	0.406694	−	0.913564i	$-0.366681\pi$
$467$	17.5775	0.813389	0.406694	+	0.913564i	$0.366681\pi$
$468$	0	0
$469$	27.3279	1.26188
$470$	0	0
$471$	2.57908	0.118838
$472$	0	0
$473$	−4.76196	−0.218955
$474$	0	0
$475$	0	0
$476$	0	0
$477$	18.6965	0.856053
$478$	0	0
$479$	−22.7372	−1.03889	−0.519446	−	0.854504i	$-0.673861\pi$
$479$	−22.7372	−1.03889	−0.519446	+	0.854504i	$0.673861\pi$
$480$	0	0
$481$	−0.0898462	−0.00409664
$482$	0	0
$483$	22.3678	1.01777
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−22.1981	−1.00589	−0.502946	−	0.864318i	$-0.667751\pi$
$487$	−22.1981	−1.00589	−0.502946	+	0.864318i	$0.667751\pi$
$488$	0	0
$489$	11.1582	0.504589
$490$	0	0
$491$	24.9984	1.12816	0.564081	−	0.825719i	$-0.309231\pi$
$491$	24.9984	1.12816	0.564081	+	0.825719i	$0.309231\pi$
$492$	0	0
$493$	−1.55991	−0.0702547
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−45.8786	−2.05794
$498$	0	0
$499$	−11.6574	−0.521855	−0.260928	−	0.965358i	$-0.584028\pi$
$499$	−11.6574	−0.521855	−0.260928	+	0.965358i	$0.584028\pi$
$500$	0	0
$501$	5.74043	0.256463
$502$	0	0
$503$	11.7504	0.523924	0.261962	−	0.965078i	$-0.415630\pi$
$503$	11.7504	0.523924	0.261962	+	0.965078i	$0.415630\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	4.12819	0.183339
$508$	0	0
$509$	29.9569	1.32781	0.663907	−	0.747815i	$-0.268897\pi$
$509$	29.9569	1.32781	0.663907	+	0.747815i	$0.268897\pi$
$510$	0	0
$511$	57.5168	2.54439
$512$	0	0
$513$	−2.97923	−0.131536
$514$	0	0
$515$	0	0
$516$	0	0
$517$	9.11982	0.401089
$518$	0	0
$519$	8.81788	0.387062
$520$	0	0
$521$	−15.1198	−0.662411	−0.331206	−	0.943559i	$-0.607455\pi$
$521$	−15.1198	−0.662411	−0.331206	+	0.943559i	$0.607455\pi$
$522$	0	0
$523$	15.6498	0.684316	0.342158	−	0.939642i	$-0.388842\pi$
$523$	15.6498	0.684316	0.342158	+	0.939642i	$0.388842\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.738835	0.0321842
$528$	0	0
$529$	58.3455	2.53676
$530$	0	0
$531$	−39.7388	−1.72452
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.32028	0.0569741
$538$	0	0
$539$	−15.1198	−0.651257
$540$	0	0
$541$	−27.2995	−1.17370	−0.586849	−	0.809697i	$-0.699632\pi$
$541$	−27.2995	−1.17370	−0.586849	+	0.809697i	$0.699632\pi$
$542$	0	0
$543$	2.01758	0.0865825
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−33.9968	−1.45360	−0.726799	−	0.686850i	$-0.758993\pi$
$547$	−33.9968	−1.45360	−0.726799	+	0.686850i	$0.758993\pi$
$548$	0	0
$549$	−20.5791	−0.878294
$550$	0	0
$551$	−6.24960	−0.266242
$552$	0	0
$553$	28.9984	1.23314
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−33.8786	−1.43548	−0.717741	−	0.696310i	$-0.754824\pi$
$557$	−33.8786	−1.43548	−0.717741	+	0.696310i	$0.754824\pi$
$558$	0	0
$559$	11.1582	0.471940
$560$	0	0
$561$	−0.124602	−0.00526068
$562$	0	0
$563$	32.2995	1.36126	0.680631	−	0.732626i	$-0.261706\pi$
$563$	32.2995	1.36126	0.680631	+	0.732626i	$0.261706\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−31.6689	−1.32997
$568$	0	0
$569$	−6.73883	−0.282507	−0.141253	−	0.989973i	$-0.545113\pi$
$569$	−6.73883	−0.282507	−0.141253	+	0.989973i	$0.545113\pi$
$570$	0	0
$571$	−34.4193	−1.44040	−0.720202	−	0.693764i	$-0.755951\pi$
$571$	−34.4193	−1.44040	−0.720202	+	0.693764i	$0.755951\pi$
$572$	0	0
$573$	1.07067	0.0447281
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−28.1773	−1.17304	−0.586519	−	0.809936i	$-0.699502\pi$
$577$	−28.1773	−1.17304	−0.586519	+	0.809936i	$0.699502\pi$
$578$	0	0
$579$	−4.39939	−0.182832
$580$	0	0
$581$	35.5775	1.47600
$582$	0	0
$583$	6.57589	0.272346
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−34.6174	−1.42881	−0.714407	−	0.699730i	$-0.753303\pi$
$587$	−34.6174	−1.42881	−0.714407	+	0.699730i	$0.753303\pi$
$588$	0	0
$589$	2.96006	0.121967
$590$	0	0
$591$	7.23804	0.297733
$592$	0	0
$593$	3.99840	0.164195	0.0820974	−	0.996624i	$-0.473838\pi$
$593$	3.99840	0.164195	0.0820974	+	0.996624i	$0.473838\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−4.75801	−0.194732
$598$	0	0
$599$	42.5759	1.73960	0.869802	−	0.493401i	$-0.164247\pi$
$599$	42.5759	1.73960	0.869802	+	0.493401i	$0.164247\pi$
$600$	0	0
$601$	−18.4577	−0.752904	−0.376452	−	0.926436i	$-0.622856\pi$
$601$	−18.4577	−0.752904	−0.376452	+	0.926436i	$0.622856\pi$
$602$	0	0
$603$	15.6398	0.636901
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−12.5791	−0.510569	−0.255285	−	0.966866i	$-0.582169\pi$
$607$	−12.5791	−0.510569	−0.255285	+	0.966866i	$0.582169\pi$
$608$	0	0
$609$	15.4992	0.628059
$610$	0	0
$611$	−21.3694	−0.864514
$612$	0	0
$613$	26.5759	1.07339	0.536695	−	0.843776i	$-0.319673\pi$
$613$	26.5759	1.07339	0.536695	+	0.843776i	$0.319673\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	37.8386	1.52333	0.761663	−	0.647973i	$-0.224383\pi$
$617$	37.8386	1.52333	0.761663	+	0.647973i	$0.224383\pi$
$618$	0	0
$619$	−30.1566	−1.21209	−0.606047	−	0.795429i	$-0.707246\pi$
$619$	−30.1566	−1.21209	−0.606047	+	0.795429i	$0.707246\pi$
$620$	0	0
$621$	26.8702	1.07826
$622$	0	0
$623$	−19.4593	−0.779619
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−0.499202	−0.0199362
$628$	0	0
$629$	0.00996876	0.000397480 0
$630$	0	0
$631$	25.4992	1.01511	0.507554	−	0.861620i	$-0.330550\pi$
$631$	25.4992	1.01511	0.507554	+	0.861620i	$0.330550\pi$
$632$	0	0
$633$	8.58988	0.341417
$634$	0	0
$635$	0	0
$636$	0	0
$637$	35.4285	1.40373
$638$	0	0
$639$	−26.2564	−1.03869
$640$	0	0
$641$	39.1965	1.54817	0.774084	−	0.633082i	$-0.218211\pi$
$641$	39.1965	1.54817	0.774084	+	0.633082i	$0.218211\pi$
$642$	0	0
$643$	−36.3778	−1.43460	−0.717300	−	0.696764i	$-0.754622\pi$
$643$	−36.3778	−1.43460	−0.717300	+	0.696764i	$0.754622\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−10.6897	−0.420255	−0.210128	−	0.977674i	$-0.567388\pi$
$647$	−10.6897	−0.420255	−0.210128	+	0.977674i	$0.567388\pi$
$648$	0	0
$649$	−13.9769	−0.548640
$650$	0	0
$651$	−7.34104	−0.287718
$652$	0	0
$653$	−36.0383	−1.41029	−0.705145	−	0.709063i	$-0.749118\pi$
$653$	−36.0383	−1.41029	−0.705145	+	0.709063i	$0.749118\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	32.9169	1.28421
$658$	0	0
$659$	−17.1889	−0.669584	−0.334792	−	0.942292i	$-0.608666\pi$
$659$	−17.1889	−0.669584	−0.334792	+	0.942292i	$0.608666\pi$
$660$	0	0
$661$	48.9054	1.90220	0.951099	−	0.308886i	$-0.0999560\pi$
$661$	48.9054	1.90220	0.951099	+	0.308886i	$0.0999560\pi$
$662$	0	0
$663$	0.291964	0.0113390
$664$	0	0
$665$	0	0
$666$	0	0
$667$	56.3662	2.18251
$668$	0	0
$669$	−4.70049	−0.181731
$670$	0	0
$671$	−7.23804	−0.279421
$672$	0	0
$673$	6.57908	0.253605	0.126802	−	0.991928i	$-0.459529\pi$
$673$	6.57908	0.253605	0.126802	+	0.991928i	$0.459529\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−38.3087	−1.47232	−0.736162	−	0.676806i	$-0.763364\pi$
$677$	−38.3087	−1.47232	−0.736162	+	0.676806i	$0.763364\pi$
$678$	0	0
$679$	87.8523	3.37146
$680$	0	0
$681$	−0.182117	−0.00697874
$682$	0	0
$683$	−7.54074	−0.288538	−0.144269	−	0.989538i	$-0.546083\pi$
$683$	−7.54074	−0.288538	−0.144269	+	0.989538i	$0.546083\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	4.20129	0.160289
$688$	0	0
$689$	−15.4085	−0.587018
$690$	0	0
$691$	−28.4193	−1.08112	−0.540561	−	0.841305i	$-0.681788\pi$
$691$	−28.4193	−1.08112	−0.540561	+	0.841305i	$0.681788\pi$
$692$	0	0
$693$	−12.4992	−0.474805
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−2.55831	−0.0967643
$700$	0	0
$701$	−24.2596	−0.916271	−0.458136	−	0.888882i	$-0.651483\pi$
$701$	−24.2596	−0.916271	−0.458136	+	0.888882i	$0.651483\pi$
$702$	0	0
$703$	0.0399387	0.00150632
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−26.4193	−0.993601
$708$	0	0
$709$	21.7372	0.816359	0.408180	−	0.912902i	$-0.366164\pi$
$709$	21.7372	0.816359	0.408180	+	0.912902i	$0.366164\pi$
$710$	0	0
$711$	16.5958	0.622392
$712$	0	0
$713$	−26.6973	−0.999822
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1.54994	0.0578835
$718$	0	0
$719$	48.9361	1.82501	0.912504	−	0.409067i	$-0.134146\pi$
$719$	48.9361	1.82501	0.912504	+	0.409067i	$0.134146\pi$
$720$	0	0
$721$	31.3794	1.16863
$722$	0	0
$723$	−10.4193	−0.387499
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−4.29874	−0.159432	−0.0797158	−	0.996818i	$-0.525401\pi$
$727$	−4.29874	−0.159432	−0.0797158	+	0.996818i	$0.525401\pi$
$728$	0	0
$729$	−13.4768	−0.499142
$730$	0	0
$731$	−1.23804	−0.0457905
$732$	0	0
$733$	12.0415	0.444764	0.222382	−	0.974960i	$-0.428617\pi$
$733$	12.0415	0.444764	0.222382	+	0.974960i	$0.428617\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.50080	0.202624
$738$	0	0
$739$	−6.61742	−0.243426	−0.121713	−	0.992565i	$-0.538839\pi$
$739$	−6.61742	−0.243426	−0.121713	+	0.992565i	$0.538839\pi$
$740$	0	0
$741$	1.16972	0.0429709
$742$	0	0
$743$	−47.6158	−1.74686	−0.873428	−	0.486954i	$-0.838108\pi$
$743$	−47.6158	−1.74686	−0.873428	+	0.486954i	$0.838108\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	20.3610	0.744972
$748$	0	0
$749$	−71.1074	−2.59821
$750$	0	0
$751$	25.6574	0.936250	0.468125	−	0.883662i	$-0.344930\pi$
$751$	25.6574	0.936250	0.468125	+	0.883662i	$0.344930\pi$
$752$	0	0
$753$	−7.73724	−0.281961
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−8.83865	−0.321246	−0.160623	−	0.987016i	$-0.551350\pi$
$757$	−8.83865	−0.321246	−0.160623	+	0.987016i	$0.551350\pi$
$758$	0	0
$759$	4.50239	0.163426
$760$	0	0
$761$	−9.40776	−0.341031	−0.170516	−	0.985355i	$-0.554543\pi$
$761$	−9.40776	−0.341031	−0.170516	+	0.985355i	$0.554543\pi$
$762$	0	0
$763$	−22.7488	−0.823562
$764$	0	0
$765$	0	0
$766$	0	0
$767$	32.7504	1.18255
$768$	0	0
$769$	7.32788	0.264250	0.132125	−	0.991233i	$-0.457820\pi$
$769$	7.32788	0.264250	0.132125	+	0.991233i	$0.457820\pi$
$770$	0	0
$771$	−5.74043	−0.206737
$772$	0	0
$773$	−54.4369	−1.95796	−0.978980	−	0.203958i	$-0.934619\pi$
$773$	−54.4369	−1.95796	−0.978980	+	0.203958i	$0.934619\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−0.0990493	−0.00355337
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−9.23485	−0.330449
$782$	0	0
$783$	18.6190	0.665389
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−27.6705	−0.986348	−0.493174	−	0.869931i	$-0.664163\pi$
$787$	−27.6705	−0.986348	−0.493174	+	0.869931i	$0.664163\pi$
$788$	0	0
$789$	1.99681	0.0710883
$790$	0	0
$791$	28.6174	1.01752
$792$	0	0
$793$	16.9601	0.602269
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−21.5906	−0.764780	−0.382390	−	0.924001i	$-0.624899\pi$
$797$	−21.5906	−0.764780	−0.382390	+	0.924001i	$0.624899\pi$
$798$	0	0
$799$	2.37101	0.0838804
$800$	0	0
$801$	−11.1366	−0.393491
$802$	0	0
$803$	11.5775	0.408561
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−12.0623	−0.424613
$808$	0	0
$809$	9.72963	0.342076	0.171038	−	0.985264i	$-0.445288\pi$
$809$	9.72963	0.342076	0.171038	+	0.985264i	$0.445288\pi$
$810$	0	0
$811$	38.7280	1.35993	0.679963	−	0.733247i	$-0.261996\pi$
$811$	38.7280	1.35993	0.679963	+	0.733247i	$0.261996\pi$
$812$	0	0
$813$	12.4685	0.437288
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−4.96006	−0.173531
$818$	0	0
$819$	29.2879	1.02340
$820$	0	0
$821$	−30.2780	−1.05671	−0.528354	−	0.849024i	$-0.677191\pi$
$821$	−30.2780	−1.05671	−0.528354	+	0.849024i	$0.677191\pi$
$822$	0	0
$823$	−7.93770	−0.276691	−0.138345	−	0.990384i	$-0.544178\pi$
$823$	−7.93770	−0.276691	−0.138345	+	0.990384i	$0.544178\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	30.2496	1.05188	0.525941	−	0.850521i	$-0.323713\pi$
$827$	30.2496	1.05188	0.525941	+	0.850521i	$0.323713\pi$
$828$	0	0
$829$	−40.6765	−1.41275	−0.706377	−	0.707836i	$-0.749672\pi$
$829$	−40.6765	−1.41275	−0.706377	+	0.707836i	$0.749672\pi$
$830$	0	0
$831$	12.8219	0.444787
$832$	0	0
$833$	−3.93092	−0.136198
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−8.81871	−0.304819
$838$	0	0
$839$	−45.9553	−1.58655	−0.793276	−	0.608862i	$-0.791626\pi$
$839$	−45.9553	−1.58655	−0.793276	+	0.608862i	$0.791626\pi$
$840$	0	0
$841$	10.0575	0.346811
$842$	0	0
$843$	9.87859	0.340237
$844$	0	0
$845$	0	0
$846$	0	0
$847$	48.0691	1.65167
$848$	0	0
$849$	−15.0982	−0.518170
$850$	0	0
$851$	−0.360214	−0.0123480
$852$	0	0
$853$	−17.2196	−0.589589	−0.294794	−	0.955561i	$-0.595251\pi$
$853$	−17.2196	−0.589589	−0.294794	+	0.955561i	$0.595251\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−17.2995	−0.590940	−0.295470	−	0.955352i	$-0.595476\pi$
$857$	−17.2995	−0.590940	−0.295470	+	0.955352i	$0.595476\pi$
$858$	0	0
$859$	−17.6190	−0.601153	−0.300577	−	0.953758i	$-0.597179\pi$
$859$	−17.6190	−0.601153	−0.300577	+	0.953758i	$0.597179\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−4.07988	−0.138881	−0.0694403	−	0.997586i	$-0.522121\pi$
$863$	−4.07988	−0.138881	−0.0694403	+	0.997586i	$0.522121\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	8.80708	0.299104
$868$	0	0
$869$	5.83705	0.198009
$870$	0	0
$871$	−12.8894	−0.436740
$872$	0	0
$873$	50.2780	1.70165
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−25.2787	−0.853602	−0.426801	−	0.904345i	$-0.640360\pi$
$877$	−25.2787	−0.853602	−0.426801	+	0.904345i	$0.640360\pi$
$878$	0	0
$879$	3.24960	0.109606
$880$	0	0
$881$	−0.0814726	−0.00274488	−0.00137244	−	0.999999i	$-0.500437\pi$
$881$	−0.0814726	−0.00274488	−0.00137244	+	0.999999i	$0.500437\pi$
$882$	0	0
$883$	19.3410	0.650878	0.325439	−	0.945563i	$-0.394488\pi$
$883$	19.3410	0.650878	0.325439	+	0.945563i	$0.394488\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	5.53914	0.185986	0.0929931	−	0.995667i	$-0.470357\pi$
$887$	5.53914	0.185986	0.0929931	+	0.995667i	$0.470357\pi$
$888$	0	0
$889$	−85.6542	−2.87275
$890$	0	0
$891$	−6.37460	−0.213557
$892$	0	0
$893$	9.49920	0.317879
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−10.5499	−0.352252
$898$	0	0
$899$	−18.4992	−0.616983
$900$	0	0
$901$	1.70963	0.0569561
$902$	0	0
$903$	12.3011	0.409355
$904$	0	0
$905$	0	0
$906$	0	0
$907$	10.8111	0.358977	0.179488	−	0.983760i	$-0.442556\pi$
$907$	10.8111	0.358977	0.179488	+	0.983760i	$0.442556\pi$
$908$	0	0
$909$	−15.1198	−0.501493
$910$	0	0
$911$	−35.1166	−1.16347	−0.581733	−	0.813380i	$-0.697625\pi$
$911$	−35.1166	−1.16347	−0.581733	+	0.813380i	$0.697625\pi$
$912$	0	0
$913$	7.16135	0.237006
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−33.1965	−1.09625
$918$	0	0
$919$	−30.8287	−1.01694	−0.508472	−	0.861078i	$-0.669790\pi$
$919$	−30.8287	−1.01694	−0.508472	+	0.861078i	$0.669790\pi$
$920$	0	0
$921$	−6.51997	−0.214840
$922$	0	0
$923$	21.6390	0.712255
$924$	0	0
$925$	0	0
$926$	0	0
$927$	17.9585	0.589833
$928$	0	0
$929$	−45.0575	−1.47829	−0.739145	−	0.673547i	$-0.764770\pi$
$929$	−45.0575	−1.47829	−0.739145	+	0.673547i	$0.764770\pi$
$930$	0	0
$931$	−15.7488	−0.516146
$932$	0	0
$933$	4.08064	0.133594
$934$	0	0
$935$	0	0
$936$	0	0
$937$	22.2464	0.726759	0.363379	−	0.931641i	$-0.381623\pi$
$937$	22.2464	0.726759	0.363379	+	0.931641i	$0.381623\pi$
$938$	0	0
$939$	5.71365	0.186458
$940$	0	0
$941$	−53.9277	−1.75799	−0.878997	−	0.476828i	$-0.841787\pi$
$941$	−53.9277	−1.75799	−0.878997	+	0.476828i	$0.841787\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.76196	0.154743	0.0773715	−	0.997002i	$-0.475347\pi$
$947$	4.76196	0.154743	0.0773715	+	0.997002i	$0.475347\pi$
$948$	0	0
$949$	−27.1282	−0.880618
$950$	0	0
$951$	−11.2088	−0.363471
$952$	0	0
$953$	4.37779	0.141811	0.0709053	−	0.997483i	$-0.477411\pi$
$953$	4.37779	0.141811	0.0709053	+	0.997483i	$0.477411\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	3.11982	0.100849
$958$	0	0
$959$	37.2480	1.20280
$960$	0	0
$961$	−22.2380	−0.717356
$962$	0	0
$963$	−40.6949	−1.31137
$964$	0	0
$965$	0	0
$966$	0	0
$967$	13.1582	0.423138	0.211569	−	0.977363i	$-0.432143\pi$
$967$	13.1582	0.423138	0.211569	+	0.977363i	$0.432143\pi$
$968$	0	0
$969$	−0.129785	−0.00416929
$970$	0	0
$971$	−25.9201	−0.831816	−0.415908	−	0.909407i	$-0.636536\pi$
$971$	−25.9201	−0.831816	−0.415908	+	0.909407i	$0.636536\pi$
$972$	0	0
$973$	−76.4960	−2.45235
$974$	0	0
$975$	0	0
$976$	0	0
$977$	31.2764	1.00062	0.500310	−	0.865846i	$-0.333219\pi$
$977$	31.2764	1.00062	0.500310	+	0.865846i	$0.333219\pi$
$978$	0	0
$979$	−3.91693	−0.125186
$980$	0	0
$981$	−13.0192	−0.415670
$982$	0	0
$983$	−26.2364	−0.836813	−0.418406	−	0.908260i	$-0.637411\pi$
$983$	−26.2364	−0.836813	−0.418406	+	0.908260i	$0.637411\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−23.5583	−0.749869
$988$	0	0
$989$	44.7356	1.42251
$990$	0	0
$991$	−39.2764	−1.24766	−0.623828	−	0.781562i	$-0.714423\pi$
$991$	−39.2764	−1.24766	−0.623828	+	0.781562i	$0.714423\pi$
$992$	0	0
$993$	17.4125	0.552570
$994$	0	0
$995$	0	0
$996$	0	0
$997$	14.4609	0.457980	0.228990	−	0.973429i	$-0.426458\pi$
$997$	14.4609	0.457980	0.228990	+	0.973429i	$0.426458\pi$
$998$	0	0
$999$	−0.118987	−0.00376458

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7600.2.a.bk.1.2		3
4.3	odd	2		950.2.a.l.1.2	yes	3
5.4	even	2		7600.2.a.bz.1.2		3
12.11	even	2		8550.2.a.ci.1.3		3
20.3	even	4		950.2.b.h.799.2		6
20.7	even	4		950.2.b.h.799.5		6
20.19	odd	2		950.2.a.j.1.2	✓	3
60.59	even	2		8550.2.a.cp.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
950.2.a.j.1.2	✓	3	20.19	odd	2
950.2.a.l.1.2	yes	3	4.3	odd	2
950.2.b.h.799.2		6	20.3	even	4
950.2.b.h.799.5		6	20.7	even	4
7600.2.a.bk.1.2		3	1.1	even	1	trivial
7600.2.a.bz.1.2		3	5.4	even	2
8550.2.a.ci.1.3		3	12.11	even	2
8550.2.a.cp.1.1		3	60.59	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)

\(f(q)\)	\(=\)	\(q-0.519969 q^{3} -4.76957 q^{7} -2.72963 q^{9} +O(q^{10})\)	\(q-0.519969 q^{3} -4.76957 q^{7} -2.72963 q^{9} -0.960061 q^{11} +2.24960 q^{13} -0.249601 q^{17} -1.00000 q^{19} +2.48003 q^{21} +9.01917 q^{23} +2.97923 q^{27} +6.24960 q^{29} -2.96006 q^{31} +0.499202 q^{33} -0.0399387 q^{37} -1.16972 q^{39} +4.96006 q^{43} -9.49920 q^{47} +15.7488 q^{49} +0.129785 q^{51} -6.84945 q^{53} +0.519969 q^{57} +14.5583 q^{59} +7.53914 q^{61} +13.0192 q^{63} -5.72963 q^{67} -4.68969 q^{69} +9.61902 q^{71} -12.0591 q^{73} +4.57908 q^{77} -6.07988 q^{79} +6.63979 q^{81} -7.45926 q^{83} -3.24960 q^{87} +4.07988 q^{89} -10.7296 q^{91} +1.53914 q^{93} -18.4193 q^{97} +2.62061 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{3} - 2 q^{7} + 5 q^{9}+O(q^{10}) \) 3 * q - 2 * q^3 - 2 * q^7 + 5 * q^9	\( 3 q - 2 q^{3} - 2 q^{7} + 5 q^{9} - 2 q^{11} - 6 q^{13} + 12 q^{17} - 3 q^{19} + 7 q^{21} + 2 q^{23} - 17 q^{27} + 6 q^{29} - 8 q^{31} - 24 q^{33} - q^{37} + 11 q^{39} + 14 q^{43} - 3 q^{47} + 9 q^{49} - 15 q^{51} - 10 q^{53} + 2 q^{57} - 6 q^{59} - 2 q^{61} + 14 q^{63} - 4 q^{67} + 6 q^{71} - 12 q^{73} - 10 q^{77} - 20 q^{79} + 23 q^{81} + 4 q^{83} + 3 q^{87} + 14 q^{89} - 19 q^{91} - 20 q^{93} - 28 q^{97} + 36 q^{99}+O(q^{100}) \) 3 * q - 2 * q^3 - 2 * q^7 + 5 * q^9 - 2 * q^11 - 6 * q^13 + 12 * q^17 - 3 * q^19 + 7 * q^21 + 2 * q^23 - 17 * q^27 + 6 * q^29 - 8 * q^31 - 24 * q^33 - q^37 + 11 * q^39 + 14 * q^43 - 3 * q^47 + 9 * q^49 - 15 * q^51 - 10 * q^53 + 2 * q^57 - 6 * q^59 - 2 * q^61 + 14 * q^63 - 4 * q^67 + 6 * q^71 - 12 * q^73 - 10 * q^77 - 20 * q^79 + 23 * q^81 + 4 * q^83 + 3 * q^87 + 14 * q^89 - 19 * q^91 - 20 * q^93 - 28 * q^97 + 36 * q^99

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