LMFDB - Embedded newform 6032.2.a.u.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6032,2,Mod(1,6032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6032, 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("6032.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6032 = 2^{4} \cdot 13 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6032.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:	$48.1657624992$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 7x^{5} + 11x^{4} + 15x^{3} - 15x^{2} - 8x + 6 $ x^7 - 2x^6 - 7x^5 + 11x^4 + 15x^3 - 15x^2 - 8x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 377)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.575509$ of defining polynomial
Character	$\chi$	$=$	6032.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.24430 q^{3} -0.495240 q^{5} -1.58211 q^{7} +2.03688 q^{9} +O(q^{10})$	$q+2.24430 q^{3} -0.495240 q^{5} -1.58211 q^{7} +2.03688 q^{9} -2.85335 q^{11} -1.00000 q^{13} -1.11147 q^{15} +1.71498 q^{17} +7.52874 q^{19} -3.55072 q^{21} +0.157618 q^{23} -4.75474 q^{25} -2.16154 q^{27} +1.00000 q^{29} +0.594242 q^{31} -6.40377 q^{33} +0.783523 q^{35} -8.86597 q^{37} -2.24430 q^{39} -6.67008 q^{41} +0.133978 q^{43} -1.00874 q^{45} -0.963672 q^{47} -4.49693 q^{49} +3.84894 q^{51} -13.8332 q^{53} +1.41309 q^{55} +16.8967 q^{57} -6.21749 q^{59} +10.0451 q^{61} -3.22256 q^{63} +0.495240 q^{65} -10.4036 q^{67} +0.353742 q^{69} +5.65407 q^{71} +9.89546 q^{73} -10.6710 q^{75} +4.51431 q^{77} -12.7478 q^{79} -10.9618 q^{81} +15.5334 q^{83} -0.849329 q^{85} +2.24430 q^{87} +4.90437 q^{89} +1.58211 q^{91} +1.33366 q^{93} -3.72853 q^{95} +10.9285 q^{97} -5.81192 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 2 q^{3} - 2 q^{5} - 7 q^{7} + 5 q^{9}+O(q^{10}) $ 7 * q - 2 * q^3 - 2 * q^5 - 7 * q^7 + 5 * q^9	$ 7 q - 2 q^{3} - 2 q^{5} - 7 q^{7} + 5 q^{9} + 3 q^{11} - 7 q^{13} - 20 q^{15} + 14 q^{17} + 12 q^{19} + q^{21} - 5 q^{23} + 5 q^{25} - 14 q^{27} + 7 q^{29} + 3 q^{31} - q^{33} + 6 q^{35} + 9 q^{37} + 2 q^{39} - 9 q^{41} - 24 q^{43} + 10 q^{45} + 9 q^{47} - 10 q^{49} - 5 q^{51} - 13 q^{53} - 7 q^{55} + 12 q^{57} - 13 q^{59} + 3 q^{61} - 2 q^{63} + 2 q^{65} + 7 q^{67} + 8 q^{69} - 22 q^{71} + 4 q^{73} + 29 q^{75} - 30 q^{77} - 48 q^{79} + 15 q^{81} + 30 q^{83} - 9 q^{85} - 2 q^{87} - 25 q^{89} + 7 q^{91} - 27 q^{93} - 23 q^{95} + 4 q^{97} + 25 q^{99}+O(q^{100}) $ 7 * q - 2 * q^3 - 2 * q^5 - 7 * q^7 + 5 * q^9 + 3 * q^11 - 7 * q^13 - 20 * q^15 + 14 * q^17 + 12 * q^19 + q^21 - 5 * q^23 + 5 * q^25 - 14 * q^27 + 7 * q^29 + 3 * q^31 - q^33 + 6 * q^35 + 9 * q^37 + 2 * q^39 - 9 * q^41 - 24 * q^43 + 10 * q^45 + 9 * q^47 - 10 * q^49 - 5 * q^51 - 13 * q^53 - 7 * q^55 + 12 * q^57 - 13 * q^59 + 3 * q^61 - 2 * q^63 + 2 * q^65 + 7 * q^67 + 8 * q^69 - 22 * q^71 + 4 * q^73 + 29 * q^75 - 30 * q^77 - 48 * q^79 + 15 * q^81 + 30 * q^83 - 9 * q^85 - 2 * q^87 - 25 * q^89 + 7 * q^91 - 27 * q^93 - 23 * q^95 + 4 * q^97 + 25 * 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
$3$	2.24430	1.29575	0.647873	−	0.761748i	$-0.275659\pi$
$3$	2.24430	1.29575	0.647873	+	0.761748i	$0.275659\pi$
$4$	0	0
$5$	−0.495240	−0.221478	−0.110739	−	0.993850i	$-0.535322\pi$
$5$	−0.495240	−0.221478	−0.110739	+	0.993850i	$0.535322\pi$
$6$	0	0
$7$	−1.58211	−0.597981	−0.298990	−	0.954256i	$-0.596650\pi$
$7$	−1.58211	−0.597981	−0.298990	+	0.954256i	$0.596650\pi$
$8$	0	0
$9$	2.03688	0.678958
$10$	0	0
$11$	−2.85335	−0.860318	−0.430159	−	0.902753i	$-0.641542\pi$
$11$	−2.85335	−0.860318	−0.430159	+	0.902753i	$0.641542\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−1.11147	−0.286979
$16$	0	0
$17$	1.71498	0.415945	0.207972	−	0.978135i	$-0.433314\pi$
$17$	1.71498	0.415945	0.207972	+	0.978135i	$0.433314\pi$
$18$	0	0
$19$	7.52874	1.72721	0.863606	−	0.504168i	$-0.168201\pi$
$19$	7.52874	1.72721	0.863606	+	0.504168i	$0.168201\pi$
$20$	0	0
$21$	−3.55072	−0.774831
$22$	0	0
$23$	0.157618	0.0328656	0.0164328	−	0.999865i	$-0.494769\pi$
$23$	0.157618	0.0328656	0.0164328	+	0.999865i	$0.494769\pi$
$24$	0	0
$25$	−4.75474	−0.950948
$26$	0	0
$27$	−2.16154	−0.415988
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	0.594242	0.106729	0.0533645	−	0.998575i	$-0.483005\pi$
$31$	0.594242	0.106729	0.0533645	+	0.998575i	$0.483005\pi$
$32$	0	0
$33$	−6.40377	−1.11475
$34$	0	0
$35$	0.783523	0.132440
$36$	0	0
$37$	−8.86597	−1.45756	−0.728778	−	0.684750i	$-0.759912\pi$
$37$	−8.86597	−1.45756	−0.728778	+	0.684750i	$0.759912\pi$
$38$	0	0
$39$	−2.24430	−0.359375
$40$	0	0
$41$	−6.67008	−1.04169	−0.520846	−	0.853651i	$-0.674383\pi$
$41$	−6.67008	−1.04169	−0.520846	+	0.853651i	$0.674383\pi$
$42$	0	0
$43$	0.133978	0.0204315	0.0102157	−	0.999948i	$-0.496748\pi$
$43$	0.133978	0.0204315	0.0102157	+	0.999948i	$0.496748\pi$
$44$	0	0
$45$	−1.00874	−0.150374
$46$	0	0
$47$	−0.963672	−0.140566	−0.0702830	−	0.997527i	$-0.522390\pi$
$47$	−0.963672	−0.140566	−0.0702830	+	0.997527i	$0.522390\pi$
$48$	0	0
$49$	−4.49693	−0.642419
$50$	0	0
$51$	3.84894	0.538959
$52$	0	0
$53$	−13.8332	−1.90013	−0.950066	−	0.312050i	$-0.898984\pi$
$53$	−13.8332	−1.90013	−0.950066	+	0.312050i	$0.898984\pi$
$54$	0	0
$55$	1.41309	0.190541
$56$	0	0
$57$	16.8967	2.23803
$58$	0	0
$59$	−6.21749	−0.809448	−0.404724	−	0.914439i	$-0.632632\pi$
$59$	−6.21749	−0.809448	−0.404724	+	0.914439i	$0.632632\pi$
$60$	0	0
$61$	10.0451	1.28615	0.643073	−	0.765805i	$-0.277659\pi$
$61$	10.0451	1.28615	0.643073	+	0.765805i	$0.277659\pi$
$62$	0	0
$63$	−3.22256	−0.406004
$64$	0	0
$65$	0.495240	0.0614269
$66$	0	0
$67$	−10.4036	−1.27100	−0.635499	−	0.772102i	$-0.719206\pi$
$67$	−10.4036	−1.27100	−0.635499	+	0.772102i	$0.719206\pi$
$68$	0	0
$69$	0.353742	0.0425855
$70$	0	0
$71$	5.65407	0.671015	0.335507	−	0.942038i	$-0.391092\pi$
$71$	5.65407	0.671015	0.335507	+	0.942038i	$0.391092\pi$
$72$	0	0
$73$	9.89546	1.15818	0.579088	−	0.815265i	$-0.303409\pi$
$73$	9.89546	1.15818	0.579088	+	0.815265i	$0.303409\pi$
$74$	0	0
$75$	−10.6710	−1.23219
$76$	0	0
$77$	4.51431	0.514454
$78$	0	0
$79$	−12.7478	−1.43424	−0.717121	−	0.696948i	$-0.754541\pi$
$79$	−12.7478	−1.43424	−0.717121	+	0.696948i	$0.754541\pi$
$80$	0	0
$81$	−10.9618	−1.21797
$82$	0	0
$83$	15.5334	1.70501	0.852505	−	0.522719i	$-0.175082\pi$
$83$	15.5334	1.70501	0.852505	+	0.522719i	$0.175082\pi$
$84$	0	0
$85$	−0.849329	−0.0921226
$86$	0	0
$87$	2.24430	0.240614
$88$	0	0
$89$	4.90437	0.519862	0.259931	−	0.965627i	$-0.416300\pi$
$89$	4.90437	0.519862	0.259931	+	0.965627i	$0.416300\pi$
$90$	0	0
$91$	1.58211	0.165850
$92$	0	0
$93$	1.33366	0.138294
$94$	0	0
$95$	−3.72853	−0.382539
$96$	0	0
$97$	10.9285	1.10962	0.554811	−	0.831976i	$-0.312791\pi$
$97$	10.9285	1.10962	0.554811	+	0.831976i	$0.312791\pi$
$98$	0	0
$99$	−5.81192	−0.584120
$100$	0	0
$101$	−11.4874	−1.14304	−0.571518	−	0.820590i	$-0.693645\pi$
$101$	−11.4874	−1.14304	−0.571518	+	0.820590i	$0.693645\pi$
$102$	0	0
$103$	−17.0344	−1.67845	−0.839223	−	0.543787i	$-0.816990\pi$
$103$	−17.0344	−1.67845	−0.839223	+	0.543787i	$0.816990\pi$
$104$	0	0
$105$	1.75846	0.171608
$106$	0	0
$107$	7.54090	0.729006	0.364503	−	0.931202i	$-0.381239\pi$
$107$	7.54090	0.729006	0.364503	+	0.931202i	$0.381239\pi$
$108$	0	0
$109$	−3.56433	−0.341401	−0.170701	−	0.985323i	$-0.554603\pi$
$109$	−3.56433	−0.341401	−0.170701	+	0.985323i	$0.554603\pi$
$110$	0	0
$111$	−19.8979	−1.88862
$112$	0	0
$113$	10.2771	0.966788	0.483394	−	0.875403i	$-0.339404\pi$
$113$	10.2771	0.966788	0.483394	+	0.875403i	$0.339404\pi$
$114$	0	0
$115$	−0.0780587	−0.00727901
$116$	0	0
$117$	−2.03688	−0.188309
$118$	0	0
$119$	−2.71329	−0.248727
$120$	0	0
$121$	−2.85838	−0.259853
$122$	0	0
$123$	−14.9696	−1.34977
$124$	0	0
$125$	4.83093	0.432092
$126$	0	0
$127$	−11.8763	−1.05385	−0.526924	−	0.849913i	$-0.676655\pi$
$127$	−11.8763	−1.05385	−0.526924	+	0.849913i	$0.676655\pi$
$128$	0	0
$129$	0.300687	0.0264740
$130$	0	0
$131$	13.5108	1.18044	0.590220	−	0.807242i	$-0.299041\pi$
$131$	13.5108	1.18044	0.590220	+	0.807242i	$0.299041\pi$
$132$	0	0
$133$	−11.9113	−1.03284
$134$	0	0
$135$	1.07048	0.0921323
$136$	0	0
$137$	−3.53668	−0.302159	−0.151080	−	0.988522i	$-0.548275\pi$
$137$	−3.53668	−0.302159	−0.151080	+	0.988522i	$0.548275\pi$
$138$	0	0
$139$	−19.4123	−1.64653	−0.823264	−	0.567659i	$-0.807849\pi$
$139$	−19.4123	−1.64653	−0.823264	+	0.567659i	$0.807849\pi$
$140$	0	0
$141$	−2.16277	−0.182138
$142$	0	0
$143$	2.85335	0.238609
$144$	0	0
$145$	−0.495240	−0.0411274
$146$	0	0
$147$	−10.0925	−0.832412
$148$	0	0
$149$	−0.161485	−0.0132294	−0.00661468	−	0.999978i	$-0.502106\pi$
$149$	−0.161485	−0.0132294	−0.00661468	+	0.999978i	$0.502106\pi$
$150$	0	0
$151$	5.85680	0.476620	0.238310	−	0.971189i	$-0.423407\pi$
$151$	5.85680	0.476620	0.238310	+	0.971189i	$0.423407\pi$
$152$	0	0
$153$	3.49321	0.282409
$154$	0	0
$155$	−0.294292	−0.0236381
$156$	0	0
$157$	−8.17951	−0.652796	−0.326398	−	0.945232i	$-0.605835\pi$
$157$	−8.17951	−0.652796	−0.326398	+	0.945232i	$0.605835\pi$
$158$	0	0
$159$	−31.0457	−2.46209
$160$	0	0
$161$	−0.249369	−0.0196530
$162$	0	0
$163$	−8.11434	−0.635564	−0.317782	−	0.948164i	$-0.602938\pi$
$163$	−8.11434	−0.635564	−0.317782	+	0.948164i	$0.602938\pi$
$164$	0	0
$165$	3.17140	0.246893
$166$	0	0
$167$	−19.3742	−1.49922	−0.749609	−	0.661881i	$-0.769759\pi$
$167$	−19.3742	−1.49922	−0.749609	+	0.661881i	$0.769759\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	15.3351	1.17270
$172$	0	0
$173$	−15.6018	−1.18618	−0.593092	−	0.805135i	$-0.702093\pi$
$173$	−15.6018	−1.18618	−0.593092	+	0.805135i	$0.702093\pi$
$174$	0	0
$175$	7.52251	0.568648
$176$	0	0
$177$	−13.9539	−1.04884
$178$	0	0
$179$	−3.51011	−0.262358	−0.131179	−	0.991359i	$-0.541876\pi$
$179$	−3.51011	−0.262358	−0.131179	+	0.991359i	$0.541876\pi$
$180$	0	0
$181$	−5.08795	−0.378184	−0.189092	−	0.981959i	$-0.560554\pi$
$181$	−5.08795	−0.378184	−0.189092	+	0.981959i	$0.560554\pi$
$182$	0	0
$183$	22.5443	1.66652
$184$	0	0
$185$	4.39078	0.322817
$186$	0	0
$187$	−4.89346	−0.357845
$188$	0	0
$189$	3.41979	0.248753
$190$	0	0
$191$	−22.6307	−1.63750	−0.818751	−	0.574148i	$-0.805333\pi$
$191$	−22.6307	−1.63750	−0.818751	+	0.574148i	$0.805333\pi$
$192$	0	0
$193$	8.01744	0.577108	0.288554	−	0.957464i	$-0.406825\pi$
$193$	8.01744	0.577108	0.288554	+	0.957464i	$0.406825\pi$
$194$	0	0
$195$	1.11147	0.0795937
$196$	0	0
$197$	10.1620	0.724012	0.362006	−	0.932176i	$-0.382092\pi$
$197$	10.1620	0.724012	0.362006	+	0.932176i	$0.382092\pi$
$198$	0	0
$199$	2.79063	0.197822	0.0989111	−	0.995096i	$-0.468464\pi$
$199$	2.79063	0.197822	0.0989111	+	0.995096i	$0.468464\pi$
$200$	0	0
$201$	−23.3487	−1.64689
$202$	0	0
$203$	−1.58211	−0.111042
$204$	0	0
$205$	3.30329	0.230712
$206$	0	0
$207$	0.321048	0.0223144
$208$	0	0
$209$	−21.4821	−1.48595
$210$	0	0
$211$	27.3046	1.87973	0.939864	−	0.341548i	$-0.110951\pi$
$211$	27.3046	1.87973	0.939864	+	0.341548i	$0.110951\pi$
$212$	0	0
$213$	12.6894	0.869465
$214$	0	0
$215$	−0.0663512	−0.00452512
$216$	0	0
$217$	−0.940156	−0.0638219
$218$	0	0
$219$	22.2084	1.50070
$220$	0	0
$221$	−1.71498	−0.115362
$222$	0	0
$223$	−5.49490	−0.367966	−0.183983	−	0.982929i	$-0.558899\pi$
$223$	−5.49490	−0.367966	−0.183983	+	0.982929i	$0.558899\pi$
$224$	0	0
$225$	−9.68481	−0.645654
$226$	0	0
$227$	−14.0352	−0.931547	−0.465774	−	0.884904i	$-0.654224\pi$
$227$	−14.0352	−0.931547	−0.465774	+	0.884904i	$0.654224\pi$
$228$	0	0
$229$	−19.0839	−1.26110	−0.630551	−	0.776148i	$-0.717171\pi$
$229$	−19.0839	−1.26110	−0.630551	+	0.776148i	$0.717171\pi$
$230$	0	0
$231$	10.1315	0.666601
$232$	0	0
$233$	−15.7384	−1.03106	−0.515529	−	0.856872i	$-0.672405\pi$
$233$	−15.7384	−1.03106	−0.515529	+	0.856872i	$0.672405\pi$
$234$	0	0
$235$	0.477249	0.0311323
$236$	0	0
$237$	−28.6099	−1.85841
$238$	0	0
$239$	13.9618	0.903116	0.451558	−	0.892242i	$-0.350868\pi$
$239$	13.9618	0.903116	0.451558	+	0.892242i	$0.350868\pi$
$240$	0	0
$241$	15.1536	0.976129	0.488065	−	0.872807i	$-0.337703\pi$
$241$	15.1536	0.976129	0.488065	+	0.872807i	$0.337703\pi$
$242$	0	0
$243$	−18.1169	−1.16220
$244$	0	0
$245$	2.22706	0.142282
$246$	0	0
$247$	−7.52874	−0.479042
$248$	0	0
$249$	34.8615	2.20926
$250$	0	0
$251$	8.49388	0.536129	0.268064	−	0.963401i	$-0.413616\pi$
$251$	8.49388	0.536129	0.268064	+	0.963401i	$0.413616\pi$
$252$	0	0
$253$	−0.449740	−0.0282749
$254$	0	0
$255$	−1.90615	−0.119368
$256$	0	0
$257$	−5.19320	−0.323943	−0.161972	−	0.986795i	$-0.551785\pi$
$257$	−5.19320	−0.323943	−0.161972	+	0.986795i	$0.551785\pi$
$258$	0	0
$259$	14.0269	0.871591
$260$	0	0
$261$	2.03688	0.126079
$262$	0	0
$263$	−11.2859	−0.695921	−0.347960	−	0.937509i	$-0.613126\pi$
$263$	−11.2859	−0.695921	−0.347960	+	0.937509i	$0.613126\pi$
$264$	0	0
$265$	6.85073	0.420837
$266$	0	0
$267$	11.0069	0.673609
$268$	0	0
$269$	−0.711393	−0.0433744	−0.0216872	−	0.999765i	$-0.506904\pi$
$269$	−0.711393	−0.0433744	−0.0216872	+	0.999765i	$0.506904\pi$
$270$	0	0
$271$	7.29807	0.443326	0.221663	−	0.975123i	$-0.428851\pi$
$271$	7.29807	0.443326	0.221663	+	0.975123i	$0.428851\pi$
$272$	0	0
$273$	3.55072	0.214900
$274$	0	0
$275$	13.5669	0.818117
$276$	0	0
$277$	26.0271	1.56381	0.781907	−	0.623394i	$-0.214247\pi$
$277$	26.0271	1.56381	0.781907	+	0.623394i	$0.214247\pi$
$278$	0	0
$279$	1.21040	0.0724646
$280$	0	0
$281$	−12.4406	−0.742145	−0.371072	−	0.928604i	$-0.621010\pi$
$281$	−12.4406	−0.742145	−0.371072	+	0.928604i	$0.621010\pi$
$282$	0	0
$283$	−7.03434	−0.418148	−0.209074	−	0.977900i	$-0.567045\pi$
$283$	−7.03434	−0.418148	−0.209074	+	0.977900i	$0.567045\pi$
$284$	0	0
$285$	−8.36794	−0.495674
$286$	0	0
$287$	10.5528	0.622912
$288$	0	0
$289$	−14.0588	−0.826990
$290$	0	0
$291$	24.5268	1.43779
$292$	0	0
$293$	4.99880	0.292033	0.146017	−	0.989282i	$-0.453355\pi$
$293$	4.99880	0.292033	0.146017	+	0.989282i	$0.453355\pi$
$294$	0	0
$295$	3.07915	0.179275
$296$	0	0
$297$	6.16763	0.357882
$298$	0	0
$299$	−0.157618	−0.00911528
$300$	0	0
$301$	−0.211968	−0.0122176
$302$	0	0
$303$	−25.7811	−1.48108
$304$	0	0
$305$	−4.97474	−0.284853
$306$	0	0
$307$	−27.4051	−1.56409	−0.782046	−	0.623221i	$-0.785824\pi$
$307$	−27.4051	−1.56409	−0.782046	+	0.623221i	$0.785824\pi$
$308$	0	0
$309$	−38.2302	−2.17484
$310$	0	0
$311$	5.56818	0.315743	0.157871	−	0.987460i	$-0.449537\pi$
$311$	5.56818	0.315743	0.157871	+	0.987460i	$0.449537\pi$
$312$	0	0
$313$	−28.0179	−1.58367	−0.791833	−	0.610737i	$-0.790873\pi$
$313$	−28.0179	−1.58367	−0.791833	+	0.610737i	$0.790873\pi$
$314$	0	0
$315$	1.59594	0.0899210
$316$	0	0
$317$	2.11400	0.118734	0.0593671	−	0.998236i	$-0.481092\pi$
$317$	2.11400	0.118734	0.0593671	+	0.998236i	$0.481092\pi$
$318$	0	0
$319$	−2.85335	−0.159757
$320$	0	0
$321$	16.9240	0.944607
$322$	0	0
$323$	12.9117	0.718425
$324$	0	0
$325$	4.75474	0.263745
$326$	0	0
$327$	−7.99943	−0.442369
$328$	0	0
$329$	1.52463	0.0840558
$330$	0	0
$331$	9.31593	0.512050	0.256025	−	0.966670i	$-0.417587\pi$
$331$	9.31593	0.512050	0.256025	+	0.966670i	$0.417587\pi$
$332$	0	0
$333$	−18.0589	−0.989620
$334$	0	0
$335$	5.15226	0.281498
$336$	0	0
$337$	26.9386	1.46744	0.733719	−	0.679453i	$-0.237783\pi$
$337$	26.9386	1.46744	0.733719	+	0.679453i	$0.237783\pi$
$338$	0	0
$339$	23.0649	1.25271
$340$	0	0
$341$	−1.69558	−0.0918209
$342$	0	0
$343$	18.1894	0.982135
$344$	0	0
$345$	−0.175187	−0.00943175
$346$	0	0
$347$	−32.8961	−1.76595	−0.882977	−	0.469416i	$-0.844464\pi$
$347$	−32.8961	−1.76595	−0.882977	+	0.469416i	$0.844464\pi$
$348$	0	0
$349$	−27.1219	−1.45180	−0.725902	−	0.687798i	$-0.758577\pi$
$349$	−27.1219	−1.45180	−0.725902	+	0.687798i	$0.758577\pi$
$350$	0	0
$351$	2.16154	0.115374
$352$	0	0
$353$	4.04308	0.215192	0.107596	−	0.994195i	$-0.465685\pi$
$353$	4.04308	0.215192	0.107596	+	0.994195i	$0.465685\pi$
$354$	0	0
$355$	−2.80012	−0.148615
$356$	0	0
$357$	−6.08944	−0.322287
$358$	0	0
$359$	25.6504	1.35377	0.676887	−	0.736087i	$-0.263328\pi$
$359$	25.6504	1.35377	0.676887	+	0.736087i	$0.263328\pi$
$360$	0	0
$361$	37.6819	1.98326
$362$	0	0
$363$	−6.41507	−0.336704
$364$	0	0
$365$	−4.90062	−0.256510
$366$	0	0
$367$	−22.8321	−1.19183	−0.595913	−	0.803049i	$-0.703210\pi$
$367$	−22.8321	−1.19183	−0.595913	+	0.803049i	$0.703210\pi$
$368$	0	0
$369$	−13.5861	−0.707265
$370$	0	0
$371$	21.8856	1.13624
$372$	0	0
$373$	5.99112	0.310208	0.155104	−	0.987898i	$-0.450429\pi$
$373$	5.99112	0.310208	0.155104	+	0.987898i	$0.450429\pi$
$374$	0	0
$375$	10.8421	0.559881
$376$	0	0
$377$	−1.00000	−0.0515026
$378$	0	0
$379$	−18.1803	−0.933861	−0.466930	−	0.884294i	$-0.654640\pi$
$379$	−18.1803	−0.933861	−0.466930	+	0.884294i	$0.654640\pi$
$380$	0	0
$381$	−26.6539	−1.36552
$382$	0	0
$383$	24.1720	1.23513	0.617566	−	0.786519i	$-0.288119\pi$
$383$	24.1720	1.23513	0.617566	+	0.786519i	$0.288119\pi$
$384$	0	0
$385$	−2.23567	−0.113940
$386$	0	0
$387$	0.272897	0.0138721
$388$	0	0
$389$	32.3571	1.64057	0.820286	−	0.571954i	$-0.193814\pi$
$389$	32.3571	1.64057	0.820286	+	0.571954i	$0.193814\pi$
$390$	0	0
$391$	0.270312	0.0136703
$392$	0	0
$393$	30.3222	1.52955
$394$	0	0
$395$	6.31323	0.317653
$396$	0	0
$397$	−20.9376	−1.05083	−0.525413	−	0.850847i	$-0.676089\pi$
$397$	−20.9376	−1.05083	−0.525413	+	0.850847i	$0.676089\pi$
$398$	0	0
$399$	−26.7325	−1.33830
$400$	0	0
$401$	1.60708	0.0802538	0.0401269	−	0.999195i	$-0.487224\pi$
$401$	1.60708	0.0802538	0.0401269	+	0.999195i	$0.487224\pi$
$402$	0	0
$403$	−0.594242	−0.0296013
$404$	0	0
$405$	5.42870	0.269754
$406$	0	0
$407$	25.2977	1.25396
$408$	0	0
$409$	24.6527	1.21900	0.609498	−	0.792788i	$-0.291371\pi$
$409$	24.6527	1.21900	0.609498	+	0.792788i	$0.291371\pi$
$410$	0	0
$411$	−7.93738	−0.391522
$412$	0	0
$413$	9.83675	0.484035
$414$	0	0
$415$	−7.69275	−0.377622
$416$	0	0
$417$	−43.5669	−2.13348
$418$	0	0
$419$	−7.91829	−0.386834	−0.193417	−	0.981117i	$-0.561957\pi$
$419$	−7.91829	−0.386834	−0.193417	+	0.981117i	$0.561957\pi$
$420$	0	0
$421$	−15.1274	−0.737263	−0.368631	−	0.929576i	$-0.620174\pi$
$421$	−15.1274	−0.737263	−0.368631	+	0.929576i	$0.620174\pi$
$422$	0	0
$423$	−1.96288	−0.0954385
$424$	0	0
$425$	−8.15430	−0.395542
$426$	0	0
$427$	−15.8925	−0.769091
$428$	0	0
$429$	6.40377	0.309177
$430$	0	0
$431$	18.0192	0.867956	0.433978	−	0.900923i	$-0.357110\pi$
$431$	18.0192	0.867956	0.433978	+	0.900923i	$0.357110\pi$
$432$	0	0
$433$	35.8821	1.72438	0.862191	−	0.506583i	$-0.169091\pi$
$433$	35.8821	1.72438	0.862191	+	0.506583i	$0.169091\pi$
$434$	0	0
$435$	−1.11147	−0.0532907
$436$	0	0
$437$	1.18666	0.0567659
$438$	0	0
$439$	11.7973	0.563057	0.281528	−	0.959553i	$-0.409159\pi$
$439$	11.7973	0.563057	0.281528	+	0.959553i	$0.409159\pi$
$440$	0	0
$441$	−9.15969	−0.436176
$442$	0	0
$443$	11.2308	0.533591	0.266796	−	0.963753i	$-0.414035\pi$
$443$	11.2308	0.533591	0.266796	+	0.963753i	$0.414035\pi$
$444$	0	0
$445$	−2.42884	−0.115138
$446$	0	0
$447$	−0.362420	−0.0171419
$448$	0	0
$449$	18.8076	0.887585	0.443793	−	0.896130i	$-0.353633\pi$
$449$	18.8076	0.887585	0.443793	+	0.896130i	$0.353633\pi$
$450$	0	0
$451$	19.0321	0.896186
$452$	0	0
$453$	13.1444	0.617578
$454$	0	0
$455$	−0.783523	−0.0367321
$456$	0	0
$457$	−29.7273	−1.39059	−0.695293	−	0.718726i	$-0.744726\pi$
$457$	−29.7273	−1.39059	−0.695293	+	0.718726i	$0.744726\pi$
$458$	0	0
$459$	−3.70701	−0.173028
$460$	0	0
$461$	12.2681	0.571381	0.285690	−	0.958322i	$-0.407777\pi$
$461$	12.2681	0.571381	0.285690	+	0.958322i	$0.407777\pi$
$462$	0	0
$463$	3.80816	0.176980	0.0884901	−	0.996077i	$-0.471796\pi$
$463$	3.80816	0.176980	0.0884901	+	0.996077i	$0.471796\pi$
$464$	0	0
$465$	−0.660480	−0.0306290
$466$	0	0
$467$	2.32664	0.107664	0.0538320	−	0.998550i	$-0.482856\pi$
$467$	2.32664	0.107664	0.0538320	+	0.998550i	$0.482856\pi$
$468$	0	0
$469$	16.4596	0.760032
$470$	0	0
$471$	−18.3573	−0.845858
$472$	0	0
$473$	−0.382286	−0.0175775
$474$	0	0
$475$	−35.7972	−1.64249
$476$	0	0
$477$	−28.1764	−1.29011
$478$	0	0
$479$	−24.0562	−1.09916	−0.549579	−	0.835442i	$-0.685212\pi$
$479$	−24.0562	−1.09916	−0.549579	+	0.835442i	$0.685212\pi$
$480$	0	0
$481$	8.86597	0.404253
$482$	0	0
$483$	−0.559658	−0.0254653
$484$	0	0
$485$	−5.41223	−0.245757
$486$	0	0
$487$	−6.53435	−0.296100	−0.148050	−	0.988980i	$-0.547300\pi$
$487$	−6.53435	−0.296100	−0.148050	+	0.988980i	$0.547300\pi$
$488$	0	0
$489$	−18.2110	−0.823530
$490$	0	0
$491$	22.8350	1.03053	0.515264	−	0.857032i	$-0.327694\pi$
$491$	22.8350	1.03053	0.515264	+	0.857032i	$0.327694\pi$
$492$	0	0
$493$	1.71498	0.0772390
$494$	0	0
$495$	2.87829	0.129370
$496$	0	0
$497$	−8.94536	−0.401254
$498$	0	0
$499$	14.2823	0.639365	0.319683	−	0.947525i	$-0.396424\pi$
$499$	14.2823	0.639365	0.319683	+	0.947525i	$0.396424\pi$
$500$	0	0
$501$	−43.4814	−1.94261
$502$	0	0
$503$	40.5832	1.80951	0.904757	−	0.425927i	$-0.140052\pi$
$503$	40.5832	1.80951	0.904757	+	0.425927i	$0.140052\pi$
$504$	0	0
$505$	5.68900	0.253157
$506$	0	0
$507$	2.24430	0.0996728
$508$	0	0
$509$	−4.61566	−0.204586	−0.102293	−	0.994754i	$-0.532618\pi$
$509$	−4.61566	−0.204586	−0.102293	+	0.994754i	$0.532618\pi$
$510$	0	0
$511$	−15.6557	−0.692567
$512$	0	0
$513$	−16.2737	−0.718500
$514$	0	0
$515$	8.43610	0.371739
$516$	0	0
$517$	2.74969	0.120931
$518$	0	0
$519$	−35.0151	−1.53699
$520$	0	0
$521$	1.67773	0.0735028	0.0367514	−	0.999324i	$-0.488299\pi$
$521$	1.67773	0.0735028	0.0367514	+	0.999324i	$0.488299\pi$
$522$	0	0
$523$	6.11351	0.267325	0.133663	−	0.991027i	$-0.457326\pi$
$523$	6.11351	0.267325	0.133663	+	0.991027i	$0.457326\pi$
$524$	0	0
$525$	16.8828	0.736824
$526$	0	0
$527$	1.01912	0.0443934
$528$	0	0
$529$	−22.9752	−0.998920
$530$	0	0
$531$	−12.6643	−0.549582
$532$	0	0
$533$	6.67008	0.288913
$534$	0	0
$535$	−3.73455	−0.161459
$536$	0	0
$537$	−7.87773	−0.339949
$538$	0	0
$539$	12.8313	0.552684
$540$	0	0
$541$	27.4294	1.17928	0.589641	−	0.807666i	$-0.299269\pi$
$541$	27.4294	1.17928	0.589641	+	0.807666i	$0.299269\pi$
$542$	0	0
$543$	−11.4189	−0.490031
$544$	0	0
$545$	1.76520	0.0756128
$546$	0	0
$547$	−3.54960	−0.151770	−0.0758850	−	0.997117i	$-0.524178\pi$
$547$	−3.54960	−0.151770	−0.0758850	+	0.997117i	$0.524178\pi$
$548$	0	0
$549$	20.4607	0.873240
$550$	0	0
$551$	7.52874	0.320735
$552$	0	0
$553$	20.1685	0.857650
$554$	0	0
$555$	9.85422	0.418288
$556$	0	0
$557$	27.2993	1.15671	0.578355	−	0.815785i	$-0.303695\pi$
$557$	27.2993	1.15671	0.578355	+	0.815785i	$0.303695\pi$
$558$	0	0
$559$	−0.133978	−0.00566667
$560$	0	0
$561$	−10.9824	−0.463676
$562$	0	0
$563$	−33.6072	−1.41638	−0.708188	−	0.706024i	$-0.750487\pi$
$563$	−33.6072	−1.41638	−0.708188	+	0.706024i	$0.750487\pi$
$564$	0	0
$565$	−5.08963	−0.214122
$566$	0	0
$567$	17.3427	0.728325
$568$	0	0
$569$	1.40995	0.0591081	0.0295541	−	0.999563i	$-0.490591\pi$
$569$	1.40995	0.0591081	0.0295541	+	0.999563i	$0.490591\pi$
$570$	0	0
$571$	12.3624	0.517352	0.258676	−	0.965964i	$-0.416714\pi$
$571$	12.3624	0.517352	0.258676	+	0.965964i	$0.416714\pi$
$572$	0	0
$573$	−50.7901	−2.12179
$574$	0	0
$575$	−0.749432	−0.0312535
$576$	0	0
$577$	−7.41545	−0.308709	−0.154355	−	0.988015i	$-0.549330\pi$
$577$	−7.41545	−0.308709	−0.154355	+	0.988015i	$0.549330\pi$
$578$	0	0
$579$	17.9935	0.747786
$580$	0	0
$581$	−24.5755	−1.01956
$582$	0	0
$583$	39.4709	1.63472
$584$	0	0
$585$	1.00874	0.0417063
$586$	0	0
$587$	32.8697	1.35668	0.678338	−	0.734750i	$-0.262700\pi$
$587$	32.8697	1.35668	0.678338	+	0.734750i	$0.262700\pi$
$588$	0	0
$589$	4.47389	0.184344
$590$	0	0
$591$	22.8066	0.938136
$592$	0	0
$593$	20.2519	0.831646	0.415823	−	0.909445i	$-0.363494\pi$
$593$	20.2519	0.831646	0.415823	+	0.909445i	$0.363494\pi$
$594$	0	0
$595$	1.34373	0.0550876
$596$	0	0
$597$	6.26300	0.256327
$598$	0	0
$599$	−27.6785	−1.13091	−0.565456	−	0.824779i	$-0.691300\pi$
$599$	−27.6785	−1.13091	−0.565456	+	0.824779i	$0.691300\pi$
$600$	0	0
$601$	3.76348	0.153516	0.0767578	−	0.997050i	$-0.475543\pi$
$601$	3.76348	0.153516	0.0767578	+	0.997050i	$0.475543\pi$
$602$	0	0
$603$	−21.1908	−0.862955
$604$	0	0
$605$	1.41559	0.0575517
$606$	0	0
$607$	−0.0746488	−0.00302990	−0.00151495	−	0.999999i	$-0.500482\pi$
$607$	−0.0746488	−0.00302990	−0.00151495	+	0.999999i	$0.500482\pi$
$608$	0	0
$609$	−3.55072	−0.143883
$610$	0	0
$611$	0.963672	0.0389860
$612$	0	0
$613$	0.526845	0.0212791	0.0106395	−	0.999943i	$-0.496613\pi$
$613$	0.526845	0.0212791	0.0106395	+	0.999943i	$0.496613\pi$
$614$	0	0
$615$	7.41356	0.298944
$616$	0	0
$617$	−27.8190	−1.11995	−0.559975	−	0.828509i	$-0.689189\pi$
$617$	−27.8190	−1.11995	−0.559975	+	0.828509i	$0.689189\pi$
$618$	0	0
$619$	4.14112	0.166446	0.0832229	−	0.996531i	$-0.473479\pi$
$619$	4.14112	0.166446	0.0832229	+	0.996531i	$0.473479\pi$
$620$	0	0
$621$	−0.340698	−0.0136717
$622$	0	0
$623$	−7.75924	−0.310867
$624$	0	0
$625$	21.3812	0.855249
$626$	0	0
$627$	−48.2123	−1.92542
$628$	0	0
$629$	−15.2050	−0.606263
$630$	0	0
$631$	−30.1358	−1.19969	−0.599843	−	0.800118i	$-0.704770\pi$
$631$	−30.1358	−1.19969	−0.599843	+	0.800118i	$0.704770\pi$
$632$	0	0
$633$	61.2797	2.43565
$634$	0	0
$635$	5.88159	0.233404
$636$	0	0
$637$	4.49693	0.178175
$638$	0	0
$639$	11.5166	0.455591
$640$	0	0
$641$	28.8048	1.13772	0.568861	−	0.822433i	$-0.307384\pi$
$641$	28.8048	1.13772	0.568861	+	0.822433i	$0.307384\pi$
$642$	0	0
$643$	19.2729	0.760051	0.380025	−	0.924976i	$-0.375915\pi$
$643$	19.2729	0.760051	0.380025	+	0.924976i	$0.375915\pi$
$644$	0	0
$645$	−0.148912	−0.00586340
$646$	0	0
$647$	−3.37536	−0.132699	−0.0663495	−	0.997796i	$-0.521135\pi$
$647$	−3.37536	−0.132699	−0.0663495	+	0.997796i	$0.521135\pi$
$648$	0	0
$649$	17.7407	0.696383
$650$	0	0
$651$	−2.10999	−0.0826970
$652$	0	0
$653$	24.3566	0.953147	0.476574	−	0.879135i	$-0.341879\pi$
$653$	24.3566	0.953147	0.476574	+	0.879135i	$0.341879\pi$
$654$	0	0
$655$	−6.69106	−0.261441
$656$	0	0
$657$	20.1558	0.786353
$658$	0	0
$659$	41.8131	1.62881	0.814403	−	0.580300i	$-0.197065\pi$
$659$	41.8131	1.62881	0.814403	+	0.580300i	$0.197065\pi$
$660$	0	0
$661$	23.8659	0.928276	0.464138	−	0.885763i	$-0.346364\pi$
$661$	23.8659	0.928276	0.464138	+	0.885763i	$0.346364\pi$
$662$	0	0
$663$	−3.84894	−0.149480
$664$	0	0
$665$	5.89894	0.228751
$666$	0	0
$667$	0.157618	0.00610299
$668$	0	0
$669$	−12.3322	−0.476790
$670$	0	0
$671$	−28.6623	−1.10649
$672$	0	0
$673$	46.3735	1.78757	0.893783	−	0.448500i	$-0.148042\pi$
$673$	46.3735	1.78757	0.893783	+	0.448500i	$0.148042\pi$
$674$	0	0
$675$	10.2776	0.395583
$676$	0	0
$677$	1.62104	0.0623017	0.0311509	−	0.999515i	$-0.490083\pi$
$677$	1.62104	0.0623017	0.0311509	+	0.999515i	$0.490083\pi$
$678$	0	0
$679$	−17.2901	−0.663533
$680$	0	0
$681$	−31.4991	−1.20705
$682$	0	0
$683$	7.17062	0.274376	0.137188	−	0.990545i	$-0.456194\pi$
$683$	7.17062	0.274376	0.137188	+	0.990545i	$0.456194\pi$
$684$	0	0
$685$	1.75151	0.0669216
$686$	0	0
$687$	−42.8301	−1.63407
$688$	0	0
$689$	13.8332	0.527002
$690$	0	0
$691$	28.4538	1.08243	0.541216	−	0.840884i	$-0.317964\pi$
$691$	28.4538	1.08243	0.541216	+	0.840884i	$0.317964\pi$
$692$	0	0
$693$	9.19509	0.349293
$694$	0	0
$695$	9.61373	0.364669
$696$	0	0
$697$	−11.4391	−0.433286
$698$	0	0
$699$	−35.3217	−1.33599
$700$	0	0
$701$	−4.03004	−0.152213	−0.0761063	−	0.997100i	$-0.524249\pi$
$701$	−4.03004	−0.152213	−0.0761063	+	0.997100i	$0.524249\pi$
$702$	0	0
$703$	−66.7496	−2.51751
$704$	0	0
$705$	1.07109	0.0403395
$706$	0	0
$707$	18.1743	0.683514
$708$	0	0
$709$	−5.44137	−0.204355	−0.102177	−	0.994766i	$-0.532581\pi$
$709$	−5.44137	−0.204355	−0.102177	+	0.994766i	$0.532581\pi$
$710$	0	0
$711$	−25.9657	−0.973791
$712$	0	0
$713$	0.0936632	0.00350772
$714$	0	0
$715$	−1.41309	−0.0528467
$716$	0	0
$717$	31.3345	1.17021
$718$	0	0
$719$	2.09293	0.0780531	0.0390266	−	0.999238i	$-0.487574\pi$
$719$	2.09293	0.0780531	0.0390266	+	0.999238i	$0.487574\pi$
$720$	0	0
$721$	26.9502	1.00368
$722$	0	0
$723$	34.0092	1.26482
$724$	0	0
$725$	−4.75474	−0.176587
$726$	0	0
$727$	−37.6516	−1.39642	−0.698210	−	0.715893i	$-0.746020\pi$
$727$	−37.6516	−1.39642	−0.698210	+	0.715893i	$0.746020\pi$
$728$	0	0
$729$	−7.77433	−0.287938
$730$	0	0
$731$	0.229770	0.00849836
$732$	0	0
$733$	0.823295	0.0304091	0.0152045	−	0.999884i	$-0.495160\pi$
$733$	0.823295	0.0304091	0.0152045	+	0.999884i	$0.495160\pi$
$734$	0	0
$735$	4.99819	0.184361
$736$	0	0
$737$	29.6850	1.09346
$738$	0	0
$739$	38.7329	1.42481	0.712406	−	0.701767i	$-0.247605\pi$
$739$	38.7329	1.42481	0.712406	+	0.701767i	$0.247605\pi$
$740$	0	0
$741$	−16.8967	−0.620717
$742$	0	0
$743$	−17.2470	−0.632732	−0.316366	−	0.948637i	$-0.602463\pi$
$743$	−17.2470	−0.632732	−0.316366	+	0.948637i	$0.602463\pi$
$744$	0	0
$745$	0.0799737	0.00293001
$746$	0	0
$747$	31.6396	1.15763
$748$	0	0
$749$	−11.9305	−0.435932
$750$	0	0
$751$	18.4996	0.675059	0.337530	−	0.941315i	$-0.390409\pi$
$751$	18.4996	0.675059	0.337530	+	0.941315i	$0.390409\pi$
$752$	0	0
$753$	19.0628	0.694687
$754$	0	0
$755$	−2.90052	−0.105561
$756$	0	0
$757$	6.50323	0.236364	0.118182	−	0.992992i	$-0.462293\pi$
$757$	6.50323	0.236364	0.118182	+	0.992992i	$0.462293\pi$
$758$	0	0
$759$	−1.00935	−0.0366371
$760$	0	0
$761$	15.2877	0.554180	0.277090	−	0.960844i	$-0.410630\pi$
$761$	15.2877	0.554180	0.277090	+	0.960844i	$0.410630\pi$
$762$	0	0
$763$	5.63916	0.204151
$764$	0	0
$765$	−1.72998	−0.0625474
$766$	0	0
$767$	6.21749	0.224501
$768$	0	0
$769$	−12.1133	−0.436816	−0.218408	−	0.975858i	$-0.570086\pi$
$769$	−12.1133	−0.436816	−0.218408	+	0.975858i	$0.570086\pi$
$770$	0	0
$771$	−11.6551	−0.419748
$772$	0	0
$773$	16.9337	0.609064	0.304532	−	0.952502i	$-0.401500\pi$
$773$	16.9337	0.609064	0.304532	+	0.952502i	$0.401500\pi$
$774$	0	0
$775$	−2.82547	−0.101494
$776$	0	0
$777$	31.4806	1.12936
$778$	0	0
$779$	−50.2173	−1.79922
$780$	0	0
$781$	−16.1331	−0.577286
$782$	0	0
$783$	−2.16154	−0.0772471
$784$	0	0
$785$	4.05082	0.144580
$786$	0	0
$787$	14.5887	0.520030	0.260015	−	0.965605i	$-0.416272\pi$
$787$	14.5887	0.520030	0.260015	+	0.965605i	$0.416272\pi$
$788$	0	0
$789$	−25.3290	−0.901737
$790$	0	0
$791$	−16.2595	−0.578121
$792$	0	0
$793$	−10.0451	−0.356713
$794$	0	0
$795$	15.3751	0.545298
$796$	0	0
$797$	32.8843	1.16482	0.582411	−	0.812895i	$-0.302110\pi$
$797$	32.8843	1.16482	0.582411	+	0.812895i	$0.302110\pi$
$798$	0	0
$799$	−1.65268	−0.0584677
$800$	0	0
$801$	9.98958	0.352965
$802$	0	0
$803$	−28.2352	−0.996399
$804$	0	0
$805$	0.123497	0.00435271
$806$	0	0
$807$	−1.59658	−0.0562022
$808$	0	0
$809$	−39.1579	−1.37672	−0.688360	−	0.725369i	$-0.741669\pi$
$809$	−39.1579	−1.37672	−0.688360	+	0.725369i	$0.741669\pi$
$810$	0	0
$811$	9.08224	0.318921	0.159460	−	0.987204i	$-0.449025\pi$
$811$	9.08224	0.318921	0.159460	+	0.987204i	$0.449025\pi$
$812$	0	0
$813$	16.3791	0.574438
$814$	0	0
$815$	4.01854	0.140763
$816$	0	0
$817$	1.00869	0.0352894
$818$	0	0
$819$	3.22256	0.112605
$820$	0	0
$821$	−2.83466	−0.0989303	−0.0494651	−	0.998776i	$-0.515752\pi$
$821$	−2.83466	−0.0989303	−0.0494651	+	0.998776i	$0.515752\pi$
$822$	0	0
$823$	8.48094	0.295627	0.147814	−	0.989015i	$-0.452776\pi$
$823$	8.48094	0.295627	0.147814	+	0.989015i	$0.452776\pi$
$824$	0	0
$825$	30.4483	1.06007
$826$	0	0
$827$	33.0701	1.14996	0.574981	−	0.818167i	$-0.305010\pi$
$827$	33.0701	1.14996	0.574981	+	0.818167i	$0.305010\pi$
$828$	0	0
$829$	10.4772	0.363889	0.181944	−	0.983309i	$-0.441761\pi$
$829$	10.4772	0.363889	0.181944	+	0.983309i	$0.441761\pi$
$830$	0	0
$831$	58.4125	2.02631
$832$	0	0
$833$	−7.71217	−0.267211
$834$	0	0
$835$	9.59486	0.332044
$836$	0	0
$837$	−1.28448	−0.0443980
$838$	0	0
$839$	18.7977	0.648967	0.324484	−	0.945891i	$-0.394809\pi$
$839$	18.7977	0.648967	0.324484	+	0.945891i	$0.394809\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−27.9204	−0.961631
$844$	0	0
$845$	−0.495240	−0.0170368
$846$	0	0
$847$	4.52228	0.155387
$848$	0	0
$849$	−15.7872	−0.541814
$850$	0	0
$851$	−1.39744	−0.0479035
$852$	0	0
$853$	−56.0222	−1.91816	−0.959082	−	0.283128i	$-0.908628\pi$
$853$	−56.0222	−1.91816	−0.959082	+	0.283128i	$0.908628\pi$
$854$	0	0
$855$	−7.59455	−0.259728
$856$	0	0
$857$	−35.8991	−1.22629	−0.613145	−	0.789970i	$-0.710096\pi$
$857$	−35.8991	−1.22629	−0.613145	+	0.789970i	$0.710096\pi$
$858$	0	0
$859$	−5.32381	−0.181646	−0.0908230	−	0.995867i	$-0.528950\pi$
$859$	−5.32381	−0.181646	−0.0908230	+	0.995867i	$0.528950\pi$
$860$	0	0
$861$	23.6836	0.807135
$862$	0	0
$863$	−0.315862	−0.0107521	−0.00537604	−	0.999986i	$-0.501711\pi$
$863$	−0.315862	−0.0107521	−0.00537604	+	0.999986i	$0.501711\pi$
$864$	0	0
$865$	7.72663	0.262713
$866$	0	0
$867$	−31.5522	−1.07157
$868$	0	0
$869$	36.3740	1.23390
$870$	0	0
$871$	10.4036	0.352511
$872$	0	0
$873$	22.2600	0.753387
$874$	0	0
$875$	−7.64306	−0.258383
$876$	0	0
$877$	6.15141	0.207718	0.103859	−	0.994592i	$-0.466881\pi$
$877$	6.15141	0.207718	0.103859	+	0.994592i	$0.466881\pi$
$878$	0	0
$879$	11.2188	0.378401
$880$	0	0
$881$	24.1350	0.813130	0.406565	−	0.913622i	$-0.366727\pi$
$881$	24.1350	0.813130	0.406565	+	0.913622i	$0.366727\pi$
$882$	0	0
$883$	−46.4091	−1.56179	−0.780896	−	0.624661i	$-0.785237\pi$
$883$	−46.4091	−1.56179	−0.780896	+	0.624661i	$0.785237\pi$
$884$	0	0
$885$	6.91053	0.232295
$886$	0	0
$887$	−0.384881	−0.0129230	−0.00646151	−	0.999979i	$-0.502057\pi$
$887$	−0.384881	−0.0129230	−0.00646151	+	0.999979i	$0.502057\pi$
$888$	0	0
$889$	18.7895	0.630181
$890$	0	0
$891$	31.2778	1.04784
$892$	0	0
$893$	−7.25524	−0.242787
$894$	0	0
$895$	1.73835	0.0581065
$896$	0	0
$897$	−0.353742	−0.0118111
$898$	0	0
$899$	0.594242	0.0198191
$900$	0	0
$901$	−23.7237	−0.790350
$902$	0	0
$903$	−0.475719	−0.0158309
$904$	0	0
$905$	2.51975	0.0837594
$906$	0	0
$907$	9.07576	0.301356	0.150678	−	0.988583i	$-0.451854\pi$
$907$	9.07576	0.301356	0.150678	+	0.988583i	$0.451854\pi$
$908$	0	0
$909$	−23.3983	−0.776074
$910$	0	0
$911$	−23.0270	−0.762919	−0.381459	−	0.924386i	$-0.624578\pi$
$911$	−23.0270	−0.762919	−0.381459	+	0.924386i	$0.624578\pi$
$912$	0	0
$913$	−44.3222	−1.46685
$914$	0	0
$915$	−11.1648	−0.369097
$916$	0	0
$917$	−21.3755	−0.705881
$918$	0	0
$919$	−36.6359	−1.20851	−0.604253	−	0.796792i	$-0.706529\pi$
$919$	−36.6359	−1.20851	−0.604253	+	0.796792i	$0.706529\pi$
$920$	0	0
$921$	−61.5052	−2.02667
$922$	0	0
$923$	−5.65407	−0.186106
$924$	0	0
$925$	42.1554	1.38606
$926$	0	0
$927$	−34.6969	−1.13960
$928$	0	0
$929$	−32.0017	−1.04994	−0.524971	−	0.851120i	$-0.675924\pi$
$929$	−32.0017	−1.04994	−0.524971	+	0.851120i	$0.675924\pi$
$930$	0	0
$931$	−33.8562	−1.10959
$932$	0	0
$933$	12.4967	0.409123
$934$	0	0
$935$	2.42343	0.0792547
$936$	0	0
$937$	−10.8955	−0.355941	−0.177970	−	0.984036i	$-0.556953\pi$
$937$	−10.8955	−0.355941	−0.177970	+	0.984036i	$0.556953\pi$
$938$	0	0
$939$	−62.8806	−2.05203
$940$	0	0
$941$	−34.1903	−1.11457	−0.557286	−	0.830321i	$-0.688157\pi$
$941$	−34.1903	−1.11457	−0.557286	+	0.830321i	$0.688157\pi$
$942$	0	0
$943$	−1.05132	−0.0342358
$944$	0	0
$945$	−1.69362	−0.0550933
$946$	0	0
$947$	41.8592	1.36024	0.680120	−	0.733101i	$-0.261928\pi$
$947$	41.8592	1.36024	0.680120	+	0.733101i	$0.261928\pi$
$948$	0	0
$949$	−9.89546	−0.321220
$950$	0	0
$951$	4.74445	0.153849
$952$	0	0
$953$	−22.2738	−0.721520	−0.360760	−	0.932659i	$-0.617483\pi$
$953$	−22.2738	−0.721520	−0.360760	+	0.932659i	$0.617483\pi$
$954$	0	0
$955$	11.2076	0.362671
$956$	0	0
$957$	−6.40377	−0.207005
$958$	0	0
$959$	5.59542	0.180686
$960$	0	0
$961$	−30.6469	−0.988609
$962$	0	0
$963$	15.3599	0.494965
$964$	0	0
$965$	−3.97056	−0.127817
$966$	0	0
$967$	16.4316	0.528403	0.264202	−	0.964467i	$-0.414892\pi$
$967$	16.4316	0.528403	0.264202	+	0.964467i	$0.414892\pi$
$968$	0	0
$969$	28.9777	0.930896
$970$	0	0
$971$	−23.0672	−0.740263	−0.370131	−	0.928979i	$-0.620687\pi$
$971$	−23.0672	−0.740263	−0.370131	+	0.928979i	$0.620687\pi$
$972$	0	0
$973$	30.7123	0.984592
$974$	0	0
$975$	10.6710	0.341747
$976$	0	0
$977$	−52.3072	−1.67346	−0.836728	−	0.547618i	$-0.815535\pi$
$977$	−52.3072	−1.67346	−0.836728	+	0.547618i	$0.815535\pi$
$978$	0	0
$979$	−13.9939	−0.447246
$980$	0	0
$981$	−7.26010	−0.231797
$982$	0	0
$983$	45.0287	1.43619	0.718096	−	0.695944i	$-0.245014\pi$
$983$	45.0287	1.43619	0.718096	+	0.695944i	$0.245014\pi$
$984$	0	0
$985$	−5.03262	−0.160353
$986$	0	0
$987$	3.42173	0.108915
$988$	0	0
$989$	0.0211173	0.000671493 0
$990$	0	0
$991$	−3.70996	−0.117851	−0.0589254	−	0.998262i	$-0.518767\pi$
$991$	−3.70996	−0.117851	−0.0589254	+	0.998262i	$0.518767\pi$
$992$	0	0
$993$	20.9077	0.663487
$994$	0	0
$995$	−1.38203	−0.0438133
$996$	0	0
$997$	12.3894	0.392376	0.196188	−	0.980566i	$-0.437144\pi$
$997$	12.3894	0.392376	0.196188	+	0.980566i	$0.437144\pi$
$998$	0	0
$999$	19.1641	0.606327

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6032.2.a.u.1.7		7
4.3	odd	2		377.2.a.e.1.7	✓	7
12.11	even	2		3393.2.a.o.1.1		7
20.19	odd	2		9425.2.a.s.1.1		7
52.51	odd	2		4901.2.a.k.1.1		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
377.2.a.e.1.7	✓	7	4.3	odd	2
3393.2.a.o.1.1		7	12.11	even	2
4901.2.a.k.1.1		7	52.51	odd	2
6032.2.a.u.1.7		7	1.1	even	1	trivial
9425.2.a.s.1.1		7	20.19	odd	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:	\( N \)	\(=\)	\( 6032 = 2^{4} \cdot 13 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6032.a (trivial)

\(f(q)\)	\(=\)	\(q+2.24430 q^{3} -0.495240 q^{5} -1.58211 q^{7} +2.03688 q^{9} +O(q^{10})\)	\(q+2.24430 q^{3} -0.495240 q^{5} -1.58211 q^{7} +2.03688 q^{9} -2.85335 q^{11} -1.00000 q^{13} -1.11147 q^{15} +1.71498 q^{17} +7.52874 q^{19} -3.55072 q^{21} +0.157618 q^{23} -4.75474 q^{25} -2.16154 q^{27} +1.00000 q^{29} +0.594242 q^{31} -6.40377 q^{33} +0.783523 q^{35} -8.86597 q^{37} -2.24430 q^{39} -6.67008 q^{41} +0.133978 q^{43} -1.00874 q^{45} -0.963672 q^{47} -4.49693 q^{49} +3.84894 q^{51} -13.8332 q^{53} +1.41309 q^{55} +16.8967 q^{57} -6.21749 q^{59} +10.0451 q^{61} -3.22256 q^{63} +0.495240 q^{65} -10.4036 q^{67} +0.353742 q^{69} +5.65407 q^{71} +9.89546 q^{73} -10.6710 q^{75} +4.51431 q^{77} -12.7478 q^{79} -10.9618 q^{81} +15.5334 q^{83} -0.849329 q^{85} +2.24430 q^{87} +4.90437 q^{89} +1.58211 q^{91} +1.33366 q^{93} -3.72853 q^{95} +10.9285 q^{97} -5.81192 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 2 q^{3} - 2 q^{5} - 7 q^{7} + 5 q^{9}+O(q^{10}) \) 7 * q - 2 * q^3 - 2 * q^5 - 7 * q^7 + 5 * q^9	\( 7 q - 2 q^{3} - 2 q^{5} - 7 q^{7} + 5 q^{9} + 3 q^{11} - 7 q^{13} - 20 q^{15} + 14 q^{17} + 12 q^{19} + q^{21} - 5 q^{23} + 5 q^{25} - 14 q^{27} + 7 q^{29} + 3 q^{31} - q^{33} + 6 q^{35} + 9 q^{37} + 2 q^{39} - 9 q^{41} - 24 q^{43} + 10 q^{45} + 9 q^{47} - 10 q^{49} - 5 q^{51} - 13 q^{53} - 7 q^{55} + 12 q^{57} - 13 q^{59} + 3 q^{61} - 2 q^{63} + 2 q^{65} + 7 q^{67} + 8 q^{69} - 22 q^{71} + 4 q^{73} + 29 q^{75} - 30 q^{77} - 48 q^{79} + 15 q^{81} + 30 q^{83} - 9 q^{85} - 2 q^{87} - 25 q^{89} + 7 q^{91} - 27 q^{93} - 23 q^{95} + 4 q^{97} + 25 q^{99}+O(q^{100}) \) 7 * q - 2 * q^3 - 2 * q^5 - 7 * q^7 + 5 * q^9 + 3 * q^11 - 7 * q^13 - 20 * q^15 + 14 * q^17 + 12 * q^19 + q^21 - 5 * q^23 + 5 * q^25 - 14 * q^27 + 7 * q^29 + 3 * q^31 - q^33 + 6 * q^35 + 9 * q^37 + 2 * q^39 - 9 * q^41 - 24 * q^43 + 10 * q^45 + 9 * q^47 - 10 * q^49 - 5 * q^51 - 13 * q^53 - 7 * q^55 + 12 * q^57 - 13 * q^59 + 3 * q^61 - 2 * q^63 + 2 * q^65 + 7 * q^67 + 8 * q^69 - 22 * q^71 + 4 * q^73 + 29 * q^75 - 30 * q^77 - 48 * q^79 + 15 * q^81 + 30 * q^83 - 9 * q^85 - 2 * q^87 - 25 * q^89 + 7 * q^91 - 27 * q^93 - 23 * q^95 + 4 * q^97 + 25 * q^99

Introduction

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Varieties

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