LMFDB - Embedded newform 7448.2.a.bj.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7448,2,Mod(1,7448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7448, 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("7448.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7448 = 2^{3} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7448.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:	$59.4725794254$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.25857.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + 9x + 3 $ x^4 - x^3 - 8x^2 + 9x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1064)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.57368$ of defining polynomial
Character	$\chi$	$=$	7448.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.30278 q^{3} +0.270905 q^{5} +2.30278 q^{9} +O(q^{10})$	$q-2.30278 q^{3} +0.270905 q^{5} +2.30278 q^{9} +2.35293 q^{11} -5.19752 q^{13} -0.623835 q^{15} +1.62383 q^{17} -1.00000 q^{19} -3.19752 q^{23} -4.92661 q^{25} +1.60555 q^{27} +6.45014 q^{29} +2.77120 q^{31} -5.41827 q^{33} +4.97676 q^{37} +11.9687 q^{39} +4.45014 q^{41} +8.01023 q^{43} +0.623835 q^{45} -6.46842 q^{47} -3.73933 q^{51} -6.84459 q^{53} +0.637421 q^{55} +2.30278 q^{57} -2.37121 q^{59} -9.61578 q^{61} -1.40804 q^{65} -10.0189 q^{67} +7.36316 q^{69} -5.67952 q^{71} +0.490059 q^{73} +11.3449 q^{75} +17.2213 q^{79} -10.6056 q^{81} +3.09561 q^{83} +0.439906 q^{85} -14.8532 q^{87} +3.81272 q^{89} -6.38144 q^{93} -0.270905 q^{95} +5.89004 q^{97} +5.41827 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} - q^{5} + 2 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 - q^5 + 2 * q^9	$ 4 q - 2 q^{3} - q^{5} + 2 q^{9} + 2 q^{11} - 2 q^{13} + 7 q^{15} - 3 q^{17} - 4 q^{19} + 6 q^{23} - 3 q^{25} - 8 q^{27} - 17 q^{31} - q^{33} + 3 q^{37} + q^{39} - 8 q^{41} + 7 q^{43} - 7 q^{45} - 5 q^{47} + 8 q^{51} - 16 q^{53} - 17 q^{55} + 2 q^{57} - 7 q^{59} + q^{61} - 10 q^{65} + 7 q^{67} - 3 q^{69} + 27 q^{71} + q^{73} + 8 q^{75} + 15 q^{79} - 28 q^{81} - 3 q^{83} + q^{85} - 26 q^{87} + 9 q^{89} + 2 q^{93} + q^{95} - 3 q^{97} + q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 - q^5 + 2 * q^9 + 2 * q^11 - 2 * q^13 + 7 * q^15 - 3 * q^17 - 4 * q^19 + 6 * q^23 - 3 * q^25 - 8 * q^27 - 17 * q^31 - q^33 + 3 * q^37 + q^39 - 8 * q^41 + 7 * q^43 - 7 * q^45 - 5 * q^47 + 8 * q^51 - 16 * q^53 - 17 * q^55 + 2 * q^57 - 7 * q^59 + q^61 - 10 * q^65 + 7 * q^67 - 3 * q^69 + 27 * q^71 + q^73 + 8 * q^75 + 15 * q^79 - 28 * q^81 - 3 * q^83 + q^85 - 26 * q^87 + 9 * q^89 + 2 * q^93 + q^95 - 3 * q^97 + 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
$3$	−2.30278	−1.32951	−0.664754	−	0.747062i	$-0.731464\pi$
$3$	−2.30278	−1.32951	−0.664754	+	0.747062i	$0.731464\pi$
$4$	0	0
$5$	0.270905	0.121153	0.0605763	−	0.998164i	$-0.480706\pi$
$5$	0.270905	0.121153	0.0605763	+	0.998164i	$0.480706\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	2.30278	0.767592
$10$	0	0
$11$	2.35293	0.709435	0.354717	−	0.934974i	$-0.384577\pi$
$11$	2.35293	0.709435	0.354717	+	0.934974i	$0.384577\pi$
$12$	0	0
$13$	−5.19752	−1.44153	−0.720766	−	0.693179i	$-0.756210\pi$
$13$	−5.19752	−1.44153	−0.720766	+	0.693179i	$0.756210\pi$
$14$	0	0
$15$	−0.623835	−0.161073
$16$	0	0
$17$	1.62383	0.393838	0.196919	−	0.980420i	$-0.436906\pi$
$17$	1.62383	0.393838	0.196919	+	0.980420i	$0.436906\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.19752	−0.666728	−0.333364	−	0.942798i	$-0.608184\pi$
$23$	−3.19752	−0.666728	−0.333364	+	0.942798i	$0.608184\pi$
$24$	0	0
$25$	−4.92661	−0.985322
$26$	0	0
$27$	1.60555	0.308988
$28$	0	0
$29$	6.45014	1.19776	0.598880	−	0.800839i	$-0.295613\pi$
$29$	6.45014	1.19776	0.598880	+	0.800839i	$0.295613\pi$
$30$	0	0
$31$	2.77120	0.497722	0.248861	−	0.968539i	$-0.419944\pi$
$31$	2.77120	0.497722	0.248861	+	0.968539i	$0.419944\pi$
$32$	0	0
$33$	−5.41827	−0.943199
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.97676	0.818175	0.409087	−	0.912495i	$-0.365847\pi$
$37$	4.97676	0.818175	0.409087	+	0.912495i	$0.365847\pi$
$38$	0	0
$39$	11.9687	1.91653
$40$	0	0
$41$	4.45014	0.694995	0.347497	−	0.937681i	$-0.387032\pi$
$41$	4.45014	0.694995	0.347497	+	0.937681i	$0.387032\pi$
$42$	0	0
$43$	8.01023	1.22155	0.610774	−	0.791805i	$-0.290858\pi$
$43$	8.01023	1.22155	0.610774	+	0.791805i	$0.290858\pi$
$44$	0	0
$45$	0.623835	0.0929958
$46$	0	0
$47$	−6.46842	−0.943516	−0.471758	−	0.881728i	$-0.656380\pi$
$47$	−6.46842	−0.943516	−0.471758	+	0.881728i	$0.656380\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−3.73933	−0.523610
$52$	0	0
$53$	−6.84459	−0.940176	−0.470088	−	0.882619i	$-0.655778\pi$
$53$	−6.84459	−0.940176	−0.470088	+	0.882619i	$0.655778\pi$
$54$	0	0
$55$	0.637421	0.0859499
$56$	0	0
$57$	2.30278	0.305010
$58$	0	0
$59$	−2.37121	−0.308706	−0.154353	−	0.988016i	$-0.549329\pi$
$59$	−2.37121	−0.308706	−0.154353	+	0.988016i	$0.549329\pi$
$60$	0	0
$61$	−9.61578	−1.23117	−0.615587	−	0.788069i	$-0.711081\pi$
$61$	−9.61578	−1.23117	−0.615587	+	0.788069i	$0.711081\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.40804	−0.174645
$66$	0	0
$67$	−10.0189	−1.22400	−0.612000	−	0.790858i	$-0.709635\pi$
$67$	−10.0189	−1.22400	−0.612000	+	0.790858i	$0.709635\pi$
$68$	0	0
$69$	7.36316	0.886420
$70$	0	0
$71$	−5.67952	−0.674035	−0.337018	−	0.941498i	$-0.609418\pi$
$71$	−5.67952	−0.674035	−0.337018	+	0.941498i	$0.609418\pi$
$72$	0	0
$73$	0.490059	0.0573571	0.0286785	−	0.999589i	$-0.490870\pi$
$73$	0.490059	0.0573571	0.0286785	+	0.999589i	$0.490870\pi$
$74$	0	0
$75$	11.3449	1.30999
$76$	0	0
$77$	0	0
$78$	0	0
$79$	17.2213	1.93755	0.968776	−	0.247939i	$-0.0797531\pi$
$79$	17.2213	1.93755	0.968776	+	0.247939i	$0.0797531\pi$
$80$	0	0
$81$	−10.6056	−1.17839
$82$	0	0
$83$	3.09561	0.339787	0.169894	−	0.985462i	$-0.445658\pi$
$83$	3.09561	0.339787	0.169894	+	0.985462i	$0.445658\pi$
$84$	0	0
$85$	0.439906	0.0477145
$86$	0	0
$87$	−14.8532	−1.59243
$88$	0	0
$89$	3.81272	0.404147	0.202074	−	0.979370i	$-0.435232\pi$
$89$	3.81272	0.404147	0.202074	+	0.979370i	$0.435232\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−6.38144	−0.661725
$94$	0	0
$95$	−0.270905	−0.0277943
$96$	0	0
$97$	5.89004	0.598043	0.299022	−	0.954246i	$-0.403340\pi$
$97$	5.89004	0.598043	0.299022	+	0.954246i	$0.403340\pi$
$98$	0	0
$99$	5.41827	0.544556
$100$	0	0
$101$	11.8031	1.17445	0.587225	−	0.809424i	$-0.300221\pi$
$101$	11.8031	1.17445	0.587225	+	0.809424i	$0.300221\pi$
$102$	0	0
$103$	8.50893	0.838409	0.419205	−	0.907892i	$-0.362309\pi$
$103$	8.50893	0.838409	0.419205	+	0.907892i	$0.362309\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.16095	0.498928	0.249464	−	0.968384i	$-0.419746\pi$
$107$	5.16095	0.498928	0.249464	+	0.968384i	$0.419746\pi$
$108$	0	0
$109$	−16.8269	−1.61172	−0.805862	−	0.592104i	$-0.798298\pi$
$109$	−16.8269	−1.61172	−0.805862	+	0.592104i	$0.798298\pi$
$110$	0	0
$111$	−11.4604	−1.08777
$112$	0	0
$113$	−16.4102	−1.54374	−0.771872	−	0.635778i	$-0.780679\pi$
$113$	−16.4102	−1.54374	−0.771872	+	0.635778i	$0.780679\pi$
$114$	0	0
$115$	−0.866225	−0.0807759
$116$	0	0
$117$	−11.9687	−1.10651
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−5.46372	−0.496702
$122$	0	0
$123$	−10.2477	−0.924001
$124$	0	0
$125$	−2.68917	−0.240527
$126$	0	0
$127$	16.5823	1.47144	0.735721	−	0.677284i	$-0.236843\pi$
$127$	16.5823	1.47144	0.735721	+	0.677284i	$0.236843\pi$
$128$	0	0
$129$	−18.4458	−1.62406
$130$	0	0
$131$	−8.62442	−0.753519	−0.376759	−	0.926311i	$-0.622962\pi$
$131$	−8.62442	−0.753519	−0.376759	+	0.926311i	$0.622962\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0.434953	0.0374348
$136$	0	0
$137$	−16.8263	−1.43757	−0.718784	−	0.695233i	$-0.755301\pi$
$137$	−16.8263	−1.43757	−0.718784	+	0.695233i	$0.755301\pi$
$138$	0	0
$139$	21.1814	1.79658	0.898292	−	0.439399i	$-0.144809\pi$
$139$	21.1814	1.79658	0.898292	+	0.439399i	$0.144809\pi$
$140$	0	0
$141$	14.8953	1.25441
$142$	0	0
$143$	−12.2294	−1.02267
$144$	0	0
$145$	1.74738	0.145112
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.70276	0.139496	0.0697478	−	0.997565i	$-0.477781\pi$
$149$	1.70276	0.139496	0.0697478	+	0.997565i	$0.477781\pi$
$150$	0	0
$151$	−2.37617	−0.193370	−0.0966848	−	0.995315i	$-0.530824\pi$
$151$	−2.37617	−0.193370	−0.0966848	+	0.995315i	$0.530824\pi$
$152$	0	0
$153$	3.73933	0.302307
$154$	0	0
$155$	0.750732	0.0603003
$156$	0	0
$157$	1.71081	0.136538	0.0682688	−	0.997667i	$-0.478252\pi$
$157$	1.71081	0.136538	0.0682688	+	0.997667i	$0.478252\pi$
$158$	0	0
$159$	15.7615	1.24997
$160$	0	0
$161$	0	0
$162$	0	0
$163$	16.1832	1.26756	0.633782	−	0.773512i	$-0.281502\pi$
$163$	16.1832	1.26756	0.633782	+	0.773512i	$0.281502\pi$
$164$	0	0
$165$	−1.46784	−0.114271
$166$	0	0
$167$	−1.30773	−0.101195	−0.0505975	−	0.998719i	$-0.516113\pi$
$167$	−1.30773	−0.101195	−0.0505975	+	0.998719i	$0.516113\pi$
$168$	0	0
$169$	14.0142	1.07801
$170$	0	0
$171$	−2.30278	−0.176098
$172$	0	0
$173$	17.1009	1.30016	0.650078	−	0.759867i	$-0.274736\pi$
$173$	17.1009	1.30016	0.650078	+	0.759867i	$0.274736\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	5.46037	0.410427
$178$	0	0
$179$	8.46037	0.632358	0.316179	−	0.948700i	$-0.397600\pi$
$179$	8.46037	0.632358	0.316179	+	0.948700i	$0.397600\pi$
$180$	0	0
$181$	−3.67031	−0.272812	−0.136406	−	0.990653i	$-0.543555\pi$
$181$	−3.67031	−0.272812	−0.136406	+	0.990653i	$0.543555\pi$
$182$	0	0
$183$	22.1430	1.63686
$184$	0	0
$185$	1.34823	0.0991240
$186$	0	0
$187$	3.82077	0.279402
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1.67399	−0.121126	−0.0605628	−	0.998164i	$-0.519290\pi$
$191$	−1.67399	−0.121126	−0.0605628	+	0.998164i	$0.519290\pi$
$192$	0	0
$193$	−7.99840	−0.575738	−0.287869	−	0.957670i	$-0.592947\pi$
$193$	−7.99840	−0.575738	−0.287869	+	0.957670i	$0.592947\pi$
$194$	0	0
$195$	3.24239	0.232192
$196$	0	0
$197$	3.26285	0.232469	0.116234	−	0.993222i	$-0.462918\pi$
$197$	3.26285	0.232469	0.116234	+	0.993222i	$0.462918\pi$
$198$	0	0
$199$	7.81330	0.553870	0.276935	−	0.960889i	$-0.410681\pi$
$199$	7.81330	0.553870	0.276935	+	0.960889i	$0.410681\pi$
$200$	0	0
$201$	23.0712	1.62732
$202$	0	0
$203$	0	0
$204$	0	0
$205$	1.20557	0.0842004
$206$	0	0
$207$	−7.36316	−0.511775
$208$	0	0
$209$	−2.35293	−0.162756
$210$	0	0
$211$	−11.5272	−0.793566	−0.396783	−	0.917912i	$-0.629873\pi$
$211$	−11.5272	−0.793566	−0.396783	+	0.917912i	$0.629873\pi$
$212$	0	0
$213$	13.0787	0.896136
$214$	0	0
$215$	2.17002	0.147994
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−1.12850	−0.0762567
$220$	0	0
$221$	−8.43991	−0.567729
$222$	0	0
$223$	−6.05015	−0.405148	−0.202574	−	0.979267i	$-0.564931\pi$
$223$	−6.05015	−0.405148	−0.202574	+	0.979267i	$0.564931\pi$
$224$	0	0
$225$	−11.3449	−0.756325
$226$	0	0
$227$	−1.42322	−0.0944625	−0.0472312	−	0.998884i	$-0.515040\pi$
$227$	−1.42322	−0.0944625	−0.0472312	+	0.998884i	$0.515040\pi$
$228$	0	0
$229$	−16.5792	−1.09559	−0.547793	−	0.836614i	$-0.684532\pi$
$229$	−16.5792	−1.09559	−0.547793	+	0.836614i	$0.684532\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−18.5807	−1.21726	−0.608632	−	0.793453i	$-0.708281\pi$
$233$	−18.5807	−1.21726	−0.608632	+	0.793453i	$0.708281\pi$
$234$	0	0
$235$	−1.75233	−0.114309
$236$	0	0
$237$	−39.6569	−2.57599
$238$	0	0
$239$	17.0972	1.10593	0.552963	−	0.833206i	$-0.313497\pi$
$239$	17.0972	1.10593	0.552963	+	0.833206i	$0.313497\pi$
$240$	0	0
$241$	4.44854	0.286556	0.143278	−	0.989683i	$-0.454236\pi$
$241$	4.44854	0.286556	0.143278	+	0.989683i	$0.454236\pi$
$242$	0	0
$243$	19.6056	1.25770
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.19752	0.330710
$248$	0	0
$249$	−7.12850	−0.451750
$250$	0	0
$251$	19.2399	1.21441	0.607205	−	0.794545i	$-0.292291\pi$
$251$	19.2399	1.21441	0.607205	+	0.794545i	$0.292291\pi$
$252$	0	0
$253$	−7.52353	−0.473000
$254$	0	0
$255$	−1.01300	−0.0634368
$256$	0	0
$257$	−29.4178	−1.83503	−0.917517	−	0.397696i	$-0.869810\pi$
$257$	−29.4178	−1.83503	−0.917517	+	0.397696i	$0.869810\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	14.8532	0.919391
$262$	0	0
$263$	−19.8047	−1.22121	−0.610604	−	0.791936i	$-0.709073\pi$
$263$	−19.8047	−1.22121	−0.610604	+	0.791936i	$0.709073\pi$
$264$	0	0
$265$	−1.85424	−0.113905
$266$	0	0
$267$	−8.77983	−0.537317
$268$	0	0
$269$	11.3903	0.694481	0.347240	−	0.937776i	$-0.387119\pi$
$269$	11.3903	0.694481	0.347240	+	0.937776i	$0.387119\pi$
$270$	0	0
$271$	5.70771	0.346719	0.173359	−	0.984859i	$-0.444538\pi$
$271$	5.70771	0.346719	0.173359	+	0.984859i	$0.444538\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−11.5920	−0.699022
$276$	0	0
$277$	−1.29414	−0.0777574	−0.0388787	−	0.999244i	$-0.512379\pi$
$277$	−1.29414	−0.0777574	−0.0388787	+	0.999244i	$0.512379\pi$
$278$	0	0
$279$	6.38144	0.382047
$280$	0	0
$281$	−18.2347	−1.08779	−0.543894	−	0.839154i	$-0.683051\pi$
$281$	−18.2347	−1.08779	−0.543894	+	0.839154i	$0.683051\pi$
$282$	0	0
$283$	−15.7582	−0.936727	−0.468364	−	0.883536i	$-0.655156\pi$
$283$	−15.7582	−0.936727	−0.468364	+	0.883536i	$0.655156\pi$
$284$	0	0
$285$	0.623835	0.0369528
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−14.3632	−0.844892
$290$	0	0
$291$	−13.5634	−0.795103
$292$	0	0
$293$	3.03347	0.177217	0.0886086	−	0.996067i	$-0.471758\pi$
$293$	3.03347	0.177217	0.0886086	+	0.996067i	$0.471758\pi$
$294$	0	0
$295$	−0.642374	−0.0374005
$296$	0	0
$297$	3.77775	0.219207
$298$	0	0
$299$	16.6191	0.961110
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−27.1798	−1.56144
$304$	0	0
$305$	−2.60497	−0.149160
$306$	0	0
$307$	−18.3671	−1.04827	−0.524133	−	0.851637i	$-0.675610\pi$
$307$	−18.3671	−1.04827	−0.524133	+	0.851637i	$0.675610\pi$
$308$	0	0
$309$	−19.5941	−1.11467
$310$	0	0
$311$	−8.12514	−0.460735	−0.230367	−	0.973104i	$-0.573993\pi$
$311$	−8.12514	−0.460735	−0.230367	+	0.973104i	$0.573993\pi$
$312$	0	0
$313$	−24.6866	−1.39537	−0.697683	−	0.716406i	$-0.745786\pi$
$313$	−24.6866	−1.39537	−0.697683	+	0.716406i	$0.745786\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−31.4361	−1.76563	−0.882814	−	0.469722i	$-0.844354\pi$
$317$	−31.4361	−1.76563	−0.882814	+	0.469722i	$0.844354\pi$
$318$	0	0
$319$	15.1767	0.849733
$320$	0	0
$321$	−11.8845	−0.663329
$322$	0	0
$323$	−1.62383	−0.0903526
$324$	0	0
$325$	25.6061	1.42037
$326$	0	0
$327$	38.7485	2.14280
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−0.995304	−0.0547068	−0.0273534	−	0.999626i	$-0.508708\pi$
$331$	−0.995304	−0.0547068	−0.0273534	+	0.999626i	$0.508708\pi$
$332$	0	0
$333$	11.4604	0.628024
$334$	0	0
$335$	−2.71417	−0.148291
$336$	0	0
$337$	−10.7885	−0.587685	−0.293843	−	0.955854i	$-0.594934\pi$
$337$	−10.7885	−0.587685	−0.293843	+	0.955854i	$0.594934\pi$
$338$	0	0
$339$	37.7890	2.05242
$340$	0	0
$341$	6.52043	0.353101
$342$	0	0
$343$	0	0
$344$	0	0
$345$	1.99472	0.107392
$346$	0	0
$347$	−26.9024	−1.44419	−0.722097	−	0.691792i	$-0.756822\pi$
$347$	−26.9024	−1.44419	−0.722097	+	0.691792i	$0.756822\pi$
$348$	0	0
$349$	−10.5731	−0.565965	−0.282982	−	0.959125i	$-0.591324\pi$
$349$	−10.5731	−0.565965	−0.282982	+	0.959125i	$0.591324\pi$
$350$	0	0
$351$	−8.34488	−0.445417
$352$	0	0
$353$	−15.0659	−0.801878	−0.400939	−	0.916105i	$-0.631316\pi$
$353$	−15.0659	−0.801878	−0.400939	+	0.916105i	$0.631316\pi$
$354$	0	0
$355$	−1.53861	−0.0816612
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−24.7153	−1.30443	−0.652213	−	0.758036i	$-0.726159\pi$
$359$	−24.7153	−1.30443	−0.652213	+	0.758036i	$0.726159\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	12.5817	0.660370
$364$	0	0
$365$	0.132760	0.00694896
$366$	0	0
$367$	−26.8541	−1.40177	−0.700885	−	0.713274i	$-0.747212\pi$
$367$	−26.8541	−1.40177	−0.700885	+	0.713274i	$0.747212\pi$
$368$	0	0
$369$	10.2477	0.533472
$370$	0	0
$371$	0	0
$372$	0	0
$373$	10.3901	0.537978	0.268989	−	0.963143i	$-0.413310\pi$
$373$	10.3901	0.537978	0.268989	+	0.963143i	$0.413310\pi$
$374$	0	0
$375$	6.19256	0.319783
$376$	0	0
$377$	−33.5247	−1.72661
$378$	0	0
$379$	10.2811	0.528107	0.264053	−	0.964508i	$-0.414941\pi$
$379$	10.2811	0.528107	0.264053	+	0.964508i	$0.414941\pi$
$380$	0	0
$381$	−38.1854	−1.95629
$382$	0	0
$383$	3.19752	0.163385	0.0816927	−	0.996658i	$-0.473967\pi$
$383$	3.19752	0.163385	0.0816927	+	0.996658i	$0.473967\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	18.4458	0.937651
$388$	0	0
$389$	15.6510	0.793538	0.396769	−	0.917919i	$-0.370131\pi$
$389$	15.6510	0.793538	0.396769	+	0.917919i	$0.370131\pi$
$390$	0	0
$391$	−5.19224	−0.262583
$392$	0	0
$393$	19.8601	1.00181
$394$	0	0
$395$	4.66535	0.234739
$396$	0	0
$397$	−12.5619	−0.630461	−0.315231	−	0.949015i	$-0.602082\pi$
$397$	−12.5619	−0.630461	−0.315231	+	0.949015i	$0.602082\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−14.8621	−0.742178	−0.371089	−	0.928597i	$-0.621016\pi$
$401$	−14.8621	−0.742178	−0.371089	+	0.928597i	$0.621016\pi$
$402$	0	0
$403$	−14.4033	−0.717481
$404$	0	0
$405$	−2.87310	−0.142766
$406$	0	0
$407$	11.7100	0.580442
$408$	0	0
$409$	−24.8718	−1.22983	−0.614915	−	0.788594i	$-0.710810\pi$
$409$	−24.8718	−1.22983	−0.614915	+	0.788594i	$0.710810\pi$
$410$	0	0
$411$	38.7472	1.91126
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0.838618	0.0411661
$416$	0	0
$417$	−48.7760	−2.38857
$418$	0	0
$419$	−15.2817	−0.746561	−0.373280	−	0.927719i	$-0.621767\pi$
$419$	−15.2817	−0.746561	−0.373280	+	0.927719i	$0.621767\pi$
$420$	0	0
$421$	16.7592	0.816794	0.408397	−	0.912804i	$-0.366088\pi$
$421$	16.7592	0.816794	0.408397	+	0.912804i	$0.366088\pi$
$422$	0	0
$423$	−14.8953	−0.724235
$424$	0	0
$425$	−8.00000	−0.388057
$426$	0	0
$427$	0	0
$428$	0	0
$429$	28.1615	1.35965
$430$	0	0
$431$	1.48143	0.0713577	0.0356789	−	0.999363i	$-0.488641\pi$
$431$	1.48143	0.0713577	0.0356789	+	0.999363i	$0.488641\pi$
$432$	0	0
$433$	−14.3618	−0.690185	−0.345092	−	0.938569i	$-0.612152\pi$
$433$	−14.3618	−0.690185	−0.345092	+	0.938569i	$0.612152\pi$
$434$	0	0
$435$	−4.02382	−0.192927
$436$	0	0
$437$	3.19752	0.152958
$438$	0	0
$439$	21.8067	1.04078	0.520390	−	0.853929i	$-0.325787\pi$
$439$	21.8067	1.04078	0.520390	+	0.853929i	$0.325787\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−15.6272	−0.742470	−0.371235	−	0.928539i	$-0.621066\pi$
$443$	−15.6272	−0.742470	−0.371235	+	0.928539i	$0.621066\pi$
$444$	0	0
$445$	1.03289	0.0489635
$446$	0	0
$447$	−3.92107	−0.185460
$448$	0	0
$449$	15.2070	0.717662	0.358831	−	0.933403i	$-0.383175\pi$
$449$	15.2070	0.717662	0.358831	+	0.933403i	$0.383175\pi$
$450$	0	0
$451$	10.4709	0.493053
$452$	0	0
$453$	5.47178	0.257086
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−14.8092	−0.692744	−0.346372	−	0.938097i	$-0.612587\pi$
$457$	−14.8092	−0.692744	−0.346372	+	0.938097i	$0.612587\pi$
$458$	0	0
$459$	2.60715	0.121691
$460$	0	0
$461$	25.3876	1.18242	0.591208	−	0.806519i	$-0.298651\pi$
$461$	25.3876	1.18242	0.591208	+	0.806519i	$0.298651\pi$
$462$	0	0
$463$	10.0502	0.467070	0.233535	−	0.972348i	$-0.424971\pi$
$463$	10.0502	0.467070	0.233535	+	0.972348i	$0.424971\pi$
$464$	0	0
$465$	−1.72877	−0.0801697
$466$	0	0
$467$	−23.0429	−1.06630	−0.533150	−	0.846021i	$-0.678992\pi$
$467$	−23.0429	−1.06630	−0.533150	+	0.846021i	$0.678992\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−3.93961	−0.181528
$472$	0	0
$473$	18.8475	0.866609
$474$	0	0
$475$	4.92661	0.226048
$476$	0	0
$477$	−15.7615	−0.721672
$478$	0	0
$479$	−41.8358	−1.91153	−0.955764	−	0.294134i	$-0.904969\pi$
$479$	−41.8358	−1.91153	−0.955764	+	0.294134i	$0.904969\pi$
$480$	0	0
$481$	−25.8668	−1.17942
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.59565	0.0724545
$486$	0	0
$487$	31.0244	1.40585	0.702925	−	0.711264i	$-0.251877\pi$
$487$	31.0244	1.40585	0.702925	+	0.711264i	$0.251877\pi$
$488$	0	0
$489$	−37.2662	−1.68524
$490$	0	0
$491$	16.5894	0.748671	0.374336	−	0.927293i	$-0.377871\pi$
$491$	16.5894	0.748671	0.374336	+	0.927293i	$0.377871\pi$
$492$	0	0
$493$	10.4740	0.471723
$494$	0	0
$495$	1.46784	0.0659744
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−33.3076	−1.49105	−0.745527	−	0.666475i	$-0.767802\pi$
$499$	−33.3076	−1.49105	−0.745527	+	0.666475i	$0.767802\pi$
$500$	0	0
$501$	3.01141	0.134540
$502$	0	0
$503$	−28.6331	−1.27668	−0.638342	−	0.769753i	$-0.720380\pi$
$503$	−28.6331	−1.27668	−0.638342	+	0.769753i	$0.720380\pi$
$504$	0	0
$505$	3.19752	0.142288
$506$	0	0
$507$	−32.2715	−1.43323
$508$	0	0
$509$	−18.2149	−0.807360	−0.403680	−	0.914900i	$-0.632269\pi$
$509$	−18.2149	−0.807360	−0.403680	+	0.914900i	$0.632269\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−1.60555	−0.0708868
$514$	0	0
$515$	2.30511	0.101575
$516$	0	0
$517$	−15.2197	−0.669363
$518$	0	0
$519$	−39.3795	−1.72857
$520$	0	0
$521$	−33.6768	−1.47541	−0.737703	−	0.675126i	$-0.764089\pi$
$521$	−33.6768	−1.47541	−0.737703	+	0.675126i	$0.764089\pi$
$522$	0	0
$523$	7.18761	0.314292	0.157146	−	0.987575i	$-0.449771\pi$
$523$	7.18761	0.314292	0.157146	+	0.987575i	$0.449771\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.49997	0.196022
$528$	0	0
$529$	−12.7759	−0.555474
$530$	0	0
$531$	−5.46037	−0.236960
$532$	0	0
$533$	−23.1297	−1.00186
$534$	0	0
$535$	1.39813	0.0604464
$536$	0	0
$537$	−19.4823	−0.840725
$538$	0	0
$539$	0	0
$540$	0	0
$541$	38.8365	1.66971	0.834857	−	0.550468i	$-0.185551\pi$
$541$	38.8365	1.66971	0.834857	+	0.550468i	$0.185551\pi$
$542$	0	0
$543$	8.45189	0.362705
$544$	0	0
$545$	−4.55850	−0.195265
$546$	0	0
$547$	18.3605	0.785040	0.392520	−	0.919743i	$-0.371603\pi$
$547$	18.3605	0.785040	0.392520	+	0.919743i	$0.371603\pi$
$548$	0	0
$549$	−22.1430	−0.945040
$550$	0	0
$551$	−6.45014	−0.274785
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−3.10468	−0.131786
$556$	0	0
$557$	−12.5940	−0.533627	−0.266813	−	0.963748i	$-0.585971\pi$
$557$	−12.5940	−0.533627	−0.266813	+	0.963748i	$0.585971\pi$
$558$	0	0
$559$	−41.6333	−1.76090
$560$	0	0
$561$	−8.79837	−0.371467
$562$	0	0
$563$	14.2181	0.599223	0.299612	−	0.954061i	$-0.403143\pi$
$563$	14.2181	0.599223	0.299612	+	0.954061i	$0.403143\pi$
$564$	0	0
$565$	−4.44562	−0.187029
$566$	0	0
$567$	0	0
$568$	0	0
$569$	37.5300	1.57334	0.786669	−	0.617375i	$-0.211804\pi$
$569$	37.5300	1.57334	0.786669	+	0.617375i	$0.211804\pi$
$570$	0	0
$571$	−25.0307	−1.04750	−0.523751	−	0.851871i	$-0.675468\pi$
$571$	−25.0307	−1.04750	−0.523751	+	0.851871i	$0.675468\pi$
$572$	0	0
$573$	3.85482	0.161037
$574$	0	0
$575$	15.7529	0.656942
$576$	0	0
$577$	−18.4070	−0.766294	−0.383147	−	0.923687i	$-0.625160\pi$
$577$	−18.4070	−0.766294	−0.383147	+	0.923687i	$0.625160\pi$
$578$	0	0
$579$	18.4185	0.765448
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−16.1048	−0.666994
$584$	0	0
$585$	−3.24239	−0.134056
$586$	0	0
$587$	4.44887	0.183624	0.0918122	−	0.995776i	$-0.470734\pi$
$587$	4.44887	0.183624	0.0918122	+	0.995776i	$0.470734\pi$
$588$	0	0
$589$	−2.77120	−0.114185
$590$	0	0
$591$	−7.51362	−0.309069
$592$	0	0
$593$	46.0241	1.88999	0.944993	−	0.327092i	$-0.106069\pi$
$593$	46.0241	1.88999	0.944993	+	0.327092i	$0.106069\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−17.9923	−0.736375
$598$	0	0
$599$	−1.37030	−0.0559888	−0.0279944	−	0.999608i	$-0.508912\pi$
$599$	−1.37030	−0.0559888	−0.0279944	+	0.999608i	$0.508912\pi$
$600$	0	0
$601$	−36.2733	−1.47962	−0.739810	−	0.672816i	$-0.765085\pi$
$601$	−36.2733	−1.47962	−0.739810	+	0.672816i	$0.765085\pi$
$602$	0	0
$603$	−23.0712	−0.939532
$604$	0	0
$605$	−1.48015	−0.0601768
$606$	0	0
$607$	−46.5770	−1.89050	−0.945252	−	0.326342i	$-0.894184\pi$
$607$	−46.5770	−1.89050	−0.945252	+	0.326342i	$0.894184\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	33.6197	1.36011
$612$	0	0
$613$	35.9234	1.45093	0.725466	−	0.688258i	$-0.241624\pi$
$613$	35.9234	1.45093	0.725466	+	0.688258i	$0.241624\pi$
$614$	0	0
$615$	−2.77615	−0.111945
$616$	0	0
$617$	−0.928209	−0.0373683	−0.0186841	−	0.999825i	$-0.505948\pi$
$617$	−0.928209	−0.0373683	−0.0186841	+	0.999825i	$0.505948\pi$
$618$	0	0
$619$	−22.7848	−0.915798	−0.457899	−	0.889004i	$-0.651398\pi$
$619$	−22.7848	−0.915798	−0.457899	+	0.889004i	$0.651398\pi$
$620$	0	0
$621$	−5.13378	−0.206011
$622$	0	0
$623$	0	0
$624$	0	0
$625$	23.9045	0.956182
$626$	0	0
$627$	5.41827	0.216385
$628$	0	0
$629$	8.08144	0.322228
$630$	0	0
$631$	−38.8209	−1.54544	−0.772718	−	0.634749i	$-0.781103\pi$
$631$	−38.8209	−1.54544	−0.772718	+	0.634749i	$0.781103\pi$
$632$	0	0
$633$	26.5446	1.05505
$634$	0	0
$635$	4.49224	0.178269
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−13.0787	−0.517384
$640$	0	0
$641$	31.9118	1.26044	0.630221	−	0.776416i	$-0.282964\pi$
$641$	31.9118	1.26044	0.630221	+	0.776416i	$0.282964\pi$
$642$	0	0
$643$	22.0069	0.867867	0.433933	−	0.900945i	$-0.357125\pi$
$643$	22.0069	0.867867	0.433933	+	0.900945i	$0.357125\pi$
$644$	0	0
$645$	−4.99706	−0.196759
$646$	0	0
$647$	−34.4428	−1.35409	−0.677044	−	0.735943i	$-0.736739\pi$
$647$	−34.4428	−1.35409	−0.677044	+	0.735943i	$0.736739\pi$
$648$	0	0
$649$	−5.57929	−0.219006
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−37.3931	−1.46331	−0.731653	−	0.681678i	$-0.761251\pi$
$653$	−37.3931	−1.46331	−0.731653	+	0.681678i	$0.761251\pi$
$654$	0	0
$655$	−2.33640	−0.0912908
$656$	0	0
$657$	1.12850	0.0440268
$658$	0	0
$659$	45.4578	1.77078	0.885391	−	0.464846i	$-0.153890\pi$
$659$	45.4578	1.77078	0.885391	+	0.464846i	$0.153890\pi$
$660$	0	0
$661$	49.8749	1.93991	0.969953	−	0.243291i	$-0.0782269\pi$
$661$	49.8749	1.93991	0.969953	+	0.243291i	$0.0782269\pi$
$662$	0	0
$663$	19.4352	0.754801
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−20.6244	−0.798581
$668$	0	0
$669$	13.9321	0.538648
$670$	0	0
$671$	−22.6253	−0.873438
$672$	0	0
$673$	−7.44620	−0.287030	−0.143515	−	0.989648i	$-0.545841\pi$
$673$	−7.44620	−0.287030	−0.143515	+	0.989648i	$0.545841\pi$
$674$	0	0
$675$	−7.90993	−0.304453
$676$	0	0
$677$	34.2138	1.31494	0.657471	−	0.753480i	$-0.271626\pi$
$677$	34.2138	1.31494	0.657471	+	0.753480i	$0.271626\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	3.27736	0.125589
$682$	0	0
$683$	−15.3747	−0.588295	−0.294148	−	0.955760i	$-0.595036\pi$
$683$	−15.3747	−0.588295	−0.294148	+	0.955760i	$0.595036\pi$
$684$	0	0
$685$	−4.55834	−0.174165
$686$	0	0
$687$	38.1782	1.45659
$688$	0	0
$689$	35.5748	1.35529
$690$	0	0
$691$	−27.2222	−1.03558	−0.517790	−	0.855508i	$-0.673245\pi$
$691$	−27.2222	−1.03558	−0.517790	+	0.855508i	$0.673245\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	5.73816	0.217661
$696$	0	0
$697$	7.22629	0.273715
$698$	0	0
$699$	42.7872	1.61836
$700$	0	0
$701$	−3.68598	−0.139217	−0.0696087	−	0.997574i	$-0.522175\pi$
$701$	−3.68598	−0.139217	−0.0696087	+	0.997574i	$0.522175\pi$
$702$	0	0
$703$	−4.97676	−0.187702
$704$	0	0
$705$	4.03522	0.151975
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−36.3095	−1.36363	−0.681816	−	0.731524i	$-0.738809\pi$
$709$	−36.3095	−1.36363	−0.681816	+	0.731524i	$0.738809\pi$
$710$	0	0
$711$	39.6569	1.48725
$712$	0	0
$713$	−8.86095	−0.331845
$714$	0	0
$715$	−3.31301	−0.123899
$716$	0	0
$717$	−39.3710	−1.47034
$718$	0	0
$719$	42.8842	1.59931	0.799655	−	0.600460i	$-0.205016\pi$
$719$	42.8842	1.59931	0.799655	+	0.600460i	$0.205016\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−10.2440	−0.380978
$724$	0	0
$725$	−31.7773	−1.18018
$726$	0	0
$727$	−27.2553	−1.01084	−0.505421	−	0.862873i	$-0.668663\pi$
$727$	−27.2553	−1.01084	−0.505421	+	0.862873i	$0.668663\pi$
$728$	0	0
$729$	−13.3305	−0.493723
$730$	0	0
$731$	13.0073	0.481092
$732$	0	0
$733$	0.585926	0.0216417	0.0108208	−	0.999941i	$-0.496556\pi$
$733$	0.585926	0.0216417	0.0108208	+	0.999941i	$0.496556\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−23.5737	−0.868348
$738$	0	0
$739$	14.2040	0.522501	0.261251	−	0.965271i	$-0.415865\pi$
$739$	14.2040	0.522501	0.261251	+	0.965271i	$0.415865\pi$
$740$	0	0
$741$	−11.9687	−0.439682
$742$	0	0
$743$	8.91622	0.327104	0.163552	−	0.986535i	$-0.447705\pi$
$743$	8.91622	0.327104	0.163552	+	0.986535i	$0.447705\pi$
$744$	0	0
$745$	0.461287	0.0169003
$746$	0	0
$747$	7.12850	0.260818
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−18.4810	−0.674381	−0.337190	−	0.941436i	$-0.609477\pi$
$751$	−18.4810	−0.674381	−0.337190	+	0.941436i	$0.609477\pi$
$752$	0	0
$753$	−44.3051	−1.61457
$754$	0	0
$755$	−0.643716	−0.0234272
$756$	0	0
$757$	−44.7629	−1.62693	−0.813467	−	0.581610i	$-0.802423\pi$
$757$	−44.7629	−1.62693	−0.813467	+	0.581610i	$0.802423\pi$
$758$	0	0
$759$	17.3250	0.628858
$760$	0	0
$761$	7.25135	0.262861	0.131431	−	0.991325i	$-0.458043\pi$
$761$	7.25135	0.262861	0.131431	+	0.991325i	$0.458043\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	1.01300	0.0366252
$766$	0	0
$767$	12.3244	0.445009
$768$	0	0
$769$	−23.4310	−0.844944	−0.422472	−	0.906376i	$-0.638838\pi$
$769$	−23.4310	−0.844944	−0.422472	+	0.906376i	$0.638838\pi$
$770$	0	0
$771$	67.7427	2.43969
$772$	0	0
$773$	49.1281	1.76701	0.883507	−	0.468418i	$-0.155176\pi$
$773$	49.1281	1.76701	0.883507	+	0.468418i	$0.155176\pi$
$774$	0	0
$775$	−13.6526	−0.490416
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−4.45014	−0.159443
$780$	0	0
$781$	−13.3635	−0.478184
$782$	0	0
$783$	10.3560	0.370094
$784$	0	0
$785$	0.463468	0.0165419
$786$	0	0
$787$	−47.7705	−1.70283	−0.851417	−	0.524489i	$-0.824256\pi$
$787$	−47.7705	−1.70283	−0.851417	+	0.524489i	$0.824256\pi$
$788$	0	0
$789$	45.6057	1.62361
$790$	0	0
$791$	0	0
$792$	0	0
$793$	49.9782	1.77478
$794$	0	0
$795$	4.26989	0.151437
$796$	0	0
$797$	27.8083	0.985022	0.492511	−	0.870306i	$-0.336079\pi$
$797$	27.8083	0.985022	0.492511	+	0.870306i	$0.336079\pi$
$798$	0	0
$799$	−10.5036	−0.371592
$800$	0	0
$801$	8.77983	0.310220
$802$	0	0
$803$	1.15307	0.0406911
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−26.2294	−0.923318
$808$	0	0
$809$	−35.9440	−1.26372	−0.631861	−	0.775081i	$-0.717709\pi$
$809$	−35.9440	−1.26372	−0.631861	+	0.775081i	$0.717709\pi$
$810$	0	0
$811$	−8.04370	−0.282453	−0.141226	−	0.989977i	$-0.545105\pi$
$811$	−8.04370	−0.282453	−0.141226	+	0.989977i	$0.545105\pi$
$812$	0	0
$813$	−13.1436	−0.460966
$814$	0	0
$815$	4.38411	0.153569
$816$	0	0
$817$	−8.01023	−0.280243
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−6.39387	−0.223147	−0.111574	−	0.993756i	$-0.535589\pi$
$821$	−6.39387	−0.223147	−0.111574	+	0.993756i	$0.535589\pi$
$822$	0	0
$823$	−11.2810	−0.393232	−0.196616	−	0.980481i	$-0.562995\pi$
$823$	−11.2810	−0.393232	−0.196616	+	0.980481i	$0.562995\pi$
$824$	0	0
$825$	26.6937	0.929355
$826$	0	0
$827$	8.05015	0.279931	0.139966	−	0.990156i	$-0.455301\pi$
$827$	8.05015	0.279931	0.139966	+	0.990156i	$0.455301\pi$
$828$	0	0
$829$	−5.11816	−0.177761	−0.0888805	−	0.996042i	$-0.528329\pi$
$829$	−5.11816	−0.177761	−0.0888805	+	0.996042i	$0.528329\pi$
$830$	0	0
$831$	2.98012	0.103379
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−0.354271	−0.0122600
$836$	0	0
$837$	4.44930	0.153790
$838$	0	0
$839$	−51.1131	−1.76462	−0.882311	−	0.470668i	$-0.844013\pi$
$839$	−51.1131	−1.76462	−0.882311	+	0.470668i	$0.844013\pi$
$840$	0	0
$841$	12.6043	0.434630
$842$	0	0
$843$	41.9903	1.44622
$844$	0	0
$845$	3.79652	0.130604
$846$	0	0
$847$	0	0
$848$	0	0
$849$	36.2876	1.24539
$850$	0	0
$851$	−15.9133	−0.545500
$852$	0	0
$853$	−19.2093	−0.657715	−0.328858	−	0.944379i	$-0.606664\pi$
$853$	−19.2093	−0.657715	−0.328858	+	0.944379i	$0.606664\pi$
$854$	0	0
$855$	−0.623835	−0.0213347
$856$	0	0
$857$	12.9049	0.440822	0.220411	−	0.975407i	$-0.429260\pi$
$857$	12.9049	0.440822	0.220411	+	0.975407i	$0.429260\pi$
$858$	0	0
$859$	10.3934	0.354619	0.177310	−	0.984155i	$-0.443261\pi$
$859$	10.3934	0.354619	0.177310	+	0.984155i	$0.443261\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	14.2161	0.483923	0.241961	−	0.970286i	$-0.422209\pi$
$863$	14.2161	0.483923	0.241961	+	0.970286i	$0.422209\pi$
$864$	0	0
$865$	4.63272	0.157517
$866$	0	0
$867$	33.0751	1.12329
$868$	0	0
$869$	40.5206	1.37457
$870$	0	0
$871$	52.0732	1.76443
$872$	0	0
$873$	13.5634	0.459053
$874$	0	0
$875$	0	0
$876$	0	0
$877$	51.1320	1.72660	0.863302	−	0.504687i	$-0.168392\pi$
$877$	51.1320	1.72660	0.863302	+	0.504687i	$0.168392\pi$
$878$	0	0
$879$	−6.98540	−0.235612
$880$	0	0
$881$	−0.353186	−0.0118991	−0.00594956	−	0.999982i	$-0.501894\pi$
$881$	−0.353186	−0.0118991	−0.00594956	+	0.999982i	$0.501894\pi$
$882$	0	0
$883$	−21.2746	−0.715947	−0.357973	−	0.933732i	$-0.616532\pi$
$883$	−21.2746	−0.715947	−0.357973	+	0.933732i	$0.616532\pi$
$884$	0	0
$885$	1.47924	0.0497243
$886$	0	0
$887$	−41.5493	−1.39509	−0.697544	−	0.716542i	$-0.745724\pi$
$887$	−41.5493	−1.39509	−0.697544	+	0.716542i	$0.745724\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−24.9541	−0.835994
$892$	0	0
$893$	6.46842	0.216457
$894$	0	0
$895$	2.29196	0.0766118
$896$	0	0
$897$	−38.2701	−1.27780
$898$	0	0
$899$	17.8746	0.596151
$900$	0	0
$901$	−11.1145	−0.370277
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−0.994306	−0.0330519
$906$	0	0
$907$	2.57470	0.0854914	0.0427457	−	0.999086i	$-0.486389\pi$
$907$	2.57470	0.0854914	0.0427457	+	0.999086i	$0.486389\pi$
$908$	0	0
$909$	27.1798	0.901498
$910$	0	0
$911$	0.611421	0.0202573	0.0101286	−	0.999949i	$-0.496776\pi$
$911$	0.611421	0.0202573	0.0101286	+	0.999949i	$0.496776\pi$
$912$	0	0
$913$	7.28375	0.241057
$914$	0	0
$915$	5.99866	0.198310
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−9.38874	−0.309706	−0.154853	−	0.987938i	$-0.549490\pi$
$919$	−9.38874	−0.309706	−0.154853	+	0.987938i	$0.549490\pi$
$920$	0	0
$921$	42.2953	1.39368
$922$	0	0
$923$	29.5194	0.971643
$924$	0	0
$925$	−24.5186	−0.806166
$926$	0	0
$927$	19.5941	0.643556
$928$	0	0
$929$	31.8152	1.04382	0.521912	−	0.852999i	$-0.325219\pi$
$929$	31.8152	1.04382	0.521912	+	0.852999i	$0.325219\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	18.7104	0.612550
$934$	0	0
$935$	1.03507	0.0338503
$936$	0	0
$937$	10.6343	0.347408	0.173704	−	0.984798i	$-0.444426\pi$
$937$	10.6343	0.347408	0.173704	+	0.984798i	$0.444426\pi$
$938$	0	0
$939$	56.8476	1.85515
$940$	0	0
$941$	−20.1082	−0.655508	−0.327754	−	0.944763i	$-0.606292\pi$
$941$	−20.1082	−0.655508	−0.327754	+	0.944763i	$0.606292\pi$
$942$	0	0
$943$	−14.2294	−0.463373
$944$	0	0
$945$	0	0
$946$	0	0
$947$	42.0376	1.36604	0.683019	−	0.730401i	$-0.260667\pi$
$947$	42.0376	1.36604	0.683019	+	0.730401i	$0.260667\pi$
$948$	0	0
$949$	−2.54709	−0.0826820
$950$	0	0
$951$	72.3903	2.34742
$952$	0	0
$953$	−4.91127	−0.159092	−0.0795458	−	0.996831i	$-0.525347\pi$
$953$	−4.91127	−0.159092	−0.0795458	+	0.996831i	$0.525347\pi$
$954$	0	0
$955$	−0.453493	−0.0146747
$956$	0	0
$957$	−34.9486	−1.12973
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−23.3205	−0.752273
$962$	0	0
$963$	11.8845	0.382973
$964$	0	0
$965$	−2.16681	−0.0697521
$966$	0	0
$967$	−21.4200	−0.688822	−0.344411	−	0.938819i	$-0.611921\pi$
$967$	−21.4200	−0.688822	−0.344411	+	0.938819i	$0.611921\pi$
$968$	0	0
$969$	3.73933	0.120124
$970$	0	0
$971$	−37.1295	−1.19154	−0.595771	−	0.803154i	$-0.703153\pi$
$971$	−37.1295	−1.19154	−0.595771	+	0.803154i	$0.703153\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−58.9652	−1.88840
$976$	0	0
$977$	−15.7895	−0.505150	−0.252575	−	0.967577i	$-0.581277\pi$
$977$	−15.7895	−0.505150	−0.252575	+	0.967577i	$0.581277\pi$
$978$	0	0
$979$	8.97105	0.286716
$980$	0	0
$981$	−38.7485	−1.23715
$982$	0	0
$983$	17.2785	0.551099	0.275550	−	0.961287i	$-0.411140\pi$
$983$	17.2785	0.551099	0.275550	+	0.961287i	$0.411140\pi$
$984$	0	0
$985$	0.883925	0.0281642
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−25.6128	−0.814441
$990$	0	0
$991$	−62.2770	−1.97829	−0.989147	−	0.146927i	$-0.953062\pi$
$991$	−62.2770	−1.97829	−0.989147	+	0.146927i	$0.953062\pi$
$992$	0	0
$993$	2.29196	0.0727332
$994$	0	0
$995$	2.11667	0.0671028
$996$	0	0
$997$	8.49709	0.269106	0.134553	−	0.990906i	$-0.457040\pi$
$997$	8.49709	0.269106	0.134553	+	0.990906i	$0.457040\pi$
$998$	0	0
$999$	7.99045	0.252807

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7448.2.a.bj.1.2		4
7.6	odd	2		1064.2.a.h.1.3	✓	4
21.20	even	2		9576.2.a.ci.1.3		4
28.27	even	2		2128.2.a.t.1.1		4
56.13	odd	2		8512.2.a.bq.1.2		4
56.27	even	2		8512.2.a.bu.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1064.2.a.h.1.3	✓	4	7.6	odd	2
2128.2.a.t.1.1		4	28.27	even	2
7448.2.a.bj.1.2		4	1.1	even	1	trivial
8512.2.a.bq.1.2		4	56.13	odd	2
8512.2.a.bu.1.4		4	56.27	even	2
9576.2.a.ci.1.3		4	21.20	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:	\( N \)	\(=\)	\( 7448 = 2^{3} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7448.a (trivial)

\(f(q)\)	\(=\)	\(q-2.30278 q^{3} +0.270905 q^{5} +2.30278 q^{9} +O(q^{10})\)	\(q-2.30278 q^{3} +0.270905 q^{5} +2.30278 q^{9} +2.35293 q^{11} -5.19752 q^{13} -0.623835 q^{15} +1.62383 q^{17} -1.00000 q^{19} -3.19752 q^{23} -4.92661 q^{25} +1.60555 q^{27} +6.45014 q^{29} +2.77120 q^{31} -5.41827 q^{33} +4.97676 q^{37} +11.9687 q^{39} +4.45014 q^{41} +8.01023 q^{43} +0.623835 q^{45} -6.46842 q^{47} -3.73933 q^{51} -6.84459 q^{53} +0.637421 q^{55} +2.30278 q^{57} -2.37121 q^{59} -9.61578 q^{61} -1.40804 q^{65} -10.0189 q^{67} +7.36316 q^{69} -5.67952 q^{71} +0.490059 q^{73} +11.3449 q^{75} +17.2213 q^{79} -10.6056 q^{81} +3.09561 q^{83} +0.439906 q^{85} -14.8532 q^{87} +3.81272 q^{89} -6.38144 q^{93} -0.270905 q^{95} +5.89004 q^{97} +5.41827 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} - q^{5} + 2 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 - q^5 + 2 * q^9	\( 4 q - 2 q^{3} - q^{5} + 2 q^{9} + 2 q^{11} - 2 q^{13} + 7 q^{15} - 3 q^{17} - 4 q^{19} + 6 q^{23} - 3 q^{25} - 8 q^{27} - 17 q^{31} - q^{33} + 3 q^{37} + q^{39} - 8 q^{41} + 7 q^{43} - 7 q^{45} - 5 q^{47} + 8 q^{51} - 16 q^{53} - 17 q^{55} + 2 q^{57} - 7 q^{59} + q^{61} - 10 q^{65} + 7 q^{67} - 3 q^{69} + 27 q^{71} + q^{73} + 8 q^{75} + 15 q^{79} - 28 q^{81} - 3 q^{83} + q^{85} - 26 q^{87} + 9 q^{89} + 2 q^{93} + q^{95} - 3 q^{97} + q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 - q^5 + 2 * q^9 + 2 * q^11 - 2 * q^13 + 7 * q^15 - 3 * q^17 - 4 * q^19 + 6 * q^23 - 3 * q^25 - 8 * q^27 - 17 * q^31 - q^33 + 3 * q^37 + q^39 - 8 * q^41 + 7 * q^43 - 7 * q^45 - 5 * q^47 + 8 * q^51 - 16 * q^53 - 17 * q^55 + 2 * q^57 - 7 * q^59 + q^61 - 10 * q^65 + 7 * q^67 - 3 * q^69 + 27 * q^71 + q^73 + 8 * q^75 + 15 * q^79 - 28 * q^81 - 3 * q^83 + q^85 - 26 * q^87 + 9 * q^89 + 2 * q^93 + q^95 - 3 * q^97 + q^99

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