LMFDB - Embedded newform 7623.2.a.bm.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7623,2,Mod(1,7623)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7623, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("7623.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7623 = 3^{2} \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7623.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:	$60.8699614608$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	7623.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.618034 q^{2} -1.61803 q^{4} -1.00000 q^{5} -1.00000 q^{7} +2.23607 q^{8} +O(q^{10})$	$q-0.618034 q^{2} -1.61803 q^{4} -1.00000 q^{5} -1.00000 q^{7} +2.23607 q^{8} +0.618034 q^{10} -3.47214 q^{13} +0.618034 q^{14} +1.85410 q^{16} +5.23607 q^{17} +6.70820 q^{19} +1.61803 q^{20} -5.70820 q^{23} -4.00000 q^{25} +2.14590 q^{26} +1.61803 q^{28} +5.00000 q^{29} -5.23607 q^{31} -5.61803 q^{32} -3.23607 q^{34} +1.00000 q^{35} -7.00000 q^{37} -4.14590 q^{38} -2.23607 q^{40} -2.47214 q^{41} -5.70820 q^{43} +3.52786 q^{46} -0.236068 q^{47} +1.00000 q^{49} +2.47214 q^{50} +5.61803 q^{52} +12.1803 q^{53} -2.23607 q^{56} -3.09017 q^{58} +11.1803 q^{59} -2.00000 q^{61} +3.23607 q^{62} -0.236068 q^{64} +3.47214 q^{65} -9.76393 q^{67} -8.47214 q^{68} -0.618034 q^{70} +2.47214 q^{71} -4.52786 q^{73} +4.32624 q^{74} -10.8541 q^{76} +14.4721 q^{79} -1.85410 q^{80} +1.52786 q^{82} +6.76393 q^{83} -5.23607 q^{85} +3.52786 q^{86} +4.47214 q^{89} +3.47214 q^{91} +9.23607 q^{92} +0.145898 q^{94} -6.70820 q^{95} +9.70820 q^{97} -0.618034 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - q^{4} - 2 q^{5} - 2 q^{7}+O(q^{10}) $ 2 * q + q^2 - q^4 - 2 * q^5 - 2 * q^7	$ 2 q + q^{2} - q^{4} - 2 q^{5} - 2 q^{7} - q^{10} + 2 q^{13} - q^{14} - 3 q^{16} + 6 q^{17} + q^{20} + 2 q^{23} - 8 q^{25} + 11 q^{26} + q^{28} + 10 q^{29} - 6 q^{31} - 9 q^{32} - 2 q^{34} + 2 q^{35} - 14 q^{37} - 15 q^{38} + 4 q^{41} + 2 q^{43} + 16 q^{46} + 4 q^{47} + 2 q^{49} - 4 q^{50} + 9 q^{52} + 2 q^{53} + 5 q^{58} - 4 q^{61} + 2 q^{62} + 4 q^{64} - 2 q^{65} - 24 q^{67} - 8 q^{68} + q^{70} - 4 q^{71} - 18 q^{73} - 7 q^{74} - 15 q^{76} + 20 q^{79} + 3 q^{80} + 12 q^{82} + 18 q^{83} - 6 q^{85} + 16 q^{86} - 2 q^{91} + 14 q^{92} + 7 q^{94} + 6 q^{97} + q^{98}+O(q^{100}) $ 2 * q + q^2 - q^4 - 2 * q^5 - 2 * q^7 - q^10 + 2 * q^13 - q^14 - 3 * q^16 + 6 * q^17 + q^20 + 2 * q^23 - 8 * q^25 + 11 * q^26 + q^28 + 10 * q^29 - 6 * q^31 - 9 * q^32 - 2 * q^34 + 2 * q^35 - 14 * q^37 - 15 * q^38 + 4 * q^41 + 2 * q^43 + 16 * q^46 + 4 * q^47 + 2 * q^49 - 4 * q^50 + 9 * q^52 + 2 * q^53 + 5 * q^58 - 4 * q^61 + 2 * q^62 + 4 * q^64 - 2 * q^65 - 24 * q^67 - 8 * q^68 + q^70 - 4 * q^71 - 18 * q^73 - 7 * q^74 - 15 * q^76 + 20 * q^79 + 3 * q^80 + 12 * q^82 + 18 * q^83 - 6 * q^85 + 16 * q^86 - 2 * q^91 + 14 * q^92 + 7 * q^94 + 6 * q^97 + q^98

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 $

$2$	−0.618034	−0.437016	−0.218508	−	0.975835i	$-0.570119\pi$
$2$	−0.618034	−0.437016	−0.218508	+	0.975835i	$0.570119\pi$
$3$	0	0
$3$	0	0
$4$	−1.61803	−0.809017
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.23607	0.790569
$9$	0	0
$10$	0.618034	0.195440
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−3.47214	−0.962997	−0.481499	−	0.876447i	$-0.659907\pi$
$13$	−3.47214	−0.962997	−0.481499	+	0.876447i	$0.659907\pi$
$14$	0.618034	0.165177
$15$	0	0
$16$	1.85410	0.463525
$17$	5.23607	1.26993	0.634967	−	0.772540i	$-0.281014\pi$
$17$	5.23607	1.26993	0.634967	+	0.772540i	$0.281014\pi$
$18$	0	0
$19$	6.70820	1.53897	0.769484	−	0.638666i	$-0.220514\pi$
$19$	6.70820	1.53897	0.769484	+	0.638666i	$0.220514\pi$
$20$	1.61803	0.361803
$21$	0	0
$22$	0	0
$23$	−5.70820	−1.19024	−0.595121	−	0.803636i	$-0.702896\pi$
$23$	−5.70820	−1.19024	−0.595121	+	0.803636i	$0.702896\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	2.14590	0.420845
$27$	0	0
$28$	1.61803	0.305780
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	0	0
$31$	−5.23607	−0.940426	−0.470213	−	0.882553i	$-0.655823\pi$
$31$	−5.23607	−0.940426	−0.470213	+	0.882553i	$0.655823\pi$
$32$	−5.61803	−0.993137
$33$	0	0
$34$	−3.23607	−0.554981
$35$	1.00000	0.169031
$36$	0	0
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	−4.14590	−0.672553
$39$	0	0
$40$	−2.23607	−0.353553
$41$	−2.47214	−0.386083	−0.193041	−	0.981191i	$-0.561835\pi$
$41$	−2.47214	−0.386083	−0.193041	+	0.981191i	$0.561835\pi$
$42$	0	0
$43$	−5.70820	−0.870493	−0.435246	−	0.900311i	$-0.643339\pi$
$43$	−5.70820	−0.870493	−0.435246	+	0.900311i	$0.643339\pi$
$44$	0	0
$45$	0	0
$46$	3.52786	0.520155
$47$	−0.236068	−0.0344341	−0.0172170	−	0.999852i	$-0.505481\pi$
$47$	−0.236068	−0.0344341	−0.0172170	+	0.999852i	$0.505481\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	2.47214	0.349613
$51$	0	0
$52$	5.61803	0.779081
$53$	12.1803	1.67310	0.836549	−	0.547892i	$-0.184569\pi$
$53$	12.1803	1.67310	0.836549	+	0.547892i	$0.184569\pi$
$54$	0	0
$55$	0	0
$56$	−2.23607	−0.298807
$57$	0	0
$58$	−3.09017	−0.405759
$59$	11.1803	1.45556	0.727778	−	0.685813i	$-0.240553\pi$
$59$	11.1803	1.45556	0.727778	+	0.685813i	$0.240553\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	3.23607	0.410981
$63$	0	0
$64$	−0.236068	−0.0295085
$65$	3.47214	0.430665
$66$	0	0
$67$	−9.76393	−1.19285	−0.596427	−	0.802667i	$-0.703414\pi$
$67$	−9.76393	−1.19285	−0.596427	+	0.802667i	$0.703414\pi$
$68$	−8.47214	−1.02740
$69$	0	0
$70$	−0.618034	−0.0738692
$71$	2.47214	0.293389	0.146694	−	0.989182i	$-0.453137\pi$
$71$	2.47214	0.293389	0.146694	+	0.989182i	$0.453137\pi$
$72$	0	0
$73$	−4.52786	−0.529946	−0.264973	−	0.964256i	$-0.585363\pi$
$73$	−4.52786	−0.529946	−0.264973	+	0.964256i	$0.585363\pi$
$74$	4.32624	0.502915
$75$	0	0
$76$	−10.8541	−1.24505
$77$	0	0
$78$	0	0
$79$	14.4721	1.62824	0.814121	−	0.580695i	$-0.197219\pi$
$79$	14.4721	1.62824	0.814121	+	0.580695i	$0.197219\pi$
$80$	−1.85410	−0.207295
$81$	0	0
$82$	1.52786	0.168724
$83$	6.76393	0.742438	0.371219	−	0.928545i	$-0.378940\pi$
$83$	6.76393	0.742438	0.371219	+	0.928545i	$0.378940\pi$
$84$	0	0
$85$	−5.23607	−0.567931
$86$	3.52786	0.380419
$87$	0	0
$88$	0	0
$89$	4.47214	0.474045	0.237023	−	0.971504i	$-0.423828\pi$
$89$	4.47214	0.474045	0.237023	+	0.971504i	$0.423828\pi$
$90$	0	0
$91$	3.47214	0.363979
$92$	9.23607	0.962927
$93$	0	0
$94$	0.145898	0.0150482
$95$	−6.70820	−0.688247
$96$	0	0
$97$	9.70820	0.985719	0.492859	−	0.870109i	$-0.335952\pi$
$97$	9.70820	0.985719	0.492859	+	0.870109i	$0.335952\pi$
$98$	−0.618034	−0.0624309
$99$	0	0
$100$	6.47214	0.647214
$101$	18.1803	1.80901	0.904506	−	0.426461i	$-0.140240\pi$
$101$	18.1803	1.80901	0.904506	+	0.426461i	$0.140240\pi$
$102$	0	0
$103$	17.4164	1.71609	0.858045	−	0.513575i	$-0.171679\pi$
$103$	17.4164	1.71609	0.858045	+	0.513575i	$0.171679\pi$
$104$	−7.76393	−0.761316
$105$	0	0
$106$	−7.52786	−0.731171
$107$	−4.23607	−0.409516	−0.204758	−	0.978813i	$-0.565641\pi$
$107$	−4.23607	−0.409516	−0.204758	+	0.978813i	$0.565641\pi$
$108$	0	0
$109$	−2.76393	−0.264737	−0.132368	−	0.991201i	$-0.542258\pi$
$109$	−2.76393	−0.264737	−0.132368	+	0.991201i	$0.542258\pi$
$110$	0	0
$111$	0	0
$112$	−1.85410	−0.175196
$113$	0.472136	0.0444148	0.0222074	−	0.999753i	$-0.492931\pi$
$113$	0.472136	0.0444148	0.0222074	+	0.999753i	$0.492931\pi$
$114$	0	0
$115$	5.70820	0.532293
$116$	−8.09017	−0.751153
$117$	0	0
$118$	−6.90983	−0.636101
$119$	−5.23607	−0.479990
$120$	0	0
$121$	0	0
$122$	1.23607	0.111908
$123$	0	0
$124$	8.47214	0.760820
$125$	9.00000	0.804984
$126$	0	0
$127$	12.6525	1.12273	0.561363	−	0.827570i	$-0.310277\pi$
$127$	12.6525	1.12273	0.561363	+	0.827570i	$0.310277\pi$
$128$	11.3820	1.00603
$129$	0	0
$130$	−2.14590	−0.188208
$131$	0.944272	0.0825014	0.0412507	−	0.999149i	$-0.486866\pi$
$131$	0.944272	0.0825014	0.0412507	+	0.999149i	$0.486866\pi$
$132$	0	0
$133$	−6.70820	−0.581675
$134$	6.03444	0.521296
$135$	0	0
$136$	11.7082	1.00397
$137$	−19.7082	−1.68379	−0.841893	−	0.539645i	$-0.818559\pi$
$137$	−19.7082	−1.68379	−0.841893	+	0.539645i	$0.818559\pi$
$138$	0	0
$139$	−14.4721	−1.22751	−0.613755	−	0.789496i	$-0.710342\pi$
$139$	−14.4721	−1.22751	−0.613755	+	0.789496i	$0.710342\pi$
$140$	−1.61803	−0.136749
$141$	0	0
$142$	−1.52786	−0.128216
$143$	0	0
$144$	0	0
$145$	−5.00000	−0.415227
$146$	2.79837	0.231595
$147$	0	0
$148$	11.3262	0.931011
$149$	5.00000	0.409616	0.204808	−	0.978802i	$-0.434343\pi$
$149$	5.00000	0.409616	0.204808	+	0.978802i	$0.434343\pi$
$150$	0	0
$151$	14.1803	1.15398	0.576990	−	0.816751i	$-0.304227\pi$
$151$	14.1803	1.15398	0.576990	+	0.816751i	$0.304227\pi$
$152$	15.0000	1.21666
$153$	0	0
$154$	0	0
$155$	5.23607	0.420571
$156$	0	0
$157$	−15.4164	−1.23036	−0.615182	−	0.788385i	$-0.710917\pi$
$157$	−15.4164	−1.23036	−0.615182	+	0.788385i	$0.710917\pi$
$158$	−8.94427	−0.711568
$159$	0	0
$160$	5.61803	0.444145
$161$	5.70820	0.449869
$162$	0	0
$163$	−22.7082	−1.77864	−0.889322	−	0.457282i	$-0.848823\pi$
$163$	−22.7082	−1.77864	−0.889322	+	0.457282i	$0.848823\pi$
$164$	4.00000	0.312348
$165$	0	0
$166$	−4.18034	−0.324457
$167$	−22.6525	−1.75290	−0.876451	−	0.481492i	$-0.840095\pi$
$167$	−22.6525	−1.75290	−0.876451	+	0.481492i	$0.840095\pi$
$168$	0	0
$169$	−0.944272	−0.0726363
$170$	3.23607	0.248195
$171$	0	0
$172$	9.23607	0.704244
$173$	−1.52786	−0.116161	−0.0580807	−	0.998312i	$-0.518498\pi$
$173$	−1.52786	−0.116161	−0.0580807	+	0.998312i	$0.518498\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	0	0
$177$	0	0
$178$	−2.76393	−0.207165
$179$	−8.94427	−0.668526	−0.334263	−	0.942480i	$-0.608487\pi$
$179$	−8.94427	−0.668526	−0.334263	+	0.942480i	$0.608487\pi$
$180$	0	0
$181$	−0.763932	−0.0567826	−0.0283913	−	0.999597i	$-0.509038\pi$
$181$	−0.763932	−0.0567826	−0.0283913	+	0.999597i	$0.509038\pi$
$182$	−2.14590	−0.159065
$183$	0	0
$184$	−12.7639	−0.940970
$185$	7.00000	0.514650
$186$	0	0
$187$	0	0
$188$	0.381966	0.0278577
$189$	0	0
$190$	4.14590	0.300775
$191$	10.7639	0.778851	0.389425	−	0.921058i	$-0.372674\pi$
$191$	10.7639	0.778851	0.389425	+	0.921058i	$0.372674\pi$
$192$	0	0
$193$	−14.6525	−1.05471	−0.527354	−	0.849646i	$-0.676816\pi$
$193$	−14.6525	−1.05471	−0.527354	+	0.849646i	$0.676816\pi$
$194$	−6.00000	−0.430775
$195$	0	0
$196$	−1.61803	−0.115574
$197$	−16.4721	−1.17359	−0.586796	−	0.809735i	$-0.699611\pi$
$197$	−16.4721	−1.17359	−0.586796	+	0.809735i	$0.699611\pi$
$198$	0	0
$199$	−3.81966	−0.270769	−0.135384	−	0.990793i	$-0.543227\pi$
$199$	−3.81966	−0.270769	−0.135384	+	0.990793i	$0.543227\pi$
$200$	−8.94427	−0.632456
$201$	0	0
$202$	−11.2361	−0.790567
$203$	−5.00000	−0.350931
$204$	0	0
$205$	2.47214	0.172661
$206$	−10.7639	−0.749959
$207$	0	0
$208$	−6.43769	−0.446374
$209$	0	0
$210$	0	0
$211$	−5.41641	−0.372881	−0.186440	−	0.982466i	$-0.559695\pi$
$211$	−5.41641	−0.372881	−0.186440	+	0.982466i	$0.559695\pi$
$212$	−19.7082	−1.35357
$213$	0	0
$214$	2.61803	0.178965
$215$	5.70820	0.389296
$216$	0	0
$217$	5.23607	0.355447
$218$	1.70820	0.115694
$219$	0	0
$220$	0	0
$221$	−18.1803	−1.22294
$222$	0	0
$223$	−6.00000	−0.401790	−0.200895	−	0.979613i	$-0.564385\pi$
$223$	−6.00000	−0.401790	−0.200895	+	0.979613i	$0.564385\pi$
$224$	5.61803	0.375371
$225$	0	0
$226$	−0.291796	−0.0194100
$227$	−2.00000	−0.132745	−0.0663723	−	0.997795i	$-0.521143\pi$
$227$	−2.00000	−0.132745	−0.0663723	+	0.997795i	$0.521143\pi$
$228$	0	0
$229$	7.23607	0.478173	0.239086	−	0.970998i	$-0.423152\pi$
$229$	7.23607	0.478173	0.239086	+	0.970998i	$0.423152\pi$
$230$	−3.52786	−0.232620
$231$	0	0
$232$	11.1803	0.734025
$233$	−14.9443	−0.979032	−0.489516	−	0.871994i	$-0.662826\pi$
$233$	−14.9443	−0.979032	−0.489516	+	0.871994i	$0.662826\pi$
$234$	0	0
$235$	0.236068	0.0153994
$236$	−18.0902	−1.17757
$237$	0	0
$238$	3.23607	0.209763
$239$	10.1246	0.654907	0.327453	−	0.944867i	$-0.393810\pi$
$239$	10.1246	0.654907	0.327453	+	0.944867i	$0.393810\pi$
$240$	0	0
$241$	−25.9443	−1.67122	−0.835609	−	0.549325i	$-0.814885\pi$
$241$	−25.9443	−1.67122	−0.835609	+	0.549325i	$0.814885\pi$
$242$	0	0
$243$	0	0
$244$	3.23607	0.207168
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−23.2918	−1.48202
$248$	−11.7082	−0.743472
$249$	0	0
$250$	−5.56231	−0.351791
$251$	−12.1246	−0.765299	−0.382649	−	0.923894i	$-0.624988\pi$
$251$	−12.1246	−0.765299	−0.382649	+	0.923894i	$0.624988\pi$
$252$	0	0
$253$	0	0
$254$	−7.81966	−0.490649
$255$	0	0
$256$	−6.56231	−0.410144
$257$	7.00000	0.436648	0.218324	−	0.975876i	$-0.429941\pi$
$257$	7.00000	0.436648	0.218324	+	0.975876i	$0.429941\pi$
$258$	0	0
$259$	7.00000	0.434959
$260$	−5.61803	−0.348416
$261$	0	0
$262$	−0.583592	−0.0360544
$263$	−26.1246	−1.61091	−0.805456	−	0.592655i	$-0.798080\pi$
$263$	−26.1246	−1.61091	−0.805456	+	0.592655i	$0.798080\pi$
$264$	0	0
$265$	−12.1803	−0.748232
$266$	4.14590	0.254201
$267$	0	0
$268$	15.7984	0.965039
$269$	−1.05573	−0.0643689	−0.0321844	−	0.999482i	$-0.510246\pi$
$269$	−1.05573	−0.0643689	−0.0321844	+	0.999482i	$0.510246\pi$
$270$	0	0
$271$	−5.29180	−0.321454	−0.160727	−	0.986999i	$-0.551384\pi$
$271$	−5.29180	−0.321454	−0.160727	+	0.986999i	$0.551384\pi$
$272$	9.70820	0.588646
$273$	0	0
$274$	12.1803	0.735841
$275$	0	0
$276$	0	0
$277$	6.47214	0.388873	0.194436	−	0.980915i	$-0.437712\pi$
$277$	6.47214	0.388873	0.194436	+	0.980915i	$0.437712\pi$
$278$	8.94427	0.536442
$279$	0	0
$280$	2.23607	0.133631
$281$	11.4721	0.684370	0.342185	−	0.939633i	$-0.388833\pi$
$281$	11.4721	0.684370	0.342185	+	0.939633i	$0.388833\pi$
$282$	0	0
$283$	13.7639	0.818181	0.409090	−	0.912494i	$-0.365846\pi$
$283$	13.7639	0.818181	0.409090	+	0.912494i	$0.365846\pi$
$284$	−4.00000	−0.237356
$285$	0	0
$286$	0	0
$287$	2.47214	0.145926
$288$	0	0
$289$	10.4164	0.612730
$290$	3.09017	0.181461
$291$	0	0
$292$	7.32624	0.428736
$293$	−16.0000	−0.934730	−0.467365	−	0.884064i	$-0.654797\pi$
$293$	−16.0000	−0.934730	−0.467365	+	0.884064i	$0.654797\pi$
$294$	0	0
$295$	−11.1803	−0.650945
$296$	−15.6525	−0.909782
$297$	0	0
$298$	−3.09017	−0.179009
$299$	19.8197	1.14620
$300$	0	0
$301$	5.70820	0.329015
$302$	−8.76393	−0.504308
$303$	0	0
$304$	12.4377	0.713351
$305$	2.00000	0.114520
$306$	0	0
$307$	20.9443	1.19535	0.597676	−	0.801737i	$-0.296091\pi$
$307$	20.9443	1.19535	0.597676	+	0.801737i	$0.296091\pi$
$308$	0	0
$309$	0	0
$310$	−3.23607	−0.183796
$311$	−9.88854	−0.560728	−0.280364	−	0.959894i	$-0.590455\pi$
$311$	−9.88854	−0.560728	−0.280364	+	0.959894i	$0.590455\pi$
$312$	0	0
$313$	24.6525	1.39344	0.696720	−	0.717343i	$-0.254642\pi$
$313$	24.6525	1.39344	0.696720	+	0.717343i	$0.254642\pi$
$314$	9.52786	0.537688
$315$	0	0
$316$	−23.4164	−1.31728
$317$	−24.1803	−1.35810	−0.679052	−	0.734091i	$-0.737609\pi$
$317$	−24.1803	−1.35810	−0.679052	+	0.734091i	$0.737609\pi$
$318$	0	0
$319$	0	0
$320$	0.236068	0.0131966
$321$	0	0
$322$	−3.52786	−0.196600
$323$	35.1246	1.95439
$324$	0	0
$325$	13.8885	0.770398
$326$	14.0344	0.777296
$327$	0	0
$328$	−5.52786	−0.305225
$329$	0.236068	0.0130148
$330$	0	0
$331$	−11.4164	−0.627503	−0.313751	−	0.949505i	$-0.601586\pi$
$331$	−11.4164	−0.627503	−0.313751	+	0.949505i	$0.601586\pi$
$332$	−10.9443	−0.600645
$333$	0	0
$334$	14.0000	0.766046
$335$	9.76393	0.533461
$336$	0	0
$337$	8.18034	0.445612	0.222806	−	0.974863i	$-0.428478\pi$
$337$	8.18034	0.445612	0.222806	+	0.974863i	$0.428478\pi$
$338$	0.583592	0.0317432
$339$	0	0
$340$	8.47214	0.459466
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−12.7639	−0.688185
$345$	0	0
$346$	0.944272	0.0507644
$347$	28.0000	1.50312	0.751559	−	0.659665i	$-0.229302\pi$
$347$	28.0000	1.50312	0.751559	+	0.659665i	$0.229302\pi$
$348$	0	0
$349$	−1.58359	−0.0847677	−0.0423839	−	0.999101i	$-0.513495\pi$
$349$	−1.58359	−0.0847677	−0.0423839	+	0.999101i	$0.513495\pi$
$350$	−2.47214	−0.132141
$351$	0	0
$352$	0	0
$353$	−24.5279	−1.30549	−0.652743	−	0.757579i	$-0.726382\pi$
$353$	−24.5279	−1.30549	−0.652743	+	0.757579i	$0.726382\pi$
$354$	0	0
$355$	−2.47214	−0.131207
$356$	−7.23607	−0.383511
$357$	0	0
$358$	5.52786	0.292157
$359$	23.4164	1.23587	0.617935	−	0.786229i	$-0.287969\pi$
$359$	23.4164	1.23587	0.617935	+	0.786229i	$0.287969\pi$
$360$	0	0
$361$	26.0000	1.36842
$362$	0.472136	0.0248149
$363$	0	0
$364$	−5.61803	−0.294465
$365$	4.52786	0.236999
$366$	0	0
$367$	−19.8885	−1.03817	−0.519087	−	0.854722i	$-0.673728\pi$
$367$	−19.8885	−1.03817	−0.519087	+	0.854722i	$0.673728\pi$
$368$	−10.5836	−0.551708
$369$	0	0
$370$	−4.32624	−0.224910
$371$	−12.1803	−0.632372
$372$	0	0
$373$	−4.65248	−0.240896	−0.120448	−	0.992720i	$-0.538433\pi$
$373$	−4.65248	−0.240896	−0.120448	+	0.992720i	$0.538433\pi$
$374$	0	0
$375$	0	0
$376$	−0.527864	−0.0272225
$377$	−17.3607	−0.894120
$378$	0	0
$379$	−31.1803	−1.60163	−0.800813	−	0.598914i	$-0.795599\pi$
$379$	−31.1803	−1.60163	−0.800813	+	0.598914i	$0.795599\pi$
$380$	10.8541	0.556804
$381$	0	0
$382$	−6.65248	−0.340370
$383$	−32.9443	−1.68337	−0.841687	−	0.539966i	$-0.818437\pi$
$383$	−32.9443	−1.68337	−0.841687	+	0.539966i	$0.818437\pi$
$384$	0	0
$385$	0	0
$386$	9.05573	0.460924
$387$	0	0
$388$	−15.7082	−0.797463
$389$	−11.0557	−0.560548	−0.280274	−	0.959920i	$-0.590425\pi$
$389$	−11.0557	−0.560548	−0.280274	+	0.959920i	$0.590425\pi$
$390$	0	0
$391$	−29.8885	−1.51153
$392$	2.23607	0.112938
$393$	0	0
$394$	10.1803	0.512878
$395$	−14.4721	−0.728172
$396$	0	0
$397$	23.1246	1.16059	0.580295	−	0.814406i	$-0.302937\pi$
$397$	23.1246	1.16059	0.580295	+	0.814406i	$0.302937\pi$
$398$	2.36068	0.118330
$399$	0	0
$400$	−7.41641	−0.370820
$401$	29.7082	1.48356	0.741778	−	0.670645i	$-0.233983\pi$
$401$	29.7082	1.48356	0.741778	+	0.670645i	$0.233983\pi$
$402$	0	0
$403$	18.1803	0.905627
$404$	−29.4164	−1.46352
$405$	0	0
$406$	3.09017	0.153363
$407$	0	0
$408$	0	0
$409$	−21.0557	−1.04114	−0.520569	−	0.853819i	$-0.674280\pi$
$409$	−21.0557	−1.04114	−0.520569	+	0.853819i	$0.674280\pi$
$410$	−1.52786	−0.0754558
$411$	0	0
$412$	−28.1803	−1.38835
$413$	−11.1803	−0.550149
$414$	0	0
$415$	−6.76393	−0.332028
$416$	19.5066	0.956389
$417$	0	0
$418$	0	0
$419$	1.18034	0.0576634	0.0288317	−	0.999584i	$-0.490821\pi$
$419$	1.18034	0.0576634	0.0288317	+	0.999584i	$0.490821\pi$
$420$	0	0
$421$	−13.0000	−0.633581	−0.316791	−	0.948495i	$-0.602605\pi$
$421$	−13.0000	−0.633581	−0.316791	+	0.948495i	$0.602605\pi$
$422$	3.34752	0.162955
$423$	0	0
$424$	27.2361	1.32270
$425$	−20.9443	−1.01595
$426$	0	0
$427$	2.00000	0.0967868
$428$	6.85410	0.331306
$429$	0	0
$430$	−3.52786	−0.170129
$431$	8.70820	0.419459	0.209730	−	0.977759i	$-0.432742\pi$
$431$	8.70820	0.419459	0.209730	+	0.977759i	$0.432742\pi$
$432$	0	0
$433$	−10.4721	−0.503259	−0.251629	−	0.967824i	$-0.580966\pi$
$433$	−10.4721	−0.503259	−0.251629	+	0.967824i	$0.580966\pi$
$434$	−3.23607	−0.155336
$435$	0	0
$436$	4.47214	0.214176
$437$	−38.2918	−1.83175
$438$	0	0
$439$	−11.1803	−0.533609	−0.266804	−	0.963751i	$-0.585968\pi$
$439$	−11.1803	−0.533609	−0.266804	+	0.963751i	$0.585968\pi$
$440$	0	0
$441$	0	0
$442$	11.2361	0.534445
$443$	−18.4721	−0.877638	−0.438819	−	0.898576i	$-0.644603\pi$
$443$	−18.4721	−0.877638	−0.438819	+	0.898576i	$0.644603\pi$
$444$	0	0
$445$	−4.47214	−0.212000
$446$	3.70820	0.175589
$447$	0	0
$448$	0.236068	0.0111532
$449$	−20.0000	−0.943858	−0.471929	−	0.881636i	$-0.656442\pi$
$449$	−20.0000	−0.943858	−0.471929	+	0.881636i	$0.656442\pi$
$450$	0	0
$451$	0	0
$452$	−0.763932	−0.0359323
$453$	0	0
$454$	1.23607	0.0580115
$455$	−3.47214	−0.162776
$456$	0	0
$457$	−15.2361	−0.712713	−0.356357	−	0.934350i	$-0.615981\pi$
$457$	−15.2361	−0.712713	−0.356357	+	0.934350i	$0.615981\pi$
$458$	−4.47214	−0.208969
$459$	0	0
$460$	−9.23607	−0.430634
$461$	−28.0000	−1.30409	−0.652045	−	0.758180i	$-0.726089\pi$
$461$	−28.0000	−1.30409	−0.652045	+	0.758180i	$0.726089\pi$
$462$	0	0
$463$	−4.81966	−0.223989	−0.111994	−	0.993709i	$-0.535724\pi$
$463$	−4.81966	−0.223989	−0.111994	+	0.993709i	$0.535724\pi$
$464$	9.27051	0.430373
$465$	0	0
$466$	9.23607	0.427853
$467$	−29.1803	−1.35031	−0.675153	−	0.737678i	$-0.735922\pi$
$467$	−29.1803	−1.35031	−0.675153	+	0.737678i	$0.735922\pi$
$468$	0	0
$469$	9.76393	0.450856
$470$	−0.145898	−0.00672977
$471$	0	0
$472$	25.0000	1.15072
$473$	0	0
$474$	0	0
$475$	−26.8328	−1.23117
$476$	8.47214	0.388320
$477$	0	0
$478$	−6.25735	−0.286205
$479$	11.7082	0.534961	0.267481	−	0.963563i	$-0.413809\pi$
$479$	11.7082	0.534961	0.267481	+	0.963563i	$0.413809\pi$
$480$	0	0
$481$	24.3050	1.10821
$482$	16.0344	0.730349
$483$	0	0
$484$	0	0
$485$	−9.70820	−0.440827
$486$	0	0
$487$	16.9443	0.767818	0.383909	−	0.923371i	$-0.374578\pi$
$487$	16.9443	0.767818	0.383909	+	0.923371i	$0.374578\pi$
$488$	−4.47214	−0.202444
$489$	0	0
$490$	0.618034	0.0279199
$491$	−18.1246	−0.817952	−0.408976	−	0.912545i	$-0.634114\pi$
$491$	−18.1246	−0.817952	−0.408976	+	0.912545i	$0.634114\pi$
$492$	0	0
$493$	26.1803	1.17910
$494$	14.3951	0.647667
$495$	0	0
$496$	−9.70820	−0.435911
$497$	−2.47214	−0.110890
$498$	0	0
$499$	−2.23607	−0.100100	−0.0500501	−	0.998747i	$-0.515938\pi$
$499$	−2.23607	−0.100100	−0.0500501	+	0.998747i	$0.515938\pi$
$500$	−14.5623	−0.651246
$501$	0	0
$502$	7.49342	0.334448
$503$	2.29180	0.102186	0.0510931	−	0.998694i	$-0.483729\pi$
$503$	2.29180	0.102186	0.0510931	+	0.998694i	$0.483729\pi$
$504$	0	0
$505$	−18.1803	−0.809015
$506$	0	0
$507$	0	0
$508$	−20.4721	−0.908304
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	0	0
$511$	4.52786	0.200301
$512$	−18.7082	−0.826794
$513$	0	0
$514$	−4.32624	−0.190822
$515$	−17.4164	−0.767459
$516$	0	0
$517$	0	0
$518$	−4.32624	−0.190084
$519$	0	0
$520$	7.76393	0.340471
$521$	−38.3050	−1.67817	−0.839085	−	0.544000i	$-0.816909\pi$
$521$	−38.3050	−1.67817	−0.839085	+	0.544000i	$0.816909\pi$
$522$	0	0
$523$	21.6525	0.946797	0.473398	−	0.880848i	$-0.343027\pi$
$523$	21.6525	0.946797	0.473398	+	0.880848i	$0.343027\pi$
$524$	−1.52786	−0.0667451
$525$	0	0
$526$	16.1459	0.703995
$527$	−27.4164	−1.19428
$528$	0	0
$529$	9.58359	0.416678
$530$	7.52786	0.326990
$531$	0	0
$532$	10.8541	0.470585
$533$	8.58359	0.371797
$534$	0	0
$535$	4.23607	0.183141
$536$	−21.8328	−0.943034
$537$	0	0
$538$	0.652476	0.0281302
$539$	0	0
$540$	0	0
$541$	36.9443	1.58836	0.794179	−	0.607684i	$-0.207901\pi$
$541$	36.9443	1.58836	0.794179	+	0.607684i	$0.207901\pi$
$542$	3.27051	0.140480
$543$	0	0
$544$	−29.4164	−1.26122
$545$	2.76393	0.118394
$546$	0	0
$547$	−14.8328	−0.634205	−0.317103	−	0.948391i	$-0.602710\pi$
$547$	−14.8328	−0.634205	−0.317103	+	0.948391i	$0.602710\pi$
$548$	31.8885	1.36221
$549$	0	0
$550$	0	0
$551$	33.5410	1.42890
$552$	0	0
$553$	−14.4721	−0.615418
$554$	−4.00000	−0.169944
$555$	0	0
$556$	23.4164	0.993077
$557$	−12.5279	−0.530823	−0.265411	−	0.964135i	$-0.585508\pi$
$557$	−12.5279	−0.530823	−0.265411	+	0.964135i	$0.585508\pi$
$558$	0	0
$559$	19.8197	0.838282
$560$	1.85410	0.0783501
$561$	0	0
$562$	−7.09017	−0.299081
$563$	34.6525	1.46043	0.730214	−	0.683219i	$-0.239420\pi$
$563$	34.6525	1.46043	0.730214	+	0.683219i	$0.239420\pi$
$564$	0	0
$565$	−0.472136	−0.0198629
$566$	−8.50658	−0.357558
$567$	0	0
$568$	5.52786	0.231944
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	−17.5279	−0.733518	−0.366759	−	0.930316i	$-0.619533\pi$
$571$	−17.5279	−0.733518	−0.366759	+	0.930316i	$0.619533\pi$
$572$	0	0
$573$	0	0
$574$	−1.52786	−0.0637718
$575$	22.8328	0.952194
$576$	0	0
$577$	7.34752	0.305881	0.152941	−	0.988235i	$-0.451126\pi$
$577$	7.34752	0.305881	0.152941	+	0.988235i	$0.451126\pi$
$578$	−6.43769	−0.267773
$579$	0	0
$580$	8.09017	0.335926
$581$	−6.76393	−0.280615
$582$	0	0
$583$	0	0
$584$	−10.1246	−0.418959
$585$	0	0
$586$	9.88854	0.408492
$587$	−46.0132	−1.89917	−0.949583	−	0.313515i	$-0.898493\pi$
$587$	−46.0132	−1.89917	−0.949583	+	0.313515i	$0.898493\pi$
$588$	0	0
$589$	−35.1246	−1.44728
$590$	6.90983	0.284473
$591$	0	0
$592$	−12.9787	−0.533422
$593$	−22.8328	−0.937631	−0.468816	−	0.883296i	$-0.655319\pi$
$593$	−22.8328	−0.937631	−0.468816	+	0.883296i	$0.655319\pi$
$594$	0	0
$595$	5.23607	0.214658
$596$	−8.09017	−0.331386
$597$	0	0
$598$	−12.2492	−0.500908
$599$	25.5279	1.04304	0.521520	−	0.853239i	$-0.325365\pi$
$599$	25.5279	1.04304	0.521520	+	0.853239i	$0.325365\pi$
$600$	0	0
$601$	−27.0000	−1.10135	−0.550676	−	0.834719i	$-0.685630\pi$
$601$	−27.0000	−1.10135	−0.550676	+	0.834719i	$0.685630\pi$
$602$	−3.52786	−0.143785
$603$	0	0
$604$	−22.9443	−0.933589
$605$	0	0
$606$	0	0
$607$	−6.81966	−0.276801	−0.138401	−	0.990376i	$-0.544196\pi$
$607$	−6.81966	−0.276801	−0.138401	+	0.990376i	$0.544196\pi$
$608$	−37.6869	−1.52841
$609$	0	0
$610$	−1.23607	−0.0500469
$611$	0.819660	0.0331599
$612$	0	0
$613$	−13.5967	−0.549167	−0.274584	−	0.961563i	$-0.588540\pi$
$613$	−13.5967	−0.549167	−0.274584	+	0.961563i	$0.588540\pi$
$614$	−12.9443	−0.522388
$615$	0	0
$616$	0	0
$617$	−32.4721	−1.30728	−0.653639	−	0.756806i	$-0.726759\pi$
$617$	−32.4721	−1.30728	−0.653639	+	0.756806i	$0.726759\pi$
$618$	0	0
$619$	−44.0689	−1.77128	−0.885639	−	0.464374i	$-0.846279\pi$
$619$	−44.0689	−1.77128	−0.885639	+	0.464374i	$0.846279\pi$
$620$	−8.47214	−0.340249
$621$	0	0
$622$	6.11146	0.245047
$623$	−4.47214	−0.179172
$624$	0	0
$625$	11.0000	0.440000
$626$	−15.2361	−0.608956
$627$	0	0
$628$	24.9443	0.995385
$629$	−36.6525	−1.46143
$630$	0	0
$631$	44.3607	1.76597	0.882985	−	0.469400i	$-0.155530\pi$
$631$	44.3607	1.76597	0.882985	+	0.469400i	$0.155530\pi$
$632$	32.3607	1.28724
$633$	0	0
$634$	14.9443	0.593513
$635$	−12.6525	−0.502098
$636$	0	0
$637$	−3.47214	−0.137571
$638$	0	0
$639$	0	0
$640$	−11.3820	−0.449912
$641$	46.5410	1.83826	0.919130	−	0.393955i	$-0.128893\pi$
$641$	46.5410	1.83826	0.919130	+	0.393955i	$0.128893\pi$
$642$	0	0
$643$	−47.9574	−1.89126	−0.945628	−	0.325250i	$-0.894552\pi$
$643$	−47.9574	−1.89126	−0.945628	+	0.325250i	$0.894552\pi$
$644$	−9.23607	−0.363952
$645$	0	0
$646$	−21.7082	−0.854098
$647$	−12.3475	−0.485431	−0.242716	−	0.970097i	$-0.578038\pi$
$647$	−12.3475	−0.485431	−0.242716	+	0.970097i	$0.578038\pi$
$648$	0	0
$649$	0	0
$650$	−8.58359	−0.336676
$651$	0	0
$652$	36.7426	1.43895
$653$	44.9443	1.75881	0.879403	−	0.476079i	$-0.157942\pi$
$653$	44.9443	1.75881	0.879403	+	0.476079i	$0.157942\pi$
$654$	0	0
$655$	−0.944272	−0.0368958
$656$	−4.58359	−0.178959
$657$	0	0
$658$	−0.145898	−0.00568770
$659$	23.5410	0.917028	0.458514	−	0.888687i	$-0.348382\pi$
$659$	23.5410	0.917028	0.458514	+	0.888687i	$0.348382\pi$
$660$	0	0
$661$	40.5410	1.57686	0.788431	−	0.615123i	$-0.210894\pi$
$661$	40.5410	1.57686	0.788431	+	0.615123i	$0.210894\pi$
$662$	7.05573	0.274229
$663$	0	0
$664$	15.1246	0.586949
$665$	6.70820	0.260133
$666$	0	0
$667$	−28.5410	−1.10511
$668$	36.6525	1.41813
$669$	0	0
$670$	−6.03444	−0.233131
$671$	0	0
$672$	0	0
$673$	−3.59675	−0.138644	−0.0693222	−	0.997594i	$-0.522084\pi$
$673$	−3.59675	−0.138644	−0.0693222	+	0.997594i	$0.522084\pi$
$674$	−5.05573	−0.194739
$675$	0	0
$676$	1.52786	0.0587640
$677$	19.3050	0.741950	0.370975	−	0.928643i	$-0.379024\pi$
$677$	19.3050	0.741950	0.370975	+	0.928643i	$0.379024\pi$
$678$	0	0
$679$	−9.70820	−0.372567
$680$	−11.7082	−0.448989
$681$	0	0
$682$	0	0
$683$	−37.0132	−1.41627	−0.708135	−	0.706078i	$-0.750463\pi$
$683$	−37.0132	−1.41627	−0.708135	+	0.706078i	$0.750463\pi$
$684$	0	0
$685$	19.7082	0.753012
$686$	0.618034	0.0235966
$687$	0	0
$688$	−10.5836	−0.403496
$689$	−42.2918	−1.61119
$690$	0	0
$691$	2.00000	0.0760836	0.0380418	−	0.999276i	$-0.487888\pi$
$691$	2.00000	0.0760836	0.0380418	+	0.999276i	$0.487888\pi$
$692$	2.47214	0.0939765
$693$	0	0
$694$	−17.3050	−0.656887
$695$	14.4721	0.548959
$696$	0	0
$697$	−12.9443	−0.490299
$698$	0.978714	0.0370449
$699$	0	0
$700$	−6.47214	−0.244624
$701$	4.11146	0.155288	0.0776438	−	0.996981i	$-0.475260\pi$
$701$	4.11146	0.155288	0.0776438	+	0.996981i	$0.475260\pi$
$702$	0	0
$703$	−46.9574	−1.77103
$704$	0	0
$705$	0	0
$706$	15.1591	0.570519
$707$	−18.1803	−0.683742
$708$	0	0
$709$	−49.7214	−1.86732	−0.933662	−	0.358154i	$-0.883406\pi$
$709$	−49.7214	−1.86732	−0.933662	+	0.358154i	$0.883406\pi$
$710$	1.52786	0.0573397
$711$	0	0
$712$	10.0000	0.374766
$713$	29.8885	1.11933
$714$	0	0
$715$	0	0
$716$	14.4721	0.540849
$717$	0	0
$718$	−14.4721	−0.540095
$719$	−7.76393	−0.289546	−0.144773	−	0.989465i	$-0.546245\pi$
$719$	−7.76393	−0.289546	−0.144773	+	0.989465i	$0.546245\pi$
$720$	0	0
$721$	−17.4164	−0.648621
$722$	−16.0689	−0.598022
$723$	0	0
$724$	1.23607	0.0459381
$725$	−20.0000	−0.742781
$726$	0	0
$727$	24.1803	0.896799	0.448400	−	0.893833i	$-0.351994\pi$
$727$	24.1803	0.896799	0.448400	+	0.893833i	$0.351994\pi$
$728$	7.76393	0.287750
$729$	0	0
$730$	−2.79837	−0.103572
$731$	−29.8885	−1.10547
$732$	0	0
$733$	8.11146	0.299603	0.149802	−	0.988716i	$-0.452136\pi$
$733$	8.11146	0.299603	0.149802	+	0.988716i	$0.452136\pi$
$734$	12.2918	0.453698
$735$	0	0
$736$	32.0689	1.18207
$737$	0	0
$738$	0	0
$739$	−34.0689	−1.25324	−0.626622	−	0.779323i	$-0.715563\pi$
$739$	−34.0689	−1.25324	−0.626622	+	0.779323i	$0.715563\pi$
$740$	−11.3262	−0.416361
$741$	0	0
$742$	7.52786	0.276357
$743$	2.81966	0.103443	0.0517216	−	0.998662i	$-0.483529\pi$
$743$	2.81966	0.103443	0.0517216	+	0.998662i	$0.483529\pi$
$744$	0	0
$745$	−5.00000	−0.183186
$746$	2.87539	0.105275
$747$	0	0
$748$	0	0
$749$	4.23607	0.154783
$750$	0	0
$751$	39.7639	1.45101	0.725503	−	0.688219i	$-0.241607\pi$
$751$	39.7639	1.45101	0.725503	+	0.688219i	$0.241607\pi$
$752$	−0.437694	−0.0159611
$753$	0	0
$754$	10.7295	0.390745
$755$	−14.1803	−0.516075
$756$	0	0
$757$	−51.7214	−1.87984	−0.939922	−	0.341388i	$-0.889103\pi$
$757$	−51.7214	−1.87984	−0.939922	+	0.341388i	$0.889103\pi$
$758$	19.2705	0.699936
$759$	0	0
$760$	−15.0000	−0.544107
$761$	27.7771	1.00692	0.503459	−	0.864019i	$-0.332060\pi$
$761$	27.7771	1.00692	0.503459	+	0.864019i	$0.332060\pi$
$762$	0	0
$763$	2.76393	0.100061
$764$	−17.4164	−0.630104
$765$	0	0
$766$	20.3607	0.735661
$767$	−38.8197	−1.40170
$768$	0	0
$769$	−13.9443	−0.502843	−0.251422	−	0.967878i	$-0.580898\pi$
$769$	−13.9443	−0.502843	−0.251422	+	0.967878i	$0.580898\pi$
$770$	0	0
$771$	0	0
$772$	23.7082	0.853277
$773$	5.47214	0.196819	0.0984095	−	0.995146i	$-0.468624\pi$
$773$	5.47214	0.196819	0.0984095	+	0.995146i	$0.468624\pi$
$774$	0	0
$775$	20.9443	0.752340
$776$	21.7082	0.779279
$777$	0	0
$778$	6.83282	0.244968
$779$	−16.5836	−0.594169
$780$	0	0
$781$	0	0
$782$	18.4721	0.660562
$783$	0	0
$784$	1.85410	0.0662179
$785$	15.4164	0.550235
$786$	0	0
$787$	−3.65248	−0.130197	−0.0650984	−	0.997879i	$-0.520736\pi$
$787$	−3.65248	−0.130197	−0.0650984	+	0.997879i	$0.520736\pi$
$788$	26.6525	0.949455
$789$	0	0
$790$	8.94427	0.318223
$791$	−0.472136	−0.0167872
$792$	0	0
$793$	6.94427	0.246598
$794$	−14.2918	−0.507197
$795$	0	0
$796$	6.18034	0.219056
$797$	2.52786	0.0895415	0.0447708	−	0.998997i	$-0.485744\pi$
$797$	2.52786	0.0895415	0.0447708	+	0.998997i	$0.485744\pi$
$798$	0	0
$799$	−1.23607	−0.0437289
$800$	22.4721	0.794510
$801$	0	0
$802$	−18.3607	−0.648338
$803$	0	0
$804$	0	0
$805$	−5.70820	−0.201188
$806$	−11.2361	−0.395774
$807$	0	0
$808$	40.6525	1.43015
$809$	−56.3050	−1.97958	−0.989788	−	0.142545i	$-0.954471\pi$
$809$	−56.3050	−1.97958	−0.989788	+	0.142545i	$0.954471\pi$
$810$	0	0
$811$	−15.2918	−0.536968	−0.268484	−	0.963284i	$-0.586523\pi$
$811$	−15.2918	−0.536968	−0.268484	+	0.963284i	$0.586523\pi$
$812$	8.09017	0.283909
$813$	0	0
$814$	0	0
$815$	22.7082	0.795434
$816$	0	0
$817$	−38.2918	−1.33966
$818$	13.0132	0.454994
$819$	0	0
$820$	−4.00000	−0.139686
$821$	−7.47214	−0.260779	−0.130390	−	0.991463i	$-0.541623\pi$
$821$	−7.47214	−0.260779	−0.130390	+	0.991463i	$0.541623\pi$
$822$	0	0
$823$	−17.1803	−0.598869	−0.299435	−	0.954117i	$-0.596798\pi$
$823$	−17.1803	−0.598869	−0.299435	+	0.954117i	$0.596798\pi$
$824$	38.9443	1.35669
$825$	0	0
$826$	6.90983	0.240424
$827$	12.3475	0.429365	0.214683	−	0.976684i	$-0.431128\pi$
$827$	12.3475	0.429365	0.214683	+	0.976684i	$0.431128\pi$
$828$	0	0
$829$	35.7771	1.24259	0.621295	−	0.783577i	$-0.286607\pi$
$829$	35.7771	1.24259	0.621295	+	0.783577i	$0.286607\pi$
$830$	4.18034	0.145102
$831$	0	0
$832$	0.819660	0.0284166
$833$	5.23607	0.181419
$834$	0	0
$835$	22.6525	0.783921
$836$	0	0
$837$	0	0
$838$	−0.729490	−0.0251998
$839$	23.5410	0.812726	0.406363	−	0.913712i	$-0.366797\pi$
$839$	23.5410	0.812726	0.406363	+	0.913712i	$0.366797\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	8.03444	0.276885
$843$	0	0
$844$	8.76393	0.301667
$845$	0.944272	0.0324839
$846$	0	0
$847$	0	0
$848$	22.5836	0.775524
$849$	0	0
$850$	12.9443	0.443985
$851$	39.9574	1.36972
$852$	0	0
$853$	29.4164	1.00720	0.503599	−	0.863937i	$-0.332009\pi$
$853$	29.4164	1.00720	0.503599	+	0.863937i	$0.332009\pi$
$854$	−1.23607	−0.0422974
$855$	0	0
$856$	−9.47214	−0.323751
$857$	0.111456	0.00380727	0.00190364	−	0.999998i	$-0.499394\pi$
$857$	0.111456	0.00380727	0.00190364	+	0.999998i	$0.499394\pi$
$858$	0	0
$859$	−40.0000	−1.36478	−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$860$	−9.23607	−0.314947
$861$	0	0
$862$	−5.38197	−0.183310
$863$	43.2361	1.47177	0.735886	−	0.677105i	$-0.236766\pi$
$863$	43.2361	1.47177	0.735886	+	0.677105i	$0.236766\pi$
$864$	0	0
$865$	1.52786	0.0519489
$866$	6.47214	0.219932
$867$	0	0
$868$	−8.47214	−0.287563
$869$	0	0
$870$	0	0
$871$	33.9017	1.14872
$872$	−6.18034	−0.209293
$873$	0	0
$874$	23.6656	0.800502
$875$	−9.00000	−0.304256
$876$	0	0
$877$	−4.58359	−0.154777	−0.0773885	−	0.997001i	$-0.524658\pi$
$877$	−4.58359	−0.154777	−0.0773885	+	0.997001i	$0.524658\pi$
$878$	6.90983	0.233195
$879$	0	0
$880$	0	0
$881$	−54.8885	−1.84924	−0.924621	−	0.380888i	$-0.875618\pi$
$881$	−54.8885	−1.84924	−0.924621	+	0.380888i	$0.875618\pi$
$882$	0	0
$883$	−14.8197	−0.498721	−0.249361	−	0.968411i	$-0.580220\pi$
$883$	−14.8197	−0.498721	−0.249361	+	0.968411i	$0.580220\pi$
$884$	29.4164	0.989381
$885$	0	0
$886$	11.4164	0.383542
$887$	10.7639	0.361417	0.180709	−	0.983537i	$-0.442161\pi$
$887$	10.7639	0.361417	0.180709	+	0.983537i	$0.442161\pi$
$888$	0	0
$889$	−12.6525	−0.424350
$890$	2.76393	0.0926472
$891$	0	0
$892$	9.70820	0.325055
$893$	−1.58359	−0.0529929
$894$	0	0
$895$	8.94427	0.298974
$896$	−11.3820	−0.380245
$897$	0	0
$898$	12.3607	0.412481
$899$	−26.1803	−0.873163
$900$	0	0
$901$	63.7771	2.12472
$902$	0	0
$903$	0	0
$904$	1.05573	0.0351130
$905$	0.763932	0.0253940
$906$	0	0
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	3.23607	0.107393
$909$	0	0
$910$	2.14590	0.0711358
$911$	14.5836	0.483176	0.241588	−	0.970379i	$-0.422332\pi$
$911$	14.5836	0.483176	0.241588	+	0.970379i	$0.422332\pi$
$912$	0	0
$913$	0	0
$914$	9.41641	0.311467
$915$	0	0
$916$	−11.7082	−0.386850
$917$	−0.944272	−0.0311826
$918$	0	0
$919$	−39.5967	−1.30618	−0.653088	−	0.757282i	$-0.726527\pi$
$919$	−39.5967	−1.30618	−0.653088	+	0.757282i	$0.726527\pi$
$920$	12.7639	0.420814
$921$	0	0
$922$	17.3050	0.569908
$923$	−8.58359	−0.282532
$924$	0	0
$925$	28.0000	0.920634
$926$	2.97871	0.0978866
$927$	0	0
$928$	−28.0902	−0.922105
$929$	−12.8885	−0.422859	−0.211430	−	0.977393i	$-0.567812\pi$
$929$	−12.8885	−0.422859	−0.211430	+	0.977393i	$0.567812\pi$
$930$	0	0
$931$	6.70820	0.219853
$932$	24.1803	0.792053
$933$	0	0
$934$	18.0344	0.590105
$935$	0	0
$936$	0	0
$937$	39.8885	1.30310	0.651551	−	0.758605i	$-0.274119\pi$
$937$	39.8885	1.30310	0.651551	+	0.758605i	$0.274119\pi$
$938$	−6.03444	−0.197032
$939$	0	0
$940$	−0.381966	−0.0124584
$941$	−60.3607	−1.96770	−0.983851	−	0.178990i	$-0.942717\pi$
$941$	−60.3607	−1.96770	−0.983851	+	0.178990i	$0.942717\pi$
$942$	0	0
$943$	14.1115	0.459532
$944$	20.7295	0.674687
$945$	0	0
$946$	0	0
$947$	5.41641	0.176010	0.0880048	−	0.996120i	$-0.471951\pi$
$947$	5.41641	0.176010	0.0880048	+	0.996120i	$0.471951\pi$
$948$	0	0
$949$	15.7214	0.510337
$950$	16.5836	0.538043
$951$	0	0
$952$	−11.7082	−0.379465
$953$	44.7771	1.45047	0.725236	−	0.688500i	$-0.241731\pi$
$953$	44.7771	1.45047	0.725236	+	0.688500i	$0.241731\pi$
$954$	0	0
$955$	−10.7639	−0.348313
$956$	−16.3820	−0.529831
$957$	0	0
$958$	−7.23607	−0.233787
$959$	19.7082	0.636411
$960$	0	0
$961$	−3.58359	−0.115600
$962$	−15.0213	−0.484306
$963$	0	0
$964$	41.9787	1.35204
$965$	14.6525	0.471680
$966$	0	0
$967$	61.1935	1.96785	0.983925	−	0.178582i	$-0.0571509\pi$
$967$	61.1935	1.96785	0.983925	+	0.178582i	$0.0571509\pi$
$968$	0	0
$969$	0	0
$970$	6.00000	0.192648
$971$	−32.1246	−1.03093	−0.515464	−	0.856911i	$-0.672380\pi$
$971$	−32.1246	−1.03093	−0.515464	+	0.856911i	$0.672380\pi$
$972$	0	0
$973$	14.4721	0.463955
$974$	−10.4721	−0.335549
$975$	0	0
$976$	−3.70820	−0.118697
$977$	−4.18034	−0.133741	−0.0668705	−	0.997762i	$-0.521301\pi$
$977$	−4.18034	−0.133741	−0.0668705	+	0.997762i	$0.521301\pi$
$978$	0	0
$979$	0	0
$980$	1.61803	0.0516862
$981$	0	0
$982$	11.2016	0.357458
$983$	−27.4164	−0.874448	−0.437224	−	0.899353i	$-0.644038\pi$
$983$	−27.4164	−0.874448	−0.437224	+	0.899353i	$0.644038\pi$
$984$	0	0
$985$	16.4721	0.524846
$986$	−16.1803	−0.515287
$987$	0	0
$988$	37.6869	1.19898
$989$	32.5836	1.03610
$990$	0	0
$991$	−29.1803	−0.926944	−0.463472	−	0.886112i	$-0.653397\pi$
$991$	−29.1803	−0.926944	−0.463472	+	0.886112i	$0.653397\pi$
$992$	29.4164	0.933972
$993$	0	0
$994$	1.52786	0.0484609
$995$	3.81966	0.121091
$996$	0	0
$997$	−26.9443	−0.853334	−0.426667	−	0.904409i	$-0.640312\pi$
$997$	−26.9443	−0.853334	−0.426667	+	0.904409i	$0.640312\pi$
$998$	1.38197	0.0437454
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.bm.1.1		2
3.2	odd	2		2541.2.a.t.1.2		2
11.10	odd	2		693.2.a.f.1.2		2
33.32	even	2		231.2.a.c.1.1	✓	2
77.76	even	2		4851.2.a.w.1.2		2
132.131	odd	2		3696.2.a.be.1.2		2
165.164	even	2		5775.2.a.be.1.2		2
231.230	odd	2		1617.2.a.p.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.a.c.1.1	✓	2	33.32	even	2
693.2.a.f.1.2		2	11.10	odd	2
1617.2.a.p.1.1		2	231.230	odd	2
2541.2.a.t.1.2		2	3.2	odd	2
3696.2.a.be.1.2		2	132.131	odd	2
4851.2.a.w.1.2		2	77.76	even	2
5775.2.a.be.1.2		2	165.164	even	2
7623.2.a.bm.1.1		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7623.a (trivial)

\(f(q)\)	\(=\)	\(q-0.618034 q^{2} -1.61803 q^{4} -1.00000 q^{5} -1.00000 q^{7} +2.23607 q^{8} +O(q^{10})\)	\(q-0.618034 q^{2} -1.61803 q^{4} -1.00000 q^{5} -1.00000 q^{7} +2.23607 q^{8} +0.618034 q^{10} -3.47214 q^{13} +0.618034 q^{14} +1.85410 q^{16} +5.23607 q^{17} +6.70820 q^{19} +1.61803 q^{20} -5.70820 q^{23} -4.00000 q^{25} +2.14590 q^{26} +1.61803 q^{28} +5.00000 q^{29} -5.23607 q^{31} -5.61803 q^{32} -3.23607 q^{34} +1.00000 q^{35} -7.00000 q^{37} -4.14590 q^{38} -2.23607 q^{40} -2.47214 q^{41} -5.70820 q^{43} +3.52786 q^{46} -0.236068 q^{47} +1.00000 q^{49} +2.47214 q^{50} +5.61803 q^{52} +12.1803 q^{53} -2.23607 q^{56} -3.09017 q^{58} +11.1803 q^{59} -2.00000 q^{61} +3.23607 q^{62} -0.236068 q^{64} +3.47214 q^{65} -9.76393 q^{67} -8.47214 q^{68} -0.618034 q^{70} +2.47214 q^{71} -4.52786 q^{73} +4.32624 q^{74} -10.8541 q^{76} +14.4721 q^{79} -1.85410 q^{80} +1.52786 q^{82} +6.76393 q^{83} -5.23607 q^{85} +3.52786 q^{86} +4.47214 q^{89} +3.47214 q^{91} +9.23607 q^{92} +0.145898 q^{94} -6.70820 q^{95} +9.70820 q^{97} -0.618034 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} - 2 q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q + q^2 - q^4 - 2 * q^5 - 2 * q^7	\( 2 q + q^{2} - q^{4} - 2 q^{5} - 2 q^{7} - q^{10} + 2 q^{13} - q^{14} - 3 q^{16} + 6 q^{17} + q^{20} + 2 q^{23} - 8 q^{25} + 11 q^{26} + q^{28} + 10 q^{29} - 6 q^{31} - 9 q^{32} - 2 q^{34} + 2 q^{35} - 14 q^{37} - 15 q^{38} + 4 q^{41} + 2 q^{43} + 16 q^{46} + 4 q^{47} + 2 q^{49} - 4 q^{50} + 9 q^{52} + 2 q^{53} + 5 q^{58} - 4 q^{61} + 2 q^{62} + 4 q^{64} - 2 q^{65} - 24 q^{67} - 8 q^{68} + q^{70} - 4 q^{71} - 18 q^{73} - 7 q^{74} - 15 q^{76} + 20 q^{79} + 3 q^{80} + 12 q^{82} + 18 q^{83} - 6 q^{85} + 16 q^{86} - 2 q^{91} + 14 q^{92} + 7 q^{94} + 6 q^{97} + q^{98}+O(q^{100}) \) 2 * q + q^2 - q^4 - 2 * q^5 - 2 * q^7 - q^10 + 2 * q^13 - q^14 - 3 * q^16 + 6 * q^17 + q^20 + 2 * q^23 - 8 * q^25 + 11 * q^26 + q^28 + 10 * q^29 - 6 * q^31 - 9 * q^32 - 2 * q^34 + 2 * q^35 - 14 * q^37 - 15 * q^38 + 4 * q^41 + 2 * q^43 + 16 * q^46 + 4 * q^47 + 2 * q^49 - 4 * q^50 + 9 * q^52 + 2 * q^53 + 5 * q^58 - 4 * q^61 + 2 * q^62 + 4 * q^64 - 2 * q^65 - 24 * q^67 - 8 * q^68 + q^70 - 4 * q^71 - 18 * q^73 - 7 * q^74 - 15 * q^76 + 20 * q^79 + 3 * q^80 + 12 * q^82 + 18 * q^83 - 6 * q^85 + 16 * q^86 - 2 * q^91 + 14 * q^92 + 7 * q^94 + 6 * q^97 + q^98

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