LMFDB - Embedded newform 7623.2.a.bj.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:	$\Z[a_1, \ldots, a_{5}]$
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			$1.61803$ 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)

$f(q)$	$=$	$q-2.23607 q^{2} +3.00000 q^{4} -1.61803 q^{5} -1.00000 q^{7} -2.23607 q^{8} +O(q^{10})$	$q-2.23607 q^{2} +3.00000 q^{4} -1.61803 q^{5} -1.00000 q^{7} -2.23607 q^{8} +3.61803 q^{10} +1.23607 q^{13} +2.23607 q^{14} -1.00000 q^{16} -1.85410 q^{17} +3.85410 q^{19} -4.85410 q^{20} -6.61803 q^{23} -2.38197 q^{25} -2.76393 q^{26} -3.00000 q^{28} -6.00000 q^{29} +8.09017 q^{31} +6.70820 q^{32} +4.14590 q^{34} +1.61803 q^{35} +2.38197 q^{37} -8.61803 q^{38} +3.61803 q^{40} +9.61803 q^{41} +3.70820 q^{43} +14.7984 q^{46} +4.47214 q^{47} +1.00000 q^{49} +5.32624 q^{50} +3.70820 q^{52} -11.2361 q^{53} +2.23607 q^{56} +13.4164 q^{58} +1.23607 q^{59} -18.0902 q^{62} -13.0000 q^{64} -2.00000 q^{65} -5.56231 q^{68} -3.61803 q^{70} -12.9443 q^{71} -0.291796 q^{73} -5.32624 q^{74} +11.5623 q^{76} +10.0000 q^{79} +1.61803 q^{80} -21.5066 q^{82} +7.52786 q^{83} +3.00000 q^{85} -8.29180 q^{86} -3.38197 q^{89} -1.23607 q^{91} -19.8541 q^{92} -10.0000 q^{94} -6.23607 q^{95} -6.00000 q^{97} -2.23607 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{4} - q^{5} - 2 q^{7}+O(q^{10}) $ 2 * q + 6 * q^4 - q^5 - 2 * q^7	$ 2 q + 6 q^{4} - q^{5} - 2 q^{7} + 5 q^{10} - 2 q^{13} - 2 q^{16} + 3 q^{17} + q^{19} - 3 q^{20} - 11 q^{23} - 7 q^{25} - 10 q^{26} - 6 q^{28} - 12 q^{29} + 5 q^{31} + 15 q^{34} + q^{35} + 7 q^{37} - 15 q^{38} + 5 q^{40} + 17 q^{41} - 6 q^{43} + 5 q^{46} + 2 q^{49} - 5 q^{50} - 6 q^{52} - 18 q^{53} - 2 q^{59} - 25 q^{62} - 26 q^{64} - 4 q^{65} + 9 q^{68} - 5 q^{70} - 8 q^{71} - 14 q^{73} + 5 q^{74} + 3 q^{76} + 20 q^{79} + q^{80} - 5 q^{82} + 24 q^{83} + 6 q^{85} - 30 q^{86} - 9 q^{89} + 2 q^{91} - 33 q^{92} - 20 q^{94} - 8 q^{95} - 12 q^{97}+O(q^{100}) $ 2 * q + 6 * q^4 - q^5 - 2 * q^7 + 5 * q^10 - 2 * q^13 - 2 * q^16 + 3 * q^17 + q^19 - 3 * q^20 - 11 * q^23 - 7 * q^25 - 10 * q^26 - 6 * q^28 - 12 * q^29 + 5 * q^31 + 15 * q^34 + q^35 + 7 * q^37 - 15 * q^38 + 5 * q^40 + 17 * q^41 - 6 * q^43 + 5 * q^46 + 2 * q^49 - 5 * q^50 - 6 * q^52 - 18 * q^53 - 2 * q^59 - 25 * q^62 - 26 * q^64 - 4 * q^65 + 9 * q^68 - 5 * q^70 - 8 * q^71 - 14 * q^73 + 5 * q^74 + 3 * q^76 + 20 * q^79 + q^80 - 5 * q^82 + 24 * q^83 + 6 * q^85 - 30 * q^86 - 9 * q^89 + 2 * q^91 - 33 * q^92 - 20 * q^94 - 8 * q^95 - 12 * q^97

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 $

$2$	−2.23607	−1.58114	−0.790569	−	0.612372i	$-0.790215\pi$
$2$	−2.23607	−1.58114	−0.790569	+	0.612372i	$0.790215\pi$
$3$	0	0
$3$	0	0
$4$	3.00000	1.50000
$5$	−1.61803	−0.723607	−0.361803	−	0.932254i	$-0.617839\pi$
$5$	−1.61803	−0.723607	−0.361803	+	0.932254i	$0.617839\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.23607	−0.790569
$9$	0	0
$10$	3.61803	1.14412
$11$	0	0
$11$	0	0
$12$	0	0
$13$	1.23607	0.342824	0.171412	−	0.985199i	$-0.445167\pi$
$13$	1.23607	0.342824	0.171412	+	0.985199i	$0.445167\pi$
$14$	2.23607	0.597614
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−1.85410	−0.449686	−0.224843	−	0.974395i	$-0.572187\pi$
$17$	−1.85410	−0.449686	−0.224843	+	0.974395i	$0.572187\pi$
$18$	0	0
$19$	3.85410	0.884192	0.442096	−	0.896968i	$-0.354235\pi$
$19$	3.85410	0.884192	0.442096	+	0.896968i	$0.354235\pi$
$20$	−4.85410	−1.08541
$21$	0	0
$22$	0	0
$23$	−6.61803	−1.37996	−0.689978	−	0.723831i	$-0.742380\pi$
$23$	−6.61803	−1.37996	−0.689978	+	0.723831i	$0.742380\pi$
$24$	0	0
$25$	−2.38197	−0.476393
$26$	−2.76393	−0.542052
$27$	0	0
$28$	−3.00000	−0.566947
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	8.09017	1.45304	0.726519	−	0.687147i	$-0.241137\pi$
$31$	8.09017	1.45304	0.726519	+	0.687147i	$0.241137\pi$
$32$	6.70820	1.18585
$33$	0	0
$34$	4.14590	0.711016
$35$	1.61803	0.273498
$36$	0	0
$37$	2.38197	0.391593	0.195796	−	0.980645i	$-0.437271\pi$
$37$	2.38197	0.391593	0.195796	+	0.980645i	$0.437271\pi$
$38$	−8.61803	−1.39803
$39$	0	0
$40$	3.61803	0.572061
$41$	9.61803	1.50208	0.751042	−	0.660254i	$-0.229551\pi$
$41$	9.61803	1.50208	0.751042	+	0.660254i	$0.229551\pi$
$42$	0	0
$43$	3.70820	0.565496	0.282748	−	0.959194i	$-0.408754\pi$
$43$	3.70820	0.565496	0.282748	+	0.959194i	$0.408754\pi$
$44$	0	0
$45$	0	0
$46$	14.7984	2.18190
$47$	4.47214	0.652328	0.326164	−	0.945313i	$-0.394244\pi$
$47$	4.47214	0.652328	0.326164	+	0.945313i	$0.394244\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	5.32624	0.753244
$51$	0	0
$52$	3.70820	0.514235
$53$	−11.2361	−1.54339	−0.771696	−	0.635991i	$-0.780591\pi$
$53$	−11.2361	−1.54339	−0.771696	+	0.635991i	$0.780591\pi$
$54$	0	0
$55$	0	0
$56$	2.23607	0.298807
$57$	0	0
$58$	13.4164	1.76166
$59$	1.23607	0.160922	0.0804612	−	0.996758i	$-0.474361\pi$
$59$	1.23607	0.160922	0.0804612	+	0.996758i	$0.474361\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	−18.0902	−2.29745
$63$	0	0
$64$	−13.0000	−1.62500
$65$	−2.00000	−0.248069
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	−5.56231	−0.674529
$69$	0	0
$70$	−3.61803	−0.432438
$71$	−12.9443	−1.53620	−0.768101	−	0.640328i	$-0.778798\pi$
$71$	−12.9443	−1.53620	−0.768101	+	0.640328i	$0.778798\pi$
$72$	0	0
$73$	−0.291796	−0.0341521	−0.0170761	−	0.999854i	$-0.505436\pi$
$73$	−0.291796	−0.0341521	−0.0170761	+	0.999854i	$0.505436\pi$
$74$	−5.32624	−0.619163
$75$	0	0
$76$	11.5623	1.32629
$77$	0	0
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	1.61803	0.180902
$81$	0	0
$82$	−21.5066	−2.37500
$83$	7.52786	0.826290	0.413145	−	0.910665i	$-0.364430\pi$
$83$	7.52786	0.826290	0.413145	+	0.910665i	$0.364430\pi$
$84$	0	0
$85$	3.00000	0.325396
$86$	−8.29180	−0.894127
$87$	0	0
$88$	0	0
$89$	−3.38197	−0.358488	−0.179244	−	0.983805i	$-0.557365\pi$
$89$	−3.38197	−0.358488	−0.179244	+	0.983805i	$0.557365\pi$
$90$	0	0
$91$	−1.23607	−0.129575
$92$	−19.8541	−2.06993
$93$	0	0
$94$	−10.0000	−1.03142
$95$	−6.23607	−0.639807
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	−2.23607	−0.225877
$99$	0	0
$100$	−7.14590	−0.714590
$101$	17.7984	1.77100	0.885502	−	0.464635i	$-0.153815\pi$
$101$	17.7984	1.77100	0.885502	+	0.464635i	$0.153815\pi$
$102$	0	0
$103$	−14.3262	−1.41161	−0.705803	−	0.708408i	$-0.749414\pi$
$103$	−14.3262	−1.41161	−0.705803	+	0.708408i	$0.749414\pi$
$104$	−2.76393	−0.271026
$105$	0	0
$106$	25.1246	2.44032
$107$	−8.79837	−0.850571	−0.425285	−	0.905059i	$-0.639826\pi$
$107$	−8.79837	−0.850571	−0.425285	+	0.905059i	$0.639826\pi$
$108$	0	0
$109$	−2.56231	−0.245424	−0.122712	−	0.992442i	$-0.539159\pi$
$109$	−2.56231	−0.245424	−0.122712	+	0.992442i	$0.539159\pi$
$110$	0	0
$111$	0	0
$112$	1.00000	0.0944911
$113$	0.763932	0.0718647	0.0359323	−	0.999354i	$-0.488560\pi$
$113$	0.763932	0.0718647	0.0359323	+	0.999354i	$0.488560\pi$
$114$	0	0
$115$	10.7082	0.998545
$116$	−18.0000	−1.67126
$117$	0	0
$118$	−2.76393	−0.254441
$119$	1.85410	0.169965
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0	0
$124$	24.2705	2.17956
$125$	11.9443	1.06833
$126$	0	0
$127$	−13.7082	−1.21641	−0.608203	−	0.793781i	$-0.708109\pi$
$127$	−13.7082	−1.21641	−0.608203	+	0.793781i	$0.708109\pi$
$128$	15.6525	1.38350
$129$	0	0
$130$	4.47214	0.392232
$131$	14.6525	1.28019	0.640096	−	0.768295i	$-0.278894\pi$
$131$	14.6525	1.28019	0.640096	+	0.768295i	$0.278894\pi$
$132$	0	0
$133$	−3.85410	−0.334193
$134$	0	0
$135$	0	0
$136$	4.14590	0.355508
$137$	14.1803	1.21151	0.605754	−	0.795652i	$-0.292872\pi$
$137$	14.1803	1.21151	0.605754	+	0.795652i	$0.292872\pi$
$138$	0	0
$139$	−19.8541	−1.68400	−0.842001	−	0.539475i	$-0.818623\pi$
$139$	−19.8541	−1.68400	−0.842001	+	0.539475i	$0.818623\pi$
$140$	4.85410	0.410246
$141$	0	0
$142$	28.9443	2.42895
$143$	0	0
$144$	0	0
$145$	9.70820	0.806222
$146$	0.652476	0.0539993
$147$	0	0
$148$	7.14590	0.587389
$149$	12.9443	1.06044	0.530218	−	0.847861i	$-0.322110\pi$
$149$	12.9443	1.06044	0.530218	+	0.847861i	$0.322110\pi$
$150$	0	0
$151$	24.4721	1.99151	0.995757	−	0.0920207i	$-0.0293326\pi$
$151$	24.4721	1.99151	0.995757	+	0.0920207i	$0.0293326\pi$
$152$	−8.61803	−0.699015
$153$	0	0
$154$	0	0
$155$	−13.0902	−1.05143
$156$	0	0
$157$	17.1246	1.36669	0.683346	−	0.730094i	$-0.260524\pi$
$157$	17.1246	1.36669	0.683346	+	0.730094i	$0.260524\pi$
$158$	−22.3607	−1.77892
$159$	0	0
$160$	−10.8541	−0.858092
$161$	6.61803	0.521574
$162$	0	0
$163$	−10.0000	−0.783260	−0.391630	−	0.920123i	$-0.628089\pi$
$163$	−10.0000	−0.783260	−0.391630	+	0.920123i	$0.628089\pi$
$164$	28.8541	2.25313
$165$	0	0
$166$	−16.8328	−1.30648
$167$	−16.7639	−1.29723	−0.648616	−	0.761116i	$-0.724652\pi$
$167$	−16.7639	−1.29723	−0.648616	+	0.761116i	$0.724652\pi$
$168$	0	0
$169$	−11.4721	−0.882472
$170$	−6.70820	−0.514496
$171$	0	0
$172$	11.1246	0.848244
$173$	−11.3820	−0.865355	−0.432677	−	0.901549i	$-0.642431\pi$
$173$	−11.3820	−0.865355	−0.432677	+	0.901549i	$0.642431\pi$
$174$	0	0
$175$	2.38197	0.180060
$176$	0	0
$177$	0	0
$178$	7.56231	0.566819
$179$	−9.79837	−0.732365	−0.366182	−	0.930543i	$-0.619335\pi$
$179$	−9.79837	−0.732365	−0.366182	+	0.930543i	$0.619335\pi$
$180$	0	0
$181$	15.8885	1.18099	0.590493	−	0.807043i	$-0.298933\pi$
$181$	15.8885	1.18099	0.590493	+	0.807043i	$0.298933\pi$
$182$	2.76393	0.204876
$183$	0	0
$184$	14.7984	1.09095
$185$	−3.85410	−0.283359
$186$	0	0
$187$	0	0
$188$	13.4164	0.978492
$189$	0	0
$190$	13.9443	1.01162
$191$	−16.0902	−1.16424	−0.582122	−	0.813102i	$-0.697777\pi$
$191$	−16.0902	−1.16424	−0.582122	+	0.813102i	$0.697777\pi$
$192$	0	0
$193$	15.1459	1.09023	0.545113	−	0.838363i	$-0.316487\pi$
$193$	15.1459	1.09023	0.545113	+	0.838363i	$0.316487\pi$
$194$	13.4164	0.963242
$195$	0	0
$196$	3.00000	0.214286
$197$	13.5279	0.963820	0.481910	−	0.876221i	$-0.339943\pi$
$197$	13.5279	0.963820	0.481910	+	0.876221i	$0.339943\pi$
$198$	0	0
$199$	19.8541	1.40742	0.703710	−	0.710487i	$-0.251525\pi$
$199$	19.8541	1.40742	0.703710	+	0.710487i	$0.251525\pi$
$200$	5.32624	0.376622
$201$	0	0
$202$	−39.7984	−2.80020
$203$	6.00000	0.421117
$204$	0	0
$205$	−15.5623	−1.08692
$206$	32.0344	2.23195
$207$	0	0
$208$	−1.23607	−0.0857059
$209$	0	0
$210$	0	0
$211$	−26.3607	−1.81474	−0.907372	−	0.420328i	$-0.861915\pi$
$211$	−26.3607	−1.81474	−0.907372	+	0.420328i	$0.861915\pi$
$212$	−33.7082	−2.31509
$213$	0	0
$214$	19.6738	1.34487
$215$	−6.00000	−0.409197
$216$	0	0
$217$	−8.09017	−0.549197
$218$	5.72949	0.388050
$219$	0	0
$220$	0	0
$221$	−2.29180	−0.154163
$222$	0	0
$223$	17.3820	1.16398	0.581991	−	0.813195i	$-0.302274\pi$
$223$	17.3820	1.16398	0.581991	+	0.813195i	$0.302274\pi$
$224$	−6.70820	−0.448211
$225$	0	0
$226$	−1.70820	−0.113628
$227$	3.52786	0.234153	0.117076	−	0.993123i	$-0.462648\pi$
$227$	3.52786	0.234153	0.117076	+	0.993123i	$0.462648\pi$
$228$	0	0
$229$	−6.18034	−0.408408	−0.204204	−	0.978928i	$-0.565461\pi$
$229$	−6.18034	−0.408408	−0.204204	+	0.978928i	$0.565461\pi$
$230$	−23.9443	−1.57884
$231$	0	0
$232$	13.4164	0.880830
$233$	−2.29180	−0.150141	−0.0750703	−	0.997178i	$-0.523918\pi$
$233$	−2.29180	−0.150141	−0.0750703	+	0.997178i	$0.523918\pi$
$234$	0	0
$235$	−7.23607	−0.472029
$236$	3.70820	0.241384
$237$	0	0
$238$	−4.14590	−0.268739
$239$	10.0344	0.649074	0.324537	−	0.945873i	$-0.394792\pi$
$239$	10.0344	0.649074	0.324537	+	0.945873i	$0.394792\pi$
$240$	0	0
$241$	−12.0000	−0.772988	−0.386494	−	0.922292i	$-0.626314\pi$
$241$	−12.0000	−0.772988	−0.386494	+	0.922292i	$0.626314\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.61803	−0.103372
$246$	0	0
$247$	4.76393	0.303122
$248$	−18.0902	−1.14873
$249$	0	0
$250$	−26.7082	−1.68918
$251$	−14.2918	−0.902090	−0.451045	−	0.892501i	$-0.648949\pi$
$251$	−14.2918	−0.902090	−0.451045	+	0.892501i	$0.648949\pi$
$252$	0	0
$253$	0	0
$254$	30.6525	1.92331
$255$	0	0
$256$	−9.00000	−0.562500
$257$	31.4508	1.96185	0.980925	−	0.194386i	$-0.0622715\pi$
$257$	31.4508	1.96185	0.980925	+	0.194386i	$0.0622715\pi$
$258$	0	0
$259$	−2.38197	−0.148008
$260$	−6.00000	−0.372104
$261$	0	0
$262$	−32.7639	−2.02416
$263$	4.14590	0.255647	0.127824	−	0.991797i	$-0.459201\pi$
$263$	4.14590	0.255647	0.127824	+	0.991797i	$0.459201\pi$
$264$	0	0
$265$	18.1803	1.11681
$266$	8.61803	0.528406
$267$	0	0
$268$	0	0
$269$	−13.4164	−0.818013	−0.409006	−	0.912532i	$-0.634125\pi$
$269$	−13.4164	−0.818013	−0.409006	+	0.912532i	$0.634125\pi$
$270$	0	0
$271$	−15.5066	−0.941958	−0.470979	−	0.882145i	$-0.656099\pi$
$271$	−15.5066	−0.941958	−0.470979	+	0.882145i	$0.656099\pi$
$272$	1.85410	0.112421
$273$	0	0
$274$	−31.7082	−1.91556
$275$	0	0
$276$	0	0
$277$	−11.0902	−0.666344	−0.333172	−	0.942866i	$-0.608119\pi$
$277$	−11.0902	−0.666344	−0.333172	+	0.942866i	$0.608119\pi$
$278$	44.3951	2.66264
$279$	0	0
$280$	−3.61803	−0.216219
$281$	−9.05573	−0.540219	−0.270110	−	0.962830i	$-0.587060\pi$
$281$	−9.05573	−0.540219	−0.270110	+	0.962830i	$0.587060\pi$
$282$	0	0
$283$	−16.2705	−0.967181	−0.483591	−	0.875294i	$-0.660668\pi$
$283$	−16.2705	−0.967181	−0.483591	+	0.875294i	$0.660668\pi$
$284$	−38.8328	−2.30430
$285$	0	0
$286$	0	0
$287$	−9.61803	−0.567735
$288$	0	0
$289$	−13.5623	−0.797783
$290$	−21.7082	−1.27475
$291$	0	0
$292$	−0.875388	−0.0512282
$293$	33.2148	1.94043	0.970214	−	0.242249i	$-0.0778851\pi$
$293$	33.2148	1.94043	0.970214	+	0.242249i	$0.0778851\pi$
$294$	0	0
$295$	−2.00000	−0.116445
$296$	−5.32624	−0.309581
$297$	0	0
$298$	−28.9443	−1.67670
$299$	−8.18034	−0.473081
$300$	0	0
$301$	−3.70820	−0.213737
$302$	−54.7214	−3.14886
$303$	0	0
$304$	−3.85410	−0.221048
$305$	0	0
$306$	0	0
$307$	−2.56231	−0.146239	−0.0731193	−	0.997323i	$-0.523295\pi$
$307$	−2.56231	−0.146239	−0.0731193	+	0.997323i	$0.523295\pi$
$308$	0	0
$309$	0	0
$310$	29.2705	1.66245
$311$	10.6525	0.604046	0.302023	−	0.953301i	$-0.402338\pi$
$311$	10.6525	0.604046	0.302023	+	0.953301i	$0.402338\pi$
$312$	0	0
$313$	−30.1803	−1.70589	−0.852947	−	0.521998i	$-0.825187\pi$
$313$	−30.1803	−1.70589	−0.852947	+	0.521998i	$0.825187\pi$
$314$	−38.2918	−2.16093
$315$	0	0
$316$	30.0000	1.68763
$317$	−11.2361	−0.631080	−0.315540	−	0.948912i	$-0.602186\pi$
$317$	−11.2361	−0.631080	−0.315540	+	0.948912i	$0.602186\pi$
$318$	0	0
$319$	0	0
$320$	21.0344	1.17586
$321$	0	0
$322$	−14.7984	−0.824681
$323$	−7.14590	−0.397608
$324$	0	0
$325$	−2.94427	−0.163319
$326$	22.3607	1.23844
$327$	0	0
$328$	−21.5066	−1.18750
$329$	−4.47214	−0.246557
$330$	0	0
$331$	−9.41641	−0.517573	−0.258786	−	0.965935i	$-0.583323\pi$
$331$	−9.41641	−0.517573	−0.258786	+	0.965935i	$0.583323\pi$
$332$	22.5836	1.23944
$333$	0	0
$334$	37.4853	2.05110
$335$	0	0
$336$	0	0
$337$	−15.9098	−0.866664	−0.433332	−	0.901234i	$-0.642662\pi$
$337$	−15.9098	−0.866664	−0.433332	+	0.901234i	$0.642662\pi$
$338$	25.6525	1.39531
$339$	0	0
$340$	9.00000	0.488094
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−8.29180	−0.447064
$345$	0	0
$346$	25.4508	1.36825
$347$	23.5066	1.26190	0.630950	−	0.775824i	$-0.282665\pi$
$347$	23.5066	1.26190	0.630950	+	0.775824i	$0.282665\pi$
$348$	0	0
$349$	8.29180	0.443850	0.221925	−	0.975064i	$-0.428766\pi$
$349$	8.29180	0.443850	0.221925	+	0.975064i	$0.428766\pi$
$350$	−5.32624	−0.284699
$351$	0	0
$352$	0	0
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	20.9443	1.11161
$356$	−10.1459	−0.537732
$357$	0	0
$358$	21.9098	1.15797
$359$	26.5623	1.40190	0.700952	−	0.713208i	$-0.252759\pi$
$359$	26.5623	1.40190	0.700952	+	0.713208i	$0.252759\pi$
$360$	0	0
$361$	−4.14590	−0.218205
$362$	−35.5279	−1.86730
$363$	0	0
$364$	−3.70820	−0.194363
$365$	0.472136	0.0247127
$366$	0	0
$367$	36.2705	1.89331	0.946653	−	0.322256i	$-0.104441\pi$
$367$	36.2705	1.89331	0.946653	+	0.322256i	$0.104441\pi$
$368$	6.61803	0.344989
$369$	0	0
$370$	8.61803	0.448030
$371$	11.2361	0.583348
$372$	0	0
$373$	−0.673762	−0.0348861	−0.0174430	−	0.999848i	$-0.505553\pi$
$373$	−0.673762	−0.0348861	−0.0174430	+	0.999848i	$0.505553\pi$
$374$	0	0
$375$	0	0
$376$	−10.0000	−0.515711
$377$	−7.41641	−0.381964
$378$	0	0
$379$	30.4721	1.56525	0.782624	−	0.622494i	$-0.213881\pi$
$379$	30.4721	1.56525	0.782624	+	0.622494i	$0.213881\pi$
$380$	−18.7082	−0.959711
$381$	0	0
$382$	35.9787	1.84083
$383$	−32.8328	−1.67768	−0.838839	−	0.544379i	$-0.816765\pi$
$383$	−32.8328	−1.67768	−0.838839	+	0.544379i	$0.816765\pi$
$384$	0	0
$385$	0	0
$386$	−33.8673	−1.72380
$387$	0	0
$388$	−18.0000	−0.913812
$389$	−10.7639	−0.545753	−0.272877	−	0.962049i	$-0.587975\pi$
$389$	−10.7639	−0.545753	−0.272877	+	0.962049i	$0.587975\pi$
$390$	0	0
$391$	12.2705	0.620546
$392$	−2.23607	−0.112938
$393$	0	0
$394$	−30.2492	−1.52393
$395$	−16.1803	−0.814121
$396$	0	0
$397$	−36.5410	−1.83394	−0.916971	−	0.398955i	$-0.869373\pi$
$397$	−36.5410	−1.83394	−0.916971	+	0.398955i	$0.869373\pi$
$398$	−44.3951	−2.22533
$399$	0	0
$400$	2.38197	0.119098
$401$	19.4164	0.969609	0.484805	−	0.874623i	$-0.338891\pi$
$401$	19.4164	0.969609	0.484805	+	0.874623i	$0.338891\pi$
$402$	0	0
$403$	10.0000	0.498135
$404$	53.3951	2.65651
$405$	0	0
$406$	−13.4164	−0.665845
$407$	0	0
$408$	0	0
$409$	−0.180340	−0.00891723	−0.00445862	−	0.999990i	$-0.501419\pi$
$409$	−0.180340	−0.00891723	−0.00445862	+	0.999990i	$0.501419\pi$
$410$	34.7984	1.71857
$411$	0	0
$412$	−42.9787	−2.11741
$413$	−1.23607	−0.0608229
$414$	0	0
$415$	−12.1803	−0.597909
$416$	8.29180	0.406539
$417$	0	0
$418$	0	0
$419$	−1.23607	−0.0603859	−0.0301929	−	0.999544i	$-0.509612\pi$
$419$	−1.23607	−0.0603859	−0.0301929	+	0.999544i	$0.509612\pi$
$420$	0	0
$421$	−14.7984	−0.721229	−0.360614	−	0.932715i	$-0.617433\pi$
$421$	−14.7984	−0.721229	−0.360614	+	0.932715i	$0.617433\pi$
$422$	58.9443	2.86936
$423$	0	0
$424$	25.1246	1.22016
$425$	4.41641	0.214227
$426$	0	0
$427$	0	0
$428$	−26.3951	−1.27586
$429$	0	0
$430$	13.4164	0.646997
$431$	3.90983	0.188330	0.0941649	−	0.995557i	$-0.469982\pi$
$431$	3.90983	0.188330	0.0941649	+	0.995557i	$0.469982\pi$
$432$	0	0
$433$	−26.7639	−1.28619	−0.643096	−	0.765785i	$-0.722350\pi$
$433$	−26.7639	−1.28619	−0.643096	+	0.765785i	$0.722350\pi$
$434$	18.0902	0.868356
$435$	0	0
$436$	−7.68692	−0.368137
$437$	−25.5066	−1.22015
$438$	0	0
$439$	4.85410	0.231674	0.115837	−	0.993268i	$-0.463045\pi$
$439$	4.85410	0.231674	0.115837	+	0.993268i	$0.463045\pi$
$440$	0	0
$441$	0	0
$442$	5.12461	0.243753
$443$	−22.6180	−1.07462	−0.537308	−	0.843386i	$-0.680559\pi$
$443$	−22.6180	−1.07462	−0.537308	+	0.843386i	$0.680559\pi$
$444$	0	0
$445$	5.47214	0.259404
$446$	−38.8673	−1.84042
$447$	0	0
$448$	13.0000	0.614192
$449$	6.11146	0.288417	0.144209	−	0.989547i	$-0.453936\pi$
$449$	6.11146	0.288417	0.144209	+	0.989547i	$0.453936\pi$
$450$	0	0
$451$	0	0
$452$	2.29180	0.107797
$453$	0	0
$454$	−7.88854	−0.370228
$455$	2.00000	0.0937614
$456$	0	0
$457$	5.41641	0.253369	0.126684	−	0.991943i	$-0.459566\pi$
$457$	5.41641	0.253369	0.126684	+	0.991943i	$0.459566\pi$
$458$	13.8197	0.645750
$459$	0	0
$460$	32.1246	1.49782
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	0	0
$463$	3.41641	0.158774	0.0793870	−	0.996844i	$-0.474704\pi$
$463$	3.41641	0.158774	0.0793870	+	0.996844i	$0.474704\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	5.12461	0.237393
$467$	−3.81966	−0.176753	−0.0883764	−	0.996087i	$-0.528168\pi$
$467$	−3.81966	−0.176753	−0.0883764	+	0.996087i	$0.528168\pi$
$468$	0	0
$469$	0	0
$470$	16.1803	0.746343
$471$	0	0
$472$	−2.76393	−0.127220
$473$	0	0
$474$	0	0
$475$	−9.18034	−0.421223
$476$	5.56231	0.254948
$477$	0	0
$478$	−22.4377	−1.02628
$479$	16.9443	0.774204	0.387102	−	0.922037i	$-0.373476\pi$
$479$	16.9443	0.774204	0.387102	+	0.922037i	$0.373476\pi$
$480$	0	0
$481$	2.94427	0.134247
$482$	26.8328	1.22220
$483$	0	0
$484$	0	0
$485$	9.70820	0.440827
$486$	0	0
$487$	14.2918	0.647623	0.323812	−	0.946122i	$-0.395036\pi$
$487$	14.2918	0.647623	0.323812	+	0.946122i	$0.395036\pi$
$488$	0	0
$489$	0	0
$490$	3.61803	0.163446
$491$	−36.5066	−1.64752	−0.823759	−	0.566940i	$-0.808127\pi$
$491$	−36.5066	−1.64752	−0.823759	+	0.566940i	$0.808127\pi$
$492$	0	0
$493$	11.1246	0.501027
$494$	−10.6525	−0.479278
$495$	0	0
$496$	−8.09017	−0.363259
$497$	12.9443	0.580630
$498$	0	0
$499$	−19.1246	−0.856135	−0.428068	−	0.903747i	$-0.640805\pi$
$499$	−19.1246	−0.856135	−0.428068	+	0.903747i	$0.640805\pi$
$500$	35.8328	1.60249
$501$	0	0
$502$	31.9574	1.42633
$503$	−5.52786	−0.246475	−0.123238	−	0.992377i	$-0.539328\pi$
$503$	−5.52786	−0.246475	−0.123238	+	0.992377i	$0.539328\pi$
$504$	0	0
$505$	−28.7984	−1.28151
$506$	0	0
$507$	0	0
$508$	−41.1246	−1.82461
$509$	15.9098	0.705191	0.352595	−	0.935776i	$-0.385299\pi$
$509$	15.9098	0.705191	0.352595	+	0.935776i	$0.385299\pi$
$510$	0	0
$511$	0.291796	0.0129083
$512$	−11.1803	−0.494106
$513$	0	0
$514$	−70.3262	−3.10196
$515$	23.1803	1.02145
$516$	0	0
$517$	0	0
$518$	5.32624	0.234021
$519$	0	0
$520$	4.47214	0.196116
$521$	−3.32624	−0.145725	−0.0728626	−	0.997342i	$-0.523213\pi$
$521$	−3.32624	−0.145725	−0.0728626	+	0.997342i	$0.523213\pi$
$522$	0	0
$523$	−19.8541	−0.868159	−0.434080	−	0.900875i	$-0.642926\pi$
$523$	−19.8541	−0.868159	−0.434080	+	0.900875i	$0.642926\pi$
$524$	43.9574	1.92029
$525$	0	0
$526$	−9.27051	−0.404213
$527$	−15.0000	−0.653410
$528$	0	0
$529$	20.7984	0.904277
$530$	−40.6525	−1.76583
$531$	0	0
$532$	−11.5623	−0.501290
$533$	11.8885	0.514950
$534$	0	0
$535$	14.2361	0.615479
$536$	0	0
$537$	0	0
$538$	30.0000	1.29339
$539$	0	0
$540$	0	0
$541$	0.673762	0.0289673	0.0144836	−	0.999895i	$-0.495390\pi$
$541$	0.673762	0.0289673	0.0144836	+	0.999895i	$0.495390\pi$
$542$	34.6738	1.48937
$543$	0	0
$544$	−12.4377	−0.533262
$545$	4.14590	0.177591
$546$	0	0
$547$	27.5279	1.17701	0.588503	−	0.808495i	$-0.299717\pi$
$547$	27.5279	1.17701	0.588503	+	0.808495i	$0.299717\pi$
$548$	42.5410	1.81726
$549$	0	0
$550$	0	0
$551$	−23.1246	−0.985142
$552$	0	0
$553$	−10.0000	−0.425243
$554$	24.7984	1.05358
$555$	0	0
$556$	−59.5623	−2.52600
$557$	−22.3607	−0.947452	−0.473726	−	0.880672i	$-0.657091\pi$
$557$	−22.3607	−0.947452	−0.473726	+	0.880672i	$0.657091\pi$
$558$	0	0
$559$	4.58359	0.193865
$560$	−1.61803	−0.0683744
$561$	0	0
$562$	20.2492	0.854162
$563$	−18.4721	−0.778508	−0.389254	−	0.921131i	$-0.627267\pi$
$563$	−18.4721	−0.778508	−0.389254	+	0.921131i	$0.627267\pi$
$564$	0	0
$565$	−1.23607	−0.0520018
$566$	36.3820	1.52925
$567$	0	0
$568$	28.9443	1.21447
$569$	−37.5967	−1.57614	−0.788069	−	0.615587i	$-0.788919\pi$
$569$	−37.5967	−1.57614	−0.788069	+	0.615587i	$0.788919\pi$
$570$	0	0
$571$	−46.6525	−1.95235	−0.976173	−	0.216995i	$-0.930374\pi$
$571$	−46.6525	−1.95235	−0.976173	+	0.216995i	$0.930374\pi$
$572$	0	0
$573$	0	0
$574$	21.5066	0.897667
$575$	15.7639	0.657401
$576$	0	0
$577$	−22.2918	−0.928020	−0.464010	−	0.885830i	$-0.653590\pi$
$577$	−22.2918	−0.928020	−0.464010	+	0.885830i	$0.653590\pi$
$578$	30.3262	1.26141
$579$	0	0
$580$	29.1246	1.20933
$581$	−7.52786	−0.312308
$582$	0	0
$583$	0	0
$584$	0.652476	0.0269996
$585$	0	0
$586$	−74.2705	−3.06809
$587$	−25.4164	−1.04905	−0.524524	−	0.851396i	$-0.675757\pi$
$587$	−25.4164	−1.04905	−0.524524	+	0.851396i	$0.675757\pi$
$588$	0	0
$589$	31.1803	1.28476
$590$	4.47214	0.184115
$591$	0	0
$592$	−2.38197	−0.0978982
$593$	−15.5066	−0.636779	−0.318389	−	0.947960i	$-0.603142\pi$
$593$	−15.5066	−0.636779	−0.318389	+	0.947960i	$0.603142\pi$
$594$	0	0
$595$	−3.00000	−0.122988
$596$	38.8328	1.59065
$597$	0	0
$598$	18.2918	0.748007
$599$	21.3820	0.873643	0.436822	−	0.899548i	$-0.356104\pi$
$599$	21.3820	0.873643	0.436822	+	0.899548i	$0.356104\pi$
$600$	0	0
$601$	−12.4721	−0.508749	−0.254375	−	0.967106i	$-0.581870\pi$
$601$	−12.4721	−0.508749	−0.254375	+	0.967106i	$0.581870\pi$
$602$	8.29180	0.337948
$603$	0	0
$604$	73.4164	2.98727
$605$	0	0
$606$	0	0
$607$	−34.9787	−1.41974	−0.709871	−	0.704332i	$-0.751247\pi$
$607$	−34.9787	−1.41974	−0.709871	+	0.704332i	$0.751247\pi$
$608$	25.8541	1.04852
$609$	0	0
$610$	0	0
$611$	5.52786	0.223633
$612$	0	0
$613$	−24.9787	−1.00888	−0.504440	−	0.863447i	$-0.668301\pi$
$613$	−24.9787	−1.00888	−0.504440	+	0.863447i	$0.668301\pi$
$614$	5.72949	0.231223
$615$	0	0
$616$	0	0
$617$	−44.0689	−1.77415	−0.887073	−	0.461629i	$-0.847265\pi$
$617$	−44.0689	−1.77415	−0.887073	+	0.461629i	$0.847265\pi$
$618$	0	0
$619$	23.6869	0.952058	0.476029	−	0.879430i	$-0.342076\pi$
$619$	23.6869	0.952058	0.476029	+	0.879430i	$0.342076\pi$
$620$	−39.2705	−1.57714
$621$	0	0
$622$	−23.8197	−0.955081
$623$	3.38197	0.135496
$624$	0	0
$625$	−7.41641	−0.296656
$626$	67.4853	2.69725
$627$	0	0
$628$	51.3738	2.05004
$629$	−4.41641	−0.176094
$630$	0	0
$631$	17.3050	0.688899	0.344450	−	0.938805i	$-0.388066\pi$
$631$	17.3050	0.688899	0.344450	+	0.938805i	$0.388066\pi$
$632$	−22.3607	−0.889460
$633$	0	0
$634$	25.1246	0.997826
$635$	22.1803	0.880200
$636$	0	0
$637$	1.23607	0.0489748
$638$	0	0
$639$	0	0
$640$	−25.3262	−1.00111
$641$	−17.2361	−0.680784	−0.340392	−	0.940284i	$-0.610560\pi$
$641$	−17.2361	−0.680784	−0.340392	+	0.940284i	$0.610560\pi$
$642$	0	0
$643$	−23.2705	−0.917699	−0.458850	−	0.888514i	$-0.651738\pi$
$643$	−23.2705	−0.917699	−0.458850	+	0.888514i	$0.651738\pi$
$644$	19.8541	0.782361
$645$	0	0
$646$	15.9787	0.628674
$647$	−27.5279	−1.08223	−0.541116	−	0.840948i	$-0.681998\pi$
$647$	−27.5279	−1.08223	−0.541116	+	0.840948i	$0.681998\pi$
$648$	0	0
$649$	0	0
$650$	6.58359	0.258230
$651$	0	0
$652$	−30.0000	−1.17489
$653$	28.3607	1.10984	0.554920	−	0.831904i	$-0.312749\pi$
$653$	28.3607	1.10984	0.554920	+	0.831904i	$0.312749\pi$
$654$	0	0
$655$	−23.7082	−0.926356
$656$	−9.61803	−0.375521
$657$	0	0
$658$	10.0000	0.389841
$659$	−19.1459	−0.745818	−0.372909	−	0.927868i	$-0.621640\pi$
$659$	−19.1459	−0.745818	−0.372909	+	0.927868i	$0.621640\pi$
$660$	0	0
$661$	34.5410	1.34349	0.671745	−	0.740782i	$-0.265545\pi$
$661$	34.5410	1.34349	0.671745	+	0.740782i	$0.265545\pi$
$662$	21.0557	0.818354
$663$	0	0
$664$	−16.8328	−0.653240
$665$	6.23607	0.241824
$666$	0	0
$667$	39.7082	1.53751
$668$	−50.2918	−1.94585
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−8.83282	−0.340480	−0.170240	−	0.985403i	$-0.554454\pi$
$673$	−8.83282	−0.340480	−0.170240	+	0.985403i	$0.554454\pi$
$674$	35.5755	1.37032
$675$	0	0
$676$	−34.4164	−1.32371
$677$	−10.3607	−0.398193	−0.199097	−	0.979980i	$-0.563801\pi$
$677$	−10.3607	−0.398193	−0.199097	+	0.979980i	$0.563801\pi$
$678$	0	0
$679$	6.00000	0.230259
$680$	−6.70820	−0.257248
$681$	0	0
$682$	0	0
$683$	−2.20163	−0.0842429	−0.0421214	−	0.999112i	$-0.513412\pi$
$683$	−2.20163	−0.0842429	−0.0421214	+	0.999112i	$0.513412\pi$
$684$	0	0
$685$	−22.9443	−0.876656
$686$	2.23607	0.0853735
$687$	0	0
$688$	−3.70820	−0.141374
$689$	−13.8885	−0.529111
$690$	0	0
$691$	6.25735	0.238041	0.119020	−	0.992892i	$-0.462025\pi$
$691$	6.25735	0.238041	0.119020	+	0.992892i	$0.462025\pi$
$692$	−34.1459	−1.29803
$693$	0	0
$694$	−52.5623	−1.99524
$695$	32.1246	1.21856
$696$	0	0
$697$	−17.8328	−0.675466
$698$	−18.5410	−0.701788
$699$	0	0
$700$	7.14590	0.270090
$701$	−9.59675	−0.362464	−0.181232	−	0.983440i	$-0.558009\pi$
$701$	−9.59675	−0.362464	−0.181232	+	0.983440i	$0.558009\pi$
$702$	0	0
$703$	9.18034	0.346243
$704$	0	0
$705$	0	0
$706$	−40.2492	−1.51480
$707$	−17.7984	−0.669377
$708$	0	0
$709$	21.0344	0.789965	0.394983	−	0.918689i	$-0.370751\pi$
$709$	21.0344	0.789965	0.394983	+	0.918689i	$0.370751\pi$
$710$	−46.8328	−1.75760
$711$	0	0
$712$	7.56231	0.283409
$713$	−53.5410	−2.00513
$714$	0	0
$715$	0	0
$716$	−29.3951	−1.09855
$717$	0	0
$718$	−59.3951	−2.21661
$719$	37.0132	1.38036	0.690179	−	0.723639i	$-0.257532\pi$
$719$	37.0132	1.38036	0.690179	+	0.723639i	$0.257532\pi$
$720$	0	0
$721$	14.3262	0.533537
$722$	9.27051	0.345013
$723$	0	0
$724$	47.6656	1.77148
$725$	14.2918	0.530784
$726$	0	0
$727$	−12.1459	−0.450466	−0.225233	−	0.974305i	$-0.572314\pi$
$727$	−12.1459	−0.450466	−0.225233	+	0.974305i	$0.572314\pi$
$728$	2.76393	0.102438
$729$	0	0
$730$	−1.05573	−0.0390742
$731$	−6.87539	−0.254295
$732$	0	0
$733$	−40.0689	−1.47998	−0.739989	−	0.672619i	$-0.765169\pi$
$733$	−40.0689	−1.47998	−0.739989	+	0.672619i	$0.765169\pi$
$734$	−81.1033	−2.99358
$735$	0	0
$736$	−44.3951	−1.63643
$737$	0	0
$738$	0	0
$739$	−0.875388	−0.0322017	−0.0161008	−	0.999870i	$-0.505125\pi$
$739$	−0.875388	−0.0322017	−0.0161008	+	0.999870i	$0.505125\pi$
$740$	−11.5623	−0.425039
$741$	0	0
$742$	−25.1246	−0.922354
$743$	−0.673762	−0.0247179	−0.0123590	−	0.999924i	$-0.503934\pi$
$743$	−0.673762	−0.0247179	−0.0123590	+	0.999924i	$0.503934\pi$
$744$	0	0
$745$	−20.9443	−0.767339
$746$	1.50658	0.0551597
$747$	0	0
$748$	0	0
$749$	8.79837	0.321486
$750$	0	0
$751$	24.0000	0.875772	0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000	0.875772	0.437886	+	0.899030i	$0.355727\pi$
$752$	−4.47214	−0.163082
$753$	0	0
$754$	16.5836	0.603939
$755$	−39.5967	−1.44107
$756$	0	0
$757$	−9.56231	−0.347548	−0.173774	−	0.984786i	$-0.555596\pi$
$757$	−9.56231	−0.347548	−0.173774	+	0.984786i	$0.555596\pi$
$758$	−68.1378	−2.47488
$759$	0	0
$760$	13.9443	0.505812
$761$	−25.4164	−0.921344	−0.460672	−	0.887570i	$-0.652392\pi$
$761$	−25.4164	−0.921344	−0.460672	+	0.887570i	$0.652392\pi$
$762$	0	0
$763$	2.56231	0.0927617
$764$	−48.2705	−1.74637
$765$	0	0
$766$	73.4164	2.65264
$767$	1.52786	0.0551680
$768$	0	0
$769$	−18.9443	−0.683148	−0.341574	−	0.939855i	$-0.610960\pi$
$769$	−18.9443	−0.683148	−0.341574	+	0.939855i	$0.610960\pi$
$770$	0	0
$771$	0	0
$772$	45.4377	1.63534
$773$	0.472136	0.0169815	0.00849077	−	0.999964i	$-0.497297\pi$
$773$	0.472136	0.0169815	0.00849077	+	0.999964i	$0.497297\pi$
$774$	0	0
$775$	−19.2705	−0.692217
$776$	13.4164	0.481621
$777$	0	0
$778$	24.0689	0.862911
$779$	37.0689	1.32813
$780$	0	0
$781$	0	0
$782$	−27.4377	−0.981170
$783$	0	0
$784$	−1.00000	−0.0357143
$785$	−27.7082	−0.988948
$786$	0	0
$787$	19.1459	0.682478	0.341239	−	0.939977i	$-0.389154\pi$
$787$	19.1459	0.682478	0.341239	+	0.939977i	$0.389154\pi$
$788$	40.5836	1.44573
$789$	0	0
$790$	36.1803	1.28724
$791$	−0.763932	−0.0271623
$792$	0	0
$793$	0	0
$794$	81.7082	2.89972
$795$	0	0
$796$	59.5623	2.11113
$797$	−21.4377	−0.759362	−0.379681	−	0.925117i	$-0.623966\pi$
$797$	−21.4377	−0.759362	−0.379681	+	0.925117i	$0.623966\pi$
$798$	0	0
$799$	−8.29180	−0.293343
$800$	−15.9787	−0.564933
$801$	0	0
$802$	−43.4164	−1.53309
$803$	0	0
$804$	0	0
$805$	−10.7082	−0.377415
$806$	−22.3607	−0.787621
$807$	0	0
$808$	−39.7984	−1.40010
$809$	2.76393	0.0971747	0.0485873	−	0.998819i	$-0.484528\pi$
$809$	2.76393	0.0971747	0.0485873	+	0.998819i	$0.484528\pi$
$810$	0	0
$811$	−35.4164	−1.24364	−0.621819	−	0.783161i	$-0.713606\pi$
$811$	−35.4164	−1.24364	−0.621819	+	0.783161i	$0.713606\pi$
$812$	18.0000	0.631676
$813$	0	0
$814$	0	0
$815$	16.1803	0.566773
$816$	0	0
$817$	14.2918	0.500007
$818$	0.403252	0.0140994
$819$	0	0
$820$	−46.6869	−1.63038
$821$	6.94427	0.242357	0.121178	−	0.992631i	$-0.461333\pi$
$821$	6.94427	0.242357	0.121178	+	0.992631i	$0.461333\pi$
$822$	0	0
$823$	−48.8328	−1.70220	−0.851102	−	0.525000i	$-0.824065\pi$
$823$	−48.8328	−1.70220	−0.851102	+	0.525000i	$0.824065\pi$
$824$	32.0344	1.11597
$825$	0	0
$826$	2.76393	0.0961695
$827$	−52.8673	−1.83837	−0.919187	−	0.393821i	$-0.871153\pi$
$827$	−52.8673	−1.83837	−0.919187	+	0.393821i	$0.871153\pi$
$828$	0	0
$829$	50.2492	1.74523	0.872614	−	0.488411i	$-0.162423\pi$
$829$	50.2492	1.74523	0.872614	+	0.488411i	$0.162423\pi$
$830$	27.2361	0.945378
$831$	0	0
$832$	−16.0689	−0.557088
$833$	−1.85410	−0.0642408
$834$	0	0
$835$	27.1246	0.938686
$836$	0	0
$837$	0	0
$838$	2.76393	0.0954784
$839$	−40.2492	−1.38956	−0.694779	−	0.719224i	$-0.744498\pi$
$839$	−40.2492	−1.38956	−0.694779	+	0.719224i	$0.744498\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	33.0902	1.14036
$843$	0	0
$844$	−79.0820	−2.72212
$845$	18.5623	0.638563
$846$	0	0
$847$	0	0
$848$	11.2361	0.385848
$849$	0	0
$850$	−9.87539	−0.338723
$851$	−15.7639	−0.540381
$852$	0	0
$853$	28.2918	0.968693	0.484346	−	0.874876i	$-0.339057\pi$
$853$	28.2918	0.968693	0.484346	+	0.874876i	$0.339057\pi$
$854$	0	0
$855$	0	0
$856$	19.6738	0.672435
$857$	10.3607	0.353914	0.176957	−	0.984219i	$-0.443375\pi$
$857$	10.3607	0.353914	0.176957	+	0.984219i	$0.443375\pi$
$858$	0	0
$859$	12.0000	0.409435	0.204717	−	0.978821i	$-0.434372\pi$
$859$	12.0000	0.409435	0.204717	+	0.978821i	$0.434372\pi$
$860$	−18.0000	−0.613795
$861$	0	0
$862$	−8.74265	−0.297776
$863$	−3.32624	−0.113226	−0.0566132	−	0.998396i	$-0.518030\pi$
$863$	−3.32624	−0.113226	−0.0566132	+	0.998396i	$0.518030\pi$
$864$	0	0
$865$	18.4164	0.626177
$866$	59.8460	2.03365
$867$	0	0
$868$	−24.2705	−0.823795
$869$	0	0
$870$	0	0
$871$	0	0
$872$	5.72949	0.194025
$873$	0	0
$874$	57.0344	1.92922
$875$	−11.9443	−0.403790
$876$	0	0
$877$	−3.52786	−0.119128	−0.0595638	−	0.998225i	$-0.518971\pi$
$877$	−3.52786	−0.119128	−0.0595638	+	0.998225i	$0.518971\pi$
$878$	−10.8541	−0.366308
$879$	0	0
$880$	0	0
$881$	4.32624	0.145755	0.0728773	−	0.997341i	$-0.476782\pi$
$881$	4.32624	0.145755	0.0728773	+	0.997341i	$0.476782\pi$
$882$	0	0
$883$	−29.5967	−0.996010	−0.498005	−	0.867174i	$-0.665934\pi$
$883$	−29.5967	−0.996010	−0.498005	+	0.867174i	$0.665934\pi$
$884$	−6.87539	−0.231244
$885$	0	0
$886$	50.5755	1.69912
$887$	−21.5967	−0.725148	−0.362574	−	0.931955i	$-0.618102\pi$
$887$	−21.5967	−0.725148	−0.362574	+	0.931955i	$0.618102\pi$
$888$	0	0
$889$	13.7082	0.459758
$890$	−12.2361	−0.410154
$891$	0	0
$892$	52.1459	1.74597
$893$	17.2361	0.576783
$894$	0	0
$895$	15.8541	0.529944
$896$	−15.6525	−0.522913
$897$	0	0
$898$	−13.6656	−0.456028
$899$	−48.5410	−1.61893
$900$	0	0
$901$	20.8328	0.694042
$902$	0	0
$903$	0	0
$904$	−1.70820	−0.0568140
$905$	−25.7082	−0.854570
$906$	0	0
$907$	29.7082	0.986445	0.493222	−	0.869903i	$-0.335819\pi$
$907$	29.7082	0.986445	0.493222	+	0.869903i	$0.335819\pi$
$908$	10.5836	0.351229
$909$	0	0
$910$	−4.47214	−0.148250
$911$	−16.9443	−0.561389	−0.280694	−	0.959797i	$-0.590565\pi$
$911$	−16.9443	−0.561389	−0.280694	+	0.959797i	$0.590565\pi$
$912$	0	0
$913$	0	0
$914$	−12.1115	−0.400611
$915$	0	0
$916$	−18.5410	−0.612613
$917$	−14.6525	−0.483867
$918$	0	0
$919$	21.5279	0.710139	0.355069	−	0.934840i	$-0.384457\pi$
$919$	21.5279	0.710139	0.355069	+	0.934840i	$0.384457\pi$
$920$	−23.9443	−0.789419
$921$	0	0
$922$	13.4164	0.441846
$923$	−16.0000	−0.526646
$924$	0	0
$925$	−5.67376	−0.186552
$926$	−7.63932	−0.251044
$927$	0	0
$928$	−40.2492	−1.32125
$929$	−57.2705	−1.87898	−0.939492	−	0.342570i	$-0.888703\pi$
$929$	−57.2705	−1.87898	−0.939492	+	0.342570i	$0.888703\pi$
$930$	0	0
$931$	3.85410	0.126313
$932$	−6.87539	−0.225211
$933$	0	0
$934$	8.54102	0.279471
$935$	0	0
$936$	0	0
$937$	24.0689	0.786296	0.393148	−	0.919475i	$-0.371386\pi$
$937$	24.0689	0.786296	0.393148	+	0.919475i	$0.371386\pi$
$938$	0	0
$939$	0	0
$940$	−21.7082	−0.708044
$941$	−38.5066	−1.25528	−0.627639	−	0.778504i	$-0.715979\pi$
$941$	−38.5066	−1.25528	−0.627639	+	0.778504i	$0.715979\pi$
$942$	0	0
$943$	−63.6525	−2.07281
$944$	−1.23607	−0.0402306
$945$	0	0
$946$	0	0
$947$	−7.14590	−0.232210	−0.116105	−	0.993237i	$-0.537041\pi$
$947$	−7.14590	−0.232210	−0.116105	+	0.993237i	$0.537041\pi$
$948$	0	0
$949$	−0.360680	−0.0117082
$950$	20.5279	0.666012
$951$	0	0
$952$	−4.14590	−0.134369
$953$	−23.8885	−0.773826	−0.386913	−	0.922116i	$-0.626459\pi$
$953$	−23.8885	−0.773826	−0.386913	+	0.922116i	$0.626459\pi$
$954$	0	0
$955$	26.0344	0.842455
$956$	30.1033	0.973611
$957$	0	0
$958$	−37.8885	−1.22412
$959$	−14.1803	−0.457907
$960$	0	0
$961$	34.4508	1.11132
$962$	−6.58359	−0.212264
$963$	0	0
$964$	−36.0000	−1.15948
$965$	−24.5066	−0.788895
$966$	0	0
$967$	48.3607	1.55517	0.777587	−	0.628775i	$-0.216443\pi$
$967$	48.3607	1.55517	0.777587	+	0.628775i	$0.216443\pi$
$968$	0	0
$969$	0	0
$970$	−21.7082	−0.697008
$971$	12.5410	0.402460	0.201230	−	0.979544i	$-0.435506\pi$
$971$	12.5410	0.402460	0.201230	+	0.979544i	$0.435506\pi$
$972$	0	0
$973$	19.8541	0.636493
$974$	−31.9574	−1.02398
$975$	0	0
$976$	0	0
$977$	58.3607	1.86712	0.933562	−	0.358417i	$-0.116683\pi$
$977$	58.3607	1.86712	0.933562	+	0.358417i	$0.116683\pi$
$978$	0	0
$979$	0	0
$980$	−4.85410	−0.155059
$981$	0	0
$982$	81.6312	2.60496
$983$	−6.94427	−0.221488	−0.110744	−	0.993849i	$-0.535323\pi$
$983$	−6.94427	−0.221488	−0.110744	+	0.993849i	$0.535323\pi$
$984$	0	0
$985$	−21.8885	−0.697427
$986$	−24.8754	−0.792194
$987$	0	0
$988$	14.2918	0.454683
$989$	−24.5410	−0.780359
$990$	0	0
$991$	−28.7639	−0.913716	−0.456858	−	0.889540i	$-0.651025\pi$
$991$	−28.7639	−0.913716	−0.456858	+	0.889540i	$0.651025\pi$
$992$	54.2705	1.72309
$993$	0	0
$994$	−28.9443	−0.918057
$995$	−32.1246	−1.01842
$996$	0	0
$997$	−58.4296	−1.85048	−0.925241	−	0.379379i	$-0.876138\pi$
$997$	−58.4296	−1.85048	−0.925241	+	0.379379i	$0.876138\pi$
$998$	42.7639	1.35367
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.bj.1.1		2
3.2	odd	2		2541.2.a.v.1.2		2
11.2	odd	10		693.2.m.a.631.1		4
11.6	odd	10		693.2.m.a.190.1		4
11.10	odd	2		7623.2.a.bk.1.2		2
33.2	even	10		231.2.j.e.169.1	✓	4
33.17	even	10		231.2.j.e.190.1	yes	4
33.32	even	2		2541.2.a.w.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.j.e.169.1	✓	4	33.2	even	10
231.2.j.e.190.1	yes	4	33.17	even	10
693.2.m.a.190.1		4	11.6	odd	10
693.2.m.a.631.1		4	11.2	odd	10
2541.2.a.v.1.2		2	3.2	odd	2
2541.2.a.w.1.1		2	33.32	even	2
7623.2.a.bj.1.1		2	1.1	even	1	trivial
7623.2.a.bk.1.2		2	11.10	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:	\( N \)	\(=\)	\( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7623.a (trivial)

\(f(q)\)	\(=\)	\(q-2.23607 q^{2} +3.00000 q^{4} -1.61803 q^{5} -1.00000 q^{7} -2.23607 q^{8} +O(q^{10})\)	\(q-2.23607 q^{2} +3.00000 q^{4} -1.61803 q^{5} -1.00000 q^{7} -2.23607 q^{8} +3.61803 q^{10} +1.23607 q^{13} +2.23607 q^{14} -1.00000 q^{16} -1.85410 q^{17} +3.85410 q^{19} -4.85410 q^{20} -6.61803 q^{23} -2.38197 q^{25} -2.76393 q^{26} -3.00000 q^{28} -6.00000 q^{29} +8.09017 q^{31} +6.70820 q^{32} +4.14590 q^{34} +1.61803 q^{35} +2.38197 q^{37} -8.61803 q^{38} +3.61803 q^{40} +9.61803 q^{41} +3.70820 q^{43} +14.7984 q^{46} +4.47214 q^{47} +1.00000 q^{49} +5.32624 q^{50} +3.70820 q^{52} -11.2361 q^{53} +2.23607 q^{56} +13.4164 q^{58} +1.23607 q^{59} -18.0902 q^{62} -13.0000 q^{64} -2.00000 q^{65} -5.56231 q^{68} -3.61803 q^{70} -12.9443 q^{71} -0.291796 q^{73} -5.32624 q^{74} +11.5623 q^{76} +10.0000 q^{79} +1.61803 q^{80} -21.5066 q^{82} +7.52786 q^{83} +3.00000 q^{85} -8.29180 q^{86} -3.38197 q^{89} -1.23607 q^{91} -19.8541 q^{92} -10.0000 q^{94} -6.23607 q^{95} -6.00000 q^{97} -2.23607 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{4} - q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q + 6 * q^4 - q^5 - 2 * q^7	\( 2 q + 6 q^{4} - q^{5} - 2 q^{7} + 5 q^{10} - 2 q^{13} - 2 q^{16} + 3 q^{17} + q^{19} - 3 q^{20} - 11 q^{23} - 7 q^{25} - 10 q^{26} - 6 q^{28} - 12 q^{29} + 5 q^{31} + 15 q^{34} + q^{35} + 7 q^{37} - 15 q^{38} + 5 q^{40} + 17 q^{41} - 6 q^{43} + 5 q^{46} + 2 q^{49} - 5 q^{50} - 6 q^{52} - 18 q^{53} - 2 q^{59} - 25 q^{62} - 26 q^{64} - 4 q^{65} + 9 q^{68} - 5 q^{70} - 8 q^{71} - 14 q^{73} + 5 q^{74} + 3 q^{76} + 20 q^{79} + q^{80} - 5 q^{82} + 24 q^{83} + 6 q^{85} - 30 q^{86} - 9 q^{89} + 2 q^{91} - 33 q^{92} - 20 q^{94} - 8 q^{95} - 12 q^{97}+O(q^{100}) \) 2 * q + 6 * q^4 - q^5 - 2 * q^7 + 5 * q^10 - 2 * q^13 - 2 * q^16 + 3 * q^17 + q^19 - 3 * q^20 - 11 * q^23 - 7 * q^25 - 10 * q^26 - 6 * q^28 - 12 * q^29 + 5 * q^31 + 15 * q^34 + q^35 + 7 * q^37 - 15 * q^38 + 5 * q^40 + 17 * q^41 - 6 * q^43 + 5 * q^46 + 2 * q^49 - 5 * q^50 - 6 * q^52 - 18 * q^53 - 2 * q^59 - 25 * q^62 - 26 * q^64 - 4 * q^65 + 9 * q^68 - 5 * q^70 - 8 * q^71 - 14 * q^73 + 5 * q^74 + 3 * q^76 + 20 * q^79 + q^80 - 5 * q^82 + 24 * q^83 + 6 * q^85 - 30 * q^86 - 9 * q^89 + 2 * q^91 - 33 * q^92 - 20 * q^94 - 8 * q^95 - 12 * q^97

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