LMFDB - Embedded newform 7650.2.a.do.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7650,2,Mod(1,7650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7650, 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("7650.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7650 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7650.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:	$61.0855575463$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 170)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.76156$ of defining polynomial
Character	$\chi$	$=$	7650.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -0.864641 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -0.864641 q^{7} +1.00000 q^{8} -2.00000 q^{11} +2.62620 q^{13} -0.864641 q^{14} +1.00000 q^{16} +1.00000 q^{17} -0.896916 q^{19} -2.00000 q^{22} -3.13536 q^{23} +2.62620 q^{26} -0.864641 q^{28} -9.49084 q^{29} +9.01395 q^{31} +1.00000 q^{32} +1.00000 q^{34} -10.1816 q^{37} -0.896916 q^{38} -9.52311 q^{41} -7.25240 q^{43} -2.00000 q^{44} -3.13536 q^{46} -10.4200 q^{47} -6.25240 q^{49} +2.62620 q^{52} +11.4017 q^{53} -0.864641 q^{56} -9.49084 q^{58} +4.14931 q^{59} +3.28467 q^{61} +9.01395 q^{62} +1.00000 q^{64} -1.25240 q^{67} +1.00000 q^{68} +12.5371 q^{71} +2.62620 q^{73} -10.1816 q^{74} -0.896916 q^{76} +1.72928 q^{77} -7.91087 q^{79} -9.52311 q^{82} -4.20617 q^{83} -7.25240 q^{86} -2.00000 q^{88} -4.14931 q^{89} -2.27072 q^{91} -3.13536 q^{92} -10.4200 q^{94} +6.14931 q^{97} -6.25240 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} + 3 q^{8}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^8	$ 3 q + 3 q^{2} + 3 q^{4} + 3 q^{8} - 6 q^{11} - q^{13} + 3 q^{16} + 3 q^{17} + q^{19} - 6 q^{22} - 12 q^{23} - q^{26} - 17 q^{29} + 3 q^{31} + 3 q^{32} + 3 q^{34} - 8 q^{37} + q^{38} - 16 q^{41} - 4 q^{43} - 6 q^{44} - 12 q^{46} - 15 q^{47} - q^{49} - q^{52} - 5 q^{53} - 17 q^{58} - 9 q^{59} - 9 q^{61} + 3 q^{62} + 3 q^{64} + 14 q^{67} + 3 q^{68} + q^{71} - q^{73} - 8 q^{74} + q^{76} + 4 q^{79} - 16 q^{82} - 20 q^{83} - 4 q^{86} - 6 q^{88} + 9 q^{89} - 12 q^{91} - 12 q^{92} - 15 q^{94} - 3 q^{97} - q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^8 - 6 * q^11 - q^13 + 3 * q^16 + 3 * q^17 + q^19 - 6 * q^22 - 12 * q^23 - q^26 - 17 * q^29 + 3 * q^31 + 3 * q^32 + 3 * q^34 - 8 * q^37 + q^38 - 16 * q^41 - 4 * q^43 - 6 * q^44 - 12 * q^46 - 15 * q^47 - q^49 - q^52 - 5 * q^53 - 17 * q^58 - 9 * q^59 - 9 * q^61 + 3 * q^62 + 3 * q^64 + 14 * q^67 + 3 * q^68 + q^71 - q^73 - 8 * q^74 + q^76 + 4 * q^79 - 16 * q^82 - 20 * q^83 - 4 * q^86 - 6 * q^88 + 9 * q^89 - 12 * q^91 - 12 * q^92 - 15 * q^94 - 3 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.864641	−0.326804	−0.163402	−	0.986560i	$-0.552247\pi$
$7$	−0.864641	−0.326804	−0.163402	+	0.986560i	$0.552247\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	2.62620	0.728376	0.364188	−	0.931325i	$-0.381347\pi$
$13$	2.62620	0.728376	0.364188	+	0.931325i	$0.381347\pi$
$14$	−0.864641	−0.231085
$15$	0	0
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−0.896916	−0.205767	−0.102883	−	0.994693i	$-0.532807\pi$
$19$	−0.896916	−0.205767	−0.102883	+	0.994693i	$0.532807\pi$
$20$	0	0
$21$	0	0
$22$	−2.00000	−0.426401
$23$	−3.13536	−0.653768	−0.326884	−	0.945065i	$-0.605999\pi$
$23$	−3.13536	−0.653768	−0.326884	+	0.945065i	$0.605999\pi$
$24$	0	0
$25$	0	0
$26$	2.62620	0.515040
$27$	0	0
$28$	−0.864641	−0.163402
$29$	−9.49084	−1.76240	−0.881202	−	0.472739i	$-0.843265\pi$
$29$	−9.49084	−1.76240	−0.881202	+	0.472739i	$0.843265\pi$
$30$	0	0
$31$	9.01395	1.61895	0.809477	−	0.587152i	$-0.199751\pi$
$31$	9.01395	1.61895	0.809477	+	0.587152i	$0.199751\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	1.00000	0.171499
$35$	0	0
$36$	0	0
$37$	−10.1816	−1.67384	−0.836921	−	0.547323i	$-0.815647\pi$
$37$	−10.1816	−1.67384	−0.836921	+	0.547323i	$0.815647\pi$
$38$	−0.896916	−0.145499
$39$	0	0
$40$	0	0
$41$	−9.52311	−1.48726	−0.743630	−	0.668591i	$-0.766898\pi$
$41$	−9.52311	−1.48726	−0.743630	+	0.668591i	$0.766898\pi$
$42$	0	0
$43$	−7.25240	−1.10598	−0.552990	−	0.833188i	$-0.686513\pi$
$43$	−7.25240	−1.10598	−0.552990	+	0.833188i	$0.686513\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	−3.13536	−0.462283
$47$	−10.4200	−1.51992	−0.759959	−	0.649971i	$-0.774781\pi$
$47$	−10.4200	−1.51992	−0.759959	+	0.649971i	$0.774781\pi$
$48$	0	0
$49$	−6.25240	−0.893199
$50$	0	0
$51$	0	0
$52$	2.62620	0.364188
$53$	11.4017	1.56615	0.783073	−	0.621930i	$-0.213651\pi$
$53$	11.4017	1.56615	0.783073	+	0.621930i	$0.213651\pi$
$54$	0	0
$55$	0	0
$56$	−0.864641	−0.115542
$57$	0	0
$58$	−9.49084	−1.24621
$59$	4.14931	0.540194	0.270097	−	0.962833i	$-0.412944\pi$
$59$	4.14931	0.540194	0.270097	+	0.962833i	$0.412944\pi$
$60$	0	0
$61$	3.28467	0.420559	0.210280	−	0.977641i	$-0.432563\pi$
$61$	3.28467	0.420559	0.210280	+	0.977641i	$0.432563\pi$
$62$	9.01395	1.14477
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−1.25240	−0.153005	−0.0765023	−	0.997069i	$-0.524375\pi$
$67$	−1.25240	−0.153005	−0.0765023	+	0.997069i	$0.524375\pi$
$68$	1.00000	0.121268
$69$	0	0
$70$	0	0
$71$	12.5371	1.48788	0.743938	−	0.668249i	$-0.232956\pi$
$71$	12.5371	1.48788	0.743938	+	0.668249i	$0.232956\pi$
$72$	0	0
$73$	2.62620	0.307373	0.153687	−	0.988120i	$-0.450885\pi$
$73$	2.62620	0.307373	0.153687	+	0.988120i	$0.450885\pi$
$74$	−10.1816	−1.18359
$75$	0	0
$76$	−0.896916	−0.102883
$77$	1.72928	0.197070
$78$	0	0
$79$	−7.91087	−0.890042	−0.445021	−	0.895520i	$-0.646804\pi$
$79$	−7.91087	−0.890042	−0.445021	+	0.895520i	$0.646804\pi$
$80$	0	0
$81$	0	0
$82$	−9.52311	−1.05165
$83$	−4.20617	−0.461687	−0.230843	−	0.972991i	$-0.574149\pi$
$83$	−4.20617	−0.461687	−0.230843	+	0.972991i	$0.574149\pi$
$84$	0	0
$85$	0	0
$86$	−7.25240	−0.782046
$87$	0	0
$88$	−2.00000	−0.213201
$89$	−4.14931	−0.439826	−0.219913	−	0.975519i	$-0.570577\pi$
$89$	−4.14931	−0.439826	−0.219913	+	0.975519i	$0.570577\pi$
$90$	0	0
$91$	−2.27072	−0.238036
$92$	−3.13536	−0.326884
$93$	0	0
$94$	−10.4200	−1.07474
$95$	0	0
$96$	0	0
$97$	6.14931	0.624368	0.312184	−	0.950022i	$-0.398939\pi$
$97$	6.14931	0.624368	0.312184	+	0.950022i	$0.398939\pi$
$98$	−6.25240	−0.631587
$99$	0	0
$100$	0	0
$101$	−0.206167	−0.0205144	−0.0102572	−	0.999947i	$-0.503265\pi$
$101$	−0.206167	−0.0205144	−0.0102572	+	0.999947i	$0.503265\pi$
$102$	0	0
$103$	−4.77551	−0.470545	−0.235273	−	0.971929i	$-0.575598\pi$
$103$	−4.77551	−0.470545	−0.235273	+	0.971929i	$0.575598\pi$
$104$	2.62620	0.257520
$105$	0	0
$106$	11.4017	1.10743
$107$	6.29862	0.608911	0.304456	−	0.952527i	$-0.401526\pi$
$107$	6.29862	0.608911	0.304456	+	0.952527i	$0.401526\pi$
$108$	0	0
$109$	5.55539	0.532110	0.266055	−	0.963958i	$-0.414280\pi$
$109$	5.55539	0.532110	0.266055	+	0.963958i	$0.414280\pi$
$110$	0	0
$111$	0	0
$112$	−0.864641	−0.0817009
$113$	10.6262	0.999629	0.499814	−	0.866133i	$-0.333402\pi$
$113$	10.6262	0.999629	0.499814	+	0.866133i	$0.333402\pi$
$114$	0	0
$115$	0	0
$116$	−9.49084	−0.881202
$117$	0	0
$118$	4.14931	0.381975
$119$	−0.864641	−0.0792615
$120$	0	0
$121$	−7.00000	−0.636364
$122$	3.28467	0.297380
$123$	0	0
$124$	9.01395	0.809477
$125$	0	0
$126$	0	0
$127$	14.3555	1.27384	0.636921	−	0.770929i	$-0.280208\pi$
$127$	14.3555	1.27384	0.636921	+	0.770929i	$0.280208\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	−17.0462	−1.48934	−0.744668	−	0.667435i	$-0.767392\pi$
$131$	−17.0462	−1.48934	−0.744668	+	0.667435i	$0.767392\pi$
$132$	0	0
$133$	0.775511	0.0672453
$134$	−1.25240	−0.108191
$135$	0	0
$136$	1.00000	0.0857493
$137$	14.5048	1.23923	0.619614	−	0.784907i	$-0.287289\pi$
$137$	14.5048	1.23923	0.619614	+	0.784907i	$0.287289\pi$
$138$	0	0
$139$	−14.7110	−1.24777	−0.623884	−	0.781517i	$-0.714446\pi$
$139$	−14.7110	−1.24777	−0.623884	+	0.781517i	$0.714446\pi$
$140$	0	0
$141$	0	0
$142$	12.5371	1.05209
$143$	−5.25240	−0.439227
$144$	0	0
$145$	0	0
$146$	2.62620	0.217346
$147$	0	0
$148$	−10.1816	−0.836921
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	−5.18785	−0.422181	−0.211090	−	0.977467i	$-0.567701\pi$
$151$	−5.18785	−0.422181	−0.211090	+	0.977467i	$0.567701\pi$
$152$	−0.896916	−0.0727495
$153$	0	0
$154$	1.72928	0.139349
$155$	0	0
$156$	0	0
$157$	−10.2341	−0.816768	−0.408384	−	0.912810i	$-0.633908\pi$
$157$	−10.2341	−0.816768	−0.408384	+	0.912810i	$0.633908\pi$
$158$	−7.91087	−0.629355
$159$	0	0
$160$	0	0
$161$	2.71096	0.213654
$162$	0	0
$163$	−6.47689	−0.507309	−0.253654	−	0.967295i	$-0.581633\pi$
$163$	−6.47689	−0.507309	−0.253654	+	0.967295i	$0.581633\pi$
$164$	−9.52311	−0.743630
$165$	0	0
$166$	−4.20617	−0.326462
$167$	−3.84632	−0.297637	−0.148819	−	0.988865i	$-0.547547\pi$
$167$	−3.84632	−0.297637	−0.148819	+	0.988865i	$0.547547\pi$
$168$	0	0
$169$	−6.10308	−0.469468
$170$	0	0
$171$	0	0
$172$	−7.25240	−0.552990
$173$	−23.9109	−1.81791	−0.908955	−	0.416895i	$-0.863118\pi$
$173$	−23.9109	−1.81791	−0.908955	+	0.416895i	$0.863118\pi$
$174$	0	0
$175$	0	0
$176$	−2.00000	−0.150756
$177$	0	0
$178$	−4.14931	−0.311004
$179$	14.7110	1.09955	0.549774	−	0.835313i	$-0.314714\pi$
$179$	14.7110	1.09955	0.549774	+	0.835313i	$0.314714\pi$
$180$	0	0
$181$	−23.4340	−1.74183	−0.870917	−	0.491430i	$-0.836474\pi$
$181$	−23.4340	−1.74183	−0.870917	+	0.491430i	$0.836474\pi$
$182$	−2.27072	−0.168317
$183$	0	0
$184$	−3.13536	−0.231142
$185$	0	0
$186$	0	0
$187$	−2.00000	−0.146254
$188$	−10.4200	−0.759959
$189$	0	0
$190$	0	0
$191$	−18.5048	−1.33896	−0.669480	−	0.742830i	$-0.733483\pi$
$191$	−18.5048	−1.33896	−0.669480	+	0.742830i	$0.733483\pi$
$192$	0	0
$193$	−5.45856	−0.392916	−0.196458	−	0.980512i	$-0.562944\pi$
$193$	−5.45856	−0.392916	−0.196458	+	0.980512i	$0.562944\pi$
$194$	6.14931	0.441495
$195$	0	0
$196$	−6.25240	−0.446600
$197$	−13.9388	−0.993097	−0.496548	−	0.868009i	$-0.665399\pi$
$197$	−13.9388	−0.993097	−0.496548	+	0.868009i	$0.665399\pi$
$198$	0	0
$199$	−11.2847	−0.799949	−0.399975	−	0.916526i	$-0.630981\pi$
$199$	−11.2847	−0.799949	−0.399975	+	0.916526i	$0.630981\pi$
$200$	0	0
$201$	0	0
$202$	−0.206167	−0.0145059
$203$	8.20617	0.575960
$204$	0	0
$205$	0	0
$206$	−4.77551	−0.332726
$207$	0	0
$208$	2.62620	0.182094
$209$	1.79383	0.124082
$210$	0	0
$211$	−14.8401	−1.02163	−0.510816	−	0.859690i	$-0.670657\pi$
$211$	−14.8401	−1.02163	−0.510816	+	0.859690i	$0.670657\pi$
$212$	11.4017	0.783073
$213$	0	0
$214$	6.29862	0.430565
$215$	0	0
$216$	0	0
$217$	−7.79383	−0.529080
$218$	5.55539	0.376258
$219$	0	0
$220$	0	0
$221$	2.62620	0.176657
$222$	0	0
$223$	3.30925	0.221604	0.110802	−	0.993843i	$-0.464658\pi$
$223$	3.30925	0.221604	0.110802	+	0.993843i	$0.464658\pi$
$224$	−0.864641	−0.0577712
$225$	0	0
$226$	10.6262	0.706844
$227$	24.7187	1.64063	0.820317	−	0.571909i	$-0.193797\pi$
$227$	24.7187	1.64063	0.820317	+	0.571909i	$0.193797\pi$
$228$	0	0
$229$	4.27072	0.282217	0.141109	−	0.989994i	$-0.454933\pi$
$229$	4.27072	0.282217	0.141109	+	0.989994i	$0.454933\pi$
$230$	0	0
$231$	0	0
$232$	−9.49084	−0.623104
$233$	−0.832365	−0.0545301	−0.0272650	−	0.999628i	$-0.508680\pi$
$233$	−0.832365	−0.0545301	−0.0272650	+	0.999628i	$0.508680\pi$
$234$	0	0
$235$	0	0
$236$	4.14931	0.270097
$237$	0	0
$238$	−0.864641	−0.0560463
$239$	4.71096	0.304727	0.152363	−	0.988325i	$-0.451312\pi$
$239$	4.71096	0.304727	0.152363	+	0.988325i	$0.451312\pi$
$240$	0	0
$241$	−23.4865	−1.51290	−0.756448	−	0.654054i	$-0.773067\pi$
$241$	−23.4865	−1.51290	−0.756448	+	0.654054i	$0.773067\pi$
$242$	−7.00000	−0.449977
$243$	0	0
$244$	3.28467	0.210280
$245$	0	0
$246$	0	0
$247$	−2.35548	−0.149876
$248$	9.01395	0.572387
$249$	0	0
$250$	0	0
$251$	6.29862	0.397566	0.198783	−	0.980044i	$-0.436301\pi$
$251$	6.29862	0.397566	0.198783	+	0.980044i	$0.436301\pi$
$252$	0	0
$253$	6.27072	0.394237
$254$	14.3555	0.900743
$255$	0	0
$256$	1.00000	0.0625000
$257$	1.49521	0.0932685	0.0466342	−	0.998912i	$-0.485150\pi$
$257$	1.49521	0.0932685	0.0466342	+	0.998912i	$0.485150\pi$
$258$	0	0
$259$	8.80342	0.547018
$260$	0	0
$261$	0	0
$262$	−17.0462	−1.05312
$263$	9.94315	0.613121	0.306560	−	0.951851i	$-0.400822\pi$
$263$	9.94315	0.613121	0.306560	+	0.951851i	$0.400822\pi$
$264$	0	0
$265$	0	0
$266$	0.775511	0.0475496
$267$	0	0
$268$	−1.25240	−0.0765023
$269$	−1.96772	−0.119974	−0.0599871	−	0.998199i	$-0.519106\pi$
$269$	−1.96772	−0.119974	−0.0599871	+	0.998199i	$0.519106\pi$
$270$	0	0
$271$	8.84006	0.536995	0.268498	−	0.963280i	$-0.413473\pi$
$271$	8.84006	0.536995	0.268498	+	0.963280i	$0.413473\pi$
$272$	1.00000	0.0606339
$273$	0	0
$274$	14.5048	0.876267
$275$	0	0
$276$	0	0
$277$	−19.8463	−1.19245	−0.596225	−	0.802817i	$-0.703333\pi$
$277$	−19.8463	−1.19245	−0.596225	+	0.802817i	$0.703333\pi$
$278$	−14.7110	−0.882305
$279$	0	0
$280$	0	0
$281$	11.9065	0.710282	0.355141	−	0.934813i	$-0.384433\pi$
$281$	11.9065	0.710282	0.355141	+	0.934813i	$0.384433\pi$
$282$	0	0
$283$	13.7370	0.816579	0.408289	−	0.912853i	$-0.366125\pi$
$283$	13.7370	0.816579	0.408289	+	0.912853i	$0.366125\pi$
$284$	12.5371	0.743938
$285$	0	0
$286$	−5.25240	−0.310581
$287$	8.23407	0.486042
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	2.62620	0.153687
$293$	12.1772	0.711401	0.355700	−	0.934600i	$-0.384242\pi$
$293$	12.1772	0.711401	0.355700	+	0.934600i	$0.384242\pi$
$294$	0	0
$295$	0	0
$296$	−10.1816	−0.591793
$297$	0	0
$298$	−10.0000	−0.579284
$299$	−8.23407	−0.476189
$300$	0	0
$301$	6.27072	0.361438
$302$	−5.18785	−0.298527
$303$	0	0
$304$	−0.896916	−0.0514417
$305$	0	0
$306$	0	0
$307$	23.7938	1.35799	0.678993	−	0.734145i	$-0.262417\pi$
$307$	23.7938	1.35799	0.678993	+	0.734145i	$0.262417\pi$
$308$	1.72928	0.0985350
$309$	0	0
$310$	0	0
$311$	−32.9205	−1.86675	−0.933374	−	0.358906i	$-0.883150\pi$
$311$	−32.9205	−1.86675	−0.933374	+	0.358906i	$0.883150\pi$
$312$	0	0
$313$	−29.0462	−1.64179	−0.820895	−	0.571079i	$-0.806525\pi$
$313$	−29.0462	−1.64179	−0.820895	+	0.571079i	$0.806525\pi$
$314$	−10.2341	−0.577542
$315$	0	0
$316$	−7.91087	−0.445021
$317$	−20.6864	−1.16186	−0.580931	−	0.813953i	$-0.697312\pi$
$317$	−20.6864	−1.16186	−0.580931	+	0.813953i	$0.697312\pi$
$318$	0	0
$319$	18.9817	1.06277
$320$	0	0
$321$	0	0
$322$	2.71096	0.151076
$323$	−0.896916	−0.0499058
$324$	0	0
$325$	0	0
$326$	−6.47689	−0.358722
$327$	0	0
$328$	−9.52311	−0.525826
$329$	9.00958	0.496714
$330$	0	0
$331$	4.02021	0.220971	0.110485	−	0.993878i	$-0.464759\pi$
$331$	4.02021	0.220971	0.110485	+	0.993878i	$0.464759\pi$
$332$	−4.20617	−0.230843
$333$	0	0
$334$	−3.84632	−0.210461
$335$	0	0
$336$	0	0
$337$	2.21386	0.120597	0.0602984	−	0.998180i	$-0.480795\pi$
$337$	2.21386	0.120597	0.0602984	+	0.998180i	$0.480795\pi$
$338$	−6.10308	−0.331964
$339$	0	0
$340$	0	0
$341$	−18.0279	−0.976266
$342$	0	0
$343$	11.4586	0.618704
$344$	−7.25240	−0.391023
$345$	0	0
$346$	−23.9109	−1.28546
$347$	1.37380	0.0737496	0.0368748	−	0.999320i	$-0.488260\pi$
$347$	1.37380	0.0737496	0.0368748	+	0.999320i	$0.488260\pi$
$348$	0	0
$349$	−20.8680	−1.11704	−0.558518	−	0.829492i	$-0.688630\pi$
$349$	−20.8680	−1.11704	−0.558518	+	0.829492i	$0.688630\pi$
$350$	0	0
$351$	0	0
$352$	−2.00000	−0.106600
$353$	13.7572	0.732221	0.366111	−	0.930571i	$-0.380689\pi$
$353$	13.7572	0.732221	0.366111	+	0.930571i	$0.380689\pi$
$354$	0	0
$355$	0	0
$356$	−4.14931	−0.219913
$357$	0	0
$358$	14.7110	0.777498
$359$	−9.31695	−0.491730	−0.245865	−	0.969304i	$-0.579072\pi$
$359$	−9.31695	−0.491730	−0.245865	+	0.969304i	$0.579072\pi$
$360$	0	0
$361$	−18.1955	−0.957660
$362$	−23.4340	−1.23166
$363$	0	0
$364$	−2.27072	−0.119018
$365$	0	0
$366$	0	0
$367$	12.3878	0.646636	0.323318	−	0.946290i	$-0.395202\pi$
$367$	12.3878	0.646636	0.323318	+	0.946290i	$0.395202\pi$
$368$	−3.13536	−0.163442
$369$	0	0
$370$	0	0
$371$	−9.85838	−0.511822
$372$	0	0
$373$	−23.3169	−1.20731	−0.603653	−	0.797247i	$-0.706289\pi$
$373$	−23.3169	−1.20731	−0.603653	+	0.797247i	$0.706289\pi$
$374$	−2.00000	−0.103418
$375$	0	0
$376$	−10.4200	−0.537372
$377$	−24.9248	−1.28369
$378$	0	0
$379$	28.9817	1.48869	0.744344	−	0.667796i	$-0.232762\pi$
$379$	28.9817	1.48869	0.744344	+	0.667796i	$0.232762\pi$
$380$	0	0
$381$	0	0
$382$	−18.5048	−0.946788
$383$	−14.9527	−0.764049	−0.382024	−	0.924152i	$-0.624773\pi$
$383$	−14.9527	−0.764049	−0.382024	+	0.924152i	$0.624773\pi$
$384$	0	0
$385$	0	0
$386$	−5.45856	−0.277834
$387$	0	0
$388$	6.14931	0.312184
$389$	19.3169	0.979408	0.489704	−	0.871889i	$-0.337105\pi$
$389$	19.3169	0.979408	0.489704	+	0.871889i	$0.337105\pi$
$390$	0	0
$391$	−3.13536	−0.158562
$392$	−6.25240	−0.315794
$393$	0	0
$394$	−13.9388	−0.702225
$395$	0	0
$396$	0	0
$397$	9.57560	0.480586	0.240293	−	0.970700i	$-0.422757\pi$
$397$	9.57560	0.480586	0.240293	+	0.970700i	$0.422757\pi$
$398$	−11.2847	−0.565649
$399$	0	0
$400$	0	0
$401$	−11.3169	−0.565141	−0.282571	−	0.959246i	$-0.591187\pi$
$401$	−11.3169	−0.565141	−0.282571	+	0.959246i	$0.591187\pi$
$402$	0	0
$403$	23.6724	1.17921
$404$	−0.206167	−0.0102572
$405$	0	0
$406$	8.20617	0.407265
$407$	20.3632	1.00937
$408$	0	0
$409$	−9.10308	−0.450119	−0.225059	−	0.974345i	$-0.572258\pi$
$409$	−9.10308	−0.450119	−0.225059	+	0.974345i	$0.572258\pi$
$410$	0	0
$411$	0	0
$412$	−4.77551	−0.235273
$413$	−3.58767	−0.176537
$414$	0	0
$415$	0	0
$416$	2.62620	0.128760
$417$	0	0
$418$	1.79383	0.0877392
$419$	4.02791	0.196776	0.0983881	−	0.995148i	$-0.468631\pi$
$419$	4.02791	0.196776	0.0983881	+	0.995148i	$0.468631\pi$
$420$	0	0
$421$	15.2524	0.743356	0.371678	−	0.928362i	$-0.378782\pi$
$421$	15.2524	0.743356	0.371678	+	0.928362i	$0.378782\pi$
$422$	−14.8401	−0.722403
$423$	0	0
$424$	11.4017	0.553716
$425$	0	0
$426$	0	0
$427$	−2.84006	−0.137440
$428$	6.29862	0.304456
$429$	0	0
$430$	0	0
$431$	4.45231	0.214460	0.107230	−	0.994234i	$-0.465802\pi$
$431$	4.45231	0.214460	0.107230	+	0.994234i	$0.465802\pi$
$432$	0	0
$433$	10.7110	0.514736	0.257368	−	0.966313i	$-0.417145\pi$
$433$	10.7110	0.514736	0.257368	+	0.966313i	$0.417145\pi$
$434$	−7.79383	−0.374116
$435$	0	0
$436$	5.55539	0.266055
$437$	2.81215	0.134524
$438$	0	0
$439$	5.09871	0.243348	0.121674	−	0.992570i	$-0.461174\pi$
$439$	5.09871	0.243348	0.121674	+	0.992570i	$0.461174\pi$
$440$	0	0
$441$	0	0
$442$	2.62620	0.124916
$443$	−10.8034	−0.513286	−0.256643	−	0.966506i	$-0.582616\pi$
$443$	−10.8034	−0.513286	−0.256643	+	0.966506i	$0.582616\pi$
$444$	0	0
$445$	0	0
$446$	3.30925	0.156698
$447$	0	0
$448$	−0.864641	−0.0408504
$449$	5.82174	0.274745	0.137372	−	0.990519i	$-0.456134\pi$
$449$	5.82174	0.274745	0.137372	+	0.990519i	$0.456134\pi$
$450$	0	0
$451$	19.0462	0.896852
$452$	10.6262	0.499814
$453$	0	0
$454$	24.7187	1.16010
$455$	0	0
$456$	0	0
$457$	13.7014	0.640923	0.320462	−	0.947261i	$-0.396162\pi$
$457$	13.7014	0.640923	0.320462	+	0.947261i	$0.396162\pi$
$458$	4.27072	0.199558
$459$	0	0
$460$	0	0
$461$	27.0741	1.26097	0.630484	−	0.776202i	$-0.282856\pi$
$461$	27.0741	1.26097	0.630484	+	0.776202i	$0.282856\pi$
$462$	0	0
$463$	37.3372	1.73520	0.867602	−	0.497258i	$-0.165660\pi$
$463$	37.3372	1.73520	0.867602	+	0.497258i	$0.165660\pi$
$464$	−9.49084	−0.440601
$465$	0	0
$466$	−0.832365	−0.0385586
$467$	−35.3449	−1.63556	−0.817782	−	0.575528i	$-0.804797\pi$
$467$	−35.3449	−1.63556	−0.817782	+	0.575528i	$0.804797\pi$
$468$	0	0
$469$	1.08287	0.0500024
$470$	0	0
$471$	0	0
$472$	4.14931	0.190988
$473$	14.5048	0.666931
$474$	0	0
$475$	0	0
$476$	−0.864641	−0.0396308
$477$	0	0
$478$	4.71096	0.215474
$479$	−31.0419	−1.41834	−0.709169	−	0.705038i	$-0.750930\pi$
$479$	−31.0419	−1.41834	−0.709169	+	0.705038i	$0.750930\pi$
$480$	0	0
$481$	−26.7389	−1.21919
$482$	−23.4865	−1.06978
$483$	0	0
$484$	−7.00000	−0.318182
$485$	0	0
$486$	0	0
$487$	13.1633	0.596485	0.298242	−	0.954490i	$-0.403600\pi$
$487$	13.1633	0.596485	0.298242	+	0.954490i	$0.403600\pi$
$488$	3.28467	0.148690
$489$	0	0
$490$	0	0
$491$	−24.3555	−1.09915	−0.549574	−	0.835445i	$-0.685210\pi$
$491$	−24.3555	−1.09915	−0.549574	+	0.835445i	$0.685210\pi$
$492$	0	0
$493$	−9.49084	−0.427446
$494$	−2.35548	−0.105978
$495$	0	0
$496$	9.01395	0.404738
$497$	−10.8401	−0.486243
$498$	0	0
$499$	36.2620	1.62331	0.811655	−	0.584138i	$-0.198567\pi$
$499$	36.2620	1.62331	0.811655	+	0.584138i	$0.198567\pi$
$500$	0	0
$501$	0	0
$502$	6.29862	0.281121
$503$	42.3511	1.88834	0.944171	−	0.329455i	$-0.106865\pi$
$503$	42.3511	1.88834	0.944171	+	0.329455i	$0.106865\pi$
$504$	0	0
$505$	0	0
$506$	6.27072	0.278767
$507$	0	0
$508$	14.3555	0.636921
$509$	1.70138	0.0754121	0.0377061	−	0.999289i	$-0.487995\pi$
$509$	1.70138	0.0754121	0.0377061	+	0.999289i	$0.487995\pi$
$510$	0	0
$511$	−2.27072	−0.100451
$512$	1.00000	0.0441942
$513$	0	0
$514$	1.49521	0.0659508
$515$	0	0
$516$	0	0
$517$	20.8401	0.916545
$518$	8.80342	0.386800
$519$	0	0
$520$	0	0
$521$	−3.01832	−0.132235	−0.0661175	−	0.997812i	$-0.521061\pi$
$521$	−3.01832	−0.132235	−0.0661175	+	0.997812i	$0.521061\pi$
$522$	0	0
$523$	−5.79383	−0.253347	−0.126673	−	0.991944i	$-0.540430\pi$
$523$	−5.79383	−0.253347	−0.126673	+	0.991944i	$0.540430\pi$
$524$	−17.0462	−0.744668
$525$	0	0
$526$	9.94315	0.433542
$527$	9.01395	0.392654
$528$	0	0
$529$	−13.1695	−0.572588
$530$	0	0
$531$	0	0
$532$	0.775511	0.0336226
$533$	−25.0096	−1.08329
$534$	0	0
$535$	0	0
$536$	−1.25240	−0.0540953
$537$	0	0
$538$	−1.96772	−0.0848346
$539$	12.5048	0.538620
$540$	0	0
$541$	0.258654	0.0111204	0.00556019	−	0.999985i	$-0.498230\pi$
$541$	0.258654	0.0111204	0.00556019	+	0.999985i	$0.498230\pi$
$542$	8.84006	0.379713
$543$	0	0
$544$	1.00000	0.0428746
$545$	0	0
$546$	0	0
$547$	32.4113	1.38581	0.692903	−	0.721030i	$-0.256331\pi$
$547$	32.4113	1.38581	0.692903	+	0.721030i	$0.256331\pi$
$548$	14.5048	0.619614
$549$	0	0
$550$	0	0
$551$	8.51249	0.362644
$552$	0	0
$553$	6.84006	0.290869
$554$	−19.8463	−0.843189
$555$	0	0
$556$	−14.7110	−0.623884
$557$	22.9248	0.971356	0.485678	−	0.874138i	$-0.338573\pi$
$557$	22.9248	0.971356	0.485678	+	0.874138i	$0.338573\pi$
$558$	0	0
$559$	−19.0462	−0.805570
$560$	0	0
$561$	0	0
$562$	11.9065	0.502245
$563$	−17.8863	−0.753817	−0.376909	−	0.926250i	$-0.623013\pi$
$563$	−17.8863	−0.753817	−0.376909	+	0.926250i	$0.623013\pi$
$564$	0	0
$565$	0	0
$566$	13.7370	0.577408
$567$	0	0
$568$	12.5371	0.526044
$569$	−12.3555	−0.517969	−0.258984	−	0.965882i	$-0.583388\pi$
$569$	−12.3555	−0.517969	−0.258984	+	0.965882i	$0.583388\pi$
$570$	0	0
$571$	−27.3169	−1.14318	−0.571589	−	0.820540i	$-0.693673\pi$
$571$	−27.3169	−1.14318	−0.571589	+	0.820540i	$0.693673\pi$
$572$	−5.25240	−0.219614
$573$	0	0
$574$	8.23407	0.343684
$575$	0	0
$576$	0	0
$577$	10.6339	0.442695	0.221347	−	0.975195i	$-0.428955\pi$
$577$	10.6339	0.442695	0.221347	+	0.975195i	$0.428955\pi$
$578$	1.00000	0.0415945
$579$	0	0
$580$	0	0
$581$	3.63682	0.150881
$582$	0	0
$583$	−22.8034	−0.944421
$584$	2.62620	0.108673
$585$	0	0
$586$	12.1772	0.503036
$587$	22.3757	0.923544	0.461772	−	0.886999i	$-0.347214\pi$
$587$	22.3757	0.923544	0.461772	+	0.886999i	$0.347214\pi$
$588$	0	0
$589$	−8.08476	−0.333127
$590$	0	0
$591$	0	0
$592$	−10.1816	−0.418461
$593$	−12.1695	−0.499742	−0.249871	−	0.968279i	$-0.580388\pi$
$593$	−12.1695	−0.499742	−0.249871	+	0.968279i	$0.580388\pi$
$594$	0	0
$595$	0	0
$596$	−10.0000	−0.409616
$597$	0	0
$598$	−8.23407	−0.336716
$599$	25.1387	1.02714	0.513569	−	0.858048i	$-0.328323\pi$
$599$	25.1387	1.02714	0.513569	+	0.858048i	$0.328323\pi$
$600$	0	0
$601$	−4.50479	−0.183754	−0.0918772	−	0.995770i	$-0.529287\pi$
$601$	−4.50479	−0.183754	−0.0918772	+	0.995770i	$0.529287\pi$
$602$	6.27072	0.255575
$603$	0	0
$604$	−5.18785	−0.211090
$605$	0	0
$606$	0	0
$607$	7.00626	0.284375	0.142188	−	0.989840i	$-0.454586\pi$
$607$	7.00626	0.284375	0.142188	+	0.989840i	$0.454586\pi$
$608$	−0.896916	−0.0363748
$609$	0	0
$610$	0	0
$611$	−27.3651	−1.10707
$612$	0	0
$613$	−7.46626	−0.301559	−0.150780	−	0.988567i	$-0.548178\pi$
$613$	−7.46626	−0.301559	−0.150780	+	0.988567i	$0.548178\pi$
$614$	23.7938	0.960241
$615$	0	0
$616$	1.72928	0.0696747
$617$	36.1772	1.45644	0.728220	−	0.685343i	$-0.240348\pi$
$617$	36.1772	1.45644	0.728220	+	0.685343i	$0.240348\pi$
$618$	0	0
$619$	−4.27072	−0.171655	−0.0858273	−	0.996310i	$-0.527353\pi$
$619$	−4.27072	−0.171655	−0.0858273	+	0.996310i	$0.527353\pi$
$620$	0	0
$621$	0	0
$622$	−32.9205	−1.31999
$623$	3.58767	0.143737
$624$	0	0
$625$	0	0
$626$	−29.0462	−1.16092
$627$	0	0
$628$	−10.2341	−0.408384
$629$	−10.1816	−0.405966
$630$	0	0
$631$	26.0279	1.03615	0.518077	−	0.855334i	$-0.326648\pi$
$631$	26.0279	1.03615	0.518077	+	0.855334i	$0.326648\pi$
$632$	−7.91087	−0.314677
$633$	0	0
$634$	−20.6864	−0.821561
$635$	0	0
$636$	0	0
$637$	−16.4200	−0.650585
$638$	18.9817	0.751492
$639$	0	0
$640$	0	0
$641$	−12.6831	−0.500950	−0.250475	−	0.968123i	$-0.580587\pi$
$641$	−12.6831	−0.500950	−0.250475	+	0.968123i	$0.580587\pi$
$642$	0	0
$643$	−25.1108	−0.990272	−0.495136	−	0.868815i	$-0.664882\pi$
$643$	−25.1108	−0.990272	−0.495136	+	0.868815i	$0.664882\pi$
$644$	2.71096	0.106827
$645$	0	0
$646$	−0.896916	−0.0352887
$647$	38.0635	1.49643	0.748215	−	0.663456i	$-0.230911\pi$
$647$	38.0635	1.49643	0.748215	+	0.663456i	$0.230911\pi$
$648$	0	0
$649$	−8.29862	−0.325750
$650$	0	0
$651$	0	0
$652$	−6.47689	−0.253654
$653$	45.2682	1.77148	0.885742	−	0.464179i	$-0.153650\pi$
$653$	45.2682	1.77148	0.885742	+	0.464179i	$0.153650\pi$
$654$	0	0
$655$	0	0
$656$	−9.52311	−0.371815
$657$	0	0
$658$	9.00958	0.351230
$659$	−40.7466	−1.58726	−0.793630	−	0.608400i	$-0.791812\pi$
$659$	−40.7466	−1.58726	−0.793630	+	0.608400i	$0.791812\pi$
$660$	0	0
$661$	2.53270	0.0985106	0.0492553	−	0.998786i	$-0.484315\pi$
$661$	2.53270	0.0985106	0.0492553	+	0.998786i	$0.484315\pi$
$662$	4.02021	0.156250
$663$	0	0
$664$	−4.20617	−0.163231
$665$	0	0
$666$	0	0
$667$	29.7572	1.15220
$668$	−3.84632	−0.148819
$669$	0	0
$670$	0	0
$671$	−6.56934	−0.253607
$672$	0	0
$673$	37.0818	1.42940	0.714700	−	0.699431i	$-0.246563\pi$
$673$	37.0818	1.42940	0.714700	+	0.699431i	$0.246563\pi$
$674$	2.21386	0.0852748
$675$	0	0
$676$	−6.10308	−0.234734
$677$	−44.3790	−1.70562	−0.852812	−	0.522218i	$-0.825105\pi$
$677$	−44.3790	−1.70562	−0.852812	+	0.522218i	$0.825105\pi$
$678$	0	0
$679$	−5.31695	−0.204046
$680$	0	0
$681$	0	0
$682$	−18.0279	−0.690324
$683$	7.46626	0.285688	0.142844	−	0.989745i	$-0.454375\pi$
$683$	7.46626	0.285688	0.142844	+	0.989745i	$0.454375\pi$
$684$	0	0
$685$	0	0
$686$	11.4586	0.437490
$687$	0	0
$688$	−7.25240	−0.276495
$689$	29.9431	1.14074
$690$	0	0
$691$	26.8959	1.02317	0.511584	−	0.859233i	$-0.329059\pi$
$691$	26.8959	1.02317	0.511584	+	0.859233i	$0.329059\pi$
$692$	−23.9109	−0.908955
$693$	0	0
$694$	1.37380	0.0521488
$695$	0	0
$696$	0	0
$697$	−9.52311	−0.360714
$698$	−20.8680	−0.789864
$699$	0	0
$700$	0	0
$701$	−23.7938	−0.898681	−0.449340	−	0.893361i	$-0.648341\pi$
$701$	−23.7938	−0.898681	−0.449340	+	0.893361i	$0.648341\pi$
$702$	0	0
$703$	9.13203	0.344421
$704$	−2.00000	−0.0753778
$705$	0	0
$706$	13.7572	0.517759
$707$	0.178261	0.00670419
$708$	0	0
$709$	−17.0140	−0.638972	−0.319486	−	0.947591i	$-0.603510\pi$
$709$	−17.0140	−0.638972	−0.319486	+	0.947591i	$0.603510\pi$
$710$	0	0
$711$	0	0
$712$	−4.14931	−0.155502
$713$	−28.2620	−1.05842
$714$	0	0
$715$	0	0
$716$	14.7110	0.549774
$717$	0	0
$718$	−9.31695	−0.347705
$719$	23.8665	0.890071	0.445036	−	0.895513i	$-0.353191\pi$
$719$	23.8665	0.890071	0.445036	+	0.895513i	$0.353191\pi$
$720$	0	0
$721$	4.12910	0.153776
$722$	−18.1955	−0.677168
$723$	0	0
$724$	−23.4340	−0.870917
$725$	0	0
$726$	0	0
$727$	28.3834	1.05268	0.526341	−	0.850274i	$-0.323564\pi$
$727$	28.3834	1.05268	0.526341	+	0.850274i	$0.323564\pi$
$728$	−2.27072	−0.0841584
$729$	0	0
$730$	0	0
$731$	−7.25240	−0.268240
$732$	0	0
$733$	28.2216	1.04239	0.521194	−	0.853439i	$-0.325487\pi$
$733$	28.2216	1.04239	0.521194	+	0.853439i	$0.325487\pi$
$734$	12.3878	0.457240
$735$	0	0
$736$	−3.13536	−0.115571
$737$	2.50479	0.0922652
$738$	0	0
$739$	0.226378	0.00832746	0.00416373	−	0.999991i	$-0.498675\pi$
$739$	0.226378	0.00832746	0.00416373	+	0.999991i	$0.498675\pi$
$740$	0	0
$741$	0	0
$742$	−9.85838	−0.361913
$743$	−47.6035	−1.74640	−0.873202	−	0.487359i	$-0.837960\pi$
$743$	−47.6035	−1.74640	−0.873202	+	0.487359i	$0.837960\pi$
$744$	0	0
$745$	0	0
$746$	−23.3169	−0.853694
$747$	0	0
$748$	−2.00000	−0.0731272
$749$	−5.44605	−0.198994
$750$	0	0
$751$	0.124733	0.00455157	0.00227578	−	0.999997i	$-0.499276\pi$
$751$	0.124733	0.00455157	0.00227578	+	0.999997i	$0.499276\pi$
$752$	−10.4200	−0.379979
$753$	0	0
$754$	−24.9248	−0.907709
$755$	0	0
$756$	0	0
$757$	33.7774	1.22766	0.613830	−	0.789438i	$-0.289628\pi$
$757$	33.7774	1.22766	0.613830	+	0.789438i	$0.289628\pi$
$758$	28.9817	1.05266
$759$	0	0
$760$	0	0
$761$	−11.4219	−0.414044	−0.207022	−	0.978336i	$-0.566377\pi$
$761$	−11.4219	−0.414044	−0.207022	+	0.978336i	$0.566377\pi$
$762$	0	0
$763$	−4.80342	−0.173895
$764$	−18.5048	−0.669480
$765$	0	0
$766$	−14.9527	−0.540264
$767$	10.8969	0.393465
$768$	0	0
$769$	12.1127	0.436794	0.218397	−	0.975860i	$-0.429917\pi$
$769$	12.1127	0.436794	0.218397	+	0.975860i	$0.429917\pi$
$770$	0	0
$771$	0	0
$772$	−5.45856	−0.196458
$773$	−37.4586	−1.34729	−0.673645	−	0.739055i	$-0.735272\pi$
$773$	−37.4586	−1.34729	−0.673645	+	0.739055i	$0.735272\pi$
$774$	0	0
$775$	0	0
$776$	6.14931	0.220747
$777$	0	0
$778$	19.3169	0.692546
$779$	8.54144	0.306029
$780$	0	0
$781$	−25.0741	−0.897223
$782$	−3.13536	−0.112120
$783$	0	0
$784$	−6.25240	−0.223300
$785$	0	0
$786$	0	0
$787$	−25.0664	−0.893522	−0.446761	−	0.894653i	$-0.647423\pi$
$787$	−25.0664	−0.893522	−0.446761	+	0.894653i	$0.647423\pi$
$788$	−13.9388	−0.496548
$789$	0	0
$790$	0	0
$791$	−9.18785	−0.326682
$792$	0	0
$793$	8.62620	0.306325
$794$	9.57560	0.339825
$795$	0	0
$796$	−11.2847	−0.399975
$797$	19.5510	0.692533	0.346266	−	0.938136i	$-0.387449\pi$
$797$	19.5510	0.692533	0.346266	+	0.938136i	$0.387449\pi$
$798$	0	0
$799$	−10.4200	−0.368634
$800$	0	0
$801$	0	0
$802$	−11.3169	−0.399615
$803$	−5.25240	−0.185353
$804$	0	0
$805$	0	0
$806$	23.6724	0.833826
$807$	0	0
$808$	−0.206167	−0.00725294
$809$	−20.1974	−0.710104	−0.355052	−	0.934847i	$-0.615537\pi$
$809$	−20.1974	−0.710104	−0.355052	+	0.934847i	$0.615537\pi$
$810$	0	0
$811$	28.1570	0.988726	0.494363	−	0.869255i	$-0.335401\pi$
$811$	28.1570	0.988726	0.494363	+	0.869255i	$0.335401\pi$
$812$	8.20617	0.287980
$813$	0	0
$814$	20.3632	0.713729
$815$	0	0
$816$	0	0
$817$	6.50479	0.227574
$818$	−9.10308	−0.318282
$819$	0	0
$820$	0	0
$821$	51.2759	1.78954	0.894771	−	0.446525i	$-0.147339\pi$
$821$	51.2759	1.78954	0.894771	+	0.446525i	$0.147339\pi$
$822$	0	0
$823$	52.9850	1.84694	0.923471	−	0.383669i	$-0.125340\pi$
$823$	52.9850	1.84694	0.923471	+	0.383669i	$0.125340\pi$
$824$	−4.77551	−0.166363
$825$	0	0
$826$	−3.58767	−0.124831
$827$	−39.7205	−1.38122	−0.690609	−	0.723228i	$-0.742658\pi$
$827$	−39.7205	−1.38122	−0.690609	+	0.723228i	$0.742658\pi$
$828$	0	0
$829$	23.0096	0.799156	0.399578	−	0.916699i	$-0.369157\pi$
$829$	23.0096	0.799156	0.399578	+	0.916699i	$0.369157\pi$
$830$	0	0
$831$	0	0
$832$	2.62620	0.0910470
$833$	−6.25240	−0.216633
$834$	0	0
$835$	0	0
$836$	1.79383	0.0620410
$837$	0	0
$838$	4.02791	0.139142
$839$	−2.39545	−0.0827002	−0.0413501	−	0.999145i	$-0.513166\pi$
$839$	−2.39545	−0.0827002	−0.0413501	+	0.999145i	$0.513166\pi$
$840$	0	0
$841$	61.0760	2.10607
$842$	15.2524	0.525632
$843$	0	0
$844$	−14.8401	−0.510816
$845$	0	0
$846$	0	0
$847$	6.05249	0.207966
$848$	11.4017	0.391536
$849$	0	0
$850$	0	0
$851$	31.9229	1.09430
$852$	0	0
$853$	−2.24614	−0.0769063	−0.0384532	−	0.999260i	$-0.512243\pi$
$853$	−2.24614	−0.0769063	−0.0384532	+	0.999260i	$0.512243\pi$
$854$	−2.84006	−0.0971849
$855$	0	0
$856$	6.29862	0.215283
$857$	18.3188	0.625760	0.312880	−	0.949793i	$-0.398706\pi$
$857$	18.3188	0.625760	0.312880	+	0.949793i	$0.398706\pi$
$858$	0	0
$859$	−42.1127	−1.43687	−0.718433	−	0.695596i	$-0.755140\pi$
$859$	−42.1127	−1.43687	−0.718433	+	0.695596i	$0.755140\pi$
$860$	0	0
$861$	0	0
$862$	4.45231	0.151646
$863$	−44.8313	−1.52608	−0.763038	−	0.646354i	$-0.776293\pi$
$863$	−44.8313	−1.52608	−0.763038	+	0.646354i	$0.776293\pi$
$864$	0	0
$865$	0	0
$866$	10.7110	0.363973
$867$	0	0
$868$	−7.79383	−0.264540
$869$	15.8217	0.536716
$870$	0	0
$871$	−3.28904	−0.111445
$872$	5.55539	0.188129
$873$	0	0
$874$	2.81215	0.0951226
$875$	0	0
$876$	0	0
$877$	2.29530	0.0775067	0.0387533	−	0.999249i	$-0.487661\pi$
$877$	2.29530	0.0775067	0.0387533	+	0.999249i	$0.487661\pi$
$878$	5.09871	0.172073
$879$	0	0
$880$	0	0
$881$	20.9171	0.704716	0.352358	−	0.935865i	$-0.385380\pi$
$881$	20.9171	0.704716	0.352358	+	0.935865i	$0.385380\pi$
$882$	0	0
$883$	−26.6743	−0.897662	−0.448831	−	0.893617i	$-0.648160\pi$
$883$	−26.6743	−0.897662	−0.448831	+	0.893617i	$0.648160\pi$
$884$	2.62620	0.0883286
$885$	0	0
$886$	−10.8034	−0.362948
$887$	29.1633	0.979207	0.489603	−	0.871945i	$-0.337142\pi$
$887$	29.1633	0.979207	0.489603	+	0.871945i	$0.337142\pi$
$888$	0	0
$889$	−12.4123	−0.416296
$890$	0	0
$891$	0	0
$892$	3.30925	0.110802
$893$	9.34590	0.312748
$894$	0	0
$895$	0	0
$896$	−0.864641	−0.0288856
$897$	0	0
$898$	5.82174	0.194274
$899$	−85.5500	−2.85325
$900$	0	0
$901$	11.4017	0.379846
$902$	19.0462	0.634170
$903$	0	0
$904$	10.6262	0.353422
$905$	0	0
$906$	0	0
$907$	23.2322	0.771412	0.385706	−	0.922622i	$-0.373958\pi$
$907$	23.2322	0.771412	0.385706	+	0.922622i	$0.373958\pi$
$908$	24.7187	0.820317
$909$	0	0
$910$	0	0
$911$	20.8646	0.691276	0.345638	−	0.938368i	$-0.387662\pi$
$911$	20.8646	0.691276	0.345638	+	0.938368i	$0.387662\pi$
$912$	0	0
$913$	8.41233	0.278408
$914$	13.7014	0.453201
$915$	0	0
$916$	4.27072	0.141109
$917$	14.7389	0.486720
$918$	0	0
$919$	18.2062	0.600566	0.300283	−	0.953850i	$-0.402919\pi$
$919$	18.2062	0.600566	0.300283	+	0.953850i	$0.402919\pi$
$920$	0	0
$921$	0	0
$922$	27.0741	0.891639
$923$	32.9248	1.08373
$924$	0	0
$925$	0	0
$926$	37.3372	1.22698
$927$	0	0
$928$	−9.49084	−0.311552
$929$	−17.2803	−0.566948	−0.283474	−	0.958980i	$-0.591487\pi$
$929$	−17.2803	−0.566948	−0.283474	+	0.958980i	$0.591487\pi$
$930$	0	0
$931$	5.60788	0.183791
$932$	−0.832365	−0.0272650
$933$	0	0
$934$	−35.3449	−1.15652
$935$	0	0
$936$	0	0
$937$	50.2986	1.64318	0.821592	−	0.570076i	$-0.193086\pi$
$937$	50.2986	1.64318	0.821592	+	0.570076i	$0.193086\pi$
$938$	1.08287	0.0353571
$939$	0	0
$940$	0	0
$941$	−20.7153	−0.675300	−0.337650	−	0.941272i	$-0.609632\pi$
$941$	−20.7153	−0.675300	−0.337650	+	0.941272i	$0.609632\pi$
$942$	0	0
$943$	29.8584	0.972323
$944$	4.14931	0.135049
$945$	0	0
$946$	14.5048	0.471591
$947$	8.89692	0.289111	0.144555	−	0.989497i	$-0.453825\pi$
$947$	8.89692	0.289111	0.144555	+	0.989497i	$0.453825\pi$
$948$	0	0
$949$	6.89692	0.223883
$950$	0	0
$951$	0	0
$952$	−0.864641	−0.0280232
$953$	−17.2158	−0.557673	−0.278836	−	0.960339i	$-0.589949\pi$
$953$	−17.2158	−0.557673	−0.278836	+	0.960339i	$0.589949\pi$
$954$	0	0
$955$	0	0
$956$	4.71096	0.152363
$957$	0	0
$958$	−31.0419	−1.00292
$959$	−12.5414	−0.404984
$960$	0	0
$961$	50.2514	1.62101
$962$	−26.7389	−0.862096
$963$	0	0
$964$	−23.4865	−0.756448
$965$	0	0
$966$	0	0
$967$	1.90754	0.0613424	0.0306712	−	0.999530i	$-0.490236\pi$
$967$	1.90754	0.0613424	0.0306712	+	0.999530i	$0.490236\pi$
$968$	−7.00000	−0.224989
$969$	0	0
$970$	0	0
$971$	4.97398	0.159623	0.0798113	−	0.996810i	$-0.474568\pi$
$971$	4.97398	0.159623	0.0798113	+	0.996810i	$0.474568\pi$
$972$	0	0
$973$	12.7197	0.407775
$974$	13.1633	0.421778
$975$	0	0
$976$	3.28467	0.105140
$977$	−51.2716	−1.64032	−0.820161	−	0.572132i	$-0.806116\pi$
$977$	−51.2716	−1.64032	−0.820161	+	0.572132i	$0.806116\pi$
$978$	0	0
$979$	8.29862	0.265225
$980$	0	0
$981$	0	0
$982$	−24.3555	−0.777215
$983$	−9.88296	−0.315218	−0.157609	−	0.987502i	$-0.550378\pi$
$983$	−9.88296	−0.315218	−0.157609	+	0.987502i	$0.550378\pi$
$984$	0	0
$985$	0	0
$986$	−9.49084	−0.302250
$987$	0	0
$988$	−2.35548	−0.0749378
$989$	22.7389	0.723054
$990$	0	0
$991$	−37.7807	−1.20014	−0.600072	−	0.799946i	$-0.704861\pi$
$991$	−37.7807	−1.20014	−0.600072	+	0.799946i	$0.704861\pi$
$992$	9.01395	0.286193
$993$	0	0
$994$	−10.8401	−0.343826
$995$	0	0
$996$	0	0
$997$	−8.14494	−0.257953	−0.128976	−	0.991648i	$-0.541169\pi$
$997$	−8.14494	−0.257953	−0.128976	+	0.991648i	$0.541169\pi$
$998$	36.2620	1.14785
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7650.2.a.do.1.2		3
3.2	odd	2		850.2.a.p.1.3		3
5.2	odd	4		1530.2.d.g.919.6		6
5.3	odd	4		1530.2.d.g.919.3		6
5.4	even	2		7650.2.a.dj.1.2		3
12.11	even	2		6800.2.a.bp.1.1		3
15.2	even	4		170.2.c.b.69.1	✓	6
15.8	even	4		170.2.c.b.69.6	yes	6
15.14	odd	2		850.2.a.q.1.1		3
60.23	odd	4		1360.2.e.c.1089.2		6
60.47	odd	4		1360.2.e.c.1089.5		6
60.59	even	2		6800.2.a.bk.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
170.2.c.b.69.1	✓	6	15.2	even	4
170.2.c.b.69.6	yes	6	15.8	even	4
850.2.a.p.1.3		3	3.2	odd	2
850.2.a.q.1.1		3	15.14	odd	2
1360.2.e.c.1089.2		6	60.23	odd	4
1360.2.e.c.1089.5		6	60.47	odd	4
1530.2.d.g.919.3		6	5.3	odd	4
1530.2.d.g.919.6		6	5.2	odd	4
6800.2.a.bk.1.3		3	60.59	even	2
6800.2.a.bp.1.1		3	12.11	even	2
7650.2.a.dj.1.2		3	5.4	even	2
7650.2.a.do.1.2		3	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:	\( N \)	\(=\)	\( 7650 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7650.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -0.864641 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -0.864641 q^{7} +1.00000 q^{8} -2.00000 q^{11} +2.62620 q^{13} -0.864641 q^{14} +1.00000 q^{16} +1.00000 q^{17} -0.896916 q^{19} -2.00000 q^{22} -3.13536 q^{23} +2.62620 q^{26} -0.864641 q^{28} -9.49084 q^{29} +9.01395 q^{31} +1.00000 q^{32} +1.00000 q^{34} -10.1816 q^{37} -0.896916 q^{38} -9.52311 q^{41} -7.25240 q^{43} -2.00000 q^{44} -3.13536 q^{46} -10.4200 q^{47} -6.25240 q^{49} +2.62620 q^{52} +11.4017 q^{53} -0.864641 q^{56} -9.49084 q^{58} +4.14931 q^{59} +3.28467 q^{61} +9.01395 q^{62} +1.00000 q^{64} -1.25240 q^{67} +1.00000 q^{68} +12.5371 q^{71} +2.62620 q^{73} -10.1816 q^{74} -0.896916 q^{76} +1.72928 q^{77} -7.91087 q^{79} -9.52311 q^{82} -4.20617 q^{83} -7.25240 q^{86} -2.00000 q^{88} -4.14931 q^{89} -2.27072 q^{91} -3.13536 q^{92} -10.4200 q^{94} +6.14931 q^{97} -6.25240 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^8	\( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{8} - 6 q^{11} - q^{13} + 3 q^{16} + 3 q^{17} + q^{19} - 6 q^{22} - 12 q^{23} - q^{26} - 17 q^{29} + 3 q^{31} + 3 q^{32} + 3 q^{34} - 8 q^{37} + q^{38} - 16 q^{41} - 4 q^{43} - 6 q^{44} - 12 q^{46} - 15 q^{47} - q^{49} - q^{52} - 5 q^{53} - 17 q^{58} - 9 q^{59} - 9 q^{61} + 3 q^{62} + 3 q^{64} + 14 q^{67} + 3 q^{68} + q^{71} - q^{73} - 8 q^{74} + q^{76} + 4 q^{79} - 16 q^{82} - 20 q^{83} - 4 q^{86} - 6 q^{88} + 9 q^{89} - 12 q^{91} - 12 q^{92} - 15 q^{94} - 3 q^{97} - q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^8 - 6 * q^11 - q^13 + 3 * q^16 + 3 * q^17 + q^19 - 6 * q^22 - 12 * q^23 - q^26 - 17 * q^29 + 3 * q^31 + 3 * q^32 + 3 * q^34 - 8 * q^37 + q^38 - 16 * q^41 - 4 * q^43 - 6 * q^44 - 12 * q^46 - 15 * q^47 - q^49 - q^52 - 5 * q^53 - 17 * q^58 - 9 * q^59 - 9 * q^61 + 3 * q^62 + 3 * q^64 + 14 * q^67 + 3 * q^68 + q^71 - q^73 - 8 * q^74 + q^76 + 4 * q^79 - 16 * q^82 - 20 * q^83 - 4 * q^86 - 6 * q^88 + 9 * q^89 - 12 * q^91 - 12 * q^92 - 15 * q^94 - 3 * q^97 - q^98

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