LMFDB - Embedded newform 5625.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5625,2,Mod(1,5625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5625, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5625.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5625 = 3^{2} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5625.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:	$44.9158511370$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.5125.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 6x^{2} + 7x + 11 $ x^4 - 2x^3 - 6x^2 + 7*x + 11
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 75)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.70636$ of defining polynomial
Character	$\chi$	$=$	5625.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.70636 q^{2} +5.32440 q^{4} +0.470294 q^{7} -8.99702 q^{8} +O(q^{10})$	$q-2.70636 q^{2} +5.32440 q^{4} +0.470294 q^{7} -8.99702 q^{8} +3.18148 q^{11} +0.563444 q^{13} -1.27279 q^{14} +13.7004 q^{16} +1.70636 q^{17} +3.74010 q^{19} -8.61023 q^{22} +2.26981 q^{23} -1.52488 q^{26} +2.50403 q^{28} +8.32440 q^{29} +5.43656 q^{31} -19.0842 q^{32} -4.61803 q^{34} +1.02085 q^{37} -10.1221 q^{38} -1.47214 q^{41} +6.72721 q^{43} +16.9395 q^{44} -6.14292 q^{46} +4.43358 q^{47} -6.77882 q^{49} +3.00000 q^{52} +7.05161 q^{53} -4.23125 q^{56} -22.5288 q^{58} +12.8720 q^{59} +0.126888 q^{61} -14.7133 q^{62} +24.2480 q^{64} -2.79469 q^{67} +9.08535 q^{68} -16.0456 q^{71} -9.78873 q^{73} -2.76279 q^{74} +19.9138 q^{76} +1.49623 q^{77} -4.75499 q^{79} +3.98413 q^{82} -11.6905 q^{83} -18.2063 q^{86} -28.6238 q^{88} +6.20049 q^{89} +0.264985 q^{91} +12.0853 q^{92} -11.9989 q^{94} +8.45443 q^{97} +18.3460 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} + 8 q^{4} + 2 q^{7} - 15 q^{8}+O(q^{10}) $ 4 * q - 2 * q^2 + 8 * q^4 + 2 * q^7 - 15 * q^8	$ 4 q - 2 q^{2} + 8 q^{4} + 2 q^{7} - 15 q^{8} + 7 q^{11} + q^{13} - 16 q^{14} + 4 q^{16} - 2 q^{17} + 5 q^{19} - 6 q^{22} - q^{23} - 3 q^{26} + 9 q^{28} + 20 q^{29} + 23 q^{31} - 12 q^{32} - 14 q^{34} + 2 q^{37} - 35 q^{38} + 12 q^{41} + 16 q^{43} + 29 q^{44} - 17 q^{46} - 2 q^{47} + 8 q^{49} + 12 q^{52} + 4 q^{53} - 5 q^{56} - 25 q^{58} + 15 q^{59} - 2 q^{61} - 9 q^{62} + 23 q^{64} + 2 q^{67} + 11 q^{68} + 2 q^{71} + 16 q^{73} + 19 q^{74} + 40 q^{76} - 19 q^{77} + 35 q^{79} - 6 q^{82} - 16 q^{83} - 3 q^{86} - 30 q^{88} + 35 q^{89} - 12 q^{91} + 23 q^{92} - 9 q^{94} + 12 q^{97} + q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 + 8 * q^4 + 2 * q^7 - 15 * q^8 + 7 * q^11 + q^13 - 16 * q^14 + 4 * q^16 - 2 * q^17 + 5 * q^19 - 6 * q^22 - q^23 - 3 * q^26 + 9 * q^28 + 20 * q^29 + 23 * q^31 - 12 * q^32 - 14 * q^34 + 2 * q^37 - 35 * q^38 + 12 * q^41 + 16 * q^43 + 29 * q^44 - 17 * q^46 - 2 * q^47 + 8 * q^49 + 12 * q^52 + 4 * q^53 - 5 * q^56 - 25 * q^58 + 15 * q^59 - 2 * q^61 - 9 * q^62 + 23 * q^64 + 2 * q^67 + 11 * q^68 + 2 * q^71 + 16 * q^73 + 19 * q^74 + 40 * q^76 - 19 * q^77 + 35 * q^79 - 6 * q^82 - 16 * q^83 - 3 * q^86 - 30 * q^88 + 35 * q^89 - 12 * q^91 + 23 * q^92 - 9 * q^94 + 12 * 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 $

$2$	−2.70636	−1.91369	−0.956844	−	0.290604i	$-0.906144\pi$
$2$	−2.70636	−1.91369	−0.956844	+	0.290604i	$0.906144\pi$
$3$	0	0
$3$	0	0
$4$	5.32440	2.66220
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.470294	0.177754	0.0888772	−	0.996043i	$-0.471672\pi$
$7$	0.470294	0.177754	0.0888772	+	0.996043i	$0.471672\pi$
$8$	−8.99702	−3.18093
$9$	0	0
$10$	0	0
$11$	3.18148	0.959252	0.479626	−	0.877473i	$-0.340772\pi$
$11$	3.18148	0.959252	0.479626	+	0.877473i	$0.340772\pi$
$12$	0	0
$13$	0.563444	0.156271	0.0781356	−	0.996943i	$-0.475103\pi$
$13$	0.563444	0.156271	0.0781356	+	0.996943i	$0.475103\pi$
$14$	−1.27279	−0.340166
$15$	0	0
$16$	13.7004	3.42510
$17$	1.70636	0.413854	0.206927	−	0.978356i	$-0.433654\pi$
$17$	1.70636	0.413854	0.206927	+	0.978356i	$0.433654\pi$
$18$	0	0
$19$	3.74010	0.858038	0.429019	−	0.903295i	$-0.358859\pi$
$19$	3.74010	0.858038	0.429019	+	0.903295i	$0.358859\pi$
$20$	0	0
$21$	0	0
$22$	−8.61023	−1.83571
$23$	2.26981	0.473287	0.236644	−	0.971597i	$-0.423953\pi$
$23$	2.26981	0.473287	0.236644	+	0.971597i	$0.423953\pi$
$24$	0	0
$25$	0	0
$26$	−1.52488	−0.299054
$27$	0	0
$28$	2.50403	0.473218
$29$	8.32440	1.54580	0.772901	−	0.634527i	$-0.218805\pi$
$29$	8.32440	1.54580	0.772901	+	0.634527i	$0.218805\pi$
$30$	0	0
$31$	5.43656	0.976434	0.488217	−	0.872722i	$-0.337647\pi$
$31$	5.43656	0.976434	0.488217	+	0.872722i	$0.337647\pi$
$32$	−19.0842	−3.37364
$33$	0	0
$34$	−4.61803	−0.791986
$35$	0	0
$36$	0	0
$37$	1.02085	0.167827	0.0839135	−	0.996473i	$-0.473258\pi$
$37$	1.02085	0.167827	0.0839135	+	0.996473i	$0.473258\pi$
$38$	−10.1221	−1.64202
$39$	0	0
$40$	0	0
$41$	−1.47214	−0.229909	−0.114955	−	0.993371i	$-0.536672\pi$
$41$	−1.47214	−0.229909	−0.114955	+	0.993371i	$0.536672\pi$
$42$	0	0
$43$	6.72721	1.02589	0.512945	−	0.858421i	$-0.328554\pi$
$43$	6.72721	1.02589	0.512945	+	0.858421i	$0.328554\pi$
$44$	16.9395	2.55372
$45$	0	0
$46$	−6.14292	−0.905724
$47$	4.43358	0.646703	0.323352	−	0.946279i	$-0.395190\pi$
$47$	4.43358	0.646703	0.323352	+	0.946279i	$0.395190\pi$
$48$	0	0
$49$	−6.77882	−0.968403
$50$	0	0
$51$	0	0
$52$	3.00000	0.416025
$53$	7.05161	0.968613	0.484307	−	0.874898i	$-0.339072\pi$
$53$	7.05161	0.968613	0.484307	+	0.874898i	$0.339072\pi$
$54$	0	0
$55$	0	0
$56$	−4.23125	−0.565424
$57$	0	0
$58$	−22.5288	−2.95818
$59$	12.8720	1.67579	0.837894	−	0.545833i	$-0.183787\pi$
$59$	12.8720	1.67579	0.837894	+	0.545833i	$0.183787\pi$
$60$	0	0
$61$	0.126888	0.0162464	0.00812319	−	0.999967i	$-0.497414\pi$
$61$	0.126888	0.0162464	0.00812319	+	0.999967i	$0.497414\pi$
$62$	−14.7133	−1.86859
$63$	0	0
$64$	24.2480	3.03100
$65$	0	0
$66$	0	0
$67$	−2.79469	−0.341426	−0.170713	−	0.985321i	$-0.554607\pi$
$67$	−2.79469	−0.341426	−0.170713	+	0.985321i	$0.554607\pi$
$68$	9.08535	1.10176
$69$	0	0
$70$	0	0
$71$	−16.0456	−1.90427	−0.952134	−	0.305681i	$-0.901116\pi$
$71$	−16.0456	−1.90427	−0.952134	+	0.305681i	$0.901116\pi$
$72$	0	0
$73$	−9.78873	−1.14568	−0.572842	−	0.819666i	$-0.694159\pi$
$73$	−9.78873	−1.14568	−0.572842	+	0.819666i	$0.694159\pi$
$74$	−2.76279	−0.321168
$75$	0	0
$76$	19.9138	2.28427
$77$	1.49623	0.170511
$78$	0	0
$79$	−4.75499	−0.534978	−0.267489	−	0.963561i	$-0.586194\pi$
$79$	−4.75499	−0.534978	−0.267489	+	0.963561i	$0.586194\pi$
$80$	0	0
$81$	0	0
$82$	3.98413	0.439974
$83$	−11.6905	−1.28320	−0.641599	−	0.767040i	$-0.721729\pi$
$83$	−11.6905	−1.28320	−0.641599	+	0.767040i	$0.721729\pi$
$84$	0	0
$85$	0	0
$86$	−18.2063	−1.96323
$87$	0	0
$88$	−28.6238	−3.05131
$89$	6.20049	0.657250	0.328625	−	0.944460i	$-0.393415\pi$
$89$	6.20049	0.657250	0.328625	+	0.944460i	$0.393415\pi$
$90$	0	0
$91$	0.264985	0.0277779
$92$	12.0853	1.25998
$93$	0	0
$94$	−11.9989	−1.23759
$95$	0	0
$96$	0	0
$97$	8.45443	0.858417	0.429209	−	0.903205i	$-0.358793\pi$
$97$	8.45443	0.858417	0.429209	+	0.903205i	$0.358793\pi$
$98$	18.3460	1.85322
$99$	0	0
$100$	0	0
$101$	5.83325	0.580430	0.290215	−	0.956961i	$-0.406273\pi$
$101$	5.83325	0.580430	0.290215	+	0.956961i	$0.406273\pi$
$102$	0	0
$103$	−14.0794	−1.38728	−0.693642	−	0.720320i	$-0.743995\pi$
$103$	−14.0794	−1.38728	−0.693642	+	0.720320i	$0.743995\pi$
$104$	−5.06932	−0.497088
$105$	0	0
$106$	−19.0842	−1.85362
$107$	8.61207	0.832561	0.416280	−	0.909236i	$-0.363334\pi$
$107$	8.61207	0.832561	0.416280	+	0.909236i	$0.363334\pi$
$108$	0	0
$109$	16.6875	1.59837	0.799187	−	0.601082i	$-0.205264\pi$
$109$	16.6875	1.59837	0.799187	+	0.601082i	$0.205264\pi$
$110$	0	0
$111$	0	0
$112$	6.44322	0.608827
$113$	−11.7758	−1.10778	−0.553889	−	0.832590i	$-0.686857\pi$
$113$	−11.7758	−1.10778	−0.553889	+	0.832590i	$0.686857\pi$
$114$	0	0
$115$	0	0
$116$	44.3224	4.11523
$117$	0	0
$118$	−34.8362	−3.20693
$119$	0.802492	0.0735643
$120$	0	0
$121$	−0.878197	−0.0798361
$122$	−0.343406	−0.0310905
$123$	0	0
$124$	28.9464	2.59946
$125$	0	0
$126$	0	0
$127$	1.77882	0.157845	0.0789225	−	0.996881i	$-0.474852\pi$
$127$	1.77882	0.157845	0.0789225	+	0.996881i	$0.474852\pi$
$128$	−27.4554	−2.42674
$129$	0	0
$130$	0	0
$131$	5.91860	0.517110	0.258555	−	0.965996i	$-0.416754\pi$
$131$	5.91860	0.517110	0.258555	+	0.965996i	$0.416754\pi$
$132$	0	0
$133$	1.75895	0.152520
$134$	7.56344	0.653382
$135$	0	0
$136$	−15.3522	−1.31644
$137$	5.11611	0.437098	0.218549	−	0.975826i	$-0.429868\pi$
$137$	5.11611	0.437098	0.218549	+	0.975826i	$0.429868\pi$
$138$	0	0
$139$	−8.23817	−0.698753	−0.349376	−	0.936982i	$-0.613607\pi$
$139$	−8.23817	−0.698753	−0.349376	+	0.936982i	$0.613607\pi$
$140$	0	0
$141$	0	0
$142$	43.4253	3.64417
$143$	1.79259	0.149904
$144$	0	0
$145$	0	0
$146$	26.4918	2.19248
$147$	0	0
$148$	5.43542	0.446789
$149$	10.0585	0.824027	0.412014	−	0.911178i	$-0.364826\pi$
$149$	10.0585	0.824027	0.412014	+	0.911178i	$0.364826\pi$
$150$	0	0
$151$	−11.9810	−0.974999	−0.487500	−	0.873123i	$-0.662091\pi$
$151$	−11.9810	−0.974999	−0.487500	+	0.873123i	$0.662091\pi$
$152$	−33.6498	−2.72936
$153$	0	0
$154$	−4.04934	−0.326305
$155$	0	0
$156$	0	0
$157$	19.5960	1.56393	0.781967	−	0.623319i	$-0.214216\pi$
$157$	19.5960	1.56393	0.781967	+	0.623319i	$0.214216\pi$
$158$	12.8687	1.02378
$159$	0	0
$160$	0	0
$161$	1.06748	0.0841290
$162$	0	0
$163$	11.8927	0.931505	0.465753	−	0.884915i	$-0.345784\pi$
$163$	11.8927	0.931505	0.465753	+	0.884915i	$0.345784\pi$
$164$	−7.83824	−0.612063
$165$	0	0
$166$	31.6387	2.45564
$167$	5.99378	0.463812	0.231906	−	0.972738i	$-0.425504\pi$
$167$	5.99378	0.463812	0.231906	+	0.972738i	$0.425504\pi$
$168$	0	0
$169$	−12.6825	−0.975579
$170$	0	0
$171$	0	0
$172$	35.8184	2.73112
$173$	−17.3244	−1.31715	−0.658575	−	0.752515i	$-0.728840\pi$
$173$	−17.3244	−1.31715	−0.658575	+	0.752515i	$0.728840\pi$
$174$	0	0
$175$	0	0
$176$	43.5875	3.28553
$177$	0	0
$178$	−16.7808	−1.25777
$179$	−10.5765	−0.790524	−0.395262	−	0.918568i	$-0.629346\pi$
$179$	−10.5765	−0.790524	−0.395262	+	0.918568i	$0.629346\pi$
$180$	0	0
$181$	11.9799	0.890455	0.445228	−	0.895417i	$-0.353123\pi$
$181$	11.9799	0.890455	0.445228	+	0.895417i	$0.353123\pi$
$182$	−0.717144	−0.0531583
$183$	0	0
$184$	−20.4215	−1.50549
$185$	0	0
$186$	0	0
$187$	5.42875	0.396990
$188$	23.6061	1.72165
$189$	0	0
$190$	0	0
$191$	−18.2657	−1.32166	−0.660829	−	0.750536i	$-0.729795\pi$
$191$	−18.2657	−1.32166	−0.660829	+	0.750536i	$0.729795\pi$
$192$	0	0
$193$	25.2063	1.81439	0.907194	−	0.420713i	$-0.138220\pi$
$193$	25.2063	1.81439	0.907194	+	0.420713i	$0.138220\pi$
$194$	−22.8807	−1.64274
$195$	0	0
$196$	−36.0931	−2.57808
$197$	−12.0923	−0.861539	−0.430769	−	0.902462i	$-0.641758\pi$
$197$	−12.0923	−0.861539	−0.430769	+	0.902462i	$0.641758\pi$
$198$	0	0
$199$	−0.544434	−0.0385939	−0.0192970	−	0.999814i	$-0.506143\pi$
$199$	−0.544434	−0.0385939	−0.0192970	+	0.999814i	$0.506143\pi$
$200$	0	0
$201$	0	0
$202$	−15.7869	−1.11076
$203$	3.91492	0.274773
$204$	0	0
$205$	0	0
$206$	38.1039	2.65483
$207$	0	0
$208$	7.71941	0.535245
$209$	11.8990	0.823074
$210$	0	0
$211$	−3.12207	−0.214932	−0.107466	−	0.994209i	$-0.534274\pi$
$211$	−3.12207	−0.214932	−0.107466	+	0.994209i	$0.534274\pi$
$212$	37.5456	2.57864
$213$	0	0
$214$	−23.3074	−1.59326
$215$	0	0
$216$	0	0
$217$	2.55678	0.173566
$218$	−45.1625	−3.05879
$219$	0	0
$220$	0	0
$221$	0.961440	0.0646734
$222$	0	0
$223$	0.339125	0.0227095	0.0113547	−	0.999936i	$-0.496386\pi$
$223$	0.339125	0.0227095	0.0113547	+	0.999936i	$0.496386\pi$
$224$	−8.97519	−0.599680
$225$	0	0
$226$	31.8697	2.11994
$227$	1.78847	0.118705	0.0593524	−	0.998237i	$-0.481096\pi$
$227$	1.78847	0.118705	0.0593524	+	0.998237i	$0.481096\pi$
$228$	0	0
$229$	−27.9833	−1.84919	−0.924593	−	0.380957i	$-0.875595\pi$
$229$	−27.9833	−1.84919	−0.924593	+	0.380957i	$0.875595\pi$
$230$	0	0
$231$	0	0
$232$	−74.8948	−4.91708
$233$	6.77400	0.443780	0.221890	−	0.975072i	$-0.428777\pi$
$233$	6.77400	0.443780	0.221890	+	0.975072i	$0.428777\pi$
$234$	0	0
$235$	0	0
$236$	68.5355	4.46128
$237$	0	0
$238$	−2.17183	−0.140779
$239$	6.48334	0.419373	0.209686	−	0.977769i	$-0.432756\pi$
$239$	6.48334	0.419373	0.209686	+	0.977769i	$0.432756\pi$
$240$	0	0
$241$	−7.44857	−0.479804	−0.239902	−	0.970797i	$-0.577115\pi$
$241$	−7.44857	−0.479804	−0.239902	+	0.970797i	$0.577115\pi$
$242$	2.37672	0.152781
$243$	0	0
$244$	0.675604	0.0432511
$245$	0	0
$246$	0	0
$247$	2.10734	0.134087
$248$	−48.9128	−3.10597
$249$	0	0
$250$	0	0
$251$	4.60217	0.290486	0.145243	−	0.989396i	$-0.453604\pi$
$251$	4.60217	0.290486	0.145243	+	0.989396i	$0.453604\pi$
$252$	0	0
$253$	7.22134	0.454002
$254$	−4.81414	−0.302066
$255$	0	0
$256$	25.8083	1.61302
$257$	5.79485	0.361473	0.180736	−	0.983532i	$-0.442152\pi$
$257$	5.79485	0.361473	0.180736	+	0.983532i	$0.442152\pi$
$258$	0	0
$259$	0.480101	0.0298320
$260$	0	0
$261$	0	0
$262$	−16.0179	−0.989587
$263$	−14.2989	−0.881707	−0.440854	−	0.897579i	$-0.645324\pi$
$263$	−14.2989	−0.881707	−0.440854	+	0.897579i	$0.645324\pi$
$264$	0	0
$265$	0	0
$266$	−4.76035	−0.291876
$267$	0	0
$268$	−14.8800	−0.908943
$269$	19.2205	1.17189	0.585946	−	0.810350i	$-0.300723\pi$
$269$	19.2205	1.17189	0.585946	+	0.810350i	$0.300723\pi$
$270$	0	0
$271$	−23.2433	−1.41193	−0.705964	−	0.708248i	$-0.749486\pi$
$271$	−23.2433	−1.41193	−0.705964	+	0.708248i	$0.749486\pi$
$272$	23.3778	1.41749
$273$	0	0
$274$	−13.8460	−0.836470
$275$	0	0
$276$	0	0
$277$	21.8910	1.31530	0.657651	−	0.753323i	$-0.271550\pi$
$277$	21.8910	1.31530	0.657651	+	0.753323i	$0.271550\pi$
$278$	22.2955	1.33719
$279$	0	0
$280$	0	0
$281$	−27.1138	−1.61748	−0.808738	−	0.588169i	$-0.799849\pi$
$281$	−27.1138	−1.61748	−0.808738	+	0.588169i	$0.799849\pi$
$282$	0	0
$283$	16.3074	0.969374	0.484687	−	0.874688i	$-0.338934\pi$
$283$	16.3074	0.969374	0.484687	+	0.874688i	$0.338934\pi$
$284$	−85.4334	−5.06954
$285$	0	0
$286$	−4.85139	−0.286868
$287$	−0.692337	−0.0408674
$288$	0	0
$289$	−14.0883	−0.828725
$290$	0	0
$291$	0	0
$292$	−52.1191	−3.05004
$293$	−5.46407	−0.319214	−0.159607	−	0.987181i	$-0.551023\pi$
$293$	−5.46407	−0.319214	−0.159607	+	0.987181i	$0.551023\pi$
$294$	0	0
$295$	0	0
$296$	−9.18462	−0.533845
$297$	0	0
$298$	−27.2220	−1.57693
$299$	1.27891	0.0739612
$300$	0	0
$301$	3.16377	0.182357
$302$	32.4249	1.86584
$303$	0	0
$304$	51.2409	2.93887
$305$	0	0
$306$	0	0
$307$	−21.4194	−1.22247	−0.611235	−	0.791450i	$-0.709327\pi$
$307$	−21.4194	−1.22247	−0.611235	+	0.791450i	$0.709327\pi$
$308$	7.96652	0.453935
$309$	0	0
$310$	0	0
$311$	2.85200	0.161722	0.0808610	−	0.996725i	$-0.474233\pi$
$311$	2.85200	0.161722	0.0808610	+	0.996725i	$0.474233\pi$
$312$	0	0
$313$	14.4375	0.816057	0.408029	−	0.912969i	$-0.366216\pi$
$313$	14.4375	0.816057	0.408029	+	0.912969i	$0.366216\pi$
$314$	−53.0340	−2.99288
$315$	0	0
$316$	−25.3175	−1.42422
$317$	−22.6102	−1.26992	−0.634959	−	0.772546i	$-0.718983\pi$
$317$	−22.6102	−1.26992	−0.634959	+	0.772546i	$0.718983\pi$
$318$	0	0
$319$	26.4839	1.48281
$320$	0	0
$321$	0	0
$322$	−2.88898	−0.160996
$323$	6.38197	0.355102
$324$	0	0
$325$	0	0
$326$	−32.1859	−1.78261
$327$	0	0
$328$	13.2448	0.731324
$329$	2.08508	0.114954
$330$	0	0
$331$	3.86645	0.212519	0.106260	−	0.994338i	$-0.466113\pi$
$331$	3.86645	0.212519	0.106260	+	0.994338i	$0.466113\pi$
$332$	−62.2448	−3.41613
$333$	0	0
$334$	−16.2213	−0.887592
$335$	0	0
$336$	0	0
$337$	17.4853	0.952484	0.476242	−	0.879314i	$-0.341999\pi$
$337$	17.4853	0.952484	0.476242	+	0.879314i	$0.341999\pi$
$338$	34.3235	1.86695
$339$	0	0
$340$	0	0
$341$	17.2963	0.936646
$342$	0	0
$343$	−6.48010	−0.349893
$344$	−60.5249	−3.26328
$345$	0	0
$346$	46.8861	2.52061
$347$	23.0941	1.23976	0.619879	−	0.784698i	$-0.287182\pi$
$347$	23.0941	1.23976	0.619879	+	0.784698i	$0.287182\pi$
$348$	0	0
$349$	−9.32650	−0.499236	−0.249618	−	0.968344i	$-0.580305\pi$
$349$	−9.32650	−0.499236	−0.249618	+	0.968344i	$0.580305\pi$
$350$	0	0
$351$	0	0
$352$	−60.7160	−3.23617
$353$	−31.9471	−1.70037	−0.850186	−	0.526483i	$-0.823510\pi$
$353$	−31.9471	−1.70037	−0.850186	+	0.526483i	$0.823510\pi$
$354$	0	0
$355$	0	0
$356$	33.0139	1.74973
$357$	0	0
$358$	28.6238	1.51282
$359$	9.58954	0.506117	0.253058	−	0.967451i	$-0.418564\pi$
$359$	9.58954	0.506117	0.253058	+	0.967451i	$0.418564\pi$
$360$	0	0
$361$	−5.01165	−0.263771
$362$	−32.4218	−1.70405
$363$	0	0
$364$	1.41088	0.0739503
$365$	0	0
$366$	0	0
$367$	1.06423	0.0555525	0.0277763	−	0.999614i	$-0.491157\pi$
$367$	1.06423	0.0555525	0.0277763	+	0.999614i	$0.491157\pi$
$368$	31.0973	1.62106
$369$	0	0
$370$	0	0
$371$	3.31633	0.172175
$372$	0	0
$373$	−4.12022	−0.213337	−0.106669	−	0.994295i	$-0.534018\pi$
$373$	−4.12022	−0.213337	−0.106669	+	0.994295i	$0.534018\pi$
$374$	−14.6922	−0.759714
$375$	0	0
$376$	−39.8890	−2.05712
$377$	4.69033	0.241564
$378$	0	0
$379$	10.3563	0.531967	0.265984	−	0.963978i	$-0.414303\pi$
$379$	10.3563	0.531967	0.265984	+	0.963978i	$0.414303\pi$
$380$	0	0
$381$	0	0
$382$	49.4336	2.52924
$383$	−1.19935	−0.0612839	−0.0306420	−	0.999530i	$-0.509755\pi$
$383$	−1.19935	−0.0612839	−0.0306420	+	0.999530i	$0.509755\pi$
$384$	0	0
$385$	0	0
$386$	−68.2173	−3.47217
$387$	0	0
$388$	45.0147	2.28528
$389$	−12.6043	−0.639062	−0.319531	−	0.947576i	$-0.603525\pi$
$389$	−12.6043	−0.639062	−0.319531	+	0.947576i	$0.603525\pi$
$390$	0	0
$391$	3.87311	0.195872
$392$	60.9892	3.08042
$393$	0	0
$394$	32.7261	1.64872
$395$	0	0
$396$	0	0
$397$	−1.31335	−0.0659152	−0.0329576	−	0.999457i	$-0.510493\pi$
$397$	−1.31335	−0.0659152	−0.0329576	+	0.999457i	$0.510493\pi$
$398$	1.47344	0.0738567
$399$	0	0
$400$	0	0
$401$	−16.1042	−0.804205	−0.402102	−	0.915595i	$-0.631720\pi$
$401$	−16.1042	−0.804205	−0.402102	+	0.915595i	$0.631720\pi$
$402$	0	0
$403$	3.06320	0.152589
$404$	31.0585	1.54522
$405$	0	0
$406$	−10.5952	−0.525830
$407$	3.24782	0.160988
$408$	0	0
$409$	21.1600	1.04630	0.523148	−	0.852242i	$-0.324758\pi$
$409$	21.1600	1.04630	0.523148	+	0.852242i	$0.324758\pi$
$410$	0	0
$411$	0	0
$412$	−74.9642	−3.69322
$413$	6.05361	0.297879
$414$	0	0
$415$	0	0
$416$	−10.7529	−0.527204
$417$	0	0
$418$	−32.2031	−1.57511
$419$	6.67094	0.325897	0.162948	−	0.986635i	$-0.447900\pi$
$419$	6.67094	0.325897	0.162948	+	0.986635i	$0.447900\pi$
$420$	0	0
$421$	27.2980	1.33042	0.665212	−	0.746655i	$-0.268341\pi$
$421$	27.2980	1.33042	0.665212	+	0.746655i	$0.268341\pi$
$422$	8.44944	0.411312
$423$	0	0
$424$	−63.4435	−3.08109
$425$	0	0
$426$	0	0
$427$	0.0596749	0.00288787
$428$	45.8541	2.21644
$429$	0	0
$430$	0	0
$431$	−12.2974	−0.592346	−0.296173	−	0.955134i	$-0.595710\pi$
$431$	−12.2974	−0.592346	−0.296173	+	0.955134i	$0.595710\pi$
$432$	0	0
$433$	−33.8452	−1.62649	−0.813247	−	0.581918i	$-0.802302\pi$
$433$	−33.8452	−1.62649	−0.813247	+	0.581918i	$0.802302\pi$
$434$	−6.91957	−0.332150
$435$	0	0
$436$	88.8509	4.25519
$437$	8.48930	0.406098
$438$	0	0
$439$	−20.9654	−1.00062	−0.500312	−	0.865845i	$-0.666781\pi$
$439$	−20.9654	−1.00062	−0.500312	+	0.865845i	$0.666781\pi$
$440$	0	0
$441$	0	0
$442$	−2.60200	−0.123765
$443$	−6.04847	−0.287371	−0.143686	−	0.989623i	$-0.545895\pi$
$443$	−6.04847	−0.287371	−0.143686	+	0.989623i	$0.545895\pi$
$444$	0	0
$445$	0	0
$446$	−0.917795	−0.0434588
$447$	0	0
$448$	11.4037	0.538773
$449$	−15.5896	−0.735721	−0.367860	−	0.929881i	$-0.619910\pi$
$449$	−15.5896	−0.735721	−0.367860	+	0.929881i	$0.619910\pi$
$450$	0	0
$451$	−4.68357	−0.220541
$452$	−62.6993	−2.94912
$453$	0	0
$454$	−4.84024	−0.227164
$455$	0	0
$456$	0	0
$457$	−2.50193	−0.117035	−0.0585176	−	0.998286i	$-0.518637\pi$
$457$	−2.50193	−0.117035	−0.0585176	+	0.998286i	$0.518637\pi$
$458$	75.7328	3.53876
$459$	0	0
$460$	0	0
$461$	0.153963	0.00717078	0.00358539	−	0.999994i	$-0.498859\pi$
$461$	0.153963	0.00717078	0.00358539	+	0.999994i	$0.498859\pi$
$462$	0	0
$463$	−11.6327	−0.540616	−0.270308	−	0.962774i	$-0.587125\pi$
$463$	−11.6327	−0.540616	−0.270308	+	0.962774i	$0.587125\pi$
$464$	114.048	5.29453
$465$	0	0
$466$	−18.3329	−0.849255
$467$	−0.470294	−0.0217626	−0.0108813	−	0.999941i	$-0.503464\pi$
$467$	−0.470294	−0.0217626	−0.0108813	+	0.999941i	$0.503464\pi$
$468$	0	0
$469$	−1.31433	−0.0606900
$470$	0	0
$471$	0	0
$472$	−115.809	−5.33056
$473$	21.4025	0.984087
$474$	0	0
$475$	0	0
$476$	4.27279	0.195843
$477$	0	0
$478$	−17.5463	−0.802548
$479$	−12.4114	−0.567092	−0.283546	−	0.958959i	$-0.591511\pi$
$479$	−12.4114	−0.567092	−0.283546	+	0.958959i	$0.591511\pi$
$480$	0	0
$481$	0.575193	0.0262265
$482$	20.1585	0.918196
$483$	0	0
$484$	−4.67587	−0.212539
$485$	0	0
$486$	0	0
$487$	34.5646	1.56627	0.783135	−	0.621852i	$-0.213619\pi$
$487$	34.5646	1.56627	0.783135	+	0.621852i	$0.213619\pi$
$488$	−1.14162	−0.0516786
$489$	0	0
$490$	0	0
$491$	24.8658	1.12218	0.561088	−	0.827756i	$-0.310383\pi$
$491$	24.8658	1.12218	0.561088	+	0.827756i	$0.310383\pi$
$492$	0	0
$493$	14.2044	0.639736
$494$	−5.70322	−0.256600
$495$	0	0
$496$	74.4830	3.34439
$497$	−7.54618	−0.338492
$498$	0	0
$499$	12.6037	0.564220	0.282110	−	0.959382i	$-0.408966\pi$
$499$	12.6037	0.564220	0.282110	+	0.959382i	$0.408966\pi$
$500$	0	0
$501$	0	0
$502$	−12.4551	−0.555900
$503$	−29.9927	−1.33731	−0.668655	−	0.743573i	$-0.733130\pi$
$503$	−29.9927	−1.33731	−0.668655	+	0.743573i	$0.733130\pi$
$504$	0	0
$505$	0	0
$506$	−19.5436	−0.868817
$507$	0	0
$508$	9.47116	0.420215
$509$	0.466989	0.0206989	0.0103495	−	0.999946i	$-0.496706\pi$
$509$	0.466989	0.0206989	0.0103495	+	0.999946i	$0.496706\pi$
$510$	0	0
$511$	−4.60358	−0.203651
$512$	−14.9358	−0.660074
$513$	0	0
$514$	−15.6830	−0.691746
$515$	0	0
$516$	0	0
$517$	14.1053	0.620351
$518$	−1.29933	−0.0570891
$519$	0	0
$520$	0	0
$521$	17.2819	0.757133	0.378566	−	0.925574i	$-0.376417\pi$
$521$	17.2819	0.757133	0.378566	+	0.925574i	$0.376417\pi$
$522$	0	0
$523$	40.6479	1.77741	0.888705	−	0.458480i	$-0.151606\pi$
$523$	40.6479	1.77741	0.888705	+	0.458480i	$0.151606\pi$
$524$	31.5130	1.37665
$525$	0	0
$526$	38.6980	1.68731
$527$	9.27673	0.404101
$528$	0	0
$529$	−17.8480	−0.775999
$530$	0	0
$531$	0	0
$532$	9.36533	0.406039
$533$	−0.829466	−0.0359282
$534$	0	0
$535$	0	0
$536$	25.1439	1.08605
$537$	0	0
$538$	−52.0175	−2.24264
$539$	−21.5667	−0.928943
$540$	0	0
$541$	−1.98099	−0.0851694	−0.0425847	−	0.999093i	$-0.513559\pi$
$541$	−1.98099	−0.0851694	−0.0425847	+	0.999093i	$0.513559\pi$
$542$	62.9047	2.70199
$543$	0	0
$544$	−32.5646	−1.39619
$545$	0	0
$546$	0	0
$547$	35.0543	1.49881	0.749406	−	0.662111i	$-0.230339\pi$
$547$	35.0543	1.49881	0.749406	+	0.662111i	$0.230339\pi$
$548$	27.2402	1.16364
$549$	0	0
$550$	0	0
$551$	31.1341	1.32636
$552$	0	0
$553$	−2.23625	−0.0950948
$554$	−59.2449	−2.51708
$555$	0	0
$556$	−43.8633	−1.86022
$557$	−19.2383	−0.815154	−0.407577	−	0.913171i	$-0.633626\pi$
$557$	−19.2383	−0.815154	−0.407577	+	0.913171i	$0.633626\pi$
$558$	0	0
$559$	3.79041	0.160317
$560$	0	0
$561$	0	0
$562$	73.3798	3.09534
$563$	29.1741	1.22954	0.614771	−	0.788706i	$-0.289248\pi$
$563$	29.1741	1.22954	0.614771	+	0.788706i	$0.289248\pi$
$564$	0	0
$565$	0	0
$566$	−44.1337	−1.85508
$567$	0	0
$568$	144.363	6.05734
$569$	32.9112	1.37971	0.689855	−	0.723947i	$-0.257674\pi$
$569$	32.9112	1.37971	0.689855	+	0.723947i	$0.257674\pi$
$570$	0	0
$571$	26.6349	1.11464	0.557319	−	0.830299i	$-0.311830\pi$
$571$	26.6349	1.11464	0.557319	+	0.830299i	$0.311830\pi$
$572$	9.54443	0.399073
$573$	0	0
$574$	1.87371	0.0782073
$575$	0	0
$576$	0	0
$577$	34.5156	1.43690	0.718452	−	0.695577i	$-0.244851\pi$
$577$	34.5156	1.43690	0.718452	+	0.695577i	$0.244851\pi$
$578$	38.1281	1.58592
$579$	0	0
$580$	0	0
$581$	−5.49797	−0.228094
$582$	0	0
$583$	22.4345	0.929144
$584$	88.0694	3.64434
$585$	0	0
$586$	14.7878	0.610877
$587$	22.2755	0.919408	0.459704	−	0.888072i	$-0.347955\pi$
$587$	22.2755	0.919408	0.459704	+	0.888072i	$0.347955\pi$
$588$	0	0
$589$	20.3333	0.837818
$590$	0	0
$591$	0	0
$592$	13.9861	0.574824
$593$	30.9158	1.26956	0.634779	−	0.772693i	$-0.281091\pi$
$593$	30.9158	1.26956	0.634779	+	0.772693i	$0.281091\pi$
$594$	0	0
$595$	0	0
$596$	53.5556	2.19372
$597$	0	0
$598$	−3.46119	−0.141539
$599$	24.0276	0.981742	0.490871	−	0.871232i	$-0.336679\pi$
$599$	24.0276	0.981742	0.490871	+	0.871232i	$0.336679\pi$
$600$	0	0
$601$	−32.0387	−1.30688	−0.653442	−	0.756977i	$-0.726676\pi$
$601$	−32.0387	−1.30688	−0.653442	+	0.756977i	$0.726676\pi$
$602$	−8.56231	−0.348974
$603$	0	0
$604$	−63.7915	−2.59564
$605$	0	0
$606$	0	0
$607$	45.5915	1.85050	0.925251	−	0.379356i	$-0.123855\pi$
$607$	45.5915	1.85050	0.925251	+	0.379356i	$0.123855\pi$
$608$	−71.3769	−2.89471
$609$	0	0
$610$	0	0
$611$	2.49807	0.101061
$612$	0	0
$613$	−10.8564	−0.438485	−0.219242	−	0.975670i	$-0.570359\pi$
$613$	−10.8564	−0.438485	−0.219242	+	0.975670i	$0.570359\pi$
$614$	57.9686	2.33942
$615$	0	0
$616$	−13.4616	−0.542384
$617$	16.1054	0.648380	0.324190	−	0.945992i	$-0.394908\pi$
$617$	16.1054	0.648380	0.324190	+	0.945992i	$0.394908\pi$
$618$	0	0
$619$	38.3581	1.54174	0.770872	−	0.636991i	$-0.219821\pi$
$619$	38.3581	1.54174	0.770872	+	0.636991i	$0.219821\pi$
$620$	0	0
$621$	0	0
$622$	−7.71854	−0.309485
$623$	2.91605	0.116829
$624$	0	0
$625$	0	0
$626$	−39.0732	−1.56168
$627$	0	0
$628$	104.337	4.16350
$629$	1.74194	0.0694558
$630$	0	0
$631$	22.7292	0.904833	0.452417	−	0.891807i	$-0.350562\pi$
$631$	22.7292	0.904833	0.452417	+	0.891807i	$0.350562\pi$
$632$	42.7808	1.70173
$633$	0	0
$634$	61.1915	2.43022
$635$	0	0
$636$	0	0
$637$	−3.81949	−0.151334
$638$	−71.6750	−2.83764
$639$	0	0
$640$	0	0
$641$	−26.9930	−1.06616	−0.533080	−	0.846065i	$-0.678965\pi$
$641$	−26.9930	−1.06616	−0.533080	+	0.846065i	$0.678965\pi$
$642$	0	0
$643$	−25.9062	−1.02164	−0.510821	−	0.859687i	$-0.670659\pi$
$643$	−25.9062	−1.02164	−0.510821	+	0.859687i	$0.670659\pi$
$644$	5.68367	0.223968
$645$	0	0
$646$	−17.2719	−0.679554
$647$	21.6623	0.851632	0.425816	−	0.904810i	$-0.359987\pi$
$647$	21.6623	0.851632	0.425816	+	0.904810i	$0.359987\pi$
$648$	0	0
$649$	40.9519	1.60750
$650$	0	0
$651$	0	0
$652$	63.3212	2.47985
$653$	−11.1775	−0.437411	−0.218705	−	0.975791i	$-0.570183\pi$
$653$	−11.1775	−0.437411	−0.218705	+	0.975791i	$0.570183\pi$
$654$	0	0
$655$	0	0
$656$	−20.1689	−0.787461
$657$	0	0
$658$	−5.64299	−0.219987
$659$	−13.4744	−0.524888	−0.262444	−	0.964947i	$-0.584529\pi$
$659$	−13.4744	−0.524888	−0.262444	+	0.964947i	$0.584529\pi$
$660$	0	0
$661$	−23.2622	−0.904793	−0.452397	−	0.891817i	$-0.649431\pi$
$661$	−23.2622	−0.904793	−0.452397	+	0.891817i	$0.649431\pi$
$662$	−10.4640	−0.406695
$663$	0	0
$664$	105.180	4.08176
$665$	0	0
$666$	0	0
$667$	18.8948	0.731608
$668$	31.9132	1.23476
$669$	0	0
$670$	0	0
$671$	0.403693	0.0155844
$672$	0	0
$673$	18.8392	0.726198	0.363099	−	0.931751i	$-0.381719\pi$
$673$	18.8392	0.726198	0.363099	+	0.931751i	$0.381719\pi$
$674$	−47.3215	−1.82276
$675$	0	0
$676$	−67.5268	−2.59719
$677$	42.2440	1.62357	0.811785	−	0.583957i	$-0.198496\pi$
$677$	42.2440	1.62357	0.811785	+	0.583957i	$0.198496\pi$
$678$	0	0
$679$	3.97607	0.152587
$680$	0	0
$681$	0	0
$682$	−46.8100	−1.79245
$683$	−48.9502	−1.87303	−0.936513	−	0.350633i	$-0.885966\pi$
$683$	−48.9502	−1.87303	−0.936513	+	0.350633i	$0.885966\pi$
$684$	0	0
$685$	0	0
$686$	17.5375	0.669585
$687$	0	0
$688$	92.1655	3.51378
$689$	3.97319	0.151366
$690$	0	0
$691$	−14.1917	−0.539878	−0.269939	−	0.962877i	$-0.587004\pi$
$691$	−14.1917	−0.539878	−0.269939	+	0.962877i	$0.587004\pi$
$692$	−92.2419	−3.50651
$693$	0	0
$694$	−62.5010	−2.37251
$695$	0	0
$696$	0	0
$697$	−2.51200	−0.0951487
$698$	25.2409	0.955382
$699$	0	0
$700$	0	0
$701$	9.00786	0.340222	0.170111	−	0.985425i	$-0.445587\pi$
$701$	9.00786	0.340222	0.170111	+	0.985425i	$0.445587\pi$
$702$	0	0
$703$	3.81809	0.144002
$704$	77.1444	2.90749
$705$	0	0
$706$	86.4604	3.25398
$707$	2.74334	0.103174
$708$	0	0
$709$	49.6994	1.86650	0.933249	−	0.359229i	$-0.116960\pi$
$709$	49.6994	1.86650	0.933249	+	0.359229i	$0.116960\pi$
$710$	0	0
$711$	0	0
$712$	−55.7859	−2.09067
$713$	12.3399	0.462134
$714$	0	0
$715$	0	0
$716$	−56.3134	−2.10453
$717$	0	0
$718$	−25.9528	−0.968549
$719$	2.13706	0.0796988	0.0398494	−	0.999206i	$-0.487312\pi$
$719$	2.13706	0.0796988	0.0398494	+	0.999206i	$0.487312\pi$
$720$	0	0
$721$	−6.62145	−0.246596
$722$	13.5633	0.504775
$723$	0	0
$724$	63.7855	2.37057
$725$	0	0
$726$	0	0
$727$	−41.4634	−1.53779	−0.768895	−	0.639375i	$-0.779193\pi$
$727$	−41.4634	−1.53779	−0.768895	+	0.639375i	$0.779193\pi$
$728$	−2.38407	−0.0883596
$729$	0	0
$730$	0	0
$731$	11.4791	0.424568
$732$	0	0
$733$	32.5210	1.20119	0.600596	−	0.799553i	$-0.294930\pi$
$733$	32.5210	1.20119	0.600596	+	0.799553i	$0.294930\pi$
$734$	−2.88020	−0.106310
$735$	0	0
$736$	−43.3175	−1.59670
$737$	−8.89125	−0.327513
$738$	0	0
$739$	−35.3175	−1.29917	−0.649587	−	0.760287i	$-0.725058\pi$
$739$	−35.3175	−1.29917	−0.649587	+	0.760287i	$0.725058\pi$
$740$	0	0
$741$	0	0
$742$	−8.97519	−0.329490
$743$	36.0897	1.32400	0.662001	−	0.749503i	$-0.269708\pi$
$743$	36.0897	1.32400	0.662001	+	0.749503i	$0.269708\pi$
$744$	0	0
$745$	0	0
$746$	11.1508	0.408261
$747$	0	0
$748$	28.9048	1.05687
$749$	4.05021	0.147991
$750$	0	0
$751$	−4.62810	−0.168882	−0.0844409	−	0.996428i	$-0.526910\pi$
$751$	−4.62810	−0.168882	−0.0844409	+	0.996428i	$0.526910\pi$
$752$	60.7418	2.21502
$753$	0	0
$754$	−12.6937	−0.462279
$755$	0	0
$756$	0	0
$757$	14.6243	0.531530	0.265765	−	0.964038i	$-0.414375\pi$
$757$	14.6243	0.531530	0.265765	+	0.964038i	$0.414375\pi$
$758$	−28.0279	−1.01802
$759$	0	0
$760$	0	0
$761$	−12.1893	−0.441861	−0.220930	−	0.975290i	$-0.570909\pi$
$761$	−12.1893	−0.441861	−0.220930	+	0.975290i	$0.570909\pi$
$762$	0	0
$763$	7.84804	0.284118
$764$	−97.2538	−3.51852
$765$	0	0
$766$	3.24587	0.117278
$767$	7.25264	0.261878
$768$	0	0
$769$	13.7961	0.497500	0.248750	−	0.968568i	$-0.419980\pi$
$769$	13.7961	0.497500	0.248750	+	0.968568i	$0.419980\pi$
$770$	0	0
$771$	0	0
$772$	134.208	4.83026
$773$	−22.6539	−0.814803	−0.407402	−	0.913249i	$-0.633565\pi$
$773$	−22.6539	−0.814803	−0.407402	+	0.913249i	$0.633565\pi$
$774$	0	0
$775$	0	0
$776$	−76.0647	−2.73056
$777$	0	0
$778$	34.1117	1.22296
$779$	−5.50594	−0.197271
$780$	0	0
$781$	−51.0489	−1.82667
$782$	−10.4820	−0.374837
$783$	0	0
$784$	−92.8726	−3.31688
$785$	0	0
$786$	0	0
$787$	−47.6857	−1.69981	−0.849905	−	0.526936i	$-0.823341\pi$
$787$	−47.6857	−1.69981	−0.849905	+	0.526936i	$0.823341\pi$
$788$	−64.3841	−2.29359
$789$	0	0
$790$	0	0
$791$	−5.53811	−0.196913
$792$	0	0
$793$	0.0714945	0.00253884
$794$	3.55440	0.126141
$795$	0	0
$796$	−2.89878	−0.102745
$797$	−27.2131	−0.963938	−0.481969	−	0.876188i	$-0.660078\pi$
$797$	−27.2131	−0.963938	−0.481969	+	0.876188i	$0.660078\pi$
$798$	0	0
$799$	7.56529	0.267641
$800$	0	0
$801$	0	0
$802$	43.5838	1.53900
$803$	−31.1426	−1.09900
$804$	0	0
$805$	0	0
$806$	−8.29012	−0.292007
$807$	0	0
$808$	−52.4819	−1.84631
$809$	19.6155	0.689644	0.344822	−	0.938668i	$-0.387939\pi$
$809$	19.6155	0.689644	0.344822	+	0.938668i	$0.387939\pi$
$810$	0	0
$811$	21.0710	0.739903	0.369951	−	0.929051i	$-0.379374\pi$
$811$	21.0710	0.739903	0.369951	+	0.929051i	$0.379374\pi$
$812$	20.8446	0.731501
$813$	0	0
$814$	−8.78977	−0.308081
$815$	0	0
$816$	0	0
$817$	25.1605	0.880253
$818$	−57.2667	−2.00228
$819$	0	0
$820$	0	0
$821$	−41.4342	−1.44606	−0.723031	−	0.690815i	$-0.757252\pi$
$821$	−41.4342	−1.44606	−0.723031	+	0.690815i	$0.757252\pi$
$822$	0	0
$823$	6.04093	0.210574	0.105287	−	0.994442i	$-0.466424\pi$
$823$	6.04093	0.210574	0.105287	+	0.994442i	$0.466424\pi$
$824$	126.673	4.41285
$825$	0	0
$826$	−16.3833	−0.570047
$827$	2.99361	0.104098	0.0520491	−	0.998645i	$-0.483425\pi$
$827$	2.99361	0.104098	0.0520491	+	0.998645i	$0.483425\pi$
$828$	0	0
$829$	26.8530	0.932642	0.466321	−	0.884616i	$-0.345579\pi$
$829$	26.8530	0.932642	0.466321	+	0.884616i	$0.345579\pi$
$830$	0	0
$831$	0	0
$832$	13.6624	0.473658
$833$	−11.5671	−0.400777
$834$	0	0
$835$	0	0
$836$	63.3552	2.19119
$837$	0	0
$838$	−18.0540	−0.623665
$839$	−12.4595	−0.430150	−0.215075	−	0.976598i	$-0.569000\pi$
$839$	−12.4595	−0.430150	−0.215075	+	0.976598i	$0.569000\pi$
$840$	0	0
$841$	40.2956	1.38950
$842$	−73.8783	−2.54601
$843$	0	0
$844$	−16.6231	−0.572191
$845$	0	0
$846$	0	0
$847$	−0.413011	−0.0141912
$848$	96.6099	3.31760
$849$	0	0
$850$	0	0
$851$	2.31714	0.0794304
$852$	0	0
$853$	−5.69736	−0.195074	−0.0975369	−	0.995232i	$-0.531096\pi$
$853$	−5.69736	−0.195074	−0.0975369	+	0.995232i	$0.531096\pi$
$854$	−0.161502	−0.00552648
$855$	0	0
$856$	−77.4830	−2.64831
$857$	7.41982	0.253456	0.126728	−	0.991937i	$-0.459552\pi$
$857$	7.41982	0.253456	0.126728	+	0.991937i	$0.459552\pi$
$858$	0	0
$859$	−5.68253	−0.193885	−0.0969427	−	0.995290i	$-0.530906\pi$
$859$	−5.68253	−0.193885	−0.0969427	+	0.995290i	$0.530906\pi$
$860$	0	0
$861$	0	0
$862$	33.2813	1.13356
$863$	34.5747	1.17693	0.588467	−	0.808521i	$-0.299732\pi$
$863$	34.5747	1.17693	0.588467	+	0.808521i	$0.299732\pi$
$864$	0	0
$865$	0	0
$866$	91.5973	3.11260
$867$	0	0
$868$	13.6133	0.462066
$869$	−15.1279	−0.513179
$870$	0	0
$871$	−1.57465	−0.0533550
$872$	−150.138	−5.08431
$873$	0	0
$874$	−22.9751	−0.777145
$875$	0	0
$876$	0	0
$877$	−5.30013	−0.178973	−0.0894863	−	0.995988i	$-0.528523\pi$
$877$	−5.30013	−0.178973	−0.0894863	+	0.995988i	$0.528523\pi$
$878$	56.7399	1.91488
$879$	0	0
$880$	0	0
$881$	−7.61288	−0.256484	−0.128242	−	0.991743i	$-0.540933\pi$
$881$	−7.61288	−0.256484	−0.128242	+	0.991743i	$0.540933\pi$
$882$	0	0
$883$	−52.8334	−1.77799	−0.888993	−	0.457921i	$-0.848594\pi$
$883$	−52.8334	−1.77799	−0.888993	+	0.457921i	$0.848594\pi$
$884$	5.11909	0.172174
$885$	0	0
$886$	16.3693	0.549939
$887$	−7.70778	−0.258802	−0.129401	−	0.991592i	$-0.541305\pi$
$887$	−7.70778	−0.258802	−0.129401	+	0.991592i	$0.541305\pi$
$888$	0	0
$889$	0.836570	0.0280577
$890$	0	0
$891$	0	0
$892$	1.80563	0.0604571
$893$	16.5820	0.554896
$894$	0	0
$895$	0	0
$896$	−12.9121	−0.431363
$897$	0	0
$898$	42.1912	1.40794
$899$	45.2560	1.50937
$900$	0	0
$901$	12.0326	0.400864
$902$	12.6754	0.422046
$903$	0	0
$904$	105.947	3.52376
$905$	0	0
$906$	0	0
$907$	−11.3735	−0.377650	−0.188825	−	0.982011i	$-0.560468\pi$
$907$	−11.3735	−0.377650	−0.188825	+	0.982011i	$0.560468\pi$
$908$	9.52251	0.316015
$909$	0	0
$910$	0	0
$911$	18.5896	0.615902	0.307951	−	0.951402i	$-0.400357\pi$
$911$	18.5896	0.615902	0.307951	+	0.951402i	$0.400357\pi$
$912$	0	0
$913$	−37.1931	−1.23091
$914$	6.77112	0.223969
$915$	0	0
$916$	−148.994	−4.92290
$917$	2.78348	0.0919187
$918$	0	0
$919$	7.64799	0.252284	0.126142	−	0.992012i	$-0.459741\pi$
$919$	7.64799	0.252284	0.126142	+	0.992012i	$0.459741\pi$
$920$	0	0
$921$	0	0
$922$	−0.416680	−0.0137226
$923$	−9.04083	−0.297582
$924$	0	0
$925$	0	0
$926$	31.4822	1.03457
$927$	0	0
$928$	−158.865	−5.21498
$929$	30.9487	1.01539	0.507696	−	0.861536i	$-0.330497\pi$
$929$	30.9487	1.01539	0.507696	+	0.861536i	$0.330497\pi$
$930$	0	0
$931$	−25.3535	−0.830927
$932$	36.0675	1.18143
$933$	0	0
$934$	1.27279	0.0416468
$935$	0	0
$936$	0	0
$937$	31.3694	1.02480	0.512398	−	0.858748i	$-0.328757\pi$
$937$	31.3694	1.02480	0.512398	+	0.858748i	$0.328757\pi$
$938$	3.55704	0.116142
$939$	0	0
$940$	0	0
$941$	3.61810	0.117947	0.0589733	−	0.998260i	$-0.481217\pi$
$941$	3.61810	0.117947	0.0589733	+	0.998260i	$0.481217\pi$
$942$	0	0
$943$	−3.34146	−0.108813
$944$	176.351	5.73974
$945$	0	0
$946$	−57.9229	−1.88323
$947$	11.3216	0.367902	0.183951	−	0.982935i	$-0.441111\pi$
$947$	11.3216	0.367902	0.183951	+	0.982935i	$0.441111\pi$
$948$	0	0
$949$	−5.51540	−0.179038
$950$	0	0
$951$	0	0
$952$	−7.22004	−0.234003
$953$	17.2182	0.557752	0.278876	−	0.960327i	$-0.410038\pi$
$953$	17.2182	0.557752	0.278876	+	0.960327i	$0.410038\pi$
$954$	0	0
$955$	0	0
$956$	34.5199	1.11645
$957$	0	0
$958$	33.5898	1.08524
$959$	2.40608	0.0776962
$960$	0	0
$961$	−1.44386	−0.0465762
$962$	−1.55668	−0.0501894
$963$	0	0
$964$	−39.6591	−1.27733
$965$	0	0
$966$	0	0
$967$	−29.9064	−0.961723	−0.480862	−	0.876796i	$-0.659676\pi$
$967$	−29.9064	−0.961723	−0.480862	+	0.876796i	$0.659676\pi$
$968$	7.90115	0.253953
$969$	0	0
$970$	0	0
$971$	25.1927	0.808472	0.404236	−	0.914655i	$-0.367537\pi$
$971$	25.1927	0.808472	0.404236	+	0.914655i	$0.367537\pi$
$972$	0	0
$973$	−3.87436	−0.124206
$974$	−93.5443	−2.99735
$975$	0	0
$976$	1.73842	0.0556455
$977$	−57.0948	−1.82663	−0.913313	−	0.407259i	$-0.866485\pi$
$977$	−57.0948	−1.82663	−0.913313	+	0.407259i	$0.866485\pi$
$978$	0	0
$979$	19.7267	0.630469
$980$	0	0
$981$	0	0
$982$	−67.2957	−2.14749
$983$	26.4168	0.842566	0.421283	−	0.906929i	$-0.361580\pi$
$983$	26.4168	0.842566	0.421283	+	0.906929i	$0.361580\pi$
$984$	0	0
$985$	0	0
$986$	−38.4423	−1.22425
$987$	0	0
$988$	11.2203	0.356965
$989$	15.2695	0.485541
$990$	0	0
$991$	−22.5276	−0.715612	−0.357806	−	0.933796i	$-0.616475\pi$
$991$	−22.5276	−0.715612	−0.357806	+	0.933796i	$0.616475\pi$
$992$	−103.752	−3.29414
$993$	0	0
$994$	20.4227	0.647768
$995$	0	0
$996$	0	0
$997$	−24.1043	−0.763392	−0.381696	−	0.924288i	$-0.624660\pi$
$997$	−24.1043	−0.763392	−0.381696	+	0.924288i	$0.624660\pi$
$998$	−34.1103	−1.07974
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.i.1.1		4
3.2	odd	2		1875.2.a.h.1.4		4
5.4	even	2		5625.2.a.n.1.4		4
15.2	even	4		1875.2.b.c.1249.8		8
15.8	even	4		1875.2.b.c.1249.1		8
15.14	odd	2		1875.2.a.e.1.1		4
25.6	even	5		225.2.h.c.136.2		8
25.21	even	5		225.2.h.c.91.2		8
75.8	even	20		375.2.i.b.199.4		16
75.17	even	20		375.2.i.b.199.1		16
75.29	odd	10		375.2.g.b.76.2		8
75.44	odd	10		375.2.g.b.301.2		8
75.47	even	20		375.2.i.b.49.4		16
75.53	even	20		375.2.i.b.49.1		16
75.56	odd	10		75.2.g.b.61.1	yes	8
75.71	odd	10		75.2.g.b.16.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.g.b.16.1	✓	8	75.71	odd	10
75.2.g.b.61.1	yes	8	75.56	odd	10
225.2.h.c.91.2		8	25.21	even	5
225.2.h.c.136.2		8	25.6	even	5
375.2.g.b.76.2		8	75.29	odd	10
375.2.g.b.301.2		8	75.44	odd	10
375.2.i.b.49.1		16	75.53	even	20
375.2.i.b.49.4		16	75.47	even	20
375.2.i.b.199.1		16	75.17	even	20
375.2.i.b.199.4		16	75.8	even	20
1875.2.a.e.1.1		4	15.14	odd	2
1875.2.a.h.1.4		4	3.2	odd	2
1875.2.b.c.1249.1		8	15.8	even	4
1875.2.b.c.1249.8		8	15.2	even	4
5625.2.a.i.1.1		4	1.1	even	1	trivial
5625.2.a.n.1.4		4	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5625 = 3^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5625.a (trivial)

\(f(q)\)	\(=\)	\(q-2.70636 q^{2} +5.32440 q^{4} +0.470294 q^{7} -8.99702 q^{8} +O(q^{10})\)	\(q-2.70636 q^{2} +5.32440 q^{4} +0.470294 q^{7} -8.99702 q^{8} +3.18148 q^{11} +0.563444 q^{13} -1.27279 q^{14} +13.7004 q^{16} +1.70636 q^{17} +3.74010 q^{19} -8.61023 q^{22} +2.26981 q^{23} -1.52488 q^{26} +2.50403 q^{28} +8.32440 q^{29} +5.43656 q^{31} -19.0842 q^{32} -4.61803 q^{34} +1.02085 q^{37} -10.1221 q^{38} -1.47214 q^{41} +6.72721 q^{43} +16.9395 q^{44} -6.14292 q^{46} +4.43358 q^{47} -6.77882 q^{49} +3.00000 q^{52} +7.05161 q^{53} -4.23125 q^{56} -22.5288 q^{58} +12.8720 q^{59} +0.126888 q^{61} -14.7133 q^{62} +24.2480 q^{64} -2.79469 q^{67} +9.08535 q^{68} -16.0456 q^{71} -9.78873 q^{73} -2.76279 q^{74} +19.9138 q^{76} +1.49623 q^{77} -4.75499 q^{79} +3.98413 q^{82} -11.6905 q^{83} -18.2063 q^{86} -28.6238 q^{88} +6.20049 q^{89} +0.264985 q^{91} +12.0853 q^{92} -11.9989 q^{94} +8.45443 q^{97} +18.3460 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 8 q^{4} + 2 q^{7} - 15 q^{8}+O(q^{10}) \) 4 * q - 2 * q^2 + 8 * q^4 + 2 * q^7 - 15 * q^8	\( 4 q - 2 q^{2} + 8 q^{4} + 2 q^{7} - 15 q^{8} + 7 q^{11} + q^{13} - 16 q^{14} + 4 q^{16} - 2 q^{17} + 5 q^{19} - 6 q^{22} - q^{23} - 3 q^{26} + 9 q^{28} + 20 q^{29} + 23 q^{31} - 12 q^{32} - 14 q^{34} + 2 q^{37} - 35 q^{38} + 12 q^{41} + 16 q^{43} + 29 q^{44} - 17 q^{46} - 2 q^{47} + 8 q^{49} + 12 q^{52} + 4 q^{53} - 5 q^{56} - 25 q^{58} + 15 q^{59} - 2 q^{61} - 9 q^{62} + 23 q^{64} + 2 q^{67} + 11 q^{68} + 2 q^{71} + 16 q^{73} + 19 q^{74} + 40 q^{76} - 19 q^{77} + 35 q^{79} - 6 q^{82} - 16 q^{83} - 3 q^{86} - 30 q^{88} + 35 q^{89} - 12 q^{91} + 23 q^{92} - 9 q^{94} + 12 q^{97} + q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 + 8 * q^4 + 2 * q^7 - 15 * q^8 + 7 * q^11 + q^13 - 16 * q^14 + 4 * q^16 - 2 * q^17 + 5 * q^19 - 6 * q^22 - q^23 - 3 * q^26 + 9 * q^28 + 20 * q^29 + 23 * q^31 - 12 * q^32 - 14 * q^34 + 2 * q^37 - 35 * q^38 + 12 * q^41 + 16 * q^43 + 29 * q^44 - 17 * q^46 - 2 * q^47 + 8 * q^49 + 12 * q^52 + 4 * q^53 - 5 * q^56 - 25 * q^58 + 15 * q^59 - 2 * q^61 - 9 * q^62 + 23 * q^64 + 2 * q^67 + 11 * q^68 + 2 * q^71 + 16 * q^73 + 19 * q^74 + 40 * q^76 - 19 * q^77 + 35 * q^79 - 6 * q^82 - 16 * q^83 - 3 * q^86 - 30 * q^88 + 35 * q^89 - 12 * q^91 + 23 * q^92 - 9 * q^94 + 12 * q^97 + q^98

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