LMFDB - Embedded newform 5625.2.a.s.1.8

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:	$1$
Dimension:	$8$
Coefficient field:	8.8.6152203125.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 8x^{6} + 20x^{5} + 26x^{4} - 35x^{3} - 27x^{2} + 16x + 1 $ x^8 - 3x^7 - 8x^6 + 20x^5 + 26x^4 - 35x^3 - 27x^2 + 16*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 625)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$3.01367$ 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)

$f(q)$	$=$	$q+2.01367 q^{2} +2.05487 q^{4} +0.369971 q^{7} +0.110485 q^{8} +O(q^{10})$	$q+2.01367 q^{2} +2.05487 q^{4} +0.369971 q^{7} +0.110485 q^{8} +1.74633 q^{11} +1.11622 q^{13} +0.745000 q^{14} -3.88725 q^{16} -5.48800 q^{17} -3.75219 q^{19} +3.51653 q^{22} -7.24619 q^{23} +2.24770 q^{26} +0.760242 q^{28} -4.19284 q^{29} +0.305684 q^{31} -8.04862 q^{32} -11.0510 q^{34} -9.21956 q^{37} -7.55568 q^{38} +4.18641 q^{41} +7.17118 q^{43} +3.58847 q^{44} -14.5914 q^{46} -0.810273 q^{47} -6.86312 q^{49} +2.29369 q^{52} +3.91508 q^{53} +0.0408762 q^{56} -8.44299 q^{58} +1.85738 q^{59} +9.68874 q^{61} +0.615546 q^{62} -8.43275 q^{64} -12.4701 q^{67} -11.2771 q^{68} -11.6767 q^{71} +3.43137 q^{73} -18.5652 q^{74} -7.71026 q^{76} +0.646091 q^{77} +5.69346 q^{79} +8.43005 q^{82} -7.13371 q^{83} +14.4404 q^{86} +0.192943 q^{88} -1.52999 q^{89} +0.412970 q^{91} -14.8900 q^{92} -1.63162 q^{94} +6.39742 q^{97} -13.8201 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 5 q^{2} + 11 q^{4} + 10 q^{7} - 15 q^{8}+O(q^{10}) $ 8 * q - 5 * q^2 + 11 * q^4 + 10 * q^7 - 15 * q^8	$ 8 q - 5 q^{2} + 11 q^{4} + 10 q^{7} - 15 q^{8} - q^{11} + 10 q^{13} + 8 q^{14} + 13 q^{16} - 15 q^{17} - 10 q^{19} - 5 q^{22} - 30 q^{23} - 11 q^{26} - 5 q^{28} - 10 q^{29} - 9 q^{31} - 30 q^{32} + 7 q^{34} - 10 q^{37} - 20 q^{38} + 4 q^{41} + 18 q^{44} - 9 q^{46} - 30 q^{47} - 4 q^{49} + 5 q^{52} - 10 q^{53} - 30 q^{58} + 5 q^{59} + 6 q^{61} - 10 q^{62} - 9 q^{64} + 10 q^{67} - 40 q^{68} + 9 q^{71} + 18 q^{74} - 10 q^{76} - 5 q^{77} - 20 q^{79} - 45 q^{82} - 40 q^{83} + 24 q^{86} - 40 q^{88} + 5 q^{89} + 6 q^{91} - 15 q^{92} + 47 q^{94} + 30 q^{98}+O(q^{100}) $ 8 * q - 5 * q^2 + 11 * q^4 + 10 * q^7 - 15 * q^8 - q^11 + 10 * q^13 + 8 * q^14 + 13 * q^16 - 15 * q^17 - 10 * q^19 - 5 * q^22 - 30 * q^23 - 11 * q^26 - 5 * q^28 - 10 * q^29 - 9 * q^31 - 30 * q^32 + 7 * q^34 - 10 * q^37 - 20 * q^38 + 4 * q^41 + 18 * q^44 - 9 * q^46 - 30 * q^47 - 4 * q^49 + 5 * q^52 - 10 * q^53 - 30 * q^58 + 5 * q^59 + 6 * q^61 - 10 * q^62 - 9 * q^64 + 10 * q^67 - 40 * q^68 + 9 * q^71 + 18 * q^74 - 10 * q^76 - 5 * q^77 - 20 * q^79 - 45 * q^82 - 40 * q^83 + 24 * q^86 - 40 * q^88 + 5 * q^89 + 6 * q^91 - 15 * q^92 + 47 * q^94 + 30 * 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 $

$2$	2.01367	1.42388	0.711940	−	0.702240i	$-0.247817\pi$
$2$	2.01367	1.42388	0.711940	+	0.702240i	$0.247817\pi$
$3$	0	0
$3$	0	0
$4$	2.05487	1.02743
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.369971	0.139836	0.0699180	−	0.997553i	$-0.477726\pi$
$7$	0.369971	0.139836	0.0699180	+	0.997553i	$0.477726\pi$
$8$	0.110485	0.0390623
$9$	0	0
$10$	0	0
$11$	1.74633	0.526538	0.263269	−	0.964722i	$-0.415199\pi$
$11$	1.74633	0.526538	0.263269	+	0.964722i	$0.415199\pi$
$12$	0	0
$13$	1.11622	0.309584	0.154792	−	0.987947i	$-0.450529\pi$
$13$	1.11622	0.309584	0.154792	+	0.987947i	$0.450529\pi$
$14$	0.745000	0.199110
$15$	0	0
$16$	−3.88725	−0.971814
$17$	−5.48800	−1.33103	−0.665517	−	0.746382i	$-0.731789\pi$
$17$	−5.48800	−1.33103	−0.665517	+	0.746382i	$0.731789\pi$
$18$	0	0
$19$	−3.75219	−0.860812	−0.430406	−	0.902635i	$-0.641630\pi$
$19$	−3.75219	−0.860812	−0.430406	+	0.902635i	$0.641630\pi$
$20$	0	0
$21$	0	0
$22$	3.51653	0.749726
$23$	−7.24619	−1.51094	−0.755468	−	0.655186i	$-0.772590\pi$
$23$	−7.24619	−1.51094	−0.755468	+	0.655186i	$0.772590\pi$
$24$	0	0
$25$	0	0
$26$	2.24770	0.440811
$27$	0	0
$28$	0.760242	0.143672
$29$	−4.19284	−0.778590	−0.389295	−	0.921113i	$-0.627281\pi$
$29$	−4.19284	−0.778590	−0.389295	+	0.921113i	$0.627281\pi$
$30$	0	0
$31$	0.305684	0.0549024	0.0274512	−	0.999623i	$-0.491261\pi$
$31$	0.305684	0.0549024	0.0274512	+	0.999623i	$0.491261\pi$
$32$	−8.04862	−1.42281
$33$	0	0
$34$	−11.0510	−1.89523
$35$	0	0
$36$	0	0
$37$	−9.21956	−1.51569	−0.757843	−	0.652436i	$-0.773747\pi$
$37$	−9.21956	−1.51569	−0.757843	+	0.652436i	$0.773747\pi$
$38$	−7.55568	−1.22569
$39$	0	0
$40$	0	0
$41$	4.18641	0.653807	0.326904	−	0.945058i	$-0.393995\pi$
$41$	4.18641	0.653807	0.326904	+	0.945058i	$0.393995\pi$
$42$	0	0
$43$	7.17118	1.09359	0.546797	−	0.837265i	$-0.315847\pi$
$43$	7.17118	1.09359	0.546797	+	0.837265i	$0.315847\pi$
$44$	3.58847	0.540983
$45$	0	0
$46$	−14.5914	−2.15139
$47$	−0.810273	−0.118190	−0.0590952	−	0.998252i	$-0.518822\pi$
$47$	−0.810273	−0.118190	−0.0590952	+	0.998252i	$0.518822\pi$
$48$	0	0
$49$	−6.86312	−0.980446
$50$	0	0
$51$	0	0
$52$	2.29369	0.318077
$53$	3.91508	0.537778	0.268889	−	0.963171i	$-0.413343\pi$
$53$	3.91508	0.537778	0.268889	+	0.963171i	$0.413343\pi$
$54$	0	0
$55$	0	0
$56$	0.0408762	0.00546231
$57$	0	0
$58$	−8.44299	−1.10862
$59$	1.85738	0.241810	0.120905	−	0.992664i	$-0.461420\pi$
$59$	1.85738	0.241810	0.120905	+	0.992664i	$0.461420\pi$
$60$	0	0
$61$	9.68874	1.24052	0.620258	−	0.784398i	$-0.287028\pi$
$61$	9.68874	1.24052	0.620258	+	0.784398i	$0.287028\pi$
$62$	0.615546	0.0781744
$63$	0	0
$64$	−8.43275	−1.05409
$65$	0	0
$66$	0	0
$67$	−12.4701	−1.52346	−0.761730	−	0.647894i	$-0.775650\pi$
$67$	−12.4701	−1.52346	−0.761730	+	0.647894i	$0.775650\pi$
$68$	−11.2771	−1.36755
$69$	0	0
$70$	0	0
$71$	−11.6767	−1.38577	−0.692885	−	0.721048i	$-0.743661\pi$
$71$	−11.6767	−1.38577	−0.692885	+	0.721048i	$0.743661\pi$
$72$	0	0
$73$	3.43137	0.401611	0.200806	−	0.979631i	$-0.435644\pi$
$73$	3.43137	0.401611	0.200806	+	0.979631i	$0.435644\pi$
$74$	−18.5652	−2.15816
$75$	0	0
$76$	−7.71026	−0.884427
$77$	0.646091	0.0736290
$78$	0	0
$79$	5.69346	0.640564	0.320282	−	0.947322i	$-0.396222\pi$
$79$	5.69346	0.640564	0.320282	+	0.947322i	$0.396222\pi$
$80$	0	0
$81$	0	0
$82$	8.43005	0.930943
$83$	−7.13371	−0.783027	−0.391513	−	0.920172i	$-0.628048\pi$
$83$	−7.13371	−0.783027	−0.391513	+	0.920172i	$0.628048\pi$
$84$	0	0
$85$	0	0
$86$	14.4404	1.55715
$87$	0	0
$88$	0.192943	0.0205678
$89$	−1.52999	−0.162179	−0.0810894	−	0.996707i	$-0.525840\pi$
$89$	−1.52999	−0.162179	−0.0810894	+	0.996707i	$0.525840\pi$
$90$	0	0
$91$	0.412970	0.0432911
$92$	−14.8900	−1.55239
$93$	0	0
$94$	−1.63162	−0.168289
$95$	0	0
$96$	0	0
$97$	6.39742	0.649559	0.324780	−	0.945790i	$-0.394710\pi$
$97$	6.39742	0.649559	0.324780	+	0.945790i	$0.394710\pi$
$98$	−13.8201	−1.39604
$99$	0	0
$100$	0	0
$101$	12.2487	1.21879	0.609396	−	0.792866i	$-0.291412\pi$
$101$	12.2487	1.21879	0.609396	+	0.792866i	$0.291412\pi$
$102$	0	0
$103$	−7.66730	−0.755482	−0.377741	−	0.925911i	$-0.623299\pi$
$103$	−7.66730	−0.755482	−0.377741	+	0.925911i	$0.623299\pi$
$104$	0.123326	0.0120931
$105$	0	0
$106$	7.88369	0.765731
$107$	0.758003	0.0732789	0.0366394	−	0.999329i	$-0.488335\pi$
$107$	0.758003	0.0732789	0.0366394	+	0.999329i	$0.488335\pi$
$108$	0	0
$109$	−3.26839	−0.313055	−0.156528	−	0.987674i	$-0.550030\pi$
$109$	−3.26839	−0.313055	−0.156528	+	0.987674i	$0.550030\pi$
$110$	0	0
$111$	0	0
$112$	−1.43817	−0.135895
$113$	−0.847957	−0.0797691	−0.0398846	−	0.999204i	$-0.512699\pi$
$113$	−0.847957	−0.0797691	−0.0398846	+	0.999204i	$0.512699\pi$
$114$	0	0
$115$	0	0
$116$	−8.61572	−0.799950
$117$	0	0
$118$	3.74015	0.344309
$119$	−2.03040	−0.186127
$120$	0	0
$121$	−7.95034	−0.722758
$122$	19.5099	1.76635
$123$	0	0
$124$	0.628139	0.0564086
$125$	0	0
$126$	0	0
$127$	14.3225	1.27092	0.635459	−	0.772135i	$-0.280811\pi$
$127$	14.3225	1.27092	0.635459	+	0.772135i	$0.280811\pi$
$128$	−0.883546	−0.0780951
$129$	0	0
$130$	0	0
$131$	16.6266	1.45267	0.726334	−	0.687342i	$-0.241222\pi$
$131$	16.6266	1.45267	0.726334	+	0.687342i	$0.241222\pi$
$132$	0	0
$133$	−1.38820	−0.120373
$134$	−25.1106	−2.16923
$135$	0	0
$136$	−0.606340	−0.0519932
$137$	3.07189	0.262449	0.131224	−	0.991353i	$-0.458109\pi$
$137$	3.07189	0.262449	0.131224	+	0.991353i	$0.458109\pi$
$138$	0	0
$139$	−15.7200	−1.33335	−0.666677	−	0.745347i	$-0.732284\pi$
$139$	−15.7200	−1.33335	−0.666677	+	0.745347i	$0.732284\pi$
$140$	0	0
$141$	0	0
$142$	−23.5130	−1.97317
$143$	1.94929	0.163008
$144$	0	0
$145$	0	0
$146$	6.90964	0.571846
$147$	0	0
$148$	−18.9450	−1.55727
$149$	−14.8504	−1.21660	−0.608298	−	0.793709i	$-0.708147\pi$
$149$	−14.8504	−1.21660	−0.608298	+	0.793709i	$0.708147\pi$
$150$	0	0
$151$	−0.712013	−0.0579428	−0.0289714	−	0.999580i	$-0.509223\pi$
$151$	−0.712013	−0.0579428	−0.0289714	+	0.999580i	$0.509223\pi$
$152$	−0.414560	−0.0336253
$153$	0	0
$154$	1.30102	0.104839
$155$	0	0
$156$	0	0
$157$	−22.0704	−1.76141	−0.880704	−	0.473667i	$-0.842930\pi$
$157$	−22.0704	−1.76141	−0.880704	+	0.473667i	$0.842930\pi$
$158$	11.4647	0.912086
$159$	0	0
$160$	0	0
$161$	−2.68088	−0.211283
$162$	0	0
$163$	−13.1619	−1.03092	−0.515460	−	0.856914i	$-0.672379\pi$
$163$	−13.1619	−1.03092	−0.515460	+	0.856914i	$0.672379\pi$
$164$	8.60251	0.671744
$165$	0	0
$166$	−14.3649	−1.11494
$167$	−10.4081	−0.805403	−0.402702	−	0.915331i	$-0.631929\pi$
$167$	−10.4081	−0.805403	−0.402702	+	0.915331i	$0.631929\pi$
$168$	0	0
$169$	−11.7540	−0.904158
$170$	0	0
$171$	0	0
$172$	14.7358	1.12360
$173$	9.19247	0.698890	0.349445	−	0.936957i	$-0.386370\pi$
$173$	9.19247	0.698890	0.349445	+	0.936957i	$0.386370\pi$
$174$	0	0
$175$	0	0
$176$	−6.78842	−0.511697
$177$	0	0
$178$	−3.08090	−0.230923
$179$	21.6873	1.62099	0.810494	−	0.585747i	$-0.199199\pi$
$179$	21.6873	1.62099	0.810494	+	0.585747i	$0.199199\pi$
$180$	0	0
$181$	10.7680	0.800378	0.400189	−	0.916433i	$-0.368944\pi$
$181$	10.7680	0.800378	0.400189	+	0.916433i	$0.368944\pi$
$182$	0.831586	0.0616413
$183$	0	0
$184$	−0.800594	−0.0590206
$185$	0	0
$186$	0	0
$187$	−9.58385	−0.700840
$188$	−1.66500	−0.121433
$189$	0	0
$190$	0	0
$191$	3.83941	0.277810	0.138905	−	0.990306i	$-0.455642\pi$
$191$	3.83941	0.277810	0.138905	+	0.990306i	$0.455642\pi$
$192$	0	0
$193$	−10.5334	−0.758208	−0.379104	−	0.925354i	$-0.623768\pi$
$193$	−10.5334	−0.758208	−0.379104	+	0.925354i	$0.623768\pi$
$194$	12.8823	0.924894
$195$	0	0
$196$	−14.1028	−1.00734
$197$	9.65302	0.687749	0.343875	−	0.939016i	$-0.388260\pi$
$197$	9.65302	0.687749	0.343875	+	0.939016i	$0.388260\pi$
$198$	0	0
$199$	−15.8462	−1.12331	−0.561654	−	0.827372i	$-0.689835\pi$
$199$	−15.8462	−1.12331	−0.561654	+	0.827372i	$0.689835\pi$
$200$	0	0
$201$	0	0
$202$	24.6649	1.73541
$203$	−1.55123	−0.108875
$204$	0	0
$205$	0	0
$206$	−15.4394	−1.07572
$207$	0	0
$208$	−4.33904	−0.300858
$209$	−6.55256	−0.453250
$210$	0	0
$211$	7.25106	0.499184	0.249592	−	0.968351i	$-0.419704\pi$
$211$	7.25106	0.499184	0.249592	+	0.968351i	$0.419704\pi$
$212$	8.04498	0.552531
$213$	0	0
$214$	1.52637	0.104340
$215$	0	0
$216$	0	0
$217$	0.113094	0.00767733
$218$	−6.58147	−0.445753
$219$	0	0
$220$	0	0
$221$	−6.12583	−0.412068
$222$	0	0
$223$	22.8653	1.53118	0.765588	−	0.643331i	$-0.222448\pi$
$223$	22.8653	1.53118	0.765588	+	0.643331i	$0.222448\pi$
$224$	−2.97776	−0.198960
$225$	0	0
$226$	−1.70751	−0.113582
$227$	0.665418	0.0441654	0.0220827	−	0.999756i	$-0.492970\pi$
$227$	0.665418	0.0441654	0.0220827	+	0.999756i	$0.492970\pi$
$228$	0	0
$229$	−23.4732	−1.55115	−0.775576	−	0.631254i	$-0.782541\pi$
$229$	−23.4732	−1.55115	−0.775576	+	0.631254i	$0.782541\pi$
$230$	0	0
$231$	0	0
$232$	−0.463245	−0.0304135
$233$	−17.8293	−1.16803	−0.584017	−	0.811742i	$-0.698520\pi$
$233$	−17.8293	−1.16803	−0.584017	+	0.811742i	$0.698520\pi$
$234$	0	0
$235$	0	0
$236$	3.81667	0.248444
$237$	0	0
$238$	−4.08856	−0.265022
$239$	23.8706	1.54406	0.772030	−	0.635587i	$-0.219242\pi$
$239$	23.8706	1.54406	0.772030	+	0.635587i	$0.219242\pi$
$240$	0	0
$241$	−4.21325	−0.271399	−0.135700	−	0.990750i	$-0.543328\pi$
$241$	−4.21325	−0.271399	−0.135700	+	0.990750i	$0.543328\pi$
$242$	−16.0094	−1.02912
$243$	0	0
$244$	19.9091	1.27455
$245$	0	0
$246$	0	0
$247$	−4.18828	−0.266494
$248$	0.0337734	0.00214461
$249$	0	0
$250$	0	0
$251$	−19.5741	−1.23551	−0.617755	−	0.786371i	$-0.711958\pi$
$251$	−19.5741	−1.23551	−0.617755	+	0.786371i	$0.711958\pi$
$252$	0	0
$253$	−12.6542	−0.795565
$254$	28.8408	1.80963
$255$	0	0
$256$	15.0863	0.942896
$257$	18.4169	1.14881	0.574407	−	0.818570i	$-0.305233\pi$
$257$	18.4169	1.14881	0.574407	+	0.818570i	$0.305233\pi$
$258$	0	0
$259$	−3.41097	−0.211948
$260$	0	0
$261$	0	0
$262$	33.4804	2.06843
$263$	4.32450	0.266660	0.133330	−	0.991072i	$-0.457433\pi$
$263$	4.32450	0.266660	0.133330	+	0.991072i	$0.457433\pi$
$264$	0	0
$265$	0	0
$266$	−2.79538	−0.171396
$267$	0	0
$268$	−25.6243	−1.56526
$269$	−4.73895	−0.288939	−0.144469	−	0.989509i	$-0.546148\pi$
$269$	−4.73895	−0.288939	−0.144469	+	0.989509i	$0.546148\pi$
$270$	0	0
$271$	9.96528	0.605348	0.302674	−	0.953094i	$-0.402121\pi$
$271$	9.96528	0.605348	0.302674	+	0.953094i	$0.402121\pi$
$272$	21.3332	1.29352
$273$	0	0
$274$	6.18577	0.373696
$275$	0	0
$276$	0	0
$277$	−17.6092	−1.05804	−0.529018	−	0.848611i	$-0.677440\pi$
$277$	−17.6092	−1.05804	−0.529018	+	0.848611i	$0.677440\pi$
$278$	−31.6549	−1.89854
$279$	0	0
$280$	0	0
$281$	25.4964	1.52099	0.760494	−	0.649345i	$-0.224957\pi$
$281$	25.4964	1.52099	0.760494	+	0.649345i	$0.224957\pi$
$282$	0	0
$283$	−16.1004	−0.957072	−0.478536	−	0.878068i	$-0.658832\pi$
$283$	−16.1004	−0.957072	−0.478536	+	0.878068i	$0.658832\pi$
$284$	−23.9941	−1.42379
$285$	0	0
$286$	3.92523	0.232104
$287$	1.54885	0.0914258
$288$	0	0
$289$	13.1181	0.771654
$290$	0	0
$291$	0	0
$292$	7.05101	0.412629
$293$	−24.9049	−1.45496	−0.727481	−	0.686128i	$-0.759309\pi$
$293$	−24.9049	−1.45496	−0.727481	+	0.686128i	$0.759309\pi$
$294$	0	0
$295$	0	0
$296$	−1.01862	−0.0592062
$297$	0	0
$298$	−29.9039	−1.73229
$299$	−8.08836	−0.467762
$300$	0	0
$301$	2.65313	0.152924
$302$	−1.43376	−0.0825035
$303$	0	0
$304$	14.5857	0.836549
$305$	0	0
$306$	0	0
$307$	1.74743	0.0997311	0.0498655	−	0.998756i	$-0.484121\pi$
$307$	1.74743	0.0997311	0.0498655	+	0.998756i	$0.484121\pi$
$308$	1.32763	0.0756489
$309$	0	0
$310$	0	0
$311$	−18.3262	−1.03919	−0.519593	−	0.854414i	$-0.673916\pi$
$311$	−18.3262	−1.03919	−0.519593	+	0.854414i	$0.673916\pi$
$312$	0	0
$313$	3.30758	0.186955	0.0934777	−	0.995621i	$-0.470202\pi$
$313$	3.30758	0.186955	0.0934777	+	0.995621i	$0.470202\pi$
$314$	−44.4425	−2.50803
$315$	0	0
$316$	11.6993	0.658137
$317$	0.999043	0.0561118	0.0280559	−	0.999606i	$-0.491068\pi$
$317$	0.999043	0.0561118	0.0280559	+	0.999606i	$0.491068\pi$
$318$	0	0
$319$	−7.32207	−0.409957
$320$	0	0
$321$	0	0
$322$	−5.39842	−0.300842
$323$	20.5920	1.14577
$324$	0	0
$325$	0	0
$326$	−26.5037	−1.46790
$327$	0	0
$328$	0.462534	0.0255392
$329$	−0.299778	−0.0165273
$330$	0	0
$331$	−14.2009	−0.780555	−0.390277	−	0.920697i	$-0.627621\pi$
$331$	−14.2009	−0.780555	−0.390277	+	0.920697i	$0.627621\pi$
$332$	−14.6588	−0.804508
$333$	0	0
$334$	−20.9585	−1.14680
$335$	0	0
$336$	0	0
$337$	13.1530	0.716489	0.358245	−	0.933628i	$-0.383375\pi$
$337$	13.1530	0.716489	0.358245	+	0.933628i	$0.383375\pi$
$338$	−23.6688	−1.28741
$339$	0	0
$340$	0	0
$341$	0.533824	0.0289082
$342$	0	0
$343$	−5.12896	−0.276938
$344$	0.792306	0.0427183
$345$	0	0
$346$	18.5106	0.995136
$347$	−13.5474	−0.727265	−0.363633	−	0.931542i	$-0.618464\pi$
$347$	−13.5474	−0.727265	−0.363633	+	0.931542i	$0.618464\pi$
$348$	0	0
$349$	−32.0976	−1.71814	−0.859072	−	0.511854i	$-0.828959\pi$
$349$	−32.0976	−1.71814	−0.859072	+	0.511854i	$0.828959\pi$
$350$	0	0
$351$	0	0
$352$	−14.0555	−0.749162
$353$	−18.3122	−0.974662	−0.487331	−	0.873217i	$-0.662029\pi$
$353$	−18.3122	−0.974662	−0.487331	+	0.873217i	$0.662029\pi$
$354$	0	0
$355$	0	0
$356$	−3.14393	−0.166628
$357$	0	0
$358$	43.6711	2.30809
$359$	14.4364	0.761927	0.380963	−	0.924590i	$-0.375592\pi$
$359$	14.4364	0.761927	0.380963	+	0.924590i	$0.375592\pi$
$360$	0	0
$361$	−4.92106	−0.259003
$362$	21.6832	1.13964
$363$	0	0
$364$	0.848599	0.0444787
$365$	0	0
$366$	0	0
$367$	10.4130	0.543555	0.271777	−	0.962360i	$-0.412389\pi$
$367$	10.4130	0.543555	0.271777	+	0.962360i	$0.412389\pi$
$368$	28.1678	1.46835
$369$	0	0
$370$	0	0
$371$	1.44847	0.0752008
$372$	0	0
$373$	10.0225	0.518944	0.259472	−	0.965751i	$-0.416451\pi$
$373$	10.0225	0.518944	0.259472	+	0.965751i	$0.416451\pi$
$374$	−19.2987	−0.997912
$375$	0	0
$376$	−0.0895228	−0.00461679
$377$	−4.68014	−0.241039
$378$	0	0
$379$	14.2995	0.734516	0.367258	−	0.930119i	$-0.380297\pi$
$379$	14.2995	0.734516	0.367258	+	0.930119i	$0.380297\pi$
$380$	0	0
$381$	0	0
$382$	7.73131	0.395568
$383$	−5.03705	−0.257381	−0.128690	−	0.991685i	$-0.541077\pi$
$383$	−5.03705	−0.257381	−0.128690	+	0.991685i	$0.541077\pi$
$384$	0	0
$385$	0	0
$386$	−21.2107	−1.07960
$387$	0	0
$388$	13.1458	0.667379
$389$	−10.1326	−0.513743	−0.256872	−	0.966446i	$-0.582692\pi$
$389$	−10.1326	−0.513743	−0.256872	+	0.966446i	$0.582692\pi$
$390$	0	0
$391$	39.7671	2.01111
$392$	−0.758270	−0.0382984
$393$	0	0
$394$	19.4380	0.979272
$395$	0	0
$396$	0	0
$397$	19.6679	0.987104	0.493552	−	0.869716i	$-0.335698\pi$
$397$	19.6679	0.987104	0.493552	+	0.869716i	$0.335698\pi$
$398$	−31.9091	−1.59946
$399$	0	0
$400$	0	0
$401$	23.0931	1.15321	0.576606	−	0.817022i	$-0.304377\pi$
$401$	23.0931	1.15321	0.576606	+	0.817022i	$0.304377\pi$
$402$	0	0
$403$	0.341211	0.0169969
$404$	25.1695	1.25223
$405$	0	0
$406$	−3.12366	−0.155025
$407$	−16.1004	−0.798066
$408$	0	0
$409$	38.6338	1.91032	0.955159	−	0.296093i	$-0.0956838\pi$
$409$	38.6338	1.91032	0.955159	+	0.296093i	$0.0956838\pi$
$410$	0	0
$411$	0	0
$412$	−15.7553	−0.776207
$413$	0.687177	0.0338138
$414$	0	0
$415$	0	0
$416$	−8.98405	−0.440479
$417$	0	0
$418$	−13.1947	−0.645373
$419$	10.7891	0.527082	0.263541	−	0.964648i	$-0.415110\pi$
$419$	10.7891	0.527082	0.263541	+	0.964648i	$0.415110\pi$
$420$	0	0
$421$	32.4550	1.58176	0.790880	−	0.611971i	$-0.209623\pi$
$421$	32.4550	1.58176	0.790880	+	0.611971i	$0.209623\pi$
$422$	14.6013	0.710778
$423$	0	0
$424$	0.432557	0.0210068
$425$	0	0
$426$	0	0
$427$	3.58456	0.173469
$428$	1.55760	0.0752892
$429$	0	0
$430$	0	0
$431$	−13.7920	−0.664339	−0.332170	−	0.943220i	$-0.607781\pi$
$431$	−13.7920	−0.664339	−0.332170	+	0.943220i	$0.607781\pi$
$432$	0	0
$433$	−21.1120	−1.01458	−0.507290	−	0.861776i	$-0.669353\pi$
$433$	−21.1120	−1.01458	−0.507290	+	0.861776i	$0.669353\pi$
$434$	0.227734	0.0109316
$435$	0	0
$436$	−6.71612	−0.321644
$437$	27.1891	1.30063
$438$	0	0
$439$	36.3457	1.73468	0.867342	−	0.497713i	$-0.165827\pi$
$439$	36.3457	1.73468	0.867342	+	0.497713i	$0.165827\pi$
$440$	0	0
$441$	0	0
$442$	−12.3354	−0.586735
$443$	−6.38810	−0.303508	−0.151754	−	0.988418i	$-0.548492\pi$
$443$	−6.38810	−0.303508	−0.151754	+	0.988418i	$0.548492\pi$
$444$	0	0
$445$	0	0
$446$	46.0432	2.18021
$447$	0	0
$448$	−3.11988	−0.147400
$449$	35.1628	1.65943	0.829717	−	0.558185i	$-0.188502\pi$
$449$	35.1628	1.65943	0.829717	+	0.558185i	$0.188502\pi$
$450$	0	0
$451$	7.31084	0.344254
$452$	−1.74244	−0.0819575
$453$	0	0
$454$	1.33993	0.0628862
$455$	0	0
$456$	0	0
$457$	22.2994	1.04312	0.521561	−	0.853214i	$-0.325350\pi$
$457$	22.2994	1.04312	0.521561	+	0.853214i	$0.325350\pi$
$458$	−47.2673	−2.20865
$459$	0	0
$460$	0	0
$461$	1.88541	0.0878122	0.0439061	−	0.999036i	$-0.486020\pi$
$461$	1.88541	0.0878122	0.0439061	+	0.999036i	$0.486020\pi$
$462$	0	0
$463$	−12.9152	−0.600218	−0.300109	−	0.953905i	$-0.597023\pi$
$463$	−12.9152	−0.600218	−0.300109	+	0.953905i	$0.597023\pi$
$464$	16.2986	0.756645
$465$	0	0
$466$	−35.9023	−1.66314
$467$	2.61816	0.121154	0.0605769	−	0.998164i	$-0.480706\pi$
$467$	2.61816	0.121154	0.0605769	+	0.998164i	$0.480706\pi$
$468$	0	0
$469$	−4.61357	−0.213035
$470$	0	0
$471$	0	0
$472$	0.205212	0.00944566
$473$	12.5232	0.575819
$474$	0	0
$475$	0	0
$476$	−4.17221	−0.191233
$477$	0	0
$478$	48.0675	2.19855
$479$	−6.73316	−0.307646	−0.153823	−	0.988098i	$-0.549159\pi$
$479$	−6.73316	−0.307646	−0.153823	+	0.988098i	$0.549159\pi$
$480$	0	0
$481$	−10.2911	−0.469233
$482$	−8.48409	−0.386440
$483$	0	0
$484$	−16.3369	−0.742586
$485$	0	0
$486$	0	0
$487$	−5.55795	−0.251855	−0.125927	−	0.992039i	$-0.540191\pi$
$487$	−5.55795	−0.251855	−0.125927	+	0.992039i	$0.540191\pi$
$488$	1.07046	0.0484574
$489$	0	0
$490$	0	0
$491$	5.55199	0.250558	0.125279	−	0.992122i	$-0.460017\pi$
$491$	5.55199	0.250558	0.125279	+	0.992122i	$0.460017\pi$
$492$	0	0
$493$	23.0103	1.03633
$494$	−8.43381	−0.379455
$495$	0	0
$496$	−1.18827	−0.0533549
$497$	−4.32005	−0.193781
$498$	0	0
$499$	19.2580	0.862107	0.431054	−	0.902326i	$-0.358142\pi$
$499$	19.2580	0.862107	0.431054	+	0.902326i	$0.358142\pi$
$500$	0	0
$501$	0	0
$502$	−39.4159	−1.75922
$503$	−31.1565	−1.38920	−0.694600	−	0.719396i	$-0.744419\pi$
$503$	−31.1565	−1.38920	−0.694600	+	0.719396i	$0.744419\pi$
$504$	0	0
$505$	0	0
$506$	−25.4814	−1.13279
$507$	0	0
$508$	29.4309	1.30578
$509$	21.2250	0.940782	0.470391	−	0.882458i	$-0.344113\pi$
$509$	21.2250	0.940782	0.470391	+	0.882458i	$0.344113\pi$
$510$	0	0
$511$	1.26951	0.0561597
$512$	32.1460	1.42067
$513$	0	0
$514$	37.0855	1.63577
$515$	0	0
$516$	0	0
$517$	−1.41500	−0.0622317
$518$	−6.86858	−0.301788
$519$	0	0
$520$	0	0
$521$	−24.0095	−1.05188	−0.525938	−	0.850523i	$-0.676286\pi$
$521$	−24.0095	−1.05188	−0.525938	+	0.850523i	$0.676286\pi$
$522$	0	0
$523$	22.8190	0.997804	0.498902	−	0.866658i	$-0.333737\pi$
$523$	22.8190	0.997804	0.498902	+	0.866658i	$0.333737\pi$
$524$	34.1654	1.49252
$525$	0	0
$526$	8.70813	0.379692
$527$	−1.67759	−0.0730770
$528$	0	0
$529$	29.5073	1.28293
$530$	0	0
$531$	0	0
$532$	−2.85257	−0.123675
$533$	4.67296	0.202408
$534$	0	0
$535$	0	0
$536$	−1.37775	−0.0595098
$537$	0	0
$538$	−9.54268	−0.411414
$539$	−11.9853	−0.516242
$540$	0	0
$541$	27.5072	1.18263	0.591314	−	0.806442i	$-0.298609\pi$
$541$	27.5072	1.18263	0.591314	+	0.806442i	$0.298609\pi$
$542$	20.0668	0.861943
$543$	0	0
$544$	44.1708	1.89381
$545$	0	0
$546$	0	0
$547$	0.243224	0.0103995	0.00519975	−	0.999986i	$-0.498345\pi$
$547$	0.243224	0.0103995	0.00519975	+	0.999986i	$0.498345\pi$
$548$	6.31232	0.269649
$549$	0	0
$550$	0	0
$551$	15.7323	0.670220
$552$	0	0
$553$	2.10642	0.0895739
$554$	−35.4592	−1.50652
$555$	0	0
$556$	−32.3025	−1.36993
$557$	−27.7280	−1.17487	−0.587436	−	0.809271i	$-0.699862\pi$
$557$	−27.7280	−1.17487	−0.587436	+	0.809271i	$0.699862\pi$
$558$	0	0
$559$	8.00463	0.338560
$560$	0	0
$561$	0	0
$562$	51.3414	2.16570
$563$	12.4049	0.522805	0.261403	−	0.965230i	$-0.415815\pi$
$563$	12.4049	0.522805	0.261403	+	0.965230i	$0.415815\pi$
$564$	0	0
$565$	0	0
$566$	−32.4210	−1.36276
$567$	0	0
$568$	−1.29010	−0.0541313
$569$	26.6074	1.11544	0.557720	−	0.830029i	$-0.311676\pi$
$569$	26.6074	1.11544	0.557720	+	0.830029i	$0.311676\pi$
$570$	0	0
$571$	12.9219	0.540765	0.270382	−	0.962753i	$-0.412850\pi$
$571$	12.9219	0.540765	0.270382	+	0.962753i	$0.412850\pi$
$572$	4.00553	0.167480
$573$	0	0
$574$	3.11888	0.130179
$575$	0	0
$576$	0	0
$577$	−28.3432	−1.17994	−0.589971	−	0.807425i	$-0.700861\pi$
$577$	−28.3432	−1.17994	−0.589971	+	0.807425i	$0.700861\pi$
$578$	26.4156	1.09874
$579$	0	0
$580$	0	0
$581$	−2.63927	−0.109495
$582$	0	0
$583$	6.83702	0.283160
$584$	0.379114	0.0156878
$585$	0	0
$586$	−50.1503	−2.07169
$587$	−8.38885	−0.346245	−0.173123	−	0.984900i	$-0.555386\pi$
$587$	−8.38885	−0.346245	−0.173123	+	0.984900i	$0.555386\pi$
$588$	0	0
$589$	−1.14698	−0.0472606
$590$	0	0
$591$	0	0
$592$	35.8388	1.47297
$593$	30.9031	1.26904	0.634518	−	0.772908i	$-0.281199\pi$
$593$	30.9031	1.26904	0.634518	+	0.772908i	$0.281199\pi$
$594$	0	0
$595$	0	0
$596$	−30.5157	−1.24997
$597$	0	0
$598$	−16.2873	−0.666037
$599$	−32.6384	−1.33357	−0.666784	−	0.745251i	$-0.732329\pi$
$599$	−32.6384	−1.33357	−0.666784	+	0.745251i	$0.732329\pi$
$600$	0	0
$601$	16.9351	0.690796	0.345398	−	0.938456i	$-0.387744\pi$
$601$	16.9351	0.690796	0.345398	+	0.938456i	$0.387744\pi$
$602$	5.34253	0.217745
$603$	0	0
$604$	−1.46309	−0.0595324
$605$	0	0
$606$	0	0
$607$	−36.3044	−1.47355	−0.736775	−	0.676138i	$-0.763652\pi$
$607$	−36.3044	−1.47355	−0.736775	+	0.676138i	$0.763652\pi$
$608$	30.2000	1.22477
$609$	0	0
$610$	0	0
$611$	−0.904445	−0.0365899
$612$	0	0
$613$	30.7941	1.24376	0.621881	−	0.783112i	$-0.286369\pi$
$613$	30.7941	1.24376	0.621881	+	0.783112i	$0.286369\pi$
$614$	3.51875	0.142005
$615$	0	0
$616$	0.0713833	0.00287611
$617$	−35.3238	−1.42208	−0.711042	−	0.703150i	$-0.751776\pi$
$617$	−35.3238	−1.42208	−0.711042	+	0.703150i	$0.751776\pi$
$618$	0	0
$619$	3.95098	0.158803	0.0794017	−	0.996843i	$-0.474699\pi$
$619$	3.95098	0.158803	0.0794017	+	0.996843i	$0.474699\pi$
$620$	0	0
$621$	0	0
$622$	−36.9030	−1.47967
$623$	−0.566053	−0.0226784
$624$	0	0
$625$	0	0
$626$	6.66037	0.266202
$627$	0	0
$628$	−45.3517	−1.80973
$629$	50.5969	2.01743
$630$	0	0
$631$	13.5268	0.538494	0.269247	−	0.963071i	$-0.413225\pi$
$631$	13.5268	0.538494	0.269247	+	0.963071i	$0.413225\pi$
$632$	0.629040	0.0250219
$633$	0	0
$634$	2.01174	0.0798965
$635$	0	0
$636$	0	0
$637$	−7.66077	−0.303531
$638$	−14.7442	−0.583730
$639$	0	0
$640$	0	0
$641$	36.1269	1.42693	0.713464	−	0.700692i	$-0.247125\pi$
$641$	36.1269	1.42693	0.713464	+	0.700692i	$0.247125\pi$
$642$	0	0
$643$	−7.35135	−0.289909	−0.144954	−	0.989438i	$-0.546304\pi$
$643$	−7.35135	−0.289909	−0.144954	+	0.989438i	$0.546304\pi$
$644$	−5.50886	−0.217080
$645$	0	0
$646$	41.4655	1.63144
$647$	−43.7004	−1.71804	−0.859020	−	0.511943i	$-0.828926\pi$
$647$	−43.7004	−1.71804	−0.859020	+	0.511943i	$0.828926\pi$
$648$	0	0
$649$	3.24359	0.127322
$650$	0	0
$651$	0	0
$652$	−27.0459	−1.05920
$653$	35.8445	1.40270	0.701352	−	0.712815i	$-0.252580\pi$
$653$	35.8445	1.40270	0.701352	+	0.712815i	$0.252580\pi$
$654$	0	0
$655$	0	0
$656$	−16.2736	−0.635379
$657$	0	0
$658$	−0.603653	−0.0235329
$659$	−0.0683150	−0.00266117	−0.00133059	−	0.999999i	$-0.500424\pi$
$659$	−0.0683150	−0.00266117	−0.00133059	+	0.999999i	$0.500424\pi$
$660$	0	0
$661$	−26.3573	−1.02518	−0.512590	−	0.858634i	$-0.671314\pi$
$661$	−26.3573	−1.02518	−0.512590	+	0.858634i	$0.671314\pi$
$662$	−28.5960	−1.11142
$663$	0	0
$664$	−0.788166	−0.0305868
$665$	0	0
$666$	0	0
$667$	30.3821	1.17640
$668$	−21.3873	−0.827498
$669$	0	0
$670$	0	0
$671$	16.9197	0.653179
$672$	0	0
$673$	−5.07703	−0.195705	−0.0978525	−	0.995201i	$-0.531197\pi$
$673$	−5.07703	−0.195705	−0.0978525	+	0.995201i	$0.531197\pi$
$674$	26.4858	1.02019
$675$	0	0
$676$	−24.1530	−0.928962
$677$	−38.5508	−1.48163	−0.740814	−	0.671710i	$-0.765560\pi$
$677$	−38.5508	−1.48163	−0.740814	+	0.671710i	$0.765560\pi$
$678$	0	0
$679$	2.36686	0.0908318
$680$	0	0
$681$	0	0
$682$	1.07494	0.0411618
$683$	−36.6133	−1.40097	−0.700484	−	0.713668i	$-0.747032\pi$
$683$	−36.6133	−1.40097	−0.700484	+	0.713668i	$0.747032\pi$
$684$	0	0
$685$	0	0
$686$	−10.3280	−0.394326
$687$	0	0
$688$	−27.8762	−1.06277
$689$	4.37010	0.166488
$690$	0	0
$691$	29.3768	1.11755	0.558774	−	0.829320i	$-0.311272\pi$
$691$	29.3768	1.11755	0.558774	+	0.829320i	$0.311272\pi$
$692$	18.8893	0.718063
$693$	0	0
$694$	−27.2801	−1.03554
$695$	0	0
$696$	0	0
$697$	−22.9750	−0.870240
$698$	−64.6339	−2.44643
$699$	0	0
$700$	0	0
$701$	0.566147	0.0213831	0.0106915	−	0.999943i	$-0.496597\pi$
$701$	0.566147	0.0213831	0.0106915	+	0.999943i	$0.496597\pi$
$702$	0	0
$703$	34.5936	1.30472
$704$	−14.7264	−0.555020
$705$	0	0
$706$	−36.8748	−1.38780
$707$	4.53167	0.170431
$708$	0	0
$709$	0.657257	0.0246838	0.0123419	−	0.999924i	$-0.496071\pi$
$709$	0.657257	0.0246838	0.0123419	+	0.999924i	$0.496071\pi$
$710$	0	0
$711$	0	0
$712$	−0.169041	−0.00633507
$713$	−2.21504	−0.0829540
$714$	0	0
$715$	0	0
$716$	44.5646	1.66546
$717$	0	0
$718$	29.0702	1.08489
$719$	−35.1052	−1.30920	−0.654601	−	0.755975i	$-0.727163\pi$
$719$	−35.1052	−1.30920	−0.654601	+	0.755975i	$0.727163\pi$
$720$	0	0
$721$	−2.83668	−0.105644
$722$	−9.90939	−0.368789
$723$	0	0
$724$	22.1268	0.822336
$725$	0	0
$726$	0	0
$727$	−41.2180	−1.52869	−0.764346	−	0.644807i	$-0.776938\pi$
$727$	−41.2180	−1.52869	−0.764346	+	0.644807i	$0.776938\pi$
$728$	0.0456269	0.00169105
$729$	0	0
$730$	0	0
$731$	−39.3554	−1.45561
$732$	0	0
$733$	−40.1082	−1.48143	−0.740715	−	0.671820i	$-0.765513\pi$
$733$	−40.1082	−1.48143	−0.740715	+	0.671820i	$0.765513\pi$
$734$	20.9684	0.773957
$735$	0	0
$736$	58.3218	2.14977
$737$	−21.7768	−0.802160
$738$	0	0
$739$	−28.5434	−1.04999	−0.524993	−	0.851106i	$-0.675932\pi$
$739$	−28.5434	−1.04999	−0.524993	+	0.851106i	$0.675932\pi$
$740$	0	0
$741$	0	0
$742$	2.91674	0.107077
$743$	−29.6851	−1.08904	−0.544520	−	0.838748i	$-0.683288\pi$
$743$	−29.6851	−1.08904	−0.544520	+	0.838748i	$0.683288\pi$
$744$	0	0
$745$	0	0
$746$	20.1820	0.738914
$747$	0	0
$748$	−19.6935	−0.720067
$749$	0.280439	0.0102470
$750$	0	0
$751$	45.2113	1.64978	0.824892	−	0.565290i	$-0.191236\pi$
$751$	45.2113	1.64978	0.824892	+	0.565290i	$0.191236\pi$
$752$	3.14974	0.114859
$753$	0	0
$754$	−9.42425	−0.343211
$755$	0	0
$756$	0	0
$757$	5.69813	0.207102	0.103551	−	0.994624i	$-0.466980\pi$
$757$	5.69813	0.207102	0.103551	+	0.994624i	$0.466980\pi$
$758$	28.7945	1.04586
$759$	0	0
$760$	0	0
$761$	41.6303	1.50910	0.754548	−	0.656245i	$-0.227856\pi$
$761$	41.6303	1.50910	0.754548	+	0.656245i	$0.227856\pi$
$762$	0	0
$763$	−1.20921	−0.0437764
$764$	7.88948	0.285431
$765$	0	0
$766$	−10.1429	−0.366480
$767$	2.07325	0.0748607
$768$	0	0
$769$	−15.9261	−0.574308	−0.287154	−	0.957884i	$-0.592709\pi$
$769$	−15.9261	−0.574308	−0.287154	+	0.957884i	$0.592709\pi$
$770$	0	0
$771$	0	0
$772$	−21.6447	−0.779009
$773$	47.8320	1.72040	0.860198	−	0.509959i	$-0.170340\pi$
$773$	47.8320	1.72040	0.860198	+	0.509959i	$0.170340\pi$
$774$	0	0
$775$	0	0
$776$	0.706817	0.0253732
$777$	0	0
$778$	−20.4037	−0.731509
$779$	−15.7082	−0.562805
$780$	0	0
$781$	−20.3914	−0.729661
$782$	80.0778	2.86358
$783$	0	0
$784$	26.6787	0.952811
$785$	0	0
$786$	0	0
$787$	40.5561	1.44567	0.722834	−	0.691021i	$-0.242839\pi$
$787$	40.5561	1.44567	0.722834	+	0.691021i	$0.242839\pi$
$788$	19.8357	0.706617
$789$	0	0
$790$	0	0
$791$	−0.313720	−0.0111546
$792$	0	0
$793$	10.8148	0.384044
$794$	39.6047	1.40552
$795$	0	0
$796$	−32.5619	−1.15413
$797$	−27.8316	−0.985847	−0.492924	−	0.870073i	$-0.664072\pi$
$797$	−27.8316	−0.985847	−0.492924	+	0.870073i	$0.664072\pi$
$798$	0	0
$799$	4.44678	0.157316
$800$	0	0
$801$	0	0
$802$	46.5018	1.64204
$803$	5.99230	0.211464
$804$	0	0
$805$	0	0
$806$	0.687086	0.0242016
$807$	0	0
$808$	1.35330	0.0476088
$809$	8.24706	0.289951	0.144976	−	0.989435i	$-0.453690\pi$
$809$	8.24706	0.289951	0.144976	+	0.989435i	$0.453690\pi$
$810$	0	0
$811$	−18.3560	−0.644567	−0.322283	−	0.946643i	$-0.604450\pi$
$811$	−18.3560	−0.644567	−0.322283	+	0.946643i	$0.604450\pi$
$812$	−3.18757	−0.111862
$813$	0	0
$814$	−32.4209	−1.13635
$815$	0	0
$816$	0	0
$817$	−26.9076	−0.941379
$818$	77.7957	2.72006
$819$	0	0
$820$	0	0
$821$	15.8635	0.553640	0.276820	−	0.960922i	$-0.410719\pi$
$821$	15.8635	0.553640	0.276820	+	0.960922i	$0.410719\pi$
$822$	0	0
$823$	−32.4705	−1.13185	−0.565925	−	0.824457i	$-0.691481\pi$
$823$	−32.4705	−1.13185	−0.565925	+	0.824457i	$0.691481\pi$
$824$	−0.847120	−0.0295108
$825$	0	0
$826$	1.38375	0.0481468
$827$	−28.7946	−1.00129	−0.500643	−	0.865654i	$-0.666903\pi$
$827$	−28.7946	−1.00129	−0.500643	+	0.865654i	$0.666903\pi$
$828$	0	0
$829$	−8.59224	−0.298421	−0.149210	−	0.988805i	$-0.547673\pi$
$829$	−8.59224	−0.298421	−0.149210	+	0.988805i	$0.547673\pi$
$830$	0	0
$831$	0	0
$832$	−9.41283	−0.326331
$833$	37.6648	1.30501
$834$	0	0
$835$	0	0
$836$	−13.4646	−0.465684
$837$	0	0
$838$	21.7257	0.750501
$839$	47.0541	1.62449	0.812244	−	0.583318i	$-0.198246\pi$
$839$	47.0541	1.62449	0.812244	+	0.583318i	$0.198246\pi$
$840$	0	0
$841$	−11.4201	−0.393797
$842$	65.3537	2.25224
$843$	0	0
$844$	14.9000	0.512878
$845$	0	0
$846$	0	0
$847$	−2.94140	−0.101068
$848$	−15.2189	−0.522620
$849$	0	0
$850$	0	0
$851$	66.8067	2.29011
$852$	0	0
$853$	−18.6891	−0.639904	−0.319952	−	0.947434i	$-0.603667\pi$
$853$	−18.6891	−0.639904	−0.319952	+	0.947434i	$0.603667\pi$
$854$	7.21811	0.246999
$855$	0	0
$856$	0.0837478	0.00286244
$857$	−3.06228	−0.104606	−0.0523028	−	0.998631i	$-0.516656\pi$
$857$	−3.06228	−0.104606	−0.0523028	+	0.998631i	$0.516656\pi$
$858$	0	0
$859$	16.9664	0.578885	0.289442	−	0.957195i	$-0.406530\pi$
$859$	16.9664	0.578885	0.289442	+	0.957195i	$0.406530\pi$
$860$	0	0
$861$	0	0
$862$	−27.7726	−0.945939
$863$	21.8969	0.745380	0.372690	−	0.927956i	$-0.378435\pi$
$863$	21.8969	0.745380	0.372690	+	0.927956i	$0.378435\pi$
$864$	0	0
$865$	0	0
$866$	−42.5127	−1.44464
$867$	0	0
$868$	0.232393	0.00788795
$869$	9.94264	0.337281
$870$	0	0
$871$	−13.9194	−0.471640
$872$	−0.361108	−0.0122287
$873$	0	0
$874$	54.7499	1.85194
$875$	0	0
$876$	0	0
$877$	46.8320	1.58141	0.790703	−	0.612200i	$-0.209715\pi$
$877$	46.8320	1.58141	0.790703	+	0.612200i	$0.209715\pi$
$878$	73.1882	2.46998
$879$	0	0
$880$	0	0
$881$	−0.0281377	−0.000947982 0	−0.000473991	−	1.00000i	$-0.500151\pi$
$881$	−0.0281377	−0.000947982 0	−0.000473991		1.00000i	$0.500151\pi$
$882$	0	0
$883$	−29.2717	−0.985070	−0.492535	−	0.870293i	$-0.663930\pi$
$883$	−29.2717	−0.985070	−0.492535	+	0.870293i	$0.663930\pi$
$884$	−12.5878	−0.423372
$885$	0	0
$886$	−12.8635	−0.432159
$887$	−19.8797	−0.667494	−0.333747	−	0.942663i	$-0.608313\pi$
$887$	−19.8797	−0.667494	−0.333747	+	0.942663i	$0.608313\pi$
$888$	0	0
$889$	5.29892	0.177720
$890$	0	0
$891$	0	0
$892$	46.9852	1.57318
$893$	3.04030	0.101740
$894$	0	0
$895$	0	0
$896$	−0.326887	−0.0109205
$897$	0	0
$898$	70.8062	2.36283
$899$	−1.28168	−0.0427465
$900$	0	0
$901$	−21.4860	−0.715801
$902$	14.7216	0.490177
$903$	0	0
$904$	−0.0936864	−0.00311596
$905$	0	0
$906$	0	0
$907$	−14.2958	−0.474685	−0.237343	−	0.971426i	$-0.576276\pi$
$907$	−14.2958	−0.474685	−0.237343	+	0.971426i	$0.576276\pi$
$908$	1.36735	0.0453770
$909$	0	0
$910$	0	0
$911$	−19.4005	−0.642766	−0.321383	−	0.946949i	$-0.604148\pi$
$911$	−19.4005	−0.642766	−0.321383	+	0.946949i	$0.604148\pi$
$912$	0	0
$913$	−12.4578	−0.412293
$914$	44.9036	1.48528
$915$	0	0
$916$	−48.2343	−1.59371
$917$	6.15135	0.203135
$918$	0	0
$919$	4.29914	0.141816	0.0709078	−	0.997483i	$-0.477410\pi$
$919$	4.29914	0.141816	0.0709078	+	0.997483i	$0.477410\pi$
$920$	0	0
$921$	0	0
$922$	3.79659	0.125034
$923$	−13.0338	−0.429013
$924$	0	0
$925$	0	0
$926$	−26.0069	−0.854639
$927$	0	0
$928$	33.7465	1.10778
$929$	−43.0317	−1.41182	−0.705912	−	0.708300i	$-0.749462\pi$
$929$	−43.0317	−1.41182	−0.705912	+	0.708300i	$0.749462\pi$
$930$	0	0
$931$	25.7517	0.843979
$932$	−36.6368	−1.20008
$933$	0	0
$934$	5.27211	0.172509
$935$	0	0
$936$	0	0
$937$	16.9808	0.554740	0.277370	−	0.960763i	$-0.410537\pi$
$937$	16.9808	0.554740	0.277370	+	0.960763i	$0.410537\pi$
$938$	−9.29020	−0.303336
$939$	0	0
$940$	0	0
$941$	18.0448	0.588242	0.294121	−	0.955768i	$-0.404973\pi$
$941$	18.0448	0.588242	0.294121	+	0.955768i	$0.404973\pi$
$942$	0	0
$943$	−30.3355	−0.987860
$944$	−7.22011	−0.234995
$945$	0	0
$946$	25.2176	0.819896
$947$	−7.20872	−0.234252	−0.117126	−	0.993117i	$-0.537368\pi$
$947$	−7.20872	−0.234252	−0.117126	+	0.993117i	$0.537368\pi$
$948$	0	0
$949$	3.83017	0.124333
$950$	0	0
$951$	0	0
$952$	−0.224329	−0.00727053
$953$	−13.8000	−0.447026	−0.223513	−	0.974701i	$-0.571753\pi$
$953$	−13.8000	−0.447026	−0.223513	+	0.974701i	$0.571753\pi$
$954$	0	0
$955$	0	0
$956$	49.0509	1.58642
$957$	0	0
$958$	−13.5584	−0.438051
$959$	1.13651	0.0366998
$960$	0	0
$961$	−30.9066	−0.996986
$962$	−20.7228	−0.668131
$963$	0	0
$964$	−8.65767	−0.278845
$965$	0	0
$966$	0	0
$967$	14.8956	0.479010	0.239505	−	0.970895i	$-0.423015\pi$
$967$	14.8956	0.479010	0.239505	+	0.970895i	$0.423015\pi$
$968$	−0.878391	−0.0282326
$969$	0	0
$970$	0	0
$971$	−39.7938	−1.27704	−0.638522	−	0.769604i	$-0.720454\pi$
$971$	−39.7938	−1.27704	−0.638522	+	0.769604i	$0.720454\pi$
$972$	0	0
$973$	−5.81595	−0.186451
$974$	−11.1919	−0.358611
$975$	0	0
$976$	−37.6626	−1.20555
$977$	−30.9627	−0.990584	−0.495292	−	0.868726i	$-0.664939\pi$
$977$	−30.9627	−0.990584	−0.495292	+	0.868726i	$0.664939\pi$
$978$	0	0
$979$	−2.67187	−0.0853933
$980$	0	0
$981$	0	0
$982$	11.1799	0.356764
$983$	27.7549	0.885243	0.442621	−	0.896709i	$-0.354049\pi$
$983$	27.7549	0.885243	0.442621	+	0.896709i	$0.354049\pi$
$984$	0	0
$985$	0	0
$986$	46.3351	1.47561
$987$	0	0
$988$	−8.60636	−0.273805
$989$	−51.9637	−1.65235
$990$	0	0
$991$	15.0180	0.477062	0.238531	−	0.971135i	$-0.423334\pi$
$991$	15.0180	0.477062	0.238531	+	0.971135i	$0.423334\pi$
$992$	−2.46033	−0.0781156
$993$	0	0
$994$	−8.69915	−0.275920
$995$	0	0
$996$	0	0
$997$	−17.9138	−0.567335	−0.283668	−	0.958923i	$-0.591551\pi$
$997$	−17.9138	−0.567335	−0.283668	+	0.958923i	$0.591551\pi$
$998$	38.7793	1.22754
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.s.1.8		8
3.2	odd	2		625.2.a.g.1.1	yes	8
5.4	even	2		5625.2.a.be.1.1		8
12.11	even	2		10000.2.a.be.1.2		8
15.2	even	4		625.2.b.d.624.4		16
15.8	even	4		625.2.b.d.624.13		16
15.14	odd	2		625.2.a.e.1.8	✓	8
60.59	even	2		10000.2.a.bn.1.7		8
75.2	even	20		625.2.e.j.124.2		32
75.8	even	20		625.2.e.k.374.2		32
75.11	odd	10		625.2.d.m.501.1		16
75.14	odd	10		625.2.d.q.501.4		16
75.17	even	20		625.2.e.k.374.7		32
75.23	even	20		625.2.e.j.124.7		32
75.29	odd	10		625.2.d.p.376.1		16
75.38	even	20		625.2.e.j.499.2		32
75.41	odd	10		625.2.d.m.126.1		16
75.44	odd	10		625.2.d.p.251.1		16
75.47	even	20		625.2.e.k.249.2		32
75.53	even	20		625.2.e.k.249.7		32
75.56	odd	10		625.2.d.n.251.4		16
75.59	odd	10		625.2.d.q.126.4		16
75.62	even	20		625.2.e.j.499.7		32
75.71	odd	10		625.2.d.n.376.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
625.2.a.e.1.8	✓	8	15.14	odd	2
625.2.a.g.1.1	yes	8	3.2	odd	2
625.2.b.d.624.4		16	15.2	even	4
625.2.b.d.624.13		16	15.8	even	4
625.2.d.m.126.1		16	75.41	odd	10
625.2.d.m.501.1		16	75.11	odd	10
625.2.d.n.251.4		16	75.56	odd	10
625.2.d.n.376.4		16	75.71	odd	10
625.2.d.p.251.1		16	75.44	odd	10
625.2.d.p.376.1		16	75.29	odd	10
625.2.d.q.126.4		16	75.59	odd	10
625.2.d.q.501.4		16	75.14	odd	10
625.2.e.j.124.2		32	75.2	even	20
625.2.e.j.124.7		32	75.23	even	20
625.2.e.j.499.2		32	75.38	even	20
625.2.e.j.499.7		32	75.62	even	20
625.2.e.k.249.2		32	75.47	even	20
625.2.e.k.249.7		32	75.53	even	20
625.2.e.k.374.2		32	75.8	even	20
625.2.e.k.374.7		32	75.17	even	20
5625.2.a.s.1.8		8	1.1	even	1	trivial
5625.2.a.be.1.1		8	5.4	even	2
10000.2.a.be.1.2		8	12.11	even	2
10000.2.a.bn.1.7		8	60.59	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)

\(f(q)\)	\(=\)	\(q+2.01367 q^{2} +2.05487 q^{4} +0.369971 q^{7} +0.110485 q^{8} +O(q^{10})\)	\(q+2.01367 q^{2} +2.05487 q^{4} +0.369971 q^{7} +0.110485 q^{8} +1.74633 q^{11} +1.11622 q^{13} +0.745000 q^{14} -3.88725 q^{16} -5.48800 q^{17} -3.75219 q^{19} +3.51653 q^{22} -7.24619 q^{23} +2.24770 q^{26} +0.760242 q^{28} -4.19284 q^{29} +0.305684 q^{31} -8.04862 q^{32} -11.0510 q^{34} -9.21956 q^{37} -7.55568 q^{38} +4.18641 q^{41} +7.17118 q^{43} +3.58847 q^{44} -14.5914 q^{46} -0.810273 q^{47} -6.86312 q^{49} +2.29369 q^{52} +3.91508 q^{53} +0.0408762 q^{56} -8.44299 q^{58} +1.85738 q^{59} +9.68874 q^{61} +0.615546 q^{62} -8.43275 q^{64} -12.4701 q^{67} -11.2771 q^{68} -11.6767 q^{71} +3.43137 q^{73} -18.5652 q^{74} -7.71026 q^{76} +0.646091 q^{77} +5.69346 q^{79} +8.43005 q^{82} -7.13371 q^{83} +14.4404 q^{86} +0.192943 q^{88} -1.52999 q^{89} +0.412970 q^{91} -14.8900 q^{92} -1.63162 q^{94} +6.39742 q^{97} -13.8201 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 5 q^{2} + 11 q^{4} + 10 q^{7} - 15 q^{8}+O(q^{10}) \) 8 * q - 5 * q^2 + 11 * q^4 + 10 * q^7 - 15 * q^8	\( 8 q - 5 q^{2} + 11 q^{4} + 10 q^{7} - 15 q^{8} - q^{11} + 10 q^{13} + 8 q^{14} + 13 q^{16} - 15 q^{17} - 10 q^{19} - 5 q^{22} - 30 q^{23} - 11 q^{26} - 5 q^{28} - 10 q^{29} - 9 q^{31} - 30 q^{32} + 7 q^{34} - 10 q^{37} - 20 q^{38} + 4 q^{41} + 18 q^{44} - 9 q^{46} - 30 q^{47} - 4 q^{49} + 5 q^{52} - 10 q^{53} - 30 q^{58} + 5 q^{59} + 6 q^{61} - 10 q^{62} - 9 q^{64} + 10 q^{67} - 40 q^{68} + 9 q^{71} + 18 q^{74} - 10 q^{76} - 5 q^{77} - 20 q^{79} - 45 q^{82} - 40 q^{83} + 24 q^{86} - 40 q^{88} + 5 q^{89} + 6 q^{91} - 15 q^{92} + 47 q^{94} + 30 q^{98}+O(q^{100}) \) 8 * q - 5 * q^2 + 11 * q^4 + 10 * q^7 - 15 * q^8 - q^11 + 10 * q^13 + 8 * q^14 + 13 * q^16 - 15 * q^17 - 10 * q^19 - 5 * q^22 - 30 * q^23 - 11 * q^26 - 5 * q^28 - 10 * q^29 - 9 * q^31 - 30 * q^32 + 7 * q^34 - 10 * q^37 - 20 * q^38 + 4 * q^41 + 18 * q^44 - 9 * q^46 - 30 * q^47 - 4 * q^49 + 5 * q^52 - 10 * q^53 - 30 * q^58 + 5 * q^59 + 6 * q^61 - 10 * q^62 - 9 * q^64 + 10 * q^67 - 40 * q^68 + 9 * q^71 + 18 * q^74 - 10 * q^76 - 5 * q^77 - 20 * q^79 - 45 * q^82 - 40 * q^83 + 24 * q^86 - 40 * q^88 + 5 * q^89 + 6 * q^91 - 15 * q^92 + 47 * q^94 + 30 * q^98

Introduction

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