LMFDB - Embedded newform 225.10.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,10,Mod(1,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 10, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("225.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	225.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:	$115.883063137$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.49740556.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 45x^{2} + 304 $ x^4 - 45*x^2 + 304
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{2}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 5)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.87724$ of defining polynomial
Character	$\chi$	$=$	225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-41.3193 q^{2} +1195.29 q^{4} +5315.22 q^{7} -28233.0 q^{8} +O(q^{10})$	$q-41.3193 q^{2} +1195.29 q^{4} +5315.22 q^{7} -28233.0 q^{8} -10426.2 q^{11} +79655.1 q^{13} -219621. q^{14} +554581. q^{16} +313750. q^{17} -246945. q^{19} +430806. q^{22} +721761. q^{23} -3.29130e6 q^{26} +6.35321e6 q^{28} -2.56903e6 q^{29} -3.29543e6 q^{31} -8.45965e6 q^{32} -1.29639e7 q^{34} -1.40463e7 q^{37} +1.02036e7 q^{38} -1.70412e7 q^{41} -2.92261e7 q^{43} -1.24624e7 q^{44} -2.98227e7 q^{46} +4.10316e7 q^{47} -1.21021e7 q^{49} +9.52108e7 q^{52} -5.67230e7 q^{53} -1.50065e8 q^{56} +1.06150e8 q^{58} -1.60408e8 q^{59} +5.33033e7 q^{61} +1.36165e8 q^{62} +6.56012e7 q^{64} +2.80916e8 q^{67} +3.75022e8 q^{68} +8.97228e7 q^{71} +7.60225e7 q^{73} +5.80383e8 q^{74} -2.95170e8 q^{76} -5.54178e7 q^{77} +4.10672e8 q^{79} +7.04131e8 q^{82} -5.21969e8 q^{83} +1.20760e9 q^{86} +2.94364e8 q^{88} -2.37312e8 q^{89} +4.23384e8 q^{91} +8.62712e8 q^{92} -1.69540e9 q^{94} +6.03778e8 q^{97} +5.00050e8 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 1368 q^{4}+O(q^{10}) $ 4 * q + 1368 * q^4	$ 4 q + 1368 q^{4} - 109968 q^{11} - 424536 q^{14} + 1631264 q^{16} + 636880 q^{19} - 6618768 q^{26} - 3531720 q^{29} - 10587712 q^{31} - 26434624 q^{34} + 16788552 q^{41} + 20638944 q^{44} - 61250072 q^{46} + 46921028 q^{49} - 315178080 q^{56} - 460829040 q^{59} + 360490568 q^{61} - 134995072 q^{64} + 47611872 q^{71} + 1176861744 q^{74} - 1168489440 q^{76} + 728043520 q^{79} + 2375904552 q^{86} - 1582700760 q^{89} + 473322528 q^{91} - 3327101704 q^{94}+O(q^{100}) $ 4 * q + 1368 * q^4 - 109968 * q^11 - 424536 * q^14 + 1631264 * q^16 + 636880 * q^19 - 6618768 * q^26 - 3531720 * q^29 - 10587712 * q^31 - 26434624 * q^34 + 16788552 * q^41 + 20638944 * q^44 - 61250072 * q^46 + 46921028 * q^49 - 315178080 * q^56 - 460829040 * q^59 + 360490568 * q^61 - 134995072 * q^64 + 47611872 * q^71 + 1176861744 * q^74 - 1168489440 * q^76 + 728043520 * q^79 + 2375904552 * q^86 - 1582700760 * q^89 + 473322528 * q^91 - 3327101704 * q^94

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 $

$2$	−41.3193	−1.82607	−0.913037	−	0.407877i	$-0.866269\pi$
$2$	−41.3193	−1.82607	−0.913037	+	0.407877i	$0.866269\pi$
$3$	0	0
$3$	0	0
$4$	1195.29	2.33455
$5$	0	0
$5$	0	0
$6$	0	0
$7$	5315.22	0.836719	0.418360	−	0.908281i	$-0.362605\pi$
$7$	5315.22	0.836719	0.418360	+	0.908281i	$0.362605\pi$
$8$	−28233.0	−2.43698
$9$	0	0
$10$	0	0
$11$	−10426.2	−0.214714	−0.107357	−	0.994221i	$-0.534239\pi$
$11$	−10426.2	−0.214714	−0.107357	+	0.994221i	$0.534239\pi$
$12$	0	0
$13$	79655.1	0.773515	0.386758	−	0.922181i	$-0.373595\pi$
$13$	79655.1	0.773515	0.386758	+	0.922181i	$0.373595\pi$
$14$	−219621.	−1.52791
$15$	0	0
$16$	554581.	2.11556
$17$	313750.	0.911095	0.455547	−	0.890212i	$-0.349444\pi$
$17$	313750.	0.911095	0.455547	+	0.890212i	$0.349444\pi$
$18$	0	0
$19$	−246945.	−0.434719	−0.217360	−	0.976092i	$-0.569744\pi$
$19$	−246945.	−0.434719	−0.217360	+	0.976092i	$0.569744\pi$
$20$	0	0
$21$	0	0
$22$	430806.	0.392084
$23$	721761.	0.537797	0.268898	−	0.963169i	$-0.413340\pi$
$23$	721761.	0.537797	0.268898	+	0.963169i	$0.413340\pi$
$24$	0	0
$25$	0	0
$26$	−3.29130e6	−1.41250
$27$	0	0
$28$	6.35321e6	1.95336
$29$	−2.56903e6	−0.674493	−0.337247	−	0.941416i	$-0.609496\pi$
$29$	−2.56903e6	−0.674493	−0.337247	+	0.941416i	$0.609496\pi$
$30$	0	0
$31$	−3.29543e6	−0.640891	−0.320445	−	0.947267i	$-0.603833\pi$
$31$	−3.29543e6	−0.640891	−0.320445	+	0.947267i	$0.603833\pi$
$32$	−8.45965e6	−1.42619
$33$	0	0
$34$	−1.29639e7	−1.66373
$35$	0	0
$36$	0	0
$37$	−1.40463e7	−1.23212	−0.616060	−	0.787699i	$-0.711272\pi$
$37$	−1.40463e7	−1.23212	−0.616060	+	0.787699i	$0.711272\pi$
$38$	1.02036e7	0.793830
$39$	0	0
$40$	0	0
$41$	−1.70412e7	−0.941830	−0.470915	−	0.882178i	$-0.656076\pi$
$41$	−1.70412e7	−0.941830	−0.470915	+	0.882178i	$0.656076\pi$
$42$	0	0
$43$	−2.92261e7	−1.30365	−0.651827	−	0.758368i	$-0.725997\pi$
$43$	−2.92261e7	−1.30365	−0.651827	+	0.758368i	$0.725997\pi$
$44$	−1.24624e7	−0.501260
$45$	0	0
$46$	−2.98227e7	−0.982056
$47$	4.10316e7	1.22653	0.613264	−	0.789878i	$-0.289856\pi$
$47$	4.10316e7	1.22653	0.613264	+	0.789878i	$0.289856\pi$
$48$	0	0
$49$	−1.21021e7	−0.299901
$50$	0	0
$51$	0	0
$52$	9.52108e7	1.80581
$53$	−5.67230e7	−0.987456	−0.493728	−	0.869616i	$-0.664366\pi$
$53$	−5.67230e7	−0.987456	−0.493728	+	0.869616i	$0.664366\pi$
$54$	0	0
$55$	0	0
$56$	−1.50065e8	−2.03907
$57$	0	0
$58$	1.06150e8	1.23167
$59$	−1.60408e8	−1.72342	−0.861710	−	0.507402i	$-0.830606\pi$
$59$	−1.60408e8	−1.72342	−0.861710	+	0.507402i	$0.830606\pi$
$60$	0	0
$61$	5.33033e7	0.492912	0.246456	−	0.969154i	$-0.420734\pi$
$61$	5.33033e7	0.492912	0.246456	+	0.969154i	$0.420734\pi$
$62$	1.36165e8	1.17031
$63$	0	0
$64$	6.56012e7	0.488767
$65$	0	0
$66$	0	0
$67$	2.80916e8	1.70310	0.851548	−	0.524276i	$-0.175664\pi$
$67$	2.80916e8	1.70310	0.851548	+	0.524276i	$0.175664\pi$
$68$	3.75022e8	2.12699
$69$	0	0
$70$	0	0
$71$	8.97228e7	0.419025	0.209513	−	0.977806i	$-0.432812\pi$
$71$	8.97228e7	0.419025	0.209513	+	0.977806i	$0.432812\pi$
$72$	0	0
$73$	7.60225e7	0.313321	0.156660	−	0.987653i	$-0.449927\pi$
$73$	7.60225e7	0.313321	0.156660	+	0.987653i	$0.449927\pi$
$74$	5.80383e8	2.24994
$75$	0	0
$76$	−2.95170e8	−1.01487
$77$	−5.54178e7	−0.179656
$78$	0	0
$79$	4.10672e8	1.18624	0.593120	−	0.805114i	$-0.297896\pi$
$79$	4.10672e8	1.18624	0.593120	+	0.805114i	$0.297896\pi$
$80$	0	0
$81$	0	0
$82$	7.04131e8	1.71985
$83$	−5.21969e8	−1.20724	−0.603620	−	0.797272i	$-0.706276\pi$
$83$	−5.21969e8	−1.20724	−0.603620	+	0.797272i	$0.706276\pi$
$84$	0	0
$85$	0	0
$86$	1.20760e9	2.38057
$87$	0	0
$88$	2.94364e8	0.523254
$89$	−2.37312e8	−0.400926	−0.200463	−	0.979701i	$-0.564245\pi$
$89$	−2.37312e8	−0.400926	−0.200463	+	0.979701i	$0.564245\pi$
$90$	0	0
$91$	4.23384e8	0.647215
$92$	8.62712e8	1.25551
$93$	0	0
$94$	−1.69540e9	−2.23973
$95$	0	0
$96$	0	0
$97$	6.03778e8	0.692476	0.346238	−	0.938147i	$-0.387459\pi$
$97$	6.03778e8	0.692476	0.346238	+	0.938147i	$0.387459\pi$
$98$	5.00050e8	0.547641
$99$	0	0
$100$	0	0
$101$	2.03606e8	0.194690	0.0973451	−	0.995251i	$-0.468965\pi$
$101$	2.03606e8	0.194690	0.0973451	+	0.995251i	$0.468965\pi$
$102$	0	0
$103$	−9.15893e8	−0.801821	−0.400910	−	0.916117i	$-0.631306\pi$
$103$	−9.15893e8	−0.801821	−0.400910	+	0.916117i	$0.631306\pi$
$104$	−2.24890e9	−1.88504
$105$	0	0
$106$	2.34376e9	1.80317
$107$	1.66237e9	1.22603	0.613014	−	0.790072i	$-0.289957\pi$
$107$	1.66237e9	1.22603	0.613014	+	0.790072i	$0.289957\pi$
$108$	0	0
$109$	1.73161e9	1.17498	0.587491	−	0.809231i	$-0.300116\pi$
$109$	1.73161e9	1.17498	0.587491	+	0.809231i	$0.300116\pi$
$110$	0	0
$111$	0	0
$112$	2.94772e9	1.77013
$113$	6.30956e8	0.364038	0.182019	−	0.983295i	$-0.441737\pi$
$113$	6.30956e8	0.364038	0.182019	+	0.983295i	$0.441737\pi$
$114$	0	0
$115$	0	0
$116$	−3.07073e9	−1.57464
$117$	0	0
$118$	6.62794e9	3.14709
$119$	1.66765e9	0.762331
$120$	0	0
$121$	−2.24924e9	−0.953898
$122$	−2.20246e9	−0.900094
$123$	0	0
$124$	−3.93898e9	−1.49619
$125$	0	0
$126$	0	0
$127$	−2.12104e9	−0.723490	−0.361745	−	0.932277i	$-0.617819\pi$
$127$	−2.12104e9	−0.723490	−0.361745	+	0.932277i	$0.617819\pi$
$128$	1.62074e9	0.533664
$129$	0	0
$130$	0	0
$131$	−5.57686e9	−1.65451	−0.827254	−	0.561828i	$-0.810098\pi$
$131$	−5.57686e9	−1.65451	−0.827254	+	0.561828i	$0.810098\pi$
$132$	0	0
$133$	−1.31257e9	−0.363738
$134$	−1.16072e10	−3.10998
$135$	0	0
$136$	−8.85810e9	−2.22032
$137$	−2.57317e9	−0.624059	−0.312029	−	0.950072i	$-0.601009\pi$
$137$	−2.57317e9	−0.624059	−0.312029	+	0.950072i	$0.601009\pi$
$138$	0	0
$139$	−1.62297e9	−0.368761	−0.184380	−	0.982855i	$-0.559028\pi$
$139$	−1.62297e9	−0.368761	−0.184380	+	0.982855i	$0.559028\pi$
$140$	0	0
$141$	0	0
$142$	−3.70729e9	−0.765171
$143$	−8.30504e8	−0.166085
$144$	0	0
$145$	0	0
$146$	−3.14120e9	−0.572147
$147$	0	0
$148$	−1.67893e10	−2.87644
$149$	5.98422e9	0.994647	0.497324	−	0.867565i	$-0.334316\pi$
$149$	5.98422e9	0.994647	0.497324	+	0.867565i	$0.334316\pi$
$150$	0	0
$151$	−5.95089e9	−0.931505	−0.465753	−	0.884915i	$-0.654216\pi$
$151$	−5.95089e9	−0.931505	−0.465753	+	0.884915i	$0.654216\pi$
$152$	6.97200e9	1.05940
$153$	0	0
$154$	2.28983e9	0.328064
$155$	0	0
$156$	0	0
$157$	2.94325e9	0.386615	0.193308	−	0.981138i	$-0.438078\pi$
$157$	2.94325e9	0.386615	0.193308	+	0.981138i	$0.438078\pi$
$158$	−1.69687e10	−2.16616
$159$	0	0
$160$	0	0
$161$	3.83632e9	0.449985
$162$	0	0
$163$	−2.69823e9	−0.299389	−0.149694	−	0.988732i	$-0.547829\pi$
$163$	−2.69823e9	−0.299389	−0.149694	+	0.988732i	$0.547829\pi$
$164$	−2.03691e10	−2.19875
$165$	0	0
$166$	2.15674e10	2.20451
$167$	−1.47132e10	−1.46380	−0.731902	−	0.681410i	$-0.761367\pi$
$167$	−1.47132e10	−1.46380	−0.731902	+	0.681410i	$0.761367\pi$
$168$	0	0
$169$	−4.25956e9	−0.401675
$170$	0	0
$171$	0	0
$172$	−3.49336e10	−3.04344
$173$	−1.56605e8	−0.0132923	−0.00664613	−	0.999978i	$-0.502116\pi$
$173$	−1.56605e8	−0.0132923	−0.00664613	+	0.999978i	$0.502116\pi$
$174$	0	0
$175$	0	0
$176$	−5.78220e9	−0.454241
$177$	0	0
$178$	9.80557e9	0.732121
$179$	5.35516e9	0.389883	0.194941	−	0.980815i	$-0.437548\pi$
$179$	5.35516e9	0.389883	0.194941	+	0.980815i	$0.437548\pi$
$180$	0	0
$181$	−1.52107e9	−0.105341	−0.0526704	−	0.998612i	$-0.516773\pi$
$181$	−1.52107e9	−0.105341	−0.0526704	+	0.998612i	$0.516773\pi$
$182$	−1.74940e10	−1.18186
$183$	0	0
$184$	−2.03775e10	−1.31060
$185$	0	0
$186$	0	0
$187$	−3.27123e9	−0.195625
$188$	4.90445e10	2.86339
$189$	0	0
$190$	0	0
$191$	9.28266e9	0.504687	0.252344	−	0.967638i	$-0.418799\pi$
$191$	9.28266e9	0.504687	0.252344	+	0.967638i	$0.418799\pi$
$192$	0	0
$193$	−6.94351e9	−0.360223	−0.180111	−	0.983646i	$-0.557646\pi$
$193$	−6.94351e9	−0.360223	−0.180111	+	0.983646i	$0.557646\pi$
$194$	−2.49477e10	−1.26451
$195$	0	0
$196$	−1.44655e10	−0.700132
$197$	3.60722e10	1.70637	0.853187	−	0.521606i	$-0.174667\pi$
$197$	3.60722e10	1.70637	0.853187	+	0.521606i	$0.174667\pi$
$198$	0	0
$199$	2.25173e10	1.01784	0.508918	−	0.860815i	$-0.330045\pi$
$199$	2.25173e10	1.01784	0.508918	+	0.860815i	$0.330045\pi$
$200$	0	0
$201$	0	0
$202$	−8.41286e9	−0.355519
$203$	−1.36549e10	−0.564362
$204$	0	0
$205$	0	0
$206$	3.78441e10	1.46418
$207$	0	0
$208$	4.41753e10	1.63642
$209$	2.57471e9	0.0933404
$210$	0	0
$211$	3.62300e10	1.25834	0.629169	−	0.777268i	$-0.283395\pi$
$211$	3.62300e10	1.25834	0.629169	+	0.777268i	$0.283395\pi$
$212$	−6.78003e10	−2.30526
$213$	0	0
$214$	−6.86879e10	−2.23882
$215$	0	0
$216$	0	0
$217$	−1.75159e10	−0.536246
$218$	−7.15490e10	−2.14560
$219$	0	0
$220$	0	0
$221$	2.49918e10	0.704746
$222$	0	0
$223$	−4.93834e10	−1.33724	−0.668619	−	0.743605i	$-0.733114\pi$
$223$	−4.93834e10	−1.33724	−0.668619	+	0.743605i	$0.733114\pi$
$224$	−4.49649e10	−1.19332
$225$	0	0
$226$	−2.60707e10	−0.664760
$227$	3.27710e10	0.819169	0.409585	−	0.912272i	$-0.365674\pi$
$227$	3.27710e10	0.819169	0.409585	+	0.912272i	$0.365674\pi$
$228$	0	0
$229$	1.35586e10	0.325803	0.162902	−	0.986642i	$-0.447915\pi$
$229$	1.35586e10	0.325803	0.162902	+	0.986642i	$0.447915\pi$
$230$	0	0
$231$	0	0
$232$	7.25313e10	1.64373
$233$	−7.99837e9	−0.177787	−0.0888935	−	0.996041i	$-0.528333\pi$
$233$	−7.99837e9	−0.177787	−0.0888935	+	0.996041i	$0.528333\pi$
$234$	0	0
$235$	0	0
$236$	−1.91733e11	−4.02340
$237$	0	0
$238$	−6.89062e10	−1.39207
$239$	−8.30208e10	−1.64587	−0.822937	−	0.568133i	$-0.807666\pi$
$239$	−8.30208e10	−1.64587	−0.822937	+	0.568133i	$0.807666\pi$
$240$	0	0
$241$	−6.04789e10	−1.15485	−0.577427	−	0.816442i	$-0.695943\pi$
$241$	−6.04789e10	−1.15485	−0.577427	+	0.816442i	$0.695943\pi$
$242$	9.29372e10	1.74189
$243$	0	0
$244$	6.37128e10	1.15073
$245$	0	0
$246$	0	0
$247$	−1.96704e10	−0.336262
$248$	9.30398e10	1.56184
$249$	0	0
$250$	0	0
$251$	−1.04682e11	−1.66471	−0.832354	−	0.554244i	$-0.813007\pi$
$251$	−1.04682e11	−1.66471	−0.832354	+	0.554244i	$0.813007\pi$
$252$	0	0
$253$	−7.52525e9	−0.115473
$254$	8.76400e10	1.32115
$255$	0	0
$256$	−1.00556e11	−1.46328
$257$	−7.92751e10	−1.13354	−0.566771	−	0.823875i	$-0.691808\pi$
$257$	−7.92751e10	−1.13354	−0.566771	+	0.823875i	$0.691808\pi$
$258$	0	0
$259$	−7.46590e10	−1.03094
$260$	0	0
$261$	0	0
$262$	2.30432e11	3.02126
$263$	−1.04629e8	−0.00134850	−0.000674250	−	1.00000i	$-0.500215\pi$
$263$	−1.04629e8	−0.00134850	−0.000674250		1.00000i	$0.500215\pi$
$264$	0	0
$265$	0	0
$266$	5.42344e10	0.664213
$267$	0	0
$268$	3.35775e11	3.97596
$269$	7.38735e9	0.0860208	0.0430104	−	0.999075i	$-0.486305\pi$
$269$	7.38735e9	0.0860208	0.0430104	+	0.999075i	$0.486305\pi$
$270$	0	0
$271$	1.27706e11	1.43831	0.719153	−	0.694852i	$-0.244530\pi$
$271$	1.27706e11	1.43831	0.719153	+	0.694852i	$0.244530\pi$
$272$	1.74000e11	1.92748
$273$	0	0
$274$	1.06322e11	1.13958
$275$	0	0
$276$	0	0
$277$	3.16563e10	0.323074	0.161537	−	0.986867i	$-0.448355\pi$
$277$	3.16563e10	0.323074	0.161537	+	0.986867i	$0.448355\pi$
$278$	6.70602e10	0.673385
$279$	0	0
$280$	0	0
$281$	9.50309e10	0.909257	0.454628	−	0.890681i	$-0.349772\pi$
$281$	9.50309e10	0.909257	0.454628	+	0.890681i	$0.349772\pi$
$282$	0	0
$283$	4.91862e10	0.455832	0.227916	−	0.973681i	$-0.426809\pi$
$283$	4.91862e10	0.455832	0.227916	+	0.973681i	$0.426809\pi$
$284$	1.07245e11	0.978234
$285$	0	0
$286$	3.43159e10	0.303283
$287$	−9.05777e10	−0.788048
$288$	0	0
$289$	−2.01488e10	−0.169906
$290$	0	0
$291$	0	0
$292$	9.08688e10	0.731462
$293$	−3.53889e10	−0.280519	−0.140260	−	0.990115i	$-0.544794\pi$
$293$	−3.53889e10	−0.280519	−0.140260	+	0.990115i	$0.544794\pi$
$294$	0	0
$295$	0	0
$296$	3.96568e11	3.00265
$297$	0	0
$298$	−2.47264e11	−1.81630
$299$	5.74920e10	0.415994
$300$	0	0
$301$	−1.55343e11	−1.09079
$302$	2.45887e11	1.70100
$303$	0	0
$304$	−1.36951e11	−0.919675
$305$	0	0
$306$	0	0
$307$	−8.30301e10	−0.533474	−0.266737	−	0.963769i	$-0.585945\pi$
$307$	−8.30301e10	−0.533474	−0.266737	+	0.963769i	$0.585945\pi$
$308$	−6.62402e10	−0.419414
$309$	0	0
$310$	0	0
$311$	−2.13935e10	−0.129676	−0.0648382	−	0.997896i	$-0.520653\pi$
$311$	−2.13935e10	−0.129676	−0.0648382	+	0.997896i	$0.520653\pi$
$312$	0	0
$313$	−2.56558e11	−1.51090	−0.755452	−	0.655204i	$-0.772582\pi$
$313$	−2.56558e11	−1.51090	−0.755452	+	0.655204i	$0.772582\pi$
$314$	−1.21613e11	−0.705988
$315$	0	0
$316$	4.90871e11	2.76933
$317$	−1.26112e11	−0.701436	−0.350718	−	0.936481i	$-0.614062\pi$
$317$	−1.26112e11	−0.701436	−0.350718	+	0.936481i	$0.614062\pi$
$318$	0	0
$319$	2.67853e10	0.144823
$320$	0	0
$321$	0	0
$322$	−1.58514e11	−0.821706
$323$	−7.74790e10	−0.396071
$324$	0	0
$325$	0	0
$326$	1.11489e11	0.546706
$327$	0	0
$328$	4.81124e11	2.29522
$329$	2.18092e11	1.02626
$330$	0	0
$331$	1.71300e11	0.784389	0.392194	−	0.919882i	$-0.371716\pi$
$331$	1.71300e11	0.784389	0.392194	+	0.919882i	$0.371716\pi$
$332$	−6.23904e11	−2.81836
$333$	0	0
$334$	6.07939e11	2.67301
$335$	0	0
$336$	0	0
$337$	−4.70320e11	−1.98637	−0.993183	−	0.116569i	$-0.962810\pi$
$337$	−4.70320e11	−1.98637	−0.993183	+	0.116569i	$0.962810\pi$
$338$	1.76002e11	0.733487
$339$	0	0
$340$	0	0
$341$	3.43589e10	0.137608
$342$	0	0
$343$	−2.78813e11	−1.08765
$344$	8.25140e11	3.17698
$345$	0	0
$346$	6.47083e9	0.0242727
$347$	2.86642e11	1.06135	0.530674	−	0.847576i	$-0.321939\pi$
$347$	2.86642e11	1.06135	0.530674	+	0.847576i	$0.321939\pi$
$348$	0	0
$349$	−3.54310e11	−1.27841	−0.639203	−	0.769038i	$-0.720736\pi$
$349$	−3.54310e11	−1.27841	−0.639203	+	0.769038i	$0.720736\pi$
$350$	0	0
$351$	0	0
$352$	8.82023e10	0.306223
$353$	−2.09491e11	−0.718092	−0.359046	−	0.933320i	$-0.616898\pi$
$353$	−2.09491e11	−0.718092	−0.359046	+	0.933320i	$0.616898\pi$
$354$	0	0
$355$	0	0
$356$	−2.83656e11	−0.935981
$357$	0	0
$358$	−2.21272e11	−0.711955
$359$	−2.88081e11	−0.915355	−0.457678	−	0.889118i	$-0.651319\pi$
$359$	−2.88081e11	−0.915355	−0.457678	+	0.889118i	$0.651319\pi$
$360$	0	0
$361$	−2.61706e11	−0.811019
$362$	6.28498e10	0.192360
$363$	0	0
$364$	5.06066e11	1.51095
$365$	0	0
$366$	0	0
$367$	−1.33587e10	−0.0384387	−0.0192193	−	0.999815i	$-0.506118\pi$
$367$	−1.33587e10	−0.0384387	−0.0192193	+	0.999815i	$0.506118\pi$
$368$	4.00275e11	1.13774
$369$	0	0
$370$	0	0
$371$	−3.01495e11	−0.826224
$372$	0	0
$373$	−4.17763e11	−1.11748	−0.558741	−	0.829342i	$-0.688716\pi$
$373$	−4.17763e11	−1.11748	−0.558741	+	0.829342i	$0.688716\pi$
$374$	1.35165e11	0.357226
$375$	0	0
$376$	−1.15844e12	−2.98903
$377$	−2.04636e11	−0.521731
$378$	0	0
$379$	2.63110e11	0.655031	0.327515	−	0.944846i	$-0.393789\pi$
$379$	2.63110e11	0.655031	0.327515	+	0.944846i	$0.393789\pi$
$380$	0	0
$381$	0	0
$382$	−3.83553e11	−0.921596
$383$	−7.84146e10	−0.186210	−0.0931049	−	0.995656i	$-0.529679\pi$
$383$	−7.84146e10	−0.186210	−0.0931049	+	0.995656i	$0.529679\pi$
$384$	0	0
$385$	0	0
$386$	2.86901e11	0.657793
$387$	0	0
$388$	7.21688e11	1.61662
$389$	−1.66007e11	−0.367581	−0.183791	−	0.982965i	$-0.558837\pi$
$389$	−1.66007e11	−0.367581	−0.183791	+	0.982965i	$0.558837\pi$
$390$	0	0
$391$	2.26452e11	0.489984
$392$	3.41678e11	0.730852
$393$	0	0
$394$	−1.49048e12	−3.11596
$395$	0	0
$396$	0	0
$397$	4.09536e11	0.827437	0.413719	−	0.910405i	$-0.364230\pi$
$397$	4.09536e11	0.827437	0.413719	+	0.910405i	$0.364230\pi$
$398$	−9.30402e11	−1.85865
$399$	0	0
$400$	0	0
$401$	7.92645e10	0.153084	0.0765419	−	0.997066i	$-0.475612\pi$
$401$	7.92645e10	0.153084	0.0765419	+	0.997066i	$0.475612\pi$
$402$	0	0
$403$	−2.62498e11	−0.495739
$404$	2.43367e11	0.454513
$405$	0	0
$406$	5.64213e11	1.03057
$407$	1.46450e11	0.264554
$408$	0	0
$409$	6.85827e11	1.21188	0.605940	−	0.795510i	$-0.292797\pi$
$409$	6.85827e11	1.21188	0.605940	+	0.795510i	$0.292797\pi$
$410$	0	0
$411$	0	0
$412$	−1.09476e12	−1.87189
$413$	−8.52601e11	−1.44202
$414$	0	0
$415$	0	0
$416$	−6.73854e11	−1.10318
$417$	0	0
$418$	−1.06385e11	−0.170447
$419$	−3.31879e11	−0.526037	−0.263018	−	0.964791i	$-0.584718\pi$
$419$	−3.31879e11	−0.526037	−0.263018	+	0.964791i	$0.584718\pi$
$420$	0	0
$421$	−6.30694e11	−0.978475	−0.489237	−	0.872151i	$-0.662725\pi$
$421$	−6.30694e11	−0.978475	−0.489237	+	0.872151i	$0.662725\pi$
$422$	−1.49700e12	−2.29782
$423$	0	0
$424$	1.60146e12	2.40641
$425$	0	0
$426$	0	0
$427$	2.83319e11	0.412429
$428$	1.98701e12	2.86222
$429$	0	0
$430$	0	0
$431$	−7.70903e11	−1.07610	−0.538049	−	0.842913i	$-0.680839\pi$
$431$	−7.70903e11	−1.07610	−0.538049	+	0.842913i	$0.680839\pi$
$432$	0	0
$433$	−1.07829e12	−1.47415	−0.737075	−	0.675811i	$-0.763794\pi$
$433$	−1.07829e12	−1.47415	−0.737075	+	0.675811i	$0.763794\pi$
$434$	7.23746e11	0.979225
$435$	0	0
$436$	2.06977e12	2.74305
$437$	−1.78235e11	−0.233791
$438$	0	0
$439$	−9.56419e11	−1.22902	−0.614508	−	0.788910i	$-0.710646\pi$
$439$	−9.56419e11	−1.22902	−0.614508	+	0.788910i	$0.710646\pi$
$440$	0	0
$441$	0	0
$442$	−1.03264e12	−1.28692
$443$	2.06392e11	0.254611	0.127305	−	0.991864i	$-0.459367\pi$
$443$	2.06392e11	0.254611	0.127305	+	0.991864i	$0.459367\pi$
$444$	0	0
$445$	0	0
$446$	2.04049e12	2.44190
$447$	0	0
$448$	3.48685e11	0.408961
$449$	−1.75939e11	−0.204293	−0.102147	−	0.994769i	$-0.532571\pi$
$449$	−1.75939e11	−0.204293	−0.102147	+	0.994769i	$0.532571\pi$
$450$	0	0
$451$	1.77676e11	0.202224
$452$	7.54174e11	0.849863
$453$	0	0
$454$	−1.35408e12	−1.49586
$455$	0	0
$456$	0	0
$457$	2.98819e11	0.320469	0.160234	−	0.987079i	$-0.448775\pi$
$457$	2.98819e11	0.320469	0.160234	+	0.987079i	$0.448775\pi$
$458$	−5.60233e11	−0.594941
$459$	0	0
$460$	0	0
$461$	−4.39422e11	−0.453135	−0.226567	−	0.973995i	$-0.572750\pi$
$461$	−4.39422e11	−0.453135	−0.226567	+	0.973995i	$0.572750\pi$
$462$	0	0
$463$	1.16254e12	1.17569	0.587846	−	0.808973i	$-0.299976\pi$
$463$	1.16254e12	1.17569	0.587846	+	0.808973i	$0.299976\pi$
$464$	−1.42473e12	−1.42693
$465$	0	0
$466$	3.30487e11	0.324652
$467$	−1.65923e11	−0.161429	−0.0807144	−	0.996737i	$-0.525720\pi$
$467$	−1.65923e11	−0.161429	−0.0807144	+	0.996737i	$0.525720\pi$
$468$	0	0
$469$	1.49313e12	1.42501
$470$	0	0
$471$	0	0
$472$	4.52879e12	4.19994
$473$	3.04718e11	0.279913
$474$	0	0
$475$	0	0
$476$	1.99332e12	1.77970
$477$	0	0
$478$	3.43036e12	3.00549
$479$	−1.33180e11	−0.115592	−0.0577960	−	0.998328i	$-0.518407\pi$
$479$	−1.33180e11	−0.115592	−0.0577960	+	0.998328i	$0.518407\pi$
$480$	0	0
$481$	−1.11886e12	−0.953064
$482$	2.49895e12	2.10885
$483$	0	0
$484$	−2.68849e12	−2.22692
$485$	0	0
$486$	0	0
$487$	−1.43276e12	−1.15423	−0.577116	−	0.816662i	$-0.695822\pi$
$487$	−1.43276e12	−1.15423	−0.577116	+	0.816662i	$0.695822\pi$
$488$	−1.50491e12	−1.20122
$489$	0	0
$490$	0	0
$491$	1.05688e12	0.820655	0.410327	−	0.911938i	$-0.365414\pi$
$491$	1.05688e12	0.820655	0.410327	+	0.911938i	$0.365414\pi$
$492$	0	0
$493$	−8.06032e11	−0.614527
$494$	8.12769e11	0.614039
$495$	0	0
$496$	−1.82758e12	−1.35584
$497$	4.76896e11	0.350607
$498$	0	0
$499$	1.77897e11	0.128445	0.0642223	−	0.997936i	$-0.479543\pi$
$499$	1.77897e11	0.128445	0.0642223	+	0.997936i	$0.479543\pi$
$500$	0	0
$501$	0	0
$502$	4.32537e12	3.03988
$503$	1.58467e12	1.10378	0.551889	−	0.833917i	$-0.313907\pi$
$503$	1.58467e12	1.10378	0.551889	+	0.833917i	$0.313907\pi$
$504$	0	0
$505$	0	0
$506$	3.10939e11	0.210861
$507$	0	0
$508$	−2.53525e12	−1.68902
$509$	1.93674e12	1.27891	0.639456	−	0.768827i	$-0.279159\pi$
$509$	1.93674e12	1.27891	0.639456	+	0.768827i	$0.279159\pi$
$510$	0	0
$511$	4.04076e11	0.262162
$512$	3.32508e12	2.13839
$513$	0	0
$514$	3.27560e12	2.06993
$515$	0	0
$516$	0	0
$517$	−4.27805e11	−0.263353
$518$	3.08486e12	1.88257
$519$	0	0
$520$	0	0
$521$	−2.72419e12	−1.61982	−0.809912	−	0.586552i	$-0.800485\pi$
$521$	−2.72419e12	−1.61982	−0.809912	+	0.586552i	$0.800485\pi$
$522$	0	0
$523$	−4.86784e11	−0.284497	−0.142249	−	0.989831i	$-0.545433\pi$
$523$	−4.86784e11	−0.284497	−0.142249	+	0.989831i	$0.545433\pi$
$524$	−6.66596e12	−3.86253
$525$	0	0
$526$	4.32320e9	0.00246246
$527$	−1.03394e12	−0.583912
$528$	0	0
$529$	−1.28021e12	−0.710775
$530$	0	0
$531$	0	0
$532$	−1.56889e12	−0.849164
$533$	−1.35742e12	−0.728520
$534$	0	0
$535$	0	0
$536$	−7.93109e12	−4.15041
$537$	0	0
$538$	−3.05240e11	−0.157080
$539$	1.26179e11	0.0643929
$540$	0	0
$541$	1.98233e11	0.0994920	0.0497460	−	0.998762i	$-0.484159\pi$
$541$	1.98233e11	0.0994920	0.0497460	+	0.998762i	$0.484159\pi$
$542$	−5.27675e12	−2.62645
$543$	0	0
$544$	−2.65421e12	−1.29939
$545$	0	0
$546$	0	0
$547$	4.79532e11	0.229021	0.114510	−	0.993422i	$-0.463470\pi$
$547$	4.79532e11	0.229021	0.114510	+	0.993422i	$0.463470\pi$
$548$	−3.07567e12	−1.45689
$549$	0	0
$550$	0	0
$551$	6.34408e11	0.293215
$552$	0	0
$553$	2.18281e12	0.992550
$554$	−1.30802e12	−0.589957
$555$	0	0
$556$	−1.93992e12	−0.860889
$557$	1.20856e11	0.0532011	0.0266006	−	0.999646i	$-0.491532\pi$
$557$	1.20856e11	0.0532011	0.0266006	+	0.999646i	$0.491532\pi$
$558$	0	0
$559$	−2.32801e12	−1.00840
$560$	0	0
$561$	0	0
$562$	−3.92662e12	−1.66037
$563$	−1.01468e12	−0.425637	−0.212818	−	0.977092i	$-0.568264\pi$
$563$	−1.01468e12	−0.425637	−0.212818	+	0.977092i	$0.568264\pi$
$564$	0	0
$565$	0	0
$566$	−2.03234e12	−0.832382
$567$	0	0
$568$	−2.53314e12	−1.02116
$569$	−1.12630e12	−0.450454	−0.225227	−	0.974306i	$-0.572312\pi$
$569$	−1.12630e12	−0.450454	−0.225227	+	0.974306i	$0.572312\pi$
$570$	0	0
$571$	−2.75478e12	−1.08449	−0.542243	−	0.840221i	$-0.682425\pi$
$571$	−2.75478e12	−1.08449	−0.542243	+	0.840221i	$0.682425\pi$
$572$	−9.92691e11	−0.387732
$573$	0	0
$574$	3.74261e12	1.43903
$575$	0	0
$576$	0	0
$577$	4.14763e12	1.55779	0.778895	−	0.627155i	$-0.215781\pi$
$577$	4.14763e12	1.55779	0.778895	+	0.627155i	$0.215781\pi$
$578$	8.32535e11	0.310261
$579$	0	0
$580$	0	0
$581$	−2.77438e12	−1.01012
$582$	0	0
$583$	5.91408e11	0.212021
$584$	−2.14634e12	−0.763557
$585$	0	0
$586$	1.46224e12	0.512248
$587$	−1.38017e12	−0.479800	−0.239900	−	0.970798i	$-0.577115\pi$
$587$	−1.38017e12	−0.479800	−0.239900	+	0.970798i	$0.577115\pi$
$588$	0	0
$589$	8.13789e11	0.278608
$590$	0	0
$591$	0	0
$592$	−7.78980e12	−2.60663
$593$	6.90406e11	0.229276	0.114638	−	0.993407i	$-0.463429\pi$
$593$	6.90406e11	0.229276	0.114638	+	0.993407i	$0.463429\pi$
$594$	0	0
$595$	0	0
$596$	7.15286e12	2.32205
$597$	0	0
$598$	−2.37553e12	−0.759635
$599$	−1.24239e12	−0.394309	−0.197155	−	0.980372i	$-0.563170\pi$
$599$	−1.24239e12	−0.394309	−0.197155	+	0.980372i	$0.563170\pi$
$600$	0	0
$601$	2.01140e12	0.628873	0.314437	−	0.949278i	$-0.398184\pi$
$601$	2.01140e12	0.628873	0.314437	+	0.949278i	$0.398184\pi$
$602$	6.41867e12	1.99187
$603$	0	0
$604$	−7.11302e12	−2.17464
$605$	0	0
$606$	0	0
$607$	−2.73579e12	−0.817963	−0.408981	−	0.912543i	$-0.634116\pi$
$607$	−2.73579e12	−0.817963	−0.408981	+	0.912543i	$0.634116\pi$
$608$	2.08907e12	0.619992
$609$	0	0
$610$	0	0
$611$	3.26838e12	0.948738
$612$	0	0
$613$	1.63707e12	0.468269	0.234135	−	0.972204i	$-0.424774\pi$
$613$	1.63707e12	0.468269	0.234135	+	0.972204i	$0.424774\pi$
$614$	3.43075e12	0.974162
$615$	0	0
$616$	1.56461e12	0.437817
$617$	1.22267e12	0.339646	0.169823	−	0.985475i	$-0.445680\pi$
$617$	1.22267e12	0.339646	0.169823	+	0.985475i	$0.445680\pi$
$618$	0	0
$619$	−5.78784e12	−1.58456	−0.792280	−	0.610158i	$-0.791106\pi$
$619$	−5.78784e12	−1.58456	−0.792280	+	0.610158i	$0.791106\pi$
$620$	0	0
$621$	0	0
$622$	8.83967e11	0.236799
$623$	−1.26136e12	−0.335463
$624$	0	0
$625$	0	0
$626$	1.06008e13	2.75902
$627$	0	0
$628$	3.51803e12	0.902571
$629$	−4.40702e12	−1.12258
$630$	0	0
$631$	2.00341e12	0.503081	0.251540	−	0.967847i	$-0.419063\pi$
$631$	2.00341e12	0.503081	0.251540	+	0.967847i	$0.419063\pi$
$632$	−1.15945e13	−2.89084
$633$	0	0
$634$	5.21084e12	1.28087
$635$	0	0
$636$	0	0
$637$	−9.63992e11	−0.231978
$638$	−1.10675e12	−0.264458
$639$	0	0
$640$	0	0
$641$	−2.75478e12	−0.644505	−0.322253	−	0.946654i	$-0.604440\pi$
$641$	−2.75478e12	−0.644505	−0.322253	+	0.946654i	$0.604440\pi$
$642$	0	0
$643$	7.38857e12	1.70455	0.852277	−	0.523091i	$-0.175221\pi$
$643$	7.38857e12	1.70455	0.852277	+	0.523091i	$0.175221\pi$
$644$	4.58550e12	1.05051
$645$	0	0
$646$	3.20138e12	0.723254
$647$	4.12045e12	0.924432	0.462216	−	0.886767i	$-0.347054\pi$
$647$	4.12045e12	0.924432	0.462216	+	0.886767i	$0.347054\pi$
$648$	0	0
$649$	1.67245e12	0.370043
$650$	0	0
$651$	0	0
$652$	−3.22517e12	−0.698937
$653$	6.51407e12	1.40198	0.700992	−	0.713170i	$-0.252741\pi$
$653$	6.51407e12	1.40198	0.700992	+	0.713170i	$0.252741\pi$
$654$	0	0
$655$	0	0
$656$	−9.45073e12	−1.99250
$657$	0	0
$658$	−9.01141e12	−1.87403
$659$	1.74145e12	0.359688	0.179844	−	0.983695i	$-0.442441\pi$
$659$	1.74145e12	0.359688	0.179844	+	0.983695i	$0.442441\pi$
$660$	0	0
$661$	−1.39849e12	−0.284939	−0.142469	−	0.989799i	$-0.545504\pi$
$661$	−1.39849e12	−0.284939	−0.142469	+	0.989799i	$0.545504\pi$
$662$	−7.07800e12	−1.43235
$663$	0	0
$664$	1.47368e13	2.94202
$665$	0	0
$666$	0	0
$667$	−1.85422e12	−0.362740
$668$	−1.75865e13	−3.41732
$669$	0	0
$670$	0	0
$671$	−5.55753e11	−0.105835
$672$	0	0
$673$	−7.23077e12	−1.35868	−0.679340	−	0.733824i	$-0.737734\pi$
$673$	−7.23077e12	−1.35868	−0.679340	+	0.733824i	$0.737734\pi$
$674$	1.94333e13	3.62725
$675$	0	0
$676$	−5.09140e12	−0.937728
$677$	1.09854e12	0.200986	0.100493	−	0.994938i	$-0.467958\pi$
$677$	1.09854e12	0.200986	0.100493	+	0.994938i	$0.467958\pi$
$678$	0	0
$679$	3.20921e12	0.579408
$680$	0	0
$681$	0	0
$682$	−1.41969e12	−0.251283
$683$	1.51781e12	0.266885	0.133442	−	0.991057i	$-0.457397\pi$
$683$	1.51781e12	0.266885	0.133442	+	0.991057i	$0.457397\pi$
$684$	0	0
$685$	0	0
$686$	1.15204e13	1.98613
$687$	0	0
$688$	−1.62082e13	−2.75796
$689$	−4.51828e12	−0.763812
$690$	0	0
$691$	1.51489e12	0.252772	0.126386	−	0.991981i	$-0.459662\pi$
$691$	1.51489e12	0.252772	0.126386	+	0.991981i	$0.459662\pi$
$692$	−1.87188e11	−0.0310314
$693$	0	0
$694$	−1.18439e13	−1.93810
$695$	0	0
$696$	0	0
$697$	−5.34668e12	−0.858097
$698$	1.46398e13	2.33446
$699$	0	0
$700$	0	0
$701$	−7.62368e12	−1.19243	−0.596216	−	0.802824i	$-0.703330\pi$
$701$	−7.62368e12	−1.19243	−0.596216	+	0.802824i	$0.703330\pi$
$702$	0	0
$703$	3.46866e12	0.535627
$704$	−6.83975e11	−0.104945
$705$	0	0
$706$	8.65605e12	1.31129
$707$	1.08221e12	0.162901
$708$	0	0
$709$	1.00657e13	1.49602	0.748010	−	0.663687i	$-0.231009\pi$
$709$	1.00657e13	1.49602	0.748010	+	0.663687i	$0.231009\pi$
$710$	0	0
$711$	0	0
$712$	6.70002e12	0.977049
$713$	−2.37851e12	−0.344669
$714$	0	0
$715$	0	0
$716$	6.40096e12	0.910199
$717$	0	0
$718$	1.19033e13	1.67151
$719$	−8.94464e12	−1.24820	−0.624098	−	0.781346i	$-0.714534\pi$
$719$	−8.94464e12	−1.24820	−0.624098	+	0.781346i	$0.714534\pi$
$720$	0	0
$721$	−4.86817e12	−0.670899
$722$	1.08135e13	1.48098
$723$	0	0
$724$	−1.81812e12	−0.245923
$725$	0	0
$726$	0	0
$727$	1.22168e13	1.62201	0.811006	−	0.585038i	$-0.198921\pi$
$727$	1.22168e13	1.62201	0.811006	+	0.585038i	$0.198921\pi$
$728$	−1.19534e13	−1.57725
$729$	0	0
$730$	0	0
$731$	−9.16968e12	−1.18775
$732$	0	0
$733$	1.44452e13	1.84823	0.924114	−	0.382118i	$-0.124805\pi$
$733$	1.44452e13	1.84823	0.924114	+	0.382118i	$0.124805\pi$
$734$	5.51974e11	0.0701919
$735$	0	0
$736$	−6.10584e12	−0.767000
$737$	−2.92889e12	−0.365679
$738$	0	0
$739$	−1.11591e13	−1.37635	−0.688175	−	0.725545i	$-0.741588\pi$
$739$	−1.11591e13	−1.37635	−0.688175	+	0.725545i	$0.741588\pi$
$740$	0	0
$741$	0	0
$742$	1.24576e13	1.50875
$743$	9.35882e12	1.12660	0.563302	−	0.826251i	$-0.309531\pi$
$743$	9.35882e12	1.12660	0.563302	+	0.826251i	$0.309531\pi$
$744$	0	0
$745$	0	0
$746$	1.72617e13	2.04061
$747$	0	0
$748$	−3.91007e12	−0.456696
$749$	8.83585e12	1.02584
$750$	0	0
$751$	−1.28609e13	−1.47533	−0.737667	−	0.675165i	$-0.764072\pi$
$751$	−1.28609e13	−1.47533	−0.737667	+	0.675165i	$0.764072\pi$
$752$	2.27553e13	2.59480
$753$	0	0
$754$	8.45543e12	0.952719
$755$	0	0
$756$	0	0
$757$	1.42250e13	1.57442	0.787211	−	0.616684i	$-0.211524\pi$
$757$	1.42250e13	1.57442	0.787211	+	0.616684i	$0.211524\pi$
$758$	−1.08716e13	−1.19613
$759$	0	0
$760$	0	0
$761$	−1.65722e13	−1.79122	−0.895610	−	0.444841i	$-0.853260\pi$
$761$	−1.65722e13	−1.79122	−0.895610	+	0.444841i	$0.853260\pi$
$762$	0	0
$763$	9.20389e12	0.983130
$764$	1.10954e13	1.17822
$765$	0	0
$766$	3.24004e12	0.340033
$767$	−1.27773e13	−1.33309
$768$	0	0
$769$	2.81269e12	0.290037	0.145019	−	0.989429i	$-0.453676\pi$
$769$	2.81269e12	0.290037	0.145019	+	0.989429i	$0.453676\pi$
$770$	0	0
$771$	0	0
$772$	−8.29949e12	−0.840957
$773$	8.57982e12	0.864311	0.432156	−	0.901799i	$-0.357753\pi$
$773$	8.57982e12	0.864311	0.432156	+	0.901799i	$0.357753\pi$
$774$	0	0
$775$	0	0
$776$	−1.70465e13	−1.68755
$777$	0	0
$778$	6.85930e12	0.671230
$779$	4.20824e12	0.409432
$780$	0	0
$781$	−9.35472e11	−0.0899707
$782$	−9.35687e12	−0.894746
$783$	0	0
$784$	−6.71158e12	−0.634458
$785$	0	0
$786$	0	0
$787$	−1.75771e12	−0.163328	−0.0816642	−	0.996660i	$-0.526024\pi$
$787$	−1.75771e12	−0.163328	−0.0816642	+	0.996660i	$0.526024\pi$
$788$	4.31166e13	3.98361
$789$	0	0
$790$	0	0
$791$	3.35367e12	0.304597
$792$	0	0
$793$	4.24588e12	0.381275
$794$	−1.69218e13	−1.51096
$795$	0	0
$796$	2.69147e13	2.37619
$797$	2.27115e13	1.99381	0.996905	−	0.0786130i	$-0.0250491\pi$
$797$	2.27115e13	1.99381	0.996905	+	0.0786130i	$0.0250491\pi$
$798$	0	0
$799$	1.28737e13	1.11748
$800$	0	0
$801$	0	0
$802$	−3.27516e12	−0.279542
$803$	−7.92629e11	−0.0672744
$804$	0	0
$805$	0	0
$806$	1.08462e13	0.905255
$807$	0	0
$808$	−5.74840e12	−0.474456
$809$	9.95176e12	0.816829	0.408415	−	0.912796i	$-0.366082\pi$
$809$	9.95176e12	0.816829	0.408415	+	0.912796i	$0.366082\pi$
$810$	0	0
$811$	−7.01292e12	−0.569253	−0.284626	−	0.958639i	$-0.591870\pi$
$811$	−7.01292e12	−0.569253	−0.284626	+	0.958639i	$0.591870\pi$
$812$	−1.63216e13	−1.31753
$813$	0	0
$814$	−6.05121e12	−0.483095
$815$	0	0
$816$	0	0
$817$	7.21723e12	0.566724
$818$	−2.83379e13	−2.21298
$819$	0	0
$820$	0	0
$821$	1.20992e13	0.929420	0.464710	−	0.885463i	$-0.346159\pi$
$821$	1.20992e13	0.929420	0.464710	+	0.885463i	$0.346159\pi$
$822$	0	0
$823$	−1.21261e13	−0.921346	−0.460673	−	0.887570i	$-0.652392\pi$
$823$	−1.21261e13	−0.921346	−0.460673	+	0.887570i	$0.652392\pi$
$824$	2.58584e13	1.95402
$825$	0	0
$826$	3.52289e13	2.63323
$827$	−1.32495e13	−0.984973	−0.492486	−	0.870320i	$-0.663912\pi$
$827$	−1.32495e13	−0.984973	−0.492486	+	0.870320i	$0.663912\pi$
$828$	0	0
$829$	−1.87106e13	−1.37592	−0.687960	−	0.725749i	$-0.741493\pi$
$829$	−1.87106e13	−1.37592	−0.687960	+	0.725749i	$0.741493\pi$
$830$	0	0
$831$	0	0
$832$	5.22548e12	0.378069
$833$	−3.79702e12	−0.273238
$834$	0	0
$835$	0	0
$836$	3.07752e12	0.217908
$837$	0	0
$838$	1.37130e13	0.960582
$839$	−1.21346e13	−0.845469	−0.422734	−	0.906254i	$-0.638930\pi$
$839$	−1.21346e13	−0.845469	−0.422734	+	0.906254i	$0.638930\pi$
$840$	0	0
$841$	−7.90725e12	−0.545059
$842$	2.60599e13	1.78677
$843$	0	0
$844$	4.33053e13	2.93765
$845$	0	0
$846$	0	0
$847$	−1.19552e13	−0.798145
$848$	−3.14575e13	−2.08902
$849$	0	0
$850$	0	0
$851$	−1.01380e13	−0.662630
$852$	0	0
$853$	−2.34098e13	−1.51401	−0.757004	−	0.653411i	$-0.773337\pi$
$853$	−2.34098e13	−1.51401	−0.757004	+	0.653411i	$0.773337\pi$
$854$	−1.17065e13	−0.753126
$855$	0	0
$856$	−4.69336e13	−2.98780
$857$	1.61008e13	1.01961	0.509805	−	0.860290i	$-0.329718\pi$
$857$	1.61008e13	1.01961	0.509805	+	0.860290i	$0.329718\pi$
$858$	0	0
$859$	−1.34163e13	−0.840745	−0.420373	−	0.907352i	$-0.638101\pi$
$859$	−1.34163e13	−0.840745	−0.420373	+	0.907352i	$0.638101\pi$
$860$	0	0
$861$	0	0
$862$	3.18532e13	1.96504
$863$	−2.03784e13	−1.25061	−0.625304	−	0.780381i	$-0.715025\pi$
$863$	−2.03784e13	−1.25061	−0.625304	+	0.780381i	$0.715025\pi$
$864$	0	0
$865$	0	0
$866$	4.45544e13	2.69191
$867$	0	0
$868$	−2.09366e13	−1.25189
$869$	−4.28176e12	−0.254703
$870$	0	0
$871$	2.23764e13	1.31737
$872$	−4.88886e13	−2.86341
$873$	0	0
$874$	7.36456e12	0.426919
$875$	0	0
$876$	0	0
$877$	−3.55027e12	−0.202658	−0.101329	−	0.994853i	$-0.532309\pi$
$877$	−3.55027e12	−0.202658	−0.101329	+	0.994853i	$0.532309\pi$
$878$	3.95186e13	2.24428
$879$	0	0
$880$	0	0
$881$	2.33443e13	1.30554	0.652769	−	0.757557i	$-0.273607\pi$
$881$	2.33443e13	1.30554	0.652769	+	0.757557i	$0.273607\pi$
$882$	0	0
$883$	8.70030e12	0.481627	0.240814	−	0.970571i	$-0.422586\pi$
$883$	8.70030e12	0.481627	0.240814	+	0.970571i	$0.422586\pi$
$884$	2.98724e13	1.64526
$885$	0	0
$886$	−8.52799e12	−0.464938
$887$	2.98568e13	1.61952	0.809761	−	0.586760i	$-0.199597\pi$
$887$	2.98568e13	1.61952	0.809761	+	0.586760i	$0.199597\pi$
$888$	0	0
$889$	−1.12738e13	−0.605358
$890$	0	0
$891$	0	0
$892$	−5.90273e13	−3.12184
$893$	−1.01325e13	−0.533196
$894$	0	0
$895$	0	0
$896$	8.61458e12	0.446527
$897$	0	0
$898$	7.26969e12	0.373055
$899$	8.46604e12	0.432277
$900$	0	0
$901$	−1.77968e13	−0.899666
$902$	−7.34144e12	−0.369277
$903$	0	0
$904$	−1.78138e13	−0.887153
$905$	0	0
$906$	0	0
$907$	−2.14293e13	−1.05142	−0.525709	−	0.850664i	$-0.676200\pi$
$907$	−2.14293e13	−1.05142	−0.525709	+	0.850664i	$0.676200\pi$
$908$	3.91708e13	1.91239
$909$	0	0
$910$	0	0
$911$	−3.43474e13	−1.65219	−0.826097	−	0.563529i	$-0.809443\pi$
$911$	−3.43474e13	−1.65219	−0.826097	+	0.563529i	$0.809443\pi$
$912$	0	0
$913$	5.44218e12	0.259212
$914$	−1.23470e13	−0.585200
$915$	0	0
$916$	1.62064e13	0.760603
$917$	−2.96422e13	−1.38436
$918$	0	0
$919$	2.05847e13	0.951975	0.475987	−	0.879452i	$-0.342091\pi$
$919$	2.05847e13	0.951975	0.475987	+	0.879452i	$0.342091\pi$
$920$	0	0
$921$	0	0
$922$	1.81566e13	0.827458
$923$	7.14688e12	0.324122
$924$	0	0
$925$	0	0
$926$	−4.80354e13	−2.14690
$927$	0	0
$928$	2.17331e13	0.961955
$929$	−1.78605e13	−0.786726	−0.393363	−	0.919383i	$-0.628688\pi$
$929$	−1.78605e13	−0.786726	−0.393363	+	0.919383i	$0.628688\pi$
$930$	0	0
$931$	2.98855e12	0.130373
$932$	−9.56035e12	−0.415052
$933$	0	0
$934$	6.85583e12	0.294781
$935$	0	0
$936$	0	0
$937$	1.00969e13	0.427917	0.213959	−	0.976843i	$-0.431364\pi$
$937$	1.00969e13	0.427917	0.213959	+	0.976843i	$0.431364\pi$
$938$	−6.16950e13	−2.60218
$939$	0	0
$940$	0	0
$941$	2.40159e12	0.0998496	0.0499248	−	0.998753i	$-0.484102\pi$
$941$	2.40159e12	0.0998496	0.0499248	+	0.998753i	$0.484102\pi$
$942$	0	0
$943$	−1.22997e13	−0.506513
$944$	−8.89591e13	−3.64600
$945$	0	0
$946$	−1.25908e13	−0.511142
$947$	−2.09609e13	−0.846907	−0.423453	−	0.905918i	$-0.639182\pi$
$947$	−2.09609e13	−0.846907	−0.423453	+	0.905918i	$0.639182\pi$
$948$	0	0
$949$	6.05558e12	0.242358
$950$	0	0
$951$	0	0
$952$	−4.70827e13	−1.85779
$953$	−1.23636e12	−0.0485543	−0.0242771	−	0.999705i	$-0.507728\pi$
$953$	−1.23636e12	−0.0485543	−0.0242771	+	0.999705i	$0.507728\pi$
$954$	0	0
$955$	0	0
$956$	−9.92338e13	−3.84237
$957$	0	0
$958$	5.50289e12	0.211080
$959$	−1.36769e13	−0.522162
$960$	0	0
$961$	−1.55798e13	−0.589259
$962$	4.62305e13	1.74037
$963$	0	0
$964$	−7.22897e13	−2.69606
$965$	0	0
$966$	0	0
$967$	1.93044e13	0.709966	0.354983	−	0.934873i	$-0.384487\pi$
$967$	1.93044e13	0.709966	0.354983	+	0.934873i	$0.384487\pi$
$968$	6.35028e13	2.32463
$969$	0	0
$970$	0	0
$971$	2.95198e12	0.106568	0.0532840	−	0.998579i	$-0.483031\pi$
$971$	2.95198e12	0.106568	0.0532840	+	0.998579i	$0.483031\pi$
$972$	0	0
$973$	−8.62646e12	−0.308549
$974$	5.92007e13	2.10771
$975$	0	0
$976$	2.95610e13	1.04279
$977$	−5.39705e13	−1.89509	−0.947547	−	0.319617i	$-0.896446\pi$
$977$	−5.39705e13	−1.89509	−0.947547	+	0.319617i	$0.896446\pi$
$978$	0	0
$979$	2.47427e12	0.0860845
$980$	0	0
$981$	0	0
$982$	−4.36697e13	−1.49858
$983$	−2.52782e13	−0.863487	−0.431743	−	0.901996i	$-0.642101\pi$
$983$	−2.52782e13	−0.863487	−0.431743	+	0.901996i	$0.642101\pi$
$984$	0	0
$985$	0	0
$986$	3.33047e13	1.12217
$987$	0	0
$988$	−2.35118e13	−0.785019
$989$	−2.10942e13	−0.701101
$990$	0	0
$991$	3.62973e13	1.19548	0.597741	−	0.801689i	$-0.296065\pi$
$991$	3.62973e13	1.19548	0.597741	+	0.801689i	$0.296065\pi$
$992$	2.78781e13	0.914032
$993$	0	0
$994$	−1.97050e13	−0.640234
$995$	0	0
$996$	0	0
$997$	−5.75313e13	−1.84406	−0.922032	−	0.387114i	$-0.873472\pi$
$997$	−5.75313e13	−1.84406	−0.922032	+	0.387114i	$0.873472\pi$
$998$	−7.35058e12	−0.234549
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.10.a.s.1.1		4
3.2	odd	2		25.10.a.e.1.4		4
5.2	odd	4		45.10.b.b.19.1		4
5.3	odd	4		45.10.b.b.19.4		4
5.4	even	2	inner	225.10.a.s.1.4		4
12.11	even	2		400.10.a.ba.1.2		4
15.2	even	4		5.10.b.a.4.4	yes	4
15.8	even	4		5.10.b.a.4.1	✓	4
15.14	odd	2		25.10.a.e.1.1		4
60.23	odd	4		80.10.c.c.49.2		4
60.47	odd	4		80.10.c.c.49.3		4
60.59	even	2		400.10.a.ba.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.10.b.a.4.1	✓	4	15.8	even	4
5.10.b.a.4.4	yes	4	15.2	even	4
25.10.a.e.1.1		4	15.14	odd	2
25.10.a.e.1.4		4	3.2	odd	2
45.10.b.b.19.1		4	5.2	odd	4
45.10.b.b.19.4		4	5.3	odd	4
80.10.c.c.49.2		4	60.23	odd	4
80.10.c.c.49.3		4	60.47	odd	4
225.10.a.s.1.1		4	1.1	even	1	trivial
225.10.a.s.1.4		4	5.4	even	2	inner
400.10.a.ba.1.2		4	12.11	even	2
400.10.a.ba.1.3		4	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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	225.a (trivial)

\(f(q)\)	\(=\)	\(q-41.3193 q^{2} +1195.29 q^{4} +5315.22 q^{7} -28233.0 q^{8} +O(q^{10})\)	\(q-41.3193 q^{2} +1195.29 q^{4} +5315.22 q^{7} -28233.0 q^{8} -10426.2 q^{11} +79655.1 q^{13} -219621. q^{14} +554581. q^{16} +313750. q^{17} -246945. q^{19} +430806. q^{22} +721761. q^{23} -3.29130e6 q^{26} +6.35321e6 q^{28} -2.56903e6 q^{29} -3.29543e6 q^{31} -8.45965e6 q^{32} -1.29639e7 q^{34} -1.40463e7 q^{37} +1.02036e7 q^{38} -1.70412e7 q^{41} -2.92261e7 q^{43} -1.24624e7 q^{44} -2.98227e7 q^{46} +4.10316e7 q^{47} -1.21021e7 q^{49} +9.52108e7 q^{52} -5.67230e7 q^{53} -1.50065e8 q^{56} +1.06150e8 q^{58} -1.60408e8 q^{59} +5.33033e7 q^{61} +1.36165e8 q^{62} +6.56012e7 q^{64} +2.80916e8 q^{67} +3.75022e8 q^{68} +8.97228e7 q^{71} +7.60225e7 q^{73} +5.80383e8 q^{74} -2.95170e8 q^{76} -5.54178e7 q^{77} +4.10672e8 q^{79} +7.04131e8 q^{82} -5.21969e8 q^{83} +1.20760e9 q^{86} +2.94364e8 q^{88} -2.37312e8 q^{89} +4.23384e8 q^{91} +8.62712e8 q^{92} -1.69540e9 q^{94} +6.03778e8 q^{97} +5.00050e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 1368 q^{4}+O(q^{10}) \) 4 * q + 1368 * q^4	\( 4 q + 1368 q^{4} - 109968 q^{11} - 424536 q^{14} + 1631264 q^{16} + 636880 q^{19} - 6618768 q^{26} - 3531720 q^{29} - 10587712 q^{31} - 26434624 q^{34} + 16788552 q^{41} + 20638944 q^{44} - 61250072 q^{46} + 46921028 q^{49} - 315178080 q^{56} - 460829040 q^{59} + 360490568 q^{61} - 134995072 q^{64} + 47611872 q^{71} + 1176861744 q^{74} - 1168489440 q^{76} + 728043520 q^{79} + 2375904552 q^{86} - 1582700760 q^{89} + 473322528 q^{91} - 3327101704 q^{94}+O(q^{100}) \) 4 * q + 1368 * q^4 - 109968 * q^11 - 424536 * q^14 + 1631264 * q^16 + 636880 * q^19 - 6618768 * q^26 - 3531720 * q^29 - 10587712 * q^31 - 26434624 * q^34 + 16788552 * q^41 + 20638944 * q^44 - 61250072 * q^46 + 46921028 * q^49 - 315178080 * q^56 - 460829040 * q^59 + 360490568 * q^61 - 134995072 * q^64 + 47611872 * q^71 + 1176861744 * q^74 - 1168489440 * q^76 + 728043520 * q^79 + 2375904552 * q^86 - 1582700760 * q^89 + 473322528 * q^91 - 3327101704 * q^94

Introduction

L-functions

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