LMFDB - Embedded newform 225.10.a.q.1.2

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:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 469x^{2} + 4449x - 5580 $ x^4 - x^3 - 469x^2 + 4449x - 5580
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$13.6993$ 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)

$f(q)$	$=$	$q-20.9703 q^{2} -72.2477 q^{4} +3575.78 q^{7} +12251.8 q^{8} +O(q^{10})$	$q-20.9703 q^{2} -72.2477 q^{4} +3575.78 q^{7} +12251.8 q^{8} +74517.5 q^{11} -34478.5 q^{13} -74985.1 q^{14} -219933. q^{16} -372733. q^{17} -835114. q^{19} -1.56265e6 q^{22} -31807.7 q^{23} +723024. q^{26} -258342. q^{28} +6.73894e6 q^{29} -4.04277e6 q^{31} -1.66087e6 q^{32} +7.81631e6 q^{34} -6.29831e6 q^{37} +1.75126e7 q^{38} -1.10746e7 q^{41} +1.42409e7 q^{43} -5.38372e6 q^{44} +667017. q^{46} +2.76408e7 q^{47} -2.75674e7 q^{49} +2.49100e6 q^{52} +8.30946e7 q^{53} +4.38099e7 q^{56} -1.41317e8 q^{58} -1.04437e8 q^{59} +3.99467e7 q^{61} +8.47779e7 q^{62} +1.47435e8 q^{64} +1.98351e8 q^{67} +2.69291e7 q^{68} -4.52976e7 q^{71} -3.64162e8 q^{73} +1.32077e8 q^{74} +6.03351e7 q^{76} +2.66458e8 q^{77} -4.55156e8 q^{79} +2.32238e8 q^{82} +3.16569e7 q^{83} -2.98635e8 q^{86} +9.12975e8 q^{88} -2.77919e8 q^{89} -1.23288e8 q^{91} +2.29804e6 q^{92} -5.79636e8 q^{94} +1.06978e9 q^{97} +5.78096e8 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{2} + 597 q^{4} - 9834 q^{7} + 7671 q^{8}+O(q^{10}) $ 4 * q - 3 * q^2 + 597 * q^4 - 9834 * q^7 + 7671 * q^8	$ 4 q - 3 q^{2} + 597 q^{4} - 9834 q^{7} + 7671 q^{8} + 35994 q^{11} - 79998 q^{13} + 208182 q^{14} - 752815 q^{16} + 667878 q^{17} - 425792 q^{19} + 143934 q^{22} + 1031232 q^{23} + 438762 q^{26} - 3265578 q^{28} - 36786 q^{29} + 237044 q^{31} + 2932239 q^{32} + 4062194 q^{34} - 15750594 q^{37} + 48524592 q^{38} - 46660044 q^{41} - 67170720 q^{43} - 37277946 q^{44} - 4832036 q^{46} + 48243420 q^{47} - 25664800 q^{49} - 114924078 q^{52} + 198376482 q^{53} + 33890610 q^{56} - 439780704 q^{58} + 118263018 q^{59} - 178713880 q^{61} + 93716088 q^{62} + 6068513 q^{64} - 16141548 q^{67} + 611464794 q^{68} + 78445332 q^{71} - 514053252 q^{73} - 761690898 q^{74} + 549348672 q^{76} + 780875028 q^{77} - 431961140 q^{79} - 241116378 q^{82} + 557494176 q^{83} - 498821196 q^{86} + 392593074 q^{88} + 178691112 q^{89} + 107377164 q^{91} - 1048179156 q^{92} + 360839912 q^{94} + 840904752 q^{97} - 3438128823 q^{98}+O(q^{100}) $ 4 * q - 3 * q^2 + 597 * q^4 - 9834 * q^7 + 7671 * q^8 + 35994 * q^11 - 79998 * q^13 + 208182 * q^14 - 752815 * q^16 + 667878 * q^17 - 425792 * q^19 + 143934 * q^22 + 1031232 * q^23 + 438762 * q^26 - 3265578 * q^28 - 36786 * q^29 + 237044 * q^31 + 2932239 * q^32 + 4062194 * q^34 - 15750594 * q^37 + 48524592 * q^38 - 46660044 * q^41 - 67170720 * q^43 - 37277946 * q^44 - 4832036 * q^46 + 48243420 * q^47 - 25664800 * q^49 - 114924078 * q^52 + 198376482 * q^53 + 33890610 * q^56 - 439780704 * q^58 + 118263018 * q^59 - 178713880 * q^61 + 93716088 * q^62 + 6068513 * q^64 - 16141548 * q^67 + 611464794 * q^68 + 78445332 * q^71 - 514053252 * q^73 - 761690898 * q^74 + 549348672 * q^76 + 780875028 * q^77 - 431961140 * q^79 - 241116378 * q^82 + 557494176 * q^83 - 498821196 * q^86 + 392593074 * q^88 + 178691112 * q^89 + 107377164 * q^91 - 1048179156 * q^92 + 360839912 * q^94 + 840904752 * q^97 - 3438128823 * 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 $

$2$	−20.9703	−0.926764	−0.463382	−	0.886159i	$-0.653364\pi$
$2$	−20.9703	−0.926764	−0.463382	+	0.886159i	$0.653364\pi$
$3$	0	0
$3$	0	0
$4$	−72.2477	−0.141109
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3575.78	0.562898	0.281449	−	0.959576i	$-0.409185\pi$
$7$	3575.78	0.562898	0.281449	+	0.959576i	$0.409185\pi$
$8$	12251.8	1.05754
$9$	0	0
$10$	0	0
$11$	74517.5	1.53458	0.767292	−	0.641297i	$-0.221603\pi$
$11$	74517.5	1.53458	0.767292	+	0.641297i	$0.221603\pi$
$12$	0	0
$13$	−34478.5	−0.334814	−0.167407	−	0.985888i	$-0.553539\pi$
$13$	−34478.5	−0.334814	−0.167407	+	0.985888i	$0.553539\pi$
$14$	−74985.1	−0.521674
$15$	0	0
$16$	−219933.	−0.838979
$17$	−372733.	−1.08237	−0.541187	−	0.840902i	$-0.682025\pi$
$17$	−372733.	−1.08237	−0.541187	+	0.840902i	$0.682025\pi$
$18$	0	0
$19$	−835114.	−1.47013	−0.735063	−	0.677998i	$-0.762848\pi$
$19$	−835114.	−1.47013	−0.735063	+	0.677998i	$0.762848\pi$
$20$	0	0
$21$	0	0
$22$	−1.56265e6	−1.42220
$23$	−31807.7	−0.0237005	−0.0118503	−	0.999930i	$-0.503772\pi$
$23$	−31807.7	−0.0237005	−0.0118503	+	0.999930i	$0.503772\pi$
$24$	0	0
$25$	0	0
$26$	723024.	0.310294
$27$	0	0
$28$	−258342.	−0.0794299
$29$	6.73894e6	1.76930	0.884649	−	0.466258i	$-0.154398\pi$
$29$	6.73894e6	1.76930	0.884649	+	0.466258i	$0.154398\pi$
$30$	0	0
$31$	−4.04277e6	−0.786232	−0.393116	−	0.919489i	$-0.628603\pi$
$31$	−4.04277e6	−0.786232	−0.393116	+	0.919489i	$0.628603\pi$
$32$	−1.66087e6	−0.280003
$33$	0	0
$34$	7.81631e6	1.00311
$35$	0	0
$36$	0	0
$37$	−6.29831e6	−0.552480	−0.276240	−	0.961089i	$-0.589088\pi$
$37$	−6.29831e6	−0.552480	−0.276240	+	0.961089i	$0.589088\pi$
$38$	1.75126e7	1.36246
$39$	0	0
$40$	0	0
$41$	−1.10746e7	−0.612072	−0.306036	−	0.952020i	$-0.599003\pi$
$41$	−1.10746e7	−0.612072	−0.306036	+	0.952020i	$0.599003\pi$
$42$	0	0
$43$	1.42409e7	0.635226	0.317613	−	0.948220i	$-0.397119\pi$
$43$	1.42409e7	0.635226	0.317613	+	0.948220i	$0.397119\pi$
$44$	−5.38372e6	−0.216543
$45$	0	0
$46$	667017.	0.0219648
$47$	2.76408e7	0.826249	0.413124	−	0.910675i	$-0.364437\pi$
$47$	2.76408e7	0.826249	0.413124	+	0.910675i	$0.364437\pi$
$48$	0	0
$49$	−2.75674e7	−0.683146
$50$	0	0
$51$	0	0
$52$	2.49100e6	0.0472452
$53$	8.30946e7	1.44654	0.723272	−	0.690564i	$-0.242637\pi$
$53$	8.30946e7	1.44654	0.723272	+	0.690564i	$0.242637\pi$
$54$	0	0
$55$	0	0
$56$	4.38099e7	0.595286
$57$	0	0
$58$	−1.41317e8	−1.63972
$59$	−1.04437e8	−1.12207	−0.561034	−	0.827793i	$-0.689596\pi$
$59$	−1.04437e8	−1.12207	−0.561034	+	0.827793i	$0.689596\pi$
$60$	0	0
$61$	3.99467e7	0.369400	0.184700	−	0.982795i	$-0.440869\pi$
$61$	3.99467e7	0.369400	0.184700	+	0.982795i	$0.440869\pi$
$62$	8.47779e7	0.728652
$63$	0	0
$64$	1.47435e8	1.09848
$65$	0	0
$66$	0	0
$67$	1.98351e8	1.20254	0.601268	−	0.799047i	$-0.294662\pi$
$67$	1.98351e8	1.20254	0.601268	+	0.799047i	$0.294662\pi$
$68$	2.69291e7	0.152733
$69$	0	0
$70$	0	0
$71$	−4.52976e7	−0.211550	−0.105775	−	0.994390i	$-0.533732\pi$
$71$	−4.52976e7	−0.211550	−0.105775	+	0.994390i	$0.533732\pi$
$72$	0	0
$73$	−3.64162e8	−1.50087	−0.750433	−	0.660947i	$-0.770155\pi$
$73$	−3.64162e8	−1.50087	−0.750433	+	0.660947i	$0.770155\pi$
$74$	1.32077e8	0.512018
$75$	0	0
$76$	6.03351e7	0.207448
$77$	2.66458e8	0.863815
$78$	0	0
$79$	−4.55156e8	−1.31473	−0.657367	−	0.753570i	$-0.728330\pi$
$79$	−4.55156e8	−1.31473	−0.657367	+	0.753570i	$0.728330\pi$
$80$	0	0
$81$	0	0
$82$	2.32238e8	0.567246
$83$	3.16569e7	0.0732178	0.0366089	−	0.999330i	$-0.488344\pi$
$83$	3.16569e7	0.0732178	0.0366089	+	0.999330i	$0.488344\pi$
$84$	0	0
$85$	0	0
$86$	−2.98635e8	−0.588704
$87$	0	0
$88$	9.12975e8	1.62288
$89$	−2.77919e8	−0.469530	−0.234765	−	0.972052i	$-0.575432\pi$
$89$	−2.77919e8	−0.469530	−0.234765	+	0.972052i	$0.575432\pi$
$90$	0	0
$91$	−1.23288e8	−0.188466
$92$	2.29804e6	0.00334435
$93$	0	0
$94$	−5.79636e8	−0.765737
$95$	0	0
$96$	0	0
$97$	1.06978e9	1.22694	0.613468	−	0.789720i	$-0.289774\pi$
$97$	1.06978e9	1.22694	0.613468	+	0.789720i	$0.289774\pi$
$98$	5.78096e8	0.633115
$99$	0	0
$100$	0	0
$101$	5.28777e8	0.505623	0.252811	−	0.967516i	$-0.418645\pi$
$101$	5.28777e8	0.505623	0.252811	+	0.967516i	$0.418645\pi$
$102$	0	0
$103$	1.45019e9	1.26957	0.634786	−	0.772688i	$-0.281088\pi$
$103$	1.45019e9	1.26957	0.634786	+	0.772688i	$0.281088\pi$
$104$	−4.22425e8	−0.354079
$105$	0	0
$106$	−1.74252e9	−1.34060
$107$	−1.18511e8	−0.0874043	−0.0437021	−	0.999045i	$-0.513915\pi$
$107$	−1.18511e8	−0.0874043	−0.0437021	+	0.999045i	$0.513915\pi$
$108$	0	0
$109$	1.47459e9	1.00058	0.500289	−	0.865859i	$-0.333227\pi$
$109$	1.47459e9	1.00058	0.500289	+	0.865859i	$0.333227\pi$
$110$	0	0
$111$	0	0
$112$	−7.86434e8	−0.472260
$113$	−1.73649e9	−1.00189	−0.500943	−	0.865480i	$-0.667013\pi$
$113$	−1.73649e9	−1.00189	−0.500943	+	0.865480i	$0.667013\pi$
$114$	0	0
$115$	0	0
$116$	−4.86873e8	−0.249663
$117$	0	0
$118$	2.19006e9	1.03989
$119$	−1.33281e9	−0.609267
$120$	0	0
$121$	3.19490e9	1.35495
$122$	−8.37694e8	−0.342347
$123$	0	0
$124$	2.92081e8	0.110944
$125$	0	0
$126$	0	0
$127$	2.07925e9	0.709235	0.354618	−	0.935011i	$-0.384611\pi$
$127$	2.07925e9	0.709235	0.354618	+	0.935011i	$0.384611\pi$
$128$	−2.24138e9	−0.738025
$129$	0	0
$130$	0	0
$131$	−6.49740e9	−1.92761	−0.963804	−	0.266612i	$-0.914096\pi$
$131$	−6.49740e9	−1.92761	−0.963804	+	0.266612i	$0.914096\pi$
$132$	0	0
$133$	−2.98619e9	−0.827532
$134$	−4.15948e9	−1.11447
$135$	0	0
$136$	−4.56666e9	−1.14465
$137$	2.45780e9	0.596080	0.298040	−	0.954553i	$-0.403667\pi$
$137$	2.45780e9	0.596080	0.298040	+	0.954553i	$0.403667\pi$
$138$	0	0
$139$	−5.53229e9	−1.25701	−0.628505	−	0.777806i	$-0.716333\pi$
$139$	−5.53229e9	−1.25701	−0.628505	+	0.777806i	$0.716333\pi$
$140$	0	0
$141$	0	0
$142$	9.49903e8	0.196057
$143$	−2.56925e9	−0.513801
$144$	0	0
$145$	0	0
$146$	7.63657e9	1.39095
$147$	0	0
$148$	4.55038e8	0.0779597
$149$	−1.06402e10	−1.76852	−0.884261	−	0.466994i	$-0.845337\pi$
$149$	−1.06402e10	−1.76852	−0.884261	+	0.466994i	$0.845337\pi$
$150$	0	0
$151$	5.40470e9	0.846009	0.423005	−	0.906128i	$-0.360975\pi$
$151$	5.40470e9	0.846009	0.423005	+	0.906128i	$0.360975\pi$
$152$	−1.02317e10	−1.55472
$153$	0	0
$154$	−5.58770e9	−0.800553
$155$	0	0
$156$	0	0
$157$	−7.89002e9	−1.03641	−0.518203	−	0.855258i	$-0.673399\pi$
$157$	−7.89002e9	−1.03641	−0.518203	+	0.855258i	$0.673399\pi$
$158$	9.54474e9	1.21845
$159$	0	0
$160$	0	0
$161$	−1.13738e8	−0.0133410
$162$	0	0
$163$	−1.43411e10	−1.59125	−0.795627	−	0.605787i	$-0.792858\pi$
$163$	−1.43411e10	−1.59125	−0.795627	+	0.605787i	$0.792858\pi$
$164$	8.00118e8	0.0863688
$165$	0	0
$166$	−6.63853e8	−0.0678556
$167$	1.43038e10	1.42308	0.711539	−	0.702647i	$-0.247999\pi$
$167$	1.43038e10	1.42308	0.711539	+	0.702647i	$0.247999\pi$
$168$	0	0
$169$	−9.41573e9	−0.887900
$170$	0	0
$171$	0	0
$172$	−1.02887e9	−0.0896360
$173$	3.74604e9	0.317955	0.158977	−	0.987282i	$-0.449180\pi$
$173$	3.74604e9	0.317955	0.158977	+	0.987282i	$0.449180\pi$
$174$	0	0
$175$	0	0
$176$	−1.63889e10	−1.28749
$177$	0	0
$178$	5.82804e9	0.435144
$179$	−2.22667e9	−0.162113	−0.0810565	−	0.996710i	$-0.525829\pi$
$179$	−2.22667e9	−0.162113	−0.0810565	+	0.996710i	$0.525829\pi$
$180$	0	0
$181$	−2.08908e10	−1.44678	−0.723388	−	0.690442i	$-0.757416\pi$
$181$	−2.08908e10	−1.44678	−0.723388	+	0.690442i	$0.757416\pi$
$182$	2.58538e9	0.174664
$183$	0	0
$184$	−3.89703e8	−0.0250642
$185$	0	0
$186$	0	0
$187$	−2.77751e10	−1.66100
$188$	−1.99699e9	−0.116591
$189$	0	0
$190$	0	0
$191$	−1.10380e9	−0.0600125	−0.0300062	−	0.999550i	$-0.509553\pi$
$191$	−1.10380e9	−0.0600125	−0.0300062	+	0.999550i	$0.509553\pi$
$192$	0	0
$193$	−2.76093e10	−1.43234	−0.716171	−	0.697925i	$-0.754107\pi$
$193$	−2.76093e10	−1.43234	−0.716171	+	0.697925i	$0.754107\pi$
$194$	−2.24336e10	−1.13708
$195$	0	0
$196$	1.99168e9	0.0963979
$197$	−3.05799e10	−1.44656	−0.723282	−	0.690553i	$-0.757367\pi$
$197$	−3.05799e10	−1.44656	−0.723282	+	0.690553i	$0.757367\pi$
$198$	0	0
$199$	7.92786e9	0.358358	0.179179	−	0.983816i	$-0.442656\pi$
$199$	7.92786e9	0.358358	0.179179	+	0.983816i	$0.442656\pi$
$200$	0	0
$201$	0	0
$202$	−1.10886e10	−0.468593
$203$	2.40970e10	0.995934
$204$	0	0
$205$	0	0
$206$	−3.04109e10	−1.17659
$207$	0	0
$208$	7.58298e9	0.280902
$209$	−6.22306e10	−2.25603
$210$	0	0
$211$	−3.97409e9	−0.138028	−0.0690139	−	0.997616i	$-0.521985\pi$
$211$	−3.97409e9	−0.138028	−0.0690139	+	0.997616i	$0.521985\pi$
$212$	−6.00340e9	−0.204120
$213$	0	0
$214$	2.48521e9	0.0810031
$215$	0	0
$216$	0	0
$217$	−1.44561e10	−0.442569
$218$	−3.09225e10	−0.927299
$219$	0	0
$220$	0	0
$221$	1.28513e10	0.362394
$222$	0	0
$223$	4.90677e10	1.32869	0.664345	−	0.747426i	$-0.268711\pi$
$223$	4.90677e10	1.32869	0.664345	+	0.747426i	$0.268711\pi$
$224$	−5.93893e9	−0.157613
$225$	0	0
$226$	3.64146e10	0.928512
$227$	4.97991e10	1.24482	0.622408	−	0.782693i	$-0.286154\pi$
$227$	4.97991e10	1.24482	0.622408	+	0.782693i	$0.286154\pi$
$228$	0	0
$229$	−3.84531e10	−0.923999	−0.462000	−	0.886880i	$-0.652868\pi$
$229$	−3.84531e10	−0.923999	−0.462000	+	0.886880i	$0.652868\pi$
$230$	0	0
$231$	0	0
$232$	8.25644e10	1.87110
$233$	5.45563e9	0.121267	0.0606336	−	0.998160i	$-0.480688\pi$
$233$	5.45563e9	0.121267	0.0606336	+	0.998160i	$0.480688\pi$
$234$	0	0
$235$	0	0
$236$	7.54531e9	0.158334
$237$	0	0
$238$	2.79494e10	0.564646
$239$	5.86338e10	1.16240	0.581202	−	0.813759i	$-0.302582\pi$
$239$	5.86338e10	1.16240	0.581202	+	0.813759i	$0.302582\pi$
$240$	0	0
$241$	−2.28707e10	−0.436719	−0.218360	−	0.975868i	$-0.570071\pi$
$241$	−2.28707e10	−0.436719	−0.218360	+	0.975868i	$0.570071\pi$
$242$	−6.69980e10	−1.25572
$243$	0	0
$244$	−2.88606e9	−0.0521256
$245$	0	0
$246$	0	0
$247$	2.87935e10	0.492219
$248$	−4.95313e10	−0.831471
$249$	0	0
$250$	0	0
$251$	1.14619e10	0.182275	0.0911374	−	0.995838i	$-0.470950\pi$
$251$	1.14619e10	0.182275	0.0911374	+	0.995838i	$0.470950\pi$
$252$	0	0
$253$	−2.37023e9	−0.0363704
$254$	−4.36025e10	−0.657293
$255$	0	0
$256$	−2.84843e10	−0.414501
$257$	7.03335e10	1.00569	0.502844	−	0.864377i	$-0.332287\pi$
$257$	7.03335e10	1.00569	0.502844	+	0.864377i	$0.332287\pi$
$258$	0	0
$259$	−2.25214e10	−0.310990
$260$	0	0
$261$	0	0
$262$	1.36252e11	1.78644
$263$	−1.11602e11	−1.43838	−0.719188	−	0.694815i	$-0.755486\pi$
$263$	−1.11602e11	−1.43838	−0.719188	+	0.694815i	$0.755486\pi$
$264$	0	0
$265$	0	0
$266$	6.26212e10	0.766926
$267$	0	0
$268$	−1.43304e10	−0.169688
$269$	6.35228e10	0.739680	0.369840	−	0.929095i	$-0.379412\pi$
$269$	6.35228e10	0.739680	0.369840	+	0.929095i	$0.379412\pi$
$270$	0	0
$271$	−7.97056e8	−0.00897691	−0.00448845	−	0.999990i	$-0.501429\pi$
$271$	−7.97056e8	−0.00897691	−0.00448845	+	0.999990i	$0.501429\pi$
$272$	8.19764e10	0.908090
$273$	0	0
$274$	−5.15408e10	−0.552425
$275$	0	0
$276$	0	0
$277$	−3.68479e10	−0.376057	−0.188029	−	0.982164i	$-0.560210\pi$
$277$	−3.68479e10	−0.376057	−0.188029	+	0.982164i	$0.560210\pi$
$278$	1.16014e11	1.16495
$279$	0	0
$280$	0	0
$281$	−7.86433e10	−0.752460	−0.376230	−	0.926526i	$-0.622780\pi$
$281$	−7.86433e10	−0.752460	−0.376230	+	0.926526i	$0.622780\pi$
$282$	0	0
$283$	−3.08266e10	−0.285685	−0.142842	−	0.989745i	$-0.545624\pi$
$283$	−3.08266e10	−0.285685	−0.142842	+	0.989745i	$0.545624\pi$
$284$	3.27265e9	0.0298515
$285$	0	0
$286$	5.38779e10	0.476172
$287$	−3.96005e10	−0.344534
$288$	0	0
$289$	2.03420e10	0.171535
$290$	0	0
$291$	0	0
$292$	2.63099e10	0.211785
$293$	6.04362e10	0.479064	0.239532	−	0.970889i	$-0.423006\pi$
$293$	6.04362e10	0.479064	0.239532	+	0.970889i	$0.423006\pi$
$294$	0	0
$295$	0	0
$296$	−7.71658e10	−0.584268
$297$	0	0
$298$	2.23127e11	1.63900
$299$	1.09668e9	0.00793526
$300$	0	0
$301$	5.09222e10	0.357568
$302$	−1.13338e11	−0.784051
$303$	0	0
$304$	1.83670e11	1.23341
$305$	0	0
$306$	0	0
$307$	−2.37829e11	−1.52807	−0.764034	−	0.645176i	$-0.776784\pi$
$307$	−2.37829e11	−1.52807	−0.764034	+	0.645176i	$0.776784\pi$
$308$	−1.92510e10	−0.121892
$309$	0	0
$310$	0	0
$311$	1.05237e11	0.637892	0.318946	−	0.947773i	$-0.396671\pi$
$311$	1.05237e11	0.637892	0.318946	+	0.947773i	$0.396671\pi$
$312$	0	0
$313$	−1.21699e11	−0.716702	−0.358351	−	0.933587i	$-0.616661\pi$
$313$	−1.21699e11	−0.716702	−0.358351	+	0.933587i	$0.616661\pi$
$314$	1.65456e11	0.960504
$315$	0	0
$316$	3.28840e10	0.185521
$317$	8.05511e10	0.448028	0.224014	−	0.974586i	$-0.428084\pi$
$317$	8.05511e10	0.448028	0.224014	+	0.974586i	$0.428084\pi$
$318$	0	0
$319$	5.02169e11	2.71514
$320$	0	0
$321$	0	0
$322$	2.38511e9	0.0123639
$323$	3.11275e11	1.59123
$324$	0	0
$325$	0	0
$326$	3.00738e11	1.47472
$327$	0	0
$328$	−1.35685e11	−0.647290
$329$	9.88376e10	0.465094
$330$	0	0
$331$	−2.15934e11	−0.988769	−0.494385	−	0.869243i	$-0.664607\pi$
$331$	−2.15934e11	−0.988769	−0.494385	+	0.869243i	$0.664607\pi$
$332$	−2.28714e9	−0.0103317
$333$	0	0
$334$	−2.99955e11	−1.31886
$335$	0	0
$336$	0	0
$337$	−1.13654e10	−0.0480010	−0.0240005	−	0.999712i	$-0.507640\pi$
$337$	−1.13654e10	−0.0480010	−0.0240005	+	0.999712i	$0.507640\pi$
$338$	1.97450e11	0.822873
$339$	0	0
$340$	0	0
$341$	−3.01257e11	−1.20654
$342$	0	0
$343$	−2.42871e11	−0.947440
$344$	1.74477e11	0.671776
$345$	0	0
$346$	−7.85555e10	−0.294669
$347$	−1.23817e11	−0.458455	−0.229228	−	0.973373i	$-0.573620\pi$
$347$	−1.23817e11	−0.458455	−0.229228	+	0.973373i	$0.573620\pi$
$348$	0	0
$349$	−1.76035e11	−0.635161	−0.317580	−	0.948231i	$-0.602870\pi$
$349$	−1.76035e11	−0.635161	−0.317580	+	0.948231i	$0.602870\pi$
$350$	0	0
$351$	0	0
$352$	−1.23764e11	−0.429688
$353$	−5.17450e11	−1.77371	−0.886853	−	0.462051i	$-0.847114\pi$
$353$	−5.17450e11	−1.77371	−0.886853	+	0.462051i	$0.847114\pi$
$354$	0	0
$355$	0	0
$356$	2.00790e10	0.0662549
$357$	0	0
$358$	4.66939e10	0.150240
$359$	2.86436e11	0.910128	0.455064	−	0.890459i	$-0.349616\pi$
$359$	2.86436e11	0.910128	0.455064	+	0.890459i	$0.349616\pi$
$360$	0	0
$361$	3.74728e11	1.16127
$362$	4.38086e11	1.34082
$363$	0	0
$364$	8.90726e9	0.0265943
$365$	0	0
$366$	0	0
$367$	−1.20491e11	−0.346704	−0.173352	−	0.984860i	$-0.555460\pi$
$367$	−1.20491e11	−0.346704	−0.173352	+	0.984860i	$0.555460\pi$
$368$	6.99559e9	0.0198842
$369$	0	0
$370$	0	0
$371$	2.97128e11	0.814257
$372$	0	0
$373$	−4.05791e11	−1.08546	−0.542728	−	0.839909i	$-0.682609\pi$
$373$	−4.05791e11	−1.08546	−0.542728	+	0.839909i	$0.682609\pi$
$374$	5.82452e11	1.53935
$375$	0	0
$376$	3.38651e11	0.873790
$377$	−2.32349e11	−0.592386
$378$	0	0
$379$	−2.12253e11	−0.528418	−0.264209	−	0.964465i	$-0.585111\pi$
$379$	−2.12253e11	−0.528418	−0.264209	+	0.964465i	$0.585111\pi$
$380$	0	0
$381$	0	0
$382$	2.31470e10	0.0556174
$383$	−4.96937e10	−0.118007	−0.0590034	−	0.998258i	$-0.518792\pi$
$383$	−4.96937e10	−0.118007	−0.0590034	+	0.998258i	$0.518792\pi$
$384$	0	0
$385$	0	0
$386$	5.78974e11	1.32744
$387$	0	0
$388$	−7.72892e10	−0.173131
$389$	−2.74472e11	−0.607750	−0.303875	−	0.952712i	$-0.598280\pi$
$389$	−2.74472e11	−0.607750	−0.303875	+	0.952712i	$0.598280\pi$
$390$	0	0
$391$	1.18558e10	0.0256528
$392$	−3.37751e11	−0.722453
$393$	0	0
$394$	6.41268e11	1.34062
$395$	0	0
$396$	0	0
$397$	2.17106e10	0.0438646	0.0219323	−	0.999759i	$-0.493018\pi$
$397$	2.17106e10	0.0438646	0.0219323	+	0.999759i	$0.493018\pi$
$398$	−1.66249e11	−0.332113
$399$	0	0
$400$	0	0
$401$	−2.16300e11	−0.417741	−0.208871	−	0.977943i	$-0.566979\pi$
$401$	−2.16300e11	−0.417741	−0.208871	+	0.977943i	$0.566979\pi$
$402$	0	0
$403$	1.39389e11	0.263242
$404$	−3.82029e10	−0.0713478
$405$	0	0
$406$	−5.05320e11	−0.922996
$407$	−4.69334e11	−0.847827
$408$	0	0
$409$	−2.30916e11	−0.408037	−0.204019	−	0.978967i	$-0.565400\pi$
$409$	−2.30916e11	−0.408037	−0.204019	+	0.978967i	$0.565400\pi$
$410$	0	0
$411$	0	0
$412$	−1.04773e11	−0.179148
$413$	−3.73443e11	−0.631610
$414$	0	0
$415$	0	0
$416$	5.72645e10	0.0937488
$417$	0	0
$418$	1.30499e12	2.09081
$419$	−2.38946e11	−0.378736	−0.189368	−	0.981906i	$-0.560644\pi$
$419$	−2.38946e11	−0.378736	−0.189368	+	0.981906i	$0.560644\pi$
$420$	0	0
$421$	4.09581e11	0.635434	0.317717	−	0.948186i	$-0.397084\pi$
$421$	4.09581e11	0.635434	0.317717	+	0.948186i	$0.397084\pi$
$422$	8.33377e10	0.127919
$423$	0	0
$424$	1.01806e12	1.52977
$425$	0	0
$426$	0	0
$427$	1.42841e11	0.207935
$428$	8.56217e9	0.0123335
$429$	0	0
$430$	0	0
$431$	−5.29879e11	−0.739655	−0.369828	−	0.929100i	$-0.620583\pi$
$431$	−5.29879e11	−0.739655	−0.369828	+	0.929100i	$0.620583\pi$
$432$	0	0
$433$	1.70997e11	0.233772	0.116886	−	0.993145i	$-0.462709\pi$
$433$	1.70997e11	0.233772	0.116886	+	0.993145i	$0.462709\pi$
$434$	3.03147e11	0.410157
$435$	0	0
$436$	−1.06535e11	−0.141190
$437$	2.65631e10	0.0348427
$438$	0	0
$439$	−2.23652e11	−0.287398	−0.143699	−	0.989621i	$-0.545900\pi$
$439$	−2.23652e11	−0.287398	−0.143699	+	0.989621i	$0.545900\pi$
$440$	0	0
$441$	0	0
$442$	−2.69495e11	−0.335854
$443$	−8.32773e11	−1.02733	−0.513665	−	0.857991i	$-0.671712\pi$
$443$	−8.32773e11	−1.02733	−0.513665	+	0.857991i	$0.671712\pi$
$444$	0	0
$445$	0	0
$446$	−1.02896e12	−1.23138
$447$	0	0
$448$	5.27195e11	0.618330
$449$	5.62842e11	0.653549	0.326774	−	0.945102i	$-0.394038\pi$
$449$	5.62842e11	0.653549	0.326774	+	0.945102i	$0.394038\pi$
$450$	0	0
$451$	−8.25254e11	−0.939276
$452$	1.25457e11	0.141375
$453$	0	0
$454$	−1.04430e12	−1.15365
$455$	0	0
$456$	0	0
$457$	7.43811e11	0.797700	0.398850	−	0.917016i	$-0.369409\pi$
$457$	7.43811e11	0.797700	0.398850	+	0.917016i	$0.369409\pi$
$458$	8.06372e11	0.856329
$459$	0	0
$460$	0	0
$461$	9.90060e11	1.02096	0.510478	−	0.859891i	$-0.329468\pi$
$461$	9.90060e11	1.02096	0.510478	+	0.859891i	$0.329468\pi$
$462$	0	0
$463$	7.40358e11	0.748734	0.374367	−	0.927281i	$-0.377860\pi$
$463$	7.40358e11	0.748734	0.374367	+	0.927281i	$0.377860\pi$
$464$	−1.48212e12	−1.48440
$465$	0	0
$466$	−1.14406e11	−0.112386
$467$	1.86063e11	0.181023	0.0905114	−	0.995895i	$-0.471150\pi$
$467$	1.86063e11	0.181023	0.0905114	+	0.995895i	$0.471150\pi$
$468$	0	0
$469$	7.09261e11	0.676906
$470$	0	0
$471$	0	0
$472$	−1.27954e12	−1.18663
$473$	1.06119e12	0.974808
$474$	0	0
$475$	0	0
$476$	9.62926e10	0.0859729
$477$	0	0
$478$	−1.22957e12	−1.07727
$479$	−5.77280e11	−0.501045	−0.250522	−	0.968111i	$-0.580602\pi$
$479$	−5.77280e11	−0.501045	−0.250522	+	0.968111i	$0.580602\pi$
$480$	0	0
$481$	2.17156e11	0.184978
$482$	4.79604e11	0.404736
$483$	0	0
$484$	−2.30824e11	−0.191196
$485$	0	0
$486$	0	0
$487$	2.33534e12	1.88135	0.940675	−	0.339310i	$-0.110194\pi$
$487$	2.33534e12	1.88135	0.940675	+	0.339310i	$0.110194\pi$
$488$	4.89421e11	0.390655
$489$	0	0
$490$	0	0
$491$	−9.07091e11	−0.704343	−0.352171	−	0.935936i	$-0.614557\pi$
$491$	−9.07091e11	−0.704343	−0.352171	+	0.935936i	$0.614557\pi$
$492$	0	0
$493$	−2.51183e12	−1.91504
$494$	−6.03808e11	−0.456171
$495$	0	0
$496$	8.89139e11	0.659633
$497$	−1.61974e11	−0.119081
$498$	0	0
$499$	−1.13701e12	−0.820937	−0.410469	−	0.911875i	$-0.634635\pi$
$499$	−1.13701e12	−0.820937	−0.410469	+	0.911875i	$0.634635\pi$
$500$	0	0
$501$	0	0
$502$	−2.40360e11	−0.168926
$503$	−1.85806e11	−0.129421	−0.0647103	−	0.997904i	$-0.520612\pi$
$503$	−1.85806e11	−0.129421	−0.0647103	+	0.997904i	$0.520612\pi$
$504$	0	0
$505$	0	0
$506$	4.97044e10	0.0337068
$507$	0	0
$508$	−1.50221e11	−0.100079
$509$	−7.38201e11	−0.487466	−0.243733	−	0.969842i	$-0.578372\pi$
$509$	−7.38201e11	−0.487466	−0.243733	+	0.969842i	$0.578372\pi$
$510$	0	0
$511$	−1.30216e12	−0.844834
$512$	1.74491e12	1.12217
$513$	0	0
$514$	−1.47491e12	−0.932035
$515$	0	0
$516$	0	0
$517$	2.05972e12	1.26795
$518$	4.72279e11	0.288214
$519$	0	0
$520$	0	0
$521$	1.17044e11	0.0695955	0.0347977	−	0.999394i	$-0.488921\pi$
$521$	1.17044e11	0.0695955	0.0347977	+	0.999394i	$0.488921\pi$
$522$	0	0
$523$	1.53771e12	0.898707	0.449354	−	0.893354i	$-0.351654\pi$
$523$	1.53771e12	0.898707	0.449354	+	0.893354i	$0.351654\pi$
$524$	4.69422e11	0.272002
$525$	0	0
$526$	2.34033e12	1.33304
$527$	1.50687e12	0.850998
$528$	0	0
$529$	−1.80014e12	−0.999438
$530$	0	0
$531$	0	0
$532$	2.15745e11	0.116772
$533$	3.81838e11	0.204930
$534$	0	0
$535$	0	0
$536$	2.43017e12	1.27173
$537$	0	0
$538$	−1.33209e12	−0.685509
$539$	−2.05425e12	−1.04834
$540$	0	0
$541$	−2.24717e12	−1.12784	−0.563922	−	0.825828i	$-0.690708\pi$
$541$	−2.24717e12	−1.12784	−0.563922	+	0.825828i	$0.690708\pi$
$542$	1.67145e10	0.00831947
$543$	0	0
$544$	6.19063e11	0.303068
$545$	0	0
$546$	0	0
$547$	−2.72880e12	−1.30325	−0.651625	−	0.758541i	$-0.725913\pi$
$547$	−2.72880e12	−1.30325	−0.651625	+	0.758541i	$0.725913\pi$
$548$	−1.77571e11	−0.0841122
$549$	0	0
$550$	0	0
$551$	−5.62779e12	−2.60109
$552$	0	0
$553$	−1.62754e12	−0.740062
$554$	7.72711e11	0.348516
$555$	0	0
$556$	3.99695e11	0.177375
$557$	−1.50494e12	−0.662477	−0.331238	−	0.943547i	$-0.607466\pi$
$557$	−1.50494e12	−0.662477	−0.331238	+	0.943547i	$0.607466\pi$
$558$	0	0
$559$	−4.91004e11	−0.212683
$560$	0	0
$561$	0	0
$562$	1.64917e12	0.697353
$563$	1.28466e12	0.538892	0.269446	−	0.963015i	$-0.413159\pi$
$563$	1.28466e12	0.538892	0.269446	+	0.963015i	$0.413159\pi$
$564$	0	0
$565$	0	0
$566$	6.46443e11	0.264762
$567$	0	0
$568$	−5.54979e11	−0.223722
$569$	−4.36230e12	−1.74466	−0.872329	−	0.488920i	$-0.837391\pi$
$569$	−4.36230e12	−1.74466	−0.872329	+	0.488920i	$0.837391\pi$
$570$	0	0
$571$	−2.22894e12	−0.877475	−0.438738	−	0.898615i	$-0.644574\pi$
$571$	−2.22894e12	−0.877475	−0.438738	+	0.898615i	$0.644574\pi$
$572$	1.85623e11	0.0725018
$573$	0	0
$574$	8.30434e11	0.319302
$575$	0	0
$576$	0	0
$577$	−2.37578e12	−0.892307	−0.446153	−	0.894956i	$-0.647206\pi$
$577$	−2.37578e12	−0.892307	−0.446153	+	0.894956i	$0.647206\pi$
$578$	−4.26576e11	−0.158972
$579$	0	0
$580$	0	0
$581$	1.13198e11	0.0412142
$582$	0	0
$583$	6.19200e12	2.21984
$584$	−4.46165e12	−1.58722
$585$	0	0
$586$	−1.26736e12	−0.443979
$587$	−4.85338e12	−1.68722	−0.843612	−	0.536954i	$-0.819575\pi$
$587$	−4.85338e12	−1.68722	−0.843612	+	0.536954i	$0.819575\pi$
$588$	0	0
$589$	3.37617e12	1.15586
$590$	0	0
$591$	0	0
$592$	1.38521e12	0.463519
$593$	−3.40807e12	−1.13178	−0.565891	−	0.824480i	$-0.691468\pi$
$593$	−3.40807e12	−1.13178	−0.565891	+	0.824480i	$0.691468\pi$
$594$	0	0
$595$	0	0
$596$	7.68728e11	0.249554
$597$	0	0
$598$	−2.29978e10	−0.00735412
$599$	9.36596e11	0.297257	0.148628	−	0.988893i	$-0.452514\pi$
$599$	9.36596e11	0.297257	0.148628	+	0.988893i	$0.452514\pi$
$600$	0	0
$601$	2.48082e12	0.775639	0.387820	−	0.921735i	$-0.373228\pi$
$601$	2.48082e12	0.775639	0.387820	+	0.921735i	$0.373228\pi$
$602$	−1.06785e12	−0.331381
$603$	0	0
$604$	−3.90477e11	−0.119379
$605$	0	0
$606$	0	0
$607$	−5.87176e12	−1.75557	−0.877787	−	0.479052i	$-0.840981\pi$
$607$	−5.87176e12	−1.75557	−0.877787	+	0.479052i	$0.840981\pi$
$608$	1.38702e12	0.411639
$609$	0	0
$610$	0	0
$611$	−9.53015e11	−0.276640
$612$	0	0
$613$	3.44781e11	0.0986215	0.0493108	−	0.998783i	$-0.484298\pi$
$613$	3.44781e11	0.0986215	0.0493108	+	0.998783i	$0.484298\pi$
$614$	4.98734e12	1.41616
$615$	0	0
$616$	3.26460e12	0.913518
$617$	3.81915e12	1.06092	0.530461	−	0.847709i	$-0.322019\pi$
$617$	3.81915e12	1.06092	0.530461	+	0.847709i	$0.322019\pi$
$618$	0	0
$619$	8.91644e11	0.244109	0.122055	−	0.992523i	$-0.461052\pi$
$619$	8.91644e11	0.244109	0.122055	+	0.992523i	$0.461052\pi$
$620$	0	0
$621$	0	0
$622$	−2.20685e12	−0.591176
$623$	−9.93779e11	−0.264298
$624$	0	0
$625$	0	0
$626$	2.55207e12	0.664213
$627$	0	0
$628$	5.70036e11	0.146246
$629$	2.34759e12	0.597990
$630$	0	0
$631$	2.72749e12	0.684907	0.342453	−	0.939535i	$-0.388742\pi$
$631$	2.72749e12	0.684907	0.342453	+	0.939535i	$0.388742\pi$
$632$	−5.57649e12	−1.39038
$633$	0	0
$634$	−1.68918e12	−0.415216
$635$	0	0
$636$	0	0
$637$	9.50483e11	0.228727
$638$	−1.05306e13	−2.51629
$639$	0	0
$640$	0	0
$641$	−5.88474e12	−1.37678	−0.688392	−	0.725339i	$-0.741683\pi$
$641$	−5.88474e12	−1.37678	−0.688392	+	0.725339i	$0.741683\pi$
$642$	0	0
$643$	8.44268e11	0.194774	0.0973870	−	0.995247i	$-0.468952\pi$
$643$	8.44268e11	0.194774	0.0973870	+	0.995247i	$0.468952\pi$
$644$	8.21728e9	0.00188253
$645$	0	0
$646$	−6.52751e12	−1.47469
$647$	−1.54662e12	−0.346988	−0.173494	−	0.984835i	$-0.555506\pi$
$647$	−1.54662e12	−0.346988	−0.173494	+	0.984835i	$0.555506\pi$
$648$	0	0
$649$	−7.78235e12	−1.72191
$650$	0	0
$651$	0	0
$652$	1.03611e12	0.224540
$653$	7.20565e12	1.55083	0.775414	−	0.631453i	$-0.217541\pi$
$653$	7.20565e12	1.55083	0.775414	+	0.631453i	$0.217541\pi$
$654$	0	0
$655$	0	0
$656$	2.43568e12	0.513516
$657$	0	0
$658$	−2.07265e12	−0.431032
$659$	1.96856e12	0.406597	0.203299	−	0.979117i	$-0.434834\pi$
$659$	1.96856e12	0.406597	0.203299	+	0.979117i	$0.434834\pi$
$660$	0	0
$661$	−8.75497e12	−1.78381	−0.891904	−	0.452225i	$-0.850630\pi$
$661$	−8.75497e12	−1.78381	−0.891904	+	0.452225i	$0.850630\pi$
$662$	4.52819e12	0.916356
$663$	0	0
$664$	3.87855e11	0.0774306
$665$	0	0
$666$	0	0
$667$	−2.14351e11	−0.0419332
$668$	−1.03342e12	−0.200809
$669$	0	0
$670$	0	0
$671$	2.97673e12	0.566876
$672$	0	0
$673$	−1.73755e12	−0.326489	−0.163244	−	0.986586i	$-0.552196\pi$
$673$	−1.73755e12	−0.326489	−0.163244	+	0.986586i	$0.552196\pi$
$674$	2.38336e11	0.0444856
$675$	0	0
$676$	6.80265e11	0.125290
$677$	1.87762e12	0.343526	0.171763	−	0.985138i	$-0.445054\pi$
$677$	1.87762e12	0.343526	0.171763	+	0.985138i	$0.445054\pi$
$678$	0	0
$679$	3.82530e12	0.690640
$680$	0	0
$681$	0	0
$682$	6.31743e12	1.11818
$683$	−3.03387e11	−0.0533463	−0.0266731	−	0.999644i	$-0.508491\pi$
$683$	−3.03387e11	−0.0533463	−0.0266731	+	0.999644i	$0.508491\pi$
$684$	0	0
$685$	0	0
$686$	5.09306e12	0.878053
$687$	0	0
$688$	−3.13204e12	−0.532941
$689$	−2.86498e12	−0.484323
$690$	0	0
$691$	7.73855e12	1.29124	0.645622	−	0.763657i	$-0.276598\pi$
$691$	7.73855e12	1.29124	0.645622	+	0.763657i	$0.276598\pi$
$692$	−2.70643e11	−0.0448662
$693$	0	0
$694$	2.59647e12	0.424880
$695$	0	0
$696$	0	0
$697$	4.12789e12	0.662491
$698$	3.69149e12	0.588644
$699$	0	0
$700$	0	0
$701$	1.29003e12	0.201776	0.100888	−	0.994898i	$-0.467832\pi$
$701$	1.29003e12	0.201776	0.100888	+	0.994898i	$0.467832\pi$
$702$	0	0
$703$	5.25981e12	0.812215
$704$	1.09865e13	1.68570
$705$	0	0
$706$	1.08511e13	1.64381
$707$	1.89079e12	0.284614
$708$	0	0
$709$	1.80911e12	0.268879	0.134440	−	0.990922i	$-0.457077\pi$
$709$	1.80911e12	0.268879	0.134440	+	0.990922i	$0.457077\pi$
$710$	0	0
$711$	0	0
$712$	−3.40502e12	−0.496546
$713$	1.28591e11	0.0186341
$714$	0	0
$715$	0	0
$716$	1.60872e11	0.0228756
$717$	0	0
$718$	−6.00664e12	−0.843473
$719$	8.24498e11	0.115056	0.0575280	−	0.998344i	$-0.481678\pi$
$719$	8.24498e11	0.115056	0.0575280	+	0.998344i	$0.481678\pi$
$720$	0	0
$721$	5.18556e12	0.714640
$722$	−7.85816e12	−1.07623
$723$	0	0
$724$	1.50931e12	0.204153
$725$	0	0
$726$	0	0
$727$	−4.26121e12	−0.565754	−0.282877	−	0.959156i	$-0.591289\pi$
$727$	−4.26121e12	−0.565754	−0.282877	+	0.959156i	$0.591289\pi$
$728$	−1.51050e12	−0.199310
$729$	0	0
$730$	0	0
$731$	−5.30804e12	−0.687552
$732$	0	0
$733$	−1.17006e13	−1.49707	−0.748535	−	0.663095i	$-0.769242\pi$
$733$	−1.17006e13	−1.49707	−0.748535	+	0.663095i	$0.769242\pi$
$734$	2.52674e12	0.321313
$735$	0	0
$736$	5.28287e10	0.00663620
$737$	1.47806e13	1.84539
$738$	0	0
$739$	1.30351e13	1.60773	0.803866	−	0.594811i	$-0.202773\pi$
$739$	1.30351e13	1.60773	0.803866	+	0.594811i	$0.202773\pi$
$740$	0	0
$741$	0	0
$742$	−6.23086e12	−0.754624
$743$	8.72982e12	1.05089	0.525443	−	0.850829i	$-0.323900\pi$
$743$	8.72982e12	1.05089	0.525443	+	0.850829i	$0.323900\pi$
$744$	0	0
$745$	0	0
$746$	8.50954e12	1.00596
$747$	0	0
$748$	2.00669e12	0.234381
$749$	−4.23770e11	−0.0491997
$750$	0	0
$751$	−1.03083e13	−1.18252	−0.591259	−	0.806482i	$-0.701369\pi$
$751$	−1.03083e13	−1.18252	−0.591259	+	0.806482i	$0.701369\pi$
$752$	−6.07914e12	−0.693206
$753$	0	0
$754$	4.87242e12	0.549001
$755$	0	0
$756$	0	0
$757$	5.42609e12	0.600559	0.300280	−	0.953851i	$-0.402920\pi$
$757$	5.42609e12	0.600559	0.300280	+	0.953851i	$0.402920\pi$
$758$	4.45101e12	0.489719
$759$	0	0
$760$	0	0
$761$	−9.12828e12	−0.986639	−0.493319	−	0.869848i	$-0.664217\pi$
$761$	−9.12828e12	−0.986639	−0.493319	+	0.869848i	$0.664217\pi$
$762$	0	0
$763$	5.27280e12	0.563223
$764$	7.97473e10	0.00846829
$765$	0	0
$766$	1.04209e12	0.109364
$767$	3.60082e12	0.375684
$768$	0	0
$769$	−5.29627e12	−0.546137	−0.273069	−	0.961995i	$-0.588039\pi$
$769$	−5.29627e12	−0.546137	−0.273069	+	0.961995i	$0.588039\pi$
$770$	0	0
$771$	0	0
$772$	1.99471e12	0.202116
$773$	1.45554e13	1.46628	0.733138	−	0.680080i	$-0.238055\pi$
$773$	1.45554e13	1.46628	0.733138	+	0.680080i	$0.238055\pi$
$774$	0	0
$775$	0	0
$776$	1.31068e13	1.29753
$777$	0	0
$778$	5.75575e12	0.563240
$779$	9.24860e12	0.899823
$780$	0	0
$781$	−3.37546e12	−0.324641
$782$	−2.48619e11	−0.0237741
$783$	0	0
$784$	6.06299e12	0.573145
$785$	0	0
$786$	0	0
$787$	1.36878e13	1.27188	0.635941	−	0.771737i	$-0.280612\pi$
$787$	1.36878e13	1.27188	0.635941	+	0.771737i	$0.280612\pi$
$788$	2.20932e12	0.204123
$789$	0	0
$790$	0	0
$791$	−6.20930e12	−0.563960
$792$	0	0
$793$	−1.37730e12	−0.123680
$794$	−4.55277e11	−0.0406522
$795$	0	0
$796$	−5.72770e11	−0.0505675
$797$	1.44438e13	1.26800	0.633999	−	0.773334i	$-0.281413\pi$
$797$	1.44438e13	1.26800	0.633999	+	0.773334i	$0.281413\pi$
$798$	0	0
$799$	−1.03026e13	−0.894311
$800$	0	0
$801$	0	0
$802$	4.53588e12	0.387147
$803$	−2.71364e13	−2.30321
$804$	0	0
$805$	0	0
$806$	−2.92302e12	−0.243963
$807$	0	0
$808$	6.47849e12	0.534715
$809$	2.28499e13	1.87550	0.937749	−	0.347314i	$-0.112906\pi$
$809$	2.28499e13	1.87550	0.937749	+	0.347314i	$0.112906\pi$
$810$	0	0
$811$	−2.16202e13	−1.75495	−0.877477	−	0.479618i	$-0.840775\pi$
$811$	−2.16202e13	−1.75495	−0.877477	+	0.479618i	$0.840775\pi$
$812$	−1.74095e12	−0.140535
$813$	0	0
$814$	9.84206e12	0.785735
$815$	0	0
$816$	0	0
$817$	−1.18927e13	−0.933863
$818$	4.84238e12	0.378154
$819$	0	0
$820$	0	0
$821$	9.90438e12	0.760822	0.380411	−	0.924817i	$-0.375782\pi$
$821$	9.90438e12	0.760822	0.380411	+	0.924817i	$0.375782\pi$
$822$	0	0
$823$	1.33090e13	1.01122	0.505610	−	0.862762i	$-0.331267\pi$
$823$	1.33090e13	1.01122	0.505610	+	0.862762i	$0.331267\pi$
$824$	1.77675e13	1.34262
$825$	0	0
$826$	7.83119e12	0.585353
$827$	−9.27779e12	−0.689715	−0.344858	−	0.938655i	$-0.612073\pi$
$827$	−9.27779e12	−0.689715	−0.344858	+	0.938655i	$0.612073\pi$
$828$	0	0
$829$	−1.67554e13	−1.23214	−0.616071	−	0.787691i	$-0.711276\pi$
$829$	−1.67554e13	−1.23214	−0.616071	+	0.787691i	$0.711276\pi$
$830$	0	0
$831$	0	0
$832$	−5.08334e12	−0.367785
$833$	1.02753e13	0.739419
$834$	0	0
$835$	0	0
$836$	4.49602e12	0.318346
$837$	0	0
$838$	5.01076e12	0.350998
$839$	−1.08548e13	−0.756295	−0.378147	−	0.925745i	$-0.623439\pi$
$839$	−1.08548e13	−0.756295	−0.378147	+	0.925745i	$0.623439\pi$
$840$	0	0
$841$	3.09062e13	2.13041
$842$	−8.58902e12	−0.588897
$843$	0	0
$844$	2.87119e11	0.0194769
$845$	0	0
$846$	0	0
$847$	1.14243e13	0.762699
$848$	−1.82753e13	−1.21362
$849$	0	0
$850$	0	0
$851$	2.00335e11	0.0130940
$852$	0	0
$853$	1.55815e12	0.100772	0.0503858	−	0.998730i	$-0.483955\pi$
$853$	1.55815e12	0.100772	0.0503858	+	0.998730i	$0.483955\pi$
$854$	−2.99541e12	−0.192706
$855$	0	0
$856$	−1.45198e12	−0.0924334
$857$	−1.18714e13	−0.751778	−0.375889	−	0.926665i	$-0.622663\pi$
$857$	−1.18714e13	−0.751778	−0.375889	+	0.926665i	$0.622663\pi$
$858$	0	0
$859$	8.41923e12	0.527598	0.263799	−	0.964578i	$-0.415024\pi$
$859$	8.41923e12	0.527598	0.263799	+	0.964578i	$0.415024\pi$
$860$	0	0
$861$	0	0
$862$	1.11117e13	0.685486
$863$	−1.09630e13	−0.672790	−0.336395	−	0.941721i	$-0.609208\pi$
$863$	−1.09630e13	−0.672790	−0.336395	+	0.941721i	$0.609208\pi$
$864$	0	0
$865$	0	0
$866$	−3.58585e12	−0.216651
$867$	0	0
$868$	1.04442e12	0.0624504
$869$	−3.39170e13	−2.01757
$870$	0	0
$871$	−6.83886e12	−0.402626
$872$	1.80664e13	1.05815
$873$	0	0
$874$	−5.57036e11	−0.0322910
$875$	0	0
$876$	0	0
$877$	1.49209e12	0.0851723	0.0425861	−	0.999093i	$-0.486440\pi$
$877$	1.49209e12	0.0851723	0.0425861	+	0.999093i	$0.486440\pi$
$878$	4.69005e12	0.266350
$879$	0	0
$880$	0	0
$881$	−5.94363e12	−0.332399	−0.166200	−	0.986092i	$-0.553150\pi$
$881$	−5.94363e12	−0.332399	−0.166200	+	0.986092i	$0.553150\pi$
$882$	0	0
$883$	−1.57021e13	−0.869230	−0.434615	−	0.900616i	$-0.643116\pi$
$883$	−1.57021e13	−0.869230	−0.434615	+	0.900616i	$0.643116\pi$
$884$	−9.28476e11	−0.0511370
$885$	0	0
$886$	1.74635e13	0.952092
$887$	−2.07953e13	−1.12800	−0.564001	−	0.825774i	$-0.690738\pi$
$887$	−2.07953e13	−1.12800	−0.564001	+	0.825774i	$0.690738\pi$
$888$	0	0
$889$	7.43495e12	0.399227
$890$	0	0
$891$	0	0
$892$	−3.54503e12	−0.187490
$893$	−2.30833e13	−1.21469
$894$	0	0
$895$	0	0
$896$	−8.01469e12	−0.415433
$897$	0	0
$898$	−1.18029e13	−0.605685
$899$	−2.72440e13	−1.39108
$900$	0	0
$901$	−3.09721e13	−1.56570
$902$	1.73058e13	0.870487
$903$	0	0
$904$	−2.12751e13	−1.05953
$905$	0	0
$906$	0	0
$907$	5.16199e12	0.253270	0.126635	−	0.991949i	$-0.459582\pi$
$907$	5.16199e12	0.253270	0.126635	+	0.991949i	$0.459582\pi$
$908$	−3.59787e12	−0.175655
$909$	0	0
$910$	0	0
$911$	1.23036e13	0.591834	0.295917	−	0.955214i	$-0.404375\pi$
$911$	1.23036e13	0.591834	0.295917	+	0.955214i	$0.404375\pi$
$912$	0	0
$913$	2.35899e12	0.112359
$914$	−1.55979e13	−0.739280
$915$	0	0
$916$	2.77815e12	0.130384
$917$	−2.32333e13	−1.08505
$918$	0	0
$919$	−6.62689e12	−0.306471	−0.153236	−	0.988190i	$-0.548969\pi$
$919$	−6.62689e12	−0.306471	−0.153236	+	0.988190i	$0.548969\pi$
$920$	0	0
$921$	0	0
$922$	−2.07618e13	−0.946185
$923$	1.56179e12	0.0708299
$924$	0	0
$925$	0	0
$926$	−1.55255e13	−0.693900
$927$	0	0
$928$	−1.11925e13	−0.495408
$929$	1.50849e12	0.0664466	0.0332233	−	0.999448i	$-0.489423\pi$
$929$	1.50849e12	0.0664466	0.0332233	+	0.999448i	$0.489423\pi$
$930$	0	0
$931$	2.30219e13	1.00431
$932$	−3.94157e11	−0.0171119
$933$	0	0
$934$	−3.90179e12	−0.167765
$935$	0	0
$936$	0	0
$937$	−2.83761e13	−1.20261	−0.601305	−	0.799019i	$-0.705352\pi$
$937$	−2.83761e13	−1.20261	−0.601305	+	0.799019i	$0.705352\pi$
$938$	−1.48734e13	−0.627332
$939$	0	0
$940$	0	0
$941$	−2.17067e13	−0.902488	−0.451244	−	0.892401i	$-0.649020\pi$
$941$	−2.17067e13	−0.902488	−0.451244	+	0.892401i	$0.649020\pi$
$942$	0	0
$943$	3.52260e11	0.0145064
$944$	2.29691e13	0.941391
$945$	0	0
$946$	−2.22535e13	−0.903417
$947$	1.50628e13	0.608600	0.304300	−	0.952576i	$-0.401578\pi$
$947$	1.50628e13	0.608600	0.304300	+	0.952576i	$0.401578\pi$
$948$	0	0
$949$	1.25558e13	0.502511
$950$	0	0
$951$	0	0
$952$	−1.63294e13	−0.644323
$953$	3.10734e13	1.22031	0.610156	−	0.792281i	$-0.291107\pi$
$953$	3.10734e13	1.22031	0.610156	+	0.792281i	$0.291107\pi$
$954$	0	0
$955$	0	0
$956$	−4.23616e12	−0.164026
$957$	0	0
$958$	1.21057e13	0.464350
$959$	8.78857e12	0.335532
$960$	0	0
$961$	−1.00957e13	−0.381839
$962$	−4.55383e12	−0.171431
$963$	0	0
$964$	1.65235e12	0.0616249
$965$	0	0
$966$	0	0
$967$	−1.56729e13	−0.576408	−0.288204	−	0.957569i	$-0.593058\pi$
$967$	−1.56729e13	−0.576408	−0.288204	+	0.957569i	$0.593058\pi$
$968$	3.91434e13	1.43291
$969$	0	0
$970$	0	0
$971$	−2.46434e13	−0.889641	−0.444820	−	0.895620i	$-0.646733\pi$
$971$	−2.46434e13	−0.889641	−0.444820	+	0.895620i	$0.646733\pi$
$972$	0	0
$973$	−1.97823e13	−0.707568
$974$	−4.89727e13	−1.74357
$975$	0	0
$976$	−8.78562e12	−0.309919
$977$	1.86706e13	0.655591	0.327796	−	0.944749i	$-0.393694\pi$
$977$	1.86706e13	0.655591	0.327796	+	0.944749i	$0.393694\pi$
$978$	0	0
$979$	−2.07098e13	−0.720534
$980$	0	0
$981$	0	0
$982$	1.90219e13	0.652759
$983$	−2.04996e13	−0.700252	−0.350126	−	0.936703i	$-0.613861\pi$
$983$	−2.04996e13	−0.700252	−0.350126	+	0.936703i	$0.613861\pi$
$984$	0	0
$985$	0	0
$986$	5.26737e13	1.77479
$987$	0	0
$988$	−2.08027e12	−0.0694565
$989$	−4.52970e11	−0.0150552
$990$	0	0
$991$	−6.90837e12	−0.227533	−0.113766	−	0.993508i	$-0.536292\pi$
$991$	−6.90837e12	−0.227533	−0.113766	+	0.993508i	$0.536292\pi$
$992$	6.71453e12	0.220147
$993$	0	0
$994$	3.39665e12	0.110360
$995$	0	0
$996$	0	0
$997$	−2.80773e13	−0.899969	−0.449984	−	0.893036i	$-0.648570\pi$
$997$	−2.80773e13	−0.899969	−0.449984	+	0.893036i	$0.648570\pi$
$998$	2.38433e13	0.760815
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.10.a.q.1.2		4
3.2	odd	2		75.10.a.l.1.3		4
5.2	odd	4		45.10.b.c.19.3		8
5.3	odd	4		45.10.b.c.19.6		8
5.4	even	2		225.10.a.u.1.3		4
15.2	even	4		15.10.b.a.4.6	yes	8
15.8	even	4		15.10.b.a.4.3	✓	8
15.14	odd	2		75.10.a.i.1.2		4
60.23	odd	4		240.10.f.c.49.2		8
60.47	odd	4		240.10.f.c.49.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.10.b.a.4.3	✓	8	15.8	even	4
15.10.b.a.4.6	yes	8	15.2	even	4
45.10.b.c.19.3		8	5.2	odd	4
45.10.b.c.19.6		8	5.3	odd	4
75.10.a.i.1.2		4	15.14	odd	2
75.10.a.l.1.3		4	3.2	odd	2
225.10.a.q.1.2		4	1.1	even	1	trivial
225.10.a.u.1.3		4	5.4	even	2
240.10.f.c.49.2		8	60.23	odd	4
240.10.f.c.49.6		8	60.47	odd	4

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)

\(f(q)\)	\(=\)	\(q-20.9703 q^{2} -72.2477 q^{4} +3575.78 q^{7} +12251.8 q^{8} +O(q^{10})\)	\(q-20.9703 q^{2} -72.2477 q^{4} +3575.78 q^{7} +12251.8 q^{8} +74517.5 q^{11} -34478.5 q^{13} -74985.1 q^{14} -219933. q^{16} -372733. q^{17} -835114. q^{19} -1.56265e6 q^{22} -31807.7 q^{23} +723024. q^{26} -258342. q^{28} +6.73894e6 q^{29} -4.04277e6 q^{31} -1.66087e6 q^{32} +7.81631e6 q^{34} -6.29831e6 q^{37} +1.75126e7 q^{38} -1.10746e7 q^{41} +1.42409e7 q^{43} -5.38372e6 q^{44} +667017. q^{46} +2.76408e7 q^{47} -2.75674e7 q^{49} +2.49100e6 q^{52} +8.30946e7 q^{53} +4.38099e7 q^{56} -1.41317e8 q^{58} -1.04437e8 q^{59} +3.99467e7 q^{61} +8.47779e7 q^{62} +1.47435e8 q^{64} +1.98351e8 q^{67} +2.69291e7 q^{68} -4.52976e7 q^{71} -3.64162e8 q^{73} +1.32077e8 q^{74} +6.03351e7 q^{76} +2.66458e8 q^{77} -4.55156e8 q^{79} +2.32238e8 q^{82} +3.16569e7 q^{83} -2.98635e8 q^{86} +9.12975e8 q^{88} -2.77919e8 q^{89} -1.23288e8 q^{91} +2.29804e6 q^{92} -5.79636e8 q^{94} +1.06978e9 q^{97} +5.78096e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{2} + 597 q^{4} - 9834 q^{7} + 7671 q^{8}+O(q^{10}) \) 4 * q - 3 * q^2 + 597 * q^4 - 9834 * q^7 + 7671 * q^8	\( 4 q - 3 q^{2} + 597 q^{4} - 9834 q^{7} + 7671 q^{8} + 35994 q^{11} - 79998 q^{13} + 208182 q^{14} - 752815 q^{16} + 667878 q^{17} - 425792 q^{19} + 143934 q^{22} + 1031232 q^{23} + 438762 q^{26} - 3265578 q^{28} - 36786 q^{29} + 237044 q^{31} + 2932239 q^{32} + 4062194 q^{34} - 15750594 q^{37} + 48524592 q^{38} - 46660044 q^{41} - 67170720 q^{43} - 37277946 q^{44} - 4832036 q^{46} + 48243420 q^{47} - 25664800 q^{49} - 114924078 q^{52} + 198376482 q^{53} + 33890610 q^{56} - 439780704 q^{58} + 118263018 q^{59} - 178713880 q^{61} + 93716088 q^{62} + 6068513 q^{64} - 16141548 q^{67} + 611464794 q^{68} + 78445332 q^{71} - 514053252 q^{73} - 761690898 q^{74} + 549348672 q^{76} + 780875028 q^{77} - 431961140 q^{79} - 241116378 q^{82} + 557494176 q^{83} - 498821196 q^{86} + 392593074 q^{88} + 178691112 q^{89} + 107377164 q^{91} - 1048179156 q^{92} + 360839912 q^{94} + 840904752 q^{97} - 3438128823 q^{98}+O(q^{100}) \) 4 * q - 3 * q^2 + 597 * q^4 - 9834 * q^7 + 7671 * q^8 + 35994 * q^11 - 79998 * q^13 + 208182 * q^14 - 752815 * q^16 + 667878 * q^17 - 425792 * q^19 + 143934 * q^22 + 1031232 * q^23 + 438762 * q^26 - 3265578 * q^28 - 36786 * q^29 + 237044 * q^31 + 2932239 * q^32 + 4062194 * q^34 - 15750594 * q^37 + 48524592 * q^38 - 46660044 * q^41 - 67170720 * q^43 - 37277946 * q^44 - 4832036 * q^46 + 48243420 * q^47 - 25664800 * q^49 - 114924078 * q^52 + 198376482 * q^53 + 33890610 * q^56 - 439780704 * q^58 + 118263018 * q^59 - 178713880 * q^61 + 93716088 * q^62 + 6068513 * q^64 - 16141548 * q^67 + 611464794 * q^68 + 78445332 * q^71 - 514053252 * q^73 - 761690898 * q^74 + 549348672 * q^76 + 780875028 * q^77 - 431961140 * q^79 - 241116378 * q^82 + 557494176 * q^83 - 498821196 * q^86 + 392593074 * q^88 + 178691112 * q^89 + 107377164 * q^91 - 1048179156 * q^92 + 360839912 * q^94 + 840904752 * q^97 - 3438128823 * q^98

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