LMFDB - Embedded newform 945.2.a.n.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [945,2,Mod(1,945)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(945, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("945.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 945 = 3^{3} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	945.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:	$7.54586299101$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.144344.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + 5x + 9 $ x^4 - x^3 - 8x^2 + 5x + 9
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.80834$ of defining polynomial
Character	$\chi$	$=$	945.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.80834 q^{2} +5.88678 q^{4} +1.00000 q^{5} +1.00000 q^{7} +10.9154 q^{8} +O(q^{10})$	\(q+2.80834 q^{2} +5.88678 q^{4} +1.00000 q^{5} +1.00000 q^{7} +10.9154 q^{8} +2.80834 q^{10} -2.07844 q^{11} -5.69512 q^{13} +2.80834 q^{14} +18.8806 q^{16} -3.22029 q^{17} -3.29872 q^{19} +5.88678 q^{20} -5.83697 q^{22} -3.22029 q^{23} +1.00000 q^{25} -15.9938 q^{26} +5.88678 q^{28} -9.77356 q^{29} -0.396397 q^{31} +31.1924 q^{32} -9.04366 q^{34} +1.00000 q^{35} +7.61668 q^{37} -9.26395 q^{38} +10.9154 q^{40} +9.99385 q^{41} +2.60360 q^{43} -12.2353 q^{44} -9.04366 q^{46} +4.76048 q^{47} +1.00000 q^{49} +2.80834 q^{50} -33.5259 q^{52} +1.14185 q^{53} -2.07844 q^{55} +10.9154 q^{56} -27.4475 q^{58} -5.61668 q^{59} -1.22029 q^{61} -1.11322 q^{62} +49.8377 q^{64} -5.69512 q^{65} -1.55328 q^{67} -18.9571 q^{68} +2.80834 q^{70} +2.31376 q^{71} +3.69512 q^{73} +21.3902 q^{74} -19.4189 q^{76} -2.07844 q^{77} +10.0131 q^{79} +18.8806 q^{80} +28.0661 q^{82} -6.39024 q^{83} -3.22029 q^{85} +7.31180 q^{86} -22.6870 q^{88} -6.85620 q^{89} -5.69512 q^{91} -18.9571 q^{92} +13.3691 q^{94} -3.29872 q^{95} +14.0573 q^{97} +2.80834 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} + 9 q^{4} + 4 q^{5} + 4 q^{7} + 6 q^{8}+O(q^{10}) $ 4 * q + q^2 + 9 * q^4 + 4 * q^5 + 4 * q^7 + 6 * q^8	$ 4 q + q^{2} + 9 q^{4} + 4 q^{5} + 4 q^{7} + 6 q^{8} + q^{10} - 4 q^{11} + 2 q^{13} + q^{14} + 19 q^{16} + 4 q^{19} + 9 q^{20} + 10 q^{22} + 4 q^{25} - 22 q^{26} + 9 q^{28} - 10 q^{29} + 6 q^{31} + 23 q^{32} - 13 q^{34} + 4 q^{35} + 10 q^{37} - q^{38} + 6 q^{40} - 2 q^{41} + 18 q^{43} - 36 q^{44} - 13 q^{46} + 18 q^{47} + 4 q^{49} + q^{50} - 34 q^{52} - 4 q^{53} - 4 q^{55} + 6 q^{56} - 14 q^{58} - 2 q^{59} + 8 q^{61} - 19 q^{62} + 54 q^{64} + 2 q^{65} + 10 q^{67} + 13 q^{68} + q^{70} - 8 q^{71} - 10 q^{73} + 36 q^{74} - 5 q^{76} - 4 q^{77} + 12 q^{79} + 19 q^{80} + 24 q^{82} + 24 q^{83} - 16 q^{86} + 4 q^{88} - 8 q^{89} + 2 q^{91} + 13 q^{92} - 38 q^{94} + 4 q^{95} + 10 q^{97} + q^{98}+O(q^{100}) $ 4 * q + q^2 + 9 * q^4 + 4 * q^5 + 4 * q^7 + 6 * q^8 + q^10 - 4 * q^11 + 2 * q^13 + q^14 + 19 * q^16 + 4 * q^19 + 9 * q^20 + 10 * q^22 + 4 * q^25 - 22 * q^26 + 9 * q^28 - 10 * q^29 + 6 * q^31 + 23 * q^32 - 13 * q^34 + 4 * q^35 + 10 * q^37 - q^38 + 6 * q^40 - 2 * q^41 + 18 * q^43 - 36 * q^44 - 13 * q^46 + 18 * q^47 + 4 * q^49 + q^50 - 34 * q^52 - 4 * q^53 - 4 * q^55 + 6 * q^56 - 14 * q^58 - 2 * q^59 + 8 * q^61 - 19 * q^62 + 54 * q^64 + 2 * q^65 + 10 * q^67 + 13 * q^68 + q^70 - 8 * q^71 - 10 * q^73 + 36 * q^74 - 5 * q^76 - 4 * q^77 + 12 * q^79 + 19 * q^80 + 24 * q^82 + 24 * q^83 - 16 * q^86 + 4 * q^88 - 8 * q^89 + 2 * q^91 + 13 * q^92 - 38 * q^94 + 4 * q^95 + 10 * q^97 + q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.80834	1.98580	0.992899	−	0.118964i	$-0.0379572\pi$
$2$	2.80834	1.98580	0.992899	+	0.118964i	$0.0379572\pi$
$3$	0	0
$3$	0	0
$4$	5.88678	2.94339
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	10.9154	3.85918
$9$	0	0
$10$	2.80834	0.888076
$11$	−2.07844	−0.626673	−0.313337	−	0.949642i	$-0.601447\pi$
$11$	−2.07844	−0.626673	−0.313337	+	0.949642i	$0.601447\pi$
$12$	0	0
$13$	−5.69512	−1.57954	−0.789771	−	0.613401i	$-0.789801\pi$
$13$	−5.69512	−1.57954	−0.789771	+	0.613401i	$0.789801\pi$
$14$	2.80834	0.750561
$15$	0	0
$16$	18.8806	4.72016
$17$	−3.22029	−0.781034	−0.390517	−	0.920596i	$-0.627704\pi$
$17$	−3.22029	−0.781034	−0.390517	+	0.920596i	$0.627704\pi$
$18$	0	0
$19$	−3.29872	−0.756779	−0.378390	−	0.925646i	$-0.623522\pi$
$19$	−3.29872	−0.756779	−0.378390	+	0.925646i	$0.623522\pi$
$20$	5.88678	1.31632
$21$	0	0
$22$	−5.83697	−1.24445
$23$	−3.22029	−0.671476	−0.335738	−	0.941955i	$-0.608986\pi$
$23$	−3.22029	−0.671476	−0.335738	+	0.941955i	$0.608986\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−15.9938	−3.13665
$27$	0	0
$28$	5.88678	1.11250
$29$	−9.77356	−1.81490	−0.907452	−	0.420155i	$-0.861976\pi$
$29$	−9.77356	−1.81490	−0.907452	+	0.420155i	$0.861976\pi$
$30$	0	0
$31$	−0.396397	−0.0711951	−0.0355975	−	0.999366i	$-0.511333\pi$
$31$	−0.396397	−0.0711951	−0.0355975	+	0.999366i	$0.511333\pi$
$32$	31.1924	5.51410
$33$	0	0
$34$	−9.04366	−1.55097
$35$	1.00000	0.169031
$36$	0	0
$37$	7.61668	1.25217	0.626087	−	0.779753i	$-0.284655\pi$
$37$	7.61668	1.25217	0.626087	+	0.779753i	$0.284655\pi$
$38$	−9.26395	−1.50281
$39$	0	0
$40$	10.9154	1.72588
$41$	9.99385	1.56078	0.780388	−	0.625295i	$-0.215021\pi$
$41$	9.99385	1.56078	0.780388	+	0.625295i	$0.215021\pi$
$42$	0	0
$43$	2.60360	0.397046	0.198523	−	0.980096i	$-0.436386\pi$
$43$	2.60360	0.397046	0.198523	+	0.980096i	$0.436386\pi$
$44$	−12.2353	−1.84454
$45$	0	0
$46$	−9.04366	−1.33341
$47$	4.76048	0.694388	0.347194	−	0.937793i	$-0.387135\pi$
$47$	4.76048	0.694388	0.347194	+	0.937793i	$0.387135\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	2.80834	0.397159
$51$	0	0
$52$	−33.5259	−4.64921
$53$	1.14185	0.156845	0.0784223	−	0.996920i	$-0.475012\pi$
$53$	1.14185	0.156845	0.0784223	+	0.996920i	$0.475012\pi$
$54$	0	0
$55$	−2.07844	−0.280257
$56$	10.9154	1.45863
$57$	0	0
$58$	−27.4475	−3.60403
$59$	−5.61668	−0.731230	−0.365615	−	0.930766i	$-0.619141\pi$
$59$	−5.61668	−0.731230	−0.365615	+	0.930766i	$0.619141\pi$
$60$	0	0
$61$	−1.22029	−0.156242	−0.0781208	−	0.996944i	$-0.524892\pi$
$61$	−1.22029	−0.156242	−0.0781208	+	0.996944i	$0.524892\pi$
$62$	−1.11322	−0.141379
$63$	0	0
$64$	49.8377	6.22972
$65$	−5.69512	−0.706393
$66$	0	0
$67$	−1.55328	−0.189763	−0.0948815	−	0.995489i	$-0.530247\pi$
$67$	−1.55328	−0.189763	−0.0948815	+	0.995489i	$0.530247\pi$
$68$	−18.9571	−2.29889
$69$	0	0
$70$	2.80834	0.335661
$71$	2.31376	0.274593	0.137296	−	0.990530i	$-0.456159\pi$
$71$	2.31376	0.274593	0.137296	+	0.990530i	$0.456159\pi$
$72$	0	0
$73$	3.69512	0.432481	0.216241	−	0.976340i	$-0.430620\pi$
$73$	3.69512	0.432481	0.216241	+	0.976340i	$0.430620\pi$
$74$	21.3902	2.48657
$75$	0	0
$76$	−19.4189	−2.22750
$77$	−2.07844	−0.236860
$78$	0	0
$79$	10.0131	1.12656	0.563280	−	0.826266i	$-0.309539\pi$
$79$	10.0131	1.12656	0.563280	+	0.826266i	$0.309539\pi$
$80$	18.8806	2.11092
$81$	0	0
$82$	28.0661	3.09939
$83$	−6.39024	−0.701420	−0.350710	−	0.936484i	$-0.614060\pi$
$83$	−6.39024	−0.701420	−0.350710	+	0.936484i	$0.614060\pi$
$84$	0	0
$85$	−3.22029	−0.349289
$86$	7.31180	0.788452
$87$	0	0
$88$	−22.6870	−2.41844
$89$	−6.85620	−0.726756	−0.363378	−	0.931642i	$-0.618377\pi$
$89$	−6.85620	−0.726756	−0.363378	+	0.931642i	$0.618377\pi$
$90$	0	0
$91$	−5.69512	−0.597011
$92$	−18.9571	−1.97642
$93$	0	0
$94$	13.3691	1.37891
$95$	−3.29872	−0.338442
$96$	0	0
$97$	14.0573	1.42730	0.713649	−	0.700504i	$-0.247041\pi$
$97$	14.0573	1.42730	0.713649	+	0.700504i	$0.247041\pi$
$98$	2.80834	0.283685
$99$	0	0
$100$	5.88678	0.588678
$101$	−6.22029	−0.618942	−0.309471	−	0.950909i	$-0.600152\pi$
$101$	−6.22029	−0.618942	−0.309471	+	0.950909i	$0.600152\pi$
$102$	0	0
$103$	11.1376	1.09742	0.548712	−	0.836011i	$-0.315118\pi$
$103$	11.1376	1.09742	0.548712	+	0.836011i	$0.315118\pi$
$104$	−62.1646	−6.09574
$105$	0	0
$106$	3.20669	0.311461
$107$	4.15688	0.401861	0.200930	−	0.979606i	$-0.435604\pi$
$107$	4.15688	0.401861	0.200930	+	0.979606i	$0.435604\pi$
$108$	0	0
$109$	12.0765	1.15672	0.578359	−	0.815783i	$-0.303693\pi$
$109$	12.0765	1.15672	0.578359	+	0.815783i	$0.303693\pi$
$110$	−5.83697	−0.556533
$111$	0	0
$112$	18.8806	1.78405
$113$	−18.9285	−1.78064	−0.890321	−	0.455333i	$-0.849520\pi$
$113$	−18.9285	−1.78064	−0.890321	+	0.455333i	$0.849520\pi$
$114$	0	0
$115$	−3.22029	−0.300293
$116$	−57.5348	−5.34197
$117$	0	0
$118$	−15.7736	−1.45207
$119$	−3.22029	−0.295203
$120$	0	0
$121$	−6.68009	−0.607281
$122$	−3.42698	−0.310264
$123$	0	0
$124$	−2.33350	−0.209555
$125$	1.00000	0.0894427
$126$	0	0
$127$	−8.81774	−0.782447	−0.391224	−	0.920296i	$-0.627948\pi$
$127$	−8.81774	−0.782447	−0.391224	+	0.920296i	$0.627948\pi$
$128$	77.5765	6.85686
$129$	0	0
$130$	−15.9938	−1.40275
$131$	−10.4095	−0.909480	−0.454740	−	0.890624i	$-0.650268\pi$
$131$	−10.4095	−0.909480	−0.454740	+	0.890624i	$0.650268\pi$
$132$	0	0
$133$	−3.29872	−0.286036
$134$	−4.36213	−0.376831
$135$	0	0
$136$	−35.1507	−3.01415
$137$	−1.14185	−0.0975545	−0.0487772	−	0.998810i	$-0.515532\pi$
$137$	−1.14185	−0.0975545	−0.0487772	+	0.998810i	$0.515532\pi$
$138$	0	0
$139$	−4.44057	−0.376644	−0.188322	−	0.982107i	$-0.560305\pi$
$139$	−4.44057	−0.376644	−0.188322	+	0.982107i	$0.560305\pi$
$140$	5.88678	0.497524
$141$	0	0
$142$	6.49782	0.545285
$143$	11.8370	0.989857
$144$	0	0
$145$	−9.77356	−0.811650
$146$	10.3772	0.858820
$147$	0	0
$148$	44.8377	3.68564
$149$	−1.40255	−0.114901	−0.0574507	−	0.998348i	$-0.518297\pi$
$149$	−1.40255	−0.114901	−0.0574507	+	0.998348i	$0.518297\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	−36.0069	−2.92055
$153$	0	0
$154$	−5.83697	−0.470356
$155$	−0.396397	−0.0318394
$156$	0	0
$157$	3.93044	0.313683	0.156842	−	0.987624i	$-0.449869\pi$
$157$	3.93044	0.313683	0.156842	+	0.987624i	$0.449869\pi$
$158$	28.1201	2.23712
$159$	0	0
$160$	31.1924	2.46598
$161$	−3.22029	−0.253794
$162$	0	0
$163$	20.5279	1.60787	0.803934	−	0.594718i	$-0.202736\pi$
$163$	20.5279	1.60787	0.803934	+	0.594718i	$0.202736\pi$
$164$	58.8316	4.59398
$165$	0	0
$166$	−17.9460	−1.39288
$167$	1.62976	0.126115	0.0630574	−	0.998010i	$-0.479915\pi$
$167$	1.62976	0.126115	0.0630574	+	0.998010i	$0.479915\pi$
$168$	0	0
$169$	19.4344	1.49496
$170$	−9.04366	−0.693617
$171$	0	0
$172$	15.3268	1.16866
$173$	−12.1700	−0.925265	−0.462632	−	0.886550i	$-0.653095\pi$
$173$	−12.1700	−0.925265	−0.462632	+	0.886550i	$0.653095\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	−39.2422	−2.95800
$177$	0	0
$178$	−19.2546	−1.44319
$179$	8.77161	0.655621	0.327810	−	0.944744i	$-0.393689\pi$
$179$	8.77161	0.655621	0.327810	+	0.944744i	$0.393689\pi$
$180$	0	0
$181$	−19.0896	−1.41892	−0.709458	−	0.704748i	$-0.751060\pi$
$181$	−19.0896	−1.41892	−0.709458	+	0.704748i	$0.751060\pi$
$182$	−15.9938	−1.18554
$183$	0	0
$184$	−35.1507	−2.59135
$185$	7.61668	0.559990
$186$	0	0
$187$	6.69317	0.489453
$188$	28.0239	2.04385
$189$	0	0
$190$	−9.26395	−0.672077
$191$	−6.67589	−0.483050	−0.241525	−	0.970395i	$-0.577648\pi$
$191$	−6.67589	−0.483050	−0.241525	+	0.970395i	$0.577648\pi$
$192$	0	0
$193$	−18.6236	−1.34056	−0.670278	−	0.742110i	$-0.733825\pi$
$193$	−18.6236	−1.34056	−0.670278	+	0.742110i	$0.733825\pi$
$194$	39.4776	2.83432
$195$	0	0
$196$	5.88678	0.420484
$197$	−8.24840	−0.587674	−0.293837	−	0.955856i	$-0.594932\pi$
$197$	−8.24840	−0.587674	−0.293837	+	0.955856i	$0.594932\pi$
$198$	0	0
$199$	25.0854	1.77825	0.889127	−	0.457660i	$-0.151312\pi$
$199$	25.0854	1.77825	0.889127	+	0.457660i	$0.151312\pi$
$200$	10.9154	0.771836
$201$	0	0
$202$	−17.4687	−1.22909
$203$	−9.77356	−0.685970
$204$	0	0
$205$	9.99385	0.698001
$206$	31.2783	2.17926
$207$	0	0
$208$	−107.527	−7.45569
$209$	6.85620	0.474253
$210$	0	0
$211$	−19.5602	−1.34658	−0.673290	−	0.739379i	$-0.735119\pi$
$211$	−19.5602	−1.34658	−0.673290	+	0.739379i	$0.735119\pi$
$212$	6.72180	0.461655
$213$	0	0
$214$	11.6739	0.798014
$215$	2.60360	0.177564
$216$	0	0
$217$	−0.396397	−0.0269092
$218$	33.9149	2.29701
$219$	0	0
$220$	−12.2353	−0.824905
$221$	18.3399	1.23368
$222$	0	0
$223$	14.2526	0.954425	0.477212	−	0.878788i	$-0.341647\pi$
$223$	14.2526	0.954425	0.477212	+	0.878788i	$0.341647\pi$
$224$	31.1924	2.08413
$225$	0	0
$226$	−53.1577	−3.53599
$227$	25.2714	1.67732	0.838660	−	0.544655i	$-0.183339\pi$
$227$	25.2714	1.67732	0.838660	+	0.544655i	$0.183339\pi$
$228$	0	0
$229$	−23.3772	−1.54481	−0.772403	−	0.635132i	$-0.780946\pi$
$229$	−23.3772	−1.54481	−0.772403	+	0.635132i	$0.780946\pi$
$230$	−9.04366	−0.596321
$231$	0	0
$232$	−106.682	−7.00404
$233$	3.09767	0.202935	0.101468	−	0.994839i	$-0.467646\pi$
$233$	3.09767	0.202935	0.101468	+	0.994839i	$0.467646\pi$
$234$	0	0
$235$	4.76048	0.310540
$236$	−33.0642	−2.15229
$237$	0	0
$238$	−9.04366	−0.586213
$239$	−17.2334	−1.11473	−0.557367	−	0.830266i	$-0.688188\pi$
$239$	−17.2334	−1.11473	−0.557367	+	0.830266i	$0.688188\pi$
$240$	0	0
$241$	1.53404	0.0988165	0.0494082	−	0.998779i	$-0.484266\pi$
$241$	1.53404	0.0988165	0.0494082	+	0.998779i	$0.484266\pi$
$242$	−18.7600	−1.20594
$243$	0	0
$244$	−7.18355	−0.459880
$245$	1.00000	0.0638877
$246$	0	0
$247$	18.7866	1.19537
$248$	−4.32684	−0.274755
$249$	0	0
$250$	2.80834	0.177615
$251$	5.17611	0.326713	0.163357	−	0.986567i	$-0.447768\pi$
$251$	5.17611	0.326713	0.163357	+	0.986567i	$0.447768\pi$
$252$	0	0
$253$	6.69317	0.420796
$254$	−24.7632	−1.55378
$255$	0	0
$256$	118.186	7.38662
$257$	−13.8177	−0.861927	−0.430963	−	0.902369i	$-0.641826\pi$
$257$	−13.8177	−0.861927	−0.430963	+	0.902369i	$0.641826\pi$
$258$	0	0
$259$	7.61668	0.473278
$260$	−33.5259	−2.07919
$261$	0	0
$262$	−29.2334	−1.80604
$263$	−8.75433	−0.539815	−0.269908	−	0.962886i	$-0.586993\pi$
$263$	−8.75433	−0.539815	−0.269908	+	0.962886i	$0.586993\pi$
$264$	0	0
$265$	1.14185	0.0701430
$266$	−9.26395	−0.568009
$267$	0	0
$268$	−9.14380	−0.558546
$269$	14.1258	0.861264	0.430632	−	0.902528i	$-0.358291\pi$
$269$	14.1258	0.861264	0.430632	+	0.902528i	$0.358291\pi$
$270$	0	0
$271$	21.2864	1.29306	0.646529	−	0.762889i	$-0.276220\pi$
$271$	21.2864	1.29306	0.646529	+	0.762889i	$0.276220\pi$
$272$	−60.8010	−3.68660
$273$	0	0
$274$	−3.20669	−0.193723
$275$	−2.07844	−0.125335
$276$	0	0
$277$	0.352224	0.0211631	0.0105815	−	0.999944i	$-0.496632\pi$
$277$	0.352224	0.0211631	0.0105815	+	0.999944i	$0.496632\pi$
$278$	−12.4706	−0.747939
$279$	0	0
$280$	10.9154	0.652320
$281$	28.9946	1.72967	0.864837	−	0.502053i	$-0.167422\pi$
$281$	28.9946	1.72967	0.864837	+	0.502053i	$0.167422\pi$
$282$	0	0
$283$	−3.80466	−0.226163	−0.113082	−	0.993586i	$-0.536072\pi$
$283$	−3.80466	−0.226163	−0.113082	+	0.993586i	$0.536072\pi$
$284$	13.6206	0.808233
$285$	0	0
$286$	33.2422	1.96566
$287$	9.99385	0.589918
$288$	0	0
$289$	−6.62976	−0.389986
$290$	−27.4475	−1.61177
$291$	0	0
$292$	21.7524	1.27296
$293$	14.4537	0.844391	0.422196	−	0.906505i	$-0.361260\pi$
$293$	14.4537	0.844391	0.422196	+	0.906505i	$0.361260\pi$
$294$	0	0
$295$	−5.61668	−0.327016
$296$	83.1392	4.83237
$297$	0	0
$298$	−3.93884	−0.228171
$299$	18.3399	1.06062
$300$	0	0
$301$	2.60360	0.150069
$302$	22.4667	1.29282
$303$	0	0
$304$	−62.2820	−3.57212
$305$	−1.22029	−0.0698733
$306$	0	0
$307$	28.3710	1.61922	0.809610	−	0.586969i	$-0.199679\pi$
$307$	28.3710	1.61922	0.809610	+	0.586969i	$0.199679\pi$
$308$	−12.2353	−0.697172
$309$	0	0
$310$	−1.11322	−0.0632266
$311$	−20.3138	−1.15189	−0.575944	−	0.817489i	$-0.695365\pi$
$311$	−20.3138	−1.15189	−0.575944	+	0.817489i	$0.695365\pi$
$312$	0	0
$313$	13.0854	0.739629	0.369814	−	0.929106i	$-0.379421\pi$
$313$	13.0854	0.739629	0.369814	+	0.929106i	$0.379421\pi$
$314$	11.0380	0.622912
$315$	0	0
$316$	58.9448	3.31590
$317$	−9.32488	−0.523738	−0.261869	−	0.965103i	$-0.584339\pi$
$317$	−9.32488	−0.523738	−0.261869	+	0.965103i	$0.584339\pi$
$318$	0	0
$319$	20.3138	1.13735
$320$	49.8377	2.78601
$321$	0	0
$322$	−9.04366	−0.503983
$323$	10.6228	0.591070
$324$	0	0
$325$	−5.69512	−0.315909
$326$	57.6493	3.19290
$327$	0	0
$328$	109.087	6.02332
$329$	4.76048	0.262454
$330$	0	0
$331$	−12.7543	−0.701041	−0.350521	−	0.936555i	$-0.613995\pi$
$331$	−12.7543	−0.701041	−0.350521	+	0.936555i	$0.613995\pi$
$332$	−37.6180	−2.06455
$333$	0	0
$334$	4.57693	0.250438
$335$	−1.55328	−0.0848646
$336$	0	0
$337$	6.79279	0.370027	0.185014	−	0.982736i	$-0.440767\pi$
$337$	6.79279	0.370027	0.185014	+	0.982736i	$0.440767\pi$
$338$	54.5785	2.96868
$339$	0	0
$340$	−18.9571	−1.02809
$341$	0.823888	0.0446160
$342$	0	0
$343$	1.00000	0.0539949
$344$	28.4194	1.53227
$345$	0	0
$346$	−34.1774	−1.83739
$347$	19.0765	1.02408	0.512040	−	0.858962i	$-0.328890\pi$
$347$	19.0765	1.02408	0.512040	+	0.858962i	$0.328890\pi$
$348$	0	0
$349$	−21.9746	−1.17627	−0.588137	−	0.808761i	$-0.700138\pi$
$349$	−21.9746	−1.17627	−0.588137	+	0.808761i	$0.700138\pi$
$350$	2.80834	0.150112
$351$	0	0
$352$	−64.8316	−3.45554
$353$	−13.8431	−0.736795	−0.368397	−	0.929668i	$-0.620093\pi$
$353$	−13.8431	−0.736795	−0.368397	+	0.929668i	$0.620093\pi$
$354$	0	0
$355$	2.31376	0.122802
$356$	−40.3610	−2.13913
$357$	0	0
$358$	24.6337	1.30193
$359$	−31.3991	−1.65718	−0.828591	−	0.559854i	$-0.810857\pi$
$359$	−31.3991	−1.65718	−0.828591	+	0.559854i	$0.810857\pi$
$360$	0	0
$361$	−8.11841	−0.427285
$362$	−53.6100	−2.81768
$363$	0	0
$364$	−33.5259	−1.75724
$365$	3.69512	0.193412
$366$	0	0
$367$	28.7543	1.50096	0.750482	−	0.660891i	$-0.229821\pi$
$367$	28.7543	1.50096	0.750482	+	0.660891i	$0.229821\pi$
$368$	−60.8010	−3.16947
$369$	0	0
$370$	21.3902	1.11203
$371$	1.14185	0.0592817
$372$	0	0
$373$	−30.6537	−1.58719	−0.793594	−	0.608448i	$-0.791792\pi$
$373$	−30.6537	−1.58719	−0.793594	+	0.608448i	$0.791792\pi$
$374$	18.7967	0.971954
$375$	0	0
$376$	51.9626	2.67977
$377$	55.6616	2.86672
$378$	0	0
$379$	−24.1577	−1.24090	−0.620448	−	0.784248i	$-0.713049\pi$
$379$	−24.1577	−1.24090	−0.620448	+	0.784248i	$0.713049\pi$
$380$	−19.4189	−0.996167
$381$	0	0
$382$	−18.7482	−0.959240
$383$	12.3579	0.631461	0.315730	−	0.948849i	$-0.397750\pi$
$383$	12.3579	0.631461	0.315730	+	0.948849i	$0.397750\pi$
$384$	0	0
$385$	−2.07844	−0.105927
$386$	−52.3015	−2.66207
$387$	0	0
$388$	82.7520	4.20109
$389$	3.00693	0.152457	0.0762286	−	0.997090i	$-0.475712\pi$
$389$	3.00693	0.152457	0.0762286	+	0.997090i	$0.475712\pi$
$390$	0	0
$391$	10.3702	0.524445
$392$	10.9154	0.551311
$393$	0	0
$394$	−23.1643	−1.16700
$395$	10.0131	0.503813
$396$	0	0
$397$	−0.862353	−0.0432803	−0.0216401	−	0.999766i	$-0.506889\pi$
$397$	−0.862353	−0.0432803	−0.0216401	+	0.999766i	$0.506889\pi$
$398$	70.4483	3.53125
$399$	0	0
$400$	18.8806	0.944031
$401$	−13.5171	−0.675010	−0.337505	−	0.941324i	$-0.609583\pi$
$401$	−13.5171	−0.675010	−0.337505	+	0.941324i	$0.609583\pi$
$402$	0	0
$403$	2.25753	0.112456
$404$	−36.6175	−1.82179
$405$	0	0
$406$	−27.4475	−1.36220
$407$	−15.8308	−0.784704
$408$	0	0
$409$	22.1577	1.09563	0.547813	−	0.836601i	$-0.315461\pi$
$409$	22.1577	1.09563	0.547813	+	0.836601i	$0.315461\pi$
$410$	28.0661	1.38609
$411$	0	0
$412$	65.5649	3.23015
$413$	−5.61668	−0.276379
$414$	0	0
$415$	−6.39024	−0.313685
$416$	−177.645	−8.70975
$417$	0	0
$418$	19.2546	0.941771
$419$	27.1515	1.32644	0.663219	−	0.748426i	$-0.269190\pi$
$419$	27.1515	1.32644	0.663219	+	0.748426i	$0.269190\pi$
$420$	0	0
$421$	−34.2595	−1.66971	−0.834854	−	0.550472i	$-0.814448\pi$
$421$	−34.2595	−1.66971	−0.834854	+	0.550472i	$0.814448\pi$
$422$	−54.9317	−2.67403
$423$	0	0
$424$	12.4637	0.605291
$425$	−3.22029	−0.156207
$426$	0	0
$427$	−1.22029	−0.0590537
$428$	24.4706	1.18283
$429$	0	0
$430$	7.31180	0.352607
$431$	18.2926	0.881122	0.440561	−	0.897723i	$-0.354780\pi$
$431$	18.2926	0.881122	0.440561	+	0.897723i	$0.354780\pi$
$432$	0	0
$433$	−20.8027	−0.999714	−0.499857	−	0.866108i	$-0.666614\pi$
$433$	−20.8027	−0.999714	−0.499857	+	0.866108i	$0.666614\pi$
$434$	−1.11322	−0.0534362
$435$	0	0
$436$	71.0916	3.40467
$437$	10.6228	0.508159
$438$	0	0
$439$	4.21833	0.201330	0.100665	−	0.994920i	$-0.467903\pi$
$439$	4.21833	0.201330	0.100665	+	0.994920i	$0.467903\pi$
$440$	−22.6870	−1.08156
$441$	0	0
$442$	51.5048	2.44983
$443$	8.14380	0.386924	0.193462	−	0.981108i	$-0.438028\pi$
$443$	8.14380	0.386924	0.193462	+	0.981108i	$0.438028\pi$
$444$	0	0
$445$	−6.85620	−0.325015
$446$	40.0262	1.89529
$447$	0	0
$448$	49.8377	2.35461
$449$	17.8047	0.840254	0.420127	−	0.907465i	$-0.361986\pi$
$449$	17.8047	0.840254	0.420127	+	0.907465i	$0.361986\pi$
$450$	0	0
$451$	−20.7716	−0.978097
$452$	−111.428	−5.24113
$453$	0	0
$454$	70.9707	3.33082
$455$	−5.69512	−0.266991
$456$	0	0
$457$	4.09572	0.191590	0.0957948	−	0.995401i	$-0.469461\pi$
$457$	4.09572	0.191590	0.0957948	+	0.995401i	$0.469461\pi$
$458$	−65.6511	−3.06767
$459$	0	0
$460$	−18.9571	−0.883880
$461$	20.7566	0.966730	0.483365	−	0.875419i	$-0.339414\pi$
$461$	20.7566	0.966730	0.483365	+	0.875419i	$0.339414\pi$
$462$	0	0
$463$	−3.23112	−0.150163	−0.0750814	−	0.997177i	$-0.523922\pi$
$463$	−3.23112	−0.150163	−0.0750814	+	0.997177i	$0.523922\pi$
$464$	−184.531	−8.56664
$465$	0	0
$466$	8.69932	0.402988
$467$	−3.27754	−0.151666	−0.0758332	−	0.997121i	$-0.524162\pi$
$467$	−3.27754	−0.151666	−0.0758332	+	0.997121i	$0.524162\pi$
$468$	0	0
$469$	−1.55328	−0.0717237
$470$	13.3691	0.616669
$471$	0	0
$472$	−61.3084	−2.82195
$473$	−5.41143	−0.248818
$474$	0	0
$475$	−3.29872	−0.151356
$476$	−18.9571	−0.868898
$477$	0	0
$478$	−48.3972	−2.21364
$479$	−0.383317	−0.0175142	−0.00875711	−	0.999962i	$-0.502788\pi$
$479$	−0.383317	−0.0175142	−0.00875711	+	0.999962i	$0.502788\pi$
$480$	0	0
$481$	−43.3779	−1.97786
$482$	4.30812	0.196229
$483$	0	0
$484$	−39.3242	−1.78746
$485$	14.0573	0.638307
$486$	0	0
$487$	30.2164	1.36923	0.684617	−	0.728903i	$-0.259969\pi$
$487$	30.2164	1.36923	0.684617	+	0.728903i	$0.259969\pi$
$488$	−13.3199	−0.602964
$489$	0	0
$490$	2.80834	0.126868
$491$	−21.6862	−0.978686	−0.489343	−	0.872091i	$-0.662763\pi$
$491$	−21.6862	−0.978686	−0.489343	+	0.872091i	$0.662763\pi$
$492$	0	0
$493$	31.4737	1.41750
$494$	52.7593	2.37375
$495$	0	0
$496$	−7.48423	−0.336052
$497$	2.31376	0.103786
$498$	0	0
$499$	−16.3653	−0.732612	−0.366306	−	0.930495i	$-0.619378\pi$
$499$	−16.3653	−0.732612	−0.366306	+	0.930495i	$0.619378\pi$
$500$	5.88678	0.263265
$501$	0	0
$502$	14.5363	0.648786
$503$	24.2595	1.08168	0.540839	−	0.841126i	$-0.318107\pi$
$503$	24.2595	1.08168	0.540839	+	0.841126i	$0.318107\pi$
$504$	0	0
$505$	−6.22029	−0.276799
$506$	18.7967	0.835615
$507$	0	0
$508$	−51.9081	−2.30305
$509$	−17.0681	−0.756530	−0.378265	−	0.925697i	$-0.623479\pi$
$509$	−17.0681	−0.756530	−0.378265	+	0.925697i	$0.623479\pi$
$510$	0	0
$511$	3.69512	0.163463
$512$	176.753	7.81146
$513$	0	0
$514$	−38.8049	−1.71161
$515$	11.1376	0.490783
$516$	0	0
$517$	−9.89437	−0.435154
$518$	21.3902	0.939833
$519$	0	0
$520$	−62.1646	−2.72610
$521$	13.8681	0.607571	0.303785	−	0.952740i	$-0.401749\pi$
$521$	13.8681	0.607571	0.303785	+	0.952740i	$0.401749\pi$
$522$	0	0
$523$	−10.8239	−0.473296	−0.236648	−	0.971596i	$-0.576049\pi$
$523$	−10.8239	−0.473296	−0.236648	+	0.971596i	$0.576049\pi$
$524$	−61.2783	−2.67696
$525$	0	0
$526$	−24.5851	−1.07196
$527$	1.27651	0.0556058
$528$	0	0
$529$	−12.6298	−0.549120
$530$	3.20669	0.139290
$531$	0	0
$532$	−19.4189	−0.841915
$533$	−56.9162	−2.46531
$534$	0	0
$535$	4.15688	0.179718
$536$	−16.9546	−0.732329
$537$	0	0
$538$	39.6700	1.71030
$539$	−2.07844	−0.0895247
$540$	0	0
$541$	9.71016	0.417472	0.208736	−	0.977972i	$-0.433065\pi$
$541$	9.71016	0.417472	0.208736	+	0.977972i	$0.433065\pi$
$542$	59.7795	2.56775
$543$	0	0
$544$	−100.449	−4.30670
$545$	12.0765	0.517300
$546$	0	0
$547$	13.8619	0.592692	0.296346	−	0.955081i	$-0.404232\pi$
$547$	13.8619	0.592692	0.296346	+	0.955081i	$0.404232\pi$
$548$	−6.72180	−0.287141
$549$	0	0
$550$	−5.83697	−0.248889
$551$	32.2403	1.37348
$552$	0	0
$553$	10.0131	0.425799
$554$	0.989165	0.0420256
$555$	0	0
$556$	−26.1407	−1.10861
$557$	2.60976	0.110579	0.0552894	−	0.998470i	$-0.482392\pi$
$557$	2.60976	0.110579	0.0552894	+	0.998470i	$0.482392\pi$
$558$	0	0
$559$	−14.8278	−0.627151
$560$	18.8806	0.797852
$561$	0	0
$562$	81.4268	3.43478
$563$	−17.2311	−0.726205	−0.363103	−	0.931749i	$-0.618283\pi$
$563$	−17.2311	−0.726205	−0.363103	+	0.931749i	$0.618283\pi$
$564$	0	0
$565$	−18.9285	−0.796327
$566$	−10.6848	−0.449114
$567$	0	0
$568$	25.2556	1.05970
$569$	−10.3522	−0.433988	−0.216994	−	0.976173i	$-0.569625\pi$
$569$	−10.3522	−0.433988	−0.216994	+	0.976173i	$0.569625\pi$
$570$	0	0
$571$	−28.0891	−1.17549	−0.587747	−	0.809045i	$-0.699985\pi$
$571$	−28.0891	−1.17549	−0.587747	+	0.809045i	$0.699985\pi$
$572$	69.6816	2.91354
$573$	0	0
$574$	28.0661	1.17146
$575$	−3.22029	−0.134295
$576$	0	0
$577$	−11.4425	−0.476359	−0.238179	−	0.971221i	$-0.576551\pi$
$577$	−11.4425	−0.476359	−0.238179	+	0.971221i	$0.576551\pi$
$578$	−18.6186	−0.774433
$579$	0	0
$580$	−57.5348	−2.38900
$581$	−6.39024	−0.265112
$582$	0	0
$583$	−2.37326	−0.0982903
$584$	40.3338	1.66902
$585$	0	0
$586$	40.5908	1.67679
$587$	45.1769	1.86465	0.932325	−	0.361622i	$-0.117777\pi$
$587$	45.1769	1.86465	0.932325	+	0.361622i	$0.117777\pi$
$588$	0	0
$589$	1.30761	0.0538790
$590$	−15.7736	−0.649387
$591$	0	0
$592$	143.808	5.91046
$593$	−16.2622	−0.667809	−0.333905	−	0.942607i	$-0.608366\pi$
$593$	−16.2622	−0.667809	−0.333905	+	0.942607i	$0.608366\pi$
$594$	0	0
$595$	−3.22029	−0.132019
$596$	−8.25651	−0.338200
$597$	0	0
$598$	51.5048	2.10619
$599$	−20.3310	−0.830704	−0.415352	−	0.909661i	$-0.636342\pi$
$599$	−20.3310	−0.830704	−0.415352	+	0.909661i	$0.636342\pi$
$600$	0	0
$601$	−40.6105	−1.65654	−0.828269	−	0.560331i	$-0.810674\pi$
$601$	−40.6105	−1.65654	−0.828269	+	0.560331i	$0.810674\pi$
$602$	7.31180	0.298007
$603$	0	0
$604$	47.0942	1.91624
$605$	−6.68009	−0.271584
$606$	0	0
$607$	−37.5048	−1.52227	−0.761135	−	0.648593i	$-0.775358\pi$
$607$	−37.5048	−1.52227	−0.761135	+	0.648593i	$0.775358\pi$
$608$	−102.895	−4.17295
$609$	0	0
$610$	−3.42698	−0.138754
$611$	−27.1115	−1.09681
$612$	0	0
$613$	34.5156	1.39407	0.697036	−	0.717036i	$-0.254502\pi$
$613$	34.5156	1.39407	0.697036	+	0.717036i	$0.254502\pi$
$614$	79.6755	3.21544
$615$	0	0
$616$	−22.6870	−0.914086
$617$	34.6801	1.39617	0.698084	−	0.716016i	$-0.254036\pi$
$617$	34.6801	1.39617	0.698084	+	0.716016i	$0.254036\pi$
$618$	0	0
$619$	30.4495	1.22387	0.611933	−	0.790909i	$-0.290392\pi$
$619$	30.4495	1.22387	0.611933	+	0.790909i	$0.290392\pi$
$620$	−2.33350	−0.0937158
$621$	0	0
$622$	−57.0480	−2.28742
$623$	−6.85620	−0.274688
$624$	0	0
$625$	1.00000	0.0400000
$626$	36.7482	1.46875
$627$	0	0
$628$	23.1376	0.923293
$629$	−24.5279	−0.977991
$630$	0	0
$631$	−6.58437	−0.262120	−0.131060	−	0.991374i	$-0.541838\pi$
$631$	−6.58437	−0.262120	−0.131060	+	0.991374i	$0.541838\pi$
$632$	109.297	4.34760
$633$	0	0
$634$	−26.1875	−1.04004
$635$	−8.81774	−0.349921
$636$	0	0
$637$	−5.69512	−0.225649
$638$	57.0480	2.25855
$639$	0	0
$640$	77.5765	3.06648
$641$	24.8377	0.981032	0.490516	−	0.871432i	$-0.336808\pi$
$641$	24.8377	0.981032	0.490516	+	0.871432i	$0.336808\pi$
$642$	0	0
$643$	23.6428	0.932383	0.466191	−	0.884684i	$-0.345626\pi$
$643$	23.6428	0.932383	0.466191	+	0.884684i	$0.345626\pi$
$644$	−18.9571	−0.747015
$645$	0	0
$646$	29.8325	1.17375
$647$	−23.3276	−0.917103	−0.458552	−	0.888668i	$-0.651632\pi$
$647$	−23.3276	−0.917103	−0.458552	+	0.888668i	$0.651632\pi$
$648$	0	0
$649$	11.6739	0.458242
$650$	−15.9938	−0.627330
$651$	0	0
$652$	120.843	4.73258
$653$	22.0881	0.864374	0.432187	−	0.901784i	$-0.357742\pi$
$653$	22.0881	0.864374	0.432187	+	0.901784i	$0.357742\pi$
$654$	0	0
$655$	−10.4095	−0.406732
$656$	188.690	7.36711
$657$	0	0
$658$	13.3691	0.521180
$659$	−33.6355	−1.31025	−0.655126	−	0.755520i	$-0.727384\pi$
$659$	−33.6355	−1.31025	−0.655126	+	0.755520i	$0.727384\pi$
$660$	0	0
$661$	6.15688	0.239475	0.119737	−	0.992806i	$-0.461795\pi$
$661$	6.15688	0.239475	0.119737	+	0.992806i	$0.461795\pi$
$662$	−35.8185	−1.39213
$663$	0	0
$664$	−69.7521	−2.70691
$665$	−3.29872	−0.127919
$666$	0	0
$667$	31.4737	1.21866
$668$	9.59406	0.371205
$669$	0	0
$670$	−4.36213	−0.168524
$671$	2.53629	0.0979124
$672$	0	0
$673$	−39.1253	−1.50817	−0.754086	−	0.656776i	$-0.771920\pi$
$673$	−39.1253	−1.50817	−0.754086	+	0.656776i	$0.771920\pi$
$674$	19.0765	0.734799
$675$	0	0
$676$	114.406	4.40024
$677$	39.4041	1.51442	0.757211	−	0.653170i	$-0.226561\pi$
$677$	39.4041	1.51442	0.757211	+	0.653170i	$0.226561\pi$
$678$	0	0
$679$	14.0573	0.539468
$680$	−35.1507	−1.34797
$681$	0	0
$682$	2.31376	0.0885984
$683$	35.9746	1.37653	0.688265	−	0.725459i	$-0.258373\pi$
$683$	35.9746	1.37653	0.688265	+	0.725459i	$0.258373\pi$
$684$	0	0
$685$	−1.14185	−0.0436277
$686$	2.80834	0.107223
$687$	0	0
$688$	49.1577	1.87412
$689$	−6.50295	−0.247743
$690$	0	0
$691$	−34.1799	−1.30026	−0.650132	−	0.759822i	$-0.725286\pi$
$691$	−34.1799	−1.30026	−0.650132	+	0.759822i	$0.725286\pi$
$692$	−71.6419	−2.72342
$693$	0	0
$694$	53.5733	2.03361
$695$	−4.44057	−0.168440
$696$	0	0
$697$	−32.1830	−1.21902
$698$	−61.7122	−2.33584
$699$	0	0
$700$	5.88678	0.222499
$701$	−19.7623	−0.746411	−0.373206	−	0.927749i	$-0.621741\pi$
$701$	−19.7623	−0.746411	−0.373206	+	0.927749i	$0.621741\pi$
$702$	0	0
$703$	−25.1253	−0.947620
$704$	−103.585	−3.90400
$705$	0	0
$706$	−38.8762	−1.46313
$707$	−6.22029	−0.233938
$708$	0	0
$709$	29.2573	1.09878	0.549390	−	0.835566i	$-0.314860\pi$
$709$	29.2573	1.09878	0.549390	+	0.835566i	$0.314860\pi$
$710$	6.49782	0.243859
$711$	0	0
$712$	−74.8382	−2.80468
$713$	1.27651	0.0478058
$714$	0	0
$715$	11.8370	0.442678
$716$	51.6365	1.92975
$717$	0	0
$718$	−88.1795	−3.29083
$719$	−25.0765	−0.935195	−0.467598	−	0.883941i	$-0.654880\pi$
$719$	−25.0765	−0.935195	−0.467598	+	0.883941i	$0.654880\pi$
$720$	0	0
$721$	11.1376	0.414788
$722$	−22.7993	−0.848501
$723$	0	0
$724$	−112.376	−4.17642
$725$	−9.77356	−0.362981
$726$	0	0
$727$	21.3019	0.790044	0.395022	−	0.918672i	$-0.370737\pi$
$727$	21.3019	0.790044	0.395022	+	0.918672i	$0.370737\pi$
$728$	−62.1646	−2.30397
$729$	0	0
$730$	10.3772	0.384076
$731$	−8.38434	−0.310106
$732$	0	0
$733$	−29.9477	−1.10614	−0.553072	−	0.833133i	$-0.686545\pi$
$733$	−29.9477	−1.10614	−0.553072	+	0.833133i	$0.686545\pi$
$734$	80.7520	2.98061
$735$	0	0
$736$	−100.449	−3.70258
$737$	3.22839	0.118919
$738$	0	0
$739$	23.7255	0.872756	0.436378	−	0.899763i	$-0.356261\pi$
$739$	23.7255	0.872756	0.436378	+	0.899763i	$0.356261\pi$
$740$	44.8377	1.64827
$741$	0	0
$742$	3.20669	0.117721
$743$	7.98769	0.293040	0.146520	−	0.989208i	$-0.453193\pi$
$743$	7.98769	0.293040	0.146520	+	0.989208i	$0.453193\pi$
$744$	0	0
$745$	−1.40255	−0.0513855
$746$	−86.0860	−3.15183
$747$	0	0
$748$	39.4012	1.44065
$749$	4.15688	0.151889
$750$	0	0
$751$	11.9869	0.437409	0.218704	−	0.975791i	$-0.429817\pi$
$751$	11.9869	0.437409	0.218704	+	0.975791i	$0.429817\pi$
$752$	89.8809	3.27762
$753$	0	0
$754$	156.317	5.69272
$755$	8.00000	0.291150
$756$	0	0
$757$	5.83082	0.211925	0.105962	−	0.994370i	$-0.466208\pi$
$757$	5.83082	0.211925	0.105962	+	0.994370i	$0.466208\pi$
$758$	−67.8429	−2.46417
$759$	0	0
$760$	−36.0069	−1.30611
$761$	−46.3153	−1.67893	−0.839464	−	0.543415i	$-0.817131\pi$
$761$	−46.3153	−1.67893	−0.839464	+	0.543415i	$0.817131\pi$
$762$	0	0
$763$	12.0765	0.437198
$764$	−39.2995	−1.42181
$765$	0	0
$766$	34.7053	1.25395
$767$	31.9877	1.15501
$768$	0	0
$769$	14.3145	0.516195	0.258098	−	0.966119i	$-0.416904\pi$
$769$	14.3145	0.516195	0.258098	+	0.966119i	$0.416904\pi$
$770$	−5.83697	−0.210350
$771$	0	0
$772$	−109.633	−3.94578
$773$	35.9885	1.29442	0.647208	−	0.762314i	$-0.275937\pi$
$773$	35.9885	1.29442	0.647208	+	0.762314i	$0.275937\pi$
$774$	0	0
$775$	−0.396397	−0.0142390
$776$	153.441	5.50820
$777$	0	0
$778$	8.44448	0.302749
$779$	−32.9670	−1.18116
$780$	0	0
$781$	−4.80901	−0.172080
$782$	29.1232	1.04144
$783$	0	0
$784$	18.8806	0.674308
$785$	3.93044	0.140283
$786$	0	0
$787$	−11.0493	−0.393865	−0.196933	−	0.980417i	$-0.563098\pi$
$787$	−11.0493	−0.393865	−0.196933	+	0.980417i	$0.563098\pi$
$788$	−48.5565	−1.72975
$789$	0	0
$790$	28.1201	1.00047
$791$	−18.9285	−0.673020
$792$	0	0
$793$	6.94967	0.246790
$794$	−2.42178	−0.0859459
$795$	0	0
$796$	147.672	5.23410
$797$	−5.50398	−0.194961	−0.0974804	−	0.995237i	$-0.531078\pi$
$797$	−5.50398	−0.194961	−0.0974804	+	0.995237i	$0.531078\pi$
$798$	0	0
$799$	−15.3301	−0.542340
$800$	31.1924	1.10282
$801$	0	0
$802$	−37.9605	−1.34043
$803$	−7.68009	−0.271024
$804$	0	0
$805$	−3.22029	−0.113500
$806$	6.33992	0.223314
$807$	0	0
$808$	−67.8969	−2.38861
$809$	8.50910	0.299164	0.149582	−	0.988749i	$-0.452207\pi$
$809$	8.50910	0.299164	0.149582	+	0.988749i	$0.452207\pi$
$810$	0	0
$811$	−7.73359	−0.271563	−0.135781	−	0.990739i	$-0.543355\pi$
$811$	−7.73359	−0.271563	−0.135781	+	0.990739i	$0.543355\pi$
$812$	−57.5348	−2.01908
$813$	0	0
$814$	−44.4583	−1.55826
$815$	20.5279	0.719061
$816$	0	0
$817$	−8.58857	−0.300476
$818$	62.2263	2.17569
$819$	0	0
$820$	58.8316	2.05449
$821$	−20.3276	−0.709439	−0.354719	−	0.934973i	$-0.615424\pi$
$821$	−20.3276	−0.709439	−0.354719	+	0.934973i	$0.615424\pi$
$822$	0	0
$823$	20.4383	0.712435	0.356218	−	0.934403i	$-0.384066\pi$
$823$	20.4383	0.712435	0.356218	+	0.934403i	$0.384066\pi$
$824$	121.572	4.23516
$825$	0	0
$826$	−15.7736	−0.548832
$827$	−29.9885	−1.04280	−0.521401	−	0.853312i	$-0.674590\pi$
$827$	−29.9885	−1.04280	−0.521401	+	0.853312i	$0.674590\pi$
$828$	0	0
$829$	−32.9374	−1.14396	−0.571981	−	0.820267i	$-0.693825\pi$
$829$	−32.9374	−1.14396	−0.571981	+	0.820267i	$0.693825\pi$
$830$	−17.9460	−0.622914
$831$	0	0
$832$	−283.832	−9.84011
$833$	−3.22029	−0.111576
$834$	0	0
$835$	1.62976	0.0564003
$836$	40.3610	1.39591
$837$	0	0
$838$	76.2507	2.63404
$839$	10.1841	0.351593	0.175796	−	0.984427i	$-0.443750\pi$
$839$	10.1841	0.351593	0.175796	+	0.984427i	$0.443750\pi$
$840$	0	0
$841$	66.5225	2.29388
$842$	−96.2124	−3.31570
$843$	0	0
$844$	−115.147	−3.96351
$845$	19.4344	0.668564
$846$	0	0
$847$	−6.68009	−0.229531
$848$	21.5588	0.740331
$849$	0	0
$850$	−9.04366	−0.310195
$851$	−24.5279	−0.840805
$852$	0	0
$853$	−30.9497	−1.05970	−0.529848	−	0.848092i	$-0.677751\pi$
$853$	−30.9497	−1.05970	−0.529848	+	0.848092i	$0.677751\pi$
$854$	−3.42698	−0.117269
$855$	0	0
$856$	45.3740	1.55085
$857$	−22.7975	−0.778747	−0.389373	−	0.921080i	$-0.627308\pi$
$857$	−22.7975	−0.778747	−0.389373	+	0.921080i	$0.627308\pi$
$858$	0	0
$859$	29.9100	1.02052	0.510259	−	0.860021i	$-0.329550\pi$
$859$	29.9100	1.02052	0.510259	+	0.860021i	$0.329550\pi$
$860$	15.3268	0.522641
$861$	0	0
$862$	51.3718	1.74973
$863$	12.1053	0.412070	0.206035	−	0.978545i	$-0.433944\pi$
$863$	12.1053	0.412070	0.206035	+	0.978545i	$0.433944\pi$
$864$	0	0
$865$	−12.1700	−0.413791
$866$	−58.4211	−1.98523
$867$	0	0
$868$	−2.33350	−0.0792043
$869$	−20.8116	−0.705985
$870$	0	0
$871$	8.84610	0.299739
$872$	131.820	4.46398
$873$	0	0
$874$	29.8325	1.00910
$875$	1.00000	0.0338062
$876$	0	0
$877$	25.5160	0.861615	0.430808	−	0.902444i	$-0.358229\pi$
$877$	25.5160	0.861615	0.430808	+	0.902444i	$0.358229\pi$
$878$	11.8465	0.399801
$879$	0	0
$880$	−39.2422	−1.32286
$881$	−40.9946	−1.38114	−0.690572	−	0.723264i	$-0.742641\pi$
$881$	−40.9946	−1.38114	−0.690572	+	0.723264i	$0.742641\pi$
$882$	0	0
$883$	−13.5989	−0.457640	−0.228820	−	0.973469i	$-0.573487\pi$
$883$	−13.5989	−0.457640	−0.228820	+	0.973469i	$0.573487\pi$
$884$	107.963	3.63119
$885$	0	0
$886$	22.8706	0.768352
$887$	−30.2896	−1.01702	−0.508512	−	0.861055i	$-0.669804\pi$
$887$	−30.2896	−1.01702	−0.508512	+	0.861055i	$0.669804\pi$
$888$	0	0
$889$	−8.81774	−0.295737
$890$	−19.2546	−0.645414
$891$	0	0
$892$	83.9019	2.80924
$893$	−15.7035	−0.525498
$894$	0	0
$895$	8.77161	0.293203
$896$	77.5765	2.59165
$897$	0	0
$898$	50.0015	1.66857
$899$	3.87421	0.129212
$900$	0	0
$901$	−3.67707	−0.122501
$902$	−58.3338	−1.94230
$903$	0	0
$904$	−206.612	−6.87182
$905$	−19.0896	−0.634559
$906$	0	0
$907$	10.0935	0.335148	0.167574	−	0.985859i	$-0.446407\pi$
$907$	10.0935	0.335148	0.167574	+	0.985859i	$0.446407\pi$
$908$	148.767	4.93701
$909$	0	0
$910$	−15.9938	−0.530191
$911$	13.5570	0.449164	0.224582	−	0.974455i	$-0.427898\pi$
$911$	13.5570	0.449164	0.224582	+	0.974455i	$0.427898\pi$
$912$	0	0
$913$	13.2817	0.439561
$914$	11.5022	0.380458
$915$	0	0
$916$	−137.616	−4.54697
$917$	−10.4095	−0.343751
$918$	0	0
$919$	−46.9251	−1.54791	−0.773957	−	0.633238i	$-0.781726\pi$
$919$	−46.9251	−1.54791	−0.773957	+	0.633238i	$0.781726\pi$
$920$	−35.1507	−1.15889
$921$	0	0
$922$	58.2915	1.91973
$923$	−13.1771	−0.433731
$924$	0	0
$925$	7.61668	0.250435
$926$	−9.07409	−0.298193
$927$	0	0
$928$	−304.861	−10.0076
$929$	−39.6105	−1.29958	−0.649790	−	0.760114i	$-0.725143\pi$
$929$	−39.6105	−1.29958	−0.649790	+	0.760114i	$0.725143\pi$
$930$	0	0
$931$	−3.29872	−0.108111
$932$	18.2353	0.597318
$933$	0	0
$934$	−9.20445	−0.301179
$935$	6.69317	0.218890
$936$	0	0
$937$	−46.2926	−1.51231	−0.756156	−	0.654391i	$-0.772925\pi$
$937$	−46.2926	−1.51231	−0.756156	+	0.654391i	$0.772925\pi$
$938$	−4.36213	−0.142429
$939$	0	0
$940$	28.0239	0.914039
$941$	49.1315	1.60164	0.800820	−	0.598904i	$-0.204397\pi$
$941$	49.1315	1.60164	0.800820	+	0.598904i	$0.204397\pi$
$942$	0	0
$943$	−32.1830	−1.04802
$944$	−106.046	−3.45152
$945$	0	0
$946$	−15.1971	−0.494102
$947$	−8.51379	−0.276661	−0.138330	−	0.990386i	$-0.544174\pi$
$947$	−8.51379	−0.276661	−0.138330	+	0.990386i	$0.544174\pi$
$948$	0	0
$949$	−21.0442	−0.683123
$950$	−9.26395	−0.300562
$951$	0	0
$952$	−35.1507	−1.13924
$953$	1.07649	0.0348708	0.0174354	−	0.999848i	$-0.494450\pi$
$953$	1.07649	0.0348708	0.0174354	+	0.999848i	$0.494450\pi$
$954$	0	0
$955$	−6.67589	−0.216027
$956$	−101.449	−3.28110
$957$	0	0
$958$	−1.07649	−0.0347797
$959$	−1.14185	−0.0368721
$960$	0	0
$961$	−30.8429	−0.994931
$962$	−121.820	−3.92764
$963$	0	0
$964$	9.03058	0.290855
$965$	−18.6236	−0.599515
$966$	0	0
$967$	19.1276	0.615102	0.307551	−	0.951532i	$-0.400491\pi$
$967$	19.1276	0.615102	0.307551	+	0.951532i	$0.400491\pi$
$968$	−72.9159	−2.34361
$969$	0	0
$970$	39.4776	1.26755
$971$	−57.6794	−1.85102	−0.925510	−	0.378723i	$-0.876363\pi$
$971$	−57.6794	−1.85102	−0.925510	+	0.378723i	$0.876363\pi$
$972$	0	0
$973$	−4.44057	−0.142358
$974$	84.8579	2.71902
$975$	0	0
$976$	−23.0398	−0.737485
$977$	−4.84014	−0.154850	−0.0774249	−	0.996998i	$-0.524670\pi$
$977$	−4.84014	−0.154850	−0.0774249	+	0.996998i	$0.524670\pi$
$978$	0	0
$979$	14.2502	0.455438
$980$	5.88678	0.188046
$981$	0	0
$982$	−60.9024	−1.94347
$983$	26.5710	0.847485	0.423742	−	0.905783i	$-0.360716\pi$
$983$	26.5710	0.847485	0.423742	+	0.905783i	$0.360716\pi$
$984$	0	0
$985$	−8.24840	−0.262816
$986$	88.3888	2.81487
$987$	0	0
$988$	110.593	3.51843
$989$	−8.38434	−0.266607
$990$	0	0
$991$	21.5725	0.685273	0.342637	−	0.939468i	$-0.388680\pi$
$991$	21.5725	0.685273	0.342637	+	0.939468i	$0.388680\pi$
$992$	−12.3646	−0.392576
$993$	0	0
$994$	6.49782	0.206098
$995$	25.0854	0.795260
$996$	0	0
$997$	−53.8708	−1.70611	−0.853053	−	0.521824i	$-0.825252\pi$
$997$	−53.8708	−1.70611	−0.853053	+	0.521824i	$0.825252\pi$
$998$	−45.9594	−1.45482
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	945.2.a.n.1.4	yes	4
3.2	odd	2		945.2.a.m.1.1	✓	4
5.4	even	2		4725.2.a.bo.1.1		4
7.6	odd	2		6615.2.a.bh.1.4		4
15.14	odd	2		4725.2.a.bx.1.4		4
21.20	even	2		6615.2.a.be.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
945.2.a.m.1.1	✓	4	3.2	odd	2
945.2.a.n.1.4	yes	4	1.1	even	1	trivial
4725.2.a.bo.1.1		4	5.4	even	2
4725.2.a.bx.1.4		4	15.14	odd	2
6615.2.a.be.1.1		4	21.20	even	2
6615.2.a.bh.1.4		4	7.6	odd	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:	\( N \)	\(=\)	\( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	945.a (trivial)

\(f(q)\)	\(=\)	\(q+2.80834 q^{2} +5.88678 q^{4} +1.00000 q^{5} +1.00000 q^{7} +10.9154 q^{8} +O(q^{10})\)	\(q+2.80834 q^{2} +5.88678 q^{4} +1.00000 q^{5} +1.00000 q^{7} +10.9154 q^{8} +2.80834 q^{10} -2.07844 q^{11} -5.69512 q^{13} +2.80834 q^{14} +18.8806 q^{16} -3.22029 q^{17} -3.29872 q^{19} +5.88678 q^{20} -5.83697 q^{22} -3.22029 q^{23} +1.00000 q^{25} -15.9938 q^{26} +5.88678 q^{28} -9.77356 q^{29} -0.396397 q^{31} +31.1924 q^{32} -9.04366 q^{34} +1.00000 q^{35} +7.61668 q^{37} -9.26395 q^{38} +10.9154 q^{40} +9.99385 q^{41} +2.60360 q^{43} -12.2353 q^{44} -9.04366 q^{46} +4.76048 q^{47} +1.00000 q^{49} +2.80834 q^{50} -33.5259 q^{52} +1.14185 q^{53} -2.07844 q^{55} +10.9154 q^{56} -27.4475 q^{58} -5.61668 q^{59} -1.22029 q^{61} -1.11322 q^{62} +49.8377 q^{64} -5.69512 q^{65} -1.55328 q^{67} -18.9571 q^{68} +2.80834 q^{70} +2.31376 q^{71} +3.69512 q^{73} +21.3902 q^{74} -19.4189 q^{76} -2.07844 q^{77} +10.0131 q^{79} +18.8806 q^{80} +28.0661 q^{82} -6.39024 q^{83} -3.22029 q^{85} +7.31180 q^{86} -22.6870 q^{88} -6.85620 q^{89} -5.69512 q^{91} -18.9571 q^{92} +13.3691 q^{94} -3.29872 q^{95} +14.0573 q^{97} +2.80834 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} + 9 q^{4} + 4 q^{5} + 4 q^{7} + 6 q^{8}+O(q^{10}) \) 4 * q + q^2 + 9 * q^4 + 4 * q^5 + 4 * q^7 + 6 * q^8	\( 4 q + q^{2} + 9 q^{4} + 4 q^{5} + 4 q^{7} + 6 q^{8} + q^{10} - 4 q^{11} + 2 q^{13} + q^{14} + 19 q^{16} + 4 q^{19} + 9 q^{20} + 10 q^{22} + 4 q^{25} - 22 q^{26} + 9 q^{28} - 10 q^{29} + 6 q^{31} + 23 q^{32} - 13 q^{34} + 4 q^{35} + 10 q^{37} - q^{38} + 6 q^{40} - 2 q^{41} + 18 q^{43} - 36 q^{44} - 13 q^{46} + 18 q^{47} + 4 q^{49} + q^{50} - 34 q^{52} - 4 q^{53} - 4 q^{55} + 6 q^{56} - 14 q^{58} - 2 q^{59} + 8 q^{61} - 19 q^{62} + 54 q^{64} + 2 q^{65} + 10 q^{67} + 13 q^{68} + q^{70} - 8 q^{71} - 10 q^{73} + 36 q^{74} - 5 q^{76} - 4 q^{77} + 12 q^{79} + 19 q^{80} + 24 q^{82} + 24 q^{83} - 16 q^{86} + 4 q^{88} - 8 q^{89} + 2 q^{91} + 13 q^{92} - 38 q^{94} + 4 q^{95} + 10 q^{97} + q^{98}+O(q^{100}) \) 4 * q + q^2 + 9 * q^4 + 4 * q^5 + 4 * q^7 + 6 * q^8 + q^10 - 4 * q^11 + 2 * q^13 + q^14 + 19 * q^16 + 4 * q^19 + 9 * q^20 + 10 * q^22 + 4 * q^25 - 22 * q^26 + 9 * q^28 - 10 * q^29 + 6 * q^31 + 23 * q^32 - 13 * q^34 + 4 * q^35 + 10 * q^37 - q^38 + 6 * q^40 - 2 * q^41 + 18 * q^43 - 36 * q^44 - 13 * q^46 + 18 * q^47 + 4 * q^49 + q^50 - 34 * q^52 - 4 * q^53 - 4 * q^55 + 6 * q^56 - 14 * q^58 - 2 * q^59 + 8 * q^61 - 19 * q^62 + 54 * q^64 + 2 * q^65 + 10 * q^67 + 13 * q^68 + q^70 - 8 * q^71 - 10 * q^73 + 36 * q^74 - 5 * q^76 - 4 * q^77 + 12 * q^79 + 19 * q^80 + 24 * q^82 + 24 * q^83 - 16 * q^86 + 4 * q^88 - 8 * q^89 + 2 * q^91 + 13 * q^92 - 38 * q^94 + 4 * q^95 + 10 * q^97 + q^98

Introduction

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