LMFDB - Embedded newform 2925.2.a.be.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2925,2,Mod(1,2925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2925, 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("2925.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2925 = 3^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2925.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:	$23.3562425912$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 975)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	2925.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.67513 q^{2} +5.15633 q^{4} -2.28726 q^{7} -8.44358 q^{8} +O(q^{10})$	$q-2.67513 q^{2} +5.15633 q^{4} -2.28726 q^{7} -8.44358 q^{8} +0.130933 q^{11} -1.00000 q^{13} +6.11871 q^{14} +12.2750 q^{16} -7.96239 q^{17} -5.11871 q^{19} -0.350262 q^{22} +5.50659 q^{23} +2.67513 q^{26} -11.7938 q^{28} +3.00000 q^{29} -7.79384 q^{31} -15.9502 q^{32} +21.3004 q^{34} +4.80606 q^{37} +13.6932 q^{38} -11.4314 q^{41} -2.93207 q^{43} +0.675131 q^{44} -14.7308 q^{46} -2.67513 q^{47} -1.76845 q^{49} -5.15633 q^{52} +8.50659 q^{53} +19.3127 q^{56} -8.02539 q^{58} +10.4133 q^{59} +5.18664 q^{61} +20.8496 q^{62} +18.1187 q^{64} -6.05571 q^{67} -41.0567 q^{68} +15.0435 q^{71} -0.932071 q^{73} -12.8568 q^{74} -26.3938 q^{76} -0.299477 q^{77} +8.85685 q^{79} +30.5804 q^{82} -6.80114 q^{83} +7.84367 q^{86} -1.10554 q^{88} +8.12601 q^{89} +2.28726 q^{91} +28.3938 q^{92} +7.15633 q^{94} +18.9624 q^{97} +4.73084 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 5 q^{4} - q^{7} - 9 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + 5 * q^4 - q^7 - 9 * q^8	$ 3 q - 3 q^{2} + 5 q^{4} - q^{7} - 9 q^{8} + 5 q^{11} - 3 q^{13} - 3 q^{14} + 5 q^{16} - 13 q^{17} + 6 q^{19} + 9 q^{22} - 4 q^{23} + 3 q^{26} - 9 q^{28} + 9 q^{29} + 3 q^{31} - 11 q^{32} + 17 q^{34} + 14 q^{37} + 8 q^{38} + 8 q^{41} - 3 q^{44} - 22 q^{46} - 3 q^{47} + 6 q^{49} - 5 q^{52} + 5 q^{53} + 37 q^{56} - 9 q^{58} + 17 q^{59} + 3 q^{61} + 19 q^{62} + 33 q^{64} - q^{67} - 39 q^{68} + 2 q^{71} + 6 q^{73} - 8 q^{74} - 26 q^{76} - 21 q^{77} - 4 q^{79} + 26 q^{82} - 7 q^{83} + 34 q^{86} - 23 q^{88} + 16 q^{89} + q^{91} + 32 q^{92} + 11 q^{94} + 46 q^{97} - 8 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 5 * q^4 - q^7 - 9 * q^8 + 5 * q^11 - 3 * q^13 - 3 * q^14 + 5 * q^16 - 13 * q^17 + 6 * q^19 + 9 * q^22 - 4 * q^23 + 3 * q^26 - 9 * q^28 + 9 * q^29 + 3 * q^31 - 11 * q^32 + 17 * q^34 + 14 * q^37 + 8 * q^38 + 8 * q^41 - 3 * q^44 - 22 * q^46 - 3 * q^47 + 6 * q^49 - 5 * q^52 + 5 * q^53 + 37 * q^56 - 9 * q^58 + 17 * q^59 + 3 * q^61 + 19 * q^62 + 33 * q^64 - q^67 - 39 * q^68 + 2 * q^71 + 6 * q^73 - 8 * q^74 - 26 * q^76 - 21 * q^77 - 4 * q^79 + 26 * q^82 - 7 * q^83 + 34 * q^86 - 23 * q^88 + 16 * q^89 + q^91 + 32 * q^92 + 11 * q^94 + 46 * q^97 - 8 * 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 $

$2$	−2.67513	−1.89160	−0.945802	−	0.324745i	$-0.894721\pi$
$2$	−2.67513	−1.89160	−0.945802	+	0.324745i	$0.894721\pi$
$3$	0	0
$3$	0	0
$4$	5.15633	2.57816
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.28726	−0.864502	−0.432251	−	0.901753i	$-0.642281\pi$
$7$	−2.28726	−0.864502	−0.432251	+	0.901753i	$0.642281\pi$
$8$	−8.44358	−2.98526
$9$	0	0
$10$	0	0
$11$	0.130933	0.0394777	0.0197388	−	0.999805i	$-0.493717\pi$
$11$	0.130933	0.0394777	0.0197388	+	0.999805i	$0.493717\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	6.11871	1.63530
$15$	0	0
$16$	12.2750	3.06876
$17$	−7.96239	−1.93116	−0.965581	−	0.260101i	$-0.916244\pi$
$17$	−7.96239	−1.93116	−0.965581	+	0.260101i	$0.916244\pi$
$18$	0	0
$19$	−5.11871	−1.17431	−0.587157	−	0.809473i	$-0.699753\pi$
$19$	−5.11871	−1.17431	−0.587157	+	0.809473i	$0.699753\pi$
$20$	0	0
$21$	0	0
$22$	−0.350262	−0.0746761
$23$	5.50659	1.14820	0.574101	−	0.818784i	$-0.305352\pi$
$23$	5.50659	1.14820	0.574101	+	0.818784i	$0.305352\pi$
$24$	0	0
$25$	0	0
$26$	2.67513	0.524636
$27$	0	0
$28$	−11.7938	−2.22883
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	−7.79384	−1.39982	−0.699908	−	0.714233i	$-0.746776\pi$
$31$	−7.79384	−1.39982	−0.699908	+	0.714233i	$0.746776\pi$
$32$	−15.9502	−2.81962
$33$	0	0
$34$	21.3004	3.65299
$35$	0	0
$36$	0	0
$37$	4.80606	0.790112	0.395056	−	0.918657i	$-0.370725\pi$
$37$	4.80606	0.790112	0.395056	+	0.918657i	$0.370725\pi$
$38$	13.6932	2.22134
$39$	0	0
$40$	0	0
$41$	−11.4314	−1.78528	−0.892640	−	0.450771i	$-0.851149\pi$
$41$	−11.4314	−1.78528	−0.892640	+	0.450771i	$0.851149\pi$
$42$	0	0
$43$	−2.93207	−0.447137	−0.223568	−	0.974688i	$-0.571771\pi$
$43$	−2.93207	−0.447137	−0.223568	+	0.974688i	$0.571771\pi$
$44$	0.675131	0.101780
$45$	0	0
$46$	−14.7308	−2.17194
$47$	−2.67513	−0.390208	−0.195104	−	0.980783i	$-0.562504\pi$
$47$	−2.67513	−0.390208	−0.195104	+	0.980783i	$0.562504\pi$
$48$	0	0
$49$	−1.76845	−0.252636
$50$	0	0
$51$	0	0
$52$	−5.15633	−0.715054
$53$	8.50659	1.16847	0.584235	−	0.811585i	$-0.301395\pi$
$53$	8.50659	1.16847	0.584235	+	0.811585i	$0.301395\pi$
$54$	0	0
$55$	0	0
$56$	19.3127	2.58076
$57$	0	0
$58$	−8.02539	−1.05379
$59$	10.4133	1.35569	0.677846	−	0.735204i	$-0.262914\pi$
$59$	10.4133	1.35569	0.677846	+	0.735204i	$0.262914\pi$
$60$	0	0
$61$	5.18664	0.664082	0.332041	−	0.943265i	$-0.392263\pi$
$61$	5.18664	0.664082	0.332041	+	0.943265i	$0.392263\pi$
$62$	20.8496	2.64790
$63$	0	0
$64$	18.1187	2.26484
$65$	0	0
$66$	0	0
$67$	−6.05571	−0.739823	−0.369911	−	0.929067i	$-0.620612\pi$
$67$	−6.05571	−0.739823	−0.369911	+	0.929067i	$0.620612\pi$
$68$	−41.0567	−4.97885
$69$	0	0
$70$	0	0
$71$	15.0435	1.78533	0.892667	−	0.450717i	$-0.148832\pi$
$71$	15.0435	1.78533	0.892667	+	0.450717i	$0.148832\pi$
$72$	0	0
$73$	−0.932071	−0.109091	−0.0545454	−	0.998511i	$-0.517371\pi$
$73$	−0.932071	−0.109091	−0.0545454	+	0.998511i	$0.517371\pi$
$74$	−12.8568	−1.49458
$75$	0	0
$76$	−26.3938	−3.02757
$77$	−0.299477	−0.0341285
$78$	0	0
$79$	8.85685	0.996473	0.498237	−	0.867041i	$-0.333981\pi$
$79$	8.85685	0.996473	0.498237	+	0.867041i	$0.333981\pi$
$80$	0	0
$81$	0	0
$82$	30.5804	3.37704
$83$	−6.80114	−0.746522	−0.373261	−	0.927726i	$-0.621760\pi$
$83$	−6.80114	−0.746522	−0.373261	+	0.927726i	$0.621760\pi$
$84$	0	0
$85$	0	0
$86$	7.84367	0.845805
$87$	0	0
$88$	−1.10554	−0.117851
$89$	8.12601	0.861355	0.430678	−	0.902506i	$-0.358275\pi$
$89$	8.12601	0.861355	0.430678	+	0.902506i	$0.358275\pi$
$90$	0	0
$91$	2.28726	0.239770
$92$	28.3938	2.96025
$93$	0	0
$94$	7.15633	0.738119
$95$	0	0
$96$	0	0
$97$	18.9624	1.92534	0.962669	−	0.270680i	$-0.0872485\pi$
$97$	18.9624	1.92534	0.962669	+	0.270680i	$0.0872485\pi$
$98$	4.73084	0.477887
$99$	0	0
$100$	0	0
$101$	3.54420	0.352661	0.176330	−	0.984331i	$-0.443577\pi$
$101$	3.54420	0.352661	0.176330	+	0.984331i	$0.443577\pi$
$102$	0	0
$103$	14.0811	1.38745	0.693726	−	0.720239i	$-0.255968\pi$
$103$	14.0811	1.38745	0.693726	+	0.720239i	$0.255968\pi$
$104$	8.44358	0.827961
$105$	0	0
$106$	−22.7562	−2.21028
$107$	2.64974	0.256160	0.128080	−	0.991764i	$-0.459119\pi$
$107$	2.64974	0.256160	0.128080	+	0.991764i	$0.459119\pi$
$108$	0	0
$109$	−9.31994	−0.892689	−0.446344	−	0.894861i	$-0.647274\pi$
$109$	−9.31994	−0.892689	−0.446344	+	0.894861i	$0.647274\pi$
$110$	0	0
$111$	0	0
$112$	−28.0762	−2.65295
$113$	−13.3503	−1.25589	−0.627943	−	0.778259i	$-0.716103\pi$
$113$	−13.3503	−1.25589	−0.627943	+	0.778259i	$0.716103\pi$
$114$	0	0
$115$	0	0
$116$	15.4690	1.43626
$117$	0	0
$118$	−27.8568	−2.56443
$119$	18.2120	1.66949
$120$	0	0
$121$	−10.9829	−0.998442
$122$	−13.8749	−1.25618
$123$	0	0
$124$	−40.1876	−3.60895
$125$	0	0
$126$	0	0
$127$	8.73084	0.774737	0.387368	−	0.921925i	$-0.373384\pi$
$127$	8.73084	0.774737	0.387368	+	0.921925i	$0.373384\pi$
$128$	−16.5696	−1.46456
$129$	0	0
$130$	0	0
$131$	−19.0435	−1.66384	−0.831919	−	0.554897i	$-0.812757\pi$
$131$	−19.0435	−1.66384	−0.831919	+	0.554897i	$0.812757\pi$
$132$	0	0
$133$	11.7078	1.01520
$134$	16.1998	1.39945
$135$	0	0
$136$	67.2311	5.76502
$137$	−7.66291	−0.654687	−0.327343	−	0.944905i	$-0.606153\pi$
$137$	−7.66291	−0.654687	−0.327343	+	0.944905i	$0.606153\pi$
$138$	0	0
$139$	7.61213	0.645652	0.322826	−	0.946458i	$-0.395367\pi$
$139$	7.61213	0.645652	0.322826	+	0.946458i	$0.395367\pi$
$140$	0	0
$141$	0	0
$142$	−40.2433	−3.37714
$143$	−0.130933	−0.0109491
$144$	0	0
$145$	0	0
$146$	2.49341	0.206356
$147$	0	0
$148$	24.7816	2.03704
$149$	−0.887166	−0.0726795	−0.0363397	−	0.999339i	$-0.511570\pi$
$149$	−0.887166	−0.0726795	−0.0363397	+	0.999339i	$0.511570\pi$
$150$	0	0
$151$	2.93700	0.239009	0.119505	−	0.992834i	$-0.461869\pi$
$151$	2.93700	0.239009	0.119505	+	0.992834i	$0.461869\pi$
$152$	43.2203	3.50563
$153$	0	0
$154$	0.801139	0.0645576
$155$	0	0
$156$	0	0
$157$	15.7816	1.25951	0.629755	−	0.776793i	$-0.283155\pi$
$157$	15.7816	1.25951	0.629755	+	0.776793i	$0.283155\pi$
$158$	−23.6932	−1.88493
$159$	0	0
$160$	0	0
$161$	−12.5950	−0.992624
$162$	0	0
$163$	−17.4314	−1.36533	−0.682665	−	0.730732i	$-0.739179\pi$
$163$	−17.4314	−1.36533	−0.682665	+	0.730732i	$0.739179\pi$
$164$	−58.9438	−4.60274
$165$	0	0
$166$	18.1939	1.41212
$167$	1.91890	0.148489	0.0742444	−	0.997240i	$-0.476346\pi$
$167$	1.91890	0.148489	0.0742444	+	0.997240i	$0.476346\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	−15.1187	−1.15279
$173$	7.93207	0.603064	0.301532	−	0.953456i	$-0.402502\pi$
$173$	7.93207	0.603064	0.301532	+	0.953456i	$0.402502\pi$
$174$	0	0
$175$	0	0
$176$	1.60720	0.121147
$177$	0	0
$178$	−21.7381	−1.62934
$179$	−14.9175	−1.11499	−0.557493	−	0.830182i	$-0.688236\pi$
$179$	−14.9175	−1.11499	−0.557493	+	0.830182i	$0.688236\pi$
$180$	0	0
$181$	−23.5804	−1.75272	−0.876358	−	0.481659i	$-0.840034\pi$
$181$	−23.5804	−1.75272	−0.876358	+	0.481659i	$0.840034\pi$
$182$	−6.11871	−0.453549
$183$	0	0
$184$	−46.4953	−3.42768
$185$	0	0
$186$	0	0
$187$	−1.04254	−0.0762378
$188$	−13.7938	−1.00602
$189$	0	0
$190$	0	0
$191$	−8.38787	−0.606925	−0.303463	−	0.952843i	$-0.598143\pi$
$191$	−8.38787	−0.606925	−0.303463	+	0.952843i	$0.598143\pi$
$192$	0	0
$193$	18.1563	1.30692	0.653460	−	0.756961i	$-0.273317\pi$
$193$	18.1563	1.30692	0.653460	+	0.756961i	$0.273317\pi$
$194$	−50.7269	−3.64198
$195$	0	0
$196$	−9.11871	−0.651337
$197$	1.19394	0.0850645	0.0425322	−	0.999095i	$-0.486457\pi$
$197$	1.19394	0.0850645	0.0425322	+	0.999095i	$0.486457\pi$
$198$	0	0
$199$	0.700523	0.0496588	0.0248294	−	0.999692i	$-0.492096\pi$
$199$	0.700523	0.0496588	0.0248294	+	0.999692i	$0.492096\pi$
$200$	0	0
$201$	0	0
$202$	−9.48119	−0.667095
$203$	−6.86177	−0.481602
$204$	0	0
$205$	0	0
$206$	−37.6688	−2.62451
$207$	0	0
$208$	−12.2750	−0.851121
$209$	−0.670206	−0.0463591
$210$	0	0
$211$	16.9829	1.16915	0.584574	−	0.811340i	$-0.301262\pi$
$211$	16.9829	1.16915	0.584574	+	0.811340i	$0.301262\pi$
$212$	43.8627	3.01250
$213$	0	0
$214$	−7.08840	−0.484553
$215$	0	0
$216$	0	0
$217$	17.8265	1.21014
$218$	24.9321	1.68861
$219$	0	0
$220$	0	0
$221$	7.96239	0.535608
$222$	0	0
$223$	−0.806063	−0.0539780	−0.0269890	−	0.999636i	$-0.508592\pi$
$223$	−0.806063	−0.0539780	−0.0269890	+	0.999636i	$0.508592\pi$
$224$	36.4821	2.43757
$225$	0	0
$226$	35.7137	2.37564
$227$	−7.68243	−0.509900	−0.254950	−	0.966954i	$-0.582059\pi$
$227$	−7.68243	−0.509900	−0.254950	+	0.966954i	$0.582059\pi$
$228$	0	0
$229$	−4.44851	−0.293966	−0.146983	−	0.989139i	$-0.546956\pi$
$229$	−4.44851	−0.293966	−0.146983	+	0.989139i	$0.546956\pi$
$230$	0	0
$231$	0	0
$232$	−25.3307	−1.66305
$233$	10.1260	0.663377	0.331688	−	0.943389i	$-0.392382\pi$
$233$	10.1260	0.663377	0.331688	+	0.943389i	$0.392382\pi$
$234$	0	0
$235$	0	0
$236$	53.6942	3.49519
$237$	0	0
$238$	−48.7196	−3.15802
$239$	19.4363	1.25723	0.628615	−	0.777717i	$-0.283622\pi$
$239$	19.4363	1.25723	0.628615	+	0.777717i	$0.283622\pi$
$240$	0	0
$241$	−6.46898	−0.416703	−0.208352	−	0.978054i	$-0.566810\pi$
$241$	−6.46898	−0.416703	−0.208352	+	0.978054i	$0.566810\pi$
$242$	29.3806	1.88866
$243$	0	0
$244$	26.7440	1.71211
$245$	0	0
$246$	0	0
$247$	5.11871	0.325696
$248$	65.8080	4.17881
$249$	0	0
$250$	0	0
$251$	17.7685	1.12153	0.560767	−	0.827973i	$-0.310506\pi$
$251$	17.7685	1.12153	0.560767	+	0.827973i	$0.310506\pi$
$252$	0	0
$253$	0.720992	0.0453283
$254$	−23.3561	−1.46549
$255$	0	0
$256$	8.08840	0.505525
$257$	4.81924	0.300616	0.150308	−	0.988639i	$-0.451974\pi$
$257$	4.81924	0.300616	0.150308	+	0.988639i	$0.451974\pi$
$258$	0	0
$259$	−10.9927	−0.683054
$260$	0	0
$261$	0	0
$262$	50.9438	3.14732
$263$	1.30536	0.0804917	0.0402459	−	0.999190i	$-0.487186\pi$
$263$	1.30536	0.0804917	0.0402459	+	0.999190i	$0.487186\pi$
$264$	0	0
$265$	0	0
$266$	−31.3199	−1.92035
$267$	0	0
$268$	−31.2252	−1.90738
$269$	8.79877	0.536470	0.268235	−	0.963353i	$-0.413560\pi$
$269$	8.79877	0.536470	0.268235	+	0.963353i	$0.413560\pi$
$270$	0	0
$271$	17.4264	1.05858	0.529290	−	0.848441i	$-0.322458\pi$
$271$	17.4264	1.05858	0.529290	+	0.848441i	$0.322458\pi$
$272$	−97.7386	−5.92627
$273$	0	0
$274$	20.4993	1.23841
$275$	0	0
$276$	0	0
$277$	11.7381	0.705276	0.352638	−	0.935760i	$-0.385285\pi$
$277$	11.7381	0.705276	0.352638	+	0.935760i	$0.385285\pi$
$278$	−20.3634	−1.22132
$279$	0	0
$280$	0	0
$281$	14.5647	0.868855	0.434428	−	0.900707i	$-0.356951\pi$
$281$	14.5647	0.868855	0.434428	+	0.900707i	$0.356951\pi$
$282$	0	0
$283$	26.1016	1.55158	0.775789	−	0.630993i	$-0.217352\pi$
$283$	26.1016	1.55158	0.775789	+	0.630993i	$0.217352\pi$
$284$	77.5691	4.60288
$285$	0	0
$286$	0.350262	0.0207114
$287$	26.1465	1.54338
$288$	0	0
$289$	46.3996	2.72939
$290$	0	0
$291$	0	0
$292$	−4.80606	−0.281254
$293$	−9.07381	−0.530098	−0.265049	−	0.964235i	$-0.585388\pi$
$293$	−9.07381	−0.530098	−0.265049	+	0.964235i	$0.585388\pi$
$294$	0	0
$295$	0	0
$296$	−40.5804	−2.35869
$297$	0	0
$298$	2.37328	0.137481
$299$	−5.50659	−0.318454
$300$	0	0
$301$	6.70640	0.386551
$302$	−7.85685	−0.452111
$303$	0	0
$304$	−62.8324	−3.60369
$305$	0	0
$306$	0	0
$307$	−11.3806	−0.649524	−0.324762	−	0.945796i	$-0.605284\pi$
$307$	−11.3806	−0.649524	−0.324762	+	0.945796i	$0.605284\pi$
$308$	−1.54420	−0.0879889
$309$	0	0
$310$	0	0
$311$	−9.22425	−0.523059	−0.261530	−	0.965195i	$-0.584227\pi$
$311$	−9.22425	−0.523059	−0.261530	+	0.965195i	$0.584227\pi$
$312$	0	0
$313$	18.7685	1.06086	0.530428	−	0.847730i	$-0.322031\pi$
$313$	18.7685	1.06086	0.530428	+	0.847730i	$0.322031\pi$
$314$	−42.2179	−2.38249
$315$	0	0
$316$	45.6688	2.56907
$317$	−10.5442	−0.592221	−0.296111	−	0.955154i	$-0.595690\pi$
$317$	−10.5442	−0.592221	−0.296111	+	0.955154i	$0.595690\pi$
$318$	0	0
$319$	0.392798	0.0219924
$320$	0	0
$321$	0	0
$322$	33.6932	1.87765
$323$	40.7572	2.26779
$324$	0	0
$325$	0	0
$326$	46.6312	2.58266
$327$	0	0
$328$	96.5217	5.32952
$329$	6.11871	0.337336
$330$	0	0
$331$	4.95651	0.272434	0.136217	−	0.990679i	$-0.456506\pi$
$331$	4.95651	0.272434	0.136217	+	0.990679i	$0.456506\pi$
$332$	−35.0689	−1.92466
$333$	0	0
$334$	−5.13330	−0.280882
$335$	0	0
$336$	0	0
$337$	−32.6082	−1.77628	−0.888140	−	0.459573i	$-0.848002\pi$
$337$	−32.6082	−1.77628	−0.888140	+	0.459573i	$0.848002\pi$
$338$	−2.67513	−0.145508
$339$	0	0
$340$	0	0
$341$	−1.02047	−0.0552614
$342$	0	0
$343$	20.0557	1.08291
$344$	24.7572	1.33482
$345$	0	0
$346$	−21.2193	−1.14076
$347$	22.1622	1.18973	0.594865	−	0.803826i	$-0.297206\pi$
$347$	22.1622	1.18973	0.594865	+	0.803826i	$0.297206\pi$
$348$	0	0
$349$	−4.34297	−0.232474	−0.116237	−	0.993222i	$-0.537083\pi$
$349$	−4.34297	−0.232474	−0.116237	+	0.993222i	$0.537083\pi$
$350$	0	0
$351$	0	0
$352$	−2.08840	−0.111312
$353$	6.96827	0.370883	0.185442	−	0.982655i	$-0.440628\pi$
$353$	6.96827	0.370883	0.185442	+	0.982655i	$0.440628\pi$
$354$	0	0
$355$	0	0
$356$	41.9003	2.22071
$357$	0	0
$358$	39.9062	2.10911
$359$	15.0122	0.792315	0.396157	−	0.918183i	$-0.370343\pi$
$359$	15.0122	0.792315	0.396157	+	0.918183i	$0.370343\pi$
$360$	0	0
$361$	7.20123	0.379012
$362$	63.0806	3.31544
$363$	0	0
$364$	11.7938	0.618165
$365$	0	0
$366$	0	0
$367$	27.0738	1.41324	0.706621	−	0.707593i	$-0.250219\pi$
$367$	27.0738	1.41324	0.706621	+	0.707593i	$0.250219\pi$
$368$	67.5936	3.52356
$369$	0	0
$370$	0	0
$371$	−19.4568	−1.01014
$372$	0	0
$373$	3.49341	0.180882	0.0904410	−	0.995902i	$-0.471172\pi$
$373$	3.49341	0.180882	0.0904410	+	0.995902i	$0.471172\pi$
$374$	2.78892	0.144212
$375$	0	0
$376$	22.5877	1.16487
$377$	−3.00000	−0.154508
$378$	0	0
$379$	26.9937	1.38657	0.693286	−	0.720663i	$-0.256162\pi$
$379$	26.9937	1.38657	0.693286	+	0.720663i	$0.256162\pi$
$380$	0	0
$381$	0	0
$382$	22.4387	1.14806
$383$	−5.52118	−0.282119	−0.141059	−	0.990001i	$-0.545051\pi$
$383$	−5.52118	−0.282119	−0.141059	+	0.990001i	$0.545051\pi$
$384$	0	0
$385$	0	0
$386$	−48.5705	−2.47218
$387$	0	0
$388$	97.7762	4.96384
$389$	15.9003	0.806179	0.403090	−	0.915161i	$-0.367936\pi$
$389$	15.9003	0.806179	0.403090	+	0.915161i	$0.367936\pi$
$390$	0	0
$391$	−43.8456	−2.21737
$392$	14.9321	0.754183
$393$	0	0
$394$	−3.19394	−0.160908
$395$	0	0
$396$	0	0
$397$	−17.1998	−0.863234	−0.431617	−	0.902057i	$-0.642057\pi$
$397$	−17.1998	−0.863234	−0.431617	+	0.902057i	$0.642057\pi$
$398$	−1.87399	−0.0939347
$399$	0	0
$400$	0	0
$401$	−19.0435	−0.950987	−0.475493	−	0.879719i	$-0.657730\pi$
$401$	−19.0435	−0.950987	−0.475493	+	0.879719i	$0.657730\pi$
$402$	0	0
$403$	7.79384	0.388239
$404$	18.2750	0.909217
$405$	0	0
$406$	18.3561	0.911000
$407$	0.629270	0.0311918
$408$	0	0
$409$	12.4182	0.614040	0.307020	−	0.951703i	$-0.400668\pi$
$409$	12.4182	0.614040	0.307020	+	0.951703i	$0.400668\pi$
$410$	0	0
$411$	0	0
$412$	72.6067	3.57708
$413$	−23.8178	−1.17200
$414$	0	0
$415$	0	0
$416$	15.9502	0.782021
$417$	0	0
$418$	1.79289	0.0876931
$419$	−6.14174	−0.300043	−0.150022	−	0.988683i	$-0.547934\pi$
$419$	−6.14174	−0.300043	−0.150022	+	0.988683i	$0.547934\pi$
$420$	0	0
$421$	−17.6121	−0.858363	−0.429181	−	0.903218i	$-0.641198\pi$
$421$	−17.6121	−0.858363	−0.429181	+	0.903218i	$0.641198\pi$
$422$	−45.4314	−2.21156
$423$	0	0
$424$	−71.8261	−3.48818
$425$	0	0
$426$	0	0
$427$	−11.8632	−0.574100
$428$	13.6629	0.660422
$429$	0	0
$430$	0	0
$431$	3.65703	0.176153	0.0880765	−	0.996114i	$-0.471928\pi$
$431$	3.65703	0.176153	0.0880765	+	0.996114i	$0.471928\pi$
$432$	0	0
$433$	9.28963	0.446431	0.223216	−	0.974769i	$-0.428345\pi$
$433$	9.28963	0.446431	0.223216	+	0.974769i	$0.428345\pi$
$434$	−47.6883	−2.28911
$435$	0	0
$436$	−48.0567	−2.30150
$437$	−28.1866	−1.34835
$438$	0	0
$439$	27.0581	1.29141	0.645706	−	0.763586i	$-0.276563\pi$
$439$	27.0581	1.29141	0.645706	+	0.763586i	$0.276563\pi$
$440$	0	0
$441$	0	0
$442$	−21.3004	−1.01316
$443$	−7.09428	−0.337059	−0.168530	−	0.985697i	$-0.553902\pi$
$443$	−7.09428	−0.337059	−0.168530	+	0.985697i	$0.553902\pi$
$444$	0	0
$445$	0	0
$446$	2.15633	0.102105
$447$	0	0
$448$	−41.4422	−1.95796
$449$	14.7612	0.696622	0.348311	−	0.937379i	$-0.386755\pi$
$449$	14.7612	0.696622	0.348311	+	0.937379i	$0.386755\pi$
$450$	0	0
$451$	−1.49674	−0.0704786
$452$	−68.8383	−3.23788
$453$	0	0
$454$	20.5515	0.964529
$455$	0	0
$456$	0	0
$457$	2.28233	0.106763	0.0533815	−	0.998574i	$-0.483000\pi$
$457$	2.28233	0.106763	0.0533815	+	0.998574i	$0.483000\pi$
$458$	11.9003	0.556066
$459$	0	0
$460$	0	0
$461$	10.5647	0.492046	0.246023	−	0.969264i	$-0.420876\pi$
$461$	10.5647	0.492046	0.246023	+	0.969264i	$0.420876\pi$
$462$	0	0
$463$	−6.18172	−0.287289	−0.143644	−	0.989629i	$-0.545882\pi$
$463$	−6.18172	−0.287289	−0.143644	+	0.989629i	$0.545882\pi$
$464$	36.8251	1.70956
$465$	0	0
$466$	−27.0884	−1.25485
$467$	−6.32250	−0.292570	−0.146285	−	0.989242i	$-0.546732\pi$
$467$	−6.32250	−0.292570	−0.146285	+	0.989242i	$0.546732\pi$
$468$	0	0
$469$	13.8510	0.639578
$470$	0	0
$471$	0	0
$472$	−87.9253	−4.04709
$473$	−0.383904	−0.0176519
$474$	0	0
$475$	0	0
$476$	93.9072	4.30423
$477$	0	0
$478$	−51.9946	−2.37818
$479$	−3.33075	−0.152186	−0.0760929	−	0.997101i	$-0.524245\pi$
$479$	−3.33075	−0.152186	−0.0760929	+	0.997101i	$0.524245\pi$
$480$	0	0
$481$	−4.80606	−0.219138
$482$	17.3054	0.788237
$483$	0	0
$484$	−56.6312	−2.57414
$485$	0	0
$486$	0	0
$487$	8.44358	0.382615	0.191308	−	0.981530i	$-0.438727\pi$
$487$	8.44358	0.382615	0.191308	+	0.981530i	$0.438727\pi$
$488$	−43.7938	−1.98245
$489$	0	0
$490$	0	0
$491$	23.3112	1.05202	0.526011	−	0.850478i	$-0.323687\pi$
$491$	23.3112	1.05202	0.526011	+	0.850478i	$0.323687\pi$
$492$	0	0
$493$	−23.8872	−1.07582
$494$	−13.6932	−0.616088
$495$	0	0
$496$	−95.6697	−4.29570
$497$	−34.4083	−1.54343
$498$	0	0
$499$	−31.8651	−1.42648	−0.713239	−	0.700921i	$-0.752773\pi$
$499$	−31.8651	−1.42648	−0.713239	+	0.700921i	$0.752773\pi$
$500$	0	0
$501$	0	0
$502$	−47.5329	−2.12150
$503$	9.43136	0.420524	0.210262	−	0.977645i	$-0.432568\pi$
$503$	9.43136	0.420524	0.210262	+	0.977645i	$0.432568\pi$
$504$	0	0
$505$	0	0
$506$	−1.92875	−0.0857433
$507$	0	0
$508$	45.0191	1.99740
$509$	−37.4821	−1.66137	−0.830684	−	0.556745i	$-0.812050\pi$
$509$	−37.4821	−1.66137	−0.830684	+	0.556745i	$0.812050\pi$
$510$	0	0
$511$	2.13189	0.0943092
$512$	11.5017	0.508306
$513$	0	0
$514$	−12.8921	−0.568646
$515$	0	0
$516$	0	0
$517$	−0.350262	−0.0154045
$518$	29.4069	1.29207
$519$	0	0
$520$	0	0
$521$	−15.9248	−0.697677	−0.348839	−	0.937183i	$-0.613424\pi$
$521$	−15.9248	−0.697677	−0.348839	+	0.937183i	$0.613424\pi$
$522$	0	0
$523$	2.27645	0.0995424	0.0497712	−	0.998761i	$-0.484151\pi$
$523$	2.27645	0.0995424	0.0497712	+	0.998761i	$0.484151\pi$
$524$	−98.1944	−4.28964
$525$	0	0
$526$	−3.49200	−0.152258
$527$	62.0576	2.70327
$528$	0	0
$529$	7.32250	0.318370
$530$	0	0
$531$	0	0
$532$	60.3693	2.61734
$533$	11.4314	0.495147
$534$	0	0
$535$	0	0
$536$	51.1319	2.20856
$537$	0	0
$538$	−23.5379	−1.01479
$539$	−0.231548	−0.00997348
$540$	0	0
$541$	19.3317	0.831135	0.415567	−	0.909562i	$-0.363583\pi$
$541$	19.3317	0.831135	0.415567	+	0.909562i	$0.363583\pi$
$542$	−46.6180	−2.00241
$543$	0	0
$544$	127.001	5.44514
$545$	0	0
$546$	0	0
$547$	−40.9438	−1.75063	−0.875316	−	0.483552i	$-0.839347\pi$
$547$	−40.9438	−1.75063	−0.875316	+	0.483552i	$0.839347\pi$
$548$	−39.5125	−1.68789
$549$	0	0
$550$	0	0
$551$	−15.3561	−0.654194
$552$	0	0
$553$	−20.2579	−0.861453
$554$	−31.4010	−1.33410
$555$	0	0
$556$	39.2506	1.66460
$557$	36.4953	1.54636	0.773178	−	0.634189i	$-0.218666\pi$
$557$	36.4953	1.54636	0.773178	+	0.634189i	$0.218666\pi$
$558$	0	0
$559$	2.93207	0.124013
$560$	0	0
$561$	0	0
$562$	−38.9624	−1.64353
$563$	−1.04349	−0.0439779	−0.0219890	−	0.999758i	$-0.507000\pi$
$563$	−1.04349	−0.0439779	−0.0219890	+	0.999758i	$0.507000\pi$
$564$	0	0
$565$	0	0
$566$	−69.8251	−2.93497
$567$	0	0
$568$	−127.021	−5.32968
$569$	39.4807	1.65512	0.827559	−	0.561378i	$-0.189729\pi$
$569$	39.4807	1.65512	0.827559	+	0.561378i	$0.189729\pi$
$570$	0	0
$571$	35.4920	1.48529	0.742647	−	0.669683i	$-0.233570\pi$
$571$	35.4920	1.48529	0.742647	+	0.669683i	$0.233570\pi$
$572$	−0.675131	−0.0282286
$573$	0	0
$574$	−69.9452	−2.91946
$575$	0	0
$576$	0	0
$577$	−36.0625	−1.50130	−0.750652	−	0.660698i	$-0.770260\pi$
$577$	−36.0625	−1.50130	−0.750652	+	0.660698i	$0.770260\pi$
$578$	−124.125	−5.16292
$579$	0	0
$580$	0	0
$581$	15.5560	0.645370
$582$	0	0
$583$	1.11379	0.0461284
$584$	7.87002	0.325664
$585$	0	0
$586$	24.2736	1.00273
$587$	22.4944	0.928442	0.464221	−	0.885719i	$-0.346334\pi$
$587$	22.4944	0.928442	0.464221	+	0.885719i	$0.346334\pi$
$588$	0	0
$589$	39.8945	1.64382
$590$	0	0
$591$	0	0
$592$	58.9946	2.42466
$593$	28.8627	1.18525	0.592625	−	0.805478i	$-0.298092\pi$
$593$	28.8627	1.18525	0.592625	+	0.805478i	$0.298092\pi$
$594$	0	0
$595$	0	0
$596$	−4.57452	−0.187379
$597$	0	0
$598$	14.7308	0.602389
$599$	−2.62927	−0.107429	−0.0537145	−	0.998556i	$-0.517106\pi$
$599$	−2.62927	−0.107429	−0.0537145	+	0.998556i	$0.517106\pi$
$600$	0	0
$601$	−31.6048	−1.28919	−0.644594	−	0.764525i	$-0.722974\pi$
$601$	−31.6048	−1.28919	−0.644594	+	0.764525i	$0.722974\pi$
$602$	−17.9405	−0.731200
$603$	0	0
$604$	15.1441	0.616205
$605$	0	0
$606$	0	0
$607$	13.9149	0.564790	0.282395	−	0.959298i	$-0.408871\pi$
$607$	13.9149	0.564790	0.282395	+	0.959298i	$0.408871\pi$
$608$	81.6444	3.31112
$609$	0	0
$610$	0	0
$611$	2.67513	0.108224
$612$	0	0
$613$	25.4471	1.02780	0.513899	−	0.857851i	$-0.328201\pi$
$613$	25.4471	1.02780	0.513899	+	0.857851i	$0.328201\pi$
$614$	30.4445	1.22864
$615$	0	0
$616$	2.52865	0.101882
$617$	−30.6497	−1.23391	−0.616956	−	0.786998i	$-0.711634\pi$
$617$	−30.6497	−1.23391	−0.616956	+	0.786998i	$0.711634\pi$
$618$	0	0
$619$	−7.99412	−0.321311	−0.160655	−	0.987011i	$-0.551361\pi$
$619$	−7.99412	−0.321311	−0.160655	+	0.987011i	$0.551361\pi$
$620$	0	0
$621$	0	0
$622$	24.6761	0.989421
$623$	−18.5863	−0.744643
$624$	0	0
$625$	0	0
$626$	−50.2081	−2.00672
$627$	0	0
$628$	81.3752	3.24722
$629$	−38.2677	−1.52583
$630$	0	0
$631$	41.3463	1.64597	0.822985	−	0.568063i	$-0.192307\pi$
$631$	41.3463	1.64597	0.822985	+	0.568063i	$0.192307\pi$
$632$	−74.7835	−2.97473
$633$	0	0
$634$	28.2071	1.12025
$635$	0	0
$636$	0	0
$637$	1.76845	0.0700686
$638$	−1.05079	−0.0416010
$639$	0	0
$640$	0	0
$641$	−3.26187	−0.128836	−0.0644180	−	0.997923i	$-0.520519\pi$
$641$	−3.26187	−0.128836	−0.0644180	+	0.997923i	$0.520519\pi$
$642$	0	0
$643$	−6.82065	−0.268980	−0.134490	−	0.990915i	$-0.542940\pi$
$643$	−6.82065	−0.268980	−0.134490	+	0.990915i	$0.542940\pi$
$644$	−64.9438	−2.55915
$645$	0	0
$646$	−109.031	−4.28976
$647$	30.9478	1.21668	0.608342	−	0.793675i	$-0.291835\pi$
$647$	30.9478	1.21668	0.608342	+	0.793675i	$0.291835\pi$
$648$	0	0
$649$	1.36344	0.0535195
$650$	0	0
$651$	0	0
$652$	−89.8818	−3.52004
$653$	−16.9003	−0.661361	−0.330681	−	0.943743i	$-0.607278\pi$
$653$	−16.9003	−0.661361	−0.330681	+	0.943743i	$0.607278\pi$
$654$	0	0
$655$	0	0
$656$	−140.320	−5.47859
$657$	0	0
$658$	−16.3684	−0.638105
$659$	−4.79621	−0.186834	−0.0934170	−	0.995627i	$-0.529779\pi$
$659$	−4.79621	−0.186834	−0.0934170	+	0.995627i	$0.529779\pi$
$660$	0	0
$661$	−0.105540	−0.00410503	−0.00205251	−	0.999998i	$-0.500653\pi$
$661$	−0.105540	−0.00410503	−0.00205251	+	0.999998i	$0.500653\pi$
$662$	−13.2593	−0.515338
$663$	0	0
$664$	57.4260	2.22856
$665$	0	0
$666$	0	0
$667$	16.5198	0.639648
$668$	9.89446	0.382828
$669$	0	0
$670$	0	0
$671$	0.679100	0.0262164
$672$	0	0
$673$	3.86670	0.149050	0.0745251	−	0.997219i	$-0.476256\pi$
$673$	3.86670	0.149050	0.0745251	+	0.997219i	$0.476256\pi$
$674$	87.2311	3.36002
$675$	0	0
$676$	5.15633	0.198320
$677$	−30.1622	−1.15923	−0.579614	−	0.814891i	$-0.696797\pi$
$677$	−30.1622	−1.15923	−0.579614	+	0.814891i	$0.696797\pi$
$678$	0	0
$679$	−43.3719	−1.66446
$680$	0	0
$681$	0	0
$682$	2.72989	0.104533
$683$	17.9805	0.688004	0.344002	−	0.938969i	$-0.388217\pi$
$683$	17.9805	0.688004	0.344002	+	0.938969i	$0.388217\pi$
$684$	0	0
$685$	0	0
$686$	−53.6516	−2.04843
$687$	0	0
$688$	−35.9913	−1.37216
$689$	−8.50659	−0.324075
$690$	0	0
$691$	31.4363	1.19589	0.597946	−	0.801536i	$-0.295984\pi$
$691$	31.4363	1.19589	0.597946	+	0.801536i	$0.295984\pi$
$692$	40.9003	1.55480
$693$	0	0
$694$	−59.2868	−2.25050
$695$	0	0
$696$	0	0
$697$	91.0210	3.44766
$698$	11.6180	0.439748
$699$	0	0
$700$	0	0
$701$	−5.32724	−0.201207	−0.100604	−	0.994927i	$-0.532077\pi$
$701$	−5.32724	−0.201207	−0.100604	+	0.994927i	$0.532077\pi$
$702$	0	0
$703$	−24.6009	−0.927839
$704$	2.37233	0.0894105
$705$	0	0
$706$	−18.6410	−0.701564
$707$	−8.10650	−0.304876
$708$	0	0
$709$	27.8046	1.04423	0.522113	−	0.852876i	$-0.325144\pi$
$709$	27.8046	1.04423	0.522113	+	0.852876i	$0.325144\pi$
$710$	0	0
$711$	0	0
$712$	−68.6126	−2.57137
$713$	−42.9175	−1.60727
$714$	0	0
$715$	0	0
$716$	−76.9194	−2.87461
$717$	0	0
$718$	−40.1596	−1.49874
$719$	0.992706	0.0370217	0.0185108	−	0.999829i	$-0.494107\pi$
$719$	0.992706	0.0370217	0.0185108	+	0.999829i	$0.494107\pi$
$720$	0	0
$721$	−32.2071	−1.19946
$722$	−19.2642	−0.716941
$723$	0	0
$724$	−121.588	−4.51879
$725$	0	0
$726$	0	0
$727$	−40.0468	−1.48525	−0.742627	−	0.669705i	$-0.766420\pi$
$727$	−40.0468	−1.48525	−0.742627	+	0.669705i	$0.766420\pi$
$728$	−19.3127	−0.715774
$729$	0	0
$730$	0	0
$731$	23.3463	0.863494
$732$	0	0
$733$	−30.8627	−1.13994	−0.569970	−	0.821665i	$-0.693045\pi$
$733$	−30.8627	−1.13994	−0.569970	+	0.821665i	$0.693045\pi$
$734$	−72.4260	−2.67329
$735$	0	0
$736$	−87.8310	−3.23749
$737$	−0.792890	−0.0292065
$738$	0	0
$739$	34.1309	1.25553	0.627763	−	0.778404i	$-0.283971\pi$
$739$	34.1309	1.25553	0.627763	+	0.778404i	$0.283971\pi$
$740$	0	0
$741$	0	0
$742$	52.0494	1.91079
$743$	14.1916	0.520638	0.260319	−	0.965523i	$-0.416172\pi$
$743$	14.1916	0.520638	0.260319	+	0.965523i	$0.416172\pi$
$744$	0	0
$745$	0	0
$746$	−9.34534	−0.342157
$747$	0	0
$748$	−5.37565	−0.196553
$749$	−6.06063	−0.221451
$750$	0	0
$751$	13.4050	0.489156	0.244578	−	0.969630i	$-0.421351\pi$
$751$	13.4050	0.489156	0.244578	+	0.969630i	$0.421351\pi$
$752$	−32.8373	−1.19745
$753$	0	0
$754$	8.02539	0.292268
$755$	0	0
$756$	0	0
$757$	5.76116	0.209393	0.104696	−	0.994504i	$-0.466613\pi$
$757$	5.76116	0.209393	0.104696	+	0.994504i	$0.466613\pi$
$758$	−72.2116	−2.62284
$759$	0	0
$760$	0	0
$761$	9.05334	0.328183	0.164092	−	0.986445i	$-0.447531\pi$
$761$	9.05334	0.328183	0.164092	+	0.986445i	$0.447531\pi$
$762$	0	0
$763$	21.3171	0.771731
$764$	−43.2506	−1.56475
$765$	0	0
$766$	14.7699	0.533657
$767$	−10.4133	−0.376001
$768$	0	0
$769$	−38.8383	−1.40054	−0.700272	−	0.713876i	$-0.746938\pi$
$769$	−38.8383	−1.40054	−0.700272	+	0.713876i	$0.746938\pi$
$770$	0	0
$771$	0	0
$772$	93.6199	3.36945
$773$	23.7381	0.853801	0.426901	−	0.904299i	$-0.359605\pi$
$773$	23.7381	0.853801	0.426901	+	0.904299i	$0.359605\pi$
$774$	0	0
$775$	0	0
$776$	−160.111	−5.74763
$777$	0	0
$778$	−42.5355	−1.52497
$779$	58.5139	2.09648
$780$	0	0
$781$	1.96968	0.0704808
$782$	117.293	4.19438
$783$	0	0
$784$	−21.7078	−0.775279
$785$	0	0
$786$	0	0
$787$	19.1803	0.683704	0.341852	−	0.939754i	$-0.388946\pi$
$787$	19.1803	0.683704	0.341852	+	0.939754i	$0.388946\pi$
$788$	6.15633	0.219310
$789$	0	0
$790$	0	0
$791$	30.5355	1.08572
$792$	0	0
$793$	−5.18664	−0.184183
$794$	46.0118	1.63290
$795$	0	0
$796$	3.61213	0.128028
$797$	−51.8627	−1.83707	−0.918536	−	0.395337i	$-0.870628\pi$
$797$	−51.8627	−1.83707	−0.918536	+	0.395337i	$0.870628\pi$
$798$	0	0
$799$	21.3004	0.753555
$800$	0	0
$801$	0	0
$802$	50.9438	1.79889
$803$	−0.122039	−0.00430665
$804$	0	0
$805$	0	0
$806$	−20.8496	−0.734394
$807$	0	0
$808$	−29.9257	−1.05278
$809$	18.1866	0.639408	0.319704	−	0.947517i	$-0.396416\pi$
$809$	18.1866	0.639408	0.319704	+	0.947517i	$0.396416\pi$
$810$	0	0
$811$	4.71133	0.165437	0.0827185	−	0.996573i	$-0.473640\pi$
$811$	4.71133	0.165437	0.0827185	+	0.996573i	$0.473640\pi$
$812$	−35.3815	−1.24165
$813$	0	0
$814$	−1.68338	−0.0590024
$815$	0	0
$816$	0	0
$817$	15.0084	0.525079
$818$	−33.2203	−1.16152
$819$	0	0
$820$	0	0
$821$	33.8046	1.17979	0.589895	−	0.807480i	$-0.299169\pi$
$821$	33.8046	1.17979	0.589895	+	0.807480i	$0.299169\pi$
$822$	0	0
$823$	−2.82653	−0.0985267	−0.0492633	−	0.998786i	$-0.515687\pi$
$823$	−2.82653	−0.0985267	−0.0492633	+	0.998786i	$0.515687\pi$
$824$	−118.895	−4.14190
$825$	0	0
$826$	63.7158	2.21696
$827$	−39.9560	−1.38941	−0.694704	−	0.719296i	$-0.744465\pi$
$827$	−39.9560	−1.38941	−0.694704	+	0.719296i	$0.744465\pi$
$828$	0	0
$829$	1.30280	0.0452482	0.0226241	−	0.999744i	$-0.492798\pi$
$829$	1.30280	0.0452482	0.0226241	+	0.999744i	$0.492798\pi$
$830$	0	0
$831$	0	0
$832$	−18.1187	−0.628153
$833$	14.0811	0.487881
$834$	0	0
$835$	0	0
$836$	−3.45580	−0.119521
$837$	0	0
$838$	16.4299	0.567563
$839$	17.9551	0.619879	0.309939	−	0.950756i	$-0.399691\pi$
$839$	17.9551	0.619879	0.309939	+	0.950756i	$0.399691\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	47.1147	1.62368
$843$	0	0
$844$	87.5691	3.01425
$845$	0	0
$846$	0	0
$847$	25.1206	0.863155
$848$	104.419	3.58575
$849$	0	0
$850$	0	0
$851$	26.4650	0.907209
$852$	0	0
$853$	−43.3258	−1.48345	−0.741724	−	0.670705i	$-0.765992\pi$
$853$	−43.3258	−1.48345	−0.741724	+	0.670705i	$0.765992\pi$
$854$	31.7356	1.08597
$855$	0	0
$856$	−22.3733	−0.764703
$857$	15.5223	0.530232	0.265116	−	0.964216i	$-0.414590\pi$
$857$	15.5223	0.530232	0.265116	+	0.964216i	$0.414590\pi$
$858$	0	0
$859$	14.0957	0.480939	0.240469	−	0.970657i	$-0.422699\pi$
$859$	14.0957	0.480939	0.240469	+	0.970657i	$0.422699\pi$
$860$	0	0
$861$	0	0
$862$	−9.78304	−0.333212
$863$	46.6105	1.58664	0.793320	−	0.608804i	$-0.208351\pi$
$863$	46.6105	1.58664	0.793320	+	0.608804i	$0.208351\pi$
$864$	0	0
$865$	0	0
$866$	−24.8510	−0.844470
$867$	0	0
$868$	91.9194	3.11995
$869$	1.15965	0.0393384
$870$	0	0
$871$	6.05571	0.205190
$872$	78.6937	2.66491
$873$	0	0
$874$	75.4030	2.55054
$875$	0	0
$876$	0	0
$877$	−33.3357	−1.12567	−0.562833	−	0.826571i	$-0.690289\pi$
$877$	−33.3357	−1.12567	−0.562833	+	0.826571i	$0.690289\pi$
$878$	−72.3839	−2.44284
$879$	0	0
$880$	0	0
$881$	−22.1793	−0.747241	−0.373621	−	0.927582i	$-0.621884\pi$
$881$	−22.1793	−0.747241	−0.373621	+	0.927582i	$0.621884\pi$
$882$	0	0
$883$	−27.4109	−0.922450	−0.461225	−	0.887283i	$-0.652590\pi$
$883$	−27.4109	−0.922450	−0.461225	+	0.887283i	$0.652590\pi$
$884$	41.0567	1.38089
$885$	0	0
$886$	18.9781	0.637582
$887$	−1.33170	−0.0447142	−0.0223571	−	0.999750i	$-0.507117\pi$
$887$	−1.33170	−0.0447142	−0.0223571	+	0.999750i	$0.507117\pi$
$888$	0	0
$889$	−19.9697	−0.669762
$890$	0	0
$891$	0	0
$892$	−4.15633	−0.139164
$893$	13.6932	0.458226
$894$	0	0
$895$	0	0
$896$	37.8989	1.26611
$897$	0	0
$898$	−39.4880	−1.31773
$899$	−23.3815	−0.779818
$900$	0	0
$901$	−67.7328	−2.25651
$902$	4.00397	0.133318
$903$	0	0
$904$	112.724	3.74915
$905$	0	0
$906$	0	0
$907$	−31.1754	−1.03516	−0.517581	−	0.855634i	$-0.673167\pi$
$907$	−31.1754	−1.03516	−0.517581	+	0.855634i	$0.673167\pi$
$908$	−39.6131	−1.31461
$909$	0	0
$910$	0	0
$911$	−9.56722	−0.316976	−0.158488	−	0.987361i	$-0.550662\pi$
$911$	−9.56722	−0.316976	−0.158488	+	0.987361i	$0.550662\pi$
$912$	0	0
$913$	−0.890491	−0.0294709
$914$	−6.10554	−0.201953
$915$	0	0
$916$	−22.9380	−0.757891
$917$	43.5574	1.43839
$918$	0	0
$919$	−35.8397	−1.18224	−0.591121	−	0.806583i	$-0.701315\pi$
$919$	−35.8397	−1.18224	−0.591121	+	0.806583i	$0.701315\pi$
$920$	0	0
$921$	0	0
$922$	−28.2619	−0.930755
$923$	−15.0435	−0.495163
$924$	0	0
$925$	0	0
$926$	16.5369	0.543436
$927$	0	0
$928$	−47.8505	−1.57077
$929$	−19.7137	−0.646785	−0.323393	−	0.946265i	$-0.604823\pi$
$929$	−19.7137	−0.646785	−0.323393	+	0.946265i	$0.604823\pi$
$930$	0	0
$931$	9.05220	0.296674
$932$	52.2130	1.71029
$933$	0	0
$934$	16.9135	0.553427
$935$	0	0
$936$	0	0
$937$	−57.2736	−1.87105	−0.935524	−	0.353263i	$-0.885072\pi$
$937$	−57.2736	−1.87105	−0.935524	+	0.353263i	$0.885072\pi$
$938$	−37.0532	−1.20983
$939$	0	0
$940$	0	0
$941$	25.2447	0.822954	0.411477	−	0.911420i	$-0.365013\pi$
$941$	25.2447	0.822954	0.411477	+	0.911420i	$0.365013\pi$
$942$	0	0
$943$	−62.9478	−2.04986
$944$	127.823	4.16029
$945$	0	0
$946$	1.02699	0.0333904
$947$	5.04254	0.163860	0.0819302	−	0.996638i	$-0.473892\pi$
$947$	5.04254	0.163860	0.0819302	+	0.996638i	$0.473892\pi$
$948$	0	0
$949$	0.932071	0.0302563
$950$	0	0
$951$	0	0
$952$	−153.775	−4.98387
$953$	15.4426	0.500236	0.250118	−	0.968215i	$-0.419531\pi$
$953$	15.4426	0.500236	0.250118	+	0.968215i	$0.419531\pi$
$954$	0	0
$955$	0	0
$956$	100.220	3.24134
$957$	0	0
$958$	8.91019	0.287875
$959$	17.5271	0.565978
$960$	0	0
$961$	29.7440	0.959484
$962$	12.8568	0.414521
$963$	0	0
$964$	−33.3561	−1.07433
$965$	0	0
$966$	0	0
$967$	−30.2979	−0.974314	−0.487157	−	0.873314i	$-0.661966\pi$
$967$	−30.2979	−0.974314	−0.487157	+	0.873314i	$0.661966\pi$
$968$	92.7347	2.98060
$969$	0	0
$970$	0	0
$971$	40.7367	1.30730	0.653652	−	0.756795i	$-0.273236\pi$
$971$	40.7367	1.30730	0.653652	+	0.756795i	$0.273236\pi$
$972$	0	0
$973$	−17.4109	−0.558168
$974$	−22.5877	−0.723756
$975$	0	0
$976$	63.6662	2.03791
$977$	−53.1608	−1.70076	−0.850382	−	0.526166i	$-0.823629\pi$
$977$	−53.1608	−1.70076	−0.850382	+	0.526166i	$0.823629\pi$
$978$	0	0
$979$	1.06396	0.0340043
$980$	0	0
$981$	0	0
$982$	−62.3606	−1.99001
$983$	−11.1831	−0.356687	−0.178343	−	0.983968i	$-0.557074\pi$
$983$	−11.1831	−0.356687	−0.178343	+	0.983968i	$0.557074\pi$
$984$	0	0
$985$	0	0
$986$	63.9013	2.03503
$987$	0	0
$988$	26.3938	0.839697
$989$	−16.1457	−0.513404
$990$	0	0
$991$	−57.9208	−1.83992	−0.919958	−	0.392018i	$-0.871777\pi$
$991$	−57.9208	−1.83992	−0.919958	+	0.392018i	$0.871777\pi$
$992$	124.313	3.94695
$993$	0	0
$994$	92.0468	2.91955
$995$	0	0
$996$	0	0
$997$	−27.4241	−0.868529	−0.434265	−	0.900785i	$-0.642992\pi$
$997$	−27.4241	−0.868529	−0.434265	+	0.900785i	$0.642992\pi$
$998$	85.2433	2.69833
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2925.2.a.be.1.1		3
3.2	odd	2		975.2.a.q.1.3	yes	3
5.2	odd	4		2925.2.c.y.2224.1		6
5.3	odd	4		2925.2.c.y.2224.6		6
5.4	even	2		2925.2.a.bk.1.3		3
15.2	even	4		975.2.c.k.274.6		6
15.8	even	4		975.2.c.k.274.1		6
15.14	odd	2		975.2.a.m.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
975.2.a.m.1.1	✓	3	15.14	odd	2
975.2.a.q.1.3	yes	3	3.2	odd	2
975.2.c.k.274.1		6	15.8	even	4
975.2.c.k.274.6		6	15.2	even	4
2925.2.a.be.1.1		3	1.1	even	1	trivial
2925.2.a.bk.1.3		3	5.4	even	2
2925.2.c.y.2224.1		6	5.2	odd	4
2925.2.c.y.2224.6		6	5.3	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:	\( N \)	\(=\)	\( 2925 = 3^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2925.a (trivial)

\(f(q)\)	\(=\)	\(q-2.67513 q^{2} +5.15633 q^{4} -2.28726 q^{7} -8.44358 q^{8} +O(q^{10})\)	\(q-2.67513 q^{2} +5.15633 q^{4} -2.28726 q^{7} -8.44358 q^{8} +0.130933 q^{11} -1.00000 q^{13} +6.11871 q^{14} +12.2750 q^{16} -7.96239 q^{17} -5.11871 q^{19} -0.350262 q^{22} +5.50659 q^{23} +2.67513 q^{26} -11.7938 q^{28} +3.00000 q^{29} -7.79384 q^{31} -15.9502 q^{32} +21.3004 q^{34} +4.80606 q^{37} +13.6932 q^{38} -11.4314 q^{41} -2.93207 q^{43} +0.675131 q^{44} -14.7308 q^{46} -2.67513 q^{47} -1.76845 q^{49} -5.15633 q^{52} +8.50659 q^{53} +19.3127 q^{56} -8.02539 q^{58} +10.4133 q^{59} +5.18664 q^{61} +20.8496 q^{62} +18.1187 q^{64} -6.05571 q^{67} -41.0567 q^{68} +15.0435 q^{71} -0.932071 q^{73} -12.8568 q^{74} -26.3938 q^{76} -0.299477 q^{77} +8.85685 q^{79} +30.5804 q^{82} -6.80114 q^{83} +7.84367 q^{86} -1.10554 q^{88} +8.12601 q^{89} +2.28726 q^{91} +28.3938 q^{92} +7.15633 q^{94} +18.9624 q^{97} +4.73084 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 5 q^{4} - q^{7} - 9 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 5 * q^4 - q^7 - 9 * q^8	\( 3 q - 3 q^{2} + 5 q^{4} - q^{7} - 9 q^{8} + 5 q^{11} - 3 q^{13} - 3 q^{14} + 5 q^{16} - 13 q^{17} + 6 q^{19} + 9 q^{22} - 4 q^{23} + 3 q^{26} - 9 q^{28} + 9 q^{29} + 3 q^{31} - 11 q^{32} + 17 q^{34} + 14 q^{37} + 8 q^{38} + 8 q^{41} - 3 q^{44} - 22 q^{46} - 3 q^{47} + 6 q^{49} - 5 q^{52} + 5 q^{53} + 37 q^{56} - 9 q^{58} + 17 q^{59} + 3 q^{61} + 19 q^{62} + 33 q^{64} - q^{67} - 39 q^{68} + 2 q^{71} + 6 q^{73} - 8 q^{74} - 26 q^{76} - 21 q^{77} - 4 q^{79} + 26 q^{82} - 7 q^{83} + 34 q^{86} - 23 q^{88} + 16 q^{89} + q^{91} + 32 q^{92} + 11 q^{94} + 46 q^{97} - 8 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 5 * q^4 - q^7 - 9 * q^8 + 5 * q^11 - 3 * q^13 - 3 * q^14 + 5 * q^16 - 13 * q^17 + 6 * q^19 + 9 * q^22 - 4 * q^23 + 3 * q^26 - 9 * q^28 + 9 * q^29 + 3 * q^31 - 11 * q^32 + 17 * q^34 + 14 * q^37 + 8 * q^38 + 8 * q^41 - 3 * q^44 - 22 * q^46 - 3 * q^47 + 6 * q^49 - 5 * q^52 + 5 * q^53 + 37 * q^56 - 9 * q^58 + 17 * q^59 + 3 * q^61 + 19 * q^62 + 33 * q^64 - q^67 - 39 * q^68 + 2 * q^71 + 6 * q^73 - 8 * q^74 - 26 * q^76 - 21 * q^77 - 4 * q^79 + 26 * q^82 - 7 * q^83 + 34 * q^86 - 23 * q^88 + 16 * q^89 + q^91 + 32 * q^92 + 11 * q^94 + 46 * q^97 - 8 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more