LMFDB - Embedded newform 2925.2.a.bh.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:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 195)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.34292$ 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)

$f(q)$	$=$	$q-2.48929 q^{2} +4.19656 q^{4} +1.19656 q^{7} -5.46787 q^{8} +O(q^{10})$	$q-2.48929 q^{2} +4.19656 q^{4} +1.19656 q^{7} -5.46787 q^{8} +1.19656 q^{11} -1.00000 q^{13} -2.97858 q^{14} +5.21798 q^{16} +6.17513 q^{17} +6.97858 q^{19} -2.97858 q^{22} +4.17513 q^{23} +2.48929 q^{26} +5.02142 q^{28} -6.00000 q^{29} -2.97858 q^{31} -2.05333 q^{32} -15.3717 q^{34} -7.78202 q^{37} -17.3717 q^{38} +6.17513 q^{41} +9.95715 q^{43} +5.02142 q^{44} -10.3931 q^{46} -1.02142 q^{47} -5.56825 q^{49} -4.19656 q^{52} +10.1751 q^{53} -6.54262 q^{56} +14.9357 q^{58} -5.37169 q^{59} +12.5682 q^{61} +7.41454 q^{62} -5.32464 q^{64} -9.37169 q^{67} +25.9143 q^{68} +5.19656 q^{71} +11.9572 q^{73} +19.3717 q^{74} +29.2860 q^{76} +1.43175 q^{77} -1.78202 q^{79} -15.3717 q^{82} -5.37169 q^{83} -24.7862 q^{86} -6.54262 q^{88} -10.1751 q^{89} -1.19656 q^{91} +17.5212 q^{92} +2.54262 q^{94} +1.82487 q^{97} +13.8610 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 8 q^{4} - q^{7} + 6 q^{8}+O(q^{10}) $ 3 * q + 8 * q^4 - q^7 + 6 * q^8	$ 3 q + 8 q^{4} - q^{7} + 6 q^{8} - q^{11} - 3 q^{13} + 6 q^{14} + 26 q^{16} - q^{17} + 6 q^{19} + 6 q^{22} - 7 q^{23} + 30 q^{28} - 18 q^{29} + 6 q^{31} + 22 q^{32} - 22 q^{34} - 13 q^{37} - 28 q^{38} - q^{41} + 30 q^{44} - 22 q^{46} - 18 q^{47} + 12 q^{49} - 8 q^{52} + 11 q^{53} + 16 q^{56} + 8 q^{59} + 9 q^{61} + 28 q^{62} + 30 q^{64} - 4 q^{67} + 18 q^{68} + 11 q^{71} + 6 q^{73} + 34 q^{74} + 4 q^{76} + 33 q^{77} + 5 q^{79} - 22 q^{82} + 8 q^{83} - 56 q^{86} + 16 q^{88} - 11 q^{89} + q^{91} + 2 q^{92} - 28 q^{94} + 25 q^{97} + 10 q^{98}+O(q^{100}) $ 3 * q + 8 * q^4 - q^7 + 6 * q^8 - q^11 - 3 * q^13 + 6 * q^14 + 26 * q^16 - q^17 + 6 * q^19 + 6 * q^22 - 7 * q^23 + 30 * q^28 - 18 * q^29 + 6 * q^31 + 22 * q^32 - 22 * q^34 - 13 * q^37 - 28 * q^38 - q^41 + 30 * q^44 - 22 * q^46 - 18 * q^47 + 12 * q^49 - 8 * q^52 + 11 * q^53 + 16 * q^56 + 8 * q^59 + 9 * q^61 + 28 * q^62 + 30 * q^64 - 4 * q^67 + 18 * q^68 + 11 * q^71 + 6 * q^73 + 34 * q^74 + 4 * q^76 + 33 * q^77 + 5 * q^79 - 22 * q^82 + 8 * q^83 - 56 * q^86 + 16 * q^88 - 11 * q^89 + q^91 + 2 * q^92 - 28 * q^94 + 25 * q^97 + 10 * 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 $

$2$	−2.48929	−1.76019	−0.880096	−	0.474795i	$-0.842522\pi$
$2$	−2.48929	−1.76019	−0.880096	+	0.474795i	$0.842522\pi$
$3$	0	0
$3$	0	0
$4$	4.19656	2.09828
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.19656	0.452256	0.226128	−	0.974098i	$-0.427393\pi$
$7$	1.19656	0.452256	0.226128	+	0.974098i	$0.427393\pi$
$8$	−5.46787	−1.93318
$9$	0	0
$10$	0	0
$11$	1.19656	0.360776	0.180388	−	0.983596i	$-0.442265\pi$
$11$	1.19656	0.360776	0.180388	+	0.983596i	$0.442265\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−2.97858	−0.796058
$15$	0	0
$16$	5.21798	1.30450
$17$	6.17513	1.49769	0.748845	−	0.662745i	$-0.230609\pi$
$17$	6.17513	1.49769	0.748845	+	0.662745i	$0.230609\pi$
$18$	0	0
$19$	6.97858	1.60100	0.800498	−	0.599336i	$-0.204569\pi$
$19$	6.97858	1.60100	0.800498	+	0.599336i	$0.204569\pi$
$20$	0	0
$21$	0	0
$22$	−2.97858	−0.635035
$23$	4.17513	0.870576	0.435288	−	0.900291i	$-0.356647\pi$
$23$	4.17513	0.870576	0.435288	+	0.900291i	$0.356647\pi$
$24$	0	0
$25$	0	0
$26$	2.48929	0.488190
$27$	0	0
$28$	5.02142	0.948960
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	−2.97858	−0.534968	−0.267484	−	0.963562i	$-0.586192\pi$
$31$	−2.97858	−0.534968	−0.267484	+	0.963562i	$0.586192\pi$
$32$	−2.05333	−0.362980
$33$	0	0
$34$	−15.3717	−2.63622
$35$	0	0
$36$	0	0
$37$	−7.78202	−1.27936	−0.639678	−	0.768643i	$-0.720932\pi$
$37$	−7.78202	−1.27936	−0.639678	+	0.768643i	$0.720932\pi$
$38$	−17.3717	−2.81806
$39$	0	0
$40$	0	0
$41$	6.17513	0.964394	0.482197	−	0.876063i	$-0.339839\pi$
$41$	6.17513	0.964394	0.482197	+	0.876063i	$0.339839\pi$
$42$	0	0
$43$	9.95715	1.51845	0.759226	−	0.650827i	$-0.225578\pi$
$43$	9.95715	1.51845	0.759226	+	0.650827i	$0.225578\pi$
$44$	5.02142	0.757008
$45$	0	0
$46$	−10.3931	−1.53238
$47$	−1.02142	−0.148990	−0.0744949	−	0.997221i	$-0.523734\pi$
$47$	−1.02142	−0.148990	−0.0744949	+	0.997221i	$0.523734\pi$
$48$	0	0
$49$	−5.56825	−0.795464
$50$	0	0
$51$	0	0
$52$	−4.19656	−0.581958
$53$	10.1751	1.39766	0.698831	−	0.715287i	$-0.253704\pi$
$53$	10.1751	1.39766	0.698831	+	0.715287i	$0.253704\pi$
$54$	0	0
$55$	0	0
$56$	−6.54262	−0.874294
$57$	0	0
$58$	14.9357	1.96116
$59$	−5.37169	−0.699335	−0.349667	−	0.936874i	$-0.613705\pi$
$59$	−5.37169	−0.699335	−0.349667	+	0.936874i	$0.613705\pi$
$60$	0	0
$61$	12.5682	1.60920	0.804600	−	0.593818i	$-0.202380\pi$
$61$	12.5682	1.60920	0.804600	+	0.593818i	$0.202380\pi$
$62$	7.41454	0.941647
$63$	0	0
$64$	−5.32464	−0.665579
$65$	0	0
$66$	0	0
$67$	−9.37169	−1.14493	−0.572467	−	0.819928i	$-0.694014\pi$
$67$	−9.37169	−1.14493	−0.572467	+	0.819928i	$0.694014\pi$
$68$	25.9143	3.14257
$69$	0	0
$70$	0	0
$71$	5.19656	0.616718	0.308359	−	0.951270i	$-0.400220\pi$
$71$	5.19656	0.616718	0.308359	+	0.951270i	$0.400220\pi$
$72$	0	0
$73$	11.9572	1.39948	0.699740	−	0.714398i	$-0.253299\pi$
$73$	11.9572	1.39948	0.699740	+	0.714398i	$0.253299\pi$
$74$	19.3717	2.25191
$75$	0	0
$76$	29.2860	3.35933
$77$	1.43175	0.163163
$78$	0	0
$79$	−1.78202	−0.200493	−0.100246	−	0.994963i	$-0.531963\pi$
$79$	−1.78202	−0.200493	−0.100246	+	0.994963i	$0.531963\pi$
$80$	0	0
$81$	0	0
$82$	−15.3717	−1.69752
$83$	−5.37169	−0.589620	−0.294810	−	0.955556i	$-0.595256\pi$
$83$	−5.37169	−0.589620	−0.294810	+	0.955556i	$0.595256\pi$
$84$	0	0
$85$	0	0
$86$	−24.7862	−2.67277
$87$	0	0
$88$	−6.54262	−0.697445
$89$	−10.1751	−1.07856	−0.539281	−	0.842126i	$-0.681304\pi$
$89$	−10.1751	−1.07856	−0.539281	+	0.842126i	$0.681304\pi$
$90$	0	0
$91$	−1.19656	−0.125433
$92$	17.5212	1.82671
$93$	0	0
$94$	2.54262	0.262251
$95$	0	0
$96$	0	0
$97$	1.82487	0.185287	0.0926435	−	0.995699i	$-0.470468\pi$
$97$	1.82487	0.185287	0.0926435	+	0.995699i	$0.470468\pi$
$98$	13.8610	1.40017
$99$	0	0
$100$	0	0
$101$	10.3503	1.02989	0.514945	−	0.857223i	$-0.327812\pi$
$101$	10.3503	1.02989	0.514945	+	0.857223i	$0.327812\pi$
$102$	0	0
$103$	−18.7434	−1.84684	−0.923420	−	0.383790i	$-0.874619\pi$
$103$	−18.7434	−1.84684	−0.923420	+	0.383790i	$0.874619\pi$
$104$	5.46787	0.536168
$105$	0	0
$106$	−25.3288	−2.46016
$107$	−18.5682	−1.79506	−0.897530	−	0.440953i	$-0.854641\pi$
$107$	−18.5682	−1.79506	−0.897530	+	0.440953i	$0.854641\pi$
$108$	0	0
$109$	8.39312	0.803915	0.401957	−	0.915658i	$-0.368330\pi$
$109$	8.39312	0.803915	0.401957	+	0.915658i	$0.368330\pi$
$110$	0	0
$111$	0	0
$112$	6.24361	0.589966
$113$	7.95715	0.748546	0.374273	−	0.927319i	$-0.377892\pi$
$113$	7.95715	0.748546	0.374273	+	0.927319i	$0.377892\pi$
$114$	0	0
$115$	0	0
$116$	−25.1793	−2.33784
$117$	0	0
$118$	13.3717	1.23096
$119$	7.38890	0.677340
$120$	0	0
$121$	−9.56825	−0.869841
$122$	−31.2860	−2.83250
$123$	0	0
$124$	−12.4998	−1.12251
$125$	0	0
$126$	0	0
$127$	10.3931	0.922240	0.461120	−	0.887338i	$-0.347448\pi$
$127$	10.3931	0.922240	0.461120	+	0.887338i	$0.347448\pi$
$128$	17.3612	1.53453
$129$	0	0
$130$	0	0
$131$	6.39312	0.558569	0.279285	−	0.960208i	$-0.409903\pi$
$131$	6.39312	0.558569	0.279285	+	0.960208i	$0.409903\pi$
$132$	0	0
$133$	8.35027	0.724060
$134$	23.3288	2.01531
$135$	0	0
$136$	−33.7648	−2.89531
$137$	16.7434	1.43048	0.715242	−	0.698877i	$-0.246316\pi$
$137$	16.7434	1.43048	0.715242	+	0.698877i	$0.246316\pi$
$138$	0	0
$139$	5.78202	0.490424	0.245212	−	0.969469i	$-0.421142\pi$
$139$	5.78202	0.490424	0.245212	+	0.969469i	$0.421142\pi$
$140$	0	0
$141$	0	0
$142$	−12.9357	−1.08554
$143$	−1.19656	−0.100061
$144$	0	0
$145$	0	0
$146$	−29.7648	−2.46335
$147$	0	0
$148$	−32.6577	−2.68445
$149$	−15.3461	−1.25720	−0.628599	−	0.777730i	$-0.716371\pi$
$149$	−15.3461	−1.25720	−0.628599	+	0.777730i	$0.716371\pi$
$150$	0	0
$151$	−8.58546	−0.698675	−0.349337	−	0.936997i	$-0.613593\pi$
$151$	−8.58546	−0.698675	−0.349337	+	0.936997i	$0.613593\pi$
$152$	−38.1579	−3.09502
$153$	0	0
$154$	−3.56404	−0.287198
$155$	0	0
$156$	0	0
$157$	−2.78623	−0.222365	−0.111183	−	0.993800i	$-0.535464\pi$
$157$	−2.78623	−0.222365	−0.111183	+	0.993800i	$0.535464\pi$
$158$	4.43596	0.352906
$159$	0	0
$160$	0	0
$161$	4.99579	0.393723
$162$	0	0
$163$	−8.76060	−0.686183	−0.343091	−	0.939302i	$-0.611474\pi$
$163$	−8.76060	−0.686183	−0.343091	+	0.939302i	$0.611474\pi$
$164$	25.9143	2.02357
$165$	0	0
$166$	13.3717	1.03784
$167$	−17.3717	−1.34426	−0.672131	−	0.740432i	$-0.734621\pi$
$167$	−17.3717	−1.34426	−0.672131	+	0.740432i	$0.734621\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	41.7858	3.18614
$173$	−7.95715	−0.604971	−0.302486	−	0.953154i	$-0.597816\pi$
$173$	−7.95715	−0.604971	−0.302486	+	0.953154i	$0.597816\pi$
$174$	0	0
$175$	0	0
$176$	6.24361	0.470630
$177$	0	0
$178$	25.3288	1.89848
$179$	15.5640	1.16331	0.581655	−	0.813435i	$-0.302405\pi$
$179$	15.5640	1.16331	0.581655	+	0.813435i	$0.302405\pi$
$180$	0	0
$181$	15.7820	1.17307	0.586534	−	0.809925i	$-0.300492\pi$
$181$	15.7820	1.17307	0.586534	+	0.809925i	$0.300492\pi$
$182$	2.97858	0.220787
$183$	0	0
$184$	−22.8291	−1.68298
$185$	0	0
$186$	0	0
$187$	7.38890	0.540330
$188$	−4.28646	−0.312622
$189$	0	0
$190$	0	0
$191$	−10.7434	−0.777364	−0.388682	−	0.921372i	$-0.627070\pi$
$191$	−10.7434	−0.777364	−0.388682	+	0.921372i	$0.627070\pi$
$192$	0	0
$193$	−9.73917	−0.701041	−0.350521	−	0.936555i	$-0.613995\pi$
$193$	−9.73917	−0.701041	−0.350521	+	0.936555i	$0.613995\pi$
$194$	−4.54262	−0.326141
$195$	0	0
$196$	−23.3675	−1.66911
$197$	−9.56404	−0.681410	−0.340705	−	0.940170i	$-0.610666\pi$
$197$	−9.56404	−0.681410	−0.340705	+	0.940170i	$0.610666\pi$
$198$	0	0
$199$	5.95715	0.422291	0.211146	−	0.977455i	$-0.432281\pi$
$199$	5.95715	0.422291	0.211146	+	0.977455i	$0.432281\pi$
$200$	0	0
$201$	0	0
$202$	−25.7648	−1.81281
$203$	−7.17935	−0.503891
$204$	0	0
$205$	0	0
$206$	46.6577	3.25080
$207$	0	0
$208$	−5.21798	−0.361802
$209$	8.35027	0.577600
$210$	0	0
$211$	23.9143	1.64633	0.823164	−	0.567803i	$-0.192206\pi$
$211$	23.9143	1.64633	0.823164	+	0.567803i	$0.192206\pi$
$212$	42.7005	2.93269
$213$	0	0
$214$	46.2217	3.15965
$215$	0	0
$216$	0	0
$217$	−3.56404	−0.241943
$218$	−20.8929	−1.41504
$219$	0	0
$220$	0	0
$221$	−6.17513	−0.415385
$222$	0	0
$223$	−2.62831	−0.176004	−0.0880022	−	0.996120i	$-0.528048\pi$
$223$	−2.62831	−0.176004	−0.0880022	+	0.996120i	$0.528048\pi$
$224$	−2.45692	−0.164160
$225$	0	0
$226$	−19.8077	−1.31759
$227$	15.7648	1.04635	0.523174	−	0.852226i	$-0.324748\pi$
$227$	15.7648	1.04635	0.523174	+	0.852226i	$0.324748\pi$
$228$	0	0
$229$	8.74338	0.577779	0.288890	−	0.957362i	$-0.406714\pi$
$229$	8.74338	0.577779	0.288890	+	0.957362i	$0.406714\pi$
$230$	0	0
$231$	0	0
$232$	32.8072	2.15390
$233$	−2.17513	−0.142498	−0.0712489	−	0.997459i	$-0.522698\pi$
$233$	−2.17513	−0.142498	−0.0712489	+	0.997459i	$0.522698\pi$
$234$	0	0
$235$	0	0
$236$	−22.5426	−1.46740
$237$	0	0
$238$	−18.3931	−1.19225
$239$	−2.80344	−0.181340	−0.0906698	−	0.995881i	$-0.528901\pi$
$239$	−2.80344	−0.181340	−0.0906698	+	0.995881i	$0.528901\pi$
$240$	0	0
$241$	−6.00000	−0.386494	−0.193247	−	0.981150i	$-0.561902\pi$
$241$	−6.00000	−0.386494	−0.193247	+	0.981150i	$0.561902\pi$
$242$	23.8181	1.53109
$243$	0	0
$244$	52.7434	3.37655
$245$	0	0
$246$	0	0
$247$	−6.97858	−0.444036
$248$	16.2865	1.03419
$249$	0	0
$250$	0	0
$251$	23.9143	1.50946	0.754729	−	0.656037i	$-0.227768\pi$
$251$	23.9143	1.50946	0.754729	+	0.656037i	$0.227768\pi$
$252$	0	0
$253$	4.99579	0.314083
$254$	−25.8715	−1.62332
$255$	0	0
$256$	−32.5678	−2.03549
$257$	−19.9572	−1.24489	−0.622447	−	0.782662i	$-0.713861\pi$
$257$	−19.9572	−1.24489	−0.622447	+	0.782662i	$0.713861\pi$
$258$	0	0
$259$	−9.31163	−0.578597
$260$	0	0
$261$	0	0
$262$	−15.9143	−0.983189
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	0	0
$265$	0	0
$266$	−20.7862	−1.27449
$267$	0	0
$268$	−39.3288	−2.40239
$269$	2.35027	0.143298	0.0716492	−	0.997430i	$-0.477174\pi$
$269$	2.35027	0.143298	0.0716492	+	0.997430i	$0.477174\pi$
$270$	0	0
$271$	10.9786	0.666901	0.333451	−	0.942768i	$-0.391787\pi$
$271$	10.9786	0.666901	0.333451	+	0.942768i	$0.391787\pi$
$272$	32.2217	1.95373
$273$	0	0
$274$	−41.6791	−2.51793
$275$	0	0
$276$	0	0
$277$	−1.21377	−0.0729283	−0.0364642	−	0.999335i	$-0.511609\pi$
$277$	−1.21377	−0.0729283	−0.0364642	+	0.999335i	$0.511609\pi$
$278$	−14.3931	−0.863242
$279$	0	0
$280$	0	0
$281$	−11.9572	−0.713304	−0.356652	−	0.934237i	$-0.616082\pi$
$281$	−11.9572	−0.713304	−0.356652	+	0.934237i	$0.616082\pi$
$282$	0	0
$283$	29.8715	1.77567	0.887837	−	0.460158i	$-0.152207\pi$
$283$	29.8715	1.77567	0.887837	+	0.460158i	$0.152207\pi$
$284$	21.8077	1.29405
$285$	0	0
$286$	2.97858	0.176127
$287$	7.38890	0.436153
$288$	0	0
$289$	21.1323	1.24308
$290$	0	0
$291$	0	0
$292$	50.1789	2.93650
$293$	0.777809	0.0454401	0.0227200	−	0.999742i	$-0.492767\pi$
$293$	0.777809	0.0454401	0.0227200	+	0.999742i	$0.492767\pi$
$294$	0	0
$295$	0	0
$296$	42.5510	2.47323
$297$	0	0
$298$	38.2008	2.21291
$299$	−4.17513	−0.241454
$300$	0	0
$301$	11.9143	0.686729
$302$	21.3717	1.22980
$303$	0	0
$304$	36.4141	2.08849
$305$	0	0
$306$	0	0
$307$	−0.760597	−0.0434095	−0.0217048	−	0.999764i	$-0.506909\pi$
$307$	−0.760597	−0.0434095	−0.0217048	+	0.999764i	$0.506909\pi$
$308$	6.00842	0.342362
$309$	0	0
$310$	0	0
$311$	23.1281	1.31147	0.655736	−	0.754990i	$-0.272358\pi$
$311$	23.1281	1.31147	0.655736	+	0.754990i	$0.272358\pi$
$312$	0	0
$313$	33.9143	1.91695	0.958475	−	0.285176i	$-0.0920522\pi$
$313$	33.9143	1.91695	0.958475	+	0.285176i	$0.0920522\pi$
$314$	6.93573	0.391406
$315$	0	0
$316$	−7.47835	−0.420690
$317$	9.64973	0.541983	0.270991	−	0.962582i	$-0.412648\pi$
$317$	9.64973	0.541983	0.270991	+	0.962582i	$0.412648\pi$
$318$	0	0
$319$	−7.17935	−0.401966
$320$	0	0
$321$	0	0
$322$	−12.4360	−0.693029
$323$	43.0937	2.39780
$324$	0	0
$325$	0	0
$326$	21.8077	1.20781
$327$	0	0
$328$	−33.7648	−1.86435
$329$	−1.22219	−0.0673816
$330$	0	0
$331$	15.3288	0.842550	0.421275	−	0.906933i	$-0.361583\pi$
$331$	15.3288	0.842550	0.421275	+	0.906933i	$0.361583\pi$
$332$	−22.5426	−1.23719
$333$	0	0
$334$	43.2432	2.36616
$335$	0	0
$336$	0	0
$337$	22.3503	1.21750	0.608748	−	0.793363i	$-0.291672\pi$
$337$	22.3503	1.21750	0.608748	+	0.793363i	$0.291672\pi$
$338$	−2.48929	−0.135399
$339$	0	0
$340$	0	0
$341$	−3.56404	−0.193004
$342$	0	0
$343$	−15.0386	−0.812010
$344$	−54.4444	−2.93544
$345$	0	0
$346$	19.8077	1.06487
$347$	−5.78202	−0.310395	−0.155198	−	0.987883i	$-0.549601\pi$
$347$	−5.78202	−0.310395	−0.155198	+	0.987883i	$0.549601\pi$
$348$	0	0
$349$	−27.5212	−1.47318	−0.736588	−	0.676342i	$-0.763564\pi$
$349$	−27.5212	−1.47318	−0.736588	+	0.676342i	$0.763564\pi$
$350$	0	0
$351$	0	0
$352$	−2.45692	−0.130955
$353$	−28.7434	−1.52986	−0.764928	−	0.644116i	$-0.777225\pi$
$353$	−28.7434	−1.52986	−0.764928	+	0.644116i	$0.777225\pi$
$354$	0	0
$355$	0	0
$356$	−42.7005	−2.26312
$357$	0	0
$358$	−38.7434	−2.04765
$359$	12.5855	0.664235	0.332118	−	0.943238i	$-0.392237\pi$
$359$	12.5855	0.664235	0.332118	+	0.943238i	$0.392237\pi$
$360$	0	0
$361$	29.7005	1.56319
$362$	−39.2860	−2.06483
$363$	0	0
$364$	−5.02142	−0.263194
$365$	0	0
$366$	0	0
$367$	27.9143	1.45712	0.728558	−	0.684985i	$-0.240191\pi$
$367$	27.9143	1.45712	0.728558	+	0.684985i	$0.240191\pi$
$368$	21.7858	1.13566
$369$	0	0
$370$	0	0
$371$	12.1751	0.632102
$372$	0	0
$373$	2.35027	0.121692	0.0608462	−	0.998147i	$-0.480620\pi$
$373$	2.35027	0.121692	0.0608462	+	0.998147i	$0.480620\pi$
$374$	−18.3931	−0.951085
$375$	0	0
$376$	5.58500	0.288025
$377$	6.00000	0.309016
$378$	0	0
$379$	24.5510	1.26110	0.630551	−	0.776148i	$-0.282829\pi$
$379$	24.5510	1.26110	0.630551	+	0.776148i	$0.282829\pi$
$380$	0	0
$381$	0	0
$382$	26.7434	1.36831
$383$	5.80765	0.296757	0.148379	−	0.988931i	$-0.452595\pi$
$383$	5.80765	0.296757	0.148379	+	0.988931i	$0.452595\pi$
$384$	0	0
$385$	0	0
$386$	24.2436	1.23397
$387$	0	0
$388$	7.65815	0.388784
$389$	−13.6497	−0.692069	−0.346034	−	0.938222i	$-0.612472\pi$
$389$	−13.6497	−0.692069	−0.346034	+	0.938222i	$0.612472\pi$
$390$	0	0
$391$	25.7820	1.30385
$392$	30.4464	1.53778
$393$	0	0
$394$	23.8077	1.19941
$395$	0	0
$396$	0	0
$397$	12.1323	0.608902	0.304451	−	0.952528i	$-0.401527\pi$
$397$	12.1323	0.608902	0.304451	+	0.952528i	$0.401527\pi$
$398$	−14.8291	−0.743314
$399$	0	0
$400$	0	0
$401$	37.4439	1.86986	0.934930	−	0.354832i	$-0.115462\pi$
$401$	37.4439	1.86986	0.934930	+	0.354832i	$0.115462\pi$
$402$	0	0
$403$	2.97858	0.148373
$404$	43.4355	2.16100
$405$	0	0
$406$	17.8715	0.886946
$407$	−9.31163	−0.461561
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	0	0
$412$	−78.6577	−3.87519
$413$	−6.42754	−0.316279
$414$	0	0
$415$	0	0
$416$	2.05333	0.100673
$417$	0	0
$418$	−20.7862	−1.01669
$419$	−3.17935	−0.155321	−0.0776606	−	0.996980i	$-0.524745\pi$
$419$	−3.17935	−0.155321	−0.0776606	+	0.996980i	$0.524745\pi$
$420$	0	0
$421$	−16.3074	−0.794775	−0.397388	−	0.917651i	$-0.630083\pi$
$421$	−16.3074	−0.794775	−0.397388	+	0.917651i	$0.630083\pi$
$422$	−59.5296	−2.89786
$423$	0	0
$424$	−55.6363	−2.70194
$425$	0	0
$426$	0	0
$427$	15.0386	0.727771
$428$	−77.9227	−3.76654
$429$	0	0
$430$	0	0
$431$	−4.58546	−0.220874	−0.110437	−	0.993883i	$-0.535225\pi$
$431$	−4.58546	−0.220874	−0.110437	+	0.993883i	$0.535225\pi$
$432$	0	0
$433$	38.3503	1.84300	0.921498	−	0.388383i	$-0.126966\pi$
$433$	38.3503	1.84300	0.921498	+	0.388383i	$0.126966\pi$
$434$	8.87192	0.425866
$435$	0	0
$436$	35.2222	1.68684
$437$	29.1365	1.39379
$438$	0	0
$439$	7.73917	0.369371	0.184685	−	0.982798i	$-0.440873\pi$
$439$	7.73917	0.369371	0.184685	+	0.982798i	$0.440873\pi$
$440$	0	0
$441$	0	0
$442$	15.3717	0.731157
$443$	34.9185	1.65903	0.829514	−	0.558485i	$-0.188617\pi$
$443$	34.9185	1.65903	0.829514	+	0.558485i	$0.188617\pi$
$444$	0	0
$445$	0	0
$446$	6.54262	0.309802
$447$	0	0
$448$	−6.37123	−0.301012
$449$	6.17513	0.291423	0.145711	−	0.989327i	$-0.453453\pi$
$449$	6.17513	0.291423	0.145711	+	0.989327i	$0.453453\pi$
$450$	0	0
$451$	7.38890	0.347930
$452$	33.3927	1.57066
$453$	0	0
$454$	−39.2432	−1.84177
$455$	0	0
$456$	0	0
$457$	−1.38890	−0.0649702	−0.0324851	−	0.999472i	$-0.510342\pi$
$457$	−1.38890	−0.0649702	−0.0324851	+	0.999472i	$0.510342\pi$
$458$	−21.7648	−1.01700
$459$	0	0
$460$	0	0
$461$	−28.4826	−1.32657	−0.663283	−	0.748369i	$-0.730837\pi$
$461$	−28.4826	−1.32657	−0.663283	+	0.748369i	$0.730837\pi$
$462$	0	0
$463$	−16.3759	−0.761053	−0.380526	−	0.924770i	$-0.624257\pi$
$463$	−16.3759	−0.761053	−0.380526	+	0.924770i	$0.624257\pi$
$464$	−31.3079	−1.45343
$465$	0	0
$466$	5.41454	0.250824
$467$	25.7476	1.19146	0.595728	−	0.803186i	$-0.296864\pi$
$467$	25.7476	1.19146	0.595728	+	0.803186i	$0.296864\pi$
$468$	0	0
$469$	−11.2138	−0.517804
$470$	0	0
$471$	0	0
$472$	29.3717	1.35194
$473$	11.9143	0.547820
$474$	0	0
$475$	0	0
$476$	31.0080	1.42125
$477$	0	0
$478$	6.97858	0.319193
$479$	7.58967	0.346781	0.173391	−	0.984853i	$-0.444528\pi$
$479$	7.58967	0.346781	0.173391	+	0.984853i	$0.444528\pi$
$480$	0	0
$481$	7.78202	0.354830
$482$	14.9357	0.680304
$483$	0	0
$484$	−40.1537	−1.82517
$485$	0	0
$486$	0	0
$487$	−4.41033	−0.199851	−0.0999255	−	0.994995i	$-0.531860\pi$
$487$	−4.41033	−0.199851	−0.0999255	+	0.994995i	$0.531860\pi$
$488$	−68.7215	−3.11088
$489$	0	0
$490$	0	0
$491$	0.0856914	0.00386720	0.00193360	−	0.999998i	$-0.499385\pi$
$491$	0.0856914	0.00386720	0.00193360	+	0.999998i	$0.499385\pi$
$492$	0	0
$493$	−37.0508	−1.66868
$494$	17.3717	0.781589
$495$	0	0
$496$	−15.5422	−0.697863
$497$	6.21798	0.278915
$498$	0	0
$499$	−17.7220	−0.793344	−0.396672	−	0.917960i	$-0.629835\pi$
$499$	−17.7220	−0.793344	−0.396672	+	0.917960i	$0.629835\pi$
$500$	0	0
$501$	0	0
$502$	−59.5296	−2.65694
$503$	−8.70054	−0.387938	−0.193969	−	0.981008i	$-0.562136\pi$
$503$	−8.70054	−0.387938	−0.193969	+	0.981008i	$0.562136\pi$
$504$	0	0
$505$	0	0
$506$	−12.4360	−0.552846
$507$	0	0
$508$	43.6153	1.93512
$509$	33.3545	1.47841	0.739206	−	0.673480i	$-0.235201\pi$
$509$	33.3545	1.47841	0.739206	+	0.673480i	$0.235201\pi$
$510$	0	0
$511$	14.3074	0.632923
$512$	46.3482	2.04832
$513$	0	0
$514$	49.6791	2.19125
$515$	0	0
$516$	0	0
$517$	−1.22219	−0.0537519
$518$	23.1793	1.01844
$519$	0	0
$520$	0	0
$521$	−18.7005	−0.819285	−0.409643	−	0.912246i	$-0.634347\pi$
$521$	−18.7005	−0.819285	−0.409643	+	0.912246i	$0.634347\pi$
$522$	0	0
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	26.8291	1.17203
$525$	0	0
$526$	−19.9143	−0.868305
$527$	−18.3931	−0.801217
$528$	0	0
$529$	−5.56825	−0.242098
$530$	0	0
$531$	0	0
$532$	35.0424	1.51928
$533$	−6.17513	−0.267475
$534$	0	0
$535$	0	0
$536$	51.2432	2.21337
$537$	0	0
$538$	−5.85050	−0.252233
$539$	−6.66273	−0.286984
$540$	0	0
$541$	−41.5296	−1.78550	−0.892749	−	0.450555i	$-0.851226\pi$
$541$	−41.5296	−1.78550	−0.892749	+	0.450555i	$0.851226\pi$
$542$	−27.3288	−1.17387
$543$	0	0
$544$	−12.6796	−0.543632
$545$	0	0
$546$	0	0
$547$	7.91431	0.338391	0.169196	−	0.985582i	$-0.445883\pi$
$547$	7.91431	0.338391	0.169196	+	0.985582i	$0.445883\pi$
$548$	70.2646	3.00155
$549$	0	0
$550$	0	0
$551$	−41.8715	−1.78378
$552$	0	0
$553$	−2.13229	−0.0906742
$554$	3.02142	0.128368
$555$	0	0
$556$	24.2646	1.02905
$557$	42.7005	1.80928	0.904640	−	0.426177i	$-0.140140\pi$
$557$	42.7005	1.80928	0.904640	+	0.426177i	$0.140140\pi$
$558$	0	0
$559$	−9.95715	−0.421143
$560$	0	0
$561$	0	0
$562$	29.7648	1.25555
$563$	1.04706	0.0441282	0.0220641	−	0.999757i	$-0.492976\pi$
$563$	1.04706	0.0441282	0.0220641	+	0.999757i	$0.492976\pi$
$564$	0	0
$565$	0	0
$566$	−74.3587	−3.12553
$567$	0	0
$568$	−28.4141	−1.19223
$569$	−16.7778	−0.703362	−0.351681	−	0.936120i	$-0.614390\pi$
$569$	−16.7778	−0.703362	−0.351681	+	0.936120i	$0.614390\pi$
$570$	0	0
$571$	−20.6111	−0.862548	−0.431274	−	0.902221i	$-0.641936\pi$
$571$	−20.6111	−0.862548	−0.431274	+	0.902221i	$0.641936\pi$
$572$	−5.02142	−0.209956
$573$	0	0
$574$	−18.3931	−0.767714
$575$	0	0
$576$	0	0
$577$	−1.38890	−0.0578208	−0.0289104	−	0.999582i	$-0.509204\pi$
$577$	−1.38890	−0.0578208	−0.0289104	+	0.999582i	$0.509204\pi$
$578$	−52.6044	−2.18805
$579$	0	0
$580$	0	0
$581$	−6.42754	−0.266659
$582$	0	0
$583$	12.1751	0.504243
$584$	−65.3801	−2.70545
$585$	0	0
$586$	−1.93619	−0.0799833
$587$	−0.935731	−0.0386218	−0.0193109	−	0.999814i	$-0.506147\pi$
$587$	−0.935731	−0.0386218	−0.0193109	+	0.999814i	$0.506147\pi$
$588$	0	0
$589$	−20.7862	−0.856482
$590$	0	0
$591$	0	0
$592$	−40.6064	−1.66891
$593$	−0.478807	−0.0196622	−0.00983112	−	0.999952i	$-0.503129\pi$
$593$	−0.478807	−0.0196622	−0.00983112	+	0.999952i	$0.503129\pi$
$594$	0	0
$595$	0	0
$596$	−64.4006	−2.63795
$597$	0	0
$598$	10.3931	0.425006
$599$	−29.0852	−1.18839	−0.594195	−	0.804321i	$-0.702529\pi$
$599$	−29.0852	−1.18839	−0.594195	+	0.804321i	$0.702529\pi$
$600$	0	0
$601$	11.4318	0.466311	0.233155	−	0.972439i	$-0.425095\pi$
$601$	11.4318	0.466311	0.233155	+	0.972439i	$0.425095\pi$
$602$	−29.6582	−1.20878
$603$	0	0
$604$	−36.0294	−1.46601
$605$	0	0
$606$	0	0
$607$	−27.9143	−1.13301	−0.566503	−	0.824059i	$-0.691704\pi$
$607$	−27.9143	−1.13301	−0.566503	+	0.824059i	$0.691704\pi$
$608$	−14.3293	−0.581130
$609$	0	0
$610$	0	0
$611$	1.02142	0.0413223
$612$	0	0
$613$	4.65394	0.187971	0.0939855	−	0.995574i	$-0.470039\pi$
$613$	4.65394	0.187971	0.0939855	+	0.995574i	$0.470039\pi$
$614$	1.89334	0.0764092
$615$	0	0
$616$	−7.82862	−0.315424
$617$	−15.9572	−0.642411	−0.321205	−	0.947010i	$-0.604088\pi$
$617$	−15.9572	−0.642411	−0.321205	+	0.947010i	$0.604088\pi$
$618$	0	0
$619$	1.02142	0.0410545	0.0205272	−	0.999789i	$-0.493466\pi$
$619$	1.02142	0.0410545	0.0205272	+	0.999789i	$0.493466\pi$
$620$	0	0
$621$	0	0
$622$	−57.5725	−2.30845
$623$	−12.1751	−0.487786
$624$	0	0
$625$	0	0
$626$	−84.4225	−3.37420
$627$	0	0
$628$	−11.6926	−0.466585
$629$	−48.0550	−1.91608
$630$	0	0
$631$	−20.4998	−0.816083	−0.408041	−	0.912963i	$-0.633788\pi$
$631$	−20.4998	−0.816083	−0.408041	+	0.912963i	$0.633788\pi$
$632$	9.74384	0.387589
$633$	0	0
$634$	−24.0210	−0.953994
$635$	0	0
$636$	0	0
$637$	5.56825	0.220622
$638$	17.8715	0.707538
$639$	0	0
$640$	0	0
$641$	−38.2646	−1.51136	−0.755680	−	0.654941i	$-0.772693\pi$
$641$	−38.2646	−1.51136	−0.755680	+	0.654941i	$0.772693\pi$
$642$	0	0
$643$	33.1109	1.30577	0.652883	−	0.757459i	$-0.273560\pi$
$643$	33.1109	1.30577	0.652883	+	0.757459i	$0.273560\pi$
$644$	20.9651	0.826141
$645$	0	0
$646$	−107.273	−4.22058
$647$	−16.9614	−0.666820	−0.333410	−	0.942782i	$-0.608199\pi$
$647$	−16.9614	−0.666820	−0.333410	+	0.942782i	$0.608199\pi$
$648$	0	0
$649$	−6.42754	−0.252303
$650$	0	0
$651$	0	0
$652$	−36.7643	−1.43980
$653$	−19.1709	−0.750216	−0.375108	−	0.926981i	$-0.622394\pi$
$653$	−19.1709	−0.750216	−0.375108	+	0.926981i	$0.622394\pi$
$654$	0	0
$655$	0	0
$656$	32.2217	1.25805
$657$	0	0
$658$	3.04239	0.118605
$659$	−37.8715	−1.47526	−0.737631	−	0.675204i	$-0.764056\pi$
$659$	−37.8715	−1.47526	−0.737631	+	0.675204i	$0.764056\pi$
$660$	0	0
$661$	24.3931	0.948782	0.474391	−	0.880314i	$-0.342668\pi$
$661$	24.3931	0.948782	0.474391	+	0.880314i	$0.342668\pi$
$662$	−38.1579	−1.48305
$663$	0	0
$664$	29.3717	1.13984
$665$	0	0
$666$	0	0
$667$	−25.0508	−0.969971
$668$	−72.9013	−2.82064
$669$	0	0
$670$	0	0
$671$	15.0386	0.580560
$672$	0	0
$673$	21.1281	0.814428	0.407214	−	0.913333i	$-0.366500\pi$
$673$	21.1281	0.814428	0.407214	+	0.913333i	$0.366500\pi$
$674$	−55.6363	−2.14303
$675$	0	0
$676$	4.19656	0.161406
$677$	−15.3973	−0.591767	−0.295884	−	0.955224i	$-0.595614\pi$
$677$	−15.3973	−0.591767	−0.295884	+	0.955224i	$0.595614\pi$
$678$	0	0
$679$	2.18356	0.0837972
$680$	0	0
$681$	0	0
$682$	8.87192	0.339723
$683$	30.0722	1.15068	0.575341	−	0.817914i	$-0.304869\pi$
$683$	30.0722	1.15068	0.575341	+	0.817914i	$0.304869\pi$
$684$	0	0
$685$	0	0
$686$	37.4355	1.42929
$687$	0	0
$688$	51.9562	1.98081
$689$	−10.1751	−0.387642
$690$	0	0
$691$	−8.14950	−0.310022	−0.155011	−	0.987913i	$-0.549541\pi$
$691$	−8.14950	−0.310022	−0.155011	+	0.987913i	$0.549541\pi$
$692$	−33.3927	−1.26940
$693$	0	0
$694$	14.3931	0.546355
$695$	0	0
$696$	0	0
$697$	38.1323	1.44436
$698$	68.5082	2.59307
$699$	0	0
$700$	0	0
$701$	28.6921	1.08369	0.541843	−	0.840480i	$-0.317727\pi$
$701$	28.6921	1.08369	0.541843	+	0.840480i	$0.317727\pi$
$702$	0	0
$703$	−54.3074	−2.04824
$704$	−6.37123	−0.240125
$705$	0	0
$706$	71.5506	2.69284
$707$	12.3847	0.465774
$708$	0	0
$709$	12.3074	0.462215	0.231108	−	0.972928i	$-0.425765\pi$
$709$	12.3074	0.462215	0.231108	+	0.972928i	$0.425765\pi$
$710$	0	0
$711$	0	0
$712$	55.6363	2.08506
$713$	−12.4360	−0.465730
$714$	0	0
$715$	0	0
$716$	65.3154	2.44095
$717$	0	0
$718$	−31.3288	−1.16918
$719$	28.7862	1.07355	0.536773	−	0.843727i	$-0.319643\pi$
$719$	28.7862	1.07355	0.536773	+	0.843727i	$0.319643\pi$
$720$	0	0
$721$	−22.4275	−0.835245
$722$	−73.9332	−2.75151
$723$	0	0
$724$	66.2302	2.46142
$725$	0	0
$726$	0	0
$727$	−34.3931	−1.27557	−0.637785	−	0.770214i	$-0.720149\pi$
$727$	−34.3931	−1.27557	−0.637785	+	0.770214i	$0.720149\pi$
$728$	6.54262	0.242485
$729$	0	0
$730$	0	0
$731$	61.4868	2.27417
$732$	0	0
$733$	29.0042	1.07129	0.535647	−	0.844442i	$-0.320068\pi$
$733$	29.0042	1.07129	0.535647	+	0.844442i	$0.320068\pi$
$734$	−69.4868	−2.56480
$735$	0	0
$736$	−8.57292	−0.316002
$737$	−11.2138	−0.413065
$738$	0	0
$739$	−6.27804	−0.230941	−0.115471	−	0.993311i	$-0.536838\pi$
$739$	−6.27804	−0.230941	−0.115471	+	0.993311i	$0.536838\pi$
$740$	0	0
$741$	0	0
$742$	−30.3074	−1.11262
$743$	12.2352	0.448866	0.224433	−	0.974490i	$-0.427947\pi$
$743$	12.2352	0.448866	0.224433	+	0.974490i	$0.427947\pi$
$744$	0	0
$745$	0	0
$746$	−5.85050	−0.214202
$747$	0	0
$748$	31.0080	1.13376
$749$	−22.2180	−0.811827
$750$	0	0
$751$	−28.8757	−1.05369	−0.526844	−	0.849962i	$-0.676625\pi$
$751$	−28.8757	−1.05369	−0.526844	+	0.849962i	$0.676625\pi$
$752$	−5.32976	−0.194357
$753$	0	0
$754$	−14.9357	−0.543927
$755$	0	0
$756$	0	0
$757$	−30.3503	−1.10310	−0.551550	−	0.834142i	$-0.685963\pi$
$757$	−30.3503	−1.10310	−0.551550	+	0.834142i	$0.685963\pi$
$758$	−61.1146	−2.21978
$759$	0	0
$760$	0	0
$761$	27.1709	0.984945	0.492473	−	0.870328i	$-0.336093\pi$
$761$	27.1709	0.984945	0.492473	+	0.870328i	$0.336093\pi$
$762$	0	0
$763$	10.0428	0.363575
$764$	−45.0852	−1.63113
$765$	0	0
$766$	−14.4569	−0.522350
$767$	5.37169	0.193961
$768$	0	0
$769$	−38.3503	−1.38295	−0.691473	−	0.722402i	$-0.743038\pi$
$769$	−38.3503	−1.38295	−0.691473	+	0.722402i	$0.743038\pi$
$770$	0	0
$771$	0	0
$772$	−40.8710	−1.47098
$773$	−50.2646	−1.80789	−0.903946	−	0.427647i	$-0.859342\pi$
$773$	−50.2646	−1.80789	−0.903946	+	0.427647i	$0.859342\pi$
$774$	0	0
$775$	0	0
$776$	−9.97812	−0.358194
$777$	0	0
$778$	33.9781	1.21817
$779$	43.0937	1.54399
$780$	0	0
$781$	6.21798	0.222497
$782$	−64.1789	−2.29503
$783$	0	0
$784$	−29.0550	−1.03768
$785$	0	0
$786$	0	0
$787$	13.2860	0.473595	0.236797	−	0.971559i	$-0.423902\pi$
$787$	13.2860	0.473595	0.236797	+	0.971559i	$0.423902\pi$
$788$	−40.1360	−1.42979
$789$	0	0
$790$	0	0
$791$	9.52119	0.338535
$792$	0	0
$793$	−12.5682	−0.446312
$794$	−30.2008	−1.07179
$795$	0	0
$796$	24.9995	0.886085
$797$	9.82487	0.348015	0.174007	−	0.984744i	$-0.444328\pi$
$797$	9.82487	0.348015	0.174007	+	0.984744i	$0.444328\pi$
$798$	0	0
$799$	−6.30742	−0.223141
$800$	0	0
$801$	0	0
$802$	−93.2087	−3.29131
$803$	14.3074	0.504898
$804$	0	0
$805$	0	0
$806$	−7.41454	−0.261166
$807$	0	0
$808$	−56.5939	−1.99097
$809$	9.91431	0.348569	0.174284	−	0.984695i	$-0.444239\pi$
$809$	9.91431	0.348569	0.174284	+	0.984695i	$0.444239\pi$
$810$	0	0
$811$	−36.5855	−1.28469	−0.642345	−	0.766416i	$-0.722038\pi$
$811$	−36.5855	−1.28469	−0.642345	+	0.766416i	$0.722038\pi$
$812$	−30.1285	−1.05730
$813$	0	0
$814$	23.1793	0.812436
$815$	0	0
$816$	0	0
$817$	69.4868	2.43103
$818$	34.8500	1.21850
$819$	0	0
$820$	0	0
$821$	34.4741	1.20316	0.601578	−	0.798814i	$-0.294539\pi$
$821$	34.4741	1.20316	0.601578	+	0.798814i	$0.294539\pi$
$822$	0	0
$823$	−13.2566	−0.462097	−0.231048	−	0.972942i	$-0.574216\pi$
$823$	−13.2566	−0.462097	−0.231048	+	0.972942i	$0.574216\pi$
$824$	102.486	3.57028
$825$	0	0
$826$	16.0000	0.556711
$827$	28.1495	0.978854	0.489427	−	0.872044i	$-0.337206\pi$
$827$	28.1495	0.978854	0.489427	+	0.872044i	$0.337206\pi$
$828$	0	0
$829$	16.3418	0.567576	0.283788	−	0.958887i	$-0.408409\pi$
$829$	16.3418	0.567576	0.283788	+	0.958887i	$0.408409\pi$
$830$	0	0
$831$	0	0
$832$	5.32464	0.184599
$833$	−34.3847	−1.19136
$834$	0	0
$835$	0	0
$836$	35.0424	1.21197
$837$	0	0
$838$	7.91431	0.273395
$839$	30.3675	1.04840	0.524201	−	0.851595i	$-0.324364\pi$
$839$	30.3675	1.04840	0.524201	+	0.851595i	$0.324364\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	40.5939	1.39896
$843$	0	0
$844$	100.358	3.45446
$845$	0	0
$846$	0	0
$847$	−11.4490	−0.393391
$848$	53.0937	1.82324
$849$	0	0
$850$	0	0
$851$	−32.4910	−1.11378
$852$	0	0
$853$	42.1407	1.44287	0.721435	−	0.692482i	$-0.243483\pi$
$853$	42.1407	1.44287	0.721435	+	0.692482i	$0.243483\pi$
$854$	−37.4355	−1.28102
$855$	0	0
$856$	101.529	3.47018
$857$	−2.17513	−0.0743012	−0.0371506	−	0.999310i	$-0.511828\pi$
$857$	−2.17513	−0.0743012	−0.0371506	+	0.999310i	$0.511828\pi$
$858$	0	0
$859$	18.5682	0.633541	0.316770	−	0.948502i	$-0.397402\pi$
$859$	18.5682	0.633541	0.316770	+	0.948502i	$0.397402\pi$
$860$	0	0
$861$	0	0
$862$	11.4145	0.388781
$863$	33.7220	1.14791	0.573954	−	0.818887i	$-0.305409\pi$
$863$	33.7220	1.14791	0.573954	+	0.818887i	$0.305409\pi$
$864$	0	0
$865$	0	0
$866$	−95.4649	−3.24403
$867$	0	0
$868$	−14.9567	−0.507663
$869$	−2.13229	−0.0723330
$870$	0	0
$871$	9.37169	0.317548
$872$	−45.8924	−1.55411
$873$	0	0
$874$	−72.5292	−2.45334
$875$	0	0
$876$	0	0
$877$	−43.4868	−1.46844	−0.734222	−	0.678910i	$-0.762453\pi$
$877$	−43.4868	−1.46844	−0.734222	+	0.678910i	$0.762453\pi$
$878$	−19.2650	−0.650164
$879$	0	0
$880$	0	0
$881$	−26.7005	−0.899564	−0.449782	−	0.893138i	$-0.648498\pi$
$881$	−26.7005	−0.899564	−0.449782	+	0.893138i	$0.648498\pi$
$882$	0	0
$883$	−20.2990	−0.683116	−0.341558	−	0.939861i	$-0.610955\pi$
$883$	−20.2990	−0.683116	−0.341558	+	0.939861i	$0.610955\pi$
$884$	−25.9143	−0.871593
$885$	0	0
$886$	−86.9223	−2.92021
$887$	36.0550	1.21061	0.605305	−	0.795994i	$-0.293051\pi$
$887$	36.0550	1.21061	0.605305	+	0.795994i	$0.293051\pi$
$888$	0	0
$889$	12.4360	0.417089
$890$	0	0
$891$	0	0
$892$	−11.0298	−0.369307
$893$	−7.12808	−0.238532
$894$	0	0
$895$	0	0
$896$	20.7737	0.694000
$897$	0	0
$898$	−15.3717	−0.512960
$899$	17.8715	0.596047
$900$	0	0
$901$	62.8328	2.09326
$902$	−18.3931	−0.612424
$903$	0	0
$904$	−43.5087	−1.44708
$905$	0	0
$906$	0	0
$907$	7.26504	0.241232	0.120616	−	0.992699i	$-0.461513\pi$
$907$	7.26504	0.241232	0.120616	+	0.992699i	$0.461513\pi$
$908$	66.1579	2.19553
$909$	0	0
$910$	0	0
$911$	6.65769	0.220579	0.110290	−	0.993899i	$-0.464822\pi$
$911$	6.65769	0.220579	0.110290	+	0.993899i	$0.464822\pi$
$912$	0	0
$913$	−6.42754	−0.212721
$914$	3.45738	0.114360
$915$	0	0
$916$	36.6921	1.21234
$917$	7.64973	0.252616
$918$	0	0
$919$	−27.1831	−0.896688	−0.448344	−	0.893861i	$-0.647986\pi$
$919$	−27.1831	−0.896688	−0.448344	+	0.893861i	$0.647986\pi$
$920$	0	0
$921$	0	0
$922$	70.9013	2.33501
$923$	−5.19656	−0.171047
$924$	0	0
$925$	0	0
$926$	40.7643	1.33960
$927$	0	0
$928$	12.3200	0.404423
$929$	15.3973	0.505170	0.252585	−	0.967575i	$-0.418719\pi$
$929$	15.3973	0.505170	0.252585	+	0.967575i	$0.418719\pi$
$930$	0	0
$931$	−38.8585	−1.27353
$932$	−9.12808	−0.299000
$933$	0	0
$934$	−64.0932	−2.09719
$935$	0	0
$936$	0	0
$937$	−1.12808	−0.0368527	−0.0184264	−	0.999830i	$-0.505866\pi$
$937$	−1.12808	−0.0368527	−0.0184264	+	0.999830i	$0.505866\pi$
$938$	27.9143	0.911434
$939$	0	0
$940$	0	0
$941$	30.1407	0.982559	0.491280	−	0.871002i	$-0.336529\pi$
$941$	30.1407	0.982559	0.491280	+	0.871002i	$0.336529\pi$
$942$	0	0
$943$	25.7820	0.839578
$944$	−28.0294	−0.912279
$945$	0	0
$946$	−29.6582	−0.964270
$947$	−20.0294	−0.650868	−0.325434	−	0.945565i	$-0.605510\pi$
$947$	−20.0294	−0.650868	−0.325434	+	0.945565i	$0.605510\pi$
$948$	0	0
$949$	−11.9572	−0.388146
$950$	0	0
$951$	0	0
$952$	−40.4015	−1.30942
$953$	−43.2259	−1.40023	−0.700113	−	0.714032i	$-0.746867\pi$
$953$	−43.2259	−1.40023	−0.700113	+	0.714032i	$0.746867\pi$
$954$	0	0
$955$	0	0
$956$	−11.7648	−0.380501
$957$	0	0
$958$	−18.8929	−0.610401
$959$	20.0344	0.646945
$960$	0	0
$961$	−22.1281	−0.713809
$962$	−19.3717	−0.624568
$963$	0	0
$964$	−25.1793	−0.810972
$965$	0	0
$966$	0	0
$967$	−57.6875	−1.85511	−0.927553	−	0.373691i	$-0.878092\pi$
$967$	−57.6875	−1.85511	−0.927553	+	0.373691i	$0.878092\pi$
$968$	52.3179	1.68156
$969$	0	0
$970$	0	0
$971$	19.5296	0.626735	0.313368	−	0.949632i	$-0.398543\pi$
$971$	19.5296	0.626735	0.313368	+	0.949632i	$0.398543\pi$
$972$	0	0
$973$	6.91852	0.221798
$974$	10.9786	0.351776
$975$	0	0
$976$	65.5809	2.09919
$977$	−40.3074	−1.28955	−0.644774	−	0.764373i	$-0.723049\pi$
$977$	−40.3074	−1.28955	−0.644774	+	0.764373i	$0.723049\pi$
$978$	0	0
$979$	−12.1751	−0.389119
$980$	0	0
$981$	0	0
$982$	−0.213311	−0.00680702
$983$	32.2008	1.02705	0.513523	−	0.858076i	$-0.328340\pi$
$983$	32.2008	1.02705	0.513523	+	0.858076i	$0.328340\pi$
$984$	0	0
$985$	0	0
$986$	92.2302	2.93721
$987$	0	0
$988$	−29.2860	−0.931712
$989$	41.5725	1.32193
$990$	0	0
$991$	26.4826	0.841246	0.420623	−	0.907235i	$-0.361811\pi$
$991$	26.4826	0.841246	0.420623	+	0.907235i	$0.361811\pi$
$992$	6.11599	0.194183
$993$	0	0
$994$	−15.4783	−0.490943
$995$	0	0
$996$	0	0
$997$	−35.1365	−1.11278	−0.556392	−	0.830920i	$-0.687815\pi$
$997$	−35.1365	−1.11278	−0.556392	+	0.830920i	$0.687815\pi$
$998$	44.1151	1.39644
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2925.2.a.bh.1.1		3
3.2	odd	2		975.2.a.o.1.3		3
5.2	odd	4		2925.2.c.w.2224.2		6
5.3	odd	4		2925.2.c.w.2224.5		6
5.4	even	2		585.2.a.n.1.3		3
15.2	even	4		975.2.c.i.274.5		6
15.8	even	4		975.2.c.i.274.2		6
15.14	odd	2		195.2.a.e.1.1	✓	3
20.19	odd	2		9360.2.a.dd.1.2		3
60.59	even	2		3120.2.a.bj.1.2		3
65.64	even	2		7605.2.a.bx.1.1		3
105.104	even	2		9555.2.a.bq.1.1		3
195.194	odd	2		2535.2.a.bc.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.2.a.e.1.1	✓	3	15.14	odd	2
585.2.a.n.1.3		3	5.4	even	2
975.2.a.o.1.3		3	3.2	odd	2
975.2.c.i.274.2		6	15.8	even	4
975.2.c.i.274.5		6	15.2	even	4
2535.2.a.bc.1.3		3	195.194	odd	2
2925.2.a.bh.1.1		3	1.1	even	1	trivial
2925.2.c.w.2224.2		6	5.2	odd	4
2925.2.c.w.2224.5		6	5.3	odd	4
3120.2.a.bj.1.2		3	60.59	even	2
7605.2.a.bx.1.1		3	65.64	even	2
9360.2.a.dd.1.2		3	20.19	odd	2
9555.2.a.bq.1.1		3	105.104	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2925 = 3^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2925.a (trivial)

\(f(q)\)	\(=\)	\(q-2.48929 q^{2} +4.19656 q^{4} +1.19656 q^{7} -5.46787 q^{8} +O(q^{10})\)	\(q-2.48929 q^{2} +4.19656 q^{4} +1.19656 q^{7} -5.46787 q^{8} +1.19656 q^{11} -1.00000 q^{13} -2.97858 q^{14} +5.21798 q^{16} +6.17513 q^{17} +6.97858 q^{19} -2.97858 q^{22} +4.17513 q^{23} +2.48929 q^{26} +5.02142 q^{28} -6.00000 q^{29} -2.97858 q^{31} -2.05333 q^{32} -15.3717 q^{34} -7.78202 q^{37} -17.3717 q^{38} +6.17513 q^{41} +9.95715 q^{43} +5.02142 q^{44} -10.3931 q^{46} -1.02142 q^{47} -5.56825 q^{49} -4.19656 q^{52} +10.1751 q^{53} -6.54262 q^{56} +14.9357 q^{58} -5.37169 q^{59} +12.5682 q^{61} +7.41454 q^{62} -5.32464 q^{64} -9.37169 q^{67} +25.9143 q^{68} +5.19656 q^{71} +11.9572 q^{73} +19.3717 q^{74} +29.2860 q^{76} +1.43175 q^{77} -1.78202 q^{79} -15.3717 q^{82} -5.37169 q^{83} -24.7862 q^{86} -6.54262 q^{88} -10.1751 q^{89} -1.19656 q^{91} +17.5212 q^{92} +2.54262 q^{94} +1.82487 q^{97} +13.8610 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 8 q^{4} - q^{7} + 6 q^{8}+O(q^{10}) \) 3 * q + 8 * q^4 - q^7 + 6 * q^8	\( 3 q + 8 q^{4} - q^{7} + 6 q^{8} - q^{11} - 3 q^{13} + 6 q^{14} + 26 q^{16} - q^{17} + 6 q^{19} + 6 q^{22} - 7 q^{23} + 30 q^{28} - 18 q^{29} + 6 q^{31} + 22 q^{32} - 22 q^{34} - 13 q^{37} - 28 q^{38} - q^{41} + 30 q^{44} - 22 q^{46} - 18 q^{47} + 12 q^{49} - 8 q^{52} + 11 q^{53} + 16 q^{56} + 8 q^{59} + 9 q^{61} + 28 q^{62} + 30 q^{64} - 4 q^{67} + 18 q^{68} + 11 q^{71} + 6 q^{73} + 34 q^{74} + 4 q^{76} + 33 q^{77} + 5 q^{79} - 22 q^{82} + 8 q^{83} - 56 q^{86} + 16 q^{88} - 11 q^{89} + q^{91} + 2 q^{92} - 28 q^{94} + 25 q^{97} + 10 q^{98}+O(q^{100}) \) 3 * q + 8 * q^4 - q^7 + 6 * q^8 - q^11 - 3 * q^13 + 6 * q^14 + 26 * q^16 - q^17 + 6 * q^19 + 6 * q^22 - 7 * q^23 + 30 * q^28 - 18 * q^29 + 6 * q^31 + 22 * q^32 - 22 * q^34 - 13 * q^37 - 28 * q^38 - q^41 + 30 * q^44 - 22 * q^46 - 18 * q^47 + 12 * q^49 - 8 * q^52 + 11 * q^53 + 16 * q^56 + 8 * q^59 + 9 * q^61 + 28 * q^62 + 30 * q^64 - 4 * q^67 + 18 * q^68 + 11 * q^71 + 6 * q^73 + 34 * q^74 + 4 * q^76 + 33 * q^77 + 5 * q^79 - 22 * q^82 + 8 * q^83 - 56 * q^86 + 16 * q^88 - 11 * q^89 + q^91 + 2 * q^92 - 28 * q^94 + 25 * q^97 + 10 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more