LMFDB - Embedded newform 4275.2.a.be.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4275,2,Mod(1,4275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4275, 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("4275.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4275 = 3^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4275.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:	$34.1360468641$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.621.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1425)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.14510$ of defining polynomial
Character	$\chi$	$=$	4275.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.14510 q^{2} +2.60147 q^{4} -2.74657 q^{7} +1.29021 q^{8} +3.74657 q^{11} -6.29021 q^{13} -5.89167 q^{14} -2.43531 q^{16} +1.00000 q^{19} +8.03677 q^{22} +0.543637 q^{23} -13.4931 q^{26} -7.14510 q^{28} -3.00000 q^{29} +1.45636 q^{31} -7.80440 q^{32} -5.20293 q^{37} +2.14510 q^{38} +12.5299 q^{41} -8.00000 q^{43} +9.74657 q^{44} +1.16616 q^{46} -11.7833 q^{47} +0.543637 q^{49} -16.3638 q^{52} -5.58041 q^{53} -3.54364 q^{56} -6.43531 q^{58} -5.25343 q^{59} -8.49314 q^{61} +3.12405 q^{62} -11.8706 q^{64} +2.83384 q^{67} -7.83384 q^{71} -7.58041 q^{73} -11.1608 q^{74} +2.60147 q^{76} -10.2902 q^{77} -7.52991 q^{79} +26.8779 q^{82} -2.25343 q^{83} -17.1608 q^{86} +4.83384 q^{88} -4.49314 q^{89} +17.2765 q^{91} +1.41425 q^{92} -25.2765 q^{94} +16.8706 q^{97} +1.16616 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{4} - 9 q^{8} + 3 q^{11} - 6 q^{13} - 3 q^{14} + 12 q^{16} + 3 q^{19} + 3 q^{22} - 3 q^{23} - 24 q^{26} - 15 q^{28} - 9 q^{29} + 9 q^{31} - 18 q^{32} - 12 q^{37} - 24 q^{43} + 21 q^{44} + 21 q^{46}+ \cdots + 21 q^{98}+O(q^{100}) $ 3 * q + 6 * q^4 - 9 * q^8 + 3 * q^11 - 6 * q^13 - 3 * q^14 + 12 * q^16 + 3 * q^19 + 3 * q^22 - 3 * q^23 - 24 * q^26 - 15 * q^28 - 9 * q^29 + 9 * q^31 - 18 * q^32 - 12 * q^37 - 24 * q^43 + 21 * q^44 + 21 * q^46 - 6 * q^47 - 3 * q^49 + 6 * q^52 + 9 * q^53 - 6 * q^56 - 24 * q^59 - 9 * q^61 - 21 * q^62 + 3 * q^64 - 9 * q^67 - 6 * q^71 + 3 * q^73 + 18 * q^74 + 6 * q^76 - 18 * q^77 + 15 * q^79 + 33 * q^82 - 15 * q^83 - 3 * q^88 + 3 * q^89 + 6 * q^91 - 39 * q^92 - 30 * q^94 + 12 * q^97 + 21 * q^98

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 $

$2$	2.14510	1.51682	0.758408	−	0.651780i	$-0.225977\pi$
$2$	2.14510	1.51682	0.758408	+	0.651780i	$0.225977\pi$
$3$	0	0
$3$	0	0
$4$	2.60147	1.30073
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.74657	−1.03811	−0.519053	−	0.854742i	$-0.673715\pi$
$7$	−2.74657	−1.03811	−0.519053	+	0.854742i	$0.673715\pi$
$8$	1.29021	0.456156
$9$	0	0
$10$	0	0
$11$	3.74657	1.12963	0.564816	−	0.825217i	$-0.308947\pi$
$11$	3.74657	1.12963	0.564816	+	0.825217i	$0.308947\pi$
$12$	0	0
$13$	−6.29021	−1.74459	−0.872295	−	0.488981i	$-0.837369\pi$
$13$	−6.29021	−1.74459	−0.872295	+	0.488981i	$0.837369\pi$
$14$	−5.89167	−1.57462
$15$	0	0
$16$	−2.43531	−0.608827
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	8.03677	1.71345
$23$	0.543637	0.113356	0.0566781	−	0.998393i	$-0.481949\pi$
$23$	0.543637	0.113356	0.0566781	+	0.998393i	$0.481949\pi$
$24$	0	0
$25$	0	0
$26$	−13.4931	−2.64622
$27$	0	0
$28$	−7.14510	−1.35030
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	1.45636	0.261570	0.130785	−	0.991411i	$-0.458250\pi$
$31$	1.45636	0.261570	0.130785	+	0.991411i	$0.458250\pi$
$32$	−7.80440	−1.37964
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5.20293	−0.855357	−0.427678	−	0.903931i	$-0.640668\pi$
$37$	−5.20293	−0.855357	−0.427678	+	0.903931i	$0.640668\pi$
$38$	2.14510	0.347982
$39$	0	0
$40$	0	0
$41$	12.5299	1.95684	0.978422	−	0.206618i	$-0.0662459\pi$
$41$	12.5299	1.95684	0.978422	+	0.206618i	$0.0662459\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	9.74657	1.46935
$45$	0	0
$46$	1.16616	0.171941
$47$	−11.7833	−1.71878	−0.859389	−	0.511323i	$-0.829156\pi$
$47$	−11.7833	−1.71878	−0.859389	+	0.511323i	$0.829156\pi$
$48$	0	0
$49$	0.543637	0.0776624
$50$	0	0
$51$	0	0
$52$	−16.3638	−2.26924
$53$	−5.58041	−0.766528	−0.383264	−	0.923639i	$-0.625200\pi$
$53$	−5.58041	−0.766528	−0.383264	+	0.923639i	$0.625200\pi$
$54$	0	0
$55$	0	0
$56$	−3.54364	−0.473538
$57$	0	0
$58$	−6.43531	−0.844997
$59$	−5.25343	−0.683939	−0.341969	−	0.939711i	$-0.611094\pi$
$59$	−5.25343	−0.683939	−0.341969	+	0.939711i	$0.611094\pi$
$60$	0	0
$61$	−8.49314	−1.08743	−0.543717	−	0.839268i	$-0.682984\pi$
$61$	−8.49314	−1.08743	−0.543717	+	0.839268i	$0.682984\pi$
$62$	3.12405	0.396754
$63$	0	0
$64$	−11.8706	−1.48383
$65$	0	0
$66$	0	0
$67$	2.83384	0.346209	0.173104	−	0.984903i	$-0.444620\pi$
$67$	2.83384	0.346209	0.173104	+	0.984903i	$0.444620\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−7.83384	−0.929706	−0.464853	−	0.885388i	$-0.653893\pi$
$71$	−7.83384	−0.929706	−0.464853	+	0.885388i	$0.653893\pi$
$72$	0	0
$73$	−7.58041	−0.887220	−0.443610	−	0.896220i	$-0.646302\pi$
$73$	−7.58041	−0.887220	−0.443610	+	0.896220i	$0.646302\pi$
$74$	−11.1608	−1.29742
$75$	0	0
$76$	2.60147	0.298409
$77$	−10.2902	−1.17268
$78$	0	0
$79$	−7.52991	−0.847181	−0.423591	−	0.905854i	$-0.639230\pi$
$79$	−7.52991	−0.847181	−0.423591	+	0.905854i	$0.639230\pi$
$80$	0	0
$81$	0	0
$82$	26.8779	2.96817
$83$	−2.25343	−0.247346	−0.123673	−	0.992323i	$-0.539467\pi$
$83$	−2.25343	−0.247346	−0.123673	+	0.992323i	$0.539467\pi$
$84$	0	0
$85$	0	0
$86$	−17.1608	−1.85050
$87$	0	0
$88$	4.83384	0.515289
$89$	−4.49314	−0.476272	−0.238136	−	0.971232i	$-0.576536\pi$
$89$	−4.49314	−0.476272	−0.238136	+	0.971232i	$0.576536\pi$
$90$	0	0
$91$	17.2765	1.81107
$92$	1.41425	0.147446
$93$	0	0
$94$	−25.2765	−2.60707
$95$	0	0
$96$	0	0
$97$	16.8706	1.71295	0.856476	−	0.516187i	$-0.172649\pi$
$97$	16.8706	1.71295	0.856476	+	0.516187i	$0.172649\pi$
$98$	1.16616	0.117800
$99$	0	0
$100$	0	0
$101$	13.2765	1.32106	0.660529	−	0.750800i	$-0.270332\pi$
$101$	13.2765	1.32106	0.660529	+	0.750800i	$0.270332\pi$
$102$	0	0
$103$	0.253432	0.0249714	0.0124857	−	0.999922i	$-0.496026\pi$
$103$	0.253432	0.0249714	0.0124857	+	0.999922i	$0.496026\pi$
$104$	−8.11566	−0.795806
$105$	0	0
$106$	−11.9706	−1.16268
$107$	2.23970	0.216520	0.108260	−	0.994123i	$-0.465472\pi$
$107$	2.23970	0.216520	0.108260	+	0.994123i	$0.465472\pi$
$108$	0	0
$109$	−1.20293	−0.115220	−0.0576100	−	0.998339i	$-0.518348\pi$
$109$	−1.20293	−0.115220	−0.0576100	+	0.998339i	$0.518348\pi$
$110$	0	0
$111$	0	0
$112$	6.68874	0.632027
$113$	−12.5436	−1.18001	−0.590003	−	0.807401i	$-0.700873\pi$
$113$	−12.5436	−1.18001	−0.590003	+	0.807401i	$0.700873\pi$
$114$	0	0
$115$	0	0
$116$	−7.80440	−0.724620
$117$	0	0
$118$	−11.2692	−1.03741
$119$	0	0
$120$	0	0
$121$	3.03677	0.276070
$122$	−18.2186	−1.64944
$123$	0	0
$124$	3.78868	0.340233
$125$	0	0
$126$	0	0
$127$	12.4426	1.10411	0.552053	−	0.833809i	$-0.313845\pi$
$127$	12.4426	1.10411	0.552053	+	0.833809i	$0.313845\pi$
$128$	−9.85490	−0.871058
$129$	0	0
$130$	0	0
$131$	−18.9495	−1.65563	−0.827813	−	0.561005i	$-0.810415\pi$
$131$	−18.9495	−1.65563	−0.827813	+	0.561005i	$0.810415\pi$
$132$	0	0
$133$	−2.74657	−0.238158
$134$	6.07888	0.525136
$135$	0	0
$136$	0	0
$137$	21.3921	1.82765	0.913827	−	0.406104i	$-0.133113\pi$
$137$	21.3921	1.82765	0.913827	+	0.406104i	$0.133113\pi$
$138$	0	0
$139$	20.3133	1.72295	0.861474	−	0.507802i	$-0.169542\pi$
$139$	20.3133	1.72295	0.861474	+	0.507802i	$0.169542\pi$
$140$	0	0
$141$	0	0
$142$	−16.8044	−1.41019
$143$	−23.5667	−1.97075
$144$	0	0
$145$	0	0
$146$	−16.2608	−1.34575
$147$	0	0
$148$	−13.5352	−1.11259
$149$	−4.69607	−0.384717	−0.192358	−	0.981325i	$-0.561614\pi$
$149$	−4.69607	−0.384717	−0.192358	+	0.981325i	$0.561614\pi$
$150$	0	0
$151$	1.59414	0.129729	0.0648645	−	0.997894i	$-0.479338\pi$
$151$	1.59414	0.129729	0.0648645	+	0.997894i	$0.479338\pi$
$152$	1.29021	0.104649
$153$	0	0
$154$	−22.0735	−1.77874
$155$	0	0
$156$	0	0
$157$	13.1103	1.04632	0.523159	−	0.852235i	$-0.324753\pi$
$157$	13.1103	1.04632	0.523159	+	0.852235i	$0.324753\pi$
$158$	−16.1524	−1.28502
$159$	0	0
$160$	0	0
$161$	−1.49314	−0.117676
$162$	0	0
$163$	7.73284	0.605683	0.302841	−	0.953041i	$-0.402065\pi$
$163$	7.73284	0.605683	0.302841	+	0.953041i	$0.402065\pi$
$164$	32.5961	2.54533
$165$	0	0
$166$	−4.83384	−0.375179
$167$	−9.32698	−0.721743	−0.360872	−	0.932615i	$-0.617521\pi$
$167$	−9.32698	−0.721743	−0.360872	+	0.932615i	$0.617521\pi$
$168$	0	0
$169$	26.5667	2.04359
$170$	0	0
$171$	0	0
$172$	−20.8117	−1.58688
$173$	−5.17455	−0.393414	−0.196707	−	0.980462i	$-0.563025\pi$
$173$	−5.17455	−0.393414	−0.196707	+	0.980462i	$0.563025\pi$
$174$	0	0
$175$	0	0
$176$	−9.12405	−0.687751
$177$	0	0
$178$	−9.63824	−0.722417
$179$	−15.3270	−1.14559	−0.572796	−	0.819698i	$-0.694141\pi$
$179$	−15.3270	−1.14559	−0.572796	+	0.819698i	$0.694141\pi$
$180$	0	0
$181$	−10.4059	−0.773462	−0.386731	−	0.922193i	$-0.626396\pi$
$181$	−10.4059	−0.773462	−0.386731	+	0.922193i	$0.626396\pi$
$182$	37.0598	2.74706
$183$	0	0
$184$	0.701404	0.0517082
$185$	0	0
$186$	0	0
$187$	0	0
$188$	−30.6540	−2.23567
$189$	0	0
$190$	0	0
$191$	6.54364	0.473481	0.236740	−	0.971573i	$-0.423921\pi$
$191$	6.54364	0.473481	0.236740	+	0.971573i	$0.423921\pi$
$192$	0	0
$193$	24.9863	1.79855	0.899276	−	0.437382i	$-0.144094\pi$
$193$	24.9863	1.79855	0.899276	+	0.437382i	$0.144094\pi$
$194$	36.1892	2.59823
$195$	0	0
$196$	1.41425	0.101018
$197$	8.79707	0.626765	0.313383	−	0.949627i	$-0.398538\pi$
$197$	8.79707	0.626765	0.313383	+	0.949627i	$0.398538\pi$
$198$	0	0
$199$	8.74657	0.620028	0.310014	−	0.950732i	$-0.399666\pi$
$199$	8.74657	0.620028	0.310014	+	0.950732i	$0.399666\pi$
$200$	0	0
$201$	0	0
$202$	28.4794	2.00380
$203$	8.23970	0.578314
$204$	0	0
$205$	0	0
$206$	0.543637	0.0378770
$207$	0	0
$208$	15.3186	1.06215
$209$	3.74657	0.259156
$210$	0	0
$211$	−12.7182	−0.875556	−0.437778	−	0.899083i	$-0.644234\pi$
$211$	−12.7182	−0.875556	−0.437778	+	0.899083i	$0.644234\pi$
$212$	−14.5172	−0.997049
$213$	0	0
$214$	4.80440	0.328422
$215$	0	0
$216$	0	0
$217$	−4.00000	−0.271538
$218$	−2.58041	−0.174767
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	10.5436	0.706054	0.353027	−	0.935613i	$-0.385152\pi$
$223$	10.5436	0.706054	0.353027	+	0.935613i	$0.385152\pi$
$224$	21.4353	1.43221
$225$	0	0
$226$	−26.9074	−1.78985
$227$	7.83384	0.519950	0.259975	−	0.965615i	$-0.416286\pi$
$227$	7.83384	0.519950	0.259975	+	0.965615i	$0.416286\pi$
$228$	0	0
$229$	−23.6035	−1.55976	−0.779880	−	0.625929i	$-0.784720\pi$
$229$	−23.6035	−1.55976	−0.779880	+	0.625929i	$0.784720\pi$
$230$	0	0
$231$	0	0
$232$	−3.87062	−0.254118
$233$	−23.7833	−1.55810	−0.779049	−	0.626963i	$-0.784298\pi$
$233$	−23.7833	−1.55810	−0.779049	+	0.626963i	$0.784298\pi$
$234$	0	0
$235$	0	0
$236$	−13.6666	−0.889621
$237$	0	0
$238$	0	0
$239$	1.27648	0.0825685	0.0412843	−	0.999147i	$-0.486855\pi$
$239$	1.27648	0.0825685	0.0412843	+	0.999147i	$0.486855\pi$
$240$	0	0
$241$	21.4931	1.38449	0.692247	−	0.721660i	$-0.256621\pi$
$241$	21.4931	1.38449	0.692247	+	0.721660i	$0.256621\pi$
$242$	6.51419	0.418748
$243$	0	0
$244$	−22.0946	−1.41446
$245$	0	0
$246$	0	0
$247$	−6.29021	−0.400236
$248$	1.87901	0.119317
$249$	0	0
$250$	0	0
$251$	29.9725	1.89185	0.945925	−	0.324385i	$-0.105157\pi$
$251$	29.9725	1.89185	0.945925	+	0.324385i	$0.105157\pi$
$252$	0	0
$253$	2.03677	0.128051
$254$	26.6907	1.67473
$255$	0	0
$256$	2.60147	0.162592
$257$	19.0735	1.18978	0.594888	−	0.803809i	$-0.297197\pi$
$257$	19.0735	1.18978	0.594888	+	0.803809i	$0.297197\pi$
$258$	0	0
$259$	14.2902	0.887950
$260$	0	0
$261$	0	0
$262$	−40.6486	−2.51128
$263$	11.6456	0.718096	0.359048	−	0.933319i	$-0.383101\pi$
$263$	11.6456	0.718096	0.359048	+	0.933319i	$0.383101\pi$
$264$	0	0
$265$	0	0
$266$	−5.89167	−0.361242
$267$	0	0
$268$	7.37214	0.450325
$269$	−26.1471	−1.59422	−0.797108	−	0.603836i	$-0.793638\pi$
$269$	−26.1471	−1.59422	−0.797108	+	0.603836i	$0.793638\pi$
$270$	0	0
$271$	15.8338	0.961837	0.480919	−	0.876765i	$-0.340303\pi$
$271$	15.8338	0.961837	0.480919	+	0.876765i	$0.340303\pi$
$272$	0	0
$273$	0	0
$274$	45.8883	2.77222
$275$	0	0
$276$	0	0
$277$	−20.6677	−1.24180	−0.620900	−	0.783889i	$-0.713233\pi$
$277$	−20.6677	−1.24180	−0.620900	+	0.783889i	$0.713233\pi$
$278$	43.5740	2.61340
$279$	0	0
$280$	0	0
$281$	−1.35536	−0.0808541	−0.0404270	−	0.999182i	$-0.512872\pi$
$281$	−1.35536	−0.0808541	−0.0404270	+	0.999182i	$0.512872\pi$
$282$	0	0
$283$	−28.1471	−1.67317	−0.836586	−	0.547836i	$-0.815452\pi$
$283$	−28.1471	−1.67317	−0.836586	+	0.547836i	$0.815452\pi$
$284$	−20.3795	−1.20930
$285$	0	0
$286$	−50.5530	−2.98926
$287$	−34.4143	−2.03141
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	−19.7202	−1.15404
$293$	−8.47009	−0.494828	−0.247414	−	0.968910i	$-0.579581\pi$
$293$	−8.47009	−0.494828	−0.247414	+	0.968910i	$0.579581\pi$
$294$	0	0
$295$	0	0
$296$	−6.71285	−0.390176
$297$	0	0
$298$	−10.0735	−0.583545
$299$	−3.41959	−0.197760
$300$	0	0
$301$	21.9725	1.26648
$302$	3.41959	0.196775
$303$	0	0
$304$	−2.43531	−0.139674
$305$	0	0
$306$	0	0
$307$	19.3133	1.10227	0.551133	−	0.834418i	$-0.314196\pi$
$307$	19.3133	1.10227	0.551133	+	0.834418i	$0.314196\pi$
$308$	−26.7696	−1.52534
$309$	0	0
$310$	0	0
$311$	−20.5804	−1.16701	−0.583504	−	0.812110i	$-0.698319\pi$
$311$	−20.5804	−1.16701	−0.583504	+	0.812110i	$0.698319\pi$
$312$	0	0
$313$	−22.4426	−1.26853	−0.634266	−	0.773115i	$-0.718698\pi$
$313$	−22.4426	−1.26853	−0.634266	+	0.773115i	$0.718698\pi$
$314$	28.1230	1.58707
$315$	0	0
$316$	−19.5888	−1.10196
$317$	11.1745	0.627625	0.313813	−	0.949485i	$-0.398394\pi$
$317$	11.1745	0.627625	0.313813	+	0.949485i	$0.398394\pi$
$318$	0	0
$319$	−11.2397	−0.629303
$320$	0	0
$321$	0	0
$322$	−3.20293	−0.178492
$323$	0	0
$324$	0	0
$325$	0	0
$326$	16.5877	0.918710
$327$	0	0
$328$	16.1662	0.892627
$329$	32.3638	1.78427
$330$	0	0
$331$	−25.9358	−1.42556	−0.712779	−	0.701388i	$-0.752564\pi$
$331$	−25.9358	−1.42556	−0.712779	+	0.701388i	$0.752564\pi$
$332$	−5.86223	−0.321731
$333$	0	0
$334$	−20.0073	−1.09475
$335$	0	0
$336$	0	0
$337$	−15.0873	−0.821856	−0.410928	−	0.911668i	$-0.634795\pi$
$337$	−15.0873	−0.821856	−0.410928	+	0.911668i	$0.634795\pi$
$338$	56.9883	3.09975
$339$	0	0
$340$	0	0
$341$	5.45636	0.295479
$342$	0	0
$343$	17.7328	0.957483
$344$	−10.3216	−0.556506
$345$	0	0
$346$	−11.0999	−0.596736
$347$	−6.62252	−0.355516	−0.177758	−	0.984074i	$-0.556884\pi$
$347$	−6.62252	−0.355516	−0.177758	+	0.984074i	$0.556884\pi$
$348$	0	0
$349$	9.07355	0.485696	0.242848	−	0.970064i	$-0.421918\pi$
$349$	9.07355	0.485696	0.242848	+	0.970064i	$0.421918\pi$
$350$	0	0
$351$	0	0
$352$	−29.2397	−1.55848
$353$	20.1471	1.07232	0.536161	−	0.844116i	$-0.319874\pi$
$353$	20.1471	1.07232	0.536161	+	0.844116i	$0.319874\pi$
$354$	0	0
$355$	0	0
$356$	−11.6887	−0.619502
$357$	0	0
$358$	−32.8779	−1.73765
$359$	−15.8843	−0.838344	−0.419172	−	0.907907i	$-0.637679\pi$
$359$	−15.8843	−0.838344	−0.419172	+	0.907907i	$0.637679\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−22.3216	−1.17320
$363$	0	0
$364$	44.9442	2.35571
$365$	0	0
$366$	0	0
$367$	−24.8853	−1.29900	−0.649500	−	0.760361i	$-0.725022\pi$
$367$	−24.8853	−1.29900	−0.649500	+	0.760361i	$0.725022\pi$
$368$	−1.32392	−0.0690143
$369$	0	0
$370$	0	0
$371$	15.3270	0.795737
$372$	0	0
$373$	−11.6677	−0.604130	−0.302065	−	0.953287i	$-0.597676\pi$
$373$	−11.6677	−0.604130	−0.302065	+	0.953287i	$0.597676\pi$
$374$	0	0
$375$	0	0
$376$	−15.2029	−0.784031
$377$	18.8706	0.971886
$378$	0	0
$379$	6.29021	0.323106	0.161553	−	0.986864i	$-0.448350\pi$
$379$	6.29021	0.323106	0.161553	+	0.986864i	$0.448350\pi$
$380$	0	0
$381$	0	0
$382$	14.0368	0.718184
$383$	12.0652	0.616501	0.308250	−	0.951305i	$-0.400257\pi$
$383$	12.0652	0.616501	0.308250	+	0.951305i	$0.400257\pi$
$384$	0	0
$385$	0	0
$386$	53.5981	2.72807
$387$	0	0
$388$	43.8883	2.22809
$389$	14.3638	0.728271	0.364136	−	0.931346i	$-0.381364\pi$
$389$	14.3638	0.728271	0.364136	+	0.931346i	$0.381364\pi$
$390$	0	0
$391$	0	0
$392$	0.701404	0.0354262
$393$	0	0
$394$	18.8706	0.950688
$395$	0	0
$396$	0	0
$397$	31.0966	1.56069	0.780347	−	0.625347i	$-0.215043\pi$
$397$	31.0966	1.56069	0.780347	+	0.625347i	$0.215043\pi$
$398$	18.7623	0.940468
$399$	0	0
$400$	0	0
$401$	0.543637	0.0271479	0.0135740	−	0.999908i	$-0.495679\pi$
$401$	0.543637	0.0271479	0.0135740	+	0.999908i	$0.495679\pi$
$402$	0	0
$403$	−9.16082	−0.456333
$404$	34.5383	1.71834
$405$	0	0
$406$	17.6750	0.877196
$407$	−19.4931	−0.966239
$408$	0	0
$409$	26.4648	1.30860	0.654299	−	0.756236i	$-0.272964\pi$
$409$	26.4648	1.30860	0.654299	+	0.756236i	$0.272964\pi$
$410$	0	0
$411$	0	0
$412$	0.659294	0.0324811
$413$	14.4289	0.710000
$414$	0	0
$415$	0	0
$416$	49.0913	2.40690
$417$	0	0
$418$	8.03677	0.393091
$419$	6.18920	0.302362	0.151181	−	0.988506i	$-0.451692\pi$
$419$	6.18920	0.302362	0.151181	+	0.988506i	$0.451692\pi$
$420$	0	0
$421$	15.2765	0.744530	0.372265	−	0.928126i	$-0.378581\pi$
$421$	15.2765	0.744530	0.372265	+	0.928126i	$0.378581\pi$
$422$	−27.2818	−1.32806
$423$	0	0
$424$	−7.19988	−0.349657
$425$	0	0
$426$	0	0
$427$	23.3270	1.12887
$428$	5.82651	0.281635
$429$	0	0
$430$	0	0
$431$	−13.5583	−0.653080	−0.326540	−	0.945183i	$-0.605883\pi$
$431$	−13.5583	−0.653080	−0.326540	+	0.945183i	$0.605883\pi$
$432$	0	0
$433$	29.6823	1.42644	0.713221	−	0.700939i	$-0.247236\pi$
$433$	29.6823	1.42644	0.713221	+	0.700939i	$0.247236\pi$
$434$	−8.58041	−0.411873
$435$	0	0
$436$	−3.12938	−0.149870
$437$	0.543637	0.0260057
$438$	0	0
$439$	34.0093	1.62318	0.811588	−	0.584230i	$-0.198603\pi$
$439$	34.0093	1.62318	0.811588	+	0.584230i	$0.198603\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−16.3416	−0.776414	−0.388207	−	0.921572i	$-0.626906\pi$
$443$	−16.3416	−0.776414	−0.388207	+	0.921572i	$0.626906\pi$
$444$	0	0
$445$	0	0
$446$	22.6172	1.07095
$447$	0	0
$448$	32.6035	1.54037
$449$	−16.8990	−0.797513	−0.398757	−	0.917057i	$-0.630558\pi$
$449$	−16.8990	−0.797513	−0.398757	+	0.917057i	$0.630558\pi$
$450$	0	0
$451$	46.9442	2.21051
$452$	−32.6318	−1.53487
$453$	0	0
$454$	16.8044	0.788669
$455$	0	0
$456$	0	0
$457$	−16.8622	−0.788782	−0.394391	−	0.918943i	$-0.629044\pi$
$457$	−16.8622	−0.788782	−0.394391	+	0.918943i	$0.629044\pi$
$458$	−50.6318	−2.36587
$459$	0	0
$460$	0	0
$461$	24.4648	1.13944	0.569719	−	0.821840i	$-0.307052\pi$
$461$	24.4648	1.13944	0.569719	+	0.821840i	$0.307052\pi$
$462$	0	0
$463$	14.7550	0.685721	0.342861	−	0.939386i	$-0.388604\pi$
$463$	14.7550	0.685721	0.342861	+	0.939386i	$0.388604\pi$
$464$	7.30592	0.339169
$465$	0	0
$466$	−51.0177	−2.36335
$467$	12.3270	0.570425	0.285212	−	0.958464i	$-0.407936\pi$
$467$	12.3270	0.570425	0.285212	+	0.958464i	$0.407936\pi$
$468$	0	0
$469$	−7.78334	−0.359401
$470$	0	0
$471$	0	0
$472$	−6.77801	−0.311983
$473$	−29.9725	−1.37814
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	2.73818	0.125241
$479$	19.0084	0.868516	0.434258	−	0.900789i	$-0.357011\pi$
$479$	19.0084	0.868516	0.434258	+	0.900789i	$0.357011\pi$
$480$	0	0
$481$	32.7275	1.49225
$482$	46.1050	2.10002
$483$	0	0
$484$	7.90006	0.359094
$485$	0	0
$486$	0	0
$487$	−33.2765	−1.50790	−0.753951	−	0.656931i	$-0.771854\pi$
$487$	−33.2765	−1.50790	−0.753951	+	0.656931i	$0.771854\pi$
$488$	−10.9579	−0.496040
$489$	0	0
$490$	0	0
$491$	−28.7275	−1.29645	−0.648227	−	0.761447i	$-0.724489\pi$
$491$	−28.7275	−1.29645	−0.648227	+	0.761447i	$0.724489\pi$
$492$	0	0
$493$	0	0
$494$	−13.4931	−0.607085
$495$	0	0
$496$	−3.54669	−0.159251
$497$	21.5162	0.965133
$498$	0	0
$499$	−24.4878	−1.09622	−0.548112	−	0.836405i	$-0.684653\pi$
$499$	−24.4878	−1.09622	−0.548112	+	0.836405i	$0.684653\pi$
$500$	0	0
$501$	0	0
$502$	64.2942	2.86959
$503$	−7.30393	−0.325666	−0.162833	−	0.986654i	$-0.552063\pi$
$503$	−7.30393	−0.325666	−0.162833	+	0.986654i	$0.552063\pi$
$504$	0	0
$505$	0	0
$506$	4.36909	0.194230
$507$	0	0
$508$	32.3691	1.43615
$509$	−33.3784	−1.47947	−0.739736	−	0.672897i	$-0.765050\pi$
$509$	−33.3784	−1.47947	−0.739736	+	0.672897i	$0.765050\pi$
$510$	0	0
$511$	20.8201	0.921028
$512$	25.2902	1.11768
$513$	0	0
$514$	40.9147	1.80467
$515$	0	0
$516$	0	0
$517$	−44.1471	−1.94159
$518$	30.6540	1.34686
$519$	0	0
$520$	0	0
$521$	4.08727	0.179067	0.0895334	−	0.995984i	$-0.471462\pi$
$521$	4.08727	0.179067	0.0895334	+	0.995984i	$0.471462\pi$
$522$	0	0
$523$	−27.9304	−1.22131	−0.610656	−	0.791896i	$-0.709094\pi$
$523$	−27.9304	−1.22131	−0.610656	+	0.791896i	$0.709094\pi$
$524$	−49.2965	−2.15353
$525$	0	0
$526$	24.9809	1.08922
$527$	0	0
$528$	0	0
$529$	−22.7045	−0.987150
$530$	0	0
$531$	0	0
$532$	−7.14510	−0.309779
$533$	−78.8157	−3.41389
$534$	0	0
$535$	0	0
$536$	3.65624	0.157925
$537$	0	0
$538$	−56.0882	−2.41813
$539$	2.03677	0.0877301
$540$	0	0
$541$	26.7917	1.15187	0.575933	−	0.817497i	$-0.304639\pi$
$541$	26.7917	1.15187	0.575933	+	0.817497i	$0.304639\pi$
$542$	33.9652	1.45893
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	9.26716	0.396235	0.198118	−	0.980178i	$-0.436517\pi$
$547$	9.26716	0.396235	0.198118	+	0.980178i	$0.436517\pi$
$548$	55.6509	2.37729
$549$	0	0
$550$	0	0
$551$	−3.00000	−0.127804
$552$	0	0
$553$	20.6814	0.879463
$554$	−44.3343	−1.88358
$555$	0	0
$556$	52.8442	2.24109
$557$	−32.5804	−1.38048	−0.690238	−	0.723582i	$-0.742494\pi$
$557$	−32.5804	−1.38048	−0.690238	+	0.723582i	$0.742494\pi$
$558$	0	0
$559$	50.3216	2.12838
$560$	0	0
$561$	0	0
$562$	−2.90739	−0.122641
$563$	33.3270	1.40456	0.702282	−	0.711899i	$-0.252164\pi$
$563$	33.3270	1.40456	0.702282	+	0.711899i	$0.252164\pi$
$564$	0	0
$565$	0	0
$566$	−60.3784	−2.53789
$567$	0	0
$568$	−10.1073	−0.424091
$569$	27.3859	1.14808	0.574038	−	0.818829i	$-0.305376\pi$
$569$	27.3859	1.14808	0.574038	+	0.818829i	$0.305376\pi$
$570$	0	0
$571$	−43.2995	−1.81203	−0.906014	−	0.423247i	$-0.860890\pi$
$571$	−43.2995	−1.81203	−0.906014	+	0.423247i	$0.860890\pi$
$572$	−61.3079	−2.56341
$573$	0	0
$574$	−73.8221	−3.08128
$575$	0	0
$576$	0	0
$577$	−47.3721	−1.97213	−0.986064	−	0.166366i	$-0.946797\pi$
$577$	−47.3721	−1.97213	−0.986064	+	0.166366i	$0.946797\pi$
$578$	−36.4667	−1.51682
$579$	0	0
$580$	0	0
$581$	6.18920	0.256771
$582$	0	0
$583$	−20.9074	−0.865896
$584$	−9.78029	−0.404711
$585$	0	0
$586$	−18.1692	−0.750563
$587$	−29.0230	−1.19791	−0.598955	−	0.800783i	$-0.704417\pi$
$587$	−29.0230	−1.19791	−0.598955	+	0.800783i	$0.704417\pi$
$588$	0	0
$589$	1.45636	0.0600084
$590$	0	0
$591$	0	0
$592$	12.6707	0.520764
$593$	30.8706	1.26770	0.633852	−	0.773454i	$-0.281473\pi$
$593$	30.8706	1.26770	0.633852	+	0.773454i	$0.281473\pi$
$594$	0	0
$595$	0	0
$596$	−12.2167	−0.500414
$597$	0	0
$598$	−7.33537	−0.299966
$599$	−17.1608	−0.701172	−0.350586	−	0.936531i	$-0.614018\pi$
$599$	−17.1608	−0.701172	−0.350586	+	0.936531i	$0.614018\pi$
$600$	0	0
$601$	−13.2029	−0.538559	−0.269279	−	0.963062i	$-0.586786\pi$
$601$	−13.2029	−0.538559	−0.269279	+	0.963062i	$0.586786\pi$
$602$	47.1334	1.92101
$603$	0	0
$604$	4.14709	0.168743
$605$	0	0
$606$	0	0
$607$	16.7603	0.680279	0.340140	−	0.940375i	$-0.389526\pi$
$607$	16.7603	0.680279	0.340140	+	0.940375i	$0.389526\pi$
$608$	−7.80440	−0.316510
$609$	0	0
$610$	0	0
$611$	74.1196	2.99856
$612$	0	0
$613$	−32.7780	−1.32389	−0.661946	−	0.749552i	$-0.730269\pi$
$613$	−32.7780	−1.32389	−0.661946	+	0.749552i	$0.730269\pi$
$614$	41.4289	1.67193
$615$	0	0
$616$	−13.2765	−0.534925
$617$	−33.4510	−1.34669	−0.673344	−	0.739329i	$-0.735143\pi$
$617$	−33.4510	−1.34669	−0.673344	+	0.739329i	$0.735143\pi$
$618$	0	0
$619$	39.3731	1.58254	0.791269	−	0.611469i	$-0.209421\pi$
$619$	39.3731	1.58254	0.791269	+	0.611469i	$0.209421\pi$
$620$	0	0
$621$	0	0
$622$	−44.1471	−1.77014
$623$	12.3407	0.494420
$624$	0	0
$625$	0	0
$626$	−48.1418	−1.92413
$627$	0	0
$628$	34.1060	1.36098
$629$	0	0
$630$	0	0
$631$	40.5804	1.61548	0.807740	−	0.589538i	$-0.200690\pi$
$631$	40.5804	1.61548	0.807740	+	0.589538i	$0.200690\pi$
$632$	−9.71513	−0.386447
$633$	0	0
$634$	23.9706	0.951992
$635$	0	0
$636$	0	0
$637$	−3.41959	−0.135489
$638$	−24.1103	−0.954537
$639$	0	0
$640$	0	0
$641$	6.81172	0.269047	0.134523	−	0.990910i	$-0.457050\pi$
$641$	6.81172	0.269047	0.134523	+	0.990910i	$0.457050\pi$
$642$	0	0
$643$	14.3868	0.567360	0.283680	−	0.958919i	$-0.408445\pi$
$643$	14.3868	0.567360	0.283680	+	0.958919i	$0.408445\pi$
$644$	−3.88434	−0.153065
$645$	0	0
$646$	0	0
$647$	0.137775	0.00541649	0.00270825	−	0.999996i	$-0.499138\pi$
$647$	0.137775	0.00541649	0.00270825	+	0.999996i	$0.499138\pi$
$648$	0	0
$649$	−19.6823	−0.772599
$650$	0	0
$651$	0	0
$652$	20.1167	0.787832
$653$	−11.1019	−0.434452	−0.217226	−	0.976121i	$-0.569701\pi$
$653$	−11.1019	−0.434452	−0.217226	+	0.976121i	$0.569701\pi$
$654$	0	0
$655$	0	0
$656$	−30.5142	−1.19138
$657$	0	0
$658$	69.4236	2.70641
$659$	29.9725	1.16756	0.583782	−	0.811910i	$-0.301572\pi$
$659$	29.9725	1.16756	0.583782	+	0.811910i	$0.301572\pi$
$660$	0	0
$661$	−42.5530	−1.65512	−0.827559	−	0.561379i	$-0.810271\pi$
$661$	−42.5530	−1.65512	−0.827559	+	0.561379i	$0.810271\pi$
$662$	−55.6349	−2.16231
$663$	0	0
$664$	−2.90739	−0.112829
$665$	0	0
$666$	0	0
$667$	−1.63091	−0.0631491
$668$	−24.2638	−0.938795
$669$	0	0
$670$	0	0
$671$	−31.8201	−1.22840
$672$	0	0
$673$	−12.6961	−0.489397	−0.244699	−	0.969599i	$-0.578689\pi$
$673$	−12.6961	−0.489397	−0.244699	+	0.969599i	$0.578689\pi$
$674$	−32.3638	−1.24661
$675$	0	0
$676$	69.1123	2.65817
$677$	−21.3784	−0.821639	−0.410819	−	0.911717i	$-0.634757\pi$
$677$	−21.3784	−0.821639	−0.410819	+	0.911717i	$0.634757\pi$
$678$	0	0
$679$	−46.3363	−1.77822
$680$	0	0
$681$	0	0
$682$	11.7045	0.448187
$683$	24.0652	0.920828	0.460414	−	0.887704i	$-0.347701\pi$
$683$	24.0652	0.920828	0.460414	+	0.887704i	$0.347701\pi$
$684$	0	0
$685$	0	0
$686$	38.0388	1.45233
$687$	0	0
$688$	19.4825	0.742762
$689$	35.1019	1.33728
$690$	0	0
$691$	29.6677	1.12861	0.564306	−	0.825566i	$-0.309144\pi$
$691$	29.6677	1.12861	0.564306	+	0.825566i	$0.309144\pi$
$692$	−13.4614	−0.511726
$693$	0	0
$694$	−14.2060	−0.539252
$695$	0	0
$696$	0	0
$697$	0	0
$698$	19.4637	0.736712
$699$	0	0
$700$	0	0
$701$	−6.65396	−0.251317	−0.125658	−	0.992074i	$-0.540104\pi$
$701$	−6.65396	−0.251317	−0.125658	+	0.992074i	$0.540104\pi$
$702$	0	0
$703$	−5.20293	−0.196232
$704$	−44.4741	−1.67618
$705$	0	0
$706$	43.2176	1.62652
$707$	−36.4648	−1.37140
$708$	0	0
$709$	−29.0735	−1.09188	−0.545940	−	0.837824i	$-0.683827\pi$
$709$	−29.0735	−1.09188	−0.545940	+	0.837824i	$0.683827\pi$
$710$	0	0
$711$	0	0
$712$	−5.79707	−0.217254
$713$	0.791733	0.0296506
$714$	0	0
$715$	0	0
$716$	−39.8726	−1.49011
$717$	0	0
$718$	−34.0735	−1.27161
$719$	−12.6739	−0.472659	−0.236329	−	0.971673i	$-0.575944\pi$
$719$	−12.6739	−0.472659	−0.236329	+	0.971673i	$0.575944\pi$
$720$	0	0
$721$	−0.696068	−0.0259229
$722$	2.14510	0.0798325
$723$	0	0
$724$	−27.0705	−1.00607
$725$	0	0
$726$	0	0
$727$	−13.2260	−0.490524	−0.245262	−	0.969457i	$-0.578874\pi$
$727$	−13.2260	−0.490524	−0.245262	+	0.969457i	$0.578874\pi$
$728$	22.2902	0.826130
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−33.0735	−1.22160	−0.610800	−	0.791785i	$-0.709152\pi$
$733$	−33.0735	−1.22160	−0.610800	+	0.791785i	$0.709152\pi$
$734$	−53.3815	−1.97035
$735$	0	0
$736$	−4.24276	−0.156390
$737$	10.6172	0.391089
$738$	0	0
$739$	−27.2260	−1.00152	−0.500762	−	0.865585i	$-0.666947\pi$
$739$	−27.2260	−1.00152	−0.500762	+	0.865585i	$0.666947\pi$
$740$	0	0
$741$	0	0
$742$	32.8779	1.20699
$743$	51.5751	1.89211	0.946053	−	0.324012i	$-0.105032\pi$
$743$	51.5751	1.89211	0.946053	+	0.324012i	$0.105032\pi$
$744$	0	0
$745$	0	0
$746$	−25.0284	−0.916354
$747$	0	0
$748$	0	0
$749$	−6.15150	−0.224771
$750$	0	0
$751$	16.7971	0.612934	0.306467	−	0.951881i	$-0.400853\pi$
$751$	16.7971	0.612934	0.306467	+	0.951881i	$0.400853\pi$
$752$	28.6961	1.04644
$753$	0	0
$754$	40.4794	1.47417
$755$	0	0
$756$	0	0
$757$	34.6402	1.25902	0.629510	−	0.776992i	$-0.283256\pi$
$757$	34.6402	1.25902	0.629510	+	0.776992i	$0.283256\pi$
$758$	13.4931	0.490093
$759$	0	0
$760$	0	0
$761$	−33.8294	−1.22632	−0.613158	−	0.789960i	$-0.710101\pi$
$761$	−33.8294	−1.22632	−0.613158	+	0.789960i	$0.710101\pi$
$762$	0	0
$763$	3.30393	0.119610
$764$	17.0230	0.615872
$765$	0	0
$766$	25.8810	0.935119
$767$	33.0452	1.19319
$768$	0	0
$769$	21.7550	0.784504	0.392252	−	0.919858i	$-0.371696\pi$
$769$	21.7550	0.784504	0.392252	+	0.919858i	$0.371696\pi$
$770$	0	0
$771$	0	0
$772$	65.0009	2.33943
$773$	−1.77495	−0.0638405	−0.0319203	−	0.999490i	$-0.510162\pi$
$773$	−1.77495	−0.0638405	−0.0319203	+	0.999490i	$0.510162\pi$
$774$	0	0
$775$	0	0
$776$	21.7666	0.781374
$777$	0	0
$778$	30.8117	1.10465
$779$	12.5299	0.448931
$780$	0	0
$781$	−29.3500	−1.05023
$782$	0	0
$783$	0	0
$784$	−1.32392	−0.0472830
$785$	0	0
$786$	0	0
$787$	−19.8937	−0.709132	−0.354566	−	0.935031i	$-0.615371\pi$
$787$	−19.8937	−0.709132	−0.354566	+	0.935031i	$0.615371\pi$
$788$	22.8853	0.815254
$789$	0	0
$790$	0	0
$791$	34.4520	1.22497
$792$	0	0
$793$	53.4236	1.89713
$794$	66.7054	2.36729
$795$	0	0
$796$	22.7539	0.806490
$797$	−7.36909	−0.261027	−0.130513	−	0.991447i	$-0.541663\pi$
$797$	−7.36909	−0.261027	−0.130513	+	0.991447i	$0.541663\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	1.16616	0.0411785
$803$	−28.4005	−1.00223
$804$	0	0
$805$	0	0
$806$	−19.6509	−0.692174
$807$	0	0
$808$	17.1294	0.602610
$809$	10.4205	0.366366	0.183183	−	0.983079i	$-0.441360\pi$
$809$	10.4205	0.366366	0.183183	+	0.983079i	$0.441360\pi$
$810$	0	0
$811$	46.9074	1.64714	0.823571	−	0.567214i	$-0.191979\pi$
$811$	46.9074	1.64714	0.823571	+	0.567214i	$0.191979\pi$
$812$	21.4353	0.752232
$813$	0	0
$814$	−41.8148	−1.46561
$815$	0	0
$816$	0	0
$817$	−8.00000	−0.279885
$818$	56.7696	1.98490
$819$	0	0
$820$	0	0
$821$	−35.5392	−1.24033	−0.620164	−	0.784472i	$-0.712934\pi$
$821$	−35.5392	−1.24033	−0.620164	+	0.784472i	$0.712934\pi$
$822$	0	0
$823$	−17.7603	−0.619085	−0.309542	−	0.950886i	$-0.600176\pi$
$823$	−17.7603	−0.619085	−0.309542	+	0.950886i	$0.600176\pi$
$824$	0.326979	0.0113909
$825$	0	0
$826$	30.9515	1.07694
$827$	−30.7843	−1.07047	−0.535237	−	0.844702i	$-0.679778\pi$
$827$	−30.7843	−1.07047	−0.535237	+	0.844702i	$0.679778\pi$
$828$	0	0
$829$	−23.7098	−0.823475	−0.411738	−	0.911302i	$-0.635078\pi$
$829$	−23.7098	−0.823475	−0.411738	+	0.911302i	$0.635078\pi$
$830$	0	0
$831$	0	0
$832$	74.6686	2.58867
$833$	0	0
$834$	0	0
$835$	0	0
$836$	9.74657	0.337092
$837$	0	0
$838$	13.2765	0.458628
$839$	−34.1662	−1.17955	−0.589773	−	0.807569i	$-0.700783\pi$
$839$	−34.1662	−1.17955	−0.589773	+	0.807569i	$0.700783\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	32.7696	1.12932
$843$	0	0
$844$	−33.0859	−1.13886
$845$	0	0
$846$	0	0
$847$	−8.34071	−0.286590
$848$	13.5900	0.466683
$849$	0	0
$850$	0	0
$851$	−2.82851	−0.0969600
$852$	0	0
$853$	28.7780	0.985340	0.492670	−	0.870216i	$-0.336021\pi$
$853$	28.7780	0.985340	0.492670	+	0.870216i	$0.336021\pi$
$854$	50.0388	1.71229
$855$	0	0
$856$	2.88968	0.0987672
$857$	−47.2849	−1.61522	−0.807610	−	0.589717i	$-0.799239\pi$
$857$	−47.2849	−1.61522	−0.807610	+	0.589717i	$0.799239\pi$
$858$	0	0
$859$	−23.0221	−0.785505	−0.392752	−	0.919644i	$-0.628477\pi$
$859$	−23.0221	−0.785505	−0.392752	+	0.919644i	$0.628477\pi$
$860$	0	0
$861$	0	0
$862$	−29.0839	−0.990603
$863$	56.0545	1.90812	0.954058	−	0.299621i	$-0.0968601\pi$
$863$	56.0545	1.90812	0.954058	+	0.299621i	$0.0968601\pi$
$864$	0	0
$865$	0	0
$866$	63.6717	2.16365
$867$	0	0
$868$	−10.4059	−0.353198
$869$	−28.2113	−0.957004
$870$	0	0
$871$	−17.8255	−0.603992
$872$	−1.55203	−0.0525583
$873$	0	0
$874$	1.16616	0.0394459
$875$	0	0
$876$	0	0
$877$	3.37748	0.114049	0.0570247	−	0.998373i	$-0.481839\pi$
$877$	3.37748	0.114049	0.0570247	+	0.998373i	$0.481839\pi$
$878$	72.9535	2.46206
$879$	0	0
$880$	0	0
$881$	−36.0314	−1.21393	−0.606965	−	0.794729i	$-0.707613\pi$
$881$	−36.0314	−1.21393	−0.606965	+	0.794729i	$0.707613\pi$
$882$	0	0
$883$	6.23970	0.209983	0.104991	−	0.994473i	$-0.466518\pi$
$883$	6.23970	0.209983	0.104991	+	0.994473i	$0.466518\pi$
$884$	0	0
$885$	0	0
$886$	−35.0545	−1.17768
$887$	40.2942	1.35295	0.676473	−	0.736467i	$-0.263507\pi$
$887$	40.2942	1.35295	0.676473	+	0.736467i	$0.263507\pi$
$888$	0	0
$889$	−34.1745	−1.14618
$890$	0	0
$891$	0	0
$892$	27.4289	0.918388
$893$	−11.7833	−0.394315
$894$	0	0
$895$	0	0
$896$	27.0671	0.904250
$897$	0	0
$898$	−36.2501	−1.20968
$899$	−4.36909	−0.145717
$900$	0	0
$901$	0	0
$902$	100.700	3.35294
$903$	0	0
$904$	−16.1839	−0.538267
$905$	0	0
$906$	0	0
$907$	−1.59414	−0.0529325	−0.0264662	−	0.999650i	$-0.508425\pi$
$907$	−1.59414	−0.0529325	−0.0264662	+	0.999650i	$0.508425\pi$
$908$	20.3795	0.676317
$909$	0	0
$910$	0	0
$911$	6.58880	0.218297	0.109148	−	0.994025i	$-0.465188\pi$
$911$	6.58880	0.218297	0.109148	+	0.994025i	$0.465188\pi$
$912$	0	0
$913$	−8.44264	−0.279410
$914$	−36.1712	−1.19644
$915$	0	0
$916$	−61.4036	−2.02883
$917$	52.0461	1.71871
$918$	0	0
$919$	−13.3270	−0.439616	−0.219808	−	0.975543i	$-0.570543\pi$
$919$	−13.3270	−0.439616	−0.219808	+	0.975543i	$0.570543\pi$
$920$	0	0
$921$	0	0
$922$	52.4794	1.72832
$923$	49.2765	1.62196
$924$	0	0
$925$	0	0
$926$	31.6509	1.04011
$927$	0	0
$928$	23.4132	0.768576
$929$	22.0735	0.724210	0.362105	−	0.932137i	$-0.382058\pi$
$929$	22.0735	0.724210	0.362105	+	0.932137i	$0.382058\pi$
$930$	0	0
$931$	0.543637	0.0178170
$932$	−61.8715	−2.02667
$933$	0	0
$934$	26.4426	0.865229
$935$	0	0
$936$	0	0
$937$	13.5436	0.442451	0.221226	−	0.975223i	$-0.428994\pi$
$937$	13.5436	0.442451	0.221226	+	0.975223i	$0.428994\pi$
$938$	−16.6961	−0.545146
$939$	0	0
$940$	0	0
$941$	−3.96323	−0.129197	−0.0645987	−	0.997911i	$-0.520577\pi$
$941$	−3.96323	−0.129197	−0.0645987	+	0.997911i	$0.520577\pi$
$942$	0	0
$943$	6.81172	0.221820
$944$	12.7937	0.416400
$945$	0	0
$946$	−64.2942	−2.09038
$947$	24.3784	0.792192	0.396096	−	0.918209i	$-0.370365\pi$
$947$	24.3784	0.792192	0.396096	+	0.918209i	$0.370365\pi$
$948$	0	0
$949$	47.6823	1.54783
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−41.4226	−1.34181	−0.670906	−	0.741543i	$-0.734094\pi$
$953$	−41.4226	−1.34181	−0.670906	+	0.741543i	$0.734094\pi$
$954$	0	0
$955$	0	0
$956$	3.32071	0.107400
$957$	0	0
$958$	40.7750	1.31738
$959$	−58.7550	−1.89730
$960$	0	0
$961$	−28.8790	−0.931581
$962$	70.2039	2.26346
$963$	0	0
$964$	55.9137	1.80086
$965$	0	0
$966$	0	0
$967$	34.5888	1.11230	0.556150	−	0.831082i	$-0.312278\pi$
$967$	34.5888	1.11230	0.556150	+	0.831082i	$0.312278\pi$
$968$	3.91806	0.125931
$969$	0	0
$970$	0	0
$971$	−9.48475	−0.304380	−0.152190	−	0.988351i	$-0.548633\pi$
$971$	−9.48475	−0.304380	−0.152190	+	0.988351i	$0.548633\pi$
$972$	0	0
$973$	−55.7917	−1.78860
$974$	−71.3815	−2.28721
$975$	0	0
$976$	20.6834	0.662060
$977$	−46.7275	−1.49495	−0.747473	−	0.664293i	$-0.768733\pi$
$977$	−46.7275	−1.49495	−0.747473	+	0.664293i	$0.768733\pi$
$978$	0	0
$979$	−16.8338	−0.538012
$980$	0	0
$981$	0	0
$982$	−61.6234	−1.96648
$983$	−21.2344	−0.677271	−0.338636	−	0.940918i	$-0.609965\pi$
$983$	−21.2344	−0.677271	−0.338636	+	0.940918i	$0.609965\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	−16.3638	−0.520600
$989$	−4.34910	−0.138293
$990$	0	0
$991$	29.7466	0.944931	0.472465	−	0.881349i	$-0.343364\pi$
$991$	29.7466	0.944931	0.472465	+	0.881349i	$0.343364\pi$
$992$	−11.3660	−0.360872
$993$	0	0
$994$	46.1544	1.46393
$995$	0	0
$996$	0	0
$997$	−4.84850	−0.153553	−0.0767767	−	0.997048i	$-0.524463\pi$
$997$	−4.84850	−0.153553	−0.0767767	+	0.997048i	$0.524463\pi$
$998$	−52.5288	−1.66277
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4275.2.a.be.1.3		3
3.2	odd	2		1425.2.a.u.1.1	✓	3
5.4	even	2		4275.2.a.bh.1.1		3
15.2	even	4		1425.2.c.p.799.2		6
15.8	even	4		1425.2.c.p.799.5		6
15.14	odd	2		1425.2.a.v.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1425.2.a.u.1.1	✓	3	3.2	odd	2
1425.2.a.v.1.3	yes	3	15.14	odd	2
1425.2.c.p.799.2		6	15.2	even	4
1425.2.c.p.799.5		6	15.8	even	4
4275.2.a.be.1.3		3	1.1	even	1	trivial
4275.2.a.bh.1.1		3	5.4	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:	\( N \)	\(=\)	\( 4275 = 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4275.a (trivial)

\(f(q)\)	\(=\)	\(q+2.14510 q^{2} +2.60147 q^{4} -2.74657 q^{7} +1.29021 q^{8} +3.74657 q^{11} -6.29021 q^{13} -5.89167 q^{14} -2.43531 q^{16} +1.00000 q^{19} +8.03677 q^{22} +0.543637 q^{23} -13.4931 q^{26} -7.14510 q^{28} -3.00000 q^{29} +1.45636 q^{31} -7.80440 q^{32} -5.20293 q^{37} +2.14510 q^{38} +12.5299 q^{41} -8.00000 q^{43} +9.74657 q^{44} +1.16616 q^{46} -11.7833 q^{47} +0.543637 q^{49} -16.3638 q^{52} -5.58041 q^{53} -3.54364 q^{56} -6.43531 q^{58} -5.25343 q^{59} -8.49314 q^{61} +3.12405 q^{62} -11.8706 q^{64} +2.83384 q^{67} -7.83384 q^{71} -7.58041 q^{73} -11.1608 q^{74} +2.60147 q^{76} -10.2902 q^{77} -7.52991 q^{79} +26.8779 q^{82} -2.25343 q^{83} -17.1608 q^{86} +4.83384 q^{88} -4.49314 q^{89} +17.2765 q^{91} +1.41425 q^{92} -25.2765 q^{94} +16.8706 q^{97} +1.16616 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{4} - 9 q^{8} + 3 q^{11} - 6 q^{13} - 3 q^{14} + 12 q^{16} + 3 q^{19} + 3 q^{22} - 3 q^{23} - 24 q^{26} - 15 q^{28} - 9 q^{29} + 9 q^{31} - 18 q^{32} - 12 q^{37} - 24 q^{43} + 21 q^{44} + 21 q^{46}+ \cdots + 21 q^{98}+O(q^{100}) \) 3 * q + 6 * q^4 - 9 * q^8 + 3 * q^11 - 6 * q^13 - 3 * q^14 + 12 * q^16 + 3 * q^19 + 3 * q^22 - 3 * q^23 - 24 * q^26 - 15 * q^28 - 9 * q^29 + 9 * q^31 - 18 * q^32 - 12 * q^37 - 24 * q^43 + 21 * q^44 + 21 * q^46 - 6 * q^47 - 3 * q^49 + 6 * q^52 + 9 * q^53 - 6 * q^56 - 24 * q^59 - 9 * q^61 - 21 * q^62 + 3 * q^64 - 9 * q^67 - 6 * q^71 + 3 * q^73 + 18 * q^74 + 6 * q^76 - 18 * q^77 + 15 * q^79 + 33 * q^82 - 15 * q^83 - 3 * q^88 + 3 * q^89 + 6 * q^91 - 39 * q^92 - 30 * q^94 + 12 * q^97 + 21 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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