LMFDB - Embedded newform 8025.2.a.bb.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8025,2,Mod(1,8025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8025, 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("8025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8025 = 3 \cdot 5^{2} \cdot 107 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8025.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:	$64.0799476221$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 9x^{5} + 24x^{4} + 13x^{3} - 47x^{2} + 19x - 2 $ x^7 - 3x^6 - 9x^5 + 24x^4 + 13x^3 - 47x^2 + 19x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 321)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.180939$ of defining polynomial
Character	$\chi$	$=$	8025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.28249 q^{2} +1.00000 q^{3} +3.20977 q^{4} -2.28249 q^{6} +1.42306 q^{7} -2.76128 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.28249 q^{2} +1.00000 q^{3} +3.20977 q^{4} -2.28249 q^{6} +1.42306 q^{7} -2.76128 q^{8} +1.00000 q^{9} +4.88649 q^{11} +3.20977 q^{12} +3.16696 q^{13} -3.24813 q^{14} -0.116926 q^{16} +4.47350 q^{17} -2.28249 q^{18} +2.08118 q^{19} +1.42306 q^{21} -11.1534 q^{22} -4.11652 q^{23} -2.76128 q^{24} -7.22855 q^{26} +1.00000 q^{27} +4.56771 q^{28} -4.56498 q^{29} -5.34123 q^{31} +5.78945 q^{32} +4.88649 q^{33} -10.2107 q^{34} +3.20977 q^{36} -5.47418 q^{37} -4.75027 q^{38} +3.16696 q^{39} -2.82598 q^{41} -3.24813 q^{42} +2.10739 q^{43} +15.6845 q^{44} +9.39592 q^{46} +3.59287 q^{47} -0.116926 q^{48} -4.97489 q^{49} +4.47350 q^{51} +10.1652 q^{52} +7.24552 q^{53} -2.28249 q^{54} -3.92949 q^{56} +2.08118 q^{57} +10.4195 q^{58} +13.1036 q^{59} +6.92445 q^{61} +12.1913 q^{62} +1.42306 q^{63} -12.9805 q^{64} -11.1534 q^{66} -8.57666 q^{67} +14.3589 q^{68} -4.11652 q^{69} +3.90715 q^{71} -2.76128 q^{72} +11.4655 q^{73} +12.4948 q^{74} +6.68009 q^{76} +6.95379 q^{77} -7.22855 q^{78} +14.4418 q^{79} +1.00000 q^{81} +6.45028 q^{82} +13.2751 q^{83} +4.56771 q^{84} -4.81010 q^{86} -4.56498 q^{87} -13.4930 q^{88} -3.03398 q^{89} +4.50678 q^{91} -13.2131 q^{92} -5.34123 q^{93} -8.20069 q^{94} +5.78945 q^{96} -12.7850 q^{97} +11.3551 q^{98} +4.88649 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{3} + 14 q^{4} - 6 q^{7} - 3 q^{8} + 7 q^{9}+O(q^{10}) $ 7 * q + 7 * q^3 + 14 * q^4 - 6 * q^7 - 3 * q^8 + 7 * q^9	$ 7 q + 7 q^{3} + 14 q^{4} - 6 q^{7} - 3 q^{8} + 7 q^{9} + 4 q^{11} + 14 q^{12} - 6 q^{13} + 12 q^{14} + 32 q^{16} + 10 q^{17} + 8 q^{19} - 6 q^{21} - 10 q^{22} - 6 q^{23} - 3 q^{24} + 7 q^{26} + 7 q^{27} - 8 q^{28} + 16 q^{31} - 6 q^{32} + 4 q^{33} - 11 q^{34} + 14 q^{36} - 10 q^{37} + 13 q^{38} - 6 q^{39} - 2 q^{41} + 12 q^{42} - 2 q^{43} + 2 q^{44} - 30 q^{46} - 16 q^{47} + 32 q^{48} + 17 q^{49} + 10 q^{51} + 23 q^{52} + 16 q^{53} + 30 q^{56} + 8 q^{57} + 56 q^{58} + 20 q^{59} + 2 q^{61} + 52 q^{62} - 6 q^{63} + 43 q^{64} - 10 q^{66} - 30 q^{67} + 61 q^{68} - 6 q^{69} + 32 q^{71} - 3 q^{72} + 12 q^{73} - q^{74} - 49 q^{76} + 46 q^{77} + 7 q^{78} + 36 q^{79} + 7 q^{81} - 2 q^{82} + 10 q^{83} - 8 q^{84} - 20 q^{86} + 14 q^{88} - 4 q^{89} + 12 q^{91} - 10 q^{92} + 16 q^{93} - 26 q^{94} - 6 q^{96} - 24 q^{97} - 8 q^{98} + 4 q^{99}+O(q^{100}) $ 7 * q + 7 * q^3 + 14 * q^4 - 6 * q^7 - 3 * q^8 + 7 * q^9 + 4 * q^11 + 14 * q^12 - 6 * q^13 + 12 * q^14 + 32 * q^16 + 10 * q^17 + 8 * q^19 - 6 * q^21 - 10 * q^22 - 6 * q^23 - 3 * q^24 + 7 * q^26 + 7 * q^27 - 8 * q^28 + 16 * q^31 - 6 * q^32 + 4 * q^33 - 11 * q^34 + 14 * q^36 - 10 * q^37 + 13 * q^38 - 6 * q^39 - 2 * q^41 + 12 * q^42 - 2 * q^43 + 2 * q^44 - 30 * q^46 - 16 * q^47 + 32 * q^48 + 17 * q^49 + 10 * q^51 + 23 * q^52 + 16 * q^53 + 30 * q^56 + 8 * q^57 + 56 * q^58 + 20 * q^59 + 2 * q^61 + 52 * q^62 - 6 * q^63 + 43 * q^64 - 10 * q^66 - 30 * q^67 + 61 * q^68 - 6 * q^69 + 32 * q^71 - 3 * q^72 + 12 * q^73 - q^74 - 49 * q^76 + 46 * q^77 + 7 * q^78 + 36 * q^79 + 7 * q^81 - 2 * q^82 + 10 * q^83 - 8 * q^84 - 20 * q^86 + 14 * q^88 - 4 * q^89 + 12 * q^91 - 10 * q^92 + 16 * q^93 - 26 * q^94 - 6 * q^96 - 24 * q^97 - 8 * q^98 + 4 * q^99

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 $

$2$	−2.28249	−1.61397	−0.806983	−	0.590575i	$-0.798901\pi$
$2$	−2.28249	−1.61397	−0.806983	+	0.590575i	$0.798901\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	3.20977	1.60488
$5$	0	0
$5$	0	0
$6$	−2.28249	−0.931823
$7$	1.42306	0.537868	0.268934	−	0.963159i	$-0.413329\pi$
$7$	1.42306	0.537868	0.268934	+	0.963159i	$0.413329\pi$
$8$	−2.76128	−0.976262
$9$	1.00000	0.333333
$10$	0	0
$11$	4.88649	1.47333	0.736667	−	0.676256i	$-0.236399\pi$
$11$	4.88649	1.47333	0.736667	+	0.676256i	$0.236399\pi$
$12$	3.20977	0.926580
$13$	3.16696	0.878356	0.439178	−	0.898400i	$-0.355270\pi$
$13$	3.16696	0.878356	0.439178	+	0.898400i	$0.355270\pi$
$14$	−3.24813	−0.868100
$15$	0	0
$16$	−0.116926	−0.0292316
$17$	4.47350	1.08498	0.542492	−	0.840061i	$-0.317481\pi$
$17$	4.47350	1.08498	0.542492	+	0.840061i	$0.317481\pi$
$18$	−2.28249	−0.537988
$19$	2.08118	0.477454	0.238727	−	0.971087i	$-0.423270\pi$
$19$	2.08118	0.477454	0.238727	+	0.971087i	$0.423270\pi$
$20$	0	0
$21$	1.42306	0.310538
$22$	−11.1534	−2.37791
$23$	−4.11652	−0.858353	−0.429177	−	0.903221i	$-0.641196\pi$
$23$	−4.11652	−0.858353	−0.429177	+	0.903221i	$0.641196\pi$
$24$	−2.76128	−0.563645
$25$	0	0
$26$	−7.22855	−1.41764
$27$	1.00000	0.192450
$28$	4.56771	0.863215
$29$	−4.56498	−0.847696	−0.423848	−	0.905733i	$-0.639321\pi$
$29$	−4.56498	−0.847696	−0.423848	+	0.905733i	$0.639321\pi$
$30$	0	0
$31$	−5.34123	−0.959314	−0.479657	−	0.877456i	$-0.659239\pi$
$31$	−5.34123	−0.959314	−0.479657	+	0.877456i	$0.659239\pi$
$32$	5.78945	1.02344
$33$	4.88649	0.850630
$34$	−10.2107	−1.75113
$35$	0	0
$36$	3.20977	0.534961
$37$	−5.47418	−0.899949	−0.449975	−	0.893041i	$-0.648567\pi$
$37$	−5.47418	−0.899949	−0.449975	+	0.893041i	$0.648567\pi$
$38$	−4.75027	−0.770595
$39$	3.16696	0.507119
$40$	0	0
$41$	−2.82598	−0.441344	−0.220672	−	0.975348i	$-0.570825\pi$
$41$	−2.82598	−0.441344	−0.220672	+	0.975348i	$0.570825\pi$
$42$	−3.24813	−0.501198
$43$	2.10739	0.321374	0.160687	−	0.987005i	$-0.448629\pi$
$43$	2.10739	0.321374	0.160687	+	0.987005i	$0.448629\pi$
$44$	15.6845	2.36453
$45$	0	0
$46$	9.39592	1.38535
$47$	3.59287	0.524073	0.262037	−	0.965058i	$-0.415606\pi$
$47$	3.59287	0.524073	0.262037	+	0.965058i	$0.415606\pi$
$48$	−0.116926	−0.0168769
$49$	−4.97489	−0.710698
$50$	0	0
$51$	4.47350	0.626416
$52$	10.1652	1.40966
$53$	7.24552	0.995249	0.497624	−	0.867393i	$-0.334206\pi$
$53$	7.24552	0.995249	0.497624	+	0.867393i	$0.334206\pi$
$54$	−2.28249	−0.310608
$55$	0	0
$56$	−3.92949	−0.525100
$57$	2.08118	0.275658
$58$	10.4195	1.36815
$59$	13.1036	1.70595	0.852975	−	0.521952i	$-0.174796\pi$
$59$	13.1036	1.70595	0.852975	+	0.521952i	$0.174796\pi$
$60$	0	0
$61$	6.92445	0.886585	0.443292	−	0.896377i	$-0.353810\pi$
$61$	6.92445	0.886585	0.443292	+	0.896377i	$0.353810\pi$
$62$	12.1913	1.54830
$63$	1.42306	0.179289
$64$	−12.9805	−1.62257
$65$	0	0
$66$	−11.1534	−1.37289
$67$	−8.57666	−1.04781	−0.523903	−	0.851778i	$-0.675525\pi$
$67$	−8.57666	−1.04781	−0.523903	+	0.851778i	$0.675525\pi$
$68$	14.3589	1.74127
$69$	−4.11652	−0.495570
$70$	0	0
$71$	3.90715	0.463693	0.231847	−	0.972752i	$-0.425523\pi$
$71$	3.90715	0.463693	0.231847	+	0.972752i	$0.425523\pi$
$72$	−2.76128	−0.325421
$73$	11.4655	1.34194	0.670969	−	0.741485i	$-0.265878\pi$
$73$	11.4655	1.34194	0.670969	+	0.741485i	$0.265878\pi$
$74$	12.4948	1.45249
$75$	0	0
$76$	6.68009	0.766259
$77$	6.95379	0.792458
$78$	−7.22855	−0.818472
$79$	14.4418	1.62483	0.812417	−	0.583076i	$-0.198151\pi$
$79$	14.4418	1.62483	0.812417	+	0.583076i	$0.198151\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	6.45028	0.712315
$83$	13.2751	1.45713	0.728564	−	0.684978i	$-0.240188\pi$
$83$	13.2751	1.45713	0.728564	+	0.684978i	$0.240188\pi$
$84$	4.56771	0.498377
$85$	0	0
$86$	−4.81010	−0.518687
$87$	−4.56498	−0.489418
$88$	−13.4930	−1.43836
$89$	−3.03398	−0.321601	−0.160801	−	0.986987i	$-0.551408\pi$
$89$	−3.03398	−0.321601	−0.160801	+	0.986987i	$0.551408\pi$
$90$	0	0
$91$	4.50678	0.472439
$92$	−13.2131	−1.37756
$93$	−5.34123	−0.553860
$94$	−8.20069	−0.845836
$95$	0	0
$96$	5.78945	0.590884
$97$	−12.7850	−1.29812	−0.649060	−	0.760738i	$-0.724837\pi$
$97$	−12.7850	−1.29812	−0.649060	+	0.760738i	$0.724837\pi$
$98$	11.3551	1.14704
$99$	4.88649	0.491111
$100$	0	0
$101$	8.79802	0.875436	0.437718	−	0.899112i	$-0.355787\pi$
$101$	8.79802	0.875436	0.437718	+	0.899112i	$0.355787\pi$
$102$	−10.2107	−1.01101
$103$	−13.9992	−1.37939	−0.689693	−	0.724102i	$-0.742255\pi$
$103$	−13.9992	−1.37939	−0.689693	+	0.724102i	$0.742255\pi$
$104$	−8.74487	−0.857505
$105$	0	0
$106$	−16.5378	−1.60630
$107$	−1.00000	−0.0966736
$107$	−1.00000	−0.0966736
$108$	3.20977	0.308860
$109$	18.2956	1.75240	0.876199	−	0.481950i	$-0.160071\pi$
$109$	18.2956	1.75240	0.876199	+	0.481950i	$0.160071\pi$
$110$	0	0
$111$	−5.47418	−0.519586
$112$	−0.166394	−0.0157227
$113$	13.5227	1.27211	0.636056	−	0.771643i	$-0.280565\pi$
$113$	13.5227	1.27211	0.636056	+	0.771643i	$0.280565\pi$
$114$	−4.75027	−0.444903
$115$	0	0
$116$	−14.6525	−1.36045
$117$	3.16696	0.292785
$118$	−29.9090	−2.75334
$119$	6.36608	0.583578
$120$	0	0
$121$	12.8778	1.17071
$122$	−15.8050	−1.43092
$123$	−2.82598	−0.254810
$124$	−17.1441	−1.53959
$125$	0	0
$126$	−3.24813	−0.289367
$127$	−0.912956	−0.0810117	−0.0405059	−	0.999179i	$-0.512897\pi$
$127$	−0.912956	−0.0810117	−0.0405059	+	0.999179i	$0.512897\pi$
$128$	18.0490	1.59532
$129$	2.10739	0.185545
$130$	0	0
$131$	−11.2875	−0.986196	−0.493098	−	0.869974i	$-0.664136\pi$
$131$	−11.2875	−0.986196	−0.493098	+	0.869974i	$0.664136\pi$
$132$	15.6845	1.36516
$133$	2.96165	0.256807
$134$	19.5762	1.69112
$135$	0	0
$136$	−12.3526	−1.05923
$137$	−17.7005	−1.51225	−0.756126	−	0.654426i	$-0.772910\pi$
$137$	−17.7005	−1.51225	−0.756126	+	0.654426i	$0.772910\pi$
$138$	9.39592	0.799834
$139$	3.80176	0.322461	0.161231	−	0.986917i	$-0.448454\pi$
$139$	3.80176	0.322461	0.161231	+	0.986917i	$0.448454\pi$
$140$	0	0
$141$	3.59287	0.302574
$142$	−8.91803	−0.748385
$143$	15.4753	1.29411
$144$	−0.116926	−0.00974387
$145$	0	0
$146$	−26.1700	−2.16584
$147$	−4.97489	−0.410322
$148$	−17.5708	−1.44431
$149$	11.7181	0.959983	0.479992	−	0.877273i	$-0.340640\pi$
$149$	11.7181	0.959983	0.479992	+	0.877273i	$0.340640\pi$
$150$	0	0
$151$	−0.220916	−0.0179779	−0.00898893	−	0.999960i	$-0.502861\pi$
$151$	−0.220916	−0.0179779	−0.00898893	+	0.999960i	$0.502861\pi$
$152$	−5.74672	−0.466120
$153$	4.47350	0.361661
$154$	−15.8720	−1.27900
$155$	0	0
$156$	10.1652	0.813867
$157$	−18.6920	−1.49179	−0.745893	−	0.666066i	$-0.767977\pi$
$157$	−18.6920	−1.49179	−0.745893	+	0.666066i	$0.767977\pi$
$158$	−32.9634	−2.62243
$159$	7.24552	0.574607
$160$	0	0
$161$	−5.85807	−0.461680
$162$	−2.28249	−0.179329
$163$	−4.87061	−0.381496	−0.190748	−	0.981639i	$-0.561091\pi$
$163$	−4.87061	−0.381496	−0.190748	+	0.981639i	$0.561091\pi$
$164$	−9.07075	−0.708307
$165$	0	0
$166$	−30.3002	−2.35175
$167$	−11.8748	−0.918898	−0.459449	−	0.888204i	$-0.651953\pi$
$167$	−11.8748	−0.918898	−0.459449	+	0.888204i	$0.651953\pi$
$168$	−3.92949	−0.303166
$169$	−2.97039	−0.228491
$170$	0	0
$171$	2.08118	0.159151
$172$	6.76424	0.515768
$173$	23.1617	1.76095	0.880475	−	0.474093i	$-0.157224\pi$
$173$	23.1617	1.76095	0.880475	+	0.474093i	$0.157224\pi$
$174$	10.4195	0.789903
$175$	0	0
$176$	−0.571360	−0.0430679
$177$	13.1036	0.984931
$178$	6.92503	0.519053
$179$	1.88266	0.140717	0.0703584	−	0.997522i	$-0.477586\pi$
$179$	1.88266	0.140717	0.0703584	+	0.997522i	$0.477586\pi$
$180$	0	0
$181$	3.20095	0.237925	0.118962	−	0.992899i	$-0.462043\pi$
$181$	3.20095	0.237925	0.118962	+	0.992899i	$0.462043\pi$
$182$	−10.2867	−0.762500
$183$	6.92445	0.511870
$184$	11.3669	0.837977
$185$	0	0
$186$	12.1913	0.893911
$187$	21.8597	1.59854
$188$	11.5323	0.841077
$189$	1.42306	0.103513
$190$	0	0
$191$	1.38181	0.0999840	0.0499920	−	0.998750i	$-0.484080\pi$
$191$	1.38181	0.0999840	0.0499920	+	0.998750i	$0.484080\pi$
$192$	−12.9805	−0.936789
$193$	4.94962	0.356281	0.178141	−	0.984005i	$-0.442992\pi$
$193$	4.94962	0.356281	0.178141	+	0.984005i	$0.442992\pi$
$194$	29.1816	2.09512
$195$	0	0
$196$	−15.9682	−1.14059
$197$	−1.49673	−0.106638	−0.0533189	−	0.998578i	$-0.516980\pi$
$197$	−1.49673	−0.106638	−0.0533189	+	0.998578i	$0.516980\pi$
$198$	−11.1534	−0.792636
$199$	−8.27670	−0.586720	−0.293360	−	0.956002i	$-0.594773\pi$
$199$	−8.27670	−0.586720	−0.293360	+	0.956002i	$0.594773\pi$
$200$	0	0
$201$	−8.57666	−0.604951
$202$	−20.0814	−1.41292
$203$	−6.49626	−0.455948
$204$	14.3589	1.00532
$205$	0	0
$206$	31.9532	2.22628
$207$	−4.11652	−0.286118
$208$	−0.370301	−0.0256757
$209$	10.1697	0.703450
$210$	0	0
$211$	−23.2266	−1.59898	−0.799491	−	0.600678i	$-0.794898\pi$
$211$	−23.2266	−1.59898	−0.799491	+	0.600678i	$0.794898\pi$
$212$	23.2564	1.59726
$213$	3.90715	0.267713
$214$	2.28249	0.156028
$215$	0	0
$216$	−2.76128	−0.187882
$217$	−7.60092	−0.515984
$218$	−41.7595	−2.82831
$219$	11.4655	0.774769
$220$	0	0
$221$	14.1674	0.953002
$222$	12.4948	0.838594
$223$	15.3134	1.02546	0.512730	−	0.858550i	$-0.328634\pi$
$223$	15.3134	1.02546	0.512730	+	0.858550i	$0.328634\pi$
$224$	8.23876	0.550475
$225$	0	0
$226$	−30.8655	−2.05314
$227$	−16.4357	−1.09088	−0.545438	−	0.838151i	$-0.683637\pi$
$227$	−16.4357	−1.09088	−0.545438	+	0.838151i	$0.683637\pi$
$228$	6.68009	0.442400
$229$	2.84827	0.188219	0.0941096	−	0.995562i	$-0.470000\pi$
$229$	2.84827	0.188219	0.0941096	+	0.995562i	$0.470000\pi$
$230$	0	0
$231$	6.95379	0.457526
$232$	12.6052	0.827573
$233$	−23.9343	−1.56799	−0.783994	−	0.620768i	$-0.786821\pi$
$233$	−23.9343	−1.56799	−0.783994	+	0.620768i	$0.786821\pi$
$234$	−7.22855	−0.472545
$235$	0	0
$236$	42.0597	2.73785
$237$	14.4418	0.938099
$238$	−14.5305	−0.941874
$239$	−14.0141	−0.906495	−0.453248	−	0.891385i	$-0.649735\pi$
$239$	−14.0141	−0.906495	−0.453248	+	0.891385i	$0.649735\pi$
$240$	0	0
$241$	−29.4278	−1.89561	−0.947805	−	0.318851i	$-0.896703\pi$
$241$	−29.4278	−1.89561	−0.947805	+	0.318851i	$0.896703\pi$
$242$	−29.3935	−1.88949
$243$	1.00000	0.0641500
$244$	22.2259	1.42287
$245$	0	0
$246$	6.45028	0.411255
$247$	6.59099	0.419375
$248$	14.7487	0.936541
$249$	13.2751	0.841273
$250$	0	0
$251$	9.23123	0.582670	0.291335	−	0.956621i	$-0.405901\pi$
$251$	9.23123	0.582670	0.291335	+	0.956621i	$0.405901\pi$
$252$	4.56771	0.287738
$253$	−20.1153	−1.26464
$254$	2.08381	0.130750
$255$	0	0
$256$	−15.2357	−0.952232
$257$	27.9285	1.74213	0.871067	−	0.491164i	$-0.163429\pi$
$257$	27.9285	1.74213	0.871067	+	0.491164i	$0.163429\pi$
$258$	−4.81010	−0.299464
$259$	−7.79011	−0.484054
$260$	0	0
$261$	−4.56498	−0.282565
$262$	25.7637	1.59169
$263$	9.71037	0.598767	0.299383	−	0.954133i	$-0.403219\pi$
$263$	9.71037	0.598767	0.299383	+	0.954133i	$0.403219\pi$
$264$	−13.4930	−0.830437
$265$	0	0
$266$	−6.75993	−0.414478
$267$	−3.03398	−0.185676
$268$	−27.5291	−1.68161
$269$	−0.196454	−0.0119780	−0.00598900	−	0.999982i	$-0.501906\pi$
$269$	−0.196454	−0.0119780	−0.00598900	+	0.999982i	$0.501906\pi$
$270$	0	0
$271$	10.8653	0.660023	0.330011	−	0.943977i	$-0.392947\pi$
$271$	10.8653	0.660023	0.330011	+	0.943977i	$0.392947\pi$
$272$	−0.523071	−0.0317158
$273$	4.50678	0.272763
$274$	40.4011	2.44072
$275$	0	0
$276$	−13.2131	−0.795333
$277$	8.64846	0.519636	0.259818	−	0.965658i	$-0.416337\pi$
$277$	8.64846	0.519636	0.259818	+	0.965658i	$0.416337\pi$
$278$	−8.67749	−0.520441
$279$	−5.34123	−0.319771
$280$	0	0
$281$	27.1433	1.61923	0.809617	−	0.586958i	$-0.199675\pi$
$281$	27.1433	1.61923	0.809617	+	0.586958i	$0.199675\pi$
$282$	−8.20069	−0.488344
$283$	23.7111	1.40948	0.704740	−	0.709466i	$-0.251064\pi$
$283$	23.7111	1.40948	0.704740	+	0.709466i	$0.251064\pi$
$284$	12.5410	0.744174
$285$	0	0
$286$	−35.3223	−2.08865
$287$	−4.02156	−0.237385
$288$	5.78945	0.341147
$289$	3.01223	0.177190
$290$	0	0
$291$	−12.7850	−0.749469
$292$	36.8017	2.15366
$293$	27.0101	1.57795	0.788975	−	0.614426i	$-0.210612\pi$
$293$	27.0101	1.57795	0.788975	+	0.614426i	$0.210612\pi$
$294$	11.3551	0.662245
$295$	0	0
$296$	15.1158	0.878586
$297$	4.88649	0.283543
$298$	−26.7464	−1.54938
$299$	−13.0368	−0.753939
$300$	0	0
$301$	2.99895	0.172857
$302$	0.504238	0.0290156
$303$	8.79802	0.505433
$304$	−0.243344	−0.0139568
$305$	0	0
$306$	−10.2107	−0.583709
$307$	−3.67259	−0.209606	−0.104803	−	0.994493i	$-0.533421\pi$
$307$	−3.67259	−0.209606	−0.104803	+	0.994493i	$0.533421\pi$
$308$	22.3201	1.27180
$309$	−13.9992	−0.796389
$310$	0	0
$311$	−21.6941	−1.23016	−0.615080	−	0.788465i	$-0.710876\pi$
$311$	−21.6941	−1.23016	−0.615080	+	0.788465i	$0.710876\pi$
$312$	−8.74487	−0.495081
$313$	−9.29453	−0.525358	−0.262679	−	0.964883i	$-0.584606\pi$
$313$	−9.29453	−0.525358	−0.262679	+	0.964883i	$0.584606\pi$
$314$	42.6644	2.40769
$315$	0	0
$316$	46.3550	2.60767
$317$	20.0952	1.12866	0.564328	−	0.825550i	$-0.309135\pi$
$317$	20.0952	1.12866	0.564328	+	0.825550i	$0.309135\pi$
$318$	−16.5378	−0.927396
$319$	−22.3068	−1.24894
$320$	0	0
$321$	−1.00000	−0.0558146
$322$	13.3710	0.745136
$323$	9.31014	0.518030
$324$	3.20977	0.178320
$325$	0	0
$326$	11.1171	0.615721
$327$	18.2956	1.01175
$328$	7.80334	0.430868
$329$	5.11288	0.281882
$330$	0	0
$331$	25.7523	1.41547	0.707736	−	0.706477i	$-0.249717\pi$
$331$	25.7523	1.41547	0.707736	+	0.706477i	$0.249717\pi$
$332$	42.6099	2.33852
$333$	−5.47418	−0.299983
$334$	27.1041	1.48307
$335$	0	0
$336$	−0.166394	−0.00907752
$337$	−19.7729	−1.07710	−0.538549	−	0.842594i	$-0.681027\pi$
$337$	−19.7729	−1.07710	−0.538549	+	0.842594i	$0.681027\pi$
$338$	6.77988	0.368777
$339$	13.5227	0.734454
$340$	0	0
$341$	−26.0999	−1.41339
$342$	−4.75027	−0.256865
$343$	−17.0410	−0.920129
$344$	−5.81911	−0.313745
$345$	0	0
$346$	−52.8663	−2.84211
$347$	35.2495	1.89229	0.946145	−	0.323743i	$-0.104941\pi$
$347$	35.2495	1.89229	0.946145	+	0.323743i	$0.104941\pi$
$348$	−14.6525	−0.785458
$349$	−17.9946	−0.963228	−0.481614	−	0.876383i	$-0.659949\pi$
$349$	−17.9946	−0.963228	−0.481614	+	0.876383i	$0.659949\pi$
$350$	0	0
$351$	3.16696	0.169040
$352$	28.2901	1.50787
$353$	−17.2244	−0.916763	−0.458382	−	0.888755i	$-0.651571\pi$
$353$	−17.2244	−0.916763	−0.458382	+	0.888755i	$0.651571\pi$
$354$	−29.9090	−1.58964
$355$	0	0
$356$	−9.73837	−0.516132
$357$	6.36608	0.336929
$358$	−4.29716	−0.227112
$359$	23.9216	1.26253	0.631267	−	0.775566i	$-0.282535\pi$
$359$	23.9216	1.26253	0.631267	+	0.775566i	$0.282535\pi$
$360$	0	0
$361$	−14.6687	−0.772037
$362$	−7.30614	−0.384002
$363$	12.8778	0.675911
$364$	14.4657	0.758210
$365$	0	0
$366$	−15.8050	−0.826140
$367$	20.1656	1.05263	0.526317	−	0.850288i	$-0.323573\pi$
$367$	20.1656	1.05263	0.526317	+	0.850288i	$0.323573\pi$
$368$	0.481330	0.0250910
$369$	−2.82598	−0.147115
$370$	0	0
$371$	10.3108	0.535312
$372$	−17.1441	−0.888881
$373$	−6.83635	−0.353973	−0.176986	−	0.984213i	$-0.556635\pi$
$373$	−6.83635	−0.353973	−0.176986	+	0.984213i	$0.556635\pi$
$374$	−49.8947	−2.57999
$375$	0	0
$376$	−9.92093	−0.511633
$377$	−14.4571	−0.744579
$378$	−3.24813	−0.167066
$379$	12.5326	0.643759	0.321879	−	0.946781i	$-0.395685\pi$
$379$	12.5326	0.643759	0.321879	+	0.946781i	$0.395685\pi$
$380$	0	0
$381$	−0.912956	−0.0467721
$382$	−3.15396	−0.161371
$383$	−21.8290	−1.11541	−0.557704	−	0.830040i	$-0.688317\pi$
$383$	−21.8290	−1.11541	−0.557704	+	0.830040i	$0.688317\pi$
$384$	18.0490	0.921061
$385$	0	0
$386$	−11.2975	−0.575026
$387$	2.10739	0.107125
$388$	−41.0369	−2.08333
$389$	−33.5774	−1.70244	−0.851220	−	0.524808i	$-0.824137\pi$
$389$	−33.5774	−1.70244	−0.851220	+	0.524808i	$0.824137\pi$
$390$	0	0
$391$	−18.4153	−0.931299
$392$	13.7371	0.693828
$393$	−11.2875	−0.569381
$394$	3.41628	0.172110
$395$	0	0
$396$	15.6845	0.788176
$397$	−3.51375	−0.176350	−0.0881751	−	0.996105i	$-0.528103\pi$
$397$	−3.51375	−0.176350	−0.0881751	+	0.996105i	$0.528103\pi$
$398$	18.8915	0.946945
$399$	2.96165	0.148268
$400$	0	0
$401$	−13.7752	−0.687902	−0.343951	−	0.938988i	$-0.611765\pi$
$401$	−13.7752	−0.687902	−0.343951	+	0.938988i	$0.611765\pi$
$402$	19.5762	0.976370
$403$	−16.9155	−0.842619
$404$	28.2396	1.40497
$405$	0	0
$406$	14.8277	0.735885
$407$	−26.7495	−1.32593
$408$	−12.3526	−0.611546
$409$	14.7520	0.729441	0.364721	−	0.931117i	$-0.381164\pi$
$409$	14.7520	0.729441	0.364721	+	0.931117i	$0.381164\pi$
$410$	0	0
$411$	−17.7005	−0.873099
$412$	−44.9343	−2.21375
$413$	18.6473	0.917575
$414$	9.39592	0.461784
$415$	0	0
$416$	18.3349	0.898945
$417$	3.80176	0.186173
$418$	−23.2121	−1.13534
$419$	4.55485	0.222519	0.111259	−	0.993791i	$-0.464512\pi$
$419$	4.55485	0.222519	0.111259	+	0.993791i	$0.464512\pi$
$420$	0	0
$421$	7.41767	0.361515	0.180758	−	0.983528i	$-0.442145\pi$
$421$	7.41767	0.361515	0.180758	+	0.983528i	$0.442145\pi$
$422$	53.0144	2.58070
$423$	3.59287	0.174691
$424$	−20.0069	−0.971623
$425$	0	0
$426$	−8.91803	−0.432080
$427$	9.85394	0.476865
$428$	−3.20977	−0.155150
$429$	15.4753	0.747155
$430$	0	0
$431$	30.3492	1.46187	0.730934	−	0.682448i	$-0.239085\pi$
$431$	30.3492	1.46187	0.730934	+	0.682448i	$0.239085\pi$
$432$	−0.116926	−0.00562562
$433$	−32.5631	−1.56488	−0.782440	−	0.622726i	$-0.786025\pi$
$433$	−32.5631	−1.56488	−0.782440	+	0.622726i	$0.786025\pi$
$434$	17.3490	0.832780
$435$	0	0
$436$	58.7245	2.81239
$437$	−8.56719	−0.409825
$438$	−26.1700	−1.25045
$439$	14.9446	0.713265	0.356633	−	0.934245i	$-0.383925\pi$
$439$	14.9446	0.713265	0.356633	+	0.934245i	$0.383925\pi$
$440$	0	0
$441$	−4.97489	−0.236899
$442$	−32.3369	−1.53811
$443$	31.2094	1.48281	0.741403	−	0.671061i	$-0.234161\pi$
$443$	31.2094	1.48281	0.741403	+	0.671061i	$0.234161\pi$
$444$	−17.5708	−0.833875
$445$	0	0
$446$	−34.9526	−1.65506
$447$	11.7181	0.554247
$448$	−18.4721	−0.872726
$449$	9.44876	0.445915	0.222957	−	0.974828i	$-0.428429\pi$
$449$	9.44876	0.445915	0.222957	+	0.974828i	$0.428429\pi$
$450$	0	0
$451$	−13.8092	−0.650248
$452$	43.4048	2.04159
$453$	−0.220916	−0.0103795
$454$	37.5143	1.76063
$455$	0	0
$456$	−5.74672	−0.269115
$457$	39.2266	1.83494	0.917471	−	0.397803i	$-0.130227\pi$
$457$	39.2266	1.83494	0.917471	+	0.397803i	$0.130227\pi$
$458$	−6.50116	−0.303779
$459$	4.47350	0.208805
$460$	0	0
$461$	23.0884	1.07534	0.537668	−	0.843157i	$-0.319305\pi$
$461$	23.0884	1.07534	0.537668	+	0.843157i	$0.319305\pi$
$462$	−15.8720	−0.738431
$463$	−22.6692	−1.05353	−0.526763	−	0.850012i	$-0.676594\pi$
$463$	−22.6692	−1.05353	−0.526763	+	0.850012i	$0.676594\pi$
$464$	0.533767	0.0247795
$465$	0	0
$466$	54.6299	2.53068
$467$	−34.2270	−1.58384	−0.791918	−	0.610628i	$-0.790917\pi$
$467$	−34.2270	−1.58384	−0.791918	+	0.610628i	$0.790917\pi$
$468$	10.1652	0.469886
$469$	−12.2051	−0.563581
$470$	0	0
$471$	−18.6920	−0.861283
$472$	−36.1829	−1.66545
$473$	10.2978	0.473491
$474$	−32.9634	−1.51406
$475$	0	0
$476$	20.4336	0.936574
$477$	7.24552	0.331750
$478$	31.9870	1.46305
$479$	−1.19842	−0.0547572	−0.0273786	−	0.999625i	$-0.508716\pi$
$479$	−1.19842	−0.0547572	−0.0273786	+	0.999625i	$0.508716\pi$
$480$	0	0
$481$	−17.3365	−0.790476
$482$	67.1686	3.05945
$483$	−5.85807	−0.266551
$484$	41.3348	1.87886
$485$	0	0
$486$	−2.28249	−0.103536
$487$	−13.6476	−0.618433	−0.309216	−	0.950992i	$-0.600067\pi$
$487$	−13.6476	−0.618433	−0.309216	+	0.950992i	$0.600067\pi$
$488$	−19.1204	−0.865539
$489$	−4.87061	−0.220257
$490$	0	0
$491$	−27.2368	−1.22918	−0.614590	−	0.788846i	$-0.710679\pi$
$491$	−27.2368	−1.22918	−0.614590	+	0.788846i	$0.710679\pi$
$492$	−9.07075	−0.408941
$493$	−20.4215	−0.919737
$494$	−15.0439	−0.676856
$495$	0	0
$496$	0.624531	0.0280423
$497$	5.56012	0.249406
$498$	−30.3002	−1.35779
$499$	−33.6869	−1.50803	−0.754015	−	0.656857i	$-0.771886\pi$
$499$	−33.6869	−1.50803	−0.754015	+	0.656857i	$0.771886\pi$
$500$	0	0
$501$	−11.8748	−0.530526
$502$	−21.0702	−0.940409
$503$	11.2872	0.503270	0.251635	−	0.967822i	$-0.419032\pi$
$503$	11.2872	0.503270	0.251635	+	0.967822i	$0.419032\pi$
$504$	−3.92949	−0.175033
$505$	0	0
$506$	45.9131	2.04109
$507$	−2.97039	−0.131920
$508$	−2.93038	−0.130014
$509$	−7.38174	−0.327190	−0.163595	−	0.986528i	$-0.552309\pi$
$509$	−7.38174	−0.327190	−0.163595	+	0.986528i	$0.552309\pi$
$510$	0	0
$511$	16.3162	0.721785
$512$	−1.32267	−0.0584545
$513$	2.08118	0.0918861
$514$	−63.7466	−2.81174
$515$	0	0
$516$	6.76424	0.297779
$517$	17.5565	0.772135
$518$	17.7809	0.781246
$519$	23.1617	1.01668
$520$	0	0
$521$	13.8239	0.605636	0.302818	−	0.953048i	$-0.402073\pi$
$521$	13.8239	0.605636	0.302818	+	0.953048i	$0.402073\pi$
$522$	10.4195	0.456051
$523$	12.9549	0.566478	0.283239	−	0.959049i	$-0.408591\pi$
$523$	12.9549	0.566478	0.283239	+	0.959049i	$0.408591\pi$
$524$	−36.2304	−1.58273
$525$	0	0
$526$	−22.1638	−0.966389
$527$	−23.8940	−1.04084
$528$	−0.571360	−0.0248653
$529$	−6.05428	−0.263230
$530$	0	0
$531$	13.1036	0.568650
$532$	9.50620	0.412146
$533$	−8.94977	−0.387657
$534$	6.92503	0.299675
$535$	0	0
$536$	23.6826	1.02293
$537$	1.88266	0.0812428
$538$	0.448404	0.0193321
$539$	−24.3098	−1.04710
$540$	0	0
$541$	14.8154	0.636962	0.318481	−	0.947929i	$-0.396827\pi$
$541$	14.8154	0.636962	0.318481	+	0.947929i	$0.396827\pi$
$542$	−24.8001	−1.06525
$543$	3.20095	0.137366
$544$	25.8991	1.11042
$545$	0	0
$546$	−10.2867	−0.440230
$547$	−27.3809	−1.17072	−0.585362	−	0.810772i	$-0.699048\pi$
$547$	−27.3809	−1.17072	−0.585362	+	0.810772i	$0.699048\pi$
$548$	−56.8143	−2.42699
$549$	6.92445	0.295528
$550$	0	0
$551$	−9.50053	−0.404736
$552$	11.3669	0.483806
$553$	20.5517	0.873946
$554$	−19.7400	−0.838674
$555$	0	0
$556$	12.2028	0.517513
$557$	−1.66529	−0.0705604	−0.0352802	−	0.999377i	$-0.511232\pi$
$557$	−1.66529	−0.0705604	−0.0352802	+	0.999377i	$0.511232\pi$
$558$	12.1913	0.516100
$559$	6.67402	0.282281
$560$	0	0
$561$	21.8597	0.922919
$562$	−61.9544	−2.61339
$563$	2.60241	0.109678	0.0548392	−	0.998495i	$-0.482535\pi$
$563$	2.60241	0.109678	0.0548392	+	0.998495i	$0.482535\pi$
$564$	11.5323	0.485596
$565$	0	0
$566$	−54.1204	−2.27485
$567$	1.42306	0.0597631
$568$	−10.7887	−0.452686
$569$	−22.3848	−0.938420	−0.469210	−	0.883087i	$-0.655461\pi$
$569$	−22.3848	−0.938420	−0.469210	+	0.883087i	$0.655461\pi$
$570$	0	0
$571$	−7.00987	−0.293354	−0.146677	−	0.989184i	$-0.546858\pi$
$571$	−7.00987	−0.293354	−0.146677	+	0.989184i	$0.546858\pi$
$572$	49.6722	2.07690
$573$	1.38181	0.0577258
$574$	9.17917	0.383131
$575$	0	0
$576$	−12.9805	−0.540855
$577$	−42.0160	−1.74915	−0.874575	−	0.484891i	$-0.838859\pi$
$577$	−42.0160	−1.74915	−0.874575	+	0.484891i	$0.838859\pi$
$578$	−6.87539	−0.285978
$579$	4.94962	0.205699
$580$	0	0
$581$	18.8913	0.783742
$582$	29.1816	1.20962
$583$	35.4052	1.46633
$584$	−31.6596	−1.31008
$585$	0	0
$586$	−61.6504	−2.54676
$587$	−3.98133	−0.164327	−0.0821636	−	0.996619i	$-0.526183\pi$
$587$	−3.98133	−0.164327	−0.0821636	+	0.996619i	$0.526183\pi$
$588$	−15.9682	−0.658519
$589$	−11.1160	−0.458029
$590$	0	0
$591$	−1.49673	−0.0615674
$592$	0.640076	0.0263070
$593$	6.74921	0.277157	0.138578	−	0.990351i	$-0.455747\pi$
$593$	6.74921	0.277157	0.138578	+	0.990351i	$0.455747\pi$
$594$	−11.1534	−0.457629
$595$	0	0
$596$	37.6124	1.54066
$597$	−8.27670	−0.338743
$598$	29.7565	1.21683
$599$	34.7455	1.41966	0.709831	−	0.704372i	$-0.248771\pi$
$599$	34.7455	1.41966	0.709831	+	0.704372i	$0.248771\pi$
$600$	0	0
$601$	−19.7488	−0.805572	−0.402786	−	0.915294i	$-0.631958\pi$
$601$	−19.7488	−0.805572	−0.402786	+	0.915294i	$0.631958\pi$
$602$	−6.84508	−0.278985
$603$	−8.57666	−0.349269
$604$	−0.709088	−0.0288524
$605$	0	0
$606$	−20.0814	−0.815751
$607$	43.6806	1.77294	0.886470	−	0.462787i	$-0.153151\pi$
$607$	43.6806	1.77294	0.886470	+	0.462787i	$0.153151\pi$
$608$	12.0489	0.488646
$609$	−6.49626	−0.263242
$610$	0	0
$611$	11.3784	0.460323
$612$	14.3589	0.580424
$613$	12.7241	0.513920	0.256960	−	0.966422i	$-0.417279\pi$
$613$	12.7241	0.513920	0.256960	+	0.966422i	$0.417279\pi$
$614$	8.38266	0.338297
$615$	0	0
$616$	−19.2014	−0.773647
$617$	−3.76414	−0.151539	−0.0757693	−	0.997125i	$-0.524141\pi$
$617$	−3.76414	−0.151539	−0.0757693	+	0.997125i	$0.524141\pi$
$618$	31.9532	1.28534
$619$	8.29468	0.333391	0.166696	−	0.986008i	$-0.446690\pi$
$619$	8.29468	0.333391	0.166696	+	0.986008i	$0.446690\pi$
$620$	0	0
$621$	−4.11652	−0.165190
$622$	49.5166	1.98544
$623$	−4.31755	−0.172979
$624$	−0.370301	−0.0148239
$625$	0	0
$626$	21.2147	0.847909
$627$	10.1697	0.406137
$628$	−59.9971	−2.39414
$629$	−24.4888	−0.976430
$630$	0	0
$631$	16.0876	0.640439	0.320219	−	0.947343i	$-0.396243\pi$
$631$	16.0876	0.640439	0.320219	+	0.947343i	$0.396243\pi$
$632$	−39.8781	−1.58626
$633$	−23.2266	−0.923173
$634$	−45.8670	−1.82161
$635$	0	0
$636$	23.2564	0.922178
$637$	−15.7553	−0.624246
$638$	50.9150	2.01574
$639$	3.90715	0.154564
$640$	0	0
$641$	−37.4670	−1.47986	−0.739929	−	0.672685i	$-0.765141\pi$
$641$	−37.4670	−1.47986	−0.739929	+	0.672685i	$0.765141\pi$
$642$	2.28249	0.0900828
$643$	−6.93554	−0.273511	−0.136755	−	0.990605i	$-0.543667\pi$
$643$	−6.93554	−0.273511	−0.136755	+	0.990605i	$0.543667\pi$
$644$	−18.8030	−0.740944
$645$	0	0
$646$	−21.2503	−0.836083
$647$	−28.9566	−1.13840	−0.569200	−	0.822199i	$-0.692747\pi$
$647$	−28.9566	−1.13840	−0.569200	+	0.822199i	$0.692747\pi$
$648$	−2.76128	−0.108474
$649$	64.0309	2.51343
$650$	0	0
$651$	−7.60092	−0.297903
$652$	−15.6335	−0.612257
$653$	16.4862	0.645153	0.322577	−	0.946543i	$-0.395451\pi$
$653$	16.4862	0.645153	0.322577	+	0.946543i	$0.395451\pi$
$654$	−41.7595	−1.63293
$655$	0	0
$656$	0.330432	0.0129012
$657$	11.4655	0.447313
$658$	−11.6701	−0.454948
$659$	−39.3302	−1.53209	−0.766044	−	0.642788i	$-0.777778\pi$
$659$	−39.3302	−1.53209	−0.766044	+	0.642788i	$0.777778\pi$
$660$	0	0
$661$	44.7178	1.73932	0.869661	−	0.493649i	$-0.164337\pi$
$661$	44.7178	1.73932	0.869661	+	0.493649i	$0.164337\pi$
$662$	−58.7793	−2.28452
$663$	14.1674	0.550216
$664$	−36.6562	−1.42254
$665$	0	0
$666$	12.4948	0.484162
$667$	18.7918	0.727623
$668$	−38.1153	−1.47472
$669$	15.3134	0.592049
$670$	0	0
$671$	33.8363	1.30624
$672$	8.23876	0.317817
$673$	−50.5907	−1.95013	−0.975064	−	0.221923i	$-0.928766\pi$
$673$	−50.5907	−1.95013	−0.975064	+	0.221923i	$0.928766\pi$
$674$	45.1314	1.73840
$675$	0	0
$676$	−9.53425	−0.366702
$677$	28.5273	1.09639	0.548196	−	0.836350i	$-0.315315\pi$
$677$	28.5273	1.09639	0.548196	+	0.836350i	$0.315315\pi$
$678$	−30.8655	−1.18538
$679$	−18.1939	−0.698216
$680$	0	0
$681$	−16.4357	−0.629817
$682$	59.5728	2.28116
$683$	−8.26197	−0.316135	−0.158068	−	0.987428i	$-0.550526\pi$
$683$	−8.26197	−0.316135	−0.158068	+	0.987428i	$0.550526\pi$
$684$	6.68009	0.255420
$685$	0	0
$686$	38.8960	1.48506
$687$	2.84827	0.108668
$688$	−0.246410	−0.00939428
$689$	22.9462	0.874182
$690$	0	0
$691$	−10.4034	−0.395762	−0.197881	−	0.980226i	$-0.563406\pi$
$691$	−10.4034	−0.395762	−0.197881	+	0.980226i	$0.563406\pi$
$692$	74.3436	2.82612
$693$	6.95379	0.264153
$694$	−80.4566	−3.05409
$695$	0	0
$696$	12.6052	0.477800
$697$	−12.6420	−0.478852
$698$	41.0725	1.55462
$699$	−23.9343	−0.905279
$700$	0	0
$701$	8.75164	0.330545	0.165273	−	0.986248i	$-0.447150\pi$
$701$	8.75164	0.330545	0.165273	+	0.986248i	$0.447150\pi$
$702$	−7.22855	−0.272824
$703$	−11.3927	−0.429685
$704$	−63.4293	−2.39058
$705$	0	0
$706$	39.3146	1.47962
$707$	12.5201	0.470868
$708$	42.0597	1.58070
$709$	3.94287	0.148078	0.0740388	−	0.997255i	$-0.476411\pi$
$709$	3.94287	0.148078	0.0740388	+	0.997255i	$0.476411\pi$
$710$	0	0
$711$	14.4418	0.541612
$712$	8.37768	0.313967
$713$	21.9873	0.823430
$714$	−14.5305	−0.543791
$715$	0	0
$716$	6.04291	0.225834
$717$	−14.0141	−0.523365
$718$	−54.6009	−2.03769
$719$	28.6919	1.07003	0.535014	−	0.844843i	$-0.320306\pi$
$719$	28.6919	1.07003	0.535014	+	0.844843i	$0.320306\pi$
$720$	0	0
$721$	−19.9218	−0.741927
$722$	33.4812	1.24604
$723$	−29.4278	−1.09443
$724$	10.2743	0.381841
$725$	0	0
$726$	−29.3935	−1.09090
$727$	−21.9695	−0.814804	−0.407402	−	0.913249i	$-0.633565\pi$
$727$	−21.9695	−0.814804	−0.407402	+	0.913249i	$0.633565\pi$
$728$	−12.4445	−0.461224
$729$	1.00000	0.0370370
$730$	0	0
$731$	9.42742	0.348686
$732$	22.2259	0.821492
$733$	−2.53720	−0.0937137	−0.0468568	−	0.998902i	$-0.514920\pi$
$733$	−2.53720	−0.0937137	−0.0468568	+	0.998902i	$0.514920\pi$
$734$	−46.0277	−1.69891
$735$	0	0
$736$	−23.8324	−0.878473
$737$	−41.9098	−1.54377
$738$	6.45028	0.237438
$739$	−39.6464	−1.45842	−0.729208	−	0.684293i	$-0.760111\pi$
$739$	−39.6464	−1.45842	−0.729208	+	0.684293i	$0.760111\pi$
$740$	0	0
$741$	6.59099	0.242126
$742$	−23.5344	−0.863975
$743$	34.6940	1.27280	0.636399	−	0.771360i	$-0.280423\pi$
$743$	34.6940	1.27280	0.636399	+	0.771360i	$0.280423\pi$
$744$	14.7487	0.540712
$745$	0	0
$746$	15.6039	0.571300
$747$	13.2751	0.485709
$748$	70.1647	2.56548
$749$	−1.42306	−0.0519976
$750$	0	0
$751$	49.2540	1.79730	0.898652	−	0.438663i	$-0.144548\pi$
$751$	49.2540	1.79730	0.898652	+	0.438663i	$0.144548\pi$
$752$	−0.420101	−0.0153195
$753$	9.23123	0.336405
$754$	32.9982	1.20172
$755$	0	0
$756$	4.56771	0.166126
$757$	0.680849	0.0247459	0.0123729	−	0.999923i	$-0.496061\pi$
$757$	0.680849	0.0247459	0.0123729	+	0.999923i	$0.496061\pi$
$758$	−28.6057	−1.03900
$759$	−20.1153	−0.730141
$760$	0	0
$761$	−44.2846	−1.60532	−0.802658	−	0.596440i	$-0.796581\pi$
$761$	−44.2846	−1.60532	−0.802658	+	0.596440i	$0.796581\pi$
$762$	2.08381	0.0754886
$763$	26.0358	0.942558
$764$	4.43528	0.160463
$765$	0	0
$766$	49.8244	1.80023
$767$	41.4987	1.49843
$768$	−15.2357	−0.549772
$769$	39.5297	1.42548	0.712738	−	0.701430i	$-0.247455\pi$
$769$	39.5297	1.42548	0.712738	+	0.701430i	$0.247455\pi$
$770$	0	0
$771$	27.9285	1.00582
$772$	15.8871	0.571790
$773$	46.6154	1.67664	0.838320	−	0.545179i	$-0.183538\pi$
$773$	46.6154	1.67664	0.838320	+	0.545179i	$0.183538\pi$
$774$	−4.81010	−0.172896
$775$	0	0
$776$	35.3030	1.26730
$777$	−7.79011	−0.279468
$778$	76.6401	2.74768
$779$	−5.88137	−0.210722
$780$	0	0
$781$	19.0923	0.683175
$782$	42.0327	1.50308
$783$	−4.56498	−0.163139
$784$	0.581696	0.0207748
$785$	0	0
$786$	25.7637	0.918961
$787$	24.3147	0.866726	0.433363	−	0.901219i	$-0.357327\pi$
$787$	24.3147	0.866726	0.433363	+	0.901219i	$0.357327\pi$
$788$	−4.80417	−0.171141
$789$	9.71037	0.345698
$790$	0	0
$791$	19.2437	0.684228
$792$	−13.4930	−0.479453
$793$	21.9294	0.778737
$794$	8.02011	0.284623
$795$	0	0
$796$	−26.5663	−0.941617
$797$	1.32298	0.0468624	0.0234312	−	0.999725i	$-0.492541\pi$
$797$	1.32298	0.0468624	0.0234312	+	0.999725i	$0.492541\pi$
$798$	−6.75993	−0.239299
$799$	16.0727	0.568611
$800$	0	0
$801$	−3.03398	−0.107200
$802$	31.4418	1.11025
$803$	56.0262	1.97712
$804$	−27.5291	−0.970876
$805$	0	0
$806$	38.6094	1.35996
$807$	−0.196454	−0.00691550
$808$	−24.2938	−0.854654
$809$	−20.8050	−0.731464	−0.365732	−	0.930720i	$-0.619181\pi$
$809$	−20.8050	−0.731464	−0.365732	+	0.930720i	$0.619181\pi$
$810$	0	0
$811$	3.24470	0.113937	0.0569684	−	0.998376i	$-0.481857\pi$
$811$	3.24470	0.113937	0.0569684	+	0.998376i	$0.481857\pi$
$812$	−20.8515	−0.731744
$813$	10.8653	0.381064
$814$	61.0556	2.14000
$815$	0	0
$816$	−0.523071	−0.0183111
$817$	4.38585	0.153441
$818$	−33.6714	−1.17729
$819$	4.50678	0.157480
$820$	0	0
$821$	20.6114	0.719343	0.359671	−	0.933079i	$-0.382889\pi$
$821$	20.6114	0.719343	0.359671	+	0.933079i	$0.382889\pi$
$822$	40.4011	1.40915
$823$	6.61634	0.230631	0.115316	−	0.993329i	$-0.463212\pi$
$823$	6.61634	0.230631	0.115316	+	0.993329i	$0.463212\pi$
$824$	38.6559	1.34664
$825$	0	0
$826$	−42.5624	−1.48093
$827$	−30.1896	−1.04980	−0.524898	−	0.851165i	$-0.675896\pi$
$827$	−30.1896	−1.04980	−0.524898	+	0.851165i	$0.675896\pi$
$828$	−13.2131	−0.459186
$829$	51.7419	1.79707	0.898535	−	0.438902i	$-0.144632\pi$
$829$	51.7419	1.79707	0.898535	+	0.438902i	$0.144632\pi$
$830$	0	0
$831$	8.64846	0.300012
$832$	−41.1088	−1.42519
$833$	−22.2552	−0.771096
$834$	−8.67749	−0.300477
$835$	0	0
$836$	32.6422	1.12895
$837$	−5.34123	−0.184620
$838$	−10.3964	−0.359138
$839$	44.2570	1.52792	0.763961	−	0.645263i	$-0.223252\pi$
$839$	44.2570	1.52792	0.763961	+	0.645263i	$0.223252\pi$
$840$	0	0
$841$	−8.16093	−0.281411
$842$	−16.9308	−0.583473
$843$	27.1433	0.934866
$844$	−74.5519	−2.56618
$845$	0	0
$846$	−8.20069	−0.281945
$847$	18.3260	0.629688
$848$	−0.847192	−0.0290927
$849$	23.7111	0.813763
$850$	0	0
$851$	22.5345	0.772474
$852$	12.5410	0.429649
$853$	8.43451	0.288792	0.144396	−	0.989520i	$-0.453876\pi$
$853$	8.43451	0.288792	0.144396	+	0.989520i	$0.453876\pi$
$854$	−22.4915	−0.769644
$855$	0	0
$856$	2.76128	0.0943788
$857$	−6.43102	−0.219679	−0.109840	−	0.993949i	$-0.535034\pi$
$857$	−6.43102	−0.219679	−0.109840	+	0.993949i	$0.535034\pi$
$858$	−35.3223	−1.20588
$859$	−43.1580	−1.47253	−0.736265	−	0.676693i	$-0.763412\pi$
$859$	−43.1580	−1.47253	−0.736265	+	0.676693i	$0.763412\pi$
$860$	0	0
$861$	−4.02156	−0.137054
$862$	−69.2717	−2.35940
$863$	−18.1532	−0.617943	−0.308972	−	0.951071i	$-0.599985\pi$
$863$	−18.1532	−0.617943	−0.308972	+	0.951071i	$0.599985\pi$
$864$	5.78945	0.196961
$865$	0	0
$866$	74.3249	2.52566
$867$	3.01223	0.102301
$868$	−24.3972	−0.828094
$869$	70.5700	2.39392
$870$	0	0
$871$	−27.1619	−0.920346
$872$	−50.5193	−1.71080
$873$	−12.7850	−0.432706
$874$	19.5545	0.661443
$875$	0	0
$876$	36.8017	1.24341
$877$	35.1860	1.18815	0.594074	−	0.804411i	$-0.297519\pi$
$877$	35.1860	1.18815	0.594074	+	0.804411i	$0.297519\pi$
$878$	−34.1108	−1.15119
$879$	27.0101	0.911029
$880$	0	0
$881$	−37.5707	−1.26579	−0.632895	−	0.774238i	$-0.718133\pi$
$881$	−37.5707	−1.26579	−0.632895	+	0.774238i	$0.718133\pi$
$882$	11.3551	0.382347
$883$	25.9545	0.873438	0.436719	−	0.899598i	$-0.356140\pi$
$883$	25.9545	0.873438	0.436719	+	0.899598i	$0.356140\pi$
$884$	45.4740	1.52946
$885$	0	0
$886$	−71.2353	−2.39320
$887$	−53.1373	−1.78417	−0.892087	−	0.451863i	$-0.850760\pi$
$887$	−53.1373	−1.78417	−0.892087	+	0.451863i	$0.850760\pi$
$888$	15.1158	0.507252
$889$	−1.29919	−0.0435736
$890$	0	0
$891$	4.88649	0.163704
$892$	49.1524	1.64574
$893$	7.47738	0.250221
$894$	−26.7464	−0.894535
$895$	0	0
$896$	25.6849	0.858073
$897$	−13.0368	−0.435287
$898$	−21.5667	−0.719691
$899$	24.3826	0.813207
$900$	0	0
$901$	32.4129	1.07983
$902$	31.5193	1.04948
$903$	2.99895	0.0997989
$904$	−37.3401	−1.24191
$905$	0	0
$906$	0.504238	0.0167522
$907$	11.8705	0.394154	0.197077	−	0.980388i	$-0.436855\pi$
$907$	11.8705	0.394154	0.197077	+	0.980388i	$0.436855\pi$
$908$	−52.7548	−1.75073
$909$	8.79802	0.291812
$910$	0	0
$911$	−55.6998	−1.84542	−0.922708	−	0.385499i	$-0.874029\pi$
$911$	−55.6998	−1.84542	−0.922708	+	0.385499i	$0.874029\pi$
$912$	−0.243344	−0.00805794
$913$	64.8685	2.14683
$914$	−89.5343	−2.96153
$915$	0	0
$916$	9.14230	0.302070
$917$	−16.0629	−0.530443
$918$	−10.2107	−0.337004
$919$	16.6284	0.548521	0.274260	−	0.961655i	$-0.411567\pi$
$919$	16.6284	0.548521	0.274260	+	0.961655i	$0.411567\pi$
$920$	0	0
$921$	−3.67259	−0.121016
$922$	−52.6992	−1.73556
$923$	12.3738	0.407287
$924$	22.3201	0.734276
$925$	0	0
$926$	51.7422	1.70035
$927$	−13.9992	−0.459795
$928$	−26.4288	−0.867566
$929$	−43.6576	−1.43236	−0.716179	−	0.697916i	$-0.754111\pi$
$929$	−43.6576	−1.43236	−0.716179	+	0.697916i	$0.754111\pi$
$930$	0	0
$931$	−10.3536	−0.339326
$932$	−76.8236	−2.51644
$933$	−21.6941	−0.710233
$934$	78.1228	2.55626
$935$	0	0
$936$	−8.74487	−0.285835
$937$	23.7562	0.776080	0.388040	−	0.921643i	$-0.373152\pi$
$937$	23.7562	0.776080	0.388040	+	0.921643i	$0.373152\pi$
$938$	27.8581	0.909600
$939$	−9.29453	−0.303315
$940$	0	0
$941$	−21.2155	−0.691605	−0.345803	−	0.938307i	$-0.612393\pi$
$941$	−21.2155	−0.691605	−0.345803	+	0.938307i	$0.612393\pi$
$942$	42.6644	1.39008
$943$	11.6332	0.378829
$944$	−1.53216	−0.0498676
$945$	0	0
$946$	−23.5045	−0.764199
$947$	31.6461	1.02836	0.514181	−	0.857682i	$-0.328096\pi$
$947$	31.6461	1.02836	0.514181	+	0.857682i	$0.328096\pi$
$948$	46.3550	1.50554
$949$	36.3108	1.17870
$950$	0	0
$951$	20.0952	0.651630
$952$	−17.5786	−0.569725
$953$	−7.25490	−0.235009	−0.117505	−	0.993072i	$-0.537490\pi$
$953$	−7.25490	−0.235009	−0.117505	+	0.993072i	$0.537490\pi$
$954$	−16.5378	−0.535432
$955$	0	0
$956$	−44.9819	−1.45482
$957$	−22.3068	−0.721075
$958$	2.73538	0.0883762
$959$	−25.1889	−0.813391
$960$	0	0
$961$	−2.47123	−0.0797171
$962$	39.5704	1.27580
$963$	−1.00000	−0.0322245
$964$	−94.4563	−3.04223
$965$	0	0
$966$	13.3710	0.430205
$967$	1.48348	0.0477056	0.0238528	−	0.999715i	$-0.492407\pi$
$967$	1.48348	0.0477056	0.0238528	+	0.999715i	$0.492407\pi$
$968$	−35.5594	−1.14292
$969$	9.31014	0.299085
$970$	0	0
$971$	9.26912	0.297460	0.148730	−	0.988878i	$-0.452481\pi$
$971$	9.26912	0.297460	0.148730	+	0.988878i	$0.452481\pi$
$972$	3.20977	0.102953
$973$	5.41015	0.173441
$974$	31.1506	0.998129
$975$	0	0
$976$	−0.809651	−0.0259163
$977$	36.2163	1.15866	0.579331	−	0.815092i	$-0.303314\pi$
$977$	36.2163	1.15866	0.579331	+	0.815092i	$0.303314\pi$
$978$	11.1171	0.355487
$979$	−14.8255	−0.473826
$980$	0	0
$981$	18.2956	0.584133
$982$	62.1678	1.98386
$983$	26.0638	0.831307	0.415653	−	0.909523i	$-0.363553\pi$
$983$	26.0638	0.831307	0.415653	+	0.909523i	$0.363553\pi$
$984$	7.80334	0.248762
$985$	0	0
$986$	46.6118	1.48442
$987$	5.11288	0.162745
$988$	21.1556	0.673048
$989$	−8.67511	−0.275853
$990$	0	0
$991$	−50.7658	−1.61263	−0.806314	−	0.591488i	$-0.798541\pi$
$991$	−50.7658	−1.61263	−0.806314	+	0.591488i	$0.798541\pi$
$992$	−30.9228	−0.981800
$993$	25.7523	0.817223
$994$	−12.6909	−0.402532
$995$	0	0
$996$	42.6099	1.35015
$997$	3.45070	0.109285	0.0546424	−	0.998506i	$-0.482598\pi$
$997$	3.45070	0.109285	0.0546424	+	0.998506i	$0.482598\pi$
$998$	76.8900	2.43391
$999$	−5.47418	−0.173195

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8025.2.a.bb.1.2		7
5.4	even	2		321.2.a.d.1.6	✓	7
15.14	odd	2		963.2.a.e.1.2		7
20.19	odd	2		5136.2.a.bi.1.7		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
321.2.a.d.1.6	✓	7	5.4	even	2
963.2.a.e.1.2		7	15.14	odd	2
5136.2.a.bi.1.7		7	20.19	odd	2
8025.2.a.bb.1.2		7	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8025 = 3 \cdot 5^{2} \cdot 107 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8025.a (trivial)

\(f(q)\)	\(=\)	\(q-2.28249 q^{2} +1.00000 q^{3} +3.20977 q^{4} -2.28249 q^{6} +1.42306 q^{7} -2.76128 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.28249 q^{2} +1.00000 q^{3} +3.20977 q^{4} -2.28249 q^{6} +1.42306 q^{7} -2.76128 q^{8} +1.00000 q^{9} +4.88649 q^{11} +3.20977 q^{12} +3.16696 q^{13} -3.24813 q^{14} -0.116926 q^{16} +4.47350 q^{17} -2.28249 q^{18} +2.08118 q^{19} +1.42306 q^{21} -11.1534 q^{22} -4.11652 q^{23} -2.76128 q^{24} -7.22855 q^{26} +1.00000 q^{27} +4.56771 q^{28} -4.56498 q^{29} -5.34123 q^{31} +5.78945 q^{32} +4.88649 q^{33} -10.2107 q^{34} +3.20977 q^{36} -5.47418 q^{37} -4.75027 q^{38} +3.16696 q^{39} -2.82598 q^{41} -3.24813 q^{42} +2.10739 q^{43} +15.6845 q^{44} +9.39592 q^{46} +3.59287 q^{47} -0.116926 q^{48} -4.97489 q^{49} +4.47350 q^{51} +10.1652 q^{52} +7.24552 q^{53} -2.28249 q^{54} -3.92949 q^{56} +2.08118 q^{57} +10.4195 q^{58} +13.1036 q^{59} +6.92445 q^{61} +12.1913 q^{62} +1.42306 q^{63} -12.9805 q^{64} -11.1534 q^{66} -8.57666 q^{67} +14.3589 q^{68} -4.11652 q^{69} +3.90715 q^{71} -2.76128 q^{72} +11.4655 q^{73} +12.4948 q^{74} +6.68009 q^{76} +6.95379 q^{77} -7.22855 q^{78} +14.4418 q^{79} +1.00000 q^{81} +6.45028 q^{82} +13.2751 q^{83} +4.56771 q^{84} -4.81010 q^{86} -4.56498 q^{87} -13.4930 q^{88} -3.03398 q^{89} +4.50678 q^{91} -13.2131 q^{92} -5.34123 q^{93} -8.20069 q^{94} +5.78945 q^{96} -12.7850 q^{97} +11.3551 q^{98} +4.88649 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{3} + 14 q^{4} - 6 q^{7} - 3 q^{8} + 7 q^{9}+O(q^{10}) \) 7 * q + 7 * q^3 + 14 * q^4 - 6 * q^7 - 3 * q^8 + 7 * q^9	\( 7 q + 7 q^{3} + 14 q^{4} - 6 q^{7} - 3 q^{8} + 7 q^{9} + 4 q^{11} + 14 q^{12} - 6 q^{13} + 12 q^{14} + 32 q^{16} + 10 q^{17} + 8 q^{19} - 6 q^{21} - 10 q^{22} - 6 q^{23} - 3 q^{24} + 7 q^{26} + 7 q^{27} - 8 q^{28} + 16 q^{31} - 6 q^{32} + 4 q^{33} - 11 q^{34} + 14 q^{36} - 10 q^{37} + 13 q^{38} - 6 q^{39} - 2 q^{41} + 12 q^{42} - 2 q^{43} + 2 q^{44} - 30 q^{46} - 16 q^{47} + 32 q^{48} + 17 q^{49} + 10 q^{51} + 23 q^{52} + 16 q^{53} + 30 q^{56} + 8 q^{57} + 56 q^{58} + 20 q^{59} + 2 q^{61} + 52 q^{62} - 6 q^{63} + 43 q^{64} - 10 q^{66} - 30 q^{67} + 61 q^{68} - 6 q^{69} + 32 q^{71} - 3 q^{72} + 12 q^{73} - q^{74} - 49 q^{76} + 46 q^{77} + 7 q^{78} + 36 q^{79} + 7 q^{81} - 2 q^{82} + 10 q^{83} - 8 q^{84} - 20 q^{86} + 14 q^{88} - 4 q^{89} + 12 q^{91} - 10 q^{92} + 16 q^{93} - 26 q^{94} - 6 q^{96} - 24 q^{97} - 8 q^{98} + 4 q^{99}+O(q^{100}) \) 7 * q + 7 * q^3 + 14 * q^4 - 6 * q^7 - 3 * q^8 + 7 * q^9 + 4 * q^11 + 14 * q^12 - 6 * q^13 + 12 * q^14 + 32 * q^16 + 10 * q^17 + 8 * q^19 - 6 * q^21 - 10 * q^22 - 6 * q^23 - 3 * q^24 + 7 * q^26 + 7 * q^27 - 8 * q^28 + 16 * q^31 - 6 * q^32 + 4 * q^33 - 11 * q^34 + 14 * q^36 - 10 * q^37 + 13 * q^38 - 6 * q^39 - 2 * q^41 + 12 * q^42 - 2 * q^43 + 2 * q^44 - 30 * q^46 - 16 * q^47 + 32 * q^48 + 17 * q^49 + 10 * q^51 + 23 * q^52 + 16 * q^53 + 30 * q^56 + 8 * q^57 + 56 * q^58 + 20 * q^59 + 2 * q^61 + 52 * q^62 - 6 * q^63 + 43 * q^64 - 10 * q^66 - 30 * q^67 + 61 * q^68 - 6 * q^69 + 32 * q^71 - 3 * q^72 + 12 * q^73 - q^74 - 49 * q^76 + 46 * q^77 + 7 * q^78 + 36 * q^79 + 7 * q^81 - 2 * q^82 + 10 * q^83 - 8 * q^84 - 20 * q^86 + 14 * q^88 - 4 * q^89 + 12 * q^91 - 10 * q^92 + 16 * q^93 - 26 * q^94 - 6 * q^96 - 24 * q^97 - 8 * q^98 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more