LMFDB - Embedded newform 3025.2.a.bj.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3025,2,Mod(1,3025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3025 = 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3025.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:	$24.1547466114$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 7x^{6} + 18x^{5} + 16x^{4} - 30x^{3} - 12x^{2} + 12x + 4 $ x^8 - 3x^7 - 7x^6 + 18x^5 + 16x^4 - 30x^3 - 12x^2 + 12*x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 275)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.20828$ of defining polynomial
Character	$\chi$	$=$	3025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.208285 q^{2} +1.89427 q^{3} -1.95662 q^{4} +0.394547 q^{6} +1.28881 q^{7} -0.824104 q^{8} +0.588246 q^{9} +O(q^{10})$	\(q+0.208285 q^{2} +1.89427 q^{3} -1.95662 q^{4} +0.394547 q^{6} +1.28881 q^{7} -0.824104 q^{8} +0.588246 q^{9} -3.70635 q^{12} -5.45014 q^{13} +0.268441 q^{14} +3.74159 q^{16} +3.93141 q^{17} +0.122523 q^{18} -2.24549 q^{19} +2.44136 q^{21} -0.711128 q^{23} -1.56107 q^{24} -1.13518 q^{26} -4.56851 q^{27} -2.52172 q^{28} -3.65814 q^{29} -6.46413 q^{31} +2.42752 q^{32} +0.818854 q^{34} -1.15097 q^{36} +8.26346 q^{37} -0.467702 q^{38} -10.3240 q^{39} -12.6329 q^{41} +0.508498 q^{42} +1.31169 q^{43} -0.148117 q^{46} +4.51506 q^{47} +7.08756 q^{48} -5.33896 q^{49} +7.44714 q^{51} +10.6638 q^{52} -9.71917 q^{53} -0.951551 q^{54} -1.06212 q^{56} -4.25356 q^{57} -0.761936 q^{58} +8.91391 q^{59} +1.84131 q^{61} -1.34638 q^{62} +0.758139 q^{63} -6.97756 q^{64} -4.30232 q^{67} -7.69227 q^{68} -1.34707 q^{69} +10.1573 q^{71} -0.484776 q^{72} -12.0048 q^{73} +1.72115 q^{74} +4.39356 q^{76} -2.15034 q^{78} -12.8117 q^{79} -10.4187 q^{81} -2.63124 q^{82} -12.5432 q^{83} -4.77680 q^{84} +0.273205 q^{86} -6.92949 q^{87} -6.28392 q^{89} -7.02422 q^{91} +1.39140 q^{92} -12.2448 q^{93} +0.940420 q^{94} +4.59838 q^{96} -0.303404 q^{97} -1.11202 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 5 q^{2} + q^{3} + 9 q^{4} - q^{6} - 8 q^{7} - 12 q^{8} + 9 q^{9}+O(q^{10}) $ 8 * q - 5 * q^2 + q^3 + 9 * q^4 - q^6 - 8 * q^7 - 12 * q^8 + 9 * q^9	$ 8 q - 5 q^{2} + q^{3} + 9 q^{4} - q^{6} - 8 q^{7} - 12 q^{8} + 9 q^{9} + 3 q^{12} - 9 q^{13} + 9 q^{14} + 23 q^{16} - 19 q^{17} - 22 q^{18} - q^{19} - 5 q^{21} + 2 q^{23} - q^{24} - 2 q^{26} - 2 q^{27} - 9 q^{28} - 7 q^{29} - 5 q^{31} - 29 q^{32} + 10 q^{34} - 16 q^{36} - 8 q^{37} - 37 q^{38} + q^{39} + 41 q^{42} - 14 q^{43} + 20 q^{46} + 11 q^{47} - 27 q^{48} - 12 q^{49} + 25 q^{51} + 7 q^{52} + 11 q^{53} - 30 q^{54} - 10 q^{56} + 2 q^{57} + 27 q^{58} + 17 q^{59} + 2 q^{61} - 25 q^{62} - 41 q^{63} + 30 q^{64} + 7 q^{67} - 66 q^{68} + 17 q^{71} + 19 q^{72} - 34 q^{73} + 6 q^{74} + 31 q^{76} - 17 q^{78} - 23 q^{79} - 4 q^{81} - 17 q^{82} - 41 q^{83} - 83 q^{84} + q^{86} - 25 q^{87} - 11 q^{89} - 7 q^{91} + 33 q^{92} - 59 q^{93} - 50 q^{94} + 61 q^{96} + 2 q^{97} - 26 q^{98}+O(q^{100}) $ 8 * q - 5 * q^2 + q^3 + 9 * q^4 - q^6 - 8 * q^7 - 12 * q^8 + 9 * q^9 + 3 * q^12 - 9 * q^13 + 9 * q^14 + 23 * q^16 - 19 * q^17 - 22 * q^18 - q^19 - 5 * q^21 + 2 * q^23 - q^24 - 2 * q^26 - 2 * q^27 - 9 * q^28 - 7 * q^29 - 5 * q^31 - 29 * q^32 + 10 * q^34 - 16 * q^36 - 8 * q^37 - 37 * q^38 + q^39 + 41 * q^42 - 14 * q^43 + 20 * q^46 + 11 * q^47 - 27 * q^48 - 12 * q^49 + 25 * q^51 + 7 * q^52 + 11 * q^53 - 30 * q^54 - 10 * q^56 + 2 * q^57 + 27 * q^58 + 17 * q^59 + 2 * q^61 - 25 * q^62 - 41 * q^63 + 30 * q^64 + 7 * q^67 - 66 * q^68 + 17 * q^71 + 19 * q^72 - 34 * q^73 + 6 * q^74 + 31 * q^76 - 17 * q^78 - 23 * q^79 - 4 * q^81 - 17 * q^82 - 41 * q^83 - 83 * q^84 + q^86 - 25 * q^87 - 11 * q^89 - 7 * q^91 + 33 * q^92 - 59 * q^93 - 50 * q^94 + 61 * q^96 + 2 * q^97 - 26 * 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 $

$2$	0.208285	0.147280	0.0736399	−	0.997285i	$-0.476538\pi$
$2$	0.208285	0.147280	0.0736399	+	0.997285i	$0.476538\pi$
$3$	1.89427	1.09366	0.546828	−	0.837245i	$-0.315835\pi$
$3$	1.89427	1.09366	0.546828	+	0.837245i	$0.315835\pi$
$4$	−1.95662	−0.978309
$5$	0	0
$5$	0	0
$6$	0.394547	0.161073
$7$	1.28881	0.487126	0.243563	−	0.969885i	$-0.421684\pi$
$7$	1.28881	0.487126	0.243563	+	0.969885i	$0.421684\pi$
$8$	−0.824104	−0.291365
$9$	0.588246	0.196082
$10$	0	0
$11$	0	0
$11$	0	0
$12$	−3.70635	−1.06993
$13$	−5.45014	−1.51160	−0.755798	−	0.654804i	$-0.772751\pi$
$13$	−5.45014	−1.51160	−0.755798	+	0.654804i	$0.772751\pi$
$14$	0.268441	0.0717438
$15$	0	0
$16$	3.74159	0.935397
$17$	3.93141	0.953508	0.476754	−	0.879037i	$-0.341813\pi$
$17$	3.93141	0.953508	0.476754	+	0.879037i	$0.341813\pi$
$18$	0.122523	0.0288789
$19$	−2.24549	−0.515151	−0.257575	−	0.966258i	$-0.582924\pi$
$19$	−2.24549	−0.515151	−0.257575	+	0.966258i	$0.582924\pi$
$20$	0	0
$21$	2.44136	0.532748
$22$	0	0
$23$	−0.711128	−0.148280	−0.0741402	−	0.997248i	$-0.523621\pi$
$23$	−0.711128	−0.148280	−0.0741402	+	0.997248i	$0.523621\pi$
$24$	−1.56107	−0.318653
$25$	0	0
$26$	−1.13518	−0.222628
$27$	−4.56851	−0.879209
$28$	−2.52172	−0.476559
$29$	−3.65814	−0.679300	−0.339650	−	0.940552i	$-0.610309\pi$
$29$	−3.65814	−0.679300	−0.339650	+	0.940552i	$0.610309\pi$
$30$	0	0
$31$	−6.46413	−1.16099	−0.580496	−	0.814263i	$-0.697141\pi$
$31$	−6.46413	−1.16099	−0.580496	+	0.814263i	$0.697141\pi$
$32$	2.42752	0.429130
$33$	0	0
$34$	0.818854	0.140432
$35$	0	0
$36$	−1.15097	−0.191829
$37$	8.26346	1.35850	0.679252	−	0.733905i	$-0.262304\pi$
$37$	8.26346	1.35850	0.679252	+	0.733905i	$0.262304\pi$
$38$	−0.467702	−0.0758712
$39$	−10.3240	−1.65317
$40$	0	0
$41$	−12.6329	−1.97293	−0.986464	−	0.163981i	$-0.947566\pi$
$41$	−12.6329	−1.97293	−0.986464	+	0.163981i	$0.947566\pi$
$42$	0.508498	0.0784629
$43$	1.31169	0.200031	0.100015	−	0.994986i	$-0.468111\pi$
$43$	1.31169	0.200031	0.100015	+	0.994986i	$0.468111\pi$
$44$	0	0
$45$	0	0
$46$	−0.148117	−0.0218387
$47$	4.51506	0.658590	0.329295	−	0.944227i	$-0.393189\pi$
$47$	4.51506	0.658590	0.329295	+	0.944227i	$0.393189\pi$
$48$	7.08756	1.02300
$49$	−5.33896	−0.762708
$50$	0	0
$51$	7.44714	1.04281
$52$	10.6638	1.47881
$53$	−9.71917	−1.33503	−0.667515	−	0.744596i	$-0.732642\pi$
$53$	−9.71917	−1.33503	−0.667515	+	0.744596i	$0.732642\pi$
$54$	−0.951551	−0.129490
$55$	0	0
$56$	−1.06212	−0.141931
$57$	−4.25356	−0.563397
$58$	−0.761936	−0.100047
$59$	8.91391	1.16049	0.580246	−	0.814441i	$-0.302956\pi$
$59$	8.91391	1.16049	0.580246	+	0.814441i	$0.302956\pi$
$60$	0	0
$61$	1.84131	0.235755	0.117878	−	0.993028i	$-0.462391\pi$
$61$	1.84131	0.235755	0.117878	+	0.993028i	$0.462391\pi$
$62$	−1.34638	−0.170991
$63$	0.758139	0.0955166
$64$	−6.97756	−0.872194
$65$	0	0
$66$	0	0
$67$	−4.30232	−0.525612	−0.262806	−	0.964849i	$-0.584648\pi$
$67$	−4.30232	−0.525612	−0.262806	+	0.964849i	$0.584648\pi$
$68$	−7.69227	−0.932825
$69$	−1.34707	−0.162168
$70$	0	0
$71$	10.1573	1.20545	0.602727	−	0.797947i	$-0.294081\pi$
$71$	10.1573	1.20545	0.602727	+	0.797947i	$0.294081\pi$
$72$	−0.484776	−0.0571314
$73$	−12.0048	−1.40506	−0.702529	−	0.711655i	$-0.747946\pi$
$73$	−12.0048	−1.40506	−0.702529	+	0.711655i	$0.747946\pi$
$74$	1.72115	0.200080
$75$	0	0
$76$	4.39356	0.503976
$77$	0	0
$78$	−2.15034	−0.243478
$79$	−12.8117	−1.44143	−0.720713	−	0.693233i	$-0.756186\pi$
$79$	−12.8117	−1.44143	−0.720713	+	0.693233i	$0.756186\pi$
$80$	0	0
$81$	−10.4187	−1.15763
$82$	−2.63124	−0.290572
$83$	−12.5432	−1.37680	−0.688400	−	0.725331i	$-0.741687\pi$
$83$	−12.5432	−1.37680	−0.688400	+	0.725331i	$0.741687\pi$
$84$	−4.77680	−0.521192
$85$	0	0
$86$	0.273205	0.0294605
$87$	−6.92949	−0.742920
$88$	0	0
$89$	−6.28392	−0.666095	−0.333047	−	0.942910i	$-0.608077\pi$
$89$	−6.28392	−0.666095	−0.333047	+	0.942910i	$0.608077\pi$
$90$	0	0
$91$	−7.02422	−0.736338
$92$	1.39140	0.145064
$93$	−12.2448	−1.26972
$94$	0.940420	0.0969969
$95$	0	0
$96$	4.59838	0.469320
$97$	−0.303404	−0.0308060	−0.0154030	−	0.999881i	$-0.504903\pi$
$97$	−0.303404	−0.0308060	−0.0154030	+	0.999881i	$0.504903\pi$
$98$	−1.11202	−0.112331
$99$	0	0
$100$	0	0
$101$	6.17812	0.614746	0.307373	−	0.951589i	$-0.400550\pi$
$101$	6.17812	0.614746	0.307373	+	0.951589i	$0.400550\pi$
$102$	1.55113	0.153585
$103$	7.97879	0.786173	0.393087	−	0.919501i	$-0.371407\pi$
$103$	7.97879	0.786173	0.393087	+	0.919501i	$0.371407\pi$
$104$	4.49148	0.440426
$105$	0	0
$106$	−2.02436	−0.196623
$107$	−10.0452	−0.971109	−0.485555	−	0.874206i	$-0.661382\pi$
$107$	−10.0452	−0.971109	−0.485555	+	0.874206i	$0.661382\pi$
$108$	8.93882	0.860138
$109$	−7.63832	−0.731619	−0.365809	−	0.930690i	$-0.619208\pi$
$109$	−7.63832	−0.731619	−0.365809	+	0.930690i	$0.619208\pi$
$110$	0	0
$111$	15.6532	1.48574
$112$	4.82221	0.455656
$113$	19.6030	1.84410	0.922049	−	0.387073i	$-0.126514\pi$
$113$	19.6030	1.84410	0.922049	+	0.387073i	$0.126514\pi$
$114$	−0.885952	−0.0829770
$115$	0	0
$116$	7.15758	0.664565
$117$	−3.20602	−0.296397
$118$	1.85663	0.170917
$119$	5.06686	0.464478
$120$	0	0
$121$	0	0
$122$	0.383517	0.0347220
$123$	−23.9301	−2.15770
$124$	12.6478	1.13581
$125$	0	0
$126$	0.157909	0.0140677
$127$	−17.1652	−1.52316	−0.761582	−	0.648069i	$-0.775577\pi$
$127$	−17.1652	−1.52316	−0.761582	+	0.648069i	$0.775577\pi$
$128$	−6.30837	−0.557586
$129$	2.48469	0.218765
$130$	0	0
$131$	−10.0223	−0.875655	−0.437828	−	0.899059i	$-0.644252\pi$
$131$	−10.0223	−0.875655	−0.437828	+	0.899059i	$0.644252\pi$
$132$	0	0
$133$	−2.89402	−0.250943
$134$	−0.896109	−0.0774120
$135$	0	0
$136$	−3.23989	−0.277819
$137$	3.52732	0.301359	0.150680	−	0.988583i	$-0.451854\pi$
$137$	3.52732	0.301359	0.150680	+	0.988583i	$0.451854\pi$
$138$	−0.280574	−0.0238840
$139$	8.77740	0.744489	0.372245	−	0.928135i	$-0.378588\pi$
$139$	8.77740	0.744489	0.372245	+	0.928135i	$0.378588\pi$
$140$	0	0
$141$	8.55274	0.720270
$142$	2.11562	0.177539
$143$	0	0
$144$	2.20097	0.183414
$145$	0	0
$146$	−2.50042	−0.206937
$147$	−10.1134	−0.834140
$148$	−16.1684	−1.32904
$149$	8.39676	0.687889	0.343945	−	0.938990i	$-0.388237\pi$
$149$	8.39676	0.687889	0.343945	+	0.938990i	$0.388237\pi$
$150$	0	0
$151$	8.53688	0.694722	0.347361	−	0.937732i	$-0.387078\pi$
$151$	8.53688	0.694722	0.347361	+	0.937732i	$0.387078\pi$
$152$	1.85052	0.150097
$153$	2.31264	0.186966
$154$	0	0
$155$	0	0
$156$	20.2002	1.61731
$157$	−12.6495	−1.00954	−0.504771	−	0.863253i	$-0.668423\pi$
$157$	−12.6495	−1.00954	−0.504771	+	0.863253i	$0.668423\pi$
$158$	−2.66848	−0.212293
$159$	−18.4107	−1.46006
$160$	0	0
$161$	−0.916511	−0.0722312
$162$	−2.17006	−0.170496
$163$	8.40910	0.658652	0.329326	−	0.944216i	$-0.393179\pi$
$163$	8.40910	0.658652	0.329326	+	0.944216i	$0.393179\pi$
$164$	24.7177	1.93013
$165$	0	0
$166$	−2.61257	−0.202775
$167$	−19.1251	−1.47994	−0.739971	−	0.672639i	$-0.765161\pi$
$167$	−19.1251	−1.47994	−0.739971	+	0.672639i	$0.765161\pi$
$168$	−2.01193	−0.155224
$169$	16.7040	1.28492
$170$	0	0
$171$	−1.32090	−0.101012
$172$	−2.56648	−0.195692
$173$	12.2387	0.930492	0.465246	−	0.885181i	$-0.345966\pi$
$173$	12.2387	0.930492	0.465246	+	0.885181i	$0.345966\pi$
$174$	−1.44331	−0.109417
$175$	0	0
$176$	0	0
$177$	16.8853	1.26918
$178$	−1.30885	−0.0981022
$179$	3.20029	0.239201	0.119601	−	0.992822i	$-0.461839\pi$
$179$	3.20029	0.239201	0.119601	+	0.992822i	$0.461839\pi$
$180$	0	0
$181$	2.16695	0.161068	0.0805340	−	0.996752i	$-0.474337\pi$
$181$	2.16695	0.161068	0.0805340	+	0.996752i	$0.474337\pi$
$182$	−1.46304	−0.108448
$183$	3.48793	0.257835
$184$	0.586043	0.0432037
$185$	0	0
$186$	−2.55040	−0.187005
$187$	0	0
$188$	−8.83425	−0.644304
$189$	−5.88795	−0.428286
$190$	0	0
$191$	1.76304	0.127569	0.0637846	−	0.997964i	$-0.479683\pi$
$191$	1.76304	0.127569	0.0637846	+	0.997964i	$0.479683\pi$
$192$	−13.2174	−0.953880
$193$	−15.4073	−1.10904	−0.554519	−	0.832171i	$-0.687098\pi$
$193$	−15.4073	−1.10904	−0.554519	+	0.832171i	$0.687098\pi$
$194$	−0.0631944	−0.00453710
$195$	0	0
$196$	10.4463	0.746164
$197$	−1.78727	−0.127338	−0.0636689	−	0.997971i	$-0.520280\pi$
$197$	−1.78727	−0.127338	−0.0636689	+	0.997971i	$0.520280\pi$
$198$	0	0
$199$	−19.6066	−1.38988	−0.694938	−	0.719070i	$-0.744568\pi$
$199$	−19.6066	−1.38988	−0.694938	+	0.719070i	$0.744568\pi$
$200$	0	0
$201$	−8.14974	−0.574839
$202$	1.28681	0.0905396
$203$	−4.71466	−0.330904
$204$	−14.5712	−1.02019
$205$	0	0
$206$	1.66186	0.115787
$207$	−0.418318	−0.0290751
$208$	−20.3922	−1.41394
$209$	0	0
$210$	0	0
$211$	16.7173	1.15087	0.575434	−	0.817848i	$-0.304833\pi$
$211$	16.7173	1.15087	0.575434	+	0.817848i	$0.304833\pi$
$212$	19.0167	1.30607
$213$	19.2407	1.31835
$214$	−2.09227	−0.143025
$215$	0	0
$216$	3.76492	0.256171
$217$	−8.33106	−0.565549
$218$	−1.59095	−0.107753
$219$	−22.7403	−1.53665
$220$	0	0
$221$	−21.4267	−1.44132
$222$	3.26033	0.218819
$223$	−5.98151	−0.400552	−0.200276	−	0.979740i	$-0.564184\pi$
$223$	−5.98151	−0.400552	−0.200276	+	0.979740i	$0.564184\pi$
$224$	3.12863	0.209040
$225$	0	0
$226$	4.08302	0.271598
$227$	−22.2492	−1.47673	−0.738364	−	0.674402i	$-0.764401\pi$
$227$	−22.2492	−1.47673	−0.738364	+	0.674402i	$0.764401\pi$
$228$	8.32258	0.551176
$229$	5.97262	0.394682	0.197341	−	0.980335i	$-0.436769\pi$
$229$	5.97262	0.394682	0.197341	+	0.980335i	$0.436769\pi$
$230$	0	0
$231$	0	0
$232$	3.01469	0.197924
$233$	17.7349	1.16185	0.580927	−	0.813956i	$-0.302690\pi$
$233$	17.7349	1.16185	0.580927	+	0.813956i	$0.302690\pi$
$234$	−0.667766	−0.0436532
$235$	0	0
$236$	−17.4411	−1.13532
$237$	−24.2687	−1.57642
$238$	1.05535	0.0684082
$239$	19.1157	1.23649	0.618246	−	0.785985i	$-0.287844\pi$
$239$	19.1157	1.23649	0.618246	+	0.785985i	$0.287844\pi$
$240$	0	0
$241$	1.79437	0.115586	0.0577929	−	0.998329i	$-0.481594\pi$
$241$	1.79437	0.115586	0.0577929	+	0.998329i	$0.481594\pi$
$242$	0	0
$243$	−6.03029	−0.386843
$244$	−3.60274	−0.230642
$245$	0	0
$246$	−4.98428	−0.317786
$247$	12.2382	0.778700
$248$	5.32711	0.338272
$249$	−23.7603	−1.50575
$250$	0	0
$251$	10.5701	0.667182	0.333591	−	0.942718i	$-0.391740\pi$
$251$	10.5701	0.667182	0.333591	+	0.942718i	$0.391740\pi$
$252$	−1.48339	−0.0934447
$253$	0	0
$254$	−3.57525	−0.224331
$255$	0	0
$256$	12.6412	0.790073
$257$	−4.49921	−0.280653	−0.140326	−	0.990105i	$-0.544815\pi$
$257$	−4.49921	−0.280653	−0.140326	+	0.990105i	$0.544815\pi$
$258$	0.517524	0.0322196
$259$	10.6501	0.661763
$260$	0	0
$261$	−2.15189	−0.133198
$262$	−2.08750	−0.128966
$263$	8.87966	0.547543	0.273772	−	0.961795i	$-0.411729\pi$
$263$	8.87966	0.547543	0.273772	+	0.961795i	$0.411729\pi$
$264$	0	0
$265$	0	0
$266$	−0.602780	−0.0369588
$267$	−11.9034	−0.728478
$268$	8.41800	0.514211
$269$	−16.8065	−1.02471	−0.512355	−	0.858773i	$-0.671227\pi$
$269$	−16.8065	−1.02471	−0.512355	+	0.858773i	$0.671227\pi$
$270$	0	0
$271$	13.5327	0.822051	0.411025	−	0.911624i	$-0.365171\pi$
$271$	13.5327	0.822051	0.411025	+	0.911624i	$0.365171\pi$
$272$	14.7097	0.891908
$273$	−13.3057	−0.805300
$274$	0.734688	0.0443841
$275$	0	0
$276$	2.63569	0.158650
$277$	−20.4683	−1.22982	−0.614910	−	0.788597i	$-0.710808\pi$
$277$	−20.4683	−1.22982	−0.614910	+	0.788597i	$0.710808\pi$
$278$	1.82820	0.109648
$279$	−3.80250	−0.227649
$280$	0	0
$281$	10.0103	0.597163	0.298582	−	0.954384i	$-0.403487\pi$
$281$	10.0103	0.597163	0.298582	+	0.954384i	$0.403487\pi$
$282$	1.78141	0.106081
$283$	2.67790	0.159185	0.0795923	−	0.996828i	$-0.474638\pi$
$283$	2.67790	0.159185	0.0795923	+	0.996828i	$0.474638\pi$
$284$	−19.8740	−1.17931
$285$	0	0
$286$	0	0
$287$	−16.2815	−0.961064
$288$	1.42798	0.0841446
$289$	−1.54399	−0.0908232
$290$	0	0
$291$	−0.574727	−0.0336911
$292$	23.4888	1.37458
$293$	−25.2903	−1.47748	−0.738738	−	0.673993i	$-0.764578\pi$
$293$	−25.2903	−1.47748	−0.738738	+	0.673993i	$0.764578\pi$
$294$	−2.10647	−0.122852
$295$	0	0
$296$	−6.80995	−0.395820
$297$	0	0
$298$	1.74892	0.101312
$299$	3.87575	0.224140
$300$	0	0
$301$	1.69052	0.0974402
$302$	1.77810	0.102318
$303$	11.7030	0.672320
$304$	−8.40169	−0.481870
$305$	0	0
$306$	0.481688	0.0275362
$307$	8.53224	0.486960	0.243480	−	0.969906i	$-0.421711\pi$
$307$	8.53224	0.486960	0.243480	+	0.969906i	$0.421711\pi$
$308$	0	0
$309$	15.1140	0.859803
$310$	0	0
$311$	−1.10401	−0.0626029	−0.0313015	−	0.999510i	$-0.509965\pi$
$311$	−1.10401	−0.0626029	−0.0313015	+	0.999510i	$0.509965\pi$
$312$	8.50806	0.481674
$313$	5.59145	0.316048	0.158024	−	0.987435i	$-0.449488\pi$
$313$	5.59145	0.316048	0.158024	+	0.987435i	$0.449488\pi$
$314$	−2.63471	−0.148685
$315$	0	0
$316$	25.0676	1.41016
$317$	−28.3491	−1.59224	−0.796121	−	0.605137i	$-0.793118\pi$
$317$	−28.3491	−1.59224	−0.796121	+	0.605137i	$0.793118\pi$
$318$	−3.83467	−0.215038
$319$	0	0
$320$	0	0
$321$	−19.0283	−1.06206
$322$	−0.190896	−0.0106382
$323$	−8.82795	−0.491200
$324$	20.3854	1.13252
$325$	0	0
$326$	1.75149	0.0970060
$327$	−14.4690	−0.800139
$328$	10.4108	0.574841
$329$	5.81908	0.320816
$330$	0	0
$331$	18.6052	1.02263	0.511317	−	0.859392i	$-0.329158\pi$
$331$	18.6052	1.02263	0.511317	+	0.859392i	$0.329158\pi$
$332$	24.5423	1.34694
$333$	4.86095	0.266378
$334$	−3.98346	−0.217965
$335$	0	0
$336$	9.13455	0.498330
$337$	29.4734	1.60552	0.802758	−	0.596305i	$-0.203365\pi$
$337$	29.4734	1.60552	0.802758	+	0.596305i	$0.203365\pi$
$338$	3.47920	0.189243
$339$	37.1334	2.01681
$340$	0	0
$341$	0	0
$342$	−0.275124	−0.0148770
$343$	−15.9026	−0.858661
$344$	−1.08097	−0.0582819
$345$	0	0
$346$	2.54914	0.137043
$347$	17.6969	0.950021	0.475010	−	0.879980i	$-0.342444\pi$
$347$	17.6969	0.950021	0.475010	+	0.879980i	$0.342444\pi$
$348$	13.5584	0.726805
$349$	−1.56340	−0.0836867	−0.0418434	−	0.999124i	$-0.513323\pi$
$349$	−1.56340	−0.0836867	−0.0418434	+	0.999124i	$0.513323\pi$
$350$	0	0
$351$	24.8990	1.32901
$352$	0	0
$353$	1.66213	0.0884663	0.0442331	−	0.999021i	$-0.485916\pi$
$353$	1.66213	0.0884663	0.0442331	+	0.999021i	$0.485916\pi$
$354$	3.51696	0.186924
$355$	0	0
$356$	12.2952	0.651646
$357$	9.59798	0.507979
$358$	0.666573	0.0352295
$359$	31.2164	1.64754	0.823770	−	0.566923i	$-0.191866\pi$
$359$	31.2164	1.64754	0.823770	+	0.566923i	$0.191866\pi$
$360$	0	0
$361$	−13.9578	−0.734620
$362$	0.451343	0.0237220
$363$	0	0
$364$	13.7437	0.720366
$365$	0	0
$366$	0.726483	0.0379739
$367$	9.32833	0.486935	0.243467	−	0.969909i	$-0.421715\pi$
$367$	9.32833	0.486935	0.243467	+	0.969909i	$0.421715\pi$
$368$	−2.66075	−0.138701
$369$	−7.43125	−0.386855
$370$	0	0
$371$	−12.5262	−0.650328
$372$	23.9584	1.24218
$373$	9.77497	0.506129	0.253064	−	0.967449i	$-0.418562\pi$
$373$	9.77497	0.506129	0.253064	+	0.967449i	$0.418562\pi$
$374$	0	0
$375$	0	0
$376$	−3.72088	−0.191890
$377$	19.9374	1.02683
$378$	−1.22637	−0.0630778
$379$	13.4548	0.691128	0.345564	−	0.938395i	$-0.387688\pi$
$379$	13.4548	0.691128	0.345564	+	0.938395i	$0.387688\pi$
$380$	0	0
$381$	−32.5154	−1.66582
$382$	0.367215	0.0187884
$383$	3.91429	0.200011	0.100005	−	0.994987i	$-0.468114\pi$
$383$	3.91429	0.200011	0.100005	+	0.994987i	$0.468114\pi$
$384$	−11.9497	−0.609807
$385$	0	0
$386$	−3.20910	−0.163339
$387$	0.771596	0.0392224
$388$	0.593645	0.0301378
$389$	−13.3279	−0.675749	−0.337875	−	0.941191i	$-0.609708\pi$
$389$	−13.3279	−0.675749	−0.337875	+	0.941191i	$0.609708\pi$
$390$	0	0
$391$	−2.79574	−0.141386
$392$	4.39986	0.222226
$393$	−18.9850	−0.957665
$394$	−0.372261	−0.0187543
$395$	0	0
$396$	0	0
$397$	8.80209	0.441764	0.220882	−	0.975300i	$-0.429106\pi$
$397$	8.80209	0.441764	0.220882	+	0.975300i	$0.429106\pi$
$398$	−4.08376	−0.204701
$399$	−5.48204	−0.274445
$400$	0	0
$401$	16.4060	0.819279	0.409639	−	0.912248i	$-0.365655\pi$
$401$	16.4060	0.819279	0.409639	+	0.912248i	$0.365655\pi$
$402$	−1.69747	−0.0846621
$403$	35.2304	1.75495
$404$	−12.0882	−0.601412
$405$	0	0
$406$	−0.981993	−0.0487355
$407$	0	0
$408$	−6.13722	−0.303838
$409$	11.8116	0.584044	0.292022	−	0.956412i	$-0.405672\pi$
$409$	11.8116	0.584044	0.292022	+	0.956412i	$0.405672\pi$
$410$	0	0
$411$	6.68168	0.329583
$412$	−15.6114	−0.769120
$413$	11.4884	0.565306
$414$	−0.0871293	−0.00428217
$415$	0	0
$416$	−13.2303	−0.648671
$417$	16.6267	0.814215
$418$	0	0
$419$	26.3901	1.28924	0.644620	−	0.764503i	$-0.277015\pi$
$419$	26.3901	1.28924	0.644620	+	0.764503i	$0.277015\pi$
$420$	0	0
$421$	11.3694	0.554111	0.277056	−	0.960854i	$-0.410641\pi$
$421$	11.3694	0.554111	0.277056	+	0.960854i	$0.410641\pi$
$422$	3.48197	0.169500
$423$	2.65597	0.129138
$424$	8.00961	0.388981
$425$	0	0
$426$	4.00755	0.194166
$427$	2.37310	0.114843
$428$	19.6547	0.950044
$429$	0	0
$430$	0	0
$431$	−18.0081	−0.867418	−0.433709	−	0.901053i	$-0.642795\pi$
$431$	−18.0081	−0.867418	−0.433709	+	0.901053i	$0.642795\pi$
$432$	−17.0935	−0.822409
$433$	−38.2609	−1.83870	−0.919352	−	0.393437i	$-0.871286\pi$
$433$	−38.2609	−1.83870	−0.919352	+	0.393437i	$0.871286\pi$
$434$	−1.73523	−0.0832939
$435$	0	0
$436$	14.9453	0.715749
$437$	1.59683	0.0763867
$438$	−4.73647	−0.226317
$439$	8.67958	0.414254	0.207127	−	0.978314i	$-0.433589\pi$
$439$	8.67958	0.414254	0.207127	+	0.978314i	$0.433589\pi$
$440$	0	0
$441$	−3.14062	−0.149553
$442$	−4.46287	−0.212277
$443$	−13.3956	−0.636445	−0.318222	−	0.948016i	$-0.603086\pi$
$443$	−13.3956	−0.636445	−0.318222	+	0.948016i	$0.603086\pi$
$444$	−30.6273	−1.45351
$445$	0	0
$446$	−1.24586	−0.0589931
$447$	15.9057	0.752314
$448$	−8.99277	−0.424868
$449$	−19.5312	−0.921736	−0.460868	−	0.887469i	$-0.652462\pi$
$449$	−19.5312	−0.921736	−0.460868	+	0.887469i	$0.652462\pi$
$450$	0	0
$451$	0	0
$452$	−38.3556	−1.80410
$453$	16.1711	0.759786
$454$	−4.63417	−0.217492
$455$	0	0
$456$	3.50537	0.164154
$457$	25.7806	1.20597	0.602983	−	0.797754i	$-0.293979\pi$
$457$	25.7806	1.20597	0.602983	+	0.797754i	$0.293979\pi$
$458$	1.24401	0.0581287
$459$	−17.9607	−0.838333
$460$	0	0
$461$	−19.0542	−0.887441	−0.443720	−	0.896165i	$-0.646342\pi$
$461$	−19.0542	−0.887441	−0.443720	+	0.896165i	$0.646342\pi$
$462$	0	0
$463$	20.0792	0.933161	0.466580	−	0.884479i	$-0.345486\pi$
$463$	20.0792	0.933161	0.466580	+	0.884479i	$0.345486\pi$
$464$	−13.6872	−0.635414
$465$	0	0
$466$	3.69392	0.171118
$467$	16.5057	0.763793	0.381896	−	0.924205i	$-0.375271\pi$
$467$	16.5057	0.763793	0.381896	+	0.924205i	$0.375271\pi$
$468$	6.27296	0.289968
$469$	−5.54489	−0.256039
$470$	0	0
$471$	−23.9616	−1.10409
$472$	−7.34599	−0.338127
$473$	0	0
$474$	−5.05481	−0.232175
$475$	0	0
$476$	−9.91390	−0.454403
$477$	−5.71726	−0.261775
$478$	3.98151	0.182110
$479$	−5.13076	−0.234430	−0.117215	−	0.993107i	$-0.537397\pi$
$479$	−5.13076	−0.234430	−0.117215	+	0.993107i	$0.537397\pi$
$480$	0	0
$481$	−45.0370	−2.05351
$482$	0.373741	0.0170234
$483$	−1.73612	−0.0789960
$484$	0	0
$485$	0	0
$486$	−1.25602	−0.0569741
$487$	16.7394	0.758535	0.379267	−	0.925287i	$-0.376176\pi$
$487$	16.7394	0.758535	0.379267	+	0.925287i	$0.376176\pi$
$488$	−1.51743	−0.0686908
$489$	15.9291	0.720338
$490$	0	0
$491$	0.771466	0.0348158	0.0174079	−	0.999848i	$-0.494459\pi$
$491$	0.771466	0.0348158	0.0174079	+	0.999848i	$0.494459\pi$
$492$	46.8220	2.11090
$493$	−14.3817	−0.647717
$494$	2.54904	0.114687
$495$	0	0
$496$	−24.1861	−1.08599
$497$	13.0909	0.587208
$498$	−4.94890	−0.221766
$499$	17.4872	0.782835	0.391418	−	0.920213i	$-0.371985\pi$
$499$	17.4872	0.782835	0.391418	+	0.920213i	$0.371985\pi$
$500$	0	0
$501$	−36.2280	−1.61855
$502$	2.20160	0.0982623
$503$	−39.1725	−1.74661	−0.873307	−	0.487169i	$-0.838029\pi$
$503$	−39.1725	−1.74661	−0.873307	+	0.487169i	$0.838029\pi$
$504$	−0.624786	−0.0278302
$505$	0	0
$506$	0	0
$507$	31.6419	1.40526
$508$	33.5857	1.49012
$509$	3.24292	0.143740	0.0718700	−	0.997414i	$-0.477103\pi$
$509$	3.24292	0.143740	0.0718700	+	0.997414i	$0.477103\pi$
$510$	0	0
$511$	−15.4720	−0.684440
$512$	15.2497	0.673948
$513$	10.2585	0.452925
$514$	−0.937117	−0.0413345
$515$	0	0
$516$	−4.86159	−0.214020
$517$	0	0
$518$	2.21825	0.0974642
$519$	23.1834	1.01764
$520$	0	0
$521$	11.6613	0.510892	0.255446	−	0.966823i	$-0.417778\pi$
$521$	11.6613	0.510892	0.255446	+	0.966823i	$0.417778\pi$
$522$	−0.448205	−0.0196174
$523$	−5.11076	−0.223478	−0.111739	−	0.993738i	$-0.535642\pi$
$523$	−5.11076	−0.223478	−0.111739	+	0.993738i	$0.535642\pi$
$524$	19.6099	0.856661
$525$	0	0
$526$	1.84950	0.0806420
$527$	−25.4132	−1.10701
$528$	0	0
$529$	−22.4943	−0.978013
$530$	0	0
$531$	5.24357	0.227552
$532$	5.66249	0.245500
$533$	68.8511	2.98227
$534$	−2.47930	−0.107290
$535$	0	0
$536$	3.54556	0.153145
$537$	6.06221	0.261603
$538$	−3.50054	−0.150919
$539$	0	0
$540$	0	0
$541$	−36.2308	−1.55769	−0.778843	−	0.627219i	$-0.784193\pi$
$541$	−36.2308	−1.55769	−0.778843	+	0.627219i	$0.784193\pi$
$542$	2.81865	0.121071
$543$	4.10478	0.176153
$544$	9.54360	0.409178
$545$	0	0
$546$	−2.77138	−0.118604
$547$	−19.1552	−0.819019	−0.409510	−	0.912306i	$-0.634300\pi$
$547$	−19.1552	−0.819019	−0.409510	+	0.912306i	$0.634300\pi$
$548$	−6.90161	−0.294822
$549$	1.08314	0.0462274
$550$	0	0
$551$	8.21432	0.349942
$552$	1.11012	0.0472499
$553$	−16.5119	−0.702156
$554$	−4.26324	−0.181128
$555$	0	0
$556$	−17.1740	−0.728341
$557$	−26.6904	−1.13091	−0.565454	−	0.824780i	$-0.691299\pi$
$557$	−26.6904	−1.13091	−0.565454	+	0.824780i	$0.691299\pi$
$558$	−0.792003	−0.0335282
$559$	−7.14889	−0.302366
$560$	0	0
$561$	0	0
$562$	2.08499	0.0879500
$563$	34.0871	1.43660	0.718300	−	0.695734i	$-0.244921\pi$
$563$	34.0871	1.43660	0.718300	+	0.695734i	$0.244921\pi$
$564$	−16.7344	−0.704647
$565$	0	0
$566$	0.557766	0.0234447
$567$	−13.4278	−0.563913
$568$	−8.37070	−0.351227
$569$	22.8260	0.956917	0.478458	−	0.878110i	$-0.341196\pi$
$569$	22.8260	0.956917	0.478458	+	0.878110i	$0.341196\pi$
$570$	0	0
$571$	36.9818	1.54764	0.773820	−	0.633406i	$-0.218344\pi$
$571$	36.9818	1.54764	0.773820	+	0.633406i	$0.218344\pi$
$572$	0	0
$573$	3.33967	0.139517
$574$	−3.39118	−0.141545
$575$	0	0
$576$	−4.10452	−0.171022
$577$	−28.9194	−1.20393	−0.601964	−	0.798523i	$-0.705615\pi$
$577$	−28.9194	−1.20393	−0.601964	+	0.798523i	$0.705615\pi$
$578$	−0.321591	−0.0133764
$579$	−29.1855	−1.21291
$580$	0	0
$581$	−16.1659	−0.670675
$582$	−0.119707	−0.00496202
$583$	0	0
$584$	9.89322	0.409385
$585$	0	0
$586$	−5.26759	−0.217602
$587$	−7.22366	−0.298152	−0.149076	−	0.988826i	$-0.547630\pi$
$587$	−7.22366	−0.298152	−0.149076	+	0.988826i	$0.547630\pi$
$588$	19.7881	0.816047
$589$	14.5151	0.598086
$590$	0	0
$591$	−3.38557	−0.139264
$592$	30.9184	1.27074
$593$	−10.9657	−0.450308	−0.225154	−	0.974323i	$-0.572289\pi$
$593$	−10.9657	−0.450308	−0.225154	+	0.974323i	$0.572289\pi$
$594$	0	0
$595$	0	0
$596$	−16.4292	−0.672968
$597$	−37.1402	−1.52005
$598$	0.807260	0.0330113
$599$	5.79798	0.236899	0.118449	−	0.992960i	$-0.462208\pi$
$599$	5.79798	0.236899	0.118449	+	0.992960i	$0.462208\pi$
$600$	0	0
$601$	8.14192	0.332116	0.166058	−	0.986116i	$-0.446896\pi$
$601$	8.14192	0.332116	0.166058	+	0.986116i	$0.446896\pi$
$602$	0.352111	0.0143510
$603$	−2.53082	−0.103063
$604$	−16.7034	−0.679652
$605$	0	0
$606$	2.43756	0.0990192
$607$	26.6628	1.08221	0.541104	−	0.840956i	$-0.318006\pi$
$607$	26.6628	1.08221	0.541104	+	0.840956i	$0.318006\pi$
$608$	−5.45098	−0.221066
$609$	−8.93083	−0.361895
$610$	0	0
$611$	−24.6077	−0.995522
$612$	−4.52495	−0.182910
$613$	−10.3240	−0.416983	−0.208492	−	0.978024i	$-0.566855\pi$
$613$	−10.3240	−0.416983	−0.208492	+	0.978024i	$0.566855\pi$
$614$	1.77714	0.0717194
$615$	0	0
$616$	0	0
$617$	−8.79766	−0.354180	−0.177090	−	0.984195i	$-0.556668\pi$
$617$	−8.79766	−0.354180	−0.177090	+	0.984195i	$0.556668\pi$
$618$	3.14801	0.126632
$619$	−13.6153	−0.547245	−0.273622	−	0.961837i	$-0.588222\pi$
$619$	−13.6153	−0.547245	−0.273622	+	0.961837i	$0.588222\pi$
$620$	0	0
$621$	3.24879	0.130369
$622$	−0.229950	−0.00922014
$623$	−8.09881	−0.324472
$624$	−38.6282	−1.54637
$625$	0	0
$626$	1.16462	0.0465474
$627$	0	0
$628$	24.7503	0.987643
$629$	32.4871	1.29534
$630$	0	0
$631$	−26.2420	−1.04468	−0.522339	−	0.852738i	$-0.674941\pi$
$631$	−26.2420	−1.04468	−0.522339	+	0.852738i	$0.674941\pi$
$632$	10.5582	0.419981
$633$	31.6671	1.25865
$634$	−5.90469	−0.234505
$635$	0	0
$636$	36.0227	1.42839
$637$	29.0981	1.15291
$638$	0	0
$639$	5.97501	0.236368
$640$	0	0
$641$	40.0975	1.58376	0.791878	−	0.610680i	$-0.209104\pi$
$641$	40.0975	1.58376	0.791878	+	0.610680i	$0.209104\pi$
$642$	−3.96332	−0.156420
$643$	−2.62100	−0.103362	−0.0516811	−	0.998664i	$-0.516458\pi$
$643$	−2.62100	−0.103362	−0.0516811	+	0.998664i	$0.516458\pi$
$644$	1.79326	0.0706644
$645$	0	0
$646$	−1.83873	−0.0723438
$647$	−30.1911	−1.18693	−0.593467	−	0.804858i	$-0.702241\pi$
$647$	−30.1911	−1.18693	−0.593467	+	0.804858i	$0.702241\pi$
$648$	8.58610	0.337294
$649$	0	0
$650$	0	0
$651$	−15.7812	−0.618516
$652$	−16.4534	−0.644365
$653$	−36.4106	−1.42486	−0.712429	−	0.701744i	$-0.752405\pi$
$653$	−36.4106	−1.42486	−0.712429	+	0.701744i	$0.752405\pi$
$654$	−3.01368	−0.117844
$655$	0	0
$656$	−47.2671	−1.84547
$657$	−7.06179	−0.275507
$658$	1.21203	0.0472497
$659$	32.9001	1.28161	0.640803	−	0.767706i	$-0.278602\pi$
$659$	32.9001	1.28161	0.640803	+	0.767706i	$0.278602\pi$
$660$	0	0
$661$	31.0455	1.20753	0.603766	−	0.797162i	$-0.293666\pi$
$661$	31.0455	1.20753	0.603766	+	0.797162i	$0.293666\pi$
$662$	3.87518	0.150613
$663$	−40.5880	−1.57631
$664$	10.3369	0.401151
$665$	0	0
$666$	1.01246	0.0392321
$667$	2.60140	0.100727
$668$	37.4204	1.44784
$669$	−11.3306	−0.438065
$670$	0	0
$671$	0	0
$672$	5.92645	0.228618
$673$	4.63076	0.178503	0.0892514	−	0.996009i	$-0.471553\pi$
$673$	4.63076	0.178503	0.0892514	+	0.996009i	$0.471553\pi$
$674$	6.13886	0.236460
$675$	0	0
$676$	−32.6834	−1.25705
$677$	−22.1844	−0.852615	−0.426307	−	0.904578i	$-0.640186\pi$
$677$	−22.1844	−0.852615	−0.426307	+	0.904578i	$0.640186\pi$
$678$	7.73432	0.297035
$679$	−0.391031	−0.0150064
$680$	0	0
$681$	−42.1458	−1.61503
$682$	0	0
$683$	42.5540	1.62828	0.814142	−	0.580665i	$-0.197207\pi$
$683$	42.5540	1.62828	0.814142	+	0.580665i	$0.197207\pi$
$684$	2.58450	0.0988207
$685$	0	0
$686$	−3.31228	−0.126463
$687$	11.3137	0.431646
$688$	4.90780	0.187108
$689$	52.9708	2.01803
$690$	0	0
$691$	−3.77491	−0.143604	−0.0718022	−	0.997419i	$-0.522875\pi$
$691$	−3.77491	−0.143604	−0.0718022	+	0.997419i	$0.522875\pi$
$692$	−23.9465	−0.910308
$693$	0	0
$694$	3.68600	0.139919
$695$	0	0
$696$	5.71062	0.216461
$697$	−49.6651	−1.88120
$698$	−0.325632	−0.0123254
$699$	33.5947	1.27067
$700$	0	0
$701$	0.857774	0.0323977	0.0161988	−	0.999869i	$-0.494844\pi$
$701$	0.857774	0.0323977	0.0161988	+	0.999869i	$0.494844\pi$
$702$	5.18609	0.195736
$703$	−18.5555	−0.699834
$704$	0	0
$705$	0	0
$706$	0.346197	0.0130293
$707$	7.96245	0.299459
$708$	−33.0381	−1.24165
$709$	24.7463	0.929368	0.464684	−	0.885477i	$-0.346168\pi$
$709$	24.7463	0.929368	0.464684	+	0.885477i	$0.346168\pi$
$710$	0	0
$711$	−7.53642	−0.282638
$712$	5.17861	0.194076
$713$	4.59682	0.172152
$714$	1.99912	0.0748150
$715$	0	0
$716$	−6.26175	−0.234012
$717$	36.2102	1.35230
$718$	6.50192	0.242649
$719$	−17.4888	−0.652223	−0.326112	−	0.945331i	$-0.605739\pi$
$719$	−17.4888	−0.652223	−0.326112	+	0.945331i	$0.605739\pi$
$720$	0	0
$721$	10.2832	0.382965
$722$	−2.90720	−0.108195
$723$	3.39902	0.126411
$724$	−4.23989	−0.157574
$725$	0	0
$726$	0	0
$727$	13.8835	0.514909	0.257455	−	0.966290i	$-0.417116\pi$
$727$	13.8835	0.514909	0.257455	+	0.966290i	$0.417116\pi$
$728$	5.78868	0.214543
$729$	19.8331	0.734561
$730$	0	0
$731$	5.15680	0.190731
$732$	−6.82454	−0.252242
$733$	30.2489	1.11727	0.558634	−	0.829414i	$-0.311326\pi$
$733$	30.2489	1.11727	0.558634	+	0.829414i	$0.311326\pi$
$734$	1.94295	0.0717156
$735$	0	0
$736$	−1.72628	−0.0636315
$737$	0	0
$738$	−1.54782	−0.0569759
$739$	24.4971	0.901139	0.450570	−	0.892741i	$-0.351221\pi$
$739$	24.4971	0.901139	0.450570	+	0.892741i	$0.351221\pi$
$740$	0	0
$741$	23.1825	0.851629
$742$	−2.60902	−0.0957801
$743$	−6.64540	−0.243796	−0.121898	−	0.992543i	$-0.538898\pi$
$743$	−6.64540	−0.243796	−0.121898	+	0.992543i	$0.538898\pi$
$744$	10.0910	0.369953
$745$	0	0
$746$	2.03598	0.0745425
$747$	−7.37851	−0.269966
$748$	0	0
$749$	−12.9464	−0.473052
$750$	0	0
$751$	−18.0627	−0.659116	−0.329558	−	0.944135i	$-0.606900\pi$
$751$	−18.0627	−0.659116	−0.329558	+	0.944135i	$0.606900\pi$
$752$	16.8935	0.616043
$753$	20.0227	0.729667
$754$	4.15266	0.151231
$755$	0	0
$756$	11.5205	0.418995
$757$	10.5847	0.384708	0.192354	−	0.981326i	$-0.438388\pi$
$757$	10.5847	0.384708	0.192354	+	0.981326i	$0.438388\pi$
$758$	2.80244	0.101789
$759$	0	0
$760$	0	0
$761$	−7.74918	−0.280908	−0.140454	−	0.990087i	$-0.544856\pi$
$761$	−7.74918	−0.280908	−0.140454	+	0.990087i	$0.544856\pi$
$762$	−6.77247	−0.245341
$763$	−9.84438	−0.356390
$764$	−3.44960	−0.124802
$765$	0	0
$766$	0.815287	0.0294575
$767$	−48.5821	−1.75420
$768$	23.9458	0.864068
$769$	23.9339	0.863078	0.431539	−	0.902094i	$-0.357971\pi$
$769$	23.9339	0.863078	0.431539	+	0.902094i	$0.357971\pi$
$770$	0	0
$771$	−8.52270	−0.306938
$772$	30.1461	1.08498
$773$	34.7483	1.24981	0.624904	−	0.780701i	$-0.285138\pi$
$773$	34.7483	1.24981	0.624904	+	0.780701i	$0.285138\pi$
$774$	0.160712	0.00577667
$775$	0	0
$776$	0.250036	0.00897578
$777$	20.1741	0.723740
$778$	−2.77599	−0.0995242
$779$	28.3670	1.01635
$780$	0	0
$781$	0	0
$782$	−0.582310	−0.0208234
$783$	16.7122	0.597246
$784$	−19.9762	−0.713435
$785$	0	0
$786$	−3.95428	−0.141045
$787$	−8.78020	−0.312980	−0.156490	−	0.987680i	$-0.550018\pi$
$787$	−8.78020	−0.312980	−0.156490	+	0.987680i	$0.550018\pi$
$788$	3.49700	0.124576
$789$	16.8204	0.598823
$790$	0	0
$791$	25.2647	0.898308
$792$	0	0
$793$	−10.0354	−0.356367
$794$	1.83334	0.0650629
$795$	0	0
$796$	38.3626	1.35973
$797$	49.5332	1.75455	0.877277	−	0.479984i	$-0.159357\pi$
$797$	49.5332	1.75455	0.877277	+	0.479984i	$0.159357\pi$
$798$	−1.14183	−0.0404202
$799$	17.7506	0.627970
$800$	0	0
$801$	−3.69649	−0.130609
$802$	3.41713	0.120663
$803$	0	0
$804$	15.9459	0.562370
$805$	0	0
$806$	7.33796	0.258469
$807$	−31.8360	−1.12068
$808$	−5.09142	−0.179115
$809$	39.6243	1.39312	0.696559	−	0.717500i	$-0.254714\pi$
$809$	39.6243	1.39312	0.696559	+	0.717500i	$0.254714\pi$
$810$	0	0
$811$	50.0156	1.75628	0.878142	−	0.478400i	$-0.158783\pi$
$811$	50.0156	1.75628	0.878142	+	0.478400i	$0.158783\pi$
$812$	9.22479	0.323727
$813$	25.6345	0.899040
$814$	0	0
$815$	0	0
$816$	27.8641	0.975440
$817$	−2.94539	−0.103046
$818$	2.46017	0.0860178
$819$	−4.13196	−0.144383
$820$	0	0
$821$	−40.6697	−1.41938	−0.709691	−	0.704513i	$-0.751166\pi$
$821$	−40.6697	−1.41938	−0.709691	+	0.704513i	$0.751166\pi$
$822$	1.39169	0.0485409
$823$	29.7474	1.03693	0.518464	−	0.855099i	$-0.326504\pi$
$823$	29.7474	1.03693	0.518464	+	0.855099i	$0.326504\pi$
$824$	−6.57535	−0.229063
$825$	0	0
$826$	2.39285	0.0832581
$827$	−46.1861	−1.60605	−0.803024	−	0.595947i	$-0.796777\pi$
$827$	−46.1861	−1.60605	−0.803024	+	0.595947i	$0.796777\pi$
$828$	0.818488	0.0284444
$829$	−48.9648	−1.70062	−0.850309	−	0.526284i	$-0.823585\pi$
$829$	−48.9648	−1.70062	−0.850309	+	0.526284i	$0.823585\pi$
$830$	0	0
$831$	−38.7724	−1.34500
$832$	38.0287	1.31841
$833$	−20.9897	−0.727248
$834$	3.46310	0.119917
$835$	0	0
$836$	0	0
$837$	29.5314	1.02075
$838$	5.49666	0.189879
$839$	−46.0659	−1.59037	−0.795185	−	0.606367i	$-0.792626\pi$
$839$	−46.0659	−1.59037	−0.795185	+	0.606367i	$0.792626\pi$
$840$	0	0
$841$	−15.6180	−0.538552
$842$	2.36808	0.0816093
$843$	18.9621	0.653091
$844$	−32.7094	−1.12590
$845$	0	0
$846$	0.553198	0.0190193
$847$	0	0
$848$	−36.3651	−1.24878
$849$	5.07266	0.174093
$850$	0	0
$851$	−5.87638	−0.201440
$852$	−37.6467	−1.28975
$853$	21.3990	0.732688	0.366344	−	0.930479i	$-0.380609\pi$
$853$	21.3990	0.732688	0.366344	+	0.930479i	$0.380609\pi$
$854$	0.494282	0.0169140
$855$	0	0
$856$	8.27831	0.282947
$857$	10.0178	0.342201	0.171100	−	0.985254i	$-0.445268\pi$
$857$	10.0178	0.342201	0.171100	+	0.985254i	$0.445268\pi$
$858$	0	0
$859$	−31.7860	−1.08452	−0.542261	−	0.840210i	$-0.682432\pi$
$859$	−31.7860	−1.08452	−0.542261	+	0.840210i	$0.682432\pi$
$860$	0	0
$861$	−30.8414	−1.05107
$862$	−3.75081	−0.127753
$863$	−27.5120	−0.936519	−0.468260	−	0.883591i	$-0.655119\pi$
$863$	−27.5120	−0.936519	−0.468260	+	0.883591i	$0.655119\pi$
$864$	−11.0902	−0.377295
$865$	0	0
$866$	−7.96918	−0.270804
$867$	−2.92474	−0.0993292
$868$	16.3007	0.553282
$869$	0	0
$870$	0	0
$871$	23.4483	0.794514
$872$	6.29477	0.213168
$873$	−0.178476	−0.00604050
$874$	0.332596	0.0112502
$875$	0	0
$876$	44.4941	1.50332
$877$	27.3382	0.923146	0.461573	−	0.887102i	$-0.347285\pi$
$877$	27.3382	0.923146	0.461573	+	0.887102i	$0.347285\pi$
$878$	1.80783	0.0610112
$879$	−47.9066	−1.61585
$880$	0	0
$881$	−48.8428	−1.64555	−0.822777	−	0.568364i	$-0.807576\pi$
$881$	−48.8428	−1.64555	−0.822777	+	0.568364i	$0.807576\pi$
$882$	−0.654144	−0.0220262
$883$	−48.1529	−1.62047	−0.810237	−	0.586102i	$-0.800662\pi$
$883$	−48.1529	−1.62047	−0.810237	+	0.586102i	$0.800662\pi$
$884$	41.9239	1.41005
$885$	0	0
$886$	−2.79011	−0.0937354
$887$	−58.3806	−1.96023	−0.980114	−	0.198436i	$-0.936414\pi$
$887$	−58.3806	−1.96023	−0.980114	+	0.198436i	$0.936414\pi$
$888$	−12.8999	−0.432891
$889$	−22.1227	−0.741972
$890$	0	0
$891$	0	0
$892$	11.7035	0.391863
$893$	−10.1385	−0.339273
$894$	3.31292	0.110801
$895$	0	0
$896$	−8.13031	−0.271615
$897$	7.34170	0.245132
$898$	−4.06806	−0.135753
$899$	23.6467	0.788661
$900$	0	0
$901$	−38.2101	−1.27296
$902$	0	0
$903$	3.20230	0.106566
$904$	−16.1549	−0.537305
$905$	0	0
$906$	3.36820	0.111901
$907$	45.0528	1.49595	0.747977	−	0.663725i	$-0.231025\pi$
$907$	45.0528	1.49595	0.747977	+	0.663725i	$0.231025\pi$
$908$	43.5331	1.44470
$909$	3.63425	0.120541
$910$	0	0
$911$	6.18824	0.205025	0.102513	−	0.994732i	$-0.467312\pi$
$911$	6.18824	0.205025	0.102513	+	0.994732i	$0.467312\pi$
$912$	−15.9150	−0.527000
$913$	0	0
$914$	5.36971	0.177614
$915$	0	0
$916$	−11.6861	−0.386121
$917$	−12.9169	−0.426554
$918$	−3.74094	−0.123469
$919$	−8.32170	−0.274508	−0.137254	−	0.990536i	$-0.543828\pi$
$919$	−8.32170	−0.274508	−0.137254	+	0.990536i	$0.543828\pi$
$920$	0	0
$921$	16.1623	0.532567
$922$	−3.96869	−0.130702
$923$	−55.3589	−1.82216
$924$	0	0
$925$	0	0
$926$	4.18220	0.137436
$927$	4.69349	0.154154
$928$	−8.88022	−0.291508
$929$	10.6402	0.349094	0.174547	−	0.984649i	$-0.444154\pi$
$929$	10.6402	0.349094	0.174547	+	0.984649i	$0.444154\pi$
$930$	0	0
$931$	11.9886	0.392910
$932$	−34.7005	−1.13665
$933$	−2.09130	−0.0684660
$934$	3.43789	0.112491
$935$	0	0
$936$	2.64210	0.0863596
$937$	−38.1741	−1.24709	−0.623547	−	0.781786i	$-0.714309\pi$
$937$	−38.1741	−1.24709	−0.623547	+	0.781786i	$0.714309\pi$
$938$	−1.15492	−0.0377094
$939$	10.5917	0.345647
$940$	0	0
$941$	40.5095	1.32057	0.660286	−	0.751014i	$-0.270435\pi$
$941$	40.5095	1.32057	0.660286	+	0.751014i	$0.270435\pi$
$942$	−4.99083	−0.162610
$943$	8.98360	0.292546
$944$	33.3522	1.08552
$945$	0	0
$946$	0	0
$947$	−25.2006	−0.818909	−0.409455	−	0.912330i	$-0.634281\pi$
$947$	−25.2006	−0.818909	−0.409455	+	0.912330i	$0.634281\pi$
$948$	47.4846	1.54223
$949$	65.4280	2.12388
$950$	0	0
$951$	−53.7007	−1.74136
$952$	−4.17562	−0.135333
$953$	−20.3828	−0.660262	−0.330131	−	0.943935i	$-0.607093\pi$
$953$	−20.3828	−0.660262	−0.330131	+	0.943935i	$0.607093\pi$
$954$	−1.19082	−0.0385542
$955$	0	0
$956$	−37.4021	−1.20967
$957$	0	0
$958$	−1.06866	−0.0345268
$959$	4.54606	0.146800
$960$	0	0
$961$	10.7850	0.347902
$962$	−9.38053	−0.302441
$963$	−5.90906	−0.190417
$964$	−3.51090	−0.113079
$965$	0	0
$966$	−0.361607	−0.0116345
$967$	4.49928	0.144687	0.0723436	−	0.997380i	$-0.476952\pi$
$967$	4.49928	0.144687	0.0723436	+	0.997380i	$0.476952\pi$
$968$	0	0
$969$	−16.7225	−0.537204
$970$	0	0
$971$	34.2299	1.09849	0.549245	−	0.835662i	$-0.314915\pi$
$971$	34.2299	1.09849	0.549245	+	0.835662i	$0.314915\pi$
$972$	11.7990	0.378452
$973$	11.3124	0.362660
$974$	3.48657	0.111717
$975$	0	0
$976$	6.88942	0.220525
$977$	−43.8652	−1.40337	−0.701686	−	0.712486i	$-0.747569\pi$
$977$	−43.8652	−1.40337	−0.701686	+	0.712486i	$0.747569\pi$
$978$	3.31779	0.106091
$979$	0	0
$980$	0	0
$981$	−4.49321	−0.143457
$982$	0.160685	0.00512766
$983$	17.9413	0.572238	0.286119	−	0.958194i	$-0.407635\pi$
$983$	17.9413	0.572238	0.286119	+	0.958194i	$0.407635\pi$
$984$	19.7209	0.628678
$985$	0	0
$986$	−2.99548	−0.0953956
$987$	11.0229	0.350862
$988$	−23.9455	−0.761809
$989$	−0.932779	−0.0296607
$990$	0	0
$991$	−33.5351	−1.06528	−0.532638	−	0.846343i	$-0.678799\pi$
$991$	−33.5351	−1.06528	−0.532638	+	0.846343i	$0.678799\pi$
$992$	−15.6918	−0.498216
$993$	35.2432	1.11841
$994$	2.72664	0.0864838
$995$	0	0
$996$	46.4897	1.47308
$997$	22.1964	0.702966	0.351483	−	0.936194i	$-0.385678\pi$
$997$	22.1964	0.702966	0.351483	+	0.936194i	$0.385678\pi$
$998$	3.64232	0.115296
$999$	−37.7517	−1.19441

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3025.2.a.bj.1.6		8
5.4	even	2		3025.2.a.bm.1.3		8
11.2	odd	10		275.2.h.e.26.3	yes	16
11.6	odd	10		275.2.h.e.201.3	yes	16
11.10	odd	2		3025.2.a.bn.1.3		8
55.2	even	20		275.2.z.c.224.4		32
55.13	even	20		275.2.z.c.224.5		32
55.17	even	20		275.2.z.c.124.5		32
55.24	odd	10		275.2.h.c.26.2	✓	16
55.28	even	20		275.2.z.c.124.4		32
55.39	odd	10		275.2.h.c.201.2	yes	16
55.54	odd	2		3025.2.a.bi.1.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
275.2.h.c.26.2	✓	16	55.24	odd	10
275.2.h.c.201.2	yes	16	55.39	odd	10
275.2.h.e.26.3	yes	16	11.2	odd	10
275.2.h.e.201.3	yes	16	11.6	odd	10
275.2.z.c.124.4		32	55.28	even	20
275.2.z.c.124.5		32	55.17	even	20
275.2.z.c.224.4		32	55.2	even	20
275.2.z.c.224.5		32	55.13	even	20
3025.2.a.bi.1.6		8	55.54	odd	2
3025.2.a.bj.1.6		8	1.1	even	1	trivial
3025.2.a.bm.1.3		8	5.4	even	2
3025.2.a.bn.1.3		8	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3025 = 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3025.a (trivial)

\(f(q)\)	\(=\)	\(q+0.208285 q^{2} +1.89427 q^{3} -1.95662 q^{4} +0.394547 q^{6} +1.28881 q^{7} -0.824104 q^{8} +0.588246 q^{9} +O(q^{10})\)	\(q+0.208285 q^{2} +1.89427 q^{3} -1.95662 q^{4} +0.394547 q^{6} +1.28881 q^{7} -0.824104 q^{8} +0.588246 q^{9} -3.70635 q^{12} -5.45014 q^{13} +0.268441 q^{14} +3.74159 q^{16} +3.93141 q^{17} +0.122523 q^{18} -2.24549 q^{19} +2.44136 q^{21} -0.711128 q^{23} -1.56107 q^{24} -1.13518 q^{26} -4.56851 q^{27} -2.52172 q^{28} -3.65814 q^{29} -6.46413 q^{31} +2.42752 q^{32} +0.818854 q^{34} -1.15097 q^{36} +8.26346 q^{37} -0.467702 q^{38} -10.3240 q^{39} -12.6329 q^{41} +0.508498 q^{42} +1.31169 q^{43} -0.148117 q^{46} +4.51506 q^{47} +7.08756 q^{48} -5.33896 q^{49} +7.44714 q^{51} +10.6638 q^{52} -9.71917 q^{53} -0.951551 q^{54} -1.06212 q^{56} -4.25356 q^{57} -0.761936 q^{58} +8.91391 q^{59} +1.84131 q^{61} -1.34638 q^{62} +0.758139 q^{63} -6.97756 q^{64} -4.30232 q^{67} -7.69227 q^{68} -1.34707 q^{69} +10.1573 q^{71} -0.484776 q^{72} -12.0048 q^{73} +1.72115 q^{74} +4.39356 q^{76} -2.15034 q^{78} -12.8117 q^{79} -10.4187 q^{81} -2.63124 q^{82} -12.5432 q^{83} -4.77680 q^{84} +0.273205 q^{86} -6.92949 q^{87} -6.28392 q^{89} -7.02422 q^{91} +1.39140 q^{92} -12.2448 q^{93} +0.940420 q^{94} +4.59838 q^{96} -0.303404 q^{97} -1.11202 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 5 q^{2} + q^{3} + 9 q^{4} - q^{6} - 8 q^{7} - 12 q^{8} + 9 q^{9}+O(q^{10}) \) 8 * q - 5 * q^2 + q^3 + 9 * q^4 - q^6 - 8 * q^7 - 12 * q^8 + 9 * q^9	\( 8 q - 5 q^{2} + q^{3} + 9 q^{4} - q^{6} - 8 q^{7} - 12 q^{8} + 9 q^{9} + 3 q^{12} - 9 q^{13} + 9 q^{14} + 23 q^{16} - 19 q^{17} - 22 q^{18} - q^{19} - 5 q^{21} + 2 q^{23} - q^{24} - 2 q^{26} - 2 q^{27} - 9 q^{28} - 7 q^{29} - 5 q^{31} - 29 q^{32} + 10 q^{34} - 16 q^{36} - 8 q^{37} - 37 q^{38} + q^{39} + 41 q^{42} - 14 q^{43} + 20 q^{46} + 11 q^{47} - 27 q^{48} - 12 q^{49} + 25 q^{51} + 7 q^{52} + 11 q^{53} - 30 q^{54} - 10 q^{56} + 2 q^{57} + 27 q^{58} + 17 q^{59} + 2 q^{61} - 25 q^{62} - 41 q^{63} + 30 q^{64} + 7 q^{67} - 66 q^{68} + 17 q^{71} + 19 q^{72} - 34 q^{73} + 6 q^{74} + 31 q^{76} - 17 q^{78} - 23 q^{79} - 4 q^{81} - 17 q^{82} - 41 q^{83} - 83 q^{84} + q^{86} - 25 q^{87} - 11 q^{89} - 7 q^{91} + 33 q^{92} - 59 q^{93} - 50 q^{94} + 61 q^{96} + 2 q^{97} - 26 q^{98}+O(q^{100}) \) 8 * q - 5 * q^2 + q^3 + 9 * q^4 - q^6 - 8 * q^7 - 12 * q^8 + 9 * q^9 + 3 * q^12 - 9 * q^13 + 9 * q^14 + 23 * q^16 - 19 * q^17 - 22 * q^18 - q^19 - 5 * q^21 + 2 * q^23 - q^24 - 2 * q^26 - 2 * q^27 - 9 * q^28 - 7 * q^29 - 5 * q^31 - 29 * q^32 + 10 * q^34 - 16 * q^36 - 8 * q^37 - 37 * q^38 + q^39 + 41 * q^42 - 14 * q^43 + 20 * q^46 + 11 * q^47 - 27 * q^48 - 12 * q^49 + 25 * q^51 + 7 * q^52 + 11 * q^53 - 30 * q^54 - 10 * q^56 + 2 * q^57 + 27 * q^58 + 17 * q^59 + 2 * q^61 - 25 * q^62 - 41 * q^63 + 30 * q^64 + 7 * q^67 - 66 * q^68 + 17 * q^71 + 19 * q^72 - 34 * q^73 + 6 * q^74 + 31 * q^76 - 17 * q^78 - 23 * q^79 - 4 * q^81 - 17 * q^82 - 41 * q^83 - 83 * q^84 + q^86 - 25 * q^87 - 11 * q^89 - 7 * q^91 + 33 * q^92 - 59 * q^93 - 50 * q^94 + 61 * q^96 + 2 * q^97 - 26 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more