LMFDB - Embedded newform 8048.2.a.q.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8048,2,Mod(1,8048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8048, 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("8048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8048 = 2^{4} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8048.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:	$64.2636035467$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5 x^{11} - 13 x^{10} + 94 x^{9} + 15 x^{8} - 616 x^{7} + 368 x^{6} + 1643 x^{5} - 1463 x^{4} + \cdots - 208 $ x^12 - 5x^11 - 13x^10 + 94x^9 + 15x^8 - 616x^7 + 368x^6 + 1643x^5 - 1463x^4 - 1556x^3 + 1284x^2 + 576*x - 208
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1006)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-2.17256$ of defining polynomial
Character	$\chi$	$=$	8048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.17256 q^{3} +2.95208 q^{5} -2.27467 q^{7} +1.72004 q^{9} +O(q^{10})$	$q+2.17256 q^{3} +2.95208 q^{5} -2.27467 q^{7} +1.72004 q^{9} -3.76405 q^{11} -4.15916 q^{13} +6.41358 q^{15} +4.07953 q^{17} +0.0709948 q^{19} -4.94187 q^{21} +1.46890 q^{23} +3.71477 q^{25} -2.78081 q^{27} -7.74424 q^{29} -4.27274 q^{31} -8.17765 q^{33} -6.71502 q^{35} +2.89623 q^{37} -9.03604 q^{39} -4.41094 q^{41} -5.20433 q^{43} +5.07768 q^{45} +3.09752 q^{47} -1.82586 q^{49} +8.86304 q^{51} +0.690667 q^{53} -11.1118 q^{55} +0.154241 q^{57} -9.44782 q^{59} +11.2047 q^{61} -3.91252 q^{63} -12.2782 q^{65} -12.1917 q^{67} +3.19127 q^{69} -7.39192 q^{71} +0.601864 q^{73} +8.07057 q^{75} +8.56199 q^{77} -10.3921 q^{79} -11.2016 q^{81} +10.8034 q^{83} +12.0431 q^{85} -16.8249 q^{87} -7.29923 q^{89} +9.46073 q^{91} -9.28281 q^{93} +0.209582 q^{95} -13.8744 q^{97} -6.47431 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 5 q^{3} + 5 q^{5} - 8 q^{7} + 15 q^{9}+O(q^{10}) $ 12 * q - 5 * q^3 + 5 * q^5 - 8 * q^7 + 15 * q^9	$ 12 q - 5 q^{3} + 5 q^{5} - 8 q^{7} + 15 q^{9} - 18 q^{11} + 4 q^{13} - 2 q^{15} - 2 q^{17} - 6 q^{19} + q^{21} - 13 q^{23} + q^{25} - 8 q^{27} + 20 q^{29} - 7 q^{31} - 8 q^{33} - q^{35} + 10 q^{37} - 7 q^{39} + 2 q^{41} - 8 q^{43} - 7 q^{45} - 12 q^{47} + 4 q^{49} - 2 q^{51} + 12 q^{53} - 8 q^{55} - 10 q^{57} - 6 q^{59} - 10 q^{61} - 7 q^{63} + 4 q^{65} - 7 q^{67} - 12 q^{69} - 22 q^{71} - 23 q^{73} + 34 q^{75} - 19 q^{77} - 13 q^{79} - 28 q^{81} - q^{83} - 28 q^{85} + 22 q^{87} - 3 q^{89} + 21 q^{91} - 33 q^{93} + 2 q^{95} - 70 q^{97} + 6 q^{99}+O(q^{100}) $ 12 * q - 5 * q^3 + 5 * q^5 - 8 * q^7 + 15 * q^9 - 18 * q^11 + 4 * q^13 - 2 * q^15 - 2 * q^17 - 6 * q^19 + q^21 - 13 * q^23 + q^25 - 8 * q^27 + 20 * q^29 - 7 * q^31 - 8 * q^33 - q^35 + 10 * q^37 - 7 * q^39 + 2 * q^41 - 8 * q^43 - 7 * q^45 - 12 * q^47 + 4 * q^49 - 2 * q^51 + 12 * q^53 - 8 * q^55 - 10 * q^57 - 6 * q^59 - 10 * q^61 - 7 * q^63 + 4 * q^65 - 7 * q^67 - 12 * q^69 - 22 * q^71 - 23 * q^73 + 34 * q^75 - 19 * q^77 - 13 * q^79 - 28 * q^81 - q^83 - 28 * q^85 + 22 * q^87 - 3 * q^89 + 21 * q^91 - 33 * q^93 + 2 * q^95 - 70 * q^97 + 6 * 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 $

$2$	0	0
$2$	0	0
$3$	2.17256	1.25433	0.627165	−	0.778886i	$-0.284215\pi$
$3$	2.17256	1.25433	0.627165	+	0.778886i	$0.284215\pi$
$4$	0	0
$5$	2.95208	1.32021	0.660105	−	0.751174i	$-0.270512\pi$
$5$	2.95208	1.32021	0.660105	+	0.751174i	$0.270512\pi$
$6$	0	0
$7$	−2.27467	−0.859746	−0.429873	−	0.902889i	$-0.641442\pi$
$7$	−2.27467	−0.859746	−0.429873	+	0.902889i	$0.641442\pi$
$8$	0	0
$9$	1.72004	0.573345
$10$	0	0
$11$	−3.76405	−1.13490	−0.567452	−	0.823406i	$-0.692071\pi$
$11$	−3.76405	−1.13490	−0.567452	+	0.823406i	$0.692071\pi$
$12$	0	0
$13$	−4.15916	−1.15354	−0.576772	−	0.816905i	$-0.695688\pi$
$13$	−4.15916	−1.15354	−0.576772	+	0.816905i	$0.695688\pi$
$14$	0	0
$15$	6.41358	1.65598
$16$	0	0
$17$	4.07953	0.989431	0.494715	−	0.869055i	$-0.335272\pi$
$17$	4.07953	0.989431	0.494715	+	0.869055i	$0.335272\pi$
$18$	0	0
$19$	0.0709948	0.0162873	0.00814367	−	0.999967i	$-0.497408\pi$
$19$	0.0709948	0.0162873	0.00814367	+	0.999967i	$0.497408\pi$
$20$	0	0
$21$	−4.94187	−1.07841
$22$	0	0
$23$	1.46890	0.306286	0.153143	−	0.988204i	$-0.451060\pi$
$23$	1.46890	0.306286	0.153143	+	0.988204i	$0.451060\pi$
$24$	0	0
$25$	3.71477	0.742953
$26$	0	0
$27$	−2.78081	−0.535166
$28$	0	0
$29$	−7.74424	−1.43807	−0.719035	−	0.694974i	$-0.755416\pi$
$29$	−7.74424	−1.43807	−0.719035	+	0.694974i	$0.755416\pi$
$30$	0	0
$31$	−4.27274	−0.767407	−0.383704	−	0.923456i	$-0.625352\pi$
$31$	−4.27274	−0.767407	−0.383704	+	0.923456i	$0.625352\pi$
$32$	0	0
$33$	−8.17765	−1.42355
$34$	0	0
$35$	−6.71502	−1.13504
$36$	0	0
$37$	2.89623	0.476138	0.238069	−	0.971248i	$-0.423486\pi$
$37$	2.89623	0.476138	0.238069	+	0.971248i	$0.423486\pi$
$38$	0	0
$39$	−9.03604	−1.44692
$40$	0	0
$41$	−4.41094	−0.688873	−0.344437	−	0.938810i	$-0.611930\pi$
$41$	−4.41094	−0.688873	−0.344437	+	0.938810i	$0.611930\pi$
$42$	0	0
$43$	−5.20433	−0.793653	−0.396827	−	0.917894i	$-0.629889\pi$
$43$	−5.20433	−0.793653	−0.396827	+	0.917894i	$0.629889\pi$
$44$	0	0
$45$	5.07768	0.756936
$46$	0	0
$47$	3.09752	0.451820	0.225910	−	0.974148i	$-0.427465\pi$
$47$	3.09752	0.451820	0.225910	+	0.974148i	$0.427465\pi$
$48$	0	0
$49$	−1.82586	−0.260837
$50$	0	0
$51$	8.86304	1.24107
$52$	0	0
$53$	0.690667	0.0948704	0.0474352	−	0.998874i	$-0.484895\pi$
$53$	0.690667	0.0948704	0.0474352	+	0.998874i	$0.484895\pi$
$54$	0	0
$55$	−11.1118	−1.49831
$56$	0	0
$57$	0.154241	0.0204297
$58$	0	0
$59$	−9.44782	−1.23000	−0.615000	−	0.788527i	$-0.710844\pi$
$59$	−9.44782	−1.23000	−0.615000	+	0.788527i	$0.710844\pi$
$60$	0	0
$61$	11.2047	1.43461	0.717305	−	0.696759i	$-0.245375\pi$
$61$	11.2047	1.43461	0.717305	+	0.696759i	$0.245375\pi$
$62$	0	0
$63$	−3.91252	−0.492931
$64$	0	0
$65$	−12.2782	−1.52292
$66$	0	0
$67$	−12.1917	−1.48945	−0.744726	−	0.667370i	$-0.767420\pi$
$67$	−12.1917	−1.48945	−0.744726	+	0.667370i	$0.767420\pi$
$68$	0	0
$69$	3.19127	0.384184
$70$	0	0
$71$	−7.39192	−0.877260	−0.438630	−	0.898668i	$-0.644536\pi$
$71$	−7.39192	−0.877260	−0.438630	+	0.898668i	$0.644536\pi$
$72$	0	0
$73$	0.601864	0.0704429	0.0352214	−	0.999380i	$-0.488786\pi$
$73$	0.601864	0.0704429	0.0352214	+	0.999380i	$0.488786\pi$
$74$	0	0
$75$	8.07057	0.931909
$76$	0	0
$77$	8.56199	0.975730
$78$	0	0
$79$	−10.3921	−1.16920	−0.584602	−	0.811320i	$-0.698749\pi$
$79$	−10.3921	−1.16920	−0.584602	+	0.811320i	$0.698749\pi$
$80$	0	0
$81$	−11.2016	−1.24462
$82$	0	0
$83$	10.8034	1.18582	0.592911	−	0.805268i	$-0.297978\pi$
$83$	10.8034	1.18582	0.592911	+	0.805268i	$0.297978\pi$
$84$	0	0
$85$	12.0431	1.30626
$86$	0	0
$87$	−16.8249	−1.80381
$88$	0	0
$89$	−7.29923	−0.773717	−0.386859	−	0.922139i	$-0.626440\pi$
$89$	−7.29923	−0.773717	−0.386859	+	0.922139i	$0.626440\pi$
$90$	0	0
$91$	9.46073	0.991754
$92$	0	0
$93$	−9.28281	−0.962583
$94$	0	0
$95$	0.209582	0.0215027
$96$	0	0
$97$	−13.8744	−1.40873	−0.704367	−	0.709836i	$-0.748769\pi$
$97$	−13.8744	−1.40873	−0.704367	+	0.709836i	$0.748769\pi$
$98$	0	0
$99$	−6.47431	−0.650692
$100$	0	0
$101$	7.20555	0.716979	0.358490	−	0.933534i	$-0.383292\pi$
$101$	7.20555	0.716979	0.358490	+	0.933534i	$0.383292\pi$
$102$	0	0
$103$	14.0056	1.38001	0.690005	−	0.723805i	$-0.257608\pi$
$103$	14.0056	1.38001	0.690005	+	0.723805i	$0.257608\pi$
$104$	0	0
$105$	−14.5888	−1.42372
$106$	0	0
$107$	13.5370	1.30867	0.654336	−	0.756204i	$-0.272948\pi$
$107$	13.5370	1.30867	0.654336	+	0.756204i	$0.272948\pi$
$108$	0	0
$109$	−4.89039	−0.468414	−0.234207	−	0.972187i	$-0.575249\pi$
$109$	−4.89039	−0.468414	−0.234207	+	0.972187i	$0.575249\pi$
$110$	0	0
$111$	6.29225	0.597234
$112$	0	0
$113$	−1.05701	−0.0994349	−0.0497174	−	0.998763i	$-0.515832\pi$
$113$	−1.05701	−0.0994349	−0.0497174	+	0.998763i	$0.515832\pi$
$114$	0	0
$115$	4.33630	0.404362
$116$	0	0
$117$	−7.15390	−0.661379
$118$	0	0
$119$	−9.27960	−0.850659
$120$	0	0
$121$	3.16810	0.288009
$122$	0	0
$123$	−9.58305	−0.864075
$124$	0	0
$125$	−3.79411	−0.339355
$126$	0	0
$127$	6.60494	0.586094	0.293047	−	0.956098i	$-0.405331\pi$
$127$	6.60494	0.586094	0.293047	+	0.956098i	$0.405331\pi$
$128$	0	0
$129$	−11.3067	−0.995503
$130$	0	0
$131$	7.57750	0.662049	0.331025	−	0.943622i	$-0.392606\pi$
$131$	7.57750	0.662049	0.331025	+	0.943622i	$0.392606\pi$
$132$	0	0
$133$	−0.161490	−0.0140030
$134$	0	0
$135$	−8.20915	−0.706532
$136$	0	0
$137$	10.4312	0.891200	0.445600	−	0.895232i	$-0.352990\pi$
$137$	10.4312	0.891200	0.445600	+	0.895232i	$0.352990\pi$
$138$	0	0
$139$	3.53662	0.299972	0.149986	−	0.988688i	$-0.452077\pi$
$139$	3.53662	0.299972	0.149986	+	0.988688i	$0.452077\pi$
$140$	0	0
$141$	6.72956	0.566731
$142$	0	0
$143$	15.6553	1.30916
$144$	0	0
$145$	−22.8616	−1.89855
$146$	0	0
$147$	−3.96680	−0.327176
$148$	0	0
$149$	2.97823	0.243986	0.121993	−	0.992531i	$-0.461071\pi$
$149$	2.97823	0.243986	0.121993	+	0.992531i	$0.461071\pi$
$150$	0	0
$151$	15.5292	1.26375	0.631873	−	0.775072i	$-0.282287\pi$
$151$	15.5292	1.26375	0.631873	+	0.775072i	$0.282287\pi$
$152$	0	0
$153$	7.01693	0.567285
$154$	0	0
$155$	−12.6135	−1.01314
$156$	0	0
$157$	−8.97304	−0.716127	−0.358063	−	0.933697i	$-0.616563\pi$
$157$	−8.97304	−0.716127	−0.358063	+	0.933697i	$0.616563\pi$
$158$	0	0
$159$	1.50052	0.118999
$160$	0	0
$161$	−3.34126	−0.263328
$162$	0	0
$163$	10.8632	0.850872	0.425436	−	0.904988i	$-0.360121\pi$
$163$	10.8632	0.850872	0.425436	+	0.904988i	$0.360121\pi$
$164$	0	0
$165$	−24.1411	−1.87938
$166$	0	0
$167$	−5.38570	−0.416758	−0.208379	−	0.978048i	$-0.566819\pi$
$167$	−5.38570	−0.416758	−0.208379	+	0.978048i	$0.566819\pi$
$168$	0	0
$169$	4.29861	0.330663
$170$	0	0
$171$	0.122114	0.00933827
$172$	0	0
$173$	−11.2394	−0.854514	−0.427257	−	0.904130i	$-0.640520\pi$
$173$	−11.2394	−0.854514	−0.427257	+	0.904130i	$0.640520\pi$
$174$	0	0
$175$	−8.44988	−0.638751
$176$	0	0
$177$	−20.5260	−1.54283
$178$	0	0
$179$	−10.5460	−0.788242	−0.394121	−	0.919059i	$-0.628951\pi$
$179$	−10.5460	−0.788242	−0.394121	+	0.919059i	$0.628951\pi$
$180$	0	0
$181$	−12.0062	−0.892411	−0.446206	−	0.894930i	$-0.647225\pi$
$181$	−12.0062	−0.892411	−0.446206	+	0.894930i	$0.647225\pi$
$182$	0	0
$183$	24.3429	1.79948
$184$	0	0
$185$	8.54991	0.628602
$186$	0	0
$187$	−15.3556	−1.12291
$188$	0	0
$189$	6.32542	0.460107
$190$	0	0
$191$	−15.0034	−1.08561	−0.542804	−	0.839859i	$-0.682637\pi$
$191$	−15.0034	−1.08561	−0.542804	+	0.839859i	$0.682637\pi$
$192$	0	0
$193$	−9.97796	−0.718229	−0.359115	−	0.933293i	$-0.616921\pi$
$193$	−9.97796	−0.718229	−0.359115	+	0.933293i	$0.616921\pi$
$194$	0	0
$195$	−26.6751	−1.91024
$196$	0	0
$197$	13.1347	0.935810	0.467905	−	0.883779i	$-0.345009\pi$
$197$	13.1347	0.935810	0.467905	+	0.883779i	$0.345009\pi$
$198$	0	0
$199$	19.2859	1.36714	0.683570	−	0.729885i	$-0.260426\pi$
$199$	19.2859	1.36714	0.683570	+	0.729885i	$0.260426\pi$
$200$	0	0
$201$	−26.4872	−1.86827
$202$	0	0
$203$	17.6156	1.23637
$204$	0	0
$205$	−13.0214	−0.909457
$206$	0	0
$207$	2.52656	0.175608
$208$	0	0
$209$	−0.267228	−0.0184846
$210$	0	0
$211$	−13.6507	−0.939750	−0.469875	−	0.882733i	$-0.655701\pi$
$211$	−13.6507	−0.939750	−0.469875	+	0.882733i	$0.655701\pi$
$212$	0	0
$213$	−16.0594	−1.10037
$214$	0	0
$215$	−15.3636	−1.04779
$216$	0	0
$217$	9.71910	0.659775
$218$	0	0
$219$	1.30759	0.0883586
$220$	0	0
$221$	−16.9674	−1.14135
$222$	0	0
$223$	15.9032	1.06496	0.532478	−	0.846444i	$-0.321261\pi$
$223$	15.9032	1.06496	0.532478	+	0.846444i	$0.321261\pi$
$224$	0	0
$225$	6.38953	0.425969
$226$	0	0
$227$	13.3751	0.887739	0.443869	−	0.896092i	$-0.353605\pi$
$227$	13.3751	0.887739	0.443869	+	0.896092i	$0.353605\pi$
$228$	0	0
$229$	−8.12094	−0.536647	−0.268323	−	0.963329i	$-0.586470\pi$
$229$	−8.12094	−0.536647	−0.268323	+	0.963329i	$0.586470\pi$
$230$	0	0
$231$	18.6015	1.22389
$232$	0	0
$233$	−16.8419	−1.10335	−0.551673	−	0.834060i	$-0.686010\pi$
$233$	−16.8419	−1.10335	−0.551673	+	0.834060i	$0.686010\pi$
$234$	0	0
$235$	9.14412	0.596497
$236$	0	0
$237$	−22.5775	−1.46657
$238$	0	0
$239$	−23.4014	−1.51371	−0.756857	−	0.653581i	$-0.773266\pi$
$239$	−23.4014	−1.51371	−0.756857	+	0.653581i	$0.773266\pi$
$240$	0	0
$241$	11.4351	0.736602	0.368301	−	0.929707i	$-0.379940\pi$
$241$	11.4351	0.736602	0.368301	+	0.929707i	$0.379940\pi$
$242$	0	0
$243$	−15.9937	−1.02600
$244$	0	0
$245$	−5.39008	−0.344360
$246$	0	0
$247$	−0.295279	−0.0187881
$248$	0	0
$249$	23.4710	1.48741
$250$	0	0
$251$	−23.8822	−1.50743	−0.753716	−	0.657200i	$-0.771741\pi$
$251$	−23.8822	−1.50743	−0.753716	+	0.657200i	$0.771741\pi$
$252$	0	0
$253$	−5.52901	−0.347606
$254$	0	0
$255$	26.1644	1.63848
$256$	0	0
$257$	−28.5971	−1.78384	−0.891918	−	0.452196i	$-0.850641\pi$
$257$	−28.5971	−1.78384	−0.891918	+	0.452196i	$0.850641\pi$
$258$	0	0
$259$	−6.58799	−0.409358
$260$	0	0
$261$	−13.3204	−0.824510
$262$	0	0
$263$	−11.2589	−0.694254	−0.347127	−	0.937818i	$-0.612843\pi$
$263$	−11.2589	−0.694254	−0.347127	+	0.937818i	$0.612843\pi$
$264$	0	0
$265$	2.03890	0.125249
$266$	0	0
$267$	−15.8581	−0.970497
$268$	0	0
$269$	−13.5146	−0.824000	−0.412000	−	0.911184i	$-0.635170\pi$
$269$	−13.5146	−0.824000	−0.412000	+	0.911184i	$0.635170\pi$
$270$	0	0
$271$	−24.4097	−1.48279	−0.741393	−	0.671071i	$-0.765834\pi$
$271$	−24.4097	−1.48279	−0.741393	+	0.671071i	$0.765834\pi$
$272$	0	0
$273$	20.5540	1.24399
$274$	0	0
$275$	−13.9826	−0.843181
$276$	0	0
$277$	23.6156	1.41893	0.709463	−	0.704743i	$-0.248938\pi$
$277$	23.6156	1.41893	0.709463	+	0.704743i	$0.248938\pi$
$278$	0	0
$279$	−7.34927	−0.439989
$280$	0	0
$281$	12.0594	0.719403	0.359701	−	0.933067i	$-0.382879\pi$
$281$	12.0594	0.719403	0.359701	+	0.933067i	$0.382879\pi$
$282$	0	0
$283$	5.53198	0.328842	0.164421	−	0.986390i	$-0.447424\pi$
$283$	5.53198	0.328842	0.164421	+	0.986390i	$0.447424\pi$
$284$	0	0
$285$	0.455331	0.0269715
$286$	0	0
$287$	10.0335	0.592256
$288$	0	0
$289$	−0.357448	−0.0210263
$290$	0	0
$291$	−30.1431	−1.76702
$292$	0	0
$293$	−30.8590	−1.80280	−0.901401	−	0.432984i	$-0.857461\pi$
$293$	−30.8590	−1.80280	−0.901401	+	0.432984i	$0.857461\pi$
$294$	0	0
$295$	−27.8907	−1.62386
$296$	0	0
$297$	10.4671	0.607363
$298$	0	0
$299$	−6.10938	−0.353315
$300$	0	0
$301$	11.8382	0.682340
$302$	0	0
$303$	15.6545	0.899329
$304$	0	0
$305$	33.0771	1.89399
$306$	0	0
$307$	−9.18514	−0.524224	−0.262112	−	0.965037i	$-0.584419\pi$
$307$	−9.18514	−0.524224	−0.262112	+	0.965037i	$0.584419\pi$
$308$	0	0
$309$	30.4280	1.73099
$310$	0	0
$311$	6.86295	0.389162	0.194581	−	0.980886i	$-0.437665\pi$
$311$	6.86295	0.389162	0.194581	+	0.980886i	$0.437665\pi$
$312$	0	0
$313$	20.7676	1.17385	0.586927	−	0.809640i	$-0.300338\pi$
$313$	20.7676	1.17385	0.586927	+	0.809640i	$0.300338\pi$
$314$	0	0
$315$	−11.5501	−0.650772
$316$	0	0
$317$	25.2237	1.41670	0.708351	−	0.705860i	$-0.249439\pi$
$317$	25.2237	1.41670	0.708351	+	0.705860i	$0.249439\pi$
$318$	0	0
$319$	29.1497	1.63207
$320$	0	0
$321$	29.4100	1.64151
$322$	0	0
$323$	0.289626	0.0161152
$324$	0	0
$325$	−15.4503	−0.857029
$326$	0	0
$327$	−10.6247	−0.587546
$328$	0	0
$329$	−7.04585	−0.388450
$330$	0	0
$331$	22.0207	1.21037	0.605183	−	0.796086i	$-0.293100\pi$
$331$	22.0207	1.21037	0.605183	+	0.796086i	$0.293100\pi$
$332$	0	0
$333$	4.98162	0.272991
$334$	0	0
$335$	−35.9908	−1.96639
$336$	0	0
$337$	−16.7935	−0.914799	−0.457400	−	0.889261i	$-0.651219\pi$
$337$	−16.7935	−0.914799	−0.457400	+	0.889261i	$0.651219\pi$
$338$	0	0
$339$	−2.29642	−0.124724
$340$	0	0
$341$	16.0828	0.870934
$342$	0	0
$343$	20.0760	1.08400
$344$	0	0
$345$	9.42089	0.507204
$346$	0	0
$347$	36.5786	1.96364	0.981821	−	0.189810i	$-0.0607871\pi$
$347$	36.5786	1.96364	0.981821	+	0.189810i	$0.0607871\pi$
$348$	0	0
$349$	25.0074	1.33862	0.669308	−	0.742985i	$-0.266590\pi$
$349$	25.0074	1.33862	0.669308	+	0.742985i	$0.266590\pi$
$350$	0	0
$351$	11.5658	0.617337
$352$	0	0
$353$	−6.22531	−0.331340	−0.165670	−	0.986181i	$-0.552979\pi$
$353$	−6.22531	−0.331340	−0.165670	+	0.986181i	$0.552979\pi$
$354$	0	0
$355$	−21.8215	−1.15817
$356$	0	0
$357$	−20.1605	−1.06701
$358$	0	0
$359$	2.01000	0.106084	0.0530418	−	0.998592i	$-0.483108\pi$
$359$	2.01000	0.106084	0.0530418	+	0.998592i	$0.483108\pi$
$360$	0	0
$361$	−18.9950	−0.999735
$362$	0	0
$363$	6.88290	0.361259
$364$	0	0
$365$	1.77675	0.0929994
$366$	0	0
$367$	24.3677	1.27198	0.635992	−	0.771696i	$-0.280591\pi$
$367$	24.3677	1.27198	0.635992	+	0.771696i	$0.280591\pi$
$368$	0	0
$369$	−7.58698	−0.394962
$370$	0	0
$371$	−1.57104	−0.0815645
$372$	0	0
$373$	7.52067	0.389405	0.194703	−	0.980862i	$-0.437626\pi$
$373$	7.52067	0.389405	0.194703	+	0.980862i	$0.437626\pi$
$374$	0	0
$375$	−8.24294	−0.425664
$376$	0	0
$377$	32.2095	1.65888
$378$	0	0
$379$	−9.49357	−0.487652	−0.243826	−	0.969819i	$-0.578403\pi$
$379$	−9.49357	−0.487652	−0.243826	+	0.969819i	$0.578403\pi$
$380$	0	0
$381$	14.3497	0.735156
$382$	0	0
$383$	1.00480	0.0513428	0.0256714	−	0.999670i	$-0.491828\pi$
$383$	1.00480	0.0513428	0.0256714	+	0.999670i	$0.491828\pi$
$384$	0	0
$385$	25.2757	1.28817
$386$	0	0
$387$	−8.95163	−0.455037
$388$	0	0
$389$	5.22571	0.264954	0.132477	−	0.991186i	$-0.457707\pi$
$389$	5.22571	0.264954	0.132477	+	0.991186i	$0.457707\pi$
$390$	0	0
$391$	5.99241	0.303049
$392$	0	0
$393$	16.4626	0.830429
$394$	0	0
$395$	−30.6783	−1.54359
$396$	0	0
$397$	20.0977	1.00868	0.504338	−	0.863506i	$-0.331736\pi$
$397$	20.0977	1.00868	0.504338	+	0.863506i	$0.331736\pi$
$398$	0	0
$399$	−0.350848	−0.0175644
$400$	0	0
$401$	7.22647	0.360873	0.180436	−	0.983587i	$-0.442249\pi$
$401$	7.22647	0.360873	0.180436	+	0.983587i	$0.442249\pi$
$402$	0	0
$403$	17.7710	0.885238
$404$	0	0
$405$	−33.0680	−1.64316
$406$	0	0
$407$	−10.9016	−0.540371
$408$	0	0
$409$	25.2004	1.24608	0.623039	−	0.782191i	$-0.285898\pi$
$409$	25.2004	1.24608	0.623039	+	0.782191i	$0.285898\pi$
$410$	0	0
$411$	22.6625	1.11786
$412$	0	0
$413$	21.4907	1.05749
$414$	0	0
$415$	31.8924	1.56553
$416$	0	0
$417$	7.68354	0.376265
$418$	0	0
$419$	29.2968	1.43124	0.715621	−	0.698488i	$-0.246144\pi$
$419$	29.2968	1.43124	0.715621	+	0.698488i	$0.246144\pi$
$420$	0	0
$421$	−25.1418	−1.22534	−0.612669	−	0.790340i	$-0.709904\pi$
$421$	−25.1418	−1.22534	−0.612669	+	0.790340i	$0.709904\pi$
$422$	0	0
$423$	5.32784	0.259049
$424$	0	0
$425$	15.1545	0.735101
$426$	0	0
$427$	−25.4870	−1.23340
$428$	0	0
$429$	34.0121	1.64212
$430$	0	0
$431$	−13.1839	−0.635047	−0.317524	−	0.948250i	$-0.602851\pi$
$431$	−13.1839	−0.635047	−0.317524	+	0.948250i	$0.602851\pi$
$432$	0	0
$433$	−20.5916	−0.989570	−0.494785	−	0.869015i	$-0.664753\pi$
$433$	−20.5916	−0.989570	−0.494785	+	0.869015i	$0.664753\pi$
$434$	0	0
$435$	−49.6683	−2.38141
$436$	0	0
$437$	0.104284	0.00498859
$438$	0	0
$439$	−8.48073	−0.404763	−0.202381	−	0.979307i	$-0.564868\pi$
$439$	−8.48073	−0.404763	−0.202381	+	0.979307i	$0.564868\pi$
$440$	0	0
$441$	−3.14054	−0.149550
$442$	0	0
$443$	−19.4282	−0.923062	−0.461531	−	0.887124i	$-0.652700\pi$
$443$	−19.4282	−0.923062	−0.461531	+	0.887124i	$0.652700\pi$
$444$	0	0
$445$	−21.5479	−1.02147
$446$	0	0
$447$	6.47040	0.306039
$448$	0	0
$449$	5.68914	0.268487	0.134244	−	0.990948i	$-0.457140\pi$
$449$	5.68914	0.268487	0.134244	+	0.990948i	$0.457140\pi$
$450$	0	0
$451$	16.6030	0.781806
$452$	0	0
$453$	33.7381	1.58515
$454$	0	0
$455$	27.9288	1.30932
$456$	0	0
$457$	−3.43175	−0.160530	−0.0802652	−	0.996774i	$-0.525577\pi$
$457$	−3.43175	−0.160530	−0.0802652	+	0.996774i	$0.525577\pi$
$458$	0	0
$459$	−11.3444	−0.529510
$460$	0	0
$461$	−31.7501	−1.47875	−0.739375	−	0.673294i	$-0.764879\pi$
$461$	−31.7501	−1.47875	−0.739375	+	0.673294i	$0.764879\pi$
$462$	0	0
$463$	36.3790	1.69067	0.845337	−	0.534233i	$-0.179400\pi$
$463$	36.3790	1.69067	0.845337	+	0.534233i	$0.179400\pi$
$464$	0	0
$465$	−27.4036	−1.27081
$466$	0	0
$467$	−25.4963	−1.17983	−0.589913	−	0.807467i	$-0.700838\pi$
$467$	−25.4963	−1.17983	−0.589913	+	0.807467i	$0.700838\pi$
$468$	0	0
$469$	27.7321	1.28055
$470$	0	0
$471$	−19.4945	−0.898260
$472$	0	0
$473$	19.5894	0.900721
$474$	0	0
$475$	0.263729	0.0121007
$476$	0	0
$477$	1.18797	0.0543935
$478$	0	0
$479$	28.5631	1.30508	0.652541	−	0.757753i	$-0.273703\pi$
$479$	28.5631	1.30508	0.652541	+	0.757753i	$0.273703\pi$
$480$	0	0
$481$	−12.0459	−0.549246
$482$	0	0
$483$	−7.25911	−0.330301
$484$	0	0
$485$	−40.9584	−1.85982
$486$	0	0
$487$	31.5828	1.43115	0.715576	−	0.698535i	$-0.246165\pi$
$487$	31.5828	1.43115	0.715576	+	0.698535i	$0.246165\pi$
$488$	0	0
$489$	23.6010	1.06728
$490$	0	0
$491$	−15.6934	−0.708234	−0.354117	−	0.935201i	$-0.615219\pi$
$491$	−15.6934	−0.708234	−0.354117	+	0.935201i	$0.615219\pi$
$492$	0	0
$493$	−31.5928	−1.42287
$494$	0	0
$495$	−19.1127	−0.859050
$496$	0	0
$497$	16.8142	0.754221
$498$	0	0
$499$	−0.419539	−0.0187811	−0.00939056	−	0.999956i	$-0.502989\pi$
$499$	−0.419539	−0.0187811	−0.00939056	+	0.999956i	$0.502989\pi$
$500$	0	0
$501$	−11.7008	−0.522752
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	21.2714	0.946563
$506$	0	0
$507$	9.33901	0.414760
$508$	0	0
$509$	24.0019	1.06386	0.531932	−	0.846787i	$-0.321466\pi$
$509$	24.0019	1.06386	0.531932	+	0.846787i	$0.321466\pi$
$510$	0	0
$511$	−1.36904	−0.0605630
$512$	0	0
$513$	−0.197423	−0.00871643
$514$	0	0
$515$	41.3455	1.82190
$516$	0	0
$517$	−11.6592	−0.512772
$518$	0	0
$519$	−24.4183	−1.07184
$520$	0	0
$521$	−16.1644	−0.708173	−0.354087	−	0.935213i	$-0.615208\pi$
$521$	−16.1644	−0.708173	−0.354087	+	0.935213i	$0.615208\pi$
$522$	0	0
$523$	−9.74258	−0.426013	−0.213007	−	0.977051i	$-0.568326\pi$
$523$	−9.74258	−0.426013	−0.213007	+	0.977051i	$0.568326\pi$
$524$	0	0
$525$	−18.3579	−0.801205
$526$	0	0
$527$	−17.4308	−0.759297
$528$	0	0
$529$	−20.8423	−0.906189
$530$	0	0
$531$	−16.2506	−0.705215
$532$	0	0
$533$	18.3458	0.794645
$534$	0	0
$535$	39.9623	1.72772
$536$	0	0
$537$	−22.9118	−0.988716
$538$	0	0
$539$	6.87263	0.296025
$540$	0	0
$541$	−44.9960	−1.93453	−0.967265	−	0.253769i	$-0.918330\pi$
$541$	−44.9960	−1.93453	−0.967265	+	0.253769i	$0.918330\pi$
$542$	0	0
$543$	−26.0842	−1.11938
$544$	0	0
$545$	−14.4368	−0.618405
$546$	0	0
$547$	16.5024	0.705591	0.352796	−	0.935700i	$-0.385231\pi$
$547$	16.5024	0.705591	0.352796	+	0.935700i	$0.385231\pi$
$548$	0	0
$549$	19.2724	0.822527
$550$	0	0
$551$	−0.549801	−0.0234223
$552$	0	0
$553$	23.6387	1.00522
$554$	0	0
$555$	18.5752	0.788474
$556$	0	0
$557$	32.0419	1.35766	0.678830	−	0.734296i	$-0.262487\pi$
$557$	32.0419	1.35766	0.678830	+	0.734296i	$0.262487\pi$
$558$	0	0
$559$	21.6456	0.915513
$560$	0	0
$561$	−33.3609	−1.40850
$562$	0	0
$563$	−23.7630	−1.00149	−0.500745	−	0.865595i	$-0.666941\pi$
$563$	−23.7630	−1.00149	−0.500745	+	0.865595i	$0.666941\pi$
$564$	0	0
$565$	−3.12037	−0.131275
$566$	0	0
$567$	25.4799	1.07006
$568$	0	0
$569$	−28.9854	−1.21513	−0.607566	−	0.794269i	$-0.707854\pi$
$569$	−28.9854	−1.21513	−0.607566	+	0.794269i	$0.707854\pi$
$570$	0	0
$571$	−14.2743	−0.597361	−0.298680	−	0.954353i	$-0.596546\pi$
$571$	−14.2743	−0.597361	−0.298680	+	0.954353i	$0.596546\pi$
$572$	0	0
$573$	−32.5959	−1.36171
$574$	0	0
$575$	5.45661	0.227556
$576$	0	0
$577$	5.43424	0.226230	0.113115	−	0.993582i	$-0.463917\pi$
$577$	5.43424	0.226230	0.113115	+	0.993582i	$0.463917\pi$
$578$	0	0
$579$	−21.6778	−0.900897
$580$	0	0
$581$	−24.5741	−1.01951
$582$	0	0
$583$	−2.59971	−0.107669
$584$	0	0
$585$	−21.1189	−0.873158
$586$	0	0
$587$	10.1865	0.420441	0.210221	−	0.977654i	$-0.432582\pi$
$587$	10.1865	0.420441	0.210221	+	0.977654i	$0.432582\pi$
$588$	0	0
$589$	−0.303343	−0.0124990
$590$	0	0
$591$	28.5360	1.17382
$592$	0	0
$593$	5.33079	0.218909	0.109455	−	0.993992i	$-0.465090\pi$
$593$	5.33079	0.218909	0.109455	+	0.993992i	$0.465090\pi$
$594$	0	0
$595$	−27.3941	−1.12305
$596$	0	0
$597$	41.8998	1.71485
$598$	0	0
$599$	−33.0217	−1.34923	−0.674615	−	0.738169i	$-0.735690\pi$
$599$	−33.0217	−1.34923	−0.674615	+	0.738169i	$0.735690\pi$
$600$	0	0
$601$	9.28797	0.378864	0.189432	−	0.981894i	$-0.439335\pi$
$601$	9.28797	0.378864	0.189432	+	0.981894i	$0.439335\pi$
$602$	0	0
$603$	−20.9701	−0.853970
$604$	0	0
$605$	9.35248	0.380232
$606$	0	0
$607$	24.1958	0.982076	0.491038	−	0.871138i	$-0.336618\pi$
$607$	24.1958	0.982076	0.491038	+	0.871138i	$0.336618\pi$
$608$	0	0
$609$	38.2711	1.55082
$610$	0	0
$611$	−12.8831	−0.521194
$612$	0	0
$613$	36.6839	1.48165	0.740825	−	0.671698i	$-0.234435\pi$
$613$	36.6839	1.48165	0.740825	+	0.671698i	$0.234435\pi$
$614$	0	0
$615$	−28.2899	−1.14076
$616$	0	0
$617$	16.9141	0.680935	0.340468	−	0.940256i	$-0.389415\pi$
$617$	16.9141	0.680935	0.340468	+	0.940256i	$0.389415\pi$
$618$	0	0
$619$	36.7149	1.47570	0.737849	−	0.674966i	$-0.235842\pi$
$619$	36.7149	1.47570	0.737849	+	0.674966i	$0.235842\pi$
$620$	0	0
$621$	−4.08472	−0.163914
$622$	0	0
$623$	16.6034	0.665200
$624$	0	0
$625$	−29.7743	−1.19097
$626$	0	0
$627$	−0.580571	−0.0231858
$628$	0	0
$629$	11.8153	0.471106
$630$	0	0
$631$	32.7698	1.30455	0.652273	−	0.757984i	$-0.273816\pi$
$631$	32.7698	1.30455	0.652273	+	0.757984i	$0.273816\pi$
$632$	0	0
$633$	−29.6569	−1.17876
$634$	0	0
$635$	19.4983	0.773767
$636$	0	0
$637$	7.59404	0.300887
$638$	0	0
$639$	−12.7144	−0.502973
$640$	0	0
$641$	12.7505	0.503615	0.251807	−	0.967777i	$-0.418975\pi$
$641$	12.7505	0.503615	0.251807	+	0.967777i	$0.418975\pi$
$642$	0	0
$643$	−39.3367	−1.55129	−0.775644	−	0.631171i	$-0.782575\pi$
$643$	−39.3367	−1.55129	−0.775644	+	0.631171i	$0.782575\pi$
$644$	0	0
$645$	−33.3784	−1.31427
$646$	0	0
$647$	36.0296	1.41647	0.708236	−	0.705976i	$-0.249491\pi$
$647$	36.0296	1.41647	0.708236	+	0.705976i	$0.249491\pi$
$648$	0	0
$649$	35.5621	1.39593
$650$	0	0
$651$	21.1154	0.827576
$652$	0	0
$653$	4.16702	0.163068	0.0815341	−	0.996671i	$-0.474018\pi$
$653$	4.16702	0.163068	0.0815341	+	0.996671i	$0.474018\pi$
$654$	0	0
$655$	22.3694	0.874044
$656$	0	0
$657$	1.03523	0.0403881
$658$	0	0
$659$	−36.5938	−1.42549	−0.712746	−	0.701422i	$-0.752549\pi$
$659$	−36.5938	−1.42549	−0.712746	+	0.701422i	$0.752549\pi$
$660$	0	0
$661$	19.9987	0.777858	0.388929	−	0.921268i	$-0.372845\pi$
$661$	19.9987	0.777858	0.388929	+	0.921268i	$0.372845\pi$
$662$	0	0
$663$	−36.8628	−1.43163
$664$	0	0
$665$	−0.476731	−0.0184869
$666$	0	0
$667$	−11.3755	−0.440461
$668$	0	0
$669$	34.5507	1.33581
$670$	0	0
$671$	−42.1750	−1.62815
$672$	0	0
$673$	−21.6791	−0.835669	−0.417834	−	0.908523i	$-0.637211\pi$
$673$	−21.6791	−0.835669	−0.417834	+	0.908523i	$0.637211\pi$
$674$	0	0
$675$	−10.3300	−0.397604
$676$	0	0
$677$	4.00327	0.153858	0.0769291	−	0.997037i	$-0.475489\pi$
$677$	4.00327	0.153858	0.0769291	+	0.997037i	$0.475489\pi$
$678$	0	0
$679$	31.5598	1.21115
$680$	0	0
$681$	29.0583	1.11352
$682$	0	0
$683$	−3.38705	−0.129602	−0.0648009	−	0.997898i	$-0.520641\pi$
$683$	−3.38705	−0.129602	−0.0648009	+	0.997898i	$0.520641\pi$
$684$	0	0
$685$	30.7938	1.17657
$686$	0	0
$687$	−17.6433	−0.673132
$688$	0	0
$689$	−2.87260	−0.109437
$690$	0	0
$691$	−7.88938	−0.300126	−0.150063	−	0.988676i	$-0.547948\pi$
$691$	−7.88938	−0.300126	−0.150063	+	0.988676i	$0.547948\pi$
$692$	0	0
$693$	14.7269	0.559430
$694$	0	0
$695$	10.4404	0.396026
$696$	0	0
$697$	−17.9946	−0.681593
$698$	0	0
$699$	−36.5900	−1.38396
$700$	0	0
$701$	45.0481	1.70144	0.850721	−	0.525618i	$-0.176166\pi$
$701$	45.0481	1.70144	0.850721	+	0.525618i	$0.176166\pi$
$702$	0	0
$703$	0.205618	0.00775502
$704$	0	0
$705$	19.8662	0.748204
$706$	0	0
$707$	−16.3903	−0.616420
$708$	0	0
$709$	47.7983	1.79510	0.897552	−	0.440909i	$-0.145344\pi$
$709$	47.7983	1.79510	0.897552	+	0.440909i	$0.145344\pi$
$710$	0	0
$711$	−17.8748	−0.670357
$712$	0	0
$713$	−6.27622	−0.235046
$714$	0	0
$715$	46.2157	1.72837
$716$	0	0
$717$	−50.8411	−1.89870
$718$	0	0
$719$	−16.8127	−0.627008	−0.313504	−	0.949587i	$-0.601503\pi$
$719$	−16.8127	−0.627008	−0.313504	+	0.949587i	$0.601503\pi$
$720$	0	0
$721$	−31.8581	−1.18646
$722$	0	0
$723$	24.8436	0.923943
$724$	0	0
$725$	−28.7680	−1.06842
$726$	0	0
$727$	−24.0606	−0.892357	−0.446178	−	0.894944i	$-0.647215\pi$
$727$	−24.0606	−0.892357	−0.446178	+	0.894944i	$0.647215\pi$
$728$	0	0
$729$	−1.14269	−0.0423218
$730$	0	0
$731$	−21.2312	−0.785265
$732$	0	0
$733$	25.0347	0.924678	0.462339	−	0.886703i	$-0.347010\pi$
$733$	25.0347	0.924678	0.462339	+	0.886703i	$0.347010\pi$
$734$	0	0
$735$	−11.7103	−0.431941
$736$	0	0
$737$	45.8902	1.69039
$738$	0	0
$739$	−47.5982	−1.75093	−0.875464	−	0.483283i	$-0.839444\pi$
$739$	−47.5982	−1.75093	−0.875464	+	0.483283i	$0.839444\pi$
$740$	0	0
$741$	−0.641512	−0.0235665
$742$	0	0
$743$	−24.9658	−0.915905	−0.457953	−	0.888977i	$-0.651417\pi$
$743$	−24.9658	−0.915905	−0.457953	+	0.888977i	$0.651417\pi$
$744$	0	0
$745$	8.79197	0.322113
$746$	0	0
$747$	18.5822	0.679886
$748$	0	0
$749$	−30.7923	−1.12512
$750$	0	0
$751$	−46.0437	−1.68016	−0.840078	−	0.542465i	$-0.817491\pi$
$751$	−46.0437	−1.68016	−0.840078	+	0.542465i	$0.817491\pi$
$752$	0	0
$753$	−51.8857	−1.89082
$754$	0	0
$755$	45.8433	1.66841
$756$	0	0
$757$	−18.0158	−0.654797	−0.327398	−	0.944886i	$-0.606172\pi$
$757$	−18.0158	−0.654797	−0.327398	+	0.944886i	$0.606172\pi$
$758$	0	0
$759$	−12.0121	−0.436013
$760$	0	0
$761$	24.2402	0.878706	0.439353	−	0.898315i	$-0.355208\pi$
$761$	24.2402	0.878706	0.439353	+	0.898315i	$0.355208\pi$
$762$	0	0
$763$	11.1240	0.402717
$764$	0	0
$765$	20.7145	0.748936
$766$	0	0
$767$	39.2950	1.41886
$768$	0	0
$769$	−52.7055	−1.90061	−0.950305	−	0.311322i	$-0.899228\pi$
$769$	−52.7055	−1.90061	−0.950305	+	0.311322i	$0.899228\pi$
$770$	0	0
$771$	−62.1290	−2.23752
$772$	0	0
$773$	20.8695	0.750625	0.375312	−	0.926898i	$-0.377535\pi$
$773$	20.8695	0.750625	0.375312	+	0.926898i	$0.377535\pi$
$774$	0	0
$775$	−15.8722	−0.570148
$776$	0	0
$777$	−14.3128	−0.513470
$778$	0	0
$779$	−0.313154	−0.0112199
$780$	0	0
$781$	27.8236	0.995607
$782$	0	0
$783$	21.5352	0.769606
$784$	0	0
$785$	−26.4891	−0.945438
$786$	0	0
$787$	5.10961	0.182138	0.0910690	−	0.995845i	$-0.470972\pi$
$787$	5.10961	0.182138	0.0910690	+	0.995845i	$0.470972\pi$
$788$	0	0
$789$	−24.4607	−0.870824
$790$	0	0
$791$	2.40435	0.0854887
$792$	0	0
$793$	−46.6020	−1.65489
$794$	0	0
$795$	4.42965	0.157103
$796$	0	0
$797$	−43.8568	−1.55349	−0.776744	−	0.629817i	$-0.783130\pi$
$797$	−43.8568	−1.55349	−0.776744	+	0.629817i	$0.783130\pi$
$798$	0	0
$799$	12.6364	0.447044
$800$	0	0
$801$	−12.5549	−0.443607
$802$	0	0
$803$	−2.26545	−0.0799460
$804$	0	0
$805$	−9.86367	−0.347649
$806$	0	0
$807$	−29.3613	−1.03357
$808$	0	0
$809$	−19.9817	−0.702520	−0.351260	−	0.936278i	$-0.614247\pi$
$809$	−19.9817	−0.702520	−0.351260	+	0.936278i	$0.614247\pi$
$810$	0	0
$811$	42.1392	1.47971	0.739855	−	0.672767i	$-0.234894\pi$
$811$	42.1392	1.47971	0.739855	+	0.672767i	$0.234894\pi$
$812$	0	0
$813$	−53.0317	−1.85990
$814$	0	0
$815$	32.0691	1.12333
$816$	0	0
$817$	−0.369481	−0.0129265
$818$	0	0
$819$	16.2728	0.568617
$820$	0	0
$821$	31.5529	1.10120	0.550601	−	0.834768i	$-0.314398\pi$
$821$	31.5529	1.10120	0.550601	+	0.834768i	$0.314398\pi$
$822$	0	0
$823$	−33.2212	−1.15802	−0.579010	−	0.815321i	$-0.696561\pi$
$823$	−33.2212	−1.15802	−0.579010	+	0.815321i	$0.696561\pi$
$824$	0	0
$825$	−30.3781	−1.05763
$826$	0	0
$827$	−2.87330	−0.0999144	−0.0499572	−	0.998751i	$-0.515908\pi$
$827$	−2.87330	−0.0999144	−0.0499572	+	0.998751i	$0.515908\pi$
$828$	0	0
$829$	0.692198	0.0240410	0.0120205	−	0.999928i	$-0.496174\pi$
$829$	0.692198	0.0240410	0.0120205	+	0.999928i	$0.496174\pi$
$830$	0	0
$831$	51.3065	1.77980
$832$	0	0
$833$	−7.44865	−0.258080
$834$	0	0
$835$	−15.8990	−0.550208
$836$	0	0
$837$	11.8817	0.410690
$838$	0	0
$839$	−47.9433	−1.65519	−0.827594	−	0.561328i	$-0.810291\pi$
$839$	−47.9433	−1.65519	−0.827594	+	0.561328i	$0.810291\pi$
$840$	0	0
$841$	30.9732	1.06804
$842$	0	0
$843$	26.1998	0.902369
$844$	0	0
$845$	12.6898	0.436544
$846$	0	0
$847$	−7.20639	−0.247615
$848$	0	0
$849$	12.0186	0.412476
$850$	0	0
$851$	4.25427	0.145835
$852$	0	0
$853$	0.623385	0.0213443	0.0106721	−	0.999943i	$-0.496603\pi$
$853$	0.623385	0.0213443	0.0106721	+	0.999943i	$0.496603\pi$
$854$	0	0
$855$	0.360489	0.0123285
$856$	0	0
$857$	4.21547	0.143998	0.0719989	−	0.997405i	$-0.477062\pi$
$857$	4.21547	0.143998	0.0719989	+	0.997405i	$0.477062\pi$
$858$	0	0
$859$	10.2033	0.348134	0.174067	−	0.984734i	$-0.444309\pi$
$859$	10.2033	0.348134	0.174067	+	0.984734i	$0.444309\pi$
$860$	0	0
$861$	21.7983	0.742885
$862$	0	0
$863$	8.16555	0.277959	0.138979	−	0.990295i	$-0.455618\pi$
$863$	8.16555	0.277959	0.138979	+	0.990295i	$0.455618\pi$
$864$	0	0
$865$	−33.1795	−1.12814
$866$	0	0
$867$	−0.776578	−0.0263740
$868$	0	0
$869$	39.1165	1.32694
$870$	0	0
$871$	50.7072	1.71815
$872$	0	0
$873$	−23.8645	−0.807691
$874$	0	0
$875$	8.63036	0.291759
$876$	0	0
$877$	27.9994	0.945471	0.472736	−	0.881204i	$-0.343267\pi$
$877$	27.9994	0.945471	0.472736	+	0.881204i	$0.343267\pi$
$878$	0	0
$879$	−67.0432	−2.26131
$880$	0	0
$881$	−38.1167	−1.28418	−0.642092	−	0.766627i	$-0.721933\pi$
$881$	−38.1167	−1.28418	−0.642092	+	0.766627i	$0.721933\pi$
$882$	0	0
$883$	−25.8919	−0.871333	−0.435667	−	0.900108i	$-0.643487\pi$
$883$	−25.8919	−0.871333	−0.435667	+	0.900108i	$0.643487\pi$
$884$	0	0
$885$	−60.5943	−2.03686
$886$	0	0
$887$	39.0846	1.31233	0.656167	−	0.754616i	$-0.272177\pi$
$887$	39.0846	1.31233	0.656167	+	0.754616i	$0.272177\pi$
$888$	0	0
$889$	−15.0241	−0.503892
$890$	0	0
$891$	42.1634	1.41253
$892$	0	0
$893$	0.219908	0.00735894
$894$	0	0
$895$	−31.1325	−1.04064
$896$	0	0
$897$	−13.2730	−0.443173
$898$	0	0
$899$	33.0891	1.10358
$900$	0	0
$901$	2.81760	0.0938677
$902$	0	0
$903$	25.7192	0.855880
$904$	0	0
$905$	−35.4431	−1.17817
$906$	0	0
$907$	0.299783	0.00995413	0.00497706	−	0.999988i	$-0.498416\pi$
$907$	0.299783	0.00995413	0.00497706	+	0.999988i	$0.498416\pi$
$908$	0	0
$909$	12.3938	0.411077
$910$	0	0
$911$	34.4979	1.14297	0.571484	−	0.820613i	$-0.306368\pi$
$911$	34.4979	1.14297	0.571484	+	0.820613i	$0.306368\pi$
$912$	0	0
$913$	−40.6644	−1.34580
$914$	0	0
$915$	71.8620	2.37569
$916$	0	0
$917$	−17.2363	−0.569194
$918$	0	0
$919$	14.0910	0.464820	0.232410	−	0.972618i	$-0.425339\pi$
$919$	14.0910	0.464820	0.232410	+	0.972618i	$0.425339\pi$
$920$	0	0
$921$	−19.9553	−0.657550
$922$	0	0
$923$	30.7442	1.01196
$924$	0	0
$925$	10.7588	0.353748
$926$	0	0
$927$	24.0901	0.791222
$928$	0	0
$929$	30.3466	0.995639	0.497820	−	0.867281i	$-0.334134\pi$
$929$	30.3466	0.995639	0.497820	+	0.867281i	$0.334134\pi$
$930$	0	0
$931$	−0.129627	−0.00424834
$932$	0	0
$933$	14.9102	0.488138
$934$	0	0
$935$	−45.3308	−1.48248
$936$	0	0
$937$	−50.3209	−1.64391	−0.821956	−	0.569551i	$-0.807117\pi$
$937$	−50.3209	−1.64391	−0.821956	+	0.569551i	$0.807117\pi$
$938$	0	0
$939$	45.1189	1.47240
$940$	0	0
$941$	40.1557	1.30904	0.654519	−	0.756046i	$-0.272871\pi$
$941$	40.1557	1.30904	0.654519	+	0.756046i	$0.272871\pi$
$942$	0	0
$943$	−6.47922	−0.210992
$944$	0	0
$945$	18.6731	0.607438
$946$	0	0
$947$	−32.1993	−1.04634	−0.523168	−	0.852229i	$-0.675250\pi$
$947$	−32.1993	−1.04634	−0.523168	+	0.852229i	$0.675250\pi$
$948$	0	0
$949$	−2.50325	−0.0812589
$950$	0	0
$951$	54.8000	1.77701
$952$	0	0
$953$	56.1294	1.81821	0.909105	−	0.416567i	$-0.136767\pi$
$953$	56.1294	1.81821	0.909105	+	0.416567i	$0.136767\pi$
$954$	0	0
$955$	−44.2912	−1.43323
$956$	0	0
$957$	63.3297	2.04716
$958$	0	0
$959$	−23.7277	−0.766206
$960$	0	0
$961$	−12.7437	−0.411086
$962$	0	0
$963$	23.2841	0.750320
$964$	0	0
$965$	−29.4557	−0.948213
$966$	0	0
$967$	−17.2033	−0.553220	−0.276610	−	0.960982i	$-0.589211\pi$
$967$	−17.2033	−0.553220	−0.276610	+	0.960982i	$0.589211\pi$
$968$	0	0
$969$	0.629230	0.0202138
$970$	0	0
$971$	−53.2550	−1.70904	−0.854518	−	0.519422i	$-0.826147\pi$
$971$	−53.2550	−1.70904	−0.854518	+	0.519422i	$0.826147\pi$
$972$	0	0
$973$	−8.04466	−0.257900
$974$	0	0
$975$	−33.5668	−1.07500
$976$	0	0
$977$	−46.4439	−1.48587	−0.742936	−	0.669362i	$-0.766567\pi$
$977$	−46.4439	−1.48587	−0.742936	+	0.669362i	$0.766567\pi$
$978$	0	0
$979$	27.4747	0.878095
$980$	0	0
$981$	−8.41164	−0.268563
$982$	0	0
$983$	6.95339	0.221779	0.110889	−	0.993833i	$-0.464630\pi$
$983$	6.95339	0.221779	0.110889	+	0.993833i	$0.464630\pi$
$984$	0	0
$985$	38.7747	1.23547
$986$	0	0
$987$	−15.3076	−0.487245
$988$	0	0
$989$	−7.64463	−0.243085
$990$	0	0
$991$	−59.2699	−1.88277	−0.941385	−	0.337333i	$-0.890475\pi$
$991$	−59.2699	−1.88277	−0.941385	+	0.337333i	$0.890475\pi$
$992$	0	0
$993$	47.8413	1.51820
$994$	0	0
$995$	56.9335	1.80491
$996$	0	0
$997$	−9.51003	−0.301186	−0.150593	−	0.988596i	$-0.548118\pi$
$997$	−9.51003	−0.301186	−0.150593	+	0.988596i	$0.548118\pi$
$998$	0	0
$999$	−8.05386	−0.254813

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.q.1.10		12
4.3	odd	2		1006.2.a.j.1.3	✓	12
12.11	even	2		9054.2.a.bi.1.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1006.2.a.j.1.3	✓	12	4.3	odd	2
8048.2.a.q.1.10		12	1.1	even	1	trivial
9054.2.a.bi.1.2		12	12.11	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:	\( N \)	\(=\)	\( 8048 = 2^{4} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8048.a (trivial)

\(f(q)\)	\(=\)	\(q+2.17256 q^{3} +2.95208 q^{5} -2.27467 q^{7} +1.72004 q^{9} +O(q^{10})\)	\(q+2.17256 q^{3} +2.95208 q^{5} -2.27467 q^{7} +1.72004 q^{9} -3.76405 q^{11} -4.15916 q^{13} +6.41358 q^{15} +4.07953 q^{17} +0.0709948 q^{19} -4.94187 q^{21} +1.46890 q^{23} +3.71477 q^{25} -2.78081 q^{27} -7.74424 q^{29} -4.27274 q^{31} -8.17765 q^{33} -6.71502 q^{35} +2.89623 q^{37} -9.03604 q^{39} -4.41094 q^{41} -5.20433 q^{43} +5.07768 q^{45} +3.09752 q^{47} -1.82586 q^{49} +8.86304 q^{51} +0.690667 q^{53} -11.1118 q^{55} +0.154241 q^{57} -9.44782 q^{59} +11.2047 q^{61} -3.91252 q^{63} -12.2782 q^{65} -12.1917 q^{67} +3.19127 q^{69} -7.39192 q^{71} +0.601864 q^{73} +8.07057 q^{75} +8.56199 q^{77} -10.3921 q^{79} -11.2016 q^{81} +10.8034 q^{83} +12.0431 q^{85} -16.8249 q^{87} -7.29923 q^{89} +9.46073 q^{91} -9.28281 q^{93} +0.209582 q^{95} -13.8744 q^{97} -6.47431 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 5 q^{3} + 5 q^{5} - 8 q^{7} + 15 q^{9}+O(q^{10}) \) 12 * q - 5 * q^3 + 5 * q^5 - 8 * q^7 + 15 * q^9	\( 12 q - 5 q^{3} + 5 q^{5} - 8 q^{7} + 15 q^{9} - 18 q^{11} + 4 q^{13} - 2 q^{15} - 2 q^{17} - 6 q^{19} + q^{21} - 13 q^{23} + q^{25} - 8 q^{27} + 20 q^{29} - 7 q^{31} - 8 q^{33} - q^{35} + 10 q^{37} - 7 q^{39} + 2 q^{41} - 8 q^{43} - 7 q^{45} - 12 q^{47} + 4 q^{49} - 2 q^{51} + 12 q^{53} - 8 q^{55} - 10 q^{57} - 6 q^{59} - 10 q^{61} - 7 q^{63} + 4 q^{65} - 7 q^{67} - 12 q^{69} - 22 q^{71} - 23 q^{73} + 34 q^{75} - 19 q^{77} - 13 q^{79} - 28 q^{81} - q^{83} - 28 q^{85} + 22 q^{87} - 3 q^{89} + 21 q^{91} - 33 q^{93} + 2 q^{95} - 70 q^{97} + 6 q^{99}+O(q^{100}) \) 12 * q - 5 * q^3 + 5 * q^5 - 8 * q^7 + 15 * q^9 - 18 * q^11 + 4 * q^13 - 2 * q^15 - 2 * q^17 - 6 * q^19 + q^21 - 13 * q^23 + q^25 - 8 * q^27 + 20 * q^29 - 7 * q^31 - 8 * q^33 - q^35 + 10 * q^37 - 7 * q^39 + 2 * q^41 - 8 * q^43 - 7 * q^45 - 12 * q^47 + 4 * q^49 - 2 * q^51 + 12 * q^53 - 8 * q^55 - 10 * q^57 - 6 * q^59 - 10 * q^61 - 7 * q^63 + 4 * q^65 - 7 * q^67 - 12 * q^69 - 22 * q^71 - 23 * q^73 + 34 * q^75 - 19 * q^77 - 13 * q^79 - 28 * q^81 - q^83 - 28 * q^85 + 22 * q^87 - 3 * q^89 + 21 * q^91 - 33 * q^93 + 2 * q^95 - 70 * q^97 + 6 * q^99

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