LMFDB - Embedded newform 3952.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3952,2,Mod(1,3952)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3952, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3952.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3952 = 2^{4} \cdot 13 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3952.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:	$31.5568788788$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 494)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.80194$ of defining polynomial
Character	$\chi$	$=$	3952.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.24698 q^{3} +1.55496 q^{5} +3.60388 q^{7} +7.54288 q^{9} +O(q^{10})$	$q-3.24698 q^{3} +1.55496 q^{5} +3.60388 q^{7} +7.54288 q^{9} +5.18598 q^{11} -1.00000 q^{13} -5.04892 q^{15} -5.96077 q^{17} +1.00000 q^{19} -11.7017 q^{21} -4.29590 q^{23} -2.58211 q^{25} -14.7506 q^{27} +2.91185 q^{29} -6.04892 q^{31} -16.8388 q^{33} +5.60388 q^{35} -6.71379 q^{37} +3.24698 q^{39} -9.83877 q^{41} -5.16421 q^{43} +11.7289 q^{45} -11.0315 q^{47} +5.98792 q^{49} +19.3545 q^{51} +3.82908 q^{53} +8.06398 q^{55} -3.24698 q^{57} -13.5254 q^{59} -0.615957 q^{61} +27.1836 q^{63} -1.55496 q^{65} +0.987918 q^{67} +13.9487 q^{69} +4.71917 q^{71} +6.59179 q^{73} +8.38404 q^{75} +18.6896 q^{77} -10.7681 q^{79} +25.2664 q^{81} -3.91723 q^{83} -9.26875 q^{85} -9.45473 q^{87} +0.789856 q^{89} -3.60388 q^{91} +19.6407 q^{93} +1.55496 q^{95} -15.9976 q^{97} +39.1172 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 5 q^{3} + 5 q^{5} + 2 q^{7} + 4 q^{9}+O(q^{10}) $ 3 * q - 5 * q^3 + 5 * q^5 + 2 * q^7 + 4 * q^9	$ 3 q - 5 q^{3} + 5 q^{5} + 2 q^{7} + 4 q^{9} + q^{11} - 3 q^{13} - 6 q^{15} - 5 q^{17} + 3 q^{19} - 8 q^{21} + q^{23} - 2 q^{25} - 8 q^{27} + 5 q^{29} - 9 q^{31} - 18 q^{33} + 8 q^{35} - 12 q^{37} + 5 q^{39} + 3 q^{41} - 4 q^{43} + 2 q^{45} - 8 q^{47} - q^{49} + 13 q^{51} + q^{53} - 10 q^{55} - 5 q^{57} - 6 q^{59} - 12 q^{61} + 26 q^{63} - 5 q^{65} - 16 q^{67} + 10 q^{69} + 3 q^{71} - 8 q^{73} + 15 q^{75} + 10 q^{77} - 12 q^{79} + 27 q^{81} - 5 q^{83} - 20 q^{85} - 6 q^{87} - 21 q^{89} - 2 q^{91} + 22 q^{93} + 5 q^{95} - 7 q^{97} + 55 q^{99}+O(q^{100}) $ 3 * q - 5 * q^3 + 5 * q^5 + 2 * q^7 + 4 * q^9 + q^11 - 3 * q^13 - 6 * q^15 - 5 * q^17 + 3 * q^19 - 8 * q^21 + q^23 - 2 * q^25 - 8 * q^27 + 5 * q^29 - 9 * q^31 - 18 * q^33 + 8 * q^35 - 12 * q^37 + 5 * q^39 + 3 * q^41 - 4 * q^43 + 2 * q^45 - 8 * q^47 - q^49 + 13 * q^51 + q^53 - 10 * q^55 - 5 * q^57 - 6 * q^59 - 12 * q^61 + 26 * q^63 - 5 * q^65 - 16 * q^67 + 10 * q^69 + 3 * q^71 - 8 * q^73 + 15 * q^75 + 10 * q^77 - 12 * q^79 + 27 * q^81 - 5 * q^83 - 20 * q^85 - 6 * q^87 - 21 * q^89 - 2 * q^91 + 22 * q^93 + 5 * q^95 - 7 * q^97 + 55 * 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
$3$	−3.24698	−1.87464	−0.937322	−	0.348464i	$-0.886703\pi$
$3$	−3.24698	−1.87464	−0.937322	+	0.348464i	$0.886703\pi$
$4$	0	0
$5$	1.55496	0.695398	0.347699	−	0.937606i	$-0.386963\pi$
$5$	1.55496	0.695398	0.347699	+	0.937606i	$0.386963\pi$
$6$	0	0
$7$	3.60388	1.36214	0.681068	−	0.732220i	$-0.261516\pi$
$7$	3.60388	1.36214	0.681068	+	0.732220i	$0.261516\pi$
$8$	0	0
$9$	7.54288	2.51429
$10$	0	0
$11$	5.18598	1.56363	0.781816	−	0.623509i	$-0.214294\pi$
$11$	5.18598	1.56363	0.781816	+	0.623509i	$0.214294\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−5.04892	−1.30362
$16$	0	0
$17$	−5.96077	−1.44570	−0.722850	−	0.691005i	$-0.757168\pi$
$17$	−5.96077	−1.44570	−0.722850	+	0.691005i	$0.757168\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−11.7017	−2.55352
$22$	0	0
$23$	−4.29590	−0.895756	−0.447878	−	0.894095i	$-0.647820\pi$
$23$	−4.29590	−0.895756	−0.447878	+	0.894095i	$0.647820\pi$
$24$	0	0
$25$	−2.58211	−0.516421
$26$	0	0
$27$	−14.7506	−2.83876
$28$	0	0
$29$	2.91185	0.540718	0.270359	−	0.962760i	$-0.412858\pi$
$29$	2.91185	0.540718	0.270359	+	0.962760i	$0.412858\pi$
$30$	0	0
$31$	−6.04892	−1.08642	−0.543209	−	0.839598i	$-0.682791\pi$
$31$	−6.04892	−1.08642	−0.543209	+	0.839598i	$0.682791\pi$
$32$	0	0
$33$	−16.8388	−2.93125
$34$	0	0
$35$	5.60388	0.947228
$36$	0	0
$37$	−6.71379	−1.10374	−0.551870	−	0.833930i	$-0.686086\pi$
$37$	−6.71379	−1.10374	−0.551870	+	0.833930i	$0.686086\pi$
$38$	0	0
$39$	3.24698	0.519933
$40$	0	0
$41$	−9.83877	−1.53656	−0.768279	−	0.640115i	$-0.778887\pi$
$41$	−9.83877	−1.53656	−0.768279	+	0.640115i	$0.778887\pi$
$42$	0	0
$43$	−5.16421	−0.787535	−0.393767	−	0.919210i	$-0.628828\pi$
$43$	−5.16421	−0.787535	−0.393767	+	0.919210i	$0.628828\pi$
$44$	0	0
$45$	11.7289	1.74843
$46$	0	0
$47$	−11.0315	−1.60910	−0.804552	−	0.593882i	$-0.797594\pi$
$47$	−11.0315	−1.60910	−0.804552	+	0.593882i	$0.797594\pi$
$48$	0	0
$49$	5.98792	0.855417
$50$	0	0
$51$	19.3545	2.71017
$52$	0	0
$53$	3.82908	0.525965	0.262983	−	0.964801i	$-0.415294\pi$
$53$	3.82908	0.525965	0.262983	+	0.964801i	$0.415294\pi$
$54$	0	0
$55$	8.06398	1.08735
$56$	0	0
$57$	−3.24698	−0.430073
$58$	0	0
$59$	−13.5254	−1.76086	−0.880430	−	0.474177i	$-0.842746\pi$
$59$	−13.5254	−1.76086	−0.880430	+	0.474177i	$0.842746\pi$
$60$	0	0
$61$	−0.615957	−0.0788652	−0.0394326	−	0.999222i	$-0.512555\pi$
$61$	−0.615957	−0.0788652	−0.0394326	+	0.999222i	$0.512555\pi$
$62$	0	0
$63$	27.1836	3.42481
$64$	0	0
$65$	−1.55496	−0.192869
$66$	0	0
$67$	0.987918	0.120693	0.0603467	−	0.998177i	$-0.480779\pi$
$67$	0.987918	0.120693	0.0603467	+	0.998177i	$0.480779\pi$
$68$	0	0
$69$	13.9487	1.67922
$70$	0	0
$71$	4.71917	0.560062	0.280031	−	0.959991i	$-0.409655\pi$
$71$	4.71917	0.560062	0.280031	+	0.959991i	$0.409655\pi$
$72$	0	0
$73$	6.59179	0.771511	0.385756	−	0.922601i	$-0.373941\pi$
$73$	6.59179	0.771511	0.385756	+	0.922601i	$0.373941\pi$
$74$	0	0
$75$	8.38404	0.968106
$76$	0	0
$77$	18.6896	2.12988
$78$	0	0
$79$	−10.7681	−1.21150	−0.605752	−	0.795653i	$-0.707128\pi$
$79$	−10.7681	−1.21150	−0.605752	+	0.795653i	$0.707128\pi$
$80$	0	0
$81$	25.2664	2.80737
$82$	0	0
$83$	−3.91723	−0.429972	−0.214986	−	0.976617i	$-0.568971\pi$
$83$	−3.91723	−0.429972	−0.214986	+	0.976617i	$0.568971\pi$
$84$	0	0
$85$	−9.26875	−1.00534
$86$	0	0
$87$	−9.45473	−1.01365
$88$	0	0
$89$	0.789856	0.0837246	0.0418623	−	0.999123i	$-0.486671\pi$
$89$	0.789856	0.0837246	0.0418623	+	0.999123i	$0.486671\pi$
$90$	0	0
$91$	−3.60388	−0.377789
$92$	0	0
$93$	19.6407	2.03665
$94$	0	0
$95$	1.55496	0.159535
$96$	0	0
$97$	−15.9976	−1.62431	−0.812155	−	0.583441i	$-0.801706\pi$
$97$	−15.9976	−1.62431	−0.812155	+	0.583441i	$0.801706\pi$
$98$	0	0
$99$	39.1172	3.93143
$100$	0	0
$101$	5.26205	0.523593	0.261797	−	0.965123i	$-0.415685\pi$
$101$	5.26205	0.523593	0.261797	+	0.965123i	$0.415685\pi$
$102$	0	0
$103$	−0.396125	−0.0390313	−0.0195157	−	0.999810i	$-0.506212\pi$
$103$	−0.396125	−0.0390313	−0.0195157	+	0.999810i	$0.506212\pi$
$104$	0	0
$105$	−18.1957	−1.77572
$106$	0	0
$107$	0.488582	0.0472330	0.0236165	−	0.999721i	$-0.492482\pi$
$107$	0.488582	0.0472330	0.0236165	+	0.999721i	$0.492482\pi$
$108$	0	0
$109$	2.71379	0.259934	0.129967	−	0.991518i	$-0.458513\pi$
$109$	2.71379	0.259934	0.129967	+	0.991518i	$0.458513\pi$
$110$	0	0
$111$	21.7995	2.06912
$112$	0	0
$113$	−17.8780	−1.68182	−0.840910	−	0.541174i	$-0.817980\pi$
$113$	−17.8780	−1.68182	−0.840910	+	0.541174i	$0.817980\pi$
$114$	0	0
$115$	−6.67994	−0.622908
$116$	0	0
$117$	−7.54288	−0.697339
$118$	0	0
$119$	−21.4819	−1.96924
$120$	0	0
$121$	15.8944	1.44495
$122$	0	0
$123$	31.9463	2.88050
$124$	0	0
$125$	−11.7899	−1.05452
$126$	0	0
$127$	1.60388	0.142321	0.0711605	−	0.997465i	$-0.477330\pi$
$127$	1.60388	0.142321	0.0711605	+	0.997465i	$0.477330\pi$
$128$	0	0
$129$	16.7681	1.47635
$130$	0	0
$131$	19.3599	1.69148	0.845740	−	0.533595i	$-0.179159\pi$
$131$	19.3599	1.69148	0.845740	+	0.533595i	$0.179159\pi$
$132$	0	0
$133$	3.60388	0.312496
$134$	0	0
$135$	−22.9366	−1.97407
$136$	0	0
$137$	−6.61596	−0.565239	−0.282620	−	0.959232i	$-0.591203\pi$
$137$	−6.61596	−0.565239	−0.282620	+	0.959232i	$0.591203\pi$
$138$	0	0
$139$	−0.933624	−0.0791890	−0.0395945	−	0.999216i	$-0.512607\pi$
$139$	−0.933624	−0.0791890	−0.0395945	+	0.999216i	$0.512607\pi$
$140$	0	0
$141$	35.8189	3.01650
$142$	0	0
$143$	−5.18598	−0.433673
$144$	0	0
$145$	4.52781	0.376014
$146$	0	0
$147$	−19.4426	−1.60360
$148$	0	0
$149$	−6.13706	−0.502768	−0.251384	−	0.967887i	$-0.580886\pi$
$149$	−6.13706	−0.502768	−0.251384	+	0.967887i	$0.580886\pi$
$150$	0	0
$151$	−11.2838	−0.918264	−0.459132	−	0.888368i	$-0.651839\pi$
$151$	−11.2838	−0.918264	−0.459132	+	0.888368i	$0.651839\pi$
$152$	0	0
$153$	−44.9614	−3.63491
$154$	0	0
$155$	−9.40581	−0.755493
$156$	0	0
$157$	−2.41550	−0.192778	−0.0963890	−	0.995344i	$-0.530729\pi$
$157$	−2.41550	−0.192778	−0.0963890	+	0.995344i	$0.530729\pi$
$158$	0	0
$159$	−12.4330	−0.985998
$160$	0	0
$161$	−15.4819	−1.22014
$162$	0	0
$163$	4.63102	0.362730	0.181365	−	0.983416i	$-0.441949\pi$
$163$	4.63102	0.362730	0.181365	+	0.983416i	$0.441949\pi$
$164$	0	0
$165$	−26.1836	−2.03839
$166$	0	0
$167$	18.0344	1.39555	0.697774	−	0.716318i	$-0.254174\pi$
$167$	18.0344	1.39555	0.697774	+	0.716318i	$0.254174\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	7.54288	0.576818
$172$	0	0
$173$	−22.2107	−1.68865	−0.844325	−	0.535831i	$-0.819999\pi$
$173$	−22.2107	−1.68865	−0.844325	+	0.535831i	$0.819999\pi$
$174$	0	0
$175$	−9.30559	−0.703436
$176$	0	0
$177$	43.9168	3.30099
$178$	0	0
$179$	22.0248	1.64621	0.823104	−	0.567891i	$-0.192241\pi$
$179$	22.0248	1.64621	0.823104	+	0.567891i	$0.192241\pi$
$180$	0	0
$181$	12.3134	0.915244	0.457622	−	0.889147i	$-0.348701\pi$
$181$	12.3134	0.915244	0.457622	+	0.889147i	$0.348701\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	0	0
$185$	−10.4397	−0.767539
$186$	0	0
$187$	−30.9124	−2.26054
$188$	0	0
$189$	−53.1594	−3.86678
$190$	0	0
$191$	−9.21313	−0.666639	−0.333319	−	0.942814i	$-0.608169\pi$
$191$	−9.21313	−0.666639	−0.333319	+	0.942814i	$0.608169\pi$
$192$	0	0
$193$	14.2784	1.02778	0.513892	−	0.857855i	$-0.328203\pi$
$193$	14.2784	1.02778	0.513892	+	0.857855i	$0.328203\pi$
$194$	0	0
$195$	5.04892	0.361560
$196$	0	0
$197$	11.7802	0.839302	0.419651	−	0.907685i	$-0.362152\pi$
$197$	11.7802	0.839302	0.419651	+	0.907685i	$0.362152\pi$
$198$	0	0
$199$	−5.88040	−0.416850	−0.208425	−	0.978038i	$-0.566834\pi$
$199$	−5.88040	−0.416850	−0.208425	+	0.978038i	$0.566834\pi$
$200$	0	0
$201$	−3.20775	−0.226257
$202$	0	0
$203$	10.4940	0.736532
$204$	0	0
$205$	−15.2989	−1.06852
$206$	0	0
$207$	−32.4034	−2.25219
$208$	0	0
$209$	5.18598	0.358722
$210$	0	0
$211$	10.5894	0.729004	0.364502	−	0.931203i	$-0.381239\pi$
$211$	10.5894	0.729004	0.364502	+	0.931203i	$0.381239\pi$
$212$	0	0
$213$	−15.3230	−1.04992
$214$	0	0
$215$	−8.03013	−0.547650
$216$	0	0
$217$	−21.7995	−1.47985
$218$	0	0
$219$	−21.4034	−1.44631
$220$	0	0
$221$	5.96077	0.400965
$222$	0	0
$223$	1.04652	0.0700805	0.0350402	−	0.999386i	$-0.488844\pi$
$223$	1.04652	0.0700805	0.0350402	+	0.999386i	$0.488844\pi$
$224$	0	0
$225$	−19.4765	−1.29843
$226$	0	0
$227$	−7.10992	−0.471902	−0.235951	−	0.971765i	$-0.575820\pi$
$227$	−7.10992	−0.471902	−0.235951	+	0.971765i	$0.575820\pi$
$228$	0	0
$229$	−16.6679	−1.10144	−0.550722	−	0.834689i	$-0.685647\pi$
$229$	−16.6679	−1.10144	−0.550722	+	0.834689i	$0.685647\pi$
$230$	0	0
$231$	−60.6848	−3.99277
$232$	0	0
$233$	−10.5187	−0.689104	−0.344552	−	0.938767i	$-0.611969\pi$
$233$	−10.5187	−0.689104	−0.344552	+	0.938767i	$0.611969\pi$
$234$	0	0
$235$	−17.1535	−1.11897
$236$	0	0
$237$	34.9638	2.27114
$238$	0	0
$239$	−0.548253	−0.0354636	−0.0177318	−	0.999843i	$-0.505644\pi$
$239$	−0.548253	−0.0354636	−0.0177318	+	0.999843i	$0.505644\pi$
$240$	0	0
$241$	17.5743	1.13206	0.566031	−	0.824384i	$-0.308478\pi$
$241$	17.5743	1.13206	0.566031	+	0.824384i	$0.308478\pi$
$242$	0	0
$243$	−37.7875	−2.42407
$244$	0	0
$245$	9.31096	0.594856
$246$	0	0
$247$	−1.00000	−0.0636285
$248$	0	0
$249$	12.7192	0.806045
$250$	0	0
$251$	−4.09783	−0.258653	−0.129327	−	0.991602i	$-0.541282\pi$
$251$	−4.09783	−0.258653	−0.129327	+	0.991602i	$0.541282\pi$
$252$	0	0
$253$	−22.2784	−1.40063
$254$	0	0
$255$	30.0954	1.88465
$256$	0	0
$257$	−6.65950	−0.415408	−0.207704	−	0.978192i	$-0.566599\pi$
$257$	−6.65950	−0.415408	−0.207704	+	0.978192i	$0.566599\pi$
$258$	0	0
$259$	−24.1957	−1.50345
$260$	0	0
$261$	21.9638	1.35952
$262$	0	0
$263$	−20.3763	−1.25645	−0.628227	−	0.778030i	$-0.716219\pi$
$263$	−20.3763	−1.25645	−0.628227	+	0.778030i	$0.716219\pi$
$264$	0	0
$265$	5.95407	0.365755
$266$	0	0
$267$	−2.56465	−0.156954
$268$	0	0
$269$	−20.9638	−1.27818	−0.639091	−	0.769131i	$-0.720689\pi$
$269$	−20.9638	−1.27818	−0.639091	+	0.769131i	$0.720689\pi$
$270$	0	0
$271$	−9.40821	−0.571508	−0.285754	−	0.958303i	$-0.592244\pi$
$271$	−9.40821	−0.571508	−0.285754	+	0.958303i	$0.592244\pi$
$272$	0	0
$273$	11.7017	0.708220
$274$	0	0
$275$	−13.3907	−0.807492
$276$	0	0
$277$	22.0194	1.32302	0.661508	−	0.749938i	$-0.269917\pi$
$277$	22.0194	1.32302	0.661508	+	0.749938i	$0.269917\pi$
$278$	0	0
$279$	−45.6262	−2.73157
$280$	0	0
$281$	24.4373	1.45781	0.728903	−	0.684617i	$-0.240031\pi$
$281$	24.4373	1.45781	0.728903	+	0.684617i	$0.240031\pi$
$282$	0	0
$283$	−15.9323	−0.947077	−0.473538	−	0.880773i	$-0.657023\pi$
$283$	−15.9323	−0.947077	−0.473538	+	0.880773i	$0.657023\pi$
$284$	0	0
$285$	−5.04892	−0.299072
$286$	0	0
$287$	−35.4577	−2.09300
$288$	0	0
$289$	18.5308	1.09005
$290$	0	0
$291$	51.9439	3.04501
$292$	0	0
$293$	15.5013	0.905593	0.452796	−	0.891614i	$-0.350426\pi$
$293$	15.5013	0.905593	0.452796	+	0.891614i	$0.350426\pi$
$294$	0	0
$295$	−21.0315	−1.22450
$296$	0	0
$297$	−76.4965	−4.43878
$298$	0	0
$299$	4.29590	0.248438
$300$	0	0
$301$	−18.6112	−1.07273
$302$	0	0
$303$	−17.0858	−0.981551
$304$	0	0
$305$	−0.957787	−0.0548427
$306$	0	0
$307$	10.0871	0.575700	0.287850	−	0.957675i	$-0.407060\pi$
$307$	10.0871	0.575700	0.287850	+	0.957675i	$0.407060\pi$
$308$	0	0
$309$	1.28621	0.0731698
$310$	0	0
$311$	6.15644	0.349100	0.174550	−	0.984648i	$-0.444153\pi$
$311$	6.15644	0.349100	0.174550	+	0.984648i	$0.444153\pi$
$312$	0	0
$313$	14.4101	0.814508	0.407254	−	0.913315i	$-0.366486\pi$
$313$	14.4101	0.814508	0.407254	+	0.913315i	$0.366486\pi$
$314$	0	0
$315$	42.2693	2.38161
$316$	0	0
$317$	−20.5241	−1.15275	−0.576374	−	0.817186i	$-0.695533\pi$
$317$	−20.5241	−1.15275	−0.576374	+	0.817186i	$0.695533\pi$
$318$	0	0
$319$	15.1008	0.845484
$320$	0	0
$321$	−1.58642	−0.0885452
$322$	0	0
$323$	−5.96077	−0.331666
$324$	0	0
$325$	2.58211	0.143229
$326$	0	0
$327$	−8.81163	−0.487284
$328$	0	0
$329$	−39.7560	−2.19182
$330$	0	0
$331$	22.4155	1.23207	0.616034	−	0.787720i	$-0.288739\pi$
$331$	22.4155	1.23207	0.616034	+	0.787720i	$0.288739\pi$
$332$	0	0
$333$	−50.6413	−2.77513
$334$	0	0
$335$	1.53617	0.0839300
$336$	0	0
$337$	25.1293	1.36888	0.684440	−	0.729069i	$-0.260047\pi$
$337$	25.1293	1.36888	0.684440	+	0.729069i	$0.260047\pi$
$338$	0	0
$339$	58.0495	3.15282
$340$	0	0
$341$	−31.3696	−1.69876
$342$	0	0
$343$	−3.64742	−0.196942
$344$	0	0
$345$	21.6896	1.16773
$346$	0	0
$347$	20.3370	1.09175	0.545875	−	0.837867i	$-0.316197\pi$
$347$	20.3370	1.09175	0.545875	+	0.837867i	$0.316197\pi$
$348$	0	0
$349$	7.12498	0.381392	0.190696	−	0.981649i	$-0.438926\pi$
$349$	7.12498	0.381392	0.190696	+	0.981649i	$0.438926\pi$
$350$	0	0
$351$	14.7506	0.787330
$352$	0	0
$353$	22.4155	1.19306	0.596528	−	0.802592i	$-0.296546\pi$
$353$	22.4155	1.19306	0.596528	+	0.802592i	$0.296546\pi$
$354$	0	0
$355$	7.33811	0.389466
$356$	0	0
$357$	69.7512	3.69163
$358$	0	0
$359$	6.29829	0.332411	0.166206	−	0.986091i	$-0.446848\pi$
$359$	6.29829	0.332411	0.166206	+	0.986091i	$0.446848\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−51.6088	−2.70876
$364$	0	0
$365$	10.2500	0.536508
$366$	0	0
$367$	−1.90754	−0.0995729	−0.0497864	−	0.998760i	$-0.515854\pi$
$367$	−1.90754	−0.0995729	−0.0497864	+	0.998760i	$0.515854\pi$
$368$	0	0
$369$	−74.2127	−3.86336
$370$	0	0
$371$	13.7995	0.716437
$372$	0	0
$373$	9.57540	0.495795	0.247898	−	0.968786i	$-0.420260\pi$
$373$	9.57540	0.495795	0.247898	+	0.968786i	$0.420260\pi$
$374$	0	0
$375$	38.2814	1.97684
$376$	0	0
$377$	−2.91185	−0.149968
$378$	0	0
$379$	−11.1400	−0.572226	−0.286113	−	0.958196i	$-0.592363\pi$
$379$	−11.1400	−0.572226	−0.286113	+	0.958196i	$0.592363\pi$
$380$	0	0
$381$	−5.20775	−0.266801
$382$	0	0
$383$	14.0489	0.717866	0.358933	−	0.933363i	$-0.383141\pi$
$383$	14.0489	0.717866	0.358933	+	0.933363i	$0.383141\pi$
$384$	0	0
$385$	29.0616	1.48112
$386$	0	0
$387$	−38.9530	−1.98009
$388$	0	0
$389$	9.95646	0.504813	0.252406	−	0.967621i	$-0.418778\pi$
$389$	9.95646	0.504813	0.252406	+	0.967621i	$0.418778\pi$
$390$	0	0
$391$	25.6069	1.29499
$392$	0	0
$393$	−62.8611	−3.17092
$394$	0	0
$395$	−16.7439	−0.842478
$396$	0	0
$397$	16.0683	0.806445	0.403222	−	0.915102i	$-0.367890\pi$
$397$	16.0683	0.806445	0.403222	+	0.915102i	$0.367890\pi$
$398$	0	0
$399$	−11.7017	−0.585818
$400$	0	0
$401$	4.59717	0.229572	0.114786	−	0.993390i	$-0.463382\pi$
$401$	4.59717	0.229572	0.114786	+	0.993390i	$0.463382\pi$
$402$	0	0
$403$	6.04892	0.301318
$404$	0	0
$405$	39.2881	1.95224
$406$	0	0
$407$	−34.8176	−1.72584
$408$	0	0
$409$	−1.56033	−0.0771536	−0.0385768	−	0.999256i	$-0.512282\pi$
$409$	−1.56033	−0.0771536	−0.0385768	+	0.999256i	$0.512282\pi$
$410$	0	0
$411$	21.4819	1.05962
$412$	0	0
$413$	−48.7439	−2.39853
$414$	0	0
$415$	−6.09113	−0.299002
$416$	0	0
$417$	3.03146	0.148451
$418$	0	0
$419$	−12.4590	−0.608664	−0.304332	−	0.952566i	$-0.598433\pi$
$419$	−12.4590	−0.608664	−0.304332	+	0.952566i	$0.598433\pi$
$420$	0	0
$421$	4.71379	0.229736	0.114868	−	0.993381i	$-0.463355\pi$
$421$	4.71379	0.229736	0.114868	+	0.993381i	$0.463355\pi$
$422$	0	0
$423$	−83.2089	−4.04576
$424$	0	0
$425$	15.3913	0.746590
$426$	0	0
$427$	−2.21983	−0.107425
$428$	0	0
$429$	16.8388	0.812984
$430$	0	0
$431$	−15.0707	−0.725929	−0.362965	−	0.931803i	$-0.618235\pi$
$431$	−15.0707	−0.725929	−0.362965	+	0.931803i	$0.618235\pi$
$432$	0	0
$433$	18.4263	0.885509	0.442755	−	0.896643i	$-0.354001\pi$
$433$	18.4263	0.885509	0.442755	+	0.896643i	$0.354001\pi$
$434$	0	0
$435$	−14.7017	−0.704893
$436$	0	0
$437$	−4.29590	−0.205501
$438$	0	0
$439$	38.1909	1.82275	0.911376	−	0.411575i	$-0.135021\pi$
$439$	38.1909	1.82275	0.911376	+	0.411575i	$0.135021\pi$
$440$	0	0
$441$	45.1661	2.15077
$442$	0	0
$443$	31.3599	1.48995	0.744976	−	0.667091i	$-0.232461\pi$
$443$	31.3599	1.48995	0.744976	+	0.667091i	$0.232461\pi$
$444$	0	0
$445$	1.22819	0.0582219
$446$	0	0
$447$	19.9269	0.942511
$448$	0	0
$449$	−15.5104	−0.731979	−0.365989	−	0.930619i	$-0.619269\pi$
$449$	−15.5104	−0.731979	−0.365989	+	0.930619i	$0.619269\pi$
$450$	0	0
$451$	−51.0237	−2.40261
$452$	0	0
$453$	36.6383	1.72142
$454$	0	0
$455$	−5.60388	−0.262714
$456$	0	0
$457$	−33.8538	−1.58362	−0.791808	−	0.610770i	$-0.790860\pi$
$457$	−33.8538	−1.58362	−0.791808	+	0.610770i	$0.790860\pi$
$458$	0	0
$459$	87.9251	4.10399
$460$	0	0
$461$	16.9989	0.791719	0.395860	−	0.918311i	$-0.370447\pi$
$461$	16.9989	0.791719	0.395860	+	0.918311i	$0.370447\pi$
$462$	0	0
$463$	33.5254	1.55806	0.779029	−	0.626988i	$-0.215712\pi$
$463$	33.5254	1.55806	0.779029	+	0.626988i	$0.215712\pi$
$464$	0	0
$465$	30.5405	1.41628
$466$	0	0
$467$	24.8659	1.15066	0.575329	−	0.817922i	$-0.304874\pi$
$467$	24.8659	1.15066	0.575329	+	0.817922i	$0.304874\pi$
$468$	0	0
$469$	3.56033	0.164401
$470$	0	0
$471$	7.84309	0.361390
$472$	0	0
$473$	−26.7815	−1.23141
$474$	0	0
$475$	−2.58211	−0.118475
$476$	0	0
$477$	28.8823	1.32243
$478$	0	0
$479$	43.5120	1.98811	0.994057	−	0.108859i	$-0.0347197\pi$
$479$	43.5120	1.98811	0.994057	+	0.108859i	$0.0347197\pi$
$480$	0	0
$481$	6.71379	0.306123
$482$	0	0
$483$	50.2693	2.28733
$484$	0	0
$485$	−24.8756	−1.12954
$486$	0	0
$487$	9.41657	0.426705	0.213353	−	0.976975i	$-0.431562\pi$
$487$	9.41657	0.426705	0.213353	+	0.976975i	$0.431562\pi$
$488$	0	0
$489$	−15.0368	−0.679989
$490$	0	0
$491$	−1.67755	−0.0757066	−0.0378533	−	0.999283i	$-0.512052\pi$
$491$	−1.67755	−0.0757066	−0.0378533	+	0.999283i	$0.512052\pi$
$492$	0	0
$493$	−17.3569	−0.781715
$494$	0	0
$495$	60.8256	2.73391
$496$	0	0
$497$	17.0073	0.762881
$498$	0	0
$499$	−39.4077	−1.76413	−0.882066	−	0.471126i	$-0.843848\pi$
$499$	−39.4077	−1.76413	−0.882066	+	0.471126i	$0.843848\pi$
$500$	0	0
$501$	−58.5575	−2.61615
$502$	0	0
$503$	1.86725	0.0832565	0.0416282	−	0.999133i	$-0.486745\pi$
$503$	1.86725	0.0832565	0.0416282	+	0.999133i	$0.486745\pi$
$504$	0	0
$505$	8.18226	0.364106
$506$	0	0
$507$	−3.24698	−0.144203
$508$	0	0
$509$	−20.8853	−0.925725	−0.462862	−	0.886430i	$-0.653178\pi$
$509$	−20.8853	−0.925725	−0.462862	+	0.886430i	$0.653178\pi$
$510$	0	0
$511$	23.7560	1.05090
$512$	0	0
$513$	−14.7506	−0.651256
$514$	0	0
$515$	−0.615957	−0.0271423
$516$	0	0
$517$	−57.2089	−2.51605
$518$	0	0
$519$	72.1178	3.16562
$520$	0	0
$521$	−6.01075	−0.263336	−0.131668	−	0.991294i	$-0.542033\pi$
$521$	−6.01075	−0.263336	−0.131668	+	0.991294i	$0.542033\pi$
$522$	0	0
$523$	−25.2325	−1.10334	−0.551670	−	0.834062i	$-0.686009\pi$
$523$	−25.2325	−1.10334	−0.551670	+	0.834062i	$0.686009\pi$
$524$	0	0
$525$	30.2150	1.31869
$526$	0	0
$527$	36.0562	1.57063
$528$	0	0
$529$	−4.54527	−0.197620
$530$	0	0
$531$	−102.021	−4.42732
$532$	0	0
$533$	9.83877	0.426165
$534$	0	0
$535$	0.759725	0.0328458
$536$	0	0
$537$	−71.5139	−3.08605
$538$	0	0
$539$	31.0532	1.33756
$540$	0	0
$541$	15.3459	0.659771	0.329885	−	0.944021i	$-0.392990\pi$
$541$	15.3459	0.659771	0.329885	+	0.944021i	$0.392990\pi$
$542$	0	0
$543$	−39.9812	−1.71576
$544$	0	0
$545$	4.21983	0.180758
$546$	0	0
$547$	1.97525	0.0844554	0.0422277	−	0.999108i	$-0.486555\pi$
$547$	1.97525	0.0844554	0.0422277	+	0.999108i	$0.486555\pi$
$548$	0	0
$549$	−4.64609	−0.198290
$550$	0	0
$551$	2.91185	0.124049
$552$	0	0
$553$	−38.8068	−1.65023
$554$	0	0
$555$	33.8974	1.43886
$556$	0	0
$557$	−22.6920	−0.961492	−0.480746	−	0.876860i	$-0.659634\pi$
$557$	−22.6920	−0.961492	−0.480746	+	0.876860i	$0.659634\pi$
$558$	0	0
$559$	5.16421	0.218423
$560$	0	0
$561$	100.372	4.23771
$562$	0	0
$563$	−22.4373	−0.945618	−0.472809	−	0.881165i	$-0.656760\pi$
$563$	−22.4373	−0.945618	−0.472809	+	0.881165i	$0.656760\pi$
$564$	0	0
$565$	−27.7995	−1.16954
$566$	0	0
$567$	91.0568	3.82403
$568$	0	0
$569$	23.9711	1.00492	0.502459	−	0.864601i	$-0.332429\pi$
$569$	23.9711	1.00492	0.502459	+	0.864601i	$0.332429\pi$
$570$	0	0
$571$	9.84787	0.412121	0.206060	−	0.978539i	$-0.433936\pi$
$571$	9.84787	0.412121	0.206060	+	0.978539i	$0.433936\pi$
$572$	0	0
$573$	29.9148	1.24971
$574$	0	0
$575$	11.0925	0.462587
$576$	0	0
$577$	−25.2922	−1.05293	−0.526464	−	0.850198i	$-0.676482\pi$
$577$	−25.2922	−1.05293	−0.526464	+	0.850198i	$0.676482\pi$
$578$	0	0
$579$	−46.3618	−1.92673
$580$	0	0
$581$	−14.1172	−0.585681
$582$	0	0
$583$	19.8576	0.822416
$584$	0	0
$585$	−11.7289	−0.484929
$586$	0	0
$587$	−41.2922	−1.70431	−0.852155	−	0.523289i	$-0.824705\pi$
$587$	−41.2922	−1.70431	−0.852155	+	0.523289i	$0.824705\pi$
$588$	0	0
$589$	−6.04892	−0.249241
$590$	0	0
$591$	−38.2500	−1.57339
$592$	0	0
$593$	−13.8043	−0.566876	−0.283438	−	0.958991i	$-0.591475\pi$
$593$	−13.8043	−0.566876	−0.283438	+	0.958991i	$0.591475\pi$
$594$	0	0
$595$	−33.4034	−1.36941
$596$	0	0
$597$	19.0935	0.781446
$598$	0	0
$599$	−9.84309	−0.402178	−0.201089	−	0.979573i	$-0.564448\pi$
$599$	−9.84309	−0.402178	−0.201089	+	0.979573i	$0.564448\pi$
$600$	0	0
$601$	14.2198	0.580039	0.290020	−	0.957021i	$-0.406338\pi$
$601$	14.2198	0.580039	0.290020	+	0.957021i	$0.406338\pi$
$602$	0	0
$603$	7.45175	0.303459
$604$	0	0
$605$	24.7151	1.00481
$606$	0	0
$607$	−19.3056	−0.783590	−0.391795	−	0.920053i	$-0.628146\pi$
$607$	−19.3056	−0.783590	−0.391795	+	0.920053i	$0.628146\pi$
$608$	0	0
$609$	−34.0737	−1.38073
$610$	0	0
$611$	11.0315	0.446285
$612$	0	0
$613$	36.6058	1.47849	0.739247	−	0.673434i	$-0.235182\pi$
$613$	36.6058	1.47849	0.739247	+	0.673434i	$0.235182\pi$
$614$	0	0
$615$	49.6752	2.00310
$616$	0	0
$617$	6.14138	0.247242	0.123621	−	0.992329i	$-0.460549\pi$
$617$	6.14138	0.247242	0.123621	+	0.992329i	$0.460549\pi$
$618$	0	0
$619$	−10.5080	−0.422351	−0.211175	−	0.977448i	$-0.567729\pi$
$619$	−10.5080	−0.422351	−0.211175	+	0.977448i	$0.567729\pi$
$620$	0	0
$621$	63.3672	2.54284
$622$	0	0
$623$	2.84654	0.114044
$624$	0	0
$625$	−5.42221	−0.216888
$626$	0	0
$627$	−16.8388	−0.672476
$628$	0	0
$629$	40.0194	1.59568
$630$	0	0
$631$	−38.7090	−1.54098	−0.770491	−	0.637451i	$-0.779989\pi$
$631$	−38.7090	−1.54098	−0.770491	+	0.637451i	$0.779989\pi$
$632$	0	0
$633$	−34.3836	−1.36662
$634$	0	0
$635$	2.49396	0.0989698
$636$	0	0
$637$	−5.98792	−0.237250
$638$	0	0
$639$	35.5961	1.40816
$640$	0	0
$641$	−46.7767	−1.84757	−0.923784	−	0.382913i	$-0.874921\pi$
$641$	−46.7767	−1.84757	−0.923784	+	0.382913i	$0.874921\pi$
$642$	0	0
$643$	30.6853	1.21011	0.605055	−	0.796183i	$-0.293151\pi$
$643$	30.6853	1.21011	0.605055	+	0.796183i	$0.293151\pi$
$644$	0	0
$645$	26.0737	1.02665
$646$	0	0
$647$	43.3846	1.70563	0.852813	−	0.522216i	$-0.174895\pi$
$647$	43.3846	1.70563	0.852813	+	0.522216i	$0.174895\pi$
$648$	0	0
$649$	−70.1426	−2.75334
$650$	0	0
$651$	70.7827	2.77419
$652$	0	0
$653$	−24.6025	−0.962772	−0.481386	−	0.876509i	$-0.659866\pi$
$653$	−24.6025	−0.962772	−0.481386	+	0.876509i	$0.659866\pi$
$654$	0	0
$655$	30.1038	1.17625
$656$	0	0
$657$	49.7211	1.93980
$658$	0	0
$659$	−25.3013	−0.985598	−0.492799	−	0.870143i	$-0.664026\pi$
$659$	−25.3013	−0.985598	−0.492799	+	0.870143i	$0.664026\pi$
$660$	0	0
$661$	20.2983	0.789512	0.394756	−	0.918786i	$-0.370829\pi$
$661$	20.2983	0.789512	0.394756	+	0.918786i	$0.370829\pi$
$662$	0	0
$663$	−19.3545	−0.751667
$664$	0	0
$665$	5.60388	0.217309
$666$	0	0
$667$	−12.5090	−0.484351
$668$	0	0
$669$	−3.39804	−0.131376
$670$	0	0
$671$	−3.19434	−0.123316
$672$	0	0
$673$	10.8552	0.418436	0.209218	−	0.977869i	$-0.432908\pi$
$673$	10.8552	0.418436	0.209218	+	0.977869i	$0.432908\pi$
$674$	0	0
$675$	38.0877	1.46600
$676$	0	0
$677$	27.0204	1.03848	0.519240	−	0.854628i	$-0.326215\pi$
$677$	27.0204	1.03848	0.519240	+	0.854628i	$0.326215\pi$
$678$	0	0
$679$	−57.6534	−2.21253
$680$	0	0
$681$	23.0858	0.884648
$682$	0	0
$683$	−26.8659	−1.02800	−0.513998	−	0.857791i	$-0.671836\pi$
$683$	−26.8659	−1.02800	−0.513998	+	0.857791i	$0.671836\pi$
$684$	0	0
$685$	−10.2875	−0.393067
$686$	0	0
$687$	54.1202	2.06481
$688$	0	0
$689$	−3.82908	−0.145877
$690$	0	0
$691$	−20.9855	−0.798327	−0.399164	−	0.916880i	$-0.630699\pi$
$691$	−20.9855	−0.798327	−0.399164	+	0.916880i	$0.630699\pi$
$692$	0	0
$693$	140.974	5.35514
$694$	0	0
$695$	−1.45175	−0.0550679
$696$	0	0
$697$	58.6467	2.22140
$698$	0	0
$699$	34.1540	1.29182
$700$	0	0
$701$	−29.2922	−1.10635	−0.553175	−	0.833065i	$-0.686584\pi$
$701$	−29.2922	−1.10635	−0.553175	+	0.833065i	$0.686584\pi$
$702$	0	0
$703$	−6.71379	−0.253215
$704$	0	0
$705$	55.6969	2.09767
$706$	0	0
$707$	18.9638	0.713205
$708$	0	0
$709$	−11.9758	−0.449762	−0.224881	−	0.974386i	$-0.572199\pi$
$709$	−11.9758	−0.449762	−0.224881	+	0.974386i	$0.572199\pi$
$710$	0	0
$711$	−81.2223	−3.04608
$712$	0	0
$713$	25.9855	0.973166
$714$	0	0
$715$	−8.06398	−0.301576
$716$	0	0
$717$	1.78017	0.0664816
$718$	0	0
$719$	−47.0025	−1.75290	−0.876449	−	0.481495i	$-0.840094\pi$
$719$	−47.0025	−1.75290	−0.876449	+	0.481495i	$0.840094\pi$
$720$	0	0
$721$	−1.42758	−0.0531660
$722$	0	0
$723$	−57.0635	−2.12221
$724$	0	0
$725$	−7.51871	−0.279238
$726$	0	0
$727$	10.7832	0.399925	0.199962	−	0.979804i	$-0.435918\pi$
$727$	10.7832	0.399925	0.199962	+	0.979804i	$0.435918\pi$
$728$	0	0
$729$	46.8961	1.73689
$730$	0	0
$731$	30.7827	1.13854
$732$	0	0
$733$	−16.9554	−0.626262	−0.313131	−	0.949710i	$-0.601378\pi$
$733$	−16.9554	−0.626262	−0.313131	+	0.949710i	$0.601378\pi$
$734$	0	0
$735$	−30.2325	−1.11514
$736$	0	0
$737$	5.12333	0.188720
$738$	0	0
$739$	25.3817	0.933679	0.466840	−	0.884342i	$-0.345393\pi$
$739$	25.3817	0.933679	0.466840	+	0.884342i	$0.345393\pi$
$740$	0	0
$741$	3.24698	0.119281
$742$	0	0
$743$	8.47517	0.310924	0.155462	−	0.987842i	$-0.450313\pi$
$743$	8.47517	0.310924	0.155462	+	0.987842i	$0.450313\pi$
$744$	0	0
$745$	−9.54288	−0.349624
$746$	0	0
$747$	−29.5472	−1.08108
$748$	0	0
$749$	1.76079	0.0643379
$750$	0	0
$751$	−35.0315	−1.27832	−0.639158	−	0.769075i	$-0.720717\pi$
$751$	−35.0315	−1.27832	−0.639158	+	0.769075i	$0.720717\pi$
$752$	0	0
$753$	13.3056	0.484882
$754$	0	0
$755$	−17.5459	−0.638559
$756$	0	0
$757$	−37.0374	−1.34615	−0.673074	−	0.739575i	$-0.735026\pi$
$757$	−37.0374	−1.34615	−0.673074	+	0.739575i	$0.735026\pi$
$758$	0	0
$759$	72.3376	2.62569
$760$	0	0
$761$	−12.4155	−0.450062	−0.225031	−	0.974352i	$-0.572248\pi$
$761$	−12.4155	−0.450062	−0.225031	+	0.974352i	$0.572248\pi$
$762$	0	0
$763$	9.78017	0.354066
$764$	0	0
$765$	−69.9130	−2.52771
$766$	0	0
$767$	13.5254	0.488375
$768$	0	0
$769$	31.8103	1.14711	0.573554	−	0.819168i	$-0.305564\pi$
$769$	31.8103	1.14711	0.573554	+	0.819168i	$0.305564\pi$
$770$	0	0
$771$	21.6233	0.778742
$772$	0	0
$773$	−9.54480	−0.343302	−0.171651	−	0.985158i	$-0.554910\pi$
$773$	−9.54480	−0.343302	−0.171651	+	0.985158i	$0.554910\pi$
$774$	0	0
$775$	15.6189	0.561049
$776$	0	0
$777$	78.5628	2.81843
$778$	0	0
$779$	−9.83877	−0.352511
$780$	0	0
$781$	24.4735	0.875731
$782$	0	0
$783$	−42.9517	−1.53497
$784$	0	0
$785$	−3.75600	−0.134058
$786$	0	0
$787$	38.5763	1.37509	0.687547	−	0.726139i	$-0.258687\pi$
$787$	38.5763	1.37509	0.687547	+	0.726139i	$0.258687\pi$
$788$	0	0
$789$	66.1613	2.35541
$790$	0	0
$791$	−64.4301	−2.29087
$792$	0	0
$793$	0.615957	0.0218733
$794$	0	0
$795$	−19.3327	−0.685661
$796$	0	0
$797$	42.2717	1.49734	0.748671	−	0.662942i	$-0.230692\pi$
$797$	42.2717	1.49734	0.748671	+	0.662942i	$0.230692\pi$
$798$	0	0
$799$	65.7560	2.32628
$800$	0	0
$801$	5.95779	0.210508
$802$	0	0
$803$	34.1849	1.20636
$804$	0	0
$805$	−24.0737	−0.848485
$806$	0	0
$807$	68.0689	2.39614
$808$	0	0
$809$	−13.2537	−0.465975	−0.232987	−	0.972480i	$-0.574850\pi$
$809$	−13.2537	−0.465975	−0.232987	+	0.972480i	$0.574850\pi$
$810$	0	0
$811$	8.98313	0.315440	0.157720	−	0.987484i	$-0.449586\pi$
$811$	8.98313	0.315440	0.157720	+	0.987484i	$0.449586\pi$
$812$	0	0
$813$	30.5483	1.07137
$814$	0	0
$815$	7.20105	0.252242
$816$	0	0
$817$	−5.16421	−0.180673
$818$	0	0
$819$	−27.1836	−0.949871
$820$	0	0
$821$	−21.9302	−0.765368	−0.382684	−	0.923879i	$-0.625000\pi$
$821$	−21.9302	−0.765368	−0.382684	+	0.923879i	$0.625000\pi$
$822$	0	0
$823$	12.0567	0.420270	0.210135	−	0.977672i	$-0.432610\pi$
$823$	12.0567	0.420270	0.210135	+	0.977672i	$0.432610\pi$
$824$	0	0
$825$	43.4795	1.51376
$826$	0	0
$827$	3.49529	0.121543	0.0607715	−	0.998152i	$-0.480644\pi$
$827$	3.49529	0.121543	0.0607715	+	0.998152i	$0.480644\pi$
$828$	0	0
$829$	33.8532	1.17577	0.587886	−	0.808944i	$-0.299960\pi$
$829$	33.8532	1.17577	0.587886	+	0.808944i	$0.299960\pi$
$830$	0	0
$831$	−71.4965	−2.48019
$832$	0	0
$833$	−35.6926	−1.23668
$834$	0	0
$835$	28.0428	0.970461
$836$	0	0
$837$	89.2253	3.08408
$838$	0	0
$839$	−38.2959	−1.32212	−0.661061	−	0.750333i	$-0.729893\pi$
$839$	−38.2959	−1.32212	−0.661061	+	0.750333i	$0.729893\pi$
$840$	0	0
$841$	−20.5211	−0.707624
$842$	0	0
$843$	−79.3473	−2.73287
$844$	0	0
$845$	1.55496	0.0534922
$846$	0	0
$847$	57.2814	1.96821
$848$	0	0
$849$	51.7318	1.77543
$850$	0	0
$851$	28.8418	0.988683
$852$	0	0
$853$	−9.33619	−0.319665	−0.159833	−	0.987144i	$-0.551095\pi$
$853$	−9.33619	−0.319665	−0.159833	+	0.987144i	$0.551095\pi$
$854$	0	0
$855$	11.7289	0.401118
$856$	0	0
$857$	−47.7405	−1.63078	−0.815392	−	0.578910i	$-0.803478\pi$
$857$	−47.7405	−1.63078	−0.815392	+	0.578910i	$0.803478\pi$
$858$	0	0
$859$	−10.3913	−0.354548	−0.177274	−	0.984162i	$-0.556728\pi$
$859$	−10.3913	−0.354548	−0.177274	+	0.984162i	$0.556728\pi$
$860$	0	0
$861$	115.130	3.92364
$862$	0	0
$863$	19.0398	0.648123	0.324061	−	0.946036i	$-0.394952\pi$
$863$	19.0398	0.648123	0.324061	+	0.946036i	$0.394952\pi$
$864$	0	0
$865$	−34.5368	−1.17429
$866$	0	0
$867$	−60.1691	−2.04345
$868$	0	0
$869$	−55.8431	−1.89435
$870$	0	0
$871$	−0.987918	−0.0334743
$872$	0	0
$873$	−120.668	−4.08399
$874$	0	0
$875$	−42.4892	−1.43640
$876$	0	0
$877$	35.8684	1.21119	0.605595	−	0.795773i	$-0.292935\pi$
$877$	35.8684	1.21119	0.605595	+	0.795773i	$0.292935\pi$
$878$	0	0
$879$	−50.3323	−1.69766
$880$	0	0
$881$	−37.3631	−1.25880	−0.629398	−	0.777083i	$-0.716698\pi$
$881$	−37.3631	−1.25880	−0.629398	+	0.777083i	$0.716698\pi$
$882$	0	0
$883$	19.6823	0.662363	0.331182	−	0.943567i	$-0.392553\pi$
$883$	19.6823	0.662363	0.331182	+	0.943567i	$0.392553\pi$
$884$	0	0
$885$	68.2887	2.29550
$886$	0	0
$887$	−35.6534	−1.19712	−0.598562	−	0.801077i	$-0.704261\pi$
$887$	−35.6534	−1.19712	−0.598562	+	0.801077i	$0.704261\pi$
$888$	0	0
$889$	5.78017	0.193861
$890$	0	0
$891$	131.031	4.38970
$892$	0	0
$893$	−11.0315	−0.369154
$894$	0	0
$895$	34.2476	1.14477
$896$	0	0
$897$	−13.9487	−0.465733
$898$	0	0
$899$	−17.6136	−0.587445
$900$	0	0
$901$	−22.8243	−0.760388
$902$	0	0
$903$	60.4301	2.01099
$904$	0	0
$905$	19.1468	0.636460
$906$	0	0
$907$	16.5757	0.550386	0.275193	−	0.961389i	$-0.411258\pi$
$907$	16.5757	0.550386	0.275193	+	0.961389i	$0.411258\pi$
$908$	0	0
$909$	39.6910	1.31647
$910$	0	0
$911$	28.4999	0.944245	0.472122	−	0.881533i	$-0.343488\pi$
$911$	28.4999	0.944245	0.472122	+	0.881533i	$0.343488\pi$
$912$	0	0
$913$	−20.3147	−0.672318
$914$	0	0
$915$	3.10992	0.102811
$916$	0	0
$917$	69.7706	2.30403
$918$	0	0
$919$	53.5297	1.76578	0.882891	−	0.469577i	$-0.155594\pi$
$919$	53.5297	1.76578	0.882891	+	0.469577i	$0.155594\pi$
$920$	0	0
$921$	−32.7525	−1.07923
$922$	0	0
$923$	−4.71917	−0.155333
$924$	0	0
$925$	17.3357	0.569995
$926$	0	0
$927$	−2.98792	−0.0981361
$928$	0	0
$929$	12.5133	0.410549	0.205275	−	0.978704i	$-0.434191\pi$
$929$	12.5133	0.410549	0.205275	+	0.978704i	$0.434191\pi$
$930$	0	0
$931$	5.98792	0.196246
$932$	0	0
$933$	−19.9898	−0.654438
$934$	0	0
$935$	−48.0676	−1.57198
$936$	0	0
$937$	44.3096	1.44753	0.723766	−	0.690045i	$-0.242409\pi$
$937$	44.3096	1.44753	0.723766	+	0.690045i	$0.242409\pi$
$938$	0	0
$939$	−46.7894	−1.52691
$940$	0	0
$941$	43.4094	1.41511	0.707553	−	0.706660i	$-0.249799\pi$
$941$	43.4094	1.41511	0.707553	+	0.706660i	$0.249799\pi$
$942$	0	0
$943$	42.2664	1.37638
$944$	0	0
$945$	−82.6607	−2.68895
$946$	0	0
$947$	−18.3338	−0.595768	−0.297884	−	0.954602i	$-0.596281\pi$
$947$	−18.3338	−0.595768	−0.297884	+	0.954602i	$0.596281\pi$
$948$	0	0
$949$	−6.59179	−0.213979
$950$	0	0
$951$	66.6413	2.16099
$952$	0	0
$953$	−1.43237	−0.0463990	−0.0231995	−	0.999731i	$-0.507385\pi$
$953$	−1.43237	−0.0463990	−0.0231995	+	0.999731i	$0.507385\pi$
$954$	0	0
$955$	−14.3260	−0.463579
$956$	0	0
$957$	−49.0320	−1.58498
$958$	0	0
$959$	−23.8431	−0.769933
$960$	0	0
$961$	5.58940	0.180303
$962$	0	0
$963$	3.68532	0.118758
$964$	0	0
$965$	22.2024	0.714720
$966$	0	0
$967$	43.6233	1.40283	0.701415	−	0.712753i	$-0.252552\pi$
$967$	43.6233	1.40283	0.701415	+	0.712753i	$0.252552\pi$
$968$	0	0
$969$	19.3545	0.621756
$970$	0	0
$971$	−33.2325	−1.06648	−0.533241	−	0.845963i	$-0.679026\pi$
$971$	−33.2325	−1.06648	−0.533241	+	0.845963i	$0.679026\pi$
$972$	0	0
$973$	−3.36467	−0.107866
$974$	0	0
$975$	−8.38404	−0.268504
$976$	0	0
$977$	−39.5579	−1.26557	−0.632785	−	0.774327i	$-0.718088\pi$
$977$	−39.5579	−1.26557	−0.632785	+	0.774327i	$0.718088\pi$
$978$	0	0
$979$	4.09618	0.130914
$980$	0	0
$981$	20.4698	0.653550
$982$	0	0
$983$	6.38703	0.203715	0.101857	−	0.994799i	$-0.467522\pi$
$983$	6.38703	0.203715	0.101857	+	0.994799i	$0.467522\pi$
$984$	0	0
$985$	18.3177	0.583649
$986$	0	0
$987$	129.087	4.10888
$988$	0	0
$989$	22.1849	0.705439
$990$	0	0
$991$	−1.76079	−0.0559333	−0.0279667	−	0.999609i	$-0.508903\pi$
$991$	−1.76079	−0.0559333	−0.0279667	+	0.999609i	$0.508903\pi$
$992$	0	0
$993$	−72.7827	−2.30969
$994$	0	0
$995$	−9.14377	−0.289877
$996$	0	0
$997$	−39.5362	−1.25212	−0.626062	−	0.779774i	$-0.715334\pi$
$997$	−39.5362	−1.25212	−0.626062	+	0.779774i	$0.715334\pi$
$998$	0	0
$999$	99.0326	3.13325

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3952.2.a.m.1.1		3
4.3	odd	2		494.2.a.g.1.3	✓	3
12.11	even	2		4446.2.a.bd.1.2		3
52.51	odd	2		6422.2.a.q.1.3		3
76.75	even	2		9386.2.a.u.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
494.2.a.g.1.3	✓	3	4.3	odd	2
3952.2.a.m.1.1		3	1.1	even	1	trivial
4446.2.a.bd.1.2		3	12.11	even	2
6422.2.a.q.1.3		3	52.51	odd	2
9386.2.a.u.1.1		3	76.75	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:	\( N \)	\(=\)	\( 3952 = 2^{4} \cdot 13 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3952.a (trivial)

\(f(q)\)	\(=\)	\(q-3.24698 q^{3} +1.55496 q^{5} +3.60388 q^{7} +7.54288 q^{9} +O(q^{10})\)	\(q-3.24698 q^{3} +1.55496 q^{5} +3.60388 q^{7} +7.54288 q^{9} +5.18598 q^{11} -1.00000 q^{13} -5.04892 q^{15} -5.96077 q^{17} +1.00000 q^{19} -11.7017 q^{21} -4.29590 q^{23} -2.58211 q^{25} -14.7506 q^{27} +2.91185 q^{29} -6.04892 q^{31} -16.8388 q^{33} +5.60388 q^{35} -6.71379 q^{37} +3.24698 q^{39} -9.83877 q^{41} -5.16421 q^{43} +11.7289 q^{45} -11.0315 q^{47} +5.98792 q^{49} +19.3545 q^{51} +3.82908 q^{53} +8.06398 q^{55} -3.24698 q^{57} -13.5254 q^{59} -0.615957 q^{61} +27.1836 q^{63} -1.55496 q^{65} +0.987918 q^{67} +13.9487 q^{69} +4.71917 q^{71} +6.59179 q^{73} +8.38404 q^{75} +18.6896 q^{77} -10.7681 q^{79} +25.2664 q^{81} -3.91723 q^{83} -9.26875 q^{85} -9.45473 q^{87} +0.789856 q^{89} -3.60388 q^{91} +19.6407 q^{93} +1.55496 q^{95} -15.9976 q^{97} +39.1172 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 5 q^{3} + 5 q^{5} + 2 q^{7} + 4 q^{9}+O(q^{10}) \) 3 * q - 5 * q^3 + 5 * q^5 + 2 * q^7 + 4 * q^9	\( 3 q - 5 q^{3} + 5 q^{5} + 2 q^{7} + 4 q^{9} + q^{11} - 3 q^{13} - 6 q^{15} - 5 q^{17} + 3 q^{19} - 8 q^{21} + q^{23} - 2 q^{25} - 8 q^{27} + 5 q^{29} - 9 q^{31} - 18 q^{33} + 8 q^{35} - 12 q^{37} + 5 q^{39} + 3 q^{41} - 4 q^{43} + 2 q^{45} - 8 q^{47} - q^{49} + 13 q^{51} + q^{53} - 10 q^{55} - 5 q^{57} - 6 q^{59} - 12 q^{61} + 26 q^{63} - 5 q^{65} - 16 q^{67} + 10 q^{69} + 3 q^{71} - 8 q^{73} + 15 q^{75} + 10 q^{77} - 12 q^{79} + 27 q^{81} - 5 q^{83} - 20 q^{85} - 6 q^{87} - 21 q^{89} - 2 q^{91} + 22 q^{93} + 5 q^{95} - 7 q^{97} + 55 q^{99}+O(q^{100}) \) 3 * q - 5 * q^3 + 5 * q^5 + 2 * q^7 + 4 * q^9 + q^11 - 3 * q^13 - 6 * q^15 - 5 * q^17 + 3 * q^19 - 8 * q^21 + q^23 - 2 * q^25 - 8 * q^27 + 5 * q^29 - 9 * q^31 - 18 * q^33 + 8 * q^35 - 12 * q^37 + 5 * q^39 + 3 * q^41 - 4 * q^43 + 2 * q^45 - 8 * q^47 - q^49 + 13 * q^51 + q^53 - 10 * q^55 - 5 * q^57 - 6 * q^59 - 12 * q^61 + 26 * q^63 - 5 * q^65 - 16 * q^67 + 10 * q^69 + 3 * q^71 - 8 * q^73 + 15 * q^75 + 10 * q^77 - 12 * q^79 + 27 * q^81 - 5 * q^83 - 20 * q^85 - 6 * q^87 - 21 * q^89 - 2 * q^91 + 22 * q^93 + 5 * q^95 - 7 * q^97 + 55 * q^99

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