LMFDB - Embedded newform 4016.2.a.m.1.21

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4016,2,Mod(1,4016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4016, 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("4016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4016 = 2^{4} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4016.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:	$32.0679214517$
Analytic rank:	$0$
Dimension:	$23$
Twist minimal:	no (minimal twist has level 2008)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.21
Character	$\chi$	$=$	4016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.19806 q^{3} -2.92729 q^{5} -4.80027 q^{7} +7.22760 q^{9} +O(q^{10})$	$q+3.19806 q^{3} -2.92729 q^{5} -4.80027 q^{7} +7.22760 q^{9} -5.38924 q^{11} +3.67192 q^{13} -9.36166 q^{15} -1.72483 q^{17} +7.21200 q^{19} -15.3516 q^{21} -4.61309 q^{23} +3.56904 q^{25} +13.5201 q^{27} +8.38123 q^{29} +2.29695 q^{31} -17.2351 q^{33} +14.0518 q^{35} -1.44327 q^{37} +11.7430 q^{39} +2.85580 q^{41} +8.84907 q^{43} -21.1573 q^{45} -5.70173 q^{47} +16.0426 q^{49} -5.51610 q^{51} +10.4814 q^{53} +15.7759 q^{55} +23.0644 q^{57} +8.60011 q^{59} +4.17805 q^{61} -34.6944 q^{63} -10.7488 q^{65} +13.8966 q^{67} -14.7530 q^{69} +2.80527 q^{71} +5.31177 q^{73} +11.4140 q^{75} +25.8698 q^{77} -2.35219 q^{79} +21.5554 q^{81} -9.40978 q^{83} +5.04907 q^{85} +26.8037 q^{87} -11.2350 q^{89} -17.6262 q^{91} +7.34577 q^{93} -21.1116 q^{95} +3.06930 q^{97} -38.9513 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - 2 q^{3} + 8 q^{5} - 2 q^{7} + 45 q^{9}+O(q^{10}) $ 23 * q - 2 * q^3 + 8 * q^5 - 2 * q^7 + 45 * q^9	$ 23 q - 2 q^{3} + 8 q^{5} - 2 q^{7} + 45 q^{9} - 8 q^{11} + 8 q^{13} - 7 q^{15} + 19 q^{17} + 9 q^{19} + 9 q^{21} - 21 q^{23} + 65 q^{25} - 5 q^{27} + 10 q^{29} + 9 q^{31} + 34 q^{33} - 12 q^{35} + 11 q^{37} + 9 q^{39} + 35 q^{41} + 9 q^{43} + 29 q^{45} - 37 q^{47} + 77 q^{49} + 17 q^{51} + 38 q^{53} + 20 q^{55} + 51 q^{57} - 17 q^{59} - 22 q^{63} + 41 q^{65} - 9 q^{67} + 8 q^{69} - 13 q^{71} + 41 q^{73} - 25 q^{75} + 36 q^{77} + 36 q^{79} + 127 q^{81} - 29 q^{83} + 34 q^{85} - 10 q^{87} + 36 q^{89} + 6 q^{91} + 36 q^{93} - 25 q^{95} + 40 q^{97} - 19 q^{99}+O(q^{100}) $ 23 * q - 2 * q^3 + 8 * q^5 - 2 * q^7 + 45 * q^9 - 8 * q^11 + 8 * q^13 - 7 * q^15 + 19 * q^17 + 9 * q^19 + 9 * q^21 - 21 * q^23 + 65 * q^25 - 5 * q^27 + 10 * q^29 + 9 * q^31 + 34 * q^33 - 12 * q^35 + 11 * q^37 + 9 * q^39 + 35 * q^41 + 9 * q^43 + 29 * q^45 - 37 * q^47 + 77 * q^49 + 17 * q^51 + 38 * q^53 + 20 * q^55 + 51 * q^57 - 17 * q^59 - 22 * q^63 + 41 * q^65 - 9 * q^67 + 8 * q^69 - 13 * q^71 + 41 * q^73 - 25 * q^75 + 36 * q^77 + 36 * q^79 + 127 * q^81 - 29 * q^83 + 34 * q^85 - 10 * q^87 + 36 * q^89 + 6 * q^91 + 36 * q^93 - 25 * q^95 + 40 * q^97 - 19 * 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
$3$	3.19806	1.84640	0.923201	−	0.384317i	$-0.125563\pi$
$3$	3.19806	1.84640	0.923201	+	0.384317i	$0.125563\pi$
$4$	0	0
$5$	−2.92729	−1.30912	−0.654562	−	0.756008i	$-0.727147\pi$
$5$	−2.92729	−1.30912	−0.654562	+	0.756008i	$0.727147\pi$
$6$	0	0
$7$	−4.80027	−1.81433	−0.907166	−	0.420774i	$-0.861759\pi$
$7$	−4.80027	−1.81433	−0.907166	+	0.420774i	$0.861759\pi$
$8$	0	0
$9$	7.22760	2.40920
$10$	0	0
$11$	−5.38924	−1.62492	−0.812458	−	0.583020i	$-0.801871\pi$
$11$	−5.38924	−1.62492	−0.812458	+	0.583020i	$0.801871\pi$
$12$	0	0
$13$	3.67192	1.01841	0.509203	−	0.860646i	$-0.329940\pi$
$13$	3.67192	1.01841	0.509203	+	0.860646i	$0.329940\pi$
$14$	0	0
$15$	−9.36166	−2.41717
$16$	0	0
$17$	−1.72483	−0.418332	−0.209166	−	0.977880i	$-0.567075\pi$
$17$	−1.72483	−0.418332	−0.209166	+	0.977880i	$0.567075\pi$
$18$	0	0
$19$	7.21200	1.65455	0.827273	−	0.561800i	$-0.189891\pi$
$19$	7.21200	1.65455	0.827273	+	0.561800i	$0.189891\pi$
$20$	0	0
$21$	−15.3516	−3.34999
$22$	0	0
$23$	−4.61309	−0.961896	−0.480948	−	0.876749i	$-0.659707\pi$
$23$	−4.61309	−0.961896	−0.480948	+	0.876749i	$0.659707\pi$
$24$	0	0
$25$	3.56904	0.713807
$26$	0	0
$27$	13.5201	2.60195
$28$	0	0
$29$	8.38123	1.55636	0.778178	−	0.628044i	$-0.216144\pi$
$29$	8.38123	1.55636	0.778178	+	0.628044i	$0.216144\pi$
$30$	0	0
$31$	2.29695	0.412544	0.206272	−	0.978495i	$-0.433867\pi$
$31$	2.29695	0.412544	0.206272	+	0.978495i	$0.433867\pi$
$32$	0	0
$33$	−17.2351	−3.00025
$34$	0	0
$35$	14.0518	2.37519
$36$	0	0
$37$	−1.44327	−0.237272	−0.118636	−	0.992938i	$-0.537852\pi$
$37$	−1.44327	−0.237272	−0.118636	+	0.992938i	$0.537852\pi$
$38$	0	0
$39$	11.7430	1.88039
$40$	0	0
$41$	2.85580	0.446000	0.223000	−	0.974818i	$-0.428415\pi$
$41$	2.85580	0.446000	0.223000	+	0.974818i	$0.428415\pi$
$42$	0	0
$43$	8.84907	1.34947	0.674735	−	0.738060i	$-0.264258\pi$
$43$	8.84907	1.34947	0.674735	+	0.738060i	$0.264258\pi$
$44$	0	0
$45$	−21.1573	−3.15394
$46$	0	0
$47$	−5.70173	−0.831683	−0.415842	−	0.909437i	$-0.636513\pi$
$47$	−5.70173	−0.831683	−0.415842	+	0.909437i	$0.636513\pi$
$48$	0	0
$49$	16.0426	2.29180
$50$	0	0
$51$	−5.51610	−0.772409
$52$	0	0
$53$	10.4814	1.43973	0.719863	−	0.694117i	$-0.244205\pi$
$53$	10.4814	1.43973	0.719863	+	0.694117i	$0.244205\pi$
$54$	0	0
$55$	15.7759	2.12722
$56$	0	0
$57$	23.0644	3.05496
$58$	0	0
$59$	8.60011	1.11964	0.559819	−	0.828615i	$-0.310871\pi$
$59$	8.60011	1.11964	0.559819	+	0.828615i	$0.310871\pi$
$60$	0	0
$61$	4.17805	0.534945	0.267472	−	0.963566i	$-0.413812\pi$
$61$	4.17805	0.534945	0.267472	+	0.963566i	$0.413812\pi$
$62$	0	0
$63$	−34.6944	−4.37109
$64$	0	0
$65$	−10.7488	−1.33322
$66$	0	0
$67$	13.8966	1.69773	0.848867	−	0.528606i	$-0.177285\pi$
$67$	13.8966	1.69773	0.848867	+	0.528606i	$0.177285\pi$
$68$	0	0
$69$	−14.7530	−1.77605
$70$	0	0
$71$	2.80527	0.332925	0.166462	−	0.986048i	$-0.446766\pi$
$71$	2.80527	0.332925	0.166462	+	0.986048i	$0.446766\pi$
$72$	0	0
$73$	5.31177	0.621695	0.310848	−	0.950460i	$-0.399387\pi$
$73$	5.31177	0.621695	0.310848	+	0.950460i	$0.399387\pi$
$74$	0	0
$75$	11.4140	1.31798
$76$	0	0
$77$	25.8698	2.94814
$78$	0	0
$79$	−2.35219	−0.264642	−0.132321	−	0.991207i	$-0.542243\pi$
$79$	−2.35219	−0.264642	−0.132321	+	0.991207i	$0.542243\pi$
$80$	0	0
$81$	21.5554	2.39505
$82$	0	0
$83$	−9.40978	−1.03286	−0.516429	−	0.856330i	$-0.672739\pi$
$83$	−9.40978	−1.03286	−0.516429	+	0.856330i	$0.672739\pi$
$84$	0	0
$85$	5.04907	0.547648
$86$	0	0
$87$	26.8037	2.87366
$88$	0	0
$89$	−11.2350	−1.19091	−0.595453	−	0.803391i	$-0.703027\pi$
$89$	−11.2350	−1.19091	−0.595453	+	0.803391i	$0.703027\pi$
$90$	0	0
$91$	−17.6262	−1.84773
$92$	0	0
$93$	7.34577	0.761721
$94$	0	0
$95$	−21.1116	−2.16601
$96$	0	0
$97$	3.06930	0.311640	0.155820	−	0.987785i	$-0.450198\pi$
$97$	3.06930	0.311640	0.155820	+	0.987785i	$0.450198\pi$
$98$	0	0
$99$	−38.9513	−3.91475
$100$	0	0
$101$	6.15938	0.612882	0.306441	−	0.951890i	$-0.400862\pi$
$101$	6.15938	0.612882	0.306441	+	0.951890i	$0.400862\pi$
$102$	0	0
$103$	0.0697355	0.00687124	0.00343562	−	0.999994i	$-0.498906\pi$
$103$	0.0697355	0.00687124	0.00343562	+	0.999994i	$0.498906\pi$
$104$	0	0
$105$	44.9385	4.38555
$106$	0	0
$107$	−19.4452	−1.87984	−0.939921	−	0.341393i	$-0.889101\pi$
$107$	−19.4452	−1.87984	−0.939921	+	0.341393i	$0.889101\pi$
$108$	0	0
$109$	−12.9318	−1.23864	−0.619321	−	0.785138i	$-0.712592\pi$
$109$	−12.9318	−1.23864	−0.619321	+	0.785138i	$0.712592\pi$
$110$	0	0
$111$	−4.61567	−0.438100
$112$	0	0
$113$	0.124030	0.0116677	0.00583387	−	0.999983i	$-0.498143\pi$
$113$	0.124030	0.0116677	0.00583387	+	0.999983i	$0.498143\pi$
$114$	0	0
$115$	13.5039	1.25924
$116$	0	0
$117$	26.5391	2.45355
$118$	0	0
$119$	8.27963	0.758992
$120$	0	0
$121$	18.0439	1.64035
$122$	0	0
$123$	9.13301	0.823496
$124$	0	0
$125$	4.18884	0.374662
$126$	0	0
$127$	−2.30304	−0.204361	−0.102181	−	0.994766i	$-0.532582\pi$
$127$	−2.30304	−0.204361	−0.102181	+	0.994766i	$0.532582\pi$
$128$	0	0
$129$	28.2999	2.49166
$130$	0	0
$131$	−2.04288	−0.178487	−0.0892437	−	0.996010i	$-0.528445\pi$
$131$	−2.04288	−0.178487	−0.0892437	+	0.996010i	$0.528445\pi$
$132$	0	0
$133$	−34.6196	−3.00190
$134$	0	0
$135$	−39.5774	−3.40628
$136$	0	0
$137$	−7.07129	−0.604141	−0.302071	−	0.953286i	$-0.597678\pi$
$137$	−7.07129	−0.604141	−0.302071	+	0.953286i	$0.597678\pi$
$138$	0	0
$139$	−1.85271	−0.157145	−0.0785725	−	0.996908i	$-0.525036\pi$
$139$	−1.85271	−0.157145	−0.0785725	+	0.996908i	$0.525036\pi$
$140$	0	0
$141$	−18.2345	−1.53562
$142$	0	0
$143$	−19.7888	−1.65482
$144$	0	0
$145$	−24.5343	−2.03746
$146$	0	0
$147$	51.3052	4.23158
$148$	0	0
$149$	−0.321930	−0.0263736	−0.0131868	−	0.999913i	$-0.504198\pi$
$149$	−0.321930	−0.0263736	−0.0131868	+	0.999913i	$0.504198\pi$
$150$	0	0
$151$	6.99687	0.569397	0.284699	−	0.958617i	$-0.408106\pi$
$151$	6.99687	0.569397	0.284699	+	0.958617i	$0.408106\pi$
$152$	0	0
$153$	−12.4664	−1.00785
$154$	0	0
$155$	−6.72383	−0.540071
$156$	0	0
$157$	−0.735912	−0.0587322	−0.0293661	−	0.999569i	$-0.509349\pi$
$157$	−0.735912	−0.0587322	−0.0293661	+	0.999569i	$0.509349\pi$
$158$	0	0
$159$	33.5200	2.65831
$160$	0	0
$161$	22.1441	1.74520
$162$	0	0
$163$	−7.91265	−0.619767	−0.309883	−	0.950775i	$-0.600290\pi$
$163$	−7.91265	−0.619767	−0.309883	+	0.950775i	$0.600290\pi$
$164$	0	0
$165$	50.4522	3.92770
$166$	0	0
$167$	−9.23080	−0.714301	−0.357151	−	0.934047i	$-0.616252\pi$
$167$	−9.23080	−0.714301	−0.357151	+	0.934047i	$0.616252\pi$
$168$	0	0
$169$	0.482963	0.0371510
$170$	0	0
$171$	52.1255	3.98614
$172$	0	0
$173$	−13.5569	−1.03071	−0.515357	−	0.856976i	$-0.672341\pi$
$173$	−13.5569	−1.03071	−0.515357	+	0.856976i	$0.672341\pi$
$174$	0	0
$175$	−17.1323	−1.29508
$176$	0	0
$177$	27.5037	2.06730
$178$	0	0
$179$	1.27782	0.0955091	0.0477546	−	0.998859i	$-0.484793\pi$
$179$	1.27782	0.0955091	0.0477546	+	0.998859i	$0.484793\pi$
$180$	0	0
$181$	12.5561	0.933286	0.466643	−	0.884446i	$-0.345463\pi$
$181$	12.5561	0.933286	0.466643	+	0.884446i	$0.345463\pi$
$182$	0	0
$183$	13.3617	0.987723
$184$	0	0
$185$	4.22488	0.310619
$186$	0	0
$187$	9.29549	0.679754
$188$	0	0
$189$	−64.9003	−4.72080
$190$	0	0
$191$	25.7230	1.86125	0.930624	−	0.365977i	$-0.119265\pi$
$191$	25.7230	1.86125	0.930624	+	0.365977i	$0.119265\pi$
$192$	0	0
$193$	1.33661	0.0962115	0.0481057	−	0.998842i	$-0.484682\pi$
$193$	1.33661	0.0962115	0.0481057	+	0.998842i	$0.484682\pi$
$194$	0	0
$195$	−34.3752	−2.46166
$196$	0	0
$197$	24.6503	1.75626	0.878132	−	0.478419i	$-0.158790\pi$
$197$	24.6503	1.75626	0.878132	+	0.478419i	$0.158790\pi$
$198$	0	0
$199$	16.9664	1.20272	0.601359	−	0.798979i	$-0.294626\pi$
$199$	16.9664	1.20272	0.601359	+	0.798979i	$0.294626\pi$
$200$	0	0
$201$	44.4420	3.13470
$202$	0	0
$203$	−40.2322	−2.82375
$204$	0	0
$205$	−8.35975	−0.583870
$206$	0	0
$207$	−33.3416	−2.31740
$208$	0	0
$209$	−38.8672	−2.68850
$210$	0	0
$211$	−2.54109	−0.174936	−0.0874680	−	0.996167i	$-0.527878\pi$
$211$	−2.54109	−0.174936	−0.0874680	+	0.996167i	$0.527878\pi$
$212$	0	0
$213$	8.97144	0.614713
$214$	0	0
$215$	−25.9038	−1.76662
$216$	0	0
$217$	−11.0260	−0.748491
$218$	0	0
$219$	16.9874	1.14790
$220$	0	0
$221$	−6.33341	−0.426032
$222$	0	0
$223$	6.43875	0.431171	0.215585	−	0.976485i	$-0.430834\pi$
$223$	6.43875	0.431171	0.215585	+	0.976485i	$0.430834\pi$
$224$	0	0
$225$	25.7956	1.71971
$226$	0	0
$227$	5.56731	0.369515	0.184758	−	0.982784i	$-0.440850\pi$
$227$	5.56731	0.369515	0.184758	+	0.982784i	$0.440850\pi$
$228$	0	0
$229$	16.1586	1.06779	0.533894	−	0.845551i	$-0.320728\pi$
$229$	16.1586	1.06779	0.533894	+	0.845551i	$0.320728\pi$
$230$	0	0
$231$	82.7332	5.44344
$232$	0	0
$233$	18.5402	1.21461	0.607304	−	0.794470i	$-0.292251\pi$
$233$	18.5402	1.21461	0.607304	+	0.794470i	$0.292251\pi$
$234$	0	0
$235$	16.6906	1.08878
$236$	0	0
$237$	−7.52244	−0.488635
$238$	0	0
$239$	−17.9072	−1.15832	−0.579161	−	0.815214i	$-0.696620\pi$
$239$	−17.9072	−1.15832	−0.579161	+	0.815214i	$0.696620\pi$
$240$	0	0
$241$	18.1717	1.17054	0.585271	−	0.810838i	$-0.300988\pi$
$241$	18.1717	1.17054	0.585271	+	0.810838i	$0.300988\pi$
$242$	0	0
$243$	28.3752	1.82027
$244$	0	0
$245$	−46.9613	−3.00025
$246$	0	0
$247$	26.4819	1.68500
$248$	0	0
$249$	−30.0931	−1.90707
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	24.8610	1.56300
$254$	0	0
$255$	16.1472	1.01118
$256$	0	0
$257$	20.3496	1.26937	0.634687	−	0.772769i	$-0.281129\pi$
$257$	20.3496	1.26937	0.634687	+	0.772769i	$0.281129\pi$
$258$	0	0
$259$	6.92809	0.430491
$260$	0	0
$261$	60.5762	3.74958
$262$	0	0
$263$	−6.76300	−0.417024	−0.208512	−	0.978020i	$-0.566862\pi$
$263$	−6.76300	−0.417024	−0.208512	+	0.978020i	$0.566862\pi$
$264$	0	0
$265$	−30.6820	−1.88478
$266$	0	0
$267$	−35.9302	−2.19889
$268$	0	0
$269$	17.6565	1.07654	0.538269	−	0.842773i	$-0.319078\pi$
$269$	17.6565	1.07654	0.538269	+	0.842773i	$0.319078\pi$
$270$	0	0
$271$	−25.3932	−1.54253	−0.771263	−	0.636516i	$-0.780375\pi$
$271$	−25.3932	−1.54253	−0.771263	+	0.636516i	$0.780375\pi$
$272$	0	0
$273$	−56.3696	−3.41165
$274$	0	0
$275$	−19.2344	−1.15988
$276$	0	0
$277$	−2.73668	−0.164431	−0.0822157	−	0.996615i	$-0.526200\pi$
$277$	−2.73668	−0.164431	−0.0822157	+	0.996615i	$0.526200\pi$
$278$	0	0
$279$	16.6014	0.993900
$280$	0	0
$281$	−7.22414	−0.430956	−0.215478	−	0.976509i	$-0.569131\pi$
$281$	−7.22414	−0.430956	−0.215478	+	0.976509i	$0.569131\pi$
$282$	0	0
$283$	−1.16498	−0.0692509	−0.0346255	−	0.999400i	$-0.511024\pi$
$283$	−1.16498	−0.0692509	−0.0346255	+	0.999400i	$0.511024\pi$
$284$	0	0
$285$	−67.5163	−3.99932
$286$	0	0
$287$	−13.7086	−0.809192
$288$	0	0
$289$	−14.0250	−0.824999
$290$	0	0
$291$	9.81582	0.575414
$292$	0	0
$293$	−4.16693	−0.243435	−0.121717	−	0.992565i	$-0.538840\pi$
$293$	−4.16693	−0.243435	−0.121717	+	0.992565i	$0.538840\pi$
$294$	0	0
$295$	−25.1750	−1.46575
$296$	0	0
$297$	−72.8632	−4.22795
$298$	0	0
$299$	−16.9389	−0.979601
$300$	0	0
$301$	−42.4779	−2.44839
$302$	0	0
$303$	19.6981	1.13163
$304$	0	0
$305$	−12.2304	−0.700310
$306$	0	0
$307$	−2.75953	−0.157495	−0.0787473	−	0.996895i	$-0.525092\pi$
$307$	−2.75953	−0.157495	−0.0787473	+	0.996895i	$0.525092\pi$
$308$	0	0
$309$	0.223019	0.0126871
$310$	0	0
$311$	−5.83186	−0.330695	−0.165347	−	0.986235i	$-0.552875\pi$
$311$	−5.83186	−0.330695	−0.165347	+	0.986235i	$0.552875\pi$
$312$	0	0
$313$	−33.2999	−1.88222	−0.941110	−	0.338101i	$-0.890215\pi$
$313$	−33.2999	−1.88222	−0.941110	+	0.338101i	$0.890215\pi$
$314$	0	0
$315$	101.561	5.72230
$316$	0	0
$317$	1.36161	0.0764754	0.0382377	−	0.999269i	$-0.487826\pi$
$317$	1.36161	0.0764754	0.0382377	+	0.999269i	$0.487826\pi$
$318$	0	0
$319$	−45.1684	−2.52895
$320$	0	0
$321$	−62.1871	−3.47094
$322$	0	0
$323$	−12.4394	−0.692149
$324$	0	0
$325$	13.1052	0.726946
$326$	0	0
$327$	−41.3567	−2.28703
$328$	0	0
$329$	27.3698	1.50895
$330$	0	0
$331$	18.9575	1.04200	0.521000	−	0.853557i	$-0.325559\pi$
$331$	18.9575	1.04200	0.521000	+	0.853557i	$0.325559\pi$
$332$	0	0
$333$	−10.4314	−0.571637
$334$	0	0
$335$	−40.6793	−2.22255
$336$	0	0
$337$	27.1185	1.47724	0.738619	−	0.674123i	$-0.235478\pi$
$337$	27.1185	1.47724	0.738619	+	0.674123i	$0.235478\pi$
$338$	0	0
$339$	0.396655	0.0215434
$340$	0	0
$341$	−12.3788	−0.670348
$342$	0	0
$343$	−43.4068	−2.34375
$344$	0	0
$345$	43.1862	2.32507
$346$	0	0
$347$	34.8231	1.86940	0.934701	−	0.355434i	$-0.115667\pi$
$347$	34.8231	1.86940	0.934701	+	0.355434i	$0.115667\pi$
$348$	0	0
$349$	18.9510	1.01442	0.507212	−	0.861821i	$-0.330676\pi$
$349$	18.9510	1.01442	0.507212	+	0.861821i	$0.330676\pi$
$350$	0	0
$351$	49.6448	2.64984
$352$	0	0
$353$	6.85117	0.364651	0.182325	−	0.983238i	$-0.441638\pi$
$353$	6.85117	0.364651	0.182325	+	0.983238i	$0.441638\pi$
$354$	0	0
$355$	−8.21186	−0.435840
$356$	0	0
$357$	26.4788	1.40141
$358$	0	0
$359$	−1.76471	−0.0931381	−0.0465690	−	0.998915i	$-0.514829\pi$
$359$	−1.76471	−0.0931381	−0.0465690	+	0.998915i	$0.514829\pi$
$360$	0	0
$361$	33.0130	1.73752
$362$	0	0
$363$	57.7054	3.02875
$364$	0	0
$365$	−15.5491	−0.813876
$366$	0	0
$367$	−29.1014	−1.51908	−0.759540	−	0.650460i	$-0.774576\pi$
$367$	−29.1014	−1.51908	−0.759540	+	0.650460i	$0.774576\pi$
$368$	0	0
$369$	20.6406	1.07450
$370$	0	0
$371$	−50.3133	−2.61214
$372$	0	0
$373$	−2.27340	−0.117712	−0.0588561	−	0.998266i	$-0.518745\pi$
$373$	−2.27340	−0.117712	−0.0588561	+	0.998266i	$0.518745\pi$
$374$	0	0
$375$	13.3962	0.691776
$376$	0	0
$377$	30.7752	1.58500
$378$	0	0
$379$	−19.6516	−1.00943	−0.504717	−	0.863285i	$-0.668403\pi$
$379$	−19.6516	−1.00943	−0.504717	+	0.863285i	$0.668403\pi$
$380$	0	0
$381$	−7.36525	−0.377333
$382$	0	0
$383$	−0.297677	−0.0152106	−0.00760530	−	0.999971i	$-0.502421\pi$
$383$	−0.297677	−0.0152106	−0.00760530	+	0.999971i	$0.502421\pi$
$384$	0	0
$385$	−75.7284	−3.85948
$386$	0	0
$387$	63.9576	3.25115
$388$	0	0
$389$	11.4500	0.580540	0.290270	−	0.956945i	$-0.406255\pi$
$389$	11.4500	0.580540	0.290270	+	0.956945i	$0.406255\pi$
$390$	0	0
$391$	7.95678	0.402392
$392$	0	0
$393$	−6.53326	−0.329560
$394$	0	0
$395$	6.88554	0.346449
$396$	0	0
$397$	26.5892	1.33447	0.667236	−	0.744847i	$-0.267477\pi$
$397$	26.5892	1.33447	0.667236	+	0.744847i	$0.267477\pi$
$398$	0	0
$399$	−110.715	−5.54271
$400$	0	0
$401$	34.8482	1.74024	0.870119	−	0.492841i	$-0.164042\pi$
$401$	34.8482	1.74024	0.870119	+	0.492841i	$0.164042\pi$
$402$	0	0
$403$	8.43419	0.420137
$404$	0	0
$405$	−63.0991	−3.13542
$406$	0	0
$407$	7.77813	0.385548
$408$	0	0
$409$	29.0693	1.43738	0.718691	−	0.695329i	$-0.244741\pi$
$409$	29.0693	1.43738	0.718691	+	0.695329i	$0.244741\pi$
$410$	0	0
$411$	−22.6144	−1.11549
$412$	0	0
$413$	−41.2828	−2.03140
$414$	0	0
$415$	27.5452	1.35214
$416$	0	0
$417$	−5.92509	−0.290153
$418$	0	0
$419$	14.1926	0.693352	0.346676	−	0.937985i	$-0.387310\pi$
$419$	14.1926	0.693352	0.346676	+	0.937985i	$0.387310\pi$
$420$	0	0
$421$	−24.1196	−1.17552	−0.587758	−	0.809037i	$-0.699989\pi$
$421$	−24.1196	−1.17552	−0.587758	+	0.809037i	$0.699989\pi$
$422$	0	0
$423$	−41.2099	−2.00369
$424$	0	0
$425$	−6.15597	−0.298608
$426$	0	0
$427$	−20.0558	−0.970567
$428$	0	0
$429$	−63.2859	−3.05547
$430$	0	0
$431$	20.4134	0.983278	0.491639	−	0.870799i	$-0.336398\pi$
$431$	20.4134	0.983278	0.491639	+	0.870799i	$0.336398\pi$
$432$	0	0
$433$	−27.6525	−1.32890	−0.664448	−	0.747335i	$-0.731333\pi$
$433$	−27.6525	−1.32890	−0.664448	+	0.747335i	$0.731333\pi$
$434$	0	0
$435$	−78.4623	−3.76198
$436$	0	0
$437$	−33.2696	−1.59150
$438$	0	0
$439$	32.0362	1.52900	0.764501	−	0.644622i	$-0.222985\pi$
$439$	32.0362	1.52900	0.764501	+	0.644622i	$0.222985\pi$
$440$	0	0
$441$	115.949	5.52140
$442$	0	0
$443$	−6.96275	−0.330810	−0.165405	−	0.986226i	$-0.552893\pi$
$443$	−6.96275	−0.330810	−0.165405	+	0.986226i	$0.552893\pi$
$444$	0	0
$445$	32.8881	1.55904
$446$	0	0
$447$	−1.02955	−0.0486962
$448$	0	0
$449$	−36.8740	−1.74019	−0.870096	−	0.492883i	$-0.835943\pi$
$449$	−36.8740	−1.74019	−0.870096	+	0.492883i	$0.835943\pi$
$450$	0	0
$451$	−15.3906	−0.724713
$452$	0	0
$453$	22.3764	1.05134
$454$	0	0
$455$	51.5970	2.41890
$456$	0	0
$457$	4.30357	0.201313	0.100656	−	0.994921i	$-0.467906\pi$
$457$	4.30357	0.201313	0.100656	+	0.994921i	$0.467906\pi$
$458$	0	0
$459$	−23.3199	−1.08848
$460$	0	0
$461$	15.2115	0.708470	0.354235	−	0.935157i	$-0.384741\pi$
$461$	15.2115	0.708470	0.354235	+	0.935157i	$0.384741\pi$
$462$	0	0
$463$	−26.8908	−1.24972	−0.624860	−	0.780737i	$-0.714844\pi$
$463$	−26.8908	−1.24972	−0.624860	+	0.780737i	$0.714844\pi$
$464$	0	0
$465$	−21.5032	−0.997188
$466$	0	0
$467$	−40.2376	−1.86197	−0.930987	−	0.365051i	$-0.881051\pi$
$467$	−40.2376	−1.86197	−0.930987	+	0.365051i	$0.881051\pi$
$468$	0	0
$469$	−66.7072	−3.08025
$470$	0	0
$471$	−2.35349	−0.108443
$472$	0	0
$473$	−47.6897	−2.19278
$474$	0	0
$475$	25.7399	1.18103
$476$	0	0
$477$	75.7551	3.46859
$478$	0	0
$479$	27.4465	1.25406	0.627032	−	0.778994i	$-0.284270\pi$
$479$	27.4465	1.25406	0.627032	+	0.778994i	$0.284270\pi$
$480$	0	0
$481$	−5.29957	−0.241640
$482$	0	0
$483$	70.8182	3.22234
$484$	0	0
$485$	−8.98474	−0.407976
$486$	0	0
$487$	1.35262	0.0612929	0.0306464	−	0.999530i	$-0.490243\pi$
$487$	1.35262	0.0612929	0.0306464	+	0.999530i	$0.490243\pi$
$488$	0	0
$489$	−25.3052	−1.14434
$490$	0	0
$491$	−3.87149	−0.174718	−0.0873589	−	0.996177i	$-0.527843\pi$
$491$	−3.87149	−0.174718	−0.0873589	+	0.996177i	$0.527843\pi$
$492$	0	0
$493$	−14.4562	−0.651073
$494$	0	0
$495$	114.022	5.12489
$496$	0	0
$497$	−13.4661	−0.604036
$498$	0	0
$499$	23.9416	1.07177	0.535887	−	0.844290i	$-0.319977\pi$
$499$	23.9416	1.07177	0.535887	+	0.844290i	$0.319977\pi$
$500$	0	0
$501$	−29.5207	−1.31889
$502$	0	0
$503$	16.8380	0.750771	0.375386	−	0.926869i	$-0.377510\pi$
$503$	16.8380	0.750771	0.375386	+	0.926869i	$0.377510\pi$
$504$	0	0
$505$	−18.0303	−0.802339
$506$	0	0
$507$	1.54455	0.0685957
$508$	0	0
$509$	−21.8072	−0.966586	−0.483293	−	0.875459i	$-0.660559\pi$
$509$	−21.8072	−0.966586	−0.483293	+	0.875459i	$0.660559\pi$
$510$	0	0
$511$	−25.4979	−1.12796
$512$	0	0
$513$	97.5073	4.30505
$514$	0	0
$515$	−0.204136	−0.00899532
$516$	0	0
$517$	30.7280	1.35141
$518$	0	0
$519$	−43.3559	−1.90311
$520$	0	0
$521$	−25.2240	−1.10508	−0.552541	−	0.833486i	$-0.686342\pi$
$521$	−25.2240	−1.10508	−0.552541	+	0.833486i	$0.686342\pi$
$522$	0	0
$523$	−6.26041	−0.273749	−0.136874	−	0.990588i	$-0.543706\pi$
$523$	−6.26041	−0.273749	−0.136874	+	0.990588i	$0.543706\pi$
$524$	0	0
$525$	−54.7903	−2.39124
$526$	0	0
$527$	−3.96183	−0.172580
$528$	0	0
$529$	−1.71938	−0.0747555
$530$	0	0
$531$	62.1582	2.69743
$532$	0	0
$533$	10.4862	0.454210
$534$	0	0
$535$	56.9219	2.46095
$536$	0	0
$537$	4.08656	0.176348
$538$	0	0
$539$	−86.4573	−3.72398
$540$	0	0
$541$	−38.8614	−1.67078	−0.835392	−	0.549655i	$-0.814759\pi$
$541$	−38.8614	−1.67078	−0.835392	+	0.549655i	$0.814759\pi$
$542$	0	0
$543$	40.1551	1.72322
$544$	0	0
$545$	37.8552	1.62154
$546$	0	0
$547$	−23.4884	−1.00429	−0.502145	−	0.864783i	$-0.667456\pi$
$547$	−23.4884	−1.00429	−0.502145	+	0.864783i	$0.667456\pi$
$548$	0	0
$549$	30.1973	1.28879
$550$	0	0
$551$	60.4455	2.57506
$552$	0	0
$553$	11.2911	0.480148
$554$	0	0
$555$	13.5114	0.573528
$556$	0	0
$557$	24.3134	1.03019	0.515096	−	0.857133i	$-0.327756\pi$
$557$	24.3134	1.03019	0.515096	+	0.857133i	$0.327756\pi$
$558$	0	0
$559$	32.4930	1.37431
$560$	0	0
$561$	29.7276	1.25510
$562$	0	0
$563$	27.9079	1.17618	0.588090	−	0.808796i	$-0.299880\pi$
$563$	27.9079	1.17618	0.588090	+	0.808796i	$0.299880\pi$
$564$	0	0
$565$	−0.363072	−0.0152745
$566$	0	0
$567$	−103.472	−4.34541
$568$	0	0
$569$	−7.44862	−0.312262	−0.156131	−	0.987736i	$-0.549902\pi$
$569$	−7.44862	−0.312262	−0.156131	+	0.987736i	$0.549902\pi$
$570$	0	0
$571$	−45.6466	−1.91025	−0.955125	−	0.296202i	$-0.904280\pi$
$571$	−45.6466	−1.91025	−0.955125	+	0.296202i	$0.904280\pi$
$572$	0	0
$573$	82.2636	3.43661
$574$	0	0
$575$	−16.4643	−0.686609
$576$	0	0
$577$	2.85988	0.119058	0.0595292	−	0.998227i	$-0.481040\pi$
$577$	2.85988	0.119058	0.0595292	+	0.998227i	$0.481040\pi$
$578$	0	0
$579$	4.27457	0.177645
$580$	0	0
$581$	45.1695	1.87395
$582$	0	0
$583$	−56.4865	−2.33943
$584$	0	0
$585$	−77.6878	−3.21200
$586$	0	0
$587$	−16.4355	−0.678365	−0.339183	−	0.940721i	$-0.610150\pi$
$587$	−16.4355	−0.678365	−0.339183	+	0.940721i	$0.610150\pi$
$588$	0	0
$589$	16.5656	0.682573
$590$	0	0
$591$	78.8333	3.24277
$592$	0	0
$593$	38.4253	1.57794	0.788970	−	0.614432i	$-0.210615\pi$
$593$	38.4253	1.57794	0.788970	+	0.614432i	$0.210615\pi$
$594$	0	0
$595$	−24.2369	−0.993616
$596$	0	0
$597$	54.2597	2.22070
$598$	0	0
$599$	36.6870	1.49899	0.749494	−	0.662011i	$-0.230297\pi$
$599$	36.6870	1.49899	0.749494	+	0.662011i	$0.230297\pi$
$600$	0	0
$601$	31.9725	1.30418	0.652092	−	0.758140i	$-0.273892\pi$
$601$	31.9725	1.30418	0.652092	+	0.758140i	$0.273892\pi$
$602$	0	0
$603$	100.439	4.09018
$604$	0	0
$605$	−52.8196	−2.14742
$606$	0	0
$607$	−15.8570	−0.643616	−0.321808	−	0.946805i	$-0.604291\pi$
$607$	−15.8570	−0.643616	−0.321808	+	0.946805i	$0.604291\pi$
$608$	0	0
$609$	−128.665	−5.21377
$610$	0	0
$611$	−20.9363	−0.846991
$612$	0	0
$613$	−8.37441	−0.338239	−0.169120	−	0.985596i	$-0.554092\pi$
$613$	−8.37441	−0.338239	−0.169120	+	0.985596i	$0.554092\pi$
$614$	0	0
$615$	−26.7350	−1.07806
$616$	0	0
$617$	15.9710	0.642967	0.321483	−	0.946915i	$-0.395819\pi$
$617$	15.9710	0.642967	0.321483	+	0.946915i	$0.395819\pi$
$618$	0	0
$619$	1.36153	0.0547246	0.0273623	−	0.999626i	$-0.491289\pi$
$619$	1.36153	0.0547246	0.0273623	+	0.999626i	$0.491289\pi$
$620$	0	0
$621$	−62.3697	−2.50281
$622$	0	0
$623$	53.9309	2.16070
$624$	0	0
$625$	−30.1072	−1.20429
$626$	0	0
$627$	−124.300	−4.96405
$628$	0	0
$629$	2.48939	0.0992586
$630$	0	0
$631$	−46.4940	−1.85090	−0.925448	−	0.378875i	$-0.876311\pi$
$631$	−46.4940	−1.85090	−0.925448	+	0.378875i	$0.876311\pi$
$632$	0	0
$633$	−8.12657	−0.323002
$634$	0	0
$635$	6.74166	0.267535
$636$	0	0
$637$	58.9070	2.33398
$638$	0	0
$639$	20.2754	0.802083
$640$	0	0
$641$	−3.48222	−0.137540	−0.0687698	−	0.997633i	$-0.521907\pi$
$641$	−3.48222	−0.137540	−0.0687698	+	0.997633i	$0.521907\pi$
$642$	0	0
$643$	20.4276	0.805587	0.402793	−	0.915291i	$-0.368039\pi$
$643$	20.4276	0.805587	0.402793	+	0.915291i	$0.368039\pi$
$644$	0	0
$645$	−82.8420	−3.26190
$646$	0	0
$647$	−1.24285	−0.0488614	−0.0244307	−	0.999702i	$-0.507777\pi$
$647$	−1.24285	−0.0488614	−0.0244307	+	0.999702i	$0.507777\pi$
$648$	0	0
$649$	−46.3480	−1.81932
$650$	0	0
$651$	−35.2617	−1.38201
$652$	0	0
$653$	−28.8096	−1.12741	−0.563703	−	0.825977i	$-0.690624\pi$
$653$	−28.8096	−1.12741	−0.563703	+	0.825977i	$0.690624\pi$
$654$	0	0
$655$	5.98011	0.233662
$656$	0	0
$657$	38.3913	1.49779
$658$	0	0
$659$	31.3383	1.22077	0.610383	−	0.792106i	$-0.291015\pi$
$659$	31.3383	1.22077	0.610383	+	0.792106i	$0.291015\pi$
$660$	0	0
$661$	38.4169	1.49424	0.747122	−	0.664687i	$-0.231435\pi$
$661$	38.4169	1.49424	0.747122	+	0.664687i	$0.231435\pi$
$662$	0	0
$663$	−20.2547	−0.786626
$664$	0	0
$665$	101.342	3.92986
$666$	0	0
$667$	−38.6634	−1.49705
$668$	0	0
$669$	20.5915	0.796114
$670$	0	0
$671$	−22.5165	−0.869240
$672$	0	0
$673$	−5.83603	−0.224962	−0.112481	−	0.993654i	$-0.535880\pi$
$673$	−5.83603	−0.224962	−0.112481	+	0.993654i	$0.535880\pi$
$674$	0	0
$675$	48.2539	1.85729
$676$	0	0
$677$	3.14410	0.120838	0.0604188	−	0.998173i	$-0.480756\pi$
$677$	3.14410	0.120838	0.0604188	+	0.998173i	$0.480756\pi$
$678$	0	0
$679$	−14.7335	−0.565419
$680$	0	0
$681$	17.8046	0.682274
$682$	0	0
$683$	−26.2041	−1.00267	−0.501337	−	0.865252i	$-0.667158\pi$
$683$	−26.2041	−1.00267	−0.501337	+	0.865252i	$0.667158\pi$
$684$	0	0
$685$	20.6997	0.790896
$686$	0	0
$687$	51.6761	1.97157
$688$	0	0
$689$	38.4867	1.46622
$690$	0	0
$691$	18.9073	0.719268	0.359634	−	0.933094i	$-0.382902\pi$
$691$	18.9073	0.719268	0.359634	+	0.933094i	$0.382902\pi$
$692$	0	0
$693$	186.977	7.10265
$694$	0	0
$695$	5.42343	0.205722
$696$	0	0
$697$	−4.92575	−0.186576
$698$	0	0
$699$	59.2926	2.24265
$700$	0	0
$701$	−8.31603	−0.314092	−0.157046	−	0.987591i	$-0.550197\pi$
$701$	−8.31603	−0.314092	−0.157046	+	0.987591i	$0.550197\pi$
$702$	0	0
$703$	−10.4089	−0.392578
$704$	0	0
$705$	53.3777	2.01032
$706$	0	0
$707$	−29.5667	−1.11197
$708$	0	0
$709$	15.4149	0.578918	0.289459	−	0.957190i	$-0.406525\pi$
$709$	15.4149	0.578918	0.289459	+	0.957190i	$0.406525\pi$
$710$	0	0
$711$	−17.0007	−0.637575
$712$	0	0
$713$	−10.5960	−0.396824
$714$	0	0
$715$	57.9276	2.16637
$716$	0	0
$717$	−57.2684	−2.13873
$718$	0	0
$719$	19.3400	0.721261	0.360631	−	0.932709i	$-0.382562\pi$
$719$	19.3400	0.721261	0.360631	+	0.932709i	$0.382562\pi$
$720$	0	0
$721$	−0.334749	−0.0124667
$722$	0	0
$723$	58.1142	2.16129
$724$	0	0
$725$	29.9129	1.11094
$726$	0	0
$727$	13.7468	0.509842	0.254921	−	0.966962i	$-0.417951\pi$
$727$	13.7468	0.509842	0.254921	+	0.966962i	$0.417951\pi$
$728$	0	0
$729$	26.0794	0.965905
$730$	0	0
$731$	−15.2631	−0.564526
$732$	0	0
$733$	19.0695	0.704348	0.352174	−	0.935935i	$-0.385443\pi$
$733$	19.0695	0.704348	0.352174	+	0.935935i	$0.385443\pi$
$734$	0	0
$735$	−150.185	−5.53967
$736$	0	0
$737$	−74.8918	−2.75867
$738$	0	0
$739$	34.6903	1.27610	0.638052	−	0.769994i	$-0.279741\pi$
$739$	34.6903	1.27610	0.638052	+	0.769994i	$0.279741\pi$
$740$	0	0
$741$	84.6906	3.11119
$742$	0	0
$743$	22.6162	0.829707	0.414854	−	0.909888i	$-0.363833\pi$
$743$	22.6162	0.829707	0.414854	+	0.909888i	$0.363833\pi$
$744$	0	0
$745$	0.942384	0.0345263
$746$	0	0
$747$	−68.0102	−2.48836
$748$	0	0
$749$	93.3424	3.41066
$750$	0	0
$751$	52.0665	1.89993	0.949966	−	0.312352i	$-0.101117\pi$
$751$	52.0665	1.89993	0.949966	+	0.312352i	$0.101117\pi$
$752$	0	0
$753$	3.19806	0.116544
$754$	0	0
$755$	−20.4819	−0.745412
$756$	0	0
$757$	31.6587	1.15066	0.575328	−	0.817923i	$-0.304874\pi$
$757$	31.6587	1.15066	0.575328	+	0.817923i	$0.304874\pi$
$758$	0	0
$759$	79.5072	2.88593
$760$	0	0
$761$	−25.4569	−0.922810	−0.461405	−	0.887190i	$-0.652655\pi$
$761$	−25.4569	−0.922810	−0.461405	+	0.887190i	$0.652655\pi$
$762$	0	0
$763$	62.0761	2.24731
$764$	0	0
$765$	36.4927	1.31940
$766$	0	0
$767$	31.5789	1.14025
$768$	0	0
$769$	−45.3287	−1.63459	−0.817297	−	0.576217i	$-0.804528\pi$
$769$	−45.3287	−1.63459	−0.817297	+	0.576217i	$0.804528\pi$
$770$	0	0
$771$	65.0793	2.34377
$772$	0	0
$773$	−13.4972	−0.485460	−0.242730	−	0.970094i	$-0.578043\pi$
$773$	−13.4972	−0.485460	−0.242730	+	0.970094i	$0.578043\pi$
$774$	0	0
$775$	8.19788	0.294477
$776$	0	0
$777$	22.1565	0.794859
$778$	0	0
$779$	20.5960	0.737929
$780$	0	0
$781$	−15.1183	−0.540975
$782$	0	0
$783$	113.315	4.04956
$784$	0	0
$785$	2.15423	0.0768878
$786$	0	0
$787$	−16.3260	−0.581961	−0.290980	−	0.956729i	$-0.593981\pi$
$787$	−16.3260	−0.581961	−0.290980	+	0.956729i	$0.593981\pi$
$788$	0	0
$789$	−21.6285	−0.769994
$790$	0	0
$791$	−0.595377	−0.0211692
$792$	0	0
$793$	15.3415	0.544791
$794$	0	0
$795$	−98.1229	−3.48006
$796$	0	0
$797$	21.3033	0.754602	0.377301	−	0.926091i	$-0.376852\pi$
$797$	21.3033	0.754602	0.377301	+	0.926091i	$0.376852\pi$
$798$	0	0
$799$	9.83449	0.347919
$800$	0	0
$801$	−81.2019	−2.86913
$802$	0	0
$803$	−28.6264	−1.01020
$804$	0	0
$805$	−64.8222	−2.28468
$806$	0	0
$807$	56.4667	1.98772
$808$	0	0
$809$	−31.4143	−1.10447	−0.552233	−	0.833690i	$-0.686224\pi$
$809$	−31.4143	−1.10447	−0.552233	+	0.833690i	$0.686224\pi$
$810$	0	0
$811$	−2.76533	−0.0971038	−0.0485519	−	0.998821i	$-0.515461\pi$
$811$	−2.76533	−0.0971038	−0.0485519	+	0.998821i	$0.515461\pi$
$812$	0	0
$813$	−81.2090	−2.84812
$814$	0	0
$815$	23.1626	0.811352
$816$	0	0
$817$	63.8195	2.23276
$818$	0	0
$819$	−127.395	−4.45154
$820$	0	0
$821$	−45.1540	−1.57589	−0.787943	−	0.615748i	$-0.788854\pi$
$821$	−45.1540	−1.57589	−0.787943	+	0.615748i	$0.788854\pi$
$822$	0	0
$823$	27.6763	0.964736	0.482368	−	0.875969i	$-0.339777\pi$
$823$	27.6763	0.964736	0.482368	+	0.875969i	$0.339777\pi$
$824$	0	0
$825$	−61.5128	−2.14160
$826$	0	0
$827$	−28.4482	−0.989241	−0.494621	−	0.869109i	$-0.664693\pi$
$827$	−28.4482	−0.989241	−0.494621	+	0.869109i	$0.664693\pi$
$828$	0	0
$829$	−14.9443	−0.519035	−0.259518	−	0.965738i	$-0.583564\pi$
$829$	−14.9443	−0.519035	−0.259518	+	0.965738i	$0.583564\pi$
$830$	0	0
$831$	−8.75208	−0.303606
$832$	0	0
$833$	−27.6707	−0.958732
$834$	0	0
$835$	27.0213	0.935109
$836$	0	0
$837$	31.0550	1.07342
$838$	0	0
$839$	−12.9976	−0.448728	−0.224364	−	0.974505i	$-0.572030\pi$
$839$	−12.9976	−0.448728	−0.224364	+	0.974505i	$0.572030\pi$
$840$	0	0
$841$	41.2451	1.42224
$842$	0	0
$843$	−23.1032	−0.795718
$844$	0	0
$845$	−1.41377	−0.0486353
$846$	0	0
$847$	−86.6154	−2.97614
$848$	0	0
$849$	−3.72568	−0.127865
$850$	0	0
$851$	6.65794	0.228231
$852$	0	0
$853$	−0.644261	−0.0220591	−0.0110295	−	0.999939i	$-0.503511\pi$
$853$	−0.644261	−0.0220591	−0.0110295	+	0.999939i	$0.503511\pi$
$854$	0	0
$855$	−152.587	−5.21835
$856$	0	0
$857$	15.9250	0.543988	0.271994	−	0.962299i	$-0.412317\pi$
$857$	15.9250	0.543988	0.271994	+	0.962299i	$0.412317\pi$
$858$	0	0
$859$	−12.5819	−0.429288	−0.214644	−	0.976692i	$-0.568859\pi$
$859$	−12.5819	−0.429288	−0.214644	+	0.976692i	$0.568859\pi$
$860$	0	0
$861$	−43.8409	−1.49409
$862$	0	0
$863$	−11.2425	−0.382699	−0.191350	−	0.981522i	$-0.561286\pi$
$863$	−11.2425	−0.382699	−0.191350	+	0.981522i	$0.561286\pi$
$864$	0	0
$865$	39.6851	1.34933
$866$	0	0
$867$	−44.8527	−1.52328
$868$	0	0
$869$	12.6765	0.430020
$870$	0	0
$871$	51.0270	1.72898
$872$	0	0
$873$	22.1837	0.750804
$874$	0	0
$875$	−20.1076	−0.679760
$876$	0	0
$877$	−5.12869	−0.173184	−0.0865918	−	0.996244i	$-0.527598\pi$
$877$	−5.12869	−0.173184	−0.0865918	+	0.996244i	$0.527598\pi$
$878$	0	0
$879$	−13.3261	−0.449478
$880$	0	0
$881$	14.5038	0.488644	0.244322	−	0.969694i	$-0.421435\pi$
$881$	14.5038	0.488644	0.244322	+	0.969694i	$0.421435\pi$
$882$	0	0
$883$	−39.9041	−1.34288	−0.671440	−	0.741059i	$-0.734324\pi$
$883$	−39.9041	−1.34288	−0.671440	+	0.741059i	$0.734324\pi$
$884$	0	0
$885$	−80.5113	−2.70636
$886$	0	0
$887$	−13.3383	−0.447856	−0.223928	−	0.974606i	$-0.571888\pi$
$887$	−13.3383	−0.447856	−0.223928	+	0.974606i	$0.571888\pi$
$888$	0	0
$889$	11.0552	0.370779
$890$	0	0
$891$	−116.167	−3.89175
$892$	0	0
$893$	−41.1209	−1.37606
$894$	0	0
$895$	−3.74057	−0.125033
$896$	0	0
$897$	−54.1716	−1.80874
$898$	0	0
$899$	19.2512	0.642065
$900$	0	0
$901$	−18.0785	−0.602283
$902$	0	0
$903$	−135.847	−4.52071
$904$	0	0
$905$	−36.7553	−1.22179
$906$	0	0
$907$	11.3278	0.376134	0.188067	−	0.982156i	$-0.439778\pi$
$907$	11.3278	0.376134	0.188067	+	0.982156i	$0.439778\pi$
$908$	0	0
$909$	44.5176	1.47656
$910$	0	0
$911$	20.3661	0.674759	0.337379	−	0.941369i	$-0.390459\pi$
$911$	20.3661	0.674759	0.337379	+	0.941369i	$0.390459\pi$
$912$	0	0
$913$	50.7115	1.67831
$914$	0	0
$915$	−39.1135	−1.29305
$916$	0	0
$917$	9.80638	0.323835
$918$	0	0
$919$	57.7336	1.90446	0.952229	−	0.305386i	$-0.0987855\pi$
$919$	57.7336	1.90446	0.952229	+	0.305386i	$0.0987855\pi$
$920$	0	0
$921$	−8.82514	−0.290798
$922$	0	0
$923$	10.3007	0.339053
$924$	0	0
$925$	−5.15109	−0.169367
$926$	0	0
$927$	0.504021	0.0165542
$928$	0	0
$929$	−52.4653	−1.72133	−0.860666	−	0.509170i	$-0.829952\pi$
$929$	−52.4653	−1.72133	−0.860666	+	0.509170i	$0.829952\pi$
$930$	0	0
$931$	115.699	3.79189
$932$	0	0
$933$	−18.6507	−0.610595
$934$	0	0
$935$	−27.2106	−0.889882
$936$	0	0
$937$	9.41296	0.307508	0.153754	−	0.988109i	$-0.450864\pi$
$937$	9.41296	0.307508	0.153754	+	0.988109i	$0.450864\pi$
$938$	0	0
$939$	−106.495	−3.47533
$940$	0	0
$941$	18.0113	0.587152	0.293576	−	0.955936i	$-0.405155\pi$
$941$	18.0113	0.587152	0.293576	+	0.955936i	$0.405155\pi$
$942$	0	0
$943$	−13.1741	−0.429006
$944$	0	0
$945$	189.982	6.18012
$946$	0	0
$947$	−42.1272	−1.36895	−0.684476	−	0.729036i	$-0.739969\pi$
$947$	−42.1272	−1.36895	−0.684476	+	0.729036i	$0.739969\pi$
$948$	0	0
$949$	19.5044	0.633138
$950$	0	0
$951$	4.35450	0.141204
$952$	0	0
$953$	12.6199	0.408798	0.204399	−	0.978888i	$-0.434476\pi$
$953$	12.6199	0.408798	0.204399	+	0.978888i	$0.434476\pi$
$954$	0	0
$955$	−75.2986	−2.43661
$956$	0	0
$957$	−144.452	−4.66945
$958$	0	0
$959$	33.9441	1.09611
$960$	0	0
$961$	−25.7240	−0.829808
$962$	0	0
$963$	−140.542	−4.52892
$964$	0	0
$965$	−3.91266	−0.125953
$966$	0	0
$967$	−41.0372	−1.31967	−0.659834	−	0.751411i	$-0.729373\pi$
$967$	−41.0372	−1.31967	−0.659834	+	0.751411i	$0.729373\pi$
$968$	0	0
$969$	−39.7821	−1.27799
$970$	0	0
$971$	−7.68841	−0.246733	−0.123366	−	0.992361i	$-0.539369\pi$
$971$	−7.68841	−0.246733	−0.123366	+	0.992361i	$0.539369\pi$
$972$	0	0
$973$	8.89352	0.285113
$974$	0	0
$975$	41.9113	1.34223
$976$	0	0
$977$	−21.2544	−0.679988	−0.339994	−	0.940428i	$-0.610425\pi$
$977$	−21.2544	−0.679988	−0.339994	+	0.940428i	$0.610425\pi$
$978$	0	0
$979$	60.5479	1.93512
$980$	0	0
$981$	−93.4659	−2.98414
$982$	0	0
$983$	−25.1262	−0.801402	−0.400701	−	0.916209i	$-0.631233\pi$
$983$	−25.1262	−0.801402	−0.400701	+	0.916209i	$0.631233\pi$
$984$	0	0
$985$	−72.1587	−2.29917
$986$	0	0
$987$	87.5305	2.78613
$988$	0	0
$989$	−40.8216	−1.29805
$990$	0	0
$991$	22.2599	0.707110	0.353555	−	0.935414i	$-0.384973\pi$
$991$	22.2599	0.707110	0.353555	+	0.935414i	$0.384973\pi$
$992$	0	0
$993$	60.6274	1.92395
$994$	0	0
$995$	−49.6657	−1.57451
$996$	0	0
$997$	22.2690	0.705267	0.352633	−	0.935762i	$-0.385286\pi$
$997$	22.2690	0.705267	0.352633	+	0.935762i	$0.385286\pi$
$998$	0	0
$999$	−19.5132	−0.617371

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.m.1.21		23
4.3	odd	2		2008.2.a.d.1.3	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.d.1.3	✓	23	4.3	odd	2
4016.2.a.m.1.21		23	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4016 = 2^{4} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4016.a (trivial)

\(f(q)\)	\(=\)	\(q+3.19806 q^{3} -2.92729 q^{5} -4.80027 q^{7} +7.22760 q^{9} +O(q^{10})\)	\(q+3.19806 q^{3} -2.92729 q^{5} -4.80027 q^{7} +7.22760 q^{9} -5.38924 q^{11} +3.67192 q^{13} -9.36166 q^{15} -1.72483 q^{17} +7.21200 q^{19} -15.3516 q^{21} -4.61309 q^{23} +3.56904 q^{25} +13.5201 q^{27} +8.38123 q^{29} +2.29695 q^{31} -17.2351 q^{33} +14.0518 q^{35} -1.44327 q^{37} +11.7430 q^{39} +2.85580 q^{41} +8.84907 q^{43} -21.1573 q^{45} -5.70173 q^{47} +16.0426 q^{49} -5.51610 q^{51} +10.4814 q^{53} +15.7759 q^{55} +23.0644 q^{57} +8.60011 q^{59} +4.17805 q^{61} -34.6944 q^{63} -10.7488 q^{65} +13.8966 q^{67} -14.7530 q^{69} +2.80527 q^{71} +5.31177 q^{73} +11.4140 q^{75} +25.8698 q^{77} -2.35219 q^{79} +21.5554 q^{81} -9.40978 q^{83} +5.04907 q^{85} +26.8037 q^{87} -11.2350 q^{89} -17.6262 q^{91} +7.34577 q^{93} -21.1116 q^{95} +3.06930 q^{97} -38.9513 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 2 q^{3} + 8 q^{5} - 2 q^{7} + 45 q^{9}+O(q^{10}) \) 23 * q - 2 * q^3 + 8 * q^5 - 2 * q^7 + 45 * q^9	\( 23 q - 2 q^{3} + 8 q^{5} - 2 q^{7} + 45 q^{9} - 8 q^{11} + 8 q^{13} - 7 q^{15} + 19 q^{17} + 9 q^{19} + 9 q^{21} - 21 q^{23} + 65 q^{25} - 5 q^{27} + 10 q^{29} + 9 q^{31} + 34 q^{33} - 12 q^{35} + 11 q^{37} + 9 q^{39} + 35 q^{41} + 9 q^{43} + 29 q^{45} - 37 q^{47} + 77 q^{49} + 17 q^{51} + 38 q^{53} + 20 q^{55} + 51 q^{57} - 17 q^{59} - 22 q^{63} + 41 q^{65} - 9 q^{67} + 8 q^{69} - 13 q^{71} + 41 q^{73} - 25 q^{75} + 36 q^{77} + 36 q^{79} + 127 q^{81} - 29 q^{83} + 34 q^{85} - 10 q^{87} + 36 q^{89} + 6 q^{91} + 36 q^{93} - 25 q^{95} + 40 q^{97} - 19 q^{99}+O(q^{100}) \) 23 * q - 2 * q^3 + 8 * q^5 - 2 * q^7 + 45 * q^9 - 8 * q^11 + 8 * q^13 - 7 * q^15 + 19 * q^17 + 9 * q^19 + 9 * q^21 - 21 * q^23 + 65 * q^25 - 5 * q^27 + 10 * q^29 + 9 * q^31 + 34 * q^33 - 12 * q^35 + 11 * q^37 + 9 * q^39 + 35 * q^41 + 9 * q^43 + 29 * q^45 - 37 * q^47 + 77 * q^49 + 17 * q^51 + 38 * q^53 + 20 * q^55 + 51 * q^57 - 17 * q^59 - 22 * q^63 + 41 * q^65 - 9 * q^67 + 8 * q^69 - 13 * q^71 + 41 * q^73 - 25 * q^75 + 36 * q^77 + 36 * q^79 + 127 * q^81 - 29 * q^83 + 34 * q^85 - 10 * q^87 + 36 * q^89 + 6 * q^91 + 36 * q^93 - 25 * q^95 + 40 * q^97 - 19 * q^99

Introduction

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