LMFDB - Embedded newform 4016.2.a.j.1.12

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:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 3 x^{13} - 27 x^{12} + 79 x^{11} + 274 x^{10} - 747 x^{9} - 1422 x^{8} + 3287 x^{7} + \cdots - 196 $ x^14 - 3x^13 - 27x^12 + 79x^11 + 274x^10 - 747x^9 - 1422x^8 + 3287x^7 + 4161x^6 - 6861x^5 - 6676x^4 + 5599x^3 + 4627x^2 - 359*x - 196
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1004)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$-1.93078$ of defining polynomial
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)

$f(q)$	$=$	$q+1.93078 q^{3} +3.14832 q^{5} -3.83969 q^{7} +0.727926 q^{9} +O(q^{10})$	$q+1.93078 q^{3} +3.14832 q^{5} -3.83969 q^{7} +0.727926 q^{9} -4.86467 q^{11} +4.27481 q^{13} +6.07872 q^{15} -6.83344 q^{17} -3.54240 q^{19} -7.41361 q^{21} +5.84778 q^{23} +4.91190 q^{25} -4.38688 q^{27} -9.26464 q^{29} -6.84177 q^{31} -9.39262 q^{33} -12.0886 q^{35} -5.31841 q^{37} +8.25373 q^{39} +1.64473 q^{41} +1.15452 q^{43} +2.29174 q^{45} -5.03188 q^{47} +7.74323 q^{49} -13.1939 q^{51} +9.73778 q^{53} -15.3155 q^{55} -6.83961 q^{57} -2.57616 q^{59} +3.42282 q^{61} -2.79501 q^{63} +13.4585 q^{65} +13.5967 q^{67} +11.2908 q^{69} +8.49921 q^{71} -9.42486 q^{73} +9.48382 q^{75} +18.6788 q^{77} -7.49365 q^{79} -10.6539 q^{81} -5.19812 q^{83} -21.5138 q^{85} -17.8880 q^{87} -4.74655 q^{89} -16.4139 q^{91} -13.2100 q^{93} -11.1526 q^{95} -0.648252 q^{97} -3.54112 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 3 q^{3} - 2 q^{5} - 8 q^{7} + 21 q^{9}+O(q^{10}) $ 14 * q - 3 * q^3 - 2 * q^5 - 8 * q^7 + 21 * q^9	$ 14 q - 3 q^{3} - 2 q^{5} - 8 q^{7} + 21 q^{9} - 9 q^{11} - q^{13} - 14 q^{15} - 27 q^{19} - 3 q^{21} - 13 q^{23} + 26 q^{25} - 15 q^{27} - 25 q^{31} + 16 q^{33} - 21 q^{35} - q^{37} - 33 q^{39} + 10 q^{41} - 35 q^{43} - 4 q^{45} - 6 q^{47} + 36 q^{49} - 48 q^{51} - q^{53} - 41 q^{55} + 14 q^{57} - 30 q^{59} + 3 q^{61} - 31 q^{63} + 7 q^{65} - 22 q^{67} - 17 q^{69} - 6 q^{71} + 5 q^{73} - 4 q^{75} - 14 q^{77} - 56 q^{79} + 26 q^{81} + 28 q^{83} - 23 q^{85} - 11 q^{87} - 24 q^{89} - 38 q^{91} - 55 q^{93} + 4 q^{95} + 6 q^{97} - 12 q^{99}+O(q^{100}) $ 14 * q - 3 * q^3 - 2 * q^5 - 8 * q^7 + 21 * q^9 - 9 * q^11 - q^13 - 14 * q^15 - 27 * q^19 - 3 * q^21 - 13 * q^23 + 26 * q^25 - 15 * q^27 - 25 * q^31 + 16 * q^33 - 21 * q^35 - q^37 - 33 * q^39 + 10 * q^41 - 35 * q^43 - 4 * q^45 - 6 * q^47 + 36 * q^49 - 48 * q^51 - q^53 - 41 * q^55 + 14 * q^57 - 30 * q^59 + 3 * q^61 - 31 * q^63 + 7 * q^65 - 22 * q^67 - 17 * q^69 - 6 * q^71 + 5 * q^73 - 4 * q^75 - 14 * q^77 - 56 * q^79 + 26 * q^81 + 28 * q^83 - 23 * q^85 - 11 * q^87 - 24 * q^89 - 38 * q^91 - 55 * q^93 + 4 * q^95 + 6 * q^97 - 12 * 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
$3$	1.93078	1.11474	0.557369	−	0.830265i	$-0.311811\pi$
$3$	1.93078	1.11474	0.557369	+	0.830265i	$0.311811\pi$
$4$	0	0
$5$	3.14832	1.40797	0.703985	−	0.710215i	$-0.251402\pi$
$5$	3.14832	1.40797	0.703985	+	0.710215i	$0.251402\pi$
$6$	0	0
$7$	−3.83969	−1.45127	−0.725634	−	0.688081i	$-0.758453\pi$
$7$	−3.83969	−1.45127	−0.725634	+	0.688081i	$0.758453\pi$
$8$	0	0
$9$	0.727926	0.242642
$10$	0	0
$11$	−4.86467	−1.46675	−0.733376	−	0.679823i	$-0.762057\pi$
$11$	−4.86467	−1.46675	−0.733376	+	0.679823i	$0.762057\pi$
$12$	0	0
$13$	4.27481	1.18562	0.592809	−	0.805343i	$-0.298019\pi$
$13$	4.27481	1.18562	0.592809	+	0.805343i	$0.298019\pi$
$14$	0	0
$15$	6.07872	1.56952
$16$	0	0
$17$	−6.83344	−1.65735	−0.828677	−	0.559728i	$-0.810906\pi$
$17$	−6.83344	−1.65735	−0.828677	+	0.559728i	$0.810906\pi$
$18$	0	0
$19$	−3.54240	−0.812682	−0.406341	−	0.913721i	$-0.633196\pi$
$19$	−3.54240	−0.812682	−0.406341	+	0.913721i	$0.633196\pi$
$20$	0	0
$21$	−7.41361	−1.61778
$22$	0	0
$23$	5.84778	1.21935	0.609673	−	0.792653i	$-0.291301\pi$
$23$	5.84778	1.21935	0.609673	+	0.792653i	$0.291301\pi$
$24$	0	0
$25$	4.91190	0.982380
$26$	0	0
$27$	−4.38688	−0.844256
$28$	0	0
$29$	−9.26464	−1.72040	−0.860200	−	0.509956i	$-0.829662\pi$
$29$	−9.26464	−1.72040	−0.860200	+	0.509956i	$0.829662\pi$
$30$	0	0
$31$	−6.84177	−1.22882	−0.614409	−	0.788988i	$-0.710606\pi$
$31$	−6.84177	−1.22882	−0.614409	+	0.788988i	$0.710606\pi$
$32$	0	0
$33$	−9.39262	−1.63505
$34$	0	0
$35$	−12.0886	−2.04334
$36$	0	0
$37$	−5.31841	−0.874342	−0.437171	−	0.899378i	$-0.644020\pi$
$37$	−5.31841	−0.874342	−0.437171	+	0.899378i	$0.644020\pi$
$38$	0	0
$39$	8.25373	1.32165
$40$	0	0
$41$	1.64473	0.256863	0.128432	−	0.991718i	$-0.459006\pi$
$41$	1.64473	0.256863	0.128432	+	0.991718i	$0.459006\pi$
$42$	0	0
$43$	1.15452	0.176063	0.0880315	−	0.996118i	$-0.471942\pi$
$43$	1.15452	0.176063	0.0880315	+	0.996118i	$0.471942\pi$
$44$	0	0
$45$	2.29174	0.341633
$46$	0	0
$47$	−5.03188	−0.733975	−0.366987	−	0.930226i	$-0.619611\pi$
$47$	−5.03188	−0.733975	−0.366987	+	0.930226i	$0.619611\pi$
$48$	0	0
$49$	7.74323	1.10618
$50$	0	0
$51$	−13.1939	−1.84752
$52$	0	0
$53$	9.73778	1.33759	0.668794	−	0.743448i	$-0.266811\pi$
$53$	9.73778	1.33759	0.668794	+	0.743448i	$0.266811\pi$
$54$	0	0
$55$	−15.3155	−2.06514
$56$	0	0
$57$	−6.83961	−0.905928
$58$	0	0
$59$	−2.57616	−0.335388	−0.167694	−	0.985839i	$-0.553632\pi$
$59$	−2.57616	−0.335388	−0.167694	+	0.985839i	$0.553632\pi$
$60$	0	0
$61$	3.42282	0.438247	0.219124	−	0.975697i	$-0.429680\pi$
$61$	3.42282	0.438247	0.219124	+	0.975697i	$0.429680\pi$
$62$	0	0
$63$	−2.79501	−0.352138
$64$	0	0
$65$	13.4585	1.66932
$66$	0	0
$67$	13.5967	1.66111	0.830553	−	0.556939i	$-0.188024\pi$
$67$	13.5967	1.66111	0.830553	+	0.556939i	$0.188024\pi$
$68$	0	0
$69$	11.2908	1.35925
$70$	0	0
$71$	8.49921	1.00867	0.504335	−	0.863508i	$-0.331738\pi$
$71$	8.49921	1.00867	0.504335	+	0.863508i	$0.331738\pi$
$72$	0	0
$73$	−9.42486	−1.10310	−0.551548	−	0.834143i	$-0.685963\pi$
$73$	−9.42486	−1.10310	−0.551548	+	0.834143i	$0.685963\pi$
$74$	0	0
$75$	9.48382	1.09510
$76$	0	0
$77$	18.6788	2.12865
$78$	0	0
$79$	−7.49365	−0.843101	−0.421551	−	0.906805i	$-0.638514\pi$
$79$	−7.49365	−0.843101	−0.421551	+	0.906805i	$0.638514\pi$
$80$	0	0
$81$	−10.6539	−1.18377
$82$	0	0
$83$	−5.19812	−0.570568	−0.285284	−	0.958443i	$-0.592088\pi$
$83$	−5.19812	−0.570568	−0.285284	+	0.958443i	$0.592088\pi$
$84$	0	0
$85$	−21.5138	−2.33350
$86$	0	0
$87$	−17.8880	−1.91780
$88$	0	0
$89$	−4.74655	−0.503133	−0.251567	−	0.967840i	$-0.580946\pi$
$89$	−4.74655	−0.503133	−0.251567	+	0.967840i	$0.580946\pi$
$90$	0	0
$91$	−16.4139	−1.72065
$92$	0	0
$93$	−13.2100	−1.36981
$94$	0	0
$95$	−11.1526	−1.14423
$96$	0	0
$97$	−0.648252	−0.0658200	−0.0329100	−	0.999458i	$-0.510477\pi$
$97$	−0.648252	−0.0658200	−0.0329100	+	0.999458i	$0.510477\pi$
$98$	0	0
$99$	−3.54112	−0.355896
$100$	0	0
$101$	8.64728	0.860437	0.430218	−	0.902725i	$-0.358437\pi$
$101$	8.64728	0.860437	0.430218	+	0.902725i	$0.358437\pi$
$102$	0	0
$103$	−15.1777	−1.49550	−0.747750	−	0.663981i	$-0.768866\pi$
$103$	−15.1777	−1.49550	−0.747750	+	0.663981i	$0.768866\pi$
$104$	0	0
$105$	−23.3404	−2.27779
$106$	0	0
$107$	−6.53504	−0.631766	−0.315883	−	0.948798i	$-0.602301\pi$
$107$	−6.53504	−0.631766	−0.315883	+	0.948798i	$0.602301\pi$
$108$	0	0
$109$	0.746297	0.0714823	0.0357412	−	0.999361i	$-0.488621\pi$
$109$	0.746297	0.0714823	0.0357412	+	0.999361i	$0.488621\pi$
$110$	0	0
$111$	−10.2687	−0.974662
$112$	0	0
$113$	8.87252	0.834656	0.417328	−	0.908756i	$-0.362967\pi$
$113$	8.87252	0.834656	0.417328	+	0.908756i	$0.362967\pi$
$114$	0	0
$115$	18.4107	1.71680
$116$	0	0
$117$	3.11174	0.287681
$118$	0	0
$119$	26.2383	2.40526
$120$	0	0
$121$	12.6650	1.15136
$122$	0	0
$123$	3.17561	0.286336
$124$	0	0
$125$	−0.277368	−0.0248086
$126$	0	0
$127$	−8.04581	−0.713950	−0.356975	−	0.934114i	$-0.616192\pi$
$127$	−8.04581	−0.713950	−0.356975	+	0.934114i	$0.616192\pi$
$128$	0	0
$129$	2.22913	0.196264
$130$	0	0
$131$	17.0825	1.49250	0.746251	−	0.665665i	$-0.231852\pi$
$131$	17.0825	1.49250	0.746251	+	0.665665i	$0.231852\pi$
$132$	0	0
$133$	13.6017	1.17942
$134$	0	0
$135$	−13.8113	−1.18869
$136$	0	0
$137$	−9.71501	−0.830010	−0.415005	−	0.909819i	$-0.636220\pi$
$137$	−9.71501	−0.830010	−0.415005	+	0.909819i	$0.636220\pi$
$138$	0	0
$139$	13.6149	1.15480	0.577401	−	0.816461i	$-0.304067\pi$
$139$	13.6149	1.15480	0.577401	+	0.816461i	$0.304067\pi$
$140$	0	0
$141$	−9.71547	−0.818190
$142$	0	0
$143$	−20.7955	−1.73901
$144$	0	0
$145$	−29.1680	−2.42227
$146$	0	0
$147$	14.9505	1.23310
$148$	0	0
$149$	10.6473	0.872262	0.436131	−	0.899883i	$-0.356348\pi$
$149$	10.6473	0.872262	0.436131	+	0.899883i	$0.356348\pi$
$150$	0	0
$151$	−15.5522	−1.26562	−0.632812	−	0.774306i	$-0.718099\pi$
$151$	−15.5522	−1.26562	−0.632812	+	0.774306i	$0.718099\pi$
$152$	0	0
$153$	−4.97424	−0.402143
$154$	0	0
$155$	−21.5401	−1.73014
$156$	0	0
$157$	−1.32340	−0.105619	−0.0528093	−	0.998605i	$-0.516818\pi$
$157$	−1.32340	−0.105619	−0.0528093	+	0.998605i	$0.516818\pi$
$158$	0	0
$159$	18.8015	1.49106
$160$	0	0
$161$	−22.4537	−1.76960
$162$	0	0
$163$	15.2082	1.19120	0.595600	−	0.803281i	$-0.296914\pi$
$163$	15.2082	1.19120	0.595600	+	0.803281i	$0.296914\pi$
$164$	0	0
$165$	−29.5709	−2.30209
$166$	0	0
$167$	17.1243	1.32511	0.662557	−	0.749011i	$-0.269471\pi$
$167$	17.1243	1.32511	0.662557	+	0.749011i	$0.269471\pi$
$168$	0	0
$169$	5.27399	0.405692
$170$	0	0
$171$	−2.57860	−0.197191
$172$	0	0
$173$	−18.9555	−1.44116	−0.720579	−	0.693373i	$-0.756124\pi$
$173$	−18.9555	−1.44116	−0.720579	+	0.693373i	$0.756124\pi$
$174$	0	0
$175$	−18.8602	−1.42570
$176$	0	0
$177$	−4.97401	−0.373870
$178$	0	0
$179$	4.06300	0.303683	0.151841	−	0.988405i	$-0.451480\pi$
$179$	4.06300	0.303683	0.151841	+	0.988405i	$0.451480\pi$
$180$	0	0
$181$	−3.25474	−0.241923	−0.120961	−	0.992657i	$-0.538598\pi$
$181$	−3.25474	−0.241923	−0.120961	+	0.992657i	$0.538598\pi$
$182$	0	0
$183$	6.60873	0.488531
$184$	0	0
$185$	−16.7441	−1.23105
$186$	0	0
$187$	33.2424	2.43093
$188$	0	0
$189$	16.8443	1.22524
$190$	0	0
$191$	−12.7547	−0.922897	−0.461449	−	0.887167i	$-0.652670\pi$
$191$	−12.7547	−0.922897	−0.461449	+	0.887167i	$0.652670\pi$
$192$	0	0
$193$	5.22017	0.375756	0.187878	−	0.982192i	$-0.439839\pi$
$193$	5.22017	0.375756	0.187878	+	0.982192i	$0.439839\pi$
$194$	0	0
$195$	25.9854	1.86085
$196$	0	0
$197$	−21.8507	−1.55680	−0.778400	−	0.627768i	$-0.783969\pi$
$197$	−21.8507	−1.55680	−0.778400	+	0.627768i	$0.783969\pi$
$198$	0	0
$199$	20.8344	1.47691	0.738457	−	0.674300i	$-0.235555\pi$
$199$	20.8344	1.47691	0.738457	+	0.674300i	$0.235555\pi$
$200$	0	0
$201$	26.2524	1.85170
$202$	0	0
$203$	35.5734	2.49676
$204$	0	0
$205$	5.17813	0.361656
$206$	0	0
$207$	4.25675	0.295865
$208$	0	0
$209$	17.2326	1.19200
$210$	0	0
$211$	−12.7423	−0.877216	−0.438608	−	0.898678i	$-0.644528\pi$
$211$	−12.7423	−0.877216	−0.438608	+	0.898678i	$0.644528\pi$
$212$	0	0
$213$	16.4101	1.12440
$214$	0	0
$215$	3.63480	0.247891
$216$	0	0
$217$	26.2703	1.78334
$218$	0	0
$219$	−18.1974	−1.22966
$220$	0	0
$221$	−29.2117	−1.96499
$222$	0	0
$223$	1.93665	0.129688	0.0648438	−	0.997895i	$-0.479345\pi$
$223$	1.93665	0.129688	0.0648438	+	0.997895i	$0.479345\pi$
$224$	0	0
$225$	3.57550	0.238367
$226$	0	0
$227$	13.6666	0.907084	0.453542	−	0.891235i	$-0.350160\pi$
$227$	13.6666	0.907084	0.453542	+	0.891235i	$0.350160\pi$
$228$	0	0
$229$	10.6320	0.702585	0.351292	−	0.936266i	$-0.385742\pi$
$229$	10.6320	0.702585	0.351292	+	0.936266i	$0.385742\pi$
$230$	0	0
$231$	36.0648	2.37289
$232$	0	0
$233$	−15.5682	−1.01991	−0.509954	−	0.860202i	$-0.670337\pi$
$233$	−15.5682	−1.01991	−0.509954	+	0.860202i	$0.670337\pi$
$234$	0	0
$235$	−15.8419	−1.03341
$236$	0	0
$237$	−14.4686	−0.939837
$238$	0	0
$239$	16.0540	1.03845	0.519223	−	0.854639i	$-0.326221\pi$
$239$	16.0540	1.03845	0.519223	+	0.854639i	$0.326221\pi$
$240$	0	0
$241$	29.1005	1.87453	0.937264	−	0.348622i	$-0.113350\pi$
$241$	29.1005	1.87453	0.937264	+	0.348622i	$0.113350\pi$
$242$	0	0
$243$	−7.40973	−0.475334
$244$	0	0
$245$	24.3781	1.55746
$246$	0	0
$247$	−15.1431	−0.963531
$248$	0	0
$249$	−10.0364	−0.636034
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−28.4475	−1.78848
$254$	0	0
$255$	−41.5386	−2.60125
$256$	0	0
$257$	−2.68611	−0.167555	−0.0837773	−	0.996484i	$-0.526698\pi$
$257$	−2.68611	−0.167555	−0.0837773	+	0.996484i	$0.526698\pi$
$258$	0	0
$259$	20.4211	1.26890
$260$	0	0
$261$	−6.74397	−0.417441
$262$	0	0
$263$	−27.4502	−1.69265	−0.846325	−	0.532666i	$-0.821190\pi$
$263$	−27.4502	−1.69265	−0.846325	+	0.532666i	$0.821190\pi$
$264$	0	0
$265$	30.6576	1.88328
$266$	0	0
$267$	−9.16456	−0.560862
$268$	0	0
$269$	−27.6365	−1.68503	−0.842515	−	0.538673i	$-0.818926\pi$
$269$	−27.6365	−1.68503	−0.842515	+	0.538673i	$0.818926\pi$
$270$	0	0
$271$	−9.45401	−0.574290	−0.287145	−	0.957887i	$-0.592706\pi$
$271$	−9.45401	−0.574290	−0.287145	+	0.957887i	$0.592706\pi$
$272$	0	0
$273$	−31.6918	−1.91807
$274$	0	0
$275$	−23.8948	−1.44091
$276$	0	0
$277$	26.7540	1.60749	0.803746	−	0.594973i	$-0.202837\pi$
$277$	26.7540	1.60749	0.803746	+	0.594973i	$0.202837\pi$
$278$	0	0
$279$	−4.98030	−0.298163
$280$	0	0
$281$	6.33284	0.377785	0.188893	−	0.981998i	$-0.439510\pi$
$281$	6.33284	0.377785	0.188893	+	0.981998i	$0.439510\pi$
$282$	0	0
$283$	1.13448	0.0674379	0.0337190	−	0.999431i	$-0.489265\pi$
$283$	1.13448	0.0674379	0.0337190	+	0.999431i	$0.489265\pi$
$284$	0	0
$285$	−21.5333	−1.27552
$286$	0	0
$287$	−6.31525	−0.372777
$288$	0	0
$289$	29.6959	1.74682
$290$	0	0
$291$	−1.25163	−0.0733721
$292$	0	0
$293$	18.0018	1.05168	0.525839	−	0.850584i	$-0.323751\pi$
$293$	18.0018	1.05168	0.525839	+	0.850584i	$0.323751\pi$
$294$	0	0
$295$	−8.11057	−0.472216
$296$	0	0
$297$	21.3407	1.23831
$298$	0	0
$299$	24.9982	1.44568
$300$	0	0
$301$	−4.43301	−0.255514
$302$	0	0
$303$	16.6960	0.959162
$304$	0	0
$305$	10.7761	0.617039
$306$	0	0
$307$	0.456641	0.0260619	0.0130310	−	0.999915i	$-0.495852\pi$
$307$	0.456641	0.0260619	0.0130310	+	0.999915i	$0.495852\pi$
$308$	0	0
$309$	−29.3048	−1.66709
$310$	0	0
$311$	−25.5265	−1.44748	−0.723739	−	0.690074i	$-0.757578\pi$
$311$	−25.5265	−1.44748	−0.723739	+	0.690074i	$0.757578\pi$
$312$	0	0
$313$	−17.8588	−1.00944	−0.504721	−	0.863283i	$-0.668404\pi$
$313$	−17.8588	−1.00944	−0.504721	+	0.863283i	$0.668404\pi$
$314$	0	0
$315$	−8.79958	−0.495800
$316$	0	0
$317$	−12.8881	−0.723868	−0.361934	−	0.932204i	$-0.617883\pi$
$317$	−12.8881	−0.723868	−0.361934	+	0.932204i	$0.617883\pi$
$318$	0	0
$319$	45.0694	2.52340
$320$	0	0
$321$	−12.6178	−0.704254
$322$	0	0
$323$	24.2068	1.34690
$324$	0	0
$325$	20.9974	1.16473
$326$	0	0
$327$	1.44094	0.0796841
$328$	0	0
$329$	19.3209	1.06519
$330$	0	0
$331$	−7.43979	−0.408928	−0.204464	−	0.978874i	$-0.565545\pi$
$331$	−7.43979	−0.408928	−0.204464	+	0.978874i	$0.565545\pi$
$332$	0	0
$333$	−3.87141	−0.212152
$334$	0	0
$335$	42.8069	2.33879
$336$	0	0
$337$	−27.4954	−1.49777	−0.748885	−	0.662700i	$-0.769410\pi$
$337$	−27.4954	−1.49777	−0.748885	+	0.662700i	$0.769410\pi$
$338$	0	0
$339$	17.1309	0.930423
$340$	0	0
$341$	33.2829	1.80237
$342$	0	0
$343$	−2.85378	−0.154090
$344$	0	0
$345$	35.5470	1.91379
$346$	0	0
$347$	−5.87258	−0.315256	−0.157628	−	0.987499i	$-0.550385\pi$
$347$	−5.87258	−0.315256	−0.157628	+	0.987499i	$0.550385\pi$
$348$	0	0
$349$	6.51686	0.348839	0.174420	−	0.984671i	$-0.444195\pi$
$349$	6.51686	0.348839	0.174420	+	0.984671i	$0.444195\pi$
$350$	0	0
$351$	−18.7531	−1.00097
$352$	0	0
$353$	10.4191	0.554551	0.277276	−	0.960790i	$-0.410568\pi$
$353$	10.4191	0.554551	0.277276	+	0.960790i	$0.410568\pi$
$354$	0	0
$355$	26.7582	1.42018
$356$	0	0
$357$	50.6605	2.68124
$358$	0	0
$359$	3.82249	0.201744	0.100872	−	0.994899i	$-0.467837\pi$
$359$	3.82249	0.201744	0.100872	+	0.994899i	$0.467837\pi$
$360$	0	0
$361$	−6.45140	−0.339548
$362$	0	0
$363$	24.4533	1.28347
$364$	0	0
$365$	−29.6724	−1.55313
$366$	0	0
$367$	−13.5501	−0.707307	−0.353653	−	0.935377i	$-0.615061\pi$
$367$	−13.5501	−0.707307	−0.353653	+	0.935377i	$0.615061\pi$
$368$	0	0
$369$	1.19724	0.0623258
$370$	0	0
$371$	−37.3901	−1.94120
$372$	0	0
$373$	−30.3593	−1.57195	−0.785973	−	0.618261i	$-0.787837\pi$
$373$	−30.3593	−1.57195	−0.785973	+	0.618261i	$0.787837\pi$
$374$	0	0
$375$	−0.535538	−0.0276551
$376$	0	0
$377$	−39.6046	−2.03974
$378$	0	0
$379$	31.5403	1.62012	0.810059	−	0.586348i	$-0.199435\pi$
$379$	31.5403	1.62012	0.810059	+	0.586348i	$0.199435\pi$
$380$	0	0
$381$	−15.5347	−0.795868
$382$	0	0
$383$	−15.2140	−0.777397	−0.388698	−	0.921365i	$-0.627075\pi$
$383$	−15.2140	−0.777397	−0.388698	+	0.921365i	$0.627075\pi$
$384$	0	0
$385$	58.8068	2.99707
$386$	0	0
$387$	0.840407	0.0427203
$388$	0	0
$389$	27.3005	1.38419	0.692095	−	0.721806i	$-0.256688\pi$
$389$	27.3005	1.38419	0.692095	+	0.721806i	$0.256688\pi$
$390$	0	0
$391$	−39.9605	−2.02089
$392$	0	0
$393$	32.9825	1.66375
$394$	0	0
$395$	−23.5924	−1.18706
$396$	0	0
$397$	−0.298908	−0.0150018	−0.00750088	−	0.999972i	$-0.502388\pi$
$397$	−0.298908	−0.0150018	−0.00750088	+	0.999972i	$0.502388\pi$
$398$	0	0
$399$	26.2620	1.31474
$400$	0	0
$401$	31.8775	1.59188	0.795942	−	0.605372i	$-0.206976\pi$
$401$	31.8775	1.59188	0.795942	+	0.605372i	$0.206976\pi$
$402$	0	0
$403$	−29.2473	−1.45691
$404$	0	0
$405$	−33.5419	−1.66671
$406$	0	0
$407$	25.8723	1.28244
$408$	0	0
$409$	−4.03841	−0.199686	−0.0998432	−	0.995003i	$-0.531834\pi$
$409$	−4.03841	−0.199686	−0.0998432	+	0.995003i	$0.531834\pi$
$410$	0	0
$411$	−18.7576	−0.925244
$412$	0	0
$413$	9.89167	0.486737
$414$	0	0
$415$	−16.3653	−0.803343
$416$	0	0
$417$	26.2874	1.28730
$418$	0	0
$419$	−3.55230	−0.173541	−0.0867706	−	0.996228i	$-0.527655\pi$
$419$	−3.55230	−0.173541	−0.0867706	+	0.996228i	$0.527655\pi$
$420$	0	0
$421$	27.4017	1.33548	0.667740	−	0.744395i	$-0.267262\pi$
$421$	27.4017	1.33548	0.667740	+	0.744395i	$0.267262\pi$
$422$	0	0
$423$	−3.66283	−0.178093
$424$	0	0
$425$	−33.5652	−1.62815
$426$	0	0
$427$	−13.1426	−0.636014
$428$	0	0
$429$	−40.1517	−1.93854
$430$	0	0
$431$	−39.0457	−1.88076	−0.940382	−	0.340120i	$-0.889532\pi$
$431$	−39.0457	−1.88076	−0.940382	+	0.340120i	$0.889532\pi$
$432$	0	0
$433$	−26.4366	−1.27046	−0.635232	−	0.772321i	$-0.719096\pi$
$433$	−26.4366	−1.27046	−0.635232	+	0.772321i	$0.719096\pi$
$434$	0	0
$435$	−56.3172	−2.70020
$436$	0	0
$437$	−20.7152	−0.990942
$438$	0	0
$439$	−30.0417	−1.43381	−0.716907	−	0.697169i	$-0.754443\pi$
$439$	−30.0417	−1.43381	−0.716907	+	0.697169i	$0.754443\pi$
$440$	0	0
$441$	5.63650	0.268405
$442$	0	0
$443$	12.5254	0.595101	0.297550	−	0.954706i	$-0.403830\pi$
$443$	12.5254	0.595101	0.297550	+	0.954706i	$0.403830\pi$
$444$	0	0
$445$	−14.9436	−0.708396
$446$	0	0
$447$	20.5577	0.972344
$448$	0	0
$449$	−7.42656	−0.350481	−0.175241	−	0.984526i	$-0.556070\pi$
$449$	−7.42656	−0.350481	−0.175241	+	0.984526i	$0.556070\pi$
$450$	0	0
$451$	−8.00105	−0.376755
$452$	0	0
$453$	−30.0280	−1.41084
$454$	0	0
$455$	−51.6763	−2.42262
$456$	0	0
$457$	−37.9074	−1.77323	−0.886616	−	0.462507i	$-0.846950\pi$
$457$	−37.9074	−1.77323	−0.886616	+	0.462507i	$0.846950\pi$
$458$	0	0
$459$	29.9775	1.39923
$460$	0	0
$461$	−39.6679	−1.84752	−0.923758	−	0.382976i	$-0.874899\pi$
$461$	−39.6679	−1.84752	−0.923758	+	0.382976i	$0.874899\pi$
$462$	0	0
$463$	19.8041	0.920375	0.460188	−	0.887822i	$-0.347782\pi$
$463$	19.8041	0.920375	0.460188	+	0.887822i	$0.347782\pi$
$464$	0	0
$465$	−41.5892	−1.92865
$466$	0	0
$467$	4.13021	0.191123	0.0955616	−	0.995424i	$-0.469535\pi$
$467$	4.13021	0.191123	0.0955616	+	0.995424i	$0.469535\pi$
$468$	0	0
$469$	−52.2073	−2.41071
$470$	0	0
$471$	−2.55519	−0.117737
$472$	0	0
$473$	−5.61637	−0.258241
$474$	0	0
$475$	−17.3999	−0.798363
$476$	0	0
$477$	7.08838	0.324555
$478$	0	0
$479$	38.7649	1.77121	0.885607	−	0.464435i	$-0.153742\pi$
$479$	38.7649	1.77121	0.885607	+	0.464435i	$0.153742\pi$
$480$	0	0
$481$	−22.7352	−1.03664
$482$	0	0
$483$	−43.3532	−1.97264
$484$	0	0
$485$	−2.04090	−0.0926726
$486$	0	0
$487$	13.5913	0.615880	0.307940	−	0.951406i	$-0.400360\pi$
$487$	13.5913	0.615880	0.307940	+	0.951406i	$0.400360\pi$
$488$	0	0
$489$	29.3638	1.32788
$490$	0	0
$491$	38.0202	1.71583	0.857914	−	0.513793i	$-0.171760\pi$
$491$	38.0202	1.71583	0.857914	+	0.513793i	$0.171760\pi$
$492$	0	0
$493$	63.3094	2.85131
$494$	0	0
$495$	−11.1486	−0.501090
$496$	0	0
$497$	−32.6343	−1.46385
$498$	0	0
$499$	−18.0935	−0.809976	−0.404988	−	0.914322i	$-0.632724\pi$
$499$	−18.0935	−0.809976	−0.404988	+	0.914322i	$0.632724\pi$
$500$	0	0
$501$	33.0632	1.47716
$502$	0	0
$503$	40.9696	1.82674	0.913371	−	0.407128i	$-0.133470\pi$
$503$	40.9696	1.82674	0.913371	+	0.407128i	$0.133470\pi$
$504$	0	0
$505$	27.2244	1.21147
$506$	0	0
$507$	10.1829	0.452240
$508$	0	0
$509$	−10.1340	−0.449183	−0.224591	−	0.974453i	$-0.572105\pi$
$509$	−10.1340	−0.449183	−0.224591	+	0.974453i	$0.572105\pi$
$510$	0	0
$511$	36.1886	1.60089
$512$	0	0
$513$	15.5401	0.686112
$514$	0	0
$515$	−47.7841	−2.10562
$516$	0	0
$517$	24.4784	1.07656
$518$	0	0
$519$	−36.5989	−1.60651
$520$	0	0
$521$	−10.9744	−0.480795	−0.240398	−	0.970674i	$-0.577278\pi$
$521$	−10.9744	−0.480795	−0.240398	+	0.970674i	$0.577278\pi$
$522$	0	0
$523$	−20.8886	−0.913396	−0.456698	−	0.889622i	$-0.650968\pi$
$523$	−20.8886	−0.913396	−0.456698	+	0.889622i	$0.650968\pi$
$524$	0	0
$525$	−36.4149	−1.58928
$526$	0	0
$527$	46.7529	2.03659
$528$	0	0
$529$	11.1966	0.486807
$530$	0	0
$531$	−1.87525	−0.0813791
$532$	0	0
$533$	7.03090	0.304542
$534$	0	0
$535$	−20.5744	−0.889508
$536$	0	0
$537$	7.84477	0.338527
$538$	0	0
$539$	−37.6682	−1.62249
$540$	0	0
$541$	−26.3216	−1.13165	−0.565826	−	0.824525i	$-0.691443\pi$
$541$	−26.3216	−1.13165	−0.565826	+	0.824525i	$0.691443\pi$
$542$	0	0
$543$	−6.28420	−0.269681
$544$	0	0
$545$	2.34958	0.100645
$546$	0	0
$547$	−21.1442	−0.904060	−0.452030	−	0.892003i	$-0.649300\pi$
$547$	−21.1442	−0.904060	−0.452030	+	0.892003i	$0.649300\pi$
$548$	0	0
$549$	2.49156	0.106337
$550$	0	0
$551$	32.8191	1.39814
$552$	0	0
$553$	28.7733	1.22356
$554$	0	0
$555$	−32.3291	−1.37230
$556$	0	0
$557$	−10.7696	−0.456324	−0.228162	−	0.973623i	$-0.573272\pi$
$557$	−10.7696	−0.456324	−0.228162	+	0.973623i	$0.573272\pi$
$558$	0	0
$559$	4.93536	0.208744
$560$	0	0
$561$	64.1839	2.70985
$562$	0	0
$563$	−0.994648	−0.0419194	−0.0209597	−	0.999780i	$-0.506672\pi$
$563$	−0.994648	−0.0419194	−0.0209597	+	0.999780i	$0.506672\pi$
$564$	0	0
$565$	27.9335	1.17517
$566$	0	0
$567$	40.9077	1.71796
$568$	0	0
$569$	−38.4887	−1.61353	−0.806764	−	0.590873i	$-0.798783\pi$
$569$	−38.4887	−1.61353	−0.806764	+	0.590873i	$0.798783\pi$
$570$	0	0
$571$	3.14387	0.131567	0.0657835	−	0.997834i	$-0.479045\pi$
$571$	3.14387	0.131567	0.0657835	+	0.997834i	$0.479045\pi$
$572$	0	0
$573$	−24.6266	−1.02879
$574$	0	0
$575$	28.7237	1.19786
$576$	0	0
$577$	16.2434	0.676222	0.338111	−	0.941106i	$-0.390212\pi$
$577$	16.2434	0.676222	0.338111	+	0.941106i	$0.390212\pi$
$578$	0	0
$579$	10.0790	0.418870
$580$	0	0
$581$	19.9592	0.828047
$582$	0	0
$583$	−47.3711	−1.96191
$584$	0	0
$585$	9.79676	0.405046
$586$	0	0
$587$	26.4340	1.09105	0.545525	−	0.838095i	$-0.316330\pi$
$587$	26.4340	1.09105	0.545525	+	0.838095i	$0.316330\pi$
$588$	0	0
$589$	24.2363	0.998639
$590$	0	0
$591$	−42.1890	−1.73543
$592$	0	0
$593$	−40.1659	−1.64942	−0.824708	−	0.565558i	$-0.808661\pi$
$593$	−40.1659	−1.64942	−0.824708	+	0.565558i	$0.808661\pi$
$594$	0	0
$595$	82.6065	3.38654
$596$	0	0
$597$	40.2268	1.64637
$598$	0	0
$599$	−39.2964	−1.60561	−0.802804	−	0.596243i	$-0.796659\pi$
$599$	−39.2964	−1.60561	−0.802804	+	0.596243i	$0.796659\pi$
$600$	0	0
$601$	−26.5526	−1.08310	−0.541552	−	0.840667i	$-0.682163\pi$
$601$	−26.5526	−1.08310	−0.541552	+	0.840667i	$0.682163\pi$
$602$	0	0
$603$	9.89742	0.403054
$604$	0	0
$605$	39.8734	1.62108
$606$	0	0
$607$	−15.1919	−0.616621	−0.308311	−	0.951286i	$-0.599764\pi$
$607$	−15.1919	−0.616621	−0.308311	+	0.951286i	$0.599764\pi$
$608$	0	0
$609$	68.6845	2.78324
$610$	0	0
$611$	−21.5103	−0.870214
$612$	0	0
$613$	−37.2149	−1.50309	−0.751547	−	0.659680i	$-0.770692\pi$
$613$	−37.2149	−1.50309	−0.751547	+	0.659680i	$0.770692\pi$
$614$	0	0
$615$	9.99784	0.403152
$616$	0	0
$617$	26.2042	1.05494	0.527472	−	0.849573i	$-0.323140\pi$
$617$	26.2042	1.05494	0.527472	+	0.849573i	$0.323140\pi$
$618$	0	0
$619$	−19.0265	−0.764738	−0.382369	−	0.924010i	$-0.624892\pi$
$619$	−19.0265	−0.764738	−0.382369	+	0.924010i	$0.624892\pi$
$620$	0	0
$621$	−25.6535	−1.02944
$622$	0	0
$623$	18.2253	0.730181
$624$	0	0
$625$	−25.4327	−1.01731
$626$	0	0
$627$	33.2724	1.32877
$628$	0	0
$629$	36.3431	1.44909
$630$	0	0
$631$	38.9710	1.55141	0.775705	−	0.631095i	$-0.217394\pi$
$631$	38.9710	1.55141	0.775705	+	0.631095i	$0.217394\pi$
$632$	0	0
$633$	−24.6026	−0.977867
$634$	0	0
$635$	−25.3308	−1.00522
$636$	0	0
$637$	33.1008	1.31150
$638$	0	0
$639$	6.18679	0.244746
$640$	0	0
$641$	33.3419	1.31692	0.658462	−	0.752614i	$-0.271207\pi$
$641$	33.3419	1.31692	0.658462	+	0.752614i	$0.271207\pi$
$642$	0	0
$643$	28.7346	1.13318	0.566591	−	0.823999i	$-0.308262\pi$
$643$	28.7346	1.13318	0.566591	+	0.823999i	$0.308262\pi$
$644$	0	0
$645$	7.01802	0.276334
$646$	0	0
$647$	29.7614	1.17004	0.585022	−	0.811018i	$-0.301086\pi$
$647$	29.7614	1.17004	0.585022	+	0.811018i	$0.301086\pi$
$648$	0	0
$649$	12.5322	0.491931
$650$	0	0
$651$	50.7223	1.98796
$652$	0	0
$653$	−10.0227	−0.392217	−0.196109	−	0.980582i	$-0.562830\pi$
$653$	−10.0227	−0.392217	−0.196109	+	0.980582i	$0.562830\pi$
$654$	0	0
$655$	53.7810	2.10140
$656$	0	0
$657$	−6.86060	−0.267657
$658$	0	0
$659$	−18.1633	−0.707540	−0.353770	−	0.935332i	$-0.615100\pi$
$659$	−18.1633	−0.707540	−0.353770	+	0.935332i	$0.615100\pi$
$660$	0	0
$661$	14.7321	0.573011	0.286506	−	0.958079i	$-0.407506\pi$
$661$	14.7321	0.573011	0.286506	+	0.958079i	$0.407506\pi$
$662$	0	0
$663$	−56.4014	−2.19045
$664$	0	0
$665$	42.8225	1.66059
$666$	0	0
$667$	−54.1776	−2.09777
$668$	0	0
$669$	3.73925	0.144568
$670$	0	0
$671$	−16.6509	−0.642800
$672$	0	0
$673$	15.6631	0.603766	0.301883	−	0.953345i	$-0.402385\pi$
$673$	15.6631	0.603766	0.301883	+	0.953345i	$0.402385\pi$
$674$	0	0
$675$	−21.5479	−0.829380
$676$	0	0
$677$	13.5230	0.519731	0.259866	−	0.965645i	$-0.416322\pi$
$677$	13.5230	0.519731	0.259866	+	0.965645i	$0.416322\pi$
$678$	0	0
$679$	2.48909	0.0955224
$680$	0	0
$681$	26.3873	1.01116
$682$	0	0
$683$	21.0996	0.807355	0.403678	−	0.914901i	$-0.367732\pi$
$683$	21.0996	0.807355	0.403678	+	0.914901i	$0.367732\pi$
$684$	0	0
$685$	−30.5859	−1.16863
$686$	0	0
$687$	20.5282	0.783198
$688$	0	0
$689$	41.6272	1.58587
$690$	0	0
$691$	−3.84854	−0.146405	−0.0732027	−	0.997317i	$-0.523322\pi$
$691$	−3.84854	−0.146405	−0.0732027	+	0.997317i	$0.523322\pi$
$692$	0	0
$693$	13.5968	0.516500
$694$	0	0
$695$	42.8640	1.62593
$696$	0	0
$697$	−11.2392	−0.425713
$698$	0	0
$699$	−30.0588	−1.13693
$700$	0	0
$701$	−10.1258	−0.382446	−0.191223	−	0.981547i	$-0.561245\pi$
$701$	−10.1258	−0.382446	−0.191223	+	0.981547i	$0.561245\pi$
$702$	0	0
$703$	18.8399	0.710562
$704$	0	0
$705$	−30.5874	−1.15199
$706$	0	0
$707$	−33.2029	−1.24872
$708$	0	0
$709$	27.4286	1.03010	0.515051	−	0.857160i	$-0.327773\pi$
$709$	27.4286	1.03010	0.515051	+	0.857160i	$0.327773\pi$
$710$	0	0
$711$	−5.45482	−0.204572
$712$	0	0
$713$	−40.0092	−1.49836
$714$	0	0
$715$	−65.4709	−2.44847
$716$	0	0
$717$	30.9968	1.15760
$718$	0	0
$719$	43.0963	1.60722	0.803611	−	0.595154i	$-0.202909\pi$
$719$	43.0963	1.60722	0.803611	+	0.595154i	$0.202909\pi$
$720$	0	0
$721$	58.2775	2.17037
$722$	0	0
$723$	56.1867	2.08961
$724$	0	0
$725$	−45.5070	−1.69009
$726$	0	0
$727$	−18.6485	−0.691633	−0.345816	−	0.938302i	$-0.612398\pi$
$727$	−18.6485	−0.691633	−0.345816	+	0.938302i	$0.612398\pi$
$728$	0	0
$729$	17.6551	0.653893
$730$	0	0
$731$	−7.88936	−0.291799
$732$	0	0
$733$	27.8600	1.02903	0.514516	−	0.857481i	$-0.327972\pi$
$733$	27.8600	1.02903	0.514516	+	0.857481i	$0.327972\pi$
$734$	0	0
$735$	47.0689	1.73616
$736$	0	0
$737$	−66.1436	−2.43643
$738$	0	0
$739$	−35.1893	−1.29446	−0.647230	−	0.762295i	$-0.724073\pi$
$739$	−35.1893	−1.29446	−0.647230	+	0.762295i	$0.724073\pi$
$740$	0	0
$741$	−29.2380	−1.07409
$742$	0	0
$743$	−18.0429	−0.661930	−0.330965	−	0.943643i	$-0.607374\pi$
$743$	−18.0429	−0.661930	−0.330965	+	0.943643i	$0.607374\pi$
$744$	0	0
$745$	33.5211	1.22812
$746$	0	0
$747$	−3.78385	−0.138444
$748$	0	0
$749$	25.0926	0.916862
$750$	0	0
$751$	9.19104	0.335386	0.167693	−	0.985839i	$-0.446368\pi$
$751$	9.19104	0.335386	0.167693	+	0.985839i	$0.446368\pi$
$752$	0	0
$753$	1.93078	0.0703617
$754$	0	0
$755$	−48.9634	−1.78196
$756$	0	0
$757$	−20.0507	−0.728755	−0.364377	−	0.931251i	$-0.618718\pi$
$757$	−20.0507	−0.728755	−0.364377	+	0.931251i	$0.618718\pi$
$758$	0	0
$759$	−54.9260	−1.99369
$760$	0	0
$761$	0.875988	0.0317546	0.0158773	−	0.999874i	$-0.494946\pi$
$761$	0.875988	0.0317546	0.0158773	+	0.999874i	$0.494946\pi$
$762$	0	0
$763$	−2.86555	−0.103740
$764$	0	0
$765$	−15.6605	−0.566206
$766$	0	0
$767$	−11.0126	−0.397642
$768$	0	0
$769$	−28.1107	−1.01370	−0.506850	−	0.862034i	$-0.669190\pi$
$769$	−28.1107	−1.01370	−0.506850	+	0.862034i	$0.669190\pi$
$770$	0	0
$771$	−5.18629	−0.186780
$772$	0	0
$773$	10.2829	0.369852	0.184926	−	0.982752i	$-0.440795\pi$
$773$	10.2829	0.369852	0.184926	+	0.982752i	$0.440795\pi$
$774$	0	0
$775$	−33.6061	−1.20717
$776$	0	0
$777$	39.4287	1.41450
$778$	0	0
$779$	−5.82628	−0.208748
$780$	0	0
$781$	−41.3458	−1.47947
$782$	0	0
$783$	40.6429	1.45246
$784$	0	0
$785$	−4.16647	−0.148708
$786$	0	0
$787$	41.7674	1.48885	0.744424	−	0.667708i	$-0.232724\pi$
$787$	41.7674	1.48885	0.744424	+	0.667708i	$0.232724\pi$
$788$	0	0
$789$	−53.0004	−1.88686
$790$	0	0
$791$	−34.0677	−1.21131
$792$	0	0
$793$	14.6319	0.519594
$794$	0	0
$795$	59.1932	2.09937
$796$	0	0
$797$	−7.50122	−0.265707	−0.132853	−	0.991136i	$-0.542414\pi$
$797$	−7.50122	−0.265707	−0.132853	+	0.991136i	$0.542414\pi$
$798$	0	0
$799$	34.3850	1.21646
$800$	0	0
$801$	−3.45514	−0.122081
$802$	0	0
$803$	45.8488	1.61797
$804$	0	0
$805$	−70.6913	−2.49154
$806$	0	0
$807$	−53.3602	−1.87837
$808$	0	0
$809$	32.5066	1.14287	0.571435	−	0.820647i	$-0.306387\pi$
$809$	32.5066	1.14287	0.571435	+	0.820647i	$0.306387\pi$
$810$	0	0
$811$	−20.6695	−0.725805	−0.362902	−	0.931827i	$-0.618214\pi$
$811$	−20.6695	−0.725805	−0.362902	+	0.931827i	$0.618214\pi$
$812$	0	0
$813$	−18.2536	−0.640183
$814$	0	0
$815$	47.8803	1.67717
$816$	0	0
$817$	−4.08978	−0.143083
$818$	0	0
$819$	−11.9481	−0.417502
$820$	0	0
$821$	−48.1509	−1.68048	−0.840239	−	0.542216i	$-0.817585\pi$
$821$	−48.1509	−1.68048	−0.840239	+	0.542216i	$0.817585\pi$
$822$	0	0
$823$	19.8412	0.691621	0.345810	−	0.938304i	$-0.387604\pi$
$823$	19.8412	0.691621	0.345810	+	0.938304i	$0.387604\pi$
$824$	0	0
$825$	−46.1356	−1.60624
$826$	0	0
$827$	33.2001	1.15448	0.577241	−	0.816574i	$-0.304129\pi$
$827$	33.2001	1.15448	0.577241	+	0.816574i	$0.304129\pi$
$828$	0	0
$829$	17.5150	0.608322	0.304161	−	0.952621i	$-0.401624\pi$
$829$	17.5150	0.608322	0.304161	+	0.952621i	$0.401624\pi$
$830$	0	0
$831$	51.6562	1.79193
$832$	0	0
$833$	−52.9129	−1.83332
$834$	0	0
$835$	53.9126	1.86572
$836$	0	0
$837$	30.0141	1.03744
$838$	0	0
$839$	−34.2266	−1.18163	−0.590816	−	0.806806i	$-0.701194\pi$
$839$	−34.2266	−1.18163	−0.590816	+	0.806806i	$0.701194\pi$
$840$	0	0
$841$	56.8336	1.95978
$842$	0	0
$843$	12.2273	0.421132
$844$	0	0
$845$	16.6042	0.571202
$846$	0	0
$847$	−48.6296	−1.67093
$848$	0	0
$849$	2.19044	0.0751757
$850$	0	0
$851$	−31.1009	−1.06613
$852$	0	0
$853$	36.3278	1.24384	0.621920	−	0.783081i	$-0.286353\pi$
$853$	36.3278	1.24384	0.621920	+	0.783081i	$0.286353\pi$
$854$	0	0
$855$	−8.11826	−0.277639
$856$	0	0
$857$	−33.7156	−1.15170	−0.575852	−	0.817554i	$-0.695329\pi$
$857$	−33.7156	−1.15170	−0.575852	+	0.817554i	$0.695329\pi$
$858$	0	0
$859$	3.99703	0.136377	0.0681884	−	0.997672i	$-0.478278\pi$
$859$	3.99703	0.136377	0.0681884	+	0.997672i	$0.478278\pi$
$860$	0	0
$861$	−12.1934	−0.415549
$862$	0	0
$863$	−22.2514	−0.757446	−0.378723	−	0.925510i	$-0.623637\pi$
$863$	−22.2514	−0.757446	−0.378723	+	0.925510i	$0.623637\pi$
$864$	0	0
$865$	−59.6779	−2.02911
$866$	0	0
$867$	57.3365	1.94725
$868$	0	0
$869$	36.4541	1.23662
$870$	0	0
$871$	58.1235	1.96944
$872$	0	0
$873$	−0.471879	−0.0159707
$874$	0	0
$875$	1.06501	0.0360038
$876$	0	0
$877$	−55.5150	−1.87461	−0.937305	−	0.348511i	$-0.886687\pi$
$877$	−55.5150	−1.87461	−0.937305	+	0.348511i	$0.886687\pi$
$878$	0	0
$879$	34.7576	1.17235
$880$	0	0
$881$	39.3996	1.32741	0.663703	−	0.747996i	$-0.268984\pi$
$881$	39.3996	1.32741	0.663703	+	0.747996i	$0.268984\pi$
$882$	0	0
$883$	−36.7126	−1.23548	−0.617739	−	0.786383i	$-0.711951\pi$
$883$	−36.7126	−1.23548	−0.617739	+	0.786383i	$0.711951\pi$
$884$	0	0
$885$	−15.6598	−0.526397
$886$	0	0
$887$	−45.0772	−1.51354	−0.756772	−	0.653679i	$-0.773225\pi$
$887$	−45.0772	−1.51354	−0.756772	+	0.653679i	$0.773225\pi$
$888$	0	0
$889$	30.8934	1.03613
$890$	0	0
$891$	51.8277	1.73629
$892$	0	0
$893$	17.8249	0.596488
$894$	0	0
$895$	12.7916	0.427576
$896$	0	0
$897$	48.2660	1.61156
$898$	0	0
$899$	63.3866	2.11406
$900$	0	0
$901$	−66.5426	−2.21685
$902$	0	0
$903$	−8.55918	−0.284832
$904$	0	0
$905$	−10.2470	−0.340620
$906$	0	0
$907$	−21.8078	−0.724117	−0.362058	−	0.932155i	$-0.617926\pi$
$907$	−21.8078	−0.724117	−0.362058	+	0.932155i	$0.617926\pi$
$908$	0	0
$909$	6.29458	0.208778
$910$	0	0
$911$	−23.3251	−0.772794	−0.386397	−	0.922333i	$-0.626280\pi$
$911$	−23.3251	−0.772794	−0.386397	+	0.922333i	$0.626280\pi$
$912$	0	0
$913$	25.2871	0.836882
$914$	0	0
$915$	20.8064	0.687837
$916$	0	0
$917$	−65.5914	−2.16602
$918$	0	0
$919$	−54.6285	−1.80203	−0.901015	−	0.433789i	$-0.857176\pi$
$919$	−54.6285	−1.80203	−0.901015	+	0.433789i	$0.857176\pi$
$920$	0	0
$921$	0.881676	0.0290522
$922$	0	0
$923$	36.3325	1.19590
$924$	0	0
$925$	−26.1235	−0.858936
$926$	0	0
$927$	−11.0482	−0.362871
$928$	0	0
$929$	9.10790	0.298821	0.149410	−	0.988775i	$-0.452262\pi$
$929$	9.10790	0.298821	0.149410	+	0.988775i	$0.452262\pi$
$930$	0	0
$931$	−27.4296	−0.898970
$932$	0	0
$933$	−49.2862	−1.61356
$934$	0	0
$935$	104.658	3.42267
$936$	0	0
$937$	51.4244	1.67996	0.839981	−	0.542616i	$-0.182566\pi$
$937$	51.4244	1.67996	0.839981	+	0.542616i	$0.182566\pi$
$938$	0	0
$939$	−34.4816	−1.12526
$940$	0	0
$941$	−13.1380	−0.428287	−0.214144	−	0.976802i	$-0.568696\pi$
$941$	−13.1380	−0.428287	−0.214144	+	0.976802i	$0.568696\pi$
$942$	0	0
$943$	9.61801	0.313206
$944$	0	0
$945$	53.0311	1.72510
$946$	0	0
$947$	−3.06019	−0.0994427	−0.0497214	−	0.998763i	$-0.515833\pi$
$947$	−3.06019	−0.0994427	−0.0497214	+	0.998763i	$0.515833\pi$
$948$	0	0
$949$	−40.2895	−1.30785
$950$	0	0
$951$	−24.8841	−0.806923
$952$	0	0
$953$	32.9996	1.06896	0.534480	−	0.845181i	$-0.320507\pi$
$953$	32.9996	1.06896	0.534480	+	0.845181i	$0.320507\pi$
$954$	0	0
$955$	−40.1558	−1.29941
$956$	0	0
$957$	87.0192	2.81293
$958$	0	0
$959$	37.3027	1.20457
$960$	0	0
$961$	15.8098	0.509995
$962$	0	0
$963$	−4.75703	−0.153293
$964$	0	0
$965$	16.4348	0.529054
$966$	0	0
$967$	12.0743	0.388283	0.194142	−	0.980974i	$-0.437808\pi$
$967$	12.0743	0.388283	0.194142	+	0.980974i	$0.437808\pi$
$968$	0	0
$969$	46.7381	1.50144
$970$	0	0
$971$	−1.98101	−0.0635735	−0.0317868	−	0.999495i	$-0.510120\pi$
$971$	−1.98101	−0.0635735	−0.0317868	+	0.999495i	$0.510120\pi$
$972$	0	0
$973$	−52.2770	−1.67593
$974$	0	0
$975$	40.5415	1.29837
$976$	0	0
$977$	0.0563775	0.00180368	0.000901839	−	1.00000i	$-0.499713\pi$
$977$	0.0563775	0.00180368	0.000901839		1.00000i	$0.499713\pi$
$978$	0	0
$979$	23.0904	0.737972
$980$	0	0
$981$	0.543249	0.0173446
$982$	0	0
$983$	42.1061	1.34298	0.671488	−	0.741015i	$-0.265655\pi$
$983$	42.1061	1.34298	0.671488	+	0.741015i	$0.265655\pi$
$984$	0	0
$985$	−68.7930	−2.19193
$986$	0	0
$987$	37.3044	1.18741
$988$	0	0
$989$	6.75140	0.214682
$990$	0	0
$991$	−56.4892	−1.79444	−0.897219	−	0.441586i	$-0.854416\pi$
$991$	−56.4892	−1.79444	−0.897219	+	0.441586i	$0.854416\pi$
$992$	0	0
$993$	−14.3646	−0.455848
$994$	0	0
$995$	65.5934	2.07945
$996$	0	0
$997$	−6.06366	−0.192038	−0.0960191	−	0.995379i	$-0.530611\pi$
$997$	−6.06366	−0.192038	−0.0960191	+	0.995379i	$0.530611\pi$
$998$	0	0
$999$	23.3313	0.738168

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.j.1.12		14
4.3	odd	2		1004.2.a.b.1.3	✓	14
12.11	even	2		9036.2.a.m.1.2		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1004.2.a.b.1.3	✓	14	4.3	odd	2
4016.2.a.j.1.12		14	1.1	even	1	trivial
9036.2.a.m.1.2		14	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:	\( N \)	\(=\)	\( 4016 = 2^{4} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4016.a (trivial)

\(f(q)\)	\(=\)	\(q+1.93078 q^{3} +3.14832 q^{5} -3.83969 q^{7} +0.727926 q^{9} +O(q^{10})\)	\(q+1.93078 q^{3} +3.14832 q^{5} -3.83969 q^{7} +0.727926 q^{9} -4.86467 q^{11} +4.27481 q^{13} +6.07872 q^{15} -6.83344 q^{17} -3.54240 q^{19} -7.41361 q^{21} +5.84778 q^{23} +4.91190 q^{25} -4.38688 q^{27} -9.26464 q^{29} -6.84177 q^{31} -9.39262 q^{33} -12.0886 q^{35} -5.31841 q^{37} +8.25373 q^{39} +1.64473 q^{41} +1.15452 q^{43} +2.29174 q^{45} -5.03188 q^{47} +7.74323 q^{49} -13.1939 q^{51} +9.73778 q^{53} -15.3155 q^{55} -6.83961 q^{57} -2.57616 q^{59} +3.42282 q^{61} -2.79501 q^{63} +13.4585 q^{65} +13.5967 q^{67} +11.2908 q^{69} +8.49921 q^{71} -9.42486 q^{73} +9.48382 q^{75} +18.6788 q^{77} -7.49365 q^{79} -10.6539 q^{81} -5.19812 q^{83} -21.5138 q^{85} -17.8880 q^{87} -4.74655 q^{89} -16.4139 q^{91} -13.2100 q^{93} -11.1526 q^{95} -0.648252 q^{97} -3.54112 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 3 q^{3} - 2 q^{5} - 8 q^{7} + 21 q^{9}+O(q^{10}) \) 14 * q - 3 * q^3 - 2 * q^5 - 8 * q^7 + 21 * q^9	\( 14 q - 3 q^{3} - 2 q^{5} - 8 q^{7} + 21 q^{9} - 9 q^{11} - q^{13} - 14 q^{15} - 27 q^{19} - 3 q^{21} - 13 q^{23} + 26 q^{25} - 15 q^{27} - 25 q^{31} + 16 q^{33} - 21 q^{35} - q^{37} - 33 q^{39} + 10 q^{41} - 35 q^{43} - 4 q^{45} - 6 q^{47} + 36 q^{49} - 48 q^{51} - q^{53} - 41 q^{55} + 14 q^{57} - 30 q^{59} + 3 q^{61} - 31 q^{63} + 7 q^{65} - 22 q^{67} - 17 q^{69} - 6 q^{71} + 5 q^{73} - 4 q^{75} - 14 q^{77} - 56 q^{79} + 26 q^{81} + 28 q^{83} - 23 q^{85} - 11 q^{87} - 24 q^{89} - 38 q^{91} - 55 q^{93} + 4 q^{95} + 6 q^{97} - 12 q^{99}+O(q^{100}) \) 14 * q - 3 * q^3 - 2 * q^5 - 8 * q^7 + 21 * q^9 - 9 * q^11 - q^13 - 14 * q^15 - 27 * q^19 - 3 * q^21 - 13 * q^23 + 26 * q^25 - 15 * q^27 - 25 * q^31 + 16 * q^33 - 21 * q^35 - q^37 - 33 * q^39 + 10 * q^41 - 35 * q^43 - 4 * q^45 - 6 * q^47 + 36 * q^49 - 48 * q^51 - q^53 - 41 * q^55 + 14 * q^57 - 30 * q^59 + 3 * q^61 - 31 * q^63 + 7 * q^65 - 22 * q^67 - 17 * q^69 - 6 * q^71 + 5 * q^73 - 4 * q^75 - 14 * q^77 - 56 * q^79 + 26 * q^81 + 28 * q^83 - 23 * q^85 - 11 * q^87 - 24 * q^89 - 38 * q^91 - 55 * q^93 + 4 * q^95 + 6 * q^97 - 12 * q^99

Introduction

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