LMFDB - Embedded newform 4016.2.a.i.1.7

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 17 x^{10} + 49 x^{9} + 106 x^{8} - 277 x^{7} - 317 x^{6} + 644 x^{5} + 537 x^{4} + \cdots + 104 $ x^12 - 3x^11 - 17x^10 + 49x^9 + 106x^8 - 277x^7 - 317x^6 + 644x^5 + 537x^4 - 601x^3 - 462x^2 + 136*x + 104
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2008)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-0.589492$ 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)

$f(q)$	$=$	$q+0.589492 q^{3} -1.64401 q^{5} +1.73430 q^{7} -2.65250 q^{9} +O(q^{10})$	$q+0.589492 q^{3} -1.64401 q^{5} +1.73430 q^{7} -2.65250 q^{9} +4.09011 q^{11} -2.54482 q^{13} -0.969131 q^{15} -0.508848 q^{17} -0.571278 q^{19} +1.02236 q^{21} -4.07913 q^{23} -2.29723 q^{25} -3.33210 q^{27} +3.73475 q^{29} +2.84465 q^{31} +2.41109 q^{33} -2.85121 q^{35} +5.86102 q^{37} -1.50015 q^{39} +5.85981 q^{41} -11.2577 q^{43} +4.36074 q^{45} +1.87093 q^{47} -3.99220 q^{49} -0.299962 q^{51} -12.1668 q^{53} -6.72418 q^{55} -0.336764 q^{57} -12.3577 q^{59} +0.487198 q^{61} -4.60024 q^{63} +4.18371 q^{65} +6.83355 q^{67} -2.40461 q^{69} -1.66547 q^{71} +5.33920 q^{73} -1.35420 q^{75} +7.09348 q^{77} -8.87711 q^{79} +5.99325 q^{81} -3.35616 q^{83} +0.836552 q^{85} +2.20161 q^{87} -14.8319 q^{89} -4.41349 q^{91} +1.67690 q^{93} +0.939188 q^{95} -10.5046 q^{97} -10.8490 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{3} + 5 q^{5} - 5 q^{7} + 7 q^{9}+O(q^{10}) $ 12 * q - 3 * q^3 + 5 * q^5 - 5 * q^7 + 7 * q^9	$ 12 q - 3 q^{3} + 5 q^{5} - 5 q^{7} + 7 q^{9} - 10 q^{11} + 3 q^{13} - 11 q^{15} + 2 q^{17} - 15 q^{19} + 3 q^{21} - 20 q^{23} - 3 q^{25} - 15 q^{27} + 6 q^{29} - 14 q^{31} - 6 q^{33} - 16 q^{35} + 5 q^{37} - 21 q^{39} - 21 q^{43} + 10 q^{45} - 27 q^{47} - 13 q^{49} - 19 q^{51} + 22 q^{53} - 24 q^{55} + q^{57} - 23 q^{59} + 4 q^{61} - 21 q^{63} - q^{65} - 26 q^{67} + 10 q^{69} - 23 q^{71} - 8 q^{73} - 16 q^{75} + 22 q^{77} - 37 q^{79} - 20 q^{81} - 30 q^{83} + 2 q^{85} - 16 q^{87} + 3 q^{89} - 8 q^{91} + 20 q^{93} - 33 q^{95} - 4 q^{97} - 15 q^{99}+O(q^{100}) $ 12 * q - 3 * q^3 + 5 * q^5 - 5 * q^7 + 7 * q^9 - 10 * q^11 + 3 * q^13 - 11 * q^15 + 2 * q^17 - 15 * q^19 + 3 * q^21 - 20 * q^23 - 3 * q^25 - 15 * q^27 + 6 * q^29 - 14 * q^31 - 6 * q^33 - 16 * q^35 + 5 * q^37 - 21 * q^39 - 21 * q^43 + 10 * q^45 - 27 * q^47 - 13 * q^49 - 19 * q^51 + 22 * q^53 - 24 * q^55 + q^57 - 23 * q^59 + 4 * q^61 - 21 * q^63 - q^65 - 26 * q^67 + 10 * q^69 - 23 * q^71 - 8 * q^73 - 16 * q^75 + 22 * q^77 - 37 * q^79 - 20 * q^81 - 30 * q^83 + 2 * q^85 - 16 * q^87 + 3 * q^89 - 8 * q^91 + 20 * q^93 - 33 * q^95 - 4 * q^97 - 15 * 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
$3$	0.589492	0.340343	0.170172	−	0.985414i	$-0.445568\pi$
$3$	0.589492	0.340343	0.170172	+	0.985414i	$0.445568\pi$
$4$	0	0
$5$	−1.64401	−0.735224	−0.367612	−	0.929979i	$-0.619825\pi$
$5$	−1.64401	−0.735224	−0.367612	+	0.929979i	$0.619825\pi$
$6$	0	0
$7$	1.73430	0.655505	0.327752	−	0.944764i	$-0.393709\pi$
$7$	1.73430	0.655505	0.327752	+	0.944764i	$0.393709\pi$
$8$	0	0
$9$	−2.65250	−0.884166
$10$	0	0
$11$	4.09011	1.23321	0.616607	−	0.787271i	$-0.288507\pi$
$11$	4.09011	1.23321	0.616607	+	0.787271i	$0.288507\pi$
$12$	0	0
$13$	−2.54482	−0.705806	−0.352903	−	0.935660i	$-0.614805\pi$
$13$	−2.54482	−0.705806	−0.352903	+	0.935660i	$0.614805\pi$
$14$	0	0
$15$	−0.969131	−0.250229
$16$	0	0
$17$	−0.508848	−0.123414	−0.0617069	−	0.998094i	$-0.519654\pi$
$17$	−0.508848	−0.123414	−0.0617069	+	0.998094i	$0.519654\pi$
$18$	0	0
$19$	−0.571278	−0.131060	−0.0655301	−	0.997851i	$-0.520874\pi$
$19$	−0.571278	−0.131060	−0.0655301	+	0.997851i	$0.520874\pi$
$20$	0	0
$21$	1.02236	0.223097
$22$	0	0
$23$	−4.07913	−0.850557	−0.425279	−	0.905063i	$-0.639824\pi$
$23$	−4.07913	−0.850557	−0.425279	+	0.905063i	$0.639824\pi$
$24$	0	0
$25$	−2.29723	−0.459445
$26$	0	0
$27$	−3.33210	−0.641263
$28$	0	0
$29$	3.73475	0.693526	0.346763	−	0.937953i	$-0.387281\pi$
$29$	3.73475	0.693526	0.346763	+	0.937953i	$0.387281\pi$
$30$	0	0
$31$	2.84465	0.510914	0.255457	−	0.966820i	$-0.417774\pi$
$31$	2.84465	0.510914	0.255457	+	0.966820i	$0.417774\pi$
$32$	0	0
$33$	2.41109	0.419716
$34$	0	0
$35$	−2.85121	−0.481943
$36$	0	0
$37$	5.86102	0.963545	0.481773	−	0.876296i	$-0.339993\pi$
$37$	5.86102	0.963545	0.481773	+	0.876296i	$0.339993\pi$
$38$	0	0
$39$	−1.50015	−0.240216
$40$	0	0
$41$	5.85981	0.915148	0.457574	−	0.889171i	$-0.348718\pi$
$41$	5.85981	0.915148	0.457574	+	0.889171i	$0.348718\pi$
$42$	0	0
$43$	−11.2577	−1.71679	−0.858394	−	0.512990i	$-0.828538\pi$
$43$	−11.2577	−1.71679	−0.858394	+	0.512990i	$0.828538\pi$
$44$	0	0
$45$	4.36074	0.650061
$46$	0	0
$47$	1.87093	0.272904	0.136452	−	0.990647i	$-0.456430\pi$
$47$	1.87093	0.272904	0.136452	+	0.990647i	$0.456430\pi$
$48$	0	0
$49$	−3.99220	−0.570314
$50$	0	0
$51$	−0.299962	−0.0420031
$52$	0	0
$53$	−12.1668	−1.67124	−0.835622	−	0.549304i	$-0.814893\pi$
$53$	−12.1668	−1.67124	−0.835622	+	0.549304i	$0.814893\pi$
$54$	0	0
$55$	−6.72418	−0.906689
$56$	0	0
$57$	−0.336764	−0.0446055
$58$	0	0
$59$	−12.3577	−1.60884	−0.804418	−	0.594064i	$-0.797523\pi$
$59$	−12.3577	−1.60884	−0.804418	+	0.594064i	$0.797523\pi$
$60$	0	0
$61$	0.487198	0.0623793	0.0311897	−	0.999513i	$-0.490070\pi$
$61$	0.487198	0.0623793	0.0311897	+	0.999513i	$0.490070\pi$
$62$	0	0
$63$	−4.60024	−0.579575
$64$	0	0
$65$	4.18371	0.518926
$66$	0	0
$67$	6.83355	0.834851	0.417425	−	0.908711i	$-0.362933\pi$
$67$	6.83355	0.834851	0.417425	+	0.908711i	$0.362933\pi$
$68$	0	0
$69$	−2.40461	−0.289481
$70$	0	0
$71$	−1.66547	−0.197655	−0.0988277	−	0.995105i	$-0.531509\pi$
$71$	−1.66547	−0.197655	−0.0988277	+	0.995105i	$0.531509\pi$
$72$	0	0
$73$	5.33920	0.624906	0.312453	−	0.949933i	$-0.398849\pi$
$73$	5.33920	0.624906	0.312453	+	0.949933i	$0.398849\pi$
$74$	0	0
$75$	−1.35420	−0.156369
$76$	0	0
$77$	7.09348	0.808378
$78$	0	0
$79$	−8.87711	−0.998753	−0.499376	−	0.866385i	$-0.666438\pi$
$79$	−8.87711	−0.998753	−0.499376	+	0.866385i	$0.666438\pi$
$80$	0	0
$81$	5.99325	0.665917
$82$	0	0
$83$	−3.35616	−0.368386	−0.184193	−	0.982890i	$-0.558967\pi$
$83$	−3.35616	−0.368386	−0.184193	+	0.982890i	$0.558967\pi$
$84$	0	0
$85$	0.836552	0.0907368
$86$	0	0
$87$	2.20161	0.236037
$88$	0	0
$89$	−14.8319	−1.57217	−0.786087	−	0.618115i	$-0.787897\pi$
$89$	−14.8319	−1.57217	−0.786087	+	0.618115i	$0.787897\pi$
$90$	0	0
$91$	−4.41349	−0.462659
$92$	0	0
$93$	1.67690	0.173886
$94$	0	0
$95$	0.939188	0.0963587
$96$	0	0
$97$	−10.5046	−1.06658	−0.533289	−	0.845933i	$-0.679044\pi$
$97$	−10.5046	−1.06658	−0.533289	+	0.845933i	$0.679044\pi$
$98$	0	0
$99$	−10.8490	−1.09037
$100$	0	0
$101$	7.32333	0.728699	0.364349	−	0.931262i	$-0.381291\pi$
$101$	7.32333	0.728699	0.364349	+	0.931262i	$0.381291\pi$
$102$	0	0
$103$	15.1083	1.48867	0.744333	−	0.667809i	$-0.232768\pi$
$103$	15.1083	1.48867	0.744333	+	0.667809i	$0.232768\pi$
$104$	0	0
$105$	−1.68077	−0.164026
$106$	0	0
$107$	0.626758	0.0605909	0.0302955	−	0.999541i	$-0.490355\pi$
$107$	0.626758	0.0605909	0.0302955	+	0.999541i	$0.490355\pi$
$108$	0	0
$109$	−2.61175	−0.250161	−0.125080	−	0.992147i	$-0.539919\pi$
$109$	−2.61175	−0.250161	−0.125080	+	0.992147i	$0.539919\pi$
$110$	0	0
$111$	3.45502	0.327936
$112$	0	0
$113$	0.616665	0.0580110	0.0290055	−	0.999579i	$-0.490766\pi$
$113$	0.616665	0.0580110	0.0290055	+	0.999579i	$0.490766\pi$
$114$	0	0
$115$	6.70613	0.625350
$116$	0	0
$117$	6.75013	0.624050
$118$	0	0
$119$	−0.882497	−0.0808983
$120$	0	0
$121$	5.72899	0.520817
$122$	0	0
$123$	3.45431	0.311465
$124$	0	0
$125$	11.9967	1.07302
$126$	0	0
$127$	−4.30621	−0.382114	−0.191057	−	0.981579i	$-0.561192\pi$
$127$	−4.30621	−0.382114	−0.191057	+	0.981579i	$0.561192\pi$
$128$	0	0
$129$	−6.63634	−0.584297
$130$	0	0
$131$	−16.0023	−1.39813	−0.699064	−	0.715059i	$-0.746400\pi$
$131$	−16.0023	−1.39813	−0.699064	+	0.715059i	$0.746400\pi$
$132$	0	0
$133$	−0.990769	−0.0859106
$134$	0	0
$135$	5.47801	0.471472
$136$	0	0
$137$	−5.46282	−0.466720	−0.233360	−	0.972390i	$-0.574972\pi$
$137$	−5.46282	−0.466720	−0.233360	+	0.972390i	$0.574972\pi$
$138$	0	0
$139$	−5.64183	−0.478534	−0.239267	−	0.970954i	$-0.576907\pi$
$139$	−5.64183	−0.478534	−0.239267	+	0.970954i	$0.576907\pi$
$140$	0	0
$141$	1.10290	0.0928810
$142$	0	0
$143$	−10.4086	−0.870410
$144$	0	0
$145$	−6.13998	−0.509897
$146$	0	0
$147$	−2.35337	−0.194102
$148$	0	0
$149$	4.78146	0.391712	0.195856	−	0.980633i	$-0.437251\pi$
$149$	4.78146	0.391712	0.195856	+	0.980633i	$0.437251\pi$
$150$	0	0
$151$	−12.0277	−0.978802	−0.489401	−	0.872059i	$-0.662785\pi$
$151$	−12.0277	−0.978802	−0.489401	+	0.872059i	$0.662785\pi$
$152$	0	0
$153$	1.34972	0.109118
$154$	0	0
$155$	−4.67664	−0.375636
$156$	0	0
$157$	−5.62596	−0.449001	−0.224500	−	0.974474i	$-0.572075\pi$
$157$	−5.62596	−0.449001	−0.224500	+	0.974474i	$0.572075\pi$
$158$	0	0
$159$	−7.17226	−0.568797
$160$	0	0
$161$	−7.07444	−0.557544
$162$	0	0
$163$	−5.41786	−0.424359	−0.212180	−	0.977231i	$-0.568056\pi$
$163$	−5.41786	−0.424359	−0.212180	+	0.977231i	$0.568056\pi$
$164$	0	0
$165$	−3.96385	−0.308585
$166$	0	0
$167$	−16.7773	−1.29827	−0.649133	−	0.760675i	$-0.724868\pi$
$167$	−16.7773	−1.29827	−0.649133	+	0.760675i	$0.724868\pi$
$168$	0	0
$169$	−6.52389	−0.501838
$170$	0	0
$171$	1.51532	0.115879
$172$	0	0
$173$	8.56627	0.651281	0.325641	−	0.945494i	$-0.394420\pi$
$173$	8.56627	0.651281	0.325641	+	0.945494i	$0.394420\pi$
$174$	0	0
$175$	−3.98409	−0.301169
$176$	0	0
$177$	−7.28476	−0.547556
$178$	0	0
$179$	−5.51111	−0.411920	−0.205960	−	0.978560i	$-0.566032\pi$
$179$	−5.51111	−0.411920	−0.205960	+	0.978560i	$0.566032\pi$
$180$	0	0
$181$	−10.3730	−0.771021	−0.385511	−	0.922703i	$-0.625975\pi$
$181$	−10.3730	−0.771021	−0.385511	+	0.922703i	$0.625975\pi$
$182$	0	0
$183$	0.287199	0.0212304
$184$	0	0
$185$	−9.63558	−0.708422
$186$	0	0
$187$	−2.08124	−0.152196
$188$	0	0
$189$	−5.77887	−0.420351
$190$	0	0
$191$	1.08104	0.0782214	0.0391107	−	0.999235i	$-0.487548\pi$
$191$	1.08104	0.0782214	0.0391107	+	0.999235i	$0.487548\pi$
$192$	0	0
$193$	−17.8444	−1.28447	−0.642234	−	0.766508i	$-0.721992\pi$
$193$	−17.8444	−1.28447	−0.642234	+	0.766508i	$0.721992\pi$
$194$	0	0
$195$	2.46626	0.176613
$196$	0	0
$197$	27.3323	1.94735	0.973674	−	0.227946i	$-0.0732011\pi$
$197$	27.3323	1.94735	0.973674	+	0.227946i	$0.0732011\pi$
$198$	0	0
$199$	−15.2058	−1.07791	−0.538956	−	0.842334i	$-0.681181\pi$
$199$	−15.2058	−1.07791	−0.538956	+	0.842334i	$0.681181\pi$
$200$	0	0
$201$	4.02832	0.284136
$202$	0	0
$203$	6.47719	0.454610
$204$	0	0
$205$	−9.63359	−0.672839
$206$	0	0
$207$	10.8199	0.752034
$208$	0	0
$209$	−2.33659	−0.161625
$210$	0	0
$211$	25.5033	1.75572	0.877861	−	0.478916i	$-0.158970\pi$
$211$	25.5033	1.75572	0.877861	+	0.478916i	$0.158970\pi$
$212$	0	0
$213$	−0.981783	−0.0672707
$214$	0	0
$215$	18.5078	1.26222
$216$	0	0
$217$	4.93348	0.334907
$218$	0	0
$219$	3.14742	0.212683
$220$	0	0
$221$	1.29493	0.0871062
$222$	0	0
$223$	−19.3257	−1.29415	−0.647073	−	0.762428i	$-0.724007\pi$
$223$	−19.3257	−1.29415	−0.647073	+	0.762428i	$0.724007\pi$
$224$	0	0
$225$	6.09339	0.406226
$226$	0	0
$227$	1.30809	0.0868210	0.0434105	−	0.999057i	$-0.486178\pi$
$227$	1.30809	0.0868210	0.0434105	+	0.999057i	$0.486178\pi$
$228$	0	0
$229$	−0.311156	−0.0205618	−0.0102809	−	0.999947i	$-0.503273\pi$
$229$	−0.311156	−0.0205618	−0.0102809	+	0.999947i	$0.503273\pi$
$230$	0	0
$231$	4.18155	0.275126
$232$	0	0
$233$	16.3971	1.07421	0.537105	−	0.843515i	$-0.319518\pi$
$233$	16.3971	1.07421	0.537105	+	0.843515i	$0.319518\pi$
$234$	0	0
$235$	−3.07584	−0.200645
$236$	0	0
$237$	−5.23298	−0.339919
$238$	0	0
$239$	−28.2366	−1.82647	−0.913235	−	0.407432i	$-0.866424\pi$
$239$	−28.2366	−1.82647	−0.913235	+	0.407432i	$0.866424\pi$
$240$	0	0
$241$	−1.66607	−0.107321	−0.0536606	−	0.998559i	$-0.517089\pi$
$241$	−1.66607	−0.107321	−0.0536606	+	0.998559i	$0.517089\pi$
$242$	0	0
$243$	13.5293	0.867904
$244$	0	0
$245$	6.56322	0.419308
$246$	0	0
$247$	1.45380	0.0925031
$248$	0	0
$249$	−1.97843	−0.125378
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	−16.6841	−1.04892
$254$	0	0
$255$	0.493141	0.0308817
$256$	0	0
$257$	−7.35106	−0.458547	−0.229273	−	0.973362i	$-0.573635\pi$
$257$	−7.35106	−0.458547	−0.229273	+	0.973362i	$0.573635\pi$
$258$	0	0
$259$	10.1648	0.631608
$260$	0	0
$261$	−9.90643	−0.613193
$262$	0	0
$263$	3.37180	0.207914	0.103957	−	0.994582i	$-0.466850\pi$
$263$	3.37180	0.207914	0.103957	+	0.994582i	$0.466850\pi$
$264$	0	0
$265$	20.0024	1.22874
$266$	0	0
$267$	−8.74327	−0.535079
$268$	0	0
$269$	−25.8277	−1.57474	−0.787371	−	0.616479i	$-0.788558\pi$
$269$	−25.8277	−1.57474	−0.787371	+	0.616479i	$0.788558\pi$
$270$	0	0
$271$	−12.2605	−0.744772	−0.372386	−	0.928078i	$-0.621460\pi$
$271$	−12.2605	−0.744772	−0.372386	+	0.928078i	$0.621460\pi$
$272$	0	0
$273$	−2.60171	−0.157463
$274$	0	0
$275$	−9.39591	−0.566595
$276$	0	0
$277$	20.1083	1.20819	0.604096	−	0.796912i	$-0.293534\pi$
$277$	20.1083	1.20819	0.604096	+	0.796912i	$0.293534\pi$
$278$	0	0
$279$	−7.54543	−0.451733
$280$	0	0
$281$	−25.0444	−1.49402	−0.747011	−	0.664812i	$-0.768512\pi$
$281$	−25.0444	−1.49402	−0.747011	+	0.664812i	$0.768512\pi$
$282$	0	0
$283$	5.42835	0.322682	0.161341	−	0.986899i	$-0.448418\pi$
$283$	5.42835	0.322682	0.161341	+	0.986899i	$0.448418\pi$
$284$	0	0
$285$	0.553644	0.0327950
$286$	0	0
$287$	10.1627	0.599884
$288$	0	0
$289$	−16.7411	−0.984769
$290$	0	0
$291$	−6.19236	−0.363003
$292$	0	0
$293$	2.46212	0.143839	0.0719194	−	0.997410i	$-0.477088\pi$
$293$	2.46212	0.143839	0.0719194	+	0.997410i	$0.477088\pi$
$294$	0	0
$295$	20.3162	1.18285
$296$	0	0
$297$	−13.6287	−0.790815
$298$	0	0
$299$	10.3806	0.600328
$300$	0	0
$301$	−19.5243	−1.12536
$302$	0	0
$303$	4.31704	0.248008
$304$	0	0
$305$	−0.800959	−0.0458628
$306$	0	0
$307$	5.34249	0.304912	0.152456	−	0.988310i	$-0.451282\pi$
$307$	5.34249	0.304912	0.152456	+	0.988310i	$0.451282\pi$
$308$	0	0
$309$	8.90622	0.506657
$310$	0	0
$311$	27.3660	1.55178	0.775892	−	0.630866i	$-0.217300\pi$
$311$	27.3660	1.55178	0.775892	+	0.630866i	$0.217300\pi$
$312$	0	0
$313$	−14.1819	−0.801607	−0.400804	−	0.916164i	$-0.631269\pi$
$313$	−14.1819	−0.801607	−0.400804	+	0.916164i	$0.631269\pi$
$314$	0	0
$315$	7.56284	0.426118
$316$	0	0
$317$	17.2768	0.970360	0.485180	−	0.874414i	$-0.338754\pi$
$317$	17.2768	0.970360	0.485180	+	0.874414i	$0.338754\pi$
$318$	0	0
$319$	15.2755	0.855266
$320$	0	0
$321$	0.369468	0.0206217
$322$	0	0
$323$	0.290694	0.0161746
$324$	0	0
$325$	5.84603	0.324279
$326$	0	0
$327$	−1.53961	−0.0851405
$328$	0	0
$329$	3.24477	0.178890
$330$	0	0
$331$	−11.4261	−0.628036	−0.314018	−	0.949417i	$-0.601675\pi$
$331$	−11.4261	−0.628036	−0.314018	+	0.949417i	$0.601675\pi$
$332$	0	0
$333$	−15.5463	−0.851934
$334$	0	0
$335$	−11.2344	−0.613802
$336$	0	0
$337$	−6.23019	−0.339380	−0.169690	−	0.985498i	$-0.554277\pi$
$337$	−6.23019	−0.339380	−0.169690	+	0.985498i	$0.554277\pi$
$338$	0	0
$339$	0.363519	0.0197437
$340$	0	0
$341$	11.6349	0.630066
$342$	0	0
$343$	−19.0638	−1.02935
$344$	0	0
$345$	3.95321	0.212834
$346$	0	0
$347$	−20.9146	−1.12275	−0.561377	−	0.827560i	$-0.689728\pi$
$347$	−20.9146	−1.12275	−0.561377	+	0.827560i	$0.689728\pi$
$348$	0	0
$349$	12.0494	0.644992	0.322496	−	0.946571i	$-0.395478\pi$
$349$	12.0494	0.644992	0.322496	+	0.946571i	$0.395478\pi$
$350$	0	0
$351$	8.47960	0.452608
$352$	0	0
$353$	26.8302	1.42803	0.714013	−	0.700133i	$-0.246876\pi$
$353$	26.8302	1.42803	0.714013	+	0.700133i	$0.246876\pi$
$354$	0	0
$355$	2.73806	0.145321
$356$	0	0
$357$	−0.520225	−0.0275332
$358$	0	0
$359$	24.1297	1.27352	0.636759	−	0.771063i	$-0.280274\pi$
$359$	24.1297	1.27352	0.636759	+	0.771063i	$0.280274\pi$
$360$	0	0
$361$	−18.6736	−0.982823
$362$	0	0
$363$	3.37719	0.177257
$364$	0	0
$365$	−8.77771	−0.459446
$366$	0	0
$367$	−13.0304	−0.680181	−0.340090	−	0.940393i	$-0.610458\pi$
$367$	−13.0304	−0.680181	−0.340090	+	0.940393i	$0.610458\pi$
$368$	0	0
$369$	−15.5431	−0.809143
$370$	0	0
$371$	−21.1010	−1.09551
$372$	0	0
$373$	36.2665	1.87781	0.938905	−	0.344176i	$-0.111842\pi$
$373$	36.2665	1.87781	0.938905	+	0.344176i	$0.111842\pi$
$374$	0	0
$375$	7.07197	0.365195
$376$	0	0
$377$	−9.50427	−0.489495
$378$	0	0
$379$	−21.4907	−1.10390	−0.551951	−	0.833877i	$-0.686116\pi$
$379$	−21.4907	−1.10390	−0.551951	+	0.833877i	$0.686116\pi$
$380$	0	0
$381$	−2.53847	−0.130050
$382$	0	0
$383$	14.8407	0.758323	0.379161	−	0.925331i	$-0.376213\pi$
$383$	14.8407	0.758323	0.379161	+	0.925331i	$0.376213\pi$
$384$	0	0
$385$	−11.6618	−0.594339
$386$	0	0
$387$	29.8611	1.51793
$388$	0	0
$389$	27.3939	1.38892	0.694462	−	0.719529i	$-0.255642\pi$
$389$	27.3939	1.38892	0.694462	+	0.719529i	$0.255642\pi$
$390$	0	0
$391$	2.07566	0.104971
$392$	0	0
$393$	−9.43324	−0.475844
$394$	0	0
$395$	14.5941	0.734307
$396$	0	0
$397$	3.49025	0.175171	0.0875853	−	0.996157i	$-0.472085\pi$
$397$	3.49025	0.175171	0.0875853	+	0.996157i	$0.472085\pi$
$398$	0	0
$399$	−0.584050	−0.0292391
$400$	0	0
$401$	−2.39429	−0.119565	−0.0597826	−	0.998211i	$-0.519041\pi$
$401$	−2.39429	−0.119565	−0.0597826	+	0.998211i	$0.519041\pi$
$402$	0	0
$403$	−7.23912	−0.360606
$404$	0	0
$405$	−9.85297	−0.489598
$406$	0	0
$407$	23.9722	1.18826
$408$	0	0
$409$	−21.1020	−1.04342	−0.521712	−	0.853121i	$-0.674707\pi$
$409$	−21.1020	−1.04342	−0.521712	+	0.853121i	$0.674707\pi$
$410$	0	0
$411$	−3.22029	−0.158845
$412$	0	0
$413$	−21.4320	−1.05460
$414$	0	0
$415$	5.51756	0.270846
$416$	0	0
$417$	−3.32581	−0.162866
$418$	0	0
$419$	31.5480	1.54122	0.770610	−	0.637307i	$-0.219952\pi$
$419$	31.5480	1.54122	0.770610	+	0.637307i	$0.219952\pi$
$420$	0	0
$421$	−31.8126	−1.55045	−0.775225	−	0.631685i	$-0.782364\pi$
$421$	−31.8126	−1.55045	−0.775225	+	0.631685i	$0.782364\pi$
$422$	0	0
$423$	−4.96265	−0.241292
$424$	0	0
$425$	1.16894	0.0567019
$426$	0	0
$427$	0.844949	0.0408899
$428$	0	0
$429$	−6.13578	−0.296238
$430$	0	0
$431$	−4.95630	−0.238737	−0.119368	−	0.992850i	$-0.538087\pi$
$431$	−4.95630	−0.238737	−0.119368	+	0.992850i	$0.538087\pi$
$432$	0	0
$433$	7.77913	0.373841	0.186920	−	0.982375i	$-0.440149\pi$
$433$	7.77913	0.373841	0.186920	+	0.982375i	$0.440149\pi$
$434$	0	0
$435$	−3.61947	−0.173540
$436$	0	0
$437$	2.33032	0.111474
$438$	0	0
$439$	−12.4381	−0.593639	−0.296819	−	0.954934i	$-0.595926\pi$
$439$	−12.4381	−0.593639	−0.296819	+	0.954934i	$0.595926\pi$
$440$	0	0
$441$	10.5893	0.504252
$442$	0	0
$443$	−17.9688	−0.853725	−0.426863	−	0.904316i	$-0.640381\pi$
$443$	−17.9688	−0.853725	−0.426863	+	0.904316i	$0.640381\pi$
$444$	0	0
$445$	24.3838	1.15590
$446$	0	0
$447$	2.81863	0.133317
$448$	0	0
$449$	38.0020	1.79342	0.896712	−	0.442615i	$-0.145949\pi$
$449$	38.0020	1.79342	0.896712	+	0.442615i	$0.145949\pi$
$450$	0	0
$451$	23.9673	1.12857
$452$	0	0
$453$	−7.09024	−0.333129
$454$	0	0
$455$	7.25582	0.340158
$456$	0	0
$457$	−1.03627	−0.0484745	−0.0242373	−	0.999706i	$-0.507716\pi$
$457$	−1.03627	−0.0484745	−0.0242373	+	0.999706i	$0.507716\pi$
$458$	0	0
$459$	1.69553	0.0791408
$460$	0	0
$461$	12.8968	0.600665	0.300333	−	0.953835i	$-0.402902\pi$
$461$	12.8968	0.600665	0.300333	+	0.953835i	$0.402902\pi$
$462$	0	0
$463$	−6.45985	−0.300215	−0.150107	−	0.988670i	$-0.547962\pi$
$463$	−6.45985	−0.300215	−0.150107	+	0.988670i	$0.547962\pi$
$464$	0	0
$465$	−2.75684	−0.127845
$466$	0	0
$467$	−12.2719	−0.567876	−0.283938	−	0.958843i	$-0.591641\pi$
$467$	−12.2719	−0.567876	−0.283938	+	0.958843i	$0.591641\pi$
$468$	0	0
$469$	11.8514	0.547249
$470$	0	0
$471$	−3.31646	−0.152814
$472$	0	0
$473$	−46.0454	−2.11717
$474$	0	0
$475$	1.31236	0.0602150
$476$	0	0
$477$	32.2726	1.47766
$478$	0	0
$479$	11.5251	0.526597	0.263299	−	0.964714i	$-0.415190\pi$
$479$	11.5251	0.526597	0.263299	+	0.964714i	$0.415190\pi$
$480$	0	0
$481$	−14.9152	−0.680076
$482$	0	0
$483$	−4.17033	−0.189756
$484$	0	0
$485$	17.2696	0.784174
$486$	0	0
$487$	2.73887	0.124110	0.0620550	−	0.998073i	$-0.480235\pi$
$487$	2.73887	0.124110	0.0620550	+	0.998073i	$0.480235\pi$
$488$	0	0
$489$	−3.19378	−0.144428
$490$	0	0
$491$	9.01001	0.406616	0.203308	−	0.979115i	$-0.434831\pi$
$491$	9.01001	0.406616	0.203308	+	0.979115i	$0.434831\pi$
$492$	0	0
$493$	−1.90042	−0.0855907
$494$	0	0
$495$	17.8359	0.801664
$496$	0	0
$497$	−2.88844	−0.129564
$498$	0	0
$499$	29.1977	1.30707	0.653534	−	0.756897i	$-0.273286\pi$
$499$	29.1977	1.30707	0.653534	+	0.756897i	$0.273286\pi$
$500$	0	0
$501$	−9.89007	−0.441856
$502$	0	0
$503$	16.1313	0.719261	0.359630	−	0.933095i	$-0.382903\pi$
$503$	16.1313	0.719261	0.359630	+	0.933095i	$0.382903\pi$
$504$	0	0
$505$	−12.0396	−0.535757
$506$	0	0
$507$	−3.84578	−0.170797
$508$	0	0
$509$	−7.17646	−0.318091	−0.159045	−	0.987271i	$-0.550842\pi$
$509$	−7.17646	−0.318091	−0.159045	+	0.987271i	$0.550842\pi$
$510$	0	0
$511$	9.25979	0.409629
$512$	0	0
$513$	1.90356	0.0840441
$514$	0	0
$515$	−24.8382	−1.09450
$516$	0	0
$517$	7.65232	0.336549
$518$	0	0
$519$	5.04975	0.221659
$520$	0	0
$521$	11.9199	0.522219	0.261110	−	0.965309i	$-0.415912\pi$
$521$	11.9199	0.522219	0.261110	+	0.965309i	$0.415912\pi$
$522$	0	0
$523$	7.20013	0.314840	0.157420	−	0.987532i	$-0.449682\pi$
$523$	7.20013	0.314840	0.157420	+	0.987532i	$0.449682\pi$
$524$	0	0
$525$	−2.34859	−0.102501
$526$	0	0
$527$	−1.44749	−0.0630539
$528$	0	0
$529$	−6.36071	−0.276553
$530$	0	0
$531$	32.7788	1.42248
$532$	0	0
$533$	−14.9122	−0.645917
$534$	0	0
$535$	−1.03040	−0.0445479
$536$	0	0
$537$	−3.24876	−0.140194
$538$	0	0
$539$	−16.3285	−0.703319
$540$	0	0
$541$	−9.17266	−0.394363	−0.197182	−	0.980367i	$-0.563179\pi$
$541$	−9.17266	−0.394363	−0.197182	+	0.980367i	$0.563179\pi$
$542$	0	0
$543$	−6.11482	−0.262412
$544$	0	0
$545$	4.29375	0.183924
$546$	0	0
$547$	10.7102	0.457937	0.228968	−	0.973434i	$-0.426465\pi$
$547$	10.7102	0.457937	0.228968	+	0.973434i	$0.426465\pi$
$548$	0	0
$549$	−1.29229	−0.0551537
$550$	0	0
$551$	−2.13358	−0.0908937
$552$	0	0
$553$	−15.3956	−0.654687
$554$	0	0
$555$	−5.68009	−0.241107
$556$	0	0
$557$	25.1285	1.06473	0.532364	−	0.846516i	$-0.321304\pi$
$557$	25.1285	1.06473	0.532364	+	0.846516i	$0.321304\pi$
$558$	0	0
$559$	28.6489	1.21172
$560$	0	0
$561$	−1.22688	−0.0517988
$562$	0	0
$563$	22.4243	0.945072	0.472536	−	0.881311i	$-0.343339\pi$
$563$	22.4243	0.945072	0.472536	+	0.881311i	$0.343339\pi$
$564$	0	0
$565$	−1.01380	−0.0426511
$566$	0	0
$567$	10.3941	0.436512
$568$	0	0
$569$	25.0341	1.04948	0.524742	−	0.851262i	$-0.324162\pi$
$569$	25.0341	1.04948	0.524742	+	0.851262i	$0.324162\pi$
$570$	0	0
$571$	17.3023	0.724077	0.362038	−	0.932163i	$-0.382081\pi$
$571$	17.3023	0.724077	0.362038	+	0.932163i	$0.382081\pi$
$572$	0	0
$573$	0.637265	0.0266221
$574$	0	0
$575$	9.37068	0.390785
$576$	0	0
$577$	−18.3503	−0.763933	−0.381967	−	0.924176i	$-0.624753\pi$
$577$	−18.3503	−0.763933	−0.381967	+	0.924176i	$0.624753\pi$
$578$	0	0
$579$	−10.5191	−0.437160
$580$	0	0
$581$	−5.82059	−0.241479
$582$	0	0
$583$	−49.7637	−2.06100
$584$	0	0
$585$	−11.0973	−0.458817
$586$	0	0
$587$	−5.91329	−0.244068	−0.122034	−	0.992526i	$-0.538942\pi$
$587$	−5.91329	−0.244068	−0.122034	+	0.992526i	$0.538942\pi$
$588$	0	0
$589$	−1.62509	−0.0669605
$590$	0	0
$591$	16.1122	0.662767
$592$	0	0
$593$	10.6209	0.436146	0.218073	−	0.975932i	$-0.430023\pi$
$593$	10.6209	0.436146	0.218073	+	0.975932i	$0.430023\pi$
$594$	0	0
$595$	1.45083	0.0594784
$596$	0	0
$597$	−8.96370	−0.366860
$598$	0	0
$599$	−3.58670	−0.146549	−0.0732743	−	0.997312i	$-0.523345\pi$
$599$	−3.58670	−0.146549	−0.0732743	+	0.997312i	$0.523345\pi$
$600$	0	0
$601$	3.15378	0.128645	0.0643226	−	0.997929i	$-0.479511\pi$
$601$	3.15378	0.128645	0.0643226	+	0.997929i	$0.479511\pi$
$602$	0	0
$603$	−18.1260	−0.738147
$604$	0	0
$605$	−9.41852	−0.382917
$606$	0	0
$607$	−10.4734	−0.425104	−0.212552	−	0.977150i	$-0.568177\pi$
$607$	−10.4734	−0.425104	−0.212552	+	0.977150i	$0.568177\pi$
$608$	0	0
$609$	3.81825	0.154723
$610$	0	0
$611$	−4.76119	−0.192617
$612$	0	0
$613$	−32.5995	−1.31668	−0.658341	−	0.752720i	$-0.728742\pi$
$613$	−32.5995	−1.31668	−0.658341	+	0.752720i	$0.728742\pi$
$614$	0	0
$615$	−5.67892	−0.228996
$616$	0	0
$617$	19.3155	0.777611	0.388805	−	0.921320i	$-0.372888\pi$
$617$	19.3155	0.777611	0.388805	+	0.921320i	$0.372888\pi$
$618$	0	0
$619$	−33.9733	−1.36550	−0.682751	−	0.730651i	$-0.739216\pi$
$619$	−33.9733	−1.36550	−0.682751	+	0.730651i	$0.739216\pi$
$620$	0	0
$621$	13.5921	0.545431
$622$	0	0
$623$	−25.7229	−1.03057
$624$	0	0
$625$	−8.23661	−0.329465
$626$	0	0
$627$	−1.37740	−0.0550081
$628$	0	0
$629$	−2.98237	−0.118915
$630$	0	0
$631$	38.0263	1.51380	0.756901	−	0.653530i	$-0.226712\pi$
$631$	38.0263	1.51380	0.756901	+	0.653530i	$0.226712\pi$
$632$	0	0
$633$	15.0340	0.597548
$634$	0	0
$635$	7.07946	0.280940
$636$	0	0
$637$	10.1594	0.402531
$638$	0	0
$639$	4.41767	0.174760
$640$	0	0
$641$	−10.1595	−0.401276	−0.200638	−	0.979665i	$-0.564302\pi$
$641$	−10.1595	−0.401276	−0.200638	+	0.979665i	$0.564302\pi$
$642$	0	0
$643$	24.1849	0.953760	0.476880	−	0.878968i	$-0.341768\pi$
$643$	24.1849	0.953760	0.476880	+	0.878968i	$0.341768\pi$
$644$	0	0
$645$	10.9102	0.429590
$646$	0	0
$647$	1.13665	0.0446864	0.0223432	−	0.999750i	$-0.492887\pi$
$647$	1.13665	0.0446864	0.0223432	+	0.999750i	$0.492887\pi$
$648$	0	0
$649$	−50.5443	−1.98404
$650$	0	0
$651$	2.90825	0.113983
$652$	0	0
$653$	34.6707	1.35677	0.678385	−	0.734706i	$-0.262680\pi$
$653$	34.6707	1.35677	0.678385	+	0.734706i	$0.262680\pi$
$654$	0	0
$655$	26.3080	1.02794
$656$	0	0
$657$	−14.1622	−0.552521
$658$	0	0
$659$	−30.5915	−1.19168	−0.595838	−	0.803105i	$-0.703180\pi$
$659$	−30.5915	−1.19168	−0.595838	+	0.803105i	$0.703180\pi$
$660$	0	0
$661$	5.34636	0.207949	0.103975	−	0.994580i	$-0.466844\pi$
$661$	5.34636	0.207949	0.103975	+	0.994580i	$0.466844\pi$
$662$	0	0
$663$	0.763349	0.0296460
$664$	0	0
$665$	1.62884	0.0631635
$666$	0	0
$667$	−15.2345	−0.589884
$668$	0	0
$669$	−11.3924	−0.440454
$670$	0	0
$671$	1.99269	0.0769271
$672$	0	0
$673$	−33.6187	−1.29591	−0.647953	−	0.761680i	$-0.724375\pi$
$673$	−33.6187	−1.29591	−0.647953	+	0.761680i	$0.724375\pi$
$674$	0	0
$675$	7.65459	0.294625
$676$	0	0
$677$	50.7626	1.95096	0.975482	−	0.220081i	$-0.0706320\pi$
$677$	50.7626	1.95096	0.975482	+	0.220081i	$0.0706320\pi$
$678$	0	0
$679$	−18.2181	−0.699147
$680$	0	0
$681$	0.771109	0.0295489
$682$	0	0
$683$	−46.8224	−1.79161	−0.895806	−	0.444446i	$-0.853400\pi$
$683$	−46.8224	−1.79161	−0.895806	+	0.444446i	$0.853400\pi$
$684$	0	0
$685$	8.98093	0.343144
$686$	0	0
$687$	−0.183424	−0.00699805
$688$	0	0
$689$	30.9624	1.17957
$690$	0	0
$691$	2.92976	0.111453	0.0557266	−	0.998446i	$-0.482252\pi$
$691$	2.92976	0.111453	0.0557266	+	0.998446i	$0.482252\pi$
$692$	0	0
$693$	−18.8155	−0.714740
$694$	0	0
$695$	9.27524	0.351830
$696$	0	0
$697$	−2.98175	−0.112942
$698$	0	0
$699$	9.66597	0.365600
$700$	0	0
$701$	27.1605	1.02584	0.512918	−	0.858437i	$-0.328564\pi$
$701$	27.1605	1.02584	0.512918	+	0.858437i	$0.328564\pi$
$702$	0	0
$703$	−3.34827	−0.126282
$704$	0	0
$705$	−1.81318	−0.0682883
$706$	0	0
$707$	12.7009	0.477665
$708$	0	0
$709$	29.4590	1.10635	0.553177	−	0.833064i	$-0.313415\pi$
$709$	29.4590	1.10635	0.553177	+	0.833064i	$0.313415\pi$
$710$	0	0
$711$	23.5465	0.883064
$712$	0	0
$713$	−11.6037	−0.434562
$714$	0	0
$715$	17.1118	0.639947
$716$	0	0
$717$	−16.6452	−0.621627
$718$	0	0
$719$	30.6677	1.14371	0.571856	−	0.820354i	$-0.306224\pi$
$719$	30.6677	1.14371	0.571856	+	0.820354i	$0.306224\pi$
$720$	0	0
$721$	26.2024	0.975827
$722$	0	0
$723$	−0.982137	−0.0365261
$724$	0	0
$725$	−8.57958	−0.318637
$726$	0	0
$727$	−44.0907	−1.63523	−0.817616	−	0.575763i	$-0.804705\pi$
$727$	−44.0907	−1.63523	−0.817616	+	0.575763i	$0.804705\pi$
$728$	0	0
$729$	−10.0044	−0.370532
$730$	0	0
$731$	5.72848	0.211875
$732$	0	0
$733$	−1.08603	−0.0401135	−0.0200568	−	0.999799i	$-0.506385\pi$
$733$	−1.08603	−0.0401135	−0.0200568	+	0.999799i	$0.506385\pi$
$734$	0	0
$735$	3.86896	0.142709
$736$	0	0
$737$	27.9500	1.02955
$738$	0	0
$739$	−23.0348	−0.847348	−0.423674	−	0.905815i	$-0.639260\pi$
$739$	−23.0348	−0.847348	−0.423674	+	0.905815i	$0.639260\pi$
$740$	0	0
$741$	0.857004	0.0314828
$742$	0	0
$743$	20.7319	0.760579	0.380289	−	0.924868i	$-0.375824\pi$
$743$	20.7319	0.760579	0.380289	+	0.924868i	$0.375824\pi$
$744$	0	0
$745$	−7.86077	−0.287996
$746$	0	0
$747$	8.90220	0.325715
$748$	0	0
$749$	1.08699	0.0397176
$750$	0	0
$751$	−18.1406	−0.661962	−0.330981	−	0.943637i	$-0.607380\pi$
$751$	−18.1406	−0.661962	−0.330981	+	0.943637i	$0.607380\pi$
$752$	0	0
$753$	−0.589492	−0.0214823
$754$	0	0
$755$	19.7737	0.719639
$756$	0	0
$757$	9.70616	0.352776	0.176388	−	0.984321i	$-0.443559\pi$
$757$	9.70616	0.352776	0.176388	+	0.984321i	$0.443559\pi$
$758$	0	0
$759$	−9.83513	−0.356993
$760$	0	0
$761$	26.3158	0.953947	0.476974	−	0.878918i	$-0.341734\pi$
$761$	26.3158	0.953947	0.476974	+	0.878918i	$0.341734\pi$
$762$	0	0
$763$	−4.52957	−0.163982
$764$	0	0
$765$	−2.21895	−0.0802265
$766$	0	0
$767$	31.4481	1.13553
$768$	0	0
$769$	7.91694	0.285492	0.142746	−	0.989759i	$-0.454407\pi$
$769$	7.91694	0.285492	0.142746	+	0.989759i	$0.454407\pi$
$770$	0	0
$771$	−4.33339	−0.156063
$772$	0	0
$773$	−1.36909	−0.0492426	−0.0246213	−	0.999697i	$-0.507838\pi$
$773$	−1.36909	−0.0492426	−0.0246213	+	0.999697i	$0.507838\pi$
$774$	0	0
$775$	−6.53480	−0.234737
$776$	0	0
$777$	5.99205	0.214964
$778$	0	0
$779$	−3.34758	−0.119940
$780$	0	0
$781$	−6.81197	−0.243751
$782$	0	0
$783$	−12.4446	−0.444733
$784$	0	0
$785$	9.24914	0.330116
$786$	0	0
$787$	−32.3153	−1.15192	−0.575958	−	0.817480i	$-0.695371\pi$
$787$	−32.3153	−1.15192	−0.575958	+	0.817480i	$0.695371\pi$
$788$	0	0
$789$	1.98765	0.0707622
$790$	0	0
$791$	1.06948	0.0380265
$792$	0	0
$793$	−1.23983	−0.0440277
$794$	0	0
$795$	11.7913	0.418193
$796$	0	0
$797$	36.6911	1.29966	0.649832	−	0.760078i	$-0.274839\pi$
$797$	36.6911	1.29966	0.649832	+	0.760078i	$0.274839\pi$
$798$	0	0
$799$	−0.952022	−0.0336801
$800$	0	0
$801$	39.3415	1.39006
$802$	0	0
$803$	21.8379	0.770643
$804$	0	0
$805$	11.6305	0.409920
$806$	0	0
$807$	−15.2252	−0.535953
$808$	0	0
$809$	−4.60315	−0.161838	−0.0809191	−	0.996721i	$-0.525786\pi$
$809$	−4.60315	−0.161838	−0.0809191	+	0.996721i	$0.525786\pi$
$810$	0	0
$811$	36.1865	1.27068	0.635341	−	0.772232i	$-0.280860\pi$
$811$	36.1865	1.27068	0.635341	+	0.772232i	$0.280860\pi$
$812$	0	0
$813$	−7.22746	−0.253478
$814$	0	0
$815$	8.90702	0.311999
$816$	0	0
$817$	6.43130	0.225003
$818$	0	0
$819$	11.7068	0.409068
$820$	0	0
$821$	−12.4577	−0.434777	−0.217388	−	0.976085i	$-0.569754\pi$
$821$	−12.4577	−0.434777	−0.217388	+	0.976085i	$0.569754\pi$
$822$	0	0
$823$	16.0034	0.557843	0.278921	−	0.960314i	$-0.410023\pi$
$823$	16.0034	0.557843	0.278921	+	0.960314i	$0.410023\pi$
$824$	0	0
$825$	−5.53881	−0.192837
$826$	0	0
$827$	8.98894	0.312576	0.156288	−	0.987712i	$-0.450047\pi$
$827$	8.98894	0.312576	0.156288	+	0.987712i	$0.450047\pi$
$828$	0	0
$829$	35.8946	1.24667	0.623336	−	0.781954i	$-0.285777\pi$
$829$	35.8946	1.24667	0.623336	+	0.781954i	$0.285777\pi$
$830$	0	0
$831$	11.8537	0.411200
$832$	0	0
$833$	2.03142	0.0703846
$834$	0	0
$835$	27.5820	0.954516
$836$	0	0
$837$	−9.47866	−0.327630
$838$	0	0
$839$	−29.8844	−1.03172	−0.515862	−	0.856671i	$-0.672528\pi$
$839$	−29.8844	−1.03172	−0.515862	+	0.856671i	$0.672528\pi$
$840$	0	0
$841$	−15.0516	−0.519021
$842$	0	0
$843$	−14.7634	−0.508480
$844$	0	0
$845$	10.7254	0.368963
$846$	0	0
$847$	9.93580	0.341398
$848$	0	0
$849$	3.19997	0.109823
$850$	0	0
$851$	−23.9078	−0.819550
$852$	0	0
$853$	3.46126	0.118511	0.0592556	−	0.998243i	$-0.481127\pi$
$853$	3.46126	0.118511	0.0592556	+	0.998243i	$0.481127\pi$
$854$	0	0
$855$	−2.49120	−0.0851971
$856$	0	0
$857$	−29.7625	−1.01667	−0.508334	−	0.861160i	$-0.669738\pi$
$857$	−29.7625	−1.01667	−0.508334	+	0.861160i	$0.669738\pi$
$858$	0	0
$859$	−49.6376	−1.69361	−0.846807	−	0.531900i	$-0.821478\pi$
$859$	−49.6376	−1.69361	−0.846807	+	0.531900i	$0.821478\pi$
$860$	0	0
$861$	5.99082	0.204166
$862$	0	0
$863$	−36.9726	−1.25856	−0.629280	−	0.777178i	$-0.716650\pi$
$863$	−36.9726	−1.25856	−0.629280	+	0.777178i	$0.716650\pi$
$864$	0	0
$865$	−14.0830	−0.478838
$866$	0	0
$867$	−9.86873	−0.335160
$868$	0	0
$869$	−36.3083	−1.23168
$870$	0	0
$871$	−17.3901	−0.589243
$872$	0	0
$873$	27.8634	0.943033
$874$	0	0
$875$	20.8059	0.703369
$876$	0	0
$877$	1.37324	0.0463712	0.0231856	−	0.999731i	$-0.492619\pi$
$877$	1.37324	0.0463712	0.0231856	+	0.999731i	$0.492619\pi$
$878$	0	0
$879$	1.45140	0.0489546
$880$	0	0
$881$	−5.60472	−0.188828	−0.0944140	−	0.995533i	$-0.530098\pi$
$881$	−5.60472	−0.188828	−0.0944140	+	0.995533i	$0.530098\pi$
$882$	0	0
$883$	12.9846	0.436967	0.218483	−	0.975841i	$-0.429889\pi$
$883$	12.9846	0.436967	0.218483	+	0.975841i	$0.429889\pi$
$884$	0	0
$885$	11.9762	0.402577
$886$	0	0
$887$	39.0821	1.31225	0.656124	−	0.754653i	$-0.272195\pi$
$887$	39.0821	1.31225	0.656124	+	0.754653i	$0.272195\pi$
$888$	0	0
$889$	−7.46827	−0.250478
$890$	0	0
$891$	24.5130	0.821218
$892$	0	0
$893$	−1.06882	−0.0357668
$894$	0	0
$895$	9.06033	0.302853
$896$	0	0
$897$	6.11931	0.204318
$898$	0	0
$899$	10.6241	0.354332
$900$	0	0
$901$	6.19108	0.206255
$902$	0	0
$903$	−11.5094	−0.383010
$904$	0	0
$905$	17.0534	0.566874
$906$	0	0
$907$	−23.8692	−0.792564	−0.396282	−	0.918129i	$-0.629700\pi$
$907$	−23.8692	−0.792564	−0.396282	+	0.918129i	$0.629700\pi$
$908$	0	0
$909$	−19.4251	−0.644291
$910$	0	0
$911$	−41.5234	−1.37573	−0.687866	−	0.725838i	$-0.741452\pi$
$911$	−41.5234	−1.37573	−0.687866	+	0.725838i	$0.741452\pi$
$912$	0	0
$913$	−13.7270	−0.454299
$914$	0	0
$915$	−0.472159	−0.0156091
$916$	0	0
$917$	−27.7529	−0.916480
$918$	0	0
$919$	44.0231	1.45219	0.726094	−	0.687595i	$-0.241334\pi$
$919$	44.0231	1.45219	0.726094	+	0.687595i	$0.241334\pi$
$920$	0	0
$921$	3.14935	0.103775
$922$	0	0
$923$	4.23833	0.139506
$924$	0	0
$925$	−13.4641	−0.442696
$926$	0	0
$927$	−40.0748	−1.31623
$928$	0	0
$929$	49.7981	1.63382	0.816911	−	0.576764i	$-0.195685\pi$
$929$	49.7981	1.63382	0.816911	+	0.576764i	$0.195685\pi$
$930$	0	0
$931$	2.28065	0.0747454
$932$	0	0
$933$	16.1320	0.528139
$934$	0	0
$935$	3.42159	0.111898
$936$	0	0
$937$	50.9096	1.66314	0.831572	−	0.555416i	$-0.187441\pi$
$937$	50.9096	1.66314	0.831572	+	0.555416i	$0.187441\pi$
$938$	0	0
$939$	−8.36010	−0.272822
$940$	0	0
$941$	−27.4115	−0.893591	−0.446795	−	0.894636i	$-0.647435\pi$
$941$	−27.4115	−0.893591	−0.446795	+	0.894636i	$0.647435\pi$
$942$	0	0
$943$	−23.9029	−0.778386
$944$	0	0
$945$	9.50053	0.309052
$946$	0	0
$947$	−12.2655	−0.398574	−0.199287	−	0.979941i	$-0.563863\pi$
$947$	−12.2655	−0.398574	−0.199287	+	0.979941i	$0.563863\pi$
$948$	0	0
$949$	−13.5873	−0.441063
$950$	0	0
$951$	10.1845	0.330255
$952$	0	0
$953$	57.3152	1.85662	0.928311	−	0.371805i	$-0.121261\pi$
$953$	57.3152	1.85662	0.928311	+	0.371805i	$0.121261\pi$
$954$	0	0
$955$	−1.77724	−0.0575103
$956$	0	0
$957$	9.00481	0.291084
$958$	0	0
$959$	−9.47417	−0.305937
$960$	0	0
$961$	−22.9080	−0.738967
$962$	0	0
$963$	−1.66247	−0.0535725
$964$	0	0
$965$	29.3364	0.944372
$966$	0	0
$967$	−18.1190	−0.582668	−0.291334	−	0.956621i	$-0.594099\pi$
$967$	−18.1190	−0.582668	−0.291334	+	0.956621i	$0.594099\pi$
$968$	0	0
$969$	0.171362	0.00550493
$970$	0	0
$971$	34.3665	1.10287	0.551437	−	0.834217i	$-0.314080\pi$
$971$	34.3665	1.10287	0.551437	+	0.834217i	$0.314080\pi$
$972$	0	0
$973$	−9.78464	−0.313681
$974$	0	0
$975$	3.44619	0.110366
$976$	0	0
$977$	−4.08584	−0.130718	−0.0653588	−	0.997862i	$-0.520819\pi$
$977$	−4.08584	−0.130718	−0.0653588	+	0.997862i	$0.520819\pi$
$978$	0	0
$979$	−60.6640	−1.93883
$980$	0	0
$981$	6.92768	0.221184
$982$	0	0
$983$	15.5433	0.495753	0.247877	−	0.968792i	$-0.420267\pi$
$983$	15.5433	0.495753	0.247877	+	0.968792i	$0.420267\pi$
$984$	0	0
$985$	−44.9346	−1.43174
$986$	0	0
$987$	1.91276	0.0608839
$988$	0	0
$989$	45.9218	1.46023
$990$	0	0
$991$	19.3910	0.615975	0.307987	−	0.951390i	$-0.400345\pi$
$991$	19.3910	0.615975	0.307987	+	0.951390i	$0.400345\pi$
$992$	0	0
$993$	−6.73560	−0.213748
$994$	0	0
$995$	24.9985	0.792506
$996$	0	0
$997$	−12.0315	−0.381042	−0.190521	−	0.981683i	$-0.561018\pi$
$997$	−12.0315	−0.381042	−0.190521	+	0.981683i	$0.561018\pi$
$998$	0	0
$999$	−19.5295	−0.617886

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.i.1.7		12
4.3	odd	2		2008.2.a.b.1.6	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.b.1.6	✓	12	4.3	odd	2
4016.2.a.i.1.7		12	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)

\(f(q)\)	\(=\)	\(q+0.589492 q^{3} -1.64401 q^{5} +1.73430 q^{7} -2.65250 q^{9} +O(q^{10})\)	\(q+0.589492 q^{3} -1.64401 q^{5} +1.73430 q^{7} -2.65250 q^{9} +4.09011 q^{11} -2.54482 q^{13} -0.969131 q^{15} -0.508848 q^{17} -0.571278 q^{19} +1.02236 q^{21} -4.07913 q^{23} -2.29723 q^{25} -3.33210 q^{27} +3.73475 q^{29} +2.84465 q^{31} +2.41109 q^{33} -2.85121 q^{35} +5.86102 q^{37} -1.50015 q^{39} +5.85981 q^{41} -11.2577 q^{43} +4.36074 q^{45} +1.87093 q^{47} -3.99220 q^{49} -0.299962 q^{51} -12.1668 q^{53} -6.72418 q^{55} -0.336764 q^{57} -12.3577 q^{59} +0.487198 q^{61} -4.60024 q^{63} +4.18371 q^{65} +6.83355 q^{67} -2.40461 q^{69} -1.66547 q^{71} +5.33920 q^{73} -1.35420 q^{75} +7.09348 q^{77} -8.87711 q^{79} +5.99325 q^{81} -3.35616 q^{83} +0.836552 q^{85} +2.20161 q^{87} -14.8319 q^{89} -4.41349 q^{91} +1.67690 q^{93} +0.939188 q^{95} -10.5046 q^{97} -10.8490 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{3} + 5 q^{5} - 5 q^{7} + 7 q^{9}+O(q^{10}) \) 12 * q - 3 * q^3 + 5 * q^5 - 5 * q^7 + 7 * q^9	\( 12 q - 3 q^{3} + 5 q^{5} - 5 q^{7} + 7 q^{9} - 10 q^{11} + 3 q^{13} - 11 q^{15} + 2 q^{17} - 15 q^{19} + 3 q^{21} - 20 q^{23} - 3 q^{25} - 15 q^{27} + 6 q^{29} - 14 q^{31} - 6 q^{33} - 16 q^{35} + 5 q^{37} - 21 q^{39} - 21 q^{43} + 10 q^{45} - 27 q^{47} - 13 q^{49} - 19 q^{51} + 22 q^{53} - 24 q^{55} + q^{57} - 23 q^{59} + 4 q^{61} - 21 q^{63} - q^{65} - 26 q^{67} + 10 q^{69} - 23 q^{71} - 8 q^{73} - 16 q^{75} + 22 q^{77} - 37 q^{79} - 20 q^{81} - 30 q^{83} + 2 q^{85} - 16 q^{87} + 3 q^{89} - 8 q^{91} + 20 q^{93} - 33 q^{95} - 4 q^{97} - 15 q^{99}+O(q^{100}) \) 12 * q - 3 * q^3 + 5 * q^5 - 5 * q^7 + 7 * q^9 - 10 * q^11 + 3 * q^13 - 11 * q^15 + 2 * q^17 - 15 * q^19 + 3 * q^21 - 20 * q^23 - 3 * q^25 - 15 * q^27 + 6 * q^29 - 14 * q^31 - 6 * q^33 - 16 * q^35 + 5 * q^37 - 21 * q^39 - 21 * q^43 + 10 * q^45 - 27 * q^47 - 13 * q^49 - 19 * q^51 + 22 * q^53 - 24 * q^55 + q^57 - 23 * q^59 + 4 * q^61 - 21 * q^63 - q^65 - 26 * q^67 + 10 * q^69 - 23 * q^71 - 8 * q^73 - 16 * q^75 + 22 * q^77 - 37 * q^79 - 20 * q^81 - 30 * q^83 + 2 * q^85 - 16 * q^87 + 3 * q^89 - 8 * q^91 + 20 * q^93 - 33 * q^95 - 4 * q^97 - 15 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more