LMFDB - Embedded newform 4004.2.m.c.2157.28

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4004,2,Mod(2157,4004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4004, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4004.2157");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4004.m (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$31.9721009693$
Analytic rank:	$0$
Dimension:	$36$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2157.28
Character	$\chi$	$=$	4004.2157
Dual form			4004.2.m.c.2157.27

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.47001 q^{3} +1.22404i q^{5} -1.00000i q^{7} -0.839061 q^{9} +O(q^{10})$	$q+1.47001 q^{3} +1.22404i q^{5} -1.00000i q^{7} -0.839061 q^{9} +1.00000i q^{11} +(-3.29392 + 1.46633i) q^{13} +1.79935i q^{15} +5.50110 q^{17} -4.73877i q^{19} -1.47001i q^{21} +8.72579 q^{23} +3.50174 q^{25} -5.64347 q^{27} -1.63307 q^{29} -5.96177i q^{31} +1.47001i q^{33} +1.22404 q^{35} +9.25792i q^{37} +(-4.84210 + 2.15552i) q^{39} +6.84298i q^{41} +5.93685 q^{43} -1.02704i q^{45} -10.8026i q^{47} -1.00000 q^{49} +8.08669 q^{51} +3.59354 q^{53} -1.22404 q^{55} -6.96605i q^{57} -2.97310i q^{59} +3.03907 q^{61} +0.839061i q^{63} +(-1.79484 - 4.03187i) q^{65} +12.8053i q^{67} +12.8270 q^{69} +12.0729i q^{71} -1.51173i q^{73} +5.14760 q^{75} +1.00000 q^{77} +6.58964 q^{79} -5.77879 q^{81} +2.25080i q^{83} +6.73355i q^{85} -2.40063 q^{87} +9.64863i q^{89} +(1.46633 + 3.29392i) q^{91} -8.76388i q^{93} +5.80042 q^{95} +5.06930i q^{97} -0.839061i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q - 4 q^{3} + 40 q^{9}+O(q^{10}) $ 36 * q - 4 * q^3 + 40 * q^9	$ 36 q - 4 q^{3} + 40 q^{9} - 4 q^{17} + 8 q^{23} - 80 q^{25} + 8 q^{27} + 8 q^{29} - 24 q^{39} + 32 q^{43} - 36 q^{49} - 20 q^{51} + 12 q^{53} + 32 q^{61} - 24 q^{65} + 80 q^{69} - 36 q^{75} + 36 q^{77} + 16 q^{79} + 132 q^{81} + 8 q^{91} + 56 q^{95}+O(q^{100}) $ 36 * q - 4 * q^3 + 40 * q^9 - 4 * q^17 + 8 * q^23 - 80 * q^25 + 8 * q^27 + 8 * q^29 - 24 * q^39 + 32 * q^43 - 36 * q^49 - 20 * q^51 + 12 * q^53 + 32 * q^61 - 24 * q^65 + 80 * q^69 - 36 * q^75 + 36 * q^77 + 16 * q^79 + 132 * q^81 + 8 * q^91 + 56 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4004\mathbb{Z}\right)^\times$.

$n$	$365$	$925$	$2003$	$3433$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	1.47001			0.848713			0.424356	−	0.905495i	$-0.360500\pi$
$3$	1.47001			0.848713			0.424356	+	0.905495i	$0.360500\pi$
$4$		0			0
$5$			1.22404i			0.547406i	0.961814	+	0.273703i	$0.0882485\pi$
$5$			1.22404i			0.547406i	−0.961814	+	0.273703i	$0.911752\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	−0.839061			−0.279687
$10$		0			0
$11$			1.00000i			0.301511i
$11$			1.00000i			0.301511i
$12$		0			0
$13$	−3.29392	+	1.46633i	−0.913568	+	0.406686i
$13$	−3.29392	+	1.46633i	−0.913568	+	0.406686i
$14$		0			0
$15$			1.79935i			0.464590i
$16$		0			0
$17$	5.50110			1.33421			0.667107	−	0.744962i	$-0.267533\pi$
$17$	5.50110			1.33421			0.667107	+	0.744962i	$0.267533\pi$
$18$		0			0
$19$		−	4.73877i		−	1.08715i	−0.839361	−	0.543574i	$-0.817071\pi$
$19$		−	4.73877i		−	1.08715i	0.839361	−	0.543574i	$-0.182929\pi$
$20$		0			0
$21$		−	1.47001i		−	0.320783i
$22$		0			0
$23$	8.72579			1.81945			0.909726	−	0.415209i	$-0.136292\pi$
$23$	8.72579			1.81945			0.909726	+	0.415209i	$0.136292\pi$
$24$		0			0
$25$	3.50174			0.700347
$26$		0			0
$27$	−5.64347			−1.08609
$28$		0			0
$29$	−1.63307			−0.303253			−0.151627	−	0.988438i	$-0.548451\pi$
$29$	−1.63307			−0.303253			−0.151627	+	0.988438i	$0.548451\pi$
$30$		0			0
$31$		−	5.96177i		−	1.07077i	−0.844610	−	0.535383i	$-0.820167\pi$
$31$		−	5.96177i		−	1.07077i	0.844610	−	0.535383i	$-0.179833\pi$
$32$		0			0
$33$			1.47001i			0.255896i
$34$		0			0
$35$	1.22404			0.206900
$36$		0			0
$37$			9.25792i			1.52199i	0.648757	+	0.760996i	$0.275289\pi$
$37$			9.25792i			1.52199i	−0.648757	+	0.760996i	$0.724711\pi$
$38$		0			0
$39$	−4.84210	+	2.15552i	−0.775357	+	0.345159i
$40$		0			0
$41$			6.84298i			1.06869i	0.845265	+	0.534347i	$0.179442\pi$
$41$			6.84298i			1.06869i	−0.845265	+	0.534347i	$0.820558\pi$
$42$		0			0
$43$	5.93685			0.905362			0.452681	−	0.891673i	$-0.350468\pi$
$43$	5.93685			0.905362			0.452681	+	0.891673i	$0.350468\pi$
$44$		0			0
$45$		−	1.02704i		−	0.153102i
$46$		0			0
$47$		−	10.8026i		−	1.57572i	−0.615852	−	0.787862i	$-0.711188\pi$
$47$		−	10.8026i		−	1.57572i	0.615852	−	0.787862i	$-0.288812\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	8.08669			1.13236
$52$		0			0
$53$	3.59354			0.493611			0.246805	−	0.969065i	$-0.420619\pi$
$53$	3.59354			0.493611			0.246805	+	0.969065i	$0.420619\pi$
$54$		0			0
$55$	−1.22404			−0.165049
$56$		0			0
$57$		−	6.96605i		−	0.922676i
$58$		0			0
$59$		−	2.97310i		−	0.387065i	−0.981094	−	0.193532i	$-0.938006\pi$
$59$		−	2.97310i		−	0.387065i	0.981094	−	0.193532i	$-0.0619945\pi$
$60$		0			0
$61$	3.03907			0.389113			0.194556	−	0.980891i	$-0.437673\pi$
$61$	3.03907			0.389113			0.194556	+	0.980891i	$0.437673\pi$
$62$		0			0
$63$			0.839061i			0.105712i
$64$		0			0
$65$	−1.79484	−	4.03187i	−0.222622	−	0.500092i
$66$		0			0
$67$			12.8053i			1.56442i	0.623016	+	0.782209i	$0.285907\pi$
$67$			12.8053i			1.56442i	−0.623016	+	0.782209i	$0.714093\pi$
$68$		0			0
$69$	12.8270			1.54419
$70$		0			0
$71$			12.0729i			1.43279i	0.697693	+	0.716397i	$0.254210\pi$
$71$			12.0729i			1.43279i	−0.697693	+	0.716397i	$0.745790\pi$
$72$		0			0
$73$		−	1.51173i		−	0.176934i	−0.996079	−	0.0884671i	$-0.971803\pi$
$73$		−	1.51173i		−	0.176934i	0.996079	−	0.0884671i	$-0.0281968\pi$
$74$		0			0
$75$	5.14760			0.594393
$76$		0			0
$77$	1.00000			0.113961
$78$		0			0
$79$	6.58964			0.741393			0.370696	−	0.928754i	$-0.379119\pi$
$79$	6.58964			0.741393			0.370696	+	0.928754i	$0.379119\pi$
$80$		0			0
$81$	−5.77879			−0.642088
$82$		0			0
$83$			2.25080i			0.247057i	0.992341	+	0.123529i	$0.0394210\pi$
$83$			2.25080i			0.247057i	−0.992341	+	0.123529i	$0.960579\pi$
$84$		0			0
$85$			6.73355i			0.730356i
$86$		0			0
$87$	−2.40063			−0.257375
$88$		0			0
$89$			9.64863i			1.02275i	0.859357	+	0.511376i	$0.170864\pi$
$89$			9.64863i			1.02275i	−0.859357	+	0.511376i	$0.829136\pi$
$90$		0			0
$91$	1.46633	+	3.29392i	0.153713	+	0.345296i
$92$		0			0
$93$		−	8.76388i		−	0.908772i
$94$		0			0
$95$	5.80042			0.595111
$96$		0			0
$97$			5.06930i			0.514709i	0.966317	+	0.257354i	$0.0828508\pi$
$97$			5.06930i			0.514709i	−0.966317	+	0.257354i	$0.917149\pi$
$98$		0			0
$99$		−	0.839061i		−	0.0843288i
$100$		0			0
$101$	19.9511			1.98520			0.992602	−	0.121413i	$-0.0387425\pi$
$101$	19.9511			1.98520			0.992602	+	0.121413i	$0.0387425\pi$
$102$		0			0
$103$	6.64635			0.654884			0.327442	−	0.944871i	$-0.393813\pi$
$103$	6.64635			0.654884			0.327442	+	0.944871i	$0.393813\pi$
$104$		0			0
$105$	1.79935			0.175599
$106$		0			0
$107$	1.27199			0.122968			0.0614840	−	0.998108i	$-0.480417\pi$
$107$	1.27199			0.122968			0.0614840	+	0.998108i	$0.480417\pi$
$108$		0			0
$109$			10.4180i			0.997862i	0.866642	+	0.498931i	$0.166274\pi$
$109$			10.4180i			0.997862i	−0.866642	+	0.498931i	$0.833726\pi$
$110$		0			0
$111$			13.6093i			1.29173i
$112$		0			0
$113$	17.5094			1.64715			0.823573	−	0.567210i	$-0.191977\pi$
$113$	17.5094			1.64715			0.823573	+	0.567210i	$0.191977\pi$
$114$		0			0
$115$			10.6807i			0.995978i
$116$		0			0
$117$	2.76380	−	1.23034i	0.255513	−	0.113745i
$118$		0			0
$119$		−	5.50110i		−	0.504285i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$			10.0593i			0.907014i
$124$		0			0
$125$			10.4064i			0.930780i
$126$		0			0
$127$	−19.8793			−1.76400			−0.882001	−	0.471247i	$-0.843804\pi$
$127$	−19.8793			−1.76400			−0.882001	+	0.471247i	$0.843804\pi$
$128$		0			0
$129$	8.72725			0.768392
$130$		0			0
$131$	14.1726			1.23826			0.619132	−	0.785287i	$-0.287484\pi$
$131$	14.1726			1.23826			0.619132	+	0.785287i	$0.287484\pi$
$132$		0			0
$133$	−4.73877			−0.410903
$134$		0			0
$135$		−	6.90781i		−	0.594530i
$136$		0			0
$137$			7.90227i			0.675136i	0.941301	+	0.337568i	$0.109604\pi$
$137$			7.90227i			0.675136i	−0.941301	+	0.337568i	$0.890396\pi$
$138$		0			0
$139$	−5.73294			−0.486261			−0.243131	−	0.969994i	$-0.578174\pi$
$139$	−5.73294			−0.486261			−0.243131	+	0.969994i	$0.578174\pi$
$140$		0			0
$141$		−	15.8800i		−	1.33734i
$142$		0			0
$143$	−1.46633	−	3.29392i	−0.122620	−	0.275451i
$144$		0			0
$145$		−	1.99894i		−	0.166003i
$146$		0			0
$147$	−1.47001			−0.121245
$148$		0			0
$149$		−	11.6548i		−	0.954802i	−0.878686	−	0.477401i	$-0.841579\pi$
$149$		−	11.6548i		−	0.954802i	0.878686	−	0.477401i	$-0.158421\pi$
$150$		0			0
$151$		−	17.0042i		−	1.38378i	−0.722002	−	0.691891i	$-0.756778\pi$
$151$		−	17.0042i		−	1.38378i	0.722002	−	0.691891i	$-0.243222\pi$
$152$		0			0
$153$	−4.61576			−0.373162
$154$		0			0
$155$	7.29742			0.586143
$156$		0			0
$157$	−3.28773			−0.262389			−0.131195	−	0.991357i	$-0.541881\pi$
$157$	−3.28773			−0.262389			−0.131195	+	0.991357i	$0.541881\pi$
$158$		0			0
$159$	5.28255			0.418934
$160$		0			0
$161$		−	8.72579i		−	0.687688i
$162$		0			0
$163$		−	6.27858i		−	0.491776i	−0.969298	−	0.245888i	$-0.920920\pi$
$163$		−	6.27858i		−	0.491776i	0.969298	−	0.245888i	$-0.0790796\pi$
$164$		0			0
$165$	−1.79935			−0.140079
$166$		0			0
$167$		−	10.7574i		−	0.832431i	−0.909266	−	0.416216i	$-0.863356\pi$
$167$		−	10.7574i		−	0.832431i	0.909266	−	0.416216i	$-0.136644\pi$
$168$		0			0
$169$	8.69977	−	9.65992i	0.669213	−	0.743071i
$170$		0			0
$171$			3.97612i			0.304061i
$172$		0			0
$173$	−6.08041			−0.462285			−0.231143	−	0.972920i	$-0.574246\pi$
$173$	−6.08041			−0.462285			−0.231143	+	0.972920i	$0.574246\pi$
$174$		0			0
$175$		−	3.50174i		−	0.264706i
$176$		0			0
$177$		−	4.37050i		−	0.328507i
$178$		0			0
$179$	19.8805			1.48594			0.742970	−	0.669325i	$-0.233416\pi$
$179$	19.8805			1.48594			0.742970	+	0.669325i	$0.233416\pi$
$180$		0			0
$181$	4.71326			0.350334			0.175167	−	0.984539i	$-0.443954\pi$
$181$	4.71326			0.350334			0.175167	+	0.984539i	$0.443954\pi$
$182$		0			0
$183$	4.46747			0.330245
$184$		0			0
$185$	−11.3320			−0.833147
$186$		0			0
$187$			5.50110i			0.402280i
$188$		0			0
$189$			5.64347i			0.410502i
$190$		0			0
$191$	−12.2115			−0.883590			−0.441795	−	0.897116i	$-0.645658\pi$
$191$	−12.2115			−0.883590			−0.441795	+	0.897116i	$0.645658\pi$
$192$		0			0
$193$			13.8500i			0.996944i	0.866906	+	0.498472i	$0.166105\pi$
$193$			13.8500i			0.996944i	−0.866906	+	0.498472i	$0.833895\pi$
$194$		0			0
$195$	−2.63843	−	5.92691i	−0.188942	−	0.424435i
$196$		0			0
$197$		−	8.42331i		−	0.600136i	−0.953918	−	0.300068i	$-0.902991\pi$
$197$		−	8.42331i		−	0.600136i	0.953918	−	0.300068i	$-0.0970094\pi$
$198$		0			0
$199$	−23.6650			−1.67756			−0.838782	−	0.544467i	$-0.816732\pi$
$199$	−23.6650			−1.67756			−0.838782	+	0.544467i	$0.816732\pi$
$200$		0			0
$201$			18.8240i			1.32774i
$202$		0			0
$203$			1.63307i			0.114619i
$204$		0			0
$205$	−8.37605			−0.585009
$206$		0			0
$207$	−7.32147			−0.508877
$208$		0			0
$209$	4.73877			0.327787
$210$		0			0
$211$	15.1796			1.04501			0.522503	−	0.852638i	$-0.324998\pi$
$211$	15.1796			1.04501			0.522503	+	0.852638i	$0.324998\pi$
$212$		0			0
$213$			17.7474i			1.21603i
$214$		0			0
$215$			7.26692i			0.495600i
$216$		0			0
$217$	−5.96177			−0.404711
$218$		0			0
$219$		−	2.22226i		−	0.150166i
$220$		0			0
$221$	−18.1202	+	8.06641i	−1.21889	+	0.542606i
$222$		0			0
$223$		−	23.3712i		−	1.56505i	−0.622620	−	0.782524i	$-0.713932\pi$
$223$		−	23.3712i		−	1.56505i	0.622620	−	0.782524i	$-0.286068\pi$
$224$		0			0
$225$	−2.93817			−0.195878
$226$		0			0
$227$		−	12.8633i		−	0.853771i	−0.904306	−	0.426885i	$-0.859611\pi$
$227$		−	12.8633i		−	0.853771i	0.904306	−	0.426885i	$-0.140389\pi$
$228$		0			0
$229$			21.1385i			1.39687i	0.715673	+	0.698435i	$0.246120\pi$
$229$			21.1385i			1.39687i	−0.715673	+	0.698435i	$0.753880\pi$
$230$		0			0
$231$	1.47001			0.0967198
$232$		0			0
$233$	−25.8076			−1.69071			−0.845355	−	0.534206i	$-0.820611\pi$
$233$	−25.8076			−1.69071			−0.845355	+	0.534206i	$0.820611\pi$
$234$		0			0
$235$	13.2228			0.862560
$236$		0			0
$237$	9.68686			0.629229
$238$		0			0
$239$			5.78204i			0.374009i	0.982359	+	0.187004i	$0.0598779\pi$
$239$			5.78204i			0.374009i	−0.982359	+	0.187004i	$0.940122\pi$
$240$		0			0
$241$			3.45665i			0.222662i	0.993783	+	0.111331i	$0.0355114\pi$
$241$			3.45665i			0.222662i	−0.993783	+	0.111331i	$0.964489\pi$
$242$		0			0
$243$	8.43551			0.541138
$244$		0			0
$245$		−	1.22404i		−	0.0782008i
$246$		0			0
$247$	6.94858	+	15.6091i	0.442128	+	0.993183i
$248$		0			0
$249$			3.30870i			0.209680i
$250$		0			0
$251$	18.2778			1.15368			0.576842	−	0.816856i	$-0.304285\pi$
$251$	18.2778			1.15368			0.576842	+	0.816856i	$0.304285\pi$
$252$		0			0
$253$			8.72579i			0.548585i
$254$		0			0
$255$			9.89840i			0.619862i
$256$		0			0
$257$	−7.98557			−0.498126			−0.249063	−	0.968487i	$-0.580123\pi$
$257$	−7.98557			−0.498126			−0.249063	+	0.968487i	$0.580123\pi$
$258$		0			0
$259$	9.25792			0.575259
$260$		0			0
$261$	1.37024			0.0848160
$262$		0			0
$263$	−26.7116			−1.64711			−0.823553	−	0.567239i	$-0.808011\pi$
$263$	−26.7116			−1.64711			−0.823553	+	0.567239i	$0.808011\pi$
$264$		0			0
$265$			4.39862i			0.270205i
$266$		0			0
$267$			14.1836i			0.868023i
$268$		0			0
$269$	24.2396			1.47792			0.738959	−	0.673751i	$-0.235318\pi$
$269$	24.2396			1.47792			0.738959	+	0.673751i	$0.235318\pi$
$270$		0			0
$271$		−	26.7339i		−	1.62397i	−0.583680	−	0.811984i	$-0.698388\pi$
$271$		−	26.7339i		−	1.62397i	0.583680	−	0.811984i	$-0.301612\pi$
$272$		0			0
$273$	2.15552	+	4.84210i	0.130458	+	0.293057i
$274$		0			0
$275$			3.50174i			0.211163i
$276$		0			0
$277$	3.97508			0.238839			0.119420	−	0.992844i	$-0.461897\pi$
$277$	3.97508			0.238839			0.119420	+	0.992844i	$0.461897\pi$
$278$		0			0
$279$			5.00229i			0.299479i
$280$		0			0
$281$		−	22.9799i		−	1.37086i	−0.728137	−	0.685432i	$-0.759614\pi$
$281$		−	22.9799i		−	1.37086i	0.728137	−	0.685432i	$-0.240386\pi$
$282$		0			0
$283$	14.7483			0.876693			0.438347	−	0.898806i	$-0.355564\pi$
$283$	14.7483			0.876693			0.438347	+	0.898806i	$0.355564\pi$
$284$		0			0
$285$	8.52670			0.505078
$286$		0			0
$287$	6.84298			0.403928
$288$		0			0
$289$	13.2621			0.780125
$290$		0			0
$291$			7.45193i			0.436840i
$292$		0			0
$293$		−	9.61531i		−	0.561733i	−0.959747	−	0.280866i	$-0.909378\pi$
$293$		−	9.61531i		−	0.561733i	0.959747	−	0.280866i	$-0.0906217\pi$
$294$		0			0
$295$	3.63918			0.211881
$296$		0			0
$297$		−	5.64347i		−	0.327467i
$298$		0			0
$299$	−28.7420	+	12.7949i	−1.66219	+	0.739946i
$300$		0			0
$301$		−	5.93685i		−	0.342195i
$302$		0			0
$303$	29.3283			1.68487
$304$		0			0
$305$			3.71993i			0.213003i
$306$		0			0
$307$			28.7531i			1.64103i	0.571627	+	0.820514i	$0.306313\pi$
$307$			28.7531i			1.64103i	−0.571627	+	0.820514i	$0.693687\pi$
$308$		0			0
$309$	9.77022			0.555808
$310$		0			0
$311$	−13.8294			−0.784196			−0.392098	−	0.919923i	$-0.628251\pi$
$311$	−13.8294			−0.784196			−0.392098	+	0.919923i	$0.628251\pi$
$312$		0			0
$313$	−0.104190			−0.00588916			−0.00294458	−	0.999996i	$-0.500937\pi$
$313$	−0.104190			−0.00588916			−0.00294458	+	0.999996i	$0.500937\pi$
$314$		0			0
$315$	−1.02704			−0.0578672
$316$		0			0
$317$			1.44379i			0.0810915i	0.999178	+	0.0405458i	$0.0129097\pi$
$317$			1.44379i			0.0810915i	−0.999178	+	0.0405458i	$0.987090\pi$
$318$		0			0
$319$		−	1.63307i		−	0.0914343i
$320$		0			0
$321$	1.86984			0.104364
$322$		0			0
$323$		−	26.0684i		−	1.45049i
$324$		0			0
$325$	−11.5344	+	5.13469i	−0.639815	+	0.284821i
$326$		0			0
$327$			15.3146i			0.846898i
$328$		0			0
$329$	−10.8026			−0.595567
$330$		0			0
$331$		−	12.1057i		−	0.665392i	−0.943034	−	0.332696i	$-0.892042\pi$
$331$		−	12.1057i		−	0.665392i	0.943034	−	0.332696i	$-0.107958\pi$
$332$		0			0
$333$		−	7.76796i		−	0.425681i
$334$		0			0
$335$	−15.6742			−0.856371
$336$		0			0
$337$	−11.4835			−0.625544			−0.312772	−	0.949828i	$-0.601258\pi$
$337$	−11.4835			−0.625544			−0.312772	+	0.949828i	$0.601258\pi$
$338$		0			0
$339$	25.7391			1.39795
$340$		0			0
$341$	5.96177			0.322848
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$			15.7007i			0.845299i
$346$		0			0
$347$	−29.1793			−1.56643			−0.783213	−	0.621754i	$-0.786420\pi$
$347$	−29.1793			−1.56643			−0.783213	+	0.621754i	$0.786420\pi$
$348$		0			0
$349$		−	15.1279i		−	0.809779i	−0.914366	−	0.404890i	$-0.867310\pi$
$349$		−	15.1279i		−	0.809779i	0.914366	−	0.404890i	$-0.132690\pi$
$350$		0			0
$351$	18.5891	−	8.27517i	0.992214	−	0.441696i
$352$		0			0
$353$		−	1.06327i		−	0.0565920i	−0.999600	−	0.0282960i	$-0.990992\pi$
$353$		−	1.06327i		−	0.0565920i	0.999600	−	0.0282960i	$-0.00900810\pi$
$354$		0			0
$355$	−14.7777			−0.784319
$356$		0			0
$357$		−	8.08669i		−	0.427993i
$358$		0			0
$359$		−	10.4559i		−	0.551842i	−0.961180	−	0.275921i	$-0.911017\pi$
$359$		−	10.4559i		−	0.551842i	0.961180	−	0.275921i	$-0.0889829\pi$
$360$		0			0
$361$	−3.45591			−0.181890
$362$		0			0
$363$	−1.47001			−0.0771557
$364$		0			0
$365$	1.85041			0.0968548
$366$		0			0
$367$	3.62200			0.189067			0.0945334	−	0.995522i	$-0.469864\pi$
$367$	3.62200			0.189067			0.0945334	+	0.995522i	$0.469864\pi$
$368$		0			0
$369$		−	5.74168i		−	0.298900i
$370$		0			0
$371$		−	3.59354i		−	0.186567i
$372$		0			0
$373$	−6.87493			−0.355971			−0.177985	−	0.984033i	$-0.556958\pi$
$373$	−6.87493			−0.355971			−0.177985	+	0.984033i	$0.556958\pi$
$374$		0			0
$375$			15.2976i			0.789964i
$376$		0			0
$377$	5.37919	−	2.39461i	0.277043	−	0.123329i
$378$		0			0
$379$			11.9962i			0.616201i	0.951354	+	0.308101i	$0.0996934\pi$
$379$			11.9962i			0.616201i	−0.951354	+	0.308101i	$0.900307\pi$
$380$		0			0
$381$	−29.2228			−1.49713
$382$		0			0
$383$			37.3087i			1.90639i	0.302359	+	0.953194i	$0.402226\pi$
$383$			37.3087i			1.90639i	−0.302359	+	0.953194i	$0.597774\pi$
$384$		0			0
$385$			1.22404i			0.0623827i
$386$		0			0
$387$	−4.98138			−0.253218
$388$		0			0
$389$	−26.4583			−1.34149			−0.670745	−	0.741688i	$-0.734026\pi$
$389$	−26.4583			−1.34149			−0.670745	+	0.741688i	$0.734026\pi$
$390$		0			0
$391$	48.0014			2.42754
$392$		0			0
$393$	20.8339			1.05093
$394$		0			0
$395$			8.06596i			0.405843i
$396$		0			0
$397$		−	16.8355i		−	0.844951i	−0.906374	−	0.422475i	$-0.861161\pi$
$397$		−	16.8355i		−	0.844951i	0.906374	−	0.422475i	$-0.138839\pi$
$398$		0			0
$399$	−6.96605			−0.348739
$400$		0			0
$401$		−	7.98670i		−	0.398837i	−0.979914	−	0.199418i	$-0.936095\pi$
$401$		−	7.98670i		−	0.398837i	0.979914	−	0.199418i	$-0.0639053\pi$
$402$		0			0
$403$	8.74190	+	19.6376i	0.435465	+	0.978217i
$404$		0			0
$405$		−	7.07345i		−	0.351483i
$406$		0			0
$407$	−9.25792			−0.458898
$408$		0			0
$409$		−	22.8324i		−	1.12899i	−0.825436	−	0.564495i	$-0.809071\pi$
$409$		−	22.8324i		−	1.12899i	0.825436	−	0.564495i	$-0.190929\pi$
$410$		0			0
$411$			11.6164i			0.572996i
$412$		0			0
$413$	−2.97310			−0.146297
$414$		0			0
$415$	−2.75506			−0.135240
$416$		0			0
$417$	−8.42749			−0.412696
$418$		0			0
$419$	−7.30919			−0.357077			−0.178539	−	0.983933i	$-0.557137\pi$
$419$	−7.30919			−0.357077			−0.178539	+	0.983933i	$0.557137\pi$
$420$		0			0
$421$			3.92648i			0.191365i	0.995412	+	0.0956824i	$0.0305033\pi$
$421$			3.92648i			0.191365i	−0.995412	+	0.0956824i	$0.969497\pi$
$422$		0			0
$423$			9.06405i			0.440709i
$424$		0			0
$425$	19.2634			0.934412
$426$		0			0
$427$		−	3.03907i		−	0.147071i
$428$		0			0
$429$	−2.15552	−	4.84210i	−0.104069	−	0.233779i
$430$		0			0
$431$		−	6.73121i		−	0.324231i	−0.986772	−	0.162116i	$-0.948168\pi$
$431$		−	6.73121i		−	0.324231i	0.986772	−	0.162116i	$-0.0518317\pi$
$432$		0			0
$433$	−19.2354			−0.924395			−0.462198	−	0.886777i	$-0.652939\pi$
$433$	−19.2354			−0.924395			−0.462198	+	0.886777i	$0.652939\pi$
$434$		0			0
$435$		−	2.93846i		−	0.140888i
$436$		0			0
$437$		−	41.3495i		−	1.97801i
$438$		0			0
$439$	33.7190			1.60932			0.804660	−	0.593735i	$-0.202347\pi$
$439$	33.7190			1.60932			0.804660	+	0.593735i	$0.202347\pi$
$440$		0			0
$441$	0.839061			0.0399553
$442$		0			0
$443$	12.0171			0.570948			0.285474	−	0.958387i	$-0.407849\pi$
$443$	12.0171			0.570948			0.285474	+	0.958387i	$0.407849\pi$
$444$		0			0
$445$	−11.8103			−0.559861
$446$		0			0
$447$		−	17.1328i		−	0.810352i
$448$		0			0
$449$		−	19.5615i		−	0.923162i	−0.887098	−	0.461581i	$-0.847282\pi$
$449$		−	19.5615i		−	0.923162i	0.887098	−	0.461581i	$-0.152718\pi$
$450$		0			0
$451$	−6.84298			−0.322223
$452$		0			0
$453$		−	24.9964i		−	1.17443i
$454$		0			0
$455$	−4.03187	+	1.79484i	−0.189017	+	0.0841433i
$456$		0			0
$457$			33.4151i			1.56309i	0.623847	+	0.781547i	$0.285569\pi$
$457$			33.4151i			1.56309i	−0.623847	+	0.781547i	$0.714431\pi$
$458$		0			0
$459$	−31.0453			−1.44907
$460$		0			0
$461$			16.6872i			0.777201i	0.921406	+	0.388601i	$0.127041\pi$
$461$			16.6872i			0.777201i	−0.921406	+	0.388601i	$0.872959\pi$
$462$		0			0
$463$		−	21.9389i		−	1.01959i	−0.860297	−	0.509793i	$-0.829722\pi$
$463$		−	21.9389i		−	1.01959i	0.860297	−	0.509793i	$-0.170278\pi$
$464$		0			0
$465$	10.7273			0.497467
$466$		0			0
$467$	−9.82381			−0.454591			−0.227296	−	0.973826i	$-0.572988\pi$
$467$	−9.82381			−0.454591			−0.227296	+	0.973826i	$0.572988\pi$
$468$		0			0
$469$	12.8053			0.591294
$470$		0			0
$471$	−4.83300			−0.222693
$472$		0			0
$473$			5.93685i			0.272977i
$474$		0			0
$475$		−	16.5939i		−	0.761381i
$476$		0			0
$477$	−3.01520			−0.138057
$478$		0			0
$479$		−	15.6626i		−	0.715641i	−0.933790	−	0.357820i	$-0.883520\pi$
$479$		−	15.6626i		−	0.715641i	0.933790	−	0.357820i	$-0.116480\pi$
$480$		0			0
$481$	−13.5751	−	30.4948i	−0.618973	−	1.39044i
$482$		0			0
$483$		−	12.8270i		−	0.583650i
$484$		0			0
$485$	−6.20500			−0.281755
$486$		0			0
$487$		−	5.04650i		−	0.228679i	−0.993442	−	0.114339i	$-0.963525\pi$
$487$		−	5.04650i		−	0.228679i	0.993442	−	0.114339i	$-0.0364751\pi$
$488$		0			0
$489$		−	9.22959i		−	0.417377i
$490$		0			0
$491$	0.766239			0.0345799			0.0172899	−	0.999851i	$-0.494496\pi$
$491$	0.766239			0.0345799			0.0172899	+	0.999851i	$0.494496\pi$
$492$		0			0
$493$	−8.98368			−0.404605
$494$		0			0
$495$	1.02704			0.0461621
$496$		0			0
$497$	12.0729			0.541545
$498$		0			0
$499$		−	15.0008i		−	0.671526i	−0.941946	−	0.335763i	$-0.891006\pi$
$499$		−	15.0008i		−	0.671526i	0.941946	−	0.335763i	$-0.108994\pi$
$500$		0			0
$501$		−	15.8135i		−	0.706495i
$502$		0			0
$503$	3.97436			0.177208			0.0886039	−	0.996067i	$-0.471759\pi$
$503$	3.97436			0.177208			0.0886039	+	0.996067i	$0.471759\pi$
$504$		0			0
$505$			24.4208i			1.08671i
$506$		0			0
$507$	12.7888	−	14.2002i	0.567970	−	0.630653i
$508$		0			0
$509$			14.8919i			0.660070i	0.943969	+	0.330035i	$0.107061\pi$
$509$			14.8919i			0.660070i	−0.943969	+	0.330035i	$0.892939\pi$
$510$		0			0
$511$	−1.51173			−0.0668748
$512$		0			0
$513$			26.7431i			1.18074i
$514$		0			0
$515$			8.13537i			0.358487i
$516$		0			0
$517$	10.8026			0.475098
$518$		0			0
$519$	−8.93828			−0.392347
$520$		0			0
$521$	−36.7258			−1.60898			−0.804492	−	0.593963i	$-0.797562\pi$
$521$	−36.7258			−1.60898			−0.804492	+	0.593963i	$0.797562\pi$
$522$		0			0
$523$	19.8183			0.866595			0.433297	−	0.901251i	$-0.357350\pi$
$523$	19.8183			0.866595			0.433297	+	0.901251i	$0.357350\pi$
$524$		0			0
$525$		−	5.14760i		−	0.224660i
$526$		0			0
$527$		−	32.7963i		−	1.42863i
$528$		0			0
$529$	53.1393			2.31041
$530$		0			0
$531$			2.49461i			0.108257i
$532$		0			0
$533$	−10.0340	−	22.5402i	−0.434623	−	0.976325i
$534$		0			0
$535$			1.55696i			0.0673134i
$536$		0			0
$537$	29.2246			1.26114
$538$		0			0
$539$		−	1.00000i		−	0.0430730i
$540$		0			0
$541$			5.15187i			0.221496i	0.993849	+	0.110748i	$0.0353247\pi$
$541$			5.15187i			0.221496i	−0.993849	+	0.110748i	$0.964675\pi$
$542$		0			0
$543$	6.92855			0.297333
$544$		0			0
$545$	−12.7520			−0.546235
$546$		0			0
$547$	24.8419			1.06216			0.531082	−	0.847321i	$-0.321786\pi$
$547$	24.8419			1.06216			0.531082	+	0.847321i	$0.321786\pi$
$548$		0			0
$549$	−2.54996			−0.108830
$550$		0			0
$551$			7.73873i			0.329681i
$552$		0			0
$553$		−	6.58964i		−	0.280220i
$554$		0			0
$555$	−16.6582			−0.707102
$556$		0			0
$557$			20.5865i			0.872279i	0.899879	+	0.436140i	$0.143655\pi$
$557$			20.5865i			0.872279i	−0.899879	+	0.436140i	$0.856345\pi$
$558$		0			0
$559$	−19.5555	+	8.70537i	−0.827109	+	0.368198i
$560$		0			0
$561$			8.08669i			0.341420i
$562$		0			0
$563$	−25.8005			−1.08736			−0.543681	−	0.839292i	$-0.682970\pi$
$563$	−25.8005			−1.08736			−0.543681	+	0.839292i	$0.682970\pi$
$564$		0			0
$565$			21.4321i			0.901657i
$566$		0			0
$567$			5.77879i			0.242686i
$568$		0			0
$569$	14.4320			0.605019			0.302510	−	0.953146i	$-0.402176\pi$
$569$	14.4320			0.605019			0.302510	+	0.953146i	$0.402176\pi$
$570$		0			0
$571$	−4.42797			−0.185305			−0.0926525	−	0.995699i	$-0.529535\pi$
$571$	−4.42797			−0.185305			−0.0926525	+	0.995699i	$0.529535\pi$
$572$		0			0
$573$	−17.9510			−0.749914
$574$		0			0
$575$	30.5554			1.27425
$576$		0			0
$577$			9.57811i			0.398742i	0.979924	+	0.199371i	$0.0638899\pi$
$577$			9.57811i			0.398742i	−0.979924	+	0.199371i	$0.936110\pi$
$578$		0			0
$579$			20.3597i			0.846119i
$580$		0			0
$581$	2.25080			0.0933788
$582$		0			0
$583$			3.59354i			0.148829i
$584$		0			0
$585$	1.50598	+	3.38299i	0.0622645	+	0.139869i
$586$		0			0
$587$		−	3.66683i		−	0.151346i	−0.997133	−	0.0756730i	$-0.975889\pi$
$587$		−	3.66683i		−	0.151346i	0.997133	−	0.0756730i	$-0.0241105\pi$
$588$		0			0
$589$	−28.2514			−1.16408
$590$		0			0
$591$		−	12.3824i		−	0.509343i
$592$		0			0
$593$		−	24.4602i		−	1.00446i	−0.864734	−	0.502229i	$-0.832513\pi$
$593$		−	24.4602i		−	1.00446i	0.864734	−	0.502229i	$-0.167487\pi$
$594$		0			0
$595$	6.73355			0.276049
$596$		0			0
$597$	−34.7878			−1.42377
$598$		0			0
$599$	13.5442			0.553400			0.276700	−	0.960956i	$-0.410759\pi$
$599$	13.5442			0.553400			0.276700	+	0.960956i	$0.410759\pi$
$600$		0			0
$601$	29.5453			1.20518			0.602589	−	0.798051i	$-0.294136\pi$
$601$	29.5453			1.20518			0.602589	+	0.798051i	$0.294136\pi$
$602$		0			0
$603$		−	10.7444i		−	0.437547i
$604$		0			0
$605$		−	1.22404i		−	0.0497641i
$606$		0			0
$607$	18.4428			0.748571			0.374286	−	0.927313i	$-0.377888\pi$
$607$	18.4428			0.748571			0.374286	+	0.927313i	$0.377888\pi$
$608$		0			0
$609$			2.40063i			0.0972786i
$610$		0			0
$611$	15.8402	+	35.5829i	0.640824	+	1.43953i
$612$		0			0
$613$			0.177819i			0.00718204i	0.999994	+	0.00359102i	$0.00114306\pi$
$613$			0.177819i			0.00718204i	−0.999994	+	0.00359102i	$0.998857\pi$
$614$		0			0
$615$	−12.3129			−0.496504
$616$		0			0
$617$		−	22.4751i		−	0.904815i	−0.891811	−	0.452407i	$-0.850565\pi$
$617$		−	22.4751i		−	0.904815i	0.891811	−	0.452407i	$-0.149435\pi$
$618$		0			0
$619$			21.2262i			0.853155i	0.904451	+	0.426577i	$0.140281\pi$
$619$			21.2262i			0.853155i	−0.904451	+	0.426577i	$0.859719\pi$
$620$		0			0
$621$	−49.2437			−1.97608
$622$		0			0
$623$	9.64863			0.386564
$624$		0			0
$625$	4.77083			0.190833
$626$		0			0
$627$	6.96605			0.278197
$628$		0			0
$629$			50.9287i			2.03066i
$630$		0			0
$631$			44.3687i			1.76629i	0.469102	+	0.883144i	$0.344578\pi$
$631$			44.3687i			1.76629i	−0.469102	+	0.883144i	$0.655422\pi$
$632$		0			0
$633$	22.3142			0.886909
$634$		0			0
$635$		−	24.3330i		−	0.965625i
$636$		0			0
$637$	3.29392	−	1.46633i	0.130510	−	0.0580980i
$638$		0			0
$639$		−	10.1299i		−	0.400734i
$640$		0			0
$641$	−41.6766			−1.64613			−0.823064	−	0.567949i	$-0.807737\pi$
$641$	−41.6766			−1.64613			−0.823064	+	0.567949i	$0.807737\pi$
$642$		0			0
$643$			4.67552i			0.184385i	0.995741	+	0.0921923i	$0.0293875\pi$
$643$			4.67552i			0.184385i	−0.995741	+	0.0921923i	$0.970613\pi$
$644$		0			0
$645$			10.6825i			0.420622i
$646$		0			0
$647$	10.0175			0.393829			0.196915	−	0.980421i	$-0.436908\pi$
$647$	10.0175			0.393829			0.196915	+	0.980421i	$0.436908\pi$
$648$		0			0
$649$	2.97310			0.116704
$650$		0			0
$651$	−8.76388			−0.343483
$652$		0			0
$653$	−27.6735			−1.08295			−0.541473	−	0.840718i	$-0.682133\pi$
$653$	−27.6735			−1.08295			−0.541473	+	0.840718i	$0.682133\pi$
$654$		0			0
$655$			17.3478i			0.677833i
$656$		0			0
$657$			1.26843i			0.0494862i
$658$		0			0
$659$	−2.69784			−0.105093			−0.0525465	−	0.998618i	$-0.516734\pi$
$659$	−2.69784			−0.105093			−0.0525465	+	0.998618i	$0.516734\pi$
$660$		0			0
$661$		−	18.0359i		−	0.701515i	−0.936466	−	0.350758i	$-0.885924\pi$
$661$		−	18.0359i		−	0.701515i	0.936466	−	0.350758i	$-0.114076\pi$
$662$		0			0
$663$	−26.6369	+	11.8577i	−1.03449	+	0.460516i
$664$		0			0
$665$		−	5.80042i		−	0.224931i
$666$		0			0
$667$	−14.2498			−0.551755
$668$		0			0
$669$		−	34.3559i		−	1.32828i
$670$		0			0
$671$			3.03907i			0.117322i
$672$		0			0
$673$	12.0042			0.462728			0.231364	−	0.972867i	$-0.425681\pi$
$673$	12.0042			0.462728			0.231364	+	0.972867i	$0.425681\pi$
$674$		0			0
$675$	−19.7619			−0.760637
$676$		0			0
$677$	−6.77083			−0.260224			−0.130112	−	0.991499i	$-0.541534\pi$
$677$	−6.77083			−0.260224			−0.130112	+	0.991499i	$0.541534\pi$
$678$		0			0
$679$	5.06930			0.194542
$680$		0			0
$681$		−	18.9093i		−	0.724606i
$682$		0			0
$683$			19.3666i			0.741042i	0.928824	+	0.370521i	$0.120821\pi$
$683$			19.3666i			0.741042i	−0.928824	+	0.370521i	$0.879179\pi$
$684$		0			0
$685$	−9.67266			−0.369573
$686$		0			0
$687$			31.0739i			1.18554i
$688$		0			0
$689$	−11.8368	+	5.26931i	−0.450947	+	0.200745i
$690$		0			0
$691$			21.5895i			0.821304i	0.911792	+	0.410652i	$0.134699\pi$
$691$			21.5895i			0.821304i	−0.911792	+	0.410652i	$0.865301\pi$
$692$		0			0
$693$	−0.839061			−0.0318733
$694$		0			0
$695$		−	7.01732i		−	0.266182i
$696$		0			0
$697$			37.6439i			1.42587i
$698$		0			0
$699$	−37.9374			−1.43493
$700$		0			0
$701$	−28.5044			−1.07660			−0.538299	−	0.842754i	$-0.680933\pi$
$701$	−28.5044			−1.07660			−0.538299	+	0.842754i	$0.680933\pi$
$702$		0			0
$703$	43.8711			1.65463
$704$		0			0
$705$	19.4377			0.732065
$706$		0			0
$707$		−	19.9511i		−	0.750337i
$708$		0			0
$709$			28.9901i			1.08875i	0.838843	+	0.544373i	$0.183233\pi$
$709$			28.9901i			1.08875i	−0.838843	+	0.544373i	$0.816767\pi$
$710$		0			0
$711$	−5.52911			−0.207358
$712$		0			0
$713$		−	52.0211i		−	1.94821i
$714$		0			0
$715$	4.03187	−	1.79484i	0.150783	−	0.0671231i
$716$		0			0
$717$			8.49967i			0.317426i
$718$		0			0
$719$	−36.9074			−1.37641			−0.688206	−	0.725515i	$-0.741602\pi$
$719$	−36.9074			−1.37641			−0.688206	+	0.725515i	$0.741602\pi$
$720$		0			0
$721$		−	6.64635i		−	0.247523i
$722$		0			0
$723$			5.08132i			0.188976i
$724$		0			0
$725$	−5.71858			−0.212383
$726$		0			0
$727$	−1.96710			−0.0729555			−0.0364778	−	0.999334i	$-0.511614\pi$
$727$	−1.96710			−0.0729555			−0.0364778	+	0.999334i	$0.511614\pi$
$728$		0			0
$729$	29.7367			1.10136
$730$		0			0
$731$	32.6592			1.20795
$732$		0			0
$733$		−	52.4302i		−	1.93655i	−0.249883	−	0.968276i	$-0.580392\pi$
$733$		−	52.4302i		−	1.93655i	0.249883	−	0.968276i	$-0.419608\pi$
$734$		0			0
$735$		−	1.79935i		−	0.0663700i
$736$		0			0
$737$	−12.8053			−0.471690
$738$		0			0
$739$			47.7388i			1.75610i	0.478569	+	0.878050i	$0.341156\pi$
$739$			47.7388i			1.75610i	−0.478569	+	0.878050i	$0.658844\pi$
$740$		0			0
$741$	10.2145	+	22.9456i	0.375239	+	0.842927i
$742$		0			0
$743$		−	14.1915i		−	0.520635i	−0.965523	−	0.260317i	$-0.916173\pi$
$743$		−	14.1915i		−	0.520635i	0.965523	−	0.260317i	$-0.0838272\pi$
$744$		0			0
$745$	14.2659			0.522664
$746$		0			0
$747$		−	1.88856i		−	0.0690986i
$748$		0			0
$749$		−	1.27199i		−	0.0464775i
$750$		0			0
$751$	−25.1886			−0.919144			−0.459572	−	0.888141i	$-0.651997\pi$
$751$	−25.1886			−0.919144			−0.459572	+	0.888141i	$0.651997\pi$
$752$		0			0
$753$	26.8686			0.979146
$754$		0			0
$755$	20.8138			0.757491
$756$		0			0
$757$	−45.9404			−1.66973			−0.834867	−	0.550452i	$-0.814455\pi$
$757$	−45.9404			−1.66973			−0.834867	+	0.550452i	$0.814455\pi$
$758$		0			0
$759$			12.8270i			0.465591i
$760$		0			0
$761$		−	16.6725i		−	0.604379i	−0.953248	−	0.302189i	$-0.902282\pi$
$761$		−	16.6725i		−	0.604379i	0.953248	−	0.302189i	$-0.0977175\pi$
$762$		0			0
$763$	10.4180			0.377156
$764$		0			0
$765$		−	5.64986i		−	0.204271i
$766$		0			0
$767$	4.35954	+	9.79315i	0.157414	+	0.353610i
$768$		0			0
$769$		−	1.35079i		−	0.0487108i	−0.999703	−	0.0243554i	$-0.992247\pi$
$769$		−	1.35079i		−	0.0487108i	0.999703	−	0.0243554i	$-0.00775334\pi$
$770$		0			0
$771$	−11.7389			−0.422766
$772$		0			0
$773$		−	37.8255i		−	1.36049i	−0.732986	−	0.680243i	$-0.761874\pi$
$773$		−	37.8255i		−	1.36049i	0.732986	−	0.680243i	$-0.238126\pi$
$774$		0			0
$775$		−	20.8765i		−	0.749907i
$776$		0			0
$777$	13.6093			0.488229
$778$		0			0
$779$	32.4273			1.16183
$780$		0			0
$781$	−12.0729			−0.432004
$782$		0			0
$783$	9.21618			0.329359
$784$		0			0
$785$		−	4.02430i		−	0.143633i
$786$		0			0
$787$			39.9390i			1.42367i	0.702345	+	0.711837i	$0.252136\pi$
$787$			39.9390i			1.42367i	−0.702345	+	0.711837i	$0.747864\pi$
$788$		0			0
$789$	−39.2664			−1.39792
$790$		0			0
$791$		−	17.5094i		−	0.622563i
$792$		0			0
$793$	−10.0104	+	4.45627i	−0.355481	+	0.158247i
$794$		0			0
$795$			6.46603i			0.229327i
$796$		0			0
$797$	−8.19805			−0.290390			−0.145195	−	0.989403i	$-0.546381\pi$
$797$	−8.19805			−0.290390			−0.145195	+	0.989403i	$0.546381\pi$
$798$		0			0
$799$		−	59.4263i		−	2.10235i
$800$		0			0
$801$		−	8.09579i		−	0.286051i
$802$		0			0
$803$	1.51173			0.0533477
$804$		0			0
$805$	10.6807			0.376444
$806$		0			0
$807$	35.6326			1.25433
$808$		0			0
$809$	−7.49162			−0.263391			−0.131696	−	0.991290i	$-0.542042\pi$
$809$	−7.49162			−0.263391			−0.131696	+	0.991290i	$0.542042\pi$
$810$		0			0
$811$		−	38.9513i		−	1.36777i	−0.729592	−	0.683883i	$-0.760290\pi$
$811$		−	38.9513i		−	1.36777i	0.729592	−	0.683883i	$-0.239710\pi$
$812$		0			0
$813$		−	39.2992i		−	1.37828i
$814$		0			0
$815$	7.68520			0.269201
$816$		0			0
$817$		−	28.1334i		−	0.984262i
$818$		0			0
$819$	−1.23034	−	2.76380i	−0.0429915	−	0.0965749i
$820$		0			0
$821$		−	28.3852i		−	0.990650i	−0.868708	−	0.495325i	$-0.835049\pi$
$821$		−	28.3852i		−	0.990650i	0.868708	−	0.495325i	$-0.164951\pi$
$822$		0			0
$823$	12.5508			0.437494			0.218747	−	0.975782i	$-0.429803\pi$
$823$	12.5508			0.437494			0.218747	+	0.975782i	$0.429803\pi$
$824$		0			0
$825$			5.14760i			0.179216i
$826$		0			0
$827$		−	33.6844i		−	1.17132i	−0.810556	−	0.585661i	$-0.800835\pi$
$827$		−	33.6844i		−	1.17132i	0.810556	−	0.585661i	$-0.199165\pi$
$828$		0			0
$829$	−16.1072			−0.559425			−0.279712	−	0.960084i	$-0.590239\pi$
$829$	−16.1072			−0.559425			−0.279712	+	0.960084i	$0.590239\pi$
$830$		0			0
$831$	5.84342			0.202706
$832$		0			0
$833$	−5.50110			−0.190602
$834$		0			0
$835$	13.1674			0.455677
$836$		0			0
$837$			33.6451i			1.16294i
$838$		0			0
$839$			17.7746i			0.613648i	0.951766	+	0.306824i	$0.0992664\pi$
$839$			17.7746i			0.613648i	−0.951766	+	0.306824i	$0.900734\pi$
$840$		0			0
$841$	−26.3331			−0.908037
$842$		0			0
$843$		−	33.7807i		−	1.16347i
$844$		0			0
$845$	11.8241	+	10.6488i	0.406761	+	0.366331i
$846$		0			0
$847$			1.00000i			0.0343604i
$848$		0			0
$849$	21.6801			0.744061
$850$		0			0
$851$			80.7826i			2.76919i
$852$		0			0
$853$			45.6140i			1.56179i	0.624660	+	0.780897i	$0.285237\pi$
$853$			45.6140i			1.56179i	−0.624660	+	0.780897i	$0.714763\pi$
$854$		0			0
$855$	−4.86691			−0.166445
$856$		0			0
$857$	35.2859			1.20534			0.602672	−	0.797989i	$-0.294103\pi$
$857$	35.2859			1.20534			0.602672	+	0.797989i	$0.294103\pi$
$858$		0			0
$859$	−34.7602			−1.18600			−0.593002	−	0.805201i	$-0.702057\pi$
$859$	−34.7602			−1.18600			−0.593002	+	0.805201i	$0.702057\pi$
$860$		0			0
$861$	10.0593			0.342819
$862$		0			0
$863$			57.4635i			1.95608i	0.208418	+	0.978040i	$0.433169\pi$
$863$			57.4635i			1.95608i	−0.208418	+	0.978040i	$0.566831\pi$
$864$		0			0
$865$		−	7.44264i		−	0.253057i
$866$		0			0
$867$	19.4955			0.662102
$868$		0			0
$869$			6.58964i			0.223538i
$870$		0			0
$871$	−18.7768	−	42.1796i	−0.636227	−	1.42920i
$872$		0			0
$873$		−	4.25345i		−	0.143957i
$874$		0			0
$875$	10.4064			0.351802
$876$		0			0
$877$		−	46.1340i		−	1.55784i	−0.627126	−	0.778918i	$-0.715769\pi$
$877$		−	46.1340i		−	1.55784i	0.627126	−	0.778918i	$-0.284231\pi$
$878$		0			0
$879$		−	14.1346i		−	0.476749i
$880$		0			0
$881$	−8.24591			−0.277812			−0.138906	−	0.990306i	$-0.544359\pi$
$881$	−8.24591			−0.277812			−0.138906	+	0.990306i	$0.544359\pi$
$882$		0			0
$883$	34.0479			1.14580			0.572902	−	0.819624i	$-0.305817\pi$
$883$	34.0479			1.14580			0.572902	+	0.819624i	$0.305817\pi$
$884$		0			0
$885$	5.34965			0.179826
$886$		0			0
$887$	−27.8768			−0.936012			−0.468006	−	0.883725i	$-0.655027\pi$
$887$	−27.8768			−0.936012			−0.468006	+	0.883725i	$0.655027\pi$
$888$		0			0
$889$			19.8793i			0.666730i
$890$		0			0
$891$		−	5.77879i		−	0.193597i
$892$		0			0
$893$	−51.1911			−1.71304
$894$		0			0
$895$			24.3345i			0.813412i
$896$		0			0
$897$	−42.2511	+	18.8086i	−1.41072	+	0.628001i
$898$		0			0
$899$			9.73598i			0.324713i
$900$		0			0
$901$	19.7684			0.658582
$902$		0			0
$903$		−	8.72725i		−	0.290425i
$904$		0			0
$905$			5.76920i			0.191775i
$906$		0			0
$907$	−36.1458			−1.20020			−0.600101	−	0.799924i	$-0.704873\pi$
$907$	−36.1458			−1.20020			−0.600101	+	0.799924i	$0.704873\pi$
$908$		0			0
$909$	−16.7402			−0.555236
$910$		0			0
$911$	−40.7508			−1.35013			−0.675067	−	0.737756i	$-0.735885\pi$
$911$	−40.7508			−1.35013			−0.675067	+	0.737756i	$0.735885\pi$
$912$		0			0
$913$	−2.25080			−0.0744905
$914$		0			0
$915$			5.46834i			0.180778i
$916$		0			0
$917$		−	14.1726i		−	0.468020i
$918$		0			0
$919$	−50.4729			−1.66495			−0.832474	−	0.554065i	$-0.813076\pi$
$919$	−50.4729			−1.66495			−0.832474	+	0.554065i	$0.813076\pi$
$920$		0			0
$921$			42.2675i			1.39276i
$922$		0			0
$923$	−17.7029	−	39.7672i	−0.582697	−	1.30895i
$924$		0			0
$925$			32.4188i			1.06592i
$926$		0			0
$927$	−5.57669			−0.183163
$928$		0			0
$929$			46.9254i			1.53957i	0.638303	+	0.769785i	$0.279637\pi$
$929$			46.9254i			1.53957i	−0.638303	+	0.769785i	$0.720363\pi$
$930$		0			0
$931$			4.73877i			0.155307i
$932$		0			0
$933$	−20.3295			−0.665557
$934$		0			0
$935$	−6.73355			−0.220211
$936$		0			0
$937$	15.1692			0.495556			0.247778	−	0.968817i	$-0.420300\pi$
$937$	15.1692			0.495556			0.247778	+	0.968817i	$0.420300\pi$
$938$		0			0
$939$	−0.153161			−0.00499821
$940$		0			0
$941$		−	12.8971i		−	0.420434i	−0.977655	−	0.210217i	$-0.932583\pi$
$941$		−	12.8971i		−	0.420434i	0.977655	−	0.210217i	$-0.0674171\pi$
$942$		0			0
$943$			59.7104i			1.94444i
$944$		0			0
$945$	−6.90781			−0.224711
$946$		0			0
$947$			37.5642i			1.22067i	0.792142	+	0.610337i	$0.208966\pi$
$947$			37.5642i			1.22067i	−0.792142	+	0.610337i	$0.791034\pi$
$948$		0			0
$949$	2.21669	+	4.97950i	0.0719567	+	0.161641i
$950$		0			0
$951$			2.12240i			0.0688234i
$952$		0			0
$953$	−17.6680			−0.572322			−0.286161	−	0.958181i	$-0.592379\pi$
$953$	−17.6680			−0.572322			−0.286161	+	0.958181i	$0.592379\pi$
$954$		0			0
$955$		−	14.9473i		−	0.483682i
$956$		0			0
$957$		−	2.40063i		−	0.0776014i
$958$		0			0
$959$	7.90227			0.255177
$960$		0			0
$961$	−4.54269			−0.146538
$962$		0			0
$963$	−1.06728			−0.0343926
$964$		0			0
$965$	−16.9529			−0.545733
$966$		0			0
$967$		−	33.5231i		−	1.07803i	−0.842296	−	0.539015i	$-0.818797\pi$
$967$		−	33.5231i		−	1.07803i	0.842296	−	0.539015i	$-0.181203\pi$
$968$		0			0
$969$		−	38.3210i		−	1.23105i
$970$		0			0
$971$	12.4946			0.400972			0.200486	−	0.979697i	$-0.435748\pi$
$971$	12.4946			0.400972			0.200486	+	0.979697i	$0.435748\pi$
$972$		0			0
$973$			5.73294i			0.183790i
$974$		0			0
$975$	−16.9558	+	7.54806i	−0.543019	+	0.241731i
$976$		0			0
$977$		−	4.30267i		−	0.137655i	−0.997629	−	0.0688274i	$-0.978074\pi$
$977$		−	4.30267i		−	0.137655i	0.997629	−	0.0688274i	$-0.0219258\pi$
$978$		0			0
$979$	−9.64863			−0.308372
$980$		0			0
$981$		−	8.74133i		−	0.279089i
$982$		0			0
$983$			41.5835i			1.32631i	0.748483	+	0.663154i	$0.230782\pi$
$983$			41.5835i			1.32631i	−0.748483	+	0.663154i	$0.769218\pi$
$984$		0			0
$985$	10.3104			0.328518
$986$		0			0
$987$	−15.8800			−0.505465
$988$		0			0
$989$	51.8037			1.64726
$990$		0			0
$991$	23.4747			0.745698			0.372849	−	0.927892i	$-0.378381\pi$
$991$	23.4747			0.745698			0.372849	+	0.927892i	$0.378381\pi$
$992$		0			0
$993$		−	17.7956i		−	0.564726i
$994$		0			0
$995$		−	28.9668i		−	0.918308i
$996$		0			0
$997$	3.36340			0.106520			0.0532599	−	0.998581i	$-0.483039\pi$
$997$	3.36340			0.106520			0.0532599	+	0.998581i	$0.483039\pi$
$998$		0			0
$999$		−	52.2468i		−	1.65301i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.m.c.2157.28	yes	36
13.12	even	2	inner	4004.2.m.c.2157.27	✓	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.m.c.2157.27	✓	36	13.12	even	2	inner
4004.2.m.c.2157.28	yes	36	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:	\( N \)	\(=\)	\( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4004.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.47001 q^{3} +1.22404i q^{5} -1.00000i q^{7} -0.839061 q^{9} +O(q^{10})\)	\(q+1.47001 q^{3} +1.22404i q^{5} -1.00000i q^{7} -0.839061 q^{9} +1.00000i q^{11} +(-3.29392 + 1.46633i) q^{13} +1.79935i q^{15} +5.50110 q^{17} -4.73877i q^{19} -1.47001i q^{21} +8.72579 q^{23} +3.50174 q^{25} -5.64347 q^{27} -1.63307 q^{29} -5.96177i q^{31} +1.47001i q^{33} +1.22404 q^{35} +9.25792i q^{37} +(-4.84210 + 2.15552i) q^{39} +6.84298i q^{41} +5.93685 q^{43} -1.02704i q^{45} -10.8026i q^{47} -1.00000 q^{49} +8.08669 q^{51} +3.59354 q^{53} -1.22404 q^{55} -6.96605i q^{57} -2.97310i q^{59} +3.03907 q^{61} +0.839061i q^{63} +(-1.79484 - 4.03187i) q^{65} +12.8053i q^{67} +12.8270 q^{69} +12.0729i q^{71} -1.51173i q^{73} +5.14760 q^{75} +1.00000 q^{77} +6.58964 q^{79} -5.77879 q^{81} +2.25080i q^{83} +6.73355i q^{85} -2.40063 q^{87} +9.64863i q^{89} +(1.46633 + 3.29392i) q^{91} -8.76388i q^{93} +5.80042 q^{95} +5.06930i q^{97} -0.839061i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 4 q^{3} + 40 q^{9}+O(q^{10}) \) 36 * q - 4 * q^3 + 40 * q^9	\( 36 q - 4 q^{3} + 40 q^{9} - 4 q^{17} + 8 q^{23} - 80 q^{25} + 8 q^{27} + 8 q^{29} - 24 q^{39} + 32 q^{43} - 36 q^{49} - 20 q^{51} + 12 q^{53} + 32 q^{61} - 24 q^{65} + 80 q^{69} - 36 q^{75} + 36 q^{77} + 16 q^{79} + 132 q^{81} + 8 q^{91} + 56 q^{95}+O(q^{100}) \) 36 * q - 4 * q^3 + 40 * q^9 - 4 * q^17 + 8 * q^23 - 80 * q^25 + 8 * q^27 + 8 * q^29 - 24 * q^39 + 32 * q^43 - 36 * q^49 - 20 * q^51 + 12 * q^53 + 32 * q^61 - 24 * q^65 + 80 * q^69 - 36 * q^75 + 36 * q^77 + 16 * q^79 + 132 * q^81 + 8 * q^91 + 56 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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