LMFDB - Embedded newform 4004.2.m.c.2157.9

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			2157.9
Character	$\chi$	$=$	4004.2157
Dual form			4004.2.m.c.2157.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.64159 q^{3} -4.02832i q^{5} -1.00000i q^{7} -0.305195 q^{9} +O(q^{10})$	$q-1.64159 q^{3} -4.02832i q^{5} -1.00000i q^{7} -0.305195 q^{9} +1.00000i q^{11} +(3.58935 + 0.341402i) q^{13} +6.61284i q^{15} +2.93822 q^{17} -5.26423i q^{19} +1.64159i q^{21} -6.16583 q^{23} -11.2274 q^{25} +5.42576 q^{27} +2.51264 q^{29} -9.67831i q^{31} -1.64159i q^{33} -4.02832 q^{35} -0.752499i q^{37} +(-5.89223 - 0.560440i) q^{39} -2.05391i q^{41} +3.54942 q^{43} +1.22942i q^{45} -10.3141i q^{47} -1.00000 q^{49} -4.82335 q^{51} +7.99742 q^{53} +4.02832 q^{55} +8.64168i q^{57} -3.00461i q^{59} +6.61169 q^{61} +0.305195i q^{63} +(1.37528 - 14.4591i) q^{65} -8.37934i q^{67} +10.1217 q^{69} +7.73832i q^{71} -6.96682i q^{73} +18.4307 q^{75} +1.00000 q^{77} -11.5285 q^{79} -7.99127 q^{81} -13.4835i q^{83} -11.8361i q^{85} -4.12471 q^{87} +12.5207i q^{89} +(0.341402 - 3.58935i) q^{91} +15.8878i q^{93} -21.2060 q^{95} +12.2072i q^{97} -0.305195i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.64159			−0.947770			−0.473885	−	0.880587i	$-0.657149\pi$
$3$	−1.64159			−0.947770			−0.473885	+	0.880587i	$0.657149\pi$
$4$		0			0
$5$		−	4.02832i		−	1.80152i	−0.434317	−	0.900760i	$-0.643010\pi$
$5$		−	4.02832i		−	1.80152i	0.434317	−	0.900760i	$-0.356990\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	−0.305195			−0.101732
$10$		0			0
$11$			1.00000i			0.301511i
$11$			1.00000i			0.301511i
$12$		0			0
$13$	3.58935	+	0.341402i	0.995507	+	0.0946878i
$13$	3.58935	+	0.341402i	0.995507	+	0.0946878i
$14$		0			0
$15$			6.61284i			1.70743i
$16$		0			0
$17$	2.93822			0.712624			0.356312	−	0.934367i	$-0.384034\pi$
$17$	2.93822			0.712624			0.356312	+	0.934367i	$0.384034\pi$
$18$		0			0
$19$		−	5.26423i		−	1.20770i	−0.797099	−	0.603848i	$-0.793633\pi$
$19$		−	5.26423i		−	1.20770i	0.797099	−	0.603848i	$-0.206367\pi$
$20$		0			0
$21$			1.64159i			0.358223i
$22$		0			0
$23$	−6.16583			−1.28566			−0.642832	−	0.766007i	$-0.722241\pi$
$23$	−6.16583			−1.28566			−0.642832	+	0.766007i	$0.722241\pi$
$24$		0			0
$25$	−11.2274			−2.24548
$26$		0			0
$27$	5.42576			1.04419
$28$		0			0
$29$	2.51264			0.466585			0.233292	−	0.972407i	$-0.425050\pi$
$29$	2.51264			0.466585			0.233292	+	0.972407i	$0.425050\pi$
$30$		0			0
$31$		−	9.67831i		−	1.73827i	−0.494571	−	0.869137i	$-0.664675\pi$
$31$		−	9.67831i		−	1.73827i	0.494571	−	0.869137i	$-0.335325\pi$
$32$		0			0
$33$		−	1.64159i		−	0.285763i
$34$		0			0
$35$	−4.02832			−0.680911
$36$		0			0
$37$		−	0.752499i		−	0.123710i	−0.998085	−	0.0618550i	$-0.980298\pi$
$37$		−	0.752499i		−	0.123710i	0.998085	−	0.0618550i	$-0.0197016\pi$
$38$		0			0
$39$	−5.89223	−	0.560440i	−0.943512	−	0.0897422i
$40$		0			0
$41$		−	2.05391i		−	0.320766i	−0.987055	−	0.160383i	$-0.948727\pi$
$41$		−	2.05391i		−	0.320766i	0.987055	−	0.160383i	$-0.0512730\pi$
$42$		0			0
$43$	3.54942			0.541281			0.270641	−	0.962680i	$-0.412764\pi$
$43$	3.54942			0.541281			0.270641	+	0.962680i	$0.412764\pi$
$44$		0			0
$45$			1.22942i			0.183272i
$46$		0			0
$47$		−	10.3141i		−	1.50447i	−0.658894	−	0.752236i	$-0.728976\pi$
$47$		−	10.3141i		−	1.50447i	0.658894	−	0.752236i	$-0.271024\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−4.82335			−0.675404
$52$		0			0
$53$	7.99742			1.09853			0.549265	−	0.835648i	$-0.314908\pi$
$53$	7.99742			1.09853			0.549265	+	0.835648i	$0.314908\pi$
$54$		0			0
$55$	4.02832			0.543179
$56$		0			0
$57$			8.64168i			1.14462i
$58$		0			0
$59$		−	3.00461i		−	0.391167i	−0.980687	−	0.195583i	$-0.937340\pi$
$59$		−	3.00461i		−	0.391167i	0.980687	−	0.195583i	$-0.0626600\pi$
$60$		0			0
$61$	6.61169			0.846540			0.423270	−	0.906004i	$-0.360882\pi$
$61$	6.61169			0.846540			0.423270	+	0.906004i	$0.360882\pi$
$62$		0			0
$63$			0.305195i			0.0384510i
$64$		0			0
$65$	1.37528	−	14.4591i	0.170582	−	1.79343i
$66$		0			0
$67$		−	8.37934i		−	1.02370i	−0.859075	−	0.511850i	$-0.828960\pi$
$67$		−	8.37934i		−	1.02370i	0.859075	−	0.511850i	$-0.171040\pi$
$68$		0			0
$69$	10.1217			1.21851
$70$		0			0
$71$			7.73832i			0.918370i	0.888341	+	0.459185i	$0.151858\pi$
$71$			7.73832i			0.918370i	−0.888341	+	0.459185i	$0.848142\pi$
$72$		0			0
$73$		−	6.96682i		−	0.815404i	−0.913115	−	0.407702i	$-0.866330\pi$
$73$		−	6.96682i		−	0.815404i	0.913115	−	0.407702i	$-0.133670\pi$
$74$		0			0
$75$	18.4307			2.12819
$76$		0			0
$77$	1.00000			0.113961
$78$		0			0
$79$	−11.5285			−1.29706			−0.648529	−	0.761190i	$-0.724615\pi$
$79$	−11.5285			−1.29706			−0.648529	+	0.761190i	$0.724615\pi$
$80$		0			0
$81$	−7.99127			−0.887919
$82$		0			0
$83$		−	13.4835i		−	1.48000i	−0.672606	−	0.740001i	$-0.734825\pi$
$83$		−	13.4835i		−	1.48000i	0.672606	−	0.740001i	$-0.265175\pi$
$84$		0			0
$85$		−	11.8361i		−	1.28381i
$86$		0			0
$87$	−4.12471			−0.442215
$88$		0			0
$89$			12.5207i			1.32719i	0.748093	+	0.663594i	$0.230970\pi$
$89$			12.5207i			1.32719i	−0.748093	+	0.663594i	$0.769030\pi$
$90$		0			0
$91$	0.341402	−	3.58935i	0.0357886	−	0.376266i
$92$		0			0
$93$			15.8878i			1.64749i
$94$		0			0
$95$	−21.2060			−2.17569
$96$		0			0
$97$			12.2072i			1.23945i	0.784817	+	0.619727i	$0.212757\pi$
$97$			12.2072i			1.23945i	−0.784817	+	0.619727i	$0.787243\pi$
$98$		0			0
$99$		−	0.305195i		−	0.0306733i
$100$		0			0
$101$	−7.87840			−0.783930			−0.391965	−	0.919980i	$-0.628205\pi$
$101$	−7.87840			−0.783930			−0.391965	+	0.919980i	$0.628205\pi$
$102$		0			0
$103$	−10.6895			−1.05327			−0.526634	−	0.850092i	$-0.676546\pi$
$103$	−10.6895			−1.05327			−0.526634	+	0.850092i	$0.676546\pi$
$104$		0			0
$105$	6.61284			0.645347
$106$		0			0
$107$	−5.68314			−0.549410			−0.274705	−	0.961529i	$-0.588580\pi$
$107$	−5.68314			−0.549410			−0.274705	+	0.961529i	$0.588580\pi$
$108$		0			0
$109$			6.60605i			0.632745i	0.948635	+	0.316372i	$0.102465\pi$
$109$			6.60605i			0.632745i	−0.948635	+	0.316372i	$0.897535\pi$
$110$		0			0
$111$			1.23529i			0.117249i
$112$		0			0
$113$	17.9010			1.68399			0.841993	−	0.539489i	$-0.181383\pi$
$113$	17.9010			1.68399			0.841993	+	0.539489i	$0.181383\pi$
$114$		0			0
$115$			24.8379i			2.31615i
$116$		0			0
$117$	−1.09545	−	0.104194i	−0.101275	−	0.00963275i
$118$		0			0
$119$		−	2.93822i		−	0.269347i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$			3.37166i			0.304013i
$124$		0			0
$125$			25.0859i			2.24375i
$126$		0			0
$127$	1.56725			0.139071			0.0695354	−	0.997579i	$-0.477848\pi$
$127$	1.56725			0.139071			0.0695354	+	0.997579i	$0.477848\pi$
$128$		0			0
$129$	−5.82668			−0.513010
$130$		0			0
$131$	19.0267			1.66237			0.831187	−	0.555994i	$-0.187662\pi$
$131$	19.0267			1.66237			0.831187	+	0.555994i	$0.187662\pi$
$132$		0			0
$133$	−5.26423			−0.456466
$134$		0			0
$135$		−	21.8567i		−	1.88113i
$136$		0			0
$137$			9.38227i			0.801582i	0.916170	+	0.400791i	$0.131265\pi$
$137$			9.38227i			0.801582i	−0.916170	+	0.400791i	$0.868735\pi$
$138$		0			0
$139$	−19.9484			−1.69200			−0.846002	−	0.533180i	$-0.820997\pi$
$139$	−19.9484			−1.69200			−0.846002	+	0.533180i	$0.820997\pi$
$140$		0			0
$141$			16.9315i			1.42589i
$142$		0			0
$143$	−0.341402	+	3.58935i	−0.0285494	+	0.300157i
$144$		0			0
$145$		−	10.1217i		−	0.840562i
$146$		0			0
$147$	1.64159			0.135396
$148$		0			0
$149$			3.70508i			0.303532i	0.988416	+	0.151766i	$0.0484960\pi$
$149$			3.70508i			0.303532i	−0.988416	+	0.151766i	$0.951504\pi$
$150$		0			0
$151$		−	1.48027i		−	0.120462i	−0.998184	−	0.0602312i	$-0.980816\pi$
$151$		−	1.48027i		−	0.120462i	0.998184	−	0.0602312i	$-0.0191838\pi$
$152$		0			0
$153$	−0.896732			−0.0724965
$154$		0			0
$155$	−38.9873			−3.13154
$156$		0			0
$157$	−12.3875			−0.988626			−0.494313	−	0.869284i	$-0.664580\pi$
$157$	−12.3875			−0.988626			−0.494313	+	0.869284i	$0.664580\pi$
$158$		0			0
$159$	−13.1285			−1.04115
$160$		0			0
$161$			6.16583i			0.485935i
$162$		0			0
$163$			12.4673i			0.976512i	0.872700	+	0.488256i	$0.162367\pi$
$163$			12.4673i			0.976512i	−0.872700	+	0.488256i	$0.837633\pi$
$164$		0			0
$165$	−6.61284			−0.514809
$166$		0			0
$167$			3.38570i			0.261993i	0.991383	+	0.130997i	$0.0418177\pi$
$167$			3.38570i			0.261993i	−0.991383	+	0.130997i	$0.958182\pi$
$168$		0			0
$169$	12.7669	+	2.45082i	0.982068	+	0.188525i
$170$		0			0
$171$			1.60662i			0.122861i
$172$		0			0
$173$	0.526945			0.0400629			0.0200314	−	0.999799i	$-0.493623\pi$
$173$	0.526945			0.0400629			0.0200314	+	0.999799i	$0.493623\pi$
$174$		0			0
$175$			11.2274i			0.848710i
$176$		0			0
$177$			4.93232i			0.370736i
$178$		0			0
$179$	−8.84103			−0.660809			−0.330405	−	0.943839i	$-0.607185\pi$
$179$	−8.84103			−0.660809			−0.330405	+	0.943839i	$0.607185\pi$
$180$		0			0
$181$	−12.6176			−0.937859			−0.468929	−	0.883236i	$-0.655360\pi$
$181$	−12.6176			−0.937859			−0.468929	+	0.883236i	$0.655360\pi$
$182$		0			0
$183$	−10.8537			−0.802325
$184$		0			0
$185$	−3.03131			−0.222866
$186$		0			0
$187$			2.93822i			0.214864i
$188$		0			0
$189$		−	5.42576i		−	0.394666i
$190$		0			0
$191$	−9.28352			−0.671732			−0.335866	−	0.941910i	$-0.609029\pi$
$191$	−9.28352			−0.671732			−0.335866	+	0.941910i	$0.609029\pi$
$192$		0			0
$193$			1.60103i			0.115245i	0.998338	+	0.0576225i	$0.0183520\pi$
$193$			1.60103i			0.115245i	−0.998338	+	0.0576225i	$0.981648\pi$
$194$		0			0
$195$	−2.25763	+	23.7358i	−0.161672	+	1.69976i
$196$		0			0
$197$		−	17.9711i		−	1.28039i	−0.768213	−	0.640194i	$-0.778854\pi$
$197$		−	17.9711i		−	1.28039i	0.768213	−	0.640194i	$-0.221146\pi$
$198$		0			0
$199$	8.13459			0.576645			0.288323	−	0.957533i	$-0.406902\pi$
$199$	8.13459			0.576645			0.288323	+	0.957533i	$0.406902\pi$
$200$		0			0
$201$			13.7554i			0.970232i
$202$		0			0
$203$		−	2.51264i		−	0.176352i
$204$		0			0
$205$	−8.27379			−0.577867
$206$		0			0
$207$	1.88178			0.130793
$208$		0			0
$209$	5.26423			0.364134
$210$		0			0
$211$	21.6911			1.49328			0.746638	−	0.665231i	$-0.231667\pi$
$211$	21.6911			1.49328			0.746638	+	0.665231i	$0.231667\pi$
$212$		0			0
$213$		−	12.7031i		−	0.870403i
$214$		0			0
$215$		−	14.2982i		−	0.975129i
$216$		0			0
$217$	−9.67831			−0.657006
$218$		0			0
$219$			11.4366i			0.772816i
$220$		0			0
$221$	10.5463	+	1.00311i	0.709422	+	0.0674768i
$222$		0			0
$223$			23.0277i			1.54205i	0.636806	+	0.771024i	$0.280255\pi$
$223$			23.0277i			1.54205i	−0.636806	+	0.771024i	$0.719745\pi$
$224$		0			0
$225$	3.42654			0.228436
$226$		0			0
$227$		−	1.16344i		−	0.0772201i	−0.999254	−	0.0386100i	$-0.987707\pi$
$227$		−	1.16344i		−	0.0772201i	0.999254	−	0.0386100i	$-0.0122930\pi$
$228$		0			0
$229$			0.913327i			0.0603543i	0.999545	+	0.0301772i	$0.00960715\pi$
$229$			0.913327i			0.0603543i	−0.999545	+	0.0301772i	$0.990393\pi$
$230$		0			0
$231$	−1.64159			−0.108008
$232$		0			0
$233$	−6.51944			−0.427103			−0.213551	−	0.976932i	$-0.568503\pi$
$233$	−6.51944			−0.427103			−0.213551	+	0.976932i	$0.568503\pi$
$234$		0			0
$235$	−41.5487			−2.71034
$236$		0			0
$237$	18.9250			1.22931
$238$		0			0
$239$		−	28.2270i		−	1.82585i	−0.408124	−	0.912927i	$-0.633817\pi$
$239$		−	28.2270i		−	1.82585i	0.408124	−	0.912927i	$-0.366183\pi$
$240$		0			0
$241$			6.11291i			0.393767i	0.980427	+	0.196884i	$0.0630821\pi$
$241$			6.11291i			0.393767i	−0.980427	+	0.196884i	$0.936918\pi$
$242$		0			0
$243$	−3.15893			−0.202645
$244$		0			0
$245$			4.02832i			0.257360i
$246$		0			0
$247$	1.79722	−	18.8952i	0.114354	−	1.20227i
$248$		0			0
$249$			22.1343i			1.40270i
$250$		0			0
$251$	−1.60455			−0.101278			−0.0506392	−	0.998717i	$-0.516126\pi$
$251$	−1.60455			−0.101278			−0.0506392	+	0.998717i	$0.516126\pi$
$252$		0			0
$253$		−	6.16583i		−	0.387642i
$254$		0			0
$255$			19.4300i			1.21675i
$256$		0			0
$257$	7.52738			0.469545			0.234773	−	0.972050i	$-0.424565\pi$
$257$	7.52738			0.469545			0.234773	+	0.972050i	$0.424565\pi$
$258$		0			0
$259$	−0.752499			−0.0467580
$260$		0			0
$261$	−0.766844			−0.0474665
$262$		0			0
$263$	20.9839			1.29392			0.646961	−	0.762523i	$-0.276039\pi$
$263$	20.9839			1.29392			0.646961	+	0.762523i	$0.276039\pi$
$264$		0			0
$265$		−	32.2162i		−	1.97903i
$266$		0			0
$267$		−	20.5538i		−	1.25787i
$268$		0			0
$269$	−12.7451			−0.777085			−0.388542	−	0.921431i	$-0.627021\pi$
$269$	−12.7451			−0.777085			−0.388542	+	0.921431i	$0.627021\pi$
$270$		0			0
$271$			16.6891i			1.01379i	0.862007	+	0.506896i	$0.169207\pi$
$271$			16.6891i			1.01379i	−0.862007	+	0.506896i	$0.830793\pi$
$272$		0			0
$273$	−0.560440	+	5.89223i	−0.0339194	+	0.356614i
$274$		0			0
$275$		−	11.2274i		−	0.677036i
$276$		0			0
$277$	−31.0363			−1.86479			−0.932396	−	0.361439i	$-0.882285\pi$
$277$	−31.0363			−1.86479			−0.932396	+	0.361439i	$0.882285\pi$
$278$		0			0
$279$			2.95377i			0.176838i
$280$		0			0
$281$		−	29.9221i		−	1.78500i	−0.451047	−	0.892500i	$-0.648949\pi$
$281$		−	29.9221i		−	1.78500i	0.451047	−	0.892500i	$-0.351051\pi$
$282$		0			0
$283$	−2.97429			−0.176803			−0.0884015	−	0.996085i	$-0.528176\pi$
$283$	−2.97429			−0.176803			−0.0884015	+	0.996085i	$0.528176\pi$
$284$		0			0
$285$	34.8115			2.06205
$286$		0			0
$287$	−2.05391			−0.121238
$288$		0			0
$289$	−8.36684			−0.492167
$290$		0			0
$291$		−	20.0392i		−	1.17472i
$292$		0			0
$293$			20.8555i			1.21839i	0.793020	+	0.609195i	$0.208507\pi$
$293$			20.8555i			1.21839i	−0.793020	+	0.609195i	$0.791493\pi$
$294$		0			0
$295$	−12.1035			−0.704695
$296$		0			0
$297$			5.42576i			0.314835i
$298$		0			0
$299$	−22.1313	−	2.10502i	−1.27989	−	0.121737i
$300$		0			0
$301$		−	3.54942i		−	0.204585i
$302$		0			0
$303$	12.9331			0.742986
$304$		0			0
$305$		−	26.6340i		−	1.52506i
$306$		0			0
$307$			19.1950i			1.09552i	0.836636	+	0.547759i	$0.184519\pi$
$307$			19.1950i			1.09552i	−0.836636	+	0.547759i	$0.815481\pi$
$308$		0			0
$309$	17.5478			0.998257
$310$		0			0
$311$	27.2009			1.54242			0.771211	−	0.636580i	$-0.219652\pi$
$311$	27.2009			1.54242			0.771211	+	0.636580i	$0.219652\pi$
$312$		0			0
$313$	0.807068			0.0456182			0.0228091	−	0.999740i	$-0.492739\pi$
$313$	0.807068			0.0456182			0.0228091	+	0.999740i	$0.492739\pi$
$314$		0			0
$315$	1.22942			0.0692702
$316$		0			0
$317$		−	11.3870i		−	0.639559i	−0.947492	−	0.319779i	$-0.896391\pi$
$317$		−	11.3870i		−	0.639559i	0.947492	−	0.319779i	$-0.103609\pi$
$318$		0			0
$319$			2.51264i			0.140681i
$320$		0			0
$321$	9.32936			0.520714
$322$		0			0
$323$		−	15.4675i		−	0.860634i
$324$		0			0
$325$	−40.2990	−	3.83304i	−2.23539	−	0.212619i
$326$		0			0
$327$		−	10.8444i		−	0.599697i
$328$		0			0
$329$	−10.3141			−0.568637
$330$		0			0
$331$			34.2151i			1.88063i	0.340305	+	0.940315i	$0.389470\pi$
$331$			34.2151i			1.88063i	−0.340305	+	0.940315i	$0.610530\pi$
$332$		0			0
$333$			0.229659i			0.0125852i
$334$		0			0
$335$	−33.7547			−1.84422
$336$		0			0
$337$	−2.44148			−0.132996			−0.0664980	−	0.997787i	$-0.521183\pi$
$337$	−2.44148			−0.132996			−0.0664980	+	0.997787i	$0.521183\pi$
$338$		0			0
$339$	−29.3860			−1.59603
$340$		0			0
$341$	9.67831			0.524110
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$		−	40.7736i		−	2.19518i
$346$		0			0
$347$	−20.2875			−1.08909			−0.544546	−	0.838731i	$-0.683298\pi$
$347$	−20.2875			−1.08909			−0.544546	+	0.838731i	$0.683298\pi$
$348$		0			0
$349$			22.7273i			1.21656i	0.793721	+	0.608282i	$0.208141\pi$
$349$			22.7273i			1.21656i	−0.793721	+	0.608282i	$0.791859\pi$
$350$		0			0
$351$	19.4750	+	1.85236i	1.03950	+	0.0988719i
$352$		0			0
$353$			16.1117i			0.857537i	0.903414	+	0.428768i	$0.141052\pi$
$353$			16.1117i			0.857537i	−0.903414	+	0.428768i	$0.858948\pi$
$354$		0			0
$355$	31.1724			1.65446
$356$		0			0
$357$			4.82335i			0.255279i
$358$		0			0
$359$		−	25.8362i		−	1.36358i	−0.731546	−	0.681792i	$-0.761201\pi$
$359$		−	25.8362i		−	1.36358i	0.731546	−	0.681792i	$-0.238799\pi$
$360$		0			0
$361$	−8.71209			−0.458531
$362$		0			0
$363$	1.64159			0.0861609
$364$		0			0
$365$	−28.0646			−1.46897
$366$		0			0
$367$	−26.5198			−1.38432			−0.692160	−	0.721744i	$-0.743341\pi$
$367$	−26.5198			−1.38432			−0.692160	+	0.721744i	$0.743341\pi$
$368$		0			0
$369$			0.626842i			0.0326321i
$370$		0			0
$371$		−	7.99742i		−	0.415206i
$372$		0			0
$373$	−11.0852			−0.573969			−0.286985	−	0.957935i	$-0.592653\pi$
$373$	−11.0852			−0.573969			−0.286985	+	0.957935i	$0.592653\pi$
$374$		0			0
$375$		−	41.1806i		−	2.12656i
$376$		0			0
$377$	9.01873	+	0.857818i	0.464488	+	0.0441799i
$378$		0			0
$379$			28.8736i			1.48314i	0.670876	+	0.741570i	$0.265918\pi$
$379$			28.8736i			1.48314i	−0.670876	+	0.741570i	$0.734082\pi$
$380$		0			0
$381$	−2.57277			−0.131807
$382$		0			0
$383$		−	14.5209i		−	0.741981i	−0.928637	−	0.370990i	$-0.879018\pi$
$383$		−	14.5209i		−	0.741981i	0.928637	−	0.370990i	$-0.120982\pi$
$384$		0			0
$385$		−	4.02832i		−	0.205302i
$386$		0			0
$387$	−1.08327			−0.0550655
$388$		0			0
$389$	19.0845			0.967624			0.483812	−	0.875172i	$-0.339252\pi$
$389$	19.0845			0.967624			0.483812	+	0.875172i	$0.339252\pi$
$390$		0			0
$391$	−18.1166			−0.916195
$392$		0			0
$393$	−31.2340			−1.57555
$394$		0			0
$395$			46.4405i			2.33668i
$396$		0			0
$397$		−	26.3360i		−	1.32177i	−0.750488	−	0.660884i	$-0.770181\pi$
$397$		−	26.3360i		−	1.32177i	0.750488	−	0.660884i	$-0.229819\pi$
$398$		0			0
$399$	8.64168			0.432625
$400$		0			0
$401$		−	24.7685i		−	1.23688i	−0.785831	−	0.618441i	$-0.787765\pi$
$401$		−	24.7685i		−	1.23688i	0.785831	−	0.618441i	$-0.212235\pi$
$402$		0			0
$403$	3.30419	−	34.7388i	0.164593	−	1.73046i
$404$		0			0
$405$			32.1914i			1.59960i
$406$		0			0
$407$	0.752499			0.0373000
$408$		0			0
$409$		−	4.99360i		−	0.246917i	−0.992350	−	0.123459i	$-0.960601\pi$
$409$		−	4.99360i		−	0.246917i	0.992350	−	0.123459i	$-0.0393987\pi$
$410$		0			0
$411$		−	15.4018i		−	0.759715i
$412$		0			0
$413$	−3.00461			−0.147847
$414$		0			0
$415$	−54.3157			−2.66625
$416$		0			0
$417$	32.7471			1.60363
$418$		0			0
$419$	−11.8262			−0.577750			−0.288875	−	0.957367i	$-0.593281\pi$
$419$	−11.8262			−0.577750			−0.288875	+	0.957367i	$0.593281\pi$
$420$		0			0
$421$			13.6380i			0.664675i	0.943160	+	0.332338i	$0.107837\pi$
$421$			13.6380i			0.664675i	−0.943160	+	0.332338i	$0.892163\pi$
$422$		0			0
$423$			3.14782i			0.153052i
$424$		0			0
$425$	−32.9885			−1.60018
$426$		0			0
$427$		−	6.61169i		−	0.319962i
$428$		0			0
$429$	0.560440	−	5.89223i	0.0270583	−	0.284480i
$430$		0			0
$431$		−	0.439608i		−	0.0211752i	−0.999944	−	0.0105876i	$-0.996630\pi$
$431$		−	0.439608i		−	0.0211752i	0.999944	−	0.0105876i	$-0.00337020\pi$
$432$		0			0
$433$	−2.37775			−0.114268			−0.0571338	−	0.998367i	$-0.518196\pi$
$433$	−2.37775			−0.114268			−0.0571338	+	0.998367i	$0.518196\pi$
$434$		0			0
$435$			16.6157i			0.796660i
$436$		0			0
$437$			32.4583i			1.55269i
$438$		0			0
$439$	12.5617			0.599538			0.299769	−	0.954012i	$-0.403090\pi$
$439$	12.5617			0.599538			0.299769	+	0.954012i	$0.403090\pi$
$440$		0			0
$441$	0.305195			0.0145331
$442$		0			0
$443$	−38.6964			−1.83852			−0.919261	−	0.393648i	$-0.871213\pi$
$443$	−38.6964			−1.83852			−0.919261	+	0.393648i	$0.871213\pi$
$444$		0			0
$445$	50.4373			2.39096
$446$		0			0
$447$		−	6.08221i		−	0.287678i
$448$		0			0
$449$		−	6.18606i		−	0.291938i	−0.989289	−	0.145969i	$-0.953370\pi$
$449$		−	6.18606i		−	0.291938i	0.989289	−	0.145969i	$-0.0466300\pi$
$450$		0			0
$451$	2.05391			0.0967146
$452$		0			0
$453$			2.42999i			0.114171i
$454$		0			0
$455$	−14.4591	−	1.37528i	−0.677851	−	0.0644739i
$456$		0			0
$457$		−	10.2781i		−	0.480787i	−0.970676	−	0.240393i	$-0.922724\pi$
$457$		−	10.2781i		−	0.480787i	0.970676	−	0.240393i	$-0.0772765\pi$
$458$		0			0
$459$	15.9421			0.744114
$460$		0			0
$461$		−	34.8204i		−	1.62175i	−0.585222	−	0.810873i	$-0.698993\pi$
$461$		−	34.8204i		−	1.62175i	0.585222	−	0.810873i	$-0.301007\pi$
$462$		0			0
$463$			21.4380i			0.996309i	0.867088	+	0.498154i	$0.165989\pi$
$463$			21.4380i			0.996309i	−0.867088	+	0.498154i	$0.834011\pi$
$464$		0			0
$465$	64.0011			2.96798
$466$		0			0
$467$	26.6020			1.23099			0.615496	−	0.788140i	$-0.288956\pi$
$467$	26.6020			1.23099			0.615496	+	0.788140i	$0.288956\pi$
$468$		0			0
$469$	−8.37934			−0.386922
$470$		0			0
$471$	20.3351			0.936991
$472$		0			0
$473$			3.54942i			0.163202i
$474$		0			0
$475$			59.1035i			2.71185i
$476$		0			0
$477$	−2.44077			−0.111755
$478$		0			0
$479$		−	22.7042i		−	1.03738i	−0.854962	−	0.518690i	$-0.826420\pi$
$479$		−	22.7042i		−	1.03738i	0.854962	−	0.518690i	$-0.173580\pi$
$480$		0			0
$481$	0.256904	−	2.70098i	0.0117138	−	0.123154i
$482$		0			0
$483$		−	10.1217i		−	0.460555i
$484$		0			0
$485$	49.1746			2.23290
$486$		0			0
$487$		−	17.9735i		−	0.814455i	−0.913327	−	0.407228i	$-0.866496\pi$
$487$		−	17.9735i		−	0.814455i	0.913327	−	0.407228i	$-0.133504\pi$
$488$		0			0
$489$		−	20.4661i		−	0.925509i
$490$		0			0
$491$	−21.9664			−0.991328			−0.495664	−	0.868514i	$-0.665075\pi$
$491$	−21.9664			−0.991328			−0.495664	+	0.868514i	$0.665075\pi$
$492$		0			0
$493$	7.38269			0.332500
$494$		0			0
$495$	−1.22942			−0.0552585
$496$		0			0
$497$	7.73832			0.347111
$498$		0			0
$499$			30.3035i			1.35657i	0.734798	+	0.678286i	$0.237277\pi$
$499$			30.3035i			1.35657i	−0.734798	+	0.678286i	$0.762723\pi$
$500$		0			0
$501$		−	5.55792i		−	0.248309i
$502$		0			0
$503$	−22.4016			−0.998839			−0.499419	−	0.866360i	$-0.666453\pi$
$503$	−22.4016			−0.998839			−0.499419	+	0.866360i	$0.666453\pi$
$504$		0			0
$505$			31.7367i			1.41227i
$506$		0			0
$507$	−20.9579	−	4.02323i	−0.930775	−	0.178678i
$508$		0			0
$509$		−	20.1063i		−	0.891198i	−0.895233	−	0.445599i	$-0.852991\pi$
$509$		−	20.1063i		−	0.891198i	0.895233	−	0.445599i	$-0.147009\pi$
$510$		0			0
$511$	−6.96682			−0.308194
$512$		0			0
$513$		−	28.5624i		−	1.26106i
$514$		0			0
$515$			43.0608i			1.89749i
$516$		0			0
$517$	10.3141			0.453615
$518$		0			0
$519$	−0.865025			−0.0379704
$520$		0			0
$521$	41.7507			1.82913			0.914565	−	0.404439i	$-0.132533\pi$
$521$	41.7507			1.82913			0.914565	+	0.404439i	$0.132533\pi$
$522$		0			0
$523$	−2.53254			−0.110740			−0.0553700	−	0.998466i	$-0.517634\pi$
$523$	−2.53254			−0.110740			−0.0553700	+	0.998466i	$0.517634\pi$
$524$		0			0
$525$		−	18.4307i		−	0.804382i
$526$		0			0
$527$		−	28.4370i		−	1.23874i
$528$		0			0
$529$	15.0174			0.652932
$530$		0			0
$531$			0.916992i			0.0397940i
$532$		0			0
$533$	0.701207	−	7.37219i	0.0303726	−	0.319325i
$534$		0			0
$535$			22.8935i			0.989773i
$536$		0			0
$537$	14.5133			0.626295
$538$		0			0
$539$		−	1.00000i		−	0.0430730i
$540$		0			0
$541$			37.6168i			1.61727i	0.588308	+	0.808637i	$0.299794\pi$
$541$			37.6168i			1.61727i	−0.588308	+	0.808637i	$0.700206\pi$
$542$		0			0
$543$	20.7129			0.888875
$544$		0			0
$545$	26.6113			1.13990
$546$		0			0
$547$	17.2805			0.738860			0.369430	−	0.929259i	$-0.379553\pi$
$547$	17.2805			0.738860			0.369430	+	0.929259i	$0.379553\pi$
$548$		0			0
$549$	−2.01785			−0.0861199
$550$		0			0
$551$		−	13.2271i		−	0.563493i
$552$		0			0
$553$			11.5285i			0.490242i
$554$		0			0
$555$	4.97615			0.211226
$556$		0			0
$557$			29.4108i			1.24618i	0.782152	+	0.623088i	$0.214122\pi$
$557$			29.4108i			1.24618i	−0.782152	+	0.623088i	$0.785878\pi$
$558$		0			0
$559$	12.7401	+	1.21178i	0.538849	+	0.0512527i
$560$		0			0
$561$		−	4.82335i		−	0.203642i
$562$		0			0
$563$	26.1535			1.10224			0.551119	−	0.834426i	$-0.314201\pi$
$563$	26.1535			1.10224			0.551119	+	0.834426i	$0.314201\pi$
$564$		0			0
$565$		−	72.1110i		−	3.03373i
$566$		0			0
$567$			7.99127i			0.335602i
$568$		0			0
$569$	23.8334			0.999149			0.499574	−	0.866271i	$-0.333490\pi$
$569$	23.8334			0.999149			0.499574	+	0.866271i	$0.333490\pi$
$570$		0			0
$571$	−27.8518			−1.16556			−0.582781	−	0.812629i	$-0.698036\pi$
$571$	−27.8518			−1.16556			−0.582781	+	0.812629i	$0.698036\pi$
$572$		0			0
$573$	15.2397			0.636647
$574$		0			0
$575$	69.2261			2.88693
$576$		0			0
$577$			4.16564i			0.173418i	0.996234	+	0.0867089i	$0.0276350\pi$
$577$			4.16564i			0.173418i	−0.996234	+	0.0867089i	$0.972365\pi$
$578$		0			0
$579$		−	2.62823i		−	0.109226i
$580$		0			0
$581$	−13.4835			−0.559388
$582$		0			0
$583$			7.99742i			0.331219i
$584$		0			0
$585$	−0.419727	+	4.41284i	−0.0173536	+	0.182448i
$586$		0			0
$587$		−	7.64129i		−	0.315390i	−0.987488	−	0.157695i	$-0.949594\pi$
$587$		−	7.64129i		−	0.315390i	0.987488	−	0.157695i	$-0.0504063\pi$
$588$		0			0
$589$	−50.9488			−2.09931
$590$		0			0
$591$			29.5011i			1.21351i
$592$		0			0
$593$			4.71886i			0.193780i	0.995295	+	0.0968902i	$0.0308896\pi$
$593$			4.71886i			0.193780i	−0.995295	+	0.0968902i	$0.969110\pi$
$594$		0			0
$595$	−11.8361			−0.485233
$596$		0			0
$597$	−13.3536			−0.546527
$598$		0			0
$599$	−12.8076			−0.523303			−0.261651	−	0.965162i	$-0.584267\pi$
$599$	−12.8076			−0.523303			−0.261651	+	0.965162i	$0.584267\pi$
$600$		0			0
$601$	41.9380			1.71069			0.855343	−	0.518063i	$-0.173347\pi$
$601$	41.9380			1.71069			0.855343	+	0.518063i	$0.173347\pi$
$602$		0			0
$603$			2.55733i			0.104143i
$604$		0			0
$605$			4.02832i			0.163775i
$606$		0			0
$607$	−19.2450			−0.781129			−0.390564	−	0.920576i	$-0.627720\pi$
$607$	−19.2450			−0.781129			−0.390564	+	0.920576i	$0.627720\pi$
$608$		0			0
$609$			4.12471i			0.167142i
$610$		0			0
$611$	3.52126	−	37.0211i	0.142455	−	1.49771i
$612$		0			0
$613$			4.20708i			0.169922i	0.996384	+	0.0849612i	$0.0270766\pi$
$613$			4.20708i			0.169922i	−0.996384	+	0.0849612i	$0.972923\pi$
$614$		0			0
$615$	13.5821			0.547685
$616$		0			0
$617$		−	2.71815i		−	0.109428i	−0.998502	−	0.0547142i	$-0.982575\pi$
$617$		−	2.71815i		−	0.109428i	0.998502	−	0.0547142i	$-0.0174248\pi$
$618$		0			0
$619$		−	21.8252i		−	0.877231i	−0.898675	−	0.438615i	$-0.855469\pi$
$619$		−	21.8252i		−	0.877231i	0.898675	−	0.438615i	$-0.144531\pi$
$620$		0			0
$621$	−33.4543			−1.34248
$622$		0			0
$623$	12.5207			0.501630
$624$		0			0
$625$	44.9171			1.79668
$626$		0			0
$627$	−8.64168			−0.345116
$628$		0			0
$629$		−	2.21101i		−	0.0881588i
$630$		0			0
$631$		−	38.0053i		−	1.51297i	−0.654013	−	0.756484i	$-0.726916\pi$
$631$		−	38.0053i		−	1.51297i	0.654013	−	0.756484i	$-0.273084\pi$
$632$		0			0
$633$	−35.6078			−1.41528
$634$		0			0
$635$		−	6.31338i		−	0.250539i
$636$		0			0
$637$	−3.58935	−	0.341402i	−0.142215	−	0.0135268i
$638$		0			0
$639$		−	2.36170i		−	0.0934273i
$640$		0			0
$641$	−1.15512			−0.0456246			−0.0228123	−	0.999740i	$-0.507262\pi$
$641$	−1.15512			−0.0456246			−0.0228123	+	0.999740i	$0.507262\pi$
$642$		0			0
$643$			15.6633i			0.617699i	0.951111	+	0.308849i	$0.0999439\pi$
$643$			15.6633i			0.617699i	−0.951111	+	0.308849i	$0.900056\pi$
$644$		0			0
$645$			23.4717i			0.924199i
$646$		0			0
$647$	12.0538			0.473884			0.236942	−	0.971524i	$-0.423855\pi$
$647$	12.0538			0.473884			0.236942	+	0.971524i	$0.423855\pi$
$648$		0			0
$649$	3.00461			0.117941
$650$		0			0
$651$	15.8878			0.622691
$652$		0			0
$653$	−15.7742			−0.617291			−0.308646	−	0.951177i	$-0.599876\pi$
$653$	−15.7742			−0.617291			−0.308646	+	0.951177i	$0.599876\pi$
$654$		0			0
$655$		−	76.6458i		−	2.99480i
$656$		0			0
$657$			2.12624i			0.0829525i
$658$		0			0
$659$	14.1915			0.552821			0.276411	−	0.961040i	$-0.410855\pi$
$659$	14.1915			0.552821			0.276411	+	0.961040i	$0.410855\pi$
$660$		0			0
$661$		−	31.8285i		−	1.23798i	−0.785397	−	0.618992i	$-0.787541\pi$
$661$		−	31.8285i		−	1.23798i	0.785397	−	0.618992i	$-0.212459\pi$
$662$		0			0
$663$	−17.3127	−	1.64670i	−0.672369	−	0.0639525i
$664$		0			0
$665$			21.2060i			0.822333i
$666$		0			0
$667$	−15.4925			−0.599871
$668$		0			0
$669$		−	37.8019i		−	1.46151i
$670$		0			0
$671$			6.61169i			0.255241i
$672$		0			0
$673$	40.2144			1.55015			0.775076	−	0.631868i	$-0.217712\pi$
$673$	40.2144			1.55015			0.775076	+	0.631868i	$0.217712\pi$
$674$		0			0
$675$	−60.9171			−2.34470
$676$		0			0
$677$	−30.5715			−1.17496			−0.587480	−	0.809239i	$-0.699880\pi$
$677$	−30.5715			−1.17496			−0.587480	+	0.809239i	$0.699880\pi$
$678$		0			0
$679$	12.2072			0.468470
$680$		0			0
$681$			1.90988i			0.0731869i
$682$		0			0
$683$		−	1.56824i		−	0.0600071i	−0.999550	−	0.0300036i	$-0.990448\pi$
$683$		−	1.56824i		−	0.0600071i	0.999550	−	0.0300036i	$-0.00955186\pi$
$684$		0			0
$685$	37.7948			1.44407
$686$		0			0
$687$		−	1.49930i		−	0.0572020i
$688$		0			0
$689$	28.7056	+	2.73033i	1.09359	+	0.104017i
$690$		0			0
$691$		−	25.0856i		−	0.954302i	−0.878821	−	0.477151i	$-0.841669\pi$
$691$		−	25.0856i		−	0.954302i	0.878821	−	0.477151i	$-0.158331\pi$
$692$		0			0
$693$	−0.305195			−0.0115934
$694$		0			0
$695$			80.3587i			3.04818i
$696$		0			0
$697$		−	6.03483i		−	0.228586i
$698$		0			0
$699$	10.7022			0.404795
$700$		0			0
$701$	38.1712			1.44171			0.720854	−	0.693087i	$-0.243750\pi$
$701$	38.1712			1.44171			0.720854	+	0.693087i	$0.243750\pi$
$702$		0			0
$703$	−3.96133			−0.149404
$704$		0			0
$705$	68.2057			2.56878
$706$		0			0
$707$			7.87840i			0.296298i
$708$		0			0
$709$			4.27537i			0.160565i	0.996772	+	0.0802824i	$0.0255822\pi$
$709$			4.27537i			0.160565i	−0.996772	+	0.0802824i	$0.974418\pi$
$710$		0			0
$711$	3.51844			0.131952
$712$		0			0
$713$			59.6748i			2.23484i
$714$		0			0
$715$	14.4591	+	1.37528i	0.540738	+	0.0514324i
$716$		0			0
$717$			46.3371i			1.73049i
$718$		0			0
$719$	3.95838			0.147623			0.0738113	−	0.997272i	$-0.476484\pi$
$719$	3.95838			0.147623			0.0738113	+	0.997272i	$0.476484\pi$
$720$		0			0
$721$			10.6895i			0.398098i
$722$		0			0
$723$		−	10.0349i		−	0.373201i
$724$		0			0
$725$	−28.2103			−1.04770
$726$		0			0
$727$	−46.2417			−1.71501			−0.857505	−	0.514475i	$-0.827987\pi$
$727$	−46.2417			−1.71501			−0.857505	+	0.514475i	$0.827987\pi$
$728$		0			0
$729$	29.1595			1.07998
$730$		0			0
$731$	10.4290			0.385730
$732$		0			0
$733$		−	10.5451i		−	0.389491i	−0.980854	−	0.194746i	$-0.937612\pi$
$733$		−	10.5451i		−	0.389491i	0.980854	−	0.194746i	$-0.0623881\pi$
$734$		0			0
$735$		−	6.61284i		−	0.243918i
$736$		0			0
$737$	8.37934			0.308657
$738$		0			0
$739$			9.83102i			0.361640i	0.983516	+	0.180820i	$0.0578751\pi$
$739$			9.83102i			0.361640i	−0.983516	+	0.180820i	$0.942125\pi$
$740$		0			0
$741$	−2.95028	+	31.0180i	−0.108381	+	1.13948i
$742$		0			0
$743$			32.1412i			1.17915i	0.807715	+	0.589573i	$0.200704\pi$
$743$			32.1412i			1.17915i	−0.807715	+	0.589573i	$0.799296\pi$
$744$		0			0
$745$	14.9252			0.546819
$746$		0			0
$747$			4.11509i			0.150563i
$748$		0			0
$749$			5.68314i			0.207657i
$750$		0			0
$751$	35.6313			1.30020			0.650102	−	0.759847i	$-0.274726\pi$
$751$	35.6313			1.30020			0.650102	+	0.759847i	$0.274726\pi$
$752$		0			0
$753$	2.63401			0.0959886
$754$		0			0
$755$	−5.96299			−0.217016
$756$		0			0
$757$	15.1061			0.549042			0.274521	−	0.961581i	$-0.411481\pi$
$757$	15.1061			0.549042			0.274521	+	0.961581i	$0.411481\pi$
$758$		0			0
$759$			10.1217i			0.367396i
$760$		0			0
$761$			35.0444i			1.27036i	0.772365	+	0.635179i	$0.219074\pi$
$761$			35.0444i			1.27036i	−0.772365	+	0.635179i	$0.780926\pi$
$762$		0			0
$763$	6.60605			0.239155
$764$		0			0
$765$			3.61232i			0.130604i
$766$		0			0
$767$	1.02578	−	10.7846i	0.0370387	−	0.389409i
$768$		0			0
$769$			32.3235i			1.16562i	0.812610	+	0.582808i	$0.198046\pi$
$769$			32.3235i			1.16562i	−0.812610	+	0.582808i	$0.801954\pi$
$770$		0			0
$771$	−12.3568			−0.445021
$772$		0			0
$773$		−	24.4241i		−	0.878475i	−0.898371	−	0.439238i	$-0.855249\pi$
$773$		−	24.4241i		−	0.878475i	0.898371	−	0.439238i	$-0.144751\pi$
$774$		0			0
$775$			108.662i			3.90325i
$776$		0			0
$777$	1.23529			0.0443158
$778$		0			0
$779$	−10.8122			−0.387388
$780$		0			0
$781$	−7.73832			−0.276899
$782$		0			0
$783$	13.6330			0.487202
$784$		0			0
$785$			49.9006i			1.78103i
$786$		0			0
$787$			22.8365i			0.814032i	0.913421	+	0.407016i	$0.133431\pi$
$787$			22.8365i			0.814032i	−0.913421	+	0.407016i	$0.866569\pi$
$788$		0			0
$789$	−34.4469			−1.22634
$790$		0			0
$791$		−	17.9010i		−	0.636487i
$792$		0			0
$793$	23.7317	+	2.25724i	0.842736	+	0.0801569i
$794$		0			0
$795$			52.8857i			1.87566i
$796$		0			0
$797$	3.92551			0.139049			0.0695244	−	0.997580i	$-0.477852\pi$
$797$	3.92551			0.139049			0.0695244	+	0.997580i	$0.477852\pi$
$798$		0			0
$799$		−	30.3052i		−	1.07212i
$800$		0			0
$801$		−	3.82125i		−	0.135017i
$802$		0			0
$803$	6.96682			0.245854
$804$		0			0
$805$	24.8379			0.875422
$806$		0			0
$807$	20.9222			0.736498
$808$		0			0
$809$	28.1282			0.988936			0.494468	−	0.869196i	$-0.335363\pi$
$809$	28.1282			0.988936			0.494468	+	0.869196i	$0.335363\pi$
$810$		0			0
$811$		−	1.42614i		−	0.0500785i	−0.999686	−	0.0250392i	$-0.992029\pi$
$811$		−	1.42614i		−	0.0500785i	0.999686	−	0.0250392i	$-0.00797107\pi$
$812$		0			0
$813$		−	27.3966i		−	0.960842i
$814$		0			0
$815$	50.2222			1.75921
$816$		0			0
$817$		−	18.6850i		−	0.653704i
$818$		0			0
$819$	−0.104194	+	1.09545i	−0.00364084	+	0.0382782i
$820$		0			0
$821$		−	50.3377i		−	1.75680i	−0.477929	−	0.878398i	$-0.658613\pi$
$821$		−	50.3377i		−	1.75680i	0.477929	−	0.878398i	$-0.341387\pi$
$822$		0			0
$823$	14.1494			0.493217			0.246608	−	0.969115i	$-0.420684\pi$
$823$	14.1494			0.493217			0.246608	+	0.969115i	$0.420684\pi$
$824$		0			0
$825$			18.4307i			0.641675i
$826$		0			0
$827$			55.1288i			1.91702i	0.285063	+	0.958509i	$0.407986\pi$
$827$			55.1288i			1.91702i	−0.285063	+	0.958509i	$0.592014\pi$
$828$		0			0
$829$	44.6656			1.55130			0.775651	−	0.631162i	$-0.217422\pi$
$829$	44.6656			1.55130			0.775651	+	0.631162i	$0.217422\pi$
$830$		0			0
$831$	50.9488			1.76739
$832$		0			0
$833$	−2.93822			−0.101803
$834$		0			0
$835$	13.6387			0.471986
$836$		0			0
$837$		−	52.5122i		−	1.81509i
$838$		0			0
$839$		−	34.1203i		−	1.17796i	−0.808147	−	0.588981i	$-0.799529\pi$
$839$		−	34.1203i		−	1.17796i	0.808147	−	0.588981i	$-0.200471\pi$
$840$		0			0
$841$	−22.6867			−0.782299
$842$		0			0
$843$			49.1196i			1.69177i
$844$		0			0
$845$	9.87269	−	51.4291i	0.339631	−	1.76922i
$846$		0			0
$847$			1.00000i			0.0343604i
$848$		0			0
$849$	4.88255			0.167569
$850$		0			0
$851$			4.63978i			0.159050i
$852$		0			0
$853$		−	12.9940i		−	0.444907i	−0.974943	−	0.222453i	$-0.928594\pi$
$853$		−	12.9940i		−	0.444907i	0.974943	−	0.222453i	$-0.0714065\pi$
$854$		0			0
$855$	6.47197			0.221337
$856$		0			0
$857$	−4.38591			−0.149820			−0.0749099	−	0.997190i	$-0.523867\pi$
$857$	−4.38591			−0.149820			−0.0749099	+	0.997190i	$0.523867\pi$
$858$		0			0
$859$	29.7069			1.01359			0.506793	−	0.862068i	$-0.330831\pi$
$859$	29.7069			1.01359			0.506793	+	0.862068i	$0.330831\pi$
$860$		0			0
$861$	3.37166			0.114906
$862$		0			0
$863$			2.65575i			0.0904029i	0.998978	+	0.0452015i	$0.0143930\pi$
$863$			2.65575i			0.0904029i	−0.998978	+	0.0452015i	$0.985607\pi$
$864$		0			0
$865$		−	2.12270i		−	0.0721741i
$866$		0			0
$867$	13.7349			0.466461
$868$		0			0
$869$		−	11.5285i		−	0.391078i
$870$		0			0
$871$	2.86072	−	30.0764i	0.0969318	−	1.01910i
$872$		0			0
$873$		−	3.72558i		−	0.126092i
$874$		0			0
$875$	25.0859			0.848057
$876$		0			0
$877$		−	50.9455i		−	1.72031i	−0.510035	−	0.860154i	$-0.670367\pi$
$877$		−	50.9455i		−	1.72031i	0.510035	−	0.860154i	$-0.329633\pi$
$878$		0			0
$879$		−	34.2361i		−	1.15475i
$880$		0			0
$881$	−22.9597			−0.773531			−0.386765	−	0.922178i	$-0.626408\pi$
$881$	−22.9597			−0.773531			−0.386765	+	0.922178i	$0.626408\pi$
$882$		0			0
$883$	−24.3442			−0.819247			−0.409624	−	0.912255i	$-0.634340\pi$
$883$	−24.3442			−0.819247			−0.409624	+	0.912255i	$0.634340\pi$
$884$		0			0
$885$	19.8690			0.667888
$886$		0			0
$887$	31.8587			1.06971			0.534855	−	0.844944i	$-0.320366\pi$
$887$	31.8587			1.06971			0.534855	+	0.844944i	$0.320366\pi$
$888$		0			0
$889$		−	1.56725i		−	0.0525638i
$890$		0			0
$891$		−	7.99127i		−	0.267718i
$892$		0			0
$893$	−54.2960			−1.81694
$894$		0			0
$895$			35.6145i			1.19046i
$896$		0			0
$897$	36.3305	+	3.45558i	1.21304	+	0.115378i
$898$		0			0
$899$		−	24.3181i		−	0.811053i
$900$		0			0
$901$	23.4982			0.782839
$902$		0			0
$903$			5.82668i			0.193900i
$904$		0			0
$905$			50.8277i			1.68957i
$906$		0			0
$907$	15.8384			0.525904			0.262952	−	0.964809i	$-0.415304\pi$
$907$	15.8384			0.525904			0.262952	+	0.964809i	$0.415304\pi$
$908$		0			0
$909$	2.40445			0.0797506
$910$		0			0
$911$	23.5367			0.779806			0.389903	−	0.920856i	$-0.372509\pi$
$911$	23.5367			0.779806			0.389903	+	0.920856i	$0.372509\pi$
$912$		0			0
$913$	13.4835			0.446237
$914$		0			0
$915$			43.7220i			1.44540i
$916$		0			0
$917$		−	19.0267i		−	0.628318i
$918$		0			0
$919$	−18.5021			−0.610326			−0.305163	−	0.952300i	$-0.598711\pi$
$919$	−18.5021			−0.610326			−0.305163	+	0.952300i	$0.598711\pi$
$920$		0			0
$921$		−	31.5103i		−	1.03830i
$922$		0			0
$923$	−2.64188	+	27.7756i	−0.0869584	+	0.914244i
$924$		0			0
$925$			8.44859i			0.277788i
$926$		0			0
$927$	3.26239			0.107151
$928$		0			0
$929$			14.4522i			0.474161i	0.971490	+	0.237080i	$0.0761905\pi$
$929$			14.4522i			0.474161i	−0.971490	+	0.237080i	$0.923810\pi$
$930$		0			0
$931$			5.26423i			0.172528i
$932$		0			0
$933$	−44.6526			−1.46186
$934$		0			0
$935$	11.8361			0.387082
$936$		0			0
$937$	−4.34329			−0.141889			−0.0709445	−	0.997480i	$-0.522601\pi$
$937$	−4.34329			−0.141889			−0.0709445	+	0.997480i	$0.522601\pi$
$938$		0			0
$939$	−1.32487			−0.0432355
$940$		0			0
$941$		−	36.0074i		−	1.17381i	−0.809657	−	0.586904i	$-0.800347\pi$
$941$		−	36.0074i		−	1.17381i	0.809657	−	0.586904i	$-0.199653\pi$
$942$		0			0
$943$			12.6640i			0.412397i
$944$		0			0
$945$	−21.8567			−0.710999
$946$		0			0
$947$			12.1759i			0.395665i	0.980236	+	0.197833i	$0.0633902\pi$
$947$			12.1759i			0.395665i	−0.980236	+	0.197833i	$0.936610\pi$
$948$		0			0
$949$	2.37848	−	25.0064i	0.0772088	−	0.811740i
$950$		0			0
$951$			18.6928i			0.606155i
$952$		0			0
$953$	−50.2748			−1.62856			−0.814281	−	0.580471i	$-0.802868\pi$
$953$	−50.2748			−1.62856			−0.814281	+	0.580471i	$0.802868\pi$
$954$		0			0
$955$			37.3970i			1.21014i
$956$		0			0
$957$		−	4.12471i		−	0.133333i
$958$		0			0
$959$	9.38227			0.302969
$960$		0			0
$961$	−62.6696			−2.02160
$962$		0			0
$963$	1.73447			0.0558924
$964$		0			0
$965$	6.44948			0.207616
$966$		0			0
$967$		−	0.959615i		−	0.0308591i	−0.999881	−	0.0154296i	$-0.995088\pi$
$967$		−	0.959615i		−	0.0308591i	0.999881	−	0.0154296i	$-0.00491158\pi$
$968$		0			0
$969$			25.3912i			0.815683i
$970$		0			0
$971$	−48.8700			−1.56831			−0.784156	−	0.620564i	$-0.786904\pi$
$971$	−48.8700			−1.56831			−0.784156	+	0.620564i	$0.786904\pi$
$972$		0			0
$973$			19.9484i			0.639517i
$974$		0			0
$975$	66.1543	+	6.29227i	2.11863	+	0.201514i
$976$		0			0
$977$		−	9.10587i		−	0.291323i	−0.989335	−	0.145661i	$-0.953469\pi$
$977$		−	9.10587i		−	0.291323i	0.989335	−	0.145661i	$-0.0465310\pi$
$978$		0			0
$979$	−12.5207			−0.400162
$980$		0			0
$981$		−	2.01613i		−	0.0643702i
$982$		0			0
$983$		−	29.1578i		−	0.929990i	−0.885313	−	0.464995i	$-0.846056\pi$
$983$		−	29.1578i		−	0.929990i	0.885313	−	0.464995i	$-0.153944\pi$
$984$		0			0
$985$	−72.3934			−2.30665
$986$		0			0
$987$	16.9315			0.538937
$988$		0			0
$989$	−21.8851			−0.695906
$990$		0			0
$991$	−45.7293			−1.45264			−0.726319	−	0.687358i	$-0.758770\pi$
$991$	−45.7293			−1.45264			−0.726319	+	0.687358i	$0.758770\pi$
$992$		0			0
$993$		−	56.1670i		−	1.78241i
$994$		0			0
$995$		−	32.7687i		−	1.03884i
$996$		0			0
$997$	16.3468			0.517709			0.258854	−	0.965916i	$-0.416655\pi$
$997$	16.3468			0.517709			0.258854	+	0.965916i	$0.416655\pi$
$998$		0			0
$999$		−	4.08288i		−	0.129177i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.m.c.2157.9	✓	36
13.12	even	2	inner	4004.2.m.c.2157.10	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.m.c.2157.9	✓	36	1.1	even	1	trivial
4004.2.m.c.2157.10	yes	36	13.12	even	2	inner

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)

\(f(q)\)	\(=\)	\(q-1.64159 q^{3} -4.02832i q^{5} -1.00000i q^{7} -0.305195 q^{9} +O(q^{10})\)	\(q-1.64159 q^{3} -4.02832i q^{5} -1.00000i q^{7} -0.305195 q^{9} +1.00000i q^{11} +(3.58935 + 0.341402i) q^{13} +6.61284i q^{15} +2.93822 q^{17} -5.26423i q^{19} +1.64159i q^{21} -6.16583 q^{23} -11.2274 q^{25} +5.42576 q^{27} +2.51264 q^{29} -9.67831i q^{31} -1.64159i q^{33} -4.02832 q^{35} -0.752499i q^{37} +(-5.89223 - 0.560440i) q^{39} -2.05391i q^{41} +3.54942 q^{43} +1.22942i q^{45} -10.3141i q^{47} -1.00000 q^{49} -4.82335 q^{51} +7.99742 q^{53} +4.02832 q^{55} +8.64168i q^{57} -3.00461i q^{59} +6.61169 q^{61} +0.305195i q^{63} +(1.37528 - 14.4591i) q^{65} -8.37934i q^{67} +10.1217 q^{69} +7.73832i q^{71} -6.96682i q^{73} +18.4307 q^{75} +1.00000 q^{77} -11.5285 q^{79} -7.99127 q^{81} -13.4835i q^{83} -11.8361i q^{85} -4.12471 q^{87} +12.5207i q^{89} +(0.341402 - 3.58935i) q^{91} +15.8878i q^{93} -21.2060 q^{95} +12.2072i q^{97} -0.305195i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 4 q^{3} + 40 q^{9}+O(q^{10}) \) 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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