LMFDB - Embedded newform 3872.2.c.i.1937.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3872,2,Mod(1937,3872)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3872, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3872.1937");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3872 = 2^{5} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3872.c (of order $2$, degree $1$, not 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:	$30.9180756626$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{18} - 2 x^{16} - 2 x^{15} - 4 x^{14} - 4 x^{13} + 12 x^{12} + 16 x^{11} + 32 x^{9} + \cdots + 1024 $ x^20 - x^18 - 2x^16 - 2x^15 - 4x^14 - 4x^13 + 12x^12 + 16x^11 + 32x^9 + 48x^8 - 32x^7 - 64x^6 - 64x^5 - 128x^4 - 256*x^2 + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 88)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1937.15
Root			$-1.30820 + 0.537231i$ of defining polynomial
Character	$\chi$	$=$	3872.1937
Dual form			3872.2.c.i.1937.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.28633i q^{3} -1.58193i q^{5} -1.86825 q^{7} +1.34535 q^{9} -0.322072i q^{13} +2.03488 q^{15} -4.79250 q^{17} -3.38629i q^{19} -2.40318i q^{21} -3.08044 q^{23} +2.49750 q^{25} +5.58956i q^{27} +10.4131i q^{29} +2.82067 q^{31} +2.95543i q^{35} +3.57225i q^{37} +0.414291 q^{39} +4.49931 q^{41} -3.61796i q^{43} -2.12826i q^{45} +12.7265 q^{47} -3.50965 q^{49} -6.16474i q^{51} -5.65725i q^{53} +4.35588 q^{57} +5.38428i q^{59} -10.9289i q^{61} -2.51346 q^{63} -0.509496 q^{65} -10.9885i q^{67} -3.96246i q^{69} +11.1359 q^{71} +12.6094 q^{73} +3.21261i q^{75} +5.97994 q^{79} -3.15396 q^{81} -0.139643i q^{83} +7.58140i q^{85} -13.3946 q^{87} +4.28990 q^{89} +0.601711i q^{91} +3.62831i q^{93} -5.35686 q^{95} +12.3942 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 10 q^{7} - 10 q^{9} + 4 q^{15} - 2 q^{17} + 4 q^{23} - 2 q^{25} - 2 q^{31} + 28 q^{39} + 2 q^{41} - 2 q^{47} - 2 q^{49} - 22 q^{57} - 30 q^{63} + 18 q^{65} + 34 q^{71} - 2 q^{73} - 58 q^{79} - 12 q^{81}+ \cdots + 10 q^{97}+O(q^{100}) $ 20 * q + 10 * q^7 - 10 * q^9 + 4 * q^15 - 2 * q^17 + 4 * q^23 - 2 * q^25 - 2 * q^31 + 28 * q^39 + 2 * q^41 - 2 * q^47 - 2 * q^49 - 22 * q^57 - 30 * q^63 + 18 * q^65 + 34 * q^71 - 2 * q^73 - 58 * q^79 - 12 * q^81 + 34 * q^87 - 8 * q^89 - 92 * q^95 + 10 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3872\mathbb{Z}\right)^\times$.

$n$	$485$	$1695$	$2785$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	1.28633i	0.742663i	0.928500	+	0.371332i	$0.121099\pi$
$3$	1.28633i	0.742663i	−0.928500	+	0.371332i	$0.878901\pi$
$4$	0	0
$5$	− 1.58193i	− 0.707460i	−0.935348	−	0.353730i	$-0.884913\pi$
$5$	− 1.58193i	− 0.707460i	0.935348	−	0.353730i	$-0.115087\pi$
$6$	0	0
$7$	−1.86825	−0.706131	−0.353066	−	0.935599i	$-0.614861\pi$
$7$	−1.86825	−0.706131	−0.353066	+	0.935599i	$0.614861\pi$
$8$	0	0
$9$	1.34535	0.448452
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	− 0.322072i	− 0.0893268i	−0.999002	−	0.0446634i	$-0.985778\pi$
$13$	− 0.322072i	− 0.0893268i	0.999002	−	0.0446634i	$-0.0142215\pi$
$14$	0	0
$15$	2.03488	0.525404
$16$	0	0
$17$	−4.79250	−1.16235	−0.581177	−	0.813778i	$-0.697407\pi$
$17$	−4.79250	−1.16235	−0.581177	+	0.813778i	$0.697407\pi$
$18$	0	0
$19$	− 3.38629i	− 0.776867i	−0.921477	−	0.388434i	$-0.873016\pi$
$19$	− 3.38629i	− 0.776867i	0.921477	−	0.388434i	$-0.126984\pi$
$20$	0	0
$21$	− 2.40318i	− 0.524418i
$22$	0	0
$23$	−3.08044	−0.642316	−0.321158	−	0.947026i	$-0.604072\pi$
$23$	−3.08044	−0.642316	−0.321158	+	0.947026i	$0.604072\pi$
$24$	0	0
$25$	2.49750	0.499500
$26$	0	0
$27$	5.58956i	1.07571i
$28$	0	0
$29$	10.4131i	1.93366i	0.255422	+	0.966830i	$0.417785\pi$
$29$	10.4131i	1.93366i	−0.255422	+	0.966830i	$0.582215\pi$
$30$	0	0
$31$	2.82067	0.506607	0.253303	−	0.967387i	$-0.418483\pi$
$31$	2.82067	0.506607	0.253303	+	0.967387i	$0.418483\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.95543i	0.499560i
$36$	0	0
$37$	3.57225i	0.587274i	0.955917	+	0.293637i	$0.0948657\pi$
$37$	3.57225i	0.587274i	−0.955917	+	0.293637i	$0.905134\pi$
$38$	0	0
$39$	0.414291	0.0663397
$40$	0	0
$41$	4.49931	0.702674	0.351337	−	0.936249i	$-0.385727\pi$
$41$	4.49931	0.702674	0.351337	+	0.936249i	$0.385727\pi$
$42$	0	0
$43$	− 3.61796i	− 0.551734i	−0.961196	−	0.275867i	$-0.911035\pi$
$43$	− 3.61796i	− 0.551734i	0.961196	−	0.275867i	$-0.0889649\pi$
$44$	0	0
$45$	− 2.12826i	− 0.317262i
$46$	0	0
$47$	12.7265	1.85635	0.928174	−	0.372147i	$-0.121378\pi$
$47$	12.7265	1.85635	0.928174	+	0.372147i	$0.121378\pi$
$48$	0	0
$49$	−3.50965	−0.501378
$50$	0	0
$51$	− 6.16474i	− 0.863237i
$52$	0	0
$53$	− 5.65725i	− 0.777083i	−0.921431	−	0.388541i	$-0.872979\pi$
$53$	− 5.65725i	− 0.777083i	0.921431	−	0.388541i	$-0.127021\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.35588	0.576951
$58$	0	0
$59$	5.38428i	0.700973i	0.936568	+	0.350487i	$0.113984\pi$
$59$	5.38428i	0.700973i	−0.936568	+	0.350487i	$0.886016\pi$
$60$	0	0
$61$	− 10.9289i	− 1.39931i	−0.714482	−	0.699654i	$-0.753338\pi$
$61$	− 10.9289i	− 1.39931i	0.714482	−	0.699654i	$-0.246662\pi$
$62$	0	0
$63$	−2.51346	−0.316666
$64$	0	0
$65$	−0.509496	−0.0631951
$66$	0	0
$67$	− 10.9885i	− 1.34246i	−0.741249	−	0.671230i	$-0.765766\pi$
$67$	− 10.9885i	− 1.34246i	0.741249	−	0.671230i	$-0.234234\pi$
$68$	0	0
$69$	− 3.96246i	− 0.477024i
$70$	0	0
$71$	11.1359	1.32159	0.660795	−	0.750566i	$-0.270219\pi$
$71$	11.1359	1.32159	0.660795	+	0.750566i	$0.270219\pi$
$72$	0	0
$73$	12.6094	1.47582	0.737912	−	0.674897i	$-0.235812\pi$
$73$	12.6094	1.47582	0.737912	+	0.674897i	$0.235812\pi$
$74$	0	0
$75$	3.21261i	0.370961i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	5.97994	0.672796	0.336398	−	0.941720i	$-0.390791\pi$
$79$	5.97994	0.672796	0.336398	+	0.941720i	$0.390791\pi$
$80$	0	0
$81$	−3.15396	−0.350440
$82$	0	0
$83$	− 0.139643i	− 0.0153279i	−0.999971	−	0.00766393i	$-0.997560\pi$
$83$	− 0.139643i	− 0.0153279i	0.999971	−	0.00766393i	$-0.00243953\pi$
$84$	0	0
$85$	7.58140i	0.822318i
$86$	0	0
$87$	−13.3946	−1.43606
$88$	0	0
$89$	4.28990	0.454729	0.227364	−	0.973810i	$-0.426989\pi$
$89$	4.28990	0.454729	0.227364	+	0.973810i	$0.426989\pi$
$90$	0	0
$91$	0.601711i	0.0630765i
$92$	0	0
$93$	3.62831i	0.376238i
$94$	0	0
$95$	−5.35686	−0.549602
$96$	0	0
$97$	12.3942	1.25844	0.629218	−	0.777229i	$-0.283375\pi$
$97$	12.3942	1.25844	0.629218	+	0.777229i	$0.283375\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	15.3627i	1.52865i	0.644831	+	0.764325i	$0.276928\pi$
$101$	15.3627i	1.52865i	−0.644831	+	0.764325i	$0.723072\pi$
$102$	0	0
$103$	−8.17691	−0.805695	−0.402848	−	0.915267i	$-0.631980\pi$
$103$	−8.17691	−0.805695	−0.402848	+	0.915267i	$0.631980\pi$
$104$	0	0
$105$	−3.80166	−0.371005
$106$	0	0
$107$	− 5.03234i	− 0.486494i	−0.969964	−	0.243247i	$-0.921787\pi$
$107$	− 5.03234i	− 0.486494i	0.969964	−	0.243247i	$-0.0782126\pi$
$108$	0	0
$109$	10.1839i	0.975439i	0.873000	+	0.487720i	$0.162171\pi$
$109$	10.1839i	0.975439i	−0.873000	+	0.487720i	$0.837829\pi$
$110$	0	0
$111$	−4.59509	−0.436147
$112$	0	0
$113$	8.05191	0.757460	0.378730	−	0.925507i	$-0.376361\pi$
$113$	8.05191	0.757460	0.378730	+	0.925507i	$0.376361\pi$
$114$	0	0
$115$	4.87303i	0.454413i
$116$	0	0
$117$	− 0.433302i	− 0.0400588i
$118$	0	0
$119$	8.95359	0.820774
$120$	0	0
$121$	0	0
$122$	0	0
$123$	5.78760i	0.521850i
$124$	0	0
$125$	− 11.8605i	− 1.06084i
$126$	0	0
$127$	−13.0462	−1.15766	−0.578830	−	0.815448i	$-0.696490\pi$
$127$	−13.0462	−1.15766	−0.578830	+	0.815448i	$0.696490\pi$
$128$	0	0
$129$	4.65389	0.409752
$130$	0	0
$131$	− 11.5519i	− 1.00930i	−0.863325	−	0.504648i	$-0.831622\pi$
$131$	− 11.5519i	− 1.00930i	0.863325	−	0.504648i	$-0.168378\pi$
$132$	0	0
$133$	6.32642i	0.548570i
$134$	0	0
$135$	8.84229	0.761023
$136$	0	0
$137$	18.0192	1.53948	0.769742	−	0.638355i	$-0.220385\pi$
$137$	18.0192	1.53948	0.769742	+	0.638355i	$0.220385\pi$
$138$	0	0
$139$	6.70490i	0.568702i	0.958720	+	0.284351i	$0.0917781\pi$
$139$	6.70490i	0.568702i	−0.958720	+	0.284351i	$0.908222\pi$
$140$	0	0
$141$	16.3705i	1.37864i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	16.4727	1.36799
$146$	0	0
$147$	− 4.51457i	− 0.372355i
$148$	0	0
$149$	4.00757i	0.328313i	0.986434	+	0.164156i	$0.0524902\pi$
$149$	4.00757i	0.328313i	−0.986434	+	0.164156i	$0.947510\pi$
$150$	0	0
$151$	−2.33413	−0.189948	−0.0949742	−	0.995480i	$-0.530277\pi$
$151$	−2.33413	−0.189948	−0.0949742	+	0.995480i	$0.530277\pi$
$152$	0	0
$153$	−6.44762	−0.521259
$154$	0	0
$155$	− 4.46210i	− 0.358404i
$156$	0	0
$157$	7.94584i	0.634147i	0.948401	+	0.317073i	$0.102700\pi$
$157$	7.94584i	0.634147i	−0.948401	+	0.317073i	$0.897300\pi$
$158$	0	0
$159$	7.27709	0.577111
$160$	0	0
$161$	5.75502	0.453559
$162$	0	0
$163$	− 5.61529i	− 0.439823i	−0.975520	−	0.219912i	$-0.929423\pi$
$163$	− 5.61529i	− 0.439823i	0.975520	−	0.219912i	$-0.0705769\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	18.0271	1.39498	0.697489	−	0.716595i	$-0.254300\pi$
$167$	18.0271	1.39498	0.697489	+	0.716595i	$0.254300\pi$
$168$	0	0
$169$	12.8963	0.992021
$170$	0	0
$171$	− 4.55576i	− 0.348387i
$172$	0	0
$173$	− 15.0682i	− 1.14562i	−0.819689	−	0.572809i	$-0.805854\pi$
$173$	− 15.0682i	− 1.14562i	0.819689	−	0.572809i	$-0.194146\pi$
$174$	0	0
$175$	−4.66595	−0.352713
$176$	0	0
$177$	−6.92596	−0.520587
$178$	0	0
$179$	22.2870i	1.66581i	0.553419	+	0.832903i	$0.313323\pi$
$179$	22.2870i	1.66581i	−0.553419	+	0.832903i	$0.686677\pi$
$180$	0	0
$181$	− 11.5115i	− 0.855643i	−0.903863	−	0.427822i	$-0.859281\pi$
$181$	− 11.5115i	− 0.855643i	0.903863	−	0.427822i	$-0.140719\pi$
$182$	0	0
$183$	14.0582	1.03921
$184$	0	0
$185$	5.65104	0.415473
$186$	0	0
$187$	0	0
$188$	0	0
$189$	− 10.4427i	− 0.759594i
$190$	0	0
$191$	10.7973	0.781262	0.390631	−	0.920547i	$-0.372257\pi$
$191$	10.7973	0.781262	0.390631	+	0.920547i	$0.372257\pi$
$192$	0	0
$193$	−9.05654	−0.651904	−0.325952	−	0.945386i	$-0.605685\pi$
$193$	−9.05654	−0.651904	−0.325952	+	0.945386i	$0.605685\pi$
$194$	0	0
$195$	− 0.655379i	− 0.0469327i
$196$	0	0
$197$	− 8.74114i	− 0.622780i	−0.950282	−	0.311390i	$-0.899205\pi$
$197$	− 8.74114i	− 0.622780i	0.950282	−	0.311390i	$-0.100795\pi$
$198$	0	0
$199$	−2.55580	−0.181176	−0.0905878	−	0.995888i	$-0.528875\pi$
$199$	−2.55580	−0.181176	−0.0905878	+	0.995888i	$0.528875\pi$
$200$	0	0
$201$	14.1349	0.996996
$202$	0	0
$203$	− 19.4542i	− 1.36542i
$204$	0	0
$205$	− 7.11759i	− 0.497114i
$206$	0	0
$207$	−4.14428	−0.288048
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 3.31568i	− 0.228261i	−0.993466	−	0.114130i	$-0.963592\pi$
$211$	− 3.31568i	− 0.228261i	0.993466	−	0.114130i	$-0.0364082\pi$
$212$	0	0
$213$	14.3245i	0.981497i
$214$	0	0
$215$	−5.72335	−0.390330
$216$	0	0
$217$	−5.26971	−0.357731
$218$	0	0
$219$	16.2199i	1.09604i
$220$	0	0
$221$	1.54353i	0.103829i
$222$	0	0
$223$	14.9051	0.998121	0.499061	−	0.866567i	$-0.333678\pi$
$223$	14.9051	0.998121	0.499061	+	0.866567i	$0.333678\pi$
$224$	0	0
$225$	3.36003	0.224002
$226$	0	0
$227$	− 1.32270i	− 0.0877904i	−0.999036	−	0.0438952i	$-0.986023\pi$
$227$	− 1.32270i	− 0.0877904i	0.999036	−	0.0438952i	$-0.0139768\pi$
$228$	0	0
$229$	23.6029i	1.55972i	0.625953	+	0.779861i	$0.284710\pi$
$229$	23.6029i	1.55972i	−0.625953	+	0.779861i	$0.715290\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2.47082	−0.161869	−0.0809343	−	0.996719i	$-0.525790\pi$
$233$	−2.47082	−0.161869	−0.0809343	+	0.996719i	$0.525790\pi$
$234$	0	0
$235$	− 20.1324i	− 1.31329i
$236$	0	0
$237$	7.69218i	0.499661i
$238$	0	0
$239$	15.8198	1.02330	0.511648	−	0.859195i	$-0.329035\pi$
$239$	15.8198	1.02330	0.511648	+	0.859195i	$0.329035\pi$
$240$	0	0
$241$	3.22511	0.207748	0.103874	−	0.994590i	$-0.466876\pi$
$241$	3.22511	0.207748	0.103874	+	0.994590i	$0.466876\pi$
$242$	0	0
$243$	12.7117i	0.815453i
$244$	0	0
$245$	5.55201i	0.354705i
$246$	0	0
$247$	−1.09063	−0.0693951
$248$	0	0
$249$	0.179628	0.0113834
$250$	0	0
$251$	− 0.345844i	− 0.0218295i	−0.999940	−	0.0109147i	$-0.996526\pi$
$251$	− 0.345844i	− 0.0218295i	0.999940	−	0.0109147i	$-0.00347434\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−9.75218	−0.610705
$256$	0	0
$257$	−0.421872	−0.0263157	−0.0131578	−	0.999913i	$-0.504188\pi$
$257$	−0.421872	−0.0263157	−0.0131578	+	0.999913i	$0.504188\pi$
$258$	0	0
$259$	− 6.67385i	− 0.414693i
$260$	0	0
$261$	14.0093i	0.867153i
$262$	0	0
$263$	−2.35352	−0.145124	−0.0725621	−	0.997364i	$-0.523118\pi$
$263$	−2.35352	−0.145124	−0.0725621	+	0.997364i	$0.523118\pi$
$264$	0	0
$265$	−8.94936	−0.549755
$266$	0	0
$267$	5.51823i	0.337710i
$268$	0	0
$269$	23.0569i	1.40580i	0.711287	+	0.702901i	$0.248112\pi$
$269$	23.0569i	1.40580i	−0.711287	+	0.702901i	$0.751888\pi$
$270$	0	0
$271$	4.37644	0.265850	0.132925	−	0.991126i	$-0.457563\pi$
$271$	4.37644	0.265850	0.132925	+	0.991126i	$0.457563\pi$
$272$	0	0
$273$	−0.773999	−0.0468446
$274$	0	0
$275$	0	0
$276$	0	0
$277$	6.26661i	0.376524i	0.982119	+	0.188262i	$0.0602854\pi$
$277$	6.26661i	0.376524i	−0.982119	+	0.188262i	$0.939715\pi$
$278$	0	0
$279$	3.79480	0.227189
$280$	0	0
$281$	−11.1852	−0.667250	−0.333625	−	0.942706i	$-0.608272\pi$
$281$	−11.1852	−0.667250	−0.333625	+	0.942706i	$0.608272\pi$
$282$	0	0
$283$	31.9805i	1.90104i	0.310662	+	0.950521i	$0.399449\pi$
$283$	31.9805i	1.90104i	−0.310662	+	0.950521i	$0.600551\pi$
$284$	0	0
$285$	− 6.89069i	− 0.408169i
$286$	0	0
$287$	−8.40583	−0.496180
$288$	0	0
$289$	5.96810	0.351065
$290$	0	0
$291$	15.9430i	0.934594i
$292$	0	0
$293$	− 7.66484i	− 0.447785i	−0.974614	−	0.223892i	$-0.928124\pi$
$293$	− 7.66484i	− 0.447785i	0.974614	−	0.223892i	$-0.0718764\pi$
$294$	0	0
$295$	8.51754	0.495911
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0.992124i	0.0573760i
$300$	0	0
$301$	6.75925i	0.389597i
$302$	0	0
$303$	−19.7616	−1.13527
$304$	0	0
$305$	−17.2888	−0.989954
$306$	0	0
$307$	− 3.57511i	− 0.204042i	−0.994782	−	0.102021i	$-0.967469\pi$
$307$	− 3.57511i	− 0.204042i	0.994782	−	0.102021i	$-0.0325309\pi$
$308$	0	0
$309$	− 10.5182i	− 0.598360i
$310$	0	0
$311$	0.0783210	0.00444118	0.00222059	−	0.999998i	$-0.499293\pi$
$311$	0.0783210	0.00444118	0.00222059	+	0.999998i	$0.499293\pi$
$312$	0	0
$313$	−8.94774	−0.505756	−0.252878	−	0.967498i	$-0.581377\pi$
$313$	−8.94774	−0.505756	−0.252878	+	0.967498i	$0.581377\pi$
$314$	0	0
$315$	3.97611i	0.224028i
$316$	0	0
$317$	− 24.4832i	− 1.37512i	−0.726129	−	0.687558i	$-0.758683\pi$
$317$	− 24.4832i	− 1.37512i	0.726129	−	0.687558i	$-0.241317\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	6.47324	0.361301
$322$	0	0
$323$	16.2288i	0.902994i
$324$	0	0
$325$	− 0.804377i	− 0.0446188i
$326$	0	0
$327$	−13.0998	−0.724423
$328$	0	0
$329$	−23.7762	−1.31083
$330$	0	0
$331$	24.2083i	1.33061i	0.746572	+	0.665305i	$0.231698\pi$
$331$	24.2083i	1.33061i	−0.746572	+	0.665305i	$0.768302\pi$
$332$	0	0
$333$	4.80594i	0.263364i
$334$	0	0
$335$	−17.3830	−0.949737
$336$	0	0
$337$	−22.3046	−1.21501	−0.607505	−	0.794315i	$-0.707830\pi$
$337$	−22.3046	−1.21501	−0.607505	+	0.794315i	$0.707830\pi$
$338$	0	0
$339$	10.3574i	0.562537i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	19.6346	1.06017
$344$	0	0
$345$	−6.26833	−0.337475
$346$	0	0
$347$	− 16.8395i	− 0.903991i	−0.892020	−	0.451996i	$-0.850712\pi$
$347$	− 16.8395i	− 0.903991i	0.892020	−	0.451996i	$-0.149288\pi$
$348$	0	0
$349$	− 5.97392i	− 0.319777i	−0.987135	−	0.159888i	$-0.948887\pi$
$349$	− 5.97392i	− 0.319777i	0.987135	−	0.159888i	$-0.0511134\pi$
$350$	0	0
$351$	1.80024	0.0960899
$352$	0	0
$353$	−6.69469	−0.356322	−0.178161	−	0.984001i	$-0.557015\pi$
$353$	−6.69469	−0.356322	−0.178161	+	0.984001i	$0.557015\pi$
$354$	0	0
$355$	− 17.6162i	− 0.934973i
$356$	0	0
$357$	11.5173i	0.609559i
$358$	0	0
$359$	20.1204	1.06191	0.530957	−	0.847399i	$-0.321833\pi$
$359$	20.1204	1.06191	0.530957	+	0.847399i	$0.321833\pi$
$360$	0	0
$361$	7.53307	0.396477
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 19.9472i	− 1.04409i
$366$	0	0
$367$	−7.45942	−0.389378	−0.194689	−	0.980865i	$-0.562370\pi$
$367$	−7.45942	−0.389378	−0.194689	+	0.980865i	$0.562370\pi$
$368$	0	0
$369$	6.05317	0.315115
$370$	0	0
$371$	10.5691i	0.548723i
$372$	0	0
$373$	1.53907i	0.0796902i	0.999206	+	0.0398451i	$0.0126864\pi$
$373$	1.53907i	0.0796902i	−0.999206	+	0.0398451i	$0.987314\pi$
$374$	0	0
$375$	15.2565	0.787844
$376$	0	0
$377$	3.35376	0.172728
$378$	0	0
$379$	5.35013i	0.274818i	0.990514	+	0.137409i	$0.0438774\pi$
$379$	5.35013i	0.274818i	−0.990514	+	0.137409i	$0.956123\pi$
$380$	0	0
$381$	− 16.7817i	− 0.859751i
$382$	0	0
$383$	9.59330	0.490195	0.245097	−	0.969498i	$-0.421180\pi$
$383$	9.59330	0.490195	0.245097	+	0.969498i	$0.421180\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 4.86744i	− 0.247426i
$388$	0	0
$389$	− 11.2345i	− 0.569614i	−0.958585	−	0.284807i	$-0.908071\pi$
$389$	− 11.2345i	− 0.569614i	0.958585	−	0.284807i	$-0.0919294\pi$
$390$	0	0
$391$	14.7630	0.746598
$392$	0	0
$393$	14.8596	0.749566
$394$	0	0
$395$	− 9.45984i	− 0.475976i
$396$	0	0
$397$	9.83583i	0.493646i	0.969061	+	0.246823i	$0.0793866\pi$
$397$	9.83583i	0.493646i	−0.969061	+	0.246823i	$0.920613\pi$
$398$	0	0
$399$	−8.13787	−0.407403
$400$	0	0
$401$	10.7477	0.536714	0.268357	−	0.963319i	$-0.413519\pi$
$401$	10.7477	0.536714	0.268357	+	0.963319i	$0.413519\pi$
$402$	0	0
$403$	− 0.908459i	− 0.0452536i
$404$	0	0
$405$	4.98933i	0.247922i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	6.33056	0.313026	0.156513	−	0.987676i	$-0.449975\pi$
$409$	6.33056	0.313026	0.156513	+	0.987676i	$0.449975\pi$
$410$	0	0
$411$	23.1786i	1.14332i
$412$	0	0
$413$	− 10.0592i	− 0.494979i
$414$	0	0
$415$	−0.220906	−0.0108438
$416$	0	0
$417$	−8.62471	−0.422354
$418$	0	0
$419$	− 8.63496i	− 0.421846i	−0.977503	−	0.210923i	$-0.932353\pi$
$419$	− 8.63496i	− 0.421846i	0.977503	−	0.210923i	$-0.0676469\pi$
$420$	0	0
$421$	29.4762i	1.43658i	0.695743	+	0.718291i	$0.255075\pi$
$421$	29.4762i	1.43658i	−0.695743	+	0.718291i	$0.744925\pi$
$422$	0	0
$423$	17.1216	0.832482
$424$	0	0
$425$	−11.9693	−0.580596
$426$	0	0
$427$	20.4180i	0.988095i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−21.6684	−1.04373	−0.521864	−	0.853029i	$-0.674763\pi$
$431$	−21.6684	−1.04373	−0.521864	+	0.853029i	$0.674763\pi$
$432$	0	0
$433$	−11.5153	−0.553392	−0.276696	−	0.960957i	$-0.589240\pi$
$433$	−11.5153	−0.553392	−0.276696	+	0.960957i	$0.589240\pi$
$434$	0	0
$435$	21.1894i	1.01595i
$436$	0	0
$437$	10.4312i	0.498994i
$438$	0	0
$439$	35.9563	1.71610	0.858049	−	0.513567i	$-0.171676\pi$
$439$	35.9563	1.71610	0.858049	+	0.513567i	$0.171676\pi$
$440$	0	0
$441$	−4.72172	−0.224844
$442$	0	0
$443$	− 27.8156i	− 1.32156i	−0.750581	−	0.660779i	$-0.770226\pi$
$443$	− 27.8156i	− 1.32156i	0.750581	−	0.660779i	$-0.229774\pi$
$444$	0	0
$445$	− 6.78632i	− 0.321702i
$446$	0	0
$447$	−5.15506	−0.243826
$448$	0	0
$449$	−13.0847	−0.617505	−0.308753	−	0.951142i	$-0.599912\pi$
$449$	−13.0847	−0.617505	−0.308753	+	0.951142i	$0.599912\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	− 3.00246i	− 0.141068i
$454$	0	0
$455$	0.951864	0.0446241
$456$	0	0
$457$	6.63015	0.310145	0.155073	−	0.987903i	$-0.450439\pi$
$457$	6.63015	0.310145	0.155073	+	0.987903i	$0.450439\pi$
$458$	0	0
$459$	− 26.7880i	− 1.25036i
$460$	0	0
$461$	− 8.43029i	− 0.392638i	−0.980540	−	0.196319i	$-0.937101\pi$
$461$	− 8.43029i	− 0.392638i	0.980540	−	0.196319i	$-0.0628988\pi$
$462$	0	0
$463$	−9.16056	−0.425727	−0.212864	−	0.977082i	$-0.568279\pi$
$463$	−9.16056	−0.425727	−0.212864	+	0.977082i	$0.568279\pi$
$464$	0	0
$465$	5.73973	0.266173
$466$	0	0
$467$	− 11.5736i	− 0.535560i	−0.963480	−	0.267780i	$-0.913710\pi$
$467$	− 11.5736i	− 0.535560i	0.963480	−	0.267780i	$-0.0862900\pi$
$468$	0	0
$469$	20.5293i	0.947954i
$470$	0	0
$471$	−10.2210	−0.470957
$472$	0	0
$473$	0	0
$474$	0	0
$475$	− 8.45726i	− 0.388046i
$476$	0	0
$477$	− 7.61101i	− 0.348484i
$478$	0	0
$479$	19.1582	0.875360	0.437680	−	0.899131i	$-0.355800\pi$
$479$	19.1582	0.875360	0.437680	+	0.899131i	$0.355800\pi$
$480$	0	0
$481$	1.15052	0.0524593
$482$	0	0
$483$	7.40286i	0.336842i
$484$	0	0
$485$	− 19.6067i	− 0.890293i
$486$	0	0
$487$	−42.4387	−1.92308	−0.961540	−	0.274664i	$-0.911433\pi$
$487$	−42.4387	−1.92308	−0.961540	+	0.274664i	$0.911433\pi$
$488$	0	0
$489$	7.22312	0.326641
$490$	0	0
$491$	39.2592i	1.77174i	0.463930	+	0.885872i	$0.346439\pi$
$491$	39.2592i	1.77174i	−0.463930	+	0.885872i	$0.653561\pi$
$492$	0	0
$493$	− 49.9047i	− 2.24759i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−20.8047	−0.933217
$498$	0	0
$499$	− 22.6637i	− 1.01457i	−0.861779	−	0.507284i	$-0.830649\pi$
$499$	− 22.6637i	− 1.01457i	0.861779	−	0.507284i	$-0.169351\pi$
$500$	0	0
$501$	23.1888i	1.03600i
$502$	0	0
$503$	15.5389	0.692847	0.346424	−	0.938078i	$-0.387396\pi$
$503$	15.5389	0.692847	0.346424	+	0.938078i	$0.387396\pi$
$504$	0	0
$505$	24.3028	1.08146
$506$	0	0
$507$	16.5889i	0.736737i
$508$	0	0
$509$	− 13.4499i	− 0.596156i	−0.954542	−	0.298078i	$-0.903654\pi$
$509$	− 13.4499i	− 0.596156i	0.954542	−	0.298078i	$-0.0963455\pi$
$510$	0	0
$511$	−23.5576	−1.04213
$512$	0	0
$513$	18.9278	0.835685
$514$	0	0
$515$	12.9353i	0.569997i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	19.3827	0.850808
$520$	0	0
$521$	−6.69298	−0.293225	−0.146612	−	0.989194i	$-0.546837\pi$
$521$	−6.69298	−0.293225	−0.146612	+	0.989194i	$0.546837\pi$
$522$	0	0
$523$	23.7587i	1.03889i	0.854503	+	0.519447i	$0.173862\pi$
$523$	23.7587i	1.03889i	−0.854503	+	0.519447i	$0.826138\pi$
$524$	0	0
$525$	− 6.00196i	− 0.261947i
$526$	0	0
$527$	−13.5181	−0.588856
$528$	0	0
$529$	−13.5109	−0.587431
$530$	0	0
$531$	7.24376i	0.314353i
$532$	0	0
$533$	− 1.44910i	− 0.0627677i
$534$	0	0
$535$	−7.96080	−0.344175
$536$	0	0
$537$	−28.6684	−1.23713
$538$	0	0
$539$	0	0
$540$	0	0
$541$	28.5717i	1.22839i	0.789153	+	0.614197i	$0.210520\pi$
$541$	28.5717i	1.22839i	−0.789153	+	0.614197i	$0.789480\pi$
$542$	0	0
$543$	14.8076	0.635455
$544$	0	0
$545$	16.1102	0.690084
$546$	0	0
$547$	− 30.2265i	− 1.29239i	−0.763172	−	0.646195i	$-0.776359\pi$
$547$	− 30.2265i	− 1.29239i	0.763172	−	0.646195i	$-0.223641\pi$
$548$	0	0
$549$	− 14.7033i	− 0.627522i
$550$	0	0
$551$	35.2616	1.50220
$552$	0	0
$553$	−11.1720	−0.475083
$554$	0	0
$555$	7.26911i	0.308556i
$556$	0	0
$557$	7.94247i	0.336533i	0.985742	+	0.168267i	$0.0538170\pi$
$557$	7.94247i	0.336533i	−0.985742	+	0.168267i	$0.946183\pi$
$558$	0	0
$559$	−1.16525	−0.0492846
$560$	0	0
$561$	0	0
$562$	0	0
$563$	17.7730i	0.749043i	0.927218	+	0.374521i	$0.122193\pi$
$563$	17.7730i	0.749043i	−0.927218	+	0.374521i	$0.877807\pi$
$564$	0	0
$565$	− 12.7375i	− 0.535872i
$566$	0	0
$567$	5.89237	0.247456
$568$	0	0
$569$	−28.3627	−1.18903	−0.594513	−	0.804086i	$-0.702655\pi$
$569$	−28.3627	−1.18903	−0.594513	+	0.804086i	$0.702655\pi$
$570$	0	0
$571$	11.9221i	0.498923i	0.968385	+	0.249461i	$0.0802535\pi$
$571$	11.9221i	0.498923i	−0.968385	+	0.249461i	$0.919746\pi$
$572$	0	0
$573$	13.8888i	0.580215i
$574$	0	0
$575$	−7.69340	−0.320837
$576$	0	0
$577$	−24.5999	−1.02411	−0.512053	−	0.858954i	$-0.671115\pi$
$577$	−24.5999	−1.02411	−0.512053	+	0.858954i	$0.671115\pi$
$578$	0	0
$579$	− 11.6497i	− 0.484145i
$580$	0	0
$581$	0.260889i	0.0108235i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−0.685452	−0.0283400
$586$	0	0
$587$	12.0954i	0.499229i	0.968345	+	0.249614i	$0.0803039\pi$
$587$	12.0954i	0.499229i	−0.968345	+	0.249614i	$0.919696\pi$
$588$	0	0
$589$	− 9.55159i	− 0.393566i
$590$	0	0
$591$	11.2440	0.462516
$592$	0	0
$593$	1.77063	0.0727109	0.0363554	−	0.999339i	$-0.488425\pi$
$593$	1.77063	0.0727109	0.0363554	+	0.999339i	$0.488425\pi$
$594$	0	0
$595$	− 14.1639i	− 0.580665i
$596$	0	0
$597$	− 3.28760i	− 0.134552i
$598$	0	0
$599$	14.3212	0.585149	0.292574	−	0.956243i	$-0.405488\pi$
$599$	14.3212	0.585149	0.292574	+	0.956243i	$0.405488\pi$
$600$	0	0
$601$	−20.9923	−0.856293	−0.428147	−	0.903709i	$-0.640833\pi$
$601$	−20.9923	−0.856293	−0.428147	+	0.903709i	$0.640833\pi$
$602$	0	0
$603$	− 14.7835i	− 0.602029i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−34.0000	−1.38002	−0.690009	−	0.723800i	$-0.742394\pi$
$607$	−34.0000	−1.38002	−0.690009	+	0.723800i	$0.742394\pi$
$608$	0	0
$609$	25.0245	1.01405
$610$	0	0
$611$	− 4.09885i	− 0.165822i
$612$	0	0
$613$	− 13.3983i	− 0.541152i	−0.962699	−	0.270576i	$-0.912786\pi$
$613$	− 13.3983i	− 0.541152i	0.962699	−	0.270576i	$-0.0872141\pi$
$614$	0	0
$615$	9.15557	0.369188
$616$	0	0
$617$	24.0930	0.969946	0.484973	−	0.874529i	$-0.338829\pi$
$617$	24.0930	0.969946	0.484973	+	0.874529i	$0.338829\pi$
$618$	0	0
$619$	2.41918i	0.0972349i	0.998817	+	0.0486175i	$0.0154815\pi$
$619$	2.41918i	0.0972349i	−0.998817	+	0.0486175i	$0.984518\pi$
$620$	0	0
$621$	− 17.2183i	− 0.690946i
$622$	0	0
$623$	−8.01461	−0.321098
$624$	0	0
$625$	−6.27497	−0.250999
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 17.1200i	− 0.682620i
$630$	0	0
$631$	−40.5047	−1.61247	−0.806234	−	0.591596i	$-0.798498\pi$
$631$	−40.5047	−1.61247	−0.806234	+	0.591596i	$0.798498\pi$
$632$	0	0
$633$	4.26506	0.169521
$634$	0	0
$635$	20.6381i	0.818998i
$636$	0	0
$637$	1.13036i	0.0447865i
$638$	0	0
$639$	14.9818	0.592670
$640$	0	0
$641$	−22.2703	−0.879624	−0.439812	−	0.898090i	$-0.644955\pi$
$641$	−22.2703	−0.879624	−0.439812	+	0.898090i	$0.644955\pi$
$642$	0	0
$643$	− 24.9678i	− 0.984633i	−0.870416	−	0.492316i	$-0.836150\pi$
$643$	− 24.9678i	− 0.984633i	0.870416	−	0.492316i	$-0.163850\pi$
$644$	0	0
$645$	− 7.36212i	− 0.289883i
$646$	0	0
$647$	12.3518	0.485599	0.242799	−	0.970077i	$-0.421934\pi$
$647$	12.3518	0.485599	0.242799	+	0.970077i	$0.421934\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	− 6.77858i	− 0.265674i
$652$	0	0
$653$	4.78361i	0.187197i	0.995610	+	0.0935987i	$0.0298371\pi$
$653$	4.78361i	0.187197i	−0.995610	+	0.0935987i	$0.970163\pi$
$654$	0	0
$655$	−18.2743	−0.714036
$656$	0	0
$657$	16.9642	0.661835
$658$	0	0
$659$	34.4552i	1.34219i	0.741374	+	0.671093i	$0.234175\pi$
$659$	34.4552i	1.34219i	−0.741374	+	0.671093i	$0.765825\pi$
$660$	0	0
$661$	− 39.9505i	− 1.55389i	−0.629567	−	0.776947i	$-0.716767\pi$
$661$	− 39.9505i	− 1.55389i	0.629567	−	0.776947i	$-0.283233\pi$
$662$	0	0
$663$	−1.98549	−0.0771102
$664$	0	0
$665$	10.0079	0.388092
$666$	0	0
$667$	− 32.0768i	− 1.24202i
$668$	0	0
$669$	19.1729i	0.741268i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−10.8179	−0.417000	−0.208500	−	0.978022i	$-0.566858\pi$
$673$	−10.8179	−0.417000	−0.208500	+	0.978022i	$0.566858\pi$
$674$	0	0
$675$	13.9599i	0.537318i
$676$	0	0
$677$	− 37.4282i	− 1.43848i	−0.694761	−	0.719241i	$-0.744490\pi$
$677$	− 37.4282i	− 1.43848i	0.694761	−	0.719241i	$-0.255510\pi$
$678$	0	0
$679$	−23.1554	−0.888621
$680$	0	0
$681$	1.70142	0.0651987
$682$	0	0
$683$	− 24.1362i	− 0.923545i	−0.886998	−	0.461772i	$-0.847214\pi$
$683$	− 24.1362i	− 0.923545i	0.886998	−	0.461772i	$-0.152786\pi$
$684$	0	0
$685$	− 28.5051i	− 1.08912i
$686$	0	0
$687$	−30.3611	−1.15835
$688$	0	0
$689$	−1.82204	−0.0694143
$690$	0	0
$691$	− 44.8113i	− 1.70470i	−0.522969	−	0.852352i	$-0.675176\pi$
$691$	− 44.8113i	− 1.70470i	0.522969	−	0.852352i	$-0.324824\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	10.6067	0.402334
$696$	0	0
$697$	−21.5630	−0.816756
$698$	0	0
$699$	− 3.17829i	− 0.120214i
$700$	0	0
$701$	− 22.9120i	− 0.865376i	−0.901544	−	0.432688i	$-0.857565\pi$
$701$	− 22.9120i	− 0.865376i	0.901544	−	0.432688i	$-0.142435\pi$
$702$	0	0
$703$	12.0967	0.456234
$704$	0	0
$705$	25.8969	0.975333
$706$	0	0
$707$	− 28.7014i	− 1.07943i
$708$	0	0
$709$	− 39.8708i	− 1.49738i	−0.662920	−	0.748690i	$-0.730683\pi$
$709$	− 39.8708i	− 1.49738i	0.662920	−	0.748690i	$-0.269317\pi$
$710$	0	0
$711$	8.04515	0.301717
$712$	0	0
$713$	−8.68889	−0.325402
$714$	0	0
$715$	0	0
$716$	0	0
$717$	20.3495i	0.759965i
$718$	0	0
$719$	36.2649	1.35245	0.676227	−	0.736693i	$-0.263614\pi$
$719$	36.2649	1.35245	0.676227	+	0.736693i	$0.263614\pi$
$720$	0	0
$721$	15.2765	0.568927
$722$	0	0
$723$	4.14856i	0.154287i
$724$	0	0
$725$	26.0067i	0.965864i
$726$	0	0
$727$	−4.42926	−0.164272	−0.0821360	−	0.996621i	$-0.526174\pi$
$727$	−4.42926	−0.164272	−0.0821360	+	0.996621i	$0.526174\pi$
$728$	0	0
$729$	−25.8133	−0.956046
$730$	0	0
$731$	17.3391i	0.641309i
$732$	0	0
$733$	− 27.2599i	− 1.00687i	−0.864034	−	0.503434i	$-0.832070\pi$
$733$	− 27.2599i	− 1.00687i	0.864034	−	0.503434i	$-0.167930\pi$
$734$	0	0
$735$	−7.14172	−0.263426
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 1.16933i	− 0.0430146i	−0.999769	−	0.0215073i	$-0.993153\pi$
$739$	− 1.16933i	− 0.0430146i	0.999769	−	0.0215073i	$-0.00684651\pi$
$740$	0	0
$741$	− 1.40291i	− 0.0515372i
$742$	0	0
$743$	39.6565	1.45486	0.727429	−	0.686183i	$-0.240715\pi$
$743$	39.6565	1.45486	0.727429	+	0.686183i	$0.240715\pi$
$744$	0	0
$745$	6.33969	0.232268
$746$	0	0
$747$	− 0.187870i	− 0.00687380i
$748$	0	0
$749$	9.40165i	0.343529i
$750$	0	0
$751$	6.61284	0.241306	0.120653	−	0.992695i	$-0.461501\pi$
$751$	6.61284	0.241306	0.120653	+	0.992695i	$0.461501\pi$
$752$	0	0
$753$	0.444870	0.0162120
$754$	0	0
$755$	3.69242i	0.134381i
$756$	0	0
$757$	20.6836i	0.751759i	0.926668	+	0.375880i	$0.122659\pi$
$757$	20.6836i	0.751759i	−0.926668	+	0.375880i	$0.877341\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−45.5490	−1.65115	−0.825575	−	0.564293i	$-0.809149\pi$
$761$	−45.5490	−1.65115	−0.825575	+	0.564293i	$0.809149\pi$
$762$	0	0
$763$	− 19.0260i	− 0.688788i
$764$	0	0
$765$	10.1997i	0.368770i
$766$	0	0
$767$	1.73413	0.0626157
$768$	0	0
$769$	19.1817	0.691711	0.345856	−	0.938288i	$-0.387589\pi$
$769$	19.1817	0.691711	0.345856	+	0.938288i	$0.387589\pi$
$770$	0	0
$771$	− 0.542667i	− 0.0195437i
$772$	0	0
$773$	− 50.1410i	− 1.80345i	−0.432314	−	0.901723i	$-0.642303\pi$
$773$	− 50.1410i	− 1.80345i	0.432314	−	0.901723i	$-0.357697\pi$
$774$	0	0
$775$	7.04463	0.253050
$776$	0	0
$777$	8.58477	0.307977
$778$	0	0
$779$	− 15.2360i	− 0.545885i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−58.2045	−2.08006
$784$	0	0
$785$	12.5697	0.448633
$786$	0	0
$787$	− 22.8144i	− 0.813246i	−0.913596	−	0.406623i	$-0.866706\pi$
$787$	− 22.8144i	− 0.813246i	0.913596	−	0.406623i	$-0.133294\pi$
$788$	0	0
$789$	− 3.02740i	− 0.107778i
$790$	0	0
$791$	−15.0430	−0.534866
$792$	0	0
$793$	−3.51991	−0.124996
$794$	0	0
$795$	− 11.5118i	− 0.408283i
$796$	0	0
$797$	26.6673i	0.944604i	0.881437	+	0.472302i	$0.156577\pi$
$797$	26.6673i	0.944604i	−0.881437	+	0.472302i	$0.843423\pi$
$798$	0	0
$799$	−60.9917	−2.15773
$800$	0	0
$801$	5.77144	0.203924
$802$	0	0
$803$	0	0
$804$	0	0
$805$	− 9.10403i	− 0.320875i
$806$	0	0
$807$	−29.6588	−1.04404
$808$	0	0
$809$	38.5665	1.35592	0.677962	−	0.735097i	$-0.262863\pi$
$809$	38.5665	1.35592	0.677962	+	0.735097i	$0.262863\pi$
$810$	0	0
$811$	− 16.8283i	− 0.590922i	−0.955355	−	0.295461i	$-0.904527\pi$
$811$	− 16.8283i	− 0.590922i	0.955355	−	0.295461i	$-0.0954733\pi$
$812$	0	0
$813$	5.62954i	0.197437i
$814$	0	0
$815$	−8.88299	−0.311157
$816$	0	0
$817$	−12.2514	−0.428624
$818$	0	0
$819$	0.809515i	0.0282867i
$820$	0	0
$821$	1.25448i	0.0437816i	0.999760	+	0.0218908i	$0.00696862\pi$
$821$	1.25448i	0.0437816i	−0.999760	+	0.0218908i	$0.993031\pi$
$822$	0	0
$823$	−50.2585	−1.75190	−0.875950	−	0.482402i	$-0.839764\pi$
$823$	−50.2585	−1.75190	−0.875950	+	0.482402i	$0.839764\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	23.6885i	0.823732i	0.911245	+	0.411866i	$0.135123\pi$
$827$	23.6885i	0.823732i	−0.911245	+	0.411866i	$0.864877\pi$
$828$	0	0
$829$	33.4106i	1.16040i	0.814475	+	0.580199i	$0.197025\pi$
$829$	33.4106i	1.16040i	−0.814475	+	0.580199i	$0.802975\pi$
$830$	0	0
$831$	−8.06092	−0.279630
$832$	0	0
$833$	16.8200	0.582779
$834$	0	0
$835$	− 28.5176i	− 0.986892i
$836$	0	0
$837$	15.7663i	0.544963i
$838$	0	0
$839$	−18.8019	−0.649113	−0.324557	−	0.945866i	$-0.605215\pi$
$839$	−18.8019	−0.649113	−0.324557	+	0.945866i	$0.605215\pi$
$840$	0	0
$841$	−79.4321	−2.73904
$842$	0	0
$843$	− 14.3878i	− 0.495542i
$844$	0	0
$845$	− 20.4010i	− 0.701815i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−41.1374	−1.41183
$850$	0	0
$851$	− 11.0041i	− 0.377215i
$852$	0	0
$853$	43.9013i	1.50315i	0.659647	+	0.751575i	$0.270706\pi$
$853$	43.9013i	1.50315i	−0.659647	+	0.751575i	$0.729294\pi$
$854$	0	0
$855$	−7.20688	−0.246470
$856$	0	0
$857$	35.4141	1.20972	0.604861	−	0.796331i	$-0.293229\pi$
$857$	35.4141	1.20972	0.604861	+	0.796331i	$0.293229\pi$
$858$	0	0
$859$	− 36.4697i	− 1.24433i	−0.782886	−	0.622165i	$-0.786253\pi$
$859$	− 36.4697i	− 1.24433i	0.782886	−	0.622165i	$-0.213747\pi$
$860$	0	0
$861$	− 10.8127i	− 0.368495i
$862$	0	0
$863$	−24.0325	−0.818075	−0.409037	−	0.912518i	$-0.634135\pi$
$863$	−24.0325	−0.818075	−0.409037	+	0.912518i	$0.634135\pi$
$864$	0	0
$865$	−23.8369	−0.810479
$866$	0	0
$867$	7.67694i	0.260723i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−3.53910	−0.119918
$872$	0	0
$873$	16.6745	0.564347
$874$	0	0
$875$	22.1584i	0.749090i
$876$	0	0
$877$	18.6541i	0.629904i	0.949108	+	0.314952i	$0.101988\pi$
$877$	18.6541i	0.629904i	−0.949108	+	0.314952i	$0.898012\pi$
$878$	0	0
$879$	9.85952	0.332553
$880$	0	0
$881$	32.4553	1.09345	0.546723	−	0.837314i	$-0.315875\pi$
$881$	32.4553	1.09345	0.546723	+	0.837314i	$0.315875\pi$
$882$	0	0
$883$	13.0200i	0.438157i	0.975707	+	0.219079i	$0.0703051\pi$
$883$	13.0200i	0.438157i	−0.975707	+	0.219079i	$0.929695\pi$
$884$	0	0
$885$	10.9564i	0.368294i
$886$	0	0
$887$	19.1455	0.642843	0.321422	−	0.946936i	$-0.395839\pi$
$887$	19.1455	0.642843	0.321422	+	0.946936i	$0.395839\pi$
$888$	0	0
$889$	24.3735	0.817460
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 43.0955i	− 1.44214i
$894$	0	0
$895$	35.2564	1.17849
$896$	0	0
$897$	−1.27620	−0.0426110
$898$	0	0
$899$	29.3718i	0.979605i
$900$	0	0
$901$	27.1124i	0.903245i
$902$	0	0
$903$	−8.69462	−0.289339
$904$	0	0
$905$	−18.2104	−0.605333
$906$	0	0
$907$	− 53.3243i	− 1.77060i	−0.465017	−	0.885302i	$-0.653952\pi$
$907$	− 53.3243i	− 1.77060i	0.465017	−	0.885302i	$-0.346048\pi$
$908$	0	0
$909$	20.6683i	0.685525i
$910$	0	0
$911$	−47.3265	−1.56800	−0.783999	−	0.620762i	$-0.786823\pi$
$911$	−47.3265	−1.56800	−0.783999	+	0.620762i	$0.786823\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	− 22.2391i	− 0.735202i
$916$	0	0
$917$	21.5818i	0.712695i
$918$	0	0
$919$	55.9089	1.84427	0.922133	−	0.386874i	$-0.126445\pi$
$919$	55.9089	1.84427	0.922133	+	0.386874i	$0.126445\pi$
$920$	0	0
$921$	4.59877	0.151535
$922$	0	0
$923$	− 3.58657i	− 0.118054i
$924$	0	0
$925$	8.92170i	0.293344i
$926$	0	0
$927$	−11.0008	−0.361315
$928$	0	0
$929$	20.2505	0.664396	0.332198	−	0.943210i	$-0.392210\pi$
$929$	20.2505	0.664396	0.332198	+	0.943210i	$0.392210\pi$
$930$	0	0
$931$	11.8847i	0.389504i
$932$	0	0
$933$	0.100747i	0.00329830i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	40.6651	1.32847	0.664235	−	0.747524i	$-0.268757\pi$
$937$	40.6651	1.32847	0.664235	+	0.747524i	$0.268757\pi$
$938$	0	0
$939$	− 11.5097i	− 0.375606i
$940$	0	0
$941$	2.23992i	0.0730194i	0.999333	+	0.0365097i	$0.0116240\pi$
$941$	2.23992i	0.0730194i	−0.999333	+	0.0365097i	$0.988376\pi$
$942$	0	0
$943$	−13.8598	−0.451339
$944$	0	0
$945$	−16.5196	−0.537382
$946$	0	0
$947$	14.7173i	0.478247i	0.970989	+	0.239123i	$0.0768601\pi$
$947$	14.7173i	0.478247i	−0.970989	+	0.239123i	$0.923140\pi$
$948$	0	0
$949$	− 4.06115i	− 0.131831i
$950$	0	0
$951$	31.4935	1.02125
$952$	0	0
$953$	18.3058	0.592984	0.296492	−	0.955035i	$-0.404183\pi$
$953$	18.3058	0.592984	0.296492	+	0.955035i	$0.404183\pi$
$954$	0	0
$955$	− 17.0805i	− 0.552712i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−33.6643	−1.08708
$960$	0	0
$961$	−23.0438	−0.743349
$962$	0	0
$963$	− 6.77028i	− 0.218169i
$964$	0	0
$965$	14.3268i	0.461196i
$966$	0	0
$967$	7.98210	0.256687	0.128343	−	0.991730i	$-0.459034\pi$
$967$	7.98210	0.256687	0.128343	+	0.991730i	$0.459034\pi$
$968$	0	0
$969$	−20.8756	−0.670620
$970$	0	0
$971$	35.5820i	1.14188i	0.820992	+	0.570940i	$0.193421\pi$
$971$	35.5820i	1.14188i	−0.820992	+	0.570940i	$0.806579\pi$
$972$	0	0
$973$	− 12.5264i	− 0.401578i
$974$	0	0
$975$	1.03469	0.0331367
$976$	0	0
$977$	−38.5825	−1.23436	−0.617181	−	0.786821i	$-0.711726\pi$
$977$	−38.5825	−1.23436	−0.617181	+	0.786821i	$0.711726\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	13.7009i	0.437437i
$982$	0	0
$983$	−33.6443	−1.07309	−0.536543	−	0.843873i	$-0.680270\pi$
$983$	−33.6443	−1.07309	−0.536543	+	0.843873i	$0.680270\pi$
$984$	0	0
$985$	−13.8279	−0.440592
$986$	0	0
$987$	− 30.5841i	− 0.973502i
$988$	0	0
$989$	11.1449i	0.354387i
$990$	0	0
$991$	−34.6130	−1.09952	−0.549758	−	0.835324i	$-0.685280\pi$
$991$	−34.6130	−1.09952	−0.549758	+	0.835324i	$0.685280\pi$
$992$	0	0
$993$	−31.1399	−0.988194
$994$	0	0
$995$	4.04309i	0.128174i
$996$	0	0
$997$	14.2720i	0.452000i	0.974127	+	0.226000i	$0.0725649\pi$
$997$	14.2720i	0.452000i	−0.974127	+	0.226000i	$0.927435\pi$
$998$	0	0
$999$	−19.9673	−0.631737

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3872.2.c.i.1937.15		20
4.3	odd	2		968.2.c.i.485.17		20
8.3	odd	2		968.2.c.i.485.18		20
8.5	even	2	inner	3872.2.c.i.1937.6		20
11.2	odd	10		352.2.w.a.81.8		40
11.6	odd	10		352.2.w.a.113.3		40
11.10	odd	2		3872.2.c.h.1937.15		20
44.3	odd	10		968.2.o.d.493.6		40
44.7	even	10		968.2.o.j.269.3		40
44.15	odd	10		968.2.o.d.269.8		40
44.19	even	10		968.2.o.j.493.5		40
44.27	odd	10		968.2.o.i.245.3		40
44.31	odd	10		968.2.o.i.565.1		40
44.35	even	10		88.2.o.a.37.10	yes	40
44.39	even	10		88.2.o.a.69.8	yes	40
44.43	even	2		968.2.c.h.485.4		20
88.3	odd	10		968.2.o.d.493.8		40
88.13	odd	10		352.2.w.a.81.3		40
88.19	even	10		968.2.o.j.493.3		40
88.21	odd	2		3872.2.c.h.1937.6		20
88.27	odd	10		968.2.o.i.245.1		40
88.35	even	10		88.2.o.a.37.8	✓	40
88.43	even	2		968.2.c.h.485.3		20
88.51	even	10		968.2.o.j.269.5		40
88.59	odd	10		968.2.o.d.269.6		40
88.61	odd	10		352.2.w.a.113.8		40
88.75	odd	10		968.2.o.i.565.3		40
88.83	even	10		88.2.o.a.69.10	yes	40
132.35	odd	10		792.2.br.b.37.1		40
132.83	odd	10		792.2.br.b.685.3		40
264.35	odd	10		792.2.br.b.37.3		40
264.83	odd	10		792.2.br.b.685.1		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.2.o.a.37.8	✓	40	88.35	even	10
88.2.o.a.37.10	yes	40	44.35	even	10
88.2.o.a.69.8	yes	40	44.39	even	10
88.2.o.a.69.10	yes	40	88.83	even	10
352.2.w.a.81.3		40	88.13	odd	10
352.2.w.a.81.8		40	11.2	odd	10
352.2.w.a.113.3		40	11.6	odd	10
352.2.w.a.113.8		40	88.61	odd	10
792.2.br.b.37.1		40	132.35	odd	10
792.2.br.b.37.3		40	264.35	odd	10
792.2.br.b.685.1		40	264.83	odd	10
792.2.br.b.685.3		40	132.83	odd	10
968.2.c.h.485.3		20	88.43	even	2
968.2.c.h.485.4		20	44.43	even	2
968.2.c.i.485.17		20	4.3	odd	2
968.2.c.i.485.18		20	8.3	odd	2
968.2.o.d.269.6		40	88.59	odd	10
968.2.o.d.269.8		40	44.15	odd	10
968.2.o.d.493.6		40	44.3	odd	10
968.2.o.d.493.8		40	88.3	odd	10
968.2.o.i.245.1		40	88.27	odd	10
968.2.o.i.245.3		40	44.27	odd	10
968.2.o.i.565.1		40	44.31	odd	10
968.2.o.i.565.3		40	88.75	odd	10
968.2.o.j.269.3		40	44.7	even	10
968.2.o.j.269.5		40	88.51	even	10
968.2.o.j.493.3		40	88.19	even	10
968.2.o.j.493.5		40	44.19	even	10
3872.2.c.h.1937.6		20	88.21	odd	2
3872.2.c.h.1937.15		20	11.10	odd	2
3872.2.c.i.1937.6		20	8.5	even	2	inner
3872.2.c.i.1937.15		20	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:	\( N \)	\(=\)	\( 3872 = 2^{5} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3872.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.28633i q^{3} -1.58193i q^{5} -1.86825 q^{7} +1.34535 q^{9} -0.322072i q^{13} +2.03488 q^{15} -4.79250 q^{17} -3.38629i q^{19} -2.40318i q^{21} -3.08044 q^{23} +2.49750 q^{25} +5.58956i q^{27} +10.4131i q^{29} +2.82067 q^{31} +2.95543i q^{35} +3.57225i q^{37} +0.414291 q^{39} +4.49931 q^{41} -3.61796i q^{43} -2.12826i q^{45} +12.7265 q^{47} -3.50965 q^{49} -6.16474i q^{51} -5.65725i q^{53} +4.35588 q^{57} +5.38428i q^{59} -10.9289i q^{61} -2.51346 q^{63} -0.509496 q^{65} -10.9885i q^{67} -3.96246i q^{69} +11.1359 q^{71} +12.6094 q^{73} +3.21261i q^{75} +5.97994 q^{79} -3.15396 q^{81} -0.139643i q^{83} +7.58140i q^{85} -13.3946 q^{87} +4.28990 q^{89} +0.601711i q^{91} +3.62831i q^{93} -5.35686 q^{95} +12.3942 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 10 q^{7} - 10 q^{9} + 4 q^{15} - 2 q^{17} + 4 q^{23} - 2 q^{25} - 2 q^{31} + 28 q^{39} + 2 q^{41} - 2 q^{47} - 2 q^{49} - 22 q^{57} - 30 q^{63} + 18 q^{65} + 34 q^{71} - 2 q^{73} - 58 q^{79} - 12 q^{81}+ \cdots + 10 q^{97}+O(q^{100}) \) 20 * q + 10 * q^7 - 10 * q^9 + 4 * q^15 - 2 * q^17 + 4 * q^23 - 2 * q^25 - 2 * q^31 + 28 * q^39 + 2 * q^41 - 2 * q^47 - 2 * q^49 - 22 * q^57 - 30 * q^63 + 18 * q^65 + 34 * q^71 - 2 * q^73 - 58 * q^79 - 12 * q^81 + 34 * q^87 - 8 * q^89 - 92 * q^95 + 10 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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