LMFDB - Embedded newform 6864.2.a.bn.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6864,2,Mod(1,6864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6864, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6864.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6864.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$54.8093159474$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.404.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x - 1 $ x^3 - x^2 - 5*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3432)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.65544$ of defining polynomial
Character	$\chi$	$=$	6864.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -2.39593 q^{5} -1.05137 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -2.39593 q^{5} -1.05137 q^{7} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{13} +2.39593 q^{15} -5.96633 q^{17} +5.05137 q^{19} +1.05137 q^{21} -9.15412 q^{23} +0.740489 q^{25} -1.00000 q^{27} -2.65544 q^{29} +9.70682 q^{31} +1.00000 q^{33} +2.51902 q^{35} -4.51902 q^{37} -1.00000 q^{39} +3.57040 q^{41} -7.96633 q^{43} -2.39593 q^{45} +1.20814 q^{47} -5.89461 q^{49} +5.96633 q^{51} -0.689115 q^{53} +2.39593 q^{55} -5.05137 q^{57} -11.4136 q^{59} -6.89461 q^{61} -1.05137 q^{63} -2.39593 q^{65} +14.2258 q^{67} +9.15412 q^{69} +8.10275 q^{71} +2.25951 q^{73} -0.740489 q^{75} +1.05137 q^{77} +11.4473 q^{79} +1.00000 q^{81} -15.4136 q^{83} +14.2949 q^{85} +2.65544 q^{87} -11.8096 q^{89} -1.05137 q^{91} -9.70682 q^{93} -12.1027 q^{95} -14.1027 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 4 q^{5} + 6 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 - 4 * q^5 + 6 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} - 4 q^{5} + 6 q^{7} + 3 q^{9} - 3 q^{11} + 3 q^{13} + 4 q^{15} + 6 q^{19} - 6 q^{21} + 5 q^{25} - 3 q^{27} - 2 q^{29} + 14 q^{31} + 3 q^{33} + 2 q^{35} - 8 q^{37} - 3 q^{39} - 4 q^{41} - 6 q^{43} - 4 q^{45} + 10 q^{47} + 7 q^{49} - 14 q^{53} + 4 q^{55} - 6 q^{57} - 4 q^{59} + 4 q^{61} + 6 q^{63} - 4 q^{65} + 22 q^{67} + 6 q^{71} + 4 q^{73} - 5 q^{75} - 6 q^{77} + 22 q^{79} + 3 q^{81} - 16 q^{83} - 14 q^{85} + 2 q^{87} - 2 q^{89} + 6 q^{91} - 14 q^{93} - 18 q^{95} - 24 q^{97} - 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 4 * q^5 + 6 * q^7 + 3 * q^9 - 3 * q^11 + 3 * q^13 + 4 * q^15 + 6 * q^19 - 6 * q^21 + 5 * q^25 - 3 * q^27 - 2 * q^29 + 14 * q^31 + 3 * q^33 + 2 * q^35 - 8 * q^37 - 3 * q^39 - 4 * q^41 - 6 * q^43 - 4 * q^45 + 10 * q^47 + 7 * q^49 - 14 * q^53 + 4 * q^55 - 6 * q^57 - 4 * q^59 + 4 * q^61 + 6 * q^63 - 4 * q^65 + 22 * q^67 + 6 * q^71 + 4 * q^73 - 5 * q^75 - 6 * q^77 + 22 * q^79 + 3 * q^81 - 16 * q^83 - 14 * q^85 + 2 * q^87 - 2 * q^89 + 6 * q^91 - 14 * q^93 - 18 * q^95 - 24 * q^97 - 3 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−2.39593	−1.07149	−0.535747	−	0.844379i	$-0.679970\pi$
$5$	−2.39593	−1.07149	−0.535747	+	0.844379i	$0.679970\pi$
$6$	0	0
$7$	−1.05137	−0.397382	−0.198691	−	0.980062i	$-0.563669\pi$
$7$	−1.05137	−0.397382	−0.198691	+	0.980062i	$0.563669\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	2.39593	0.618627
$16$	0	0
$17$	−5.96633	−1.44705	−0.723523	−	0.690300i	$-0.757479\pi$
$17$	−5.96633	−1.44705	−0.723523	+	0.690300i	$0.757479\pi$
$18$	0	0
$19$	5.05137	1.15886	0.579432	−	0.815020i	$-0.303274\pi$
$19$	5.05137	1.15886	0.579432	+	0.815020i	$0.303274\pi$
$20$	0	0
$21$	1.05137	0.229429
$22$	0	0
$23$	−9.15412	−1.90877	−0.954383	−	0.298584i	$-0.903486\pi$
$23$	−9.15412	−1.90877	−0.954383	+	0.298584i	$0.903486\pi$
$24$	0	0
$25$	0.740489	0.148098
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−2.65544	−0.493103	−0.246552	−	0.969130i	$-0.579297\pi$
$29$	−2.65544	−0.493103	−0.246552	+	0.969130i	$0.579297\pi$
$30$	0	0
$31$	9.70682	1.74340	0.871698	−	0.490044i	$-0.163019\pi$
$31$	9.70682	1.74340	0.871698	+	0.490044i	$0.163019\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	2.51902	0.425792
$36$	0	0
$37$	−4.51902	−0.742922	−0.371461	−	0.928448i	$-0.621143\pi$
$37$	−4.51902	−0.742922	−0.371461	+	0.928448i	$0.621143\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	3.57040	0.557602	0.278801	−	0.960349i	$-0.410063\pi$
$41$	3.57040	0.557602	0.278801	+	0.960349i	$0.410063\pi$
$42$	0	0
$43$	−7.96633	−1.21485	−0.607427	−	0.794376i	$-0.707798\pi$
$43$	−7.96633	−1.21485	−0.607427	+	0.794376i	$0.707798\pi$
$44$	0	0
$45$	−2.39593	−0.357164
$46$	0	0
$47$	1.20814	0.176225	0.0881124	−	0.996111i	$-0.471917\pi$
$47$	1.20814	0.176225	0.0881124	+	0.996111i	$0.471917\pi$
$48$	0	0
$49$	−5.89461	−0.842087
$50$	0	0
$51$	5.96633	0.835453
$52$	0	0
$53$	−0.689115	−0.0946573	−0.0473286	−	0.998879i	$-0.515071\pi$
$53$	−0.689115	−0.0946573	−0.0473286	+	0.998879i	$0.515071\pi$
$54$	0	0
$55$	2.39593	0.323067
$56$	0	0
$57$	−5.05137	−0.669071
$58$	0	0
$59$	−11.4136	−1.48593	−0.742964	−	0.669331i	$-0.766581\pi$
$59$	−11.4136	−1.48593	−0.742964	+	0.669331i	$0.766581\pi$
$60$	0	0
$61$	−6.89461	−0.882765	−0.441382	−	0.897319i	$-0.645512\pi$
$61$	−6.89461	−0.882765	−0.441382	+	0.897319i	$0.645512\pi$
$62$	0	0
$63$	−1.05137	−0.132461
$64$	0	0
$65$	−2.39593	−0.297179
$66$	0	0
$67$	14.2258	1.73796	0.868981	−	0.494845i	$-0.164775\pi$
$67$	14.2258	1.73796	0.868981	+	0.494845i	$0.164775\pi$
$68$	0	0
$69$	9.15412	1.10203
$70$	0	0
$71$	8.10275	0.961619	0.480810	−	0.876825i	$-0.340343\pi$
$71$	8.10275	0.961619	0.480810	+	0.876825i	$0.340343\pi$
$72$	0	0
$73$	2.25951	0.264456	0.132228	−	0.991219i	$-0.457787\pi$
$73$	2.25951	0.264456	0.132228	+	0.991219i	$0.457787\pi$
$74$	0	0
$75$	−0.740489	−0.0855044
$76$	0	0
$77$	1.05137	0.119815
$78$	0	0
$79$	11.4473	1.28792	0.643961	−	0.765058i	$-0.277290\pi$
$79$	11.4473	1.28792	0.643961	+	0.765058i	$0.277290\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−15.4136	−1.69187	−0.845933	−	0.533289i	$-0.820956\pi$
$83$	−15.4136	−1.69187	−0.845933	+	0.533289i	$0.820956\pi$
$84$	0	0
$85$	14.2949	1.55050
$86$	0	0
$87$	2.65544	0.284693
$88$	0	0
$89$	−11.8096	−1.25181	−0.625906	−	0.779899i	$-0.715271\pi$
$89$	−11.8096	−1.25181	−0.625906	+	0.779899i	$0.715271\pi$
$90$	0	0
$91$	−1.05137	−0.110214
$92$	0	0
$93$	−9.70682	−1.00655
$94$	0	0
$95$	−12.1027	−1.24172
$96$	0	0
$97$	−14.1027	−1.43192	−0.715959	−	0.698143i	$-0.754010\pi$
$97$	−14.1027	−1.43192	−0.715959	+	0.698143i	$0.754010\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	−9.17446	−0.912893	−0.456447	−	0.889751i	$-0.650878\pi$
$101$	−9.17446	−0.912893	−0.456447	+	0.889751i	$0.650878\pi$
$102$	0	0
$103$	−2.62177	−0.258331	−0.129165	−	0.991623i	$-0.541230\pi$
$103$	−2.62177	−0.258331	−0.129165	+	0.991623i	$0.541230\pi$
$104$	0	0
$105$	−2.51902	−0.245831
$106$	0	0
$107$	−6.37559	−0.616352	−0.308176	−	0.951329i	$-0.599719\pi$
$107$	−6.37559	−0.616352	−0.308176	+	0.951329i	$0.599719\pi$
$108$	0	0
$109$	−1.22147	−0.116995	−0.0584977	−	0.998288i	$-0.518631\pi$
$109$	−1.22147	−0.116995	−0.0584977	+	0.998288i	$0.518631\pi$
$110$	0	0
$111$	4.51902	0.428926
$112$	0	0
$113$	13.4136	1.26185	0.630924	−	0.775844i	$-0.282676\pi$
$113$	13.4136	1.26185	0.630924	+	0.775844i	$0.282676\pi$
$114$	0	0
$115$	21.9327	2.04523
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	6.27284	0.575031
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−3.57040	−0.321932
$124$	0	0
$125$	10.2055	0.912807
$126$	0	0
$127$	−8.06908	−0.716015	−0.358007	−	0.933719i	$-0.616544\pi$
$127$	−8.06908	−0.716015	−0.358007	+	0.933719i	$0.616544\pi$
$128$	0	0
$129$	7.96633	0.701396
$130$	0	0
$131$	18.3082	1.59960	0.799799	−	0.600267i	$-0.204939\pi$
$131$	18.3082	1.59960	0.799799	+	0.600267i	$0.204939\pi$
$132$	0	0
$133$	−5.31088	−0.460512
$134$	0	0
$135$	2.39593	0.206209
$136$	0	0
$137$	−18.2258	−1.55714	−0.778569	−	0.627559i	$-0.784054\pi$
$137$	−18.2258	−1.55714	−0.778569	+	0.627559i	$0.784054\pi$
$138$	0	0
$139$	4.65544	0.394869	0.197435	−	0.980316i	$-0.436739\pi$
$139$	4.65544	0.394869	0.197435	+	0.980316i	$0.436739\pi$
$140$	0	0
$141$	−1.20814	−0.101743
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	6.36226	0.528357
$146$	0	0
$147$	5.89461	0.486179
$148$	0	0
$149$	11.7759	0.964719	0.482359	−	0.875973i	$-0.339780\pi$
$149$	11.7759	0.964719	0.482359	+	0.875973i	$0.339780\pi$
$150$	0	0
$151$	22.6351	1.84202	0.921009	−	0.389541i	$-0.127366\pi$
$151$	22.6351	1.84202	0.921009	+	0.389541i	$0.127366\pi$
$152$	0	0
$153$	−5.96633	−0.482349
$154$	0	0
$155$	−23.2569	−1.86804
$156$	0	0
$157$	0.259511	0.0207112	0.0103556	−	0.999946i	$-0.496704\pi$
$157$	0.259511	0.0207112	0.0103556	+	0.999946i	$0.496704\pi$
$158$	0	0
$159$	0.689115	0.0546504
$160$	0	0
$161$	9.62441	0.758510
$162$	0	0
$163$	−2.60143	−0.203760	−0.101880	−	0.994797i	$-0.532486\pi$
$163$	−2.60143	−0.203760	−0.101880	+	0.994797i	$0.532486\pi$
$164$	0	0
$165$	−2.39593	−0.186523
$166$	0	0
$167$	9.06471	0.701448	0.350724	−	0.936479i	$-0.385936\pi$
$167$	9.06471	0.701448	0.350724	+	0.936479i	$0.385936\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	5.05137	0.386288
$172$	0	0
$173$	−17.8990	−1.36083	−0.680417	−	0.732825i	$-0.738201\pi$
$173$	−17.8990	−1.36083	−0.680417	+	0.732825i	$0.738201\pi$
$174$	0	0
$175$	−0.778532	−0.0588515
$176$	0	0
$177$	11.4136	0.857901
$178$	0	0
$179$	12.8813	0.962792	0.481396	−	0.876503i	$-0.340130\pi$
$179$	12.8813	0.962792	0.481396	+	0.876503i	$0.340130\pi$
$180$	0	0
$181$	−5.84324	−0.434324	−0.217162	−	0.976136i	$-0.569680\pi$
$181$	−5.84324	−0.434324	−0.217162	+	0.976136i	$0.569680\pi$
$182$	0	0
$183$	6.89461	0.509664
$184$	0	0
$185$	10.8273	0.796036
$186$	0	0
$187$	5.96633	0.436301
$188$	0	0
$189$	1.05137	0.0764762
$190$	0	0
$191$	9.82991	0.711267	0.355634	−	0.934625i	$-0.384265\pi$
$191$	9.82991	0.711267	0.355634	+	0.934625i	$0.384265\pi$
$192$	0	0
$193$	−17.0868	−1.22993	−0.614967	−	0.788553i	$-0.710831\pi$
$193$	−17.0868	−1.22993	−0.614967	+	0.788553i	$0.710831\pi$
$194$	0	0
$195$	2.39593	0.171576
$196$	0	0
$197$	−5.74049	−0.408993	−0.204496	−	0.978867i	$-0.565556\pi$
$197$	−5.74049	−0.408993	−0.204496	+	0.978867i	$0.565556\pi$
$198$	0	0
$199$	−16.2055	−1.14878	−0.574389	−	0.818583i	$-0.694760\pi$
$199$	−16.2055	−1.14878	−0.574389	+	0.818583i	$0.694760\pi$
$200$	0	0
$201$	−14.2258	−1.00341
$202$	0	0
$203$	2.79186	0.195950
$204$	0	0
$205$	−8.55442	−0.597467
$206$	0	0
$207$	−9.15412	−0.636256
$208$	0	0
$209$	−5.05137	−0.349411
$210$	0	0
$211$	−5.27721	−0.363298	−0.181649	−	0.983363i	$-0.558144\pi$
$211$	−5.27721	−0.363298	−0.181649	+	0.983363i	$0.558144\pi$
$212$	0	0
$213$	−8.10275	−0.555191
$214$	0	0
$215$	19.0868	1.30171
$216$	0	0
$217$	−10.2055	−0.692794
$218$	0	0
$219$	−2.25951	−0.152684
$220$	0	0
$221$	−5.96633	−0.401339
$222$	0	0
$223$	20.0824	1.34482	0.672409	−	0.740180i	$-0.265260\pi$
$223$	20.0824	1.34482	0.672409	+	0.740180i	$0.265260\pi$
$224$	0	0
$225$	0.740489	0.0493660
$226$	0	0
$227$	5.25687	0.348911	0.174455	−	0.984665i	$-0.444184\pi$
$227$	5.25687	0.348911	0.174455	+	0.984665i	$0.444184\pi$
$228$	0	0
$229$	−16.7919	−1.10964	−0.554819	−	0.831971i	$-0.687212\pi$
$229$	−16.7919	−1.10964	−0.554819	+	0.831971i	$0.687212\pi$
$230$	0	0
$231$	−1.05137	−0.0691753
$232$	0	0
$233$	5.93092	0.388548	0.194274	−	0.980947i	$-0.437765\pi$
$233$	5.93092	0.388548	0.194274	+	0.980947i	$0.437765\pi$
$234$	0	0
$235$	−2.89461	−0.188824
$236$	0	0
$237$	−11.4473	−0.743582
$238$	0	0
$239$	20.3977	1.31942	0.659708	−	0.751522i	$-0.270680\pi$
$239$	20.3977	1.31942	0.659708	+	0.751522i	$0.270680\pi$
$240$	0	0
$241$	17.1001	1.10151	0.550757	−	0.834665i	$-0.314339\pi$
$241$	17.1001	1.10151	0.550757	+	0.834665i	$0.314339\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	14.1231	0.902291
$246$	0	0
$247$	5.05137	0.321411
$248$	0	0
$249$	15.4136	0.976799
$250$	0	0
$251$	0.997361	0.0629528	0.0314764	−	0.999504i	$-0.489979\pi$
$251$	0.997361	0.0629528	0.0314764	+	0.999504i	$0.489979\pi$
$252$	0	0
$253$	9.15412	0.575515
$254$	0	0
$255$	−14.2949	−0.895182
$256$	0	0
$257$	29.3463	1.83057	0.915286	−	0.402806i	$-0.131965\pi$
$257$	29.3463	1.83057	0.915286	+	0.402806i	$0.131965\pi$
$258$	0	0
$259$	4.75118	0.295224
$260$	0	0
$261$	−2.65544	−0.164368
$262$	0	0
$263$	25.2029	1.55407	0.777037	−	0.629454i	$-0.216722\pi$
$263$	25.2029	1.55407	0.777037	+	0.629454i	$0.216722\pi$
$264$	0	0
$265$	1.65107	0.101425
$266$	0	0
$267$	11.8096	0.722734
$268$	0	0
$269$	4.68912	0.285900	0.142950	−	0.989730i	$-0.454341\pi$
$269$	4.68912	0.285900	0.142950	+	0.989730i	$0.454341\pi$
$270$	0	0
$271$	−0.465007	−0.0282472	−0.0141236	−	0.999900i	$-0.504496\pi$
$271$	−0.465007	−0.0282472	−0.0141236	+	0.999900i	$0.504496\pi$
$272$	0	0
$273$	1.05137	0.0636321
$274$	0	0
$275$	−0.740489	−0.0446532
$276$	0	0
$277$	20.3082	1.22020	0.610102	−	0.792323i	$-0.291128\pi$
$277$	20.3082	1.22020	0.610102	+	0.792323i	$0.291128\pi$
$278$	0	0
$279$	9.70682	0.581132
$280$	0	0
$281$	17.4003	1.03801	0.519007	−	0.854770i	$-0.326302\pi$
$281$	17.4003	1.03801	0.519007	+	0.854770i	$0.326302\pi$
$282$	0	0
$283$	−31.1692	−1.85282	−0.926408	−	0.376522i	$-0.877120\pi$
$283$	−31.1692	−1.85282	−0.926408	+	0.376522i	$0.877120\pi$
$284$	0	0
$285$	12.1027	0.716905
$286$	0	0
$287$	−3.75382	−0.221581
$288$	0	0
$289$	18.5971	1.09394
$290$	0	0
$291$	14.1027	0.826718
$292$	0	0
$293$	19.8432	1.15925	0.579627	−	0.814882i	$-0.303198\pi$
$293$	19.8432	1.15925	0.579627	+	0.814882i	$0.303198\pi$
$294$	0	0
$295$	27.3463	1.59216
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	−9.15412	−0.529397
$300$	0	0
$301$	8.37559	0.482761
$302$	0	0
$303$	9.17446	0.527059
$304$	0	0
$305$	16.5190	0.945876
$306$	0	0
$307$	13.5704	0.774503	0.387252	−	0.921974i	$-0.373424\pi$
$307$	13.5704	0.774503	0.387252	+	0.921974i	$0.373424\pi$
$308$	0	0
$309$	2.62177	0.149147
$310$	0	0
$311$	24.3216	1.37915	0.689575	−	0.724214i	$-0.257797\pi$
$311$	24.3216	1.37915	0.689575	+	0.724214i	$0.257797\pi$
$312$	0	0
$313$	18.6084	1.05181	0.525906	−	0.850543i	$-0.323727\pi$
$313$	18.6084	1.05181	0.525906	+	0.850543i	$0.323727\pi$
$314$	0	0
$315$	2.51902	0.141931
$316$	0	0
$317$	25.6395	1.44006	0.720028	−	0.693945i	$-0.244129\pi$
$317$	25.6395	1.44006	0.720028	+	0.693945i	$0.244129\pi$
$318$	0	0
$319$	2.65544	0.148676
$320$	0	0
$321$	6.37559	0.355851
$322$	0	0
$323$	−30.1382	−1.67693
$324$	0	0
$325$	0.740489	0.0410750
$326$	0	0
$327$	1.22147	0.0675474
$328$	0	0
$329$	−1.27020	−0.0700286
$330$	0	0
$331$	−33.7422	−1.85464	−0.927320	−	0.374269i	$-0.877894\pi$
$331$	−33.7422	−1.85464	−0.927320	+	0.374269i	$0.877894\pi$
$332$	0	0
$333$	−4.51902	−0.247641
$334$	0	0
$335$	−34.0841	−1.86222
$336$	0	0
$337$	12.1701	0.662947	0.331474	−	0.943464i	$-0.392454\pi$
$337$	12.1701	0.662947	0.331474	+	0.943464i	$0.392454\pi$
$338$	0	0
$339$	−13.4136	−0.728529
$340$	0	0
$341$	−9.70682	−0.525654
$342$	0	0
$343$	13.5571	0.732013
$344$	0	0
$345$	−21.9327	−1.18081
$346$	0	0
$347$	−23.4490	−1.25881	−0.629405	−	0.777077i	$-0.716701\pi$
$347$	−23.4490	−1.25881	−0.629405	+	0.777077i	$0.716701\pi$
$348$	0	0
$349$	20.8946	1.11846	0.559231	−	0.829012i	$-0.311096\pi$
$349$	20.8946	1.11846	0.559231	+	0.829012i	$0.311096\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	3.97966	0.211816	0.105908	−	0.994376i	$-0.466225\pi$
$353$	3.97966	0.211816	0.105908	+	0.994376i	$0.466225\pi$
$354$	0	0
$355$	−19.4136	−1.03037
$356$	0	0
$357$	−6.27284	−0.331994
$358$	0	0
$359$	−36.9433	−1.94980	−0.974898	−	0.222654i	$-0.928528\pi$
$359$	−36.9433	−1.94980	−0.974898	+	0.222654i	$0.928528\pi$
$360$	0	0
$361$	6.51638	0.342967
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−5.41363	−0.283363
$366$	0	0
$367$	14.8946	0.777492	0.388746	−	0.921345i	$-0.372908\pi$
$367$	14.8946	0.777492	0.388746	+	0.921345i	$0.372908\pi$
$368$	0	0
$369$	3.57040	0.185867
$370$	0	0
$371$	0.724518	0.0376151
$372$	0	0
$373$	−10.2462	−0.530527	−0.265264	−	0.964176i	$-0.585459\pi$
$373$	−10.2462	−0.530527	−0.265264	+	0.964176i	$0.585459\pi$
$374$	0	0
$375$	−10.2055	−0.527010
$376$	0	0
$377$	−2.65544	−0.136762
$378$	0	0
$379$	35.6395	1.83068	0.915338	−	0.402686i	$-0.131923\pi$
$379$	35.6395	1.83068	0.915338	+	0.402686i	$0.131923\pi$
$380$	0	0
$381$	8.06908	0.413391
$382$	0	0
$383$	23.9327	1.22290	0.611451	−	0.791282i	$-0.290586\pi$
$383$	23.9327	1.22290	0.611451	+	0.791282i	$0.290586\pi$
$384$	0	0
$385$	−2.51902	−0.128381
$386$	0	0
$387$	−7.96633	−0.404951
$388$	0	0
$389$	−6.45168	−0.327113	−0.163556	−	0.986534i	$-0.552297\pi$
$389$	−6.45168	−0.327113	−0.163556	+	0.986534i	$0.552297\pi$
$390$	0	0
$391$	54.6165	2.76207
$392$	0	0
$393$	−18.3082	−0.923529
$394$	0	0
$395$	−27.4270	−1.38000
$396$	0	0
$397$	−4.51902	−0.226803	−0.113402	−	0.993549i	$-0.536175\pi$
$397$	−4.51902	−0.226803	−0.113402	+	0.993549i	$0.536175\pi$
$398$	0	0
$399$	5.31088	0.265877
$400$	0	0
$401$	1.33123	0.0664782	0.0332391	−	0.999447i	$-0.489418\pi$
$401$	1.33123	0.0664782	0.0332391	+	0.999447i	$0.489418\pi$
$402$	0	0
$403$	9.70682	0.483531
$404$	0	0
$405$	−2.39593	−0.119055
$406$	0	0
$407$	4.51902	0.224000
$408$	0	0
$409$	14.4650	0.715249	0.357624	−	0.933866i	$-0.383587\pi$
$409$	14.4650	0.715249	0.357624	+	0.933866i	$0.383587\pi$
$410$	0	0
$411$	18.2258	0.899014
$412$	0	0
$413$	12.0000	0.590481
$414$	0	0
$415$	36.9300	1.81282
$416$	0	0
$417$	−4.65544	−0.227978
$418$	0	0
$419$	−0.156762	−0.00765833	−0.00382916	−	0.999993i	$-0.501219\pi$
$419$	−0.156762	−0.00765833	−0.00382916	+	0.999993i	$0.501219\pi$
$420$	0	0
$421$	15.8973	0.774785	0.387392	−	0.921915i	$-0.373376\pi$
$421$	15.8973	0.774785	0.387392	+	0.921915i	$0.373376\pi$
$422$	0	0
$423$	1.20814	0.0587416
$424$	0	0
$425$	−4.41800	−0.214305
$426$	0	0
$427$	7.24882	0.350795
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	1.69175	0.0814889	0.0407445	−	0.999170i	$-0.487027\pi$
$431$	1.69175	0.0814889	0.0407445	+	0.999170i	$0.487027\pi$
$432$	0	0
$433$	−9.16745	−0.440560	−0.220280	−	0.975437i	$-0.570697\pi$
$433$	−9.16745	−0.440560	−0.220280	+	0.975437i	$0.570697\pi$
$434$	0	0
$435$	−6.36226	−0.305047
$436$	0	0
$437$	−46.2409	−2.21200
$438$	0	0
$439$	18.5208	0.883947	0.441974	−	0.897028i	$-0.354278\pi$
$439$	18.5208	0.883947	0.441974	+	0.897028i	$0.354278\pi$
$440$	0	0
$441$	−5.89461	−0.280696
$442$	0	0
$443$	−5.03804	−0.239365	−0.119682	−	0.992812i	$-0.538188\pi$
$443$	−5.03804	−0.239365	−0.119682	+	0.992812i	$0.538188\pi$
$444$	0	0
$445$	28.2949	1.34131
$446$	0	0
$447$	−11.7759	−0.556981
$448$	0	0
$449$	−2.70946	−0.127867	−0.0639336	−	0.997954i	$-0.520365\pi$
$449$	−2.70946	−0.127867	−0.0639336	+	0.997954i	$0.520365\pi$
$450$	0	0
$451$	−3.57040	−0.168123
$452$	0	0
$453$	−22.6351	−1.06349
$454$	0	0
$455$	2.51902	0.118094
$456$	0	0
$457$	19.8973	0.930754	0.465377	−	0.885113i	$-0.345919\pi$
$457$	19.8973	0.930754	0.465377	+	0.885113i	$0.345919\pi$
$458$	0	0
$459$	5.96633	0.278484
$460$	0	0
$461$	−27.4624	−1.27905	−0.639525	−	0.768770i	$-0.720869\pi$
$461$	−27.4624	−1.27905	−0.639525	+	0.768770i	$0.720869\pi$
$462$	0	0
$463$	0.0470050	0.00218451	0.00109225	−	0.999999i	$-0.499652\pi$
$463$	0.0470050	0.00218451	0.00109225	+	0.999999i	$0.499652\pi$
$464$	0	0
$465$	23.2569	1.07851
$466$	0	0
$467$	26.2188	1.21326	0.606631	−	0.794983i	$-0.292520\pi$
$467$	26.2188	1.21326	0.606631	+	0.794983i	$0.292520\pi$
$468$	0	0
$469$	−14.9567	−0.690635
$470$	0	0
$471$	−0.259511	−0.0119576
$472$	0	0
$473$	7.96633	0.366292
$474$	0	0
$475$	3.74049	0.171625
$476$	0	0
$477$	−0.689115	−0.0315524
$478$	0	0
$479$	30.6351	1.39975	0.699877	−	0.714264i	$-0.253238\pi$
$479$	30.6351	1.39975	0.699877	+	0.714264i	$0.253238\pi$
$480$	0	0
$481$	−4.51902	−0.206050
$482$	0	0
$483$	−9.62441	−0.437926
$484$	0	0
$485$	33.7892	1.53429
$486$	0	0
$487$	−10.3553	−0.469241	−0.234621	−	0.972087i	$-0.575385\pi$
$487$	−10.3553	−0.469241	−0.234621	+	0.972087i	$0.575385\pi$
$488$	0	0
$489$	2.60143	0.117641
$490$	0	0
$491$	−9.48098	−0.427871	−0.213935	−	0.976848i	$-0.568628\pi$
$491$	−9.48098	−0.427871	−0.213935	+	0.976848i	$0.568628\pi$
$492$	0	0
$493$	15.8432	0.713544
$494$	0	0
$495$	2.39593	0.107689
$496$	0	0
$497$	−8.51902	−0.382130
$498$	0	0
$499$	−44.2879	−1.98260	−0.991299	−	0.131626i	$-0.957980\pi$
$499$	−44.2879	−1.98260	−0.991299	+	0.131626i	$0.957980\pi$
$500$	0	0
$501$	−9.06471	−0.404981
$502$	0	0
$503$	−36.1115	−1.61013	−0.805066	−	0.593186i	$-0.797870\pi$
$503$	−36.1115	−1.61013	−0.805066	+	0.593186i	$0.797870\pi$
$504$	0	0
$505$	21.9814	0.978159
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	7.80957	0.346153	0.173076	−	0.984908i	$-0.444629\pi$
$509$	7.80957	0.346153	0.173076	+	0.984908i	$0.444629\pi$
$510$	0	0
$511$	−2.37559	−0.105090
$512$	0	0
$513$	−5.05137	−0.223024
$514$	0	0
$515$	6.28158	0.276800
$516$	0	0
$517$	−1.20814	−0.0531338
$518$	0	0
$519$	17.8990	0.785678
$520$	0	0
$521$	−18.7512	−0.821504	−0.410752	−	0.911747i	$-0.634734\pi$
$521$	−18.7512	−0.821504	−0.410752	+	0.911747i	$0.634734\pi$
$522$	0	0
$523$	2.92829	0.128045	0.0640225	−	0.997948i	$-0.479607\pi$
$523$	2.92829	0.128045	0.0640225	+	0.997948i	$0.479607\pi$
$524$	0	0
$525$	0.778532	0.0339779
$526$	0	0
$527$	−57.9140	−2.52278
$528$	0	0
$529$	60.7980	2.64339
$530$	0	0
$531$	−11.4136	−0.495309
$532$	0	0
$533$	3.57040	0.154651
$534$	0	0
$535$	15.2755	0.660417
$536$	0	0
$537$	−12.8813	−0.555868
$538$	0	0
$539$	5.89461	0.253899
$540$	0	0
$541$	−27.0380	−1.16246	−0.581228	−	0.813741i	$-0.697428\pi$
$541$	−27.0380	−1.16246	−0.581228	+	0.813741i	$0.697428\pi$
$542$	0	0
$543$	5.84324	0.250757
$544$	0	0
$545$	2.92656	0.125360
$546$	0	0
$547$	−37.5588	−1.60590	−0.802949	−	0.596048i	$-0.796737\pi$
$547$	−37.5588	−1.60590	−0.802949	+	0.596048i	$0.796737\pi$
$548$	0	0
$549$	−6.89461	−0.294255
$550$	0	0
$551$	−13.4136	−0.571440
$552$	0	0
$553$	−12.0354	−0.511797
$554$	0	0
$555$	−10.8273	−0.459592
$556$	0	0
$557$	16.1161	0.682860	0.341430	−	0.939907i	$-0.389089\pi$
$557$	16.1161	0.682860	0.341430	+	0.939907i	$0.389089\pi$
$558$	0	0
$559$	−7.96633	−0.336940
$560$	0	0
$561$	−5.96633	−0.251899
$562$	0	0
$563$	2.10275	0.0886203	0.0443101	−	0.999018i	$-0.485891\pi$
$563$	2.10275	0.0886203	0.0443101	+	0.999018i	$0.485891\pi$
$564$	0	0
$565$	−32.1382	−1.35206
$566$	0	0
$567$	−1.05137	−0.0441536
$568$	0	0
$569$	25.9663	1.08857	0.544283	−	0.838902i	$-0.316802\pi$
$569$	25.9663	1.08857	0.544283	+	0.838902i	$0.316802\pi$
$570$	0	0
$571$	−6.54742	−0.274001	−0.137000	−	0.990571i	$-0.543746\pi$
$571$	−6.54742	−0.274001	−0.137000	+	0.990571i	$0.543746\pi$
$572$	0	0
$573$	−9.82991	−0.410650
$574$	0	0
$575$	−6.77853	−0.282684
$576$	0	0
$577$	−24.3082	−1.01197	−0.505983	−	0.862544i	$-0.668870\pi$
$577$	−24.3082	−1.01197	−0.505983	+	0.862544i	$0.668870\pi$
$578$	0	0
$579$	17.0868	0.710102
$580$	0	0
$581$	16.2055	0.672317
$582$	0	0
$583$	0.689115	0.0285402
$584$	0	0
$585$	−2.39593	−0.0990596
$586$	0	0
$587$	3.00264	0.123932	0.0619661	−	0.998078i	$-0.480263\pi$
$587$	3.00264	0.123932	0.0619661	+	0.998078i	$0.480263\pi$
$588$	0	0
$589$	49.0328	2.02036
$590$	0	0
$591$	5.74049	0.236132
$592$	0	0
$593$	−32.7733	−1.34584	−0.672918	−	0.739717i	$-0.734959\pi$
$593$	−32.7733	−1.34584	−0.672918	+	0.739717i	$0.734959\pi$
$594$	0	0
$595$	−15.0293	−0.616141
$596$	0	0
$597$	16.2055	0.663247
$598$	0	0
$599$	−14.0354	−0.573471	−0.286736	−	0.958010i	$-0.592570\pi$
$599$	−14.0354	−0.573471	−0.286736	+	0.958010i	$0.592570\pi$
$600$	0	0
$601$	25.3730	1.03498	0.517492	−	0.855688i	$-0.326866\pi$
$601$	25.3730	1.03498	0.517492	+	0.855688i	$0.326866\pi$
$602$	0	0
$603$	14.2258	0.579321
$604$	0	0
$605$	−2.39593	−0.0974085
$606$	0	0
$607$	45.5855	1.85026	0.925128	−	0.379655i	$-0.123957\pi$
$607$	45.5855	1.85026	0.925128	+	0.379655i	$0.123957\pi$
$608$	0	0
$609$	−2.79186	−0.113132
$610$	0	0
$611$	1.20814	0.0488760
$612$	0	0
$613$	5.71383	0.230779	0.115390	−	0.993320i	$-0.463188\pi$
$613$	5.71383	0.230779	0.115390	+	0.993320i	$0.463188\pi$
$614$	0	0
$615$	8.55442	0.344948
$616$	0	0
$617$	−19.4606	−0.783456	−0.391728	−	0.920081i	$-0.628123\pi$
$617$	−19.4606	−0.783456	−0.391728	+	0.920081i	$0.628123\pi$
$618$	0	0
$619$	11.3259	0.455228	0.227614	−	0.973751i	$-0.426908\pi$
$619$	11.3259	0.455228	0.227614	+	0.973751i	$0.426908\pi$
$620$	0	0
$621$	9.15412	0.367342
$622$	0	0
$623$	12.4163	0.497447
$624$	0	0
$625$	−28.1541	−1.12616
$626$	0	0
$627$	5.05137	0.201732
$628$	0	0
$629$	26.9620	1.07504
$630$	0	0
$631$	11.7068	0.466041	0.233021	−	0.972472i	$-0.425139\pi$
$631$	11.7068	0.466041	0.233021	+	0.972472i	$0.425139\pi$
$632$	0	0
$633$	5.27721	0.209750
$634$	0	0
$635$	19.3330	0.767205
$636$	0	0
$637$	−5.89461	−0.233553
$638$	0	0
$639$	8.10275	0.320540
$640$	0	0
$641$	22.7245	0.897564	0.448782	−	0.893641i	$-0.351858\pi$
$641$	22.7245	0.897564	0.448782	+	0.893641i	$0.351858\pi$
$642$	0	0
$643$	29.4340	1.16076	0.580381	−	0.814345i	$-0.302904\pi$
$643$	29.4340	1.16076	0.580381	+	0.814345i	$0.302904\pi$
$644$	0	0
$645$	−19.0868	−0.751541
$646$	0	0
$647$	−44.3170	−1.74228	−0.871140	−	0.491034i	$-0.836619\pi$
$647$	−44.3170	−1.74228	−0.871140	+	0.491034i	$0.836619\pi$
$648$	0	0
$649$	11.4136	0.448024
$650$	0	0
$651$	10.2055	0.399985
$652$	0	0
$653$	−18.7245	−0.732747	−0.366374	−	0.930468i	$-0.619401\pi$
$653$	−18.7245	−0.732747	−0.366374	+	0.930468i	$0.619401\pi$
$654$	0	0
$655$	−43.8653	−1.71396
$656$	0	0
$657$	2.25951	0.0881519
$658$	0	0
$659$	−9.72716	−0.378916	−0.189458	−	0.981889i	$-0.560673\pi$
$659$	−9.72716	−0.378916	−0.189458	+	0.981889i	$0.560673\pi$
$660$	0	0
$661$	49.7980	1.93692	0.968458	−	0.249176i	$-0.0801599\pi$
$661$	49.7980	1.93692	0.968458	+	0.249176i	$0.0801599\pi$
$662$	0	0
$663$	5.96633	0.231713
$664$	0	0
$665$	12.7245	0.493436
$666$	0	0
$667$	24.3082	0.941219
$668$	0	0
$669$	−20.0824	−0.776431
$670$	0	0
$671$	6.89461	0.266164
$672$	0	0
$673$	6.41627	0.247329	0.123665	−	0.992324i	$-0.460535\pi$
$673$	6.41627	0.247329	0.123665	+	0.992324i	$0.460535\pi$
$674$	0	0
$675$	−0.740489	−0.0285015
$676$	0	0
$677$	20.7315	0.796777	0.398389	−	0.917217i	$-0.369570\pi$
$677$	20.7315	0.796777	0.398389	+	0.917217i	$0.369570\pi$
$678$	0	0
$679$	14.8273	0.569018
$680$	0	0
$681$	−5.25687	−0.201444
$682$	0	0
$683$	−6.85921	−0.262460	−0.131230	−	0.991352i	$-0.541893\pi$
$683$	−6.85921	−0.262460	−0.131230	+	0.991352i	$0.541893\pi$
$684$	0	0
$685$	43.6679	1.66846
$686$	0	0
$687$	16.7919	0.640650
$688$	0	0
$689$	−0.689115	−0.0262532
$690$	0	0
$691$	−39.8183	−1.51476	−0.757380	−	0.652975i	$-0.773521\pi$
$691$	−39.8183	−1.51476	−0.757380	+	0.652975i	$0.773521\pi$
$692$	0	0
$693$	1.05137	0.0399384
$694$	0	0
$695$	−11.1541	−0.423100
$696$	0	0
$697$	−21.3021	−0.806876
$698$	0	0
$699$	−5.93092	−0.224328
$700$	0	0
$701$	30.1985	1.14058	0.570291	−	0.821443i	$-0.306831\pi$
$701$	30.1985	1.14058	0.570291	+	0.821443i	$0.306831\pi$
$702$	0	0
$703$	−22.8273	−0.860947
$704$	0	0
$705$	2.89461	0.109017
$706$	0	0
$707$	9.64579	0.362767
$708$	0	0
$709$	−29.7219	−1.11623	−0.558114	−	0.829764i	$-0.688475\pi$
$709$	−29.7219	−1.11623	−0.558114	+	0.829764i	$0.688475\pi$
$710$	0	0
$711$	11.4473	0.429308
$712$	0	0
$713$	−88.8574	−3.32774
$714$	0	0
$715$	2.39593	0.0896028
$716$	0	0
$717$	−20.3977	−0.761765
$718$	0	0
$719$	9.44904	0.352390	0.176195	−	0.984355i	$-0.443621\pi$
$719$	9.44904	0.352390	0.176195	+	0.984355i	$0.443621\pi$
$720$	0	0
$721$	2.75646	0.102656
$722$	0	0
$723$	−17.1001	−0.635960
$724$	0	0
$725$	−1.96633	−0.0730276
$726$	0	0
$727$	44.3436	1.64461	0.822307	−	0.569043i	$-0.192686\pi$
$727$	44.3436	1.64461	0.822307	+	0.569043i	$0.192686\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	47.5297	1.75795
$732$	0	0
$733$	−8.77325	−0.324047	−0.162024	−	0.986787i	$-0.551802\pi$
$733$	−8.77325	−0.324047	−0.162024	+	0.986787i	$0.551802\pi$
$734$	0	0
$735$	−14.1231	−0.520938
$736$	0	0
$737$	−14.2258	−0.524015
$738$	0	0
$739$	31.3863	1.15456	0.577282	−	0.816545i	$-0.304114\pi$
$739$	31.3863	1.15456	0.577282	+	0.816545i	$0.304114\pi$
$740$	0	0
$741$	−5.05137	−0.185567
$742$	0	0
$743$	17.5164	0.642614	0.321307	−	0.946975i	$-0.395878\pi$
$743$	17.5164	0.642614	0.321307	+	0.946975i	$0.395878\pi$
$744$	0	0
$745$	−28.2142	−1.03369
$746$	0	0
$747$	−15.4136	−0.563955
$748$	0	0
$749$	6.70313	0.244927
$750$	0	0
$751$	−48.1382	−1.75659	−0.878293	−	0.478123i	$-0.841317\pi$
$751$	−48.1382	−1.75659	−0.878293	+	0.478123i	$0.841317\pi$
$752$	0	0
$753$	−0.997361	−0.0363458
$754$	0	0
$755$	−54.2322	−1.97371
$756$	0	0
$757$	19.0868	0.693721	0.346860	−	0.937917i	$-0.387248\pi$
$757$	19.0868	0.693721	0.346860	+	0.937917i	$0.387248\pi$
$758$	0	0
$759$	−9.15412	−0.332274
$760$	0	0
$761$	35.1895	1.27562	0.637810	−	0.770194i	$-0.279841\pi$
$761$	35.1895	1.27562	0.637810	+	0.770194i	$0.279841\pi$
$762$	0	0
$763$	1.28422	0.0464919
$764$	0	0
$765$	14.2949	0.516834
$766$	0	0
$767$	−11.4136	−0.412122
$768$	0	0
$769$	−43.1088	−1.55454	−0.777272	−	0.629164i	$-0.783397\pi$
$769$	−43.1088	−1.55454	−0.777272	+	0.629164i	$0.783397\pi$
$770$	0	0
$771$	−29.3463	−1.05688
$772$	0	0
$773$	12.8069	0.460633	0.230317	−	0.973116i	$-0.426024\pi$
$773$	12.8069	0.460633	0.230317	+	0.973116i	$0.426024\pi$
$774$	0	0
$775$	7.18780	0.258193
$776$	0	0
$777$	−4.75118	−0.170448
$778$	0	0
$779$	18.0354	0.646185
$780$	0	0
$781$	−8.10275	−0.289939
$782$	0	0
$783$	2.65544	0.0948978
$784$	0	0
$785$	−0.621770	−0.0221919
$786$	0	0
$787$	8.50569	0.303195	0.151598	−	0.988442i	$-0.451558\pi$
$787$	8.50569	0.303195	0.151598	+	0.988442i	$0.451558\pi$
$788$	0	0
$789$	−25.2029	−0.897245
$790$	0	0
$791$	−14.1027	−0.501436
$792$	0	0
$793$	−6.89461	−0.244835
$794$	0	0
$795$	−1.65107	−0.0585575
$796$	0	0
$797$	0.129413	0.00458404	0.00229202	−	0.999997i	$-0.499270\pi$
$797$	0.129413	0.00458404	0.00229202	+	0.999997i	$0.499270\pi$
$798$	0	0
$799$	−7.20814	−0.255006
$800$	0	0
$801$	−11.8096	−0.417270
$802$	0	0
$803$	−2.25951	−0.0797364
$804$	0	0
$805$	−23.0594	−0.812738
$806$	0	0
$807$	−4.68912	−0.165065
$808$	0	0
$809$	31.9610	1.12369	0.561845	−	0.827242i	$-0.310092\pi$
$809$	31.9610	1.12369	0.561845	+	0.827242i	$0.310092\pi$
$810$	0	0
$811$	−36.4243	−1.27903	−0.639516	−	0.768778i	$-0.720865\pi$
$811$	−36.4243	−1.27903	−0.639516	+	0.768778i	$0.720865\pi$
$812$	0	0
$813$	0.465007	0.0163085
$814$	0	0
$815$	6.23285	0.218327
$816$	0	0
$817$	−40.2409	−1.40785
$818$	0	0
$819$	−1.05137	−0.0367380
$820$	0	0
$821$	−3.09206	−0.107913	−0.0539567	−	0.998543i	$-0.517183\pi$
$821$	−3.09206	−0.107913	−0.0539567	+	0.998543i	$0.517183\pi$
$822$	0	0
$823$	11.1001	0.386925	0.193463	−	0.981108i	$-0.438028\pi$
$823$	11.1001	0.386925	0.193463	+	0.981108i	$0.438028\pi$
$824$	0	0
$825$	0.740489	0.0257805
$826$	0	0
$827$	−31.6998	−1.10231	−0.551155	−	0.834403i	$-0.685813\pi$
$827$	−31.6998	−1.10231	−0.551155	+	0.834403i	$0.685813\pi$
$828$	0	0
$829$	−13.0328	−0.452647	−0.226323	−	0.974052i	$-0.572671\pi$
$829$	−13.0328	−0.452647	−0.226323	+	0.974052i	$0.572671\pi$
$830$	0	0
$831$	−20.3082	−0.704485
$832$	0	0
$833$	35.1692	1.21854
$834$	0	0
$835$	−21.7184	−0.751597
$836$	0	0
$837$	−9.70682	−0.335517
$838$	0	0
$839$	−2.93529	−0.101338	−0.0506688	−	0.998716i	$-0.516135\pi$
$839$	−2.93529	−0.101338	−0.0506688	+	0.998716i	$0.516135\pi$
$840$	0	0
$841$	−21.9486	−0.756849
$842$	0	0
$843$	−17.4003	−0.599298
$844$	0	0
$845$	−2.39593	−0.0824226
$846$	0	0
$847$	−1.05137	−0.0361256
$848$	0	0
$849$	31.1692	1.06972
$850$	0	0
$851$	41.3677	1.41807
$852$	0	0
$853$	39.4757	1.35162	0.675811	−	0.737075i	$-0.263794\pi$
$853$	39.4757	1.35162	0.675811	+	0.737075i	$0.263794\pi$
$854$	0	0
$855$	−12.1027	−0.413905
$856$	0	0
$857$	42.4534	1.45018	0.725090	−	0.688654i	$-0.241798\pi$
$857$	42.4534	1.45018	0.725090	+	0.688654i	$0.241798\pi$
$858$	0	0
$859$	38.1329	1.30108	0.650538	−	0.759473i	$-0.274543\pi$
$859$	38.1329	1.30108	0.650538	+	0.759473i	$0.274543\pi$
$860$	0	0
$861$	3.75382	0.127930
$862$	0	0
$863$	18.3489	0.624605	0.312302	−	0.949983i	$-0.398900\pi$
$863$	18.3489	0.624605	0.312302	+	0.949983i	$0.398900\pi$
$864$	0	0
$865$	42.8847	1.45812
$866$	0	0
$867$	−18.5971	−0.631589
$868$	0	0
$869$	−11.4473	−0.388323
$870$	0	0
$871$	14.2258	0.482024
$872$	0	0
$873$	−14.1027	−0.477306
$874$	0	0
$875$	−10.7298	−0.362733
$876$	0	0
$877$	−28.7599	−0.971154	−0.485577	−	0.874194i	$-0.661390\pi$
$877$	−28.7599	−0.971154	−0.485577	+	0.874194i	$0.661390\pi$
$878$	0	0
$879$	−19.8432	−0.669296
$880$	0	0
$881$	−41.5251	−1.39902	−0.699508	−	0.714624i	$-0.746598\pi$
$881$	−41.5251	−1.39902	−0.699508	+	0.714624i	$0.746598\pi$
$882$	0	0
$883$	1.03804	0.0349329	0.0174664	−	0.999847i	$-0.494440\pi$
$883$	1.03804	0.0349329	0.0174664	+	0.999847i	$0.494440\pi$
$884$	0	0
$885$	−27.3463	−0.919235
$886$	0	0
$887$	−40.1329	−1.34753	−0.673765	−	0.738946i	$-0.735324\pi$
$887$	−40.1329	−1.34753	−0.673765	+	0.738946i	$0.735324\pi$
$888$	0	0
$889$	8.48362	0.284531
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	6.10275	0.204221
$894$	0	0
$895$	−30.8627	−1.03163
$896$	0	0
$897$	9.15412	0.305647
$898$	0	0
$899$	−25.7759	−0.859674
$900$	0	0
$901$	4.11149	0.136973
$902$	0	0
$903$	−8.37559	−0.278722
$904$	0	0
$905$	14.0000	0.465376
$906$	0	0
$907$	18.2676	0.606565	0.303282	−	0.952901i	$-0.401917\pi$
$907$	18.2676	0.606565	0.303282	+	0.952901i	$0.401917\pi$
$908$	0	0
$909$	−9.17446	−0.304298
$910$	0	0
$911$	−33.5430	−1.11133	−0.555665	−	0.831406i	$-0.687536\pi$
$911$	−33.5430	−1.11133	−0.555665	+	0.831406i	$0.687536\pi$
$912$	0	0
$913$	15.4136	0.510117
$914$	0	0
$915$	−16.5190	−0.546102
$916$	0	0
$917$	−19.2488	−0.635652
$918$	0	0
$919$	−14.6200	−0.482271	−0.241135	−	0.970492i	$-0.577520\pi$
$919$	−14.6200	−0.482271	−0.241135	+	0.970492i	$0.577520\pi$
$920$	0	0
$921$	−13.5704	−0.447160
$922$	0	0
$923$	8.10275	0.266705
$924$	0	0
$925$	−3.34629	−0.110025
$926$	0	0
$927$	−2.62177	−0.0861102
$928$	0	0
$929$	12.2879	0.403153	0.201577	−	0.979473i	$-0.435394\pi$
$929$	12.2879	0.403153	0.201577	+	0.979473i	$0.435394\pi$
$930$	0	0
$931$	−29.7759	−0.975865
$932$	0	0
$933$	−24.3216	−0.796253
$934$	0	0
$935$	−14.2949	−0.467494
$936$	0	0
$937$	0.930015	0.0303823	0.0151911	−	0.999885i	$-0.495164\pi$
$937$	0.930015	0.0303823	0.0151911	+	0.999885i	$0.495164\pi$
$938$	0	0
$939$	−18.6084	−0.607263
$940$	0	0
$941$	−35.3277	−1.15165	−0.575825	−	0.817573i	$-0.695319\pi$
$941$	−35.3277	−1.15165	−0.575825	+	0.817573i	$0.695319\pi$
$942$	0	0
$943$	−32.6838	−1.06433
$944$	0	0
$945$	−2.51902	−0.0819438
$946$	0	0
$947$	−7.65107	−0.248626	−0.124313	−	0.992243i	$-0.539673\pi$
$947$	−7.65107	−0.248626	−0.124313	+	0.992243i	$0.539673\pi$
$948$	0	0
$949$	2.25951	0.0733468
$950$	0	0
$951$	−25.6395	−0.831417
$952$	0	0
$953$	56.3507	1.82538	0.912688	−	0.408656i	$-0.134002\pi$
$953$	56.3507	1.82538	0.912688	+	0.408656i	$0.134002\pi$
$954$	0	0
$955$	−23.5518	−0.762118
$956$	0	0
$957$	−2.65544	−0.0858383
$958$	0	0
$959$	19.1622	0.618779
$960$	0	0
$961$	63.2223	2.03943
$962$	0	0
$963$	−6.37559	−0.205451
$964$	0	0
$965$	40.9388	1.31787
$966$	0	0
$967$	−40.5625	−1.30440	−0.652201	−	0.758046i	$-0.726154\pi$
$967$	−40.5625	−1.30440	−0.652201	+	0.758046i	$0.726154\pi$
$968$	0	0
$969$	30.1382	0.968177
$970$	0	0
$971$	−9.33296	−0.299509	−0.149754	−	0.988723i	$-0.547848\pi$
$971$	−9.33296	−0.299509	−0.149754	+	0.988723i	$0.547848\pi$
$972$	0	0
$973$	−4.89461	−0.156914
$974$	0	0
$975$	−0.740489	−0.0237146
$976$	0	0
$977$	−51.5634	−1.64966	−0.824829	−	0.565382i	$-0.808729\pi$
$977$	−51.5634	−1.64966	−0.824829	+	0.565382i	$0.808729\pi$
$978$	0	0
$979$	11.8096	0.377435
$980$	0	0
$981$	−1.22147	−0.0389985
$982$	0	0
$983$	−27.3463	−0.872211	−0.436106	−	0.899896i	$-0.643643\pi$
$983$	−27.3463	−0.872211	−0.436106	+	0.899896i	$0.643643\pi$
$984$	0	0
$985$	13.7538	0.438233
$986$	0	0
$987$	1.27020	0.0404310
$988$	0	0
$989$	72.9247	2.31887
$990$	0	0
$991$	18.8679	0.599360	0.299680	−	0.954040i	$-0.403120\pi$
$991$	18.8679	0.599360	0.299680	+	0.954040i	$0.403120\pi$
$992$	0	0
$993$	33.7422	1.07078
$994$	0	0
$995$	38.8273	1.23091
$996$	0	0
$997$	−3.41891	−0.108278	−0.0541390	−	0.998533i	$-0.517241\pi$
$997$	−3.41891	−0.108278	−0.0541390	+	0.998533i	$0.517241\pi$
$998$	0	0
$999$	4.51902	0.142975

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.bn.1.2		3
4.3	odd	2		3432.2.a.n.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3432.2.a.n.1.2	✓	3	4.3	odd	2
6864.2.a.bn.1.2		3	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:	\( N \)	\(=\)	\( 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6864.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.39593 q^{5} -1.05137 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.39593 q^{5} -1.05137 q^{7} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{13} +2.39593 q^{15} -5.96633 q^{17} +5.05137 q^{19} +1.05137 q^{21} -9.15412 q^{23} +0.740489 q^{25} -1.00000 q^{27} -2.65544 q^{29} +9.70682 q^{31} +1.00000 q^{33} +2.51902 q^{35} -4.51902 q^{37} -1.00000 q^{39} +3.57040 q^{41} -7.96633 q^{43} -2.39593 q^{45} +1.20814 q^{47} -5.89461 q^{49} +5.96633 q^{51} -0.689115 q^{53} +2.39593 q^{55} -5.05137 q^{57} -11.4136 q^{59} -6.89461 q^{61} -1.05137 q^{63} -2.39593 q^{65} +14.2258 q^{67} +9.15412 q^{69} +8.10275 q^{71} +2.25951 q^{73} -0.740489 q^{75} +1.05137 q^{77} +11.4473 q^{79} +1.00000 q^{81} -15.4136 q^{83} +14.2949 q^{85} +2.65544 q^{87} -11.8096 q^{89} -1.05137 q^{91} -9.70682 q^{93} -12.1027 q^{95} -14.1027 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 4 q^{5} + 6 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 - 4 * q^5 + 6 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} - 4 q^{5} + 6 q^{7} + 3 q^{9} - 3 q^{11} + 3 q^{13} + 4 q^{15} + 6 q^{19} - 6 q^{21} + 5 q^{25} - 3 q^{27} - 2 q^{29} + 14 q^{31} + 3 q^{33} + 2 q^{35} - 8 q^{37} - 3 q^{39} - 4 q^{41} - 6 q^{43} - 4 q^{45} + 10 q^{47} + 7 q^{49} - 14 q^{53} + 4 q^{55} - 6 q^{57} - 4 q^{59} + 4 q^{61} + 6 q^{63} - 4 q^{65} + 22 q^{67} + 6 q^{71} + 4 q^{73} - 5 q^{75} - 6 q^{77} + 22 q^{79} + 3 q^{81} - 16 q^{83} - 14 q^{85} + 2 q^{87} - 2 q^{89} + 6 q^{91} - 14 q^{93} - 18 q^{95} - 24 q^{97} - 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 4 * q^5 + 6 * q^7 + 3 * q^9 - 3 * q^11 + 3 * q^13 + 4 * q^15 + 6 * q^19 - 6 * q^21 + 5 * q^25 - 3 * q^27 - 2 * q^29 + 14 * q^31 + 3 * q^33 + 2 * q^35 - 8 * q^37 - 3 * q^39 - 4 * q^41 - 6 * q^43 - 4 * q^45 + 10 * q^47 + 7 * q^49 - 14 * q^53 + 4 * q^55 - 6 * q^57 - 4 * q^59 + 4 * q^61 + 6 * q^63 - 4 * q^65 + 22 * q^67 + 6 * q^71 + 4 * q^73 - 5 * q^75 - 6 * q^77 + 22 * q^79 + 3 * q^81 - 16 * q^83 - 14 * q^85 + 2 * q^87 - 2 * q^89 + 6 * q^91 - 14 * q^93 - 18 * q^95 - 24 * q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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