LMFDB - Embedded newform 2672.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2672,2,Mod(1,2672)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2672, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2672.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2672 = 2^{4} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2672.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:	$21.3360274201$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.826865.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 6x - 1 $ x^5 - 2x^4 - 5x^3 + 6x^2 + 6x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 668)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.147687$ of defining polynomial
Character	$\chi$	$=$	2672.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.12588 q^{3} +2.00000 q^{5} +0.516539 q^{7} +6.77110 q^{9} +O(q^{10})$	$q-3.12588 q^{3} +2.00000 q^{5} +0.516539 q^{7} +6.77110 q^{9} -0.538350 q^{11} -0.295373 q^{13} -6.25175 q^{15} -6.99507 q^{17} +4.06647 q^{19} -1.61464 q^{21} -0.781327 q^{23} -1.00000 q^{25} -11.7880 q^{27} +0.811912 q^{29} -3.07629 q^{31} +1.68282 q^{33} +1.03308 q^{35} -3.95638 q^{37} +0.923300 q^{39} +0.743318 q^{41} -0.590746 q^{43} +13.5422 q^{45} +3.47572 q^{47} -6.73319 q^{49} +21.8657 q^{51} +6.69970 q^{53} -1.07670 q^{55} -12.7113 q^{57} +8.91427 q^{59} +4.63992 q^{61} +3.49753 q^{63} -0.590746 q^{65} -6.55274 q^{67} +2.44233 q^{69} -12.4655 q^{71} +7.77640 q^{73} +3.12588 q^{75} -0.278079 q^{77} +4.88529 q^{79} +16.5345 q^{81} +1.11388 q^{83} -13.9901 q^{85} -2.53794 q^{87} -1.79742 q^{89} -0.152572 q^{91} +9.61609 q^{93} +8.13294 q^{95} -17.1192 q^{97} -3.64522 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 3 q^{3} + 10 q^{5} - 9 q^{7} + 6 q^{9}+O(q^{10}) $ 5 * q - 3 * q^3 + 10 * q^5 - 9 * q^7 + 6 * q^9	$ 5 q - 3 q^{3} + 10 q^{5} - 9 q^{7} + 6 q^{9} - 5 q^{11} - 4 q^{13} - 6 q^{15} - 2 q^{17} - 5 q^{19} - 4 q^{21} - 6 q^{23} - 5 q^{25} - 12 q^{27} - 5 q^{29} - 9 q^{31} - 8 q^{33} - 18 q^{35} + 8 q^{37} - 4 q^{41} - 8 q^{43} + 12 q^{45} - 13 q^{47} + 14 q^{49} - 2 q^{53} - 10 q^{55} - 12 q^{57} - 4 q^{59} + 11 q^{61} + q^{63} - 8 q^{65} - 28 q^{67} - 16 q^{69} - 2 q^{71} + 8 q^{73} + 3 q^{75} - 12 q^{77} + 10 q^{79} - 15 q^{81} - 2 q^{83} - 4 q^{85} - 4 q^{87} - 17 q^{89} + 12 q^{91} - 12 q^{93} - 10 q^{95} - 27 q^{97} - 3 q^{99}+O(q^{100}) $ 5 * q - 3 * q^3 + 10 * q^5 - 9 * q^7 + 6 * q^9 - 5 * q^11 - 4 * q^13 - 6 * q^15 - 2 * q^17 - 5 * q^19 - 4 * q^21 - 6 * q^23 - 5 * q^25 - 12 * q^27 - 5 * q^29 - 9 * q^31 - 8 * q^33 - 18 * q^35 + 8 * q^37 - 4 * q^41 - 8 * q^43 + 12 * q^45 - 13 * q^47 + 14 * q^49 - 2 * q^53 - 10 * q^55 - 12 * q^57 - 4 * q^59 + 11 * q^61 + q^63 - 8 * q^65 - 28 * q^67 - 16 * q^69 - 2 * q^71 + 8 * q^73 + 3 * q^75 - 12 * q^77 + 10 * q^79 - 15 * q^81 - 2 * q^83 - 4 * q^85 - 4 * q^87 - 17 * q^89 + 12 * q^91 - 12 * q^93 - 10 * q^95 - 27 * 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
$3$	−3.12588	−1.80472	−0.902362	−	0.430978i	$-0.858169\pi$
$3$	−3.12588	−1.80472	−0.902362	+	0.430978i	$0.858169\pi$
$4$	0	0
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	0.516539	0.195233	0.0976167	−	0.995224i	$-0.468878\pi$
$7$	0.516539	0.195233	0.0976167	+	0.995224i	$0.468878\pi$
$8$	0	0
$9$	6.77110	2.25703
$10$	0	0
$11$	−0.538350	−0.162319	−0.0811593	−	0.996701i	$-0.525862\pi$
$11$	−0.538350	−0.162319	−0.0811593	+	0.996701i	$0.525862\pi$
$12$	0	0
$13$	−0.295373	−0.0819218	−0.0409609	−	0.999161i	$-0.513042\pi$
$13$	−0.295373	−0.0819218	−0.0409609	+	0.999161i	$0.513042\pi$
$14$	0	0
$15$	−6.25175	−1.61420
$16$	0	0
$17$	−6.99507	−1.69655	−0.848277	−	0.529553i	$-0.822360\pi$
$17$	−6.99507	−1.69655	−0.848277	+	0.529553i	$0.822360\pi$
$18$	0	0
$19$	4.06647	0.932912	0.466456	−	0.884544i	$-0.345531\pi$
$19$	4.06647	0.932912	0.466456	+	0.884544i	$0.345531\pi$
$20$	0	0
$21$	−1.61464	−0.352342
$22$	0	0
$23$	−0.781327	−0.162918	−0.0814590	−	0.996677i	$-0.525958\pi$
$23$	−0.781327	−0.162918	−0.0814590	+	0.996677i	$0.525958\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−11.7880	−2.26860
$28$	0	0
$29$	0.811912	0.150768	0.0753841	−	0.997155i	$-0.475982\pi$
$29$	0.811912	0.150768	0.0753841	+	0.997155i	$0.475982\pi$
$30$	0	0
$31$	−3.07629	−0.552517	−0.276259	−	0.961083i	$-0.589095\pi$
$31$	−3.07629	−0.552517	−0.276259	+	0.961083i	$0.589095\pi$
$32$	0	0
$33$	1.68282	0.292941
$34$	0	0
$35$	1.03308	0.174622
$36$	0	0
$37$	−3.95638	−0.650424	−0.325212	−	0.945641i	$-0.605436\pi$
$37$	−3.95638	−0.650424	−0.325212	+	0.945641i	$0.605436\pi$
$38$	0	0
$39$	0.923300	0.147846
$40$	0	0
$41$	0.743318	0.116087	0.0580434	−	0.998314i	$-0.481514\pi$
$41$	0.743318	0.116087	0.0580434	+	0.998314i	$0.481514\pi$
$42$	0	0
$43$	−0.590746	−0.0900880	−0.0450440	−	0.998985i	$-0.514343\pi$
$43$	−0.590746	−0.0900880	−0.0450440	+	0.998985i	$0.514343\pi$
$44$	0	0
$45$	13.5422	2.01875
$46$	0	0
$47$	3.47572	0.506986	0.253493	−	0.967337i	$-0.418420\pi$
$47$	3.47572	0.506986	0.253493	+	0.967337i	$0.418420\pi$
$48$	0	0
$49$	−6.73319	−0.961884
$50$	0	0
$51$	21.8657	3.06181
$52$	0	0
$53$	6.69970	0.920274	0.460137	−	0.887848i	$-0.347800\pi$
$53$	6.69970	0.920274	0.460137	+	0.887848i	$0.347800\pi$
$54$	0	0
$55$	−1.07670	−0.145182
$56$	0	0
$57$	−12.7113	−1.68365
$58$	0	0
$59$	8.91427	1.16054	0.580269	−	0.814425i	$-0.302947\pi$
$59$	8.91427	1.16054	0.580269	+	0.814425i	$0.302947\pi$
$60$	0	0
$61$	4.63992	0.594081	0.297041	−	0.954865i	$-0.404000\pi$
$61$	4.63992	0.594081	0.297041	+	0.954865i	$0.404000\pi$
$62$	0	0
$63$	3.49753	0.440648
$64$	0	0
$65$	−0.590746	−0.0732731
$66$	0	0
$67$	−6.55274	−0.800544	−0.400272	−	0.916396i	$-0.631084\pi$
$67$	−6.55274	−0.800544	−0.400272	+	0.916396i	$0.631084\pi$
$68$	0	0
$69$	2.44233	0.294022
$70$	0	0
$71$	−12.4655	−1.47938	−0.739691	−	0.672947i	$-0.765028\pi$
$71$	−12.4655	−1.47938	−0.739691	+	0.672947i	$0.765028\pi$
$72$	0	0
$73$	7.77640	0.910158	0.455079	−	0.890451i	$-0.349611\pi$
$73$	7.77640	0.910158	0.455079	+	0.890451i	$0.349611\pi$
$74$	0	0
$75$	3.12588	0.360945
$76$	0	0
$77$	−0.278079	−0.0316900
$78$	0	0
$79$	4.88529	0.549638	0.274819	−	0.961496i	$-0.411382\pi$
$79$	4.88529	0.549638	0.274819	+	0.961496i	$0.411382\pi$
$80$	0	0
$81$	16.5345	1.83716
$82$	0	0
$83$	1.11388	0.122264	0.0611321	−	0.998130i	$-0.480529\pi$
$83$	1.11388	0.122264	0.0611321	+	0.998130i	$0.480529\pi$
$84$	0	0
$85$	−13.9901	−1.51744
$86$	0	0
$87$	−2.53794	−0.272095
$88$	0	0
$89$	−1.79742	−0.190527	−0.0952633	−	0.995452i	$-0.530369\pi$
$89$	−1.79742	−0.190527	−0.0952633	+	0.995452i	$0.530369\pi$
$90$	0	0
$91$	−0.152572	−0.0159939
$92$	0	0
$93$	9.61609	0.997142
$94$	0	0
$95$	8.13294	0.834422
$96$	0	0
$97$	−17.1192	−1.73819	−0.869097	−	0.494641i	$-0.835300\pi$
$97$	−17.1192	−1.73819	−0.869097	+	0.494641i	$0.835300\pi$
$98$	0	0
$99$	−3.64522	−0.366358
$100$	0	0
$101$	−14.0281	−1.39585	−0.697926	−	0.716170i	$-0.745894\pi$
$101$	−14.0281	−1.39585	−0.697926	+	0.716170i	$0.745894\pi$
$102$	0	0
$103$	−16.7553	−1.65094	−0.825472	−	0.564443i	$-0.809091\pi$
$103$	−16.7553	−1.65094	−0.825472	+	0.564443i	$0.809091\pi$
$104$	0	0
$105$	−3.22927	−0.315145
$106$	0	0
$107$	−9.47360	−0.915847	−0.457924	−	0.888992i	$-0.651407\pi$
$107$	−9.47360	−0.915847	−0.457924	+	0.888992i	$0.651407\pi$
$108$	0	0
$109$	20.3467	1.94886	0.974429	−	0.224695i	$-0.0721384\pi$
$109$	20.3467	1.94886	0.974429	+	0.224695i	$0.0721384\pi$
$110$	0	0
$111$	12.3671	1.17384
$112$	0	0
$113$	16.8270	1.58295	0.791476	−	0.611200i	$-0.209313\pi$
$113$	16.8270	1.58295	0.791476	+	0.611200i	$0.209313\pi$
$114$	0	0
$115$	−1.56265	−0.145718
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	−3.61322	−0.331224
$120$	0	0
$121$	−10.7102	−0.973653
$122$	0	0
$123$	−2.32352	−0.209505
$124$	0	0
$125$	−12.0000	−1.07331
$126$	0	0
$127$	−9.52781	−0.845456	−0.422728	−	0.906257i	$-0.638927\pi$
$127$	−9.52781	−0.845456	−0.422728	+	0.906257i	$0.638927\pi$
$128$	0	0
$129$	1.84660	0.162584
$130$	0	0
$131$	16.5809	1.44868	0.724339	−	0.689444i	$-0.242145\pi$
$131$	16.5809	1.44868	0.724339	+	0.689444i	$0.242145\pi$
$132$	0	0
$133$	2.10049	0.182136
$134$	0	0
$135$	−23.5760	−2.02909
$136$	0	0
$137$	3.53903	0.302360	0.151180	−	0.988506i	$-0.451693\pi$
$137$	3.53903	0.302360	0.151180	+	0.988506i	$0.451693\pi$
$138$	0	0
$139$	9.40022	0.797316	0.398658	−	0.917100i	$-0.369476\pi$
$139$	9.40022	0.797316	0.398658	+	0.917100i	$0.369476\pi$
$140$	0	0
$141$	−10.8647	−0.914971
$142$	0	0
$143$	0.159014	0.0132974
$144$	0	0
$145$	1.62382	0.134851
$146$	0	0
$147$	21.0471	1.73594
$148$	0	0
$149$	−18.5809	−1.52221	−0.761103	−	0.648631i	$-0.775342\pi$
$149$	−18.5809	−1.52221	−0.761103	+	0.648631i	$0.775342\pi$
$150$	0	0
$151$	−17.2644	−1.40495	−0.702477	−	0.711706i	$-0.747923\pi$
$151$	−17.2644	−1.40495	−0.702477	+	0.711706i	$0.747923\pi$
$152$	0	0
$153$	−47.3643	−3.82918
$154$	0	0
$155$	−6.15257	−0.494186
$156$	0	0
$157$	−7.19863	−0.574513	−0.287256	−	0.957854i	$-0.592743\pi$
$157$	−7.19863	−0.574513	−0.287256	+	0.957854i	$0.592743\pi$
$158$	0	0
$159$	−20.9424	−1.66084
$160$	0	0
$161$	−0.403586	−0.0318070
$162$	0	0
$163$	−23.6948	−1.85592	−0.927959	−	0.372683i	$-0.878438\pi$
$163$	−23.6948	−1.85592	−0.927959	+	0.372683i	$0.878438\pi$
$164$	0	0
$165$	3.36563	0.262014
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−12.9128	−0.993289
$170$	0	0
$171$	27.5345	2.10561
$172$	0	0
$173$	−19.5278	−1.48467	−0.742336	−	0.670028i	$-0.766282\pi$
$173$	−19.5278	−1.48467	−0.742336	+	0.670028i	$0.766282\pi$
$174$	0	0
$175$	−0.516539	−0.0390467
$176$	0	0
$177$	−27.8649	−2.09445
$178$	0	0
$179$	10.5038	0.785092	0.392546	−	0.919732i	$-0.371594\pi$
$179$	10.5038	0.785092	0.392546	+	0.919732i	$0.371594\pi$
$180$	0	0
$181$	8.36184	0.621531	0.310765	−	0.950487i	$-0.399415\pi$
$181$	8.36184	0.621531	0.310765	+	0.950487i	$0.399415\pi$
$182$	0	0
$183$	−14.5038	−1.07215
$184$	0	0
$185$	−7.91275	−0.581757
$186$	0	0
$187$	3.76580	0.275382
$188$	0	0
$189$	−6.08895	−0.442906
$190$	0	0
$191$	−9.43844	−0.682942	−0.341471	−	0.939892i	$-0.610925\pi$
$191$	−9.43844	−0.682942	−0.341471	+	0.939892i	$0.610925\pi$
$192$	0	0
$193$	−16.3291	−1.17540	−0.587698	−	0.809080i	$-0.699966\pi$
$193$	−16.3291	−1.17540	−0.587698	+	0.809080i	$0.699966\pi$
$194$	0	0
$195$	1.84660	0.132238
$196$	0	0
$197$	−17.3460	−1.23585	−0.617926	−	0.786237i	$-0.712027\pi$
$197$	−17.3460	−1.23585	−0.617926	+	0.786237i	$0.712027\pi$
$198$	0	0
$199$	−6.46097	−0.458006	−0.229003	−	0.973426i	$-0.573547\pi$
$199$	−6.46097	−0.458006	−0.229003	+	0.973426i	$0.573547\pi$
$200$	0	0
$201$	20.4830	1.44476
$202$	0	0
$203$	0.419384	0.0294350
$204$	0	0
$205$	1.48664	0.103831
$206$	0	0
$207$	−5.29044	−0.367711
$208$	0	0
$209$	−2.18918	−0.151429
$210$	0	0
$211$	2.41035	0.165935	0.0829677	−	0.996552i	$-0.473560\pi$
$211$	2.41035	0.165935	0.0829677	+	0.996552i	$0.473560\pi$
$212$	0	0
$213$	38.9656	2.66988
$214$	0	0
$215$	−1.18149	−0.0805771
$216$	0	0
$217$	−1.58902	−0.107870
$218$	0	0
$219$	−24.3080	−1.64259
$220$	0	0
$221$	2.06616	0.138985
$222$	0	0
$223$	0.798106	0.0534452	0.0267226	−	0.999643i	$-0.491493\pi$
$223$	0.798106	0.0534452	0.0267226	+	0.999643i	$0.491493\pi$
$224$	0	0
$225$	−6.77110	−0.451406
$226$	0	0
$227$	10.4979	0.696769	0.348385	−	0.937352i	$-0.386730\pi$
$227$	10.4979	0.696769	0.348385	+	0.937352i	$0.386730\pi$
$228$	0	0
$229$	11.6916	0.772602	0.386301	−	0.922373i	$-0.373753\pi$
$229$	11.6916	0.772602	0.386301	+	0.922373i	$0.373753\pi$
$230$	0	0
$231$	0.869240	0.0571918
$232$	0	0
$233$	−23.8858	−1.56481	−0.782404	−	0.622771i	$-0.786007\pi$
$233$	−23.8858	−1.56481	−0.782404	+	0.622771i	$0.786007\pi$
$234$	0	0
$235$	6.95145	0.453462
$236$	0	0
$237$	−15.2708	−0.991946
$238$	0	0
$239$	−21.8051	−1.41046	−0.705228	−	0.708981i	$-0.749155\pi$
$239$	−21.8051	−1.41046	−0.705228	+	0.708981i	$0.749155\pi$
$240$	0	0
$241$	17.4951	1.12696	0.563481	−	0.826129i	$-0.309462\pi$
$241$	17.4951	1.12696	0.563481	+	0.826129i	$0.309462\pi$
$242$	0	0
$243$	−16.3207	−1.04697
$244$	0	0
$245$	−13.4664	−0.860335
$246$	0	0
$247$	−1.20113	−0.0764258
$248$	0	0
$249$	−3.48185	−0.220653
$250$	0	0
$251$	1.26889	0.0800917	0.0400458	−	0.999198i	$-0.487250\pi$
$251$	1.26889	0.0800917	0.0400458	+	0.999198i	$0.487250\pi$
$252$	0	0
$253$	0.420628	0.0264446
$254$	0	0
$255$	43.7314	2.73857
$256$	0	0
$257$	26.5168	1.65407	0.827036	−	0.562149i	$-0.190025\pi$
$257$	26.5168	1.65407	0.827036	+	0.562149i	$0.190025\pi$
$258$	0	0
$259$	−2.04362	−0.126985
$260$	0	0
$261$	5.49753	0.340289
$262$	0	0
$263$	14.2975	0.881622	0.440811	−	0.897600i	$-0.354691\pi$
$263$	14.2975	0.881622	0.440811	+	0.897600i	$0.354691\pi$
$264$	0	0
$265$	13.3994	0.823118
$266$	0	0
$267$	5.61852	0.343848
$268$	0	0
$269$	−18.0183	−1.09859	−0.549297	−	0.835627i	$-0.685104\pi$
$269$	−18.0183	−1.09859	−0.549297	+	0.835627i	$0.685104\pi$
$270$	0	0
$271$	−5.40925	−0.328589	−0.164294	−	0.986411i	$-0.552535\pi$
$271$	−5.40925	−0.328589	−0.164294	+	0.986411i	$0.552535\pi$
$272$	0	0
$273$	0.476920	0.0288645
$274$	0	0
$275$	0.538350	0.0324637
$276$	0	0
$277$	0.475355	0.0285613	0.0142806	−	0.999898i	$-0.495454\pi$
$277$	0.475355	0.0285613	0.0142806	+	0.999898i	$0.495454\pi$
$278$	0	0
$279$	−20.8298	−1.24705
$280$	0	0
$281$	−9.23067	−0.550655	−0.275328	−	0.961350i	$-0.588786\pi$
$281$	−9.23067	−0.550655	−0.275328	+	0.961350i	$0.588786\pi$
$282$	0	0
$283$	−22.6333	−1.34541	−0.672704	−	0.739911i	$-0.734867\pi$
$283$	−22.6333	−1.34541	−0.672704	+	0.739911i	$0.734867\pi$
$284$	0	0
$285$	−25.4226	−1.50590
$286$	0	0
$287$	0.383953	0.0226640
$288$	0	0
$289$	31.9310	1.87829
$290$	0	0
$291$	53.5126	3.13696
$292$	0	0
$293$	−10.0171	−0.585207	−0.292604	−	0.956234i	$-0.594522\pi$
$293$	−10.0171	−0.585207	−0.292604	+	0.956234i	$0.594522\pi$
$294$	0	0
$295$	17.8285	1.03802
$296$	0	0
$297$	6.34606	0.368236
$298$	0	0
$299$	0.230783	0.0133465
$300$	0	0
$301$	−0.305143	−0.0175882
$302$	0	0
$303$	43.8502	2.51913
$304$	0	0
$305$	9.27984	0.531362
$306$	0	0
$307$	−16.0119	−0.913849	−0.456925	−	0.889505i	$-0.651049\pi$
$307$	−16.0119	−0.913849	−0.456925	+	0.889505i	$0.651049\pi$
$308$	0	0
$309$	52.3748	2.97950
$310$	0	0
$311$	10.3613	0.587537	0.293768	−	0.955877i	$-0.405091\pi$
$311$	10.3613	0.587537	0.293768	+	0.955877i	$0.405091\pi$
$312$	0	0
$313$	9.36002	0.529059	0.264530	−	0.964378i	$-0.414783\pi$
$313$	9.36002	0.529059	0.264530	+	0.964378i	$0.414783\pi$
$314$	0	0
$315$	6.99507	0.394127
$316$	0	0
$317$	12.8595	0.722260	0.361130	−	0.932515i	$-0.382391\pi$
$317$	12.8595	0.722260	0.361130	+	0.932515i	$0.382391\pi$
$318$	0	0
$319$	−0.437093	−0.0244725
$320$	0	0
$321$	29.6133	1.65285
$322$	0	0
$323$	−28.4452	−1.58273
$324$	0	0
$325$	0.295373	0.0163844
$326$	0	0
$327$	−63.6012	−3.51715
$328$	0	0
$329$	1.79535	0.0989806
$330$	0	0
$331$	2.81289	0.154611	0.0773053	−	0.997007i	$-0.475368\pi$
$331$	2.81289	0.154611	0.0773053	+	0.997007i	$0.475368\pi$
$332$	0	0
$333$	−26.7890	−1.46803
$334$	0	0
$335$	−13.1055	−0.716029
$336$	0	0
$337$	19.3021	1.05145	0.525725	−	0.850654i	$-0.323794\pi$
$337$	19.3021	1.05145	0.525725	+	0.850654i	$0.323794\pi$
$338$	0	0
$339$	−52.5992	−2.85679
$340$	0	0
$341$	1.65612	0.0896839
$342$	0	0
$343$	−7.09373	−0.383025
$344$	0	0
$345$	4.88466	0.262981
$346$	0	0
$347$	−28.4120	−1.52524	−0.762618	−	0.646849i	$-0.776086\pi$
$347$	−28.4120	−1.52524	−0.762618	+	0.646849i	$0.776086\pi$
$348$	0	0
$349$	7.74264	0.414454	0.207227	−	0.978293i	$-0.433556\pi$
$349$	7.74264	0.414454	0.207227	+	0.978293i	$0.433556\pi$
$350$	0	0
$351$	3.48185	0.185847
$352$	0	0
$353$	19.3204	1.02832	0.514161	−	0.857694i	$-0.328104\pi$
$353$	19.3204	1.02832	0.514161	+	0.857694i	$0.328104\pi$
$354$	0	0
$355$	−24.9310	−1.32320
$356$	0	0
$357$	11.2945	0.597768
$358$	0	0
$359$	2.57278	0.135786	0.0678932	−	0.997693i	$-0.478372\pi$
$359$	2.57278	0.135786	0.0678932	+	0.997693i	$0.478372\pi$
$360$	0	0
$361$	−2.46383	−0.129675
$362$	0	0
$363$	33.4787	1.75718
$364$	0	0
$365$	15.5528	0.814070
$366$	0	0
$367$	23.1615	1.20902	0.604509	−	0.796598i	$-0.293369\pi$
$367$	23.1615	1.20902	0.604509	+	0.796598i	$0.293369\pi$
$368$	0	0
$369$	5.03308	0.262012
$370$	0	0
$371$	3.46065	0.179668
$372$	0	0
$373$	20.9233	1.08337	0.541684	−	0.840582i	$-0.317787\pi$
$373$	20.9233	1.08337	0.541684	+	0.840582i	$0.317787\pi$
$374$	0	0
$375$	37.5105	1.93703
$376$	0	0
$377$	−0.239817	−0.0123512
$378$	0	0
$379$	−30.4945	−1.56640	−0.783198	−	0.621773i	$-0.786413\pi$
$379$	−30.4945	−1.56640	−0.783198	+	0.621773i	$0.786413\pi$
$380$	0	0
$381$	29.7827	1.52582
$382$	0	0
$383$	−27.8997	−1.42561	−0.712804	−	0.701363i	$-0.752575\pi$
$383$	−27.8997	−1.42561	−0.712804	+	0.701363i	$0.752575\pi$
$384$	0	0
$385$	−0.556158	−0.0283444
$386$	0	0
$387$	−4.00000	−0.203331
$388$	0	0
$389$	18.9087	0.958707	0.479353	−	0.877622i	$-0.340871\pi$
$389$	18.9087	0.958707	0.479353	+	0.877622i	$0.340871\pi$
$390$	0	0
$391$	5.46544	0.276399
$392$	0	0
$393$	−51.8298	−2.61447
$394$	0	0
$395$	9.77058	0.491611
$396$	0	0
$397$	−23.2771	−1.16824	−0.584122	−	0.811666i	$-0.698561\pi$
$397$	−23.2771	−1.16824	−0.584122	+	0.811666i	$0.698561\pi$
$398$	0	0
$399$	−6.56587	−0.328705
$400$	0	0
$401$	−23.3692	−1.16700	−0.583500	−	0.812113i	$-0.698317\pi$
$401$	−23.3692	−1.16700	−0.583500	+	0.812113i	$0.698317\pi$
$402$	0	0
$403$	0.908652	0.0452632
$404$	0	0
$405$	33.0689	1.64321
$406$	0	0
$407$	2.12992	0.105576
$408$	0	0
$409$	−13.5940	−0.672178	−0.336089	−	0.941830i	$-0.609104\pi$
$409$	−13.5940	−0.672178	−0.336089	+	0.941830i	$0.609104\pi$
$410$	0	0
$411$	−11.0626	−0.545676
$412$	0	0
$413$	4.60456	0.226576
$414$	0	0
$415$	2.22776	0.109356
$416$	0	0
$417$	−29.3839	−1.43894
$418$	0	0
$419$	−20.6740	−1.00999	−0.504996	−	0.863121i	$-0.668506\pi$
$419$	−20.6740	−1.00999	−0.504996	+	0.863121i	$0.668506\pi$
$420$	0	0
$421$	33.9625	1.65523	0.827614	−	0.561297i	$-0.189698\pi$
$421$	33.9625	1.65523	0.827614	+	0.561297i	$0.189698\pi$
$422$	0	0
$423$	23.5345	1.14428
$424$	0	0
$425$	6.99507	0.339311
$426$	0	0
$427$	2.39670	0.115984
$428$	0	0
$429$	−0.497059	−0.0239982
$430$	0	0
$431$	4.35940	0.209985	0.104992	−	0.994473i	$-0.466518\pi$
$431$	4.35940	0.209985	0.104992	+	0.994473i	$0.466518\pi$
$432$	0	0
$433$	−8.95196	−0.430204	−0.215102	−	0.976592i	$-0.569008\pi$
$433$	−8.95196	−0.430204	−0.215102	+	0.976592i	$0.569008\pi$
$434$	0	0
$435$	−5.07587	−0.243369
$436$	0	0
$437$	−3.17724	−0.151988
$438$	0	0
$439$	−30.0473	−1.43408	−0.717039	−	0.697033i	$-0.754503\pi$
$439$	−30.0473	−1.43408	−0.717039	+	0.697033i	$0.754503\pi$
$440$	0	0
$441$	−45.5911	−2.17100
$442$	0	0
$443$	14.2252	0.675858	0.337929	−	0.941172i	$-0.390274\pi$
$443$	14.2252	0.675858	0.337929	+	0.941172i	$0.390274\pi$
$444$	0	0
$445$	−3.59485	−0.170412
$446$	0	0
$447$	58.0815	2.74716
$448$	0	0
$449$	−5.54334	−0.261606	−0.130803	−	0.991408i	$-0.541756\pi$
$449$	−5.54334	−0.261606	−0.130803	+	0.991408i	$0.541756\pi$
$450$	0	0
$451$	−0.400165	−0.0188431
$452$	0	0
$453$	53.9663	2.53556
$454$	0	0
$455$	−0.305143	−0.0143053
$456$	0	0
$457$	−11.2195	−0.524826	−0.262413	−	0.964956i	$-0.584518\pi$
$457$	−11.2195	−0.524826	−0.262413	+	0.964956i	$0.584518\pi$
$458$	0	0
$459$	82.4577	3.84880
$460$	0	0
$461$	−1.92450	−0.0896328	−0.0448164	−	0.998995i	$-0.514270\pi$
$461$	−1.92450	−0.0896328	−0.0448164	+	0.998995i	$0.514270\pi$
$462$	0	0
$463$	−26.4339	−1.22849	−0.614244	−	0.789116i	$-0.710539\pi$
$463$	−26.4339	−1.22849	−0.614244	+	0.789116i	$0.710539\pi$
$464$	0	0
$465$	19.2322	0.891871
$466$	0	0
$467$	31.5897	1.46180	0.730898	−	0.682486i	$-0.239101\pi$
$467$	31.5897	1.46180	0.730898	+	0.682486i	$0.239101\pi$
$468$	0	0
$469$	−3.38474	−0.156293
$470$	0	0
$471$	22.5020	1.03684
$472$	0	0
$473$	0.318028	0.0146230
$474$	0	0
$475$	−4.06647	−0.186582
$476$	0	0
$477$	45.3643	2.07709
$478$	0	0
$479$	−12.8045	−0.585052	−0.292526	−	0.956258i	$-0.594496\pi$
$479$	−12.8045	−0.585052	−0.292526	+	0.956258i	$0.594496\pi$
$480$	0	0
$481$	1.16861	0.0532839
$482$	0	0
$483$	1.26156	0.0574029
$484$	0	0
$485$	−34.2385	−1.55469
$486$	0	0
$487$	38.2482	1.73319	0.866596	−	0.499011i	$-0.166303\pi$
$487$	38.2482	1.73319	0.866596	+	0.499011i	$0.166303\pi$
$488$	0	0
$489$	74.0669	3.34942
$490$	0	0
$491$	38.4176	1.73376	0.866881	−	0.498514i	$-0.166121\pi$
$491$	38.4176	1.73376	0.866881	+	0.498514i	$0.166121\pi$
$492$	0	0
$493$	−5.67938	−0.255786
$494$	0	0
$495$	−7.29044	−0.327681
$496$	0	0
$497$	−6.43891	−0.288825
$498$	0	0
$499$	3.61884	0.162001	0.0810007	−	0.996714i	$-0.474188\pi$
$499$	3.61884	0.162001	0.0810007	+	0.996714i	$0.474188\pi$
$500$	0	0
$501$	−3.12588	−0.139654
$502$	0	0
$503$	31.3565	1.39812	0.699059	−	0.715064i	$-0.253602\pi$
$503$	31.3565	1.39812	0.699059	+	0.715064i	$0.253602\pi$
$504$	0	0
$505$	−28.0563	−1.24849
$506$	0	0
$507$	40.3637	1.79261
$508$	0	0
$509$	22.9722	1.01822	0.509111	−	0.860701i	$-0.329974\pi$
$509$	22.9722	1.01822	0.509111	+	0.860701i	$0.329974\pi$
$510$	0	0
$511$	4.01681	0.177693
$512$	0	0
$513$	−47.9354	−2.11640
$514$	0	0
$515$	−33.5105	−1.47665
$516$	0	0
$517$	−1.87116	−0.0822934
$518$	0	0
$519$	61.0415	2.67942
$520$	0	0
$521$	12.1707	0.533210	0.266605	−	0.963806i	$-0.414098\pi$
$521$	12.1707	0.533210	0.266605	+	0.963806i	$0.414098\pi$
$522$	0	0
$523$	−6.62991	−0.289906	−0.144953	−	0.989439i	$-0.546303\pi$
$523$	−6.62991	−0.289906	−0.144953	+	0.989439i	$0.546303\pi$
$524$	0	0
$525$	1.61464	0.0704685
$526$	0	0
$527$	21.5188	0.937375
$528$	0	0
$529$	−22.3895	−0.973458
$530$	0	0
$531$	60.3593	2.61937
$532$	0	0
$533$	−0.219556	−0.00951003
$534$	0	0
$535$	−18.9472	−0.819159
$536$	0	0
$537$	−32.8336	−1.41688
$538$	0	0
$539$	3.62481	0.156132
$540$	0	0
$541$	−14.6963	−0.631842	−0.315921	−	0.948785i	$-0.602313\pi$
$541$	−14.6963	−0.631842	−0.315921	+	0.948785i	$0.602313\pi$
$542$	0	0
$543$	−26.1381	−1.12169
$544$	0	0
$545$	40.6934	1.74311
$546$	0	0
$547$	−40.9360	−1.75030	−0.875148	−	0.483856i	$-0.839236\pi$
$547$	−40.9360	−1.75030	−0.875148	+	0.483856i	$0.839236\pi$
$548$	0	0
$549$	31.4174	1.34086
$550$	0	0
$551$	3.30162	0.140654
$552$	0	0
$553$	2.52344	0.107308
$554$	0	0
$555$	24.7343	1.04991
$556$	0	0
$557$	41.2803	1.74910	0.874552	−	0.484932i	$-0.161156\pi$
$557$	41.2803	1.74910	0.874552	+	0.484932i	$0.161156\pi$
$558$	0	0
$559$	0.174491	0.00738017
$560$	0	0
$561$	−11.7714	−0.496989
$562$	0	0
$563$	−15.4103	−0.649467	−0.324733	−	0.945806i	$-0.605275\pi$
$563$	−15.4103	−0.649467	−0.324733	+	0.945806i	$0.605275\pi$
$564$	0	0
$565$	33.6540	1.41584
$566$	0	0
$567$	8.54069	0.358675
$568$	0	0
$569$	11.2870	0.473177	0.236588	−	0.971610i	$-0.423971\pi$
$569$	11.2870	0.473177	0.236588	+	0.971610i	$0.423971\pi$
$570$	0	0
$571$	−1.50980	−0.0631830	−0.0315915	−	0.999501i	$-0.510058\pi$
$571$	−1.50980	−0.0631830	−0.0315915	+	0.999501i	$0.510058\pi$
$572$	0	0
$573$	29.5034	1.23252
$574$	0	0
$575$	0.781327	0.0325836
$576$	0	0
$577$	24.9800	1.03993	0.519966	−	0.854187i	$-0.325945\pi$
$577$	24.9800	1.03993	0.519966	+	0.854187i	$0.325945\pi$
$578$	0	0
$579$	51.0428	2.12127
$580$	0	0
$581$	0.575363	0.0238701
$582$	0	0
$583$	−3.60678	−0.149378
$584$	0	0
$585$	−4.00000	−0.165380
$586$	0	0
$587$	−20.9400	−0.864287	−0.432144	−	0.901805i	$-0.642243\pi$
$587$	−20.9400	−0.864287	−0.432144	+	0.901805i	$0.642243\pi$
$588$	0	0
$589$	−12.5096	−0.515450
$590$	0	0
$591$	54.2214	2.23037
$592$	0	0
$593$	−32.8904	−1.35065	−0.675323	−	0.737522i	$-0.735996\pi$
$593$	−32.8904	−1.35065	−0.675323	+	0.737522i	$0.735996\pi$
$594$	0	0
$595$	−7.22645	−0.296256
$596$	0	0
$597$	20.1962	0.826575
$598$	0	0
$599$	18.3583	0.750098	0.375049	−	0.927005i	$-0.377626\pi$
$599$	18.3583	0.750098	0.375049	+	0.927005i	$0.377626\pi$
$600$	0	0
$601$	−45.8253	−1.86925	−0.934626	−	0.355633i	$-0.884265\pi$
$601$	−45.8253	−1.86925	−0.934626	+	0.355633i	$0.884265\pi$
$602$	0	0
$603$	−44.3692	−1.80685
$604$	0	0
$605$	−21.4204	−0.870861
$606$	0	0
$607$	−17.6455	−0.716210	−0.358105	−	0.933681i	$-0.616577\pi$
$607$	−17.6455	−0.716210	−0.358105	+	0.933681i	$0.616577\pi$
$608$	0	0
$609$	−1.31094	−0.0531221
$610$	0	0
$611$	−1.02664	−0.0415332
$612$	0	0
$613$	2.27761	0.0919919	0.0459960	−	0.998942i	$-0.485354\pi$
$613$	2.27761	0.0919919	0.0459960	+	0.998942i	$0.485354\pi$
$614$	0	0
$615$	−4.64704	−0.187387
$616$	0	0
$617$	1.33514	0.0537507	0.0268753	−	0.999639i	$-0.491444\pi$
$617$	1.33514	0.0537507	0.0268753	+	0.999639i	$0.491444\pi$
$618$	0	0
$619$	−16.6808	−0.670458	−0.335229	−	0.942137i	$-0.608814\pi$
$619$	−16.6808	−0.670458	−0.335229	+	0.942137i	$0.608814\pi$
$620$	0	0
$621$	9.21027	0.369595
$622$	0	0
$623$	−0.928440	−0.0371971
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	6.84312	0.273288
$628$	0	0
$629$	27.6751	1.10348
$630$	0	0
$631$	3.60704	0.143594	0.0717971	−	0.997419i	$-0.477127\pi$
$631$	3.60704	0.143594	0.0717971	+	0.997419i	$0.477127\pi$
$632$	0	0
$633$	−7.53445	−0.299468
$634$	0	0
$635$	−19.0556	−0.756199
$636$	0	0
$637$	1.98880	0.0787992
$638$	0	0
$639$	−84.4050	−3.33901
$640$	0	0
$641$	10.2139	0.403426	0.201713	−	0.979445i	$-0.435349\pi$
$641$	10.2139	0.403426	0.201713	+	0.979445i	$0.435349\pi$
$642$	0	0
$643$	−2.82338	−0.111343	−0.0556717	−	0.998449i	$-0.517730\pi$
$643$	−2.82338	−0.111343	−0.0556717	+	0.998449i	$0.517730\pi$
$644$	0	0
$645$	3.69320	0.145420
$646$	0	0
$647$	41.4408	1.62921	0.814604	−	0.580018i	$-0.196954\pi$
$647$	41.4408	1.62921	0.814604	+	0.580018i	$0.196954\pi$
$648$	0	0
$649$	−4.79900	−0.188377
$650$	0	0
$651$	4.96708	0.194675
$652$	0	0
$653$	−43.7584	−1.71240	−0.856200	−	0.516645i	$-0.827181\pi$
$653$	−43.7584	−1.71240	−0.856200	+	0.516645i	$0.827181\pi$
$654$	0	0
$655$	33.1618	1.29574
$656$	0	0
$657$	52.6547	2.05426
$658$	0	0
$659$	−7.47042	−0.291006	−0.145503	−	0.989358i	$-0.546480\pi$
$659$	−7.47042	−0.291006	−0.145503	+	0.989358i	$0.546480\pi$
$660$	0	0
$661$	28.3065	1.10099	0.550497	−	0.834837i	$-0.314438\pi$
$661$	28.3065	1.10099	0.550497	+	0.834837i	$0.314438\pi$
$662$	0	0
$663$	−6.45854	−0.250829
$664$	0	0
$665$	4.20098	0.162907
$666$	0	0
$667$	−0.634369	−0.0245629
$668$	0	0
$669$	−2.49478	−0.0964538
$670$	0	0
$671$	−2.49790	−0.0964305
$672$	0	0
$673$	−31.8789	−1.22884	−0.614421	−	0.788979i	$-0.710610\pi$
$673$	−31.8789	−1.22884	−0.614421	+	0.788979i	$0.710610\pi$
$674$	0	0
$675$	11.7880	0.453719
$676$	0	0
$677$	34.6470	1.33159	0.665797	−	0.746133i	$-0.268092\pi$
$677$	34.6470	1.33159	0.665797	+	0.746133i	$0.268092\pi$
$678$	0	0
$679$	−8.84275	−0.339354
$680$	0	0
$681$	−32.8151	−1.25748
$682$	0	0
$683$	−7.53159	−0.288188	−0.144094	−	0.989564i	$-0.546027\pi$
$683$	−7.53159	−0.288188	−0.144094	+	0.989564i	$0.546027\pi$
$684$	0	0
$685$	7.07806	0.270439
$686$	0	0
$687$	−36.5464	−1.39433
$688$	0	0
$689$	−1.97891	−0.0753905
$690$	0	0
$691$	34.3094	1.30519	0.652596	−	0.757706i	$-0.273680\pi$
$691$	34.3094	1.30519	0.652596	+	0.757706i	$0.273680\pi$
$692$	0	0
$693$	−1.88290	−0.0715254
$694$	0	0
$695$	18.8004	0.713141
$696$	0	0
$697$	−5.19956	−0.196947
$698$	0	0
$699$	74.6639	2.82405
$700$	0	0
$701$	0.0251855	0.000951242 0	0.000475621	−	1.00000i	$-0.499849\pi$
$701$	0.0251855	0.000951242 0	0.000475621		1.00000i	$0.499849\pi$
$702$	0	0
$703$	−16.0885	−0.606789
$704$	0	0
$705$	−21.7294	−0.818375
$706$	0	0
$707$	−7.24608	−0.272517
$708$	0	0
$709$	−28.2131	−1.05957	−0.529783	−	0.848133i	$-0.677727\pi$
$709$	−28.2131	−1.05957	−0.529783	+	0.848133i	$0.677727\pi$
$710$	0	0
$711$	33.0788	1.24055
$712$	0	0
$713$	2.40359	0.0900150
$714$	0	0
$715$	0.318028	0.0118936
$716$	0	0
$717$	68.1600	2.54548
$718$	0	0
$719$	24.5507	0.915585	0.457792	−	0.889059i	$-0.348640\pi$
$719$	24.5507	0.915585	0.457792	+	0.889059i	$0.348640\pi$
$720$	0	0
$721$	−8.65474	−0.322319
$722$	0	0
$723$	−54.6877	−2.03386
$724$	0	0
$725$	−0.811912	−0.0301537
$726$	0	0
$727$	−16.2103	−0.601208	−0.300604	−	0.953749i	$-0.597188\pi$
$727$	−16.2103	−0.601208	−0.300604	+	0.953749i	$0.597188\pi$
$728$	0	0
$729$	1.41315	0.0523389
$730$	0	0
$731$	4.13231	0.152839
$732$	0	0
$733$	−16.0087	−0.591296	−0.295648	−	0.955297i	$-0.595535\pi$
$733$	−16.0087	−0.591296	−0.295648	+	0.955297i	$0.595535\pi$
$734$	0	0
$735$	42.0942	1.55267
$736$	0	0
$737$	3.52767	0.129943
$738$	0	0
$739$	11.9536	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$739$	11.9536	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$740$	0	0
$741$	3.75457	0.137928
$742$	0	0
$743$	−38.6259	−1.41705	−0.708524	−	0.705686i	$-0.750639\pi$
$743$	−38.6259	−1.41705	−0.708524	+	0.705686i	$0.750639\pi$
$744$	0	0
$745$	−37.1618	−1.36150
$746$	0	0
$747$	7.54219	0.275954
$748$	0	0
$749$	−4.89348	−0.178804
$750$	0	0
$751$	−21.6589	−0.790346	−0.395173	−	0.918607i	$-0.629315\pi$
$751$	−21.6589	−0.790346	−0.395173	+	0.918607i	$0.629315\pi$
$752$	0	0
$753$	−3.96639	−0.144543
$754$	0	0
$755$	−34.5287	−1.25663
$756$	0	0
$757$	43.6016	1.58473	0.792364	−	0.610048i	$-0.208850\pi$
$757$	43.6016	1.58473	0.792364	+	0.610048i	$0.208850\pi$
$758$	0	0
$759$	−1.31483	−0.0477253
$760$	0	0
$761$	32.5442	1.17973	0.589863	−	0.807504i	$-0.299182\pi$
$761$	32.5442	1.17973	0.589863	+	0.807504i	$0.299182\pi$
$762$	0	0
$763$	10.5099	0.380482
$764$	0	0
$765$	−94.7286	−3.42492
$766$	0	0
$767$	−2.63303	−0.0950734
$768$	0	0
$769$	−11.9886	−0.432319	−0.216159	−	0.976358i	$-0.569353\pi$
$769$	−11.9886	−0.432319	−0.216159	+	0.976358i	$0.569353\pi$
$770$	0	0
$771$	−82.8882	−2.98515
$772$	0	0
$773$	31.3355	1.12706	0.563530	−	0.826096i	$-0.309443\pi$
$773$	31.3355	1.12706	0.563530	+	0.826096i	$0.309443\pi$
$774$	0	0
$775$	3.07629	0.110503
$776$	0	0
$777$	6.38811	0.229172
$778$	0	0
$779$	3.02268	0.108299
$780$	0	0
$781$	6.71080	0.240131
$782$	0	0
$783$	−9.57080	−0.342032
$784$	0	0
$785$	−14.3973	−0.513860
$786$	0	0
$787$	−9.39669	−0.334956	−0.167478	−	0.985876i	$-0.553562\pi$
$787$	−9.39669	−0.334956	−0.167478	+	0.985876i	$0.553562\pi$
$788$	0	0
$789$	−44.6922	−1.59109
$790$	0	0
$791$	8.69181	0.309045
$792$	0	0
$793$	−1.37051	−0.0486682
$794$	0	0
$795$	−41.8848	−1.48550
$796$	0	0
$797$	21.7756	0.771330	0.385665	−	0.922639i	$-0.373972\pi$
$797$	21.7756	0.771330	0.385665	+	0.922639i	$0.373972\pi$
$798$	0	0
$799$	−24.3129	−0.860129
$800$	0	0
$801$	−12.1705	−0.430025
$802$	0	0
$803$	−4.18642	−0.147736
$804$	0	0
$805$	−0.807172	−0.0284491
$806$	0	0
$807$	56.3229	1.98266
$808$	0	0
$809$	−1.83916	−0.0646614	−0.0323307	−	0.999477i	$-0.510293\pi$
$809$	−1.83916	−0.0646614	−0.0323307	+	0.999477i	$0.510293\pi$
$810$	0	0
$811$	−7.27579	−0.255488	−0.127744	−	0.991807i	$-0.540774\pi$
$811$	−7.27579	−0.255488	−0.127744	+	0.991807i	$0.540774\pi$
$812$	0	0
$813$	16.9087	0.593012
$814$	0	0
$815$	−47.3895	−1.65998
$816$	0	0
$817$	−2.40225	−0.0840441
$818$	0	0
$819$	−1.03308	−0.0360987
$820$	0	0
$821$	−45.0675	−1.57287	−0.786434	−	0.617674i	$-0.788075\pi$
$821$	−45.0675	−1.57287	−0.786434	+	0.617674i	$0.788075\pi$
$822$	0	0
$823$	8.71724	0.303864	0.151932	−	0.988391i	$-0.451451\pi$
$823$	8.71724	0.303864	0.151932	+	0.988391i	$0.451451\pi$
$824$	0	0
$825$	−1.68282	−0.0585881
$826$	0	0
$827$	8.18072	0.284471	0.142236	−	0.989833i	$-0.454571\pi$
$827$	8.18072	0.284471	0.142236	+	0.989833i	$0.454571\pi$
$828$	0	0
$829$	−43.1349	−1.49814	−0.749068	−	0.662493i	$-0.769498\pi$
$829$	−43.1349	−1.49814	−0.749068	+	0.662493i	$0.769498\pi$
$830$	0	0
$831$	−1.48590	−0.0515453
$832$	0	0
$833$	47.0991	1.63189
$834$	0	0
$835$	2.00000	0.0692129
$836$	0	0
$837$	36.2632	1.25344
$838$	0	0
$839$	37.2048	1.28445	0.642226	−	0.766515i	$-0.278011\pi$
$839$	37.2048	1.28445	0.642226	+	0.766515i	$0.278011\pi$
$840$	0	0
$841$	−28.3408	−0.977269
$842$	0	0
$843$	28.8539	0.993782
$844$	0	0
$845$	−25.8255	−0.888425
$846$	0	0
$847$	−5.53222	−0.190089
$848$	0	0
$849$	70.7488	2.42809
$850$	0	0
$851$	3.09123	0.105966
$852$	0	0
$853$	11.9553	0.409341	0.204671	−	0.978831i	$-0.434388\pi$
$853$	11.9553	0.409341	0.204671	+	0.978831i	$0.434388\pi$
$854$	0	0
$855$	55.0689	1.88332
$856$	0	0
$857$	−2.28383	−0.0780142	−0.0390071	−	0.999239i	$-0.512420\pi$
$857$	−2.28383	−0.0780142	−0.0390071	+	0.999239i	$0.512420\pi$
$858$	0	0
$859$	16.8387	0.574529	0.287265	−	0.957851i	$-0.407254\pi$
$859$	16.8387	0.574529	0.287265	+	0.957851i	$0.407254\pi$
$860$	0	0
$861$	−1.20019	−0.0409023
$862$	0	0
$863$	31.2591	1.06407	0.532036	−	0.846721i	$-0.321427\pi$
$863$	31.2591	1.06407	0.532036	+	0.846721i	$0.321427\pi$
$864$	0	0
$865$	−39.0556	−1.32793
$866$	0	0
$867$	−99.8123	−3.38980
$868$	0	0
$869$	−2.63000	−0.0892165
$870$	0	0
$871$	1.93550	0.0655820
$872$	0	0
$873$	−115.916	−3.92316
$874$	0	0
$875$	−6.19847	−0.209546
$876$	0	0
$877$	−6.86671	−0.231872	−0.115936	−	0.993257i	$-0.536987\pi$
$877$	−6.86671	−0.231872	−0.115936	+	0.993257i	$0.536987\pi$
$878$	0	0
$879$	31.3123	1.05614
$880$	0	0
$881$	−9.06820	−0.305515	−0.152758	−	0.988264i	$-0.548815\pi$
$881$	−9.06820	−0.305515	−0.152758	+	0.988264i	$0.548815\pi$
$882$	0	0
$883$	2.28390	0.0768594	0.0384297	−	0.999261i	$-0.487764\pi$
$883$	2.28390	0.0768594	0.0384297	+	0.999261i	$0.487764\pi$
$884$	0	0
$885$	−55.7298	−1.87334
$886$	0	0
$887$	31.8293	1.06872	0.534361	−	0.845256i	$-0.320552\pi$
$887$	31.8293	1.06872	0.534361	+	0.845256i	$0.320552\pi$
$888$	0	0
$889$	−4.92148	−0.165061
$890$	0	0
$891$	−8.90133	−0.298206
$892$	0	0
$893$	14.1339	0.472974
$894$	0	0
$895$	21.0076	0.702208
$896$	0	0
$897$	−0.721399	−0.0240868
$898$	0	0
$899$	−2.49767	−0.0833021
$900$	0	0
$901$	−46.8648	−1.56129
$902$	0	0
$903$	0.953840	0.0317418
$904$	0	0
$905$	16.7237	0.555914
$906$	0	0
$907$	54.0598	1.79502	0.897512	−	0.440989i	$-0.145372\pi$
$907$	54.0598	1.79502	0.897512	+	0.440989i	$0.145372\pi$
$908$	0	0
$909$	−94.9859	−3.15048
$910$	0	0
$911$	−6.19867	−0.205371	−0.102686	−	0.994714i	$-0.532744\pi$
$911$	−6.19867	−0.205371	−0.102686	+	0.994714i	$0.532744\pi$
$912$	0	0
$913$	−0.599658	−0.0198458
$914$	0	0
$915$	−29.0076	−0.958963
$916$	0	0
$917$	8.56467	0.282830
$918$	0	0
$919$	−6.53624	−0.215611	−0.107805	−	0.994172i	$-0.534382\pi$
$919$	−6.53624	−0.215611	−0.107805	+	0.994172i	$0.534382\pi$
$920$	0	0
$921$	50.0513	1.64925
$922$	0	0
$923$	3.68197	0.121194
$924$	0	0
$925$	3.95638	0.130085
$926$	0	0
$927$	−113.451	−3.72623
$928$	0	0
$929$	21.4372	0.703331	0.351666	−	0.936126i	$-0.385615\pi$
$929$	21.4372	0.703331	0.351666	+	0.936126i	$0.385615\pi$
$930$	0	0
$931$	−27.3803	−0.897353
$932$	0	0
$933$	−32.3882	−1.06034
$934$	0	0
$935$	7.53159	0.246309
$936$	0	0
$937$	−1.58083	−0.0516434	−0.0258217	−	0.999667i	$-0.508220\pi$
$937$	−1.58083	−0.0516434	−0.0258217	+	0.999667i	$0.508220\pi$
$938$	0	0
$939$	−29.2582	−0.954807
$940$	0	0
$941$	10.6995	0.348794	0.174397	−	0.984675i	$-0.444202\pi$
$941$	10.6995	0.348794	0.174397	+	0.984675i	$0.444202\pi$
$942$	0	0
$943$	−0.580775	−0.0189126
$944$	0	0
$945$	−12.1779	−0.396147
$946$	0	0
$947$	−29.2664	−0.951030	−0.475515	−	0.879708i	$-0.657738\pi$
$947$	−29.2664	−0.951030	−0.475515	+	0.879708i	$0.657738\pi$
$948$	0	0
$949$	−2.29694	−0.0745618
$950$	0	0
$951$	−40.1971	−1.30348
$952$	0	0
$953$	−45.0155	−1.45820	−0.729098	−	0.684410i	$-0.760060\pi$
$953$	−45.0155	−1.45820	−0.729098	+	0.684410i	$0.760060\pi$
$954$	0	0
$955$	−18.8769	−0.610842
$956$	0	0
$957$	1.36630	0.0441661
$958$	0	0
$959$	1.82805	0.0590307
$960$	0	0
$961$	−21.5365	−0.694725
$962$	0	0
$963$	−64.1466	−2.06710
$964$	0	0
$965$	−32.6583	−1.05131
$966$	0	0
$967$	−61.7031	−1.98424	−0.992118	−	0.125304i	$-0.960009\pi$
$967$	−61.7031	−1.98424	−0.992118	+	0.125304i	$0.960009\pi$
$968$	0	0
$969$	88.9162	2.85640
$970$	0	0
$971$	11.4233	0.366590	0.183295	−	0.983058i	$-0.441324\pi$
$971$	11.4233	0.366590	0.183295	+	0.983058i	$0.441324\pi$
$972$	0	0
$973$	4.85558	0.155663
$974$	0	0
$975$	−0.923300	−0.0295693
$976$	0	0
$977$	−3.28909	−0.105227	−0.0526137	−	0.998615i	$-0.516755\pi$
$977$	−3.28909	−0.105227	−0.0526137	+	0.998615i	$0.516755\pi$
$978$	0	0
$979$	0.967644	0.0309260
$980$	0	0
$981$	137.769	4.39864
$982$	0	0
$983$	−15.4572	−0.493009	−0.246505	−	0.969142i	$-0.579282\pi$
$983$	−15.4572	−0.493009	−0.246505	+	0.969142i	$0.579282\pi$
$984$	0	0
$985$	−34.6920	−1.10538
$986$	0	0
$987$	−5.61203	−0.178633
$988$	0	0
$989$	0.461566	0.0146770
$990$	0	0
$991$	21.3319	0.677630	0.338815	−	0.940853i	$-0.389974\pi$
$991$	21.3319	0.677630	0.338815	+	0.940853i	$0.389974\pi$
$992$	0	0
$993$	−8.79276	−0.279030
$994$	0	0
$995$	−12.9219	−0.409653
$996$	0	0
$997$	−36.9757	−1.17103	−0.585516	−	0.810661i	$-0.699108\pi$
$997$	−36.9757	−1.17103	−0.585516	+	0.810661i	$0.699108\pi$
$998$	0	0
$999$	46.6377	1.47555

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2672.2.a.j.1.1		5
4.3	odd	2		668.2.a.b.1.5	✓	5
12.11	even	2		6012.2.a.d.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
668.2.a.b.1.5	✓	5	4.3	odd	2
2672.2.a.j.1.1		5	1.1	even	1	trivial
6012.2.a.d.1.2		5	12.11	even	2

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:	\( N \)	\(=\)	\( 2672 = 2^{4} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2672.a (trivial)

\(f(q)\)	\(=\)	\(q-3.12588 q^{3} +2.00000 q^{5} +0.516539 q^{7} +6.77110 q^{9} +O(q^{10})\)	\(q-3.12588 q^{3} +2.00000 q^{5} +0.516539 q^{7} +6.77110 q^{9} -0.538350 q^{11} -0.295373 q^{13} -6.25175 q^{15} -6.99507 q^{17} +4.06647 q^{19} -1.61464 q^{21} -0.781327 q^{23} -1.00000 q^{25} -11.7880 q^{27} +0.811912 q^{29} -3.07629 q^{31} +1.68282 q^{33} +1.03308 q^{35} -3.95638 q^{37} +0.923300 q^{39} +0.743318 q^{41} -0.590746 q^{43} +13.5422 q^{45} +3.47572 q^{47} -6.73319 q^{49} +21.8657 q^{51} +6.69970 q^{53} -1.07670 q^{55} -12.7113 q^{57} +8.91427 q^{59} +4.63992 q^{61} +3.49753 q^{63} -0.590746 q^{65} -6.55274 q^{67} +2.44233 q^{69} -12.4655 q^{71} +7.77640 q^{73} +3.12588 q^{75} -0.278079 q^{77} +4.88529 q^{79} +16.5345 q^{81} +1.11388 q^{83} -13.9901 q^{85} -2.53794 q^{87} -1.79742 q^{89} -0.152572 q^{91} +9.61609 q^{93} +8.13294 q^{95} -17.1192 q^{97} -3.64522 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 3 q^{3} + 10 q^{5} - 9 q^{7} + 6 q^{9}+O(q^{10}) \) 5 * q - 3 * q^3 + 10 * q^5 - 9 * q^7 + 6 * q^9	\( 5 q - 3 q^{3} + 10 q^{5} - 9 q^{7} + 6 q^{9} - 5 q^{11} - 4 q^{13} - 6 q^{15} - 2 q^{17} - 5 q^{19} - 4 q^{21} - 6 q^{23} - 5 q^{25} - 12 q^{27} - 5 q^{29} - 9 q^{31} - 8 q^{33} - 18 q^{35} + 8 q^{37} - 4 q^{41} - 8 q^{43} + 12 q^{45} - 13 q^{47} + 14 q^{49} - 2 q^{53} - 10 q^{55} - 12 q^{57} - 4 q^{59} + 11 q^{61} + q^{63} - 8 q^{65} - 28 q^{67} - 16 q^{69} - 2 q^{71} + 8 q^{73} + 3 q^{75} - 12 q^{77} + 10 q^{79} - 15 q^{81} - 2 q^{83} - 4 q^{85} - 4 q^{87} - 17 q^{89} + 12 q^{91} - 12 q^{93} - 10 q^{95} - 27 q^{97} - 3 q^{99}+O(q^{100}) \) 5 * q - 3 * q^3 + 10 * q^5 - 9 * q^7 + 6 * q^9 - 5 * q^11 - 4 * q^13 - 6 * q^15 - 2 * q^17 - 5 * q^19 - 4 * q^21 - 6 * q^23 - 5 * q^25 - 12 * q^27 - 5 * q^29 - 9 * q^31 - 8 * q^33 - 18 * q^35 + 8 * q^37 - 4 * q^41 - 8 * q^43 + 12 * q^45 - 13 * q^47 + 14 * q^49 - 2 * q^53 - 10 * q^55 - 12 * q^57 - 4 * q^59 + 11 * q^61 + q^63 - 8 * q^65 - 28 * q^67 - 16 * q^69 - 2 * q^71 + 8 * q^73 + 3 * q^75 - 12 * q^77 + 10 * q^79 - 15 * q^81 - 2 * q^83 - 4 * q^85 - 4 * q^87 - 17 * q^89 + 12 * q^91 - 12 * q^93 - 10 * q^95 - 27 * q^97 - 3 * q^99

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