LMFDB - Embedded newform 7605.2.a.cx.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7605,2,Mod(1,7605)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7605, 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("7605.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7605 = 3^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7605.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:	$60.7262307372$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} - 18 x^{10} + 16 x^{9} + 118 x^{8} - 90 x^{7} - 339 x^{6} + 212 x^{5} + 388 x^{4} + \cdots + 7 $ x^12 - x^11 - 18x^10 + 16x^9 + 118x^8 - 90x^7 - 339x^6 + 212x^5 + 388x^4 - 183x^3 - 91x^2 + 21x + 7
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$1.67318$ of defining polynomial
Character	$\chi$	$=$	7605.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.67318 q^{2} +0.799537 q^{4} -1.00000 q^{5} +2.20111 q^{7} -2.00859 q^{8} +O(q^{10})$	$q+1.67318 q^{2} +0.799537 q^{4} -1.00000 q^{5} +2.20111 q^{7} -2.00859 q^{8} -1.67318 q^{10} +5.87787 q^{11} +3.68285 q^{14} -4.95981 q^{16} -2.98518 q^{17} +3.00315 q^{19} -0.799537 q^{20} +9.83475 q^{22} +3.33124 q^{23} +1.00000 q^{25} +1.75987 q^{28} +3.00604 q^{29} -7.61735 q^{31} -4.28148 q^{32} -4.99476 q^{34} -2.20111 q^{35} +10.5340 q^{37} +5.02481 q^{38} +2.00859 q^{40} +0.0937812 q^{41} +5.55346 q^{43} +4.69958 q^{44} +5.57377 q^{46} +8.31052 q^{47} -2.15512 q^{49} +1.67318 q^{50} -8.39923 q^{53} -5.87787 q^{55} -4.42113 q^{56} +5.02964 q^{58} -7.52009 q^{59} -13.5698 q^{61} -12.7452 q^{62} +2.75593 q^{64} +10.7165 q^{67} -2.38676 q^{68} -3.68285 q^{70} +8.88350 q^{71} -4.62382 q^{73} +17.6253 q^{74} +2.40113 q^{76} +12.9378 q^{77} -9.51410 q^{79} +4.95981 q^{80} +0.156913 q^{82} -3.47128 q^{83} +2.98518 q^{85} +9.29195 q^{86} -11.8063 q^{88} -4.95616 q^{89} +2.66345 q^{92} +13.9050 q^{94} -3.00315 q^{95} +11.8578 q^{97} -3.60591 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + q^{2} + 13 q^{4} - 12 q^{5} - 12 q^{7} + 3 q^{8}+O(q^{10}) $ 12 * q + q^2 + 13 * q^4 - 12 * q^5 - 12 * q^7 + 3 * q^8	$ 12 q + q^{2} + 13 q^{4} - 12 q^{5} - 12 q^{7} + 3 q^{8} - q^{10} - 6 q^{11} + 18 q^{14} + 11 q^{16} + 14 q^{17} - 13 q^{20} - 12 q^{22} + 28 q^{23} + 12 q^{25} - 26 q^{28} + 24 q^{29} - 2 q^{31} + 7 q^{32} - 6 q^{34} + 12 q^{35} - 8 q^{37} + 8 q^{38} - 3 q^{40} - 4 q^{41} + 4 q^{43} - 30 q^{44} + 9 q^{46} + 22 q^{47} + 8 q^{49} + q^{50} + 36 q^{53} + 6 q^{55} + 32 q^{56} + 8 q^{58} - 28 q^{59} - 38 q^{61} + 26 q^{62} - 31 q^{64} - 22 q^{67} + 46 q^{68} - 18 q^{70} + 24 q^{71} - 12 q^{73} + 26 q^{74} + 17 q^{76} + 28 q^{77} + 20 q^{79} - 11 q^{80} - 36 q^{82} - 26 q^{83} - 14 q^{85} + 74 q^{86} - 34 q^{88} - 2 q^{89} + 86 q^{92} - 16 q^{94} - 26 q^{97} - 49 q^{98}+O(q^{100}) $ 12 * q + q^2 + 13 * q^4 - 12 * q^5 - 12 * q^7 + 3 * q^8 - q^10 - 6 * q^11 + 18 * q^14 + 11 * q^16 + 14 * q^17 - 13 * q^20 - 12 * q^22 + 28 * q^23 + 12 * q^25 - 26 * q^28 + 24 * q^29 - 2 * q^31 + 7 * q^32 - 6 * q^34 + 12 * q^35 - 8 * q^37 + 8 * q^38 - 3 * q^40 - 4 * q^41 + 4 * q^43 - 30 * q^44 + 9 * q^46 + 22 * q^47 + 8 * q^49 + q^50 + 36 * q^53 + 6 * q^55 + 32 * q^56 + 8 * q^58 - 28 * q^59 - 38 * q^61 + 26 * q^62 - 31 * q^64 - 22 * q^67 + 46 * q^68 - 18 * q^70 + 24 * q^71 - 12 * q^73 + 26 * q^74 + 17 * q^76 + 28 * q^77 + 20 * q^79 - 11 * q^80 - 36 * q^82 - 26 * q^83 - 14 * q^85 + 74 * q^86 - 34 * q^88 - 2 * q^89 + 86 * q^92 - 16 * q^94 - 26 * q^97 - 49 * q^98

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 $

$2$	1.67318	1.18312	0.591559	−	0.806262i	$-0.298513\pi$
$2$	1.67318	1.18312	0.591559	+	0.806262i	$0.298513\pi$
$3$	0	0
$3$	0	0
$4$	0.799537	0.399768
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.20111	0.831941	0.415970	−	0.909378i	$-0.363442\pi$
$7$	2.20111	0.831941	0.415970	+	0.909378i	$0.363442\pi$
$8$	−2.00859	−0.710145
$9$	0	0
$10$	−1.67318	−0.529106
$11$	5.87787	1.77225	0.886123	−	0.463451i	$-0.153389\pi$
$11$	5.87787	1.77225	0.886123	+	0.463451i	$0.153389\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	3.68285	0.984284
$15$	0	0
$16$	−4.95981	−1.23995
$17$	−2.98518	−0.724014	−0.362007	−	0.932175i	$-0.617908\pi$
$17$	−2.98518	−0.724014	−0.362007	+	0.932175i	$0.617908\pi$
$18$	0	0
$19$	3.00315	0.688969	0.344485	−	0.938792i	$-0.388054\pi$
$19$	3.00315	0.688969	0.344485	+	0.938792i	$0.388054\pi$
$20$	−0.799537	−0.178782
$21$	0	0
$22$	9.83475	2.09678
$23$	3.33124	0.694612	0.347306	−	0.937752i	$-0.387097\pi$
$23$	3.33124	0.694612	0.347306	+	0.937752i	$0.387097\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	1.75987	0.332584
$29$	3.00604	0.558207	0.279103	−	0.960261i	$-0.409963\pi$
$29$	3.00604	0.558207	0.279103	+	0.960261i	$0.409963\pi$
$30$	0	0
$31$	−7.61735	−1.36812	−0.684058	−	0.729427i	$-0.739787\pi$
$31$	−7.61735	−1.36812	−0.684058	+	0.729427i	$0.739787\pi$
$32$	−4.28148	−0.756867
$33$	0	0
$34$	−4.99476	−0.856594
$35$	−2.20111	−0.372055
$36$	0	0
$37$	10.5340	1.73178	0.865889	−	0.500236i	$-0.166754\pi$
$37$	10.5340	1.73178	0.865889	+	0.500236i	$0.166754\pi$
$38$	5.02481	0.815132
$39$	0	0
$40$	2.00859	0.317586
$41$	0.0937812	0.0146462	0.00732308	−	0.999973i	$-0.497669\pi$
$41$	0.0937812	0.0146462	0.00732308	+	0.999973i	$0.497669\pi$
$42$	0	0
$43$	5.55346	0.846895	0.423447	−	0.905921i	$-0.360820\pi$
$43$	5.55346	0.846895	0.423447	+	0.905921i	$0.360820\pi$
$44$	4.69958	0.708488
$45$	0	0
$46$	5.57377	0.821808
$47$	8.31052	1.21221	0.606107	−	0.795383i	$-0.292730\pi$
$47$	8.31052	1.21221	0.606107	+	0.795383i	$0.292730\pi$
$48$	0	0
$49$	−2.15512	−0.307875
$50$	1.67318	0.236624
$51$	0	0
$52$	0	0
$53$	−8.39923	−1.15372	−0.576862	−	0.816842i	$-0.695723\pi$
$53$	−8.39923	−1.15372	−0.576862	+	0.816842i	$0.695723\pi$
$54$	0	0
$55$	−5.87787	−0.792572
$56$	−4.42113	−0.590798
$57$	0	0
$58$	5.02964	0.660425
$59$	−7.52009	−0.979032	−0.489516	−	0.871994i	$-0.662826\pi$
$59$	−7.52009	−0.979032	−0.489516	+	0.871994i	$0.662826\pi$
$60$	0	0
$61$	−13.5698	−1.73743	−0.868716	−	0.495310i	$-0.835054\pi$
$61$	−13.5698	−1.73743	−0.868716	+	0.495310i	$0.835054\pi$
$62$	−12.7452	−1.61864
$63$	0	0
$64$	2.75593	0.344491
$65$	0	0
$66$	0	0
$67$	10.7165	1.30923	0.654615	−	0.755963i	$-0.272831\pi$
$67$	10.7165	1.30923	0.654615	+	0.755963i	$0.272831\pi$
$68$	−2.38676	−0.289438
$69$	0	0
$70$	−3.68285	−0.440185
$71$	8.88350	1.05428	0.527139	−	0.849779i	$-0.323265\pi$
$71$	8.88350	1.05428	0.527139	+	0.849779i	$0.323265\pi$
$72$	0	0
$73$	−4.62382	−0.541177	−0.270589	−	0.962695i	$-0.587218\pi$
$73$	−4.62382	−0.541177	−0.270589	+	0.962695i	$0.587218\pi$
$74$	17.6253	2.04890
$75$	0	0
$76$	2.40113	0.275428
$77$	12.9378	1.47440
$78$	0	0
$79$	−9.51410	−1.07042	−0.535210	−	0.844719i	$-0.679768\pi$
$79$	−9.51410	−1.07042	−0.535210	+	0.844719i	$0.679768\pi$
$80$	4.95981	0.554524
$81$	0	0
$82$	0.156913	0.0173281
$83$	−3.47128	−0.381023	−0.190511	−	0.981685i	$-0.561015\pi$
$83$	−3.47128	−0.381023	−0.190511	+	0.981685i	$0.561015\pi$
$84$	0	0
$85$	2.98518	0.323789
$86$	9.29195	1.00198
$87$	0	0
$88$	−11.8063	−1.25855
$89$	−4.95616	−0.525352	−0.262676	−	0.964884i	$-0.584605\pi$
$89$	−4.95616	−0.525352	−0.262676	+	0.964884i	$0.584605\pi$
$90$	0	0
$91$	0	0
$92$	2.66345	0.277684
$93$	0	0
$94$	13.9050	1.43419
$95$	−3.00315	−0.308117
$96$	0	0
$97$	11.8578	1.20397	0.601987	−	0.798506i	$-0.294376\pi$
$97$	11.8578	1.20397	0.601987	+	0.798506i	$0.294376\pi$
$98$	−3.60591	−0.364252
$99$	0	0
$100$	0.799537	0.0799537
$101$	8.12602	0.808569	0.404285	−	0.914633i	$-0.367520\pi$
$101$	8.12602	0.808569	0.404285	+	0.914633i	$0.367520\pi$
$102$	0	0
$103$	9.31734	0.918064	0.459032	−	0.888420i	$-0.348196\pi$
$103$	9.31734	0.918064	0.459032	+	0.888420i	$0.348196\pi$
$104$	0	0
$105$	0	0
$106$	−14.0534	−1.36499
$107$	10.0810	0.974569	0.487285	−	0.873243i	$-0.337987\pi$
$107$	10.0810	0.974569	0.487285	+	0.873243i	$0.337987\pi$
$108$	0	0
$109$	10.0112	0.958896	0.479448	−	0.877570i	$-0.340837\pi$
$109$	10.0112	0.958896	0.479448	+	0.877570i	$0.340837\pi$
$110$	−9.83475	−0.937706
$111$	0	0
$112$	−10.9171	−1.03157
$113$	19.6705	1.85045	0.925224	−	0.379422i	$-0.123877\pi$
$113$	19.6705	1.85045	0.925224	+	0.379422i	$0.123877\pi$
$114$	0	0
$115$	−3.33124	−0.310640
$116$	2.40344	0.223153
$117$	0	0
$118$	−12.5825	−1.15831
$119$	−6.57071	−0.602336
$120$	0	0
$121$	23.5494	2.14085
$122$	−22.7047	−2.05559
$123$	0	0
$124$	−6.09035	−0.546930
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	0.688946	0.0611341	0.0305670	−	0.999533i	$-0.490269\pi$
$127$	0.688946	0.0611341	0.0305670	+	0.999533i	$0.490269\pi$
$128$	13.1741	1.16444
$129$	0	0
$130$	0	0
$131$	16.9409	1.48013	0.740067	−	0.672533i	$-0.234794\pi$
$131$	16.9409	1.48013	0.740067	+	0.672533i	$0.234794\pi$
$132$	0	0
$133$	6.61025	0.573182
$134$	17.9307	1.54897
$135$	0	0
$136$	5.99602	0.514155
$137$	13.3796	1.14310	0.571548	−	0.820569i	$-0.306343\pi$
$137$	13.3796	1.14310	0.571548	+	0.820569i	$0.306343\pi$
$138$	0	0
$139$	−17.4455	−1.47971	−0.739854	−	0.672768i	$-0.765105\pi$
$139$	−17.4455	−1.47971	−0.739854	+	0.672768i	$0.765105\pi$
$140$	−1.75987	−0.148736
$141$	0	0
$142$	14.8637	1.24734
$143$	0	0
$144$	0	0
$145$	−3.00604	−0.249638
$146$	−7.73649	−0.640277
$147$	0	0
$148$	8.42232	0.692310
$149$	5.78235	0.473709	0.236854	−	0.971545i	$-0.423884\pi$
$149$	5.78235	0.473709	0.236854	+	0.971545i	$0.423884\pi$
$150$	0	0
$151$	−0.801937	−0.0652607	−0.0326304	−	0.999467i	$-0.510388\pi$
$151$	−0.801937	−0.0652607	−0.0326304	+	0.999467i	$0.510388\pi$
$152$	−6.03210	−0.489268
$153$	0	0
$154$	21.6473	1.74439
$155$	7.61735	0.611840
$156$	0	0
$157$	−7.63125	−0.609040	−0.304520	−	0.952506i	$-0.598496\pi$
$157$	−7.63125	−0.609040	−0.304520	+	0.952506i	$0.598496\pi$
$158$	−15.9188	−1.26643
$159$	0	0
$160$	4.28148	0.338481
$161$	7.33243	0.577876
$162$	0	0
$163$	−1.90400	−0.149133	−0.0745664	−	0.997216i	$-0.523757\pi$
$163$	−1.90400	−0.149133	−0.0745664	+	0.997216i	$0.523757\pi$
$164$	0.0749815	0.00585507
$165$	0	0
$166$	−5.80809	−0.450795
$167$	5.90128	0.456655	0.228328	−	0.973584i	$-0.426674\pi$
$167$	5.90128	0.456655	0.228328	+	0.973584i	$0.426674\pi$
$168$	0	0
$169$	0	0
$170$	4.99476	0.383080
$171$	0	0
$172$	4.44020	0.338562
$173$	4.84337	0.368235	0.184117	−	0.982904i	$-0.441057\pi$
$173$	4.84337	0.368235	0.184117	+	0.982904i	$0.441057\pi$
$174$	0	0
$175$	2.20111	0.166388
$176$	−29.1532	−2.19750
$177$	0	0
$178$	−8.29256	−0.621554
$179$	19.8739	1.48544	0.742722	−	0.669600i	$-0.233534\pi$
$179$	19.8739	1.48544	0.742722	+	0.669600i	$0.233534\pi$
$180$	0	0
$181$	−2.40011	−0.178399	−0.0891993	−	0.996014i	$-0.528431\pi$
$181$	−2.40011	−0.178399	−0.0891993	+	0.996014i	$0.528431\pi$
$182$	0	0
$183$	0	0
$184$	−6.69111	−0.493275
$185$	−10.5340	−0.774475
$186$	0	0
$187$	−17.5465	−1.28313
$188$	6.64457	0.484605
$189$	0	0
$190$	−5.02481	−0.364538
$191$	−5.83817	−0.422435	−0.211218	−	0.977439i	$-0.567743\pi$
$191$	−5.83817	−0.422435	−0.211218	+	0.977439i	$0.567743\pi$
$192$	0	0
$193$	4.99325	0.359422	0.179711	−	0.983719i	$-0.442484\pi$
$193$	4.99325	0.359422	0.179711	+	0.983719i	$0.442484\pi$
$194$	19.8402	1.42444
$195$	0	0
$196$	−1.72310	−0.123079
$197$	−10.5751	−0.753447	−0.376724	−	0.926326i	$-0.622949\pi$
$197$	−10.5751	−0.753447	−0.376724	+	0.926326i	$0.622949\pi$
$198$	0	0
$199$	14.7003	1.04208	0.521040	−	0.853532i	$-0.325544\pi$
$199$	14.7003	1.04208	0.521040	+	0.853532i	$0.325544\pi$
$200$	−2.00859	−0.142029
$201$	0	0
$202$	13.5963	0.956633
$203$	6.61661	0.464395
$204$	0	0
$205$	−0.0937812	−0.00654996
$206$	15.5896	1.08618
$207$	0	0
$208$	0	0
$209$	17.6521	1.22102
$210$	0	0
$211$	−16.0465	−1.10468	−0.552342	−	0.833617i	$-0.686266\pi$
$211$	−16.0465	−1.10468	−0.552342	+	0.833617i	$0.686266\pi$
$212$	−6.71550	−0.461222
$213$	0	0
$214$	16.8674	1.15303
$215$	−5.55346	−0.378743
$216$	0	0
$217$	−16.7666	−1.13819
$218$	16.7505	1.13449
$219$	0	0
$220$	−4.69958	−0.316845
$221$	0	0
$222$	0	0
$223$	−29.1697	−1.95335	−0.976675	−	0.214723i	$-0.931115\pi$
$223$	−29.1697	−1.95335	−0.976675	+	0.214723i	$0.931115\pi$
$224$	−9.42401	−0.629668
$225$	0	0
$226$	32.9124	2.18930
$227$	−18.4618	−1.22535	−0.612675	−	0.790335i	$-0.709907\pi$
$227$	−18.4618	−1.22535	−0.612675	+	0.790335i	$0.709907\pi$
$228$	0	0
$229$	23.4947	1.55258	0.776288	−	0.630378i	$-0.217100\pi$
$229$	23.4947	1.55258	0.776288	+	0.630378i	$0.217100\pi$
$230$	−5.57377	−0.367524
$231$	0	0
$232$	−6.03790	−0.396408
$233$	3.70678	0.242839	0.121419	−	0.992601i	$-0.461255\pi$
$233$	3.70678	0.242839	0.121419	+	0.992601i	$0.461255\pi$
$234$	0	0
$235$	−8.31052	−0.542119
$236$	−6.01258	−0.391386
$237$	0	0
$238$	−10.9940	−0.712635
$239$	14.1927	0.918051	0.459025	−	0.888423i	$-0.348199\pi$
$239$	14.1927	0.918051	0.459025	+	0.888423i	$0.348199\pi$
$240$	0	0
$241$	3.71915	0.239572	0.119786	−	0.992800i	$-0.461779\pi$
$241$	3.71915	0.239572	0.119786	+	0.992800i	$0.461779\pi$
$242$	39.4024	2.53288
$243$	0	0
$244$	−10.8495	−0.694571
$245$	2.15512	0.137686
$246$	0	0
$247$	0	0
$248$	15.3002	0.971561
$249$	0	0
$250$	−1.67318	−0.105821
$251$	−17.8109	−1.12421	−0.562106	−	0.827065i	$-0.690009\pi$
$251$	−17.8109	−1.12421	−0.562106	+	0.827065i	$0.690009\pi$
$252$	0	0
$253$	19.5806	1.23102
$254$	1.15273	0.0723288
$255$	0	0
$256$	16.5309	1.03318
$257$	−4.06775	−0.253739	−0.126870	−	0.991919i	$-0.540493\pi$
$257$	−4.06775	−0.253739	−0.126870	+	0.991919i	$0.540493\pi$
$258$	0	0
$259$	23.1865	1.44074
$260$	0	0
$261$	0	0
$262$	28.3452	1.75117
$263$	−21.2603	−1.31097	−0.655483	−	0.755210i	$-0.727535\pi$
$263$	−21.2603	−1.31097	−0.655483	+	0.755210i	$0.727535\pi$
$264$	0	0
$265$	8.39923	0.515961
$266$	11.0602	0.678142
$267$	0	0
$268$	8.56824	0.523389
$269$	23.1780	1.41319	0.706595	−	0.707618i	$-0.250230\pi$
$269$	23.1780	1.41319	0.706595	+	0.707618i	$0.250230\pi$
$270$	0	0
$271$	31.3674	1.90543	0.952716	−	0.303862i	$-0.0982761\pi$
$271$	31.3674	1.90543	0.952716	+	0.303862i	$0.0982761\pi$
$272$	14.8060	0.897743
$273$	0	0
$274$	22.3865	1.35242
$275$	5.87787	0.354449
$276$	0	0
$277$	9.82232	0.590166	0.295083	−	0.955472i	$-0.404653\pi$
$277$	9.82232	0.590166	0.295083	+	0.955472i	$0.404653\pi$
$278$	−29.1895	−1.75067
$279$	0	0
$280$	4.42113	0.264213
$281$	−14.9335	−0.890860	−0.445430	−	0.895317i	$-0.646949\pi$
$281$	−14.9335	−0.890860	−0.445430	+	0.895317i	$0.646949\pi$
$282$	0	0
$283$	−7.76373	−0.461506	−0.230753	−	0.973012i	$-0.574119\pi$
$283$	−7.76373	−0.461506	−0.230753	+	0.973012i	$0.574119\pi$
$284$	7.10269	0.421467
$285$	0	0
$286$	0	0
$287$	0.206423	0.0121847
$288$	0	0
$289$	−8.08867	−0.475804
$290$	−5.02964	−0.295351
$291$	0	0
$292$	−3.69692	−0.216346
$293$	−13.4541	−0.785997	−0.392998	−	0.919539i	$-0.628562\pi$
$293$	−13.4541	−0.785997	−0.392998	+	0.919539i	$0.628562\pi$
$294$	0	0
$295$	7.52009	0.437836
$296$	−21.1585	−1.22981
$297$	0	0
$298$	9.67492	0.560453
$299$	0	0
$300$	0	0
$301$	12.2238	0.704566
$302$	−1.34179	−0.0772111
$303$	0	0
$304$	−14.8951	−0.854290
$305$	13.5698	0.777003
$306$	0	0
$307$	−13.8497	−0.790442	−0.395221	−	0.918586i	$-0.629332\pi$
$307$	−13.8497	−0.790442	−0.395221	+	0.918586i	$0.629332\pi$
$308$	10.3443	0.589420
$309$	0	0
$310$	12.7452	0.723879
$311$	10.7515	0.609662	0.304831	−	0.952406i	$-0.401400\pi$
$311$	10.7515	0.609662	0.304831	+	0.952406i	$0.401400\pi$
$312$	0	0
$313$	29.0893	1.64422	0.822112	−	0.569326i	$-0.192796\pi$
$313$	29.0893	1.64422	0.822112	+	0.569326i	$0.192796\pi$
$314$	−12.7685	−0.720567
$315$	0	0
$316$	−7.60687	−0.427920
$317$	−14.8613	−0.834692	−0.417346	−	0.908748i	$-0.637040\pi$
$317$	−14.8613	−0.834692	−0.417346	+	0.908748i	$0.637040\pi$
$318$	0	0
$319$	17.6691	0.989279
$320$	−2.75593	−0.154061
$321$	0	0
$322$	12.2685	0.683696
$323$	−8.96495	−0.498823
$324$	0	0
$325$	0	0
$326$	−3.18574	−0.176442
$327$	0	0
$328$	−0.188368	−0.0104009
$329$	18.2924	1.00849
$330$	0	0
$331$	−1.24530	−0.0684478	−0.0342239	−	0.999414i	$-0.510896\pi$
$331$	−1.24530	−0.0684478	−0.0342239	+	0.999414i	$0.510896\pi$
$332$	−2.77542	−0.152321
$333$	0	0
$334$	9.87392	0.540277
$335$	−10.7165	−0.585505
$336$	0	0
$337$	14.1736	0.772086	0.386043	−	0.922481i	$-0.373842\pi$
$337$	14.1736	0.772086	0.386043	+	0.922481i	$0.373842\pi$
$338$	0	0
$339$	0	0
$340$	2.38676	0.129441
$341$	−44.7738	−2.42464
$342$	0	0
$343$	−20.1514	−1.08807
$344$	−11.1546	−0.601418
$345$	0	0
$346$	8.10384	0.435665
$347$	25.4917	1.36846	0.684232	−	0.729264i	$-0.260137\pi$
$347$	25.4917	1.36846	0.684232	+	0.729264i	$0.260137\pi$
$348$	0	0
$349$	−35.1516	−1.88162	−0.940810	−	0.338935i	$-0.889933\pi$
$349$	−35.1516	−1.88162	−0.940810	+	0.338935i	$0.889933\pi$
$350$	3.68285	0.196857
$351$	0	0
$352$	−25.1660	−1.34135
$353$	−7.86270	−0.418489	−0.209245	−	0.977863i	$-0.567100\pi$
$353$	−7.86270	−0.418489	−0.209245	+	0.977863i	$0.567100\pi$
$354$	0	0
$355$	−8.88350	−0.471487
$356$	−3.96263	−0.210019
$357$	0	0
$358$	33.2526	1.75746
$359$	27.0340	1.42680	0.713401	−	0.700756i	$-0.247154\pi$
$359$	27.0340	1.42680	0.713401	+	0.700756i	$0.247154\pi$
$360$	0	0
$361$	−9.98110	−0.525321
$362$	−4.01582	−0.211067
$363$	0	0
$364$	0	0
$365$	4.62382	0.242022
$366$	0	0
$367$	−28.4607	−1.48564	−0.742818	−	0.669493i	$-0.766511\pi$
$367$	−28.4607	−1.48564	−0.742818	+	0.669493i	$0.766511\pi$
$368$	−16.5223	−0.861287
$369$	0	0
$370$	−17.6253	−0.916295
$371$	−18.4876	−0.959830
$372$	0	0
$373$	11.5026	0.595585	0.297792	−	0.954631i	$-0.403750\pi$
$373$	11.5026	0.595585	0.297792	+	0.954631i	$0.403750\pi$
$374$	−29.3585	−1.51809
$375$	0	0
$376$	−16.6925	−0.860848
$377$	0	0
$378$	0	0
$379$	5.74009	0.294848	0.147424	−	0.989073i	$-0.452902\pi$
$379$	5.74009	0.294848	0.147424	+	0.989073i	$0.452902\pi$
$380$	−2.40113	−0.123175
$381$	0	0
$382$	−9.76833	−0.499791
$383$	−10.0615	−0.514121	−0.257060	−	0.966395i	$-0.582754\pi$
$383$	−10.0615	−0.514121	−0.257060	+	0.966395i	$0.582754\pi$
$384$	0	0
$385$	−12.9378	−0.659373
$386$	8.35462	0.425239
$387$	0	0
$388$	9.48072	0.481311
$389$	11.3900	0.577495	0.288748	−	0.957405i	$-0.406761\pi$
$389$	11.3900	0.577495	0.288748	+	0.957405i	$0.406761\pi$
$390$	0	0
$391$	−9.94437	−0.502909
$392$	4.32876	0.218636
$393$	0	0
$394$	−17.6941	−0.891417
$395$	9.51410	0.478706
$396$	0	0
$397$	−16.5495	−0.830596	−0.415298	−	0.909685i	$-0.636323\pi$
$397$	−16.5495	−0.830596	−0.415298	+	0.909685i	$0.636323\pi$
$398$	24.5963	1.23290
$399$	0	0
$400$	−4.95981	−0.247991
$401$	30.9846	1.54730	0.773648	−	0.633616i	$-0.218430\pi$
$401$	30.9846	1.54730	0.773648	+	0.633616i	$0.218430\pi$
$402$	0	0
$403$	0	0
$404$	6.49705	0.323241
$405$	0	0
$406$	11.0708	0.549434
$407$	61.9175	3.06913
$408$	0	0
$409$	36.2934	1.79459	0.897297	−	0.441426i	$-0.145527\pi$
$409$	36.2934	1.79459	0.897297	+	0.441426i	$0.145527\pi$
$410$	−0.156913	−0.00774938
$411$	0	0
$412$	7.44955	0.367013
$413$	−16.5525	−0.814496
$414$	0	0
$415$	3.47128	0.170399
$416$	0	0
$417$	0	0
$418$	29.5352	1.44461
$419$	20.4467	0.998889	0.499444	−	0.866346i	$-0.333537\pi$
$419$	20.4467	0.998889	0.499444	+	0.866346i	$0.333537\pi$
$420$	0	0
$421$	5.03925	0.245598	0.122799	−	0.992432i	$-0.460813\pi$
$421$	5.03925	0.245598	0.122799	+	0.992432i	$0.460813\pi$
$422$	−26.8487	−1.30697
$423$	0	0
$424$	16.8706	0.819311
$425$	−2.98518	−0.144803
$426$	0	0
$427$	−29.8686	−1.44544
$428$	8.06015	0.389602
$429$	0	0
$430$	−9.29195	−0.448097
$431$	−33.0261	−1.59081	−0.795406	−	0.606077i	$-0.792743\pi$
$431$	−33.0261	−1.59081	−0.795406	+	0.606077i	$0.792743\pi$
$432$	0	0
$433$	−2.17568	−0.104556	−0.0522782	−	0.998633i	$-0.516648\pi$
$433$	−2.17568	−0.104556	−0.0522782	+	0.998633i	$0.516648\pi$
$434$	−28.0536	−1.34662
$435$	0	0
$436$	8.00430	0.383336
$437$	10.0042	0.478567
$438$	0	0
$439$	12.0779	0.576449	0.288224	−	0.957563i	$-0.406935\pi$
$439$	12.0779	0.576449	0.288224	+	0.957563i	$0.406935\pi$
$440$	11.8063	0.562841
$441$	0	0
$442$	0	0
$443$	−7.75110	−0.368266	−0.184133	−	0.982901i	$-0.558948\pi$
$443$	−7.75110	−0.368266	−0.184133	+	0.982901i	$0.558948\pi$
$444$	0	0
$445$	4.95616	0.234945
$446$	−48.8063	−2.31104
$447$	0	0
$448$	6.06609	0.286596
$449$	6.71779	0.317032	0.158516	−	0.987356i	$-0.449329\pi$
$449$	6.71779	0.317032	0.158516	+	0.987356i	$0.449329\pi$
$450$	0	0
$451$	0.551234	0.0259566
$452$	15.7273	0.739750
$453$	0	0
$454$	−30.8899	−1.44973
$455$	0	0
$456$	0	0
$457$	−35.1739	−1.64536	−0.822682	−	0.568502i	$-0.807523\pi$
$457$	−35.1739	−1.64536	−0.822682	+	0.568502i	$0.807523\pi$
$458$	39.3110	1.83688
$459$	0	0
$460$	−2.66345	−0.124184
$461$	−17.0505	−0.794120	−0.397060	−	0.917793i	$-0.629969\pi$
$461$	−17.0505	−0.794120	−0.397060	+	0.917793i	$0.629969\pi$
$462$	0	0
$463$	−25.1017	−1.16657	−0.583287	−	0.812266i	$-0.698234\pi$
$463$	−25.1017	−1.16657	−0.583287	+	0.812266i	$0.698234\pi$
$464$	−14.9094	−0.692151
$465$	0	0
$466$	6.20211	0.287307
$467$	−0.113343	−0.00524488	−0.00262244	−	0.999997i	$-0.500835\pi$
$467$	−0.113343	−0.00524488	−0.00262244	+	0.999997i	$0.500835\pi$
$468$	0	0
$469$	23.5882	1.08920
$470$	−13.9050	−0.641391
$471$	0	0
$472$	15.1048	0.695254
$473$	32.6425	1.50090
$474$	0	0
$475$	3.00315	0.137794
$476$	−5.25353	−0.240795
$477$	0	0
$478$	23.7470	1.08616
$479$	10.2305	0.467442	0.233721	−	0.972304i	$-0.424910\pi$
$479$	10.2305	0.467442	0.233721	+	0.972304i	$0.424910\pi$
$480$	0	0
$481$	0	0
$482$	6.22281	0.283441
$483$	0	0
$484$	18.8286	0.855845
$485$	−11.8578	−0.538434
$486$	0	0
$487$	−19.9235	−0.902819	−0.451409	−	0.892317i	$-0.649079\pi$
$487$	−19.9235	−0.902819	−0.451409	+	0.892317i	$0.649079\pi$
$488$	27.2562	1.23383
$489$	0	0
$490$	3.60591	0.162898
$491$	38.1677	1.72249	0.861243	−	0.508193i	$-0.169686\pi$
$491$	38.1677	1.72249	0.861243	+	0.508193i	$0.169686\pi$
$492$	0	0
$493$	−8.97357	−0.404149
$494$	0	0
$495$	0	0
$496$	37.7806	1.69640
$497$	19.5536	0.877097
$498$	0	0
$499$	13.8474	0.619893	0.309947	−	0.950754i	$-0.399689\pi$
$499$	13.8474	0.619893	0.309947	+	0.950754i	$0.399689\pi$
$500$	−0.799537	−0.0357564
$501$	0	0
$502$	−29.8008	−1.33008
$503$	11.5990	0.517172	0.258586	−	0.965988i	$-0.416744\pi$
$503$	11.5990	0.517172	0.258586	+	0.965988i	$0.416744\pi$
$504$	0	0
$505$	−8.12602	−0.361603
$506$	32.7619	1.45645
$507$	0	0
$508$	0.550838	0.0244395
$509$	15.3222	0.679143	0.339572	−	0.940580i	$-0.389718\pi$
$509$	15.3222	0.679143	0.339572	+	0.940580i	$0.389718\pi$
$510$	0	0
$511$	−10.1775	−0.450228
$512$	1.31087	0.0579330
$513$	0	0
$514$	−6.80608	−0.300203
$515$	−9.31734	−0.410571
$516$	0	0
$517$	48.8482	2.14834
$518$	38.7952	1.70456
$519$	0	0
$520$	0	0
$521$	29.2960	1.28348	0.641740	−	0.766922i	$-0.278213\pi$
$521$	29.2960	1.28348	0.641740	+	0.766922i	$0.278213\pi$
$522$	0	0
$523$	8.91749	0.389934	0.194967	−	0.980810i	$-0.437540\pi$
$523$	8.91749	0.389934	0.194967	+	0.980810i	$0.437540\pi$
$524$	13.5449	0.591711
$525$	0	0
$526$	−35.5723	−1.55103
$527$	22.7392	0.990535
$528$	0	0
$529$	−11.9028	−0.517514
$530$	14.0534	0.610443
$531$	0	0
$532$	5.28514	0.229140
$533$	0	0
$534$	0	0
$535$	−10.0810	−0.435841
$536$	−21.5251	−0.929743
$537$	0	0
$538$	38.7811	1.67197
$539$	−12.6675	−0.545629
$540$	0	0
$541$	35.9141	1.54407	0.772034	−	0.635582i	$-0.219240\pi$
$541$	35.9141	1.54407	0.772034	+	0.635582i	$0.219240\pi$
$542$	52.4833	2.25435
$543$	0	0
$544$	12.7810	0.547982
$545$	−10.0112	−0.428831
$546$	0	0
$547$	−17.1490	−0.733238	−0.366619	−	0.930371i	$-0.619485\pi$
$547$	−17.1490	−0.733238	−0.366619	+	0.930371i	$0.619485\pi$
$548$	10.6975	0.456973
$549$	0	0
$550$	9.83475	0.419355
$551$	9.02757	0.384587
$552$	0	0
$553$	−20.9416	−0.890526
$554$	16.4345	0.698236
$555$	0	0
$556$	−13.9483	−0.591540
$557$	10.6737	0.452257	0.226129	−	0.974097i	$-0.427393\pi$
$557$	10.6737	0.452257	0.226129	+	0.974097i	$0.427393\pi$
$558$	0	0
$559$	0	0
$560$	10.9171	0.461331
$561$	0	0
$562$	−24.9865	−1.05399
$563$	8.69238	0.366340	0.183170	−	0.983081i	$-0.441364\pi$
$563$	8.69238	0.366340	0.183170	+	0.983081i	$0.441364\pi$
$564$	0	0
$565$	−19.6705	−0.827545
$566$	−12.9901	−0.546016
$567$	0	0
$568$	−17.8433	−0.748690
$569$	−8.20176	−0.343835	−0.171918	−	0.985111i	$-0.554996\pi$
$569$	−8.20176	−0.343835	−0.171918	+	0.985111i	$0.554996\pi$
$570$	0	0
$571$	−6.86423	−0.287259	−0.143630	−	0.989632i	$-0.545877\pi$
$571$	−6.86423	−0.287259	−0.143630	+	0.989632i	$0.545877\pi$
$572$	0	0
$573$	0	0
$574$	0.345382	0.0144160
$575$	3.33124	0.138922
$576$	0	0
$577$	−24.6348	−1.02556	−0.512780	−	0.858520i	$-0.671384\pi$
$577$	−24.6348	−1.02556	−0.512780	+	0.858520i	$0.671384\pi$
$578$	−13.5338	−0.562933
$579$	0	0
$580$	−2.40344	−0.0997973
$581$	−7.64067	−0.316989
$582$	0	0
$583$	−49.3696	−2.04468
$584$	9.28738	0.384314
$585$	0	0
$586$	−22.5111	−0.929927
$587$	−22.1699	−0.915048	−0.457524	−	0.889197i	$-0.651264\pi$
$587$	−22.1699	−0.915048	−0.457524	+	0.889197i	$0.651264\pi$
$588$	0	0
$589$	−22.8760	−0.942590
$590$	12.5825	0.518012
$591$	0	0
$592$	−52.2467	−2.14732
$593$	40.4785	1.66225	0.831127	−	0.556083i	$-0.187696\pi$
$593$	40.4785	1.66225	0.831127	+	0.556083i	$0.187696\pi$
$594$	0	0
$595$	6.57071	0.269373
$596$	4.62320	0.189374
$597$	0	0
$598$	0	0
$599$	−23.9642	−0.979153	−0.489576	−	0.871960i	$-0.662849\pi$
$599$	−23.9642	−0.979153	−0.489576	+	0.871960i	$0.662849\pi$
$600$	0	0
$601$	−2.15250	−0.0878024	−0.0439012	−	0.999036i	$-0.513979\pi$
$601$	−2.15250	−0.0878024	−0.0439012	+	0.999036i	$0.513979\pi$
$602$	20.4526	0.833585
$603$	0	0
$604$	−0.641178	−0.0260892
$605$	−23.5494	−0.957418
$606$	0	0
$607$	14.0772	0.571375	0.285687	−	0.958323i	$-0.407778\pi$
$607$	14.0772	0.571375	0.285687	+	0.958323i	$0.407778\pi$
$608$	−12.8579	−0.521458
$609$	0	0
$610$	22.7047	0.919287
$611$	0	0
$612$	0	0
$613$	−31.4908	−1.27190	−0.635950	−	0.771730i	$-0.719392\pi$
$613$	−31.4908	−1.27190	−0.635950	+	0.771730i	$0.719392\pi$
$614$	−23.1730	−0.935187
$615$	0	0
$616$	−25.9868	−1.04704
$617$	48.4698	1.95132	0.975660	−	0.219289i	$-0.0703738\pi$
$617$	48.4698	1.95132	0.975660	+	0.219289i	$0.0703738\pi$
$618$	0	0
$619$	−24.5608	−0.987181	−0.493590	−	0.869695i	$-0.664316\pi$
$619$	−24.5608	−0.987181	−0.493590	+	0.869695i	$0.664316\pi$
$620$	6.09035	0.244594
$621$	0	0
$622$	17.9892	0.721303
$623$	−10.9090	−0.437062
$624$	0	0
$625$	1.00000	0.0400000
$626$	48.6717	1.94531
$627$	0	0
$628$	−6.10147	−0.243475
$629$	−31.4459	−1.25383
$630$	0	0
$631$	−24.4290	−0.972503	−0.486251	−	0.873819i	$-0.661636\pi$
$631$	−24.4290	−0.972503	−0.486251	+	0.873819i	$0.661636\pi$
$632$	19.1100	0.760153
$633$	0	0
$634$	−24.8656	−0.987539
$635$	−0.688946	−0.0273400
$636$	0	0
$637$	0	0
$638$	29.5636	1.17043
$639$	0	0
$640$	−13.1741	−0.520753
$641$	−42.4077	−1.67500	−0.837501	−	0.546436i	$-0.815984\pi$
$641$	−42.4077	−1.67500	−0.837501	+	0.546436i	$0.815984\pi$
$642$	0	0
$643$	−44.7911	−1.76639	−0.883195	−	0.469006i	$-0.844612\pi$
$643$	−44.7911	−1.76639	−0.883195	+	0.469006i	$0.844612\pi$
$644$	5.86254	0.231017
$645$	0	0
$646$	−15.0000	−0.590167
$647$	24.9327	0.980204	0.490102	−	0.871665i	$-0.336959\pi$
$647$	24.9327	0.980204	0.490102	+	0.871665i	$0.336959\pi$
$648$	0	0
$649$	−44.2021	−1.73508
$650$	0	0
$651$	0	0
$652$	−1.52232	−0.0596186
$653$	−28.6239	−1.12014	−0.560069	−	0.828446i	$-0.689225\pi$
$653$	−28.6239	−1.12014	−0.560069	+	0.828446i	$0.689225\pi$
$654$	0	0
$655$	−16.9409	−0.661936
$656$	−0.465137	−0.0181606
$657$	0	0
$658$	30.6064	1.19316
$659$	28.5647	1.11272	0.556361	−	0.830941i	$-0.312197\pi$
$659$	28.5647	1.11272	0.556361	+	0.830941i	$0.312197\pi$
$660$	0	0
$661$	−39.4211	−1.53330	−0.766652	−	0.642063i	$-0.778079\pi$
$661$	−39.4211	−1.53330	−0.766652	+	0.642063i	$0.778079\pi$
$662$	−2.08361	−0.0809819
$663$	0	0
$664$	6.97240	0.270582
$665$	−6.61025	−0.256335
$666$	0	0
$667$	10.0138	0.387737
$668$	4.71829	0.182556
$669$	0	0
$670$	−17.9307	−0.692722
$671$	−79.7614	−3.07916
$672$	0	0
$673$	−37.5953	−1.44919	−0.724596	−	0.689174i	$-0.757974\pi$
$673$	−37.5953	−1.44919	−0.724596	+	0.689174i	$0.757974\pi$
$674$	23.7150	0.913469
$675$	0	0
$676$	0	0
$677$	−46.1459	−1.77353	−0.886765	−	0.462221i	$-0.847053\pi$
$677$	−46.1459	−1.77353	−0.886765	+	0.462221i	$0.847053\pi$
$678$	0	0
$679$	26.1002	1.00163
$680$	−5.99602	−0.229937
$681$	0	0
$682$	−74.9147	−2.86863
$683$	−35.5813	−1.36148	−0.680740	−	0.732525i	$-0.738342\pi$
$683$	−35.5813	−1.36148	−0.680740	+	0.732525i	$0.738342\pi$
$684$	0	0
$685$	−13.3796	−0.511208
$686$	−33.7170	−1.28732
$687$	0	0
$688$	−27.5441	−1.05011
$689$	0	0
$690$	0	0
$691$	−41.0144	−1.56026	−0.780131	−	0.625617i	$-0.784847\pi$
$691$	−41.0144	−1.56026	−0.780131	+	0.625617i	$0.784847\pi$
$692$	3.87245	0.147209
$693$	0	0
$694$	42.6522	1.61906
$695$	17.4455	0.661745
$696$	0	0
$697$	−0.279954	−0.0106040
$698$	−58.8149	−2.22618
$699$	0	0
$700$	1.75987	0.0665167
$701$	−27.7156	−1.04680	−0.523402	−	0.852086i	$-0.675337\pi$
$701$	−27.7156	−1.04680	−0.523402	+	0.852086i	$0.675337\pi$
$702$	0	0
$703$	31.6351	1.19314
$704$	16.1990	0.610522
$705$	0	0
$706$	−13.1557	−0.495122
$707$	17.8863	0.672682
$708$	0	0
$709$	5.63088	0.211472	0.105736	−	0.994394i	$-0.466280\pi$
$709$	5.63088	0.211472	0.105736	+	0.994394i	$0.466280\pi$
$710$	−14.8637	−0.557825
$711$	0	0
$712$	9.95491	0.373076
$713$	−25.3752	−0.950310
$714$	0	0
$715$	0	0
$716$	15.8899	0.593834
$717$	0	0
$718$	45.2329	1.68808
$719$	−27.6971	−1.03293	−0.516464	−	0.856309i	$-0.672752\pi$
$719$	−27.6971	−1.03293	−0.516464	+	0.856309i	$0.672752\pi$
$720$	0	0
$721$	20.5085	0.763775
$722$	−16.7002	−0.621517
$723$	0	0
$724$	−1.91897	−0.0713181
$725$	3.00604	0.111641
$726$	0	0
$727$	17.1083	0.634510	0.317255	−	0.948340i	$-0.397239\pi$
$727$	17.1083	0.634510	0.317255	+	0.948340i	$0.397239\pi$
$728$	0	0
$729$	0	0
$730$	7.73649	0.286340
$731$	−16.5781	−0.613163
$732$	0	0
$733$	−3.64370	−0.134583	−0.0672915	−	0.997733i	$-0.521436\pi$
$733$	−3.64370	−0.134583	−0.0672915	+	0.997733i	$0.521436\pi$
$734$	−47.6199	−1.75768
$735$	0	0
$736$	−14.2627	−0.525729
$737$	62.9903	2.32028
$738$	0	0
$739$	−10.5246	−0.387155	−0.193577	−	0.981085i	$-0.562009\pi$
$739$	−10.5246	−0.387155	−0.193577	+	0.981085i	$0.562009\pi$
$740$	−8.42232	−0.309610
$741$	0	0
$742$	−30.9332	−1.13559
$743$	−29.4629	−1.08089	−0.540445	−	0.841379i	$-0.681744\pi$
$743$	−29.4629	−1.08089	−0.540445	+	0.841379i	$0.681744\pi$
$744$	0	0
$745$	−5.78235	−0.211849
$746$	19.2460	0.704647
$747$	0	0
$748$	−14.0291	−0.512955
$749$	22.1894	0.810784
$750$	0	0
$751$	43.5122	1.58778	0.793892	−	0.608058i	$-0.208051\pi$
$751$	43.5122	1.58778	0.793892	+	0.608058i	$0.208051\pi$
$752$	−41.2187	−1.50309
$753$	0	0
$754$	0	0
$755$	0.801937	0.0291855
$756$	0	0
$757$	−0.869490	−0.0316022	−0.0158011	−	0.999875i	$-0.505030\pi$
$757$	−0.869490	−0.0316022	−0.0158011	+	0.999875i	$0.505030\pi$
$758$	9.60421	0.348841
$759$	0	0
$760$	6.03210	0.218807
$761$	−45.0860	−1.63437	−0.817184	−	0.576377i	$-0.804466\pi$
$761$	−45.0860	−1.63437	−0.817184	+	0.576377i	$0.804466\pi$
$762$	0	0
$763$	22.0357	0.797745
$764$	−4.66784	−0.168876
$765$	0	0
$766$	−16.8348	−0.608265
$767$	0	0
$768$	0	0
$769$	−16.9400	−0.610873	−0.305437	−	0.952212i	$-0.598802\pi$
$769$	−16.9400	−0.610873	−0.305437	+	0.952212i	$0.598802\pi$
$770$	−21.6473	−0.780116
$771$	0	0
$772$	3.99229	0.143686
$773$	39.7998	1.43150	0.715750	−	0.698356i	$-0.246085\pi$
$773$	39.7998	1.43150	0.715750	+	0.698356i	$0.246085\pi$
$774$	0	0
$775$	−7.61735	−0.273623
$776$	−23.8174	−0.854996
$777$	0	0
$778$	19.0575	0.683245
$779$	0.281639	0.0100908
$780$	0	0
$781$	52.2161	1.86844
$782$	−16.6387	−0.595000
$783$	0	0
$784$	10.6890	0.381750
$785$	7.63125	0.272371
$786$	0	0
$787$	−4.30295	−0.153384	−0.0766918	−	0.997055i	$-0.524436\pi$
$787$	−4.30295	−0.153384	−0.0766918	+	0.997055i	$0.524436\pi$
$788$	−8.45521	−0.301205
$789$	0	0
$790$	15.9188	0.566366
$791$	43.2970	1.53946
$792$	0	0
$793$	0	0
$794$	−27.6903	−0.982694
$795$	0	0
$796$	11.7535	0.416590
$797$	38.9711	1.38043	0.690214	−	0.723605i	$-0.257516\pi$
$797$	38.9711	1.38043	0.690214	+	0.723605i	$0.257516\pi$
$798$	0	0
$799$	−24.8084	−0.877660
$800$	−4.28148	−0.151373
$801$	0	0
$802$	51.8428	1.83063
$803$	−27.1782	−0.959099
$804$	0	0
$805$	−7.33243	−0.258434
$806$	0	0
$807$	0	0
$808$	−16.3219	−0.574201
$809$	48.3036	1.69826	0.849132	−	0.528180i	$-0.177125\pi$
$809$	48.3036	1.69826	0.849132	+	0.528180i	$0.177125\pi$
$810$	0	0
$811$	−28.0369	−0.984508	−0.492254	−	0.870452i	$-0.663827\pi$
$811$	−28.0369	−0.984508	−0.492254	+	0.870452i	$0.663827\pi$
$812$	5.29022	0.185650
$813$	0	0
$814$	103.599	3.63115
$815$	1.90400	0.0666942
$816$	0	0
$817$	16.6779	0.583485
$818$	60.7255	2.12322
$819$	0	0
$820$	−0.0749815	−0.00261847
$821$	18.1825	0.634574	0.317287	−	0.948329i	$-0.397228\pi$
$821$	18.1825	0.634574	0.317287	+	0.948329i	$0.397228\pi$
$822$	0	0
$823$	−43.6571	−1.52179	−0.760896	−	0.648874i	$-0.775240\pi$
$823$	−43.6571	−1.52179	−0.760896	+	0.648874i	$0.775240\pi$
$824$	−18.7147	−0.651959
$825$	0	0
$826$	−27.6954	−0.963645
$827$	−5.89278	−0.204912	−0.102456	−	0.994738i	$-0.532670\pi$
$827$	−5.89278	−0.204912	−0.102456	+	0.994738i	$0.532670\pi$
$828$	0	0
$829$	4.42371	0.153642	0.0768209	−	0.997045i	$-0.475523\pi$
$829$	4.42371	0.153642	0.0768209	+	0.997045i	$0.475523\pi$
$830$	5.80809	0.201602
$831$	0	0
$832$	0	0
$833$	6.43344	0.222905
$834$	0	0
$835$	−5.90128	−0.204222
$836$	14.1135	0.488126
$837$	0	0
$838$	34.2111	1.18180
$839$	13.7390	0.474324	0.237162	−	0.971470i	$-0.423783\pi$
$839$	13.7390	0.474324	0.237162	+	0.971470i	$0.423783\pi$
$840$	0	0
$841$	−19.9637	−0.688405
$842$	8.43158	0.290571
$843$	0	0
$844$	−12.8297	−0.441618
$845$	0	0
$846$	0	0
$847$	51.8347	1.78106
$848$	41.6586	1.43056
$849$	0	0
$850$	−4.99476	−0.171319
$851$	35.0913	1.20291
$852$	0	0
$853$	41.4971	1.42083	0.710417	−	0.703781i	$-0.248506\pi$
$853$	41.4971	1.42083	0.710417	+	0.703781i	$0.248506\pi$
$854$	−49.9755	−1.71013
$855$	0	0
$856$	−20.2487	−0.692085
$857$	−21.1626	−0.722901	−0.361450	−	0.932391i	$-0.617718\pi$
$857$	−21.1626	−0.722901	−0.361450	+	0.932391i	$0.617718\pi$
$858$	0	0
$859$	46.5958	1.58983	0.794914	−	0.606722i	$-0.207516\pi$
$859$	46.5958	1.58983	0.794914	+	0.606722i	$0.207516\pi$
$860$	−4.44020	−0.151409
$861$	0	0
$862$	−55.2587	−1.88212
$863$	−29.3444	−0.998895	−0.499448	−	0.866344i	$-0.666464\pi$
$863$	−29.3444	−0.998895	−0.499448	+	0.866344i	$0.666464\pi$
$864$	0	0
$865$	−4.84337	−0.164680
$866$	−3.64030	−0.123702
$867$	0	0
$868$	−13.4055	−0.455013
$869$	−55.9227	−1.89705
$870$	0	0
$871$	0	0
$872$	−20.1084	−0.680955
$873$	0	0
$874$	16.7389	0.566201
$875$	−2.20111	−0.0744110
$876$	0	0
$877$	−51.1148	−1.72602	−0.863012	−	0.505184i	$-0.831425\pi$
$877$	−51.1148	−1.72602	−0.863012	+	0.505184i	$0.831425\pi$
$878$	20.2086	0.682007
$879$	0	0
$880$	29.1532	0.982753
$881$	1.51537	0.0510540	0.0255270	−	0.999674i	$-0.491874\pi$
$881$	1.51537	0.0510540	0.0255270	+	0.999674i	$0.491874\pi$
$882$	0	0
$883$	−15.8608	−0.533760	−0.266880	−	0.963730i	$-0.585993\pi$
$883$	−15.8608	−0.533760	−0.266880	+	0.963730i	$0.585993\pi$
$884$	0	0
$885$	0	0
$886$	−12.9690	−0.435702
$887$	29.0167	0.974285	0.487142	−	0.873323i	$-0.338039\pi$
$887$	29.0167	0.974285	0.487142	+	0.873323i	$0.338039\pi$
$888$	0	0
$889$	1.51644	0.0508599
$890$	8.29256	0.277967
$891$	0	0
$892$	−23.3223	−0.780888
$893$	24.9577	0.835179
$894$	0	0
$895$	−19.8739	−0.664311
$896$	28.9977	0.968745
$897$	0	0
$898$	11.2401	0.375087
$899$	−22.8980	−0.763692
$900$	0	0
$901$	25.0733	0.835311
$902$	0.922314	0.0307097
$903$	0	0
$904$	−39.5101	−1.31409
$905$	2.40011	0.0797823
$906$	0	0
$907$	−35.3710	−1.17448	−0.587238	−	0.809414i	$-0.699785\pi$
$907$	−35.3710	−1.17448	−0.587238	+	0.809414i	$0.699785\pi$
$908$	−14.7609	−0.489856
$909$	0	0
$910$	0	0
$911$	19.4023	0.642827	0.321413	−	0.946939i	$-0.395842\pi$
$911$	19.4023	0.642827	0.321413	+	0.946939i	$0.395842\pi$
$912$	0	0
$913$	−20.4038	−0.675266
$914$	−58.8523	−1.94666
$915$	0	0
$916$	18.7849	0.620671
$917$	37.2888	1.23138
$918$	0	0
$919$	−34.2070	−1.12838	−0.564192	−	0.825644i	$-0.690812\pi$
$919$	−34.2070	−1.12838	−0.564192	+	0.825644i	$0.690812\pi$
$920$	6.69111	0.220599
$921$	0	0
$922$	−28.5285	−0.939537
$923$	0	0
$924$	0	0
$925$	10.5340	0.346356
$926$	−41.9997	−1.38019
$927$	0	0
$928$	−12.8703	−0.422488
$929$	22.2676	0.730576	0.365288	−	0.930895i	$-0.380971\pi$
$929$	22.2676	0.730576	0.365288	+	0.930895i	$0.380971\pi$
$930$	0	0
$931$	−6.47215	−0.212116
$932$	2.96370	0.0970793
$933$	0	0
$934$	−0.189643	−0.00620532
$935$	17.5465	0.573833
$936$	0	0
$937$	−14.7527	−0.481950	−0.240975	−	0.970531i	$-0.577467\pi$
$937$	−14.7527	−0.481950	−0.240975	+	0.970531i	$0.577467\pi$
$938$	39.4673	1.28865
$939$	0	0
$940$	−6.64457	−0.216722
$941$	38.4869	1.25464	0.627318	−	0.778763i	$-0.284153\pi$
$941$	38.4869	1.25464	0.627318	+	0.778763i	$0.284153\pi$
$942$	0	0
$943$	0.312408	0.0101734
$944$	37.2982	1.21395
$945$	0	0
$946$	54.6169	1.77575
$947$	−43.5707	−1.41586	−0.707929	−	0.706284i	$-0.750370\pi$
$947$	−43.5707	−1.41586	−0.707929	+	0.706284i	$0.750370\pi$
$948$	0	0
$949$	0	0
$950$	5.02481	0.163026
$951$	0	0
$952$	13.1979	0.427746
$953$	17.9696	0.582093	0.291047	−	0.956709i	$-0.405997\pi$
$953$	17.9696	0.582093	0.291047	+	0.956709i	$0.405997\pi$
$954$	0	0
$955$	5.83817	0.188919
$956$	11.3476	0.367008
$957$	0	0
$958$	17.1174	0.553039
$959$	29.4499	0.950987
$960$	0	0
$961$	27.0240	0.871742
$962$	0	0
$963$	0	0
$964$	2.97360	0.0957731
$965$	−4.99325	−0.160738
$966$	0	0
$967$	−12.0395	−0.387163	−0.193581	−	0.981084i	$-0.562010\pi$
$967$	−12.0395	−0.387163	−0.193581	+	0.981084i	$0.562010\pi$
$968$	−47.3011	−1.52032
$969$	0	0
$970$	−19.8402	−0.637030
$971$	−40.1570	−1.28870	−0.644350	−	0.764730i	$-0.722872\pi$
$971$	−40.1570	−1.28870	−0.644350	+	0.764730i	$0.722872\pi$
$972$	0	0
$973$	−38.3994	−1.23103
$974$	−33.3356	−1.06814
$975$	0	0
$976$	67.3036	2.15434
$977$	−5.99845	−0.191907	−0.0959537	−	0.995386i	$-0.530590\pi$
$977$	−5.99845	−0.191907	−0.0959537	+	0.995386i	$0.530590\pi$
$978$	0	0
$979$	−29.1317	−0.931053
$980$	1.72310	0.0550424
$981$	0	0
$982$	63.8616	2.03790
$983$	−26.6303	−0.849374	−0.424687	−	0.905340i	$-0.639616\pi$
$983$	−26.6303	−0.849374	−0.424687	+	0.905340i	$0.639616\pi$
$984$	0	0
$985$	10.5751	0.336952
$986$	−15.0144	−0.478156
$987$	0	0
$988$	0	0
$989$	18.4999	0.588263
$990$	0	0
$991$	−13.0648	−0.415017	−0.207508	−	0.978233i	$-0.566535\pi$
$991$	−13.0648	−0.415017	−0.207508	+	0.978233i	$0.566535\pi$
$992$	32.6136	1.03548
$993$	0	0
$994$	32.7166	1.03771
$995$	−14.7003	−0.466032
$996$	0	0
$997$	50.5777	1.60181	0.800907	−	0.598789i	$-0.204351\pi$
$997$	50.5777	1.60181	0.800907	+	0.598789i	$0.204351\pi$
$998$	23.1692	0.733407
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7605.2.a.cx.1.9	yes	12
3.2	odd	2		7605.2.a.cv.1.4	✓	12
13.12	even	2		7605.2.a.cw.1.4	yes	12
39.38	odd	2		7605.2.a.cy.1.9	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7605.2.a.cv.1.4	✓	12	3.2	odd	2
7605.2.a.cw.1.4	yes	12	13.12	even	2
7605.2.a.cx.1.9	yes	12	1.1	even	1	trivial
7605.2.a.cy.1.9	yes	12	39.38	odd	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:	\( N \)	\(=\)	\( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7605.a (trivial)

\(f(q)\)	\(=\)	\(q+1.67318 q^{2} +0.799537 q^{4} -1.00000 q^{5} +2.20111 q^{7} -2.00859 q^{8} +O(q^{10})\)	\(q+1.67318 q^{2} +0.799537 q^{4} -1.00000 q^{5} +2.20111 q^{7} -2.00859 q^{8} -1.67318 q^{10} +5.87787 q^{11} +3.68285 q^{14} -4.95981 q^{16} -2.98518 q^{17} +3.00315 q^{19} -0.799537 q^{20} +9.83475 q^{22} +3.33124 q^{23} +1.00000 q^{25} +1.75987 q^{28} +3.00604 q^{29} -7.61735 q^{31} -4.28148 q^{32} -4.99476 q^{34} -2.20111 q^{35} +10.5340 q^{37} +5.02481 q^{38} +2.00859 q^{40} +0.0937812 q^{41} +5.55346 q^{43} +4.69958 q^{44} +5.57377 q^{46} +8.31052 q^{47} -2.15512 q^{49} +1.67318 q^{50} -8.39923 q^{53} -5.87787 q^{55} -4.42113 q^{56} +5.02964 q^{58} -7.52009 q^{59} -13.5698 q^{61} -12.7452 q^{62} +2.75593 q^{64} +10.7165 q^{67} -2.38676 q^{68} -3.68285 q^{70} +8.88350 q^{71} -4.62382 q^{73} +17.6253 q^{74} +2.40113 q^{76} +12.9378 q^{77} -9.51410 q^{79} +4.95981 q^{80} +0.156913 q^{82} -3.47128 q^{83} +2.98518 q^{85} +9.29195 q^{86} -11.8063 q^{88} -4.95616 q^{89} +2.66345 q^{92} +13.9050 q^{94} -3.00315 q^{95} +11.8578 q^{97} -3.60591 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + q^{2} + 13 q^{4} - 12 q^{5} - 12 q^{7} + 3 q^{8}+O(q^{10}) \) 12 * q + q^2 + 13 * q^4 - 12 * q^5 - 12 * q^7 + 3 * q^8	\( 12 q + q^{2} + 13 q^{4} - 12 q^{5} - 12 q^{7} + 3 q^{8} - q^{10} - 6 q^{11} + 18 q^{14} + 11 q^{16} + 14 q^{17} - 13 q^{20} - 12 q^{22} + 28 q^{23} + 12 q^{25} - 26 q^{28} + 24 q^{29} - 2 q^{31} + 7 q^{32} - 6 q^{34} + 12 q^{35} - 8 q^{37} + 8 q^{38} - 3 q^{40} - 4 q^{41} + 4 q^{43} - 30 q^{44} + 9 q^{46} + 22 q^{47} + 8 q^{49} + q^{50} + 36 q^{53} + 6 q^{55} + 32 q^{56} + 8 q^{58} - 28 q^{59} - 38 q^{61} + 26 q^{62} - 31 q^{64} - 22 q^{67} + 46 q^{68} - 18 q^{70} + 24 q^{71} - 12 q^{73} + 26 q^{74} + 17 q^{76} + 28 q^{77} + 20 q^{79} - 11 q^{80} - 36 q^{82} - 26 q^{83} - 14 q^{85} + 74 q^{86} - 34 q^{88} - 2 q^{89} + 86 q^{92} - 16 q^{94} - 26 q^{97} - 49 q^{98}+O(q^{100}) \) 12 * q + q^2 + 13 * q^4 - 12 * q^5 - 12 * q^7 + 3 * q^8 - q^10 - 6 * q^11 + 18 * q^14 + 11 * q^16 + 14 * q^17 - 13 * q^20 - 12 * q^22 + 28 * q^23 + 12 * q^25 - 26 * q^28 + 24 * q^29 - 2 * q^31 + 7 * q^32 - 6 * q^34 + 12 * q^35 - 8 * q^37 + 8 * q^38 - 3 * q^40 - 4 * q^41 + 4 * q^43 - 30 * q^44 + 9 * q^46 + 22 * q^47 + 8 * q^49 + q^50 + 36 * q^53 + 6 * q^55 + 32 * q^56 + 8 * q^58 - 28 * q^59 - 38 * q^61 + 26 * q^62 - 31 * q^64 - 22 * q^67 + 46 * q^68 - 18 * q^70 + 24 * q^71 - 12 * q^73 + 26 * q^74 + 17 * q^76 + 28 * q^77 + 20 * q^79 - 11 * q^80 - 36 * q^82 - 26 * q^83 - 14 * q^85 + 74 * q^86 - 34 * q^88 - 2 * q^89 + 86 * q^92 - 16 * q^94 - 26 * q^97 - 49 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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