LMFDB - Embedded newform 7605.2.a.bn.1.2

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:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2535)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.24698$ 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)

$f(q)$	$=$	$q-0.554958 q^{2} -1.69202 q^{4} -1.00000 q^{5} -4.49396 q^{7} +2.04892 q^{8} +O(q^{10})$	$q-0.554958 q^{2} -1.69202 q^{4} -1.00000 q^{5} -4.49396 q^{7} +2.04892 q^{8} +0.554958 q^{10} +2.00000 q^{11} +2.49396 q^{14} +2.24698 q^{16} -2.91185 q^{17} +2.04892 q^{19} +1.69202 q^{20} -1.10992 q^{22} -4.35690 q^{23} +1.00000 q^{25} +7.60388 q^{28} +0.713792 q^{29} -5.89977 q^{31} -5.34481 q^{32} +1.61596 q^{34} +4.49396 q^{35} +11.2078 q^{37} -1.13706 q^{38} -2.04892 q^{40} -4.21983 q^{41} -2.59179 q^{43} -3.38404 q^{44} +2.41789 q^{46} +11.6528 q^{47} +13.1957 q^{49} -0.554958 q^{50} -8.71917 q^{53} -2.00000 q^{55} -9.20775 q^{56} -0.396125 q^{58} -11.9758 q^{59} +1.36227 q^{61} +3.27413 q^{62} -1.52781 q^{64} +13.8780 q^{67} +4.92692 q^{68} -2.49396 q^{70} +12.0000 q^{71} +11.3056 q^{73} -6.21983 q^{74} -3.46681 q^{76} -8.98792 q^{77} +12.3327 q^{79} -2.24698 q^{80} +2.34183 q^{82} +3.80194 q^{83} +2.91185 q^{85} +1.43834 q^{86} +4.09783 q^{88} -1.78017 q^{89} +7.37196 q^{92} -6.46681 q^{94} -2.04892 q^{95} +1.20775 q^{97} -7.32304 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} - 3 q^{5} - 4 q^{7} - 3 q^{8}+O(q^{10}) $ 3 * q - 2 * q^2 - 3 * q^5 - 4 * q^7 - 3 * q^8	$ 3 q - 2 q^{2} - 3 q^{5} - 4 q^{7} - 3 q^{8} + 2 q^{10} + 6 q^{11} - 2 q^{14} + 2 q^{16} - 5 q^{17} - 3 q^{19} - 4 q^{22} - 9 q^{23} + 3 q^{25} + 14 q^{28} - 6 q^{29} + 5 q^{31} + 7 q^{32} + 15 q^{34} + 4 q^{35} + 16 q^{37} + 2 q^{38} + 3 q^{40} - 14 q^{41} + 20 q^{43} + 13 q^{46} + 17 q^{47} + 3 q^{49} - 2 q^{50} - 15 q^{53} - 6 q^{55} - 10 q^{56} - 10 q^{58} + 2 q^{59} - 3 q^{61} - q^{62} - 11 q^{64} + 22 q^{67} - 14 q^{68} + 2 q^{70} + 36 q^{71} - 2 q^{73} - 20 q^{74} - 7 q^{76} - 8 q^{77} - 5 q^{79} - 2 q^{80} + 28 q^{82} + 7 q^{83} + 5 q^{85} - 18 q^{86} - 6 q^{88} - 4 q^{89} - 7 q^{92} - 16 q^{94} + 3 q^{95} - 14 q^{97} - 2 q^{98}+O(q^{100}) $ 3 * q - 2 * q^2 - 3 * q^5 - 4 * q^7 - 3 * q^8 + 2 * q^10 + 6 * q^11 - 2 * q^14 + 2 * q^16 - 5 * q^17 - 3 * q^19 - 4 * q^22 - 9 * q^23 + 3 * q^25 + 14 * q^28 - 6 * q^29 + 5 * q^31 + 7 * q^32 + 15 * q^34 + 4 * q^35 + 16 * q^37 + 2 * q^38 + 3 * q^40 - 14 * q^41 + 20 * q^43 + 13 * q^46 + 17 * q^47 + 3 * q^49 - 2 * q^50 - 15 * q^53 - 6 * q^55 - 10 * q^56 - 10 * q^58 + 2 * q^59 - 3 * q^61 - q^62 - 11 * q^64 + 22 * q^67 - 14 * q^68 + 2 * q^70 + 36 * q^71 - 2 * q^73 - 20 * q^74 - 7 * q^76 - 8 * q^77 - 5 * q^79 - 2 * q^80 + 28 * q^82 + 7 * q^83 + 5 * q^85 - 18 * q^86 - 6 * q^88 - 4 * q^89 - 7 * q^92 - 16 * q^94 + 3 * q^95 - 14 * q^97 - 2 * 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 $

$2$	−0.554958	−0.392415	−0.196207	−	0.980562i	$-0.562863\pi$
$2$	−0.554958	−0.392415	−0.196207	+	0.980562i	$0.562863\pi$
$3$	0	0
$3$	0	0
$4$	−1.69202	−0.846011
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.49396	−1.69856	−0.849278	−	0.527945i	$-0.822963\pi$
$7$	−4.49396	−1.69856	−0.849278	+	0.527945i	$0.822963\pi$
$8$	2.04892	0.724402
$9$	0	0
$10$	0.554958	0.175493
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	2.49396	0.666539
$15$	0	0
$16$	2.24698	0.561745
$17$	−2.91185	−0.706228	−0.353114	−	0.935580i	$-0.614877\pi$
$17$	−2.91185	−0.706228	−0.353114	+	0.935580i	$0.614877\pi$
$18$	0	0
$19$	2.04892	0.470054	0.235027	−	0.971989i	$-0.424482\pi$
$19$	2.04892	0.470054	0.235027	+	0.971989i	$0.424482\pi$
$20$	1.69202	0.378348
$21$	0	0
$22$	−1.10992	−0.236635
$23$	−4.35690	−0.908476	−0.454238	−	0.890880i	$-0.650088\pi$
$23$	−4.35690	−0.908476	−0.454238	+	0.890880i	$0.650088\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	7.60388	1.43700
$29$	0.713792	0.132548	0.0662739	−	0.997801i	$-0.478889\pi$
$29$	0.713792	0.132548	0.0662739	+	0.997801i	$0.478889\pi$
$30$	0	0
$31$	−5.89977	−1.05963	−0.529815	−	0.848113i	$-0.677739\pi$
$31$	−5.89977	−1.05963	−0.529815	+	0.848113i	$0.677739\pi$
$32$	−5.34481	−0.944839
$33$	0	0
$34$	1.61596	0.277134
$35$	4.49396	0.759618
$36$	0	0
$37$	11.2078	1.84254	0.921271	−	0.388920i	$-0.127152\pi$
$37$	11.2078	1.84254	0.921271	+	0.388920i	$0.127152\pi$
$38$	−1.13706	−0.184456
$39$	0	0
$40$	−2.04892	−0.323962
$41$	−4.21983	−0.659027	−0.329514	−	0.944151i	$-0.606885\pi$
$41$	−4.21983	−0.659027	−0.329514	+	0.944151i	$0.606885\pi$
$42$	0	0
$43$	−2.59179	−0.395245	−0.197622	−	0.980278i	$-0.563322\pi$
$43$	−2.59179	−0.395245	−0.197622	+	0.980278i	$0.563322\pi$
$44$	−3.38404	−0.510164
$45$	0	0
$46$	2.41789	0.356499
$47$	11.6528	1.69973	0.849867	−	0.526997i	$-0.176682\pi$
$47$	11.6528	1.69973	0.849867	+	0.526997i	$0.176682\pi$
$48$	0	0
$49$	13.1957	1.88510
$50$	−0.554958	−0.0784829
$51$	0	0
$52$	0	0
$53$	−8.71917	−1.19767	−0.598835	−	0.800872i	$-0.704369\pi$
$53$	−8.71917	−1.19767	−0.598835	+	0.800872i	$0.704369\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	−9.20775	−1.23044
$57$	0	0
$58$	−0.396125	−0.0520137
$59$	−11.9758	−1.55912	−0.779561	−	0.626327i	$-0.784558\pi$
$59$	−11.9758	−1.55912	−0.779561	+	0.626327i	$0.784558\pi$
$60$	0	0
$61$	1.36227	0.174421	0.0872106	−	0.996190i	$-0.472205\pi$
$61$	1.36227	0.174421	0.0872106	+	0.996190i	$0.472205\pi$
$62$	3.27413	0.415815
$63$	0	0
$64$	−1.52781	−0.190976
$65$	0	0
$66$	0	0
$67$	13.8780	1.69547	0.847734	−	0.530422i	$-0.177966\pi$
$67$	13.8780	1.69547	0.847734	+	0.530422i	$0.177966\pi$
$68$	4.92692	0.597477
$69$	0	0
$70$	−2.49396	−0.298085
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	11.3056	1.32322	0.661609	−	0.749849i	$-0.269874\pi$
$73$	11.3056	1.32322	0.661609	+	0.749849i	$0.269874\pi$
$74$	−6.21983	−0.723041
$75$	0	0
$76$	−3.46681	−0.397671
$77$	−8.98792	−1.02427
$78$	0	0
$79$	12.3327	1.38754	0.693770	−	0.720196i	$-0.255948\pi$
$79$	12.3327	1.38754	0.693770	+	0.720196i	$0.255948\pi$
$80$	−2.24698	−0.251220
$81$	0	0
$82$	2.34183	0.258612
$83$	3.80194	0.417317	0.208658	−	0.977989i	$-0.433090\pi$
$83$	3.80194	0.417317	0.208658	+	0.977989i	$0.433090\pi$
$84$	0	0
$85$	2.91185	0.315835
$86$	1.43834	0.155100
$87$	0	0
$88$	4.09783	0.436831
$89$	−1.78017	−0.188697	−0.0943487	−	0.995539i	$-0.530077\pi$
$89$	−1.78017	−0.188697	−0.0943487	+	0.995539i	$0.530077\pi$
$90$	0	0
$91$	0	0
$92$	7.37196	0.768580
$93$	0	0
$94$	−6.46681	−0.667001
$95$	−2.04892	−0.210214
$96$	0	0
$97$	1.20775	0.122629	0.0613143	−	0.998119i	$-0.480471\pi$
$97$	1.20775	0.122629	0.0613143	+	0.998119i	$0.480471\pi$
$98$	−7.32304	−0.739739
$99$	0	0
$100$	−1.69202	−0.169202
$101$	−13.1099	−1.30449	−0.652243	−	0.758010i	$-0.726172\pi$
$101$	−13.1099	−1.30449	−0.652243	+	0.758010i	$0.726172\pi$
$102$	0	0
$103$	−0.933624	−0.0919927	−0.0459964	−	0.998942i	$-0.514646\pi$
$103$	−0.933624	−0.0919927	−0.0459964	+	0.998942i	$0.514646\pi$
$104$	0	0
$105$	0	0
$106$	4.83877	0.469983
$107$	9.53079	0.921377	0.460688	−	0.887562i	$-0.347603\pi$
$107$	9.53079	0.921377	0.460688	+	0.887562i	$0.347603\pi$
$108$	0	0
$109$	0.0935228	0.00895786	0.00447893	−	0.999990i	$-0.498574\pi$
$109$	0.0935228	0.00895786	0.00447893	+	0.999990i	$0.498574\pi$
$110$	1.10992	0.105826
$111$	0	0
$112$	−10.0978	−0.954156
$113$	6.72886	0.632998	0.316499	−	0.948593i	$-0.397493\pi$
$113$	6.72886	0.632998	0.316499	+	0.948593i	$0.397493\pi$
$114$	0	0
$115$	4.35690	0.406283
$116$	−1.20775	−0.112137
$117$	0	0
$118$	6.64609	0.611822
$119$	13.0858	1.19957
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−0.756004	−0.0684454
$123$	0	0
$124$	9.98254	0.896459
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	11.5375	1.01978
$129$	0	0
$130$	0	0
$131$	−22.2500	−1.94399	−0.971994	−	0.235005i	$-0.924489\pi$
$131$	−22.2500	−1.94399	−0.971994	+	0.235005i	$0.924489\pi$
$132$	0	0
$133$	−9.20775	−0.798413
$134$	−7.70171	−0.665326
$135$	0	0
$136$	−5.96615	−0.511593
$137$	17.3274	1.48038	0.740188	−	0.672400i	$-0.234737\pi$
$137$	17.3274	1.48038	0.740188	+	0.672400i	$0.234737\pi$
$138$	0	0
$139$	20.8509	1.76855	0.884273	−	0.466970i	$-0.154654\pi$
$139$	20.8509	1.76855	0.884273	+	0.466970i	$0.154654\pi$
$140$	−7.60388	−0.642645
$141$	0	0
$142$	−6.65950	−0.558853
$143$	0	0
$144$	0	0
$145$	−0.713792	−0.0592772
$146$	−6.27413	−0.519250
$147$	0	0
$148$	−18.9638	−1.55881
$149$	−16.0978	−1.31879	−0.659393	−	0.751798i	$-0.729187\pi$
$149$	−16.0978	−1.31879	−0.659393	+	0.751798i	$0.729187\pi$
$150$	0	0
$151$	−18.6310	−1.51617	−0.758086	−	0.652155i	$-0.773865\pi$
$151$	−18.6310	−1.51617	−0.758086	+	0.652155i	$0.773865\pi$
$152$	4.19806	0.340508
$153$	0	0
$154$	4.98792	0.401938
$155$	5.89977	0.473881
$156$	0	0
$157$	18.2392	1.45565	0.727824	−	0.685764i	$-0.240532\pi$
$157$	18.2392	1.45565	0.727824	+	0.685764i	$0.240532\pi$
$158$	−6.84415	−0.544491
$159$	0	0
$160$	5.34481	0.422545
$161$	19.5797	1.54310
$162$	0	0
$163$	−11.3056	−0.885522	−0.442761	−	0.896640i	$-0.646001\pi$
$163$	−11.3056	−0.885522	−0.442761	+	0.896640i	$0.646001\pi$
$164$	7.14005	0.557544
$165$	0	0
$166$	−2.10992	−0.163761
$167$	15.9976	1.23793	0.618966	−	0.785418i	$-0.287552\pi$
$167$	15.9976	1.23793	0.618966	+	0.785418i	$0.287552\pi$
$168$	0	0
$169$	0	0
$170$	−1.61596	−0.123938
$171$	0	0
$172$	4.38537	0.334381
$173$	19.6082	1.49078	0.745391	−	0.666627i	$-0.232263\pi$
$173$	19.6082	1.49078	0.745391	+	0.666627i	$0.232263\pi$
$174$	0	0
$175$	−4.49396	−0.339711
$176$	4.49396	0.338745
$177$	0	0
$178$	0.987918	0.0740476
$179$	−6.94438	−0.519047	−0.259524	−	0.965737i	$-0.583566\pi$
$179$	−6.94438	−0.519047	−0.259524	+	0.965737i	$0.583566\pi$
$180$	0	0
$181$	−17.7265	−1.31760	−0.658799	−	0.752319i	$-0.728935\pi$
$181$	−17.7265	−1.31760	−0.658799	+	0.752319i	$0.728935\pi$
$182$	0	0
$183$	0	0
$184$	−8.92692	−0.658101
$185$	−11.2078	−0.824010
$186$	0	0
$187$	−5.82371	−0.425872
$188$	−19.7168	−1.43799
$189$	0	0
$190$	1.13706	0.0824912
$191$	−3.60388	−0.260767	−0.130384	−	0.991464i	$-0.541621\pi$
$191$	−3.60388	−0.260767	−0.130384	+	0.991464i	$0.541621\pi$
$192$	0	0
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	−0.670251	−0.0481212
$195$	0	0
$196$	−22.3274	−1.59481
$197$	4.36658	0.311106	0.155553	−	0.987828i	$-0.450284\pi$
$197$	4.36658	0.311106	0.155553	+	0.987828i	$0.450284\pi$
$198$	0	0
$199$	−3.48427	−0.246993	−0.123497	−	0.992345i	$-0.539411\pi$
$199$	−3.48427	−0.246993	−0.123497	+	0.992345i	$0.539411\pi$
$200$	2.04892	0.144880
$201$	0	0
$202$	7.27545	0.511899
$203$	−3.20775	−0.225140
$204$	0	0
$205$	4.21983	0.294726
$206$	0.518122	0.0360993
$207$	0	0
$208$	0	0
$209$	4.09783	0.283453
$210$	0	0
$211$	−11.7313	−0.807613	−0.403806	−	0.914845i	$-0.632313\pi$
$211$	−11.7313	−0.807613	−0.403806	+	0.914845i	$0.632313\pi$
$212$	14.7530	1.01324
$213$	0	0
$214$	−5.28919	−0.361562
$215$	2.59179	0.176759
$216$	0	0
$217$	26.5133	1.79984
$218$	−0.0519012	−0.00351520
$219$	0	0
$220$	3.38404	0.228152
$221$	0	0
$222$	0	0
$223$	15.7453	1.05438	0.527190	−	0.849747i	$-0.323246\pi$
$223$	15.7453	1.05438	0.527190	+	0.849747i	$0.323246\pi$
$224$	24.0194	1.60486
$225$	0	0
$226$	−3.73423	−0.248398
$227$	−16.8364	−1.11747	−0.558735	−	0.829346i	$-0.688713\pi$
$227$	−16.8364	−1.11747	−0.558735	+	0.829346i	$0.688713\pi$
$228$	0	0
$229$	7.34050	0.485074	0.242537	−	0.970142i	$-0.422020\pi$
$229$	7.34050	0.485074	0.242537	+	0.970142i	$0.422020\pi$
$230$	−2.41789	−0.159431
$231$	0	0
$232$	1.46250	0.0960178
$233$	−21.7942	−1.42778	−0.713892	−	0.700256i	$-0.753069\pi$
$233$	−21.7942	−1.42778	−0.713892	+	0.700256i	$0.753069\pi$
$234$	0	0
$235$	−11.6528	−0.760144
$236$	20.2634	1.31903
$237$	0	0
$238$	−7.26205	−0.470728
$239$	−2.96854	−0.192019	−0.0960095	−	0.995380i	$-0.530608\pi$
$239$	−2.96854	−0.192019	−0.0960095	+	0.995380i	$0.530608\pi$
$240$	0	0
$241$	−0.972853	−0.0626670	−0.0313335	−	0.999509i	$-0.509975\pi$
$241$	−0.972853	−0.0626670	−0.0313335	+	0.999509i	$0.509975\pi$
$242$	3.88471	0.249718
$243$	0	0
$244$	−2.30499	−0.147562
$245$	−13.1957	−0.843040
$246$	0	0
$247$	0	0
$248$	−12.0881	−0.767598
$249$	0	0
$250$	0.554958	0.0350986
$251$	−24.9879	−1.57722	−0.788612	−	0.614892i	$-0.789200\pi$
$251$	−24.9879	−1.57722	−0.788612	+	0.614892i	$0.789200\pi$
$252$	0	0
$253$	−8.71379	−0.547831
$254$	−1.10992	−0.0696423
$255$	0	0
$256$	−3.34721	−0.209200
$257$	−8.07308	−0.503585	−0.251792	−	0.967781i	$-0.581020\pi$
$257$	−8.07308	−0.503585	−0.251792	+	0.967781i	$0.581020\pi$
$258$	0	0
$259$	−50.3672	−3.12966
$260$	0	0
$261$	0	0
$262$	12.3478	0.762850
$263$	−23.6450	−1.45801	−0.729007	−	0.684506i	$-0.760018\pi$
$263$	−23.6450	−1.45801	−0.729007	+	0.684506i	$0.760018\pi$
$264$	0	0
$265$	8.71917	0.535614
$266$	5.10992	0.313309
$267$	0	0
$268$	−23.4819	−1.43438
$269$	26.4892	1.61507	0.807537	−	0.589817i	$-0.200800\pi$
$269$	26.4892	1.61507	0.807537	+	0.589817i	$0.200800\pi$
$270$	0	0
$271$	−14.7181	−0.894061	−0.447031	−	0.894519i	$-0.647518\pi$
$271$	−14.7181	−0.894061	−0.447031	+	0.894519i	$0.647518\pi$
$272$	−6.54288	−0.396720
$273$	0	0
$274$	−9.61596	−0.580921
$275$	2.00000	0.120605
$276$	0	0
$277$	5.90217	0.354627	0.177313	−	0.984154i	$-0.443259\pi$
$277$	5.90217	0.354627	0.177313	+	0.984154i	$0.443259\pi$
$278$	−11.5714	−0.694004
$279$	0	0
$280$	9.20775	0.550268
$281$	−19.3840	−1.15636	−0.578178	−	0.815911i	$-0.696236\pi$
$281$	−19.3840	−1.15636	−0.578178	+	0.815911i	$0.696236\pi$
$282$	0	0
$283$	−31.9758	−1.90077	−0.950383	−	0.311082i	$-0.899309\pi$
$283$	−31.9758	−1.90077	−0.950383	+	0.311082i	$0.899309\pi$
$284$	−20.3043	−1.20484
$285$	0	0
$286$	0	0
$287$	18.9638	1.11940
$288$	0	0
$289$	−8.52111	−0.501242
$290$	0.396125	0.0232612
$291$	0	0
$292$	−19.1293	−1.11946
$293$	0.806250	0.0471016	0.0235508	−	0.999723i	$-0.492503\pi$
$293$	0.806250	0.0471016	0.0235508	+	0.999723i	$0.492503\pi$
$294$	0	0
$295$	11.9758	0.697260
$296$	22.9638	1.33474
$297$	0	0
$298$	8.93362	0.517511
$299$	0	0
$300$	0	0
$301$	11.6474	0.671346
$302$	10.3394	0.594968
$303$	0	0
$304$	4.60388	0.264050
$305$	−1.36227	−0.0780035
$306$	0	0
$307$	9.85384	0.562388	0.281194	−	0.959651i	$-0.409270\pi$
$307$	9.85384	0.562388	0.281194	+	0.959651i	$0.409270\pi$
$308$	15.2078	0.866542
$309$	0	0
$310$	−3.27413	−0.185958
$311$	8.49396	0.481648	0.240824	−	0.970569i	$-0.422582\pi$
$311$	8.49396	0.481648	0.240824	+	0.970569i	$0.422582\pi$
$312$	0	0
$313$	−12.8901	−0.728591	−0.364295	−	0.931283i	$-0.618690\pi$
$313$	−12.8901	−0.728591	−0.364295	+	0.931283i	$0.618690\pi$
$314$	−10.1220	−0.571217
$315$	0	0
$316$	−20.8672	−1.17387
$317$	−7.60819	−0.427318	−0.213659	−	0.976908i	$-0.568538\pi$
$317$	−7.60819	−0.427318	−0.213659	+	0.976908i	$0.568538\pi$
$318$	0	0
$319$	1.42758	0.0799293
$320$	1.52781	0.0854072
$321$	0	0
$322$	−10.8659	−0.605534
$323$	−5.96615	−0.331965
$324$	0	0
$325$	0	0
$326$	6.27413	0.347492
$327$	0	0
$328$	−8.64609	−0.477400
$329$	−52.3672	−2.88710
$330$	0	0
$331$	−18.2524	−1.00324	−0.501620	−	0.865088i	$-0.667262\pi$
$331$	−18.2524	−1.00324	−0.501620	+	0.865088i	$0.667262\pi$
$332$	−6.43296	−0.353055
$333$	0	0
$334$	−8.87800	−0.485783
$335$	−13.8780	−0.758236
$336$	0	0
$337$	8.26337	0.450135	0.225067	−	0.974343i	$-0.427740\pi$
$337$	8.26337	0.450135	0.225067	+	0.974343i	$0.427740\pi$
$338$	0	0
$339$	0	0
$340$	−4.92692	−0.267200
$341$	−11.7995	−0.638981
$342$	0	0
$343$	−27.8431	−1.50339
$344$	−5.31037	−0.286316
$345$	0	0
$346$	−10.8817	−0.585005
$347$	−33.4252	−1.79436	−0.897179	−	0.441667i	$-0.854387\pi$
$347$	−33.4252	−1.79436	−0.897179	+	0.441667i	$0.854387\pi$
$348$	0	0
$349$	34.8122	1.86346	0.931728	−	0.363158i	$-0.118301\pi$
$349$	34.8122	1.86346	0.931728	+	0.363158i	$0.118301\pi$
$350$	2.49396	0.133308
$351$	0	0
$352$	−10.6896	−0.569759
$353$	1.27114	0.0676561	0.0338281	−	0.999428i	$-0.489230\pi$
$353$	1.27114	0.0676561	0.0338281	+	0.999428i	$0.489230\pi$
$354$	0	0
$355$	−12.0000	−0.636894
$356$	3.01208	0.159640
$357$	0	0
$358$	3.85384	0.203682
$359$	−13.2185	−0.697646	−0.348823	−	0.937189i	$-0.613419\pi$
$359$	−13.2185	−0.697646	−0.348823	+	0.937189i	$0.613419\pi$
$360$	0	0
$361$	−14.8019	−0.779049
$362$	9.83745	0.517045
$363$	0	0
$364$	0	0
$365$	−11.3056	−0.591761
$366$	0	0
$367$	−27.5905	−1.44021	−0.720105	−	0.693865i	$-0.755907\pi$
$367$	−27.5905	−1.44021	−0.720105	+	0.693865i	$0.755907\pi$
$368$	−9.78986	−0.510332
$369$	0	0
$370$	6.21983	0.323354
$371$	39.1836	2.03431
$372$	0	0
$373$	−14.3177	−0.741341	−0.370670	−	0.928764i	$-0.620872\pi$
$373$	−14.3177	−0.741341	−0.370670	+	0.928764i	$0.620872\pi$
$374$	3.23191	0.167118
$375$	0	0
$376$	23.8756	1.23129
$377$	0	0
$378$	0	0
$379$	7.51035	0.385781	0.192890	−	0.981220i	$-0.438214\pi$
$379$	7.51035	0.385781	0.192890	+	0.981220i	$0.438214\pi$
$380$	3.46681	0.177844
$381$	0	0
$382$	2.00000	0.102329
$383$	27.0465	1.38201	0.691006	−	0.722849i	$-0.257168\pi$
$383$	27.0465	1.38201	0.691006	+	0.722849i	$0.257168\pi$
$384$	0	0
$385$	8.98792	0.458067
$386$	1.10992	0.0564933
$387$	0	0
$388$	−2.04354	−0.103745
$389$	−0.415502	−0.0210668	−0.0105334	−	0.999945i	$-0.503353\pi$
$389$	−0.415502	−0.0210668	−0.0105334	+	0.999945i	$0.503353\pi$
$390$	0	0
$391$	12.6866	0.641591
$392$	27.0368	1.36557
$393$	0	0
$394$	−2.42327	−0.122083
$395$	−12.3327	−0.620527
$396$	0	0
$397$	0.737955	0.0370369	0.0185184	−	0.999829i	$-0.494105\pi$
$397$	0.737955	0.0370369	0.0185184	+	0.999829i	$0.494105\pi$
$398$	1.93362	0.0969238
$399$	0	0
$400$	2.24698	0.112349
$401$	11.1400	0.556307	0.278154	−	0.960537i	$-0.410278\pi$
$401$	11.1400	0.556307	0.278154	+	0.960537i	$0.410278\pi$
$402$	0	0
$403$	0	0
$404$	22.1823	1.10361
$405$	0	0
$406$	1.78017	0.0883482
$407$	22.4155	1.11110
$408$	0	0
$409$	0.376747	0.0186289	0.00931447	−	0.999957i	$-0.497035\pi$
$409$	0.376747	0.0186289	0.00931447	+	0.999957i	$0.497035\pi$
$410$	−2.34183	−0.115655
$411$	0	0
$412$	1.57971	0.0778268
$413$	53.8189	2.64826
$414$	0	0
$415$	−3.80194	−0.186630
$416$	0	0
$417$	0	0
$418$	−2.27413	−0.111231
$419$	−0.968541	−0.0473163	−0.0236582	−	0.999720i	$-0.507531\pi$
$419$	−0.968541	−0.0473163	−0.0236582	+	0.999720i	$0.507531\pi$
$420$	0	0
$421$	35.4185	1.72619	0.863096	−	0.505040i	$-0.168522\pi$
$421$	35.4185	1.72619	0.863096	+	0.505040i	$0.168522\pi$
$422$	6.51035	0.316919
$423$	0	0
$424$	−17.8649	−0.867594
$425$	−2.91185	−0.141246
$426$	0	0
$427$	−6.12200	−0.296264
$428$	−16.1263	−0.779495
$429$	0	0
$430$	−1.43834	−0.0693628
$431$	−20.9772	−1.01043	−0.505217	−	0.862992i	$-0.668588\pi$
$431$	−20.9772	−1.01043	−0.505217	+	0.862992i	$0.668588\pi$
$432$	0	0
$433$	−22.3806	−1.07554	−0.537771	−	0.843091i	$-0.680734\pi$
$433$	−22.3806	−1.07554	−0.537771	+	0.843091i	$0.680734\pi$
$434$	−14.7138	−0.706285
$435$	0	0
$436$	−0.158243	−0.00757845
$437$	−8.92692	−0.427032
$438$	0	0
$439$	27.5743	1.31605	0.658026	−	0.752996i	$-0.271392\pi$
$439$	27.5743	1.31605	0.658026	+	0.752996i	$0.271392\pi$
$440$	−4.09783	−0.195357
$441$	0	0
$442$	0	0
$443$	−20.3086	−0.964889	−0.482445	−	0.875926i	$-0.660251\pi$
$443$	−20.3086	−0.964889	−0.482445	+	0.875926i	$0.660251\pi$
$444$	0	0
$445$	1.78017	0.0843880
$446$	−8.73795	−0.413754
$447$	0	0
$448$	6.86592	0.324384
$449$	−30.3129	−1.43055	−0.715277	−	0.698841i	$-0.753699\pi$
$449$	−30.3129	−1.43055	−0.715277	+	0.698841i	$0.753699\pi$
$450$	0	0
$451$	−8.43967	−0.397408
$452$	−11.3854	−0.535523
$453$	0	0
$454$	9.34349	0.438512
$455$	0	0
$456$	0	0
$457$	−5.20775	−0.243608	−0.121804	−	0.992554i	$-0.538868\pi$
$457$	−5.20775	−0.243608	−0.121804	+	0.992554i	$0.538868\pi$
$458$	−4.07367	−0.190350
$459$	0	0
$460$	−7.37196	−0.343719
$461$	3.92154	0.182644	0.0913222	−	0.995821i	$-0.470891\pi$
$461$	3.92154	0.182644	0.0913222	+	0.995821i	$0.470891\pi$
$462$	0	0
$463$	2.16554	0.100641	0.0503206	−	0.998733i	$-0.483976\pi$
$463$	2.16554	0.100641	0.0503206	+	0.998733i	$0.483976\pi$
$464$	1.60388	0.0744580
$465$	0	0
$466$	12.0949	0.560283
$467$	0.738546	0.0341758	0.0170879	−	0.999854i	$-0.494560\pi$
$467$	0.738546	0.0341758	0.0170879	+	0.999854i	$0.494560\pi$
$468$	0	0
$469$	−62.3672	−2.87985
$470$	6.46681	0.298292
$471$	0	0
$472$	−24.5375	−1.12943
$473$	−5.18359	−0.238342
$474$	0	0
$475$	2.04892	0.0940108
$476$	−22.1414	−1.01485
$477$	0	0
$478$	1.64742	0.0753511
$479$	−23.6668	−1.08136	−0.540682	−	0.841227i	$-0.681834\pi$
$479$	−23.6668	−1.08136	−0.540682	+	0.841227i	$0.681834\pi$
$480$	0	0
$481$	0	0
$482$	0.539893	0.0245914
$483$	0	0
$484$	11.8442	0.538370
$485$	−1.20775	−0.0548411
$486$	0	0
$487$	−30.9831	−1.40398	−0.701990	−	0.712187i	$-0.747705\pi$
$487$	−30.9831	−1.40398	−0.701990	+	0.712187i	$0.747705\pi$
$488$	2.79118	0.126351
$489$	0	0
$490$	7.32304	0.330821
$491$	−2.00000	−0.0902587	−0.0451294	−	0.998981i	$-0.514370\pi$
$491$	−2.00000	−0.0902587	−0.0451294	+	0.998981i	$0.514370\pi$
$492$	0	0
$493$	−2.07846	−0.0936090
$494$	0	0
$495$	0	0
$496$	−13.2567	−0.595242
$497$	−53.9275	−2.41898
$498$	0	0
$499$	−10.3830	−0.464806	−0.232403	−	0.972620i	$-0.574659\pi$
$499$	−10.3830	−0.464806	−0.232403	+	0.972620i	$0.574659\pi$
$500$	1.69202	0.0756695
$501$	0	0
$502$	13.8672	0.618926
$503$	22.6165	1.00842	0.504211	−	0.863580i	$-0.331783\pi$
$503$	22.6165	1.00842	0.504211	+	0.863580i	$0.331783\pi$
$504$	0	0
$505$	13.1099	0.583384
$506$	4.83579	0.214977
$507$	0	0
$508$	−3.38404	−0.150143
$509$	22.6112	1.00222	0.501111	−	0.865383i	$-0.332925\pi$
$509$	22.6112	1.00222	0.501111	+	0.865383i	$0.332925\pi$
$510$	0	0
$511$	−50.8068	−2.24756
$512$	−21.2174	−0.937687
$513$	0	0
$514$	4.48022	0.197614
$515$	0.933624	0.0411404
$516$	0	0
$517$	23.3056	1.02498
$518$	27.9517	1.22813
$519$	0	0
$520$	0	0
$521$	−24.6353	−1.07929	−0.539647	−	0.841892i	$-0.681442\pi$
$521$	−24.6353	−1.07929	−0.539647	+	0.841892i	$0.681442\pi$
$522$	0	0
$523$	20.2392	0.884999	0.442499	−	0.896769i	$-0.354092\pi$
$523$	20.2392	0.884999	0.442499	+	0.896769i	$0.354092\pi$
$524$	37.6474	1.64464
$525$	0	0
$526$	13.1220	0.572146
$527$	17.1793	0.748341
$528$	0	0
$529$	−4.01746	−0.174672
$530$	−4.83877	−0.210183
$531$	0	0
$532$	15.5797	0.675466
$533$	0	0
$534$	0	0
$535$	−9.53079	−0.412052
$536$	28.4349	1.22820
$537$	0	0
$538$	−14.7004	−0.633778
$539$	26.3913	1.13676
$540$	0	0
$541$	22.9691	0.987520	0.493760	−	0.869598i	$-0.335622\pi$
$541$	22.9691	0.987520	0.493760	+	0.869598i	$0.335622\pi$
$542$	8.16793	0.350843
$543$	0	0
$544$	15.5633	0.667272
$545$	−0.0935228	−0.00400608
$546$	0	0
$547$	8.70901	0.372370	0.186185	−	0.982515i	$-0.440388\pi$
$547$	8.70901	0.372370	0.186185	+	0.982515i	$0.440388\pi$
$548$	−29.3183	−1.25241
$549$	0	0
$550$	−1.10992	−0.0473270
$551$	1.46250	0.0623046
$552$	0	0
$553$	−55.4228	−2.35682
$554$	−3.27545	−0.139161
$555$	0	0
$556$	−35.2801	−1.49621
$557$	−8.45367	−0.358193	−0.179097	−	0.983831i	$-0.557317\pi$
$557$	−8.45367	−0.358193	−0.179097	+	0.983831i	$0.557317\pi$
$558$	0	0
$559$	0	0
$560$	10.0978	0.426711
$561$	0	0
$562$	10.7573	0.453771
$563$	9.95002	0.419343	0.209672	−	0.977772i	$-0.432761\pi$
$563$	9.95002	0.419343	0.209672	+	0.977772i	$0.432761\pi$
$564$	0	0
$565$	−6.72886	−0.283085
$566$	17.7453	0.745889
$567$	0	0
$568$	24.5870	1.03165
$569$	31.6426	1.32653	0.663264	−	0.748385i	$-0.269171\pi$
$569$	31.6426	1.32653	0.663264	+	0.748385i	$0.269171\pi$
$570$	0	0
$571$	4.68532	0.196074	0.0980372	−	0.995183i	$-0.468744\pi$
$571$	4.68532	0.196074	0.0980372	+	0.995183i	$0.468744\pi$
$572$	0	0
$573$	0	0
$574$	−10.5241	−0.439267
$575$	−4.35690	−0.181695
$576$	0	0
$577$	1.28621	0.0535456	0.0267728	−	0.999642i	$-0.491477\pi$
$577$	1.28621	0.0535456	0.0267728	+	0.999642i	$0.491477\pi$
$578$	4.72886	0.196695
$579$	0	0
$580$	1.20775	0.0501491
$581$	−17.0858	−0.708836
$582$	0	0
$583$	−17.4383	−0.722222
$584$	23.1642	0.958542
$585$	0	0
$586$	−0.447435	−0.0184834
$587$	45.2922	1.86941	0.934704	−	0.355427i	$-0.115664\pi$
$587$	45.2922	1.86941	0.934704	+	0.355427i	$0.115664\pi$
$588$	0	0
$589$	−12.0881	−0.498083
$590$	−6.64609	−0.273615
$591$	0	0
$592$	25.1836	1.03504
$593$	3.32544	0.136559	0.0682797	−	0.997666i	$-0.478249\pi$
$593$	3.32544	0.136559	0.0682797	+	0.997666i	$0.478249\pi$
$594$	0	0
$595$	−13.0858	−0.536464
$596$	27.2379	1.11571
$597$	0	0
$598$	0	0
$599$	17.3733	0.709853	0.354927	−	0.934894i	$-0.384506\pi$
$599$	17.3733	0.709853	0.354927	+	0.934894i	$0.384506\pi$
$600$	0	0
$601$	15.4969	0.632133	0.316067	−	0.948737i	$-0.397638\pi$
$601$	15.4969	0.632133	0.316067	+	0.948737i	$0.397638\pi$
$602$	−6.46383	−0.263446
$603$	0	0
$604$	31.5241	1.28270
$605$	7.00000	0.284590
$606$	0	0
$607$	−26.5918	−1.07933	−0.539664	−	0.841881i	$-0.681449\pi$
$607$	−26.5918	−1.07933	−0.539664	+	0.841881i	$0.681449\pi$
$608$	−10.9511	−0.444125
$609$	0	0
$610$	0.756004	0.0306097
$611$	0	0
$612$	0	0
$613$	−11.7065	−0.472821	−0.236410	−	0.971653i	$-0.575971\pi$
$613$	−11.7065	−0.472821	−0.236410	+	0.971653i	$0.575971\pi$
$614$	−5.46847	−0.220689
$615$	0	0
$616$	−18.4155	−0.741982
$617$	−8.04785	−0.323994	−0.161997	−	0.986791i	$-0.551794\pi$
$617$	−8.04785	−0.323994	−0.161997	+	0.986791i	$0.551794\pi$
$618$	0	0
$619$	8.30260	0.333710	0.166855	−	0.985981i	$-0.446639\pi$
$619$	8.30260	0.333710	0.166855	+	0.985981i	$0.446639\pi$
$620$	−9.98254	−0.400909
$621$	0	0
$622$	−4.71379	−0.189006
$623$	8.00000	0.320513
$624$	0	0
$625$	1.00000	0.0400000
$626$	7.15346	0.285910
$627$	0	0
$628$	−30.8611	−1.23149
$629$	−32.6353	−1.30126
$630$	0	0
$631$	22.7058	0.903902	0.451951	−	0.892043i	$-0.350728\pi$
$631$	22.7058	0.903902	0.451951	+	0.892043i	$0.350728\pi$
$632$	25.2687	1.00514
$633$	0	0
$634$	4.22223	0.167686
$635$	−2.00000	−0.0793676
$636$	0	0
$637$	0	0
$638$	−0.792249	−0.0313654
$639$	0	0
$640$	−11.5375	−0.456060
$641$	21.5749	0.852158	0.426079	−	0.904686i	$-0.359894\pi$
$641$	21.5749	0.852158	0.426079	+	0.904686i	$0.359894\pi$
$642$	0	0
$643$	−23.9866	−0.945939	−0.472969	−	0.881079i	$-0.656818\pi$
$643$	−23.9866	−0.945939	−0.472969	+	0.881079i	$0.656818\pi$
$644$	−33.1293	−1.30548
$645$	0	0
$646$	3.31096	0.130268
$647$	3.31873	0.130473	0.0652364	−	0.997870i	$-0.479220\pi$
$647$	3.31873	0.130473	0.0652364	+	0.997870i	$0.479220\pi$
$648$	0	0
$649$	−23.9517	−0.940185
$650$	0	0
$651$	0	0
$652$	19.1293	0.749161
$653$	3.41657	0.133701	0.0668503	−	0.997763i	$-0.478705\pi$
$653$	3.41657	0.133701	0.0668503	+	0.997763i	$0.478705\pi$
$654$	0	0
$655$	22.2500	0.869378
$656$	−9.48188	−0.370205
$657$	0	0
$658$	29.0616	1.13294
$659$	−10.6461	−0.414713	−0.207356	−	0.978265i	$-0.566486\pi$
$659$	−10.6461	−0.414713	−0.207356	+	0.978265i	$0.566486\pi$
$660$	0	0
$661$	22.8769	0.889810	0.444905	−	0.895578i	$-0.353237\pi$
$661$	22.8769	0.889810	0.444905	+	0.895578i	$0.353237\pi$
$662$	10.1293	0.393686
$663$	0	0
$664$	7.78986	0.302305
$665$	9.20775	0.357061
$666$	0	0
$667$	−3.10992	−0.120416
$668$	−27.0683	−1.04730
$669$	0	0
$670$	7.70171	0.297543
$671$	2.72455	0.105180
$672$	0	0
$673$	3.96987	0.153027	0.0765136	−	0.997069i	$-0.475621\pi$
$673$	3.96987	0.153027	0.0765136	+	0.997069i	$0.475621\pi$
$674$	−4.58583	−0.176639
$675$	0	0
$676$	0	0
$677$	12.1002	0.465050	0.232525	−	0.972590i	$-0.425301\pi$
$677$	12.1002	0.465050	0.232525	+	0.972590i	$0.425301\pi$
$678$	0	0
$679$	−5.42758	−0.208292
$680$	5.96615	0.228791
$681$	0	0
$682$	6.54825	0.250746
$683$	−10.8528	−0.415270	−0.207635	−	0.978206i	$-0.566577\pi$
$683$	−10.8528	−0.415270	−0.207635	+	0.978206i	$0.566577\pi$
$684$	0	0
$685$	−17.3274	−0.662044
$686$	15.4517	0.589950
$687$	0	0
$688$	−5.82371	−0.222027
$689$	0	0
$690$	0	0
$691$	13.2319	0.503366	0.251683	−	0.967810i	$-0.419016\pi$
$691$	13.2319	0.503366	0.251683	+	0.967810i	$0.419016\pi$
$692$	−33.1775	−1.26122
$693$	0	0
$694$	18.5496	0.704132
$695$	−20.8509	−0.790918
$696$	0	0
$697$	12.2875	0.465424
$698$	−19.3193	−0.731247
$699$	0	0
$700$	7.60388	0.287399
$701$	−31.3142	−1.18272	−0.591361	−	0.806407i	$-0.701409\pi$
$701$	−31.3142	−1.18272	−0.591361	+	0.806407i	$0.701409\pi$
$702$	0	0
$703$	22.9638	0.866094
$704$	−3.05562	−0.115163
$705$	0	0
$706$	−0.705431	−0.0265492
$707$	58.9154	2.21574
$708$	0	0
$709$	16.1715	0.607334	0.303667	−	0.952778i	$-0.401789\pi$
$709$	16.1715	0.607334	0.303667	+	0.952778i	$0.401789\pi$
$710$	6.65950	0.249926
$711$	0	0
$712$	−3.64742	−0.136693
$713$	25.7047	0.962648
$714$	0	0
$715$	0	0
$716$	11.7500	0.439119
$717$	0	0
$718$	7.33572	0.273767
$719$	−44.2586	−1.65057	−0.825283	−	0.564719i	$-0.808985\pi$
$719$	−44.2586	−1.65057	−0.825283	+	0.564719i	$0.808985\pi$
$720$	0	0
$721$	4.19567	0.156255
$722$	8.21446	0.305710
$723$	0	0
$724$	29.9936	1.11470
$725$	0.713792	0.0265096
$726$	0	0
$727$	2.28275	0.0846625	0.0423313	−	0.999104i	$-0.486522\pi$
$727$	2.28275	0.0846625	0.0423313	+	0.999104i	$0.486522\pi$
$728$	0	0
$729$	0	0
$730$	6.27413	0.232216
$731$	7.54693	0.279133
$732$	0	0
$733$	19.1051	0.705664	0.352832	−	0.935687i	$-0.385219\pi$
$733$	19.1051	0.705664	0.352832	+	0.935687i	$0.385219\pi$
$734$	15.3116	0.565160
$735$	0	0
$736$	23.2868	0.858363
$737$	27.7560	1.02241
$738$	0	0
$739$	32.6631	1.20153	0.600765	−	0.799425i	$-0.294863\pi$
$739$	32.6631	1.20153	0.600765	+	0.799425i	$0.294863\pi$
$740$	18.9638	0.697122
$741$	0	0
$742$	−21.7453	−0.798293
$743$	−49.2898	−1.80827	−0.904133	−	0.427250i	$-0.859482\pi$
$743$	−49.2898	−1.80827	−0.904133	+	0.427250i	$0.859482\pi$
$744$	0	0
$745$	16.0978	0.589779
$746$	7.94571	0.290913
$747$	0	0
$748$	9.85384	0.360292
$749$	−42.8310	−1.56501
$750$	0	0
$751$	−26.3569	−0.961777	−0.480888	−	0.876782i	$-0.659686\pi$
$751$	−26.3569	−0.961777	−0.480888	+	0.876782i	$0.659686\pi$
$752$	26.1836	0.954817
$753$	0	0
$754$	0	0
$755$	18.6310	0.678052
$756$	0	0
$757$	−33.0073	−1.19967	−0.599835	−	0.800124i	$-0.704767\pi$
$757$	−33.0073	−1.19967	−0.599835	+	0.800124i	$0.704767\pi$
$758$	−4.16793	−0.151386
$759$	0	0
$760$	−4.19806	−0.152280
$761$	−51.8297	−1.87882	−0.939412	−	0.342790i	$-0.888628\pi$
$761$	−51.8297	−1.87882	−0.939412	+	0.342790i	$0.888628\pi$
$762$	0	0
$763$	−0.420288	−0.0152154
$764$	6.09783	0.220612
$765$	0	0
$766$	−15.0097	−0.542322
$767$	0	0
$768$	0	0
$769$	−46.7851	−1.68711	−0.843556	−	0.537041i	$-0.819542\pi$
$769$	−46.7851	−1.68711	−0.843556	+	0.537041i	$0.819542\pi$
$770$	−4.98792	−0.179752
$771$	0	0
$772$	3.38404	0.121794
$773$	16.0519	0.577347	0.288673	−	0.957428i	$-0.406786\pi$
$773$	16.0519	0.577347	0.288673	+	0.957428i	$0.406786\pi$
$774$	0	0
$775$	−5.89977	−0.211926
$776$	2.47458	0.0888323
$777$	0	0
$778$	0.230586	0.00826691
$779$	−8.64609	−0.309778
$780$	0	0
$781$	24.0000	0.858788
$782$	−7.04056	−0.251770
$783$	0	0
$784$	29.6504	1.05894
$785$	−18.2392	−0.650985
$786$	0	0
$787$	5.96987	0.212803	0.106401	−	0.994323i	$-0.466067\pi$
$787$	5.96987	0.212803	0.106401	+	0.994323i	$0.466067\pi$
$788$	−7.38835	−0.263199
$789$	0	0
$790$	6.84415	0.243504
$791$	−30.2392	−1.07518
$792$	0	0
$793$	0	0
$794$	−0.409534	−0.0145338
$795$	0	0
$796$	5.89546	0.208959
$797$	−4.02475	−0.142564	−0.0712821	−	0.997456i	$-0.522709\pi$
$797$	−4.02475	−0.142564	−0.0712821	+	0.997456i	$0.522709\pi$
$798$	0	0
$799$	−33.9312	−1.20040
$800$	−5.34481	−0.188968
$801$	0	0
$802$	−6.18226	−0.218303
$803$	22.6112	0.797931
$804$	0	0
$805$	−19.5797	−0.690094
$806$	0	0
$807$	0	0
$808$	−26.8611	−0.944971
$809$	−2.08708	−0.0733779	−0.0366889	−	0.999327i	$-0.511681\pi$
$809$	−2.08708	−0.0733779	−0.0366889	+	0.999327i	$0.511681\pi$
$810$	0	0
$811$	9.78017	0.343428	0.171714	−	0.985147i	$-0.445069\pi$
$811$	9.78017	0.343428	0.171714	+	0.985147i	$0.445069\pi$
$812$	5.42758	0.190471
$813$	0	0
$814$	−12.4397	−0.436010
$815$	11.3056	0.396017
$816$	0	0
$817$	−5.31037	−0.185786
$818$	−0.209079	−0.00731027
$819$	0	0
$820$	−7.14005	−0.249341
$821$	3.23788	0.113003	0.0565014	−	0.998403i	$-0.482005\pi$
$821$	3.23788	0.113003	0.0565014	+	0.998403i	$0.482005\pi$
$822$	0	0
$823$	13.8780	0.483757	0.241878	−	0.970307i	$-0.422237\pi$
$823$	13.8780	0.483757	0.241878	+	0.970307i	$0.422237\pi$
$824$	−1.91292	−0.0666397
$825$	0	0
$826$	−29.8672	−1.03921
$827$	−5.88876	−0.204772	−0.102386	−	0.994745i	$-0.532648\pi$
$827$	−5.88876	−0.204772	−0.102386	+	0.994745i	$0.532648\pi$
$828$	0	0
$829$	22.5090	0.781771	0.390885	−	0.920439i	$-0.372169\pi$
$829$	22.5090	0.781771	0.390885	+	0.920439i	$0.372169\pi$
$830$	2.10992	0.0732363
$831$	0	0
$832$	0	0
$833$	−38.4239	−1.33131
$834$	0	0
$835$	−15.9976	−0.553620
$836$	−6.93362	−0.239804
$837$	0	0
$838$	0.537500	0.0185676
$839$	−19.5931	−0.676430	−0.338215	−	0.941069i	$-0.609823\pi$
$839$	−19.5931	−0.676430	−0.338215	+	0.941069i	$0.609823\pi$
$840$	0	0
$841$	−28.4905	−0.982431
$842$	−19.6558	−0.677383
$843$	0	0
$844$	19.8495	0.683249
$845$	0	0
$846$	0	0
$847$	31.4577	1.08090
$848$	−19.5918	−0.672785
$849$	0	0
$850$	1.61596	0.0554269
$851$	−48.8310	−1.67391
$852$	0	0
$853$	−22.0495	−0.754961	−0.377480	−	0.926018i	$-0.623209\pi$
$853$	−22.0495	−0.754961	−0.377480	+	0.926018i	$0.623209\pi$
$854$	3.39745	0.116258
$855$	0	0
$856$	19.5278	0.667447
$857$	50.3430	1.71968	0.859842	−	0.510560i	$-0.170562\pi$
$857$	50.3430	1.71968	0.859842	+	0.510560i	$0.170562\pi$
$858$	0	0
$859$	38.8179	1.32445	0.662224	−	0.749306i	$-0.269613\pi$
$859$	38.8179	1.32445	0.662224	+	0.749306i	$0.269613\pi$
$860$	−4.38537	−0.149540
$861$	0	0
$862$	11.6414	0.396509
$863$	23.0670	0.785209	0.392604	−	0.919707i	$-0.371574\pi$
$863$	23.0670	0.785209	0.392604	+	0.919707i	$0.371574\pi$
$864$	0	0
$865$	−19.6082	−0.666698
$866$	12.4203	0.422059
$867$	0	0
$868$	−44.8611	−1.52269
$869$	24.6655	0.836719
$870$	0	0
$871$	0	0
$872$	0.191621	0.00648909
$873$	0	0
$874$	4.95407	0.167574
$875$	4.49396	0.151924
$876$	0	0
$877$	−15.3190	−0.517286	−0.258643	−	0.965973i	$-0.583275\pi$
$877$	−15.3190	−0.517286	−0.258643	+	0.965973i	$0.583275\pi$
$878$	−15.3026	−0.516438
$879$	0	0
$880$	−4.49396	−0.151491
$881$	19.0556	0.642000	0.321000	−	0.947079i	$-0.395981\pi$
$881$	19.0556	0.642000	0.321000	+	0.947079i	$0.395981\pi$
$882$	0	0
$883$	9.14005	0.307587	0.153794	−	0.988103i	$-0.450851\pi$
$883$	9.14005	0.307587	0.153794	+	0.988103i	$0.450851\pi$
$884$	0	0
$885$	0	0
$886$	11.2704	0.378637
$887$	−25.1424	−0.844201	−0.422100	−	0.906549i	$-0.638707\pi$
$887$	−25.1424	−0.844201	−0.422100	+	0.906549i	$0.638707\pi$
$888$	0	0
$889$	−8.98792	−0.301445
$890$	−0.987918	−0.0331151
$891$	0	0
$892$	−26.6413	−0.892017
$893$	23.8756	0.798967
$894$	0	0
$895$	6.94438	0.232125
$896$	−51.8491	−1.73216
$897$	0	0
$898$	16.8224	0.561370
$899$	−4.21121	−0.140452
$900$	0	0
$901$	25.3889	0.845828
$902$	4.68366	0.155949
$903$	0	0
$904$	13.7869	0.458545
$905$	17.7265	0.589248
$906$	0	0
$907$	−33.0121	−1.09615	−0.548074	−	0.836430i	$-0.684639\pi$
$907$	−33.0121	−1.09615	−0.548074	+	0.836430i	$0.684639\pi$
$908$	28.4875	0.945391
$909$	0	0
$910$	0	0
$911$	44.6655	1.47983	0.739916	−	0.672699i	$-0.234865\pi$
$911$	44.6655	1.47983	0.739916	+	0.672699i	$0.234865\pi$
$912$	0	0
$913$	7.60388	0.251652
$914$	2.89008	0.0955955
$915$	0	0
$916$	−12.4203	−0.410378
$917$	99.9904	3.30197
$918$	0	0
$919$	−30.1129	−0.993333	−0.496666	−	0.867941i	$-0.665443\pi$
$919$	−30.1129	−0.993333	−0.496666	+	0.867941i	$0.665443\pi$
$920$	8.92692	0.294312
$921$	0	0
$922$	−2.17629	−0.0716724
$923$	0	0
$924$	0	0
$925$	11.2078	0.368509
$926$	−1.20178	−0.0394930
$927$	0	0
$928$	−3.81508	−0.125236
$929$	2.88530	0.0946636	0.0473318	−	0.998879i	$-0.484928\pi$
$929$	2.88530	0.0946636	0.0473318	+	0.998879i	$0.484928\pi$
$930$	0	0
$931$	27.0368	0.886097
$932$	36.8762	1.20792
$933$	0	0
$934$	−0.409862	−0.0134111
$935$	5.82371	0.190456
$936$	0	0
$937$	7.23191	0.236256	0.118128	−	0.992998i	$-0.462311\pi$
$937$	7.23191	0.236256	0.118128	+	0.992998i	$0.462311\pi$
$938$	34.6112	1.13009
$939$	0	0
$940$	19.7168	0.643090
$941$	5.10992	0.166579	0.0832893	−	0.996525i	$-0.473457\pi$
$941$	5.10992	0.166579	0.0832893	+	0.996525i	$0.473457\pi$
$942$	0	0
$943$	18.3854	0.598710
$944$	−26.9095	−0.875828
$945$	0	0
$946$	2.87667	0.0935287
$947$	−51.6335	−1.67786	−0.838932	−	0.544236i	$-0.816820\pi$
$947$	−51.6335	−1.67786	−0.838932	+	0.544236i	$0.816820\pi$
$948$	0	0
$949$	0	0
$950$	−1.13706	−0.0368912
$951$	0	0
$952$	26.8116	0.868970
$953$	16.9638	0.549510	0.274755	−	0.961514i	$-0.411403\pi$
$953$	16.9638	0.549510	0.274755	+	0.961514i	$0.411403\pi$
$954$	0	0
$955$	3.60388	0.116619
$956$	5.02284	0.162450
$957$	0	0
$958$	13.1341	0.424343
$959$	−77.8684	−2.51450
$960$	0	0
$961$	3.80731	0.122817
$962$	0	0
$963$	0	0
$964$	1.64609	0.0530169
$965$	2.00000	0.0643823
$966$	0	0
$967$	−41.7271	−1.34185	−0.670926	−	0.741524i	$-0.734103\pi$
$967$	−41.7271	−1.34185	−0.670926	+	0.741524i	$0.734103\pi$
$968$	−14.3424	−0.460983
$969$	0	0
$970$	0.670251	0.0215205
$971$	−0.572417	−0.0183697	−0.00918486	−	0.999958i	$-0.502924\pi$
$971$	−0.572417	−0.0183697	−0.00918486	+	0.999958i	$0.502924\pi$
$972$	0	0
$973$	−93.7029	−3.00398
$974$	17.1943	0.550942
$975$	0	0
$976$	3.06100	0.0979802
$977$	−41.5502	−1.32931	−0.664654	−	0.747151i	$-0.731421\pi$
$977$	−41.5502	−1.32931	−0.664654	+	0.747151i	$0.731421\pi$
$978$	0	0
$979$	−3.56033	−0.113789
$980$	22.3274	0.713221
$981$	0	0
$982$	1.10992	0.0354189
$983$	6.45952	0.206027	0.103013	−	0.994680i	$-0.467152\pi$
$983$	6.45952	0.206027	0.103013	+	0.994680i	$0.467152\pi$
$984$	0	0
$985$	−4.36658	−0.139131
$986$	1.15346	0.0367335
$987$	0	0
$988$	0	0
$989$	11.2922	0.359070
$990$	0	0
$991$	15.2790	0.485354	0.242677	−	0.970107i	$-0.421974\pi$
$991$	15.2790	0.485354	0.242677	+	0.970107i	$0.421974\pi$
$992$	31.5332	1.00118
$993$	0	0
$994$	29.9275	0.949243
$995$	3.48427	0.110459
$996$	0	0
$997$	−30.0388	−0.951337	−0.475668	−	0.879625i	$-0.657794\pi$
$997$	−30.0388	−0.951337	−0.475668	+	0.879625i	$0.657794\pi$
$998$	5.76212	0.182397
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7605.2.a.bn.1.2		3
3.2	odd	2		2535.2.a.bg.1.2	yes	3
13.12	even	2		7605.2.a.cd.1.2		3
39.38	odd	2		2535.2.a.u.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2535.2.a.u.1.2	✓	3	39.38	odd	2
2535.2.a.bg.1.2	yes	3	3.2	odd	2
7605.2.a.bn.1.2		3	1.1	even	1	trivial
7605.2.a.cd.1.2		3	13.12	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:	\( N \)	\(=\)	\( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7605.a (trivial)

\(f(q)\)	\(=\)	\(q-0.554958 q^{2} -1.69202 q^{4} -1.00000 q^{5} -4.49396 q^{7} +2.04892 q^{8} +O(q^{10})\)	\(q-0.554958 q^{2} -1.69202 q^{4} -1.00000 q^{5} -4.49396 q^{7} +2.04892 q^{8} +0.554958 q^{10} +2.00000 q^{11} +2.49396 q^{14} +2.24698 q^{16} -2.91185 q^{17} +2.04892 q^{19} +1.69202 q^{20} -1.10992 q^{22} -4.35690 q^{23} +1.00000 q^{25} +7.60388 q^{28} +0.713792 q^{29} -5.89977 q^{31} -5.34481 q^{32} +1.61596 q^{34} +4.49396 q^{35} +11.2078 q^{37} -1.13706 q^{38} -2.04892 q^{40} -4.21983 q^{41} -2.59179 q^{43} -3.38404 q^{44} +2.41789 q^{46} +11.6528 q^{47} +13.1957 q^{49} -0.554958 q^{50} -8.71917 q^{53} -2.00000 q^{55} -9.20775 q^{56} -0.396125 q^{58} -11.9758 q^{59} +1.36227 q^{61} +3.27413 q^{62} -1.52781 q^{64} +13.8780 q^{67} +4.92692 q^{68} -2.49396 q^{70} +12.0000 q^{71} +11.3056 q^{73} -6.21983 q^{74} -3.46681 q^{76} -8.98792 q^{77} +12.3327 q^{79} -2.24698 q^{80} +2.34183 q^{82} +3.80194 q^{83} +2.91185 q^{85} +1.43834 q^{86} +4.09783 q^{88} -1.78017 q^{89} +7.37196 q^{92} -6.46681 q^{94} -2.04892 q^{95} +1.20775 q^{97} -7.32304 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} - 3 q^{5} - 4 q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q - 2 * q^2 - 3 * q^5 - 4 * q^7 - 3 * q^8	\( 3 q - 2 q^{2} - 3 q^{5} - 4 q^{7} - 3 q^{8} + 2 q^{10} + 6 q^{11} - 2 q^{14} + 2 q^{16} - 5 q^{17} - 3 q^{19} - 4 q^{22} - 9 q^{23} + 3 q^{25} + 14 q^{28} - 6 q^{29} + 5 q^{31} + 7 q^{32} + 15 q^{34} + 4 q^{35} + 16 q^{37} + 2 q^{38} + 3 q^{40} - 14 q^{41} + 20 q^{43} + 13 q^{46} + 17 q^{47} + 3 q^{49} - 2 q^{50} - 15 q^{53} - 6 q^{55} - 10 q^{56} - 10 q^{58} + 2 q^{59} - 3 q^{61} - q^{62} - 11 q^{64} + 22 q^{67} - 14 q^{68} + 2 q^{70} + 36 q^{71} - 2 q^{73} - 20 q^{74} - 7 q^{76} - 8 q^{77} - 5 q^{79} - 2 q^{80} + 28 q^{82} + 7 q^{83} + 5 q^{85} - 18 q^{86} - 6 q^{88} - 4 q^{89} - 7 q^{92} - 16 q^{94} + 3 q^{95} - 14 q^{97} - 2 q^{98}+O(q^{100}) \) 3 * q - 2 * q^2 - 3 * q^5 - 4 * q^7 - 3 * q^8 + 2 * q^10 + 6 * q^11 - 2 * q^14 + 2 * q^16 - 5 * q^17 - 3 * q^19 - 4 * q^22 - 9 * q^23 + 3 * q^25 + 14 * q^28 - 6 * q^29 + 5 * q^31 + 7 * q^32 + 15 * q^34 + 4 * q^35 + 16 * q^37 + 2 * q^38 + 3 * q^40 - 14 * q^41 + 20 * q^43 + 13 * q^46 + 17 * q^47 + 3 * q^49 - 2 * q^50 - 15 * q^53 - 6 * q^55 - 10 * q^56 - 10 * q^58 + 2 * q^59 - 3 * q^61 - q^62 - 11 * q^64 + 22 * q^67 - 14 * q^68 + 2 * q^70 + 36 * q^71 - 2 * q^73 - 20 * q^74 - 7 * q^76 - 8 * q^77 - 5 * q^79 - 2 * q^80 + 28 * q^82 + 7 * q^83 + 5 * q^85 - 18 * q^86 - 6 * q^88 - 4 * q^89 - 7 * q^92 - 16 * q^94 + 3 * q^95 - 14 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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