LMFDB - Embedded newform 7605.2.a.bp.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			$0.445042$ 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)

$f(q)$	$=$	$q-0.445042 q^{2} -1.80194 q^{4} +1.00000 q^{5} -4.00000 q^{7} +1.69202 q^{8} +O(q^{10})$	$q-0.445042 q^{2} -1.80194 q^{4} +1.00000 q^{5} -4.00000 q^{7} +1.69202 q^{8} -0.445042 q^{10} +4.71379 q^{11} +1.78017 q^{14} +2.85086 q^{16} -1.95108 q^{17} -7.34481 q^{19} -1.80194 q^{20} -2.09783 q^{22} +2.02177 q^{23} +1.00000 q^{25} +7.20775 q^{28} -6.98792 q^{29} +7.82908 q^{31} -4.65279 q^{32} +0.868313 q^{34} -4.00000 q^{35} +10.3720 q^{37} +3.26875 q^{38} +1.69202 q^{40} -10.8116 q^{41} +4.27413 q^{43} -8.49396 q^{44} -0.899772 q^{46} -1.07069 q^{47} +9.00000 q^{49} -0.445042 q^{50} +8.23490 q^{53} +4.71379 q^{55} -6.76809 q^{56} +3.10992 q^{58} +13.7017 q^{59} -2.76271 q^{61} -3.48427 q^{62} -3.63102 q^{64} -11.5797 q^{67} +3.51573 q^{68} +1.78017 q^{70} -7.20775 q^{71} +1.60388 q^{73} -4.61596 q^{74} +13.2349 q^{76} -18.8552 q^{77} -8.47219 q^{79} +2.85086 q^{80} +4.81163 q^{82} +1.87263 q^{83} -1.95108 q^{85} -1.90217 q^{86} +7.97584 q^{88} -3.32975 q^{89} -3.64310 q^{92} +0.476501 q^{94} -7.34481 q^{95} +9.60388 q^{97} -4.00538 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} - q^{4} + 3 q^{5} - 12 q^{7}+O(q^{10}) $ 3 * q - q^2 - q^4 + 3 * q^5 - 12 * q^7	$ 3 q - q^{2} - q^{4} + 3 q^{5} - 12 q^{7} - q^{10} + 6 q^{11} + 4 q^{14} - 5 q^{16} - 15 q^{17} + q^{19} - q^{20} + 12 q^{22} + 3 q^{23} + 3 q^{25} + 4 q^{28} - 2 q^{29} + 13 q^{31} + 4 q^{32} + 5 q^{34} - 12 q^{35} + 2 q^{37} + 2 q^{38} - 6 q^{41} + 2 q^{43} - 16 q^{44} + 20 q^{46} + 9 q^{47} + 27 q^{49} - q^{50} + q^{53} + 6 q^{55} + 10 q^{58} + 14 q^{59} + 9 q^{61} - 23 q^{62} + 4 q^{64} + 12 q^{67} - 2 q^{68} + 4 q^{70} - 4 q^{71} - 4 q^{73} - 24 q^{74} + 16 q^{76} - 24 q^{77} - 19 q^{79} - 5 q^{80} - 12 q^{82} - 11 q^{83} - 15 q^{85} - 24 q^{86} - 14 q^{88} - 12 q^{89} - 15 q^{92} - 24 q^{94} + q^{95} + 20 q^{97} - 9 q^{98}+O(q^{100}) $ 3 * q - q^2 - q^4 + 3 * q^5 - 12 * q^7 - q^10 + 6 * q^11 + 4 * q^14 - 5 * q^16 - 15 * q^17 + q^19 - q^20 + 12 * q^22 + 3 * q^23 + 3 * q^25 + 4 * q^28 - 2 * q^29 + 13 * q^31 + 4 * q^32 + 5 * q^34 - 12 * q^35 + 2 * q^37 + 2 * q^38 - 6 * q^41 + 2 * q^43 - 16 * q^44 + 20 * q^46 + 9 * q^47 + 27 * q^49 - q^50 + q^53 + 6 * q^55 + 10 * q^58 + 14 * q^59 + 9 * q^61 - 23 * q^62 + 4 * q^64 + 12 * q^67 - 2 * q^68 + 4 * q^70 - 4 * q^71 - 4 * q^73 - 24 * q^74 + 16 * q^76 - 24 * q^77 - 19 * q^79 - 5 * q^80 - 12 * q^82 - 11 * q^83 - 15 * q^85 - 24 * q^86 - 14 * q^88 - 12 * q^89 - 15 * q^92 - 24 * q^94 + q^95 + 20 * q^97 - 9 * 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 $

$2$	−0.445042	−0.314692	−0.157346	−	0.987544i	$-0.550294\pi$
$2$	−0.445042	−0.314692	−0.157346	+	0.987544i	$0.550294\pi$
$3$	0	0
$3$	0	0
$4$	−1.80194	−0.900969
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	1.69202	0.598220
$9$	0	0
$10$	−0.445042	−0.140735
$11$	4.71379	1.42126	0.710631	−	0.703565i	$-0.248410\pi$
$11$	4.71379	1.42126	0.710631	+	0.703565i	$0.248410\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	1.78017	0.475770
$15$	0	0
$16$	2.85086	0.712714
$17$	−1.95108	−0.473207	−0.236604	−	0.971606i	$-0.576034\pi$
$17$	−1.95108	−0.473207	−0.236604	+	0.971606i	$0.576034\pi$
$18$	0	0
$19$	−7.34481	−1.68502	−0.842508	−	0.538684i	$-0.818922\pi$
$19$	−7.34481	−1.68502	−0.842508	+	0.538684i	$0.818922\pi$
$20$	−1.80194	−0.402926
$21$	0	0
$22$	−2.09783	−0.447260
$23$	2.02177	0.421568	0.210784	−	0.977533i	$-0.432398\pi$
$23$	2.02177	0.421568	0.210784	+	0.977533i	$0.432398\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	7.20775	1.36214
$29$	−6.98792	−1.29762	−0.648812	−	0.760949i	$-0.724734\pi$
$29$	−6.98792	−1.29762	−0.648812	+	0.760949i	$0.724734\pi$
$30$	0	0
$31$	7.82908	1.40615	0.703073	−	0.711118i	$-0.251811\pi$
$31$	7.82908	1.40615	0.703073	+	0.711118i	$0.251811\pi$
$32$	−4.65279	−0.822505
$33$	0	0
$34$	0.868313	0.148915
$35$	−4.00000	−0.676123
$36$	0	0
$37$	10.3720	1.70514	0.852570	−	0.522613i	$-0.175043\pi$
$37$	10.3720	1.70514	0.852570	+	0.522613i	$0.175043\pi$
$38$	3.26875	0.530261
$39$	0	0
$40$	1.69202	0.267532
$41$	−10.8116	−1.68849	−0.844246	−	0.535956i	$-0.819951\pi$
$41$	−10.8116	−1.68849	−0.844246	+	0.535956i	$0.819951\pi$
$42$	0	0
$43$	4.27413	0.651798	0.325899	−	0.945405i	$-0.394333\pi$
$43$	4.27413	0.651798	0.325899	+	0.945405i	$0.394333\pi$
$44$	−8.49396	−1.28051
$45$	0	0
$46$	−0.899772	−0.132664
$47$	−1.07069	−0.156176	−0.0780879	−	0.996946i	$-0.524881\pi$
$47$	−1.07069	−0.156176	−0.0780879	+	0.996946i	$0.524881\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	−0.445042	−0.0629384
$51$	0	0
$52$	0	0
$53$	8.23490	1.13115	0.565575	−	0.824697i	$-0.308654\pi$
$53$	8.23490	1.13115	0.565575	+	0.824697i	$0.308654\pi$
$54$	0	0
$55$	4.71379	0.635608
$56$	−6.76809	−0.904424
$57$	0	0
$58$	3.10992	0.408352
$59$	13.7017	1.78381	0.891905	−	0.452222i	$-0.149369\pi$
$59$	13.7017	1.78381	0.891905	+	0.452222i	$0.149369\pi$
$60$	0	0
$61$	−2.76271	−0.353729	−0.176864	−	0.984235i	$-0.556595\pi$
$61$	−2.76271	−0.353729	−0.176864	+	0.984235i	$0.556595\pi$
$62$	−3.48427	−0.442503
$63$	0	0
$64$	−3.63102	−0.453878
$65$	0	0
$66$	0	0
$67$	−11.5797	−1.41469	−0.707344	−	0.706870i	$-0.750107\pi$
$67$	−11.5797	−1.41469	−0.707344	+	0.706870i	$0.750107\pi$
$68$	3.51573	0.426345
$69$	0	0
$70$	1.78017	0.212771
$71$	−7.20775	−0.855403	−0.427701	−	0.903920i	$-0.640676\pi$
$71$	−7.20775	−0.855403	−0.427701	+	0.903920i	$0.640676\pi$
$72$	0	0
$73$	1.60388	0.187719	0.0938597	−	0.995585i	$-0.470079\pi$
$73$	1.60388	0.187719	0.0938597	+	0.995585i	$0.470079\pi$
$74$	−4.61596	−0.536594
$75$	0	0
$76$	13.2349	1.51815
$77$	−18.8552	−2.14875
$78$	0	0
$79$	−8.47219	−0.953196	−0.476598	−	0.879121i	$-0.658130\pi$
$79$	−8.47219	−0.953196	−0.476598	+	0.879121i	$0.658130\pi$
$80$	2.85086	0.318735
$81$	0	0
$82$	4.81163	0.531355
$83$	1.87263	0.205547	0.102774	−	0.994705i	$-0.467228\pi$
$83$	1.87263	0.205547	0.102774	+	0.994705i	$0.467228\pi$
$84$	0	0
$85$	−1.95108	−0.211625
$86$	−1.90217	−0.205116
$87$	0	0
$88$	7.97584	0.850227
$89$	−3.32975	−0.352953	−0.176476	−	0.984305i	$-0.556470\pi$
$89$	−3.32975	−0.352953	−0.176476	+	0.984305i	$0.556470\pi$
$90$	0	0
$91$	0	0
$92$	−3.64310	−0.379820
$93$	0	0
$94$	0.476501	0.0491473
$95$	−7.34481	−0.753562
$96$	0	0
$97$	9.60388	0.975126	0.487563	−	0.873088i	$-0.337886\pi$
$97$	9.60388	0.975126	0.487563	+	0.873088i	$0.337886\pi$
$98$	−4.00538	−0.404604
$99$	0	0
$100$	−1.80194	−0.180194
$101$	11.1642	1.11088	0.555440	−	0.831556i	$-0.312550\pi$
$101$	11.1642	1.11088	0.555440	+	0.831556i	$0.312550\pi$
$102$	0	0
$103$	−0.591794	−0.0583112	−0.0291556	−	0.999575i	$-0.509282\pi$
$103$	−0.591794	−0.0583112	−0.0291556	+	0.999575i	$0.509282\pi$
$104$	0	0
$105$	0	0
$106$	−3.66487	−0.355964
$107$	−5.86294	−0.566791	−0.283396	−	0.959003i	$-0.591461\pi$
$107$	−5.86294	−0.566791	−0.283396	+	0.959003i	$0.591461\pi$
$108$	0	0
$109$	6.39373	0.612408	0.306204	−	0.951966i	$-0.400941\pi$
$109$	6.39373	0.612408	0.306204	+	0.951966i	$0.400941\pi$
$110$	−2.09783	−0.200021
$111$	0	0
$112$	−11.4034	−1.07752
$113$	−7.30798	−0.687477	−0.343738	−	0.939065i	$-0.611693\pi$
$113$	−7.30798	−0.687477	−0.343738	+	0.939065i	$0.611693\pi$
$114$	0	0
$115$	2.02177	0.188531
$116$	12.5918	1.16912
$117$	0	0
$118$	−6.09783	−0.561351
$119$	7.80433	0.715422
$120$	0	0
$121$	11.2198	1.01998
$122$	1.22952	0.111316
$123$	0	0
$124$	−14.1075	−1.26689
$125$	1.00000	0.0894427
$126$	0	0
$127$	−12.4940	−1.10866	−0.554330	−	0.832297i	$-0.687025\pi$
$127$	−12.4940	−1.10866	−0.554330	+	0.832297i	$0.687025\pi$
$128$	10.9215	0.965337
$129$	0	0
$130$	0	0
$131$	−8.71379	−0.761328	−0.380664	−	0.924714i	$-0.624304\pi$
$131$	−8.71379	−0.761328	−0.380664	+	0.924714i	$0.624304\pi$
$132$	0	0
$133$	29.3793	2.54750
$134$	5.15346	0.445191
$135$	0	0
$136$	−3.30127	−0.283082
$137$	−3.37867	−0.288659	−0.144329	−	0.989530i	$-0.546103\pi$
$137$	−3.37867	−0.288659	−0.144329	+	0.989530i	$0.546103\pi$
$138$	0	0
$139$	10.8877	0.923482	0.461741	−	0.887015i	$-0.347225\pi$
$139$	10.8877	0.923482	0.461741	+	0.887015i	$0.347225\pi$
$140$	7.20775	0.609166
$141$	0	0
$142$	3.20775	0.269188
$143$	0	0
$144$	0	0
$145$	−6.98792	−0.580315
$146$	−0.713792	−0.0590738
$147$	0	0
$148$	−18.6896	−1.53628
$149$	12.1957	0.999108	0.499554	−	0.866283i	$-0.333497\pi$
$149$	12.1957	0.999108	0.499554	+	0.866283i	$0.333497\pi$
$150$	0	0
$151$	6.69202	0.544589	0.272294	−	0.962214i	$-0.412218\pi$
$151$	6.69202	0.544589	0.272294	+	0.962214i	$0.412218\pi$
$152$	−12.4276	−1.00801
$153$	0	0
$154$	8.39134	0.676193
$155$	7.82908	0.628847
$156$	0	0
$157$	−2.31767	−0.184970	−0.0924850	−	0.995714i	$-0.529481\pi$
$157$	−2.31767	−0.184970	−0.0924850	+	0.995714i	$0.529481\pi$
$158$	3.77048	0.299963
$159$	0	0
$160$	−4.65279	−0.367836
$161$	−8.08708	−0.637351
$162$	0	0
$163$	0.317667	0.0248816	0.0124408	−	0.999923i	$-0.496040\pi$
$163$	0.317667	0.0248816	0.0124408	+	0.999923i	$0.496040\pi$
$164$	19.4819	1.52128
$165$	0	0
$166$	−0.833397	−0.0646841
$167$	−19.5550	−1.51321	−0.756604	−	0.653873i	$-0.773143\pi$
$167$	−19.5550	−1.51321	−0.756604	+	0.653873i	$0.773143\pi$
$168$	0	0
$169$	0	0
$170$	0.868313	0.0665966
$171$	0	0
$172$	−7.70171	−0.587250
$173$	−13.0881	−0.995073	−0.497537	−	0.867443i	$-0.665762\pi$
$173$	−13.0881	−0.995073	−0.497537	+	0.867443i	$0.665762\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	13.4383	1.01295
$177$	0	0
$178$	1.48188	0.111071
$179$	2.31767	0.173231	0.0866153	−	0.996242i	$-0.472395\pi$
$179$	2.31767	0.173231	0.0866153	+	0.996242i	$0.472395\pi$
$180$	0	0
$181$	−4.25906	−0.316574	−0.158287	−	0.987393i	$-0.550597\pi$
$181$	−4.25906	−0.316574	−0.158287	+	0.987393i	$0.550597\pi$
$182$	0	0
$183$	0	0
$184$	3.42088	0.252191
$185$	10.3720	0.762562
$186$	0	0
$187$	−9.19700	−0.672551
$188$	1.92931	0.140710
$189$	0	0
$190$	3.26875	0.237140
$191$	4.49396	0.325171	0.162586	−	0.986694i	$-0.448017\pi$
$191$	4.49396	0.325171	0.162586	+	0.986694i	$0.448017\pi$
$192$	0	0
$193$	14.0978	1.01478	0.507392	−	0.861715i	$-0.330610\pi$
$193$	14.0978	1.01478	0.507392	+	0.861715i	$0.330610\pi$
$194$	−4.27413	−0.306864
$195$	0	0
$196$	−16.2174	−1.15839
$197$	−2.97285	−0.211807	−0.105904	−	0.994376i	$-0.533773\pi$
$197$	−2.97285	−0.211807	−0.105904	+	0.994376i	$0.533773\pi$
$198$	0	0
$199$	3.13467	0.222211	0.111105	−	0.993809i	$-0.464561\pi$
$199$	3.13467	0.222211	0.111105	+	0.993809i	$0.464561\pi$
$200$	1.69202	0.119644
$201$	0	0
$202$	−4.96854	−0.349585
$203$	27.9517	1.96182
$204$	0	0
$205$	−10.8116	−0.755117
$206$	0.263373	0.0183501
$207$	0	0
$208$	0	0
$209$	−34.6219	−2.39485
$210$	0	0
$211$	24.5284	1.68860	0.844302	−	0.535867i	$-0.180015\pi$
$211$	24.5284	1.68860	0.844302	+	0.535867i	$0.180015\pi$
$212$	−14.8388	−1.01913
$213$	0	0
$214$	2.60925	0.178365
$215$	4.27413	0.291493
$216$	0	0
$217$	−31.3163	−2.12589
$218$	−2.84548	−0.192720
$219$	0	0
$220$	−8.49396	−0.572663
$221$	0	0
$222$	0	0
$223$	−12.7922	−0.856632	−0.428316	−	0.903629i	$-0.640893\pi$
$223$	−12.7922	−0.856632	−0.428316	+	0.903629i	$0.640893\pi$
$224$	18.6112	1.24351
$225$	0	0
$226$	3.25236	0.216344
$227$	24.8267	1.64781	0.823903	−	0.566731i	$-0.191792\pi$
$227$	24.8267	1.64781	0.823903	+	0.566731i	$0.191792\pi$
$228$	0	0
$229$	9.64742	0.637519	0.318760	−	0.947836i	$-0.396734\pi$
$229$	9.64742	0.637519	0.318760	+	0.947836i	$0.396734\pi$
$230$	−0.899772	−0.0593292
$231$	0	0
$232$	−11.8237	−0.776264
$233$	−21.7409	−1.42430	−0.712148	−	0.702029i	$-0.752278\pi$
$233$	−21.7409	−1.42430	−0.712148	+	0.702029i	$0.752278\pi$
$234$	0	0
$235$	−1.07069	−0.0698440
$236$	−24.6896	−1.60716
$237$	0	0
$238$	−3.47325	−0.225138
$239$	4.07846	0.263813	0.131907	−	0.991262i	$-0.457890\pi$
$239$	4.07846	0.263813	0.131907	+	0.991262i	$0.457890\pi$
$240$	0	0
$241$	21.5579	1.38867	0.694335	−	0.719652i	$-0.255699\pi$
$241$	21.5579	1.38867	0.694335	+	0.719652i	$0.255699\pi$
$242$	−4.99330	−0.320981
$243$	0	0
$244$	4.97823	0.318699
$245$	9.00000	0.574989
$246$	0	0
$247$	0	0
$248$	13.2470	0.841184
$249$	0	0
$250$	−0.445042	−0.0281469
$251$	−0.811626	−0.0512294	−0.0256147	−	0.999672i	$-0.508154\pi$
$251$	−0.811626	−0.0512294	−0.0256147	+	0.999672i	$0.508154\pi$
$252$	0	0
$253$	9.53020	0.599159
$254$	5.56033	0.348886
$255$	0	0
$256$	2.40150	0.150094
$257$	−20.8702	−1.30185	−0.650925	−	0.759142i	$-0.725619\pi$
$257$	−20.8702	−1.30185	−0.650925	+	0.759142i	$0.725619\pi$
$258$	0	0
$259$	−41.4878	−2.57793
$260$	0	0
$261$	0	0
$262$	3.87800	0.239584
$263$	−30.5187	−1.88186	−0.940932	−	0.338595i	$-0.890048\pi$
$263$	−30.5187	−1.88186	−0.940932	+	0.338595i	$0.890048\pi$
$264$	0	0
$265$	8.23490	0.505866
$266$	−13.0750	−0.801680
$267$	0	0
$268$	20.8659	1.27459
$269$	−1.95646	−0.119287	−0.0596437	−	0.998220i	$-0.518996\pi$
$269$	−1.95646	−0.119287	−0.0596437	+	0.998220i	$0.518996\pi$
$270$	0	0
$271$	−0.428648	−0.0260385	−0.0130193	−	0.999915i	$-0.504144\pi$
$271$	−0.428648	−0.0260385	−0.0130193	+	0.999915i	$0.504144\pi$
$272$	−5.56225	−0.337261
$273$	0	0
$274$	1.50365	0.0908387
$275$	4.71379	0.284252
$276$	0	0
$277$	−23.9758	−1.44057	−0.720284	−	0.693679i	$-0.755989\pi$
$277$	−23.9758	−1.44057	−0.720284	+	0.693679i	$0.755989\pi$
$278$	−4.84548	−0.290612
$279$	0	0
$280$	−6.76809	−0.404470
$281$	4.37196	0.260809	0.130405	−	0.991461i	$-0.458372\pi$
$281$	4.37196	0.260809	0.130405	+	0.991461i	$0.458372\pi$
$282$	0	0
$283$	−20.0629	−1.19262	−0.596308	−	0.802755i	$-0.703366\pi$
$283$	−20.0629	−1.19262	−0.596308	+	0.802755i	$0.703366\pi$
$284$	12.9879	0.770691
$285$	0	0
$286$	0	0
$287$	43.2465	2.55276
$288$	0	0
$289$	−13.1933	−0.776075
$290$	3.10992	0.182621
$291$	0	0
$292$	−2.89008	−0.169129
$293$	−23.7168	−1.38555	−0.692774	−	0.721154i	$-0.743612\pi$
$293$	−23.7168	−1.38555	−0.692774	+	0.721154i	$0.743612\pi$
$294$	0	0
$295$	13.7017	0.797744
$296$	17.5496	1.02005
$297$	0	0
$298$	−5.42758	−0.314411
$299$	0	0
$300$	0	0
$301$	−17.0965	−0.985426
$302$	−2.97823	−0.171378
$303$	0	0
$304$	−20.9390	−1.20093
$305$	−2.76271	−0.158192
$306$	0	0
$307$	6.51334	0.371736	0.185868	−	0.982575i	$-0.440490\pi$
$307$	6.51334	0.371736	0.185868	+	0.982575i	$0.440490\pi$
$308$	33.9758	1.93595
$309$	0	0
$310$	−3.48427	−0.197893
$311$	−22.8659	−1.29661	−0.648304	−	0.761382i	$-0.724521\pi$
$311$	−22.8659	−1.29661	−0.648304	+	0.761382i	$0.724521\pi$
$312$	0	0
$313$	−20.3913	−1.15259	−0.576293	−	0.817243i	$-0.695501\pi$
$313$	−20.3913	−1.15259	−0.576293	+	0.817243i	$0.695501\pi$
$314$	1.03146	0.0582086
$315$	0	0
$316$	15.2664	0.858800
$317$	−18.1631	−1.02014	−0.510072	−	0.860132i	$-0.670381\pi$
$317$	−18.1631	−1.02014	−0.510072	+	0.860132i	$0.670381\pi$
$318$	0	0
$319$	−32.9396	−1.84426
$320$	−3.63102	−0.202980
$321$	0	0
$322$	3.59909	0.200569
$323$	14.3303	0.797361
$324$	0	0
$325$	0	0
$326$	−0.141375	−0.00783005
$327$	0	0
$328$	−18.2935	−1.01009
$329$	4.28275	0.236116
$330$	0	0
$331$	1.96184	0.107832	0.0539161	−	0.998545i	$-0.482830\pi$
$331$	1.96184	0.107832	0.0539161	+	0.998545i	$0.482830\pi$
$332$	−3.37435	−0.185192
$333$	0	0
$334$	8.70278	0.476195
$335$	−11.5797	−0.632667
$336$	0	0
$337$	−27.9215	−1.52098	−0.760492	−	0.649348i	$-0.775042\pi$
$337$	−27.9215	−1.52098	−0.760492	+	0.649348i	$0.775042\pi$
$338$	0	0
$339$	0	0
$340$	3.51573	0.190667
$341$	36.9047	1.99850
$342$	0	0
$343$	−8.00000	−0.431959
$344$	7.23191	0.389919
$345$	0	0
$346$	5.82477	0.313142
$347$	−11.9946	−0.643905	−0.321953	−	0.946756i	$-0.604339\pi$
$347$	−11.9946	−0.643905	−0.321953	+	0.946756i	$0.604339\pi$
$348$	0	0
$349$	−7.49694	−0.401302	−0.200651	−	0.979663i	$-0.564306\pi$
$349$	−7.49694	−0.401302	−0.200651	+	0.979663i	$0.564306\pi$
$350$	1.78017	0.0951540
$351$	0	0
$352$	−21.9323	−1.16900
$353$	−7.85623	−0.418145	−0.209073	−	0.977900i	$-0.567044\pi$
$353$	−7.85623	−0.418145	−0.209073	+	0.977900i	$0.567044\pi$
$354$	0	0
$355$	−7.20775	−0.382548
$356$	6.00000	0.317999
$357$	0	0
$358$	−1.03146	−0.0545143
$359$	−12.3720	−0.652967	−0.326484	−	0.945203i	$-0.605864\pi$
$359$	−12.3720	−0.652967	−0.326484	+	0.945203i	$0.605864\pi$
$360$	0	0
$361$	34.9463	1.83928
$362$	1.89546	0.0996232
$363$	0	0
$364$	0	0
$365$	1.60388	0.0839507
$366$	0	0
$367$	4.85517	0.253438	0.126719	−	0.991939i	$-0.459555\pi$
$367$	4.85517	0.253438	0.126719	+	0.991939i	$0.459555\pi$
$368$	5.76377	0.300457
$369$	0	0
$370$	−4.61596	−0.239972
$371$	−32.9396	−1.71014
$372$	0	0
$373$	−28.9530	−1.49913	−0.749565	−	0.661931i	$-0.769737\pi$
$373$	−28.9530	−1.49913	−0.749565	+	0.661931i	$0.769737\pi$
$374$	4.09305	0.211647
$375$	0	0
$376$	−1.81163	−0.0934275
$377$	0	0
$378$	0	0
$379$	−36.0713	−1.85286	−0.926429	−	0.376471i	$-0.877138\pi$
$379$	−36.0713	−1.85286	−0.926429	+	0.376471i	$0.877138\pi$
$380$	13.2349	0.678936
$381$	0	0
$382$	−2.00000	−0.102329
$383$	−29.7778	−1.52157	−0.760787	−	0.649002i	$-0.775187\pi$
$383$	−29.7778	−1.52157	−0.760787	+	0.649002i	$0.775187\pi$
$384$	0	0
$385$	−18.8552	−0.960948
$386$	−6.27413	−0.319345
$387$	0	0
$388$	−17.3056	−0.878558
$389$	−10.7681	−0.545964	−0.272982	−	0.962019i	$-0.588010\pi$
$389$	−10.7681	−0.545964	−0.272982	+	0.962019i	$0.588010\pi$
$390$	0	0
$391$	−3.94464	−0.199489
$392$	15.2282	0.769140
$393$	0	0
$394$	1.32304	0.0666540
$395$	−8.47219	−0.426282
$396$	0	0
$397$	−0.865921	−0.0434593	−0.0217297	−	0.999764i	$-0.506917\pi$
$397$	−0.865921	−0.0434593	−0.0217297	+	0.999764i	$0.506917\pi$
$398$	−1.39506	−0.0699280
$399$	0	0
$400$	2.85086	0.142543
$401$	−3.32975	−0.166280	−0.0831399	−	0.996538i	$-0.526495\pi$
$401$	−3.32975	−0.166280	−0.0831399	+	0.996538i	$0.526495\pi$
$402$	0	0
$403$	0	0
$404$	−20.1172	−1.00087
$405$	0	0
$406$	−12.4397	−0.617370
$407$	48.8913	2.42345
$408$	0	0
$409$	−4.41550	−0.218333	−0.109166	−	0.994024i	$-0.534818\pi$
$409$	−4.41550	−0.218333	−0.109166	+	0.994024i	$0.534818\pi$
$410$	4.81163	0.237629
$411$	0	0
$412$	1.06638	0.0525366
$413$	−54.8068	−2.69687
$414$	0	0
$415$	1.87263	0.0919236
$416$	0	0
$417$	0	0
$418$	15.4082	0.753640
$419$	−26.9831	−1.31821	−0.659106	−	0.752050i	$-0.729065\pi$
$419$	−26.9831	−1.31821	−0.659106	+	0.752050i	$0.729065\pi$
$420$	0	0
$421$	−13.6015	−0.662896	−0.331448	−	0.943474i	$-0.607537\pi$
$421$	−13.6015	−0.662896	−0.331448	+	0.943474i	$0.607537\pi$
$422$	−10.9162	−0.531391
$423$	0	0
$424$	13.9336	0.676677
$425$	−1.95108	−0.0946414
$426$	0	0
$427$	11.0508	0.534787
$428$	10.5646	0.510661
$429$	0	0
$430$	−1.90217	−0.0917306
$431$	−31.2814	−1.50677	−0.753387	−	0.657578i	$-0.771581\pi$
$431$	−31.2814	−1.50677	−0.753387	+	0.657578i	$0.771581\pi$
$432$	0	0
$433$	3.47112	0.166812	0.0834058	−	0.996516i	$-0.473420\pi$
$433$	3.47112	0.166812	0.0834058	+	0.996516i	$0.473420\pi$
$434$	13.9371	0.669001
$435$	0	0
$436$	−11.5211	−0.551761
$437$	−14.8495	−0.710349
$438$	0	0
$439$	−15.5405	−0.741707	−0.370853	−	0.928691i	$-0.620935\pi$
$439$	−15.5405	−0.741707	−0.370853	+	0.928691i	$0.620935\pi$
$440$	7.97584	0.380233
$441$	0	0
$442$	0	0
$443$	−29.6256	−1.40756	−0.703778	−	0.710420i	$-0.748505\pi$
$443$	−29.6256	−1.40756	−0.703778	+	0.710420i	$0.748505\pi$
$444$	0	0
$445$	−3.32975	−0.157845
$446$	5.69309	0.269575
$447$	0	0
$448$	14.5241	0.686199
$449$	−1.64742	−0.0777464	−0.0388732	−	0.999244i	$-0.512377\pi$
$449$	−1.64742	−0.0777464	−0.0388732	+	0.999244i	$0.512377\pi$
$450$	0	0
$451$	−50.9638	−2.39979
$452$	13.1685	0.619395
$453$	0	0
$454$	−11.0489	−0.518551
$455$	0	0
$456$	0	0
$457$	36.5483	1.70966	0.854828	−	0.518912i	$-0.173663\pi$
$457$	36.5483	1.70966	0.854828	+	0.518912i	$0.173663\pi$
$458$	−4.29350	−0.200622
$459$	0	0
$460$	−3.64310	−0.169861
$461$	22.8987	1.06650	0.533250	−	0.845958i	$-0.320971\pi$
$461$	22.8987	1.06650	0.533250	+	0.845958i	$0.320971\pi$
$462$	0	0
$463$	27.3927	1.27305	0.636523	−	0.771258i	$-0.280372\pi$
$463$	27.3927	1.27305	0.636523	+	0.771258i	$0.280372\pi$
$464$	−19.9215	−0.924834
$465$	0	0
$466$	9.67563	0.448215
$467$	−16.4741	−0.762331	−0.381165	−	0.924507i	$-0.624477\pi$
$467$	−16.4741	−0.762331	−0.381165	+	0.924507i	$0.624477\pi$
$468$	0	0
$469$	46.3188	2.13881
$470$	0.476501	0.0219793
$471$	0	0
$472$	23.1836	1.06711
$473$	20.1473	0.926376
$474$	0	0
$475$	−7.34481	−0.337003
$476$	−14.0629	−0.644573
$477$	0	0
$478$	−1.81508	−0.0830200
$479$	−7.50125	−0.342741	−0.171371	−	0.985207i	$-0.554820\pi$
$479$	−7.50125	−0.342741	−0.171371	+	0.985207i	$0.554820\pi$
$480$	0	0
$481$	0	0
$482$	−9.59419	−0.437003
$483$	0	0
$484$	−20.2174	−0.918975
$485$	9.60388	0.436090
$486$	0	0
$487$	−30.8611	−1.39845	−0.699226	−	0.714901i	$-0.746472\pi$
$487$	−30.8611	−1.39845	−0.699226	+	0.714901i	$0.746472\pi$
$488$	−4.67456	−0.211608
$489$	0	0
$490$	−4.00538	−0.180944
$491$	−5.25129	−0.236987	−0.118494	−	0.992955i	$-0.537807\pi$
$491$	−5.25129	−0.236987	−0.118494	+	0.992955i	$0.537807\pi$
$492$	0	0
$493$	13.6340	0.614045
$494$	0	0
$495$	0	0
$496$	22.3196	1.00218
$497$	28.8310	1.29325
$498$	0	0
$499$	34.5381	1.54614	0.773069	−	0.634322i	$-0.218721\pi$
$499$	34.5381	1.54614	0.773069	+	0.634322i	$0.218721\pi$
$500$	−1.80194	−0.0805851
$501$	0	0
$502$	0.361208	0.0161215
$503$	27.6819	1.23427	0.617137	−	0.786856i	$-0.288293\pi$
$503$	27.6819	1.23427	0.617137	+	0.786856i	$0.288293\pi$
$504$	0	0
$505$	11.1642	0.496801
$506$	−4.24134	−0.188551
$507$	0	0
$508$	22.5133	0.998868
$509$	39.7646	1.76254	0.881268	−	0.472617i	$-0.156690\pi$
$509$	39.7646	1.76254	0.881268	+	0.472617i	$0.156690\pi$
$510$	0	0
$511$	−6.41550	−0.283805
$512$	−22.9119	−1.01257
$513$	0	0
$514$	9.28813	0.409682
$515$	−0.591794	−0.0260776
$516$	0	0
$517$	−5.04700	−0.221967
$518$	18.4638	0.811254
$519$	0	0
$520$	0	0
$521$	20.8659	0.914153	0.457076	−	0.889427i	$-0.348897\pi$
$521$	20.8659	0.914153	0.457076	+	0.889427i	$0.348897\pi$
$522$	0	0
$523$	−6.70304	−0.293103	−0.146552	−	0.989203i	$-0.546817\pi$
$523$	−6.70304	−0.293103	−0.146552	+	0.989203i	$0.546817\pi$
$524$	15.7017	0.685932
$525$	0	0
$526$	13.5821	0.592208
$527$	−15.2752	−0.665398
$528$	0	0
$529$	−18.9124	−0.822280
$530$	−3.66487	−0.159192
$531$	0	0
$532$	−52.9396	−2.29522
$533$	0	0
$534$	0	0
$535$	−5.86294	−0.253477
$536$	−19.5931	−0.846294
$537$	0	0
$538$	0.870706	0.0375388
$539$	42.4241	1.82734
$540$	0	0
$541$	−3.98493	−0.171326	−0.0856629	−	0.996324i	$-0.527301\pi$
$541$	−3.98493	−0.171326	−0.0856629	+	0.996324i	$0.527301\pi$
$542$	0.190766	0.00819412
$543$	0	0
$544$	9.07798	0.389215
$545$	6.39373	0.273877
$546$	0	0
$547$	27.6426	1.18191	0.590957	−	0.806703i	$-0.298750\pi$
$547$	27.6426	1.18191	0.590957	+	0.806703i	$0.298750\pi$
$548$	6.08815	0.260073
$549$	0	0
$550$	−2.09783	−0.0894520
$551$	51.3250	2.18652
$552$	0	0
$553$	33.8888	1.44110
$554$	10.6703	0.453336
$555$	0	0
$556$	−19.6189	−0.832028
$557$	28.7047	1.21626	0.608128	−	0.793839i	$-0.291921\pi$
$557$	28.7047	1.21626	0.608128	+	0.793839i	$0.291921\pi$
$558$	0	0
$559$	0	0
$560$	−11.4034	−0.481882
$561$	0	0
$562$	−1.94571	−0.0820746
$563$	−35.9541	−1.51528	−0.757642	−	0.652671i	$-0.773648\pi$
$563$	−35.9541	−1.51528	−0.757642	+	0.652671i	$0.773648\pi$
$564$	0	0
$565$	−7.30798	−0.307449
$566$	8.92884	0.375307
$567$	0	0
$568$	−12.1957	−0.511719
$569$	19.2728	0.807958	0.403979	−	0.914768i	$-0.367627\pi$
$569$	19.2728	0.807958	0.403979	+	0.914768i	$0.367627\pi$
$570$	0	0
$571$	−18.2524	−0.763837	−0.381919	−	0.924196i	$-0.624737\pi$
$571$	−18.2524	−0.763837	−0.381919	+	0.924196i	$0.624737\pi$
$572$	0	0
$573$	0	0
$574$	−19.2465	−0.803334
$575$	2.02177	0.0843136
$576$	0	0
$577$	−20.2306	−0.842210	−0.421105	−	0.907012i	$-0.638358\pi$
$577$	−20.2306	−0.842210	−0.421105	+	0.907012i	$0.638358\pi$
$578$	5.87156	0.244225
$579$	0	0
$580$	12.5918	0.522846
$581$	−7.49050	−0.310758
$582$	0	0
$583$	38.8176	1.60766
$584$	2.71379	0.112298
$585$	0	0
$586$	10.5550	0.436021
$587$	9.18359	0.379047	0.189524	−	0.981876i	$-0.439306\pi$
$587$	9.18359	0.379047	0.189524	+	0.981876i	$0.439306\pi$
$588$	0	0
$589$	−57.5032	−2.36938
$590$	−6.09783	−0.251044
$591$	0	0
$592$	29.5690	1.21528
$593$	−2.22819	−0.0915009	−0.0457505	−	0.998953i	$-0.514568\pi$
$593$	−2.22819	−0.0915009	−0.0457505	+	0.998953i	$0.514568\pi$
$594$	0	0
$595$	7.80433	0.319946
$596$	−21.9758	−0.900165
$597$	0	0
$598$	0	0
$599$	−34.3564	−1.40377	−0.701883	−	0.712293i	$-0.747657\pi$
$599$	−34.3564	−1.40377	−0.701883	+	0.712293i	$0.747657\pi$
$600$	0	0
$601$	6.54586	0.267011	0.133506	−	0.991048i	$-0.457377\pi$
$601$	6.54586	0.267011	0.133506	+	0.991048i	$0.457377\pi$
$602$	7.60866	0.310106
$603$	0	0
$604$	−12.0586	−0.490658
$605$	11.2198	0.456151
$606$	0	0
$607$	−7.48188	−0.303680	−0.151840	−	0.988405i	$-0.548520\pi$
$607$	−7.48188	−0.303680	−0.151840	+	0.988405i	$0.548520\pi$
$608$	34.1739	1.38593
$609$	0	0
$610$	1.22952	0.0497819
$611$	0	0
$612$	0	0
$613$	35.8625	1.44847	0.724236	−	0.689553i	$-0.242193\pi$
$613$	35.8625	1.44847	0.724236	+	0.689553i	$0.242193\pi$
$614$	−2.89871	−0.116982
$615$	0	0
$616$	−31.9033	−1.28542
$617$	−38.8587	−1.56439	−0.782197	−	0.623031i	$-0.785901\pi$
$617$	−38.8587	−1.56439	−0.782197	+	0.623031i	$0.785901\pi$
$618$	0	0
$619$	43.4312	1.74565	0.872823	−	0.488037i	$-0.162287\pi$
$619$	43.4312	1.74565	0.872823	+	0.488037i	$0.162287\pi$
$620$	−14.1075	−0.566572
$621$	0	0
$622$	10.1763	0.408032
$623$	13.3190	0.533614
$624$	0	0
$625$	1.00000	0.0400000
$626$	9.07500	0.362710
$627$	0	0
$628$	4.17629	0.166652
$629$	−20.2366	−0.806884
$630$	0	0
$631$	−38.9051	−1.54879	−0.774395	−	0.632703i	$-0.781946\pi$
$631$	−38.9051	−1.54879	−0.774395	+	0.632703i	$0.781946\pi$
$632$	−14.3351	−0.570221
$633$	0	0
$634$	8.08336	0.321031
$635$	−12.4940	−0.495808
$636$	0	0
$637$	0	0
$638$	14.6595	0.580375
$639$	0	0
$640$	10.9215	0.431712
$641$	30.2150	1.19342	0.596711	−	0.802456i	$-0.296474\pi$
$641$	30.2150	1.19342	0.596711	+	0.802456i	$0.296474\pi$
$642$	0	0
$643$	9.01208	0.355402	0.177701	−	0.984085i	$-0.443134\pi$
$643$	9.01208	0.355402	0.177701	+	0.984085i	$0.443134\pi$
$644$	14.5724	0.574234
$645$	0	0
$646$	−6.37760	−0.250923
$647$	12.0925	0.475404	0.237702	−	0.971338i	$-0.423606\pi$
$647$	12.0925	0.475404	0.237702	+	0.971338i	$0.423606\pi$
$648$	0	0
$649$	64.5870	2.53526
$650$	0	0
$651$	0	0
$652$	−0.572417	−0.0224176
$653$	17.6093	0.689103	0.344552	−	0.938767i	$-0.388031\pi$
$653$	17.6093	0.689103	0.344552	+	0.938767i	$0.388031\pi$
$654$	0	0
$655$	−8.71379	−0.340476
$656$	−30.8224	−1.20341
$657$	0	0
$658$	−1.90600	−0.0743037
$659$	2.83579	0.110467	0.0552333	−	0.998473i	$-0.482410\pi$
$659$	2.83579	0.110467	0.0552333	+	0.998473i	$0.482410\pi$
$660$	0	0
$661$	−18.0683	−0.702775	−0.351388	−	0.936230i	$-0.614290\pi$
$661$	−18.0683	−0.702775	−0.351388	+	0.936230i	$0.614290\pi$
$662$	−0.873099	−0.0339340
$663$	0	0
$664$	3.16852	0.122963
$665$	29.3793	1.13928
$666$	0	0
$667$	−14.1280	−0.547037
$668$	35.2368	1.36335
$669$	0	0
$670$	5.15346	0.199095
$671$	−13.0228	−0.502741
$672$	0	0
$673$	−15.4819	−0.596783	−0.298391	−	0.954444i	$-0.596450\pi$
$673$	−15.4819	−0.596783	−0.298391	+	0.954444i	$0.596450\pi$
$674$	12.4263	0.478641
$675$	0	0
$676$	0	0
$677$	−7.57434	−0.291105	−0.145553	−	0.989351i	$-0.546496\pi$
$677$	−7.57434	−0.291105	−0.145553	+	0.989351i	$0.546496\pi$
$678$	0	0
$679$	−38.4155	−1.47425
$680$	−3.30127	−0.126598
$681$	0	0
$682$	−16.4241	−0.628912
$683$	−40.5924	−1.55322	−0.776612	−	0.629979i	$-0.783064\pi$
$683$	−40.5924	−1.55322	−0.776612	+	0.629979i	$0.783064\pi$
$684$	0	0
$685$	−3.37867	−0.129092
$686$	3.56033	0.135934
$687$	0	0
$688$	12.1849	0.464546
$689$	0	0
$690$	0	0
$691$	−22.0844	−0.840131	−0.420066	−	0.907494i	$-0.637993\pi$
$691$	−22.0844	−0.840131	−0.420066	+	0.907494i	$0.637993\pi$
$692$	23.5840	0.896530
$693$	0	0
$694$	5.33811	0.202632
$695$	10.8877	0.412994
$696$	0	0
$697$	21.0944	0.799006
$698$	3.33645	0.126287
$699$	0	0
$700$	7.20775	0.272427
$701$	−1.59312	−0.0601714	−0.0300857	−	0.999547i	$-0.509578\pi$
$701$	−1.59312	−0.0601714	−0.0300857	+	0.999547i	$0.509578\pi$
$702$	0	0
$703$	−76.1801	−2.87319
$704$	−17.1159	−0.645079
$705$	0	0
$706$	3.49635	0.131587
$707$	−44.6568	−1.67949
$708$	0	0
$709$	−20.9422	−0.786503	−0.393251	−	0.919431i	$-0.628650\pi$
$709$	−20.9422	−0.786503	−0.393251	+	0.919431i	$0.628650\pi$
$710$	3.20775	0.120385
$711$	0	0
$712$	−5.63401	−0.211143
$713$	15.8286	0.592786
$714$	0	0
$715$	0	0
$716$	−4.17629	−0.156075
$717$	0	0
$718$	5.50604	0.205484
$719$	7.56033	0.281953	0.140976	−	0.990013i	$-0.454976\pi$
$719$	7.56033	0.281953	0.140976	+	0.990013i	$0.454976\pi$
$720$	0	0
$721$	2.36718	0.0881582
$722$	−15.5526	−0.578807
$723$	0	0
$724$	7.67456	0.285223
$725$	−6.98792	−0.259525
$726$	0	0
$727$	49.9845	1.85382	0.926911	−	0.375282i	$-0.122454\pi$
$727$	49.9845	1.85382	0.926911	+	0.375282i	$0.122454\pi$
$728$	0	0
$729$	0	0
$730$	−0.713792	−0.0264186
$731$	−8.33917	−0.308436
$732$	0	0
$733$	−0.944378	−0.0348814	−0.0174407	−	0.999848i	$-0.505552\pi$
$733$	−0.944378	−0.0348814	−0.0174407	+	0.999848i	$0.505552\pi$
$734$	−2.16075	−0.0797548
$735$	0	0
$736$	−9.40688	−0.346742
$737$	−54.5844	−2.01064
$738$	0	0
$739$	−11.6829	−0.429763	−0.214882	−	0.976640i	$-0.568937\pi$
$739$	−11.6829	−0.429763	−0.214882	+	0.976640i	$0.568937\pi$
$740$	−18.6896	−0.687044
$741$	0	0
$742$	14.6595	0.538167
$743$	−34.2252	−1.25560	−0.627801	−	0.778374i	$-0.716045\pi$
$743$	−34.2252	−1.25560	−0.627801	+	0.778374i	$0.716045\pi$
$744$	0	0
$745$	12.1957	0.446815
$746$	12.8853	0.471764
$747$	0	0
$748$	16.5724	0.605948
$749$	23.4517	0.856908
$750$	0	0
$751$	9.81269	0.358070	0.179035	−	0.983843i	$-0.442702\pi$
$751$	9.81269	0.358070	0.179035	+	0.983843i	$0.442702\pi$
$752$	−3.05238	−0.111309
$753$	0	0
$754$	0	0
$755$	6.69202	0.243548
$756$	0	0
$757$	9.91079	0.360214	0.180107	−	0.983647i	$-0.442356\pi$
$757$	9.91079	0.360214	0.180107	+	0.983647i	$0.442356\pi$
$758$	16.0532	0.583080
$759$	0	0
$760$	−12.4276	−0.450796
$761$	−18.1280	−0.657138	−0.328569	−	0.944480i	$-0.606566\pi$
$761$	−18.1280	−0.657138	−0.328569	+	0.944480i	$0.606566\pi$
$762$	0	0
$763$	−25.5749	−0.925875
$764$	−8.09783	−0.292969
$765$	0	0
$766$	13.2524	0.478827
$767$	0	0
$768$	0	0
$769$	−25.0804	−0.904422	−0.452211	−	0.891911i	$-0.649365\pi$
$769$	−25.0804	−0.904422	−0.452211	+	0.891911i	$0.649365\pi$
$770$	8.39134	0.302403
$771$	0	0
$772$	−25.4034	−0.914289
$773$	33.6588	1.21062	0.605311	−	0.795989i	$-0.293049\pi$
$773$	33.6588	1.21062	0.605311	+	0.795989i	$0.293049\pi$
$774$	0	0
$775$	7.82908	0.281229
$776$	16.2500	0.583340
$777$	0	0
$778$	4.79225	0.171810
$779$	79.4094	2.84514
$780$	0	0
$781$	−33.9758	−1.21575
$782$	1.75553	0.0627776
$783$	0	0
$784$	25.6577	0.916346
$785$	−2.31767	−0.0827211
$786$	0	0
$787$	36.8224	1.31258	0.656288	−	0.754510i	$-0.272126\pi$
$787$	36.8224	1.31258	0.656288	+	0.754510i	$0.272126\pi$
$788$	5.35690	0.190832
$789$	0	0
$790$	3.77048	0.134148
$791$	29.2319	1.03937
$792$	0	0
$793$	0	0
$794$	0.385371	0.0136763
$795$	0	0
$796$	−5.64848	−0.200205
$797$	−52.5779	−1.86241	−0.931203	−	0.364502i	$-0.881239\pi$
$797$	−52.5779	−1.86241	−0.931203	+	0.364502i	$0.881239\pi$
$798$	0	0
$799$	2.08900	0.0739035
$800$	−4.65279	−0.164501
$801$	0	0
$802$	1.48188	0.0523269
$803$	7.56033	0.266798
$804$	0	0
$805$	−8.08708	−0.285032
$806$	0	0
$807$	0	0
$808$	18.8901	0.664551
$809$	−1.60388	−0.0563893	−0.0281946	−	0.999602i	$-0.508976\pi$
$809$	−1.60388	−0.0563893	−0.0281946	+	0.999602i	$0.508976\pi$
$810$	0	0
$811$	25.9758	0.912135	0.456067	−	0.889945i	$-0.349258\pi$
$811$	25.9758	0.912135	0.456067	+	0.889945i	$0.349258\pi$
$812$	−50.3672	−1.76754
$813$	0	0
$814$	−21.7587	−0.762641
$815$	0.317667	0.0111274
$816$	0	0
$817$	−31.3927	−1.09829
$818$	1.96508	0.0687075
$819$	0	0
$820$	19.4819	0.680337
$821$	29.9168	1.04410	0.522051	−	0.852914i	$-0.325167\pi$
$821$	29.9168	1.04410	0.522051	+	0.852914i	$0.325167\pi$
$822$	0	0
$823$	−8.77287	−0.305803	−0.152902	−	0.988241i	$-0.548862\pi$
$823$	−8.77287	−0.305803	−0.152902	+	0.988241i	$0.548862\pi$
$824$	−1.00133	−0.0348829
$825$	0	0
$826$	24.3913	0.848683
$827$	−52.3913	−1.82183	−0.910913	−	0.412599i	$-0.864621\pi$
$827$	−52.3913	−1.82183	−0.910913	+	0.412599i	$0.864621\pi$
$828$	0	0
$829$	−34.5894	−1.20134	−0.600670	−	0.799497i	$-0.705099\pi$
$829$	−34.5894	−1.20134	−0.600670	+	0.799497i	$0.705099\pi$
$830$	−0.833397	−0.0289276
$831$	0	0
$832$	0	0
$833$	−17.5597	−0.608409
$834$	0	0
$835$	−19.5550	−0.676727
$836$	62.3866	2.15768
$837$	0	0
$838$	12.0086	0.414831
$839$	14.1473	0.488421	0.244210	−	0.969722i	$-0.421471\pi$
$839$	14.1473	0.488421	0.244210	+	0.969722i	$0.421471\pi$
$840$	0	0
$841$	19.8310	0.683828
$842$	6.05323	0.208608
$843$	0	0
$844$	−44.1987	−1.52138
$845$	0	0
$846$	0	0
$847$	−44.8793	−1.54207
$848$	23.4765	0.806186
$849$	0	0
$850$	0.868313	0.0297829
$851$	20.9697	0.718833
$852$	0	0
$853$	7.51466	0.257297	0.128649	−	0.991690i	$-0.458936\pi$
$853$	7.51466	0.257297	0.128649	+	0.991690i	$0.458936\pi$
$854$	−4.91808	−0.168293
$855$	0	0
$856$	−9.92021	−0.339066
$857$	−12.5724	−0.429466	−0.214733	−	0.976673i	$-0.568888\pi$
$857$	−12.5724	−0.429466	−0.214733	+	0.976673i	$0.568888\pi$
$858$	0	0
$859$	−48.7794	−1.66433	−0.832166	−	0.554526i	$-0.812900\pi$
$859$	−48.7794	−1.66433	−0.832166	+	0.554526i	$0.812900\pi$
$860$	−7.70171	−0.262626
$861$	0	0
$862$	13.9215	0.474170
$863$	−27.9560	−0.951633	−0.475816	−	0.879545i	$-0.657847\pi$
$863$	−27.9560	−0.951633	−0.475816	+	0.879545i	$0.657847\pi$
$864$	0	0
$865$	−13.0881	−0.445010
$866$	−1.54480	−0.0524943
$867$	0	0
$868$	56.4301	1.91536
$869$	−39.9361	−1.35474
$870$	0	0
$871$	0	0
$872$	10.8183	0.366355
$873$	0	0
$874$	6.60866	0.223541
$875$	−4.00000	−0.135225
$876$	0	0
$877$	−13.2620	−0.447827	−0.223914	−	0.974609i	$-0.571883\pi$
$877$	−13.2620	−0.447827	−0.223914	+	0.974609i	$0.571883\pi$
$878$	6.91617	0.233409
$879$	0	0
$880$	13.4383	0.453006
$881$	−14.8358	−0.499830	−0.249915	−	0.968268i	$-0.580403\pi$
$881$	−14.8358	−0.499830	−0.249915	+	0.968268i	$0.580403\pi$
$882$	0	0
$883$	2.36718	0.0796618	0.0398309	−	0.999206i	$-0.487318\pi$
$883$	2.36718	0.0796618	0.0398309	+	0.999206i	$0.487318\pi$
$884$	0	0
$885$	0	0
$886$	13.1847	0.442947
$887$	−0.873953	−0.0293445	−0.0146722	−	0.999892i	$-0.504670\pi$
$887$	−0.873953	−0.0293445	−0.0146722	+	0.999892i	$0.504670\pi$
$888$	0	0
$889$	49.9758	1.67614
$890$	1.48188	0.0496727
$891$	0	0
$892$	23.0508	0.771799
$893$	7.86400	0.263159
$894$	0	0
$895$	2.31767	0.0774711
$896$	−43.6862	−1.45945
$897$	0	0
$898$	0.733169	0.0244662
$899$	−54.7090	−1.82465
$900$	0	0
$901$	−16.0670	−0.535268
$902$	22.6810	0.755195
$903$	0	0
$904$	−12.3653	−0.411262
$905$	−4.25906	−0.141576
$906$	0	0
$907$	19.4168	0.644725	0.322363	−	0.946616i	$-0.395523\pi$
$907$	19.4168	0.644725	0.322363	+	0.946616i	$0.395523\pi$
$908$	−44.7362	−1.48462
$909$	0	0
$910$	0	0
$911$	−11.8479	−0.392537	−0.196269	−	0.980550i	$-0.562882\pi$
$911$	−11.8479	−0.392537	−0.196269	+	0.980550i	$0.562882\pi$
$912$	0	0
$913$	8.82717	0.292137
$914$	−16.2655	−0.538015
$915$	0	0
$916$	−17.3840	−0.574385
$917$	34.8552	1.15102
$918$	0	0
$919$	39.8756	1.31537	0.657687	−	0.753291i	$-0.271535\pi$
$919$	39.8756	1.31537	0.657687	+	0.753291i	$0.271535\pi$
$920$	3.42088	0.112783
$921$	0	0
$922$	−10.1909	−0.335619
$923$	0	0
$924$	0	0
$925$	10.3720	0.341028
$926$	−12.1909	−0.400617
$927$	0	0
$928$	32.5133	1.06730
$929$	57.3879	1.88284	0.941418	−	0.337243i	$-0.109494\pi$
$929$	57.3879	1.88284	0.941418	+	0.337243i	$0.109494\pi$
$930$	0	0
$931$	−66.1033	−2.16645
$932$	39.1758	1.28325
$933$	0	0
$934$	7.33167	0.239899
$935$	−9.19700	−0.300774
$936$	0	0
$937$	3.80433	0.124282	0.0621410	−	0.998067i	$-0.480207\pi$
$937$	3.80433	0.124282	0.0621410	+	0.998067i	$0.480207\pi$
$938$	−20.6138	−0.673065
$939$	0	0
$940$	1.92931	0.0629272
$941$	−40.7042	−1.32692	−0.663460	−	0.748212i	$-0.730913\pi$
$941$	−40.7042	−1.32692	−0.663460	+	0.748212i	$0.730913\pi$
$942$	0	0
$943$	−21.8586	−0.711815
$944$	39.0616	1.27135
$945$	0	0
$946$	−8.96641	−0.291523
$947$	−22.7006	−0.737672	−0.368836	−	0.929495i	$-0.620244\pi$
$947$	−22.7006	−0.737672	−0.368836	+	0.929495i	$0.620244\pi$
$948$	0	0
$949$	0	0
$950$	3.26875	0.106052
$951$	0	0
$952$	13.2051	0.427980
$953$	22.4999	0.728844	0.364422	−	0.931234i	$-0.381267\pi$
$953$	22.4999	0.728844	0.364422	+	0.931234i	$0.381267\pi$
$954$	0	0
$955$	4.49396	0.145421
$956$	−7.34913	−0.237688
$957$	0	0
$958$	3.33837	0.107858
$959$	13.5147	0.436411
$960$	0	0
$961$	30.2946	0.977244
$962$	0	0
$963$	0	0
$964$	−38.8461	−1.25115
$965$	14.0978	0.453825
$966$	0	0
$967$	−14.2547	−0.458402	−0.229201	−	0.973379i	$-0.573611\pi$
$967$	−14.2547	−0.458402	−0.229201	+	0.973379i	$0.573611\pi$
$968$	18.9842	0.610175
$969$	0	0
$970$	−4.27413	−0.137234
$971$	39.3927	1.26417	0.632085	−	0.774899i	$-0.282199\pi$
$971$	39.3927	1.26417	0.632085	+	0.774899i	$0.282199\pi$
$972$	0	0
$973$	−43.5508	−1.39617
$974$	13.7345	0.440082
$975$	0	0
$976$	−7.87608	−0.252107
$977$	−30.6021	−0.979047	−0.489524	−	0.871990i	$-0.662829\pi$
$977$	−30.6021	−0.979047	−0.489524	+	0.871990i	$0.662829\pi$
$978$	0	0
$979$	−15.6957	−0.501638
$980$	−16.2174	−0.518047
$981$	0	0
$982$	2.33704	0.0745781
$983$	36.4325	1.16202	0.581008	−	0.813898i	$-0.302659\pi$
$983$	36.4325	1.16202	0.581008	+	0.813898i	$0.302659\pi$
$984$	0	0
$985$	−2.97285	−0.0947230
$986$	−6.06770	−0.193235
$987$	0	0
$988$	0	0
$989$	8.64130	0.274777
$990$	0	0
$991$	12.7084	0.403696	0.201848	−	0.979417i	$-0.435305\pi$
$991$	12.7084	0.403696	0.201848	+	0.979417i	$0.435305\pi$
$992$	−36.4271	−1.15656
$993$	0	0
$994$	−12.8310	−0.406975
$995$	3.13467	0.0993757
$996$	0	0
$997$	9.07979	0.287560	0.143780	−	0.989610i	$-0.454074\pi$
$997$	9.07979	0.287560	0.143780	+	0.989610i	$0.454074\pi$
$998$	−15.3709	−0.486557
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7605.2.a.bp.1.2		3
3.2	odd	2		2535.2.a.be.1.2	yes	3
13.12	even	2		7605.2.a.ca.1.2		3
39.38	odd	2		2535.2.a.x.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2535.2.a.x.1.2	✓	3	39.38	odd	2
2535.2.a.be.1.2	yes	3	3.2	odd	2
7605.2.a.bp.1.2		3	1.1	even	1	trivial
7605.2.a.ca.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)

\(f(q)\)	\(=\)	\(q-0.445042 q^{2} -1.80194 q^{4} +1.00000 q^{5} -4.00000 q^{7} +1.69202 q^{8} +O(q^{10})\)	\(q-0.445042 q^{2} -1.80194 q^{4} +1.00000 q^{5} -4.00000 q^{7} +1.69202 q^{8} -0.445042 q^{10} +4.71379 q^{11} +1.78017 q^{14} +2.85086 q^{16} -1.95108 q^{17} -7.34481 q^{19} -1.80194 q^{20} -2.09783 q^{22} +2.02177 q^{23} +1.00000 q^{25} +7.20775 q^{28} -6.98792 q^{29} +7.82908 q^{31} -4.65279 q^{32} +0.868313 q^{34} -4.00000 q^{35} +10.3720 q^{37} +3.26875 q^{38} +1.69202 q^{40} -10.8116 q^{41} +4.27413 q^{43} -8.49396 q^{44} -0.899772 q^{46} -1.07069 q^{47} +9.00000 q^{49} -0.445042 q^{50} +8.23490 q^{53} +4.71379 q^{55} -6.76809 q^{56} +3.10992 q^{58} +13.7017 q^{59} -2.76271 q^{61} -3.48427 q^{62} -3.63102 q^{64} -11.5797 q^{67} +3.51573 q^{68} +1.78017 q^{70} -7.20775 q^{71} +1.60388 q^{73} -4.61596 q^{74} +13.2349 q^{76} -18.8552 q^{77} -8.47219 q^{79} +2.85086 q^{80} +4.81163 q^{82} +1.87263 q^{83} -1.95108 q^{85} -1.90217 q^{86} +7.97584 q^{88} -3.32975 q^{89} -3.64310 q^{92} +0.476501 q^{94} -7.34481 q^{95} +9.60388 q^{97} -4.00538 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} - q^{4} + 3 q^{5} - 12 q^{7}+O(q^{10}) \) 3 * q - q^2 - q^4 + 3 * q^5 - 12 * q^7	\( 3 q - q^{2} - q^{4} + 3 q^{5} - 12 q^{7} - q^{10} + 6 q^{11} + 4 q^{14} - 5 q^{16} - 15 q^{17} + q^{19} - q^{20} + 12 q^{22} + 3 q^{23} + 3 q^{25} + 4 q^{28} - 2 q^{29} + 13 q^{31} + 4 q^{32} + 5 q^{34} - 12 q^{35} + 2 q^{37} + 2 q^{38} - 6 q^{41} + 2 q^{43} - 16 q^{44} + 20 q^{46} + 9 q^{47} + 27 q^{49} - q^{50} + q^{53} + 6 q^{55} + 10 q^{58} + 14 q^{59} + 9 q^{61} - 23 q^{62} + 4 q^{64} + 12 q^{67} - 2 q^{68} + 4 q^{70} - 4 q^{71} - 4 q^{73} - 24 q^{74} + 16 q^{76} - 24 q^{77} - 19 q^{79} - 5 q^{80} - 12 q^{82} - 11 q^{83} - 15 q^{85} - 24 q^{86} - 14 q^{88} - 12 q^{89} - 15 q^{92} - 24 q^{94} + q^{95} + 20 q^{97} - 9 q^{98}+O(q^{100}) \) 3 * q - q^2 - q^4 + 3 * q^5 - 12 * q^7 - q^10 + 6 * q^11 + 4 * q^14 - 5 * q^16 - 15 * q^17 + q^19 - q^20 + 12 * q^22 + 3 * q^23 + 3 * q^25 + 4 * q^28 - 2 * q^29 + 13 * q^31 + 4 * q^32 + 5 * q^34 - 12 * q^35 + 2 * q^37 + 2 * q^38 - 6 * q^41 + 2 * q^43 - 16 * q^44 + 20 * q^46 + 9 * q^47 + 27 * q^49 - q^50 + q^53 + 6 * q^55 + 10 * q^58 + 14 * q^59 + 9 * q^61 - 23 * q^62 + 4 * q^64 + 12 * q^67 - 2 * q^68 + 4 * q^70 - 4 * q^71 - 4 * q^73 - 24 * q^74 + 16 * q^76 - 24 * q^77 - 19 * q^79 - 5 * q^80 - 12 * q^82 - 11 * q^83 - 15 * q^85 - 24 * q^86 - 14 * q^88 - 12 * q^89 - 15 * q^92 - 24 * q^94 + q^95 + 20 * q^97 - 9 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more