LMFDB - Embedded newform 7605.2.a.bm.1.1

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			1.1
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)

$f(q)$	$=$	$q-2.69202 q^{2} +5.24698 q^{4} +1.00000 q^{5} +1.69202 q^{7} -8.74094 q^{8} +O(q^{10})$	$q-2.69202 q^{2} +5.24698 q^{4} +1.00000 q^{5} +1.69202 q^{7} -8.74094 q^{8} -2.69202 q^{10} +4.60388 q^{11} -4.55496 q^{14} +13.0368 q^{16} -5.38404 q^{17} -4.96077 q^{19} +5.24698 q^{20} -12.3937 q^{22} -6.76271 q^{23} +1.00000 q^{25} +8.87800 q^{28} -3.64310 q^{29} +5.59179 q^{31} -17.6136 q^{32} +14.4940 q^{34} +1.69202 q^{35} +3.86294 q^{37} +13.3545 q^{38} -8.74094 q^{40} +10.8998 q^{41} -9.91185 q^{43} +24.1564 q^{44} +18.2054 q^{46} -4.80194 q^{47} -4.13706 q^{49} -2.69202 q^{50} +9.99761 q^{53} +4.60388 q^{55} -14.7899 q^{56} +9.80731 q^{58} -10.6136 q^{59} +7.44504 q^{61} -15.0532 q^{62} +21.3424 q^{64} -10.8726 q^{67} -28.2500 q^{68} -4.55496 q^{70} +5.03923 q^{71} -0.917231 q^{73} -10.3991 q^{74} -26.0291 q^{76} +7.78986 q^{77} -5.11960 q^{79} +13.0368 q^{80} -29.3424 q^{82} +15.7168 q^{83} -5.38404 q^{85} +26.6829 q^{86} -40.2422 q^{88} -3.85086 q^{89} -35.4838 q^{92} +12.9269 q^{94} -4.96077 q^{95} -5.27413 q^{97} +11.1371 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 11 q^{4} + 3 q^{5} - 12 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + 11 * q^4 + 3 * q^5 - 12 * q^8	$ 3 q - 3 q^{2} + 11 q^{4} + 3 q^{5} - 12 q^{8} - 3 q^{10} + 5 q^{11} - 14 q^{14} + 11 q^{16} - 6 q^{17} - 2 q^{19} + 11 q^{20} - 5 q^{22} - 3 q^{23} + 3 q^{25} + 7 q^{28} - 15 q^{29} - 11 q^{31} - 22 q^{32} + 34 q^{34} + 17 q^{37} - 5 q^{38} - 12 q^{40} + 10 q^{41} - 26 q^{43} + 23 q^{44} - 4 q^{46} - 10 q^{47} - 7 q^{49} - 3 q^{50} - 11 q^{53} + 5 q^{55} - 21 q^{56} + 22 q^{58} - q^{59} + 22 q^{61} - 17 q^{62} - 16 q^{67} - 36 q^{68} - 14 q^{70} + 28 q^{71} + 4 q^{73} + 4 q^{74} - 12 q^{76} + 6 q^{79} + 11 q^{80} - 24 q^{82} - 5 q^{83} - 6 q^{85} + 12 q^{86} - 34 q^{88} + 2 q^{89} - 18 q^{92} + 10 q^{94} - 2 q^{95} - 5 q^{97} + 28 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 11 * q^4 + 3 * q^5 - 12 * q^8 - 3 * q^10 + 5 * q^11 - 14 * q^14 + 11 * q^16 - 6 * q^17 - 2 * q^19 + 11 * q^20 - 5 * q^22 - 3 * q^23 + 3 * q^25 + 7 * q^28 - 15 * q^29 - 11 * q^31 - 22 * q^32 + 34 * q^34 + 17 * q^37 - 5 * q^38 - 12 * q^40 + 10 * q^41 - 26 * q^43 + 23 * q^44 - 4 * q^46 - 10 * q^47 - 7 * q^49 - 3 * q^50 - 11 * q^53 + 5 * q^55 - 21 * q^56 + 22 * q^58 - q^59 + 22 * q^61 - 17 * q^62 - 16 * q^67 - 36 * q^68 - 14 * q^70 + 28 * q^71 + 4 * q^73 + 4 * q^74 - 12 * q^76 + 6 * q^79 + 11 * q^80 - 24 * q^82 - 5 * q^83 - 6 * q^85 + 12 * q^86 - 34 * q^88 + 2 * q^89 - 18 * q^92 + 10 * q^94 - 2 * q^95 - 5 * q^97 + 28 * 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 $

$2$	−2.69202	−1.90355	−0.951773	−	0.306802i	$-0.900741\pi$
$2$	−2.69202	−1.90355	−0.951773	+	0.306802i	$0.900741\pi$
$3$	0	0
$3$	0	0
$4$	5.24698	2.62349
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.69202	0.639524	0.319762	−	0.947498i	$-0.396397\pi$
$7$	1.69202	0.639524	0.319762	+	0.947498i	$0.396397\pi$
$8$	−8.74094	−3.09039
$9$	0	0
$10$	−2.69202	−0.851292
$11$	4.60388	1.38812	0.694060	−	0.719917i	$-0.255820\pi$
$11$	4.60388	1.38812	0.694060	+	0.719917i	$0.255820\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−4.55496	−1.21736
$15$	0	0
$16$	13.0368	3.25921
$17$	−5.38404	−1.30582	−0.652911	−	0.757435i	$-0.726452\pi$
$17$	−5.38404	−1.30582	−0.652911	+	0.757435i	$0.726452\pi$
$18$	0	0
$19$	−4.96077	−1.13808	−0.569039	−	0.822310i	$-0.692685\pi$
$19$	−4.96077	−1.13808	−0.569039	+	0.822310i	$0.692685\pi$
$20$	5.24698	1.17326
$21$	0	0
$22$	−12.3937	−2.64235
$23$	−6.76271	−1.41012	−0.705061	−	0.709147i	$-0.749080\pi$
$23$	−6.76271	−1.41012	−0.705061	+	0.709147i	$0.749080\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	8.87800	1.67778
$29$	−3.64310	−0.676507	−0.338254	−	0.941055i	$-0.609836\pi$
$29$	−3.64310	−0.676507	−0.338254	+	0.941055i	$0.609836\pi$
$30$	0	0
$31$	5.59179	1.00432	0.502158	−	0.864776i	$-0.332540\pi$
$31$	5.59179	1.00432	0.502158	+	0.864776i	$0.332540\pi$
$32$	−17.6136	−3.11367
$33$	0	0
$34$	14.4940	2.48569
$35$	1.69202	0.286004
$36$	0	0
$37$	3.86294	0.635063	0.317531	−	0.948248i	$-0.397146\pi$
$37$	3.86294	0.635063	0.317531	+	0.948248i	$0.397146\pi$
$38$	13.3545	2.16639
$39$	0	0
$40$	−8.74094	−1.38206
$41$	10.8998	1.70226	0.851129	−	0.524956i	$-0.175918\pi$
$41$	10.8998	1.70226	0.851129	+	0.524956i	$0.175918\pi$
$42$	0	0
$43$	−9.91185	−1.51154	−0.755772	−	0.654835i	$-0.772738\pi$
$43$	−9.91185	−1.51154	−0.755772	+	0.654835i	$0.772738\pi$
$44$	24.1564	3.64172
$45$	0	0
$46$	18.2054	2.68423
$47$	−4.80194	−0.700435	−0.350217	−	0.936668i	$-0.613892\pi$
$47$	−4.80194	−0.700435	−0.350217	+	0.936668i	$0.613892\pi$
$48$	0	0
$49$	−4.13706	−0.591009
$50$	−2.69202	−0.380709
$51$	0	0
$52$	0	0
$53$	9.99761	1.37328	0.686638	−	0.726999i	$-0.259086\pi$
$53$	9.99761	1.37328	0.686638	+	0.726999i	$0.259086\pi$
$54$	0	0
$55$	4.60388	0.620786
$56$	−14.7899	−1.97638
$57$	0	0
$58$	9.80731	1.28776
$59$	−10.6136	−1.38177	−0.690884	−	0.722965i	$-0.742779\pi$
$59$	−10.6136	−1.38177	−0.690884	+	0.722965i	$0.742779\pi$
$60$	0	0
$61$	7.44504	0.953240	0.476620	−	0.879109i	$-0.341862\pi$
$61$	7.44504	0.953240	0.476620	+	0.879109i	$0.341862\pi$
$62$	−15.0532	−1.91176
$63$	0	0
$64$	21.3424	2.66780
$65$	0	0
$66$	0	0
$67$	−10.8726	−1.32830	−0.664151	−	0.747598i	$-0.731207\pi$
$67$	−10.8726	−1.32830	−0.664151	+	0.747598i	$0.731207\pi$
$68$	−28.2500	−3.42581
$69$	0	0
$70$	−4.55496	−0.544422
$71$	5.03923	0.598046	0.299023	−	0.954246i	$-0.403339\pi$
$71$	5.03923	0.598046	0.299023	+	0.954246i	$0.403339\pi$
$72$	0	0
$73$	−0.917231	−0.107354	−0.0536769	−	0.998558i	$-0.517094\pi$
$73$	−0.917231	−0.107354	−0.0536769	+	0.998558i	$0.517094\pi$
$74$	−10.3991	−1.20887
$75$	0	0
$76$	−26.0291	−2.98574
$77$	7.78986	0.887736
$78$	0	0
$79$	−5.11960	−0.576001	−0.288000	−	0.957630i	$-0.592990\pi$
$79$	−5.11960	−0.576001	−0.288000	+	0.957630i	$0.592990\pi$
$80$	13.0368	1.45756
$81$	0	0
$82$	−29.3424	−3.24033
$83$	15.7168	1.72514	0.862570	−	0.505938i	$-0.168853\pi$
$83$	15.7168	1.72514	0.862570	+	0.505938i	$0.168853\pi$
$84$	0	0
$85$	−5.38404	−0.583981
$86$	26.6829	2.87729
$87$	0	0
$88$	−40.2422	−4.28983
$89$	−3.85086	−0.408190	−0.204095	−	0.978951i	$-0.565425\pi$
$89$	−3.85086	−0.408190	−0.204095	+	0.978951i	$0.565425\pi$
$90$	0	0
$91$	0	0
$92$	−35.4838	−3.69944
$93$	0	0
$94$	12.9269	1.33331
$95$	−4.96077	−0.508964
$96$	0	0
$97$	−5.27413	−0.535506	−0.267753	−	0.963488i	$-0.586281\pi$
$97$	−5.27413	−0.535506	−0.267753	+	0.963488i	$0.586281\pi$
$98$	11.1371	1.12501
$99$	0	0
$100$	5.24698	0.524698
$101$	−4.37435	−0.435265	−0.217632	−	0.976031i	$-0.569833\pi$
$101$	−4.37435	−0.435265	−0.217632	+	0.976031i	$0.569833\pi$
$102$	0	0
$103$	−11.8019	−1.16288	−0.581440	−	0.813589i	$-0.697510\pi$
$103$	−11.8019	−1.16288	−0.581440	+	0.813589i	$0.697510\pi$
$104$	0	0
$105$	0	0
$106$	−26.9138	−2.61410
$107$	15.4698	1.49552	0.747761	−	0.663968i	$-0.231129\pi$
$107$	15.4698	1.49552	0.747761	+	0.663968i	$0.231129\pi$
$108$	0	0
$109$	−4.76809	−0.456700	−0.228350	−	0.973579i	$-0.573333\pi$
$109$	−4.76809	−0.456700	−0.228350	+	0.973579i	$0.573333\pi$
$110$	−12.3937	−1.18170
$111$	0	0
$112$	22.0586	2.08434
$113$	1.28083	0.120490	0.0602452	−	0.998184i	$-0.480812\pi$
$113$	1.28083	0.120490	0.0602452	+	0.998184i	$0.480812\pi$
$114$	0	0
$115$	−6.76271	−0.630626
$116$	−19.1153	−1.77481
$117$	0	0
$118$	28.5719	2.63026
$119$	−9.10992	−0.835105
$120$	0	0
$121$	10.1957	0.926879
$122$	−20.0422	−1.81454
$123$	0	0
$124$	29.3400	2.63481
$125$	1.00000	0.0894427
$126$	0	0
$127$	0.461435	0.0409458	0.0204729	−	0.999790i	$-0.493483\pi$
$127$	0.461435	0.0409458	0.0204729	+	0.999790i	$0.493483\pi$
$128$	−22.2271	−1.96462
$129$	0	0
$130$	0	0
$131$	−3.10454	−0.271245	−0.135622	−	0.990761i	$-0.543303\pi$
$131$	−3.10454	−0.271245	−0.135622	+	0.990761i	$0.543303\pi$
$132$	0	0
$133$	−8.39373	−0.727829
$134$	29.2693	2.52849
$135$	0	0
$136$	47.0616	4.03550
$137$	3.75063	0.320438	0.160219	−	0.987082i	$-0.448780\pi$
$137$	3.75063	0.320438	0.160219	+	0.987082i	$0.448780\pi$
$138$	0	0
$139$	−2.34721	−0.199088	−0.0995438	−	0.995033i	$-0.531738\pi$
$139$	−2.34721	−0.199088	−0.0995438	+	0.995033i	$0.531738\pi$
$140$	8.87800	0.750328
$141$	0	0
$142$	−13.5657	−1.13841
$143$	0	0
$144$	0	0
$145$	−3.64310	−0.302543
$146$	2.46921	0.204353
$147$	0	0
$148$	20.2687	1.66608
$149$	−14.7114	−1.20520	−0.602602	−	0.798042i	$-0.705869\pi$
$149$	−14.7114	−1.20520	−0.602602	+	0.798042i	$0.705869\pi$
$150$	0	0
$151$	−4.43535	−0.360944	−0.180472	−	0.983580i	$-0.557763\pi$
$151$	−4.43535	−0.360944	−0.180472	+	0.983580i	$0.557763\pi$
$152$	43.3618	3.51711
$153$	0	0
$154$	−20.9705	−1.68985
$155$	5.59179	0.449144
$156$	0	0
$157$	−2.56704	−0.204872	−0.102436	−	0.994740i	$-0.532664\pi$
$157$	−2.56704	−0.204872	−0.102436	+	0.994740i	$0.532664\pi$
$158$	13.7821	1.09644
$159$	0	0
$160$	−17.6136	−1.39247
$161$	−11.4426	−0.901807
$162$	0	0
$163$	−13.1957	−1.03356	−0.516782	−	0.856117i	$-0.672870\pi$
$163$	−13.1957	−1.03356	−0.516782	+	0.856117i	$0.672870\pi$
$164$	57.1909	4.46586
$165$	0	0
$166$	−42.3099	−3.28388
$167$	−2.65817	−0.205695	−0.102848	−	0.994697i	$-0.532795\pi$
$167$	−2.65817	−0.205695	−0.102848	+	0.994697i	$0.532795\pi$
$168$	0	0
$169$	0	0
$170$	14.4940	1.11164
$171$	0	0
$172$	−52.0073	−3.96552
$173$	−16.4058	−1.24731	−0.623655	−	0.781699i	$-0.714353\pi$
$173$	−16.4058	−1.24731	−0.623655	+	0.781699i	$0.714353\pi$
$174$	0	0
$175$	1.69202	0.127905
$176$	60.0200	4.52418
$177$	0	0
$178$	10.3666	0.777008
$179$	−16.4547	−1.22988	−0.614942	−	0.788572i	$-0.710821\pi$
$179$	−16.4547	−1.22988	−0.614942	+	0.788572i	$0.710821\pi$
$180$	0	0
$181$	−9.46681	−0.703663	−0.351831	−	0.936063i	$-0.614441\pi$
$181$	−9.46681	−0.703663	−0.351831	+	0.936063i	$0.614441\pi$
$182$	0	0
$183$	0	0
$184$	59.1124	4.35783
$185$	3.86294	0.284009
$186$	0	0
$187$	−24.7875	−1.81264
$188$	−25.1957	−1.83758
$189$	0	0
$190$	13.3545	0.968837
$191$	−16.4252	−1.18849	−0.594243	−	0.804286i	$-0.702548\pi$
$191$	−16.4252	−1.18849	−0.594243	+	0.804286i	$0.702548\pi$
$192$	0	0
$193$	1.62565	0.117017	0.0585083	−	0.998287i	$-0.481366\pi$
$193$	1.62565	0.117017	0.0585083	+	0.998287i	$0.481366\pi$
$194$	14.1981	1.01936
$195$	0	0
$196$	−21.7071	−1.55051
$197$	−6.51035	−0.463843	−0.231922	−	0.972734i	$-0.574501\pi$
$197$	−6.51035	−0.463843	−0.231922	+	0.972734i	$0.574501\pi$
$198$	0	0
$199$	11.9119	0.844409	0.422204	−	0.906501i	$-0.361256\pi$
$199$	11.9119	0.844409	0.422204	+	0.906501i	$0.361256\pi$
$200$	−8.74094	−0.618078
$201$	0	0
$202$	11.7759	0.828546
$203$	−6.16421	−0.432643
$204$	0	0
$205$	10.8998	0.761273
$206$	31.7711	2.21360
$207$	0	0
$208$	0	0
$209$	−22.8388	−1.57979
$210$	0	0
$211$	−16.1782	−1.11375	−0.556877	−	0.830595i	$-0.688001\pi$
$211$	−16.1782	−1.11375	−0.556877	+	0.830595i	$0.688001\pi$
$212$	52.4572	3.60278
$213$	0	0
$214$	−41.6450	−2.84680
$215$	−9.91185	−0.675983
$216$	0	0
$217$	9.46144	0.642284
$218$	12.8358	0.869349
$219$	0	0
$220$	24.1564	1.62863
$221$	0	0
$222$	0	0
$223$	−5.23729	−0.350715	−0.175357	−	0.984505i	$-0.556108\pi$
$223$	−5.23729	−0.350715	−0.175357	+	0.984505i	$0.556108\pi$
$224$	−29.8025	−1.99127
$225$	0	0
$226$	−3.44803	−0.229359
$227$	−14.0925	−0.935349	−0.467675	−	0.883901i	$-0.654908\pi$
$227$	−14.0925	−0.935349	−0.467675	+	0.883901i	$0.654908\pi$
$228$	0	0
$229$	22.6679	1.49793	0.748967	−	0.662607i	$-0.230550\pi$
$229$	22.6679	1.49793	0.748967	+	0.662607i	$0.230550\pi$
$230$	18.2054	1.20043
$231$	0	0
$232$	31.8442	2.09067
$233$	−14.7802	−0.968281	−0.484140	−	0.874990i	$-0.660868\pi$
$233$	−14.7802	−0.968281	−0.484140	+	0.874990i	$0.660868\pi$
$234$	0	0
$235$	−4.80194	−0.313244
$236$	−55.6892	−3.62506
$237$	0	0
$238$	24.5241	1.58966
$239$	20.8243	1.34701	0.673506	−	0.739182i	$-0.264788\pi$
$239$	20.8243	1.34701	0.673506	+	0.739182i	$0.264788\pi$
$240$	0	0
$241$	−5.14244	−0.331254	−0.165627	−	0.986188i	$-0.552965\pi$
$241$	−5.14244	−0.331254	−0.165627	+	0.986188i	$0.552965\pi$
$242$	−27.4470	−1.76436
$243$	0	0
$244$	39.0640	2.50082
$245$	−4.13706	−0.264307
$246$	0	0
$247$	0	0
$248$	−48.8775	−3.10373
$249$	0	0
$250$	−2.69202	−0.170258
$251$	−0.269815	−0.0170306	−0.00851528	−	0.999964i	$-0.502711\pi$
$251$	−0.269815	−0.0170306	−0.00851528	+	0.999964i	$0.502711\pi$
$252$	0	0
$253$	−31.1347	−1.95742
$254$	−1.24219	−0.0779422
$255$	0	0
$256$	17.1511	1.07194
$257$	−8.67456	−0.541104	−0.270552	−	0.962705i	$-0.587206\pi$
$257$	−8.67456	−0.541104	−0.270552	+	0.962705i	$0.587206\pi$
$258$	0	0
$259$	6.53617	0.406138
$260$	0	0
$261$	0	0
$262$	8.35749	0.516327
$263$	−22.6950	−1.39943	−0.699717	−	0.714420i	$-0.746691\pi$
$263$	−22.6950	−1.39943	−0.699717	+	0.714420i	$0.746691\pi$
$264$	0	0
$265$	9.99761	0.614148
$266$	22.5961	1.38546
$267$	0	0
$268$	−57.0484	−3.48479
$269$	−2.79656	−0.170509	−0.0852547	−	0.996359i	$-0.527170\pi$
$269$	−2.79656	−0.170509	−0.0852547	+	0.996359i	$0.527170\pi$
$270$	0	0
$271$	1.14914	0.0698056	0.0349028	−	0.999391i	$-0.488888\pi$
$271$	1.14914	0.0698056	0.0349028	+	0.999391i	$0.488888\pi$
$272$	−70.1909	−4.25595
$273$	0	0
$274$	−10.0968	−0.609968
$275$	4.60388	0.277624
$276$	0	0
$277$	2.36121	0.141871	0.0709356	−	0.997481i	$-0.477402\pi$
$277$	2.36121	0.141871	0.0709356	+	0.997481i	$0.477402\pi$
$278$	6.31873	0.378972
$279$	0	0
$280$	−14.7899	−0.883863
$281$	22.8170	1.36115	0.680574	−	0.732679i	$-0.261730\pi$
$281$	22.8170	1.36115	0.680574	+	0.732679i	$0.261730\pi$
$282$	0	0
$283$	22.3351	1.32769	0.663843	−	0.747872i	$-0.268924\pi$
$283$	22.3351	1.32769	0.663843	+	0.747872i	$0.268924\pi$
$284$	26.4407	1.56897
$285$	0	0
$286$	0	0
$287$	18.4426	1.08864
$288$	0	0
$289$	11.9879	0.705172
$290$	9.80731	0.575905
$291$	0	0
$292$	−4.81269	−0.281641
$293$	−24.6474	−1.43992	−0.719959	−	0.694017i	$-0.755839\pi$
$293$	−24.6474	−1.43992	−0.719959	+	0.694017i	$0.755839\pi$
$294$	0	0
$295$	−10.6136	−0.617946
$296$	−33.7657	−1.96259
$297$	0	0
$298$	39.6034	2.29416
$299$	0	0
$300$	0	0
$301$	−16.7711	−0.966668
$302$	11.9401	0.687074
$303$	0	0
$304$	−64.6728	−3.70924
$305$	7.44504	0.426302
$306$	0	0
$307$	6.14675	0.350814	0.175407	−	0.984496i	$-0.443876\pi$
$307$	6.14675	0.350814	0.175407	+	0.984496i	$0.443876\pi$
$308$	40.8732	2.32897
$309$	0	0
$310$	−15.0532	−0.854966
$311$	11.1032	0.629605	0.314803	−	0.949157i	$-0.398062\pi$
$311$	11.1032	0.629605	0.314803	+	0.949157i	$0.398062\pi$
$312$	0	0
$313$	−4.18359	−0.236470	−0.118235	−	0.992986i	$-0.537724\pi$
$313$	−4.18359	−0.236470	−0.118235	+	0.992986i	$0.537724\pi$
$314$	6.91053	0.389984
$315$	0	0
$316$	−26.8625	−1.51113
$317$	−0.631023	−0.0354418	−0.0177209	−	0.999843i	$-0.505641\pi$
$317$	−0.631023	−0.0354418	−0.0177209	+	0.999843i	$0.505641\pi$
$318$	0	0
$319$	−16.7724	−0.939074
$320$	21.3424	1.19308
$321$	0	0
$322$	30.8039	1.71663
$323$	26.7090	1.48613
$324$	0	0
$325$	0	0
$326$	35.5230	1.96744
$327$	0	0
$328$	−95.2742	−5.26064
$329$	−8.12498	−0.447945
$330$	0	0
$331$	18.0653	0.992959	0.496480	−	0.868048i	$-0.334626\pi$
$331$	18.0653	0.992959	0.496480	+	0.868048i	$0.334626\pi$
$332$	82.4656	4.52589
$333$	0	0
$334$	7.15585	0.391551
$335$	−10.8726	−0.594035
$336$	0	0
$337$	−17.0562	−0.929111	−0.464556	−	0.885544i	$-0.653786\pi$
$337$	−17.0562	−0.929111	−0.464556	+	0.885544i	$0.653786\pi$
$338$	0	0
$339$	0	0
$340$	−28.2500	−1.53207
$341$	25.7439	1.39411
$342$	0	0
$343$	−18.8442	−1.01749
$344$	86.6389	4.67126
$345$	0	0
$346$	44.1648	2.37431
$347$	−9.36658	−0.502825	−0.251412	−	0.967880i	$-0.580895\pi$
$347$	−9.36658	−0.502825	−0.251412	+	0.967880i	$0.580895\pi$
$348$	0	0
$349$	7.03684	0.376673	0.188337	−	0.982105i	$-0.439690\pi$
$349$	7.03684	0.376673	0.188337	+	0.982105i	$0.439690\pi$
$350$	−4.55496	−0.243473
$351$	0	0
$352$	−81.0907	−4.32215
$353$	−12.7922	−0.680863	−0.340431	−	0.940269i	$-0.610573\pi$
$353$	−12.7922	−0.680863	−0.340431	+	0.940269i	$0.610573\pi$
$354$	0	0
$355$	5.03923	0.267454
$356$	−20.2054	−1.07088
$357$	0	0
$358$	44.2965	2.34114
$359$	−24.0834	−1.27107	−0.635536	−	0.772072i	$-0.719221\pi$
$359$	−24.0834	−1.27107	−0.635536	+	0.772072i	$0.719221\pi$
$360$	0	0
$361$	5.60925	0.295224
$362$	25.4849	1.33945
$363$	0	0
$364$	0	0
$365$	−0.917231	−0.0480101
$366$	0	0
$367$	11.5821	0.604581	0.302290	−	0.953216i	$-0.402249\pi$
$367$	11.5821	0.604581	0.302290	+	0.953216i	$0.402249\pi$
$368$	−88.1643	−4.59588
$369$	0	0
$370$	−10.3991	−0.540624
$371$	16.9162	0.878244
$372$	0	0
$373$	20.3521	1.05379	0.526896	−	0.849930i	$-0.323356\pi$
$373$	20.3521	1.05379	0.526896	+	0.849930i	$0.323356\pi$
$374$	66.7284	3.45044
$375$	0	0
$376$	41.9734	2.16462
$377$	0	0
$378$	0	0
$379$	−3.42865	−0.176118	−0.0880589	−	0.996115i	$-0.528066\pi$
$379$	−3.42865	−0.176118	−0.0880589	+	0.996115i	$0.528066\pi$
$380$	−26.0291	−1.33526
$381$	0	0
$382$	44.2170	2.26234
$383$	−15.7778	−0.806207	−0.403103	−	0.915154i	$-0.632068\pi$
$383$	−15.7778	−0.806207	−0.403103	+	0.915154i	$0.632068\pi$
$384$	0	0
$385$	7.78986	0.397008
$386$	−4.37627	−0.222746
$387$	0	0
$388$	−27.6732	−1.40490
$389$	36.6939	1.86046	0.930228	−	0.366981i	$-0.119609\pi$
$389$	36.6939	1.86046	0.930228	+	0.366981i	$0.119609\pi$
$390$	0	0
$391$	36.4107	1.84137
$392$	36.1618	1.82645
$393$	0	0
$394$	17.5260	0.882948
$395$	−5.11960	−0.257595
$396$	0	0
$397$	−28.9004	−1.45047	−0.725234	−	0.688503i	$-0.758268\pi$
$397$	−28.9004	−1.45047	−0.725234	+	0.688503i	$0.758268\pi$
$398$	−32.0670	−1.60737
$399$	0	0
$400$	13.0368	0.651842
$401$	24.2620	1.21159	0.605794	−	0.795621i	$-0.292855\pi$
$401$	24.2620	1.21159	0.605794	+	0.795621i	$0.292855\pi$
$402$	0	0
$403$	0	0
$404$	−22.9521	−1.14191
$405$	0	0
$406$	16.5942	0.823556
$407$	17.7845	0.881544
$408$	0	0
$409$	−8.81461	−0.435854	−0.217927	−	0.975965i	$-0.569930\pi$
$409$	−8.81461	−0.435854	−0.217927	+	0.975965i	$0.569930\pi$
$410$	−29.3424	−1.44912
$411$	0	0
$412$	−61.9245	−3.05080
$413$	−17.9584	−0.883674
$414$	0	0
$415$	15.7168	0.771506
$416$	0	0
$417$	0	0
$418$	61.4825	3.00721
$419$	28.6722	1.40073	0.700364	−	0.713786i	$-0.253021\pi$
$419$	28.6722	1.40073	0.700364	+	0.713786i	$0.253021\pi$
$420$	0	0
$421$	−25.4416	−1.23995	−0.619973	−	0.784623i	$-0.712857\pi$
$421$	−25.4416	−1.23995	−0.619973	+	0.784623i	$0.712857\pi$
$422$	43.5521	2.12008
$423$	0	0
$424$	−87.3885	−4.24396
$425$	−5.38404	−0.261164
$426$	0	0
$427$	12.5972	0.609620
$428$	81.1697	3.92349
$429$	0	0
$430$	26.6829	1.28676
$431$	−19.1347	−0.921685	−0.460842	−	0.887482i	$-0.652453\pi$
$431$	−19.1347	−0.921685	−0.460842	+	0.887482i	$0.652453\pi$
$432$	0	0
$433$	33.0954	1.59047	0.795233	−	0.606304i	$-0.207349\pi$
$433$	33.0954	1.59047	0.795233	+	0.606304i	$0.207349\pi$
$434$	−25.4704	−1.22262
$435$	0	0
$436$	−25.0180	−1.19815
$437$	33.5483	1.60483
$438$	0	0
$439$	−18.5797	−0.886761	−0.443381	−	0.896333i	$-0.646221\pi$
$439$	−18.5797	−0.886761	−0.443381	+	0.896333i	$0.646221\pi$
$440$	−40.2422	−1.91847
$441$	0	0
$442$	0	0
$443$	32.3588	1.53741	0.768707	−	0.639601i	$-0.220901\pi$
$443$	32.3588	1.53741	0.768707	+	0.639601i	$0.220901\pi$
$444$	0	0
$445$	−3.85086	−0.182548
$446$	14.0989	0.667602
$447$	0	0
$448$	36.1118	1.70612
$449$	23.5472	1.11126	0.555630	−	0.831429i	$-0.312477\pi$
$449$	23.5472	1.11126	0.555630	+	0.831429i	$0.312477\pi$
$450$	0	0
$451$	50.1812	2.36294
$452$	6.72050	0.316106
$453$	0	0
$454$	37.9372	1.78048
$455$	0	0
$456$	0	0
$457$	−5.50066	−0.257310	−0.128655	−	0.991689i	$-0.541066\pi$
$457$	−5.50066	−0.257310	−0.128655	+	0.991689i	$0.541066\pi$
$458$	−61.0224	−2.85139
$459$	0	0
$460$	−35.4838	−1.65444
$461$	−11.4916	−0.535216	−0.267608	−	0.963528i	$-0.586233\pi$
$461$	−11.4916	−0.535216	−0.267608	+	0.963528i	$0.586233\pi$
$462$	0	0
$463$	27.3056	1.26900	0.634499	−	0.772924i	$-0.281206\pi$
$463$	27.3056	1.26900	0.634499	+	0.772924i	$0.281206\pi$
$464$	−47.4946	−2.20488
$465$	0	0
$466$	39.7885	1.84317
$467$	28.6655	1.32648	0.663240	−	0.748407i	$-0.269181\pi$
$467$	28.6655	1.32648	0.663240	+	0.748407i	$0.269181\pi$
$468$	0	0
$469$	−18.3967	−0.849481
$470$	12.9269	0.596274
$471$	0	0
$472$	92.7725	4.27020
$473$	−45.6329	−2.09820
$474$	0	0
$475$	−4.96077	−0.227616
$476$	−47.7995	−2.19089
$477$	0	0
$478$	−56.0595	−2.56410
$479$	−10.6950	−0.488667	−0.244334	−	0.969691i	$-0.578569\pi$
$479$	−10.6950	−0.488667	−0.244334	+	0.969691i	$0.578569\pi$
$480$	0	0
$481$	0	0
$482$	13.8436	0.630557
$483$	0	0
$484$	53.4965	2.43166
$485$	−5.27413	−0.239486
$486$	0	0
$487$	31.9560	1.44806	0.724032	−	0.689766i	$-0.242287\pi$
$487$	31.9560	1.44806	0.724032	+	0.689766i	$0.242287\pi$
$488$	−65.0767	−2.94588
$489$	0	0
$490$	11.1371	0.503121
$491$	−18.1981	−0.821267	−0.410634	−	0.911800i	$-0.634692\pi$
$491$	−18.1981	−0.821267	−0.410634	+	0.911800i	$0.634692\pi$
$492$	0	0
$493$	19.6146	0.883398
$494$	0	0
$495$	0	0
$496$	72.8993	3.27328
$497$	8.52648	0.382465
$498$	0	0
$499$	17.3840	0.778217	0.389108	−	0.921192i	$-0.372783\pi$
$499$	17.3840	0.778217	0.389108	+	0.921192i	$0.372783\pi$
$500$	5.24698	0.234652
$501$	0	0
$502$	0.726347	0.0324185
$503$	−18.5133	−0.825469	−0.412734	−	0.910851i	$-0.635426\pi$
$503$	−18.5133	−0.825469	−0.412734	+	0.910851i	$0.635426\pi$
$504$	0	0
$505$	−4.37435	−0.194656
$506$	83.8152	3.72604
$507$	0	0
$508$	2.42114	0.107421
$509$	−10.9618	−0.485875	−0.242937	−	0.970042i	$-0.578111\pi$
$509$	−10.9618	−0.485875	−0.242937	+	0.970042i	$0.578111\pi$
$510$	0	0
$511$	−1.55197	−0.0686553
$512$	−1.71678	−0.0758715
$513$	0	0
$514$	23.3521	1.03002
$515$	−11.8019	−0.520056
$516$	0	0
$517$	−22.1075	−0.972288
$518$	−17.5955	−0.773103
$519$	0	0
$520$	0	0
$521$	−21.0881	−0.923888	−0.461944	−	0.886909i	$-0.652848\pi$
$521$	−21.0881	−0.923888	−0.461944	+	0.886909i	$0.652848\pi$
$522$	0	0
$523$	−27.0237	−1.18166	−0.590832	−	0.806795i	$-0.701200\pi$
$523$	−27.0237	−1.18166	−0.590832	+	0.806795i	$0.701200\pi$
$524$	−16.2895	−0.711608
$525$	0	0
$526$	61.0954	2.66389
$527$	−30.1065	−1.31146
$528$	0	0
$529$	22.7342	0.988445
$530$	−26.9138	−1.16906
$531$	0	0
$532$	−44.0417	−1.90945
$533$	0	0
$534$	0	0
$535$	15.4698	0.668818
$536$	95.0370	4.10497
$537$	0	0
$538$	7.52840	0.324572
$539$	−19.0465	−0.820392
$540$	0	0
$541$	−40.7332	−1.75126	−0.875628	−	0.482986i	$-0.839552\pi$
$541$	−40.7332	−1.75126	−0.875628	+	0.482986i	$0.839552\pi$
$542$	−3.09352	−0.132878
$543$	0	0
$544$	94.8322	4.06590
$545$	−4.76809	−0.204242
$546$	0	0
$547$	−17.1943	−0.735177	−0.367588	−	0.929989i	$-0.619816\pi$
$547$	−17.1943	−0.735177	−0.367588	+	0.929989i	$0.619816\pi$
$548$	19.6795	0.840665
$549$	0	0
$550$	−12.3937	−0.528470
$551$	18.0726	0.769919
$552$	0	0
$553$	−8.66248	−0.368366
$554$	−6.35642	−0.270058
$555$	0	0
$556$	−12.3157	−0.522304
$557$	14.6450	0.620530	0.310265	−	0.950650i	$-0.399582\pi$
$557$	14.6450	0.620530	0.310265	+	0.950650i	$0.399582\pi$
$558$	0	0
$559$	0	0
$560$	22.0586	0.932146
$561$	0	0
$562$	−61.4239	−2.59101
$563$	−13.9989	−0.589985	−0.294992	−	0.955500i	$-0.595317\pi$
$563$	−13.9989	−0.589985	−0.294992	+	0.955500i	$0.595317\pi$
$564$	0	0
$565$	1.28083	0.0538850
$566$	−60.1266	−2.52731
$567$	0	0
$568$	−44.0476	−1.84820
$569$	−37.6340	−1.57770	−0.788850	−	0.614586i	$-0.789323\pi$
$569$	−37.6340	−1.57770	−0.788850	+	0.614586i	$0.789323\pi$
$570$	0	0
$571$	4.93661	0.206591	0.103295	−	0.994651i	$-0.467061\pi$
$571$	4.93661	0.206591	0.103295	+	0.994651i	$0.467061\pi$
$572$	0	0
$573$	0	0
$574$	−49.6480	−2.07227
$575$	−6.76271	−0.282024
$576$	0	0
$577$	10.0368	0.417839	0.208919	−	0.977933i	$-0.433005\pi$
$577$	10.0368	0.417839	0.208919	+	0.977933i	$0.433005\pi$
$578$	−32.2717	−1.34233
$579$	0	0
$580$	−19.1153	−0.793719
$581$	26.5931	1.10327
$582$	0	0
$583$	46.0277	1.90627
$584$	8.01746	0.331765
$585$	0	0
$586$	66.3514	2.74095
$587$	43.3903	1.79091	0.895454	−	0.445154i	$-0.146851\pi$
$587$	43.3903	1.79091	0.895454	+	0.445154i	$0.146851\pi$
$588$	0	0
$589$	−27.7396	−1.14299
$590$	28.5719	1.17629
$591$	0	0
$592$	50.3605	2.06980
$593$	−36.4534	−1.49696	−0.748481	−	0.663156i	$-0.769216\pi$
$593$	−36.4534	−1.49696	−0.748481	+	0.663156i	$0.769216\pi$
$594$	0	0
$595$	−9.10992	−0.373470
$596$	−77.1904	−3.16184
$597$	0	0
$598$	0	0
$599$	−9.77777	−0.399509	−0.199755	−	0.979846i	$-0.564014\pi$
$599$	−9.77777	−0.399509	−0.199755	+	0.979846i	$0.564014\pi$
$600$	0	0
$601$	26.2373	1.07024	0.535121	−	0.844776i	$-0.320266\pi$
$601$	26.2373	1.07024	0.535121	+	0.844776i	$0.320266\pi$
$602$	45.1481	1.84010
$603$	0	0
$604$	−23.2722	−0.946933
$605$	10.1957	0.414513
$606$	0	0
$607$	−40.2409	−1.63333	−0.816663	−	0.577115i	$-0.804179\pi$
$607$	−40.2409	−1.63333	−0.816663	+	0.577115i	$0.804179\pi$
$608$	87.3769	3.54360
$609$	0	0
$610$	−20.0422	−0.811485
$611$	0	0
$612$	0	0
$613$	−35.8780	−1.44910	−0.724549	−	0.689223i	$-0.757952\pi$
$613$	−35.8780	−1.44910	−0.724549	+	0.689223i	$0.757952\pi$
$614$	−16.5472	−0.667790
$615$	0	0
$616$	−68.0907	−2.74345
$617$	−17.7144	−0.713154	−0.356577	−	0.934266i	$-0.616056\pi$
$617$	−17.7144	−0.713154	−0.356577	+	0.934266i	$0.616056\pi$
$618$	0	0
$619$	−34.7948	−1.39852	−0.699260	−	0.714868i	$-0.746487\pi$
$619$	−34.7948	−1.39852	−0.699260	+	0.714868i	$0.746487\pi$
$620$	29.3400	1.17832
$621$	0	0
$622$	−29.8901	−1.19848
$623$	−6.51573	−0.261047
$624$	0	0
$625$	1.00000	0.0400000
$626$	11.2623	0.450132
$627$	0	0
$628$	−13.4692	−0.537480
$629$	−20.7982	−0.829279
$630$	0	0
$631$	40.9855	1.63161	0.815804	−	0.578328i	$-0.196295\pi$
$631$	40.9855	1.63161	0.815804	+	0.578328i	$0.196295\pi$
$632$	44.7502	1.78007
$633$	0	0
$634$	1.69873	0.0674650
$635$	0.461435	0.0183115
$636$	0	0
$637$	0	0
$638$	45.1517	1.78757
$639$	0	0
$640$	−22.2271	−0.878604
$641$	36.8049	1.45371	0.726854	−	0.686793i	$-0.240982\pi$
$641$	36.8049	1.45371	0.726854	+	0.686793i	$0.240982\pi$
$642$	0	0
$643$	4.28621	0.169032	0.0845158	−	0.996422i	$-0.473066\pi$
$643$	4.28621	0.169032	0.0845158	+	0.996422i	$0.473066\pi$
$644$	−60.0393	−2.36588
$645$	0	0
$646$	−71.9012	−2.82892
$647$	12.0750	0.474717	0.237359	−	0.971422i	$-0.423718\pi$
$647$	12.0750	0.474717	0.237359	+	0.971422i	$0.423718\pi$
$648$	0	0
$649$	−48.8635	−1.91806
$650$	0	0
$651$	0	0
$652$	−69.2374	−2.71155
$653$	−17.7084	−0.692984	−0.346492	−	0.938053i	$-0.612627\pi$
$653$	−17.7084	−0.692984	−0.346492	+	0.938053i	$0.612627\pi$
$654$	0	0
$655$	−3.10454	−0.121304
$656$	142.099	5.54802
$657$	0	0
$658$	21.8726	0.852684
$659$	−28.7222	−1.11886	−0.559428	−	0.828879i	$-0.688979\pi$
$659$	−28.7222	−1.11886	−0.559428	+	0.828879i	$0.688979\pi$
$660$	0	0
$661$	38.1215	1.48276	0.741378	−	0.671088i	$-0.234173\pi$
$661$	38.1215	1.48276	0.741378	+	0.671088i	$0.234173\pi$
$662$	−48.6322	−1.89014
$663$	0	0
$664$	−137.379	−5.33135
$665$	−8.39373	−0.325495
$666$	0	0
$667$	24.6373	0.953958
$668$	−13.9474	−0.539640
$669$	0	0
$670$	29.2693	1.13077
$671$	34.2760	1.32321
$672$	0	0
$673$	30.3159	1.16859	0.584295	−	0.811541i	$-0.301371\pi$
$673$	30.3159	1.16859	0.584295	+	0.811541i	$0.301371\pi$
$674$	45.9157	1.76861
$675$	0	0
$676$	0	0
$677$	6.45042	0.247910	0.123955	−	0.992288i	$-0.460442\pi$
$677$	6.45042	0.247910	0.123955	+	0.992288i	$0.460442\pi$
$678$	0	0
$679$	−8.92394	−0.342469
$680$	47.0616	1.80473
$681$	0	0
$682$	−69.3032	−2.65376
$683$	−5.29590	−0.202642	−0.101321	−	0.994854i	$-0.532307\pi$
$683$	−5.29590	−0.202642	−0.101321	+	0.994854i	$0.532307\pi$
$684$	0	0
$685$	3.75063	0.143304
$686$	50.7289	1.93684
$687$	0	0
$688$	−129.219	−4.92644
$689$	0	0
$690$	0	0
$691$	2.84117	0.108083	0.0540415	−	0.998539i	$-0.482790\pi$
$691$	2.84117	0.108083	0.0540415	+	0.998539i	$0.482790\pi$
$692$	−86.0810	−3.27231
$693$	0	0
$694$	25.2150	0.957150
$695$	−2.34721	−0.0890346
$696$	0	0
$697$	−58.6848	−2.22285
$698$	−18.9433	−0.717015
$699$	0	0
$700$	8.87800	0.335557
$701$	−47.9415	−1.81073	−0.905363	−	0.424639i	$-0.860401\pi$
$701$	−47.9415	−1.81073	−0.905363	+	0.424639i	$0.860401\pi$
$702$	0	0
$703$	−19.1631	−0.722752
$704$	98.2579	3.70323
$705$	0	0
$706$	34.4370	1.29605
$707$	−7.40150	−0.278362
$708$	0	0
$709$	21.5786	0.810403	0.405202	−	0.914227i	$-0.367201\pi$
$709$	21.5786	0.810403	0.405202	+	0.914227i	$0.367201\pi$
$710$	−13.5657	−0.509112
$711$	0	0
$712$	33.6601	1.26147
$713$	−37.8157	−1.41621
$714$	0	0
$715$	0	0
$716$	−86.3376	−3.22659
$717$	0	0
$718$	64.8329	2.41954
$719$	−15.2597	−0.569089	−0.284544	−	0.958663i	$-0.591842\pi$
$719$	−15.2597	−0.569089	−0.284544	+	0.958663i	$0.591842\pi$
$720$	0	0
$721$	−19.9691	−0.743689
$722$	−15.1002	−0.561972
$723$	0	0
$724$	−49.6722	−1.84605
$725$	−3.64310	−0.135301
$726$	0	0
$727$	−6.21446	−0.230481	−0.115241	−	0.993338i	$-0.536764\pi$
$727$	−6.21446	−0.230481	−0.115241	+	0.993338i	$0.536764\pi$
$728$	0	0
$729$	0	0
$730$	2.46921	0.0913894
$731$	53.3658	1.97381
$732$	0	0
$733$	−21.9191	−0.809602	−0.404801	−	0.914405i	$-0.632659\pi$
$733$	−21.9191	−0.809602	−0.404801	+	0.914405i	$0.632659\pi$
$734$	−31.1793	−1.15085
$735$	0	0
$736$	119.115	4.39065
$737$	−50.0562	−1.84384
$738$	0	0
$739$	−0.853248	−0.0313872	−0.0156936	−	0.999877i	$-0.504996\pi$
$739$	−0.853248	−0.0313872	−0.0156936	+	0.999877i	$0.504996\pi$
$740$	20.2687	0.745094
$741$	0	0
$742$	−45.5387	−1.67178
$743$	−12.0670	−0.442694	−0.221347	−	0.975195i	$-0.571045\pi$
$743$	−12.0670	−0.442694	−0.221347	+	0.975195i	$0.571045\pi$
$744$	0	0
$745$	−14.7114	−0.538984
$746$	−54.7883	−2.00594
$747$	0	0
$748$	−130.059	−4.75544
$749$	26.1752	0.956422
$750$	0	0
$751$	−36.9976	−1.35006	−0.675031	−	0.737789i	$-0.735870\pi$
$751$	−36.9976	−1.35006	−0.675031	+	0.737789i	$0.735870\pi$
$752$	−62.6021	−2.28286
$753$	0	0
$754$	0	0
$755$	−4.43535	−0.161419
$756$	0	0
$757$	23.3769	0.849647	0.424823	−	0.905276i	$-0.360336\pi$
$757$	23.3769	0.849647	0.424823	+	0.905276i	$0.360336\pi$
$758$	9.23000	0.335249
$759$	0	0
$760$	43.3618	1.57290
$761$	−41.8374	−1.51661	−0.758303	−	0.651902i	$-0.773971\pi$
$761$	−41.8374	−1.51661	−0.758303	+	0.651902i	$0.773971\pi$
$762$	0	0
$763$	−8.06770	−0.292070
$764$	−86.1826	−3.11798
$765$	0	0
$766$	42.4741	1.53465
$767$	0	0
$768$	0	0
$769$	41.2107	1.48610	0.743049	−	0.669238i	$-0.233379\pi$
$769$	41.2107	1.48610	0.743049	+	0.669238i	$0.233379\pi$
$770$	−20.9705	−0.755723
$771$	0	0
$772$	8.52973	0.306992
$773$	20.9995	0.755300	0.377650	−	0.925948i	$-0.376732\pi$
$773$	20.9995	0.755300	0.377650	+	0.925948i	$0.376732\pi$
$774$	0	0
$775$	5.59179	0.200863
$776$	46.1008	1.65492
$777$	0	0
$778$	−98.7809	−3.54147
$779$	−54.0713	−1.93730
$780$	0	0
$781$	23.2000	0.830161
$782$	−98.0184	−3.50513
$783$	0	0
$784$	−53.9342	−1.92622
$785$	−2.56704	−0.0916216
$786$	0	0
$787$	−27.5996	−0.983818	−0.491909	−	0.870647i	$-0.663701\pi$
$787$	−27.5996	−0.983818	−0.491909	+	0.870647i	$0.663701\pi$
$788$	−34.1597	−1.21689
$789$	0	0
$790$	13.7821	0.490345
$791$	2.16719	0.0770566
$792$	0	0
$793$	0	0
$794$	77.8004	2.76103
$795$	0	0
$796$	62.5013	2.21530
$797$	19.5483	0.692435	0.346217	−	0.938154i	$-0.387466\pi$
$797$	19.5483	0.692435	0.346217	+	0.938154i	$0.387466\pi$
$798$	0	0
$799$	25.8538	0.914643
$800$	−17.6136	−0.622734
$801$	0	0
$802$	−65.3139	−2.30632
$803$	−4.22282	−0.149020
$804$	0	0
$805$	−11.4426	−0.403300
$806$	0	0
$807$	0	0
$808$	38.2360	1.34514
$809$	−9.87933	−0.347339	−0.173669	−	0.984804i	$-0.555562\pi$
$809$	−9.87933	−0.347339	−0.173669	+	0.984804i	$0.555562\pi$
$810$	0	0
$811$	−9.01938	−0.316713	−0.158357	−	0.987382i	$-0.550620\pi$
$811$	−9.01938	−0.316713	−0.158357	+	0.987382i	$0.550620\pi$
$812$	−32.3435	−1.13503
$813$	0	0
$814$	−47.8762	−1.67806
$815$	−13.1957	−0.462224
$816$	0	0
$817$	49.1704	1.72026
$818$	23.7291	0.829669
$819$	0	0
$820$	57.1909	1.99719
$821$	−11.3357	−0.395619	−0.197810	−	0.980240i	$-0.563383\pi$
$821$	−11.3357	−0.395619	−0.197810	+	0.980240i	$0.563383\pi$
$822$	0	0
$823$	−15.5840	−0.543225	−0.271612	−	0.962407i	$-0.587557\pi$
$823$	−15.5840	−0.543225	−0.271612	+	0.962407i	$0.587557\pi$
$824$	103.160	3.59375
$825$	0	0
$826$	48.3443	1.68211
$827$	2.33108	0.0810595	0.0405297	−	0.999178i	$-0.487095\pi$
$827$	2.33108	0.0810595	0.0405297	+	0.999178i	$0.487095\pi$
$828$	0	0
$829$	−40.6467	−1.41172	−0.705859	−	0.708353i	$-0.749439\pi$
$829$	−40.6467	−1.41172	−0.705859	+	0.708353i	$0.749439\pi$
$830$	−42.3099	−1.46860
$831$	0	0
$832$	0	0
$833$	22.2741	0.771753
$834$	0	0
$835$	−2.65817	−0.0919898
$836$	−119.835	−4.14457
$837$	0	0
$838$	−77.1861	−2.66635
$839$	−31.2529	−1.07897	−0.539486	−	0.841995i	$-0.681381\pi$
$839$	−31.2529	−1.07897	−0.539486	+	0.841995i	$0.681381\pi$
$840$	0	0
$841$	−15.7278	−0.542338
$842$	68.4893	2.36030
$843$	0	0
$844$	−84.8867	−2.92192
$845$	0	0
$846$	0	0
$847$	17.2513	0.592761
$848$	130.337	4.47580
$849$	0	0
$850$	14.4940	0.497139
$851$	−26.1239	−0.895516
$852$	0	0
$853$	41.4034	1.41763	0.708813	−	0.705396i	$-0.249231\pi$
$853$	41.4034	1.41763	0.708813	+	0.705396i	$0.249231\pi$
$854$	−33.9119	−1.16044
$855$	0	0
$856$	−135.221	−4.62174
$857$	−43.7168	−1.49334	−0.746668	−	0.665197i	$-0.768348\pi$
$857$	−43.7168	−1.49334	−0.746668	+	0.665197i	$0.768348\pi$
$858$	0	0
$859$	47.8112	1.63130	0.815648	−	0.578549i	$-0.196381\pi$
$859$	47.8112	1.63130	0.815648	+	0.578549i	$0.196381\pi$
$860$	−52.0073	−1.77343
$861$	0	0
$862$	51.5109	1.75447
$863$	36.7590	1.25129	0.625645	−	0.780108i	$-0.284836\pi$
$863$	36.7590	1.25129	0.625645	+	0.780108i	$0.284836\pi$
$864$	0	0
$865$	−16.4058	−0.557814
$866$	−89.0936	−3.02753
$867$	0	0
$868$	49.6440	1.68503
$869$	−23.5700	−0.799558
$870$	0	0
$871$	0	0
$872$	41.6775	1.41138
$873$	0	0
$874$	−90.3126	−3.05487
$875$	1.69202	0.0572008
$876$	0	0
$877$	−17.1699	−0.579785	−0.289892	−	0.957059i	$-0.593619\pi$
$877$	−17.1699	−0.579785	−0.289892	+	0.957059i	$0.593619\pi$
$878$	50.0170	1.68799
$879$	0	0
$880$	60.0200	2.02327
$881$	−1.39553	−0.0470167	−0.0235084	−	0.999724i	$-0.507484\pi$
$881$	−1.39553	−0.0470167	−0.0235084	+	0.999724i	$0.507484\pi$
$882$	0	0
$883$	38.8374	1.30698	0.653492	−	0.756933i	$-0.273303\pi$
$883$	38.8374	1.30698	0.653492	+	0.756933i	$0.273303\pi$
$884$	0	0
$885$	0	0
$886$	−87.1106	−2.92654
$887$	−5.84654	−0.196308	−0.0981539	−	0.995171i	$-0.531294\pi$
$887$	−5.84654	−0.196308	−0.0981539	+	0.995171i	$0.531294\pi$
$888$	0	0
$889$	0.780758	0.0261858
$890$	10.3666	0.347489
$891$	0	0
$892$	−27.4800	−0.920097
$893$	23.8213	0.797150
$894$	0	0
$895$	−16.4547	−0.550021
$896$	−37.6088	−1.25642
$897$	0	0
$898$	−63.3895	−2.11534
$899$	−20.3715	−0.679427
$900$	0	0
$901$	−53.8275	−1.79326
$902$	−135.089	−4.49797
$903$	0	0
$904$	−11.1957	−0.372362
$905$	−9.46681	−0.314687
$906$	0	0
$907$	−40.1159	−1.33203	−0.666013	−	0.745940i	$-0.732000\pi$
$907$	−40.1159	−1.33203	−0.666013	+	0.745940i	$0.732000\pi$
$908$	−73.9428	−2.45388
$909$	0	0
$910$	0	0
$911$	13.5536	0.449052	0.224526	−	0.974468i	$-0.427917\pi$
$911$	13.5536	0.449052	0.224526	+	0.974468i	$0.427917\pi$
$912$	0	0
$913$	72.3581	2.39470
$914$	14.8079	0.489802
$915$	0	0
$916$	118.938	3.92982
$917$	−5.25295	−0.173468
$918$	0	0
$919$	−40.6152	−1.33977	−0.669886	−	0.742464i	$-0.733657\pi$
$919$	−40.6152	−1.33977	−0.669886	+	0.742464i	$0.733657\pi$
$920$	59.1124	1.94888
$921$	0	0
$922$	30.9355	1.01881
$923$	0	0
$924$	0	0
$925$	3.86294	0.127013
$926$	−73.5072	−2.41560
$927$	0	0
$928$	64.1680	2.10642
$929$	−8.96508	−0.294135	−0.147067	−	0.989126i	$-0.546983\pi$
$929$	−8.96508	−0.294135	−0.147067	+	0.989126i	$0.546983\pi$
$930$	0	0
$931$	20.5230	0.672615
$932$	−77.5512	−2.54028
$933$	0	0
$934$	−77.1680	−2.52502
$935$	−24.7875	−0.810637
$936$	0	0
$937$	6.65386	0.217372	0.108686	−	0.994076i	$-0.465336\pi$
$937$	6.65386	0.217372	0.108686	+	0.994076i	$0.465336\pi$
$938$	49.5244	1.61703
$939$	0	0
$940$	−25.1957	−0.821792
$941$	−28.5754	−0.931531	−0.465766	−	0.884908i	$-0.654221\pi$
$941$	−28.5754	−0.931531	−0.465766	+	0.884908i	$0.654221\pi$
$942$	0	0
$943$	−73.7120	−2.40039
$944$	−138.367	−4.50347
$945$	0	0
$946$	122.845	3.99403
$947$	22.1758	0.720617	0.360309	−	0.932833i	$-0.382671\pi$
$947$	22.1758	0.720617	0.360309	+	0.932833i	$0.382671\pi$
$948$	0	0
$949$	0	0
$950$	13.3545	0.433277
$951$	0	0
$952$	79.6292	2.58080
$953$	21.4282	0.694127	0.347063	−	0.937842i	$-0.387179\pi$
$953$	21.4282	0.694127	0.347063	+	0.937842i	$0.387179\pi$
$954$	0	0
$955$	−16.4252	−0.531507
$956$	109.265	3.53387
$957$	0	0
$958$	28.7912	0.930201
$959$	6.34614	0.204928
$960$	0	0
$961$	0.268159	0.00865029
$962$	0	0
$963$	0	0
$964$	−26.9823	−0.869041
$965$	1.62565	0.0523314
$966$	0	0
$967$	5.26768	0.169397	0.0846987	−	0.996407i	$-0.473007\pi$
$967$	5.26768	0.169397	0.0846987	+	0.996407i	$0.473007\pi$
$968$	−89.1197	−2.86442
$969$	0	0
$970$	14.1981	0.455872
$971$	33.0525	1.06070	0.530352	−	0.847777i	$-0.322060\pi$
$971$	33.0525	1.06070	0.530352	+	0.847777i	$0.322060\pi$
$972$	0	0
$973$	−3.97152	−0.127321
$974$	−86.0262	−2.75646
$975$	0	0
$976$	97.0598	3.10681
$977$	9.16480	0.293208	0.146604	−	0.989195i	$-0.453166\pi$
$977$	9.16480	0.293208	0.146604	+	0.989195i	$0.453166\pi$
$978$	0	0
$979$	−17.7289	−0.566617
$980$	−21.7071	−0.693407
$981$	0	0
$982$	48.9896	1.56332
$983$	0.202374	0.00645473	0.00322737	−	0.999995i	$-0.498973\pi$
$983$	0.202374	0.00645473	0.00322737	+	0.999995i	$0.498973\pi$
$984$	0	0
$985$	−6.51035	−0.207437
$986$	−52.8030	−1.68159
$987$	0	0
$988$	0	0
$989$	67.0310	2.13146
$990$	0	0
$991$	−58.7743	−1.86703	−0.933514	−	0.358541i	$-0.883274\pi$
$991$	−58.7743	−1.86703	−0.933514	+	0.358541i	$0.883274\pi$
$992$	−98.4914	−3.12711
$993$	0	0
$994$	−22.9535	−0.728040
$995$	11.9119	0.377631
$996$	0	0
$997$	43.9396	1.39158	0.695790	−	0.718245i	$-0.255054\pi$
$997$	43.9396	1.39158	0.695790	+	0.718245i	$0.255054\pi$
$998$	−46.7982	−1.48137
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7605.2.a.bm.1.1		3
3.2	odd	2		2535.2.a.bh.1.3	yes	3
13.12	even	2		7605.2.a.ce.1.3		3
39.38	odd	2		2535.2.a.t.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2535.2.a.t.1.1	✓	3	39.38	odd	2
2535.2.a.bh.1.3	yes	3	3.2	odd	2
7605.2.a.bm.1.1		3	1.1	even	1	trivial
7605.2.a.ce.1.3		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)

\(f(q)\)	\(=\)	\(q-2.69202 q^{2} +5.24698 q^{4} +1.00000 q^{5} +1.69202 q^{7} -8.74094 q^{8} +O(q^{10})\)	\(q-2.69202 q^{2} +5.24698 q^{4} +1.00000 q^{5} +1.69202 q^{7} -8.74094 q^{8} -2.69202 q^{10} +4.60388 q^{11} -4.55496 q^{14} +13.0368 q^{16} -5.38404 q^{17} -4.96077 q^{19} +5.24698 q^{20} -12.3937 q^{22} -6.76271 q^{23} +1.00000 q^{25} +8.87800 q^{28} -3.64310 q^{29} +5.59179 q^{31} -17.6136 q^{32} +14.4940 q^{34} +1.69202 q^{35} +3.86294 q^{37} +13.3545 q^{38} -8.74094 q^{40} +10.8998 q^{41} -9.91185 q^{43} +24.1564 q^{44} +18.2054 q^{46} -4.80194 q^{47} -4.13706 q^{49} -2.69202 q^{50} +9.99761 q^{53} +4.60388 q^{55} -14.7899 q^{56} +9.80731 q^{58} -10.6136 q^{59} +7.44504 q^{61} -15.0532 q^{62} +21.3424 q^{64} -10.8726 q^{67} -28.2500 q^{68} -4.55496 q^{70} +5.03923 q^{71} -0.917231 q^{73} -10.3991 q^{74} -26.0291 q^{76} +7.78986 q^{77} -5.11960 q^{79} +13.0368 q^{80} -29.3424 q^{82} +15.7168 q^{83} -5.38404 q^{85} +26.6829 q^{86} -40.2422 q^{88} -3.85086 q^{89} -35.4838 q^{92} +12.9269 q^{94} -4.96077 q^{95} -5.27413 q^{97} +11.1371 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 11 q^{4} + 3 q^{5} - 12 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 11 * q^4 + 3 * q^5 - 12 * q^8	\( 3 q - 3 q^{2} + 11 q^{4} + 3 q^{5} - 12 q^{8} - 3 q^{10} + 5 q^{11} - 14 q^{14} + 11 q^{16} - 6 q^{17} - 2 q^{19} + 11 q^{20} - 5 q^{22} - 3 q^{23} + 3 q^{25} + 7 q^{28} - 15 q^{29} - 11 q^{31} - 22 q^{32} + 34 q^{34} + 17 q^{37} - 5 q^{38} - 12 q^{40} + 10 q^{41} - 26 q^{43} + 23 q^{44} - 4 q^{46} - 10 q^{47} - 7 q^{49} - 3 q^{50} - 11 q^{53} + 5 q^{55} - 21 q^{56} + 22 q^{58} - q^{59} + 22 q^{61} - 17 q^{62} - 16 q^{67} - 36 q^{68} - 14 q^{70} + 28 q^{71} + 4 q^{73} + 4 q^{74} - 12 q^{76} + 6 q^{79} + 11 q^{80} - 24 q^{82} - 5 q^{83} - 6 q^{85} + 12 q^{86} - 34 q^{88} + 2 q^{89} - 18 q^{92} + 10 q^{94} - 2 q^{95} - 5 q^{97} + 28 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 11 * q^4 + 3 * q^5 - 12 * q^8 - 3 * q^10 + 5 * q^11 - 14 * q^14 + 11 * q^16 - 6 * q^17 - 2 * q^19 + 11 * q^20 - 5 * q^22 - 3 * q^23 + 3 * q^25 + 7 * q^28 - 15 * q^29 - 11 * q^31 - 22 * q^32 + 34 * q^34 + 17 * q^37 - 5 * q^38 - 12 * q^40 + 10 * q^41 - 26 * q^43 + 23 * q^44 - 4 * q^46 - 10 * q^47 - 7 * q^49 - 3 * q^50 - 11 * q^53 + 5 * q^55 - 21 * q^56 + 22 * q^58 - q^59 + 22 * q^61 - 17 * q^62 - 16 * q^67 - 36 * q^68 - 14 * q^70 + 28 * q^71 + 4 * q^73 + 4 * q^74 - 12 * q^76 + 6 * q^79 + 11 * q^80 - 24 * q^82 - 5 * q^83 - 6 * q^85 + 12 * q^86 - 34 * q^88 + 2 * q^89 - 18 * q^92 + 10 * q^94 - 2 * q^95 - 5 * q^97 + 28 * q^98

Introduction

L-functions

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Database

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