LMFDB - Embedded newform 7605.2.a.bt.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:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 195)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	7605.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.67513 q^{2} +0.806063 q^{4} -1.00000 q^{5} -3.63752 q^{7} +2.00000 q^{8} +O(q^{10})$	$q-1.67513 q^{2} +0.806063 q^{4} -1.00000 q^{5} -3.63752 q^{7} +2.00000 q^{8} +1.67513 q^{10} +1.61213 q^{11} +6.09332 q^{14} -4.96239 q^{16} +3.28726 q^{17} +7.11871 q^{19} -0.806063 q^{20} -2.70052 q^{22} -5.73813 q^{23} +1.00000 q^{25} -2.93207 q^{28} -7.44358 q^{29} +6.76845 q^{31} +4.31265 q^{32} -5.50659 q^{34} +3.63752 q^{35} +11.9248 q^{37} -11.9248 q^{38} -2.00000 q^{40} -5.13093 q^{41} -7.09332 q^{43} +1.29948 q^{44} +9.61213 q^{46} -1.67513 q^{47} +6.23155 q^{49} -1.67513 q^{50} +1.42548 q^{53} -1.61213 q^{55} -7.27504 q^{56} +12.4690 q^{58} +11.5696 q^{59} +6.76845 q^{61} -11.3380 q^{62} +2.70052 q^{64} -5.40597 q^{67} +2.64974 q^{68} -6.09332 q^{70} -3.83146 q^{71} +6.67513 q^{73} -19.9756 q^{74} +5.73813 q^{76} -5.86414 q^{77} +2.45580 q^{79} +4.96239 q^{80} +8.59498 q^{82} -2.96239 q^{83} -3.28726 q^{85} +11.8822 q^{86} +3.22425 q^{88} +1.18172 q^{89} -4.62530 q^{92} +2.80606 q^{94} -7.11871 q^{95} -9.01810 q^{97} -10.4387 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{4} - 3 q^{5} + 5 q^{7} + 6 q^{8}+O(q^{10}) $ 3 * q + 2 * q^4 - 3 * q^5 + 5 * q^7 + 6 * q^8	$ 3 q + 2 q^{4} - 3 q^{5} + 5 q^{7} + 6 q^{8} + 4 q^{11} + 12 q^{14} - 4 q^{16} + 4 q^{17} - 2 q^{20} + 12 q^{22} - 8 q^{23} + 3 q^{25} - 6 q^{29} + 9 q^{31} - 8 q^{32} + 4 q^{34} - 5 q^{35} + 14 q^{37} - 14 q^{38} - 6 q^{40} - 20 q^{41} - 15 q^{43} + 24 q^{44} + 28 q^{46} + 30 q^{49} + 16 q^{53} - 4 q^{55} + 10 q^{56} + 6 q^{58} + 10 q^{59} + 9 q^{61} + 2 q^{62} - 12 q^{64} + 11 q^{67} + 18 q^{68} - 12 q^{70} + 4 q^{71} + 15 q^{73} - 8 q^{74} + 8 q^{76} + 17 q^{79} + 4 q^{80} - 14 q^{82} + 2 q^{83} - 4 q^{85} - 10 q^{86} + 8 q^{88} - 22 q^{89} + 28 q^{92} + 8 q^{94} + q^{97} - 2 q^{98}+O(q^{100}) $ 3 * q + 2 * q^4 - 3 * q^5 + 5 * q^7 + 6 * q^8 + 4 * q^11 + 12 * q^14 - 4 * q^16 + 4 * q^17 - 2 * q^20 + 12 * q^22 - 8 * q^23 + 3 * q^25 - 6 * q^29 + 9 * q^31 - 8 * q^32 + 4 * q^34 - 5 * q^35 + 14 * q^37 - 14 * q^38 - 6 * q^40 - 20 * q^41 - 15 * q^43 + 24 * q^44 + 28 * q^46 + 30 * q^49 + 16 * q^53 - 4 * q^55 + 10 * q^56 + 6 * q^58 + 10 * q^59 + 9 * q^61 + 2 * q^62 - 12 * q^64 + 11 * q^67 + 18 * q^68 - 12 * q^70 + 4 * q^71 + 15 * q^73 - 8 * q^74 + 8 * q^76 + 17 * q^79 + 4 * q^80 - 14 * q^82 + 2 * q^83 - 4 * q^85 - 10 * q^86 + 8 * q^88 - 22 * q^89 + 28 * q^92 + 8 * q^94 + 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 $

$2$	−1.67513	−1.18450	−0.592248	−	0.805756i	$-0.701760\pi$
$2$	−1.67513	−1.18450	−0.592248	+	0.805756i	$0.701760\pi$
$3$	0	0
$3$	0	0
$4$	0.806063	0.403032
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−3.63752	−1.37485	−0.687427	−	0.726254i	$-0.741260\pi$
$7$	−3.63752	−1.37485	−0.687427	+	0.726254i	$0.741260\pi$
$8$	2.00000	0.707107
$9$	0	0
$10$	1.67513	0.529723
$11$	1.61213	0.486075	0.243037	−	0.970017i	$-0.421856\pi$
$11$	1.61213	0.486075	0.243037	+	0.970017i	$0.421856\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	6.09332	1.62851
$15$	0	0
$16$	−4.96239	−1.24060
$17$	3.28726	0.797277	0.398639	−	0.917108i	$-0.369483\pi$
$17$	3.28726	0.797277	0.398639	+	0.917108i	$0.369483\pi$
$18$	0	0
$19$	7.11871	1.63314	0.816572	−	0.577243i	$-0.195871\pi$
$19$	7.11871	1.63314	0.816572	+	0.577243i	$0.195871\pi$
$20$	−0.806063	−0.180241
$21$	0	0
$22$	−2.70052	−0.575754
$23$	−5.73813	−1.19648	−0.598242	−	0.801316i	$-0.704134\pi$
$23$	−5.73813	−1.19648	−0.598242	+	0.801316i	$0.704134\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	−2.93207	−0.554109
$29$	−7.44358	−1.38224	−0.691119	−	0.722741i	$-0.742882\pi$
$29$	−7.44358	−1.38224	−0.691119	+	0.722741i	$0.742882\pi$
$30$	0	0
$31$	6.76845	1.21565	0.607825	−	0.794071i	$-0.292042\pi$
$31$	6.76845	1.21565	0.607825	+	0.794071i	$0.292042\pi$
$32$	4.31265	0.762376
$33$	0	0
$34$	−5.50659	−0.944372
$35$	3.63752	0.614853
$36$	0	0
$37$	11.9248	1.96042	0.980211	−	0.197957i	$-0.0634307\pi$
$37$	11.9248	1.96042	0.980211	+	0.197957i	$0.0634307\pi$
$38$	−11.9248	−1.93445
$39$	0	0
$40$	−2.00000	−0.316228
$41$	−5.13093	−0.801317	−0.400659	−	0.916227i	$-0.631219\pi$
$41$	−5.13093	−0.801317	−0.400659	+	0.916227i	$0.631219\pi$
$42$	0	0
$43$	−7.09332	−1.08172	−0.540861	−	0.841112i	$-0.681901\pi$
$43$	−7.09332	−1.08172	−0.540861	+	0.841112i	$0.681901\pi$
$44$	1.29948	0.195903
$45$	0	0
$46$	9.61213	1.41723
$47$	−1.67513	−0.244343	−0.122171	−	0.992509i	$-0.538986\pi$
$47$	−1.67513	−0.244343	−0.122171	+	0.992509i	$0.538986\pi$
$48$	0	0
$49$	6.23155	0.890221
$50$	−1.67513	−0.236899
$51$	0	0
$52$	0	0
$53$	1.42548	0.195805	0.0979027	−	0.995196i	$-0.468787\pi$
$53$	1.42548	0.195805	0.0979027	+	0.995196i	$0.468787\pi$
$54$	0	0
$55$	−1.61213	−0.217379
$56$	−7.27504	−0.972168
$57$	0	0
$58$	12.4690	1.63726
$59$	11.5696	1.50623	0.753116	−	0.657887i	$-0.228550\pi$
$59$	11.5696	1.50623	0.753116	+	0.657887i	$0.228550\pi$
$60$	0	0
$61$	6.76845	0.866611	0.433306	−	0.901247i	$-0.357347\pi$
$61$	6.76845	0.866611	0.433306	+	0.901247i	$0.357347\pi$
$62$	−11.3380	−1.43993
$63$	0	0
$64$	2.70052	0.337565
$65$	0	0
$66$	0	0
$67$	−5.40597	−0.660445	−0.330222	−	0.943903i	$-0.607124\pi$
$67$	−5.40597	−0.660445	−0.330222	+	0.943903i	$0.607124\pi$
$68$	2.64974	0.321328
$69$	0	0
$70$	−6.09332	−0.728291
$71$	−3.83146	−0.454710	−0.227355	−	0.973812i	$-0.573008\pi$
$71$	−3.83146	−0.454710	−0.227355	+	0.973812i	$0.573008\pi$
$72$	0	0
$73$	6.67513	0.781265	0.390632	−	0.920547i	$-0.372256\pi$
$73$	6.67513	0.781265	0.390632	+	0.920547i	$0.372256\pi$
$74$	−19.9756	−2.32211
$75$	0	0
$76$	5.73813	0.658209
$77$	−5.86414	−0.668281
$78$	0	0
$79$	2.45580	0.276299	0.138150	−	0.990411i	$-0.455885\pi$
$79$	2.45580	0.276299	0.138150	+	0.990411i	$0.455885\pi$
$80$	4.96239	0.554812
$81$	0	0
$82$	8.59498	0.949157
$83$	−2.96239	−0.325164	−0.162582	−	0.986695i	$-0.551982\pi$
$83$	−2.96239	−0.325164	−0.162582	+	0.986695i	$0.551982\pi$
$84$	0	0
$85$	−3.28726	−0.356553
$86$	11.8822	1.28130
$87$	0	0
$88$	3.22425	0.343707
$89$	1.18172	0.125262	0.0626309	−	0.998037i	$-0.480051\pi$
$89$	1.18172	0.125262	0.0626309	+	0.998037i	$0.480051\pi$
$90$	0	0
$91$	0	0
$92$	−4.62530	−0.482221
$93$	0	0
$94$	2.80606	0.289423
$95$	−7.11871	−0.730365
$96$	0	0
$97$	−9.01810	−0.915649	−0.457825	−	0.889043i	$-0.651371\pi$
$97$	−9.01810	−0.915649	−0.457825	+	0.889043i	$0.651371\pi$
$98$	−10.4387	−1.05446
$99$	0	0
$100$	0.806063	0.0806063
$101$	−9.73813	−0.968981	−0.484490	−	0.874797i	$-0.660995\pi$
$101$	−9.73813	−0.968981	−0.484490	+	0.874797i	$0.660995\pi$
$102$	0	0
$103$	−11.4060	−1.12386	−0.561932	−	0.827184i	$-0.689942\pi$
$103$	−11.4060	−1.12386	−0.561932	+	0.827184i	$0.689942\pi$
$104$	0	0
$105$	0	0
$106$	−2.38787	−0.231931
$107$	5.21203	0.503866	0.251933	−	0.967745i	$-0.418934\pi$
$107$	5.21203	0.503866	0.251933	+	0.967745i	$0.418934\pi$
$108$	0	0
$109$	−5.77575	−0.553216	−0.276608	−	0.960983i	$-0.589210\pi$
$109$	−5.77575	−0.553216	−0.276608	+	0.960983i	$0.589210\pi$
$110$	2.70052	0.257485
$111$	0	0
$112$	18.0508	1.70564
$113$	5.35026	0.503310	0.251655	−	0.967817i	$-0.419025\pi$
$113$	5.35026	0.503310	0.251655	+	0.967817i	$0.419025\pi$
$114$	0	0
$115$	5.73813	0.535084
$116$	−6.00000	−0.557086
$117$	0	0
$118$	−19.3806	−1.78413
$119$	−11.9575	−1.09614
$120$	0	0
$121$	−8.40105	−0.763732
$122$	−11.3380	−1.02650
$123$	0	0
$124$	5.45580	0.489945
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−0.801139	−0.0710896	−0.0355448	−	0.999368i	$-0.511317\pi$
$127$	−0.801139	−0.0710896	−0.0355448	+	0.999368i	$0.511317\pi$
$128$	−13.1490	−1.16222
$129$	0	0
$130$	0	0
$131$	11.1065	0.970379	0.485189	−	0.874409i	$-0.338751\pi$
$131$	11.1065	0.970379	0.485189	+	0.874409i	$0.338751\pi$
$132$	0	0
$133$	−25.8945	−2.24533
$134$	9.05571	0.782294
$135$	0	0
$136$	6.57452	0.563760
$137$	−19.9878	−1.70767	−0.853836	−	0.520543i	$-0.825730\pi$
$137$	−19.9878	−1.70767	−0.853836	+	0.520543i	$0.825730\pi$
$138$	0	0
$139$	5.12601	0.434782	0.217391	−	0.976085i	$-0.430245\pi$
$139$	5.12601	0.434782	0.217391	+	0.976085i	$0.430245\pi$
$140$	2.93207	0.247805
$141$	0	0
$142$	6.41819	0.538603
$143$	0	0
$144$	0	0
$145$	7.44358	0.618156
$146$	−11.1817	−0.925406
$147$	0	0
$148$	9.61213	0.790112
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	0	0
$151$	20.2071	1.64443	0.822216	−	0.569176i	$-0.192738\pi$
$151$	20.2071	1.64443	0.822216	+	0.569176i	$0.192738\pi$
$152$	14.2374	1.15481
$153$	0	0
$154$	9.82321	0.791577
$155$	−6.76845	−0.543655
$156$	0	0
$157$	6.83146	0.545210	0.272605	−	0.962126i	$-0.412115\pi$
$157$	6.83146	0.545210	0.272605	+	0.962126i	$0.412115\pi$
$158$	−4.11379	−0.327275
$159$	0	0
$160$	−4.31265	−0.340945
$161$	20.8726	1.64499
$162$	0	0
$163$	1.66196	0.130175	0.0650873	−	0.997880i	$-0.479267\pi$
$163$	1.66196	0.130175	0.0650873	+	0.997880i	$0.479267\pi$
$164$	−4.13586	−0.322956
$165$	0	0
$166$	4.96239	0.385156
$167$	−19.0132	−1.47128	−0.735642	−	0.677371i	$-0.763119\pi$
$167$	−19.0132	−1.47128	−0.735642	+	0.677371i	$0.763119\pi$
$168$	0	0
$169$	0	0
$170$	5.50659	0.422336
$171$	0	0
$172$	−5.71767	−0.435968
$173$	−3.21203	−0.244206	−0.122103	−	0.992517i	$-0.538964\pi$
$173$	−3.21203	−0.244206	−0.122103	+	0.992517i	$0.538964\pi$
$174$	0	0
$175$	−3.63752	−0.274971
$176$	−8.00000	−0.603023
$177$	0	0
$178$	−1.97953	−0.148372
$179$	−26.4568	−1.97747	−0.988735	−	0.149674i	$-0.952178\pi$
$179$	−26.4568	−1.97747	−0.988735	+	0.149674i	$0.952178\pi$
$180$	0	0
$181$	7.38058	0.548594	0.274297	−	0.961645i	$-0.411555\pi$
$181$	7.38058	0.548594	0.274297	+	0.961645i	$0.411555\pi$
$182$	0	0
$183$	0	0
$184$	−11.4763	−0.846042
$185$	−11.9248	−0.876727
$186$	0	0
$187$	5.29948	0.387536
$188$	−1.35026	−0.0984780
$189$	0	0
$190$	11.9248	0.865114
$191$	5.56959	0.403001	0.201501	−	0.979488i	$-0.435418\pi$
$191$	5.56959	0.403001	0.201501	+	0.979488i	$0.435418\pi$
$192$	0	0
$193$	−14.5247	−1.04551	−0.522755	−	0.852483i	$-0.675096\pi$
$193$	−14.5247	−1.04551	−0.522755	+	0.852483i	$0.675096\pi$
$194$	15.1065	1.08458
$195$	0	0
$196$	5.02302	0.358787
$197$	−24.8627	−1.77140	−0.885698	−	0.464262i	$-0.846320\pi$
$197$	−24.8627	−1.77140	−0.885698	+	0.464262i	$0.846320\pi$
$198$	0	0
$199$	−13.2374	−0.938376	−0.469188	−	0.883098i	$-0.655453\pi$
$199$	−13.2374	−0.938376	−0.469188	+	0.883098i	$0.655453\pi$
$200$	2.00000	0.141421
$201$	0	0
$202$	16.3127	1.14775
$203$	27.0762	1.90038
$204$	0	0
$205$	5.13093	0.358360
$206$	19.1065	1.33121
$207$	0	0
$208$	0	0
$209$	11.4763	0.793830
$210$	0	0
$211$	3.18664	0.219378	0.109689	−	0.993966i	$-0.465015\pi$
$211$	3.18664	0.219378	0.109689	+	0.993966i	$0.465015\pi$
$212$	1.14903	0.0789158
$213$	0	0
$214$	−8.73084	−0.596828
$215$	7.09332	0.483760
$216$	0	0
$217$	−24.6204	−1.67134
$218$	9.67513	0.655283
$219$	0	0
$220$	−1.29948	−0.0876107
$221$	0	0
$222$	0	0
$223$	12.9624	0.868026	0.434013	−	0.900907i	$-0.357097\pi$
$223$	12.9624	0.868026	0.434013	+	0.900907i	$0.357097\pi$
$224$	−15.6873	−1.04816
$225$	0	0
$226$	−8.96239	−0.596169
$227$	14.7127	0.976519	0.488260	−	0.872698i	$-0.337632\pi$
$227$	14.7127	0.976519	0.488260	+	0.872698i	$0.337632\pi$
$228$	0	0
$229$	4.49341	0.296933	0.148467	−	0.988917i	$-0.452566\pi$
$229$	4.49341	0.296933	0.148467	+	0.988917i	$0.452566\pi$
$230$	−9.61213	−0.633805
$231$	0	0
$232$	−14.8872	−0.977390
$233$	7.78892	0.510269	0.255135	−	0.966906i	$-0.417880\pi$
$233$	7.78892	0.510269	0.255135	+	0.966906i	$0.417880\pi$
$234$	0	0
$235$	1.67513	0.109273
$236$	9.32582	0.607059
$237$	0	0
$238$	20.0303	1.29837
$239$	1.93937	0.125447	0.0627236	−	0.998031i	$-0.480021\pi$
$239$	1.93937	0.125447	0.0627236	+	0.998031i	$0.480021\pi$
$240$	0	0
$241$	29.6834	1.91207	0.956037	−	0.293245i	$-0.0947352\pi$
$241$	29.6834	1.91207	0.956037	+	0.293245i	$0.0947352\pi$
$242$	14.0729	0.904637
$243$	0	0
$244$	5.45580	0.349272
$245$	−6.23155	−0.398119
$246$	0	0
$247$	0	0
$248$	13.5369	0.859594
$249$	0	0
$250$	1.67513	0.105945
$251$	0.412311	0.0260248	0.0130124	−	0.999915i	$-0.495858\pi$
$251$	0.412311	0.0260248	0.0130124	+	0.999915i	$0.495858\pi$
$252$	0	0
$253$	−9.25060	−0.581580
$254$	1.34201	0.0842054
$255$	0	0
$256$	16.6253	1.03908
$257$	26.5623	1.65691	0.828455	−	0.560055i	$-0.189220\pi$
$257$	26.5623	1.65691	0.828455	+	0.560055i	$0.189220\pi$
$258$	0	0
$259$	−43.3766	−2.69529
$260$	0	0
$261$	0	0
$262$	−18.6048	−1.14941
$263$	19.0762	1.17629	0.588144	−	0.808756i	$-0.299859\pi$
$263$	19.0762	1.17629	0.588144	+	0.808756i	$0.299859\pi$
$264$	0	0
$265$	−1.42548	−0.0875668
$266$	43.3766	2.65959
$267$	0	0
$268$	−4.35756	−0.266180
$269$	−20.3453	−1.24048	−0.620239	−	0.784413i	$-0.712964\pi$
$269$	−20.3453	−1.24048	−0.620239	+	0.784413i	$0.712964\pi$
$270$	0	0
$271$	12.5877	0.764648	0.382324	−	0.924028i	$-0.375124\pi$
$271$	12.5877	0.764648	0.382324	+	0.924028i	$0.375124\pi$
$272$	−16.3127	−0.989100
$273$	0	0
$274$	33.4821	2.02273
$275$	1.61213	0.0972149
$276$	0	0
$277$	7.61213	0.457368	0.228684	−	0.973501i	$-0.426558\pi$
$277$	7.61213	0.457368	0.228684	+	0.973501i	$0.426558\pi$
$278$	−8.58673	−0.514998
$279$	0	0
$280$	7.27504	0.434767
$281$	10.1441	0.605147	0.302573	−	0.953126i	$-0.402154\pi$
$281$	10.1441	0.605147	0.302573	+	0.953126i	$0.402154\pi$
$282$	0	0
$283$	−17.6982	−1.05205	−0.526023	−	0.850470i	$-0.676317\pi$
$283$	−17.6982	−1.05205	−0.526023	+	0.850470i	$0.676317\pi$
$284$	−3.08840	−0.183263
$285$	0	0
$286$	0	0
$287$	18.6639	1.10169
$288$	0	0
$289$	−6.19394	−0.364349
$290$	−12.4690	−0.732203
$291$	0	0
$292$	5.38058	0.314875
$293$	20.5623	1.20126	0.600631	−	0.799526i	$-0.294916\pi$
$293$	20.5623	1.20126	0.600631	+	0.799526i	$0.294916\pi$
$294$	0	0
$295$	−11.5696	−0.673608
$296$	23.8496	1.38623
$297$	0	0
$298$	3.35026	0.194075
$299$	0	0
$300$	0	0
$301$	25.8021	1.48721
$302$	−33.8496	−1.94782
$303$	0	0
$304$	−35.3258	−2.02607
$305$	−6.76845	−0.387560
$306$	0	0
$307$	15.3453	0.875805	0.437902	−	0.899023i	$-0.355722\pi$
$307$	15.3453	0.875805	0.437902	+	0.899023i	$0.355722\pi$
$308$	−4.72687	−0.269338
$309$	0	0
$310$	11.3380	0.643958
$311$	3.56959	0.202413	0.101206	−	0.994865i	$-0.467730\pi$
$311$	3.56959	0.202413	0.101206	+	0.994865i	$0.467730\pi$
$312$	0	0
$313$	15.1441	0.855996	0.427998	−	0.903780i	$-0.359219\pi$
$313$	15.1441	0.855996	0.427998	+	0.903780i	$0.359219\pi$
$314$	−11.4436	−0.645799
$315$	0	0
$316$	1.97953	0.111357
$317$	−28.0870	−1.57752	−0.788761	−	0.614700i	$-0.789277\pi$
$317$	−28.0870	−1.57752	−0.788761	+	0.614700i	$0.789277\pi$
$318$	0	0
$319$	−12.0000	−0.671871
$320$	−2.70052	−0.150964
$321$	0	0
$322$	−34.9643	−1.94848
$323$	23.4010	1.30207
$324$	0	0
$325$	0	0
$326$	−2.78400	−0.154191
$327$	0	0
$328$	−10.2619	−0.566617
$329$	6.09332	0.335936
$330$	0	0
$331$	12.2750	0.674697	0.337349	−	0.941380i	$-0.390470\pi$
$331$	12.2750	0.674697	0.337349	+	0.941380i	$0.390470\pi$
$332$	−2.38787	−0.131052
$333$	0	0
$334$	31.8496	1.74273
$335$	5.40597	0.295360
$336$	0	0
$337$	−0.931116	−0.0507211	−0.0253606	−	0.999678i	$-0.508073\pi$
$337$	−0.931116	−0.0507211	−0.0253606	+	0.999678i	$0.508073\pi$
$338$	0	0
$339$	0	0
$340$	−2.64974	−0.143702
$341$	10.9116	0.590896
$342$	0	0
$343$	2.79526	0.150930
$344$	−14.1866	−0.764892
$345$	0	0
$346$	5.38058	0.289262
$347$	−33.6483	−1.80634	−0.903168	−	0.429287i	$-0.858765\pi$
$347$	−33.6483	−1.80634	−0.903168	+	0.429287i	$0.858765\pi$
$348$	0	0
$349$	−8.94525	−0.478828	−0.239414	−	0.970918i	$-0.576955\pi$
$349$	−8.94525	−0.478828	−0.239414	+	0.970918i	$0.576955\pi$
$350$	6.09332	0.325702
$351$	0	0
$352$	6.95254	0.370572
$353$	−0.150446	−0.00800741	−0.00400370	−	0.999992i	$-0.501274\pi$
$353$	−0.150446	−0.00800741	−0.00400370	+	0.999992i	$0.501274\pi$
$354$	0	0
$355$	3.83146	0.203353
$356$	0.952539	0.0504845
$357$	0	0
$358$	44.3185	2.34231
$359$	15.8070	0.834263	0.417131	−	0.908846i	$-0.363036\pi$
$359$	15.8070	0.834263	0.417131	+	0.908846i	$0.363036\pi$
$360$	0	0
$361$	31.6761	1.66716
$362$	−12.3634	−0.649808
$363$	0	0
$364$	0	0
$365$	−6.67513	−0.349392
$366$	0	0
$367$	16.8677	0.880484	0.440242	−	0.897879i	$-0.354893\pi$
$367$	16.8677	0.880484	0.440242	+	0.897879i	$0.354893\pi$
$368$	28.4749	1.48435
$369$	0	0
$370$	19.9756	1.03848
$371$	−5.18523	−0.269204
$372$	0	0
$373$	−7.50166	−0.388421	−0.194211	−	0.980960i	$-0.562215\pi$
$373$	−7.50166	−0.388421	−0.194211	+	0.980960i	$0.562215\pi$
$374$	−8.87732	−0.459035
$375$	0	0
$376$	−3.35026	−0.172777
$377$	0	0
$378$	0	0
$379$	−13.5950	−0.698327	−0.349164	−	0.937062i	$-0.613534\pi$
$379$	−13.5950	−0.698327	−0.349164	+	0.937062i	$0.613534\pi$
$380$	−5.73813	−0.294360
$381$	0	0
$382$	−9.32979	−0.477354
$383$	1.22662	0.0626775	0.0313388	−	0.999509i	$-0.490023\pi$
$383$	1.22662	0.0626775	0.0313388	+	0.999509i	$0.490023\pi$
$384$	0	0
$385$	5.86414	0.298864
$386$	24.3307	1.23840
$387$	0	0
$388$	−7.26916	−0.369036
$389$	3.42548	0.173679	0.0868395	−	0.996222i	$-0.472323\pi$
$389$	3.42548	0.173679	0.0868395	+	0.996222i	$0.472323\pi$
$390$	0	0
$391$	−18.8627	−0.953929
$392$	12.4631	0.629481
$393$	0	0
$394$	41.6483	2.09821
$395$	−2.45580	−0.123565
$396$	0	0
$397$	9.66784	0.485215	0.242607	−	0.970125i	$-0.421997\pi$
$397$	9.66784	0.485215	0.242607	+	0.970125i	$0.421997\pi$
$398$	22.1744	1.11150
$399$	0	0
$400$	−4.96239	−0.248119
$401$	28.4387	1.42016	0.710079	−	0.704122i	$-0.248659\pi$
$401$	28.4387	1.42016	0.710079	+	0.704122i	$0.248659\pi$
$402$	0	0
$403$	0	0
$404$	−7.84955	−0.390530
$405$	0	0
$406$	−45.3561	−2.25099
$407$	19.2243	0.952911
$408$	0	0
$409$	16.5310	0.817407	0.408703	−	0.912667i	$-0.365981\pi$
$409$	16.5310	0.817407	0.408703	+	0.912667i	$0.365981\pi$
$410$	−8.59498	−0.424476
$411$	0	0
$412$	−9.19394	−0.452953
$413$	−42.0846	−2.07085
$414$	0	0
$415$	2.96239	0.145418
$416$	0	0
$417$	0	0
$418$	−19.2243	−0.940289
$419$	27.2687	1.33216	0.666082	−	0.745879i	$-0.267970\pi$
$419$	27.2687	1.33216	0.666082	+	0.745879i	$0.267970\pi$
$420$	0	0
$421$	28.3806	1.38318	0.691592	−	0.722288i	$-0.256909\pi$
$421$	28.3806	1.38318	0.691592	+	0.722288i	$0.256909\pi$
$422$	−5.33804	−0.259852
$423$	0	0
$424$	2.85097	0.138455
$425$	3.28726	0.159455
$426$	0	0
$427$	−24.6204	−1.19146
$428$	4.20123	0.203074
$429$	0	0
$430$	−11.8822	−0.573013
$431$	5.68735	0.273950	0.136975	−	0.990575i	$-0.456262\pi$
$431$	5.68735	0.273950	0.136975	+	0.990575i	$0.456262\pi$
$432$	0	0
$433$	−2.28726	−0.109919	−0.0549593	−	0.998489i	$-0.517503\pi$
$433$	−2.28726	−0.109919	−0.0549593	+	0.998489i	$0.517503\pi$
$434$	41.2424	1.97970
$435$	0	0
$436$	−4.65562	−0.222964
$437$	−40.8481	−1.95403
$438$	0	0
$439$	−0.918898	−0.0438566	−0.0219283	−	0.999760i	$-0.506981\pi$
$439$	−0.918898	−0.0438566	−0.0219283	+	0.999760i	$0.506981\pi$
$440$	−3.22425	−0.153710
$441$	0	0
$442$	0	0
$443$	−10.3611	−0.492269	−0.246135	−	0.969236i	$-0.579161\pi$
$443$	−10.3611	−0.492269	−0.246135	+	0.969236i	$0.579161\pi$
$444$	0	0
$445$	−1.18172	−0.0560188
$446$	−21.7137	−1.02817
$447$	0	0
$448$	−9.82321	−0.464103
$449$	24.3634	1.14978	0.574891	−	0.818230i	$-0.305045\pi$
$449$	24.3634	1.14978	0.574891	+	0.818230i	$0.305045\pi$
$450$	0	0
$451$	−8.27171	−0.389500
$452$	4.31265	0.202850
$453$	0	0
$454$	−24.6458	−1.15668
$455$	0	0
$456$	0	0
$457$	27.9805	1.30887	0.654436	−	0.756117i	$-0.272906\pi$
$457$	27.9805	1.30887	0.654436	+	0.756117i	$0.272906\pi$
$458$	−7.52705	−0.351716
$459$	0	0
$460$	4.62530	0.215656
$461$	−13.1211	−0.611110	−0.305555	−	0.952174i	$-0.598842\pi$
$461$	−13.1211	−0.611110	−0.305555	+	0.952174i	$0.598842\pi$
$462$	0	0
$463$	−36.5550	−1.69886	−0.849428	−	0.527705i	$-0.823053\pi$
$463$	−36.5550	−1.69886	−0.849428	+	0.527705i	$0.823053\pi$
$464$	36.9380	1.71480
$465$	0	0
$466$	−13.0475	−0.604412
$467$	−10.4264	−0.482478	−0.241239	−	0.970466i	$-0.577554\pi$
$467$	−10.4264	−0.482478	−0.241239	+	0.970466i	$0.577554\pi$
$468$	0	0
$469$	19.6643	0.908014
$470$	−2.80606	−0.129434
$471$	0	0
$472$	23.1392	1.06507
$473$	−11.4353	−0.525797
$474$	0	0
$475$	7.11871	0.326629
$476$	−9.63847	−0.441779
$477$	0	0
$478$	−3.24869	−0.148592
$479$	0.556417	0.0254233	0.0127117	−	0.999919i	$-0.495954\pi$
$479$	0.556417	0.0254233	0.0127117	+	0.999919i	$0.495954\pi$
$480$	0	0
$481$	0	0
$482$	−49.7235	−2.26485
$483$	0	0
$484$	−6.77178	−0.307808
$485$	9.01810	0.409491
$486$	0	0
$487$	9.30536	0.421666	0.210833	−	0.977522i	$-0.432382\pi$
$487$	9.30536	0.421666	0.210833	+	0.977522i	$0.432382\pi$
$488$	13.5369	0.612787
$489$	0	0
$490$	10.4387	0.471571
$491$	44.1949	1.99449	0.997244	−	0.0741948i	$-0.0236387\pi$
$491$	44.1949	1.99449	0.997244	+	0.0741948i	$0.0236387\pi$
$492$	0	0
$493$	−24.4690	−1.10203
$494$	0	0
$495$	0	0
$496$	−33.5877	−1.50813
$497$	13.9370	0.625160
$498$	0	0
$499$	−25.7539	−1.15290	−0.576451	−	0.817132i	$-0.695563\pi$
$499$	−25.7539	−1.15290	−0.576451	+	0.817132i	$0.695563\pi$
$500$	−0.806063	−0.0360483
$501$	0	0
$502$	−0.690674	−0.0308263
$503$	43.5633	1.94239	0.971195	−	0.238287i	$-0.0765860\pi$
$503$	43.5633	1.94239	0.971195	+	0.238287i	$0.0765860\pi$
$504$	0	0
$505$	9.73813	0.433341
$506$	15.4960	0.688880
$507$	0	0
$508$	−0.645769	−0.0286514
$509$	−4.52373	−0.200511	−0.100255	−	0.994962i	$-0.531966\pi$
$509$	−4.52373	−0.200511	−0.100255	+	0.994962i	$0.531966\pi$
$510$	0	0
$511$	−24.2809	−1.07412
$512$	−1.55149	−0.0685669
$513$	0	0
$514$	−44.4953	−1.96260
$515$	11.4060	0.502607
$516$	0	0
$517$	−2.70052	−0.118769
$518$	72.6615	3.19256
$519$	0	0
$520$	0	0
$521$	−11.1309	−0.487655	−0.243828	−	0.969819i	$-0.578403\pi$
$521$	−11.1309	−0.487655	−0.243828	+	0.969819i	$0.578403\pi$
$522$	0	0
$523$	10.8872	0.476063	0.238031	−	0.971257i	$-0.423498\pi$
$523$	10.8872	0.476063	0.238031	+	0.971257i	$0.423498\pi$
$524$	8.95254	0.391094
$525$	0	0
$526$	−31.9551	−1.39331
$527$	22.2496	0.969210
$528$	0	0
$529$	9.92619	0.431574
$530$	2.38787	0.103723
$531$	0	0
$532$	−20.8726	−0.904941
$533$	0	0
$534$	0	0
$535$	−5.21203	−0.225336
$536$	−10.8119	−0.467005
$537$	0	0
$538$	34.0811	1.46934
$539$	10.0460	0.432714
$540$	0	0
$541$	−19.6761	−0.845941	−0.422971	−	0.906143i	$-0.639013\pi$
$541$	−19.6761	−0.845941	−0.422971	+	0.906143i	$0.639013\pi$
$542$	−21.0860	−0.905722
$543$	0	0
$544$	14.1768	0.607825
$545$	5.77575	0.247406
$546$	0	0
$547$	−0.630225	−0.0269465	−0.0134732	−	0.999909i	$-0.504289\pi$
$547$	−0.630225	−0.0269465	−0.0134732	+	0.999909i	$0.504289\pi$
$548$	−16.1114	−0.688246
$549$	0	0
$550$	−2.70052	−0.115151
$551$	−52.9887	−2.25740
$552$	0	0
$553$	−8.93303	−0.379871
$554$	−12.7513	−0.541751
$555$	0	0
$556$	4.13189	0.175231
$557$	−29.5778	−1.25325	−0.626627	−	0.779320i	$-0.715565\pi$
$557$	−29.5778	−1.25325	−0.626627	+	0.779320i	$0.715565\pi$
$558$	0	0
$559$	0	0
$560$	−18.0508	−0.762785
$561$	0	0
$562$	−16.9927	−0.716794
$563$	−4.44851	−0.187482	−0.0937411	−	0.995597i	$-0.529883\pi$
$563$	−4.44851	−0.187482	−0.0937411	+	0.995597i	$0.529883\pi$
$564$	0	0
$565$	−5.35026	−0.225087
$566$	29.6467	1.24614
$567$	0	0
$568$	−7.66291	−0.321529
$569$	−3.20616	−0.134409	−0.0672045	−	0.997739i	$-0.521408\pi$
$569$	−3.20616	−0.134409	−0.0672045	+	0.997739i	$0.521408\pi$
$570$	0	0
$571$	23.0336	0.963928	0.481964	−	0.876191i	$-0.339924\pi$
$571$	23.0336	0.963928	0.481964	+	0.876191i	$0.339924\pi$
$572$	0	0
$573$	0	0
$574$	−31.2644	−1.30495
$575$	−5.73813	−0.239297
$576$	0	0
$577$	42.8529	1.78399	0.891994	−	0.452047i	$-0.149306\pi$
$577$	42.8529	1.78399	0.891994	+	0.452047i	$0.149306\pi$
$578$	10.3757	0.431570
$579$	0	0
$580$	6.00000	0.249136
$581$	10.7757	0.447053
$582$	0	0
$583$	2.29806	0.0951760
$584$	13.3503	0.552438
$585$	0	0
$586$	−34.4445	−1.42289
$587$	42.7997	1.76653	0.883267	−	0.468871i	$-0.155339\pi$
$587$	42.7997	1.76653	0.883267	+	0.468871i	$0.155339\pi$
$588$	0	0
$589$	48.1827	1.98533
$590$	19.3806	0.797886
$591$	0	0
$592$	−59.1754	−2.43209
$593$	−34.0362	−1.39770	−0.698850	−	0.715269i	$-0.746304\pi$
$593$	−34.0362	−1.39770	−0.698850	+	0.715269i	$0.746304\pi$
$594$	0	0
$595$	11.9575	0.490208
$596$	−1.61213	−0.0660353
$597$	0	0
$598$	0	0
$599$	−9.38295	−0.383377	−0.191688	−	0.981456i	$-0.561396\pi$
$599$	−9.38295	−0.383377	−0.191688	+	0.981456i	$0.561396\pi$
$600$	0	0
$601$	0.282333	0.0115166	0.00575831	−	0.999983i	$-0.498167\pi$
$601$	0.282333	0.0115166	0.00575831	+	0.999983i	$0.498167\pi$
$602$	−43.2219	−1.76159
$603$	0	0
$604$	16.2882	0.662758
$605$	8.40105	0.341551
$606$	0	0
$607$	−40.0059	−1.62379	−0.811894	−	0.583804i	$-0.801563\pi$
$607$	−40.0059	−1.62379	−0.811894	+	0.583804i	$0.801563\pi$
$608$	30.7005	1.24507
$609$	0	0
$610$	11.3380	0.459064
$611$	0	0
$612$	0	0
$613$	−27.3307	−1.10388	−0.551939	−	0.833884i	$-0.686112\pi$
$613$	−27.3307	−1.10388	−0.551939	+	0.833884i	$0.686112\pi$
$614$	−25.7054	−1.03739
$615$	0	0
$616$	−11.7283	−0.472546
$617$	−36.4749	−1.46842	−0.734211	−	0.678921i	$-0.762448\pi$
$617$	−36.4749	−1.46842	−0.734211	+	0.678921i	$0.762448\pi$
$618$	0	0
$619$	28.9610	1.16404	0.582020	−	0.813175i	$-0.302263\pi$
$619$	28.9610	1.16404	0.582020	+	0.813175i	$0.302263\pi$
$620$	−5.45580	−0.219110
$621$	0	0
$622$	−5.97953	−0.239757
$623$	−4.29852	−0.172217
$624$	0	0
$625$	1.00000	0.0400000
$626$	−25.3684	−1.01392
$627$	0	0
$628$	5.50659	0.219737
$629$	39.1998	1.56300
$630$	0	0
$631$	−32.1041	−1.27805	−0.639023	−	0.769188i	$-0.720661\pi$
$631$	−32.1041	−1.27805	−0.639023	+	0.769188i	$0.720661\pi$
$632$	4.91160	0.195373
$633$	0	0
$634$	47.0494	1.86857
$635$	0.801139	0.0317922
$636$	0	0
$637$	0	0
$638$	20.1016	0.795829
$639$	0	0
$640$	13.1490	0.519761
$641$	10.9478	0.432412	0.216206	−	0.976348i	$-0.430632\pi$
$641$	10.9478	0.432412	0.216206	+	0.976348i	$0.430632\pi$
$642$	0	0
$643$	20.3742	0.803482	0.401741	−	0.915753i	$-0.368405\pi$
$643$	20.3742	0.803482	0.401741	+	0.915753i	$0.368405\pi$
$644$	16.8246	0.662983
$645$	0	0
$646$	−39.1998	−1.54230
$647$	19.2896	0.758354	0.379177	−	0.925324i	$-0.376207\pi$
$647$	19.2896	0.758354	0.379177	+	0.925324i	$0.376207\pi$
$648$	0	0
$649$	18.6516	0.732141
$650$	0	0
$651$	0	0
$652$	1.33964	0.0524645
$653$	38.5524	1.50867	0.754337	−	0.656487i	$-0.227959\pi$
$653$	38.5524	1.50867	0.754337	+	0.656487i	$0.227959\pi$
$654$	0	0
$655$	−11.1065	−0.433967
$656$	25.4617	0.994112
$657$	0	0
$658$	−10.2071	−0.397915
$659$	−20.5599	−0.800901	−0.400451	−	0.916318i	$-0.631146\pi$
$659$	−20.5599	−0.800901	−0.400451	+	0.916318i	$0.631146\pi$
$660$	0	0
$661$	45.1622	1.75661	0.878303	−	0.478104i	$-0.158676\pi$
$661$	45.1622	1.75661	0.878303	+	0.478104i	$0.158676\pi$
$662$	−20.5623	−0.799176
$663$	0	0
$664$	−5.92478	−0.229926
$665$	25.8945	1.00414
$666$	0	0
$667$	42.7123	1.65383
$668$	−15.3258	−0.592974
$669$	0	0
$670$	−9.05571	−0.349853
$671$	10.9116	0.421238
$672$	0	0
$673$	−0.987781	−0.0380762	−0.0190381	−	0.999819i	$-0.506060\pi$
$673$	−0.987781	−0.0380762	−0.0190381	+	0.999819i	$0.506060\pi$
$674$	1.55974	0.0600790
$675$	0	0
$676$	0	0
$677$	17.8740	0.686953	0.343477	−	0.939161i	$-0.388395\pi$
$677$	17.8740	0.686953	0.343477	+	0.939161i	$0.388395\pi$
$678$	0	0
$679$	32.8035	1.25888
$680$	−6.57452	−0.252121
$681$	0	0
$682$	−18.2784	−0.699915
$683$	31.3987	1.20144	0.600718	−	0.799461i	$-0.294881\pi$
$683$	31.3987	1.20144	0.600718	+	0.799461i	$0.294881\pi$
$684$	0	0
$685$	19.9878	0.763694
$686$	−4.68243	−0.178776
$687$	0	0
$688$	35.1998	1.34198
$689$	0	0
$690$	0	0
$691$	22.2447	0.846229	0.423115	−	0.906076i	$-0.360937\pi$
$691$	22.2447	0.846229	0.423115	+	0.906076i	$0.360937\pi$
$692$	−2.58910	−0.0984230
$693$	0	0
$694$	56.3653	2.13960
$695$	−5.12601	−0.194441
$696$	0	0
$697$	−16.8667	−0.638872
$698$	14.9845	0.567170
$699$	0	0
$700$	−2.93207	−0.110822
$701$	−4.27645	−0.161519	−0.0807597	−	0.996734i	$-0.525735\pi$
$701$	−4.27645	−0.161519	−0.0807597	+	0.996734i	$0.525735\pi$
$702$	0	0
$703$	84.8891	3.20165
$704$	4.35359	0.164082
$705$	0	0
$706$	0.252016	0.00948475
$707$	35.4227	1.33221
$708$	0	0
$709$	41.8627	1.57219	0.786094	−	0.618107i	$-0.212100\pi$
$709$	41.8627	1.57219	0.786094	+	0.618107i	$0.212100\pi$
$710$	−6.41819	−0.240870
$711$	0	0
$712$	2.36344	0.0885735
$713$	−38.8383	−1.45451
$714$	0	0
$715$	0	0
$716$	−21.3258	−0.796983
$717$	0	0
$718$	−26.4788	−0.988181
$719$	35.9838	1.34197	0.670985	−	0.741471i	$-0.265872\pi$
$719$	35.9838	1.34197	0.670985	+	0.741471i	$0.265872\pi$
$720$	0	0
$721$	41.4894	1.54515
$722$	−53.0616	−1.97475
$723$	0	0
$724$	5.94921	0.221101
$725$	−7.44358	−0.276448
$726$	0	0
$727$	−25.2095	−0.934968	−0.467484	−	0.884002i	$-0.654839\pi$
$727$	−25.2095	−0.934968	−0.467484	+	0.884002i	$0.654839\pi$
$728$	0	0
$729$	0	0
$730$	11.1817	0.413854
$731$	−23.3176	−0.862432
$732$	0	0
$733$	11.9199	0.440270	0.220135	−	0.975469i	$-0.429350\pi$
$733$	11.9199	0.440270	0.220135	+	0.975469i	$0.429350\pi$
$734$	−28.2555	−1.04293
$735$	0	0
$736$	−24.7466	−0.912171
$737$	−8.71511	−0.321025
$738$	0	0
$739$	−19.7842	−0.727773	−0.363886	−	0.931443i	$-0.618550\pi$
$739$	−19.7842	−0.727773	−0.363886	+	0.931443i	$0.618550\pi$
$740$	−9.61213	−0.353349
$741$	0	0
$742$	8.68594	0.318871
$743$	24.2760	0.890600	0.445300	−	0.895381i	$-0.353097\pi$
$743$	24.2760	0.890600	0.445300	+	0.895381i	$0.353097\pi$
$744$	0	0
$745$	2.00000	0.0732743
$746$	12.5663	0.460084
$747$	0	0
$748$	4.27171	0.156189
$749$	−18.9589	−0.692742
$750$	0	0
$751$	18.1563	0.662534	0.331267	−	0.943537i	$-0.392524\pi$
$751$	18.1563	0.662534	0.331267	+	0.943537i	$0.392524\pi$
$752$	8.31265	0.303131
$753$	0	0
$754$	0	0
$755$	−20.2071	−0.735412
$756$	0	0
$757$	−42.8178	−1.55624	−0.778120	−	0.628116i	$-0.783827\pi$
$757$	−42.8178	−1.55624	−0.778120	+	0.628116i	$0.783827\pi$
$758$	22.7734	0.827166
$759$	0	0
$760$	−14.2374	−0.516446
$761$	18.6253	0.675167	0.337583	−	0.941296i	$-0.390391\pi$
$761$	18.6253	0.675167	0.337583	+	0.941296i	$0.390391\pi$
$762$	0	0
$763$	21.0094	0.760591
$764$	4.48944	0.162422
$765$	0	0
$766$	−2.05475	−0.0742413
$767$	0	0
$768$	0	0
$769$	−4.25790	−0.153544	−0.0767718	−	0.997049i	$-0.524461\pi$
$769$	−4.25790	−0.153544	−0.0767718	+	0.997049i	$0.524461\pi$
$770$	−9.82321	−0.354004
$771$	0	0
$772$	−11.7078	−0.421374
$773$	22.3879	0.805236	0.402618	−	0.915368i	$-0.368100\pi$
$773$	22.3879	0.805236	0.402618	+	0.915368i	$0.368100\pi$
$774$	0	0
$775$	6.76845	0.243130
$776$	−18.0362	−0.647462
$777$	0	0
$778$	−5.73813	−0.205722
$779$	−36.5256	−1.30867
$780$	0	0
$781$	−6.17679	−0.221023
$782$	31.5975	1.12993
$783$	0	0
$784$	−30.9234	−1.10441
$785$	−6.83146	−0.243825
$786$	0	0
$787$	−6.92715	−0.246926	−0.123463	−	0.992349i	$-0.539400\pi$
$787$	−6.92715	−0.246926	−0.123463	+	0.992349i	$0.539400\pi$
$788$	−20.0409	−0.713929
$789$	0	0
$790$	4.11379	0.146362
$791$	−19.4617	−0.691978
$792$	0	0
$793$	0	0
$794$	−16.1949	−0.574735
$795$	0	0
$796$	−10.6702	−0.378195
$797$	−3.23647	−0.114642	−0.0573209	−	0.998356i	$-0.518256\pi$
$797$	−3.23647	−0.114642	−0.0573209	+	0.998356i	$0.518256\pi$
$798$	0	0
$799$	−5.50659	−0.194809
$800$	4.31265	0.152475
$801$	0	0
$802$	−47.6385	−1.68217
$803$	10.7612	0.379753
$804$	0	0
$805$	−20.8726	−0.735662
$806$	0	0
$807$	0	0
$808$	−19.4763	−0.685173
$809$	−19.3620	−0.680732	−0.340366	−	0.940293i	$-0.610551\pi$
$809$	−19.3620	−0.680732	−0.340366	+	0.940293i	$0.610551\pi$
$810$	0	0
$811$	17.8510	0.626832	0.313416	−	0.949616i	$-0.398527\pi$
$811$	17.8510	0.626832	0.313416	+	0.949616i	$0.398527\pi$
$812$	21.8251	0.765911
$813$	0	0
$814$	−32.2031	−1.12872
$815$	−1.66196	−0.0582158
$816$	0	0
$817$	−50.4953	−1.76661
$818$	−27.6916	−0.968215
$819$	0	0
$820$	4.13586	0.144430
$821$	52.9643	1.84847	0.924233	−	0.381828i	$-0.124705\pi$
$821$	52.9643	1.84847	0.924233	+	0.381828i	$0.124705\pi$
$822$	0	0
$823$	−1.08840	−0.0379391	−0.0189696	−	0.999820i	$-0.506039\pi$
$823$	−1.08840	−0.0379391	−0.0189696	+	0.999820i	$0.506039\pi$
$824$	−22.8119	−0.794692
$825$	0	0
$826$	70.4972	2.45291
$827$	39.5247	1.37441	0.687204	−	0.726465i	$-0.258838\pi$
$827$	39.5247	1.37441	0.687204	+	0.726465i	$0.258838\pi$
$828$	0	0
$829$	−55.2795	−1.91994	−0.959968	−	0.280109i	$-0.909629\pi$
$829$	−55.2795	−1.91994	−0.959968	+	0.280109i	$0.909629\pi$
$830$	−4.96239	−0.172247
$831$	0	0
$832$	0	0
$833$	20.4847	0.709753
$834$	0	0
$835$	19.0132	0.657978
$836$	9.25060	0.319939
$837$	0	0
$838$	−45.6786	−1.57794
$839$	40.2882	1.39090	0.695452	−	0.718573i	$-0.255204\pi$
$839$	40.2882	1.39090	0.695452	+	0.718573i	$0.255204\pi$
$840$	0	0
$841$	26.4069	0.910584
$842$	−47.5412	−1.63838
$843$	0	0
$844$	2.56864	0.0884161
$845$	0	0
$846$	0	0
$847$	30.5590	1.05002
$848$	−7.07381	−0.242916
$849$	0	0
$850$	−5.50659	−0.188874
$851$	−68.4260	−2.34561
$852$	0	0
$853$	14.9370	0.511433	0.255717	−	0.966752i	$-0.417689\pi$
$853$	14.9370	0.511433	0.255717	+	0.966752i	$0.417689\pi$
$854$	41.2424	1.41128
$855$	0	0
$856$	10.4241	0.356287
$857$	22.8651	0.781057	0.390528	−	0.920591i	$-0.372292\pi$
$857$	22.8651	0.781057	0.390528	+	0.920591i	$0.372292\pi$
$858$	0	0
$859$	20.6444	0.704376	0.352188	−	0.935929i	$-0.385438\pi$
$859$	20.6444	0.704376	0.352188	+	0.935929i	$0.385438\pi$
$860$	5.71767	0.194971
$861$	0	0
$862$	−9.52705	−0.324493
$863$	−31.3357	−1.06668	−0.533339	−	0.845901i	$-0.679063\pi$
$863$	−31.3357	−1.06668	−0.533339	+	0.845901i	$0.679063\pi$
$864$	0	0
$865$	3.21203	0.109212
$866$	3.83146	0.130198
$867$	0	0
$868$	−19.8456	−0.673603
$869$	3.95906	0.134302
$870$	0	0
$871$	0	0
$872$	−11.5515	−0.391183
$873$	0	0
$874$	68.4260	2.31454
$875$	3.63752	0.122971
$876$	0	0
$877$	−11.7988	−0.398416	−0.199208	−	0.979957i	$-0.563837\pi$
$877$	−11.7988	−0.398416	−0.199208	+	0.979957i	$0.563837\pi$
$878$	1.53927	0.0519480
$879$	0	0
$880$	8.00000	0.269680
$881$	17.5271	0.590501	0.295251	−	0.955420i	$-0.404597\pi$
$881$	17.5271	0.590501	0.295251	+	0.955420i	$0.404597\pi$
$882$	0	0
$883$	38.9838	1.31191	0.655955	−	0.754800i	$-0.272266\pi$
$883$	38.9838	1.31191	0.655955	+	0.754800i	$0.272266\pi$
$884$	0	0
$885$	0	0
$886$	17.3561	0.583091
$887$	−22.9600	−0.770922	−0.385461	−	0.922724i	$-0.625958\pi$
$887$	−22.9600	−0.770922	−0.385461	+	0.922724i	$0.625958\pi$
$888$	0	0
$889$	2.91416	0.0977377
$890$	1.97953	0.0663541
$891$	0	0
$892$	10.4485	0.349842
$893$	−11.9248	−0.399047
$894$	0	0
$895$	26.4568	0.884352
$896$	47.8299	1.59788
$897$	0	0
$898$	−40.8119	−1.36191
$899$	−50.3815	−1.68032
$900$	0	0
$901$	4.68594	0.156111
$902$	13.8562	0.461361
$903$	0	0
$904$	10.7005	0.355894
$905$	−7.38058	−0.245339
$906$	0	0
$907$	−21.3952	−0.710415	−0.355207	−	0.934788i	$-0.615590\pi$
$907$	−21.3952	−0.710415	−0.355207	+	0.934788i	$0.615590\pi$
$908$	11.8594	0.393568
$909$	0	0
$910$	0	0
$911$	−1.33567	−0.0442528	−0.0221264	−	0.999755i	$-0.507044\pi$
$911$	−1.33567	−0.0442528	−0.0221264	+	0.999755i	$0.507044\pi$
$912$	0	0
$913$	−4.77575	−0.158054
$914$	−46.8710	−1.55035
$915$	0	0
$916$	3.62198	0.119673
$917$	−40.4001	−1.33413
$918$	0	0
$919$	11.8046	0.389399	0.194700	−	0.980863i	$-0.437627\pi$
$919$	11.8046	0.389399	0.194700	+	0.980863i	$0.437627\pi$
$920$	11.4763	0.378361
$921$	0	0
$922$	21.9795	0.723857
$923$	0	0
$924$	0	0
$925$	11.9248	0.392084
$926$	61.2344	2.01229
$927$	0	0
$928$	−32.1016	−1.05379
$929$	−44.5355	−1.46116	−0.730581	−	0.682826i	$-0.760751\pi$
$929$	−44.5355	−1.46116	−0.730581	+	0.682826i	$0.760751\pi$
$930$	0	0
$931$	44.3606	1.45386
$932$	6.27836	0.205655
$933$	0	0
$934$	17.4657	0.571494
$935$	−5.29948	−0.173311
$936$	0	0
$937$	48.5618	1.58645	0.793223	−	0.608931i	$-0.208401\pi$
$937$	48.5618	1.58645	0.793223	+	0.608931i	$0.208401\pi$
$938$	−32.9403	−1.07554
$939$	0	0
$940$	1.35026	0.0440407
$941$	−22.0787	−0.719746	−0.359873	−	0.933001i	$-0.617180\pi$
$941$	−22.0787	−0.719746	−0.359873	+	0.933001i	$0.617180\pi$
$942$	0	0
$943$	29.4420	0.958763
$944$	−57.4128	−1.86863
$945$	0	0
$946$	19.1557	0.622805
$947$	31.1730	1.01299	0.506493	−	0.862244i	$-0.330941\pi$
$947$	31.1730	1.01299	0.506493	+	0.862244i	$0.330941\pi$
$948$	0	0
$949$	0	0
$950$	−11.9248	−0.386891
$951$	0	0
$952$	−23.9149	−0.775087
$953$	−46.1886	−1.49619	−0.748097	−	0.663589i	$-0.769032\pi$
$953$	−46.1886	−1.49619	−0.748097	+	0.663589i	$0.769032\pi$
$954$	0	0
$955$	−5.56959	−0.180228
$956$	1.56325	0.0505592
$957$	0	0
$958$	−0.932071	−0.0301139
$959$	72.7059	2.34780
$960$	0	0
$961$	14.8119	0.477805
$962$	0	0
$963$	0	0
$964$	23.9267	0.770627
$965$	14.5247	0.467566
$966$	0	0
$967$	9.17347	0.294999	0.147499	−	0.989062i	$-0.452878\pi$
$967$	9.17347	0.294999	0.147499	+	0.989062i	$0.452878\pi$
$968$	−16.8021	−0.540040
$969$	0	0
$970$	−15.1065	−0.485040
$971$	−7.73813	−0.248329	−0.124164	−	0.992262i	$-0.539625\pi$
$971$	−7.73813	−0.248329	−0.124164	+	0.992262i	$0.539625\pi$
$972$	0	0
$973$	−18.6460	−0.597762
$974$	−15.5877	−0.499462
$975$	0	0
$976$	−33.5877	−1.07512
$977$	29.4664	0.942714	0.471357	−	0.881942i	$-0.343764\pi$
$977$	29.4664	0.942714	0.471357	+	0.881942i	$0.343764\pi$
$978$	0	0
$979$	1.90508	0.0608866
$980$	−5.02302	−0.160455
$981$	0	0
$982$	−74.0322	−2.36246
$983$	−31.1754	−0.994340	−0.497170	−	0.867653i	$-0.665627\pi$
$983$	−31.1754	−0.994340	−0.497170	+	0.867653i	$0.665627\pi$
$984$	0	0
$985$	24.8627	0.792192
$986$	40.9887	1.30535
$987$	0	0
$988$	0	0
$989$	40.7024	1.29426
$990$	0	0
$991$	32.0567	1.01831	0.509157	−	0.860674i	$-0.329957\pi$
$991$	32.0567	1.01831	0.509157	+	0.860674i	$0.329957\pi$
$992$	29.1900	0.926782
$993$	0	0
$994$	−23.3463	−0.740499
$995$	13.2374	0.419655
$996$	0	0
$997$	53.7654	1.70277	0.851384	−	0.524543i	$-0.175764\pi$
$997$	53.7654	1.70277	0.851384	+	0.524543i	$0.175764\pi$
$998$	43.1411	1.36561
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7605.2.a.bt.1.1		3
3.2	odd	2		2535.2.a.z.1.3		3
13.4	even	6		585.2.j.g.406.1		6
13.10	even	6		585.2.j.g.451.1		6
13.12	even	2		7605.2.a.bu.1.3		3
39.17	odd	6		195.2.i.e.16.3	✓	6
39.23	odd	6		195.2.i.e.61.3	yes	6
39.38	odd	2		2535.2.a.y.1.1		3
195.17	even	12		975.2.bb.j.874.2		12
195.23	even	12		975.2.bb.j.724.2		12
195.62	even	12		975.2.bb.j.724.5		12
195.134	odd	6		975.2.i.m.601.1		6
195.173	even	12		975.2.bb.j.874.5		12
195.179	odd	6		975.2.i.m.451.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.2.i.e.16.3	✓	6	39.17	odd	6
195.2.i.e.61.3	yes	6	39.23	odd	6
585.2.j.g.406.1		6	13.4	even	6
585.2.j.g.451.1		6	13.10	even	6
975.2.i.m.451.1		6	195.179	odd	6
975.2.i.m.601.1		6	195.134	odd	6
975.2.bb.j.724.2		12	195.23	even	12
975.2.bb.j.724.5		12	195.62	even	12
975.2.bb.j.874.2		12	195.17	even	12
975.2.bb.j.874.5		12	195.173	even	12
2535.2.a.y.1.1		3	39.38	odd	2
2535.2.a.z.1.3		3	3.2	odd	2
7605.2.a.bt.1.1		3	1.1	even	1	trivial
7605.2.a.bu.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)

\(f(q)\)	\(=\)	\(q-1.67513 q^{2} +0.806063 q^{4} -1.00000 q^{5} -3.63752 q^{7} +2.00000 q^{8} +O(q^{10})\)	\(q-1.67513 q^{2} +0.806063 q^{4} -1.00000 q^{5} -3.63752 q^{7} +2.00000 q^{8} +1.67513 q^{10} +1.61213 q^{11} +6.09332 q^{14} -4.96239 q^{16} +3.28726 q^{17} +7.11871 q^{19} -0.806063 q^{20} -2.70052 q^{22} -5.73813 q^{23} +1.00000 q^{25} -2.93207 q^{28} -7.44358 q^{29} +6.76845 q^{31} +4.31265 q^{32} -5.50659 q^{34} +3.63752 q^{35} +11.9248 q^{37} -11.9248 q^{38} -2.00000 q^{40} -5.13093 q^{41} -7.09332 q^{43} +1.29948 q^{44} +9.61213 q^{46} -1.67513 q^{47} +6.23155 q^{49} -1.67513 q^{50} +1.42548 q^{53} -1.61213 q^{55} -7.27504 q^{56} +12.4690 q^{58} +11.5696 q^{59} +6.76845 q^{61} -11.3380 q^{62} +2.70052 q^{64} -5.40597 q^{67} +2.64974 q^{68} -6.09332 q^{70} -3.83146 q^{71} +6.67513 q^{73} -19.9756 q^{74} +5.73813 q^{76} -5.86414 q^{77} +2.45580 q^{79} +4.96239 q^{80} +8.59498 q^{82} -2.96239 q^{83} -3.28726 q^{85} +11.8822 q^{86} +3.22425 q^{88} +1.18172 q^{89} -4.62530 q^{92} +2.80606 q^{94} -7.11871 q^{95} -9.01810 q^{97} -10.4387 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{4} - 3 q^{5} + 5 q^{7} + 6 q^{8}+O(q^{10}) \) 3 * q + 2 * q^4 - 3 * q^5 + 5 * q^7 + 6 * q^8	\( 3 q + 2 q^{4} - 3 q^{5} + 5 q^{7} + 6 q^{8} + 4 q^{11} + 12 q^{14} - 4 q^{16} + 4 q^{17} - 2 q^{20} + 12 q^{22} - 8 q^{23} + 3 q^{25} - 6 q^{29} + 9 q^{31} - 8 q^{32} + 4 q^{34} - 5 q^{35} + 14 q^{37} - 14 q^{38} - 6 q^{40} - 20 q^{41} - 15 q^{43} + 24 q^{44} + 28 q^{46} + 30 q^{49} + 16 q^{53} - 4 q^{55} + 10 q^{56} + 6 q^{58} + 10 q^{59} + 9 q^{61} + 2 q^{62} - 12 q^{64} + 11 q^{67} + 18 q^{68} - 12 q^{70} + 4 q^{71} + 15 q^{73} - 8 q^{74} + 8 q^{76} + 17 q^{79} + 4 q^{80} - 14 q^{82} + 2 q^{83} - 4 q^{85} - 10 q^{86} + 8 q^{88} - 22 q^{89} + 28 q^{92} + 8 q^{94} + q^{97} - 2 q^{98}+O(q^{100}) \) 3 * q + 2 * q^4 - 3 * q^5 + 5 * q^7 + 6 * q^8 + 4 * q^11 + 12 * q^14 - 4 * q^16 + 4 * q^17 - 2 * q^20 + 12 * q^22 - 8 * q^23 + 3 * q^25 - 6 * q^29 + 9 * q^31 - 8 * q^32 + 4 * q^34 - 5 * q^35 + 14 * q^37 - 14 * q^38 - 6 * q^40 - 20 * q^41 - 15 * q^43 + 24 * q^44 + 28 * q^46 + 30 * q^49 + 16 * q^53 - 4 * q^55 + 10 * q^56 + 6 * q^58 + 10 * q^59 + 9 * q^61 + 2 * q^62 - 12 * q^64 + 11 * q^67 + 18 * q^68 - 12 * q^70 + 4 * q^71 + 15 * q^73 - 8 * q^74 + 8 * q^76 + 17 * q^79 + 4 * q^80 - 14 * q^82 + 2 * q^83 - 4 * q^85 - 10 * q^86 + 8 * q^88 - 22 * q^89 + 28 * q^92 + 8 * q^94 + q^97 - 2 * q^98

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