LMFDB - Embedded newform 7448.2.a.bn.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7448,2,Mod(1,7448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7448, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7448.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7448 = 2^{3} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7448.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:	$59.4725794254$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.98211824.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 6x^{4} + 15x^{3} + 8x^{2} - 9x + 1 $ x^6 - 3x^5 - 6x^4 + 15x^3 + 8x^2 - 9*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.04143$ of defining polynomial
Character	$\chi$	$=$	7448.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.04143 q^{3} +3.17676 q^{5} +1.16742 q^{9} +O(q^{10})$	$q+2.04143 q^{3} +3.17676 q^{5} +1.16742 q^{9} +2.64156 q^{11} +5.60178 q^{13} +6.48513 q^{15} +7.48513 q^{17} -1.00000 q^{19} +4.87939 q^{23} +5.09182 q^{25} -3.74108 q^{27} +2.60909 q^{29} -3.41086 q^{31} +5.39255 q^{33} -9.05242 q^{37} +11.4356 q^{39} -10.4198 q^{41} +3.77689 q^{43} +3.70862 q^{45} +7.21782 q^{47} +15.2803 q^{51} -9.74609 q^{53} +8.39160 q^{55} -2.04143 q^{57} +3.30969 q^{59} -2.35314 q^{61} +17.7955 q^{65} +13.8674 q^{67} +9.96091 q^{69} +8.57828 q^{71} -14.1861 q^{73} +10.3946 q^{75} -16.8431 q^{79} -11.1394 q^{81} -6.51760 q^{83} +23.7785 q^{85} +5.32626 q^{87} -12.1886 q^{89} -6.96302 q^{93} -3.17676 q^{95} -7.06695 q^{97} +3.08381 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{3} + q^{5} + 3 q^{9}+O(q^{10}) $ 6 * q + 3 * q^3 + q^5 + 3 * q^9	$ 6 q + 3 q^{3} + q^{5} + 3 q^{9} + 3 q^{11} + 6 q^{13} - 8 q^{15} - 2 q^{17} - 6 q^{19} + 2 q^{23} + 5 q^{25} + 6 q^{27} - q^{29} + 14 q^{31} + 19 q^{33} - 7 q^{37} + 22 q^{39} - 21 q^{41} + q^{43} - 4 q^{45} + 23 q^{47} + 2 q^{51} - 15 q^{53} + 2 q^{55} - 3 q^{57} + 25 q^{59} + 21 q^{61} + 6 q^{65} + 2 q^{67} - 20 q^{69} + 13 q^{71} - 2 q^{73} + 24 q^{75} - 17 q^{79} - 2 q^{81} + 4 q^{83} + 40 q^{85} + 30 q^{87} - q^{89} + 6 q^{93} - q^{95} - 13 q^{97} + 41 q^{99}+O(q^{100}) $ 6 * q + 3 * q^3 + q^5 + 3 * q^9 + 3 * q^11 + 6 * q^13 - 8 * q^15 - 2 * q^17 - 6 * q^19 + 2 * q^23 + 5 * q^25 + 6 * q^27 - q^29 + 14 * q^31 + 19 * q^33 - 7 * q^37 + 22 * q^39 - 21 * q^41 + q^43 - 4 * q^45 + 23 * q^47 + 2 * q^51 - 15 * q^53 + 2 * q^55 - 3 * q^57 + 25 * q^59 + 21 * q^61 + 6 * q^65 + 2 * q^67 - 20 * q^69 + 13 * q^71 - 2 * q^73 + 24 * q^75 - 17 * q^79 - 2 * q^81 + 4 * q^83 + 40 * q^85 + 30 * q^87 - q^89 + 6 * q^93 - q^95 - 13 * q^97 + 41 * q^99

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	0
$2$	0	0
$3$	2.04143	1.17862	0.589309	−	0.807908i	$-0.299400\pi$
$3$	2.04143	1.17862	0.589309	+	0.807908i	$0.299400\pi$
$4$	0	0
$5$	3.17676	1.42069	0.710346	−	0.703853i	$-0.248539\pi$
$5$	3.17676	1.42069	0.710346	+	0.703853i	$0.248539\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.16742	0.389140
$10$	0	0
$11$	2.64156	0.796460	0.398230	−	0.917286i	$-0.369625\pi$
$11$	2.64156	0.796460	0.398230	+	0.917286i	$0.369625\pi$
$12$	0	0
$13$	5.60178	1.55365	0.776827	−	0.629714i	$-0.216828\pi$
$13$	5.60178	1.55365	0.776827	+	0.629714i	$0.216828\pi$
$14$	0	0
$15$	6.48513	1.67445
$16$	0	0
$17$	7.48513	1.81541	0.907705	−	0.419609i	$-0.137833\pi$
$17$	7.48513	1.81541	0.907705	+	0.419609i	$0.137833\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.87939	1.01742	0.508711	−	0.860937i	$-0.330122\pi$
$23$	4.87939	1.01742	0.508711	+	0.860937i	$0.330122\pi$
$24$	0	0
$25$	5.09182	1.01836
$26$	0	0
$27$	−3.74108	−0.719970
$28$	0	0
$29$	2.60909	0.484495	0.242248	−	0.970214i	$-0.422115\pi$
$29$	2.60909	0.484495	0.242248	+	0.970214i	$0.422115\pi$
$30$	0	0
$31$	−3.41086	−0.612608	−0.306304	−	0.951934i	$-0.599092\pi$
$31$	−3.41086	−0.612608	−0.306304	+	0.951934i	$0.599092\pi$
$32$	0	0
$33$	5.39255	0.938722
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−9.05242	−1.48821	−0.744104	−	0.668064i	$-0.767123\pi$
$37$	−9.05242	−1.48821	−0.744104	+	0.668064i	$0.767123\pi$
$38$	0	0
$39$	11.4356	1.83116
$40$	0	0
$41$	−10.4198	−1.62730	−0.813652	−	0.581352i	$-0.802524\pi$
$41$	−10.4198	−1.62730	−0.813652	+	0.581352i	$0.802524\pi$
$42$	0	0
$43$	3.77689	0.575971	0.287986	−	0.957635i	$-0.407014\pi$
$43$	3.77689	0.575971	0.287986	+	0.957635i	$0.407014\pi$
$44$	0	0
$45$	3.70862	0.552848
$46$	0	0
$47$	7.21782	1.05283	0.526413	−	0.850229i	$-0.323536\pi$
$47$	7.21782	1.05283	0.526413	+	0.850229i	$0.323536\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	15.2803	2.13967
$52$	0	0
$53$	−9.74609	−1.33873	−0.669364	−	0.742935i	$-0.733433\pi$
$53$	−9.74609	−1.33873	−0.669364	+	0.742935i	$0.733433\pi$
$54$	0	0
$55$	8.39160	1.13152
$56$	0	0
$57$	−2.04143	−0.270394
$58$	0	0
$59$	3.30969	0.430885	0.215443	−	0.976516i	$-0.430881\pi$
$59$	3.30969	0.430885	0.215443	+	0.976516i	$0.430881\pi$
$60$	0	0
$61$	−2.35314	−0.301289	−0.150644	−	0.988588i	$-0.548135\pi$
$61$	−2.35314	−0.301289	−0.150644	+	0.988588i	$0.548135\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	17.7955	2.20726
$66$	0	0
$67$	13.8674	1.69417	0.847086	−	0.531457i	$-0.178355\pi$
$67$	13.8674	1.69417	0.847086	+	0.531457i	$0.178355\pi$
$68$	0	0
$69$	9.96091	1.19915
$70$	0	0
$71$	8.57828	1.01805	0.509027	−	0.860750i	$-0.330005\pi$
$71$	8.57828	1.01805	0.509027	+	0.860750i	$0.330005\pi$
$72$	0	0
$73$	−14.1861	−1.66035	−0.830176	−	0.557502i	$-0.811760\pi$
$73$	−14.1861	−1.66035	−0.830176	+	0.557502i	$0.811760\pi$
$74$	0	0
$75$	10.3946	1.20026
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−16.8431	−1.89500	−0.947500	−	0.319756i	$-0.896399\pi$
$79$	−16.8431	−1.89500	−0.947500	+	0.319756i	$0.896399\pi$
$80$	0	0
$81$	−11.1394	−1.23771
$82$	0	0
$83$	−6.51760	−0.715399	−0.357700	−	0.933837i	$-0.616439\pi$
$83$	−6.51760	−0.715399	−0.357700	+	0.933837i	$0.616439\pi$
$84$	0	0
$85$	23.7785	2.57914
$86$	0	0
$87$	5.32626	0.571035
$88$	0	0
$89$	−12.1886	−1.29199	−0.645997	−	0.763340i	$-0.723558\pi$
$89$	−12.1886	−1.29199	−0.645997	+	0.763340i	$0.723558\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−6.96302	−0.722031
$94$	0	0
$95$	−3.17676	−0.325929
$96$	0	0
$97$	−7.06695	−0.717540	−0.358770	−	0.933426i	$-0.616804\pi$
$97$	−7.06695	−0.717540	−0.358770	+	0.933426i	$0.616804\pi$
$98$	0	0
$99$	3.08381	0.309935
$100$	0	0
$101$	−4.33427	−0.431276	−0.215638	−	0.976473i	$-0.569183\pi$
$101$	−4.33427	−0.431276	−0.215638	+	0.976473i	$0.569183\pi$
$102$	0	0
$103$	−12.9215	−1.27320	−0.636598	−	0.771196i	$-0.719659\pi$
$103$	−12.9215	−1.27320	−0.636598	+	0.771196i	$0.719659\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.63745	−0.254972	−0.127486	−	0.991840i	$-0.540691\pi$
$107$	−2.63745	−0.254972	−0.127486	+	0.991840i	$0.540691\pi$
$108$	0	0
$109$	−8.84895	−0.847576	−0.423788	−	0.905761i	$-0.639300\pi$
$109$	−8.84895	−0.847576	−0.423788	+	0.905761i	$0.639300\pi$
$110$	0	0
$111$	−18.4798	−1.75403
$112$	0	0
$113$	9.47520	0.891352	0.445676	−	0.895194i	$-0.352963\pi$
$113$	9.47520	0.891352	0.445676	+	0.895194i	$0.352963\pi$
$114$	0	0
$115$	15.5007	1.44544
$116$	0	0
$117$	6.53963	0.604589
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−4.02217	−0.365652
$122$	0	0
$123$	−21.2713	−1.91797
$124$	0	0
$125$	0.291699	0.0260904
$126$	0	0
$127$	−16.3048	−1.44682	−0.723410	−	0.690419i	$-0.757426\pi$
$127$	−16.3048	−1.44682	−0.723410	+	0.690419i	$0.757426\pi$
$128$	0	0
$129$	7.71025	0.678850
$130$	0	0
$131$	12.8724	1.12467	0.562335	−	0.826909i	$-0.309903\pi$
$131$	12.8724	1.12467	0.562335	+	0.826909i	$0.309903\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−11.8845	−1.02286
$136$	0	0
$137$	−4.67817	−0.399683	−0.199841	−	0.979828i	$-0.564043\pi$
$137$	−4.67817	−0.399683	−0.199841	+	0.979828i	$0.564043\pi$
$138$	0	0
$139$	−3.18023	−0.269744	−0.134872	−	0.990863i	$-0.543062\pi$
$139$	−3.18023	−0.269744	−0.134872	+	0.990863i	$0.543062\pi$
$140$	0	0
$141$	14.7346	1.24088
$142$	0	0
$143$	14.7974	1.23742
$144$	0	0
$145$	8.28845	0.688318
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0.273874	0.0224366	0.0112183	−	0.999937i	$-0.496429\pi$
$149$	0.273874	0.0224366	0.0112183	+	0.999937i	$0.496429\pi$
$150$	0	0
$151$	22.1226	1.80032	0.900158	−	0.435564i	$-0.143451\pi$
$151$	22.1226	1.80032	0.900158	+	0.435564i	$0.143451\pi$
$152$	0	0
$153$	8.73829	0.706449
$154$	0	0
$155$	−10.8355	−0.870327
$156$	0	0
$157$	1.37859	0.110024	0.0550119	−	0.998486i	$-0.482480\pi$
$157$	1.37859	0.110024	0.0550119	+	0.998486i	$0.482480\pi$
$158$	0	0
$159$	−19.8959	−1.57785
$160$	0	0
$161$	0	0
$162$	0	0
$163$	2.05078	0.160630	0.0803149	−	0.996770i	$-0.474407\pi$
$163$	2.05078	0.160630	0.0803149	+	0.996770i	$0.474407\pi$
$164$	0	0
$165$	17.1308	1.33363
$166$	0	0
$167$	3.51297	0.271842	0.135921	−	0.990720i	$-0.456601\pi$
$167$	3.51297	0.271842	0.135921	+	0.990720i	$0.456601\pi$
$168$	0	0
$169$	18.3799	1.41384
$170$	0	0
$171$	−1.16742	−0.0892749
$172$	0	0
$173$	23.5896	1.79349	0.896744	−	0.442551i	$-0.145926\pi$
$173$	23.5896	1.79349	0.896744	+	0.442551i	$0.145926\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	6.75649	0.507849
$178$	0	0
$179$	8.93373	0.667738	0.333869	−	0.942619i	$-0.391646\pi$
$179$	8.93373	0.667738	0.333869	+	0.942619i	$0.391646\pi$
$180$	0	0
$181$	−25.1244	−1.86748	−0.933741	−	0.357950i	$-0.883476\pi$
$181$	−25.1244	−1.86748	−0.933741	+	0.357950i	$0.883476\pi$
$182$	0	0
$183$	−4.80376	−0.355104
$184$	0	0
$185$	−28.7574	−2.11428
$186$	0	0
$187$	19.7724	1.44590
$188$	0	0
$189$	0	0
$190$	0	0
$191$	18.3297	1.32629	0.663147	−	0.748489i	$-0.269220\pi$
$191$	18.3297	1.32629	0.663147	+	0.748489i	$0.269220\pi$
$192$	0	0
$193$	−0.833709	−0.0600117	−0.0300059	−	0.999550i	$-0.509553\pi$
$193$	−0.833709	−0.0600117	−0.0300059	+	0.999550i	$0.509553\pi$
$194$	0	0
$195$	36.3283	2.60152
$196$	0	0
$197$	4.66174	0.332135	0.166068	−	0.986114i	$-0.446893\pi$
$197$	4.66174	0.332135	0.166068	+	0.986114i	$0.446893\pi$
$198$	0	0
$199$	−14.3836	−1.01963	−0.509814	−	0.860285i	$-0.670286\pi$
$199$	−14.3836	−1.01963	−0.509814	+	0.860285i	$0.670286\pi$
$200$	0	0
$201$	28.3092	1.99678
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−33.1013	−2.31190
$206$	0	0
$207$	5.69630	0.395920
$208$	0	0
$209$	−2.64156	−0.182720
$210$	0	0
$211$	−21.9757	−1.51287	−0.756436	−	0.654067i	$-0.773061\pi$
$211$	−21.9757	−1.51287	−0.756436	+	0.654067i	$0.773061\pi$
$212$	0	0
$213$	17.5119	1.19990
$214$	0	0
$215$	11.9983	0.818277
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−28.9598	−1.95692
$220$	0	0
$221$	41.9300	2.82052
$222$	0	0
$223$	2.44849	0.163963	0.0819814	−	0.996634i	$-0.473875\pi$
$223$	2.44849	0.163963	0.0819814	+	0.996634i	$0.473875\pi$
$224$	0	0
$225$	5.94430	0.396287
$226$	0	0
$227$	8.70936	0.578061	0.289030	−	0.957320i	$-0.406667\pi$
$227$	8.70936	0.578061	0.289030	+	0.957320i	$0.406667\pi$
$228$	0	0
$229$	18.2921	1.20878	0.604388	−	0.796690i	$-0.293418\pi$
$229$	18.2921	1.20878	0.604388	+	0.796690i	$0.293418\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4.96121	−0.325020	−0.162510	−	0.986707i	$-0.551959\pi$
$233$	−4.96121	−0.325020	−0.162510	+	0.986707i	$0.551959\pi$
$234$	0	0
$235$	22.9293	1.49574
$236$	0	0
$237$	−34.3840	−2.23348
$238$	0	0
$239$	−4.42911	−0.286495	−0.143248	−	0.989687i	$-0.545755\pi$
$239$	−4.42911	−0.286495	−0.143248	+	0.989687i	$0.545755\pi$
$240$	0	0
$241$	17.8698	1.15109	0.575547	−	0.817769i	$-0.304789\pi$
$241$	17.8698	1.15109	0.575547	+	0.817769i	$0.304789\pi$
$242$	0	0
$243$	−11.5170	−0.738817
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−5.60178	−0.356433
$248$	0	0
$249$	−13.3052	−0.843183
$250$	0	0
$251$	−11.4381	−0.721968	−0.360984	−	0.932572i	$-0.617559\pi$
$251$	−11.4381	−0.721968	−0.360984	+	0.932572i	$0.617559\pi$
$252$	0	0
$253$	12.8892	0.810336
$254$	0	0
$255$	48.5420	3.03982
$256$	0	0
$257$	−2.18402	−0.136235	−0.0681176	−	0.997677i	$-0.521699\pi$
$257$	−2.18402	−0.136235	−0.0681176	+	0.997677i	$0.521699\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	3.04590	0.188537
$262$	0	0
$263$	19.3509	1.19322	0.596612	−	0.802530i	$-0.296513\pi$
$263$	19.3509	1.19322	0.596612	+	0.802530i	$0.296513\pi$
$264$	0	0
$265$	−30.9610	−1.90192
$266$	0	0
$267$	−24.8822	−1.52277
$268$	0	0
$269$	−11.5091	−0.701725	−0.350862	−	0.936427i	$-0.614112\pi$
$269$	−11.5091	−0.701725	−0.350862	+	0.936427i	$0.614112\pi$
$270$	0	0
$271$	7.84158	0.476342	0.238171	−	0.971223i	$-0.423452\pi$
$271$	7.84158	0.476342	0.238171	+	0.971223i	$0.423452\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	13.4503	0.811086
$276$	0	0
$277$	−12.3647	−0.742925	−0.371463	−	0.928448i	$-0.621144\pi$
$277$	−12.3647	−0.742925	−0.371463	+	0.928448i	$0.621144\pi$
$278$	0	0
$279$	−3.98191	−0.238391
$280$	0	0
$281$	7.02949	0.419344	0.209672	−	0.977772i	$-0.432760\pi$
$281$	7.02949	0.419344	0.209672	+	0.977772i	$0.432760\pi$
$282$	0	0
$283$	31.3214	1.86186	0.930931	−	0.365195i	$-0.118998\pi$
$283$	31.3214	1.86186	0.930931	+	0.365195i	$0.118998\pi$
$284$	0	0
$285$	−6.48513	−0.384146
$286$	0	0
$287$	0	0
$288$	0	0
$289$	39.0271	2.29571
$290$	0	0
$291$	−14.4266	−0.845705
$292$	0	0
$293$	−11.0475	−0.645402	−0.322701	−	0.946501i	$-0.604591\pi$
$293$	−11.0475	−0.645402	−0.322701	+	0.946501i	$0.604591\pi$
$294$	0	0
$295$	10.5141	0.612155
$296$	0	0
$297$	−9.88227	−0.573427
$298$	0	0
$299$	27.3333	1.58072
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−8.84809	−0.508310
$304$	0	0
$305$	−7.47536	−0.428038
$306$	0	0
$307$	−8.24429	−0.470527	−0.235263	−	0.971932i	$-0.575595\pi$
$307$	−8.24429	−0.470527	−0.235263	+	0.971932i	$0.575595\pi$
$308$	0	0
$309$	−26.3783	−1.50061
$310$	0	0
$311$	14.6155	0.828766	0.414383	−	0.910103i	$-0.363997\pi$
$311$	14.6155	0.828766	0.414383	+	0.910103i	$0.363997\pi$
$312$	0	0
$313$	−0.0179410	−0.00101408	−0.000507041	−	1.00000i	$-0.500161\pi$
$313$	−0.0179410	−0.00101408	−0.000507041		1.00000i	$0.500161\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−21.4897	−1.20698	−0.603491	−	0.797370i	$-0.706224\pi$
$317$	−21.4897	−1.20698	−0.603491	+	0.797370i	$0.706224\pi$
$318$	0	0
$319$	6.89205	0.385881
$320$	0	0
$321$	−5.38416	−0.300515
$322$	0	0
$323$	−7.48513	−0.416484
$324$	0	0
$325$	28.5233	1.58219
$326$	0	0
$327$	−18.0645	−0.998968
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−15.1332	−0.831797	−0.415898	−	0.909411i	$-0.636533\pi$
$331$	−15.1332	−0.831797	−0.415898	+	0.909411i	$0.636533\pi$
$332$	0	0
$333$	−10.5680	−0.579122
$334$	0	0
$335$	44.0534	2.40689
$336$	0	0
$337$	−25.4252	−1.38500	−0.692500	−	0.721418i	$-0.743491\pi$
$337$	−25.4252	−1.38500	−0.692500	+	0.721418i	$0.743491\pi$
$338$	0	0
$339$	19.3429	1.05056
$340$	0	0
$341$	−9.00998	−0.487918
$342$	0	0
$343$	0	0
$344$	0	0
$345$	31.6435	1.70363
$346$	0	0
$347$	−7.49458	−0.402330	−0.201165	−	0.979557i	$-0.564473\pi$
$347$	−7.49458	−0.402330	−0.201165	+	0.979557i	$0.564473\pi$
$348$	0	0
$349$	6.60122	0.353355	0.176678	−	0.984269i	$-0.443465\pi$
$349$	6.60122	0.353355	0.176678	+	0.984269i	$0.443465\pi$
$350$	0	0
$351$	−20.9567	−1.11858
$352$	0	0
$353$	22.8798	1.21777	0.608885	−	0.793259i	$-0.291617\pi$
$353$	22.8798	1.21777	0.608885	+	0.793259i	$0.291617\pi$
$354$	0	0
$355$	27.2512	1.44634
$356$	0	0
$357$	0	0
$358$	0	0
$359$	19.6109	1.03502	0.517511	−	0.855676i	$-0.326858\pi$
$359$	19.6109	1.03502	0.517511	+	0.855676i	$0.326858\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−8.21097	−0.430964
$364$	0	0
$365$	−45.0657	−2.35885
$366$	0	0
$367$	−0.544256	−0.0284099	−0.0142050	−	0.999899i	$-0.504522\pi$
$367$	−0.544256	−0.0284099	−0.0142050	+	0.999899i	$0.504522\pi$
$368$	0	0
$369$	−12.1643	−0.633249
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−13.2016	−0.683553	−0.341776	−	0.939781i	$-0.611029\pi$
$373$	−13.2016	−0.683553	−0.341776	+	0.939781i	$0.611029\pi$
$374$	0	0
$375$	0.595483	0.0307506
$376$	0	0
$377$	14.6155	0.752738
$378$	0	0
$379$	−20.8032	−1.06859	−0.534293	−	0.845299i	$-0.679422\pi$
$379$	−20.8032	−1.06859	−0.534293	+	0.845299i	$0.679422\pi$
$380$	0	0
$381$	−33.2851	−1.70525
$382$	0	0
$383$	11.9617	0.611215	0.305607	−	0.952158i	$-0.401141\pi$
$383$	11.9617	0.611215	0.305607	+	0.952158i	$0.401141\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.40923	0.224134
$388$	0	0
$389$	18.7881	0.952597	0.476298	−	0.879284i	$-0.341978\pi$
$389$	18.7881	0.952597	0.476298	+	0.879284i	$0.341978\pi$
$390$	0	0
$391$	36.5228	1.84704
$392$	0	0
$393$	26.2781	1.32556
$394$	0	0
$395$	−53.5066	−2.69221
$396$	0	0
$397$	25.9270	1.30124	0.650618	−	0.759405i	$-0.274510\pi$
$397$	25.9270	1.30124	0.650618	+	0.759405i	$0.274510\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.15411	0.0576333	0.0288166	−	0.999585i	$-0.490826\pi$
$401$	1.15411	0.0576333	0.0288166	+	0.999585i	$0.490826\pi$
$402$	0	0
$403$	−19.1069	−0.951781
$404$	0	0
$405$	−35.3872	−1.75840
$406$	0	0
$407$	−23.9125	−1.18530
$408$	0	0
$409$	21.6754	1.07178	0.535890	−	0.844288i	$-0.319976\pi$
$409$	21.6754	1.07178	0.535890	+	0.844288i	$0.319976\pi$
$410$	0	0
$411$	−9.55013	−0.471073
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−20.7049	−1.01636
$416$	0	0
$417$	−6.49221	−0.317925
$418$	0	0
$419$	−19.9806	−0.976117	−0.488058	−	0.872811i	$-0.662295\pi$
$419$	−19.9806	−0.976117	−0.488058	+	0.872811i	$0.662295\pi$
$420$	0	0
$421$	−30.0972	−1.46685	−0.733425	−	0.679771i	$-0.762079\pi$
$421$	−30.0972	−1.46685	−0.733425	+	0.679771i	$0.762079\pi$
$422$	0	0
$423$	8.42623	0.409697
$424$	0	0
$425$	38.1129	1.84875
$426$	0	0
$427$	0	0
$428$	0	0
$429$	30.2079	1.45845
$430$	0	0
$431$	−31.7226	−1.52802	−0.764012	−	0.645202i	$-0.776773\pi$
$431$	−31.7226	−1.52802	−0.764012	+	0.645202i	$0.776773\pi$
$432$	0	0
$433$	4.35424	0.209252	0.104626	−	0.994512i	$-0.466636\pi$
$433$	4.35424	0.209252	0.104626	+	0.994512i	$0.466636\pi$
$434$	0	0
$435$	16.9203	0.811264
$436$	0	0
$437$	−4.87939	−0.233413
$438$	0	0
$439$	−19.4310	−0.927390	−0.463695	−	0.885995i	$-0.653477\pi$
$439$	−19.4310	−0.927390	−0.463695	+	0.885995i	$0.653477\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	6.70100	0.318374	0.159187	−	0.987248i	$-0.449113\pi$
$443$	6.70100	0.318374	0.159187	+	0.987248i	$0.449113\pi$
$444$	0	0
$445$	−38.7204	−1.83552
$446$	0	0
$447$	0.559093	0.0264442
$448$	0	0
$449$	8.79763	0.415186	0.207593	−	0.978215i	$-0.433437\pi$
$449$	8.79763	0.415186	0.207593	+	0.978215i	$0.433437\pi$
$450$	0	0
$451$	−27.5246	−1.29608
$452$	0	0
$453$	45.1617	2.12188
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−32.8664	−1.53743	−0.768713	−	0.639594i	$-0.779102\pi$
$457$	−32.8664	−1.53743	−0.768713	+	0.639594i	$0.779102\pi$
$458$	0	0
$459$	−28.0024	−1.30704
$460$	0	0
$461$	20.5086	0.955180	0.477590	−	0.878583i	$-0.341510\pi$
$461$	20.5086	0.955180	0.477590	+	0.878583i	$0.341510\pi$
$462$	0	0
$463$	21.6865	1.00786	0.503928	−	0.863745i	$-0.331888\pi$
$463$	21.6865	1.00786	0.503928	+	0.863745i	$0.331888\pi$
$464$	0	0
$465$	−22.1198	−1.02578
$466$	0	0
$467$	12.7524	0.590110	0.295055	−	0.955480i	$-0.404662\pi$
$467$	12.7524	0.590110	0.295055	+	0.955480i	$0.404662\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	2.81430	0.129676
$472$	0	0
$473$	9.97689	0.458738
$474$	0	0
$475$	−5.09182	−0.233629
$476$	0	0
$477$	−11.3778	−0.520953
$478$	0	0
$479$	16.3071	0.745088	0.372544	−	0.928014i	$-0.378486\pi$
$479$	16.3071	0.745088	0.372544	+	0.928014i	$0.378486\pi$
$480$	0	0
$481$	−50.7096	−2.31216
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−22.4500	−1.01940
$486$	0	0
$487$	4.75145	0.215309	0.107654	−	0.994188i	$-0.465666\pi$
$487$	4.75145	0.215309	0.107654	+	0.994188i	$0.465666\pi$
$488$	0	0
$489$	4.18652	0.189321
$490$	0	0
$491$	−13.8998	−0.627290	−0.313645	−	0.949540i	$-0.601550\pi$
$491$	−13.8998	−0.627290	−0.313645	+	0.949540i	$0.601550\pi$
$492$	0	0
$493$	19.5293	0.879558
$494$	0	0
$495$	9.79653	0.440321
$496$	0	0
$497$	0	0
$498$	0	0
$499$	18.6230	0.833681	0.416841	−	0.908980i	$-0.363137\pi$
$499$	18.6230	0.833681	0.416841	+	0.908980i	$0.363137\pi$
$500$	0	0
$501$	7.17148	0.320398
$502$	0	0
$503$	14.0137	0.624840	0.312420	−	0.949944i	$-0.398860\pi$
$503$	14.0137	0.624840	0.312420	+	0.949944i	$0.398860\pi$
$504$	0	0
$505$	−13.7690	−0.612710
$506$	0	0
$507$	37.5213	1.66638
$508$	0	0
$509$	12.6112	0.558980	0.279490	−	0.960149i	$-0.409835\pi$
$509$	12.6112	0.558980	0.279490	+	0.960149i	$0.409835\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	3.74108	0.165172
$514$	0	0
$515$	−41.0486	−1.80882
$516$	0	0
$517$	19.0663	0.838534
$518$	0	0
$519$	48.1565	2.11384
$520$	0	0
$521$	4.03072	0.176589	0.0882945	−	0.996094i	$-0.471858\pi$
$521$	4.03072	0.176589	0.0882945	+	0.996094i	$0.471858\pi$
$522$	0	0
$523$	−44.0062	−1.92426	−0.962129	−	0.272596i	$-0.912118\pi$
$523$	−44.0062	−1.92426	−0.962129	+	0.272596i	$0.912118\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−25.5307	−1.11214
$528$	0	0
$529$	0.808435	0.0351493
$530$	0	0
$531$	3.86380	0.167675
$532$	0	0
$533$	−58.3696	−2.52827
$534$	0	0
$535$	−8.37856	−0.362237
$536$	0	0
$537$	18.2375	0.787008
$538$	0	0
$539$	0	0
$540$	0	0
$541$	32.6699	1.40459	0.702295	−	0.711886i	$-0.252159\pi$
$541$	32.6699	1.40459	0.702295	+	0.711886i	$0.252159\pi$
$542$	0	0
$543$	−51.2896	−2.20105
$544$	0	0
$545$	−28.1110	−1.20414
$546$	0	0
$547$	−17.1209	−0.732038	−0.366019	−	0.930607i	$-0.619280\pi$
$547$	−17.1209	−0.732038	−0.366019	+	0.930607i	$0.619280\pi$
$548$	0	0
$549$	−2.74710	−0.117244
$550$	0	0
$551$	−2.60909	−0.111151
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−58.7061	−2.49193
$556$	0	0
$557$	−14.2650	−0.604427	−0.302213	−	0.953240i	$-0.597726\pi$
$557$	−14.2650	−0.604427	−0.302213	+	0.953240i	$0.597726\pi$
$558$	0	0
$559$	21.1573	0.894860
$560$	0	0
$561$	40.3639	1.70416
$562$	0	0
$563$	29.2283	1.23182	0.615912	−	0.787815i	$-0.288788\pi$
$563$	29.2283	1.23182	0.615912	+	0.787815i	$0.288788\pi$
$564$	0	0
$565$	30.1005	1.26634
$566$	0	0
$567$	0	0
$568$	0	0
$569$	44.4290	1.86256	0.931280	−	0.364305i	$-0.118693\pi$
$569$	44.4290	1.86256	0.931280	+	0.364305i	$0.118693\pi$
$570$	0	0
$571$	1.46048	0.0611192	0.0305596	−	0.999533i	$-0.490271\pi$
$571$	1.46048	0.0611192	0.0305596	+	0.999533i	$0.490271\pi$
$572$	0	0
$573$	37.4188	1.56319
$574$	0	0
$575$	24.8450	1.03611
$576$	0	0
$577$	−10.6800	−0.444616	−0.222308	−	0.974977i	$-0.571359\pi$
$577$	−10.6800	−0.444616	−0.222308	+	0.974977i	$0.571359\pi$
$578$	0	0
$579$	−1.70196	−0.0707309
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−25.7449	−1.06624
$584$	0	0
$585$	20.7749	0.858935
$586$	0	0
$587$	−18.5240	−0.764569	−0.382285	−	0.924045i	$-0.624863\pi$
$587$	−18.5240	−0.764569	−0.382285	+	0.924045i	$0.624863\pi$
$588$	0	0
$589$	3.41086	0.140542
$590$	0	0
$591$	9.51660	0.391461
$592$	0	0
$593$	−37.3332	−1.53309	−0.766546	−	0.642189i	$-0.778026\pi$
$593$	−37.3332	−1.53309	−0.766546	+	0.642189i	$0.778026\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−29.3631	−1.20175
$598$	0	0
$599$	37.5271	1.53331	0.766657	−	0.642057i	$-0.221919\pi$
$599$	37.5271	1.53331	0.766657	+	0.642057i	$0.221919\pi$
$600$	0	0
$601$	38.2640	1.56082	0.780411	−	0.625267i	$-0.215010\pi$
$601$	38.2640	1.56082	0.780411	+	0.625267i	$0.215010\pi$
$602$	0	0
$603$	16.1891	0.659270
$604$	0	0
$605$	−12.7775	−0.519479
$606$	0	0
$607$	26.1112	1.05982	0.529910	−	0.848054i	$-0.322226\pi$
$607$	26.1112	1.05982	0.529910	+	0.848054i	$0.322226\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	40.4326	1.63573
$612$	0	0
$613$	−27.5292	−1.11190	−0.555948	−	0.831217i	$-0.687645\pi$
$613$	−27.5292	−1.11190	−0.555948	+	0.831217i	$0.687645\pi$
$614$	0	0
$615$	−67.5739	−2.72484
$616$	0	0
$617$	−19.0678	−0.767641	−0.383821	−	0.923408i	$-0.625392\pi$
$617$	−19.0678	−0.767641	−0.383821	+	0.923408i	$0.625392\pi$
$618$	0	0
$619$	41.8369	1.68157	0.840783	−	0.541373i	$-0.182095\pi$
$619$	41.8369	1.68157	0.840783	+	0.541373i	$0.182095\pi$
$620$	0	0
$621$	−18.2542	−0.732514
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−24.5325	−0.981298
$626$	0	0
$627$	−5.39255	−0.215358
$628$	0	0
$629$	−67.7585	−2.70171
$630$	0	0
$631$	−23.9463	−0.953286	−0.476643	−	0.879097i	$-0.658146\pi$
$631$	−23.9463	−0.953286	−0.476643	+	0.879097i	$0.658146\pi$
$632$	0	0
$633$	−44.8619	−1.78310
$634$	0	0
$635$	−51.7966	−2.05548
$636$	0	0
$637$	0	0
$638$	0	0
$639$	10.0145	0.396166
$640$	0	0
$641$	18.2101	0.719258	0.359629	−	0.933095i	$-0.382903\pi$
$641$	18.2101	0.719258	0.359629	+	0.933095i	$0.382903\pi$
$642$	0	0
$643$	−29.5698	−1.16612	−0.583060	−	0.812429i	$-0.698145\pi$
$643$	−29.5698	−1.16612	−0.583060	+	0.812429i	$0.698145\pi$
$644$	0	0
$645$	24.4936	0.964436
$646$	0	0
$647$	−22.2490	−0.874698	−0.437349	−	0.899292i	$-0.644083\pi$
$647$	−22.2490	−0.874698	−0.437349	+	0.899292i	$0.644083\pi$
$648$	0	0
$649$	8.74275	0.343183
$650$	0	0
$651$	0	0
$652$	0	0
$653$	32.4918	1.27150	0.635751	−	0.771894i	$-0.280691\pi$
$653$	32.4918	1.27150	0.635751	+	0.771894i	$0.280691\pi$
$654$	0	0
$655$	40.8927	1.59781
$656$	0	0
$657$	−16.5611	−0.646110
$658$	0	0
$659$	20.1618	0.785391	0.392695	−	0.919669i	$-0.371543\pi$
$659$	20.1618	0.785391	0.392695	+	0.919669i	$0.371543\pi$
$660$	0	0
$661$	29.5870	1.15080	0.575400	−	0.817872i	$-0.304846\pi$
$661$	29.5870	1.15080	0.575400	+	0.817872i	$0.304846\pi$
$662$	0	0
$663$	85.5971	3.32431
$664$	0	0
$665$	0	0
$666$	0	0
$667$	12.7307	0.492937
$668$	0	0
$669$	4.99840	0.193249
$670$	0	0
$671$	−6.21595	−0.239964
$672$	0	0
$673$	9.97995	0.384699	0.192349	−	0.981326i	$-0.438389\pi$
$673$	9.97995	0.384699	0.192349	+	0.981326i	$0.438389\pi$
$674$	0	0
$675$	−19.0489	−0.733192
$676$	0	0
$677$	−3.68285	−0.141543	−0.0707717	−	0.997493i	$-0.522546\pi$
$677$	−3.68285	−0.141543	−0.0707717	+	0.997493i	$0.522546\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	17.7795	0.681313
$682$	0	0
$683$	33.7957	1.29316	0.646579	−	0.762847i	$-0.276199\pi$
$683$	33.7957	1.29316	0.646579	+	0.762847i	$0.276199\pi$
$684$	0	0
$685$	−14.8614	−0.567826
$686$	0	0
$687$	37.3419	1.42468
$688$	0	0
$689$	−54.5954	−2.07992
$690$	0	0
$691$	−11.1423	−0.423873	−0.211936	−	0.977283i	$-0.567977\pi$
$691$	−11.1423	−0.423873	−0.211936	+	0.977283i	$0.567977\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−10.1029	−0.383223
$696$	0	0
$697$	−77.9937	−2.95422
$698$	0	0
$699$	−10.1279	−0.383074
$700$	0	0
$701$	−36.7249	−1.38708	−0.693539	−	0.720419i	$-0.743950\pi$
$701$	−36.7249	−1.38708	−0.693539	+	0.720419i	$0.743950\pi$
$702$	0	0
$703$	9.05242	0.341418
$704$	0	0
$705$	46.8085	1.76291
$706$	0	0
$707$	0	0
$708$	0	0
$709$	15.4053	0.578560	0.289280	−	0.957245i	$-0.406584\pi$
$709$	15.4053	0.578560	0.289280	+	0.957245i	$0.406584\pi$
$710$	0	0
$711$	−19.6630	−0.737421
$712$	0	0
$713$	−16.6429	−0.623282
$714$	0	0
$715$	47.0079	1.75800
$716$	0	0
$717$	−9.04170	−0.337668
$718$	0	0
$719$	32.0902	1.19676	0.598380	−	0.801212i	$-0.295811\pi$
$719$	32.0902	1.19676	0.598380	+	0.801212i	$0.295811\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	36.4798	1.35670
$724$	0	0
$725$	13.2850	0.493393
$726$	0	0
$727$	−41.0731	−1.52332	−0.761658	−	0.647979i	$-0.775614\pi$
$727$	−41.0731	−1.52332	−0.761658	+	0.647979i	$0.775614\pi$
$728$	0	0
$729$	9.90703	0.366927
$730$	0	0
$731$	28.2705	1.04562
$732$	0	0
$733$	24.7871	0.915534	0.457767	−	0.889072i	$-0.348649\pi$
$733$	24.7871	0.915534	0.457767	+	0.889072i	$0.348649\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	36.6315	1.34934
$738$	0	0
$739$	−38.0986	−1.40148	−0.700740	−	0.713417i	$-0.747147\pi$
$739$	−38.0986	−1.40148	−0.700740	+	0.713417i	$0.747147\pi$
$740$	0	0
$741$	−11.4356	−0.420098
$742$	0	0
$743$	−33.2083	−1.21830	−0.609148	−	0.793057i	$-0.708488\pi$
$743$	−33.2083	−1.21830	−0.609148	+	0.793057i	$0.708488\pi$
$744$	0	0
$745$	0.870031	0.0318755
$746$	0	0
$747$	−7.60878	−0.278391
$748$	0	0
$749$	0	0
$750$	0	0
$751$	37.8448	1.38098	0.690488	−	0.723344i	$-0.257396\pi$
$751$	37.8448	1.38098	0.690488	+	0.723344i	$0.257396\pi$
$752$	0	0
$753$	−23.3501	−0.850925
$754$	0	0
$755$	70.2784	2.55769
$756$	0	0
$757$	19.5426	0.710289	0.355145	−	0.934811i	$-0.384432\pi$
$757$	19.5426	0.710289	0.355145	+	0.934811i	$0.384432\pi$
$758$	0	0
$759$	26.3123	0.955077
$760$	0	0
$761$	−29.9515	−1.08574	−0.542870	−	0.839817i	$-0.682662\pi$
$761$	−29.9515	−1.08574	−0.542870	+	0.839817i	$0.682662\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	27.7595	1.00365
$766$	0	0
$767$	18.5402	0.669447
$768$	0	0
$769$	−15.2561	−0.550150	−0.275075	−	0.961423i	$-0.588703\pi$
$769$	−15.2561	−0.550150	−0.275075	+	0.961423i	$0.588703\pi$
$770$	0	0
$771$	−4.45851	−0.160569
$772$	0	0
$773$	15.2561	0.548724	0.274362	−	0.961626i	$-0.411533\pi$
$773$	15.2561	0.548724	0.274362	+	0.961626i	$0.411533\pi$
$774$	0	0
$775$	−17.3675	−0.623858
$776$	0	0
$777$	0	0
$778$	0	0
$779$	10.4198	0.373329
$780$	0	0
$781$	22.6600	0.810839
$782$	0	0
$783$	−9.76079	−0.348822
$784$	0	0
$785$	4.37947	0.156310
$786$	0	0
$787$	−1.08333	−0.0386166	−0.0193083	−	0.999814i	$-0.506146\pi$
$787$	−1.08333	−0.0386166	−0.0193083	+	0.999814i	$0.506146\pi$
$788$	0	0
$789$	39.5033	1.40636
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−13.1818	−0.468098
$794$	0	0
$795$	−63.2046	−2.24164
$796$	0	0
$797$	−17.9472	−0.635722	−0.317861	−	0.948137i	$-0.602965\pi$
$797$	−17.9472	−0.635722	−0.317861	+	0.948137i	$0.602965\pi$
$798$	0	0
$799$	54.0263	1.91131
$800$	0	0
$801$	−14.2293	−0.502767
$802$	0	0
$803$	−37.4733	−1.32240
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−23.4951	−0.827066
$808$	0	0
$809$	26.3254	0.925551	0.462775	−	0.886476i	$-0.346854\pi$
$809$	26.3254	0.925551	0.462775	+	0.886476i	$0.346854\pi$
$810$	0	0
$811$	52.2631	1.83521	0.917603	−	0.397498i	$-0.130122\pi$
$811$	52.2631	1.83521	0.917603	+	0.397498i	$0.130122\pi$
$812$	0	0
$813$	16.0080	0.561425
$814$	0	0
$815$	6.51485	0.228205
$816$	0	0
$817$	−3.77689	−0.132137
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−0.737155	−0.0257269	−0.0128634	−	0.999917i	$-0.504095\pi$
$821$	−0.737155	−0.0257269	−0.0128634	+	0.999917i	$0.504095\pi$
$822$	0	0
$823$	38.7222	1.34977	0.674885	−	0.737922i	$-0.264193\pi$
$823$	38.7222	1.34977	0.674885	+	0.737922i	$0.264193\pi$
$824$	0	0
$825$	27.4579	0.955961
$826$	0	0
$827$	−49.5257	−1.72218	−0.861088	−	0.508456i	$-0.830216\pi$
$827$	−49.5257	−1.72218	−0.861088	+	0.508456i	$0.830216\pi$
$828$	0	0
$829$	−34.9798	−1.21490	−0.607450	−	0.794358i	$-0.707808\pi$
$829$	−34.9798	−1.21490	−0.607450	+	0.794358i	$0.707808\pi$
$830$	0	0
$831$	−25.2417	−0.875625
$832$	0	0
$833$	0	0
$834$	0	0
$835$	11.1599	0.386204
$836$	0	0
$837$	12.7603	0.441060
$838$	0	0
$839$	26.6963	0.921660	0.460830	−	0.887488i	$-0.347552\pi$
$839$	26.6963	0.921660	0.460830	+	0.887488i	$0.347552\pi$
$840$	0	0
$841$	−22.1927	−0.765264
$842$	0	0
$843$	14.3502	0.494246
$844$	0	0
$845$	58.3887	2.00863
$846$	0	0
$847$	0	0
$848$	0	0
$849$	63.9403	2.19442
$850$	0	0
$851$	−44.1703	−1.51414
$852$	0	0
$853$	−12.4509	−0.426312	−0.213156	−	0.977018i	$-0.568374\pi$
$853$	−12.4509	−0.426312	−0.213156	+	0.977018i	$0.568374\pi$
$854$	0	0
$855$	−3.70862	−0.126832
$856$	0	0
$857$	21.5641	0.736615	0.368307	−	0.929704i	$-0.379937\pi$
$857$	21.5641	0.736615	0.368307	+	0.929704i	$0.379937\pi$
$858$	0	0
$859$	25.1796	0.859117	0.429558	−	0.903039i	$-0.358669\pi$
$859$	25.1796	0.859117	0.429558	+	0.903039i	$0.358669\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−5.39527	−0.183657	−0.0918286	−	0.995775i	$-0.529271\pi$
$863$	−5.39527	−0.183657	−0.0918286	+	0.995775i	$0.529271\pi$
$864$	0	0
$865$	74.9387	2.54799
$866$	0	0
$867$	79.6710	2.70577
$868$	0	0
$869$	−44.4921	−1.50929
$870$	0	0
$871$	77.6821	2.63216
$872$	0	0
$873$	−8.25010	−0.279224
$874$	0	0
$875$	0	0
$876$	0	0
$877$	6.02445	0.203431	0.101716	−	0.994814i	$-0.467567\pi$
$877$	6.02445	0.203431	0.101716	+	0.994814i	$0.467567\pi$
$878$	0	0
$879$	−22.5527	−0.760682
$880$	0	0
$881$	−36.2622	−1.22170	−0.610852	−	0.791745i	$-0.709173\pi$
$881$	−36.2622	−1.22170	−0.610852	+	0.791745i	$0.709173\pi$
$882$	0	0
$883$	14.3386	0.482533	0.241267	−	0.970459i	$-0.422437\pi$
$883$	14.3386	0.482533	0.241267	+	0.970459i	$0.422437\pi$
$884$	0	0
$885$	21.4638	0.721497
$886$	0	0
$887$	20.0310	0.672575	0.336287	−	0.941759i	$-0.390829\pi$
$887$	20.0310	0.672575	0.336287	+	0.941759i	$0.390829\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−29.4253	−0.985786
$892$	0	0
$893$	−7.21782	−0.241535
$894$	0	0
$895$	28.3803	0.948650
$896$	0	0
$897$	55.7988	1.86307
$898$	0	0
$899$	−8.89923	−0.296806
$900$	0	0
$901$	−72.9507	−2.43034
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−79.8143	−2.65312
$906$	0	0
$907$	−15.2833	−0.507475	−0.253737	−	0.967273i	$-0.581660\pi$
$907$	−15.2833	−0.507475	−0.253737	+	0.967273i	$0.581660\pi$
$908$	0	0
$909$	−5.05992	−0.167827
$910$	0	0
$911$	41.1998	1.36501	0.682506	−	0.730880i	$-0.260890\pi$
$911$	41.1998	1.36501	0.682506	+	0.730880i	$0.260890\pi$
$912$	0	0
$913$	−17.2166	−0.569787
$914$	0	0
$915$	−15.2604	−0.504493
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−27.0599	−0.892625	−0.446312	−	0.894877i	$-0.647263\pi$
$919$	−27.0599	−0.892625	−0.446312	+	0.894877i	$0.647263\pi$
$920$	0	0
$921$	−16.8301	−0.554571
$922$	0	0
$923$	48.0536	1.58170
$924$	0	0
$925$	−46.0933	−1.51554
$926$	0	0
$927$	−15.0849	−0.495452
$928$	0	0
$929$	−52.3083	−1.71618	−0.858090	−	0.513500i	$-0.828349\pi$
$929$	−52.3083	−1.71618	−0.858090	+	0.513500i	$0.828349\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	29.8364	0.976799
$934$	0	0
$935$	62.8122	2.05418
$936$	0	0
$937$	−17.8565	−0.583345	−0.291673	−	0.956518i	$-0.594212\pi$
$937$	−17.8565	−0.583345	−0.291673	+	0.956518i	$0.594212\pi$
$938$	0	0
$939$	−0.0366251	−0.00119522
$940$	0	0
$941$	20.6849	0.674310	0.337155	−	0.941449i	$-0.390535\pi$
$941$	20.6849	0.674310	0.337155	+	0.941449i	$0.390535\pi$
$942$	0	0
$943$	−50.8424	−1.65566
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−32.7059	−1.06280	−0.531399	−	0.847122i	$-0.678334\pi$
$947$	−32.7059	−1.06280	−0.531399	+	0.847122i	$0.678334\pi$
$948$	0	0
$949$	−79.4671	−2.57961
$950$	0	0
$951$	−43.8696	−1.42257
$952$	0	0
$953$	51.9928	1.68421	0.842107	−	0.539311i	$-0.181315\pi$
$953$	51.9928	1.68421	0.842107	+	0.539311i	$0.181315\pi$
$954$	0	0
$955$	58.2293	1.88425
$956$	0	0
$957$	14.0696	0.454806
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−19.3660	−0.624711
$962$	0	0
$963$	−3.07902	−0.0992199
$964$	0	0
$965$	−2.64850	−0.0852581
$966$	0	0
$967$	46.7892	1.50464	0.752320	−	0.658798i	$-0.228935\pi$
$967$	46.7892	1.50464	0.752320	+	0.658798i	$0.228935\pi$
$968$	0	0
$969$	−15.2803	−0.490875
$970$	0	0
$971$	−15.2980	−0.490936	−0.245468	−	0.969405i	$-0.578942\pi$
$971$	−15.2980	−0.490936	−0.245468	+	0.969405i	$0.578942\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	58.2281	1.86479
$976$	0	0
$977$	44.6983	1.43002	0.715012	−	0.699112i	$-0.246421\pi$
$977$	44.6983	1.43002	0.715012	+	0.699112i	$0.246421\pi$
$978$	0	0
$979$	−32.1970	−1.02902
$980$	0	0
$981$	−10.3305	−0.329826
$982$	0	0
$983$	−23.2120	−0.740347	−0.370174	−	0.928963i	$-0.620702\pi$
$983$	−23.2120	−0.740347	−0.370174	+	0.928963i	$0.620702\pi$
$984$	0	0
$985$	14.8092	0.471862
$986$	0	0
$987$	0	0
$988$	0	0
$989$	18.4289	0.586006
$990$	0	0
$991$	−46.9794	−1.49235	−0.746174	−	0.665751i	$-0.768111\pi$
$991$	−46.9794	−1.49235	−0.746174	+	0.665751i	$0.768111\pi$
$992$	0	0
$993$	−30.8933	−0.980370
$994$	0	0
$995$	−45.6933	−1.44858
$996$	0	0
$997$	−32.9126	−1.04235	−0.521176	−	0.853449i	$-0.674506\pi$
$997$	−32.9126	−1.04235	−0.521176	+	0.853449i	$0.674506\pi$
$998$	0	0
$999$	33.8658	1.07147

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7448.2.a.bn.1.5	yes	6
7.6	odd	2		7448.2.a.bm.1.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7448.2.a.bm.1.2	✓	6	7.6	odd	2
7448.2.a.bn.1.5	yes	6	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7448 = 2^{3} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7448.a (trivial)

\(f(q)\)	\(=\)	\(q+2.04143 q^{3} +3.17676 q^{5} +1.16742 q^{9} +O(q^{10})\)	\(q+2.04143 q^{3} +3.17676 q^{5} +1.16742 q^{9} +2.64156 q^{11} +5.60178 q^{13} +6.48513 q^{15} +7.48513 q^{17} -1.00000 q^{19} +4.87939 q^{23} +5.09182 q^{25} -3.74108 q^{27} +2.60909 q^{29} -3.41086 q^{31} +5.39255 q^{33} -9.05242 q^{37} +11.4356 q^{39} -10.4198 q^{41} +3.77689 q^{43} +3.70862 q^{45} +7.21782 q^{47} +15.2803 q^{51} -9.74609 q^{53} +8.39160 q^{55} -2.04143 q^{57} +3.30969 q^{59} -2.35314 q^{61} +17.7955 q^{65} +13.8674 q^{67} +9.96091 q^{69} +8.57828 q^{71} -14.1861 q^{73} +10.3946 q^{75} -16.8431 q^{79} -11.1394 q^{81} -6.51760 q^{83} +23.7785 q^{85} +5.32626 q^{87} -12.1886 q^{89} -6.96302 q^{93} -3.17676 q^{95} -7.06695 q^{97} +3.08381 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{3} + q^{5} + 3 q^{9}+O(q^{10}) \) 6 * q + 3 * q^3 + q^5 + 3 * q^9	\( 6 q + 3 q^{3} + q^{5} + 3 q^{9} + 3 q^{11} + 6 q^{13} - 8 q^{15} - 2 q^{17} - 6 q^{19} + 2 q^{23} + 5 q^{25} + 6 q^{27} - q^{29} + 14 q^{31} + 19 q^{33} - 7 q^{37} + 22 q^{39} - 21 q^{41} + q^{43} - 4 q^{45} + 23 q^{47} + 2 q^{51} - 15 q^{53} + 2 q^{55} - 3 q^{57} + 25 q^{59} + 21 q^{61} + 6 q^{65} + 2 q^{67} - 20 q^{69} + 13 q^{71} - 2 q^{73} + 24 q^{75} - 17 q^{79} - 2 q^{81} + 4 q^{83} + 40 q^{85} + 30 q^{87} - q^{89} + 6 q^{93} - q^{95} - 13 q^{97} + 41 q^{99}+O(q^{100}) \) 6 * q + 3 * q^3 + q^5 + 3 * q^9 + 3 * q^11 + 6 * q^13 - 8 * q^15 - 2 * q^17 - 6 * q^19 + 2 * q^23 + 5 * q^25 + 6 * q^27 - q^29 + 14 * q^31 + 19 * q^33 - 7 * q^37 + 22 * q^39 - 21 * q^41 + q^43 - 4 * q^45 + 23 * q^47 + 2 * q^51 - 15 * q^53 + 2 * q^55 - 3 * q^57 + 25 * q^59 + 21 * q^61 + 6 * q^65 + 2 * q^67 - 20 * q^69 + 13 * q^71 - 2 * q^73 + 24 * q^75 - 17 * q^79 - 2 * q^81 + 4 * q^83 + 40 * q^85 + 30 * q^87 - q^89 + 6 * q^93 - q^95 - 13 * q^97 + 41 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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