LMFDB - Embedded newform 7448.2.a.bw.1.13

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:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{13} - 27 x^{12} + 46 x^{11} + 286 x^{10} - 386 x^{9} - 1525 x^{8} + 1414 x^{7} + \cdots + 392 $ x^14 - 2x^13 - 27x^12 + 46x^11 + 286x^10 - 386x^9 - 1525x^8 + 1414x^7 + 4390x^6 - 1910x^5 - 6603x^4 - 610x^3 + 4031x^2 + 2352*x + 392
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			1.13
Root			$3.00074$ 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)

$f(q)$	$=$	$q+3.00074 q^{3} +1.81892 q^{5} +6.00446 q^{9} +O(q^{10})$	$q+3.00074 q^{3} +1.81892 q^{5} +6.00446 q^{9} +3.65489 q^{11} +1.31266 q^{13} +5.45812 q^{15} -1.00288 q^{17} -1.00000 q^{19} -2.54981 q^{23} -1.69153 q^{25} +9.01562 q^{27} -1.89214 q^{29} +7.63310 q^{31} +10.9674 q^{33} +5.65915 q^{37} +3.93896 q^{39} +3.71907 q^{41} +2.53347 q^{43} +10.9216 q^{45} -7.38310 q^{47} -3.00940 q^{51} -9.27345 q^{53} +6.64796 q^{55} -3.00074 q^{57} +5.58703 q^{59} -6.40629 q^{61} +2.38762 q^{65} +8.36219 q^{67} -7.65132 q^{69} +0.250341 q^{71} +12.2029 q^{73} -5.07583 q^{75} +8.76638 q^{79} +9.04018 q^{81} -12.0104 q^{83} -1.82417 q^{85} -5.67784 q^{87} -7.62041 q^{89} +22.9050 q^{93} -1.81892 q^{95} +11.9095 q^{97} +21.9457 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 2 q^{3} + 2 q^{5} + 16 q^{9}+O(q^{10}) $ 14 * q + 2 * q^3 + 2 * q^5 + 16 * q^9	$ 14 q + 2 q^{3} + 2 q^{5} + 16 q^{9} - 6 q^{11} + 16 q^{13} - 4 q^{15} + 20 q^{17} - 14 q^{19} + 4 q^{23} + 16 q^{25} + 20 q^{27} + 6 q^{29} + 34 q^{33} - 6 q^{37} + 8 q^{39} + 46 q^{41} - 18 q^{43} - 10 q^{47} - 4 q^{51} - 2 q^{53} + 28 q^{55} - 2 q^{57} - 22 q^{59} + 26 q^{61} + 8 q^{65} - 12 q^{67} + 48 q^{69} + 18 q^{71} + 28 q^{73} - 24 q^{75} - 10 q^{79} - 2 q^{81} + 8 q^{83} - 16 q^{85} + 16 q^{87} + 78 q^{89} - 2 q^{95} + 54 q^{97} + 26 q^{99}+O(q^{100}) $ 14 * q + 2 * q^3 + 2 * q^5 + 16 * q^9 - 6 * q^11 + 16 * q^13 - 4 * q^15 + 20 * q^17 - 14 * q^19 + 4 * q^23 + 16 * q^25 + 20 * q^27 + 6 * q^29 + 34 * q^33 - 6 * q^37 + 8 * q^39 + 46 * q^41 - 18 * q^43 - 10 * q^47 - 4 * q^51 - 2 * q^53 + 28 * q^55 - 2 * q^57 - 22 * q^59 + 26 * q^61 + 8 * q^65 - 12 * q^67 + 48 * q^69 + 18 * q^71 + 28 * q^73 - 24 * q^75 - 10 * q^79 - 2 * q^81 + 8 * q^83 - 16 * q^85 + 16 * q^87 + 78 * q^89 - 2 * q^95 + 54 * q^97 + 26 * 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
$3$	3.00074	1.73248	0.866240	−	0.499628i	$-0.166530\pi$
$3$	3.00074	1.73248	0.866240	+	0.499628i	$0.166530\pi$
$4$	0	0
$5$	1.81892	0.813446	0.406723	−	0.913551i	$-0.366671\pi$
$5$	1.81892	0.813446	0.406723	+	0.913551i	$0.366671\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	6.00446	2.00149
$10$	0	0
$11$	3.65489	1.10199	0.550996	−	0.834508i	$-0.314248\pi$
$11$	3.65489	1.10199	0.550996	+	0.834508i	$0.314248\pi$
$12$	0	0
$13$	1.31266	0.364066	0.182033	−	0.983292i	$-0.441732\pi$
$13$	1.31266	0.364066	0.182033	+	0.983292i	$0.441732\pi$
$14$	0	0
$15$	5.45812	1.40928
$16$	0	0
$17$	−1.00288	−0.243235	−0.121618	−	0.992577i	$-0.538808\pi$
$17$	−1.00288	−0.243235	−0.121618	+	0.992577i	$0.538808\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.54981	−0.531672	−0.265836	−	0.964018i	$-0.585648\pi$
$23$	−2.54981	−0.531672	−0.265836	+	0.964018i	$0.585648\pi$
$24$	0	0
$25$	−1.69153	−0.338305
$26$	0	0
$27$	9.01562	1.73506
$28$	0	0
$29$	−1.89214	−0.351362	−0.175681	−	0.984447i	$-0.556213\pi$
$29$	−1.89214	−0.351362	−0.175681	+	0.984447i	$0.556213\pi$
$30$	0	0
$31$	7.63310	1.37095	0.685473	−	0.728098i	$-0.259596\pi$
$31$	7.63310	1.37095	0.685473	+	0.728098i	$0.259596\pi$
$32$	0	0
$33$	10.9674	1.90918
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.65915	0.930359	0.465179	−	0.885216i	$-0.345990\pi$
$37$	5.65915	0.930359	0.465179	+	0.885216i	$0.345990\pi$
$38$	0	0
$39$	3.93896	0.630738
$40$	0	0
$41$	3.71907	0.580821	0.290410	−	0.956902i	$-0.406208\pi$
$41$	3.71907	0.580821	0.290410	+	0.956902i	$0.406208\pi$
$42$	0	0
$43$	2.53347	0.386351	0.193175	−	0.981164i	$-0.438121\pi$
$43$	2.53347	0.386351	0.193175	+	0.981164i	$0.438121\pi$
$44$	0	0
$45$	10.9216	1.62810
$46$	0	0
$47$	−7.38310	−1.07694	−0.538468	−	0.842646i	$-0.680997\pi$
$47$	−7.38310	−1.07694	−0.538468	+	0.842646i	$0.680997\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−3.00940	−0.421400
$52$	0	0
$53$	−9.27345	−1.27381	−0.636903	−	0.770944i	$-0.719785\pi$
$53$	−9.27345	−1.27381	−0.636903	+	0.770944i	$0.719785\pi$
$54$	0	0
$55$	6.64796	0.896411
$56$	0	0
$57$	−3.00074	−0.397458
$58$	0	0
$59$	5.58703	0.727369	0.363684	−	0.931522i	$-0.381519\pi$
$59$	5.58703	0.727369	0.363684	+	0.931522i	$0.381519\pi$
$60$	0	0
$61$	−6.40629	−0.820241	−0.410120	−	0.912031i	$-0.634513\pi$
$61$	−6.40629	−0.820241	−0.410120	+	0.912031i	$0.634513\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.38762	0.296148
$66$	0	0
$67$	8.36219	1.02160	0.510802	−	0.859699i	$-0.329349\pi$
$67$	8.36219	1.02160	0.510802	+	0.859699i	$0.329349\pi$
$68$	0	0
$69$	−7.65132	−0.921111
$70$	0	0
$71$	0.250341	0.0297101	0.0148550	−	0.999890i	$-0.495271\pi$
$71$	0.250341	0.0297101	0.0148550	+	0.999890i	$0.495271\pi$
$72$	0	0
$73$	12.2029	1.42825	0.714123	−	0.700020i	$-0.246826\pi$
$73$	12.2029	1.42825	0.714123	+	0.700020i	$0.246826\pi$
$74$	0	0
$75$	−5.07583	−0.586107
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.76638	0.986295	0.493148	−	0.869946i	$-0.335846\pi$
$79$	8.76638	0.986295	0.493148	+	0.869946i	$0.335846\pi$
$80$	0	0
$81$	9.04018	1.00446
$82$	0	0
$83$	−12.0104	−1.31831	−0.659154	−	0.752008i	$-0.729086\pi$
$83$	−12.0104	−1.31831	−0.659154	+	0.752008i	$0.729086\pi$
$84$	0	0
$85$	−1.82417	−0.197859
$86$	0	0
$87$	−5.67784	−0.608728
$88$	0	0
$89$	−7.62041	−0.807762	−0.403881	−	0.914812i	$-0.632339\pi$
$89$	−7.62041	−0.807762	−0.403881	+	0.914812i	$0.632339\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	22.9050	2.37514
$94$	0	0
$95$	−1.81892	−0.186617
$96$	0	0
$97$	11.9095	1.20922	0.604612	−	0.796520i	$-0.293328\pi$
$97$	11.9095	1.20922	0.604612	+	0.796520i	$0.293328\pi$
$98$	0	0
$99$	21.9457	2.20562
$100$	0	0
$101$	7.32317	0.728683	0.364341	−	0.931265i	$-0.381294\pi$
$101$	7.32317	0.728683	0.364341	+	0.931265i	$0.381294\pi$
$102$	0	0
$103$	−4.50560	−0.443950	−0.221975	−	0.975052i	$-0.571250\pi$
$103$	−4.50560	−0.443950	−0.221975	+	0.975052i	$0.571250\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.01626	0.678287	0.339144	−	0.940735i	$-0.389863\pi$
$107$	7.01626	0.678287	0.339144	+	0.940735i	$0.389863\pi$
$108$	0	0
$109$	14.8524	1.42261	0.711303	−	0.702886i	$-0.248106\pi$
$109$	14.8524	1.42261	0.711303	+	0.702886i	$0.248106\pi$
$110$	0	0
$111$	16.9817	1.61183
$112$	0	0
$113$	7.76933	0.730877	0.365438	−	0.930835i	$-0.380919\pi$
$113$	7.76933	0.730877	0.365438	+	0.930835i	$0.380919\pi$
$114$	0	0
$115$	−4.63790	−0.432487
$116$	0	0
$117$	7.88182	0.728674
$118$	0	0
$119$	0	0
$120$	0	0
$121$	2.35824	0.214386
$122$	0	0
$123$	11.1600	1.00626
$124$	0	0
$125$	−12.1714	−1.08864
$126$	0	0
$127$	−13.0190	−1.15525	−0.577624	−	0.816303i	$-0.696020\pi$
$127$	−13.0190	−1.15525	−0.577624	+	0.816303i	$0.696020\pi$
$128$	0	0
$129$	7.60230	0.669345
$130$	0	0
$131$	−9.11959	−0.796782	−0.398391	−	0.917216i	$-0.630431\pi$
$131$	−9.11959	−0.796782	−0.398391	+	0.917216i	$0.630431\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	16.3987	1.41138
$136$	0	0
$137$	−10.9471	−0.935277	−0.467638	−	0.883920i	$-0.654895\pi$
$137$	−10.9471	−0.935277	−0.467638	+	0.883920i	$0.654895\pi$
$138$	0	0
$139$	−4.12834	−0.350161	−0.175081	−	0.984554i	$-0.556019\pi$
$139$	−4.12834	−0.350161	−0.175081	+	0.984554i	$0.556019\pi$
$140$	0	0
$141$	−22.1548	−1.86577
$142$	0	0
$143$	4.79763	0.401198
$144$	0	0
$145$	−3.44166	−0.285814
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3.87772	0.317675	0.158838	−	0.987305i	$-0.449225\pi$
$149$	3.87772	0.317675	0.158838	+	0.987305i	$0.449225\pi$
$150$	0	0
$151$	−3.76480	−0.306375	−0.153187	−	0.988197i	$-0.548954\pi$
$151$	−3.76480	−0.306375	−0.153187	+	0.988197i	$0.548954\pi$
$152$	0	0
$153$	−6.02178	−0.486832
$154$	0	0
$155$	13.8840	1.11519
$156$	0	0
$157$	−17.5862	−1.40353	−0.701767	−	0.712406i	$-0.747605\pi$
$157$	−17.5862	−1.40353	−0.701767	+	0.712406i	$0.747605\pi$
$158$	0	0
$159$	−27.8272	−2.20684
$160$	0	0
$161$	0	0
$162$	0	0
$163$	2.93699	0.230043	0.115021	−	0.993363i	$-0.463306\pi$
$163$	2.93699	0.230043	0.115021	+	0.993363i	$0.463306\pi$
$164$	0	0
$165$	19.9488	1.55301
$166$	0	0
$167$	3.96456	0.306787	0.153393	−	0.988165i	$-0.450980\pi$
$167$	3.96456	0.306787	0.153393	+	0.988165i	$0.450980\pi$
$168$	0	0
$169$	−11.2769	−0.867456
$170$	0	0
$171$	−6.00446	−0.459173
$172$	0	0
$173$	11.3357	0.861835	0.430918	−	0.902391i	$-0.358190\pi$
$173$	11.3357	0.861835	0.430918	+	0.902391i	$0.358190\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	16.7652	1.26015
$178$	0	0
$179$	−2.52290	−0.188570	−0.0942852	−	0.995545i	$-0.530057\pi$
$179$	−2.52290	−0.188570	−0.0942852	+	0.995545i	$0.530057\pi$
$180$	0	0
$181$	−25.2264	−1.87507	−0.937533	−	0.347896i	$-0.886896\pi$
$181$	−25.2264	−1.87507	−0.937533	+	0.347896i	$0.886896\pi$
$182$	0	0
$183$	−19.2236	−1.42105
$184$	0	0
$185$	10.2935	0.756797
$186$	0	0
$187$	−3.66543	−0.268043
$188$	0	0
$189$	0	0
$190$	0	0
$191$	15.9720	1.15569	0.577846	−	0.816146i	$-0.303893\pi$
$191$	15.9720	1.15569	0.577846	+	0.816146i	$0.303893\pi$
$192$	0	0
$193$	−16.1246	−1.16067	−0.580336	−	0.814377i	$-0.697079\pi$
$193$	−16.1246	−1.16067	−0.580336	+	0.814377i	$0.697079\pi$
$194$	0	0
$195$	7.16465	0.513071
$196$	0	0
$197$	−5.24033	−0.373358	−0.186679	−	0.982421i	$-0.559772\pi$
$197$	−5.24033	−0.373358	−0.186679	+	0.982421i	$0.559772\pi$
$198$	0	0
$199$	−25.0219	−1.77375	−0.886876	−	0.462007i	$-0.847129\pi$
$199$	−25.0219	−1.77375	−0.886876	+	0.462007i	$0.847129\pi$
$200$	0	0
$201$	25.0928	1.76991
$202$	0	0
$203$	0	0
$204$	0	0
$205$	6.76469	0.472467
$206$	0	0
$207$	−15.3102	−1.06413
$208$	0	0
$209$	−3.65489	−0.252814
$210$	0	0
$211$	−28.0679	−1.93227	−0.966137	−	0.258029i	$-0.916927\pi$
$211$	−28.0679	−1.93227	−0.966137	+	0.258029i	$0.916927\pi$
$212$	0	0
$213$	0.751210	0.0514721
$214$	0	0
$215$	4.60819	0.314276
$216$	0	0
$217$	0	0
$218$	0	0
$219$	36.6179	2.47441
$220$	0	0
$221$	−1.31645	−0.0885537
$222$	0	0
$223$	−4.82425	−0.323056	−0.161528	−	0.986868i	$-0.551642\pi$
$223$	−4.82425	−0.323056	−0.161528	+	0.986868i	$0.551642\pi$
$224$	0	0
$225$	−10.1567	−0.677113
$226$	0	0
$227$	1.15784	0.0768488	0.0384244	−	0.999262i	$-0.487766\pi$
$227$	1.15784	0.0768488	0.0384244	+	0.999262i	$0.487766\pi$
$228$	0	0
$229$	20.2100	1.33551	0.667756	−	0.744380i	$-0.267255\pi$
$229$	20.2100	1.33551	0.667756	+	0.744380i	$0.267255\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4.39890	0.288182	0.144091	−	0.989564i	$-0.453974\pi$
$233$	4.39890	0.288182	0.144091	+	0.989564i	$0.453974\pi$
$234$	0	0
$235$	−13.4293	−0.876029
$236$	0	0
$237$	26.3057	1.70874
$238$	0	0
$239$	2.95704	0.191275	0.0956374	−	0.995416i	$-0.469511\pi$
$239$	2.95704	0.191275	0.0956374	+	0.995416i	$0.469511\pi$
$240$	0	0
$241$	24.9497	1.60715	0.803575	−	0.595204i	$-0.202929\pi$
$241$	24.9497	1.60715	0.803575	+	0.595204i	$0.202929\pi$
$242$	0	0
$243$	0.0804042	0.00515793
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.31266	−0.0835225
$248$	0	0
$249$	−36.0400	−2.28394
$250$	0	0
$251$	3.15732	0.199288	0.0996441	−	0.995023i	$-0.468230\pi$
$251$	3.15732	0.199288	0.0996441	+	0.995023i	$0.468230\pi$
$252$	0	0
$253$	−9.31928	−0.585898
$254$	0	0
$255$	−5.47386	−0.342786
$256$	0	0
$257$	23.0510	1.43788	0.718940	−	0.695072i	$-0.244628\pi$
$257$	23.0510	1.43788	0.718940	+	0.695072i	$0.244628\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−11.3613	−0.703247
$262$	0	0
$263$	−5.03340	−0.310373	−0.155187	−	0.987885i	$-0.549598\pi$
$263$	−5.03340	−0.310373	−0.155187	+	0.987885i	$0.549598\pi$
$264$	0	0
$265$	−16.8677	−1.03617
$266$	0	0
$267$	−22.8669	−1.39943
$268$	0	0
$269$	3.33507	0.203343	0.101672	−	0.994818i	$-0.467581\pi$
$269$	3.33507	0.203343	0.101672	+	0.994818i	$0.467581\pi$
$270$	0	0
$271$	−15.3231	−0.930812	−0.465406	−	0.885097i	$-0.654092\pi$
$271$	−15.3231	−0.930812	−0.465406	+	0.885097i	$0.654092\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−6.18234	−0.372809
$276$	0	0
$277$	8.67376	0.521156	0.260578	−	0.965453i	$-0.416087\pi$
$277$	8.67376	0.521156	0.260578	+	0.965453i	$0.416087\pi$
$278$	0	0
$279$	45.8327	2.74393
$280$	0	0
$281$	−0.0359612	−0.00214526	−0.00107263	−	0.999999i	$-0.500341\pi$
$281$	−0.0359612	−0.00214526	−0.00107263	+	0.999999i	$0.500341\pi$
$282$	0	0
$283$	25.1191	1.49317	0.746587	−	0.665288i	$-0.231691\pi$
$283$	25.1191	1.49317	0.746587	+	0.665288i	$0.231691\pi$
$284$	0	0
$285$	−5.45812	−0.323311
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−15.9942	−0.940837
$290$	0	0
$291$	35.7373	2.09496
$292$	0	0
$293$	−2.68183	−0.156674	−0.0783372	−	0.996927i	$-0.524961\pi$
$293$	−2.68183	−0.156674	−0.0783372	+	0.996927i	$0.524961\pi$
$294$	0	0
$295$	10.1624	0.591675
$296$	0	0
$297$	32.9511	1.91202
$298$	0	0
$299$	−3.34703	−0.193564
$300$	0	0
$301$	0	0
$302$	0	0
$303$	21.9750	1.26243
$304$	0	0
$305$	−11.6525	−0.667222
$306$	0	0
$307$	24.1094	1.37600	0.687999	−	0.725712i	$-0.258489\pi$
$307$	24.1094	1.37600	0.687999	+	0.725712i	$0.258489\pi$
$308$	0	0
$309$	−13.5202	−0.769135
$310$	0	0
$311$	−29.6242	−1.67984	−0.839918	−	0.542713i	$-0.817397\pi$
$311$	−29.6242	−1.67984	−0.839918	+	0.542713i	$0.817397\pi$
$312$	0	0
$313$	13.1006	0.740487	0.370243	−	0.928935i	$-0.379274\pi$
$313$	13.1006	0.740487	0.370243	+	0.928935i	$0.379274\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−28.9498	−1.62598	−0.812992	−	0.582275i	$-0.802163\pi$
$317$	−28.9498	−1.62598	−0.812992	+	0.582275i	$0.802163\pi$
$318$	0	0
$319$	−6.91558	−0.387198
$320$	0	0
$321$	21.0540	1.17512
$322$	0	0
$323$	1.00288	0.0558019
$324$	0	0
$325$	−2.22040	−0.123166
$326$	0	0
$327$	44.5684	2.46464
$328$	0	0
$329$	0	0
$330$	0	0
$331$	26.0410	1.43134	0.715671	−	0.698437i	$-0.246121\pi$
$331$	26.0410	1.43134	0.715671	+	0.698437i	$0.246121\pi$
$332$	0	0
$333$	33.9802	1.86210
$334$	0	0
$335$	15.2102	0.831020
$336$	0	0
$337$	−3.70637	−0.201899	−0.100949	−	0.994892i	$-0.532188\pi$
$337$	−3.70637	−0.201899	−0.100949	+	0.994892i	$0.532188\pi$
$338$	0	0
$339$	23.3138	1.26623
$340$	0	0
$341$	27.8982	1.51077
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−13.9172	−0.749274
$346$	0	0
$347$	−15.6983	−0.842727	−0.421363	−	0.906892i	$-0.638448\pi$
$347$	−15.6983	−0.842727	−0.421363	+	0.906892i	$0.638448\pi$
$348$	0	0
$349$	−5.20003	−0.278351	−0.139176	−	0.990268i	$-0.544445\pi$
$349$	−5.20003	−0.278351	−0.139176	+	0.990268i	$0.544445\pi$
$350$	0	0
$351$	11.8344	0.631676
$352$	0	0
$353$	5.33630	0.284023	0.142011	−	0.989865i	$-0.454643\pi$
$353$	5.33630	0.284023	0.142011	+	0.989865i	$0.454643\pi$
$354$	0	0
$355$	0.455351	0.0241675
$356$	0	0
$357$	0	0
$358$	0	0
$359$	26.9516	1.42245	0.711225	−	0.702964i	$-0.248140\pi$
$359$	26.9516	1.42245	0.711225	+	0.702964i	$0.248140\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	7.07648	0.371419
$364$	0	0
$365$	22.1962	1.16180
$366$	0	0
$367$	−5.45871	−0.284942	−0.142471	−	0.989799i	$-0.545505\pi$
$367$	−5.45871	−0.284942	−0.142471	+	0.989799i	$0.545505\pi$
$368$	0	0
$369$	22.3310	1.16251
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−17.8737	−0.925468	−0.462734	−	0.886497i	$-0.653131\pi$
$373$	−17.8737	−0.925468	−0.462734	+	0.886497i	$0.653131\pi$
$374$	0	0
$375$	−36.5231	−1.88605
$376$	0	0
$377$	−2.48374	−0.127919
$378$	0	0
$379$	32.8054	1.68510	0.842549	−	0.538620i	$-0.181054\pi$
$379$	32.8054	1.68510	0.842549	+	0.538620i	$0.181054\pi$
$380$	0	0
$381$	−39.0667	−2.00145
$382$	0	0
$383$	9.99560	0.510751	0.255376	−	0.966842i	$-0.417801\pi$
$383$	9.99560	0.510751	0.255376	+	0.966842i	$0.417801\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	15.2121	0.773276
$388$	0	0
$389$	−34.8525	−1.76709	−0.883547	−	0.468343i	$-0.844851\pi$
$389$	−34.8525	−1.76709	−0.883547	+	0.468343i	$0.844851\pi$
$390$	0	0
$391$	2.55716	0.129321
$392$	0	0
$393$	−27.3655	−1.38041
$394$	0	0
$395$	15.9454	0.802298
$396$	0	0
$397$	−12.9221	−0.648542	−0.324271	−	0.945964i	$-0.605119\pi$
$397$	−12.9221	−0.648542	−0.324271	+	0.945964i	$0.605119\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−31.9937	−1.59769	−0.798845	−	0.601537i	$-0.794555\pi$
$401$	−31.9937	−1.59769	−0.798845	+	0.601537i	$0.794555\pi$
$402$	0	0
$403$	10.0197	0.499115
$404$	0	0
$405$	16.4434	0.817078
$406$	0	0
$407$	20.6836	1.02525
$408$	0	0
$409$	5.31073	0.262599	0.131299	−	0.991343i	$-0.458085\pi$
$409$	5.31073	0.262599	0.131299	+	0.991343i	$0.458085\pi$
$410$	0	0
$411$	−32.8495	−1.62035
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−21.8459	−1.07237
$416$	0	0
$417$	−12.3881	−0.606648
$418$	0	0
$419$	−30.6349	−1.49661	−0.748306	−	0.663353i	$-0.769133\pi$
$419$	−30.6349	−1.49661	−0.748306	+	0.663353i	$0.769133\pi$
$420$	0	0
$421$	19.7833	0.964179	0.482089	−	0.876122i	$-0.339878\pi$
$421$	19.7833	0.964179	0.482089	+	0.876122i	$0.339878\pi$
$422$	0	0
$423$	−44.3315	−2.15547
$424$	0	0
$425$	1.69640	0.0822877
$426$	0	0
$427$	0	0
$428$	0	0
$429$	14.3965	0.695068
$430$	0	0
$431$	−13.8503	−0.667144	−0.333572	−	0.942725i	$-0.608254\pi$
$431$	−13.8503	−0.667144	−0.333572	+	0.942725i	$0.608254\pi$
$432$	0	0
$433$	−18.5690	−0.892370	−0.446185	−	0.894941i	$-0.647218\pi$
$433$	−18.5690	−0.892370	−0.446185	+	0.894941i	$0.647218\pi$
$434$	0	0
$435$	−10.3275	−0.495168
$436$	0	0
$437$	2.54981	0.121974
$438$	0	0
$439$	22.0519	1.05248	0.526241	−	0.850336i	$-0.323601\pi$
$439$	22.0519	1.05248	0.526241	+	0.850336i	$0.323601\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−6.52561	−0.310041	−0.155021	−	0.987911i	$-0.549544\pi$
$443$	−6.52561	−0.310041	−0.155021	+	0.987911i	$0.549544\pi$
$444$	0	0
$445$	−13.8609	−0.657071
$446$	0	0
$447$	11.6360	0.550366
$448$	0	0
$449$	34.3922	1.62307	0.811534	−	0.584306i	$-0.198633\pi$
$449$	34.3922	1.62307	0.811534	+	0.584306i	$0.198633\pi$
$450$	0	0
$451$	13.5928	0.640060
$452$	0	0
$453$	−11.2972	−0.530788
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−7.59781	−0.355411	−0.177705	−	0.984084i	$-0.556867\pi$
$457$	−7.59781	−0.355411	−0.177705	+	0.984084i	$0.556867\pi$
$458$	0	0
$459$	−9.04162	−0.422027
$460$	0	0
$461$	15.4373	0.718985	0.359492	−	0.933148i	$-0.382950\pi$
$461$	15.4373	0.718985	0.359492	+	0.933148i	$0.382950\pi$
$462$	0	0
$463$	−32.5376	−1.51215	−0.756076	−	0.654484i	$-0.772886\pi$
$463$	−32.5376	−1.51215	−0.756076	+	0.654484i	$0.772886\pi$
$464$	0	0
$465$	41.6624	1.93205
$466$	0	0
$467$	42.6245	1.97243	0.986213	−	0.165480i	$-0.0529173\pi$
$467$	42.6245	1.97243	0.986213	+	0.165480i	$0.0529173\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−52.7718	−2.43160
$472$	0	0
$473$	9.25957	0.425755
$474$	0	0
$475$	1.69153	0.0776125
$476$	0	0
$477$	−55.6821	−2.54951
$478$	0	0
$479$	−33.4787	−1.52968	−0.764840	−	0.644220i	$-0.777182\pi$
$479$	−33.4787	−1.52968	−0.764840	+	0.644220i	$0.777182\pi$
$480$	0	0
$481$	7.42854	0.338712
$482$	0	0
$483$	0	0
$484$	0	0
$485$	21.6624	0.983639
$486$	0	0
$487$	−14.7697	−0.669280	−0.334640	−	0.942346i	$-0.608615\pi$
$487$	−14.7697	−0.669280	−0.334640	+	0.942346i	$0.608615\pi$
$488$	0	0
$489$	8.81315	0.398545
$490$	0	0
$491$	−24.3921	−1.10080	−0.550400	−	0.834901i	$-0.685525\pi$
$491$	−24.3921	−1.10080	−0.550400	+	0.834901i	$0.685525\pi$
$492$	0	0
$493$	1.89760	0.0854636
$494$	0	0
$495$	39.9174	1.79416
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−3.70369	−0.165800	−0.0829001	−	0.996558i	$-0.526418\pi$
$499$	−3.70369	−0.165800	−0.0829001	+	0.996558i	$0.526418\pi$
$500$	0	0
$501$	11.8966	0.531502
$502$	0	0
$503$	4.56977	0.203756	0.101878	−	0.994797i	$-0.467515\pi$
$503$	4.56977	0.203756	0.101878	+	0.994797i	$0.467515\pi$
$504$	0	0
$505$	13.3203	0.592744
$506$	0	0
$507$	−33.8392	−1.50285
$508$	0	0
$509$	−26.9805	−1.19589	−0.597945	−	0.801537i	$-0.704016\pi$
$509$	−26.9805	−1.19589	−0.597945	+	0.801537i	$0.704016\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−9.01562	−0.398049
$514$	0	0
$515$	−8.19534	−0.361130
$516$	0	0
$517$	−26.9844	−1.18677
$518$	0	0
$519$	34.0154	1.49311
$520$	0	0
$521$	−10.9015	−0.477603	−0.238801	−	0.971068i	$-0.576754\pi$
$521$	−10.9015	−0.477603	−0.238801	+	0.971068i	$0.576754\pi$
$522$	0	0
$523$	9.41464	0.411674	0.205837	−	0.978586i	$-0.434008\pi$
$523$	9.41464	0.411674	0.205837	+	0.978586i	$0.434008\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−7.65511	−0.333462
$528$	0	0
$529$	−16.4985	−0.717325
$530$	0	0
$531$	33.5471	1.45582
$532$	0	0
$533$	4.88187	0.211457
$534$	0	0
$535$	12.7620	0.551750
$536$	0	0
$537$	−7.57058	−0.326694
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−4.31267	−0.185416	−0.0927081	−	0.995693i	$-0.529552\pi$
$541$	−4.31267	−0.185416	−0.0927081	+	0.995693i	$0.529552\pi$
$542$	0	0
$543$	−75.6981	−3.24851
$544$	0	0
$545$	27.0154	1.15721
$546$	0	0
$547$	−29.3028	−1.25290	−0.626449	−	0.779462i	$-0.715492\pi$
$547$	−29.3028	−1.25290	−0.626449	+	0.779462i	$0.715492\pi$
$548$	0	0
$549$	−38.4663	−1.64170
$550$	0	0
$551$	1.89214	0.0806081
$552$	0	0
$553$	0	0
$554$	0	0
$555$	30.8883	1.31114
$556$	0	0
$557$	24.6041	1.04251	0.521254	−	0.853402i	$-0.325464\pi$
$557$	24.6041	1.04251	0.521254	+	0.853402i	$0.325464\pi$
$558$	0	0
$559$	3.32559	0.140657
$560$	0	0
$561$	−10.9990	−0.464379
$562$	0	0
$563$	28.2943	1.19246	0.596232	−	0.802812i	$-0.296664\pi$
$563$	28.2943	1.19246	0.596232	+	0.802812i	$0.296664\pi$
$564$	0	0
$565$	14.1318	0.594529
$566$	0	0
$567$	0	0
$568$	0	0
$569$	29.3344	1.22976	0.614882	−	0.788619i	$-0.289204\pi$
$569$	29.3344	1.22976	0.614882	+	0.788619i	$0.289204\pi$
$570$	0	0
$571$	−8.47496	−0.354666	−0.177333	−	0.984151i	$-0.556747\pi$
$571$	−8.47496	−0.354666	−0.177333	+	0.984151i	$0.556747\pi$
$572$	0	0
$573$	47.9278	2.00221
$574$	0	0
$575$	4.31307	0.179867
$576$	0	0
$577$	47.1325	1.96215	0.981076	−	0.193621i	$-0.0620234\pi$
$577$	47.1325	1.96215	0.981076	+	0.193621i	$0.0620234\pi$
$578$	0	0
$579$	−48.3857	−2.01084
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−33.8935	−1.40372
$584$	0	0
$585$	14.3364	0.592737
$586$	0	0
$587$	−19.6848	−0.812479	−0.406239	−	0.913767i	$-0.633160\pi$
$587$	−19.6848	−0.812479	−0.406239	+	0.913767i	$0.633160\pi$
$588$	0	0
$589$	−7.63310	−0.314516
$590$	0	0
$591$	−15.7249	−0.646836
$592$	0	0
$593$	38.5343	1.58241	0.791206	−	0.611549i	$-0.209453\pi$
$593$	38.5343	1.58241	0.791206	+	0.611549i	$0.209453\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−75.0842	−3.07299
$598$	0	0
$599$	−9.77277	−0.399304	−0.199652	−	0.979867i	$-0.563981\pi$
$599$	−9.77277	−0.399304	−0.199652	+	0.979867i	$0.563981\pi$
$600$	0	0
$601$	6.12489	0.249840	0.124920	−	0.992167i	$-0.460133\pi$
$601$	6.12489	0.249840	0.124920	+	0.992167i	$0.460133\pi$
$602$	0	0
$603$	50.2104	2.04473
$604$	0	0
$605$	4.28946	0.174391
$606$	0	0
$607$	6.72528	0.272971	0.136485	−	0.990642i	$-0.456419\pi$
$607$	6.72528	0.272971	0.136485	+	0.990642i	$0.456419\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−9.69150	−0.392076
$612$	0	0
$613$	29.1063	1.17559	0.587796	−	0.809009i	$-0.299996\pi$
$613$	29.1063	1.17559	0.587796	+	0.809009i	$0.299996\pi$
$614$	0	0
$615$	20.2991	0.818539
$616$	0	0
$617$	−14.5304	−0.584971	−0.292486	−	0.956270i	$-0.594482\pi$
$617$	−14.5304	−0.584971	−0.292486	+	0.956270i	$0.594482\pi$
$618$	0	0
$619$	−32.1373	−1.29171	−0.645854	−	0.763461i	$-0.723498\pi$
$619$	−32.1373	−1.29171	−0.645854	+	0.763461i	$0.723498\pi$
$620$	0	0
$621$	−22.9881	−0.922481
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−13.6811	−0.547245
$626$	0	0
$627$	−10.9674	−0.437996
$628$	0	0
$629$	−5.67547	−0.226296
$630$	0	0
$631$	−30.9281	−1.23123	−0.615614	−	0.788048i	$-0.711092\pi$
$631$	−30.9281	−1.23123	−0.615614	+	0.788048i	$0.711092\pi$
$632$	0	0
$633$	−84.2246	−3.34763
$634$	0	0
$635$	−23.6805	−0.939733
$636$	0	0
$637$	0	0
$638$	0	0
$639$	1.50317	0.0594643
$640$	0	0
$641$	−18.2586	−0.721172	−0.360586	−	0.932726i	$-0.617423\pi$
$641$	−18.2586	−0.721172	−0.360586	+	0.932726i	$0.617423\pi$
$642$	0	0
$643$	39.0623	1.54047	0.770234	−	0.637761i	$-0.220139\pi$
$643$	39.0623	1.54047	0.770234	+	0.637761i	$0.220139\pi$
$644$	0	0
$645$	13.8280	0.544476
$646$	0	0
$647$	−30.5021	−1.19916	−0.599581	−	0.800314i	$-0.704666\pi$
$647$	−30.5021	−1.19916	−0.599581	+	0.800314i	$0.704666\pi$
$648$	0	0
$649$	20.4200	0.801554
$650$	0	0
$651$	0	0
$652$	0	0
$653$	34.5806	1.35324	0.676622	−	0.736330i	$-0.263443\pi$
$653$	34.5806	1.35324	0.676622	+	0.736330i	$0.263443\pi$
$654$	0	0
$655$	−16.5878	−0.648140
$656$	0	0
$657$	73.2721	2.85862
$658$	0	0
$659$	−38.7784	−1.51059	−0.755296	−	0.655384i	$-0.772507\pi$
$659$	−38.7784	−1.51059	−0.755296	+	0.655384i	$0.772507\pi$
$660$	0	0
$661$	−22.0108	−0.856120	−0.428060	−	0.903750i	$-0.640803\pi$
$661$	−22.0108	−0.856120	−0.428060	+	0.903750i	$0.640803\pi$
$662$	0	0
$663$	−3.95031	−0.153418
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4.82461	0.186810
$668$	0	0
$669$	−14.4763	−0.559688
$670$	0	0
$671$	−23.4143	−0.903898
$672$	0	0
$673$	−37.0202	−1.42703	−0.713513	−	0.700642i	$-0.752897\pi$
$673$	−37.0202	−1.42703	−0.713513	+	0.700642i	$0.752897\pi$
$674$	0	0
$675$	−15.2502	−0.586979
$676$	0	0
$677$	−3.01583	−0.115908	−0.0579539	−	0.998319i	$-0.518458\pi$
$677$	−3.01583	−0.115908	−0.0579539	+	0.998319i	$0.518458\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	3.47439	0.133139
$682$	0	0
$683$	15.8112	0.604999	0.302500	−	0.953150i	$-0.402179\pi$
$683$	15.8112	0.604999	0.302500	+	0.953150i	$0.402179\pi$
$684$	0	0
$685$	−19.9120	−0.760798
$686$	0	0
$687$	60.6449	2.31375
$688$	0	0
$689$	−12.1729	−0.463750
$690$	0	0
$691$	13.4643	0.512206	0.256103	−	0.966650i	$-0.417561\pi$
$691$	13.4643	0.512206	0.256103	+	0.966650i	$0.417561\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7.50913	−0.284838
$696$	0	0
$697$	−3.72979	−0.141276
$698$	0	0
$699$	13.2000	0.499269
$700$	0	0
$701$	0.718720	0.0271457	0.0135728	−	0.999908i	$-0.495679\pi$
$701$	0.718720	0.0271457	0.0135728	+	0.999908i	$0.495679\pi$
$702$	0	0
$703$	−5.65915	−0.213439
$704$	0	0
$705$	−40.2978	−1.51770
$706$	0	0
$707$	0	0
$708$	0	0
$709$	31.0145	1.16477	0.582386	−	0.812912i	$-0.302119\pi$
$709$	31.0145	1.16477	0.582386	+	0.812912i	$0.302119\pi$
$710$	0	0
$711$	52.6374	1.97406
$712$	0	0
$713$	−19.4630	−0.728893
$714$	0	0
$715$	8.72651	0.326353
$716$	0	0
$717$	8.87331	0.331380
$718$	0	0
$719$	−1.18227	−0.0440913	−0.0220456	−	0.999757i	$-0.507018\pi$
$719$	−1.18227	−0.0440913	−0.0220456	+	0.999757i	$0.507018\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	74.8676	2.78436
$724$	0	0
$725$	3.20061	0.118868
$726$	0	0
$727$	−24.1781	−0.896715	−0.448357	−	0.893854i	$-0.647991\pi$
$727$	−24.1781	−0.896715	−0.448357	+	0.893854i	$0.647991\pi$
$728$	0	0
$729$	−26.8793	−0.995529
$730$	0	0
$731$	−2.54078	−0.0939741
$732$	0	0
$733$	38.4994	1.42201	0.711003	−	0.703189i	$-0.248241\pi$
$733$	38.4994	1.42201	0.711003	+	0.703189i	$0.248241\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	30.5629	1.12580
$738$	0	0
$739$	18.9223	0.696067	0.348033	−	0.937482i	$-0.386850\pi$
$739$	18.9223	0.696067	0.348033	+	0.937482i	$0.386850\pi$
$740$	0	0
$741$	−3.93896	−0.144701
$742$	0	0
$743$	−24.2692	−0.890350	−0.445175	−	0.895444i	$-0.646859\pi$
$743$	−24.2692	−0.890350	−0.445175	+	0.895444i	$0.646859\pi$
$744$	0	0
$745$	7.05327	0.258412
$746$	0	0
$747$	−72.1158	−2.63858
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−4.67341	−0.170535	−0.0852675	−	0.996358i	$-0.527175\pi$
$751$	−4.67341	−0.170535	−0.0852675	+	0.996358i	$0.527175\pi$
$752$	0	0
$753$	9.47431	0.345263
$754$	0	0
$755$	−6.84787	−0.249219
$756$	0	0
$757$	−0.0105890	−0.000384864 0	−0.000192432	−	1.00000i	$-0.500061\pi$
$757$	−0.0105890	−0.000384864 0	−0.000192432		1.00000i	$0.500061\pi$
$758$	0	0
$759$	−27.9648	−1.01506
$760$	0	0
$761$	25.2766	0.916277	0.458138	−	0.888881i	$-0.348516\pi$
$761$	25.2766	0.916277	0.458138	+	0.888881i	$0.348516\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−10.9531	−0.396012
$766$	0	0
$767$	7.33386	0.264811
$768$	0	0
$769$	32.4999	1.17198	0.585988	−	0.810320i	$-0.300707\pi$
$769$	32.4999	1.17198	0.585988	+	0.810320i	$0.300707\pi$
$770$	0	0
$771$	69.1701	2.49110
$772$	0	0
$773$	−1.99291	−0.0716801	−0.0358400	−	0.999358i	$-0.511411\pi$
$773$	−1.99291	−0.0716801	−0.0358400	+	0.999358i	$0.511411\pi$
$774$	0	0
$775$	−12.9116	−0.463798
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3.71907	−0.133249
$780$	0	0
$781$	0.914971	0.0327402
$782$	0	0
$783$	−17.0589	−0.609634
$784$	0	0
$785$	−31.9880	−1.14170
$786$	0	0
$787$	27.8560	0.992959	0.496480	−	0.868048i	$-0.334626\pi$
$787$	27.8560	0.992959	0.496480	+	0.868048i	$0.334626\pi$
$788$	0	0
$789$	−15.1040	−0.537715
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−8.40927	−0.298622
$794$	0	0
$795$	−50.6156	−1.79515
$796$	0	0
$797$	18.3619	0.650412	0.325206	−	0.945643i	$-0.394566\pi$
$797$	18.3619	0.650412	0.325206	+	0.945643i	$0.394566\pi$
$798$	0	0
$799$	7.40439	0.261949
$800$	0	0
$801$	−45.7564	−1.61672
$802$	0	0
$803$	44.6005	1.57392
$804$	0	0
$805$	0	0
$806$	0	0
$807$	10.0077	0.352288
$808$	0	0
$809$	4.09120	0.143839	0.0719195	−	0.997410i	$-0.477088\pi$
$809$	4.09120	0.143839	0.0719195	+	0.997410i	$0.477088\pi$
$810$	0	0
$811$	−47.0793	−1.65318	−0.826589	−	0.562806i	$-0.809722\pi$
$811$	−47.0793	−1.65318	−0.826589	+	0.562806i	$0.809722\pi$
$812$	0	0
$813$	−45.9807	−1.61261
$814$	0	0
$815$	5.34215	0.187127
$816$	0	0
$817$	−2.53347	−0.0886350
$818$	0	0
$819$	0	0
$820$	0	0
$821$	38.6140	1.34764	0.673819	−	0.738896i	$-0.264653\pi$
$821$	38.6140	1.34764	0.673819	+	0.738896i	$0.264653\pi$
$822$	0	0
$823$	10.3798	0.361819	0.180909	−	0.983500i	$-0.442096\pi$
$823$	10.3798	0.361819	0.180909	+	0.983500i	$0.442096\pi$
$824$	0	0
$825$	−18.5516	−0.645885
$826$	0	0
$827$	−24.9574	−0.867854	−0.433927	−	0.900948i	$-0.642872\pi$
$827$	−24.9574	−0.867854	−0.433927	+	0.900948i	$0.642872\pi$
$828$	0	0
$829$	21.3589	0.741826	0.370913	−	0.928668i	$-0.379045\pi$
$829$	21.3589	0.741826	0.370913	+	0.928668i	$0.379045\pi$
$830$	0	0
$831$	26.0277	0.902892
$832$	0	0
$833$	0	0
$834$	0	0
$835$	7.21122	0.249555
$836$	0	0
$837$	68.8172	2.37867
$838$	0	0
$839$	−18.5519	−0.640484	−0.320242	−	0.947336i	$-0.603764\pi$
$839$	−18.5519	−0.640484	−0.320242	+	0.947336i	$0.603764\pi$
$840$	0	0
$841$	−25.4198	−0.876544
$842$	0	0
$843$	−0.107910	−0.00371663
$844$	0	0
$845$	−20.5118	−0.705629
$846$	0	0
$847$	0	0
$848$	0	0
$849$	75.3759	2.58689
$850$	0	0
$851$	−14.4298	−0.494646
$852$	0	0
$853$	27.8896	0.954923	0.477462	−	0.878653i	$-0.341557\pi$
$853$	27.8896	0.954923	0.477462	+	0.878653i	$0.341557\pi$
$854$	0	0
$855$	−10.9216	−0.373512
$856$	0	0
$857$	−23.6056	−0.806350	−0.403175	−	0.915123i	$-0.632093\pi$
$857$	−23.6056	−0.806350	−0.403175	+	0.915123i	$0.632093\pi$
$858$	0	0
$859$	47.3371	1.61512	0.807561	−	0.589784i	$-0.200787\pi$
$859$	47.3371	1.61512	0.807561	+	0.589784i	$0.200787\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−36.3453	−1.23721	−0.618604	−	0.785703i	$-0.712301\pi$
$863$	−36.3453	−1.23721	−0.618604	+	0.785703i	$0.712301\pi$
$864$	0	0
$865$	20.6187	0.701057
$866$	0	0
$867$	−47.9946	−1.62998
$868$	0	0
$869$	32.0402	1.08689
$870$	0	0
$871$	10.9767	0.371931
$872$	0	0
$873$	71.5100	2.42025
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−49.0762	−1.65719	−0.828593	−	0.559852i	$-0.810858\pi$
$877$	−49.0762	−1.65719	−0.828593	+	0.559852i	$0.810858\pi$
$878$	0	0
$879$	−8.04750	−0.271435
$880$	0	0
$881$	6.17463	0.208028	0.104014	−	0.994576i	$-0.466831\pi$
$881$	6.17463	0.208028	0.104014	+	0.994576i	$0.466831\pi$
$882$	0	0
$883$	25.5930	0.861274	0.430637	−	0.902525i	$-0.358289\pi$
$883$	25.5930	0.861274	0.430637	+	0.902525i	$0.358289\pi$
$884$	0	0
$885$	30.4946	1.02507
$886$	0	0
$887$	−51.4044	−1.72599	−0.862995	−	0.505212i	$-0.831414\pi$
$887$	−51.4044	−1.72599	−0.862995	+	0.505212i	$0.831414\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	33.0409	1.10691
$892$	0	0
$893$	7.38310	0.247066
$894$	0	0
$895$	−4.58896	−0.153392
$896$	0	0
$897$	−10.0436	−0.335346
$898$	0	0
$899$	−14.4429	−0.481699
$900$	0	0
$901$	9.30019	0.309834
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−45.8849	−1.52527
$906$	0	0
$907$	−42.0706	−1.39693	−0.698466	−	0.715643i	$-0.746134\pi$
$907$	−42.0706	−1.39693	−0.698466	+	0.715643i	$0.746134\pi$
$908$	0	0
$909$	43.9717	1.45845
$910$	0	0
$911$	−16.5026	−0.546755	−0.273377	−	0.961907i	$-0.588141\pi$
$911$	−16.5026	−0.546755	−0.273377	+	0.961907i	$0.588141\pi$
$912$	0	0
$913$	−43.8966	−1.45276
$914$	0	0
$915$	−34.9662	−1.15595
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−46.7116	−1.54087	−0.770437	−	0.637516i	$-0.779962\pi$
$919$	−46.7116	−1.54087	−0.770437	+	0.637516i	$0.779962\pi$
$920$	0	0
$921$	72.3462	2.38389
$922$	0	0
$923$	0.328613	0.0108164
$924$	0	0
$925$	−9.57260	−0.314745
$926$	0	0
$927$	−27.0537	−0.888561
$928$	0	0
$929$	20.8881	0.685318	0.342659	−	0.939460i	$-0.388673\pi$
$929$	20.8881	0.685318	0.342659	+	0.939460i	$0.388673\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−88.8948	−2.91028
$934$	0	0
$935$	−6.66713	−0.218039
$936$	0	0
$937$	6.06312	0.198074	0.0990368	−	0.995084i	$-0.468424\pi$
$937$	6.06312	0.198074	0.0990368	+	0.995084i	$0.468424\pi$
$938$	0	0
$939$	39.3114	1.28288
$940$	0	0
$941$	18.0592	0.588715	0.294357	−	0.955695i	$-0.404894\pi$
$941$	18.0592	0.588715	0.294357	+	0.955695i	$0.404894\pi$
$942$	0	0
$943$	−9.48291	−0.308806
$944$	0	0
$945$	0	0
$946$	0	0
$947$	45.4630	1.47735	0.738675	−	0.674061i	$-0.235452\pi$
$947$	45.4630	1.47735	0.738675	+	0.674061i	$0.235452\pi$
$948$	0	0
$949$	16.0183	0.519977
$950$	0	0
$951$	−86.8710	−2.81699
$952$	0	0
$953$	30.9775	1.00346	0.501730	−	0.865024i	$-0.332697\pi$
$953$	30.9775	1.00346	0.501730	+	0.865024i	$0.332697\pi$
$954$	0	0
$955$	29.0518	0.940093
$956$	0	0
$957$	−20.7519	−0.670814
$958$	0	0
$959$	0	0
$960$	0	0
$961$	27.2642	0.879491
$962$	0	0
$963$	42.1289	1.35758
$964$	0	0
$965$	−29.3293	−0.944144
$966$	0	0
$967$	−55.9852	−1.80036	−0.900181	−	0.435516i	$-0.856566\pi$
$967$	−55.9852	−1.80036	−0.900181	+	0.435516i	$0.856566\pi$
$968$	0	0
$969$	3.00940	0.0966758
$970$	0	0
$971$	−61.9835	−1.98915	−0.994573	−	0.104043i	$-0.966822\pi$
$971$	−61.9835	−1.98915	−0.994573	+	0.104043i	$0.966822\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−6.66284	−0.213382
$976$	0	0
$977$	0.699708	0.0223856	0.0111928	−	0.999937i	$-0.496437\pi$
$977$	0.699708	0.0223856	0.0111928	+	0.999937i	$0.496437\pi$
$978$	0	0
$979$	−27.8518	−0.890146
$980$	0	0
$981$	89.1809	2.84733
$982$	0	0
$983$	41.8843	1.33590	0.667950	−	0.744206i	$-0.267172\pi$
$983$	41.8843	1.33590	0.667950	+	0.744206i	$0.267172\pi$
$984$	0	0
$985$	−9.53175	−0.303707
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−6.45987	−0.205412
$990$	0	0
$991$	−3.98266	−0.126513	−0.0632567	−	0.997997i	$-0.520149\pi$
$991$	−3.98266	−0.126513	−0.0632567	+	0.997997i	$0.520149\pi$
$992$	0	0
$993$	78.1424	2.47977
$994$	0	0
$995$	−45.5128	−1.44285
$996$	0	0
$997$	30.7912	0.975166	0.487583	−	0.873077i	$-0.337879\pi$
$997$	30.7912	0.975166	0.487583	+	0.873077i	$0.337879\pi$
$998$	0	0
$999$	51.0208	1.61423

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7448.2.a.bw.1.13	yes	14
7.6	odd	2		7448.2.a.bv.1.2	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7448.2.a.bv.1.2	✓	14	7.6	odd	2
7448.2.a.bw.1.13	yes	14	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)

\(f(q)\)	\(=\)	\(q+3.00074 q^{3} +1.81892 q^{5} +6.00446 q^{9} +O(q^{10})\)	\(q+3.00074 q^{3} +1.81892 q^{5} +6.00446 q^{9} +3.65489 q^{11} +1.31266 q^{13} +5.45812 q^{15} -1.00288 q^{17} -1.00000 q^{19} -2.54981 q^{23} -1.69153 q^{25} +9.01562 q^{27} -1.89214 q^{29} +7.63310 q^{31} +10.9674 q^{33} +5.65915 q^{37} +3.93896 q^{39} +3.71907 q^{41} +2.53347 q^{43} +10.9216 q^{45} -7.38310 q^{47} -3.00940 q^{51} -9.27345 q^{53} +6.64796 q^{55} -3.00074 q^{57} +5.58703 q^{59} -6.40629 q^{61} +2.38762 q^{65} +8.36219 q^{67} -7.65132 q^{69} +0.250341 q^{71} +12.2029 q^{73} -5.07583 q^{75} +8.76638 q^{79} +9.04018 q^{81} -12.0104 q^{83} -1.82417 q^{85} -5.67784 q^{87} -7.62041 q^{89} +22.9050 q^{93} -1.81892 q^{95} +11.9095 q^{97} +21.9457 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 2 q^{3} + 2 q^{5} + 16 q^{9}+O(q^{10}) \) 14 * q + 2 * q^3 + 2 * q^5 + 16 * q^9	\( 14 q + 2 q^{3} + 2 q^{5} + 16 q^{9} - 6 q^{11} + 16 q^{13} - 4 q^{15} + 20 q^{17} - 14 q^{19} + 4 q^{23} + 16 q^{25} + 20 q^{27} + 6 q^{29} + 34 q^{33} - 6 q^{37} + 8 q^{39} + 46 q^{41} - 18 q^{43} - 10 q^{47} - 4 q^{51} - 2 q^{53} + 28 q^{55} - 2 q^{57} - 22 q^{59} + 26 q^{61} + 8 q^{65} - 12 q^{67} + 48 q^{69} + 18 q^{71} + 28 q^{73} - 24 q^{75} - 10 q^{79} - 2 q^{81} + 8 q^{83} - 16 q^{85} + 16 q^{87} + 78 q^{89} - 2 q^{95} + 54 q^{97} + 26 q^{99}+O(q^{100}) \) 14 * q + 2 * q^3 + 2 * q^5 + 16 * q^9 - 6 * q^11 + 16 * q^13 - 4 * q^15 + 20 * q^17 - 14 * q^19 + 4 * q^23 + 16 * q^25 + 20 * q^27 + 6 * q^29 + 34 * q^33 - 6 * q^37 + 8 * q^39 + 46 * q^41 - 18 * q^43 - 10 * q^47 - 4 * q^51 - 2 * q^53 + 28 * q^55 - 2 * q^57 - 22 * q^59 + 26 * q^61 + 8 * q^65 - 12 * q^67 + 48 * q^69 + 18 * q^71 + 28 * q^73 - 24 * q^75 - 10 * q^79 - 2 * q^81 + 8 * q^83 - 16 * q^85 + 16 * q^87 + 78 * q^89 - 2 * q^95 + 54 * q^97 + 26 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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