LMFDB - Embedded newform 4275.2.a.bs.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4275,2,Mod(1,4275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4275, 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("4275.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4275 = 3^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4275.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:	$34.1360468641$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.16717036.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 10x^{4} + 26x^{2} - 19 $ x^6 - 10x^4 + 26x^2 - 19
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.52185$ of defining polynomial
Character	$\chi$	$=$	4275.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.52185 q^{2} +0.316031 q^{4} -1.48028 q^{7} -2.56275 q^{8} -0.730913 q^{11} +2.32850 q^{13} -2.25276 q^{14} -4.53219 q^{16} +6.58733 q^{17} -1.00000 q^{19} -1.11234 q^{22} -2.31279 q^{23} +3.54362 q^{26} -0.467814 q^{28} -5.60645 q^{29} +3.84822 q^{31} -1.77181 q^{32} +10.0249 q^{34} -1.67150 q^{37} -1.52185 q^{38} -3.29366 q^{41} -7.69643 q^{43} -0.230991 q^{44} -3.51972 q^{46} -3.54362 q^{47} -4.80877 q^{49} +0.735877 q^{52} -0.480952 q^{53} +3.79358 q^{56} -8.53219 q^{58} +0.249961 q^{59} -3.73588 q^{61} +5.85641 q^{62} +6.36794 q^{64} -11.2162 q^{67} +2.08180 q^{68} -0.249961 q^{71} -16.6175 q^{73} -2.54378 q^{74} -0.316031 q^{76} +1.08195 q^{77} +9.54465 q^{79} -5.01247 q^{82} -6.81832 q^{83} -11.7128 q^{86} +1.87315 q^{88} -10.7320 q^{89} -3.44682 q^{91} -0.730913 q^{92} -5.39287 q^{94} +2.32850 q^{97} -7.31824 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 8 q^{4} - 16 q^{13} - 6 q^{19} - 10 q^{22} - 30 q^{28} + 2 q^{31} - 12 q^{34} - 40 q^{37} - 4 q^{43} - 30 q^{46} + 10 q^{49} - 20 q^{52} - 24 q^{58} + 2 q^{61} + 26 q^{64} - 34 q^{67} - 22 q^{73}+ \cdots - 16 q^{97}+O(q^{100}) $ 6 * q + 8 * q^4 - 16 * q^13 - 6 * q^19 - 10 * q^22 - 30 * q^28 + 2 * q^31 - 12 * q^34 - 40 * q^37 - 4 * q^43 - 30 * q^46 + 10 * q^49 - 20 * q^52 - 24 * q^58 + 2 * q^61 + 26 * q^64 - 34 * q^67 - 22 * q^73 - 8 * q^76 - 6 * q^79 + 6 * q^82 - 82 * q^88 - 44 * q^91 + 52 * q^94 - 16 * q^97

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 $

$2$	1.52185	1.07611	0.538056	−	0.842909i	$-0.319159\pi$
$2$	1.52185	1.07611	0.538056	+	0.842909i	$0.319159\pi$
$3$	0	0
$3$	0	0
$4$	0.316031	0.158015
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.48028	−0.559493	−0.279746	−	0.960074i	$-0.590250\pi$
$7$	−1.48028	−0.559493	−0.279746	+	0.960074i	$0.590250\pi$
$8$	−2.56275	−0.906069
$9$	0	0
$10$	0	0
$11$	−0.730913	−0.220379	−0.110189	−	0.993911i	$-0.535146\pi$
$11$	−0.730913	−0.220379	−0.110189	+	0.993911i	$0.535146\pi$
$12$	0	0
$13$	2.32850	0.645809	0.322904	−	0.946432i	$-0.395341\pi$
$13$	2.32850	0.645809	0.322904	+	0.946432i	$0.395341\pi$
$14$	−2.25276	−0.602076
$15$	0	0
$16$	−4.53219	−1.13305
$17$	6.58733	1.59766	0.798831	−	0.601556i	$-0.205452\pi$
$17$	6.58733	1.59766	0.798831	+	0.601556i	$0.205452\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	−1.11234	−0.237152
$23$	−2.31279	−0.482250	−0.241125	−	0.970494i	$-0.577516\pi$
$23$	−2.31279	−0.482250	−0.241125	+	0.970494i	$0.577516\pi$
$24$	0	0
$25$	0	0
$26$	3.54362	0.694962
$27$	0	0
$28$	−0.467814	−0.0884085
$29$	−5.60645	−1.04109	−0.520546	−	0.853834i	$-0.674272\pi$
$29$	−5.60645	−1.04109	−0.520546	+	0.853834i	$0.674272\pi$
$30$	0	0
$31$	3.84822	0.691160	0.345580	−	0.938389i	$-0.387682\pi$
$31$	3.84822	0.691160	0.345580	+	0.938389i	$0.387682\pi$
$32$	−1.77181	−0.313215
$33$	0	0
$34$	10.0249	1.71926
$35$	0	0
$36$	0	0
$37$	−1.67150	−0.274794	−0.137397	−	0.990516i	$-0.543874\pi$
$37$	−1.67150	−0.274794	−0.137397	+	0.990516i	$0.543874\pi$
$38$	−1.52185	−0.246877
$39$	0	0
$40$	0	0
$41$	−3.29366	−0.514384	−0.257192	−	0.966360i	$-0.582797\pi$
$41$	−3.29366	−0.514384	−0.257192	+	0.966360i	$0.582797\pi$
$42$	0	0
$43$	−7.69643	−1.17370	−0.586848	−	0.809697i	$-0.699631\pi$
$43$	−7.69643	−1.17370	−0.586848	+	0.809697i	$0.699631\pi$
$44$	−0.230991	−0.0348232
$45$	0	0
$46$	−3.51972	−0.518955
$47$	−3.54362	−0.516891	−0.258445	−	0.966026i	$-0.583210\pi$
$47$	−3.54362	−0.516891	−0.258445	+	0.966026i	$0.583210\pi$
$48$	0	0
$49$	−4.80877	−0.686968
$50$	0	0
$51$	0	0
$52$	0.735877	0.102048
$53$	−0.480952	−0.0660638	−0.0330319	−	0.999454i	$-0.510516\pi$
$53$	−0.480952	−0.0660638	−0.0330319	+	0.999454i	$0.510516\pi$
$54$	0	0
$55$	0	0
$56$	3.79358	0.506939
$57$	0	0
$58$	−8.53219	−1.12033
$59$	0.249961	0.0325421	0.0162711	−	0.999868i	$-0.494821\pi$
$59$	0.249961	0.0325421	0.0162711	+	0.999868i	$0.494821\pi$
$60$	0	0
$61$	−3.73588	−0.478330	−0.239165	−	0.970979i	$-0.576874\pi$
$61$	−3.73588	−0.478330	−0.239165	+	0.970979i	$0.576874\pi$
$62$	5.85641	0.743765
$63$	0	0
$64$	6.36794	0.795992
$65$	0	0
$66$	0	0
$67$	−11.2162	−1.37027	−0.685136	−	0.728415i	$-0.740257\pi$
$67$	−11.2162	−1.37027	−0.685136	+	0.728415i	$0.740257\pi$
$68$	2.08180	0.252455
$69$	0	0
$70$	0	0
$71$	−0.249961	−0.0296649	−0.0148325	−	0.999890i	$-0.504721\pi$
$71$	−0.249961	−0.0296649	−0.0148325	+	0.999890i	$0.504721\pi$
$72$	0	0
$73$	−16.6175	−1.94494	−0.972469	−	0.233034i	$-0.925135\pi$
$73$	−16.6175	−1.94494	−0.972469	+	0.233034i	$0.925135\pi$
$74$	−2.54378	−0.295708
$75$	0	0
$76$	−0.316031	−0.0362512
$77$	1.08195	0.123300
$78$	0	0
$79$	9.54465	1.07386	0.536929	−	0.843628i	$-0.319584\pi$
$79$	9.54465	1.07386	0.536929	+	0.843628i	$0.319584\pi$
$80$	0	0
$81$	0	0
$82$	−5.01247	−0.553534
$83$	−6.81832	−0.748408	−0.374204	−	0.927347i	$-0.622084\pi$
$83$	−6.81832	−0.748408	−0.374204	+	0.927347i	$0.622084\pi$
$84$	0	0
$85$	0	0
$86$	−11.7128	−1.26303
$87$	0	0
$88$	1.87315	0.199678
$89$	−10.7320	−1.13758	−0.568792	−	0.822481i	$-0.692589\pi$
$89$	−10.7320	−1.13758	−0.568792	+	0.822481i	$0.692589\pi$
$90$	0	0
$91$	−3.44682	−0.361325
$92$	−0.730913	−0.0762029
$93$	0	0
$94$	−5.39287	−0.556232
$95$	0	0
$96$	0	0
$97$	2.32850	0.236423	0.118211	−	0.992988i	$-0.462284\pi$
$97$	2.32850	0.236423	0.118211	+	0.992988i	$0.462284\pi$
$98$	−7.31824	−0.739254
$99$	0	0
$100$	0	0
$101$	−16.2184	−1.61379	−0.806893	−	0.590697i	$-0.798853\pi$
$101$	−16.2184	−1.61379	−0.806893	+	0.590697i	$0.798853\pi$
$102$	0	0
$103$	−7.21616	−0.711029	−0.355514	−	0.934671i	$-0.615694\pi$
$103$	−7.21616	−0.711029	−0.355514	+	0.934671i	$0.615694\pi$
$104$	−5.96735	−0.585147
$105$	0	0
$106$	−0.731937	−0.0710920
$107$	7.79919	0.753976	0.376988	−	0.926218i	$-0.376960\pi$
$107$	7.79919	0.753976	0.376988	+	0.926218i	$0.376960\pi$
$108$	0	0
$109$	4.63206	0.443671	0.221836	−	0.975084i	$-0.428795\pi$
$109$	4.63206	0.443671	0.221836	+	0.975084i	$0.428795\pi$
$110$	0	0
$111$	0	0
$112$	6.70890	0.633931
$113$	−16.9493	−1.59445	−0.797226	−	0.603680i	$-0.793700\pi$
$113$	−16.9493	−1.59445	−0.797226	+	0.603680i	$0.793700\pi$
$114$	0	0
$115$	0	0
$116$	−1.77181	−0.164509
$117$	0	0
$118$	0.380403	0.0350190
$119$	−9.75108	−0.893880
$120$	0	0
$121$	−10.4658	−0.951433
$122$	−5.68545	−0.514736
$123$	0	0
$124$	1.21616	0.109214
$125$	0	0
$126$	0	0
$127$	15.2411	1.35243	0.676214	−	0.736705i	$-0.263620\pi$
$127$	15.2411	1.35243	0.676214	+	0.736705i	$0.263620\pi$
$128$	13.2347	1.16979
$129$	0	0
$130$	0	0
$131$	16.9493	1.48086	0.740432	−	0.672131i	$-0.234621\pi$
$131$	16.9493	1.48086	0.740432	+	0.672131i	$0.234621\pi$
$132$	0	0
$133$	1.48028	0.128356
$134$	−17.0693	−1.47456
$135$	0	0
$136$	−16.8817	−1.44759
$137$	14.6365	1.25048	0.625239	−	0.780433i	$-0.285001\pi$
$137$	14.6365	1.25048	0.625239	+	0.780433i	$0.285001\pi$
$138$	0	0
$139$	−0.519721	−0.0440822	−0.0220411	−	0.999757i	$-0.507016\pi$
$139$	−0.519721	−0.0440822	−0.0220411	+	0.999757i	$0.507016\pi$
$140$	0	0
$141$	0	0
$142$	−0.380403	−0.0319227
$143$	−1.70193	−0.142322
$144$	0	0
$145$	0	0
$146$	−25.2894	−2.09297
$147$	0	0
$148$	−0.528247	−0.0434216
$149$	−5.00545	−0.410062	−0.205031	−	0.978755i	$-0.565730\pi$
$149$	−5.00545	−0.410062	−0.205031	+	0.978755i	$0.565730\pi$
$150$	0	0
$151$	11.6175	0.945423	0.472711	−	0.881217i	$-0.343275\pi$
$151$	11.6175	0.945423	0.472711	+	0.881217i	$0.343275\pi$
$152$	2.56275	0.207867
$153$	0	0
$154$	1.64657	0.132685
$155$	0	0
$156$	0	0
$157$	−17.2805	−1.37914	−0.689568	−	0.724221i	$-0.742200\pi$
$157$	−17.2805	−1.37914	−0.689568	+	0.724221i	$0.742200\pi$
$158$	14.5255	1.15559
$159$	0	0
$160$	0	0
$161$	3.42357	0.269815
$162$	0	0
$163$	−9.48028	−0.742553	−0.371276	−	0.928522i	$-0.621080\pi$
$163$	−9.48028	−0.742553	−0.371276	+	0.928522i	$0.621080\pi$
$164$	−1.04090	−0.0812806
$165$	0	0
$166$	−10.3765	−0.805370
$167$	−0.749883	−0.0580277	−0.0290138	−	0.999579i	$-0.509237\pi$
$167$	−0.749883	−0.0580277	−0.0290138	+	0.999579i	$0.509237\pi$
$168$	0	0
$169$	−7.57811	−0.582931
$170$	0	0
$171$	0	0
$172$	−2.43231	−0.185462
$173$	−13.1557	−1.00021	−0.500104	−	0.865965i	$-0.666705\pi$
$173$	−13.1557	−1.00021	−0.500104	+	0.865965i	$0.666705\pi$
$174$	0	0
$175$	0	0
$176$	3.31263	0.249699
$177$	0	0
$178$	−16.3324	−1.22417
$179$	5.37546	0.401781	0.200890	−	0.979614i	$-0.435616\pi$
$179$	5.37546	0.401781	0.200890	+	0.979614i	$0.435616\pi$
$180$	0	0
$181$	2.73588	0.203356	0.101678	−	0.994817i	$-0.467579\pi$
$181$	2.73588	0.203356	0.101678	+	0.994817i	$0.467579\pi$
$182$	−5.24555	−0.388826
$183$	0	0
$184$	5.92710	0.436952
$185$	0	0
$186$	0	0
$187$	−4.81476	−0.352090
$188$	−1.11989	−0.0816767
$189$	0	0
$190$	0	0
$191$	3.77462	0.273122	0.136561	−	0.990632i	$-0.456395\pi$
$191$	3.77462	0.273122	0.136561	+	0.990632i	$0.456395\pi$
$192$	0	0
$193$	−10.3036	−0.741667	−0.370833	−	0.928699i	$-0.620928\pi$
$193$	−10.3036	−0.741667	−0.370833	+	0.928699i	$0.620928\pi$
$194$	3.54362	0.254417
$195$	0	0
$196$	−1.51972	−0.108552
$197$	9.63103	0.686182	0.343091	−	0.939302i	$-0.388526\pi$
$197$	9.63103	0.686182	0.343091	+	0.939302i	$0.388526\pi$
$198$	0	0
$199$	7.48028	0.530263	0.265131	−	0.964212i	$-0.414585\pi$
$199$	7.48028	0.530263	0.265131	+	0.964212i	$0.414585\pi$
$200$	0	0
$201$	0	0
$202$	−24.6819	−1.73661
$203$	8.29911	0.582484
$204$	0	0
$205$	0	0
$206$	−10.9819	−0.765146
$207$	0	0
$208$	−10.5532	−0.731731
$209$	0.730913	0.0505583
$210$	0	0
$211$	−7.54465	−0.519395	−0.259698	−	0.965690i	$-0.583623\pi$
$211$	−7.54465	−0.519395	−0.259698	+	0.965690i	$0.583623\pi$
$212$	−0.151996	−0.0104391
$213$	0	0
$214$	11.8692	0.811362
$215$	0	0
$216$	0	0
$217$	−5.69643	−0.386699
$218$	7.04931	0.477439
$219$	0	0
$220$	0	0
$221$	15.3386	1.03178
$222$	0	0
$223$	−1.19123	−0.0797703	−0.0398852	−	0.999204i	$-0.512699\pi$
$223$	−1.19123	−0.0797703	−0.0398852	+	0.999204i	$0.512699\pi$
$224$	2.62278	0.175242
$225$	0	0
$226$	−25.7943	−1.71581
$227$	3.41371	0.226576	0.113288	−	0.993562i	$-0.463862\pi$
$227$	3.41371	0.226576	0.113288	+	0.993562i	$0.463862\pi$
$228$	0	0
$229$	7.24109	0.478504	0.239252	−	0.970957i	$-0.423098\pi$
$229$	7.24109	0.478504	0.239252	+	0.970957i	$0.423098\pi$
$230$	0	0
$231$	0	0
$232$	14.3679	0.943301
$233$	−4.04355	−0.264901	−0.132451	−	0.991190i	$-0.542285\pi$
$233$	−4.04355	−0.264901	−0.132451	+	0.991190i	$0.542285\pi$
$234$	0	0
$235$	0	0
$236$	0.0789954	0.00514216
$237$	0	0
$238$	−14.8397	−0.961914
$239$	26.4694	1.71216	0.856080	−	0.516843i	$-0.172893\pi$
$239$	26.4694	1.71216	0.856080	+	0.516843i	$0.172893\pi$
$240$	0	0
$241$	−14.4323	−0.929667	−0.464833	−	0.885398i	$-0.653886\pi$
$241$	−14.4323	−0.929667	−0.464833	+	0.885398i	$0.653886\pi$
$242$	−15.9273	−1.02385
$243$	0	0
$244$	−1.18065	−0.0755835
$245$	0	0
$246$	0	0
$247$	−2.32850	−0.148159
$248$	−9.86202	−0.626239
$249$	0	0
$250$	0	0
$251$	7.54923	0.476503	0.238252	−	0.971203i	$-0.423426\pi$
$251$	7.54923	0.476503	0.238252	+	0.971203i	$0.423426\pi$
$252$	0	0
$253$	1.69045	0.106278
$254$	23.1947	1.45536
$255$	0	0
$256$	7.40533	0.462833
$257$	10.2320	0.638257	0.319128	−	0.947712i	$-0.396610\pi$
$257$	10.2320	0.638257	0.319128	+	0.947712i	$0.396610\pi$
$258$	0	0
$259$	2.47429	0.153745
$260$	0	0
$261$	0	0
$262$	25.7943	1.59357
$263$	0.730913	0.0450700	0.0225350	−	0.999746i	$-0.492826\pi$
$263$	0.730913	0.0450700	0.0225350	+	0.999746i	$0.492826\pi$
$264$	0	0
$265$	0	0
$266$	2.25276	0.138126
$267$	0	0
$268$	−3.54465	−0.216524
$269$	2.42373	0.147777	0.0738887	−	0.997266i	$-0.476459\pi$
$269$	2.42373	0.147777	0.0738887	+	0.997266i	$0.476459\pi$
$270$	0	0
$271$	2.13727	0.129830	0.0649150	−	0.997891i	$-0.479322\pi$
$271$	2.13727	0.129830	0.0649150	+	0.997891i	$0.479322\pi$
$272$	−29.8550	−1.81022
$273$	0	0
$274$	22.2745	1.34565
$275$	0	0
$276$	0	0
$277$	−17.4323	−1.04741	−0.523703	−	0.851901i	$-0.675450\pi$
$277$	−17.4323	−1.04741	−0.523703	+	0.851901i	$0.675450\pi$
$278$	−0.790938	−0.0474373
$279$	0	0
$280$	0	0
$281$	27.2003	1.62263	0.811316	−	0.584608i	$-0.198752\pi$
$281$	27.2003	1.62263	0.811316	+	0.584608i	$0.198752\pi$
$282$	0	0
$283$	−6.43231	−0.382361	−0.191181	−	0.981555i	$-0.561232\pi$
$283$	−6.43231	−0.382361	−0.191181	+	0.981555i	$0.561232\pi$
$284$	−0.0789954	−0.00468751
$285$	0	0
$286$	−2.59008	−0.153155
$287$	4.87554	0.287794
$288$	0	0
$289$	26.3929	1.55252
$290$	0	0
$291$	0	0
$292$	−5.25166	−0.307330
$293$	9.40004	0.549156	0.274578	−	0.961565i	$-0.411462\pi$
$293$	9.40004	0.549156	0.274578	+	0.961565i	$0.411462\pi$
$294$	0	0
$295$	0	0
$296$	4.28365	0.248982
$297$	0	0
$298$	−7.61755	−0.441273
$299$	−5.38532	−0.311441
$300$	0	0
$301$	11.3929	0.656674
$302$	17.6802	1.01738
$303$	0	0
$304$	4.53219	0.259939
$305$	0	0
$306$	0	0
$307$	6.99147	0.399025	0.199512	−	0.979895i	$-0.436064\pi$
$307$	6.99147	0.399025	0.199512	+	0.979895i	$0.436064\pi$
$308$	0.341931	0.0194833
$309$	0	0
$310$	0	0
$311$	23.3877	1.32620	0.663098	−	0.748533i	$-0.269241\pi$
$311$	23.3877	1.32620	0.663098	+	0.748533i	$0.269241\pi$
$312$	0	0
$313$	−7.84822	−0.443607	−0.221804	−	0.975091i	$-0.571194\pi$
$313$	−7.84822	−0.443607	−0.221804	+	0.975091i	$0.571194\pi$
$314$	−26.2984	−1.48410
$315$	0	0
$316$	3.01640	0.169686
$317$	−16.3194	−0.916591	−0.458295	−	0.888800i	$-0.651540\pi$
$317$	−16.3194	−0.916591	−0.458295	+	0.888800i	$0.651540\pi$
$318$	0	0
$319$	4.09783	0.229434
$320$	0	0
$321$	0	0
$322$	5.21017	0.290351
$323$	−6.58733	−0.366529
$324$	0	0
$325$	0	0
$326$	−14.4276	−0.799069
$327$	0	0
$328$	8.44084	0.466067
$329$	5.24555	0.289197
$330$	0	0
$331$	22.2016	1.22031	0.610156	−	0.792281i	$-0.291107\pi$
$331$	22.2016	1.22031	0.610156	+	0.792281i	$0.291107\pi$
$332$	−2.15480	−0.118260
$333$	0	0
$334$	−1.14121	−0.0624442
$335$	0	0
$336$	0	0
$337$	−9.92112	−0.540438	−0.270219	−	0.962799i	$-0.587096\pi$
$337$	−9.92112	−0.540438	−0.270219	+	0.962799i	$0.587096\pi$
$338$	−11.5328	−0.627299
$339$	0	0
$340$	0	0
$341$	−2.81271	−0.152317
$342$	0	0
$343$	17.4803	0.943846
$344$	19.7240	1.06345
$345$	0	0
$346$	−20.0210	−1.07634
$347$	−2.54378	−0.136557	−0.0682786	−	0.997666i	$-0.521751\pi$
$347$	−2.54378	−0.136557	−0.0682786	+	0.997666i	$0.521751\pi$
$348$	0	0
$349$	−21.8817	−1.17130	−0.585650	−	0.810564i	$-0.699161\pi$
$349$	−21.8817	−1.17130	−0.585650	+	0.810564i	$0.699161\pi$
$350$	0	0
$351$	0	0
$352$	1.29504	0.0690259
$353$	8.28925	0.441192	0.220596	−	0.975365i	$-0.429200\pi$
$353$	8.28925	0.441192	0.220596	+	0.975365i	$0.429200\pi$
$354$	0	0
$355$	0	0
$356$	−3.39163	−0.179756
$357$	0	0
$358$	8.18065	0.432361
$359$	2.54378	0.134256	0.0671278	−	0.997744i	$-0.478616\pi$
$359$	2.54378	0.134256	0.0671278	+	0.997744i	$0.478616\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	4.16360	0.218834
$363$	0	0
$364$	−1.08930	−0.0570950
$365$	0	0
$366$	0	0
$367$	−29.0104	−1.51433	−0.757166	−	0.653223i	$-0.773416\pi$
$367$	−29.0104	−1.51433	−0.757166	+	0.653223i	$0.773416\pi$
$368$	10.4820	0.546412
$369$	0	0
$370$	0	0
$371$	0.711943	0.0369622
$372$	0	0
$373$	−24.9606	−1.29241	−0.646204	−	0.763164i	$-0.723645\pi$
$373$	−24.9606	−1.29241	−0.646204	+	0.763164i	$0.723645\pi$
$374$	−7.32735	−0.378888
$375$	0	0
$376$	9.08142	0.468339
$377$	−13.0546	−0.672346
$378$	0	0
$379$	0.553177	0.0284148	0.0142074	−	0.999899i	$-0.495477\pi$
$379$	0.553177	0.0284148	0.0142074	+	0.999899i	$0.495477\pi$
$380$	0	0
$381$	0	0
$382$	5.74440	0.293909
$383$	−2.67369	−0.136619	−0.0683096	−	0.997664i	$-0.521761\pi$
$383$	−2.67369	−0.136619	−0.0683096	+	0.997664i	$0.521761\pi$
$384$	0	0
$385$	0	0
$386$	−15.6805	−0.798116
$387$	0	0
$388$	0.735877	0.0373585
$389$	7.20730	0.365425	0.182712	−	0.983166i	$-0.441512\pi$
$389$	7.20730	0.365425	0.182712	+	0.983166i	$0.441512\pi$
$390$	0	0
$391$	−15.2351	−0.770472
$392$	12.3237	0.622440
$393$	0	0
$394$	14.6570	0.738409
$395$	0	0
$396$	0	0
$397$	−10.0729	−0.505544	−0.252772	−	0.967526i	$-0.581342\pi$
$397$	−10.0729	−0.505544	−0.252772	+	0.967526i	$0.581342\pi$
$398$	11.3839	0.570622
$399$	0	0
$400$	0	0
$401$	25.2006	1.25846	0.629229	−	0.777220i	$-0.283371\pi$
$401$	25.2006	1.25846	0.629229	+	0.777220i	$0.283371\pi$
$402$	0	0
$403$	8.96056	0.446357
$404$	−5.12550	−0.255003
$405$	0	0
$406$	12.6300	0.626817
$407$	1.22172	0.0605586
$408$	0	0
$409$	−29.6425	−1.46573	−0.732863	−	0.680376i	$-0.761816\pi$
$409$	−29.6425	−1.46573	−0.732863	+	0.680376i	$0.761816\pi$
$410$	0	0
$411$	0	0
$412$	−2.28053	−0.112354
$413$	−0.370012	−0.0182071
$414$	0	0
$415$	0	0
$416$	−4.12566	−0.202277
$417$	0	0
$418$	1.11234	0.0544064
$419$	34.4806	1.68449	0.842243	−	0.539098i	$-0.181235\pi$
$419$	34.4806	1.68449	0.842243	+	0.539098i	$0.181235\pi$
$420$	0	0
$421$	−4.93563	−0.240548	−0.120274	−	0.992741i	$-0.538377\pi$
$421$	−4.93563	−0.240548	−0.120274	+	0.992741i	$0.538377\pi$
$422$	−11.4818	−0.558927
$423$	0	0
$424$	1.23256	0.0598584
$425$	0	0
$426$	0	0
$427$	5.53014	0.267622
$428$	2.46479	0.119140
$429$	0	0
$430$	0	0
$431$	−24.6375	−1.18675	−0.593374	−	0.804927i	$-0.702204\pi$
$431$	−24.6375	−1.18675	−0.593374	+	0.804927i	$0.702204\pi$
$432$	0	0
$433$	−11.3679	−0.546308	−0.273154	−	0.961970i	$-0.588067\pi$
$433$	−11.3679	−0.546308	−0.273154	+	0.961970i	$0.588067\pi$
$434$	−8.66912	−0.416131
$435$	0	0
$436$	1.46387	0.0701069
$437$	2.31279	0.110636
$438$	0	0
$439$	−10.0729	−0.480753	−0.240377	−	0.970680i	$-0.577271\pi$
$439$	−10.0729	−0.480753	−0.240377	+	0.970680i	$0.577271\pi$
$440$	0	0
$441$	0	0
$442$	23.3430	1.11031
$443$	24.2387	1.15161	0.575807	−	0.817586i	$-0.304688\pi$
$443$	24.2387	1.15161	0.575807	+	0.817586i	$0.304688\pi$
$444$	0	0
$445$	0	0
$446$	−1.81287	−0.0858418
$447$	0	0
$448$	−9.42632	−0.445352
$449$	28.7541	1.35699	0.678494	−	0.734606i	$-0.262633\pi$
$449$	28.7541	1.35699	0.678494	+	0.734606i	$0.262633\pi$
$450$	0	0
$451$	2.40738	0.113359
$452$	−5.35649	−0.251948
$453$	0	0
$454$	5.19516	0.243821
$455$	0	0
$456$	0	0
$457$	26.8192	1.25455	0.627274	−	0.778798i	$-0.284170\pi$
$457$	26.8192	1.25455	0.627274	+	0.778798i	$0.284170\pi$
$458$	11.0199	0.514924
$459$	0	0
$460$	0	0
$461$	−38.6442	−1.79984	−0.899919	−	0.436057i	$-0.856375\pi$
$461$	−38.6442	−1.79984	−0.899919	+	0.436057i	$0.856375\pi$
$462$	0	0
$463$	36.4033	1.69180	0.845902	−	0.533338i	$-0.179063\pi$
$463$	36.4033	1.69180	0.845902	+	0.533338i	$0.179063\pi$
$464$	25.4095	1.17961
$465$	0	0
$466$	−6.15367	−0.285063
$467$	−17.5692	−0.813007	−0.406504	−	0.913649i	$-0.633252\pi$
$467$	−17.5692	−0.813007	−0.406504	+	0.913649i	$0.633252\pi$
$468$	0	0
$469$	16.6030	0.766657
$470$	0	0
$471$	0	0
$472$	−0.640587	−0.0294854
$473$	5.62542	0.258657
$474$	0	0
$475$	0	0
$476$	−3.08164	−0.141247
$477$	0	0
$478$	40.2824	1.84247
$479$	−11.4818	−0.524618	−0.262309	−	0.964984i	$-0.584484\pi$
$479$	−11.4818	−0.524618	−0.262309	+	0.964984i	$0.584484\pi$
$480$	0	0
$481$	−3.89209	−0.177464
$482$	−21.9638	−1.00042
$483$	0	0
$484$	−3.30751	−0.150341
$485$	0	0
$486$	0	0
$487$	−29.5926	−1.34097	−0.670485	−	0.741923i	$-0.733914\pi$
$487$	−29.5926	−1.34097	−0.670485	+	0.741923i	$0.733914\pi$
$488$	9.57412	0.433400
$489$	0	0
$490$	0	0
$491$	26.3114	1.18742	0.593708	−	0.804681i	$-0.297664\pi$
$491$	26.3114	1.18742	0.593708	+	0.804681i	$0.297664\pi$
$492$	0	0
$493$	−36.9315	−1.66331
$494$	−3.54362	−0.159435
$495$	0	0
$496$	−17.4408	−0.783117
$497$	0.370012	0.0165973
$498$	0	0
$499$	−26.2831	−1.17659	−0.588296	−	0.808646i	$-0.700201\pi$
$499$	−26.2831	−1.17659	−0.588296	+	0.808646i	$0.700201\pi$
$500$	0	0
$501$	0	0
$502$	11.4888	0.512770
$503$	32.0568	1.42934	0.714672	−	0.699460i	$-0.246576\pi$
$503$	32.0568	1.42934	0.714672	+	0.699460i	$0.246576\pi$
$504$	0	0
$505$	0	0
$506$	2.57261	0.114366
$507$	0	0
$508$	4.81665	0.213704
$509$	42.6687	1.89126	0.945629	−	0.325246i	$-0.105447\pi$
$509$	42.6687	1.89126	0.945629	+	0.325246i	$0.105447\pi$
$510$	0	0
$511$	24.5986	1.08818
$512$	−15.1995	−0.671731
$513$	0	0
$514$	15.5716	0.686835
$515$	0	0
$516$	0	0
$517$	2.59008	0.113912
$518$	3.76550	0.165447
$519$	0	0
$520$	0	0
$521$	7.30838	0.320186	0.160093	−	0.987102i	$-0.448821\pi$
$521$	7.30838	0.320186	0.160093	+	0.987102i	$0.448821\pi$
$522$	0	0
$523$	4.93563	0.215820	0.107910	−	0.994161i	$-0.465584\pi$
$523$	4.93563	0.215820	0.107910	+	0.994161i	$0.465584\pi$
$524$	5.35649	0.233999
$525$	0	0
$526$	1.11234	0.0485004
$527$	25.3495	1.10424
$528$	0	0
$529$	−17.6510	−0.767435
$530$	0	0
$531$	0	0
$532$	0.467814	0.0202823
$533$	−7.66928	−0.332193
$534$	0	0
$535$	0	0
$536$	28.7442	1.24156
$537$	0	0
$538$	3.68856	0.159025
$539$	3.51480	0.151393
$540$	0	0
$541$	−20.3594	−0.875320	−0.437660	−	0.899141i	$-0.644193\pi$
$541$	−20.3594	−0.875320	−0.437660	+	0.899141i	$0.644193\pi$
$542$	3.25261	0.139711
$543$	0	0
$544$	−11.6715	−0.500412
$545$	0	0
$546$	0	0
$547$	−38.4513	−1.64406	−0.822028	−	0.569446i	$-0.807158\pi$
$547$	−38.4513	−1.64406	−0.822028	+	0.569446i	$0.807158\pi$
$548$	4.62558	0.197595
$549$	0	0
$550$	0	0
$551$	5.60645	0.238843
$552$	0	0
$553$	−14.1287	−0.600815
$554$	−26.5294	−1.12713
$555$	0	0
$556$	−0.164248	−0.00696566
$557$	−33.1767	−1.40574	−0.702872	−	0.711317i	$-0.748099\pi$
$557$	−33.1767	−1.40574	−0.702872	+	0.711317i	$0.748099\pi$
$558$	0	0
$559$	−17.9211	−0.757982
$560$	0	0
$561$	0	0
$562$	41.3948	1.74613
$563$	−39.0142	−1.64425	−0.822126	−	0.569306i	$-0.807212\pi$
$563$	−39.0142	−1.64425	−0.822126	+	0.569306i	$0.807212\pi$
$564$	0	0
$565$	0	0
$566$	−9.78902	−0.411463
$567$	0	0
$568$	0.640587	0.0268785
$569$	−13.0447	−0.546864	−0.273432	−	0.961891i	$-0.588159\pi$
$569$	−13.0447	−0.546864	−0.273432	+	0.961891i	$0.588159\pi$
$570$	0	0
$571$	29.9795	1.25460	0.627302	−	0.778776i	$-0.284159\pi$
$571$	29.9795	1.25460	0.627302	+	0.778776i	$0.284159\pi$
$572$	−0.537862	−0.0224891
$573$	0	0
$574$	7.41985	0.309698
$575$	0	0
$576$	0	0
$577$	−12.6630	−0.527167	−0.263583	−	0.964637i	$-0.584904\pi$
$577$	−12.6630	−0.527167	−0.263583	+	0.964637i	$0.584904\pi$
$578$	40.1660	1.67069
$579$	0	0
$580$	0	0
$581$	10.0930	0.418729
$582$	0	0
$583$	0.351534	0.0145590
$584$	42.5866	1.76225
$585$	0	0
$586$	14.3055	0.590953
$587$	−43.2606	−1.78556	−0.892779	−	0.450495i	$-0.851247\pi$
$587$	−43.2606	−1.78556	−0.892779	+	0.450495i	$0.851247\pi$
$588$	0	0
$589$	−3.84822	−0.158563
$590$	0	0
$591$	0	0
$592$	7.57557	0.311354
$593$	2.08180	0.0854892	0.0427446	−	0.999086i	$-0.486390\pi$
$593$	2.08180	0.0854892	0.0427446	+	0.999086i	$0.486390\pi$
$594$	0	0
$595$	0	0
$596$	−1.58188	−0.0647962
$597$	0	0
$598$	−8.19566	−0.335145
$599$	7.54923	0.308453	0.154227	−	0.988036i	$-0.450711\pi$
$599$	7.54923	0.308453	0.154227	+	0.988036i	$0.450711\pi$
$600$	0	0
$601$	−1.21017	−0.0493638	−0.0246819	−	0.999695i	$-0.507857\pi$
$601$	−1.21017	−0.0493638	−0.0246819	+	0.999695i	$0.507857\pi$
$602$	17.3382	0.706654
$603$	0	0
$604$	3.67150	0.149391
$605$	0	0
$606$	0	0
$607$	−28.9126	−1.17353	−0.586763	−	0.809759i	$-0.699598\pi$
$607$	−28.9126	−1.17353	−0.586763	+	0.809759i	$0.699598\pi$
$608$	1.77181	0.0718565
$609$	0	0
$610$	0	0
$611$	−8.25131	−0.333812
$612$	0	0
$613$	2.31997	0.0937027	0.0468514	−	0.998902i	$-0.485081\pi$
$613$	2.31997	0.0937027	0.0468514	+	0.998902i	$0.485081\pi$
$614$	10.6400	0.429395
$615$	0	0
$616$	−2.77278	−0.111718
$617$	−32.5568	−1.31069	−0.655343	−	0.755331i	$-0.727476\pi$
$617$	−32.5568	−1.31069	−0.655343	+	0.755331i	$0.727476\pi$
$618$	0	0
$619$	−5.83370	−0.234476	−0.117238	−	0.993104i	$-0.537404\pi$
$619$	−5.83370	−0.234476	−0.117238	+	0.993104i	$0.537404\pi$
$620$	0	0
$621$	0	0
$622$	35.5926	1.42713
$623$	15.8863	0.636470
$624$	0	0
$625$	0	0
$626$	−11.9438	−0.477371
$627$	0	0
$628$	−5.46118	−0.217925
$629$	−11.0107	−0.439027
$630$	0	0
$631$	−29.9710	−1.19313	−0.596563	−	0.802566i	$-0.703467\pi$
$631$	−29.9710	−1.19313	−0.596563	+	0.802566i	$0.703467\pi$
$632$	−24.4606	−0.972989
$633$	0	0
$634$	−24.8358	−0.986354
$635$	0	0
$636$	0	0
$637$	−11.1972	−0.443650
$638$	6.23628	0.246897
$639$	0	0
$640$	0	0
$641$	31.9368	1.26143	0.630714	−	0.776016i	$-0.282762\pi$
$641$	31.9368	1.26143	0.630714	+	0.776016i	$0.282762\pi$
$642$	0	0
$643$	5.99147	0.236281	0.118140	−	0.992997i	$-0.462307\pi$
$643$	5.99147	0.236281	0.118140	+	0.992997i	$0.462307\pi$
$644$	1.08195	0.0426350
$645$	0	0
$646$	−10.0249	−0.394426
$647$	23.5366	0.925319	0.462659	−	0.886536i	$-0.346895\pi$
$647$	23.5366	0.925319	0.462659	+	0.886536i	$0.346895\pi$
$648$	0	0
$649$	−0.182700	−0.00717159
$650$	0	0
$651$	0	0
$652$	−2.99606	−0.117335
$653$	−0.841853	−0.0329443	−0.0164721	−	0.999864i	$-0.505243\pi$
$653$	−0.841853	−0.0329443	−0.0164721	+	0.999864i	$0.505243\pi$
$654$	0	0
$655$	0	0
$656$	14.9275	0.582821
$657$	0	0
$658$	7.98295	0.311208
$659$	−6.08740	−0.237132	−0.118566	−	0.992946i	$-0.537830\pi$
$659$	−6.08740	−0.237132	−0.118566	+	0.992946i	$0.537830\pi$
$660$	0	0
$661$	34.1287	1.32745	0.663727	−	0.747975i	$-0.268974\pi$
$661$	34.1287	1.32745	0.663727	+	0.747975i	$0.268974\pi$
$662$	33.7876	1.31319
$663$	0	0
$664$	17.4736	0.678109
$665$	0	0
$666$	0	0
$667$	12.9665	0.502067
$668$	−0.236986	−0.00916927
$669$	0	0
$670$	0	0
$671$	2.73060	0.105414
$672$	0	0
$673$	24.5361	0.945798	0.472899	−	0.881117i	$-0.343208\pi$
$673$	24.5361	0.945798	0.472899	+	0.881117i	$0.343208\pi$
$674$	−15.0985	−0.581571
$675$	0	0
$676$	−2.39492	−0.0921121
$677$	9.23219	0.354822	0.177411	−	0.984137i	$-0.443228\pi$
$677$	9.23219	0.354822	0.177411	+	0.984137i	$0.443228\pi$
$678$	0	0
$679$	−3.44682	−0.132277
$680$	0	0
$681$	0	0
$682$	−4.28053	−0.163910
$683$	10.5010	0.401808	0.200904	−	0.979611i	$-0.435612\pi$
$683$	10.5010	0.401808	0.200904	+	0.979611i	$0.435612\pi$
$684$	0	0
$685$	0	0
$686$	26.6024	1.01568
$687$	0	0
$688$	34.8817	1.32985
$689$	−1.11989	−0.0426646
$690$	0	0
$691$	24.3036	0.924551	0.462276	−	0.886736i	$-0.347033\pi$
$691$	24.3036	0.924551	0.462276	+	0.886736i	$0.347033\pi$
$692$	−4.15760	−0.158048
$693$	0	0
$694$	−3.87126	−0.146951
$695$	0	0
$696$	0	0
$697$	−21.6964	−0.821811
$698$	−33.3006	−1.26045
$699$	0	0
$700$	0	0
$701$	−27.8111	−1.05041	−0.525206	−	0.850975i	$-0.676012\pi$
$701$	−27.8111	−1.05041	−0.525206	+	0.850975i	$0.676012\pi$
$702$	0	0
$703$	1.67150	0.0630420
$704$	−4.65441	−0.175420
$705$	0	0
$706$	12.6150	0.474772
$707$	24.0077	0.902902
$708$	0	0
$709$	−31.5781	−1.18594	−0.592970	−	0.805224i	$-0.702045\pi$
$709$	−31.5781	−1.18594	−0.592970	+	0.805224i	$0.702045\pi$
$710$	0	0
$711$	0	0
$712$	27.5033	1.03073
$713$	−8.90012	−0.333312
$714$	0	0
$715$	0	0
$716$	1.69881	0.0634876
$717$	0	0
$718$	3.87126	0.144474
$719$	6.73621	0.251218	0.125609	−	0.992080i	$-0.459912\pi$
$719$	6.73621	0.251218	0.125609	+	0.992080i	$0.459912\pi$
$720$	0	0
$721$	10.6819	0.397816
$722$	1.52185	0.0566374
$723$	0	0
$724$	0.864621	0.0321334
$725$	0	0
$726$	0	0
$727$	47.8836	1.77590	0.887952	−	0.459937i	$-0.152128\pi$
$727$	47.8836	1.77590	0.887952	+	0.459937i	$0.152128\pi$
$728$	8.83335	0.327386
$729$	0	0
$730$	0	0
$731$	−50.6989	−1.87517
$732$	0	0
$733$	17.7857	0.656931	0.328466	−	0.944516i	$-0.393469\pi$
$733$	17.7857	0.656931	0.328466	+	0.944516i	$0.393469\pi$
$734$	−44.1495	−1.62959
$735$	0	0
$736$	4.09783	0.151048
$737$	8.19803	0.301978
$738$	0	0
$739$	−17.7668	−0.653562	−0.326781	−	0.945100i	$-0.605964\pi$
$739$	−17.7668	−0.653562	−0.326781	+	0.945100i	$0.605964\pi$
$740$	0	0
$741$	0	0
$742$	1.08347	0.0397755
$743$	−49.0251	−1.79856	−0.899278	−	0.437378i	$-0.855907\pi$
$743$	−49.0251	−1.79856	−0.899278	+	0.437378i	$0.855907\pi$
$744$	0	0
$745$	0	0
$746$	−37.9863	−1.39078
$747$	0	0
$748$	−1.52161	−0.0556357
$749$	−11.5450	−0.421844
$750$	0	0
$751$	50.9066	1.85761	0.928804	−	0.370571i	$-0.120838\pi$
$751$	50.9066	1.85761	0.928804	+	0.370571i	$0.120838\pi$
$752$	16.0604	0.585661
$753$	0	0
$754$	−19.8672	−0.723519
$755$	0	0
$756$	0	0
$757$	6.08930	0.221319	0.110660	−	0.993858i	$-0.464704\pi$
$757$	6.08930	0.221319	0.110660	+	0.993858i	$0.464704\pi$
$758$	0.841853	0.0305775
$759$	0	0
$760$	0	0
$761$	52.2808	1.89518	0.947589	−	0.319493i	$-0.103512\pi$
$761$	52.2808	1.89518	0.947589	+	0.319493i	$0.103512\pi$
$762$	0	0
$763$	−6.85674	−0.248231
$764$	1.19289	0.0431574
$765$	0	0
$766$	−4.06896	−0.147017
$767$	0.582033	0.0210160
$768$	0	0
$769$	19.9606	0.719796	0.359898	−	0.932992i	$-0.382811\pi$
$769$	19.9606	0.719796	0.359898	+	0.932992i	$0.382811\pi$
$770$	0	0
$771$	0	0
$772$	−3.25624	−0.117195
$773$	−0.869934	−0.0312893	−0.0156447	−	0.999878i	$-0.504980\pi$
$773$	−0.869934	−0.0312893	−0.0156447	+	0.999878i	$0.504980\pi$
$774$	0	0
$775$	0	0
$776$	−5.96735	−0.214216
$777$	0	0
$778$	10.9684	0.393237
$779$	3.29366	0.118008
$780$	0	0
$781$	0.182700	0.00653751
$782$	−23.1856	−0.829114
$783$	0	0
$784$	21.7943	0.778367
$785$	0	0
$786$	0	0
$787$	24.7548	0.882414	0.441207	−	0.897405i	$-0.354550\pi$
$787$	24.7548	0.882414	0.441207	+	0.897405i	$0.354550\pi$
$788$	3.04370	0.108427
$789$	0	0
$790$	0	0
$791$	25.0896	0.892085
$792$	0	0
$793$	−8.69897	−0.308910
$794$	−15.3295	−0.544022
$795$	0	0
$796$	2.36400	0.0837897
$797$	38.8941	1.37770	0.688850	−	0.724904i	$-0.258116\pi$
$797$	38.8941	1.37770	0.688850	+	0.724904i	$0.258116\pi$
$798$	0	0
$799$	−23.3430	−0.825816
$800$	0	0
$801$	0	0
$802$	38.3515	1.35424
$803$	12.1460	0.428622
$804$	0	0
$805$	0	0
$806$	13.6366	0.480330
$807$	0	0
$808$	41.5636	1.46220
$809$	25.5075	0.896794	0.448397	−	0.893834i	$-0.351995\pi$
$809$	25.5075	0.896794	0.448397	+	0.893834i	$0.351995\pi$
$810$	0	0
$811$	46.3724	1.62835	0.814177	−	0.580617i	$-0.197188\pi$
$811$	46.3724	1.62835	0.814177	+	0.580617i	$0.197188\pi$
$812$	2.62278	0.0920414
$813$	0	0
$814$	1.85928	0.0651678
$815$	0	0
$816$	0	0
$817$	7.69643	0.269264
$818$	−45.1114	−1.57728
$819$	0	0
$820$	0	0
$821$	−19.2241	−0.670926	−0.335463	−	0.942053i	$-0.608893\pi$
$821$	−19.2241	−0.670926	−0.335463	+	0.942053i	$0.608893\pi$
$822$	0	0
$823$	24.6655	0.859786	0.429893	−	0.902880i	$-0.358551\pi$
$823$	24.6655	0.859786	0.429893	+	0.902880i	$0.358551\pi$
$824$	18.4932	0.644241
$825$	0	0
$826$	−0.563103	−0.0195929
$827$	34.1386	1.18712	0.593558	−	0.804791i	$-0.297723\pi$
$827$	34.1386	1.18712	0.593558	+	0.804791i	$0.297723\pi$
$828$	0	0
$829$	−6.17073	−0.214318	−0.107159	−	0.994242i	$-0.534175\pi$
$829$	−6.17073	−0.214318	−0.107159	+	0.994242i	$0.534175\pi$
$830$	0	0
$831$	0	0
$832$	14.8277	0.514059
$833$	−31.6770	−1.09754
$834$	0	0
$835$	0	0
$836$	0.230991	0.00798899
$837$	0	0
$838$	52.4743	1.81269
$839$	−40.9380	−1.41334	−0.706668	−	0.707545i	$-0.749803\pi$
$839$	−40.9380	−1.41334	−0.706668	+	0.707545i	$0.749803\pi$
$840$	0	0
$841$	2.43231	0.0838728
$842$	−7.51129	−0.258856
$843$	0	0
$844$	−2.38434	−0.0820725
$845$	0	0
$846$	0	0
$847$	15.4923	0.532320
$848$	2.17976	0.0748534
$849$	0	0
$850$	0	0
$851$	3.86584	0.132519
$852$	0	0
$853$	31.2016	1.06832	0.534162	−	0.845382i	$-0.320627\pi$
$853$	31.2016	1.06832	0.534162	+	0.845382i	$0.320627\pi$
$854$	8.41605	0.287991
$855$	0	0
$856$	−19.9874	−0.683155
$857$	34.7305	1.18637	0.593186	−	0.805065i	$-0.297870\pi$
$857$	34.7305	1.18637	0.593186	+	0.805065i	$0.297870\pi$
$858$	0	0
$859$	−31.6090	−1.07849	−0.539243	−	0.842150i	$-0.681290\pi$
$859$	−31.6090	−1.07849	−0.539243	+	0.842150i	$0.681290\pi$
$860$	0	0
$861$	0	0
$862$	−37.4946	−1.27707
$863$	−21.4738	−0.730975	−0.365488	−	0.930816i	$-0.619098\pi$
$863$	−21.4738	−0.730975	−0.365488	+	0.930816i	$0.619098\pi$
$864$	0	0
$865$	0	0
$866$	−17.3003	−0.587888
$867$	0	0
$868$	−1.80025	−0.0611044
$869$	−6.97631	−0.236655
$870$	0	0
$871$	−26.1168	−0.884933
$872$	−11.8708	−0.401997
$873$	0	0
$874$	3.51972	0.119056
$875$	0	0
$876$	0	0
$877$	−45.1931	−1.52606	−0.763032	−	0.646361i	$-0.776290\pi$
$877$	−45.1931	−1.52606	−0.763032	+	0.646361i	$0.776290\pi$
$878$	−15.3295	−0.517344
$879$	0	0
$880$	0	0
$881$	8.90923	0.300159	0.150080	−	0.988674i	$-0.452047\pi$
$881$	8.90923	0.300159	0.150080	+	0.988674i	$0.452047\pi$
$882$	0	0
$883$	9.03092	0.303915	0.151957	−	0.988387i	$-0.451442\pi$
$883$	9.03092	0.303915	0.151957	+	0.988387i	$0.451442\pi$
$884$	4.84746	0.163038
$885$	0	0
$886$	36.8877	1.23927
$887$	−15.3386	−0.515019	−0.257509	−	0.966276i	$-0.582902\pi$
$887$	−15.3386	−0.515019	−0.257509	+	0.966276i	$0.582902\pi$
$888$	0	0
$889$	−22.5611	−0.756674
$890$	0	0
$891$	0	0
$892$	−0.376464	−0.0126049
$893$	3.54362	0.118583
$894$	0	0
$895$	0	0
$896$	−19.5910	−0.654490
$897$	0	0
$898$	43.7594	1.46027
$899$	−21.5748	−0.719561
$900$	0	0
$901$	−3.16819	−0.105548
$902$	3.66368	0.121987
$903$	0	0
$904$	43.4367	1.44468
$905$	0	0
$906$	0	0
$907$	−12.6570	−0.420269	−0.210134	−	0.977673i	$-0.567390\pi$
$907$	−12.6570	−0.420269	−0.210134	+	0.977673i	$0.567390\pi$
$908$	1.07884	0.0358025
$909$	0	0
$910$	0	0
$911$	−46.8232	−1.55132	−0.775661	−	0.631149i	$-0.782584\pi$
$911$	−46.8232	−1.55132	−0.775661	+	0.631149i	$0.782584\pi$
$912$	0	0
$913$	4.98360	0.164933
$914$	40.8148	1.35003
$915$	0	0
$916$	2.28841	0.0756111
$917$	−25.0896	−0.828533
$918$	0	0
$919$	0.873147	0.0288025	0.0144012	−	0.999896i	$-0.495416\pi$
$919$	0.873147	0.0288025	0.0144012	+	0.999896i	$0.495416\pi$
$920$	0	0
$921$	0	0
$922$	−58.8107	−1.93683
$923$	−0.582033	−0.0191578
$924$	0	0
$925$	0	0
$926$	55.4004	1.82057
$927$	0	0
$928$	9.93358	0.326086
$929$	−12.1748	−0.399443	−0.199721	−	0.979853i	$-0.564004\pi$
$929$	−12.1748	−0.399443	−0.199721	+	0.979853i	$0.564004\pi$
$930$	0	0
$931$	4.80877	0.157601
$932$	−1.27789	−0.0418585
$933$	0	0
$934$	−26.7378	−0.874887
$935$	0	0
$936$	0	0
$937$	3.53423	0.115458	0.0577292	−	0.998332i	$-0.481614\pi$
$937$	3.53423	0.115458	0.0577292	+	0.998332i	$0.481614\pi$
$938$	25.2674	0.825008
$939$	0	0
$940$	0	0
$941$	33.7497	1.10021	0.550104	−	0.835096i	$-0.314588\pi$
$941$	33.7497	1.10021	0.550104	+	0.835096i	$0.314588\pi$
$942$	0	0
$943$	7.61755	0.248062
$944$	−1.13287	−0.0368718
$945$	0	0
$946$	8.56106	0.278344
$947$	23.8876	0.776244	0.388122	−	0.921608i	$-0.373124\pi$
$947$	23.8876	0.776244	0.388122	+	0.921608i	$0.373124\pi$
$948$	0	0
$949$	−38.6939	−1.25606
$950$	0	0
$951$	0	0
$952$	24.9896	0.809917
$953$	49.9779	1.61894	0.809471	−	0.587160i	$-0.199754\pi$
$953$	49.9779	1.61894	0.809471	+	0.587160i	$0.199754\pi$
$954$	0	0
$955$	0	0
$956$	8.36513	0.270548
$957$	0	0
$958$	−17.4736	−0.564548
$959$	−21.6661	−0.699634
$960$	0	0
$961$	−16.1912	−0.522298
$962$	−5.92318	−0.190971
$963$	0	0
$964$	−4.56106	−0.146902
$965$	0	0
$966$	0	0
$967$	−37.0019	−1.18990	−0.594950	−	0.803762i	$-0.702828\pi$
$967$	−37.0019	−1.18990	−0.594950	+	0.803762i	$0.702828\pi$
$968$	26.8211	0.862064
$969$	0	0
$970$	0	0
$971$	59.9599	1.92421	0.962103	−	0.272686i	$-0.0879121\pi$
$971$	59.9599	1.92421	0.962103	+	0.272686i	$0.0879121\pi$
$972$	0	0
$973$	0.769332	0.0246637
$974$	−45.0356	−1.44303
$975$	0	0
$976$	16.9317	0.541970
$977$	17.7623	0.568266	0.284133	−	0.958785i	$-0.408294\pi$
$977$	17.7623	0.568266	0.284133	+	0.958785i	$0.408294\pi$
$978$	0	0
$979$	7.84412	0.250699
$980$	0	0
$981$	0	0
$982$	40.0420	1.27779
$983$	−59.4881	−1.89738	−0.948688	−	0.316214i	$-0.897588\pi$
$983$	−59.4881	−1.89738	−0.948688	+	0.316214i	$0.897588\pi$
$984$	0	0
$985$	0	0
$986$	−56.2043	−1.78991
$987$	0	0
$988$	−0.735877	−0.0234114
$989$	17.8002	0.566014
$990$	0	0
$991$	−2.92964	−0.0930631	−0.0465316	−	0.998917i	$-0.514817\pi$
$991$	−2.92964	−0.0930631	−0.0465316	+	0.998917i	$0.514817\pi$
$992$	−6.81832	−0.216482
$993$	0	0
$994$	0.563103	0.0178605
$995$	0	0
$996$	0	0
$997$	19.0953	0.604754	0.302377	−	0.953188i	$-0.402220\pi$
$997$	19.0953	0.604754	0.302377	+	0.953188i	$0.402220\pi$
$998$	−39.9989	−1.26614
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4275.2.a.bs.1.5	yes	6
3.2	odd	2	inner	4275.2.a.bs.1.2	✓	6
5.4	even	2		4275.2.a.bt.1.2	yes	6
15.14	odd	2		4275.2.a.bt.1.5	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4275.2.a.bs.1.2	✓	6	3.2	odd	2	inner
4275.2.a.bs.1.5	yes	6	1.1	even	1	trivial
4275.2.a.bt.1.2	yes	6	5.4	even	2
4275.2.a.bt.1.5	yes	6	15.14	odd	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:	\( N \)	\(=\)	\( 4275 = 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4275.a (trivial)

\(f(q)\)	\(=\)	\(q+1.52185 q^{2} +0.316031 q^{4} -1.48028 q^{7} -2.56275 q^{8} -0.730913 q^{11} +2.32850 q^{13} -2.25276 q^{14} -4.53219 q^{16} +6.58733 q^{17} -1.00000 q^{19} -1.11234 q^{22} -2.31279 q^{23} +3.54362 q^{26} -0.467814 q^{28} -5.60645 q^{29} +3.84822 q^{31} -1.77181 q^{32} +10.0249 q^{34} -1.67150 q^{37} -1.52185 q^{38} -3.29366 q^{41} -7.69643 q^{43} -0.230991 q^{44} -3.51972 q^{46} -3.54362 q^{47} -4.80877 q^{49} +0.735877 q^{52} -0.480952 q^{53} +3.79358 q^{56} -8.53219 q^{58} +0.249961 q^{59} -3.73588 q^{61} +5.85641 q^{62} +6.36794 q^{64} -11.2162 q^{67} +2.08180 q^{68} -0.249961 q^{71} -16.6175 q^{73} -2.54378 q^{74} -0.316031 q^{76} +1.08195 q^{77} +9.54465 q^{79} -5.01247 q^{82} -6.81832 q^{83} -11.7128 q^{86} +1.87315 q^{88} -10.7320 q^{89} -3.44682 q^{91} -0.730913 q^{92} -5.39287 q^{94} +2.32850 q^{97} -7.31824 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 8 q^{4} - 16 q^{13} - 6 q^{19} - 10 q^{22} - 30 q^{28} + 2 q^{31} - 12 q^{34} - 40 q^{37} - 4 q^{43} - 30 q^{46} + 10 q^{49} - 20 q^{52} - 24 q^{58} + 2 q^{61} + 26 q^{64} - 34 q^{67} - 22 q^{73}+ \cdots - 16 q^{97}+O(q^{100}) \) 6 * q + 8 * q^4 - 16 * q^13 - 6 * q^19 - 10 * q^22 - 30 * q^28 + 2 * q^31 - 12 * q^34 - 40 * q^37 - 4 * q^43 - 30 * q^46 + 10 * q^49 - 20 * q^52 - 24 * q^58 + 2 * q^61 + 26 * q^64 - 34 * q^67 - 22 * q^73 - 8 * q^76 - 6 * q^79 + 6 * q^82 - 82 * q^88 - 44 * q^91 + 52 * q^94 - 16 * q^97

Introduction

L-functions

Modular forms

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