LMFDB - Embedded newform 7500.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7500,2,Mod(1,7500)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7500, 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("7500.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7500 = 2^{2} \cdot 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7500.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:	$59.8878015160$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.5125.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 6x^{2} + 7x + 11 $ x^4 - 2x^3 - 6x^2 + 7*x + 11
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 300)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.12233$ of defining polynomial
Character	$\chi$	$=$	7500.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.74037 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.74037 q^{7} +1.00000 q^{9} +2.32027 q^{11} +4.93831 q^{13} +3.74037 q^{17} -1.69364 q^{19} +1.74037 q^{21} +8.67867 q^{23} -1.00000 q^{27} +3.12233 q^{29} +9.56166 q^{31} -2.32027 q^{33} +5.36700 q^{37} -4.93831 q^{39} -8.72540 q^{41} +2.86270 q^{43} -8.46248 q^{47} -3.97112 q^{49} -3.74037 q^{51} -1.34449 q^{53} +1.69364 q^{57} +4.38793 q^{59} -15.3316 q^{61} -1.74037 q^{63} +9.23075 q^{67} -8.67867 q^{69} +0.0235645 q^{71} -1.02251 q^{73} -4.03813 q^{77} -0.798776 q^{79} +1.00000 q^{81} +2.37008 q^{83} -3.12233 q^{87} -4.86205 q^{89} -8.59447 q^{91} -9.56166 q^{93} -3.46151 q^{97} +2.32027 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + 4 * q^7 + 4 * q^9	$ 4 q - 4 q^{3} + 4 q^{7} + 4 q^{9} - q^{11} + 5 q^{13} + 4 q^{17} - 5 q^{19} - 4 q^{21} + 9 q^{23} - 4 q^{27} + 6 q^{29} + 11 q^{31} + q^{33} + 2 q^{37} - 5 q^{39} - 6 q^{43} + 16 q^{47} - 4 q^{49} - 4 q^{51} - 2 q^{53} + 5 q^{57} + q^{59} - 22 q^{61} + 4 q^{63} + 36 q^{67} - 9 q^{69} + 20 q^{71} + 12 q^{73} - 11 q^{77} - 3 q^{79} + 4 q^{81} + 14 q^{83} - 6 q^{87} - 15 q^{89} - 10 q^{91} - 11 q^{93} - 12 q^{97} - q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 4 * q^7 + 4 * q^9 - q^11 + 5 * q^13 + 4 * q^17 - 5 * q^19 - 4 * q^21 + 9 * q^23 - 4 * q^27 + 6 * q^29 + 11 * q^31 + q^33 + 2 * q^37 - 5 * q^39 - 6 * q^43 + 16 * q^47 - 4 * q^49 - 4 * q^51 - 2 * q^53 + 5 * q^57 + q^59 - 22 * q^61 + 4 * q^63 + 36 * q^67 - 9 * q^69 + 20 * q^71 + 12 * q^73 - 11 * q^77 - 3 * q^79 + 4 * q^81 + 14 * q^83 - 6 * q^87 - 15 * q^89 - 10 * q^91 - 11 * q^93 - 12 * q^97 - 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.74037	−0.657797	−0.328899	−	0.944365i	$-0.606677\pi$
$7$	−1.74037	−0.657797	−0.328899	+	0.944365i	$0.606677\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.32027	0.699589	0.349794	−	0.936827i	$-0.386251\pi$
$11$	2.32027	0.699589	0.349794	+	0.936827i	$0.386251\pi$
$12$	0	0
$13$	4.93831	1.36964	0.684820	−	0.728712i	$-0.259881\pi$
$13$	4.93831	1.36964	0.684820	+	0.728712i	$0.259881\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.74037	0.907172	0.453586	−	0.891212i	$-0.350144\pi$
$17$	3.74037	0.907172	0.453586	+	0.891212i	$0.350144\pi$
$18$	0	0
$19$	−1.69364	−0.388548	−0.194274	−	0.980947i	$-0.562235\pi$
$19$	−1.69364	−0.388548	−0.194274	+	0.980947i	$0.562235\pi$
$20$	0	0
$21$	1.74037	0.379779
$22$	0	0
$23$	8.67867	1.80963	0.904814	−	0.425806i	$-0.140009\pi$
$23$	8.67867	1.80963	0.904814	+	0.425806i	$0.140009\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.12233	0.579803	0.289901	−	0.957057i	$-0.406377\pi$
$29$	3.12233	0.579803	0.289901	+	0.957057i	$0.406377\pi$
$30$	0	0
$31$	9.56166	1.71732	0.858662	−	0.512542i	$-0.171296\pi$
$31$	9.56166	1.71732	0.858662	+	0.512542i	$0.171296\pi$
$32$	0	0
$33$	−2.32027	−0.403908
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.36700	0.882329	0.441165	−	0.897426i	$-0.354565\pi$
$37$	5.36700	0.882329	0.441165	+	0.897426i	$0.354565\pi$
$38$	0	0
$39$	−4.93831	−0.790762
$40$	0	0
$41$	−8.72540	−1.36268	−0.681339	−	0.731968i	$-0.738602\pi$
$41$	−8.72540	−1.36268	−0.681339	+	0.731968i	$0.738602\pi$
$42$	0	0
$43$	2.86270	0.436558	0.218279	−	0.975886i	$-0.429956\pi$
$43$	2.86270	0.436558	0.218279	+	0.975886i	$0.429956\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.46248	−1.23438	−0.617190	−	0.786814i	$-0.711729\pi$
$47$	−8.46248	−1.23438	−0.617190	+	0.786814i	$0.711729\pi$
$48$	0	0
$49$	−3.97112	−0.567303
$50$	0	0
$51$	−3.74037	−0.523756
$52$	0	0
$53$	−1.34449	−0.184680	−0.0923398	−	0.995728i	$-0.529435\pi$
$53$	−1.34449	−0.184680	−0.0923398	+	0.995728i	$0.529435\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.69364	0.224328
$58$	0	0
$59$	4.38793	0.571260	0.285630	−	0.958340i	$-0.407797\pi$
$59$	4.38793	0.571260	0.285630	+	0.958340i	$0.407797\pi$
$60$	0	0
$61$	−15.3316	−1.96300	−0.981502	−	0.191452i	$-0.938681\pi$
$61$	−15.3316	−1.96300	−0.981502	+	0.191452i	$0.938681\pi$
$62$	0	0
$63$	−1.74037	−0.219266
$64$	0	0
$65$	0	0
$66$	0	0
$67$	9.23075	1.12772	0.563858	−	0.825872i	$-0.309317\pi$
$67$	9.23075	1.12772	0.563858	+	0.825872i	$0.309317\pi$
$68$	0	0
$69$	−8.67867	−1.04479
$70$	0	0
$71$	0.0235645	0.00279659	0.00139830	−	0.999999i	$-0.499555\pi$
$71$	0.0235645	0.00279659	0.00139830	+	0.999999i	$0.499555\pi$
$72$	0	0
$73$	−1.02251	−0.119676	−0.0598380	−	0.998208i	$-0.519058\pi$
$73$	−1.02251	−0.119676	−0.0598380	+	0.998208i	$0.519058\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−4.03813	−0.460187
$78$	0	0
$79$	−0.798776	−0.0898693	−0.0449346	−	0.998990i	$-0.514308\pi$
$79$	−0.798776	−0.0898693	−0.0449346	+	0.998990i	$0.514308\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.37008	0.260150	0.130075	−	0.991504i	$-0.458478\pi$
$83$	2.37008	0.260150	0.130075	+	0.991504i	$0.458478\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−3.12233	−0.334749
$88$	0	0
$89$	−4.86205	−0.515376	−0.257688	−	0.966228i	$-0.582961\pi$
$89$	−4.86205	−0.515376	−0.257688	+	0.966228i	$0.582961\pi$
$90$	0	0
$91$	−8.59447	−0.900945
$92$	0	0
$93$	−9.56166	−0.991498
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−3.46151	−0.351463	−0.175731	−	0.984438i	$-0.556229\pi$
$97$	−3.46151	−0.351463	−0.175731	+	0.984438i	$0.556229\pi$
$98$	0	0
$99$	2.32027	0.233196
$100$	0	0
$101$	−7.58056	−0.754294	−0.377147	−	0.926154i	$-0.623095\pi$
$101$	−7.58056	−0.754294	−0.377147	+	0.926154i	$0.623095\pi$
$102$	0	0
$103$	10.7699	1.06119	0.530595	−	0.847626i	$-0.321969\pi$
$103$	10.7699	1.06119	0.530595	+	0.847626i	$0.321969\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.34344	−0.419896	−0.209948	−	0.977713i	$-0.567329\pi$
$107$	−4.34344	−0.419896	−0.209948	+	0.977713i	$0.567329\pi$
$108$	0	0
$109$	1.07127	0.102609	0.0513045	−	0.998683i	$-0.483662\pi$
$109$	1.07127	0.102609	0.0513045	+	0.998683i	$0.483662\pi$
$110$	0	0
$111$	−5.36700	−0.509413
$112$	0	0
$113$	18.5617	1.74613	0.873067	−	0.487601i	$-0.162128\pi$
$113$	18.5617	1.74613	0.873067	+	0.487601i	$0.162128\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.93831	0.456547
$118$	0	0
$119$	−6.50961	−0.596735
$120$	0	0
$121$	−5.61633	−0.510576
$122$	0	0
$123$	8.72540	0.786743
$124$	0	0
$125$	0	0
$126$	0	0
$127$	19.0096	1.68683	0.843414	−	0.537265i	$-0.180542\pi$
$127$	19.0096	1.68683	0.843414	+	0.537265i	$0.180542\pi$
$128$	0	0
$129$	−2.86270	−0.252047
$130$	0	0
$131$	−0.745032	−0.0650937	−0.0325469	−	0.999470i	$-0.510362\pi$
$131$	−0.745032	−0.0650937	−0.0325469	+	0.999470i	$0.510362\pi$
$132$	0	0
$133$	2.94756	0.255586
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−11.1326	−0.951119	−0.475559	−	0.879684i	$-0.657754\pi$
$137$	−11.1326	−0.951119	−0.475559	+	0.879684i	$0.657754\pi$
$138$	0	0
$139$	13.6889	1.16108	0.580539	−	0.814233i	$-0.302842\pi$
$139$	13.6889	1.16108	0.580539	+	0.814233i	$0.302842\pi$
$140$	0	0
$141$	8.46248	0.712670
$142$	0	0
$143$	11.4582	0.958185
$144$	0	0
$145$	0	0
$146$	0	0
$147$	3.97112	0.327533
$148$	0	0
$149$	−10.6283	−0.870701	−0.435351	−	0.900261i	$-0.643376\pi$
$149$	−10.6283	−0.870701	−0.435351	+	0.900261i	$0.643376\pi$
$150$	0	0
$151$	−10.4689	−0.851943	−0.425972	−	0.904737i	$-0.640068\pi$
$151$	−10.4689	−0.851943	−0.425972	+	0.904737i	$0.640068\pi$
$152$	0	0
$153$	3.74037	0.302391
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−17.2987	−1.38059	−0.690295	−	0.723528i	$-0.742519\pi$
$157$	−17.2987	−1.38059	−0.690295	+	0.723528i	$0.742519\pi$
$158$	0	0
$159$	1.34449	0.106625
$160$	0	0
$161$	−15.1041	−1.19037
$162$	0	0
$163$	−6.14688	−0.481460	−0.240730	−	0.970592i	$-0.577387\pi$
$163$	−6.14688	−0.481460	−0.240730	+	0.970592i	$0.577387\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.90018	−0.147040	−0.0735201	−	0.997294i	$-0.523423\pi$
$167$	−1.90018	−0.147040	−0.0735201	+	0.997294i	$0.523423\pi$
$168$	0	0
$169$	11.3869	0.875914
$170$	0	0
$171$	−1.69364	−0.129516
$172$	0	0
$173$	11.5859	0.880857	0.440429	−	0.897788i	$-0.354826\pi$
$173$	11.5859	0.880857	0.440429	+	0.897788i	$0.354826\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−4.38793	−0.329817
$178$	0	0
$179$	22.0477	1.64792	0.823961	−	0.566646i	$-0.191759\pi$
$179$	22.0477	1.64792	0.823961	+	0.566646i	$0.191759\pi$
$180$	0	0
$181$	20.6571	1.53543	0.767717	−	0.640790i	$-0.221393\pi$
$181$	20.6571	1.53543	0.767717	+	0.640790i	$0.221393\pi$
$182$	0	0
$183$	15.3316	1.13334
$184$	0	0
$185$	0	0
$186$	0	0
$187$	8.67867	0.634648
$188$	0	0
$189$	1.74037	0.126593
$190$	0	0
$191$	20.8321	1.50736	0.753680	−	0.657242i	$-0.228277\pi$
$191$	20.8321	1.50736	0.753680	+	0.657242i	$0.228277\pi$
$192$	0	0
$193$	4.78053	0.344110	0.172055	−	0.985087i	$-0.444959\pi$
$193$	4.78053	0.344110	0.172055	+	0.985087i	$0.444959\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	17.3520	1.23628	0.618141	−	0.786068i	$-0.287886\pi$
$197$	17.3520	1.23628	0.618141	+	0.786068i	$0.287886\pi$
$198$	0	0
$199$	11.0756	0.785129	0.392564	−	0.919724i	$-0.371588\pi$
$199$	11.0756	0.785129	0.392564	+	0.919724i	$0.371588\pi$
$200$	0	0
$201$	−9.23075	−0.651087
$202$	0	0
$203$	−5.43401	−0.381393
$204$	0	0
$205$	0	0
$206$	0	0
$207$	8.67867	0.603210
$208$	0	0
$209$	−3.92971	−0.271824
$210$	0	0
$211$	5.66902	0.390272	0.195136	−	0.980776i	$-0.437485\pi$
$211$	5.66902	0.390272	0.195136	+	0.980776i	$0.437485\pi$
$212$	0	0
$213$	−0.0235645	−0.00161461
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−16.6408	−1.12965
$218$	0	0
$219$	1.02251	0.0690950
$220$	0	0
$221$	18.4711	1.24250
$222$	0	0
$223$	−21.3805	−1.43174	−0.715872	−	0.698231i	$-0.753971\pi$
$223$	−21.3805	−1.43174	−0.715872	+	0.698231i	$0.753971\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−22.9624	−1.52407	−0.762036	−	0.647535i	$-0.775800\pi$
$227$	−22.9624	−1.52407	−0.762036	+	0.647535i	$0.775800\pi$
$228$	0	0
$229$	−28.1047	−1.85721	−0.928604	−	0.371072i	$-0.878990\pi$
$229$	−28.1047	−1.85721	−0.928604	+	0.371072i	$0.878990\pi$
$230$	0	0
$231$	4.03813	0.265689
$232$	0	0
$233$	−1.33195	−0.0872592	−0.0436296	−	0.999048i	$-0.513892\pi$
$233$	−1.33195	−0.0872592	−0.0436296	+	0.999048i	$0.513892\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0.798776	0.0518861
$238$	0	0
$239$	3.44529	0.222857	0.111429	−	0.993772i	$-0.464457\pi$
$239$	3.44529	0.222857	0.111429	+	0.993772i	$0.464457\pi$
$240$	0	0
$241$	−24.0938	−1.55202	−0.776008	−	0.630722i	$-0.782759\pi$
$241$	−24.0938	−1.55202	−0.776008	+	0.630722i	$0.782759\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−8.36372	−0.532171
$248$	0	0
$249$	−2.37008	−0.150198
$250$	0	0
$251$	−30.1621	−1.90381	−0.951907	−	0.306387i	$-0.900880\pi$
$251$	−30.1621	−1.90381	−0.951907	+	0.306387i	$0.900880\pi$
$252$	0	0
$253$	20.1369	1.26600
$254$	0	0
$255$	0	0
$256$	0	0
$257$	10.6862	0.666588	0.333294	−	0.942823i	$-0.391840\pi$
$257$	10.6862	0.666588	0.333294	+	0.942823i	$0.391840\pi$
$258$	0	0
$259$	−9.34055	−0.580394
$260$	0	0
$261$	3.12233	0.193268
$262$	0	0
$263$	9.23278	0.569318	0.284659	−	0.958629i	$-0.408120\pi$
$263$	9.23278	0.569318	0.284659	+	0.958629i	$0.408120\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	4.86205	0.297553
$268$	0	0
$269$	26.7381	1.63025	0.815124	−	0.579286i	$-0.196669\pi$
$269$	26.7381	1.63025	0.815124	+	0.579286i	$0.196669\pi$
$270$	0	0
$271$	24.1840	1.46907	0.734535	−	0.678571i	$-0.237400\pi$
$271$	24.1840	1.46907	0.734535	+	0.678571i	$0.237400\pi$
$272$	0	0
$273$	8.59447	0.520161
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−31.6079	−1.89914	−0.949568	−	0.313563i	$-0.898477\pi$
$277$	−31.6079	−1.89914	−0.949568	+	0.313563i	$0.898477\pi$
$278$	0	0
$279$	9.56166	0.572441
$280$	0	0
$281$	23.2625	1.38773	0.693863	−	0.720107i	$-0.255907\pi$
$281$	23.2625	1.38773	0.693863	+	0.720107i	$0.255907\pi$
$282$	0	0
$283$	24.8321	1.47612	0.738058	−	0.674737i	$-0.235743\pi$
$283$	24.8321	1.47612	0.738058	+	0.674737i	$0.235743\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	15.1854	0.896366
$288$	0	0
$289$	−3.00965	−0.177038
$290$	0	0
$291$	3.46151	0.202917
$292$	0	0
$293$	29.2758	1.71031	0.855156	−	0.518371i	$-0.173461\pi$
$293$	29.2758	1.71031	0.855156	+	0.518371i	$0.173461\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−2.32027	−0.134636
$298$	0	0
$299$	42.8580	2.47854
$300$	0	0
$301$	−4.98215	−0.287166
$302$	0	0
$303$	7.58056	0.435492
$304$	0	0
$305$	0	0
$306$	0	0
$307$	8.73972	0.498802	0.249401	−	0.968400i	$-0.419766\pi$
$307$	8.73972	0.498802	0.249401	+	0.968400i	$0.419766\pi$
$308$	0	0
$309$	−10.7699	−0.612678
$310$	0	0
$311$	−15.4561	−0.876436	−0.438218	−	0.898869i	$-0.644390\pi$
$311$	−15.4561	−0.876436	−0.438218	+	0.898869i	$0.644390\pi$
$312$	0	0
$313$	−1.55898	−0.0881185	−0.0440593	−	0.999029i	$-0.514029\pi$
$313$	−1.55898	−0.0881185	−0.0440593	+	0.999029i	$0.514029\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5.46256	−0.306808	−0.153404	−	0.988164i	$-0.549024\pi$
$317$	−5.46256	−0.306808	−0.153404	+	0.988164i	$0.549024\pi$
$318$	0	0
$319$	7.24467	0.405623
$320$	0	0
$321$	4.34344	0.242427
$322$	0	0
$323$	−6.33484	−0.352480
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−1.07127	−0.0592414
$328$	0	0
$329$	14.7278	0.811972
$330$	0	0
$331$	0.855804	0.0470392	0.0235196	−	0.999723i	$-0.492513\pi$
$331$	0.855804	0.0470392	0.0235196	+	0.999723i	$0.492513\pi$
$332$	0	0
$333$	5.36700	0.294110
$334$	0	0
$335$	0	0
$336$	0	0
$337$	14.8246	0.807546	0.403773	−	0.914859i	$-0.367699\pi$
$337$	14.8246	0.807546	0.403773	+	0.914859i	$0.367699\pi$
$338$	0	0
$339$	−18.5617	−1.00813
$340$	0	0
$341$	22.1857	1.20142
$342$	0	0
$343$	19.0938	1.03097
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.17069	−0.0628459	−0.0314229	−	0.999506i	$-0.510004\pi$
$347$	−1.17069	−0.0628459	−0.0314229	+	0.999506i	$0.510004\pi$
$348$	0	0
$349$	−21.4346	−1.14737	−0.573683	−	0.819077i	$-0.694486\pi$
$349$	−21.4346	−1.14737	−0.573683	+	0.819077i	$0.694486\pi$
$350$	0	0
$351$	−4.93831	−0.263587
$352$	0	0
$353$	−3.56271	−0.189624	−0.0948119	−	0.995495i	$-0.530225\pi$
$353$	−3.56271	−0.189624	−0.0948119	+	0.995495i	$0.530225\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	6.50961	0.344525
$358$	0	0
$359$	13.3912	0.706761	0.353381	−	0.935480i	$-0.385032\pi$
$359$	13.3912	0.706761	0.353381	+	0.935480i	$0.385032\pi$
$360$	0	0
$361$	−16.1316	−0.849031
$362$	0	0
$363$	5.61633	0.294781
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−1.42436	−0.0743508	−0.0371754	−	0.999309i	$-0.511836\pi$
$367$	−1.42436	−0.0743508	−0.0371754	+	0.999309i	$0.511836\pi$
$368$	0	0
$369$	−8.72540	−0.454226
$370$	0	0
$371$	2.33990	0.121482
$372$	0	0
$373$	25.5327	1.32203	0.661017	−	0.750371i	$-0.270125\pi$
$373$	25.5327	1.32203	0.661017	+	0.750371i	$0.270125\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	15.4190	0.794121
$378$	0	0
$379$	−38.0791	−1.95599	−0.977995	−	0.208628i	$-0.933100\pi$
$379$	−38.0791	−1.95599	−0.977995	+	0.208628i	$0.933100\pi$
$380$	0	0
$381$	−19.0096	−0.973890
$382$	0	0
$383$	13.6545	0.697710	0.348855	−	0.937177i	$-0.386571\pi$
$383$	13.6545	0.697710	0.348855	+	0.937177i	$0.386571\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.86270	0.145519
$388$	0	0
$389$	35.7899	1.81462	0.907310	−	0.420462i	$-0.138132\pi$
$389$	35.7899	1.81462	0.907310	+	0.420462i	$0.138132\pi$
$390$	0	0
$391$	32.4614	1.64165
$392$	0	0
$393$	0.745032	0.0375819
$394$	0	0
$395$	0	0
$396$	0	0
$397$	24.5424	1.23175	0.615873	−	0.787846i	$-0.288804\pi$
$397$	24.5424	1.23175	0.615873	+	0.787846i	$0.288804\pi$
$398$	0	0
$399$	−2.94756	−0.147562
$400$	0	0
$401$	−25.7068	−1.28373	−0.641867	−	0.766816i	$-0.721840\pi$
$401$	−25.7068	−1.28373	−0.641867	+	0.766816i	$0.721840\pi$
$402$	0	0
$403$	47.2184	2.35212
$404$	0	0
$405$	0	0
$406$	0	0
$407$	12.4529	0.617268
$408$	0	0
$409$	−18.5559	−0.917532	−0.458766	−	0.888557i	$-0.651708\pi$
$409$	−18.5559	−0.917532	−0.458766	+	0.888557i	$0.651708\pi$
$410$	0	0
$411$	11.1326	0.549129
$412$	0	0
$413$	−7.63661	−0.375773
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−13.6889	−0.670348
$418$	0	0
$419$	4.69324	0.229280	0.114640	−	0.993407i	$-0.463429\pi$
$419$	4.69324	0.229280	0.114640	+	0.993407i	$0.463429\pi$
$420$	0	0
$421$	9.22050	0.449380	0.224690	−	0.974430i	$-0.427863\pi$
$421$	9.22050	0.449380	0.224690	+	0.974430i	$0.427863\pi$
$422$	0	0
$423$	−8.46248	−0.411460
$424$	0	0
$425$	0	0
$426$	0	0
$427$	26.6825	1.29126
$428$	0	0
$429$	−11.4582	−0.553208
$430$	0	0
$431$	28.5809	1.37669	0.688346	−	0.725382i	$-0.258337\pi$
$431$	28.5809	1.37669	0.688346	+	0.725382i	$0.258337\pi$
$432$	0	0
$433$	3.63365	0.174622	0.0873110	−	0.996181i	$-0.472173\pi$
$433$	3.63365	0.174622	0.0873110	+	0.996181i	$0.472173\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−14.6986	−0.703127
$438$	0	0
$439$	−17.6073	−0.840352	−0.420176	−	0.907443i	$-0.638032\pi$
$439$	−17.6073	−0.840352	−0.420176	+	0.907443i	$0.638032\pi$
$440$	0	0
$441$	−3.97112	−0.189101
$442$	0	0
$443$	7.11807	0.338190	0.169095	−	0.985600i	$-0.445916\pi$
$443$	7.11807	0.338190	0.169095	+	0.985600i	$0.445916\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	10.6283	0.502700
$448$	0	0
$449$	−20.4121	−0.963305	−0.481653	−	0.876362i	$-0.659963\pi$
$449$	−20.4121	−0.963305	−0.481653	+	0.876362i	$0.659963\pi$
$450$	0	0
$451$	−20.2453	−0.953315
$452$	0	0
$453$	10.4689	0.491870
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−16.8375	−0.787624	−0.393812	−	0.919191i	$-0.628844\pi$
$457$	−16.8375	−0.787624	−0.393812	+	0.919191i	$0.628844\pi$
$458$	0	0
$459$	−3.74037	−0.174585
$460$	0	0
$461$	17.3249	0.806903	0.403451	−	0.915001i	$-0.367810\pi$
$461$	17.3249	0.806903	0.403451	+	0.915001i	$0.367810\pi$
$462$	0	0
$463$	8.60412	0.399867	0.199934	−	0.979809i	$-0.435927\pi$
$463$	8.60412	0.399867	0.199934	+	0.979809i	$0.435927\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0.835852	0.0386786	0.0193393	−	0.999813i	$-0.493844\pi$
$467$	0.835852	0.0386786	0.0193393	+	0.999813i	$0.493844\pi$
$468$	0	0
$469$	−16.0649	−0.741808
$470$	0	0
$471$	17.2987	0.797084
$472$	0	0
$473$	6.64225	0.305411
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−1.34449	−0.0615599
$478$	0	0
$479$	−33.4495	−1.52835	−0.764173	−	0.645012i	$-0.776853\pi$
$479$	−33.4495	−1.52835	−0.764173	+	0.645012i	$0.776853\pi$
$480$	0	0
$481$	26.5039	1.20847
$482$	0	0
$483$	15.1041	0.687260
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−24.4158	−1.10638	−0.553192	−	0.833054i	$-0.686590\pi$
$487$	−24.4158	−1.10638	−0.553192	+	0.833054i	$0.686590\pi$
$488$	0	0
$489$	6.14688	0.277971
$490$	0	0
$491$	−14.6475	−0.661032	−0.330516	−	0.943800i	$-0.607223\pi$
$491$	−14.6475	−0.661032	−0.330516	+	0.943800i	$0.607223\pi$
$492$	0	0
$493$	11.6787	0.525981
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−0.0410109	−0.00183959
$498$	0	0
$499$	−13.5655	−0.607276	−0.303638	−	0.952787i	$-0.598201\pi$
$499$	−13.5655	−0.607276	−0.303638	+	0.952787i	$0.598201\pi$
$500$	0	0
$501$	1.90018	0.0848937
$502$	0	0
$503$	−40.9282	−1.82490	−0.912449	−	0.409191i	$-0.865811\pi$
$503$	−40.9282	−1.82490	−0.912449	+	0.409191i	$0.865811\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−11.3869	−0.505709
$508$	0	0
$509$	12.1412	0.538147	0.269074	−	0.963120i	$-0.413282\pi$
$509$	12.1412	0.538147	0.269074	+	0.963120i	$0.413282\pi$
$510$	0	0
$511$	1.77955	0.0787226
$512$	0	0
$513$	1.69364	0.0747760
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−19.6353	−0.863559
$518$	0	0
$519$	−11.5859	−0.508563
$520$	0	0
$521$	26.2250	1.14894	0.574470	−	0.818526i	$-0.305208\pi$
$521$	26.2250	1.14894	0.574470	+	0.818526i	$0.305208\pi$
$522$	0	0
$523$	15.3245	0.670095	0.335048	−	0.942201i	$-0.391248\pi$
$523$	15.3245	0.670095	0.335048	+	0.942201i	$0.391248\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	35.7641	1.55791
$528$	0	0
$529$	52.3194	2.27476
$530$	0	0
$531$	4.38793	0.190420
$532$	0	0
$533$	−43.0887	−1.86638
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−22.0477	−0.951429
$538$	0	0
$539$	−9.21409	−0.396879
$540$	0	0
$541$	9.47745	0.407467	0.203734	−	0.979026i	$-0.434692\pi$
$541$	9.47745	0.407467	0.203734	+	0.979026i	$0.434692\pi$
$542$	0	0
$543$	−20.6571	−0.886483
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0.645935	0.0276182	0.0138091	−	0.999905i	$-0.495604\pi$
$547$	0.645935	0.0276182	0.0138091	+	0.999905i	$0.495604\pi$
$548$	0	0
$549$	−15.3316	−0.654335
$550$	0	0
$551$	−5.28811	−0.225281
$552$	0	0
$553$	1.39016	0.0591158
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−12.0297	−0.509716	−0.254858	−	0.966978i	$-0.582029\pi$
$557$	−12.0297	−0.509716	−0.254858	+	0.966978i	$0.582029\pi$
$558$	0	0
$559$	14.1369	0.597927
$560$	0	0
$561$	−8.67867	−0.366414
$562$	0	0
$563$	15.8832	0.669398	0.334699	−	0.942325i	$-0.391365\pi$
$563$	15.8832	0.669398	0.334699	+	0.942325i	$0.391365\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1.74037	−0.0730886
$568$	0	0
$569$	21.9304	0.919368	0.459684	−	0.888082i	$-0.347963\pi$
$569$	21.9304	0.919368	0.459684	+	0.888082i	$0.347963\pi$
$570$	0	0
$571$	−6.80835	−0.284921	−0.142460	−	0.989801i	$-0.545501\pi$
$571$	−6.80835	−0.284921	−0.142460	+	0.989801i	$0.545501\pi$
$572$	0	0
$573$	−20.8321	−0.870274
$574$	0	0
$575$	0	0
$576$	0	0
$577$	18.6273	0.775464	0.387732	−	0.921772i	$-0.373259\pi$
$577$	18.6273	0.775464	0.387732	+	0.921772i	$0.373259\pi$
$578$	0	0
$579$	−4.78053	−0.198672
$580$	0	0
$581$	−4.12481	−0.171126
$582$	0	0
$583$	−3.11958	−0.129200
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−14.2102	−0.586518	−0.293259	−	0.956033i	$-0.594740\pi$
$587$	−14.2102	−0.586518	−0.293259	+	0.956033i	$0.594740\pi$
$588$	0	0
$589$	−16.1940	−0.667262
$590$	0	0
$591$	−17.3520	−0.713767
$592$	0	0
$593$	40.7850	1.67484	0.837420	−	0.546559i	$-0.184063\pi$
$593$	40.7850	1.67484	0.837420	+	0.546559i	$0.184063\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−11.0756	−0.453294
$598$	0	0
$599$	34.8928	1.42568	0.712841	−	0.701326i	$-0.247408\pi$
$599$	34.8928	1.42568	0.712841	+	0.701326i	$0.247408\pi$
$600$	0	0
$601$	10.6287	0.433552	0.216776	−	0.976221i	$-0.430446\pi$
$601$	10.6287	0.433552	0.216776	+	0.976221i	$0.430446\pi$
$602$	0	0
$603$	9.23075	0.375905
$604$	0	0
$605$	0	0
$606$	0	0
$607$	30.5033	1.23809	0.619045	−	0.785355i	$-0.287520\pi$
$607$	30.5033	1.23809	0.619045	+	0.785355i	$0.287520\pi$
$608$	0	0
$609$	5.43401	0.220197
$610$	0	0
$611$	−41.7904	−1.69066
$612$	0	0
$613$	−12.7732	−0.515904	−0.257952	−	0.966158i	$-0.583048\pi$
$613$	−12.7732	−0.515904	−0.257952	+	0.966158i	$0.583048\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−11.9327	−0.480391	−0.240196	−	0.970725i	$-0.577212\pi$
$617$	−11.9327	−0.480391	−0.240196	+	0.970725i	$0.577212\pi$
$618$	0	0
$619$	−14.5045	−0.582987	−0.291494	−	0.956573i	$-0.594152\pi$
$619$	−14.5045	−0.582987	−0.291494	+	0.956573i	$0.594152\pi$
$620$	0	0
$621$	−8.67867	−0.348263
$622$	0	0
$623$	8.46176	0.339013
$624$	0	0
$625$	0	0
$626$	0	0
$627$	3.92971	0.156937
$628$	0	0
$629$	20.0746	0.800425
$630$	0	0
$631$	−3.66780	−0.146013	−0.0730064	−	0.997331i	$-0.523259\pi$
$631$	−3.66780	−0.146013	−0.0730064	+	0.997331i	$0.523259\pi$
$632$	0	0
$633$	−5.66902	−0.225323
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−19.6106	−0.777001
$638$	0	0
$639$	0.0235645	0.000932198 0
$640$	0	0
$641$	−13.8519	−0.547116	−0.273558	−	0.961856i	$-0.588201\pi$
$641$	−13.8519	−0.547116	−0.273558	+	0.961856i	$0.588201\pi$
$642$	0	0
$643$	−27.1641	−1.07125	−0.535624	−	0.844456i	$-0.679924\pi$
$643$	−27.1641	−1.07125	−0.535624	+	0.844456i	$0.679924\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−5.14988	−0.202463	−0.101231	−	0.994863i	$-0.532278\pi$
$647$	−5.14988	−0.202463	−0.101231	+	0.994863i	$0.532278\pi$
$648$	0	0
$649$	10.1812	0.399647
$650$	0	0
$651$	16.6408	0.652204
$652$	0	0
$653$	−29.6734	−1.16121	−0.580604	−	0.814186i	$-0.697184\pi$
$653$	−29.6734	−1.16121	−0.580604	+	0.814186i	$0.697184\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−1.02251	−0.0398920
$658$	0	0
$659$	15.7740	0.614466	0.307233	−	0.951634i	$-0.400597\pi$
$659$	15.7740	0.614466	0.307233	+	0.951634i	$0.400597\pi$
$660$	0	0
$661$	−24.0557	−0.935657	−0.467828	−	0.883819i	$-0.654963\pi$
$661$	−24.0557	−0.935657	−0.467828	+	0.883819i	$0.654963\pi$
$662$	0	0
$663$	−18.4711	−0.717357
$664$	0	0
$665$	0	0
$666$	0	0
$667$	27.0977	1.04923
$668$	0	0
$669$	21.3805	0.826618
$670$	0	0
$671$	−35.5734	−1.37330
$672$	0	0
$673$	32.0035	1.23365	0.616823	−	0.787102i	$-0.288420\pi$
$673$	32.0035	1.23365	0.616823	+	0.787102i	$0.288420\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−24.3809	−0.937035	−0.468517	−	0.883454i	$-0.655212\pi$
$677$	−24.3809	−0.937035	−0.468517	+	0.883454i	$0.655212\pi$
$678$	0	0
$679$	6.02429	0.231191
$680$	0	0
$681$	22.9624	0.879923
$682$	0	0
$683$	−11.1528	−0.426752	−0.213376	−	0.976970i	$-0.568446\pi$
$683$	−11.1528	−0.426752	−0.213376	+	0.976970i	$0.568446\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	28.1047	1.07226
$688$	0	0
$689$	−6.63949	−0.252945
$690$	0	0
$691$	10.8286	0.411941	0.205970	−	0.978558i	$-0.433965\pi$
$691$	10.8286	0.411941	0.205970	+	0.978558i	$0.433965\pi$
$692$	0	0
$693$	−4.03813	−0.153396
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−32.6362	−1.23618
$698$	0	0
$699$	1.33195	0.0503791
$700$	0	0
$701$	43.2512	1.63357	0.816787	−	0.576939i	$-0.195753\pi$
$701$	43.2512	1.63357	0.816787	+	0.576939i	$0.195753\pi$
$702$	0	0
$703$	−9.08977	−0.342827
$704$	0	0
$705$	0	0
$706$	0	0
$707$	13.1930	0.496172
$708$	0	0
$709$	−44.6052	−1.67518	−0.837592	−	0.546296i	$-0.816037\pi$
$709$	−44.6052	−1.67518	−0.837592	+	0.546296i	$0.816037\pi$
$710$	0	0
$711$	−0.798776	−0.0299564
$712$	0	0
$713$	82.9825	3.10772
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−3.44529	−0.128667
$718$	0	0
$719$	−32.0868	−1.19664	−0.598318	−	0.801259i	$-0.704164\pi$
$719$	−32.0868	−1.19664	−0.598318	+	0.801259i	$0.704164\pi$
$720$	0	0
$721$	−18.7436	−0.698047
$722$	0	0
$723$	24.0938	0.896057
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−13.6216	−0.505196	−0.252598	−	0.967571i	$-0.581285\pi$
$727$	−13.6216	−0.505196	−0.252598	+	0.967571i	$0.581285\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	10.7076	0.396033
$732$	0	0
$733$	28.4242	1.04987	0.524935	−	0.851142i	$-0.324090\pi$
$733$	28.4242	1.04987	0.524935	+	0.851142i	$0.324090\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	21.4179	0.788937
$738$	0	0
$739$	9.71457	0.357356	0.178678	−	0.983908i	$-0.442818\pi$
$739$	9.71457	0.357356	0.178678	+	0.983908i	$0.442818\pi$
$740$	0	0
$741$	8.36372	0.307249
$742$	0	0
$743$	34.2029	1.25478	0.627390	−	0.778705i	$-0.284123\pi$
$743$	34.2029	1.25478	0.627390	+	0.778705i	$0.284123\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2.37008	0.0867168
$748$	0	0
$749$	7.55917	0.276206
$750$	0	0
$751$	−12.7840	−0.466495	−0.233248	−	0.972417i	$-0.574935\pi$
$751$	−12.7840	−0.466495	−0.233248	+	0.972417i	$0.574935\pi$
$752$	0	0
$753$	30.1621	1.09917
$754$	0	0
$755$	0	0
$756$	0	0
$757$	26.6200	0.967520	0.483760	−	0.875201i	$-0.339271\pi$
$757$	26.6200	0.967520	0.483760	+	0.875201i	$0.339271\pi$
$758$	0	0
$759$	−20.1369	−0.730923
$760$	0	0
$761$	25.7137	0.932122	0.466061	−	0.884753i	$-0.345673\pi$
$761$	25.7137	0.932122	0.466061	+	0.884753i	$0.345673\pi$
$762$	0	0
$763$	−1.86440	−0.0674959
$764$	0	0
$765$	0	0
$766$	0	0
$767$	21.6689	0.782420
$768$	0	0
$769$	−38.3717	−1.38372	−0.691860	−	0.722032i	$-0.743208\pi$
$769$	−38.3717	−1.38372	−0.691860	+	0.722032i	$0.743208\pi$
$770$	0	0
$771$	−10.6862	−0.384855
$772$	0	0
$773$	−1.39884	−0.0503126	−0.0251563	−	0.999684i	$-0.508008\pi$
$773$	−1.39884	−0.0503126	−0.0251563	+	0.999684i	$0.508008\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	9.34055	0.335090
$778$	0	0
$779$	14.7777	0.529466
$780$	0	0
$781$	0.0546761	0.00195647
$782$	0	0
$783$	−3.12233	−0.111583
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−4.21023	−0.150078	−0.0750392	−	0.997181i	$-0.523908\pi$
$787$	−4.21023	−0.150078	−0.0750392	+	0.997181i	$0.523908\pi$
$788$	0	0
$789$	−9.23278	−0.328696
$790$	0	0
$791$	−32.3041	−1.14860
$792$	0	0
$793$	−75.7119	−2.68861
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−31.6196	−1.12002	−0.560012	−	0.828485i	$-0.689203\pi$
$797$	−31.6196	−1.12002	−0.560012	+	0.828485i	$0.689203\pi$
$798$	0	0
$799$	−31.6528	−1.11980
$800$	0	0
$801$	−4.86205	−0.171792
$802$	0	0
$803$	−2.37251	−0.0837240
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−26.7381	−0.941224
$808$	0	0
$809$	42.1098	1.48050	0.740252	−	0.672330i	$-0.234706\pi$
$809$	42.1098	1.48050	0.740252	+	0.672330i	$0.234706\pi$
$810$	0	0
$811$	15.1530	0.532095	0.266048	−	0.963960i	$-0.414282\pi$
$811$	15.1530	0.532095	0.266048	+	0.963960i	$0.414282\pi$
$812$	0	0
$813$	−24.1840	−0.848168
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−4.84839	−0.169624
$818$	0	0
$819$	−8.59447	−0.300315
$820$	0	0
$821$	−0.402320	−0.0140411	−0.00702053	−	0.999975i	$-0.502235\pi$
$821$	−0.402320	−0.0140411	−0.00702053	+	0.999975i	$0.502235\pi$
$822$	0	0
$823$	−0.609361	−0.0212410	−0.0106205	−	0.999944i	$-0.503381\pi$
$823$	−0.609361	−0.0212410	−0.0106205	+	0.999944i	$0.503381\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	27.4672	0.955128	0.477564	−	0.878597i	$-0.341520\pi$
$827$	27.4672	0.955128	0.477564	+	0.878597i	$0.341520\pi$
$828$	0	0
$829$	31.1543	1.08204	0.541018	−	0.841011i	$-0.318039\pi$
$829$	31.1543	1.08204	0.541018	+	0.841011i	$0.318039\pi$
$830$	0	0
$831$	31.6079	1.09647
$832$	0	0
$833$	−14.8535	−0.514642
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−9.56166	−0.330499
$838$	0	0
$839$	39.1176	1.35049	0.675245	−	0.737594i	$-0.264038\pi$
$839$	39.1176	1.35049	0.675245	+	0.737594i	$0.264038\pi$
$840$	0	0
$841$	−19.2510	−0.663829
$842$	0	0
$843$	−23.2625	−0.801204
$844$	0	0
$845$	0	0
$846$	0	0
$847$	9.77448	0.335855
$848$	0	0
$849$	−24.8321	−0.852236
$850$	0	0
$851$	46.5785	1.59669
$852$	0	0
$853$	−52.5757	−1.80016	−0.900079	−	0.435727i	$-0.856491\pi$
$853$	−52.5757	−1.80016	−0.900079	+	0.435727i	$0.856491\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−31.0291	−1.05993	−0.529967	−	0.848018i	$-0.677796\pi$
$857$	−31.0291	−1.05993	−0.529967	+	0.848018i	$0.677796\pi$
$858$	0	0
$859$	38.8450	1.32537	0.662687	−	0.748897i	$-0.269416\pi$
$859$	38.8450	1.32537	0.662687	+	0.748897i	$0.269416\pi$
$860$	0	0
$861$	−15.1854	−0.517517
$862$	0	0
$863$	−3.60246	−0.122629	−0.0613147	−	0.998118i	$-0.519529\pi$
$863$	−3.60246	−0.122629	−0.0613147	+	0.998118i	$0.519529\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	3.00965	0.102213
$868$	0	0
$869$	−1.85338	−0.0628715
$870$	0	0
$871$	45.5843	1.54456
$872$	0	0
$873$	−3.46151	−0.117154
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−31.5597	−1.06569	−0.532847	−	0.846211i	$-0.678878\pi$
$877$	−31.5597	−1.06569	−0.532847	+	0.846211i	$0.678878\pi$
$878$	0	0
$879$	−29.2758	−0.987449
$880$	0	0
$881$	6.02783	0.203083	0.101541	−	0.994831i	$-0.467623\pi$
$881$	6.02783	0.203083	0.101541	+	0.994831i	$0.467623\pi$
$882$	0	0
$883$	−3.96581	−0.133460	−0.0667300	−	0.997771i	$-0.521257\pi$
$883$	−3.96581	−0.133460	−0.0667300	+	0.997771i	$0.521257\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	55.7151	1.87073	0.935365	−	0.353684i	$-0.115071\pi$
$887$	55.7151	1.87073	0.935365	+	0.353684i	$0.115071\pi$
$888$	0	0
$889$	−33.0837	−1.10959
$890$	0	0
$891$	2.32027	0.0777321
$892$	0	0
$893$	14.3324	0.479616
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−42.8580	−1.43099
$898$	0	0
$899$	29.8547	0.995709
$900$	0	0
$901$	−5.02888	−0.167536
$902$	0	0
$903$	4.98215	0.165796
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−21.4063	−0.710785	−0.355392	−	0.934717i	$-0.615653\pi$
$907$	−21.4063	−0.710785	−0.355392	+	0.934717i	$0.615653\pi$
$908$	0	0
$909$	−7.58056	−0.251431
$910$	0	0
$911$	31.5890	1.04659	0.523295	−	0.852151i	$-0.324702\pi$
$911$	31.5890	1.04659	0.523295	+	0.852151i	$0.324702\pi$
$912$	0	0
$913$	5.49924	0.181998
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.29663	0.0428185
$918$	0	0
$919$	−18.0775	−0.596322	−0.298161	−	0.954516i	$-0.596373\pi$
$919$	−18.0775	−0.596322	−0.298161	+	0.954516i	$0.596373\pi$
$920$	0	0
$921$	−8.73972	−0.287983
$922$	0	0
$923$	0.116369	0.00383033
$924$	0	0
$925$	0	0
$926$	0	0
$927$	10.7699	0.353730
$928$	0	0
$929$	8.93003	0.292985	0.146492	−	0.989212i	$-0.453202\pi$
$929$	8.93003	0.292985	0.146492	+	0.989212i	$0.453202\pi$
$930$	0	0
$931$	6.72565	0.220424
$932$	0	0
$933$	15.4561	0.506011
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−0.0607147	−0.00198346	−0.000991732	−	1.00000i	$-0.500316\pi$
$937$	−0.0607147	−0.00198346	−0.000991732		1.00000i	$0.500316\pi$
$938$	0	0
$939$	1.55898	0.0508752
$940$	0	0
$941$	20.6449	0.673005	0.336503	−	0.941682i	$-0.390756\pi$
$941$	20.6449	0.673005	0.336503	+	0.941682i	$0.390756\pi$
$942$	0	0
$943$	−75.7249	−2.46594
$944$	0	0
$945$	0	0
$946$	0	0
$947$	51.3845	1.66977	0.834886	−	0.550423i	$-0.185534\pi$
$947$	51.3845	1.66977	0.834886	+	0.550423i	$0.185534\pi$
$948$	0	0
$949$	−5.04948	−0.163913
$950$	0	0
$951$	5.46256	0.177136
$952$	0	0
$953$	24.5442	0.795064	0.397532	−	0.917588i	$-0.369867\pi$
$953$	24.5442	0.795064	0.397532	+	0.917588i	$0.369867\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−7.24467	−0.234187
$958$	0	0
$959$	19.3747	0.625643
$960$	0	0
$961$	60.4253	1.94920
$962$	0	0
$963$	−4.34344	−0.139965
$964$	0	0
$965$	0	0
$966$	0	0
$967$	38.1844	1.22793	0.613964	−	0.789334i	$-0.289574\pi$
$967$	38.1844	1.22793	0.613964	+	0.789334i	$0.289574\pi$
$968$	0	0
$969$	6.33484	0.203504
$970$	0	0
$971$	25.8172	0.828512	0.414256	−	0.910160i	$-0.364042\pi$
$971$	25.8172	0.828512	0.414256	+	0.910160i	$0.364042\pi$
$972$	0	0
$973$	−23.8237	−0.763753
$974$	0	0
$975$	0	0
$976$	0	0
$977$	20.6259	0.659881	0.329941	−	0.944002i	$-0.392971\pi$
$977$	20.6259	0.659881	0.329941	+	0.944002i	$0.392971\pi$
$978$	0	0
$979$	−11.2813	−0.360551
$980$	0	0
$981$	1.07127	0.0342030
$982$	0	0
$983$	−22.1336	−0.705953	−0.352976	−	0.935632i	$-0.614830\pi$
$983$	−22.1336	−0.705953	−0.352976	+	0.935632i	$0.614830\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−14.7278	−0.468792
$988$	0	0
$989$	24.8445	0.790008
$990$	0	0
$991$	−1.66509	−0.0528933	−0.0264466	−	0.999650i	$-0.508419\pi$
$991$	−1.66509	−0.0528933	−0.0264466	+	0.999650i	$0.508419\pi$
$992$	0	0
$993$	−0.855804	−0.0271581
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−23.4762	−0.743497	−0.371749	−	0.928333i	$-0.621242\pi$
$997$	−23.4762	−0.743497	−0.371749	+	0.928333i	$0.621242\pi$
$998$	0	0
$999$	−5.36700	−0.169804

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7500.2.a.e.1.1		4
5.2	odd	4		7500.2.d.c.1249.5		8
5.3	odd	4		7500.2.d.c.1249.4		8
5.4	even	2		7500.2.a.f.1.4		4
25.3	odd	20		1500.2.o.b.49.4		16
25.4	even	10		1500.2.m.a.1201.2		8
25.6	even	5		300.2.m.b.61.1	✓	8
25.8	odd	20		1500.2.o.b.949.2		16
25.17	odd	20		1500.2.o.b.949.3		16
25.19	even	10		1500.2.m.a.301.2		8
25.21	even	5		300.2.m.b.241.1	yes	8
25.22	odd	20		1500.2.o.b.49.1		16
75.56	odd	10		900.2.n.b.361.2		8
75.71	odd	10		900.2.n.b.541.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.m.b.61.1	✓	8	25.6	even	5
300.2.m.b.241.1	yes	8	25.21	even	5
900.2.n.b.361.2		8	75.56	odd	10
900.2.n.b.541.2		8	75.71	odd	10
1500.2.m.a.301.2		8	25.19	even	10
1500.2.m.a.1201.2		8	25.4	even	10
1500.2.o.b.49.1		16	25.22	odd	20
1500.2.o.b.49.4		16	25.3	odd	20
1500.2.o.b.949.2		16	25.8	odd	20
1500.2.o.b.949.3		16	25.17	odd	20
7500.2.a.e.1.1		4	1.1	even	1	trivial
7500.2.a.f.1.4		4	5.4	even	2
7500.2.d.c.1249.4		8	5.3	odd	4
7500.2.d.c.1249.5		8	5.2	odd	4

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:	\( N \)	\(=\)	\( 7500 = 2^{2} \cdot 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7500.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.74037 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.74037 q^{7} +1.00000 q^{9} +2.32027 q^{11} +4.93831 q^{13} +3.74037 q^{17} -1.69364 q^{19} +1.74037 q^{21} +8.67867 q^{23} -1.00000 q^{27} +3.12233 q^{29} +9.56166 q^{31} -2.32027 q^{33} +5.36700 q^{37} -4.93831 q^{39} -8.72540 q^{41} +2.86270 q^{43} -8.46248 q^{47} -3.97112 q^{49} -3.74037 q^{51} -1.34449 q^{53} +1.69364 q^{57} +4.38793 q^{59} -15.3316 q^{61} -1.74037 q^{63} +9.23075 q^{67} -8.67867 q^{69} +0.0235645 q^{71} -1.02251 q^{73} -4.03813 q^{77} -0.798776 q^{79} +1.00000 q^{81} +2.37008 q^{83} -3.12233 q^{87} -4.86205 q^{89} -8.59447 q^{91} -9.56166 q^{93} -3.46151 q^{97} +2.32027 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + 4 * q^7 + 4 * q^9	\( 4 q - 4 q^{3} + 4 q^{7} + 4 q^{9} - q^{11} + 5 q^{13} + 4 q^{17} - 5 q^{19} - 4 q^{21} + 9 q^{23} - 4 q^{27} + 6 q^{29} + 11 q^{31} + q^{33} + 2 q^{37} - 5 q^{39} - 6 q^{43} + 16 q^{47} - 4 q^{49} - 4 q^{51} - 2 q^{53} + 5 q^{57} + q^{59} - 22 q^{61} + 4 q^{63} + 36 q^{67} - 9 q^{69} + 20 q^{71} + 12 q^{73} - 11 q^{77} - 3 q^{79} + 4 q^{81} + 14 q^{83} - 6 q^{87} - 15 q^{89} - 10 q^{91} - 11 q^{93} - 12 q^{97} - q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 4 * q^7 + 4 * q^9 - q^11 + 5 * q^13 + 4 * q^17 - 5 * q^19 - 4 * q^21 + 9 * q^23 - 4 * q^27 + 6 * q^29 + 11 * q^31 + q^33 + 2 * q^37 - 5 * q^39 - 6 * q^43 + 16 * q^47 - 4 * q^49 - 4 * q^51 - 2 * q^53 + 5 * q^57 + q^59 - 22 * q^61 + 4 * q^63 + 36 * q^67 - 9 * q^69 + 20 * q^71 + 12 * q^73 - 11 * q^77 - 3 * q^79 + 4 * q^81 + 14 * q^83 - 6 * q^87 - 15 * q^89 - 10 * q^91 - 11 * q^93 - 12 * q^97 - q^99

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