LMFDB - Embedded newform 7500.2.d.g.1249.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7500,2,Mod(1249,7500)]

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, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7500.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7500 = 2^{2} \cdot 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7500.d (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$59.8878015160$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 300)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.17
Character	$\chi$	$=$	7500.1249
Dual form			7500.2.d.g.1249.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -2.44380i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -2.44380i q^{7} -1.00000 q^{9} -0.576983 q^{11} +6.44645i q^{13} -1.89772i q^{17} -8.27470 q^{19} +2.44380 q^{21} +4.20898i q^{23} -1.00000i q^{27} -6.53865 q^{29} +4.86687 q^{31} -0.576983i q^{33} +0.218005i q^{37} -6.44645 q^{39} +6.45452 q^{41} -3.42419i q^{43} +9.61833i q^{47} +1.02783 q^{49} +1.89772 q^{51} -13.9922i q^{53} -8.27470i q^{57} +12.0530 q^{59} -4.81065 q^{61} +2.44380i q^{63} -3.87094i q^{67} -4.20898 q^{69} -6.48247 q^{71} -10.0509i q^{73} +1.41003i q^{77} +4.74470 q^{79} +1.00000 q^{81} +3.43418i q^{83} -6.53865i q^{87} +5.68638 q^{89} +15.7538 q^{91} +4.86687i q^{93} -6.58475i q^{97} +0.576983 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 24 q^{9}+O(q^{10}) $ 24 * q - 24 * q^9	$ 24 q - 24 q^{9} + 4 q^{11} - 20 q^{19} + 16 q^{21} - 16 q^{29} - 4 q^{31} + 20 q^{41} - 56 q^{49} + 16 q^{51} + 4 q^{59} + 68 q^{61} - 36 q^{69} - 12 q^{79} + 24 q^{81} - 20 q^{89} + 40 q^{91} - 4 q^{99}+O(q^{100}) $ 24 * q - 24 * q^9 + 4 * q^11 - 20 * q^19 + 16 * q^21 - 16 * q^29 - 4 * q^31 + 20 * q^41 - 56 * q^49 + 16 * q^51 + 4 * q^59 + 68 * q^61 - 36 * q^69 - 12 * q^79 + 24 * q^81 - 20 * q^89 + 40 * q^91 - 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/7500\mathbb{Z}\right)^\times$.

$n$	$2501$	$3751$	$6877$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.44380i	− 0.923670i	−0.886966	−	0.461835i	$-0.847191\pi$
$7$	− 2.44380i	− 0.923670i	0.886966	−	0.461835i	$-0.152809\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−0.576983	−0.173967	−0.0869835	−	0.996210i	$-0.527723\pi$
$11$	−0.576983	−0.173967	−0.0869835	+	0.996210i	$0.527723\pi$
$12$	0	0
$13$	6.44645i	1.78792i	0.448145	+	0.893961i	$0.352085\pi$
$13$	6.44645i	1.78792i	−0.448145	+	0.893961i	$0.647915\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1.89772i	− 0.460265i	−0.973159	−	0.230133i	$-0.926084\pi$
$17$	− 1.89772i	− 0.460265i	0.973159	−	0.230133i	$-0.0739160\pi$
$18$	0	0
$19$	−8.27470	−1.89835	−0.949174	−	0.314753i	$-0.898078\pi$
$19$	−8.27470	−1.89835	−0.949174	+	0.314753i	$0.898078\pi$
$20$	0	0
$21$	2.44380	0.533281
$22$	0	0
$23$	4.20898i	0.877633i	0.898577	+	0.438816i	$0.144602\pi$
$23$	4.20898i	0.877633i	−0.898577	+	0.438816i	$0.855398\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	−6.53865	−1.21420	−0.607098	−	0.794627i	$-0.707667\pi$
$29$	−6.53865	−1.21420	−0.607098	+	0.794627i	$0.707667\pi$
$30$	0	0
$31$	4.86687	0.874117	0.437058	−	0.899433i	$-0.356020\pi$
$31$	4.86687	0.874117	0.437058	+	0.899433i	$0.356020\pi$
$32$	0	0
$33$	− 0.576983i	− 0.100440i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.218005i	0.0358398i	0.999839	+	0.0179199i	$0.00570439\pi$
$37$	0.218005i	0.0358398i	−0.999839	+	0.0179199i	$0.994296\pi$
$38$	0	0
$39$	−6.44645	−1.03226
$40$	0	0
$41$	6.45452	1.00803	0.504014	−	0.863696i	$-0.331856\pi$
$41$	6.45452	1.00803	0.504014	+	0.863696i	$0.331856\pi$
$42$	0	0
$43$	− 3.42419i	− 0.522184i	−0.965314	−	0.261092i	$-0.915917\pi$
$43$	− 3.42419i	− 0.522184i	0.965314	−	0.261092i	$-0.0840825\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.61833i	1.40298i	0.712680	+	0.701489i	$0.247481\pi$
$47$	9.61833i	1.40298i	−0.712680	+	0.701489i	$0.752519\pi$
$48$	0	0
$49$	1.02783	0.146833
$50$	0	0
$51$	1.89772	0.265734
$52$	0	0
$53$	− 13.9922i	− 1.92198i	−0.276589	−	0.960988i	$-0.589204\pi$
$53$	− 13.9922i	− 1.92198i	0.276589	−	0.960988i	$-0.410796\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 8.27470i	− 1.09601i
$58$	0	0
$59$	12.0530	1.56917	0.784586	−	0.620020i	$-0.212876\pi$
$59$	12.0530	1.56917	0.784586	+	0.620020i	$0.212876\pi$
$60$	0	0
$61$	−4.81065	−0.615941	−0.307970	−	0.951396i	$-0.599650\pi$
$61$	−4.81065	−0.615941	−0.307970	+	0.951396i	$0.599650\pi$
$62$	0	0
$63$	2.44380i	0.307890i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 3.87094i	− 0.472910i	−0.971642	−	0.236455i	$-0.924014\pi$
$67$	− 3.87094i	− 0.472910i	0.971642	−	0.236455i	$-0.0759856\pi$
$68$	0	0
$69$	−4.20898	−0.506701
$70$	0	0
$71$	−6.48247	−0.769328	−0.384664	−	0.923057i	$-0.625683\pi$
$71$	−6.48247	−0.769328	−0.384664	+	0.923057i	$0.625683\pi$
$72$	0	0
$73$	− 10.0509i	− 1.17637i	−0.808727	−	0.588184i	$-0.799843\pi$
$73$	− 10.0509i	− 1.17637i	0.808727	−	0.588184i	$-0.200157\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.41003i	0.160688i
$78$	0	0
$79$	4.74470	0.533821	0.266910	−	0.963721i	$-0.413997\pi$
$79$	4.74470	0.533821	0.266910	+	0.963721i	$0.413997\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.43418i	0.376950i	0.982078	+	0.188475i	$0.0603544\pi$
$83$	3.43418i	0.376950i	−0.982078	+	0.188475i	$0.939646\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 6.53865i	− 0.701017i
$88$	0	0
$89$	5.68638	0.602755	0.301378	−	0.953505i	$-0.402554\pi$
$89$	5.68638	0.602755	0.301378	+	0.953505i	$0.402554\pi$
$90$	0	0
$91$	15.7538	1.65145
$92$	0	0
$93$	4.86687i	0.504671i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 6.58475i	− 0.668580i	−0.942470	−	0.334290i	$-0.891503\pi$
$97$	− 6.58475i	− 0.668580i	0.942470	−	0.334290i	$-0.108497\pi$
$98$	0	0
$99$	0.576983	0.0579890
$100$	0	0
$101$	−12.0363	−1.19766	−0.598828	−	0.800877i	$-0.704367\pi$
$101$	−12.0363	−1.19766	−0.598828	+	0.800877i	$0.704367\pi$
$102$	0	0
$103$	− 5.78624i	− 0.570135i	−0.958507	−	0.285068i	$-0.907984\pi$
$103$	− 5.78624i	− 0.570135i	0.958507	−	0.285068i	$-0.0920161\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 4.07081i	− 0.393540i	−0.980450	−	0.196770i	$-0.936955\pi$
$107$	− 4.07081i	− 0.393540i	0.980450	−	0.196770i	$-0.0630453\pi$
$108$	0	0
$109$	1.45776	0.139628	0.0698139	−	0.997560i	$-0.477759\pi$
$109$	1.45776	0.139628	0.0698139	+	0.997560i	$0.477759\pi$
$110$	0	0
$111$	−0.218005	−0.0206921
$112$	0	0
$113$	− 5.85785i	− 0.551060i	−0.961292	−	0.275530i	$-0.911147\pi$
$113$	− 5.85785i	− 0.551060i	0.961292	−	0.275530i	$-0.0888533\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 6.44645i	− 0.595974i
$118$	0	0
$119$	−4.63766	−0.425133
$120$	0	0
$121$	−10.6671	−0.969735
$122$	0	0
$123$	6.45452i	0.581985i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 15.2854i	− 1.35636i	−0.734894	−	0.678182i	$-0.762768\pi$
$127$	− 15.2854i	− 1.35636i	0.734894	−	0.678182i	$-0.237232\pi$
$128$	0	0
$129$	3.42419	0.301483
$130$	0	0
$131$	−3.75392	−0.327981	−0.163991	−	0.986462i	$-0.552437\pi$
$131$	−3.75392	−0.327981	−0.163991	+	0.986462i	$0.552437\pi$
$132$	0	0
$133$	20.2217i	1.75345i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 20.2511i	− 1.73017i	−0.501628	−	0.865084i	$-0.667265\pi$
$137$	− 20.2511i	− 1.73017i	0.501628	−	0.865084i	$-0.332735\pi$
$138$	0	0
$139$	0.928868	0.0787855	0.0393928	−	0.999224i	$-0.487458\pi$
$139$	0.928868	0.0787855	0.0393928	+	0.999224i	$0.487458\pi$
$140$	0	0
$141$	−9.61833	−0.810010
$142$	0	0
$143$	− 3.71949i	− 0.311040i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.02783i	0.0847740i
$148$	0	0
$149$	−13.1432	−1.07673	−0.538364	−	0.842712i	$-0.680958\pi$
$149$	−13.1432	−1.07673	−0.538364	+	0.842712i	$0.680958\pi$
$150$	0	0
$151$	17.5864	1.43116	0.715580	−	0.698531i	$-0.246163\pi$
$151$	17.5864	1.43116	0.715580	+	0.698531i	$0.246163\pi$
$152$	0	0
$153$	1.89772i	0.153422i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 8.68198i	− 0.692897i	−0.938069	−	0.346449i	$-0.887387\pi$
$157$	− 8.68198i	− 0.692897i	0.938069	−	0.346449i	$-0.112613\pi$
$158$	0	0
$159$	13.9922	1.10965
$160$	0	0
$161$	10.2859	0.810643
$162$	0	0
$163$	− 1.40791i	− 0.110276i	−0.998479	−	0.0551380i	$-0.982440\pi$
$163$	− 1.40791i	− 0.110276i	0.998479	−	0.0551380i	$-0.0175599\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.81475i	0.682106i	0.940044	+	0.341053i	$0.110784\pi$
$167$	8.81475i	0.682106i	−0.940044	+	0.341053i	$0.889216\pi$
$168$	0	0
$169$	−28.5567	−2.19667
$170$	0	0
$171$	8.27470	0.632782
$172$	0	0
$173$	− 7.90310i	− 0.600862i	−0.953804	−	0.300431i	$-0.902870\pi$
$173$	− 7.90310i	− 0.600862i	0.953804	−	0.300431i	$-0.0971304\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	12.0530i	0.905962i
$178$	0	0
$179$	11.4886	0.858701	0.429351	−	0.903138i	$-0.358742\pi$
$179$	11.4886	0.858701	0.429351	+	0.903138i	$0.358742\pi$
$180$	0	0
$181$	3.92307	0.291600	0.145800	−	0.989314i	$-0.453424\pi$
$181$	3.92307	0.291600	0.145800	+	0.989314i	$0.453424\pi$
$182$	0	0
$183$	− 4.81065i	− 0.355614i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1.09495i	0.0800709i
$188$	0	0
$189$	−2.44380	−0.177760
$190$	0	0
$191$	−25.5456	−1.84841	−0.924207	−	0.381893i	$-0.875272\pi$
$191$	−25.5456	−1.84841	−0.924207	+	0.381893i	$0.875272\pi$
$192$	0	0
$193$	0.421651i	0.0303511i	0.999885	+	0.0151756i	$0.00483071\pi$
$193$	0.421651i	0.0303511i	−0.999885	+	0.0151756i	$0.995169\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 7.22140i	− 0.514504i	−0.966344	−	0.257252i	$-0.917183\pi$
$197$	− 7.22140i	− 0.514504i	0.966344	−	0.257252i	$-0.0828170\pi$
$198$	0	0
$199$	−3.93505	−0.278949	−0.139474	−	0.990226i	$-0.544541\pi$
$199$	−3.93505	−0.278949	−0.139474	+	0.990226i	$0.544541\pi$
$200$	0	0
$201$	3.87094	0.273035
$202$	0	0
$203$	15.9792i	1.12152i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 4.20898i	− 0.292544i
$208$	0	0
$209$	4.77437	0.330250
$210$	0	0
$211$	−24.7610	−1.70462	−0.852308	−	0.523041i	$-0.824798\pi$
$211$	−24.7610	−1.70462	−0.852308	+	0.523041i	$0.824798\pi$
$212$	0	0
$213$	− 6.48247i	− 0.444172i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 11.8937i	− 0.807396i
$218$	0	0
$219$	10.0509	0.679176
$220$	0	0
$221$	12.2336	0.822918
$222$	0	0
$223$	19.6727i	1.31738i	0.752415	+	0.658690i	$0.228889\pi$
$223$	19.6727i	1.31738i	−0.752415	+	0.658690i	$0.771111\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 26.3748i	− 1.75055i	−0.483622	−	0.875277i	$-0.660679\pi$
$227$	− 26.3748i	− 1.75055i	0.483622	−	0.875277i	$-0.339321\pi$
$228$	0	0
$229$	16.0212	1.05871	0.529355	−	0.848400i	$-0.322434\pi$
$229$	16.0212	1.05871	0.529355	+	0.848400i	$0.322434\pi$
$230$	0	0
$231$	−1.41003	−0.0927734
$232$	0	0
$233$	− 29.3108i	− 1.92021i	−0.279632	−	0.960107i	$-0.590213\pi$
$233$	− 29.3108i	− 1.92021i	0.279632	−	0.960107i	$-0.409787\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	4.74470i	0.308201i
$238$	0	0
$239$	−25.8558	−1.67247	−0.836237	−	0.548368i	$-0.815250\pi$
$239$	−25.8558	−1.67247	−0.836237	+	0.548368i	$0.815250\pi$
$240$	0	0
$241$	−9.34158	−0.601744	−0.300872	−	0.953665i	$-0.597278\pi$
$241$	−9.34158	−0.601744	−0.300872	+	0.953665i	$0.597278\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 53.3424i	− 3.39410i
$248$	0	0
$249$	−3.43418	−0.217632
$250$	0	0
$251$	−13.5088	−0.852666	−0.426333	−	0.904566i	$-0.640195\pi$
$251$	−13.5088	−0.852666	−0.426333	+	0.904566i	$0.640195\pi$
$252$	0	0
$253$	− 2.42851i	− 0.152679i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	13.0662i	0.815044i	0.913195	+	0.407522i	$0.133607\pi$
$257$	13.0662i	0.815044i	−0.913195	+	0.407522i	$0.866393\pi$
$258$	0	0
$259$	0.532761	0.0331042
$260$	0	0
$261$	6.53865	0.404732
$262$	0	0
$263$	4.43065i	0.273206i	0.990626	+	0.136603i	$0.0436184\pi$
$263$	4.43065i	0.273206i	−0.990626	+	0.136603i	$0.956382\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	5.68638i	0.348001i
$268$	0	0
$269$	−28.0509	−1.71029	−0.855146	−	0.518387i	$-0.826533\pi$
$269$	−28.0509	−1.71029	−0.855146	+	0.518387i	$0.826533\pi$
$270$	0	0
$271$	5.98512	0.363570	0.181785	−	0.983338i	$-0.441812\pi$
$271$	5.98512	0.363570	0.181785	+	0.983338i	$0.441812\pi$
$272$	0	0
$273$	15.7538i	0.953466i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5.07072i	0.304670i	0.988329	+	0.152335i	$0.0486792\pi$
$277$	5.07072i	0.304670i	−0.988329	+	0.152335i	$0.951321\pi$
$278$	0	0
$279$	−4.86687	−0.291372
$280$	0	0
$281$	−18.1559	−1.08309	−0.541546	−	0.840671i	$-0.682161\pi$
$281$	−18.1559	−1.08309	−0.541546	+	0.840671i	$0.682161\pi$
$282$	0	0
$283$	4.69717i	0.279218i	0.990207	+	0.139609i	$0.0445845\pi$
$283$	4.69717i	0.279218i	−0.990207	+	0.139609i	$0.955416\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 15.7736i	− 0.931085i
$288$	0	0
$289$	13.3987	0.788156
$290$	0	0
$291$	6.58475	0.386005
$292$	0	0
$293$	− 9.60771i	− 0.561288i	−0.959812	−	0.280644i	$-0.909452\pi$
$293$	− 9.60771i	− 0.561288i	0.959812	−	0.280644i	$-0.0905481\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0.576983i	0.0334800i
$298$	0	0
$299$	−27.1329	−1.56914
$300$	0	0
$301$	−8.36804	−0.482326
$302$	0	0
$303$	− 12.0363i	− 0.691467i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	13.5400i	0.772771i	0.922337	+	0.386386i	$0.126277\pi$
$307$	13.5400i	0.772771i	−0.922337	+	0.386386i	$0.873723\pi$
$308$	0	0
$309$	5.78624	0.329168
$310$	0	0
$311$	−2.96725	−0.168257	−0.0841286	−	0.996455i	$-0.526811\pi$
$311$	−2.96725	−0.168257	−0.0841286	+	0.996455i	$0.526811\pi$
$312$	0	0
$313$	14.0058i	0.791652i	0.918326	+	0.395826i	$0.129542\pi$
$313$	14.0058i	0.791652i	−0.918326	+	0.395826i	$0.870458\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 27.0176i	− 1.51746i	−0.651405	−	0.758730i	$-0.725820\pi$
$317$	− 27.0176i	− 1.51746i	0.651405	−	0.758730i	$-0.274180\pi$
$318$	0	0
$319$	3.77269	0.211230
$320$	0	0
$321$	4.07081	0.227211
$322$	0	0
$323$	15.7031i	0.873743i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1.45776i	0.0806141i
$328$	0	0
$329$	23.5053	1.29589
$330$	0	0
$331$	−6.65492	−0.365787	−0.182894	−	0.983133i	$-0.558546\pi$
$331$	−6.65492	−0.365787	−0.182894	+	0.983133i	$0.558546\pi$
$332$	0	0
$333$	− 0.218005i	− 0.0119466i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	19.2028i	1.04604i	0.852320	+	0.523021i	$0.175195\pi$
$337$	19.2028i	1.04604i	−0.852320	+	0.523021i	$0.824805\pi$
$338$	0	0
$339$	5.85785	0.318155
$340$	0	0
$341$	−2.80811	−0.152067
$342$	0	0
$343$	− 19.6184i	− 1.05930i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 16.7591i	− 0.899676i	−0.893110	−	0.449838i	$-0.851482\pi$
$347$	− 16.7591i	− 0.899676i	0.893110	−	0.449838i	$-0.148518\pi$
$348$	0	0
$349$	−22.9371	−1.22780	−0.613898	−	0.789385i	$-0.710399\pi$
$349$	−22.9371	−1.22780	−0.613898	+	0.789385i	$0.710399\pi$
$350$	0	0
$351$	6.44645	0.344086
$352$	0	0
$353$	7.01413i	0.373324i	0.982424	+	0.186662i	$0.0597670\pi$
$353$	7.01413i	0.373324i	−0.982424	+	0.186662i	$0.940233\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 4.63766i	− 0.245451i
$358$	0	0
$359$	14.9199	0.787443	0.393722	−	0.919230i	$-0.371187\pi$
$359$	14.9199	0.787443	0.393722	+	0.919230i	$0.371187\pi$
$360$	0	0
$361$	49.4707	2.60372
$362$	0	0
$363$	− 10.6671i	− 0.559877i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 0.680967i	− 0.0355462i	−0.999842	−	0.0177731i	$-0.994342\pi$
$367$	− 0.680967i	− 0.0355462i	0.999842	−	0.0177731i	$-0.00565765\pi$
$368$	0	0
$369$	−6.45452	−0.336009
$370$	0	0
$371$	−34.1942	−1.77527
$372$	0	0
$373$	− 28.6371i	− 1.48277i	−0.671079	−	0.741386i	$-0.734169\pi$
$373$	− 28.6371i	− 1.48277i	0.671079	−	0.741386i	$-0.265831\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 42.1510i	− 2.17089i
$378$	0	0
$379$	21.9020	1.12503	0.562514	−	0.826787i	$-0.309834\pi$
$379$	21.9020	1.12503	0.562514	+	0.826787i	$0.309834\pi$
$380$	0	0
$381$	15.2854	0.783097
$382$	0	0
$383$	− 4.56593i	− 0.233308i	−0.993173	−	0.116654i	$-0.962783\pi$
$383$	− 4.56593i	− 0.233308i	0.993173	−	0.116654i	$-0.0372169\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.42419i	0.174061i
$388$	0	0
$389$	−21.7420	−1.10236	−0.551182	−	0.834385i	$-0.685823\pi$
$389$	−21.7420	−1.10236	−0.551182	+	0.834385i	$0.685823\pi$
$390$	0	0
$391$	7.98747	0.403944
$392$	0	0
$393$	− 3.75392i	− 0.189360i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	15.6234i	0.784114i	0.919941	+	0.392057i	$0.128236\pi$
$397$	15.6234i	0.784114i	−0.919941	+	0.392057i	$0.871764\pi$
$398$	0	0
$399$	−20.2217	−1.01235
$400$	0	0
$401$	−19.9417	−0.995841	−0.497920	−	0.867223i	$-0.665903\pi$
$401$	−19.9417	−0.995841	−0.497920	+	0.867223i	$0.665903\pi$
$402$	0	0
$403$	31.3740i	1.56285i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 0.125785i	− 0.00623495i
$408$	0	0
$409$	17.2159	0.851273	0.425636	−	0.904894i	$-0.360050\pi$
$409$	17.2159	0.851273	0.425636	+	0.904894i	$0.360050\pi$
$410$	0	0
$411$	20.2511	0.998913
$412$	0	0
$413$	− 29.4552i	− 1.44940i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0.928868i	0.0454869i
$418$	0	0
$419$	−13.1307	−0.641477	−0.320738	−	0.947168i	$-0.603931\pi$
$419$	−13.1307	−0.641477	−0.320738	+	0.947168i	$0.603931\pi$
$420$	0	0
$421$	17.0222	0.829610	0.414805	−	0.909910i	$-0.363850\pi$
$421$	17.0222	0.829610	0.414805	+	0.909910i	$0.363850\pi$
$422$	0	0
$423$	− 9.61833i	− 0.467659i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	11.7563i	0.568927i
$428$	0	0
$429$	3.71949	0.179579
$430$	0	0
$431$	−12.5147	−0.602811	−0.301405	−	0.953496i	$-0.597456\pi$
$431$	−12.5147	−0.602811	−0.301405	+	0.953496i	$0.597456\pi$
$432$	0	0
$433$	− 20.7471i	− 0.997044i	−0.866877	−	0.498522i	$-0.833876\pi$
$433$	− 20.7471i	− 0.997044i	0.866877	−	0.498522i	$-0.166124\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 34.8280i	− 1.66605i
$438$	0	0
$439$	33.0806	1.57885	0.789425	−	0.613847i	$-0.210379\pi$
$439$	33.0806	1.57885	0.789425	+	0.613847i	$0.210379\pi$
$440$	0	0
$441$	−1.02783	−0.0489443
$442$	0	0
$443$	23.4802i	1.11558i	0.829984	+	0.557788i	$0.188350\pi$
$443$	23.4802i	1.11558i	−0.829984	+	0.557788i	$0.811650\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 13.1432i	− 0.621650i
$448$	0	0
$449$	31.6965	1.49585	0.747925	−	0.663783i	$-0.231050\pi$
$449$	31.6965	1.49585	0.747925	+	0.663783i	$0.231050\pi$
$450$	0	0
$451$	−3.72415	−0.175363
$452$	0	0
$453$	17.5864i	0.826281i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 28.2267i	− 1.32039i	−0.751094	−	0.660196i	$-0.770473\pi$
$457$	− 28.2267i	− 1.32039i	0.751094	−	0.660196i	$-0.229527\pi$
$458$	0	0
$459$	−1.89772	−0.0885780
$460$	0	0
$461$	−4.94266	−0.230203	−0.115101	−	0.993354i	$-0.536719\pi$
$461$	−4.94266	−0.230203	−0.115101	+	0.993354i	$0.536719\pi$
$462$	0	0
$463$	− 18.4463i	− 0.857272i	−0.903477	−	0.428636i	$-0.858994\pi$
$463$	− 18.4463i	− 0.857272i	0.903477	−	0.428636i	$-0.141006\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 17.3011i	− 0.800600i	−0.916384	−	0.400300i	$-0.868906\pi$
$467$	− 17.3011i	− 0.800600i	0.916384	−	0.400300i	$-0.131094\pi$
$468$	0	0
$469$	−9.45980	−0.436813
$470$	0	0
$471$	8.68198	0.400044
$472$	0	0
$473$	1.97570i	0.0908427i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	13.9922i	0.640659i
$478$	0	0
$479$	34.3756	1.57066	0.785331	−	0.619076i	$-0.212493\pi$
$479$	34.3756	1.57066	0.785331	+	0.619076i	$0.212493\pi$
$480$	0	0
$481$	−1.40536	−0.0640788
$482$	0	0
$483$	10.2859i	0.468025i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	5.04208i	0.228478i	0.993453	+	0.114239i	$0.0364430\pi$
$487$	5.04208i	0.228478i	−0.993453	+	0.114239i	$0.963557\pi$
$488$	0	0
$489$	1.40791	0.0636678
$490$	0	0
$491$	−26.0163	−1.17410	−0.587051	−	0.809550i	$-0.699711\pi$
$491$	−26.0163	−1.17410	−0.587051	+	0.809550i	$0.699711\pi$
$492$	0	0
$493$	12.4085i	0.558852i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	15.8419i	0.710605i
$498$	0	0
$499$	−2.49658	−0.111762	−0.0558812	−	0.998437i	$-0.517797\pi$
$499$	−2.49658	−0.111762	−0.0558812	+	0.998437i	$0.517797\pi$
$500$	0	0
$501$	−8.81475	−0.393814
$502$	0	0
$503$	14.2600i	0.635824i	0.948120	+	0.317912i	$0.102982\pi$
$503$	14.2600i	0.635824i	−0.948120	+	0.317912i	$0.897018\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 28.5567i	− 1.26825i
$508$	0	0
$509$	15.6163	0.692179	0.346089	−	0.938202i	$-0.387509\pi$
$509$	15.6163	0.692179	0.346089	+	0.938202i	$0.387509\pi$
$510$	0	0
$511$	−24.5624	−1.08658
$512$	0	0
$513$	8.27470i	0.365337i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 5.54962i	− 0.244072i
$518$	0	0
$519$	7.90310	0.346908
$520$	0	0
$521$	14.3727	0.629678	0.314839	−	0.949145i	$-0.398049\pi$
$521$	14.3727	0.629678	0.314839	+	0.949145i	$0.398049\pi$
$522$	0	0
$523$	− 4.41486i	− 0.193048i	−0.995331	−	0.0965241i	$-0.969228\pi$
$523$	− 4.41486i	− 0.193048i	0.995331	−	0.0965241i	$-0.0307725\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 9.23597i	− 0.402325i
$528$	0	0
$529$	5.28451	0.229761
$530$	0	0
$531$	−12.0530	−0.523057
$532$	0	0
$533$	41.6087i	1.80227i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	11.4886i	0.495771i
$538$	0	0
$539$	−0.593041	−0.0255441
$540$	0	0
$541$	36.0371	1.54936	0.774679	−	0.632355i	$-0.217912\pi$
$541$	36.0371	1.54936	0.774679	+	0.632355i	$0.217912\pi$
$542$	0	0
$543$	3.92307i	0.168355i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	12.1032i	0.517497i	0.965945	+	0.258748i	$0.0833101\pi$
$547$	12.1032i	0.517497i	−0.965945	+	0.258748i	$0.916690\pi$
$548$	0	0
$549$	4.81065	0.205314
$550$	0	0
$551$	54.1054	2.30497
$552$	0	0
$553$	− 11.5951i	− 0.493074i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	24.0437i	1.01876i	0.860541	+	0.509382i	$0.170126\pi$
$557$	24.0437i	1.01876i	−0.860541	+	0.509382i	$0.829874\pi$
$558$	0	0
$559$	22.0738	0.933624
$560$	0	0
$561$	−1.09495	−0.0462290
$562$	0	0
$563$	18.9430i	0.798352i	0.916874	+	0.399176i	$0.130704\pi$
$563$	18.9430i	0.798352i	−0.916874	+	0.399176i	$0.869296\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 2.44380i	− 0.102630i
$568$	0	0
$569$	31.2220	1.30889	0.654447	−	0.756108i	$-0.272902\pi$
$569$	31.2220	1.30889	0.654447	+	0.756108i	$0.272902\pi$
$570$	0	0
$571$	−0.0391729	−0.00163934	−0.000819668	−	1.00000i	$-0.500261\pi$
$571$	−0.0391729	−0.00163934	−0.000819668		1.00000i	$0.500261\pi$
$572$	0	0
$573$	− 25.5456i	− 1.06718i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 20.1295i	− 0.838001i	−0.907986	−	0.419000i	$-0.862381\pi$
$577$	− 20.1295i	− 0.838001i	0.907986	−	0.419000i	$-0.137619\pi$
$578$	0	0
$579$	−0.421651	−0.0175232
$580$	0	0
$581$	8.39245	0.348178
$582$	0	0
$583$	8.07327i	0.334361i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 9.96937i	− 0.411480i	−0.978607	−	0.205740i	$-0.934040\pi$
$587$	− 9.96937i	− 0.411480i	0.978607	−	0.205740i	$-0.0659601\pi$
$588$	0	0
$589$	−40.2719	−1.65938
$590$	0	0
$591$	7.22140	0.297049
$592$	0	0
$593$	20.3619i	0.836163i	0.908410	+	0.418082i	$0.137297\pi$
$593$	20.3619i	0.836163i	−0.908410	+	0.418082i	$0.862703\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 3.93505i	− 0.161051i
$598$	0	0
$599$	24.1075	0.985007	0.492503	−	0.870311i	$-0.336082\pi$
$599$	24.1075	0.985007	0.492503	+	0.870311i	$0.336082\pi$
$600$	0	0
$601$	37.2054	1.51764	0.758820	−	0.651300i	$-0.225776\pi$
$601$	37.2054	1.51764	0.758820	+	0.651300i	$0.225776\pi$
$602$	0	0
$603$	3.87094i	0.157637i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 40.5752i	− 1.64690i	−0.567392	−	0.823448i	$-0.692048\pi$
$607$	− 40.5752i	− 1.64690i	0.567392	−	0.823448i	$-0.307952\pi$
$608$	0	0
$609$	−15.9792	−0.647508
$610$	0	0
$611$	−62.0041	−2.50842
$612$	0	0
$613$	− 5.13559i	− 0.207424i	−0.994607	−	0.103712i	$-0.966928\pi$
$613$	− 5.13559i	− 0.207424i	0.994607	−	0.103712i	$-0.0330721\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 16.3817i	− 0.659504i	−0.944068	−	0.329752i	$-0.893035\pi$
$617$	− 16.3817i	− 0.659504i	0.944068	−	0.329752i	$-0.106965\pi$
$618$	0	0
$619$	−18.4485	−0.741507	−0.370754	−	0.928731i	$-0.620901\pi$
$619$	−18.4485	−0.741507	−0.370754	+	0.928731i	$0.620901\pi$
$620$	0	0
$621$	4.20898	0.168900
$622$	0	0
$623$	− 13.8964i	− 0.556747i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	4.77437i	0.190670i
$628$	0	0
$629$	0.413713	0.0164958
$630$	0	0
$631$	38.5616	1.53511	0.767556	−	0.640982i	$-0.221473\pi$
$631$	38.5616	1.53511	0.767556	+	0.640982i	$0.221473\pi$
$632$	0	0
$633$	− 24.7610i	− 0.984160i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	6.62585i	0.262526i
$638$	0	0
$639$	6.48247	0.256443
$640$	0	0
$641$	18.9989	0.750412	0.375206	−	0.926941i	$-0.377572\pi$
$641$	18.9989	0.750412	0.375206	+	0.926941i	$0.377572\pi$
$642$	0	0
$643$	37.6504i	1.48479i	0.669964	+	0.742393i	$0.266309\pi$
$643$	37.6504i	1.48479i	−0.669964	+	0.742393i	$0.733691\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	5.73458i	0.225449i	0.993626	+	0.112725i	$0.0359578\pi$
$647$	5.73458i	0.225449i	−0.993626	+	0.112725i	$0.964042\pi$
$648$	0	0
$649$	−6.95440	−0.272984
$650$	0	0
$651$	11.8937	0.466150
$652$	0	0
$653$	− 40.0810i	− 1.56849i	−0.620450	−	0.784246i	$-0.713050\pi$
$653$	− 40.0810i	− 1.56849i	0.620450	−	0.784246i	$-0.286950\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	10.0509i	0.392123i
$658$	0	0
$659$	31.5748	1.22998	0.614989	−	0.788536i	$-0.289160\pi$
$659$	31.5748	1.22998	0.614989	+	0.788536i	$0.289160\pi$
$660$	0	0
$661$	31.3504	1.21939	0.609695	−	0.792636i	$-0.291292\pi$
$661$	31.3504	1.21939	0.609695	+	0.792636i	$0.291292\pi$
$662$	0	0
$663$	12.2336i	0.475112i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 27.5210i	− 1.06562i
$668$	0	0
$669$	−19.6727	−0.760589
$670$	0	0
$671$	2.77567	0.107153
$672$	0	0
$673$	− 4.47476i	− 0.172489i	−0.996274	−	0.0862446i	$-0.972513\pi$
$673$	− 4.47476i	− 0.172489i	0.996274	−	0.0862446i	$-0.0274867\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 14.0828i	− 0.541244i	−0.962686	−	0.270622i	$-0.912771\pi$
$677$	− 14.0828i	− 0.541244i	0.962686	−	0.270622i	$-0.0872294\pi$
$678$	0	0
$679$	−16.0918	−0.617548
$680$	0	0
$681$	26.3748	1.01068
$682$	0	0
$683$	11.8357i	0.452880i	0.974025	+	0.226440i	$0.0727088\pi$
$683$	11.8357i	0.452880i	−0.974025	+	0.226440i	$0.927291\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	16.0212i	0.611247i
$688$	0	0
$689$	90.2000	3.43634
$690$	0	0
$691$	−37.7571	−1.43635	−0.718173	−	0.695865i	$-0.755021\pi$
$691$	−37.7571	−1.43635	−0.718173	+	0.695865i	$0.755021\pi$
$692$	0	0
$693$	− 1.41003i	− 0.0535627i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 12.2489i	− 0.463960i
$698$	0	0
$699$	29.3108	1.10864
$700$	0	0
$701$	−39.1678	−1.47935	−0.739674	−	0.672965i	$-0.765020\pi$
$701$	−39.1678	−1.47935	−0.739674	+	0.672965i	$0.765020\pi$
$702$	0	0
$703$	− 1.80393i	− 0.0680364i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	29.4143i	1.10624i
$708$	0	0
$709$	42.4884	1.59569	0.797843	−	0.602865i	$-0.205974\pi$
$709$	42.4884	1.59569	0.797843	+	0.602865i	$0.205974\pi$
$710$	0	0
$711$	−4.74470	−0.177940
$712$	0	0
$713$	20.4846i	0.767153i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 25.8558i	− 0.965604i
$718$	0	0
$719$	−29.6979	−1.10754	−0.553772	−	0.832668i	$-0.686812\pi$
$719$	−29.6979	−1.10754	−0.553772	+	0.832668i	$0.686812\pi$
$720$	0	0
$721$	−14.1404	−0.526617
$722$	0	0
$723$	− 9.34158i	− 0.347417i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	41.9319i	1.55517i	0.628780	+	0.777583i	$0.283555\pi$
$727$	41.9319i	1.55517i	−0.628780	+	0.777583i	$0.716445\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−6.49815	−0.240343
$732$	0	0
$733$	− 29.2104i	− 1.07891i	−0.842014	−	0.539456i	$-0.818630\pi$
$733$	− 29.2104i	− 1.07891i	0.842014	−	0.539456i	$-0.181370\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.23347i	0.0822707i
$738$	0	0
$739$	−25.9202	−0.953491	−0.476746	−	0.879041i	$-0.658184\pi$
$739$	−25.9202	−0.953491	−0.476746	+	0.879041i	$0.658184\pi$
$740$	0	0
$741$	53.3424	1.95958
$742$	0	0
$743$	9.22935i	0.338592i	0.985565	+	0.169296i	$0.0541494\pi$
$743$	9.22935i	0.338592i	−0.985565	+	0.169296i	$0.945851\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 3.43418i	− 0.125650i
$748$	0	0
$749$	−9.94826	−0.363502
$750$	0	0
$751$	−37.2805	−1.36038	−0.680192	−	0.733034i	$-0.738103\pi$
$751$	−37.2805	−1.36038	−0.680192	+	0.733034i	$0.738103\pi$
$752$	0	0
$753$	− 13.5088i	− 0.492287i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 27.6758i	− 1.00589i	−0.864317	−	0.502947i	$-0.832249\pi$
$757$	− 27.6758i	− 1.00589i	0.864317	−	0.502947i	$-0.167751\pi$
$758$	0	0
$759$	2.42851	0.0881493
$760$	0	0
$761$	−13.6324	−0.494174	−0.247087	−	0.968993i	$-0.579473\pi$
$761$	−13.6324	−0.494174	−0.247087	+	0.968993i	$0.579473\pi$
$762$	0	0
$763$	− 3.56247i	− 0.128970i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	77.6992i	2.80556i
$768$	0	0
$769$	30.5109	1.10025	0.550126	−	0.835082i	$-0.314580\pi$
$769$	30.5109	1.10025	0.550126	+	0.835082i	$0.314580\pi$
$770$	0	0
$771$	−13.0662	−0.470566
$772$	0	0
$773$	8.44505i	0.303748i	0.988400	+	0.151874i	$0.0485307\pi$
$773$	8.44505i	0.303748i	−0.988400	+	0.151874i	$0.951469\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0.532761i	0.0191127i
$778$	0	0
$779$	−53.4093	−1.91359
$780$	0	0
$781$	3.74028	0.133838
$782$	0	0
$783$	6.53865i	0.233672i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 40.8557i	− 1.45635i	−0.685393	−	0.728174i	$-0.740369\pi$
$787$	− 40.8557i	− 1.45635i	0.685393	−	0.728174i	$-0.259631\pi$
$788$	0	0
$789$	−4.43065	−0.157735
$790$	0	0
$791$	−14.3154	−0.508998
$792$	0	0
$793$	− 31.0116i	− 1.10125i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	29.5521i	1.04679i	0.852090	+	0.523395i	$0.175335\pi$
$797$	29.5521i	1.04679i	−0.852090	+	0.523395i	$0.824665\pi$
$798$	0	0
$799$	18.2529	0.645742
$800$	0	0
$801$	−5.68638	−0.200918
$802$	0	0
$803$	5.79920i	0.204649i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 28.0509i	− 0.987438i
$808$	0	0
$809$	−24.8982	−0.875373	−0.437687	−	0.899128i	$-0.644202\pi$
$809$	−24.8982	−0.875373	−0.437687	+	0.899128i	$0.644202\pi$
$810$	0	0
$811$	24.3586	0.855347	0.427674	−	0.903933i	$-0.359333\pi$
$811$	24.3586	0.855347	0.427674	+	0.903933i	$0.359333\pi$
$812$	0	0
$813$	5.98512i	0.209907i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	28.3341i	0.991286i
$818$	0	0
$819$	−15.7538	−0.550484
$820$	0	0
$821$	−55.3787	−1.93273	−0.966365	−	0.257175i	$-0.917208\pi$
$821$	−55.3787	−1.93273	−0.966365	+	0.257175i	$0.917208\pi$
$822$	0	0
$823$	17.4649i	0.608789i	0.952546	+	0.304394i	$0.0984540\pi$
$823$	17.4649i	0.608789i	−0.952546	+	0.304394i	$0.901546\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 23.3838i	− 0.813136i	−0.913621	−	0.406568i	$-0.866725\pi$
$827$	− 23.3838i	− 0.813136i	0.913621	−	0.406568i	$-0.133275\pi$
$828$	0	0
$829$	−36.0766	−1.25299	−0.626497	−	0.779424i	$-0.715512\pi$
$829$	−36.0766	−1.25299	−0.626497	+	0.779424i	$0.715512\pi$
$830$	0	0
$831$	−5.07072	−0.175901
$832$	0	0
$833$	− 1.95053i	− 0.0675820i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 4.86687i	− 0.168224i
$838$	0	0
$839$	−19.0416	−0.657390	−0.328695	−	0.944436i	$-0.606609\pi$
$839$	−19.0416	−0.657390	−0.328695	+	0.944436i	$0.606609\pi$
$840$	0	0
$841$	13.7539	0.474273
$842$	0	0
$843$	− 18.1559i	− 0.625323i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	26.0683i	0.895716i
$848$	0	0
$849$	−4.69717	−0.161206
$850$	0	0
$851$	−0.917579	−0.0314542
$852$	0	0
$853$	− 18.0499i	− 0.618016i	−0.951059	−	0.309008i	$-0.900003\pi$
$853$	− 18.0499i	− 0.618016i	0.951059	−	0.309008i	$-0.0999970\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 12.4435i	− 0.425061i	−0.977154	−	0.212531i	$-0.931829\pi$
$857$	− 12.4435i	− 0.425061i	0.977154	−	0.212531i	$-0.0681705\pi$
$858$	0	0
$859$	−46.0026	−1.56959	−0.784794	−	0.619757i	$-0.787231\pi$
$859$	−46.0026	−1.56959	−0.784794	+	0.619757i	$0.787231\pi$
$860$	0	0
$861$	15.7736	0.537562
$862$	0	0
$863$	47.3472i	1.61172i	0.592109	+	0.805858i	$0.298296\pi$
$863$	47.3472i	1.61172i	−0.592109	+	0.805858i	$0.701704\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	13.3987i	0.455042i
$868$	0	0
$869$	−2.73761	−0.0928672
$870$	0	0
$871$	24.9538	0.845526
$872$	0	0
$873$	6.58475i	0.222860i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	31.9576i	1.07913i	0.841944	+	0.539565i	$0.181411\pi$
$877$	31.9576i	1.07913i	−0.841944	+	0.539565i	$0.818589\pi$
$878$	0	0
$879$	9.60771	0.324060
$880$	0	0
$881$	−7.92960	−0.267155	−0.133578	−	0.991038i	$-0.542647\pi$
$881$	−7.92960	−0.267155	−0.133578	+	0.991038i	$0.542647\pi$
$882$	0	0
$883$	− 37.2505i	− 1.25358i	−0.779188	−	0.626790i	$-0.784368\pi$
$883$	− 37.2505i	− 1.25358i	0.779188	−	0.626790i	$-0.215632\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 16.4148i	− 0.551155i	−0.961279	−	0.275577i	$-0.911131\pi$
$887$	− 16.4148i	− 0.551155i	0.961279	−	0.275577i	$-0.0888690\pi$
$888$	0	0
$889$	−37.3546	−1.25283
$890$	0	0
$891$	−0.576983	−0.0193297
$892$	0	0
$893$	− 79.5889i	− 2.66334i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 27.1329i	− 0.905943i
$898$	0	0
$899$	−31.8228	−1.06135
$900$	0	0
$901$	−26.5533	−0.884619
$902$	0	0
$903$	− 8.36804i	− 0.278471i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.85782i	0.0616880i	0.999524	+	0.0308440i	$0.00981950\pi$
$907$	1.85782i	0.0616880i	−0.999524	+	0.0308440i	$0.990181\pi$
$908$	0	0
$909$	12.0363	0.399219
$910$	0	0
$911$	−22.5064	−0.745672	−0.372836	−	0.927897i	$-0.621615\pi$
$911$	−22.5064	−0.745672	−0.372836	+	0.927897i	$0.621615\pi$
$912$	0	0
$913$	− 1.98146i	− 0.0655769i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	9.17384i	0.302947i
$918$	0	0
$919$	−20.9049	−0.689590	−0.344795	−	0.938678i	$-0.612052\pi$
$919$	−20.9049	−0.689590	−0.344795	+	0.938678i	$0.612052\pi$
$920$	0	0
$921$	−13.5400	−0.446160
$922$	0	0
$923$	− 41.7889i	− 1.37550i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	5.78624i	0.190045i
$928$	0	0
$929$	−13.9748	−0.458498	−0.229249	−	0.973368i	$-0.573627\pi$
$929$	−13.9748	−0.458498	−0.229249	+	0.973368i	$0.573627\pi$
$930$	0	0
$931$	−8.50499	−0.278740
$932$	0	0
$933$	− 2.96725i	− 0.0971434i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 32.7584i	− 1.07017i	−0.844798	−	0.535086i	$-0.820279\pi$
$937$	− 32.7584i	− 1.07017i	0.844798	−	0.535086i	$-0.179721\pi$
$938$	0	0
$939$	−14.0058	−0.457060
$940$	0	0
$941$	−25.8709	−0.843368	−0.421684	−	0.906743i	$-0.638561\pi$
$941$	−25.8709	−0.843368	−0.421684	+	0.906743i	$0.638561\pi$
$942$	0	0
$943$	27.1669i	0.884677i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	55.7717i	1.81234i	0.422918	+	0.906168i	$0.361006\pi$
$947$	55.7717i	1.81234i	−0.422918	+	0.906168i	$0.638994\pi$
$948$	0	0
$949$	64.7925	2.10325
$950$	0	0
$951$	27.0176	0.876106
$952$	0	0
$953$	24.9297i	0.807551i	0.914858	+	0.403775i	$0.132302\pi$
$953$	24.9297i	0.807551i	−0.914858	+	0.403775i	$0.867698\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	3.77269i	0.121954i
$958$	0	0
$959$	−49.4897	−1.59810
$960$	0	0
$961$	−7.31353	−0.235920
$962$	0	0
$963$	4.07081i	0.131180i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 8.68802i	− 0.279388i	−0.990195	−	0.139694i	$-0.955388\pi$
$967$	− 8.68802i	− 0.279388i	0.990195	−	0.139694i	$-0.0446119\pi$
$968$	0	0
$969$	−15.7031	−0.504456
$970$	0	0
$971$	24.6797	0.792011	0.396005	−	0.918248i	$-0.370396\pi$
$971$	24.6797	0.792011	0.396005	+	0.918248i	$0.370396\pi$
$972$	0	0
$973$	− 2.26997i	− 0.0727719i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 35.5977i	− 1.13887i	−0.822036	−	0.569435i	$-0.807162\pi$
$977$	− 35.5977i	− 1.13887i	0.822036	−	0.569435i	$-0.192838\pi$
$978$	0	0
$979$	−3.28095	−0.104860
$980$	0	0
$981$	−1.45776	−0.0465426
$982$	0	0
$983$	5.94123i	0.189496i	0.995501	+	0.0947479i	$0.0302045\pi$
$983$	5.94123i	0.189496i	−0.995501	+	0.0947479i	$0.969795\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	23.5053i	0.748182i
$988$	0	0
$989$	14.4123	0.458285
$990$	0	0
$991$	4.22674	0.134267	0.0671335	−	0.997744i	$-0.478615\pi$
$991$	4.22674	0.134267	0.0671335	+	0.997744i	$0.478615\pi$
$992$	0	0
$993$	− 6.65492i	− 0.211187i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	12.5455i	0.397319i	0.980069	+	0.198660i	$0.0636588\pi$
$997$	12.5455i	0.397319i	−0.980069	+	0.198660i	$0.936341\pi$
$998$	0	0
$999$	0.218005	0.00689738

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7500.2.d.g.1249.17		24
5.2	odd	4		7500.2.a.n.1.8		12
5.3	odd	4		7500.2.a.m.1.5		12
5.4	even	2	inner	7500.2.d.g.1249.8		24
25.2	odd	20		1500.2.m.c.601.4		24
25.9	even	10		300.2.o.a.169.6	✓	24
25.11	even	5		300.2.o.a.229.6	yes	24
25.12	odd	20		1500.2.m.c.901.4		24
25.13	odd	20		1500.2.m.d.901.3		24
25.14	even	10		1500.2.o.c.649.1		24
25.16	even	5		1500.2.o.c.349.1		24
25.23	odd	20		1500.2.m.d.601.3		24
75.11	odd	10		900.2.w.c.829.2		24
75.59	odd	10		900.2.w.c.469.2		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.o.a.169.6	✓	24	25.9	even	10
300.2.o.a.229.6	yes	24	25.11	even	5
900.2.w.c.469.2		24	75.59	odd	10
900.2.w.c.829.2		24	75.11	odd	10
1500.2.m.c.601.4		24	25.2	odd	20
1500.2.m.c.901.4		24	25.12	odd	20
1500.2.m.d.601.3		24	25.23	odd	20
1500.2.m.d.901.3		24	25.13	odd	20
1500.2.o.c.349.1		24	25.16	even	5
1500.2.o.c.649.1		24	25.14	even	10
7500.2.a.m.1.5		12	5.3	odd	4
7500.2.a.n.1.8		12	5.2	odd	4
7500.2.d.g.1249.8		24	5.4	even	2	inner
7500.2.d.g.1249.17		24	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:	\( N \)	\(=\)	\( 7500 = 2^{2} \cdot 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7500.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -2.44380i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -2.44380i q^{7} -1.00000 q^{9} -0.576983 q^{11} +6.44645i q^{13} -1.89772i q^{17} -8.27470 q^{19} +2.44380 q^{21} +4.20898i q^{23} -1.00000i q^{27} -6.53865 q^{29} +4.86687 q^{31} -0.576983i q^{33} +0.218005i q^{37} -6.44645 q^{39} +6.45452 q^{41} -3.42419i q^{43} +9.61833i q^{47} +1.02783 q^{49} +1.89772 q^{51} -13.9922i q^{53} -8.27470i q^{57} +12.0530 q^{59} -4.81065 q^{61} +2.44380i q^{63} -3.87094i q^{67} -4.20898 q^{69} -6.48247 q^{71} -10.0509i q^{73} +1.41003i q^{77} +4.74470 q^{79} +1.00000 q^{81} +3.43418i q^{83} -6.53865i q^{87} +5.68638 q^{89} +15.7538 q^{91} +4.86687i q^{93} -6.58475i q^{97} +0.576983 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{9}+O(q^{10}) \) 24 * q - 24 * q^9	\( 24 q - 24 q^{9} + 4 q^{11} - 20 q^{19} + 16 q^{21} - 16 q^{29} - 4 q^{31} + 20 q^{41} - 56 q^{49} + 16 q^{51} + 4 q^{59} + 68 q^{61} - 36 q^{69} - 12 q^{79} + 24 q^{81} - 20 q^{89} + 40 q^{91} - 4 q^{99}+O(q^{100}) \) 24 * q - 24 * q^9 + 4 * q^11 - 20 * q^19 + 16 * q^21 - 16 * q^29 - 4 * q^31 + 20 * q^41 - 56 * q^49 + 16 * q^51 + 4 * q^59 + 68 * q^61 - 36 * q^69 - 12 * q^79 + 24 * q^81 - 20 * q^89 + 40 * q^91 - 4 * q^99

Introduction

L-functions

Modular forms

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