LMFDB - Embedded newform 7500.2.d.d.1249.5

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:	$8$
Coefficient field:	8.0.324000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 9x^{6} + 26x^{4} + 24x^{2} + 1 $ x^8 + 9x^6 + 26x^4 + 24*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 300)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.5
Root			$-1.33826i$ of defining polynomial
Character	$\chi$	$=$	7500.1249
Dual form			7500.2.d.d.1249.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -0.747238i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -0.747238i q^{7} -1.00000 q^{9} +0.209057 q^{11} -2.50361i q^{13} +6.81953i q^{17} -1.24520 q^{19} +0.747238 q^{21} +3.25085i q^{23} -1.00000i q^{27} +5.18323 q^{29} +3.73968 q^{31} +0.209057i q^{33} -1.96543i q^{37} +2.50361 q^{39} +3.21373 q^{41} -12.7127i q^{43} +6.48225i q^{47} +6.44163 q^{49} -6.81953 q^{51} +4.14301i q^{53} -1.24520i q^{57} -11.7108 q^{59} -12.4794 q^{61} +0.747238i q^{63} -2.91706i q^{67} -3.25085 q^{69} +6.55008 q^{71} -2.51933i q^{73} -0.156215i q^{77} +10.2423 q^{79} +1.00000 q^{81} -5.67652i q^{83} +5.18323i q^{87} +0.921727 q^{89} -1.87080 q^{91} +3.73968i q^{93} +15.5444i q^{97} -0.209057 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^9	$ 8 q - 8 q^{9} - 2 q^{11} + 10 q^{19} - 8 q^{21} + 8 q^{29} - 18 q^{31} - 10 q^{39} + 8 q^{49} - 8 q^{51} - 2 q^{59} - 4 q^{61} + 18 q^{69} - 40 q^{71} + 6 q^{79} + 8 q^{81} - 30 q^{89} - 20 q^{91} + 2 q^{99}+O(q^{100}) $ 8 * q - 8 * q^9 - 2 * q^11 + 10 * q^19 - 8 * q^21 + 8 * q^29 - 18 * q^31 - 10 * q^39 + 8 * q^49 - 8 * q^51 - 2 * q^59 - 4 * q^61 + 18 * q^69 - 40 * q^71 + 6 * q^79 + 8 * q^81 - 30 * q^89 - 20 * q^91 + 2 * 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
$7$	− 0.747238i	− 0.282430i	−0.989979	−	0.141215i	$-0.954899\pi$
$7$	− 0.747238i	− 0.282430i	0.989979	−	0.141215i	$-0.0451008\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0.209057	0.0630330	0.0315165	−	0.999503i	$-0.489966\pi$
$11$	0.209057	0.0630330	0.0315165	+	0.999503i	$0.489966\pi$
$12$	0	0
$13$	− 2.50361i	− 0.694377i	−0.937795	−	0.347189i	$-0.887136\pi$
$13$	− 2.50361i	− 0.694377i	0.937795	−	0.347189i	$-0.112864\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.81953i	1.65398i	0.562217	+	0.826990i	$0.309949\pi$
$17$	6.81953i	1.65398i	−0.562217	+	0.826990i	$0.690051\pi$
$18$	0	0
$19$	−1.24520	−0.285670	−0.142835	−	0.989747i	$-0.545622\pi$
$19$	−1.24520	−0.285670	−0.142835	+	0.989747i	$0.545622\pi$
$20$	0	0
$21$	0.747238	0.163061
$22$	0	0
$23$	3.25085i	0.677849i	0.940813	+	0.338925i	$0.110063\pi$
$23$	3.25085i	0.677849i	−0.940813	+	0.338925i	$0.889937\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	5.18323	0.962501	0.481250	−	0.876583i	$-0.340183\pi$
$29$	5.18323	0.962501	0.481250	+	0.876583i	$0.340183\pi$
$30$	0	0
$31$	3.73968	0.671667	0.335833	−	0.941921i	$-0.390982\pi$
$31$	3.73968	0.671667	0.335833	+	0.941921i	$0.390982\pi$
$32$	0	0
$33$	0.209057i	0.0363921i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 1.96543i	− 0.323115i	−0.986863	−	0.161558i	$-0.948348\pi$
$37$	− 1.96543i	− 0.323115i	0.986863	−	0.161558i	$-0.0516517\pi$
$38$	0	0
$39$	2.50361	0.400899
$40$	0	0
$41$	3.21373	0.501900	0.250950	−	0.968000i	$-0.419257\pi$
$41$	3.21373	0.501900	0.250950	+	0.968000i	$0.419257\pi$
$42$	0	0
$43$	− 12.7127i	− 1.93866i	−0.245755	−	0.969332i	$-0.579036\pi$
$43$	− 12.7127i	− 1.93866i	0.245755	−	0.969332i	$-0.420964\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.48225i	0.945533i	0.881188	+	0.472767i	$0.156745\pi$
$47$	6.48225i	0.945533i	−0.881188	+	0.472767i	$0.843255\pi$
$48$	0	0
$49$	6.44163	0.920234
$50$	0	0
$51$	−6.81953	−0.954926
$52$	0	0
$53$	4.14301i	0.569086i	0.958663	+	0.284543i	$0.0918419\pi$
$53$	4.14301i	0.569086i	−0.958663	+	0.284543i	$0.908158\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 1.24520i	− 0.164931i
$58$	0	0
$59$	−11.7108	−1.52461	−0.762306	−	0.647217i	$-0.775933\pi$
$59$	−11.7108	−1.52461	−0.762306	+	0.647217i	$0.775933\pi$
$60$	0	0
$61$	−12.4794	−1.59782	−0.798909	−	0.601451i	$-0.794589\pi$
$61$	−12.4794	−1.59782	−0.798909	+	0.601451i	$0.794589\pi$
$62$	0	0
$63$	0.747238i	0.0941432i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 2.91706i	− 0.356375i	−0.983997	−	0.178188i	$-0.942977\pi$
$67$	− 2.91706i	− 0.356375i	0.983997	−	0.178188i	$-0.0570234\pi$
$68$	0	0
$69$	−3.25085	−0.391357
$70$	0	0
$71$	6.55008	0.777351	0.388676	−	0.921375i	$-0.372933\pi$
$71$	6.55008	0.777351	0.388676	+	0.921375i	$0.372933\pi$
$72$	0	0
$73$	− 2.51933i	− 0.294865i	−0.989072	−	0.147433i	$-0.952899\pi$
$73$	− 2.51933i	− 0.294865i	0.989072	−	0.147433i	$-0.0471010\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 0.156215i	− 0.0178024i
$78$	0	0
$79$	10.2423	1.15235	0.576175	−	0.817326i	$-0.304545\pi$
$79$	10.2423	1.15235	0.576175	+	0.817326i	$0.304545\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 5.67652i	− 0.623079i	−0.950233	−	0.311540i	$-0.899155\pi$
$83$	− 5.67652i	− 0.623079i	0.950233	−	0.311540i	$-0.100845\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	5.18323i	0.555700i
$88$	0	0
$89$	0.921727	0.0977029	0.0488514	−	0.998806i	$-0.484444\pi$
$89$	0.921727	0.0977029	0.0488514	+	0.998806i	$0.484444\pi$
$90$	0	0
$91$	−1.87080	−0.196113
$92$	0	0
$93$	3.73968i	0.387787i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	15.5444i	1.57830i	0.614202	+	0.789149i	$0.289478\pi$
$97$	15.5444i	1.57830i	−0.614202	+	0.789149i	$0.710522\pi$
$98$	0	0
$99$	−0.209057	−0.0210110
$100$	0	0
$101$	−11.0405	−1.09857	−0.549285	−	0.835635i	$-0.685100\pi$
$101$	−11.0405	−1.09857	−0.549285	+	0.835635i	$0.685100\pi$
$102$	0	0
$103$	13.8843i	1.36806i	0.729455	+	0.684029i	$0.239774\pi$
$103$	13.8843i	1.36806i	−0.729455	+	0.684029i	$0.760226\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1.02510i	− 0.0991002i	−0.998772	−	0.0495501i	$-0.984221\pi$
$107$	− 1.02510i	− 0.0991002i	0.998772	−	0.0495501i	$-0.0157788\pi$
$108$	0	0
$109$	6.70511	0.642233	0.321117	−	0.947040i	$-0.395942\pi$
$109$	6.70511	0.642233	0.321117	+	0.947040i	$0.395942\pi$
$110$	0	0
$111$	1.96543	0.186551
$112$	0	0
$113$	8.32157i	0.782827i	0.920215	+	0.391414i	$0.128014\pi$
$113$	8.32157i	0.782827i	−0.920215	+	0.391414i	$0.871986\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.50361i	0.231459i
$118$	0	0
$119$	5.09582	0.467133
$120$	0	0
$121$	−10.9563	−0.996027
$122$	0	0
$123$	3.21373i	0.289772i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	14.5363i	1.28989i	0.764231	+	0.644943i	$0.223119\pi$
$127$	14.5363i	1.28989i	−0.764231	+	0.644943i	$0.776881\pi$
$128$	0	0
$129$	12.7127	1.11929
$130$	0	0
$131$	7.42396	0.648635	0.324317	−	0.945948i	$-0.394865\pi$
$131$	7.42396	0.648635	0.324317	+	0.945948i	$0.394865\pi$
$132$	0	0
$133$	0.930465i	0.0806815i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	7.33537i	0.626703i	0.949637	+	0.313352i	$0.101452\pi$
$137$	7.33537i	0.626703i	−0.949637	+	0.313352i	$0.898548\pi$
$138$	0	0
$139$	−19.7471	−1.67493	−0.837464	−	0.546492i	$-0.815963\pi$
$139$	−19.7471	−1.67493	−0.837464	+	0.546492i	$0.815963\pi$
$140$	0	0
$141$	−6.48225	−0.545904
$142$	0	0
$143$	− 0.523398i	− 0.0437687i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	6.44163i	0.531297i
$148$	0	0
$149$	21.7551	1.78224	0.891122	−	0.453763i	$-0.149919\pi$
$149$	21.7551	1.78224	0.891122	+	0.453763i	$0.149919\pi$
$150$	0	0
$151$	10.1308	0.824432	0.412216	−	0.911086i	$-0.364755\pi$
$151$	10.1308	0.824432	0.412216	+	0.911086i	$0.364755\pi$
$152$	0	0
$153$	− 6.81953i	− 0.551327i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	9.99520i	0.797704i	0.917015	+	0.398852i	$0.130591\pi$
$157$	9.99520i	0.797704i	−0.917015	+	0.398852i	$0.869409\pi$
$158$	0	0
$159$	−4.14301	−0.328562
$160$	0	0
$161$	2.42916	0.191445
$162$	0	0
$163$	− 2.56308i	− 0.200756i	−0.994949	−	0.100378i	$-0.967995\pi$
$163$	− 2.56308i	− 0.200756i	0.994949	−	0.100378i	$-0.0320052\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	20.8512i	1.61352i	0.590882	+	0.806758i	$0.298780\pi$
$167$	20.8512i	1.61352i	−0.590882	+	0.806758i	$0.701220\pi$
$168$	0	0
$169$	6.73192	0.517840
$170$	0	0
$171$	1.24520	0.0952232
$172$	0	0
$173$	2.05496i	0.156235i	0.996944	+	0.0781177i	$0.0248910\pi$
$173$	2.05496i	0.156235i	−0.996944	+	0.0781177i	$0.975109\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 11.7108i	− 0.880235i
$178$	0	0
$179$	19.5604	1.46201	0.731006	−	0.682371i	$-0.239051\pi$
$179$	19.5604	1.46201	0.731006	+	0.682371i	$0.239051\pi$
$180$	0	0
$181$	−7.37007	−0.547813	−0.273906	−	0.961756i	$-0.588316\pi$
$181$	−7.37007	−0.547813	−0.273906	+	0.961756i	$0.588316\pi$
$182$	0	0
$183$	− 12.4794i	− 0.922501i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1.42567i	0.104255i
$188$	0	0
$189$	−0.747238	−0.0543536
$190$	0	0
$191$	−2.24520	−0.162457	−0.0812287	−	0.996695i	$-0.525884\pi$
$191$	−2.24520	−0.162457	−0.0812287	+	0.996695i	$0.525884\pi$
$192$	0	0
$193$	1.10589i	0.0796035i	0.999208	+	0.0398018i	$0.0126726\pi$
$193$	1.10589i	0.0796035i	−0.999208	+	0.0398018i	$0.987327\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 20.6366i	− 1.47029i	−0.677908	−	0.735147i	$-0.737113\pi$
$197$	− 20.6366i	− 1.47029i	0.677908	−	0.735147i	$-0.262887\pi$
$198$	0	0
$199$	12.3822	0.877749	0.438874	−	0.898548i	$-0.355377\pi$
$199$	12.3822	0.877749	0.438874	+	0.898548i	$0.355377\pi$
$200$	0	0
$201$	2.91706	0.205753
$202$	0	0
$203$	− 3.87311i	− 0.271839i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 3.25085i	− 0.225950i
$208$	0	0
$209$	−0.260319	−0.0180066
$210$	0	0
$211$	20.6274	1.42005	0.710025	−	0.704176i	$-0.248684\pi$
$211$	20.6274	1.42005	0.710025	+	0.704176i	$0.248684\pi$
$212$	0	0
$213$	6.55008i	0.448804i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 2.79443i	− 0.189698i
$218$	0	0
$219$	2.51933	0.170241
$220$	0	0
$221$	17.0735	1.14849
$222$	0	0
$223$	− 4.15464i	− 0.278215i	−0.990277	−	0.139107i	$-0.955577\pi$
$223$	− 4.15464i	− 0.278215i	0.990277	−	0.139107i	$-0.0444234\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	23.6290i	1.56831i	0.620566	+	0.784154i	$0.286903\pi$
$227$	23.6290i	1.56831i	−0.620566	+	0.784154i	$0.713097\pi$
$228$	0	0
$229$	−18.2401	−1.20534	−0.602669	−	0.797991i	$-0.705896\pi$
$229$	−18.2401	−1.20534	−0.602669	+	0.797991i	$0.705896\pi$
$230$	0	0
$231$	0.156215	0.0102782
$232$	0	0
$233$	2.26742i	0.148544i	0.997238	+	0.0742718i	$0.0236632\pi$
$233$	2.26742i	0.148544i	−0.997238	+	0.0742718i	$0.976337\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	10.2423i	0.665310i
$238$	0	0
$239$	−17.1069	−1.10655	−0.553276	−	0.832998i	$-0.686622\pi$
$239$	−17.1069	−1.10655	−0.553276	+	0.832998i	$0.686622\pi$
$240$	0	0
$241$	28.2002	1.81654	0.908268	−	0.418388i	$-0.137405\pi$
$241$	28.2002	1.81654	0.908268	+	0.418388i	$0.137405\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.11751i	0.198362i
$248$	0	0
$249$	5.67652	0.359735
$250$	0	0
$251$	23.9575	1.51219	0.756093	−	0.654464i	$-0.227106\pi$
$251$	23.9575	1.51219	0.756093	+	0.654464i	$0.227106\pi$
$252$	0	0
$253$	0.679613i	0.0427269i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	28.5421i	1.78041i	0.455561	+	0.890204i	$0.349439\pi$
$257$	28.5421i	1.78041i	−0.455561	+	0.890204i	$0.650561\pi$
$258$	0	0
$259$	−1.46865	−0.0912572
$260$	0	0
$261$	−5.18323	−0.320834
$262$	0	0
$263$	− 3.55679i	− 0.219321i	−0.993969	−	0.109660i	$-0.965024\pi$
$263$	− 3.55679i	− 0.219321i	0.993969	−	0.109660i	$-0.0349763\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0.921727i	0.0564088i
$268$	0	0
$269$	−29.2618	−1.78412	−0.892062	−	0.451913i	$-0.850742\pi$
$269$	−29.2618	−1.78412	−0.892062	+	0.451913i	$0.850742\pi$
$270$	0	0
$271$	−24.8183	−1.50760	−0.753801	−	0.657102i	$-0.771782\pi$
$271$	−24.8183	−1.50760	−0.753801	+	0.657102i	$0.771782\pi$
$272$	0	0
$273$	− 1.87080i	− 0.113226i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	17.4275i	1.04712i	0.851990	+	0.523558i	$0.175396\pi$
$277$	17.4275i	1.04712i	−0.851990	+	0.523558i	$0.824604\pi$
$278$	0	0
$279$	−3.73968	−0.223889
$280$	0	0
$281$	3.75349	0.223914	0.111957	−	0.993713i	$-0.464288\pi$
$281$	3.75349	0.223914	0.111957	+	0.993713i	$0.464288\pi$
$282$	0	0
$283$	16.3735i	0.973302i	0.873596	+	0.486651i	$0.161782\pi$
$283$	16.3735i	0.973302i	−0.873596	+	0.486651i	$0.838218\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 2.40142i	− 0.141751i
$288$	0	0
$289$	−29.5060	−1.73565
$290$	0	0
$291$	−15.5444	−0.911231
$292$	0	0
$293$	8.70991i	0.508838i	0.967094	+	0.254419i	$0.0818843\pi$
$293$	8.70991i	0.508838i	−0.967094	+	0.254419i	$0.918116\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 0.209057i	− 0.0121307i
$298$	0	0
$299$	8.13888	0.470683
$300$	0	0
$301$	−9.49939	−0.547536
$302$	0	0
$303$	− 11.0405i	− 0.634259i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	17.7123i	1.01090i	0.862857	+	0.505448i	$0.168673\pi$
$307$	17.7123i	1.01090i	−0.862857	+	0.505448i	$0.831327\pi$
$308$	0	0
$309$	−13.8843	−0.789849
$310$	0	0
$311$	20.2876	1.15040	0.575201	−	0.818012i	$-0.304924\pi$
$311$	20.2876	1.15040	0.575201	+	0.818012i	$0.304924\pi$
$312$	0	0
$313$	20.7701i	1.17400i	0.809589	+	0.586998i	$0.199690\pi$
$313$	20.7701i	1.17400i	−0.809589	+	0.586998i	$0.800310\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	16.7651i	0.941623i	0.882234	+	0.470811i	$0.156039\pi$
$317$	16.7651i	0.941623i	−0.882234	+	0.470811i	$0.843961\pi$
$318$	0	0
$319$	1.08359	0.0606694
$320$	0	0
$321$	1.02510	0.0572156
$322$	0	0
$323$	− 8.49172i	− 0.472492i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	6.70511i	0.370794i
$328$	0	0
$329$	4.84378	0.267046
$330$	0	0
$331$	−26.2157	−1.44095	−0.720474	−	0.693482i	$-0.756076\pi$
$331$	−26.2157	−1.44095	−0.720474	+	0.693482i	$0.756076\pi$
$332$	0	0
$333$	1.96543i	0.107705i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	26.0024i	1.41644i	0.705990	+	0.708221i	$0.250502\pi$
$337$	26.0024i	1.41644i	−0.705990	+	0.708221i	$0.749498\pi$
$338$	0	0
$339$	−8.32157	−0.451966
$340$	0	0
$341$	0.781806	0.0423372
$342$	0	0
$343$	− 10.0441i	− 0.542331i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 4.60134i	− 0.247013i	−0.992344	−	0.123506i	$-0.960586\pi$
$347$	− 4.60134i	− 0.247013i	0.992344	−	0.123506i	$-0.0394140\pi$
$348$	0	0
$349$	29.2108	1.56362	0.781810	−	0.623516i	$-0.214297\pi$
$349$	29.2108	1.56362	0.781810	+	0.623516i	$0.214297\pi$
$350$	0	0
$351$	−2.50361	−0.133633
$352$	0	0
$353$	− 28.1211i	− 1.49674i	−0.663284	−	0.748368i	$-0.730838\pi$
$353$	− 28.1211i	− 1.49674i	0.663284	−	0.748368i	$-0.269162\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	5.09582i	0.269699i
$358$	0	0
$359$	9.40304	0.496274	0.248137	−	0.968725i	$-0.420182\pi$
$359$	9.40304	0.496274	0.248137	+	0.968725i	$0.420182\pi$
$360$	0	0
$361$	−17.4495	−0.918393
$362$	0	0
$363$	− 10.9563i	− 0.575056i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 22.0061i	− 1.14871i	−0.818606	−	0.574355i	$-0.805253\pi$
$367$	− 22.0061i	− 1.14871i	0.818606	−	0.574355i	$-0.194747\pi$
$368$	0	0
$369$	−3.21373	−0.167300
$370$	0	0
$371$	3.09582	0.160727
$372$	0	0
$373$	18.6198i	0.964098i	0.876144	+	0.482049i	$0.160107\pi$
$373$	18.6198i	0.964098i	−0.876144	+	0.482049i	$0.839893\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 12.9768i	− 0.668339i
$378$	0	0
$379$	16.8742	0.866767	0.433384	−	0.901210i	$-0.357320\pi$
$379$	16.8742	0.866767	0.433384	+	0.901210i	$0.357320\pi$
$380$	0	0
$381$	−14.5363	−0.744716
$382$	0	0
$383$	− 9.05849i	− 0.462867i	−0.972851	−	0.231434i	$-0.925658\pi$
$383$	− 9.05849i	− 0.462867i	0.972851	−	0.231434i	$-0.0743416\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	12.7127i	0.646221i
$388$	0	0
$389$	−33.0807	−1.67726	−0.838629	−	0.544703i	$-0.816642\pi$
$389$	−33.0807	−1.67726	−0.838629	+	0.544703i	$0.816642\pi$
$390$	0	0
$391$	−22.1693	−1.12115
$392$	0	0
$393$	7.42396i	0.374489i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	21.7267i	1.09043i	0.838296	+	0.545215i	$0.183552\pi$
$397$	21.7267i	1.09043i	−0.838296	+	0.545215i	$0.816448\pi$
$398$	0	0
$399$	−0.930465	−0.0465815
$400$	0	0
$401$	20.1663	1.00706	0.503529	−	0.863978i	$-0.332035\pi$
$401$	20.1663	1.00706	0.503529	+	0.863978i	$0.332035\pi$
$402$	0	0
$403$	− 9.36272i	− 0.466390i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 0.410887i	− 0.0203669i
$408$	0	0
$409$	14.0490	0.694679	0.347340	−	0.937739i	$-0.387085\pi$
$409$	14.0490	0.694679	0.347340	+	0.937739i	$0.387085\pi$
$410$	0	0
$411$	−7.33537	−0.361827
$412$	0	0
$413$	8.75073i	0.430595i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 19.7471i	− 0.967020i
$418$	0	0
$419$	20.2614	0.989836	0.494918	−	0.868940i	$-0.335198\pi$
$419$	20.2614	0.989836	0.494918	+	0.868940i	$0.335198\pi$
$420$	0	0
$421$	4.37888	0.213413	0.106707	−	0.994291i	$-0.465969\pi$
$421$	4.37888	0.213413	0.106707	+	0.994291i	$0.465969\pi$
$422$	0	0
$423$	− 6.48225i	− 0.315178i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	9.32506i	0.451271i
$428$	0	0
$429$	0.523398	0.0252699
$430$	0	0
$431$	1.16804	0.0562627	0.0281313	−	0.999604i	$-0.491044\pi$
$431$	1.16804	0.0562627	0.0281313	+	0.999604i	$0.491044\pi$
$432$	0	0
$433$	2.24890i	0.108075i	0.998539	+	0.0540376i	$0.0172091\pi$
$433$	2.24890i	0.108075i	−0.998539	+	0.0540376i	$0.982791\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 4.04798i	− 0.193641i
$438$	0	0
$439$	−22.1178	−1.05562	−0.527812	−	0.849361i	$-0.676987\pi$
$439$	−22.1178	−1.05562	−0.527812	+	0.849361i	$0.676987\pi$
$440$	0	0
$441$	−6.44163	−0.306745
$442$	0	0
$443$	28.1534i	1.33761i	0.743439	+	0.668804i	$0.233193\pi$
$443$	28.1534i	1.33761i	−0.743439	+	0.668804i	$0.766807\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	21.7551i	1.02898i
$448$	0	0
$449$	−16.5924	−0.783044	−0.391522	−	0.920169i	$-0.628051\pi$
$449$	−16.5924	−0.783044	−0.391522	+	0.920169i	$0.628051\pi$
$450$	0	0
$451$	0.671852	0.0316363
$452$	0	0
$453$	10.1308i	0.475986i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	11.1599i	0.522039i	0.965334	+	0.261019i	$0.0840586\pi$
$457$	11.1599i	0.522039i	−0.965334	+	0.261019i	$0.915941\pi$
$458$	0	0
$459$	6.81953	0.318309
$460$	0	0
$461$	6.26603	0.291838	0.145919	−	0.989297i	$-0.453386\pi$
$461$	6.26603	0.291838	0.145919	+	0.989297i	$0.453386\pi$
$462$	0	0
$463$	− 17.0296i	− 0.791431i	−0.918373	−	0.395716i	$-0.870497\pi$
$463$	− 17.0296i	− 0.791431i	0.918373	−	0.395716i	$-0.129503\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	18.6843i	0.864606i	0.901729	+	0.432303i	$0.142299\pi$
$467$	18.6843i	0.864606i	−0.901729	+	0.432303i	$0.857701\pi$
$468$	0	0
$469$	−2.17974	−0.100651
$470$	0	0
$471$	−9.99520	−0.460555
$472$	0	0
$473$	− 2.65767i	− 0.122200i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 4.14301i	− 0.189695i
$478$	0	0
$479$	−2.94322	−0.134479	−0.0672395	−	0.997737i	$-0.521419\pi$
$479$	−2.94322	−0.134479	−0.0672395	+	0.997737i	$0.521419\pi$
$480$	0	0
$481$	−4.92068	−0.224364
$482$	0	0
$483$	2.42916i	0.110531i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	14.0157i	0.635113i	0.948239	+	0.317556i	$0.102862\pi$
$487$	14.0157i	0.635113i	−0.948239	+	0.317556i	$0.897138\pi$
$488$	0	0
$489$	2.56308	0.115906
$490$	0	0
$491$	8.94500	0.403682	0.201841	−	0.979418i	$-0.435308\pi$
$491$	8.94500	0.403682	0.201841	+	0.979418i	$0.435308\pi$
$492$	0	0
$493$	35.3472i	1.59196i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 4.89447i	− 0.219547i
$498$	0	0
$499$	2.39366	0.107155	0.0535774	−	0.998564i	$-0.482938\pi$
$499$	2.39366	0.107155	0.0535774	+	0.998564i	$0.482938\pi$
$500$	0	0
$501$	−20.8512	−0.931564
$502$	0	0
$503$	8.42403i	0.375609i	0.982206	+	0.187805i	$0.0601372\pi$
$503$	8.42403i	0.375609i	−0.982206	+	0.187805i	$0.939863\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	6.73192i	0.298975i
$508$	0	0
$509$	−32.3130	−1.43225	−0.716125	−	0.697972i	$-0.754086\pi$
$509$	−32.3130	−1.43225	−0.716125	+	0.697972i	$0.754086\pi$
$510$	0	0
$511$	−1.88254	−0.0832787
$512$	0	0
$513$	1.24520i	0.0549771i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.35516i	0.0595998i
$518$	0	0
$519$	−2.05496	−0.0902025
$520$	0	0
$521$	3.56515	0.156192	0.0780960	−	0.996946i	$-0.475116\pi$
$521$	3.56515	0.156192	0.0780960	+	0.996946i	$0.475116\pi$
$522$	0	0
$523$	− 32.4388i	− 1.41845i	−0.704982	−	0.709225i	$-0.749045\pi$
$523$	− 32.4388i	− 1.41845i	0.704982	−	0.709225i	$-0.250955\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	25.5029i	1.11092i
$528$	0	0
$529$	12.4320	0.540520
$530$	0	0
$531$	11.7108	0.508204
$532$	0	0
$533$	− 8.04593i	− 0.348508i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	19.5604i	0.844093i
$538$	0	0
$539$	1.34667	0.0580051
$540$	0	0
$541$	−11.0149	−0.473566	−0.236783	−	0.971563i	$-0.576093\pi$
$541$	−11.0149	−0.473566	−0.236783	+	0.971563i	$0.576093\pi$
$542$	0	0
$543$	− 7.37007i	− 0.316280i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	5.33728i	0.228206i	0.993469	+	0.114103i	$0.0363994\pi$
$547$	5.33728i	0.228206i	−0.993469	+	0.114103i	$0.963601\pi$
$548$	0	0
$549$	12.4794	0.532606
$550$	0	0
$551$	−6.45418	−0.274957
$552$	0	0
$553$	− 7.65345i	− 0.325458i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0.0652731i	0.00276571i	0.999999	+	0.00138286i	$0.000440177\pi$
$557$	0.0652731i	0.00276571i	−0.999999	+	0.00138286i	$0.999560\pi$
$558$	0	0
$559$	−31.8276	−1.34616
$560$	0	0
$561$	−1.42567	−0.0601919
$562$	0	0
$563$	− 28.7426i	− 1.21136i	−0.795709	−	0.605679i	$-0.792902\pi$
$563$	− 28.7426i	− 1.21136i	0.795709	−	0.605679i	$-0.207098\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 0.747238i	− 0.0313811i
$568$	0	0
$569$	8.26312	0.346408	0.173204	−	0.984886i	$-0.444588\pi$
$569$	8.26312	0.346408	0.173204	+	0.984886i	$0.444588\pi$
$570$	0	0
$571$	11.7359	0.491131	0.245565	−	0.969380i	$-0.421026\pi$
$571$	11.7359	0.491131	0.245565	+	0.969380i	$0.421026\pi$
$572$	0	0
$573$	− 2.24520i	− 0.0937948i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 21.3886i	− 0.890419i	−0.895427	−	0.445209i	$-0.853129\pi$
$577$	− 21.3886i	− 0.890419i	0.895427	−	0.445209i	$-0.146871\pi$
$578$	0	0
$579$	−1.10589	−0.0459591
$580$	0	0
$581$	−4.24171	−0.175976
$582$	0	0
$583$	0.866125i	0.0358712i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 41.2313i	− 1.70180i	−0.525328	−	0.850900i	$-0.676057\pi$
$587$	− 41.2313i	− 1.70180i	0.525328	−	0.850900i	$-0.323943\pi$
$588$	0	0
$589$	−4.65667	−0.191875
$590$	0	0
$591$	20.6366	0.848874
$592$	0	0
$593$	− 23.2238i	− 0.953685i	−0.878989	−	0.476843i	$-0.841781\pi$
$593$	− 23.2238i	− 0.953685i	0.878989	−	0.476843i	$-0.158219\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	12.3822i	0.506768i
$598$	0	0
$599$	22.6226	0.924335	0.462168	−	0.886793i	$-0.347072\pi$
$599$	22.6226	0.924335	0.462168	+	0.886793i	$0.347072\pi$
$600$	0	0
$601$	−12.9540	−0.528405	−0.264203	−	0.964467i	$-0.585109\pi$
$601$	−12.9540	−0.528405	−0.264203	+	0.964467i	$0.585109\pi$
$602$	0	0
$603$	2.91706i	0.118792i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 4.54036i	− 0.184288i	−0.995746	−	0.0921439i	$-0.970628\pi$
$607$	− 4.54036i	− 0.184288i	0.995746	−	0.0921439i	$-0.0293720\pi$
$608$	0	0
$609$	3.87311	0.156946
$610$	0	0
$611$	16.2290	0.656557
$612$	0	0
$613$	− 25.3536i	− 1.02402i	−0.858978	−	0.512012i	$-0.828900\pi$
$613$	− 25.3536i	− 1.02402i	0.858978	−	0.512012i	$-0.171100\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	45.2189i	1.82045i	0.414119	+	0.910223i	$0.364090\pi$
$617$	45.2189i	1.82045i	−0.414119	+	0.910223i	$0.635910\pi$
$618$	0	0
$619$	−31.3549	−1.26026	−0.630130	−	0.776490i	$-0.716998\pi$
$619$	−31.3549	−1.26026	−0.630130	+	0.776490i	$0.716998\pi$
$620$	0	0
$621$	3.25085	0.130452
$622$	0	0
$623$	− 0.688750i	− 0.0275942i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 0.260319i	− 0.0103961i
$628$	0	0
$629$	13.4033	0.534426
$630$	0	0
$631$	31.9468	1.27178	0.635892	−	0.771778i	$-0.280633\pi$
$631$	31.9468	1.27178	0.635892	+	0.771778i	$0.280633\pi$
$632$	0	0
$633$	20.6274i	0.819866i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 16.1274i	− 0.638989i
$638$	0	0
$639$	−6.55008	−0.259117
$640$	0	0
$641$	41.3333	1.63257	0.816284	−	0.577650i	$-0.196030\pi$
$641$	41.3333	1.63257	0.816284	+	0.577650i	$0.196030\pi$
$642$	0	0
$643$	− 33.2313i	− 1.31052i	−0.755406	−	0.655258i	$-0.772560\pi$
$643$	− 33.2313i	− 1.31052i	0.755406	−	0.655258i	$-0.227440\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 33.6916i	− 1.32455i	−0.749260	−	0.662277i	$-0.769590\pi$
$647$	− 33.6916i	− 1.32455i	0.749260	−	0.662277i	$-0.230410\pi$
$648$	0	0
$649$	−2.44822	−0.0961009
$650$	0	0
$651$	2.79443	0.109522
$652$	0	0
$653$	21.8480i	0.854978i	0.904020	+	0.427489i	$0.140602\pi$
$653$	21.8480i	0.854978i	−0.904020	+	0.427489i	$0.859398\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2.51933i	0.0982885i
$658$	0	0
$659$	11.2883	0.439731	0.219866	−	0.975530i	$-0.429438\pi$
$659$	11.2883	0.439731	0.219866	+	0.975530i	$0.429438\pi$
$660$	0	0
$661$	−36.5720	−1.42249	−0.711244	−	0.702945i	$-0.751868\pi$
$661$	−36.5720	−1.42249	−0.711244	+	0.702945i	$0.751868\pi$
$662$	0	0
$663$	17.0735i	0.663079i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	16.8499i	0.652431i
$668$	0	0
$669$	4.15464	0.160627
$670$	0	0
$671$	−2.60890	−0.100715
$672$	0	0
$673$	7.68769i	0.296339i	0.988962	+	0.148169i	$0.0473381\pi$
$673$	7.68769i	0.296339i	−0.988962	+	0.148169i	$0.952662\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	37.2965i	1.43342i	0.697370	+	0.716711i	$0.254353\pi$
$677$	37.2965i	1.43342i	−0.697370	+	0.716711i	$0.745647\pi$
$678$	0	0
$679$	11.6154	0.445758
$680$	0	0
$681$	−23.6290	−0.905464
$682$	0	0
$683$	36.4794i	1.39585i	0.716172	+	0.697924i	$0.245892\pi$
$683$	36.4794i	1.39585i	−0.716172	+	0.697924i	$0.754108\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 18.2401i	− 0.695902i
$688$	0	0
$689$	10.3725	0.395161
$690$	0	0
$691$	−32.3521	−1.23073	−0.615365	−	0.788242i	$-0.710992\pi$
$691$	−32.3521	−1.23073	−0.615365	+	0.788242i	$0.710992\pi$
$692$	0	0
$693$	0.156215i	0.00593413i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	21.9161i	0.830132i
$698$	0	0
$699$	−2.26742	−0.0857617
$700$	0	0
$701$	5.27498	0.199233	0.0996166	−	0.995026i	$-0.468238\pi$
$701$	5.27498	0.199233	0.0996166	+	0.995026i	$0.468238\pi$
$702$	0	0
$703$	2.44737i	0.0923041i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	8.24988i	0.310268i
$708$	0	0
$709$	−29.2235	−1.09751	−0.548755	−	0.835983i	$-0.684898\pi$
$709$	−29.2235	−1.09751	−0.548755	+	0.835983i	$0.684898\pi$
$710$	0	0
$711$	−10.2423	−0.384117
$712$	0	0
$713$	12.1571i	0.455289i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 17.1069i	− 0.638868i
$718$	0	0
$719$	−43.9638	−1.63957	−0.819787	−	0.572668i	$-0.805908\pi$
$719$	−43.9638	−1.63957	−0.819787	+	0.572668i	$0.805908\pi$
$720$	0	0
$721$	10.3749	0.386380
$722$	0	0
$723$	28.2002i	1.04878i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 29.9261i	− 1.10990i	−0.831884	−	0.554949i	$-0.812738\pi$
$727$	− 29.9261i	− 1.10990i	0.831884	−	0.554949i	$-0.187262\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	86.6945	3.20651
$732$	0	0
$733$	− 16.5265i	− 0.610419i	−0.952285	−	0.305210i	$-0.901273\pi$
$733$	− 16.5265i	− 0.610419i	0.952285	−	0.305210i	$-0.0987265\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 0.609831i	− 0.0224634i
$738$	0	0
$739$	50.8984	1.87233	0.936164	−	0.351564i	$-0.114350\pi$
$739$	50.8984	1.87233	0.936164	+	0.351564i	$0.114350\pi$
$740$	0	0
$741$	−3.11751	−0.114525
$742$	0	0
$743$	− 28.6937i	− 1.05267i	−0.850277	−	0.526336i	$-0.823566\pi$
$743$	− 28.6937i	− 1.05267i	0.850277	−	0.526336i	$-0.176434\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	5.67652i	0.207693i
$748$	0	0
$749$	−0.765995	−0.0279888
$750$	0	0
$751$	−0.927935	−0.0338608	−0.0169304	−	0.999857i	$-0.505389\pi$
$751$	−0.927935	−0.0338608	−0.0169304	+	0.999857i	$0.505389\pi$
$752$	0	0
$753$	23.9575i	0.873061i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 41.4243i	− 1.50559i	−0.658254	−	0.752796i	$-0.728705\pi$
$757$	− 41.4243i	− 1.50559i	0.658254	−	0.752796i	$-0.271295\pi$
$758$	0	0
$759$	−0.679613	−0.0246684
$760$	0	0
$761$	3.78721	0.137286	0.0686431	−	0.997641i	$-0.478133\pi$
$761$	3.78721	0.137286	0.0686431	+	0.997641i	$0.478133\pi$
$762$	0	0
$763$	− 5.01032i	− 0.181386i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	29.3192i	1.05866i
$768$	0	0
$769$	1.31040	0.0472542	0.0236271	−	0.999721i	$-0.492479\pi$
$769$	1.31040	0.0472542	0.0236271	+	0.999721i	$0.492479\pi$
$770$	0	0
$771$	−28.5421	−1.02792
$772$	0	0
$773$	29.1324i	1.04782i	0.851774	+	0.523910i	$0.175527\pi$
$773$	29.1324i	1.04782i	−0.851774	+	0.523910i	$0.824473\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 1.46865i	− 0.0526874i
$778$	0	0
$779$	−4.00175	−0.143378
$780$	0	0
$781$	1.36934	0.0489988
$782$	0	0
$783$	− 5.18323i	− 0.185233i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	46.6157i	1.66167i	0.556520	+	0.830835i	$0.312136\pi$
$787$	46.6157i	1.66167i	−0.556520	+	0.830835i	$0.687864\pi$
$788$	0	0
$789$	3.55679	0.126625
$790$	0	0
$791$	6.21819	0.221094
$792$	0	0
$793$	31.2435i	1.10949i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	13.7585i	0.487350i	0.969857	+	0.243675i	$0.0783531\pi$
$797$	13.7585i	0.487350i	−0.969857	+	0.243675i	$0.921647\pi$
$798$	0	0
$799$	−44.2059	−1.56389
$800$	0	0
$801$	−0.921727	−0.0325676
$802$	0	0
$803$	− 0.526684i	− 0.0185863i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 29.2618i	− 1.03006i
$808$	0	0
$809$	−9.99577	−0.351433	−0.175716	−	0.984441i	$-0.556224\pi$
$809$	−9.99577	−0.351433	−0.175716	+	0.984441i	$0.556224\pi$
$810$	0	0
$811$	18.5193	0.650300	0.325150	−	0.945663i	$-0.394585\pi$
$811$	18.5193	0.650300	0.325150	+	0.945663i	$0.394585\pi$
$812$	0	0
$813$	− 24.8183i	− 0.870415i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	15.8299i	0.553817i
$818$	0	0
$819$	1.87080	0.0653709
$820$	0	0
$821$	28.3160	0.988234	0.494117	−	0.869395i	$-0.335491\pi$
$821$	28.3160	0.988234	0.494117	+	0.869395i	$0.335491\pi$
$822$	0	0
$823$	11.9853i	0.417781i	0.977939	+	0.208890i	$0.0669852\pi$
$823$	11.9853i	0.417781i	−0.977939	+	0.208890i	$0.933015\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 13.3711i	− 0.464960i	−0.972601	−	0.232480i	$-0.925316\pi$
$827$	− 13.3711i	− 0.464960i	0.972601	−	0.232480i	$-0.0746841\pi$
$828$	0	0
$829$	16.5521	0.574879	0.287439	−	0.957799i	$-0.407196\pi$
$829$	16.5521	0.574879	0.287439	+	0.957799i	$0.407196\pi$
$830$	0	0
$831$	−17.4275	−0.604553
$832$	0	0
$833$	43.9289i	1.52205i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 3.73968i	− 0.129262i
$838$	0	0
$839$	−24.4837	−0.845273	−0.422636	−	0.906299i	$-0.638895\pi$
$839$	−24.4837	−0.845273	−0.422636	+	0.906299i	$0.638895\pi$
$840$	0	0
$841$	−2.13416	−0.0735919
$842$	0	0
$843$	3.75349i	0.129277i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	8.18696i	0.281307i
$848$	0	0
$849$	−16.3735	−0.561936
$850$	0	0
$851$	6.38933	0.219023
$852$	0	0
$853$	19.1354i	0.655185i	0.944819	+	0.327592i	$0.106237\pi$
$853$	19.1354i	0.655185i	−0.944819	+	0.327592i	$0.893763\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	7.43367i	0.253929i	0.991907	+	0.126965i	$0.0405235\pi$
$857$	7.43367i	0.253929i	−0.991907	+	0.126965i	$0.959477\pi$
$858$	0	0
$859$	−41.2172	−1.40631	−0.703157	−	0.711035i	$-0.748227\pi$
$859$	−41.2172	−1.40631	−0.703157	+	0.711035i	$0.748227\pi$
$860$	0	0
$861$	2.40142	0.0818402
$862$	0	0
$863$	− 38.1189i	− 1.29758i	−0.760966	−	0.648791i	$-0.775275\pi$
$863$	− 38.1189i	− 1.29758i	0.760966	−	0.648791i	$-0.224725\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 29.5060i	− 1.00208i
$868$	0	0
$869$	2.14123	0.0726362
$870$	0	0
$871$	−7.30318	−0.247459
$872$	0	0
$873$	− 15.5444i	− 0.526099i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 29.0253i	− 0.980115i	−0.871690	−	0.490057i	$-0.836976\pi$
$877$	− 29.0253i	− 0.980115i	0.871690	−	0.490057i	$-0.163024\pi$
$878$	0	0
$879$	−8.70991	−0.293778
$880$	0	0
$881$	−34.4248	−1.15980	−0.579901	−	0.814687i	$-0.696909\pi$
$881$	−34.4248	−1.15980	−0.579901	+	0.814687i	$0.696909\pi$
$882$	0	0
$883$	− 38.8027i	− 1.30582i	−0.757438	−	0.652908i	$-0.773549\pi$
$883$	− 38.8027i	− 1.30582i	0.757438	−	0.652908i	$-0.226451\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 5.76954i	− 0.193722i	−0.995298	−	0.0968610i	$-0.969120\pi$
$887$	− 5.76954i	− 0.193722i	0.995298	−	0.0968610i	$-0.0308802\pi$
$888$	0	0
$889$	10.8621	0.364302
$890$	0	0
$891$	0.209057	0.00700367
$892$	0	0
$893$	− 8.07173i	− 0.270110i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	8.13888i	0.271749i
$898$	0	0
$899$	19.3836	0.646480
$900$	0	0
$901$	−28.2534	−0.941257
$902$	0	0
$903$	− 9.49939i	− 0.316120i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 31.7260i	− 1.05345i	−0.850037	−	0.526723i	$-0.823421\pi$
$907$	− 31.7260i	− 1.05345i	0.850037	−	0.526723i	$-0.176579\pi$
$908$	0	0
$909$	11.0405	0.366190
$910$	0	0
$911$	−16.0331	−0.531200	−0.265600	−	0.964083i	$-0.585570\pi$
$911$	−16.0331	−0.531200	−0.265600	+	0.964083i	$0.585570\pi$
$912$	0	0
$913$	− 1.18672i	− 0.0392746i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 5.54747i	− 0.183194i
$918$	0	0
$919$	10.9062	0.359761	0.179881	−	0.983688i	$-0.442429\pi$
$919$	10.9062	0.359761	0.179881	+	0.983688i	$0.442429\pi$
$920$	0	0
$921$	−17.7123	−0.583641
$922$	0	0
$923$	− 16.3989i	− 0.539775i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 13.8843i	− 0.456019i
$928$	0	0
$929$	−22.6227	−0.742226	−0.371113	−	0.928588i	$-0.621024\pi$
$929$	−22.6227	−0.742226	−0.371113	+	0.928588i	$0.621024\pi$
$930$	0	0
$931$	−8.02115	−0.262883
$932$	0	0
$933$	20.2876i	0.664185i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	5.15582i	0.168433i	0.996447	+	0.0842166i	$0.0268388\pi$
$937$	5.15582i	0.168433i	−0.996447	+	0.0842166i	$0.973161\pi$
$938$	0	0
$939$	−20.7701	−0.677807
$940$	0	0
$941$	12.4570	0.406086	0.203043	−	0.979170i	$-0.434917\pi$
$941$	12.4570	0.406086	0.203043	+	0.979170i	$0.434917\pi$
$942$	0	0
$943$	10.4474i	0.340213i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 16.6616i	− 0.541430i	−0.962660	−	0.270715i	$-0.912740\pi$
$947$	− 16.6616i	− 0.541430i	0.962660	−	0.270715i	$-0.0872600\pi$
$948$	0	0
$949$	−6.30743	−0.204748
$950$	0	0
$951$	−16.7651	−0.543646
$952$	0	0
$953$	51.8577i	1.67983i	0.542715	+	0.839917i	$0.317396\pi$
$953$	51.8577i	1.67983i	−0.542715	+	0.839917i	$0.682604\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1.08359i	0.0350275i
$958$	0	0
$959$	5.48127	0.177000
$960$	0	0
$961$	−17.0148	−0.548864
$962$	0	0
$963$	1.02510i	0.0330334i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 12.7719i	− 0.410717i	−0.978687	−	0.205358i	$-0.934164\pi$
$967$	− 12.7719i	− 0.410717i	0.978687	−	0.205358i	$-0.0658360\pi$
$968$	0	0
$969$	8.49172	0.272793
$970$	0	0
$971$	−15.5348	−0.498534	−0.249267	−	0.968435i	$-0.580190\pi$
$971$	−15.5348	−0.498534	−0.249267	+	0.968435i	$0.580190\pi$
$972$	0	0
$973$	14.7558i	0.473049i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	33.2063i	1.06236i	0.847258	+	0.531182i	$0.178252\pi$
$977$	33.2063i	1.06236i	−0.847258	+	0.531182i	$0.821748\pi$
$978$	0	0
$979$	0.192693	0.00615851
$980$	0	0
$981$	−6.70511	−0.214078
$982$	0	0
$983$	− 49.0945i	− 1.56587i	−0.622103	−	0.782935i	$-0.713722\pi$
$983$	− 49.0945i	− 1.56587i	0.622103	−	0.782935i	$-0.286278\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	4.84378i	0.154179i
$988$	0	0
$989$	41.3270	1.31412
$990$	0	0
$991$	−25.6162	−0.813724	−0.406862	−	0.913490i	$-0.633377\pi$
$991$	−25.6162	−0.813724	−0.406862	+	0.913490i	$0.633377\pi$
$992$	0	0
$993$	− 26.2157i	− 0.831932i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	10.8299i	0.342986i	0.985185	+	0.171493i	$0.0548591\pi$
$997$	10.8299i	0.342986i	−0.985185	+	0.171493i	$0.945141\pi$
$998$	0	0
$999$	−1.96543	−0.0621835

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7500.2.d.d.1249.5		8
5.2	odd	4		7500.2.a.g.1.4		4
5.3	odd	4		7500.2.a.d.1.1		4
5.4	even	2	inner	7500.2.d.d.1249.4		8
25.2	odd	20		300.2.m.a.121.2	✓	8
25.9	even	10		1500.2.o.a.349.4		16
25.11	even	5		1500.2.o.a.649.3		16
25.12	odd	20		300.2.m.a.181.2	yes	8
25.13	odd	20		1500.2.m.b.901.1		8
25.14	even	10		1500.2.o.a.649.2		16
25.16	even	5		1500.2.o.a.349.1		16
25.23	odd	20		1500.2.m.b.601.1		8
75.2	even	20		900.2.n.a.721.1		8
75.62	even	20		900.2.n.a.181.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.m.a.121.2	✓	8	25.2	odd	20
300.2.m.a.181.2	yes	8	25.12	odd	20
900.2.n.a.181.1		8	75.62	even	20
900.2.n.a.721.1		8	75.2	even	20
1500.2.m.b.601.1		8	25.23	odd	20
1500.2.m.b.901.1		8	25.13	odd	20
1500.2.o.a.349.1		16	25.16	even	5
1500.2.o.a.349.4		16	25.9	even	10
1500.2.o.a.649.2		16	25.14	even	10
1500.2.o.a.649.3		16	25.11	even	5
7500.2.a.d.1.1		4	5.3	odd	4
7500.2.a.g.1.4		4	5.2	odd	4
7500.2.d.d.1249.4		8	5.4	even	2	inner
7500.2.d.d.1249.5		8	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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -0.747238i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -0.747238i q^{7} -1.00000 q^{9} +0.209057 q^{11} -2.50361i q^{13} +6.81953i q^{17} -1.24520 q^{19} +0.747238 q^{21} +3.25085i q^{23} -1.00000i q^{27} +5.18323 q^{29} +3.73968 q^{31} +0.209057i q^{33} -1.96543i q^{37} +2.50361 q^{39} +3.21373 q^{41} -12.7127i q^{43} +6.48225i q^{47} +6.44163 q^{49} -6.81953 q^{51} +4.14301i q^{53} -1.24520i q^{57} -11.7108 q^{59} -12.4794 q^{61} +0.747238i q^{63} -2.91706i q^{67} -3.25085 q^{69} +6.55008 q^{71} -2.51933i q^{73} -0.156215i q^{77} +10.2423 q^{79} +1.00000 q^{81} -5.67652i q^{83} +5.18323i q^{87} +0.921727 q^{89} -1.87080 q^{91} +3.73968i q^{93} +15.5444i q^{97} -0.209057 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^9	\( 8 q - 8 q^{9} - 2 q^{11} + 10 q^{19} - 8 q^{21} + 8 q^{29} - 18 q^{31} - 10 q^{39} + 8 q^{49} - 8 q^{51} - 2 q^{59} - 4 q^{61} + 18 q^{69} - 40 q^{71} + 6 q^{79} + 8 q^{81} - 30 q^{89} - 20 q^{91} + 2 q^{99}+O(q^{100}) \) 8 * q - 8 * q^9 - 2 * q^11 + 10 * q^19 - 8 * q^21 + 8 * q^29 - 18 * q^31 - 10 * q^39 + 8 * q^49 - 8 * q^51 - 2 * q^59 - 4 * q^61 + 18 * q^69 - 40 * q^71 + 6 * q^79 + 8 * q^81 - 30 * q^89 - 20 * q^91 + 2 * q^99

Introduction

L-functions

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Database

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