LMFDB - Embedded newform 2325.2.c.r.1024.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2325,2,Mod(1024,2325)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2325.1024"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2325 = 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2325.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,-14,0,2,0,0,-12,0,14,0,0,-20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$18.5652184699$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 19x^{10} + 127x^{8} + 357x^{6} + 412x^{4} + 204x^{2} + 36 $ x^12 + 19x^10 + 127x^8 + 357x^6 + 412x^4 + 204*x^2 + 36
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1024.6
Root			$-0.667396i$ of defining polynomial
Character	$\chi$	$=$	2325.1024
Dual form			2325.2.c.r.1024.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.667396i q^{2} -1.00000i q^{3} +1.55458 q^{4} -0.667396 q^{6} +3.64222i q^{7} -2.37231i q^{8} -1.00000 q^{9} -1.15709 q^{11} -1.55458i q^{12} -5.71167i q^{13} +2.43080 q^{14} +1.52589 q^{16} -1.37855i q^{17} +0.667396i q^{18} -2.28711 q^{19} +3.64222 q^{21} +0.772236i q^{22} -5.17025i q^{23} -2.37231 q^{24} -3.81195 q^{26} +1.00000i q^{27} +5.66213i q^{28} +5.51387 q^{29} -1.00000 q^{31} -5.76300i q^{32} +1.15709i q^{33} -0.920040 q^{34} -1.55458 q^{36} +2.77224i q^{37} +1.52641i q^{38} -5.71167 q^{39} -0.122562 q^{41} -2.43080i q^{42} -1.11738i q^{43} -1.79879 q^{44} -3.45060 q^{46} -8.92527i q^{47} -1.52589i q^{48} -6.26574 q^{49} -1.37855 q^{51} -8.87926i q^{52} -3.42456i q^{53} +0.667396 q^{54} +8.64048 q^{56} +2.28711i q^{57} -3.67993i q^{58} -0.542646 q^{59} +10.4883 q^{61} +0.667396i q^{62} -3.64222i q^{63} -0.794421 q^{64} +0.772236 q^{66} -15.8274i q^{67} -2.14307i q^{68} -5.17025 q^{69} +10.7253 q^{71} +2.37231i q^{72} +8.34049i q^{73} +1.85018 q^{74} -3.55550 q^{76} -4.21437i q^{77} +3.81195i q^{78} -5.83719 q^{79} +1.00000 q^{81} +0.0817974i q^{82} -16.4935i q^{83} +5.66213 q^{84} -0.745734 q^{86} -5.51387i q^{87} +2.74498i q^{88} +8.27438 q^{89} +20.8031 q^{91} -8.03757i q^{92} +1.00000i q^{93} -5.95669 q^{94} -5.76300 q^{96} +9.97701i q^{97} +4.18173i q^{98} +1.15709 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 14 q^{4} + 2 q^{6} - 12 q^{9} + 14 q^{11} - 20 q^{14} + 34 q^{16} - 34 q^{19} + 4 q^{21} + 6 q^{24} + 4 q^{26} + 16 q^{29} - 12 q^{31} + 26 q^{34} + 14 q^{36} - 8 q^{39} + 36 q^{41} - 8 q^{44} - 20 q^{46}+ \cdots - 14 q^{99}+O(q^{100}) $ 12 * q - 14 * q^4 + 2 * q^6 - 12 * q^9 + 14 * q^11 - 20 * q^14 + 34 * q^16 - 34 * q^19 + 4 * q^21 + 6 * q^24 + 4 * q^26 + 16 * q^29 - 12 * q^31 + 26 * q^34 + 14 * q^36 - 8 * q^39 + 36 * q^41 - 8 * q^44 - 20 * q^46 - 36 * q^49 - 2 * q^54 + 64 * q^56 - 64 * q^59 - 58 * q^64 + 4 * q^66 + 2 * q^69 + 2 * q^71 - 40 * q^74 + 84 * q^76 - 26 * q^79 + 12 * q^81 - 44 * q^84 - 22 * q^86 - 68 * q^89 - 4 * q^91 - 38 * q^94 - 70 * q^96 - 14 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2325\mathbb{Z}\right)^\times$.

$n$	$652$	$776$	$1801$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	− 0.667396i	− 0.471920i	−0.971763	−	0.235960i	$-0.924177\pi$
$2$	− 0.667396i	− 0.471920i	0.971763	−	0.235960i	$-0.0758235\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	1.55458	0.777291
$5$	0	0
$5$	0	0
$6$	−0.667396	−0.272463
$7$	3.64222i	1.37663i	0.725413	+	0.688314i	$0.241649\pi$
$7$	3.64222i	1.37663i	−0.725413	+	0.688314i	$0.758351\pi$
$8$	− 2.37231i	− 0.838740i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.15709	−0.348875	−0.174438	−	0.984668i	$-0.555811\pi$
$11$	−1.15709	−0.348875	−0.174438	+	0.984668i	$0.555811\pi$
$12$	− 1.55458i	− 0.448769i
$13$	− 5.71167i	− 1.58413i	−0.610435	−	0.792066i	$-0.709005\pi$
$13$	− 5.71167i	− 1.58413i	0.610435	−	0.792066i	$-0.290995\pi$
$14$	2.43080	0.649659
$15$	0	0
$16$	1.52589	0.381473
$17$	− 1.37855i	− 0.334348i	−0.985927	−	0.167174i	$-0.946536\pi$
$17$	− 1.37855i	− 0.334348i	0.985927	−	0.167174i	$-0.0534642\pi$
$18$	0.667396i	0.157307i
$19$	−2.28711	−0.524699	−0.262349	−	0.964973i	$-0.584497\pi$
$19$	−2.28711	−0.524699	−0.262349	+	0.964973i	$0.584497\pi$
$20$	0	0
$21$	3.64222	0.794797
$22$	0.772236i	0.164641i
$23$	− 5.17025i	− 1.07807i	−0.842283	−	0.539035i	$-0.818789\pi$
$23$	− 5.17025i	− 1.07807i	0.842283	−	0.539035i	$-0.181211\pi$
$24$	−2.37231	−0.484247
$25$	0	0
$26$	−3.81195	−0.747584
$27$	1.00000i	0.192450i
$28$	5.66213i	1.07004i
$29$	5.51387	1.02390	0.511950	−	0.859015i	$-0.328923\pi$
$29$	5.51387	1.02390	0.511950	+	0.859015i	$0.328923\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	− 5.76300i	− 1.01876i
$33$	1.15709i	0.201423i
$34$	−0.920040	−0.157786
$35$	0	0
$36$	−1.55458	−0.259097
$37$	2.77224i	0.455753i	0.973690	+	0.227876i	$0.0731782\pi$
$37$	2.77224i	0.455753i	−0.973690	+	0.227876i	$0.926822\pi$
$38$	1.52641i	0.247616i
$39$	−5.71167	−0.914599
$40$	0	0
$41$	−0.122562	−0.0191410	−0.00957048	−	0.999954i	$-0.503046\pi$
$41$	−0.122562	−0.0191410	−0.00957048	+	0.999954i	$0.503046\pi$
$42$	− 2.43080i	− 0.375081i
$43$	− 1.11738i	− 0.170399i	−0.996364	−	0.0851993i	$-0.972847\pi$
$43$	− 1.11738i	− 0.170399i	0.996364	−	0.0851993i	$-0.0271527\pi$
$44$	−1.79879	−0.271178
$45$	0	0
$46$	−3.45060	−0.508763
$47$	− 8.92527i	− 1.30189i	−0.759127	−	0.650943i	$-0.774374\pi$
$47$	− 8.92527i	− 1.30189i	0.759127	−	0.650943i	$-0.225626\pi$
$48$	− 1.52589i	− 0.220244i
$49$	−6.26574	−0.895106
$50$	0	0
$51$	−1.37855	−0.193036
$52$	− 8.87926i	− 1.23133i
$53$	− 3.42456i	− 0.470400i	−0.971947	−	0.235200i	$-0.924426\pi$
$53$	− 3.42456i	− 0.470400i	0.971947	−	0.235200i	$-0.0755745\pi$
$54$	0.667396	0.0908211
$55$	0	0
$56$	8.64048	1.15463
$57$	2.28711i	0.302935i
$58$	− 3.67993i	− 0.483199i
$59$	−0.542646	−0.0706465	−0.0353232	−	0.999376i	$-0.511246\pi$
$59$	−0.542646	−0.0706465	−0.0353232	+	0.999376i	$0.511246\pi$
$60$	0	0
$61$	10.4883	1.34289	0.671444	−	0.741056i	$-0.265675\pi$
$61$	10.4883	1.34289	0.671444	+	0.741056i	$0.265675\pi$
$62$	0.667396i	0.0847594i
$63$	− 3.64222i	− 0.458876i
$64$	−0.794421	−0.0993026
$65$	0	0
$66$	0.772236	0.0950557
$67$	− 15.8274i	− 1.93363i	−0.255477	−	0.966815i	$-0.582233\pi$
$67$	− 15.8274i	− 1.93363i	0.255477	−	0.966815i	$-0.417767\pi$
$68$	− 2.14307i	− 0.259886i
$69$	−5.17025	−0.622424
$70$	0	0
$71$	10.7253	1.27286	0.636428	−	0.771336i	$-0.280411\pi$
$71$	10.7253	1.27286	0.636428	+	0.771336i	$0.280411\pi$
$72$	2.37231i	0.279580i
$73$	8.34049i	0.976181i	0.872793	+	0.488090i	$0.162306\pi$
$73$	8.34049i	0.976181i	−0.872793	+	0.488090i	$0.837694\pi$
$74$	1.85018	0.215079
$75$	0	0
$76$	−3.55550	−0.407844
$77$	− 4.21437i	− 0.480272i
$78$	3.81195i	0.431618i
$79$	−5.83719	−0.656735	−0.328368	−	0.944550i	$-0.606498\pi$
$79$	−5.83719	−0.656735	−0.328368	+	0.944550i	$0.606498\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0.0817974i	0.00903301i
$83$	− 16.4935i	− 1.81039i	−0.424993	−	0.905197i	$-0.639723\pi$
$83$	− 16.4935i	− 1.81039i	0.424993	−	0.905197i	$-0.360277\pi$
$84$	5.66213	0.617789
$85$	0	0
$86$	−0.745734	−0.0804146
$87$	− 5.51387i	− 0.591149i
$88$	2.74498i	0.292616i
$89$	8.27438	0.877083	0.438541	−	0.898711i	$-0.355495\pi$
$89$	8.27438	0.877083	0.438541	+	0.898711i	$0.355495\pi$
$90$	0	0
$91$	20.8031	2.18076
$92$	− 8.03757i	− 0.837975i
$93$	1.00000i	0.103695i
$94$	−5.95669	−0.614386
$95$	0	0
$96$	−5.76300	−0.588184
$97$	9.97701i	1.01301i	0.862237	+	0.506506i	$0.169063\pi$
$97$	9.97701i	1.01301i	−0.862237	+	0.506506i	$0.830937\pi$
$98$	4.18173i	0.422418i
$99$	1.15709	0.116292
$100$	0	0
$101$	−7.18097	−0.714533	−0.357266	−	0.934003i	$-0.616291\pi$
$101$	−7.18097	−0.714533	−0.357266	+	0.934003i	$0.616291\pi$
$102$	0.920040i	0.0910976i
$103$	1.70411i	0.167911i	0.996470	+	0.0839555i	$0.0267554\pi$
$103$	1.70411i	0.167911i	−0.996470	+	0.0839555i	$0.973245\pi$
$104$	−13.5499	−1.32867
$105$	0	0
$106$	−2.28554	−0.221991
$107$	− 18.1570i	− 1.75530i	−0.479299	−	0.877652i	$-0.659109\pi$
$107$	− 18.1570i	− 1.75530i	0.479299	−	0.877652i	$-0.340891\pi$
$108$	1.55458i	0.149590i
$109$	−6.50614	−0.623176	−0.311588	−	0.950217i	$-0.600861\pi$
$109$	−6.50614	−0.623176	−0.311588	+	0.950217i	$0.600861\pi$
$110$	0	0
$111$	2.77224	0.263129
$112$	5.55763i	0.525147i
$113$	− 3.36042i	− 0.316122i	−0.987429	−	0.158061i	$-0.949476\pi$
$113$	− 3.36042i	− 0.316122i	0.987429	−	0.158061i	$-0.0505243\pi$
$114$	1.52641	0.142961
$115$	0	0
$116$	8.57176	0.795868
$117$	5.71167i	0.528044i
$118$	0.362160i	0.0333395i
$119$	5.02099	0.460273
$120$	0	0
$121$	−9.66115	−0.878286
$122$	− 6.99984i	− 0.633736i
$123$	0.122562i	0.0110510i
$124$	−1.55458	−0.139606
$125$	0	0
$126$	−2.43080	−0.216553
$127$	10.6138i	0.941823i	0.882181	+	0.470911i	$0.156075\pi$
$127$	10.6138i	0.941823i	−0.882181	+	0.470911i	$0.843925\pi$
$128$	− 10.9958i	− 0.971902i
$129$	−1.11738	−0.0983797
$130$	0	0
$131$	14.5080	1.26757	0.633786	−	0.773508i	$-0.281500\pi$
$131$	14.5080	1.26757	0.633786	+	0.773508i	$0.281500\pi$
$132$	1.79879i	0.156565i
$133$	− 8.33014i	− 0.722315i
$134$	−10.5632	−0.912519
$135$	0	0
$136$	−3.27036	−0.280431
$137$	5.31941i	0.454468i	0.973840	+	0.227234i	$0.0729682\pi$
$137$	5.31941i	0.454468i	−0.973840	+	0.227234i	$0.927032\pi$
$138$	3.45060i	0.293735i
$139$	8.27443	0.701828	0.350914	−	0.936408i	$-0.385871\pi$
$139$	8.27443	0.701828	0.350914	+	0.936408i	$0.385871\pi$
$140$	0	0
$141$	−8.92527	−0.751644
$142$	− 7.15801i	− 0.600687i
$143$	6.60891i	0.552665i
$144$	−1.52589	−0.127158
$145$	0	0
$146$	5.56641	0.460679
$147$	6.26574i	0.516789i
$148$	4.30967i	0.354253i
$149$	−15.9731	−1.30857	−0.654284	−	0.756249i	$-0.727030\pi$
$149$	−15.9731	−1.30857	−0.654284	+	0.756249i	$0.727030\pi$
$150$	0	0
$151$	7.39770	0.602016	0.301008	−	0.953622i	$-0.402677\pi$
$151$	7.39770	0.602016	0.301008	+	0.953622i	$0.402677\pi$
$152$	5.42574i	0.440086i
$153$	1.37855i	0.111449i
$154$	−2.81265	−0.226650
$155$	0	0
$156$	−8.87926	−0.710910
$157$	15.7243i	1.25493i	0.778643	+	0.627467i	$0.215908\pi$
$157$	15.7243i	1.25493i	−0.778643	+	0.627467i	$0.784092\pi$
$158$	3.89572i	0.309927i
$159$	−3.42456	−0.271585
$160$	0	0
$161$	18.8312	1.48410
$162$	− 0.667396i	− 0.0524356i
$163$	9.68816i	0.758836i	0.925225	+	0.379418i	$0.123876\pi$
$163$	9.68816i	0.758836i	−0.925225	+	0.379418i	$0.876124\pi$
$164$	−0.190533	−0.0148781
$165$	0	0
$166$	−11.0077	−0.854361
$167$	19.1075i	1.47858i	0.673385	+	0.739292i	$0.264840\pi$
$167$	19.1075i	1.47858i	−0.673385	+	0.739292i	$0.735160\pi$
$168$	− 8.64048i	− 0.666628i
$169$	−19.6232	−1.50948
$170$	0	0
$171$	2.28711	0.174900
$172$	− 1.73706i	− 0.132449i
$173$	21.1286i	1.60638i	0.595724	+	0.803189i	$0.296865\pi$
$173$	21.1286i	1.60638i	−0.595724	+	0.803189i	$0.703135\pi$
$174$	−3.67993	−0.278975
$175$	0	0
$176$	−1.76559	−0.133087
$177$	0.542646i	0.0407878i
$178$	− 5.52229i	− 0.413913i
$179$	−13.0689	−0.976816	−0.488408	−	0.872615i	$-0.662422\pi$
$179$	−13.0689	−0.976816	−0.488408	+	0.872615i	$0.662422\pi$
$180$	0	0
$181$	9.98243	0.741988	0.370994	−	0.928635i	$-0.379017\pi$
$181$	9.98243	0.741988	0.370994	+	0.928635i	$0.379017\pi$
$182$	− 13.8839i	− 1.02915i
$183$	− 10.4883i	− 0.775316i
$184$	−12.2654	−0.904221
$185$	0	0
$186$	0.667396	0.0489358
$187$	1.59511i	0.116646i
$188$	− 13.8751i	− 1.01194i
$189$	−3.64222	−0.264932
$190$	0	0
$191$	−8.69024	−0.628804	−0.314402	−	0.949290i	$-0.601804\pi$
$191$	−8.69024	−0.628804	−0.314402	+	0.949290i	$0.601804\pi$
$192$	0.794421i	0.0573324i
$193$	− 18.4643i	− 1.32909i	−0.747249	−	0.664544i	$-0.768626\pi$
$193$	− 18.4643i	− 1.32909i	0.747249	−	0.664544i	$-0.231374\pi$
$194$	6.65862	0.478061
$195$	0	0
$196$	−9.74061	−0.695758
$197$	1.73458i	0.123584i	0.998089	+	0.0617919i	$0.0196815\pi$
$197$	1.73458i	0.123584i	−0.998089	+	0.0617919i	$0.980318\pi$
$198$	− 0.772236i	− 0.0548804i
$199$	−13.7835	−0.977084	−0.488542	−	0.872540i	$-0.662471\pi$
$199$	−13.7835	−0.977084	−0.488542	+	0.872540i	$0.662471\pi$
$200$	0	0
$201$	−15.8274	−1.11638
$202$	4.79255i	0.337202i
$203$	20.0827i	1.40953i
$204$	−2.14307	−0.150045
$205$	0	0
$206$	1.13732	0.0792406
$207$	5.17025i	0.359357i
$208$	− 8.71539i	− 0.604304i
$209$	2.64639	0.183054
$210$	0	0
$211$	−8.73784	−0.601538	−0.300769	−	0.953697i	$-0.597243\pi$
$211$	−8.73784	−0.601538	−0.300769	+	0.953697i	$0.597243\pi$
$212$	− 5.32377i	− 0.365638i
$213$	− 10.7253i	− 0.734884i
$214$	−12.1179	−0.828363
$215$	0	0
$216$	2.37231	0.161416
$217$	− 3.64222i	− 0.247250i
$218$	4.34217i	0.294089i
$219$	8.34049	0.563598
$220$	0	0
$221$	−7.87384	−0.529652
$222$	− 1.85018i	− 0.124176i
$223$	10.4409i	0.699176i	0.936903	+	0.349588i	$0.113679\pi$
$223$	10.4409i	0.699176i	−0.936903	+	0.349588i	$0.886321\pi$
$224$	20.9901	1.40246
$225$	0	0
$226$	−2.24273	−0.149184
$227$	− 25.1149i	− 1.66693i	−0.552571	−	0.833466i	$-0.686353\pi$
$227$	− 25.1149i	− 1.66693i	0.552571	−	0.833466i	$-0.313647\pi$
$228$	3.55550i	0.235469i
$229$	3.58524	0.236919	0.118460	−	0.992959i	$-0.462204\pi$
$229$	3.58524	0.236919	0.118460	+	0.992959i	$0.462204\pi$
$230$	0	0
$231$	−4.21437	−0.277285
$232$	− 13.0806i	− 0.858785i
$233$	18.2989i	1.19880i	0.800449	+	0.599401i	$0.204594\pi$
$233$	18.2989i	1.19880i	−0.800449	+	0.599401i	$0.795406\pi$
$234$	3.81195	0.249195
$235$	0	0
$236$	−0.843588	−0.0549129
$237$	5.83719i	0.379166i
$238$	− 3.35099i	− 0.217212i
$239$	9.65731	0.624680	0.312340	−	0.949970i	$-0.398887\pi$
$239$	9.65731	0.624680	0.312340	+	0.949970i	$0.398887\pi$
$240$	0	0
$241$	−5.31631	−0.342454	−0.171227	−	0.985232i	$-0.554773\pi$
$241$	−5.31631	−0.342454	−0.171227	+	0.985232i	$0.554773\pi$
$242$	6.44781i	0.414481i
$243$	− 1.00000i	− 0.0641500i
$244$	16.3049	1.04381
$245$	0	0
$246$	0.0817974	0.00521521
$247$	13.0632i	0.831192i
$248$	2.37231i	0.150642i
$249$	−16.4935	−1.04523
$250$	0	0
$251$	−9.20850	−0.581235	−0.290618	−	0.956839i	$-0.593861\pi$
$251$	−9.20850	−0.581235	−0.290618	+	0.956839i	$0.593861\pi$
$252$	− 5.66213i	− 0.356680i
$253$	5.98243i	0.376112i
$254$	7.08361	0.444465
$255$	0	0
$256$	−8.92740	−0.557963
$257$	7.04525i	0.439470i	0.975560	+	0.219735i	$0.0705193\pi$
$257$	7.04525i	0.439470i	−0.975560	+	0.219735i	$0.929481\pi$
$258$	0.745734i	0.0464274i
$259$	−10.0971	−0.627402
$260$	0	0
$261$	−5.51387	−0.341300
$262$	− 9.68260i	− 0.598193i
$263$	4.35053i	0.268265i	0.990963	+	0.134133i	$0.0428248\pi$
$263$	4.35053i	0.268265i	−0.990963	+	0.134133i	$0.957175\pi$
$264$	2.74498	0.168942
$265$	0	0
$266$	−5.55951	−0.340875
$267$	− 8.27438i	− 0.506384i
$268$	− 24.6051i	− 1.50299i
$269$	20.1710	1.22985	0.614925	−	0.788586i	$-0.289186\pi$
$269$	20.1710	1.22985	0.614925	+	0.788586i	$0.289186\pi$
$270$	0	0
$271$	13.0303	0.791533	0.395767	−	0.918351i	$-0.370479\pi$
$271$	13.0303	0.791533	0.395767	+	0.918351i	$0.370479\pi$
$272$	− 2.10352i	− 0.127545i
$273$	− 20.8031i	− 1.25906i
$274$	3.55015	0.214473
$275$	0	0
$276$	−8.03757	−0.483805
$277$	− 5.93472i	− 0.356583i	−0.983978	−	0.178292i	$-0.942943\pi$
$277$	− 5.93472i	− 0.356583i	0.983978	−	0.178292i	$-0.0570570\pi$
$278$	− 5.52232i	− 0.331207i
$279$	1.00000	0.0598684
$280$	0	0
$281$	−2.56333	−0.152915	−0.0764577	−	0.997073i	$-0.524361\pi$
$281$	−2.56333	−0.152915	−0.0764577	+	0.997073i	$0.524361\pi$
$282$	5.95669i	0.354716i
$283$	− 22.5046i	− 1.33776i	−0.743371	−	0.668879i	$-0.766774\pi$
$283$	− 22.5046i	− 1.33776i	0.743371	−	0.668879i	$-0.233226\pi$
$284$	16.6733	0.989380
$285$	0	0
$286$	4.41076	0.260814
$287$	− 0.446397i	− 0.0263500i
$288$	5.76300i	0.339588i
$289$	15.0996	0.888211
$290$	0	0
$291$	9.97701	0.584863
$292$	12.9660i	0.758777i
$293$	9.86771i	0.576478i	0.957559	+	0.288239i	$0.0930697\pi$
$293$	9.86771i	0.576478i	−0.957559	+	0.288239i	$0.906930\pi$
$294$	4.18173	0.243883
$295$	0	0
$296$	6.57662	0.382258
$297$	− 1.15709i	− 0.0671411i
$298$	10.6604i	0.617540i
$299$	−29.5307	−1.70781
$300$	0	0
$301$	4.06973	0.234576
$302$	− 4.93719i	− 0.284104i
$303$	7.18097i	0.412536i
$304$	−3.48988	−0.200158
$305$	0	0
$306$	0.920040	0.0525952
$307$	13.4887i	0.769843i	0.922949	+	0.384922i	$0.125771\pi$
$307$	13.4887i	0.769843i	−0.922949	+	0.384922i	$0.874229\pi$
$308$	− 6.55158i	− 0.373311i
$309$	1.70411	0.0969435
$310$	0	0
$311$	8.33830	0.472822	0.236411	−	0.971653i	$-0.424029\pi$
$311$	8.33830	0.472822	0.236411	+	0.971653i	$0.424029\pi$
$312$	13.5499i	0.767111i
$313$	− 7.03489i	− 0.397636i	−0.980036	−	0.198818i	$-0.936290\pi$
$313$	− 7.03489i	− 0.397636i	0.980036	−	0.198818i	$-0.0637102\pi$
$314$	10.4943	0.592229
$315$	0	0
$316$	−9.07439	−0.510474
$317$	28.9660i	1.62689i	0.581640	+	0.813446i	$0.302411\pi$
$317$	28.9660i	1.62689i	−0.581640	+	0.813446i	$0.697589\pi$
$318$	2.28554i	0.128167i
$319$	−6.38004	−0.357213
$320$	0	0
$321$	−18.1570	−1.01343
$322$	− 12.5678i	− 0.700378i
$323$	3.15290i	0.175432i
$324$	1.55458	0.0863657
$325$	0	0
$326$	6.46584	0.358110
$327$	6.50614i	0.359791i
$328$	0.290756i	0.0160543i
$329$	32.5078	1.79221
$330$	0	0
$331$	4.97223	0.273299	0.136649	−	0.990619i	$-0.456367\pi$
$331$	4.97223	0.273299	0.136649	+	0.990619i	$0.456367\pi$
$332$	− 25.6405i	− 1.40720i
$333$	− 2.77224i	− 0.151918i
$334$	12.7523	0.697773
$335$	0	0
$336$	5.55763	0.303194
$337$	3.96964i	0.216240i	0.994138	+	0.108120i	$0.0344831\pi$
$337$	3.96964i	0.216240i	−0.994138	+	0.108120i	$0.965517\pi$
$338$	13.0964i	0.712352i
$339$	−3.36042	−0.182513
$340$	0	0
$341$	1.15709	0.0626599
$342$	− 1.52641i	− 0.0825386i
$343$	2.67434i	0.144401i
$344$	−2.65077	−0.142920
$345$	0	0
$346$	14.1011	0.758082
$347$	− 32.8011i	− 1.76085i	−0.474182	−	0.880427i	$-0.657256\pi$
$347$	− 32.8011i	− 1.76085i	0.474182	−	0.880427i	$-0.342744\pi$
$348$	− 8.57176i	− 0.459495i
$349$	−36.2630	−1.94111	−0.970557	−	0.240872i	$-0.922567\pi$
$349$	−36.2630	−1.94111	−0.970557	+	0.240872i	$0.922567\pi$
$350$	0	0
$351$	5.71167	0.304866
$352$	6.66830i	0.355422i
$353$	30.9661i	1.64816i	0.566473	+	0.824080i	$0.308307\pi$
$353$	30.9661i	1.64816i	−0.566473	+	0.824080i	$0.691693\pi$
$354$	0.362160	0.0192486
$355$	0	0
$356$	12.8632	0.681749
$357$	− 5.02099i	− 0.265739i
$358$	8.72213i	0.460979i
$359$	11.6777	0.616324	0.308162	−	0.951334i	$-0.400286\pi$
$359$	11.6777	0.616324	0.308162	+	0.951334i	$0.400286\pi$
$360$	0	0
$361$	−13.7691	−0.724691
$362$	− 6.66223i	− 0.350159i
$363$	9.66115i	0.507079i
$364$	32.3402	1.69509
$365$	0	0
$366$	−6.99984	−0.365887
$367$	2.52502i	0.131805i	0.997826	+	0.0659026i	$0.0209927\pi$
$367$	2.52502i	0.131805i	−0.997826	+	0.0659026i	$0.979007\pi$
$368$	− 7.88924i	− 0.411255i
$369$	0.122562	0.00638032
$370$	0	0
$371$	12.4730	0.647566
$372$	1.55458i	0.0806014i
$373$	− 15.0961i	− 0.781646i	−0.920466	−	0.390823i	$-0.872191\pi$
$373$	− 15.0961i	− 0.781646i	0.920466	−	0.390823i	$-0.127809\pi$
$374$	1.06457	0.0550475
$375$	0	0
$376$	−21.1736	−1.09194
$377$	− 31.4934i	− 1.62199i
$378$	2.43080i	0.125027i
$379$	19.1633	0.984354	0.492177	−	0.870495i	$-0.336201\pi$
$379$	19.1633	0.984354	0.492177	+	0.870495i	$0.336201\pi$
$380$	0	0
$381$	10.6138	0.543761
$382$	5.79983i	0.296745i
$383$	2.27917i	0.116460i	0.998303	+	0.0582301i	$0.0185457\pi$
$383$	2.27917i	0.116460i	−0.998303	+	0.0582301i	$0.981454\pi$
$384$	−10.9958	−0.561128
$385$	0	0
$386$	−12.3230	−0.627223
$387$	1.11738i	0.0567995i
$388$	15.5101i	0.787405i
$389$	14.5865	0.739563	0.369782	−	0.929119i	$-0.379432\pi$
$389$	14.5865	0.739563	0.369782	+	0.929119i	$0.379432\pi$
$390$	0	0
$391$	−7.12745	−0.360451
$392$	14.8643i	0.750761i
$393$	− 14.5080i	− 0.731833i
$394$	1.15765	0.0583217
$395$	0	0
$396$	1.79879	0.0903926
$397$	33.0931i	1.66089i	0.557098	+	0.830447i	$0.311915\pi$
$397$	33.0931i	1.66089i	−0.557098	+	0.830447i	$0.688085\pi$
$398$	9.19903i	0.461106i
$399$	−8.33014	−0.417029
$400$	0	0
$401$	10.6123	0.529953	0.264977	−	0.964255i	$-0.414636\pi$
$401$	10.6123	0.529953	0.264977	+	0.964255i	$0.414636\pi$
$402$	10.5632i	0.526843i
$403$	5.71167i	0.284519i
$404$	−11.1634	−0.555400
$405$	0	0
$406$	13.4031	0.665185
$407$	− 3.20772i	− 0.159001i
$408$	3.27036i	0.161907i
$409$	28.1971	1.39426	0.697128	−	0.716947i	$-0.254461\pi$
$409$	28.1971	1.39426	0.697128	+	0.716947i	$0.254461\pi$
$410$	0	0
$411$	5.31941	0.262387
$412$	2.64918i	0.130516i
$413$	− 1.97643i	− 0.0972540i
$414$	3.45060	0.169588
$415$	0	0
$416$	−32.9164	−1.61386
$417$	− 8.27443i	− 0.405201i
$418$	− 1.76619i	− 0.0863871i
$419$	−30.2128	−1.47599	−0.737996	−	0.674805i	$-0.764228\pi$
$419$	−30.2128	−1.47599	−0.737996	+	0.674805i	$0.764228\pi$
$420$	0	0
$421$	−18.4246	−0.897959	−0.448980	−	0.893542i	$-0.648212\pi$
$421$	−18.4246	−0.897959	−0.448980	+	0.893542i	$0.648212\pi$
$422$	5.83160i	0.283878i
$423$	8.92527i	0.433962i
$424$	−8.12414	−0.394543
$425$	0	0
$426$	−7.15801	−0.346807
$427$	38.2006i	1.84866i
$428$	− 28.2266i	− 1.36438i
$429$	6.60891	0.319081
$430$	0	0
$431$	−13.6428	−0.657152	−0.328576	−	0.944478i	$-0.606569\pi$
$431$	−13.6428	−0.657152	−0.328576	+	0.944478i	$0.606569\pi$
$432$	1.52589i	0.0734145i
$433$	33.6141i	1.61539i	0.589599	+	0.807696i	$0.299286\pi$
$433$	33.6141i	1.61539i	−0.589599	+	0.807696i	$0.700714\pi$
$434$	−2.43080	−0.116682
$435$	0	0
$436$	−10.1143	−0.484389
$437$	11.8249i	0.565662i
$438$	− 5.56641i	− 0.265973i
$439$	21.9603	1.04811	0.524054	−	0.851685i	$-0.324419\pi$
$439$	21.9603	1.04811	0.524054	+	0.851685i	$0.324419\pi$
$440$	0	0
$441$	6.26574	0.298369
$442$	5.25497i	0.249953i
$443$	5.44404i	0.258654i	0.991602	+	0.129327i	$0.0412817\pi$
$443$	5.44404i	0.258654i	−0.991602	+	0.129327i	$0.958718\pi$
$444$	4.30967	0.204528
$445$	0	0
$446$	6.96824	0.329956
$447$	15.9731i	0.755502i
$448$	− 2.89345i	− 0.136703i
$449$	−36.8483	−1.73898	−0.869489	−	0.493952i	$-0.835552\pi$
$449$	−36.8483	−1.73898	−0.869489	+	0.493952i	$0.835552\pi$
$450$	0	0
$451$	0.141815	0.00667781
$452$	− 5.22406i	− 0.245719i
$453$	− 7.39770i	− 0.347574i
$454$	−16.7616	−0.786659
$455$	0	0
$456$	5.42574	0.254084
$457$	− 5.03603i	− 0.235575i	−0.993039	−	0.117788i	$-0.962420\pi$
$457$	− 5.03603i	− 0.235575i	0.993039	−	0.117788i	$-0.0375802\pi$
$458$	− 2.39277i	− 0.111807i
$459$	1.37855	0.0643453
$460$	0	0
$461$	−22.5649	−1.05095	−0.525476	−	0.850808i	$-0.676113\pi$
$461$	−22.5649	−1.05095	−0.525476	+	0.850808i	$0.676113\pi$
$462$	2.81265i	0.130856i
$463$	− 7.92517i	− 0.368314i	−0.982897	−	0.184157i	$-0.941045\pi$
$463$	− 7.92517i	− 0.368314i	0.982897	−	0.184157i	$-0.0589555\pi$
$464$	8.41357	0.390590
$465$	0	0
$466$	12.2126	0.565739
$467$	− 10.7488i	− 0.497397i	−0.968581	−	0.248698i	$-0.919997\pi$
$467$	− 10.7488i	− 0.497397i	0.968581	−	0.248698i	$-0.0800028\pi$
$468$	8.87926i	0.410444i
$469$	57.6470	2.66189
$470$	0	0
$471$	15.7243	0.724536
$472$	1.28733i	0.0592540i
$473$	1.29291i	0.0594479i
$474$	3.89572	0.178936
$475$	0	0
$476$	7.80554	0.357766
$477$	3.42456i	0.156800i
$478$	− 6.44525i	− 0.294799i
$479$	31.1215	1.42198	0.710989	−	0.703203i	$-0.248248\pi$
$479$	31.1215	1.42198	0.710989	+	0.703203i	$0.248248\pi$
$480$	0	0
$481$	15.8341	0.721973
$482$	3.54809i	0.161611i
$483$	− 18.8312i	− 0.856847i
$484$	−15.0190	−0.682684
$485$	0	0
$486$	−0.667396	−0.0302737
$487$	22.1648i	1.00438i	0.864756	+	0.502192i	$0.167473\pi$
$487$	22.1648i	1.00438i	−0.864756	+	0.502192i	$0.832527\pi$
$488$	− 24.8815i	− 1.12633i
$489$	9.68816	0.438114
$490$	0	0
$491$	38.3744	1.73181	0.865905	−	0.500208i	$-0.166743\pi$
$491$	38.3744	1.73181	0.865905	+	0.500208i	$0.166743\pi$
$492$	0.190533i	0.00858988i
$493$	− 7.60116i	− 0.342339i
$494$	8.71834	0.392256
$495$	0	0
$496$	−1.52589	−0.0685146
$497$	39.0638i	1.75225i
$498$	11.0077i	0.493266i
$499$	3.14013	0.140572	0.0702858	−	0.997527i	$-0.477609\pi$
$499$	3.14013	0.140572	0.0702858	+	0.997527i	$0.477609\pi$
$500$	0	0
$501$	19.1075	0.853660
$502$	6.14572i	0.274297i
$503$	− 26.4210i	− 1.17805i	−0.808114	−	0.589027i	$-0.799511\pi$
$503$	− 26.4210i	− 1.17805i	0.808114	−	0.589027i	$-0.200489\pi$
$504$	−8.64048	−0.384878
$505$	0	0
$506$	3.99265	0.177495
$507$	19.6232i	0.871496i
$508$	16.5000i	0.732070i
$509$	27.5147	1.21957	0.609785	−	0.792567i	$-0.291256\pi$
$509$	27.5147	1.21957	0.609785	+	0.792567i	$0.291256\pi$
$510$	0	0
$511$	−30.3779	−1.34384
$512$	− 16.0335i	− 0.708588i
$513$	− 2.28711i	− 0.100978i
$514$	4.70197	0.207395
$515$	0	0
$516$	−1.73706	−0.0764697
$517$	10.3273i	0.454196i
$518$	6.73875i	0.296084i
$519$	21.1286	0.927443
$520$	0	0
$521$	30.2824	1.32670	0.663349	−	0.748310i	$-0.269135\pi$
$521$	30.2824	1.32670	0.663349	+	0.748310i	$0.269135\pi$
$522$	3.67993i	0.161066i
$523$	42.8200i	1.87239i	0.351486	+	0.936193i	$0.385676\pi$
$523$	42.8200i	1.87239i	−0.351486	+	0.936193i	$0.614324\pi$
$524$	22.5539	0.985273
$525$	0	0
$526$	2.90353	0.126600
$527$	1.37855i	0.0600507i
$528$	1.76559i	0.0768376i
$529$	−3.73144	−0.162236
$530$	0	0
$531$	0.542646	0.0235488
$532$	− 12.9499i	− 0.561449i
$533$	0.700034i	0.0303218i
$534$	−5.52229	−0.238973
$535$	0	0
$536$	−37.5477	−1.62181
$537$	13.0689i	0.563965i
$538$	− 13.4621i	− 0.580391i
$539$	7.25001	0.312280
$540$	0	0
$541$	−10.9585	−0.471142	−0.235571	−	0.971857i	$-0.575696\pi$
$541$	−10.9585	−0.471142	−0.235571	+	0.971857i	$0.575696\pi$
$542$	− 8.69636i	− 0.373541i
$543$	− 9.98243i	− 0.428387i
$544$	−7.94460	−0.340622
$545$	0	0
$546$	−13.8839	−0.594177
$547$	11.8536i	0.506822i	0.967359	+	0.253411i	$0.0815524\pi$
$547$	11.8536i	0.506822i	−0.967359	+	0.253411i	$0.918448\pi$
$548$	8.26947i	0.353254i
$549$	−10.4883	−0.447629
$550$	0	0
$551$	−12.6108	−0.537239
$552$	12.2654i	0.522052i
$553$	− 21.2603i	− 0.904080i
$554$	−3.96081	−0.168279
$555$	0	0
$556$	12.8633	0.545525
$557$	− 31.2361i	− 1.32351i	−0.749718	−	0.661757i	$-0.769811\pi$
$557$	− 31.2361i	− 1.32351i	0.749718	−	0.661757i	$-0.230189\pi$
$558$	− 0.667396i	− 0.0282531i
$559$	−6.38210	−0.269934
$560$	0	0
$561$	1.59511	0.0673455
$562$	1.71075i	0.0721638i
$563$	16.1583i	0.680990i	0.940246	+	0.340495i	$0.110595\pi$
$563$	16.1583i	0.680990i	−0.940246	+	0.340495i	$0.889405\pi$
$564$	−13.8751	−0.584246
$565$	0	0
$566$	−15.0195	−0.631315
$567$	3.64222i	0.152959i
$568$	− 25.4437i	− 1.06760i
$569$	8.39847	0.352082	0.176041	−	0.984383i	$-0.443671\pi$
$569$	8.39847	0.352082	0.176041	+	0.984383i	$0.443671\pi$
$570$	0	0
$571$	−5.39179	−0.225640	−0.112820	−	0.993615i	$-0.535988\pi$
$571$	−5.39179	−0.225640	−0.112820	+	0.993615i	$0.535988\pi$
$572$	10.2741i	0.429582i
$573$	8.69024i	0.363040i
$574$	−0.297924	−0.0124351
$575$	0	0
$576$	0.794421	0.0331009
$577$	− 14.7206i	− 0.612828i	−0.951898	−	0.306414i	$-0.900871\pi$
$577$	− 14.7206i	− 0.612828i	0.951898	−	0.306414i	$-0.0991292\pi$
$578$	− 10.0774i	− 0.419165i
$579$	−18.4643	−0.767349
$580$	0	0
$581$	60.0728	2.49224
$582$	− 6.65862i	− 0.276008i
$583$	3.96252i	0.164111i
$584$	19.7863	0.818761
$585$	0	0
$586$	6.58567	0.272052
$587$	− 41.7243i	− 1.72215i	−0.508481	−	0.861073i	$-0.669793\pi$
$587$	− 41.7243i	− 1.72215i	0.508481	−	0.861073i	$-0.330207\pi$
$588$	9.74061i	0.401696i
$589$	2.28711	0.0942387
$590$	0	0
$591$	1.73458	0.0713511
$592$	4.23013i	0.173857i
$593$	40.4139i	1.65960i	0.558059	+	0.829801i	$0.311546\pi$
$593$	40.4139i	1.65960i	−0.558059	+	0.829801i	$0.688454\pi$
$594$	−0.772236	−0.0316852
$595$	0	0
$596$	−24.8315	−1.01714
$597$	13.7835i	0.564120i
$598$	19.7087i	0.805949i
$599$	−38.8888	−1.58895	−0.794477	−	0.607294i	$-0.792255\pi$
$599$	−38.8888	−1.58895	−0.794477	+	0.607294i	$0.792255\pi$
$600$	0	0
$601$	13.9675	0.569747	0.284873	−	0.958565i	$-0.408048\pi$
$601$	13.9675	0.569747	0.284873	+	0.958565i	$0.408048\pi$
$602$	− 2.71612i	− 0.110701i
$603$	15.8274i	0.644543i
$604$	11.5003	0.467942
$605$	0	0
$606$	4.79255	0.194684
$607$	− 40.8060i	− 1.65626i	−0.560534	−	0.828132i	$-0.689404\pi$
$607$	− 40.8060i	− 1.65626i	0.560534	−	0.828132i	$-0.310596\pi$
$608$	13.1806i	0.534544i
$609$	20.0827	0.813792
$610$	0	0
$611$	−50.9782	−2.06236
$612$	2.14307i	0.0866286i
$613$	26.4962i	1.07017i	0.844798	+	0.535086i	$0.179721\pi$
$613$	26.4962i	1.07017i	−0.844798	+	0.535086i	$0.820279\pi$
$614$	9.00233	0.363304
$615$	0	0
$616$	−9.99780	−0.402823
$617$	28.7271i	1.15651i	0.815856	+	0.578255i	$0.196266\pi$
$617$	28.7271i	1.15651i	−0.815856	+	0.578255i	$0.803734\pi$
$618$	− 1.13732i	− 0.0457496i
$619$	44.0519	1.77060	0.885299	−	0.465023i	$-0.153954\pi$
$619$	44.0519	1.77060	0.885299	+	0.465023i	$0.153954\pi$
$620$	0	0
$621$	5.17025	0.207475
$622$	− 5.56495i	− 0.223134i
$623$	30.1371i	1.20742i
$624$	−8.71539	−0.348895
$625$	0	0
$626$	−4.69506	−0.187652
$627$	− 2.64639i	− 0.105687i
$628$	24.4447i	0.975449i
$629$	3.82167	0.152380
$630$	0	0
$631$	−0.777435	−0.0309492	−0.0154746	−	0.999880i	$-0.504926\pi$
$631$	−0.777435	−0.0309492	−0.0154746	+	0.999880i	$0.504926\pi$
$632$	13.8476i	0.550830i
$633$	8.73784i	0.347298i
$634$	19.3318	0.767764
$635$	0	0
$636$	−5.32377	−0.211101
$637$	35.7878i	1.41797i
$638$	4.25801i	0.168576i
$639$	−10.7253	−0.424286
$640$	0	0
$641$	12.6585	0.499981	0.249991	−	0.968248i	$-0.419572\pi$
$641$	12.6585	0.499981	0.249991	+	0.968248i	$0.419572\pi$
$642$	12.1179i	0.478256i
$643$	46.7414i	1.84330i	0.388021	+	0.921651i	$0.373159\pi$
$643$	46.7414i	1.84330i	−0.388021	+	0.921651i	$0.626841\pi$
$644$	29.2746	1.15358
$645$	0	0
$646$	2.10423	0.0827899
$647$	− 19.0171i	− 0.747639i	−0.927501	−	0.373820i	$-0.878048\pi$
$647$	− 19.0171i	− 0.747639i	0.927501	−	0.373820i	$-0.121952\pi$
$648$	− 2.37231i	− 0.0931933i
$649$	0.627890	0.0246468
$650$	0	0
$651$	−3.64222	−0.142750
$652$	15.0611i	0.589836i
$653$	− 23.6344i	− 0.924885i	−0.886649	−	0.462442i	$-0.846973\pi$
$653$	− 23.6344i	− 0.924885i	0.886649	−	0.462442i	$-0.153027\pi$
$654$	4.34217	0.169792
$655$	0	0
$656$	−0.187016	−0.00730176
$657$	− 8.34049i	− 0.325394i
$658$	− 21.6956i	− 0.845781i
$659$	−0.736021	−0.0286713	−0.0143356	−	0.999897i	$-0.504563\pi$
$659$	−0.736021	−0.0286713	−0.0143356	+	0.999897i	$0.504563\pi$
$660$	0	0
$661$	23.1277	0.899561	0.449781	−	0.893139i	$-0.351502\pi$
$661$	23.1277	0.899561	0.449781	+	0.893139i	$0.351502\pi$
$662$	− 3.31845i	− 0.128975i
$663$	7.87384i	0.305794i
$664$	−39.1277	−1.51845
$665$	0	0
$666$	−1.85018	−0.0716930
$667$	− 28.5081i	− 1.10384i
$668$	29.7042i	1.14929i
$669$	10.4409	0.403670
$670$	0	0
$671$	−12.1359	−0.468500
$672$	− 20.9901i	− 0.809711i
$673$	32.9126i	1.26869i	0.773051	+	0.634344i	$0.218730\pi$
$673$	32.9126i	1.26869i	−0.773051	+	0.634344i	$0.781270\pi$
$674$	2.64932	0.102048
$675$	0	0
$676$	−30.5059	−1.17330
$677$	4.83090i	0.185667i	0.995682	+	0.0928333i	$0.0295924\pi$
$677$	4.83090i	0.185667i	−0.995682	+	0.0928333i	$0.970408\pi$
$678$	2.24273i	0.0861317i
$679$	−36.3384	−1.39454
$680$	0	0
$681$	−25.1149	−0.962403
$682$	− 0.772236i	− 0.0295705i
$683$	36.8203i	1.40889i	0.709759	+	0.704444i	$0.248804\pi$
$683$	36.8203i	1.40889i	−0.709759	+	0.704444i	$0.751196\pi$
$684$	3.55550	0.135948
$685$	0	0
$686$	1.78484	0.0681456
$687$	− 3.58524i	− 0.136785i
$688$	− 1.70500i	− 0.0650025i
$689$	−19.5600	−0.745176
$690$	0	0
$691$	25.2787	0.961646	0.480823	−	0.876818i	$-0.340338\pi$
$691$	25.2787	0.961646	0.480823	+	0.876818i	$0.340338\pi$
$692$	32.8462i	1.24862i
$693$	4.21437i	0.160091i
$694$	−21.8913	−0.830983
$695$	0	0
$696$	−13.0806	−0.495820
$697$	0.168958i	0.00639975i
$698$	24.2018i	0.916051i
$699$	18.2989	0.692128
$700$	0	0
$701$	50.8414	1.92025	0.960127	−	0.279565i	$-0.0901901\pi$
$701$	50.8414	1.92025	0.960127	+	0.279565i	$0.0901901\pi$
$702$	− 3.81195i	− 0.143873i
$703$	− 6.34041i	− 0.239133i
$704$	0.919215	0.0346442
$705$	0	0
$706$	20.6667	0.777800
$707$	− 26.1546i	− 0.983646i
$708$	0.843588i	0.0317040i
$709$	37.1976	1.39698	0.698492	−	0.715618i	$-0.253855\pi$
$709$	37.1976	1.39698	0.698492	+	0.715618i	$0.253855\pi$
$710$	0	0
$711$	5.83719	0.218912
$712$	− 19.6294i	− 0.735644i
$713$	5.17025i	0.193627i
$714$	−3.35099	−0.125407
$715$	0	0
$716$	−20.3167	−0.759270
$717$	− 9.65731i	− 0.360659i
$718$	− 7.79364i	− 0.290856i
$719$	−35.3863	−1.31969	−0.659843	−	0.751403i	$-0.729377\pi$
$719$	−35.3863	−1.31969	−0.659843	+	0.751403i	$0.729377\pi$
$720$	0	0
$721$	−6.20674	−0.231151
$722$	9.18946i	0.341996i
$723$	5.31631i	0.197716i
$724$	15.5185	0.576741
$725$	0	0
$726$	6.44781	0.239301
$727$	− 40.7826i	− 1.51254i	−0.654257	−	0.756272i	$-0.727019\pi$
$727$	− 40.7826i	− 1.51254i	0.654257	−	0.756272i	$-0.272981\pi$
$728$	− 49.3516i	− 1.82909i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−1.54036	−0.0569724
$732$	− 16.3049i	− 0.602647i
$733$	− 26.8011i	− 0.989922i	−0.868915	−	0.494961i	$-0.835182\pi$
$733$	− 26.8011i	− 0.989922i	0.868915	−	0.494961i	$-0.164818\pi$
$734$	1.68519	0.0622015
$735$	0	0
$736$	−29.7961	−1.09830
$737$	18.3138i	0.674596i
$738$	− 0.0817974i	− 0.00301100i
$739$	8.28484	0.304763	0.152381	−	0.988322i	$-0.451306\pi$
$739$	8.28484	0.304763	0.152381	+	0.988322i	$0.451306\pi$
$740$	0	0
$741$	13.0632	0.479889
$742$	− 8.32443i	− 0.305599i
$743$	24.8614i	0.912077i	0.889960	+	0.456039i	$0.150732\pi$
$743$	24.8614i	0.912077i	−0.889960	+	0.456039i	$0.849268\pi$
$744$	2.37231	0.0869733
$745$	0	0
$746$	−10.0751	−0.368874
$747$	16.4935i	0.603464i
$748$	2.47973i	0.0906677i
$749$	66.1317	2.41640
$750$	0	0
$751$	17.0776	0.623172	0.311586	−	0.950218i	$-0.399140\pi$
$751$	17.0776	0.623172	0.311586	+	0.950218i	$0.399140\pi$
$752$	− 13.6190i	− 0.496634i
$753$	9.20850i	0.335576i
$754$	−21.0186	−0.765451
$755$	0	0
$756$	−5.66213	−0.205930
$757$	− 22.6984i	− 0.824988i	−0.910960	−	0.412494i	$-0.864658\pi$
$757$	− 22.6984i	− 0.824988i	0.910960	−	0.412494i	$-0.135342\pi$
$758$	− 12.7895i	− 0.464536i
$759$	5.98243	0.217149
$760$	0	0
$761$	4.34342	0.157449	0.0787244	−	0.996896i	$-0.474915\pi$
$761$	4.34342	0.157449	0.0787244	+	0.996896i	$0.474915\pi$
$762$	− 7.08361i	− 0.256612i
$763$	− 23.6968i	− 0.857881i
$764$	−13.5097	−0.488764
$765$	0	0
$766$	1.52111	0.0549599
$767$	3.09942i	0.111913i
$768$	8.92740i	0.322140i
$769$	−2.57132	−0.0927242	−0.0463621	−	0.998925i	$-0.514763\pi$
$769$	−2.57132	−0.0927242	−0.0463621	+	0.998925i	$0.514763\pi$
$770$	0	0
$771$	7.04525	0.253728
$772$	− 28.7042i	− 1.03309i
$773$	− 12.4442i	− 0.447588i	−0.974636	−	0.223794i	$-0.928156\pi$
$773$	− 12.4442i	− 0.447588i	0.974636	−	0.223794i	$-0.0718443\pi$
$774$	0.745734	0.0268049
$775$	0	0
$776$	23.6686	0.849653
$777$	10.0971i	0.362231i
$778$	− 9.73495i	− 0.349015i
$779$	0.280313	0.0100432
$780$	0	0
$781$	−12.4101	−0.444068
$782$	4.75683i	0.170104i
$783$	5.51387i	0.197050i
$784$	−9.56084	−0.341459
$785$	0	0
$786$	−9.68260	−0.345367
$787$	37.3565i	1.33161i	0.746124	+	0.665807i	$0.231913\pi$
$787$	37.3565i	1.33161i	−0.746124	+	0.665807i	$0.768087\pi$
$788$	2.69655i	0.0960606i
$789$	4.35053	0.154883
$790$	0	0
$791$	12.2394	0.435183
$792$	− 2.74498i	− 0.0975385i
$793$	− 59.9056i	− 2.12731i
$794$	22.0862	0.783809
$795$	0	0
$796$	−21.4275	−0.759479
$797$	− 42.1378i	− 1.49260i	−0.665611	−	0.746299i	$-0.731829\pi$
$797$	− 42.1378i	− 1.49260i	0.665611	−	0.746299i	$-0.268171\pi$
$798$	5.55951i	0.196804i
$799$	−12.3040	−0.435283
$800$	0	0
$801$	−8.27438	−0.292361
$802$	− 7.08261i	− 0.250096i
$803$	− 9.65069i	− 0.340565i
$804$	−24.6051	−0.867754
$805$	0	0
$806$	3.81195	0.134270
$807$	− 20.1710i	− 0.710054i
$808$	17.0355i	0.599307i
$809$	−2.15676	−0.0758277	−0.0379138	−	0.999281i	$-0.512071\pi$
$809$	−2.15676	−0.0758277	−0.0379138	+	0.999281i	$0.512071\pi$
$810$	0	0
$811$	12.8658	0.451780	0.225890	−	0.974153i	$-0.427471\pi$
$811$	12.8658	0.451780	0.225890	+	0.974153i	$0.427471\pi$
$812$	31.2202i	1.09561i
$813$	− 13.0303i	− 0.456992i
$814$	−2.14082	−0.0750358
$815$	0	0
$816$	−2.10352	−0.0736380
$817$	2.55557i	0.0894079i
$818$	− 18.8186i	− 0.657977i
$819$	−20.8031	−0.726921
$820$	0	0
$821$	13.9740	0.487696	0.243848	−	0.969813i	$-0.421590\pi$
$821$	13.9740	0.487696	0.243848	+	0.969813i	$0.421590\pi$
$822$	− 3.55015i	− 0.123826i
$823$	− 45.9522i	− 1.60179i	−0.598802	−	0.800897i	$-0.704357\pi$
$823$	− 45.9522i	− 1.60179i	0.598802	−	0.800897i	$-0.295643\pi$
$824$	4.04269	0.140834
$825$	0	0
$826$	−1.31906	−0.0458961
$827$	6.37379i	0.221638i	0.993841	+	0.110819i	$0.0353474\pi$
$827$	6.37379i	0.221638i	−0.993841	+	0.110819i	$0.964653\pi$
$828$	8.03757i	0.279325i
$829$	21.0176	0.729970	0.364985	−	0.931013i	$-0.381074\pi$
$829$	21.0176	0.729970	0.364985	+	0.931013i	$0.381074\pi$
$830$	0	0
$831$	−5.93472	−0.205873
$832$	4.53747i	0.157308i
$833$	8.63765i	0.299277i
$834$	−5.52232	−0.191222
$835$	0	0
$836$	4.11403	0.142287
$837$	− 1.00000i	− 0.0345651i
$838$	20.1639i	0.696551i
$839$	−20.6247	−0.712045	−0.356022	−	0.934477i	$-0.615867\pi$
$839$	−20.6247	−0.712045	−0.356022	+	0.934477i	$0.615867\pi$
$840$	0	0
$841$	1.40275	0.0483708
$842$	12.2965i	0.423765i
$843$	2.56333i	0.0882857i
$844$	−13.5837	−0.467570
$845$	0	0
$846$	5.95669	0.204795
$847$	− 35.1880i	− 1.20907i
$848$	− 5.22551i	− 0.179445i
$849$	−22.5046	−0.772355
$850$	0	0
$851$	14.3331	0.491334
$852$	− 16.6733i	− 0.571219i
$853$	− 6.65175i	− 0.227752i	−0.993495	−	0.113876i	$-0.963673\pi$
$853$	− 6.65175i	− 0.227752i	0.993495	−	0.113876i	$-0.0363266\pi$
$854$	25.4949	0.872418
$855$	0	0
$856$	−43.0741	−1.47224
$857$	− 26.3351i	− 0.899590i	−0.893132	−	0.449795i	$-0.851497\pi$
$857$	− 26.3351i	− 0.899590i	0.893132	−	0.449795i	$-0.148503\pi$
$858$	− 4.41076i	− 0.150581i
$859$	−26.8919	−0.917541	−0.458770	−	0.888555i	$-0.651710\pi$
$859$	−26.8919	−0.917541	−0.458770	+	0.888555i	$0.651710\pi$
$860$	0	0
$861$	−0.446397	−0.0152132
$862$	9.10517i	0.310123i
$863$	5.04588i	0.171764i	0.996305	+	0.0858818i	$0.0273707\pi$
$863$	5.04588i	0.171764i	−0.996305	+	0.0858818i	$0.972629\pi$
$864$	5.76300	0.196061
$865$	0	0
$866$	22.4339	0.762336
$867$	− 15.0996i	− 0.512809i
$868$	− 5.66213i	− 0.192185i
$869$	6.75414	0.229119
$870$	0	0
$871$	−90.4012	−3.06313
$872$	15.4346i	0.522682i
$873$	− 9.97701i	− 0.337671i
$874$	7.89190	0.266947
$875$	0	0
$876$	12.9660	0.438080
$877$	1.50014i	0.0506560i	0.999679	+	0.0253280i	$0.00806301\pi$
$877$	1.50014i	0.0506560i	−0.999679	+	0.0253280i	$0.991937\pi$
$878$	− 14.6562i	− 0.494623i
$879$	9.86771	0.332830
$880$	0	0
$881$	−52.2328	−1.75977	−0.879884	−	0.475188i	$-0.842380\pi$
$881$	−52.2328	−1.75977	−0.879884	+	0.475188i	$0.842380\pi$
$882$	− 4.18173i	− 0.140806i
$883$	− 39.4859i	− 1.32881i	−0.747375	−	0.664403i	$-0.768686\pi$
$883$	− 39.4859i	− 1.32881i	0.747375	−	0.664403i	$-0.231314\pi$
$884$	−12.2405	−0.411694
$885$	0	0
$886$	3.63333	0.122064
$887$	48.5387i	1.62977i	0.579623	+	0.814885i	$0.303200\pi$
$887$	48.5387i	1.62977i	−0.579623	+	0.814885i	$0.696800\pi$
$888$	− 6.57662i	− 0.220697i
$889$	−38.6578	−1.29654
$890$	0	0
$891$	−1.15709	−0.0387639
$892$	16.2313i	0.543464i
$893$	20.4131i	0.683097i
$894$	10.6604	0.356537
$895$	0	0
$896$	40.0491	1.33795
$897$	29.5307i	0.986003i
$898$	24.5924i	0.820659i
$899$	−5.51387	−0.183898
$900$	0	0
$901$	−4.72094	−0.157277
$902$	− 0.0946468i	− 0.00315139i
$903$	− 4.06973i	− 0.135432i
$904$	−7.97198	−0.265144
$905$	0	0
$906$	−4.93719	−0.164027
$907$	− 47.3682i	− 1.57284i	−0.617694	−	0.786418i	$-0.711933\pi$
$907$	− 47.3682i	− 1.57284i	0.617694	−	0.786418i	$-0.288067\pi$
$908$	− 39.0431i	− 1.29569i
$909$	7.18097	0.238178
$910$	0	0
$911$	−28.7118	−0.951264	−0.475632	−	0.879644i	$-0.657781\pi$
$911$	−28.7118	−0.951264	−0.475632	+	0.879644i	$0.657781\pi$
$912$	3.48988i	0.115562i
$913$	19.0844i	0.631602i
$914$	−3.36102	−0.111173
$915$	0	0
$916$	5.57355	0.184155
$917$	52.8414i	1.74498i
$918$	− 0.920040i	− 0.0303659i
$919$	3.19601	0.105427	0.0527133	−	0.998610i	$-0.483213\pi$
$919$	3.19601	0.105427	0.0527133	+	0.998610i	$0.483213\pi$
$920$	0	0
$921$	13.4887	0.444469
$922$	15.0597i	0.495966i
$923$	− 61.2593i	− 2.01637i
$924$	−6.55158	−0.215531
$925$	0	0
$926$	−5.28923	−0.173815
$927$	− 1.70411i	− 0.0559703i
$928$	− 31.7764i	− 1.04311i
$929$	−2.28411	−0.0749392	−0.0374696	−	0.999298i	$-0.511930\pi$
$929$	−2.28411	−0.0749392	−0.0374696	+	0.999298i	$0.511930\pi$
$930$	0	0
$931$	14.3304	0.469661
$932$	28.4472i	0.931818i
$933$	− 8.33830i	− 0.272984i
$934$	−7.17373	−0.234732
$935$	0	0
$936$	13.5499	0.442892
$937$	− 9.90513i	− 0.323587i	−0.986825	−	0.161793i	$-0.948272\pi$
$937$	− 9.90513i	− 0.323587i	0.986825	−	0.161793i	$-0.0517278\pi$
$938$	− 38.4734i	− 1.25620i
$939$	−7.03489	−0.229575
$940$	0	0
$941$	−8.74045	−0.284930	−0.142465	−	0.989800i	$-0.545503\pi$
$941$	−8.74045	−0.284930	−0.142465	+	0.989800i	$0.545503\pi$
$942$	− 10.4943i	− 0.341923i
$943$	0.633676i	0.0206353i
$944$	−0.828019	−0.0269497
$945$	0	0
$946$	0.862880	0.0280547
$947$	55.8934i	1.81629i	0.418654	+	0.908146i	$0.362502\pi$
$947$	55.8934i	1.81629i	−0.418654	+	0.908146i	$0.637498\pi$
$948$	9.07439i	0.294723i
$949$	47.6381	1.54640
$950$	0	0
$951$	28.9660	0.939287
$952$	− 11.9114i	− 0.386049i
$953$	36.3033i	1.17598i	0.808869	+	0.587989i	$0.200080\pi$
$953$	36.3033i	1.17598i	−0.808869	+	0.587989i	$0.799920\pi$
$954$	2.28554	0.0739971
$955$	0	0
$956$	15.0131	0.485558
$957$	6.38004i	0.206237i
$958$	− 20.7704i	− 0.671060i
$959$	−19.3744	−0.625634
$960$	0	0
$961$	1.00000	0.0322581
$962$	− 10.5676i	− 0.340714i
$963$	18.1570i	0.585101i
$964$	−8.26465	−0.266186
$965$	0	0
$966$	−12.5678	−0.404363
$967$	− 25.7135i	− 0.826890i	−0.910529	−	0.413445i	$-0.864325\pi$
$967$	− 25.7135i	− 0.826890i	0.910529	−	0.413445i	$-0.135675\pi$
$968$	22.9193i	0.736653i
$969$	3.15290	0.101286
$970$	0	0
$971$	33.1226	1.06296	0.531478	−	0.847072i	$-0.321637\pi$
$971$	33.1226	1.06296	0.531478	+	0.847072i	$0.321637\pi$
$972$	− 1.55458i	− 0.0498633i
$973$	30.1373i	0.966156i
$974$	14.7927	0.473989
$975$	0	0
$976$	16.0040	0.512275
$977$	− 22.6735i	− 0.725388i	−0.931908	−	0.362694i	$-0.881857\pi$
$977$	− 22.6735i	− 0.725388i	0.931908	−	0.362694i	$-0.118143\pi$
$978$	− 6.46584i	− 0.206755i
$979$	−9.57419	−0.305992
$980$	0	0
$981$	6.50614	0.207725
$982$	− 25.6109i	− 0.817276i
$983$	− 9.35725i	− 0.298450i	−0.988803	−	0.149225i	$-0.952322\pi$
$983$	− 9.35725i	− 0.298450i	0.988803	−	0.149225i	$-0.0476778\pi$
$984$	0.290756	0.00926895
$985$	0	0
$986$	−5.07298	−0.161557
$987$	− 32.5078i	− 1.03473i
$988$	20.3078i	0.646079i
$989$	−5.77712	−0.183702
$990$	0	0
$991$	12.9858	0.412506	0.206253	−	0.978499i	$-0.433873\pi$
$991$	12.9858	0.412506	0.206253	+	0.978499i	$0.433873\pi$
$992$	5.76300i	0.182976i
$993$	− 4.97223i	− 0.157789i
$994$	26.0710	0.826922
$995$	0	0
$996$	−25.6405	−0.812449
$997$	3.74266i	0.118531i	0.998242	+	0.0592656i	$0.0188759\pi$
$997$	3.74266i	0.118531i	−0.998242	+	0.0592656i	$0.981124\pi$
$998$	− 2.09571i	− 0.0663385i
$999$	−2.77224	−0.0877097

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2325.2.c.r.1024.6		12
5.2	odd	4		2325.2.a.y.1.4	✓	6
5.3	odd	4		2325.2.a.bb.1.3	yes	6
5.4	even	2	inner	2325.2.c.r.1024.7		12
15.2	even	4		6975.2.a.cc.1.3		6
15.8	even	4		6975.2.a.ca.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2325.2.a.y.1.4	✓	6	5.2	odd	4
2325.2.a.bb.1.3	yes	6	5.3	odd	4
2325.2.c.r.1024.6		12	1.1	even	1	trivial
2325.2.c.r.1024.7		12	5.4	even	2	inner
6975.2.a.ca.1.4		6	15.8	even	4
6975.2.a.cc.1.3		6	15.2	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2325 = 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2325.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.667396i q^{2} -1.00000i q^{3} +1.55458 q^{4} -0.667396 q^{6} +3.64222i q^{7} -2.37231i q^{8} -1.00000 q^{9} -1.15709 q^{11} -1.55458i q^{12} -5.71167i q^{13} +2.43080 q^{14} +1.52589 q^{16} -1.37855i q^{17} +0.667396i q^{18} -2.28711 q^{19} +3.64222 q^{21} +0.772236i q^{22} -5.17025i q^{23} -2.37231 q^{24} -3.81195 q^{26} +1.00000i q^{27} +5.66213i q^{28} +5.51387 q^{29} -1.00000 q^{31} -5.76300i q^{32} +1.15709i q^{33} -0.920040 q^{34} -1.55458 q^{36} +2.77224i q^{37} +1.52641i q^{38} -5.71167 q^{39} -0.122562 q^{41} -2.43080i q^{42} -1.11738i q^{43} -1.79879 q^{44} -3.45060 q^{46} -8.92527i q^{47} -1.52589i q^{48} -6.26574 q^{49} -1.37855 q^{51} -8.87926i q^{52} -3.42456i q^{53} +0.667396 q^{54} +8.64048 q^{56} +2.28711i q^{57} -3.67993i q^{58} -0.542646 q^{59} +10.4883 q^{61} +0.667396i q^{62} -3.64222i q^{63} -0.794421 q^{64} +0.772236 q^{66} -15.8274i q^{67} -2.14307i q^{68} -5.17025 q^{69} +10.7253 q^{71} +2.37231i q^{72} +8.34049i q^{73} +1.85018 q^{74} -3.55550 q^{76} -4.21437i q^{77} +3.81195i q^{78} -5.83719 q^{79} +1.00000 q^{81} +0.0817974i q^{82} -16.4935i q^{83} +5.66213 q^{84} -0.745734 q^{86} -5.51387i q^{87} +2.74498i q^{88} +8.27438 q^{89} +20.8031 q^{91} -8.03757i q^{92} +1.00000i q^{93} -5.95669 q^{94} -5.76300 q^{96} +9.97701i q^{97} +4.18173i q^{98} +1.15709 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 14 q^{4} + 2 q^{6} - 12 q^{9} + 14 q^{11} - 20 q^{14} + 34 q^{16} - 34 q^{19} + 4 q^{21} + 6 q^{24} + 4 q^{26} + 16 q^{29} - 12 q^{31} + 26 q^{34} + 14 q^{36} - 8 q^{39} + 36 q^{41} - 8 q^{44} - 20 q^{46}+ \cdots - 14 q^{99}+O(q^{100}) \) 12 * q - 14 * q^4 + 2 * q^6 - 12 * q^9 + 14 * q^11 - 20 * q^14 + 34 * q^16 - 34 * q^19 + 4 * q^21 + 6 * q^24 + 4 * q^26 + 16 * q^29 - 12 * q^31 + 26 * q^34 + 14 * q^36 - 8 * q^39 + 36 * q^41 - 8 * q^44 - 20 * q^46 - 36 * q^49 - 2 * q^54 + 64 * q^56 - 64 * q^59 - 58 * q^64 + 4 * q^66 + 2 * q^69 + 2 * q^71 - 40 * q^74 + 84 * q^76 - 26 * q^79 + 12 * q^81 - 44 * q^84 - 22 * q^86 - 68 * q^89 - 4 * q^91 - 38 * q^94 - 70 * q^96 - 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more